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Titanic Magnetoresistance in WTe 2 23 Sep 2014 Mazhar N Ali Department of Chemistry Princeton University 08544PrincetonNew JerseyUSA Jun Xiong Department of Physics Joseph Henry Laboratories Princeton University Princeton New Jer-sey 08544USA Steven Flynn Department of Chemistry Princeton University 08544PrincetonNew JerseyUSA Quinn Gibson Department of Chemistry Princeton University 08544PrincetonNew JerseyUSA Leslie Schoop Department of Chemistry Princeton University 08544PrincetonNew JerseyUSA Neel Haldolaarachchige Department of Chemistry Princeton University 08544PrincetonNew JerseyUSA N P Ong Department of Physics Joseph Henry Laboratories Princeton University Princeton New Jer-sey 08544USA Jing Tao Department of Physics Brookhaven National Laboratory 11973UptonNew YorkUSA R J Cava Department of Chemistry Princeton University 08544PrincetonNew JerseyUSA Titanic Magnetoresistance in WTe 2 23 Sep 2014 Magnetoresistance is the change of a material's electrical resistance in response to an applied magnetic field. In addition to its intrinsic scientific interest, it is a technologically important property, placing it in "Pasteur's quadrant" of research value: materials with large magnetorsistance have found use as magnetic sensors 1 , in magnetic memory 2 , hard drives 3 , transistors 4 , and are the subject of frequent study in the field of spintronics 5, 6 . Here we report the observation of an extremely large one-dimensional positive magnetoresistance (XMR) in the layered transition metal dichalcogenide (TMD) WTe 2 ; 452,700% at 4.5 Kelvin in a magnetic field of 14.7 Tesla, and 2.5 million% at 0.4 Kelvin in 45 Tesla, with no saturation. The XMR is highly anisotropic, maximized in the crystallographic direction where small pockets of holes and electrons are found in the electronic structure. The determination of the origin of this effect and the fabrication of nanostructures and devices based on the XMR of WTe 2 will represent a significant new direction in the study and uses of magnetoresistivity.Large magnetoresistance (MR) is an uncommon property, mostly displayed by magnetic 1 compounds. Giant Magnetoresistance (GMR) 7,8 and Colossal Magnetoresistance (CMR) 9, 10 , are exhibited by thin film metals and manganese-based perovskites, for example. In contrast, ordinary magnetoresistance, a relatively weak effect, is commonly found in nonmagnetic compounds and elements 11 . Magnetic materials typically show negative magnetoresistances (where MR is defined as ρ(H)−ρ(0) ρ(0) , is typically reported as a percent figure, and ρ(H) is the resistivity in an applied magnetic field, H). Positive magnetoresistance is seen in both metals and semiconductors, where it is usually at the level of a few percent for metals; semiconducting silver chalcogenides show magnetoresistances up to 350%, comparable with CMR materials 12 . For single carrier type semiconductors, the MR behaves like (1+µH 2 ) where µ is the carrier mobility; high mobility semiconductors can therefore exhibit relatively large effects 13 . WTe 2 is a TMD crystallizing in a distorted version of the common layered MoS 2 structure type 14 . TMDs are known to display many interesting properties, such as catalysis of chemical reactions 15 , charge density waves (CDW) 16 , superconductivity 17 , have been exfoliated to fabricate interesting nanosctructures 18,19 , and now, we report, can display extremely large magnetoresistance. In the layered TMD compounds, metal layers are sandwiched between adjacent chalcogenide layers; this dichalogenide sandwich stacks along the c-axis of the hexagonal structure, with van der Waals bonding between layers. Due to this anisotropic bonding, layered TMDs are typically electronically two-dimensional. In WTe 2 , however, there is an additional structural distortion − tungsten chains are formed within the dichalcogenide layers along the a-axis of the orthorhombic unit cell, making the compound structurally one-dimensional ( Figure 1a). WTe 2 is semi-metallic and has previously been investigated for thermoelectric applications in solid solutions with WSe 2 2 and MoTe 2 20 . Here we report the discovery of an extremely large positive magnetoresistance (XMR) in WTe 2 of up to 452,700% at 4.5 Kelvin in an applied field of 14.7 Tesla (T) when the current direction is along the tungsten chains (a-axis) and the magnetic field is applied perpendicular to the dichalcogenide layers, along the c-axis. The magnetoresistance is still increasing at 45 T, the highest field in our measurements, where it has a value of 2.5 million percent with no saturation. It is especially surprising that this XMR is present in a non-magnetic, non-semiconducting system. WTe 2 has a highly anisotropic electronic structure, with small pockets of holes and electrons in the directions where the XMR is maximized. The XMR is very anisotropic; largest along the chain direction, and dropping by more than 90% when the magnetic field is applied in other directions, making this the largest one-dimensional magnetoresistance ever reported. The effect becomes significant at temperatures below ≈ 150 K; with the temperature of the "turn on" increasing with the magnitude of the applied magnetic field. Electron diffraction patterns taken at low temperatures indicate that the origin of the observed effect is not linked to the onset of a charge density wave or a Peierls-like distortion in the tungsten chains. The temperature dependent resistivity under various applied magnetic fields (µ 0 H up to 14.7 T) is presented in Figure 2 on both logarithmic (main panel) and linear scales (lower inset). In zerofield, the room temperature resistivity is 0.6 mΩcm and falls to 1.9 µΩcm by 2 K, yielding a RRR of ≈ 370. When a field is applied, the resistivity of the sample essentially follows the zero-field curve until it is cooled close to the "turn on" temperature, T * , (taken as the minimum in the resistivity) below which the resistivity begins to dramatically increase. The magnetoresistance effect at low temperatures is extremely large, reaching 452,700% by 4.5 K in a 14.7 T field. The "turn on" temperature is shifted to higher temperature (at the rate of T. This behavior may be due to the interference between oscillations arising from the hole and electron pockets (see below) as their waves beat against each other. Figure 3b shows the 4.5 K MR field-dependence on the angle of the applied field to c-axis. When the field is aligned parallel to the c-axis the XMR effect is maximized with an ρ(14. The band structure shows the electron and hole pockets along the Γ -X direction that make WTe 2 a semimetal, as well as a potential second set of electron and hole pockets forming along Z -U. The Γ -X direction in reciprocal space corresponds to the a-axis in real space, or along the tungsten chains. Since the Z -U direction is parallel to the Γ -X direction, but shifted along k z into the perpendicular face of the Brillouin zone, this potential second crossing would change the pockets into tubes in the Fermi surface. Future ARPES study and more detailed transport analysis and characterization of the quantum oscillations will be needed to determine the details of the Fermi surface and form a basis for understanding the observed XMR. With a positive magnetoresistance this large, and the one-dimensional behavior of the XMR there are few comparable systems to WTe 2 . The closest case appears to be high purity Bismuth, a semimetal with pockets in its Fermi surface, 11,22,23 which also shows extremely large positive 5 magnetoresistance 24 comparable in magnitude to what is reported here. WTe 2 is different from Bi in a number of fundamental ways; for example, WTe 2 is structurally one-dimensional and shows an extreme one-dimensional anisotropy in its XMR, the XMR in WTe 2 has a sharper "turn on" temperature than is seen in Bi, WTe 2 is a semimetal with extremely small overlap between valence band and conduction band states, resembling an excitonic insulator 21 , and ordinary purity WTe 2 shows the effect. Particularly important from a materials development perspective is the fact that WTe 2 is a layered TMD that can be easily exfoliated and therefore form the basis for thin films and advanced nanostructures similar to MoS 2 19 . The single crystals made in this study were crudely exfoliated by using double sided tape, and thicknesses down to a few microns were easily achieved. Evaporation growth and subsequent annealing to make single crystal thin films may enhance the MR effect seen here due to higher crystal quality. Hybrid structures of various kinds, such as layering WTe 2 with magnetic films, combined with the XMR effect may be useful in devices such as highly sensitive low temperature magnetic field sensors or high field temperature sensors in cryogenics. In particular, the onedimensional aspect of the anisotropic MR in WTe 2 may be useful in low temperature magnetic field sensing and, especially, orienting. In fact, recently it was reported that below 20 K or above 5 T 25, 26 the materials currently used for temperature or field measurements are prone to large degrees of error. In contrast, however, this regime is actually where WTe 2 performs best. The ease with which this system can be exfoliated as well as the effect that even small changes in the Fermi level may have on the properties makes it an ideal candidate for electron gating experiments. Careful chemical doping and intercalation of WTe 2 may also be key in elucidating the cause of the XMR 6 and potentially unlocking further properties of interest. Methods High quality single crystals of WTe 2 were grown via Br 2 vapor transport. Tungsten powder was ground together with purified Tellurium, pressed into a pellet, and heated in an evacuated quartz tube at 700 • C, homogenized, then reheated at 750 • C for 2 days each. This final pellet was ground into a fine powder and a temperature gradient of 100 • C between 750 • C -650 • C was used for crystal growth, with a Br 2 concentration of ≈ 3 mg/ml in a sealed quartz tube for 1 week. Optimal crystals were obtained under these conditions; crystals grown at higher temperatures showed substantially lower residual resistivity ratios (RRR) ρ(300K) ρ(2K) and degraded magnetoresistance. The need to employ low temperature synthesis to avoid defect formation that degrades properties is frequently observed in TMDs, for example in TiSe 2 27 . The crystals grew as thin ribbons (Figure 1b), with the long direction being the W-W chain direction and the larger flat faces being perpendicular to the stacking direction of the layers. WTe 2 crystals were structurally and chemically characterized by powder-XRD to confirm bulk purity, single crystal XRD to determine crystal growth orientation, SEM-EDX for chemical analysis, and TEM to search for a low temperature phase transition. For general electronic characterization, SQUID magnetometer measurements revealed weak diamagnetism typical of a metal, and thermopower measurements confirm a previously reported n-p type crossover at 65 K in a 0 T field and n-type Hall effect behavior down to 2 K 28 . In resistivity measurements, electrical anisotropy was found; the tungsten chain direction (the a-axis) had the lowest resistivity. ≈ 4.4 K/T, upper inset) as larger fields are applied, implying a competition between dominating scattering mechanisms (phonon scattering and an as-yet unidentified mechanism responsible for the XMR). TEM electron diffraction studies (Figure 1c) at low temperature in a ≈ 2.8 T field (the field in the TEM at the sample position) show no evidence of a structural phase transition or the onset of a charge density wave to accompany the onset of the XMR effect. Figure 3a 3ashows the field-dependence of the XMR at various temperatures. The upper inset is a zoom of the main panel to clarify the XMR at higher temperatures while the lower inset shows the Shubnikov de Haas (SdH) quantum oscillations. The high quality of the crystals is shown not only in the high RRR, but also by the SdH oscillations inFigure 3b. The oscillations have been extracted after fitting a 2nd order polynomial to the 4.5 K parallel field measurement and subtracting that as background. They become visible by 6 T, begin increasing in amplitude, and then become dampened around 10.5 T before reemerging with much larger amplitude near 12.3 ( 452,700%). As the field is rotated to align parallel to either the a-axis or the b-axis, the MR effect is greatly diminished and dies like the cosine of the angle; this large anisotropy could be due to the very anisotropic Fermi4 surface of WTe 2 and scattering rates. Measurements up to 45 T at 0.4 K show an XMR of ρFigure 3c). The magnetic field dependence of the resistivity is close to quadratic at intermediate fields (1-11 T), but displays a different field dependence at both higher and lower applied magnetic fields. Our electronic structure calculations show WTe 2 to be a semimetal where the valence band and conduction bands barely cross the Fermi energy at different places in the Brillouin zone (Figure 4a), an electronic structure that is reminiscent of that of the excitonic insulator TiSe 2 21 . The detailed shape of the Fermi surface is (Figure 4b) very sensitive to the position of the Fermi level. Powder x-ray diffraction patterns were collected using Cu Kα radiation on a Bruker D8 Focus diffractometer with a graphite monochromator. Single crystal x-ray diffraction data was collected on a Bruker APEX II using Mo Kα radiation (λ = 0.71073Å) at 100 K. Scanning electron microscopy and energy dispersive x-ray analysis were carried out on a FEI Quanta 200 FEG Environmental-SEM and was used to determine the composition of the crystals. Electron diffraction was carried out in a JEOL 3000F transmission electron microscope equipped with a Gatan liquid-helium cooling stage.The magnetoresistance of WTe 2 samples was measured using the 4-point probe method in a Quantum Design PPMS and with a Delta-mode method by a Keithley 6221 current source meter and a 2182A nanovoltmeter. The high-field dependent data were taken at 4.5 K up to 14.7 T in an American Magnetics superconducting magnet. Resistivity measurements up to 45 T were performed at the National Magnet Laboratory. Magnetic susceptibility was measured as a function of temperature in the 2 -300 K range at applied fields of 0.3 and 3 Tesla. The electronic structure calculations were performed in the framework of density functional theory using the WIEN2K 29 code with a full-potential linearized augmented plane-wave and local orbitals [FP-LAPW + lo] basis 30 together with the Perdew Burke Ernzerhof (PBE) parametrization of the generalized gradient approximation (GGA) as the exchange-correlation functional. The plane wave cut-off parameter R M T K M AX was set to 8 and the Brillouin zone was sampled by 10000 k-points. The convergence was confirmed by increasing both R M T K M AX and the number of k points until no change in the total energy was observed. The Fermi surface was plotted with the program Xcrysden. 8 Figure 1 ( 81color online): Structural considerations. Panel a) the crystal structure of WTe 2 , showing the layered nature, typical of what is seen for transition metal dichalcogenides, and also the chains of W atoms along the a-axis distorting the ideal hexagonal net. All distances are inÅ. Panel b) a typical crystal of WTe 2 , crystallographic directions marked. The XMR exists when the current (I) flows along a and the field is parallel to c (see below). Panel c) electron diffraction images looking down the [021] zone showing the reciprocal lattice at room temperature and low temperature. The data indicate that there has been no structural transition in WTe 2 upon cooling (effective magnetic field at the sample in the TEM is about 2.8 T). Figure 2 ( 2color online): The temperature and field dependence of the extremely large magnetoresistance in WTe 2 . Main panellog of resistivity vs temperature. Lower inset, resistivity vs. temperature showing turn on of the effect. T* is defined as the temperature where the resistivity is a minimum -an approximation of the turn on temperature of the XMR. Upper inset, the linear dependence of T* on magnetic field, the slope is equal to 4.4 K/T. Figure 3 ( 3color online): Field and angular dependence of the XMR in WTe 2 . Panel a) the field dependence of the XMR in WTe 2 with the current along the W-W chains and the applied field parallel to the c-axis from 0 -9 T at representative temperatures. Upper inset shows detail of the magnetoresistance at higher temperature. Lower inset shows the detail of the quantum oscillations at 4.5 K. This demonstrates the high quality of the crystal and suggests that carriers in the hole and electron pockets (Figure 4) 'beat' against each other at intermediate field values. Panel b) the angular dependence of the XMR at 4.5 K shows that the effect is one-dimensional; it is maximized when H is parallel to c and goes to 0 when H is perpendicular to c. The main panel shows the MR as the applied field is rotated to be parallel to a and the inset shows the same effect when the field is rotated to be parallel to b. Panel c) The XMR of WTe 2 up to 45 T. The m = 1.96 line is from a fit of ρ(H) ρ(0) α H m from 1 -11 T, m = 1.56 line is from a fit of ρ(H) ρ(0) α H m from 11 -45 T. m = 1 at low field is a guide to the eye. Figure 4 ( 4color online): The electronic structure of WTe 2 , calculated including spin orbit coupling. a) The energy vs. wavevector relationships for high symmetry directions in the orthorhombic brillouin zone. Note the semimetal character due to the crossing of the hole and electron bands near Γ. b) Detail of the calculated electronic structure in the Γ -X and Γ -Z directions and similar detail showing the possible crossing of the states between Z and U. c) The Fermi surface of WTe 2 showing electron (yellow) and hole (blue) pockets displaced from the Γ point and aligned along the chain direction. FIG. 1. FIG. 3. Competing Interests The authors declare that they have no competing financial interests.Correspondence Correspondence and requests for materials should be addressed to Mazhar N. Ali (email: mnali@princeton.edu). A review of magnetic sensors. J E Lenz, Proceedings of the IEEE. 78Lenz, J. E. A review of magnetic sensors. Proceedings of the IEEE 78, 973-989 (1990). Giant magnetoresistance of manganese oxides with a layered perovskite structure. Y Moritomo, K H Asamitsu, A & Y , T , Nature. 380Moritomo Y., K. H., Asamitsu A. & Y., T. Giant magnetoresistance of manganese oxides with a layered perovskite structure. Nature 380, 141-144 (1996). . J Daughton, Applications, Journal of Magnetism and Magnetic Materials. 192Daughton, J. GMR applications. Journal of Magnetism and Magnetic Materials 192, 334-342 (1999). Coulomb Blockade Anisotropic Magnetoresistance Effect in a GaMnAs Single-Electron Transistor. J Wunderlich, Phys. Rev. Lett. 9777201Wunderlich, J. et al. Coulomb Blockade Anisotropic Magnetoresistance Effect in a GaMnAs Single-Electron Transistor. Phys. Rev. Lett. 97, 077201 (2006). Spintronics: a spin-based electronics vision for the future. 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Neutron scattering study of the charge-density wave transitions in 2H-TaSe 2 and 2H-NbSe 2. D E Moncton, J D Axe, F J Disalvo, Phys. Rev. B. 16Moncton, D. E., Axe, J. D. & DiSalvo, F. J. Neutron scattering study of the charge-density wave transitions in 2H-TaSe 2 and 2H-NbSe 2 . Phys. Rev. B 16, 801-819 (1977). Superconductivity and Magnetoresistance in NbSe 2. R C Morris, R V Coleman, R Bhandari, Phys. Rev. B. 5Morris, R. C., Coleman, R. V. & Bhandari, R. Superconductivity and Magnetoresistance in NbSe 2 . Phys. Rev. B 5, 895-901 (1972). Realization and electrical characterization of ultrathin crystals of layered transition-metal dichalcogenides. A Ayari, E Cobas, O Ogundadegbe, M S Fuhrer, Journal of Applied Physics. 10114507Ayari, A., Cobas, E., Ogundadegbe, O. & Fuhrer, M. S. Realization and electrical character- ization of ultrathin crystals of layered transition-metal dichalcogenides. Journal of Applied Physics 101, 014507 (2007). Single-layer MoS 2 transistors. B Radisavljevic, A Radenovic, J Brivio, V Giacometti, A Kis, Nature nanotechnology. 6Radisavljevic, B., Radenovic, A., Brivio, J., Giacometti, V. & Kis, A. Single-layer MoS 2 transistors. Nature nanotechnology 6, 147-150 (2011). Electrical Properties of the MoTe 2 -WTe 2 and MoSe 2 -WSe 2. E Revolinsky, D Beerntsen, Revolinsky, E. & Beerntsen, D. Electrical Properties of the MoTe 2 -WTe 2 and MoSe 2 -WSe 2 . Systems. Journal of Applied Physics. 35Systems. Journal of Applied Physics 35, 2086-2089 (1964). Photoemission of bands above the Fermi level: The excitonic insulator phase transition in 1T-TiSe 2. T Pillo, Phys. Rev. B. 61Pillo, T. et al. Photoemission of bands above the Fermi level: The excitonic insulator phase transition in 1T-TiSe 2 . Phys. Rev. B 61, 16213-16222 (2000). Transport properties of bismuth in quantizing magnetic fields. J Mangez, J Issi, J Heremans, Physical Review B. 144381Mangez, J., Issi, J. & Heremans, J. Transport properties of bismuth in quantizing magnetic fields. Physical Review B 14, 4381 (1976). Magnetoacoustic measurement of fermi surface in bismuth. M Giura, R Marcon, T Papa, F Wanderlingh, Il Nuovo Cimento B Series. 10Giura, M., Marcon, R., Papa, T. & Wanderlingh, F. Magnetoacoustic measurement of fermi surface in bismuth. Il Nuovo Cimento B Series 10 51, 150-168 (1967). The magnetoresistance of bismuth crystals at low temperatures. P B Alers, R T Webber, Phys. Rev. 91Alers, P. B. & Webber, R. T. The magnetoresistance of bismuth crystals at low temperatures. Phys. Rev. 91, 1060-1065 (1953). Experimental Techniques for Low-Temperature Measurements : Cryostat Design, Material Properties and Superconductor Critical-Current Testing: Cryostat Design, Material Properties and Superconductor Critical-Current Testing. J Ekin, OUPOxfordEkin, J. Experimental Techniques for Low-Temperature Measurements : Cryostat Design, Material Properties and Superconductor Critical-Current Testing: Cryostat Design, Material Properties and Superconductor Critical-Current Testing (OUP Oxford, 2006). Hall sensors for extreme temperatures. Jakub Jankowsi, S E , .-A Oszwaldowski, M , Sensors. 11Jakub Jankowsi, S. E.-A. & Oszwaldowski, M. Hall sensors for extreme temperatures. Sensors 11, 876-885 (2011). Electronic properties and superlattice formation in the semimetal TiSe 2. Di Salvo, F J Moncton, D E Waszczak, J V , Phys. Rev. B. 14Di Salvo, F. J., Moncton, D. E. & Waszczak, J. V. Electronic properties and superlattice formation in the semimetal TiSe 2 . Phys. Rev. B 14, 4321-4328 (1976). Electrical Properties of Tungsten-Ditelluride WTe 2. S Kabashima, Journal of the Physical Society of Japan. 21Kabashima, S. Electrical Properties of Tungsten-Ditelluride WTe 2 . Journal of the Physical Society of Japan 21, 945-948 (1966). An Augmented Plane Wave+ Local Orbitals Program for calculating Crystal Properties. P Blaha, K Schwarz, G Madsen, D Kvasnicka, J Luitz, Wien2k, AustriaTechnische Universität WienBlaha, P., Schwarz, K., Madsen, G., Kvasnicka, D. & Luitz, J. WIEN2k, An Augmented Plane Wave+ Local Orbitals Program for calculating Crystal Properties, Technische Universität Wien, Austria (2001). . D J Singh & Nordström, L Planewaves, Pseudopotentials Method, SpringerNew York2nd ed.D. J. Singh & Nordström, L. Planewaves, Pseudopotentials, and the LAPW Method, Springer, New York, 2nd ed. (2006).

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{'abstract': 'Magnetoresistance is the change of a material\'s electrical resistance in response to an applied magnetic field. In addition to its intrinsic scientific interest, it is a technologically important property, placing it in "Pasteur\'s quadrant" of research value: materials with large magnetorsistance have found use as magnetic sensors 1 , in magnetic memory 2 , hard drives 3 , transistors 4 , and are the subject of frequent study in the field of spintronics 5, 6 . Here we report the observation of an extremely large one-dimensional positive magnetoresistance (XMR) in the layered transition metal dichalcogenide (TMD) WTe 2 ; 452,700% at 4.5 Kelvin in a magnetic field of 14.7 Tesla, and 2.5 million% at 0.4 Kelvin in 45 Tesla, with no saturation. The XMR is highly anisotropic, maximized in the crystallographic direction where small pockets of holes and electrons are found in the electronic structure. The determination of the origin of this effect and the fabrication of nanostructures and devices based on the XMR of WTe 2 will represent a significant new direction in the study and uses of magnetoresistivity.Large magnetoresistance (MR) is an uncommon property, mostly displayed by magnetic', 'arxivid': '1405.0973', 'author': ['Mazhar N Ali \nDepartment of Chemistry\nPrinceton University\n08544PrincetonNew JerseyUSA\n', 'Jun Xiong \nDepartment of Physics\nJoseph Henry Laboratories\nPrinceton University\nPrinceton New Jer-sey 08544USA\n', 'Steven Flynn \nDepartment of Chemistry\nPrinceton University\n08544PrincetonNew JerseyUSA\n', 'Quinn Gibson \nDepartment of Chemistry\nPrinceton University\n08544PrincetonNew JerseyUSA\n', 'Leslie Schoop \nDepartment of Chemistry\nPrinceton University\n08544PrincetonNew JerseyUSA\n', 'Neel Haldolaarachchige \nDepartment of Chemistry\nPrinceton University\n08544PrincetonNew JerseyUSA\n', 'N P Ong \nDepartment of Physics\nJoseph Henry Laboratories\nPrinceton University\nPrinceton New Jer-sey 08544USA\n', 'Jing Tao \nDepartment of Physics\nBrookhaven National Laboratory\n11973UptonNew YorkUSA\n', 'R J Cava \nDepartment of Chemistry\nPrinceton University\n08544PrincetonNew JerseyUSA\n'], 'authoraffiliation': ['Department of Chemistry\nPrinceton University\n08544PrincetonNew JerseyUSA', 'Department of Physics\nJoseph Henry Laboratories\nPrinceton University\nPrinceton New Jer-sey 08544USA', 'Department of Chemistry\nPrinceton University\n08544PrincetonNew JerseyUSA', 'Department of Chemistry\nPrinceton University\n08544PrincetonNew JerseyUSA', 'Department of Chemistry\nPrinceton University\n08544PrincetonNew JerseyUSA', 'Department of Chemistry\nPrinceton University\n08544PrincetonNew JerseyUSA', 'Department of Physics\nJoseph Henry Laboratories\nPrinceton University\nPrinceton New Jer-sey 08544USA', 'Department of Physics\nBrookhaven National Laboratory\n11973UptonNew YorkUSA', 'Department of Chemistry\nPrinceton University\n08544PrincetonNew JerseyUSA'], 'corpusid': 119180228, 'doi': '10.1038/nature13763', 'github_urls': [], 'n_tokens_mistral': 7901, 'n_tokens_neox': 6624, 'n_words': 4117, 'pdfsha': '73ee152cb30a965ac39f1d9e17aa125cce8b055f', 'pdfurls': ['https://arxiv.org/pdf/1405.0973v3.pdf'], 'title': ['Titanic Magnetoresistance in WTe 2', 'Titanic Magnetoresistance in WTe 2'], 'venue': []}

arxiv

Bayesian inference of neutron-star observables based on effective nuclear interactions Jia Zhou Shanghai Institute of Applied Physics Chinese Academy of Sciences 201800ShanghaiChina University of Chinese Academy of Sciences 100049BeijingChina Jun Xu Shanghai Institute of Applied Physics Chinese Academy of Sciences 201800ShanghaiChina School of Physics Science and Engineering Tongji University 200092ShanghaiChina Shanghai Advanced Research Institute Chinese Academy of Sciences 201210ShanghaiChina Panagiota Papakonstantinou Rare Isotope Science Project Institute for Basic Science 34000DaejeonKorea Bayesian inference of neutron-star observables based on effective nuclear interactions Based on the Skyrme-Hartree-Fock model (SHF) as well as its extension (the Korea-IBS-Daegu-SKKU (KIDS) model) and the relativistic mean-field (RMF) model, we have studied the constraints on the parameters of the nuclear matter equation of state (EOS) from adopted astrophysical observables using a Bayesian approach. While the masses and radii of neutron stars generally favors a stiff isoscalar EOS and a moderately soft nuclear symmetry energy, model dependence on the constraints is observed and mostly originates from the incorporation of higher-order EOS parameters and difference between relativistic and non-relativistic models. At twice saturation density, the value of the symmetry energy is constrained to be 48 +15 −11 MeV in the standard SHF model, 48 +8−15MeV in the KIDS model, and 48 +5 −6 MeV in the RMF model, around their maximum a posteriori values within 68% confidence intervals. Our study helps to obtain a robust constraint on nuclear matter EOS, and meanwhile, to understand the model dependence of the results. Based on the Skyrme-Hartree-Fock model (SHF) as well as its extension (the Korea-IBS-Daegu-SKKU (KIDS) model) and the relativistic mean-field (RMF) model, we have studied the constraints on the parameters of the nuclear matter equation of state (EOS) from adopted astrophysical observables using a Bayesian approach. While the masses and radii of neutron stars generally favors a stiff isoscalar EOS and a moderately soft nuclear symmetry energy, model dependence on the constraints is observed and mostly originates from the incorporation of higher-order EOS parameters and difference between relativistic and non-relativistic models. At twice saturation density, the value of the symmetry energy is constrained to be 48 +15 −11 MeV in the standard SHF model, 48 +8 −15 MeV in the KIDS model, and 48 +5 −6 MeV in the RMF model, around their maximum a posteriori values within 68% confidence intervals. Our study helps to obtain a robust constraint on nuclear matter EOS, and meanwhile, to understand the model dependence of the results. I. INTRODUCTION Compact stars are natural laboratories for investigating properties of dense nuclear matter. Observables of neutron stars, such as their masses, radii, as well as the gravitational waves emitted from the mergers of binary stars, are helpful for understanding the equation of state (EOS) of nuclear matter in both the isoscalar and isovector channels [1,2], or in other words, the binding energy per nucleon E SN M in isospin symmetric nuclear matter and the energy excess due to the finite isospin asymmetry characterized by the nuclear symmetry energy E sym . For example, the mass of the neutron star is determined by the stiffness of the nuclear matter EOS, and the radius of the neutron star is closely related to the nuclear symmetry energy [3]. Thanks to the pioneer studies by nuclear physicists, E SN M (ρ) and E sym (ρ) around the saturation density ρ 0 are better constrained, compared to those at suprasaturation densities. For instance, the incompressibility K 0 characterizing the stiffness of E SN M (ρ) is constrained within 220 − 260 MeV from studies on the isoscalar giant monopole resonance (ISGMR) [4][5][6][7][8], and the value E 0 sym and the slope parameter L of the nuclear symmetry energy at the saturation density are constrained respectively within E 0 sym = 31.7 ± 3.2 MeV and L = 58.7 ± 28.1 MeV from surveying dozens of analyses [9,10]. Neutron star observables may help to constrain better higher-order EOS parameters characterizing E SN M and E sym at suprasaturation densities. It is encouraging to see that data on neutron star properties have been emerging in recent years, providing opportunities to constrain the nuclear matter EOS in the multimessage era of nuclear physics. From relativistic Shapiro time delay, the mass of PSR J0740+6620 was * Correspond to junxu@tongji.edu.cn measured to be 2.14 +0. 10 −0.09 M ⊙ [11] and later refined to be 2.08 +0.07 −0.07 M ⊙ [12], providing a large maximum mass to rule out soft EOS of neutron star matter. Based on data collected by Neutron Star Interior Composition Explorer (NICER), the radius of PSR J0740+6620 was further measured to be R = 13.7 +2.6 −1.5 km in Ref. [13] and R = 12.39 +1. 30 −0.98 km in Ref. [14]. For canonical neutron stars, their radii are estimated to be within R 1.4 = 10.62 − 12.83 km inferred from photospheric radius expansion bursts and thermal emissions [15]. The more recent measurements of PSR J0035+451 by NICER gave a mass of 1.44 +0. 15 −0.14 M ⊙ and a radius of R = 13.02 +1.24 −1.06 km in Ref. [16], and a mass of 1.34 +0.15 −0.16 M ⊙ and a radius of R = 12.71 +1.14 −1.19 km in Ref. [17], with the deduced radius slightly larger than that from Ref. [15] while there are significant overlaps. Besides, the analysis of GW170817 by the LIGO/Virgo Collaboration has found that the tidal deformability from the neutron star merger is constrained within Λ 1.4 = 190 +390 −120 [18] for canonical neutron stars. Observations of neutron stars have been used to constrain the nuclear matter EOS based on various models, e.g., the parameterized EOSs such as those directly using EOS parameters [19][20][21], using polytropic EOSs [22][23][24][25][26], and speed-of-sound models [27,28], and non-parametric models such as those using spectral methods [29] or Gaussian processes [30], as well as the chiral effective field theory [31][32][33][34]. In the present study, we investigate the constraints on the EOS from neutron star observables based on widely used effective nuclear models, i.e., the Skyrme-Hartree-Fock (SHF) model as well as its extension (the Korea-IBS-Daegu-SKKU (KIDS) model), and the relativistic mean-field (RMF) model. Compared with the parameterized EOS, these phenological models, which start from an effective nuclear interaction or a Lagrangian and give the nuclear matter EOS based on the mean-field approximation, have a more clear and better-defined theoretical basis. Another advantage of employing effective nuclear interaction models is that one can study neutron stars, heavy-ion reactions, and nuclear structures based on the same model with well-developed approaches, thus helpful for constraining the nuclear force and the nuclear matter EOS from high to low densities. In order to constrain quantitatively different EOS parameters and investigate their correlations under the constraints of multiple neutron star observables, we employ the Bayesian analysis in the present study. In previous studies [35][36][37], coefficients in these effective models have been successfully expressed inversely in terms of EOS parameters, so we are able to do Bayesian sampling in the space of EOS parameters rather than in that of model coefficients, making the Bayesian analysis more effective. We have also investigated the effect of the neutron star crust on the observables as well as its impact on the constraint of EOS parameters. While considerable model dependence on the final constraints is observed, a stiff E SN M and a moderately soft E sym at suprasaturation densities are favored by the adopted astrophysical observables based on the SHF model as well as its extension and the RMF model. The rest part of this manuscript is organized as follows. Section II reviews briefly the theoretical framework, including the formulism of the SHF model as well as its extension and the RMF model, the way to calculate neutron star observables based on effective interactions, and the Bayesian analysis method. Section III discusses the detailed constraints and correlations on EOS parameters from neutron star observables after a sensitivity analysis. Finally, we conclude and outlook in Sec. IV. II. THEORETICAL FRAMEWORK In the present study, compact stars are assumed to consist of only nucleons and leptons, and their properties are obtained from the EOS of neutron star matter based on the non-relativistic SHF model and the RMF model. We choose a standard energy-density functional (EDF) of the SHF model as in Ref. [35] as well as an extension for the density-dependent term, which was named as the KIDS model [38]. For the RMF model, we choose the Lagrangian form as in Ref. [37]. The chosen EDFs of the SHF, KIDS, and RMF models allow us to express model parameters inversely in terms of EOS parameters, so we are able to change a single physics quantity at one time while keeping the values of other quantities unchanged. The core-crust transition density is consistently calculated for a given set of EOS parameters based on the effective nuclear interaction, and different EOSs are used in the liquid core, inner crust, and outer crust of the neutron star, from which the neutron star properties can be obtained by solving the Tolman-Oppenheimer-Volkoff (TOV) equation as well as the coupled differential equation for the calculation of the tidal deformability. The EOS parameters are then constrained by comparing the resulting neutron star properties with the adopted astrophysical observables based on a Bayesian approach. A. Definition of EOS parameters In this subsection, we briefly remind the reader the definition of the EOS parameters that characterize the density dependence of E SN M (ρ) and E sym (ρ). The binding energy per nucleon in isospin asymmetric nuclear matter with nucleon density ρ = ρ n + ρ p and isospin asymmetry δ = (ρ n − ρ p ) ρ can be expressed as E(ρ, δ) = E SN M (ρ) + E sym (ρ)δ 2 + O(δ 4 ),(1) where the symmetry energy is defined as E sym (ρ) = 1 2 ∂ 2 E(ρ, δ) ∂δ 2 δ=0 .(2) The higher-order δ terms are generally much smaller, so the EOS is mostly dominated by E SN M (ρ) and E sym (ρ). Both E SN M (ρ) and E sym (ρ) contain contributions from the kinetic part and the potential part. While the kinetic part is calculated from the quasi-particle assumption, the potential part depends on the EDFs. Around the saturation density ρ 0 , E SN M (ρ) and E sym (ρ) can be expanded in the power of χ = ρ−ρ0 3ρ0 as E SN M (ρ) = E SN M (ρ 0 ) + K 0 2! χ 2 + Q 0 3! χ 3 + O(χ 4 ), E sym (ρ) = E sym (ρ 0 ) + Lχ + K sym 2! χ 2 + Q sym 3! χ 3 + O(χ 4 ). In the above, the linear term in the expansion of E SN M (ρ) vanishes due to zero pressure of SNM at ρ 0 . The independent EOS parameters relevant in the present study are the saturation density ρ 0 , the binding energy E 0 , the incompressibility K 0 , and the skewness parameter Q 0 of SNM at ρ 0 , the symmetry energy E 0 sym and its slope parameter L, curvature parameter K sym , and skewness parameter Q sym at ρ 0 , and they are defined respectively as ∂E SN M (ρ) ∂ρ ρ=ρ0 = 0,(3)E 0 ≡ E SN M (ρ 0 ),(4)K 0 = 9ρ 2 0 ∂ 2 E SN M (ρ) ∂ρ 2 ρ=ρ0 ,(5)Q 0 = 27ρ 3 0 ∂ 3 E SN M (ρ) ∂ρ 3 ρ=ρ0 ,(6)E 0 sym ≡ E sym (ρ 0 ),(7)L = 3ρ 0 ∂E sym (ρ) ∂ρ ρ=ρ0 ,(8)K sym = 9ρ 2 0 ∂ 2 E sym (ρ) ∂ρ 2 ρ=ρ0 ,(9)Q sym = 27ρ 3 0 ∂ 3 E sym (ρ) ∂ρ 3 ρ=ρ0 .(10) Neglecting the spin-orbit interaction, the effective interaction between nucleons at ⃗ r 1 and ⃗ r 2 in the standard SHF model can be expressed as v SHF (⃗ r 1 , ⃗ r 2 ) = t 0 (1 + x 0 P σ )δ(⃗ r) + 1 2 t 1 (1 + x 1 P σ )[ ⃗ k ′2 δ(⃗ r) + δ(⃗ r) ⃗ k 2 ] + t 2 (1 + x 2 P σ ) ⃗ k ′ ⋅ δ(⃗ r) ⃗ k + 1 6 t 3 (1 + x 3 P σ )ρ α ( ⃗ R)δ(⃗ r).(11) In the above, ⃗ r = ⃗ r 1 − ⃗ r 2 and ⃗ R = (⃗ r 1 + ⃗ r 2 ) 2 are respectively the relative and central coordinates for the two nucleons, ⃗ k = (∇ 1 − ∇ 2 ) 2i is the relative momentum operator and ⃗ k ′ is its complex conjugate acting on the left, and P σ = (1 + ⃗ σ 1 ⋅ ⃗ σ 2 ) 2 is the spin exchange operator. Based on the Hartree-Fock approach, the above effective interaction leads to the following energy density for uniform nuclear matter = k + 0 + SHF ρ + ef f ,(12) where the kinetic energy density k as well as the potential energy density 0 from the zero-range interaction, SHF ρ from the density-dependent interaction, and ef f from the momentum-dependent interaction can be expressed respectively as k = τ 2m , 0 = t 0 4 [(2 + x 0 )ρ 2 − (2x 0 + 1)(ρ 2 n + ρ 2 p )], SHF ρ = t 3 ρ α 24 [(2 + x 3 )ρ 2 − (2x 3 + 1)(ρ 2 n + ρ 2 p )], ef f = 1 8 [t 2 (2x 2 + 1) − t 1 (2x 1 + 1)](τ n ρ n + τ p ρ p ) + 1 8 [t 1 (2 + x 1 ) + t 2 (2 + x 2 )]τ ρ, with m being the bare nucleon mass and τ = τ n +τ p being the total kinetic density. For nucleons with isospin index q = n, p in a cold static nuclear matter, the kinetic density is τ q = p 5 F q 10π 2 , with p F q = (3π 2 ρ q ) 1 3 being the Fermi momentum. The parameters t 0 , t 1 , t 2 , t 3 , x 0 , x 1 , x 2 , x 3 , and α can be solved inversely from the macroscopic quantities [35], i.e., the saturation density ρ 0 , the binding energy E 0 at ρ 0 , the incompressibility K 0 , the isoscalar and isovector nucleon effective mass m ⋆ s and m ⋆ v at the Fermi momentum in normal nuclear matter, the value E 0 sym and the slope parameter L of the symmetry energy at ρ 0 , and the isoscalar and isovector density gradient coefficient G S and G V . As an extension of the above standard SHF EDF, the density-dependent term in the effective interaction [Eq. (11)] is replaced by the following form in the KIDS model v KIDS ρ (⃗ r 1 , ⃗ r 2 ) = 1 6 3 i=1 (t 3i + y 3i P σ )ρ i 3 ( ⃗ R)δ(⃗ r),(13) and the energy density is modified accordingly to = k + 0 + KIDS ρ + ef f ,(14) where KIDS ρ = 3 i=1 1 16 t 3i ρ 2+i 3 − 1 48 (t 3i + 2y 3i )ρ i 3 ρ 2 3(15) is the contribution from the density-dependent interaction, with ρ 3 = ρ n − ρ p being the isovector density. Compared to the standard SHF model, the additional coefficients in the KIDS model, i.e., t 3i and y 3i , allow us to vary more individual EOS parameters, i.e., Q 0 , K sym , and Q sym as shown in Ref. [36]. C. Relativistic mean-field model In the present study, we take the following Lagrangian form of the RMF model L = L nm + L σ + L ω + L ρ + L ωρ ,(16) with L nm =ψ(iγ µ ∂ µ − m)ψ + g σ σψψ − g ωψ γ µ ω µ ψ, − g ρ 2ψ γ µ ⃗ ρ µ ⃗ τ ψ, L σ = 1 2 (∂ µ σ∂ µ σ − m 2 σ σ 2 ) − A 3 σ 3 − B 4 σ 4 , L ω = − 1 4 F µν F µν + 1 2 m 2 ω ω µ ω µ + C 4 (g 2 ω ω µ ω µ ) 2 , L ρ = − 1 4 ⃗ B µν ⃗ B µν + 1 2 m 2 ρ ⃗ ρ µ ⃗ ρ µ , L ωρ = 1 2 α ′ 3 g 2 ω g 2 ρ ω µ ω µ ⃗ ρ µ ⃗ ρ µ . In the above, L nm is the contribution from the kinetic part of nucleons as well as its coupling to σ, ω, and ρ mesons, with ψ, σ, ω µ , and ⃗ ρ µ being the fields of nucleons and corresponding mesons, L σ , L ω , and L ρ are free and self-interacting terms of σ, ω, and ρ mesons, respectively, with ⃗ τ being the Pauli matrices, and L ωρ represents the cross interaction term between ω and ρ mesons. The antisymmetric field tensors F µν and ⃗ B µν are defined as F µν = ∂ ν ω µ − ∂ µ ω ν and ⃗ B µν = ∂ ν ⃗ ρ µ − ∂ µ ⃗ ρ ν − g ρ (⃗ ρ µ × ⃗ ρ ν ) . Based on the mean-field approximation, the meson fields are treated as classical fields, and applying the Euler-Lagrange equations leads to the following coupling equations for these fields m 2 σ σ = g σ ρ s − Aσ 2 − Bσ 3 ,(17)m 2 ω ω 0 = g ω ρ − Cg 4 ω ω 3 0 − α ′ 3 g 2 ω g 2 ρ ρ 2 0(3) ω 0 ,(18)m 2 ρ ρ 0(3) = 1 2 g ρ ρ 3 − α ′ 3 g 2 ω g 2 ρ ρ 0(3) ω 2 0 ,(19) where σ, ω 0 , and ρ 0(3) are expectation values of the meson fields at the ground state, with the subscript '0' representing the time component in the Dirac space, and the subscript '3' representing the z component in the Pauli space. ρ s = ρ sn + ρ sp is the scalar density, with the contribution from nucleons of isospin index q expressed as ρ sq = 2 m ⋆ q p 2 + m ⋆ q 2 d 3 p (2π) 3 ,(20) where m ⋆ q = m − g σ σ is the Dirac nucleon effective mass, different from the non-relativistic p-mass in the SHF model (see, e.g., Ref. [39]). The energy density can be expressed as = RM F k + 1 2 m 2 σ σ 2 + A 3 σ 3 + B 4 σ 4 − 1 2 m 2 ω ω 2 0 + g ω ω 0 ρ − C 4 (g 2 ω ω 2 0 ) 2 − 1 2 m 2 ρ ρ 2 0(3) + g ρ 2 ρ 0(3) ρ 3 − 1 2 α ′ 3 g 2 ω g 2 ρ ω 2 0 ρ 2 0(3) , (21) where RM F k = 2 q p 2 + m ⋆ q 2 − m d 3 p (2π) 3(22) is the kinetic energy contribution. The EOS in the RMF model is determined by g 2 σ m 2 σ , g 2 ω m 2 ω , g 2 ρ m 2 ρ , A, B, α ′ 3 and C. As shown in Ref. [37], the former 6 parameters can be expressed in terms of ρ 0 , E 0 , K 0 , E 0 sym , L, and m ⋆ s for a given C. In the present study, we are able to vary another independent EOS parameter Q 0 by adjusting the value of C, so there are totally 7 independent EOS parameters in the RMF model. For arbitrary values of these EOS parameters, the field equations [Eqs. (17)- (19)] do not necessarily have solutions in asymmetric nuclear matter at high densities, especially for ω 0 . However, we noticed that all RMF parameterization sets in Ref. [40] have C > 0 and α ′ 3 > 0, which guarantee that the field equations have solutions, and this condition is then used to rule out unphysical EOS parameter sets. In addition, the square of the coupling constants (g 2 σ , g 2 ω , and g 2 ρ ) calculated from the macroscopic physics quantities should be positive, adding to the limitation of the parameter space. For the quantitative limits of the parameter space from the present EDF of the RMF model, we refer the reader to Appendix A. D. Neutron star observables In the present study, we assume that the neutron star from the center to the surface contains the liquid core of uniform neutron star matter, the inner crust consists of nuclear pasta phase, and the outer crust is composed of ion lattice and relativistic electron gas. The neutron star matter contains neutrons, protons, electrons, and possibly muons if the charge chemical potential is large enough. The fraction of each component is determined by the β-equilibrium and the charge-neutrality condition. The total energy density of neutron star matter can be expressed as V = + ρm + l ,(23) where is obtained from the standard SHF [Eq. (12)], KIDS [Eq. (14)], or RMF [Eq. (21)] model, and l is the energy density of electrons and muons by assuming that they are free massive Fermions. The pressure of neutron star matter can be calculated through the relation P = P nuc + P l(24) where P nuc = q µ q ρ q −(25) is the pressure from nucleons, with the chemical potential for nucleons of isospin q obtained from µ q = ∂ ∂ρ q , and P l is the pressure from leptons. The thermodynamic consistency relation is satisfied for the global neutron star matter and for each component. The transition density ρ t between the liquid core and the inner crust is self-consistently determined with a thermodynamical approach as detailed in Refs. [41,42], i.e., below the transition density the system is unstable and satisfies the relation ∂µ n ∂ρ n ∂µ p ∂ρ p − ∂µ n ∂ρ p 2 < 0.(26) The EOS of the inner crust is parameterized as P = a + bV γ ,(27) where the default value of γ is taken to be 4 3 [43][44][45] while results from other values are compared in order to investigate the effect of the crust EOS on the constraints of the EOS parameters, and a and b are determined by the continuity condition [41] of the EOS at ρ t and at the boundary between the inner crust and the outer crust, with the density in the latter case taken to be ρ out = 2.46 × 10 −4 fm −3 . In the outer crust, we use the BPS EOS [46,47] in the density range 6.93 × 10 −13 fm −3 < ρ < ρ out , and we use the FMT EOS [46] in the density range of 4.73 × 10 −15 fm −3 < ρ < 6.93 × 10 −13 fm −3 . For special parameter sets that the neutron star matter is always stable, there is no core-crust transition, and we use the EOS of neutron star matter in the whole density range. More consistent studies using unified EOSs from core to crust can be found in Refs. [48][49][50][51]. With the EOS from high to low densities constructed above, the mass and radius of a neutron star can be calculated through the TOV equation dP (r) dr = − M (r)[V (r) + P (r)] r 2 1 + 4πP (r)r 3 M (r) × 1 − 2M (r) r −1 ,(28) where M (r) is the gravitational mass inside the radius r of the compact star and can be obtained from the integral of the following equation dM (r) dr = 4πr 2 V (r).(29) The tidal deformability Λ of compact stars during their merge is related to the love number k 2 through the relation Λ = 2 3 k 2 β −5 , with the latter given by [52][53][54] k 2 = 8 5 (1 − 2β) 2 [2 − y R + 2β(y R − 1)] × {2β[6 − 3y R + 3β(5y R − 8)] + 4β 3 [13 − 11y R + β(3y R − 2) + 2β 2 (1 + y R )] + 3(1 − 2β) 2 [2 − y R + 2β(y R − 1)]ln(1 − 2β)} −1 .(30) In the above, β ≡ M R is the compactness of the neutron star, and y R ≡ y(R) is the solution at the star surface to the first-order differential equation r dy(r) dr + y(r) 2 + y(r)F (r) + r 2 Q(r) = 0,(31) with F (r) = r − 4πr 3 [V (r) − P (r)] r − 2M (r) , Q(r) = 4πr 5V (r) + 9P (r) + V (r)+P (r) ∂P (r) ∂V (r) − 6 4πr 2 r − 2M (r) − 4 M (r) + 4πr 3 P (r) r 2 (1 − 2M (r) r) 2 .(32) For a given central density ρ(r = 0), the above equations can be solved from the center (r = 0) to the surface (r = R) where the density is lower than ρ cut ∼ 4.73 × 10 −15 fm −3 . For special parameter sets that the neutron star matter is always stable but the pressure becomes negative at low densities, the above equations are solved from the center to where the pressure becomes zero. E. Bayesian analysis To obtain the probability distribution functions (PDFs) of EOS parameters under the constraints of astrophysical observables, we employ the Bayesian approach, and the analysis method can be formally expressed as the Bayes' theorem P (M D) = P (D M )P (M ) ∫ P (D M )P (M )dM ,(33) where P (M D) is the posterior probability for the model M given the data set D, P (D M ) is the likelihood function or the conditional probability for a given theoretical model M to predict correctly the data D, and P (M ) denotes the prior probability of the model M before being confronted with the data. The denominator of the right-hand side of the above equation is the normalization constant. Since the coefficients in the standard SHF, KIDS, and RMF model can now be expressed in terms of physics quantities, we vary the physics quantities as model parameters in the Bayesian analysis. Due to the different numbers of coefficients in different models, the numbers of independent model parameters are also different. Table I lists the default values of model parameters in each model as well as their prior ranges. In the sensitivity analysis, we will check with the sensitivity of a single model parameter within its prior range to astrophysical observables, with the values of other model parameters fixed at their default values. In the Bayesian analysis, we will vary all independent model parameters within their prior ranges. We try to set the default values of model parameters to be the same so that the model dependence can be investigated on the same basis. In the standard SHF model, we try to get a two-solar mass neutron star by setting the default value of K 0 as the upper limit from its prior range obtained from studies on IS-GMR [4][5][6][7][8], while values of other quantities are taken as the default ones in the MSL0 force [35]. Q 0 , K sym , and Q sym are not independent quantities in the standard SHF model but are calculated from other quantities. In the KIDS model, Q 0 , K sym , and Q sym can be varied independently, while we set their default values as those calculated from the default parameter set for the standard SHF model. Although Q 0 is an independent variable in the RMF model, the parameter space is limited, so the largest available value of Q 0 is chosen as the default value. For the Dirac isoscalar effective mass in the RMF model, we set its default value to be m ⋆ s = 0.73m so that it corresponds effectively to the same non-relativistic isoscalar effective mass [39,55] as in SHF and KIDS models. The Dirac isovector effective mass is then m ⋆ v = 0.73m from the present RMF Lagrangian without δ-meson coupling. For the standard SHF model, there are totally 10 independent variables, and we choose to vary EOS parameters p 1 = K 0 uniformly within 220 − 260 MeV from IS-GMR studies [4][5][6][7][8], and p 2 = E 0 sym and p 3 = L uniformly within 28.5 − 34.9 MeV and 30 − 90 MeV, respectively, according to Refs. [9,10]. For the KIDS model, there are totally 13 independent variables, and we choose to vary higher-order EOS parameters p 4 = Q 0 , p 5 = K sym , and p 6 = Q sym uniformly within their prior ranges obtained based on analyses of terrestrial nuclear experiments and EDFs [56,57], in additional to those in the standard SHF model. For the RMF model, there are totally 7 independent variables, and we choose to vary EOS parameters p 1 = K 0 , p 2 = E 0 sym , p 3 = L, and p 4 = Q 0 . We also vary m ⋆ s m and m ⋆ v m in the standard SHF and KIDS models within their empirical ranges. In non-relativistic models, especially for KIDS [58], we expect that the effective masses are decoupled from the nuclear matter EOS, to be confirmed by the results. For the Dirac effect mass m ⋆ s m 0.5 ∼ 0.9 m ⋆ v m 0.7 0.7 0.73 ⋆ 0.5 ∼ 0.9 G S (MeVfm 5 ) 132 132 - - G V (MeVfm 5 ) 5 5 - - in the RMF model, it is expected to be closely related to the EOS, and we will vary it in the same empirical range. Results of representative astrophysical observables from a certain model parameter set are compared with data sets, for which we choose the radius d exp = Π i=1 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 2πσ i exp − (d th i − d exp i ) 2 2σ 2 i × Θ(M max − 2.08M ⊙ )Θ(1 − c s ) ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ ,(34) where the Heavyside functions put solid constraints that the maximum mass of the neutron star should be larger than 2.08M ⊙ and the speed of sound c s = ∂P ∂ inside a neutron star should be smaller than the speed of light, otherwise the likelihood function is zero. σ i are estimated from uncertainty ranges for the astrophysical data. In the case of asymmetric uncertainties, we use different values of σ i for d th i > d exp i and d th i < d exp i . Table II lists the values of d exp i as well as the corresponding uncertainty ranges to be used in the Bayesian analysis. III. RESULTS AND DISCUSSIONS We first do sensitivity analysis by changing each individual EOS parameter within its prior range and thus get a global picture how the resulting astrophysical observables change with these EOS parameters. Then, we vary all EOS parameters within their prior ranges and obtain the constraints on these EOS parameters as well as their correlations from the astrophysical data based on the Bayesian approach. We will also discuss the posterior EOS from the resulting constrained EOS parameters. served in Refs. [41,42], while it is not very sensitive to other EOS parameters. Both the radius and the tidal deformability of a canonical neutron star increase with increasing L but are not very sensitive to other EOS parameters. The maximum mass of a neutron star is found to be moderately sensitive to L but is not very sensitive to other EOS parameters. We note that the weak sensitivity of the maximum mass of a neutron star to K 0 is due to the small prior range of K 0 constrained by ISGMR, and the resulting Q 0 , which increases with increasing K 0 according to Eq. (3) and Fig. 16 of Ref. [36], also has a small range. If a neutron star with mass M = 2.08M ⊙ can be achieved, its radius becomes sensitive to most EOS parameters. We have also compared results with different EOSs for crust. Using a soft EOS (γ = 1) for inner crust increases the radius of a canonical neutron star by 1 − 2 km compared with a stiff EOS (γ = 1.5). Using the EOS of the neutron star matter as that for the crust, or identically by setting ρ t = 0, the radius of a canonical neutron star becomes even smaller. The EOS of the crust has a smaller effect on the radius of heavier neutron stars, and has a minor effect on the tidal deformability and the maximum mass of a neutron star. Figures 2 displays similar content as Fig. 1 but for the KIDS model, and additional dependencies on Q 0 , K sym , and Q sym are shown. It is seen that the core-crust transition density is most sensitivity to K sym rather than L. We note that K sym increases linearly with increasing L in the standard SHF model as shown by Eq. (4) of Ref. [36]. Both the radius and the tidal deformability of a canonical neutron star is sensitive to L, K sym , as well as higher-order EOS parameters Q 0 and Q sym . The maximum mass is again insensitive to K 0 , but most sensitive to Q 0 , and moderately sensitive to E 0 sym , L, and Q sym , within their prior ranges. Again, if a neutron star with mass M = 2.08M ⊙ can be achieved, its radius becomes sensitive to most EOS parameters. The crust EOS has larger effects on the radius of a canonical neutron star, smaller effects on the radius of a heavy neutron star, and minor effects on the tidal deformability and the maximum mass of a neutron star. In the case of ρ t = 0, the kinks for R 1.4 and Λ 1.4 at larger L are from the negative pressure at low densities of neutron star matter, so the TOV equation is solved until the pressure is zero rather than a density cut, as mentioned in Sec. II D. Figures 3 displays similar content as Fig. 1 but for the RMF model, and the individual variables K 0 , Q 0 , E 0 sym , L, and m ⋆ s m are varied independently within their limited parameter space with other parameters fixed at their default values. For example, the available ranges of K 0 , Q 0 , and m ⋆ s m are much smaller than the prior ranges as shown in Appendix A. Here K sym is not an independent variable, and the transition density decreases almost linearly with increasing L, similar to the standard SHF model. Incorporating Q 0 as an independent variable, the radius, the tidal deformability, and the maximum mass of a neutron star become sensitive to Q 0 , similar to the KIDS model. The moderate sensitivities of most astrophysical observables to the Dirac effective mass m ⋆ s m is a special feature in the RMF model compared to nonrelativistic models. The considerable sensitivity of the neutron-star radius and the less sensitivity of the tidal deformability and maximum mass to the crust EOS are also observed in Fig. 3. B. Constraints on EOS parameters For the default scenario by considering γ = 4 3 for the EOS of the inner crust, we now display in Fig. 4 the posterior correlated PDFs between lower-order and higher-order EOS parameters in both the isoscalar and isovector channels, as well as the correlated PDFs between the isoscalar and isovector EOS parameters. In the standard SHF model, it is seen that a smaller L is generally associated with a larger K 0 , due to the constraint of a large M max but a small R 1.4 . A slightly positive correlation between L and E 0 sym is observed in the same model, likely due to the opposite dependence of neutron star radii, Λ 1.4 , and M max on L and E 0 sym , as shown in Fig. 1. In the KIDS model, where both lowerorder and higher-order EOS parameters can be varied independently, there are no non-trivial correlations, except for the slightly negative correlation between L and K sym , likely due to the similar dependence of neutron star radii and Λ 1.4 on L and K sym , as shown in Fig. 2. In both the KIDS and RMF models, the sharp cut on the correlated PDF in the (Q 0 , K 0 ) plane is from the crite- rion M max > 2.08M ⊙ in the definition of the likelihood function [Eq. (34)]. Similar to the situation in the KIDS model, there are no significant correlations between K 0 and Q 0 , K 0 and L, or E 0 sym and L in the RMF model. Integrating over all the other variables leads to the onedimensional PDF of each individual physics quantity. We compare in Fig. 5 the posterior PDFs of EOS parameters and nucleon effective masses in the three models from the constraints of astrophysical observables using different crust EOSs. Q 0 , K sym , and Q sym in the standard SHF model are not independent variables, but are constrained through the posterior PDFs of other EOS parameters. Comparing the default scenario (γ = 4 3), without considering crust (ρ t = 0) may lead to significantly different constraints on most EOS parameters, depending on the chosen EDFs. A too soft EOS for the inner crust (γ = 1) may lead to a smaller L and/or a larger K sym , compared with results from γ = 4 3 and 1.5. Basically, the astrophysical observables do not put much constraint on K 0 and E 0 sym . On the other hand, a small L is favored by the small R 1.4 , while a large K sym is favored by the large R 2.08 , in all three models for the default case of γ = 4 3. The constraint on K sym is roughly consistent with −200 < K sym < 0 MeV extracted in Refs. [59,60] based on the KIDS EDF. A large Q sym is favored by the neutron star radii in the standard SHF model, while Q sym is not much constrained in the KIDS and RMF model. The constraint of M max favors a large Q 0 in the KIDS and RMF model. Due to the limited prior ranges of Q 0 and K sym in the standard SHF model and K sym in the RMF model, which are calculated from other variables, the corresponding posterior PDFs of these higher-order EOS parameters are narrower compared to those in the KIDS model. In the standard SHF model there are some constraints on the non-relativistic p-mass of nucleons, since these effective masses are related to higher-order EOS parameters, e.g., m ⋆ s is related to Q 0 and K sym according to Eqs. (3)(4)(5) in Ref. [36]. In the KIDS model, where higher-order EOS parameters can be varied independently, there are almost no constraints on these non-relativistic effective masses. In the RMF model, where the Dirac mass is closely related to the EOS, the constraint of M max favors a smaller Dirac effective mass of nucleons, corresponding to a stiffer E SN M . C. Constraints on EOS The parameters of effective models are constrained by the astrophysical observables through the Bayesian analysis, resulting in the constraints on the EOS of nuclear matter characterized by E SN M (ρ) and E sym (ρ) according to the EDF. Based on the three effective models, we compare the prior and posterior probability distributions of E SN M (ρ) and E sym (ρ) in Fig. 6, with the prior distribution obtained based on parameter ranges in Table I, and the posterior distribution from the Bayesian anal-ysis under the constraints of astrophysical observables. Since the major constraints from the adopted astrophysical observables are on the EOS around and above the saturation density, these figures are plotted in the density range from 0.5ρ 0 to 3ρ 0 . One sees that the prior distributions of both E SN M (ρ) and E sym (ρ) are broader in the KIDS model than in the standard SHF model, due to the larger parameter space in the KIDS model. While a broad prior distribution of E SN M (ρ) is seen in the RMF model, that of E sym (ρ) is very different from the other two models. A large neutron star mass favors a stiffer E SN M (ρ), especially for the KIDS model and the RMF model, where Q 0 is incorporated as an independent variable. The posterior E SN M (ρ) is even stiffer in the RMF model than in the KIDS model, since in the former case the resulting smaller Dirac effective mass also stiffens the EOS. While a very stiff E sym (ρ) is still favored to support a heavy neutron star in the standard SHF model, the radius data mostly favors a moderately soft E sym (ρ) at suprasaturation densities, corresponding to a small L and a large K sym from Fig. 5, based on all three models. The resulting soft E sym (ρ) is qualitatively consistent with results from other studies based on nucleonic models [19,31,[61][62][63], where different astrophysical observables are adopted. The symmetry energy at suprasaturation densities is of special interest for the nuclear physics community, and we compare its values at ρ = 1.5ρ 0 and 2ρ 0 from the constraints of astrophysical observables based on the three effective models in Fig. 7. As expected, the constraint on E sym (1.5ρ 0 ) is stronger than that on E sym (2ρ 0 ). One sees that the RMF model gives the most stringent constraint of the symmetry energy at suprasaturation densities, mostly due to the narrow prior range of E sym , compared to the other two models. Interestingly, despite of the different widths of the posterior PDFs, E sym (1.5ρ 0 ) peak around 38 MeV and E sym (2ρ 0 ) peak around 48 MeV for all three models. Within 68% confidence intervals, we obtain E sym (1.5ρ 0 ) = 38 +6 −5 MeV in the standard SHF model, E sym (1.5ρ 0 ) = 38 +6 −5 MeV in the KIDS model, and E sym (1.5ρ 0 ) = 38 +4 −4 MeV in the RMF model, and we obtain E sym (2ρ 0 ) = 48 +15 −11 MeV in the standard SHF model, E sym (2ρ 0 ) = 48 +8 −15 MeV in the KIDS model, and E sym (2ρ 0 ) = 48 +5 −6 MeV in the RMF model. Our constraints of E sym (2ρ 0 ) are in good agreement with the fiducial value of about 47 MeV (see Fig. 1 of Ref. [21] and corresponding discussions). IV. SUMMARY AND OUTLOOK Based on three effective nuclear interactions, we have studied the constraints on the EOS of both isoscalar and isovector channels from adopted astrophysical observables using the Bayesian approach. In all three models, i.e., the standard SHF model, the KIDS model, and the RMF model, a stiff isoscalar EOS is favored by the heavy mass of PSR J0740+6620. While a soft symmetry energy with a small L is favored by the empirical radii of canonical neutron stars, K sym > −200 MeV is favored by the radius of PSR J0740+6620. Due to the limit number of independent parameters in the SHF model, higherorder EOS parameters are related to lower-order ones, and correlation between EOS parameters are observed under the astrophysical constraints. With higher-order EOS parameters incorporated as independent variables, there are almost no such correlations between different EOS parameters in the KIDS and RMF models. The resulting smaller Dirac effective mass in the relativistic model further stiffens the isoscalar EOS compared to the non-relativistic models. In the RMF model, the parameter space is intrinsically limited in order to get physical solutions of model coefficients, and this leads to a different and actually more narrow constraint on the symmetry energy at suprasaturation densities. The symmetry energy at twice saturation density is constrained to be 48 +15 −11 MeV in the standard SHF model, 48 +8 −15 MeV in the KIDS model, and 48 +5 −6 MeV in the RMF model, within their 68% confidence intervals, and these values are in good agreement with those from state-of-art studies. In the present study, three models with different numbers of free parameters and EDF forms are compared. While there are some model dependencies, the constraints from the adopted astrophysical observables on the EOS, especially on the E sym (ρ) at ρ = 1 − 2ρ 0 , are robust and less sensitive to model details. Generally, increasing the number of parameters enhances the flexibility of the model and further enables it to explain better the data. On the other hand, a model with less number of free parameters but has a stronger prediction power is always favored. With the limited astrophysical observables adopted in the present study, although we are unable to judge the effectiveness of the three models, some lessons have been learnt. On the other hand, while all three models are nucleonic models, one can consider them as effective models to mimic the high-density EOS with hyperon or quark degrees of freedom. Non-parameterized models, e.g., studies using Gaussian processes [30], are free from the possible hadron-quark phase transition at high densities. Furthermore, it will be of great interest to constrain the EOS parameters from not only astrophysical observables but also nuclear structure data, e.g., neutron-skin thickness and nucleus resonances, based on different models. In that case, we can constrain the EOS from high to low densities, and have a deeper understanding on the performance of EDFs from effective nuclear interactions. Table I in the RMF model. 1 = 1R 1.4 of a canonical neutron star with M = 1.4M ⊙ , the radius d exp 2 = R 2.08 of PSR J0740+6620 with M = 2.08M ⊙ , and the tidal deformability d exp 3 = Λ 1.4 of a canonical neutron star. The likelihood function, which describes quantitatively how well the theoretical results d th 1,2,3,... reproduces the corresponding observables d exp 1,2,3,... , is defined as P [D(d 1 , d 2 , d 3 , ...) M (p 1 , p 2 , p 3 , ...)] FIG. 1 . 1Dependence of the core-crust transition density ρt (first row), the radius R1.4 (second row) and the tidal deformability Λ1.4 (third row) of a canonical neutron star, the maximum mass of the neutron star Mmax M⊙ (fourth row), and the radius R2.08 (fifth row) of a neutron star with mass M = 2.08M⊙ individually on K0, E 0 sym , L, m ⋆ s m, and m ⋆ v m within their prior ranges, with the values of other parameters fixed at their default values as inTable I, based on the standard SHF model. Results from different values of EOS coefficients γ for the inner crust are compared, together with those without considering the crust (ρt = 0). Figure 1 FIG. 2 . 12displays extensively how the core-crust transition density and relevant astrophysical observables change with each individual EOS parameters in the standard SHF model. The transition density is seen to decrease almost linearly with increasing L, as already ob-Similar toFig. 1but for the KIDS model showing dependence of observables individually on K0, Q0, E 0 sym , L, Ksym, Qsym, m ⋆ s m, and m ⋆ v m within their prior ranges. FIG. 3 . 3Similar to Fig. 1 but for the RMF model showing dependence of observables individually on K0, Q0, E 0 sym , L, m ⋆ s m, and m ⋆ v m within their prior ranges. FIG. 4 . 4Posterior correlated PDFs in the (L, K0) and (L, E 0 sym ) planes in the standard SHF model (top row), in the (Q0, K0), (L, K0), (L, E 0 sym ), and (L, Ksym) planes in the KIDS model (middle row), and in the (Q0, K0), (L, K0), and (L, E 0 sym ) planes in the RMF model (bottom row) from the constraints of astrophysical observables. FIG. 5 . 5Comparison of the posterior PDFs of K0, Q0, E 0 sym , L, Ksym, Qsym, m ⋆ s , and m ⋆ v in the standard SHF model (top row), the KIDS model (middle row), and the RMF model (bottom row) from the constraints of astrophysical observables with different crust EOSs. Results from different crust EOSs with different coefficients γ are compared, and the prior PDF of each individual quantity is also displayed. FIG. 6 . 6Probability distributions of E SN M (ρ) and Esym(ρ) from parameter ranges inTable I (prior) and under the constraints of astrophysical observables (posterior) in the standard SHF model (top), the KIDS model (middle), and the RMF model (bottom). FIG. 7 . 7Posterior and prior PDFs of Esym(ρ) at ρ = 1.5ρ0 (a) and 2ρ0 (b) in the standard SHF model, the KIDS model, and the RMF model. FIG. 8 . 8Illustration of the limited parameter space in the K0 − Q0 plane (a), the m ⋆ s m − Q0 plane (b), and the E 0 sym − L plane (c) with other parameters set as their default values in TABLE I . IDefault values of macroscopic quantities in the standard SHF, KIDS, and RMF models used in the present study. Quantities with asterisk are not independent ones but are calculated from other independent quantities. For inde- pendent quantities, they are varied within their prior ranges in the sensitivity analysis and Bayesian analysis. SHF KIDS RMF prior range ρ0 (fm −3 ) 0.16 0.16 0.16 - E0 (MeV) −16 −16 −16 - K0 (MeV) 260 260 260 220 ∼ 260 [4-8] Q0 (MeV) −323 ⋆ −323 −389 −800 ∼ 400 [56, 57] E 0 sym (MeV) 30 30 30 28.5 ∼ 34.9 [9, 10] L (MeV) 60 60 60 30 ∼ 90 [9, 10] Ksym (MeV) −105 ⋆ −105 -127 ⋆ −400 ∼ 100 [56, 57] Qsym (MeV) 214 ⋆ 214 474 ⋆ -200 ∼ 800 [56, 57] m ⋆ s m 0.8 0.8 0.73 TABLE II . IIMost probable values and uncertainties of adopted astrophysical observables for the Bayesian analysis.R1.4 (km) 11.725 ± 1.105 [15] R2.08 (km) 13.7 +2.6 −1.5 [13] and 12.39 +1.30 −0.98 [14] Λ1.4 190 +390 −120 [18] Mmax > 2.08M⊙ [12] cs < 1 . 0 4 0 . 0 8 ( b ) E s y m ( 2 ρ 0 ) ( M e V ) t h i c k : p o s t e r i o r t h i n : p r i o r Appendix A: Limited parameter space for theRMF modelIn the present study on neutron stars using the Lagrangian form as Eq.(16)in the RMF model, we set additional constraints of C > 0 and α ′ 3 > 0, otherwise the field equations [Eqs. 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{'abstract': 'Based on the Skyrme-Hartree-Fock model (SHF) as well as its extension (the Korea-IBS-Daegu-SKKU (KIDS) model) and the relativistic mean-field (RMF) model, we have studied the constraints on the parameters of the nuclear matter equation of state (EOS) from adopted astrophysical observables using a Bayesian approach. While the masses and radii of neutron stars generally favors a stiff isoscalar EOS and a moderately soft nuclear symmetry energy, model dependence on the constraints is observed and mostly originates from the incorporation of higher-order EOS parameters and difference between relativistic and non-relativistic models. At twice saturation density, the value of the symmetry energy is constrained to be 48 +15 −11 MeV in the standard SHF model, 48 +8−15MeV in the KIDS model, and 48 +5 −6 MeV in the RMF model, around their maximum a posteriori values within 68% confidence intervals. Our study helps to obtain a robust constraint on nuclear matter EOS, and meanwhile, to understand the model dependence of the results.', 'arxivid': '2301.07904', 'author': ['Jia Zhou \nShanghai Institute of Applied Physics\nChinese Academy of Sciences\n201800ShanghaiChina\n\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n', 'Jun Xu \nShanghai Institute of Applied Physics\nChinese Academy of Sciences\n201800ShanghaiChina\n\nSchool of Physics Science and Engineering\nTongji University\n200092ShanghaiChina\n\nShanghai Advanced Research Institute\nChinese Academy of Sciences\n201210ShanghaiChina\n', 'Panagiota Papakonstantinou \nRare Isotope Science Project\nInstitute for Basic Science\n34000DaejeonKorea\n'], 'authoraffiliation': ['Shanghai Institute of Applied Physics\nChinese Academy of Sciences\n201800ShanghaiChina', 'University of Chinese Academy of Sciences\n100049BeijingChina', 'Shanghai Institute of Applied Physics\nChinese Academy of Sciences\n201800ShanghaiChina', 'School of Physics Science and Engineering\nTongji University\n200092ShanghaiChina', 'Shanghai Advanced Research Institute\nChinese Academy of Sciences\n201210ShanghaiChina', 'Rare Isotope Science Project\nInstitute for Basic Science\n34000DaejeonKorea'], 'corpusid': 255999919, 'doi': '10.1103/physrevc.107.055803', 'github_urls': [], 'n_tokens_mistral': 26570, 'n_tokens_neox': 21929, 'n_words': 12341, 'pdfsha': '4a318f73f8c9888f3fa31c59f8907ce7535edb14', 'pdfurls': ['https://export.arxiv.org/pdf/2301.07904v2.pdf'], 'title': ['Bayesian inference of neutron-star observables based on effective nuclear interactions', 'Bayesian inference of neutron-star observables based on effective nuclear interactions'], 'venue': []}

arxiv

Feynman graphs and related Hopf algebras ccsd-00010155, version 1 -14 Oct 2005 G H E Duchamp Institut Galilée UMR 7030 LIPN CNRS 99 Av. J.-B. ClementF-93430VilletaneuseFrance P Blasiak pawel.blasiak@ifj.edu.pl H. Niewodniczański Institute of Nuclear Physics Polish Academy of Sciences ul. Eliasza-Radzikowskiego 15231342KrakówPLPoland A Horzela H. Niewodniczański Institute of Nuclear Physics Polish Academy of Sciences ul. Eliasza-Radzikowskiego 15231342KrakówPLPoland K A Penson penson@lptl.jussieu.fr Laboratoire de Physique Théorique de la Matière Condensée UMR 7600 Université Pierre et Marie Curie CNRS Tour 24 -2ièmeét., 4 pl. Jussieu75252, Cedex 05ParisFrance A I Solomon a.i.solomon@open.ac.uk Laboratoire de Physique Théorique de la Matière Condensée UMR 7600 Université Pierre et Marie Curie CNRS Tour 24 -2ièmeét., 4 pl. Jussieu75252, Cedex 05ParisFrance Physics and Astronomy Department Milton The Open University MK7 6AAKeynesUnited Kingdom Feynman graphs and related Hopf algebras ccsd-00010155, version 1 -14 Oct 2005 In a recent series of communications we have shown that the reordering problem of bosons leads to certain combinatorial structures. These structures may be associated with a certain graphical description. In this paper, we show that there is a Hopf Algebra structure associated with this problem which is, in a certain sense, unique. Introduction In a relatively recent paper Bender, Brody and Meister [3] introduced a special Field Theory described by a product formula (a kind of Hadamard product for two exponential generating functions -EGF) in the purpose of proving that any sequence of numbers could be described by a suitable set of rules applied to some type of Feynman graphs. Inspired by this idea, we have worked out combinatorial consequences of the product and exponential formulas in a recent series of papers [12,13,14,15,16,17,18]. Here, we consider two aspects of the product formula for formal power series applied to combinatorial field theories. Firstly, we remark that the case when the functions involved in the product formula have a constant term (equal to one) is of special interest as often these functions give rise to substitutional groups. The groups arising from the normal ordering problem of boson strings are naturally associated with explicit vector fields, or their conjugates, in the case when there is only one annihilation operator [14,17]. We also consider one-parameter groups of operators when several annihilators are present. Secondly, we discuss the Feynman-type graph representation resulting from the product formula. We show that there is a correspondence between the packed integer matrices of the theory of noncommutative symmetric functions and the labelled version of these Feynman-type graphs. We thus obtain a new Hopf algebra structure over the space of matrix quasisymmetric functions that is a natural cocommutative Hopf algebra structure on the space of diagrams themselves which originates from the formal doubling of variables in the product formula. Aknowledgements We would like here to express our gratitude to Jean-Louis Loday, Jean-Bernard Zuber, Jean-Yves Thibon and Florent Hivert for stimulating interactions on this subject. Single and double exponentials 2.1. One parameter groups and the connected graph theorem 2.1.1. Substitutions The Weyl algebra W is the C-associative algebra (with unit) defined by two generators a and a + and the unique relation [a, a + ] = 1. This algebra is of Gelfand-Kirillov dimension 2 and has a basis consisting of the following family (a + ) k a l k,l≥0 . It is known that it is impossible to represent faithfully a, a + by bounded operators in a Banach space, but one often uses the representation a → d dx ; a + → x as operators acting "on the line" or, better said, on the space of polynomials C[x]. Through this representation (faithful and coined under the name "Bargmann-Fock"), one sees that we can define a grading on W by the weight function w(a) = −1; w(a + ) = 1. A homogeneous operator (under this grading) Ω ∈ W is then of the form Ω = k,l; k−l=e c(k, l)(a + ) k a l(1) According to whether the excess e is positive or negative, the normal ordering of Ω n reads N (Ω n ) = (a + ) ne ∞ k=0 S Ω (n, k)(a + ) k a k or ∞ k=0 S Ω (n, k)(a + ) k a k (a) n|e|(2) We get combinatorial quantities with two indices i.e. an infinite N × N matrix {S Ω (n, k)} n,k≥0 which we will call the generalized Stirling matrix of Ω. In fact, it is easily checked that, if the coefficients c(k, l) of Ω are non-negative integers, so are the entries (S Ω (n, k)) of this matrix. Let us give some examples of these generalized Stirling matrices. For Ω = a + a, one gets the usual matrix of the Stirling numbers of the second kind                 1 0 0 0 0 0 0 · · · 0 1 0 0 0 0 0 · · · 0 1 1 0 0 0 0 · · · 0 1 3 1 0 0 0 · · · 0 1 7 6 1 0 0 · · · 0 1 15 25 10 1 0 · · · 0 1 31 90 65 15 For Ω = a + aa + + a + , we obtain                 1 0 0 0 0 0 0 · · · 2 1 0 0 0 0 0 · · · 6 6 1 0 0 0 0 · · · 24 36 12 1 0 0 0 · · · 120 240 120 20 1 0 0 · · · 720 1800 1200 300 30 1 0 · · · 5040 15120 12600 4200 630 42 1 · · · . . . . . . . . . . . . . . . . . . . . . . . .(4) and for w = a + aaa + a +             1 0 0 0 0 0 0 0 0 · · · 2 4 1 0 0 0 0 0 0 · · Let Ω = k,l≥0 c(k, l)(a + ) k a l (finite supported sum) be a general term of W in normal form and let us call a dominant term, the sum of monomials with maximum length k + l. It is not difficult to prove that, if Ω is homogeneous, the dominant term consists of a single monomial c(k 0 , l 0 )(a + ) k 0 a l 0 . Thus, the dominant term of Ω n must be c(k 0 , l 0 ) n (a + ) nk 0 a nl 0 . Then, for example in the case when e = k 0 − l 0 ≥ 0, in the generalized Stirling matrix of Ω, the rightmost non-zero coefficient of the line n has address (n, n.l 0 ) and bears the coefficient c(k 0 , l 0 ) n . All these matrices are row-finite and triangular iff l 0 = 1 (which means that no monomial possesses more than one a). Remark 2.1 i) There is a beautiful combinatorial expression of the normal form of w n in case w is a string in a and a + . The normal form of w is N (w) = k≥0 r(B, k)(a + ) r−k a s−k (6) where r(B, k) is the kth rook number of a certain board B constructed after w (see [13,25], and r = |w| a + ; s = |w| a are the number of occurences of a + and a in w. ii) To each matrix M ∈ C N×N of this kind and more generally "row finite" matrices (which means that, for each n, the family (M(n, k)) k∈N is finite supported), one can associate a transformation of EGFs (see [14,17]) f →f such that, if f = n≥0 a n z n n! thenf = n≥0 b n z n n! (with b n = k≥0 M(n, k)a k ). iii) It can be shown that, if no monomial of Ω possesses more than one a, the action of the transformation induced by Ω (through the Bargmann-Fock representation) can be expressed in terms of vector fields or their conjugates, thus the one-parameter group e λΩ acts by substitutions and products [14,17]. 2.1.2. Combinatorial matrices and one-parameter groups One can also draw generalized Stirling matrices from another source, namely from the combinatorial graph theory. Let C be a class of graphs such that Γ ∈ C iff every connected component of Γ is in C For these classes of graphs, one has the exponential formula [9,23,21] saying roughly that EGF(all graphs) = e EGF(Connected Graphs) This implies, in particular, that the matrix M(n, k) = number of graphs with n vertices and having k connected components (9) is the matrix of a substitution (see [14,17]). One can prove, using a Zariski-like argument (a polynomial vanishing for every integer vanishes everywhere), that, if M is such a matrix (with identity diagonal) then, all its powers (positive, negative and fractional) are substitution matrices and form a one-parameter group of substitutions, thus coming from a vector field on the line which can be computed. But no nice combinatorial principle seems to emerge. For example, beginning with the Stirling substitution z → e z − 1, we know that there is a unique one-parameter group of substitutions s λ (z) such that, for λ integer, one has the value (s 2 (z) ↔ partitionofpartitions) s 2 (z) = e e z −1 − 1 ; s 3 (z) = e e e z −1 −1 − 1 ; s −1 (z) = ln(1 + z)(10) but we have no nice description of this group nor of the vector field generating it. A product formula The Hadamard product of two sequences (a n ) n≥0 ; (b n ) n≥0 is given by the pointwise product (a n b n ) n≥0 . We can at once transfer this law on EGFs by n≥0 a n x n n! ⊙ exp n≥0 b n x n n! := n≥0 a n b n x n n!(11) In the following, we will omit the subscript (in ⊙ exp ) as this will be the only kind of Hadamard product under consideration. But, it is not difficult to check that the family (y ∂ ∂x ) n n! x m m! n,m∈N(12) is summable in C[[x, y]] (the space of formal power series in x and y) as we have (y ∂ ∂x ) n n! x m m! = 0 if n > m y n x m−n n!(m−n)! otherwise(13) and therefore, for F (x) = n≥0 a n x n n! and G(x) = n≥0 b n x n n! one gets the product formula (F ⊙ G)(x) := F (y ∂ ∂x )G(x)| x=0 = n≥0 a n b n y n n!(14) With this product, the set of series forms a commutative associative algebra with unit, which is actually the product algebra C N . The double exponential formula The case F (0) = G(0) = 1 will be of special interest in our study. Every series with constant term 1 can be represented by an exponential exp( n≥1 L n x n n! ) which can be expanded using Bell polynomials and Faà di Bruno coefficients. Let us now recall some facts about these combinatorial notions. We still consider the alphabet L = {L 1 , L 2 , · · ·} = {L i } i≥1 , then the complete Bell polynomials [7] are defined by exp( m≥1 L m x m m! ) = n≥0 Y n (L) x n n!(15) We will denote alternatively Y n (L 1 , · · · L n ) for Y n (L) as this polynomial is independent from the subalphabet (L m ) m>n . We know [7] that Y n (L) = Y n (L 1 , · · · L n ) = ||α||=n ((α))L α = ||α||=n ((α))L α 1 1 L α 2 2 · · · L αn n(16) where α = (α 1 , α 2 , · · · α n ) is an integral vector, ||α|| := m j=1 jα j , L α = L α 1 1 L α 2 2 · · · L αn n is the multiindex standard notation and ((α)) = ||α||! (1!) α 1 (2!) α 2 · · · (n!) αn (α 1 )! · · · (α n )!(17) is the Faà di Bruno coefficient [8,20] which will be interpreted, in the next section, as enumerating structures called set partitions. Combining (15) and (16) one gets exp m≥1 L m (y ∂ ∂x ) m m! exp n≥1 V n x n n! x=0 = k≥0 y k k! ||α||=||β|| ((α))((β))L α V β(18) Formula (18) will be called in the sequel the double exponential formula. Monomial expansion of the double exponential formula In this paragraph, we will use unordered and ordered set partitions. By an unordered partition P of the set X we mean a finite subset P ⊂ (P(X) − {∅}) (P(X) is the set of all subsets of X [7]) such that Y ∈P Y = Xand(Y 1 , Y 2 ∈ P, Y 1 = Y 2 =⇒ Y 1 ∩ Y 2 = ∅)(19) this explains why without any convention the classical Stirling number of second the kind S(0, 0) equals 1. The elements of P are called blocks. Following Comtet ([8] p 39), we will say that a partition P is of type α = (α 1 , α 2 , · · · , α m ) iff there is no j-block for j > m and α j j-block(s) for each j ≤ m. This implies in particular that the set X is of cardinality ||α|| := m j=1 jα j . Here one can see easily that the number of blocks of a partition of type α is |α| = m j=1 α j . An ordered partition of type α of the set X is just a partition in which the blocks are labelled from 1 to |α|. In other words, one could say that an ordered partition is a list of subsets and an unordered partition is a set of subsets. To every ordered partition P = (B 1 , B 2 , · · · , B |α| ) corresponds an unordered one Φ p (P ) = {B 1 , B 2 , · · · , B |α| } where Φ p is the "forgetful" function which forgets the order. Now to a pair (P (1) , P (2) ) of ordered partitions of the same set (call it X) P (1) = (B(1)1 , B(1) 2 , · · · , B (1) k 1 ) P (2) = (B(2) 1 , B 2 , · · · , B k 2 ) (20) one can associate the intersection matrix IM o (P (1) , P (2) ) such that the entry of address Formally, IM o (P (1) , P (2) ) is the matrix of size k 1 × k 2 such that IM o (P (1) , P (2) )[i, j] = card(B (1) i ∩ B (2) j )(22) The matrices obtained in such a way form the set of packed matrices defined in [19] as, indeed, one sees that every packed matrix can be obtained through the matching procedure illustrated above. If we consider now a pair of unordered partitions (Q (1) , Q (2) ), we cannot associate to them a single matrix but rather a class of matrices obtained from the preimages of (Q (1) , Q (2) ) under Φ p × Φ p . In a compact formulation, the set of matrices so obtained is IM 0 (P (1) , P (2) ) Φp(P (1) )=Q (1) ; Φp(P (2) )=Q (2) This is the orbit of one of them under permutation of lines and columns. The correspondence which, to a pair of unordered partitions, associates a class of matrices (under permutations of lines and columns) will be denoted IM u . Thus, one gets a commutative diagram of mappings Pairsoforderedpartitions Φp×Φp − −−− → Pairsofunorderedpartitions IMo     IMu Packedmatrices Class − −− → Classesofpackedmatrices Dgo     Dgu Labelleddiagrams Φ d − −− → Diagrams(26) The scheme presented above shows how to associate to a pair of ordered (resp. unordered) set partitions, a packed matrix (resp. a class of packed matrices). The packed matrices can be alternatively represented by labelled diagrams which are bipartite multigraphs built from two sets of vertices being a column of white spots (WS) and column of black spots (BS) as shown below. Let us explain how to associate to a (drawn) diagram a packed matrix. The white (resp. black) spots are labelled from 1 to r (resp. 1 to c) from top to bottom and the number of lines from the i-th white spot to the j-th black spot is exactly the entry a ij of the matrix. Conversely, a packed matrix of dimension r × c being given, one draws r white spots (resp. c black spots) and (with the labelling as above) join the i-th white spot to the j-th black by a ij lines. This gives exactly the one-to-one correspondence between (drawn) diagrams and packed matrices. In the sequel, we set Diag u := Dg u • IM u and Diag o := Dg o • IM o for the mappings which associate diagrams to pairs of partitions. Now, the multiplicity of a diagram D is the number of pairs (P (1) , P (2) ) of unordered partitions such that Dg u (IM u (P (1) , P (2) )) = D. Let us call bitype of a diagram D the pair (α(P (1) , α(P (2) ) where Dg u (IM u (P (1) , P (2) )) = D (remark that it does not depend on the choosen premiage inside the formula) and we will refer it as the bitype (α(D), β(D))) of D. In a similar way α(D) (resp. β(D)) will be called the left (resp. the right) type of D. Diagrammatic expansion of the double exponential formula The main interest of the expansion (27) is that we can impose (at least) two types of rules on the diagrams • on the diagrams themselves (selection rules) : on the outgoing degrees, ingoing degrees, total or partial weights (the graph is supposed oriented from white to black spots) • on the set of diagrams (composition and decomposition rules): product and coproduct on the space of diagrams. We have already such a structure on the space of monomials (i.e. the polynomials). The (usual) product of polynomials is well known and amounts to the addition of the multidegrees. The (usual) coproduct is given by the substitution of a "doubled" variable to each variable [4,20]. For example, with P = x 2 y 3 , we first form (x 1 + x 2 ) 2 (y 1 + y 2 ) 3 , expand and then separate (on the left) the "1" labelled variables and (on the right) the "2" labelled. As P = x 2 1 y 3 1 + 3x 2 1 y 2 1 y 2 + 3x 2 1 y 1 y 2 2 + x 2 1 y 3 2 + 2x 1 y 3 1 x 2 + 6x 1 y 2 1 x 2 y 2 + 6x 1 y 1 x 2 y 2 2 + 2x 1 x 2 y 3 2 + y 3 1 x 2 2 + 3y 2 1 x 2 2 y 2 + 3y 1 x 2 2 y 2 2 + x 2 2 y 3 2 (29) one gets, with ∆ the coproduct operator, ∆(P ) = x 2 y 3 ⊗ 1 + 3x 2 y 2 ⊗ y + 3x 2 y ⊗ y 2 + x 2 ⊗ y 3 + 2xy 3 ⊗ x + 6xy 2 ⊗ xy + 6xy ⊗ xy 2 + 2x ⊗ xy 3 + y 3 ⊗ x 2 + 3y 2 ⊗ x 2 y + 3y ⊗ x 2 y 2 + 1 ⊗ x 2 y 3 .(30) The space of polynomials with product and coproduct (and other items like neutrals, co-neutrals and antipode, which will be made more precise in the next paragraph) is endowed with the structure of a Hopf algebra. The last consideration suggests the following question: Is it possible to structure the (spaces of ) diagrams into a Hopf algebra ? Is it possible that this structure be compatible, in some sense, with the mapping (D, L, V, y) → m(D, L, V, y) ? Answer is yes. To establish it, we have to proceed in three steps. Second step. -We remark that, if d 1 ⋆ d 2 = d 1 d 2(32) denotes the superposition of the diagrams, then m(d 1 ⋆ d 2 , L, V, z) = m(d 1 , L, V, z)m(d 2 , L, V, z).(33) The law (32) makes sense as well for labelled and unlabelled diagrams. In the first case, it amounts to computing the blockdiagonal product of packed matrices. Indeed, for M 1 , M 2 being packed matrices, one has Dg o M 1 0 0 M 2 = Dg o (M 1 ) ⋆ Dg o (M 2 ).(34) This product yields the product of monomials in the following way. From D a diagram and all the other parameters fixed, with the setting of (28), we get a polynomial. The product (32) is associative with unit (the empty diagram), it is compatible with the arrow Φ d and so defines the product on Diag which, in turn is compatible with the product of monomials. Remark 3.1 One sees easily that the labelled diagram (resp. diagrams) form monoids thus the spaces LDiag C and Diag C are algebras of these monoids [2,1]. Labelleddiagrams 2 Φ d ×Φ d − −−− → Admissible coproducts For the coproduct on LDiag, we have several possibilities: (i) split with respect to the white spots (two ways : by intervals and by subsets) (ii) split with respect to the black spots (two ways : by intervals and by subsets) (iii) split with respect to the edges The discussion goes as follows: i) (3) does not give a nice identity with the monomials (when applying d → m(d, ?, ?, ?)) nor do (2) and (3) by intervals. ii) (2) and (3) are essentially the same (because of the WS ↔ BS symmetry). In fact (2) and (3) by subsets give a good representation and, moreover, they are appropriate for several physics models. In the next section, we develop the possibility (1) and (2) by subsets. Hopf algebra structures associated with ∆ W S and ∆ BS The philosophy of bi-and Hopf algebras thru representation theory Let A is a k-algebra (k is a field as R or C). In this paragraph, we consider associative algebras with unit (AAU). A representation of A is here a pair (V, ρ V ) where V is a k-vector space and ρ V : A → End k (V ) a morphism of k-algebras (AAU). One can make operations with representations as direct sums and quotient of a representation by a sub-representation (a sub-representation is a subspace which is closed under the action of A). In general, one does not know how to endow the tensor product (of two representations) and the dual (of a representation) with the structure of representation. It is however classical in two cases: groups and Lie algebras. If G is a group, a representation of G is a pair (V, ρ V ) where V is a k-vector space and ρ V : G → Aut k (V ) a morphism of groups. If G is a Lie algebra, a representation of G is a pair (V, ρ V ) where V is a k-vector space and ρ V : G → End k (V ) a morphism of Lie algebras (i.e. ρ V ([u, v]) = ρ V (u)ρ V (v) − ρ V (v)ρ V (u) ). These two cases enter the scheme of (AAU) as a representation of a group can be extended uniquely as a representation of its algebra kG and a representation of a Lie algebra as a representation of U k (G), its envelopping algebra. These two constructions (kG and U k (G)) are (AAU). For the sake of readibility let us denote in all cases ρ V (g)(u) by g.u (g ∈ G and u ∈ V ). If G is a group and V, W two representations, we construct a representation of G on V ⊗ W by g.(u ⊗ v) = g.u ⊗ g.v(36) If G is a Lie algebra and V, W two representations, we construct a representation of G on V ⊗ W by g.(u ⊗ v) = g.u ⊗ v + u ⊗ g.v(37) This can be rephrased in saying that the action of g in the first case (group) is g ⊗ g and in the second (Lie algebra) g ⊗ 1 + 1 ⊗ g (1 is here for the appropriate identity mapping). In the two cases, it amounts to give a linear mapping ∆ : A → A ⊗ A which will be called a coproduct. One can show [6] that, if we want that this new operation enjoy "nice" properties (associativity of the tensor product etc...), one has to suppose that this coproduct is a morphism of (AAU) (A ⊗ A has received the structure of -non twisted -tensor product of algebras), that it is coassociative with a counit [6]. Let us make these requirements more precise. The first says that for all x, y ∈ A one has ∆(xy) = ∆(x)∆(y), the second that the two compositions A ∆ −→ A ⊗ A ∆⊗1 A −→ A ⊗ A ⊗ AandA ∆ −→ A ⊗ A 1 A ⊗∆ −→ A ⊗ A ⊗ A (38) are equal, the third says that there is a mapping (linear form) ε : A → k such that the compositions A ∆ −→ A ⊗ A ∆⊗ε −→ A ⊗ k nat −→ AandA ∆ −→ A ⊗ A ε⊗∆ −→ k ⊗ A nat −→ A (39) (where nat is for the natural mappings) are equal to the identity 1 A . An algebra (AAU) together with a coproduct ∆ and a counity ε which fulfills the three requirements above is called a bialgebra. If, moreover one wants to have a nice dualization of the representations (i.e. nice structures for the duals V * = Hom(V, k)), it should exist an element of Hom(A, A) such that the compositions A ∆ −→ A ⊗ A α⊗1 A −→ A ⊗ A µ −→ AandA ∆ −→ A ⊗ A 1 A ⊗α −→ A ⊗ A µ −→ A(40) are equal to e A ε (where e A denotes the unit of A). When a bialgebra possesses such an element (unique), it is called the antipode and the bialgebra a Hopf algebra. For more details and connections to physics, one can consult [6]. One can prove that the bialgebras constructed below possess an antipode and then are Hopf algebras. Bialgebra structures on LDiag and Diag The space spanned by the packed matrices has already received a structure of Hopf algebra, the algebra MQSym [19]. We briefly review the structure of this Hopf algebra. We describe in details ∆ W S as the other coproduct is actually got by the same process but applied on the columns instead of the lines. Let M be a packed matrix of dimensions k 1 ×k 2 for every subset X ∈ [1..k 1 ] we consider the matrix π X (M) := pack(M[X, [1..k 2 ]]), the restriction to the lines of X and then packed (with this restriction to the lines, we only need to perform a horizontal packing). Thus, the coproduct ∆ W S reads ∆ W S (M) = X+Y =[1..k 1 ] π X (M) ⊗ π Y (M)(41) To avoid confusion we will call the supporting space H W S (= MQSym). We keep the (total) grading of MQSym by the total weight (i.e. the sum of the coefficients) of the matrices. The packed matrices are a linear basis of H W S = MQSym, thus every element expresses uniquely x = M packed λ M (x)M(42) The coproduct above is cocommutative and with counit λ ∅ where ∅ is the void matrix corresponding to the void diagram. This particular matrix will be denoted 1 H W S . For example, with the packed matrix above one has ∆ W S (    2 0 0 2 1 1    ) =    2 0 0 2 1 1    ⊗ 1 H W S + 2 ⊗ 0 2 1 1 + 2 ⊗ 2 0 1 1 + 1 1 ⊗ 2 0 0 2 + 2 0 0 2 ⊗ 1 1 + 2 0 1 1 ⊗ 2 + 0 2 1 1 ⊗ 2 + 1 H W S ⊗    2 0 0 2 1 1    This coproduct is compatible with the usual coproduct on the monomials for the constant alphabet V = 1 N defined by V n = 1 for all n ≥ 0. Then, using Sweedler's notation, for this particular V, if ∆ W S (d) = d (1) ⊗ d (2) , one has m(d, L ′ + L ′′ , 1 N , z) = m(d (1) , L ′ , 1 N , z)m(d (2) , L ′′ , 1 N , z) Thus, one sees easily that, with this structure (product with unit, coproduct and the counit), LDiag C is a bialgebra graded in finite dimensions and then a Hopf algebra. The arrow LDiag C → Diag C endows Diag C with a structure of Hopf algebra. Conclusion The structure of the Hopf algebras LDiag C , Diag C , by a theorem of Cartier, Milnor and Moore [5,22], is that of envelopping algebras of their primitive elements (Diag C , being commutative, is thus an algebra of polynomials). Moreover, it appears that the structure described above is the starting point for a series of connections with mathematical and physical Hopf algebras. The coproduct ∆ BS is the cristallisation (q = 1) of a one-parameter deformation of coproducts (all coassociative) on LDiag C ≃ MQSym, the other end (q = 0) being an infinitesimal coproduct isomorphic to ∆ MQSym . Recently, FQSym (a subalgebra of MQSym) has been established by Foissy [10] as a case in a family of Hopf algebras of decorated planar trees which is strongly related to other Hopf algebras like Connes-Kreimer's and Connes-Moscovici's [10,11]. P (i, j) is the number of elements of the intersection of the block i of the first partition and the block j of the second. For example with partitions of X = {1, (1) = ({1, 2, 5}, {3, 4, 6}) P (2) = ({1, 2}, {3, 4}, {5, 6}), one gets {1, 2} {3, 4} {5, 6} {1, For example, with (Q ( 1 ) with1, Q (2) ) = ( {{1, 2, 5}, {3, 4, 6}} , {{1, 2}, {3, 4}, {5, {5, 6}, {3, 4}, {1, 2}) are the preimages of Q (2) . The set of matrices so obtained reads IM u (M) D, L, V, y) := L α(D) V β(D) y |D| .(28) First step. -Let Diag C (resp. LDiag C ) be the C-vector space freely generated by the diagrams (resp. labelled diagrams) i.e. stage, we have a linear mapping (linear arrow) LDiag C → Diag C provided by the linear extension of Φ d and an arrow (linear, by construction) m(., L, V, z) : Diag C → C[L ∪ V ∪ {z}] provided by the linear extension of m(., L, V, z). Rational series and their languages. J Berstel, C Reutenauer, EATCS Monographs on Theoretical Computer Science. SpringerJ. Berstel, C. Reutenauer, Rational series and their languages EATCS Monographs on Theoretical Computer Science, Springer (1988). C Bertelle, G H E Duchamp, K Khatatneh, arXiv:cs.MA/0502081Tables, Memorized Semirings and Applications. C. Bertelle, G. H. E. Duchamp, K. Khatatneh, Tables, Memorized Semirings and Applications, arXiv: cs.MA/0502081. Quantum field theory of partitions. C M Bender, D C Brody, B K Meister, J. Math. Phys. 40C. M. Bender, D. C. Brody, and B. K. Meister, Quantum field theory of partitions, J. Math. Phys. Vol 40 (1999) . N Bourbaki, Chapitre Algèbre, Hermann Iii, Bourbaki N., Algèbre, chapitre III, Hermann ?? Sopus Lie. P Cartier, Séminaire, 2ème année, Faculté des Sciences de Paris. Cartier P., Séminaire "Sopus Lie", 2ème année, Faculté des Sciences de Paris (1955-56) A guide to quantum groups. V Chari, A Pressley, CambridgeV. Chari, A. Pressley, A guide to quantum groups. Cambridge (1994). . L Comtet, Analyse Combinatoire, PUF, Tome. 1L. Comtet, Analyse Combinatoire, PUF, Tome 1 (1970) . L Comtet, Analyse Combinatoire. 2L. Comtet, Analyse Combinatoire, PUF, Tome 2 (1970) . P Flajolet, P. Flajolet, http://algo.inria.fr/flajolet/ Isomorphisme entre l'algèbre des fonctions quasi-symétriques libres et une algèbre de Hopf des arbres enracinés décorés plans. L Foissy, personal communicationL. Foissy, Isomorphisme entre l'algèbre des fonctions quasi-symétriques libres et une algèbre de Hopf des arbres enracinés décorés plans, personal communication. Les algèbres de Hopf des arbres enracinés decorés, PhD Memoir. L Foissy, Reims UniversityL. Foissy, Les algèbres de Hopf des arbres enracinés decorés, PhD Memoir, Reims University (2002). Combinatorial Physics, Normal Order and Model Feynman Graphs. A I Solomon, P Blasiak, G Duchamp, A Horzela, K A Penson, arXiv:quant-ph/0310174Proceedings of the Symposium 'Symmetries in Science XIII. B. Gruber, G. Marmo and N. Yoshinagathe Symposium 'Symmetries in Science XIIIBregenz, AustriaKluwer Academic Publishers527A.I. Solomon, P. Blasiak, G. Duchamp, A. Horzela and K.A. Penson, Combinatorial Physics, Normal Order and Model Feynman Graphs, Proceedings of the Symposium 'Symmetries in Science XIII', Bregenz, Austria, 2003, B. Gruber, G. Marmo and N. Yoshinaga (eds.), p.527 (Kluwer Academic Publishers 2004) arXiv : quant-ph/0310174 Normal Order: Combinatorial Graphs Quantum Theory and Symmetries. A I Solomon, G Duchamp, P Blasiak, A Horzela, K A Penson, arXiv:quant-ph/0402082Proceedings of the 3rd International Symposium. P.C. Argyres, T.J. Hodges, F. Mansouri, J.J. Scanio, P. Suranyi, and L.C.R. Wijewardhanathe 3rd International SymposiumWorld Scientific Publishing398A.I. Solomon, G. Duchamp, P. Blasiak, A. Horzela and K.A. Penson, Normal Order: Combinatorial Graphs Quantum Theory and Symmetries, Proceedings of the 3rd International Symposium P.C. Argyres, T.J. Hodges, F. Mansouri, J.J. Scanio, P. Suranyi, and L.C.R. Wijewardhana (eds.), p.398 (World Scientific Publishing 2004) arXiv:quant-ph/0402082 One-parameter groups and combinatorial physics. G Duchamp, A I Solomon, K A Penson, A Horzela, P Lasiak, arXiv:quant-ph/04011262Proceedings of the Symposium Third International Workshop on Contemporary Problems in Mathematical Physics (COPROMAPH3). J. Govaerts, M. N. Hounkonnou and A. Z. Msezanethe Symposium Third International Workshop on Contemporary Problems in Mathematical Physics (COPROMAPH3)Porto-Novo, BeninWorld Scientific Publishing436G. Duchamp, A.I. Solomon, K.A. Penson, A. Horzela and P. B lasiak, One-parameter groups and combinatorial physics, Proceedings of the Symposium Third International Workshop on Contemporary Problems in Mathematical Physics (COPROMAPH3) (Porto- Novo, Benin, Nov. 2003), J. Govaerts, M. N. Hounkonnou and A. Z. Msezane (eds.), p.436 (World Scientific Publishing 2004) arXiv: quant-ph/04011262 Partition functions and graphs: A combinatorial approach. A I Solomon, G Duchamp, P Blasiak, A Horzela, K A Penson, arXiv:quant-ph/0409082Proceedings of the XI International Conference on Symmetry Methods in Physics (SYMPHYS-11). C. Burdik, O. Navratil, and S. Postathe XI International Conference on Symmetry Methods in Physics (SYMPHYS-11)Prague, Czech Republic; DubnaJINR PublishersA.I. Solomon, G. Duchamp, P. Blasiak, A. Horzela and K. A. Penson, Partition functions and graphs: A combinatorial approach, Proceedings of the XI International Conference on Symmetry Methods in Physics (SYMPHYS-11) (Prague, Czech Republic, June 2004), C. Burdik, O. Navratil, and S. Posta (eds.) (JINR Publishers, Dubna, 2004) arXiv:quant-ph/0409082 A product formula and combinatorial field theory. A Horzela, P Blasiak, G Duchamp, K A Penson, A I Solomon, arXiv:quant-ph/0409152Proceedings of the XI International Conference on Symmetry Methods in Physics (SYMPHYS-11). C. Burdik, O. Navratil, and S. Postathe XI International Conference on Symmetry Methods in Physics (SYMPHYS-11)Prague, Czech Republic; DubnaJINR PublishersA. Horzela, P. Blasiak, G. Duchamp, K. A. Penson and A.I. Solomon, A product formula and combinatorial field theory, Proceedings of the XI International Conference on Symmetry Methods in Physics (SYMPHYS-11) (Prague, Czech Republic, June 2004), C. Burdik, O. Navratil, and S. Posta (eds.) (JINR Publishers, Dubna) arXiv:quant-ph/0409152 Boson normal ordering via substitutions and Sheffer-Type Polynomials. P Blasiak, A Horzela, K A Penson, G H E Duchamp, A I Solomon, Phys. Lett. A. 338108P. Blasiak, A. Horzela, K. A. Penson, G. H. E. Duchamp, A.I. Solomon, Boson normal ordering via substitutions and Sheffer-Type Polynomials, Phys. Lett. A 338 (2005) 108 Some useful formula for bosonic operators. P Blasiak, K A Penson, A I Solomon, A Horzela, G H E Duchamp, Jour. Math. Phys. 4652110P. Blasiak, K. A. Penson, A.I. Solomon, A. Horzela, G. H. E. Duchamp, Some useful formula for bosonic operators, Jour. Math. Phys. 46 052110 (2005). Non commutative functions VI: Free quasisymmetric functions and related algebras. G Duchamp, F Hivert, J Y Thibon, International Journal of Algebra and Computation. 125G. Duchamp, F. Hivert, J. Y. Thibon, Non commutative functions VI: Free quasi- symmetric functions and related algebras, International Journal of Algebra and Computation Vol 12, No 5 (2002). S A Joni, G.-C Rota, Colgebras and Bialgebras in Combinatorics. 61S. A. Joni, G.-C.Rota, Colgebras and Bialgebras in Combinatorics, Studies in Applied Mathematics 61, 93-139 (1979). The art of computer programming Tome I. D Knuth, Addison-WesleyD. Knuth, The art of computer programming Tome I. Addison-Wesley (1981) On the structure of Hopf Algebras. J Milnor, J Moore, Ann. of Math. 81J. Milnor, J. Moore, On the structure of Hopf Algebras, Ann. of Math. 81 (1965), 211-264. . R P Stanley, Enumerative Combinatorics. 2Cambridge Univ. PressR. P. Stanley, Enumerative Combinatorics, vol. 2, Cambridge Univ. Press (2004). Hopf algebra structures in particle physics. S , arXiv:hep-th/0310124Eur. Phys. J. C. 33S. Weinzierl, Hopf algebra structures in particle physics, Eur. Phys. J. C 33 (2004) arXiv: hep-th/0310124 Rook numbers and the normal ordering problem. A Varvak, arXiv:math.CO/0402376PreprintA. Varvak, Rook numbers and the normal ordering problem, Preprint arXiv: math. CO/0402376

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{'abstract': 'In a recent series of communications we have shown that the reordering problem of bosons leads to certain combinatorial structures. These structures may be associated with a certain graphical description. In this paper, we show that there is a Hopf Algebra structure associated with this problem which is, in a certain sense, unique.', 'arxivid': 'cs/0510041', 'author': ['G H E Duchamp \nInstitut Galilée\nUMR 7030\nLIPN\nCNRS\n99 Av. J.-B. ClementF-93430VilletaneuseFrance\n', 'P Blasiak pawel.blasiak@ifj.edu.pl \nH. Niewodniczański Institute of Nuclear Physics\nPolish Academy of Sciences\nul. Eliasza-Radzikowskiego 15231342KrakówPLPoland\n', 'A Horzela \nH. Niewodniczański Institute of Nuclear Physics\nPolish Academy of Sciences\nul. Eliasza-Radzikowskiego 15231342KrakówPLPoland\n', 'K A Penson penson@lptl.jussieu.fr \nLaboratoire de Physique Théorique de la Matière Condensée\nUMR 7600\nUniversité Pierre et Marie Curie\nCNRS\nTour 24 -2ièmeét., 4 pl. Jussieu75252, Cedex 05ParisFrance\n', 'A I Solomon a.i.solomon@open.ac.uk \nLaboratoire de Physique Théorique de la Matière Condensée\nUMR 7600\nUniversité Pierre et Marie Curie\nCNRS\nTour 24 -2ièmeét., 4 pl. Jussieu75252, Cedex 05ParisFrance\n', '\nPhysics and Astronomy Department Milton\nThe Open University\nMK7 6AAKeynesUnited Kingdom\n', 'G H E Duchamp \nInstitut Galilée\nUMR 7030\nLIPN\nCNRS\n99 Av. J.-B. ClementF-93430VilletaneuseFrance\n', 'P Blasiak pawel.blasiak@ifj.edu.pl \nH. Niewodniczański Institute of Nuclear Physics\nPolish Academy of Sciences\nul. Eliasza-Radzikowskiego 15231342KrakówPLPoland\n', 'A Horzela \nH. Niewodniczański Institute of Nuclear Physics\nPolish Academy of Sciences\nul. Eliasza-Radzikowskiego 15231342KrakówPLPoland\n', 'K A Penson penson@lptl.jussieu.fr \nLaboratoire de Physique Théorique de la Matière Condensée\nUMR 7600\nUniversité Pierre et Marie Curie\nCNRS\nTour 24 -2ièmeét., 4 pl. Jussieu75252, Cedex 05ParisFrance\n', 'A I Solomon a.i.solomon@open.ac.uk \nLaboratoire de Physique Théorique de la Matière Condensée\nUMR 7600\nUniversité Pierre et Marie Curie\nCNRS\nTour 24 -2ièmeét., 4 pl. Jussieu75252, Cedex 05ParisFrance\n', '\nPhysics and Astronomy Department Milton\nThe Open University\nMK7 6AAKeynesUnited Kingdom\n'], 'authoraffiliation': ['Institut Galilée\nUMR 7030\nLIPN\nCNRS\n99 Av. J.-B. ClementF-93430VilletaneuseFrance', 'H. Niewodniczański Institute of Nuclear Physics\nPolish Academy of Sciences\nul. Eliasza-Radzikowskiego 15231342KrakówPLPoland', 'H. Niewodniczański Institute of Nuclear Physics\nPolish Academy of Sciences\nul. Eliasza-Radzikowskiego 15231342KrakówPLPoland', 'Laboratoire de Physique Théorique de la Matière Condensée\nUMR 7600\nUniversité Pierre et Marie Curie\nCNRS\nTour 24 -2ièmeét., 4 pl. Jussieu75252, Cedex 05ParisFrance', 'Laboratoire de Physique Théorique de la Matière Condensée\nUMR 7600\nUniversité Pierre et Marie Curie\nCNRS\nTour 24 -2ièmeét., 4 pl. Jussieu75252, Cedex 05ParisFrance', 'Physics and Astronomy Department Milton\nThe Open University\nMK7 6AAKeynesUnited Kingdom', 'Institut Galilée\nUMR 7030\nLIPN\nCNRS\n99 Av. J.-B. ClementF-93430VilletaneuseFrance', 'H. Niewodniczański Institute of Nuclear Physics\nPolish Academy of Sciences\nul. Eliasza-Radzikowskiego 15231342KrakówPLPoland', 'H. Niewodniczański Institute of Nuclear Physics\nPolish Academy of Sciences\nul. Eliasza-Radzikowskiego 15231342KrakówPLPoland', 'Laboratoire de Physique Théorique de la Matière Condensée\nUMR 7600\nUniversité Pierre et Marie Curie\nCNRS\nTour 24 -2ièmeét., 4 pl. Jussieu75252, Cedex 05ParisFrance', 'Laboratoire de Physique Théorique de la Matière Condensée\nUMR 7600\nUniversité Pierre et Marie Curie\nCNRS\nTour 24 -2ièmeét., 4 pl. Jussieu75252, Cedex 05ParisFrance', 'Physics and Astronomy Department Milton\nThe Open University\nMK7 6AAKeynesUnited Kingdom'], 'corpusid': 8699665, 'doi': '10.1088/1742-6596/30/1/014', 'github_urls': [], 'n_tokens_mistral': 11972, 'n_tokens_neox': 10445, 'n_words': 6115, 'pdfsha': 'c6493f9f5f845005fb900794c557d1d306ef8791', 'pdfurls': ['https://web.archive.org/web/20120624225237/http:/hal.archives-ouvertes.fr/docs/00/03/83/86/PDF/GHED-Myczkowce_f.pdf'], 'title': ['Feynman graphs and related Hopf algebras', 'Feynman graphs and related Hopf algebras', 'Feynman graphs and related Hopf algebras', 'Feynman graphs and related Hopf algebras'], 'venue': []}

arxiv

Research Opportunities in Sociotechnical Interventions for Health Disparity Reduction Research Opportunities in Sociotechnical Interventions for Health Disparity Reduction Sponsored by the Computing Community Consortium RESEARCH OPPORTUNITIES IN SOCIOTECHNICAL INTERVENTIONS FOR HEALTH DISPARITY REDUCTION 1. Overview .1 2. Developing Equity-centered Intervention Strategies and Implementation Approaches . 3 2.1 Upstream Interventions . 3 2.2 Equity-Centered Intervention Uptake and Study Recruitment .5 2.3 Equity-Centered Engagement/Adherence and Study Retention . 6 3. Sociotechnical Black Boxes . 7 3.1 Participatory Methods for Study and Technology Design . 8 3.2 Understanding Data Quality in Existing Systems .9 3.3 Designing Dosing Schemes .10 4. Sociotechnical Systems to Inform Theory .11 4.1 Building Better Theories: New Opportunities .11 4.2 Tailoring and Optimization of Sociotechnical Systems .13 5. Multidimensional Evaluation to Reduce Health Disparities at the Population Level .13 5.1 Improving Measurement and Methods for Multidimensional Evaluation June 2019 Syed Haider Eric Hekler Predrag Klasnja Donna Spruit-Metz Katie Siek Tiffany Veinot Beth Mynatt Research Opportunities in Sociotechnical Interventions for Health Disparity Reduction Research Opportunities in Sociotechnical Interventions for Health Disparity Reduction Sponsored by the Computing Community Consortium RESEARCH OPPORTUNITIES IN SOCIOTECHNICAL INTERVENTIONS FOR HEALTH DISPARITY REDUCTION 1. Overview .1 2. Developing Equity-centered Intervention Strategies and Implementation Approaches . 3 2.1 Upstream Interventions . 3 2.2 Equity-Centered Intervention Uptake and Study Recruitment .5 2.3 Equity-Centered Engagement/Adherence and Study Retention . 6 3. Sociotechnical Black Boxes . 7 3.1 Participatory Methods for Study and Technology Design . 8 3.2 Understanding Data Quality in Existing Systems .9 3.3 Designing Dosing Schemes .10 4. Sociotechnical Systems to Inform Theory .11 4.1 Building Better Theories: New Opportunities .11 4.2 Tailoring and Optimization of Sociotechnical Systems .13 5. Multidimensional Evaluation to Reduce Health Disparities at the Population Level .13 5.1 Improving Measurement and Methods for Multidimensional Evaluation June 2019 The authors gratefully acknowledge the contributions of the workshop planning committee members to the design and facilitation of the workshop, which formed the basis of this document. Committee members included Heather Cole-Lewis, 1. Overview The implicit and explicit biases built into our computing systems [1] are becoming increasingly clear -they impact everything from targeting of advertisements [2] to how we are identified as people [3]. These biases disproportionately affect marginalized groups -people who are excluded from mainstream social, economic, cultural, or political life [4] -more acutely. While these biases can affect all aspects of our lives, from leisure [5] to criminal justice [6] to personal finances [7], they are all the more critical in the context of health and healthcare due to their significant personal and societal implications. In this interdisciplinary workshop, we explored how to design and build health systems for diverse populations through the following disciplinary lenses. Human computer interaction (HCI) researchers address the growing need to empower lay populations to manage their health by designing, developing, and deploying novel sociotechnical interventions [8,9]. As research in interactive systems in healthcare has matured, computing and health informatics researchers have increasingly drawn upon social and behavioral science theories [10] to design, develop, and analyze sociotechnical systems. Health informatics researchers focus on basic research concerning patient information needs [11] and healthcare-oriented topics such as implementation of technologies in healthcare contexts, technical standards, health policy, impacts on healthcare quality, and access to, and uptake of, technologies [12]. Researchers also concentrate on the development of analytical techniques and algorithms focused on applied clinical problems such as illness diagnosis and prognosis. Behavioral medicine researchers explore psychosocial mechanisms underlying health behavior -from determinants of behavior to how behavior is changed. Additionally, behavioral medicine research has a longstanding research focus on health disparities. At the same time, behavioral medicine researchers have traditionally developed health behavior theories and models through participant self-report or by utilizing commodity systems to evaluate the theory at scale. Health disparity researchers investigate the prevalence and underlying correlates of health disparities, typically using observational study methods originating in epidemiology, such as cohort and case-control designs. Additionally, clinical epidemiologists contribute methods in the areas of research synthesis, with a recent focus on equity-focused systematic reviews that can inform intervention design [13][14][15]. Critically, reduction of health disparities (see box 1) through socio-technical interventions requires the knowledge and methods of each of these fields. Because health disparities are rooted in a variety of social, behavioral, economic, and healthcare-based factors, there is a need for researchers to consider the insights and research methods offered by each of these fields when designing and deploying interventions [16]. Furthermore, designing interventions that will be engaging to, and usable by, health disparity populations is a prerequisite for intervention impact -critical insights about which can be provided from different perspectives in each of these fields. Moreover, because interventions that could work well for health disparity populations may not be available to or readily adopted by them, there is a need to consider policy and implementation issues such as integration with healthcare systems and workflows, technology platforms, and incentives [16,17] -challenges which researchers from these four fields are best positioned to tackle. There is also a need to incorporate understanding of the mechanisms driving different health disparities into design, implementation, and evaluation. Assessing the equity impact of interventions [18] in the context of specific studies, or across studies, is also critical. The four themes were explored through two short multidisciplinary panels and coordinated discussions followed by summarizing presentations to ensure that researchers from different disciplines had the opportunity to listen, learn, and share with each other. The researchers identified major research challenges and opportunities within each theme, specifically the need to: ◗ Develop and evaluate equity-centered intervention strategies and implementation approaches. Prevailing intervention strategies, which often focus on individual patient effort, behavior and choice may be less effective for marginalized populations -supporting a greater focus on upstream and multi-level interventions. Furthermore, existing approaches for implementing systems (e.g., promoting uptake and ongoing usage) tend to favor advantaged groups. There is a need for new approaches that can ensure equitable outcomes, as well as uptake and usage, of effective interventions. Upstream Interventions The extension of the World Health Organization's model on health disparities, shown in Figure 1 LGBT people [20,21], whereas residential segregation is a fundamental cause of health disparities among African Americans [22,23]. Accordingly, different intervention foci and strategies may be needed to influence social hierarchies depending on the group that is targeted in an intervention. Workshop participants discussed the key capabilities of technology which can facilitate upstream interventions. [77] information. [26] and experiences of racism in health care [27]. Lower trust in technology may also play a role in the uptake of sociotechnical interventions, as trust is an antecedent to technology adoption and use [28][29][30]. Community-based participatory research (CBPR), an approach emerging from the public health field, has been successfully applied in many observational and interventional studies, including those using sociotechnical interventions [31]. CBPR Equity-Centered Engagement/ Adherence and Study Retention Workshop attendees discussed difficulties with both differential engagement/adherence with socio-technical interventions and study retention, which was defined as a research subject continuing in the study until the last data collection point. Workshop participants also noted that, in field studies of sociotechnical interventions, it may be the case that subjects engage with and adhere to an intervention while still dropping out of a study. Workshop attendees identified challenges in defining active engagement with sociotechnical interventions since the term "engagement" is not well defined. Some researchers defined engagement as usage, and others as more of a subjective experience. Subjective engagement has been linked to ongoing use of sociotechnical interventions in health [32]. Despite this conceptual distinction, much research has focused on engagement operationalized as intervention usage levels. When operationalized as usage, a number of engagementfocused studies have shown that people with less formal education (an indicator of SES) use sociotechnical interventions less than those with more education, regardless of the intervention's level of structure [33][34][35][36][37][38][39][40]. Similarly, study retention, focused on completion of all points of data collection, is characterized by lower completion among those with less formal education. At the same time, published papers typically do not report on dropout rates and the demographics of non-users, less-engaged users, and study dropouts. Workshop participants highlighted the importance of using technology design strategies for sociotechnical interventions that can assist in reaching those who most need them. Participants' successful experiences supported a process involving needs assessment, participatory technology design, and community partnerships. These partnerships were most successful when faithful to the principles of CBPR, including equity between academic and community partners. As Cortés found in urban communities [41], such models may also align with the expectations of marginalized groups for research involvement. Community involvement in developing strategies for promoting intervention engagement was also believed valuable. When planning evaluation studies, participants advocated an experimental design that includes a specific protocol for engagement and retention. They also recommended it is unclear how long that change will be sustained. We use the term "black box" to highlight these ambiguities. Sociotechnical Black Boxes Although improved health outcomes as a result of a sociotechnical intervention are exciting, they are not enough -we must also understand the mechanisms behind the change and potential "side effects" so that we can reproduce the changes and continue improving on them. We outline the topics discussed by Participatory Methods for Study and Technology Design Understanding Data Quality in Existing Systems In order to identify how sociotechnical black boxes work, researchers must consider the quality of the data generated in a socio-technical system. Workshop participants approached data quality from two viewpoints -methodology and provenance. Methodology refers to a study's design and how it can impact data quality generated from participants, instruments, systems, and study components. Provenance refers to the quality of data streams that people, technology, and inferences (e.g., machine learning) generate. Computing researchers often investigate novel interactions, technologies, and infrastructures by conducting pilot studies [46] -which are not always recognized by health-oriented fields because of their small size, short duration, or lack of statistical power. In addition, although there are computing researchers investigating how to assist marginalized groups in improving their health, most studies are fairly short and difficult to compare [9]. Researchers, however, stressed the importance of pilot studies and their important role in helping to ensure that starting conditions for interventions are correct. This view is not unique to computing; indeed, public health researchers have advocated for treating pilot studies as an integral part of the scientific process [47]. In addition, computing and behavioral medicine researchers have encouraged their communities to better report on data and contributions to identify causal effects of behavior change [48]. When conducting larger studies, particularly to assess health outcomes, a challenge is the difficulty researchers encounter when recruiting and retaining marginalized participants; thus smaller study samples may remain common. Two promising ways to strengthen conclusions with smaller samples and thereby overcome the research-to-practice gap include adopting models that iteratively design sociotechnical systems that are eventually sustainable without researchers [49] and modularizing sociotechnical systems into the bare components to identify their effectiveness -even on smaller sample sizes (e.g., agile science [50]). During study analysis, researchers are strongly encouraged to consider the provenance of the data streams that people, technology, and inferences generate. There is also a need to improve our ability to account for the impact of complex social relationships on data collection and use in some groups (e.g., parent-child; patient-provider) and for the impact of user characteristics (e.g., age, health literacy) and environments (e.g., rural vs. urban areas) on data quality [51,52]. When workshop participants discussed all of the ways in which researchers can collect data -from instruments to data streams -participants raised questions about how much data to collect in a given study. This is important in relation to both user burden and future-proofing the set of measures in the event of novel research questions which may emerge over time. In general, lower user burden is associated with more successful data collection -a phenomenon which may be amplified with marginalized groups, such as people with low SES [51]. This concern would tend to favor collection of less data. Indeed, for a given goal or set of questions, a small set of measurements, taken infrequently and with a focus on trends or absolute accuracy, may be all that is required. However, future-proofing may favor collection of more Designing Dosing Schemes We must consider the ideal or actual "dosage" of Sociotechnical Systems to Inform Theory Building Better Theories: New Opportunities Traditionally, use of theory has been one-way -an A promising area of interdisciplinary research for theory development is just-in-time adaptive systems [56,57]. In this approach, depending on one's dynamic behavior and context, the sociobehavioral model is updated along with the sociotechnical systems' interactions with the world, thus creating a dynamic system (green, dashed arrows in Figure 3) that can adapt and provide relevant information to the user and research teams. In adaptive systems researchers must address many challenges: ◗ Multidimensional Evaluation to Reduce Health Disparities at the Population Level The multidisciplinary workshop provided participants with ample time to share experiences designing, implementing, and evaluating studies at various levels of granularityfrom individuals to families to communities and, finally, to the population level. Workshop participants discussed the need to measure multiple dimensions -contextual, structural, and social determinants of health (e.g., Figure 1, page 5) -to better evaluate changes with respect to health disparities. Improving Measurement and Methods for Multidimensional Evaluation For each project, researchers and intervention designers must decide upon the "right" set of factors to measure -balancing trade-offs in practicality, comprehensiveness, strategic value, and risk. Researchers may select measures to produce novel findings that are informative (e.g., describe what is happening or why) and/or actionable (e.g., lead to design or adoption of new interventions). Researchers may also adapt measures from a well-understood intervention to ensure that it works as they scale it up or roll it out to practice. A caveat is that researchers should not limit themselves to what can easily be measured. Indeed, "real world" success may be especially difficult to assess. Measurement With regard to what to measure, it is important to measure health equity-relevant outcomes, which may assess different types of equity (e.g., healthcare access, financial access, health behaviors, health literacy, healthcare quality, health-related outcomes). It is also important to know how well a specific intervention actually implemented a theory, and to know what parts of an intervention generate treatment effects. To this end, researchers must evaluate the quality of prototype systems, intervention usage, and theoretical mechanisms of action (mediators) at different stages of the posited causal pathways, as well as the outcomes that the intervention is intended to influence. Methods Like technological interventions, upstream and multilevel interventions may also fail to fit well into the existing Randomized Controlled Trial (RCT) paradigm. Implementation may be "nonlinear, iterative, and adaptive" Therefore, participants advocated additional research on how to best facilitate re-use [49,50]. In a health disparity context, it may be necessary to facilitate re-use for older technologies that are more widely used in low-SES groups, such as interactive voice response, SMS, and 2-G telephones [51]. In addition to collecting data for analysis, data can be used to design predictive models that provide opportunities to intervene to amplify the treatment effect (outcome), or mitigate intervention risks. However, these models are only useful if we can characterize highly complex behavior systems through interactions between sub-systems and dynamics over time. Workshop participants challenged the scientific community to consider: ◗ How can we better measure intervention effects across micro, meso, and macro levels? How can we effectively account for interactions between different levels of outcomes? ◗ How can criteria specifically related to marginalized groups and the social factors that drive disparities be incorporated into sociotechnical interventions as optimization and/or tailoring criteria? ◗ How can we best facilitate re-use of measures, data and technical frameworks specifically for marginalized populations? ◗ How can we characterize complex systems in predictive models regarding disparities? Assessing Equity Impacts and Unintended Consequences Since Additionally, for interventions that already exist (e.g., self-tracking of eating, exercise, or symptoms) there is a need for equity-focused, interdisciplinary systematic reviews that aggregate individual patient data across studies. One benefit of this approach would be the ability to include marginalized participants from studies which originally lacked statistical power for HTE analyses. Open questions include: ◗ What, if any, "universal" intervention types and designs perform better in marginalized groups than the general population? Ethics of Conducting Sociotechnical Research with Marginalized Groups Responsibility Workshop participants emphasized the view that (2) creating circumstances that enable more people and groups to help themselves, and then share their learning and insights with others. Return of Results Risk-Benefit Ratio Informed Consent With For a given intervention researchers should consider asking themselves: ◗ Who can easily access the intervention? What barriers might marginalized groups face in trying to access it? ◗ What are the potential unintended consequences of my research, particularly for marginalized populations? ◗ What is the impact of the intervention on health equity? ◗ Was there heterogeneity of treatment effect? If so, for whom, and how much was the effect size difference? ◗ Who engaged with the intervention? What was the impact of any differential engagement levels on intervention effects? ◗ Is there differential dropout or abandonment of the intervention in marginalized groups? ◗ What would the outcomes be in marginalized groups who were not reached? The consortium or coordinated national centers could have annual "themes" to drive collective action (e.g. "measuring stress") and teams could contribute measurement tools and data with respect to study design, recruitment, retention, sociobehavioral models, and dosing related to specific populations. The consortium/national centers could put researchers into cohorts who are working with similar populations, dosing, or theoretical constructs to build on each others' successes and failures and improve translation of the research from pilot to community impact. At the end of each year, the consortium/national centers would converge on a standard metric or tool that could then be broadly adopted. The consortium would also emphasize team science and promote the next generation of interdisciplinary researchers in these areas by building a pipeline of underrepresented scholars and highly represented allies from undergraduate to early career researchers. The consortium would need resources devoted to both research and sustaining community engagement by involving stakeholders throughout research. They would also incentivize data sharing and community engagement. The Future of Sociotechnical Systems to Address Health Disparities Studies funded by the National Institutes of Health have been formally investigating how to address health disparities for almost three decades [76]. However, health disparities persist -suggesting the need for fresh approaches. To that end, in this workshop report, we highlight computing, health informatics, behavioral medicine, and health disparities research challenges that cut across disciplines and federal funding agencies. We also stress the many opportunities that emerge from these challenges. They are summarized in the table below. 4 https://md2k.org/ Challenge Opportunity Marginalized groups are understudied because of difficulty with recruitment, retention, or trust issues. Ensure researchers have resources to build and maintain community-based research collaborations. Develop and evaluate methods of recruitment, technology uptake, and study retention for studies that work with marginalized communities. Current sociotechnical interventions focus on "downstream" interventions where a participant manages a set of issues specific to themselves. Downstream interventions do not address the social origins of health disparities. Support is needed to develop upstream and multi-level interventions to reduce health disparities by impacting community, social, economic, and political factors. When we create sociotechnical interventions that have an impact on outcomes, it is not clear what part of the sociotechnical intervention initiated and maintained that change. Encourage funding agencies to continue supporting broader impacts in research to ensure researcher are addressing issues that are important to communities. Emphasize the need for pilot studies and iterative design to ensure initial conditions are correct. Evaluate the "dose" of sociotechnical systems to better understand the frequency of use, as well as the dosing contexts and infrastructure support available. Current behavioral theories and models often do not account for sociotechnical systems and are not representative of marginalized populations. Document instruments, data streams, and mappings between sociotechnical systems and theories used. Develop dynamic new theories that can account for future sociotechnical systems and capture the social contexts of marginalized populations. Researchers must measure multiple dimensions of social determinants of health to evaluate impact at the population level, but there is a lack of dynamic theories, study designs, or metrics to capture the changing technological and contextual landscape of marginalized populations. Create and document equity-relevant metrics that can capture appropriate levels of detail to contextualize user groups and interventions. Develop, evaluate, share, and validate study designs and theories for interventions. By designing to improve health disparities, researchers may introduce unintended consequences (e.g., everyone benefits and thus the disparities stay the same or worsen). Establish research processes that check on what groups, data, or resources are unaccounted for and monitor unintended consequences. Ensure data collection about unintended consequences. Engender a research culture in which learning, sharing, and disclosing failures are encouraged. Based on past treatment in research, some marginalized groups may have less trust in research. These trust issues are exacerbated when it is unclear how study participation or data access -especially in commodity products -is scoped. Produce systems that assist researchers in identifying ethical issues and proactively assess risks with benefits. Researchers in multiple disciplines are encountering similar issues in their research endeavors to address health disparities, but continue working in their disciplinary silossometimes reinventing each others' approaches or solving the same problems. Develop a consortium or national centers to address health disparities that bring researchers from multiple disciplines together with partners to address the research to practice gap. We also encourage the scientific enterprise to better align incentives (e.g., funding, resources, tenure, publication) with helping people -especially those who are marginalized. Although there are alternative funding models that could be promising to encourage people to address health disparities (e.g., funding people and not projects [77]), we also acknowledge that with the dearth of underrepresented groups in research -especially computing -these models may not adequately support innovation in sociotechnical interventions for health disparity reduction. Workshop participants recognized a broader need to align the scientific enterprise with helping people. Specific to academic research, an easier mechanism for aligning incentives is to add a fourth "impact" pillar for hiring, promotion, tenure, and merit reviews that goes beyond the traditional pillars of research, teaching, and service [78]. The Computing Community Consortium (CCC) sponsored a two-day workshop titled Sociotechnical Interventions for Health Disparity Reduction in collaboration with the leadership of the Society for Behavioral Medicine's (SBM) 39th Annual Meeting on Monday, April 9 and Tuesday, April 10, 2018 in New Orleans, Louisiana. The workshop's goal was to bring together leading researchers in computing, health informatics, behavioral medicine, and health disparities to develop an integrative research agenda focused on sociotechnical interventions to reduce health disparities and improve the health of marginalized populations. The workshop was informed by four themes: ◗ Theory to Design and Implementation: Sociotechnical interventions that reduce health disparities require interdisciplinary knowledge to inform intervention design and implementation because health disparities are rooted in social, behavioral, economic, and healthcarebased factors. ◗ Sociotechnical System Blackboxes: As research in interactive systems in healthcare has matured, computing and health informatics researchers have increasingly drawn upon social and behavioral science theories to design, develop, and analyze sociotechnical systems. However, we do not always know why or how sociotechnical interventions "work." ◗ Sociotechnical Systems to Inform Theory: Behavioral medicine researchers have traditionally developed health behavior theories and models through participant self-report or by utilizing commodity systems to evaluate the theory at scale. However, data collected by sociotechnical systems can be leveraged more consistently to help develop existing behavioral science theories or extend new theories. ◗ Multidimensional Evaluation to Reduce Health Disparities at the Population Level: Sociotechnical interventions hold promise for reducing disparities and improving the health of marginalized populations, however interventions can generate unintended consequences that exacerbate disparities. There is a need to proactively evaluate equity impacts of sociotechnical interventions, at all phases of design and implementation. Such technologies incorporate and allow for: (1) social coordination; (2) communication mediation; (3) optimizing resource distribution; (4) framing and supporting decisions; (5) educating; and (6) improving access to Figure 1 . 1Extension of the World Health Organization's model on Health Disparities closely monitoring system engagement and intervening quickly to re-engage participants if necessary. It was also thought helpful to predict when disengagement might happen, and proactively use strategies for re-engagement. Given the aforementioned challenges, workshop participants saw a need for research specifically to develop and test existing and emerging methods of engagement and retention for marginalized groups. Workshop participants identified the following open questions in need of further research: ◗ What is the meaning of engagement with sociotechnical interventions from the point of view of different marginalized groups? ◗ What current and new engagement/retention strategies are possible to reach marginalized populations? When are they effective? How are they effective? What are their costs and benefits? ◗ How and when should we re-engage marginalized people who have ceased to use sociotechnical interventions or dropped out of evaluation studies? ◗ For sociotechnical interventions that engage groups or networks, how should we measure group dynamics to assess engagement? ◗ What ethical frameworks apply to the engagement/ retention of marginalized participants? What limits should researchers observe in encouraging engagement/ retention? Workshop participants identified some needed resources, including increased multidisciplinary research on problems of recruitment/uptake and retention/ engagement. Workshop participants also believed that research funding opportunities should allow for more resources to be devoted to recruitment and retention and that ethics boards should be educated to understand that methods of recruitment and retention require iteration and refinement over time. Finally, workshop attendees identified a need for a mechanism for sharing effective recruitment and retention strategies with different populations (e.g., advertising methods and keywords). workshop participants and indicate the interplay between sociotechnical systems and sociobehavioral theory (if used) in Figure 2. As Figure 2 shows, the top three factors which we need to understand to "open" black boxes are (1) identifying how participants are involved in the intervention design; (2) understanding data quality from a hardware, software, and human perspectives; and (3) identifying the appropriate dosing of intervention use. Discussed less here, but as important, is integration with theory. These factors are important to understand independent of research aims -whether they are informative (where they describe what is happening) or actionable (that lead to design or adoption of new interventions). sociotechnical systems for their users. Dosage refers to the frequency and intensity of user experience with sociotechnical systems -including use of technological features, interactions between system users, and exposure to theory-informed behavior change techniques. As part of this, the context in which a technology is employed is relevant to dosing because exposure to elements of the social and community context may create enabling or constraining conditions for intervention effectiveness; in this sense, a certain "dose" of community walkability and safety from crime may make it more possible to benefit from physical activity more often. Or, a certain amount of social support -whether included in the design of a technology or not -may be needed to benefit from a mental health intervention. Within this framework, researchers encouraged more research regarding:◗ Understanding how often a dose of a sociotechnical system should be given (e.g., daily, as needed, in a structured program of a pre-specified length) and what mechanism to use for to administer a "dose" -which will change depending on one's context and experiences.◗ Evaluating and reporting on the burden-engagement trade-off of different dosage schemes and of different parts of sociotechnical systems (e.g., participant burden using the system remotely or in-person; research burden managing the data streams). individual or community's behavior informs an abstracted understanding of what is happening to create a theory (A→B). Theories can be extended or new theories developed as prior ones no longer express what is observed (black arrows in Figure 3). However, this approach is insufficient when technology is taken into account because behavior is dynamic and changes depending on context and time; moreover, technologies can both create and capture variance. In addition, researchers attempt to map abstracted sociobehavioral theory onto concrete technology interactions (blue arrow in Figure 3). A common example is displaying one's history of past actions (e.g., a food or activity log) as evidence of past performance as part of feedback provision in an intervention, then logging how many times a user accesses the history screen as evidence on a participant's reflection on past history. However, it is often unclear how well such measures truly map onto theoretical constructs. Accordingly, researchers must identify ways to map sociobehavioral theory appropriately onto sociotechnical systems and evaluate the scope of technology mapping in theoretical constructs.To facilitate theory development, participants identified a need for new methodologies that can learn both new "hypotheses" and construct new theories or extend current theories using data, and adapt as more data and data types emerge. Moreover, these theories need to be specifically developed to reflect the experiences of marginalized users and the under-resourced contexts in which many are more likely to reside. These theories also need to explicitly address the meso-and macro-level factors from which disparities emerge (see section 2.1). ◗ To show a change, these measures must be assessed longitudinally. Furthermore, researchers must understand and document for whom the intervention has an effect, and in what circumstances (moderators).Measure selection is complicated in health disparity contexts because health disparities emerge from social conditions at multiple levels (macro, meso, micro). It may be necessary to develop upstream interventions or synchronous multilevel interventions (see Section 2 andFigure 1 on page 5). Furthermore, an intervention operating at any level may have effects at another level (e.g., policies to provide women with access to education may increase their power in intimate relationships, thus empowering them to insist upon condom use and reduce their HIV risk).An open area of research is to identify what level of analysis should be used in measurement. Theory can guide selection of these measurements in evaluation of interventions, however, it is difficult to ground evaluations of upstream and/or multilevel interventions in theory due to the lack of maturity in available frameworks. For example, similar to sociobehavioral theories, upstream and multi-level theories lack dynamism and do not necessarily account for bi-directional relationships between different levels of social conditions or between social conditions and individual characteristics and behavior. Mechanisms and drivers of change may also be unclear. The lack of theoretical guidance makes evaluation challenging in this context; thus there is a need to improve upstream and multi-level theories to improve measurement, and vice versa. Participants also noted that measures used in disparity research tend to focus on deficits and barriers, rather than resilience and facilitators. Attendees advocated measurement of a broader range of phenomena in our studies, with an emphasis on developing a fuller picture of marginalized groups and individuals. Workshop participants encouraged researchers to share what they measure -including challenges and strengths, while also working towards a widely accepted set of metrics to assess intervention impact at different levels. Despite the plethora of sociotechnical measurement options, there is a need for expanded technical capabilities to measure a wider range of equityrelevant characteristics, such as culture and patient goals. Furthermore, there is an opportunity for developing better measurement methods that leverage new data sources, such as characterizing digital phenotypes based on patterns of user interactions with interventions, and trajectories of usage over time. A recurrent challenge, however, is that patient-centered measures are often individualized, not standardized, which makes reproducibility difficult. This is especially true in marginalized groups that are less studied; thus, researchers have less baseline and longitudinal data. What is the relative value of different measures for improving predictions? What are the relative costs and benefits of using different measures? ◗ How can health-equity-relevant phenomena such as resilience, culture, context, and patient-centered goals be more effectively measured? How can we balance individualization and standardization in creating these measures? ◗ What patterns of user interaction with interventions exist? How do they vary for different marginalized groups? How do they change over time? How do these patterns influence intervention effectiveness, if at all? What can they tell us about when interventions should end? ◗ How can we assess the quality of a theory's operationalization within a socio-technical intervention? ◗ To improve extant theories and models, how can we better measure mechanisms of action (mediators) and identify groups/settings in which interventions are effective (moderators)? How can we measure multi-level outcomes within individual studies? [ 60 ] 60, leading to differences in interventions over time or across sites. Evaluations may be complicated because of a need to collect data about effects at different levels (e.g., both community norm change and individual behavior). Furthermore, it is not always clear what the "active ingredients" of interventions are, and how those ingredients interact -key requirements for understanding generalizability and translation. Workshop participants identified a need for using more varied existing study designs to find the intervention components that do or don't work. Relatedly, there is a need to further apply and develop new adaptive trial methods such as multiphase optimization strategy (MOST) and sequential multiple assignment randomized trial (SMART)[61].Optimization and tailoring criteria specifically related to marginalized groups and the social factors that drive disparities would increase the applicability of these evolving methods to disparity research.Participants also noted that follow-up, including long after an intervention, is a critical missing piece in prior research. Such longitudinal follow-up may prove more difficult to conduct in low-SES groups due to less home ownership and, potentially, more contingent and precarious employment. Accordingly, there is a need for further study of methods for retaining these groups in research (see Section 2.2).Participants also highlighted the need for greater understanding regarding implementation of successful interventions in new settings, and with new marginalized groups. A related issue of concern is the potential for effect modification based on contextual factors for research with marginalized groups. Workshop participants thus voiced a need for cross-cutting studies with multiple comparisons across different marginalized groups. Translation also involves potential re-use of measures, data, and technical frameworks. marginalized groups are not often included in research (see Section 2.1), technology models may make assumptions that are not valid for them. Consequently, marginalized groups may miss potential benefits of an intervention -potentially worsening disparities. Workshop attendees argued for a research process which continuously questions who may be left out in the design, implementation, and evaluation of socio-technical interventions to monitor the potential unintended consequences. To begin, there is a need for any intervention to specifically measure intervention outcomes. Furthermore, there is a need for expanded effort to assess the equity impacts of both existing and emerging "universal" informatics interventions that are intended for all, rather than just a disparity population. For prospective studies of new interventions, it is important for outcome evaluations to include planned heterogeneity of treatment effect (HTE) analyses. This requires recruitment of diverse samples, possibly oversampling from marginalized groups to ensure statistical power for such analyses. Such analyses may involve planned moderation, stratified, or subgroup analyses. Workshop participants noted that an area of ambiguity in such analyses concerned how to evaluate intersectionality and overlapping disparities in intervention evaluation participants. It is also important for intervention studies to include a qualitative component to assess possible unintended consequences in relation to health equity. ◗ How can we effectively evaluate the effects of interventions in situations of overlapping disparities? ◗ What are the equity-related consequences of previous socio-technical interventions in health? ◗ What are the health equity impacts of consumer health technologies that are now in wide usage (e.g., selftracking, patient portals)? What are the population-level impacts of any identified inequities? For progress on unintended consequences, participants identified a need to create a research culture in which learning -including failures -are embraced. As part of this, workshop attendees wished to see the development of venues in which unintended consequences can be openly discussed, including panels and workshops to discuss equity-related lessons learned in projects. Furthermore, participants wanted to encourage the development of scholarship on unintended consequences, including explicit sections on unintended consequences in publications, complete papers on unintended consequences, and systematic reviews of equity-related unintended consequences of socio-technical interventions in health. Funders also have a role to play. Funders should support mechanisms to adjust interventions as they are implemented, allowing for a more iterative approach to research. In addition, there is a need for funding research on, and reporting about, equity-related unintended consequences as part of grant progress reports. evaluation of sociotechnical interventions must have a clear vision of success -and one that is not fully bound by the methods and tools that we use in our research. Unfortunately, research and academic priorities are frequently misaligned with marginalized groups' priorities. This creates a challenge in establishing the direction of research, including whether research questions are generated by the needs and concerns of the community or whether researchers are trying to solve "problems" that only exist in theory, but are not critical to addressing the key causes of disparities. Consequently, some participants advocated an emphasis on "real world" success, or impacts on people's lives outside of the academy. To define real world success, participants advocated dialogue from diverse stakeholders. As a starting place, participants noted that success would likely involve: (1) people in the greatest need receiving the support they require; and ◗ Enhance participatory methods for designing, studying, and evaluating technology. To ensure that we effectively address real problems, marginalized groups should be involved in choosing intervention priorities and designing and evaluating interventions.While researchers currently use participatory methods, there is a need to evaluate and improve these methods.Critically, there is a need to develop and support mechanisms for building capacity for marginalized communities to meaningfully participate in health research on socio-technical systems.◗ Build dynamic and multilevel theories for designing interventions. Existing sociobehavioral theories typically Health disparities are differences in disease prevalance, incidence, morbidity and/or mortality in one group as compared to the general population. In Western countries, groups which experience disparities in health outcomes include:◗ People of lower socio-economic status (SES) based on income, wealth, education, and occupation;◗ Racial and ethnic minority groups including African Americans, Latinos, Native Hawaiians/Pacific Islanders, and Indigenous peoples; ◗ Rural and urban residents; ◗ Lesbian, gay, bisexual and transgender (LGBT) people; ◗ People with disabilities; and ◗ Men or women (varies by health issue). Box 1: The definition of health disparities. do not account for the dynamism of new types of sociotechnical systems. Moreover, few available theories have been developed with marginalized populations in mind, including the social and economic conditions that contribute to marginalization. Development of new theories can be facilitated by better mapping of theories onto sociotechnical systems and with new methodologies that can learn both new "hypotheses" and construct new theories or extend current theories using data. ◗ Advance methods for dosing, tailoring, and optimizing sociotechnical interventions. Little is known about how much usage of sociotechnical systems is needed to gain benefits from them and what doses of other aspects of the intervention context are needed to benefit from interventions (e.g., neighborhood walkability). Particularly for marginalized groups, we also know little about what aspects of interventions should change based on individual or contextual characteristics. Knowledge in this area can be gained through greater support for pilot studies, developing methods to strengthen conclusions based on small-N studies, and studies which tailor to characteristics relevant to health disparities. ◗ Evaluate systems via multiple dimensions to reduce health disparities at the population level. It is important for any sociotechnical intervention to be evaluated in relation to its impacts on health equity. Interventions should also be assessed at multiple levels where applicable (micro, meso, macro). Researchers should ask themselves equity-related questions (see Box 2 on page 18) in relation to any intervention studies and plan studies in which differential uptake, engagement and outcomes can be assessed. It is also important for researchers to examine potential unintended consequencesparticularly through qualitative research. There is also a need for research and tools to assist researchers in evaluating the ethical implications of studies that gather data from marginalized participants, especially those that use third-party platforms and that capture social and community contexts.For sociotechnical interventions to reduce health disparities, it is critical that intervention strategiesactivities or features that aim to improve some predetermined health-related outcome -are grounded in an understanding of health disparities and the ways in which inequity can emerge at all stages of the intervention cycle, from access to effectiveness. This means that◗ Create interdisciplinary bridges to continue collaborating. There is a need for development of a consortium or national centers to address health disparities with sociotechnical systems that creates a collaborative network of researchers, industry, providers, payers, and communities to aid in scaling sociotechnical interventions. This consortium should create reusable components, share algorithms and data, and develop approaches for transferability and robust partnerships. This multidisciplinary workshop sought to take stock of prior successes and failures, of accumulated learnings and persistent challenges. In addition, workshop attendees sought to identify knowledge gaps and opportunities for advancement through research. 2. Developing Equity-centered Intervention Strategies and Implementation Approaches interventionists must understand what populations experience disparities for a given health outcome, the antecedents of those disparities, and potential theoretical pathways by which those disparities can be reduced. Workshop participants specifically advocated for the further development of "upstream interventions," described in section 2.1, to achieve this. In addition, interventions can only have an effect if they are adopted and used; or, in a research context, that participants are recruited and then remain in a study. Because marginalized groups are less likely to do these things, there is a need for equity- centered implementation approaches focused on adoption/ recruitment and usage/retention. Workshop participants expanded on defining a science of engagement and retention (section 2.2) by discussing how to integrate marginalized people into the study and technology design processes. Researchers acknowledged that existing tools and methods used for research have implied criteria for success (e.g., significant differences; usability), thus tools, methods, and criteria for success are conflated. Researchers mitigated these effects by actively collaborating with the community where the study was being completed to investigate real-world problems which lead to broader, more contextualized measurements that could benefit the community,increase researcher's broader impacts, and strengthen their community collaborations. If research communities do not support addressing validated real-world problems, researchers run the risk of solving the wrong (or artificially easier) problems and not having a clear sense of dynamic risks and unintended consequences.Researchers reported many ways to collaborate with communities -from university-led centers or initiatives in communities (e.g., a center for rural engagement, 1 community-centers in assisted housing neighborhoods 2 ) to including community members as part of the research team[31] to remotely creating a community via social media[42]. Likewise, methods to involve communitymembers varied -from including community members in research ideation to data collection, analysis, and dissemination. Researchers, who include community members in research, had to balance multiple tensionsespecially when considering what is valued in scholarly research versus what is most beneficial to the community. A continued effort by funders to encourage and support true broader impacts in research will assist researchers in addressing this tension.Researchers also discussed how to engage community members in the research process when some parts of research (e.g., research ethics training to deliver novel interventions or evaluate data; dissemination for publication) is time-consuming and an academicallyoriented burden. In these cases, researchers emphasized the need to ensure research benefits not only the researcher, but the community, and to adequately compensate community members at an equitable level to researchers, since they are part of the research team.Although there are many participatory methods -from CBPR to Action Research[43] to participatory design [44] -researchers noted challenges specifically for marginalized groups that the research community needs to explore further: ◗ How do we build capacity for communities to meaningfully participate? How do we mitigate power issues that may be perceived when working with people of varying backgrounds and experiences? ◗ How do we evaluate the effectiveness of participatory methods with marginalized groups? ◗ How do we provide a safe space for not knowing, learning, iterating, and reporting failure in participatory design when working with marginalized groups?Researchers acknowledged the immense value of working with marginalized groups. Participants reported that formally trained researcher team members who were marginalized community members ("community research liaisons") assisted with recruitment, retention, and trust formation. So although participatory methods to involve community members are important, it is more so important to ensure there are funded programs to encourage, mentor, advocate for, and train marginalized groups to become formally trained practitioners and researchers in these research areas. As part of this, there is a need for expanded opportunities to train and support community research liaisons, including facilitating dialogue between liaisons. Creating a more inclusive research community will improve, strengthen, and push innovation to benefit everyone[45]. they should be designed, and when different options should be used, for which group, and in what context.◗ Investigating missed doses because non-use or inactivity within each dose does not necessarily imply that the intervention has failed or that change is complete. ◗ Understanding what mix of dosing of technological and non-technological elements are needed to achieve a given outcome; for example, ways in which community or social network characteristics may moderate intervention effectiveness. ◗ What role can novel dosing schemes such as pulsed, decreasing or event-based schemes play in sociotechnical interventions for marginalized groups? Once dosing schemes are established, more research must be done to investigate their reproducibility in new contexts. Reproducibility is especially difficult in sociotechnical intervention research because norms and technology move quickly while infrastructure availability and technology adoption among marginalized groups tends to lag behind those with greater resources. We were encouraged by federal initiatives, such as the FCC-NCI Broadband Cancer Collaboration, 3 to enhance infrastructure for groups that experience health disparities. We must be able to model, interpret, and communicate about dosing schemes to better understand how For example, given their implications for engagement, when should researchers designing self-monitoring interventions utilize sensing versus self-report? How much in-person, virtual, or asynchronous communication with others (e.g., healthcare providers, patient peers) is needed to have an intervention effect? In addition, we must be able to effectively communicate dosing information to researchers in other disciplines and lay community members. Dissemination must be more than writing up results and throwing them over the wall -we must actively pursue new mediums to communicate findings to people with various backgrounds. Researchers could envision tools to assist researchers and practitioners in deciding on the optimal dose of intervention components for specific health outcomes in specific populations. In addition, researchers should embrace succinct visual and multimedia communication techniques to justify dosing to non-experts and lay populations all the while emphasizing how the study design meets their community-based research goals. Despite the aforementioned shortcomings of existing sociobehavioral theories, researchers are encouraged to select theories and define theoretical constructs for use in study design; designing interactions within the sociotechnical intervention itself; or analysis of data[10].implicit theory to which all research team members may not have agreed. When defining a theory, researchers should also cite the origin of the theory and how, if at all, the target populations of the sociotechnical intervention differ from the population in which the theory was developed.One of the biggest challenges of developing sociotechnical health interventions is that although there are many sociobehavioral theories available [53], they are often dated because they do not account for new types of sociotechnical systems [10, 50] and are not necessarily representative of marginalized populations. Indeed, workshop participants could only identify two health behavior theories or models that were developed specifically with marginalized populations -the Theory of Positive Deviance [54, 55] and the Reserve Capacity Model [54]. Therefore, often used, but dated sociobehavioral theories may have limited predictive power for marginalized groups in a digital age. Additionally, new types of data (e.g., continuous sensor monitoring, social media streams) cannot be automatically used to validate theories that were created before this type of continuous data was available. By identifying an explicit theory and related constructs, researchers articulate their assumptions and provide a record to explain how an intervention worked. Although choosing an explicit theory forces the research team to make assumptions, not doing so may mean relying on by researchers (e.g., Fitbit, Apple ResearchKit). In such cases, it may not be entirely clear whether patients can opt out of data collection, whether stored data are deidentified, where data are held, and whether researchers or participants can delete said data. Furthermore, research participants' privacy and security will be subject to the platform's practices, and data may be vulnerable to sharing with unknown additional parties. Such difficulties may be particularly risky for study participants with stigmatized identities or conditions, such as transgender people, people with HIV/AIDS or mental health conditions[64]. More research must be done to create models that researchers in any discipline can use to assess risks of study participation for marginalized groups.Furthermore, sociotechnical interventions are increasingly gathering and using contextual data, which includes participant demographics, place of residence, geolocation, call and text logs, social networks, patterns of technology usage, keystrokes, and biometrics. Some just-in-time adaptive interventions focus on delivering support and information when needed, based on data gathered from geolocation and sensing technologies[64][65][66][67]. For instance, contextual data such as call and text logs and geolocation have been used to sense a person's mental well-being[68,69]. Furthermore, healthcare organizations are increasingly collecting social determinants of health data and incorporating these data into electronic health records[70,71]. More and more clinical prediction algorithms are incorporating contextual variables, such as whether a patient lives in a high-poverty neighborhood, into their models[72,73]. The ethical implications of gathering this growing amount of data are unclear, and they may represent greater risks for marginalized individuals.Sociotechnical interventions typically involve the collection of a great deal of data about their users/study participants. Studies may involve the deployment of third-party platforms that are not owned and controlled an increasing amount of data collected about people in their everyday lives and as part of studies, questions regarding informed consent become increasingly complicated. Secondary use of data may involve data, such as digital traces from such a large group of people, that make traditional methods of obtaining informed consent become infeasible. Furthermore, if study participants share accounts and technology, then the definition of a "participant" may have to be further specified.People may also find themselves participating in research, especially social media platforms[74], without intending to do so. To illustrate, patients may unwittingly find their contributions in online patient communities used for research, and details on secondary usage are often buried in terms of use and privacy policies that users infrequently review[75]. Contextual data may involve others who have not consented to be part of a study, such as Facebook friends or conversation partners on a smartphone. In each of these cases, there is a potential impact on the autonomy of all study participants, but the impact may be greater in individuals with low health literacy or people who experience stigma and discrimination in their daily lives, such as those with substance use disorders.With the aforementioned ethical issues in mind, participants identified the following open questions: ◗ How can we include more diverse perspectives in discussions of research ethics for sociotechnical health interventions? ◗ How do we ensure the data we give back to study participants and groups is actionable, meaningful, and understandable? What is the impact of returning data to participants upon their willingness to engage in research in the future? ◗ What are the risks of capturing contextual data for marginalized individuals and groups? How can these risks be mitigated or managed? ◗ What are the ethical implications of use of contextual data to tailor interventions, diagnose health conditions, and identify health risks? Box 2: What should intervention researchers consider when designing interventions? ◗ What models of informed consent can be used for largescale secondary use? How can informed consent models take unwitting research participants into account, if they constitute a participants' context? Workshop attendees also identified a need for safe and encouraging spaces for discussing ethical dilemmas, sharing experiences and exemplars, and garnering resources. Workshop participants saw value in developing a system to assist researchers, participants and companies in assessing ethical issues, including tools for proactive risk assessment and for balancing the inherent trade-offs in choices. Another type of tool would assist study participants, researchers, and companies in developing privacy literacy. Mechanisms for reconciliation and remediation in the event of ethical breaches or harms were also desired. 6. Interdisciplinary Bridges Many diverse populations are affected by health disparities; thus different, adaptable sociotechnical intervention approaches are necessary to help address the needs of individuals, communities, and populations. Currently, researchers are largely developing separate approaches from scratch and in relative isolation or small interdisciplinary teams, which makes it difficult to create scalable progress and larger real-world impacts. Without interventions that can scale up, our solutions are of reduced effectiveness, limited only to those who can afford them or happen to be in the right geographic area. We recommend the development of a consortium or national centers to address health disparities with sociotechnical systems that creates a collaborative network of researchers, industry, providers, payers, and communities to aid in scaling sociotechnical interventions -similar to the NIH funded Center of Excellence for Mobile Sensor Data-to-Knowledge (MD2K). 4 This consortium would act collectively to "raise all boats" by creating reusable components, sharing algorithms and data, developing approaches for transferability and robust partnerships, and developing the science of recruitment and retention of underserved populations in pilot and longitudinal studies. https://rural.indiana.edu/ 2 http://www.denverbridgeproject.org/ Figure 2. Factors to consider when exploring sociotechnical black boxes https://www.fcc.gov/health/cancer L Street, NW, Suite 800 Washington, DC 20036 P: 202 234 2111 F: 202 667 1066 www.cra.org cccinfo@cra.org When Discrimination Is Baked Into Algorithms. 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{'abstract': 'The authors gratefully acknowledge the contributions of the workshop planning committee members to the design and facilitation of the workshop, which formed the basis of this document. Committee members included Heather Cole-Lewis,', 'arxivid': '1908.01035', 'author': ['Syed Haider ', 'Eric Hekler ', 'Predrag Klasnja ', 'Donna Spruit-Metz ', 'Katie Siek ', 'Tiffany Veinot ', 'Beth Mynatt '], 'authoraffiliation': [], 'corpusid': 199441991, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 23789, 'n_tokens_neox': 20629, 'n_words': 13339, 'pdfsha': '22e302a89658d33a942ca1e9e34121025d5fd4cf', 'pdfurls': ['https://arxiv.org/pdf/1908.01035v2.pdf'], 'title': ['Research Opportunities in Sociotechnical Interventions for Health Disparity Reduction Research Opportunities in Sociotechnical Interventions for Health Disparity Reduction Sponsored by the Computing Community Consortium RESEARCH OPPORTUNITIES IN SOCIOTECHNICAL INTERVENTIONS FOR HEALTH DISPARITY REDUCTION 1. Overview ......................................................................................................................................................................1 2. Developing Equity-centered Intervention Strategies and Implementation Approaches ..................................... 3 2.1 Upstream Interventions ...................................................................................................................................... 3 2.2 Equity-Centered Intervention Uptake and Study Recruitment ........................................................................5 2.3 Equity-Centered Engagement/Adherence and Study Retention ...................................................................... 6 3. Sociotechnical Black Boxes ...................................................................................................................................... 7 3.1 Participatory Methods for Study and Technology Design ................................................................................. 8 3.2 Understanding Data Quality in Existing Systems .............................................................................................9 3.3 Designing Dosing Schemes ...............................................................................................................................10 4. Sociotechnical Systems to Inform Theory ...............................................................................................................11 4.1 Building Better Theories: New Opportunities ....................................................................................................11 4.2 Tailoring and Optimization of Sociotechnical Systems ...................................................................................13 5. Multidimensional Evaluation to Reduce Health Disparities at the Population Level ..........................................13 5.1 Improving Measurement and Methods for Multidimensional Evaluation', 'Research Opportunities in Sociotechnical Interventions for Health Disparity Reduction Research Opportunities in Sociotechnical Interventions for Health Disparity Reduction Sponsored by the Computing Community Consortium RESEARCH OPPORTUNITIES IN SOCIOTECHNICAL INTERVENTIONS FOR HEALTH DISPARITY REDUCTION 1. Overview ......................................................................................................................................................................1 2. Developing Equity-centered Intervention Strategies and Implementation Approaches ..................................... 3 2.1 Upstream Interventions ...................................................................................................................................... 3 2.2 Equity-Centered Intervention Uptake and Study Recruitment ........................................................................5 2.3 Equity-Centered Engagement/Adherence and Study Retention ...................................................................... 6 3. Sociotechnical Black Boxes ...................................................................................................................................... 7 3.1 Participatory Methods for Study and Technology Design ................................................................................. 8 3.2 Understanding Data Quality in Existing Systems .............................................................................................9 3.3 Designing Dosing Schemes ...............................................................................................................................10 4. Sociotechnical Systems to Inform Theory ...............................................................................................................11 4.1 Building Better Theories: New Opportunities ....................................................................................................11 4.2 Tailoring and Optimization of Sociotechnical Systems ...................................................................................13 5. Multidimensional Evaluation to Reduce Health Disparities at the Population Level ..........................................13 5.1 Improving Measurement and Methods for Multidimensional Evaluation'], 'venue': []}

arxiv

Highly degenerate photonic flat bands arising from complete graph configurations Hanyu Wang College of Advanced Interdisciplinary Studies National University of Defense Technology 410073ChangshaChina Biao Yang yangbiaocam@nudt.edu.cn College of Advanced Interdisciplinary Studies National University of Defense Technology 410073ChangshaChina Department of Physics Center for Metamaterials Research The Hong Kong University of Science and Technology Hong KongChina Wei Xu College of Advanced Interdisciplinary Studies National University of Defense Technology 410073ChangshaChina Yuancheng Fan Department of Physics Center for Metamaterials Research The Hong Kong University of Science and Technology Hong KongChina Qinghua Guo Department of Physics Center for Metamaterials Research The Hong Kong University of Science and Technology Hong KongChina Zhihong Zhu College of Advanced Interdisciplinary Studies National University of Defense Technology 410073ChangshaChina C T Chan phchan@ust.hk†these Department of Physics Center for Metamaterials Research The Hong Kong University of Science and Technology Hong KongChina Highly degenerate photonic flat bands arising from complete graph configurations *Correspondence to: authors contributed equally to this work. Abstract Inspired by complete graph theory, we demonstrate that a metallic claw "metaatom" structure can carry a high number of nearly degenerate resonant modes. A photonic meta-crystal composing of a lattice of such meta-atoms exhibits a large number of flat bands that are squeezed into a narrow frequency window, and these flat bands can be designed to locate in a wide complete 3D bandgap. The degeneracy dimension (N f ) of the flat bands is determined by the number of branches (N b ) of the metallic claw with N f =N b -3, which is geometrically related to the complete graph theory. Different from those flat bands emerging from special lattice arrangements (e.g., Kagome lattice), the isolated flat bands here are insensitive to lattice perturbations. The proposed mechanism offers a new platform for realizing various dispersion-less phenomena and a new paradigm to realize high density of states and spectra compressing. Introduction Flat bands [1][2][3][4][5] refer to that spectral bands are dispersion-less or nearly so and their energy spectrum ( ) are almost independent of momentum . In photonics, the realization of flat bands has been long pursued [6] for enhancing light-matter interaction with slow light [7][8][9] and wave localization [10,11], or it offers platforms for other applications such as distortion-free imaging and pulse buffering in nonlinear optics [12,13]. Typically, flat bands are found in the Dice [1], Lieb [3], Kagome [14,15] and other lattices [16,17] due to destructive interference, where fine-tuned nearest-and next-nearest neighboring hopping parameters are the key factors. Researchers have used waveguide arrays [10,[18][19][20][21][22][23], dielectric/plasmonic resonators [24][25][26][27][28][29][30][31][32] and finetuned photonic crystals [33][34][35] to realize photonic flat bands, where high dielectric contrast or exact lattice symmetry are required. However, these mechanisms inspired by analogies with "frustrated" condensed matter systems [5,36,37] exhibit many limitations in the photonic regime where photonic bands usually arise from multiple coherent scattering rather than the hopping of local atomic orbits. Moreover, realization of full three dimension flat bands [38] is very challenging using such lattice arrangement. In this letter, we demonstrate that three-dimensional (3D) flat bands can be realized using a periodic array of claw-like metallic structures. Isolated flat bands [39,40] existing in an absolute band gap emerge in such photonic crystals, and the flat-band modes are strongly localized near the meta-atoms. Different from the previous works, the flat bands are extremely stable when lattice constant changes. In addition, we find an interesting relation between the number of claw-branches and the number of flat bands, and the relation can be understood readily when we see that the claw-like structure is a manifestation of a "complete graph" from a geometrical point of view. Recently, geometric aspects of physics have attracted a lot of attention. In photonics, various topological photonic semimetals and insulators [41] have been theoretically proposed and experimentally verified. Their properties are characterized by integers and are stable against local perturbations. For example, the number of edge states of photonic Haldane model [42,43] is directly related to its bulk integer Chern number defined in momentum (parameter) space [44]. Here, the real space geometry of the meta-atom generates interesting physics. We will see below that the electromagnetic response of the metallic claw structure can be described by a dynamic matrix that mimic the Laplacian matrix in complete graph theory [45], giving rise to highly degenerate localized modes which become the flat bands when the meta-atoms form a 3D crystal. & = 3 −1 −1 −1 −1 3 −1 −1 −1 −1 3 −1 −1 −1 −1 3 (1) where each entry indicates the connection conditions, such as 11 = 3 being the degree of vertex and 13 = −1 when ≠ . From a physics viewpoint, each entry of the matrix can also be read as the hopping term in a tight-binding model. Generally, the spectra of Laplacian matrix are given by, ( = 0, , … , ()*(2) which are highly degenerate except for the first 0-eigenvalue state with eigenvector 1, … ,1 *×( . The flat bands of our meta-crystal arise from the − 1 non-zero degenerate eigenvalues. Results: Metallic claw structure and photonic flat bands We start with a metallic claw structure with C 4 rotation symmetry. As shown in Fig. 1(b), each meta-atom consists of two perpendicularly placed split ring resonators (SRRs) which touch each other on the top, forming a four-branch claw. The whole meta-crystal is formed by arranging the claw structure in a primitive 3D tetragonal lattice. In the simulation, we assume the hosting material is air and regard metallic components as perfect electric conductors (PEC), which is a good approximation in microwave, teraherz and even far-infrared bands. Figure 1(c) schematically shows the effective electric circuit describing the electromagnetic response of the metallic claw structure, where each capacitor works as an edge in the complete graph and there are 6 possible edges. We have effectively introduced the capacitance ′ (between next-nearest branches) to distinguish with arising from those nearest neighbor branches. The CST simulated photonic band structures and density of states (DOS) [46] are shown in Fig. 1(e, f) with the first Brillouin zone (FBZ) depicted in Fig. 1(d). The structural parameters are given in the figure caption. There is a 3D bandgap between the second and fourth bands, inside which lies a third band that is flat throughout the 3D FBZ and has no crossing with other bulk bands. We dubbed it an isolated flat band. The isolated flat band has almost vanishing dispersion among the full 3D FBZ, while the flat bands found in many previously works are typically dispersion-less in some particular planes or along some particular directions. As the flat band lines in a gap with the width of 1.73GHz at around 25 GHz, it cannot hybridize with the dispersive states and spans an orthogonal state-space by itself. Relationship with complete graph The flat bands originate from a set of highly localized degenerate modes of the metaatoms. The underlying mechanism can be understood through local potential orbitals by analyzing the equivalent electric circuit consisting of capacitors and inductors (Fig. 1c). Following Chua's circuit notation [47,48], the Lagrangian of the circuit reads, ℒ = 2 * − + + + + − D + + D − & + + & − * + + F 2 * − D + + + − & + − 1 2 * + + + + + D + + & +(3) where G indicates local potential on the branch as shown in Fig. 1(b-c). Without loss of generality, we have assumed I = 0. The Euler-Lagrange equation of motion is then, * = K KL Mℒ MN O − Mℒ MN O = 2 * − + − & + F * − D + * P * (4) where * indicating external current has been set to 0 as our system is source-free. In the same way, one can obtain +,D,& = 0 and expressing in the matrix form, we get, ( + − * P &×& ) = 0(5) where, = 2 + ′ − − ′ − − 2 + ′ − − ′ − ′ − 2 + ′ − − − ′ − 2 + ′(6) and In order to verify those predictions, we show in Fig. 4 the evolution of the band structure as the number of branches changes. In the left inset, we show the metallic claw structures with the number of branches ranging from g = 3 to 6. These claws are arranged in a primitive tetragonal lattice to build the photonic meta-crystal. For simplicity, we also assume all claw-structures possesses j k rotation symmetry in each unit cell, although it is not compatible with the tetragonal Bravais lattice when g ≠ 4. The orientation of the claw structure within the unit cell does not affect the existence of the flat bands. In the right column, the corresponding band structures and DOS [46] for different g are shown. We have normalized the DOS [46] within the interval of 0 to 1 to reveal the contrast between the flat bands and dipolar pass bands. In order to clearly count the number of isolated flat bands and to check more details, different line styles are used and indicated in the figure. We find that the bands will become flatter when each unit cell gets bigger (keeping the size of claw-structure the same). Therefore, the small dispersions of the flat bands originate from the weak coupling of the localized modes between neighbouring meta-atoms. However, when the lattice constant gets too big, the gap will close as the Bragg scattering of the dipolar modes becomes too weak to sustain a complete gap. In that limit, the flat bands will intersect with the dipolar band manifold and no longer exist in a clean absolute gap. As predicated above, we observe a f = g − 3 rule for the claw structures. When g = 3, there is no flat band even though the bandgap width is larger than 1.4GHz [50]. The DOS peak at the frequency corresponds to the lower boundary of the bandgap. Isolated flat bands emerge when g > 3, and all are confined inside a narrow frequency window (from 24.24 GHz to 24.53 GHz for the structural parameters specified in Fig. 1). The number of flat bands inside the band gap increases with the number of branches in the claw, and the corresponding DOS in the flat-band frequency window grows dramatically, which can potentially facilitate applications that requires a high photonic DOS. The +I case is shown in Fig. S2(a)-(b) [49] to further illustrate this phenomena. In particular, there are 17 flat bands in total as one can check in Fig. S2(c) [49], confirming again the f = g − 3 rule. A unique feature of the metallic claw design, as the realization of a complete graph, is that the dimension of flat-band set depends on the number of branches and as such, we can arbitrarily enlarge the flat-band sub-space without shifting their operating frequency. As the spectral property of a complete graph is mainly determined by geometry and connectivity, the structural details of the meta-atoms are unimportant, and hence the flat bands and the related phenomena are robust even if the real samples deviate from the theoretical design (Fig. S3, [49]). On the other hand, their very high DOS and the high Q-factors of the high order orbitals render these complete-graph-inspired systems sensitive to environmental external fields, making them good platforms for information sensing. Conclusion In conclusion, we have designed claw-like metallic meta-atoms inspired by complete graph theory. We investigated photonic crystals composing of these meta-atoms and Figure 1a shows a simple complete graph & , where each pair of graph vertices (red dots) is connected by an edge (black line). There are vertices and a complete graph ( . Mathematically, each complete graph ( is precisely characterized by Laplacian matrix. For example, . T = * , + , D , & . The dynamic matrix shows strong resemblance to Laplacian matrix as mentioned above (Eq. 1). Different from the ideal complete graph & , there are two sets of weighted edges and ′ which are slightly different. The condition of det( and \ are degenerate states representing the two orthogonal dipole moments ( ],^) of two independent SRRs. Here, the most interesting mode D shows the symmetry of an ]`)^` orbital and is orthogonal to the ],^ orbitals. The flat band of the photonic crystal is the Bloch state comprising this D mode. This mode arises only when the two SRRs touch each other. The CST simulated electric/magnetic field distributions of the flat band from several high-symmetry k-point as shown in Fig. 2 corroborates with our circuit prediction. At Γ, electromagnetic eigen-fields on the cutting plane = 0 oscillate symmetrically. At , and , the electromagnetic eigenfields remain almost the same, which further indicates the flatness of the isolated band (derived from D ), and the mode profiles are very similar for different momentum . Similar to the spectra of a Laplacian matrix (Eq. 2), there are 3 non-zero eigenvalues in claw structure shown in Fig. 1b, with two of them being the dipole modes ( ],^) , leaving one degree of freedom to contribute to the flat band. For claws with more branches, we can predict a general relation between the number of flat bands ( f ) and number of branches ( g ) as f = g − 3. The dimension of the flat-band sub-space gets biggerwith an increasing number of branches, squeezing more and more flat bands into a narrow band of frequency. The zero-eigenvalue mode of a single meta-atom corresponds to h orbital excitation with electric/magnetic field shown inFig. S1[49]. Schematically,Figure 3shows the local orbitals giving rise to the flat bands with number of branches increasing from g = 3 to 6. Without going through a tedious derivation of the Lagrangian (See for example, Eq. S1-S4)[49], all of them can be simply solved using the corresponding Laplacian matrix of a weighted complete graph as shown in the first column, where edges with different colours represent different weights. In realistic metallic claw structures, the different weights correspond to different capacitances between pairs of branches. The level of degeneracy of those flat bands is simply determined by the differences of those capacitances. Although it is hard to make the capacitances exactly the same in practice, a symmetrical design can make the differences smaller. In the second column, we explicitly provide the potential distributions of the dipole excitations where red and light-blue colours indicate respectively positive and negative potential distributions with the size representing the amplitude. The blue arrows show the dipole magnitude and directions defined as The third column shows higher-multipole excitations. Different from the dipole excitations, the higher-multipole orbitals consist of alternating positive and negative potentials, thus the sum of potential distributions higher-multipole orbitals determines the number of flat bands ( f ). For the D case, there is no higher-multipole excitation, which agrees well with the geometric prediction of f = g − 3. From & on, the number of higher-multipole modes increases linearly with the number of branches as indicated by the orange arrow inFig. 3. found isolated flat bands confined in a narrow frequency window. Different from previous works based on lattice geometry, the flat-band mode is insensitive to structural and lattice parameter perturbations. We found a simple relation governing the number of flat bands ( f ), which can easily be understood by mapping to the corresponding weighted complete graph. The highly degenerate flat bands and the associated highDOS persist even if the lattice (translation symmetry) is destroyed as the phenomena originate from internal degrees of freedom of the metallic claws. Acknowledgments This work is supported by National Natural Science Foundation of China (Grant No. 11674396). The work in Hong Kong is supported by Research Grants Council of Hong Kong (AoE/P-02/12). FIG. 1. Schematic view of metallic claw-like structures and flat bands. (a) Complete graph & with four vertices and six edges. (b) Geometry of the claw-like metallic structure with & rotation symmetry with length =1.74mm, gap =0.615mm, diameter =0.21mm, as the building block of a primitive tetragonal photonic crystal with lattice constant =3.5mm. (c) Effective electric circuit with inductors ( ) and capacitors ( and ′). (d) Three-dimensional first Brillioun zone (FBZ). Black solid lines show the paths followed by energy bands in (f). (e) Isolated flat band (surface with red edge) located in a complete band gap with h = 0. The fourth quadrant has been cut to show the sectional view for clarity. (f) Photonic band structure along a specific path as indicated in (d), with normalized density of states (DOS) shown in the right panel. The sharp peak in DOS stems from the flat band (red line, left panel) while zero DOS corresponds to the complete band gap. FIG. 2. Eigen-fields at high symmetry points of the isolated flat band for the & metallic claw structure. The direction of arrow refers to the direction of electric (magnetic) field while its size representing the local intensity. The eigen-fields are in good agreement with our theoretical predication. FIG. 3. Complete graph analogy of the metallic claw-like structures. The left column illustrates equivalent complete graph circuits of different g (the number of branches for each "claw"), while the middle and the right columns show their corresponding eigen-potential distributions. Red (light-blue) disk represents positive (negative) electric potential, and its radius represents the absolute value of each potential. The number of eigenstates increases with g . For each single claw-like metallic structure, there are two non-collinear dipole excitations. Higher-multipole excitations appear when g ≥ 4, concomitant with the emergence of flat bands. The number of flat bands ( f ) varies with g as f = g − 3. 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Murty, Graph theory with applications (Macmillan, London, 1976), Accessed from https://nla.gov.au/nla.cat-vn2190177. . B Liu, S G Johnson, J D Joannopoulos, L Lu, Journal of Optics. 2044005B. Liu, S. G. Johnson, J. D. Joannopoulos, and L. Lu, Journal of Optics 20, 044005 (2018). O Feely, Chaos, CNN, Memristors and Beyond. WORLD SCIENTIFIC36O. Feely, in Chaos, CNN, Memristors and Beyond (WORLD SCIENTIFIC, 2012), pp. 36. . E Zhao, Annals of Physics. 399289E. Zhao, Annals of Physics 399, 289 (2018). See details in supplementary information. See details in supplementary information. . Y Yang, Nature. 565622Y. Yang et al., Nature 565, 622 (2019).

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{'abstract': 'authors contributed equally to this work. Abstract Inspired by complete graph theory, we demonstrate that a metallic claw "metaatom" structure can carry a high number of nearly degenerate resonant modes. A photonic meta-crystal composing of a lattice of such meta-atoms exhibits a large number of flat bands that are squeezed into a narrow frequency window, and these flat bands can be designed to locate in a wide complete 3D bandgap. The degeneracy dimension (N f ) of the flat bands is determined by the number of branches (N b ) of the metallic claw with N f =N b -3, which is geometrically related to the complete graph theory. Different from those flat bands emerging from special lattice arrangements (e.g., Kagome lattice), the isolated flat bands here are insensitive to lattice perturbations. The proposed mechanism offers a new platform for realizing various dispersion-less phenomena and a new paradigm to realize high density of states and spectra compressing.', 'arxivid': '1907.00333', 'author': ['Hanyu Wang \nCollege of Advanced Interdisciplinary Studies\nNational University of Defense Technology\n410073ChangshaChina\n', 'Biao Yang yangbiaocam@nudt.edu.cn \nCollege of Advanced Interdisciplinary Studies\nNational University of Defense Technology\n410073ChangshaChina\n\nDepartment of Physics\nCenter for Metamaterials Research\nThe Hong Kong University of Science and Technology\nHong KongChina\n', 'Wei Xu \nCollege of Advanced Interdisciplinary Studies\nNational University of Defense Technology\n410073ChangshaChina\n', 'Yuancheng Fan \nDepartment of Physics\nCenter for Metamaterials Research\nThe Hong Kong University of Science and Technology\nHong KongChina\n', 'Qinghua Guo \nDepartment of Physics\nCenter for Metamaterials Research\nThe Hong Kong University of Science and Technology\nHong KongChina\n', 'Zhihong Zhu \nCollege of Advanced Interdisciplinary Studies\nNational University of Defense Technology\n410073ChangshaChina\n', 'C T Chan phchan@ust.hk†these \nDepartment of Physics\nCenter for Metamaterials Research\nThe Hong Kong University of Science and Technology\nHong KongChina\n'], 'authoraffiliation': ['College of Advanced Interdisciplinary Studies\nNational University of Defense Technology\n410073ChangshaChina', 'College of Advanced Interdisciplinary Studies\nNational University of Defense Technology\n410073ChangshaChina', 'Department of Physics\nCenter for Metamaterials Research\nThe Hong Kong University of Science and Technology\nHong KongChina', 'College of Advanced Interdisciplinary Studies\nNational University of Defense Technology\n410073ChangshaChina', 'Department of Physics\nCenter for Metamaterials Research\nThe Hong Kong University of Science and Technology\nHong KongChina', 'Department of Physics\nCenter for Metamaterials Research\nThe Hong Kong University of Science and Technology\nHong KongChina', 'College of Advanced Interdisciplinary Studies\nNational University of Defense Technology\n410073ChangshaChina', 'Department of Physics\nCenter for Metamaterials Research\nThe Hong Kong University of Science and Technology\nHong KongChina'], 'corpusid': 195766767, 'doi': '10.1103/physreva.100.043841', 'github_urls': [], 'n_tokens_mistral': 8522, 'n_tokens_neox': 7154, 'n_words': 4213, 'pdfsha': '5e216bc7a7e7d5976e1365bb3fd8e5846ff5e1da', 'pdfurls': ['https://export.arxiv.org/pdf/1907.00333v1.pdf'], 'title': ['Highly degenerate photonic flat bands arising from complete graph configurations', 'Highly degenerate photonic flat bands arising from complete graph configurations'], 'venue': []}

arxiv

Primordial Black Holes and Loops in Single-Field Inflation 16 Apr 2023 Hassan Firouzjahi School of Astronomy Institute for Research in Fundamental Sciences (IPM) P. O. Box19395-5531TehranIran Antonio Riotto Department of Theoretical Physics Gravitational Wave Science Center 24 quai E. AnsermetCH-1211Geneva 4Switzerland LAPTh CNRS USMB F-74940AnnecyFrance Primordial Black Holes and Loops in Single-Field Inflation 16 Apr 2023 Using the δN formalism we calculate the one-loop correction to the large-scale power spectrum of the curvature perturbation in the standard scenario where primordial black holes are formed in the early universe thanks to a phase of ultra-slow-roll in single-field inflation. We explicitly show that one-loop corrections are negligible when the transition from the ultra-slow-roll to the slow-roll phase is smooth. We conclude that the PBH formation scenario through a ultra-slow-roll phase is viable. I. INTRODUCTION A standard scenario where Primordial Black Holes (PBHs) are generated in the early universe is by enhancing the curvature perturbation at short scales [1][2][3] during an Ultra Slow Roll (USR) period when the inflaton potential V (φ) becomes sufficiently flat. The δN formalism is a powerful approach to calculate the curvature perturbation at the end of inflation [4][5][6]. Based on the separate universe approach, it is able to capture the super-Hubble evolutions of the perturbations beyond the linear order, thus capturing the effects of non-Gaussianity and providing a useful set-up to calculate the loop corrections [7]. In this note we use the δN formalism to calculate the one-loop corrections to the very large scale (i.e. CMB scale) power spectrum of the curvature perturbation induced by short-scale curvature modes. The latter modes, as the seeds of PBHs formation, are enhanced during the USR phase which is preceded and followed by the standard Slow-Roll (SR) phases. The issue of the impact of the one-loop correction from the short modes onto the large-scale power spectrum has been actively debated in the recent literature (see for instance Refs. [8][9][10][11][12][13][14]). We show explicitly that loop corrections are suppressed in the case in which the transition between the USR and the SR phase is smooth, as originally advocated in Refs. [9,11]. The paper is organised as follows. In section II we discuss the δN formalism, loop corrections are calculated in section III and conclusions in section IV. II. THREE-PHASE δN ANALYSIS The crucial starting point of our calculation is to take into account the SR phase preceding the USR phase. As we shall see, this turns out to be essential to capture the dependence of the long CMB mode onto the short modes. We consider therefore a three-step model of inflation: SR → USR → SR. In our convention the CMB scale modes are denoted by p while those of small scales running in the loop by q. Our goal is to extend the original δN analysis in Refs. [15,16] and [17] to the case where the mode of interest p leaves the horizon during the first SR phase. An important step is that we should keep track of the decaying mode in the first SR phase. During the first SR phase the decaying mode falls off exponentially. However, during the intermediate USR phase, this decaying mode is growing and it is behind the evolution of curvature perturbations on superhorizon scales. Therefore, for a consistent treatment we have to glue the decaying modes at the transition from the first SR phase to the USR phase. For this purpose, we have to solve the second order Klein-Gordon equation in the first SR phase to keep track of the decaying mode. In the spirit, this is similar to the method employed in Ref. [17] who studied a second order differential equation in the final SR phase to keep track of the evolution of the mode perturbations. First SR phase With the above discussion in mind, let us start with the first SR phase. We assume the field starts at the initial position φ i where the CMB scale mode p leaves the horizon. The first SR phase ends at t = t s when the USR phase starts. The USR phase itself is extended in the period t s ≤ t ≤ t e . It is also important for our δN formalism that the positions of the start and the end of USR phase, φ s ≡ φ(t s ) and φ e ≡ φ(t e ), are fixed so they are not varied when performing the variation δN . To keep track of the evolution of mode function during the first SR phase, similar to Ref. [17], we expand the potential near the point φ = φ s as follows V (φ) = V (φ s ) + √ 2ǫ s V (φ s )(φ − φ s ) + η s 2 V (φ s )(φ − φ s ) 2 + · · · . (II.1) Here the SR parameters ǫ s and η s are defined in terms if the derivatives of the potential, ǫ s = 1 2 V ′ (φ s ) V (φ s ) 2 , η s = V ′′ (φ s ) V (φ s ) , (II.2) where a prime denote the derivative with respect to the field φ and we work with the convention that the reduced Planck mass is set to unity, M P = 1. Note that to leading orders in the SR parameters, ǫ s ≃φ 2 s 2M 2 P H 2 ≃ −Ḣ H 2 ts , (II.3) in which a dot represents the derivative with respect to cosmic time and H is the Hubble expansion rate during inflation. To prevent the confusion, we define the other SR parameter associated to the time derivative of ǫ as η ≡ǫ/(Hǫ). During the USR phase η ≃ −6 which should not be mistaken with η s ≪ 1. Using the number of e-olds N as the clock via dN = Hdt, the evolution of the field during the first SR phase is given by d 2 φ dN 2 + 3 dφ dN + 3 √ 2ǫ s + 3η s (φ − φ s ) ≃ 0 , 3H 2 ≃ V (φ s ) , (II.4) whose solution is φ(N ) = C 1 e −αN + C 2 e −βN + φ s − √ 2ǫ s η s , (II.5) in which C 1 and C 2 are two constants of integration and, α ≡ 1 2 −3 + 9 − 12η s ≃ −η s , β ≡ 1 2 −3 − 9 − 12η s ≃ −3 + η s . (II.6) The decaying mode is controlled mainly by the evolution of the momentum π ≡ dφ/dN . Therefore, it is better to use the phase space variables (φ, π) and then express the constants C 1 and C 2 in terms of (φ(N ), π(N )). This way, we can use the values of C 1 and C 2 to match the solution φ(N ), π(N ) with the initial conditions (φ i , π i ) to their corresponding values (φ s , π s ) at the time of the start of USR phase, t = t s . Solving C 1 and C 2 in terms of (φ(N ), π(N )), we obtain C 1 = e αN π + β φ β − α , C 2 = −e βN π + α φ β − α . (II.7) φ ≡ φ − φ s + √ 2ǫ s η s . (II.8) The above expressions for (C 1 , C 2 ) are valid for anytime during t ≤ t s . Therefore, we can relate (φ s , π s ) to (φ(N ), π(N )) for any time during the first SR phase and in particular to the initial values (φ i , π i ). More specifically, we obtain the following relation between (φ s , π s ), (φ i , π i ), N i and N s : e α(Ns−Ni) = π + β φ i π s + β φ s , (II.9) and e β(Ns−Ni) = π + α φ i π s + α φ s . (II.10) Note that , as in Ref. [17], we have choses the convention that N = 0 at the end of USR phase, so during the first SR phase and during USR phase N < 0. Also N i is the initial value of N while the time of start of USR is denoted by N s , such that N i , N s < 0. For the CMB scale modes N i ∼ −30 assuming that the USR phase happens somewhere in the middle of inflation to generate the right mass scale for PBHs formation. Also, typically, |N s | few such that |N i | ≫ |N s |. From Eq. (II.10), we see that π s + α φ s falls off exponentially to zero like e −3(Ns−Ni) 0. This is the hallmark of the decaying mode in the first SR phase. Therefore, to a very good accuracy, and using Eq. (II.3), we have π s ≃ − √ 2ǫ s . This approximation can be used safely in Eq. (II.9) to solve for N s − N i , yielding N s − N i = 1 α ln π i + β φ i π s + β φ s ≃ 1 η s ln π i + β φ −3 √ 2ǫs ηs ≃ 1 η s ln 1 + η s φ i − φ s √ 2ǫ s − η s 3 √ 2ǫ s π i + √ 2ǫ s . (II.11) Note that for practical purpose, we can ignore the last term in the big bracket above since π ≃ − √ 2ǫ s to very good accuracy as described above. Now, plugging this expression for N s − N i in Eq. (II.10), we can relate π s to the initial values (φ i , π i ) as follows π s ≃ − √ 2ǫ s + (π i + η s φ i ) 1 + η s φ i − φ s √ 2ǫ s − 3 η . (II.12) Since the last term falls off exponentially for our setup in which N s − N i ∼ 30 or so, then we can neglect for all practical purposes the contributions of δπ s in δN . The USR phase During the USR phase we haveφ + 3Hφ = 0 . (II.13) Similar to the first phase we solve for (φ(N ), π(N )) as follows π(N ) = D 1 e −3N , φ(N ) = − π(N ) 3 + D 2 , (II.14) in which D 1 and D 2 are two constants of integrations. Eliminating the constants D 1 and D 2 , we obtain the following relations, π e = π s e 3Ns , π e = π s + 3(φ s − φ e ) , (II. 15) in which π e is the value of π at the time t e . As mentioned previously, it is important to note that φ s and φ e are fixed while π s and π e can vary as functions of the initial values in phase space (φ i , π i ). In particular, from the second equation above we find δπ e = −δπ s . The relations in Eq. (II.15) can then be combined, to yield the following expression for N s : N s = 1 3 ln π e π s = 1 3 ln 1 + 3(φ s − φ e ) π s . (II.16) A crucial difference compared to the setup of Ref. [17] is that the contribution of N s in total δN is quite negligible. This is because since φ s and φ e are fixed, the only way in which N s can contribute to δN is through π s . However, π s is exponentially close to − √ 2ǫ s as given in Eq. (II.12), so basically we can neglect the contribution of δπ s in δN in our calculation. Another important point to notice is that as π falls off exponentially during the USR phase, the SR parameter ǫ falls off exponentially as well. More specifically, at the end of USR phase we have ǫ e = ǫ s e −6|Ns| . As a result, for the modes which leave the horizon during the USR phase the curvature perturbation grows as R ∝ a 3 (t) ∝ e −3N in which a(t) is the FLRW scale factor. The final SR phase Finally, the evolution of the field in the final SR phase and the contribution to δN is the same as in Ref. [17]. This time, we expand the potential around φ e as follows V (φ) = V (φ e ) + √ 2ǫ V V (φ e )(φ − φ e ) + η V 2 V (φ e )(φ − φ e ) 2 + · · · . (II.17) Here 2ǫ V ≡ (V ′ (φ e )/V (φ e )) 2 and η V ≡ V ′′ (φ e )/V (φ e ) are the SR parameters defined at the point φ e . As the potential supports a SR phase after the USR period and the system approaches its attractor SR phase shortly afterwards, we can assume ǫ V and η V to be the corresponding SR parameters during the final SR phase. We also allow for the possibilities that ǫ s = ǫ V and η s = η V . The evolution of the field is then given by d 2 φ dN 2 + 3 dφ dN + 3 √ 2ǫ V + 3η V (φ − φ e ) ≃ 0 , 3H 2 ≃ V (φ e ) . (II.18) N f = 1 η V ln − 2η V π e − 6 √ 2ǫ V M P , (II.19) with the important point that the dependence of N f to the initial values (φ i , π i ) comes only through π e . However, as mentioned previously, with fixed values of φ s and φ e , from Eq. (II. 16) we see that δπ e = −δπ s . Since the dependence of π s to the initial values (φ i , π i ) are exponentially suppressed, we conclude that the contribution of the final duration of inflation N f to δN associated to initial conditions (φ i , π i ) are exponentially suppressed as well. As mentioned before, we allow for the possibility that ǫ s = ǫ V and η s = η V . However, in a natural situation where the shapes of the potential before and after the USR phase are similar, we typically expect that ǫ s ∼ ǫ V and η s ∼ η V . However, it is important to realize that there can be a large hierarchy between ǫ e = π 2 e /2 and ǫ 2 V . This hierarchy is a measure of the sharpness of the transition. Following the convention of [17] we define the sharpness parameter h as follows h ≡ 6 √ 2ǫ V π e = −6 ǫ V ǫ e . (II.20) For a sharp transition we have h ≤ −6 while for a smooth transition we have h → 0. The curvature perturbation from the δN formalism At this stage, we are ready to calculate the curvature perturbation at non-linear order through the δN formalism. The total number of e-folds is given by N (φ i , π i ) ≡ (N f − N i ). Using Eq. (II.11) for N i with N s given in Eq. (II. 16) we obtain N (φ i , π i ) ≃ 1 η s ln 1 + η s √ 2ǫ s M P (φ i − φ s ) − 1 3 ln 1 + 3(φ s − φ e ) π s (φ i , π i ) + 1 η V ln − 2η V π e − 6 √ 2ǫ V M P . (II.21) There are a number of important differences compared to the result of Ref. [17]. The first difference is that we have the first term above coming from the first SR phase. Second, the variation is with respect to δφ i which leaves the horizon at the initial flat hypersurface in the first SR phase while φ s and φ e are held fixed. Also note that the dependence to π i comes only via π s which is exponentially suppressed. As discussed before, the leading contribution to δN comes from the first term in Eq. (II.21) which is the usual contribution from single field SR inflation. This is consistent with the intuition that the long CMB scale modes which have left the horizon during the early SR phase should be largely unaffected during the subsequent evolution of the inflationary background. Having the explicit function N (φ i ) we can proceed with the δN analysis and expand around the background value φ i as follows δN = N ′ δφ + N ′′ 2 δφ 2 + N ′′′ 3! δφ 3 + · · · , (II.22) where by δφ we actually mean δφ i calculated at the time of horizon crossing on a flat hypersurface in the first SR phase. Also a prime denotes the derivative with respect to φ. The curvature perturbation is R = −δN and correspondingly, the power spectrum P R at the linear order is P R (p) = N ′2 P δφ = N ′2 H 2 2p 3 . (II.23) On the other hand, from Eq. (II.21) we see that to the leading order Defining the dimensionless power spectrum via P R = (p 3 /2π 2 )P R (p), the power spectrum of CMB scale modes which leave the horizon during the first phase of inflation is given by N ′ ≃ 1 √ 2ǫ s 1 + η s √ 2ǫ s M P (φ i − φ s ) −1 ≃ 1 √ 2ǫ s + O(η s ) .P R ≃ N ′2 P δφ ≃ H 2 8π 2 ǫ s + O e −3(Ns−Ni) . (II.25) At this stage, it is also instructive to look at the bispectrum and the f N L parameter. Defining the bispectrum as B R (k 1 , k 2 , k 3 ) ≡ (2π) 3 δ 3 (k 1 + k 2 + k 3 ) R(k 1 )R(k 2 )R(k 3 ) , the f N L parameter is related to B R (k 1 , k 2 , k 3 ) via [7] B R (k 1 , k 2 , k 3 ) = 6 5 f N L P R (k 1 )P R (k 2 ) + P R (k 1 )P R (k 3 ) + P R (k 2 )P R (k 3 ) . (II.26) With the expansion of δN as given in Eq. (II.22) we obtain [7] f N L = 5 6 N ′′ N ′2 . (II.27) With the logic explained above, the dominant contribution in N ′′ comes from the first SR phase of inflation with N ′′ ≃ −η s 2ǫ s 1 + η s √ 2ǫ s M P (φ i − φ s ) −2 , (II.28) yielding f N L ≃ − 5 6 η s + O e −3(Ns−Ni) . (II.29) The above results for the power spectrum and bispectrum confirm the physical expectation that at the tree level the CMB scale modes are largely unaffected by the small scale modes while the latter modes experiences a growth during the intermediate USR phase. Also, for later reference we note that N ′′′ ≃ 2η 2 s (2ǫ s ) 3 2 1 + η s √ 2ǫ s M P (φ i − φ s ) −δN p = N ′ δφ p + N ′′ 2 d 3 q (2π) 3 δφ q δφ p−q + N ′′′ 3! d 3 q 1 (2π) 3 d 3 q 2 (2π) 3 δφ q1 δφ q2 δφ p−q1−q2 . (III.32) Correspondingly, the two-point correlation function is given by the following series δN p1 δN p2 ≡ (2π) 3 δ 3 (p 1 + p 2 ) (A 1 + A 2 + A 3 + · · · ) , (III.33) in which A i are given as follow. In constructing the terms A i , we note that since p ≪ q, the leading terms are only those which contain at least one component δφ p1 multiplied by convolution integrals over d 3 q j . For example, a term like d 3 q 1 d 3 q 2 δφ q1 δφ p1−q1 φ q2 δφ p1−q2 is subleading as this term does not lead to the extra factor p −3 1 as required for the scaling of power spectrum. Starting with A 1 we have A 1 ≡ N ′2 δφ p δφ −p . (III.34) In particular, note that at the tree-level in which δφ p δφ −p = (H 2 /2p 3 ) the term A 1 is the power spectrum as given in Eq. (II.23). However, there are loop corrections in A 1 as well. This is because in δN formalism, δφ p δφ −p should be calculated on a flat hypersurface as an initial input value. Suppose we go to the flat gauge in which R = −(H/φ)δφ. Expanding the potential around its background value, and going to decoupling limit where the metric takes its unperturbed FLRW form, we have the interactions V ′′′ δφ 3 and V ′′′′ δφ 4 . In the SR setup or in a case with a smooth transition from a USR phase to a SR phase, both V ′′′ and V ′′′′ are SR suppressed. Therefore, the corresponding interactions V ′′′ δφ 3 and V ′′′′ δφ 4 contains additional factors of SR parameters and can be neglected to leading order. However, in a sharp transition in which |h| ≫ 1, there can be large loop corrections from these interactions. The analysis for the loop corrections in A 1 for the limit of sharp transition were performed specifically in [13] which we do not repeat here. We comment that the large interactions from V ′′′ and V ′′′′ contain the derivatives of the low-roll parameterη =ǫ/Hǫ. So in a sharp transition from a USR phase to a SR phase,η ∼ δ(t − t e ) which can induce large loop corrections as in Ref. [8]. However, as just mentioned and already advocated in Refs. [9,11], in the limit of a smooth transition, the interactions with the vertices V ′′′ and V ′′′′ come with extra factor of SR parameter and their contributions are suppressed compared to dangerous loop corrections calculated in the limit of sharp transition of Refs. [8] and [13]. The term A 2 is given by A 2 ≡ N ′ N ′′ d 3 q (2π) 3 δφ p1 δφ q δφ −p1−q . (III.35) It depends on the intrinsic bispectrum of the fluctuations δφ and can yield a non-zero value if there is intrinsic non-Gaussianity in the system. This happens when we have δφ 3 interaction in the model, that is an interaction term in the Lagrangian of the form L 3 ∼ a 3 V ′′′ δφ 3 . (III.36) For a smooth USR to SR transition, as long as we have a SR potential, that is V ′′′ small, the non-Gaussianities would be always small. Of course, if the potential is not smooth around the transition between the USR and the SR phase, this may yield large V ′′′ and loop corrections may be large. To calculate A 2 one can use the standard in-in analysis involving the typical interaction L 3 given above. Here we present an alternative derivation of A 2 as follows. Since p 1 ≪ q, we can use the long-short decomposition as employed in Ref. [18] and write Now, using the relations qτ * ≃ −1 and ǫ(τ ) = ǫ e τ /τ e 6 during the USR phase, we obtain δφ p1 δφ q δφ −p1−q ≃ δφ p1 d dφ δφ q δφ −q t * δφ −p1 = δφ p1 δφ −p1 d dφ δφ q δφ −q t * ,(A 2 ≃ N ′′ N ′ H 2 τ 3 e √ 2ǫ e δφ p1 δφ −p1 d 3 q (2π) 3 ∼ N ′′ N ′ H 2 3 √ 2ǫ e δφ p1 δφ −p1 , (III.41) where the integration is limited to modes which leave the horizon during the USR period, q s ≤ q ≤ q e . Using the expression for power spectrum given in Eq. (II.23), we finally obtain P loop R (p)| A2 ∼ η s e −3|Ns| P R (p) P short R , (III.42) in which P short R ≡ (H 2 /8π 2 ǫ e ) is the power spectrum associated to short modes which leave the horizon during the USR phase. Using a direct in-in analysis with the action (III.36) (see Ref. [17] for details about the action L 3 ) one obtains a result similar to Eq. (III.42). We see that the loop corrections induced from intrinsic non-Gaussianity is too small to be dangerous. More specifically, compared to Refs. [8,10] we have the suppression factor η s e −3|Ns| which make the loop contribution (III.42) harmless. Finally, the term A 3 is given by A 3 ≡ N ′ N ′′′ |δφ p | 2 d 3 q (2π) 3 |δφ(q, τ = 0)| 2 = N ′′′ N ′ P R (p) d 3 q (2π) 3 |δφ(q, τ = 0)| 2 . (III.43) To calculate the integral above, we note that R is smooth across the transitions while δφ(q) is not. This is becauseǫ undergoes jumps at the start and end points of USR phase. Using the analysis of Ref. [13], the curvature perturbation in the final phase is given by R (3) k = H M P 4ǫ(τ )k 3 α(3) k (1 + ikτ )e −ikτ + β Again, this contribution is suppressed by a factor η 2 s compared to what we expect from large loop corrections such as obtained in Ref. [8]. While we presented the specific forms of A 1 , A 2 and A 3 , one can proceed with the remaining terms in Eq. (III.33). However, associated to higher orders of A n , there will be n-th derivative of N which makes the contribution of A n more SR suppressed compared to the first few terms. IV. CONCLUSIONS Using the δN formalism, we have computed the correction to the large-scale power spectrum from the short modes which are amplified during the USR phase. Accounting carefully for the initial SR phase, we have shown that the loop corrections are negligible when the transition between the USR and the SR phase is smooth, as advocated in Refs. [9,11]. of the second and third terms in Eq. (II.21) contain the additional factors e −3(Ns−Ni) and e −3Ni e −ηV N f , respectively. However, in our setup in which N i ∼ −30 and N s ∼ few these contributions are exponentially suppressed compared to the contribution from the first term as given in Eq. (II.24). 1 − ikτ e ) 2 (1 + ikτ i ) 2 e 2ik(τe−τi) − i(2k 3 τ 3 i + 3ik 2 τ 2 i + 3i)(4ik 3 τ 3 e − hk 2 τ 2 e − h) ikτ i ) 2 (h + hk 2 τ 2 e + 4ik 3 τ 3 e )e −2ikτi + ih(1 + ikτ e ) 2 (3i + 3ik 2 τ 2 i + 2k 3 τ 3 i )e −2ikτe .Using R = −(H/φ)δφ, at the end of inflation we have |δφ(q, the integral and using the value of N ′′′ calculated in previous section we obtainP loop R (p)| A3 ≃ η 2 s |N s |P R (p)P short R .(III.46) III.37) in which t * represents the time in which the small scale mode q leaves the horizon during the USR phase. Since δφ is frozen on superhorizon scale we can relate δφ at the time of end of inflation (or at the end of attractor phase in the final SR regime) to its value at horizon crossing. On the other hand, on a flat hypersurface δφ = − √ 2ǫR = H √ 2ǫdt so we can trade dφ by dt and obtainδφ p1 δφ q δφ −p1−q ≃ δφ p1 δφ −p1 1 2ǫ(t * )a(t * )H d dτ δφ q δφ −q τ * , (III.38)in which τ is the conformal time with dτ = dt/a(t). On the other hand, to good accuracyδφ(τ ) = H 2q 3 (1 + iqτ )e −iqτ , (III.39) and hence d dτ δφ q δφ −q = H 2 τ q . (III.40) AcknowledgementsA.R. is supported by the Boninchi Foundation for the project "PBHs in the Era of GW Astronomy". H. F. thanks Mohammad Hossein Namjoo for useful discussions. . P Ivanov, P Naselsky, I Novikov, Phys. Rev. D. 507173P. Ivanov, P. Naselsky and I. Novikov, Phys. Rev. D 50, 7173 (1994). . J García-Bellido, A D Linde, D Wands, astro-ph/9605094Phys. Rev. D. 546040J. García-Bellido, A.D. Linde and D. Wands, Phys. Rev. D 54 (1996) 6040 [astro-ph/9605094]. . P Ivanov, astro-ph/9708224Phys. Rev. D. 577145P. Ivanov, Phys. Rev. D 57, 7145 (1998) [astro-ph/9708224]. . D S Salopek, J R Bond, Phys. Rev. D. 42D. S. Salopek and J. R. Bond, Phys. Rev. D 42 (1990), 3936-3962. . M Sasaki, E D Stewart, astro-ph/9507001Prog. Theor. Phys. 95M. Sasaki and E. D. Stewart, Prog. Theor. Phys. 95 (1996), 71-78 [astro-ph/9507001]. Delta N Formalism in Cosmological Perturbation Theory. A A Abolhasani, H Firouzjahi, A Naruko, M Sasaki, WSP. A. A. Abolhasani, H. Firouzjahi, A. Naruko and M. Sasaki, "Delta N Formalism in Cosmological Perturbation Theory," WSP, 2019, . C T Byrnes, K Koyama, M Sasaki, D Wands, hep-th/0705.4096JCAP. 1127C. T. Byrnes, K. Koyama, M. Sasaki and D. Wands, JCAP 11 (2007), 027 [hep-th/0705.4096]. . J Kristiano, J Yokoyama, hep-th/2211.03395J. Kristiano and J. Yokoyama, [hep-th/2211.03395]. . A Riotto, ph.CO/2301.00599A. Riotto, [astro-ph.CO/2301.00599]. . J Kristiano, J Yokoyama, hep-th/2303.00341J. Kristiano and J. Yokoyama, [hep-th/2303.00341]. . A Riotto, ph.CO/2303.01727A. Riotto, [astro-ph.CO/2303.01727]. . S Choudhury, M R Gangopadhyay, M Sami, astro-ph.CO/2301.10000S. Choudhury, M. R. Gangopadhyay and M. Sami, [astro-ph.CO/2301.10000]. . H Firouzjahi, ph.CO/2303.12025H. Firouzjahi, [astro-ph.CO/2303.12025]. . H Motohashi, Y Tada, astro-ph.CO/2303.16035H. Motohashi and Y. Tada, [astro-ph.CO/2303.16035]. . M H Namjoo, H Firouzjahi, M Sasaki, hep-th/1301.5699Europhys. Lett. 10139001M. H. Namjoo, H. Firouzjahi and M. Sasaki, Europhys. Lett. 101, 39001 (2013) [hep-th/1301.5699]. . X Chen, H Firouzjahi, E Komatsu, M H Namjoo, M Sasaki, astro-ph.CO/1308.5341JCAP. 1239X. Chen, H. Firouzjahi, E. Komatsu, M. H. Namjoo and M. Sasaki, JCAP 12, 039 (2013) [astro-ph.CO/1308.5341]. . Y F Cai, X Chen, M H Namjoo, M Sasaki, D G Wang, Z Wang, astro-ph.CO/1712.09998JCAP. 0512Y. F. Cai, X. Chen, M. H. Namjoo, M. Sasaki, D. G. Wang and Z. Wang, JCAP 05, 012 (2018) [astro-ph.CO/1712.09998]. . J M Maldacena, astro-ph/0210603JHEP. 030513J. M. Maldacena, JHEP 0305, 013 (2003) [astro-ph/0210603].

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{'abstract': 'Using the δN formalism we calculate the one-loop correction to the large-scale power spectrum of the curvature perturbation in the standard scenario where primordial black holes are formed in the early universe thanks to a phase of ultra-slow-roll in single-field inflation. We explicitly show that one-loop corrections are negligible when the transition from the ultra-slow-roll to the slow-roll phase is smooth. We conclude that the PBH formation scenario through a ultra-slow-roll phase is viable.', 'arxivid': '2304.07801', 'author': ['Hassan Firouzjahi \nSchool of Astronomy\nInstitute for Research in Fundamental Sciences (IPM)\nP. O. Box19395-5531TehranIran\n', 'Antonio Riotto \nDepartment of Theoretical Physics\nGravitational Wave Science Center\n24 quai E. AnsermetCH-1211Geneva 4Switzerland\n\nLAPTh\nCNRS\nUSMB\nF-74940AnnecyFrance\n'], 'authoraffiliation': ['School of Astronomy\nInstitute for Research in Fundamental Sciences (IPM)\nP. O. Box19395-5531TehranIran', 'Department of Theoretical Physics\nGravitational Wave Science Center\n24 quai E. AnsermetCH-1211Geneva 4Switzerland', 'LAPTh\nCNRS\nUSMB\nF-74940AnnecyFrance'], 'corpusid': 258179268, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 9187, 'n_tokens_neox': 7967, 'n_words': 4941, 'pdfsha': '79e7a07cad4809736454cc75ec994263f702abb3', 'pdfurls': ['https://export.arxiv.org/pdf/2304.07801v1.pdf'], 'title': ['Primordial Black Holes and Loops in Single-Field Inflation', 'Primordial Black Holes and Loops in Single-Field Inflation'], 'venue': []}

arxiv

Meta-Learning Parameterized Skills Haotian Fu Shangqun Yu Saket Tiwari Michael Littman George Konidaris Meta-Learning Parameterized Skills We propose a novel parameterized skill-learning algorithm that aims to learn transferable parameterized skills and synthesize them into a new action space that supports efficient learning in long-horizon tasks. We propose to leverage off-policy Meta-RL combined with a trajectorycentric smoothness term to learn a set of parameterized skills. Our agent can use these learned skills to construct a three-level hierarchical framework that models a Temporally-extended Parameterized Action Markov Decision Process. We empirically demonstrate that the proposed algorithms enable an agent to solve a set of difficult long-horizon (obstacle-course and robot manipulation) tasks. Introduction To improve Reinforcement Learning (RL)'s generalization to novel tasks, meta-Reinforcement Learning (meta-RL) learns a meta-policy from a large number of tasks that aims to quickly adapt to a new task within the same distribution. Off-policy meta-RL methods [48; 34; 19; 12] normally train a context-encoder that takes in a few collected trajectories/transitions on a new task as input and output latent parameters that function as a descriptor of the current task. That descriptor is fed into the policy as an additional input to generate actions. Compared to On-policy meta-RL methods [16; 57; 64], off-policy methods generally have much higher sample efficiency and better or comparable overall performance [43; 64; 48] on tasks whose differences vary smoothly and can be described by a single vector (e.g., tasks change between different goal velocity for a halfcheetah)-a setting also known as Hidden-parameter MDPs (HiP-MDPs) [13; 30]. However, for tasks with more diverse variations (e.g., tasks change between pull the mug, press the button, open the door, etc., see Figure 1), off-policy methods fail to generalize well compared to on-policy methods and methods based on fine-tuning [59; 61], even given a much larger number of adaptation steps. This makes offpolicy methods hard to to apply to realistic problems despite their superiority on HiP-MDP environments. However, fast adaptation of an entire policy to a new task is not the only possible form of generalization that we may want RL agents to display. Another approach is learning reusable high-level skills [55], which enable an agent to explore efficiently and solve hard long-horizon tasks using hierachical methods. In realistic tasks, we want skills that are flexible-able to be efficiently adapted to many different situations. For example, a skill that opens a door should be adjustable to many different types of doors and handles, from office doors to microwave doors. The most flexible skills are parametrized: discrete skills augmented with continuous parameters that adjust their behavior, thereby making them more likely to be reusable in new tasks because they are flexible enough to be applied in diverse situations. Finding the appropriate parametrization of a skill abtracted from the primitive action space in such settings is still an open question. We propose that the problem of learning parameterized skills is very similar to the HiP-MDP setting, in which off-policy meta-RL methods successfully generalize. Specifically, by leveraging Off-policy Meta-RL, we propose to learn parameterized skills [10]-both the skills themselves and the parameter space-as well as a high-level control policy that will use the learned parameterized skills as the new action space and perform on new tasks. Our contributions are: 1. We propose a novel three-level hierarchical RL framework combining off-policy Meta-RL and Parameterized Action MDP (PAMDP) algorithms to model Temporally-extend PAMDP problems, which can be used to solve long-horizon tasks. 2. For low-level policy learning, we propose a novel trajectory-centric smoothness training objective for learning parameterized skills capable of expressing diverse behaviors with a smooth parameter space. 3. For high-level and mid-level policy learning, we propose a novel hierarchical actor-critic algorithm that, given the learned parameterized action space, exhibits better performance compared to previous PAMDP algorithms. 4. Using the proposed algorithm, we are able to solve a set of difficult long-horizon ant obstacle course tasks, as well as long-horizon robotic manipulation tasks, and we demonstrate the importance of smoothness for learning a new action space and the ef- Figure 1. Left: Off-policy Meta-RL. The meta-policy π takes in the state as well as a latent vector z as input. On a new task, the context encoder φ will try to find the latent vector corresponding to the current task from a few trajectories τ . Mid: Off-policy Meta-RL in two different scenarios. Right: Leveraging off-policy Meta-RL to learn parameterized skills. fectiveness of the different components of our algorithm independently. A video of the learned policy can be found at https://youtu.be/T55bN2M2wAQ. Background A Parameterized Action Markov Decision Process (PAMDP) [36] is defined by the tuple {S, H, T, R, γ}, where the parameterized action space H can be defined as: H = {(k, z k )|z k ∈ Z k for all k ∈ {1, · · · , K}}, where z k is the corresponding continuous parameter set for each discrete action k. Here, z k is the continuous parameter corresponding to k, and K is the total number of discrete actions. At each step, the agent must select both a discrete action k and a continuous parameter z k . Thus, we have the dynamic transition function T (s |s, k, z k ) and the reward function R(r|s, k, z k ). A practical example is a football game, where the player needs to choose between kick the ball or move to some position (discrete), as well as the direction the player wants to kick the ball to or the specific position the player wants to move to (continuous). Most previous work assumes the primitive action space is parameterized, or a set of predefined parameterized skills are given. Our work makes an attempt to learn/synthesize the parameterized action space from scratch. HiP-MDPs model the variations in the transition dynamics and reward functions by assigning each task a hidden parameter θ, drawn from the distribution P Ω . The agent neither observes θ nor has access to the distribution P Ω that generates the task family. For a given task, parameterized by θ ∈ Θ, the stochastic dynamics are given by T (s |s, a; θ) and the deterministic reward function by R(s, a; θ Off-policy Meta-RL (OPML), shown in Figure 1, learns a meta-policy π(a|s, z) that is shared across all the tasks from the same distribution, as well as a context encoder φ(z|τ ) that maps collected transitions τ = {s 1 , a 1 , r 1 , s 2 , · · · , s n } to a task encoding z. Meta-Learning Parameterized Skills In general, we want the agent to learn a set of parameterized skills suitable to be used as the parameterized action space in a PAMDP, for which the agent will in turn learn a high-level control policy to solve new tasks. We show the overall three-level hierarchical framework of our proposed algorithm in Figure 2. We use this hierarchical framework to model a Temporally-extended PAMDP (TPAMDP). At the beginning of one episode, the agent receives a state from the environment. The state will be passed to the high-level policy π h first, which will output the discrete skill label k. Then the skill label and the state will be fed into the mid-level policy π m , which will output the skill parameter z corresponding to skill k. The agent will then choose the low-level policy π k corresponding to the skill label k as the current executing policy, which will take the state and skill parameter z as input and output primitive actions. The low-level policy π k will interact with the environment for T steps, after which the high-level policy will receive a new state and carry out the same process to choose the skill label and the corresponding parameters again. Overall, for a TPAMDP, we have a high-level policy and a mid-level policy that solve a new task by mapping the states to parameterized skill pairs (k, z)-learning in the high-level temporally extended parameterized action space. Each discrete skill label k corresponds to a low-level skill-conditioned policy network π k (a|s, z), which takes the continuous skill parameter z as an additional input. As the low-level policies are fixed, they can be treated as part of the environment during the training of high and mid-level policies. In Sec. 3.1, we introduce how our agent learn the low-level policy. In Sec. 3.2, we explain how our agent learns the high-level and mid-level policies. π m (z | s, k) Mid-level policy: π k (a | s, z) Low-level policies: , k = 1,2,⋯ Q(s, k, z) Critic: Skill parameter: Figure 2. Meta-learning parameterized skills: a three-level hierarchical framework modeling a TPAMDP. The learned parameterized skills are treated as a parameterized action space for the high-and mid-level policies, while each of the skills is actually a temporally abstraction of the low-level policy on the primitive action space. Off-policy Meta-RL for Parameterized Skills (MLPS) We first address how to learn the continuous parameters associated with each discrete action (skill category) to cover policies with similar and smoothly changing behaviors. To this end, we model a parameterized skill as a HiP-MDP, meaning the agent is given a set of tasks that share similar reward/dynamics structure. By modeling the parameterized skill as a HiP-MDP, the task set that we train our agent on has an underlying and potentially smoothly-varying hidden parameter that controls the distinct features of each task. Ideally, we want the agent to learn a policy that is able to solve the HiP-MDP-a robust skill-conditioned policy, and also learn a continuous representation z that smoothly approximates how the hidden parameters θ affect the agent's optimal policy on each task. Using off-policy Meta-RL, it is straightforward way to let the agent learn a skill-conditioned policy that additionally takes the continuous representation z as a input: π : S × Z → A. Then, given different values of z, the policy will output actions that can solve different tasks. By leveraging the high sample efficiencyof Off-policy Meta-RL, we can get a high-performing skill-conditioned policy quickly. We let the agent learn K different skillconditioned policies, which will be fixed as the low-level policies during the following higher-level policies' training. For practical implementation, we adopt the framework of a recent off-policy Meta-RL algorithm, PEARL [48], and train a context encoder that aims to put the collected trajectories into a latent representation, along with an actor and a critic network that both take in the latent representation as an additional input. In particular, we train a context encoder network φ : τ → z that generates latent representation z using historical transitions. Then, the generated z can be viewed as part of the state and can help the decision-making process as input to the actor network π(a|s, z) and critic network Q(s, a, z) as in PEARL. We provide more detailed algorithm and implementation information in Appendix A.1. Trajectory-Centric Smoothness In the parameterized skilllearning setting, besides the goal of learning a policy that performs well in all tasks, we also want that the continuous representation z which the policy is conditioned on is able to smoothly varying the agent's behaviors so that we can get a new smooth action space for this skill type and is reusable in other contexts. To achieve this goal, we propose the trajectory-centric smoothness training objectives for training the context encoder network. Note that instead of focusing on the difference between single transitions [14], we propose that parameterized skill learning should focus more on the overall difference between different trajectories. The learned representation of the skill should be able to encode the distinguishable features of the trajectories into its continuous parameters. Previous work shows the importance of smoothness in state representation learning [22; 60; 1]. Our case can be seen as policy representation learning, as we will use the learned representation space as the new action space, better smoothness intuitively will help the agent learn to identify the values of the continuous parameters for a new task more quickly. In Section 4.5, we empirically show how the smoothness of the learned skill parameter space will affect the overall performance of the algorithm. We propose that the agent's behavior under the skill-conditioned policy should change proportionally to the change of the continuous parameters' value. We hope to implicitly encode the semantic meanings of the underlying hidden parameters into our latent skill representation, thus improve the smoothness of the latent skill embedding space. Therefore we add another learning objective that aims to embed intermediate features of the state trajectories into the latent representation. Our main intuition is that the distance of different skills in the latent space should be proportional to the distance between their trajectories. Specifically, suppose we sample two batches of trajectories τ 1 and τ 2 from two different tasks. Then, we write the smoothness term as: 38], and κ controls the scale of the DTW distance. Instead of directly computing the Euclidean distance between two state trajectories, we use Dynamic Time Warping to align the trajectories before measuring the distance. The idea is illustrated in Figure 3. Even from the exact same state and using the same policy, the pointwise Euclidean distance between two trajectories can be large as there exists uncertainty in both the environmental dynamics and the output actions from the policy. Thus, we use a more reasonable metric that compares the overall "shape" of the two trajectories, which is more consistent with our goal of extracting the overall features of the trajectory instead of focusing on specific transitions. By minimizing the smoothness term, we obtain skill embeddings that correspond to the dynamic time warping distance of trajectories. L Smoothness := M SE[||φ(τ 1 )−φ(τ 2 )|| 2 −κDTW(τ 1 , τ 2 )],(1) Hierarchical actor-critic with Parameterized Skills (HPS) Then, given a set of low-level parameterized skills, the remaining question is how to efficiently learn high-level and mid-level control policies of our three-level hierarchical model in this temporally-extended PAMDP. As the low-level policies are fixed, the interaction between these higher-level policies and the environment is very close to a standard PAMDP. Thus a straightforward way is to directly apply PAMDP algorithms. HyAR [35] is a recently proposed algorithm that constructs a latent embedding space to model the dependency between discrete actions and continuous parameters. The discrete action along with the continuous parameters are mapped into a single latent action space, for which a policy is learned. However, learning directly in this latent embedding space means that the quality of exploration highly depends on whether the embedding space is learned properly. This problem becomes more severe as our parameterized action space are learned from data and can be quite noisy. That is, given the same state and the same parameterized skills, the distribution of the next state might have large uncertainty because executing each skill involves a large number of steps' interaction with the environment, of which the resulting trajectories could be quite noisy. Thus, learning to embed this generated action space further into some latent space may magnify the uncertainty of transitions. Another straightforward but effective approach is P-DQN [5; 58]. The P-DQN agent maintains a separate policy network for each discrete action k to output the corresponding continuous parameters, and then feed all these parameters from different discrete actions into the critic network. This makes computation highly expensive as it always has to compute all the continuous parameters for each discrete action, and is magnified when the number of discrete actions are large. In our case, to enable structured exploration at both discrete action and continuous parameter level, we propose to directly model the dependency of the discrete and continuous part of the parameterized action with two consecutive policy networks: for each decisionmaking step, we first choose the discrete action, then choose the continuous parameters conditioned on both the state and discrete action, which is in consistent with human's decision making process [45]. Concretely, as shown in Figure 2, we decompose the policy of parameterized actions as: π(k, z k |s) = π θc (z k |s, k)π θ d (k|s), where the policy network for discrete part of the action takes in state s and is parameterized by θ d , the policy network for the continuous parameter z k takes in state and the discrete action k output from π θ d and is parameterized by θ c . Compared with P-DQN, we only need to compute the continuous parameters for the discrete action we chose and thus avoid the redundancy problem. We update the policy using actor-critic framework with the maximum entropy learning objective for reinforcement learning [63; 24]. Maximum entropy RL greatly improve the exploration especially in the face of estimation error. It functions by maximizing the entropy of the policy as well as the expected return. This particularly fits our framework as the parameterized action space is learned and can be quite noisy. Further, exploration with different rates at different time periods of training is important in the long-horizon tasks as we explained in introduction. Concretely, we update the critic network Q ψ (s, k, z k ) according to: L critic = E (s,k,z k ,r,s )∼B [Q ψ (s, k, z k ) − (r + γV (s))] 2 , where B denotes the replay buffer, V (s) denotes the value network. We update the policy (actor) networks according to: L actor = E s∼B,k∼GS[π θ d (s)] D KL π θc (z k |s, k) exp(Q ψ (s, k, z k )) W ψ (s) , where W ψ (s) is the partition function that normalizes the distribution, GS denotes the gumbel-softmax distribution [28]. That is, to enable structured exploration at different levels of the action execution phase, we use the maximum entropy training objective to augment exploration for the policy of continuous parameters π θc , while we use gumbel-softmax technique to sample the discrete action to further augment the exploration for the policy of discrete action π θ d . Compared to -greedy exploration strategy, gumbel-softmax further augments structured exploration by sampling from the categorical distribution. It enables computing gradients for parameters of π θ d , of which the outputs are discrete, by leveraging the reparameterization trick [31]. We use gumbel-softmax to sample from the discrete policy network when interacting with the environment during training and also when updating the network. The latter one uses a smaller value of temperature τ (controls the exploration rate) to make the updating process smoother following the intuition in [21]. Note that HHQN [18] which focuses on multi-agent problem domain also uses a similar consecutive policy networks structure. However, they use two different Q networks to approximate the value of discrete and continuous policy which may cause high-level non-stationary problem [35], i.e. when sampling a transition from the replay buffer, the same discrete action may not lead to the same reward and next state as the continuous parameter can be different from the moment it was chosen. Thus, computing the Q-value of a discrete action without considering the continuous parameter can be quite noisy. We avoid this problem as HPS has only one critic network that measures the value of the hybrid action pair as a whole. New action space constraint For practical implementation, as we are using the learned skills as a new action space for the higher-level policies, we also need to find and add constraints to the values of the action space that the midlevel policy can choose from. For each category of skills, we first run the standard meta-test process across across all the available training tasks for multiple times and collect the value of skill parameters z. As shown in Figure 4, most of the learned representations are close to each other in the latent space, but there are always outliers that are far away from the main cluster. If we set the bounds of the value of the action space to contain all these data points, the blank area between the outliers and the main cluster, also called "unreliable areas", may deteriorate the higher-level policies, as shown in [62; 44; 35]. Thus, in practice, we rescale each dimension of the learned action space to a new bounded area by calculating the t% central range over the values of the collected data points, where t ∈ [90, 100). New action space Experiments Experimental Setup As shown in Figure The Make Coffee task requires the robot arm to push the mug under the coffee machine, press the button, return to the original position, reach the mug and pull the mug to the target position. We train the agent to learn three parameterized skills: Coffee-push, Coffee-pull and Coffee-button as well as two discrete skill: reach and return. The input states are the proprioceptive state of the robot arm, as well as the position of the mug. For the high-level and mid-level policies, we also include the label of the current subtask we want to agent to do (e.g, push, pull, etc.). Otherwise, the environment would be non-stationary (i.e., same state-action pair but different reward.) For training the three parameterized skills, the target position we want to push/pull the mug to, and the position of the button are the corresponding hidden parameters in their MDPs, and the values of them are all sampled independently from a uniform distribution. We run MLPS as well as standard Off-Policy Meta-RL (OPML) on each of the HiP-MDPs and get the skills {Goal(x 1 , x 2 ), Bridge(x 1 , x 2 ), Gather(x 1 , x 2 ), Box()} for Ant-mix and {P ush(x 1 , x 2 , x 3 , x 4 ), Reach(), Return() Button(x 1 , x 2 , x 3 , x 4 ), P ull(x 1 , x 2 , x 3 , x 4 )} for Make- coffee. For each random seed of training, we sampled the order of the subtasks (Make-coffee) as well as the hidden parameters of each subtask at the beginning of the experiment and fixed them for the rest of training and evaluation. We then used the parameterized skills learned in previous section as the new parameterized action space, and let HPS learn a solution policy for it. We give the agent sparse staged reward: a positive reward is received only when the ant has completed a subtask or it reaches the final goal, otherwise, the reward is 0. More environmental details and experiments can be found in Appendix A.3 and A.4. Overall Performance Comparison As shown in Figure 5, we compared to OPML+HPS, which means that we run OPML without the smoothness term to learn the parameterized skills and use our proposed higherlevel algorithm HPS to learn the policy. As mentioned before, we use PEARL as the OPML baseline, and we further augment it with contrastive loss as suggested by [19]. We also compared to MLPS+HyAR and MLPS+PDQN, which means that we use the same parameterized skills learned by MLPS but use different PAMDP learning algorithms to do high level policy learning. As shown in the attached video, the agent trained by our MLPS+HPS algorithm is able to successfully complete the long-horizon tasks in both cases. From Figure 5 second row, we can see that the performance drops if we replace the parameterized skills learned by MLPS with that of OFML. The performance gap is much larger than each single skill's performance gap as we will show later (Figure 7), indicating that the proposed trajectory-centric smoothness learning objective help construct a better parameterized action space ( Figure 6) which leads to better performance of high-level control policy. With the same pretrained parameterized skills, HPS learns the high-level control policy more efficiently than the other two PAMDP algorithms. In Ant obstacle course tasks, PDQN reaches similar performance in the end but took twice as many environment steps compared to HPS due to the redundancy problem we explained in Section 3.2. HyAR fails to learn a good policy possibly because our parameterized action space is learned and synthesized so the noise of high level dynamics is magnified when planning in the further generated latent action space. Quality of the learned skill parameters space We show the visualization of the learned skill parameters' embeddings in Figure 6. For each domain, We run the learned policies on 40 test tasks multiple times to collect enough successful trajectories covering the whole hiddenparameter space. The test tasks are linearly sampled from the given task distribution. Then we encode the trajectories into latent embeddings using the trained context encoder. The original dimension of the latent skill is set to be 2 in the ant domains so we just directly plot the latent embed- dings in a 2-D space. As shown in Figure 6, the embeddings generated from the trajectories of the same tasks are close together in the latent space. Moreover, we can see a strong monotonic relationship between the components of the learned latent representation and the real position of the open space in Ant-goal, as well as the coin's horizontal position in Ant-gather. A similar conclusion can also be made in the Ant-gather-two-coins domain, where there are actually two variables for different tasks unlike the other three tasks, which only have one. We can see that the two dimensions of the latent skill approximate these two variables separately, showing a linear correlation between each coin's position and the value of the latent representation. We also compared it with a visualization of the latent embedding encoded using PEARL's context encoder. Without the proposed trajectory-centric smoothness objective, the learned skill embeddings have large areas of overlap and ignore important patterns in the trajectories influenced by the changing positions of the goals. Quality of the learned skill-conditioned policies We also compare the performance of MLPS and OPML using standard meta-test in meta-RL to see how the proposed trajectory-centric smoothness objectives in MLPS will influence the low-level skill-conditioned policies' performance. For meta-testing, the test tasks are sampled from the same distribution as the training tasks. The results of meta-testing performance are shown in Figure 7. We find that the smoothness loss does not make the meta-policy's performance worse in any of the tested domains, and actually helps improve the meta-RL performance in the tasks in Ant domain. Unlike the benchmark mujoco tasks in previous meta-RL papers, the difference between optimal policies in these Ant tasks are mainly from the trajectories as a whole, instead of the terminal states/goals. In such settings, which are also common in practice, our proposed trajectory-centric smoothness objectives can help the agent encode the difference in trajectories into the latent embeddings, thus enabling the agent to quickly identify the correct embedding when adapting to a new task. The importance of smoothness of the action space In this subsection, we show how the smoothness of the learned action space (skill parameter space) will affect the performance of the policy that will use this action space. We create another coffee long-horizon task where the agent needs to constantly to push and pull the mug to different locations, such that the quality of any one of the two skills will affect the agent's final performance greatly as it has to cal- culate the skill parameter multiple times within one episode. We use MLPS and OPML to generate a set of coffee-push policies and coffee-pull policies. We choose those policies with close meta-test success rate to do the further comparison. Then we calculate the normalized smoothness loss for each of them following Equation (1). And we run HPS for each of them and compare their overall performance. We first visualize the influence of smoothness loss to the skill parameter embeddings. For push and pull skill, we set the dimension of the latent parameters as 4, so we first run Multidimensional Scaling (MDS) and then draw the scatter plot. As shown in Figure 8, the embedding with the lowest smoothness loss shows the strongest correlation with respect to the change of the real pull target position with few outliers (Color changes from yellow to blue means the target position changes from −4 to 4). And as the smoothness loss increases, more datapoints are dispersed and the correlation becomes weaker. As shown in the right two plots, bad smoothness can greatly increase the difficulty of finding the optimal policy. We find that for both skills we test, the performance of the overall algorithm decreases fast as the smoothness loss of the learned skill embeddings becomes larger, which indicates that smoothness is a very important factor to consider if we are trying to synthesize a new action space composed of skills learned on the primitive action space. Related Work Learning skills in a multi-task setting is common in prior These approaches typically assume the parameterzed skills already exist. By contrast, our three-level hierarchy policies are all learned from scratch using RL. Conclusion We propose a three-level hierarchy framework that models a temporally-extended PAMDP. We leverage off-policy Meta-RL framework to learn the skills while further aug-ment it with a trajectory-centric smoothness loss to train the trajectory encoder -aiming to improve the smoothness of the latent parameter space. We empirically show that our meta-learning parameterized skills framework enables an agent to solve two sets of complex long-horizon continuous control tasks. We also demonstrate the importance of the different components of our algorithm independently. [4] Bellman, R. and Kalaba, R. E. On adaptive control processes. Ire Transactions on Automatic Control, 4: 1-9, 1959. [5] Bester, C. J., James, S., and Konidaris, G. D. Multipass q-networks for deep reinforcement learning with parameterised action spaces. ArXiv, abs/1905.04388, 2019. [6] Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., and Zaremba, W. Openai gym, 2016. [7] Campos, V., Trott, A., Xiong, C., Socher, R., i Nieto, X. G., and Torres, J. Explore, discover and learn: Unsupervised discovery of state-covering skills. In ICML, 2020. [8] Chitnis, R., Tulsiani, S., Gupta, S., and Gupta, A. Efficient bimanual manipulation using learned task schemas. In 2020 IEEE International Conference on Robotics and Automation, ICRA 2020, Paris, France, May 31 -August 31, 2020, pp. 1149-1155. IEEE, 2020. [9] Co-Reyes, J. D., Liu, Y., Gupta, A., Eysenbach, B., Abbeel, P., and Levine, S. Self-consistent trajectory autoencoder: Hierarchical reinforcement learning with trajectory embeddings. In ICML, 2018. [13] Doshi-Velez, F. and Konidaris, G. D. Hidden parameter markov decision processes: A semiparametric regression approach for discovering latent task parametrizations. IJCAI : proceedings of the conference, 2016:1432-1440, 2016. [14] Eysenbach, B., Gupta, A., Ibarz, J., and Levine, S. Diversity is all you need: Learning skills without a reward function. In 7th International Conference on Learning Representations, ICLR 2019, New Orleans, LA, USA, May 6-9, 2019, 2019. [15] Fan, Z., Su, R., Zhang, W., and Yu, Y. Hybrid actorcritic reinforcement learning in parameterized action space. In IJCAI, 2019. [16] Finn, C., Abbeel, P., and Levine, S. Model-agnostic meta-learning for fast adaptation of deep networks. In ICML, 2017. [17] Frans, K., Ho, J., Chen, X., Abbeel, P., and Schulman, J. Meta learning shared hierarchies. ArXiv, abs/1710.09767, 2018. [18] Fu, H., Tang, H., Hao, J., Lei, Z., Chen, Y., and Fan, C. Deep multi-agent reinforcement learning with discretecontinuous hybrid action spaces. In IJCAI, 2019. [19] Fu, H., Tang [23] Goyal, A., Sodhani, S., Binas, J., Peng, X. B., Levine, S., and Bengio, Y. Reinforcement learning with competitive ensembles of information-constrained primitives. ArXiv, abs/1906.10667, 2020. [24] Haarnoja, T., Zhou, A., Abbeel, P., and Levine, S. Soft actor-critic: Off-policy maximum entropy deep reinforcement learning with a stochastic actor. In ICML, 2018. [ [58] Xiong, J., Wang, Q., Yang, Z., Sun, P., Han, L., Zheng, Y., Fu, H., Zhang, T., Liu, J., and Liu, H. Parametrized deep q-networks learning: Reinforcement learning with discrete-continuous hybrid action space. ArXiv, abs/1810.06394, 2018. [59] Yu, T., Quillen, D., He, Z., Julian, R., Hausman, K., b i 1 = {(s k , a k , r k , s k )} k=1···K ∼ B i , b i 2 = {(s k , a k , r k , s k )} k=1···K ∼ B i Sample latent embedding z i 1 ∼ φ(b i 1 ), z i target ∼ φ target (b i 2 ) Update actor and critic networks with {z i 1 , b i 1 }, and calculate L V alue end for Calculate contrastive loss L N CE with {z 1 1 , · · · , z C 1 }, {z 1 target , · · · , z C target } if calculating DT W = T rue then Sample one success trajectory from each task's replay buffer: {τ 1 suc , · · · , τ C suc } Calculate Dynamic Time Warping loss L Smoothness with {z 1 1 , · · · , z C 1 }, {z 1 target , · · · , z C target }, {τ 1 suc , · · · , τ C suc } end if Update cotext encoder network with L Skill = L V alue + αL N CE + βL Smoothness end for end while We show detailed procedures in Algorithm 1. The training procedures for the actor and critic networks are the same as in PEARL. After collecting data, for each training step, we first sample a meta batch of tasks {1, · · · , C}. Then for each task, we sample two transition batches b i 1 and b i 2 from its own replay buffer. We feed the first transition batch into the context encoder, then use the output latent embedding to calculate the RL loss L V alue and update actor and critic network parameters. This procedure is the same as in PEARL. We feed the second transition batch into the target context encoder network to get the latent embedding which will be used to calculate the auxiliary losses. After we get all the latent embeddings for tasks in the meta batch, we first calculate the contrastive loss using the latent embedding pairs from given task set. Then, if each task has collected at least one success trajectory (that is, the agent successfully reached the goal position), we will let the agent also calculate Dynamic Time Warping loss with the latent embedding and success trajectories sampled for each task in the meta batch. And we will update the context encoder network's parameters at the end of this training step. Note that one limitation of the implementation here is that for some tasks, it is possible that not all tasks in the training task set can collect a success trajectory within the given number of episodes. This will lead to the problem that the DTW is not calculated and used throughout the training process. Thus, we provide another implementation in A.1.2, which does not have such requirement and achieves similar final performance. For calculating contrastive loss, we adopt the same procedures in [33; 19], where we model the similarity score calculating function as bilinear products, i.e. z T µ W z k , where W is the learned parameter. Using the denotations in Algorithm 1, for z 1 1 , we can rewrite the InfoNCE loss as: L N CE := −E[f (z 1 1 , z 1 target ) − log 1 N C j=2 exp(f (z 1 1 , z j target )))]. And we calculate the loss use same procedures for other latent embedding {z 2 1 , · · · , z C 1 }. For calculating Dynamic Time Warping loss, given a latent embedding pair from different tasks: (z j 1 , z k target ), we draw the corresponding pair from the success trajectories set: (τ j suc , τ k suc ), and calculate the DTW loss with: L Smoothness := E τ j suc ,τ k suc M SE[||z j 1 − z k target || 2 − κDTW(τ j suc , τ k suc )],(2) where κ denotes the hyperparameter controls the scale of the DTW distance. Different from standard meta-RL setting, we assume the training task set (a fixed number of tasks) is given, whereas in [16] each time a task is randomly generated using parameters sampled from a prior distribution. Based on the intuition that the distance of different skills in the latent space should be proportional to the distance between their trajectories, we can compute DTW distance for any pair of trajectories, no matter if they succeed or not, and match the distance to their corresponding latent embeddings' distance. Thus, we do not need to wait until there's at least one success trajectory in each task's replay buffer to calculate the smoothness loss. Concretely, we provide the algorithm in Algorithm 2. Instead of modeling the context/trajectory encoder network as a product of independent Gaussian factors, we use a sequential encoder network, SNAIL [37], which uses temporal convolution and soft attention. Then, at each training step, instead of sampling two random batches of transitions, we sample two complete trajectories τ 1 , τ 2 and transform them to the same length. We compute the corresponding latent embeddings z 1 , z 2 for both of them using the context encoder, and calculate the DTW distance as well as the smoothness loss using the same equation (2). Thus we update the encoder network with the smoothness loss at every training step. We show the results comparison in Figure 9. Although MLPS with sequential encoder does not learn as fast as the original version, it achieves similar final performance. Besides, the requirement for using this version of the algorithm is a little looser. The readers can choose to apply one of the two versions of our algorithm based on the properties of their own test tasks. Algorithm 2 Parameterized Skill Learning (MLPS) Meta-training (Sequential encoder network) Input: Batch of training tasks µ i=1,··· ,M from p(µ), Initialize replay buffer B i for each training task Initialize parameters θ a and θ c for the actor and critic networks separately. Initialize parameters context encoder network φ, context encoder target network φ target while not done do for each task µ i do Roll out policy π θa , producing transitions {(s j , a j , r j , s j )} j:1···N Add tuples to execution replay buffer B i end for if there's at least one success trajectory in each task's replay buffer then calculating DT W = T rue end if for each training step do Sample a meta batch of tasks {1, · · · , C} for each task i in meta batch do Sample two trajectories and transform them to same length K: We show a comparison of different algorithms' properties in Table 2. P-DQN lacks scalability as it maintains a separate actor network for each discrete action, and have to compute all of them during both training and execution as we explained in the main text. HHQN has the problem of potential nonstationarity as we explained in the last paragraph of Section 4.2. PADDPG makes the actor output an concatenation of the discrete action and the continuous parameters for each of them together, which tends to ignore the dependency between discrete action and continuous parameters. This leads to performance drop as shown in PDQN and HyAR's original papers. HyAR don't have the above three problems but it needs to further learn a latent action space and plan based on it instead of the primitive parameterized action space. In our scenario where the parameterized action space is actually learned, the noise in the dynamics is magnified and it's hard to learn a proper latent action space. We assume this leads to HyAR's performance drop in our experiments. τ i 1 = {(s k , a k , r k , s k )} k=1···K ∼ B i , τ i 2 = {(s k , a k , r k , s k )} k=1···K ∼ B i Sample latent embedding z i 1 ∼ φ(τ i 1 ), z i target ∼ φ target (τ i 2 ) Sample transition batch b i = {(s k , a k , r k , s k )} k=1···K ∼ B A.2.2. IMPLEMENTATION DETAILS For the actor of discrete action π θ d , we use two hidden layers of MLPs with (300, 300) units, the output layer follows by a gumbel-softmax layer. For both the actor of continuous parameters π θc and critic network, we use two hidden layers of MLPs with (300, 300) units. The learning rates are all set as 3e − 4. The output of the actor of continuous parameters are stochastic the same as in SAC. Note that we fix the temperature for gumbel-softmax to be 1.0 across the whole training process, without using any decaying strategy. We also tried automatic temperature tuning as in SAC but did not get satisfactory result. We set the reward scale as 5 and the batch size as 128. Figure 10. Decision making process in the temporally-extended PAMDP. π θ d denotes the policy for discrete action and π θc denotes the policy for continuous parameters. Specifically, after we let the agent train on K different categories of tasks using MLPS, we get K different skillconditioned policies and fix them. Then we can directly let the high-level agent solve a new task by learning a policy that maps states to parameterized skill pairs (k, z k )-learning in the high-level temporal-extended action space. Each discrete skill label corresponds to a low-level skill-conditioned policy network π k (a|s, z k ), which takes the continuous skill parameter z k as an additional input. The decision making process of this new temporal-extended PAMDP is illustrated in Figure 10. Upon receiving a new observation, the agent must first choose the discrete skill label k using π θ d and then choose the corresponding skill parameter z k given the state s and k using π θc . The low-level skill-conditioned policy π k (a|s, z k ), which is learned by MLPS and fixed, takes in the observation and the skill parameter and outputs a primitive action to interact with the environment. The discrete skill label and the continuous parameter are fixed and the low level policy π k (a|s, z k ) will constantly output actions for a given number of environmental steps. Then, the last observation received from the environment is used as the new input state for the high-level policy, which will select new k and z k , and so on. A.3. Environment details and baselines We run all experiment with the mujoco simulator [56]: • Ant-box: The ant needs to push the box and walk pass a gap to reach the goal position. The position of the box is fixed. The task horizon is 500. • Ant-mix: The ant needs pass 10/15 different barriers consist of Ant-goal, Ant-bridge, Ant-gather-one-coin, Ant-box and reach the goal position. The task order as well as their specific features (door position, wind speed etc.) are all fixed. The origianl task horizon is 4000/6000. The task horizon when we do high-level learning with the skills is 10/15. The state input includes: High-level: the ant's horizontal position x and vertical position y, how many barrier it has passed. Low-level: the ant's horizontal position x and its relative vertical position y to the midlane of the current subtask, as well as the position and velocity of different joints of ant, and how many coins the ant has gathered. • Coffee-button: We adopt the same environment in MetaWorld [59]. The goal is press a button on the coffee machine. The button's position if changing across different tasks. • Coffee-push: We adopt the same environment in MetaWorld and further modify it by letting the gripper start at a position above the mug at the beginning of every episode. The goal is to push the mug to a target position under the coffee machine. The target position is changing across different tasks. • Coffee-pull: We adopt the same environment in MetaWorld and further modify it by letting the gripper start at a position above the mug at the beginning of every episode. The goal is to pull the mug to a target position under the coffee machine. The target position is changing across different tasks. • Reach: The goal is to reach the mug. This is a discrete skill. • Return: The goal is to return to the gripper's start position. This is a discrete skill. Reward Functions: • Ant-goal: R t =I{The ant has not passed the door} * ∆d Distance to door + I{door} * 10 + I{The ant has passed the door} * ∆d Distance to goal + I{goal} * 20 • Ant-bridge: R t = ∆d Distance to goal + I{goal} * 20 • Ant-gather: R t =I{The ant has not gathered the first coin} * ∆d Distance to first coin + I{first coin} * 10 + I{The ant has gathered one coin, one left} * ∆d Distance to second coin + I{second coin} * 10 I{The ant has gathered two coins} * ∆d Distance to goal + I{goal} * 20 • Ant-mix (sparse): R t = I{The ant passed a barrier} * 5 + I{goal} * 100 • Ant-mix (dense): For the dense reward used by other baselines, we use the direct combination of the dense reward we set for each specific subtask. The environment knows what the subtask is and it will give the corresponding dense reward. Moreover, we also give it the sparse reward when it passes each barrier. • Coffee-button, coffee-push, coffee-pull: same as in the original MetaWorld. The number of environment steps needed to complete the tasks (Ant-mix) and reach the final goal is around 3500 for 10b-3c, and around 5000 for 15b-4c. The results shown in the main text are averaged over three random seeds. The error bar shows one standard deviation. All experiments were run on our university's high performance computing cluster. When comparing with PDQN & HyAR & PEARL in ant obstacle course (ant-mix) domain (results shown in two plots of Figure 8), we fix the task order across different random seeds to make the environment setting consistent to all baselines. Ant-gather neighbour tasks Figure 12. Comparison results of DTW distance against pointwise euclidean distance in Coefficient of Variation. In all scenarios, we collect ten pairs of data (two categories of distance) and then compute the coefficient of variation for the ten values. performance on ant obstacle course tasks. Moreover, we do not need to consider the additional hyperparameters brought by epsilon-greedy method (final epsilon, number of decay steps) and just fix the "temparture" of gumbel-softmax to be 1.0 for all the scenarios. A.5. The difficulty of Long-horizon tasks for RL The first problem that stems from this long horizon is that a single policy neural network based on the primitive actions needs to be able to handle the distinct changes of the environment at different stages during the long execution episode (e.g., In our ant obstacle course, to reach the final goal, the ant has to move pass several gaps, obstacles, bridges), which is quite difficult. The more insidious problem is exploration. Because of the long action sequence needed, uninformed exploration methods are unlikely to be successful: In the ant obstacle course, early barriers, once mastered, should be navigated so as to maximize success probability (requiring a low exploration rate), while grappling with later barriers should involve collecting enough data to pass the barrier (high exploration rate). These two problems make learning to solve such long-horizon tasks at the level of primitive actions highly difficult. Figure 3 . 3Trajectories' Dynamic Time Warping distance compared with Pointwise Euclidean distance. Trajectory τ1 and τ2 are sampled from the task. Using Dynamic Time Warping to compute the distance (right) reveals they are quite close. However, unwarped pointwise Euclidean distance (left) ends up with the erroneous conclusion that the trajectories are very different. Figure 4 . 4Visualization of the learned representation space of one skill. All the data points are from the same skill label k but with different values of skill parameter z. 5 first row, we evaluate our algorithm on a Ant obstacle course domain built on OpenAI gym [6] and a robotic manipulation domain from MetaWorld [59]. Longhorizon tasks at the level of primitive actions are highly difficult (see Appendix A.5) and can be reduced to very short-horizon tasks with the help of skills.Ant-mix (obstacle course) have 10/15 consecutive barriers (denoted as 10-3c and 15-4c respectively in the plots) sampled from 4 categories of tasks: Ant-Goal, Ant-Bridge, Ant-Gather and Ant-box. Ant-Goal requires the agent to walk past a doorway at a position unknown and unseen to the agent, and reach the goal on the other side. Ant-Bridge requires the agent to walk across a bridge with cross wind. The speed of the wind is unknown to the agent. Ant-Gather requires the agent to gather two coins along its way to the goal position. The positions of the two coins are unknown to the agent. The agent succeeds after it reaches the goal position, which is fixed across all the tasks. The input states consist of the ant's position and other proprioceptive state, i.e., the angle/velocity of different joints. In these three tasks, the positions of the coins, the position of the doorway, and the wind speed are the corresponding hidden parameters in their MDPs, and the values of them are all sampled independently from a uniform distribution. Figure 5 . 5First row: The environments we used for Parameterized skill learning experiments. Second row: Comparison results of our method MLPS + HPS against other baselines in four scenarios . The horizontal axis denotes the number of "env" steps the high-level agent takes instead of the original environment steps. Dashed lines correspond to the maximum average return achieved by MLPS+PDQN and MLPS+HyAR after 1e6 "env" steps, as well as the max average return achieved by SAC learning from scratch using dense reward. Figure 6 . 6Visualization of learned skill embedding (best among three random seeds) of MLPS (first row) and OPML (second row), from left to right: Ant-Goal, Ant-Gather-one-coin (only one coin's position is changing), Ant-Gather-two-coins. We draw the ground-truth distribution of how the tasks are generated for ant-gather-two-coins at the bottom right corner of the last figure. Figure 7 . 7The meta-learning performance comparison on tasks in Ant and Coffee domains. Figure 8 . 8Smoothness of the learned action space and its influence. The left three figures are visualizations of the learned skill embeddings for the pull skill with different smoothness loss. Among them, the first two are generated by MLPS and the last one is generated by OPML. The right two figures show the overall performance comparisons of how the smoothness of one skill (pull, push) will affect the agent's overall performance on the long-horizon task. We keep the other skill policy fixed while testing each one of them. [ 2 ] 2Bagaria, A., Senthil, J. K., and Konidaris, G. Skill discovery for exploration and planning using deep skill graphs. In Meila, M. and Zhang, T. (eds.), Proceedings of the 38th International Conference on Machine Learning, ICML 2021, 18-24 July 2021, Virtual Event, volume 139 of Proceedings of Machine Learning Research, pp. 521-531. PMLR, 2021. [3] Barreto, A., Borsa, D., Hou, S., Comanici, G., Aygün, E., Hamel, P., Toyama, D., Hunt, J. J., Mourad, S., Silver, D., and Precup, D. The option keyboard: Combining skills in reinforcement learning. In NeurIPS, 2019. [ 10 ] 10da Silva, B. C., Konidaris, G. D., and Barto, A. G. Learning parameterized skills. 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CoRR, abs/2206.03271, 2022. doi: 10.48550/arXiv. 2206.03271. [62] Zhou, W., Bajracharya, S., and Held, D. PLAS: latent action space for offline reinforcement learning. In Kober, J., Ramos, F., and Tomlin, C. J. (eds.), 4th Conference on Robot Learning, CoRL 2020, 16-18 November 2020, Virtual Event / Cambridge, MA, USA, volume 155 of Proceedings of Machine Learning Research, pp. 1719-1735. PMLR, 2020. [63] Ziebart, B. D., Maas, A. L., Bagnell, J. A., and Dey, A. K. Maximum entropy inverse reinforcement learning. In AAAI, 2008.[64] Zintgraf, L. M., Shiarlis, K., Igl, M., Schulze, S., Gal, Y., Hofmann, K., and Whiteson, S. Varibad: A very good method for bayes-adaptive deep rl via meta-learning. ArXiv, abs/1910.08348, 2020. Meta-Learning Parameterized Skill (MLPS) Meta-training (regular encoder network) Input: Batch of training tasks µ i=1,··· ,M from p(µ), Initialize replay buffer B i for each training task Initialize parameters θ a and θ c for the actor and critic networks separately. Initialize parameters context encoder network φ, context encoder target network φ target while not done do for each task µ i do Roll out policy π θa , producing transitions {(s j , a j , r j , s j )} j:1···N Add tuples to execution replay buffer B i end for if there's at least one success trajectory in each task's replay buffer then calculating DT W = T rue end if for each training step do Sample a meta batch of tasks {1, · · · , C} for each task i in meta batch do Sample two transition batches Figure 9 . 9Comparison of different implementation strategy for MLPS on Ant-goal. • Ant-goal: The horizontal position of the doorway changes across all the tasks (uniformly sampled from [−10, 10]). The other environmental properties are fixed, including the goal's position. The task horizon is 400. The agent succeeds when it reaches the goal position (x = 0, y = 25). The state input includes the position and velocity of different joints of ant, and the ant's horizontal position x, as well as its relative vertical position y to the midlane y = 10. • Ant-bridge: The wind speed when the ant is on the bridge changes across all the tasks (uniformly sampled from [−3, 3]). The other environmental properties are fixed. The task horizon is 300. The agent succeeds when it reaches the goal position (x = 0, y = 26). The state input includes the position and velocity of different joints of ant, and the ant's horizontal position x, as well as its relative vertical position y to the midlane y = 10. • Ant-gather: The position of the first coin (Ant-gather-one-coin) or both coins (Ant-gather-two-coins) change across all the tasks (uniformly sampled from [−4.5, 4.5]). The other environmental properties are fixed 1 . The task horizon is 400. The agent succeeds only when it gathers both coins and reaches the goal position (x = 0, y = 16). The state input includes the position and velocity of different joints of ant, an indicator for how many coins the ant has gathered, and the ant's horizontal position x, as well as its relative vertical position y to the midlane y = 8. Figure 11 . 11Ant obstacle course (ant-mix) further illustration. Figure 13 .Figure 14 . 1314Left: Comparison of different metrics for calculating the distance between latent embeddings on Ant-goal. Right: Comparison of different exploration strategies for HPS on Ant obstacle course 10b-3c. Visualization of coffee-push skill with different smoothness loss (corresponding toFigure 8). arXiv:2206.03597v3 [cs.LG] 24 Feb 2023 Test TrainBad performance on tasks when the differences cannot be described by a low-dimensional vectorϕ τ z π Q ? s a L auxillary π(a | s, z) Q(s, a, z) The problem of Generalization Off-policy Meta-RL Test ⋯ ⋯ Train Excellent sample efficiency and performance on HiP-MDPs ⋯ Meta-RL Push ( ) z 1 , z 2 , z 3 Meta-RL Pull ( ) z 1 , z 2 , z 3 Train Test Make coffee The learned task encoding should indicate how the underlying hidden parameter θ changes the optimal policy in the HiP-MDP. When facing a new task, the agent interacts with the environment for a few episodes and inputs the resulting trajectories into the context encoder, from which it can infer the corresponding latent parameter to the policy. To train the context encoder, previous work uses the critic loss[48], or some auxiliary loss like the dynamics prediction [34; 12; 54] or contrastive loss[19]. where DTW stands for Dynamic Time Warping [4; DADS[51] successfully encode trajectories into a smooth latent skill pace in simple navigation tasks. However, the pure unsupervised learning setting does not allow the agent to master one complete category of high-level skill, e.g., find the coffee machine in a house, because of lack of task-specific exploration as no environmental reward is given. Thus it's hard to directly use these skills to solve long-horizon tasks.work [27; 50; 26]. da Silva et al. [10] first proposed to construct parameterized skills by analyzing the structure of policy manifold, but required labeled parameters of tasks for training. With the meta-RL setting, MLSH [17] learns fixed low-level policies during training and further finetune the high level policy on new tasks. Nam et al. [40] focus on using skilled pretrained from offline data to do better meta-RL. Some approaches also introduce multiple levels of hierarchies for skill learning [9] or planning [39]. Bar- reto et al. [3] and Qureshi et al. [46] propose a method to compose new task-relevant skills with pretrained simple skills. Goyal et al. [23] learn a high-level controller with decentralized low-level policies. However, these low-level skills are not parameterized so the generalization ability is limited. Rao et al. [49] introduce a similar three-level hierarchy of policies that also have discrete and continuous parts. However, they focus on learning skills from offline dataset and the learned skills do not involve temporal ab- straction of the actions. Another category of skill learning method is unsupervised skill discovery [32; 14; 7; 2]. In particular, A large body of recent work focuses on Deep RL problems with parameterized action spaces [36]. We have discussed PDQN [5; 58] and HyAR [35] in previous sections. PAD- DPG [25] and HPPO [15] let the actor output an concatena- tion of the discrete action and the continuous parameters for each discrete action label together. This category of methods tends to ignore the dependency between discrete action and continuous parameter, which is crucial for finding the opti- mal parameterized action. Neunert et al. [42] also considers discrete-continuous control but the settings are not standard parameterized action space, i.e., the discrete part and the continuous part of action are independent of each other. Pa- rameterized actions have also been studied in the task and motion planning (TAMP) literature [29; 11; 8; 53; 52; 41]. Table 1 . 1When computing the latent embedding z using context encoder, for coffee domain, the state component in the input trajectory only contains the first three elements (x&y&z coordinates of the gripper). For ant domain, the state component in the input trajectory only contains the first two elements (x&y coordinates of the ant) for ant-goal and ant-gather, for ant-bridge the state component in the input trajectory is the original state. Both actor network and critic network in MLPSMLPS's hyperparameters Environment # Meta-train tasks α β κ Meta batch size Embedding batch size Ant-goal 100 10 1 0.5 16 100 Ant-bridge 100 100 1 0.1 16 50 Ant-gather-one-coin 100 10 1 0.5 16 100 Ant-gather-two-coins 200 10 0.1 0.5 32 150 Coffee-push 60 0 0.1 0.5 16 100 Coffee-pull 60 0 0.1 0.5 16 100 Coffee-button 60 0 0.1 0.5 16 100 A.1.1. IMPLEMENTATION DETAILS are parameterized MLPs with 2 hidden layers of (300, 300) units. The context/trajectory encoder network is modeled as product of independent Gaussian factors, with 3 hidden layers of (400, 400, 400) units. We set the learning rate as 3e − 4. The scale of KL divergence loss is set to be 0.1. Other hyperparameters are listed in Table 1. A.1.2. ANOTHER APPROACH FOR IMPLEMENTING THE CONTEXT ENCODER AND ITS TRAINING PROCESS i Update actor and critic networks with {z i 1 , b i }, and calculate L V alue end for Calculate contrastive loss L N CE with {z 1 Calculate Dynamic Time Warping distance and Smoothness loss L Smoothness with {z 1 Update context encoder network with L Skill = L V alue + αL N CE + βL Smoothness end for end while A.2. Hierarchical actor-critic with Parameterized Skills (HPS) A.2.1. FURTHER COMPARISON WITH OTHER EXISTING RL WITH PARAMETERIZED ACTION SPACE ALGORITHMS1 , · · · , z C 1 }, {z 1 target , · · · , z C target } 1 , · · · , z C 1 }, {z 1 target , · · · , z C target }, {τ 1 1 , · · · , τ C 1 }, {τ 1 2 , · · · , τ C 2 } Table 2 . 2Comparison with other parameterized action space algorithms Algorithm Scalability Stationarity Dependence Primitive P-DQN PADDPG HHQN HyAR HPS A.2.3. TEMPORALLY-EXTENDED PAMDP π k ( ⋅ | s, z k ) Temporal − extended PAMDP Original MDPs π θ d π θ c k z k Environment Baselines: 1. Parameterized skill learning: We use the original source code for PEARL 2 , VariBAD 3 and their implementation for RL 2 . 2. Learning with learned parameterized skills: We use the original code for HyAR-TD3 4 , and their implementation for PDQN-TD3. For SAC, we use the stable-baselines3 implementation 5 .Additionally, for HyAR, we let the agent pretrain the Variational Auto-encoder 2000 steps.DTW distance Pointwise Euclidean distance 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Max number of barriers passed Ant-goal same tasks DTW distance Pointwise Euclidean distance 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Max number of barriers passed Ant-goal neighbour tasks DTW distance Pointwise Euclidean distance 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Max number of barriers passed Ant-bridge same tasks DTW distance Pointwise Euclidean distance 0.00 0.02 0.04 0.06 0.08 0.10 Max number of barriers passed Ant-bridge neighbour tasks DTW distance Pointwise Euclidean distance 0.0 0.2 0.4 0.6 0.8 Ant-gather same tasks DTW distance Pointwise Euclidean distance 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Department of Computer Science, Brown University. Correspondence to: Haotian Fu <hfu7@cs.brown.edu>. Note that in this paper, we consider the HiP-MDP setting. If we change the task order as well as the task parameters, without giving the agent these information the problem would become partially observable and extremely hard to solve. https://github.com/katerakelly/oyster 3 https://github.com/lmzintgraf/varibad 4 https://github.com/TJU-DRL-LAB/AI-Optimizer/tree/1e2a33a4a3a7a8235f1c12ea71b1ea686c071094/ self-supervised-rl/RL_with_Action_Representation/HyAR 5 https://github.com/DLR-RM/stable-baselines3 AcknowledgementThe authors would like to thank Akhil Bagaria, Sam Lobel, Anita de Mello Koch, Paul Zhiyuan Zhou and other members of Brown bigAI, as well as Tom Silver, Rohan Chitnis, Riley Simmons-Edler, Anurag Ajay for discussions and helpful feedback, and the anonymous reviewers for valuable feedback that improved the paper substantially. This research was conducted using computational resources and services at the Center for Computation and Visualization, Brown University.A.4. More Experimental resultsWe compared the difference between DTW distance and pointwise euclidean distance. For each domain (ant-goal, antbridge, ant-gather), we test two scenarios: same tasks, where we fix the hidden parameter (door position/wind speed/coin position), and calculate the distance between success trajectories that are able to solve the same task. Another scenario is neighbour tasks, where we sample 5 values from the original range of the hidden parameter with same distance from each other. For instance, for ant-goal, we sample 5 doorway position: {−9, −4.5, 0, 4.5, 9}. Then we calculate the distance between success trajectories from two neighbour tasks. Ideally, the distance of different pairs of neighbour tasks (e.g. {−9, −4.5}&{−4.5, 0}) should be similar to each other, as the actual distance between the hidden parameters are the same.We show the Coefficient of Variation of the two methods for calculating distance in different scenarios inFigure 12. In both same tasks and neighbour tasks scenarios, we expect the coefficient of variation to be small. This is because different metrics will result in different means, but the variation of the distance should be small as these distance are either calculated for the same tasks (that is, actual hidden parameter distance is fixed as 0) or for tasks with the same actual hidden parameter distance. We find that the distance calculated by DTW gets smaller variation in all scenarios which is consistent to our hypothesis. The gap between the two methods is especially large for trajectories from the same tasks, indicating that unwrapped pointwise Euclidean distance can end up with the erroneous conclusion that the trajectories are very different even though they have quite similar overall shape.We also compare the metrics for calculating the distance between z when calculating the DTW distance, shown inFigure 13Left. We find that besides directly using Euclidean distance as in Equation(2), we can also use the similarity score function f to calculate the distance between two latent embedding. And the result shows that these two metrics achieve similar results, although the performance of using similarity score drops a bit at the end of training. As shown inFigure 13Right, compared with regular epsilon-greedy strategy, our exploration strategy based on gumbel-softmax is important for HPS to achieve goodA.6. More discussion about the smoothness term for learning parameterized skills Note that in all our experiments, we use PEARL + contrastive loss as the Off-policy Meta-RL baseline. However, the proposed smoothness loss can be directly applied to other Off-policy Meta-RL algorithms as well, as it functions as an auxiliary loss to train the context encoder. Moreover, we use Dynamic Time Warping to calculate the smoothness loss and it works well for navigation tasks and robot manipulation tasks. 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J.Sodhani, S., Meier, F., Pineau, J., and Zhang, A. Block contextual mdps for continual learning. In Firoozi, R., Mehr, N., Yel, E., Antonova, R., Bohg, J., Schwager, M., and Kochenderfer, M. J. (eds.), Learning for Dy- namics and Control Conference, L4DC 2022, 23-24 of Proceedings of Machine Learning Research. Stanford, CA, USAPMLR168Stanford UniversityJune 2022, Stanford University, Stanford, CA, USA, volume 168 of Proceedings of Machine Learning Re- search, pp. 608-623. PMLR, 2022. Between mdps and semi-mdps: A framework for temporal abstraction in reinforcement learning. R S Sutton, D Precup, S Singh, Artif. Intell. 112Sutton, R. S., Precup, D., and Singh, S. Between mdps and semi-mdps: A framework for temporal abstraction in reinforcement learning. Artif. Intell., 112:181-211, 1999. Mujoco: A physics engine for model-based control. E Todorov, T Erez, Y Tassa, IEEE/RSJ. Todorov, E., Erez, T., and Tassa, Y. Mujoco: A physics engine for model-based control. In 2012 IEEE/RSJ

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{'abstract': 'We propose a novel parameterized skill-learning algorithm that aims to learn transferable parameterized skills and synthesize them into a new action space that supports efficient learning in long-horizon tasks. We propose to leverage off-policy Meta-RL combined with a trajectorycentric smoothness term to learn a set of parameterized skills. Our agent can use these learned skills to construct a three-level hierarchical framework that models a Temporally-extended Parameterized Action Markov Decision Process. We empirically demonstrate that the proposed algorithms enable an agent to solve a set of difficult long-horizon (obstacle-course and robot manipulation) tasks.', 'arxivid': '2206.03597', 'author': ['Haotian Fu ', 'Shangqun Yu ', 'Saket Tiwari ', 'Michael Littman ', 'George Konidaris '], 'authoraffiliation': [], 'corpusid': 256389894, 'doi': None, 'github_urls': ['https://github.com/katerakelly/oyster', 'https://github.com/lmzintgraf/varibad', 'https://github.com/TJU-DRL-LAB/AI-Optimizer/tree/1e2a33a4a3a7a8235f1c12ea71b1ea686c071094/', 'https://github.com/DLR-RM/stable-baselines3'], 'n_tokens_mistral': 21655, 'n_tokens_neox': 19172, 'n_words': 12364, 'pdfsha': '78839ec995beab7f5fa8ce8d549fb4cf04b33d45', 'pdfurls': ['https://export.arxiv.org/pdf/2206.03597v3.pdf'], 'title': ['Meta-Learning Parameterized Skills', 'Meta-Learning Parameterized Skills'], 'venue': []}

arxiv

Nonextensive/dissipative correspondence in relativistic hydrodynamics 23 May 2008 Takeshi Osada Faculty of Knowledge Engineering Theoretical Physics Lab Musashi Institute of Technology Setagaya-ku158-8557TokyoJapan Grzegorz Wilk The Andrzej So ltan Institute for Nuclear Studies Hoża 6900681WarsawPoland Nonextensive/dissipative correspondence in relativistic hydrodynamics 23 May 2008Progress of Theoretical Physics Supplement 1 We argue that there is profound correspondence (the nonextensive/dissipative correspondence -NexDC) between the perfect nonextensive hydrodynamics and the usual dissipative hydrodynamics which leads to simple expression for dissipative entropy current. We have recently proposed and discussed in detail 1) a nonextensive hydrodynamical model, q-hydrodynamics. It is based on the relativistic nonextensive kinetic theory (as, for example, proposed in works 2), 3) ), which automatically accounts for all kinds of possible strong intrinsic fluctuations and long-range correlations existing in systems of quark and/or hadronic matter produced relativistic heavy-ion collisions. It is expected to be in a kind of stationary state 4) rather than in exact thermal equilibrium. It is described by nonextensive statistics 5) and characterized by the non-extensive parameter q . In this approach the perfect q-hydrodynamical equations (for perfect, non-viscous in q-language q-fluid) are given by T µν q;µ = ε q (T q )u µ q u ν q − P q (T q )∆ µν q ;µ = 0.(1) Here ε q (T q ), P q (T q ) and u µ q (x) are, respectively, the nonextensive energy density and pressure (both being functions of the none-extensive temperature T q 1) ) and accompanying hydrodynamical flow four vector, whereas ∆ µν q ≡ g µν − u µ q u ν q . Note that one can always decompose tensor T µν q by using another 4-velocity field u µ (x) and obtain εu µ u ν −P ∆ µν + 2W (µ u ν) + π µν ;µ = 0,(2) where (we denote δu µ q ≡ u µ q − u µ and ∆ µν ≡ g µν − u µ u ν ) ε = ε q + 3Π,P = P q + Π, (3a) W µ = w q [1 + γ] ∆ µ λ δu λ q ,(3b)π µν = W µ W ν w q [1 + γ] 2 + Π∆ µν = w q δu <µ q δu ν> q (3c) can be interpreted as, respectively, energy density (ε), pressure (P ), energy or heat flow vector (W µ ) and shear pressure tensor (π µν ) accompanying the field u µ (x). Here w q ≡ ε q + P q , γ ≡ u µ δu µ q = − 1 2 δu qµ δu µ q , A (µ B ν) ≡ 1 2 (A µ B ν + A ν B µ ), a <µ b ν> ≡ [ 1 2 (∆ µ λ ∆ ν σ + ∆ µ σ ∆ ν λ ) − 1 3 ∆ µν ∆ λσ ]a λ b σ whereas Π ≡ 1 3 w q [γ 2 + 2γ].(4) This last quantity can be regarded as a bulk pressure. The crucial point of our work is assumption that there exists some temperature T and velocity field δu µ q satisfying the following (which we call the NexDC relations): P (T ) = P q (T q ), ε(T ) = ε q (T q ) + 3Π.(5) Let ε ≡ ε q=1 and P ≡ P q=1 be the energy density and pressure (both functions of temperature T ) defined in the usual Boltzmann-Gibbs statistics (i.e., for q = 1). Using them one can transform equation (2) into following usual dissipative hydrodynamical equation (or d-hydrodynamics): 6)-9) ε(T )u µ u ν −(P (T ) + Π)∆ µν +2W (µ u ν) +π µν ;µ = 0.(6) This completes demonstration of our conjecture that perfect q-hydrodynamics represented by Eq. (1) is equivalent to d-hydrodynamics represented by Eq. (6), which is therefore its viscous counterpart. Notice that with bulk pressure (4) and NexDC relations (5) one obtains the q-enthalpy, ε q (T q ) + P q (T q ) = ε(T ) + P (T ) [1 + γ] 2 ,(7) which can be also used in definition of γ because w ≡ T s = ε + P and 1/(γ + 1) = 1 − 3Π/w (s is the entropy density in the usual Boltzmann-Gibbs statistics). Notice that in the NexDC one has the following relations: W µ W µ = −3Πw, π µν W ν = −2ΠW µ , π µν π µν = 6Π 2 .(8) Let us consider now respective entropies. Dissipation is connected with the production of entropy and in the usual approach 7), 8) the most general off-equilibrium four-entropy current σ µ is given by σ µ = P (T )β µ + β ν (T µν eq + δT µν ) + Q µ ,(9) where β µ ≡ u µ /T , T µν eq ≡ ε(T )u µ u ν − P (T )∆ µν , δT µν ≡ −Π∆ µν + W µ u ν + W ν u µ + π µν and where Q µ = Q µ (δT µν ) is some function which characterizes the off-equilibrium state. In the case of the q-entropy current 1) the NexDC (i.e., Eqs. (4) and (7)) leads to the following off-equilibrium state: Q µ =Q µ χ ≡ χ su µ + W µ T ,(10)(with χ ≡ T Tq 1 − 3Π w − 1) which results in σ µ χ = su µ + W µ T + χ su µ + W µ T .(11) Notice that, because of the strict q-entropy conservation assumed here, when using Q µ = Q µ χ one always gets σ µ χ;µ = 0. It means that, although there is no production of q-entropy, there is some production of the usual entropy, i.e., our q-system is really dissipative in the usual meaning of this word. Let us compare now the usual causal relativistic dissipative theory as given by 7), 8) with the one emerging from our NexDC. The most general algebraic form of Q µ , calculated up to the second order in the dissipative flux, is given by 8) Q µ 2nd = −β 0 Π 2 + β 1 W ν W ν − β 2 π νλ π νλ 2T u µ − α 0 ΠW µ T + α 1 π µν W ν T ,(12) where β i=1,2,3 are the corresponding thermodynamic coefficients for the, respectively, scalar, vector and tensor dissipative contributions to the entropy current whereas α i=0,1 are the corresponding viscous/heat coupling coefficients. Correspondingly, in the NexDC one has Q µ 2nd →Γ 2nd su µ + Υ 1st W µ T ,(13) where Γ 2nd ≡ − 3β 1 2 Π − (β 0 + 6β 2 ) 2w Π 2 , Υ 1st ≡ −(α 0 + 2α 1 )Π.(14) As one can see, in this case Q µ can be expressed by polynomials in the bulk pressure Π. Therefore, it is natural to expect that the most general entropy current in the NexDC approach has the following form: Q µ full = Γ (Π)su µ + Υ (Π) W µ T ,(15) where Γ, Υ are (in general infinite) series in powers of the bulk pressure Π. In this sense the Q µ full can be regarded as being the full order dissipative current. In general one has entropy production/reduction, i.e., σ µ ;µ = 0. However in the case when Γ (Π) = Υ (Π) = χ one has σ µ χ;µ = 0 and therefore one can write the full order dissipative entropy current as being equal to Q µ full = (χ + ξ)su µ + (χ − ξ) W µ T ,(16) where Γ and Υ are determined by χ ≡ (Γ + Υ )/2 and ξ ≡ (Γ − Υ )/2. Note here that one always can express χ by κ and γ: χ = − γ (1 + γ)(1 + κ) − κ 1 + κ(17) where κ ≡ T q /T − 1. The expression Eq. (17) suggests that one can get two possible solutions for (Γ, Υ ) satisfying both χ ≡ (Γ + Υ )/2 and Eq. (10), Γ 2 ≡ T T q 1 − 3Π w − 1 , Υ 2 ≡ T − T q T q (18a) or Γ 2 ≡ T − T q T q , Υ 2 ≡ T T q 1 − 3Π w − 1 .(18b) Out of them only (18a) is acceptable because only for it u µ Q µ full ≤ 0 (i.e., entropy is maximal in the equilibrium, 8) this is because (T − T q )/T q is always positive for q ≥ 1 1) ). In this way we finally arrive at the following possible expression for the full order dissipative entropy current in the NexDC approach: σ µ full ≡ su µ + W µ T − 2T T q 1 − 1 − 3Π w su µ + 2(T − T q ) T q W µ T .(19) Limiting ourselves to situations when T /T q ≈ 1 and neglecting terms higher than O(3Π/w) 2 , one obtains that Q µ full ≈ − 3Π w − 1 4 3Π w 2 su µ .(20) Comparing Eqs. (13) and (20) one gets β 1 = 2 w , β 0 +6β 2 = 9 2w , α 0 +2α 1 = 0. Since in the Israel-Stewart theory 7) the relaxation time τ is proportional to thermodynamical coefficients β 0,1,2 , it is naturally to assume that in our NexDC case τ ∝ 1/w, i.e., it is proportional to the inverse of the enthalpy. To summarize, we have proposed to describe dissipative hydrodynamics (at least partially) by using nonextensive formulation of the usual perfect hydrodynamical model introducing a nonextensive/dissipative correspondence (NexDC). As discussed in more detail in 1) such model can be solved exactly and when compared to the usual hydrodynamical approach it reveals terms which can be interpreted as due to dissipative effects. They can be therefore expressed by the single parameter of the theory used, namely the nonextensivity parameter q. We have used this finding to propose a possible full order expression for the dissipative entropy current σ µ full . AcknowledgementsPartial support (GW) of the Ministry of Science and Higher Education under contracts 1P03B02230 and CERN/88/2006 is acknowledged. Here one can also find all necessary references to subject of nonextensivity and q-statistical mechanics. T Osada, G Wilk, arXiv:0710.1905arXiv:0805.2253Phys. Rev. C. 7744903nucl-th. nucl-thT. Osada and G. Wilk, Phys. Rev. C 77 (2008), 044903; arXiv:0710.1905 [nucl-th]. Here one can also find all necessary references to subject of nonextensivity and q-statistical mechanics. See also arXiv:0805.2253 [nucl-th]. . A Lavagno, Phys. Lett. 30113A. Lavagno, Phys. Lett. A301, 13 (2002). . J A S Lima, R Silva, A R Plastino, Phys. Rev.Lett. 862983J. A. S. Lima, R. Silva and A. R. Plastino, Phys. Rev.Lett. 86 2983 (2001). . T Kodama, H T Elze, C E Aguiar, T Koide, cond-mat/0406732T. Kodama, H. T. Elze, C. E. Aguiar and T. Koide, cond-mat/0406732; . T Kodama, J. Phys. 311051T. Kodama, J. Phys. 31, S1051 (2005). . C Tsallis, J. Stat. Phys. 52479C. Tsallis, J. Stat. Phys. 52 479 (1988); . Braz. J. Phys. 291Braz. J. Phys. 29, 1 (1999). . C Eckart, Phys.Rev. 58919C. Eckart, Phys.Rev. 58, 919 (1940). . W Israel, Ann. Phys. (N.Y.). 100310W. Israel, Ann. Phys. (N.Y.) 100, 310 (1976); . J M Stewart, Proc. R. Soc. London, Ser. A. 35759J. M. Stewart, Proc. R. Soc. London, Ser. A 357, 59 (1977); . W Israel, J M Stewart, Ann Phys. (N.Y.). 118341W. Israel and J. M. Stewart, Ann Phys. (N.Y.) 118, 341 (1979). . A Muronga, Phys. Rev.C. 6934903A. Muronga, Phys. Rev.C 69, 034903 (2004); . A Muronga, D H Rischke, ; A Muronga, arXiv:nucl-th/0407114Phys. Rev. Lett. 8862302A. Muronga and D.H. Rischke, arXiv:nucl-th/0407114; A. Muronga, Phys. Rev. Lett. 88, 062302 (2002). . U About, H Heinz, A K Song, Chaudhuri, Phys. Rev. C. 7334904About dissipative hydrodynamics, see for example, U. Heinz, H. Song and A.K. Chaud- huri, Phys. Rev. C 73, 034904 (2006); . H Song, U Heinz, arXiv:0712.3715nucl-thH. Song and U. Heinz, arXiv:0712.3715 [nucl-th]; . P Romatschke, Eur. Phys. J. C. 52203P. Romatschke, Eur. Phys. J. C 52 203 (2007); . P Romatschke, U Romatschke, Phys. Rev. Lett. 9917230P. Romatschke and U. Romatschke, Phys. Rev. Lett. 99, 17230 (2007).

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{'abstract': 'We argue that there is profound correspondence (the nonextensive/dissipative correspondence -NexDC) between the perfect nonextensive hydrodynamics and the usual dissipative hydrodynamics which leads to simple expression for dissipative entropy current.', 'arxivid': '0805.3572', 'author': ['Takeshi Osada \nFaculty of Knowledge Engineering\nTheoretical Physics Lab\nMusashi Institute of Technology\nSetagaya-ku158-8557TokyoJapan\n', 'Grzegorz Wilk \nThe Andrzej So ltan Institute for Nuclear Studies\nHoża 6900681WarsawPoland\n'], 'authoraffiliation': ['Faculty of Knowledge Engineering\nTheoretical Physics Lab\nMusashi Institute of Technology\nSetagaya-ku158-8557TokyoJapan', 'The Andrzej So ltan Institute for Nuclear Studies\nHoża 6900681WarsawPoland'], 'corpusid': 119195148, 'doi': '10.1143/ptps.174.168', 'github_urls': [], 'n_tokens_mistral': 4183, 'n_tokens_neox': 3459, 'n_words': 1955, 'pdfsha': '2d5f36396f39069e34d8801eff007bccebdcae9b', 'pdfurls': ['https://arxiv.org/pdf/0805.3572v1.pdf'], 'title': ['Nonextensive/dissipative correspondence in relativistic hydrodynamics', 'Nonextensive/dissipative correspondence in relativistic hydrodynamics'], 'venue': []}

arxiv

On Christol's conjecture 1 Jan 2020 Y Abdelaziz C Koutschan Johann Radon Institute for Computational and Applied Mathematics RICAM Altenberger Strasse 69A-4040LinzAustria J-M Maillard UMR 7600 LPTMC CNRS Université Pierre et Marie Curie Sorbonne Université Tour 23, 5èmeétage, case 121, 4 Place Jussieu75252, Cedex 05ParisFrance On Christol's conjecture 1 Jan 2020AMS Classification scheme numbers: 33C0533C2033F1068W30 Key-words: Christol's conjecturediagonals of rational functionsShimura curvescreative telescopingtelescopersD-finite seriesglobally bounded series We show that the unresolved examples of Christol's conjecture 3 F 2 ([2/9, 5/9, 8/9], [2/3, 1], x) and 3 F 2 ([1/9, 4/9, 7/9], [1/3, 1], x), are indeed diagonals of rational functions.We also show that other 3 F 2 and 4 F 3 unresolved examples of Christol's conjecture are diagonals of rational functions. Finally we give two arguments that show that it is likely that the 3 F 2 ([1/9, 4/9, 5/9], [1/3, 1], 27 · x) function is a diagonal of a rational function. Introduction There is a plethora of multiple integrals in physics: Feynman integrals, lattice Green functions, the summands of the magnetic susceptibility of the 2D Ising model [2,22], that have very specific mathematical properties. These functions are D-finite, i.e., solutions of linear differential operators with polynomial coefficients, and have series expansions with integer coefficients. Furthermore, it was also shown that the linear differential operators annihilating the summands of the magnetic susceptibility of the Ising modelχ(n), verify the specific property of being Fuchsian † operators: the critical exponents of all their singularities are given by rational numbers, and their Wronskians are N -th roots of rational functions [2,22]. It was also shown that theχ(n) functions are solutions of globally nilpotent operators [5], and that they "come from geometry" being G-operators [25]. The unifying scheme behind these seemingly sparse properties is that these functions are diagonals of rational functions [3,4]. It was shown for example in [4], that if summands of the magnetic susceptibilityχ(n) for any n have an integer coefficient series expansion reducing to algebraic series modulo any prime, it is because they are diagonals of rational functions for any integer n. In fact many problems in mathematical physics involving n-fold integrals, could be interpreted in terms of diagonals of algebraic or rational ¶ functions †. Gilles Christol has shown in [9] that for every rational function, its diagonal f (x) has the following properties: • it is globally bounded: there exist integers c and d in N * , such that d f (c x) ∈ Z [[x]] and f (x) has a radius of convergence that is non-zero in C. • D-finite: there exists a differential operator L ∈ Z[x][ d dx ], with L = 0, such that L(f ) = 0. Christol conjectured in [10] that every series that verifies these two properties is the diagonal of a rational function. In this paper [10] Christol came up with an unresolved example to his conjecture [10], and a longer list was generated by Christol and his co-authors in 2012 in [4]. In this paper we show that two of the unresolved examples of the conjecture given in [4] on page 58, namely the 3 F 2 [2/9, 5/9, 8/9], [2/3, 1], 3 6 · x and 3 F 2 [1/9, 4/9, 7/9], [1/3, 1], 3 6 · x are indeed diagonals of rational functions and provide a generalization of this result. Recalls on diagonals of rational functions and on Christol's conjecture Definition of the diagonal of a rational function The diagonal of a rational function in n variables R(x 1 , . . . , x n ) = P(x 1 , . . . , x n )/Q(x 1 , . . . , x n ), where P, Q ∈ Q[x 1 , . . . , x n ] such that Q(0, . . . , 0) = 0, is defined through its multi-Taylor expansion around (0, . . . , 0): R x 1 , . . . , x n = ∞ m1 = 0 · · · ∞ mn = 0 R m1, ..., mn · x m1 1 . . . x mn n ,(1) as the series in one variable x: Diag R x 1 , . . . , x n = ∞ m = 0 R m, m, ..., m · x m .(2) Hadamard product of algebraic functions and Christol's conjecture Recall that the Hadamard product of two series f (x) = ∞ n=0 α n · x n and g(x) = ∞ n=0 β n · x n is given by: f (x) ⋆ g(x) = ∞ n=0 α n · β n · x n .(3) Hypergeometric series of the form p F p−1 ([a 1 , . . . , a p ], [b 1 . . . , b p−1 ], x) of height h = h(a 1 , . . . , a p , b 1 . . . , b p−1 ), where the height h is given by: h = #{1 ≤ j ≤ p | b j ∈ Z} − #{1 ≤ j ≤ p | a j ∈ Z}(4) with b p = 1, can be written ‡ as the Hadamard product of h globally bounded♯ series of height 1, were shown to verify Christol's conjecture. For example, the globally ¶ Any diagonal of an algebraic function in n variables can be rewritten as the diagonal of a rational function in 2n variables: see [17]. † See [3,4] 3 F 2 ([ 1 3 , 1 3 , 1 3 ], [1, 1], x) = (1 − x) −1/3 ⋆ (1 − x) −1/3 ⋆ (1 − x) −1/3 ,(5) and can thus be written as the diagonal of the algebraic function in three variables: (1 − x) −1/3 · (1 − y) −1/3 · (1 − z) −1/3 .(6) Unlike the case of 3 F 2 ([1/3, 1/3, 1/3], [1,1], x) , the hypergeometric functions 3 F 2 ([2/9, 5/9, 8/9], [2/3, 1], x) and 3 F 2 ([1/9, 4/9, 7/9], [1/3, 1], x), while being globally bounded functions [12], were constructed in a way that avoids them being written as "simple" Hadamard products of algebraic functions. Note that a p F p−1 hypergeometric function is globally bounded without restrictions if all its parameters in the denominator are integers, while a p F p−1 hypergeometric function can be shown to be globally bounded in general, by looking at Landau functions as explained in the work of Christol [10]. Furthermore, Beukers and Heckman have shown in [7], that p F p−1 hypergeometric functions of height zero that are globally bounded are algebraic functions. Unresolved examples to the conjecture Generalized hypergeometric functions with regular singularities p F p−1 are a simple and natural testing ground for Christol's conjecture. All 2 F 1 ([a, b], [c], x) hypergeometric series that are globally bounded are diagonals of rational functions. There are two cases that fall into this category: • Either the parameter c is an integer, and the 2 F 1 function can be written as the Hadamard product of two 1 F 0 functions, which are algebraic functions, and thus are diagonals of rational functions by Furstenberg's [15] theorem ¶. • The parameter c is not an integer, and in this case the 2 F 1 function is a diagonal of a rational function if and only if it is an algebraic function † † (and consequently its series is globally bounded). Moving on to 3 F 2 hypergeometric functions, one can ask when is a 3 F 2 hypergeometric function a diagonal of a rational function? • When the parameters d and e in 3 F 2 ([a, b, c], [d, e], x) are integers, because, in this case, the 3 F 2 can be written as the Hadamard product of three 1 F 0 algebraic functions, and is thus the diagonal of a rational function, by the closure of diagonals under the Hadamard product. • When the parameters d and e in 3 F 2 ([a, b, c], [d, e], x) are rational numbers but not integers, because in this case the 3 F 2 is algebraic, and is thus a diagonal by Furstenberg's theorem. † Diagonals are closed under the Hadamard product: if two series are diagonals of rational functions, their Hadamard product is also a diagonal of a rational function. ¶ Furstenberg's theorem states that any algebraic function is the diagonal of a rational function in two variables. † † The only 2 F 1 hypergeometric functions that are globally bounded with c ∈ Q are the algebraic ones: they are the ones appearing in the list of Schwarz [13]. Hence the interesting case occurs when only one of the two parameters d or e is rational, and the other is an integer. But even in this case, a lot of the 3 F 2 functions are easily seen to be diagonals of a rational function. Suppose that a 3 F 2 ([a, b, c], [1, e], x) is globally bounded, with the parameters a, b, c, e ∈ Q \ Z, then there are six ways to write the 3 F 2 ([a, b, c], [1, e], x) function as the diagonal of a rational function. This corresponds to the six ways to write the 3 F 2 ([a, b, c], [1, e], x) as a Hadamard product of hypergeometric functions: • 2 F 1 ([a, b], [e], x) ⋆ 1 F 0 ([c], x) • 2 F 1 ([a, c], [e], x) ⋆ 1 F 0 ([b], x) • 2 F 1 ([b, c], [e], x) ⋆ 1 F 0 ([a], x) • 2 F 1 ([a, b], [1], x) ⋆ 2 F 1 ([c, 1], [e], x) • 2 F 1 ([a, c], [1], x) ⋆ 2 F 1 ([b, 1], [e], x) • 2 F 1 ([b, c], [1], x) ⋆ 2 F 1 ([a, 1], [e], x) Now 1 F 0 ([c], x) and 2 F 1 ([a, b], [1], x) are diagonals of rational functions by what we have said above. Then 3 F 2 ([a, b, c], [1, e], x) is a diagonal of rational functions if 2 9 , 5 9 ], [ 2 3 ], x , 2 F 1 [ 2 9 , 8 9 ], [ 2 3 ], x , 2 F 1 [ 5 9 , 8 9 ], [ 2 3 ], x ,(7) and nor are the 2 F 1 hypergeometric series: 2 F 1 [ 1 9 , 4 9 ], [ 1 3 ], x , 2 F 1 [ 4 9 , 7 9 ], [ 1 3 ], x , 2 F 1 [ 1 9 , 7 9 ], [ 1 3 ], x .(8) The main results The globally bounded 3 F 2 hypergeometric series 3 F 2 [ 2 9 , 5 9 , 8 9 ], [ 2 3 , 1], 27 · x , 3 F 2 [ 1 9 , 4 9 , 7 9 ], [ 1 3 , 1], 27 · x(9) are ‡ respectively the diagonals of the two algebraic functions 3 F 2 [ 2 9 , 5 9 , 8 9 ], [ 2 3 , 1], 27 · x = Diag (1 − x − y) 1/3 1 − x − y − z , (10) † † Instead of 2 F 1 ([c, 1], [e], x), or one could take any one of the three permuted versions: [4], also see [11] p. 19. † See [23] for a proof that 3 F 2 ([1/9, 4/9, 7/9], [2/3, 1], x) cannot be written as a Hadamard product. ¶ One can see this experimentally by taking the series expansion of any of the Gauss hypergeometric functions: the prime numbers in the denominators of the coefficients grow continuously. ‡ The operators annihilating the two hypergeometric functions (9) are adjoint of each other. , 2 F 1 ([b, 1], [e], x) , etc. ♯ Appendix F p.58 of4 9 , 7 9 ], [ 1 3 , 1], 27 · x = Diag (1 − x − y) 2/3 1 − x − y − z .(11) These two hypergeometric series † † (9) can be recast into series with integer coefficients 3 F 2 [ 2 9 , 5 9 , 8 9 ], [ 2 3 , 1], 3 6 · x = 1 + 120x + 47124x 2 + 23483460x 3 + · · · ,(12) and 3 F 2 [ 1 9 , 4 9 , 7 9 ], [ 1 3 , 1], 3 6 · x) = 1 + 84x + 32760x 2 + 16302000x 3 + · · ·(13) Now Denef and Lipshitz in [17] show that any power series in Q [[x 1 , . . . , x n ]], algebraic over Q(x 1 , . . . , x n ), is the diagonal of a rational function in 2n variables, and they give an algorithm to build this rational function. This means that we can construct the rational functions, whose corresponding diagonals are the From diagonals of algebraic functions to diagonals of rational functions: Denef and Lipshitz We explain a method which, for a given algebraic power series in n variables, constructs a rational function in 2n variables whose diagonal equals the diagonal of the given algebraic series. Moreover, the partial diagonal of that 2n-variable rational function, with respect to the pairs of variables (x 1 , x n+1 ), . . . , (x n−1 , x 2n ), yields the original n-variable algebraic power series. The method is described in the paper by Denef and Lipshitz [17] in the proof of their Theorem 6.2. As a running example we use the three-variable algebraic function f (x, y, z) = (1 − x − y) 1/3 1 − x − y − z ,(14) whose multi-Taylor series expansion at 0 is actually a power series in the three variables x, y, z: f (x, y, z) = 1 + 2 3 x + 2 3 y + z + 10 9 xy + 5 3 xz + 5 3 yz + 40 9 xyz + . . . (15) Note that the minimal polynomial of f is given by p(x, y, z, f ) = (x + y + z − 1) · f 3 + 1 − x − y.(16) Denef and Lipshitz's theorem is formulated forétale extensions, which basically means that the partial derivative (w.r.t. f ) of the minimal polynomial has a nonzero constant coefficient at 0. Clearly, the above polynomial p(x, y, z, f ) does not meet this criterion. However, by consideringf = f − 1, i.e. by removing the constant term of f , we can achieve anétale extension. The minimal polynomial then reads p(x, y, z, f ) = (x + y + z − 1) · (f + 1) 3 + 1 − x − y.(17) Indeed, ∂p ∂f (0, 0, 0, 0) = −3 = 0. According to the proof of Theorem 6.2 (i) in [17], the rational functionr (x, y, z, f ) = f 2 · ∂p ∂f (xf, yf, zf, f ) p(xf, yf, zf, f )(18) has the property that D r(x, y, z, f ) =f (x, y, z), and hence D r(x, y, z, f ) = f (x, y, z) for r(x, y, z, f ) =r(x, y, z, f ) + 1. Here the operator D denotes a special kind of "diagonalization" with respect to the last variable: for f (x 1 , . . . , x n , y) = a i1,...,in,j · x i1 1 · · · x in n y j ,(19) one defines D f (x 1 , . . . , x n , y) = j=i1+···+in a i1,...,in,j · x i1 1 · · · x in n .(20) In our running example we obtain: r(x, y, z, f ) = 3 f 2 · (f + 1) 2 · (xf + yf + zf − 1) 3 (f + 1) 3 · (xf + yf + zf − 1) 3 − xf − yf + 1 + 1.(21) In the second step, which is explained in the proof of Theorem 6.2(ii) of [17], one has to transform the rational function r (that has n + 1 variables) into another rational function (having 2n variables) such that its "true" (partial) diagonal gives the nvariable algebraic series f . It consists of a sequence of n − 1 elementary steps, each of which is adding one more variable. In our example, we have to do the following r 1 (x, y, z, u 1 , v 1 ) = u 1 · r(x, y, z, u 1 ) − v 1 · r(x, y, z, v 1 ) u 1 − v 1 ,(22)r 2 (x, y, z, u 1 , u 2 , v 2 ) = u 2 · r 1 (x, y, z, u 1 , u 2 ) − v 2 · r 1 (x, y, z, u 1 , v 2 ) u 2 − v 2 , and obtain with r 2 the desired rational function in six variables. Generalization of the previous result By the algorithm of Denef and Lipshitz given in the previous section, it is possible to show that the algebraic function (1 − x − y) a/b 1 − x − y − z ,(23) corresponds to the following rational function in six variables, by taking the diagonal with respect to (x, u), (y, v) and (z, w): a · u 3 v · (1 − ux − uy − uz) · (1 + u) a−1 · (1 − ux − uy − uz) a−1 (1 + u) a · (1 − ux − uy − uz) a − (1 − ux − uy) b · (u − v) · (v − w) − a · v 4 · (1 − vx − vy − vz) · ((1 + v) · (1 − vx − vy − vz)) a−1 (1 + v) a · (1 − vx − vy − vz) a − (1 − vx − vy) b · (u − v)(v − w) (24) − a · u 3 w · (1 − ux − uy − uz)((1 + u) · (1 − ux − uy − uz)) a−1 (1 + u) a · (1 − ux − uy − uz) a − (1 − ux − uy) b · (u − w) · (v − w) − aw 4 · (1 − wx − wy − wz) · (1 + w) a−1 · (1 − wx − wy − wz) a−1 (1 + w) a · (1 − wx − wy − wz) a − (1 − wx − wy) b · (u − w) · (v − w) + 1. The diagonal of the rational function (24) is annihilated by the linear differential operator of order three: b 3 x 2 (1 − 27 x) · D 3 x + b 2 x ((27 a − 135 b) · x − a + 3 b) · D 2 x (25) − b · ((9 a 2 − 63 a b + 114 b 2 ) · x + a b − b 2 ) · D x + (a − 3 b) · (a − 2 b) · (a − b), and can be expressed as the 3 F 2 hypergeometric function 3 F 2 [ 3 a − b 3 a , 2 a − b 3 a , a − b 3 a ], [ a − b a , 1], 27 · x .(26) In particular, the two hypergeometric functions 3 F 2 ([2/9, 5/9, 8/9], [2/3, 1], 27· x) and 3 F 2 ([1/9, 4/9, 7/9], [1/3, 1], 27· x) appearing in (9), correspond respectively to the parameters (a, b) = (1, 3), and (a, b) = (2, 3) in the algebraic function (23). Other values of the parameters (a, b) are not necessarily unresolved examples of Christol's conjecture. For example if we consider the parameter values a = 1 and b = 7 , we see that the diagonal of (24) being a globally bounded series, which means that it can be written as a diagonal using one of the procedures given in Section 2.3. We note that algebraic functions close to the algebraic functions appearing in (10) and (11), also give 3 F 2 or 4 F 3 hypergeometric functions as their diagonals that are unresolved examples to Christol's conjecture: 13 21 , 20 21 ], [ 6 7 , 1], 27 · 7 3 · x is a series with integer coefficients. ‡ 3 F 2 [ 3 4 , 5 12 , 1 12 ], [ 1 4 , 1], 1728 · x is a series with integer coefficients. Diag (1 − x − 2 y) 2/3 1 − x − y − z = 3 F 2 [ 1 9 , 4 9 , 7 9 ], [ 2 3 , 1], 27 · x ,(30)Diag (1 − x − 2 y) 1/3 1 − x − y − z = 3 F 2 [ 2 9 , 5 9 , 8 9 ], [ 5 6 , 1], 27 · x ,(31)Diag (1 − x) 1/3 1 − x − y − z = 4 F 3 [ 2 9 , 5 9 , 8 9 , 1 2 ], [ 1 3 , 5 6 , 1], 27 · x ,(32)Diag (1 − x − y) 1/3 1 − x − z = 4 F 3 [ 2 9 , 5 9 , 8 9 , −1 3 ], [ 1 3 , 5 6 , 1], 27 · x . (33) † 3 F 2 [ 2 7 , Proof A computer algebra proof of this result can easily be obtained using the creative telescoping program [16]: one computes the operator (25) using the program [16], and verifies that this operator does annihilate the diagonal of (23) † †. Another longer way to do it which we provide below, is through binomial sums. The denominator of the algebraic function (1 − x − y) a/b /(1 − x − y − z) can be expanded as a geometric series: (1 − x − y − z) −1 = ∞ n=0 ∞ m=0 n m · (x + y) m z n−m = ∞ n=0 ∞ m=0 ∞ l=0 n m m l · x l y m−l z n−m ,(34) while the numerator can be written as the sum: (1 − (x + y)) a/b = ∞ k=0 (−a/b) k k! · (x + y) k = ∞ k=0 k j=0 (−a/b) k k! · k j x j y k−j . (35) Multiplying these two sums (34) and (35) and re-indexing, we obtain: ∞ s=0 ∞ t=0 ∞ u=0 x s y t z u · s j=0 ∞ k=0 (−a/b) k k! · k j s + t + u − k s + t − k s + t − k s − j .(36) Now taking the coefficients corresponding to the diagonal in (36), i.e. such that s = t = u = n, we get: n j=0 ∞ k=0 (−a/b) k k! · k j 3n − k 2n − k 2n − k n − j = 2n k=0 (−a/b) k k! · 3n − k 2n − k · n j=0 k j 2n − k n − j .(37) Now recalling the Chu-Vandermonde identity which says that 2n n = n j=0 k j 2n−k n−j , we find that (37) can be written as S(n) = 2n n · 2n k=0 (−a/b) k k! · 3n − k 2n − k ,(38) and by using a computer algebra tool like Mathematica or Maple to simplify this sum into a closed form, from which we can read off the hypergeometric function representation of the diagonal. More precisely, we used creative telescoping (Zeilberger's algorithm) to prove that (38) satisfies the first-order recurrence: (a − 3 b − 3 b n) · (a − 2 b − 3 b n) · (a − b − 3 b n) · S(n) = b 2 · (n + 1) 2 · (a − b − b n) · S(n + 1).(39) Together with the initial condition S(0) = 1, we obtain the closed form S(n) = 3 3n · (b − a)/3b n · (2b − a)/3b n · (3b − a)/3b n (b − a)/b n · n! 2 . (40) † † One also needs to note that initial conditions have to be compared. Telescopers of algebraic functions versus diagonals of algebraic functions The diagonal of a rational function and a solution of a telescoper † of a rational function are close, yet distinct notions. A telescoper annihilates an n-fold integral of a rational function over all integration cycles † †. For example the 3 F 2 ([a, b, c], [d, 1], x) can be written through the well-known integral representation as: (1 − y) −1−b+d · y b · (1 − x · y 2 ) −a · (1 − z) −c ,(41) with a, b, c, d ∈ Q. Hence if one takes the parameters a, b, c, d to have the values a = 1/9, b = 4/9, c = 7/9, d = 1/3, one immediately obtains that the telescoper of the algebraic function y 4/9 (1 − y) 10/9 · (1 − x y 2 ) 1/9 · (1 − z) 7/9 , admits as a solution the hypergeometric function 3 F 2 ([ 1 9 , 4 9 , 7 9 ], [ 1 3 , 1], x). Yet the diagonal of the algebraic function (42) is equal to zero, and it is through the algebraic function (11) that we were able to obtain it as the diagonal of an algebraic function. Other 3 F 2 unresolved examples to Christol's conjecture like [10,11] 3 F 2 [ 1 9 , 4 9 , 5 9 ], [ 1 3 , 1], 27 · x ,(43) were not obtained here as diagonals of a rational function, yet they are solution of the telescoper of an algebraic function and can thus be seen as a period of an algebraic variety over a non-evanescent cycle ‡, but not necessarily as a diagonal of an algebraic function (i.e. a period over an evanescent cycle). We give two arguments in favour of the fact that the 3 F 2 hypergeometric function (43) is most probably a diagonal of an algebraic function. Diagonal: algebraic mod p If one expects 3 F 2 hypergeometric functions like (43) to be diagonals of an algebraic function, one should find [3,4] that the corresponding series expansion reduces to an algebraic series modulo any prime number p, or power of a prime number p r . In order to verify this fact on (43) we look at the series expansion of 3 + 4881796920 x 4 + 2734407111744 x 5 + 1605040007778900 x 6 + · · · which becomes modulo 2: 3 F 2 [ 1 9 , 4 9 , 5 9 ], [ 1 3 , 1], 27 2 · x = 1 + x 2 + x 128 + x 130 + x 8192 + x 8194 + x 8320 + x 8322 + x 524288 + x 524290 + x 524416 + x 524418 + x 532480 + x 532482 + x 532608 + x 532610 + O(x 600000 ) = (1 + x 2 ) · (1 + x 128 ) · (1 + x 8192 ) · (1 + x 524288 ) + O x 600000 . (45) † By "telescoper" of a rational function R(x, y, z) we denote the output of the creative telescoping program [16], applied to the transformed rational function R(x/y, y/z, z)/(yz), which is a differential operator that annihilates the diagonal of R. † † Diagonals correspond only to evanescent integration cycles over rational functions. ‡ To be totally rigorous, one has to consider the two certificates of the telescoping equation see if that the integral of the derivatives of these two certificates over that cycle are actually zero. Straightforward guessing gives the infinite product formula F (x) = (1 + x 2 ) · (1 + x 2 7 ) · (1 + x 2 13 ) · (1 + x 2 19 ) · · · (1 + x 2 6 n +1 ) · · · (46) which is solution of F (x) = (1 + x 2 ) · F (x 64 ) mod. 2,(47) i.e. 3 F 2 [ 1 9 , 4 9 , 5 9 ], [ 1 3 , 1], 27 2 · x is an algebraic function modulo 2 satisfying: F (x) = (1 + x 2 ) · F (x) 64 mod. 2(48) or: (1 + x 2 ) · F (x) 63 = 1 mod. 2.(49)· x − 1 3 = 2 · F (x) mod. 3,(50) where: F (x) = x + x 3 + x 9 + x 27 + x 81 + x 243 + x 729 + x 2187 + x 6561 + x 19683 + x 59049 + O x 60000(51) which is solution of x + F (x 3 ) = F (x) mod. 3,(52) i.e. F (x) is an algebraic function modulo 3: x + F (x) 3 = F (x) mod. 3.(53) ·x) functions. We recall the algorithm of Denef and Lipshitz and apply it to the algebraic function (1−x−y) 1/3 /(1−x−y −z) in the first subsection below, and then we give the rational function and a generalization of the result in the second subsection. Finally, we give a second proof of the general result using binomial sums. Shimura curve[24]. For more details please refer to Appendix A.† † The hypergeometric function 3 F 2 ([2/9, 5/9, 8/9], [2/3, 1], 27x) can be rewritten as the Hadamard product 2 F 1 [ 2 9 , 5 9 ], [ 2 3 ], 27 x ⋆ (1 − x) −8/9 with 2 F 1 [ 2 9 , 5 9 ], [ 2 3 ], 27 x being associated with a is given by the globally bounded † series(27)3 F 2 [ 2 7 , 13 21 , 20 21 ], [ 6 7 , 1], 27 x = 1 + 260 49 x + 188190 2401 x 2 + · · · (27) with the 2 F 1 series 2 F 1 [ 13 21 , 20 21 ], [ 6 7 ], 27 x , 2 F 1 [ 2 7 , 20 21 ], [ 6 7 ], 27 x , 2 F 1 [ 2 7 , 13 21 ], [ 6 7 ], 27 x , being series that are not globally bounded. Hence the hypergeometric series (27) cannot be easily written as a Hadamard product, as explained in Section 2.3. In contrast, for a = 3 and b = 4 the diagonal of (24) which is given by the globally bounded ‡ series (28) 3 F 2 [ 3 4 , 5 12 , 1 12 ], [ 1 4 , 1], 27 x = 1 + 45 16 x + 41769 1024 x 2 + · · · (28) with the 2 F 1 series 2 F 1 [ 5 12 , 1 12 ], [ 1 4 ], 27 x , · x = 1 + 60 x + 20475 x 2 + 9373650 x 3F 2 [ 1 9 , 4 9 , 5 9 ], [ 1 3 , 1], 27 2 Modulo 3 we have the following expansion3 F 2 [ 1 9 , 4 9 , 5 9 ], [ 1 3 , 1], 27 2 F 1 ([c, 1], [e], x) or 2 F 1 ([a, b], [e], x) † †, are diagonals of rational functions, i.e. if and only if they are algebraic functions, since e ∈ Q \ Z. Now 2 F 1 ([c, 1], [e], x)cannot be an algebraic functions for e ∈ Q by Goursat[6]. Hence if one of2 F 1 ([a, b], [e], x), 2 F 1 ([b, c], [e], x), or 2 F 1 ([a, c], [e], x) is an algebraic function, then 3 F 2 ([a, b, c], [1, e], x)is the diagonal of a rational function. Now taking the two examples given in[4] or by G. Christol in[11]♯ that we are looking at here, we see that neither 3 F 2 ([2/9, 5/9, 8/9], [2/3, 1], x), nor 3 F 2 ([1/9, 4/9, 7/9], [2/3, 1], x), can be obtained as diagonals of rational functions through Hadamard products † since the three 2 F 1 hypergeometric series are not globally bounded ¶:2 F 1 [ Acknowledgments. J-M. M. and Y.A. would like to thank G. Christol for many enlightening discussions on diagonals of rational functions. We would like to thank A. Bostan for useful discussions on creative telescoping. Y.A. would like to thank A. Bostan for many explanations on Christol's conjecture including the details of 2.3 in an enlightening email correspondence[1]. J-M. M. would like to thank the School of Mathematics and Statistics of Melbourne University where part of this paper was written for hospitality. J-M. M. would like to thank A.J. Guttmann for many discussions on D-finite series. We would like to thank A.J. Guttmann for proofreading the paper. Y.A. would like to thank J. Voight for providing enlightening explanations on Shimura curves and for providing reference[20]. C.K. was supported by the Austrian Science Fund (FWF): P29467-N32. Y. A. was supported by the Austrian Science Fund (FWF): F5011-N15. Y. A. would like to thank the RICAM for hosting him on several occasions. Y.A. would like to thank Elaine and Rob for their hospitality in Linz, and for the great discussions he had with them there. Y.A. would like to thank his parents and his family for all their support and love. We thank the Research Institute for Symbolic Computation for giving us access to the RISC software packages.Unlike the situation on the 3 F 2 [ 1 9 , 4 9 ,7 9], [1 3, 1], 27 2 · x hypergeometric series, it is less obvious how to obtain the 3 F 2 [ 1 9 , 4 9 ,5 9], [1 3, 1], 27 2 · x as the diagonal of a rational function.It is however possible to obtain the solution of 3 F 2 [ 1 9 , 4 9 ,5 9], [1 3, 1], 27 2 · x , as a telescoper of an algebraic function, and this solution is an algebraic function modulo p.4.2.A relation between 3 F 2 ([1/9, 4/9, 5/9],[1/3, 1], 27 · x) and a 4 F 3 diagonal of an algebraic functionThe diagonal of the product of algebraic functionsis given by the 4 F 3 hypergeometric function H which is the Hadamard product of 3 F 2 ([1/9, 4/9, 7/9], [1/3, 1], 27 · x) and (1 − x) −5/9 :This 4 F 3 hypergeometric series (55) can also be written as the Hadamard product:So even though we did not find a rational (or algebraic) function whose diagonal is given by (43), knowing that 3 F 2 [ 1 9 , 4 9 ,7 9], [1 3, 1], 27 · x is the diagonal of a rational function, we see that the Hadamard product of (43) with a simple algebraic function (1 − x) −7/9 is actually a diagonal of an algebraic (or rational) function. This suggests but does not prove, that 3 F 2 [ 1 9 , 4 9 ,5 9], [ 1 3 , 1], 27 · x could also be a diagonal of a rational function.ConclusionBecause of the crucial role played by diagonals of rational functions in physics[3,4], Christol's conjecture is an important open problem.We have shown that the hypergeometric series 3 F 2 ([2/9, 5/9, 8/9], [2/3, 1], x) and 3 F 2 ([1/9, 4/9, 7/9], [1/3, 1], x) appearing in[4,11]are diagonals of rational functions. We did so by first finding two algebraic functions whose diagonals were given by these two hypergeometric functions, and through an algorithm outlined in the paper[17], we were able to recover the rational functions whose diagonals are given by these twoWe were also able to give a generalization of this result, and obtain other unresolved examples of Christol's conjecture as diagonals of rational functions. Furthermore, even though we were not able to write the 3 F 2 ([1/9, 4/9, 5/9], [1/3, 1], 27 · x) given by Christol in[10], as a diagonal of a rational function, we gave two arguments that suggested that it was likely to be so. More generally, we believe after writing the 3 F 2 ([2/9, 5/9, 8/9], [2/3, 1], x) and 3 F 2 ([1/9, 4/9, 7/9], [1/3, 1], x) as diagonal of rational functions, that it is likely that the other 3 F 2 unresolved examples of Christol's conjecture are diagonals of rational functions.Appendix A. Counterexamples and links with Shimura curvesThe Gauss hypergeometric function appearing on the left in(9)3 F 2 [ 2 9,can be seen as the Hadamard product of a Gauss hypergeometric function and an algebraic function given by:Now the Gauss hypergeometric function 2 F 1 [ 2 9 , 5 9 ], [2 3], 27 x happens to be a hypergeometric function corresponding to an automorphic form associated with a Shimura curve[24,18,19]. One has the identity:,The Gauss hypergeometric function 2 F 1 [ 1 36 ,19 36], [8 9], x can be also expressed as:Now the Gauss hypergeometric function 2 F 1 ([ 1 36 , 13 36 ], [8 9], x) which occurs in p.14 of[21], corresponds to a hypergeometric function related to a Shimura curve since it has exponent differences † (1/9, 1/2, 1/3), and these exponent differences are listed in the exhaustive list of hypergeometric functions that are associated with Shimura curves appearing inTable 1of[20]. Other 3 F 2 functions that are unresolved examples to Christol's conjecture that we found to be related to 2 F 1 hypergeometric functions related to Shimura curves are given by:1], 27 x), and the three globally bounded 3 F 2 hypergeometric series (A.6), (A.7) and (A.8), we were not able to write the other examples given in this section as a Hadamard product involving a 2 F 1 hypergeometric function associated to a Shimura curve. In any case, since the class of potential counterexamples formulated by Christol is infinite, while the list of Shimura inTable 1of[20]is finite, a list of 3 F 2 functions both related to Shimura curves and to Christol's conjecture is bound to be finite. † See[8]p.10 for a definition of exponent difference. Message to Youssef Abdelaziz. A Bostan, Trivialement diagonaleA. Bostan "Re: "Trivialement diagonale"". Message to Youssef Abdelaziz. 16 March 2019. Maillard and Zenine High order Fuchsian equations for the square lattice Ising model:χ (5). A Bostan, S Boukraa, A J Guttmann, S Hassani, I Jensen, J-M , J. Phys. A. 42275209A. Bostan, S. Boukraa, A.J. Guttmann, S. Hassani, I. Jensen, J-M. Maillard and Zenine High order Fuchsian equations for the square lattice Ising model:χ (5) J. Phys. A 42 275209, (2009), http://arxiv.org/abs/0904.1601 Ising n-fold integrals as diagonals of rational functions and integrality of series expansions. 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Christol Globally bounded solutions of differential equations Analytic number theory (Tokyo, 1988) (Lecture Notes in Math. vol 1434) (Berlin: Springer) pp. 45-64 http://dx.doi.org/10.1007/BFb0097124 H A Schwarz Ueber Diejenigen Fälle, welchen die Gaussische hypergeometrische Reihe eine algebraische Function ihres vierten Elementes darstellt Journal für die reine und angewandte Mathematik. 187344H.A. Schwarz Ueber diejenigen Fälle, in welchen die Gaussische hypergeometrische Reihe eine algebraische Function ihres vierten Elementes darstellt Journal für die reine und angewandte Mathematik (Crelle's Journal), Volume 1873 (75) Jan 1, 1873, 44 pages HolonomicFunctions Package version 1.7.1 (09-Oct-2013) written by Christoph Koutschan. Linz, AustriaJohannes Kepler UniversityResearch Institute for Symbolic Computation (RISC)HolonomicFunctions Package version 1.7.1 (09-Oct-2013) written by Christoph Koutschan, Copyright 2007-2013, Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Linz, Austria Algebraic functions over finite fields. H Furstenberg, J. Algebra. 7H. Furstenberg Algebraic functions over finite fields J. Algebra 7, 271-277 (1969) . 10.1016/0021-8693(67)90061-0http://dx.doi.org/10.1016/0021-8693(67)90061-0 HolonomicFunctions Package version 1.7.1 (09-Oct-2013) written by Christoph Koutschan. Linz, AustriaJohannes Kepler UniversityResearch Institute for Symbolic Computation (RISC)HolonomicFunctions Package version 1.7.1 (09-Oct-2013) written by Christoph Koutschan, Copyright 2007-2013, Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Linz, Austria Algebraic power series and diagonals. J Denef, L Lipshitz, 10.1016/0022-314X(87)90095-3J. Number Theory. 26J. Denef, L. Lipshitz, Algebraic power series and diagonals, J. Number Theory 26 46-67 http://dx.doi.org/10.1016/0022-314X(87)90095-3 Shimura curves of genus at most two. J Voight, Math. Comp. 78J. Voight, Shimura curves of genus at most two, Math. Comp. 78 pp 1155-1172, (2009). Three lectures on Shimura curves. J Voight, J. Voight, Three lectures on Shimura curves, 16th april (2006). Commensurability classes of arithmetic triangle groups. K Takeuchi, K. Takeuchi, Commensurability classes of arithmetic triangle groups, 1977 Belyi functions for hyperbolic Hypergeometric-to-Heun transformations. M Van Hoeij, R Viduñas, arXiv:1212.3803v3Journal of Algebra. 441M. van Hoeij and R. Viduñas, Belyi functions for hyperbolic Hypergeometric-to-Heun transformations, Journal of Algebra, Volume 441, 1 November 2015, Pages 609-659, (2015) arXiv:1212.3803v3 High order Fuchsian equations for the square lattice Ising model:χ (6). S Boukraa, S Hassani, I Jensen, J-M Maillard, N Zenine, J. Phys. A. 43115201S. Boukraa, S. Hassani, I. Jensen, J-M. Maillard and N. Zenine High order Fuchsian equations for the square lattice Ising model:χ (6) J. Phys. A 43 115201, (2010), http://arxiv.org/abs/0912.4968 Roques Hadamard products of algebraic functions. T Rivoal, J , Journal of Number Theory. 145T. Rivoal, J. Roques Hadamard products of algebraic functions, Journal of Number Theory, 145:579-603, (2014). Y Abdelaziz, S Boukraa, C Koutschan, J-M Maillard, arXiv:1910.10761v1Heun functions and diagonals of rational functions (unabridged version). Y. Abdelaziz, S. Boukraa, C. Koutschan, J-M. Maillard, Heun functions and diagonals of rational functions (unabridged version), arXiv:1910.10761v1 3-528-06317-3G-functions and geometry Aspects of Mathematics E13 (Braunschweig: Friedr.Vieweg and Sohn. Y. André G-functions and geometry Aspects of Mathematics E13 (Braunschweig: Friedr.Vieweg and Sohn) ISBN 3-528-06317-3

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{'abstract': "We show that the unresolved examples of Christol's conjecture 3 F 2 ([2/9, 5/9, 8/9], [2/3, 1], x) and 3 F 2 ([1/9, 4/9, 7/9], [1/3, 1], x), are indeed diagonals of rational functions.We also show that other 3 F 2 and 4 F 3 unresolved examples of Christol's conjecture are diagonals of rational functions. Finally we give two arguments that show that it is likely that the 3 F 2 ([1/9, 4/9, 5/9], [1/3, 1], 27 · x) function is a diagonal of a rational function.", 'arxivid': '1912.10259', 'author': ['Y Abdelaziz ', 'C Koutschan \nJohann Radon Institute for Computational and Applied Mathematics\nRICAM\nAltenberger Strasse 69A-4040LinzAustria\n', 'J-M Maillard ', '\nUMR 7600\nLPTMC\nCNRS\nUniversité Pierre et Marie Curie\nSorbonne Université\nTour 23, 5èmeétage, case 121, 4 Place Jussieu75252, Cedex 05ParisFrance\n'], 'authoraffiliation': ['Johann Radon Institute for Computational and Applied Mathematics\nRICAM\nAltenberger Strasse 69A-4040LinzAustria', 'UMR 7600\nLPTMC\nCNRS\nUniversité Pierre et Marie Curie\nSorbonne Université\nTour 23, 5èmeétage, case 121, 4 Place Jussieu75252, Cedex 05ParisFrance'], 'corpusid': 209445064, 'doi': '10.1088/1751-8121/ab82dc', 'github_urls': [], 'n_tokens_mistral': 14240, 'n_tokens_neox': 11984, 'n_words': 6860, 'pdfsha': 'ddd27764769380c2e57b34e1bd4dcb080617d302', 'pdfurls': ['https://arxiv.org/pdf/1912.10259v2.pdf'], 'title': ["On Christol's conjecture", "On Christol's conjecture"], 'venue': []}

arxiv

Probabilistic load forecasting for the low voltage network: forecast fusion and daily peaks Ciaran Gilbert Department of Electronic and Electrical Engineering University of Strathclyde G1 1XQGlasgowUK Jethro Browell School of Mathematics and Statistics University of Glasgow G12 8TAUK Bruce Stephen Department of Electronic and Electrical Engineering University of Strathclyde G1 1XQGlasgowUK Probabilistic load forecasting for the low voltage network: forecast fusion and daily peaks Low voltageload forecastingdemand forecastingsmart metersprobabilistic forecastingforecast combination Short-term forecasts of energy consumption are invaluable for the operation of energy systems, including low voltage electricity networks. However, network loads are challenging to predict when highly desegregated to small numbers of customers, which may be dominated by individual behaviours rather than the smooth profiles associated with aggregate consumption. Furthermore, distribution networks are challenged almost entirely by peak loads, and tasks such as scheduling storage and/or demand flexibility maybe be driven by predicted peak demand, a feature that is often poorly characterised by general-purpose forecasting methods.Here we propose an approach to predict the timing and level of daily peak demand, and a data fusion procedure for combining conventional and peak forecasts to produce a general-purpose probabilistic forecast with improved performance during peaks. The proposed approach is demonstrated using real smart meter data and a hypothetical low voltage network hierarchy comprising feeders, secondary and primary substations. Fusing state-of-the-art probabilistic load forecasts with peak forecasts is found to improve performance overall, particularly at smart-meter and feeder levels and during peak hours, where improvement in terms of CRPS exceeds 10%. (Jethro Browell) gence of household level forecasting in the context of hierarchical modelling; an opportunity provided by newly available smart meter datasets.Forecasting at Low Voltage (LV) levels poses a different challenge to the conventional load forecasting problem at the transmission level. As electricity is aggregated group behaviours emerge which tend to change slowly and are therefore relatively predictable. Disaggregated demand at the household level is much more changeable and influenced by individual behaviours and processes, as shown inFigure 1. The effect of the signal to noise ratio at the various voltage levels is discussed in much of the following literature where it is suggested that new approaches to forecasting are required and that should be developed with end-use in mind.Apart from the challenge of the lower signal to noise ratio at the household level, there are also challenges relating to the large number of nodes in LV networks where forecasts may be required, limited coverage of monitoring, data quality, and data privacy. These constraints affect the applicable methodologies for forecasting; for instance the models must be computationally efficient and the input features may not be location specific. Challenges and opportunities for low voltage forecasts are discussed in an extensive review of the literature[3,4], where the authors outline recommendations for future research; such as the need for probabilistic forecasting, handling limited observability, robust forecast verification, and avoiding widespread single-source data-bias in research projects.The smart meter roll-out in Great Britain and around the globe presents new opportunities in household load forecasting. This area has received the most attention in the literature Introduction Distribution networks are in a state of transition, with their original remit of 'fit and forget' now supplanted by the potential of massive increase in utilisation from electrified transport and heat, and two-way power flow from embedded renewable generation. This has led to the emerging necessity of the Distribution System Operator (DSO), who holds responsibility for balancing power flows under the transmission network [1]. While balancing has been commonplace at transmissionlevel for decades and the requisite forecasting and dispatch capabilities well understood, there is not a direct translation from transmission to distribution. Going down the voltage levels in power networks makes individual, low diversity, demand behaviours less predictable and hence unsuited to the methods used at transmission and regional levels. Short-term forecasts of load at all levels of the distribution network will be essential to coordinate flexibility services from distributed energy resources. Load forecasting on the transmission network is a highly active area of research, and has been a mature technology for decades. Research in recent years has been focused on probabilistic forecasts, which communicate the uncertainty associated with a forecast to end-users. There is a growing appetite for such forecasts in industry, which are now used by both Transmission System Operators (TSOs) and energy traders in operational decision making. Probabilistic load forecasting is extensively reviewed in [2], where the authors highlight the emer-around LV forecasting, which has mainly focused on deterministic forecasting [5,6]. The high penalisation of phase or 'timing' errors by traditional point-wise deterministic metrics like mean absolute error is highlighted in [7], where a new evaluation measure based on temporal permutation is proposed. The problem highlighted is that typical evaluation metrics tend to reward a smoother forecast on average compared to a forecast that may better represent the underlying process, that misses the precise timing of a sharp increase in demand. Therefore, the importance of predicting peaks in household (and LV) electricity demand has been discussed extensively in the literature [4]. The volatility of household demand necessitates a probabilistic approach to forecasting. Univariate probabilistic forecasts are typically communicated as full density forecasts, which are the most flexible for use in decision making, or in the form of multiple discrete quantiles at various probability levels. Interval forecasts with a specific coverage probability are also common [8]. In [9] density forecasts are obtained using Kernel Density Estimation (KDE), but conditional on information such as time-of-day, with a boundary correction applied to account for the positive nature of demand. Similarly, beta kernels are used in [10] to address the same problem, with a focus on building a scalable forecasting algorithm. Full density forecasts are generated as a benchmark model in [11], where the conditional density is assumed to be Gaussian with variance conditional by time-of-day; this is compared to non-parametric forecasts produced using an LSTM network for quantile regression, which outperforms the conditional density approach for the quantiles considered. A quantile regression approach based on boosting with additive models is demonstrated in [12], where the additive models are flexible and benefit from automatic feature selection by nature of the component-wise boosting procedure. Importantly, the quantile forecasts at the smart meter level are shown to be more skilful than an advanced parametric approach based on the Gaussian distribution. Bernstein polynomials have also been proposed for producing non-parametric density forecasts in [13] which show improvement over Gaussian and Gaussian mixture density forecasts. Finally, multivariate forecasts are generated for a hierarchy of smart meters in [14], where a coherency constraint is placed on the samples of the multivariate distribution, i.e. lower levels must sum to higher levels of the hierarchy. This literature may give the impression that only non-parametric densities are suitable for household load forecasting, however, there have been no studies examining alternatives to the Gaussian distribution until the present work. Some of the works already discussed (e.g. [12,14]) also forecast low-and medium-voltage levels, i.e. feeder and substation load. however, the networks are hypothetical, in that they are generated from aggregated smart meter data. As discussed in [4], this necessarily excludes important elements on the distribution network such as commercial loads, street lighting, embedded generation, electrical losses, dependencies between customers, etc. However, there is a shortage of opensource data sets that contain a satisfactory amount of nodes due to a lack of widespread monitoring. An alternative option is to generate a synthetic LV dataset from smart meter data and basic information on the local network architecture [15]. In [16] several probabilistic methods (quantile regression, KDE) are evaluated using a dataset comprising of 100 real LV feeders where there was no clear best forecaster at all feeders, however, autoregressive type models performed well, and (forecast) temperature was shown to have negligible influence on the forecast skill. Peak demand is typically the limiting factor in the capacity of distribution networks, set by the maximum power a cable of transformer can handle. Additionally, for a lot of flexibility applications the main goal is to reduce, flatten, or shift the daily peak demand in the LV network. Therefore, day-ahead forecasts of the daily maximum at the different nodes are valuable from both utilities perspective (e.g. in setting dynamic prices) and the consumers perspective (e.g. for scheduling battery or EV charging) [1]. The level of the daily peak is only half of the issue however, forecasting the time-of-peak is also relevant. There is little published in this specific area for the LV network to the best of our knowledge; related work [17] focuses on Extreme Value Theory and peaks over a defined threshold, which are by definition rare. In this paper, we consider a four-level hierarchy: a primary substation (33kV-11kV), secondary substations (11kV-4151kV), feeders (415kV), and households (230V, single phase). Methods for generating sharp and calibrated probabilistic forecasts of demand for the day-ahead are developed, including non-Gaussian parametric density forecasting at the household level, which is innovative in and of itself. We also investigate probabilistic daily peak forecasting, as in the daily maximum average energy demand, in terms of both peak intensity and timing. Finally, a method for combining (or blending) the daily bivariate (level and timing) peak forecasts and the halfhourly demand forecasts is described, which is termed forecast fusion. While methods for combining forecasts of varying spatial and/or temporal resolution have been proposed, combining forecasts of related yet distinct quantities has not been explored before, to the best of our knowledge. The case study presented is based on a hypothetical network generated using the Low Carbon London dataset, and the proposed forecasts are robustly verified against benchmark models. Probabilistic load forecasting is introduced in Section 2, followed by the concept of forecast fusion in Section 3. Stateof-the-art methods for conventional day-ahead load forecasting are presented in Section 4.1, followed by proposed methods for daily peak intensity and timing forecasting in 4.2 and 4.3, respectively. An extensive, fully reproducible, case study based on the Low Carbon London dataset [18] and a hypothetical LV network is then presented where forecasts for load at household, feeder, secondary substation and primary substation levels are analysed in detail. Finally, brief conclusions are drawn in Section 6. Probabilistic Forecasting In this section the probabilistic forecasting framework is formalised and the flexible statistical learning framework employed to generate forecasts throughout the work is introduced. We are typically interested in the the predictive density or cumulative distribution function (CDF) of random variable Y t at time t. The predictive CDF is defined aŝ F t (y t ) = P(Y t ≤ y t )(1) whereF is a strictly increasing function. Here we consider time in half-hour periods, as this is the resolution of electricity metering in Great Britain, but other this is not a restriction on the methodology. A density forecast provides maximum flexibility as quantiles or intervals can easily be extracted from the forecast, and no need to approximation are necessary, such as interpolating between quantiles. Additionally, the full distribution may be described by a smaller number of parameters, which may be functions of explanatory variables. One drawback is that a suitable conditional parametric family must be found for the forecast. Kernel density estimation provides a non-parametric alternative but can be less flexible and more computationally demanding and therefore less scalable -critical for DSOs where the number of assets is large. 2.1. Generalised Additive Models for Location, Scale, and Shape Generalised Additive Models for Location, Scale, and Shape (GAMLSS) [19], are semi-parametric models. This is because a parametric distribution is assumed for the target variable, and the parameters that define the assumed distribution may depend on non-parametric smooth functions of explanatory variables. The framework is an extension of the more familiar Generalised Additive Model (GAM) [20], such that any parameter of the distribution can be a function of input features, not just the conditional mean. If we have observations y, in this case demand at a particular location on the LV network, the conditional density typically f (y|θ) depends on up to four parameters; these are the location (θ 1 ), scale (θ 2 ), and shape parameters (θ 3 , θ 4 ). An additive regression model is generated for each distribution parameter θ i for i = 1, . . . , 4. Let x i be the pool of N i input features in the sub-model for θ i , and g i (·) the link function, then the model formulation of a GAMLSS is g i (θ i ) = β 0,θi + Ni n=1 f n,θi (x i,n ), i = 1, . . . , 4(2) where the function f n,θi is the effect of explanatory variable n on the distribution parameter θ i , which can be linear or non-linear functions, such as penalised smoothing splines, linear coefficients, surfaces, etc; β 0,θi are the intercepts of each sub-model. These models may be estimated numerically using a combination of maximum likelihood, and successive back-fitting of the predictor functions for each parameter [19]. In Figure 2 four example density forecasting models are visualised in a fan plot, where probability intervals are extracted from the conditional distribution at each lead time. Load forecasts are shown at different levels of aggregation. In general, forecast uncertainty is greater the further load is disaggregated. In particular, the possibility of large peaks is clearly quantified by high quantiles of the household-level forecast, which would not be captured by point forecasts. Importantly, demand can approach zero in households although in our case study framework cannot be negative (netdemand, demand less embedded generation, is reserved for future work). This removes some parametric families for the forecasts from consideration, such as the Gaussian distribution. In Figure 2, the Generalised Beta Prime distribution [21] is used for disaggregate demand and the Gaussian distribution is used for the three other aggregated levels. This small example is indicative of the approach throughout, which was to model the aggregate and household levels in the network distinctly; the forecasting models at the household level have to be simple, computationally efficient, and be suitable for right-skewed data. Forecast Fusion In this paper, we hypothesise that better forecast skill can be achieved by fusing a bespoke forecast of the daily peaks in demand, in terms of both the timing and intensity of the event, with a state-of-the-art half-hourly resolution forecast, both produced one day-ahead. This technique is very similar to forecast combination (or blending, expert mixtures, etc.), but distinct in that we are combining forecasts of two different types: the demand for a particular half-hour and day y d,h and the daily peak in demand y (p) d . A forecast of the timing of the peak, i.e. the number of half-hours from midnight until the daily peak h (p) d , is also used in the method. Consider the fusion of the two forecasts as a linear combination of the two distribution functions, F d,h (y) = (1 − w d,h )F d,h (y) + w d,hF (p) d (y)(3) with weights w d,h that may also be forecasts,or otherwise chosen, for each half-hour. This formulation respects the fact that a probabilistic forecast of the daily peak intensity (maximum value) is not sufficient in most applications and that the timing of the peak is also relevant. In this paper the weights are derived from probabilistic forecasts of the time-of-peak. If we define the random variable H = {1, 2, ..., 48} to be the number of half-hours to each daily maximum in demand from midnight h (p) d , then the weights are found via a forecast probability mass function w d,h =f d,h (h) = P(H = h)(4) since forecasts are typically issued for discrete blocks of energy. Therefore, estimating the weights is re-framed into a discrete time-to-event prediction problem. Note that the sum of the probabilitiesf d,h (h) over the total number of blocks in each day should be 1. If the conventional forecast were a prediction of load conditional on there not being a peak at d, h, this could be interpreted as a statement of the law of total probability. While it may be possible to produce such a forecast, we choose to proceed using conventional day-ahead forecasts so that the fusion method is as applicable as possible, Forecast fusion is related to techniques found in probabilistic forecast combination [22,23], blending [24], expert mixtures [25], linear pools [26,27], and so on. Empirically it was found that important concepts in forecast combination, such as re-calibration of a linearly combined forecasts, were not necessary in the following case study. A beta-transformation of the combined forecast, following [26], reduced the forecast skill in cross-validation; further work on forecast combination may yield benefits here. However, all the aforementioned methods are based on combining forecasts of the same type (e.g. hourly resolution, day-ahead forecasts), hence the distinct terminology here, which highlights the similarity with the broader topic of data fusion. Data fusion can be defined as the combination of multiple sources to obtain improved information in terms of quality, expense, and/or relevancy [28]. This is typically applied to combining data from multiple sensors and sources. In [29] multimodel forecast combination is discussed in terms of data fusion, however, the method entails combining forecasts of the same variable much like the literature discussed previously. In this work, we propose the fusion of conventional load forecasts with forecasts of peak load. These two forecasts exist in different temporal domains, hence this is a fusion problem not the usual practice of forecast combination or blending. We hypothesise that this will improve forecast skill overall as well as at peaks specifically, similar to how reconciliation of temporal hierarchies have been found to improve the skill across the different temporal domains [30]. Day-ahead Load Forecasting In this section, the approach for generating the day-ahead probabilistic forecasts of half-hourly load, peak intensity and peak timing are detailed. Throughout, different approaches are used depending on the level of aggregation in the LV network, including model specifications and input features. The household level is treated as one group, and feeder level and above as the other, referred to as 'aggregated' levels. This prevents the framework from becoming too complex but is not a strict constraint. Common to all base probabilistic forecasts generated (the half-hourly forecastsF d,h (y), the daily peak intensity fore-castsF (p) d (y), and the peak timing forecastsf d,h (h)) is the GAM framework, where both half-hourly and peak demand forecasts utilise GAMLSS. Half-hourly Forecasts The half-hourly forecasts at the aggregated levels of the LV network are described by f d,h (y|θ 1d,h ; θ 2d,h ) where we assume the conditional distribution is Gaussian. The model is formulated as follows at each aggregate node g 1 (θ 1d,h ) = β 0 + β 1 y d−1,h + β 2 y d−7,h + β 3 y (p) d−1 + 48 j=1 α j H j (h)y d−1,h + 48 j=1 γ j H j (h)y (p) d−1 + f pvc (h, D(d)) + f pbc (d)(5) and for the scale parameter, the formula is reduced to only depend on time-of-day for robustness g 2 (θ 2d,h ) = β 0 + f pb (h)(6) where f pvc (·) is a varying coefficient penalised spline, f pbc (·) is a cyclic penalised spline, and f pb (·) is a penalised spline. There are two dummy variables for each half of the day H j (h), and period of the week D(d) which is split into day type (Weekday, Saturday, and Sunday). So the load forecast is dependent on lags of the demand, a lag of peak demand, and interactions between yesterday's lags for each half-hour of the day. We also model the diurnal trend via the varying coefficient spline which changes according to day-type. Finally the annual seasonality is included, although in practice this spline is constrained to be very smooth to prevent interpolation of the data (in the case study we only have one year of data). Other formulations were tested; a full exploratory analysis can be found in the supplementary material [18]. However, this model formulation produced skilful forecasts relative to the benchmarks averaged over all time periods as well as during daily peaks, thanks to the interaction terms and the simple formula for the scale parameter. Due to the complexity and sheer number of households, the half-hourly forecast models for the household level have to be more simple and robust than those at the aggregate levels. This is to prevent over-fitting and for computational efficiency. They are given by f d,h (y|θ 1d,h ; θ 2d,h ; θ 3d,h ; θ 4d,h ) where we assume the conditional distribution follows the Generalised Beta Prime distribution. The model is formulated as g 1 (θ 1d,h ) =β 0 + β 1 y d−1,h + β 2 y d−7,h + 7 j=1 α j D j (d)+ f pb (h) + f pbc (d)(7) and for the scale parameter, the formula is similar to the aggregated levels g 2 (θ 2d,h ) = β 0 + f pb (h)(8) and the two shape parameters of the distribution are constants to be estimated. Here the dummy variable D j (d) is for each of the 7 periods of the week. Daily Peak Intensity For the daily peak intensity forecasting, data exploration revealed seasonal trends and a high correlation in the lag dependency variables, as shown in Figures 3 and 4 respectively, in the hypothetical LV network. Albeit the strength of the relationships are much weaker at the household level, again due to the low signal to noise ratio. An important consideration when creating the forecasting models here is the reduced size of the data set, since there is only one data point per day. Therefore, we reduce the number of features and categories in each formulation compared to half-hourly forecasting. The peak intensity forecasts at the aggregated levels of the LV network are described by f (p) d (y|θ 1d ; θ 2d ) where we assume the conditional distribution is Gaussian. The model is formulated as follows at each aggregate node g 1 (θ 1d ) =β 0 + β 1 y (p) d−1 + β 2 y (p) d−7 + β 3 σ yd−1 + f pbc (d)+ β 4 D 1 (d) + β 5 D 2 (d)(9) and for the scale parameter, the formula is g 2 (θ 2d ) = β 0 + β 1 D 1 (d) + β 2 D 2 (d) + β 3 σ yd−1(10) where period of the week D i (d) is now reduced simply to either weekday or weekend categories, and σ yd−1 is yesterday's standard deviation of the half-hourly demand time-series. Figure 3: Example time series of the daily peak intensity for primary substation (ps1), a secondary substation (ss1), a feeder (ss1 fdr1), and a household (N1174), from the hypothetical LV network. The peak demand shows seasonality at the aggregated levels, but again is more volatile at this household. : Lag dependency plots of the the daily peak intensity at four levels in a hypothetical LV network. This includes all nodes at each level and shows the motivation for using autoregressive based models for peak intensity forecasting, especially at the aggregate levels. An important feature at the household level for model robustness is the 'empty house' feature I(d). We include lags of this variable, which defined an empty house by look at runlengths of the daily standard deviation of the half-hourly demand. If this value dropped to approximately zero for a period of at least 7 days then the empty house feature is active. Note that the household had to have at least 30 days of emptiness for the feature to be included. This is because we want to identify houses here that are regularly empty for a reasonable period of time, rather than capture things such as holidays, public holidays, etc. which are reserved for future work. So, the daily peak intensity forecasts at the household level are described by f d (y|θ 1d ; θ 2d ; θ 3d ; θ 4d ) where we assume the conditional distribution follows the Generalised Beta Prime distribution. The model is formulated as follows at each household and for g 2 (θ 2d ) g 1 (θ 1d ) =β 0 + β 1 y (p) d−1 + β 2 y (p) d−7 + f pb (d)+ β 3 D 1 (d) + β 4 D 2 (d) + β 5 I(d − 1)(11)= β 0 + β 1 I(d − 1), g 3 (θ 3d ) = β 0 + β 1 I(d − 1) , the formula for the fourth moment of the distribution is kept constant. So the scale and shape parameters are only dependent on the empty house feature to make the forecasts more robust to overfitting. Daily Peak Timing A key component of the forecast fusion method is the weighting. In this study we choose to forecast the weights by defining them as the probability of the peak demand timing over the discrete blocks of energy in a day. Histograms showing the distribution of the peak timing at different levels on a hypothetical LV network are shown in Figure 5, which shows that as expected the time of the daily peaks become more variable as demand becomes disaggregate. The timing of the peak demand, specially at the higher aggregations, is dependent on the time-of-year; it is widely understood during the winter the peak daily demand tends to be earlier in the evening (and the level of the peak becomes higher, an example of which is shown in Figure 3) on the GB network. However, complex seasonal interactions were observed in the time-of-peak data, especially at the feeder and secondary substation levels of aggregation. The process is framed as a discrete time-to-event problem, where with suitable transformations of the time-of-daily peak time series h (p) d , a GAM framework can be applied to generate forecasts [31]. This means we can leverage the powerful smoothing capabilities of a GAM to capture complex seasonal interactions between input features at the daily peak timing. The framework is relatively unique in terms of time-to-event or survival analysis, in that due to the framework there must be an event (i.e. peak) for every subject (i.e. day) for each experiment (i.e. node), and the domain of the peak timing in the analysis is h (p) d ∈ {1, 2, ..., 48}. The discrete hazard function is the conditional probability at time interval h which gives the conditional probability of a peak in interval h given that the peak happens at time H ≥ h. To model this hazard rate, we use a time varying linear predictor with a GAM framework λ(h|x) = P(H = h|H ≥ h, x)(12)λ(h|x) = g −1 β 0h + N n=1 f n (x nh )(13) where the link function here g(·) is the logit link. In this case for the aggregated levels in this case we use a tensor interaction term for the period of day and day of year as the input features as well as a dummy variable for the day type (weekday/weekend). At the household level a more simple approach is needed for computational efficiency; two separate smooth splines for each of the input features are used. The discrete survival function is defined as S (h|x) = P(H > h|x) = h s=1 (1 − λ(h|x))(14) from which the discrete cumulative distribution function is found F(h) = 1 − S (h), and then the probability mass function for each day f h (h) = P(H = h) is easily calculated. For estimating the regression model, an indicator variable for each daily recorded value of h (p) is defined as the target variable which is zero for each discrete block of time until the event is recorded where the indicator variable is 1. For more information on the data transformation the reader is referred to [32,33]. Note that the final peak timing probability forecast could similarly be obtained via a simple logistic regression/classification method, but the discrete time-to-event setting gives the proposed method a stronger theoretical foundation. In Figure 6 four probability forecasts are shown for the same LV nodes as Figure 2 and for the same day; this constitutes the weights w d,h , i.e. the last component required to fuse the half-hourly and daily peak intensity forecasts together. 6 Figure 7: Diagram of the hypothetical LV network hierarchy used in the case study where the central node represents the primary substation and so on down to the household level at the end nodes. The size of the primary substation is small compared to reality in the GB distribution network due to the limited number of smart meters used. However, as shown in Figure 1, group behaviours begin to emerge quickly at high levels of aggregation. Case Study The methodology is tested using data from the Low Carbon London trial [34]. Households which are on a variable price tariff are first removed, and then households which have regular communication issues and/or suspect data are removed, and finally only households which have a complete record of measurements during 2013 are retained. This represents quite a strict data cleaning process in which we are left with 742 smart meters; however, addressing challenges due to missing-data and forecasting dynamic price tariff households are beyond the scope of this study. Data from this experiment is anonymised which means location specific effects, such as temperature, are necessarily excluded from the analysis. The households are sampled (without replacement) to create a hypothetical LV network, albeit given the amount of the available smart meters, the primary substation level (i.e. the top aggregation) is smaller than perhaps you would find in practice. However, as shown in Figure 1, group behaviours begin to emerge quickly. The sampling process was configured whereby each secondary substation comprises of between 4-7 feeders and each feeder contains around 16-45 smart meters. The hypothetical LV network is illustrated in Figure 7 showing all the different levels of aggregation. As discussed in [4] there is a severe lack of real open-access LV network data available. Therefore, this approach is effectively a compromise because we implicitly cannot account for street furniture, embedded generation, and potential correlations between nodes in a real life network. For the regression problems, the coverage of the dataset is January 2013 to December 2013 inclusive. To compare the forecast performance, each month is partitioned into thee blocks of approximately 10 days, depending on month. The first two blocks of every month compromise the training data, on which cross-validation is also carried out to generate out of sample forecasts covering the full year. The last block in each month constitutes the testing data, i.e. not used anywhere for model estimation or tuning. All methods are implemented in R [35] using the package ProbCast [36], developed by the authors for the modelling, evaluation, and visualisation of probabilistic forecasts. The wrapper functions for GAMLSS for the regression models [19] are used extensively here. A two pronged approach is necessary to assess probabilistic forecasts performance; calibration (or reliability) is the necessary condition that predicted probabilities are unbiased, and sharpness is a measure of forecast uncertainty, i.e. the spread of the predictive distribution. Combined, these allow for the ranking of competing forecasting methods by sharpness subject to calibration [37]. For the full predictive cumulative distributions in this case study, the Continuous Ranked Probability Score (CRPS) [37,38] is used to measure both sharpness and calibration crps = 1 N N t=1 ∞ −∞ {F t (y) − 1(y ≥ y t )} 2 dy(15) where 1(·) is the indicator function. The CRPS for a single forecast observation pair is therefore the area between the squared difference of the forecast and observation CDF, where the latter is a step-function from 0 to 1 at the observed value. For the discrete probability forecasts of the peak timing, the Ranked Probability Score (RPS) is used for verification, which is the discrete form of CRPS [39]. The Probability Integral Transform (PIT) histogram is used to verify the calibration of the forecasts u t =F(y t ) (16) whereby, if the forecasts are well calibrated and the sample is sufficiently large, u ∼ U(0, 1), which is inspected visually via a histogram with a certain number of (typically 20 [39]) bins. One limitation with this approach is that is becomes time consuming to verify all nodes in a large network. So, in this case the calibration checked within grouped network levels (e.g. smart meters, feeders, etc.). Reliability diagrams [40] are used as an alternative to check the calibration of quantiles of the distribution at individual nodes in the supplementary material [18]. The calibration of the discrete probability forecasts is not presented here for brevity, but are shown in the supplementary material via multi-category reliability diagrams [18,41]. Benchmarks & Skill Scores A set of benchmark models have been chosen for case study, with models chosen depending on the forecasting task and level in the LV network. There a limited literature on benchmarks for probabilistic forecasts in general, and none established for LV load forecasting. For the half-hourly forecasts a simple auto-regressive model is used and implemented in the GAMLSS framework for the aggregated levels g 1 (θ 1d,h ) = β 0 + β 1 y d−1,h + β 2 y d−7,h + f pb (h)(17) with and the scale parameter and distribution family the matching the proposed advanced model. At the household level three benchmark models are employed; two are based on Kernel Density Estimation (KDE) using zero-truncated Guassian kernels. In the first separate KDE models are defined for each for each half-hour of the day, and in the second separate KDE models are defined for each half-hour and day type (weekday, Saturday, Sunday). Both of these are similar to a previously published method [9]. The third is a simplified version of the GAMLSS model used at the household level, with the same Generalised Beta Prime family, where only the location and scale parameters vary smoothly with time-of-day. For the peak intensity forecasts a very simple autoregressive based GAMLSS model is used for the aggregated levels, with the Gaussian conditional distribution and formula g 1 (θ 1d ) = β 0 + β 1 y (p) d−1 + β 2 y (p) d−7(18) where the scale parameter is a constant. At the household level, again three benchmark models are employed. The first is a very simple unconditional KDE estimate, the second is a KDE estimate conditional on day type (weekday/weekend), and finally a simple location-only autoregressive GAMLSS model, i.e. the same as Equation (18) except using the Generalised Beta Prime family as the conditional distribution. For the peak timing probability forecasts the same simple seasonal climatology model is used at all points of the network. This is a competitive benchmark because seasonality is the only effect used in our more advanced model. Forecast evaluation is reported via the relative change of the score of the proposed modelS to a benchmarkS re f via skill scores. If the perfect score is zero, as in the cases considered here, then the skill score is skill = 1 −S S re f (19) and in the following the terms skill score, percentage improvement, and relative change are used interchangeably. Bootstrap re-sampling is used as a simple non-parametric method for estimating the significance in forecast improvement [40]. Forecast target times are sampled with replacement with a length equal to the original length of the set, and then skill scores are calculated. This process is repeated a large number of times until the sampling variation of the result is determined, which are then typically presented via boxplots [40]. Daily Peak Intensity Evaluation In Figure 8 the skill of the peak intensity forecasts is demonstrated at the household level against three benchmarks, an unconditional KDE, KDE conditional on day type (weekday/weekend), and a simple GAMLSS model, based on autoregressive features for the location parameter only. The full model is used as a component in the fused forecasts, and as you can see on average results in improved forecasts of over 15% relative to the most simple benchmark in testing. This validates the motivation for using a bespoke model for predicting the peaks alone. There is some variability in the skill between households and this is skewed towards improved skill, as detailed in [18] and discussed in Section 5.5. Some households show over a 50% improvement in CRPS in cross-validation and testing compared to the unconditional KDE estimate and few showing negative skill below -5%. The improvement between the two GAMLSS models validates the inclusion of the smooth day of year, day type (weekday/weekend), and empty house features. The skill scores are similar across all levels of the network from Feeder to Primary Substation, in contrast to peak timing and half-hourly forecasts which are generally more skilful at higher levels of aggregation than lower. The breakdown by individual nodes is presented in [18] and we find that skill is positive and statistically significant in the majority of cases on the test data, but also note that the sample variation is greater due to the reduced data volume. This is also reflected in how the nodes where Fusion shows the greatest/least skill are not the same between the cross-validation and test results, although the overall behaviour is similar. Daily Peak Timing Evaluation In this subsection we evaluate in more detail the weights used in the fusion forecast, w d,h , which is defined as the probability of the daily peak at each node occurring in each discrete block of energy throughout the day. At the aggregate levels, the skill scores of the discrete time-to-event probability forecasts, where the reference is seasonal climatology, are range from 0% (for four feeders) to over 20% for the Primary Substation. Detailed results may be found in [18]. There is a clear trend for greater improvement at higher levels in the hierarchy. This is because the peak timing is less variable and more smoothly dependent on seasonal, which is easily modelled. Whereas, even at the feeder level during testing some of the nodes have similar or less skill than the benchmark, although there is a skew generally for positive skill scores. At the feeder levels the time of peak is clearly dependent on more complex behaviours than can be described by time of day and day of year. Additionally, the general change in skill between cross-validation and testing is worth further investigation at all the levels; ideally more data would be available when learning complex seasonal interactions between day of year and time of day which may be leading to overfitting. This problem is also evident at the smart meter level.The time-to-event based forecast is only marginally more skilful in testing than the benchmark with a CRPS skill score of under 0.5%. This highlights the difficulty in predicting the peak timing at the household level and that the predictions of the time-to-event based model are close to seasonal climatology on average. The variability in skill has been investigated, shown in [18]. There is variation in skill between households, which ranges from ±10%; some households are apparently more predictable than others. Teasing out relationships between node type, the skill of the timing probability forecasts, and drivers of predictability is an interesting aspect of future work. Stakeholders could use this information to gauge locations for the provision of flexibility in the LV network. Finally, further analysis of the forecasts could give an understanding as to which nodes are likely to reach their daily peak at the same time as the more aggregated nodes higher up the network. This information could be valuable for revealing which households or nodes to leverage for peak demand shifting via (for example) time-of-use tariffs. Forecast Fusion Evaluation In the following subsections we first evaluate the forecast fusion methodology. To this end, we evaluate the forecasts using CRPS skill scores averaged over all time periods, and also inspect the calibration directly. However, to demonstrate the improved forecast skill for the daily peak demand we also retrospectively select periods where the daily peak demand is recorded and evaluate the fused forecast averaged over these time periods only. Due to the double penalty effect [7], an advanced (and smoother) forecast might produce improved skill on average compared to a benchmark, but fail predict peak demand well. Therefore, the structure of this evaluation is aimed to demonstrated the skill of the forecast on average and during the daily peak demand. Aggregate Levels At the aggregate levels, the two advanced forecasting methods show similar improved skill over the benchmark at the primary substation (ps), secondary substation (ss), and feeder (fdr) levels in the network. This is true for both crossvalidation and testing, as shown in Figure 9. So we can conclude that the forecast fusion method is at least as skilful as the full half-hourly model. This is encouraging since the latter is a key component in the fused forecast. More generally, we can see the skill improvements possible from adding seasonal features and interactions in the regression model, as evidenced by the ≈10% improvement in forecast skill during testing across the aggregated network compared to the simple autoregressive benchmark. Rather than metrics at each level of the network, in Figure in improvement between nodes, especially at the feeder level which is hidden in Figure 9. This plot also emphasises that during cross-validation the improvement is increasing with the aggregation (or voltage) level. However, during testing the skill scores are far closer between the levels which could indicate that the testing data is more difficult to predict at the aggregate levels, the benchmark models are improved relative to the advanced models with more training data, or the advanced models are over-fitting slightly, or a combination of all three. If we evaluate the forecasts only during the periods when the daily peak demand is recorded the skill of the three forecasting methods looks very different. Figure 11 shows that the full model is similar in skill during testing to the benchmark. In fact, during the data exploration and tuning of the models, it proved difficult to find a suitable feature set which performed equally well to the benchmark model during the peak half-hours, as evidenced in the supplementary material [18]. However, the Fusion method is again ≈10% better at forecasting the daily peak during testing across the aggregated network compared to the benchmark. This is a key result of the paper and shows that by fusing a bespoke forecast of the daily peaks to a state-of-the-art half-hourly forecasts, it is possible to achieve skilful forecasts on average and during the daily peak demand. We have also investigated the variation in forecast performance across individual nodes of the aggregated network. While there is again variability between nodes, all show improvement from forecast fusion of between 5% and 15%. Bootstrap skill scores by node are illustrated in [18]. The calibration of the advanced GAMLSS and fusion models is shown in Figure 12 where essentially the average calibration at the aggregated levels is shown. Importantly, the Fusion method is at least as well calibrated as the Full model, if not marginally better calibrated. Clearly, the right tail of the distribution could be improved at most of the levels to account for large peaks in demand. However, this is reserved for future work. Additionally, the calibration of both models at the feeder level is relatively poor on average. Using distribution free regression approaches might be beneficial here as well as accounting for holiday and special events. The key result here is that the forecasts are reasonably well calibrated and the fusion methodology did not introduce calibration issues via the linear combination, an issue that is widely discussed in forecast combination [26]. Household Level Recall that there are three benchmarks for household level forecasts, two variations on KDE and a very simple GAMLSS model based only on time-of-day. An interesting result, shown in Figure 13 is that using autoregressive and seasonal terms it is possible to achieve better skill than simple benchmarks, by 3-4% during testing in this case study. As well, the Fusion methodology is marginally better than the advanced GAMLSS model in both cross-validation and testing. The Simple GAMLSS model is the worst performing out of the methods tested. At some nodes (8 out of 742) the Full model failed to converge and the performance of the resulting forecasts was very poor during cross-validation. Further inspection revealed issues at these households such as structural changes in the time-series and so on. At these 8 nodes, the Simple model was therefore used in place of the Full. Additional detail on this can be found in the supplementary material [18]. As one would expect, the relative performance of the benchmark and Fusion methods varies between households. Unlike the aggregated level where there was consistent improvement at all nodes for the proposed Fusion methodology, at the household level forecast fusion provides improvement for 80% of households relative to KDE1, and 70% relative to the sophisticated Full GAMLSS benchmark. This is due to the diversity of behaviours at the household level and perhaps at some nodes the model is imply over-parameterised given the information in the time series which suggests perhaps boosting or regularisation would be beneficial in the model fitting if computationally feasible. However, the density is clearly skewed toward improvement as you would expect from Figure 13. Supplementary results on this topic are provided in [18]. Again, when evaluated during the periods when the daily peak demand is recorded the ranking of the different forecasters is very different. As shown in Figure 14, the simple and advanced GAMLSS models are now significantly worse than even the simple time-of-day KDE benchmark by ≈1-5% during testing, which demonstrates the double penalisation effect reported in the literature. However, the fusion model remains the most skilful forecast, and is significantly better than the more advanced KDE model averaged over both cross-validation and testing. Although the skill score is not as large as the aggregated levels detailed above, there is still a large and significant improvement from the full model, which is one of the inputs to the fused forecast, of approximately 6% during testing. In terms of calibration, the PIT histogram for all the households is shown in Figure 15. Although it is not possible to distinguish individual nodes in this case the plot shows that both the advanced GAMLSS and forecast fusion method are reasonably well calibrated across this level of the hierarchy, with some evidence of over-confidence. however, given the nature of smart meter demand and that we are using a parametric assumption for the predictive distribution, the calibration is better than expected. There are some differences between the calibration of the GAMLSS and Fusion forecast now however, with the right tail of the distribution going from too narrow to too wide on average. This indicates a possible area of improvement for the forecasts. Results Summary and Discussion To summarise, at aggregated and household levels, fusion of peak and conventional forecasts provides a significant improvement in CRPS relative to both simple and advanced benchmarks methods during peak hours. The improvement relative to advanced methods ranges from 6% to 9%. This comes with no penalty to overall performance, which is also improved but only marginally. Average CRPS is reported for each aggregation level in Table 1, though we refer the interested reader to the earlier figures and supplementary material where we verify that this improvement is largely consistent across individual households, feeders and substations, which differ in size and variability. Finally, we have investigated whether there is any relationship between the variability of load (at substations, feeders and households) and forecast improvement of the fusion method relative to benchmarks. We have compared the skill scores with the coefficient of variation for all individual substations, feeders and households, illustrated in Figure 16. There is no apparent relationship between variability and skill for any aggregate level, which all have positive skill. Of course we have few examples of primary and secondary substations, but skill at these levels is comparable to individual feeders. For households, however, we observe a negative correlation between variability and forecast skill, although there is a large amount of variation, and positive skill for 80% of households. Furthermore, only households with a relatively low coefficient of variation exhibit very high forecast skill. This highlights the importance of considering forecast skill for individual households, as 'average' performance across multiple households will mask this variation in forecast performance. For most applications, we believe that significant improvement at 80% of households is more than sufficient to justify a slight increase in the complexity of the forecasting process. Conclusions Forecasting methods that are effective across all voltage levels of distribution networks will be essential as Distribution Network Operators take on new responsibilities for managing energy balancing and ancillary services. This paper presents a novel approach to probabilistic load forecasting that addresses deficiencies of existing methods caused by peak loads, and is shown to improve forecast skill across distribution networks from household to primary substation level by as much as 10% overall, and more during peaks. The skill of forecasts during peaks is particularly important in many use-cases, including network constraint management, peak shaving, and battery and demand response scheduling. The approach we propose combines forecasts of daily peak timing and intensity with conventional load forecasts. By forecasting peaks specifically, we compensate for the tendency of conventional methods to be too smooth and under-forecast peaks. Probabilistic forecasts are combined or 'fused' using a simple weighting scheme inspired by the more general practice of data fusion. A comprehensive case study based on open data is presented where we find that while sophisticated methods for conventional forecasts may provide skill overall with respect to competitive benchmarks, they add little value during peaks. Fusion of conventional forecast with a peak forecast marginally improves performance overall, and greatly improves performance during peaks. Average improvement during peaks ranged from 6% at household and Primary substation level, and 8-9% at Feeder and Secondary substation level in our case study. Additionally, we have proposed a method for producing parametric density forecasts at the household level based on the Generalised Beta-Prime distribution and the GAMLSS framework. This is in contrast to non-parametric methods that have dominated the literature to date and are far less parsimonious. The proposed method produced an average improvement of approximately 5% relative to methods based on Kernel Density Estimation across the 742 households in our case study. However, forecasting capabilities require further development to meet the expected future needs of DSOs. Not least, consideration should be given to embedded generation, storage, and demand response. Furthermore, to be of maximum practical use, forecasting models should be applicable to feeders/substations they have not been trained on, known as 'global' forecasting models as proposed in [42]. As distribution net-works feature tens-of-thousands of LV feeders, the use of domain adaptation via transfer learning would take a pre-trained model and adapt it to any feeder given minimal adjustment, but certainly not retraining. Development of forecasting models that are adaptive to track changes in load behaviours, structural breaks in particular, should also be considered. Another aspect to consider and potentially exploit is the hierarchical nature of electricity demand. Encoding this structure in forecasting models can help improve accuracy and enable more coordinated decisions at different levels of the network. Supplementary Figures for "Probabilistic load forecasting for the low voltage network: forecast fusion and daily peaks" Ciaran Gilbert, Jethro Browell, Bruce Stephen This document contains supplementary results to support the analysis presented in the main paper. These are primarily breakdowns of forecast evaluation by individual secondary substations, feeders, and households. Their purpose is to illustrate how improvements in forecast performance is distributed across individual households/feeders/substations. For substations and feeders, forecast improvement is very consistent and of a similar magnitude across most substations/feeders. The Fusion method improves performance at 80% of households compared to KDE1, and 70% of households compared the 'Full' half-hourly benchmark GAMLSS model. Node Figure 1: Skill scores averages of the advanced peak intensity forecasts at the aggregated nodes of the network, where unconditional climatology is the reference model. The sample distribution is found via bootstrap averages, and is naturally wider compared to the other evaluation plots because of the reduced sample size at each node. The forecast skill is posttive and similar around the aggregated network on average. : Density of the skill scores for peak timing forecasts at all the smart meter nodes where the reference model is seasonal climatology. There is variable improvement around the households in terms of predicting the daily peak timing. In testing the average skill is close to 0%. Figure 1 : 1Half-hourly demand during one week in 2013 averaged across an increasing number of households from 1 to 1000. Group behaviours become more apparent as the aggregation level increases. Figure 2 : 2Day-ahead example probabilistic forecasts as fan plots for the primary substation (ps1 top left), a secondary substation (ss1 top right), a feeder (ss1 fdr1, bottom left), and a household (N1174) on 17-10-2013. The widest and lightest coloured interval has a coverage probability of 98% and the measurement is overlaid in black for reference. Figure 4 4Figure 4: Lag dependency plots of the the daily peak intensity at four levels in a hypothetical LV network. This includes all nodes at each level and shows the motivation for using autoregressive based models for peak intensity forecasting, especially at the aggregate levels. Figure 5 : 5Histograms of the daily peak timing at the four different aggregations of the hypothetical LV network. The peak timing becomes more dispersed at the lower aggregations on the network. At the primary substation level the peak timing during the analysis was consistently in the evening, except one data point which corresponds to the 'turkey peak' during Christmas day. Figure 6 : 6Day-ahead example probability forecasts for the time of the daily peak at the same four nodes asFigure 2and for the same day. The measurement is represented by the dashed red vertical line in each panel for reference. These forecasts correspond to the weights used in the forecast fusion for each discrete time period of this day. Figure 8 : 8Skill scores averages of the peak intensity forecasts at the household level of the network relative to KDE1, where the Full model is a component of the Fusion model. The sample distribution is found via bootstrap averages. The peak intensity forecasts at the household level are as skilful as those at the aggregate levels relative to the benchmark. Figure 9 : 910 the skill scores of the Fusion method are plotted at each single node of the aggregated network. Clearly there is variability Skill scores averages of the three half-hourly forecasting methods employed at the aggregated levels of the network relative to the 'Simple' model. The sample distribution is found via bootstrap averages, where all available samples are included at the the primary substation (ps), secondary substation (ss), and feeder (fdr) levels. Figure 10 : 10Skill scores averages of the Fusion forecasting method at the aggregated levels of the network relative to the 'Simple' model. The sample distribution is found via bootstrap averages, where all available samples are included at each node. Figure 11 : 11Skill scores averages of the three half-hourly forecasting methods employed at the aggregated levels of the network relative to the 'Simple' model. The sample distribution is found via bootstrap averages, where only samples which correspond to the daily peak demand are included at the the primary substation (ps), secondary substation (ss), and feeder (fdr) levels Figure 12 : 12PIT histograms of two forecast models at the aggregated levels of the network. Note that except from the primary substation (ps) level, these histograms show the average calibration of all the nodes. The fused forecast is similarly calibrated at all levels and across both data partitions. Figure 13 : 13Skill score averages of the five half-hourly forecasting methods employed at the household level of the network relative to KDE1. The sample distribution is found via bootstrap averages, where all available samples are included. Figure 14 :Figure 15 : 1415Skill scores averages of the Fusion forecasting method at the household level of the network during peaks relative to KDE1. The sample distribution is found via bootstrap averages, using only samples which correspond to the daily peak demand are included at each node. PIT histograms of two forecast models at the household level of the network, which show the average calibration of all the nodes. The fused forecast is similarly calibrated at all levels and across both data partitions, except for the right tail of the two methods. Figure 16 : 16Forecast skill of the fusion forecast relative to benchmark (KDE1 for households, Simple half-hourly forecast for aggregations) against coefficient of variation for each substation, feeder and household. Figure 2 :Figure 3 :Figure 4 234Density of the skill scores at the household level where the reference model is KDE1. There is variable improvement across households in terms of predicting the daily intensity, a clear positive skew for both GAMLSS models. Skill scores averages of the time to peak forecasts at the aggregated nodes of the network, where seasonal climatology is the reference model. The sample distribution is found via bootstrap averages. As expected, it becomes more difficult to predict the peak timing at the lower voltage levels. Figure 5 :Figure 6 : 56Skill scores averages of the Fusion forecasting method at the aggregated levels of the network relative to the Simple model. The sample distribution is found via bootstrap averages, where only samples which correspond to the daily peak demand are included at each node. Density of the skill scores at all the smart meter nodes relative to KDE1. Although there is variable improvement around the households the distribution is clearly skewed in favour of the fusion forecasts in both cross validation and testing. Table 1 : 1Summary of the mean CRPS from the test set for the Simple and Advanced (Adv.) half-hourly forecasts, and Fusion forecasts. Scores are presented for All periods and Peak periods. Skill scores are for the Fusion forecasts relative to the Advanced benchmark. Aggregation Simple Adv. Fusion SkillAll Time Primary 7.04 6.32 6.31 0.2% Secondary 1.92 1.77 1.77 0.2% Feeder 0.69 0.64 0.64 0.0% Household 0.08 0.08 0.08 0.4% Peaks Only Primary 8.94 8.80 8.30 5.7% Secondary 3.08 3.09 2.81 9.0% Feeder 1.54 1.53 1.40 8.2% Household 0.47 0.47 0.44 6.0% AcknowledgementsThe authors would like to thank Stephen Haben for many insightful discussions on this topic which have informed this work and UK Power Networks for provision of the Low Carbon London dataset, available at https://data.london.gov.uk/dataset/smartmeter-energy-usedata-in-london-households, or the pre-processed version used here[18]. This study is fully reproducible, all data and R code associated with this work are available in[18]. The authors were funded by the EPSRC project Analytical Middleware for Informed Distribution Networks (AMIDiNe, EP/S030131/1) and the Innovation Fellowship held by JB (EP/R023484/1 and EP/R023484/2). Transition from Distribution Network Operator to Distribution System Operator. 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{'abstract': 'Short-term forecasts of energy consumption are invaluable for the operation of energy systems, including low voltage electricity networks. However, network loads are challenging to predict when highly desegregated to small numbers of customers, which may be dominated by individual behaviours rather than the smooth profiles associated with aggregate consumption. Furthermore, distribution networks are challenged almost entirely by peak loads, and tasks such as scheduling storage and/or demand flexibility maybe be driven by predicted peak demand, a feature that is often poorly characterised by general-purpose forecasting methods.Here we propose an approach to predict the timing and level of daily peak demand, and a data fusion procedure for combining conventional and peak forecasts to produce a general-purpose probabilistic forecast with improved performance during peaks. The proposed approach is demonstrated using real smart meter data and a hypothetical low voltage network hierarchy comprising feeders, secondary and primary substations. Fusing state-of-the-art probabilistic load forecasts with peak forecasts is found to improve performance overall, particularly at smart-meter and feeder levels and during peak hours, where improvement in terms of CRPS exceeds 10%. (Jethro Browell) gence of household level forecasting in the context of hierarchical modelling; an opportunity provided by newly available smart meter datasets.Forecasting at Low Voltage (LV) levels poses a different challenge to the conventional load forecasting problem at the transmission level. As electricity is aggregated group behaviours emerge which tend to change slowly and are therefore relatively predictable. Disaggregated demand at the household level is much more changeable and influenced by individual behaviours and processes, as shown inFigure 1. The effect of the signal to noise ratio at the various voltage levels is discussed in much of the following literature where it is suggested that new approaches to forecasting are required and that should be developed with end-use in mind.Apart from the challenge of the lower signal to noise ratio at the household level, there are also challenges relating to the large number of nodes in LV networks where forecasts may be required, limited coverage of monitoring, data quality, and data privacy. These constraints affect the applicable methodologies for forecasting; for instance the models must be computationally efficient and the input features may not be location specific. Challenges and opportunities for low voltage forecasts are discussed in an extensive review of the literature[3,4], where the authors outline recommendations for future research; such as the need for probabilistic forecasting, handling limited observability, robust forecast verification, and avoiding widespread single-source data-bias in research projects.The smart meter roll-out in Great Britain and around the globe presents new opportunities in household load forecasting. This area has received the most attention in the literature', 'arxivid': '2206.11745', 'author': ['Ciaran Gilbert \nDepartment of Electronic and Electrical Engineering\nUniversity of Strathclyde\nG1 1XQGlasgowUK\n', 'Jethro Browell \nSchool of Mathematics and Statistics\nUniversity of Glasgow\nG12 8TAUK\n', 'Bruce Stephen \nDepartment of Electronic and Electrical Engineering\nUniversity of Strathclyde\nG1 1XQGlasgowUK\n'], 'authoraffiliation': ['Department of Electronic and Electrical Engineering\nUniversity of Strathclyde\nG1 1XQGlasgowUK', 'School of Mathematics and Statistics\nUniversity of Glasgow\nG12 8TAUK', 'Department of Electronic and Electrical Engineering\nUniversity of Strathclyde\nG1 1XQGlasgowUK'], 'corpusid': 249953676, 'doi': '10.1016/j.segan.2023.100998', 'github_urls': [], 'n_tokens_mistral': 20404, 'n_tokens_neox': 17515, 'n_words': 11660, 'pdfsha': 'd9c4318d90c638d4c999bf93c5056def29cdbbda', 'pdfurls': ['https://export.arxiv.org/pdf/2206.11745v2.pdf'], 'title': ['Probabilistic load forecasting for the low voltage network: forecast fusion and daily peaks', 'Probabilistic load forecasting for the low voltage network: forecast fusion and daily peaks'], 'venue': []}

arxiv

Human Health Indicator Prediction from Gait Video Ziqing Li Tsinghua University Haidian District100084BeijingChina Xuexin Yu Tsinghua University Haidian District100084BeijingChina Xiaocong Lian Tsinghua University Haidian District100084BeijingChina Yifeng Wang HIT Campus of University Town of Shenzhen 518055ShenzhenChina Xiangyang Ji Tsinghua University Haidian District100084BeijingChina Human Health Indicator Prediction from Gait Video A R T I C L E I N F OBMI prediction Age prediction Gait video analysis Pose estimation A B S T R A C T Body Mass Index (BMI), age, height and weight are important indicators of human health conditions, which can provide useful information for plenty of practical purposes, such as health care, monitoring and re-identification. Most existing methods of health indicator prediction mainly use front-view body or face images. These inputs are hard to be obtained in daily life and often lead to the lack of robustness for the models, considering their strict requirements on view and pose. In this paper, we propose to employ gait videos to predict health indicators, which are more prevalent in surveillance and home monitoring scenarios. However, the study of health indicator prediction from gait videos using deep learning was hindered due to the small amount of open-sourced data. To address this issue, we analyse the similarity and relationship between pose estimation and health indicator prediction tasks, and then propose a paradigm enabling deep learning for small health indicator datasets by pre-training on the pose estimation task. Furthermore, to better suit the health indicator prediction task, we bring forward Global-Local Aware aNd Centrosymmetric Encoder (GLANCE) module. It first extracts local and global features by progressive convolutions and then fuses multi-level features by a centrosymmetric double-path hourglass structure in two different ways.Experiments demonstrate that the proposed paradigm achieves state-of-the-art results for predicting health indicators on MoVi, and that the GLANCE module is also beneficial for pose estimation on 3DPW. Introduction The Body Mass Index (BMI), height, weight and age are important indicators of one's health conditions, where BMI is defined as . (1) For example, BMI, as an integrated variable of weight and height, is found to be related to cancers, unstable angina, myocardial infarction, type II diabetes and cardiovascular disease Renehan, Tyson, Egger, Heller and Zwahlen (2008); Wolk, Berger, Lennon, Brilakis and Somers (2003); Meigs, Wilson, Fox, Vasan, Nathan, Sullivan and D'Agostino (2006). Moreover, age is a widely recognized indicator for health status evaluation and almost all disease diagnoses. Therefore, accurately predicting BMI, height, weight and age in a contactless manner can be helpful for daily monitoring of health conditions and primary screening for diseases. In addition, these health indicators are beneficial for the police department as they are frequently used in surveillance, forensics and re-identification applicationsKlontz and Jain (2013). Most existing works on health indicator prediction are conducted on the frontal-view images of body or face using manual feature extraction and deep learning methods Velardo, Dugelay, Paleari and Ariano (2012); Pascali, Giorgi, Bastiani, Buzzigoli, Henriquez, Matuszewski, Morales and Colantonio (2016); Wen and Guo (2013); Kocabey, Camurcu, Ofli, Aytar, Marin, Torralba and Weber (2017) ;Yousaf, Hussein and Sultani (2021). Traditional methods using manual ⋆⋆ lizq21@mails.tsinghua.edu.cn (Z.Li) * Corresponding author ORCID(s): feature extraction emphasizes on domain knowledge, constraining the upper limit of the algorithm performance to human capability Velardo et al. (2012); Pascali et al. (2016); Wen and Guo (2013). Meanwhile, deep learning methods pushes boundaries by letting the network learn features itself. However, these methods are sensitive to viewpoint change and some require complex pre-processing steps Jin, Huang, Xiong, Pang, Wang and Ding (2022). In this work, we introduce an end-to-end deep learning algorithm to predict health indicators from gait videos. Compared with existing approaches, our scheme has three advantages. First, it directly extracts related features from captured gait videos in an end-to-end manner, without needing prior knowledge. Second, gait videos are more convenient and easy to be obtained in real-life scenarios, such as surveillance and short videos. Predicting health status from gait videos is promising for home health monitoring in the Internet of Things era. Third, the methods based on gait videos are less sensitive to view changes than those based on frontal body or facial images. However, the datasets containing human gait with health status indicators are either relatively small or not available to the public, which would make it difficult to directly train a well-functioning deep learning network. Fortunately, monocular-view video-based pose estimation can provide useful information for health indicator prediction Rosso, Agostini, Takeda, Tadano and Gastaldi; Shin, Chung, Kistler, Fitschen, Wilund and Sosnoff; van der Straaten, De Baets, Jonkers and Timmermans; Windham, Griswold, Wang, Kucharska-Newton, Demerath, Gabriel, Pompeii, Butler, Wagenknecht, Kritchevsky and Mosley. It is also a rather developed field of study with many large-scale and publicly-available datasets Ionescu, Papava, Olaru and Sminchisescu (2013); Mehta, Sotnychenko, Mueller, Xu, Srid-har, Pons-Moll and Theobalt (2018);Varol, Romero, Martin, Mahmood, Black, Laptev and Schmid (2017). Based on the relationship of both tasks, transfer learning is introduced to the health status prediction task to overcome the issue of insufficient data. The backbone of the model is first pretrained on pose estimation datasets and then transferred into the health status prediction task. Our main contributions are summarized as follows: • To the best of our knowledge, we make the first attempt to use deep learning on gait videos to predict health indicators. • We present a paradigm enabling deep learning for a small health indicator dataset by pre-training on the pose estimation task. • We propose a global-local aware and centrosymmetric encoder (GLANCE) to extract spatial features, which focuses on the extraction and integration of multi-level features. • Experiments demonstrate that the proposed method achieves state-of-the-art results for predicting health indicators on MoVi Ghorbani, Mahdaviani, Thaler, Kording, Cook, Blohm and Troje (2021), and that the GLANCE module is also beneficial for pose estimation on 3DPW. The rest of this paper is organized as follows. Section 2 summarizes related works, including health indicators prediction, the relationship between pose estimation and health indicator prediction, and transfer learning. Section 3 first elaborates the proposed global-local aware and centrosymmetric encoder (GLANCE) module, and then describes the architecture and pipeline of the proposed GlanceNet. Section 4 discusses implementing details briefly and demonstrates the superior performance of the proposed GlanceNet compared with the state-of-the-art method on the MoVi. Furthermore, the ablation study is investigated in this section. Finally, the summary of the paper is presented in Section 5. Related work Currently, most methods estimate health indicators from facial and body images, which can be roughly categorized into conventional and deep learning methods according to their way of extracting features. Conventional methods extract features in a computational and manual way relying on prior domain knowledge Velardo et al. (2012); Pascali et al. (2016); Wen and Guo (2013). For example, Wen and Guo Wen and Guo (2013) detect keypoints in frontal-view face image and use their coordinates to calculate pre-defined features, cheekbone to jaw width (CJWR), width tupper facial height ratio (WHR), perimeter to area ratio (PAR) etc. These computational features serve as input to the support vector regression for BMI prediction. Whereas deep learning methods automatically learns how to extract features. With the rapid development of deep learning techniques, these deep learning methods have outperformed the conventional ones in both the face image and front body image prediction tasks. Kocabey et al.Kocabey et al. (2017) use a pre-trained backbone network to extract features from face images. Improving on their work, Yousaf et al.Yousaf et al. (2021) proposed Region aware Global Average Pooling (Reg-GAP), which pools the feature maps from pre-trained backbone networks according to their corresponding face regions, eye, nose, eyebrow, lips, etc. From the perspective of ergonomics and medicine, various studies have validated that gait pose could reflect human inner health statusRosso et al.; Shin et al.; van der Straaten et al.; Windham et al.. For instance, Zhong et al.Zhong, Rau and Yan calculated several features of gait with wearable sensors and found that pre-frail older adults showed a decrease in speed and increases in RMS and step irregularity significantly compared with the non-frail counterparts. Calvache et al.Calvache, Bernal, Guarín, Aguía, Orjuela-Cañón and Perdomo (2020) have successfully utilized pose estimation methods to predict the balance and physical equilibrium of the human body from videos, in order to prevent falls. Moccia et al.Moccia, Migliorelli, Carnielli and Frontoni (2020) uses one infant's limb joint information from pose estimation to assess its cognitive development. Therefore, for health status evaluation, it is worth looking into the research of pose estimation. Among the pose estimation methods, the one most relative to health indicator prediction objectives is the monocularview video-based pose estimation. One of the most influential works in this domain is the introduction of a parametric model Skinned Multi-person Linear Model (SMPL) Loper, Mahmood, Romero, Pons-Moll and Black (2015). The information on body shape and pose variation are summarized and reduced to pose, shape and camera parameters in the SMPL model. Using this parametric model, one can easily reconstruct a realistic human body mesh. Plenty of pose estimation studies are carried out based on the SMPL model.For example, VIBE Kocabas, Athanasiou and Black is an important benchmark, which is the first one to utilize adversarial learning to incorporate prior knowledge into pose estimation from video. Some works do not rely on parametric model and try to regress mesh vertices and joints coordinates directly from images Cao, Simon, Wei and Sheikh (2017); Sun, Xiao, Liu and Wang (2019); Lin, . METRO Lin et al. (2021) uses transformer to attend to the interactions between joints and vertices, in order to accurately reconstruct human body from an image. Compared with health status prediction from gait videos, pose estimation is a relatively developed area of research and provides many large-scale and publicly-available datasets Ionescu et al. (2013); Mehta et al. (2018); Varol et al. (2017). This makes it possible to train deep neural networks firstly on pose estimation datasets and then transfer learned weights into health status prediction models. Transfer learning can take advantage of similarities between tasks, such as applying a model trained on bicycles to motorcycles, or a model trained on cats to dogs. Since there is little difference in features between bikes and motorcycles, cats and dogs, transfer learning can often achieve good results, especially in the absence of a certain number of data, and the use of sufficient similar data can largely compensate for this deficiency. In today's research, the weights pre-trained on ImageNet Krizhevsky, Sutskever and Hinton (2012) have been widely transferred to many fields, e.g., Transferring GANs Wang, Wu, Herranz, van de Weijer, Gonzalez-Garcia and Raducanu (2018) achieve good image style transfer based on limited data using the weights pre-trained on ImageNet. DWGAN Fu, Liu, Yu, Chen and Wang (2021) achieves favorable results on the image deblurring problem and overcomes the problem of insufficient data. Inspired by these tasks and previous studies on the correlation between human posture estimation and health status, we propose to transfer the weights of human pose estimation to human health status prediction. Methodology Motivated by insufficient datasets for human gait with health status indicators, we introduce transfer learning to the health status prediction task. Therefore, the proposed GlanceNet, shown in Figure 1, is divided into two phases. In phase I, we perform training on a large dataset for the human pose estimation task. In phase II, the well-trained encoder in the previous phase serves as a feature extractor for the tiny health indicator prediction dataset. To take full advantage of the pose estimation task, the network architecture of phase I follows the VIBE Kocabas et al. model, which mainly consists of a spatial-temporal encoder (a ResNet and a GRU unit) and a SMPL Generator. Here, we choose the SMPL model as optimization objective for the pose estimation task. The main reason is that the parameters of the SMPL model include information about body shape and joint locations, which is directly related to health indicators like height and weight. Considering characters of videos in space and time, a spatial-temporal encoder is indispensable to extract features from intra-and inter-frame. In addition, to improve the prediction of health indicators, we propose a spatial encoder, Global-Local Aware aNd Centrosymmetric Encoder (GLANCE), to extract and integrate local and global spatial features. The spatial and temporal encoders are further discussed in Section 3.1 and 3.2, respectively. Then the overall pipeline of our GlanceNet is explained in detail in Section 3.3. Spatial Encoder: GLANCE For health indicator prediction, it is of great importance to extract feature representation with local-and global-awareness from each frame. For example, height is the length between the top of the head and the feet, which is calculated across global human body features. The estimation of weight needs to obtain the joint position and body information, which are closely related to both local and global information. To meet the above requirements, we elaborately design a spatial encoder, GLANCE. As shown at the top of Figure 1, the GLANCE module can be further divided into feature extractor and centrosymmetric fusion components, which bear the responsibilities of extracting global-local features and fusing multilevel features, respectively. As shown in Figure 1, the feature extractor is composed of ResNet and three convolutions, which is inspired by Kim et al.Kim, Kook, Sun, Kang and Ko (2018) and Artacho et al.Artacho and Savakis (2020). Here, ResNet is used to extract local features and global features are extracted progressively by the following three convolutions, and all the feature maps from every convolutional neural network (CNN) layer are concatenated to form a representation with both global and local information. In addition, we also use dilated convolutions to quickly enlarge the receptive field without introducing more CNN layers. In fact, compared to algorithms like transformers, three convolutions following the ResNet extract features to be relative local scale. Nevertheless, convolutions are comparably more lightweight and requires fewer computation resources, which is advantageous considering the size of input videos. To fuse the stacked features from the feature extractor, we propose a centrosymmetric fusion, as shown in Figure 1. In centrosymmetric fusion, the two basic building blocks are an hourglass channel-wise architecture and a global average pooling with depth-wise convolution, where the former is responsible for channel-wise feature fusion and the latter accomplishes feature integration in space. Concretely, in the top path, since features first go into global average pooling with depth-wise convolution and then hourglass network, the features are fused at the global level due to losing most local information in global average pooling process. Therefore, the obtained features enhance the global information yet may ignore local information. To avoid the drawback, the bottom path is designed. When the features first walk through the hourglass network, some local features may be emphasized since current features contain all local information. The output features also highlight some local information after global average pooling with depth-wise convolution. The two paths in centrosymmetric fusion complement each other and explore the incorporation of global and local information. Temporal Encoder Temporal information among frames is significant for health indicator prediction tasks since it provides information associated with motion velocity and acceleration, which is closely related to some heath indicators like age. Moreover, when the viewpoint changes and certain limbs maybe occluded in some frames of gait videos, temporal information is beneficial for making more informed judgements by comprehending the whole video sequence. Inspired by the success of the Gated Reccurent Unit (GRU) in machine translation task by Cho et al.Cho, van Merrienboer, Bahdanau and Bengio (2014), we use bi-directional GRU as the temporal encoder, which makes the feature vectors corresponding to different timestamp learn from each other and the model more robust to viewpoint changes and occlusion. Figure 1: The architecture and pipeline of the proposed GlanceNet, which consists of two phases. In phase I, the model is trained on a large pose estimation dataset to generate SMPL parameters for every frame in the input video, using a spatial-temporal encoder and a SMPL generator. In phase II, the spatial-temporal encoder trained in phase I is employed to extract features for the small MoVi dataset. The features are then downsampled and fed into a SVM to predict health indicators. The spatial encoder GLANCE in the spatialtemporal encoder is designed to extract and incorporate local and global information. Overall Pipeline: GlanceNet In this section, we will expand on the description of the pipeline GlanceNet in Figure 1 and walk through the phase I and phase II design in detail. In phase I, we train the network on the pose estimation task with large publicly-available datasets. We directly feed every input video frame into the spatial encoder GLANCE, which outputs a feature vector composing of spatial information. Then the spatial feature vectors of all frames are ordered to a time sequence and fed into the temporal encoder GRU for temporal information learning, producing a stream of feature vectors of both spatial and temporal information. Finally, the feature vector of every frame is fed into the SMPL generator to produce SMPL parameters for every frame. The SMPL generator and the loss function follow the same design as VIBE Kocabas et al.. The SMPL generator contains a pre-trained regressor and a generative adversarial network to incorporate prior knowledge about the human body in AMASS dataset. In phase II, we will predict health indicators from gait videos. The dataset for phase II is minimal in size. Therefore, considering the similarity between pose estimation and health indicator prediction tasks stated in Section 1, the pretrained spatial-temporal encoder trained in phase I is used to extract features for the health indicator prediction task. For the output of the spatial-temporal encoder, only the feature vector of the last frame is utilized since it is expected to contain information of the whole video sequence considering the GRU design and the health indicator prediction task predicts one label for the entire video sequence instead of for every frame. Also, due to the small dataset of the health indicator prediction task, we first downsample the original features using average pooling operation, and then feed it into Support Vector Machine (SVM) regressor, which is good at handling datasets with small sample yet high-dimensional input. Experiments and Results In this section, We first describe the experimental setup. Subsequently, we compare our method with state-of-the-art methods on MoVi dataset. Finally, ablation studies are conducted to verify the effectiveness of the proposed GLANCE module. Experimental Setup For the pose estimation task in phase I, three pose estimation datasets PennAction Zhou, Barnes, Lu, Yang and Li (2019), PoseTrack Andriluka, Iqbal, Insafutdinov, Pishchulin, Milan, Gall and Schiele (2018) and 3DPW Von Marcard, Henschel, Black, Rosenhahn and Pons-Moll (2018) are used for training and evaluation is performed on 3DPW. The three datasets contain video recordings of people doing daily activities. PennAction and PoseTrack have 2D ground truth keypoint annotations. 3DPW dataset has 3D ground truth keypoint labels as well as annotation of SMPL parameters. In phase II, the health indicator prediction task is trained and tested on MoVi dataset Ghorbani et al. (2021). MoVi contains gait video sequences and there are a total of 87 people with available health indicator annotations. These health indicators include age, weight, height and BMI, whose statistical information is shown in Table 4. The strategy of 5-fold cross-validation is used, and training and testing set ratios are 4:1. The proposed model is implemented in PyTorch. The model in phase I is trained on eight NVIDIA GeForce RTX3090 GPUs with a batch size of 24. Its weights are initialized and optimized by the kaiming method He, Zhang, Ren and Sun (2015) and the Adam algorithm Kingma and Ba (2014) with 1 = 0.900, respectively. During 30 epochs, the initial learning rate is 1 − 3, then is divided by 10 times after 5 epochs. In phase II, the SVM regressor is only trained on the 3DPW dataset as the parameters of spatial-temporal encoder are transferred from phase I. For health indicator prediction, phase II, the models are evaluated by the Mean Absolute Percentage Error (MAPE) and Mean Absolute Error (MAE) of each health indicator. To demonstrate the effectiveness of the proposed GLANCE module in pose estimation task, common pose estimation metrics, Procrustes-Aligned Mean Per Joint Position Error (PA-MPJPE), Mean Per Joint Position Error (MPJPE) and Per Vertex Error (PVE), are used to evaluate the performance of models in phase I. In addition, we introduce a new metric, LimbLen Error, to reflect the relationship between pose estimation and health indicator prediction. Specially, the metric calculates the error of the total length of limbs between prediction and ground truth, where the length of a limb is defined as the distance between adjacent joint locations in 3D coordinates. Therefore, LimbLen is a factor to bridge through both tasks. Comparisons with state-of-the-art methods Since our method is the first to use gait videos to predict relevant health indicators, there are no other video-based approaches to be compared against it. Currently, as discussed in Section 1, most related works mainly use single-frame image to predict health status. One of the most recent methods is the attention guided end-to-end BMI estimation network Jin et al. (2022), which estimates the health indicators from a frontal-view instance image. To be a fair comparison, we first select front-view images of subjects in stance position from the videos of the MoVi dataset according to the requirements Jin et al. (2022). Then these selected images are pre-processed to remove the background and segment human figures before feeding into the pre-trained model released officially. Table 1 shows the corresponding quantitative comparison between both methods, where the imagebased model refers to the guided end-to-end BMI estimation network Jin et al. (2022). Our method outperforms the image-based model Jin et al. (2022) in MAE and MAPE for the BMI indicator, which indicates that our method extracts more effective features from videos than the image-based model. Moreover, we test sensitivity to the view angle of posture in images for the image-based model, whose results are shown in Figure 3. With the viewpoint changing from frontview to side-view, the prediction error of the image-based model becomes gradually larger. The results indicate that the image-based model is susceptible to viewpoint changes in posture, which increases the difficulty of image captures. The minimum and maximum MAEs of the image-based model are 3.9924 and 5.7677 respectively, while the MAE of our model is only 0.2600 for the same video sequence. The result validates that our model can obtain more effective and robust features from the entire video sequence, taking the information from many different angles into account by spatialtemporal encoder. Among the testing results, we find that the image-based model always overestimates BMI for the people wearing black clothes, as shown in Figure 2. A possible reason is the weight is overestimated, considering the colour of clothes is the same as the processed background. Instead, our method can predict BMI accurately in this case. Therefore, the proposed approach is more applicable than the image-based model. Ablation Study In this section, we investigate the effectiveness of the proposed GLANCE module, which is divided into feature extractor and centrosymmetric fusion. The ablation study results of health indicator prediction are presented in Table 2. Compared with baseline model with only ResNet, which is actually VIBEKocabas et al., the performance is progressively enhanced by adding the feature extractor and the centrosymmetric fusion components. For the pose estimation task, the same conclusion is obtained from Table 3. The gain indicates the effectiveness of the GLANCE module. Concretely, the feature extractor brings about global features, building upon the local feature map of ResNet. Then, the fusion component encourages local and global features to complement each other, further improving performance of the model. Moreover, we also conduct some qualitative experiments to validate the effectiveness of the GLANCE module. Some cases of the variations of LimbLen over time are presented in Figure 4. The predictions of our model stay more closely to the targeted value than the one without GLANCE, which mainly attributes to extraction and integration of GLANCE module for multi-level features. This observation explains the excellent performance of our model on the health indicator prediction task since the LimbLen is directly related to health indicators. LimbLen error, bridging phase I and phase II, testifies the relationship between the two tasks by its resonant movements with the traditional pose estimation metric in Figure 5. The better the performance in pose estimation, the better it is for health indicator prediction. In Figure 5, the MPJPE of our model throughout a video is always less than the one without GLANCE. The reason for the GLANCE module to succeed is its ability to incorporate both global and local information and let each party learn from each other. In the case of occlusion of a joint due to viewpoint change, the global information of the relative position of a limb can help predict the location of the occluded joint using adjacent joint coordinates. The incorporation of global and local information helps the model cope with extreme circumstances and maintain competitive performance. Summary In this paper, we develop a paradigm for health indicator prediction from gait videos in small datasets. The spatialtemporal encoder for video feature extraction is first pretrained on pose estimation, which is a related task to health indicator prediction. Then the pre-trained encoder is used for feature extraction in health indicator prediction. We also de- sign the GLANCE module emphasizing global and local information extraction and fusion, whose effectiveness is validated on both tasks. A new pose estimation metric LimbLen Error is proposed to bridge the gap between tasks and show their connections. Experimental results show that our proposed paradigm is viable, and that the GLANCE module can improve performances in both tasks. Appendix Declaration of Competing Interest:We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome. We confirm that the manuscript has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship but are not listed. We further confirm that the order of authors listed in the manuscript has been approved by all of us. We confirm that we have given due consideration to the protection of intellectual property associated with this work and that there are no impediments to publication, including the timing of publication, with respect to intellectual property. In so doing we confirm that we have followed the regulations of our institutions concerning intellectual property. We understand that the Corresponding Author is the sole contact for the Editorial process (including Editorial Manager and direct communications with the office). He/she is responsible for communicating with the other authors about progress, submissions of revisions, and final approval of proofs. We confirm that we have provided a current, correct email address which is accessible by the Corresponding Author. Credit authorship contribution statement: Ziqing Li: conceptualization, software, data analysis, writing the original draft. Xuexin Yu: conceptualization, editing, data analysis. Xiaocong Lian: Xiaocong Lian: conceptualization, editing, supervision. Yifeng Wang: conceptualization Xiangyang Ji: conceptualization, supervision, project administration. Figure 2 : 2Examples of different methods of estimating BMI Figure 4 : 4The effectiveness of the GLANCE module for LimbLen variations in time. Figure 5 : 5The effectiveness of the GLANCE module for MPJPE variations in time. Table 1 1Quantitative comparison for health indicators prediction.Models Metrics Health Indicators BMI Age Height Weight Image-based model Jin et al. (2022) MAE ↓ 2.61 - - - MAPE ↓ 11.45% - - - Our Model MAE ↓ 2.29 2.09 5.16 6.96 MAPE ↓ 9.86% 9.20% 3.09% 10.72% Table 2 2Ablation study for health indicator prediction.Modules Metrics Health Indicators ResNet Extractor Fusion BMI Age Height Weight ✓ MAE ↓ 2.6 3.0 7.5 8.9 MAPE ↓ 11.2% 13.4% 4.5% 13.9% ✓ ✓ MAE ↓ 2.5 2.3 5.8 7.7 MAPE ↓ 10.7% 10.3% 3.5% 11.8% ✓ ✓ ✓ MAE ↓ 2.3 2.1 5.2 7.0 MAPE ↓ 9.9% 9.2% 3.1% 10.7% Table 3 Ablation study for pose estimation. The model with only the Resnet module as encoder is equivalent to VIBEKocabas et al.. Modules Metrics ResNet Extractor Fusion MPJPE ↓ PA-MPJPE ↓ PVE ↓ LimbLen Error ↓ ✓ 111.4 70.3 129.6 333.5 ✓ ✓ 109.5 69.7 131.4 299.1 ✓ ✓ ✓ 105.9 67.2 127.2 292.8 Table 4 Health indicator statistics. Health Indicators Average Standard Deviation Age 21.75 3.77 Height 168.84 8.93 Weight 64.87 11.07 BMI 22.71 3.21 The effect of viewpoint on image-based methodJin et al. (2022) … … 3.9224 4.2128 3.9212 4.3619 4.5759 5.091 4.7543 5.394 5.1852 5.0382 4.9466 4.9432 5.7677 MAE Figure 3: 0 2 4 6 8 10 12 14 Frames 4.35 4.40 4.45 4.50 4.55 Limb Length (m) w/ GLANCE w/o GLANCE GT 0 2 4 6 8 10 12 14 Frames 4.54 4.56 4.58 4.60 4.62 4.64 Limb Length (m) w/ GLANCE w/o GLANCE GT 0 2 4 6 8 10 12 14 Frames 4.30 4.35 4.40 4.45 4.50 4.55 Limb Length (m) w/ GLANCE w/o GLANCE GT Posetrack: A benchmark for human pose estimation and tracking. M Andriluka, U Iqbal, E Insafutdinov, L Pishchulin, A Milan, J Gall, B Schiele, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionAndriluka, M., Iqbal, U., Insafutdinov, E., Pishchulin, L., Milan, A., Gall, J., Schiele, B., 2018. 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{'abstract': 'A B S T R A C T Body Mass Index (BMI), age, height and weight are important indicators of human health conditions, which can provide useful information for plenty of practical purposes, such as health care, monitoring and re-identification. Most existing methods of health indicator prediction mainly use front-view body or face images. These inputs are hard to be obtained in daily life and often lead to the lack of robustness for the models, considering their strict requirements on view and pose. In this paper, we propose to employ gait videos to predict health indicators, which are more prevalent in surveillance and home monitoring scenarios. However, the study of health indicator prediction from gait videos using deep learning was hindered due to the small amount of open-sourced data. To address this issue, we analyse the similarity and relationship between pose estimation and health indicator prediction tasks, and then propose a paradigm enabling deep learning for small health indicator datasets by pre-training on the pose estimation task. Furthermore, to better suit the health indicator prediction task, we bring forward Global-Local Aware aNd Centrosymmetric Encoder (GLANCE) module. It first extracts local and global features by progressive convolutions and then fuses multi-level features by a centrosymmetric double-path hourglass structure in two different ways.Experiments demonstrate that the proposed paradigm achieves state-of-the-art results for predicting health indicators on MoVi, and that the GLANCE module is also beneficial for pose estimation on 3DPW.', 'arxivid': '2212.12948', 'author': ['Ziqing Li \nTsinghua University\nHaidian District100084BeijingChina\n', 'Xuexin Yu \nTsinghua University\nHaidian District100084BeijingChina\n', 'Xiaocong Lian \nTsinghua University\nHaidian District100084BeijingChina\n', 'Yifeng Wang \nHIT Campus of University Town of Shenzhen\n518055ShenzhenChina\n', 'Xiangyang Ji \nTsinghua University\nHaidian District100084BeijingChina\n'], 'authoraffiliation': ['Tsinghua University\nHaidian District100084BeijingChina', 'Tsinghua University\nHaidian District100084BeijingChina', 'Tsinghua University\nHaidian District100084BeijingChina', 'HIT Campus of University Town of Shenzhen\n518055ShenzhenChina', 'Tsinghua University\nHaidian District100084BeijingChina'], 'corpusid': 255125002, 'doi': '10.48550/arxiv.2212.12948', 'github_urls': [], 'n_tokens_mistral': 14038, 'n_tokens_neox': 12149, 'n_words': 7100, 'pdfsha': 'c1d901cf2a65100e0bb9f0c894461ae7256b377c', 'pdfurls': ['https://export.arxiv.org/pdf/2212.12948v1.pdf'], 'title': ['Human Health Indicator Prediction from Gait Video', 'Human Health Indicator Prediction from Gait Video'], 'venue': []}

arxiv

Scalable Gaussian Process Inference with Stan Till Hoffmann Jukka-Pekka Onnela Harvard T H Chan School of Public Health School of Public Health Harvard T H Chan School of Public Health School of Public Health Scalable Gaussian Process Inference with Stan Till Hoffmann Jukka-Pekka Onnela Gaussian processFourier transformsparse approximationStanPython Gaussian processes (GPs) are sophisticated distributions to model functional data. Whilst theoretically appealing, they are computationally cumbersome except for small datasets. We implement two methods for scaling GP inference in Stan: First, a general sparse approximation using a directed acyclic dependency graph. Second, a fast, exact method for regularly spaced data modeled by GPs with stationary kernels using the fast Fourier transform. Based on benchmark experiments, we offer guidance for practitioners to decide between different methods and parameterizations. We consider two real-world examples to illustrate the package. The implementation follows Stan's design and exposes performant inference through a familiar interface. Full posterior inference for ten thousand data points is feasible on a laptop in less than 20 seconds. and Zheng 2016). It implements many common likelihood functions, e.g., Poisson for count data, Bernoulli for classification, and Student-t for robust regression, but hierarchical models cannot be easily constructed. GPyTorch (Gardner, Pleiss, Bindel, Weinberger, and Wilson 2018) offers similar functionality for PyTorch (Paszke, Gross, Introduction Gaussian processes (GPs) are flexible non-parametric models for functions with applications in time series analysis (Roberts, Osborne, Ebden, Reece, Gibson, and Aigrain 2013), geospatial statistics (Krige 1951), robotics (Deisenroth, Fox, and Rasmussen 2015), and beyond. More formally, a GP is a distribution over functions f (x) such that any finite set of n values f = f (x) evaluated at x = {x 1 , . . . , x n } follows a multivariate normal distribution (Rasmussen and Williams 2006). The distribution is thus fully specified by its mean µ (x) = E (f (x)) and covariance kernel k (x, x ′ ) = cov (f (x) , f (x ′ )). A range of problem-specific kernels have been developed, such as squared exponential and Matérn kernels to model local correlations and sinusoidal kernels to capture periodic signals (Duvenaud 2014, Chapter 2). In general, evaluating the likelihood requires inverting the covariance matrix K obtained by evaluating the kernel k at all pairs of observed locations x. Unfortunately, the computational cost of inverting K scales as O n 3 , making GPs prohibitively expensive save for relatively small datasets. Diverse schemes have been developed to approximate the likelihood for larger datasets, such as low-rank approximations of the covariance matrix (Hensman, Fusi, and Lawrence 2013), nearest-neighbor approximations (Wu, Pleiss, and Cunningham 2022), and Fourier methods (Hensman, Durrande, and Solin 2017;Greengard 2021). Numerous packages provide implementations in different programming languages, but most of them focus exclusively on GPs which makes it difficult to integrate GPs in larger models. For example, GPflow (Matthews, van der Wilk, Nickson, Fujii, Boukouvalas, León-Villagrá, Ghahramani, and Hensman 2017) is tegration into larger models. We consider a benchmark problem and discuss the importance of different parameterizations for performant inference in Section 5. Furthermore, we demonstrate the utility of both approaches with two examples: Inferring the density of trees in a 50 ha plot in Panama (Condit, Perez, Aguilar, Lao, Foster, and Hubbell 2019) and predicting passenger numbers on the London Underground transportation network. We summarize our contributions in Section 7 and discuss how the package can be employed to build more complex models. As Stan does not have a package repository, we have published the library as a Python package gptools-stan on PyPI. The package includes the Stan library code and provides utility functions to integrate with the popular Stan interface cmdstanpy. The library can also be obtained directly from https://github.com/onnela-lab/gptools. Extensive technical documentation and examples are available at https://gptools-stan.readthedocs.org. Introduction to Stan Stan is a probabilistic programming framework (Carpenter et al. 2017), comprising both a concise R-like syntax to declare probabilistic models and an efficient Hamilton Monte Carlo algorithm to draw posterior samples (Betancourt 2018). Readers familiar with Stan may skip to Section 3. Each Stan program consists of blocks to declare inputs, parameters, and the probabilistic model. For a concrete example, consider a linear regression model with n × p design matrix X, coefficient vector θ with p elements, outcome vector y with n elements, and observation noise variance σ 2 , i.e., y ∼ Normal Xθ, σ 2 . The corresponding Stan program is shown below. // Text file "linear.stan" containing the model definition. The data block defines inputs required to evaluate the likelihood of the model; parameters declares parameters of the model including any constraints, such as the noise scale being non-negative. Finally, the model block declares priors for parameters and the observation model of outcomes y given covariates X and parameters θ and σ. To illustrate the analysis workflow in Python, we generated synthetic data using the numpy package by sampling from the prior predictive distribution with n = 100 observations and p = 3 covariates. We fix the random number generator seed for reproducibility. >>> import numpy as np >>> np.random.seed(0) >>> X = np.random.normal(0, 1, (100, 3)) >>> theta = np.random.normal(0, 1, 3) >>> sigma = np.random.gamma(2, 2) >>> y = np.random.normal(X @ theta, sigma) >>> print(f"coefficients: {theta}") >>> print(f"observation noise scale: {sigma}") coefficients: [-1.307 1.658 -0.118] observation noise scale: 1.867 We used the cmdstanpy interface to compile the above model, draw posterior samples, and report summary statistics. >>> import cmdstanpy >>> model = cmdstanpy.CmdStanModel(stan_file="linear.stan") >>> fit = model.sample(data={"n": n, "p": p, "X": X, "y": y}, seed=0) >>> print(fit.summary()) Gaussian processes with structured dependencies The joint distribution of observations f may be expressed as the product of conditional distributions p (f ) = p (f 1 ) n j=2 p (f j | f j−1 , . . . , f 1 ) . (1) The conditional structure in Equation (1) can be encoded by a directed acyclic graph (DAG) whose nodes represent observations such that a directed edge exists from a node j to each of its predecessors P j = {j − 1, . . . , 1}; the ordering is arbitrary. If two observations do not depend on one another, the corresponding edge can be removed from the DAG to reduce the computational cost. In particular, evaluating each factor of Equation (1) requires inverting a matrix with size equal to the number of predecessors of the corresponding node-a substantial saving if the graph is sparse. For example, nearest-neighbor methods, a special case, reduce the asymptotic runtime to O nq 3 by retaining only edges from each node to at most q of its nearest predecessors. This approach can yield excellent approximations provided that the neighborhoods are large enough and that the kernel only models local correlations (Wu et al. 2022). For example, nearest-neighbor methods are not suitable for periodic kernels but can be approximated by structured dependencies if the period is known, such as diurnal or yearly patterns. Algorithm 1 Evaluate the log likelihood of the Gaussian process realization f given its mean µ, locations of observations x, covariance kernel k, and the dependency graph encoded as a set of predecessors P. 1: function gp_graph_lpdf(f | µ, x, k, P) 2: L ← normal_lpdf(f 1 | µ 1 , k (x 1 , x 1 )) ▷ MarginalΣ ← k (x P i , x P i ) ▷ Covariance among predecessors of i. 5: s ← k (x i , x Pi ) ▷ Covariance between i and its predecessors. 6: ν ← s ⊺ Σ −1 µ P i ▷ Conditional mean. 7: τ 2 = k (x i , x i ) − s ⊺ Σ −1 s ▷ Conditional variance. 8: L ← L + normal_lpdf(f i | ν, τ 2 ) ▷ Conditional log likelihood for i given P i . 9: end for 10: return L 11: end function Pseudocode to approximate the likelihood of a GP realization f using structured dependencies is shown in Algorithm 1 for a general kernel k. The algorithm approximates the log likelihood iteratively by evaluating the conditional mean and variance for each node i given its predecessors P i in lines 4-7; the conditional distributions are available in closed form for multivariate normal distributions (Gelman, Carlin, S, Dunson, Vehtari, and Rubin 2013, Appendix A1). The evaluation of likelihood contributions can be further accelerated by parallelizing the loop in line 3. We implemented a custom distribution in Stan such that a GP with squared exponential kernel on a DAG embedded in a p-dimensional space can be specified as f~gp_graph_exp_quad_cov(loc, x, sigma, length_scale, edges); where vector[n] loc is the prior mean, and array[n] vector [p] x is an array of locations in p dimensions for each of the n nodes of the graph. The parameters real sigma and real length_scale control the marginal scale and smoothness of the kernel which is defined as (Duvenaud 2014, Chapter 2) k x, x ′ = sigma 2 × exp − |x − x ′ | 2 2 × length_scale 2 . The larger the length scale the more slowly the GP varies because even points with substantial separation |x − x ′ | remain highly correlated. The graph is encoded by the edge list array[,] int edges, a two-dimensional array of integer node labels comprising predecessors and successors in the first and second row, respectively. For example, {{1, 2, 3}, {2, 3, 4}} represents the directed acyclic line graph of four nodes 1 ← 2 ← 3 ← 4. Following Stan's indexing convention, node labels start at one. Similar distributions are provided for the matern32 and matern52 kernels. Gaussian processes in Fourier space We can use Fourier methods to evaluate the likelihood efficiently if three conditions are satisfied (Rasmussen and Williams 2006, Appendix B). First, we need to consider observation points x on a regular grid to reap the computational benefits of the fast Fourier transform (FFT) (Press, Teukolsky, Vetterling, and Flannery 2007). Second, the kernel must be stationary, i.e., k (x, x ′ ) = k (x − x ′ ) such that the correlation only depends on the separation between observations. Third, the kernel must be n-periodic because the FFT is subject to periodic boundary conditions, i.e., k (x + n, x ′ ) = k (x − x ′ ), where n is the number of observations. These conditions may seem overly restrictive. However, in many settings, data naturally form a regular grid, e.g., financial time series with fixed sampling interval (Hoffmann, Peel, Lambiotte, and Jones 2020), resampled or binned time series (Flaxman, Wilson, Neill, Nickisch, and Smola 2015), or rasterized images (Tipping and Bishop 2002). Likewise, stationary kernels, such as squared exponential and Matérn kernels, are common choices for modeling functional data using GPs. Finally, the effect of periodic boundary conditions can be attenuated by padding the domain, as discussed in more detail in Section 6.2 and Appendix A. Because the Fourier transform is a linear operator and f is multivariate normal, the discrete Fourier coefficientsf ξ = n−1 j=0 exp − 2πiξj n f j are also multivariate normal, where ξ is the (discrete) frequency, f j is the GP at the j th grid point, and i is the imaginary number. Assuming µ (x) = 0 for simplicity, the mean of Fourier coefficients is zero and their expected complex-conjugate product at two different frequencies ξ and ξ ′ is E f ξfξ ′ = n−1 j=0 n−1 j ′ =0 exp − 2πi n jξ − j ′ ξ ′ k j − j ′ = n−1 j ′ =0 exp − 2πij ′ n ξ − ξ ′ n−1−j ′ ∆=−j ′ exp − 2πi∆ n k (∆) , where we changed variables to j = ∆ + j ′ in the second line. The argument of the inner sum is n-periodic, and we may shift the limits of summation to [0..n − 1] without changing the sum. The change of limits decouples the two sums. The first is a sum-representation of the Kronecker delta nδ ξξ ′ ; the second is the Fourier transform of the kernelk. We obtain E f ξfξ ′ = nδ ξξ ′k ξ . Algorithm 2 Evaluate the log likelihood of the Gaussian process realization f given its mean µ and Fourier-transformed covariance kernelk. Range indexing is inclusive on the left and exclusive on the right, i.e., f a:b = {f a , . . . , f b−1 }. 1: function gp_rfft_lpdf(f | µ,k) 2: z ← |rfft(f − µ)| ▷ Modulus of centered real FFT with ⌊n/2⌋ + 1 elements. 3: L ← normal_lpdf(z 0 | 0, nk 0 ) ▷ Real zero-frequency term. 4: if n mod 2 = 1 then 5: m ← n+1 2 ▷ Index following highest-frequency complex coefficient. L ← L + 2 × normal_lpdf(z 1:m | 0, nk 1:m ) ▷ Complex oscillatory terms. 11: return L 12: end function Fourier coefficients of different frequencies are thus independent with variance nk ξ . Subject to careful bookkeeping, we can evaluate the likelihood exactly, as illustrated in Algorithm 2. Because f is real, we use the real FFT (RFFT) for efficiency. It comprises ⌊n/2⌋ + 1 complex coefficients because just under half the coefficients are redundant (Press et al. 2007, Chapter 12.3). The zero-frequency termf 0 and, for even n, the Nyquist frequency termf n/2 are real (see lines 3 and 8 of Algorithm 2, respectively). The complex coefficients contribute twice in line 10 to account for the redundant terms omitted by the RFFT. We implemented a custom distribution in Stan such that a GP on a grid can be specified as f~gp_rfft(loc, cov_rfft); where vector[n] loc is the prior mean and vector[n %/% 2 + 1] cov_rfft is the RFFT of the kernel evaluated on the grid (%/% denotes floor division in Stan). Fortunately, the RFFT of common kernels, such as the squared exponential kernel and Matérn kernels, can be evaluated directly in the Fourier domain (Rasmussen and Williams 2006, Chapter 4), as shown in Figure 1 (see Appendix B for details). Evaluating the kernel in the Fourier domain also obviates the need for small "nugget" variance or "jitter" typically required for numerical stability (Neal 1997). We thus only need to evaluate one Fourier transform, that of the signal, to evaluate the likelihood. The library provides the following functions to evaluate Fourier-domain kernels gp_periodic_exp_quad_cov_rfft(n, sigma, length_scale, period) gp_periodic_matern_cov_rfft (n, nu, sigma, length_scale, period) where period is the size of the domain, n is the number of grid points with spacing period / n, and nu is the smoothness parameter of the Matérn kernel. The parameters sigma and length_scale have the same meaning as in Section 3. Equivalent functions, which we discuss further in Section 6.2, are provided for two-dimensional grids. We implemented the Fourierdomain kernels by naively discretizing frequencies. This approach works well if the number of grid points is large and the correlation length is small compared with the size of the domain. Panel (a) shows the periodic and non-periodic (standard) versions of the squared exponential kernel (blue) and Matérn 3 /2 kernel (orange) with σ = 1 and ℓ = 0.2. The two versions are hardly distinguishable for x < ℓ. Padding may need to be introduced if non-periodic signals are modeled with periodic kernels. The power spectrum of the two periodic kernels is shown in panel (b). The squared exponential kernel has negligible power for all but the first few frequencies, explaining why it is often considered too smooth to "represent natural phenomena" (Handcock and Stein 1993). Panels (c) and (d) show realizations of GPs with squared exponential and Matérn 3 /2 kernels, respectively. Different line styles correspond to approximations with a different number of Fourier modes. For the squared exponential kernel, the number of modes can be reduced by an order of magnitude without substantially affecting realizations. The Matérn kernel requires more modes due to its heavy-tailed power spectrum. More sophisticated methods may be required otherwise (Borovitskiy, Terenin, Mostowsky, and Deisenroth 2020). Benchmark and the importance of parameterizations We consider a simple benchmark problem to study the performance of different methods and compare them with standard Gaussian process inference which inverts the kernel. The model comprises a one-dimensional zero-mean Gaussian process prior with squared exponential kernel and an independent normal observation model with variance κ 2 , i.e., f ∼ MultivariateNormal (0, K) y ∼ Normal f , κ 2 .(2) We used a marginal kernel scale σ = 1 and unit correlation length ℓ = 1 to evaluate the covariance matrix K on an integer grid, i.e., x = {0, . . . , n − 1}. Employing the cmdstanpy interface, we drew 100 posterior samples from a single chain after 100 warmup samples. Warmup samples are used to adapt the sampler for efficient exploration of the posterior (Homan and Gelman 2014). Default values were used for all other parameters. We considered different dataset sizes between n = 2 4 and n = 2 14 and allocated a maximum computational budget of one minute for all n, i.e., the sampler was terminated if it did not complete after 60 seconds. The mean runtime as a function of dataset size is shown in panels (a) and (b) of Figure 2 for small (κ = 0.1) and large (κ = 10) noise scales as solid lines, respectively. As expected, the runtime of the standard approach grows rapidly as n increases. We observed an empirical runtime scaling of n ≈2.5 for the standard approach, not dissimilar from the expected asymptotic scaling of O n 3 . Exploring models with more than a few hundred data points is prohibitively expensive-even for this simple setup. For the graph-based approach, we used the five nearest predecessors (q = 5) to construct a dependency graph. The method is comparatively slow for small datasets but outperforms the standard approach as n grows. The Fourier approach has the best performance irrespective of dataset size but is limited to observations on a grid. The model in Equation (2) However, if the data are weak (large κ), they cannot independently constrain each element of f and the GP prior dominates the posterior. The resulting correlation among elements of f frustrates the sampler, especially if the correlation length is large. We can overcome this challenge by employing a non-centered parameterization such that the parameters of the model are uncorrelated under the prior (Papaspiliopoulos et al. 2007). Here, we reparameterize the model in terms of a white noise vector z of the same size as f and obtain realizations of the GP f = ϕ −1 (z, µ, K) using an inverse transform ϕ −1 which must be selected carefully to ensure f follows the desired distribution. We chose the inverse transform for consistency with the FFT: The forward transform maps to the Fourier domain, and the inverse transform , respectively. Independent of parameterization and dataset size, Fourier methods offer the best performance when observations are regularly spaced. For datasets exceeding a few hundred observations, graph methods are faster than the standard approach which requires inversion of the covariance matrix. The dotted horizontal line represents the maximum computational budget of 60 s. As shown in panel (c) for the Fourier approach, the runtime of the centered and non-centered parameterizations increases and decreases, respectively, as the data become less informative. While different parameterizations are primarily a performance concern for drawing posterior samples, they have important consequences for variational posterior approximations. Panel (d) shows the difference ∆ in log posterior density (p.d.) between the non-centered and centered parameterization on 20% held-out data for n = 1,024 using the Fourier approach. If the model is fit with a variational mean-field approximation, the non-centered parameterization offers higher log posterior scores than the centered parameterization when the data are weak and vice versa. Bootstrapped standard errors are smaller than the size of markers in panel (d). maps to real space. The reparameterized model is z ∼ Normal (0, 1) f = ϕ −1 (z, 0, K) y ∼ Normal f , κ 2 .(3) We implemented the following transforms for the graph-based and Fourier approaches. f = gp_inv_graph_exp_quad_cov(z, loc, x, sigma, length_scale, edges); f = gp_inv_rfft(z, loc, cov_rfft); where vector[n] z are the non-centered white noise parameters and all other parameters are as described previously. For the standard method, we implemented the non-centered parameterization as f = Lz (Papaspiliopoulos et al. 2007), where L is the Cholesky decomposition of the covariance matrix K such that K = LL ⊺ . Algorithm 3 Transform white noise z to a Gaussian process realization f given its mean µ, locations of observations x, covariance kernel k, and the dependency graph encoded as a set of predecessors P. 1: function gp_inv_graph(z, µ, x, k, P) 2: . . . ▷ Compute conditional mean ν and variance τ 2 as in Algorithm 1. f 1 ← µ 1 + k (x 1 , x 1 )z 1 ▷ Sample 5: f i ← ν + τ z i ▷ Sample from the conditional distribution given P i . 6: end for 7: return f 8: end function Algorithm 3 implements the transform for approximate GPs using structured dependencies and closely follows Algorithm 1. Instead of evaluating the log likelihood iteratively given the conditional distribution, the algorithm draws a sample sequentially by transforming white noise to the target distribution given previous samples in lines 2 and 5. Unlike Algorithm 1 the loop in line 3 cannot be parallelized because the conditional mean and variance depend on the results of previous iterations. The transformation from white noise to a GP realization using Fourier methods is illustrated in Algorithm 4. As in Algorithm 2, we account for the real zero-frequency term and, for even n, Nyquist frequency term in lines 3 and 8, respectively. Line 10 constructs the ⌊(n − 1) /2⌋ complex Fourier coefficients from z 1:m (real part) and z m+(n+1) mod 2:n (imaginary part). The (n + 1) mod 2 term in the index accounts for the presence of the Nyquist frequency for even n. The omission of redundant terms in the RFFT is addressed by dividing the complex coefficients by √ 2. As shown in panels (a) and (b) of Figure 2, the non-centered parameterization (dashed lines) is more performant than the centered parameterization (solid lines) if the noise scale is large and vice versa. Panel (c) further illustrates the importance of choosing the right parameterization: The runtime differs by up to a factor of five as we vary the noise scale κ. The higher-frequency terms of smooth GPs have low power, as shown in panel (b) of Figure 1. We can further Algorithm 4 Transform white noise z to a Gaussian process realization f given its mean µ and Fourier-transformed covariance kernelk. Range indexing is inclusive on the left and exclusive on the right, i.e., f a:b = {f a , . . . , f b−1 }. end if 10:f 1:m ← τ 1:m z 1:m + i × z m+(n+1) mod 2:n / √ 2 ▷ Complex oscillatory terms. 11: f ← inv_rfft(f , n) ▷ Inverse RFFT returning a vector with n elements. 12: return f 13: end function improve performance of the non-centered parameterization by discarding all but the first few low-frequency terms. This approach is particularly effective for the squared exponential kernel because the power spectrum decays rapidly with increasing frequency. For the example shown in panel (c), a GP using only the first five Fourier modes is indistinguishable from the GP considering all 51 modes, reducing the dimensionality of the parameter space by an order of magnitude. GPs with Matérn kernels typically require more Fourier modes because the power spectrum has a relatively heavy tail. While parameterization is primarily a performance concern for Hamiltonian Monte Carlo samplers, it can have a substantial impact on the predictive ability of models if variational mean-field inference is used. Variational approximations of the posterior tend to assign low probability mass to regions of the parameter space where the full posterior has low mass (Bishop 2006, Chapter 10.1). Consequently, variational approximations are too narrow if the posterior is highly correlated, and we expect predictions to be overconfident. To test this hypothesis, we sampled synthetic data with n = 1,024 data points from the prior predictive distribution and fitted the models in Equations (2) and (3) to 80% of each synthetic dataset using variational inference (Kucukelbir et al. 2017). Stan's ADVI implementation approximates the posterior by a product of independent normal distributions, one for each parameter. The variational approximation is optimized in an unconstrained space. Constrained parameters are obtained by applying a transform, e.g., an exponential transform to obtain a positive parameter such as the length scale ℓ. We evaluated the predictive ability of the fitted models by evaluating the log posterior density on the 20% held-out GP realizations, i.e., log p (f test | y train ). To compare the parameterizations, we approximated the log posterior difference ∆ = log p f test = ϕ −1 (z test ) | y train − log p (f test | y train ) using a Gaussian kernel density estimator (Bishop 2006, Chapter 2.5.1). As shown in panel (d) of Figure 2, the non-centered parameterization makes better predictions than the centered parameterization when the noise scale is large and vice versa. We also fitted the two parameterizations by drawing posterior samples using Stan's Hamiltonian sampler and evaluated the log posterior difference. Different parameterizations did not affect the predictive performance but had a significant impact on runtime. Illustrations Passengers on the London Underground transportation network Millions of people use the London Underground transportation network, commonly referred to as the "Tube", to travel across the city each day (Transport for London 2019). The number of passengers using each station is affected by various factors, including how connected it is and which zone the station is in (the network comprises nine transport zones). In addition to these fixed effects, we also expect passenger numbers to be affected by smooth spatial effects, e.g., due to variability in population density. The spatial effect can naturally be modeled as a GP with structured dependencies induced by the transport network itself. We converted the undirected graph to a directed acyclic graph in two steps. First, we assigned an integer label to each node. Second, we added an edge from node i to node j if i < j and the corresponding edge exists in the undirected graph. The order of nodes is arbitrary because the joint probability in Equation (1) can be factorized in any order. We collected network data from the Transport for London open data API (Transport for London 2022) and obtained the average daily number of entries and exits at each station in 2019 (Transport for London 2019), as shown in panel (a) of Figure 3. The model includes fixed effects for each degree and zone, mildly regularized by half-t priors with two degrees of freedom (Gelman, Jakulin, Pittau, and Su 2008). We truncated the degree and zone of each station at five and six, respectively, because only few stations exceed these values. The overall number of passengers is captured by a scalar µ, and we used the GP to explain any residual effects. A squared exponential kernel was employed for the covariance, and we used a half-t prior for the marginal scale. The correlation length of the kernel is not identifiable if it is smaller than the smallest distance between stations (0.16 km) or larger than the extent of the transportation network (62 km) (Trangucci, Betancourt, and Vehtari 2016). We thus used a log-uniform prior on the interval [0.32 km, 31 km] to suppress extreme length scales. All distances were evaluated in the Ordnance Survey National Grid projection (epsg:27700). A non-centered parameterization was used because the residual effects are not strongly identified by the data after controlling for zone and degree. We used a log-normal observation model (rather than a model for count data) because passenger data are heavy-tailed and reported as daily averages. (d) show the effect of degree and zone on passenger numbers, respectively. Central stations with larger degree tend to have more passengers. Termini with degree one have uncharacteristically many passengers because they serve as interchanges for longer-distance trains. Panel (c) shows the GP which explains residual variations in passenger volumes after controlling for zone and degree. On the one hand, stations in the Hainault loop have relatively few passengers because they are served infrequently compared with nearby stations. On the other hand, Canary Wharf has surprisingly many entries and exits because it is a busy financial center in London. Processing complete, no problems detected. We employed the default configuration of cmdstanpy to draw posterior samples, resulting in four independent chains with 2,000 samples each. The print( fig.diagnose()) call evaluates and reports a suite of diagnostics to identify potential problems and assess convergence. For example, the splitR statistic compares samples both within and between chains to determine whether they are likely to have mixed well (Vehtari, Gelman, Simpson, Carpenter, and Bürkner 2021). Tree depth, divergence, and Bayesian fraction of missing information (BFMI) are technical diagnostics to assess whether the sampler was able to explore the posterior distribution; the effective sample size estimates the number of independent samples drawn which may be smaller than 2,000 due to autocorrelation within each chain (Betancourt 2018). Here, the samples satisfied all posterior checks offered by cmdstan. Panel (b) shows the effect of degree on passenger numbers on the log scale. They tend to increase with the degree of a station as they offer passengers a variety of travel options. Termini with degree one are an exception: Their passenger numbers are uncharacteristically large because they serve as stepping stones to longer-distance travel beyond the Tube network. Unsurprisingly, central stations in zones one to three tend to have more passengers than stations in the suburbs (zones four and above), as shown in panel (d). The GP captures any residuals that cannot be explained by the degree of the station or the zone it is located in, as shown in panel (c). For example, on the one hand, Canary Wharf has the largest residual effect. It is one of London's financial centers, and the station serves tens of thousands of commuters each day despite being a station without an interchange. On the other hand, stations in the north of the Hainault loop have the largest negative residual effect because the stations are served by only three trains an hour (Transport for London 2020). Passengers divert to nearby stations that are served by twelve trains an hour (Transport for London 2020). Comparing the model with a model without GP effects using the log posterior predictive distribution on held-out data, we observe no significant difference after bootstrapping errors. Nevertheless, this example illustrates how our package can be used to easily construct GPs with structured dependencies. Density of T. panamensis on a 50 ha plot in Panama To illustrate the use of Fourier methods, we consider the density of T. panamensis trees during the 2015 census of the 50 ha Barro Colorado plot in Panama (Condit et al. 2019). The plot is divided into quadrants of 20 m side length. As shown in panel (a) of Figure 4, the data comprise the frequency of trees within each quadrant, i.e., a matrix of count data with shape (25, 50). The observed tree frequency counts y were modeled by a negativebinomial distribution to account for possible overdispersion. We used a latent GP to model the log-mean of this distribution and capture the tree density. We employed a Matérn kernel with smoothness parameter ν = 3 /2, half-t prior for the marginal scale and overdispersion parameter, and log-uniform prior for the correlation length as in Section 6.1. Because the quadrants are regularly spaced, the likelihood can be evaluated exactly using Fourier methods, as discussed in Section 4. However, unlike the FFT, trees are not subject to periodic boundary conditions. To mitigate this issue and reduce correlation between opposing sides of the plot, we padded the matrix with ten additional quadrants in each dimension (corresponding to 200 m) resulting in a matrix with shape (35, 60). Despite increasing the number of latent variables by almost 70%, the method is faster than the standard approach which inverts the covariance matrix. Processing complete, no problems detected. Despite the noisy, masked observations, the model was able to learn a smooth estimate of the density of trees, as shown in panel (c). The posterior median of the correlation length of 60 m was well below the padding of 200 m introduced to attenuate the effect of periodic boundary conditions, as shown in panel (b). We used a scaled mean-squared error (SMSE) to evaluate the model on the held-out data and compare it with the simpler approach of smoothing the data with a two-dimensional Gaussian filter. The SMSE is S y,ŷ = expf = 1 m m j=1 y i − expf i 2 max (y i , 1) , where the sum is over m test points andf is the posterior median of the latent GP. We divided each term by the observed count y i (or one if the count was zero) to ensure the measure was not dominated by large counts because the sampling variance of a Poisson count process (without overdispersion) is equal to its mean. A simple method to estimate the number of trees in held-out quadrants is to apply a Gaussian filter to the data and compare the two methods. The Gaussian filter estimate iŝ y λ = g λ * (b • y) g λ * b , where * denotes convolution, • denotes the elementwise product, g λ is a Gaussian filter with smoothing scale λ, and b is the binary mask indicating which data are available for training. Gaussian filters "blur" the data locally such that adjacent elements of the smoothed signal can inform one another (Lindeberg 1990). For large λ, estimates are approximated by the sample mean, and, for small λ, they are dominated by local noise. Panel (d) of Figure 4 shows the SMSE for the Gaussian filter as a function of smoothing scale and the SMSE achieved by the GP model. The latter achieves a lower SMSE than the former for all smoothing scales, illustrating the utility of GPs for modeling spatial effects. Unlike in Section 6.1, it was not possible to use the posterior predictive distribution for evaluation because the Gaussian filter is not a generative model. Discussion We implemented two popular approaches for scaling GPs to larger datasets in Stan: The sparse approximation with structured dependencies discussed in Section 3 and the exact Fourier approach in Section 4 which is applicable to data on a grid. For centered parameterizations, the likelihood can be evaluated or approximated directly. For non-centered parameterizations, we sample standard normal random variables z and use the inverse transform to obtain a GP sample f = ϕ −1 (z). Given different parameterizations and approaches, which should be used in practice? As discussed in Section 5, a non-centered parameterization is appropriate if the data are weak, and a centered parameterization is preferable if the data are strong. Choosing the right parameterization ensures parameters are relatively uncorrelated under the posterior distribution which accelerates inference. For variational mean-field approximations, choosing the right parameterization is even more important: It affects the quality of the approximation, as discussed in Section 5. Most variational approaches use a centered parameterization (Hensman et al. 2013;Wu et al. 2022), and their approximations may be improved by considering non-centered parameterizations. If the data are very strong, the benefits of GPs may be outweighed by their complexity because the likelihood dominates the GP prior. If in doubt, we suggest using a non-centered parameterization, as we have done in Section 6, because GPs are typically employed when the data are not sufficiently informative for simpler approaches to succeed. Figure 5: The optimal inference approach and package depends on the data. Choosing an appropriate inference approach and package depends on the data at hand, as shown in Figure 5. Here, we only consider general purpose packages that can handle arbitrary likelihoods and facilitate the use of GPs as parts of larger models. At one extreme, posterior samples can be obtained with most inference frameworks if the data are small. At the other, if the data do not fit in memory, evaluating the likelihood repeatedly as part of a Monte Carlo sampler is not feasible-even if the likelihood can be approximated. ADVI (e.g., using GPyTorch and Pyro) is a viable approach although at the cost of considering a narrower set of posteriors (Kucukelbir et al. 2017). If the data are low-dimensional and form a regular grid, the Fourier methods in Section 4 are suitable. Basis function approximations (currently implemented in the R package brms which builds on Stan) may be appropriate for low-dimensional data with irregular spacing. The implementation represents the GP as a linear superposition of eigenfunctions of the Laplace operator with Dirichlet boundary conditions (Riutort-Mayol et al. 2022). The approach is similar to the Fourier methods presented here because Fourier modes are eigenfunctions of the Laplace operator although with periodic boundary conditions. The unique advantage of Fourier methods is that the likelihood can be evaluated exactly in O (n log n) if observations form a grid. In higher dimensional spaces, sparse approximations using structured dependencies can approximate the posterior, as discussed in Section 3. Padding may be required to attenuate the effect of periodic boundary conditions inherent to the fast Fourier transform. The necessary amount of padding depends on the kernel. We have found one to two correlation lengths ℓ to be sufficient for squared exponential and Matérn 3 /2 kernels (see Appendix A for details). However, the correlation length is often not known a priori, and finding the "right" amount of padding that appropriately balances performance and the need for non-periodic boundary conditions may be an iterative process. For example, we can start with a small amount of padding and increase it until the posterior stabilizes. Fourier methods may also be appropriate if the density of observation points is relatively homogeneous. In particular, we may consider a latent GP g on a grid and use it to predict the GP of interest f at each observation point, i.e., p (f | g) = n j=1 p (f j | g) . This method reduces the computational cost because elements of f are conditionally independent given g at the regularly spaced "inducing points" (Hensman et al. 2013). We hope that our library and the illustrations in Section 6 will accelerate the development of models employing GPs in Stan. Integrating GP approximations with Stan's ecosystem, rather than developing a bespoke GP library, will allow practitioners to leverage the framework's flexibility and the shared knowledge of the engaged Stan community. Computational details The results in this paper were obtained using Python 3.10.0, cmdstanpy 1.0.8, and cmdstan 2.30.1. All experiments were run on a single core of a 2020 MacBook Pro with an Apple Silicon M1 chip and 16 GB of RAM. (d) Figure 6: A small amount of padding is sufficient to attenuate the effect of periodic boundary conditions. Panel (a) shows a realization of the benchmark model in Equation (2), and panel (c) shows the true latent GP f together with posterior means for different models, including the true generative model and Fourier methods with varying amounts of padding w. The effect of periodic boundary conditions is stark for the fit without padding, but it disappears quickly with increasing padding. Panels (b) and (d) show the log posterior density evaluated at the true f n , which corresponds to the held-out data point y n , as a function of padding for squared exponential and Matérn 3 /2 kernels, respectively. The horizontal line corresponds to the log posterior density obtained using the true generative model. Padding with one length scale ℓ is sufficient to approximate the true model well. A. Effect of padding for Fourier methods We considered a simulation study to examine the effect of periodic boundary conditions inherent to Fourier methods and assess the amount of padding required to balance model misspecification and performance concerns in two steps. First, we generated m = 100 synthetic datasets each comprising n = 128 observations on an integer grid according to the benchmark model in Equation (2) with marginal kernel scale σ = 1 and observation noise κ = 1. We used a correlation length ℓ = 16 large enough for periodic boundary conditions to have an effect on the inference. For each dataset, we fitted the standard non-centered model, i.e., the true generative model, and Fourier-based Gaussian processes with varying amounts of padding w. We evaluate both models by holding out the last data point y n , which should be most severely affected by periodic boundary conditions, and approximating the log posterior density log p (f n | y <n ) of the corresponding element of the latent GP using a Gaussian kernel density estimator (Bishop 2006, Chapter 2.5.1). Even a small amount of padding, such as one correlation length, is sufficient to attenuate the effect of periodic boundary conditions, as shown in panels (b) and (d) of Figure 6 for squared exponential and Matérn 3 /2 kernels, respectively. Simulation-based calibration is a technique to validate a Bayesian inference pipeline (Talts, Betancourt, Simpson, Vehtari, and Gelman 2018). For synthetic data generated from the model, the rank of the true parameter value among posterior samples should have a uniform distribution. For each combination of the different paddings and two kernels, we evaluated the rank across m synthetic datasets and evaluated the p value of the Kolmogorov-Smirnov test by comparing with a discrete uniform reference distribution. If no padding is used, the null hypothesis that the ranks are uniform can be confidently rejected (p value < 10 −3 ), but the ranks are not inconsistent with a uniform distribution for w > ℓ (p value > 0.3 at w = ℓ in our simulations). B. Kernels in the real and Fourier domains B.1. Squared exponential kernel The non-periodic squared exponential kernel is defined as k x, x ′ = σ 2 exp − (x − x ′ ) 2 2ℓ 2 , where σ is the marginal scale and ℓ is the correlation length. Its discrete power spectrum on a periodic domain of size L isk ξ = √ 2πnσ 2 ℓ L exp −2 πξℓ L 2 , where n is the number of grid points and ξ ∈ [0..n − 1] is the discrete frequency. B.2. Matérn kernel The non-periodic Matérn kernel is defined as k ν x, x ′ = σ 2 2 1−ν Γ(ν) ζ ν K ν (ζ) , where ζ = √ 2ν |x − x ′ | ℓ is a rescaled distance, ν is a smoothness parameter, Γ denotes the gamma function, |x − x ′ | is the Euclidean distance between x and x ′ , and K ν denotes the modified Bessel function of the second kind. For ν = 3 /2 and ν = 5 /2, the kernel simplifies to k3 /2 = σ 2 1 + √ 3 |x − x ′ | ℓ exp − √ 3 |x − x ′ | ℓ k5 /2 = σ 2 1 + √ 5 |x − x ′ | ℓ + 5 |x − x ′ | 2 3ℓ 2 exp − √ 5 |x − x ′ | ℓ . ▷ Index following highest-frequency complex coefficient.8: L ← normal_lpdf(z m | 0, nk m )▷ Real Nyquist-frequency term. Figure 1 : 1Gaussian processes can often be captured by a small number of Fourier modes. employs the natural centered parameterization(Papaspiliopoulos, Roberts, and Sköld 2007), i.e., each observation y i is independent given the corresponding latent f i . This parameterization works well if the data are informative (small κ) because each observation y i constrains the corresponding latent parameter f i . The elements of f are thus relatively uncorrelated under the posterior, and the Hamiltonian sampler can explore the distribution efficiently(Homan and Gelman 2014). Figure 2 : 2Different approaches, parameterizations, as well as the informativeness of the data substantially affect runtimes. Centered parameterizations are preferable when the data are strong (small observation noise scale κ), and non-centered parameterizations are superior when the data are weak (large κ), as shown in panels (a) and (b) Figure 3 : 3A Gaussian process on the London Underground transportation network identifies idiosyncrasies of transport use. Panel (a) shows the daily average number of entries and exits for each of the 267 stations in 2019 together with the transport network. Empty nodes denote held-out data. Panels (b) and } Figure 4 : 4A Gaussian process based on the two-dimensional Fourier transform can accurately predict the frequency of trees. Panel (a) shows the number of T. panamensis trees per 20 m quadrant collected during the 2015 census of the 50 ha Barro Colorado plot in Panama (Condit et al. 2019) as a heat map. Gray quadrants indicate the 20% held-out test data. Panel (c) shows the posterior median of the expected tree frequency, recovering the held-out data and smoothing the empirical frequencies. Posterior samples of the correlation length and marginal scale of the Matérn kernel are shown in panel (b). Correlation length samples are well below the padding (200 m) introduced to mitigate the effect of periodic boundary conditions. Panel (d) shows the scaled mean squared error with bootstrapped errors on the held-out data under the Gaussian process model and a Gaussian filter with variable smoothing scale. The 90% marginal posterior intervals for all parameters are consistent with the values used to generate the data. Having gained some intuition for Stan and cmdstanpy, we consider two approaches to scalable GP inference and their implementation in Stan in the following two sections.5% 50% 95% ... theta[1] -1.535 -1.229 -0.932 ... theta[2] 1.439 1.754 2.062 ... theta[3] -0.385 -0.062 0.266 ... sigma 1.715 1.921 2.170 ... ... log likelihood for the first node.3: for i ∈ [2..n] do 4: first observation from the marginal distribution.3: for i ∈ [2..n] do 4: We fitted the model to 80% of the passenger data using the below Python code, withholding 20% of the stations uniformly at random for later evaluation. Held-out data are encoded as -1 in the Stan model.>>> from gptools.stan import compile_model >>> import json >>> import numpy as np >>> # Load station locations, edges, passenger numbers, apply training mask. >>> with open("tube-stan.json") as fp: ... data = json.load(fp) >>> train_mask = np.random.binomial(1, 0.8, data["num_stations"]) >>> data["passengers"] = np.where(train_mask, data["passengers"], -1) >>> # Compile model and fit it. >>> model = compile_model(stan_file="tube.stan") >>> fit = model.sample(data) >>> print(fit.diagnose()) Processing csv files: ... Checking E-BFMI -sampler transitions HMC potential energy. E-BFMI satisfactory. Effective sample size satisfactory. Split R-hat values satisfactory all parameters.parameters { vector[num_stations] z; real mu; real<lower=0> sigma, kappa; real<lower=log(0.32), upper=log(31)> log_length_scale; vector[num_zones] zone_effect; vector[num_degrees] degree_effect; } transformed parameters { real length_scale = exp(log_length_scale); vector[num_stations] f = gp_inv_graph_exp_quad_cov( z, zeros_vector(num_stations), station_locations, sigma, length_scale, edge_index); vector[num_stations] log_mean = mu + f + one_hot_zones * zone_effect + one_hot_degrees * degree_effect; } model { z~std_normal(); sigma~student_t(2, 0, 1); zone_effect~student_t(2, 0, 1); degree_effect~student_t(2, 0, 1); kappa~student_t(2, 0, 1); for (i in 1:num_stations) { if (passengers[i] > 0) { log(passengers[i])~normal(log_mean[i], kappa); } } } Checking sampler transitions treedepth. Treedepth satisfactory for all transitions. Checking sampler transitions for divergences. No divergent transitions found. We fitted the model to 80% of the quadrants chosen uniformly at random using the below Python code, withholding the remainder for evaluation.Split R-hat values satisfactory all parameters.functions { #include gptools/util.stan #include gptools/fft.stan } data { int num_rows, num_cols, num_rows_padded, num_cols_padded; array[num_rows, num_cols] int frequency; } parameters { matrix[num_rows_padded, num_cols_padded] z; real mu; real<lower=0> sigma, kappa; real<lower=log(2), upper=log(28)> log_length_scale; } transformed parameters { real<lower=0> length_scale = exp(log_length_scale); matrix[num_rows_padded, num_cols_padded %/% 2 + 1] rfft2_cov = gp_periodic_matern_cov_rfft2(1.5, num_rows_padded, num_cols_padded, sigma, [length_scale, length_scale]', [num_rows_padded, num_cols_padded]'); matrix[num_rows_padded, num_cols_padded] f = gp_inv_rfft2( z, rep_matrix(mu, num_rows_padded, num_cols_padded), rfft2_cov); } model { to_vector(z)~std_normal(); mu~student_t(2, 0, 1); sigma~student_t(2, 0, 1); kappa~student_t(2, 0, 1); for (i in 1:num_rows) { for (j in 1:num_cols) { if (frequency[i, j] >= 0) { frequency[i, j]~neg_binomial_2(exp(f[i, j]), 1 / kappa); } } } } >>> from gptools.stan import compile_model >>> import numpy as np >>> # Load tree frequency matrix, define padding, apply training mask. >>> frequency = np.loadtxt("tachve.csv", delimiter=",", dtype=int) >>> num_rows, num_cols = frequency.shape >>> padding = 10 >>> train_mask = np.random.binomial(1, 0.8, frequency.shape) >>> data = { ... "num_rows": num_rows, ... "num_rows_padded": num_rows + padding, ... "num_cols": num_cols, ... "num_cols_padded": num_cols + padding, ... "frequency": np.where(train_mask, frequency, -1), ... } >>> # Compile model and fit it. >>> model = compile_model(stan_file="trees.stan") >>> fit = model.sample(data) >>> print(fit.diagnose()) Processing csv files: ... Checking sampler transitions treedepth. Treedepth satisfactory for all transitions. Checking sampler transitions for divergences. No divergent transitions found. Checking E-BFMI -sampler transitions HMC potential energy. E-BFMI satisfactory. Effective sample size satisfactory. AcknowledgmentsWe thank Philip Greengard, Mike Lawrence, and Aki Vehtari for comments on the manuscript and Brian Ward for answering numerous questions about cmdstanpy.It reduces to the Laplace kernel for ν = 1/2. 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{'abstract': "Gaussian processes (GPs) are sophisticated distributions to model functional data. Whilst theoretically appealing, they are computationally cumbersome except for small datasets. We implement two methods for scaling GP inference in Stan: First, a general sparse approximation using a directed acyclic dependency graph. Second, a fast, exact method for regularly spaced data modeled by GPs with stationary kernels using the fast Fourier transform. Based on benchmark experiments, we offer guidance for practitioners to decide between different methods and parameterizations. We consider two real-world examples to illustrate the package. The implementation follows Stan's design and exposes performant inference through a familiar interface. Full posterior inference for ten thousand data points is feasible on a laptop in less than 20 seconds. and Zheng 2016). It implements many common likelihood functions, e.g., Poisson for count data, Bernoulli for classification, and Student-t for robust regression, but hierarchical models cannot be easily constructed. GPyTorch (Gardner, Pleiss, Bindel, Weinberger, and Wilson 2018) offers similar functionality for PyTorch (Paszke, Gross,", 'arxivid': '2301.08836', 'author': ['Harvard T H Chan \nSchool of Public Health\nSchool of Public Health\n\n', 'Harvard T H Chan \nSchool of Public Health\nSchool of Public Health\n\n'], 'authoraffiliation': ['School of Public Health\nSchool of Public Health\n', 'School of Public Health\nSchool of Public Health\n'], 'corpusid': 256105357, 'doi': None, 'github_urls': ['https://github.com/onnela-lab/gptools.', 'https://github.com/SheffieldML/GPy.'], 'n_tokens_mistral': 19402, 'n_tokens_neox': 16766, 'n_words': 9824, 'pdfsha': 'a4ce46de2a10b48bbdee56de097910f4de4bde5c', 'pdfurls': ['https://export.arxiv.org/pdf/2301.08836v3.pdf'], 'title': ['Scalable Gaussian Process Inference with Stan Till Hoffmann Jukka-Pekka Onnela', 'Scalable Gaussian Process Inference with Stan Till Hoffmann Jukka-Pekka Onnela'], 'venue': []}

arxiv

Closed String Tachyon Driving f (R) Cosmology 13 Oct 2016 Peng Wang pengw@scu.edu.cn†wu˙houwen@g.harvard.edu‡hyanga@scu.edu.cn Center for Theoretical Physics College of Physical Science and Technology Sichuan University 610064ChengduPR China Houwen Wu Center for Theoretical Physics College of Physical Science and Technology Sichuan University 610064ChengduPR China Center for the Fundamental Laws of Nature Harvard University 02138CambridgeMAUSA Haitang Yang Center for Theoretical Physics College of Physical Science and Technology Sichuan University 610064ChengduPR China Closed String Tachyon Driving f (R) Cosmology 13 Oct 2016* Electronic address: 1 Contents To study quantum effects on the bulk tachyon dynamics, we replace R with f (R) in the lowenergy effective action that couples gravity, the dilaton, and the bulk closed string tachyon of bosonic closed string theory and study properties of their classical solutions. The α ′ corrections of the graviton-dilaton-tachyon system are implemented in the f (R). We obtain the tachyon-induced rolling solutions and show that the string metric does not need to remain fixed in some cases.The singular behavior of more classical solutions are investigated and found to be modified by quantum effects. In particular, there could exist some classical solutions, in which the tachyon field rolls down from a maximum of the tachyon potential while the dilaton expectation value is always bounded from above during the rolling process. I. INTRODUCTION In the standard cosmological model, the Friedmann equations are derived from Hilbert action, coupling the metric with ad hoc matter sources. In string theory, a low energy effective action arises from the requirement of quantum conformal invariance. The tree level of this action couples the metric, the dilaton and the axion. This action, implying the dynamics of gravitational field and other background fields, has been used in many areas. One of the most important applications is to develop a string cosmology [1,2]. There exist two ways to improve this string effective action [3,4]. The first one is α ′controlled expansion, which includes higher derivatives of the metric and background fields. Another one is string coupling g s -controlled expansion (higher-genus expansion) and which reflects the higher loop string interactions. The α ′ -controlled expansion, which becomes significant in Planck and high curvature region, had been discussed in [5,6]. When curvature and energy scale grow up, the usual simplest perturbative expansion of the string effective action becomes no-go theorem. In this condition, the higher order expansions should be taken into account. A good review of string cosmology is given by [7] and references therein. The open string tachyon rolls down from the perturbative unstable vacuum to an excited state of closed string carrying the energy of the D-brane. On the other hand, the instabilities associated with closed string tachyon is a much more difficult problem since the action is non-polynomial. Some progresses have been achieved in the calculation of the effective potential [8][9][10][11] by truncating the polynomial to quintic order. Those calculations indicate that there do exist a local minimum and several saddle points in the closed string tachyon and dilaton effective potential. The conjecture that the action has to vanish at the closed string field theory (CSFT) vacuum, made in [8,12], is also implied. At the closed string vacuum, parallel to the conclusions in open string theory, one is led to believe that spacetime itself ceases to exist. To make these conjectures more reliable, higher order calculations or more desirable analytical methods are necessary. For these reasons, studying the string effective action that couples the metric, the dilaton and the tachyon is of importance. One can expect that this action will reveal some features of string theory and related problems. Some progresses have been made in recent developments. The author of [13] discussed the solutions of the effective action with tachyon and B-field and analyzed the solutions in AdS 3 background. In [14], Brandenberger et.al. found a nonsingular and static tachyon condensation by discussing an effective theory with a non-vanishing dilaton potential. The quantum effects of graviton-dilaton-tachyon system is investigated in [15] where it shows that the singular behavior of classical solutions should be modified by quantum effects. The author of [16] uses some constraints to fix the form of action without computing the beta functions. Some other progresses of graviton-dilatontachyon system refer to [17][18][19][20][21][22][23][24][25][26]. The aim of this paper is an extension of [8,12] to study the α ′ corrections of the gravitondilaton-tachyon system. The α ′ correction should also satisfy the quantum conformal invariance and then brings higher derivative terms of gravitational field as well as other background fields. To preserve the covariance and gauge invariance, the same expansion order of α ′ could lead to different effective actions [7,27]. To simplify the story, we only implement corrections to the graviton by replacing R with f (R). This could be accomplished by integrating out other massive fields in the action. In [12], the rolling process triggered by the tachyon was investigated for tachyonic potentials. It was found that during the rolling process, the string metric did not evolve while the dilaton rolled to strong coupling. In the framework of f (R) gravity, our analysis shows some interesting results. In section III, we find for tachyon-induced rolling solutions that time evolution of the string metric could be nontrivial in some cases. In section IV, assuming the Hubble parameter of the string metric is constant, we obtain a set of solutions, in which the tachyon field rolls down from the top of some tachyon potential while the string coupling is always finite during the rolling process. This paper is organized as follows. In section II, we derive the equations of motion of graviton, dilaton, and tachyon field in f (R) theories. In section III, we define tachyoninduced rolling solutions and solve the equations of motion for them. More classical solutions are discussed in section IV. The section V is our conclusion. II. COUPLED SYSTEM OF FIELDS The low energy effective action for the metric, the dilaton, and the tachyon is given by S = 1 2κ 2 d d+1 x √ −ge −2Φ f (R) + 4 (∇Φ) 2 − (∇T ) 2 − 2V (T ) ,(1) where g µν is string metric, Φ is dilaton, T is tachyon with potential V (T ), and f (R) is the generalized gravity. The number of spatial dimensions is d. Since here we focus on the tachyon driving solutions, the dilaton potential is set to zero for simplicity. Varying the action (1) with respect to g µν , Φ, and T , we find that the equations of motion for graviton, dilaton and tachyon are F (R) R µν − ∇ µ T ∇ ν T + 4g µν [F (R) − 1] (∇Φ) 2 − 4g µν ∇ α Φ∇ α F (R) −2g µν [F (R) − 1] ∇ 2 Φ + g µν ∇ 2 F (R) + 2F (R) ∇ ν ∇ µ Φ −∇ ν ∇ µ F (R) − 4 [F (R) − 1] ∇ ν Φ∇ µ Φ + 2∇ µ Φ∇ ν F (R) + 2∇ ν Φ∇ µ F (R) = 0,(2)1 2 [F (R) R − f (R)] + 2 {d [F (R) − 1] + 1} (∇Φ) 2 − 2d∇ α Φ∇ α F (R) − {d [F (R) − 1] + 1} ∇ 2 Φ + 1 2 d∇ 2 F (R) + V (T ) = 0,(3)∇ 2 T − 2∇ µ Φ∇ µ T − V ′ (T ) = 0,(4) where F (R) ≡ df (R)/dR. As in [12], we make the following ansatz ds 2 = −dt 2 + a (t) 2 δ ij dx i dx j , Φ = Φ (t) ,(5)T = T (t) . For the string metric in eqn. (5), the Ricci scalar is R = 2dḢ + d (d + 1) H 2 ,(6) where H =˙a a . In this case, the 00 and ij components of gravitational equations become F (R) dḢ + dH 2 +Ṫ 2 − 2Φ + 2 [F (R) − 1]Φ −Ḟ (R) dH = 0,(7)F (R) Ḣ + dH 2 − 4 [F (R) − 1]Φ 2 + 4ΦḞ (R) + 2 [F (R) − 1]Φ +2 [F (R) (d − 1) − d] HΦ −F (R) − (d − 1) HḞ (R) = 0,(8) which can be rearranged into two equivalent equations 1 2 F (R) (d − 1)Ḣ + 1 2Ṫ 2 + 2 [F (R) − 1]Φ 2 − 2ΦḞ (R) −F (R)Φ + F (R) HΦ + 1 2F (R) − 1 2 HḞ (R) = 0,(9)1 2 F (R) (d − 1) dH 2 − 2d [F (R) − 1]Φ 2 + 2dΦḞ (R) + {d [F (R) − 1] + 1}Φ + [d (F (R) − 1) − 2F (R) + 1]Φ − 1 2 (d − 2)Ḟ (R) dH − 1 2 dF (R) − 1 2Ṫ 2 = 0. (10) The equations of motion for the dilaton and the tachyon are 1 2 [F (R) R − f (R)] − 2 [d (F (R) − 1) + 1]Φ 2 + 2dΦḞ (R) + {d [F (R) − 1] + 1} Φ + dHΦ − 1 2 d F (R) + dHḞ (R) + V (T ) = 0,(11)T + dH − 2Φ Ṫ + V ′ (T ) = 0.(12) Note that there are four differential equations for three unknown dynamical variables: a (t), Φ (t), and T (t). Therefore, one of the four equations should be redundant. In fact, it shows in the appendix that eqns. (7), (8), and (12) could guarantee that eqn. (11) holds wheneveṙ T = 0. III. TACHYON-DRIVEN ROLLING SOLUTIONS In this section, we consider a general class of potentials V (T ) for a tachyon T that has a local maximum at T = 0, which can be written as V (T ) = V (0) − 1 2 m 2 T 2 + O T 3 .(13) In [12] where f (R) = R, the rolling solutions driven by the tachyon were discussed. The ansatzes for T (t) and Φ (t) were assumed to be T (t) = e mt + n≥2 t n e nmt , Φ (t) = n≥2 φ n e nmt ,(14)H (t) = n≥2 h n e nγt , where T → 0 for t → −∞, and Φ (t) has exponentials subleading to e mt since the tachyon drives the rolling in the very early time. For the case with f (R) = R, eqn. (10) becomes 1 2 (d − 1) dH 2 = 1 2Ṫ 2 −Φ + dHΦ,(15) which gives that H → 0 when t → −∞, which is consistent with the ansatz for H (t) proposed in eqn. (14). In fact, it showed in [12] that H (t) vanished identically for the tachyon-driven rolling solutions. For the case with a general form of f (R), we build an analogous tachyon-driven rolling solution: T (t) = e γt + n≥2 t n e nγt , Φ (t) = n≥2 φ n e nγt ,(16) where γ is a positive real number, and the first term in T (t) is the solution to the linearized tachyon equation of motion. The arbitrary constant multiplying the first term in T (t) can be absorbed, as we did, by a redefinition of time. Unlike in the case with f (R) = R, eqn. H (t) = H 0 + n≥1 h n e nγt .(17) If the rolling process is solely trigger by the tachyon filed, one needs that H 0 = h 1 = 0. As we show below, for such ansatz nontrivial H (t) could exist in some cases. However to explore more possibilities, we now do not require H 0 = h 1 = 0. Plugging T (t) and Φ (t) from eqns. (16) into the tachyon equation (12), we find γ 2 + dH 0 γ − m 2 = 0,(18) which gives γ = m 1 + d 2 H 2 0 4m 2 − dH 0 2m . The corresponding Ricci scalar is R = R 0 + n≥1 r n e nγt ,(19) where R 0 = d (d + 1) H 2 0 and r n depends on h m≤n . We assume f (R) is analytic along the real axis except certain poles. In the rest of this section, we will use eqns. (7), (8) and (12) to solve for R 0 , h n , φ n and t n . And eqn. (11) could be used to determine the value of V (0). Note that in [12] where f (R) = R, it showed that V (0) = 0, R 0 = 0 and h n = 0. A. Analytic at R 0 Now f (R) is assumed to be analytic at R 0 . Thus, we can expand f (R) at R = R 0 : f (R) = f (R 0 ) + l≥1 α l (R − R 0 ) l .(20) Eqn. (19) gives F (R) = F (R 0 ) + O (e γt ). Thus, the leading terms of the 00 and ij components of gravitational equations (7) and (8) both become F (R 0 ) dH 2 0 = 0. Given R 0 = d (d + 1) H 2 0 , one finds that either F (R 0 ) = 0 or R 0 = 0. The tachyon equation (12) is trivial at the leading order. The leading order of the dilation equation (11) gives V (0) = − f (R 0 ) 2 ,(21) where we use F (R 0 ) R 0 = 0. Note that eqn. (21) must be satisfied to admit the tachyon rolling solution. For example, if f (R) = R − Λ where Λ is the cosmological constant and R 0 = 0, one has V (0) = Λ 2 . In what follows, we will solve for h n , φ n and t n in the two cases with R 0 = 0 and F (R 0 ) = 0. 1. R 0 = 0 If f (R) is analytic at R = 0, F (R) = 1 + l≥2 lα l R l−1 . Plugging eqns. (16) and (17) with H 0 = 0 into the equations of motion, one could determine the coefficients t n , φ n and h n order by order. Note that γ = m for H 0 = 0. At O (e mt ) , eqns. (7) and (8) give dmh 1 e mt = 0, 1 − 4dm 2 α 2 mh 1 e mt = 0,(22) which leads to h 1 = 0. Tachyon equation is also trivial at this order. At O (e 2mt ) ,eqns. (7) and (8) give m 2 + 2dmh 2 − 8m 2 φ 2 e 2mt = 0, 2m 1 − 16dm 2 α 2 h 2 e 2mt = 0,(23) which yield φ 2 = 1 8 + dh 2 4m , h 2 = 0 or α 2 = 1 16dm 2 .(24) Note that we have h 2 = 0 if α 2 = 1 16dm 2 . Now we prove by induction that H(t) vanishes identically if α 2 = 1 4dN 2 m 2 for N = 2, 3 · · · . First assume that h 2 = h 3 = · · · = h N = 0. SinceΦ (t) ∼ e 2mt , eqn. (8) gives (N + 1) 1 − 4dα 2 (N + 1) 2 m 2 mh N +1 e (N +1)mt + O e (N +2)mt = 0,(25)which yields 1 − 4dα 2 (N + 1) 2 m 2 h N +1 = 0. Therefore, if α 2 = 1 4dN 2 m 2 for N = 2, 3 · · · , h N +1 = 0. On the other hand, if α 2 = 1 4dN 2 m 2 for some positive integer N > 1, h N could be nonzero. Therefore, there might be solutions with nonzero H(t) in such cases. Consider an example with f (R) = R + R 2 16dm 2 + l≥3 α l R l and V (T ) = − 1 2 m 2 T 2 , we find solutions up to O (e 4mt ) T (t) = e mt + e 3mt 16 + O e 4mt , Φ (t) = 1 8 + dh 4m e 2mt + O e 4mt ,(26)H (t) = he 2mt + O e 4mt , where h is a free parameter. As far as the tachyon rolling ansatz is concerned, the parameter h could be arbitrary. 2. F (R 0 ) = 0 Suppose F (R) is analytic at R 0 where F (R 0 ) = 0. Then, one can expand F (R) at R 0 : F (R) = l≥1 β l (R − R 0 ) l ,(27) where we assume β 1 = 0 for simplicity. Plugging eqns. (16) and (17) into eqns. (7), (8), and (12) gives the recurrence relations for h n , φ n ,and t n with necessary initial values. In fact, after putting the ansatzes into eqns. (7), (8), and (12), one has at O (e nγt ) that 2nγ (nγ + dH 0 ) φ n − d 2 H 2 0 β 1 h n = G n (h i<n , φ i<n , t i<n ) , 2nγ (nγ + dH 0 ) φ n + 2nγd n 2 γ 2 − H 2 0 + (d − 1) H 0 nγ β 1 h n = F n (h i<n , φ i<n , t i<n ) ,(28)γ n 2 − 1 γ + dH 0 (n − 1) t n = H n (h i<n , φ i<n , t i<n ) , where G n , F n and H n are functions of only h i<n , φ i<n , and t i<n . Solving the above equations for h n , φ n , and t n , we find that the recurrence relations for h n , φ n ,and t n for n ≥ 2 are φ n = G n (h i<n , φ i<n , t i<n ) 2nγ (nγ + dH 0 ) + dH 2 0 nγ + dH 0 F n (h i<n , φ i<n , t i<n ) − G n (h i<n , φ i<n , t i<n ) 2n 2 γ 2 [nγ + (d − 1) H 0 ] , h n = F n (h i<n , φ i<n , t i<n ) − G n (h i<n , φ i<n , t i<n ) 2dn 2 γ 2 β 1 [nγ + (d − 1) H 0 ] ,(29)t n = 1 γ H n (h i<n , φ i<n , t i<n ) (n 2 − 1) γ + dH 0 (n − 1) . (29) to find values of h n , φ n ,and t n . For example, we have for n = 2 that φ 2 = 1 8 , h 2 = γ 4dβ 1 [(d + 1) H 2 0 − 2dH 0 γ − 4γ 2 ] , t 2 = 0,(30) where h 2 is generally not zero. B. Pole at R 0 We now study the scenario in which f (R) has a pole of order |L| at R 0 . Therefore, the Laurent series expansion of f (R) at R 0 is f (R) = l≥L α l (R − R 0 ) l ,(31) where α L = 0 and L ≤ −1 is some negative integer. We now consider two cases: R 0 = 0 and R 0 = 0. 1. R 0 = 0 Since H 0 = 0, the ansatz for H (t) (17) becomes H (t) = n≥N h n e nmt ,(32) where h N e N mt is assumed to be the nonzero leading term. If H (t) = 0, one has some integer N ≥ 1 for which h N = 0. Plugging eqn. (32) into R = 2dḢ + d (d + 1) H 2 gives for nonzero integer l that R l = r l N e N lmt + O e (N l+1)mt ,(33) where r N = 2dNmh N . Thus, we have F (R) = l≥L lα l R l−1 = Lr L−1 N α L e N (L−1)mt + O e [N (L−1)+1]mt .(34) Given eqn. (34), the ij components of gravitational equation (8) becomes −Lr l−1 N α L N 2 (L − 1) 2 m 2 e N (L−1)mt + O e [N (L−1)+1]mt = 0, where gives h L−1 N α L = 0. Since α L = 0, one obtains h N = 0 and hence a contradiction, which means H (t) = 0. However f (R) blows up at R = 0, and hence the tachyon rolling ansatzes (16) and (17) do not solve the equations of motion (7), (8) and (12) in this case. 2. R 0 = 0 The ansatz for H (t) is H (t) = H 0 + n≥N h n e nγt ,(35) where h N e N γt is the first nonzero term in the series. Using eqn. (31), one has f (R) = α L r L N e N Lγt + O e (N L+1)γt ,(36) where r N = 2d [Nγ + (d + 1) H 0 ] h N . Therefore, eqn. (7) gives [H 0 − N (L − 1) γ] dH 0 Lr L−1 N α L e N (L−1)γt + O e (N (L−1)+1)γt = 0.(37) IV. COSMOLOGICAL SOLUTIONS In section III, we considered a special class of solutions to the equations of motion, whose ansatzes are given in eqns. (16) and (17). For the case with f (R) = R, the string metric remains fixed for such solutions. However for some more general form of f (R), there might exist tachyon-driven rolling solutions with nonzero H (t). It appears that the behavior of the classical solutions is richer in the f (R) theory. In this section, we study the classical solutions beyond the ansatzes (16) and (17) H → −H, R → R, Φ → Φ, T → −T.(38) However, the equations of motion might become nonlinear higher order differential equations, which are difficult to solve. To investigate the properties of their solutions, we consider two simple scenarios, the one with constant H, and the other with constant T . A. Constant H If we assume H (t) = H 0 which is a constant, there are three independent equations of motion for Φ (t) and T (t). Therefore, one of these equations determines the form of the tachyon potential V (T ), which depends on H 0 . In other words, only some particular potentials V (T ) admit the classical solutions with H (t) = H 0 . In this case, the ij components of gravitational equation (8) becomes dF (R 0 ) H 2 0 − 4 [F (R 0 ) − 1]Φ 2 + 2 [F (R 0 ) − 1]Φ + 2dH 0 [F (R 0 ) − 1]Φ − 2H 0 F (R 0 )Φ = 0,(39) where R (t) = R 0 = d (d + 1) H 2 0 . To solve this equation, we can introduce the variable τ = dH 0 t. In what follows, the dot denotes t-derivative, and the prime denotes τ -derivative. If F (R 0 ) = 1 and H 0 = 0, eqn. (39) becomes trivial and hence there are only two independent equations of motion for Φ (t) and T (t). In this case, Φ (t) and T (t) can be solved for any V (T ), and the properties of these solutions have been discussed in [12]. It has been found that the string coupling always became divergent at some time. If F (R 0 ) = 1 and H 0 = 0, the solution to eqn. (39) is Φ (τ ) = τ − τ 0 2 ,(40) where the integration constant τ 0 is a constant time translation and can be set to zero for simplicity from now on. The dilaton Φ goes to infinity at t = +∞. This solution evolves to a singular configuration with a strongly coupled background at infinite string time. The dilaton equation (11) gives V (T ) = 1 2 [f (R 0 ) − R 0 ]. However, eqn. (7) becomeṡ T 2 = −dH 2 0 < 0,(41) which contradicts the tachyon field T being real. Thus, there are no classical solutions with constant H in this case. If F (R 0 ) = 1, solving eqn. (39) for Φ gives Φ ± (τ ) = 1 2 ln e τ 1 ± γe Aτ + C Φ ,(42) where C Φ is an integration constant, (11), one could solve it for V (T (τ )): A = 1 + F (R 0 ) d[F (R 0 )−1] , and γ = |1 − F (R 0 )|. Plugging Φ (t) into the dilaton equationV (τ ) d 2 H 2 0 = − F (R 0 ) R 0 − f (R 0 ) 2d 2 H 2 0 + sgn (F (R 0 ) − 1) ρAe Aτ ±1 + γe Aτ ,(43)where ρ = F (R 0 )[dF (R 0 )−d+1] 2d , and sgn (x) is the sign function with sgn (x) = x |x| . Plugging Φ (t) into the gravitational equation (7) gives T ′2 ± = α ±1 + βe Aτ (1 ± γe Aτ ) 2 ,(44)where B = d + F (R 0 ) − 3dF (R 0 ) + 2dF 2 (R 0 ), α = F (R 0 )(d+1)−d d , and β = B 2d|1−F (R 0 )| . When F (R 0 ) = d d+1 , one has that α = A = 0. In this case, T stays constant, and Φ (τ ) = τ 2 + C Φ ,(45) which is the linear dilaton solution. When F (R 0 ) = d d+1 , we list the sign of T ′2 ± for the possible values of F (R 0 ) in F ± = −1+3d± √ 1−6d+d 2 4d such that B = 0 when F (R 0 ) = F ± . Note that 0 < F − < F + < d d+1 . When T ′2 ± ≥ 0, we integrate eqn. (44) and obtain where C T is an integration constant. In principle, one could use eqns. (43) and (46) to find V (T ). T ± (τ ) = 2 √ α A Re γ − β γ arctanh γ γ − β ±1 + βe Aτ − arctanh ±1 + βe Aτ +C T ,(46)F (R 0 ) > d d+1 F (R 0 ) < d d+1 T ′2 + T ′2 + > 0 T ′2 + < 0 T ′2 − T ′2 − ≥ (<) 0 for e Aτ ≥ (<) 2dγ B T ′2 − ≥ (<) 0 for e Aτ ≤ (>) 2dγ BF (R 0 ) > d d+1 F + ≤ F (R 0 ) < d d+1 or F (R 0 ) ≤ F − F − < F (R 0 ) < F + T ′2 + T ′2 + > 0 T ′2 + < 0 T ′2 + ≥ (<) 0 for e Aτ ≥ (<) 2dγ |B| T ′2 − T ′2 − ≥ (<) 0 for e Aτ ≥ (<) 2dγ |B| T ′2 − ≥ (<) 0 for e Aτ ≤ (>) 2dγ |B| T ′2 − < 0 When τ → −∞, Φ ± (τ ) always goes to −∞. Defining τ = τ a such that γe Aτa = 1, one finds that Φ − (τ a ) = +∞. As τ → +∞, one has for Φ ± (τ ) that Φ ± (τ ) →          +∞, for A < 1 C Φ , for A = 1 −∞, for A > 1 .(47) For Φ + (τ ) with A < 1 and Φ − (τ ), the string coupling always diverges at some time. However for Φ + (τ ) with A ≥ 1, the string coupling always stay finite. In FIG. 1(a), we plot Φ + (τ ) for F (R 0 ) = 2. Note that A ≥ 1 implies that F (R 0 ) ≤ 0 or F (R 0 ) > 1. TABLEs I and II show that T ′2 + < 0 when F (R 0 ) ≤ 0. Thus when A ≥ 1, the solution T + (τ ) only exists for F (R 0 ) > 1. When F (R 0 ) > 1, eqns. (43) and (46) shows that as τ → −∞ T + (τ ) ∼ √ ατ , dV (T ) dT ∼ d 2 H 2 0 ρA 2 √ α e Aτ , and d 2 V (T ) dT 2 ∼ d 2 H 2 0 ρA 4 α e Aτ , and as τ → +∞ T + (τ ) ∼ C T , dV (T ) dT ∼ d 2 H 2 0 ρA 2 √ αβγ e −Aτ /2 ∼ 0, and d 2 V (T ) dT 2 ∼ −d 2 H 2 0 ρA 4 2αβ < 0, where we use α > 0, β > 0, and ρ > 0 for F (R 0 ) > 1. Therefore when τ = +∞, the tachyon field T stays at a maximum of V (T ). Since T + (τ ) goes to −∞ as τ → −∞, one finds for T → −∞ that the tachyon potential becomes V (T ) ∼ − F (R 0 ) R 0 − f (R 0 ) 2 + ρAd 2 H 2 0 e AT / √ α .(48) In FIG. 1(b), we plot V (T ) for F (R 0 ) = 2. Recalling that τ = dH 0 t, we find for H 0 < 0 and (7) and (8) for H (t) and Φ (t). The dilaton equation (11) impose constraints on the integration constants. This scenario could provide some insights into the possible final state of bulk tachyon condensation. The case with f (R) = R has been discussed in [15], where the classical solutions always were found to evolve from or to singular configurations. We here investigate the singular behavior of the solutions in the case with f (R) = R + α 2 R 2 , in which the perturbation method is used to find solutions. The perturbation method gives how these solutions are altered for non-zero but small α. In doing so, we assume that the altered solutions can be Taylor expanded in α. In addition, it turns out that the forms of the solutions depend on the sign of V (T 0 ). We calculate the perturbative solutions to O (α) for V (T 0 ) = 0 and discuss the singular behavior of the solutions for V (T 0 ) > 0 and V (T 0 ) < 0. Substituting the Taylor expansions of H (t) and Φ (t) in powers of α H (t) = H 0 (t) + αH 1 (t) + O α 2 , Φ (t) = Φ 0 (t) + αΦ 1 (t) + O α 2 ,(49) into the gravitational equations (7) and (8), one finds dḢ 0 + dH 2 0 − 2Φ 0 = 0, H 0 + dH 2 0 − 2H 0Φ0 = 0, dḢ 1 + 2dH 0 H 1 − 2Φ 1 = F (t) ,(50)H 1 + 2dH 0 H 1 − 2H 1Φ0 − 2H 0Φ1 = G (t) , where we define and Φ + (t) with α = 0 evolve from (to) big bang (big crunch) at t = t 0 , while they have a very weakly string coupled background. However for α > 0, a + (t) goes to infinity at t = t 0 , while the string coupling becomes divergent. Note that these perturbative solution are valid when α (t − t 0 ) −2 ≪ 1. Therefore, the higher order corrections are necessary to study the singular behavior of classical solutions. However, our analysis gives a sense of how quantum effects modify the singular behavior of classical solutions. F (t) ≡ −d R 0 Ḣ 0 + H 2 0 + 2R 0Φ0 −Ṙ 0 H 0 , G (t) ≡ −R 0 Ḣ 0 + dH 2 0 + 4R 0Φ Α 0 Α 1 Α 1 10 5 0 5 10 t For V (T 0 ) > 0, the leading terms of the solutions are H ± 0 (t) = ± C √ d 1 sinh C (t − t 0 ) , Φ ± 0 (t) = 1 2 ± √ d − 1 ln sinh C 2 (t − t 0 ) − 1 2 ± √ d + 1 ln cosh C 2 (t − t 0 ) + Φ 0 ,(55) where C = 2V (T 0 ), and t 0 and Φ 0 are integration constants. For V (T 0 ) < 0, the leading terms are H ± 0 (t) = ± C √ d 1 sin C (t − t 0 ) , Φ ± 0 (t) = 1 2 ± √ d − 1 ln sin C 2 (t − t 0 ) − 1 2 ± √ d + 1 ln cos C 2 (t − t 0 ) + Φ 0 ,(56) where C = −2V (T 0 ), and t 0 and Φ 0 are integration constants. These solutions have a singularity at t = t 0 . Around t = t 0 , their singular behaviors are the same as in the case with V (T 0 ) = 0. Therefore, the last two equations in eqns. (50) give that the singular behaviors of H 1 (t) and Φ 1 (t) at t = t 0 are also the same as in the case with V (T 0 ) = 0. V. CONCLUSION In [12], the tachyon-induced rolling solutions have been considered using the low-energy effective field equations, which were derived from the effective action (1) with f (R) = R. To gain some insight into quantum effects on the tachyon dynamics, in this paper we investigated the behavior of the classical solutions of the low-energy effective action (1) of the graviton-dilaton-tachyon system, in which quantum corrections are included only in f (R) for simplicity. After the equations of motion were obtained in section II, we solved them for the tachyon-induced rolling ansatzes (16) and (17) in section III. Finally, more classical solutions were discussed in section IV. In [12] where f (R) = R, it showed that H (t) vanished identically for the tachyon-induced rolling solutions. For more general forms of f (R), we found in subsection III A that there were some cases in which H (t) could be nonzero. Moreover, we solved the equations of motion assuming H (t) = H 0 which was a constant. When F (R 0 ) = 1 and H 0 = 0, the properties of classical solutions have been discussed in [12], and it has been found that the dilaton always rolled toward stronger coupling. However for F (R 0 ) > 1, we found that there existed some solutions in which the string coupling could always stay finite. In the case of f (R) = R+ α 2 R 2 , we also solved the equations of motion assuming T (t) = T 0 , whose scenario is related to the possible final state of bulk tachyon condensation. It turned out for some solutions that higher order terms in f (R) could dramatically change their singular behavior. Since we only used an effective model, the f (R) gravity theory, to investigate quantum effects on tachyon dynamics, one might not take our analysis too seriously. However, our analysis suggests that quantum corrections should be important to understand tachyon dynamics. Appendix A: Appendix On defining G 1 ≡ F (R) dḢ + dH 2 +Ṫ 2 − 2Φ + 2 (F (R) − 1)Φ −Ḟ (R) dH,(A1)G 2 ≡ F (R) Ḣ + dH 2 − 4 [F (R) − 1]Φ 2 + 4ΦḞ (R) + 2 [F (R) − 1]Φ +2 [F (R) (d − 1) − d] HΦ −F (R) − (d − 1) HḞ (R) ,(A2)G 3 ≡ 1 2 [F (R) R − f (R)] − 2 {d [F (R) − 1] + 1}Φ 2 + 2dΦḞ (R) (A3) + {d [F (R) − 1] + 1} Φ + dHΦ − 1 2 d F (R) + dHḞ (R) + V (T ) , G 4 ≡T + dH − 2Φ Ṫ + V ′ (T ) ,(A4) the equations of motion (7), (8), (11), and (12) become G i = 0. Supposing that G 1 = G 2 = G 3 = 0, we now show that this leads to G 4 = 0 wheneverṪ = 0. Differentiating G 1 = 0 with respect to time giveṡ TT = −Ḟ (R) dḢ + dH 2 2 − F (R) dḦ + 2dHḢ 2 + ... Φ − [F (R) − 1]Φ −Ḟ (R) 2 dḢ − [F (R) − 1]Φ +Ḟ (R)Φ −F (R) 2 dH,(A5) where we use R = 2dḢ + d (d + 1) H 2 . By multiplying G 4 byṪ , eliminatingṪT through eqn. (A5) andṪ 2 through the equation of motion G 1 = 0, we finḋ T G 4 − dH − 2Φ G 1 − 1 2 dG 1 dt = −Ḟ (R) dḢ + dH 2 2 − F (R) dḦ + 2dHḢ 2 + ... Φ − [F (R) − 1]Φ −Ḟ (R) 2 dḢ − [F (R) − 1]Φ +Ḟ (R)Φ −F (R) 2 dH (A6) − dH − 2Φ F (R) dḢ + dH 2 − 2Φ + 2 (F (R) − 1)Φ −Ḟ (R) dH + V ′ (T )Ṫ = 0. Differentiating G 3 = 0 with respect to time giveṡ F (R) R 2 − 4 [d (F (R) − 1) + 1]ΦΦ − 2dḞ (R)Φ 2 + 2dΦḞ (R) + 2dΦF (R) + dḞ (R) Φ + dHΦ + {d [F (R) − 1] + 1} ... Φ + dḢΦ + dHΦ − 1 2 d ... F (R) + dḢḞ (R) + dHF (R) = −V ′ (T )Ṫ .(A7) Differentiating G 2 = 0 with respect to time and multiplying by d/2 on both sides gives Comparing eqn. (A10) with eqn. (A4), we obtain the relation for G 1 , G 2 , G 3 and G 4 dH − 2Φ G 1 + 1 2 d [F (R) − 1] ... Φ − d 2 ... F (R) − 4d [F (R) − 1]ΦΦ − 2dḞ (R)Φ 2 + 2dΦḞ (R) +2dΦF (R) + dḞ (R)Φ + d 2 HΦḞ (R) − d 2 2ḢḞ (R) −dG 1 dt − d 2 dG 2 dt − dHG 2 + dG 3 dt −Ṫ G 4 = 0,(A11) which shows that G 4 = 0 if G 1 = G 2 = G 3 = 0 andṪ = 0. The instabilities associated with open string tachyon is well understood presently. The stable vacuum is the vacuum of closed string without open string excitations and D-branes. ( 10 ) 10fails to require that H → 0 when t → −∞. There might exist possible solutions with H (t) which does not go to zero when t → −∞. Thus, one might consider a more general ansatz for H (t): From eqns.(16), one obtains that t 1 = 1 and φ 1 = 0. SinceḢ, F (R) ∼ O (e γt ) anḋT 2 , Φ ∼ O (e 2γt ) ,all terms of eqn. (9) except 1 2F (R) are at O (e 2γt ). GivenF (R) = 2dβ 1 h 1 γ 3 e γt + O (e 2γt ), one has h 1 = 0 since β 1 = 0 by assumption. After the initial conditions h 1 = 0, φ 1 = 0, and t 1 = 1 are obtained, one could use the recurrence relations If Nγ + (d + 1) H 0 = 0 and H 0 − N (L − 1) γ = 0 for any positive integer N, eqn. (37) gives α L = 0, which means H (t) = H 0 . However, f (R) blows up at R = R 0 , and hence the tachyon rolling ansatz does not solve the equations of motion. If Nγ + (d + 1) H 0 = 0 or H 0 − N (L − 1) γ = 0 some N, the coefficient of e N (L−1)γt in eqn. (37) is zero without making α L = 0. In principle, one could put the ansatz into the equations of motion andsolve for h n , φ n , and t n . However, one could not obtain the recurrence relations as in the F (R 0 ) = 0 case. One might need other means to find values of h n , φ n ,and t n . . We are still interested in the form of solutions in eqns. (5), in which the metric is the spatially flat FRW metric. The equations of motion then reduce to eqns. (7), (8), (11), and (12). Note that if V (T ) = V (−T ), the solutions are invariant under the "time-reversal" transformations t → −t, for which FIG. 1 : 1Plots of Φ + (τ ) and V (T ) for F (R 0 ) = 2, where we have C Φ = 0. F (R 0 ) 0> 1, that the solution T + (t) in eqn. (46) describes the scenario in which the tachyon field is at a maximum of V (T ) at t = −∞ and begins to roll down V (T ) afterwards. At t = ∞, T + (t) goes to −∞, and V (T ) is given in eqn. (48). Moreover, this tachyon rolling down scenario is free of singularities since the Ricci scalar is constant, and the string coupling is always finite. B. Constant T If T = T 0 such that V ′ (T 0 ) = 0, the tachyon equation (12) becomes trivial. As a result, one could solve the gravitational equations FIG. 2 : 2Plots of a ± (t) and Φ ± (t), where we have d = 3, α = 1, t 0 = 0, Φ 0 = 0, a 0 = 1 and c = 0.where a 0 is a constant. The O (α) corrections would not change the asymptotic behavior of the solutions at t = ±∞. Around t = t 0 , the O (α) corrections could dramatically change the asymptotic behavior of the these solutions. We plot a ± (t) and Φ ± (t)in FIG. 2, wherewe have d = 3, α = 1, t 0 = 0, Φ 0 = 0, a 0 = 1 and c = 0. For example, the solutions a + (t) = [F (R) (d − 1) − d]ḢΦ − d [F (R) (d − 1) − d] HΦ − dḢḞ (dHΦḞ (R) − d 2 F (R) HḢ + dḞ (R) dH 2 (F (R) − 1)Φ −F (R) + F (R) Ḣ + dH 2 − (d − 1) HḞ (R) (A10) −4 [F (R) − 1]Φ 2 + 4ΦḞ (R) + 2 [F (R) (d − 1) − d] HΦ . TABLE I for d < 6 and TABLE II for d ≥ 6, where we define TABLE I : IThe sign of T ′2 ± for d < 6. TABLE II : IIThe sign of T ′2 ± for d ≥ 6. 0 − 4Φ 0Ṙ0 − 2R 0Φ0 − 2 (d − 1) R 0 H 0Φ0 +R 0 + (d − 1) H 0Ṙ0 .(51) AcknowledgmentsWe are grateful to Jun Tao and Zheng Sun for useful discussions. This work is supported in part by NSFC(Grant No. 11005016, 11175039 and 11375121).If V (T 0 ) = 0, solving eqns. (50) and using eqn.(11)to constrain the integration constants, one finds that the solutions to O (α) arewhere c and Φ 0 and are integration constants, and we haveIntegrating H ± (t), we find The Pre-big bang scenario in string cosmology. M Gasperini, G Veneziano, arXiv:hep-th/0207130Phys. Rept. 373M. Gasperini and G. Veneziano, "The Pre-big bang scenario in string cosmology," Phys. Rept. 373, 1 (2003) [arXiv:hep-th/0207130]. String gas cosmology. T Battefeld, S Watson, arXiv:hep-th/0510022Rev. Mod. Phys. 78435T. Battefeld and S. Watson, "String gas cosmology," Rev. Mod. 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Quant. Grav. 3035005gr-qcC. Escamilla-Rivera, G. Garcia-Jimenez, O. Loaiza-Brito and O. Obregon, "Closed String Tachyon: Inflation and Cosmological Collapse," Class. Quant. Grav. 30, 035005 (2013) [arXiv:1110.6223 [gr-qc]]. Order alpha-prime (Two Loop) Equivalence of the String Equations of Motion and the Sigma Model Weyl Invariance Conditions: Dependence on the Dilaton and the Antisymmetric Tensor. R R Metsaev, A A Tseytlin, Nucl. Phys. B. 293385R. R. Metsaev and A. A. Tseytlin, "Order alpha-prime (Two Loop) Equivalence of the String Equations of Motion and the Sigma Model Weyl Invariance Conditions: Dependence on the Dilaton and the Antisymmetric Tensor," Nucl. Phys. B 293, 385 (1987). A New Type of Isotropic Cosmological Models Without Singularity. A A Starobinsky, Phys. Lett. B. 9199A. A. Starobinsky, "A New Type of Isotropic Cosmological Models Without Singularity," Phys. Lett. B 91, 99 (1980). Curvature quintessence. S Capozziello, arXiv:gr-qc/0201033Int. J. Mod. Phys. 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{'abstract': 'To study quantum effects on the bulk tachyon dynamics, we replace R with f (R) in the lowenergy effective action that couples gravity, the dilaton, and the bulk closed string tachyon of bosonic closed string theory and study properties of their classical solutions. The α ′ corrections of the graviton-dilaton-tachyon system are implemented in the f (R). We obtain the tachyon-induced rolling solutions and show that the string metric does not need to remain fixed in some cases.The singular behavior of more classical solutions are investigated and found to be modified by quantum effects. In particular, there could exist some classical solutions, in which the tachyon field rolls down from a maximum of the tachyon potential while the dilaton expectation value is always bounded from above during the rolling process.', 'arxivid': '1610.03952', 'author': ['Peng Wang pengw@scu.edu.cn†wu˙houwen@g.harvard.edu‡hyanga@scu.edu.cn \nCenter for Theoretical Physics\nCollege of Physical Science and Technology\nSichuan University\n610064ChengduPR China\n', 'Houwen Wu \nCenter for Theoretical Physics\nCollege of Physical Science and Technology\nSichuan University\n610064ChengduPR China\n\nCenter for the Fundamental Laws of Nature\nHarvard University\n02138CambridgeMAUSA\n', 'Haitang Yang \nCenter for Theoretical Physics\nCollege of Physical Science and Technology\nSichuan University\n610064ChengduPR China\n'], 'authoraffiliation': ['Center for Theoretical Physics\nCollege of Physical Science and Technology\nSichuan University\n610064ChengduPR China', 'Center for Theoretical Physics\nCollege of Physical Science and Technology\nSichuan University\n610064ChengduPR China', 'Center for the Fundamental Laws of Nature\nHarvard University\n02138CambridgeMAUSA', 'Center for Theoretical Physics\nCollege of Physical Science and Technology\nSichuan University\n610064ChengduPR China'], 'corpusid': 118398977, 'doi': '10.1088/1475-7516/2018/05/034', 'github_urls': [], 'n_tokens_mistral': 16849, 'n_tokens_neox': 14172, 'n_words': 7916, 'pdfsha': '9d50f8df910c9bac276abf843a06a2e0cda27aa4', 'pdfurls': ['https://arxiv.org/pdf/1610.03952v1.pdf'], 'title': ['Closed String Tachyon Driving f (R) Cosmology', 'Closed String Tachyon Driving f (R) Cosmology'], 'venue': []}

arxiv

Exploration and Visualization in the Web of Big Linked Data: A Survey of the State of the Art * 29 Jan 2016 Nikos Bikakis Timos Sellis Swinburne Univ. of Technology NTU Athens & Australia Athena R C Greece Timos Sellis Swinburne Univ. of Technology NTU Athens & Australia Exploration and Visualization in the Web of Big Linked Data: A Survey of the State of the Art * 29 Jan 2016Visual analyticsbig data challengesdata explorationlarge databasesvisual explorationsemantic webvisualization toolsscalability Data exploration and visualization systems are of great importance in the Big Data era. Exploring and visualizing very large datasets has become a major research challenge, of which scalability is a vital requirement. In this survey, we describe the major prerequisites and challenges that should be addressed by the modern exploration and visualization systems. Considering these challenges, we present how state-of-the-art approaches from the Database and Information Visualization communities attempt to handle them. Finally, we survey the systems developed by Semantic Web community in the context of the Web of Linked Data, and discuss to which extent these satisfy the contemporary requirements. INTRODUCTION The purpose of data exploration and visualization is to offer ways for information perception and manipulation, as well as knowledge extraction and inference [68,56]. Data visualization 1 provides users with an intuitive means to explore the content of the data, identify interesting patterns, infer correlations and causalities, and supports sense-making activities. Data exploration and visualization systems are of great importance in the Big Data era, in which the volume and heterogeneity of available information make it difficult for humans to manually explore and analyse data. Most traditional systems cannot handle the large size of many contemporary datasets. Exploring and visualizing large datasets has become a major research challenge [24,119,55,103,140,49]. Therefore, modern systems have to take into account scalability, as a vital requirement. Dealing with scalability, modern systems have to address numerous issues related to storage, access, rendering/presentation, interaction, etc. In the Web of Data (WoD) context, following the abundance of Linked Data, several recent efforts have offered tools and techniques for exploration and visualization in many different domains * This paper appears in 6th International Workshop on Linked Web Data Management (LWDM 2016). 1 Throughout this paper we use the term "visualization" referring to visual data exploration. [35]. However, most of these approaches fail to take into account issues related to performance and scalability. In this work, we describe the major requirement and challenges that should be addressed by the modern exploration and visualization systems. Additionally, we refer to state-of-the-art approaches from the Database and Information Visualization communities, which attempt to handle some of these challenges. Further, we describe the systems that have been developed in the context of WoD, and discuss to which extent they satisfy the contemporary requirements. CHALLENGES Most traditional exploration and visualization systems operate in an offline way, limited to accessing static sets of preprocessed data. Additionally, they restrict themselves to dealing with small dataset sizes, which can be easily handled and explored with conventional data management and (visual) explorations techniques. On the other hand, nowadays, the Big Data era has realized the availability of the great number and variety of very large datasets that are dynamic in nature. For example, most data sources offer query or API endpoints for online access and updates; in other cases (e.g., scientific databases), new data is constantly arrived (e.g., on a daily/hour basis). Beyond these, modern systems should operate on exploratory context. In an exploration scenario, it common that users are interesting in finding something interesting and useful without previously know what exactly are searching for, until the time they identify it. In this case, users perform a sequence of operations (e.g., queries), in which the result of each operation determine the formulation of the next operation. Finally, an increasingly large number of diverse users (i.e., different preferences, skills, etc.) explore and analyse data in a plethora of different scenarios. Therefore, some of the major challenges that should be dealt with by modern systems, are posed by the: (1) Large size and the dynamic nature of data in conjunction with the exploration-driven setting; and (2) Variety of tasks and users. Large & Dynamic Data in Exploration-driven Setting. One of the major challenges in exploration and visualization is related to the size that characterizes most contemporary datasets. A second challenge is related to the availability of query and API endpoints for online data access and retrieval, as well as the cases where that data is received in a stream fashion. The later pose the challenge of handling large sets of data in a dynamic setting, and as a result, a preprocessing phase (e.g., traditional indexing) is prevented. In this respect, modern visualization and exploration systems must be able to efficiently and effectively handle billion objects dynamic datasets throughout an exploratory scenario. Therefore, scalable and efficient data structures and algorithms have to be developed. Crucial issues related to storage, access, management, presentation, interaction (e.g., pan, zoom, search, drill-down), etc. over large dynamic datasets have to be handled. Scalability has become a major challenge for the modern systems. Beyond these, systems have to efficiently operate on machines with limited computational and memory resources (e.g., laptops). In a "conventional" setting (e.g., explore a small fragment of a preprocessed dataset), most of the aforementioned issues can be handled by the traditional systems that provide database exploration and analysis, such as Tableau 2 (previously know as Polaris [124]), DEVise [98], Spotfire [3], VisDB [81], Lumira 3 , QlikView 4 , Datawatch 5 , etc. However, in a "modern" setting, when a large part (or the whole) of a billion objects dynamic dataset has to be explored, the aforementioned traditional database-oriented systems cannot be adopted. In conjunction with performance issues, modern systems have to address challenges related to visual presentation and interaction issues. Particularly, systems should be able to present, as well as, offer ways to "easily" explore large datasets. Handling a large number of data objects is a challenging task; modern systems have to "squeeze a billion records into a million pixels" [119]. Even, in much smaller datasets, offering a dataset overview is extremely difficult; in both cases information overloading is a common issue. As aslo stated in the visual information seeking mantra: "overview first, zoom and filter, then details on demand" [118], gaining overview is crucial in the visual exploration scenario. Based on the aforementioned, it follows that a basic requirement of the modern systems is to develop methods that provide summaries and abstractions over the enormous number of data objects. In order to tackle both performance and presentations issues, a large number of systems adopt approximation techniques (a.k.a. data reduction techniques) in which partial results are computed. Considering the existing approaches, most of them are based on: (1) sampling and filtering [46,105,2,69,17]; or/and (2) aggregation (e.g., binning, clustering) [42,25,74,73,97,138,96,1,15,71]. In this respect, some modern database-oriented systems adopt approximation techniques using query-based approaches (e.g., query translation, query rewriting) [17,74,73]. In order to improve efficiency several systems adopt incremental (a.k.a. progressive) techniques. In these techniques the results/visual elements are computed/constructed incrementally based on user interaction or as time progresses (e.g., [123,25]). Numerous recent systems integrate incremental and approximate techniques, in these approaches, approximate answers are computed incrementally over progressively larger samples of the data [46,2,69]. The dynamic setting prevents modern systems from preprocessed the data. Additionally, it is common in exploration scenarios only a small fragment of data to be accessed by the user. In this context, an adaptive indexing approach [67] is used in [144], where the indexes are created incrementally and adaptively throughout exploration. Similarly, in [25] the hierarchy tree is incrementally constructed based on user's interaction. Finally, in some approaches, parallel architectures are adopted; e.g., [41,78,77,69]. To sum up, modern systems should provide scalable techniques that on-the-fly effectively (i.e., in a way that can be easily explored) handle a large number of data objects over an exploration scenario, using a limited number of resources Variety of Tasks & Users. The requirement of scalable, on-the-2 tableau.com 3 sap-lumira.com 4 clickview.com 5 datawatch.com fly exploration and analysis must be coupled with the diversity of preferences and requirements posed by different users and tasks. Therefore, the modern systems should provide the user with the ability to customize the exploration experience based on her preferences and the requirements posed by the examined task. For example, systems should allow the user to: (1) organize data into different ways, according to the type of information or the level of detail she wishes to explore (e.g., [25]); (2) modify approximation criteria, thresholds, sampling rates, etc. (e.g., [78]); (3) define her own operations for data manipulation and analysis (e.g., aggregation, statistical, filtering functions), etc. Furthermore, systems should automatically adjust their parameters, by taking into account the environment setting (e.g., screen resolution, memory size) [74,25,73]. Beyond the personalization, modern systems should provide mechanisms that assist the user and reduce the effort needed on their part. In this direction, several approaches have been recently developed. In what follows, we mention some of the most common ones. Several systems assist users by recommending visualization that seems to be more useful or capture surprising and/or interesting data; e.g., [139,134,82]. Other approaches help users to discover interest areas in the dataset; by capturing user interests, they guide her to interesting data parts; e.g., [37]. Finally, in other cases systems provide explanations regarding data trends and anomalies; e.g., [141]. EXPLORATION & VISUALIZATION SYSTEMS This section reviews works related to exploration and visualization in the WoD. A large number of works studying issues related to WoD visual exploration and analysis have been proposed in the literature [35,101,4]. In what follows, we classify these works into the following categories: (1) Browsers and exploratory systems (Section 3.1), (2) Generic visualization systems (Section 3.2), (3) Domain, vocabulary & device-specific visualization systems (Section 3.3), (4) Graph-based visualization systems (Section 3.4), (5) Ontology visualization systems (Section 3.5), and (6) Visualization libraries (Section 3.6). Browsers & Exploratory Systems WoD browsers have been the first systems developed for WoD utilization and analysis [35,4]. Similarly to the traditional ones, WoD browsers provide the functionality for link navigation and representation of WoD resources and their properties; thus enabling browsing and exploration of WoD in a most intuitive way. WoD browsers mainly use tabular views and links to provide navigation over the WoD resources. Haystack [111] is one of the first WoD browsers, it exploits stylesheets in order to customize the data presentation. Similarly, Disco 6 renders all information related to a particular RDF resource as HTML table with property-value pairs. Noadster [113] performs property-based data clustering in order to structure the results. Piggy Bank [66] is a Web browser plug-in, that allows users to convert HTML content into RDF. LESS [13] allows users to create their own Web-based templates in order to aggregate and display WoD. Tabulator [21] another WoD browser, additionally provides maps and timeline visualizations. LENA [87] provides different views of data, following user's criteria that are expressed as SPARQL queries. Visor [110] provides a multi-pivot approach for exploring graphs, allowing users to explore multiple nodes at a time, as well as to connect points of interest. Finally, in the context SynopsViz [26,25] 2014 N, T, H C, P, T, TL ✓ ✓ ✓ ✓ ✓ ✓ generic Web Vis Wizard [131] 2014 N, T, S B, C, M, P, PC, SG ✓ ✓ generic Web LinkDaViz [129] 2015 N, T, S B, C, S, M, P ✓ ✓ generic Web ViCoMap [112] 2015 , Humboldt [86] and gFacet [57] provide faceted navigation over WoD resources. Explorator [7] is a WoD exploratory tool that allows users to browse a dataset by combining search and facets. VisiNav [53] is a system that allows users to pose expressive exploratory-based queries. The system is built on top of following concepts: keyword search, object focus, path traversal, and facet selection. Information Workbench (IWB) [52] is a generic platform for semantic data management offering several back-end (e.g., triple store) and front-end tools. Regarding the front-end, IWB offers a flexible user interface for data exploration and visualization. Marbles 7 formats RDF triples using the Fresnel vocabulary (a vocabulary for rendering RDF resources as HTML). Also, it retrieves information about a resource by accessing Semantic Web indexes and search engines. Finally, URI Burner 8 is a service which retrieves data about resources. For the requested resources, it generates an RDF graph by exploiting existing ontologies and other knowledge from the Web. Generic Visualization Systems In the context of WoD visual exploration, there is a large number of generic visualization frameworks, that offer a wide range of visualization types and operations. Next, we outline the best known systems in this category. In Table 1 we provide an overview and compare several generic visualization systems. The Year column presents the released date. The Data Types column specifies the supported data types. The Vis. Types column presents the types of visualizations that are provided. The Recomm. column indicates systems that offer recommendation mechanisms for visualization settings (e.g., appropriate visualization type, visualization parameters). The Preferences column captures the ability of the users to apply data (e.g., filter, aggregate) or visual (e.g., increase abstraction) operations. The Statistics column captures the provision of statistics about the visualized data. The Sampling column indicates systems that exploit techniques based on sampling and/or filtering. The Aggregation column indicates systems that exploit techniques based on aggregation (e.g., binning, clustering). The Incr. column indicates systems that adopt incremental techniques; i.e., the results/visualization are computed/generated based on user interaction or as time progresses. Finally, the Disk column indicates systems that use external memory (e.g., file, database) to perform operations during runtime (i.e., not just initially load data from disk). 7 mes.github.io/marbles 8 linkeddata.uriburner.com Rhizomer [30] provides WoD exploration based on a overview, zoom and filter workflow. Rhizomer offers various types of visualizations such as maps, timelines, treemaps and charts. VizBoard [135,136,109] is an information visualization workbench for WoD build on top of a mashup platform. VizBoard presents datasets in a dashboard-like, composite, and interactive visualization. Additionally, the system provides visualization recommendations. Payola [84] is a generic framework for WoD visualization and analysis. The framework offers a variety of domain-specific (e.g., public procurement) analysis plugins (i.e., analyzers), as well as several visualization techniques (e.g., graphs, tables). In addition, Payola offers collaborative features for users to create and share analyzers. In Payola the visualizations can be customized according to ontologies used in the resulting data. The Linked Data Visualization Model (LDVM) [29] provides an abstract visualization process for WoD datasets. LDVM enables the connection of different datasets with various kinds of visualizations in a dynamic way. The visualization process follows a four stage workflow: Source data, Analytical abstraction, Visualization abstraction, and View. LDVM considers several visualization techniques, e.g., circle, sunburst, treemap, etc. Finally, the LDVM has been adopted in several use cases [85]. Vis Wizard [131] is a Webbased visualization system, which exploits data semantics to simplify the process of setting up visualizations. Vis Wizard is able to analyse multiple datasets using brushing and linking methods. Similarly, Linked Data Visualization Wizard (LDVizWiz) [11] provides a semi-automatic way for the production of possible visualization for WoD datasets. In a same context, LinkDaViz [129] finds the suitable visualizations for a give part of a dataset. The framework uses heuristic data analysis and a visualization model in order to facilitate automatic binding between data and visualization options. Balloon Synopsis [117] provides a WoD visualizer based on HTML and JavaScript. It adopts a node-centric visualization approach in a tile design. Additionally, it supports automatic information enhancement of the local RDF data by accessing either remote SPARQL endpoints or performing federated queries over endpoints using the Balloon Fusion service [116]. Balloon Synopsis offers customizable filters, namely ontology templates, for the users to handle and transform (e.g., filter, merge) input data. LODWheel [126] is a Web-based visualizing tool which combines JavaScript libraries (e.g., MooWheel, JQPlot) in order to visualize RDF data in charts and graphs. SemLens [59] is a visual tool that combines scatter plots and semantic lenses, offering visual discovery of correlations and patterns in data. Objects are arranged in a scatter plot and are analysed using user-defined semantic lenses. ViCoMap [112] combines WoD statistical analysis and visualization, in a Web-based tool, which offers correlation analysis and data visualization on maps. Finally, SynopsViz [26,25] is a Web-based visualization tool built on top of a generic tree-based model. The adopted model performs a hierarchical aggregation, allowing efficient personalized multilevel exploration over large datasets. In order to provide scalability under different exploration scenarios, the model offers a method that incrementally constructs the hierarchy based on user's interaction, as well as a method that enables dynamic and efficient adaptation of the hierarchy to the user's preferences. Domain, Vocabulary & Device-specific Visualization Systems In this section, we present systems that target visualization needs for specific types of data and domains, RDF vocabularies or devices. Several systems focus on visualizing and exploring geo-spatial data. Map4rdf [92] is a faceted browsing tool that enables RDF datasets to be visualized on an OSM or Google Map. Facete [122] is an exploration and visualization tool for SPARQL accessible data, offering faceted filtering functionalities. SexTant [20] and Spacetime [133] focus on visualizing and exploring time-evolving geo-spatial data. The LinkedGeoData Browser [121] is a faceted browser and editor which is developed in the context of LinkedGeo-Data project. Finally, in the same context DBpedia Atlas [132] offers exploration over the DBpedia dataset by exploiting the dataset's spatial data. Furthermore, in the context of linked university data, VISUalization Playground (VISU) [6] is an interactive tool for specifying and creating visualizations using the contents of linked university data cloud. Particularly, VISU offers a novel SPARQL interface for creating data visualizations. Query results from selected SPARQL endpoints are visualized with Google Charts. A variety of systems target multidimensional WoD modelled with the Data Cube vocabulary. CubeViz [43,114] is a faceted browser for exploring statistical data. The tool provides data visualizations using different types of charts (i.e., line, bar, column, area and pie). The Payola Data Cube Vocabulary [60] adopts the LDVM stages [29] in order to visualize RDF data described by the Data Cube vocabulary. The same types of charts as in CubeViz are provided in this tool. The OpenCube Toolkit [75] offers several tools related to statistical WoD. For example, OpenCube Browser explores RDF data cubes by presenting a two-dimensional table. Additionally, the OpenCube Map View offers interactive map-based visualizations of RDF data cubes based on their geo-spatial dimension. The Linked Data Cubes Explorer (LDCE) [79] allows users to explore and analyse statistical datasets. Finally, [106] offers several map and chart visualizations of demographic, social and statistical linked cube data. Regarding device-specific systems, DBpedia Mobile [18] is a location-aware mobile application for exploring and visualizing DBpedia resources. Who's Who [32] is an application for exploring and visualizing information focusing on several issues that appear in the mobile environment. For example, the application considers the usability and data processing challenges related to the small display size and limited resources of the mobile devices. Graph-based Visualization Systems A large number of systems visualize WoD datasets adopting a graph-based (a.k.a., node-link) approach [102]. In Table 2 we provide an overview and compare several graph-based visualization systems. Table 2 is structured in a similar way to Table 1. Additionally, in this table the Keyword column indicates systems that provide keyword search functionality. The Filter column indicates systems that provide mechanisms for data filtering. Note that, Table 2 also includes the ontology visualization systems (Section 3.5) that follow a node-link approach (indicated by using the term "ontology" in the Domain column). RelFinder [58] is a Web-based tool that offers interactive discovery and visualization of relationships (i.e., connections) between selected WoD resources. Fenfire [54] and Lodlive [31] are exploratory tools that allow users to browse WoD using interactive graphs. Starting from a given URI, the user can explore WoD by following the links. LODeX [19] is a tool that generates a representative summary of a WoD source. The tool takes as input a SPARQL endpoint and generates a visual (graph-based) summary of the WoD source, accompanied by statistical and structural information of the source. IsaViz [108] allows users to zoom and navigate over the RDF graph, and also it offers several "edit" operations (e.g., delete/add/rename nodes and edges). In the same context, graphVizdb [23,22] is built on top of spatial and database techniques offering interactive visualization over very large (RDF) graphs. ZoomRDF [142] employs a space-optimized visualization algorithm in order to increase the number of resources which are displayed. Trisolda [38] proposes a hierarchical RDF graph visualization. It adopts clustering techniques in order to merge graph nodes. Paged Graph Visualization (PGV) [36] utilizes a Ferris-Wheel approach to display nodes with high degree. RDF graph visualizer [115] adopts a node-centric approach to visualize RDF graphs. Rather than trying to visualize the whole graph, nodes of interest (i.e., staring nodes) are discovered by searching over nodes labels; then the user can interactively navigate over the graph. RDF-Gravity 9 visualizes RDF and OWL data. It offers filtering, keyword search and editing the graph layout. Also, the nodes can be displayed in different colors and shapes based on their RDF types. A different approach has been adopted in [127], where sampling techniques have been exploited. Finally, Gephi [15] is a generic tool that offers several visualization and analysis features over graph data. Ontology Visualization Systems The problems of ontology visualization and exploration have been extensively studied in several research areas (e.g., biology, chemistry). In what follows we focus on graph-based ontology visualization systems that have been developed in the WoD context [47,40,51,91,80]. In most systems, ontologies are visualized following the node-link paradigm [100,99,64,104,27,45,65,94,5,89,125] 10,11 . On the other hand, CropCircles [137] uses a geometric containment approach, representing the class hierarchy as a set of concentric circles. Furthermore, hybrids approaches are adopted in other works. Knoocks [88] combines containment-based and node-link approaches. In this work, ontologies are visualized as nested blocks where each block is depicted as a rectangle containing a sub-branch shown as tree map. Finally, OntoTrix [14] and NodeTrix [61] use node-link and adjacency matrix representations. Visualization Libraries Finally, there is a variety of Javascript libraries which allow WoD visualizations to be embedded in Web pages. Sgvizler [120] is a JavaScript wrapper for visualizing SPARQL results. Sgvizler allows users to specify SPARQL Select queries directly into HTML elements. Sgvizler uses Google Charts to generate the output, offering numerous visualizations types such as charts, treemaps, graphs, DISCUSSION In this section we discuss to which extent the systems developed in the WoD context fulfilled the nowadays requirements, focussing on performance and scalability issues, availability of personalized services facilities for assisting users through exploration. As previously mentioned, most of WoD exploration and visualization systems do not handle issues related to performance and scalability. They basically adopt traditional techniques in order to handle small sets of data. As we can observe from Table 1, generic systems support several types of data (e.g., numeric, temporal, graph, spatial) and provide a plethora of visualization types. Additionally, an increasing number of recent systems (e.g., LinkDaViz, Vis Wizard, LDVizWiz, LDVM) focus on providing recommendation mechanisms. Particularity, these systems mainly recommend the most suitable visualization technique by considering the type of input data. Regarding visual scalability, as we can see in Table 1, none of the systems, with the exceptions of SynopsViz and VizBoard cases, adopt approximation techniques (i.e., sampling/filtering, aggregation). Hence, the existing approaches assume that all the examined data objects can be presented on the screen and handled by traditional visualization techniques. Due to this assumption, the current systems restrict their applicability to small sets of data. In conjunction with the limited visual scalability, most of the existing systems (except for SynopsViz) do not exploit external memory during runtime. Particularly, they initially load all the examined objects in main memory, assuming that the main memory is large enough. An alternative approach is adopted by the SynopsViz system, which incrementally retrieves data and generates visualiza-tions based on user interaction. As a result, each time, only a part of the examined dataset needs to be loaded in main memory. The graph-based exploration and visualization systems are presented in Table 2. These systems are of great importance in WoD, due to the graph structure of the RDF data model. Although several systems offer sampling or aggregation mechanisms, most of these systems load the whole graph in main memory. Given the large memory requirements of graph layout algorithms in order to draw a large graph, the current WoD systems are restricted to handle small sized graphs. In order to be able to handle large graphs, modern WoD systems should adopt more sophisticated techniques similar to those proposed by the information visualization community. Particularly, state-of-the-art systems for exploring large graphs utilize hierarchical aggregation approaches where the graph is recursively decomposed into smaller sub-graphs (in most cases using clustering and partitioning) that form a hierarchy of abstraction layers [93,10,95,9,8,1,143,12,15,71,130]. Other approaches adopt edge bundling techniques which aggregate graph edges to bundles [48,44,107,90,34,63]. Beyond hierarchical approaches, WoD systems should also consider disk-based implementations, such as [22,1,72,127,130]. To sum up, WoD community should consider scalability and performance as vital requirements for the development of the future exploration and visualization systems. Handing large datasets is crucial in the Big Data era. Therefore, in what follows we summarize some possible directions for the future WoD exploration and visualization systems. Approximation techniques such as sampling and aggregation that have been widely used in systems from database and information visualization communities, have to be adopted and adjusted to WoD data and requirements. Systems should be integrated with disk structures, retrieving data dynamically during runtime. Also caching and prefetching techniques may be exploited; e.g., [128,76,70,16,33,83,39]. Data structures and indexes should be developed focusing on WoD tasks and data, such as Nanocubes [96] in the context of spatio-temporal data exploration, and HETree [25] in numeric and temporal datasets. Finally, considering users' perspective, beyond visualization recommendations, modern WoD systems should provide more sophisticated mechanisms that capture users' preferences and assist them throughout large data exploration and analysis tasks. c 2016 , 2016Copyright is with the authors. Published in the Workshop Proceedings of the EDBT/ICDT 2016 Joint Conference (March 15, 2016, Bordeaux, France) on CEUR-WS.org (ISSN 1613-0073). Distribution of this paper is permitted under the terms of the Creative Commons license CCby-nc-nd 4.0 LWDM '16 March 15, 2016, Bordeaux, France ⋆ N: Numeric, T: Temporal, S: Spatial, H: Hierarchical (tree), G: Graph (network) ⋆⋆ B: bubble chart, C: chart, CI: circles, G: graph, M: map, P: pie, PC: parallel coordinates, S: scatter, SG: streamgraph, T: treemap, TL: timeline, TR: tree of faceted browsing, /facet [62] Table 1 : 1Generic Visualization Systems System Year Data Types ⋆ Vis. Types ⋆⋆ Recomm. Preferences Statistics Sampling Aggregation Incr. Disk Domain App. TypeRhizomer [30] 2006 N, T, S, H, G C, M, T, TL ✓ Table 2 : 2Graph-based Visualization Systemstimelines, etc. Visualbox[50] provides an environment where users can build and debug SPARQL queries in order to retrieve WoD; then, a set of visualization templates is provided to visualize results. Visualbox uses several visualization libraries like Google Charts and D3[28], offering 14 visualization types.System Year Keyword Filter Sampling Aggregation Incr. Disk Domain App. Type RDF-Gravity 9 2003 ✓ ✓ generic Desktop IsaViz [108] 2003 ✓ ✓ generic Desktop RDF graph visualizer [115] 2004 ✓ generic Desktop GrOWL [89] 2007 ✓ ✓ ✓ ontology Desktop NodeTrix [61] 2007 ✓ ontology Desktop PGV [36] 2007 ✓ ✓ generic Desktop Fenfire [54] 2008 generic Desktop Gephi [15] 2009 ✓ ✓ ✓ generic Desktop Trisolda [38] 2010 ✓ ✓ ✓ generic Desktop Cytospace [127] 2010 ✓ ✓ ✓ ✓ ✓ generic Desktop FlexViz [45] 2010 ✓ ✓ ontology Web RelFinder [58] 2010 generic Web ZoomRDF [142] 2010 ✓ ✓ ✓ generic Desktop KC-Viz [104] 2011 ✓ ontology Desktop LODWheel [126] 2011 ✓ ✓ generic Web GLOW [64] 2012 ✓ ✓ ontology Desktop Lodlive [31] 2012 ✓ generic Web OntoTrix [14] 2013 ✓ ✓ ontology Desktop LODeX [19] 2014 ✓ ✓ generic Web VOWL 2 [100, 99] 2014 ontology Web graphVizdb [23, 22] 2015 ✓ ✓ ✓ ✓ generic Web www4.wiwiss.fu-berlin.de/bizer/ng4j/disco semweb.salzburgresearch.at/apps/rdf-gravity 10 protegewiki.stanford.edu/wiki/OntoGraf 11 protegewiki.stanford.edu/wiki/OWLViz ASK-GraphView: A Large Scale Graph Visualization System. 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{'abstract': 'Data exploration and visualization systems are of great importance in the Big Data era. Exploring and visualizing very large datasets has become a major research challenge, of which scalability is a vital requirement. In this survey, we describe the major prerequisites and challenges that should be addressed by the modern exploration and visualization systems. Considering these challenges, we present how state-of-the-art approaches from the Database and Information Visualization communities attempt to handle them. Finally, we survey the systems developed by Semantic Web community in the context of the Web of Linked Data, and discuss to which extent these satisfy the contemporary requirements.', 'arxivid': '1601.08059', 'author': ['Nikos Bikakis \nTimos Sellis Swinburne Univ. of Technology\nNTU Athens &\nAustralia\n', 'Athena R C Greece \nTimos Sellis Swinburne Univ. of Technology\nNTU Athens &\nAustralia\n'], 'authoraffiliation': ['Timos Sellis Swinburne Univ. of Technology\nNTU Athens &\nAustralia', 'Timos Sellis Swinburne Univ. of Technology\nNTU Athens &\nAustralia'], 'corpusid': 8434895, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 20990, 'n_tokens_neox': 18738, 'n_words': 9934, 'pdfsha': 'a979fec7e4803af4068cd83f75eb15bb09d79989', 'pdfurls': ['https://arxiv.org/pdf/1601.08059v1.pdf'], 'title': ['Exploration and Visualization in the Web of Big Linked Data: A Survey of the State of the Art *', 'Exploration and Visualization in the Web of Big Linked Data: A Survey of the State of the Art *'], 'venue': []}

arxiv

A new absolute magnitude calibration with 2MASS for cataclysmic variables 18 Jan 2007 T Ak Faculty of Sciences Department of Astronomy and Space Sciences Istanbul University 34119 University IstanbulTurkey S Bilir Faculty of Sciences Department of Astronomy and Space Sciences Istanbul University 34119 University IstanbulTurkey S Ak A Retter Faculty of Sciences Department of Astronomy and Space Sciences Istanbul University 34119 University IstanbulTurkey P.O. Box 426460850ShohamIsrael A new absolute magnitude calibration with 2MASS for cataclysmic variables 18 Jan 20079780Gm stars: cataclysmic binaries9710Vm stars: distancesparallaxes Using reliable trigonometric measurements, we find that the absolute magnitude of cataclysmic variables depends on the orbital period and de-reddened (J − H) 0 and (H −K s ) 0 colours of 2MASS (Two Micron All Sky Survey) photometric system. The calibration equation covers the ranges 0.032 d < P orb ≤ 0.454 d , −0.08 < (J − H) 0 ≤ 1.54, −0.03 < (H − K s ) 0 ≤ 0.56 and 2.0 < M J < 11.7; It is based on trigonometric parallaxes with relative errors of (σ π /π) ≤ 0.4. By using the period-luminositycolours (PLCs) relation, we estimated the distances of cataclysmic variables with orbital periods and 2MASS observations and compared them with distances found from other methods. We suggest that the PLCs relation can be a useful statistical tool to estimate the distances of cataclysmic variables. Introduction Cataclysmic variables (hereafter referred to as CVs) are short period interacting binary stars in which a red-dwarf, the secondary star, overflows its Roche lobe and transfers matter to a white dwarf typically via an accretion disc. A bright spot is formed in the location where the matter stream impacts the accretion disc. In CVs that have strongly magnetic white dwarf pimaries the accreting matter can not construct an accretion disc, instead, the accretion is maintained through accretion columns above the magnetic poles of the white dwarf. For a detailed description of the CV phenomenon and its subclasses see Warner (1995) and Hellier (2001). Although distances of CVs are needed to improve and constrain physical models, reliable distance measurements can be only found for a few systems. Different methods have been used to measure CV distances (Thorstensen 2003). However, all but trigonometric parallaxes yield rough distance estimates. The most promising method for determining CV distances has been to make use of the properties of the secondary star (Bailey 1981;Sproats, Howell & Mason 1996). This method assumes that all K-band emission originates from the secondary star. Almost all secondary stars in CVs lie on or near the ZAMS (Zero Age Main Sequence) for near-solar metallicity within the uncertainties (Warner, 1995;Beuermann et al., 1998;Beuermann, 2000;Kolb & Baraffe, 2000). However, if one tries to measure the absolute magnitude of the secondary star, contributions to the light from the other components contaminate the results. Therefore, only a lower limit to the distance can be obtained by using this method due to the effects of the disc and the irradiated area of the secondary star (Berriman, Szkody & Capps 1985). Almost all distance estimation methods attempts to use properties of a component of the system such as surface brightness of the secondary star (Bailey 1981), spectra of the white dwarf (Sion et al. 1995;Urban & Sion 2006) and M V − P orb relationship of dwarf novae at outburst (Warner 1995;Harrison et al. 2004). However, none of them can give a distance as precise as trigonometric parallax due to the contaminations from the other components and the lack of the information about the individual contributions of the components to the observed light. It is accepted that the most reliable distances are obtained from trigonometric parallaxes. However, this trigonometric parallax method can only be applied to the closest objects due to observational constraints. First precise trigonometric parallax measurements of the brightest CVs came from Hipparcos Satellite (Duerbeck 1999). Trigonometric parallaxes of some CVs were measured by using Hubble Space Telecope's Fine Guidence Sensor (McArthur et al. 1999, 2001Beuermann et al. 2003a,b;Harrison et al. 2004), as well. In addition, Thorstensen (2003) measured trigonometric parallaxes of 14 CVs from groundbased observations. A relationship of absolute magnitude in any wavelength interval with the orbital period and at least one colour of the system can be a very useful tool to estimate distances of binary stars. This method was applied for W UMatype binary stars (Rucinski & Duerbeck 1997;Rucinski 2004). If there is such a relationship for CVs as well, their distances can be statistically estimated to a certain precision. For this, 2MASS magnitudes and colours can be a good choice since most of the light in these photometric bands comes from the secondary star and the effect of the interstellar reddening for J, H and K bands is weaker than that in visual wavelengths. Although we can not measure the absolute magnitude of the secondary star to a good precision, at least, we know that the secondary star in a CV can be considered as the least-active component of the system. We should state that our aim is not to measure the absolute magnitudes of the secondary stars in CVs. In this study, we first estimate the systemic Jband absolute magnitudes M J of the closest CVs using reliable trigonometric parallaxes by assuming that the light in J, H and K bands comes from the system as a whole. Then, we find the dependence of the absolute magnitude on the orbital period P orb and de-reddened colours (J − H) 0 and (H − K) 0 to derive an absolute magnitude calibration for CVs with 2MASS photometric system. The Data Our data sample consists of CVs with trigonometric parallax (π) errors smaller than (σ π /π)≤0.4. The 27 systems listed in Table 1 include CVs with orbital periods shorter than ∼12 hr since a CV with orbital period longer than this limit possibly contains a secondary star on its way to becoming a red giant (Hellier 2001). Dwarf novae and nova-like systems were selected from Duerbeck (1999), McArthur et al. (1999McArthur et al. ( , 2001, Beuermann et al. (2003a,b), Thorstensen (2003) and Harrison et al. (2004). Duerbeck (1999) obtained trigonometric parallaxes of four novae however, we included only two of them in our sample. The orbital period of T CrB (P orb ∼228 days), which is classified as a recurrent nova in Downes et al. (2001) catalogue, is longer than our upper limit. In addition, a comparison of Tables 1 and 2 in Duerbeck (1999) shows that the distance of HR Del evaluated from the shell expansion parallax method is very different from its distance found from trigonometric parallax. Thus, we excluded these two systems from our sample. Table 1 lists the systems used in the analysis. Our data sample consists of 14 dwarf novae, 11 nova-like stars and two novae. These are CVs with the most precise distance estimates ever found in the literature. J, H and K s magnitudes were taken from the Point-Source Catalogue and Atlas (Cutri et al. 2003;Skrutskie et al. 2006) which is based on the 2MASS (Two Micron All Sky Survey) observations. The 2MASS photometric system comprises Johnson's J (1.25 µm) and H (1.65 µm) bands with the addition of K s (2.17 µm) band, which is bluer than Johnson's K-band. Although we study the most closest CVs ever known, the total interstellar absorption in the direction of the star should be taken into account. We used the equations of Fiorucci & Munari (2003) for the determination of the total absorption for J, H and K s bands, i.e. A J = 0.887 × E(B − V ), A H = 0.565 × E(B − V ) and A Ks = 0.382 × E(B − V ), respectively. Fortunately, the E(B − V ) colour excesses of many CVs were estimated by Bruch & Engel (1994) and Harrison et al. (2004). Bruch & Engel's catalogue includes the systems whose E(B − V ) values were given in Verbunt (1987) and La Dous (1991). Our primary E(B − V ) source is Harrison et al. (2004). If we did not find the E(B − V ) value of a star in their study, we returned to Bruch & Engel (1994). Unfortunately, colour excesses of GP Com, GW Lib, EF Eri and V893 Sco were not given in the sources mentioned above. Thus, we calculated their colour excesses from Schlegel, Finkbeiner & Davis (1998) maps by using NASA Extragalactic Database 1 . Since these are relatively close systems, the colour excesses found from Schlegel et al. (1998) need to be reduced according to the stellar distance. In order to do this, we used the E ∞ (B − V ) colour excess in the Galactic latitude (b) and longitude (l) for the model from Schlegel et al. (1998). The total absorption for the model was evaluated from A ∞ (b) = 3.1E ∞ (B − V ). (1) The total absorption for the distance d to the star is calculated as following (Bahcall & Soneira, 1980) A d (b) = A ∞ (b) 1 − exp − | d sin(b) | H ,(2) where H is the scaleheight for the interstellar dust which is adopted as 100 pc. Finally, the colour excess for a star at the distance d is estimated from E d (B − V ) = A d (b) / 3.1.(3) Once we obtained the apparent magnitudes (J, H and K s ), total absorption (A J , A H and A Ks ) and distance d = 1/π for a CV, the absolute magnitudes (M J , M H and M Ks ) of the system were easily calculated from the well known distance-modulus formula, i.e. Table 1. M J = J − 5 log d + 5 − A J . The calculated M J values are listed in Analysis We used the data in Table 1 to derive an absolute magnitude calibration for CVs with 2MASS. In order to find the dependence of the absolute magnitude Table 1 The data sample. Types and orbital periods (P orb ) were taken from Downes et al. (2001). DN, NL and N denote dwarf nova, nova-like star and nova, respectively. J, H and Ks magnitudes were collected from the 2MASS Point Source Catalogue (Cutri et al., 2003). π denotes parallax, E(B − V ) colour excess and M J absolute magnitude in J-band. Name Type were estimated as R = 0.96 and s = 0.88, respectively. It should be emphasized that the calibration equation covers the ranges 0.032 d < P orb ≤ 0.454 d , −0.08 < (J − H) 0 ≤ 1.54, −0.03 < (H − K s ) 0 ≤ 0.56 and 2.0 < M J < 11.7; It is based on trigonometric parallaxes with relative errors (σ π /π) ≤ 0.4. A comparison of the fit values of absolute magnitudes, M Jc , calculated from this period-luminosity-colours (PLCs) relation with the observed M J absolute magnitudes is shown in Figure 1. P orb J J − H H − Ks π σπ/π E(B − V ) M J (days)(mas) Application to other cataclysmic variables We collected the list of cataclysmic variables with E(B − V ) values from Bruch & Engel (1994). Then, we selected systems with orbital periods shorter than 12 hr from Downes et al.'s catalogue 2 (Downes et al. 2001) and with 2MASS observations. Systems in the application limits of the PLCs relation (see, Section 3) are listed in Table 3. We also collected their distances, when available, from the literature. For example, Urban & Sion (2006) give distances of many dwarf novae estimated from the M v − P orb relation. Most of the distances mentioned in Table 3 were taken from Urban & Sion (2006). E(B − V ) values and distances of EZ Del, LL Lyr, RY Ser, CH UMa and SDSS J081327.07+452833.0 were taken from Thorstensen et al. (2004), as well. Distances of several systems were calculated by Sproats et al. (1996) using Bailey's method (Bailey 1981). However, their E(B − V ) values were not listed in Bruch & Engel (1994), except for UV Per and AR And. Among them, we selected the systems located above the Galactic latitude | b | ≥ 20 o and assumed E(B − V ) = 0 for them. This is a reasonable assumption since in these latitudes the interstellar absorption is very small. We have taken the distances from Sproats et al. (1996) calculated under the assumption that the percentage of the K-band light contributed by the secondary star is 50, i.e. K%=50. Table 3 were determined by the shell parallax method (Cohen 1985). For V1500 Cyg, we calculated the average distance of the nova collected from different sources. Selvelli (2004) calculated the E(B − V ) colour excesses and distances of some old novae from the UV spectra. We included these novae with orbital periods and 2MASS observations in Table 3. Finally, Patterson (1984) listed distances of many CVs based on various methods, such as detection of the secondary, proper motion, M v − EW (H β ) relation, interstellar absorption, position in the Galaxy, etc. Note that none of the distances in Table 3 were determined by the trigonometric parallax method. Distances of V1500 Cyg and V533 Her in The absolute magnitudes (M Jc ) for the systems listed in Table 3 were calculated by using the calibration equation given above. The distances (d c ) were evaluated from the distance modulus formula with absolute magnitude M Jc and interstellar absorption A J . Figure 2 compares the distances obtained from the the PLCs relation with those collected from the literature. Discussion We have suggested an absolute magnitude calibration for CVs based on the trigonometric parallaxes and 2MASS observations. The calibration equation covers a wide range of periods and colours. The mean error in the absolute magnitude M J is 0.22. However, there is a considerable scatter for some systems. Large deviations from the PLCs relation are seen mostly in faint systems. The source of the deviations can not be the colour excess E(B − V ), since the deviated systems are located in the middle and higher Galactic latitudes (| b | ≥ 27 o ) and they are relatively close objects. Also, since they are relatively close objects, it is unlikely that they are due to parallax errors. It is therefore more likely that these deviations come from the intrinsic properties of the systems. Although for the systems GW Lib and EF Eri the magnitude flags in 2MASS are (ABC) and (DBC), respectively, which indicates low-quality observations, the fact that the largest deviations come from systems with the faintest absolute magnitudes suggests that here the disc makes a more dominant contribution relative to the donor star than in systems with brighter donor stars. Indeed, the nova-like system EF Eri has possibly a substellar secondary and the deviation of this system from the PLCs relation can then be attributed to its very faint substellar component. Other deviated systems (GW Lib, T Leo, VY Aqr and Z Cam) are all dwarf novae. Disc or donor activity or a third component of the Hellier & van Zyl (2005), v: Prinja et al. (2000), w: Ringwald & Naylor (1998), x: Thorstensen & Taylor (1998), y: Gaensicke et al. (2000), z: Thorstensen et al. (2004), aa: Thoroughgood et al. (2005), bb : Littlefair et. al. (2001), cc: Hessman (1988), dd: Warner (1995), ee: Staude et al. (2001) binary system can affect the magnitude of the system. We used AAVSO's Light Curve Generator 3 to find the activity stage of these systems during the 2MASS observations. Unfortunately, only Z Cam has enough visual observations to conclude that its activity stage. We found that this system was at the beginning of the decline from an outburst during the 2MASS observations. Name Type P orb (d) J J − H H − Ks E(B − V ) M Jc d(pc) dc(pc) (1) (2) (3) (4) (5) (6) (7) (8) (9)(10) As for the non or less-deviating dwarf novae from the PLCs relation, only SU UMa was found in a superoutburst maximum from the AAVSO's Light Curve Generator. Other dwarf novae were either in quiescence (WZ Sge, EX Hya, V893 Sco, YZ Cnc, U Gem, SS Aur, RU Peg) or in the decline branch from an outburst (AH Her and SS Cyg) like Z Cam during the 2MASS obser-vations. These systems are located in or very near the 99% confidence limit without regarding their activity stage. So, it is possible to say that activity stage does not affect the location of a CV in the PLCs relation. However, it seems that finding considerable deviations from the PLCs relation mostly for some dwarf novae is not a coincidence (Interestingly note that the novalike star GP Com, in which a CO white dwarf is acreeting from a helium degenerate (Morales-Rueda et al. 2003), obeys the relation very well). We compared the distances obtained from the PLCs relation with those found by various other methods in Figure 2. Although Figure 2 shows that the distances found from the PLCs relation are generally somewhat smaller than those found by other methods, for the nova-like systems the PLCs relation yields distances longer than other methods. The distance inferred from the trigonometric parallax of HR Del is very different than found from its shell parallax (760 pc, Duerbeck 1999). This is why we did not include this system in our data sample listed in Table 1. Its distance found from the PLCs relation is, bf however, much closer to that obtained from the shell parallax. In view of the above results, we suggest that the PLCs relation can be a useful statistical tool to calculate the distances of CVs from their 2MASS observations since the PLCs relation has been calibrated with the most reliable distance estimation method (trigonometric parallax). Distances calculated from the PLCs relation can give clues for astrometric observations of these systems, as well. Finally, it should be stated that future astrometric observations of CVs such as GAIA and SIM missions, will refine the PLCs relation. (2004), b: Beuermann et al. (2003a), c: McArthur et al. (1999), d: McArthur et al. (2001), e: Beuermann et al. (2003b), f: Thorstensen (2003), g: Duerbeck (1999), h: Bruch & Engel (1994), i: Schlegel et al. (1998), j: La Dous (1991) M J on the orbital period P orb , and colours (J − H) 0 and (H − K s ) 0 , we used a fit equation in the following form M J = a + b log P orb (day) + c (J − H) 0 + d (H − K) 0 , whose least square coefficients and their 1-σ errors are given in Table 2. The subscript '0' indicates de-reddened magnitudes. The M J calibration that utilizes de-reddened colour indices (J − H) 0 and (H − K s ) 0 and the periods can be used to predict individual values within an error of about ± 0.22 mag. The correlation coefficient and standard deviation of the calibration Fig. 1 . 1A comparison of the absolute magnitudes (M Jc ) calculated from the PLCs relation with the estimated M J absolute magnitudes in Table 1. Upper and lower confidence limits of 99% are shown with dotted lines. The bottom panel displays the residuals from the fit. The diagonal line represents the equal values. Largely scattered systems are shown. Fig. 2 . 2A comparison of the distances (d c ) calculated using the absolute magnitudes (M Jc ) from the PLCs relation, with these distances (d) estimated from other methods. The diagonal line represents the equal values. The symbols •, • and + denote dwarf novae, nova-like systems and novae respectively. The type of the cataclysmic variable J0813+4528 is unknown and it is shown by × in the figure. Table 2 2Coefficients of the calibration equation.Coefficient a b c d -0.894 -5.721 2.598 7.380 σ ±0.522 ±0.705 ±0.610 ±1.711 Table 3 3Absolute magnitudes M Jc and distances d c calculated from the PLCs relation found in this study. Columns 1-7 are as inTable 1. The letter d denotes distances collected from the literature. The type of the cataclysmic variable J0813+4528 is unknown. Table 3 -continued 3a: assumed value (see text), b:Sproats et al. (1996), c: Bruch & Engel (1994, d:Urban & Sion (2006), e: Selvelli (2004), f: Saito & Baptista (2006) g: Nadalin & Sion (2001) h: Ramsay et al. (2004), i: Verbunt et al. (1997) j: Wagner et al. (1998) k: Wood et al. (1992) l:Patterson (1984), m:Gerke et al. (2006), n:Taylor et al. (1998), o:Gaensicke & Koester (1999), p:Groot et al. (2001), q: Esenoglu (1997, r:Cohen (1985), s:Duerbeck (1981), t:Gill & O'Brien (2000), u:Name Type P orb (d) J J − H H − Ks E(B − V ) M Jc d(pc) dc(pc) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) TW Vir DN 0.1827 13.35 0.27 0.14 0 c 5.07 500 d 452 +77 −94 DQ Her N 0.1936 13.60 0.32 0.20 0.05 e 5.35 363 e 437 +94 −119 UX UMa NL 0.1967 12.76 0.35 0.14 0.02 c 5.08 340 l 340 +67 −83 V345 Pav NL 0.1981 12.15 0.39 0.09 0 a 4.80 295 +52 −62 BT Mon N 0.3338 14.40 0.44 0.24 0.20 e 4.28 1914 e 972 +286 −406 FO Aqr NL 0.2021 12.87 0.13 0.24 0 c 5.16 349 +74 −93 T Aur N 0.2044 16.14 0.28 0.16 0.30 e 4.31 991 e 772 +108 −127 V446 Her N 0.207 15.39 0.59 0.11 0.25 e 4.80 1762 e 1185 +224 −277 RX And DN 0.2099 12.45 0.71 0.19 0.02 c 6.16 200 d 180 +55 −78 HR Del N 0.2142 12.32 0.05 0.06 0.16 e 3.12 673 e 648 +28 −28 PQ Gem NL 0.2164 13.49 0.29 0.20 0 a 5.18 459 +108 −142 HL CMa DN 0.2168 11.64 0.19 0.22 0 c 4.99 80 l 214 +47 −60 AY Psc DN 0.2173 14.52 0.50 0.02 0 a 4.31 565 b 1101 +179 −214 EZ Del DN 0.2234 15.08 0.30 0.08 0.16 z 3.82 1000 z 1679 +212 −243 V347 Pup NL 0.2319 13.13 0.65 0.19 0.05 aa 5.73 470 aa 296 +89 −126 DO Leo DN 0.2345 15.93 0.73 -0.02 0 a 4.45 878 b 1971 +397 −496 TX Col NL 0.2383 13.63 0.26 0.20 0.05 c 4.73 591 +135 −175 AH Eri DN 0.2391 15.92 0.58 0.40 0 a 7.12 160 b 576 +232 −390 LL Lyr DN 0.2491 15.42 0.53 0.25 0.06 z 5.63 960 z 887 +277 −402 XY Ari NL 0.2527 16.14 1.88 0.90 3.7 bb 5.93 270 bb 243 +82 −123 TZ Per DN 0.2629 13.09 0.58 0.11 0.27 c 4.17 435 d 545 +116 −147 BV Pup DN 0.265 13.34 0.45 0.10 0 c 4.30 630 d 642 +145 −188 TT Crt DN 0.2684 13.87 0.48 0.21 0 d 5.16 500 d 554 +165 −235 V426 Oph DN 0.2853 11.00 0.50 0.17 0.08 c 4.58 202 cc 186 +51 −69 J0813+4528 CV 0.289 15.94 0.68 0.11 0.05 z 4.64 2100 z 1779 +499 −695 EM Cyg DN 0.2909 11.74 0.40 0.18 0.03 c 4.50 350 d 277 +75 −102 AC Cnc NL 0.3005 13.08 0.38 0.10 0 c 3.84 400 l 704 +161 −208 RY Ser DN 0.3009 13.70 0.63 0.18 0.39 z 4.21 620 z 673 +174 −235 V363 Aur NL 0.3212 13.30 0.33 0.16 0.13 c 3.69 900 dd 790 +187 −245 V1309 Ori NL 0.3326 14.22 0.56 0.15 0 a 4.40 550 ee 920 +279 −399 CH UMa DN 0.3432 12.71 0.50 0.17 0.06 z 4.17 480 z 498 +146 −208 SY Cnc DN 0.38 11.27 0.22 0.13 0 c 3.04 300 l 443 +104 −137 Q Cyg N 0.4202 13.54 0.29 0.14 0.44 c 2.12 2188 e 1610 +306 −376 http://nedwww.ipac.caltech.edu/forms/calculator.html http://archive.stsci.edu/prepds/cvcat/ http://www.aavso.org/data/lcg/ AcknowledgmentsWe thank the anonymous referee for a thorough report and useful comments that helped improving an early version of the paper. 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{'abstract': 'Using reliable trigonometric measurements, we find that the absolute magnitude of cataclysmic variables depends on the orbital period and de-reddened (J − H) 0 and (H −K s ) 0 colours of 2MASS (Two Micron All Sky Survey) photometric system. The calibration equation covers the ranges 0.032 d < P orb ≤ 0.454 d , −0.08 < (J − H) 0 ≤ 1.54, −0.03 < (H − K s ) 0 ≤ 0.56 and 2.0 < M J < 11.7; It is based on trigonometric parallaxes with relative errors of (σ π /π) ≤ 0.4. By using the period-luminositycolours (PLCs) relation, we estimated the distances of cataclysmic variables with orbital periods and 2MASS observations and compared them with distances found from other methods. We suggest that the PLCs relation can be a useful statistical tool to estimate the distances of cataclysmic variables.', 'arxivid': 'astro-ph/0701531', 'author': ['T Ak \nFaculty of Sciences\nDepartment of Astronomy and Space Sciences\nIstanbul University\n34119 University\nIstanbulTurkey\n', 'S Bilir \nFaculty of Sciences\nDepartment of Astronomy and Space Sciences\nIstanbul University\n34119 University\nIstanbulTurkey\n', 'S Ak ', 'A Retter \nFaculty of Sciences\nDepartment of Astronomy and Space Sciences\nIstanbul University\n34119 University\nIstanbulTurkey\n\nP.O. Box 426460850ShohamIsrael\n'], 'authoraffiliation': ['Faculty of Sciences\nDepartment of Astronomy and Space Sciences\nIstanbul University\n34119 University\nIstanbulTurkey', 'Faculty of Sciences\nDepartment of Astronomy and Space Sciences\nIstanbul University\n34119 University\nIstanbulTurkey', 'Faculty of Sciences\nDepartment of Astronomy and Space Sciences\nIstanbul University\n34119 University\nIstanbulTurkey', 'P.O. Box 426460850ShohamIsrael'], 'corpusid': 15409754, 'doi': '10.1016/j.newast.2007.01.002', 'github_urls': [], 'n_tokens_mistral': 12526, 'n_tokens_neox': 10082, 'n_words': 5124, 'pdfsha': '3d23562e0466a3dfb982642065faeb75e75d685f', 'pdfurls': ['https://arxiv.org/pdf/astro-ph/0701531v1.pdf'], 'title': ['A new absolute magnitude calibration with 2MASS for cataclysmic variables', 'A new absolute magnitude calibration with 2MASS for cataclysmic variables'], 'venue': []}

arxiv

VegaProf: Profiling Vega Visualizations J / Yang Vegaprof J Yang University of Washington WAUSA Sigma Computing A Bäuerle Sigma Computing CAUSA D Moritz Carnegie Mellon University PAUSA Ç Demiralp Sigma Computing CAUSA VegaProf: Profiling Vega Visualizations CCS ConceptsHuman-centered computing → Visualization toolkits;Software and its engineering → Domain specific languages; loadData() -340ms initScales() -60ms renderMark() -10ms 1 -Flame Graph 3 -Dataflow Graph 2 -DSL SpecificationFigure 1: VegaProf records low-level execution times and encodes them in coordinated visualizations corresponding to different abstraction levels of Vega's DSL execution. (1) A flame graph provides an overview of where the major amount of time is spent. (2) In contrast to previous debugging approaches, which relied on trial and error-based changes of the visualization specification, VegaProf maps performance measures directly to the Vega specification. (3) To analyze performance on different levels, VegaProf augments the dataflow graph with performance measures.AbstractVega is a popular domain-specific language (DSL) for visualization specification. At runtime, Vega's DSL is first transformed into a dataflow graph and then functions to render visualization primitives. While the Vega abstraction of implementation details simplifies visualization creation, it also makes Vega visualizations challenging to debug and profile without adequate tools. Our formative interviews with three practitioners at Sigma Computing showed that existing developer tools are not suited for visualization profiling as they are disconnected from the semantics of the Vega DSL specification and its resulting dataflow graph. We introduce VegaProf, the first performance profiler for Vega visualizations. VegaProf effectively instruments the Vega library by associating the declarative specification with its compilation and execution. Using interactive visualizations, VegaProf enables visualization engineers to interactively profile visualization performance at three abstraction levels: function, dataflow graph, and visualization specification. Our evaluation through two use cases and feedback from five visualization engineers at Sigma Computing shows that VegaProf makes visualization profiling tractable and actionable. Introduction Domain-specific languages (DSLs) such as Vega [Veg22a], gglpot2 [Wic16], and D3 [BOH11] simplify and accelerate visualization creation by abstracting implementation details from practitioners. Using intermediate representations (IRs), DSLs also open opportunities for automated optimizations, which otherwise require deep domain knowledge and programming expertise. For example, Vega parses a visualization specification that is written in JSON-whose format has been informed by the grammar of graphics [Wil12]-into a dataflow graph. The resulting dataflow graph, in turn, triggers the execution of low-level visualization rendering functions. This successive creation of IRs to execute DSL code is also called the lowering process. arXiv:2212.13670v1 [cs.HC] 28 Dec 2022 However, the abstraction that DSLs provide through the lowering process comes at a price. One limitation that Vega trades in for its simplicity is the limited access to code execution [HSH15]; there is no direct mapping between the DSL specification, the dataflow graph, and measured function performance. Since Vega specifications are parsed and then finally rendered by the underlying framework, there is currently no way of connecting the elements defined in the DSL and the functions executed to generate visualization primitives. Vega is often developed in a browser environment, which typically provide integrated profiling instruments. However, because of the missing connection between Vega's IRs, these instruments cannot provide such a multi-level performance trace. In turn, one might know which rendering function slowed down the visualization drawing, but this knowledge cannot be mapped back to parts of the DSL or its resulting dataflow graph. As a result of this missing link between performance measures and elements of the DSL, we found that visualization engineers often have to use trial-and-error procedures to fix their performance problems. This calls for dedicated visualization profiling tools that operate under the premise of DSLs, visualization parsers, and their rendering engines. To support visualization engineers discover and resolve performance problems of their Vega visualizations, we present VegaProf (video demo available online), the first profiler for the Vega visualization grammar. We enable such debugging capabilities by tracing execution time measured at a function execution level all the way back to the DSL, creating a bidirectional mapping between performance issues on the function level, the dataflow graph, and the user-defined or computer-generated DSL code. Based on this mapping, we provide timings for DSL segments, nodes of the resulting dataflow graph, as well as rendering functions generated from dataflow graph nodes. In addition to the mere recording of performance measures, we further present interactive profiling visualizations within a familiar developer tool, the Vega Editor. Bridging the gap of current profiling tools through a direct-manipulation visualization interface [HHN85], practitioners can trace performance bottlenecks from rendering functions all the way back to the DSL. Our evaluation with five visualization engineers shows that this form of performance tracing and visualization introduces means to reason about and resolve performance bottlenecks in a way that was not possible before. VegaProf is available as open-source software on GitHub. In summary, we contribute VegaProf, an interactive profiler enabling visualization practitioners to quickly explore the time performance of Vega visualizations at different IRs of the Vega DSL. We demonstrate VegaProf's usage workflow with two use cases of visualization-based profiling and evaluate VegaProf by eliciting feedback from five visualization engineers through a user study. Our findings from the user study show the necessity for and usability of VegaProf. In addition, these findings indicate the effectiveness of encoding performance measurements in a flame graph visualization linked to the underlying visualization specification. While VegaProf is the first profiler dedicated to Vega, the presented concepts could be applied to any visualization DSL. We release VegaProf as open-source software to support future research and applications. Related Work Our work relates to prior research on both visualization debugging and dataflow system profiling. VegaProf adds a new element to both lines of earlier work as it is the first time-performance profiler for DSL-specified visualizations. Through interactive visualizations, VegaProf enables profiling across all underlying abstraction levels of the DSL. Debugging Visualizations While data visualization has a long history of tools [KWHH17, WQM * 17, MWN * 18] and grammars [Veg22a, SMWH16, BOH11] for visualization specification, research into linting and debugging visualizations is nascent. McNutt and Kindlmann [MK18] introduce one of the first visualization linters. Their linter checks a predefined set of rules on a given visualization and returns the list of failed rules with explanations. This postprocessing approach is disconnected from the development workflow and does not localize errors for rendered visualizations directly in their specifications. In contrast, VisuaLint [HCS20] annotates visualizations in situ with red marks. These marks are akin to conventional linting-error visualizations in IDEs, but cannot be traced back to the visualization specification. To rectify defective visualization designs, VizLinter [CSX * 21] highlights flaws directly in the visualization specification. VizLinter maps flaws to the DSL code while suggesting potential fixes. Since interactions can be particularly hard to debug, Hoffswell et al. [HSH16] propose debugging techniques specifically designed for reactive visualizations. Prior to their work, users could only use the JavaScript console to traverse the system internals, which required existing knowledge and was hard to track changes. To provide the needed detail to the visualization engineer, they track state through interactions and map that to a visual debugging interface. As these works focus on discovering errors in the visualization specification rather than providing performance insights, they target a different problem space than VegaProf. With VegaProf, we provide the first performance profiler for visualizations specified with a DSL. Profiling Dataflow Systems A bidirectional coupling of the dataflow graph with the underlying DSL code and visualization rendering functions is central to interactive profiling in VegaProf. Dataflow graphs are a common abstraction used by myriad tools and DSLs across domains beyond data visualization (e.g., PyTorch, TensorFlow, Spark, Flink, Naiad, SQL). Earlier work presents profiling tools to help discover performance issues in dataflow systems [GLB20]. For example, Perfopticon [MHHH15] shows the runtime distribution of individual query operators and per-worker execution traces. Similar to our approach, Perfopticon also maps the profiling result to user input. [YGP20] embeds neural network training performance predictions in the code editor to help machine learning practitioners tune hyperparameters. Umlaut [SHH21], another profiling instrument for the deep learning domain, provides heuristics and error messages by analyzing the program structure and model behavior. Mapping performance directly to code has been a common paradigm in recent research [CLRG19]. However, none of the approaches discussed in this paragraph target visualization engineers or consider dataflow graphs as a profiling entity. Beischl et al. [BKB * 21] propose a multi-level performance profiling technique specifically for dataflow-based systems. We build on this approach and adopt it for data visualization, where different abstraction levels and, more importantly, user-facing specification code need to be considered. Additionally, we provide an interactive, visualization-based interface for mitigating performance problems, whereas Beischl et al. focus on the technical aspects of performance profiling. Formative Interviews To assess the needs of visualization engineers, we interviewed 3 professional visualization engineers at Sigma Computing. For all participants, the programmatic generation of Vega visualizations is part of their daily work. We conducted and recorded our interviews in a semi-structured manner via online video conferencing software. A list of predefined topics and questions was covered, while at the same time leaving room for open discussion. We specifically asked our interviewees about their performance optimization needs, their current practice and tools, and potential improvements. Performance Optimization We discovered that performance issues "they usually have large impacts" (P1). Our interviewees attributed this to the fact that performance issues are currently hard to localize, debug, and fix. This manifests in the fact that "performance issues are often neglected to fix" (P2) and "once they find the causes, they always tell the customers not to perform such operations" (P1). Furthermore, visualization engineers typically "limit the data input to Vega spec to be less than 25k [data points] to prevent a lot of slow rendering issues" (P3). Moving forward, these cannot be the go-to solutions, especially since interviewees "have seen a lot of questions regarding visualization performance" (P1). Current Tooling To reason about poor performance, visualization engineers typically "have zoom meetings with customers to talk about problematic visualizations" (P1). Then, they often "simulate the configurations" (P1 and P3) and test them in a sandbox environment. As such, they do not have specialized tooling for performance debugging, but instead rely on "the Vega Editor in combination with Chrome's devtools" (All participants). However, the problem with this is that "devtools can only tell you that the issues are caused by Vega function calls, but it can't help with locating them in the spec" (P1). Potential Improvements When asked about what tooling could improve their situation, interviewees asked for a "breakdown of the transforms, mark rendering, etc. that can immediately indicate which lines [in the DSL] caused the issue" (P3). Thus, what professional visualization engineers are asking for is a mapping from performance issues back to the visualization specification and the IRs of the underlying DSL representation. Therefore, adequate developer tooling for performance profiling must surface this mapping effectively. Design Goals Based on our insights from the aforementioned formative interviews, we distilled the following requirements for a successful Vega performance profiler: Bidirectional timing mapping. To help visualization engineers discover the root cause of performance bottlenecks, it is not sufficient to just measure function execution times. Instead, to be able to make informed decisions about performance optimizations, practitioners need a bidirectional mapping that connects these timings to the DSL, its associated dataflow graph, and individual rendering function calls. Multi-level profiling insights. Once a bidirectional mapping of timings between function execution and the DSL exists, practitioners need to be able to investigate the results of this mapping. Visualization instruments that surface timing measurements can support this investigation. Using such visualizations, practitioners can trace performance measurements through the IRs used in the lowering pipeline of a DSL. Familiar development environment. Finally, we want to support visualization engineers in a familiar environment. visualization engineers adopt a tool only if the burden of entry does not outweigh its benefits. Therefore, we want to provide these visual insights for visualization performance debugging in a familiar environment, in our case, the Vega Editor [Veg22b]. Bidirectional Profiling Map Like other profiling instruments, we measure function execution times. To do this, we hook into how Vega parses specifications and instantiates dataflow operators and record the runtime when operators' evaluation functions are executed. While recording function execution times is well-established for time-profiling, effective profiling instruments for a visualization DSL rely on a bidirectional mapping of execution time measurements and DSL segments with semantic meanings. Only with such a mapping can visualization engineers put performance bottlenecks in the context of the IRs that the DSL gets transformed into (cf. Figure 2). To realize such a mapping, we annotate the nodes of the dataflow graph description as they are created when parsing the visualization specification. This way, we are also able to reverse this mapping, associating dataflow graph nodes with the corresponding lines of DSL code. Once the dataflow graph description is transformed into a dataflow runtime where the nodes represent functions to be DSL Specification Dataflow Graph Function Execution "type": "rect", "from": {"data":"table"}, "encode": { ... loadData() -340ms initScales() -60ms renderMark() -10ms Lowering Annotation of lower levels. Backtracking Mapping execution time to higher levels. Figure 2: During the lowering process, the DSL specification is parsed into a computation graph and then functions to be evaluated eventually. We hook into this lowering process and add annotations that indicate which element on a higher-level IR corresponds to what part of the lowered representation, e.g., what part of the spec corresponds to which data flow graph nodes. Based on these annotations, we can track measured function evaluation times back to higher levels, i.e., assigning time-measurements to the nodes of the dataflow graph and visualization specification. evaluated, we further annotate these functions with the respective dataflow graph nodes to realize such a mapping for this second lowering process. This way, our measurements of function execution time can be mapped back to the node of the dataflow graph that triggered the execution. By chaining the aforementioned annotations, we are able to assign the execution of individual functions not only to nodes in the dataflow graph but also to lines and segments of DSL code. Using this bidirectional mapping, execution times can be traced from the function level all the way to the highest level of operation, namely the DSL specification for the visualization. Altogether, this directly addresses our first design goal of creating a bidirectional mapping between Vega's IRs. While we only use this inter-IR indexing approach to provide better profiling instruments, it could be helpful for other introspection tools such as educating about Vega's lowering process or dataflow debugging as well. In theory, this bidirectional profiling map can be visualized and analyzed in any environment. We integrated this bidirectional mapping into a well-established visualization development tool, the Vega Editor. In the following section, we explain how this technology is used for visually supporting visualization engineer's performance analysis needs. Visual Performance Inspection On the basis of the information obtained through the aforementioned bidirectional mapping of profiling results, we provide a visual interface that enables visualization engineers to take action and improve the performance of their visualization designs. We implemented this interactive performance profiling interface as an extension to the Vega Editor to place visualization engineers in a familiar environment. A new performance tab provides a performance flame graph (cf. Figure 3 (B)) and augments the dataflow graph (cf. Figure 3 (C)) as well as the DSL specification editor (cf. Figure 3 (A)). All three of those components are connected through brushing and linking techniques using our bidirectional profiling map (cf. Section 5) as the underlying data source. This way, we map selections, mouseover events, and zoom transitions that happen in one of the three views to the other two. Throughout our visualizations, such interactions are indicated through blue highlights. Hereby, hovered items are assigned a semi-transparent blue highlight, whereas selected items are highlighted in full blue consistently across all visualizations. Aside from these main views, the Vega Editor further displays the visualization that results from the provided specification. If the visualization is interactive, the resulting profiling and operator states from interaction events are recorded as multiple pulses. The first pulse marks the initial rendering of the visualization. Subsequent pulses are added whenever the visualization needs to be updated based on user interaction. Pulses can be selected from the pulse table, updating the flame graph and dataflow graph to only show the operators being re-evaluated and their timings. In addition, pulses augment the dataflow graph via the node tooltips by providing insights of how data changes in the individual nodes along every pulses. By default, we show profiling results for the initial rendering pulse for both static and interactive charts. This way, visualization engineers can use our visualizations and profiling results not only to debug the initial rendering process but also to improve interaction performance. Directly above the pulse selection, we prominently show the total runtime of the selected pulse. This way, users have an anchor to put all the timings in the visual interface into context. DSL Specification Editor The DSL specification editor is prominently positioned at the left edge of the Vega Editor (cf. Figure 3 (A)), marking a natural entry point for visualization engineers. It represents the highest level of abstraction for VegaProf, directly connecting performance profiles to the DSL code that defines the visualization. Since this level can be directly influenced by visualization engineers, it is often where their time-performance analysis begins. To map function execution times to blocks of the DSL specification, we consider different levels of ranges in the specification. These blocks directly map to JSON's hierarchical object structure in the Vega specification. For example, a user would specify both the x-axis and y-axis blocks under the axis block. In Vega, these blocks are the units that visualization engineers would associate with visual components. Hence, this is the level at which they would make edits, e.g., changing the type of mark that is used for rendering or how data is mapped to these marks. As such, this way of clustering parts of the DSL naturally aligns with how Vega users understand and modify the specifications. Hovering over one of these blocks of DSL code highlights the respective specification segment. When clicking on such a high- lighted block of code, it is selected for further inspection. As mentioned at the beginning of this Section, highlights and selections are transferred to the according elements in the dataflow graph and the performance flame graph. Maybe even more importantly, we also implemented the reverse linking directions from the flame graph and the dataflow graph. As a result, one can easily interpret, the high-level responsibility of a certain element of the flame chart and the dataflow graph by inspecting the highlighted block in the specification. Meanwhile, visualization engineers get insight into how much time individual blocks of the visualization specification require during rendering. For example, investigating elements that require a large portion of the rendering time in the flame graph scrolls to the corresponding code segment of the DSL and highlights it. This makes such performance measures insightful and actionable, as visualization engineers can make direct adjustments to relevant parts of the DSL code. A -DSL Specification Editor B -Performance Flame Graph C -Dataflow View Dataflow View As a first IR of the Vega visualization grammar, the DSL specification provided by the visualization engineer is transformed into a dataflow graph. With the dataflow view, our visual performance inspection interface also allows for analysis at this more detailed level. The dataflow view contains a visualization of the parsed dataflow graph that Vega's DSL gets transformed into (cf. Figure 3 (C)). It can be color-coded by node type or, more conveniently for performance analysis, by node runtime. We use D3's interpo-lateReds color scale to encode node runtime since red is often used as an alarm color, again drawing attention to the nodes that are most performance-intensive during rendering. Whenever a node is selected from this graph visualization, the dataflow graph gets transformed to show only the subgraph with connections to the selected node based on dependency. A zoom-in animation further puts the focus on selected nodes. If a selection comes from any other visualization, such as the DSL editor or the performance flame graph, we analyze the nodes that are involved in the selected subset of performance analysis elements and employ a filtering and zooming similar to the one for direct node selection. This interaction concept further embraces the combined analysis of Vega's different IRs, similar to how interaction with the DSL specification editor is mapped to all other visualizations. To directly surface the most time-consuming nodes, VegaProf further includes a table of all nodes, positioned next to the dataflow graph. This table is based on the dataflow graph and ordered by node execution time, placing the most performance-intensive nodes at the top. With this ordering, this tabular visualization can be used as an entry-point of the analysis on the dataflow level as it guides the visualization engineer's attention directly to nodes of interest. Performance Flame Graph Positioned directly above the dataflow graph, the performance flame graph (cf. Figure 3 (B)) functions as an intermediate representation between the dataflow graph and the DSL specification editor. The flame graph is defined by its different levels of aggregation, going from coarse performance elements to more fine-grained ones. To symbolize this aggregation structure, coarser levels are colored in grey and light blue, whereas a dark blue coloring is employed for the most detailed performance analysis levels. The most detailed level in this flame graph directly represents nodes of the dataflow graph. However, the flame graph also visualizes the hierarchical structure of the Vega DSL at its higher levels, connecting the two other views in one visualization. Hovering over and selecting elements in the flame graph works just as it does in our other visualizations. The flame graph additionally zooms into selected elements to provide more detailed information about a selection. Similar to the dataflow graph, this zooming and highlighting might also be triggered by events from other visualizations. Altogether, the flame graph, with its different levels of performance aggregation and linked interaction concepts, serves as a bridge between both the specification editor and the dataflow view. Use Cases This section describes two example use cases for VegaProf. It demonstrates how VegaProf can help visualization engineers in discovering and resolving performance problems of their Vega visualizations. These use cases highlight how connecting different IRs help debug performance and, specifically, how a direct linking of performance bottlenecks to Vega's specification make such analyses actionable. Visualization Design Decisions Mary is a visualization engineer in the data analysis team of a large airline. She wants to analyze the effect of flight distance on the delay of flights based on a dataset that contains information on three million flights. She considers a scatter plot to visualize the data. When she specifies the scatter plot in Vega, she notices that the visualization she has created is too slow to be usable. Having heard of VegaProf, Mary loads her data and visualization specification in the Vega Editor and analyzes the performance of her visualization. Through an investigation of the Flame Graph, she immediately notices that mark rendering takes most of the overall visualization generation time. Looking at the connected location in the visualization specification, she notices that rendering individual scatter marks for millions of flights is just too slow to sustain interactivity. Since Mary is looking after general trends rather than individual flights anyway, she decides to get rid of these marks and instead render a heatmap for binned results. Next, Mary notices that loading the data was rather slow. Using the dataflow graph, she locates an operation that copies part of the data during the transformation stage. As the relevant part of the Vega specification is highlighted when she hovers the corresponding dataflow node, she identifies the problem and is able to modify the transformation code so that data processing becomes much more performant. After these modifications, Mary notices that while much faster than before, data processing is still her main performance bottleneck. The final step she could take is to pre-aggregate data instead of binning it at present time. However, since the data frequently changes, she decides against it and accepts the initial loading time because of the data transformation. Offloading Computational-Intensive Operations Alice works as a visualization engineer for a software company. Her team is responsible for implementing product features around UI-based visualization authoring. Within their product, users can create various types of visualizations to explore data in a cloud data warehouse (CDW) without programming expertise. Alice's team uses Vega as the underlying technology to specify visualizations through the product's UI. By default, Vega requires all data to be loaded and processed in the client's browser. However, it is computationally impossible to query the CDW for the raw data and transfer it to the browser's memory for processing in Vega. Therefore, Alice decides to preprocess the data with SQL queries so that the query result is ready to be directly mapped to visual channels without further Vega transforms. However, her testing visualization is not fast in its initial rendering and does not seamlessly respond to user interactions. When Alice inspects VegaProf's pulses and dataflow graph visualizations, she finds out that both the initial rendering and every interaction trigger a request to the CDW, blocking the entire dataflow graph. Since all the data processing is done as a pre-process step in SQL, each interactive selection re-executes the whole data pipeline. In turn, Vega is blocked waiting for data that could have been cached. Knowing that the transforms parameterized by interactions are fast enough to be executed in the browser, she moves these interaction-based data transformations into the Vega specification, keeping only the underlying computational-intensive operations on the backend. These time-consuming operations are only executed once and cached in the dataflow graph. As a result, the initial rendering time for the chart is acceptable, while interactions are significantly smoother. Expert Interviews To further evaluate the usability of VegaProf, we conducted a qualitative user study with five visualization engineers at Sigma Computing. In the following, we describe the study setup and then report our observations gathered during these interviews. Finally, we summarize feedback on the usability of VegaProf elicited from participants through a post-study questionnaire. Interview Setup We now provide details on the setup of our study, including our participant pool, and the procedure and data used. Figure 4: During our evaluation, we used a scatter plot with binned aggregation as an example. At the beginning, the rendering time was about 600 ms. The individual point marks did not add substantial value to the visualization; in fact, they even obstructed the aggregated heatmap visualization. Thus, removing them did not undermine the message the visualization aims to communicate while reducing the mark rendering time to almost zero. Furthermore, the data transformation was specified in a suboptimal way that required Vega to copy data. Restructuring the data transformation further saved about 200 ms without changing the visualization. Participants. Our interviewees worked with Vega-generated visualizations daily, although they had different levels of background knowledge of Vega's internal dataflow. As such, they well represented VegaProf's target audience. While we had to collect our data opportunistically because of the limited availability of our participants, field studies like ours excel at capturing how visualization engineers actually work. Procedure. To understand the affordances and limitations of Ve-gaProf, we had 30-minute long think-aloud sessions with each interviewee individually. During the interviews, we first gave a quick tutorial of VegaProf, before participants could experiment with the profiler themselves. For this experimentation, our participants got access to VegaProf with a visualization specification preloaded. Our participants were asked to explore the profiler based on two guiding questions, namely how can the mark rendering be improved without harming the message of the visualization? and how can the data processing be improved?. We also encouraged them to share a specification from their recent work and show us how they would use VegaProf to inspect it. Finally, after the main think-aloud session, we sent interviewees an online questionnaire to evaluate the usefulness of a visualization profiler to their job in general, and their ratings of each VegaProf's specific feature. Data and specification. The specification we used for this evaluation renders a scatter plot with binned aggregation (cf. Figure 4) as described in Section 7.1. It includes three million data points and supports panning and zooming to re-calculate the aggregation with new buckets. Naturally, rendering or aggregating a large number of data points is prone to performance issues. Study Observations and Discussion In the following, we outline the main findings of our interviews. Initial performance improvements. With the help of the flame graph, all participants correctly identified the point marks as the most time-consuming components. They located the relevant part in the specification by hovering on the flame graph and stated that they found the feature useful "it is impressive that you can highlight the spec [from the flame chart]" (P3). Based on the highlighted region in the specification, all participants recognized that a scatter plot might not be the most efficient visually and computationally and removed it. Highlighting relevant parts of the visualization specification was one of the most well-received features of VegaProf as the current way of debugging slow specifications is to "just guess which part is the cause and modify it so see if it solves the issue or not" (P1). VegaProf greatly simplifies this laborious process as it connects profiling measurements back to the visualization specification. One participant underlined the importance of the flame graph to their workflow, since without VegaProf "we could separate the data transform out and profile it programmatically, but there was no way to do that for everything else, [including] the marks, the rendering, etc." (P5). Data transformation performance. As participants inspected the resulting performance after this first edit, they discovered that the runtime for rendering the marks was greatly reduced. Subsequently, they recognized that, with the modified specification, the most time-consuming operations came from the transformations that were used for processing the data. Participants were able to select the relevant dataflow nodes responsible for the performance bottleneck, however, most of them lacked the background knowledge about how nodes are instantiated from the specification through parsing and compilation. Specifically, they were not able to infer from the node name relay that the performance bottleneck was caused by a unnecessary data copy operation. P5 managed to solve the task by removing unnecessary operations, although it required a hint from our side: "Now that you told me a transform in the spec can be expanded to multiple operations, I can see it in the flame chart and everything makes sense to me" (P5). Some participants couldn't come up with a solution addressing this performance bottleneck. After we explained how to reconstruct the data transformation pipeline, our participants acknowledged that knowing the Vega internals and inspecting the dataflow graph would help optimize specification authoring: "I'm surprised that doing this can save so much [execution] time!" (P1). We believe that the above findings also suggest that the data pipeline development in Vega can be further optimized, for example, to avoid unnecessary data copying and streaming regardless of how users structure the specifications. Visualization usage. Overall, we observed that participants spent most of their time exploring the flame chart "because it exactly tells you what part of rendering is taking up all the time" (P2). When our participants decided to temporarily focus on part of the flame chart after initial exploration, they typically inspected the highlighted segments of the specification. Participants spent less time inspecting the dataflow graph. This might be partly due to the fact that most participants had a lack of understanding of the node names and "so far have just been using Vega as a black box [...] assuming that it would work well" (P2). Dataflow graph usability. Initially, even participants who had frequently debugged Vega visualizations with the dataflow graph before found the flame chart more helpful than the dataflow graph. They focused more on the dataflow graph only after being reminded of the connection between the flame chart and dataflow graph. While this underlines the importance of our flame graph visualization, it also raises questions about the usefulness of the dataflow graph for debugging purposes. One of our participants noted that "we used a lot of the Chrome Devtools and their entire interface is basically only the flame chart" (P3), attributing their focused view partly to previous habits. However, they also mentioned that "the connection between the spec and dataflow graph, and the structural features [in the dataflow graph] could be really helpful to understand what goes on behind the scenes for Vega" (P4). Further research targeted directly at visualizing dataflow systems in a more understandable way, including explanations of individual nodes, might help visualization engineers make better use of the dataflow graph as an IR for debugging. Participant-provided specifications. After the guided exploration, three participants asked us to directly explore specifications they recently worked on in VegaProf. We observed how they used VegaProf to validate or reject their assumptions about a given specification. P2 showed us a scatter plot with categorical data. At first, they were surprised that "the axes took the longest to render and then the marks were comparatively shorter [...] that's not what I would have guessed initially" (P2). Then, they realized that the dataset they used was relatively small while the categorical variables being mapped to the axes had a high cardinality. P3 shared a Sankey diagram that they have been working on for Sigma Computing's product with us. During the development, they frequently inspected the dataflow graph to understand and debug the connection between nodes. Concluding that "I'm not surprised that the "linkpath" and "datajoin" operations took the most time" (P4), they verified that the performance conformed with their mental model for such diagram. Finally, P5 wanted to explore a visualization automatically generated from a Sigma Workbook [GSU * 22]. They exported the specification of a simple chart to test their mental model of how it was implemented. As expected, it was implemented well, such that "it's so fast that the axis take half of the rendering time" (P5). Figure 5: After the main think-aloud session, our study participants rated VegaProf on a five-point Likert scale. We separated these questions into three main topics. First, we asked about the general value of a visualization profiler and VegaProf specifically. Second, participants gave feedback about the perceived usefulness of VegaProf's functionality. Third, we evaluated individual components of VegaProf. Overall, our participants saw great value in VegaProf and its visualizations. Usefulness Questionnaire Upon completion of the main interview study, we sent our participants a link to an online form with ten questions to be answered. Participants were able to provide feedback on their experience and takeaways from our interview session on a 5-point Likert scale. The results of this evaluation can be seen in Figure 5. Questions one (I think a visualization profiler is an important tool), two (A visualization profiler would be helpful to my work), and three (If I ever have to profile visualizations, I would use the presented profiler) were targeted at the general value of a visualization profiler and VegaProf, specifically. Overall, participants found visualization profilers useful and noted that they might be helpful to their work. This confirms the findings from our formative interviews. Furthermore, most of them would like to use VegaProf for their visualization profiling needs, affirming the usefulness of our profiler implementation. Questions four (Overall, I like the functionality of the presented profiler) and five (The fact that different levels of profiling are linked is very helpful) were to evaluate the functionality of Veg-aProf. Regarding the way VegaProf functions, our participants all liked its functionality, indicating that it indeed provides tooling that was not available before. Our participants also liked the way we coordinate Vega's intermediate representations via visual interaction, supporting our architectural design choices. Finally, questions six to ten were designed to assess the usability of VegaProf's individual components (Overall, I like the visualizations used to present profiling results, Visualizing which part of the specification can be attributed to profiling results is very helpful, Visualizing profiling results in the dataflow graph is very helpful, The flame graph is very helpful, and I like the fact that I can select pulses and this way debug interactive visualizations). Overall, our participants also liked the individual visualizations that VegaProf provides. They were especially fond of the visualization of performance results mapped to the DSL specification and the flame graph, which can provide an overview of performance results. Participants were torn on the usefulness of interactive pulses. One reason for that might be that we did not focus on them for the interactive evaluation session, however. The most controversial of the visualizations was the dataflow graph. While some found it valuable, others did not see it as beneficial for their workflows. We discuss potential reasons for this discrepancy in the previous subsection. Overall, our participants rated VegaProf very positively, underlining the importance of this line of work, its architectural design choices, and most of the visualization decisions we made. Discussion VegaProf is the first profiler for the Vega visualization grammar. Through its multi-level performance tracing approach, VegaProf provides previously unavailable background for performance problems, making performance debugging tractable and actionable. In the following, we will discuss our findings made during the development and evaluation of VegaProf. Lastly, while our studies and experiments show that VegaProf can be helpful for profiling Vega visualizations, we also identified several promising directions for future research. Direct visualization connection. Surprisingly, our study participants did not interact much with the rendered visualization. However, there might still be a case to be made for dataflow graph nodes to be connected to the scene graph that renders the visualizations. With such a connection, components of the resulting visualization could be linked to the visualizations included in our profiler. Profiling dataflow systems. While VegaProf focuses on the Vega DSL, the underlying approach and visual interaction design can readily apply to other DSLs. Many DSLs go through a similar lowering process and use a dataflow graph as an intermediate representation. Further generalization of our approach could enable performance profiling for a wide array of these systems, broadening the availability of accessible profiling even further. Performance at scale. Our evaluation shows how our approach enables performance improvements for individual visualizations. However, DSLs are also frequently used for visualization generation at scale. As a result, thousands of users can use automatically generated visualizations, e.g., in web applications under different browser and cloud configurations. Future research could extend our work to help visualization engineers profile the distributed performance of visualizations. Improvement recommendations. While mapping performance issues to the DSL proved to be an important step towards providing visualization engineers with more powerful profiling tools, the next step would be to suggest ways of improving performance. Here, knowledge from other visualizations or performance improvement sessions could be used to provide suggestions proactively. Using approaches such as the described performance at scale in combination with machine learning methods could enable such recommendations. In-browser profiling. Based on the insights of our formative interviews, we provide our profiler as an extension to the Vega Editor. While this is a common place for Vega visualization experiments, debugging is even more commonly done in the browser. We developed our visualizations in the Vega Editor because of its extensibility and flexibility in accessing Vega internals and providing rich visualizations. If the same tooling could be provided directly in browser profiling tools, fixing performance issues might become even more accessible. Scope of evaluation. While our studies were conducted with professional visualization engineers, they all work at the same institution. In this context, they work with a spectrum of visualizations targeted at the business intelligence domain, creating visualizations for large data sets and user bases. This is a common use case for DSL-based visualization grammars; however, we recognize that our findings might not transfer to all usage scenarios, such as one-off visualizations, small data sets, and specific user bases. Future work might evaluate the usefulness of our approach in such settings. Offloading decisions. A common pattern we saw during our evaluation was that visualization engineers are willing to offload certain aspects of the data transformation to a dedicated backend. However, they often do not know which parts are worth the effort. With VegaProf, such offloading decisions become much easier, as data transformation performance can be inspected in detail and tailored for specific designs. In turn, VegaProf could be combined with tools like VegaPlus [YJY * 22], VegaFusion [KMM22], and other backends or data processors to move time-consuming parts of the visualization process to dedicated services. Conclusion We introduced VegaProf, the first profiler enabling in-depth analysis of visualization performance bottlenecks. The design of Ve-gaProf is informed by formative interviews that surfaced the difficulty of Vega visualization performance debugging. VegaProf brings visual profiling affordances to Vega's different IRs by hooking into the lowering process. This way, the presented visualizations surface profiling results directly on the dataflow graph and visualization specification. We demonstrated the usefulness of Ve-gaProf through two use cases and reported feedback elicited from five visualization engineers, utilizing one of the use cases as a probe. In this evaluation, visualization engineers were able to locate and address performance bottlenecks through our linked visualization of Vega's IRs. VegaProf replaces the state of the art of either debugging Vega's performance through trial and error procedures or unnecessarily limiting data set sizes. While our work marks the first endeavor into the domain of visualization profiling, we hope that future research will broaden the applicability of our approach. In particular, our instrumentation for bidirectional mapping and corresponding visual interaction design coupling IRs can benefit developer tools for dataflow systems at large. Finally, we advocate for designing future visualization DSLs with their developer tools, such as debuggers and profilers, in mind. This co-design approach would simplify and accelerate the development of introspection tools for DSLs, further enhancing the developer experience in using them. Figure 3 : 3Our visual inspection tools were implemented in a familiar development environment -the Vega Editor. (A) We highlight selected regions of DSL code when inspecting performance bottlenecks. (B) A flame graph depicts the measurement of rendering function execution time. (C) In Vega's dataflow graph, we highlight nodes that contribute to selected timing measurements. Note how hovering over the flame graph highlights the corresponding elements in the dataflow graph and DSL editor. Battle et al. propose StreamTrace [BFD * 16], disentangling SQL queries as a series of intermediate queries to help developers debug the behavior of their queries. In a similar vein, Grust et al. [GKRS11] link intermediate query results directly to the SQL code that generated them instead of using representative visualizations. Other approaches from the deep learning domain [GGGP21, WCLR22,CLC * 22] focus on program code performance optimization. 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X., GROSSMAN T., PEKHIMENKO G.: Skyline: In- teractive in-editor computational performance profiling for deep neural network training. In Proceedings of the 33rd Annual ACM Symposium on User Interface Software and Technology (2020), pp. 126-139. 3 Demonstration of vegaplus: Optimizing declarative visualization languages. Yang J Joo, H K S Yerramreddy S, Li S, Moritz D, Battle L, Proceedings of the 2022 International Conference on Management of Data (2022). the 2022 International Conference on Management of Data (2022)YJY * 22[YJY * 22] YANG J., JOO H. K., YERRAMREDDY S. S., LI S., MORITZ D., BATTLE L.: Demonstration of vegaplus: Optimizing declarative vi- sualization languages. In Proceedings of the 2022 International Confer- ence on Management of Data (2022), pp. 2425-2428. 9

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{'abstract': "loadData() -340ms initScales() -60ms renderMark() -10ms 1 -Flame Graph 3 -Dataflow Graph 2 -DSL SpecificationFigure 1: VegaProf records low-level execution times and encodes them in coordinated visualizations corresponding to different abstraction levels of Vega's DSL execution. (1) A flame graph provides an overview of where the major amount of time is spent. (2) In contrast to previous debugging approaches, which relied on trial and error-based changes of the visualization specification, VegaProf maps performance measures directly to the Vega specification. (3) To analyze performance on different levels, VegaProf augments the dataflow graph with performance measures.AbstractVega is a popular domain-specific language (DSL) for visualization specification. At runtime, Vega's DSL is first transformed into a dataflow graph and then functions to render visualization primitives. While the Vega abstraction of implementation details simplifies visualization creation, it also makes Vega visualizations challenging to debug and profile without adequate tools. Our formative interviews with three practitioners at Sigma Computing showed that existing developer tools are not suited for visualization profiling as they are disconnected from the semantics of the Vega DSL specification and its resulting dataflow graph. We introduce VegaProf, the first performance profiler for Vega visualizations. VegaProf effectively instruments the Vega library by associating the declarative specification with its compilation and execution. Using interactive visualizations, VegaProf enables visualization engineers to interactively profile visualization performance at three abstraction levels: function, dataflow graph, and visualization specification. Our evaluation through two use cases and feedback from five visualization engineers at Sigma Computing shows that VegaProf makes visualization profiling tractable and actionable.", 'arxivid': '2212.13670', 'author': ['J / Yang ', 'Vegaprof ', 'J Yang \nUniversity of Washington\nWAUSA\n\nSigma Computing\n\n', 'A Bäuerle \nSigma Computing\nCAUSA\n', 'D Moritz \nCarnegie Mellon University\nPAUSA\n', 'Ç Demiralp \nSigma Computing\nCAUSA\n'], 'authoraffiliation': ['University of Washington\nWAUSA', 'Sigma Computing\n', 'Sigma Computing\nCAUSA', 'Carnegie Mellon University\nPAUSA', 'Sigma Computing\nCAUSA'], 'corpusid': 255186489, 'doi': '10.48550/arxiv.2212.13670', 'github_urls': [], 'n_tokens_mistral': 14338, 'n_tokens_neox': 12595, 'n_words': 8786, 'pdfsha': '29486f109a5cd45285c3a64650fb5b907bf81d37', 'pdfurls': ['https://export.arxiv.org/pdf/2212.13670v1.pdf'], 'title': ['VegaProf: Profiling Vega Visualizations', 'VegaProf: Profiling Vega Visualizations'], 'venue': []}

arxiv

Benchmarking the Ising Universality Class in 3 ≤ d < 4 dimensions 3 Apr 2023 Claudio Bonanno INFN Sezione di Firenze Via G. Sansone 150019Sesto Fiorentino (FI)Italy Andrea Cappelli bandrea.cappelli@fi.infn.it INFN Sezione di Firenze Via G. Sansone 150019Sesto Fiorentino (FI)Italy Mikhail Kompaniets cm.kompaniets@spbu.rudokudas@rikkyo.ac.jp Saint Petersburg State University 7/9 Universitetskaya Embankment199034St. PetersburgRussia Bogoliubov Laboratory of Theoretical Physics JINR Satoshi Okuda Department of Physics Rikkyo University Toshima 171-8501TokyoJapan Kay Jörg Wiese ewiese@lpt.ens.fr Laboratoire de Physique de l'Ećole Normale Supérieure Université PSL CNRS Sorbonne Université Université Paris-Diderot Sorbonne Paris Cité 24 rue Lhomond75005ParisFrance Joliot-Curie 141980DubnaRussia Benchmarking the Ising Universality Class in 3 ≤ d < 4 dimensions 3 Apr 2023 The Ising critical exponents η, ν and ω are determined up to one-per-thousand relative error in the whole range of dimensions 3 ≤ d < 4, using numerical conformalbootstrap techniques. A detailed comparison is made with results by the resummed epsilon expansion in varying dimension, the analytic bootstrap, Monte Carlo and nonperturbative renormalization-group methods, finding very good overall agreement. Precise conformal field theory data of scaling dimensions and structure constants are obtained as functions of dimension, improving on earlier findings, and providing benchmarks in 3 ≤ d < 4. a claudio.bonanno@fi.infn.it Many approaches to critical phenomena obtain results in continuous space dimension, although physically relevant dimensions are integer. Most notable is the perturbative renormalization group in d = 4− dimensions[1][2][3][4]. This is not merely a technical issue: quantities as functions of real d can clarify features that are harder to see at discrete values. E.g., one can follow the topology of the renormalization-group (RG) flow as a function of dimension and find instances where the universality class changes at non-integer values. This proved particularly useful for systems with long-range interactions[5][6][7]or disorder[8][9][10][11][12][13].The recent very precise numerical conformal bootstrap [14-16] has been formulated in continuous dimension[17,18], in particular for the Ising model in its whole range 4 > d ≥ 2[19][20][21]. The interest lies in understanding how the strongly interacting Ising conformal field theory connects to a free scalar in d = 4 and to the integrable fully-solvable model in d = 2[22,23]. Analytic bootstrap approaches which use the dimension as a tunable parameter were also developed[24][25][26][27][28][29][30][31][32]. Initially, the non-unitarity of the theory in noninteger dimensions[33]was thought to hamper the numerical methods involving positive quantities. These concerns have been overcome by de facto never observing problems for the quantities of interest, as explained later.In this paper, we extend the numerical approach of Ref.[20] using a single correlator, the SDPB [34] routine for determining the unitarity domain, and the Extremal Functional Method[35,36]for solving the bootstrap equations. We obtain improved results for the scaling dimensions in 4 > d ≥ 3 by a denser scanning of the unitary region near the Ising point, i.e., the kink. The latter gets parametrically sharper as d approaches 4, allowing for its better identification. The conformal spectrum in dimensions 4 > d ≥ 2.6 has also been obtained in Ref.[21] via the advanced navigator bootstrap technique[37]. We use these very precise results in combination with ours to obtain a consistent description of the low-lying spectrum.The achieved precision allows us to perform a detailed comparison with state-of-the-art epsilon expansion in two regimes: for d close to 4, the series is directly compared to bootstrap data, using the necessary finer scale for the latter; for intermediate values between 4 and 3 (included), the divergent perturbative series is resummed using well-established methods involving the Borel transform[38][39][40][41].The analysis is done on the dimensions of the conformal fields σ, , , corresponding to spin, energy and subleading energy. They determine the critical exponents η, ν, ω. The precision of our bootstrap data is summarized by the (mostly) d-independent value of the relative error Err(γ)/γ = O(10 −3 ) for the anomalous dimensions γ of the conformal fields σ and . As the anomalous dimensions are very small for d ≈ 4, the precision for the conformal dimensions ∆ σ , ∆ is even higher in this region. Regarding the subleading energy, the relative error Err(∆ )/∆ stays at three digits, as explained later. Some of the structure constants are determined with a higher O(10 −4 ) accuracy.We compare our data with recent results of the analytic bootstrap[27][28][29][30][31][32], Monte Carlo simulations[42][43][44]and the non-perturbative RG[45,46]. We find that the data by all methods agree very well. This is rather rewarding given the achieved precision. Besides Introduction confirming the high quality of conformal-bootstrap results, our analysis provides a reference point for further analytic and numerical methods aiming at exploring critical phenomena in varying dimensions. The outline of this paper is the following. In Sec. 2 we summarize our bootstrap protocol [20] and present the results for the three main conformal dimensions mentioned above, together with their polynomial fits as a function of dimension and the estimation of errors. In Sec. 3 we briefly recall the properties of the epsilon expansion and resummation techniques. We then compare its predictions with our bootstrap data and the results by other methods, and authors. A detailed analysis of all issues is presented. In Sec. 4, we report the numerical bootstrap data for scaling dimensions of structure constants and other conformal fields, and compare them to the existing epsilon expansion. In the conclusions in Sec. 5 we discuss open questions. Conformal bootstrap in non-integer dimension The aim of this section is to summarize our procedure for deriving conformal data of scaling dimensions and structure constants, as a function of the space-time dimension 4 > d ≥ 2. We first discuss the conformal dimensions of three main fields O = σ, , . Our goal is to provide a polynomial description of ∆ O as a function of y = 4 − d, by performing a best fit of the data obtained at several values of d 1 . Our results are finally compared to those obtained from the resummed epsilon expansion in Section 3. Summary of numerical methods The conformal dimensions and structure constants of the critical Ising model as a function of d are computed in the setup of Ref. [20], which we shortly summarize for the reader's convenience. We consider a single 4-point correlator σ(x 1 )σ(x 2 )σ(x 3 )σ(x 4 ) , where σ(x) is the primary scalar field with lowest dimension, denoted ∆ σ . We truncate the functional bootstrap equation to 190 components 2 . The unitarity condition for this equation is determined through the SDPB algorithm [34], leading to a bound in the (∆ σ , ∆ ) plane; next, the Extremal Functional Method (EFM) [35,36] is used to solve the equations on this boundary. We use the generalization of these numerical methods to non-integer dimensions developed in Ref. [20], and detailed in its Appendix A. Our 1-correlator numerical bootstrap approach has been surpassed by more recent implementations [16,19,21,47,48], but we find it convenient for determining the low-lying spectrum with modest computing resources. The complete determination of the conformal data for one value of d requires about 20 hours on 256 cores, corresponding to 5000 core hours. This simple setting allows us to evaluate the spectrum for various dimensions d. The first crucial step is to locate the Ising critical point in parameter space. To this end, we adopt the twofold strategy of Ref. [20], consisting in searching the kink on the unitarity boundary in the (∆ σ , ∆ ) plane and, at the same time, minimizing the central charge c [15]. This procedure allow us to determine for each value of d an interval of values for ∆ σ , ∆ and c, that we take as the Ising conformal theory, accompanied by an estimate of the uncertainty. This procedure is displayed in Fig. 1, where we show the identification of the Ising point for d = 3, 3.25, 3.5 and 3.75. The gray area in the plots indicates the chosen errors for ∆ σ , ∆ and c, which are roughly determined by the mismatch between the positions of the minimum and the kink. As a conservative choice, we consider an interval of four data points for each value of d. The precision is greater than in Ref. [20], because we perform a finer scan of the ∆ σ values around the kink. We observe that the kink and the minimum get sharper for d → 4, as shown by the four pairs of plots drawn on the same scale in Fig. 1; this is convenient in our approach, since it leads to an increased precision when anomalous dimensions are smaller. In Fig. 2, we show the point d = 3.875, not considered in the earlier work. It is necessary for studying the region of d → 4. Here the curves are so steep that magnified scales are needed. Once the Ising point is determined, we obtain the rest of the conformal data as follows. The solution of the bootstrap equations gives a spectrum of conformal dimensions ∆ O and structure constants f σσO as a function of ∆ σ ; they are divided into different sets characterized by the spin = 0, 2, 4, . . . of the operator O. The estimation of ∆ O and f σσO is obtained by taking the central value of such quantities for ∆ σ varying in the interval previously identified as the Ising point (grey areas in Figs. 1 and 2). The error is obtained from their dispersion. It is interesting to point out that, although we largely improved the precision of our results for 4 > d > 3 with respect to Ref. [20], we observe no signs of trouble associated to nonunitarity in our bootstrap spectrum. On general grounds, non-unitarity contributions are expected to appear for non-integer values of d due to the presence of negative-norm states [33]. However, these occur at very high order in the OPE expansion of the correlator σσσσ , thus we may argue that they have numerically negligible structure constants. As a matter of fact, their presence does not seem to yield problems in solving the bootstrap equations with our method. This conclusion was also reached by recent 3-correlator bootstrap studies of the critical O(N ) models [18] and the Ising model [21] in non-integer space dimensions using the navigator method [37]. Analysis of conformal dimensions of the three leading fields for 4 > d ≥ 3 In Tab. 1 we present our results for the conformal dimensions ∆ O in 4 > d > 3 along with those of Ref. [20] for 3 ≥ d > 2, also employed in the following. Our implementation of the bootstrap determines with high precision the conformal dimensions and structure constants for the first few low-lying operators with = 0, 2 and 4: We employ an improved fit method for ∆ O (y) that uses orthogonal polynomials [49]: the idea is to expresses the n th -order polynomial fit function f n (y) in terms of orthogonal polynomials P k (y) of degree k = 0, 1, . . . , n, instead of a parameterization in terms of monomials, 1, y, y 2 , . . . , y n . To this aim we write O =0 = σ, , , O =2 = T and O =4 = C [20]. d ∆ σ ∆ ∆ ∆ ∆ T ∆ C ∆f n (y) = n k=0 α k P k (y), P r (y)P s (y) ∝ 14 i=1 P r (y i )P s (y i ) ∝ δ rs , (2.1) where y i are the values in Tab. 1. This method is equivalent to the naive one, but is numerically more stable and the fit parameters α k can be determined with improved precision and less statistical noise. The optimal degree n for the fitting polynomial is not known a priori and is determined in the following way: The fit with weights proportional to the inverse square of errors is done for several values of n, and the least chi-square χ 2 min is found as a function of n. At a given order n, adding a further term α n+1 P n+1 results in a negligible change of χ 2 min and the best fit yields a result for α n+1 which is compatible with zero within errors. This identifies n as the degree of the optimal polynomial. Finally, we use the results of our best fit for {α k } to assign an error to f n (y) in the whole range of 4 > d ≥ 3. Details on the fitting procedure and the computation of errors can be found in App. A. In this section we focus on the three leading operators σ, and (corresponding to φ, φ 2 and φ 4 in the φ 4 field theory), which are determined with very good precision. The analysis of higher-dimensional operators is postponed to Sec. 4.2. Instead of working with conformal dimensions, we consider the anomalous dimensions γ σ = ∆ σ − d − 2 2 , γ = ∆ − (d − 2), γ = ∆ − 2(d − 2). (2.2) They are related to the Ising critical exponents η, ν and ω by η = 2γ σ , 1 ν = 2 − γ , ω = d − 4 + γ . (2.3) The vanishing of anomalous dimensions in the free theory (d = 4) is assumed in the following fits. Our analysis starts by comparing the old [20] and new data for 4 > d > 3. In Fig. 3 the new results (blue circles) show much smaller errors than the earlier findings (red crosses), due to a more accurate localization of the Ising point, as explained above. In these and later figures we report the differences (γ O − fit) between data and fitting polynomial, because simpler plots would not capture the small errors involved (note that the abscissas of the three plots differ by factors of ten). The explicit form of the best fitting polynomials are provided in Sec. 3. [20] (red crosses) and new (blue circles) bootstrap data for γ σ , γ , γ , minus the corresponding best fits. The plots use the same scales as in Ref. [20]. Next, we compare these results with those recently obtained by solving the 3-correlator bootstrap with the navigator method [21]. In Fig. 4 our data, given in earlier figures (blue circles), are shown on a finer scale, together with the estimated error of the fit (cyan shaded area). The red triangles are the navigator values: they come with no errors and thus cannot be directly used for the fits 4 . A first observation is the fairly good agreement between the two different bootstrap approaches at our level of precision. We propose to estimate the error of navigator data as follows. We suppose that they are roughly of the same size as those found in other 3-correlator studies at d = 3 (rigorous bounds) [48,50], which are plotted in Fig. 4 as black diamonds (γ σ and γ ), and a grey rightward triangle (γ ). Assuming these very small uncertainties for each value of d, there seems to be a negative offset with respect to our data, in particular for ε . This could be a systematic error due to our approximate identification of the Ising point within the unitarity region (Section 2.1), while the navigator method rigorously determines it within a unitarity island [37]. However, other explanations are possible. In conclusion, taking into account these considerations, we enlarge the error estimate of 4 Earlier results of Ref. [19] are not considered here due to their large errors. our fits to the shaded gray bands in Figs. 4, which correspond to the following bounds: Err(γ σ ) γ σ ≈ Err(γ ) γ 1 × 10 −3 , Err(∆ ) ∆ 0.5 × 10 −3 , 3.875 ≥ d ≥ 3. (2.4) Given the small value of anomalous dimensions for d → 4, these imply extremely low absolute errors, Err(γ σ ) = O(10 −6 ) and Err(γ ) = O(10 −5 ) in this range, as spelled out in the following sections. This allows us to give a precise comparison to other methods, as a benchmark for the Ising universality class in non-integer dimensions. Figure 4: Plot of bootstrap data for γ σ , γ , γ minus the best fit values.The shaded area represents the error obtained from the χ 2 minimization of the fitting polynomial. The red triangles are results from Ref. [21] using the navigator method in a 3-correlator bootstrap setup (no error bars). Black diamonds and grey rightward triangle for d = 3 represent respectively results by Ref. [48] (γ σ and γ ) and Ref. [50] (γ ); these data points are slightly displaced around d = 3 to improve readability. The gray shaded bands represents the error bounds reported in Eq. (2.4). Comparison with the epsilon expansion in 4 > d ≥ 3 In this section, we recall some features of the epsilon expansion and the resummation methods employed for it. We compare unresummed and resummed series with the bootstrap results for γ σ . Then, the analysis is extended to γ and γ . Warm-up analysis of the anomalous dimensions γ σ We start with a brief summary of the properties of the perturbative expansion of the φ 4 field theory in d = 4 − y, which describes the Ising universality class. This is a textbook subject [51] but we would like to single out a few aspects that are important in the following comparison with bootstrap results in varying dimensions 5 . The β-function β(g, y) and the anomalous dimensions γ O (g), where O = φ, φ 2 , φ 4 , take the following form, in the Minimal Subtraction (MS) [51,52] renormalization scheme, β(g, y) = −yg + n+1 k=2 β k g k , γ O (g) = n k=1 γ O,k g k . (3.1) The numerical coefficients β k , γ O,k were computed up to order n = 6 in Ref. [40], and n = 7 in Ref. [53]. While results up to order n = 15 are known for a subclass of Feynman diagrams believed to give the dominant contribution, they are not used here [40,54]. The coefficients of the β-function (3.1) grow exponentially with k, and their asymptotic behavior can be estimated from the contribution of instanton field configurations [51] β k ∼ k→∞ C (−a) k k b k! . (3.2) Similar behaviors are found for the coefficients γ O,k . The parameters a, b, C depend on the quantity considered. One finds that the known values of the coefficients up to order n = 7 grow very fast with n but have not yet reached their asymptotic values (3.2) [40,54]. The behavior (3.2) can be understood as follows: The perturbative series has a vanishing radius of convergence in the complex g plane, because real negative values of g correspond to an upside-down potential and an action not bounded from below. This fact can be exemplified by the simple zero-dimension path integral (see App. B.1): I(g) = ∞ −∞ dx √ 2π e − x 2 2 −gx 4 = ∞ k=0 a k (−g) k , a k = (4k)! 2 2k (2k)!k! ∼ k→∞ 2 4k √ 2πk × k! . (3.3) This is the generating function counting the number of vacuum Feynman diagrams. The asymptotic behavior of a k can be found by a saddle-point analysis of the integral. In field theory the corresponding saddle point is given by instantons [51] 6 . The solution of the fixed-point equation β(g, y) = 0 gives g = g(y) by perturbative inversion around g = y = 0; this is used to rewrite the anomalous dimensions as a series in y, γ O (y) = n k=1 γ O,k y k . (3.4) This is again a divergent series of asymptotic form (3.2), with suitable parameters a, b and C. The ratio of two consecutive terms in the series (3.4) can be estimated from (3.2) as, γ O,k y/γ O,k−1 ≈ −aky, which is larger than one for y > 1/|ak|. A simple conclusion is that the more terms are present in the perturbative series (3.4), the sooner it diverges as a series in y. We can draw two main conclusions: i ) As it stands, the perturbative series (3.4) is basically useless for physical dimension y = 1, apart from the first couple of terms, and resummation methods are necessary for extracting precise values of anomalous dimensions. The resummation is based on the Borel transform, followed by a conformal mapping, as will be explained later, and further discussed in App. B.1. This procedure gives resummed finite expressions γ O (y). ii ) For dimensions close to d = 4, i.e., y 1, there is an optimal number of terms n opt (y), for each y value, for which the distance between the series and the resummed function γ O (y), | γ O (y) − nopt 1 γ O,k y k |, is minimal before growing again. The perturbative anomalous dimensions γ O may differ from results obtained by other methods, such as the lattice formulation of the path-integral for the Ising model, or by the bootstrap. These differences are non-analytic, e.g., δγ O (y) ∼ exp(−A/y). Within the resummation procedure, these terms may change according to how the inverse Borel transform is performed [55]. Before discussing the resummation methods in the next section, a first comparison of the perturbative expansion and the bootstrap data for γ σ clarifies the issues at stake. The perturbative series is [40,53] γ σ (y) = 0.00925926y 2 + 0.00934499y 3 − 0.00416439y 4 + 0.0128282y 5 −0.0406363y 6 + 0.15738y 7 , (epsilon expansion). (3.5) The best polynomial fit of bootstrap data in Tab. 1 using the methods outlined in Sec. 2.2 is 7 γ σ (y) = 0.009306473y 2 + 0.008899908y 3 − 0.001435107y 4 + 0.001788710y 5 −0.000533980y 6 + 0.000128667y 7 , (conformal bootstrap). (3.6) The two polynomials (3.5) and (3.6) have different meanings, although their first two coefficients are close. On one hand the Feynman-diagram series is exact, but has a vanishing radius of convergence. On the other hand, the numerical bootstrap data in Tab. 1 should converge to exact non-perturbative results upon increasing the numerical precision. The collection of these values for any dimension d = 4 − y gives the exact function γ ex σ (y), which however cannot be expressed in terms of a simple polynomial. Therefore, the fit (3.6) gives approximated values around γ ex σ (y), whose precision is a priori limited. Nonetheless, this description is sufficient at the present level of numerical accuracy. In Fig. 5 we show the difference between the perturbative series (3.5) and the bootstrap fit (3.6) for 4 > d ≥ 3. Color lines correspond to the series (3.5) truncated at different orders n = 2, 3, . . . , 7 (cf. color legend in the plot). One sees that, the higher the order n ≥ 4, the sooner the perturbative series diverges from the bootstrap data (corresponding to the zero horizontal line in Fig. 5). The tiny errors of bootstrap points cannot be seen at this scale, thus showing that the unresummed perturbative series cannot be used for a precise determination of critical exponents in d = 3, as stated in point ii ) above. Yet, the lower terms n = 2, 3 may provide crude estimates. 6 shows the other regime, close to four dimensions. Only the bootstrap point for d = 3.875 is present in this range, but we also show results of Ref. [21] for d ≥ 3.8, which match very well while lacking error bars, as discussed earlier 8 . In contrast to the d ≈ 3 region, we observe that the truncated perturbative series shows a different behavior. At any given y value, upon increasing the perturbative order up to an optimal value n opt ∼ 1/y, the perturbative series approaches the zero horizontal line (with a cyan error band), before starting to diverge. Namely, it matches the exact bootstrap value γ ex σ (y), within numerical errors. Therefore, the comparison between non-perturbative bootstrap results and unresummed epsilon expansion for γ σ (y) is extremely good in the region 4 > d > 3.8, with precision Err(γ σ ) ≈ 1 × 10 −6 , i.e., Err(γ σ )/γ σ < 1 × 10 −3 . According to the previous discussion, we conclude that we do not see any non-perturbative difference for d → 4. Figure 6: Comparison of γ σ data minus best fit in the region 4 > d > 3.8, between bootstrap (blue circle) and unresummed epsilon expansion (3.5) with different truncations of the perturbative series (cf. Fig. 5). The red triangles are the results of the bootstrap navigator method [21]. The cyan shaded area is the fit error. We remark that the epsilon expansion can also be obtained by analytic solution of the bootstrap equations around d = 4, assuming a perturbative expansion near the free theory [24,25,27,28,[30][31][32]. Thus, is our comparison in Fig. 6 tautological? It is not, because the bootstrap identity is a set of consistency conditions that depends on the kind of quantities they act on. Our numerical solution does not assume any perturbative expansion, i.e., it is an independent solution of the bootstrap constraints. That without any perturbative input, our conformal bootstrap results accurately reproduce perturbative predictions close to d = 4 is non-trivial. A natural question is how our numerical bootstrap approach can reproduce the perturbative series, i.e., in which regime the two polynomials (3.5) and (3.6) may agree beyond the O(y 3 ) term. As said earlier, the bootstrap polynomial (3.6) is approximated, it can at most describe a band of values around γ ex σ (y). While the size Err(γ σ ) of this band stays finite in the whole range 0 < y < 1 (see plots), that of the epsilon expansion is expanding in y and can be finite only for y < y max ∼ O(1/n), n being the perturbative order. We expect that, upon running the bootstrap for several points y i , with 0 < y i < y max 1, and by performing best fits with polynomials limited to such a small interval, one may find that the two expressions (3.5) and (3.6) match order by order, i.e., the epsilon expansion is fully recovered. Bootstrap data versus resummed perturbative results Precise estimates of the critical exponents have been obtained over the years by refining the resummation techniques applied to the epsilon expansion series [2][3][4]40,41,51,56,57]. In this work, we use the methods of Refs. [40,41] extended to dimension 4 > d ≥ 3. Let us briefly recall the main steps involved [51]. The Borel transform B γ O (t) of the perturbative expansion for the anomalous dimension γ O (3.4) is defined by removing the factorial growth from the series, B γ O (t) = n k=1 γ O,k k! t k . (3.7) One infers from the asymptotic behavior (3.2) that this function has a singularity B γ O (t) ∼ (1 + ta) −b−1 and a corresponding finite radius of convergence. The resummed quantity is defined by the inverse Borel transform, At this point, one can only make educated guesses on these singularities, that translate into (physical) ansatzes for γ O (y). γ O (y) = ∞ 0 dt e −t B γ O (yt). In practice, one assumes that the only singularity of B γ O (t) lies at t = −1/a real and negative, and that it is a branch cut extending to t = −∞. Using a conformal mapping t(z), this branch cut is mapped onto the unit circle, with the start of the branch cut mapped onto z = −1, and t = −∞ to z = 1, preserving the origin z = t = 0. As long as there are no other singularities, B(t(z)) has a radius of convergence one in z. As t = ∞ corresponds to z = 1, this allows one to perform the inverse Borel transform (3.8). Details on this procedure can be found in App. B.1. This general idea can be improved in several ways, allowing one to introduce a set of free parameters. The latter are determined such that the final result is the least sensible to their variation. Apart from providing a robust resummation scheme, the parameter uncertainty implies an estimate of the resummation error. These methods have been improved over the years by taking into account the phenomenology of critical phenomena [51]. In our work, the resummed data are obtained by extending the setup of Refs. [38,40,41] from d = 3 to non-integer dimensions. A complete account of these methods is too long to be presented here; nonetheless, we provide some introductory material that will allow the reader to assess the original works. In App. B.1, the resummation is worked out in a toy model, where one can compare it with the exact result. In App. B.2, instead, a "reader's guide" to Ref. [40] is presented, together with the values of the resummation parameters used here. Let us also mention that another option for the analytic continuation is to use Hypergeometric functions, for which the inverse Borel transform can be written as a Meijer-G function [56]. One drawback of this approach is the possibility for spurious poles on the integration contour. As here we could not give justice to their influence, we exclude this resummation method. Figure 7 shows the fitted bootstrap data (blue points) of γ σ (y) already reported in Fig. 4, now compared to the resummed epsilon-expansion values of Tab. 2 (green squares) 10 . The agreement between these two results is very good, especially for d ≥ 3.5, where the unresummed series (magenta line) is already diverging, and greatly improves on earlier studies [2,3] analyzed in [20]. Let us remark that resummed γ σ (y) values have been obtained for non-integer dimensions down to d = 2, still finding agreement with bootstrap data, although with larger uncertainties. Finally, Fig. 7 shows the latest Monte Carlo results in d = 3 (yellow rhombus), that match extremely well the bootstrap points. Further d = 3 results by these and other methods are summarized in a later figure. Finally, Fig. 7 and later plots for the dimensions γ ε and γ ε also report a solid red curve linearly interpolating the navigator points of Ref. [21] obtained for 4 > d ≥ 3. This allows one to assess the negligible difference between the two sets of bootstrap data in the comparison to the epsilon-expansion. (7) 3.9276(5) 3.25 0.63386 (8) 1.5458 (4) 3.873 (2) 3 0.5181 (3) 1.4108 (12) 3.820(7) Table 2: Conformal dimensions of σ, and field from resummed perturbative expansion, obtained according to the methods of [40]. We now extend the previous analysis to the energy field . The best fit of the conformal bootstrap data is γ (y) = 0.333441601y + 0.114095325y 2 − 0.083458310y 3 +0.081381007y 4 − 0.045296977y 5 + 0.014290102y 6 −0.001741325y 7 , (conformal bootstrap). (3.9) The epsilon-expansion series reads [40,53] γ (y) = 0.333333y + 0.117284y 2 − 0.124527y 3 + 0.30685y 4 − 0.95124y 5 +3.57258y 6 − 15.2869y 7 , (epsilon expansion). (3.10) One remarks the agreement, within errors, of the first two coefficients of this series; this corrects less precise results of [20] (cf. Fig. 6b there). The comparison for d → 4 before resummation is shown in Fig. 8. As for Fig. 7, the truncated perturbative series for γ are plotted. Their curves approach the bootstrap fit (horizontal zero axis with cyan error band) with better and better precision. Note the remarkable quality of the navigator method (red triangles) [21]. Altogether, the agreement for d → 4 is found with high precision, Err(γ ) = 3 × 10 −5 and Err(γ )/γ = 1 × 10 −3 . Probably there is a slight underestimation of the error. Let us remark that this resummation procedure is honest, as it does not use the exact d = 2 conformal dimension as an input, with which it could be improved. The comparison with another method, called Self-Consistent (SC) resummation 11 is presented in Fig. 10, where we plot data of Tab. 3. In this case, the Borel transform is done on the perturbative series of 1/ν 3 , instead of 1/ν = 2 − γ e : this choice is motivated by a match with the d = 2 conformal field theory, that is achieved through comparing the n dependence of the O(n)-symmetric φ 4 theory [41]. We conclude that adding information of the exact results in d = 2 improves the resummation of the perturbative series (for this particular critical exponent). A similar constraint does not seem to be possible for the other critical exponents, as discussed in Ref. [41]. Summarizing, the bootstrap and epsilon-expansion results agree very well: for d → 4 the unresummed series fits perfectly, for 4 > d ≥ 3 there is remarkable agreement, keeping in mind that the resummation error is roughly one order of magnitude larger than that of bootstrap and Monte Carlo results. A comparison of all d = 3 results available in the literature for γ σ and γ is given in Figs. 11 and 12. The corresponding numerical values are in Tab. 4. Besides data already discussed (drawn in earlier colors), we report recent results of the non-perturbative renormalization group [45] (brown downward triangle). The central value is given by our fit of the bootstrap data with error given by the cyan band, not by the mean value of all results. The Figs. 11 and 12 respect our convention of plotting the two anomalous dimensions on scales differing by one order of magnitude, roughly equal to the ratio of their actual value. Finally, Tab. 4 and Figs. 11, 12 report also the results of other 3-correlator bootstrap approaches, using EFM [48] and the navigator method [50], and paying particular attention to error estimates (cf. rigorous bounds). We also remark that the results obtained by perturbative expansions directly in d = 3 [3,4] are consistent with bootstrap results too, but have one order of magnitude larger errors and are therefore not plotted in Figs. 11 and 12. [20] (blue circle), 3-correlator bootstrap with rigorous bounds [48] (black pentagon), Monte Carlo [44] (yellow rhombus), Borel-resummed epsilon expansion [40] (green square), Self-Consistent resummed epsilon expansion [41] (red star), non-perturbative renormalization group [45] (brown downward triangle), bootstrap navigator method [21] (red upward triangle). We now analyze the subleading Z 2 -even scalar field , which is related to the critical exponent ω = ∆ − d = d − 4 + γ . The best fit of our data gives 12 : γ (y) = 2.000178549y − 0.518006835y 2 + 0.721996645y 3 −0.684437170y 4 + 0.447648598y 5 − 0.162903635y 6 +0.026155257y 7 , (conformal bootstrap). (3.11) The large errors of the earlier analysis [20] have been reduced, as explained earlier (see 12 The fit again assumes γ = 0 for d = 4. [20] (blue circle), 3-correlator bootstrap with rigorous bounds [48] (black pentagon), Monte Carlo [44] (yellow rhombus), Borel-resummed epsilon expansion [40] (green square), Self-Consistent resummed epsilon expansion [41] (red star), non-perturbative renormalization group [45] In Fig. 13 we show the difference between the data and the bootstrap best fit (3.11). The overall error of the fit for γ is estimated to be less than 2.0 × 10 −3 in the whole range. The relative error is Err(γ )/γ = 1 × 10 −3 for d = 3 but goes up to 13 1 × 10 −2 for d = 3.875. The comparison with Monte Carlo [42,44] in d = 3, and the resummed epsilon-expansion series are also shown, finding again good agreement at the coarser scale (note a factor of 10 w.r.t. Fig. 9). A systematic difference between bootstrap and epsilon-expansion points is seen for d → 3, similar to what was found for γ in Fig. 9. Such a drift is smaller for the navigator results [21] (red line) than for our data, for 4 > d Structure constants and scaling dimensions of higher fields In this section we analyze further bootstrap data. The structure constants (OPE coefficients) of low-lying fields σ, , , T are very precise, the error being on the fifth decimal, thus better than those of the corresponding conformal dimensions presented earlier. Next we discuss subleading and spinful fields, , T , C, C , presenting results for both dimensions and structure constants. Some of them are good, others are not completely correct, showing the limits of our numerical bootstrap approach. Structure constants in 4 > d ≥ 3 Tab. 5 reports all data for structure constants: those for 4 > d > 3 are new results, the ones for 3 ≥ d > 2 are taken from [20]. The central charge c is obtained from the structure constant f σσT of the energy-momentum tensor T by f 2 σσT = d 4(d − 1) ∆ 2 σ c . (4.1) For f σσO , we adopt the by-now standard normalization of [21,48]. The relation with the earlier normalization f σσO of Ref. [15] is f 2 σσO = d−2 2 (d − 2) f 2 σσO , (4.2) where (x) ≡ Γ(x + )/Γ(x) is the Pochhammer symbol. (5) [20]. The central charge c and the structure constants f σσ and f σσ are determined with very high accuracy: their dependence on y = 4 − d is obtained with the fit method of Sec. 3.1, assuming the exact d = 4 value. The resulting polynomials are reported together with the available epsilon-expansion series [30,31,58,59]: The comparison with other conformal bootstrap results is as follows: The best 3-correlator determination in d = 3 [48] is shown as a black pentagon in the figures. Data from the navigator method are unfortunately only available for f σσ [21]. The agreement among different numerical setups is extremely good. Moreover, as already observed for scaling dimensions, the unresummed epsilon expansion captures the d → 4 behavior, and it does it very well, since the lower-order terms of the respective polynomials (4.3)-(4.8) are equal within errors. For f σσ , the results of the resummed epsilon expansion, reported in Tab. 6, are also shown, determined by earlier methods: the 4 th -order series (4.6) only allows for a precise agreement down to d ≈ 3.6, given the fine scale of Fig. 17. For the remaining quantities, the epsilon expansion is either too short for a resummation, or not alternating. Figure 17: Comparison of f σσ and f σσ minus best fit: bootstrap (blue circles), unresummed epsilon expansion [30,31,58,59] (magenta solid curve), 3-correlator bootstrap at d = 3 [48] (black pentagon). For f σσ we also report the resummed epsilon expansion (green squares) and bootstrap navigator results [21] (red triangles). c(y) = 1 − 0.015415049y 2 − 0.026663929y 3 − 0.004992140y 4 − 0. Higher fields T and C The analysis of the fields T ( = 2) and C ( = 4) is done along the same lines. The fit polynomials for ∆ T and ∆ C , obtained as before, are ∆ T (y) = 6 − 0.567900778y + 0.1779633663y 2 − 0.806164966y 3 +1.749534636y 4 − 1.684842086y 5 + 0.765011179y 6 −0.126284231y 7 , (conformal bootstrap), (4.9) ∆ C (y) = 6 − 1.001598184y + 0.030791232y 2 −0.033868719y 3 + 0.041665026y 4 − 0.002907562y 5 −0.006602770y 6 , (conformal bootstrap). (4.10) They are shown in Fig. 18, along with the bootstrap results of [21] (red triangles) and the available epsilon-expansion series (magenta solid lines) [27,32,58,59]: ∆ T (y) = 6 − 0.5555556y, (epsilon expansion), (4.11) ∆ C (y) = 6 − y + 0.01296296y 2 + 0.01198731y 3 −0.006591585y 4 , (epsilon expansion). (4.12) As shown by the cyan band, representing our fitting error, the scaling dimensions of these fields are determined with an accuracy comparable to that achieved for the low-lying = 0 states: Err(∆ T ) ≈ 10 −2 and Err(∆ C ) ≈ 3 × 10 −3 , meaning that Err(∆ T )/∆ T ≈ 10 −3 and Err(∆ C )/∆ C ≈ 5 × 10 −4 . Within our precision, we observe very good agreement with the results of [21] (especially for T ). Furthermore, the unresummed epsilon expansion is again in agreement with the bootstrap results for d → 4. Overall, the picture is consistent with the = 0 case discussed earlier 14 . The corresponding structure constants are given by the polynomial fits f σσT (y) = 0.026278214y − 0.012019512y 2 − 0.016779681y 3 +0.025762223y 4 − 0.018571573y 5 + 0.006902659y 6 −0.001000504y 7 , (conformal bootstrap), (4.13) f σσC (y) = 0.16903085 − 0.122480930y + 0.077087613y 2 − 0.591032947y 3 +1.331591787y 4 − 1.231373513y 5 + 0.512308476y 6 −0.079520247y 7 , (conformal bootstrap). (4.14) They can be compared to the available epsilon expansions [27,32,[58][59][60]: f σσT (y) = 0.02635231y − 0.013176155y 2 , (epsilon expansion), (4.15) f σσC (y) = 0.16903085 − 0.12244675y + 0.02131741y 2 +0.002168567y 3 − 0.0019760553y 4 , (epsilon expansion). (4.16) The comparison is shown in Fig. 19. Also in this case we observe good agreement between the conformal bootstrap polynomials and the epsilon expansion series up to O(y 3 ) terms. Subleading fields and C The numerical 1-correlator bootstrap approach used in this paper is known to have a limited precision for states higher up in the conformal spectrum, in particular for our approximation to 190 components of the truncated bootstrap equations. In this section, we show that our identification of ( = 0) and C ( = 4) has some problems, especially for d → 4. We explain these difficulties by using the epsilon expansion for conformal dimensions and structure constants, as well as the 3-correlator bootstrap data [21] in varying dimensions, which are definitely more accurate for the higher spectrum than our results. We think that these aspects are worth discussing, especially because the y = 4 − d dependence plays a crucial role. We start our analysis from the subleading twist = 4 operator C , for which we find the following best fit polynomial: ∆ C (y) = 8 − 0.827053961y − 0.055211344y 2 + 0.053430207y 3 +0.010354264y 4 − 0.003205703y 5 , (conformal bootstrap). (4.17) These data are shown in Fig. 20 (left part). It turns out that C is degenerate at d = 4 with another field with same dimension and spin, called C 2 . Their dimensions are known to leading order in the epsilon expansion, ∆ C (y) = 8 − 1.555556y, (4.18) ∆ C 2 (y) = 8 − 0.833333y, (epsilon expansion), (4.19) and are plotted in Fig. 20 with magenta dashed and solid lines, respectively. Near these lines, the navigator bootstrap results [21] are plotted with gold and red triangles. One sees that our results start at d → 4 very close to C 2 (see first coefficient in polynomials (4.17) and (4.18)) and end up near C at d = 3. Therefore, the state we found is a mixture of C and C 2 : better numerical precision would be needed for disentangling the two states near d → 4, obtained, e.g., by increasing the number of components approximating the bootstrap equations. The fit of the structure constant is given by f σσC (y) = 0.006871047y − 0.005215834y 2 − 0.003223129y 3 +0.005087571y 4 − 0.001393464y 5 , (conformal bootstrap), (4.20) and plotted in the right part of Fig. 20. The epsilon-expansion results for C and C 2 read, f σσC (y) = 0.001543806y, (4.21) f σσC 2 (y) = 0.006458202y, (epsilon expansion), (4.22) and are shown as magenta dashed and solid lines on the right of Fig. 20. These perturbative data show a remarkable fact: for d < 4 the state of higher dimension C 2 has a larger structure constant, contrary to the standard behavior of f σσO decreasing fast with ∆ O . It is thus clear that, close to d = 4, C 2 gives the dominant contribution to a putative mixed C -C 2 state. This suggests the reason why our results with limited precision start close to C 2 . The analysis is confirmed by the bootstrap result for the structure constant in (4.20): for d → 4 it fits the perturbative behavior of f σσC 2 , as seen in the right plot of Fig. 20. In conclusion, our subleading = 4 state is identified as C 2 for d → 4, but gradually approaches C in d = 3. Another problematic identification concerns the field (corresponding to φ 6 in the φ 4 theory). The best fit of bootstrap data gives ∆ (y) = 2.313321845y − 1.678645012y 2 + 0.336440006y 3 +0.090959178y 4 , (conformal bootstrap), (4.23) while the leading epsilon-expansion result reads [24,25,60]: ∆ (y) = 2y − 4.759259y 2 , (epsilon expansion). (4.24) For the structure constant we find f σσ (y) = 0.002851280y 2 − 0.003188068y 3 + 0.001218496y 4 −0.000161879y 5 , (conformal bootstrap); (4.25) f σσ (y) = 0.006901444y 2 , (epsilon expansion). (4.26) It is apparent that our bootstrap results do not match the leading perturbative expansion for d → 4. The corresponding plots are shown in Fig. 21, where the disagreement with bootstrap results from Ref. [21] (red triangles) is also seen. and [21]. The solid and dashed magenta lines are the corresponding leading-order epsilon expansion, which agree with the navigator results, but not ours. Let us investigate the possibility of another mixing of states near d → 4. In this case there is no degenerate field with at d = 4. However, the next subleading one ∼ 2 φ 4 in the φ 4 theory is present at higher dimension ∆ ≤ 8. The epsilon expansion and navigator results for this field are also shown in Fig. 21 (left part, gold downward triangles). We remark that a mixing of and was shown to take place at d = 2.8, i.e., rather far from d = 4 [21]. We suppose that the limited resolution of our data finds a state which is a mixture of and also for d → 4, but we cannot be certain of this. As for C and C 2 , support for this argument could come from a comparison of the corresponding structure constants f σσ and f σσ . Unfortunately, the epsilon expansion of the latter is not available, so we cannot get a definite explanation of our ∆ data. Conclusions In this paper we obtained the conformal dimensions and structure constants of the critical Ising CFT as a function of varying dimension 4 > d ≥ 3 by using the numerical conformal bootstrap approach. Our main result is the precise determination of the anomalous dimensions of the σ, , fields, which are related to the Ising critical exponents η, ν, ω. Our relatively simple 1correlator bootstrap setup is able to compute the d-dependence of these quantities with up to one-per-thousand relative accuracy; therefore, our findings can be used as a benchmark for future studies in non-integer space dimension. For these low-lying states of the conformal spectrum, our results are in very good agreement with those of more advanced 3-correlator bootstrap techniques [21,37,47,48], with a small offset included in the error estimate. We presented a detailed comparison of available predictions from different methods. For d → 4, our results agree with those from unresummed perturbation theory. This shows two things: that non-perturbative differences, which might effect the bootstrap program or the resummed series, are negligible for d → 4. The other non-trivial result is that both approaches agree on the same analytic continuation in dimension. A possible explanation of this correspondence is provided by the analytical bootstrap, which on one hand reproduces the epsilon expansion, and on the other hand uses the same ingredients as the numerical bootstrap. For 3 ≤ d < 4, but away from d = 4, the bootstrap data agree very well with other results, obtained by resummation techniques of the perturbative series, Monte Carlo simulations, and other bootstrap approaches. In the whole 4 > d ≥ 3 range we find overall consistency among the different approaches; improvements are needed by adding further terms to the perturbative series in d = 3, as the current state of the art still shows a O(10 −3 ), O(10 −2 ) discrepancy, respectively for ν and ω, and in general much larger error bars than bootstrap and Monte Carlo results. We were able to compute bootstrap data for the conformal dimensions of higher-order fields in 4 > d ≥ 3, including the lowest-lying spinful fields T ( = 2) and C ( = 4), with a precision comparable to that of spinless operators. The central charge and OPE coefficients of low-lying fields were obtained with even higher precision than that of the corresponding anomalous dimensions. The structure constants agree well with those of the 3-correlator bootstrap, where available (mostly in d = 3), and with perturbation theory for d → 4. A possible future development is to improve current bootstrap results in the region 3 > d ≥ 2, in order to better understand how the d = 3 theory approaches the d = 2 Virasoro minimal model. To this aim, it is important to go beyond the lowest-lying states and precisely probe higher-dimensional and higher-spin fields. Improved 3-correlator bootstrap protocols, such as the recently proposed navigator method, may be well suited here. Appendix A Orthogonal polynomial regression Standard polynomial regression of the data set S ≡ {x i , y i , ∆y i } N i=1 is achieved by minimizing χ 2 = N k=1 y k − f (x k ) ∆y k 2 , (A.1) with respect to the parameters {c i } d i=0 of the fit function, f n (x) = n r=0 c r x r . (A.2) The degree n of the polynomial is not known a priori. A smarter fit is obtained by changing the basis in which the polynomial is expressed: B naive = 1, x, x 2 , . . . , x d → B ortho = {P 0 (x), P 1 (x), P 2 (x), . . . , P d (x)} , (A.3) where the polynomials P k (x) (of degree k) are chosen to be orthogonal on the independent variables of the dataset S, i.e.: P r (x)P s (x) S = 1 N N k=1 P r (x k )P s (x k ) = k 2 r δ rs , (A.4) where k r are constants. With this choice, the fit function becomes f n (x) = n r=0 α r P r (x). (A.5) The best fit is obtained by minimizing χ 2 in Eq. (A.1). The advantage of the orthogonal polynomial regression is that the coefficients α r do not depend on the α s with s > r, i.e., adding higher-degree polynomials r > n to f n (x) does not change the value of α r with r ≤ n within the statistical errors [49]. Thus, this procedure is better suited to assess the optimal degree of the polynomial. The expression of the polynomials P r (x) is known in the literature. In this work, we follow the conventions of Ref. [49]. We start by fixing the r = 0 and r = 1 polynomials as P 0 (x) = 1, P 1 (x) = 2(x − a 1 ), a 1 = 1 N N k=1 x k ≡ x. (A.6) Higher-order polynomials with r ≥ 2 are obtained through the recursive relation [49], P r+1 (x) = 2(x − a r+1 )P r (x) − b r P r−1 (x), (A.7) where the coefficients a r+1 and b r are given by a r+1 = N k=1 x k P 2 r (x k ) N k=1 P 2 r (x k ) , b r = N k=1 P 2 r (x k ) N k=1 P 2 r−1 (x k ) . (A. 8) In this work, we find the best fitting polynomial for γ O and f σσO as a function of y = 4 − d. We always assume their known analytic value for d = 4, for example γ O (d = 4) = 0. To enforce such constraint, it is sufficient to use as fit function h n (x) = f n (x) − f n (0) = n r=1α r [P r (x) − P r (0)] . (A.9) Finally, we reconstruct the original expansion in the naive basis by summing all equal monomials among every P r (x) included in the fit function: h n (x) = n r=1α r [P r (x) − P r (0)] = n r=1c r x r , (A.10) wherec r = n l=rα l d r P l (x) dx r x=0 . (A.11) Once the two expansions are properly matched, the coefficients obtained from orthogonal polynomials agree with those obtained using a standard polynomial fit. The advantage of orthogonal polynomials resides in their improved numerical stability, which results in an improved precision in the computation of the c i . Finally, once the best fitting polynomial is obtained, we assign an error to our best fit function h n (x) through standard error propagation, via the so-called parameter covariance matrix, C ij ≡ Cov(α i ,α j ). (A.12) Let us define v i (x) as the gradient of the fit function with respect to the i th fit parameter, v i (x) = ∂h n (x | α) ∂α i . (A.13) The error on the best fitting polynomial is Err(h n )(x) = v T (x)Cv(x) = C ij v i (x)v j (x). (A. 14) The best fit of γ O (y) via orthogonal polynomial regression was done by using the curve_fit routine from the standard Python library scipy. In this appendix, we discuss the perturbative expansion of a toy model in dimension zero: I(g) ≡ ∞ −∞ dx √ 2π e − x 2 2 −gx 4 . (B.1) Its perturbative expansion is I(g) = ∞ n=0 a n (−g) n , a n = (4n)! 2 2n (2n)!n! ∼ n→∞ 2 4n √ 2πn × n! . (B.2) The analytic continuation of the integral (B.1) from Re(g) > 0 to the full complex plane is given by a second-kind modified Bessel K-function: I(g) = 1 4 √ πg e 1 32g K1 4 1 32g . (B.3) Using the asymptotic behavior of K1 4 (z) for z → ∞, one sees that the exponential prefactor is canceled, and the series (B.2) recovered. Note that I(g) has a cut on the whole negative real axis, see Fig. 22. In field theory, the divergent series is analytically continued without the knowledge of its exact expression. Let us explain the strategy on the example of integral (B.1). The basic idea [51] to obtain a convergent series out of Eq. (B.2), is to divide each term by n!, defining the Borel transform I B (t) of the series. In a second step, one reconstructs the original series via an integral transform: In our example we know the analytic expression in terms of the first-kind complete elliptic integral function I B (t) ≡I B (t) = 2K elliptic 1 2 − 1 2 √ 16t+1 π 4 √ 16t + 1 . (B.5) The Borel transform I B (t) has a finite radius of convergence, denoted by −t bc (equal to 1/16 in our example). As a consequence, the start of the branch cut is moved from g = 0 to t = t bc < 0, see figure 22. Since the radius of convergence of I B (t) is still finite, the integral transform (B.4) does not work as written. One first has to continue I B (t) to the domain 0 ≤ t < ∞. This can be achieved by replacing the known truncated series via a converging Padé approximant, leading to a Padé-Borel resummation. A more powerful strategy is to use a conformal mapping. The most common ansatz is to assume that at t = t bc < 0 a cut-singularity starts, which extends on the negative real axis to t = −∞. One first maps the complex plane with the expected branch cut of I B (t) onto the inside of the unit-circle: z = 1 − t/t bc − 1 1 − t/t bc + 1 ⇐⇒ t = −4t bc z (z − 1) 2 . (B.6) Next one constructs a series in z by expanding both sides in this variable: f (z) ≡ ∞ n=0 c n z n = ∞ n=0 a n (−t(z)) n n! = I B (t(z)) . (B.7) This series is expected to converge for |z| < 1, a fact we can check for our example (but which is difficult to prove in general): f (z) = 1 − 3z 4 + 9z 2 64 − 51z 3 256 + 1353z 4 16384 − 7347z 5 65536 + 61617z 6 1048576 + O(z 7 ) . (B.8) Given n terms in the original series, we know f (z) up to the same order. Using this approximation for f (z), we finally obtain: I(g) = ∞ 0 dt e −t I B (tg) = 1 g ∞ 0 dt e −t/g I B (t) = 1 g 1 0 dz t (z) e −t(z)/g f (z) . (B.9) The result of this resummation is shown on Fig. 23. First, in black is the analytic result (B.3). Next are the first three orders in several expansions, using the same color code for order 1 (blue), 2 (green), and 3 (red): first the direct expansion in g (dotted), then in solid the resummed expansion (B.9). Dashed, we show a large-g expansion obtained by changing variables gx 4 → y in the integral (B.1), and then expanding the integrand in powers of 1/ √ g: I(g) = 1 2 √ 2π 4 √ g ∞ 0 dy e − √ y 2 √ g −y y 3 4 = 1 2 √ 2π 4 √ g Γ 1 4 − 2 3 Γ 7 4 √ g + Γ 5 4 8g + O g − 5 4 . (B.10) B.2 Details on the resummation method This appendix aims at providing a "reader's guide" to the analysis in Ref. [40], which determines the resummed series for the d = 3 critical exponents η, ν −1 and ω (related, respectively, to γ σ , γ and γ ). The same methods are used in our work, by a simple generalization to varying dimension 4 > d ≥ 3. This guide, together with the introduction in the main text and the example in App. B.1, should provide enough information to follow the discussion in Ref. [40]. In particular, we are interested in its Sec. V. Let us denote the equations in Ref. [40] by double parentheses, e.g., Eq. ((25)), to avoid confusion with our numbering. The resummation procedure with Borel transform and conformal mapping goes along the lines described in our Sec. 3.2 and App. B.1. The perturbative series of a critical exponent f (ε) in (−2ε) = D − 4 (cf. our y = 4 − d) is defined in Eq. ((25)) of [40]: f (ε) = ∞ k=0 f k (−2ε) k , f k ∼ C f k! a k k b f as k → ∞. (B.11) With respect to our notation (cf. our Eq. (3.2)), the negative sign of a is included in the power of epsilon, and the exponent of the power-law behavior earlier denoted by b is now b f . The values for the parameters (a, b f ) are given in Eq. ((26)) for the λφ 4 theory with O(n) symmetry, n = 1 being the case of interest, and they are determined by the known asymptotic behavior of the beta function. With respect to the definition given here in Eq. (3.7), in [40] the Borel transform is replaced by the more general Borel-Leroy transform, defined as follows (cf. Eq. ( (27)) in [40]): B b f (x) = ∞ k=0 f k Γ(k + b + 1) (−x) k , (B.12) where b is a free parameter. The function B b f (x) behaves as B b f (x) ∼ (1 + ax) b−b f −1 around x = −1/a. The function B b f (x) is then modified in three ways in order to define the inverse transform and improve its convergence. The first step is the conformal mapping ((29)) already described in App. B.1, involving the known parameter a. The second step is the addition of the powerlaw prefactor in ( (30)) with a second free parameter λ. The third step is the "homographic transformation" ε = h q (ε ) defined in ( (32)) which introduces a third free parameter q. The resummed epsilon-expansion seriesf (x) is finally obtained from the inverse Borel transform of the modified function B b,λ, f •hq reported in ( (33)) of [40], f (ε) = ∞ 0 t b e −t B b,λ, f •hq 2εt 1 − qε dt . (B.13) It depends on three free parameters: b, λ and q ( being the perturbative order considered, = 6 here). Let us briefly mention how these are determined. The behavior off (ε) ≡f b,λ,q (ε) is studied in the cubic range (B.14) The optimal values of the parameters are chosen according to the principle of "minimal sensitivity" (w.r.t. varying the parameters) and "fastest apparent convergence" (w.r.t. increasing the perturbative order by one, − 1 → ). These dependences are taken into account by a proper definition of the error function E f (b, λ, q) that is given in Eq. ( (36)). The global minimum of the errorĒ f = E f (b,λ,q) in the cubic range (B.14) identifies the optimal values b =b, λ =λ and q =q. The final estimate for the critical exponents is obtained from the inverse Borel transform (B.13) with these parameters. The optimization procedure is done independently for each dimension d = 4 − 2ε. The results forb,λ and q are reported in Tab. 7 for the resummations of η, ν −1 and ω at the d values considered. Note the mild dependence of the parameters on d. We remark that this brief outline brushes over many fine details discussed in Ref. [40], but which are crucial for achieving high-quality results, as well as the comparison with other methods developed in the extensive literature. More technical information can be found in Ref. [40] and its supplementary material, available in arXiv:1705.06483. Table 7: Optimal variational parameters used here in the resummation procedure for the critical exponents η, ν −1 and ω, as a function of 4 > d ≥ 3. The Self Consistent resummation of perturbative data used to obtain results shown in Figs. 10, 11, 12, 14 and 17, instead, does not involve the optimization of free parameters introduced before. As explained in Sec. III of [41], the asymptotic behavior (B.11) is fitted from the perturbative series, thus finding the position of the singularity x = −1/a of the Borel transform. Such fit is done for several values of the free parameter α, defined in Eq. ((44)) of [41] and analogous to b f in (B.11), varied in the range −6 ≤ α < 6 in steps of 0.2. For each α, the value of a obtained from the fit is used in the conformal mapping (B.6) (t bc ≡ a) and the resulting function is Borel inverted, giving the resummed series. The best estimate of the resummed quantity with this procedure is obtained through the mean over all the values of α employed, while the error bars represent the maximal and minimal values obtained varying α. Since only one parameter is varied, this error estimate is less reliable than that determined by the methods of Ref. [40] described earlier. Figure 1 : 1Determination of the Ising critical point for d = 3, 3.25, 3.5, 3.75 (d = 3 data from Ref. [20]). Left plots: Identification of the kink; the blue points correspond to the solutions of the bootstrap equations. Right plots: position of the c minimum. The grey shaded areas represent the estimated errors on ∆ σ , ∆ and c. Figure 2 : 2Determination of the Ising point for d = 3.875, as in Fig. 1. Note the magnified scale on both axis with respect to those of Fig. 1. Figure 3 : 3Old Figure 5 : 5Comparison of γ σ bootstrap data with unresummed epsilon expansion (3.5) in the region 4 > d > 3 for truncations of the series to order n = 2, . . . , 7 (see color legend). All quantities have been subtracted by the best fit values (see(3.6)). Fig. Fig. 6 shows the other regime, close to four dimensions. Only the bootstrap point for d = 3.875 is present in this range, but we also show results of Ref. [21] for d ≥ 3.8, which match very well while lacking error bars, as discussed earlier 8 . In contrast to the d ≈ 3 region, we observe that the truncated perturbative series shows a different behavior. At any given y value, upon increasing the perturbative order up to an optimal value n opt ∼ 1/y, the perturbative series approaches the zero horizontal line (with a cyan error band), before starting to diverge. Namely, it matches the exact bootstrap value γ ex σ (y), within numerical errors. γ O (y) in(3.4) and γ O (y) in(3.8) have the same perturbative expansion; however, the latter should be better behaved if B γ O (t) is suitably continued analytically outside the original disc |t| < 1/|a| to a region including the real positive axis 9 . Such analytic continuation in principle requires the knowledge of all singularities of B γ O (t) in the complex t-plane. Figure 7 : 7Comparison of γ σ data minus best-fit values: bootstrap (blue circles), Borel-resummed epsilon expansion[40] (green squares), unresummed high-order epsilon expansion (magenta solid curve), d = 3 Monte Carlo[44] (yellow rhombus Figure 9 9presents a comparison with the resummed perturbative series (Tab. 2): the agreement is again very good for 4 > d ≥ 3.5; there is a small O(10 −3 ) deviation from the bootstrap and Monte Carlo results [44] (yellow rhombus) in d = 3. Figure 8 : 8Comparison of the γ data minus the best fit in the region 4 > d > 3.8. Our bootstrap point is the blue circle with error bar; the triangles are obtained by the navigator method[21]; the different truncations of the perturbative series are as inFig. 5. The cyan shaded area is the fit error. Figure 9 : 9Comparison of γ data minus best fit: bootstrap (blue circles), Borel-resummed epsilon expansion[40] (green squares), unresummed epsilon expansion (magenta solid curve), d = 3 Monte Carlo[44] (yellow rhombus). We also plot a solid red line linearly interpolating results of Ref.[21] for 4 > d ≥ 3. The cyan shaded area is the fit error as in earlier plots. Figure 10 : 10Comparison of γ minus best fit: bootstrap (blue circles), Self-Consistent resummed epsilon expansion[41] (red stars), unresummed epsilon expansion (magenta solid curve), d = 3 Monte Carlo[44] (yellow rhombus). Figure 11 : 11Summary of up-to-date predictions for γ σ at d = 3 (minus best fit): 1-correlator bootstrap Figure 12 : 12Summary of up-to-date predictions for γ in d = 3 (minus best fit): 1-correlator bootstrap Fig. 3 ) 3. The epsilon-expansion series is[40,53], γ (y) = 2y − 0.62963y 2 + 1.61822y 3 − 5.23514y 4 + 20.7498y5 −93.1113y 6 + 458.7424y 7 , (epsilon expansion).(3.12) Figure 13 : 13Comparison of γ data minus best fit: bootstrap (blue circles), Borel-resummed epsilon expansion[40] (green squares), unresummed epsilon expansion (magenta solid curve), d = 3 Monte Carlo[42] (yellow rhombus). We also plot a solid red line linearly interpolating results of Ref.[21] for 4 > d ≥ 3. ≥ 3 . 5 .Figure 14 : 3514Further values of ∆ in d = 3 found in the literature are reported in Tab. 4 and plotted in Fig. 14. A zoom over the region close to d = 4 is drawn in Fig. 15, showing the same features as in Figs. 6 and 8.We conclude this section by stressing the very good overall agreement of bootstrap and resummed epsilon expansion. The study in varying dimensions clarifies the different behavior of quantities in the perturbative and non-perturbative regimes. Summary of up-to-date predictions for γ in d = 3 (minus our best fit, from bottom to top): 1-correlator bootstrap[20] (blue circle), 3-correlator bootstrap[48] (black pentagon), Monte Carlo[44] (yellow rhombus), Borel-resummed epsilon expansion[40] (green square), Self-Consistent resummed epsilon expansion[41] (red star), non-perturbative renormalization group[45] (brown downward triangle), bootstrap navigator method with rigorous bounds[50] (grey rightward triangle), bootstrap navigator method[21] (red upward triangles). Figure 15 : 15Comparison of the γ data minus the best fit in the region 4 > d > 3.8. Our bootstrap point is the blue circle with error bar; the triangles are obtained by the navigator method[21]; the different truncations of the perturbative series are as inFig. 5. The cyan shaded area is the fit error. 0.235465537y − 0.170275458y 2 + 0.096635030y 3 − 0.113371408y 4 +0.100586943y 5 − 0.054667196y 6 + 0.016161292y 7 − 0.001992399y 8 ,(conformal bootstrap), (4.5) f σσ (y) = √ 2 − 0.235702y − 0.168047y 2 + 0.103680y 3 − 0.224776y 4 , (epsilon expansion), (4.6) f σσ (y) = 0.136221303y − 0.118250195y 2 + 0.067116467y 3 − 0.058700794y 4 +0.037159615y 5 − 0.012211017y 6 + 0.001647332y 7 (conformal bootstrap), (4.7) f σσ (y) = 0.1360828y + 0.11844240525y 2 , (epsilon expansion). (4.8) We remark: i ) the excellent agreement between the first few terms of the conformal bootstrap and epsilon-expansion series, and ii ) the need of a high-order O(y 7 , y 8 ) polynomial for precise fits. The corresponding curves are shown in Figs. 16 and 17. Note that c, f σσ and f σσ were determined with strikingly small (relative) errors, respectively O(10 −5 ), O(10 −4 ) and O(10 −4 ) over the entire d range. Figure 16 : 16Comparison of c data minus best fit: bootstrap (blue circles), unresummed epsilon expansion[58,59] (magenta solid curve), 3-correlator bootstrap at d = 3[48] (black pentagon). Figure 18 : 18Comparison of scaling dimensions minus best fit for T , C fields: bootstrap (blue round points), navigator method[21] (triangle red points), 3-correlator bootstrap at d = 3[48] (black pentagon) and unresummed epsilon expansion[27,32,58,59] (magenta solid line). Figure 19 : 19Behavior of structure constants f σσT and f σσC (round blue points) compared with 3correlator bootstrap at d = 3[48] (black pentagon) and epsilon expansion (magenta solid line)[58,59]. Figure 20 : 20Scaling dimension and structure constant of would-be C operator in our bootstrap spectrum (blue circles). Upward red and downward gold triangles represent navigator results for C and C 2[21]. The dashed and solid magenta lines are the corresponding leading-order epsilon expansion. Figure 21 : 21Scaling dimension and structure constant of the would-be operator in our bootstrap spectrum (blue circles). Upward red and downward gold triangles represent navigator results for Figure 22 : 22The branch cut in I(g) (top left) and I B (t) (top right). While the former starts at g = 0, the latter is moved to g = −1/16. The lower plot shows I B (t(z)), which now has a branch-cut singularity at |z| = 1. (We set I B (t(z)) to 0 outside the disc |z| ≥ 1.)B Resummation of perturbative seriesB.1 Toy model example Figure 23 : 23Left: function I(g) (black, thick, dot-dashed) and its diverse approximations. Dotted for the series expansion at order 1 (blue), 2 (green), and 3 (red). Solid for the resummed series at the same order. Dashed for the large-g expansion (same color code). Right: deviation of the resummed series (B.9) from the exact result (B.5) for g = 10 as a function of n, assuming one knows t bc only approximately. In blue for t bc = −1/16 (the exact result), in red t bc = −1/32 (a conservative guess), in black t bc = −1/1000 (much too small). Resummation with t bc = −1/15 (green) does not work. We see that conform to expectations, taking a too small value for −t bc , the series converges more slowly, while taking a too large value of −t bc the series does not converge. −t I B (tg). (B.4) (b, λ, q) ∈ [0, 40] × [0, 4.5] × [0, 0.8]. Table 1 : 1Conformal dimensions of the first few low-lying states for 4 > d > 2. Exact values for d = 2, 4 are given in bold, results for 3 ≥ d > 2 are taken from Ref.[20]. goal of this section is to determine the behavior of ∆ O as a function of the variable y = 4 − d, by finding the best fitting polynomial that describes the data in Tab. 1. We use all available values, but focus on the range 4 > d ≥ 3 where results are more precise and allow for a comparison with other approaches. The points for 3 > d ≥ 2 are mainly used for stabilizing the higher powers of the fitting polynomials 3 .The ). We also plot a solid red line linearly interpolating results of Ref.[21] for 4 > d ≥ 3. Note that data points are slightly displaced around the same d values (d = 3.875, d = 3.75, d = 3.5, d = 3.25 and d = 3) to improve readability. Results from earlier work[3] have been omitted due to their large error bars.d ∆ σ ∆ ∆ 3.875 0.937662197(7) 1.91831086(14) 3.9924550(11) 3.75 0.8757158(3) 1.839419(4) 3.97529(3) 3.5 0.753393(10) 1.68854 Table 3 : 3Conformal dimension of field from resummed perturbative expansion, obtained according to the methods of[41]. (brown downward triangle), bootstrap navigator method[21] (red upward triangle).d = 3 Ising critical indices ∆ σ ∆ ∆ Bootstrap (1 correlator) 0.518155(15) 1.41270(15) 3.8305(15) Bootstrap (3 correlators) 0.5181489(10) 1.412625(10) 3.8297(2) Borel resummed epsilon expansion 0.5181(3) 1.4107(13) 3.820(7) SC Borel resummed epsilon expansion 0.5178(2) 1.4122(15) 3.827(13) Monte Carlo 0.51814(2) 1.41265(13) 3.832(6) Non-perturbative RG 0.5179(3) 1.41270(50) 3.832(14) Navigator (rigorous bounds) 0.518157(35) 1.41265(36) 3.8295(6) Table 4 : 4Comparison of d = 3 results for the conformal dimensions of low-lying fields: 1-correlator Table 5 : 5Structure constants of the first few low-lying states for 4 > d > 2. The exact values for d = 2, 4 are given in bold, results for 3 ≥ d > 2 are taken from Table 6 : 6Structure constant f σσ from resummed perturbative expansion, obtained according to the methods of[41]. Note that is the energy field, the next-to-lowest scalar primary field, not to be confused with the deviation from four dimensions denoted by y.2 This corresponds to the standard bootstrap parameter Λ = 18, which counts the number of derivatives in the approximation of the functional basis. Note that the lower quality of 3 > d > 2 data is due to the coarse scanning of ∆ σ values, not to an intrinsic limitation of the numerical bootstrap approach[20]. An up-to-date discussion of epsilon expansion can be found in Refs.[38][39][40][41]. We refer to these works for a proof of the following statements and appropriate referencing.6 There is growing consensus that the large-order behavior is governed by an instanton rather than a renormalon[54]. If one could go to much higher orders in the series expansion (e.g., 20-loop order) one could apply methods of resurgence and trans-series[55]. Note that the best-fit polynomial (3.6) starts with an O(y 2 ) term, because the linear term vanishes within errors. If a linear term is included in the fit procedure, it leads to a coefficient three orders of magnitude smaller than the quadratic term. Therefore, the conformal bootstrap implies γ σ (y) = O(y 2 ) close to d = 4, in agreement with perturbation theory. Note that the red triangles are not used in our fit of bootstrap data. In particular, a real negative value of the parameter a in (3.2), i.e., a perturbative series (3.4) of definite sign, is problematic. Resummations in this section use the 6-loop results, that were verified in several independent works[40,41,53]. We do not use the 7-loop results of Ref.[53], since they were not yet checked independently. Past experience, e.g., with the 5-loop results, teaches us that involved perturbative calculations require confirmation. See Ref.[41] for a detailed discussion of this approach. 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{'abstract': 'The Ising critical exponents η, ν and ω are determined up to one-per-thousand relative error in the whole range of dimensions 3 ≤ d < 4, using numerical conformalbootstrap techniques. A detailed comparison is made with results by the resummed epsilon expansion in varying dimension, the analytic bootstrap, Monte Carlo and nonperturbative renormalization-group methods, finding very good overall agreement. Precise conformal field theory data of scaling dimensions and structure constants are obtained as functions of dimension, improving on earlier findings, and providing benchmarks in 3 ≤ d < 4. a claudio.bonanno@fi.infn.it Many approaches to critical phenomena obtain results in continuous space dimension, although physically relevant dimensions are integer. Most notable is the perturbative renormalization group in d = 4− dimensions[1][2][3][4]. This is not merely a technical issue: quantities as functions of real d can clarify features that are harder to see at discrete values. E.g., one can follow the topology of the renormalization-group (RG) flow as a function of dimension and find instances where the universality class changes at non-integer values. This proved particularly useful for systems with long-range interactions[5][6][7]or disorder[8][9][10][11][12][13].The recent very precise numerical conformal bootstrap [14-16] has been formulated in continuous dimension[17,18], in particular for the Ising model in its whole range 4 > d ≥ 2[19][20][21]. The interest lies in understanding how the strongly interacting Ising conformal field theory connects to a free scalar in d = 4 and to the integrable fully-solvable model in d = 2[22,23]. Analytic bootstrap approaches which use the dimension as a tunable parameter were also developed[24][25][26][27][28][29][30][31][32]. Initially, the non-unitarity of the theory in noninteger dimensions[33]was thought to hamper the numerical methods involving positive quantities. These concerns have been overcome by de facto never observing problems for the quantities of interest, as explained later.In this paper, we extend the numerical approach of Ref.[20] using a single correlator, the SDPB [34] routine for determining the unitarity domain, and the Extremal Functional Method[35,36]for solving the bootstrap equations. We obtain improved results for the scaling dimensions in 4 > d ≥ 3 by a denser scanning of the unitary region near the Ising point, i.e., the kink. The latter gets parametrically sharper as d approaches 4, allowing for its better identification. The conformal spectrum in dimensions 4 > d ≥ 2.6 has also been obtained in Ref.[21] via the advanced navigator bootstrap technique[37]. We use these very precise results in combination with ours to obtain a consistent description of the low-lying spectrum.The achieved precision allows us to perform a detailed comparison with state-of-the-art epsilon expansion in two regimes: for d close to 4, the series is directly compared to bootstrap data, using the necessary finer scale for the latter; for intermediate values between 4 and 3 (included), the divergent perturbative series is resummed using well-established methods involving the Borel transform[38][39][40][41].The analysis is done on the dimensions of the conformal fields σ, , , corresponding to spin, energy and subleading energy. They determine the critical exponents η, ν, ω. The precision of our bootstrap data is summarized by the (mostly) d-independent value of the relative error Err(γ)/γ = O(10 −3 ) for the anomalous dimensions γ of the conformal fields σ and . As the anomalous dimensions are very small for d ≈ 4, the precision for the conformal dimensions ∆ σ , ∆ is even higher in this region. Regarding the subleading energy, the relative error Err(∆ )/∆ stays at three digits, as explained later. Some of the structure constants are determined with a higher O(10 −4 ) accuracy.We compare our data with recent results of the analytic bootstrap[27][28][29][30][31][32], Monte Carlo simulations[42][43][44]and the non-perturbative RG[45,46]. We find that the data by all methods agree very well. This is rather rewarding given the achieved precision. Besides', 'arxivid': '2210.03051', 'author': ['Claudio Bonanno \nINFN\nSezione di Firenze\nVia G. Sansone 150019Sesto Fiorentino (FI)Italy\n', 'Andrea Cappelli bandrea.cappelli@fi.infn.it \nINFN\nSezione di Firenze\nVia G. Sansone 150019Sesto Fiorentino (FI)Italy\n', 'Mikhail Kompaniets cm.kompaniets@spbu.rudokudas@rikkyo.ac.jp \nSaint Petersburg State University\n7/9 Universitetskaya Embankment199034St. PetersburgRussia\n\nBogoliubov Laboratory of Theoretical Physics\nJINR\n\n', 'Satoshi Okuda \nDepartment of Physics\nRikkyo University Toshima\n171-8501TokyoJapan\n', "Kay Jörg Wiese ewiese@lpt.ens.fr \nLaboratoire de Physique de l'Ećole Normale Supérieure\nUniversité PSL\nCNRS\nSorbonne Université\nUniversité Paris-Diderot\nSorbonne Paris Cité\n24 rue Lhomond75005ParisFrance\n", '\nJoliot-Curie\n141980DubnaRussia\n'], 'authoraffiliation': ['INFN\nSezione di Firenze\nVia G. Sansone 150019Sesto Fiorentino (FI)Italy', 'INFN\nSezione di Firenze\nVia G. Sansone 150019Sesto Fiorentino (FI)Italy', 'Saint Petersburg State University\n7/9 Universitetskaya Embankment199034St. PetersburgRussia', 'Bogoliubov Laboratory of Theoretical Physics\nJINR\n', 'Department of Physics\nRikkyo University Toshima\n171-8501TokyoJapan', "Laboratoire de Physique de l'Ećole Normale Supérieure\nUniversité PSL\nCNRS\nSorbonne Université\nUniversité Paris-Diderot\nSorbonne Paris Cité\n24 rue Lhomond75005ParisFrance", 'Joliot-Curie\n141980DubnaRussia'], 'corpusid': 252734896, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 31809, 'n_tokens_neox': 25949, 'n_words': 14730, 'pdfsha': 'c45f0e193189921bccee0f592c91a8190a884a81', 'pdfurls': ['https://export.arxiv.org/pdf/2210.03051v3.pdf'], 'title': ['Benchmarking the Ising Universality Class in 3 ≤ d < 4 dimensions', 'Benchmarking the Ising Universality Class in 3 ≤ d < 4 dimensions'], 'venue': []}

arxiv

LHC Benchmark Scenarios for the Real Higgs Singlet Extension of the Standard Model 25 May 2016 Tania Robens TU Dresden Institut für Kern-und Teilchenphysik Zellescher Weg 19D-01069DresdenGermany Tim Stefaniak Department of Physics Santa Cruz Institute for Particle Physics University of California 95064Santa CruzCAUSA LHC Benchmark Scenarios for the Real Higgs Singlet Extension of the Standard Model 25 May 2016(Dated: May 26, 2016)They have also been presented in the framework of the LHC Higgs Cross Section Working Group. * Electronic address: Tania.Robens@tu-dresden.de † Electronic address: tistefan@ucsc.edu 1 We present benchmark scenarios for searches for an additional Higgs state in the real Higgs singlet extension of the Standard Model in Run 2 of the LHC. The scenarios are selected such that they fulfill all relevant current theoretical and experimental constraints, but can potentially be discovered at the current LHC run. We take into account the results presented in earlier work and update the experimental constraints from relevant LHC Higgs searches and signal rate measurements. The benchmark scenarios are given separately for the low mass and high mass region, i.e. the mass range where the additional Higgs state is lighter or heavier than the discovered Higgs state at around 125 GeV. I. INTRODUCTION The first run of the LHC at center-of-mass (CM) energies of 7 and 8 TeV has been completed in 2015. Its remarkable success is highlighted by the breakthrough discovery of a scalar boson in July 2012 and the measurements of its coupling properties, which thus far are well compatible with the interpretation in terms of the Higgs boson of the Standard Model (SM) Higgs mechanism [1][2][3][4][5]. The combination of the Higgs mass measurements performed by ATLAS and CMS yields [6] m H = 125.09 ± 0.21 (stat.) ± 0.11 (syst.) GeV. (1) If the discovered particle is indeed the Higgs boson of the SM, its mass measurement determines the last unknown ingredient of this model, as all other properties of the electroweak sector then follow directly from theory. In the coming years a thorough investigation of the Higgs boson's properties is needed in order to identify whether the SM Higgs sector is indeed complete, or instead, the structure of a more involved Higgs sector is realized. This includes detailed and accurate measurements of its coupling strengths and CP structure at the LHC and ultimately at future experimental facilities for Higgs boson precision studies. Complementary to this, collider searches for additional Higgs bosons need to be continued over the full accessible mass range. The discovery of another Higgs boson would inevitably prove the existence of a non-minimal Higgs sector. In this work we consider the simplest extension of the SM Higgs sector, where an additional real scalar field is added, which is neutral under all quantum numbers of the SM gauge groups [7,8] and acquires a vacuum expectation value (VEV). This model has been widely studied in the literature , also in the context of electroweak higher order corrections [53,54] or offshell and interference effects [33,34,[55][56][57][58][59]. Here, we present an update of the exploration of the model parameter space presented in Ref. [38], where we take the latest experimental constraints into account. As before, we consider masses of the second (nonstandard) Higgs boson in the whole mass range up to 1 TeV. This minimal setup can be interpreted as a limiting case for more generic BSM scenarios, e.g. models with additional gauge sectors [60] or additional matter content [61,62]. Experimental searches for the model have been presented in [63 -70]. As in Ref. [38] we take the following theoretical and experimental constraints into account: bounds from perturbative unitarity and electroweak (EW) precision measurements, in particular focussing on higher order corrections to the W boson mass [32]; perturbativity, vacuum stability and correct minimization of the model up to a high energy scale using renormalization group (RG) evolved couplings; exclusion limits from Higgs searches at the LEP, Tevatron and LHC experiments via the public tool HiggsBounds [71][72][73][74][75], and compatibility of the model with the signal strength measurements of the discovered Higgs state using HiggsSignals [76] (cf. also Ref. [77]). We separate the discussion of the parameter space into two different mass regions: (i) the high mass region, m H ∈ [130, 1000] GeV, where the lighter Higgs boson h is interpreted as the discovered Higgs state; (ii) the low mass region, m h ∈ [1, 120] GeV, where the heavier Higgs boson H is interpreted as the discovered Higgs state. We find that the most severe constraints in the whole parameter space for the second Higgs mass m H 250 GeV are mostly given by limits from collider searches for a SM Higgs boson as well as by the LHC Higgs boson signal strength measurements. For m H 250 GeV limits from higher order contributions to the W boson mass prevail, followed by the requirement of perturbativity of the couplings. For the remaining viable parameter space we present predictions for signal cross sections of the yet undiscovered second Higgs boson for the LHC at a CM energy of 14 TeV, discussing both the SM Higgs decay signatures and the novel Higgs-to-Higgs decay mode H → hh. For both the high mass and low mass region we present a variety of benchmark scenarios. These are designed to render a maximal direct production rate for the collider signature of interest. Whenever kinematically accessible we give two different benchmark points for each mass, for which the Higgs-to-Higgs decay H → hh is maximal or minimal, respectively. The paper is organized as follows: In Section II we briefly review the model and the chosen parametrization. In Section III we review the constraints that are taken into account and in particular discuss the impact of the new constraints on the parameter space. In Section IV we provide benchmark points and planes discussed above. We summarize and conclude in Section V. II. THE MODEL In the following we briefly review the main features of the real Higgs singlet extension of the SM that are important for the benchmark choices. More details about the model can e.g. be found in Refs. [29,32,38,54] and references therein. A. Potential and couplings The real Higgs singlet extension of the SM [7,8,78] contains a complex SU (2) L doublet, in the following denoted by Φ, and in additional a real scalar S which is a singlet under the SM gauge group. The most general renormalizable Lagrangian compatible with an additional Z 2 symmetry is then given by L s = (D µ Φ) † D µ Φ + ∂ µ S∂ µ S − V (Φ, S) ,(2) 3 II The model with the scalar potential V (Φ, S) = −m 2 Φ † Φ − µ 2 S 2 + Φ † Φ S 2 λ 1 λ 3 2 λ 3 2 λ 2 Φ † Φ S 2 = −m 2 Φ † Φ − µ 2 S 2 + λ 1 (Φ † Φ) 2 + λ 2 S 4 + λ 3 Φ † ΦS 2 .(3) The implicitly imposed Z 2 symmetry forbids all linear or cubic terms of the singlet field S in the potential. We assume that both Higgs fields Φ and S have a non-zero vacuum expectation value (VEV), denoted by v and x, respectively. In the unitary gauge, the Higgs fields are given by Φ ≡ 0 h+v √ 2 , S ≡ h + x √ 2 .(4) After diagonalization of the mass matrix we obtain the mass eigenstates h and H with mass eigenvalues given by m 2 h = λ 1 v 2 + λ 2 x 2 − (λ 1 v 2 − λ 2 x 2 ) 2 + (λ 3 xv) 2 ,(5)m 2 H = λ 1 v 2 + λ 2 x 2 + (λ 1 v 2 − λ 2 x 2 ) 2 + (λ 3 xv) 2 ,(6) and m 2 h ≤ m 2 H by convention. The gauge and mass eigenstates are related via the mixing matrix h H = cos α − sin α sin α cos α h h ,(7) where the mixing angle − π 2 ≤ α ≤ π 2 is given by sin 2α = λ 3 xv (λ 1 v 2 − λ 2 x 2 ) 2 + (λ 3 xv) 2 ,(8)cos 2α = λ 2 x 2 − λ 1 v 2 (λ 1 v 2 − λ 2 x 2 ) 2 + (λ 3 xv) 2 .(9) It follows from Eq. (7) that the light (heavy) Higgs boson couplings to SM particles are suppressed by cos α (sin α). If kinematically allowed, the additional decay channel H → hh is present. Its partial decay width at leading order (LO) is given by [7,78] Γ H→hh = |µ | 2 8πm H 1 − 4m 2 h m 2 H ,(10) where the coupling strength µ of the H → hh decay reads µ = − sin (2α) 2vx (sin αv + cos α x) m 2 h + m 2 H 2 .(11) 4 Next-to-leading order (NLO) corrections to the H → hh decay width for this model have been calculated recently in Ref. [54]. The branching ratios of the heavy Higgs mass eigenstate m H are then given by BR H→hh = Γ H→hh Γ tot ,(12)BR H→SM = sin 2 α × Γ SM,H→SM Γ tot ,(13) where Γ SM, H→SM is the partial decay width of the SM Higgs boson and H → SM represents any SM Higgs decay mode. The total width is then Γ tot = sin 2 α × Γ SM, tot + Γ H→hh ,(14) where Γ SM, tot denotes the total width of the SM Higgs boson with mass m H . The suppression by sin 2 α directly follows from the suppression of all SM-like couplings, cf. Eq. (7). For µ = 0, the decay H → hh vanishes and we recover the SM Higgs boson branching ratios. For the collider phenomenology of the model two features are important: • the suppression of the production cross section of the two Higgs states induced by the mixing, which is given by sin 2 α (cos 2 α) for the heavy (light) Higgs, respectively; • the suppression of the Higgs decay modes to SM particles, which is realized if the competing decay mode H → hh is kinematically accessible. For the high mass (low mass) scenario, i.e. the case where the light (heavy) Higgs boson is identified with the discovered Higgs state at ∼ 125 GeV, | sin α| = 0 (1) corresponds to the complete decoupling of the second Higgs boson and therefore the SM-like scenario. B. Model parameters At the Lagrangian level, the model has five free parameters, λ 1 , λ 2 , λ 3 , v, x,(15) while the values of the additional parameters µ 2 , m 2 are fixed by the minimization conditions. A more intuitive basis, where the free model parameters are represented by physical (i.e. observable) quantities, is given by 1 m h , m H , sin α, v, tan β ≡ v x .(16) The vacuum expectation value of the Higgs doublet Φ is given by the SM value v ∼ 246 GeV, and one of the Higgs masses is fixed to m h/H = 125.09 GeV, eliminating two of the five parameters. We are thus left with only three independent parameters, m ≡ m H/h , sin α, tan β ,(17) where the latter enters the collider phenomenology only through the heavy Higgs decay mode into the lighter Higgs, H → hh. Note that from a collider perspective, for cases where the decay mode H → hh is kinematically allowed, the input parameter tan β could be replaced by either the total width of the heavier state, Γ(H), the branching ratio BR (H → hh), or the partial decay width of this channel, Γ(H → hh), respectively, rendering the following viable parameter choices besides Eq. (17): m ≡ m H/h , sin α, Γ(H) ,(18)m ≡ m H/h , sin α, BR(H → hh) ,(19)m ≡ m H/h , sin α, Γ(H → hh) .(20) If the insertion starts on the Lagrangian level (via e.g. FeynRules [79], SARAH [80,81] or similar), also the Lagrangian parameters as such can be used as input values, but then care must be taken to correctly translate these into the phenomenologically viable parameter regions. III. CONSTRAINTS In this section we list all theoretical and experimental constraints that we take into account, and give an overview over the impact of these constraints on the parameter space. We refer the reader to Ref. [38] for details on the implementation of these constraints. With respect to Ref. [38] we update the experimental limits from LHC Higgs searches, leading to a change in the allowed parameter space especially in the lower mass range, m H ∈ [130, 250] GeV. We also include constraints from the combined ATLAS and CMS Higgs signal strength [82], rendering a significantly stronger limit on the mixing angle. However, this limit is still not as strong as the constraint from the W boson mass measurement in most of the parameter space. A. Theoretical Constraints We consider the following theoretical constraints in the selection of the benchmark scenarios: • vacuum stability and minimization of model up to a scale µ run = 4 × 10 10 GeV, • perturbative unitarity of the 2 → 2 S-matrix for (W + W − , ZZ, hh, hH, HH) initial and final states, • perturbativity of the couplings in the potential, |λ i | ≤ 4 π, up to a high energy scale, µ run = 4 × 10 10 GeV, employing one-loop renormalization group equations (RGEs) [83]. B. Experimental Constraints The following experimental constraints are taken into account at the 95% C.L.: • agreement with electroweak precision observables, employing the oblique parameters S, T, U [84][85][86][87] and using the results from the global fit from the GFitter Group [88], • agreement with the observed W boson mass [89][90][91], M W = 80.385 ± 0.015 GeV, employing the NLO calculation presented in Ref. [32], • agreement with limits from direct Higgs searches at LEP, Tevatron, and the LHC using HiggsBounds (version 4.3.1) [71][72][73][74][75]. With respect to the results presented in Ref. [38], limits from the following searches have been included here: -ATLAS search for H → W W [92], -ATLAS search for H → ZZ [70], combination of ATLAS searches for H → hh → bbτ τ, γγW W * , γγbb, bbbb [67], • Agreement with the observed signal strengths of the 125 GeV Higgs boson, using HiggsSignals (version 1.4.0) [76], and using the results from the ATLAS and CMS combination of the LHC Run 1 data, µ = 1.09 ± 0.11 [82], leading to | sin α| ≤ 0.36 (21) for the heavy Higgs mass range m H 150 GeV (high mass range, m h ∼ 125 GeV), and | sin α| ≥ 0.87 (22) for the light Higgs mass range m h 100 GeV (low-mass range, m H ∼ 125 GeV). In these mass regions potential signal overlap with the SM-like Higgs at 125 GeV can be neglected. For Higgs masses in the range [100, 150] GeV we employ HiggsSignals using observables from the individual Higgs channels, which enables to approximately take into account a potential signal overlap [76], see also Ref. [38] for details. -CMS search for H → V V (V = W ± , Z) [66], -CMS search for H → hh → 4τ , C. Allowed Parameter Regions and Sensitivity of the Constraints High mass region The importance of the different constraints on the mixing angle sin α in the high mass region, where m h ∼ 125 GeV, is summarized in Figure 1. Recall that this angle is responsible for the global suppression of the production cross section with respect to the SM prediction at the same Higgs mass. We see that in the lower mass region, m H 250 GeV, the most important constraints stem from direct Higgs searches [66,70,[94][95][96] and the combined Higgs signal strength [82], whereas for higher masses, m H ∈ [250 GeV; 800 GeV], the W boson mass becomes the strongest constraint [32]. Requiring perturbativity of the couplings yields the upper limit on | sin α| for very heavy Higgs bosons, m H ≥ 800 GeV. The updated combined signal strength reduces the maximally allowed mixing angle from previously | sin α| 0.50 [38] Fig. 2. We see that the updated constraints yield stronger limits in particular for m H ≤ 250 GeV as well as for m H 400 GeV. We supplement this comparison by giving a detailed list in Tab. I of the LHC Higgs search channels that have been applied by HiggsBounds in the various mass regions. 2 The relatively strong constraints on the mixing angle lead to a significant suppression of the direct production rates of the heavy Higgs boson at LHC run 2. Fig. 3 shows the predicted production cross section at 14 TeV after all constraints have been taken into account. The production cross sections rapidly decrease with higher masses m H due to both the stronger constraints on the mixing angle (cf. Fig. 1) and a reduction of the available phase space for higher masses. The cross section for direct production in gluon fusion and successive decay into SM final states ranges from about 10 pb at lower masses to about 10 fb for masses around 800 GeV. Note that in order to obtain the predictions for a particular SM decay mode, H → XX, these numbers need to be multiplied by a factor of Note that these plots were obtained using a simple rescaling of production cross section of a SM Higgs boson of the same mass as given in Ref. [23], i.e. contributions due to interference with the additional scalar are not included. Tools which can handle these have been presented e.g. in Refs. [55,56,58,59]. These studies, however, focus on effects on the line-shape of the heavy scalar boson after a possible discovery. Moreover, thus far, their calculations neglect additional higher order corrections, whereas these have been calculated to great precision for the SM Higgs boson and are included in Fig. 3 [23]. For the future, it would be desirable to perform a dedicated study of interference effects including higher order corrections for the benchmark points presented in this work in order to estimate their effects (and the systematic uncertainty introduced here by neglecting them). Low mass region In the low mass region, where the heavier Higgs state takes the role of the discovered Higgs boson, m H ∼ 125 GeV, the parameter space is extremely constrained by the Higgs signal strength and exclusion limits from LEP Higgs searches [89]. The updated experimental results do not change the limits presented in Ref. [38]. We review these limits in Tab. II. Note that in the low mass region the couplings of the heavy Higgs boson at 125 GeV become SM-like for | sin α| = 1. Tab. III gives the direct production cross section in gluon fusion for the undiscovered light Higgs state at a 8 and 14 TeV LHC, respectively. Again, the production cross section stems from a simple rescaling of the corresponding cross section for a SM Higgs boson of that mass [23,98]. In the second column we give the lower limit on sin α stemming from exclusion limits from LEP or LHC Higgs searches (evaluated with HiggsBounds). If the lower limit on sin α obtained from the Higgs signal rates (evaluated with HiggsSignals) results in stricter limits, they are displayed in the third column. The fourth column displays the upper limit on tan β that stems from perturbative unitarity in the complete decoupling case (| sin α| = 1). In the fifth column we give the tan β value for which Γ H→hh = 0 is obtained given the maximal mixing angle allowed by the Higgs exclusion limits (second column). At this tan β value, the | sin α| limit obtained from the Higgs signal rates (third column) is abrogated. The table is taken from Ref. [38]. Intermediate mass region The intermediate mass region, where both Higgs bosons have masses between 120 GeV and 130 GeV, was originally discussed in Ref. [38]. In this mass region the observed Higgs signal at 125 GeV may be due to a signal overlap of both Higgs bosons, depending on the mass separation and the mass resolution of the experimental analysis. We show the allowed parameter space in the (m h , m H ) and (m h , sin α) plane from the updated fit in Fig. 4. The updated signal strength observables in HiggsSignals-1.4.0 yield only marginal improvements in the constrained parameter space, while the updated limits from direct Higgs searches are irrelevant in this mass region. IV. BENCHMARK SCENARIOS FOR LHC RUN 2 The benchmark scenarios that are presented in this section are chosen such that they feature the maximally allowed production cross section at the LHC. We first present the benchmark scenarios for the high mass region, where the light Higgs plays the role of the discovered SM-like Higgs at 125 GeV, and then turn to the low mass range, where the heavy Higgs state is the SM-like Higgs boson. 3 A. High mass region We distinguish between two different search channels: • Higgs decays into SM particles: Maximizing the production cross section corresponds to maximizing the parameter [29] κ ≡ σ σ SM × BR(H → SM) = sin 4 α Γ SM,tot Γ tot . In general, following Eq. (13) κ ≡ σ σ SM × BR(H → hh) = sin 2 α Γ H→hh Γ tot , is maximized to obtain the largest possible signal yield. Figure 5 shows the allowed range of these two quantities, after all constraints have been taken into account. For the Higgs decay channel into SM particles, we see that searches from CMS pose important constraints for m H 400 GeV. For the Higgs-to-Higgs decay channel H → hh, on the other hand, both ATLAS [67] and CMS [100,101] searches are not yet sensitive enough to exclude points that are not already in conflict with other constraints. We quantify the benchmark scenarios for both signal channels in this regime by considering the maximally allowed mixing angle together with the maximal and minimal branching ratio for the decay H → hh, respectively. While these maximal and minimal points define benchmark points, all BR(H → hh) values in between are in principle allowed. Therefore, an interpolation between the minimal and maximal values defines a higherdimensional benchmark scenario (benchmark slope or plane), where the additional third parameter (cf. Eq. (17)-(20)) is floating. We furthermore distinguish scenarios for which the H → hh on-shell decay mode is kinematically allowed or forbidden. As we neglect all other triple and quartic Higgs selfcouplings apart from µ , and work in the on-shell approximation, tan β only influences the collider phenomenology for regions in parameter space where the decay H → hh is kinematically allowed, i.e. for heavy Higgs masses m H ≥ 2m h ≈ 250 GeV. For lower masses tan β is irrelevant for the phenomenology considered here. However, to be consistent, we recommend to still keep the values within the respective parameter regions allowed by perturbativity and perturbative unitarity. Benchmark scenarios for both cases are given in Tab. IV and V, respectively. Parameter ranges which are not explicitly listed can to a first approximation be linearly interpolated. In addition, we also list exemplary benchmark points for this mass region in Tables VI and VII, where we additionally give the predictions for other relevant decay modes. Whenever kinematically accessible, we provide two benchmark points for every heavy Higgs mass, representing the maximal and minimal branching ratio for the H → hh decay, respectively. 4 The mixing angle is always chosen such that the production rate of the additional scalar is maximized. a and b). Reference production cross sections have been taken from the upcoming CERN Yellow Report 4 by the LHC Higgs Cross Section Working Group [104]. B. Low mass region For the case that the heavier Higgs boson is taken to be the discovered SM-like Higgs boson with m H ∼ 125 GeV, | sin α| = 1 corresponds to the SM limit, and deviations from this value parametrize the new physics contributions. As in the high mass region, the following channels are interesting: • Direct production of the lighter Higgs state h and successive decay into SM particles, • Decay of the SM-like Higgs boson H into the lighter Higgs states, H → hh. For the direct production of the light Higgs state smaller | sin α| values are of interest, as the cross section scales with cos 2 α. We provide the minimally allowed values for | sin α| in Tab. II. Tab. III lists the respective direct production cross sections at 8 and 14 TeV. These values can directly be used as benchmark scenarios for collider searches for direct light Higgs production. For the second channel -the decay of the SM-like Higgs into two lighter Higgs stateswe list maximal branching ratios for the decay H → hh in Tab. VIII. As long as the decay H → hh is kinematically accessible, the maximal value of its branching ratio, BR(H → hh) 0.259, is not dependent on the light Higgs mass. The lighter Higgs bosons then decay further according to the branching ratios of a SM Higgs of the respective mass. A first experimental search of this signature with the light Higgs boson decaying into τ lepton pairs in the mass range m h ∈ [5,15] GeV has already been performed by the CMS experiment [93]. We present benchmark points for fixed masses in Tab. IX. Here, | sin α| values closer to unity are needed in order to obtain maximal branching ratios for this channel, which in turn leads to the reduction of direct production for the lighter state by almost an order of magnitude with respect to the values presented in Tab. III. Again, we recommend to scan over tan β between the values of scenario a and b (thus defining a higher dimensional benchmark scenario) in order to obtain a range of possible branching ratios. a and b). In scenario b we have tan β = − cot α. The | sin α| values have been optimized for scenario a, which in turn leads to a suppression of direct production for the lighter state. For direct production of the lighter scalar, the parameters in Tab. II and III should be used. For BHM50 -BHM10, the production cross section for the SM like Higgs is σ(gg → H) = 49.66 pb. Reference production cross sections have been taken from the upcoming CERN Yellow Report 4 by the LHC Higgs Cross Section Working Group [104]. In this paper we have revisited and updated the constraints on the parameter space of the real scalar singlet extension of the SM. In comparison with the previous results presented in Ref. [38], the most important improvements have been made in the constraints from new results in LHC searches for a heavy Higgs boson decaying into vector boson final states, as well as from the ATLAS and CMS combination of the signal strength of the discovered Higgs state. We found that these modify our previous findings in the mass range 130 GeV ≤ m H ≤ 250 GeV, where now the direct Higgs searches as well as the ATLAS and CMS signal strength combination render the strongest constraints on the parameter space. Based on these updated results, we have provided benchmark scenarios for both the high mass and low mass region for upcoming LHC searches. Hereby, we pursued the philosophy of selecting those points which feature a maximal discovery potential in a dedicated collider search of the corresponding signature. We provided predictions of production cross sections for the LHC at 14 TeV, and supplemented these with information about the branching fractions of the relevant decay modes. We encourage the experimental collaborations to make use of these benchmark scenarios in the current and upcoming LHC runs. BR(H → XX)/BR(H → SM), where BR(H → SM) is the sum over all branching ratios of Higgs decays into SM particles according to Eq.(13). Taking into account the current design strategy for the LHC run (cf. e.g. Ref.[97]) and expecting an integrated luminosity of about 100 fb −1 and 300 fb −1 before the shutdowns in 2019 and 2023, respectively, this translates into the fact that at least O (10 3 ) heavy Higgs bosons could be produced in that mass range in optimistic scenarios. For the hh final state, on the other hand, cross sections are about an order of magnitude lower. A comparison of current exclusion limits from LHC H → hh searches with the predictions in the viable parameter space will be given in Section IV. Higgs signal rate with SM particles in the final state for the LHC at 14 TeV. (b) Heavy Higgs signal rate with light Higgs bosons in the final state for the LHC at 14 TeV. FIG. 3: LHC signal rates of the heavy Higgs boson H decaying into SM particles (a) or into two light Higgs bosons, H → hh, (b), in dependence of the heavy Higgs mass, m H , for a center-ofmass (CM) energy of 14 TeV.Shown are regions which are still allowed after all constraints are taken into account: Red and yellow regions correspond to agreement with the Higgs signal strength measurements at the 1σ and 2σ level, respectively, blue points comply with direct experimental searches but do not agree with the Higgs signal strength within 2σ. Light gray points denote scan points that are excluded by either perturbative unitarity, perturbativity of the couplings, RGE running or the W boson mass, while dark gray points denote regions in parameter space that obey these constraints but are excluded by direct searches. (a) (m h , m H ) plane. (b) (m h , sin α) plane. FIG. 4: Parameter space for the intermediate mass region after taking all constraints into account. The color coding follows Fig. 3. Higgs signal rate with SM particles in the final state. We display the observed and expected 95% C.L. limits from the CMS combination of SM Higgs searches [95] as well as from the H → V V (V = W, Z) search [66]. (b) Heavy Higgs signal rate with light Higgs bosons in the final state. We display the current expected and observed 95% C.L. limits from the ATLAS H → hh search (combination of various final states) [67] and CMS H → hh searches with γγbb [100] and bbbb [101] final states. FIG. 5: Collider signal rates of the heavy Higgs boson H decaying into SM particles (a) or into two light Higgs bosons, H → hh, (b), in dependence of the heavy Higgs mass, m H . The color coding is the same as inFig. 3. The rates are normalized to the inclusive SM Higgs production cross section at the corresponding mass value[23,102,103]. Fixed parameters M h = 125.1 GeV or M H = 125.1 GeV. Irrelevant parameters tan β whenever channel H → hh kinematically not accessible. additional comments predictions at LO, factorized production and decay; a,b signify maximal and minimal BR(H → hh); for b, sin α < 0; any values for tan β between scenario a and b are allowed. Production cross sections at 14 TeV [pb] and branching fractions BHM300 a,b Spectrum M H =300 GeV, | sin α| = 0.31, tan β (a) = 0.79, tan β (b) = 0.79σ(gg → h) 44.91 σ(gg → H) 1.09 BR(H → hh) 0.41 (a), 0.17 (b) BR(H → W W ) 0.41 (a), 0.57 (b) BR(H → ZZ) 0.18 (a), 0.25 (b) BHM400 a,b Spectrum M H =400 GeV, | sin α| = 0.26, tan β (a) = 0.58, tan β (b) = 0.59 σ(gg → h) 46.32 σ(gg → H) 0.76 BR(H → hh) 0.32 (a), 0.20 (b) BR(H → W W ) 0.40 (a), 0.47 (b) BR(H → ZZ) 0.18 (a), 0.22 (b) BR(H → tt) 0.10 (a), 0.12 (b) BHM500 a,b Spectrum M H =500 GeV, | sin α| = 0.24, tan β (a) = 0.44, tan β (b) = 0.46 σ(gg → h) 46.82 σ(gg → H) 0.31 BR(H → hh) 0.26 (a), 0.19 (b) BR(H → W W ) 0.41 (a), 0.44 (b) BR(H → ZZ) 0.19 (a), 0.21 (b) BR(H → tt)0.14 (a), 0.16 (b)TABLE VI: Benchmark scenarios for the high mass region for fixed masses and | sin α|, floating tan β (between scenariosa and b). Reference production cross sections have been taken from the upcoming CERN Yellow Report 4 by the LHC Higgs Cross Section Working Group[104].Production cross sections at 14 TeV [pb] and branching fractions (continued ) BHM600 a,b Spectrum M H =600 GeV, | sin α| = 0.22, tan β (a) = 0.37, tan β (b) = 0.38 σ(gg → h) 47.28 σ(gg → H) 0.12 BR(H → hh) 0.25 (a), 0.19 (b) BR(H → W W ) 0.41 (a), 0.45 (b) BR(H → ZZ) 0.21 (a), 0.22 (b) BR(H → tt) 0.13 (a), 0.14 (b) BHM700 a,b Spectrum M H =700 GeV, | sin α| = 0.21, tan β (a) = 0.31, tan β (b) = 0.32 σ(gg → h) 47.49 σ(gg → H) 0.050 BR(H → hh) 0.24 (a), 0.19 (b) BR(H → W W ) 0.44 (a), 0.47 (b) BR(H → ZZ) 0.22 (a), 0.23 (b) BR(H → tt) 0.10 (a), 0.11 (b) BHM800 a,b Spectrum M H =800 GeV, | sin α| = 0.2, tan β (a) = 0.25, tan β (b) = 0.27 σ(gg → h) 47.69 σ(gg → H) 0.022 BR(H → hh) 0.23 (a), 0.19 (b) BR(H → W W ) 0.46 (a), 0.48 (b) BR(H → ZZ) 0.23 (a), 0.24 (b) BR(H → tt) 0.08 (a), 0.09 (b) BHM200 Spectrum M H =200 GeV, | sin α| = 0.29, tan β = 1.19 σ(gg → h) 45.50 σ(gg → H) 1.74 BR(H → SM) as for a SM Higgs boson with mass of 200 GeV M h =60 GeV, | sin α| = 0.9997, tan β (a) = 3.48, tan β (b) = 0.025 σ(gg → h) 0.10 σ(gg → H) 49.65 BR(H → hh) 0.26 (a), 0 (b) BR(H → SM) rescaled by 0.74 (a), as in SM (b) BHM50 a,b Spectrum M h =50 GeV, | sin α| = 0.9998, tan β (a) = 3.25, tan β (b) = 0.020 σ(gg → h) 0.098 BR(H → hh) 0.26 (a), 0 (b) BR(H → SM) rescaled by 0.74 (a), as in SM (b) BHM40 a,b Spectrum M h =40 GeV, | sin α| = 0.9998, tan β (a) = 3.13, tan β (b) = 0.020 σ(gg → h) 0.16 BR(H → hh) 0.26 (a), 0 (b) BR(H → SM) rescaled by 0.74 (a), as in SM (b) BHM30 a,b Spectrum M h =30 GeV, | sin α| = 0.9998, tan β (a) = 3.16, tan β (b) = 0.020 σ(gg → h) 0.31 BR(H → hh) 0.26 (a), 0 (b) BR(H → SM) rescaled by 0.74 (a), as in SM (b) BHM20 a,b Spectrum M h =20 GeV, | sin α| = 0.9998, tan β (a) = 3.23, tan β (b) = 0.020 σ(gg → h) 0.90 BR(H → hh) 0.26 (a), 0 (b) BR(H → SM) rescaled by 0.74 (a), as in SM (b) BHM10 a,b Spectrum M h =10 GeV, | sin α| = 0.9998, tan β (a) = 3.29, tan β (b) = 0.020 σ(gg → h) 2.98 BR(H → hh) 0.26 (a), 0 (b) BR(H → SM) rescaled by 0.74 (a), as in SM (b) to | sin α| 0.36. The updated limits from LHC HiggsFIG. 1: Maximal allowed values for | sin α| in the high mass region, m H ∈ [130, 1000] GeV, from NLO calculations of the W boson mass (red, solid ) [32], electroweak precision observables (EWPOs) tested via the oblique parameters S, T and U (orange, dashed ), perturbativity of the RG-evolved coupling λ 1 (blue, dotted ), evaluated for an exemplary choice tan β = 0.1, perturbative unitarity (grey, dash-dotted ), direct LHC Higgs searches (green, dashed ), and the Higgs signal strength (magenta, dash-dotted ). FIG. 2: Comparison of the | sin α| limit obtained from the LHC Higgs searches with SM final states as presented in Ref. [38] (red) with the updated analysis (green).searches in channels with vector boson final states also generally lead to stronger constraints, except in the region m H ∈ [260, 300] GeV, where a statistical upward fluctuation in the CMS H → ZZ → 4 channel[66] leads to a slightly weaker limit than previously observed. A comparison of previously presented limits from LHC Higgs searches with the current status is displayed in0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 200 300 400 500 600 700 800 900 1000 | sinα | (upper limit) m H [GeV] W boson mass EW observables (S,T,U) λ 1 perturbativity (tanβ=0.1) perturbative unitarity (tanβ=0.1) LHC SM Higgs searches Higgs signal rates 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 200 300 400 500 600 700 800 900 1000 | sinα | (upper limit) m H [GeV] LHC searches from [1] updated results TABLE I : IList of LHC Higgs search channels that are applied by HiggsBounds in the high mass region, yielding the upper limit on | sin α| shown in Figs. 1 and 2. m h [ hGeV] | sin α| min, HB | sin α| min, HS (tan β) max (tan β) no H→hh120 0.410 0.918 8.4 - 110 0.819 0.932 9.3 - 100 0.852 0.891 10.1 - 90 0.901 - 11.2 - 80 0.974 - 12.6 - 70 0.985 - 14.4 - 60 0.978 0.996 16.8 0.21 50 0.981 0.998 20.2 0.20 40 0.984 0.998 25.2 0.18 30 0.988 0.998 33.6 0.16 20 0.993 0.998 50.4 0.12 10 0.997 0.998 100.8 0.08 TABLE II : IILimits on sin α and tan β in the low mass scenario for various light Higgs masses m h and tan β = 1. TABLE III : IIIMaximally allowed cross section for light Higgs production in gluon fusion, σ gg = cos 2 α max × σ gg,SM , at the LHC at CM energies of 8 and 14 TeV after all current constraints have been taken into account, corresponding to the mixing angles from Tab. II. This is an updated version of Tab. V in Ref.[38]. m H [ HGeV] | sin α| max tan β max m H [GeV] | sin α| max tan β max130 0.42 1.79 195 0.28 1.22 135 0.38 1.73 200 0.29 1.19 140 0.36 1.69 210 0.28 1.14 145 0.35 1.62 215 0.33 1.12 150 0.34 1.57 220 0.34 1.10 160 0.36 1.49 230 0.35 1.05 180 0.30 1.32 235 0.34 1.03 185 0.27 1.28 240 0.31 1.00 190 0.29 1.26 245 0.28 0.98 TABLE IV : IVBenchmark points for mass ranges where the onshell decay H → hh is kinematically forbidden. Maximal values of tan β were calculated at the maximal mixing angle, and should be applied for consistency reasons.m H [GeV] | sin α| max BR H→hhmin BR H→hh max m H [GeV] | sin α| max BR H→hh min BR H→hh max 255 0.31 0.09 0.27 430 0.25 0.19 0.30 260 0.34 0.11 0.33 470 0.24 0.19 0.28 265 0.33 0.13 0.36 520 0.23 0.19 0.26 280 0.32 0.17 0.40 590 0.22 0.19 0.25 290 0.31 0.18 0.40 665 0.21 0.19 0.24 305 0.30 0.20 0.40 770 0.20 0.19 0.23 325 0.29 0.21 0.40 875 0.19 0.19 0.22 345 0.28 0.22 0.39 920 0.18 0.19 0.22 365 0.27 0.21 0.36 975 0.17 0.19 0.21 395 0.26 0.20 0.32 1000 0.17 0.19 0.21 TABLE V : VMaximal and minimal allowed branching ratios of the decay H → hh, taken at the maximally allowed value of | sin α|. Note that mininal values for the BR(H → hh) stem from sin α ≤ 0.Benchmark Scenarios for the Real Singlet Main Features real singlet extension, with two vevs and no hidden sector interaction with heavy Higgs H and light Higgs h. TABLE VII : VIIBenchmark scenarios for the high mass region for fixed masses and | sin α|, floating tan β (between scenarios TABLE VIII : VIIIMaximal branching ratios for H → hh. This BR can always be zero for the choice tan β = − cot α. TABLE IX : IXLow mass benchmark scenarios for the Higgs-to-Higgs decay signature for fixed masses and | sin α|, floating tan β (between scenarios Note that even if the Z 2 symmetry is not imposed, the parameters of the model relevant for the collider phenomenology considered here can always be chosen in terms of the masses, a mixing angle, and an additional parameter determining the H → hh decay channel. HiggsBounds selects the most sensitive channel by comparing the expected exclusion limits first. In a second step, the predicted signal strength is confronted with the observed exclusion limit only of this selected channel. 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{'abstract': 'We present benchmark scenarios for searches for an additional Higgs state in the real Higgs singlet extension of the Standard Model in Run 2 of the LHC. The scenarios are selected such that they fulfill all relevant current theoretical and experimental constraints, but can potentially be discovered at the current LHC run. We take into account the results presented in earlier work and update the experimental constraints from relevant LHC Higgs searches and signal rate measurements. The benchmark scenarios are given separately for the low mass and high mass region, i.e. the mass range where the additional Higgs state is lighter or heavier than the discovered Higgs state at around 125 GeV.', 'arxivid': '1601.07880', 'author': ['Tania Robens \nTU Dresden\nInstitut für Kern-und Teilchenphysik\nZellescher Weg 19D-01069DresdenGermany\n', 'Tim Stefaniak \nDepartment of Physics\nSanta Cruz Institute for Particle Physics\nUniversity of California\n95064Santa CruzCAUSA\n'], 'authoraffiliation': ['TU Dresden\nInstitut für Kern-und Teilchenphysik\nZellescher Weg 19D-01069DresdenGermany', 'Department of Physics\nSanta Cruz Institute for Particle Physics\nUniversity of California\n95064Santa CruzCAUSA'], 'corpusid': 119205773, 'doi': '10.1140/epjc/s10052-016-4115-8', 'github_urls': [], 'n_tokens_mistral': 20703, 'n_tokens_neox': 16338, 'n_words': 8707, 'pdfsha': 'a3d3a0ac8becaa45f2f4c45f9914c179b84a9cbe', 'pdfurls': ['https://arxiv.org/pdf/1601.07880v2.pdf'], 'title': ['LHC Benchmark Scenarios for the Real Higgs Singlet Extension of the Standard Model', 'LHC Benchmark Scenarios for the Real Higgs Singlet Extension of the Standard Model'], 'venue': []}

arxiv

Kondo effects and shot noise enhancement in a laterally coupled double quantum dot 21 Dec 2010 Toshihiro Kubo JST, ICORP Quantum Spin Information Project Atsugi-shi243-0198KanagawaJapan NTT Basic Research Laboratories NTT Corporation Atsugi-shi243-0198KanagawaJapan Yasuhiro Tokura JST, ICORP Quantum Spin Information Project Atsugi-shi243-0198KanagawaJapan NTT Basic Research Laboratories NTT Corporation Atsugi-shi243-0198KanagawaJapan Seigo Tarucha JST, ICORP Quantum Spin Information Project Atsugi-shi243-0198KanagawaJapan Department of Applied Physics University of Tokyo Bunkyo-ku113-8656Hongo, TokyoJapan Kondo effects and shot noise enhancement in a laterally coupled double quantum dot 21 Dec 2010(Dated: December 23, 2010)APS/123-QEDnumbers: 7363Kv7215Qm7323-b7340Gk The spin and orbital Kondo effects and the related shot noise for a laterally coupled double quantum dot are studied taking account of coherent indirect coupling via a reservoir. We calculate the linear conductance and shot noise for various charge states to distinguish between the spin and orbital Kondo effects. We find that a novel antiferromagnetic exchange coupling can be generated by the coherent indirect coupling, and it works to suppress the spin Kondo effect when each quantum dot holds just one electron. We also show that we can capture the feature of the pseudospin Kondo effect from the shot noise measurement. I. INTRODUCTION The Kondo effect was discovered many years ago in metals with dilute magnetic impurities and has long been studied as one of the most important many-body correlations in condensed matter physics 1,2 . We have obtained the physical understanding in equilibrium Kondo systems using the powerful methods such as exact solution and numerical renormalization group (RG) approach 3,4 More recently it has been predicted that the Kondo effect occurs in semiconductor quantum dots (QDs) 5,6 , and indeed, it has been observed in transport measurements for various kinds of QDs 7 . In a single QD system, the Kondo effect gives rise to the enhancement of the conductance, and the conductance reaches the value of 2e 2 /h at the unitary limit 8,9 . Since then the Kondo effect in QDs has been attracting a lot of new interests associated with extended degrees of freedom, such as tunnel coupling to reservoirs, the number of trapped electrons in a QD, and the number of Kondo channels. By tuning these parameters, various aspects of the Kondo effect have been revealed including enhancement induced by state degeneracy 10 , the unitary limit 11 , and the nonequilibrium Kondo effect 12 . Therefore, QDs are regarded as artificial Kondo systems, in which various theoretical approximations can be tested to acquire a better understanding of strongly correlated electron systems. In particular, the nonequilibrium Kondo problem is not yet solved completely despite the large number of theoretical studies. The nonequilibrium magnetization of the QD was revisited using the Schwinger-Keldysh perturbation formalism 13 . When the large bias voltage or a magnetic field is applied, the transport through the QD was studied by the perturbative RG approach [14][15][16] (so-called poor man's scaling developed by Anderson 17 ). Using the real-time perturbation theory in Schwinger-Keldysh formulation, the universal properties that the perturbative series of any average in the steady state satisfies the equilibrium Callan-Symanzik equations 18 . By the real-time RG in frequency space, the nonequilibrium anisotropic Kondo model was examined in the weak coupling regime, where the maximum of bias voltage and magnetic field is larger than the Kondo temperature 19 . In the framework of the same approach, the dynamical spin-spin correlation function was calculated in nonequilibrium Kondo systems describing spin and/or orbital fluctuations 20 . Using the generalized flow equation approach to include a magnetic field similar to the real-time RG performed by Schoeller et al., the spin-spin correlation function, the T -matrix, and the magnetization were calculated as a function of applied magnetic field 21 . By a nonequilibrium RG method, the real-time evolution of spin and current in the anisotropic Kondo model were investigated at a finite magnetic field and bias voltage 22 . Recently, in not only the single QD but also the double quantum dot (DQD) systems, the Kondo effects have been studied. In particular, the interplay between the Kondo effect and inter-dot correlation was discussed [23][24][25][26] . It is theoretically predicted that the two-channel Kondo model realized in a DQD system exhibits a non-Fermi liquid quantum critical point 27 . Such a two-channel Kondo problem was experimentally investigated 28 . Moreover, the Kondo problem is more intriguing in DQDs than in single QDs because of the competition between the spin Kondo effect and the orbital (pseudospin) Kondo effect [29][30][31][32][33] . In a DQD, the pseudospin state is represented as a state with an electron in either of two capacitively coupled QDs but separately contacted to a pair of reserviors (see Fig. 1(a)). It has been predicted that the SU (4) Kondo effect will provide a novel feature for a highly symmetric DQD configuration 31,34 . However, it is difficult to realize the SU (4) condition experimentally since the intra-dot Coulomb interaction is usually larger than the inter-dot Coulomb interaction. The pseudospin Kondo effect is only defined in DQDs, and has recently been confirmed experimentally, but not in reference to the interplay with the spin Kondo effect 35 . In contrast with an ideal DQD as shown in Fig. 1(a), most experiments are performed for DQDs with an integrated reservoir (see Fig. 1(b)). In such DQDs, the pseudospin-dependent linewidth function is induced by the coherent indirect coupling FIG. 1: Schematic diagrams of laterally coupled DQDs with a separated drain reservoir. s12 is the minimum distance that electrons propagate in the source reservoir. (a) The source and drain reservoirs are both completely separated, namely there is no coupling between two QDs via the reservoirs. This situation corresponds to s12 → ∞. (b) The source reservoir is common and the drain reservoir is separated. (c) There is maximal coherence between two QDs via the reservoirs. This condition corresponds to s12 = 0. via the integrated reservoir 36 . The effect of the indirect coupling on the spin Kondo effect in DQDs with integrated reservoirs have been discussed only where the indirect coupling is at its maximum value as shown in Fig. 1(c) [37][38][39] . However, most of the actual experimental conditions correspond to an intermediate condition (for example 40 ), and so it is important to study the effects of indirect coupling on the Kondo effect. Moreover, theoretical studies often focus on a situation where two QDs are energetically aligned. The pseudospin Kondo effect strongly depends on the charge states in the DQD. Therefore, it is useful to employ the entire charge state diagram to capture the signature of the pseudospin Kondo effect. In this paper, we investigate the effects of coherent indirect coupling via a reservoir on the Kondo effects in a laterally coupled DQD. We employ the finite Coulomb interaction slave-boson mean-field theory (SBMFT) 41 using the nonequilibrium Green's function method. This approach allows us to take account of the coherence between two QDs nonperturbatively. To characterize the indirect coupling, we introduce a coherent indirect coupling parameter α 42,43 . For finite α, the pseudospin Kondo effect is suppressed since the linewidth function depends on the pseudospin and the SU (2) symmetry is broken 36 . Here we newly find that the coherent indirect coupling leads to novel antiferromagnetic kinetic exchange coupling between two local spins in QDs via the reservoir. This kinetic exchange coupling via the reservoir competes with the Kondo exchange coupling, and hence the spin Kondo effect is suppressed when each QD holds just one electron. Such a phenomenon can occur in parallel but not series coupled DQDs. Then, we examine the shot noise to devise a new approach for characterizing the pseudospin Kondo effect. To distinguish between the spin Kondo effect and pseudospin Kondo effect can be difficult in standard conductance measurements. We find that the shot noise experiment can provide a clear contrast between them. The shot noise has recently been discussed extensively in relation to charge fluctuations in mesoscopic systems 44 . The current noise S(ω) is defined by a Fourier transform of S(t, t ′ ) = {δI(t), δI(t ′ )} , where δI(t) ≡ I(t) − I(t) is the amount by which the current deviates from its average value. The equilibrium zero-frequency current noise S(0) cannot carry additional information beyond the conductance. In contrast, the nonequilibrium zero-frequency shot noise can provide information on charge fluctuations, which is not accessible in conventional transport measurements. The source-drain bias voltage dependence of the shot noise through a QD in the spin Kondo regime has been studied theoretically 45 . Here, the pseudospin Kondo effect is generally promoted by the charge fluctuation, so we examine the shot noise in the charge stability diagram, and find that it is strongly enhanced in the pseudospin Kondo regime. This paper is organized as follows. In Sec. II, a standard tunneling Hamiltonian is employed to describe a laterally coupled DQD. We introduce the notion of the coherent indirect coupling for the source reservoir. We provide the expressions of the linear conductance and the zero-frequency shot noise at zero temperature using the nonequilibrium Green's function method. We discuss the numerical results for the linear conductance and zero-frequency shot noise at zero temperature in Sec. III. In particular, we derive the new antiferromagnetic kinetic exchange coupling induced by a coherent indirect coupling via the reservoir. We show that the spin-spin correlation is antiferromagnetic. All our results are summarized in Sec. IV. In Appendix A, we provide the detailed derivation of the effective spin-spin Hamiltonian with an antiferromagnetic kinetic exchange coupling induced by a coherent indirect coupling using the 4th-order Rayleigh-Schrödinger degenerate perturbation theory. II. MODEL AND FORMULATION We consider a DQD tunnel coupled to one common source reservoir and two drain reservoirs as shown in Fig. 1(b). We assume only a single energy level for each QD. The Hamiltonian represents the sum of the following terms: H = H R + H DQD + H T . The Hamiltonian of the Fermi liquid reservoirs is H R = ν∈{S,D1,D2} k σ∈{↑,↓} ǫ νk a νkσ † a νkσ ,(1) where ǫ νk is the electron energy with wave number k in the reservoir ν and the operator a νkσ (a νkσ † ) annihilates (creates) an electron with spin σ in the reservoirs. H DQD describes an isolated DQD, H DQD = 2 i=1 σ∈{↑,↓} ǫ i n iσ + 2 i=1 U i n i↑ n i↓ + V inter σ∈{↑,↓} σ ′ ∈{↑,↓} n 1σ n 2σ ′ ,(2) where ǫ i is the energy level of the ith QD, U i is the on-site Coulomb interaction in the ith QD, and V inter is the inter-dot Coulomb interaction. Here the following notations are introduced: c iσ (c iσ † ) is an annihilation (creation) operator of an electron in the ith QD with spin σ and n iσ = c iσ † c iσ is its number operator. The tunneling Hamiltonian between the QDs and source and drain reservoirs is given by H T = k 2 i=1 σ∈{↑,↓} t (i) Sk a Skσ † c iσ + t Dik a Dikσ † c iσ + H.c. .(3) We take account of the propagation process of electrons in the source reservoir. This leads to coherent indirect coupling between two QDs via the source reservoir 42,43 , which is characterized by a parameter α. The linewidth function matrices are then given by Γ S σ = Γ S 1 α α 1 , Γ D1 σ = Γ D 1 0 0 0 , Γ D2 σ = Γ D 0 0 0 1 ,(4) where the boldface notation indicates a matrix whose basis is the localized state in each QD. Here we assume that |t (1) Sk | 2 = |t (2) Sk | 2 ≡ |t Sk | 2 and |t D1k | 2 = |t D2k | 2 ≡ |t Dk | 2 . The linewidth function is defined by Γ ν (ǫ) ≡ (2π/ ) k |t νk | 2 δ(ǫ − ǫ νk ), and we neglected its energy dependence in the wide-band limit, namely Γ ν (ǫ) = Γ ν . α is a function of the propagation length s 12 of the electrons in the reservoir 43 , and in general, |α| ≤ 1. The condition s 12 = 0 is equivalent to α = 1 46 . The importance of the sign of the coherent indirect coupling parameter was pointed out by S. A. Gurvitz 47 . The wave number dependence of the tunneling amplitude t (j) νk is usually neglected in the theoretical treatment. However, in case the two QDs are indirectly coupled via the source reservoir as assumed here, the wave number dependence of the tunneling amplitude plays an important role in generating an indirect hopping between the QDs. As explained later, such an indirect hopping process causes an antiferromagnetic kinetic exchange coupling. The mechanism is similar to that by a direct inter-dot coupling mechanism 25 , however, for the indirect inter-dot coupling, the exchange coupling constant includes information of coherence in the source reservoir. We use the finite Coulomb interaction SBMFT 41 to investigate the linear conductance and shot noise through DQDs. In this approach, the slave-boson operators are replaced by nonfluctuating average values, leading to a noninteracting resonant tunneling model, whose 28 unknown parameters have to be determined self-consistently. The result obtained with this method agrees fairly well with a numerical Lanczos calculation and a numerical renormalization group calculation for a tunnel-coupled DQD 25,48,49 . The tunneling current through a DQD can be expressed in terms of the transmission matrix 50 , I = e h 2 i=1 σ∈{↑,↓} dǫ[f S (ǫ) − f Di (ǫ)]Tr {T iσ (ǫ)} .(5) Here the transmission matrix is defined as T iσ (ǫ) = G r σ (ǫ)Γ S σ G a σ (ǫ)Γ Di σ using the retarded (advanced) Green's function G r σ (ǫ) (G a σ (ǫ)) of the DQD, and f ν (ǫ) = 1/(1 + e (ǫ−µν )/kB T ) is the Fermi-Dirac distribution function in the reservoir ν at temperature T . Within the framework of the finite Coulomb interaction SBMFT, the retarded Green's function is given by G r σ (ǫ) = ǫ−ǫ1 + i 2Γ 11,σ i 2Γ 12,σ i 2Γ 21,σ ǫ−ǫ2 + i 2Γ 22,σ −1 ,(6) whereǫ i andΓ ij,σ are the renormalized energy level of the ith QD and the (i, j) matrix element of the linewidth function matrix for spin σ. Such renormalizations indicate the Coulomb interaction effects. The advanced Green's function is obtained from the retarded Green's function: G a σ (ǫ) = [G r σ (ǫ)] † . The source and drain reservoirs have chemical potentials µ S = µ + eV SD /2 and µ Di = µ − eV SD /2 with the source-drain bias voltage V SD , and µ = 0 is the origin of the energy. Here we assume that the two drain reservoirs have the same chemical potential. In the following, we focus on the zero temperature condition. Then, the linear conductance through the ith QD is given by G i = e 2 h σ∈{↑,↓} T iσ ,(7) where T iσ ≡ Tr {T iσ (0)} is the transmission probability of the conduction channel for spin σ in the ith QD. Within the framework of the SBMFT, the zero-frequency shot noise is given by the Khlus-Lesovik formula 51,52 , S(0) = e 2 π 2 i=1 σ∈{↑,↓} eVSD /2 −eVSD /2 dǫ Tr {T iσ (ǫ) [1 − T iσ (ǫ)]} = e 2 π 2 i=1 σ∈{↑,↓} eVSD /2 −eVSD /2 dǫ T iσ (ǫ)[1 − T iσ (ǫ)],(8) where T iσ (ǫ) = Tr {T iσ (ǫ)}. In our problem, although the transmission matrix has finite off-diagonal elements for α = 0, the zero-frequency shot noise can be expressed as Eq. (8) in terms of the simple summation of T iσ (ǫ)[1 − T iσ (ǫ)] for each conduction mode since the drain reservoirs are separated and there is no indirect coupling. III. THEORETICAL RESULTS A. Linear transport In the following discussions, we assume that U 1 / Γ = U 2 / Γ ≡ U/ Γ = 2V inter / Γ = 6, and Γ S = Γ D = Γ as a typical example, and to show the charge configurations, we introduce the notation (N 1 , N 2 ), where N i is the population of the ith QD. First, we consider the situation without coherent indirect coupling, namely α = 0, shown in Fig. 1(a). The total linear conductance G = G 1 + G 2 is shown in Fig. 2(a) as a function of ǫ 1 and ǫ 2 (charge stability diagram). The conductance is suppressed owing to the Coulomb blockade in the (0, 0), (2, 0), (0, 2), and (2, 2) regimes. In the (1, 0), (0, 1), (2, 1), and (1, 2) regimes, G ≃ 2e 2 /h since the conductance is enhanced as a result of the spin Kondo effects. In the (1, 1) regime, we have the double spin Kondo effect, namely spin Kondo effects in each QD, and the conductance value reaches 4e 2 /h. Without depending on the ratio between U and V inter , the linear conductance can reach 4e 2 /h at ǫ 1 / Γ = ǫ 2 / Γ = −(U/2 + V inter )/ Γ, namely the center of the (1, 1) region 53 . In Fig. 2(b), we plot the energy offset ∆ǫ(≡ ǫ 1 − ǫ 2 ) dependence of the linear conductance along the white line in Fig. 2(a). ∆ǫ = 0 corresponds to ǫ 1 = ǫ 2 = −V inter /2. For the spinless electrons, the linear conductance cannot exceed 2e 2 /h in the pseudospin Kondo regime, namely the (1, 0) − (0, 1), (2, 0) − (1, 1), (1, 1) − (0, 2), and (2, 1) − (1, 2) boundaries. However, when the spin and pseudospin degrees of freedom are entangled, G exceeds 2e 2 /h as shown in Fig. 2(b). For a large ∆ǫ, G approaches 2e 2 /h since the situation becomes equivalent to that of the spin Kondo regime in a single QD. These results are qualitatively consistent with those obtained with the numerical renormalization group method 34 . Next, we consider the effect of α. In Fig. 3(a), we show the conductance difference ∆G α between G of α = 0.5 and G of α = 0. From Fig. 3(a), we find that the linear conductance decreases only in the (1, 1) charge configuration. In this (1, 1) charge configuration, the coherent indirect coupling gives rise to antiferromagnetic kinetic exchange coupling as follows: We consider the tunneling Hamiltonian (3) as a perturbation, and we calculate the effective spinspin interaction Hamiltonian using the 4th-order Rayleigh-Schrödinger degenerate perturbation theory, namely the effective Hamiltonian is given as H α ef f = H T 1 E−H0 H T 1 E−H0 H T 1 E−H0 H T , where H 0 ≡ H R + H DQD is the unperturbed Hamiltonian. As a result, we obtain the following effective spin-spin interaction Hamiltonian: H α ef f ≃ J α S 1 · S 2 with J α = 16ǫ F α Γ πU 2 ,(9) where S i is the spin operator of the ith QD and ǫ F is the Fermi energy. Here we consider the possibility to observe this exchange coupling experimentally. To observe the Kondo effect, we usually use the QD systems in the strong coupling regime, namely large Γ, since the Kondo temperature becomes higher. Thus, we expect to be experimentally possible to verify the antiferromagnetic kinetic exchange interaction induced by the coherent indirect coupling in the strong coupling QD systems since the factor Γ/U in Eq. (9) is not small in such systems. We provide the detailed derivation of this antiferromagnetic kinetic exchange interaction in Appendix A. This kinetic exchange coupling competes with the Kondo exchange coupling. Therefore, in the (1, 1) regime, the spin Kondo effect is suppressed with the increase in |α| and hence the conductance decreases as shown in Fig. 3(b). This suppression is independent of the sign of α. In inset of Fig. 3(b), we plot the |α| dependence of the conductance when ǫ 1 / Γ = ǫ 2 / Γ = −6 as indicated by green circle in Fig. 3(a). The linear conductance decreases monotonically with increasing |α|. Similarly, we show the spin-spin correlation function S 1 · S 2 in Fig. 3 (c). We evaluate the spin-spin correlation function S 1 · S 2 using the nonequilibrium Green's functions as follows: S 1 · S 2 = 3 8π 2 dω dǫ G −+ 21,σ (ǫ)G +− 12,σ (ǫ + ω),(10) where G −+ ij,σ (ǫ) and G +− ij,σ (ǫ) are the (i, j) matrix element of the lesser and greater Green's functions for spin σ. These can be obtained from the retarded and advanced Green's functions using the Keldysh equation 54 as follows G −+ σ (ǫ) = i ν∈{S,D1,D2} f ν (ǫ)G r σ (ǫ)Γ ν σ G a σ (ǫ),(11)G +− σ (ǫ) = −i ν∈{S,D1,D2} [1 − f ν (ǫ)]G r σ (ǫ)Γ ν σ G a σ (ǫ).(12) When |α| increases, S 1 · S 2 increases negatively. This means that the antiferromagnetic kinetic exchange coupling becomes dominating as |α| increases. The Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction is well known as an indirect exchange interaction between two local spins [55][56][57] . In the RKKY interaction the exchange coupling becomes weak with changing the sign between positive and negative, and therefore changing the magnetic character between ferromagnetic and antiferromagnetic as the two local spins become separated from each other. The RKKY interaction in semiconductor QD systems has been studied both theoretically and experimentally 58,59 . Particularly when α = 1 for both source and drain reservoirs with ∆ǫ = 0, Konik discussed the RKKY-Kondo like effect in a similar type DQD 39 . In contrast we concentrate on the competition between the Kondo exchnage and J α when ∆ǫ = 0 and α = 0 for the drain reservoir. If we investigate this competition when ∆ǫ = 0 and α = 1 for both the source and drain reservoirs, we expect the single channel Kondo effect (the exchange coupling caused by the coherent indirect coupling vanishes as shown in Appendix B) since there is only a single conduction mode in such a situation, namely one of the two orbital channels is in a dark state 43 . Here, although we considered the effect of the integrated reservoir only for the source, we can expect stronger suppression of the spin Kondo effect in the (1, 1) regime when both the source and drain reservoirs are integrated with 0 < |α| = 1. It is noted that the kinetic exchange coupling induced by a coherent indirect coupling is different from the RKKY exchange coupling. The main difference between these two exchange interactions is where the dependence of the inter-dot distance is included. In our exchange interaction the coherent indirect coupling parameter α is a decision factor, and the interaction strength is proportional to |α| 2 , and the magnetic character is always antiferromagnetic. Note that α becomes small and changes the sign with increasing distance between the two local spins. In the RKKY interaction, although the wave number dependence of the response function is considered, the wave number dependence of the tunneling amplitude t (i) Sk is neglected, just like the case for an impurity as a point scatterer, to account for the oscillatory behavior of the exchange coupling with the distance between the impurities. However, it is very important to take account of the wave number dependence of the tunneling amplitude in DQD systems since the wave function of electron confined in QDs relatively spreads, and the tunnel couplings are highly anisotropic. Thus, in the present problem, we believe that it is preferable to discuss our exchange interaction in terms of the coherent indirect coupling than the RKKY exchange interaction. B. Shot noise Interplay or competition between the spin and pseudospin Kondo effects can appear in the linear transport characteristic as shown in Fig. 2(b). However, it is still difficult to distinguish their contributions. This is particularly the case in experiments, because the spin Kondo conductance observed for single QDs is usually less than 2e 2 /h. To capture the feature of the pseudospin Kondo effect, which originates with the charge fluctuation, we investigate the shot noise, which provides information on charge fluctuations. In the following, we focus on the condition where eV SD / Γ = 0.1. First, we consider the situation without coherent indirect coupling. The zero-frequency shot noise is shown in Fig. 4(a). In the (0, 0), (2, 0), (0, 2), and (2, 2) regimes, the shot noise is strongly suppressed because of the Coulomb blockade. In the (1, 0), (0, 1), (1, 1), (2, 1), and (1, 2) regimes, the shot noise is also strongly suppressed since a perfect transmission is realized by the spin Kondo effect. By contrast, the zero-frequency shot noise is enhanced at the Coulomb peaks owing to the maximum charge fluctuations in one of the two QDs. As an example, we consider the (0, 0)−(1, 0) boundary. In this situation, the transmission probabilities T 1↑ and T 1↓ of two conduction channels for the up and down spins in QD1, respectively, are T 1↑ = T 1↓ = 1/2, and thus the shot noise becomes large. Moreover, the zero-frequency shot noise is enhanced in the pseudospin Kondo regimes, because the charge fluctuation is maximal, as shown in Fig. 4(a). In the pseudospin Kondo regimes, there can be four conduction channels, for example, for the (1, 0) − (0, 1) boundary, T 1↑ = T 1↓ = T 2↑ = T 2↓ ≃ 1/2. As a result, the shot noise in the pseudospin Kondo regimes is about double that at the Coulomb peaks. Therefore, the shot noise in the charge stability diagram is maximal in the pseudospin Kondo regime, and the signature can be easily captured experimentally. It should be noted that the shot noise enhancement discussed here cannot be obtained in calculations of the mean-field level such as the Hartree-Fock approximation, and thus the many-body correlation is essential. Next, we discuss the effects of the coherent indirect coupling on the shot noise. First, in Fig. 4(b), we show the shot noise difference ∆S α between S(0) of |α| = 0.5 and S(0) of α = 0. We found that the spin Kondo effects are suppressed with |α| in the (1, 1) regime. In this regime, the transmission probabilities of all the conduction channels become smaller than 1 due to the kinetic antiferromagnetic exchange coupling induced by the coherent indirect coupling. As a result, the shot noise becomes large. We plot the QD energy dependence of the shot noise as shown in Fig. 4(c). When |α| increases, the shot noise is mainly affected in the (1, 1) regime. In Fig. 4(d), we plot the |α| dependence of the shot noise when ǫ 1 / Γ = ǫ 2 / Γ = −6 indicated by the green circle in Fig. 4(b). The value of transmission probabilities for all conduction modes are the same since we consider the condition when the two QD energies align. As shown in Fig. 3(c), the value of the transmission probability for each conduction mode are approximately equal to 1/2 at |α| ∼ 0.97 under low bias voltage since the linear conductance is proportional to the transmission probability (see Eq. (7)). Therefore, from Eq. (8), the zero-frequency shot noise becomes maximal at |α| ∼ 0.97. IV. CONCLUSIONS To conclude, we have studied the effects of inter-dot coherent indirect coupling via the reservoir on the Kondo effect and shot noise in a laterally coupled DQD using the finite-Coulomb interaction SBMFT to demonstrate the significance of many-body correlations. In particular, we found that the coherent indirect coupling gives rise to antiferromagnetic kinetic exchange coupling using the 4th-order Rayleigh-Schrödinger perturbation theory. Thus the spin Kondo effect is suppressed in the (1, 1) regime. To support that the new exchange coupling is antiferromagnetic, we estimate the spin-spin correlation function. The spin-spin correlation function increases negatively as the coherent indirect coupling parameter increases. We discussed the difference between the RKKY exchange coupling and the new antiferromagnetic exchange coupling induced by the coherent indirect coupling. Moreover, we suggested that shot noise measurement is more appropriate than conductance measurement for capturing the signature of the pseudospin Kondo effect, because the shot noise is strongly enhanced in the pseudospin Kondo regime. FIG. 2 : 2Total linear conductance G for α = 0 and U1/ Γ = U2/ Γ = 2Vinter/ Γ = 6. (a) The charge configuration is shown as (N1, N2). The white dotted line indicates the charge degeneracy line schematically. (b) ∆ǫ dependence of the linear conductance along the white line in (a). The broken, dotted, and solid lines indicate the conductance G1, G2, and the total conductance G, respectively. FIG. 3 : 3Reduction of the linear conductance caused by the coherent indirect coupling and the spin-spin correlation function. (a) ∆Gα for α = 0.5. (b) G for ǫ1 = ǫ2. The solid, broken, dotted, and dash-dotted line lines indicate α = 0, |α| = 0.5, |α| = 0.8, and |α| = 1, respectively. Inset: |α| dependence of the linear conductane at ǫ1/ Γ = ǫ2/ Γ = −6 indicated by the green circle in (a). (c) |α| dependence of the spin-spin correlation function at ǫ1/ Γ = ǫ2/ Γ = −6 indicated by the green circle in (a). FIG. 4 : 4Shot noise S(0) and shot noise difference ∆Sα for U1/ Γ = U2/ Γ = 6, Vinter/ Γ = 3, and eVSD/ Γ = 0.1. The charge configuration is shown as (N1, N2). (a) S(0) for α = 0. (b) ∆Sα for |α| = 0.5. (c) S(0) for ǫ1 = ǫ2. The solid, broken, dotted, and dash-dotted line lines indicate α = 0, |α| = 0.5, |α| = 0.8, and |α| = 1, respectively. (d) |α| dependence of the zero-frequency shot noise at ǫ1/ Γ = ǫ2/ Γ = −6 indicated by the green circle in (b). AcknowledgmentsWe thank S. A. GurvitzAppendix A: Derivation of antiferromagnetic kinetic exchange interaction by coherent indirect couplingHere we show the detailed derivation of the antiferromagnetic kinetic exchange interaction induced by the coherent indirect coupling as discussed in Sec. III. Starting from the state d 1↑ † d 2↓ † |F , where the state |F corresponds to the Fermi seas of conduction electrons in the source reservoir S with empty DQD, we consider the tunneling Hamiltonian as a perturbation and derive the effective spin-spin interaction Hamiltonian using the 4th-order Rayleigh-Schrödinger degenerate perturbation theory. Then, we consider the following process:whereis the unperturbed Hamiltonian, E is its ground state energy, and H T is the tunneling Hamiltonian. Only the source reservoir is essential for the coherent indirect coupling. Thus, in the following, we consider only the source reservoir part of the tunneling Hamiltonian and omit the index S for clarity. As a result, we obtain 32 terms that contribute to the kinetic exchange interaction. In such contributions, the most dominant contribution has the formwhere η is positive infinitesimal, and we focused on the particle-hole symmetric condition, namely ǫ 1 = ǫ 2 = −V inter − U 2 . These have one electron-hole excitation pair in the intermediate states, and this pair leads to the energy denominator of ǫ k − ǫ k ′ . In Eq. (A3), we only need to consider the low energy excitation in the vicinity of the Fermi surface because of the energy denominator ǫ k − ǫ k ′ and the condition ǫ k ′ < ǫ F < ǫ k , where ǫ F is the Fermi energy. Moreover, we can neglect ǫ k in ǫ k ± U 2 since |ǫ k | ≪ U 2 . Thus, we haveAlthough we have to estimate the wave number integration, according to the prescription given in Ref.43, the azimuthal integration gives rise to the oscillatory behavior of the coherent indirect coupling parameter with respect to the propagation length, and the radial integration isIn the wide-band limit, we neglect the energy dependence of the linewidth functions, and thus we obtainwhere we have neglected the spin-independent terms. Therefore, the exchange coupling constant isAppendix B: Effect of coherent indirect coupling for both source and drain reservoirs on spin-spin correlationIn this Appendix, we show the α dependence of the spin-spin correlation when the coherent indirect couplings are considered for both the source and drain reservoirs as discueed in Sec. III A. Then, we define the coherent indirect coupling parameter of the reservoir ν (ν ∈ {S, D}) as α ν . Then, inFig. 5, we plot the |α S | dependence of the spin-spin correlation function for various quotients between α S and α D at ǫ 1 / Γ = ǫ 2 / Γ = −6 indicated by the green circle inFig. 3 (a). It is clear that we have a stronger suppression of Kondo effect due to an antiferromagnetic kinetic exchange coupling induced by the coherent indirect couplings for both the source and drain reservoirs in comparison with the result shown inFig. 3 (c). As shown inFig. 5, the spin-spin correlation vanishes at |α S | = |α D | = 1. 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{'abstract': 'The spin and orbital Kondo effects and the related shot noise for a laterally coupled double quantum dot are studied taking account of coherent indirect coupling via a reservoir. We calculate the linear conductance and shot noise for various charge states to distinguish between the spin and orbital Kondo effects. We find that a novel antiferromagnetic exchange coupling can be generated by the coherent indirect coupling, and it works to suppress the spin Kondo effect when each quantum dot holds just one electron. We also show that we can capture the feature of the pseudospin Kondo effect from the shot noise measurement.', 'arxivid': '1012.4838', 'author': ['Toshihiro Kubo \nJST, ICORP\nQuantum Spin Information Project\nAtsugi-shi243-0198KanagawaJapan\n\nNTT Basic Research Laboratories\nNTT Corporation\nAtsugi-shi243-0198KanagawaJapan\n', 'Yasuhiro Tokura \nJST, ICORP\nQuantum Spin Information Project\nAtsugi-shi243-0198KanagawaJapan\n\nNTT Basic Research Laboratories\nNTT Corporation\nAtsugi-shi243-0198KanagawaJapan\n', 'Seigo Tarucha \nJST, ICORP\nQuantum Spin Information Project\nAtsugi-shi243-0198KanagawaJapan\n\nDepartment of Applied Physics\nUniversity of Tokyo\nBunkyo-ku113-8656Hongo, TokyoJapan\n'], 'authoraffiliation': ['JST, ICORP\nQuantum Spin Information Project\nAtsugi-shi243-0198KanagawaJapan', 'NTT Basic Research Laboratories\nNTT Corporation\nAtsugi-shi243-0198KanagawaJapan', 'JST, ICORP\nQuantum Spin Information Project\nAtsugi-shi243-0198KanagawaJapan', 'NTT Basic Research Laboratories\nNTT Corporation\nAtsugi-shi243-0198KanagawaJapan', 'JST, ICORP\nQuantum Spin Information Project\nAtsugi-shi243-0198KanagawaJapan', 'Department of Applied Physics\nUniversity of Tokyo\nBunkyo-ku113-8656Hongo, TokyoJapan'], 'corpusid': 119200914, 'doi': '10.1103/physrevb.83.115310', 'github_urls': [], 'n_tokens_mistral': 13550, 'n_tokens_neox': 11634, 'n_words': 6865, 'pdfsha': 'a601f7fd195223c77f75cb10967fede1ac6101be', 'pdfurls': ['https://arxiv.org/pdf/1012.4838v1.pdf'], 'title': ['Kondo effects and shot noise enhancement in a laterally coupled double quantum dot', 'Kondo effects and shot noise enhancement in a laterally coupled double quantum dot'], 'venue': []}

arxiv

Centrality Dependence of Direct Photon Production in √ s NN = 200 GeV Au+Au Collisions 7 Mar 2005 S S Adler Brookhaven National Laboratory Upton11973-5000NYUSA S Afanasiev Joint Institute for Nuclear Research 141980Dubna, Moscow RegionRussia C Aidala Brookhaven National Laboratory Upton11973-5000NYUSA N N Ajitanand Chemistry Department SUNY, Stony Brook Stony Brook University 11794-3400NYUSA Y Akiba KEK, High Energy Accelerator Research Organization Tsukuba-shi, Ibaraki-ken 305-0801Japan RIKEN (The Institute of Physical and Chemical Research) 351-0198WakoSaitamaJAPAN J Alexander Chemistry Department SUNY, Stony Brook Stony Brook University 11794-3400NYUSA R Amirikas Florida State University 32306TallahasseeFLUSA L Aphecetche SUBATECH (Ecole des Mines de Nantes, CNRS-IN2P3 Université de Nantes) BP 20722 -44307NantesFrance S H Aronson Brookhaven National Laboratory Upton11973-5000NYUSA R Averbeck Department of Physics and Astronomy SUNY, Stony Brook Stony Brook University 11794NYUSA T C Awes Oak Ridge National Laboratory 37831Oak RidgeTNUSA R Azmoun Department of Physics and Astronomy SUNY, Stony Brook Stony Brook University 11794NYUSA V Babintsev Institute for High Energy Physics (IHEP) ProtvinoRussia A Baldisseri Dapnia CEA Saclay, F-91191Gif-sur-YvetteFrance K N Barish University of California -Riverside 92521RiversideCAUSA P D Barnes Los Alamos National Laboratory 87545Los AlamosNMUSA B Bassalleck University of New Mexico 87131AlbuquerqueNMUSA S Bathe Institut für Kernphysik University of Muenster D-48149MuensterGermany S Batsouli and Nevis Laboratories Columbia University 10027, 10533New York, IrvingtonNY, NYUSA V Baublis Petersburg Nuclear Physics Institute PNPI GatchinaRussia A Bazilevsky Institute for High Energy Physics (IHEP) ProtvinoRussia RIKEN BNL Research Center, Brookhaven National Laboratory Upton11973-5000NYUSA S Belikov Institute for High Energy Physics (IHEP) ProtvinoRussia Iowa State University 50011AmesIAUSA Y Berdnikov St. Petersburg State Technical University St. PetersburgRussia S Bhagavatula Iowa State University 50011AmesIAUSA J G Boissevain Los Alamos National Laboratory 87545Los AlamosNMUSA H Borel Dapnia CEA Saclay, F-91191Gif-sur-YvetteFrance S Borenstein Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS-IN2P3 Route de Saclay, F-91128PalaiseauFrance M L Brooks Los Alamos National Laboratory 87545Los AlamosNMUSA D S Brown New Mexico State University Las Cruces88003NMUSA N Bruner University of New Mexico 87131AlbuquerqueNMUSA D Bucher Institut für Kernphysik University of Muenster D-48149MuensterGermany H Buesching Institut für Kernphysik University of Muenster D-48149MuensterGermany V Bumazhnov Institute for High Energy Physics (IHEP) ProtvinoRussia G Bunce Brookhaven National Laboratory Upton11973-5000NYUSA RIKEN BNL Research Center, Brookhaven National Laboratory Upton11973-5000NYUSA J M Burward-Hoy Lawrence Livermore National Laboratory 94550LivermoreCAUSA Department of Physics and Astronomy SUNY, Stony Brook Stony Brook University 11794NYUSA S Butsyk Department of Physics and Astronomy SUNY, Stony Brook Stony Brook University 11794NYUSA X Camard SUBATECH (Ecole des Mines de Nantes, CNRS-IN2P3 Université de Nantes) BP 20722 -44307NantesFrance J.-S Chai KAERI, Cyclotron Application Laboratory SeoulSouth Korea P Chand Bhabha Atomic Research Centre 400 085BombayIndia W C Chang Institute of Physics Academia Sinica 11529TaipeiTaiwan S Chernichenko Institute for High Energy Physics (IHEP) ProtvinoRussia C Y Chi and Nevis Laboratories Columbia University 10027, 10533New York, IrvingtonNY, NYUSA J Chiba KEK, High Energy Accelerator Research Organization Tsukuba-shi, Ibaraki-ken 305-0801Japan M Chiu and Nevis Laboratories Columbia University 10027, 10533New York, IrvingtonNY, NYUSA I J Choi Yonsei University 120-749SeoulKorea J Choi Kangnung National University 210-702KangnungSouth Korea R K Choudhury Bhabha Atomic Research Centre 400 085BombayIndia T Chujo Brookhaven National Laboratory Upton11973-5000NYUSA V Cianciolo Oak Ridge National Laboratory 37831Oak RidgeTNUSA Y Cobigo Dapnia CEA Saclay, F-91191Gif-sur-YvetteFrance B A Cole and Nevis Laboratories Columbia University 10027, 10533New York, IrvingtonNY, NYUSA P Constantin Iowa State University 50011AmesIAUSA D D&apos;enterria SUBATECH (Ecole des Mines de Nantes, CNRS-IN2P3 Université de Nantes) BP 20722 -44307NantesFrance G David Brookhaven National Laboratory Upton11973-5000NYUSA H Delagrange SUBATECH (Ecole des Mines de Nantes, CNRS-IN2P3 Université de Nantes) BP 20722 -44307NantesFrance A Denisov Institute for High Energy Physics (IHEP) ProtvinoRussia A Deshpande RIKEN BNL Research Center, Brookhaven National Laboratory Upton11973-5000NYUSA E J Desmond Brookhaven National Laboratory Upton11973-5000NYUSA A Devismes Department of Physics and Astronomy SUNY, Stony Brook Stony Brook University 11794NYUSA O Dietzsch Instituto de Física Universidade de São Paulo 66318, CEP05315-970São PauloCaixa PostalBrazil O Drapier Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS-IN2P3 Route de Saclay, F-91128PalaiseauFrance A Drees Department of Physics and Astronomy SUNY, Stony Brook Stony Brook University 11794NYUSA R Du Rietz Department of Physics Lund University Box 118221 00LundSESweden A Durum Institute for High Energy Physics (IHEP) ProtvinoRussia D Dutta Bhabha Atomic Research Centre 400 085BombayIndia Y V Efremenko Oak Ridge National Laboratory 37831Oak RidgeTNUSA K El Chenawi Vanderbilt University 37235NashvilleTNUSA A Enokizono Hiroshima University Kagamiyama, Higashi-Hiroshima 739-8526Japan H B D Fox RIKEN BNL Research Center, Brookhaven National Laboratory Upton11973-5000NYUSA Z Fraenkel Weizmann Institute 76100RehovotIsrael J E Frantz and Nevis Laboratories Columbia University 10027, 10533New York, IrvingtonNY, NYUSA A Franz Brookhaven National Laboratory Upton11973-5000NYUSA A D Frawley Florida State University 32306TallahasseeFLUSA S.-Y Fung University of California -Riverside 92521RiversideCAUSA S Garpman Department of Physics Lund University Box 118221 00LundSESweden T K Ghosh Vanderbilt University 37235NashvilleTNUSA A Glenn University of Tennessee 37996KnoxvilleTNUSA G Gogiberidze University of Tennessee 37996KnoxvilleTNUSA M Gonin Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS-IN2P3 Route de Saclay, F-91128PalaiseauFrance J Gosset Dapnia CEA Saclay, F-91191Gif-sur-YvetteFrance Y Goto RIKEN BNL Research Center, Brookhaven National Laboratory Upton11973-5000NYUSA R Granier De Cassagnac Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS-IN2P3 Route de Saclay, F-91128PalaiseauFrance N Grau Iowa State University 50011AmesIAUSA S V Greene Vanderbilt University 37235NashvilleTNUSA M Grosse Perdekamp RIKEN BNL Research Center, Brookhaven National Laboratory Upton11973-5000NYUSA W Guryn Brookhaven National Laboratory Upton11973-5000NYUSA H.-Å Gustafsson Department of Physics Lund University Box 118221 00LundSESweden T Hachiya Hiroshima University Kagamiyama, Higashi-Hiroshima 739-8526Japan J S Haggerty Brookhaven National Laboratory Upton11973-5000NYUSA H Hamagaki Center for Nuclear Study Graduate School of Science University of Tokyo 7-3-1 Hongo113-0033TokyoJapan A G Hansen Los Alamos National Laboratory 87545Los AlamosNMUSA E P Hartouni Lawrence Livermore National Laboratory 94550LivermoreCAUSA M Harvey Brookhaven National Laboratory Upton11973-5000NYUSA R Hayano Center for Nuclear Study Graduate School of Science University of Tokyo 7-3-1 Hongo113-0033TokyoJapan N Hayashi RIKEN (The Institute of Physical and Chemical Research) 351-0198WakoSaitamaJAPAN X He Georgia State University 30303AtlantaGAUSA M Heffner Lawrence Livermore National Laboratory 94550LivermoreCAUSA T K Hemmick Department of Physics and Astronomy SUNY, Stony Brook Stony Brook University 11794NYUSA J M Heuser Department of Physics and Astronomy SUNY, Stony Brook Stony Brook University 11794NYUSA M Hibino Advanced Research Institute for Science and Engineering Waseda University 17 Kikui-cho, Shinjuku-ku162-0044TokyoJapan J C Hill Iowa State University 50011AmesIAUSA W Holzmann Chemistry Department SUNY, Stony Brook Stony Brook University 11794-3400NYUSA K Homma Hiroshima University Kagamiyama, Higashi-Hiroshima 739-8526Japan B Hong Korea University 136-701SeoulKorea A Hoover New Mexico State University Las Cruces88003NMUSA J H Kang Yonsei University 120-749SeoulKorea S S Kapoor Bhabha Atomic Research Centre 400 085BombayIndia K Katou Advanced Research Institute for Science and Engineering Waseda University 17 Kikui-cho, Shinjuku-ku162-0044TokyoJapan S Kelly and Nevis Laboratories Columbia University 10027, 10533New York, IrvingtonNY, NYUSA B Khachaturov Weizmann Institute 76100RehovotIsrael A Khanzadeev Petersburg Nuclear Physics Institute PNPI GatchinaRussia J Kikuchi Advanced Research Institute for Science and Engineering Waseda University 17 Kikui-cho, Shinjuku-ku162-0044TokyoJapan D H Kim Myongji University Kyonggido 449-728YonginKorea D J Kim Yonsei University 120-749SeoulKorea D W Kim Kangnung National University 210-702KangnungSouth Korea E Kim System Electronics Laboratory Seoul National University SeoulSouth Korea G.-B Kim Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS-IN2P3 Route de Saclay, F-91128PalaiseauFrance H J Kim E Kistenev Brookhaven National Laboratory Upton11973-5000NYUSA Yonsei University 120-749SeoulKorea A Kiyomichi Institute of Physics University of Tsukuba 305TsukubaIbarakiJapan K Kiyoyama Nagasaki Institute of Applied Science Nagasaki-shi851-0193NagasakiJapan C Klein-Boesing Institut für Kernphysik University of Muenster D-48149MuensterGermany H Kobayashi RIKEN (The Institute of Physical and Chemical Research) 351-0198WakoSaitamaJAPAN RIKEN BNL Research Center, Brookhaven National Laboratory Upton11973-5000NYUSA L Kochenda Petersburg Nuclear Physics Institute PNPI GatchinaRussia V Kochetkov Institute for High Energy Physics (IHEP) ProtvinoRussia D Koehler University of New Mexico 87131AlbuquerqueNMUSA T Kohama Hiroshima University Kagamiyama, Higashi-Hiroshima 739-8526Japan M Kopytine Department of Physics and Astronomy SUNY, Stony Brook Stony Brook University 11794NYUSA D Kotchetkov University of California -Riverside 92521RiversideCAUSA A Kozlov Weizmann Institute 76100RehovotIsrael P J Kroon Brookhaven National Laboratory Upton11973-5000NYUSA C H Kuberg Abilene Christian University 79699AbileneTXUSA Los Alamos National Laboratory 87545Los AlamosNMUSA K Kurita RIKEN BNL Research Center, Brookhaven National Laboratory Upton11973-5000NYUSA Y Kuroki Institute of Physics University of Tsukuba 305TsukubaIbarakiJapan M J Kweon Korea University 136-701SeoulKorea Y Kwon Yonsei University 120-749SeoulKorea G S Kyle New Mexico State University Las Cruces88003NMUSA R Lacey Chemistry Department SUNY, Stony Brook Stony Brook University 11794-3400NYUSA V Ladygin Joint Institute for Nuclear Research 141980Dubna, Moscow RegionRussia J G Lajoie Iowa State University 50011AmesIAUSA A Lebedev Iowa State University 50011AmesIAUSA Russian Research Center "Kurchatov Institute" MoscowRussia S Leckey Department of Physics and Astronomy SUNY, Stony Brook Stony Brook University 11794NYUSA D M Lee Los Alamos National Laboratory 87545Los AlamosNMUSA S Lee Kangnung National University 210-702KangnungSouth Korea M J Leitch Los Alamos National Laboratory 87545Los AlamosNMUSA X H Li University of California -Riverside 92521RiversideCAUSA H Lim System Electronics Laboratory Seoul National University SeoulSouth Korea A Litvinenko Joint Institute for Nuclear Research 141980Dubna, Moscow RegionRussia M X Liu Los Alamos National Laboratory 87545Los AlamosNMUSA Y Liu IPN-Orsay Universite Paris Sud CNRS-IN2P3 BP1, F-91406OrsayFrance C F Maguire Vanderbilt University 37235NashvilleTNUSA Y I Makdisi Brookhaven National Laboratory Upton11973-5000NYUSA A Malakhov Joint Institute for Nuclear Research 141980Dubna, Moscow RegionRussia V I Manko Russian Research Center "Kurchatov Institute" MoscowRussia Y Mao Institute of Atomic Energy (CIAE) BeijingChina, People's Republic of China RIKEN (The Institute of Physical and Chemical Research) 351-0198WakoSaitamaJAPAN G Martinez SUBATECH (Ecole des Mines de Nantes, CNRS-IN2P3 Université de Nantes) BP 20722 -44307NantesFrance M D Marx Department of Physics and Astronomy SUNY, Stony Brook Stony Brook University 11794NYUSA H Masui Institute of Physics University of Tsukuba 305TsukubaIbarakiJapan F Matathias Department of Physics and Astronomy SUNY, Stony Brook Stony Brook University 11794NYUSA T Matsumoto Center for Nuclear Study Graduate School of Science University of Tokyo 7-3-1 Hongo113-0033TokyoJapan Advanced Research Institute for Science and Engineering Waseda University 17 Kikui-cho, Shinjuku-ku162-0044TokyoJapan P L Mcgaughey Los Alamos National Laboratory 87545Los AlamosNMUSA E Melnikov Institute for High Energy Physics (IHEP) ProtvinoRussia F Messer Department of Physics and Astronomy SUNY, Stony 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Upton11973-5000NYUSA F K Wohn Iowa State University 50011AmesIAUSA C L Woody Brookhaven National Laboratory Upton11973-5000NYUSA W Xie University of California -Riverside 92521RiversideCAUSA Y Yang Institute of Atomic Energy (CIAE) BeijingChina, People's Republic of China A Yanovich Institute for High Energy Physics (IHEP) ProtvinoRussia S Yokkaichi RIKEN (The Institute of Physical and Chemical Research) 351-0198WakoSaitamaJAPAN RIKEN BNL Research Center, Brookhaven National Laboratory Upton11973-5000NYUSA G R Young Oak Ridge National Laboratory 37831Oak RidgeTNUSA I E Yushmanov Russian Research Center "Kurchatov Institute" MoscowRussia W A Zajc and Nevis Laboratories Columbia University 10027, 10533New York, IrvingtonNY, NYUSA C Zhang and Nevis Laboratories Columbia University 10027, 10533New York, IrvingtonNY, NYUSA S Zhou Institute of Atomic Energy (CIAE) BeijingChina, People's Republic of China S J Zhou Weizmann Institute 76100RehovotIsrael L Zolin Joint Institute for Nuclear Research 141980Dubna, Moscow RegionRussia LPC Université Blaise Pascal CNRS-IN2P3 Clermont-Fd63177Aubiere CedexFrance Centrality Dependence of Direct Photon Production in √ s NN = 200 GeV Au+Au Collisions 16457 Mar 2005(Dated: March 30, 2022)arXiv:nucl-ex/0503003v1 (PHENIX Collaboration) 2PACS numbers: 2575Dw The first measurement of direct photons in Au+Au collisions at √ s N N = 200 GeV is presented.The direct photon signal is extracted as a function of the Au+Au collision centrality and compared to NLO pQCD calculations. The direct photon yield is shown to scale with the number of nucleonnucleon collisions for all centralities. The first measurement of direct photons in Au+Au collisions at √ s N N = 200 GeV is presented. The direct photon signal is extracted as a function of the Au+Au collision centrality and compared to NLO pQCD calculations. The direct photon yield is shown to scale with the number of nucleonnucleon collisions for all centralities. One of the most exciting observations from experiments at the Relativistic Heavy Ion Collider (RHIC) is * Deceased † PHENIX Spokesperson:zajc@nevis.columbia.edu the strong suppression of the yield of hadrons at large transverse momenta (p T > 2 GeV/c) in central Au+Au collisions, as compared to measured yields in p + p collisions scaled by the number of binary nucleon-nucleon collisions [1,2,3,4]. Such quenching was predicted to result from the energy loss of hard-scattered partons propagating through the high density matter created in heavy ion collisions [5]. It was later proposed that the observed hadron suppression could be an initial-state effect due to saturation of the initial parton distributions in large nuclei [6]. The high-p T hadron suppression was not observed in d+Au collisions [7,8]. This indicates that the suppression in Au+Au collisions is due to the extended dense matter in the final state, that is absent in d+Au collisions. Measurement of direct photon production allows more definitive discrimination between initial-and final-state suppression due to the fact that photons, once produced, are essentially unaffected by the surrounding matter. Hence photons produced directly in initial parton scatterings will not be quenched unless the initial parton distributions are suppressed in the nucleus. In fact, there may be additional direct photon yield in AA collisions [9] due to various processes such as momentum broadening of the incoming partons, additional fragmentation contributions [10,11], or additional scatterings in the thermalizing dense matter of the final state. This letter reports on direct photon production in Au+Au collisions at √ s N N = 200 GeV with data taken by the PHENIX experiment [12] during the second RHIC run (2001). This analysis used the Beam-Beam Counters (BBC, 3.0 < |η| < 3.9) and the Zero Degree Calorimeter (ZDC) for trigger and event characterization, the Electromagnetic Calorimeter (EMCal) in the two central arms (|η| ≤ 0.35) to measure the inclusive γ, π 0 , and η yields, and the tracking system of the central arms to estimate the charged particle contamination. The EMCal consists of two subsystems: six sectors of lead-scintillator sandwich calorimeter (PbSc) and two sectors of lead-glass Cherenkov calorimeter (PbGl). Located at a radial distance of about 5 m each sector covers an azimuthal interval of ∆φ ≈ 22.5 • . The fine segmentation of the EMCal (∆φ×∆η ∼ 0.01×0.01) ensures that the two photons from a decayed π 0 are well-resolved up to transverse momenta of 15-20 GeV/c. The event centrality was selected by cuts on the correlated distribution of charged particles detected in the BBCs versus energy measured in the ZDC detectors. A Glauber model Monte Carlo combined with a simulation of the BBC and ZDC responses gave an estimate of the associated number of binary collisions (N coll ) and participating nucleons (N part ) for each centrality bin (values tabulated in Ref. [3]). For this analysis a mimimum bias trigger sample of 30 × 10 6 events, also used for the previously published π 0 analysis [3], was combined with a Level-2 trigger event sample equivalent to an additional 55 × 10 6 minimum bias events. The Level-2 trigger sample was obtained by use of an EMCal software trigger on highly energetic showers equivalent to the Level-1 hardware trigger used in Ref. [13]. The threshold energy of the trigger was set at 3.5 GeV with a resulting trigger efficiency plateau at 100% for single photons above p T ≈ 5 GeV/c (6.5 GeV/c) for the PbSc (PbGl). The normalization of the Level-2 data sample relative to the minimum bias data sample is accurate to 2%. In the following, the minimum bias result refers to the combined Level-2 and minimum bias trigger samples without selection on centrality. The direct photon yield is extracted on a statistical basis, without isolation cuts, by a comparison of the inclusive photon spectra to the expected background from hadronic decays [14,15] (mainly π 0 → 2γ). Photon-like clusters are identified in the EMCal by applying appropriate Particle Identification (PID) cuts based on timeof-flight and the shower profile. The consistency of the final results obtained independently with the PbSc and PbGl, and with different PID cuts, including no PID cut, is used to check the systematic error estimates. The π 0 and η yields are determined as described in [3,16] by an invariant mass analysis of photon pairs, with the combinatorial background established by combining uncorrelated photon pairs from different events. The raw inclusive photon-candidate spectra must be corrected for charged and neutral hadron contaminations not removed by the PID cuts, as well as for photon conversions. Charged contaminants are identified by associating photon candidates in the EMCal with charged hits in the pad chamber (PC3) positioned directly in front of the EMCal. The charged contaminant spectra are subtracted from the photon-candidate spectra. The charged hadron contamination depends strongly on the PID cut and increases significantly for p T < 3 GeV/c with a contribution of 4% above 3 GeV/c for the tightest PID cut. The contamination of neutral hadrons (mainly antineutrons) is determined with a full GEANT simulation of the detector response to neutrons and anti-neutrons with input spectra based on the proton and anti-proton yields measured by PHENIX [4]. The neutral hadron contamination is found to be negligible above p T = 5 GeV/c (< 1%). The neutral photon-candidate spectra are corrected for conversions removed by the charged contaminant subtraction with a p T -independent factor (5.9-7.3% for different sectors based on simulation). The raw spectra are normalized to one unit of rapidity and full azimuth (the purely geometrical acceptance correction is ∼ 1/0.35) . The spectra are further corrected for (i) the detector response (energy resolution, dead areas), (ii) the reconstruction efficiency (PID cuts), and (iii) occupancy effects (cluster overlaps). These corrections are quantified by embedding simulated single γ's, π 0 's, or η's from a full PHENIX GEANT simulation into real events, and analyzing the merged events with the same analysis cuts used to obtain the real yields. The overall π 0 yield correction was ∼2.5 with a centrality dependence of < ∼ 25%. The losses were dominated by fiducial and asymmetry cuts. The nominal energy resolution was adjusted in the simulation by smearing the energies with a constant term of ∼ 5% for PbSc and ∼ 7% for PbGl to reproduce the measured width of the π 0 peak observed at each p T . The shape, position, and width of the π 0 peak measured for all centralities were confirmed to be well reproduced by the embedded data. The energy calibration of the EMCal was corroborated by the position of the π 0 invariant mass peak, by the energy deposit from minimum ionizing charged particles traversing the EMCal (PbSc), and by the correlation between the measured momentum of electron and positron tracks identified by the ring-imaging Cherenkov detector and the associated energy deposit in the EMCal. From these studies it is determined that the accuracy of the energy measurement was better than 1.5%. The main sources of systematic errors in the PbSc and PbGl measurements are the uncertainties in: (i) the yield extraction, (ii) the yield correction, and (iii) the energy scale. The relative contributions of these effects to the total error differ for the PbSc and PbGl (Table I). The weighted average of the two independent measurements reduces the total error. The final systematic errors on the spectra are at the level of ∼ 15 − 20% (Table I). A correction for the true mean value of the p T bin is applied to the steeply falling spectra. The completely corrected and combined PbSc and PbGl inclusive photon yields are compared to the expected yields of background photons from hadronic decays in Fig. 1 for minimum bias Au+Au collisions and for five centrality bins. The decay photon calculations are based on the measured π 0 and η spectra [16] assuming m T -scaling for all other radiative decays (η ′ ,K 0 s ,ω). The comparison is made as the ratio of measured (inclusive) γ/π 0 and calculated background γ/π 0 since this has the advantage that many uncertainties, such as the energy scale, cancel to varying extent in the ratio. Since the π 0 spectra of the background calculations are taken to be the same as the measured spectra we have R γ = γ/π 0 Measured (γ/π 0 ) Background ≈ γ Measured γ Background(1) and any significant deviation of the double ratio above unity indicates a direct photon excess. In Fig. 1 an excess is observed at high p T with a magnitude that increases with increasing centrality of the collision. The measured results are compared to NLO pQCD predictions [17], scaled by the number of binary nucleon collisions for each centrality selection. The same calculations are in agreement with the PHENIX direct photon measurement [15] for p + p collisions at the same √ s, and similar NLO pQCD calculations provide a good description of the measured π 0 production in p+p collisions [13]. The calculations were performed [15,17] with normalization and factorization scales equal to p T , the CTEQ6 [18] set of parton distribution functions, and the GRV set of fragmentation functions [19]. The direct photon spectra extracted as γ Direct = (1 − R −1 γ ) · γ Measured are shown in Fig. 2 for all nine centrality selections as well as minimum bias, and compared to the same NLO calculations. The binary collision scaled predictions are seen to provide a good description of the measured direct photon spectra (Fig. 2). The increasing ratio with centrality seen in Fig. 1 is therefore attributed to the decreasing decay background due to π 0 suppression [3]. Medium effects in AA collisions are often presented using the nuclear modification factor given as the ratio of the measured AA invariant yields to the N N -collisionscaled p + p invariant yields: R AA (p T ) = (1/N evt AA ) d 2 N AA /dp T dy N coll /σ inel pp × d 2 σ pp /dp T dy ,(2) where the N coll /σ inel pp is the average nuclear thickness function, T AA , in the centrality bin under consideration (Ref [3]). R AA (p T ) measures the deviation of AA data from an incoherent superposition of N N collisions. The centrality dependence of the high p T γ production represented as a function of the number of participating nucleons, N part , is shown by the closed circles in Fig. 3. The production in Au+Au collisions relative to p + p is characterized by the R AA (p T > 6 GeV/c) ratio of Eq. (2) as the ratio of Au+Au over the N coll -scaled p + p yields each integrated above 6 GeV/c. The direct photon p + p yields are taken as the NLO pQCD predictions described above. As noted above, the high p T direct γ production is observed to scale as the N coll -scaled p+ p γ yield prediction for all centralities. This is in sharp contrast [3] to the centrality dependence of the π 0 R AA (p T > 6 GeV/c) shown by open circles in Fig. 3 where the measured π 0 yield [13] is used as the p + p reference in Eq. (2). The observed close agreement between the measured yields and NLO calculations is in striking contrast to observations for central Pb+Pb collisions at √ s N N = 17.3 GeV [14] where the measured photon yield exceeds the N coll -scaled p + p yield by about a factor of two. The present result constrains modifications of the initial parton distributions, or of the fragmentation contributions [10,11] (in these NLO calculations the contribution is significant: ∼ 50% at 3.5 GeV/c and ∼ 35% at 10 GeV/c), or additional photon yield from thermal radiation to levels comparable to the present measurement uncertainty. In summary, the transverse momentum spectra of direct photons have been measured at mid-rapidity up to p T ≈ 13 GeV/c for nine centrality bins of Au+Au collisions at √ s N N = 200 GeV. The significance of the direct photon signal increases with collision centrality due to the increasingly suppressed π 0 production and associated decrease in the photon background from hadron decays. The direct photon spectral shapes and invariant yields are consistent with NLO pQCD predictions for p + p reactions scaled by the average number of inelastic N N collisions for each centrality class. The close agreement between measurement and the binary scaled pQCD predictions of the direct photon yield suggests that nuclear modifications of the quark and gluon distribution func-tions in the relevant region of momentum fraction x are minor. The result provides strong confirmation that the observed large suppression of high p T hadron production in central Au+Au collisions is dominantly a final-state effect due to parton energy loss in the dense produced medium, rather than an initial-state effect. FIG. 1 : 1Double ratio of measured (γ/π 0 ) Measured invariant yield ratio to the background decay (γ/π 0 ) Background ratio as a function of pT for minimum bias and for five centralities of Au+Au collisions at √ s N N = 200 GeV (0-10% is the most central). Statistical and total errors are indicated separately on each data point by the vertical bar and shaded region, respectively. The solid curves are the ratio of pQCD predictions described in the text to the background photon invariant yield based on the measured π 0 yield for each centrality class. The shaded region around the curves indicate the variation of the pQCD calculation for scale changes from pT /2 to 2pT , plus the N coll uncertainty. FIG. 2 : 2Direct γ invariant yields as a function of transverse momentum for 9 centrality selections and minimum bias Au+Au collisions at √ s N N = 200 GeV. The vertical error bar on each point indicates the total error. Arrows indicate measurements consistent with zero yield with the tail of the arrow indicating the 90% confidence level upper limit. The solid curves are pQCD predictions described in the text. FIG. 3 : 3Ratio of Au+Au yield to p+p yield normalized by the number of binary nucleon collisions as a function of centrality given by Npart for direct γ (closed circles) and π 0 (open circles) yields integrated above 6 GeV/c. The p+p direct photon yield is taken as the NLO pQCD prediction described in the text. The error bars indicate the total error excluding the error on N coll shown by the dashed lines and the scale uncertainty of the NLO calculation shown by the shaded region at the right. TABLE I : ISummary of the dominant sources of systematic errors on the π 0 and inclusive γ yields extracted independently with the PbGl and PbSc electromagnetic calorimeters. The error estimates are quoted at two pT values in central events for the PbGl and PbSc. For the combined π 0 and inclusive γ spectra and γ/π 0 ratios, the approximate statistical and systematical errors are quoted for the most peripheral and most central reactions.PbGl (Central) PbSc (Central) π 0 error source 3 GeV/c 8.5 GeV/c 3 GeV/c 8.5 GeV/c Yield extraction 8.7% 7.0% 9.8% 7.2% Yield correction 12.1% 12.0% 10.3% 12.5% Energy scale 13.8% 14.1% 10.5% 11.4% Total systematic 20.3% 19.8% 17.7% 18.4% Statistical 10.6% 32.5% 2.1% 10.5% Inclusive γ error Non-γ correction 2.4% 2.4% 3.2% 3.2% Yield correction 10.2% 12.0% 9.1% 12.5% Energy scale 15.7% 13.7% 12.4% 10.8% Total systematic 18.9% 18.4% 15.7% 16.8% Statistical 1.2% 14.1% 0.6% 4.1% γ/π 0 syst. 13.6% 12.6% 14.0% 13.4% γ/π 0 stat. 10.7% 35.4% 2.2% 11.3% Total errors PbGl and PbSc combined Peripheral Central Error 3 GeV/c 8.5 GeV/c 3 GeV/c 8.5 GeV/c π 0 syst. 13.2% 17.0% 13.9% 16.1% π 0 stat. 3.0% 35.3% 1.8% 9.6% γ syst. 11.4% 15.6% 11.5% 15.9% γ stat. 3.0% 28.8% 0.6% 3.8% γ/π 0 syst. 9.9% 13.1% 9.7% 11.2% γ/π 0 stat. 4.2% 45.6 % 1.9% 10.3% γ/π 0 bkg calc. 4% 4% . 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{'abstract': 'The first measurement of direct photons in Au+Au collisions at √ s N N = 200 GeV is presented.The direct photon signal is extracted as a function of the Au+Au collision centrality and compared to NLO pQCD calculations. The direct photon yield is shown to scale with the number of nucleonnucleon collisions for all centralities.', 'arxivid': 'nucl-ex/0503003', 'author': ['S S Adler \nBrookhaven National Laboratory\nUpton11973-5000NYUSA\n', 'S Afanasiev \nJoint Institute for Nuclear Research\n141980Dubna, Moscow RegionRussia\n', 'C Aidala \nBrookhaven National Laboratory\nUpton11973-5000NYUSA\n', 'N N Ajitanand \nChemistry Department\nSUNY, Stony Brook\nStony Brook University\n11794-3400NYUSA\n', 'Y Akiba \nKEK, High Energy Accelerator Research Organization\nTsukuba-shi, Ibaraki-ken 305-0801Japan\n\nRIKEN (The Institute of Physical and Chemical Research)\n351-0198WakoSaitamaJAPAN\n', 'J Alexander \nChemistry Department\nSUNY, Stony Brook\nStony Brook University\n11794-3400NYUSA\n', 'R Amirikas \nFlorida State University\n32306TallahasseeFLUSA\n', 'L Aphecetche \nSUBATECH (Ecole des Mines de Nantes, CNRS-IN2P3\nUniversité de Nantes)\nBP 20722 -44307NantesFrance\n', 'S H Aronson \nBrookhaven National Laboratory\nUpton11973-5000NYUSA\n', 'R Averbeck \nDepartment of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA\n', 'T C Awes \nOak Ridge National Laboratory\n37831Oak RidgeTNUSA\n', 'R Azmoun \nDepartment of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA\n', 'V Babintsev \nInstitute for High Energy Physics (IHEP)\nProtvinoRussia\n', 'A Baldisseri \nDapnia\nCEA\nSaclay, F-91191Gif-sur-YvetteFrance\n', 'K N Barish \nUniversity of California -Riverside\n92521RiversideCAUSA\n', 'P D Barnes \nLos Alamos National Laboratory\n87545Los AlamosNMUSA\n', 'B Bassalleck \nUniversity of New Mexico\n87131AlbuquerqueNMUSA\n', 'S Bathe \nInstitut für Kernphysik\nUniversity of Muenster\nD-48149MuensterGermany\n', 'S Batsouli \nand Nevis Laboratories\nColumbia University\n10027, 10533New York, IrvingtonNY, NYUSA\n', 'V Baublis \nPetersburg Nuclear Physics Institute\nPNPI\nGatchinaRussia\n', 'A Bazilevsky \nInstitute for High Energy Physics (IHEP)\nProtvinoRussia\n\nRIKEN BNL Research Center, Brookhaven National Laboratory\nUpton11973-5000NYUSA\n', 'S Belikov \nInstitute for High Energy Physics (IHEP)\nProtvinoRussia\n\nIowa State University\n50011AmesIAUSA\n', 'Y Berdnikov \nSt. Petersburg State Technical University\nSt. PetersburgRussia\n', 'S Bhagavatula \nIowa State University\n50011AmesIAUSA\n', 'J G Boissevain \nLos Alamos National Laboratory\n87545Los AlamosNMUSA\n', 'H Borel \nDapnia\nCEA\nSaclay, F-91191Gif-sur-YvetteFrance\n', 'S Borenstein \nLaboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS-IN2P3\nRoute de Saclay, F-91128PalaiseauFrance\n', 'M L Brooks \nLos Alamos National Laboratory\n87545Los AlamosNMUSA\n', 'D S Brown \nNew Mexico State University\nLas Cruces88003NMUSA\n', 'N Bruner \nUniversity of New Mexico\n87131AlbuquerqueNMUSA\n', 'D Bucher \nInstitut für Kernphysik\nUniversity of Muenster\nD-48149MuensterGermany\n', 'H Buesching \nInstitut für Kernphysik\nUniversity of Muenster\nD-48149MuensterGermany\n', 'V Bumazhnov \nInstitute for High Energy Physics (IHEP)\nProtvinoRussia\n', 'G Bunce \nBrookhaven National Laboratory\nUpton11973-5000NYUSA\n\nRIKEN BNL Research Center, Brookhaven National Laboratory\nUpton11973-5000NYUSA\n', 'J M Burward-Hoy \nLawrence Livermore National Laboratory\n94550LivermoreCAUSA\n\nDepartment of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA\n', 'S Butsyk \nDepartment of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA\n', 'X Camard \nSUBATECH (Ecole des Mines de Nantes, CNRS-IN2P3\nUniversité de Nantes)\nBP 20722 -44307NantesFrance\n', 'J.-S Chai \nKAERI, Cyclotron Application Laboratory\nSeoulSouth Korea\n', 'P Chand \nBhabha Atomic Research Centre\n400 085BombayIndia\n', 'W C Chang \nInstitute of Physics\nAcademia Sinica\n11529TaipeiTaiwan\n', 'S Chernichenko \nInstitute for High Energy Physics (IHEP)\nProtvinoRussia\n', 'C Y Chi \nand Nevis Laboratories\nColumbia University\n10027, 10533New York, IrvingtonNY, NYUSA\n', 'J Chiba \nKEK, High Energy Accelerator Research Organization\nTsukuba-shi, Ibaraki-ken 305-0801Japan\n', 'M Chiu \nand Nevis Laboratories\nColumbia University\n10027, 10533New York, IrvingtonNY, NYUSA\n', 'I J Choi \nYonsei University\n120-749SeoulKorea\n', 'J Choi \nKangnung National University\n210-702KangnungSouth Korea\n', 'R K Choudhury \nBhabha Atomic Research Centre\n400 085BombayIndia\n', 'T Chujo \nBrookhaven National Laboratory\nUpton11973-5000NYUSA\n', 'V Cianciolo \nOak Ridge National Laboratory\n37831Oak RidgeTNUSA\n', 'Y Cobigo \nDapnia\nCEA\nSaclay, F-91191Gif-sur-YvetteFrance\n', 'B A Cole \nand Nevis Laboratories\nColumbia University\n10027, 10533New York, IrvingtonNY, NYUSA\n', 'P Constantin \nIowa State University\n50011AmesIAUSA\n', 'D D&apos;enterria \nSUBATECH (Ecole des Mines de Nantes, CNRS-IN2P3\nUniversité de Nantes)\nBP 20722 -44307NantesFrance\n', 'G David \nBrookhaven National Laboratory\nUpton11973-5000NYUSA\n', 'H Delagrange \nSUBATECH (Ecole des Mines de Nantes, CNRS-IN2P3\nUniversité de Nantes)\nBP 20722 -44307NantesFrance\n', 'A Denisov \nInstitute for High Energy Physics (IHEP)\nProtvinoRussia\n', 'A Deshpande \nRIKEN BNL Research Center, Brookhaven National Laboratory\nUpton11973-5000NYUSA\n', 'E J Desmond \nBrookhaven National Laboratory\nUpton11973-5000NYUSA\n', 'A Devismes \nDepartment of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA\n', 'O Dietzsch \nInstituto de Física\nUniversidade de São Paulo\n66318, CEP05315-970São PauloCaixa PostalBrazil\n', 'O Drapier \nLaboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS-IN2P3\nRoute de Saclay, F-91128PalaiseauFrance\n', 'A Drees \nDepartment of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA\n', 'R Du Rietz \nDepartment of Physics\nLund University\nBox 118221 00LundSESweden\n', 'A Durum \nInstitute for High Energy Physics (IHEP)\nProtvinoRussia\n', 'D Dutta \nBhabha Atomic Research Centre\n400 085BombayIndia\n', 'Y V Efremenko \nOak Ridge National Laboratory\n37831Oak RidgeTNUSA\n', 'K El Chenawi \nVanderbilt University\n37235NashvilleTNUSA\n', 'A Enokizono \nHiroshima University\nKagamiyama, Higashi-Hiroshima 739-8526Japan\n', 'H ', 'B D Fox \nRIKEN BNL Research Center, Brookhaven National Laboratory\nUpton11973-5000NYUSA\n', 'Z Fraenkel \nWeizmann Institute\n76100RehovotIsrael\n', 'J E Frantz \nand Nevis Laboratories\nColumbia University\n10027, 10533New York, IrvingtonNY, NYUSA\n', 'A Franz \nBrookhaven National Laboratory\nUpton11973-5000NYUSA\n', 'A D Frawley \nFlorida State University\n32306TallahasseeFLUSA\n', 'S.-Y Fung \nUniversity of California -Riverside\n92521RiversideCAUSA\n', 'S Garpman \nDepartment of Physics\nLund University\nBox 118221 00LundSESweden\n', 'T K Ghosh \nVanderbilt University\n37235NashvilleTNUSA\n', 'A Glenn \nUniversity of Tennessee\n37996KnoxvilleTNUSA\n', 'G Gogiberidze \nUniversity of Tennessee\n37996KnoxvilleTNUSA\n', 'M Gonin \nLaboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS-IN2P3\nRoute de Saclay, F-91128PalaiseauFrance\n', 'J Gosset \nDapnia\nCEA\nSaclay, F-91191Gif-sur-YvetteFrance\n', 'Y Goto \nRIKEN BNL Research Center, Brookhaven National Laboratory\nUpton11973-5000NYUSA\n', 'R Granier De Cassagnac \nLaboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS-IN2P3\nRoute de Saclay, F-91128PalaiseauFrance\n', 'N Grau \nIowa State University\n50011AmesIAUSA\n', 'S V Greene \nVanderbilt University\n37235NashvilleTNUSA\n', 'M Grosse Perdekamp \nRIKEN BNL Research Center, Brookhaven National Laboratory\nUpton11973-5000NYUSA\n', 'W Guryn \nBrookhaven National Laboratory\nUpton11973-5000NYUSA\n', 'H.-Å Gustafsson \nDepartment of Physics\nLund University\nBox 118221 00LundSESweden\n', 'T Hachiya \nHiroshima University\nKagamiyama, Higashi-Hiroshima 739-8526Japan\n', 'J S Haggerty \nBrookhaven National Laboratory\nUpton11973-5000NYUSA\n', 'H Hamagaki \nCenter for Nuclear Study\nGraduate School of Science\nUniversity of Tokyo\n7-3-1 Hongo113-0033TokyoJapan\n', 'A G Hansen \nLos Alamos National Laboratory\n87545Los AlamosNMUSA\n', 'E P Hartouni \nLawrence Livermore National Laboratory\n94550LivermoreCAUSA\n', 'M Harvey \nBrookhaven National Laboratory\nUpton11973-5000NYUSA\n', 'R Hayano \nCenter for Nuclear Study\nGraduate School of Science\nUniversity of Tokyo\n7-3-1 Hongo113-0033TokyoJapan\n', 'N Hayashi \nRIKEN (The Institute of Physical and Chemical Research)\n351-0198WakoSaitamaJAPAN\n', 'X He \nGeorgia State University\n30303AtlantaGAUSA\n', 'M Heffner \nLawrence Livermore National Laboratory\n94550LivermoreCAUSA\n', 'T K Hemmick \nDepartment of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA\n', 'J M Heuser \nDepartment of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA\n', 'M Hibino \nAdvanced Research Institute for Science and Engineering\nWaseda University\n17 Kikui-cho, Shinjuku-ku162-0044TokyoJapan\n', 'J C Hill \nIowa State University\n50011AmesIAUSA\n', 'W Holzmann \nChemistry Department\nSUNY, Stony Brook\nStony Brook University\n11794-3400NYUSA\n', 'K Homma \nHiroshima University\nKagamiyama, Higashi-Hiroshima 739-8526Japan\n', 'B Hong \nKorea University\n136-701SeoulKorea\n', 'A Hoover \nNew Mexico State University\nLas Cruces88003NMUSA\n', 'J H Kang \nYonsei University\n120-749SeoulKorea\n', 'S S Kapoor \nBhabha Atomic Research Centre\n400 085BombayIndia\n', 'K Katou \nAdvanced Research Institute for Science and Engineering\nWaseda University\n17 Kikui-cho, Shinjuku-ku162-0044TokyoJapan\n', 'S Kelly \nand Nevis Laboratories\nColumbia University\n10027, 10533New York, IrvingtonNY, NYUSA\n', 'B Khachaturov \nWeizmann Institute\n76100RehovotIsrael\n', 'A Khanzadeev \nPetersburg Nuclear Physics Institute\nPNPI\nGatchinaRussia\n', 'J Kikuchi \nAdvanced Research Institute for Science and Engineering\nWaseda University\n17 Kikui-cho, Shinjuku-ku162-0044TokyoJapan\n', 'D H Kim \nMyongji University\nKyonggido 449-728YonginKorea\n', 'D J Kim \nYonsei University\n120-749SeoulKorea\n', 'D W Kim \nKangnung National University\n210-702KangnungSouth Korea\n', 'E Kim \nSystem Electronics Laboratory\nSeoul National University\nSeoulSouth Korea\n', 'G.-B Kim \nLaboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS-IN2P3\nRoute de Saclay, F-91128PalaiseauFrance\n', 'H J Kim ', 'E Kistenev \nBrookhaven National Laboratory\nUpton11973-5000NYUSA\n\nYonsei University\n120-749SeoulKorea\n', 'A Kiyomichi \nInstitute of Physics\nUniversity of Tsukuba\n305TsukubaIbarakiJapan\n', 'K Kiyoyama \nNagasaki Institute of Applied Science\nNagasaki-shi851-0193NagasakiJapan\n', 'C Klein-Boesing \nInstitut für Kernphysik\nUniversity of Muenster\nD-48149MuensterGermany\n', 'H Kobayashi \nRIKEN (The Institute of Physical and Chemical Research)\n351-0198WakoSaitamaJAPAN\n\nRIKEN BNL Research Center, Brookhaven National Laboratory\nUpton11973-5000NYUSA\n', 'L Kochenda \nPetersburg Nuclear Physics Institute\nPNPI\nGatchinaRussia\n', 'V Kochetkov \nInstitute for High Energy Physics (IHEP)\nProtvinoRussia\n', 'D Koehler \nUniversity of New Mexico\n87131AlbuquerqueNMUSA\n', 'T Kohama \nHiroshima University\nKagamiyama, Higashi-Hiroshima 739-8526Japan\n', 'M Kopytine \nDepartment of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA\n', 'D Kotchetkov \nUniversity of California -Riverside\n92521RiversideCAUSA\n', 'A Kozlov \nWeizmann Institute\n76100RehovotIsrael\n', 'P J Kroon \nBrookhaven National Laboratory\nUpton11973-5000NYUSA\n', 'C H Kuberg \nAbilene Christian University\n79699AbileneTXUSA\n\nLos Alamos National Laboratory\n87545Los AlamosNMUSA\n', 'K Kurita \nRIKEN BNL Research Center, Brookhaven National Laboratory\nUpton11973-5000NYUSA\n', 'Y Kuroki \nInstitute of Physics\nUniversity of Tsukuba\n305TsukubaIbarakiJapan\n', 'M J Kweon \nKorea University\n136-701SeoulKorea\n', 'Y Kwon \nYonsei University\n120-749SeoulKorea\n', 'G S Kyle \nNew Mexico State University\nLas Cruces88003NMUSA\n', 'R Lacey \nChemistry Department\nSUNY, Stony Brook\nStony Brook University\n11794-3400NYUSA\n', 'V Ladygin \nJoint Institute for Nuclear Research\n141980Dubna, Moscow RegionRussia\n', 'J G Lajoie \nIowa State University\n50011AmesIAUSA\n', 'A Lebedev \nIowa State University\n50011AmesIAUSA\n\nRussian Research Center "Kurchatov Institute"\nMoscowRussia\n', 'S Leckey \nDepartment of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA\n', 'D M Lee \nLos Alamos National Laboratory\n87545Los AlamosNMUSA\n', 'S Lee \nKangnung National University\n210-702KangnungSouth Korea\n', 'M J Leitch \nLos Alamos National Laboratory\n87545Los AlamosNMUSA\n', 'X H Li \nUniversity of California -Riverside\n92521RiversideCAUSA\n', 'H Lim \nSystem Electronics Laboratory\nSeoul National University\nSeoulSouth Korea\n', 'A Litvinenko \nJoint Institute for Nuclear Research\n141980Dubna, Moscow RegionRussia\n', 'M X Liu \nLos Alamos National Laboratory\n87545Los AlamosNMUSA\n', 'Y Liu \nIPN-Orsay\nUniversite Paris Sud\nCNRS-IN2P3\nBP1, F-91406OrsayFrance\n', 'C F Maguire \nVanderbilt University\n37235NashvilleTNUSA\n', 'Y I Makdisi \nBrookhaven National Laboratory\nUpton11973-5000NYUSA\n', 'A Malakhov \nJoint Institute for Nuclear Research\n141980Dubna, Moscow RegionRussia\n', 'V I Manko \nRussian Research Center "Kurchatov Institute"\nMoscowRussia\n', "Y Mao \nInstitute of Atomic Energy (CIAE)\nBeijingChina, People's Republic of China\n\nRIKEN (The Institute of Physical and Chemical Research)\n351-0198WakoSaitamaJAPAN\n", 'G Martinez \nSUBATECH (Ecole des Mines de Nantes, CNRS-IN2P3\nUniversité de Nantes)\nBP 20722 -44307NantesFrance\n', 'M D Marx \nDepartment of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA\n', 'H Masui \nInstitute of Physics\nUniversity of Tsukuba\n305TsukubaIbarakiJapan\n', 'F Matathias \nDepartment of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA\n', 'T Matsumoto \nCenter for Nuclear Study\nGraduate School of Science\nUniversity of Tokyo\n7-3-1 Hongo113-0033TokyoJapan\n\nAdvanced Research Institute for Science and Engineering\nWaseda University\n17 Kikui-cho, Shinjuku-ku162-0044TokyoJapan\n', 'P L Mcgaughey \nLos Alamos National Laboratory\n87545Los AlamosNMUSA\n', 'E Melnikov \nInstitute for High Energy Physics (IHEP)\nProtvinoRussia\n', 'F Messer \nDepartment of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA\n', 'Y Miake \nInstitute of Physics\nUniversity of Tsukuba\n305TsukubaIbarakiJapan\n', 'J Milan \nChemistry Department\nSUNY, Stony Brook\nStony Brook University\n11794-3400NYUSA\n', 'T E Miller \nVanderbilt University\n37235NashvilleTNUSA\n', 'A Milov \nDepartment of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA\n\nWeizmann Institute\n76100RehovotIsrael\n', 'S Mioduszewski \nBrookhaven National Laboratory\nUpton11973-5000NYUSA\n', 'R E Mischke \nLos Alamos National Laboratory\n87545Los AlamosNMUSA\n', 'G C Mishra \nGeorgia State University\n30303AtlantaGAUSA\n', 'J T Mitchell \nBrookhaven National Laboratory\nUpton11973-5000NYUSA\n', 'A K Mohanty \nBhabha Atomic Research Centre\n400 085BombayIndia\n', 'D P Morrison \nBrookhaven National Laboratory\nUpton11973-5000NYUSA\n', 'J M Moss \nLos Alamos National Laboratory\n87545Los AlamosNMUSA\n', 'F Mühlbacher \nDepartment of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA\n', 'D Mukhopadhyay \nWeizmann Institute\n76100RehovotIsrael\n', 'M Muniruzzaman \nUniversity of California -Riverside\n92521RiversideCAUSA\n', 'J Newby \nUniversity of Tennessee\n37996KnoxvilleTNUSA\n', 'P Nilsson \nDepartment of Physics\nLund University\nBox 118221 00LundSESweden\n', 'A S Nyanin \nRussian Research Center "Kurchatov Institute"\nMoscowRussia\n', 'J Nystrand \nDepartment of Physics\nLund University\nBox 118221 00LundSESweden\n', 'E O&apos;brien \nBrookhaven National Laboratory\nUpton11973-5000NYUSA\n', 'C A Ogilvie \nIowa State University\n50011AmesIAUSA\n', 'H Ohnishi \nBrookhaven National Laboratory\nUpton11973-5000NYUSA\n\nRIKEN (The Institute of Physical and Chemical Research)\n351-0198WakoSaitamaJAPAN\n', 'I D Ojha \nDepartment of Physics\nBanaras Hindu University\n221005VaranasiIndia\n\nVanderbilt University\n37235NashvilleTNUSA\n', 'K Okada \nRIKEN (The Institute of Physical and Chemical Research)\n351-0198WakoSaitamaJAPAN\n', 'M Ono \nInstitute of Physics\nUniversity of Tsukuba\n305TsukubaIbarakiJapan\n', 'V Onuchin \nInstitute for High Energy Physics (IHEP)\nProtvinoRussia\n', 'A Oskarsson \nDepartment of Physics\nLund University\nBox 118221 00LundSESweden\n', 'I Otterlund \nDepartment of Physics\nLund University\nBox 118221 00LundSESweden\n', 'K Oyama \nCenter for Nuclear Study\nGraduate School of Science\nUniversity of Tokyo\n7-3-1 Hongo113-0033TokyoJapan\n', 'K Ozawa \nCenter for Nuclear Study\nGraduate School of Science\nUniversity of Tokyo\n7-3-1 Hongo113-0033TokyoJapan\n', 'D Pal \nWeizmann Institute\n76100RehovotIsrael\n', 'A P T Palounek \nLos Alamos National Laboratory\n87545Los AlamosNMUSA\n', 'V Pantuev \nDepartment of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA\n', 'V Papavassiliou \nNew Mexico State University\nLas Cruces88003NMUSA\n', 'J Park \nSystem Electronics Laboratory\nSeoul National University\nSeoulSouth Korea\n', 'A Parmar \nUniversity of New Mexico\n87131AlbuquerqueNMUSA\n', 'S F Pate \nNew Mexico State University\nLas Cruces88003NMUSA\n', 'T Peitzmann \nInstitut für Kernphysik\nUniversity of Muenster\nD-48149MuensterGermany\n', 'J.-C Peng \nLos Alamos National Laboratory\n87545Los AlamosNMUSA\n', 'V Peresedov \nJoint Institute for Nuclear Research\n141980Dubna, Moscow RegionRussia\n', 'C Pinkenburg \nBrookhaven National Laboratory\nUpton11973-5000NYUSA\n', 'R P Pisani \nBrookhaven National Laboratory\nUpton11973-5000NYUSA\n', 'F Plasil \nOak Ridge National Laboratory\n37831Oak RidgeTNUSA\n', 'M L Purschke \nBrookhaven National Laboratory\nUpton11973-5000NYUSA\n', 'A K Purwar \nDepartment of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA\n', 'V Semenov \nInstitute for High Energy Physics (IHEP)\nProtvinoRussia\n', 'R Seto \nUniversity of California -Riverside\n92521RiversideCAUSA\n', 'M R Shaw \nAbilene Christian University\n79699AbileneTXUSA\n\nLos Alamos National Laboratory\n87545Los AlamosNMUSA\n', 'T K Shea \nBrookhaven National Laboratory\nUpton11973-5000NYUSA\n', 'T.-A Shibata \nRIKEN (The Institute of Physical and Chemical Research)\n351-0198WakoSaitamaJAPAN\n\nDepartment of Physics\nTokyo Institute of Technology\n152-8551TokyoJapan\n', 'K Shigaki \nHiroshima University\nKagamiyama, Higashi-Hiroshima 739-8526Japan\n\nKEK, High Energy Accelerator Research Organization\nTsukuba-shi, Ibaraki-ken 305-0801Japan\n', 'T Shiina \nLos Alamos National Laboratory\n87545Los AlamosNMUSA\n', 'C L Silva \nInstituto de Física\nUniversidade de São Paulo\n66318, CEP05315-970São PauloCaixa PostalBrazil\n', 'D Silvermyr \nLos Alamos National Laboratory\n87545Los AlamosNMUSA\n\nDepartment of Physics\nLund University\nBox 118221 00LundSESweden\n', 'K S Sim \nKorea University\n136-701SeoulKorea\n', 'C P Singh \nDepartment of Physics\nBanaras Hindu University\n221005VaranasiIndia\n', 'V Singh ', 'M Sivertz \nDepartment of Physics\nBanaras Hindu University\n221005VaranasiIndia\n\nBrookhaven National Laboratory\nUpton11973-5000NYUSA\n', 'A Soldatov \nInstitute for High Energy Physics (IHEP)\nProtvinoRussia\n', 'R A Soltz \nLawrence Livermore National Laboratory\n94550LivermoreCAUSA\n', 'W E Sondheim \nLos Alamos National Laboratory\n87545Los AlamosNMUSA\n', 'S P Sorensen \nUniversity of Tennessee\n37996KnoxvilleTNUSA\n', 'I V Sourikova \nBrookhaven National Laboratory\nUpton11973-5000NYUSA\n', 'F Staley \nDapnia\nCEA\nSaclay, F-91191Gif-sur-YvetteFrance\n', 'P W Stankus \nOak Ridge National Laboratory\n37831Oak RidgeTNUSA\n', 'E Stenlund \nDepartment of Physics\nLund University\nBox 118221 00LundSESweden\n', 'M Stepanov \nNew Mexico State University\nLas Cruces88003NMUSA\n', 'A Ster \nKFKI Research Institute for Particle and Nuclear Physics (RMKI)\nPOBox 49H-1525, 114BudapestHungary\n', 'S P Stoll \nBrookhaven National Laboratory\nUpton11973-5000NYUSA\n', 'T Sugitate \nHiroshima University\nKagamiyama, Higashi-Hiroshima 739-8526Japan\n', 'J P Sullivan \nLos Alamos National Laboratory\n87545Los AlamosNMUSA\n', 'E M Takagui \nInstituto de Física\nUniversidade de São Paulo\n66318, CEP05315-970São PauloCaixa PostalBrazil\n', 'A Taketani \nRIKEN (The Institute of Physical and Chemical Research)\n351-0198WakoSaitamaJAPAN\n\nRIKEN BNL Research Center, Brookhaven National Laboratory\nUpton11973-5000NYUSA\n', 'M Tamai \nAdvanced Research Institute for Science and Engineering\nWaseda University\n17 Kikui-cho, Shinjuku-ku162-0044TokyoJapan\n', 'K H Tanaka \nKEK, High Energy Accelerator Research Organization\nTsukuba-shi, Ibaraki-ken 305-0801Japan\n', 'Y Tanaka \nNagasaki Institute of Applied Science\nNagasaki-shi851-0193NagasakiJapan\n', 'K Tanida \nRIKEN (The Institute of Physical and Chemical Research)\n351-0198WakoSaitamaJAPAN\n', 'M J Tannenbaum \nBrookhaven National Laboratory\nUpton11973-5000NYUSA\n', 'P Tarján \nDebrecen University\nEgyetem tér 1H-4010DebrecenHungary\n', 'J D Tepe \nAbilene Christian University\n79699AbileneTXUSA\n\nLos Alamos National Laboratory\n87545Los AlamosNMUSA\n', 'T L Thomas \nUniversity of New Mexico\n87131AlbuquerqueNMUSA\n', 'J Tojo \nKyoto University\n606-8502KyotoJapan\n\nRIKEN (The Institute of Physical and Chemical Research)\n351-0198WakoSaitamaJAPAN\n', 'H Torii \nKyoto University\n606-8502KyotoJapan\n\nRIKEN (The Institute of Physical and Chemical Research)\n351-0198WakoSaitamaJAPAN\n', 'R S Towell \nAbilene Christian University\n79699AbileneTXUSA\n', 'I Tserruya \nWeizmann Institute\n76100RehovotIsrael\n', 'H Tsuruoka \nInstitute of Physics\nUniversity of Tsukuba\n305TsukubaIbarakiJapan\n', 'S K Tuli \nDepartment of Physics\nBanaras Hindu University\n221005VaranasiIndia\n', 'H Tydesjö \nDepartment of Physics\nLund University\nBox 118221 00LundSESweden\n', 'N Tyurin \nInstitute for High Energy Physics (IHEP)\nProtvinoRussia\n', 'H W Van Hecke \nLos Alamos National Laboratory\n87545Los AlamosNMUSA\n', 'J Velkovska \nBrookhaven National Laboratory\nUpton11973-5000NYUSA\n\nDepartment of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA\n', 'M Velkovsky \nDepartment of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA\n', 'V Veszprémi \nDebrecen University\nEgyetem tér 1H-4010DebrecenHungary\n', 'L Villatte \nUniversity of Tennessee\n37996KnoxvilleTNUSA\n', 'A A Vinogradov \nRussian Research Center "Kurchatov Institute"\nMoscowRussia\n', 'M A Volkov \nRussian Research Center "Kurchatov Institute"\nMoscowRussia\n', 'E Vznuzdaev \nPetersburg Nuclear Physics Institute\nPNPI\nGatchinaRussia\n', 'X R Wang \nGeorgia State University\n30303AtlantaGAUSA\n', 'Y Watanabe \nRIKEN (The Institute of Physical and Chemical Research)\n351-0198WakoSaitamaJAPAN\n\nRIKEN BNL Research Center, Brookhaven National Laboratory\nUpton11973-5000NYUSA\n', 'S N White \nBrookhaven National Laboratory\nUpton11973-5000NYUSA\n', 'F K Wohn \nIowa State University\n50011AmesIAUSA\n', 'C L Woody \nBrookhaven National Laboratory\nUpton11973-5000NYUSA\n', 'W Xie \nUniversity of California -Riverside\n92521RiversideCAUSA\n', "Y Yang \nInstitute of Atomic Energy (CIAE)\nBeijingChina, People's Republic of China\n", 'A Yanovich \nInstitute for High Energy Physics (IHEP)\nProtvinoRussia\n', 'S Yokkaichi \nRIKEN (The Institute of Physical and Chemical Research)\n351-0198WakoSaitamaJAPAN\n\nRIKEN BNL Research Center, Brookhaven National Laboratory\nUpton11973-5000NYUSA\n', 'G R Young \nOak Ridge National Laboratory\n37831Oak RidgeTNUSA\n', 'I E Yushmanov \nRussian Research Center "Kurchatov Institute"\nMoscowRussia\n', 'W A Zajc \nand Nevis Laboratories\nColumbia University\n10027, 10533New York, IrvingtonNY, NYUSA\n', 'C Zhang \nand Nevis Laboratories\nColumbia University\n10027, 10533New York, IrvingtonNY, NYUSA\n', "S Zhou \nInstitute of Atomic Energy (CIAE)\nBeijingChina, People's Republic of China\n", 'S J Zhou \nWeizmann Institute\n76100RehovotIsrael\n', 'L Zolin \nJoint Institute for Nuclear Research\n141980Dubna, Moscow RegionRussia\n', '\nLPC\nUniversité Blaise Pascal\nCNRS-IN2P3\nClermont-Fd63177Aubiere CedexFrance\n'], 'authoraffiliation': ['Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'Joint Institute for Nuclear Research\n141980Dubna, Moscow RegionRussia', 'Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'Chemistry Department\nSUNY, Stony Brook\nStony Brook University\n11794-3400NYUSA', 'KEK, High Energy Accelerator Research Organization\nTsukuba-shi, Ibaraki-ken 305-0801Japan', 'RIKEN (The Institute of Physical and Chemical Research)\n351-0198WakoSaitamaJAPAN', 'Chemistry Department\nSUNY, Stony Brook\nStony Brook University\n11794-3400NYUSA', 'Florida State University\n32306TallahasseeFLUSA', 'SUBATECH (Ecole des Mines de Nantes, CNRS-IN2P3\nUniversité de Nantes)\nBP 20722 -44307NantesFrance', 'Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'Department of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA', 'Oak Ridge National Laboratory\n37831Oak RidgeTNUSA', 'Department of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA', 'Institute for High Energy Physics (IHEP)\nProtvinoRussia', 'Dapnia\nCEA\nSaclay, F-91191Gif-sur-YvetteFrance', 'University of California -Riverside\n92521RiversideCAUSA', 'Los Alamos National Laboratory\n87545Los AlamosNMUSA', 'University of New Mexico\n87131AlbuquerqueNMUSA', 'Institut für Kernphysik\nUniversity of Muenster\nD-48149MuensterGermany', 'and Nevis Laboratories\nColumbia University\n10027, 10533New York, IrvingtonNY, NYUSA', 'Petersburg Nuclear Physics Institute\nPNPI\nGatchinaRussia', 'Institute for High Energy Physics (IHEP)\nProtvinoRussia', 'RIKEN BNL Research Center, Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'Institute for High Energy Physics (IHEP)\nProtvinoRussia', 'Iowa State University\n50011AmesIAUSA', 'St. Petersburg State Technical University\nSt. PetersburgRussia', 'Iowa State University\n50011AmesIAUSA', 'Los Alamos National Laboratory\n87545Los AlamosNMUSA', 'Dapnia\nCEA\nSaclay, F-91191Gif-sur-YvetteFrance', 'Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS-IN2P3\nRoute de Saclay, F-91128PalaiseauFrance', 'Los Alamos National Laboratory\n87545Los AlamosNMUSA', 'New Mexico State University\nLas Cruces88003NMUSA', 'University of New Mexico\n87131AlbuquerqueNMUSA', 'Institut für Kernphysik\nUniversity of Muenster\nD-48149MuensterGermany', 'Institut für Kernphysik\nUniversity of Muenster\nD-48149MuensterGermany', 'Institute for High Energy Physics (IHEP)\nProtvinoRussia', 'Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'RIKEN BNL Research Center, Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'Lawrence Livermore National Laboratory\n94550LivermoreCAUSA', 'Department of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA', 'Department of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA', 'SUBATECH (Ecole des Mines de Nantes, CNRS-IN2P3\nUniversité de Nantes)\nBP 20722 -44307NantesFrance', 'KAERI, Cyclotron Application Laboratory\nSeoulSouth Korea', 'Bhabha Atomic Research Centre\n400 085BombayIndia', 'Institute of Physics\nAcademia Sinica\n11529TaipeiTaiwan', 'Institute for High Energy Physics (IHEP)\nProtvinoRussia', 'and Nevis Laboratories\nColumbia University\n10027, 10533New York, IrvingtonNY, NYUSA', 'KEK, High Energy Accelerator Research Organization\nTsukuba-shi, Ibaraki-ken 305-0801Japan', 'and Nevis Laboratories\nColumbia University\n10027, 10533New York, IrvingtonNY, NYUSA', 'Yonsei University\n120-749SeoulKorea', 'Kangnung National University\n210-702KangnungSouth Korea', 'Bhabha Atomic Research Centre\n400 085BombayIndia', 'Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'Oak Ridge National Laboratory\n37831Oak RidgeTNUSA', 'Dapnia\nCEA\nSaclay, F-91191Gif-sur-YvetteFrance', 'and Nevis Laboratories\nColumbia University\n10027, 10533New York, IrvingtonNY, NYUSA', 'Iowa State University\n50011AmesIAUSA', 'SUBATECH (Ecole des Mines de Nantes, CNRS-IN2P3\nUniversité de Nantes)\nBP 20722 -44307NantesFrance', 'Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'SUBATECH (Ecole des Mines de Nantes, CNRS-IN2P3\nUniversité de Nantes)\nBP 20722 -44307NantesFrance', 'Institute for High Energy Physics (IHEP)\nProtvinoRussia', 'RIKEN BNL Research Center, Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'Department of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA', 'Instituto de Física\nUniversidade de São Paulo\n66318, CEP05315-970São PauloCaixa PostalBrazil', 'Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS-IN2P3\nRoute de Saclay, F-91128PalaiseauFrance', 'Department of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA', 'Department of Physics\nLund University\nBox 118221 00LundSESweden', 'Institute for High Energy Physics (IHEP)\nProtvinoRussia', 'Bhabha Atomic Research Centre\n400 085BombayIndia', 'Oak Ridge National Laboratory\n37831Oak RidgeTNUSA', 'Vanderbilt University\n37235NashvilleTNUSA', 'Hiroshima University\nKagamiyama, Higashi-Hiroshima 739-8526Japan', 'RIKEN BNL Research Center, Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'Weizmann Institute\n76100RehovotIsrael', 'and Nevis Laboratories\nColumbia University\n10027, 10533New York, IrvingtonNY, NYUSA', 'Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'Florida State University\n32306TallahasseeFLUSA', 'University of California -Riverside\n92521RiversideCAUSA', 'Department of Physics\nLund University\nBox 118221 00LundSESweden', 'Vanderbilt University\n37235NashvilleTNUSA', 'University of Tennessee\n37996KnoxvilleTNUSA', 'University of Tennessee\n37996KnoxvilleTNUSA', 'Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS-IN2P3\nRoute de Saclay, F-91128PalaiseauFrance', 'Dapnia\nCEA\nSaclay, F-91191Gif-sur-YvetteFrance', 'RIKEN BNL Research Center, Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS-IN2P3\nRoute de Saclay, F-91128PalaiseauFrance', 'Iowa State University\n50011AmesIAUSA', 'Vanderbilt University\n37235NashvilleTNUSA', 'RIKEN BNL Research Center, Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'Department of Physics\nLund University\nBox 118221 00LundSESweden', 'Hiroshima University\nKagamiyama, Higashi-Hiroshima 739-8526Japan', 'Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'Center for Nuclear Study\nGraduate School of Science\nUniversity of Tokyo\n7-3-1 Hongo113-0033TokyoJapan', 'Los Alamos National Laboratory\n87545Los AlamosNMUSA', 'Lawrence Livermore National Laboratory\n94550LivermoreCAUSA', 'Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'Center for Nuclear Study\nGraduate School of Science\nUniversity of Tokyo\n7-3-1 Hongo113-0033TokyoJapan', 'RIKEN (The Institute of Physical and Chemical Research)\n351-0198WakoSaitamaJAPAN', 'Georgia State University\n30303AtlantaGAUSA', 'Lawrence Livermore National Laboratory\n94550LivermoreCAUSA', 'Department of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA', 'Department of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA', 'Advanced Research Institute for Science and Engineering\nWaseda University\n17 Kikui-cho, Shinjuku-ku162-0044TokyoJapan', 'Iowa State University\n50011AmesIAUSA', 'Chemistry Department\nSUNY, Stony Brook\nStony Brook University\n11794-3400NYUSA', 'Hiroshima University\nKagamiyama, Higashi-Hiroshima 739-8526Japan', 'Korea University\n136-701SeoulKorea', 'New Mexico State University\nLas Cruces88003NMUSA', 'Yonsei University\n120-749SeoulKorea', 'Bhabha Atomic Research Centre\n400 085BombayIndia', 'Advanced Research Institute for Science and Engineering\nWaseda University\n17 Kikui-cho, Shinjuku-ku162-0044TokyoJapan', 'and Nevis Laboratories\nColumbia University\n10027, 10533New York, IrvingtonNY, NYUSA', 'Weizmann Institute\n76100RehovotIsrael', 'Petersburg Nuclear Physics Institute\nPNPI\nGatchinaRussia', 'Advanced Research Institute for Science and Engineering\nWaseda University\n17 Kikui-cho, Shinjuku-ku162-0044TokyoJapan', 'Myongji University\nKyonggido 449-728YonginKorea', 'Yonsei University\n120-749SeoulKorea', 'Kangnung National University\n210-702KangnungSouth Korea', 'System Electronics Laboratory\nSeoul National University\nSeoulSouth Korea', 'Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS-IN2P3\nRoute de Saclay, F-91128PalaiseauFrance', 'Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'Yonsei University\n120-749SeoulKorea', 'Institute of Physics\nUniversity of Tsukuba\n305TsukubaIbarakiJapan', 'Nagasaki Institute of Applied Science\nNagasaki-shi851-0193NagasakiJapan', 'Institut für Kernphysik\nUniversity of Muenster\nD-48149MuensterGermany', 'RIKEN (The Institute of Physical and Chemical Research)\n351-0198WakoSaitamaJAPAN', 'RIKEN BNL Research Center, Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'Petersburg Nuclear Physics Institute\nPNPI\nGatchinaRussia', 'Institute for High Energy Physics (IHEP)\nProtvinoRussia', 'University of New Mexico\n87131AlbuquerqueNMUSA', 'Hiroshima University\nKagamiyama, Higashi-Hiroshima 739-8526Japan', 'Department of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA', 'University of California -Riverside\n92521RiversideCAUSA', 'Weizmann Institute\n76100RehovotIsrael', 'Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'Abilene Christian University\n79699AbileneTXUSA', 'Los Alamos National Laboratory\n87545Los AlamosNMUSA', 'RIKEN BNL Research Center, Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'Institute of Physics\nUniversity of Tsukuba\n305TsukubaIbarakiJapan', 'Korea University\n136-701SeoulKorea', 'Yonsei University\n120-749SeoulKorea', 'New Mexico State University\nLas Cruces88003NMUSA', 'Chemistry Department\nSUNY, Stony Brook\nStony Brook University\n11794-3400NYUSA', 'Joint Institute for Nuclear Research\n141980Dubna, Moscow RegionRussia', 'Iowa State University\n50011AmesIAUSA', 'Iowa State University\n50011AmesIAUSA', 'Russian Research Center "Kurchatov Institute"\nMoscowRussia', 'Department of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA', 'Los Alamos National Laboratory\n87545Los AlamosNMUSA', 'Kangnung National University\n210-702KangnungSouth Korea', 'Los Alamos National Laboratory\n87545Los AlamosNMUSA', 'University of California -Riverside\n92521RiversideCAUSA', 'System Electronics Laboratory\nSeoul National University\nSeoulSouth Korea', 'Joint Institute for Nuclear Research\n141980Dubna, Moscow RegionRussia', 'Los Alamos National Laboratory\n87545Los AlamosNMUSA', 'IPN-Orsay\nUniversite Paris Sud\nCNRS-IN2P3\nBP1, F-91406OrsayFrance', 'Vanderbilt University\n37235NashvilleTNUSA', 'Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'Joint Institute for Nuclear Research\n141980Dubna, Moscow RegionRussia', 'Russian Research Center "Kurchatov Institute"\nMoscowRussia', "Institute of Atomic Energy (CIAE)\nBeijingChina, People's Republic of China", 'RIKEN (The Institute of Physical and Chemical Research)\n351-0198WakoSaitamaJAPAN', 'SUBATECH (Ecole des Mines de Nantes, CNRS-IN2P3\nUniversité de Nantes)\nBP 20722 -44307NantesFrance', 'Department of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA', 'Institute of Physics\nUniversity of Tsukuba\n305TsukubaIbarakiJapan', 'Department of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA', 'Center for Nuclear Study\nGraduate School of Science\nUniversity of Tokyo\n7-3-1 Hongo113-0033TokyoJapan', 'Advanced Research Institute for Science and Engineering\nWaseda University\n17 Kikui-cho, Shinjuku-ku162-0044TokyoJapan', 'Los Alamos National Laboratory\n87545Los AlamosNMUSA', 'Institute for High Energy Physics (IHEP)\nProtvinoRussia', 'Department of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA', 'Institute of Physics\nUniversity of Tsukuba\n305TsukubaIbarakiJapan', 'Chemistry Department\nSUNY, Stony Brook\nStony Brook University\n11794-3400NYUSA', 'Vanderbilt University\n37235NashvilleTNUSA', 'Department of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA', 'Weizmann Institute\n76100RehovotIsrael', 'Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'Los Alamos National Laboratory\n87545Los AlamosNMUSA', 'Georgia State University\n30303AtlantaGAUSA', 'Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'Bhabha Atomic Research Centre\n400 085BombayIndia', 'Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'Los Alamos National Laboratory\n87545Los AlamosNMUSA', 'Department of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA', 'Weizmann Institute\n76100RehovotIsrael', 'University of California -Riverside\n92521RiversideCAUSA', 'University of Tennessee\n37996KnoxvilleTNUSA', 'Department of Physics\nLund University\nBox 118221 00LundSESweden', 'Russian Research Center "Kurchatov Institute"\nMoscowRussia', 'Department of Physics\nLund University\nBox 118221 00LundSESweden', 'Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'Iowa State University\n50011AmesIAUSA', 'Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'RIKEN (The Institute of Physical and Chemical Research)\n351-0198WakoSaitamaJAPAN', 'Department of Physics\nBanaras Hindu University\n221005VaranasiIndia', 'Vanderbilt University\n37235NashvilleTNUSA', 'RIKEN (The Institute of Physical and Chemical Research)\n351-0198WakoSaitamaJAPAN', 'Institute of Physics\nUniversity of Tsukuba\n305TsukubaIbarakiJapan', 'Institute for High Energy Physics (IHEP)\nProtvinoRussia', 'Department of Physics\nLund University\nBox 118221 00LundSESweden', 'Department of Physics\nLund University\nBox 118221 00LundSESweden', 'Center for Nuclear Study\nGraduate School of Science\nUniversity of Tokyo\n7-3-1 Hongo113-0033TokyoJapan', 'Center for Nuclear Study\nGraduate School of Science\nUniversity of Tokyo\n7-3-1 Hongo113-0033TokyoJapan', 'Weizmann Institute\n76100RehovotIsrael', 'Los Alamos National Laboratory\n87545Los AlamosNMUSA', 'Department of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA', 'New Mexico State University\nLas Cruces88003NMUSA', 'System Electronics Laboratory\nSeoul National University\nSeoulSouth Korea', 'University of New Mexico\n87131AlbuquerqueNMUSA', 'New Mexico State University\nLas Cruces88003NMUSA', 'Institut für Kernphysik\nUniversity of Muenster\nD-48149MuensterGermany', 'Los Alamos National Laboratory\n87545Los AlamosNMUSA', 'Joint Institute for Nuclear Research\n141980Dubna, Moscow RegionRussia', 'Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'Oak Ridge National Laboratory\n37831Oak RidgeTNUSA', 'Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'Department of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA', 'Institute for High Energy Physics (IHEP)\nProtvinoRussia', 'University of California -Riverside\n92521RiversideCAUSA', 'Abilene Christian University\n79699AbileneTXUSA', 'Los Alamos National Laboratory\n87545Los AlamosNMUSA', 'Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'RIKEN (The Institute of Physical and Chemical Research)\n351-0198WakoSaitamaJAPAN', 'Department of Physics\nTokyo Institute of Technology\n152-8551TokyoJapan', 'Hiroshima University\nKagamiyama, Higashi-Hiroshima 739-8526Japan', 'KEK, High Energy Accelerator Research Organization\nTsukuba-shi, Ibaraki-ken 305-0801Japan', 'Los Alamos National Laboratory\n87545Los AlamosNMUSA', 'Instituto de Física\nUniversidade de São Paulo\n66318, CEP05315-970São PauloCaixa PostalBrazil', 'Los Alamos National Laboratory\n87545Los AlamosNMUSA', 'Department of Physics\nLund University\nBox 118221 00LundSESweden', 'Korea University\n136-701SeoulKorea', 'Department of Physics\nBanaras Hindu University\n221005VaranasiIndia', 'Department of Physics\nBanaras Hindu University\n221005VaranasiIndia', 'Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'Institute for High Energy Physics (IHEP)\nProtvinoRussia', 'Lawrence Livermore National Laboratory\n94550LivermoreCAUSA', 'Los Alamos National Laboratory\n87545Los AlamosNMUSA', 'University of Tennessee\n37996KnoxvilleTNUSA', 'Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'Dapnia\nCEA\nSaclay, F-91191Gif-sur-YvetteFrance', 'Oak Ridge National Laboratory\n37831Oak RidgeTNUSA', 'Department of Physics\nLund University\nBox 118221 00LundSESweden', 'New Mexico State University\nLas Cruces88003NMUSA', 'KFKI Research Institute for Particle and Nuclear Physics (RMKI)\nPOBox 49H-1525, 114BudapestHungary', 'Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'Hiroshima University\nKagamiyama, Higashi-Hiroshima 739-8526Japan', 'Los Alamos National Laboratory\n87545Los AlamosNMUSA', 'Instituto de Física\nUniversidade de São Paulo\n66318, CEP05315-970São PauloCaixa PostalBrazil', 'RIKEN (The Institute of Physical and Chemical Research)\n351-0198WakoSaitamaJAPAN', 'RIKEN BNL Research Center, Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'Advanced Research Institute for Science and Engineering\nWaseda University\n17 Kikui-cho, Shinjuku-ku162-0044TokyoJapan', 'KEK, High Energy Accelerator Research Organization\nTsukuba-shi, Ibaraki-ken 305-0801Japan', 'Nagasaki Institute of Applied Science\nNagasaki-shi851-0193NagasakiJapan', 'RIKEN (The Institute of Physical and Chemical Research)\n351-0198WakoSaitamaJAPAN', 'Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'Debrecen University\nEgyetem tér 1H-4010DebrecenHungary', 'Abilene Christian University\n79699AbileneTXUSA', 'Los Alamos National Laboratory\n87545Los AlamosNMUSA', 'University of New Mexico\n87131AlbuquerqueNMUSA', 'Kyoto University\n606-8502KyotoJapan', 'RIKEN (The Institute of Physical and Chemical Research)\n351-0198WakoSaitamaJAPAN', 'Kyoto University\n606-8502KyotoJapan', 'RIKEN (The Institute of Physical and Chemical Research)\n351-0198WakoSaitamaJAPAN', 'Abilene Christian University\n79699AbileneTXUSA', 'Weizmann Institute\n76100RehovotIsrael', 'Institute of Physics\nUniversity of Tsukuba\n305TsukubaIbarakiJapan', 'Department of Physics\nBanaras Hindu University\n221005VaranasiIndia', 'Department of Physics\nLund University\nBox 118221 00LundSESweden', 'Institute for High Energy Physics (IHEP)\nProtvinoRussia', 'Los Alamos National Laboratory\n87545Los AlamosNMUSA', 'Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'Department of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA', 'Department of Physics and Astronomy\nSUNY, Stony Brook\nStony Brook University\n11794NYUSA', 'Debrecen University\nEgyetem tér 1H-4010DebrecenHungary', 'University of Tennessee\n37996KnoxvilleTNUSA', 'Russian Research Center "Kurchatov Institute"\nMoscowRussia', 'Russian Research Center "Kurchatov Institute"\nMoscowRussia', 'Petersburg Nuclear Physics Institute\nPNPI\nGatchinaRussia', 'Georgia State University\n30303AtlantaGAUSA', 'RIKEN (The Institute of Physical and Chemical Research)\n351-0198WakoSaitamaJAPAN', 'RIKEN BNL Research Center, Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'Iowa State University\n50011AmesIAUSA', 'Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'University of California -Riverside\n92521RiversideCAUSA', "Institute of Atomic Energy (CIAE)\nBeijingChina, People's Republic of China", 'Institute for High Energy Physics (IHEP)\nProtvinoRussia', 'RIKEN (The Institute of Physical and Chemical Research)\n351-0198WakoSaitamaJAPAN', 'RIKEN BNL Research Center, Brookhaven National Laboratory\nUpton11973-5000NYUSA', 'Oak Ridge National Laboratory\n37831Oak RidgeTNUSA', 'Russian Research Center "Kurchatov Institute"\nMoscowRussia', 'and Nevis Laboratories\nColumbia University\n10027, 10533New York, IrvingtonNY, NYUSA', 'and Nevis Laboratories\nColumbia University\n10027, 10533New York, IrvingtonNY, NYUSA', "Institute of Atomic Energy (CIAE)\nBeijingChina, People's Republic of China", 'Weizmann Institute\n76100RehovotIsrael', 'Joint Institute for Nuclear Research\n141980Dubna, Moscow RegionRussia', 'LPC\nUniversité Blaise Pascal\nCNRS-IN2P3\nClermont-Fd63177Aubiere CedexFrance'], 'corpusid': 119403333, 'doi': '10.1103/physrevlett.94.232301', 'github_urls': [], 'n_tokens_mistral': 15379, 'n_tokens_neox': 12636, 'n_words': 5805, 'pdfsha': '4e4d2f3fe73c75895e128e6b96fa6ddaf327cbf5', 'pdfurls': ['https://export.arxiv.org/pdf/nucl-ex/0503003v1.pdf'], 'title': ['Centrality Dependence of Direct Photon Production in √ s NN = 200 GeV Au+Au Collisions', 'Centrality Dependence of Direct Photon Production in √ s NN = 200 GeV Au+Au Collisions'], 'venue': []}

arxiv

29 Jul 2013 Sivaraman Balakrishnan School of Computer Science and Statistics Department School of Computer Science Department of Statistics Carnegie TIGHT LOWER BOUNDS FOR HOMOLOGY INFERENCE Carnegie Mellon University Carnegie Mellon University Pittsburgh Mellon University Pittsburgh 15213, 15213PA, PA Alessandro Rinaldo arinaldo@cmu.edu School of Computer Science and Statistics Department School of Computer Science Department of Statistics Carnegie TIGHT LOWER BOUNDS FOR HOMOLOGY INFERENCE Carnegie Mellon University Carnegie Mellon University Pittsburgh Mellon University Pittsburgh 15213, 15213PA, PA Aarti Singh School of Computer Science and Statistics Department School of Computer Science Department of Statistics Carnegie TIGHT LOWER BOUNDS FOR HOMOLOGY INFERENCE Carnegie Mellon University Carnegie Mellon University Pittsburgh Mellon University Pittsburgh 15213, 15213PA, PA Larry Wasserman School of Computer Science and Statistics Department School of Computer Science Department of Statistics Carnegie TIGHT LOWER BOUNDS FOR HOMOLOGY INFERENCE Carnegie Mellon University Carnegie Mellon University Pittsburgh Mellon University Pittsburgh 15213, 15213PA, PA 29 Jul 2013arXiv:1307.7666v1 [stat.ML] The homology groups of a manifold are important topological invariants that provide an algebraic summary of the manifold. These groups contain rich topological information, for instance, about the connected components, holes, tunnels and sometimes the dimension of the manifold. In earlier work [1], we have considered the statistical problem of estimating the homology of a manifold from noiseless samples and from noisy samples under several different noise models. We derived upper and lower bounds on the minimax risk for this problem. In this note we revisit the noiseless case. In [1], we used Le Cam's lemma to establish the lower bound 1as n → ∞ thus establishing rate optimal asymptotic minimax bounds for the problem. The techniques we use here extend in a straightforward way to the noisy settings considered in [1]. Although, we do not consider the extension here non-asymptotic bounds are also straightforward. R n = Ω exp −nτ d for d ≥ 1 and D > d. In the noiseless case the upper bound follows from the work of [2], who show that R n = O 1 τ d exp −nτ d . In this note we use a different construction based on the direct analysis of the likelihood ratio test to show that 1. Introduction. Let M be a d-dimensional manifold embedded in R D where d ≤ D. The homology groups H(M) of M (see [3]), are an algebraic summary of the properties of M. The homology groups of a manifold describe its topological features such as its connected components, holes, tunnels, etc. In this note we study the problem of estimating the homology of a manifold M from a sample X = {X 1 , . . . , X n }. Specifically, we bound the minimax risk (1) R n ≡ inf H sup Q∈Q Q n H = H(M) where the infimum is over all estimators H of the homology of M and the supremum is over appropriately defined classes of distributions Q for Y . Note that 0 ≤ R n ≤ 1 with R n = 1 meaning that the problem is hopeless. Bounding the minimax risk is equivalent to bounding the sample complexity of the best possible estimator, defined by n(ǫ) = min n : R n ≤ ǫ where 0 < ǫ < 1. We assume that the sample X ⊂ R D constitutes a set of observations of an unknown ddimensional manifold M, with d < D, whose homology we seek to estimate. The distribution of the sample depends on the properties of the manifold M as well as on the distribution of points on M. We consider the collection Manifold Assumptions. We assume that the unknown manifold M is a d-dimensional smooth compact Riemannian manifold without boundary embedded in the compact set X = [0, 1] D . We further assume that the volume of the manifold is bounded from above by a constant which can depend on the dimensions d, D, i.e. we assume vol(M) ≤ C D,d . We will also make the further assumption that D > d. The main regularity condition we impose on M is that its condition number be not too large. The condition number κ(M) (see [2]) is 1/τ , where τ is the largest number such that the open normal bundle about M of radius r is imbedded in R D for every r < τ . For τ > 0 let M ≡ M(τ ) = M : κ(M) ≥ τ denote the set of all such manifolds with condition number no smaller than τ . A manifold with small condition number does not come too close to being self-intersecting. 1.1. Lower bounding the minimax risk. In this note we will lower bound the minimax risk by considering a related testing problem. Before describing the hypotheses we describe the null and alternate manifolds. The null manifold M 0 is a collection of m, d-spheres of radius τ , denoted S 1 , . . . , S m , with centers on one face of the unit hypercube in d + 1 dimensions (M 0 is embedded in a space of dimension D which is of dimension at least d + 1), with spacing between adjacent centers = 4τ . It is easy to see that m = O 1 (4τ ) d because the manifold must be completely in [0, 1] D , and that the manifold has condition number at least 1/τ . We will use m = Θ 1 (4τ ) d in this note. Let P 0 denote the uniform distribution on M 0 . The alternate manifolds are a collection {M 1i : i ∈ {1, . . . , m}}, where M 1i is M 0 with S i removed. Let π denote the uniform distribution on {1, . . . , m}, and P 1i denote the uniform distribution on M 1i . We need to ensure that the density p is lower bounded by a constant. Note that the total d-dimensional volume of M 0 is v d τ d m, and so p(x) ≥ 1 v d τ d m where v d is the volume of the d-dimensional unit ball. This is Ω(1) as desired. A similar argument works for M 1i . Consider the following testing problem: H 0 : X ∼ P 0 H 1 : X ∼ P 1i with i ∼ π A test T , is a measurable function of X, in particular T : X → {0, 1}, and its risk is defined as R T n . . = P H 0 (T (X) = 1) + P H 1 (T (X) = 0) The relationship between testing and estimation is standard [4]. In our case it is easy to see that the estimation minimax risk of Equation 1 satisfies, R T n ≤ 2R n and so it suffices to lower bound R T n to obtain a lower bound on R n . This relation is a straightforward consequence of the fact that H(M 0 ) = H(M 1i ) for every i (since they have different number of connected components), and so any estimator can be used in the testing problem described. The optimal test for the hypothesis testing problem described is the likelihood ratio test, T (X) = 0 if and only if L(X) ≤ 1 where L(X) = L 1 (X) L 0 (X) where L 1 (X) and L 0 (X) are likelihoods of the data under the alternate and null respectively. 1.2. Coupon collector lower bound. We begin with a theorem from [5]. Lemma 1 (Theorem 3.8 of [5]). Let the random variable X denote the number of trials for collecting each of the n types of coupons. Then for any constant c ∈ R, and m = n log n − cn, Theorem 2. For any constant δ < 1, we have R n ≥ Ω min 1 τ d exp −nτ d , δ as n → ∞. Proof. Notice that since From this we can see the probability of a Type I error → cδ, and R T n ≥ cδ, which gives R n ≥ c 2 δ as desired. m = Θ 1 (4τ ) d 3. Discussion. In this note we have established tight minimax rates for the problem of homology inference in the noiseless case. The intuition behind the construction extends to the noisy cases considered in [1] in a straightforward way. Although the bound we have shown is an asymptotic lower bound, a finite sample lower bound follows in a straightforward way by replacing the asymptotic calculation in Lemma 1 with finite sample estimates. We also expect similar constructions to be useful in establishing tight lower bounds for the problems of manifold estimation in Hausdorff distance considered in [6,7], and for the problem of estimation of persistence diagrams in bottleneck distance considered in [8]. probability distributions supported over manifolds M in M having densities p with respect to the volume form on M uniformly bounded from below by a constant a > 0, i.e. 0 < a ≤ p(x) < ∞ for all x ∈ M. lim n→∞ P(X > m) = 1 − exp (− exp (c)) 2. Main result. the theorem is implied by the statement thatn = m log m + m log 1 δ =⇒ R n ≥ cδfor some constant c. We will focus on proving this claim.Let us consider the case when samples are drawn according to P 0 . From Lemma 1 we have that if n = m log m + m log 1 δ then the probability with which we do not see a point in each of the m spheres is 1 − exp(− exp(− log 1/δ)) ≥ cδ since δ < 1, for some constant c. It is easy to see that if we do not see a point in each of the m spheres then L(X) ≥ 1 m 1 (1 − 1/m) n . . = T m,nWhen n = m log m + m log 1 δ , T m,n → 1 δ > 1 so asymptotically the likelihood ratio test always rejects the null. The asymptotic notation in both the upper and lower bounds hide constants that could depend on the dimensions d and D. Minimax rates for homology inference. Sivaraman Balakrishnan, Alessandro Rinaldo, Don Sheehy, Aarti Singh, Larry Wasserman, AISTATSSivaraman Balakrishnan, Alessandro Rinaldo, Don Sheehy, Aarti Singh, and Larry Wasserman. Minimax rates for homology inference. AISTATS, 2012. Finding the homology of submanifolds with high confidence from random samples. Partha Niyogi, Stephen Smale, Shmuel Weinberger, Discrete & Computational Geometry. 391-3Partha Niyogi, Stephen Smale, and Shmuel Weinberger. Finding the homology of submanifolds with high confidence from random samples. Discrete & Computational Geometry, 39(1-3):419-441, 2008. Algebraic Topology. Allen Hatcher, Cambridge University PressAllen Hatcher. Algebraic Topology. Cambridge University Press, 2002. Testing Statistical Hypotheses. Springer Texts in Statistics. E L Lehmann, J P Romano, SpringerE.L. Lehmann and J.P. Romano. Testing Statistical Hypotheses. Springer Texts in Statistics. Springer, 2005. Randomized Algorithms. R Motwani, P Raghavan, Cambridge University PressR. Motwani and P. Raghavan. Randomized Algorithms. Cambridge University Press, 1995. Manifold estimation and singular deconvolution under Hausdorff loss. Christopher R Genovese, Marco Perone-Pacifico, Isabella Verdinelli, Larry Wasserman, Ann. Statist. 402Christopher R. Genovese, Marco Perone-Pacifico, Isabella Verdinelli, and Larry Wasserman. Manifold estimation and singular deconvolution under Hausdorff loss. Ann. Statist., 40(2):941-963, 2012. Minimax manifold estimation. Christopher R Genovese, Marco Perone-Pacifico, Isabella Verdinelli, Larry Wasserman, Journal of Machine Learning Research. 13Christopher R. Genovese, Marco Perone-Pacifico, Isabella Verdinelli, and Larry Wasserman. Minimax manifold estimation. Journal of Machine Learning Research, 13:1263-1291, 2012. Optimal rates of convergence for persistence diagrams in topological data analysis. Frédéric Chazal, Marc Glisse, Catherine Labruère, Bertrand Michel, Frédéric Chazal, Marc Glisse, Catherine Labruère, and Bertrand Michel. Optimal rates of convergence for persistence diagrams in topological data analysis, 2013.

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{'abstract': "The homology groups of a manifold are important topological invariants that provide an algebraic summary of the manifold. These groups contain rich topological information, for instance, about the connected components, holes, tunnels and sometimes the dimension of the manifold. In earlier work [1], we have considered the statistical problem of estimating the homology of a manifold from noiseless samples and from noisy samples under several different noise models. We derived upper and lower bounds on the minimax risk for this problem. In this note we revisit the noiseless case. In [1], we used Le Cam's lemma to establish the lower bound 1as n → ∞ thus establishing rate optimal asymptotic minimax bounds for the problem. The techniques we use here extend in a straightforward way to the noisy settings considered in [1]. Although, we do not consider the extension here non-asymptotic bounds are also straightforward.", 'arxivid': '1307.7666', 'author': ['Sivaraman Balakrishnan \nSchool of Computer Science and Statistics Department\nSchool of Computer Science\nDepartment of Statistics Carnegie\nTIGHT LOWER BOUNDS FOR HOMOLOGY INFERENCE\nCarnegie Mellon University\nCarnegie Mellon University Pittsburgh\nMellon University Pittsburgh\n15213, 15213PA, PA\n', 'Alessandro Rinaldo arinaldo@cmu.edu \nSchool of Computer Science and Statistics Department\nSchool of Computer Science\nDepartment of Statistics Carnegie\nTIGHT LOWER BOUNDS FOR HOMOLOGY INFERENCE\nCarnegie Mellon University\nCarnegie Mellon University Pittsburgh\nMellon University Pittsburgh\n15213, 15213PA, PA\n', 'Aarti Singh \nSchool of Computer Science and Statistics Department\nSchool of Computer Science\nDepartment of Statistics Carnegie\nTIGHT LOWER BOUNDS FOR HOMOLOGY INFERENCE\nCarnegie Mellon University\nCarnegie Mellon University Pittsburgh\nMellon University Pittsburgh\n15213, 15213PA, PA\n', 'Larry Wasserman \nSchool of Computer Science and Statistics Department\nSchool of Computer Science\nDepartment of Statistics Carnegie\nTIGHT LOWER BOUNDS FOR HOMOLOGY INFERENCE\nCarnegie Mellon University\nCarnegie Mellon University Pittsburgh\nMellon University Pittsburgh\n15213, 15213PA, PA\n'], 'authoraffiliation': ['School of Computer Science and Statistics Department\nSchool of Computer Science\nDepartment of Statistics Carnegie\nTIGHT LOWER BOUNDS FOR HOMOLOGY INFERENCE\nCarnegie Mellon University\nCarnegie Mellon University Pittsburgh\nMellon University Pittsburgh\n15213, 15213PA, PA', 'School of Computer Science and Statistics Department\nSchool of Computer Science\nDepartment of Statistics Carnegie\nTIGHT LOWER BOUNDS FOR HOMOLOGY INFERENCE\nCarnegie Mellon University\nCarnegie Mellon University Pittsburgh\nMellon University Pittsburgh\n15213, 15213PA, PA', 'School of Computer Science and Statistics Department\nSchool of Computer Science\nDepartment of Statistics Carnegie\nTIGHT LOWER BOUNDS FOR HOMOLOGY INFERENCE\nCarnegie Mellon University\nCarnegie Mellon University Pittsburgh\nMellon University Pittsburgh\n15213, 15213PA, PA', 'School of Computer Science and Statistics Department\nSchool of Computer Science\nDepartment of Statistics Carnegie\nTIGHT LOWER BOUNDS FOR HOMOLOGY INFERENCE\nCarnegie Mellon University\nCarnegie Mellon University Pittsburgh\nMellon University Pittsburgh\n15213, 15213PA, PA'], 'corpusid': 15182145, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 3260, 'n_tokens_neox': 2859, 'n_words': 1873, 'pdfsha': '7dae767b7bef8d85627ca5960bdbf9ced6ae1cc4', 'pdfurls': ['https://arxiv.org/pdf/1307.7666v1.pdf'], 'title': [], 'venue': []}

arxiv

Stimulated emission at 1.54 µm from Erbium/Oxygen-doped silicon-based light emitting diodes Jin Hong Ministry of Education Key Laboratory of Polar Materials and Devices East China Normal University 200241ShanghaiChina Huimin Wen Department of Micro/Nano Electronics National Key Laboratory of Science and Technology on Micro/Nano Fabrication Laboratory University of Michigan-Shanghai Jiao Tong University Joint Institute Shanghai Jiao Tong University 200240ShanghaiChina Jiajing He Department of Micro/Nano Electronics National Key Laboratory of Science and Technology on Micro/Nano Fabrication Laboratory University of Michigan-Shanghai Jiao Tong University Joint Institute Shanghai Jiao Tong University 200240ShanghaiChina Jingquan Liu Department of Micro/Nano Electronics National Key Laboratory of Science and Technology on Micro/Nano Fabrication Laboratory University of Michigan-Shanghai Jiao Tong University Joint Institute Shanghai Jiao Tong University 200240ShanghaiChina Jens W Tomm Department of Micro/Nano Electronics National Key Laboratory of Science and Technology on Micro/Nano Fabrication Laboratory University of Michigan-Shanghai Jiao Tong University Joint Institute Shanghai Jiao Tong University 200240ShanghaiChina Max-Born-Institut für Nichtlineare Optik und Kurzzeitspektroskopie Max-Born-Str. 2A12489BerlinGermany Fangyu Yue Ministry of Education Key Laboratory of Polar Materials and Devices East China Normal University 200241ShanghaiChina Junhao Chu Ministry of Education Key Laboratory of Polar Materials and Devices East China Normal University 200241ShanghaiChina National Laboratory for Infrared Physics Shanghai Institute of Technical Physics Chinese Academy of Sciences 500 Yutian Road200083ShanghaiChina And Chungang Duan Ministry of Education Key Laboratory of Polar Materials and Devices East China Normal University 200241ShanghaiChina Stimulated emission at 1.54 µm from Erbium/Oxygen-doped silicon-based light emitting diodes 1 # These authors contributed equally * Corresponding authors: yaping.dan@sjtu.edu.cn (Yaping Dan), Chungang Duan), and fyyue@ee.ecnu.edu.cn (Fangyu Yue).Abstract：Silicon-based light sources including light-emitting diodes (LEDs)and laser diodes (LDs) for information transmission are urgently needed for developing monolithic integrated silicon photonics. Silicon doped by ion implantation with erbium ions (Er 3+ ) is considered a promising approach, but suffers from an extremely low quantum efficiency. Here we report an electrically pumped superlinear emission at 1.54 µm from Er/O-doped silicon planar LEDs, which are produced by applying a new deep cooling process.Stimulated emission at room temperature is realized with a low threshold current of ~6 mA (~0.8 A/cm 2 ). Time-resolved photoluminescence and photocurrent results disclose the complex carrier transfer dynamics from the silicon to Er 3+ by relaxing electrons from the indirect conduction band of the silicon. This picture differs from the frequently-assumed energy transfer by electron-hole pair recombination of the silicon host. Moreover, the amplified emission from the LEDs is likely due to a quasi-continuous Er/O-related donor band created by the deep cooling technique. This work paves a way for fabricating superluminescent diodes or efficient LDs at communication wavelengths based on rare-earth doped silicon. INTRODUCTION Silicon/Si-based light sources including lasers at telecommunication wavelengths are the bottleneck for the heterogeneous integration of photonics with complementary metal-oxide-semiconductor circuits [1][2][3][4][5][6]. Ion implantation of Erbium/Er (often with Oxygen/O) into Si is believed to be one of the most promising approaches to create Si-based light emitting devices (LEDs) at 1.54 µm [7][8][9][10][11][12][13][14][15]. However, the reported quantum efficiencies are extremely low (≈ 0.01%) at room temperature, mainly due to strong nonradiative recombination caused by the comparably large Er-related precipitates formed during the cooling process in the standard rapid thermal annealing (RTA) [3,[15][16][17]. Recently, the efficiency has been substantially improved by introducing a deep cooling (DC) technique [11,18] which can effectively mitigate the Er precipitation created during the RTA process. LEDs with a perpendicular emission structure based on the obtained material achieved a record external quantum efficiency of ~0.8% at room temperature [11]. Here, by optimizing the implantation of Er and O, the DC procedure, and the Si-based LED structure with a planar emission geometry, a near-unity quantum yield (or ~100% slope efficiency) of the photoluminescence (PL) in Er-doped Si is reached. Moreover, a super-linearly growing electroluminescence (EL) is obtained with a slope efficiency beyond 2 even at room temperature. The low threshold of ~6 mA (~0.8 A/cm 2 ) [19,20], the super-linear EL integral intensity [21][22][23], the narrowing full width at half maximum (FWHM) [24,25], and the Gaussian-like spatial emission distribution [26,27], as well as the fast radiation recombination lifetime [28][29][30][31], confirm the presence of stimulated emission involving transitions related to the Er 3+ [32][33][34]. Time-resolved PL (TR-PL) and photocurrent (PC) measurements reveal the relaxation dynamics of the nonequilibrium carriers from the Si host to the Er 3+ precipitates. The hot nonequilibrium carriers in Si first cross the intra-valley barrier (e.g., Γ or L point in k-space if the laser energy is high enough) with a time constant of ~110 ps to the bottom of the indirect conduction band (CB) (or the Δ1c point) [35,36]. The excess carriers further decay from the CB bottom to the evenly-distributed Er/O-related complexes that act as a quasi-continuous donor band with a decay time of ~30 ps. Within this donor band, the carriers resonantly excite (i.e., transfer the energy of carriers by non-radiative recombination to) the Er 3+ -4f electrons to emit at ~1.54 µm ( 4 I13/2 → 4 I15/2). MATERIALS AND METHODS A. Sample fabrication Float zone (FZ) intrinsic Si (100) [11,18], where the samples were annealed at 950°C for 5 min by means of copper coil-based electromagnetic heating and followed by a flush of high purity He (99.999%) gas cooled in liquid N2 (77 K). Its detailed description can be referred to our previous works in Ref. 11. RESULTS AND DISCUSSION Room temperature PL spectra of Er/O-doped Si (see Sample Information) at different excitation powers (λ = 405 nm; 3.06 eV) are shown in Fig. 1a. Er 3+ emission at ~1.54 µm (~0.81 eV; marked with A) is observed and becomes more pronounced when the excitation power increases. Besides the main emission peak A, there are a side shoulder and a broad tail at the high energy side denoted as B (at ~0.82 eV) and C (even beyond the bandgap of Si), respectively (see inset). The shoulder B is ascribed to the Er 3+ -related states [12] and the tail C is related to the Er/O-doping induced defect centers in the bandgap of Si. No shape change of the emission occurs when the excitation power is higher than ~5 mW. The integral intensity of the three specified emission bands A, B, and C shows a slope efficiency of near-unity (S ≈ 0.93) within two orders of magnitude of excitation power (Fig. 1b in a log-log scale). It characterizes the quasi-free carrier recombination and the high internal quantum yield of Er 3+ without efficient (Auger or thermally-related) non-radiative recombination even at room temperature. At low temperature, the PL efficiency is enhanced as expected by suppressing the Shockley-Read-Hall recombination [11,15]. What's interesting is that the shoulder B and tail C are evident and can be strengthened in comparison with the main peak A as the excitation power increases; see PL spectra (4 K) at 10 mW and 100 mW in Fig. 1c. It is clear by comparing the curve at 4 K with 300 K that elevated temperatures have strong influence on the emission processes in B and C. Fig. 1d (normalized to peak A) demonstrates that the long tail C is substantially narrowed by shifting the edge-emission toward the long wavelength side upon increase of temperature; also see the two colored areas in Fig. 1c. The narrowing of the tail C unlikely originates from the temperature-dependent absorptivity of excitation light (λ = 405 nm) as this effect has been taken into account in our measurements (see Fig. S1). Instead, it likely comes from the temperature-driven redistribution of excess carriers from the high-energy states in the Si CB to those within the Si bandgap (see more discussions later). This narrowing results in a relatively stronger emission for both shoulder B and tail C as shown in Fig. 1d. To have a deeper insight into the above process, the integral intensity of the main peak A, the side shoulder B, and the tail C are normalized to their integral intensity at 4 K, respectively. The normalized integral intensities are shown in the Arrhenius plot in Fig. 1e. While the main peak A quenches monotonously with increase of temperature, emissions of B and C first increase and then decrease at a critical temperature point of ~180 K (~15.5 meV). Moreover, the total integral PL intensity also shows a monotonous decrease but with a relatively low temperature-quenching rate at 300 K (still ~50% of 4 K, i.e., with a decrease of 3 dB). This means that excitons transfer between the states, which are involved into the generation of the emission bands A, B, and C, without being affected by strong temperature-dependent non-radiative recombination. Based on the evolutions in Fig. 1e, an activation energy of ~14.9 meV can be extracted for the main peak A (rather close to the phonon energy of ~15.5 meV) [15], and ~40.0 meV for the total emission. Fig. 1f shows the PL decay of the shows the EL spectra of the LED at different injection currents (at 300 K). Notice that the EL is similar in spectral shape to the PL from the Er-activated Si except that the tail C is weaker and cut off at the Si bandgap; see the curves at the bottom of Fig. 2c. With the injection current increase, the emission signal at To clarify further the energetic structure in the LEDs, we recorded the shortcircuit PC spectra at 300 K under light illumination from 0.6 eV to 1.3 eV in photon energy, as shown in Fig. 2c. The third-order derivative of the curve sets the cutoff at ~1.1 eV which corresponds to the Si bandgap Eg. It is necessary to point out that around 0.1 eV below the Si bandgap, the PC reaches a plateau after an exponential decline by 2 orders of magnitude, which extends to ~0.6 eV before it drops below the background noise. This PC plateau roughly lies in the same spectral range of the broad EL band, and should come from the Errelated defect (including Er 3+ -4f degenerated) states [38].Taking into account the PC and EL bands from 0.7 eV to 1.1 eV, this implies that a quasi-continuous band from 0.7 eV above the valence band (VB) to the bottom of (even above) CB is formed. Fig. 2d and e shows the main peak positions and FWHMs as a function of the injection current at 300 K and 7 K. It can be observed that the main emission peak at 1.54 µm is almost independent of the injection current, whereas the FWHM is reduced from 120 meV to 20 meV at 300 K (from 110 meV to 10 meV at 7 K) until the current approaches to ~10 mA. After a saturation region, the FWHM starts to broaden again when the injection current is higher than ~30 mA. Moreover, in Fig. 2f and g, as the injection current ramps up, the peak intensity and integral intensity distinctively show a sub-linear increase followed by super-linear ramp with a slope of S ≈ 2.7 at 300 K (~3.6 at 7 K) and S ≈ 1.5 at 300 K (almost the same at 7 K), respectively. A well-pronounced threshold is observed at ~6 mA or 0.8 A/cm 2 (refer to the integral intensity in Fig. 2g). These features along with the FWHM narrowing suggest that amplified spontaneous [21,23,24] (or stimulated [33,34]) emission occurs in our Si LEDs. The emission cross-section can be thus estimated by the following equation [30,39], !" = #ln & # $ ' ( % 4 # ⁄ ) (1) where c is the light speed, n is the refractive index Fig. 4b, which likely comes from the band-to-band emission in Si [40]. The corresponding decay profile shown in Fig. 4c suggests that the lifetime is much longer than ~12 ns, consistent with the picture presented in Fig. 4a. Notice that the streak camera used is not capable of detecting light near ~1.5 µm in wavelength, and that the time resolution is better than 10 ps; also see What needs to be emphasized again is that the results achieved are largely due to the use of the DC process in sample preparation. The process offers advantages for the optically pumped Er 3+ . These ions appear in high density in relatively uniformly sized, Er-related clusters with a diameter of ~1 nm, which is much smaller than the 5 nm diameter achieved with the standard RTA process [11]. Moreover, the spatial distribution of the small clusters is extremely even, which provides very useful conditions for the Er/O cluster to act as a broad and quasi-continuous donor band [42]. The decaying carriers from the Si host into the Er-related states excite the 4f-electron of Er 3+ , which produces the emission at ~1.54 µm. Details on material characterizations by high resolution transmission electron microscopy and X-ray photoelectron spectroscopy have been described in Ref. 11, and the striking difference of the defect-related EL signal between RTA and DC is shown in Fig. S2. CONCLUSION In conclusion, we observed amplified spontaneous (or stimulated) emissions at room temperature with a low threshold of ~0. A pair of co-axial electrodes was prepared by UV photolithography (MDA-400M, MIDAS) and metal film deposition (Nexdep, Angstrom Engineering Inc.). The internal electrode is in contact with p-type boron doping region and the external electrode in contact with the n-type P region. All the microfabrication processes were performed with home-built devices at the Center for Advanced Electronic Materials and Devices. After the Al metal wire bonding (7476D, West Bond), the devices were integrated on a PCB board. The I-V curves were taken using a digital sourcemeter (Keithley 2400) controlled by a LabVIEW script. Fig. 1 | 1PL spectra. (a) Room-temperature PL spectra from the Er/O-doped Si at different excitation powers (P). PL curves are tentatively divided into three parts, marked by A, B and C, as shown in the inset. (b) Excitation power dependence of PL integral intensity (II) at 300 K. (c) PL spectra at P = 10 mW (blue curve) and 100 mW (red curve) at 4 K. The room-temperature PL result at P = 100 mW (gray curve) is also given for comparison. (d) PL spectra at different temperatures (P = 10 mW). (e) Arrhenius plot of normalized PL integral intensity vs. inversed temperature. (f) PL decay curves at 77 K and 300 K together with the fitting results.A lateral LED structure based on the DC-processed Er/O-doped Si structure is fabricated with a planar emission geometry (in contrast to our previous perpendicular structure)[11] (Fig. 2a). We separately implanted boron/B and phosphorus/P into the Er/O-Si samples both with a peak concentration of10 19 cm -3 , forming a coaxial pn-junction diode on the Si surface. The DC process was then applied to activate the Er/O, B, and P dopants (see Sample Information). A rectifying current vs. voltage (I-V) curve was observed.Fig. 2b ~0.81 eV becomes clearer with the feature of the Er 3+ emission at ~1.54 µm, and simultaneously narrowed first and then broadened. The EL cutoff at the Si bandgap (see the blue vertical line) is due to the fact that electrons can only be electrically pumped to the CB bottom. In PL, electrons can be excited to high energy states in the CB, resulting in a long emission tail well beyond the Si bandgap. ( 3 . 347), is lifetime (14 µs), w is the FWHM (~0.10 µm) and is the emission wavelength (1.54 µm). Its maximum is approximately !" = 7.0 × 10 &'( cm # , which is even one order of magnitude larger than the values for the Er 3+ in nanocrystal-Si sensitized silica (8.0 × 10 -20 cm 2 ) [15, 22]. Fig. 2 | 2EL spectra and analysis. (a) Scheme of the LED structure. (b) Currentdependent EL spectra at 300 K. (c) PC spectrum at 300 K (upper panel). The star '*' marks the maximum of the third-order derivative and the blue line Eg (Si) = 1.10 eV. The PC shoulder at ~0.81 eV is from the Er-related defects. The lower panel shows the EL and PL spectra of the LED at 300 K. (d)-(g) Currentdependence of the EL peak energy, FWHM, intensity and integrated intensity, respectively, at 300 K (red) and 7 K (blue). The green lines are guides to the eyes. Fig. 3 3shows the spatially-resolved emission intensity of the LED surface as taken by an InGaAs camera. At an injection current lower than the threshold (Fig. 3a), the emission is weak and uniformly distributed across the entire emitter surface (300 μm in diameter) except for the region close to the center electrode (see the 2-dimensional image at the bottom), which is heated up.When the injection current is higher than the threshold, the emission is strongly enhanced with a maximum in the LED surface center; seeFig. 3b. In this case, the spatial distribution of the emission follows a Gaussian distribution if the loss caused by the center electrode is not considered (the pixels are saturated inFig. 3b). The supplementary Video.mp4 shows the evolution of the emission intensity profile as the LED is electrically pumped across the threshold current.Together with the above results including the behavior of the emission energy, FWHM, intensity, and integral intensity, this observation confirms the presence of electrically-driven stimulated emission, although it lacks a suitable cavity[32][33][34]. Fig. 3 | 3Spatial-resolved intensity distribution at the emitting surface of the LED at 1.55 m (a) below and (b) above the threshold current. The emission imaging of the surface is also shown at the bottom for comparison.We now address the electronic structure of the Er 3+ -doped Si and the carrier relaxation and transfer dynamics.Fig. 4a shows the TR-PL results under a pulsed laser excitation (λ = 760 nm, 80 MHz) at 5 K. The recorded PL signals are between 1.1 eV and 1.2 eV. On the time scale, the PL amplitudes remain nearly constant at a scale of ~12 ns, indicating that the emission has a longer lifetime. The time-integrated PL spectrum can be fitted with a Gaussian distribution with a peak centered at ~1.15 eV, as shown in Fig. 4c . 4cWhen the frequency-doubled wavelength (380 nm) excites the sample, a similar but smaller emission band at ~1.15 eV was observed; seeFig. 4d. In the time domain, the emission has two striking features shown in Fig. 4e. First, unlike the long emission lifetime excited at the wavelength of 760 nm, the emission has a short decay time of ~30 ps. Second, the emission lags behind the excitation pulse by ~110 ps. These features are likely caused by the fact that high energy photons from the 380 nm laser can excite electrons from the VB to the L (or Γ') point of the CB, which the 760 nm laser excitation cannot. With assistance of phonons, hot electrons from the L point will first transit to the CB bottom (Δ1c point) and then emit photons at ~1.15 eV via radiative recombination from band to band. The time constant of ~110 ps could be attributed to the transfer time of the carriers across the intra-valley barrier. The relatively long transit time results in a low concentration of excess carriers at the CB bottom because the barrier temporally holds the carriers in the L point. As a result, excess carriers do not generate strong spontaneous emission but rapidly (~30 ps) relax to the Er/Orelated quasi-continuous levels below the indirect CB of Si [15,16]; see more discussion later. This behavior is illustrated by the blue transients in Fig. 4e. This fast relaxing process suggests a fundamental discrepancy with the existing model of energy transfer from electron-hole pair recombination directly to the excitation of the Er 3+ -4f electrons for the 1.54 µm emission. Fig. 4 | 4TR-PL spectral results. (a) TR-PL image from the Er/O-doped Si at 5 K (left) with 760-nm excitation laser. (b) Time-integrated spectrum and Gaussian-fit. (c) Decay curves of the sample and the laser. (d) Time-integrated spectra at excitation wavelengths of 760 nm and 380 nm. (e) Decay curves at different excitation wavelengths. The analysis of all these findings results in a comprehensive picture, which is shown schematically in Fig. 5. High energy photons from the 380 nm laser first excite electrons from the valence to the L point of the CB. Due to the intervalley barriers, these carriers slowly transit to the bottom of the indirect CB (Δ1c) with a time constant of ~110 ps. Since the localized states in the quasi-continuous Er/O-related donor band have larger momentum, excess carriers at the indirect CB bottom can be readily coupled and transferred to the Er/O-related donor band with a relatively short lifetime of ~30 ps. Finally, the carriers relax to the VB via radiative and nonradiative recombination in the quasi-continuous Er/Orelated donor band. The radiative recombination broadens the emission band up to the Si bandgap and the nonradiative recombination mainly transfers energy to resonantly excite the Er 3+ . Thus the quasi-continuous band not only facilitates the rapid emission decay of ~30 ps (by 380 nm laser excitation) shown in Fig. 4e, but also serves as efficient non-radiation recombination centers to resonantly excite the Er 3+ for the 1.54 µm emission. This could be one of the reasons why the 1.54 µm emission in Er/O-doped Si is usually not effectively excited by infrared photons, but by high-energy photons, e.g., ~400 nm, which then produce the amplified emission. Although the fast relaxation of carriers within the quasi-continuous donor band cannot be directly detected, the extremely long lifetime of the Er 3+ emission at 1.54 µm (~ms) facilitates the achievement of inversion, resulting in amplified spontaneous (or stimulated) emission. The main reason why it is complicated to obtain amplified emission by optical excitation (i.e., PL) is the high reflectance loss of visible light at the surface of the polished Si matrix.Although the temperature shows no influence on the reflectivity, the loss of surface reflectivity is high (e.g., ~40% for 405 nm)[41]; also seeFig. S1. Fig. 5 | 5Scheme of the carrier relaxation dynamics. The hot-carriers in the upper states in Γ′ or L points excited by the 380 nm laser transfer to the indirect CB minimum (Δ1c) of Si with a time constant of ~110 ps. From here, a time constant of ~30 ps characterizes the transition of carriers to the distributed band created by the Er/O-related donor states. 8 A/cm 2 from the Er/O-doped Sibased LEDs that were treated with a DC process. In comparison with the standard RTA process, the DC process can effectively suppress the precipitation of Er/O related nanocrystals and form more uniformly distributed Er-O-Si compounds. The impact of the DC process on Er/O formation can effectively reduce the density of nonradiative defects in the Si bandgap and facilitate the formation of a quasi-continuous Er/O-related donor band right below the Si CB. As a result, strong room temperature PL and amplified spontaneous (or stimulated) emissions were observed in EL spectra. In particular, the quasicontinuous Er/O-related donor band not only facilitates the rapid emission decay of excess carriers but also serves as efficient recombination centers to extend the emissions up to the Si bandgap, forming a broad tail in the spectrum in addition to the widely observed Er 3+ emission at 1.54 µm. This work may pave a way for fabricating superluminescent (or laser) diodes at communication wavelengths based on rare-earth doped silicon. Funding. National Science Foundation of China (61874043, 61790583, 21703140, 61874072); Aero-Science Fund (201824X001); Special-key project of the "Innovative Research Plan", Shanghai Municipality Bureau of Education (2019-01-07-00-02-E00075). wafers (Resistivity: ≥ 10 kΩ·cm; Thickness: 500 ± 20 μm; Suzhou Resemi Semiconductor Co., Ltd, China). Er and O ions were implanted with an injection energy and dose of 200 keV and 4 × 10 15 cm -2 , and 32 keV and 10 16 cm -2 , respectively, at the Institute of Semiconductors, Chinese Academy of Sciences, China. After that, the Er/O-implanted Si samples were cleaned with ethanol and deionized water, and then immersed in a piranha solution (sulfuric acid: 30% hydrogen peroxide=3:1) for 30 min at 90°C, followed by drying with a high purity nitrogen (99.99%) stream. To form a planar pn junction, we further implanted boron/B and phosphorus/P dopants into these Si samples (B: 30 keV and 2.2 × 10 14 cm -2 ; P: 80 keV and 2.2 × 10 14 cm -which matches the peak depth of the Er in Si. A 200-nm-thick SiO2 films was then deposited on the Er/O-implanted Si samples by reactive magnetron sputtering (Delton multi-target magnetic control sputtering system, AEMD, SJTU). A deep cooling process was performed to activate the Er/O, B, and P dopants at the same time via an upgraded dilatometer (DIL 805A, TA Instruments) B . BOptical characterizations of an 80 fs-pulse) is better than 10 ps. In all setups the samples are mounted on the cold head of a Helium closed cycle cooler. This allows for temperatureA Fourier transform infrared (FTIR) spectrometer (Vertex 80v, Bruker) is employed to measure transmission, reflectance, steady-state photoluminescence/PL and electroluminescence/EL, and photocurrent/PC spectra. The focused 405-nm emission from a continuous wave/cw LD (MLL- III-405, CNI, Changchun, China) with a maximum excitation power of ~160 mW was employed as the excitation source. The effective excitation power on the sample surface was calibrated by referring to the 405 nm-transmission and reflectance of the samples at different temperatures. Different optical filters including notch filters, neutral density filters, and long-pass filters were utilized for avoiding the influence of the excitation source during the excitation power dependent PL measurements. For EL measurements, cw and pulsed (with a ns- µs pulse duration and a repetition rate of ~10 4 Hz) currents were injected into the structures. For PC measurements, a low-noise current preamplifier (SRS SR570) was employed for recording the defect-related PC signal below the bandgap of Si. Time-resolved PL (TR-PL) was carried out in order to determine the non-equilibrium carrier lifetime related to Er 3+ . This was implemented by mW-excitation with a 405 nm emitting diode laser being operated at 3 kHz as the excitation source, a fast InGaAs photodiode, and a GHz sampling oscilloscope (Agilent MSO9404A) for data accumulation and readout. Simultaneously, the PL decay curve was cross-checked by a luminescence spectrometer. TR-PL at an 80-MHz repetition rate was implemented by using a Tsunami Ti: sapphire laser (760 nm or 380 nm by using second harmonic, spot size ∼100 µm) with 80-fs excitation pulses. The maximum energy density per pulse amounts to 57 and 3.4 µJ/cm 2 at 780 and 390 nm, respectively. Detection is made by a Hamamatsu C5680 streak camera with S1 photocathode operated in synchro-scan mode. The overall temporal resolution of the setup (1/e decay adjustment from 4 K to ambient. Near Infrared EL imaging from the Er/O-Si LEDs was obtained at room temperature under optical microscope (BX53M, Olympus) equipped with a near-infrared camera (C12471-03, Hamamatsu). The recorded video is provided in the supplementary Video. ). 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General procedure for the analysis of Er 3+ cross sections. W J Miniscalco, R S Quimby, Opt. Lett. 16W. J. Miniscalco and R. S. Quimby, "General procedure for the analysis of Er 3+ cross sections," Opt. Lett. 16, 258-260 (1991). . W L Ng, M Lourenco, R Gwilliam, S Ledain, G Shao, K , W. L. Ng, M. Lourenco, R. Gwilliam, S. Ledain, G. Shao, and K. An efficient room-temperature silicon-based light-emitting diode. Homewood, Nature. 410Homewood, "An efficient room-temperature silicon-based light-emitting diode," Nature 410, 192-194 (2001). Improving light capture on crystalline silicon wafers. I Costa, D Pera, J A Silva, Mater. Lett. 127825I. Costa, D. Pera, and J. A. Silva, "Improving light capture on crystalline silicon wafers," Mater. Lett., 127825 (2020). . M Bürkle, M Lozac&apos;h, C Mcdonald, M Macias-Montero, B Alessi, D , M. Bürkle, M. Lozac'h, C. McDonald, M. Macias-Montero, B. Alessi, D. Tuning the bandgap character of quantumconfined Si-Sn alloyed nanocrystals. V Mariotti, Švrček, Adv. Funct. Mater. 301907210Mariotti, and V. Švrček, "Tuning the bandgap character of quantum- confined Si-Sn alloyed nanocrystals," Adv. Funct. Mater. 30, 1907210 (2020).

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{'abstract': 'Chungang Duan), and fyyue@ee.ecnu.edu.cn (Fangyu Yue).Abstract：Silicon-based light sources including light-emitting diodes (LEDs)and laser diodes (LDs) for information transmission are urgently needed for developing monolithic integrated silicon photonics. Silicon doped by ion implantation with erbium ions (Er 3+ ) is considered a promising approach, but suffers from an extremely low quantum efficiency. Here we report an electrically pumped superlinear emission at 1.54 µm from Er/O-doped silicon planar LEDs, which are produced by applying a new deep cooling process.Stimulated emission at room temperature is realized with a low threshold current of ~6 mA (~0.8 A/cm 2 ). Time-resolved photoluminescence and', 'arxivid': '2012.04387', 'author': ['Jin Hong \nMinistry of Education\nKey Laboratory of Polar Materials and Devices\nEast China Normal University\n200241ShanghaiChina\n', 'Huimin Wen \nDepartment of Micro/Nano Electronics\nNational Key Laboratory of Science and Technology on Micro/Nano Fabrication Laboratory\nUniversity of Michigan-Shanghai Jiao Tong University Joint Institute\nShanghai Jiao Tong University\n200240ShanghaiChina\n', 'Jiajing He \nDepartment of Micro/Nano Electronics\nNational Key Laboratory of Science and Technology on Micro/Nano Fabrication Laboratory\nUniversity of Michigan-Shanghai Jiao Tong University Joint Institute\nShanghai Jiao Tong University\n200240ShanghaiChina\n', 'Jingquan Liu \nDepartment of Micro/Nano Electronics\nNational Key Laboratory of Science and Technology on Micro/Nano Fabrication Laboratory\nUniversity of Michigan-Shanghai Jiao Tong University Joint Institute\nShanghai Jiao Tong University\n200240ShanghaiChina\n', 'Jens W Tomm \nDepartment of Micro/Nano Electronics\nNational Key Laboratory of Science and Technology on Micro/Nano Fabrication Laboratory\nUniversity of Michigan-Shanghai Jiao Tong University Joint Institute\nShanghai Jiao Tong University\n200240ShanghaiChina\n\nMax-Born-Institut für Nichtlineare Optik und Kurzzeitspektroskopie\nMax-Born-Str. 2A12489BerlinGermany\n', 'Fangyu Yue \nMinistry of Education\nKey Laboratory of Polar Materials and Devices\nEast China Normal University\n200241ShanghaiChina\n', 'Junhao Chu \nMinistry of Education\nKey Laboratory of Polar Materials and Devices\nEast China Normal University\n200241ShanghaiChina\n\nNational Laboratory for Infrared Physics\nShanghai Institute of Technical Physics\nChinese Academy of Sciences\n500 Yutian Road200083ShanghaiChina\n', 'And Chungang Duan \nMinistry of Education\nKey Laboratory of Polar Materials and Devices\nEast China Normal University\n200241ShanghaiChina\n'], 'authoraffiliation': ['Ministry of Education\nKey Laboratory of Polar Materials and Devices\nEast China Normal University\n200241ShanghaiChina', 'Department of Micro/Nano Electronics\nNational Key Laboratory of Science and Technology on Micro/Nano Fabrication Laboratory\nUniversity of Michigan-Shanghai Jiao Tong University Joint Institute\nShanghai Jiao Tong University\n200240ShanghaiChina', 'Department of Micro/Nano Electronics\nNational Key Laboratory of Science and Technology on Micro/Nano Fabrication Laboratory\nUniversity of Michigan-Shanghai Jiao Tong University Joint Institute\nShanghai Jiao Tong University\n200240ShanghaiChina', 'Department of Micro/Nano Electronics\nNational Key Laboratory of Science and Technology on Micro/Nano Fabrication Laboratory\nUniversity of Michigan-Shanghai Jiao Tong University Joint Institute\nShanghai Jiao Tong University\n200240ShanghaiChina', 'Department of Micro/Nano Electronics\nNational Key Laboratory of Science and Technology on Micro/Nano Fabrication Laboratory\nUniversity of Michigan-Shanghai Jiao Tong University Joint Institute\nShanghai Jiao Tong University\n200240ShanghaiChina', 'Max-Born-Institut für Nichtlineare Optik und Kurzzeitspektroskopie\nMax-Born-Str. 2A12489BerlinGermany', 'Ministry of Education\nKey Laboratory of Polar Materials and Devices\nEast China Normal University\n200241ShanghaiChina', 'Ministry of Education\nKey Laboratory of Polar Materials and Devices\nEast China Normal University\n200241ShanghaiChina', 'National Laboratory for Infrared Physics\nShanghai Institute of Technical Physics\nChinese Academy of Sciences\n500 Yutian Road200083ShanghaiChina', 'Ministry of Education\nKey Laboratory of Polar Materials and Devices\nEast China Normal University\n200241ShanghaiChina'], 'corpusid': 227738950, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 13871, 'n_tokens_neox': 11563, 'n_words': 6685, 'pdfsha': '84de30b99d61051532447b4fb93059fac50cd596', 'pdfurls': ['https://arxiv.org/pdf/2012.04387v1.pdf'], 'title': ['Stimulated emission at 1.54 µm from Erbium/Oxygen-doped silicon-based light emitting diodes', 'Stimulated emission at 1.54 µm from Erbium/Oxygen-doped silicon-based light emitting diodes'], 'venue': []}

arxiv

Numerische Mathematik Convergence of the Euler-Maruyama method for multidimensional SDEs with discontinuous drift and degenerate diffusion coefficient 2018 Gunther Leobacher Michaela Szölgyenyi Numerische Mathematik Convergence of the Euler-Maruyama method for multidimensional SDEs with discontinuous drift and degenerate diffusion coefficient Numer. Math 138201810.1007/s00211-017-0903-9Received: 29 December 2016 / Revised: 21 April 2017 / Published online: 20 July 2017Stochastic differential equations · Discontinuous drift · Degenerate diffusion · Euler-Maruyama method · Strong convergence rate Mathematics Subject Classification Primary 60H10 · 65C30 · 65C20; Secondary 65L20 We prove strong convergence of order 1/4 − for arbitrarily small > 0 of the Euler-Maruyama method for multidimensional stochastic differential equations (SDEs) with discontinuous drift and degenerate diffusion coefficient. The proof is based on estimating the difference between the Euler-Maruyama scheme and another numerical method, which is constructed by applying the Euler-Maruyama scheme to a transformation of the SDE we aim to solve. Introduction We consider time-homogeneous stochastic differential equations (SDEs) of the form d X t = μ(X t )dt + σ (X t )dW t , X 0 = x ,(1) where x ∈ R d is the initial value, μ : R d −→ R d is the drift and σ : R d −→ R d×d is the diffusion coefficient. The Euler-Maruyama approximation with step-size δ > 0 of the solution to (1) is given by X δ t = x + t 0 μ(X δ s )ds + t 0 σ (X δ s )dW s ,(2) with s = jδ for s ∈ [ jδ, ( j + 1)δ), j = 0, . . . , (T − δ)/δ. In particular, for t ∈ { jδ : j = 0, . . . , (T − δ)/δ}, we have X δ t+δ = X δ t + μ(X δ t )δ + σ (X δ t )(W t+δ − W t ) . For μ, σ Lipschitz, Itô [9] proved existence and uniqueness of the solution of (1). In this case the Euler-Maruyama method (2) converges with strong order 1/2 to the true solution, see [12,Theorem 10.2.2]. Higher order algorithms exist, but require stronger conditions on the coefficients. In applications, frequently SDEs with less regular coefficients appear. For example in stochastic control theory, whenever the optimal control is of bang-bang type, meaning that the strategy is of the form 1 S (X ) for a measurable set S ⊆ R d , the drift of the controlled dynamical system is discontinuous. Furthermore, there are models which involve only noisy observations of a signal that has to be filtered. After applying filtering theory the diffusion coefficient typically is degenerate in the sense that σ (x) v = 0, for some x, v ∈ R d . This motivates the study of SDEs with these kind of irregularities in the coefficients. If μ is bounded and measurable, and σ is bounded, Lipschitz, and uniformly nondegenerate, i.e. if there exists a constant c 0 > 0 such that for all x ∈ R d and all v ∈ R d it holds that σ (x) v ≥ c 0 v , Zvonkin [29] and Veretennikov [25,26] prove existence and uniqueness of a solution. Veretennikov [27] extends these results by allowing a part of the diffusion to be degenerate. In [16] existence and uniqueness of a solution for the case where the drift is discontinuous at a hyperplane, or a special hypersurface and where the diffusion coefficient is degenerate is proven, and in [23] it is shown how these results extend to the nonhomogeneous case. Currently, research on numerical methods for SDEs with irregular coefficients is highly active. Hutzenthaler et al. [8] introduce the tamed Euler-Maruyama scheme and prove strong order 1/2 convergence for SDEs with continuously differentiable and polynomially growing drift that satisfy a one-sided Lipschitz condition. Sabanis [22] proves strong convergence of the tamed Euler-Maruyama scheme from a different perspective and also considers the case of locally Lipschitz diffusion coefficient. Gyöngy [4] proves almost sure convergence of the Euler-Maruyama scheme for the case where the drift satisfies a monotonicity condition. Halidias and Kloeden [7] show that the Euler-Maruyama scheme converges strongly for SDEs with a discontinuous monotone drift coefficient. Kohatsu-Higa et al. [13] show weak convergence with rates smaller than 1 of a method where they first regularize the discontinuous drift and then apply the Euler-Maruyama scheme. Étoré and Martinez [1,2] introduce an exact simulation algorithm for one-dimensional SDEs with a drift coefficient which is discontinuous in one point, but differentiable everywhere else. For one-dimensional SDEs with piecewise Lipschitz drift and possibly degenerate diffusion coefficient, in [14] an existence and uniqueness result is proven, and a numerical method, which is based on applying the Euler-Maruyama scheme to a transformation of (1), is presented. This method converges with strong order 1/2. In [15] a (non-trivial) extension of the method is introduced, which converges with strong order 1/2 also in the multidimensional case. The paper also contains an existence and uniqueness result for the multidimensional setting under more general conditions than, e.g., the ones stated in [16]. The method introduced in [15] is the first numerical method that is proven to converge with positive strong rate for multidimensional SDEs with discontinuous drift and degenerate diffusion coefficient. It requires application of a transformation and its numerical inverse in each step, which makes the method rather slow in practice. Furthermore, the method requires specific inputs about the geometry of the discontinuity of the drift to calculate this transformation. This is a drawback, if, e.g., the method shall be applied for solving control problems, since the control is usually not explicitly known. So a method is preferred that can deal with the discontinuities in the drift automatically. First results in this direction are contained in a series of papers by Ngo and Taguchi. In [21] they show convergence of order up to 1/4 of the Euler-Maruyama method for multidimensional SDEs with discontinuous bounded drift that satisfies a onesided Lipschitz condition and with Hölder continuous, bounded, and uniformly nondegenerate diffusion coefficient. In [19] they extend this result to cases where the drift is not necessarily one-sided Lipschitz for one-dimensional SDEs, and in [20] they extend the result for one-dimensional SDEs by allowing for discontinuities also in the diffusion coefficient. For many applications, their results fail to be applicable, since they only hold for one-dimensional SDEs and their method of proof relies on uniform non-degeneracy of the diffusion coefficient. Contrasting the above, there are several delimiting results which state that even equations with infinitely often differentiable coefficients cannot always be solved approximately in finite time, even if the Euler-Maruyama method converges, cf. Hairer et al. [6], Jentzen et al. [10], Müller-Gronbah and Yaroslavtseva [18], Yaroslavtseva [28]. However, there is still a big gap between the assumptions on the coefficients under which convergence with strong convergence rate has been proven and the properties of the coefficients of the equation presented in [6]. In this paper we prove strong convergence of order 1/4− for arbitrarily small > 0 of the Euler-Maruyama method for multidimensional SDEs with discontinuous drift satisfying a piecewise Lipschitz condition and with a degenerate diffusion coefficient. Note that we do not impose a one-sided Lipschitz condition on the drift. So even for SDEs with non-degenerate diffusion coefficient, which do not have a one-sided Lipschitz drift, this result is novel. Our convergence proof is based on estimating the difference between the Euler-Maruyama scheme and the scheme presented in [15]. Close to the set of discontinuities of the drift, we have no tight estimate of this difference, so we need to study the occupation time of an Itô process with degenerate diffusion coefficient there. Away from the set of discontinuities, it is essential to estimate the probability that during one step the distance between the interpolation of the Euler-Maruyama method and the previous Euler-Maruyama step becomes greater than some threshold. This paper's result is the first one that gives strong convergence and also a strong convergence rate of a fully explicit scheme for multidimensional SDEs with discontinuous drift and degenerate diffusion coefficient, and the first one for multidimensional SDEs with discontinuous drift that does not satisfy a one-sided Lipschitz condition. Preliminaries In this section we first state the assumptions on the coefficients of SDE (1), under which the result of this paper is proven, then we study the occupation time of an Itô process close to a hypersurface, and finally we recall the transformation from [15], which is also essential for our proof. Definitions and assumptions We want to prove strong convergence of the Euler-Maruyama method for SDEs with discontinuous drift coefficient. Instead of the usual requirement of Lipschitz continuity we only assume that the drift is a piecewise Lipschitz function on the R d . (γ ) = sup n,0≤t 1 <···<t n ≤1 n k=1 γ (t k ) − γ (t k−1 ) . The intrinsic metric ρ on A is given by ρ(x, y) : = inf{ (γ ) : γ : [0, 1] −→ A is a continuous curve satisfying γ (0) = x, γ (1) = y} , where ρ(x, y) := ∞,sup x,y∈R d \ f (x) − f (y) ρ(x, y) the piecewise Lipschitz constant of f . In this paper will be a fixed C 3 -hypersurface, and we will only consider piecewise Lipschitz functions with exceptional set . In the following, L f denotes the piecewise Lipschitz constant of a function f , if f is piecewise Lipschitz, and it denotes the Lipschitz constant, if f is Lipschitz. We define the distance d(x, ) between a point x and the hypersurface by d(x, ) := inf{ x − y : y ∈ }, and for every ε > 0 we define ε : = {x ∈ R d : d(x, ) < ε}. Recall that, since ∈ C 3 , for every ξ ∈ there exists an open environment U ⊆ of ξ and a continuously differentiable function n: U −→ R d such that for every ζ ∈ U the vector n(ζ ) has length 1 and is orthogonal to the tangent space of in ζ . On a given connected open subset of the local unit normal vector n is unique up to a factor ±1. We recall a definition from differential geometry. Definition 2.4 Let ∈ R d be any set. 1. An environment ε is said to have the unique closest point property, if for every x ∈ R d with d(x, ) < ε there is a unique p ∈ with d(x, ) = x − p . Therefore, we can define a mapping p: ε −→ assigning to each x the point p(x) in closest to x. 2. is said to be of positive reach, if there exists ε > 0 such that ε has the unique closest point property. The reach of is the supremum over all such ε if such an ε exists, and 0 otherwise. Now, we give assumptions which are sufficient for the results in [15] to hold and which we need to prove the main result here. Assumption 2.1 We assume the following for the coefficients of (1): 1. μ and σ are bounded; 2. the diffusion coefficient σ is Lipschitz; 3. the drift coefficient μ is a piecewise Lipschitz function R d −→ R d . Its exceptional set is a C 3 -hypersurface of positive reach; 4. non-parallelity condition: there exists a constant c 0 > 0 such that σ (ξ) n(ξ ) ≥ c 0 for all ξ ∈ ; 5. the function α: −→ R d defined by α(ξ ) := lim h→0 μ(ξ − hn(ξ )) − μ(ξ + hn(ξ )) 2 σ (ξ) n(ξ ) 2(3) is C 3 and all derivatives up to order three are bounded. of irregularities in the coefficients. There are results in the literature, where the authors deal with a non-globally Lipschitz diffusion coefficient, see, e.g., [5], but in contributions where only Hölder continuity is required for σ , usually uniform non-degeneracy is assumed. 3. Assumption 2.1.3 is a geometrical condition which we require in order to locally flatten , i.e. to map to a hyperplane in a regular way. This is crucial in many places in [15] and here, in particular for the proof of Theorem 2.7 below. In addition to that, Assumption 2.1.3 implies that there exists a constant c 1 such that n (ξ ) ≤ c 1 for every ξ ∈ and every orthonormal vector n on , see [15, Lemma 3.10]. 4. Assumption 2.1.4 means that the diffusion coefficient must have a component orthogonal to in all ξ ∈ . This condition is significantly weaker than uniform non-degeneracy, and it is essential: in [16] we give a counterexample for the case where the non-parallelity condition does not hold. Then, even existence of a solution is not guaranteed. 5. Assumption 2.1.5 is a technical condition, which is required for our transformation method to work. Boundedness of α and α is needed for proving the local invertibility of our transform. Existence and boundedness of α and α is required for the multidimensional version of Itô's formula to hold for the transform, see [15]. Moreover, it has been shown in [15,Proposition 3.13] that α is a well-defined function on , i.e. it does not depend on the choice of the normal vector n and, in particular, on its sign. Example 2.6 Suppose is the finite and disjoint union of orientable compact C 3manifolds. Then is of positive reach by the lemma in [3], and each connected component of separates the R n into two open connected components by the Jordan-Brouwer separation theorem, see [17]. Thus R d \ is the union of finitely many disjoint open connected subsets of R d ; we can write R d \ = A 1 ∪ · · · ∪ A n . Suppose there exist bounded and Lipschitz C 3 -functions μ 1 , . . . , μ n : R d −→ R d such that μ = n k=1 1 A k μ k , and suppose that σ : R d −→ R d×d is bounded, Lipschitz, and C 3 with σ (ξ) n(ξ ) = 0 for every ξ ∈ . Then it is readily checked that μ and σ satisfy Assumption 2.1. In Sect. 4 we present a number of concrete examples which satisfy Assumption 2.1 and we perform numerical tests on the associated SDEs. Occupation time close to a hypersurface In this section we study the occupation time of an Itô process close to a C 3hypersurface. In the proof of our main theorem, the Euler-Maruyama approximation X δ in equation (2) will play the role of that Itô process. Theorem 2.7 Let be a C 3 -hypersurface of positive reach and let ε 0 > 0 be such that the closure of ε 0 has the unique closest point property. Let further X = (X t ) t≥0 be an R d -valued Itô process X t = X 0 + t 0 A s ds + t 0 B s dW s , with progressively measurable processes A = (A t ) t≥0 , B = (B t ) t≥0 , where A is R d -valued and B is R d×d -valued.∀t ∈ [0, T ] : X t (ω) ∈ ε 0 ⇒ max( A t (ω) , B t (ω) ) ≤ c AB ; 2. there exists a constant c 0 such that for almost all ω ∈ it holds that ∀t ∈ [0, T ] : X t (ω) ∈ ε 0 ⇒ n( p(X t (ω))) B t (ω)B t (ω) n( p(X t (ω))) ≥ c 0 . Then there exists a constant C such that for all 0 < ε < ε 0 /2, T 0 P {X s ∈ ε } ds ≤ Cε . For the proof we will construct a one-dimensional Itô process Y with the property that Y is close to 0, if and only if X is close to . For the construction of Y we decompose the path of X into pieces close to and pieces farther away. These pieces are then mapped to R by using a signed distance of X from and pasted together in a continuous way. A signed distance to is locally given by D( x) := n( p(x)) (x − p(x)), where n is a local unit normal vector. Lemma 2.8 For all x ∈ ε 0 it holds that D (x) = n( p(x)) . Proof Fix x ∈ ε 0 \ and consider the function h defined by h(b) := x − p(x +b) 2 . By definition of the projection map p, h has a minimum in b = 0, such that h (0) = 0. Hence from h (b) = −2(x − p(x + b)) p (x + b), we get (x − p(x)) p (x) = 0. This implies n( p(x)) p (x) = 0, since (x − p(x)) is a scalar multiple of n( p(x)). Using that D(x) = a x − p(x) for an a ∈ {−1, 1}, we compute D (x) = a x − p(x) −1 (x − p(x)) (id R d − p (x)) = a n( p(x)) (id R d − p (x)) = a n( p(x)) − n( p(x)) p (x) = an( p(x)) . For ψ ∈ R with |ψ| small we get D(x + ψn( p(x))) = n p x + ψn( p(x)) x + ψn( p(x)) − p x + ψn( p(x)) = n( p(x)) (x + ψn( p(x)) − p(x)) = D(x) + ψ , such that the directional derivative of D in direction n( p(x)) in x is 1. From this and from (4) it follows that D (x) = n( p(x)) . This also holds for x ∈ by the continuity of D . The following lemma states that for any continuous curve γ in ε 0 there is a continuous path of unit normal vectors, such that to every point of γ we can assign a signed distance in a continuous way. Proof For ξ ∈ we denote the tangent space to in ξ by tang ξ . Let S : = {a ≤ s ≤ b : ∃m : [a, s] −→ R d continuous, m(t) = 1, m(t)⊥tang p(γ (t)) ∀t ∈ [a, s]}. The set S is nonempty and its elements are bounded by b. Let s 1 := sup S. There exists an open and connected subset U ⊆ such that p(γ (s 1 )) ∈ U , and a unit normal vector n 1 : U −→ R d . Since U is open and p • γ is continuous, there exists η > 0 such that p(γ ([s 1 − η, s 1 ])) ⊆ U . By the definition of s 1 there exists s ∈ (s 1 −η, s 1 ) and m : [a, s] −→ R d continuous, with m(t) = 1 and m(t)⊥tang p(γ (t)) for all t ∈ [a, s]. Since n 1 is unique up to a factor ±1, the mapping n 1 • p • γ either coincides with m or −m on (s 1 − η, s). Without loss of generality we may assume that the former is the case. Thus we can extend m continuously to [a, s 1 ] by defining m(t) := n 1 ( p(γ (t))) for all t ∈ (s, s 1 ]. Now, if s 1 was strictly smaller than b, then we could use the same mapping n 1 • p •γ to extend m continuously beyond s 1 , contradicting the definition of s 1 . We will need the following estimate on the local time of a one-dimensional Itô process. Lemma 2.10 Let Y = (Y t ) t≥0 be an Itô process with bounded and progressively measurable coefficients = ( t ) t≥0 ,B = (B t ) t≥0 . Then sup y∈R E(L y T (Y )) ≤ 3T 2  2 ∞ + 3 2 T B s 2 ∞ 1/2 . The claim is a special case of [19,Lemma 3.2]. We give a proof for the convenience of the reader. Proof From the Meyer-Tanaka formula [11, Section 3.7, Eq. (7.9)] we have 2L y T (Y ) = |Y T − y| − |Y 0 − y| − T 0 1 {Y s >y} − 1 {Y s <y} dY s ≤ |Y T − Y 0 | + T 0 1 {Y s >y} − 1 {Y s <y} dY s ≤ T 0 s ds + T 0B s dW s + T 0 1 {Y s >y} − 1 {Y s <y}  s ds + T 0 1 {Y s >y} − 1 {Y s <y} B s dW s ≤ 2 T 0 | s |ds + T 0B s dW s + T 0 1 {Y s >y} − 1 {Y s <y} B s dW s . Using the inequality (a + b + c) 2 ≤ 3(a 2 + b 2 + c 2 ) we get 4L y T (Y ) 2 ≤ 12  2 ∞ T 2 + 3 T 0B s dW s 2 + 3 T 0 1 {Y s >y} − 1 {Y s <y} B s dW s 2 , and, using Itô's L 2 -isometry, 4E L y T (Y ) 2 ≤ 12  2 ∞ T 2 + 6 T 0B 2 s ds ≤ 12  2 ∞ T 2 + 6T B 2 ∞ . The claim now follows by applying the Cauchy-Schwarz-inequality and taking the supremum over all y ∈ R. We are ready to prove the result of this section. Proof of Theorem 2.7 Let ε 1 = ε 0 /2. Define a mapping λ : R −→ R by λ(z) = ⎧ ⎪ ⎨ ⎪ ⎩ z − 2 3ε 2 1 z 3 + 1 5ε 4 1 z 5 |z| ≤ ε 1 8ε 1 15 z > ε 1 − 8ε 1 15 z < −ε 1 . Note that λ (0) = 1 and λ (±ε 1 ) = λ (±ε 1 ) = 0, so that λ ∈ C 2 . Next we decompose the path of X : let τ 0 := inf{t ≥ 0 : X t ∈ ε 1 }. In particular we have τ 0 = 0, if X 0 ∈ ε 1 . For k ∈ N 0 , define κ k+1 := inf{t ≥ τ k : X t / ∈ 2ε 1 } ∧ T , τ k+1 := inf{t ≥ κ k+1 : X t ∈ ε 1 } ∧ T . By Lemma 2.9 there exist continuous m k : [τ k , κ k+1 ] −→ R d , with m k (t) = 1 and m k (t)⊥tang p(X t ) for all t ∈ [τ k , κ k+1 ]. Without loss of generality m 0 can be chosen such that m 0 (τ 0 ) (X τ 0 − p(X τ 0 )) ≥ 0. We construct a one-dimensional process Y as follows: Y t = ⎧ ⎪ ⎨ ⎪ ⎩ λ(m 0 (τ 0 ) (X τ 0 − p(X τ 0 ))) t ≤ τ 0 λ(m k (t) (X t − p(X t ))) t ∈ [τ k , κ k+1 ] λ(m k (κ k ) (X κ k − p(X κ k ))) t ∈ [κ k , τ k ] , where without loss of generality the m k are chosen such that λ(m k+1 (τ k+1 ) (X τ k+1 − p(X τ k+1 ))) = λ(m k (κ k ) (X κ k − p(X κ k ))) .(5) Note that by construction both sides of (5) can only take the values ±λ(ε 1 ). We have thus constructed a continuous [λ(−ε 1 ), λ(ε 1 )]-valued process Y with the property that the occupation time of Y in an environment of 0 is the same as the occupation time of X in an environment of , i.e. Y ∈ (−λ(ε), λ(ε)), iff X ∈ ε for all 0 < ε < ε 1 . To show that Y is an Itô process, we want to use Itô's formula. For this we recognize that Y , depending on its proximity to , is either constant or locally of the form Y t = λ(n( p(X t )) (X t − p(X t ))) for a suitable choice of the unit normal vector. Denote D(x) = n( p(x)) (x − p(x)). The function D is locally a signed distance to and D ∈ C 2 . This can be seen by following the proof of [3, Theorem 1]. Hence, we may apply Itô's formula to get dY t = λ (D(X t ))D (X t )A t dt + λ (D(X t ))D (X t )B t dW t + 1 2 tr B t λ (D(X t ))B t dt . By Lemma 2.8 we have D (x) = n( p(x)) , and hence (λ(D(x))) = (λ (D(x))n( p(x)) ) = λ (D(x))n( p(x))n( p(x)) + λ (D(x))n ( p(x)) . Since λ and λ are bounded by construction, n( p(x))n( p(x)) = 1, n is bounded (c.f. the remark on Assumption 2.1.3), and by Assumption 1 of the theorem, the coefficients of Y are uniformly bounded. Therefore dY t = t dt +B t dW t , with bounded and progressively measurableÂ,B. Let 0 < ε ≤ ε 1 /2. For all |z| ≤ ε, we have λ (z) ≥ 3 1 {X s ∈ ε } ds = 3 4 2 c 2 0 t 0 1 {Y s ∈(−λ(ε),λ(ε))} ds ≤ t 0 1 {Y s ∈(−λ(ε),λ(ε))} λ (D(X s )) 2 n( p(X s )) B s B s n( p(X s ))ds = t 0 1 {Y s ∈(−λ(ε),λ(ε))} d [Y ] s . By the occupation time formula [11, Chapter 3, 7.1 Theorem] for one-dimensional continuous semimartingales, we get T 0 P {X s ∈ ε } ds ≤ 4 3c 0 2 E T 0 1 {Y s ∈(−λ(ε),λ(ε))} d [Y ] s = 2 4 3c 0 2 E R 1 (−λ(ε),λ(ε)) (y)L y T (Y ) dy ≤ 4 3 3 2 c 2 0 sup y∈R E L y T (Y ) ε . The transformation The proof of convergence is based on a transformation that removes the discontinuity from the drift and makes the drift Lipschitz while preserving the Lipschitz property of the diffusion coefficient. A suitable transform is presented in [15]. We recall it here. Define G:R d −→ R d , G(x) = x +φ(x)α( p(x)) x ∈ ε 0 x x ∈ R d \ ε 0 , where ε 0 > 0 is smaller than the reach of , see Assumption 2.1.3, α is the function defined in Assumption 2.1.5, and φ(x) = n( p(x)) (x − p(x)) x − p(x) φ x − p(x) c , with positive constant c and φ(u) = (1 + u) 3 (1 − u) 3 |u| ≤ 1 0 |u| > 1. Note that G is piecewise Lipschitz with exceptional set . If c is chosen sufficiently small, see [15,Lemma 3.18], G is invertible by [15,Theorem 3.14]. Furthermore, Itô's formula holds for G and G −1 by [15,Theorem 3.19]. With this we can define a process Z = (Z t ) t≥0 by Z t = G(X t ), which solves the SDE d Z t =μ(Z t )dt +σ (Z t )dW t ,(6) wherẽ μ(z) = G (G −1 (z))μ(G −1 (z)) + 1 2 tr σ (G −1 (z)) G (G −1 (z))σ (G −1 (z)) , σ (z) = G (G −1 (z))σ (G −1 (z)) . From [15,Theorem 3.20] we know thatμ andσ are Lipschitz, and hence the solution to (6) can be approximated with strong order 1/2 using the Euler-Maruyama scheme. Main result We are ready to formulate the main result. X t − X δ t 2 1/2 ≤ Cδ 1/4− . In preparation of the proof of the main result, we prove two lemmas. E T 0 X δ s − X δ s 2 ds ≤ Cδ . Proof By the definition of the Euler-Maruyama method (2) we have E T 0 X δ s − X δ s 2 ds = T /δ−1 j=0 E ( j+1)δ jδ X δ jδ − X δ s 2 ds ≤ T δ sup t∈{ jδ: j=0,...,T /δ−1} E t+δ t X δ t − X δ s 2 ds = T δ sup t∈{ jδ: j=0,...,T /δ−1} E t+δ t X δ t − X δ t − μ(X δ t )δ − σ (X δ t )(W s − W t ) 2 ds ≤ 2T δ sup t∈{ jδ: j=0,...,T /δ−1} E t+δ t μ(X δ t )δ 2 ds + t+δ t σ (X δ t )(W s − W t ) 2 ds ≤ 2T δ μ 2 ∞ δ 3 + σ 2 ∞ sup t∈{ jδ: j=0,...,(T −δ)/δ} t+δ t E( W s − W t 2 )ds = 2T δ μ 2 ∞ δ 3 + d σ 2 ∞ sup t∈{ jδ: j=0,...,(T −δ)/δ} t+δ t (s − t)ds = 2T δ μ 2 ∞ δ 3 + d 2 σ 2 ∞ δ 2 ≤ Cδ . For all δ, ε > 0 and all j = 0, . . . , T /δ − 1, define δ,ε, j := ω ∈ : sup s∈[ jδ,( j+1)δ] X δ s (ω) − X δ s (ω) ≥ ε .(7) Lemma 3.3 Let Assumption 2.1.1 hold. Then there exists a constant C such that for all 0 < δ ≤ 1, all ε > 0, and all j = 0, . . . , T /δ − 1, it holds that P( δ,ε, j ) ≤ C exp(−ε/ σ ∞ δ 1/2 ). Proof P sup jδ≤s≤( j+1)δ X δ s − X δ s ≥ ε = P sup jδ≤s≤( j+1)δ μ(X δ s )(s − s) + σ (X δ s )(W s − W s ) ≥ ε ≤ P sup jδ≤s≤( j+1)δ μ ∞ δ + σ ∞ W s − W s ≥ ε = P sup jδ≤s≤( j+1)δ W s − W s ≥ ε − μ ∞ δ σ ∞ = P sup 0≤s≤1 W s − W 0 ≥ ε − μ ∞ δ σ ∞ δ 1/2 = P exp sup 0≤s≤1 W s ≥ exp ε − μ ∞ δ σ ∞ δ 1/2 ≤ E (exp( W 1 )) exp μ ∞ δ − ε σ ∞ δ 1/2 ≤ C exp − ε σ ∞ δ 1/2 , where we applied Doob's submartingal inequality, and in the last step used that δ ≤ 1. Now, we are ready to prove our main result. Proof of Theorem 3.1 Since G −1 is Lipschitz by the proof of [15,Theorem 3.20], E sup 0≤t≤T X t − X δ t 2 1/2 ≤ L G −1 E sup 0≤t≤T Z t − G(X δ t ) 2 1/2 ,(8) with Z = G(X ) as in (6). Let Z δ be the Euler-Maruyama approximation of Z . It holds that E sup 0≤t≤T Z t − G(X δ t ) 2 1/2 ≤ E sup 0≤t≤T Z t − Z δ t 2 1/2 + E sup 0≤t≤T Z δ t − G(X δ t ) 2 1/2 .(9) For estimating the first term in (9), recall that by [15,Theorem 3.20], the transformed SDE (6) has Lipschitz coefficients. Since the Euler-Maruyama method converges with strong order 1/2 for SDEs with Lipschitz coefficients (see [12,Theorem 10.2.2]), there exists a constant C 1 > 0 such that for sufficiently small δ > 0, E sup 0≤t≤T Z t − Z δ t 2 ≤ C 1 δ .(10) We now turn to the second term in (9), i.e. we estimate the difference between G applied to the Euler-Maruyama approximation of X and the Euler-Maruyama approximation of Z . Denote, for all τ ∈ [0, T ], u(τ ) := E sup 0≤t≤τ G(X δ t ) − Z δ t 2 . With ν(x 1 , x 2 ) = G (x 1 )μ(x 2 ) + 1 2 tr(σ (x 2 ) G (x 1 )σ (x 2 )) we have by Itô's formula, G(X δ t ) = G(X δ 0 ) + t 0 ν(X δ s , X δ s )ds + t 0 G (X δ s )σ (X δ s )dW s , so that u(τ ) = E sup 0≤t≤τ t 0 ν(X δ s , X δ s )ds + t 0 G (X δ s )σ (X δ s )dW s − t 0μ Z δ s ds − t 0σ Z δ s dW s 2 ≤ E sup 0≤t≤τ 4 t 0 ν(X δ s , X δ s ) − ν(X δ s , X δ s ) ds 2 + 4 t 0 G (X δ s )σ (X δ s ) −G (X δ s )σ (X δ s ) dW s 2 + 4 t 0 μ(G(X δ s )) −μ(Z δ s ) ds 2 + 4 t 0 σ (G(X δ s )) −σ (Z δ s ) dW s 2 . Applying the Cauchy-Schwarz inequality to the Lebesgue integrals and the ddimensional Burkholder-Davis-Gundy inequality [8,Lemma 3.7] to the Itô integrals, we obtain u(τ ) ≤ 4T E τ 0 ν(X δ s , X δ s ) − ν(X δ s , X δ s ) 2 ds + 8d E τ 0 G (X δ s )σ (X δ s ) − G (X δ s )σ (X δ s ) 2 ds + 4T E τ 0 μ(G(X δ s )) −μ(Z δ s ) 2 ds + 8d E τ 0 σ (G(X δ s )) −σ (Z δ s ) 2 ds =: 4T E 1 + 8d E 2 + 4T E 3 + 8d E 4 .(11) For estimating E 1 in (11), we will use that ν(x 1 , x 2 ) − ν(x 2 , x 2 ) 2 ≤ 2L 2 G μ 2 ∞ + 1 2 L 2 G σ 4 ∞ x 1 − x 2 2 x 1 / ∈ ε , x 1 − x 2 < ε 4 μ 2 ∞ G 2 ∞ + σ 4 ∞ G 2 ∞ otherwise . With this and the definition of c δ,ε, j in (7), we get E 1 = τ 0 E ν(X δ s , X δ s ) − ν(X δ s , X δ s ) 2 1 {X δ s / ∈ ε } 1 c δ,ε,s/δ + 1 {X δ s / ∈ ε } 1 δ,ε,s/δ + 1 {X δ s ∈ ε } ds ≤ 2L 2 G μ 2 ∞ + 1 2 L 2 G σ 4 ∞ ε 2 T + 4 μ 2 ∞ G 2 ∞ + σ 4 G 2 ∞ T 0 P( δ,ε,s/δ )ds + T 0 P({X δ s ∈ ε })ds . By Lemma 3.3, T 0 P( δ,ε,s/δ )ds ≤ C 2 exp(−ε/ σ ∞ δ 1/2 ), and by Theorem 2.7, T 0 P({X δ s ∈ ε })ds ≤ C 3 ε, for suitable constants C 2 , C 3 . In order to minimize the bound on E 1 , we choose ε such that exp(−ε/ σ ∞ δ 1/2 ) + ε is minimized for δ sufficiently small, yielding ε = − σ ∞ δ 1/2 log( σ ∞ δ 1/2 ) = σ ∞ δ 1/2−2 (−δ 2 log( σ ∞ δ 1/2 )) for arbitrarily small > 0. Hence, with C 4 = (2L 2 G μ 2 ∞ + 1 2 L 2 G σ 4 ∞ )T , C 5 = (4 μ 2 ∞ G 2 ∞ + σ 4 G 2 ∞ )C 2 , C 6 = (4 μ 2 ∞ G 2 ∞ + σ 4 G 2 ∞ )C 3 , we get E 1 ≤ C 4 ε 2 + C 5 exp − ε σ ∞ δ 1/2 + C 6 ε = C 4 σ 2 ∞ δ 1−4 (−δ 2 log( σ ∞ δ 1/2 )) 2 + C 5 σ ∞ δ 1/2 + C 6 σ ∞ δ 1/2−2 (−δ 2 log( σ ∞ δ 1/2 )) . Thus, with C 7 = C 4 σ 2 ∞ +C 5 σ ∞ +C 6 σ ∞ and for arbitrarily small fixed > 0, it holds that for sufficiently small δ E 1 ≤ C 7 δ 1/2−2 .(12) For estimating E 2 in (11), we apply Lemma 3.2 to get E 2 ≤ L 2 G σ 2 ∞ T 0 E X δ s − X δ s 2 ds ≤ L 2 G σ 2 ∞ C 8 δ .(13) For estimating E 3 , E 4 in (11), we use thatμ,σ are Lipschitz by [15,Theorem 3.20], to get E 3 ≤ L 2 μ τ 0 E G(X δ s ) − Z δ s 2 ds ≤ L 2 μ τ 0 u(s)ds ,(14)E 4 ≤ L 2 σ τ 0 E G(X δ s ) − Z δ s 2 ds ≤ L 2 σ τ 0 u(s)ds .(15) Combining the estimates (12), (13), (14), (15) with (11), we get 0 ≤ u(τ ) ≤ C 9 τ 0 u(s)ds + 4T C 7 δ 1/2−2 + 8d L 2 G σ 2 ∞ C 8 δ , with C 9 = 4T L 2 μ + 8d L 2 σ . Using that 4T C 7 δ 1/2−2 + 8d L 2 G σ 2 ∞ C 8 δ ≤ C 10 δ 1/2−2 for δ ≤ 1, and applying Gronwall's inequality yields for all τ ∈ [0, T ], u(τ ) ≤ C 10 exp(C 9 τ )δ 1/2−2 .(16) Combining (10) and (16) with (9), and the result with (8), finally yields E sup 0≤t≤T X t − X δ t 2 1/2 ≤ L G −1 C 1 δ 1/2 + L G −1 C 10 exp(C 9 T )δ 1/2−2 1/2 ≤ Cδ 1/4− , for a suitably chosen constant C, for arbitrarily small > 0, and for δ sufficiently small. Examples We ran simulations for several examples-ones of theoretical interest as well as an example coming from applications. When studying stochastic dynamical systems which include a noisy signal, then filtering this signal leads to a higher dimensional system with a degenerate diffusion coefficient. Stochastic control problems often lead to an optimal control policy which makes the drift of the system discontinuous. Examples are models with incomplete market information in mathematical finance where the rate with which cashflows are paid from a firm value process change systematically when the asset-liability ratio passes a certain threshold which then triggers a rating change. The class of equations studied here appears frequently in several areas of applied mathematics and the natural sciences. Step-function In the first example the drift is the step function μ(x 1 , 1) , and σ ≡ id R 2 . It can easily be checked that these coefficients satisfy Assumption 2.1. x 2 ) = (3(1 {x 1 ≥0} −1 {x 1 <0} ), In particular, note that the non-parallelity condition is trivially satisfied, since σ is uniformly non-degenerate. Since μ does not satisfy a one-sided Lipschitz condition, our result is the first one that gives a strong convergence rate of the Euler-Maruyama method for this example. Discontinuity along the unit circle In this example the drift has a discontinuity along the unit circle, and the diffusion coefficient is degenerate on the whole of R 2 : μ(x 1 , x 2 ) = (1, 1) x 2 1 + x 2 2 ≥ 1 (−x 1 , x 2 ) x 2 1 + x 2 2 < 1 , σ (x 1 , x 2 ) = 2 1 + x 2 1 + x 2 2 x 1 0 x 2 0 . Assumption 2.1 largely follows from Example 2.6. The non-parallelity condition is readily verified: 2 (1 + x 2 1 + x 2 2 )(x 2 1 + x 2 2 ) x 1 x 2 0 0 x 1 x 2 = 2 x 2 1 + x 2 2 (1 + x 2 1 + x 2 2 )(x 2 1 + x 2 2 ) = 1 for all points (x 1 , x 2 ) that lie on the unit circle, i.e. x 2 1 + x 2 2 = 1. Dividend maximization under incomplete information In insurance mathematics, a well-studied problem is the maximization of the expected discounted future dividend payments until the time of ruin of an insurance company, a value which serves as a risk measure. In [24] the problem is studied in a setup that allows for incomplete information about the market. This leads to a joint filtering and stochastic optimal control problem, and after solving the filtering problem, the driving dynamics are high dimensional and have a degenerate diffusion coefficient. This issue is described in more detail in [24]. Solving the stochastic optimal control problem in dimensions higher than three with the usual technique (solving an associated partial differential equation) becomes practically infeasible. Therefore, one has to resort to simulation. The SDE that has to be simulated has the coefficients corresponding processes stay within this simplex almost surely, see [24]. The function f determines the hypersurface along which the drift is discontinuous: μ(x 1 , . . . , x d ) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ ϑ d + d−1 i=1 (ϑ i − ϑ d )x i+1 −ū1 [ f (x 2 ,...,x d ),∞) (x 1 ) q d1 + d−1 j=1 (q j1 − q d1 )x j+1 . . . q d(d−1) + d−1 j=1 (q j (d−1) − q d(d−1) )x j+1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ σ (x 1 , . . . , x d ) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ β 0 . . . 0 x 2 ϑ 1 −ϑ d − d−1 j=1 (ϑ j −ϑ d )x j+1 β . . . . . . . . . . . . . . . x d ϑ d−1 −ϑ d − d−1 j=1 (ϑ j −ϑ d )x j+1 β 0 . . . 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , whereū, β, (ϑ i ) d i=1 , (q i j ) d i,= {(x 1 , . . . , x d ) : x 1 = f (x 2 , . . . , x d )}. In our simulations we choose d = 5 and f affine linear, but note that we need not restrict ourselves to affine linear f . We need to check Assumption 2.1: Since x 2 , . . . , x d ∈ [0, 1], μ, σ are bounded, and all first order derivatives of the entries of σ are bounded. Hence, σ is Lipschitz. μ is piecewise Lipschitz, and since f is affine linear, ∈ C 3 . Whether the non-parallelity condition holds depends on the choice of the parameters, but for ours the condition is satisfied. Assumption 2.1.5 can easily be checked. Note that the coefficients can be extended to the whole of R d in a way that they still satisfy our assumptions. Error estimate The L 2 -error is estimated by err k :=ēÊ X (k) T − X (k−1) T 2 1/2 , where X (k) T is the numerical approximation of X T with step size δ (k) ,Ê is an estimator of the mean value using 2 14 paths, andē is a normalizing constant so that err 1 = √ 1/4. Figure 1 shows log 2 of the estimated L 2 -error of the Euler-Maruyama approximation of X T plotted over log 2 δ (k) for the examples presented above. We observe that the theoretical convergence rate is approximately obtained for the example of a step-function and that the other examples converge at a faster rate. In particular, for the examples with degenerate diffusion coefficient, the convergence rate is not worse than for the other example. Even for the step-function example, for sufficiently small step-size the convergence rate seems to be higher than the theoretical one. Hence, it will be an interesting topic for future research to prove sharpness, or find a sharp bound. Even though the proven rate for the Euler-Maruyama method is lower than for the transformation-based method from [15], the calculations are usually faster in practice using the first method, since the simulation of a single path is faster. Table 1 confirms this claim: we observe that computation times are higher by up to two orders of magnitude for the transformation method, while the estimated error is of comparable size. For completeness, we remark that one can construct examples, where the transformation method is much faster while giving a smaller error. For example, start with prescribing the transform G(x) = x + x|x|φ(10x) and set μ(x) = 1 2 (G −1 ) (G(x)) and σ (x) = (G −1 ) (G(x)). This leads toμ(z) = 0 andσ (z) = 1. Hence, if we use the transformation method with the same G, then Z δ = Z = W and the transformation method gives the estimate G −1 (W ), which is the exact solution. Conclusion In this paper we have for the first time proven strong convergence and also a positive strong convergence rate for an explicit method (the Euler-Maruyama method) for multidimensional SDEs with discontinuous drift that has a degenerate diffusion coefficient, or with a discontinuous drift that does not satisfy a one-sided Lipschitz condition, or both. The Euler-Maruyama method has the advantage that it does not need the exact form of the set of discontinuities of the drift as an input, and that in practice, computation of one path is fast in comparison to the second method in the literature that can deal with this class of SDEs. Our numerical experiments suggest that in addition to these advantages, it even seems that the Euler-Maruyama method converges at a higher than the theoretically obtained rate for many examples and it will be a topic of future research to prove sharpness, or find a sharp bound. Research Grant "Numerical Methods for Stochastic Differential Equations with Irregular Coefficients with Applications in Risk Theory and Mathematical Finance". Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Definition 2.1 ([15, Definitions 3.1 and 3.2]) Let A ⊆ R d . 1. For a continuous curve γ : [0, 1] −→ R d , let (γ ) denote its length, Lemma 2. 9 9Let γ : [a, b] −→ ε 0 be a continuous function. Then there exists m : [a, b] −→ R d such that 1. m is continuous; 2. m(t) = 1 for all t ∈ [a, b]; 3. m(t) is orthogonal to in the point p(γ (t)) for all t ∈ [a, b]. Fig. 1 1Estimated L 2 -errors if there is no continuous curve from x to y. 2. Let f : A −→ R m be a function. We say that f is intrinsic Lipschitz, if it is Lipschitz w.r.t. the intrinsic metric on A, i.e. if there exists a constant L such that Let A ⊆ R d be open and let f : A −→ R m be a differentiable function with f < ∞. Then f is intrinsic Lipschitz with Lipschitz constant f .Definition 2.3 ([15, Definition 3.4]) A functionf : R d −→ R m is piecewise Lipschitz,if there exists a hypersurface with finitely many connected components and with the property, that the restriction f | R d \ is intrinsic Lipschitz. We call an exceptional set for f , and we call∀x, y ∈ A : f (x) − f (y) ≤ Lρ(x, y) . The prototypical examples for intrinsic Lipschitz function are given, like in the one-dimensional case, by differentiable functions with bounded derivative. Lemma 2.2 ([15, Lemma 3.8]) is a technical condition; the focus in this paper is on other typesTheorem 2.5 ([15, Theorem 3.21]) Let Assumption 2.1 hold. Then SDE (1) has a unique strong solution. Remark on Assumption 2.1: 1. For existence and uniqueness of a solution to (1), in [15, Theorem 3.21] instead of Assumption 2.1.1 only boundedness in an ε-environment of is needed. However, for the proof of our convergence result we require global boundedness. Note that other results in the literature on numerical methods for SDEs with discontinuous drift also rely on boundedness of the coefficients, cf. [19-21]. 2. Assumption 2.1.2 Let the coefficients A, B be such that 1. there exists a constant c AB such that for almost all ω ∈ it holds that j=1 are known constants. 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{'abstract': 'We prove strong convergence of order 1/4 − for arbitrarily small > 0 of the Euler-Maruyama method for multidimensional stochastic differential equations (SDEs) with discontinuous drift and degenerate diffusion coefficient. The proof is based on estimating the difference between the Euler-Maruyama scheme and another numerical method, which is constructed by applying the Euler-Maruyama scheme to a transformation of the SDE we aim to solve.', 'arxivid': '1610.07047', 'author': ['Gunther Leobacher ', 'Michaela Szölgyenyi '], 'authoraffiliation': [], 'corpusid': 4952761, 'doi': '10.1007/s00211-017-0903-9', 'github_urls': [], 'n_tokens_mistral': 17257, 'n_tokens_neox': 14551, 'n_words': 8834, 'pdfsha': '8298294def7447d583f0994451dc34c969f4e37d', 'pdfurls': None, 'title': ['Numerische Mathematik Convergence of the Euler-Maruyama method for multidimensional SDEs with discontinuous drift and degenerate diffusion coefficient', 'Numerische Mathematik Convergence of the Euler-Maruyama method for multidimensional SDEs with discontinuous drift and degenerate diffusion coefficient'], 'venue': ['Numer. Math']}

arxiv

One-loop unquenched three-gluon vertex in the Curci-Ferrari model 18 Oct 2021 Felipe Figueroa Instituto de Física Facultad de Ingeniería Laboratoire d'Annecy-le-Vieux de Physique Théorique LAPTh Université Grenoble Alpes Université Savoie Mont Blanc CNRS F-74000AnnecyFrance Marcela Peláez Universidad de la República MontevideoUruguay One-loop unquenched three-gluon vertex in the Curci-Ferrari model 18 Oct 2021(Dated: October 20, 2021)arXiv:2110.09561v1 [hep-th] In this article we study the unquenched three-gluon vertex in all momentum range going from the ultraviolet to the infrared regime using the Curci-Ferrari model at one-loop in Landau gauge as an extension of the results presented in[1]. Results are compared with recent lattice data for SU (3) in the unquenched case. This calculation is a pure prediction of the model because it does not require fixing any parameter once two-point functions are fitted. An analysis of the influence of dynamical quarks in the position of the zero crossing is presented. Due to the recent improvement of infrared lattice data for the quenched three-gluon correlation function [2] we also redo the comparison of one-loop results in this limit obtaining very good results. I. INTRODUCTION The infrared sector of QCD is usually called the Nonperturbative regime due to the fact that standard perturbation theory based on Faddeev-Popov Lagrangian presents a Landau pole in the infrared. This implies that perturbation theory cannot be applied together with this particular gauge-fixed Lagrangian to study the infrared. For these reasons different alternatives have been developed in order to approach this regime. These approaches can be classified in two different categories, the first one includes non-perturbative functional techniques, for instance, treatments based on Schwinger-Dyson (SD) [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] or Non perturbative renormalization group (NPRG) equations [19][20][21][22][23][24][25]. The other category includes approaches that focus on finding the correct gauge-fixed Lagrangian that should be used in the infrared. This line is motivated due to the fact that Faddeev-Popov procedure, generally used to fixed the gauge, is not completely justified in the infrared because of Gribov copies [26]. Therefore, until now it is not known how to build from first principles the correct gauge-fixed Lagrangian valid in the infrared. There are some interesting attempts to reach this Lagrangian based on Gribov-Zwanziger action and the refined Gribov-Zwanziger approach [27][28][29]. Lattice simulations can deal with the problem of Gribov copies so they are a good tool to obtain information about the infrared behavior of Yang-Mills theory. Two important observations of lattice simulations are, first, that the gluon propagator behaves as a massive propagator in the infrared [30][31][32][33][34][35][36][37]. Second, that the relevant expansion parameter obtained through the ghost-gluon or the three-gluon vertex does not present a Landaupole and in fact it does not become too large [33,[38][39][40]. These points have motivated us to study the infrared regime using a gauge-fixed Lagrangian with a gluon mass term [41,42]. This Lagrangian is a particular case of Curci-Ferrari Lagrangians in Landau gauge (CF) [43]. Even though we do not know how to justify CF Lagrangian from first principles it is important to observe that it can reproduce a great variety of correlation functions using the first order in perturbation theory. It is important to mention that we do not dare to try to reproduce all infrared quantities of QCD perturbatively, in particular the perturbative expansion for correlation functions involving quarks near the chiral limit fails. Other approach using CF model was proposed in [44,45] in order to explore the chiral limit. See [46] for a detail summary of the already studied properties of the model. In particular, one-loop corrections within the CF model were computed for propagators, ghost-gluon vertex and the quenched three-gluon vertex [1,42,[47][48][49]. In addition to this, two-loop corrections were studied for propagators [50,51] and the ghost-gluon vertex with a vanishing gluon momentum [52] and compared with lattice data with great accuracy. It is important to mention that vertices are obtained as a pure prediction of the model, in the sense that the free parameters are fixed by minimizing the error between propagators and the corresponding lattice data and therefore there are no free parameters left when studying vertices. The aim of this article is to extend the study of oneloop corrections for the three gluon vertex in the presence of dynamical quarks. The infrared regime of the threegluon vertex has been studied by different approaches specially in the last decade [1,16,24,[53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68][69] as it is an important ingredient to understand QCD at low energies. The three-gluon vertex is more difficult to calculate than the propagators because instead of depending on a single momentum, it depends on three independent scalars. Moreover, it has a richer tensorial decomposition so different scalar functions (associated with different tensors) have to be reproduced together. In this article, we study the effects of dynamical quarks in the three-gluon vertex using one-loop CF model. The unquenched results are compared with lattice data from [70]. Moreover, recent simulations of the quenched threegluon vertex show a better handling of the infrared regime, yielding more precise data in this limit [2]. For this reason it is worth to extend the results presented in [1] for SU (2) to SU (3) gauge-group and compare it with the newest lattice data. For both cases, quenched and unquenched, the parameters used in the plots were chosen to minimize the error of the propagators previously computed in [47,50]. In this sense, the results shown in this article are a pure prediction of the model that reproduces with great accuracy the lattice data. Due to the presence of massless ghosts, CF model also features a zero crossing as it is observed in [56,62,[65][66][67][68]. We also find that dynamical quarks shift the zero crossing in a direction that depends on the momentum configuration in a way consistent with what is observed in [60]. The article is organized as follows. In Sec. II we describe in more detail the Curci-Ferrari model in Landau gauge. We give some details on the one-loop calculations of the three-gluon vertex in Sec. III in terms of the Ball-Chiu components. In Sec. IV we present the renormalization conditions and the renormalization group equations. We present our results in Sec. V. and compare them with lattice data. At the end of the article we present the conclusions of the results. II. CURCI-FERRARI MODEL WITH QUARKS We start by introducing the Curci-Ferrari Lagrangian [43] in the presence of dynamical quarks in Euclidean space: L = 1 4 (F a µν ) 2 + ∂ µ c a (D µ c) a + ih a ∂ µ A a µ + m 2 2 (A a µ ) 2 + N f i=1ψ i (γ µ D µ + M i )ψ i ,(1) where g is the coupling constant, and the flavor index i spans the N f quark flavors. The covariant derivative D µ acting on a ghost field in the adjoint representation of SU(N) reads (D µ c) a = ∂ µ c a + gf abc A b µ c c , while when applied to a quark in the fundamental representation it reads D µ ψ = ∂ µ ψ − gt a A a µ ψ. The latin indices correspond to the SU (N ) gauge-group, the t a are the generators in the fundamental representation and the f abc are the structure constants of the group. Finally, the field strength is given by F a µν = ∂ µ A a ν − ∂ ν A a µ + gf abc A b µ A c ν . The Feynman rules associated to this Lagrangian are the standard Feynman rules for Euclidean-QCD in Landau gauge except for the gluon's free propagator, which reads A a µ A b ν 0 (p) = δ ab P ⊥ µν (p) 1 p 2 + m 2 ,(2) where we have introduced the transverse projector: P ⊥ µν (p) = δ µν − p µ p ν p 2 .(3) It is important to mention that the gluon mass term added to the Faddeev-Popov Lagrangian breaks the BRST symmetry. However, it can be shown that (1) satisfies a modified (non-nilpotent) BRST symmetry that can be used to prove its renormalizability [71]. Probably the most interesting aspect of this model is the fact that, as it has been shown in various previous articles (see [41,42,49,50,77] for instance), the addition of a gluon mass term regularizes the theory in the infrared, allowing for a perturbative treatment of the theory in this region. More specifically, it is possible to find a renormalization scheme without an infrared Landau pole for particular choices of the initial condition of the renormalization-group flow. These features have made it possible to use this model to compute various two and three-point functions to 1-loop and 2-loop order, obtaining a very good match with lattice simulations [1,42,47,48,[50][51][52]. It is important to mention that this model has also been studied at finite temperature and chemical potential in [72][73][74]. III. ONE-LOOP CALCULATION OF THE THREE GLUON VERTEX A. Tensorial structure and computation In this work we extend the one-loop computation of the gluon three-point function obtained in [1] for SU (N ) Yang-Mills theory to unquenched QCD. In order to calculate the gluon's three-point function at one-loop order we need to compute the Feynman diagrams shown in As shown in [75], the color structure of the three-gluon vertex is simply the structure constant f abc of the SU (N ) group, so it can be factored out. Furthermore, we follow the usual convention [76] of factorizing the coupling constant, and thus we define: Γ (3) A a µ A b ν A c ρ (p, k, r) = −igf abc Γ µνρ (p, k, r) . The tensor structure of Γ µνρ (p, k, r) can be easily deduced: it must depend on three Lorentz indices (one for each gluon) and on two independent momenta due to momentum conservation. As a consequence, we can have only two types of tensor structures: the ones made up of three momenta (p µ p ν p ρ , p µ p ν k ρ ,...) and the ones made up of one momentum and the Euclidean metric tensor (p µ δ νρ , k ν δ µρ ,...). It's not hard to convince oneself that there are eight possible terms of the first kind and six of the second, adding up to a total of 14 possible terms in the vertex's tensor structure. However, the vertex is symmetric under the exchange of any pair of external legs, and this ends up reducing the total number of possible independent coefficients in the vertex's tensor structure to six. A cleaner way of exploiting these symmetries is following the decomposition proposed by Ball and Chiu in [78], where they parametrize the vertex using six scalar functions: Γ µνρ (p, k, r) = A(p 2 , k 2 , r 2 )δ µν (p − k) ρ + B(p 2 , k 2 , r 2 )δ µν (p + k) ρ − C(p 2 , k 2 , r 2 )(δ µν p.k − p ν k µ )(p − k) ρ + 1 3 S(p 2 , k 2 , r 2 )(p ρ k µ r ν + p ν k ρ r µ ) + F (p 2 , k 2 , r 2 )(δ µν p.k − p ν k µ )(p ρ k.r − k ρ p.r) + H(p 2 , k 2 , r 2 ) −δ µν (p ρ k.r − k ρ p.r) + 1 3 (p ρ k µ r ν − p ν k ρ r µ ) + cyclic permutations (4) The scalar functions have the following symmetry properties: A, C and F are symmetric under permutation of the first two arguments; B is antisymmetric under permutation of the first two arguments; H is completely symmetric and S is completely antisymmetric. It is important to note that only some of these scalar functions are accessible through lattice simulations, since they have access to the vertex function only through the correlation function, i.e. the vertex contracted with the full external propagators. Since the propagators in Lan-dau gauge are transverse, the longitudinal part of the vertex function is lost in the process. In particular this means that the B and S functions are not accessible through lattice computations. We decomposed every diagram contributing to the three gluon vertex into the Ball-Chiu tensorial structure. In this way we obtained the contribution of each diagram to each of the scalar functions A, B, C, F, H and S. To perform our computations we expressed the integrals in Feynman diagrams of Fig. 1 in terms of only three Master Integrals defined following the convention on [79] as: A[m 1 ] =C d d q 1 [q 2 +m 2 1 ] (5) B[p 1 , m 1 , m 2 ] =C d d q 1 [q 2 +m 2 1 ][(q+p1) 2 +m 2 2 ] (6) C[p 1 , p 2 , m 1 , m 2 , m 3 ] =C d d q 1 [q 2 +m 2 1 ][(q+p1) 2 +m 2 2 ][(q−p2) 2 +m 2 3 ](7) whereC = 16π 2μ 2ǫ (2π) d , and the regularization scaleμ is related to the renormalization scale µ by µ 2 = 4πe −γμ2 . The A and B-Master Integrals can be solved analytically in d = 4 − 2ǫ in terms of the external momentum and the masses, but the C-Master Integral must be treated numerically except for particular kinemetics. We chose the FIRE5 algorithm [80] to perform the Master Integral reduction, thus obtaining analytic expressions for each of the scalar functions in terms of the three Master Integrals for arbitrary momentum configurations. The expressions are complicated and not very enlightening, however, the explicit expressions appear in the suplemental material of [1] for the quenched case while the quark contribution is written in [76]. In the case of one vanishing external momentum the computation becomes much simpler and the result for the quenched vertex function in this configuration is given in [1] 1 while the unquenched case is presented in the Appendix A. B. Checks Various checks for the Yang-Mills part of the result for the three-gluon vertex function had already been made in [1]. We only need to check the quark triangle diagram to test our unquenched results. To do this we compared our results to those of [76], verifying that they yield the same expressions when properly transformed to the Euclidean space. This was expected, as the quark triangle diagram is independent from the mass of the gluons and therefore its contribution in the Curci-Ferrari model is the same as in standard QCD. IV. RENORMALIZATION AND RENORMALIZATION GROUP In this section we introduce the renormalization scheme that we used in this work and we explain how we implemented renormalization-group ideas to improve our perturbative calculation. A. Renormalization To take care of the divergences appearing in the 1loop quantities we took the usual approach of redefining the fields, masses and coupling constants through renormalization factors that absorb the infinities. The renormalized quantities are defined in terms of the bare ones (denoted with a "0" subscript) as follows: A a µ 0 = Z A A a µ , ψ 0 = Z ψ ψ, c a 0 = Z c c a ,c a 0 = Z cc a , g 0 = Z g g m 2 0 = Z m 2 m 2 M 0 = Z M M(8) From now on, all expressions will refer to renormalized quantities unless explicitly stated otherwise. The renormalized propagators and the three-gluon 1PI correlation function are thus defined as: Γ (2) A a µ A b ν (p) = Z A Γ (2) A a µ A b ν ,0 (p) Γ (2) c acb (p) = Z c Γ (2) c acb ,0 (p) Γ (2) ψψ (p) = Z ψ Γ (2) ψψ,0 (p) Γ (3) A a µ A b ν A c ρ (p, r) = Z 3/2 A Γ (3) A a µ A b ν A c ρ ,0 (p, r)(9) B. Infrared Safe renormalization scheme To fix the renormalization factors we used the Infrared Safe (IS) renormalization scheme defined in [42]. It is based on a non-renormalization theorem for the gluon mass [81][82][83], and is defined by Γ ⊥ (p = µ) = m 2 + µ 2 , J(p = µ) = 1, Z m 2 Z A Z c = 1.(10) where Γ ⊥ (p) is the transversal part of Γ (2) A a µ A b ν (p) and J(p) is the ghost dressing function. To fix the renormalization of the coupling constant we used the Taylor scheme, in which the coupling constant is defined as the ghost-gluon vertex with a vanishing ghost momentum. Requiring that the renormalized vertex is finite leads to a relation among the renormalization factors Z A , Z c and Z g : Z g Z A Z c = 1(11) The divergent part of the renormalization factors for the quenched case were presented in [42]. Here we show the extension to the unquenched Curci-Ferrari model already computed in [47,84]. In d = 4 − 2ǫ they read: Z c = 1 + 3g 2 N 64π 2 ǫ Z A = 1 + g 2 96π 2 (13N − 8N f T f ) ǫ Z m 2 = 1 − g 2 192π 2 (35N − 16N f T f ) ǫ Z g = 1 − g 2 96π 2 (11N − 4N f T f ) ǫ(12) Finally, the quantity we are interested in is actually Γ µνρ as defined earlier. Since in it's definition we factorized a factor of g, the relation between the renormalized and bare quantities is the following: Γ µνρ (p, r) = Z 3/2 A Z g Γ µνρ,0 (p, r) = Z A Z c Γ µνρ,0 (p, r), where in the last equality we used equation 11. C. Renormalization Group After the renormalization procedure we obtain a finite expression for the three-gluon vertex, but it comes with the usual loop corrections of the form log( p µ ). To handle this situation we implemented the renormalization-group flow to take care of the large logarithms coming from loop corrections. First we define the β functions and anomalous dimensions of the fields: β g (g, m 2 ) = µ dg dµ g0,m 2 0 ,(13)β m 2 (g, m 2 ) = µ dm 2 dµ g0,m 2 0 ,(14)γ A (g, m 2 ) = µ d log Z A dµ g0,m 2 0 ,(15)γ c (g, m 2 ) = µ d log Z c dµ g0,m 2 0 .(16) Analogous expressions are used for the β−function of quark mass and the quark anomalous dimension. The renormalization group equation for the vertex function with n A gluon legs and n c ghost legs reads: µ∂ µ − 1 2 (n A γ A + n c γ c ) + β g ∂ g + β m 2 ∂ m 2 Γ (nA,nc) = 0,(17) This equation allows us to obtain a relation for the vertex function renormalized at scale µ 0 and the same vertex function renormalized at a different scale µ: Γ (nA,nc) ({p i }; µ, g(µ), m 2 (µ), M (µ)) = z A (µ) nA/2 z c (µ) nc/2 × Γ (nA,nc) ({p i }; µ 0 , g(µ 0 ), m 2 (µ 0 ), M (µ 0 )).(18) where g(µ), m 2 (µ) and M (µ) are obtained by integration of the β functions with initial conditions given at some scale µ 0 and z A and z c are obtained repectively from: log z A (µ, µ 0 ) = µ µ0 dµ ′ µ ′ γ A g(µ ′ ), m 2 (µ ′ ) , log z c (µ, µ 0 ) = µ µ0 dµ ′ µ ′ γ c g(µ ′ ), m 2 (µ ′ ) .(19) We can then use the non-renormalization theorems of Eq.(10) and Eq.(11) to relate the anomalous dimensions and the β functions. It is simple to check that one obtains the following relations: γ A (g, m 2 ) = 2 β m 2 m 2 − β g g ,(20)γ c (g, m 2 ) = 2β g g − β m 2 m 2 .(21) Finally we use these relations to integrate Eq. (19), obtaining analytical expressions for z A and z c in terms of the running gluon mass and coupling constant, being this feature another of the advantages of the Infrared Safe scheme: z A (µ, µ 0 ) = m 4 (µ)g 2 (µ 0 ) m 4 (µ 0 )g 2 (µ) , z c (µ, µ 0 ) = m 2 (µ 0 )g 2 (µ) m 2 (µ)g 2 (µ 0 ) .(22) We are able now to express the three-gluon vertex renormalized at scale µ 0 in terms of the same quantity using a running scale µ = p, thus avoiding the large logarithms problem. Taking into account the factor of g on the definition of Γ µνρ (p, r) this reads: Γ µνρ (p, r; µ 0 ) = g 4 (p)m 6 (µ 0 ) g 4 (µ 0 )m 6 (p) Γ µνρ (p, r; µ = p) V. RESULTS We now present our results for the different scalar functions associated to the three-gluon vertex introduced in the previous section. All our results correspond to four dimensions and the SU (3) gauge group, and we evaluate the scalar functions in different momentum configurations in order to compare them with the available lattice data. A. Fixing Parameters The only fitting parameters we need to adjust to compare our results with the lattice are the overall normalization constant of the gluon three-point function and the initial conditions of the renormalization-group flow, i.e. the values of the mass of the gluon, the mass of the quark and the coupling constant at some renormalization scale µ 0 . The initial conditions for the renormalization-group are best obtained by looking for the set of parameters (m 0 , M 0 , g 0 ) that produce the best fit between the gluon and ghost propagators computed using the Curci-Ferrari approach and the lattice data, since the lattice results are much more precise for propagators than for threepoint functions. This task was done in [1,50] for different gauge groups and renormalization schemes in the quenched case, and in [47] including dynamical quarks. For the SU (3) group and the IS scheme the initial conditions for the R-G flow at µ 0 = 1 GeV obtained are the ones listed in table I . In this work we use these values to compute the one loop three-gluon vertex, which means that up to the overall normalization constant our results are a pure prediction of the model. B. Comparison with the lattice In order to compare with the lattice, we must choose specific momentum configurations for Γ µνρ (p 1 , p 2 , p 3 ). Most available lattice data employs some of the following configurations: The Symmetric Configuration, with p 2 1 = p 2 2 = p 2 3 = p 2 and p 1 · p 2 = p 1 · p 3 = p 2 · p 3 = − p 2 2 , the Asymmetric Configuration, with p 1 = 0 and p 2 = −p 3 = p, and the Orthogonal Configuration, with p 1 · p 2 = 0, p 2 1 = p 2 2 = p 2 and p 2 3 = 2p 2 . For the quenched case, we compared our results with the lattice data from [2]. Following their definitions, in the symmetric configuration we work with the scalar functionsΓ sym 1 andΓ sym 2 : gΓ µνρ (p 1 , p 2 , p 3 ) =Γ sym 1 s 2 λ µνρ 1 (p 1 , p 2 , p 3 ) +Γ sym 2 s 2 λ µνρ 2 (p 1 , p 2 , p 3 ), (24) where λ µνρ 1 (p 1 , p 2 , p 3 ) = Γ (0) µ ′ ν ′ ρ ′ (p 1 , p 2 , p 3 )P ⊥ µ ′ µ (p 1 )P ⊥ ν ′ ν (p 2 )P ⊥ ρ ′ ρ (p 3 ), with Γ (0) µ ′ ν ′ ρ ′ (p 1 , p 2 , p 3 ) defined as the perturbative treelevel tensor of the three-gluon vertex, and λ µνρ 2 (p 1 , p 2 , p 3 ) = (p1−p2)ρ(p2−p3)µ(p3−p1)ν p 2 . On the other hand the asymmetric configuration of the vertex is parametrized by a single scalar functionΓ asym 3 defined by gΓ µνρ (p, −p, 0) =Γ asym 3 p 2 λ µνρ 3 (p, −p, 0),(25) with λ µνρ 3 (p, −p, 0) = 2p ρ P ⊥µν (p). We compared our unquenched results with the lattice data from [70]. They work in the orthogonal configuration, and define the usual scalar function G 1 , which consists on contracting the external legs of the vertex with transverse propagators and the tree-level momentum structure of the 3-gluon vertex, normalized to the same expression at tree-level. This reads: G 1 (p, k, r) = [(r − k) γ δ αβ + cyclic permutations]P ⊥ αµ (p)P ⊥ βν (k)P ⊥ γρ (r)Γ µνρ (p, k, r) [(r − k) γ δ αβ + cyclic permutations]P ⊥ αµ (p)P ⊥ βν (k)P ⊥ γρ (r)[(r − k) ρ δ µν + cyclic permutations](27) The results of the model are shown below using the different scalar functions defined in this section including the renormalization group effects. C. SU (3) Yang Mills results We first present our results for SU (3) Yang Mills theory and we compare them with lattice results from [2]. As stated before, we integrate the beta functions with initial conditions at µ 0 = 1 GeV using the initial conditions listed in table I. In Fig. 2 we show the results for the scalar func-tionΓ asym 3 in the asymmetric configuration (one vanishing momentum), and in Fig. 3 we do the same for the functionsΓ sym 1 andΓ sym 2 the symmetric configuration (all momenta equal). In all cases the agreement is very good, specially considering that the initial conditions for the renormalization-group flow were not fitted for the threepoint function but for the propagators. It is also noticeable that in all cases the different scalar functions become negative at low energies, a qualitative feature that was observed in many lattice simulations as well as in different analysis [56,62,[65][66][67][68]. While the scalar functions associated to the tree-level tensor diverges logarithmically theΓ sym 2 goes to a constant value in the infrared as stated in [2]. The simplicity of one-loop CF model allows to write the infrared behaviour ofΓ sym scalar function as a function of momentum for one vanishing momentum (asymmetric configuration). The points are lattice data from [2]. The plain red line corresponds to our 1-loop computation. Γ sym 2 ∼ g 2 N 414720π 2 20 16 √ 3Cl 2 π 3 − 33 + 189 p 2 m 2(28) where Cl 2 is the Clausen function and Cl 2 π 3 = 1.0149417. We observe that the analytical expression indicates that theΓ sym 2 goes to a constant value in the infrared and this situation is not modified with the effects of the renormalization group. These expressions show the divergent behavior forΓ 1 , that can be easily understood due to the presence of (bottom) scalar functions as a function of momentum for all momenta equal (symmetric configuration). The points are lattice data from [2]. The plain red line corresponds to our 1-loop computation. massless ghosts as stated in [1]. As observed in [2], it is also find thatΓ sym 2 is finite in the infrared. D. Unquenched QCD results If we want to include the influence of dynamical quarks to the previous computation we must add only the triangle diagram with a loop of quarks to the vertex but also include the running of the coupling obtained in the unquenched analysis [47]. The contribution of that diagram can be computed with no difficulty in arbitrary dimension and for arbitrary number of quarks. Our explicit expressions match with the ones presented in [76] when transforming to Euclidean space. In order to be more specific we show as an example the explicit bare contribution of the quark-loop diagram to the factor G 1 in d = 4 − 2ǫ dimensions in Appendix B. In Fig. 4 we show the results and their comparison with lattice data from [70]. The data available corresponds to the G 1 scalar function in the case of two mass-degenerate quark flavors (N f = 2) in the orthogonal configuration (two momenta orthogonal to each other and of equal magnitude). In this case, the agreement is still very good in the infrared but worsens in the UV. More precisely, the model and the lattice results start separating at a scale of about 2.5 GeV. This scale is of the order of magnitude of the inverse of the lattice spacing used in most lattice sim- ulations, and therefore lattice results beyond this scale are subject to hypercubic lattice spacing artefacts. Taking this fact into account and also considering that perturbation theory must work at one-loop in the UV, we suspected that the decrease in the values of G 1 after the inverse lattice spacing scale must be caused by finite lattice artefacts such lattice effects in the UV. To confirm this statement, we computed analytically the high-energy limit of G 1 , finding that it behaves in the UV as Ln ( p µ0 ) α with α = 17N −16N f T f 44N −16N f T f , which is compatible with our results. The idea behind this computation is that since the UV limit of G 1 (p, p) is equal to 1, the high-energy behavior of G 1 (µ 0 , p) must be given by z In this section we study the influence of dynamical quarks in the position of the zero crossing of the threegluon vertex. In the one-loop CF-model the influence of quarks in the renormalized vertex can be isolated as the term proportional to N f . At one loop, the quarks contribution to the renormalized vertex is proportional to: F = 3 2 δZ UQ A f inite + G UQ 1 f inite where δZ UQ A f inite and G UQ 1 f inite represent the finite part of the coefficient proportional to g 2 N f T f in the Z A -renormalization factor and in G 1 respectively. We study here the sign of the factor, F , in order to analyze in which direction the zero crossing is shifted. As F depends on the finite part of the renormalization factor, it is expected that its sign depends on the chosen renormalization scheme. Moreover, we also observe that it depends on the kinematical configuration of the vertex function. In order to do analyze this fact we simplify the expression by taking two momenta of equal norm and study the behavior of the factor F at low momentum as a function of the angle between the chosen momenta. The dependence F on the renormalization scheme of can be observed in Fig. 5 where F is depicted in the case when no contribution for the finite part of the renormalization factor is taken into account and when F is evaluated in the IS-scheme. The plot is done by settingμ = 1 GeV for the regularization scale, M = 0.36GeV for the value of the constituent quark's mass and for momentum p = 0.27GeV, as this is approximately the value of the momentum at which the zero crossing is observed. This shows that different renormalization schemes can indeed yield directions for the shift of the zero crossing caused by the quarks contribution to the vertex. Focusing now on the IS-scheme only, we analyzed the dependence of the factor F on the kinematical configuration for two particular values of the momentum p = 0.2GeV and 0.27GeV in Fig. 6. These values were chosen taken into account the observation that the zero crossing occurs for values of p below 0. 3 We considered quark masses in the region between 0.3 − 0.36GeV which correspond to the infrared values. For completeness we also studied in Fig. 7 the sign of F as a function of the quark mass for the antisymmetric (Cos(θ) = −1), symmetric (Cos(θ) = −1/2), orthogonal (Cos(θ) = 0) and parallel (Cos(θ) = 1) kinematical configurations, where it is shown again that the position of the zero crossing changes with the presence of dynamical quarks but the It is worth noting from Fig. 7 that dynamical quarks move the zero crossing towards the infrared in the symmetric configuration (as observed in [60]) while it is shifted in the opposite direction in the orthogonal configuration. Fig. 8 shows the study of the orthogonal case in detail when including the effects of the renormalization group. It can be seen that the zero crossing is moved towards the ultraviolet when including dynamical quarks as expected from the previous analysis. We conclude from this analysis that the position of the zero crossing is shifted with the presence of quarks but the direction of the shift depends on the kinematical configuration and on the renormalization scheme where the vertex is analyzed. VI. CONCLUSIONS With the aim of studying the infrared properties of the three-point correlation function, in this article, we present a one-loop calculation using Curci-Ferrari model in Landau gauge for arbitrary kinematical configuration. The results are an extension of a previous work [1] to the unquenched case. In particular, we compare the results for the vertex with the available lattice data including dynamical quarks that corresponds to the kinematical configuration with a vanishing gluon momentum and two degenerate flavors. A study of the position of the zero crossing as a function of the kinematical configuration of the vertex is done. It is observed that the position of the zero crossing is shifted due to the presence of dynamical quarks when comparing with the quenched case. However, the direction of the shift depends on the chosen renormalization scheme and on the kinematical configuration of the vertex. We also study the quenched case because some infrared properties observed by the model in the previous work were not clear in lattice simulations at that time. However, the infrared lattice study of the three-gluon vertex has improved in the last years and now error bars are good enough to understand its infrared behavior as discussed in [2]. In particular, lattice simulations show a change of sign in the deep infrared that is easily understood by Curci-Ferrari model. As it has been discussed in [1,56] it can be explained as a consequence of the diagram with a loop of massless ghosts. In the quenched case we compare the results with lattice simulations for the completely symmetric and the antisymmetric configuration. The results are really good, specially considering that the free parameters of the model were already fixed by fitting the propagators. where s = p 2 m 2 andm 2 = m 2 e γ /(4π), with γ the Euler constant. Appendix B: Contribution of dynamical quarks The contribution of dynamical quarks to the G 1 factor: g 2 N f T f 12π 2 ǫ + g 2 N f T f 12π 2 − 1 3 + 2χ A 0 M 2 + M 2 D 1 + 1 D 1 D 2 24k 2 p 2 r 2 C 0 M 2 , M 2 , M 2 , p, r S − M 2 χ + K 1 B 0 M 2 , M 2 , −p 2 + K 2 B 0 M 2 , M 2 , −r 2 + K 3 B 0 M 2 , M 2 , −k 2 (B1) where A 0 , B 0 and C 0 are the corresponding finite part of master integrals defined in Eq. ?? and χ = p 2 + r 2 + k 2 , S = p 2 r 2 + p 2 k 2 + r 2 k 2 , D 1 = k 4 + p 4 + r 4 + 10S, D 2 = k 4 + p 4 + r 4 − 2S and K 1 = k 6 2M 2 − p 2 + k 4 2M 2 9p 2 − r 2 − 9p 4 − 11p 2 r 2 − k 2 2M 2 9p 4 + 2p 2 r 2 + r 4 − 9p 6 + 10p 4 r 2 + 11p 2 r 4 − 2M 2 − p 2 p 6 + 9p 4 r 2 − 9p 2 r 4 − r 6 while K 2 and K 3 are obtained exchanging p ↔ r and p ↔ k in K 1 respectevely. FIG. 1 . 1Feynman diagrams present in the one-loop calculation of the three-gluon vertex. . Values of the masses of the quark and gluon (M0 and m0 respectively) and the coupling constant (g0) at renormalization scale µ0 =1 GeV obtained by adjusting the 2-point functions to lattice data, both for the quenched case and the unquenched case with two degenerate quark flavors. FIG. 2.Γ asym 3 FIG. 3.Γ sym 1 (top) andΓ sym 2 FIG. 4 . 4G1 scalar function as a function of momentum for two momenta orthogonal (orthogonal configuration) and two mass-degenerate quark flavors. The points are lattice data from[70]. The plain red line corresponds to our 1-loop computation. N = 3 and N f = 2. In conclusion, our one-loop computation matches the lattice results in their regime of validity and the renormalization-group prediction in the UV. 1 . 1Zero crossing and the number of flavours. upper) and in the ISscheme (lower) as a function of Cos(θ) (the cosine of the angle between the two chosen momenta) FIG. 6 .FIG. 7 . 67F as a function of Cos(θ) computed in the IS-scheme. Regions represent different values of M between 0.3−0.36GeV, while the upper region is for p = 0.2GeV and the lower region corresponds to p = 0.27GeV.direction depends on the kinematical configuration. Plots correspond to the IS-F factor as a function of the quark mass evaluated in p = 0.27GeV for different kinematical configurations, from top to bottom: the antisymmetric (Cos(θ) = −1), symmetric (Cos(θ) = −1/2), orthogonal (Cos(θ) = 0) and parallel (Cos(θ) = 1) kinematical configurations. FIG. 8 . 8G1 scalar function as a function p for in the orthogonal configuration for N f = 0 (blue) and N f = 2 (red). The position of the zero crossing is shifted towards the UV by the quark's contribution in this configuration. The vanishing momentum configuration in[1] has a misprint, the correct result is shown in the Appendix A ACKNOWLEDGMENTSThe authors would like to acknowledge the financial support from PEDECIBA and ECOS program and from the ANII-FCE-126412 project. We also thank N. 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{'abstract': 'In this article we study the unquenched three-gluon vertex in all momentum range going from the ultraviolet to the infrared regime using the Curci-Ferrari model at one-loop in Landau gauge as an extension of the results presented in[1]. Results are compared with recent lattice data for SU (3) in the unquenched case. This calculation is a pure prediction of the model because it does not require fixing any parameter once two-point functions are fitted. An analysis of the influence of dynamical quarks in the position of the zero crossing is presented. Due to the recent improvement of infrared lattice data for the quenched three-gluon correlation function [2] we also redo the comparison of one-loop results in this limit obtaining very good results.', 'arxivid': '2110.09561', 'author': ["Felipe Figueroa \nInstituto de Física\nFacultad de Ingeniería\nLaboratoire d'Annecy-le-Vieux de Physique Théorique LAPTh\nUniversité Grenoble Alpes\nUniversité Savoie Mont Blanc\nCNRS\nF-74000AnnecyFrance\n", 'Marcela Peláez \nUniversidad de la República\nMontevideoUruguay\n'], 'authoraffiliation': ["Instituto de Física\nFacultad de Ingeniería\nLaboratoire d'Annecy-le-Vieux de Physique Théorique LAPTh\nUniversité Grenoble Alpes\nUniversité Savoie Mont Blanc\nCNRS\nF-74000AnnecyFrance", 'Universidad de la República\nMontevideoUruguay'], 'corpusid': 239024702, 'doi': '10.1103/physrevd.105.094005', 'github_urls': [], 'n_tokens_mistral': 24397, 'n_tokens_neox': 18671, 'n_words': 8716, 'pdfsha': '4a4da12a4272d83552041828fe6e238d40c90fba', 'pdfurls': ['https://arxiv.org/pdf/2110.09561v1.pdf'], 'title': ['One-loop unquenched three-gluon vertex in the Curci-Ferrari model', 'One-loop unquenched three-gluon vertex in the Curci-Ferrari model'], 'venue': []}

arxiv

Unsupervised diffeomorphic cardiac image registration using parameterization of the deformation field Ameneh Sheikhjafari Department of Computing Science University of Alberta T6G 2G8EdmontonABCanada Department of Radiology and Diagnostic Imaging T6G 2G8EdmontonABCanada Deepa Krishnaswamy Department of Radiology and Diagnostic Imaging T6G 2G8EdmontonABCanada Michelle Noga Department of Radiology and Diagnostic Imaging T6G 2G8EdmontonABCanada Nilanjan Ray Department of Computing Science University of Alberta T6G 2G8EdmontonABCanada Kumaradevan Punithakumar Department of Radiology and Diagnostic Imaging T6G 2G8EdmontonABCanada Unsupervised diffeomorphic cardiac image registration using parameterization of the deformation field Preprint submitted to Medical Image Analysis *Corresponding author: (Ameneh Sheikhjafari), deepa@ualberta.ca (Deepa Krishnaswamy), mnoga@ualberta.ca (Michelle Noga), nray1@ualberta.ca (Nilanjan Ray), punithak@ualberta.ca (Kumaradevan Punithakumar) 1 Nilanjan Ray and Kumaradevan Punithakumar contributed equally as co-senior authors to this work.Deformable image registrationDiffeomorphic registration learningMoving mesh grid generationUnsupervised deep learning A B S T R A C TThis study proposes an end-to-end unsupervised diffeomorphic deformable registration framework based on moving mesh parameterization. Using this parameterization, a deformation field can be modeled with its transformation Jacobian determinant and curl of end velocity field. The new model of the deformation field has three important advantages; firstly, it relaxes the need for an explicit regularization term and the corresponding weight in the cost function. The smoothness is implicitly embedded in the solution which results in a physically plausible deformation field. Secondly, it guarantees diffeomorphism through explicit constraints applied to the transformation Jacobian determinant to keep it positive. Finally, it is suitable for cardiac data processing, since the nature of this parameterization is to define the deformation field in terms of the radial and rotational components. The effectiveness of the algorithm is investigated by evaluating the proposed method on three different data sets including 2D and 3D cardiac MRI scans. The results demonstrate that the proposed framework outperforms existing learningbased and non-learning-based methods while generating diffeomorphic transformations. Introduction Deformable image registration plays a fundamental role in a variety of medical image analyses such as image guidedsurgery Han et al. (2021), visual stabilization Sheikhjafari et al. (2015b), reconstruction Liu et al. (2021), and the construction of many other image analysis problems Krebs et al. (2019); Haskins et al. (2019); Sheikhjafari et al. (2015a). Many existing state-of-the-art deformable registration methods use traditional iterative algorithms, such as standard symmetric normalization (SyN) Wu et al. (2018) and log-domain based transformation Mansi et al. (2011). Due to the important properties such as folding-free and invertiblity Dalca et al. (2018) of diffeomorphic transformation, a wide range of researchers utilized diffeomorphisms by adding constraints to their formulation Zhang and Fletcher (2015); Avants et al. (2008); Vercauteren et al. (2008); Punithakumar et al. (2013). These traditional algorithms are computationally expensive and do not learn the features from data to be registered. Recently, Sheikhjafari et al. (2022) proposed a convolutional neural network (CNN) to model the optimization problem for deformable registration and shared the parameters through a temporal sequence. However, they still establish the displacement field via iterative optimization between images. Even though traditional deformable image registration techniques can generate promising mappings between images, most of these methods require users arXiv:2208.13275v1 [eess.IV] 28 Aug 2022 to identify parameters that match the characteristics of the problem and manually adjust regularization terms for each application to obtain accurate results. In recent years, the popularity of learning-based registration algorithms has been increasing due to the lower computational costs and execution times Krebs et al. (2018). In supervisedlearning methods, a CNN is trained using examples of medical images along with their ground truth transformations to predict the transformations directly on test images Rohé et al. (2017); Cao et al. (2017). Even though the accuracy of these approaches is considerable, their performance is highly dependent on the quality of the ground truth Sang et al. (2020). One of the most significant challenges in applying the supervised methods to medical imaging applications is that the actual ground truth of a desired neural network output is not often available. With that limitation in mind, several unsupervised learningbased image registrations have been proposed. Most of the unsupervised approaches use spatial transformer layers (STN) to warp the moving image in a differentiable way. In this way, an optimization can be performed by using a similarity metric based on the warped image Jaderberg et al. (2015);de Vos et al. (2017de Vos et al. ( , 2019; Balakrishnan et al. (2018); Sheikhjafari et al. (2018). When image registration is stated as an optimization of a similarity metric alone, it is commonly understood as an ill-posed problem. To tackle this problem, a regularization approach is commonly used. Without regularization, this may result in multiple and physically non-plausible solutions. For instance, it might lead to tissue folding and tearing of anatomical structures in images. Aside from that, while unsupervised approaches can perform well in minimizing a similarity metric between warped moving images and fixed images, important properties such as symmetry, diffeomorphism, and regularity of the retrieved deformation fields are still unclear and missing. Rohlfing (2011);Haber and Modersitzki (2004). Inspired by Punithakumar et al. (2015Punithakumar et al. ( , 2017; Chen et al. (2010), we tackle aforementioned issues with the help of moving mesh parameterization which was originally designed to generate a suitable grid for solving partial differential equations Haber and Modersitzki (2004). We propose a ConvNet method based on unsupervised learning for deformable cardiac registration, which formulates the deformation field by the moving mesh approach. This parameterization naturally leads to a formulation of diffeomorphic image registration as a constrained optimization problem. It also bypasses the need for an explicit regularization term and the corresponding weight in the cost function. Such a strategy has been adopted in the demons algorithm, where unconstrained optimization is followed by Gaussian filtering to impose a smoothness constraint. Using the moving mesh grid generation, we can define a deformation field with its transformation Jacobian determinant and curl of end velocity field which make it appealing to image registration Punithakumar et al. (2015Punithakumar et al. ( , 2017. The new formulation of the deformation field ensures diffeomorphic properties on the deformation field by explicitly applying constraints on transformation Jacobian determinant to keep it positive. Since the heart motion could be decomposed of radial expansion and twisting Garreau et al. (2006), defining the deformation field in terms of radial and rotational components makes this formulation suitable for cardiac analysis Bijnens et al. (2012). Methodology Most of the learning-based algorithms formulate the deformable registration problem as the minimization of the following equation: φ * = argmin φ L(I F , I M • φ(ξ))(1) where ξ denotes the pixel location in the image domain Ω, φ : Ω → Ω denotes the transformation function, and the dissimilarity metric is denoted by L(.). With the above formulation, introducing a regularization is necessarily to obtain a unique solution. Without regularization, this may result in multiple physically non-plausible solutions. In our setting, we tackle these issues with the help of the moving mesh parameterization. Moving Mesh Grid Generation To avoid adding extra terms to the above formulation and having a unique solution, more constraints are required to be added using a monitor function µ and curl of end velocity field γ. First a continuous monitor function is defined and constrained by: Ω µ = |Ω|.(2) The goal here is to find a transformation φ 1 : Ω → Ω, ∂Ω → ∂Ω such that the transformation Jacobian determinant J φ 1 (ξ) is equal to the monitor function µ : J φ (ξ) = det∇φ 1 (ξ) = µ(ξ).(3) To find the transformation φ 1 which satisfies 3, the following steps need to be taken, Step 1: A vector field V(ξ) is defined such that: div V(ξ) = µ(ξ) − 1.(4) Step 2: A velocity vector field based on artificial-time is then constructed from V(ξ): V t (ξ) = V(ξ) t + (1 − t)µ(ξ) , t ∈ [0, 1](5) The desire transformation φ 1 can be found by solving the following ordinary differential equation (ODE) at t = 1, φ 1 (ξ) = ψ(ξ, t = 1) where ψ(ξ,t=0) = φ 0 (ξ) ψ(ξ, t) dt = V t (ψ(ξ, t)), t ∈ [0, 1],(6) Where φ 0 (ξ) is the identity mapping and det∇φ 0 (ξ) = 1 and φ 0 (ξ) = ξ. Since the φ 1 (ξ) is the desire transformation that we are looking for, we drop the subscript and use φ(ξ) for the rest Algorithm 1: Moving Mesh based deformable registration Input: Given two 2D/3D pair of images, fixed image I F and moving image I M . The upper bound τ ub and lower bound τ lb of the transformation Jacobian determinant Output: Deformation field φ Step 1: Pass the input to the CNN to compute µ(ξ) and V(ξ); Step 2: Impose constraints from (8) for each pixel location ξ ∈ Ω : µ(ξ) ← |Ω| ξ⊂Ω µ(ξ) Step 3: Compute a curl of velocity field V(ξ) that satisfies (4) and compute the deformation field φ Step 4: Compute the loss function Step 5: Update the µ and V(ξ) using back-propagation of the paper. The main problem is how to find V(ξ) such that divV(ξ) = µ(ξ) − 1. There are different methods to solve this problem such as the div-curl system. To solve the problem with the div-curl system, we need to find the divergence and curl at each point and set up the div-curl system of equations for each point. By solving this system we can reconstruct a differentiable and invertible transformation.      divV(ξ) = µ(ξ) − 1 curlV(ξ) = γ(ξ).(7) To have a unique φ a constraint need to be applied to the div of the vector field V(ξ), 7. The generated transformation φ now can be parameterized with transformation Jacobian determinant and the curl of the end velocity field. Diffeomorphic Image Registration Using the above parameterization, the diffeomorphic image registration can be formulated as a constrained optimization problem. Let I F and I M be 2D/3D fixed and moving images/volumes, defined over Ω → R 2 /Ω → R 3 . We need to find µ(ξ) and γ(ξ) ∀ξ ∈ Ω, that optimize a similarity metric L S im between the warped moving image and fixed image, subject to the following constraints:          µ(ξ)dξ = |Ω| τ ub > µ(ξ) > τ lb (8) where the τ ub is the upper bound and τ lb is the lower bound of the transformation Jacobian determinant which were set by the user. The τ lb > 0 guarantees the diffeomorphism. Numerical Methods 2.3.1. 2D Div-curl solver We represent the deformation field by divergence and curl (div-curl) system representation Cheng (1989) (7). To find V(ξ) under the null condition we converted the (7) into a set of Poisson equations as follows and used a Fast Fourier Transform (FFT) based Poisson solver. As shown in (9) the radial component is given by F 1 and the rotational components is given by F 2 :        ∆V x = µ x − γ y = F 1 , ∆V y = µ y + γ x = F 2 ,(9) 3D Div-curl solver The div-curl system for the 3D case is given in Equation (7). Where the divergence of the deformation field represents the radial motion while the curl operator represents the rotation of the media around every point. The 3D operator directly extends from the 2D curl, where each rotational component represents the rotational motion of the deformation field about each of the three axes. As it shown in (10) the radial component is given by f 1 and the three rotational components are given by f 2 , f 3 and f 4 . For the 3D version, there ate three unknowns (V x , V y , V z ) with four scalar equations which makes this system overdetermined. Furthermore, a dummy variable θ is introduced to solve the system. (please check Liu (2006) for more details.)                                      divV = ∂V x ∂x + ∂V y ∂y + ∂V z ∂z = f 1 curl x V = ∂θ ∂x + ∂V z ∂y − ∂V y ∂z = f 2 curl y V = ∂θ ∂y + ∂V x ∂z − ∂V z ∂x = f 3 curl z V = ∂θ ∂z + ∂V y ∂x − ∂V x ∂y = f 4 .(10) Similar to the 2D version, we converted the (10) into a set of Poisson equations as follows:              ∆V x = f 1 x + f 3 z − f 4 y = F 1 , ∆V y = f 1 y + f 4 x − f 2 z = F 2 , ∆V z = f 1 z + f 2 y − f 3 x = F 3 .(11) Then the Euler method with arbitrary time steps is used to compute the transformation φ from V(ξ) via (5) and (6). For derivation and numerical implementation details, we refer the reader to Liu (2006) Data driven parameter computation Despite the traditional methods that iteratively and manually compute the parameters and update the gradient Chen et al. (2010); Punithakumar et al. (2017) which are time-consuming, we use an unsupervised CNN and back-propagation Algorithm 1. In the proposed framework, the network parameters are learnt in an unsupervised fashion and a diffeomorphic deformation field is generated by moving mesh parameterization Figure 1. As shown in Figure 1, the network takes I F and I M as input and outputs the monitor function µ(ξ) and the velocity vector filed V(ξ). Then using the curl of end velocity and a div-cur system a diffeomorphic transformation φ is computed. To establish the uniqueness of the solution the Dirichlet boundary condition is used Zhou (2006). Additionally, a diffeomorphism, which is corresponded to a positive transformation Jacobian determinant, is enforced explicitly via the monitor function Liu (2006). All of the steps are designed to be differentiable and the network parameters are learnt using stochastic gradient descent optimization. Registration To train the framework a set of pair images (I F , I M ) were given. Then using the monitor function and curl of end velocity, the desire φ was computed. Finally, the moving image was warped to have the minimum dissimilarity with fixed image I F . For each pair of image, we simultaneously calculated the forward transformation which registers the fixed image I F to moving image I M and the backward transformation which registers the moving image I M to fixed image I F . A symmetric loss function is used as follows: φ * = argmin θ,µ,γ {w × L(I F , I M • φ f ) + w × L(I M , I F • φ b )} (12) Where φ f is the forward transformation and φ b is the backward transformation. The registration process is performed pairwise on both 2D images and 3D volumes. In the cardiac data sets, the enddiastolic and end-systolic images are passed to the proposed framework as input to compute the forward transformation φ f and the reverse transformation φ b . For the 2D version the mean squared error (MSE) and for the 3D version the normalise cross correlation (NCC) is used as dissimilarity metric. Experiments We perform a series of experiments to evaluate the registration accuracy of the proposed diffeomorphic CNN method against the state-of-the-art methods. The evaluations were performed over three data sets consisting of clinical 2D cardiac MR images to assess the performance of the 2D version of our method. We also evaluated the 3D version of the proposed framework using ACDC data set in 3D. Data sets The following three data sets are considered in this study: Automated Cardiac Diagnosis Challenge (ACDC) Bernard et al. (2018b). This data set contains multiple temporal 2D short-axis cardiac cine MRI sequences acquired from 100 patients and is one of the publicly available data sets for cardiac MRI assessment. The spatial resolution varies from 1.37 to 1.68 mm 2 /pixel with a slice thickness of 5 mm to 8 mm (in average 5mm). The testing set contained 20 cases of each of the following cardiac diseases: dilated cardiomyopathy (DCM), hypertrophic cardiomyopathy (HCM), previous myocardial infarction (MINF), abnormal right ventricle (RV) and healthy (Normal). The images are cropped to a size of 128×128, and padded the third dimension to 16 for the 3D voxels. The Sunnybrook Cardiac Challenge data (SCD) Radau et al. (2009). This data set contains multiple temporal 2D short-axis cardiac cine MRI scans acquired from 45 patients. Each cine sequence includes 20 frames to cover the cardiac cycle. The data set is equally divided into 15 patient scans for training, 15 patient scans for validation, and 15 patient scans for testing. The image resolution is 256 × 256, with a pixel spacing of 1.25 mm and slice thickness of 8 mm. Left Atrium (LA). This data set includes 100 temporal 2D longaxis cine MRI steady-state sequences from the 2, 3 and 4chamber views, acquired from the University Alberta Hospital. Each cycle includes 25 or 30 frames with image resolutions 176 × 189 -256 × 208 and image spacing 1.445 − 1.795 mm. The ground truth manual segmentation is initially performed by a medical student and edited by an experienced radiologist. The 2ch, 3ch and 4ch are used in the rest of the paper to denote 2, 3 and 4-chamber sequences, respectively. The results are compared on end-diastolic and end-systolic frames. Quantitative Evaluation Metrics The proposed method is evaluated quantitatively using four metrics, namely, Dice metric (DM), Hausdorff distance (HD in mm), determinant of Jacobian of the deformation field det(J), and reliability R(d). Dice Metric The DM Dice (1945) is a segmentation-based metric to measure the similarity (overlap) between two regions, warped moving and fixed images. Where the Dice score of 1 indicates complete overlap and Dice score of 0 indicates no overlap. The DM of two regions A and B is formulated as: DM(A, B) = 2|A ∩ B| A + B(13) Hausdorff Distance The HD Huttenlocher et al. (1993) is another metric which measures the maximum deviation between two regions' contours. The HD between two contours (C A ) and C B is formulated as: HD(C A , C B ) = max(max i (min j (d(p i A , p j B ))), max j (min i (d(p i A , p j B ))))(14) where p i A , p j B denote the set of all the points in C A and C B respectively. The term d(·) denotes the Euclidean distance. Reliability: We also evaluated the performance of the proposed algorithm using a reliability function computed based on DMs for each data set. The complementary cumulative distribution function is defined for each d ∈ [0, 1] as the probability of obtaining DM higher than d overall volumes. R(d) = P r (Dice > d) = # Images segmented with DM higher than d total number of images . R(d) measures how reliable the algorithm is in yielding accuracy d. det(J): To analyze deformation regularity in different algorithms, we calculate the determinant of the Jacobian det(J) Ashburner et al. (1999). If the value of det(J) equals 1, the area remains constant after the transformation, whereas the value smaller or larger than 1 indicates the local area shrinkage or expansion, respectively. The negative value of det(J) implies that local folding and twisting have occurred, which are physically not realizable and mathematically not invertible Dalca et al. (2018). Baseline Methods We compared the performance of the proposed framework with state-of-the-art algorithms, SimpleElastix (Elastix) Marstal et al. (2016) 2D Image Registration Results Tables 1, 2 and 3 provide a summary of the results of the proposed method, the mean and standard deviations of DM, HD, the percentage of the number of pixels with negative Jacobian determinant %|J θ | < 0, and reliability R(0.75) on the held out test set on ACDC, LA, and SCD data sets, respectively. Figures 2, 3, 4 show samples of registered images on the LA, SCD, and ACDC data sets with the corresponding deformation field grid. End-systolic frame is the moving image and end-diastolic frame is the fixed image. The registered image of each row is shown in the third column. Also, the true and predicted segmentation maps are shown by the green and blue line respectively. For each new 2D pair of images, the registration process takes an average of 0.05 ± 0.03 seconds on a GPU. The ACDC data set is originally a 3D data set where a set of 2D axial slices are stacked to form a 3D volume. To evaluate the 2D version of the proposed framework on ACDC, we computed 2D metrics on each slice separately and aggregated the results over all slices to obtain the final values reported in Table 1. The presented method shows a better performance among the all compared methods in all aspects e.g., there is a noticeable difference between the obtain Dice score and Hausdorff distance. As can be seen, the improvement is not just limited to these two parameters, the Jacobian determinant is zero which means there is no folding or twisting in the transformation. This is in contrast to other methods where the determinant Jacobian is non-zero. Figure. 5 shows the end-diastolic and end-systolic images and the determinant of the Jacobian (|J θ |) with grid overlay for five example patients. As shown in all tables and Figure. 5, no negative values were observed on the test data for the proposed method which means our approach produced smooth and regular deformations. 3D Image Registration results The publicly available Automated Cardiac Diagnosis Challenge (ACDC) data set was employed for the evaluation of the proposed 3D-to-3D registration algorithm. Table 4 provides a summary of the results of the proposed method on the ACDC data set. The presented method displays a better performance among all the compared methods in all aspects e.g., there is a noticeable difference between the obtained Dice score and Hausdorff distance. Also, the higher probability values of R(0.75) proves that the proposed method is more reliable than the other compared methods since more patients have the dice score higher or equal to %0.75. In addition, similar to 2D version, the Jacobian determinant is also zero in 3D version which Table 1. Quantitative evaluation of the results for cardiac MRI registration on the 2D ACDC data set. The following metrics are reported for each method: The Dice score Dice (mean± standard deviation), Hausdorff distance HD, the percentage of the number of pixels with negative Jacobian determinant %|J θ | < 0, and reliability R(0.75). Smaller values of HD and larger values of Dice indicate more accurate results. Also the smaller %|J θ | < 0 indicates less mesh folding. The higher probability values of R(0.75) show that more patients have the dice score higher or equal to %0.75. Values that are highlighted in bold indicate the metric that gave the best performance compared to the other algorithms. Method Dice The grid deformations in the 3rd column displays the deformation from end-systole to end-diastole, while the last column displays the deformation from end-diastole to end-systole. The color represents the value of the Jacobian determinant, where red indicates values below 0, which is where mesh folding occurs. It can be seen that using the proposed method, no mesh folding occurs. means there is no mesh folding in the transformation. The registration process takes an average of 0.07 ± 0.005 seconds on a GPU to register an unseen 3D pair of images. Figure 6 displays a correlation plot, where the ground truth volume in mL is plotted against the volume from the proposed method. The clustering of the dots to the reference yellow line indicates the high agreement between the proposed method to the ground truth. The analysis produced a Pearson correlation coefficient of 0.98. HD %|J θ | < 0 R(0. Implementation and Parameters Analysis The proposed method is implemented in Python programming language using Pytorch module. The network is designed based on a UNet-style architecture Ronneberger et al. (2015) which includes a convolutional layer with 16 filters, three downsampling layers with 32,64,64 convolutional filters and a stride of two, and upsampling convolutional layers with 64,64,32,32,32,16 filters. The Adam optimization with learning rate of 5 × 10 −4 is used for all the three datasets. The proposed framework is evaluated on an NVIDIA GeForce GTX 1080 Ti GPU. To guarantee the diffeomorphism and keep the transformation determinant Jacobian positive, different activation functions are used to apply constraints on µ and V(ξ) and keep their range in (0, 1) and (−λ, +λ) respectively. Where λ can be any value in range of (1, ∞), we set λ = 10 in our experiment. Then using the two hyper-parameters, lower bound τ lb ∈ (0, 1) and upper bound τ ub ∈ (1, λ) of the transformation Jacobian determinant |J θ |, the user can control the amount of movement which directly affects the evaluation metrics. By increasing the values of τ lb and τ ub , each node in a grid (each pixel) can have a larger displacement; however, after a certain point, the results do not change significantly. We vary the precision τ lb , τ ub and set them to 0.2, 8.0 respectively. The chosen values resulted in the best Dice score and HD distance. Conclusion In this work, we build a principled connection between classical registration methods and recent learning-based approaches. We propose an end-to-end framework for diffeomorphic image registration and derive a learning algorithm that leverages a convolutional neural network and unsupervised learning for fast runtime. To achieve diffeomorphic transforms, we integrate a new parameterization of deformation fields for 2D-to-2D and 3D-to-3D diffeomorphic registration algorithm for the application of MRI cardiac registration, which describe a deformation field with its transformation Jacobian determinant and curl of the end velocity field. It also relaxes the need for an explicit regularization to produce a physically plausible result, as smoothness is implicitly embedded in the solution. Removing explicit regularization makes the need for an empirical trade-off between the similarity term and the regularization term, which may cause bias Beg et al. (2005), unnecessary. Furthermore, by directly requiring the transformation Jacobian to be positive, the deformation can be ensured to be diffeomorphic. The other desirable constraints also can be enforced within the same framework using an explicit restriction on the transformation Jacobian such as incompressibility constraint. Additionally, the proposed parameterization naturally describes a deformation field in terms of radial and rotational components, making it especially suited for processing cardiac data Noble et al. (2002). Our algorithm can infer the regis-tration of new image pairs in under a second, which is significantly faster than traditional iterative methods. Compared to recent learning-based methods, our method offers a guarantee of a diffeomorphic transform. The proposed algorithm was evaluated on end-diastolic to end-systolic cardiac cine-MRI registration on two publicly available ACDC Challenge Bernard et al. (2018a) and Sunnybrook data sets (SCD) Radau et al. (2009) as well as a set of left atrium images obtained from the Mazankowski Alberta Heart Institute. The proposed algorithm is diffeomorphic, allowing it to capture the true deformation of the cardiac tissue. Observing the percentage of voxels with a Jacobian determinant less than zero, most of the other registration methods yielded mesh folding for either the MRI data sets. The presence of mesh folding may result in the inability of these methods to capture the true anatomical motion. Fig. 1 . 1Overview of end-to-end unsupervised architecture. The ConvNet g θ (I F , I M ) takes the input fixed image(I F ) and moving image(I M ) and outputs the transformation Jacobian determinant J φ (ξ) = µ(ξ), and the vector field V(ξ). Then the diffeomorphic forward and backward transformations φ f and φ b are computed using the moving mesh approach. Finally, the moving and fixed images are warped using φ f and φ b . al. (2008) which are optimization based methods and diffeomorphic learning-based methods LPMKrebs et al. (2019) and LapIRNMok and Chung (2020). Fig. 2 .Fig. 3 .Fig. 5 . 235Samples of registered images on the left atrium data set with the corresponding deformation field grid (DF). The end-systolic (ES) frame is the moving image and end-diastolic (ED) frame is the fixed image. The warped ES of each row is shown in the third column. The last column labeled ground truth (GT) displays the true segmentation and the predicted segmentation, which are shown by the green line and blue line respectively. The 2ch, 3ch and 4ch stand for the 2, 3 and 4-chamber. Samples of registered images on the SCD with the corresponding deformation filed grid (DF). End-systolic (ES) frame is the moving image and end-diastolic (ED) frame is the fixed image. The warped ES of each row is shown in the third column. The last column labeled ground truth (GT) displays the true segmentation and the predicted segmentation, which are shown by the green line and blue line respectively.ES EDWarped ES DF GT ACDCFig. 4. Samples of registered images on the ACDC with the corresponding deformation filed grid (DF). End-systolic (ES) frame is the moving image and end-diastolic (ED) frame is the fixed image. The warped ES of each row is shown in the third column. The last column labeled ground truth (GT) displays the true segmentation and the predicted segmentation, which are shown by the green line and blue line respectively. 2D registration results for five example patients, where the first column is the end-systolic image and the second column is the end-diastolic image. Fig. 6 . 6The ground truth volume in mL plotted against the volume from the proposed method, where each patient is represented by a blue dot. The yellow dotted line indicates the y=x line for reference. The Pearson correlation coefficient calculated is 0.98, revealing a high correlation of the proposed method to the ground truth. Table 4 . 4Quantitative evaluation of the results for cardiac MRI registration on the 3D ACDC data set. The following metrics are reported for each method: The Dice score Dice (mean± standard deviation), Hausdorff distance HD, the percentage of the number of pixels with negative Jacobian determinant %|J θ | < 0, and reliability R(0.75). Smaller values of HD and larger values of Dice indicate more accurate results. Also the smaller %|J θ | < 0 indicates less mesh folding. The higher probability values of R(0.75) show that more patients have the dice score higher or equal to %0.75. Values that are highlighted in bold indicate the metric that gave the best performance compared to the other algorithms.Method Dice HD %|J θ | < 0 R(0.75) Undeformed 0.71 ± 0.145 10.1 - - DemonYoo et al. (2002) 0.80 ± 0.17 8.3 0.34 0.28 SyNAvants et al. (2008) 0.80 ± 0.091 8.1 0.17 0.51 LPMKrebs et al. (2019) 0.81 ± 0.085 7.3 0.12 0.52 LapIRN Mok and Chung (2020) 0.72 ± 0.162 7.4 0 0.35 MMKrishnaswamy (2021) 0.75 ± 0.156 7.03 0 0.56 ElastixMarstal et al. (2016) 0.83 ± 0.161 5.75 0.09 0.60 Proposed Method 0.84 ± 0.06 5.3 0 0.78 AcknowledgmentThe authors wish to thank Alberta Innovates for the AICE Concepts funding that supported this research work. High-dimensional image registration using symmetric priors. 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{'abstract': 'A B S T R A C TThis study proposes an end-to-end unsupervised diffeomorphic deformable registration framework based on moving mesh parameterization. Using this parameterization, a deformation field can be modeled with its transformation Jacobian determinant and curl of end velocity field. The new model of the deformation field has three important advantages; firstly, it relaxes the need for an explicit regularization term and the corresponding weight in the cost function. The smoothness is implicitly embedded in the solution which results in a physically plausible deformation field. Secondly, it guarantees diffeomorphism through explicit constraints applied to the transformation Jacobian determinant to keep it positive. Finally, it is suitable for cardiac data processing, since the nature of this parameterization is to define the deformation field in terms of the radial and rotational components. The effectiveness of the algorithm is investigated by evaluating the proposed method on three different data sets including 2D and 3D cardiac MRI scans. The results demonstrate that the proposed framework outperforms existing learningbased and non-learning-based methods while generating diffeomorphic transformations.', 'arxivid': '2208.13275', 'author': ['Ameneh Sheikhjafari \nDepartment of Computing Science\nUniversity of Alberta\nT6G 2G8EdmontonABCanada\n\nDepartment of Radiology and Diagnostic Imaging\nT6G 2G8EdmontonABCanada\n', 'Deepa Krishnaswamy \nDepartment of Radiology and Diagnostic Imaging\nT6G 2G8EdmontonABCanada\n', 'Michelle Noga \nDepartment of Radiology and Diagnostic Imaging\nT6G 2G8EdmontonABCanada\n', 'Nilanjan Ray \nDepartment of Computing Science\nUniversity of Alberta\nT6G 2G8EdmontonABCanada\n', 'Kumaradevan Punithakumar \nDepartment of Radiology and Diagnostic Imaging\nT6G 2G8EdmontonABCanada\n'], 'authoraffiliation': ['Department of Computing Science\nUniversity of Alberta\nT6G 2G8EdmontonABCanada', 'Department of Radiology and Diagnostic Imaging\nT6G 2G8EdmontonABCanada', 'Department of Radiology and Diagnostic Imaging\nT6G 2G8EdmontonABCanada', 'Department of Radiology and Diagnostic Imaging\nT6G 2G8EdmontonABCanada', 'Department of Computing Science\nUniversity of Alberta\nT6G 2G8EdmontonABCanada', 'Department of Radiology and Diagnostic Imaging\nT6G 2G8EdmontonABCanada'], 'corpusid': 251903210, 'doi': '10.48550/arxiv.2208.13275', 'github_urls': [], 'n_tokens_mistral': 14979, 'n_tokens_neox': 12738, 'n_words': 7719, 'pdfsha': '750a7ccfe7fcef8d397f1590cf2d1266621a452a', 'pdfurls': ['https://export.arxiv.org/pdf/2208.13275v1.pdf'], 'title': ['Unsupervised diffeomorphic cardiac image registration using parameterization of the deformation field', 'Unsupervised diffeomorphic cardiac image registration using parameterization of the deformation field'], 'venue': []}

arxiv

Confinedness of an X3.1 class solar flare occurred in NOAA 12192: Analysis from multi-instruments observations April 25, 2023 N Vasantharaju Department of Physics and Astronomy "Ettore Majorana" Università degli Studi di Catania Via S. Sofia 78I-95123CataniaItaly F Zuccarello Department of Physics and Astronomy "Ettore Majorana" Università degli Studi di Catania Via S. Sofia 78I-95123CataniaItaly INAF -Catania Astrophysical Observatory Via S. Sofia 78I-95123CataniaItaly F Ferrente Department of Physics and Astronomy "Ettore Majorana" Università degli Studi di Catania Via S. Sofia 78I-95123CataniaItaly S L Guglielmino INAF -Catania Astrophysical Observatory Via S. Sofia 78I-95123CataniaItaly Confinedness of an X3.1 class solar flare occurred in NOAA 12192: Analysis from multi-instruments observations April 25, 2023Draft version Typeset using L A T E X twocolumn style in AASTeX631Sun: flares -Sun: coronal mass ejection -Sun: magnetic fields-Sun: reconnection- Sun: non-potentiality The non-association of coronal mass ejections with high energetic flares is sparse. For this reason, the magnetic conditions required for the confinedness of major flares is a topic of active research. Using multi-instrument observations, we investigated the evolution and effects of confinedness in an X3.1 flare, which occurred in active region (AR) 12192. The decrease of net fluxes in the brightening regions, near the footpoints of the multi-sigmoidal AR in photosphere and chromosphere, indicative of flux cancellation favouring tether-cutting reconnection (TCR), is observed using the magnetic field observations of HMI/SDO and SOT/Hinode, respectively. The analysis of spectropolarimetric data obtained by the Interferometric Bidimensional Spectrometer over the brightening regions suggests untwisting of field lines, which further supports TCR. Filaments near polarity inversion line region, resulted from TCR of low lying sheared loops, undergo merging and form an elongated filament. The temperature and density differences between footpoints of the merged filament, revealed by DEM analysis, caused streaming and counter-streaming of plasma flow along the filament and unloads at its footpoints with an average velocity of ≈ 40 km s −1 . This results in decrease of mass of the filament (density decreased by > 50%), leading to its rise and expansion outwards. However, due to strong strapping flux, the filament separates itself instead of erupting. Further, the evolution of non-potential parameters describes the characteristics of confinedness of the flare. Our study suggests that the sigmoid-filament system exhibits upward catastrophe due to mass unloading, but gets suppressed by strong confinement of external poloidal field. INTRODUCTION Solar flares and Coronal Mass Ejections (CMEs) are violent explosive phenomena that occur on the Sun. If both these phenomena occur simultaneously and are directed at Earth, they can produce detrimental effects on Earth's magnetosphere and atmosphere. The active regions (ARs) with high magnetic complexity and nonpotentiality produce these explosive phenomena (Zirin & Liggett 1987;Schrijver et al. 2005), and when flares are accompanied by CMEs they are referred to as eruptive or else confined/non-eruptive. The association of flares and CMEs has been studied quite extensively and is still an active research topic. Previous findings, for exvasantharaju.naganna@dfa.unict.it ample Yashiro et al. (2006), showed that the probability of CME-flare association rate increases with the increase in flare strength and the association rate is 90% − 92% for X3.0 class or more intense flares. The magnetic flux ropes (MFRs), twisted magnetic field lines wrapped around axial magnetic field, are an essential part of CME structure and support filament/prominence plasma against gravity. There are many possible mechanisms responsible for initiating the outward motion of the MFR and they all come under three main models: ideal magnetohydrodynamic (MHD) instabilities (Török & Kliem 2003), flux rope catastrophes (van Tend & Kuperus 1978) and magnetic reconnection (Antiochos et al. 1999;Moore et al. 2001). One of the popular mechanism in the context of ideal MHD instability is helical kink instability (Török et al. 2004). Kink instability triggers when the twist of the MFR ex-ceeds the critical twist value of 2.5π (Török & Kliem 2003). Another relevant mechanism to the present study in the context of MFR catastrophe is "mass draining" or "mass unloading" effect, which perturbs the equilibrium of MFR. In this mechanism, an upward catastrophe occurs when the mass of the MFR decreases below a critical value (Jenkins et al. 2019;Zhang et al. 2021). Under the magnetic reconnection models, the tether-cutting reconnection (Moore et al. 2001) between sheared arcades can explain the formation and initiation of MFR eruption successfully. Though these mechanisms efficiently explain the initiation rise motion of the MFR, they failed to explain confined or suppressed eruptions after MFR exceeds the critical values. For example, a statistical study (Jing et al. 2018) of 36 strong flare events shows that kink instability plays a minor role in the successful eruption of MFRs. Thus, the rate at which overlying magnetic field decays with height plays an important role in determining the confinedness or successful eruption of MFR. This kind of ideal MHD instability is known as Torus instability (Bateman 1978;Kliem & Török 2006). Torus instability (TI) triggers when there is a force imbalance between the outward "hoop force" due to the curvature of the MFR and inwardly directed Lorentz force due to the overlying field. It is quantified by a dimensionless parameter, the decay index n, which indicates the rate at which overlying field declines with height. MHD simulations provide the onset TI criterion when n ≥ 1.5 (Török & Kliem 2005). In some events, even torus-unstable (n > 1.5) flux ropes fail to erupt and studies were conducted in this direction as well in determining the causes for confinedness of such events. A few notable ones are the dynamic tension force from the external toroidal field (Myers et al. 2015), the Lorentz force due to the non-axisymmetry of the flux rope (Zhong et al. 2021) and the rotation of the flux rope (Zhou et al. 2019) that all could contribute to the downward Lorentz force in confining the eruption. Andrews (2003) showed that about 40 % of M-class flares occurred during the period 1996-1999 are confined and that there are high probability of lack of CMEs association with weaker flares (less than C-class), whereas confined eruptions with more energetic flares are rare. Schmahl et al. (1990) reported about a confined X4class flare occurred in AR 4492 on 19 May 1984 using radio and X-ray observations. Few more case studies of X-class confined flares are studied by Feynman & Hundhausen (1994), Green et al. (2002), Chen et al. (2013) and Liu et al. (2014). Wang & Zhang (2007) conducted a statistical study of 104 X-class flares during 1996 -2004 and showed that confined X-class flares, constituting 10% of the sample, occur closer to AR center, while the eruptive flares are at the outskirts. Cheng et al. (2011) performed a comparative study between eruptive (three) and confined flares (six) occurred in AR 10720 and found that eruptive flares have higher decay index in low corona (< 10 Mm) than the confined ones. The AR 12192 is one of the largest, flare prolific and CME poor ARs of solar cycle 24. This AR produced about 35 major non-eruptive flares (29 M-class and 6 X-class) and one eruptive flare (M 4.0) during its disk passage from 18 to 29, October 2014. Many studies were conducted on the X3.1 confined flare event, the strongest amongst the flare series. Sun et al. (2015) and Sarkar & Srivastava (2018) studied the magnetic conditions of the AR and found that the core of the AR exhibits weak non-potentiality, small flare-related field changes and attribute strong overlying magnetic field strength for the confined nature of the flare. Inoue et al. (2016) using nonlinear force free field (NLFFF) extrapolations showed that the core of the AR 12192 is a multi-flux tube system located near the polarity inversion line (PIL) region, where the onset of flare is due to tether-cutting reconnection of low lying field lines of the multi-flux tube system. The confinedness of eruption is attributed to low sheared field lines, which are kink stable, as well as to the strong overlying field strength. Jiang et al. (2016), using simulations, suggested that an absence of flux rope resulted in confined eruption. On the contrary, Zhang et al. (2017) suggests that that confined flare was due to the complexity of the magnetic field structures. Past observational and simulation studies could successfully explain the formation of post-flare less sheared core field and stableness against kink instability, but they did not explain the formation of the observed filament and its rise motion during the long duration X3.1 flare event. Owing to the peculiar qualities and rareness of the event, we carried out a comprehensive analysis to investigate the evolution, cause and properties of confinedness of the X3.1 flare using spectropolarimetric imaging data, magnetograms and filtergrams corresponding to different layers of the solar atmosphere obtained by multi-instruments, on board different spaceand ground-based telescopes. Description of instruments and data is provided in section 2. In section 3, we detailed the analysis and results, followed by summary and discussion in section 4. OBSERVATIONS AND DATA The high spatial, temporal and spectral resolution spectropolarimetric data in the Ca II 8542Å line, used in the present analysis, were obtained by the Interferometric Bidimensional Spectropolarimeter (IBIS; Cav- Figure 1(b). Unfortunately, due to poor seeing conditions, we couldn't use the entire data set acquired during this time period. Based on the root mean square (RMS) contrast and visual inspection, we selected only 6 good scans of IBIS data, included in the time interval indicated by the dark green shaded region in Figure 1(a). The Solar Optical Telescope (SOT; Tsuneta et al. 2008;Ichimoto et al. 2008) onboard Hinode, has two filtergraph (FG) instruments called the Broadband Filter Imager (BFI) and the Narrowband Filter Imager (NFI) and a Spectro-Polarimeter (SP). We used filtergrams obtained by the BFI in the Ca II H line (3968Å) and Stokes-V/I images obtained by NFI in the Na I D1 line (5896Å). Ca II H and Na I D1 lines are sensitive to the upper and lower chromosphere, respectively. The FOV of SOT was limited to 328 x 164 for the NFI and 218 x 109 for the BFI. The blue dashed rectangle in Figure 1(b) marks the FOV of Hinode filtergraph observations used in the present study. The spatial resolutions of the NFI and BFI are about 0. 3 and 0. 2, respectively. To calibrate Na I D1 V/I data, we used B LOS magnetogram derived from Level 2 dataset of SP. The Hα data acquired from six different telescopes of the Global Oscillation Network Group (GONG; Harvey et al. 2011) were used in the analysis of filament evolution during the decay phase of the X3.1 flare. GONG provides full-disk Hα images at the cadence of 1 minute, with a pixel size of 1 . The Atmospheric Imaging Assembly (AIA; Lemen et al. 2012) on board the Solar Dynamics Observatory (SDO; Pesnell et al. 2012) produces full-disk Extreme Ultra-Violet (EUV) images in 10 wavelength bands at a high cadence of 12 s with pixel size of 0. 6. The photospheric magnetic field observations are obtained from the Helioseismic and Magnetic Imager (HMI; Schou et al. 2012) on board SDO. Both line-of-sight (LOS) and vector magnetograms (hmi.sharp cea 720s series) ob-tained at a cadence of 45 and 720 seconds, respectively, are used in this study. Geostationary Operational Environmental Satellite (GOES) provides the full solar disk integrated soft X-ray (SXR) flux, used to characterize the magnitude, onset, peak and end times of solar flares. ANALYSIS AND RESULTS 3.1. Overview of the X3.1 flare The X3.1 flare (SOL20141024T 21:41), the strongest flare produced by AR 12192, occurred at the heliographic location of S16W21. The X3.1 flare was not associated to any CME (Sun et al. 2015), similarly to any other X-class flare produced by this AR. The AR evolved into a highly complex region with a Mount Wilson class of βγδ during its flare prolific period i.e. 20 to 30, October 2014. On 24 October 2014, AR 12192 possessed multi inverse S-shaped sigmoidal loops prior to the X3.1 flare. Images acquired by AIA at 131 and 171Å wavebands are used to represent the morphological evolution of sheared structures during the X3.1 flare in the first and second row panels of Figure 2 (flare prior images are not shown). AR 12192 at the start of the X3.1 flare has multiple sheared structures resembling sigmoids of varying lengths. Two prominent sigmoidal structures are traced by red and blue dashed curves in Figure 2(a). The brightenings in low temperature channel of AIA 171Å waveband (Fig 2d) during the onset of the flare indicates that reconnection occurred in between low lying sheared loops rooted at the flare ribbons (Fig 2g). Figure 2(b) depicts the peak phase of the flare, flare loops brighten successively from lower to higher atmospheric layers, consequently plasma gets heated up to 10 -20 MK and an increase in overall brightening is observed. In the decay phase of the flare (Fig 2c & 2f), it is evident that many pre-flare sigmoidal structures (red and blue dashed curves) are still present and a few more formed as a result of reconnection (orange dashed curve). Due to the flare reconnection, most of these sigmoidal structures hold the filaments underneath and these filaments apparently merge to form a long elongated filamental structure that is shown in Figure 2(i). The bottom row panels are the Hα images obtained by GONG, depicting the chromospheric features evolution during the X3.1 flare at the same epochs as that of top two rows in Figure 2. Motivated by these observations, we studied the dynamics and non-eruptiveness of filamental structures and the mechanisms responsible for it. Weak-Field Approximation (WFA) and Changes in B LOS Under weak magnetic field limit, the first order perturbation relates the circular polarization profile (V) to the first derivative of the intensity profile (I) with respect to wavelength (Landi Degl'Innocenti 1992) : V λ = −∆λ H cosθ dI(λ)/dλ (1) where the proportionality factor (∆λ H cosθ) depends on the LOS magnetic field, B LOS = Bcosθ, with θ being the angle between the direction of the magnetic field vector and the B LOS component, and the Zeeman splitting is given by: ∆λ H = (e/4πm e c)Bλ 0 2 g ef f(2) where g ef f is the effective Landé factor and has the value of 1.1 for Ca II 8542Å, B is the magnetic field strength, λ 0 is the central wavelength of the spectral line, e is the charge of an electron, m e is the mass of an electron and c is the speed of light. We determined the chromospheric Line-of-Sight magnetic field B LOS by computing the slope of the linear regression model fit to V and −dI(λ)/dλ values obtained for each pixel at all the 25 spectral points of the Ca II 8542Å line acquired by IBIS. We applied the WFA separately to the whole line profile (8539.8 -8544.6Å) and the core (8541.8 -8542.2Å) of the line to obtain two values for B LOS . The B LOS values obtained from WFA applied to the core of the line profile, indicative of the B LOS values at chromospheric height, are used for further analysis. Kleint (2017) estimated noise in the polarization images to be 1% of I in 8542Å and considered pixels having V signal strength greater than 2% of I in deriving the B LOS value using WFA method and found that values less than ±60 Mx cm −2 have low S/N. However, we have only considered flare ribbon region for analysis, where V signal has a strength of about 6 -10% of I (Fig 3). Thus, the uncertainty in the derived chromospheric B LOS values should be less than ±60 Mx cm −2 . Figure 3(a) shows the Stokes I image obtained at the core of Ca II 8542Å line by IBIS at 21:32 UT, while the generated chromospheric LOS magnetogram using WFA and the corresponding photospheric LOS magnetogram obtained from HMI/SDO are displayed in figure 3(b) and figure 3(c), respectively, for qualitative comparison. Though the chromospheric B LOS values obtained from WFA are apparently higher than they are supposed to be, the reconstructed polarity patches are well in agreement with HMI magnetogram. To illustrate how well the WFA fits with observed profile, we considered two arbitrary pixels located in umbra (green dot) and over flare ribbon (blue dot) as shown in Figure 3 It is evident in Figure 3(h) that the signal strength of stokes V profile and the fitting of WFA with the observed profile is better than in Figure 3(f), indicating that the chromospheric B LOS value obtained in the flare ribbon region has a more reliable estimation (less noise) than in umbral region. This is true not just for this particular pixel in flare ribbon, but for all the pixels in the ribbon region, as evidenced in the Circular Polarization (CP) map (Fig 3d), where CP signals are stronger in flare ribbon region. The mean CP maps were generated by using the equation (del Toro Iniesta 2003), Inoue et al. 2016). As the QSLs are the potential sites of magnetic reconnection, the evolution of B LOS in different sub-regions over the flare ribbon during the flare at chromospheric and photospheric heights would be helpful in understanding the orientation and connectivity of field lines. CP = 1 12<Ic> 12 i=1 kV i , where < I c > is the average The chromospheric B LOS , determined from WFA over the IBIS FOV, is studied and compared with the photospheric B LOS obtained from HMI onboard SDO. Though, the strength of chromospheric B LOS determined from WFA over the IBIS FOV is found to be higher than the corresponding strength of B LOS at lower photospheric layer obtained from HMI, the comparison of behavior of temporal variation of B LOS values at these two layers can be studied effectively. Four subregions of 4×4 pixels are selected in different locations over the flare ribbon, outlined by squares of different colors in Figure 4 separately. We found that B LOS exhibits a decreasing behavior after the flare peak in three sub-regions at both photospheric (Fig 4c-e) and chromospheric heights (Fig 4h-j). Conversely, in a sub-region marked by cyan color, B LOS tends to increase (Fig 4(b) & Fig 4g). We would like to note that one of the footpoints of inverse 'S' shaped structures (Fig 2a & 2d) anchored in the western part of the flare ribbon, are co-spatial with the initial flare brightening regions (yellow filled contours in Fig 5a & 5e). The decrease of B LOS in three subregions (lie within initial flare brightening regions) can be attributed to the untwisting of field lines due to magnetic reconnection, similar to the scenario described in the Figure 8 of Kleint (2017). The more significant decrease of B LOS is at chromosphere height than at the photosphere is mostly due to the fact that the untwisting of field lines at higher chromospheric height is more prominent than near the footpoints i.e., at the photosphere. The different behavior of B LOS in subregion marked by cyan color from other subregions can be understood more clearly analysing Figure 5(a)&(e), where this subregion lies out-side of initial flare brightening regions. This indicates that the field lines in this sub-region continue to retain twisted configuration. Flux evolution at Photosphere and Chromosphere To compare the temporal evolution of B LOS flux over the initial flare brightening regions in photosphere and chromosphere, we used HMI LOS magnetograms and AIA 1600Å data obtained from SDO with Ca II H and Stokes-V/I of Na I D1 line data obtained from Hinode. The Na I D1 V/I data provide the LOS magnetic field distribution in the lower chromosphere just qualitatively. Na I D1 V/I signal values (dimensionless quantities) are in the range of -1 to +1. To obtain the B LOS quantitative distribution, calibration of V/I data with B LOS data obtained from Hinode Spectro-Polarimeter (SP) of SOT has to be performed (Bamba et al. 2013). We converted the Stokes V/I signal to magnetic field strength in gauss using the regression line equation, B LOS = 10900B L − 10.21, which is derived from the scatter plot of Stokes V/I signals and SP B LOS data obtained before the flare onset (20:30 - In Figure 5(a) and (e), HMI LOS magnetogram and calibrated V/I map are overlaid with yellow filled contours of flare brightenings observed during flare onset in AIA 1600Å and Ca II H wavebands, respectively. First, we identified the initial flare brightening regions that are co-spatial with the footpoints of the inverse sigmoidal structure (fig 2a) and then regions R1, R2 and R3 (blue squares in Fig 5a & e) are carefully defined such that they should enclose such initial flare brightening regions at both heights. The flux evolution in these three regions at photospheric and chromospheric heights are shown in Figure 5(b-d) and Figure 5(f-h), respectively. At photospheric height (Fig 5b-d), the de- crease of positive and negative flux is clearly observed during the X3.1 flare in all the panels except for region R2, where the positive B LOS flux exhibits increasing trend from the flare start time. This is possibly due to flux emergence in positive polarity of R2. Whereas at chromospheric height (Fig 5f-h), though the B LOS flux evolution trend appears to be same as that of photospheric height in these regions, flare related artifacts are more prominently visible, especially the sudden increase and decrease of positive flux in region R3 (Fig 5h). The brightening that appear at R3 region during peak time of flare indicates that the Na I D1 line core at R3 region evidently turned from absorption into emission (Maurya et al. 2012). It is worth to note here that the positive and negative fluxes ranges at two heights are significantly different and these values are obtained from two different instruments. Owing to calibration is-sues, we can not compare the absolute values of positive and negative fluxes at two heights but their decreasing behavior with time at two heights strongly suggests flux cancellation. The flux cancellation in these brightening regions most likely initiates the tether-cutting reconnection in the sheared arcade which in turn leads to the X3.1 flare. The brightening of sheared loops rooted at the flare ribbons observed in low temperature channel of AIA 171Å (Fig 2d) indicates that the shorter and lower sheared loops undergo tether-cutting reconnection. Figure 2(c) indicates that most of the higher sigmoidal structures continue to exist in their sheared form rather than getting relaxed after the flare. Therefore, it is likely that low lying sheared structures are involved in the tether-cutting reconnection, leading to the formation of filaments. Figure 7. Top row -AIA 171Å images showing the separation process of the merged sigmoid-filament system. Middle row -Maps of EM distribution at the same epochs as top row panels. Bottom row -T maps. Clearly, the distribution ofT and EM is higher around sigmoidal footpoint FP2 than near FP1. The two sub-regions (white rectangles) of size 25 × 50 arcsecs are chosen to study the differences in thermal and emission properties between the two sigmoidal footpoints. Dynamics of the filaments AR 12192 holds multi-sigmoidal structures (Fig 2(ac)) on 24 October 2014. These multi-sigmoidal structures are observed to carry filaments underneath after the peak phase of the flare and the analysis of the evolution of these filamental structures paves way in understanding the confinedness of the X3.1 flare. The filaments underlying the multi-sigmoidal structures lay one above the other and the evolution of these filaments is displayed in Figure 6 using GONG Hα and AIA 304Å images. The panels in Figure 6(a-d) report GONG Hα images, showing the merging of filaments (see the white arrows). This process leads to the formation of a merged elongated filament lying over the main polarity inversion line (Figure 6d). From Figure 6(e), the merged filament started to undergo separation along its axis. At this epoch, the merged filament started to rise and expand slowly, and during this process it underwent separation. What initiates the rise motion of the merged filament will be discussed in the next subsection 3.4.2. In Figure 6(h), the two distinctly separated filaments indicated by yellow arrows are shown. This separation process of the filament is distinctly visible in AIA 304Å waveband ( Figure 6(i-l)) as well and separated filaments are marked by yellow dashed curves in (Figure 6(l)). Based on the visual inspection, it appears that the coronal loops entered into a more relaxed energy state during the process of separation of the filament. Emission Measure and Thermal evolution Emission measure (EM) and temperature evolution of the sigmoidal structure holding the merged filament underneath, is studied by applying Differential Emission Measure (DEM) diagnostic technique to six EUV wavebands of AIA/SDO. DEM diagnostic technique al- lows us to measure the amount of emitting plasma along the LOS with respect to temperature. We used slightly modified version of DEM reconstruction routine xrt dem iterative2.pro available in Solar Software to work with AIA data, which was initially developed for X-ray Telescope data of Hinode Weber et al. 2004). Nonetheless, Cheng et al. (2012) applied this code comprehensively on AIA data to study the thermal properties of CME structures. Once DEM(T) maps are reconstructed, the EM and DEM weighted average temperature (T ) can be derived using the following equations:T = DEM (T )T dt/ DEM (T )dt EM = DEM (T )dt(3) where integration is performed within the temperature limits of 6.0 < LogT < 7.1. The reappearance of the sigmoidal structure (23:30 UT) in AIA EUV wavebands is co-temporal with the formation of the merged elongated filament in GONG Hα observations, as shown in Figure 6(d) & 2(f). DEM analysis is used to understand the rise motion and separation of the merged filament. The maps of EM andT in spatial domain are constructed to study the temporal evolution of thermal and emission properties of the sigmoidal structure during the decay phase of the flare i.e., from flare peak time to GOES X-ray flux attaining pre-flare level (24 October 21:41 UT -25 October 02:15 UT). AIA 171Å images in top row of Figure 7 are used to represent the evolution of the sigmoidal structure, while the corresponding maps of EM andT are plotted in middle and bottom row panels, respectively. It is evident from Figure 7 that the two footpoints, indicated by two white rectangles of size 25 × 50 arcsecs, of the sigmoidal structure have different temperature and EM distribution. The footpoint 1 (FP1) region appears to be at lowerT and EM distribution than footpoint 2 (FP2) region, which has relatively higher temperature and EM distribution. To further confirm the asymmetries of these parameters in the two footpoints, we computed the average values of EM andT over the two sub-regions enclosing the footpoints of the sigmoidal structure. The temporal evolution of these parameters are shown in Figure 8(ab). The time period between the reappearance of the sigmoid/formation of merged filament (24 October 23:40 UT) and the separation of the filament into two distinct filaments until GOES X-ray flux reaches pre-flare level (25 October 02:15 UT) is highlighted in gray shaded region. During this time period, the average EM andT values of FP2 (blue curves) are found to be higher than the average EM andT values of FP1 (black curves) in Figure 8(a-b). Once the EM distribution is known, the density (n) of the sigmoidal structure can be obtained using n = EM/l, where l is the width of the sigmoidal structure. As there are no STEREO observations during October 2014, the width of the sigmoidal structure is computed directly on AIA 304Å filtergrams by assuming that the depth of the sigmoidal structure along the line of sight is equal to its width. Before the separation of the filament, i.e. at 23:40 UT, the width of the sigmoidal structure near FP1 and FP2 is estimated to be respectively 8.7 Mm and 10.8 Mm, and the average EM is 4 × 10 28 cm −5 and 9 × 10 28 cm −5 , corresponding to densities of 6.7 × 10 9 cm −3 and 9 × 10 9 cm −3 , respectively. Once the filaments get separated i.e. at 2:15 UT, the widths of the sigmoid near FP1 and FP2 increased to 9.9 Mm and 11.7 Mm and the average EM reduced to 0.9 × 10 28 cm −5 and 2 × 10 28 cm −5 , corresponding to a decreased density of 3.1 × 10 9 cm −3 and 4.1 × 10 9 cm −3 , respectively. We carried out the similar exercise at the middle of the sigmoid and found that the density decreases from 8.2 × 10 9 cm −3 down to 2.3 × 10 9 cm −3 . The density of the sigmoid decreases by more than 50% during the process of filament separation. The calculated density values are consistent with past studies (Cheng et al. 2012). These results strongly indicate mass draining or mass unloading from the sigmoid-filament system. To compute the velocity of plasma flow along the filament structure, space-time or stack plots were generated using the slits AB and CD as shown in Figure 6(i). The slits AB and CD are placed on the filament to char-acterize the trajectories of the plasma flows directed towards footpoints FP1 and FP2, and corresponding space-time plots are displayed in Figure 8(c) and 8(d), respectively. Using the trajectories of plasma flows in the stack plots, projected velocities are computed by taking time derivative of smoothed height-time data. It is clear from the stack plots that streaming of plasma along the filament and unloading at its footpoints initiated right after merging of filaments i.e. around 23:45 UT. Initially, the velocity of the plasma flow is slower at FP2 ( ≈ 31 km s −1 ) than at FP1 ( ≈ 43 km s −1 ); this is most probably due to density differences between the footpoints, where FP2 is at higher density than FP1. Gradually, the flow velocity stabilizes and reaches the sigmoid footpoints with an average velocity of ≈ 40 km s −1 , which is consistent with past cases (Wang 1999). We believe that the temperature and density differences between the two footpoints of the sigmoidfilamental structure lead to streaming and counterstreaming of plasma flow (see animation) within it, which eventually leads to mass unloading at its footpoints. This draining of mass from the sigmoidfilamental structure would reduce the gravitational force acting on it, helping the subsequent ascent and expansion (Low 1999). However, the sigmoid-filamental structure undergoes splitting instead of eruption and the two filament sections start to move apart from one another, as shown in Figure 6, due to suppression of overlying fields (next section 3.5). Magnetic non-potentiality and confinedness As shown in Figure 2(c), after the flare the AR continues to hold pre-existing sigmoidal structures and form new sigmoids. This makes the AR quite different from others and therefore we decided to further study the temporal evolution of magnetic non-potential parameters using HMI vector magnetogram data. Basically, the photospheric magnetic non-potential measures are area-dependent, hence we computed the non-potential parameters by taking into account the minimum fluximbalance condition (< 4%) and the maximum field line connectivity involved in flare using AIA EUV images in the flaring area enclosed in the white dashed rectangle in Figure 9(a). The vector magnetic field map corresponding to the area enclosed by the white dashed rectangle is shown in Figure 9(b). The total absolute magnetic flux, given by Φ = Σ|B z |dA where B z is the vertical component of the magnetic field, is computed in positive and negative polarity regions and their temporal evolution is observed to be almost constant from 20:00 UT 24 October to 02:00 UT 25 October and is shown in Figure 9 in large scale conceals the flux cancellation occurring in small flare brightening sub-regions (Fig 5) by averag-ing out the small scale flux cancellation and emergence that occur in small sub-regions. This indicates that the amount of flux decrease due to cancellation process occurring at three small flare brightening regions is not such a significant decrease to reduce the average flux in a large flaring area. During the time interval of 5-6 hours, the total vertical current in positive and negative polarity regions of the flaring area, computed using Ampere's law : I = (∇ × B) z /µ o also does not show any significant variations (dashed curves in Fig 9(c)). The total unsigned flux (sum of net fluxes) in the flaring area of the X3.1 flare in AR 12192 is about 1.2 × 10 23 Mx, which is in agreement with the recent statistical study of Li et al. (2021), showing that flares occurring in ARs with the total unsigned flux greater than 1 × 10 23 Mx tend to be confined. Thus, the large absolute flux of the AR 12192 could be one of the causes for confinedness of the X3.1 flare. Magnetic shear is one of the important parameters that account for non-potentiality of magnetic field during flares. Magnetic shear (Wang et al. 1994) is defined as the product of observed transverse field strength with the shear angle. Shear angle is the angular separation between directions of observed vector transverse field (B o ) and potential vector transverse field (B p ), and is given by (Hagyard & Rabin 1986;Ambastha et al. 1993). The Weighted Shear Angle (WSA) is the ratio of summation of magnetic shear to the transverse field strength over all the pixels in the flaring area. It is computed by using W SA = |B o |∆θ/ |B o |. The temporal evolution of the average magnetic shear (solid curve) and WSA (dashed curve in blue) of the flaring area is plotted in Figure 9(d). The magnetic shear and WSA clearly exhibit rapid, step-wise enhancements during the onset of the flare and continues to remain in a strong sheared state for more than a couple of hours after the flare. The alignment of magnetic transverse field vectors (Figure 9(b)) are nearly parallel with the main PIL, further substantiating the increase of shear in flaring area in the post flare phase. Again, note that this increase of magnetic shear or WSA during the flare is not due to flux emergence in the flaring area (Fig 9c). In the past, there have been many studies showing the abrupt and irreversible increase of magnetic shear along the flaring PIL regions during major flares. Wang et al. (1994) showed impulsive magnetic shear enhancements along the flaring neutral line during six X-class flares and Wang et al. (2012) observed the rapid enhancement of magnetic shear in the localised region of PIL during an X2.2 flare occurred in NOAA 11158. This is mostly caused by the changes in photospheric magnetic fields, especially to the enhancement of horizontal magnetic fields near PIL region (Wang et al. 2012;Zuccarello et al. 2020;Vasantharaju et al. 2022), as a consequence of coronal implosion during flares (Hudson et al. 2008). However, the result could be different if the analysis is extended from localised regions of the PIL to the whole flaring area. For example, Li et al. (2000) considered the whole flaring area in three ARs to study the changes of average magnetic shear after the flares. They found that magnetic shear in flaring area decreases significantly after the flare. On the contrary, we found that the average magnetic shear in the flaring area of AR 12192 increases after the X3.1 flare. This irreversible increase of magnetic shear is consistent with the fact that no observation of eruption is found. If there were any eruptions, these would have taken away magnetic helicity (Nindos & Andrews 2004), thereby leading to less sheared postflare loops. Thus, permanent increase of magnetic shear is an effect of confinedness of the X3.1 flare. ∆θ = cos −1 (B o · B p /|B o B p |) Average α (α av ) or global α is one of the non-potential parameters used to indicate the degree of twist of magnetic field lines in an AR. It is derived from the zcomponent of the magnetic field in force-free conditions (µJ z = αB z ) and can be computed using the equation given by α av = [J z (x, y)B z (x, y)/|B z (x, y)|] (Pevtsov et al. 1994;Hagino & Sakurai 2004), where B z is the vertical magnetic field and J z is the vertical current density. The temporal evolution of α av is plotted in Figure 9(e). α av exhibits a slight increasing trend during the flare and maintains almost the same value for a couple of hours after the flare, indicating that the twistness of field lines in the AR slightly increases after the flare, which is in support of non-eruptiveness of the flare. Moreover, the magnetic transverse field vector in Figure 9(b) exhibits a swirling pattern in the upper main negative polarity region: this further corroborates the twistness present in field lines. Thus, non-decreasing α av is also a characteristic effect of confinedness of the X3.1 flare. Past studies indicate that magnetic gradient is a better proxy than magnetic shear in locating the occurrence and productivity of flares and their strength in an AR (Wang et al. 2006;Vasantharaju et al. 2018). We therefore computed the Strong Gradient PIL (SG-PIL) using an automated procedure described in Vasantharaju et al. (2018). In this procedure, vertical gradient maps and potential field are computed using a smoothed B z map and applying the threshold of potential transverse field (> 300G) and strong magnetic field gradient (> 50 G/Mm) to the zero gauss contours on smoothed B z maps. The SGPIL length evolution in time is plotted in Figure 9(f). Total SGPIL length decreases from 185 Mm (peak flare) to 155 Mm after the flare, only for a short time interval. Thereafter, strong gradients near PIL start to increase so as the SGPIL length. SGPIL segments are fragmented, scattered and not continuous in the flaring area of AR 12192. Mostly, the twisted flux rope resides above the continuous high gradients of PIL region, whereas the flaring area in AR 12192 possesses fragmented and scattered SGPILs which might indicate the AR's inability to host strong, long and continuous flux-rope capable of eruption (Vemareddy 2019). Thus, fragmented SGPILs in the AR could also be one of the causes for confinedness of the X3.1 flare. Liu et al. (2017) suggested that the degree of Net Current Neutralization (NCN) would be a better proxy than strong shear or gradients near PIL in assessing the eruptive nature of flares from an AR. The net current in each polarity has both positive and negative components. The NCN is computed for each polarity by obtaining the ratio of Direct Current (DC) and Return Current (RC) (Török et al. 2014). DC and RC are computed by integrating vertical current density values of different signs separately. The temporal evolution of |DC/RC| values in both polarity regions are plotted in Figure 9(g). The DC is found to be positive in the north polarity and negative in the south polarity. The |DC/RC| values in both polarity regions are almost equal to unity. Past studies (Liu et al. 2017;Vemareddy 2019) showed that the full current neutralization (NCN=1) is a characteristic of a non-eruptive AR, indicating the absence of direct-current channels over the PIL region, whereas an AR characterized by non-neutralization (NCN > 1.3) of currents is prone to erupt. Thus, the full current neutralization in AR 12192 for an extended time interval leads to produce many confined flares, including the X3.1 flare under study. It's worth noting that the distribution of fragmented SGPIL in the flaring area and the full current neutralization (NCN=1), both indicating absence of robust flux rope along PIL, may contribute to the confinedness of the X3.1 flare. However, we observed the appearance of sigmoid-filament structure along the main PIL and its dynamics of rise and expansion. Thus, the main contribution to the confinedness of X3.1 flare should be the stronger inward directed force from background field and not the weaker outward driving force from the inner non-potential magnetic field. So, we examined the role of background coronal magnetic field using the Potential Field Solar Surface (PFSS; Schrijver & De Rosa 2003) approximation. The lower-boundary data is provided by HMI vertical component of the magnetic field (SHARP series). The decay index is defined as, n(z) = − z B h ∂B h ∂z , where z is the geometrical height from the photospheric surface and B h is the horizontal field strength. After the coronal field extrapolation, B h as a function of z along the filament (yellow contour region in Fig 9a) is obtained. We repeated the procedure on a time interval of one hour from 20:00 UT on 24 October to 02:00 UT on 25 October. We then plotted the average decay index along the filament channel and B h as a function of z at each hour (Fig 10d-f) and we found that the decay index reaches the theoretical critical value of 1.5 (Török & Kliem 2005) beyond 80 Mm above the surface. The temporal evolution of average critical height above the filament channel (or main PIL) is plotted in Figure 9(h). Past statistical studies like Vasantharaju et al. (2018);Baumgartner et al. (2018); Li et al. (2020) showed that the ARs producing confined flares mostly tend to have high critical heights above 50 Mm, owing to strong confinement, whereas for eruptive flares, critical heights are less than 42 Mm, indicative of weaker overlying field strength. For AR 12192 during the X3.1 flare, almost constant critical height of about 80 Mm throughout the flare duration of 6-7 hours, indicates that the background magnetic field strength is strong enough to confine any possible eruption. Furthermore, Myers et al. (2015) using laboratory experiment showed that the orientation of external potential field configuration with respect to flux rope axis is necessary to determine the specific component of downward Lorentz force. The total potential magnetic field is the superposition of strapping field, running perpendicular to the flux rope axis and guide field, running toroidally along the flux rope axis. The coronal field lines rendering extrapolated using PFSS approximation at different time instants throughout the flare duration is shown in Figure 10. Figure 10(a-c) displays the potential field configuration of AR 12192 at different stages of the X3.1 flare and panels (d-f) show the corresponding variations of decay index and horizontal magnetic field strength with height. The filament axis is lies along the PIL of two main polarities (yellow contour in Fig 9a). From the PFSS plots, it appears that the direction of external poloidal magnetic field is oriented nearly perpendicular to the axial direction of filament. This indicates that the strapping force is more dominant downward force than the dynamic tension force, induced by the toroidal field. Thus, we opine that the main contributor to the downward Lorentz force towards confining the X3.1 flare would be the strong strapping field. GONG) and space-based (Hinode & SDO) instruments. The X3.1 flare (strongest among the flares produced by AR 12192) was of long duration, lasting for 5 -6 hours and occurred at a heliocentric angle of µ = 0.9. The AR holds multi sigmoidal structures prior to the start of the flare. Low lying sheared field lines underwent tethercutting reconnection during the flare, bringing minimum morphological changes to the high lying pre-flare coronal sigmoidal structures, but showing the appearance of filaments underneath these sigmoids. These sigmoidfilament systems lying one over the other exhibit dynamic behavior of merging and subsequent separation. The temperature and density differences between the footpoints of the merged sigmoid-filament system, as revealed by DEM analysis, aids in understanding the separation and non-eruptiveness of the merged filament. Confinedness of the X3.1 flare is mainly caused by the strong confinement provided by the external magnetic field rather than the weaker non-potentiality of the core AR. AR 12192, being located in the southern hemisphere, shows positive helicity and follow the dominant helicity sign rule (Pevtsov et al. 1995), but it shows inverse S-shaped sigmoids on 24 October. Generally, inverse S-shaped sigmoids predominantly appear in northern hemisphere (Rust & Kumar 1996), which makes the AR 12192 unconventionally twisted. EUV/AIA observations reveal that the AR has multi sigmoidal structure. Moreover, brightening of the flare loops with their footpoints rooted at flare ribbons observed in low temperature channel of AIA 171 A (Fig 2d) provides evidence that the shorter and lower sigmoidal loops undergo magnetic reconnection. The B LOS flux cancellation at both photospheric and chromospheric heights in the brightening regions, which are co-spatial with footpoints of low lying sheared field lines, supports the idea of tether-cutting reconnection (Moore et al. 2001) to produce the X3.1 flare and is in agreement with the numerical study of Inoue et al. (2016). Further support of tether-cutting reconnection comes from the analysis of a part of the flare ribbon area (segment identified as QSL in Inoue et al. 2016), specifically in initial brightening regions. Using spectropolarimetric data obtained by IBIS to examine the orientation of field lines during the flare, we found that the scenario resembles untwisting of field lines during the flare, as observed by Kleint (2017). As the flare progresses, flare loops brighten successively from lower to higher atmospheric layers (Zhang et al. 2017) and most of the higher sigmoidal structures con-tinue to exist in their sheared form rather than getting relaxed after the flare (Fig 2c). The tether-cutting reconnection in low lying sheared field lines leads to the formation of filaments near the PIL region (Moore et al. 2001). These filaments which are underneath the high lying sigmoids form the sigmoid-filament systems, which undergo apparent merging to form an elongated filamental structure in chromosphere, as observed in GONG Hα images, and are co-spatial with an inverse S-shaped sigmoid in the higher layers, as revealed in EUV/AIA observations. Once the filaments merge together to form a long filament, the sigmoid footpoints were found to have temperature and density differences, as shown by DEM analysis. The temperature and density differences between the sigmoid footpoints mostly cause the streaming and counter-streaming of plasma inside the filament. The average flow velocity directed towards the footpoints of the filament is found to be about 40 km s −1 , in agreement with past studies (Wang 1999), leading to a density decrease by more than 50%. The continous streaming of chromospheric material of the filament at its footpoints leads to draining of the filament mass (supplement movie). As the total mass of the filament decreases, the sigmoid holding the filament becomes unstable and consequently starts to rise and expands in upward direction . However, the sigmoid-filament system could not proceed with its outward motion, but instead it splits axially (Fig 6). We note that majority of filament eruptions are studied by considering negligible pressure and mass of filament plasma suspended by a flux rope in comparison with the dominant magnetic pressure and tension forces of the flux rope and its surroundings (Titov & Démoulin 1999). However, a few studies (Seaton et al. 2011;Jenkins et al. 2018), including this one, provide evidences for "mass-unloading" as an eruption driver or increase the height of flux rope, suggests that a modification of gravitational force due to reduction in mass may influence the stabilization of flux ropes. The filaments that are formed in between the flare ribbons along the PIL started to appear in H-alpha images around 22:30 UT (Fig 6a), only after the flare peak time (i.e., 21:41 UT) but as a result of long flare magnetic reconnection. AIA/SDO observations revealed the stratified structure of flare loops and each set of flare loops did not undergo significant ascending or descending motions after the flare peak time (Zhang et al. 2017), which is corroborated by observations of no considerable lateral separation of flare ribbons (Thalmann et al. 2015). This further substantiates the fact that the same magnetic field structure undergoes reconnection repeatedly for a long period of time, leading to the formation of filaments. Further, the sigmoidal filament structure formed after the flare peak lies along the main PIL with its footpoints rooted at the two flare ribbons on either side of the PIL (animation video), and the dynamics of filament evolution like its rise motion and separation, are all closely related to thermodynamic properties of the same set of flaring loops rooted at flare ribbons, which all get constrained under the same canopy of strong external field within the flare duration of 5-6 hours. Thus, we believe that there is an inherent association of dynamics and non-eruptiveness of filament to the occurrence and confinement nature of X3.1 flare. Regarding the causes for confinedness of the flare, magnetic reconnection in the low lying, sheared core field is supposed to reduce the constraints of overlying field lines and to allow the core field to erupt (Antiochos et al. 1999;Moore et al. 2001). However, in the present event, flare loops did not undergo ascending or descending motions after the flare peak time, suggesting that the tether-cutting reconnection failed to weaken the constraints of upper magnetic loops and to produce the eruption of formed filaments. This is not a new result, for example Zou et al. (2019) studied a confined X2.2 flare which exhibited two episodes of brightenings. They found that these brightenings correspond to two magnetic reconnections, one occurred at the null point beside the pre-existing flux rope and the other tethercutting reconnection occurred below the flux rope. However, these two magnetic reconnections failed to produce an eruption because of the strong strapping flux. Thus, although tether-cutting reconnection may act as the trigger of an eruption, it alone is less likely to produce a successful eruption. In eruptive flares, the ejection of twisted flux ropes into interplanetary space leads to less sheared post-flare loops (Forbes & Isenberg 1991). On the other hand, in a confined flare, like the one we have investigated, twist and shear of the core field is conserved with minimum changes in morphological complexity, as shown in Fig 9 & 2. These are the characteristic effects of confinedness of the flare. Further investigation of non-potentiality of the core AR 12192 suggests that the AR has fragmented and scattered high gradient PILs, which is an indication of not having a continuous, strong twisted flux-rope capable of eruption at certain instability conditions (Vemareddy 2019). This in turn is in agreement with the full neutralization condition (NCN =1) of AR 12192, indicating the absence of a direct current channel over the PIL. On the contrary, sigmoid-filament structure appeared along the main PIL of AR and exhibited dynamics of rise and expansion. Thus, the main con-tribution to the confinedness of the X3.1 flare should be the stronger inward directed force from background field and not the weaker outward driving force from the magnetic non-potentiality of core AR. AR 12192 has a mean area of more than 3500 millionths of a solar hemisphere (µsh) and a peak area of more than 4000 µsh on 24 October (Cliver et al. 2022) with a total unsigned magnetic flux (|φ|) larger than 1 × 10 23 Mx. Recent statistical studies (Li et al. 2020;Cliver et al. 2022) showed that the probability of producing eruptive flares by an AR with area above ≈ 3500 µsh and |φ| above 1 × 10 23 Mx is greatly reduced. They argued that larger the flux and area, stronger will be the confinement of the overlying magnetic field. This argument holds true even for the location of the X3.1 flare, which occurs near the center of the AR. Statistically, Baumgartner et al. (2018) showed that confined flares occur close to the AR centers, where the constraining field strength is stronger and eruptive flares occur at the periphery of ARs, where the confinement is weaker. The total flux and area of AR along with the location of the X3.1 flare indicate the increase of horizontal field strength, which decreases the decay rate of overlying field with height, suppressing eruption. The average critical height (height at which the decay index = 1.5) above the sigmoid-filament system remains constant at about ≈ 80 Mm throughout the flare duration of 5-6 hours, suggesting the strong confinement over the core of the AR (Vasantharaju et al. 2018;Baumgartner et al. 2018;Li et al. 2020). It is very difficult to point out the exact component of downward Lorentz force, generated from the interaction between external field and the erupting structure, contributing towards confining the eruption with pure observations. Given the fact that the X3.1 flare event is an on-disk event and the non-availability of STEREO observations, it is difficult to determine the exact height till which the merged filament raised before it actually got suppressed by downward acting Lorentz force. However, based on AIA 171Å and 304Å observations (low temperature channels), the filament eruption is confined in the lower corona (< 80 Mm) and the decay index of the external poloidal field does not exceed the criterion for torus instability (i.e. when n c = 1.5, H c = 80 Mm). Further, the potential field configuration at different time instants throughout the flare duration (Fig 10) provide evidences that the direction of the external poloidal magnetic field is oriented nearly perpendicular to the axial direction of filament (along the PIL). This indicates that the strapping force is a more dominant downward force than the dynamic tension force, induced by the toroidal field (Myers et al. 2015). However, we can not rule out the possibility of downward acting non-axisymmetry induced forces due to the radial magnetic field of the magnetic flux rope carrying the filament (Zhong et al. 2021). The direction of the forces induced by the radial magnetic field of the filament changes with the evolution of the filament but determining them using the observations is very hard. So, we conclude that from an observational point of view the confinedness of the X3.1 event is due to the net downward Lorentz force contributed mainly by the strapping field with the possible contribution from non-axisymmetry of the filament. More of such unique X-class confined events need to be analyzed to generalize the results reported in this work and to provide reliable input to flare/CME forecasting studies. Figure 2 . 2Images acquired by AIA at 131Å, 171Å, as well as Hα images are used to give an overview of the evolution of AR 12192 during the X3.1 flare. (a)-(c) AIA 131Å images show the morphological changes in the multi-sigmoidal system during the X3.1 flare. (d)-(f) AIA 171Å images are at almost same epochs as the top row panels, where HMI BLOS maps with contour levels of ± 500 G are overlaid. (d) provides evidence of brightenings in the low lying sheared arcade during the onset of flare. (g)-(i) GONG Hα images reporting the two flare-ribbon evolution during the flare and the resultant filament formed underneath the sigmoid, as shown in (i). The IBIS FOV is marked by a red dashed rectangle in (h). (a)&(b). WFA fits for the whole line and core of the line profile indicated by magenta and blue asterisks, respectively, are over-plotted on the observed normalized V profile (solid curve) in Figure 3(f)&(h). Figure 3 . 3Illustration of the results of WFA using a sample scan obtained by IBIS. (a) Stokes I image obtained at the core of Ca II 8542Å line by IBIS. (b) Chromospheric LOS magnetogram deduced by WFA. (c) HMI LOS magnetogram with the same FOV as of IBIS data. (d) Circular polarization (CP) map obtained from IBIS data. CP signals are predominant in the flare ribbon (marked by brown contour), which causes the reconstructed polarity patches within and around the flare ribbon in chromospheric magnetogram (b) have a better match with HMI magnetogram (c). (e) Normalised Stokes I profile of an umbral region indicated by a small green square in (a). (f) V/I profile (solid green) and the two WFA fits obtained from the derivative of Stokes I for the full profile (magenta asterisks) and only for the core profile (Blue asterisks). (g & h) -Same as (e & f), but for a small ribbon region (blue square in (a)).continuum intensity of the quiet sun region within IBIS FOV. In this method, Stokes V i images obtained at 12 wavelength positions along the Ca II 8542Å line are considered such that 6 wavelength positions are in blue wing and remaining 6 wavelength positions are from red wing of the line. To reconstruct the sign of CP signal, we multiplied Stokes V i images in blue wing with k=+1 and Stokes V i images in red wing with k=-1.As all the twisted field lines in an AR are not related to the flare it produces,Inoue et al. (2016) extensively explored the location of Quasi-Separatrix Layer (QSL;Demoulin et al. 1996) connected to X3.1 flare in AR 12192. QSL is the region of very high magnetic connectivity gradient which favours the formation of thin-current layer, where the magnetic reconnection is considered to occur relatively easy.Inoue et al. (2016) found out that QSLs of X3.1 flare correspond to boundary of flare ribbons. The IBIS FOV, indicated by red dashed rectangles inFigure 1(a) & 2(h), encloses a part of western flare ribbon which corresponds to the location QSL (refer fig 6 of (a)&(f). The average B LOS values of the 4×4 pixels in four different locations are plotted in four different panels for photosphere and chromosphere Figure 4 . 4Comparison of evolution of BLOS values at photospheric (b-e) and chromospheric heights (g-j). (a) HMI continuum image with four different locations in the flare ribbon region marked by four different colored square boxes of 4×4 pixels each. (f) Same as (a) but with Stokes I image of core Ca II 8542Å line. The colored curves in all the remaining panels represent the evolution of BLOS values averaged over the same color of square boxes indicated in (a) & (f). (h)-(j) BLOS at chromospheric height shows decreasing behaviour indicating untwisting of field lines, while corresponding locations at photospheric height (c-e) shows increasing trend. The dashed vertical red and black lines indicate the GOES peak and end times of the X3.1 flare, respectively. Figure 5 . 5Magnetic flux evolution near the footpoints of the sigmoidal structure in photosphere (b-d) and chromosphere (f-h) using HMI and Hinode BLOS data respectively. (a) HMI LOS magnetogram overlaid with the yellow color filled contours of initial flare brightening observed in AIA 1600Å waveband. (e) Na I D1 V/I map overlaid with the yellow color filled contours of initial flare brightening observed in Ca II H line. The decreasing flux content in the positive polarity (blue solid curves) and negative polarity (red solid curves) patches in the three sub-regions signifies flux cancellation. The flare artifacts observed in the chromospheric Na I D1 line camouflaged the decreasing trend of net fluxes (h). The four sub-regions identified in IBIS FOV (Fig 4a & f) are also marked in (a) and (e) to specify the location with respect to flare brightening regions. The three dashed vertical lines correspond to GOES start (black), peak (red) and end (black) timings of the X3.1 flare. 20:50 UT), where B LOS and B L are the converted LOS magnetic field strength in gauss and Stokes V/I values, respectively. Figure 6 . 6(a-d) GONG Hα images showing the merging process of the filaments (underneath sigmoids) near the PIL region. White arrows are used to guide the visualization of the merging process. (e-h) Merged filament splits axially leading to two split filaments indicated by yellow arrows. (i-l) same as (e-h) but with AIA 304Å images, where separated filaments are indicated by traced yellow curves (l). An animation of this figure is available, where AIA 304Å image sequences run from 20:00 UT on 2014 October 24 to 03:00 UT on 2014 October 25 showing the formation of sigmoid-filamental structure along the main PIL, and the subsequent expansion and separation of the structure. Figure 8 . 8(a-b) Temporal evolution of average EM andT of the regions enclosed by white rectangles, showed in Fig 7, near the two footpoints of the sigmoidal structure. Solid black and blue curves refer to footpoints FP1 and FP2, respectively. Dashed vertical red line marks the peak time of the X3.1 flare and shaded region indicates the time interval of filament separation. (c) Space-time plot of plasma flow along the slice AB (Figure 6i) directed towards FP1 and projected flow velocities are annotated against the trajectories of flows in AB. (d) Same as (c) but along the slice CD (Figure 6i), showing the flow directed towards FP2. Figure 9 . 9Temporal evolution of magnetic parameters computed within the region enclosed by the white dashed rectangle shown in (a). Automated Strong Gradient Polarity Inversion Line (SGPIL) traced in blue curves and contour of filament (Fig 6d) in yellow is overplotted on Bz in (a). (b) Vector magnetogram (transverse vectors overlaid on Bz) of the region enclosed by the white dashed rectangle in (a). (c) Net flux and current evolution. (d) Step-wise enhancements of Magnetic shear and WSA. (e) αav evolution (see text) (f) SGPIL (g) Degree of NCN signifying full current neutralization (NCN≈1) and (h) Average critical heights, when n = nc ≥ 1.5, over the filament channel. The critical heights are computed at every hour throughout 7 hours. The gray shaded region indicates the time duration of the flare as recorded by GOES. Running average (black solid curve) of αav and SGPIL measurements is over-plotted in (f) and (g) to enhance the actual variations. 4 .Figure 10 . 410SUMMARY AND DISCUSSION In this paper, we investigated the nature of confinedness of an X3.1 flare originated in AR 12192 in different layers of the solar atmosphere using the multiwavelength observations obtained from ground-(IBIS & (a-c) PFSS configuration of AR 12192 at different time instances, from pre-flare time (20:48 UT) to end time (3:00 UT), of X3.1 flare. The unvarying potential field configuration provides the robust confinement throughout the flare. (d-f) Corresponding variations of decay index and horizontal magnetic field strength with height. Decay index, n, reaches 1.5 at about 80 Mm above the photosphere. Figure 1. (a) Disk integrated GOES X-ray flux variations on 24 October 2014. The timeline of data coverage of IBIS and Hinode instruments are shown in shaded regions. (b) HMI/SDO continuum image of NOAA AR 12192 taken near the peak time of the X3.1 class flare. The dashed rectangles in red and blue indicate the FOVs of IBIS and Hinode (BFI/SOT) instruments, respectively. allini & Reardon 2006; Reardon & Cavallini 2008) at the ground-based Dunn Solar Telescope (DST). The IBIS instrument is based on a dual Fabry-Perot interferometer and mounted in the collimated beam of DST.09:00 12:00 15:00 18:00 21:00 00:00 03:00 Start Time (24-Oct-14 08:00:00) 10 -6 10 -5 10 -4 10 -3 GOES X-ray flux(Wm -2 ) (a) GOES 1−8Å IBIS data Hinode data C5.1 X3.1 00:00 03:00 C M X X10 SDO HMI 6173 24−Oct−2014 21:34:14.7 UT 150 200 250 300 350 400 X (arcsec) −350 −300 −250 −200 150 200 250 300 350 400 −350 −300 −250 −200 (b) The Ca II 8542Å line was scanned along 25 wavelength points from 8539.8 to 8544.6Å, with an average step size of 0.19Å . The pixel size is of 0. 095 and the maximum spatial resolution is about 0. 3. There are two sets of ob- servations available corresponding to two different fields of view (FOVs) over AR 12192, with a total of 144 full spectropolarimetric scans on 24 October 2014. These two sets of observations track portions of flare ribbons evolution corresponding to two different flares that oc- curred in AR 12192 on 24 October 2014. The FOV for the first set of observations corresponds to a C5.1 flare (start time -14:31 UT, peak time -15:06 UT and end time -15:54 UT). For this flare, we have IBIS observa- tions from 14:55 UT to 16:42 UT (indicated by the light green shaded region in Figure 1(a)) and not used in the present work. The FOV for the second set of observa- tions corresponds to the X3.1 flare (start time -21:07 UT, peak time -21:41 UT and end time -22:13 UT). For this flare, we have IBIS observations from 21:20 UT to 22:30 UT. This FOV is marked as red dashed rect- angle in ACKNOWLEDGEMENTSWe thank the referee for the detailed comments that definitely improved the quality of the paper. N.V. acknowledges support from the European Union's Horizon 2020 research and innovation programme under grant agreement no. 824135 (SOLARNET project) and no. 739500 (PRE-EST project). This research has been carried out in the framework of the CAESAR (Comprehensive spAce wEather Studies for the ASPIS prototype Realization) project. N.V. acknowledges financial contribution from the Agreement ASI-INAF n.2020-35-HH.0. This work was also supported by the Italian MIUR-PRIN grant 2017APKP7T, by the Università degli Studi di Catania (Piano per la Ricerca Università di Catania 2020-2022, Linea di intervento 2). SDO is a mission of NASA's Living With a Star Program. Authors thank the HMI and AIA science team for their open data policy. We also thank Dr. Christian Beck for providing the IBIS calibrated data. 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H., Criscuoli, S., Falco, M., & Murabito, M. 2020, ApJ, 889, 65

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{'abstract': 'The non-association of coronal mass ejections with high energetic flares is sparse. For this reason, the magnetic conditions required for the confinedness of major flares is a topic of active research. Using multi-instrument observations, we investigated the evolution and effects of confinedness in an X3.1 flare, which occurred in active region (AR) 12192. The decrease of net fluxes in the brightening regions, near the footpoints of the multi-sigmoidal AR in photosphere and chromosphere, indicative of flux cancellation favouring tether-cutting reconnection (TCR), is observed using the magnetic field observations of HMI/SDO and SOT/Hinode, respectively. The analysis of spectropolarimetric data obtained by the Interferometric Bidimensional Spectrometer over the brightening regions suggests untwisting of field lines, which further supports TCR. Filaments near polarity inversion line region, resulted from TCR of low lying sheared loops, undergo merging and form an elongated filament. The temperature and density differences between footpoints of the merged filament, revealed by DEM analysis, caused streaming and counter-streaming of plasma flow along the filament and unloads at its footpoints with an average velocity of ≈ 40 km s −1 . This results in decrease of mass of the filament (density decreased by > 50%), leading to its rise and expansion outwards. However, due to strong strapping flux, the filament separates itself instead of erupting. Further, the evolution of non-potential parameters describes the characteristics of confinedness of the flare. Our study suggests that the sigmoid-filament system exhibits upward catastrophe due to mass unloading, but gets suppressed by strong confinement of external poloidal field.', 'arxivid': '2304.12156', 'author': ['N Vasantharaju \nDepartment of Physics and Astronomy "Ettore Majorana"\nUniversità degli Studi di Catania\nVia S. Sofia 78I-95123CataniaItaly\n', 'F Zuccarello \nDepartment of Physics and Astronomy "Ettore Majorana"\nUniversità degli Studi di Catania\nVia S. Sofia 78I-95123CataniaItaly\n\nINAF -Catania Astrophysical Observatory\nVia S. Sofia 78I-95123CataniaItaly\n', 'F Ferrente \nDepartment of Physics and Astronomy "Ettore Majorana"\nUniversità degli Studi di Catania\nVia S. Sofia 78I-95123CataniaItaly\n', 'S L Guglielmino \nINAF -Catania Astrophysical Observatory\nVia S. Sofia 78I-95123CataniaItaly\n'], 'authoraffiliation': ['Department of Physics and Astronomy "Ettore Majorana"\nUniversità degli Studi di Catania\nVia S. Sofia 78I-95123CataniaItaly', 'Department of Physics and Astronomy "Ettore Majorana"\nUniversità degli Studi di Catania\nVia S. Sofia 78I-95123CataniaItaly', 'INAF -Catania Astrophysical Observatory\nVia S. Sofia 78I-95123CataniaItaly', 'Department of Physics and Astronomy "Ettore Majorana"\nUniversità degli Studi di Catania\nVia S. Sofia 78I-95123CataniaItaly', 'INAF -Catania Astrophysical Observatory\nVia S. Sofia 78I-95123CataniaItaly'], 'corpusid': 258298569, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 24330, 'n_tokens_neox': 20217, 'n_words': 12752, 'pdfsha': '99d052b68a5feea5682ba418660e5362238698a5', 'pdfurls': ['https://export.arxiv.org/pdf/2304.12156v1.pdf'], 'title': ['Confinedness of an X3.1 class solar flare occurred in NOAA 12192: Analysis from multi-instruments observations', 'Confinedness of an X3.1 class solar flare occurred in NOAA 12192: Analysis from multi-instruments observations'], 'venue': []}

arxiv

Formation, evolution, and survival of massive star clusters Proceedings IAU Symposium No. 316 2015 Giovanni Carraro gcarraro@eso.org ESO Alondo de Cordova 310719001Santiago de ChileChile Dept. of Physics and Astronomy University of Padova Vicolo Osservatorio 335122PadovaItaly Emanuele Dalessandro emanuele.dalessandr2@unibo.it Dept. of Physics and Astronomy University of Bologna Via Ranzani 1, 40127BolognaItaly Formation, evolution, and survival of massive star clusters Proceedings IAU Symposium No. 316 2015Open clusters: general -Open clusters: individual: NGC 6791 We present the first evidence of clear signatures of tidal distortions in the density distribution of the fascinating open cluster NGC 6791. We find that the 2D density map shows a clear elongation and an irregular distribution starting from ∼ 300 from the cluster center and two tails extending in opposite directions beyond the tidal radius. These features are aligned to both the absolute proper motion and to the Galactic centre directions. Accordingly we find that both the surface brightness and star count density profiles reveal a departure from a King model starting from ∼ 600 . These observational evidences suggest that NGC 6791 is currently undergoing mass-loss likely due to gravitational shocking and interactions with the tidal field of the Milky Way. We derive the expected mass-loss due to stellar evolution and tidal interactions and we estimate the initial cluster mass to be Mini = (1.5 − 4.0) × 10 5 M . Context and results NGC 6791 is one of the most massive open cluster (Carraro 2014) in the MW, possibly harbouring more than a stellar population (Geisler et al. 2012, Bragaglia et al. 2014.We used deep images obtained with the wide field imager MegaCam mounted at the Canada-France-Hawaii Telescope (CFHT) to cover a 2 o × 2 o area around the cluster (Dalessandro et al. 2015). The upper left panel in Fig 1 shows the obtained (g , g − r ) CMDs of the innermost region of NGC 6791 (left) and the external control field (right) including stars located at a distance r 3000 from the cluster center. Large scale 2D colour-coded surface density map of NGC6791 obtained by using the optical matched filter technique is shown in the upper right panel of Fig. 1. The contour levels span from 3σ to 40σ with irregular steps. The solid arrow represents the direction of the absolute proper motion while the dashed ones mark the direction of the Galactic Center and that perpendicular to the Galactic plane (Z=0).The observed star count density profile is shown in the upper left panel of Fig. 1 (open grey squares). The dashed line represents the density value of the background as obtained in the control field. The black filled dots are densities obtained after background subtraction. The best single-mass King model is also over-plotted to the observations (solid line). For r 600 the density profile clearly deviates from the King model following a power-law with exponent α ∼ −1.7.By using a simple analytic approach (Lamers et al. 2005), we estimated the mass likely lost by NGC 6791 during its evolution because of the effect of both stellar evolution and dynamical interactions. On this basis we estimated the cluster initial mass as a function of the dissolution time parameter t 0 . In particular the red curve (lower right panel in Fig. 1) shows the dependence for the current cluster mass (5000 M ) and its actual age of 8 Gyr. Conclusions NGC 6791 shows clear evidence of tidal features in its star distribution in the form of irregular but evident elongations and tidal tails. These features are present also in the star density and surface brightness profile and they represent clear indication of recent mass-loss. By using the simple recipes we derived the initial mass of NGC6791 to be M ini = (1.5 − 4) × 10 5 M , i.e. several tens larger than its present day mass. This finding would qualitatively explain why the cluster could have survived for such a long time contrary to the expectations of current estimates of the destruction rate of Galactic open clusters. . A Bragaglia, C Sneden, E Carretta, R Gratton, S Lucatello, P Bernath, J S A Brooke, R S Ram, ApJ. 79668Bragaglia, A., Sneden, C., Carretta, E., Gratton, R., Lucatello, S., Bernath, P., Brooke, J.S.A., Ram, R.S. ApJ, 796, 68 . G Carraro, ASPC. 482245Carraro, G. 1995, ASPC, 482, 245 . E Dalessandro, P Miocchi, G Carraro, L Jilkova, A Moitinho, MNRAS. 4491811Dalessandro, E., Miocchi, P., Carraro, G., Jilkova, L., Moitinho, A. 2015, MNRAS, 449, 1811 . D Geisler, S Villanova, G Carraro, C Pilachowski, J Cummings, C I Johnson, F Bresolin, ApJ. 75640Geisler, D., Villanova, S., Carraro, G., Pilachowski, C., Cummings, J., Johnson, C.I., Bresolin, F. 2012, ApJ, 756, 40 . H Lamers, M Gieles, N Bastian, H Baumgartd, N Kharchenko, S Portegies Zwart, A&A. 44117Lamers, H., Gieles, M., Bastian, N., Baumgartd, H., Kharchenko, N., Portegies Zwart, S. 2005, A&A, 441, 17 Upper right: star counts . Lower left: King profile fitting. 1Upper left: CMD of NGC 6791 and nearby field. Lower right: Mass at birth estimateFigure 1. Upper left: CMD of NGC 6791 and nearby field. Upper right: star counts . Lower left: King profile fitting . Lower right: Mass at birth estimate.

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{'abstract': 'We present the first evidence of clear signatures of tidal distortions in the density distribution of the fascinating open cluster NGC 6791. We find that the 2D density map shows a clear elongation and an irregular distribution starting from ∼ 300 from the cluster center and two tails extending in opposite directions beyond the tidal radius. These features are aligned to both the absolute proper motion and to the Galactic centre directions. Accordingly we find that both the surface brightness and star count density profiles reveal a departure from a King model starting from ∼ 600 . These observational evidences suggest that NGC 6791 is currently undergoing mass-loss likely due to gravitational shocking and interactions with the tidal field of the Milky Way. We derive the expected mass-loss due to stellar evolution and tidal interactions and we estimate the initial cluster mass to be Mini = (1.5 − 4.0) × 10 5 M .', 'arxivid': '1511.00073', 'author': ['Giovanni Carraro gcarraro@eso.org \nESO\nAlondo de Cordova 310719001Santiago de ChileChile\n\nDept. of Physics and Astronomy\nUniversity of Padova\nVicolo Osservatorio 335122PadovaItaly\n', 'Emanuele Dalessandro emanuele.dalessandr2@unibo.it \nDept. of Physics and Astronomy\nUniversity of Bologna\nVia Ranzani 1, 40127BolognaItaly\n'], 'authoraffiliation': ['ESO\nAlondo de Cordova 310719001Santiago de ChileChile', 'Dept. of Physics and Astronomy\nUniversity of Padova\nVicolo Osservatorio 335122PadovaItaly', 'Dept. of Physics and Astronomy\nUniversity of Bologna\nVia Ranzani 1, 40127BolognaItaly'], 'corpusid': 118478504, 'doi': '10.1093/mnras/stv395', 'github_urls': [], 'n_tokens_mistral': 1631, 'n_tokens_neox': 1355, 'n_words': 847, 'pdfsha': 'ee1465d1e8919bd4dd342db8d16d75164a318b67', 'pdfurls': ['https://arxiv.org/pdf/1511.00073v1.pdf'], 'title': ['Formation, evolution, and survival of massive star clusters Proceedings IAU Symposium No. 316', 'Formation, evolution, and survival of massive star clusters Proceedings IAU Symposium No. 316'], 'venue': []}

arxiv

A relativistic mean field study of multi-strange system 18 Jun 2014 M Ikram Department of Physics Aligarh Muslim University Aligarh -202002India. S K Singh Institute of Physics Sachivalaya MargBhubaneswar -751005India A A Usmani Department of Physics Aligarh Muslim University Aligarh -202002India. S K Patra Institute of Physics Sachivalaya MargBhubaneswar -751005India A relativistic mean field study of multi-strange system 18 Jun 2014(Dated: June 19, 2014)arXiv:1406.4612v1 [nucl-th]PACS numbers: We study the binding energies, radii, single-particle energies, spin-orbit potential and density profile for multi-strange hypernuclei in the range of light mass to superheavy region within the relativistic mean field (RMF) theory. The stability of multi-strange hypernuclei as a function of introduced hyperons (Λ and Σ) is investigated. The neutron, lambda and sigma mean potentials are presented for light to superheavy hypernuclei. The inclusion of hyperons affects the nucleon, lambda and sigma spin-orbit potentials significantly. The bubble structure of nuclei and corresponding hypernuclei is studied. The nucleon and lambda halo structure are also investigated. A large class of bound multi-strange systems formed from the combination of nucleons and hyperons (n, p, Λ, Σ + and n, p, Λ, Σ − ) is suggested in the region of superheavy hypernuclei which might be stable against the strong decay. These multi-strange systems might be produced in heavy-ion reactions.PACS numbers: I. INTRODUCTION Hypernuclei provide an opportunity to extend our knowledge from normal nucleon-nucleon (NN) interaction to hyperon-nucleon (YN) and hyperon-hyperon (YY) interactions. Many of the single-and few doublelambda hypernuclei have been observed experimentally [1][2][3][4], which confirm the existence of S = -1 and -2 systems. Available experimental data is limited for S = -2 sector, and there is no further information for S ≥ -3 system. Due to complexity of YY scattering, the production of hypernuclei with strangeness beyond S > -2 is very difficult and not only this, ambiguities also exist in theoretical understanding. It is well known that the hyperon resides at the centre of the nucleus for most of the time, and only two hyperons with opposite spin can stay in sstate. Then further injected hyperons would be sat on p-state would have less binding in comparison to s-state and because of this the production of S ≥ -3 systems is difficult. It is obvious, with increasing the strength of strangeness, the hypernuclear physics becomes more complicated. Due to complication and importance of strangeness degree of freedom in bound as well as in infinite nuclear system, this subject has been draw an attention from last few decades . The system containing a variety of multi-strange baryons has a unique feature to extend the knowledge on hypernuclear chart with strangeness of S ≥ -3 dimension. Many of the theoretical calculations on multistrange hadronic system have been made which explain the changes and effects occur on bound nuclear system due to injection of Λ hyperon [34][35][36][37][38][39]45]. In early investigations within mean field, Rufa et al. suggested the * Electronic address: ikram@iopb.res.in † Electronic address: shailesh@iopb.res.in ‡ Electronic address: anisul@iucaa.ernet.in § Electronic address: patra@iopb.res.in stability of multi-lambda hypernuclei and also discussed the pure lambda matter and lambda droplets [45]. The calculations were performed by considering NN and ΛN interactions as a whole for describing the multi-strange system. Even though, they studied the multi-lambda hypernuclei but without including the YY interaction. And the work was limited for medium-heavy spherical nuclei. In present work, we make a complete study of multistrange hypernuclei within RMF formalism incorporating YY interaction over the periodic chart from light mass to superheavy nuclei by introducing Λ as well as Σ hyperons. Informations gathered from multi-strange systems are quite useful for studying or simulating the structure of highly dense astrophysical objects. In such a system, there is a possibility of existence of bunch of lambdas which is heavier than nucleons. Not only this, the existence of all variety of hyperons alongwith nucleons is also possible inside the core of neutron star at extreme conditions. In addition to lambda, the production of sigma hypernuclei is more difficult because of Σ−hyperon has repulsive nature in nuclear matter with potential depth of 30 MeV. But the production of 4 Σ He [46,47], reflects that the many of others Σ−hypernuclei might be produced. The Λ and Σ − are appeared at high density around ten times of normal nuclear matter densities at saturation [55]. On the other way, as well as to search the Λ− and Σ−hypernuclei separately, it would be more interesting to look for the bound state of Λ and Σ with nucleons, where ΛN − ΣN coupling will play an important role for binding mechanism. In this context, the array of stable objects composed of n, p, Λ, Ξ 0 , Ξ − baryons with very high strangeness content and small net charge has been investigated in many Refs. [40][41][42] within the relativistic mean field model. Not only this, pure hyperonic bound system involving Λ, Ξ 0 , Ξ − hyperons with A ≥ 6 has also been suggested [41]. In current study, we search the bound class of multi-strange system by considering Λ, Σ + and Σ − as basic participants in ad-dition to nucleons. The possibilities of bound states of hyperons and nucleons (n, p, Λ, Σ + and n, p, Λ, Σ − ) are discussed in this paper in the mass range of superheavy region. It is to expect that the attractive nature of hyperon-hyperon interaction allows to form the bound class of multi-strange system as well as pure hyperonic matter [40][41][42]48]. Various calculations have been performed to study the hyperonic system within the RMF with effective interactions [49][50][51][52][53]. The strong strength of attractive YY interaction leads to the formation of a system having nucleons and hyperons or pure hyperonic matter inclusion of all hyperons such as Λ, Σ 0 , Σ + , Σ − , Ξ − and Ξ 0 at lower densities [55]. It has to be mention that, the presence of hyperons makes the EOS as softer, and the inclusion of strong YY interaction leads to a further softening of EOS [55]. Incorporation of YY interaction has an impact to study the bound system including hyperons as well as infinite nuclear matter system. In present work, our motive is to analyze the bulk properties like, binding energies, radii, single-particle energies, spin-orbit potential and density profile for multi-Λ as well as multi-Σ hypernuclei by continuous injection of hyperons with replacing neutrons. The stability of multistrange system as a function of introduced hyperons from light mass to superheavy region is discussed. The neutron, lambda and sigma mean potentials are investigated for light to superheavy hypernuclei. Nucleon, lambda and sigma spin-orbit potentials are also displayed for different cases of injected hyperons. The bubble structure of nuclei and their disappearance by injection of Λ's is studied. On viewing the density profile, the nucleon and lambda halo nature are reported. The bound class of strange and nonstrange baryons (n, p, Λ, Σ + and n, p, Λ, Σ − ) in the mass range of superheavy hypernuclei is predicted which might be produced in heavy ion reactions. The paper is organized as follow: the formalism of RMF model including YY interaction is given in section 2. The results are displayed in section 3. Paper is summarized in section 4. II. FORMALISM The structural properties of nuclei are described within the framework of effective mean field interactions in relativistic and non-relativistic approach. RMF model takes care the spin-orbit interaction naturally and produces quite remarkable result over the whole periodic table including superheavy region [62][63][64][65][66][67][68][69]. However, the results produced by original Walecka model was enough qualitatively, but there had some modification by Boguta and Bodmer to match the results with experimental data in quantitative way [69,70]. This implies that, for a better understanding of nuclear structure studies, it is imperative to include all the possible interactions which affect all the physical observables are being to be calculated. To find the results in quantitative way, we include the σ * and φ mesons which simulate the hyperon-hyperon interaction. Both relativistic (RMF) and non-relativistic (SHF) mean field approaches have been played an interesting as well as successful role in order to explore the hypernuclear systems [37,45,56,57]. The extension towards multi-strange system has been well investigated within RMF however, no experimental data is available for such a high strange system of new kinds as discussed in Refs. [37,40,41,45]. The Lagrangian density for multi-strange hypernuclei is discussed in Refs. [40,58]. Here, we write the Lagrangian density for multi-strange hypernuclei as given below: L = L N + L Y + L Y Y , Y = Λ, Σ ,(1)L N =ψ i {iγ µ ∂ µ − M }ψ i + 1 2 (∂ µ σ∂ µ σ − m 2 σ σ 2 ) − 1 3 g 2 σ 3 − 1 4 g 3 σ 4 − g sψi ψ i σ − 1 4 Ω µν Ω µν + 1 2 m 2 ω V µ V µ − g ωψi γ µ ψ i V µ − 1 4 B µν B µν + 1 2 m 2 ρ R µ R µ − 1 4 F µν F µν − g ρψi γ µ τ ψ i R µ − eψ i γ µ (1 − τ 3i ) 2 ψ i A µ , L Y =ψ Y {iγ µ ∂ µ − m Y }ψ Y − g σYψY ψ Y σ − g ωYψY γ µ ψ Y V µ + L ρY + L AY , L Y Y = 1 2 (∂ µ σ * ∂ µ σ * − m 2 σ * σ * 2 ) + 1 2 m 2 φ φ µ φ µ − 1 4 S µν S µν − g σ * YψY ψ Y σ * − g φYψY γ µ ψ Y φ µ ,(2)L ρΛ + L AΛ = 0 ,(3) because of Λ is neutral and isoscalar L ρΣ + L AΣ =ψ Σ {g ρΣ γ µ τ Σ . R µ + e (1 − τ 3Σ ) 2 γ µ A µ }ψ Σ ,(4) here ψ and ψ Y denote the Dirac spinors for nucleon and hyperon, whose masses are M and m Y , respectively. The quantities m σ , m ω , m ρ are the masses for σ−, ω− ρ− mesons. The field for the σ−meson is denoted by σ, ω−meson by V µ , ρ−meson by R µ . The quantities g s , g ω , g ρ , and e 2 /4π=1/137 are the coupling constants for σ−, ω−, ρ− and photon fields, respectively. We have g 2 and g 3 self-interaction coupling constants for σ−mesons. The hyperon-meson coupling constant for strange and non-strange mesons are expressed by g σY , g ωY , g σ * Y and g φY . The field tensors of the vector, isovector mesons and of the electromagnetic field are given by Ω µν = ∂ µ V ν − ∂ ν V µ , B µν = ∂ µ R ν − ∂ ν R µ , F µν = ∂ µ A ν − ∂ ν A µ , S µν = ∂ µ φ ν − ∂ ν φ µ .(5) The classical variational principle is used to solve the Lagrangian and field equations for hypernuclei are obtained. The Dirac equation with potential terms for the nucleon is [−iα.∇ + β(M + S(r)) + V (r)]ψ i = ǫ i ψ i ,(6) where S(r) is the scalar potential of nucleon written as S(r) = g σ σ(r) ,(7) and V(r) represents the vector potential of nucleon given as V (r) = g ω V 0 (r) + g ρ τ 3 R 0 (r) + e (1 − τ 3 ) 2 A 0 (r) ,(8) where subscript i = n, p in wavefunction denotes the neutron and proton, respectively. The Dirac equation for Λ−hyperon is [−iα.∇ + β m Λ + S Λ (r) + V Λ (r)]ψ Λ = ǫ Λ ψ Λ ,(9) where S Λ (r) is the scalar potential of Λ−hyperon given as S Λ (r) = g σΛ σ(r) + g σ * Λ σ * (r) ,(10) and V Λ (r) represents the vector potential of Λ−hyperon written as V Λ (r) = g ωΛ V 0 (r) + g φΛ φ(r) .(11) The Dirac equation for Σ−hyperon is [−iα.∇ + β m Σ + S Σ (r) + V Σ (r)]ψ Σ = ǫ Σ ψ Σ ,(12) where S Σ (r) is the scalar potential of Σ−hyperon given as S Σ (r) = g σΣ σ(r) + g σ * Σ σ * (r) ,(13) and V Σ (r) represents the vector potential of Σ−hyperon written as V Σ (r) = g ωΣ V 0 (r)+g ρΣ τ 3Σ R 0 (r)+e (1 − τ 3Σ ) 2 A 0 (r)+g φΣ φ(r) .(14) The Klein-Gordon equations for mesons and Coulomb fields are {− △ +m 2 σ }σ(r) = −g σ ρ s (r) − g 2 σ 2 (r) − g 3 σ 3 (r) − g σY ρ Y s (r) , {− △ +m 2 σ * }σ * (r) = g σ * Y ρ Y s (r) , {− △ +m 2 ω }V 0 (r) = g ω ρ v (r) + g ωY ρ Y v (r) , {− △ +m 2 φ }φ 0 (r) = g φY ρ Y v (r) , {− △ +m 2 ρ }R 0 3 (r) = g ρ ρ 3 (r) + g ρ Y ρ Y 3 (r) , Y = Σonly , − △ A 0 (r) = eρ c (r) + eρ Y c (r) , Y = Σonly.(15) Here ρ s , ρ Y s and ρ v , ρ Y v are the scalar and vector density for σ− and ω−field in nuclear and hypernuclear system which are expressed as ρ s (r) = i=n,pψ i (r)ψ i (r) , ρ Y s (r) = Y =Λ,Σψ Y (r)ψ Y (r) , ρ v (r) = i=n,p ψ † i (r)ψ i (r) , ρ Y v (r) = Y =Λ,Σ ψ † Y (r)ψ Y (r) .(16) The vector density ρ 3 (r) for ρ-field and charge density ρ c (r) for photon field are expressed by ρ 3 (r) = i=n,p ψ † i (r)γ 0 τ 3i ψ i (r) , ρ Y 3 (r) = Y =Σ ψ † Y (r)γ 0 τ 3Y ψ Y (r) , Y = Σonly, ρ c (r) = i=n,p ψ † i (r)γ 0 (1 − τ 3i ) 2 ψ i (r) , ρ Y c (r) = Y =Σ ψ † Y (r)γ 0 (1 − τ 3Y ) 2 ψ Y (r) , Y = Σonly. (17) The various rms radii are defined as r 2 p = 1 Z r 2 p d 3 rρ p , r 2 n = 1 N r 2 n d 3 rρ n , r 2 m = 1 A r 2 m d 3 rρ , r 2 Λ = 1 Λ r 2 Λ d 3 rρ Λ , r 2 Σ = 1 Σ r 2 Σ d 3 rρ Σ ,(18) for proton, neutron, matter, lambda and sigma rms radii, respectively and ρ p , ρ n , ρ, ρ Λ and ρ Σ are their corresponding densities. The charge rms radius can be found from the proton rms radius using the relation r ch = r 2 p + 0.64 by taking into consideration the finite size of the proton. The total energy of the system is given by E total = E part (N, Y ) + E σ + E ω + E ρ + E σ * + E φ + E c + E pair + E c.m. ,(19) where E part (N, Y ) = E part (N, Λ, Σ) is the sum of the single particle energies of nucleons (N) and hyperons (Y=Λ, Σ). The other contributions E σ , E ω , E ρ , E σ * , E φ , E c , E pair and E cm are from meson fields, Coulomb field, pairing energy and the center-of-mass energy, respectively. For present study, we use NL3* nucleon parameter set through out the calculations [71], which produces a good description of nuclear matter as well as finite nuclei including superheavy region [64,65,68]. We adopt the relative σ and ω coupling to find the numerical values of hyperon-meson coupling constants. The ratio of meson-hyperon coupling to nucleon-meson coupling is defined by R σ = g σY /g s , R ω = g ωY /g ω , R σ * = g σ * Y /g s and R φ = g φY /g ω . The relative coupling R σ , R ω for Λ and Σ are adopted from Ref. [72]. For mesonhyperon couplings, the naive quark model values are used for vector coupling constants. To incorporate the hyperon-hyperon interaction into the calculation, the relative coupling R σ * , R φ are taken from Refs. [40,58,73]. Here, we consider the coupling strength of sigma-sigma interaction same as lambda-lambda interaction as like as used by Yang, Shen [74] and Miyazaki [75]. That is, g φΛ = g φΣ = − √ 2 3 g ω from naive quark model and g σ * Λ = g σ * Σ = 0.69 from Ref. [40]. In present calculations, to take care of pairing interaction the constant gap BCS approximation is used and the centre of mass correction is included by the formula E cm = −(3/4)41A −1/3 . III. RESULTS AND DISCUSSIONS A. Binding energies and radii The stability of multi-strange hypernuclei has been studied by introducing the lambda as well as sigma hyperon by replacing the neutrons. Total and single particle energies for single-Λ and Σ hypernuclei are listed in the Tables I, II, III. Total (r t ), charge (r ch ), neutron (r n ) and hyperon (r Y ) radii are also framed in these tables. The single particle energies for s-and p-states are compared with existing data for single−Λ hypernuclei. To check the stability of bound system with high strangeness in respect of injected hyperons, the binding energy per par- I: Total and single-particle (for s-and p-state) binding energies and radii are listed for single-Λ hypernuclei. The single-particle energies are compared with available experimental data. Experimental values are given in parenthesis [4,76]. ticle (BE/A) are plotted for light to superheavy hypernuclei as displayed in Figs. 1− 6. In case of light mass region the binding energies are enhanced by introducing one or two hyperons and further it goes to reduce. However, for heavy mass region, the BE/A increases with injection of large number of hyperons and form a more bound system than their normal counter parts, for example, inclusion of one lambda increases the binding of 16 nΛ O, the binding of 51 nΛ V increases up to the addition of 8 lambdas and the number of injected lambdas for superheavy hypernuclei, 304 nΛ 120, goes to 51. These numbers of lambda hyperons form a multi-strange bound system having maximum stability. Not only lambda, we also look for the stability of multi-sigma hypernuclei. In case of Σ + , the maximum stability comes forward in comparison to Λ and Σ − hypernuclei. Because it has a repulsive sigma potential as well as enhance the repulsive Coulomb potential due to its positive charge and as a result the binding of multi-Σ + hypernuclei is less. Due to attractive Coulomb potential between Σ − and proton, the maximum stability for multi-Σ − hypernuclei is extended. For example, injection of 51 Λ's provide the maximum stability for 304 nΛ 120 and 38 Σ's for 304 nΣ + 120 while for this nuclei the number of injected Σ − 's are 70 which produce the maximum stability. By reducing the number of neutrons of nuclei, the neutron radius gradually decreases. On contrary to this, and obviously the lambda, sigma radius increases with increasing the numbers of substituted hyperons (Λ, Σ + , Σ − ). In some cases, the hyperon radius drastically increases by addition of hyperons, for example, in 16 nY O, 40 nY Ca, 51 nY V, 72 nY Ni and so on. This behaviour of radii can be explained by internal shell structure by means of single particle energy levels. The total radius of the hypernuclei initially decreases and after certain limit it goes to increase. This behaviour of r t indicates that, by addition of hyperons the size of hypernucleus goes to shrink up to a certain limit and then extend the size. B. Density profile and single particle energies The nucleon distribution can be explained by density profile which has gross information about the structure of the nucleus. Many of the bulk properties like binding energies, radii, single particle energies and density profile are affected by continuous injection of hyperons. In this regard, we plot the lambda and nucleon density distribution for some light and superheavy hypernuclei as shown in Fig. 8. The nucleon and lambda density distributions are changed by addition of lambdas to normal nuclei. The magnitude of lambda density increases due to increasing number of lambdas. By viewing the density profile, one can examine the most interesting feature of nuclei i.e. bubble structure, which is the measure of depletion of central density. Anomalous behavior of density distribution is observed for bubble nucleus. It shows a dip at the center and a hump nearby to it following a slow decreasing in density to zero at the surface. Some of the interesting examples of bubble nuclei in superheavy region are 286 114, 292 120, 304 120 as reported in Ref. [77]. The existence of bubble structure other than the spherical was first suggested by Wheeler [78] and extensively studied by Wilson [79] however, later by Siemens and Bethe [80]. The explanation on occurrence of bubble nuclei have been made using several models like, independent particle and Hartree-Fock Model [81,82]. One of the interesting thing in this context is, it is not con-fined to a particular region but have the possibility for light mass to superheavy region. One may expect that, the mechanism behind the formation of bubble structure is the depopulation of s levels and as a result due to less bound lower s levels the radius increases and subsequently, central part of density decreases [83,84]. The bubble and semi-bubble structure for superheavy and hyperheavy mass region have been reported in Refs. [83,85]. Not only depopulation of s-levels is responsible to make the hollow of central region but this may also be interpreted by s-d orbital inversion as discussed by Zhao et al. and E Khan et al. [86,87]. In quantitative way, the amount of bubble effect can be measured by calculating depletion fraction (DF) using the relation [86,87]; (DF ) α = (ρ max ) α − (ρ cen ) α (ρ max ) α × 100, where ρ max , ρ cen represent the maximum and central density, respectively and α denotes the neutron and proton. Not only syperheavy but some medium-heavy nuclei also have a good amount of depletion fraction as tabulated in Table IV. The depletion of central density is decreased or completely reduced by injection of hyperons to normal nuclei as shown in Fig. 7 fects the DF partially or completely and as a result the amount of DF becomes very small or zero. In the same time, the magnitude of nucleon density decreases because of decreasing number of neutrons. For example, for 46 Ar, the DF is 36 and this amount becomes zero by injection of 2 or more lambdas to the core nucleus. This happens because of lambda particle resides at the centre of the nucleus and attracts the surrounding nucleon towards the centre and as a result central density becomes high and the hollow part of the centre is filled by partially or completely. This is one of the most important implication of hyperon to nuclei for removing the bubble nature of nuclei. The other prospects of density profile is to analyze the halo nature of nucleon and lambda. It is one of the interesting character of the nuclei which makes differ from the normal nuclei. The halo nuclei has slowly decaying exponential tails extending beyond the size of the nucleus. To examine the halo nature of nucleon and Λ hyperon, we plot the density in logarithm scale. The halo nature is identified by wide space extension of density distribution. It is evident from Fig. 9 nΣ + 117, 304 nΣ + 120 and 286 nΣ − 114, 293 nΣ − 117, 304 nΣ − 120 as a function of substituted hyperons (Λ, Σ + and Σ − ) with replacing neutrons. Solid lines with red, black and blue color represent the 1s 1/2 level for neutron, proton and hyperon, respectively. The higher neutron (2f 7/2 ), proton (1h 11/2 ) and hyperon (1p 3/2 ) levels are represented by dashed line with red, black and blue color, respectively. Fig. 9. In general, the addition of excessive lambdas corresponding to nuclei exhibit the neutron and lambda halo nature. The reason is simple, the majority of lambda hyperons push out the nucleon towards periphery and formed nucleon halo. The successive ad- dition of hyperons by replacing the neutrons provide the deep binding of neutrons due to the symmetry energy. V Λ so , V Σ + so , V Σ − so ) Because of successive addition of Λ to the nuclei the Λ separation energy reduces and weak binding of hyperon levels leads to the halo nature of hyperon as discussed in Ref. [88]. The Λ density is found to be maximum near the center from where it pushes the nucleons towards the low density regions both at the periphery and at the center. And due to this, the bubble structure is disappeared partially or completely and also Λ and nucleon halo structure is seen. Any kinds of change in a system is directly reflected from the single particle energy levels. To analyze the impact of hyperon on single particle energy levels, we plot the first and some higher filled neutron, proton and hyperon (Λ, Σ + , Σ − ) levels as a function of substituted hyperons for superheavy region. In case of lambda hypernuclei, the first filled neutron level goes deeper and first proton level also feels the attraction. By decreasing the number of neutrons (injection of hyperons) the impact of Coulomb repulsion becomes higher and the upper proton level goes to unbound. The deep binding of neutron level is because of decresing in symmetry energy due to substitution of neutron by Λ as discussed in Ref. [89]. For Σ + case, the attraction is less and Coulomb repulsion becomes high enough because of positive charge of Σ + hyperon. On the other hand, due to Σ − hyperons the proton as well as neutron levels feel more bound. Here in this case, the proton levels are bound enough than their neutron levels as shown in Fig. 10. The lambda energy levels feel to be constant or mild attractive with the function of injected lambdas. On contrary to this, the sigma levels go towards less binding which is clearly reflected from Fig. 10. C. Spin-orbit and mean potentials In order to investigate the structural properties of nuclei, the spin-orbit interaction plays a significant role to produce the results in quantitative way. It is the beauty of RMF in which the spin-orbit splitting develops nat-urally by the exchange of scalar and vector mesons and this is not limited only for nuclei but exists in hypernuclei also [90][91][92]. However, the spin-orbit potential in hypernuclei is weaker than their normal nuclear case as demonstrated in Refs. [73,93,94]. It is clearly shown in Figs. 11, 12 that the spin-orbit potential for hyperon is weaker than their normal counter parts and these results are consistent with existing predictions [73,93,94]. In this work, we study the spin-orbit potential for nucleons (V N so ) as well as hyperons (V Λ so , V Σ + so , V Σ − so ) of hypernuclei for different cases of hyperons. The effect of large number of injected hyperons on spin-orbit potentials is significantly investigated and plotted in Figs. 11, 12 for medium to superheavy multi-strange hypernuclei. The neutron (V N ), lambda (V Λ ) and sigma (V + Σ , V − Σ ) mean potentials are also investigated and plotted in Fig. 13 for light to superheavy hypernuclei. The mean potential depth of lambda and sigma is found to be 30 MeV, which is in agreement of existing calculations [73]. The neutron potential depth for light hypernuclei lies approx 80 MeV, which reduces to around 64 MeV for superheavy hypernuclei. The shape of hyperon potential looks like to be same as neutron potential but only the amount of depth is different. It is to be notice that the neutron potential looks like as V-shape type and shows the maximum depth around 78 MeV at r=4 fm, while this amount of depth is reduced to around 65 MeV at r=0 fm. This is an indication of relatively low concentration of the particles at central region (r=0) which is the direct consequence of formation of bubble structure. D. Possible bound states of multi-strange systems The extension towards the systems of large strangeness has firstly been investigated in Refs. [40][41][42]45]. Some of the theoretical calculations have been made and suggested the bound class of objects composed of neutrons, protons, Λ's and Ξ's for light-medium nuclei [40,41]. In this context, we deal the multi-strange system with other heavy hyperon. The systems of high strangeness developed by addition of Λ's have more bound nature than their normal nuclear counter parts. As we have seen in Figs. 1−6, where the BE/A increases for a certain limit of lambda particle and shows the maximum stability for a particular system. Here, we try to suggest the stable systems having more heavier hyperons including lambdas for example, Σ + and Σ − . In this connection, we examine some combinations of n, p, Λ, Σ + and n, p, Λ, Σ − in the range of superheavy nuclei by two ways. In first one, we add the hyperons with replacing neutrons by staying the mass number as constant as shown in upper part of Fig. 14 while in second one, we add the hyperons in ordinary nuclei with increasing mass number and as a result the bound system with high strangeness is suggested. We frame the bound state in such a way that after getting the maximum stability by addition of Λ's, the Σ + and Σ − hyperons are injected which further increase the binding of the system as shown in Fig. 14. The upper portion of Fig. 14 indicates that the more deeply bound than systems of same A are presented with replacing the neutrons by Λ's and Σ's. The following possible combinations are presented as n Λ =46, with n Σ + =2, 8, 10, 14 and n Σ − =2, 6, 8, 10 by maintaining A=286 for 286 nY 114. For 304 nY 120 hypernuclei with maintaining the mass number A=304, the possible combinations are n Λ =51, with n Σ + =2, 6, 8, 10, 11 and n Σ − =2, 4, 8, 14. In other way, the possible combinations of nucleons and hyperons by adding the hyperons with increasing the mass number for 286 114+Y are given as follow; n Λ =90, with n Σ + =2, 4, 6, 8, 10 and n Σ − =2, 4, 6, 8, 14. In the same way for 304 120+Y, the combinations are given as n Λ =90, with n Σ + =2, 4,6,8,10,14,16,18,20,22 and n Σ − =2, 6, 8, 10, 12 which have increasing binding for a particular system. IV. SUMMARY AND CONCLUSIONS It is really true that, the RMF produces quite excellent result not only for normal nuclei but hypernuclei also. We demonstrate the various physical properties of hypernuclei within the RMF and also see the effects of successive addition of hyperons to nuclear bound system. In which, we expose how does the binding energies, radii, density and single particle energies are affected by continuous injection of hyperons (Λ, Σ + , Σ − ). In this paper, we study the bulk properties as well as check the stability of hypernuclei within the RMF for a wide spectrum from light mass to superheavy region. We investigate the binding energies, radii and single particle energies as a function of successive added Λ's, Σ + 's and Σ − 's. The stability of multi-strange hypernuclei is investigated as a function of added hyperons from light mass to superheavy region. A variation in achieving the maximum stability for a particular system is observed for injection of different kinds of hyperons. The study of bubble structure of the nuclei and the disappearance of bubble nature by addition of hyperons to normal nuclei is presented. The amount of depletion fraction of the nuclei is reduced by successive addition of hyperons. Removing the bubble structure of nuclei by injection of lambdas is an important implication of strange baryons to ordinary nuclei. The nucleon and hyperon halo structure is also investigated. The study of single-particle energy levels of Λ, Σ + and Σ − hypernuclei is presented. It is obsrved that the inclusion of hyperons affects the nucleon and hyperon spin-orbit potential significantly. The neutron and hyperon (Λ, Σ + , Σ − ) mean potentials are also displayed. The bound class of n, p, Λ, Σ + and n, p, Λ, Σ − are suggested within the RMF. The addition of lambda obviously increase the stability of the system but further addition of sigma hyperons leads to more stable system with increasing binding. The combinations of hyperons with nucleons for a particular system are suggested which will form the bound system with high strangeness and might be produced in heavy-ion reactions. The investi-gation on hypernuclei with high strangeness is quite welcome because these type of system might be produced in high-energy heavy ion reactions near the future. These types of study of hypernuclei with multiple strangeness will provide us a basic input for neutron as well as hyperon star studies, which are the body of current interest. FIG. 1 : 1Energy per particle is shown as a function of substituted hyperons (Λ, Σ + , Σ − ) for 16 nY N, 16 nY O and 27 nY Al, where Y indicates the injected hyperons (Λ, Σ + , Σ − ). Total (rt), charge (r ch ), neutron (rn) and hyperon (rY ) radii are also displayed as a function of substituted hyperons. Total, charge, neutron and hyperon radii are represented by solid, dashed, dot-dashed and long dashed lines with black, red, green and blue colors, respectively. FIG. 2 : 2same as Fig. 1 but for 28 nY Si, 51 nY V and 40 nY Ca. FIG. 3 : 3same as Fig. 1 but for 72 nY Ni, 89 nY Y and 90 nY Zr. FIG. 4 : 4same as Fig. 1 but for 132 nY Sn, 139 nY La and 208 nY Pb. FIG. 5 : 5same as Fig. 1 but for 286 nY 114, 294 nY 118 and 292 nY 120. FIG. 6 : 6same as Fig. 1 but for 298 nY 114, 293 nY 117 and 304 nY 120. FIG. 9 : 9The nucleon and lambda densities of Figure 8 are plotted in logarithm scale to analyze the nucleon and lambda halo nature for 286 nΛ 114, 298 nΛ 114, 293 nΛ 117, 294 nΛ 118, 292 nΛ 120, 304 nΛ 120, 16 nΛ O, 28 nΛ Si and 90 nΛ Zr. In the same way as Fig 8, the nucleon and lambda densities are represented by solid and dashed lines, respectively. FIG. 10 : 10The first occupied and higher orbits of neutron and proton are shown for 286 nΛ 114, 293 nΛ 117, 304 nΛ 120 and 286 nΣ + 114, 293 FIG. 11 : 11Radial dependence of spin-orbit potential for nucleon (V N so ) and hyperons ( FIG. 14 : 14Binding energy per baryon (BE/A) as a function of substituted hyperons for 286 nY 114 and 304 nY 120 is displayed in upper portion of the figure. Binding energy per baryon (BE/A) as a function of mass number A for 286 114+Y and 304 120+Y, where Y=Λ+Σ + , Λ+Σ − is presented in lower part of the figure. The solid circles in black color represent the binding for lambda, while the solid triangles in red color denote the binding for added Σ + 's and the stars in blue color represent the binding by added Σ − 's. TABLE TABLE II : IITotal and single-particle (for s-and p-state) binding energies and radii are listed for single-Σ hypernuclei.Hypernuclei BE B s Σ + B p Σ + r ch rt rp rn r Σ + 16 Σ + N -130.7513 -13.631 -4.0893 2.597 2.465 2.478 2.473 2.300 16 Σ + O -124.8292 -11.449 -2.4888 2.706 2.488 2.586 2.391 2.363 27 Σ + Al -227.6015 -18.490 -8.4594 2.978 2.814 2.872 2.789 2.337 28 Σ + Si -234.9222 -18.395 -8.2107 3.017 2.831 2.910 2.779 2.327 32 Σ + S -272.8755 -20.031 -9.6561 3.187 2.988 3.086 2.921 2.318 40 Σ + Ca -346.4849 -19.008 -10.769 3.457 3.288 3.365 3.239 2.594 48 Σ + Ca -430.6025 -23.637 -16.267 3.449 3.451 3.364 3.536 2.757 51 Σ + V -457.6058 -23.251 -15.742 3.561 3.488 3.475 3.523 2.746 56 Σ + Fe -503.1424 -24.068 -16.321 3.656 3.570 3.572 3.594 2.724 72 Σ + Ni -636.0653 -28.154 -21.217 3.866 3.962 3.793 4.089 2.883 89 Σ + Y -792.5601 -26.508 -20.686 4.226 4.202 4.155 4.258 3.148 90 Σ + Zr -799.4227 -26.162 -20.519 4.253 4.217 4.182 4.263 3.192 132 Σ + Sn -1128.721 -31.481 -26.615 4.683 4.823 4.624 4.955 3.455 139 Σ + La -1191.296 -29.877 -25.234 4.841 4.891 4.781 4.981 3.536 208 Σ + Pb -1665.060 -32.310 -28.540 5.494 5.598 5.443 5.709 3.932 286 Σ + 114 -2075.488 -32.596 -29.719 6.211 6.275 6.165 6.356 4.471 298 Σ + 114 -2148.451 -34.004 -30.666 6.245 6.375 6.200 6.491 4.195 293 Σ + 117 -2110.368 -32.711 -29.779 6.250 6.320 6.205 6.405 4.441 294 Σ + 118 -2112.770 -32.602 -29.695 6.259 6.324 6.214 6.407 4.459 292 Σ + 120 -2087.232 -31.939 -29.210 6.262 6.301 6.217 6.369 4.577 304 Σ + 120 -2168.554 -33.324 -30.155 6.299 6.397 6.254 6.498 4.301 TABLE III : IIITotal and single-particle (for s-and p-state) binding energies and radii are listed for single-Σ hypernuclei. Hypernuclei BE B s Σ − B p Σ − r ch rt rp rn r Σ − 16 Σ − N -131.8578 -14.551 -5.1063 2.524 2.469 2.401 2.545 2.321 16 Σ − O -131.0975 -17.268 -7.2379 2.618 2.460 2.493 2.450 2.259 27 Σ − Al -233.6156 -24.192 -13.646 2.918 2.803 2.810 2.831 2.293 28 Σ − Si -242.9184 -26.117 -15.343 2.954 2.813 2.845 2.815 2.279 32 Σ − S -281.4468 -28.488 -16.885 3.118 2.968 3.015 2.963 2.211 40 Σ − Ca -355.4514 -27.717 -19.053 3.405 3.277 3.311 3.274 2.546 48 Σ − Ca -433.1445 -25.891 -17.289 3.412 3.452 3.326 3.570 2.581 51 Σ − V -463.4586 -28.799 -20.399 3.523 3.484 3.437 3.552 2.621 56 Σ − Fe -510.6156 -31.242 -23.060 3.619 3.566 3.534 3.621 2.667 72 Σ − Ni -637.4301 -27.808 -21.023 3.836 3.970 3.762 4.120 2.922 89 Σ − Y -798.7386 -34.723 -27.787 4.196 4.198 4.125 4.277 2.919 90 Σ − Zr -806.6424 -35.459 -28.481 4.223 4.212 4.152 4.282 2.912 132 Σ − Sn -1129.498 -34.684 -29.323 4.661 4.825 4.602 4.973 3.319 139 Σ − La -1195.729 -37.227 -32.361 4.820 4.891 4.760 4.995 3.474 208 Σ − Pb -1679.528 -42.618 -38.066 5.475 5.596 5.423 5.720 3.626 286 Σ − 114 -2096.123 -47.418 -43.950 6.194 6.271 6.148 6.362 4.130 298 Σ − 114 -2166.315 -46.453 -43.118 6.228 6.374 6.183 6.499 4.208 293 Σ − 117 -2120.401 -48.304 -44.889 6.233 6.316 6.188 6.411 4.161 294 Σ − 118 -2123.234 -48.646 -45.247 6.242 6.320 6.197 6.413 4.168 292 Σ − 120 -2099.320 -49.699 -46.318 6.246 6.296 6.200 6.373 4.112 304 Σ − 120 -2189.437 -48.468 -45.219 6.282 6.395 6.237 6.505 4.250 TABLE IV : IVTotal depletion fraction (D.F.) (in %) as a measure of depletion of central density for nucleon distribution is listed for some selected bubble nuclei and their corresponding multi-strange hypernuclei. Nuclei D.F. Hypernuclei D.F. Hypernuclei D.F. Hypernuclei D.F. Hypernuclei D.F. Hypernuclei D.F.16 O 15.11 16 1Λ O 4.81 16 5Λ O 0.0 16 6Λ O 0.0 16 7Λ O 0.0 16 8Λ O 0.0 22 O 26.22 22 1Λ O 23.37 22 2Λ O 17.94 22 4Λ O 12.63 22 5Λ O 8.42 22 12Λ O 0.0 28 Si 41.45 28 1Λ Si 38.59 28 2Λ Si 33.75 28 4Λ Si 25.77 28 6Λ Si 18.35 28 7Λ Si 0.0 34 Si 15.28 34 1Λ Si 12.57 34 2Λ Si 9.82 34 6Λ Si 47.67 34 8Λ Si 49.98 34 18Λ Si 0.0 46 Ar 36.03 46 1Λ Ar 0.0 46 2Λ Ar 0.0 46 4Λ Ar 0.0 46 6Λ Ar 0.0 46 26Λ Ar 0.0 90 Zr 19.29 90 5Λ Zr 18.75 90 10Λ Zr 21.03 90 15Λ Zr 27.33 90 20Λ Zr 1.62 90 25Λ Zr 2.28 138 Ce 44.91 138 10Λ Ce 16.02 138 20Λ Ce 38.35 138 30Λ Ce 31.66 138 40Λ Ce 10.73 138 70Λ Ce 0.0 200 Hg 42.35 200 10Λ Hg 34.42 200 30Λ Hg 23.47 200 50Λ Hg 54.0 200 70Λ Hg 15.85 200 90Λ Hg 5.08 206 Hg 34.57 206 10Λ Hg 37.05 206 30Λ Hg 28.56 206 50Λ Hg 52.69 206 70Λ Hg 51.14 206 90Λ Hg 0.0 286 114 24.62 286 10Λ 114 20.57 286 20Λ 114 16.09 286 30Λ 114 11.99 286 40Λ 114 8.54 286 50Λ 114 4.19 298 114 0.0 298 10Λ 114 22.25 298 20Λ 114 22.96 298 30Λ 114 18.36 298 40Λ 114 13.04 298 50Λ 114 10.52 293 117 25.17 293 10Λ 117 22.44 293 20Λ 117 18.35 293 30Λ 117 15.83 293 40Λ 117 10.21 293 50Λ 117 9.66 294 118 25.49 294 10Λ 118 22.93 294 20Λ 118 18.18 294 30Λ 118 16.02 294 40Λ 118 10.38 294 50Λ 118 10.70 292 120 30.85 292 10Λ 120 23.59 292 20Λ 120 19.96 292 30Λ 120 14.74 292 40Λ 120 10.96 292 50Λ 120 9.78 304 120 0.89 304 10Λ 120 28.37 304 20Λ 120 21.59 304 30Λ 120 20.40 304 40Λ 120 14.14 304 50Λ 120 7.54 pernucleus as given in Table IV. In the same way, 286 114 has a big amount of DF as 24.62 and this value reduced to 4.19 by injecting 50 lambdas to 286 114 with replacing neutrons. In this way, we can say that the addition of lambdas to nuclei has the ability to remove the bubble structure partially or fully which is appeared in normal nuclei. The effect of lambdas on density profile by means of effect on depletion fraction are shown in Figs. 7, 8. Some interesting bubble nuclei like as 22 O, 34 Si, 46 Ar, 138 Ce, 200 Hg and 206 Hg are considered as to reveal the effect of lambdas on density profile. It is evident from Figs. 7, 8 and Table IV that the injection of lambdas af-, 8. For the case of 16 O, the depletion fraction is 15.11, and this amount reaches to 4.81 for 16 Λ O and ultimately becomes zero for 16 5Λ O hy- ACKNOWLEDGMENTSOne of the author (MI) would like to acknowledge the hospitality provided by Institute of Physics, Bhubaneswar during the work. . M Denysz, Nucl. Phys. 49121M. Denysz et. al. Nucl. 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{'abstract': 'We study the binding energies, radii, single-particle energies, spin-orbit potential and density profile for multi-strange hypernuclei in the range of light mass to superheavy region within the relativistic mean field (RMF) theory. The stability of multi-strange hypernuclei as a function of introduced hyperons (Λ and Σ) is investigated. The neutron, lambda and sigma mean potentials are presented for light to superheavy hypernuclei. The inclusion of hyperons affects the nucleon, lambda and sigma spin-orbit potentials significantly. The bubble structure of nuclei and corresponding hypernuclei is studied. The nucleon and lambda halo structure are also investigated. A large class of bound multi-strange systems formed from the combination of nucleons and hyperons (n, p, Λ, Σ + and n, p, Λ, Σ − ) is suggested in the region of superheavy hypernuclei which might be stable against the strong decay. These multi-strange systems might be produced in heavy-ion reactions.PACS numbers:', 'arxivid': '1406.4612', 'author': ['M Ikram \nDepartment of Physics\nAligarh Muslim University\nAligarh -202002India.\n', 'S K Singh \nInstitute of Physics\nSachivalaya MargBhubaneswar -751005India\n', 'A A Usmani \nDepartment of Physics\nAligarh Muslim University\nAligarh -202002India.\n', 'S K Patra \nInstitute of Physics\nSachivalaya MargBhubaneswar -751005India\n'], 'authoraffiliation': ['Department of Physics\nAligarh Muslim University\nAligarh -202002India.', 'Institute of Physics\nSachivalaya MargBhubaneswar -751005India', 'Department of Physics\nAligarh Muslim University\nAligarh -202002India.', 'Institute of Physics\nSachivalaya MargBhubaneswar -751005India'], 'corpusid': 119205587, 'doi': '10.1142/s0218301314500529', 'github_urls': [], 'n_tokens_mistral': 22547, 'n_tokens_neox': 18074, 'n_words': 9309, 'pdfsha': '78b5077c2706a67869e48021bdb9125ef3a1df90', 'pdfurls': ['https://arxiv.org/pdf/1406.4612v1.pdf'], 'title': ['A relativistic mean field study of multi-strange system', 'A relativistic mean field study of multi-strange system'], 'venue': []}

arxiv

First-principles study of spin orbit coupling contribution to anisotropic magnetic interaction (Dated: December 29, 2022) Di Wang National Laboratory of Solid State Microstructures and School of Physics Nanjing University 210093NanjingChina Collaborative Innovation Center of Advanced Microstructures Nanjing University 210093NanjingChina Xiangyan Bo Nanjing University of Posts and Telecommunications 210023NanjingChina Feng Tang National Laboratory of Solid State Microstructures and School of Physics Nanjing University 210093NanjingChina Collaborative Innovation Center of Advanced Microstructures Nanjing University 210093NanjingChina Xiangang Wan National Laboratory of Solid State Microstructures and School of Physics Nanjing University 210093NanjingChina Collaborative Innovation Center of Advanced Microstructures Nanjing University 210093NanjingChina First-principles study of spin orbit coupling contribution to anisotropic magnetic interaction (Dated: December 29, 2022) Anisotropic magnetic exchange interactions lead to a surprisingly rich variety of the magnetic properties. Considering the spin orbit coupling (SOC) as perturbation, we extract the general expression of a bilinear spin Hamiltonian, including isotropic exchange interaction, antisymmetric Dzyaloshinskii-Moriya (DM) interaction and symmetric Γ term. Though it is commonly believed that the magnitude of the DM and Γ interaction correspond to the first and second order of SOC strength λ respectively, we clarify that the term proportional to λ 2 also has contribution to DM interaction. Based on combining magnetic force theorem and linear-response approach, we have presented the method of calculating anisotropic magnetic interactions, which now has been implemented in the open source software WienJ. Furthermore, we introduce another method which could calculate the first and second order SOC contribution to the DM interaction separately, and overcome some shortcomings of previous methods. Our methods are successfully applied to several typical weak ferromagnets for 3d, 4d and 5d transition metal oxides. We also predict the conditions where the DM interactions proportional to λ are symmetrically forbidden while the DM interactions proportional to λ 2 are nonzero, and believe that it is widespread in certain magnetic materials. arXiv:2212.13963v1 [cond-mat.mtrl-sci] I. INTRODUCTION Magnetic properties can be typically described by a quadratic spin Hamiltonian, which is the basis of most magnetic theoretical investigations [1][2][3]. Generally, spin-orbit coupling (SOC) always exists and leads to the anisotropic magnetic interactions with low symmetry. The general form of the bilinear expression of a spin exchange Hamiltonian could be written as H = i<j J ij S i · S j + i<j D ij · [S i × S j ] + i<j S i · Γ ij · S j(1) where the first term describes the isotropic Heisenberg Hamiltonian, the second one represents the Dzyaloshinskii-Moriya (DM) [4][5][6] interaction, and the third one is marked as Γ term [5]. The antisymmetric DM interaction, which comes from the combination of low symmetry and SOC, is introduced by Dzyaloshinskii [4] and Moriya [5] in a phenomenological model and a microscopic model respectively. It is commonly believed that DM and Γ term are contributed by first and second order of SOC respectively [1,5]. Generally, DM interaction favors twisted spin structures and is constrained by the crystal symmetry. For example, when a inversion center located at the bond center of two magnetic ion sites, the DM interaction between these two magnetic ions should be zero due to its antisymmetric property [5,6]. Now the DM interaction is invoked to explain numerous interesting magnetic systems featuring non-collinear spin * The corresponding author: xgwan@nju.edu.cn. textures, such as weak ferromagnets [4,5], helimagnets [7], skyrmion formation [8][9][10] and chiral domain walls [11,12]. In addition, the DM interaction also plays an important role in multiferroic materials [13][14][15][16][17], topological magnon materials [18][19][20] and spintronics [21]. It is worth mentioning that, since DM interactions are very sensitive to small atomic displacements and symmetry restrictions, it can also be used to reveal the interplay of delicate structural distortions and complex magnetic configurations [22]. Recently, the first-principles study of magnetic exchange interactions especially DM interaction, has also attracted much interest [3,. A popular numerical method is the energy-mapping analysis [3,23,24] to estimate magnetic interactions from the energy differences of various magnetic structures. However, this approach becomes inconvenient for the complicated systems where it is not clear how many exchange interactions needs to be considered, since in some magnetic compounds the magnetic moments may couple over a variety of distances, and even the ninth-nearest-neighbor coupling plays an important role [25,26]. Meanwhile, in itinerant magnetic systems, the magnetism is not so localized and the calculated magnetic moments may depend on the magnetic configurations, which also significantly affects the accuracy of the calculated DM interactions. Another approach using total energy differences could extract DM strength by directly calculating the energies of spirals with the finite vector q [27][28][29][30][31][32]. Meanwhile, an efficient approach is proposed based on the magnetic force theorem [33][34][35][36][37][38][39][40][41][42][43][44][45][46]. Katsnelson et al. [35] have derived the expression for DM interaction term based on Green's function approach. This method is applied to a large number of magnetic materials such as the antiferromagnets with weak ferromagnetism [36], thin magnetic films [38], diluted magnetic semiconductors [39] and various other magnetic materials [41][42][43][44][45]. This Green's function approach was previously formulated in first-principles codes with direct definition of a localized orbitals basis set such as linear muffin-tin orbitals method [51]. Furthermore, they have also developed the method of calculating DM interactions using Wannier function formalism [37,40]. In addition, DM interactions could also be estimated by computing the long-wave length limit of the spin susceptibility [47], the expectation value of the spin current density [48,49], or utilizing Berry phase [50]. In this paper, up to the second-order perturbation of SOC, we revisit the general expression of anisotropic magnetic interactions. We clarify that the second order term of SOC has contribution to both the antisymmetric DM interaction and symmetric Γ interaction, and reveal that their distinction arises from different hopping processes as shown in Fig. 1 and following. Note that in these approaches using total energy differences, such as energy-mapping [3,23,24], spirals approach [28,29] and the approaches through calculating energy variations due to spin rotations [35,52,53], can get the entire DM interaction but do not distinguish the contribution from first or second order of SOC. On the other hand, one can only consider DM interactions with the first order of SOC in their perturbation schemes [36,49]. We extend the method of calculating Heisenberg interactions based on combining magnetic force theorem and linearresponse approach [25,33,[52][53][54][55] to estimate DM and Γ interactions, and the algorithm of our proposed method is now implemented in the open source called WienJ [56], as an interface to the linearized augmented plane wave (LAPW) software Wien2k [57]. Furthermore, to overcome some shortcomings of previous methods, we develop a new method that can estimate the first and second order SOC contribution to the DM exchange couplings separately. While our methods can also calculate Γ interaction, here we only present the results of Heisenberg and DM interactions since they could be compared with many previous works. We have applied our methods to several representatives of canted antiferromagnetic materials La 2 CuO 4 , Ca 2 RuO 4 and Ca 3 LiOsO 6 for 3d, 4d and 5d transition metal oxides, and the calculation results are consistent with the experiment. Particularly, we find that the DM interaction proportional to λ 2 can not be ignored in 4d transition metal oxide Ca 2 RuO 4 , and the DM interactions proportional to λ and λ 2 have the same magnitude in 5d transition metal oxide Ca 3 LiOsO 6 . As shown in the following, the DM interactions proportional to λ and λ 2 involve different exchange channels. Thus, based on the symmetry analysis, we explore the possibility that the DM interactions proportional to λ are symmetrically forbidden while the DM interactions proportional to λ 2 still exist. We believe that this case is widespread in certain magnetic materials, and our method would play more important role in these magnetic systems. II. METHOD A. Anisotropic magnetic interactions by perturbation theory We start from an effective model: H = H 0 + H t + H U + H soc = iασ ε α c + iασ c iασ + ijαβσ t ij αβ c + iασ c jβσ +U i n i,↑ n i,↓ + i λl i · s i(2) where H 0 , H t , H U and H soc represent the on-site orbital energy, the hopping term, the Hubbard U term and the SOC term, respectively. Here i, j represent the site index, while α, β represent the orbital index and σ represents the spin index. We consider the spin exchange interaction between the magnetic ions located at site A and site B. We label the ground state and the unoccupied states at site A as n and m, respectively. Similarly, the ground state and the excited states at site B are labeled as n and m , respectively. When SOC is not considered, the Heisenberg interactions H ef f = JS A · S B can be obtained by considering the hopping term as perturbations for the case of U >> t [58]. Considering the SOC term λl · s as perturbation, the first-order SOC contribution to effective spin model H (1) ef f has the expression of antisymmetric DM interaction as H (1) (1) could be written as [5] ef f = D (1) (S A × S B ) where D(D α ) (1) = −4i λt nn U m l α mn ε m − ε n t mn − m l α m n ε m − ε n t m n (3) Meanwhile, by considering perturbation theory up to the second-order SOC correction (see details in Appendix A), we find that the second-order SOC correction has the contribution to both DM term D (2) and the Γ term, where D (2) could be written as (D α ) (2) = 2 λ 2 t nn U m,m l β m n l γ mn − l γ m n l β mn (ε m − ε n ) (ε m − ε n ) t mm −2 λ 2 t nn U m1,m2 l β m1m2 l γ m2n − l γ m1m2 l β m2n (ε m1 − ε n ) (ε m2 − ε n ) t m1n +2 λ 2 t nn U m 1 ,m 2 l β m 1 m 2 l γ m 2 n − l γ m 1 m 2 l β m 2 n ε m 1 − ε n ε m 2 − ε n t m 1 n(4) Meanwhile, the expression of parameter Γ (2) could be seen in Eq. (12) of Appendix. (c). Meanwhile, the perturbation processes for the symmetric Γ term are also shown in (d) and (e) for comparison. Here GS and ES represent ground state and excited state respectively. It is worth mentioning that, the perturbation processes of DM interactions involve the hopping between GSs, which denoted by the green solid line. Meanwhile, Γ terms would only involve the hopping processes between GS and ES. Here we present schematic pictures of the exchange processes in Fig. 1. It is easy to see that, the bilinear spin exchange Hamiltonian should contain two hopping processes between two sites A and B as shown in Fig. 1. Considering up to the second-order perturbation of SOC, we find that there are several different exchange processes. Among them, the first-order SOC correction has only DM contribution D (1) , as shown in Fig. 1(a), which represents the first term of Eq. (3). When swap the sites A and B in Fig. 1(a), one can obtain the second term of Eq. (3). Meanwhile, the second-order SOC correction has not only the contribution to DM interaction, but also the contribution to Γ term as shown in Fig. 1(b)-(e). While the type-I D (2) as shown in Fig. 1 (b) represents the first term of Eq. (4), the type-II D (2) as shown in Fig. 1(c) represents the second term of Eq. (4), and the third term of Eq. (4) could be obtained by swapping the sites A and B. It is worth mentioning that, the exchange processes for DM interactions ( Fig. 1 (a)-(c)) involve the hopping between ground states, which denoted by the green solid line in Fig. 1 and t nn in Eq. (3)(4) respectively. In sharp contrast, there are only hoppings between ground states and excited states in the processes of Γ terms as shown in Fig. 1 (d)-(e). Therefore, we emphasize that the essential difference between DM and Γ term is from their different hopping processes, rather than the commonly believed different orders of SOC [1,5]. B. Magnetic interactions in the first-principles approach Firstly we present the method to calculate magnetic interactions based on the force theorem and linear-response approach [33,52,53], which could be written as the fol-lowing form [52] J αβ R l +τ ,R l +τ = nkn k f nk − f n k ε nk − ε n k ψ nk |[σ × B τ ] α | ψ n k × ψ n k [σ × B τ ] β ψ nk e i(k −k)(R l −R l )(5) This method has been successfully applied to calculate Heisenberg interactions in various magnetic materials [25,33,[52][53][54][55]. Considering the case of α = β, we extend this method to estimate DM and Γ interactions, and the algorithm of this method is now implemented in the open source called WienJ [56], as an interface to Wien2k [57]. It is worth mentioning that, the general expression of bilinear spin exchange parameter J αβ , which could be written in J, D and Γ as Eq. (1), has 9 independent components. However, one can only yield 4 out of 9 components of J αβ for a given magnetic configuration. For example, for the collinear magnetic configuration with all spin moments lying along the z-axis, only the four spin exchange parameters J xx , J yy , J xy and J yx can be estimated. Therefore, to obtain the full nine spin exchange parameters J αβ (i.e. J, D and Γ terms), one need perform different first-principles self-consistent calculations for at least three independent orientations of the magnetization [25]. Based on the self-consistent results from different spin orientations, the magnetic interactions could be calculated from Eq. (5) [25]. However, these self-consistent calculations by choosing three different spin orientations would produce 12 parameters, resulting in that sets of parameters J αβ are not necessarily unique, and naturally leading to the calculation deviation. To reduce this calculation deviation, we also propose a new method when SOC is relatively small. Firstly, we perform the standard LSDA (+U ) calculations. Based on the eigenvalues ε nk and eigenstates ψ (0) nk from LSDA (+U ) calculations, we take SOC as a purterbation, and estimate the first-order and second-order SOC corrections wavefunction ψ (1) nk and ψ (2) nk in Wien2k [57]. Then all J αβ elements can be calculated with no need to do the separate self-consistent calculations with different spin orientations. Meanwhile, this method can produce the first and second order SOC contribution to the DM and Γ interaction separately. In the following, we will apply our two methods to several typical examples corresponding to 3d, 4d and 5d transition metal oxides respectively in the next section. As a benchmark on the accuracy of our methods in calculating Heisenberg and DM interactions, we first study the famous La 2 CuO 4 , which have been studied in a number of theoretical work [36,37,53,[59][60][61][62][63]. The LSDA + U ( = 7 eV) [64] calculation is applied. Without SOC considered, the calculated Heisenberg exchange parameters have no difference between these two approaches, which are summarized in the Table I. The calculated nearest neighbor magnetic coupling J 1 are dominant with the value of about 25.76 meV. We can find that the spin exchange coupling parameters decrease rapidly with the increasing distance between two Cu ions. The next nearest neighbor magnetic coupling J 2 shows ferromagnetic behavior and is one order of magnitude smaller than J 1 . The third nearest neighbor J 3 is antiferromagnetic and almost negligible. The results agree well with the previous theoretical work [36,53]. The weak ferromagnetism of La 2 CuO 4 is originated from the canting of the magnetic moments, which can be descried by the competition of Heisenberg interaction and DM interaction. Based on the two approaches in above section, the nearest neighbor DM parameters are calculated as shown in table II. As shown in table II, the DM interactions proportional to λ 2 are negligible due to the small SOC in 3d orbital, and the DM interactions proportional to λ are almost the same as the calculated DM parameters in WienJ. According to the calculated Heisenberg and DM parameters, the value of the canting angle is estimated to be about 1.7×10 −3 , which is in a good agreement with the experimental value of 2.2-2.9×10 −3 [62,63]. For comparison with previous theoretical works, Mazurenko et al. [36] proposed the angle value of 0.7×10 −3 using Green's function technique. With the construction of a tight-binding parametrization of the Hamiltonian with SOC, Katsnelson et al. [37] calculated canting angle to be 5.0×10 −3 . Comparatively our results agree very well with the experiment and quite promising. B. Ca 2 RuO 4 As the example of 4d transition metal oxides, Ca 2 RuO 4 crystallizes in the space group Pbca and has the layered perovskite structure [65][66][67][68][69][70]. The ground state of Ca 2 RuO 4 is an antiferromagnetic spin ordering with an insulating electrical behavior [66]. A weak ferromagnetic component is induced by spin canting below the magnetic transition temperature 113K [65]. II. The calculated nearest neighbor DM interaction parameters (in meV) for La2CuO4 via the two approaches in this work. R is the radius vector from two sites of magnetic ions in units of the lattice constant. The columns D (1) and D (2) represent the calculated DM interaction proportional to λ and λ 2 . Due to the small SOC, the DM interactions proportional to λ 2 are zero with an accuracy of 0.01meV. To study its magnetic properties, we performed the LSDA + U (= 3eV ) [67] calculations for Ca 2 RuO 4 . The calculated nearest neighbor Heisenberg interaction is about 20.9 meV. Experimentally, the Heisenberg parameters were estimated via inelastic neutron scattering as 16 meV in Ref. [69] and 5.8 meV in Ref. [70], and our result 20.9 meV is closer to the first value. Meanwhile, the calculated DM interactions by the two approaches mentioned above are both presented in Table III. As shown in Table III, the calculated DM interactions in WienJ are also almost the same as the sum of DM interactions proportional to λ and λ 2 , i.e., D ≈ D(λ)+D(λ 2 ) as shown in Table III. Note that the strength of first-order SOC corrected DM interactions |D(λ)| is around 1.31-1.41 meV, while the D(λ 2 ) is about 0.44 meV, therefore the DM interactions proportional to λ 2 are non-negligible in such 4d magnetic system. The ratio of DM interactions and Heisenberg interaction is estimated to be |D| /J ≈ 0.05, which is in good agreement with the rough estimate 0.06 from experiment [68]. WienJ the second method in this work C. Ca 3 LiOsO 6 In 5d transition-metal oxides systems, the strength of SOC is expected to be stronger than 3d or 4d materials due to the large atomic number. However, in orbital singlet states with relatively large electronic gap such as 5d 3 with half-filling t 2g orbitals [71,72], the electronic structures from fully self-consistent LSDA (+U ) + SOC calculation and the ones from further one iteration of SOC calculation after LSDA (+U ) calculation have small difference (see Fig. 2 in Appendix B), indicating that the effect of SOC is still small [71,72]. As one concrete 5d 3 example, we focus on Ca 3 LiOsO 6 [72][73][74][75] with the crystal structure of K 4 CdCl 6 -type. The ground state of Ca 3 LiOsO 6 is AFM with the magnetic transition temperature 117K. Both the first-principles study and the experiment suggested that Ca 3 LiOsO 6 has a fully opened electronic gap [74]. Though the AFM ordered state has been confirmed experimentally, the magnetization curve suggests a soft magnetism with a small spontaneous magnetization. The net magnetization is about 0.02 µ B per Os 5+ ion and is suggested due to a DM interaction generated by the broken inversion symmetry [72]. We perform the LSDA + U calculations of Ca 3 LiOsO 6 with U = 2 eV [71,[76][77][78] and calculate the magnetic interactions by applying our methods. The Heisenberg interactions J 1 , J 2 and J 3 are estimated to be all AFM with the values of 13.1 meV, 5.5 meV and 1.1 meV respectively. The J 1 is the strongest spin exchange, while J 2 is slightly less than one half of J 1 , and J 3 is an order of magnitude smaller than J 1 . These properties are in consistent with the energy-mapping results, though our calculated spin parameters are slightly larger than theirs (9.9 meV, 4.1 meV and 0.63 meV for J 1 -J 3 respectively) [75]. Meanwhile, our numerical DM interactions by the two approaches mentioned in Method section are both summarized in Table IV. Since the DM interactions between the 3rd nearest neighbor and longer-range distances for Os 5+ ions are negligible, thus we only show the DM interactions for nearest neighbor and the next nearest neighbor in Table IV. It can be seen that the DM interactions proportional to λ 2 have the same order of magnitude as the one proportional to λ in Ca 3 LiOsO 6 . According to the crystal symmetry, the nearest neighbor D 1 has the form of (0, 0, D z ), and the calculated D 1 via the two approaches are summarized in Table IV. Meanwhile, there are three different directions of D 2 connected by the symmetry of threefold rotation along z-axis, as shown in last three rows of Table IV. Summarizing the DM parameters of all nearest neighbors and the isotropic spin exchange parameters J up to the 3rd nearest neighbor, the expected magnetic moment is estimated to be 0.03 µ B , which is in good agreement with the experimental value of 0.02 µ B [72]. B. Materials with the second-order SOC correction to DM interactions Here we discuss the conditions where the first order of SOC in DM interactions (i.e. D (1) ) are absent, while DM interactions proportional to the second order of SOC (i.e. D (2) ) are non-zero. According to the picture of Fig. 1 and Eq. (3),(4), we summarize these three conditions that need to be met: (1) the hopping processes between ground state and excited state (t mn and t m n ) are symmetry forbidden. (2) the hopping between ground states (t nn ) and the hopping between excited states at two sites (t mm ) are non-zero. (3) the relation of orbital angular momentum of the two magnetic ions l β m n l γ mn −l γ m n l β mn should be also nonzero. According to the first condition t mn = t m n = 0, the exchange processes in Fig. 1(a) and (c) are forbidden, therefore D (1) is constrained to zero. Meanwhile, the second and third conditions make the exchange processes in Fig. 1(b) exist, thus D (2) could be present. Based on these restrictions, one can easily predict promising candidates with only the second-order SOC correction to DM interactions according to their different combinations of crystal symmetry, Wyckoff sites and orbital occupation pattern. As a simple example, we consider the space group P m (No. 6) and put two magnetic ions A and B located at two different 1a Wyckoff Positions (x, 0, z) and (x , 0, z ). These Wyckoff Positions have a mirror symmetry. Therefore, according to the different eigenvalues of mirror operation, we can mark their states with the irreducible representation A and A respectively [79]. Here we assume that their ground states n and n both belong to representation A , while their excited states m and m all belong to representation A . Thus, the hopping between the orbitals with different representations are symmetry forbidden, i.e. t mn = t m n = 0, while the hopping t nn and t mm between the orbitals with same representation can exist. Moreover, since these two magnetic ions are not related by the crystal symmetry, l β m n l γ mn − l γ m n l β mn could also be non-zero. Therefore, this case satisfies all the three conditions we listed above to be a candidate with only the DM interactions of second order SOC. The above three conditions can be satisfied in many Wyckoff positions of various space groups, and we believe that it may be widespread in a variety of magnetic materials. IV. CONCLUSION The magnetic model plays an important role in magnetic investigations. Here we revisit the general expression of magnetic interactions, including isotropic exchange interaction, antisymmetric DM interaction and symmetric Γ term. We clarify that the term proportional to λ 2 has both contribution to DM interaction and Γ term. We find that the DM and Γ interactions can be separated from their different hopping processes rather than the orders of SOC. We present two first-principles methods to calculate the anisotropic magnetic interactions. Based on the first method, one need perform selfconsistent calculations for at least three different spin orientations to obtain the full nine exchange parameters J αβ . On the other hand, using the second method, one can estimate these magnetic exchange parameters with no need to do the separate self-consistent calculations for different spin orientations. This method can also calculate the first-order and second-order SOC contribution to DM interactions separately. We have successfully applied our methods to several typical weak ferromagnetic materials La 2 CuO 4 , Ca 2 RuO 4 and Ca 3 LiOsO 6 respectively. Furthermore, according to microscopic mechanism shown in Fig. 1, we list all three conditions which can lead to that the DM interactions proportional to λ are symmetric forbidden while the DM interactions proportional to λ 2 exist. FIG. 1 . 1Schematic pictures of exchange paths for anisotropic magnetic interactions between site A and site B. The dotted line represents the SOC excitation process, while the solid line represents the hopping process. (a) represents the DM interactions for the first order of SOC. The DM interactions for the second order of SOC have two types of perturbation processes (b) and III. RESULTSA. First-principles examples of typical materialsA. La 2 CuO 4 . The calculated DM interaction parameters (in meV) for Ca2RuO4 via the two approaches in this work. Here R is the radius vector from two sites of magnetic ions in units of the lattice constant. The columns D(1) and D(2) represent the calculated DM interaction proportional to λ and λ 2 . It can be seen that the calculated interactions via these two approaches are close, i.e. D ≈ D (1) + D(2) IV. The calculated DM interaction parameters (in meV) for Ca3LiOsO6 via the two approaches in this work. R is the radius vector from two sites of magnetic ions in units of the lattice constant. The columns D(1) and D(2) represent the calculated DM interaction proportional to λ and λ TABLE I . IThe calculated Heisenberg exchange parameters J (in meV) for La2CuO4. The calculated spin exchange parameters in the previous theoretical work are also shown for comparison.La2CuO4 J Ref. [53] Ref. [36] this work J1 27.2 29.2 25.76 J2 −3.00 −4.1, −3.9 −3.80, −3.38 J3 −0.05 0 −0.11 TABLE TABLE V. ACKNOWLEDGEMENTS R M White, Quantum Theory of Magnetism: Magnetic Properties of Materials. Springer-Verlag Berlin HeidelbergR. M. White, Quantum Theory of Magnetism: Magnetic Properties of Materials (Springer-Verlag Berlin Heidel- berg, 2007). K H J Buschow, F R Boer, Physics of Magnetism and Magnetic Materials. SpringerK. H. J. Buschow, F. R. Boer, et al., Physics of Mag- netism and Magnetic Materials (Springer, 2003). G De, B Ria, Magnetic interactions in molecules and solids. SpringerG. Coen de and B. 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{'abstract': 'Anisotropic magnetic exchange interactions lead to a surprisingly rich variety of the magnetic properties. Considering the spin orbit coupling (SOC) as perturbation, we extract the general expression of a bilinear spin Hamiltonian, including isotropic exchange interaction, antisymmetric Dzyaloshinskii-Moriya (DM) interaction and symmetric Γ term. Though it is commonly believed that the magnitude of the DM and Γ interaction correspond to the first and second order of SOC strength λ respectively, we clarify that the term proportional to λ 2 also has contribution to DM interaction. Based on combining magnetic force theorem and linear-response approach, we have presented the method of calculating anisotropic magnetic interactions, which now has been implemented in the open source software WienJ. Furthermore, we introduce another method which could calculate the first and second order SOC contribution to the DM interaction separately, and overcome some shortcomings of previous methods. Our methods are successfully applied to several typical weak ferromagnets for 3d, 4d and 5d transition metal oxides. We also predict the conditions where the DM interactions proportional to λ are symmetrically forbidden while the DM interactions proportional to λ 2 are nonzero, and believe that it is widespread in certain magnetic materials. arXiv:2212.13963v1 [cond-mat.mtrl-sci]', 'arxivid': '2212.13963', 'author': ['Di Wang \nNational Laboratory of Solid State Microstructures and School of Physics\nNanjing University\n210093NanjingChina\n\nCollaborative Innovation Center of Advanced Microstructures\nNanjing University\n210093NanjingChina\n', 'Xiangyan Bo \nNanjing University of Posts and Telecommunications\n210023NanjingChina\n', 'Feng Tang \nNational Laboratory of Solid State Microstructures and School of Physics\nNanjing University\n210093NanjingChina\n\nCollaborative Innovation Center of Advanced Microstructures\nNanjing University\n210093NanjingChina\n', 'Xiangang Wan \nNational Laboratory of Solid State Microstructures and School of Physics\nNanjing University\n210093NanjingChina\n\nCollaborative Innovation Center of Advanced Microstructures\nNanjing University\n210093NanjingChina\n'], 'authoraffiliation': ['National Laboratory of Solid State Microstructures and School of Physics\nNanjing University\n210093NanjingChina', 'Collaborative Innovation Center of Advanced Microstructures\nNanjing University\n210093NanjingChina', 'Nanjing University of Posts and Telecommunications\n210023NanjingChina', 'National Laboratory of Solid State Microstructures and School of Physics\nNanjing University\n210093NanjingChina', 'Collaborative Innovation Center of Advanced Microstructures\nNanjing University\n210093NanjingChina', 'National Laboratory of Solid State Microstructures and School of Physics\nNanjing University\n210093NanjingChina', 'Collaborative Innovation Center of Advanced Microstructures\nNanjing University\n210093NanjingChina'], 'corpusid': 255186473, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 17115, 'n_tokens_neox': 14524, 'n_words': 8223, 'pdfsha': '9ebb80b2f880facb3648eec43c30574e720bffae', 'pdfurls': ['https://export.arxiv.org/pdf/2212.13963v1.pdf'], 'title': ['First-principles study of spin orbit coupling contribution to anisotropic magnetic interaction', 'First-principles study of spin orbit coupling contribution to anisotropic magnetic interaction'], 'venue': []}

arxiv

A Data-Driven Gradient Algorithm for High-Precision Quantum Control 9 Apr 2018 Bing Wu Bing Chu David H Owens Herschel Rabitz Department of Automation Center for Quantum Information Science and Technology Tsinghua University 100084BeijingChina BNRist 100084BeijingChina School of Electronic and Computer Science University of Southampton SO17 1BJSouthamptonUK Department of Automation Department of Automatic Control and Systems Engineering Zhengzhou University 450001ZhengzhouChina The University of Sheffield Mappin StreetS1 3JDSheffieldUK Department of Chemistry Princeton University 08544PrincetonNJUSA § A Data-Driven Gradient Algorithm for High-Precision Quantum Control 9 Apr 2018(Dated: April 10, 2018)1* Electronic address: rbwu@tsinghuaeducn † Electronic address: bchu@sotonacuk ‡ Electronic address: dhowens@shefacuk § Electronic address: hrabitz@PrincetonEdu 2 In the quest to achieve scalable quantum information processing technologies, gradient-based optimal control algorithms (e.g., GRAPE) are broadly used for implementing high-precision quantum gates, but their performance is often hindered by deterministic or random errors in the system model and the control electronics. In this paper, we show that GRAPE can be taught to be more effective by jointly learning from the design model and the experimental data obtained from process tomography. The resulting data-driven gradient optimization algorithm (d-GRAPE) can in principle correct all deterministic gate errors, with a mild efficiency loss. The d-GRAPE algorithm may become more powerful with broadband controls that involve a large number of control parameters, while other algorithms usually slow down due to the increased size of the search space. These advantages are demonstrated by simulating the implementationof a two-qubit CNOT gate. * Electronic address: rbwu@tsinghua.edu.cn † Electronic address: b.chu@soton.ac.uk ‡ Electronic address: d.h.owens@shef.ac.uk § Electronic address: hrabitz@Princeton.Edu I. INTRODUCTION In practical quantum information processing, high-precision implementation of universal quantum gates (usually involving 1∼3 qubits) is vital. Although the current control technology has been able to meet the minimum requirement for quantum error correction [1] (e.g., the 0.6-1% error threshold for surface codes has been reached in superconducting circuits [2], ion-traps [3], quantum-dots [4] and nitrogen-vacancy centers in diamond [5]), the achievable precision still needs to be improved in order to reduce the resource overhead required for scalable quantum computation [6]. Towards this "last mile" target, an effective method for gate tuneup is to optimize the control pulses by following the gradient direction of the error function, which popularly has one form known as the GRAPE (GRadient Ascending Pulse Engineering) algorithm [7]. When supplied with abundant control resources, the algorithm is highly efficient in that the optimization almost always quickly converges to a global optimal solution, owing to the underlying expectation of finding an attractive trap-free optimal control landscape [8][9][10]. The GRAPE algorithm is by nature offline (or ex situ [11]) because the optimization is usually with respect to a design model identified from prior experiments, and no real online data is used during the optimization process. Thus, the systematic errors in the design model (e.g., the identified Hamiltonian and the pulse distortion by a waveform generater), as well as the uncharacterized random noises in the system and pulses, limit the control precision. Regarding these items, the designed control pulses should be immune to the systematic errors and be robust to the random noises. Online (or in situ) learning can in principle correct for the systematic errors by iteratively calibrating the control pulses based on measurement outcomes. This learning control concept can be traced back to the early 1990s in the control of molecules by training ultrafast laser pulses [12], which has been successful in hundreds of physical and chemical experiments [13]. In most applications, the control objective is with respect to a target state or the ensemble average of some quantum observable, where the control fields are updated by heuristic optimization algorithms such as a genetic algorithm [12] or evolutionary strategy [14]. Learning control for quantum gate tune-up is much more difficult than the aforementioned applications, because the full characterization of the control outcome requires process tomography that needs many more experiments to measure additional observables at high precision. In existing protocols, the extra data acquisition problem is usually bypassed via randomized benchmarking (RB) [15], which is much easier for gate error verification without having to fully reconstruct the gate matrix. Several RB-based learning algorithms have been proposed, e.g., the Nelder-Mead algorithm was used in [16] and [17], with applications to superconducting qubits. To exploit the attractive trap-free control landscape [8], gradient-based (or greedy) algorithms were also introduced to accelerate the online optimization, where extra measurements (proportional to the number of control variables) need to be done to estimate the full or partial gradient from the data [11,[18][19][20][21]. The complexity of online learning control algorithms mainly depends on the total experimental costs, while the numerical calculations on a computer is usually negligible when only a few qubits are involved. In the existing algorithms, the overall cost can be very high due to the required many iterations (mainly for RB-based optimization [11,16,17]) or the expensive measurements in each iteration (mainly for gradient-based optimization [18][19][20]). To further reduce the total experimental cost, we find that the design model, which is often used for obtaining a good initial guess for the control pulse, can play a new role in accelerating the succeeding online learning calibration process. This opportunity arises because the design model contains valuable a priori knowledge about the experimental system, which is obtained from elaborately designed offline measurements. This motivation leads to the algorithm proposed in this paper, in which the design model is embedded into the data-driven learning procedure to synthesize the gradient vector also utilizing data from process tomography. The algorithm can effectively reduce the number of iterations by predicting the gradient descent or ascent direction for quantum gate tuneup, which compensates for the increased cost of tomography. Besides, under circumstances where broad-bandwidth controls are required for noise suppression or high-speed gate operations, the total experimental cost of our algorithm may be further reduced, while the corresponding cost usually increases with other algorithms. We refer to the method presented here as a data-driven type of GRAPE algorithm, or d-GRAPE for short. The remainder of the paper is organized as follows. The d-GRAPE algorithm is described in Section II, whose effectiveness in correcting model error and control pulse distortion is demonstrated through simulations in Section III. Finally, conclusions are presented in Section IV. II. THE DATA-DRIVEN GRADIENT ALGORITHM In this section, we will present the basic procedure of the data-driven gradient algorithm. A. The quantum control model We assume that the quantum control system is closed and governed by the following Schrödinger equation:U (t) = −i H 0 + m k=1 u k (t)H k U (t),(1) where U (t) ∈ C N ×N represents the quantum gate operation on the states, with U (0) = I N , the identity matrix; and u k (t) ∈ R, k = 1. · · · , m, are the control fields imposed on the control system. The free Hamiltonian H 0 and the control Hamiltonians H k 's are N × N Hermitian matrices that steer the unitary U (t). In practice, the above Hamiltonians are never precisely known. Thus, any numerical calculation has to be based on a design model: which are sometimes called quantum gate bleedthrough [17]. For example, the distortion can be modeled by a linear filter described as follows U D (t) = −i H D,0 + m k=1 v k (t)H D,k U D (t),(2)u k (t) = D[v k (t)] = ∞ 0 h(t − τ )v k (τ )dτ,(3) where h(t) is the impulse response of the linear filter. The control pulse is distortion free only when h(t) = δ(t) is the Dirac function. In the following, we will show how to correct these errors by learning from online data. B. From GRAPE to d-GRAPE The goal of quantum gate tune-up is to find proper design control pulse sequences {v k (t)} such that the generated control {u k (t)} can lead the system propagator U (T ) as close as possible to a desired unitary matrix U f . This can be achieved by minimizing the infidelity function [22] J = 1 2N U (T ) − U f 2 ,(4) where the norm is defined as X = Tr(X † X). There are different ways of utilizing the gradient to optimize the control pulse. We illustrate the concept in the paper with the typical steepest descent algorithm that updates the control pulses in the following fashion: v k (t, + 1) = v k (t, ) − α( ) · g k (t, ),(5) where g k (t, ) = δJ δv k (t, ) is the gradient in the -th iteration and α( ) is the learning rate that is chosen as a sufficiently small positive real number. Taking U (T ) as an implicit (1) and (3), we have function of {v k (t)} throughg k (t, ) = ∞ 0 δJ δu k (t , ) δu k (t , ) δv k (t, ) dt = ∞ 0 ∆(T, ), H k (t , ) δu k (t , ) δv k (t, ) dt where H k (t, ) = U † (t, )H k U (t, ) and the inner product is defined as X, Y = Tr(X † Y ). The error matrix is ∆(T, ) = 1 2i U † f U (T, ) − U † (T, )U f . The variation term in the integral is induced by the distortion of the control pulses. In the linear case exemplified in (3), we have δu k (t , ) δv k (t, ) = h(t − t).(6) Because the true gradient function (6) can never be precisely evaluated due to the unavailability of the true model of the system, a practical operation is to ignore the pulse distortion and calculate the gradient in an offline fashion, as follows g OL k (t, ) = ∆ D (T, ), H D,k (t, ) ,(7) where H D,k (t, ) = U † D (t, )H D,k U D (t, ) and ∆ D (T, ) = 1 2i U † f U D (T, ) − U † D (T, )U f are both computed from the design model. Since the optimization is completely blind without checking the control performance with experimental data, the learning process along this gradient direction will be inevitably guided to a false solution that is optimal for the design model but not for the actual system. To find the genuine optimal control pulses, we take advantage of both of the above two approaches. The key concept is to estimate the gradient as follows: g k (t, ) = ∆ (T, ), H D,k (t, )(8) where the error matrix∆(T, ) comes form the estimation of ∆(T, ) through process tomography of U (T ), and H D,k (t, ) is calculated from the design model as in (7). In this way, the real data are employed in order to deduce whether the learning algorithm is converging to a correct solution such that ∆(T, ) = 0, and its incorporation with H D,k (t, ) provides an approximate gradient whose deviation from the real gradient depends on the accuracy of the design model. The entire learning process is shown in Fig. 1 C. Convergence analysis It is difficult to rigorously prove the convergence of the d-GRAPE to a globally optimal solution. Heuristically, d-GRAPE should converge to at least a locally optimal solution because the estimated gradient (8) can still maintain descent, although possibly not the steepest in presence of various uncertainties, as long as they are not too large. On the other hand, d-GRAPE can stop at a desired globally optimal control solution corresponding to U (T ) = U f (assuming that the tomography error is negligible), where the gradient (8) vanishes. Therefore, when the system is controllable and the control resources are sufficiently abundant [9], the well-preserved attractive character of the control landscape should assure that d-GRAPE almost always converges to the desired global optimal solution, which will be verified in the following simulation examples. In principle, d-GRAPE is able to correct for any deterministic errors in the model or in the control pulses. Its precision is limited by that of the process tomography and other random noise sources in the system. Compared with the existing online learning algorithms, d-GRAPE will be more competitive when broadband controls that involve a large number of variables are required for high precision, speed and robustness [23,24]. Under such circumstance, the experimental cost of d-GRAPE per iteration will stay invariant, but the convergence may be faster owing to increased freedom in the control. However, the RB-based algorithms are expected to be more expensive because many more iterations are needed for search in the enlarged control space, as well as for the gradient-based algorithms proposed in [18][19][20], whose experimental costs per iteration increase with the number of control parameters. III. SIMULATIONS In this section, we will show by numerical simulations how the algorithm can correct deterministic errors in the model and control pulses. A. Simulation Model We assume that the actual system is described by the following Hamiltonian: H(t) = Jσ 1 z ⊗ σ 2 z + 2 i=1 u i x (t)σ i x + u i y (t)σ i y , where J is the coupling strength between the two qubits. The design model is as follows: H D (t) = (J + δJ)σ 1 z ⊗ σ 2 z + 2 i=1 v i x (t)σ i x + v i y (t)σ i y , in which δJ represents the identification error of J in the design model. Moreover, we assume that the control pulses are distorted by a linear filter u i x,y (t) = t 0 h(t − τ )v i x,y (τ )dτ, i = 1, 2, in which the impulse response h(t) is taken as h(t) = 1 t r e −t/tr , t ≥ 0.(9) The the time constant t r characterizes the degree of pulse distortion by the steepness of the rising edge of distorted pulses. The pulses are heavily distorted when t r is long. B. Gate tuneup simulation results To demonstrate the ability of quantum gate tune-up by d-GRAPE, we test the target of a CNOT gate U f =        1 0 0 0 0 1 0 0 0 0 0 1 0 0 −1 0        . In the simulation, we set the coupling constant as J = 1 and the final time as T = 5. The time interval is evenly divided into M = 20 sub-intervals, and hence the duration of each sub-interval is ∆t = T /M . In Fig. 2, we show three cases with parametric error δJ in J and pulse distortion characterized by st r . Each case includes results from 12 different initial random guesses. We first offline optimize these fields [ i.e., following g OL k (t) in (7)] to obtain a set of candidate pulses that are close to the optimal solution. Then, starting from these pulses, we perform d-GRAPE [ i.e., followingĝ k (t) in (8)] based on the BFGS algorithm (a most popular gradient-based optimization algorithm [25]) that is more efficient than the steepest descent gradient algorithm. The estimation errors in the process tomography is simulated by injecting an additive random noise ∆U ( ) (whose Frobenius norm is ∼ 2 × 10 −5 ) to U (T, ) in each iteration. For comparison, we also run the ideal GRAPE [ i.e., following g k (t) in (6), assuming that both δJ and t r are precisely known] from the same set of initial pulses. The simulation results show that the precision of the candidate pulses obtained from offline optimization (at the beginning of the optimization process shown in the plots) is always limited by the accuracy of the design model. When the model error is relatively small [see Fig. 2(a)], the succeeding optimization based on the proposed d-GRAPE algorithm (solid curves) almost always converges to its global optimal solution that is limited by the tomography error. Compared with the ideal GRAPE optimization (see the blue dash curves), its convergence speed is only slightly reduced. When model error is not small enough (e.g., with severe pulse distortion in Fig. 2(b) or parameter deviation δJ in Fig. 2(c), fewer runs can quickly converge to the global optimal solution. Some runs still converge, but at the price of an increase in the number of iterations. We plot in Fig. 3 C. Comparison with RB-based algorithms We also tested the performance of d-GRAPE using different numbers of control pulses, and compared its performance with that of the gradient-free Nelder-Mead (NM) algorithm. The latter algorithm can be applied based on randomized benchmarking (RB) without having to use process tomography. The simulations are all based on a relatively precise model with δJ/J = 0.02. As shown in Fig. 4 IV. CONCLUSION AND DISCUSSION To summarize, we have proposed a data-driven gradient (d-GRAPE) algorithm for optimizing laboratory control pulses against deterministic errors. The entire optimization procedure essentially performs both in a reinforcement learning manner from the online data in addition to supervised learning from the design model (or offline data). Analyses and simulations exemplify the calibration ability against errors induced by pulse distortion and model uncertainty, which is in principle extendable to more general non-uniform and nonlinear errors, as long as the process tomography can be done with sufficient precision and a reasonably good design model is available. There is much room for the d-GRAPE algorithm to be improved. Several extensions of the algorithm are possible. First, extracting more knowledge from the offline model will improve the online optimization. For example, we can estimate the gradient more precisely by incorporating the pulse distortion function h(t) that can be offline identified from the waveform generator; or we can use a more sophisticated learning algorithm such as a Newton algorithm, because the Hessian matrix can be estimated based on the same use of process tomography without increasing the number of experiments. Second, combined with adaptive tomography [26][27][28], it is possible to simultaneously improve the precision of the control and the process tomography, which will further accelerate the learning process. We also remark that d-GRAPE algorithm can be extended to more general objectives, e.g., quantum state preparation problems, where the cost of state tomography is cheaper and hence can be more efficient. When the real quantum system undergoes open dynamics, we can replace the unitary propagators by open-system process matrices, but the achievable precision may be limited by the decoherence effects. These potential topics and developments will be explored in the future. that can be accessed by a computer. The free and control Hamiltonians in the design model (2) can be very close to those in the actual system (1), but they are always imprecise to some degree. The control pulses v k (t) in the design model are often chosen as piecewise-constant pulses to facilitate numerical simulation on a digital computer and in some experimental situations. Note that the (designed) control pulses v k (t) are usually not identical with the actual pulses u k (t) applied to the system, because the control signal produced by an arbitrary waveform generator (AWG) is often distorted due to various factors including electronic limitations and transmission through the control line to the qubit. Such distorted signals have rising and falling edges or other unanticipated features, Fig 1 : 1, where the explicit use of the design model is the major difference with existing model-free learning control strategies in the literature. Therefore, we refer to the algorithm as d-GRAPE. (color online). Schematic diagram of the data-driven GRAPE (d-GRAPE) optimization procedure. The gradient is estimated from both the design model and the online data (for process tomography), which can in principle correct all deterministic errors such as the pulse distortion and the model uncertainty. Fig 2 : 2(color online). Two-qubit quantum gate tuneup with the proposed d-GRAPE algorithm (solid black curves) for different model uncertainties and their comparison with ideal GRAPE algorithm (blue dashed curves). Each case include 12 runs from different initial guesses. (a) small pulse distortion t r /∆t = 0.1 and small parametric error δJ/J = 0.05; (b) large pulse distortion t r /∆t = 0.5 and small parametric error δJ/J = 0.05; (c) small pulse distortion t r /∆t = 0.1 and large parametric error δJ/J = 0.20. The ultimate control precision ∼ 2 × 10 −5 is limited by the estimation error of the process tomography (indicated by the horizontal dashed line). Fig 3 : 3(color online). The examples of optimized control pulses on the first qubit along the x-axis with δJ/J = 0.1. The pulse distortion parameters are t r /∆t = 0.1 (upper plot) and t r /∆t = 0.5 (lower plot). The blue dashed curves are the initial AWG reference signal, and the black solid curves are the corrected AWG signal optimized to the precision ∼ 2 × 10 −5 that is limited by the tomography error. The actual distorted signals with rising and fall edges are shown by red dash-dotted curves. the shapes of the corrected and actual x-axis control signals on the first qubit, showing that the d-GRAPE can correct for large pulse-distortion to achieve high-precision control without having to exactly know how the pulses are distorted. The model error we choose in the simulations are relatively large (e.g, 20% error in J and pulse distortion t r /∆t = 0.5), and even under such a bad situation d-GRAPE can still tuneup the gate to some extent. When the model error gets larger, more and more optimizations become slower, and there even exist cases that the d-GRAPE algorithm gets lost and is trapped at a local false optimum solution. Thus, d-GRAPE should not be applied with a very coarse model, because of the potential traps and the increase experimental cost on process tomography. In practice, one should improve the precision of the design model as much as possible. Based on the high-precision model, the d-GRAPE algorithm can correct the error caused by the residue model imprecision within a few iterations. Fig 4 : 4(color online). Performance comparison of the d-GRAPE algorithm and the Nelder-Mead algorithm. d-GRAPE may fail when there are only a few control parameters (M = 10), and succeed with more control parameters (M = 20 and M = 50). When M increases from 20 to 50, d-GRAPE becomes more efficient because less number of iterations are required, while the Nelder-Mead algorithm takes much more iterations to converge. , when there are few pulses to tune (M = 10), d-GRAPE is trapped over a very rugged control landscape (i.e., resulting from an insufficient number of control variables producing false landscape traps), while NM can struggle to achieve a high precision after about 2000 iterations. Given more control pulses, d-GRAPE can easily find high-precision control solutions over an almost trap-free control landscape in several tens of iterations. Correspondingly, the number of NM iterations is hundreds of times (or even over one thousand times) larger than that of d-GRAPE iterations. More importantly, the number of iterations increases for NM, but decreases for d-GRAPE, when the number of control pulses grows. 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{'abstract': 'In the quest to achieve scalable quantum information processing technologies, gradient-based optimal control algorithms (e.g., GRAPE) are broadly used for implementing high-precision quantum gates, but their performance is often hindered by deterministic or random errors in the system model and the control electronics. In this paper, we show that GRAPE can be taught to be more effective by jointly learning from the design model and the experimental data obtained from process tomography. The resulting data-driven gradient optimization algorithm (d-GRAPE) can in principle correct all deterministic gate errors, with a mild efficiency loss. The d-GRAPE algorithm may become more powerful with broadband controls that involve a large number of control parameters, while other algorithms usually slow down due to the increased size of the search space. These advantages are demonstrated by simulating the implementationof a two-qubit CNOT gate. * Electronic address: rbwu@tsinghua.edu.cn † Electronic address: b.chu@soton.ac.uk ‡ Electronic address: d.h.owens@shef.ac.uk § Electronic address: hrabitz@Princeton.Edu', 'arxivid': '1712.01780', 'author': ['Bing Wu ', 'Bing Chu ', 'David H Owens ', 'Herschel Rabitz ', '\nDepartment of Automation\nCenter for Quantum Information Science and Technology\nTsinghua University\n100084BeijingChina\n', '\nBNRist\n100084BeijingChina\n', '\nSchool of Electronic and Computer Science\nUniversity of Southampton\nSO17 1BJSouthamptonUK\n', '\nDepartment of Automation\nDepartment of Automatic Control and Systems Engineering\nZhengzhou University\n450001ZhengzhouChina\n', '\nThe University of Sheffield\nMappin StreetS1 3JDSheffieldUK\n', '\nDepartment of Chemistry\nPrinceton University\n08544PrincetonNJUSA §\n'], 'authoraffiliation': ['Department of Automation\nCenter for Quantum Information Science and Technology\nTsinghua University\n100084BeijingChina', 'BNRist\n100084BeijingChina', 'School of Electronic and Computer Science\nUniversity of Southampton\nSO17 1BJSouthamptonUK', 'Department of Automation\nDepartment of Automatic Control and Systems Engineering\nZhengzhou University\n450001ZhengzhouChina', 'The University of Sheffield\nMappin StreetS1 3JDSheffieldUK', 'Department of Chemistry\nPrinceton University\n08544PrincetonNJUSA §'], 'corpusid': 13808560, 'doi': '10.1103/physreva.97.042122', 'github_urls': [], 'n_tokens_mistral': 8800, 'n_tokens_neox': 7705, 'n_words': 4819, 'pdfsha': 'e762e17a53e91a490c180164e3f1f7f98bed28e2', 'pdfurls': ['https://arxiv.org/pdf/1712.01780v4.pdf'], 'title': ['A Data-Driven Gradient Algorithm for High-Precision Quantum Control', 'A Data-Driven Gradient Algorithm for High-Precision Quantum Control'], 'venue': []}

arxiv

Synthesis and post-annealing effects of alkaline-metal-ethylenediamine-intercalated superconductors A x (C 2 H 8 N 2 ) y Fe 2-z Se 2 (A = Li, Na) with T c = 45 K Takashi Noji Department of Applied Physics Graduate School of Engineering Tohoku University 6-6-05 Aoba, Aoba-ku980-8579Aramaki, SendaiJapan Takehiro Hatakeda Department of Applied Physics Graduate School of Engineering Tohoku University 6-6-05 Aoba, Aoba-ku980-8579Aramaki, SendaiJapan Shohei Hosono Department of Applied Physics Graduate School of Engineering Tohoku University 6-6-05 Aoba, Aoba-ku980-8579Aramaki, SendaiJapan Takayuki Kawamata Department of Applied Physics Graduate School of Engineering Tohoku University 6-6-05 Aoba, Aoba-ku980-8579Aramaki, SendaiJapan Masatsune Kato Department of Applied Physics Graduate School of Engineering Tohoku University 6-6-05 Aoba, Aoba-ku980-8579Aramaki, SendaiJapan Yoji Koike Department of Applied Physics Graduate School of Engineering Tohoku University 6-6-05 Aoba, Aoba-ku980-8579Aramaki, SendaiJapan Synthesis and post-annealing effects of alkaline-metal-ethylenediamine-intercalated superconductors A x (C 2 H 8 N 2 ) y Fe 2-z Se 2 (A = Li, Na) with T c = 45 K superconductivityintercalationFeSeetylenediaminealkaline metal New iron-based intercalation superconductors A x (C 2 H 8 N 2 ) y Fe 2-z Se 2 (A = Li, Na) with T c = 45 K have successfully been synthesized via intercalation of dissolved alkaline metal in ethylenediamine. The c-axis lengths of A x (C 2 H 8 N 2 ) y Fe 2-z Se 2 (A = Li, Na) are 20.74(7) Å and 21.9(1) Å, respectively, and are about 50 % larger than that of K x Fe 2 Se 2 , indicating that not only alkaline metal but also ethylenediamine is intercalated between the Se-Se layers of FeSe. It seems that the high-T c of A x (C 2 H 8 N 2 ) y Fe 2-z Se 2 (A = Li, Na) is caused by the possible two-dimensional electronic structure due to the large c-axis length. Through the post-annealing in an evacuated glass tube, it has been found that T c decreases with increasing post-annealing temperature and that deintercalation of EDA from the as-intercalated sample takes place at low temperatures below 250℃. Introduction The superconducting transition temperature, T c , of the iron-based chalcogenide superconductor FeSe is only 8 K [1], but dramatically increases up to ~37 K by the application of high pressure [2]. The successful synthesis of potassium-intercalated K x Fe 2-y Se 2 with T c ~31 K at ambient pressure was an exciting breakthrough [3]. Recently, it has been reported that the intercalation into FeSe of alkaline and alkaline-earth metals in liquid ammonia yields a variety of compounds with significantly enhanced T c 's of 40 -46 K [4 -6]. The c-axis length of Li 0.9 Fe 2 Se 2 (NH 3 ) 0.5 with T c = 44 K is 16.518 Å [5] and larger than 14.0367 Å in K x Fe 2 Se 2 [3], indicating that not only lithium but also ammonia is intercalated. Moreover, it has been reported that lithium-and pyridine-intercalated Li x (C 5 H 5 N) y Fe 2-z Se 2 exhibits superconductivity with T c = 45 K and that post-annealing of the intercalated sample drastically expands the c-axis length from 16.0549 Å to 23.09648 Å and increases the superconducting shielding volume fraction [7]. Here, we report on the successful synthesis of new superconductors A x (C 2 H 8 N 2 ) y Fe 2-z Se 2 (A = Li, Na) with T c = 45 K via intercalation of dissolved alkaline metal in ethylenediamine (EDA), C 2 H 8 N 2 . Post-annealing effects on the crystal structure and superconductivity are also discussed. Experimental Polycrystalline samples of FeSe were prepared by the solid-state reaction method. Starting materials were powders of Fe and Se, which were weighted stoichiometrically, mixed thoroughly and pressed into pellets. The pellets were sealed in an evacuated quartz tube and heated at 800℃ for 40 h. The obtained pellets of FeSe were pulverized into powder to use for the intercalation. Dissolved alkaline metal in EDA was intercalated into the powdery FeSe as follows. An appropriate amount of the powdery FeSe was placed in a beaker filled with 0.2 M solution of pure lithium or sodium metal dissolved in EDA. All the process was performed in an argon-filled glove box. Post-annealing of as-intercalated samples was carried out at 100 -500℃ for 60 h in an evacuated glass tube. Both FeSe and the intercalated samples were characterized by powder x-ray diffraction using Cu K α radiation. For the intercalated samples, an airtight sample holder was used. In order to observe the superconducting transition, the magnetic susceptibility, χ, was measured using a superconducting quantum interference device (SQUID) magnetometer. Measurements of the electrical resistivity, ρ, were also carried out by the standard dc four-probe method. For the ρ measurements, as-intercalated powdery samples were pressed into pellets. Then, the pellets were sintered at 200℃ for 20 h in an evacuated glass tube. Thermogravimetric (TG) measurements were performed in flowing gas of argon, using a commercial analyzer (SII Nano Technology Inc., EXSTAR DSC7020). Figure 1 shows powder x-ray diffraction patterns of as-intercalated samples of A x (C 2 H 8 N 2 ) y Fe 2-z Se 2 (A = Li, Na). The broad peak around 2θ = 20° is due to the airtight sample holder. Although there are unknown peaks, most of sharp Bragg peaks are due to the intercalation compound of A x (C 2 H 8 N 2 ) y Fe 2-z Se 2 (A = Li, Na) and the host compound of FeSe, so that they are able to be indexed based on the ThCr 2 Si 2 -type and PbO-type structures, respectively. Therefore, it is found that alkaline metal and EDA are partially intercalated into FeSe, while there remains a non-intercalated region of FeSe in the samples. The lattice constants of Li x (C 2 H 8 N 2 ) y Fe 2-z Se 2 are calculated to be a = 3.458 (6) Å and c = 20.74 (7) Å. The c-axis length of Na x (C 2 H 8 N 2 ) y Fe 2-z Se 2 is 21.9(1) Å. Since the unit cell of A x (C 2 H 8 N 2 ) y Fe 2-z Se 2 (A = Li, Na) includes two Fe layers, the distance between the neighboring Fe layers is 10.37(4) Å and 10.95(5) Å, respectively, and much larger than 5.515(1) Å of FeSe. Taking into account our previous results that the intercalation of only lithium into Fe(Se,Te) has neither effect on the superconductivity nor crystal structure [8], it is concluded that not only lithium or sodium but also EDA has been intercalated between the Se-Se layers of FeSe. Figure 2 shows the temperature dependence of χ in a magnetic field of 10 Oe on zero-field cooling (ZFC) and on field cooling (FC) for as-intercalated powdery samples. Results and discussion The first T c is observed at 45 K and the second T c is at 8 K. Taking into account the powder x-ray diffraction results, it is concluded that the first is due to bulk Fig. 1. Powder x-ray diffraction patterns of as-intercalated samples consisting of A x (C 2 H 8 N 2 ) y Fe 2-z Se 2 (A = Li, Na) and FeSe using Cu K α radiation. Indexes without and with asterisk are based on the ThCr 2 Si 2 -type and PbO-type structures, respectively. superconductivity of A x (C 2 H 8 N 2 ) y Fe 2-z Se 2 (A = Li, Na), Peaks marked by ▼ are unknown. temperature and reaches zero at 18 K. Since the intercalation of only lithium does not increase T c [8], it seems that the high-T c of A x (C 2 H 8 N 2 ) y Fe 2-z Se 2 (A = Li, Na) is caused by the possible two-dimensional electronic structure due to the large c-axis length. Figure 3 shows the TG curve on heating up to 900℃ at the rate of 1℃/min for the as-intercalated powdery sample. Three steps of mass loss are observed; 2.9 % loss below 100℃, 11.3 % loss between 100 and 200℃, and large loss above 450℃. second step between 100 and 220℃ is due to deintercalation of EDA. The third step above 500℃ may be due to unknown products including iron. Figure 5 shows Finally, Fig. 6 shows the maximum T c 's of FeSe-based superconductors obtained so far at ambient pressure as a function of the distance between the neighboring Fe layers. Interestingly, T c increases monotonically with increasing such distance and is saturated at about 45 K above 9 Å. Summary We have succeeded in synthesizing new intercalation compounds A x (C 2 H 8 N 2 ) y Fe 2-z Se 2 (A = Li, Na) via intercalation of dissolved alkaline metal in EDA. Although the samples include non-intercalated regions of FeSe, the c-axes of A x (C 2 H 8 N 2 ) y Fe 2-z Se 2 (A = Li, Na) have drastically expanded to be 20.74(7) and 21.9(1) Å, respectively, indicating that not only alkaline metal but also EDA is intercalated between the Se-Se layers of FeSe. Bulk superconductivity has been observed below 45 K in the χ measurements of as-intercalated samples of A x (C 2 H 8 N 2 ) y Fe 2-z Se 2 (A = Li, Na). Moreover, zero resistivity has been observed below 18 K for the sintered pellet sample decreases with increasing post-annealing temperature and that deintercalation of EDA from the as-intercalated sample takes place at low temperatures below 250℃ and that FeSe remains at temperatures around 400℃. It has also turned out that the maximum T c 's of FeSe-based superconductors rise with increasing distance between the neighboring Fe layers and are saturated at about 45 K above 9 Å. Figure 4 4shows powder x-ray diffraction patterns of the as-intercalated sample and samples post-annealed at various temperatures for 60 h. For samples post-annealed below 200 ℃ , diffraction peaks of Li x (C 2 H 8 N 2 ) y Fe 2-z Se 2 remain. For samples post-annealed at 250℃ and 300℃, only peaks due to FeSe are observed, and these peaks disappear for the sample post-annealed at 500℃. Accordingly, the first step of the TG curve below 100℃ may be due to desorption of EDA on the surface of grains. The Fig. 2 . 2Temperature dependence of the magnetic susceptibility, χ, in a magnetic field of 10 Oe on zero-field cooling (ZFC) and field cooling (FC) for as-intercalated powdery samples consisting of A x (C 2 H 8 N 2 ) y Fe 2-z Se 2 ((a)A = Li, (b)A = Na) and FeSe. The inset shows the temperature dependence of the electrical resistivity, ρ, for the sintered (200℃, 20 h) pellet sample consisting of Li x (C 2 H 8 N 2 ) y Fe 2-z Se 2 and FeSe. Fig. 3 . 3Thermogravimetric (TG) curve on heating at the rate of 1℃/min for the as-intercalated powdery sample consisting of Li x (C 2 H 8 N 2 ) y Fe 2-z Se 2 and FeSe. Fig. 4 . 4the temperature dependence of χ in a magnetic field of 10 Oe on ZFC and FC for as-intercalated and post-annealed (100 -500℃, 60 h) powdery samples. The T c of Li x (C 2 H 8 N 2 ) y Fe 2-z Se 2 decreases with increasing post-annealing temperature. For samples post-annealed at 250℃ and 300℃, only the superconducting transition of FeSe is observed, and the transition also disappears for the sample post-annealed at 500℃. These results are in good correspondence to the results of x-ray diffraction. Powder x-ray diffraction patterns of as-intercalated and post-annealed (100 -500℃, 60 h) samples using Cu K α radiation. Indexes without and with asterisk are based on the ThCr 2 Si 2 -type and PbO-type structures, respectively. Peaks marked by▼ are unknown. Fig. 5 . 5Temperature dependence of the magnetic susceptibility, χ, in a magnetic field of 10 Oe on zero-field cooling (ZFC) and field cooling (FC) for as-intercalated and post-annealed (100 -500℃, 60 h) powdery samples consisting of Li x (C 2 H 8 N 2 ) y Fe 2-z Se 2 and FeSe. of Li x (C 2 H 8 N 2 ) y Fe 2-z Se 2 . It seems that the high-T c of A x (C 2 H 8 N 2 ) y Fe 2-z Se 2 (A = Li, Na) is caused by the possible two-dimensional electronic structure due to the large c-axis length. Through the post-annealing in an evacuated glass tube, it has been found that T Fig. 6 . 6Relation between the maximum T c and the distance of the neighbouring Fe layers of FeSe-based superconductors. while the second is due to that of the non-intercalated region of FeSe. The inset shows the temperature dependence of ρ for the sintered (200℃, 20 h) pellet sample. Although the as-intercalated sample pelletized simply at room temperature did not show zero resistivity, it is found that the resistivity of the sintered pellet sample starts to decrease at 43 K with decreasing10 20 30 40 50 60 2 (deg.) Intensity (arb. units) Li x (C 2 H 8 N 2 ) y Fe 2-z Se 2 + FeSe N a x (C 2 H 8 N 2 ) y F e 2-z S e 2 + FeSe 002 001* 004 006 101* 103 111* 112* 200* 003* 201* 103* 211* 002 004 001* 200* 112* 111* 002* T c ( K ) cKDistance between neighboring Fe layers (Å)Li x (C 5 H 5 N) y Fe 2-z Se 2[7] FeSe 0.5 Te 0.5 [11] ] FeSe [1] Li x (C 5 H 5 N) y Fe 2-z Se 2 [7] Na x (C 2 H 8 N 2 ) y Fe 2-z Se 2 (present work) Li x (C 2 H 8 N 2 ) y Fe 2-z Se 2 (present work) K 0.87 Fe 2.19 Se 2 [12] Cs 0.8 (FeSe 0.98 ) 2 [9] Ba 0.64 Fe 2 Se 1.62 [4] Na 0.61 Fe 2 Se 1.9 [4] Li x Fe 2 Se 2-y [4] Li 0.9 (NH 3 ) 0.5 Fe 2 Se 2 [5] Rb 0.8 Fe 2 Se 2 [10] K x Fe 2 Se 2 [3] . F.-C Hsu, J.-Y Luo, K.-W Yeh, T.-K Chen, T.-W Huang, P M Wu, Y.-C Lee, Y.-L Huang, Y.-Y Chu, D.-C Yan, M.-K Wu, Proc. Natl. Acad. Sci. U.S.A. 105F.-C. Hsu, J.-Y. Luo, K.-W. Yeh, T.-K. Chen, T.-W. Huang, P. M. Wu, Y.-C. Lee, Y.-L. Huang, Y.-Y. Chu, D.-C. Yan, and M.-K. Wu, Proc. Natl. Acad. Sci. U.S.A 105 (2008) 14262 -14264. . 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Conder, J. Phys.: Condens. Matter 23 (2011) 052203/1 -4. . A F Wang, J J Ying, Y J Yan, R H Liu, X G Luo, Z Y Li, X F Wang, M Zhang, G J Ye, P Cheng, Z J Xiang, X H Chen, Phys. Rev. B. 83A. F. Wang, J. J. Ying, Y. J. Yan, R. H. Liu, X. G. Luo, Z. Y. Li, X. F. Wang, M. Zhang, G. J. Ye, P. Cheng, Z. J. Xiang, and X. H. Chen, Phys. Rev. B 83 (2011) 060512(R)/1 -4. . K.-W Yeh, T.-W Huang, Y.-L Huang, T.-K Chen, F.-C Hsu, P M Wu, Y.-C , K.-W. Yeh, T.-W. Huang, Y.-L. Huang, T.-K. Chen, F.-C. Hsu, P. M. Wu, Y.-C. . Y.-Y Lee, C.-L Chu, J.-Y Chen, D.-C Luo, M.-K Yan, Wu, Europhys. Lett. 84Lee, Y.-Y. Chu, C.-L. Chen, J.-Y. Luo, D.-C. Yan, and M.-K. Wu: Europhys. Lett. 84 (2008) 37002/1 -4. . A. -M Zhang, T. -L Xia, K Liu, W Tong, Z. -R Yang, Q. -M Zhang, Sci. Rep. 3A. -M. Zhang, T. -L. Xia, K. Liu, W. Tong, Z. -R. Yang, and Q. -M. Zhang, Sci. Rep. 3 (2013) 1216/1 -5.

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{'abstract': 'New iron-based intercalation superconductors A x (C 2 H 8 N 2 ) y Fe 2-z Se 2 (A = Li, Na) with T c = 45 K have successfully been synthesized via intercalation of dissolved alkaline metal in ethylenediamine. The c-axis lengths of A x (C 2 H 8 N 2 ) y Fe 2-z Se 2 (A = Li, Na) are 20.74(7) Å and 21.9(1) Å, respectively, and are about 50 % larger than that of K x Fe 2 Se 2 , indicating that not only alkaline metal but also ethylenediamine is intercalated between the Se-Se layers of FeSe. It seems that the high-T c of A x (C 2 H 8 N 2 ) y Fe 2-z Se 2 (A = Li, Na) is caused by the possible two-dimensional electronic structure due to the large c-axis length. Through the post-annealing in an evacuated glass tube, it has been found that T c decreases with increasing post-annealing temperature and that deintercalation of EDA from the as-intercalated sample takes place at low temperatures below 250℃.', 'arxivid': '1408.0346', 'author': ['Takashi Noji \nDepartment of Applied Physics\nGraduate School of Engineering\nTohoku University\n6-6-05 Aoba, Aoba-ku980-8579Aramaki, SendaiJapan\n', 'Takehiro Hatakeda \nDepartment of Applied Physics\nGraduate School of Engineering\nTohoku University\n6-6-05 Aoba, Aoba-ku980-8579Aramaki, SendaiJapan\n', 'Shohei Hosono \nDepartment of Applied Physics\nGraduate School of Engineering\nTohoku University\n6-6-05 Aoba, Aoba-ku980-8579Aramaki, SendaiJapan\n', 'Takayuki Kawamata \nDepartment of Applied Physics\nGraduate School of Engineering\nTohoku University\n6-6-05 Aoba, Aoba-ku980-8579Aramaki, SendaiJapan\n', 'Masatsune Kato \nDepartment of Applied Physics\nGraduate School of Engineering\nTohoku University\n6-6-05 Aoba, Aoba-ku980-8579Aramaki, SendaiJapan\n', 'Yoji Koike \nDepartment of Applied Physics\nGraduate School of Engineering\nTohoku University\n6-6-05 Aoba, Aoba-ku980-8579Aramaki, SendaiJapan\n'], 'authoraffiliation': ['Department of Applied Physics\nGraduate School of Engineering\nTohoku University\n6-6-05 Aoba, Aoba-ku980-8579Aramaki, SendaiJapan', 'Department of Applied Physics\nGraduate School of Engineering\nTohoku University\n6-6-05 Aoba, Aoba-ku980-8579Aramaki, SendaiJapan', 'Department of Applied Physics\nGraduate School of Engineering\nTohoku University\n6-6-05 Aoba, Aoba-ku980-8579Aramaki, SendaiJapan', 'Department of Applied Physics\nGraduate School of Engineering\nTohoku University\n6-6-05 Aoba, Aoba-ku980-8579Aramaki, SendaiJapan', 'Department of Applied Physics\nGraduate School of Engineering\nTohoku University\n6-6-05 Aoba, Aoba-ku980-8579Aramaki, SendaiJapan', 'Department of Applied Physics\nGraduate School of Engineering\nTohoku University\n6-6-05 Aoba, Aoba-ku980-8579Aramaki, SendaiJapan'], 'corpusid': 96420600, 'doi': '10.1016/j.physc.2014.01.007', 'github_urls': [], 'n_tokens_mistral': 6260, 'n_tokens_neox': 5311, 'n_words': 2925, 'pdfsha': '701a60fb386b24bf0e805b0c3f80577ae5518853', 'pdfurls': ['https://export.arxiv.org/pdf/1408.0346v1.pdf'], 'title': ['Synthesis and post-annealing effects of alkaline-metal-ethylenediamine-intercalated superconductors A x (C 2 H 8 N 2 ) y Fe 2-z Se 2 (A = Li, Na) with T c = 45 K', 'Synthesis and post-annealing effects of alkaline-metal-ethylenediamine-intercalated superconductors A x (C 2 H 8 N 2 ) y Fe 2-z Se 2 (A = Li, Na) with T c = 45 K'], 'venue': []}

arxiv

ULTRA BROADBAND COHERENT DIFFRACTIVE IMAGING A FAST MONOCHROMATIZATION METHOD A PREPRINT February 20, 2023 Boyang Li Institute of Optics and Precision Mechanics of Chinese Academy of Sciences 710119Xi'an, Xi'anChina ULTRA BROADBAND COHERENT DIFFRACTIVE IMAGING A FAST MONOCHROMATIZATION METHOD A PREPRINT February 20, 2023 Coherent diffractive imaging (CDI) as a lensless imaging technique, has been applied in a wide range of sciences. However it is not compatible with short pulse lasers, because it is limited with monochromatic illumination. Although there have been reports on the broadband CDI, the bandwidth is limited with 10% for robust imaging. In this article, we report a algorithm which is able to CDI robustly with at least 80% bandwidth. This algorithm is apply to published experiment data and simulated ultra-broadband data to demonstrate its ability. This technique enables us to capture the non-repeatable transient dynamics as short as the laser pulse width, which is one step forward to the ultrafast CDI. Introduction Since the first experimental demonstration of the coherent diffractive imaging (CDI) in 1999 [1], CDI has been widely utilized to biology [2,3,4,5,6,7,8,9] and material [10,11,12,13,14,15] sciences. With the emerging of ultrafast lasers, the idea of capturing ultrafast dynamics was brought to the table. The broadband CDI was first achieved in experiments recently [16], making femtosecond ultrafast CDI possible. However in the current algorithm can only recover the image with bandwidth no longer than 10%. Although there has been reports of Atto-CDI [17]. However it is ptychography that need repetitive scanning, making ultra-fast imaging impossible. In this article, we report an algorithm to recover monochromatic pattern with at least 80% bandwidth of single-shot pattern. The monochromatization problem can be expressed in linear equations [16]: Ax = b A = a 0 I + a 1 A 1 + a 2 A 2 + ...(1) Where x is the monochromatic diffraction pattern and b is the polluted pattern due to the broadband illumination. a i is the relative intensity of each wavelength. A i is the transfer function which transfer the pattern from one wavelength to another. The A i can be deduced from interpolation [16]. In the literature, A matrix is sparse, which can be save in the memory and solved later with convensional linear solvers. However, this interpolation method is not precise which causes this method limited with only 10% bandwidth. This part will be discussed in section 2 and a new expression will be come up with to break this limit. In the new expression, the A matrix is no longer sparse, and can not be saved in memory any more, meaning convensional solvers no longer available. To overcome this, we provide a gradiant iterative solver for the equation, which will be in section 3. Broadband diffraction In this section, we explain our derivation of A (referred as FFT method later) and compare our results with the benchmark [16] (referred as interpolation method). Different wave length produce different diffraction pattern. Assuming the object have the same complex refractive index, different wave length have actually the same pattern but with different zoom. Take Fig. 1(a), a 150 × 150 image of Einstein oversampled by 2, making a 300 × 300 image as an example. With correct setups, we have a Fraunhofer diffraction of Fig. 1(a), as shown in Fig. 1(b) for λ 0 . So the diffraction pattern is 300 × 300. Now to get the diffraction pattern of 2λ 0 , we double the oversampling ratio, making 600 × 600 as shown in Fig. 1(c). Since our sensor is limited in size, we are trading resolution for FOV when switching to 2λ. So the diffraction pattern is actually 300 × 300 with high resolution cropped, as shown in Fig. 1(g). It is not hard to find that the information in Fig. 1(b) is redundant. Because we are only padding 0's outside the object to increases the resolution of diffraction pattern. So we are able to derive diffraction pattern of longer wave length from shorter wave length not vise versa. Because we lost information when cropping the high frequncy pixels. Until now, we are talking about the object and amplitude. This process can also be applied to autocorrelation and intensity, because with oversampling ratio larger than 2, we are safe to pad 0's to autocorrelation as well. So now we can write down the matrix A in operator form i and matrix form A i as following: A i (x) = CROP i {FFT{PAD i [IFFT(x)]}} A i,mn = 1 W L Σ W 2 +Pi,x, L 2 +Pi,y k=− W 2 −Pi,x,l=− L 2 −Pi,y exp j2π k( x m W + 2P i,x − x n W ) + l( y m L + 2P i,y − y n N )(2) Since the oversampling ratio of pattern x is not smaller than 2, the autocorrelation of the object is constrained in W ×L image. So A can be re-written as: A i,mn = 1 W L Σ W 2 , L 2 k=− W 2 ,l=− L 2 exp j2π k( x m W + 2P i,x − x n W ) + l( y m L + 2P i,y − y n N )(3) By swapping m and n, i 's transpose T i can be expressed in the form: A T i (x) = FFT{CROP i {IFFT[PAD i (x)]}}(4) It is not hard to find that matrix A i 's rank is 0, since after the transfer, the high frequency information is lost. However, A i matrix is approximately orthogonal in low frequency domain, which means the operation T i i is an identity matrix in low frequency domain and 0 in high frequency: A T iÂi =          1 . . . 1 0 . . . 0         (5) Which is quite obvious because outer pixels are cropped and then padded back with 0's. Now we discuss the difference between FFT method and interpolation method. By taking a second look at the matrix in Eq. (2). A i,mn only have large values for: x m ≈ (W + 2P i,x ) x n W y m ≈ (L + 2P i,y ) y n L(6) Which is in good agreement with the interpolation method. However in FFT method, the matrix A is no longer sparse. It has small values all over the place, which is a huge disadvantage. Because we are no longer able to save the matrix into memory and solve the equation with existing solvers. Yet FFT method has it's unbeatable advantage: It is precise, and reliable. We can compare these two method for λ → 2λ transformation. Fig. 1(d-f) shows the result of interpolation method and Fig. 1(g-h) for FFT method. The interpolation method cannot guarentee the outer part of autocorrelation to be 0. So the reconstruction method fails to recover the image. However, the FFT method is doing a good job on this. So we will stick to the FFT method for the monochromatization. ultra broadband coherent diffractive imaging a fast monochromatization method Figure 1: Algorithm comparison of interpolation [16] and our FFT method. ultra broadband coherent diffractive imaging a fast monochromatization method A PREPRINT A PREPRINT (a) (b) (c) (d) (e) (f) (g) (h) (i) Monochromatization solver The residual and the gradiant of the linear equation in n-th iteration is: n = |∆b n | 2 = |b − Σ i a iÂi (x n )| 2 ∇ n = 2∇(∆b n ) · ∆b n = −2Σ i a i T i (∆b n )(7) Where x n here is the result value of the n-th iteration. The gradiant method is: x 0 = b x n+1 = x n − α∇ n(8) Where α is the step size which could be set to 1 from our experience. A momentum can be added to the iteration to speed up the convergance: v 0 = 0 v n+1 = (v n − ∇ n )(1 − f · dt) x n+1 = x n + v n+1 · dt(9) With parameters dt = 1 and f = 0.2 for our reconstruction. After each iteration, we force x to be non-negative to make sure the result is physically meaningful. The broadband diffraction data was taken from [16] as shown in Fig. 2(a). After only 20 iterations, we get the monochromatic pattern as shown in Fig. 2(b). The reconstructed image using 2000 iterations of HIO with shrinking-wrap each 20 iterations is in Fig. 2(c). (a) (b) (c) Figure 2: Algorithm testing using data from [16]. This algorithm runs entirely on GPU. The code was running on a machine with RTX 3060Ti. Little memory was used. It only took a few seconds to get Fig. 2(b), which is super computation friendly. Another simulation is used to test the ability of this algorithm on ultra-broadband discrete spectra. We simulated diffraction of 3,5,7,9,11 harmonics of data from MNIST database extended as a 128 × 128 image as shown in Fig 3(a) and (b). Where the oversampling of 11 harmonics is 2. The relative intensity is 0.2,0.4,0.4,0.3,0.2. Solved monochromatic pattern is shown in Fig 3(c). The reconstructed image with 1000 RAAR iterations and shrinking wrap each 20 iteration is shown in Fig 3(d). A ultra-broadband continuous spectrum illumination is also simulated and tested. The normalized spectrum is shown in Fig 4 (a) with λ c = 2.5 and ∆λ/λ c = 80%. 384 points are taken for simulation and monochromatization. The oversampling of λ = 1 is 2. The same recipes as Fig. 3(b-d) is shown in Fig 4(b-d). More lambda component slows down the convergance. 500 iterations was used for this monochromatization, which took 84 seconds. Figure 4 : 4Algorithm testing using continuous spectrum with 80% bandwidth. Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens. Jianwei Miao, Pambos Charalambous, Janos Kirz, David Sayre, 400Jianwei Miao, Pambos Charalambous, Janos Kirz, and David Sayre. Extending the methodology of x-ray crys- tallography to allow imaging of micrometre-sized non-crystalline specimens. 400(6742):342-344. Biological imaging by soft x-ray diffraction microscopy. D Shapiro, Proc. Natl Acad. Sci. USA. 102D. Shapiro. Biological imaging by soft x-ray diffraction microscopy. Proc. Natl Acad. Sci. USA, 102, 2005. Quantitative imaging of single, unstained viruses with coherent x rays. C Song, Phys. Rev. Lett. 101C. Song. Quantitative imaging of single, unstained viruses with coherent x rays. Phys. Rev. Lett., 101, 2008. Three-dimensional visualization of a human chromosome using coherent x-ray diffraction. Y Nishino, Phys. Rev. Lett. 102Y. Nishino. Three-dimensional visualization of a human chromosome using coherent x-ray diffraction. Phys. Rev. Lett., 102, 2009. High-resolution x-ray diffraction microscopy of specifically labeled yeast cells. J Nelson, Proc. Natl Acad. Sci. USA. 107J. Nelson. High-resolution x-ray diffraction microscopy of specifically labeled yeast cells. Proc. Natl Acad. Sci. USA, 107, 2010. Quantitative 3d imaging of whole, unstained cells by using x-ray diffraction microscopy. H Jiang, Proc. Natl Acad. Sci. USA. 107H. Jiang. Quantitative 3d imaging of whole, unstained cells by using x-ray diffraction microscopy. Proc. Natl Acad. Sci. USA, 107, 2010. Quantitative biological imaging by ptychographic x-ray diffraction microscopy. K Giewekemeyer, Proc. Natl Acad. Sci. USA. 107K. Giewekemeyer. Quantitative biological imaging by ptychographic x-ray diffraction microscopy. Proc. Natl Acad. Sci. USA, 107, 2010. Single mimivirus particles intercepted and imaged with an x-ray laser. M M Seibert, Nature. 470M. M. Seibert. Single mimivirus particles intercepted and imaged with an x-ray laser. Nature, 470, 2011. Imaging live cell in micro-liquid enclosure by x-ray laser diffraction. T Kimura, Nat. Commun. 5T. Kimura. Imaging live cell in micro-liquid enclosure by x-ray laser diffraction. Nat. Commun., 5, 2014. Three-dimensional mapping of a deformation field inside a nanocrystal. M A Pfeifer, G J Williams, I A Vartanyants, R Harder, I K Robinson, Nature. 442M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson. Three-dimensional mapping of a deformation field inside a nanocrystal. Nature, 442, 2006. High-resolution ab initio three-dimensional x-ray diffraction microscopy. H N Chapman, J. Opt. Soc. Am. A. 23H. N. Chapman. High-resolution ab initio three-dimensional x-ray diffraction microscopy. J. Opt. Soc. Am. A, 23, 2006. Nanoscale imaging of buried structures with elemental specificity using resonant x-ray diffraction microscopy. C Song, Phys. Rev. Lett. 100C. Song. Nanoscale imaging of buried structures with elemental specificity using resonant x-ray diffraction microscopy. Phys. Rev. Lett., 100, 2008. Coherent x-ray diffraction imaging of strain at the nanoscale. I Robinson, R Harder, Nat. Mater. 8I. Robinson and R. Harder. Coherent x-ray diffraction imaging of strain at the nanoscale. Nat. Mater., 8, 2009. Ultrafast three-dimensional imaging of lattice dynamics in individual gold nanocrystals. Science, 341, 2013. ultra broadband coherent diffractive imaging a fast monochromatization method A PREPRINT. J N Clark, L Beitra, G Xiong, A Higginbotham, D M Fritz, H T Lemke, J. N. Clark, L. Beitra, G. Xiong, A. Higginbotham, D. M. Fritz, and H. T. Lemke. Ultrafast three-dimensional imaging of lattice dynamics in individual gold nanocrystals. Science, 341, 2013. ultra broadband coherent diffractive imaging a fast monochromatization method A PREPRINT Three-dimensional coherent x-ray diffraction imaging of molten iron in mantle olivine at nanoscale resolution. H Jiang, Phys. Rev. Lett. 110H. Jiang. Three-dimensional coherent x-ray diffraction imaging of molten iron in mantle olivine at nanoscale resolution. Phys. Rev. Lett., 110, 2013. . Julius Huijts, Sara Fernandez, David Gauthier, Maria Kholodtsova, Ahmed Maghraoui, Kadda Medjoubi, Andrea Somogyi, Willem Boutu, and Hamed Merdji. Broadband coherent diffractive imaging. 1410Julius Huijts, Sara Fernandez, David Gauthier, Maria Kholodtsova, Ahmed Maghraoui, Kadda Medjoubi, Andrea Somogyi, Willem Boutu, and Hamed Merdji. Broadband coherent diffractive imaging. 14(10):618-622. Potential of attosecond coherent diffractive imaging. Arjun Rana, Jianhua Zhang, Minh Pham, Andrew Yuan, Yuanhung Lo, Huaidong Jiang, Stanley J Osher, Jianwei Miao, 12586101Arjun Rana, Jianhua Zhang, Minh Pham, Andrew Yuan, Yuanhung Lo, Huaidong Jiang, Stanley J. Osher, and Jianwei Miao. Potential of attosecond coherent diffractive imaging. 125(8):086101.

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{'abstract': 'Coherent diffractive imaging (CDI) as a lensless imaging technique, has been applied in a wide range of sciences. However it is not compatible with short pulse lasers, because it is limited with monochromatic illumination. Although there have been reports on the broadband CDI, the bandwidth is limited with 10% for robust imaging. In this article, we report a algorithm which is able to CDI robustly with at least 80% bandwidth. This algorithm is apply to published experiment data and simulated ultra-broadband data to demonstrate its ability. This technique enables us to capture the non-repeatable transient dynamics as short as the laser pulse width, which is one step forward to the ultrafast CDI.', 'arxivid': '2302.08898', 'author': ["Boyang Li \nInstitute of Optics and Precision Mechanics of Chinese Academy of Sciences\n710119Xi'an, Xi'anChina\n"], 'authoraffiliation': ["Institute of Optics and Precision Mechanics of Chinese Academy of Sciences\n710119Xi'an, Xi'anChina"], 'corpusid': 257019980, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 4546, 'n_tokens_neox': 3766, 'n_words': 2258, 'pdfsha': '3518c6d2ef10397311291bbe6db5d2ed8d76bca3', 'pdfurls': ['https://export.arxiv.org/pdf/2302.08898v1.pdf'], 'title': ['ULTRA BROADBAND COHERENT DIFFRACTIVE IMAGING A FAST MONOCHROMATIZATION METHOD A PREPRINT', 'ULTRA BROADBAND COHERENT DIFFRACTIVE IMAGING A FAST MONOCHROMATIZATION METHOD A PREPRINT'], 'venue': []}

arxiv

Benchmarking of eight recurrent neural network variants for breath phase and adventitious sound detection on a self-developed open-access lung sound database-HF_Lung_V1 Short Title: Automated lung sound analysis database † Corresponding Author Fu-Shun Hsu Graduate Institute of Biomedical Electronics and Bioinformatics National Taiwan University 10617TaipeiTaiwan Department of Critical Care Medicine Far Eastern Memorial Hospital 22060New TaipeiTaiwan Heroic Faith Medical Science Co. Ltd 23553New TaipeiTaiwan Shang-Ran Huang Heroic Faith Medical Science Co. Ltd 23553New TaipeiTaiwan Chien-Wen Huang Avalanche Computing Inc 10687TaipeiTaiwan Chao-Jung Huang Yuan-Ren Cheng Heroic Faith Medical Science Co. Ltd 23553New TaipeiTaiwan Department of Life Science College of Life Science National Taiwan University 10617TaipeiTaiwan Institute of Biomedical Sciences Academia Sinica 11529TaipeiTaiwan Chun-Chieh Chen Heroic Faith Medical Science Co. Ltd 23553New TaipeiTaiwan Avalanche Computing Inc 10687TaipeiTaiwan Jack Hsiao HCC Healthcare Group 22060New TaipeiTaiwan Chung-Wei Chen Department of Critical Care Medicine Far Eastern Memorial Hospital 22060New TaipeiTaiwan Li-Chin Chen Research Center for Information Technology Innovation Graduate Institute of Biomedical Electronics and Bioinformatics Academia Sinica 11529TaipeiTaiwan National Taiwan University No. 1, Sec. 4, Roosevelt Road Taipei 10617Taiwan Yen-Chun Lai Heroic Faith Medical Science Co. Ltd 23553New TaipeiTaiwan Bi-Fang Hsu Heroic Faith Medical Science Co. Ltd 23553New TaipeiTaiwan Nian-Jhen Lin Heroic Faith Medical Science Co. Ltd 23553New TaipeiTaiwan Wan-Lin Tsai Heroic Faith Medical Science Co. Ltd 23553New TaipeiTaiwan Yi-Lin Wu Heroic Faith Medical Science Co. Ltd 23553New TaipeiTaiwan Tzu-Ling Tseng Heroic Faith Medical Science Co. Ltd 23553New TaipeiTaiwan Ching-Ting Tseng Yi-Tsun Chen Heroic Faith Medical Science Co. Ltd 23553New TaipeiTaiwan Feipei Lai Graduate Institute of Biomedical Electronics and Bioinformatics National Taiwan University 10617TaipeiTaiwan Heroic Faith Medical Science Co. Ltd 23553New TaipeiTaiwan Feipei Lai Benchmarking of eight recurrent neural network variants for breath phase and adventitious sound detection on a self-developed open-access lung sound database-HF_Lung_V1 Short Title: Automated lung sound analysis database † Corresponding Author 1 9 Division of Pulmonary Medicine, Far Eastern Memorial Hospital, New Taipei 22060, Taiwan 2 3adventitious soundauscultationconvolutional neural networkslung soundrecurrent neural networksrespiratory monitor 5 A reliable, remote, and continuous real-time respiratory sound monitor with automated respiratory sound analysis ability is urgently required in many clinical scenarios-such as in monitoring disease progression of coronavirus disease 2019-to replace conventional auscultation with a handheld stethoscope. However, a robust computerized respiratory sound analysis algorithm has not yet been validated in practical applications. In this study, we developed a lung sound database (HF_Lung_V1) comprising 9,765 audio files of lung sounds (duration of 15 s each), 34,095 inhalation labels, 18,349 exhalation labels, 13,883 continuous adventitious sound (CAS) labels (comprising 8,457 wheeze labels, 686 stridor labels, and 4,740 rhonchi labels), and 15,606 discontinuous adventitious sound labels (all crackles). We conducted benchmark tests for long short-term memory (LSTM), gated recurrent unit (GRU), bidirectional LSTM (BiLSTM), bidirectional GRU (BiGRU), convolutional neural network (CNN)-LSTM, CNN-GRU, CNN-BiLSTM, and CNN-BiGRU models for breath phase detection and adventitious sound detection. We also conducted a performance comparison between the LSTM-based and GRU-based models, between unidirectional and bidirectional models, and between models with and without a CNN. The results revealed that these models exhibited adequate performance in lung sound analysis. The GRU-based models outperformed, in terms of F1 scores and areas under the receiver operating characteristic curves, the LSTM-based models in most of the defined tasks. Furthermore, all bidirectional models outperformed their unidirectional counterparts.Finally, the addition of a CNN improved the accuracy of lung sound analysis, especially in the CAS 4 detection tasks. Introduction Respiration is vital for the normal functioning of the human body. Therefore, clinical physicians are frequently required to examine respiratory conditions. Respiratory auscultation (Bohadana et al., 2014;Goettel & Herrmann, 2019;Sarkar et al., 2015) using a stethoscope has long been a crucial first-line physical examination. The chestpiece of a stethoscope is usually placed on a patient's chest or back for lung sound auscultation or over the patient's tracheal region for tracheal sound auscultation. During auscultation, breath cycles can be inferred, which help clinical physicians evaluate the patient's respiratory rate. In addition, pulmonary pathologies are suspected when the frequency or intensity of respiratory sounds changes or when adventitious sounds, including continuous adventitious sounds (CASs) and discontinuous adventitious sounds (DASs), are identified (Bohadana et al., 2014;Goettel & Herrmann, 2019;Pramono et al., 2017). Patients with coronavirus disease 2019 exhibit adventitious sounds ; hence, auscultation may be a useful approach for disease diagnosis (Raj et al., 2020) and disease progression tracking. However, auscultation performed using a conventional handheld stethoscope involves some limitations (Sovijärvi et al., 1997). First, the interpretation of auscultation results substantially depends on the subjectivity of the practitioners. Even experienced clinicians might not have high consensus rates in their interpretations of auscultatory manifestations (Berry et al., 2016;Grunnreis, 2016). Second, auscultation is a qualitative analysis method. Comparing auscultation results between individuals and quantifying the sound change by reviewing historical records are difficult tasks. Third, prolonged continuous monitoring of respiratory sound is almost impractical. To overcome the aforementioned limitations, computerized methods for respiratory sound recording and analyses based on traditional signal processing and machine learning have been proposed and reviewed (Gurung et al., 2011;Huq & Moussavi, 2012;Mesaros et al., 2016;Pasterkamp et al., 1997;Pramono et al., 2017). With the advent of the deep learning era, studies have developed novel deep learning-based methods for respiratory sound analysis. However, many of such studies have focused on only distinguishing healthy participants from participants with respiratory disorders (Chambres et al., 2018;Demir et al., 2020;Hosseini et al., 2020;Perna & Tagarelli, 2019;Pham et al., 2020) and distinguishing various types of normal breathing sounds from adventitious sounds (Acharya & Basu, 2020;Aykanat et al., 2017;Bardou et al., 2018;Chen et al., 2019;Grzywalski et al., 2019;Kochetov et al., 2018;Li et al., 2016). Only a few studies (Hsiao et al., 2020;Jácome et al., 2019;Liu et al., 2017;Messner et al., 2018) have explored the use of deep learning for detecting breath phases and adventitious sounds. Moreover, most previous studies on computerized lung sound analysis have been limited by insufficient data. As of writing this paper, the largest reported respiratory sound database is ICBHI 2017 Challenge (Rocha et al., 2017), which comprises 6,898 breath cycles and 10,775 events of wheezes and crackles acquired from 126 individuals. Data size plays a major role in the creation of a robust and accurate deep learning-based respiratory sound analysis algorithm (Hestness et al., 2017;Sun et al., 2017). Accordingly, the first aim of the present study was to establish a large and open-access respiratory sound database for training such algorithms for the detection of breath phase and adventitious sounds, mainly focusing on lung sounds. The second aim was to conduct a benchmark test on the established lung sound database by using eight recurrent neural network (RNN)-based models. RNNs (Elman, 1990) are effective for time-series analysis; long short-term memory (LSTM; Hochreiter & Schmidhuber, 1997) and gated recurrent unit (GRU; Cho et al., 2014) networks, which are two RNN variants, exhibit superior performance to the original RNN model. However, whether LSTM models are superior to GRU models (and vice versa) in many applications, particularly in respiratory sound analysis, is inconclusive. Bidirectional RNN models (Graves & Schmidhuber, 2005;Schuster & Paliwal, 1997) can transfer not only past information to the future but also future information to the past; these models consistently exhibit superior performance to unidirectional RNN models in many applications (Khandelwal et al., 2016;Linchuan Li et al., 2016;Parascandolo et al., 2016) as well as in breath phase and crackle detection (Messner et al., 2018). However, whether bidirectional RNN models outperform unidirectional RNN models in CAS detection has yet to be determined. Furthermore, the convolutional neural network (CNN)-RNN structure has been proven to be suitable for heart sound analysis (Deng et al., 2020), lung sound analysis (Acharya & Basu, 2020), and other tasks (Linchuan Li et al., 2016;Zhao et al., 2018). Nevertheless, the application of the CNN-RNN structure in respiratory sound detection has yet to be fully investigated. Benchmarking can enable demonstrating the reliability and goodness of a database; it can also be applied to investigate the 8 performance of the RNN variants in respiratory analysis. In summary, the aims of this study are outlined as follows: ◼ Establish the largest open-access lung sound database as of writing this paper-HF_Lung_V1 (https://gitlab.com/techsupportHF/HF_Lung_V1). ◼ Conduct a performance comparison between LSTM and GRU models, between unidirectional and bidirectional models, and between models with and without a CNN in breath phase and adventitious sound detection based on lung sound data. ◼ Discuss factors influencing model performance. Establishment of the lung sound database Data sources and patients The lung sound database was established using two sources. The first source was a database used in a datathon in Taiwan Helsinki Declaration and its later amendments or comparable ethical standards. All patients were Taiwanese and aged older than 20 years. Descriptive statistics regarding the patients' demographic data, major diagnosis, and comorbidities are presented in Table 1; however, information on the patients in the TSECC database is missing. Moreover, all 18 RCW/RCC residents were under mechanical ventilation. Sound recording Breathing lung sounds were recorded using two devices: (1) a commercial electronic stethoscope (Littmann 3200, 3M, Saint Paul, Minnesota, USA) and (2) a customized multichannel acoustic recording device (HF-Type-1) that supports the connection of eight electret microphones. The signals collected by the HF-Type-1 device were transmitted to a tablet (Surface Pro 6, Microsoft, Redmond, Washington, USA; Fig. 1). Breathing lung sounds were collected at the eight locations All lung sounds in the TSECC database were collected using the Littmann 3200 device only, where 15.8-s recordings were obtained sequentially from L1 to L8 ( Fig. 2b; Littmann 3200). One round of recording with the Littmann 3200 device entails a recording of lung sounds from L1 to L8. The TSECC database was composed of data obtained from one to three rounds of recording with the Littmann 3200 device for each patient. We recorded the lung sounds of the 18 RCW/RCC residents by using both the Littmann 3200 device and the HF-Type-1 device. The Littmann 3200 recording protocol was in accordance with that used in the TSECC database, except that data from four to five rounds of lung sound recording were collected instead. The HF-Type-1 device was used to record breath sounds at L1, L2, L4, L5, L6, and L8. One round of recording with the HF-Type-1 device entails a synchronous and continuous recording of breath sounds for 30 min ( Fig. 2b; HF-Type-1). However, the recording with the HF-Type-1 device was occasionally interrupted; in this case, the recording duration was <30 min. Voluntary deep breathing was not mandated during the recording of lung sounds. The statistics of the recordings are listed in Table 2. recorded sequentially from L1 to L8. Subsequently, all recordings were truncated to 15 s. When the HF-Type-1 device was used, sounds at L1, L2, L4, L5, L6, and L8 were recorded simultaneously. Subsequently, each 2-min signal was truncated to generate new 15-s audio files. Audio file truncation In this study, the standard duration of an audio signal used for inhalation, exhalation, and adventitious sound detection was 15 s. This duration was selected because a 15-s signal contains at least three complete breath cycles, which are adequate for a clinician to reach a clinical conclusion. Table 2 Statistics of recordings and labels of HF_Lung_V1 database. Because each audio file generated by the Littmann 3200 device had a length of 15.8 s, we cropped out the final 0.8-s signal from the files ( Fig. 2b; Littmann 3200). Moreover, we used only the first 15 s of each 2-min signal of the audio files ( Fig. 2b; HF-Type-1) generated by the HF-Type-1 device. Table 2 presents the number of truncated 15-s recordings and the total duration. Data labeling Because the data in the TSECC database contains only classification labels indicating whether a CAS or DAS exists in a recording, we attempted to label the event level of all sound recordings. Two board-certified respiratory therapists (NJL and YLW) and one board-certified nurse (WLT), with 8, 3, and 13 years of clinical experience, respectively, were recruited to label the start and end points of inhalation (I), exhalation (E), wheeze (W), stridor (S), rhonchus (R), and DAS (D) events in the recordings. They labeled the sound events by listening to the recorded breath sounds while simultaneously observing the corresponding patterns on a spectrogram by using customized labeling software (Hsu et al., 2021). The labelers were asked not to label sound events if they could not clearly identify the corresponding sound or if an incomplete event at the beginning or end of an audio file caused difficulty in identification. BFH held regular meetings to ensure that the labelers had good agreement on labeling criteria based on a few samples by judging the mean pseudo-κ value (Jácome et al., 2019). When developing artificial intelligence (AI) detection models, we combined the W, S, and R labels to form CAS labels (C). Moreover, the D labels comprised only crackles, which were not differentiated into coarse or fine crackles. The labelers were asked to label the period containing crackles but not a single explosive sound (generally less than 25 ms) of a crackle. Each recording was annotated by only one labeler; thus, the labels did not represent perfect ground truth. However, we used the labels as ground-truth labels for model training, validation, and testing. The statistics of the labels are listed in Table 2. Inhalation, exhalation, CAS, and DAS detection Framework The inhalation, exhalation, CAS, and DAS detection framework developed in this study is displayed in Fig. 3. The prominent advantage of the research framework is its modular design. Specifically, each unit of the framework can be tested separately, and the algorithms in different parts of the framework can be modified to achieve optimal overall performance. Moreover, the output of some blocks can be used for multiple purposes. For instance, the spectrogram generated by the preprocessing block can be used as the input of a model or for visualization in the user interface for real-time monitoring. Preprocessing We processed the lung sound recordings at a sampling frequency of 4 kHz. First, to eliminate the 60-Hz electrical interference and a part of the heart sound noise, we applied a high-pass filter to the recordings by setting a filter order of 10 and cut-off frequency of 80 Hz. The filtered signals were then processed using the short-time Fourier transform (STFT). In the STFT, we set a Hanning window size of 256 and hop length of 64; no additional zero-padding was applied. Thus, a 15-s sound signal could be transformed into a corresponding spectrogram with a size of 938 × 129. To obtain the spectral information regarding the lung sounds, we extracted the following features (Chamberlain et al., 2016;Messner et al., 2018): ◼ Spectrogram: We extracted 129-bin log-magnitude spectrograms. ◼ Mel frequency cepstral coefficients (MFCCs): We extracted 20 static coefficients, 20 delta coefficients (Δ), and 20 acceleration coefficients (Δ 2 ). We used 40 mel bands within a frequency range of 0-4,000 Hz. The frame width used to calculate the delta and acceleration coefficients was set to 9, which resulted in a 60-bin vector per frame. ◼ Energy summation: We computed the energy summation of four frequency bands, namely 0-250, 250-500, 500-1,000, and 0-2,000 Hz, and obtained four values per time frame. After extracting the aforementioned features, we concatenated them to form a 938 × 193 feature matrix. Subsequently, we conducted min-max normalization on each feature. The values of the normalized features ranged between 0 and 1. Deep learning models We investigated the performance of eight RNN models, namely LSTM, GRU, bidirectional We used Adam as the optimizer in the benchmark model, and we set the initial learning rate to 0.0001 with a step decay (0.2×) when the validation loss did not decrease for 10 epochs. The learning process stopped when no improvement occurred over 50 consecutive epochs. Postprocessing The prediction vectors obtained using the adopted models can be further processed for different purposes. For example, we can transform the prediction result from frames to time for real-time monitoring. The breathing duration of most humans lies within a certain range; we considered this fact in our study. Accordingly, when the prediction results obtained using the models indicated that two consecutive inhalation events occurred within a very small interval, we checked the continuity of these two events and decided whether to merge them, as illustrated in the bottom panel of Fig. 4a. For example, when the interval between the jth and ith events was smaller than T s, we computed the difference in frequency between their energy peaks (| − |). Subsequently, if the difference was below a given threshold P, the two events were merged into a single event. In the experiment, T was set to 0.5 s, and P was set to 25 Hz. After the merging process, we further assessed whether a burst event existed. If the duration of an event was shorter than 0.05 s, the event was deleted. Dataset arrangement and cross-validation We adopted fivefold cross-validation in the training dataset to train and validate the models. Moreover, we used an independent testing dataset to test the performance of the trained models. According to our preliminary experience, the acoustic patterns of the breath sounds collected from one patient at different auscultation locations or between short intervals had many similarities. To avoid potential data leakage caused by our methods of collecting and truncating the breath sound signals, we assigned all truncated recordings collected on the same day to only one of the training, validation, or testing datasets; this is because these recordings might have been collected from the same patient within a short period. The statistics of the datasets are listed in Table 3. We used only audio files containing CASs and DASs to train and test their corresponding detection models. Table 3 Statistics of the datasets and labels of the HF_Lung_V1 database. Pramono et al. (2017) clearly defined classification and detection at the segment, event, and recording levels. In this study, we performed two tasks. The first task involved performing detection at the segment level. The acoustic signal of each lung sound recording was transformed into a spectrogram. The temporal resolution of the spectrogram depended on the window size and overlap ratio of the STFT. The aforementioned parameters were fixed such that each spectrogram was a matrix of size 938 × 129. Thus, each recording contained 938 time segments (time frames), and each time segment was automatically labeled (Fig. 5b) according to the ground-truth event labels (Fig. 5a) assigned by the labelers. The output of the prediction process was a sequential prediction matrix ( The second task entailed event detection at the recording level. After completing the sequential prediction (Fig. 5c), we assembled the time segments associated with the same label into a corresponding event (Fig. 5e). We also derived the start and end times of each assembled event. The Jaccard index (JI; Jácome et al., 2019) was used to determine whether an AI inference result correctly matched the ground-truth event. For an assembled event to be designated as a TP event (orange horizontal bars in Fig. 5e), the corresponding JI value must be greater than 0.5. If the JI was between 0 and 0.5, the assembled event was designated as an FN event (yellow horizontal bars in Fig. 5e), and if it was 0, the assembled event was designated as an FP event (black horizontal bars in Fig. 5e). A TN event cannot be defined in the task of event detection. Task definition and evaluation metrics The performance of the models was evaluated using the F1 score, and that of segment detection was evaluated using the receiver operating characteristic (ROC) curve and area under the ROC curve 25 (AUC). In addition, the mean absolute percentage error (MAPE) of event detection was derived. The accuracy, positive predictive value (PPV), sensitivity, specificity, and F1 score of the models are presented in Appendix A. Hardware and software We trained the baseline models on an Ubuntu 18.04 server that was provided by the National Table 4 presents the F1 scores used to compare the eight LSTM-and GRU-based models. When a CNN was not added, the GRU models outperformed the LSTM models by 0.7%-9.5% in terms of the F1 scores. However, the CNN-GRU and CNN-BiGRU models did not outperform the CNN-LSTM and CNN-BiLSTM models in terms of the F1 scores (and vice versa). Results LSTM versus GRU models According to the ROC curves presented in Fig. 6a-d, the GRU-based models outperformed the LSTM-based models in all compared pairs, except for one pair, in terms of DAS segment detection (AUC of 0.891 for CNN-BiLSTM vs 0.889 for CNN-BiGRU). Table 4 Comparison of F1 scores between LSTM-based models and GRU-based models. The bold values indicate the higher F1 score between the compared pairs of models. Unidirectional versus bidirectional models As presented in Table 5, the bidirectional models outperformed their unidirectional counterparts in all the defined tasks by 0.4%-9.8% in terms of the F1 scores, even when the bidirectional models had fewer trainable parameters after model adjustment. Models with CNN versus those without CNN According to Table 6, the models with a CNN outperformed those without a CNN in 26 of the 32 compared pairs. The models with a CNN exhibited higher AUC values than did those without a CNN (Fig. 6a-d), Table 5 Comparison of F1 scores between the unidirectional and bidirectional models. The bold values indicate the higher F1 score between the compared pairs of models. SIMP means the number of trainable parameters is adjusted. except that BiGRU had a higher AUC value than did CNN-BiGRU in terms of inhalation detection (0.963 vs 0.961), GRU had a higher AUC value than did CNN-GRU in terms of exhalation detection (0.886 vs 0.883), and BiGRU had a higher AUC value than did CNN-BiGRU in terms of exhalation detection (0.911 vs 0.899). Moreover, compared with the LSTM, GRU, BiLSTM, and BiGRU models, the CNN-LSTM, CNN-GRU, CNN-BiLSTM, and CNN-BiGRU models exhibited flatter and lower MAPE curves over a wide range of threshold values in all event detection tasks ( Fig. 7a-d). Table 6 Comparison of F1 scores between models without and with a CNN. The bold values indicate the higher F1 score between the compared pairs of models. Discussion Benchmark results According to the F1 scores presented in Table 4, among models without a CNN, the GRU and BiGRU models consistently outperformed the LSTM and BiLSTM models in all defined tasks. However, the GRU-based models did not have superior F1 scores among models with a CNN. Regarding the ROC curves and AUC values (Fig. 6a-d), the GRU-based models consistently outperformed the other models in all but one task. Accordingly, we can conclude that GRU-based models perform slightly better than LSTM-based models in lung sound analysis. Previous studies have also compared LSTM-and GRU-based models (Chung et al., 2014;Khandelwal et al., 2016;Shewalkar, 2018). Although a concrete conclusion cannot be drawn regarding whether LSTM-based models are superior to the GRU-based models (and vice versa), GRU-based models have been reported to outperform LSTM-based models in terms of computation time (Khandelwal et al., 2016;Shewalkar, 2018). As presented in Table 5, the bidirectional models outperformed their unidirectional counterparts in all defined tasks, a finding that is consistent with several previously obtained results (Graves & Schmidhuber, 2005;Khandelwal et al., 2016;Messner et al., 2018;Parascandolo et al., 2016). A CNN can facilitate the extraction of useful features and enhance the prediction accuracy of RNN-based models. The benefits engendered by a CNN are particularly vital in CAS detection. For the models with a CNN, the F1 score improvement ranged from 26.0% to 30.3% and the AUC improvement ranged from 0.067 to 0.089 in the CAS detection tasks. Accordingly, we can infer that considerable information used in CAS detection resides in the local positional arrangement of the features. Thus, a two-dimensional CNN facilitates the extraction of the associated information. Notably, CNN-induced improvements in model performance in the inhalation, exhalation, and DAS detection tasks were not as high as those observed in the CAS detection tasks. The MAPE curves ( Fig. 7a-d) reveal that a model with a CNN has more consistent predictions over various threshold values. In our previous study (Hsiao et al., 2020), Another reason is that an exhalation label is not always available following an inhalation label in our data. Finally, we did not specifically control the sounds we recorded; for example, we did not ask patients to perform voluntary deep breathing or keep ambient noise down. The factors influencing the model performance are further discussed in the next section. Factors influencing model performance The benchmark performance of the proposed models may have been influenced by the following factors: (1) unusual breathing patterns; (2) imbalanced data; (3) low signal-to-noise ratio (SNR); (4) noisy labels, including class and attribute noise, in the database; and (5) sound overlapping. Fig. 8 displays most of the breath patterns present in the HF_Lung_V1 database. Fig. 8a illustrates the general pattern of a breath cycle in the lung sounds when the ratio of inhalation to exhalation durations is approximately 2:1 and an expiratory pause is noted (Pramono et al., 2017;Sarkar et al., 2015). Fig. 8b presents a frequent condition under which an exhalation is not completely heard by the labelers. However, because we did not ask the subjects to breath voluntarily when recording the sound, many unusual breath patterns might have been recorded, such as patterns caused by shallow breathing, fast breathing, and apnea as well as those caused by double triggering of the ventilator (Thille et al., 2006) and air trapping (Blanch et al., 2005;Miller et al., 2014). These unusual breathing patterns might confuse the labeling and learning processes and result in poor testing results. represents an identifiable exhalation event, and the black areas represent pause phases. The developed database contains imbalanced numbers of inhalation and exhalation labels (34,095 and 18,349, respectively) because not every exhalation was heard and labeled. In addition, the proposed models may possess the capability of learning the rhythmic rise and fall of breathing signals but not the capability of learning acoustic or texture features that can distinguish an inhalation from an exhalation. This may thus explain the models' poor performance in exhalation detection. However, these models are suitable for respiratory rate estimation and apnea detection as long as appropriate inhalation detection is achieved. Furthermore, for all labels, the summation of the event duration was smaller than that of the background signal duration (these factors had a ratio of approximately 1:2.5 to 1:7). The aforementioned phenomenon can be regarded as foregroundbackground class imbalance (Oksuz et al., 2020) and will be addressed in future studies. Most of the sounds in the established database were not recorded during the patients performed deep breathing; thus, the signal quality was not maximized. However, training models with such nonoptimal data increase their adaptability to real-world scenarios. Moreover, the SNR may be reduced by noise, such as human voices; music; sounds from bedside monitors, televisions, air conditioners, fans, and radios; sounds generated by mechanical ventilators; electrical noise generated by touching or moving the parts of acoustic sensors; and friction sounds generated by the rubbing of two surfaces together (e.g., rubbing clothes with the skin). A poor SNR of audio signals can lead to difficulties in labeling and prediction tasks. The features of some noise types are considerably similar to those of adventitious sounds. The poor performance of the proposed models in CAS detection can be partly attributed to the noisy environment in which the lung sounds were recorded. In particular, the sounds generated by ventilators caused numerous FP events in the CAS detection tasks. Thus, additional effort is required to develop a superior preprocessing algorithm that can filter out influential noise or to identify a strategy to ensure that models focus on learning the correct CAS features. Furthermore, the integration of active noise-canceling technology (Wu et al., 2020) or noise suppression technology (Emmanouilidou et al., 2017) into respiratory sound monitors can help reduce the noise from auscultatory signals. The sound recordings in the HF_Lung_V1 database were labeled by only one labeler; thus, some noisy labels, including class and attribute noise, may exist in the database (Zhu & Wu, 2004). These noisy labels are attributable to (1) the different hearing abilities of the labeler, which can cause differences in the labeled duration; (2) the absence of clear criteria for differentiating between target and confusing events; (3) individual human errors; (4) tendency to not label events located close to the beginning and end of a recording; and (5) confusion caused by unusual breath patterns and poor SNRs. However, deep learning models exhibit high robustness to noisy labels (Rolnick et al., 2017). Accordingly, we are currently working toward establishing better ground-truth labels. Breathing generates CASs and DASs under abnormal respiratory conditions. This means that the breathing sound, CAS, and DAS might overlap with one another during the same period. This sound overlapping, along with the data imbalance, makes the CAS and DAS detection models learn to read the rise and fall of the breathing energy and falsely identify an inhalation or exhalation as CAS or DAS, respectively. This FP detection was observed in our benchmark results. In the future, strategies must be adopted to address the problem of sound overlap. Conclusion We We also investigated the performance of eight RNN-based models in terms of inhalation, exhalation, CAS detection, and DAS detection in the HF_Lung_V1 database. We determined that the bidirectional models outperformed the unidirectional models in lung sound analysis. Furthermore, the addition of a CNN to these models further improved their performance. Future studies can develop more accurate respiratory sound analysis models. First, highly accurate ground-truth labels should be established. Second, researchers should investigate the performance of RNN-based models containing state-of-the-art convolutional layers. Third, regional CNN variants can be adopted in lung sound analysis if the labels are expanded to two-dimensional bounding boxes (Jácome et al., 2019). Fourth, wavelet-based approaches, empirical mode decomposition, and other methods that can extract different features should be investigated (Pramono et al., 2017;Pramono et al., 2019). Finally, respiratory sound monitors should be equipped with the capability of tracheal breath sound analysis (Wu et al., 2020). Conflicts of Interest The authors declare that they have no conflicts of interest relevant to this research. ( denoted by L1-L8) indicated in Fig. 2a. The auscultation locations are described in detail in the caption of Fig. 2. The two devices had a sampling rate of 4,000 Hz and a bit depth of 16 bits. The audio files were recorded in the WAVE (.wav) format. Fig. 1 . 1Customized multichannel acoustic recording device (HF-Type-1) connected to a tablet. Fig. 2 . 2Auscultation locations and lung sound recording protocol. (a) Auscultation locations (L1-L8): L1: second intercostal space (ICS) on the right midclavicular line (MCL); L2: fifth ICS on the right MCL; L3: fourth ICS on the right midaxillary line (MAL); L4: tenth ICS on the right MAL; L5: second ICS on the left MCL; L6: fifth ICS on the left MCL; L7: fourth ICS on the left MAL; and L8: tenth ICS on the left MAL. (b) A standard round of breathing lung sound recording with Littmann 3200 and HF-Type-1 devices. The white arrows represent a continuous recording, and the small red blocks represent 15-s recordings. When the Littmann 3200 device was used, 15.8-s signals were Fig. 3 . 3Pipeline of detection framework.The framework comprises three parts: preprocessing, deep learning-based modeling, and postprocessing. The preprocessing part involves signal processing and feature engineering techniques. The deep learning-based modeling part entails the use of a well-designed neural network for obtaining a sequence of classification predictions rather than a single prediction. The postprocessing part involves merging the segment prediction results and eliminating the burst event. LSTM (BiLSTM), bidirectional GRU (BiGRU), CNN-LSTM, CNN-GRU, CNN-BiLSTM, and CNN-BiGRU, in terms of inhalation, exhalation, and adventitious sound detection. Fig. 4 illustrates the detailed model structures. The outputs of the LSTM, GRU, BiLSTM, and BiGRU models were 19 938 × 1 vectors, and those of the CNN-LSTM, CNN-GRU, CNN-BiLSTM, and CNN-BiGRU models were 469 × 1 vectors. An element in these vectors was set to 1 if an inhalation, exhalation, CAS, or DAS occurred within a time segment in which the output value passed the thresholding criterion; otherwise, the element was set to 0. For a fairer comparison of the performance of the unidirectional and bidirectional models, we trained additional simplified (SIMP) BiLSTM, SIMP BiGRU, SIMP CNN-BiLSTM, and SIMP CNN-BiGRU models by adjusting the number of trainable parameters. Parameter adjustment was conducted by halving the number of cells of the LSTM and GRU layers. Fig. 4 . 4Model architectures and postprocessing for inhalation, exhalation, CAS, and DAS segment and event detection. (a) LSTM and GRU models; (b) BiLSTM and BiGRU models; and (c) CNN-LSTM, CNN-GRU, CNN-BiLSTM, and CNN-BiGRU models. Fig. 5c) of size 938 × 1 in the LSTM, GRU, BiLSTM, and BiGRU models and size 469 × 1 in the CNN-LSTM, CNN-GRU, CNN-BiLSTM, and CNN-BiGRU models. By comparing the sequential prediction with the ground-truth time segments, we could define true positive (TP; orange vertical bars in Fig. 5d), true negative (TN; green vertical bars in Fig. 5d), false positive (FP; black vertical bars in Fig. 5d), and false negative (FN; yellow vertical bars in Fig. 5d) time segments. Subsequently, the models' sensitivity and specificity in classifying the segments in each recording were computed. Fig. 5 . 5Task definition and evaluation metrics. (a) Ground-truth event labels, (b) ground-truth time segments, (c) AI inference results, (d) segment classification, (e) event detection, and (f) legend. JI: Jaccard index. Fig. 6 . 6ROC curves for (a) inhalation, (b) exhalation, (c) CAS, and (d) DAS segment detection. The corresponding AUC values are presented. Fig. 7 . 7MAPE curves for (a) inhalation, (b) exhalation, (c) CAS, and (d) DAS event detection. an attention-based encoder-decoder architecture based on ResNet and LSTM exhibited favorable performance in inhalation (F1 score of 90.4%) and exhalation (F1 score of 93.2%) segment detection tasks. However, the model was established on the basis of a very small dataset (489 recordings of 15-s-long lung sounds). Moreover, the model involves a complicated architecture; hence, it is impossible to implement real-time respiratory monitoring in devices with limited computing power, such as smartphones or medical-grade tablets. Few studies have performed event detection at the recording level by using a comparatively simple deep learning model. Messner et al. (2018) used the BiGRU model and one-dimensional labels (similar to those used in the present study) for breath phase and crackle detection. Their BiGRU model exhibited comparable performance to our models in terms of inhalation event detection (F1 scores, 87.0% vs 86.2%) and in terms of DAS event detection (F1 scores, 72.1% vs 71.4%). However, the performance of the BiGRU model differed considerably from that of our models in terms of exhalation detection (F1 scores: 84.6% vs 70.9%). One of the reasons for this discrepancy is that Messner et al. (2018) established their ground-truth labels on the basis of the gold-standard signals of a pneumotachograph. Fig. 8 . 8Patterns of normal breathing lung sounds. (a) General lung sound patterns and (b) general lung sound patterns with unidentifiable exhalations. "I" represents an identifiable inhalation event, "E" Smart Emergency and Critical Care (TSECC), 2020, under the license of Creative Commons Attribution 4.0 (CC BY 4.0), provided by the Taiwan Society of Emergency and Critical Care Medicine. Lung sound recordings in the TSECC database were acquired from 261 patients. The second source was sound recordings acquired from 18 residents of a respiratory care ward (RCW) or a respiratory care center (RCC) in Northern Taiwan between August 2018 and October 2019. The recordings were approved by the Research Ethics Review Committee of Far Eastern Memorial Hospital (case number: 107052-F). This study was conducted in accordance with the 19649 Table 1 1Demographic data of patients.RCW: respiratory care ward, RCC: respiratory care center, ARF: acute respiratory failure, CRF: chronic respiratory failure, COPD AE: chronic obstructive pulmonary disease acute exacerbation, COPD: chronic obstructive pulmonary disease, ARDS: acute respiratory distress syndrome, CKD: chronic kidney disease, AKI: acute kidney injury, CHF: chronic heart failure, DM: diabetes, HTN: hypertension, CAD: cardiovascular disease. The mean values of the age, height, weight, and BMI are presented, with the corresponding 95% CI in parentheses.Subjects from RCW/RCC (n= 18) TSECC Database (n = 261) Gender (M/F) 11/7 NA Age 67.5 (36.7, 98.3) NA Height (cm) 163.6 (147.2, 180.0) NA Weight (kg) 62.1 (38.2, 86.1) NA BMI (kg/m 2 ) 23.1 (15.6, 30.7) NA Respiratory Diseases ARF 4 (22.2%) NA CRF 8 (44.4%) NA COPD AE 1 (5.6%) NA COPD 2 (11.1%) NA Pneumonia 4 (22.2%) NA ARDS 1 (5.6%) NA Emphysema 1 (5.6%) NA Comorbidity CKD 1 (5.6%) NA AKI 3 (16.7%) NA CHF 2 (11.1%) NA DM 7 (38.9%) NA HTN 6 (33.3%) NA Malignancy 1 (5.6%) NA Arrythmia 1 (5.6%) NA CAD 1 (5.6%) NA GHz CPU with 90 GB RAM. To manage the intensive computation involved in RNN training, we implemented the training module by using the TensorFlow 2.10, CUDA 10, and CuDNN 7 programs to run the NVIDIA Titan V100 card on the TWCC server for GPU acceleration.Center for High-Performance Computing in Taiwan [Taiwan Computing Cloud (TWCC)] and was equipped with an Intel(R) Xeon(R) Gold 6154 @3.00 YCL, NJL, YLW, and BFH collected the breath sounds. NJL, YLW, and WLT established the labels.BFH and ZLT organized the labels. JH and CWC helped recruit the study participants. FSH, SRH, YRC, and FPL conceptualized the research. CWH and CCC trained the deep learning models. FSH, SRH, LCC, YTC, and CTT contributed to the final manuscript. FPL supervised the research.This study was partially funded by the Raising Children Medical Foundation, Taiwan. The authors thank the employees of Heroic Faith Medical Science Co. Ltd. who have ever partially contributed to developing the HF-Type-1 and establishing the HF_Lung_V1 database. This manuscript was edited by Wallace Academic Editing. We also thank the All Vista Healthcare Center, Ministry of Science and Technology, Taiwan for the support.Author Contributions Acknowledgments Khandelwal, S.,Lecouteux, B., & Besacier, L. (2016). Comparing GRU and LSTM for automatic speech recognition.Kochetov, K., Putin, E., Balashov, M., Filchenkov, A., & Shalyto, A. (2018). Noise Masking Recurrent Neural Network for Respiratory Sound Classification. In Artificial NeuralNetworks and Machine Learning -ICANN 2018 (pp. 208-217).Li, L.,Wu, Z., Xu, M., Meng, H. M., & Cai, L. (2016). Combining CNN and BLSTM to Extract Textual and Acoustic Features for Recognizing Stances in Mandarin Ideological Debate Competition. In INTERSPEECH (pp. 1392-1396). Li, L., Xu, W., Hong, Q., Tong, F., & Wu, J. (2016). Classification between normal and adventitious lung sounds using deep neural network. In 2016 10th International Symposium on Chinese Spoken Language Processing (ISCSLP) (pp. 1-5): IEEE. 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American journal of respiratory and critical care medicine, 156, 974-987.Appendix Tables 1-4 list the accuracy, PPV, sensitivity, specificity, and F1 scores of all models in terms of inhalation, exhalation, CAS, and DAS detection based on the HF_Lung_V1 dataset.Accuracy, PPV, sensitivity, specificity, and F1 scores of all models in inhalation detection.Segment Event Segment Event Segment Event Segment Event Segment Event Detection Detection Detection Detection Detection Detection Detection Detection Detection DetectionAppendix ATable 2.Accuracy, PPV, sensitivity, specificity, and F1 scores of all models in exhalation detection.Segment Event Segment Event Segment Event Segment Event Segment Event Detection Detection Detection Detection Detection Detection Detection Detection Detection DetectionSegment Event Segment Event Segment Event Segment Event Segment Event Detection Detection Detection Detection Detection Detection Detection Detection Detection DetectionAppendix ATable 4. Accuracy, PPV, sensitivity, specificity, and F1 scores of all models in DAS detection.Segment Event Segment Event Segment Event Segment Event Segment EventDetection Detection Detection Detection Detection Detection Detection Detection Detection DetectionApplied Sciences, 6, 162. 874-881. 724-732. Appendix A Appendix A Table 1. Models n of trainable parameters Accuracy PPV Sensitivity Specificity F1 score LSTM 300,609 0.890 NA 0.781 0.890 0.701 0.664 0.944 NA 0.739 0.761 GRU 227,265 0.899 NA 0.801 0.904 0.726 0.696 0.948 NA 0.762 0.789 BiLSTM 732,225 0.906 NA 0.814 0.885 0.750 0.772 0.951 NA 0.781 0.840 BiGRU 552,769 0.916 NA 0.836 0.898 0.773 0.800 0.956 NA 0.803 0.862 CNN-LSTM 3,448,513 0.903 NA 0.809 0.898 0.747 0.730 0.948 NA 0.776 0.811 CNN-GRU 2,605,249 0.905 NA 0.804 0.906 0.765 0.742 0.945 NA 0.784 0.820 CNN-BiLSTM 6,959,809 0.914 NA 0.822 0.902 0.791 0.803 0.950 NA 0.806 0.863 CNN-BiGRU 5,240,513 0.914 NA 0.829 0.898 0.785 0.812 0.952 NA 0.806 0.862 SIMP BiLSTM 235,073 0.906 NA 0.817 0.882 0.743 0.773 0.952 NA 0.778 0.841 SIMP BiGRU 178,113 0.915 NA 0.837 0.894 0.769 0.803 0.957 NA 0.801 0.861 SIMP CNN-BiLSTM 3,382,977 0.912 NA 0.828 0.895 0.774 0.799 0.953 NA 0.800 0.858 SIMP CNN-BiGRU 2,556,097 0.913 NA 0.830 0.889 0.774 0.810 0.953 NA 0.801 0.859 Models n of trainable parameters Accuracy PPV Sensitivity Specificity F1 score LSTM 300,609 0.855 NA 0.716 0.561 0.406 0.456 0.962 NA 0.518 0.570 GRU 227,265 0.868 NA 0.715 0.687 0.514 0.554 0.951 NA 0.598 0.656 BiLSTM 732,225 0.866 NA 0.739 0.630 0.469 0.532 0.961 NA 0.573 0.639 BiGRU 552,769 0.882 NA 0.772 0.713 0.548 0.617 0.962 NA 0.641 0.709 CNN-LSTM 3,448,513 0.864 NA 0.732 0.628 0.476 0.512 0.957 NA 0.577 0.621 CNN-GRU 2,605,249 0.863 NA 0.731 0.629 0.470 0.516 0.958 NA 0.572 0.620 CNN-BiLSTM 6,959,809 0.867 NA 0.729 0.677 0.520 0.557 0.952 NA 0.604 0.656 CNN-BiGRU 5,240,513 0.874 NA 0.747 0.693 0.533 0.600 0.956 NA 0.622 0.685 SIMP BiLSTM 235,073 0.864 NA 0.736 0.612 0.450 0.520 0.962 NA 0.558 0.624 SIMP BiGRU 178,113 0.878 NA 0.741 0.716 0.559 0.603 0.954 NA 0.637 0.700 SIMP CNN-BiLSTM 3,382,977 0.869 NA 0.737 0.667 0.513 0.569 0.955 NA 0.604 0.662 SIMP CNN-BiGRU 2,556,097 0.873 NA 0.736 0.697 0.543 0.598 0.952 NA 0.624 0.684 Models n of trainable parameters Accuracy PPV Sensitivity Specificity F1 score LSTM 300,609 0.812 NA 0.554 0.120 0.087 0.095 0.983 NA 0.151 0.122 GRU 227,265 0.812 NA 0.529 0.217 0.160 0.153 0.966 NA 0.246 0.201 BiLSTM 732,225 0.815 NA 0.579 0.155 0.119 0.167 0.980 NA 0.198 0.191 BiGRU 552,769 0.818 NA 0.574 0.237 0.176 0.227 0.969 NA 0.269 0.256 CNN-LSTM 3,448,513 0.840 NA 0.676 0.475 0.341 0.329 0.960 NA 0.453 0.425 CNN-GRU 2,605,249 0.849 NA 0.689 0.556 0.411 0.402 0.955 NA 0.515 0.498 CNN-BiLSTM 6,959,809 0.844 NA 0.686 0.443 0.369 0.419 0.959 NA 0.479 0.464 CNN-BiGRU 5,240,513 0.851 NA 0.690 0.508 0.435 0.463 0.952 NA 0.533 0.516 SIMP BiLSTM 235,073 0.814 NA 0.560 0.152 0.121 0.148 0.977 NA 0.198 0.179 SIMP BiGRU 178,113 0.814 NA 0.546 0.202 0.162 0.178 0.968 NA 0.250 0.222 SIMP CNN-BiLSTM 3,382,977 0.848 NA 0.688 0.490 0.403 0.443 0.956 NA 0.508 0.502 SIMP CNN-BiGRU 2,556,097 0.851 NA 0.699 0.499 0.423 0.475 0.955 NA 0.526 0.515 Models n of trainable parameters Accuracy PPV Sensitivity Specificity F1 score LSTM 300,609 0.800 NA 0.716 0.699 0.556 0.485 0.905 NA 0.626 0.591 GRU 227,265 0.805 NA 0.697 0.746 0.624 0.514 0.883 NA 0.659 0.625 BiLSTM 732,225 0.821 NA 0.713 0.755 0.681 0.609 0.881 NA 0.696 0.700 BiGRU 552,769 0.827 NA 0.727 0.765 0.681 0.638 0.889 NA 0.703 0.714 CNN-LSTM 3,448,513 0.813 NA 0.706 0.734 0.672 0.526 0.876 NA 0.688 0.644 CNN-GRU 2,605,249 0.815 NA 0.725 0.709 0.640 0.539 0.893 NA 0.680 0.646 CNN-BiLSTM 6,959,809 0.830 NA 0.742 0.741 0.685 0.633 0.895 NA 0.712 0.708 CNN-BiGRU 5,240,513 0.826 NA 0.731 0.718 0.683 0.626 0.889 NA 0.706 0.700 SIMP BiLSTM 235,073 0.815 NA 0.702 0.764 0.675 0.582 0.876 NA 0.688 0.689 SIMP BiGRU 178,113 0.824 NA 0.718 0.779 0.689 0.626 0.883 NA 0.703 0.713 SIMP CNN-BiLSTM 3,382,977 0.828 NA 0.751 0.727 0.662 0.624 0.902 NA 0.702 0.702 SIMP CNN-BiGRU 2,556,097 0.825 NA 0.741 0.712 0.663 0.622 0.896 NA 0.699 0.695 SIMP means the number of trainable parameters is adjusted. 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{'abstract': 'A reliable, remote, and continuous real-time respiratory sound monitor with automated respiratory sound analysis ability is urgently required in many clinical scenarios-such as in monitoring disease progression of coronavirus disease 2019-to replace conventional auscultation with a handheld stethoscope. However, a robust computerized respiratory sound analysis algorithm has not yet been validated in practical applications. In this study, we developed a lung sound database (HF_Lung_V1) comprising 9,765 audio files of lung sounds (duration of 15 s each), 34,095 inhalation labels, 18,349 exhalation labels, 13,883 continuous adventitious sound (CAS) labels (comprising 8,457 wheeze labels, 686 stridor labels, and 4,740 rhonchi labels), and 15,606 discontinuous adventitious sound labels (all crackles). We conducted benchmark tests for long short-term memory (LSTM), gated recurrent unit (GRU), bidirectional LSTM (BiLSTM), bidirectional GRU (BiGRU), convolutional neural network (CNN)-LSTM, CNN-GRU, CNN-BiLSTM, and CNN-BiGRU models for breath phase detection and adventitious sound detection. We also conducted a performance comparison between the LSTM-based and GRU-based models, between unidirectional and bidirectional models, and between models with and without a CNN. The results revealed that these models exhibited adequate performance in lung sound analysis. The GRU-based models outperformed, in terms of F1 scores and areas under the receiver operating characteristic curves, the LSTM-based models in most of the defined tasks. Furthermore, all bidirectional models outperformed their unidirectional counterparts.Finally, the addition of a CNN improved the accuracy of lung sound analysis, especially in the CAS 4 detection tasks.', 'arxivid': '2102.03049', 'author': ['Fu-Shun Hsu \nGraduate Institute of Biomedical Electronics and Bioinformatics\nNational Taiwan University\n10617TaipeiTaiwan\n\nDepartment of Critical Care Medicine\nFar Eastern Memorial Hospital\n22060New TaipeiTaiwan\n\nHeroic Faith Medical Science Co. Ltd\n23553New TaipeiTaiwan\n', 'Shang-Ran Huang \nHeroic Faith Medical Science Co. Ltd\n23553New TaipeiTaiwan\n', 'Chien-Wen Huang \nAvalanche Computing Inc\n10687TaipeiTaiwan\n', 'Chao-Jung Huang ', 'Yuan-Ren Cheng \nHeroic Faith Medical Science Co. Ltd\n23553New TaipeiTaiwan\n\nDepartment of Life Science\nCollege of Life Science\nNational Taiwan University\n10617TaipeiTaiwan\n\nInstitute of Biomedical Sciences\nAcademia Sinica\n11529TaipeiTaiwan\n', 'Chun-Chieh Chen \nHeroic Faith Medical Science Co. Ltd\n23553New TaipeiTaiwan\n\nAvalanche Computing Inc\n10687TaipeiTaiwan\n', 'Jack Hsiao \nHCC Healthcare Group\n22060New TaipeiTaiwan\n', 'Chung-Wei Chen \nDepartment of Critical Care Medicine\nFar Eastern Memorial Hospital\n22060New TaipeiTaiwan\n', 'Li-Chin Chen \nResearch Center for Information Technology Innovation\nGraduate Institute of Biomedical Electronics and Bioinformatics\nAcademia Sinica\n11529TaipeiTaiwan\n\nNational Taiwan University\nNo. 1, Sec. 4, Roosevelt Road Taipei 10617Taiwan\n', 'Yen-Chun Lai \nHeroic Faith Medical Science Co. Ltd\n23553New TaipeiTaiwan\n', 'Bi-Fang Hsu \nHeroic Faith Medical Science Co. Ltd\n23553New TaipeiTaiwan\n', 'Nian-Jhen Lin \nHeroic Faith Medical Science Co. Ltd\n23553New TaipeiTaiwan\n', 'Wan-Lin Tsai \nHeroic Faith Medical Science Co. Ltd\n23553New TaipeiTaiwan\n', 'Yi-Lin Wu \nHeroic Faith Medical Science Co. Ltd\n23553New TaipeiTaiwan\n', 'Tzu-Ling Tseng \nHeroic Faith Medical Science Co. Ltd\n23553New TaipeiTaiwan\n', 'Ching-Ting Tseng ', 'Yi-Tsun Chen \nHeroic Faith Medical Science Co. Ltd\n23553New TaipeiTaiwan\n', 'Feipei Lai \nGraduate Institute of Biomedical Electronics and Bioinformatics\nNational Taiwan University\n10617TaipeiTaiwan\n\nHeroic Faith Medical Science Co. Ltd\n23553New TaipeiTaiwan\n', 'Feipei Lai '], 'authoraffiliation': ['Graduate Institute of Biomedical Electronics and Bioinformatics\nNational Taiwan University\n10617TaipeiTaiwan', 'Department of Critical Care Medicine\nFar Eastern Memorial Hospital\n22060New TaipeiTaiwan', 'Heroic Faith Medical Science Co. Ltd\n23553New TaipeiTaiwan', 'Heroic Faith Medical Science Co. Ltd\n23553New TaipeiTaiwan', 'Avalanche Computing Inc\n10687TaipeiTaiwan', 'Heroic Faith Medical Science Co. Ltd\n23553New TaipeiTaiwan', 'Department of Life Science\nCollege of Life Science\nNational Taiwan University\n10617TaipeiTaiwan', 'Institute of Biomedical Sciences\nAcademia Sinica\n11529TaipeiTaiwan', 'Heroic Faith Medical Science Co. Ltd\n23553New TaipeiTaiwan', 'Avalanche Computing Inc\n10687TaipeiTaiwan', 'HCC Healthcare Group\n22060New TaipeiTaiwan', 'Department of Critical Care Medicine\nFar Eastern Memorial Hospital\n22060New TaipeiTaiwan', 'Research Center for Information Technology Innovation\nGraduate Institute of Biomedical Electronics and Bioinformatics\nAcademia Sinica\n11529TaipeiTaiwan', 'National Taiwan University\nNo. 1, Sec. 4, Roosevelt Road Taipei 10617Taiwan', 'Heroic Faith Medical Science Co. Ltd\n23553New TaipeiTaiwan', 'Heroic Faith Medical Science Co. Ltd\n23553New TaipeiTaiwan', 'Heroic Faith Medical Science Co. Ltd\n23553New TaipeiTaiwan', 'Heroic Faith Medical Science Co. Ltd\n23553New TaipeiTaiwan', 'Heroic Faith Medical Science Co. Ltd\n23553New TaipeiTaiwan', 'Heroic Faith Medical Science Co. Ltd\n23553New TaipeiTaiwan', 'Heroic Faith Medical Science Co. Ltd\n23553New TaipeiTaiwan', 'Graduate Institute of Biomedical Electronics and Bioinformatics\nNational Taiwan University\n10617TaipeiTaiwan', 'Heroic Faith Medical Science Co. Ltd\n23553New TaipeiTaiwan'], 'corpusid': 231839742, 'doi': '10.1371/journal.pone.0254134', 'github_urls': [], 'n_tokens_mistral': 22451, 'n_tokens_neox': 18486, 'n_words': 9630, 'pdfsha': 'a629b92beb971b6d529f87df1dca84ba9eae2047', 'pdfurls': ['https://arxiv.org/pdf/2102.03049v2.pdf'], 'title': ['Benchmarking of eight recurrent neural network variants for breath phase and adventitious sound detection on a self-developed open-access lung sound database-HF_Lung_V1 Short Title: Automated lung sound analysis database † Corresponding Author', 'Benchmarking of eight recurrent neural network variants for breath phase and adventitious sound detection on a self-developed open-access lung sound database-HF_Lung_V1 Short Title: Automated lung sound analysis database † Corresponding Author'], 'venue': []}

arxiv

Diophantine approximations with positive integers: some remarks 22 Feb 2012 Nikolay Moshchevitin Diophantine approximations with positive integers: some remarks 22 Feb 2012 We give some comments on our recent results related to W.M. Schmidt's conjecture and Diophantine exponents. This short communication is a supplement to our papers [4,6]. We consider a pair of real numbers Θ = (θ 1 , θ 2 ). We are interested in small values of the linear form ||θ 1 m 1 + θ 2 m 2 || in positive integers m 1 , m 2 . Put ψ(t) = ψ Θ (t) = min m 1 ,m 2 ∈Z, 0<max(|m 1 |,|m 2 |) t ||m 1 θ 1 + m 2 θ 2 ||, ψ * (t) = ψ * Θ (t) = min x∈Z, 0<x t max j=1,2 ||xθ j || and ψ + (t) = ψ +:Θ (t) = min m 1 ,m 2 ∈Z + , 0<max(m 1 ,m 2 ) t ||m 1 θ 1 + m 2 θ 2 ||. Recall the definitions of Diophantine exponents ω = ω(Θ) = sup{γ : lim inf t→∞ t γ ψ Θ (t) < ∞}, ω =ω(Θ) = sup{γ : lim sup t→∞ t γ ψ Θ (t) < ∞} and ω * = ω * (Θ) = sup{γ : lim inf t→∞ t γ ψ * Θ (t) < ∞}, We introduce Diophantine exponents ω + = ω + (Θ) = sup{γ : lim inf t→∞ t γ ψ +;Θ (t) < ∞}, andω + =ω + (Θ) = sup{γ : lim sup t→∞ t γ ψ +;Θ (t) < ∞}. W.M. Schmidt's theorem and its extensions Put φ = 1 + √ 5 2 = 1.618 + . In 1976 W.M. Schmidt [7] proved the following theorem. Theorem 1 (W.M. Schmidt). Let real numbers θ 1 1 , θ 2 be linearly independent over Z together with 1. Then there exists a sequence of integer two-dimensional vectors (x 1 (i), x 2 (i)) such that 1. x 1 (i), x 2 (i) > 0; 2. ||θ 1 x 1 (i) + θ 2 x 2 (i)|| · (max{x 1 (i), x 2 (i)}) φ → 0 as i → +∞. In fact W.M. Schmidt proved (see discussion in [1]) that for θ 1 , θ 2 under consideration one has the inequality ω + max ω ω − 1 ;ω − 1 +ω ω (1) from which we immediately deduce ω + (Θ) φ. From Schmidt's argument one can easily see that for θ 1 , θ 2 linearly independent together with 1 one hasω + ω ω − 1 .(2) We would like to note here that Thurnheer (see Theorem 2 from [9]) showed that for θ 1 , θ 2 linearly independent together with 1 in the case 1 2 ω * = ω * (Θ) 1(3) one has ω + ω * + 1 4ω * + ω * + 1 4ω * 2 + 1.(4) (inequality 4 is a particular case of a general result obtained by Thurnheer). A lower bound for ω + in terms of ω was obtained by the author in [4]. It was based on the original Schmidt's argument from [7]. However the choice of parameters in [4] was not optimal. Here we explain the optimal choice. From Schmidt's proof and Jarník's result ω ω(ω − 1) (see [2] and a recent paper [3]) one can easily see that ω + max g : max y,z 1: yω −1 z y ω/ω max y −ω x z −ω min x 1−g z −g ; xy −1 z g+1 1 .(5) This inequality immediately follows from Schmidt's argument, see Lemma 1 and Lemma 2 from [4]. The right hand side of (5) can be easily calculated. We divide the set A = (ω,ω) ∈ R 2 :ω 2, ω ω(ω − 1) of all admissible values of (ω,ω) into two parts: A = A 1 ∪ A 2 , A 1 = (ω,ω) ∈ R 2 : 2 ω φ 2 , ω ω(ω − 1) 3ω −ω 2 − 1 , A 2 = A \ A 1 . If (ω,ω) ∈ A 1 then ω + G(ω) = 1 2   ω + 1 ω + ω + 1 ω 2 + 4   (the function G(ω) on the right hand side decreases from G(2) = 2 to G(+∞) = φ). If (ω,ω) ∈ A 2 then ω + ω − 1 +ω ω So we get the following result. Theorem 2. Let real numbers θ 1 1 , θ 2 be linearly independent over Z together with 1. Then ω + max   1 2   ω + 1 ω + ω + 1 ω 2 + 4   ;ω − 1 +ω ω   . This theorem gives the best bound in terms of ω,ω which one can deduce from Schmidt's argument from [7]. 2 About counterexample to W.M. Schmidt's conjecture In the paper [7] W.M. Schmidt wrote that he did not know if the exponent φ in Theorem 1 may be replaced by a lagrer constant. At that time he was not able even to rule a possibility that there exists an infinite sequence (x 1 (i), x 2 (i)) ∈ Z 2 with condition 1. and such that ||θ 1 x 1 (i) + θ 2 x 2 (i)|| · (max{x 1 (i), x 2 (i)}) 2 c(Θ)(7) with some large positive c(Θ). Later in [8] he conjectured that the exponent φ may be replaced by any exponent of the form 2 − ε, ε > 0 and wrote that probably such a result should be obtained by analytical tools. It happened that this conjecture is not true. In [6] the author proved the following result. Theorem 3. Let σ = 1.94696 + be the largest real root of the equation x 4 − 2x 2 − 4x + 1 = 0. There exist real numbers θ 1 , θ 2 such that they are linearly independent over Z together with 1 and for every integer vector (m 1 , m 2 ) ∈ Z 2 with m 1 , m 2 0 and max(m 1 , m 2 ) 2 200 one has ||m 1 θ 1 + m 2 θ 2 || 1 2 300 (max(m 1 , m 2 )) σ . Here we should note that for the numbers constucted in Theorem 3 one has ω = (σ + 1) 2 (σ 2 − 1) 4σ = 3.1103 + ,ω = (σ + 1) 2 2σ = 2.2302 + . So (ω,ω) ∈ A 2 and the inequality (6) gives ω + σ + 2 σ 2 − 1 = 1.413 + . However from the proof of Theorem 3 (see [6]) it is clear that for the numbers constructed one has ω + = σ = 1.94696 + . Diophantine exponents for mildly restricted approximation. Y Bugeaud, S Kristensen, Arkiv. f. Mat. 47Y. Bugeaud, S. Kristensen, Diophantine exponents for mildly restricted approximation, Arkiv. f. Mat. 47 (2009), 243 -266. Contributionà la théorie des approximations diophantiennes linéaires et homogènes. V Jarník, Czechoslovak Math. J. 4in Russian. French summaryV. Jarník, Contributionà la théorie des approximations diophantiennes linéaires et homogènes, Czechoslovak Math. J. 4 (1954), 330 -353 (in Russian, French summary). Exponents of Diophantine approximations in dimension two. M Laurent, Canad.J.Math. 61M. Laurent, Exponents of Diophantine approximations in dimension two, Canad.J.Math. 61, 1 (2009),165 -189. N G Moshchevitin, arXiv:0904.1906Diophantine approximations with positive integers: a remark to W.M. Schmidt's theorem. preprint available atN.G. Moshchevitin, Diophantine approximations with positive integers: a remark to W.M. Schmidt's theorem, preprint available at arXiv:0904.1906 (2009). Khintchine's singular Diophantine systems and their applications. N G Moshchevitin, Russian Mathematical Surveys. 65N.G. Moshchevitin, Khintchine's singular Diophantine systems and their applications., Russian Mathematical Surveys. 65:3 43 -126 (2010); N G Moshchevitin, arXiv:1108.4435Positive integers: counterexample to W.M. Schmidt's conjecture. preprint available atN.G. Moshchevitin, Positive integers: counterexample to W.M. Schmidt's conjecture, preprint available at arXiv:1108.4435 (2011). Two questions in Diophantine approximations. W M Schmidt, Monatshefte für Mathematik. 82W.M. Schmidt, Two questions in Diophantine approximations, Monatshefte für Mathematik 82, 237 -245 (1976). Open problems in Diophantine approximations, in "Approximations Diophantiennes et nombres transcendants. W M Schmidt, Progress in Mathematics. LuminyW.M. Schmidt, Open problems in Diophantine approximations, in "Approximations Diophanti- ennes et nombres transcendants" Luminy, 1982, Progress in Mathematics, Birkhäuser, p.271 -289 (1983). On Dirichlet's theorem concerning diophantine approximations. P Thurneer, Acta Arithmetica. 54P. Thurneer, On Dirichlet's theorem concerning diophantine approximations, Acta Arithmetica 54 (1990), 241 -250.

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{'abstract': "We give some comments on our recent results related to W.M. Schmidt's conjecture and Diophantine exponents.", 'arxivid': '1201.4232', 'author': ['Nikolay Moshchevitin '], 'authoraffiliation': [], 'corpusid': 119722024, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 2864, 'n_tokens_neox': 2339, 'n_words': 1293, 'pdfsha': '89d95782bb7501951fbec11386cc7cad0a834c04', 'pdfurls': ['https://arxiv.org/pdf/1201.4232v2.pdf'], 'title': ['Diophantine approximations with positive integers: some remarks', 'Diophantine approximations with positive integers: some remarks'], 'venue': []}

arxiv

On Necessary and Sufficient Conditions for Near-Optimal Singular Stochastic Controls 25 Oct 2011 October 26, 2011 Mokhtar Hafayed Syed Abbas Petr Veverka School of Basic Sciences Laboratory of Applied Mathematics Mohamed Khider University Box 14507000Po, BiskraAlgeria Department of Mathematics Faculty of Nuclear Sciences and Physical Engineering Indian Institute of Technology Mandi Mandi-175001India EU-Czech Republic Czech Technical University Trojanova 13, Prague 120 00 On Necessary and Sufficient Conditions for Near-Optimal Singular Stochastic Controls 25 Oct 2011 October 26, 20111and phrases Near-optimal singular stochastic controlStochastic maximum principleNecessary conditionsEkeland's varia- tional principle AMS Subject Classification: 60H10 This paper is concerned with necessary and sufficient conditions for near-optimal singular stochastic controls for systems driven by a nonlinear stochastic differential equations (SDEs in short). The proof of our result is based on Ekeland's variational principle and some dilecate estimates of the state and adjoint processes. This result is a generalization of Zhou's stochastic maximum principle for near-optimaity to singular control problem. Introduction In this paper, we consider the singular stochastic control problem for systems governed by nonlinear controlled diffusion of the type dx t = f (t, x t , u t ) dt + σ (t, x t , u t ) dW t + G t dη t , x s = y,(1.1) where (W t ) t is a standard l−dimentional Brownian motion defined on the filtered probability space (Ω, F, (F t ) t , P). The minimized criteria associated with the state equation (1.1) is defined by J (s, y, u, η) = E h (x T ) + T s ℓ (t, x t , u t ) dt + T s k t dη t ,(1.2) where E denotes the expectation with respect to P, and the value function is defined as V (s, y) = inf ( u,η)∈U([s,T ]) {J (s, y, u, η)} . (1. 3) The kind of stochastic control problem has been investigated extensively, both by the Bellman's dynamic programming method [6] and by Pontryagin's maximum principle [18]. In this paper, we are concerned by th second method. Peng [17] introduced the second-order adjoint equation and obtained the global maximum principle of optimality, in which the control is present in the both drift and diffusion coefficients.Studying near-optimal controls makes a good sense as studying optimal controls from both theoretical as well as applications point of view. Many more near-optimal controls are available than optimal ones, indeed, optimal controls my not even exist in many situations, while near-optimal controls always exist. The near-optimal deterministic controls problem has been treated by many authors, including [12,20,21,15]. Recently, in an interesting paper, Zhou [22] established the second-order necessary as well as sufficient conditions for near-optimal stochastic controls for general controlled diffusion with two adjoint processes. The near-optimal control problem for systems descripted by volterra integral equations has been studied by [16]. However, Chighoub et al., [8] extended Zhou's maximum principle of near-optimality to SDEs with jumps. The similar problem for systems driven by forward-backward stochastic differential equations has been solved in [5]. For justification of establishing a theory of near-optimal controls see ( [22,20], Introduction). Singular stochastic control problem is an important and challenging class of problems in control theory, it appear in various fields like mathematical finance, problem of optimal consumption etc. Stochastic maximum principle for singular controls was considered by many authors, see for instance [1,2,7,10,11,3,13,14]. The first version of maximum principle for singular stochastic control problems was obtained by Cadenillas et al., [7]. The firstorder weak stochastic maximum principle has been studied in [3]. In [10] the authors derived stochastic maximum principle where the singular part has a linear form. Sufficient conditions for existence of optimal singular control have been obtained in [11]. The main objective of this paper is to establish necessary as well as sufficient conditions for near-optimal singular control for SDEs. The control domain is not necessarily convex. The proof of our result is based on Ekeland's variational principle [12], and some delicate estimates of the state and adjoint processes. Finally, as an illustration an example is solved explicitly. This result permit us to extend Zhou's maximum principle of near-optimality to singular control problem. The paper is organized as follows. The assumptions and statment of the control problem is given in the second section. In the third and forth section, we establish the main result of this paper. Assumptions and statement of the problem We consider stochastic optimal control of the following kind. Let T be a fixed strictly positive real number and (Ω, F, {F t } t , P) be a fixed filtered probability space satisfying the usual conditions in which a l−dimentional Brownian motion W = {W t : s ≤ t ≤ T } with s ∈ [0, T ] and W s = 0 is defined. Let A 1 be a closed convex subset of R m and A 2 := ([0, ∞)) m . Let U 1 be the class of measurable adapted processes u : [s, T ] × Ω → A 1 and U 2 is the class of measurable adapted processes η : [0, T ] × Ω → A 2 . Definition 1. An admissible control is a pair (u, η) of measurable A 1 × A 2valued, F t −adapted processes, such that 1) η is of bounded variation, nondecreasing continuous on the left with right limits and η s = 0. 2) E sup t∈[s,T ] |u t | 2 + |η T | 2 < ∞. We denote U = U 1 × U 2 , the set of all admissible controls. Since dη t may be singular with respect to Lebesgue measure dt, we call η the singular part of the control and the process u its absolutely continuous part. Throughout this paper, we also assume that (H1) f : [0, T ] × R n ×A 1 → R n , σ : [0, T ] × R n ×A 1 →M n×l (R) and ℓ : [0, T ]×R n ×A 1 → R, are measurable in (t, x, u, ) and twice continuously differentiable in x, and there exists a constant C > 0 such that, for ϕ = f, σ, ℓ : ϕ(t, x, u, ) − ϕ(t, x ′ , u, ) + ϕ x (t, x, u, ) − ϕ x (t, x ′ , u, ) ≤ C x − x ′ . (2.1) |ϕ(t, x, u, )| ≤ C (1 + |x|) . (2.2) (H2) h : R n → R is twice continuously differentiable in x, and there exists a constant C > 0 such that h(x) − h(x ′ )) + h x (x) − h x (x ′ )) ≤ C x − x ′ . (2.3) |h(x)| ≤ C (1 + |x|) . (2.4) (H3) G : [0, T ] → M n×m (R) , k : [0, T ] → ([0, ∞)) m , for each t ∈ [0, T ] : G is continuous and bounded, and k is continuous. Under the above assumptions, the SDE (1.1) has a unique strong solution x t which is given by x t = y + t s f (r, x r , u r ) dr + t s σ (r, x r , u r ) dW r + t s G r dη r , and by standard arguments it is easy to show that for any q > 0, it hold that E( sup t∈[s,T ] |x t | q ) < C (q) , where C (q) is a constant depending only on q and the functional J is well defined. For any (u, η) ∈ U and the corresponding state trajectory x, we define the first-order adjoint process Ψ t and the second-order adjoint process Q t as the ones satisfying the following two backward SDEs respectively      dΨ t = − [f * x (t, x t , u t ) Ψ t + σ * x (t, x t , u t ) K t + ℓ x (t, x t , u t )] dt + K t dW t , Ψ T = h x (x T ) , (2.5) and      dQ t = − [f * x (t, x t , u t ) Q t + Q t f * x (t, x t , u t ) + σ * x (t, x t , u t ) Q t σ * x (t, x t , u t ) +σ * x (t, x t , u t ) R t + R t σ x (t, x t , u t ) + Γ t ] dt + R t dW t , Ψ T = h xx (x T ) , (2.6) where Γ t = ℓ xx (t, x t , u t ) + n i=1 Ψ i t f i xx (t, x t , u t ) + K i t σ i xx (t, x t , u t ) . As is well known, under conditions (H1), (H2) and (H3) the first-order adjoint equation (2.5) admits one and only one F −adapted solution pair (Ψ, K) ∈ L 2 F ([0, T ] , R n )×L 2 F ([0, T ] , R n ) and the second-order adjoint equation (2.6) admits one and only one F −adapted solution pair (Q, R) ∈ L 2 F ([0, T ] , R n×n ) × L 2 F ([0, T ] , R n×n ) . Moreover, since f x , σ x , ℓ x and h x are bounded then we have the following estimate E sup s≤t≤T |Ψ t | 2 + T s |K t | 2 dt + sup s≤t≤T |Q t | 2 + T s |R t | 2 ≤ C. Define the usual Hamiltonian H (t, x, u, p, q) := −pf (t, x, u) − qσ (t, x, u) − ℓ (t, x, u) , (2.7) for (t, x, u) ∈ [s, T ] × R n × A 1 . Furthermore, we define the H functional corresponding to a given admissible pair (x, u) as follows [12] ) Let (F, ρ) be a complete metric space and f : F → R be a lower semi-continuous function which is bounded below. For a given ε > 0, suppose that u ε ∈ F satisfying f (u ε ) ≤ inf (f ) + ε, then for any λ > 0, there exists u λ ∈ F such that 1) f u λ ≤ f (u ε ) . H (x,u) (t, x, u) = H (t, x, u, Ψ t , K t − Q t σ (t, x, u)) − 1 2 σ * (t, x, u) Q t σ (t, x, u) , for (t, x, u, p, q) ∈ [s, T ] × R n × A 1 ×R n × R n , 2) ρ u λ , u ε ≤ λ. 3) f u λ ≤ f (u) + ε λ ρ u, u λ for all u ∈ F. For u, v ∈ U To apply Ekeland's variational principle to our problem, we define a distance function ρ on the space of admissible controls such that (U, d) becomes a complete metric space. To achieve this goal, we define for any (u, η) and (v, ξ) ∈ U : d ((u, η) , (v, ξ)) = d 1 (u, v) + d 2 (η, ξ) ,(2.8) where d 1 (u, v) = P⊗dt {(w, t) ∈ Ω × [0, T ] : u (w, t) = v (w, t)} ,(2.9) and [9]) Let E be a convex set in R n and let f : E → R be a locally Lipschitz function. The Clarke's generalized gradient of f at x ∈ E, denoted by ∂ x f , is a set defined by d 2 (η, ξ) = E( sup t∈[s,T ] |η t − ξ t | 2 ) 1 2 ,(2.∂ x f (x) = ξ ∈ R n : ξ, v ≤ lim y→x sup t→0 f (y + tv) − f (y) t . ∀v ∈ R and y, (y + tv) ∈ E . Necessary conditions for near-optimal singular control Our purpose in this paper is to establish second-order necessary and sufficient conditions for near-optimal singular control for systems governed by nonlinear SDEs. It is worth montioning that optimal singular controls may not even exist in many situations, while near-optimal singular controls always exists. Ekeland's variational principle [12] is applied to prove our maximum principle. The proof follows the general ideas as in ( [20,21,22]) where similar results are obtained for other class of controls. We give the definition of near-optimal control as given in Zhou ( [22], Definition 2.1 and Definition 2.2). Definition 2. For a given ε > 0 the admissible control (u ε , η ε ) is near- optimal if |J (s, y, u ε , η ε ) − V (s, y)| ≤ O (ε) ,(3.1) where O (.) is a function of ε satisfying lim ε→0 O (ε) = 0. The estimater O (ε) is called an error bound. If O (ε) = Cε δ for some δ > 0 independent of the constant C then (u ε , η ε ) is called near-optimal control with order ε δ . If O (ε) = ε the admissible control (u ε , η ε ) called ε−optimal. Our first Lemma below, is deals with the continiuity of the state processes under distance ρ Lemma 2. If x u,η t and x v,η t be the solution of the state equation (1.1) associated respectively with u and v. For any 0 < α < 1 and β > 0 satisfying αβ < 1, there exists a positive constants C = C (T, α, β) such that E( sup s≤t≤T x u,η t − x v,ξ t 2β ) ≤ Cd αβ 1 (u, v) . (3.2) Proof. First, we assume that β ≥ 1. Using Burkholder-Davis-Gundy inequality for the martingale part, we can compute, for any r ≥ s E sup s≤t≤r x u,η t − x v,ξ t 2β ≤ CE( r s f (t, x u,η t , u t ) − f t, x v,ξ t , v t 2β + r s σ (t, x u,η t , u t ) − σ t, x v,ξ t , v t 2β dt +CE |η T − ξ T | 2β , ≤ CE( r s |f (t, x u,η t , u t ) − f (t, x u,η t , v t )| 2β + r s |σ (t, x u,η t , u t ) − σ (t, x u,η t , v t )| 2β χ ut =vt (t) dt +CE( r s |f (t, x u,η t , v t ) − f (t, x u,η t , v t )| 2β + t 0 |σ (t, x u,η t , v t ) − σ (t, x u,η t , v t )| 2β +CE |η T − ξ T | 2β , now arguing as in ( [22], Lemma 3.1) taking b = 1 αβ > 1 and a > 1 such that 1 a + 1 b = 1, and applying Cauchy-Schwarz inequality, we get E r s |f (t, x u,η t , u t ) − f (t, x u,η t , v t )| 2β χ ut =vt (t) dt ≤ E r s |f (t, x u,η t , u t ) − f (t, x u,η t , v t )| 2βa dt 1 a × E r s χ ut =vt (t) dt 1 b , using definition of d 1 and linear growth condition on f we obtain E r s |f (t, x u,η t , u t ) − f (t, x u,η t , v t )| 2β χ ut =vt (t) dt ≤ C E r s 1 + |x u,η t | 2βa dt 1 a d 1 (u, v) αβ ≤ Cd 1 (u, v) αβ . Similarly, we can prove E r s |σ (t, x u,η t , u t ) − σ (t, x u,η t , v t )| 2β χ ut =vt (t) dt ≤ Cd 1 (u, v) αβ . (3.3) Therefore, by using assumption (H1), we conclued that E( sup s≤t≤r x u,η t − x v,ξ t 2β ) ≤ C E r s sup s≤r≤θ x u,η t − x v,ξ t 2β dθ + E |η T − ξ T | 2β + d 1 (u, v) αβ . Hence (3.1) follows immediately from definition 1 and Gronwall's inequality. Now we assume 0 ≤ β < 1. Since 2 α > 1 then the Cauchy-Schwarz inequality yields E( sup s≤t≤T x u,η t − x v,ξ t 2β ) ≤ E( sup s≤t≤T x u,η t − x v,ξ t 2 ) β ≤ [Cd 1 (u, v) α ] β ≤ Cd 1 (u, v) αβ . This completes the proof of Lemma 2. Lemma 3. For any 0 < α < 1 and 1 < β < 2 satisfying (1 + α) β < 2, there exist a positive constant C = C (α, β) such that for any (u, η), (v, ξ) ∈ U ([s, T ]), along with the correspending trajectories x u,η , x v,ξ and the solutions (Ψ, K, Q, R) , (Ψ ′ , K ′ , Q ′ , R ′ ) of the corresponding adjoint equations, it holds that T 0 ( Ψ t − Ψ ′ t β + K t − K ′ t β )dt ≤ Cd αβ 2 1 (u, v) . (3.4) E T 0 ( Q t − Q ′ t β + R t − R ′ t β )dt ≤ Cd αβ 2 1 (u, v) . (3.5) Proof. Since the adjoint processes are independant to singular part, we use similar argument as in Zhou ([22] Lemma 3.2). Now we are able to state and prove the necessary conditions for nearoptimal singular control for our problem, which is the main result in this paper. Let (Ψ ε , K ε ) and (Q ε , R ε ) be the solution of adjoint equations (2.5) and (2.6) respectively corresponding to (x ε , (u ε , η ε )) . Theorem 1. (Maximum principle for any near-optimal singular control). For any δ ∈ (0, 1 3 ], and any near-optimal singular control (u ε , η ε ) there exists a positive constant C = C (δ) such that for each ε > 0      −Cε δ ≤ E T s 1 2 (σ (t, x ε − Cε δ ≤ E T 0 (k t + G * t Ψ ε t )d (η − η ε ) t . (3.7) Proof. By using Ekeland's variational principle with λ = ε 2 3 , there is an admissible pair (x ε , (u ε , η ε )) such that for any (u, η) ∈ U : ρ ((u ε , η ε ) , (u ε , η ε )) ≤ ε 2 3 . (3.8) and J (s, y, u ε , η ε ) ≤ J (s, y, u ε , η ε ) + ε 1 2 d ((u, η) , (u ε , η ε )) . Notice that (u ε , η ε ) which is near-optimal for the initial cost J defined in (1.2) is optimal for the new cost J ε given by J ε (s, y, u, η) = J (s, y, u, η) + ε 1 3 d ((u, η) , (u ε , η ε )) . (3.9) then we have J ε (s, y, u ε , η ε ) ≤ J ε (s, y, u, η) for any (u, η) ∈ U ([s, T ]) , Next, we use the spike variation techniques for u ε to drive the first variational inequality and we use convex perturbation for η ε as follows First variational inequality: For any θ > 0, we define the following strong perturbation (u ε,θ t , η ε t ) ∈ U : (u ε,θ , η ε t ) = (u, η ε t ) , t ∈ [t 0 , t 0 + θ] , (u ε t , η ε t ) , otherwise. (3.10) The fact that J ε (s, y, u ε , η ε ) ≤ J ε (s, y, u ε,θ , η ε ), (3.11) and d((u ε,θ , η ε t ), (u ε , η ε t )) = d 1 (u ε,θ , u ε ) ≤ θ, imply that J(s, y, u ε,θ , η ε ) − J (s, y, u ε , η ε ) ≥ −θε 1 3 . Since the diference J(s, y, u ε,θ , η ε )) − J (s, y, u ε , η ε ) is independant to the singular part, the near-maximum condition (3.6) follows by applying similar argument as in Zhou ([22]), we get −Cε 1 3 ≤ E T s 1 2 (σ (t, x ε where Ψ ε , K ε and Q ε , R ε are the solutions of adjoint equations (2.5) and (2.6) respectively corresponding to (x ε , (u ε , η ε )) . The first variational inequality (3.6) follows from combining, (3.12) (3.13) and.(3.14) Corollary 1. Under the assumptions of Theorem 1, we have E T s H (x ε ,u ε ) (t, x ε t , u ε t )dt ≥ sup u∈U([s,T ]) E T s H (x ε ,u ε ) (t, x ε t , u t )dt − Cε δ . (3.15) Second variational inequality: To obtain the second variational inequality, we define the following convex perturbation (u ε t , η ε,θ t ) ∈ U 1 × U 2 : (u ε Using the boundness of G t , k t , Lemma3, definition 1 (η s = η ε s = η ε s = 0, E |η T − η ε T | 2 + E |η ε T − η ε T | 2 < ∞) and the fact that E sup s≤t≤T |Ψ ε t | 2 < C we have E T s (k t + G * t Ψ ε t )d(η − η ε ) t − E T s (k t + G * t Ψ ε t )d(η − η ε ) t ≤(k t + G * t Ψ ε t )dη ε t ≤ inf η∈U 2 ([s,T ]) E T s (k t + G * t Ψ ε t )dη t + Cε δ . (3.20) Sufficient near-optimality conditions In this section, we will prove that under an additional assumptions, the near-maximum condition on the Hamiltonian function is sufficient for nearoptimality. We assume: (H4) ρ is differentiable in u for ϕ = f, σ, ℓ and there is a constante C such that ϕ(t, x, u, ) − ϕ(t, x, u ′ , ) + ϕ u (t, x, u, ) − ϕ u (t, x, u ′ , ) ≤ C u − u ′ . (4.1) Theorem 2. Assume the H (t, ·, ·, Ψ ε then (3.15) and (3.20) gives E 1 0 1 2 (u ε t ) 2 + 2u ε t (1 − K ε t ) dt ≥ sup u∈[0,1] 1 2 )W t + 1, (1 − ε 1 2 ) . Hence (4.15) and (4.16) will be satisfied. Conversely, for the sufficient part, since the hamiltonian H (t, x, u, p, q) = u− qu is concave in (x, u), we use Theorem 1 to conclude that u ε t = (1− ε C 2 ) is a condidate to be ε−optimality for sufficiently small ε is indeed an ε−optimal control. 10 ) 10here P⊗dt is the product measure of P with the Lebesgue measure dt on [s, T ] . It is easy to see that (U 2 , d 2 ) is a complete metric space. Moreover, it has been shown in Yong et al.,([19] pp. 146-147) that (U 1 , d 1 ) is a complete metric space. Hence (U, d) as a product of two complete metric spaces is a complete metric space under d. Definition 1. (Clarke's generalized gradient Cε δ . (3.19) Combining (3.18) and (3.19) the proof of inequality (3.7) is complete. Corollary 2. Under the assumptions of Theorem 1, we haveE T s t , u) − σ (t, x ε t , u ε t )) * Q ε t (σ (t, x ε , u) − σ (t, x ε t , u ε t )) +Ψ ε t (f (t, x ε t , u) − f (t, x ε t , u ε t )) + K ε t (σ (t, x ε t , u) − σ (t, x ε t , u ε t )) + (ℓ (t, x ε t , u) − ℓ (t,x ε t , u ε t ))} dt, (3.6) and t , u) − σ (t, x ε t , u ε t )) * Q ε t (σ (t, x ε , u) − σ (t, x ε t , u ε t )) + Ψ ε t (f (t, x ε t , u) − f (t, x ε t , u ε t )) + K ε t (σ (t, x ε t , u) − σ (t, x ε t , u ε t )) + (ℓ (t, x ε t , u) − ℓ (t, x ε t , u ε t ))} dt.(3.12) New we are to derive an estimate for the term similar to the right hand side of the abov inquality with all the (x ε t , (u ε t , η ε t )) etc, replacing by (x ε t , (u ε t , η ε t )) etc. To this end, we use similar method as inZhou ([22]) we obtian the following estimates:E T s K ε t (σ (t, x ε t , u) − σ (t, x ε t , u ε t )) − K ε t (σ (t, x ε t , u) − σ (t, x ε t , u ε t ))] dt ≤ Cε δ ,(3.13)andE T s 1 2 (σ (t, x ε t , u) − σ (t, x ε t , u ε t )) * Q ε t (σ (t, x ε , u) − σ (t, x ε t , u ε t )) − 1 2 (σ (t, x ε t , u) − σ (t, x ε t , u ε t )) * Q ε t (σ (t, x ε , u) − σ (t, x ε t , u ε t )) + Ψ ε t (f (t, x ε t , u) − f (t, x ε t , u ε t )) − Ψ ε t (f (t, x ε t , u) − f (t, x ε t , u ε t )) + [ℓ (t, x ε t , u) − ℓ (t, x ε t , u ε t )] − [ℓ (t, x ε t , u) − ℓ (t,x ε t , u ε t )]} dt ≤ Cε δ , (3.14) t , η ε,θ t ) = (u ε t , η ε t + θ (ξ − η ε t )) , (3.16)where ξ is an arbitrary element of the set U 2 . Using the optimality of (u ε , η ε ) to the new cost, J ε we haveJ ε (s, y, u ε , η ε ) ≤ J ε (s, y, u ε t , η ε,θ t ), (3.17) a simple computation on d 2 (u ε t , η ε,θ t ) we obtain J(s, y, u ε t , η ε,θ t ) − J (s, y, u ε , η ε ) ≥ −Cθε 1 3 ≥ −Cθε δ .Finally, arguing as in ([4]) for the left-hand side of the above inequality, then t , K ε t ) is concave for a.e. t ∈ [s, T ] , P − a.s, and h is convex. Let (Ψ ε t , K ε t ) , (Q ε t , R ε t ) be the solution of the adjoint equation (2.5)-(2.6) associated with (u ε , η ε ) . If for some ε > 0 and for any (u, η) ∈ U :E T s H (x ε ,u ε ) (t, x ε t , u ε t )dt ≥ sup u∈U 1 ([s,T ]) E T s H (x ε ,u ε ) (t, x ε t , u t )dt − ε, (4.2) and E T s k t d (η − η ε ) t ≥ −Cε 1 2 ,(4.3) then we have J (s, y, u ε , η ε ) ≤ inf (u,η)∈U([s,T ]) J (s, y, u, η) + Cε 1 2 , (4.4)where C is a positive constant independent of ε. ∂ u H(t, x ε t , u ε t ) ⊂ ∂ u H (x ε ,u ε ) (t, x ε t , u ε t ) + −ε 1 2 ς ε t , ε 1 2 ς ε t . = x ε 1 , (4.18)with the corresponding trajectories x ε t = (1−ε 1 2 )W t +η ε t then the unique solution pair of the first-order adjoint equation will be (Ψ ε t , K ε t ) = (1 − ε [9]for the detailed proof of the above Proposition. As an example, the Clarke's generalized gradient of the absolute value function f : x → |x| which is continuously differentiable everywhere except at 0. Since f ′ (x) = 1 for x > 0 and, f ′ (x) = −1 for x < 0 then the Clarke's generalized gradient of f at x = 0 is given bythen the near-singular maximum condition followsNow, we are to drive an estimate for the term similar to the right hand side of (3.18) with all (x ε , (u ε , η ε )) ect., replacing by (x ε , (u ε , η ε )) ect. We first estimate the following difference:Proof. First, define the cost functional J (s, y, u, η) = J 1 (s, y, u) + J 2 (s, η) ,(4.5)whereandLet us fix ε > 0, Define a new metric d on U ([s, T ]) as follows: for any (u, η) and (v, ξ) ∈ U :andObviously d 1 is a metric on (U 1 , d 1 ), and it is a complete metric as a weighted L 1 norm. Hence (U, d) as a product of two complete metric spaces is a complete metric space under d.a simple computation shows thatwhich implies that Υ is continuous on U 1 ([s, T ]) with respct to d 1 . Now by using (4.2) and Ekeland variational principle, there exists a u ε ∈ U 1 ([s, T ]) such thatwhereThe maximum condition (4.8) implies a pointwise maximum condition namely, for a.e. t ∈ [s, T ] and P − a.s,Using Proposition A1 (Appendix), then we haveSince |u − u ε t | is not differentiable in u ε t (locally Lipschitz), then we use Proposition A1 (Appendix) we getBy using (4.9) and fact that the Clarke's generalized gradient of the sum of two functions is contained in the sum of the Clarke's generalized gradient of the two functions, we getApplying the similar method as in ([22]) for the rest of the proof, we obtian, for an arbitrary u J 1 (s, y, u ε ) ≤ J 1 (s, y, u) + Cε Now, by using (4.3) we getwhich implies that for an arbitrary ηExample 1. Consider the one-dimensional stochastic control problem:12)and the cost functional being J (s, y, u, η) = E 1 2For a given admissible pair (x ε , (u ε , η ε )), the correspending second-order adjoint equation is dQ ε t = R ε t dW t Q ε 1 = 1, (4.14)By the uniqueness of this solution, (Q ε , R ε ) = (1, 0), then for any admissible control (u, η) we haveReplacing u t = u ε t , we get.Hence by simple computation shows that ifAppendixThe following result gives some basic properties of the Clarke's gneralized gradient. Proposition A1. If f : R n → R is locally Lipschitz at x ∈ R n , then the following statements holds (1) ∂ x f (x) is nonempty, compact, and convex set in R n0 ∈ ∂ x (f ) (x) if f attains a local minimum or maximum at x (4) If f is Frêchet-differentiable at x, then ∂ x f (x) = {f ′ (x)} .(5) If f, g : R n → R are locally Lipschitz at x ∈ R d , then Since (u, η) is arbitrary, the desired result follows. Since (u, η) is arbitrary, the desired result follows. Let the assumptions of Theorem 2 holds. A sufficient conditions for an admissible pair (x ε , u ε , η ε ) to be ε−optimal is References. Corollary 3.Corollary 3. Let the assumptions of Theorem 2 holds. A sufficient condi- tions for an admissible pair (x ε , u ε , η ε ) to be ε−optimal is References A class of solvable singular stochastic control problems. L H R Alvarez, Stochastics, Stochastics Rep. 67L. H.R. Alvarez, A class of solvable singular stochastic control problems, Stochastics, Stochastics Rep., 67, (1999), 83-122. Singular stochastic control linear diffusion and optimal stopping: A class of solvable problems. L H R Alvarez, SIAM J. Contr. Optim. 39L. H.R. Alvarez, Singular stochastic control linear diffusion and opti- mal stopping: A class of solvable problems, SIAM J. Contr. Optim., 39 (2001), pp. 1697-1710. The stochastic maximum principle in optimal control of singular diffusions with non linear coefficients. S Bahlali, ROSEA Chala, ROSERandom Oper. and Stoch. Equ. S. 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{'abstract': "This paper is concerned with necessary and sufficient conditions for near-optimal singular stochastic controls for systems driven by a nonlinear stochastic differential equations (SDEs in short). The proof of our result is based on Ekeland's variational principle and some dilecate estimates of the state and adjoint processes. This result is a generalization of Zhou's stochastic maximum principle for near-optimaity to singular control problem.", 'arxivid': '1110.5553', 'author': ['Mokhtar Hafayed ', 'Syed Abbas ', 'Petr Veverka ', '\nSchool of Basic Sciences\nLaboratory of Applied Mathematics\nMohamed Khider University\nBox 14507000Po, BiskraAlgeria\n', '\nDepartment of Mathematics\nFaculty of Nuclear Sciences and Physical Engineering\nIndian Institute of Technology Mandi\nMandi-175001India\n', '\nEU-Czech Republic\nCzech Technical University\nTrojanova 13, Prague 120 00\n'], 'authoraffiliation': ['School of Basic Sciences\nLaboratory of Applied Mathematics\nMohamed Khider University\nBox 14507000Po, BiskraAlgeria', 'Department of Mathematics\nFaculty of Nuclear Sciences and Physical Engineering\nIndian Institute of Technology Mandi\nMandi-175001India', 'EU-Czech Republic\nCzech Technical University\nTrojanova 13, Prague 120 00'], 'corpusid': 10085158, 'doi': '10.1007/s11590-012-0484-6', 'github_urls': [], 'n_tokens_mistral': 11374, 'n_tokens_neox': 9842, 'n_words': 5827, 'pdfsha': 'f0c41ed57effea1ec5554699cfb86b75c6456a5e', 'pdfurls': ['https://arxiv.org/pdf/1110.5553v1.pdf'], 'title': ['On Necessary and Sufficient Conditions for Near-Optimal Singular Stochastic Controls', 'On Necessary and Sufficient Conditions for Near-Optimal Singular Stochastic Controls'], 'venue': []}

arxiv

Computing Horn Rewritings of Description Logics Ontologies Mark Kaminski Department of Computer Science University of Oxford UK Bernardo Cuenca Grau Department of Computer Science University of Oxford UK Computing Horn Rewritings of Description Logics Ontologies We study the problem of rewriting an ontology O 1 expressed in a DL L 1 into an ontology O 2 in a Horn DL L 2 such that O 1 and O 2 are equisatisfiable when extended with an arbitrary dataset. Ontologies that admit such rewritings are amenable to reasoning techniques ensuring tractability in data complexity. After showing undecidability whenever L 1 extends ALCF, we focus on devising efficiently checkable conditions that ensure existence of a Horn rewriting. By lifting existing techniques for rewriting Disjunctive Datalog programs into plain Datalog to the case of arbitrary first-order programs with function symbols, we identify a class of ontologies that admit Horn rewritings of polynomial size. Our experiments indicate that many real-world ontologies satisfy our sufficient conditions and thus admit polynomial Horn rewritings.arXiv:1504.05150v2 [cs.AI] 21 Apr 2015Proof. The claim follows by a straightforward induction on ρ.We call a node in a derivation Horn (resp. disjunctive) if it is labeled by a Horn atom (resp. a disjunction of disjunctive atoms).Proposition 19. Let P be a program, M a marking of P, and D a dataset over the predicates in P. Then Ξ M (P) ∪ D |= ⊥(s) for every ground term s over the signature of P ∪ D.Proof. The claim is a straightforward consequence of the axiomatisation of ⊥ in Ξ M (P).Theorem 7. Let M be a marking of a program P. Then Ξ M (P) is a polynomial-size Horn rewriting of P.Proof. We proceed in two steps, which together imply the theorem. We fix an arbitrary markable program P, a marking M of P, and a dataset D. W.l.o.g. we assume that D only contains predicates in P. Introduction Reasoning over ontology-enriched datasets is a key requirement in many applications of semantic technologies. Standard reasoning tasks are, however, of high worst-case complexity. Satisfiability checking is 2NEXPTIME-complete for the description logic (DL) SROIQ underpinning the standard ontology language OWL 2 and NEXPTIME-complete for SHOIN , which underpins OWL DL [Kazakov, 2008]. Reasoning is also co-NP-hard with respect to data complexity-a key measure of complexity for applications involving large amounts of instance data [Hustadt et al., 2005]. Tractability in data complexity is typically associated with Horn DLs, where ontologies correspond to first-order Horn clauses [Ortiz et al., 2011;Hustadt et al., 2005]. The more favourable computational properties of Horn DLs make them a natural choice for data-intensive applications, but they also come at the expense of a loss in expressive power. In particular, Horn DLs cannot capture disjunctive axioms, i.e., statements such as "every X is either a Y or a Z". Disjunctive axioms are common in real-world ontologies, like the NCI Thesaurus or the ontologies underpinning the European Bioinformatics Institute (EBI) linked data platform. 1 1 http://www.ebi.ac.uk/rdf/platform In this paper we are interested in Horn rewritability of description logic ontologies; that is, whether an ontology O 1 expressed in a DL L 1 can be rewritten into an ontology O 2 in a Horn DL L 2 such that O 1 and O 2 are equisatisfiable when extended with an arbitrary dataset. Ontologies that admit such Horn rewritings are amenable to more efficient reasoning techniques that ensure tractability in data complexity. Horn rewritability of DL ontologies is strongly related to the rewritability of Disjunctive Datalog programs into Datalog, where both the source and target languages for rewriting are function-free. Kaminski et al. [2014b] characterised Datalog rewritability of Disjunctive Datalog programs in terms of linearity: a restriction that requires each rule to contain at most one body atom that is IDB (i.e., whose predicate also occurs in head position in the program). It was shown that every linear Disjunctive Datalog program can be rewritten into plain Datalog (and vice versa) by means of program transposition-a polynomial transformation in which rules are "inverted" by shuffling all IDB atoms between head and body while at the same time replacing their predicates by auxiliary ones. Subsequently, Kaminski et al. [2014a] proposed the class of markable Disjunctive Datalog programs, where the linearity requirement is relaxed so that it applies only to a subset of "marked" atoms. Every markable program can be polynomially rewritten into Datalog by exploiting a variant of transposition where only marked atoms are affected. Our contributions in this paper are as follows. In Section 3, we show undecidability of Horn rewritability whenever the input ontology is expressed in ALCF. This is in consonance with the related undecidability results by Bienvenu et al. [2014] and Lutz and Wolter [2012] for Datalog rewritability and non-uniform data complexity for ALCF ontologies. In Section 4, we lift the markability condition and the transposition transformation in [Kaminski et al., 2014a] for Disjunctive Datalog to arbitrary first-order programs with function symbols. We then show that all markable first-order programs admit Horn rewritings of polynomial size. This result is rather general and has potential implications in areas such as theorem proving [Robinson and Voronkov, 2001] and knowledge compilation [Darwiche and Marquis, 2002]. The notion of markability for first-order programs can be seamlessly adapted to ontologies via the standard FOL translation of DLs [Baader et al., 2003]. This is, however, of limited practical value since Horn programs ob-tained via transposition may not be expressible using standard DL constructors. In Section 5, we introduce an alternative satisfiability-preserving translation from ALCHIF ontologies into first-order programs and show in Section 6 that the corresponding transposed programs can be translated back into Horn-ALCHIF ontologies. Finally, we focus on complexity and show that reasoning over markable L-ontologies is EXPTIME-complete in combined complexity and PTIME-complete w.r.t. data for each DL L between ELU and ALCHIF. All our results immediately extend to DLs with transitive roles (e.g., SHIF) by exploiting standard transitivity elimination techniques [Baader et al., 2003]. We have implemented markability checking and evaluated our techniques on a large ontology repository. Our results indicate that many real-world ontologies are markable and thus admit Horn rewritings of polynomial size. The proofs of all our results are delegated to the appendix. Preliminaries We assume standard first-order syntax and semantics. We treat the universal truth and falsehood ⊥ symbols as well as equality (≈) as ordinary predicates of arity one ( and ⊥) and two (≈), the meaning of which will be axiomatised. Programs A first-order rule (or just a rule) is a sentence ∀ x∀ z.[ϕ( x, z) → ψ( x)] where variables x and z are disjoint, ϕ( x, z) is a conjunction of distinct atoms over x ∪ y, and ψ( x) is a disjunction of distinct atoms over x. Formula ϕ is the body of r, and ψ is the head. Quantifiers are omitted for brevity, and safety is assumed (all variables in the rule occur in the body). We define the following sets of rules for a finite signature Σ: (i) P Σ consists of a rule P (x 1 , . . . , x n ) → (x i ) for each predicate P ∈ Σ and each 1 ≤ i ≤ n and a rule → (a) for each constant a ∈ Σ; (ii) P ⊥ Σ consists of the rule having ⊥(x) in the body and an empty head; and (iii) P ≈ Σ consists of the standard axiomatisation of ≈ as a congruence over Σ. 2 A program is a finite set of rules P = P 0 ∪ P Σ ∪ P ⊥ Σ ∪ P ≈ Σ with Σ the signature of P 0 , where we assume w.l.o.g. that the body of each rule in P 0 does not mention ⊥ or ≈, and the head is non-empty and does not mention . We omit Σ for the components of P and write P , P ⊥ and P ≈ . A rule is Horn if its head consists of at most one atom, and a program is Horn if so are all of its rules. Finally, a fact is a ground, function-free atom, and a dataset is a finite set of facts. Ontologies We assume familiarity with DLs and ontology languages [Baader et al., 2003]. A DL signature Σ consists of disjoint countable sets of concept names Σ C and role names Σ R . A role is an element of Σ R ∪ {R − | R ∈ Σ R }. The function inv is defined over roles as follows, where R ∈ Σ R : inv(R) = R − and inv(R − ) = R. W.l.o.g., we consider normalised axioms as on the left-hand side of Table 1 O such that R 1 * R 2 and inv(R 1 ) * inv(R 2 ) hold whenever R 1 R 2 is an axiom in O. We refer to the DL where only axioms of type T1-T3 are available and the use of inverse roles is disallowed as ELU. The logic ALC extends ELU with axioms T4. We then use standard naming conventions for DLs based on the presence of inverse roles (I), axioms T5 (H) and axioms T6 (F). Finally, an ontology is EL if it is both ELU and Horn. Table 1 also provides the standard translation π from normalised axioms into first-order rules, where at(R, x, y) is defined as R(x, y) if R is named and as S(y, x) if R = S − . We define π(O) as the smallest program containing π(α) for each axiom α in O. Given a dataset D, we say that O ∪ D is satisfiable iff so is π(O) ∪ D in first-order logic. Horn Rewritability Our focus is on satisfiability-preserving rewritings. Standard reasoning tasks in description logics are reducible to unsatisfiability checking [Baader et al., 2003], which makes our results practically relevant. We start by formulating our notion of rewriting in general terms. Definition 1. Let F and F be sets of rules. We say that F is a rewriting of F if it holds that F ∪ D is satisfiable iff so is F ∪ D for each dataset D over predicates from F. We are especially interested in computing Horn rewritings of ontologies-that is, rewritings where the given ontology O 1 is expressed in a DL L 1 and the rewritten ontology O 2 is in a Horn DL L 2 (where preferably L 2 ⊆ L 1 ). This is not possible in general: satisfiability checking is co-NP-complete in data complexity even for the basic logic ELU [Krisnadhi and Lutz, 2007], whereas data complexity is tractable even for highly expressive Horn languages such as Horn-SROIQ [Ortiz et al., 2011]. Horn rewritability for DLs can be formulated as a decision problem as follows: Definition 2. The (L 1 , L 2 )-Horn rewritability problem for DLs L 1 and L 2 is to decide whether a given L 1 -ontology admits a rewriting expressed in Horn-L 2 . Our first result establishes undecidability whenever the input ontology contains at-most cardinality restrictions and thus equality. This result is in consonance with the related undecidability results by Bienvenu et al. [2014] and Lutz and Wolter [2012] for Datalog rewritability and non-uniform data complexity for ALCF ontologies. Theorem 3. (L 1 , L 2 )-Horn rewritability is undecidable for L 1 = ALCF and L 2 any DL between ELU and ALCHIF. This result holds under the assumption that PTIME =NP. Intractability results in data complexity rely on the ability of non-Horn DLs to encode co-NP-hard problems, such as non-3-colourability [Krisnadhi and Lutz, 2007;Hustadt et al., 2005]. In practice, however, it can be expected that ontologies do not encode such problems. Thus, our focus from now onwards will be on identifying classes of ontologies that admit (polynomial size) Horn rewritings. Program Markability and Transposition In this section, we introduce the class of markable programs and show that every markable program can be rewritten into a Horn program by means of a polynomial transformation, which we refer to as transposition. Roughly speaking, transposition inverts the rules in a program P by moving certain atoms from head to body and vice versa while replacing their corresponding predicates with fresh ones. Markability of P ensures that we can pick a set of predicates (a marking) such that, by shuffling only atoms with a marked predicate, we obtain a Horn rewriting of P. Our results in this section generalise the results by Kaminski et al. [2014a] for Disjunctive Datalog to first-order programs with function symbols. To illustrate our definitions throughout this section, we use an example program P ex consisting of the following rules: T 1. n i=1 Ai m j=1 Cj n i=1 Ai(x) → m j=1 Cj(x) T 2. ∃R.A C at(R, x, y) ∧ A(y) → C(x) T 3. A ∃R.B A(x) → at(R, x, f (x)); A(x) → B(f (x)) T 4. A ∀R.C A(x) ∧ at(R, x, y) → C(y) T 5. S R S(x, y) → at(R, x, y) T 6. A ≤ 1 R.B A(z) ∧ at(R, z, x1) ∧ at(R, z, x2) ∧ B(x1) ∧ B(x2) → x1 ≈ x2A(x) → B(x) B(x) → C(x) ∨ D(x) C(x) → ⊥(x) D(x) → C(f (x)) Markability. The notion of markability involves a partitioning of the program's predicates into Horn and disjunctive. Intuitively, the former are those whose extension for all datasets depends only on the Horn rules in the program, whereas the latter are those whose extension may depend on a disjunctive rule. This intuition can be formalised using the standard notion of a dependency graph in Logic Programming. Definition 4. The dependency graph G P = (V, E, µ) of a program P is the smallest edge-labeled digraph such that: (i) V contains all predicates in P; (ii) r ∈ µ(P, Q) whenever r ∈ P, P is in the body of r, and Q is in the head of r; and (iii) (P, Q) ∈ E whenever µ(P, Q) = ∅. A predicate Q depends on r ∈ P if G P has a path ending in Q and involving an r-labeled edge. Predicate Q is Horn if it depends only on Horn rules; otherwise, Q is disjunctive. For instance, predicates C, D, and ⊥ are disjunctive in our example program P ex , whereas A and B are Horn. We can now introduce the notion of a marking-a subset of the disjunctive predicates in a program P ensuring that the transposition of P where only marked atoms are shuffled between head and body results in a Horn program. Definition 5. A marking of a program P is a set M of disjunctive predicates in P satisfying the following properties, where we say that an atom is marked if its predicate is in M : (i) each rule in P has at most one marked body atom; (ii) each rule in P has at most one unmarked head atom; and (iii) if Q ∈ M and P is reachable from Q in G P , then P ∈ M . We say that a program is markable if it admits a marking. Condition (i) in Def. 5 ensures that at most one atom is moved from body to head during transposition. Condition (ii) ensures that all but possibly one head atom are moved to the body. Finally, condition (iii) requires that all predicates depending on a marked predicate are also marked. We can observe that our example program P ex admits two markings: M 1 = {C, ⊥} and M 2 = {C, D, ⊥}. Markability can be efficiently checked via a 2-SAT reduction, where we assign to each predicate Q in P a propositional variable X Q and encode the constraints in Def. 5 as 2-clauses. For each rule ϕ ∧ n i=1 P i ( s i ) → m j=1 Q j ( t j ) , with ϕ the conjunction of all Horn atoms in the rule head, we include clauses (i) ¬X Pi ∨ ¬X Pj for all 1 ≤ i < j ≤ n, which enforce at most one body atom to be marked; (ii) X Qi ∨ X Qj for 1 ≤ i < j ≤ m, which ensure that at most one head atom is unmarked; and (iii) ¬X Pi ∨ X Qj for 1 ≤ i ≤ n and 1 ≤ j ≤ m, which close markings under rule dependencies. Each model of the resulting clauses yields a marking of P. Transposition. Before defining transposition, we illustrate the main intuitions using program P ex and marking M 1 . The first step to transpose P ex is to introduce fresh unary predicates C and ⊥, which stand for the negation of the marked predicates C and ⊥. To capture the intended meaning of these predicates, we introduce rules X(x) → ⊥(x) for X ∈ {A, B, C, D} and a rule ⊥(x) → ⊥(f (x)) for the unique function symbol f in P ex . The first rules mimick the usual axiomatisation of and ensure that an atom ⊥(c) holds in a Herbrand model of the transposed program whenever X(c) also holds. The last rule ensures that ⊥ holds for all terms in the Herbrand universe of the transposed programan additional requirement that is consistent with the intended meaning of ⊥, and critical to the completeness of transposition in the presence of function symbols. Finally, a rule ⊥(z) ∧ C(x) ∧ C(x) → ⊥(z) ensures that the fresh predicate C behaves like the negation of C (⊥(z) is added for safety). The key step of transposition is to invert the rules involving the marked predicates by shuffling marked atoms between head and body while replacing their predicate with the corresponding fresh one. In this way, rule B(x) → C(x) ∨ D(x) yields B(x) ∧ C(x) → D(x), and C(x) → ⊥(x) yields ⊥(x) → C(x). Additionally, rule D(x) → C(f (x)) is trans- posed as ⊥(z)∧D(x)∧C(f (x)) → ⊥(z) to ensure safety. Finally, transposition does not affect rules containing only Horn predicates, e.g., rule A(x) → B(x) is included unchanged. Definition 6. Let M be a marking of a program P. For each disjunctive predicate P in P, let P be a fresh predicate of the same arity. The M -transposition of P is the smallest program Ξ M (P) containing every rule in P involving only Horn predicates and all rules 1-6 given next, where ϕ is the conjunction of all Horn atoms in a rule, ϕ is the least conjunction of ⊥atoms making a rule safe and all P i , Q j are disjunctive: 1. ϕ ∧ϕ∧ m j=1 Q j ( t j )∧ n i=1 P i ( s i ) → Q( t) for each rule in P of the form ϕ ∧ Q( t) ∧ m j=1 Q j ( t j ) → n i=1 P i ( s i ) where Q( t) is the only marked body atom; 2. ⊥(x) ∧ ϕ ∧ m j=1 Q j ( t j ) ∧ n i=1 P i ( s i ) → ⊥(x), where x a fresh variable, for each rule in P of the form ϕ ∧ m j=1 Q j ( t j ) → n i=1 P i ( s i ), with no marked body atoms and no unmarked head atoms; 3. ϕ ∧ m j=1 Q j ( t j ) ∧ n i=1 P i ( s i ) → P ( s) for each rule in P of the form ϕ ∧ m j=1 Q j ( t j ) → P ( s) ∨ n i=1 P i ( s i ) where P ( s) is the only unmarked head atom; 4. ⊥(z) ∧ P ( x) ∧ P ( x) → ⊥(z) for marked predicate P ; 5. P (x 1 , . . . , x n ) → ⊥(x i ) for each P in P and 1 ≤ i ≤ n; 6. ⊥(x 1 ) ∧ . . . ∧ ⊥(x n ) → ⊥(f (x 1 , . . . , x n )) for each n-ary function symbol f in P. Note that rules of type 1 in Def. 6 satisfy {P 1 , . . . , P n } ⊆ M since Q ∈ M , while for rules of type 3 we have {Q 1 , . . . , Q m } ∩ M = ∅ since P / ∈ M . Clearly, P ex is unsatisfiable when extended with fact A(a). To see that Ξ M1 (P ex ) ∪ {A(a)} is also unsatisfiable, note that B(a) is derived by the unchanged rule A(x) → B(x). Fact C(a) is derived using A(x) → ⊥(x) and the transposed rule ⊥(x) → C(x). We derive D(a) using B(x)∧C(x) → D(x). But then, to derive a contradiction we need to apply rule ⊥(z)∧D(x)∧C(f (x)) → ⊥(z), which is not possible unless we derive C(f (a)). For this, we first use ⊥(x) → ⊥(f (x)), which ensures that ⊥ holds for f (a), and then ⊥(x) → C(x). Transposition yields quadratically many Horn rules. The following theorem establishes its correctness. Theorem 7. Let M be a marking of a program P. Then Ξ M (P) is a polynomial-size Horn rewriting of P. It follows that every markable set of non-Horn clauses N can be polynomially transformed into a set of Horn clauses N such that N ∪ D and N ∪ D are equisatisfiable for every set of facts D. This result is rather general and has potential applications in first-order theorem proving, as well as in knowledge compilation, where Horn clauses are especially relevant [Darwiche and Marquis, 2002;Del Val, 2005]. Markability of DL Ontologies The notion of markability is applicable to first-order programs and hence can be seamlessly adapted to ontologies via the standard translation π in Table 1. This, however, would be of limited value since the Horn programs resulting from transposition may not be expressible in Horn-ALCHIF. Consider any ontology with an axiom ∃R.A B and any marking M involving R. Rule R(x, y)∧A(y) → B(x) stemming from π would be transposed as B(x)∧A(y) → R(x, y), which cannot be captured in ALCHIF. 3 To address this limitation we introduce an alternative translation ξ from DL axioms into rules, which we illustrate using the example ontology O ex in Table 2. The key idea is to encode existential restrictions in axioms T3 as unary atoms over functional terms. For instance, axiom α 2 in O ex would yield B(x) → D(f R,D (x)), where the "successor" relation between an instance b of B and some instance of D in a Herbrand model is encoded as a term f R,D (b), instead of a binary atom of the form R(b, g(b)). This encoding has an immediate impact on markings: by marking B we are only forced to also mark D (rather than both R and D). In this way, we will be able to ensure that markings consist of unary predicates only. To compensate for the lack of binary atoms involving functional terms in Herbrand models, we introduce new rules when translating axioms T2, T4, and T6 using ξ. For instance, ξ(α 3 ) yields the following rules in addition to π(α 3 ): a rule D(f R,D (x)) → D(x) to ensure that all objects c with an R- successor f R,D (c) generated by ξ(α 2 ) are instances of D; a rule D(f R,B (x)) → D(x) , which makes sure that an object whose R-successor generated by ξ(α 4 ) is an instance of D is also an instance of D. Finally, axioms α 1 and α 5 , which involve no binary predicates, are translated as usual. Definition 8. Let O be an ontology. For each concept ∃R.B in an axiom of type T3, let f R,B be a unary function symbol, and Φ the set of all such symbols. We define ξ(O) as the smallest program containing π(α) for each axiom α in O of type T1-T2 and T4-T6, as well as the following rules: • A(x) → B(f R,B (x)) for each axiom T3; • A(f R ,Y (x)) → C(x) for each axiom T2 and R and Y s.t. f R ,Y ∈ Φ and R * R. • A(f inv(R ),Y (x)) → C(x) for each axiom T4 and R and Y s.t. f inv(R ),Y ∈ Φ and R * R. • A(x) ∧ Y (f inv(R ),Y (x)) → C(f inv(R ),Y (x)) for each ax- iom T2 and R and Y s.t. f inv(R ),Y ∈ Φ and R * R. • A(x) ∧ Y (f R ,Y (x)) → C(f R ,Y (x)) for each axiom T4 and R and Y s.t. f R ,Y ∈ Φ and R * R. • A(z) ∧ B(f R ,Y (z)) ∧ at(R, z, x) ∧ B(x) → f R ,Y (z) ≈ x for each ax. T6 and R , Y s.t. f R ,Y ∈ Φ and R * R. • A(f inv(R ),Y (x))∧B(x)∧at(R, f inv(R ),Y (x), y)∧B(y) → x ≈ y for each axiom T6 and R and Y s.t. f inv(R ),Y ∈ Φ and R * R. • A(z) ∧ B(f R 1 ,Y1 (z)) ∧ B(f R 2 ,Y2 (z)) → f R 1 ,Y1 (z) ≈ f R 2 ,Y2 (z) for each axiom T6 and f R i ,Yi ∈ Φ s.t. R i * R. • A(f inv(R 1 ),Y1 (x)) ∧ B(x) ∧ B(f R 2 ,Y2 (f inv(R 1 ),Y1 (x))) → x ≈ f R 2 ,Y2 (f inv(R 1 ),Y1 (x)) for each axiom T6 and each R i and Y i s.t. {f inv(R 1 ),Y1 , f R 2 ,Y2 } ⊆ Φ and R i * R. Note that, in contrast to the standard translation π, which introduces at most two rules per DL axiom, ξ can introduce linearly many rules in the size of the role hierarchy induced by axioms of type T5. The translation ξ(O ex ) of our example ontology O ex is given in the second column of Table 2. Clearly, O ex is unsatisfiable when extended with A(a) and E(a). We can check that ξ(O ex ) ∪ {A(a), E(a)} is also unsatisfiable. The following theorem establishes the correctness of ξ. Theorem 9. For every ontology O and dataset D over predi- Ontology Oex Rule translation ξ(Oex) Thus, we define markability of ontologies in terms of ξ rather than in terms of π. We can check that cates in O we have that O ∪D is satisfiable iff so is ξ(O)∪D.Transposition ΞM ex (ξ(Oex)) Horn DL rewriting Ψ(ΞM ex (ξ(Oex))) α1 A B C A(x) → B(x) ∨ C(x) A(x) ∧ B(x) → C(x) A B C α2 B ∃R.D B(x) → D(fR,D(x)) D(fR,D(x)) → B(x) ∃RD.D B α3 ∃R.D D R(x, y) ∧ D(y) → D(x) R(x, y) ∧ D(x) → D(y) D ∀R.D D(fR,D(x)) → D(x) D(x) → D(fR,D(x)) D ∀RDD D(fR,B(x)) → D(x) D(x) → D(fR,B(x)) D ∀RBD α4 C ∃R.B C(x) → B(fR,B(x)) ⊥(z) ∧ C(x) ∧ B(fR,B(x)) → ⊥(z) C ∃RB.B ⊥ α5 D E ⊥ D(x) ∧ E(x) → ⊥(x) E(x) ∧ ⊥(x) → D(x) E ⊥ D X(x) → ⊥(x), X ∈ {A, B, C, D, E} X ⊥ R(x1, x2) → ⊥(xi), 1 ≤ i ≤ 2 ∀R.⊥, ∃R. ⊥ ⊥(x) → ⊥(fR,Y (x)), Y ∈ {B, D} ⊥ ∃RY .⊥π(O ex ) is not markable, whereas ξ(O ex ) admits the marking M ex . Definition 11. An ontology O is markable if so is ξ(O). We conclude this section with the observation that markability of an ontology O can be efficiently checked by first computing the program ξ(O) and then exploiting the 2-SAT encoding sketched in Section 4. Rewriting Markable Ontologies It follows from the correctness of transposition in Theorem 7 and ξ in Theorem 9 that every ALCHIF ontology O admitting a marking M has a Horn rewriting of polynomial size given as the program Ξ M (ξ(O)). In what follows, we show that this rewriting can be expressed within Horn-ALCHIF. Let us consider the transposition of ξ(O ex ) via the marking M ex , which is given in the third column of Table 2. The transposition of α 1 and α 5 corresponds directly to DL axioms via the standard translation in Table 1. In contrast, the transposition of all other axioms leads to rules that have no direct correspondence in DLs. The following lemma establishes that the latter rules are restricted to the types T7-T20 specified on the left-hand side of Table 3. Lemma 12. Let O be an ontology and M a minimal marking of ξ(O). Then Ξ M (ξ(O)) contains only Horn rules of type T1-T2 and T4-T6 in Table 1 as well as type T7-T20 in Table 3. We can now specify a transformation Ψ that allows us to translate rules T7-T20 in Table 3 back into DL axioms. Definition 13. We define Ψ as the transformation mapping (i) each Horn rule r of types T1-T2 and T4-T6 in Table 1 to the DL axiom π −1 (r) (ii) each rule T7-T20 on the left-hand side of Table 3 to the DL axioms on the right-hand side. 4 Intuitively, Ψ works as follows: (i) Function-free rules are "rolled up" as usual into DL axioms (see e.g., T7). (ii) Unary atoms A(f R,Y (x)) with A = ⊥ involving a functional term are translated as either existentially or universally quantified concepts depending on whether they occur in the body or in the head (e.g., T10, T11); in contrast, atoms ⊥(f R,Y (x)) in rules ⊥(x) → ⊥(f R,Y (x)) are translated as ∃R Y .⊥, instead of ∀R Y .⊥ (see T9). (iii) Rules T15-T18, which involve ≈ in the head and roles R and R in the body, are rolled back into axioms of type T6 over the "union" of R and R , which is captured using fresh roles and role inclusions. The ontology obtained by applying Ψ to our running example is given in the last column of Table 2. Correctness of Ψ and its implications for the computation of Horn rewritings are summarised in the following lemma. A closer look at our transformations reveals that our rewritings do not introduce constructs such as inverse roles and cardinality restrictions if these were not already present in the input ontology. In contrast, fresh role inclusions may originate from cardinality restrictions in the input ontology. As a result, our approach is language-preserving: if the input O 1 is in a DL L 1 between ALC and ALCHI, then its rewriting O 2 stays in the Horn fragment of L 1 ; furthermore, if L 1 is between ALCF and ALCIF, then O 2 may contain fresh role inclusions (H). A notable exception is when O 1 is an ELU ontology, in which case axioms T2 and T3 in O 1 may yield axioms of type T4 in O 2 . The following theorem follows from these observations and Lemma 14. Theorem 15. Let L be a DL between ALC and ALCHI. Then every markable L ontology is polynomially rewritable into a Horn-L ontology. If L is between ALCF and ALCHIF, then every markable L ontology is polynomially rewritable into Horn-LH. Finally, every markable ELU ontology is polynomially rewritable into Horn-ALC. Complexity Results We next establish the complexity of satisfiability checking over markable ontologies. We first show that satisfiability checking over markable ELU ontologies is EXPTIME-hard. This implies that it is not possible to polynomially rewrite every markable ELU ontology into EL. Consequently, our rewriting approach is optimal for ELU in the sense that introducing universal restrictions (or equivalently inverse roles) in the rewriting is unavoidable. Lemma 16. Satisfiability checking over markable ELU ontologies is EXPTIME-hard. T 7. ⊥(z) ∧ B(x) ∧ R(x, y) ∧ A(y) → ⊥(z) B ∃R.A ⊥ T 8. ⊥(z) ∧ A(fR,Y (x)) ∧ B(x) → ⊥(z) B ∃RY .A ⊥ T 9. ⊥(x) → ⊥(fR,Y (x)) ⊥ ∃RY .⊥ T 10. B(x) → A(fR,Y (x)) B ∀RY .A if A = ⊥ or B = ⊥ T 11. B(fR,Y (x)) → A(x) ∃RY .B A T 12. A(x) ∧ B(fR,Y (x)) → C(fR,Y (x)) A ∃RY .B ∀RY .C T 13. ⊥(z) ∧ A(x) ∧ B(fR,Y (x)) ∧ C(fR,Y (x)) → ⊥(z) A ∃RY (B C) ⊥ T 14. B(fR,Y (x)) ∧ C(fR,Y (x)) → A(x) ∃RY (B C) A T 15. A(z) ∧ B(f R ,Y (z)) ∧ at(R, z, x) ∧ B(x) R Y S {R Y ,R} and R S {R Y ,R} and → f R ,Y (z) ≈ x A ≤1S {R Y ,R} .B T 16. A(f R ,Y (x)) ∧ B(x) ∧ at(R, f R ,Y (x), y) ∧ B(y)R Y S {R Y ,R} and R S {R Y ,R} and → x ≈ y A ≤1S {R Y ,R} .B andR Y ≡ inv(R Y ) T 17. A(z) ∧ B(fR,Y (z)) ∧ B(f R ,Z (z)) RY S {R Y ,R Z } and R Z S {R Y ,R Z } and → fR,Y (z) ≈ f R ,Z (z) A ≤1S {R Y ,R Z } .B T 18. A(fR,Y (x)) ∧ B(x) ∧ B(f R ,Z (fR,Y (x)))RY S {R Y ,R Z } and R Z S {R Y ,R Z } and → x ≈ f R ,Z (fR,Y (x)) A ≤1S {R Y ,R Z } .B andRY ≡ inv(RY ) T 19. R(x, y) → ⊥(x) ∃R. ⊥ T 20. R(x, y) → ⊥(y) ∀R.⊥ All Horn DLs from ALC to ALCHIF are EXPTIMEcomplete in combined complexity and PTIME-complete in data complexity [Krötzsch et al., 2013]. By Theorem 15, the same result holds for markable ontologies in DLs from ALC to ALCHIF. Finally, Lemma 16 shows that these complexity results also extend to markable ELU ontologies. Theorem 17. Let L be in-between ELU and ALCHIF. Satisfiability checking over markable L-ontologies is EXPTIMEcomplete and PTIME-complete w.r.t. data. Related Work Horn logics are common target languages for knowledge compilation [Darwiche and Marquis, 2002]. Selman and Kautz [1996] proposed an algorithm for compiling a set of propositional clauses into a set of Horn clauses s.t. their Horn consequences coincide. This approach was generalised to FOL by Del Val [2005], without termination guarantees. Bienvenu et al. [2014] showed undecidability of Datalog rewritability for ALCF and decidability in NEXPTIME for SHI. Cuenca Grau et al. Lutz and Wolter [2012] investigated (non-uniform) data complexity of query answering w.r.t. fixed ontologies. They studied the boundary of PTIME and co-NP-hardness and established a connection with constraint satisfaction problems. Finally, Lutz et al. [2011] studied model-theoretic rewritability of ontologies in a DL L 1 into a fragment L 2 of L 1 . These rewritings preserve models rather than just satisfiability, which severely restricts the class of rewritable ontologies; in particular, only ontologies that are "semantically Horn" can be rewritten. For instance, O = {A B C}, which is rewritable by our approach, is not Horn-rewritable according to Lutz et al. [2011]. Proof of Concept To assess the practical implications of our results, we have evaluated whether real-world ontologies are markable (and hence also polynomially Horn rewritable). We analysed 120 non-Horn ontologies extracted from the Protege Ontology Library, BioPortal (http://bioportal.bioontology.org/), the corpus by Gardiner et al. [2006], and the EBI linked data platform (http://www.ebi.ac.uk/rdf/platform). To check markability, we have implemented the 2-SAT reduction in Section 4 and a simple 2-SAT solver. We found that a total of 32 ontologies were markable and thus rewritable into a Horn ontology, including some ontologies commonly used in applications, such as ChEMBL (see http://www.ebi.ac.uk/rdf/services/chembl/) and BioPAX Reactome (http://www.ebi.ac.uk/rdf/services/reactome/). When using π as first-order logic translation, we obtained 30 markable ontologies-a strict subset of the ontologies markable using ξ. However, only 27 ontologies were rewritable to a Horn DL since in three cases the marking contained a role. Conclusion and Future Work We have presented the first practical technique for rewriting non-Horn ontologies into a Horn DL. Our rewritings are polynomial, and our experiments suggest that they are applicable to widely-used ontologies. We anticipate several directions for future work. First, we would like to conduct an extensive evaluation to assess whether the use of our rewritings can significantly speed up satisfiability checking in practice. Second, we will investigate relaxations of markability that would allow us to capture a wider range of ontologies. [ Gottlob et al., 2012] A Proofs for Section 3 Theorem 3. (L 1 , L 2 )-Horn rewritability is undecidable for L 1 = ALCF and L 2 any DL between ELU and ALCHIF. This result holds under the assumption that PTIME =NP. Proof. We adapt the undecidability proof for datalog-rewritability of ALCF in [Bienvenu et al., 2014]. Given an instance Π of the undecidable finite rectangle tiling problem, Bienvenu et al. give an ALCF ontology O 1 , signature Σ and concept name E such that the following three conditions are equivalent: • Π admits a tiling • there is a dataset D over Σ such that O 1 ∪ D is satisfiable and O 1 ∪ D |= E(a) for some a in D; • there is a dataset D over Σ such that O 1 ∪ D is satisfiable and O 1 ∪ D |= ∃x.E(x). Let S, S be fresh role names and P 1 , P 2 , P 3 fresh concept names. Let O 2 be an extension of O 1 by the following axioms. ∃S.E (∃S . ) P i P j ⊥ for 1 ≤ i < j ≤ 3 ∃S . P 1 P 2 P 3 (∃S . ) P i ∃S .P i ⊥ for 1 ≤ i ≤ 3 We next show that Π admits a tiling if and only if O 2 is not rewritable to Horn-L 2 . First, suppose Π admits a tiling. Then there is a dataset D 1 over Σ such that O 1 ∪D 1 is satisfiable and O 1 ∪D 1 |= E(a) for some a in D 1 . Given a connected undirected graph G, let D G = { S (d, d ), S (d , d) | {d, d } edge in G } and D 2 = D 1 ∪ D G ∪ { S(d, c) | d occurs in D G ∪ D 1 , c occurs in D 1 }. Then O 2 ∪ D 2 is consistent if and only if G is 3-colourable. Therefore, since 3-colourability is NP-complete in data whereas satisfiability checking w.r.t. Horn-ALCHIF ontologies is tractable in data, O 2 is not rewritable to Horn-L 2 unless PTIME = NP. Now suppose Π does not admit a tiling. Then O 2 ∪ D is unsatisfiable for every D and hence the ontology { ⊥} is a Horn-ELU rewriting of O 2 . B Proofs for Section 4 Reasoning w.r.t. programs can be realised by means of the hyperresolution calculus. In our treatment of hyperresolution we treat disjunctions of atoms as sets and hence we do not allow for duplicated atoms in a disjunction. Let r = n i=1 β i → ϕ be a rule and, for each 1 ≤ i ≤ n, let ψ i be a disjunction of atoms ψ i = χ i ∨ α i with α i a single atom. Let σ be an MGU of each β i , α i . Then the disjunction of atoms ϕ = ϕσ ∨ χ 1 ∨ · · · ∨ χ n is a hyperresolvent of r and ψ 1 , . . . , ψ n . Let P be a program, let D be a dataset, and let ϕ be a disjunction of atoms. A (hyperresolution) derivation of ϕ from P ∪ D is a pair ρ = (T, λ) where T is a tree, λ a labeling function mapping each node in T to a disjunction of atoms, and the following properties hold for each v ∈ T : 1. λ(v) = ϕ if v is the root; 2. λ(v) ∈ P ∪ D if v is a leaf; and 3. if v has children w 1 , . . . , w n , then λ(v) is a hyperresolvent of a rule r ∈ P and λ(w 1 ), . . . , λ(w n ). We write P ∪ D ϕ to denote that ϕ has a derivation from P ∪ D. Hyperresolution is sound and complete in the following sense: P ∪ D is unsatisfiable iff P ∪ D . Furthermore, if P ∪ D is satisfiable then P ∪ D α iff P ∪ D |= α for every atom α. Hyperresolution derivations satisfy the following property. Proposition 18. Let P be a program, D a dataset, and ρ a derivation from P ∪ D. Then every node in ρ is labeled by either a single Horn atom or a (possibly empty) disjunction of disjunctive atoms. 1. We show that P ∪ D |= implies Ξ M (P) ∪ D |= . For this, we consider a derivation ρ of from P ∪ D and show that for every disjunctive atom Q( s) in the label of a node in ρ, we have Ξ M (P) ∪ D |= Q( s) if Q ∈ M and otherwise Ξ M (P) ∪ D |= Q( s). This claim, in turn, is shown by first showing a more general statement and then instantiating it with ρ. 2. We show that Ξ M (P) ∪ D |= implies P ∪ D |= . Again, we first show a general claim that holds for any derivation from Ξ M (P) ∪ D and then instantiate the claim with a derivation of . In both steps we use that P and Ξ M (P) entail the same facts over Horn predicates for every dataset. We now detail the two steps formally. Step 1. Suppose P ∪ D |= . We show Ξ M (P) ∪ D |= . We begin by showing the following claim. Claim (♦). Let ϕ = Q 1 ( s 1 )∨· · ·∨Q n ( s n ) be a non-empty disjunction of facts satisfying the following properties: (i) Ξ M (P)∪ D |= Q i ( s i ) for each Q i ∈ M . (ii) ϕ is derivable from P ∪ D. Then, for each derivation ρ of ϕ from P ∪ D and each atom R( t) with R disjunctive in the label of a core node in ρ we have Ξ M (P) ∪ D |= R( t) if R ∈ M and Ξ M (P) ∪ D |= R( t) otherwise. We show the claim by induction on ρ = (T, λ). W.l.o.g., the root v of T has a disjunctive predicate in its label (otherwise, the claim is vacuous since the core of ρ contains no disjunctive nodes). For the base case, suppose v has no children labeled with disjunctive predicates. We then distinguish two subcases: • ϕ ∈ D. Then ϕ is a fact, i.e., ϕ = Q( a) for some Q and a. If Q ∈ M , the claim is immediate by assumption (i). If Q = M , the claim follows as D |= Q( a). • ϕ is obtained by a rule ψ → ϕ ∈ P where ψ is a conjunction of Horn atoms and, for some σ, ϕ = ϕ σ and P ∪ D |= ψσ. If {Q 1 , . . . , Q n } ⊆ M , the claim is immediate by assumption (i), so let us assume w.l.o.g. that Q 1 / ∈ M . By the definition of a marking, we then have {Q 2 , . . . , Q n } ⊆ M , and hence it suffices to show Ξ M (P) ∪ D |= Q 1 ( s 1 ). This follows since ψ ∧ n i=2 Q i ( s i ) → Q 1 ( s 1 ) ∈ Ξ M (P) (where s i σ = s i ), Ξ M (P) ∪ D |= n i=2 Q i ( s i ) by assumption (i), Ξ M (P) ∪ D |= ψσ since Ξ M( P) ∪ D and P ∪ D entail the same Horn atoms. For the inductive step, suppose v has children w 1 , . . . , w m in T that are labeled with disjunctive predicates. W.l.o.g., there is a rule r = ψ ∧ m i=1 R i ( t i ) → k j=1 Q j ( s j ) in P (with ψ a conjunction of Horn atoms, 0 ≤ k ≤ n, and all R i disjunctive in P) such that λ(v) is obtained by a hyperresolution step using r from ψσ and λ(w 1 ), . . . , λ(w m ) where σ is a substitution mapping every atom R i ( t i ) to a disjunct in λ(w i ). In particular, we have s j σ = s j , R i ( t i σ) ∈ λ(w i ), and P ∪ D |= ψσ. We distinguish three cases: • {Q 1 , . . . , Q k } ⊆ M and {R 1 , . . . , R m } ∩ M = ∅. Then, for every i ∈ [1, m], every marked atom in λ(w i ) also occurs in λ(v); furthermore, every unmarked atom in λ(v) occurs in λ(w i ) for some i ∈ [1, m]. By the latter statement, it suffices to show the claim for the subderivations rooted at w 1 , . . . , w m . Let i ∈ [1, m]. By the fact that every marked atom in λ(w i ) also occurs in λ(v) and assumption (i), we have Ξ M (P)∪D |= S( u) for every marked disjunct S( u) in λ(w i ). Then, we can apply the inductive hypothesis to the subderivation rooted at w i and the claim follows. • {Q 1 , . . . , Q k } ⊆ M , R 1 ∈ M , and {R 2 , . . . , R m } ∩ M = ∅ (note that R 1 ∈ M implies {R 2 , . . . , R m } ∩ M = ∅ since M is a marking). Then (a) for every i ∈ [1, m], every marked atom in λ(w i ) except for possibly R 1 ( t 1 σ) in λ(w 1 ) also occurs in λ(v), and (b) every unmarked atom in λ(v) occurs in λ(w i ) for some i ∈ [1, m]. Also, we have (c) ϕ ∧ ψ ∧ m i=2 R i ( t i ) ∧ k j=1 Q j ( s j ) → R 1 ( t 1 ) ∈ Ξ M (P). As in the preceding case, by (b), it suffices to show the claim for the subderivations rooted at w 1 , . . . , w m . For w 2 , . . . , w n , we proceed as follows. Let i ∈ [2, m]. By (a) and assumption (i), we have Ξ M (P) ∪ D |= S( u) for every marked disjunct S( u) in λ(w i ). Thus, we can apply the inductive hypothesis to the subderivation rooted at w i . In particular, we obtain Ξ M (P) ∪ D |= R i ( t i σ). In the case of w 1 , we need to show Ξ M (P) ∪ D |= R 1 ( t 1 σ) in order to apply the inductive hypothesis. This follows by (c) and assumption (i) since Ξ M (P) ∪ D |= ψσ, {Q 1 ( s 1 σ), . . . , Q k ( s k σ)} ⊆ λ(v), Ξ M (P) ∪ D |= R i ( t i σ) for i ∈ [2, m], and Ξ M (P) ∪ D |= ϕ σ. • Q 1 / ∈ M , {Q 2 , . . . , Q k } ⊆ M , and {R 1 , . . . , R m } ∩ M = ∅ (note that Q 1 / ∈ M implies {Q 2 , . . . , Q k } ⊆ M and {R 1 , . . . , R m } ∩ M = ∅). Then (a) for every i ∈ [1, m], every marked atom in λ(w i ) also occurs in λ(v), and (b) every unmarked atom in λ(v) except for possibly Q 1 ( s 1 ) (but including Q 2 (s 2 ), . . . , Q m (s m )) occurs in λ(w i ) for some i ∈ [1, m]. By (b), it suffices to show the main claim for the subderivations rooted at w 1 , . . . , w m and also that Ξ M (P) ∪ D |= Q 1 ( s 1 ). Let i ∈ [1, m]. The main claim for the subderivations follows from (a) and assumption (i), which imply that Ξ M (P) ∪ D |= S( u) for every marked disjunct S( u) in λ(w i ); as a result, we can apply the inductive hypothesis to the subderivation rooted at w i . Finally, note that ψ ∧ m i=1 R i ( t i ) ∧ k j=2 Q j ( s j ) → Q 1 ( s 1 ) ∈ Ξ M( P) (since r ∈ P). Then, Ξ M (P)∪D |= Q 1 ( s 1 ) follows from Ξ M (P)∪D |= ψσ, the inductive hypothesis (which implies Ξ M (P)∪D |= R i ( t i σ)), and the assumption (i) (which implies Ξ M (P) ∪ D |= Q j ( s j σ)). We next instantiate (♦) to show the claim in Step 1. Let ϕ = ⊥(s). We have assumed in Step 1 that P ∪ D |= so ⊥(s) is derivable from P ∪ D for some s (as can only be derived by the rule ⊥(x) → ), and hence condition (ii) in (♦) holds. Furthermore, if ⊥ ∈ M , we have Ξ M (P) ∪ D |= ⊥(s); hence, condition (i) in (♦) also holds. Now, let ρ = (T, λ) be a derivation of ⊥(s) from P ∪ D. We exploit (♦) applied to ρ to show that Ξ M (P) ∪ D |= ⊥(s). We distinguish two cases: • ⊥ / ∈ M . Since ⊥(s) labels the root of ρ we can apply (♦) to obtain Ξ M (P) ∪ D |= ⊥(s); the claim follows. • ⊥ ∈ M . Then there is a core node v in ρ such that: λ(v) contains only marked atoms and v has no successor w in T such that all atoms in λ(w) are marked. We distinguish two cases. If λ(v) ∈ D, then λ(v) = Q( b) for some Q and b. Moreover, by (♦), we have Ξ M (P) ∪ D |= Q( b). The claim follows since ⊥(z) ∧ Q( x) ∧ Q( x) → ⊥(z) ∈ Ξ M (P) and Ξ M (P) ∪ D |= ⊥(s). If λ(v) / ∈ D, then v has successors v 1 , . . . , v n (n ≥ 0) in T such that λ(v) is a hyperresolvent of λ(v 1 ), . . . , λ(v n ) and a rule in P of the form n i=1 Q i ( s i ) → m j=1 R j ( t j ), where the atoms Q i ( s i ) are resolved with λ(v i ). Since, λ(v) contains only marked atoms but λ(v 1 ), . . . , λ(v n ) all contain Horn or unmarked atoms, all Q i must be Horn or unmarked and all R j must be marked. Hence, Ξ M (P) contains a rule r = ⊥(x) ∧ ( k i=1 Q i ( s i )) ∧ ( n l=k+1 Q l ( s l )) ∧ m j=1 R j ( t j ) → ⊥(x) where, w.l.o.g., Q 1 , . . . , Q k are Horn and Q k+1 , . . . , Q n are disjunctive and unmarked. Let σ be the substitution used in the hyperresolution step deriving λ(v). By (♦), we then have Ξ M (P) ∪ D |= Q l ( s l σ) for every l ∈ [k + 1, n] and k]. Finally, we have Ξ M (P) ∪ D |= ⊥(s). The claim follows with r. Ξ M (P) ∪ D |= R j ( t j σ) for every j ∈ [1, m]. Moreover, we have λ(v i ) = Q i ( s i σ) and hence Ξ M (P) ∪ D |= Q i ( s i σ) for every i ∈ [1, Step 2. Let Ξ M (P) ∪ D |= . Then there is a derivation ρ of ⊥(s) for some s from Ξ M (P) ∪ D. The fact that P ∪ D |= ⊥(s) follows directly from Statement 1 in Claim (♣), which we show next. Claim (♣). Let ρ be a derivation from Ξ M (P) ∪ D, and let v be the root of ρ. Then: 1. If λ(v) = Q( t), then P ∪ D |= Q( t). If λ(v) = Q( t), then P ∪ D |= ¬Q( t). We show the two claims by simultaneous induction on ρ. For the base case, suppose v is the only node in ρ. We distinguish two cases: • λ(v) ∈ D. Then D |= λ(v) and the claim is immediate. • λ(v) = Q( t) where Q is Horn in P and r = (→ Q( t)) ∈ Ξ M (P). Then r ∈ P and the claim follows. For the inductive step, suppose v has children v 1 , . . . , v n and, λ(v) is a hyperresolvent of λ(v 1 ), . . . , λ(v n ) and a rule r ∈ Ξ M (P). We distinguish five cases: • r contains no disjunctive predicates, in which case the claim follows since P ∪ D and Ξ M (P) ∪ D entail the same facts over a Horn predicate. • r = ⊥(z)∧P ( x)∧P ( x) → ⊥(z). Then λ(v) = ⊥(s) for some s. Since, by the inductive hypothesis, P ∪D |= P ( t)∧P ( t) for some t, P ∪ D is inconsistent, and hence P ∪ D |= ⊥(s). • r = ϕ ∧ ϕ ∧ m j=1 Q j ( t j ) ∧ n i=1 P i ( s i ) → Q( t) where ϕ is the conjunction of all Horn atoms in r and r = ϕ ∧ Q( t) ∧ m j=1 Q j ( t j ) → n i=1 P i ( s i ) ∈ P. Then λ(v) = Q( s) for some s. For some σ, we have P ∪ D |= ϕσ, tσ = s and, for each i, j, Ξ M (P) ∪ D |= P i ( s i σ) and Ξ M (P) ∪ D |= Q j ( t j σ). Then, by the inductive hypothesis, P ∪ D |= ¬P i ( s i σ) and P ∪ D |= Q j ( t j σ). With r , we obtain P ∪ D |= ¬Q( s). • r = ⊥(x) ∧ ϕ ∧ m j=1 Q j ( t j ) ∧ n i=1 P i ( s i ) → ⊥(x) where ϕ is the conjunction of all Horn atoms in r and r = ϕ ∧ m j=1 Q j ( t j ) → n i=1 P i ( s i ) ∈ P. Then λ(v) = ⊥(s) for some s. For some σ, we then have P ∪ D |= ϕσ and, for each i, j, Ξ M (P) ∪ D |= P i ( s i σ) and Ξ M (P) ∪ D |= Q j ( t j σ). Then, by the inductive hypothesis, P ∪ D |= ¬P i ( s i σ) and P ∪ D |= Q j ( t j σ). With r , we obtain that P ∪ D is inconsistent and hence P ∪ D |= ⊥(s). • r = ϕ ∧ m j=1 Q j ( t j ) ∧ n i=1 P i ( s i ) → P ( s) where ϕ is the conjunction of all Horn atoms in r and r = ϕ ∧ m j=1 Q j ( t j ) → P ( s) ∨ n i=1 P i ( s i ) in P. Then λ(v) = P ( t) for some t. For some σ we then have P ∪ D |= ϕσ, sσ = t and, for each i, j, Ξ M (P) ∪ D |= P i ( s i σ) and Ξ M (P) ∪ D |= Q j ( t j σ). Then, by the inductive hypothesis, P ∪ D |= ¬P i ( s i σ) and P ∪ D |= Q j ( t j σ). With r , we obtain P ∪ D |= P ( t). C Proofs for Section 5 Theorem 9. For every ontology O and dataset D over predicates in O we have that O ∪ D is satisfiable iff so is ξ(O) ∪ D. Proof. For the direction from left to right, suppose I is a model of O ∪ D. We define the interpretation J such that • the domain of J extends the domain of I by one additional individual u; • J coincides with I on every concept name, role name and constant in O ∪ D, and ≈ J = ≈ I ∪ {(u, u)}; • f J R,A (v) ∈ { w ∈ A I | (v, w) ∈ R I } if the set { w ∈ A I | (v, w) ∈ R I } is nonempty and otherwise f J R,A (v) = u (if R = S − for a role name S, we write R I for (S I ) −1 ). We show that J is a model of ξ(O) ∪ D. Clearly, J satisfies D and every rule in ξ(O) of type T1-T2 and T4-T6, so it suffices to show that J satisfies the rules introduced by ξ. So, let r = ξ(α) \ π(α) for some α ∈ O. We distinguish the following cases: 1. r = A(x) → B(f R,B (x)) and α = A ∃R.B. Let v ∈ A J . It suffices to show that f J R,B (v) ∈ B J . Since I satisfies α, v has an R I -successor that is in B I , and hence f J R,B (v) ∈ B J = B I . 2. r = A(f R ,Y (x)) → C(x), α = ∃R.A C, and R * R. Let f J R ,Y (v) ∈ A J . It suffices to show v ∈ C J . By construction, we have f J R ,Y (v) = u and hence f J R ,Y (v) ∈ { w ∈ A I | (v, w) ∈ R I }. Since R * R, it follows that f J R ,Y (v) ∈ { w ∈ A I | (v, w) ∈ R I }, i.e., v ∈ (∃R.A) I . Since I satisfies α, we conclude v ∈ C I = C J . 3. r = A(x) ∧ Y (f inv(R ),Y (x)) → C(f inv(R ),Y (x)), α = ∃R.A C and R * R. Let v ∈ A J and f J inv(R ),Y (v) ∈ Y J . It suffices to show f J inv(R ),Y (v) ∈ C J . By construction, we have f J inv(R ),Y (v) = u and hence f J inv(R ),Y (v) ∈ { w ∈ Y I | (w, v) ∈ R I }. Since R * R, it follows that f J inv(R ),Y (v) ∈ { w ∈ Y I | (w, v) ∈ R I }. Since v ∈ A J = A I , we have f J inv(R ),Y (v) ∈ (∃R.A) I . Since I satisfies α, we conclude f J inv(R ),Y (v) ∈ C I = C J . 4. r = A(f inv(R ),B (x)) → C(x), α = A ∀R.C, and R * R. The claim follows similarly to Case 2. 5. r = A(x) ∧ Y (f R ,Y (x)) → C(f R ,Y (x)), α = A ∀R.C, and R * R. The claim follows similarly to Case 3. 6. r = A(z) ∧ B(f R ,Y (z)) ∧ at(R, z, x) ∧ B(x) → f R ,Y (z) ≈ x, α = A ≤1 R.B, and R * R. Let v ∈ A J , f J R ,Y (v) ∈ B J , (v, w) ∈ R J and w ∈ B J . It suffices to show f J R ,Y (v) ≈ J w. By construction, we have f J R ,Y (v) ∈ { w | (v, w ) ∈ R I } ⊆ { w | (v, w ) ∈ R I }. The claim follows since A J = A I , B J = B I , R J = R I and I satisfies α. 7. r = A(f inv(R ),Y (x)) ∧ B(x) ∧ at(R, f inv(R ),Y (x), y) ∧ B(y) → x ≈ y, α = A ≤1 R.B, and R * R. Let f J inv(R ),Y (v) ∈ A J , v ∈ B J , (f J inv(R ),Y (v), w) ∈ R J , and w ∈ B J . It suffices to show v ≈ J w. By construction, we have f J inv(R ),Y (v) ∈ { w | (v, w ) ∈ inv(R ) I } ⊆ { w | (w , v) ∈ R I }. The claim follows since A J = A I , B J = B I , R J = R I , and I satisfies α. 8. r = A(z) ∧ B(f R 1 ,Y1 (z)) ∧ B(f R 2 ,Y2 (z)) → f R 1 ,Y1 (z) ≈ f R 2 ,Y2 (z), α = A ≤1 R.B, and R i * R. The claim follows similarly to Case 6. 9. r = A(f inv(R 1 ),Y1 (x)) ∧ B(x) ∧ B(f R 2 ,Y2 (f inv(R 1 ),Y1 (x))) → x ≈ f R 2 ,Y2 (f inv(R 1 ),Y1 (x)), α = A ≤1 R.B, and R i * R. The claim follows similarly to Case 7. For the direction from right to left, suppose J is a minimal Herbrand model of ξ(O) ∪ D. We define the interpretation I such that • I coincides with J on its domain as well as on every concept name and every constant in O ∪ D; • R I = R J ∪ { (v, f J R ,Y (v)) | f R ,Y ∈ Φ, v ∈ ∆ J , f J R ,Y (v) ∈ Y J , R * R } ∪ { (f J inv(R ),Y (v), v) | f inv(R ),Y ∈ Φ, v ∈ ∆ J , f J inv(R ),Y (v) ∈ Y J , R * R }. We show that I is a model of O ∪ D. Clearly, I satisfies D and every axiom in O of type T1, so it suffices to show that I satisfies axioms of type T2-T6, which we do next. • Let ∃R.A C ∈ O. W.l.o.g., let v ∈ (∃R.A) I \ (∃R.A) J (if v ∈ (∃R.A) J the claim is immediate since π(∃R.A C) ∈ ξ(O)) . It suffices to show v ∈ C I . By construction of R I , there exists some R * R and Y such that either f R ,Y ∈ Φ and f J R ,Y (v) ∈ Y J ∩ A J or f inv(R ),Y ∈ Φ and there is some w ∈ ∆ J such that v = f J inv(R ),Y (w) ∈ Y J and w ∈ A I = A J . In the former case, v ∈ C J = C I follows since A(f R ,Y (x)) → C(x) ∈ ξ(O). In the latter case, v ∈ C I follows since A(x) ∧ Y (f inv(R ),Y (x)) → C(f inv(R ),Y (x)) ∈ ξ(O). • Let A ∃R.B ∈ O and let v ∈ A I . It suffices to show v ∈ (∃R.B) I . Since A I = A J and A(x) → B(f R,B (x)) ∈ ξ(O), we have f J R,B (v) ∈ B J = B I . Hence it suffices to show (v, f J R,B (v)) ∈ R I , which follows since f R,B ∈ Φ, f J R,B (v) ∈ B J and R * R. • Let A ∀R.C ∈ O. The claim follows analogously to the case for ∃R.A C ∈ O. • Let S R ∈ O. W.l.o.g., let (v, w) ∈ S I \ S J (if (v, w) ∈ S J we immediately obtain (v, w) ∈ R J since π(S R) ∈ ξ(O)). We show (v, w) ∈ R I . By construction of S I , there exists some R * S and Y such that either f R ,Y ∈ Φ and and (v, w), (v, u) ∈ R I . We show w ≈ I u. We distinguish the following subcases: w = f J R ,Y (v) ∈ Y J or f inv(R ),Y ∈ Φ and v = f J inv(R ),Y (w) ∈ Y J . In both cases we obtain (v, w) ∈ R I since R * S and S R ∈ O implies R * R. • Let A ≤1 R.B ∈ O. Let v ∈ A I , w, u ∈ B I ,-{(v, u), (v, w)} ⊆ R J . Then the claim is immediate since π(A ≤1 R.B) ∈ ξ(O). -(v, u) ∈ R J and (v, w) ∈ R I \ R J . By construction of R I , there exists some R * R and Y such that either f R ,Y ∈ Φ and w = f J R ,Y (v) ∈ Y J or f inv(R ),Y ∈ Φ and v = f J inv(R ),Y (w) ∈ Y J . In the former case, w ≈ I u follows since A(z) ∧ B(f R ,Y (z)) ∧ at(R, z, x) ∧ B(x) → f R ,Y (z) ≈ x ∈ ξ(O). In the latter case, the claim follows since A(f inv(R ),Y (x)) ∧ B(x) ∧ at(R, f inv(R ),Y (x), y) ∧ B(y) → x ≈ y ∈ ξ(O). -{(v, u), (v, w)} ⊆ R I \ R J . By construction of R I , there are some R 1 , R 2 * R and Y 1 , Y 2 such that we have one of the three following cases: 1. {f R 1 ,Y1 , f R 2 ,Y2 } ⊆ Φ, u = f J R 1 ,Y1 (v) ∈ Y J 1 and w = f J R 2 ,Y2 (v) ∈ Y J 2 . Then the claim follows since A(z) ∧ B(f R 1 ,Y1 (z)) ∧ B(f R 2 ,Y2 (z)) → f R 1 ,Y1 (z) ≈ f R 2 ,Y2 (z) ∈ ξ(O). 2. {f inv(R 1 ),Y1 , f R 2 ,Y2 } ⊆ Φ, v = f J inv(R 1 ),Y1 (u) ∈ Y J 1 and w = f J R 2 ,Y2 (f J inv(R 1 ),Y1 (u)) ∈ Y J 2 . Then the claim follows since A(f inv(R 1 ),Y1 (x)) ∧ B(x) ∧ B(f R 2 ,Y2 (f inv(R 1 ),Y1 (x))) → x ≈ f R 2 ,Y2 (f inv(R 1 ),Y1 (x)) ∈ ξ(O). 3. {f inv(R 1 ),Y1 , f inv(R 2 ),Y2 } ⊆ Φ, v = f J inv(R 1 ),Y1 (u) ∈ Y J 1 and v = f J inv(R 2 ),Y2 (w) ∈ Y J 2 . The claim then follows since, as J is a Herbrand model, we must have u = w and ≈ I is reflexive. Proof. Note that all non-Horn rules in ξ(O) are of type T1, i.e., have unary predicates in the head. Both claims follow from this observation and the fact that ξ(O) contains no rules with unary predicates in the body and binary predicates in the head except for rules of type T6. Thus, whenever a binary predicate P is disjunctive in ξ(O) (resp., is part of a minimal marking of ξ(O)), this is due to an axiom P (x, y) ∧ x ≈ z → P (z, y) or P (x, y) ∧ y ≈ z → P (x, z) in ξ(O) ≈ where ≈ is disjunctive (resp., marked) in ξ(O). However, predicate ≈ cannot be part of any marking since then the transitivity rule x ≈ y ∧ y ≈ z → x ≈ z in ξ(O) ≈ would have two marked body atoms. D Proofs for Section 6 Lemma 12. Let O be an ontology and M a minimal marking of ξ(O). Then Ξ M (ξ(O)) contains only Horn rules of type T1-T2 and T4-T6 in Table 1 as well as type T7-T20 in Table 3. Proof. The claim follows by a simple case analysis over the possible rule types in ξ(O) (as given in Definition 8) as well as the possible minimal markings for each rule type. The analysis exploits that minimal markings involve no binary predicates (Proposition 10 (ii)). Lemma 14. Let O be a markable ALCHIF ontology and let M be a marking of O. Then the ontology Ψ(Ξ M (ξ(O))) is a Horn rewriting of O. Proof. By Theorems 7 and 9, it suffices to show that Ψ(P) is a rewriting of P whenever P = Ξ M (ξ(O)) for some O and M . So let P be as required and let D be a dataset over the predicates in P. We show that P ∪ D is satisfiable if and only if so is π(Ψ(P)) ∪ D. For the direction from left to right, let I be a minimal Herbrand model of P. We define the interpretation J such that • J coincides with I on its domain as well as on every concept name, role name, and individual constant in P ∪ D; • R J Y = { (v, f I R,Y (v)) | v ∈ ∆ I } for each function f R,Y in P; •R J Y = (R J Y ) −1 for each roleR Y in Ψ(P); • S J {R1,R2} = R J 1 ∪ R J 2 for each role S R1,R2 in Ψ(P). We next show that J is a model of π(Ψ(P)) ∪ D. By construction, J satisfies axioms of type T1-T2 and T4-T6, so it suffices to show that J satisfies axioms of type T7-T20: T7 Let ⊥(z) ∧ B(x) ∧ R(x, y) ∧ A(y) → ⊥(z) ∈ P and B ∃R.A ⊥ ∈ Ψ(P). Let v ∈ B J ∩ (∃R.A) J = B I ∩ (∃R.A) I . By Proposition 19, we also have v ∈ ⊥ I , and hence v ∈ ⊥ I = ⊥ J . T8 Let ⊥(z) ∧ A(f R,Y (x)) ∧ B(x) → ⊥(z) ∈ P and B ∃R Y .A ⊥ ∈ Ψ(P). Let v ∈ B J ∩ (∃R Y .A) J . Then v ∈ B I and f I R,Y (v) ∈ A I . Moreover, by Proposition 19, we have v ∈ ⊥ I , and hence v ∈ ⊥ I = ⊥ J . T9 Let ⊥(x) → ⊥(f R,Y (x)) ∈ P and ⊥ ∃R Y .⊥ ∈ Ψ(P). Let v ∈ ⊥ J = ⊥ I . Then f I R,Y (v) ∈ ⊥ I , and hence v ∈ (∃R Y .⊥) J . T10 Let B(x) → A(f R,Y (x)) ∈ P and B ∀R Y .A ∈ Ψ(P). Let v ∈ B J = B I . Then f I R,Y (v) ∈ A I , and hence v ∈ (∃R Y .A) J . Moreover, since R J Y is functional by definition, we have v ∈ (∀R Y .A) J . T11 Let B(f R,Y (x)) → A(x) ∈ P and ∃R Y .B A ∈ Ψ(P). Let v ∈ (∃R Y .B) J . Then f I R,Y (v) ∈ B I , and hence v ∈ A I = A J . T12 Let A(x) ∧ B(f R,Y (x)) → C(f R,Y (x)) ∈ P and A ∃R Y .B ∀R Y .C ∈ Ψ(P). Suppose v ∈ A J ∩ (∃R Y .B) J . Then v ∈ A I and f I R,Y (v) ∈ B I . Consequently, f I R,Y (v) ∈ C I = C J , and hence v ∈ (∃R Y .C) J . Moreover, since R J Y is functional by definition, we have v ∈ (∀R Y .C) J , as required. T13 Let ⊥(z) ∧ A(x) ∧ B(f R,Y (x)) ∧ C(f R,Y (x)) → ⊥(z) ∈ P and A ∃R Y (B C) ⊥ ∈ Ψ(P). The claim follows similarly to Case T8. T14 Let B(f R,Y (x)) ∧ C(f R,Y (x)) → A(x) ∈ P and ∃R Y (B C) A ∈ Ψ(P). The claim follows similarly to Case T11. T15 Let A(z) ∧ B(f R ,Y (z)) ∧ at(R, z, x) ∧ B(x) → f R ,Y (z) ≈ x ∈ P and {R Y S {R Y ,R} , R S {R Y ,R} , A ≤1S {R Y ,R} .B} ⊆ Ψ(P) where R occurs in O. It suffices to show that J satisfies A ≤1S {R Y ,R} .B. Let v ∈ A J = A I , {u, w} ⊆ B J = B I , and {(v, u), (v, w)} ⊆ S J {R Y ,R} . We show u ≈ J w. We distinguish three cases: • {(v, u), (v, w)} ⊆ R J = R I . Then the claim follows since A(z) ∧ at(R, z, x 1 ) ∧ at(R, z, x 2 ) ∧ B(x 1 ) ∧ B(x 2 ) → x 1 ≈ x 2 ∈ P and ≈ J = ≈ I . • (v, u) ∈ R J Y and (v, w) ∈ R J = R I . By construction, u = f I R ,Y (v), and the claim follows since A(z) ∧ B(f R ,Y (z)) ∧ at(R, z, x) ∧ B(x) → f R ,Y (z) ≈ x ∈ P and ≈ J = ≈ I . Let v ∈ A J = A I , {u, w} ⊆ B J = B I , and {(v, u), (v, w)} ⊆ S J {R Y ,R} . We show u ≈ J w. We distinguish three cases: • {(v, u), (v, w)} ⊆ R J = R I . Then the claim follows since A(z) ∧ at(R, z, x 1 ) ∧ at(R, z, x 2 ) ∧ B(x 1 ) ∧ B(x 2 ) → x 1 ≈ x 2 ∈ P and ≈ J = ≈ I . • (v, u) ∈R J Y and (v, w) ∈ R J = R I . By construction, v = f I R ,Y (u), and the claim follows since A(f R ,Y (x)) ∧ B(x) ∧ at(R, f R ,Y (x), y) → x ≈ y ∈ P and ≈ J = ≈ I . • {(v, u), (v, w)} ⊆R J Y . Since I is a Herbrand model, f I R ,Y is injective and henceR J Y is functional. Therefore, we have u = w and hence u ≈ J w by reflexivity of ≈ J . T17 Let A(z) ∧ B(f R,Y (z)) ∧ B(f R ,Z (z)) → f R,Y (z) ≈ f R ,Z (z) ∈ P and {R Y S {R Y ,R Z } , R Z S {R Y ,R Z } , A ≤1S {R Y ,R Z } .B} ⊆ Ψ(P ). It suffices to show that J satisfies A ≤1S {R Y ,R Z } .B. Let v ∈ A J = A I , {u, w} ⊆ B J = B I , and {(v, u), (v, w)} ⊆ S J {R Y ,R Z } . We show u ≈ J w. W.l.o.g., we distinguish two cases: • {(v, u), (v, w)} ⊆ R J Y . Since R J Y is functional by definition, we have u = w and hence u ≈ J w by reflexivity of ≈ J . • (v, u) ∈ R J Y and (v, w) ∈ R This translation has a clear benefit for markability checking: in contrast to π(O), binary predicates in ξ(O) do not belong to any minimal marking. In particular, M ex = {B, D, ⊥} is the only minimal marking of ξ(O ex ). Proposition 10. (i) If ≈ is Horn in ξ(O) then so are all binary predicates in ξ(O). (ii) If ξ(O) is markable, then it has a marking containing only unary predicates. Lemma 14 . 14Let O be a markable ALCHIF ontology and let M be a marking of O. Then the ontology Ψ(Ξ M (ξ(O))) is a Horn rewriting of O. [2013] andKaminski et al. [2014a] proposed practical techniques for computing Datalog rewritings of SHI ontologies based on a two-step process. First, O is rewritten using a resolution calculus Ω into a Disjunc-tive Datalog program Ω(O) of exponential size [Hustadt et al., 2007]. Second, Ω(O) is rewritten into a Datalog program P. For the second step, Kaminski et al. [2014a] propose the notion of markability of a Disjunctive Datalog program and show that P can be polynomially computed from Ω(O) using transposition whenever Ω(O) is markable. In contrast to our work, Kaminski et al. [2014a] focus on Datalog as target language for rewriting (rather than Horn DLs). Furthermore, their Datalog rewritings may be exponential w.r.t. the input ontology and cannot generally be represented in DLs. Gottlob et al. [2012] showed tractability in data complexity of fact entailment for the class of first-order rules with singleatom bodies, which is sufficient to capture most DLs in the DL-Lite bool family [Artale et al., 2009]. Proposition 10 . 10(i) If ≈ is Horn in ξ(O) then so are all binary predicates in ξ(O). (ii) If ξ(O) is markable, then it has a marking containing only unary predicates. functional by definition, we have u = w and hence u ≈ J w by reflexivity of ≈ J .T16 LetA(f R ,Y (x)) ∧ B(x) ∧ at(R, f R ,Y (x), y) → x ≈ y ∈ P and {R Y S {R Y ,R} , R S {R Y ,R} , A ≤1S {R Y ,R} .B,R Y ≡ inv(R Y )} ⊆ Ψ(P) where R occurs in O.It suffices to show that J satisfies A ≤1S {R Y ,R} .B. . An ALCHIF ontology O is a finite set of DL axioms of type T1-T6 inTable 1. An ontology is Horn if it contains no axiom of type T1 satisfying m ≥ 2. Given O, we denote with * the minimal reflexive and transitive relation over roles in 2 Reflexivity of ≈ is axiomatised by the safe rule (x) → x ≈ x. Table 1 : 1Normalised DL axioms. A, B are named or ; C named or ⊥; role S is named and R is a (possibly inverse) role. Table 2 : 2Rewriting the example ELU ontology O ex into a Horn-ALC ontology using the marking M ex = {B, D, ⊥}. Table 3 : 3Transformation Ψ from transposed rules to DLs. Role namesR are fresh for every R, and S {R,R } for every {R, R }. Georg Gottlob, Marco Manna, Michael Morak, and Andreas Pieris. On the complexity of ontological reasoning under disjunctive existential rules. In MFCS, pages 1-18, 2012. [Hustadt et al., 2005] Ullrich Hustadt, Boris Motik, and Ulrike Sattler. Data complexity of reasoning in very expressive description logics. In IJCAI, pages 466-471, 2005. [Hustadt et al., 2007] Ullrich Hustadt, Boris Motik, and Ul-Yevgeny Kazakov. RIQ and SROIQ are harder than SHOIQ. In KR, pages 274-284, 2008. [Krisnadhi and Lutz, 2007] Adila Krisnadhi and Carsten Lutz. Data complexity in the EL family of description logics. In LPAR, pages 333-347, 2007. [Krötzsch et al., 2013] Markus Krötzsch, Sebastian Rudolph, and Pascal Hitzler. Complexities of Horn description logics. ACM Trans. Comput. Log., 14(1), 2013. [Lutz and Wolter, 2012] Carsten Lutz and Frank Wolter. Ortiz et al., 2011] Magdalena Ortiz, Sebastian Rudolph, and Mantas Simkus. Query answering in the Horn fragments of the description logics SHOIQ and SROIQ. In IJCAI, pages 1039-1044, 2011. [Robinson and Voronkov, 2001] Alan Robinson and Andrei Voronkov, editors. Handbook of Automated Reasoning. Elsevier, 2001. [Rudolph et al., 2008] Sebastian Rudolph, Markus Krötzsch, and Pascal Hitzler. All elephants are bigger than all mice. In DL, 2008. [Selman and Kautz, 1996] B. Selman and H. Kautz. Knowl-rike Sattler. Reasoning in description logics by a reduction to disjunctive datalog. J. Autom. Reasoning, 39(3):351- 384, 2007. [Kaminski et al., 2014a] Mark Kaminski, Yavor Nenov, and Bernardo Cuenca Grau. Computing datalog rewritings for disjunctive datalog programs and description logic ontolo- gies. In RR, pages 76-91, 2014. [Kaminski et al., 2014b] Mark Kaminski, Yavor Nenov, and Bernardo Cuenca Grau. Datalog rewritability of disjunc- tive datalog programs and its applications to ontology rea- soning. In AAAI, pages 1077-1083, 2014. [Kazakov, 2008] Non-uniform data complexity of query answering in de- scription logics. In KR, pages 297-307, 2012. [Lutz et al., 2011] Carsten Lutz, Robert Piro, and Frank Wolter. Description logic TBoxes: Model-theoretic char- acterizations and rewritability. In IJCAI, pages 983-988, 2011. [edge compilation and theory approximation. J. ACM, 43(2):193-224, 1996. Capturing such a rule would require a DL that can express products of concepts[Rudolph et al., 2008]. For succinctness, axioms resulting from T7, T8, T12, T13, T14, T16 and T18 are not given in normal form. AcknowledgmentsWork supported by the Royal Society, the EPSRC projects Score!, MaSI 3 and DBOnto, and the FP7 project Optique.Y } is nonempty for every v ∈ ∆ J ). We show that I is a model of P ∪ D. By construction, I satisfies rules of type T1-T2 and T4-T6, so it suffices to show that J satisfies rules of type T7-T20:T7 Let ⊥(z) ∧ B(x) ∧ R(x, y) ∧ A(y) → ⊥(z) ∈ P and B ∃R.A ⊥ ∈ Ψ(P). Let w ∈ ⊥ I , v ∈ B I , v ∈ A I , and (v, v ) ∈ R I . Then v ∈ (B ∃R.A) J and hence v ∈ ⊥ J . Since ⊥(x) → ∈ π(Ψ(P)), this means that J is not a model of π(Ψ(P)) ∪ D, so the claim holds vacuously.T8 Let ⊥(z) ∧ A(f R,Y (x)) ∧ B(x) → ⊥(z) ∈ P and B ∃R Y .A ⊥ ∈ Ψ(P). Let w ∈ ⊥ I , v ∈ B I , andSince ⊥(x) → ∈ π(Ψ(P)), this means that J is not a model of π(Ψ(P)) ∪ D, so the claim holds vacuously.The claim follows similarly to Case T11.Since R(x, y) → (y) ∈ π(Ψ(P)), we then have v ∈ (∃R. ) J , and hence v ∈ ⊥ J = ⊥ I . T20 Let R(x, y) → ⊥(y) ∈ P and ∀R.⊥ ∈ Ψ(P). Let (v, w) ∈ R I = R J . Since R(x, y) → (x) ∈ π(Ψ(P)), we then have v ∈ J , and hence v ∈ (∀R.⊥) J . Since (v, w) ∈ R J , we thus obtain w ∈ ⊥ J = ⊥ I .Theorem 15. Let L be a DL between ALC and ALCHI. Then every markable L ontology is polynomially rewritable into a Horn-L ontology. If L is between ALCF and ALCHIF, then every markable L ontology is polynomially rewritable into Horn-LH. Finally, every markable ELU ontology is polynomially rewritable into Horn-ALC.Proof. The claims follow from the observation that Ψ(Ξ M (ξ(O))) only introduces new axioms of type T1-T2, T4, T7-T14 and T19-T20 to ξ(O) unless O contains functionality assertions. Moreover, none of the axioms in Ψ(Ξ M (ξ(O))) \ ξ(O) contains inverse roles unless so does ξ(O). Thus, all axioms inThe claims immediately follow.E Proofs for Section 7Lemma 16. Satisfiability checking over markable ELU ontologies is EXPTIME-hard.Proof. We prove the claim by adapting the EXPTIME-hardness argument for Horn-ALC byKrötzsch et al. [2013]. Ontology O M,w is in EL except for axioms of the form H C ∀S.C. We will now encode all such axioms into ELU. Theorem 17. Let L be in-between ELU and ALCHIF. Satisfiability checking over markable L-ontologies is EXPTIMEcomplete and PTIME-complete w.r.t. data.Proof. The claim follows by Theorem 15, Lemma 16 and the results for logics between Horn-ALC and Horn-ALCHIF in[Krötzsch et al., 2013]. Ontology-based data access: A study through disjunctive datalog. References [artale, Giorgos Stoilos, and Ian Horrocks. Computing datalog rewritings beyond Horn ontologies. In IJCAI. Boris MotikCambridge University Press36The Description Logic Handbook: Theory, Implementation, and ApplicationsReferences [Artale et al., 2009] Alessandro Artale, Diego Calvanese, Roman Kontchakov, and Michael Zakharyaschev. The DL-Lite family and relations. J. Artif. Intell. Res., 36:1-69, 2009. [Baader et al., 2003] Franz Baader, Diego Calvanese, Deb- orah L. McGuinness, Daniele Nardi, and Peter F. Patel- Schneider. The Description Logic Handbook: Theory, Implementation, and Applications. Cambridge University Press, 2003. [Bienvenu et al., 2014] Meghyn Bienvenu, Balder ten Cate, Carsten Lutz, and Frank Wolter. Ontology-based data ac- cess: A study through disjunctive datalog, CSP, and MM- SNP. ACM Tans. Database Syst., 39(4):33, 2014. [Cuenca Grau et al., 2013] Bernardo Cuenca Grau, Boris Motik, Giorgos Stoilos, and Ian Horrocks. Computing dat- alog rewritings beyond Horn ontologies. In IJCAI, pages 832-838, 2013. A knowledge compilation map. Marquis ; Adnan Darwiche, Pierre Marquis, J. Artif. Intell. Res. 17and Marquis, 2002] Adnan Darwiche and Pierre Marquis. A knowledge compilation map. J. Artif. Intell. Res., 17:229-264, 2002. Alvaro Del Val. First order LUB approximations: Characterization and algorithms. Del Val, Artif. Intell. 1621-2Del Val, 2005] Alvaro Del Val. First order LUB approxi- mations: Characterization and algorithms. Artif. Intell., 162(1-2):7-48, 2005. Framework for an automated comparison of description logic reasoners. Gardiner, ISWC. Gardiner et al., 2006] Tom Gardiner, Dmitry Tsarkov, and Ian Horrocks. Framework for an automated comparison of description logic reasoners. In ISWC, pages 654-667, 2006.

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{'abstract': 'We study the problem of rewriting an ontology O 1 expressed in a DL L 1 into an ontology O 2 in a Horn DL L 2 such that O 1 and O 2 are equisatisfiable when extended with an arbitrary dataset. Ontologies that admit such rewritings are amenable to reasoning techniques ensuring tractability in data complexity. After showing undecidability whenever L 1 extends ALCF, we focus on devising efficiently checkable conditions that ensure existence of a Horn rewriting. By lifting existing techniques for rewriting Disjunctive Datalog programs into plain Datalog to the case of arbitrary first-order programs with function symbols, we identify a class of ontologies that admit Horn rewritings of polynomial size. Our experiments indicate that many real-world ontologies satisfy our sufficient conditions and thus admit polynomial Horn rewritings.arXiv:1504.05150v2 [cs.AI] 21 Apr 2015Proof. The claim follows by a straightforward induction on ρ.We call a node in a derivation Horn (resp. disjunctive) if it is labeled by a Horn atom (resp. a disjunction of disjunctive atoms).Proposition 19. Let P be a program, M a marking of P, and D a dataset over the predicates in P. Then Ξ M (P) ∪ D |= ⊥(s) for every ground term s over the signature of P ∪ D.Proof. The claim is a straightforward consequence of the axiomatisation of ⊥ in Ξ M (P).Theorem 7. Let M be a marking of a program P. Then Ξ M (P) is a polynomial-size Horn rewriting of P.Proof. We proceed in two steps, which together imply the theorem. We fix an arbitrary markable program P, a marking M of P, and a dataset D. W.l.o.g. we assume that D only contains predicates in P.', 'arxivid': '1504.05150', 'author': ['Mark Kaminski \nDepartment of Computer Science\nUniversity of Oxford\nUK\n', 'Bernardo Cuenca Grau \nDepartment of Computer Science\nUniversity of Oxford\nUK\n'], 'authoraffiliation': ['Department of Computer Science\nUniversity of Oxford\nUK', 'Department of Computer Science\nUniversity of Oxford\nUK'], 'corpusid': 3233281, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 26007, 'n_tokens_neox': 23450, 'n_words': 13703, 'pdfsha': 'f32650f8f9a5ce58687d5724a8986af822281ad4', 'pdfurls': ['https://arxiv.org/pdf/1504.05150v2.pdf'], 'title': ['Computing Horn Rewritings of Description Logics Ontologies', 'Computing Horn Rewritings of Description Logics Ontologies'], 'venue': []}

arxiv

Ringing and echoes from black bounces surrounded by the string cloud Yi Yang College of Physics Guizhou University 550025GuiyangChina Dong Liu College of Physics Guizhou University 550025GuiyangChina Zhaoyi Xu College of Physics Guizhou University 550025GuiyangChina Zheng-Wen Long College of Physics Guizhou University 550025GuiyangChina Ringing and echoes from black bounces surrounded by the string cloud In the string theory, the fundamental blocks of nature are not particles but one-dimensional strings. Therefore, a generalization of this idea is to think of it as a cloud of strings. Rodrigues et al. embedded the black bounces spacetime into the string cloud, which demonstrates that the existence of the string cloud makes the Bardeen black hole singular, while the black bounces spacetime remains regular. On the other hand, the echoes are the correction to the late stage of the quasinormal ringing for a black hole, which is caused by the deviation of the spacetime relative to the initial black hole spacetime geometry in the near-horizon region. In this work, we study the gravitational wave echoes of black bounces spacetime surrounded by a cloud of strings under scalar field and electromagnetic field perturbation to explore the effects caused by a string cloud in the near-horizon region. The ringing of the regular black hole and traversable wormhole with string cloud are presented. Our results demonstrate that the black bounce spacetime with strings cloud is characterized by gravitational wave echoes as it transitions from regular black holes to wormholes, i.e. the echoes signal will facilitate us to distinguish between black holes and the wormholes in black bounces surrounded by the string cloud. I. INTRODUCTION Recently, the LIGO and Virgo interferometers have made significant progress in the observation of gravitational waves (GWs) [1][2][3][4][5][6]. In addition, the Event Horizon Telescope has also made a breakthrough in the imaging of black hole shadows [7,8]. These results validate the predictions of general relativity (GR) about black holes (BH). It also allows physicists to test new physical features beyond GR [9][10][11][12][13][14], such as the existence of event horizons in compact objects. Gravitational wave spectroscopy plays a crucial role in the examination of new physical features beyond general relativity [15,16]. For the gravitational wave signal generated by the binary merger, its late stage always decays in the form of the ringdown. It can usually be described using a superposition of complex frequency damping exponents, which are called quasinormal modes (QNMs) [17][18][19]. The detection of QNMs can serve as a tool to test GR predictions. Therefore, this makes gravitational wave detectors (LIGO/Virgo and LISA, etc.) expected to detect some new physical features in the future, such as gravitational wave echoes and so on. Gravitational wave echoes are an important observable for probing the spacetime near the event horizon of the black hole. In addition, gravitational wave echoes are closely related to the unique characteristics of compact objects. Under the framework of general relativity, with the perturbation of black hole spacetime, it must be accompanied by the emergence of quasinormal modes. Because as long as a black hole is perturbed, it re- * yangyigz@yeah.net † dongliuvv@yeah.net ‡ zyxu@gzu.edu.cn § zwlong@gzu.edu.cn (corresponding author) sponds to the perturbation by emitting gravitational waves, and the evolution of gravitational waves can be divided into three stages [20,21]: first, a relatively short initial burst of radiation; then a longer damped oscillation, which depends entirely on the parameters of the black hole; and finally the exponentially decays over a longer period of time. Note that the three stages refer to the postmerger gravitationalwave signal. Among these three stages, people are generally most concerned about the middle quasinormal ringing stage. The QNMs of black holes have attracted extensive attention [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41]. Although there are many indirect ways to identify black holes in the universe, gravitational waves emitted by perturbed black holes will carry unique "fingerprints" that allow physicists to directly identify the existence of black holes. In particular, Ref. [42] proposes that gravitational wave echoes can be used as a new feature of exotic compact objects. Later, when people studied QNM in various spacetime backgrounds, gravitational wave echoes were analysed in the late stage of quasinormal ringing . These works make GWs echoes very important in studying the properties of compact objects. In Ref. [72], the author found a new mechanism to produce the gravitational wave echoes in the black hole spacetime. Bronnikov and Konoplya [73] found that the echoes appeared in the black hole-wormhole transition when studying the quasinormal ringing of black hole mimickers in brane worlds. In Ref. [74], the authors studied the time evolutions of external field perturbation in the asymmetric wormhole and black bounce spacetime background, they observed echoes signals from the spacetime of asymmetric wormholes and black bounce. Especially, Churilova and Stuchlik in Ref. [75] studied the quasinormal ringing of black bounce, and they found the gravitational wave echoes signal during the regular black-hole/wormhole transition. We need to pay attention that not all compact objects can show echoes signals in the late stage of quasinormal ringing. Cardoso et al. [76] pointed out that the precise observation of the late stage of quasinormal ringing allows us to distinguish different compact objects. Therefore, in our work, we plan to explore whether the string cloud will destroy the gravitational wave echoes signal in the black bounce spacetime. We hope to provide some direction for probing black bounces with strings cloud experimentally after obtaining its relevant basic properties. String theory points out that the fundamental blocks of nature are not particles but one-dimensional strings. Therefore, a generalization of this basic idea is to think of it as a cloud of strings. On the other hand, the black hole in general relativity usually has singularities, which forces theoretical physicists to constantly try to avoid the occurrence of singularities. A black hole without singularities is called a regular black hole (RBH). Bardeen was the first theoretical physicist to propose regular black hole [77]. Ayon-Beato et al. interpret it as a black hole solution for the Einstein equations under the presence of nonlinear electrodynamics [78]. Letelier proposed a black hole solution in 1979, which is surrounded by the string cloud [79]. The string cloud is a closed system, therefore its stress-energy tensor is conserved. Subsequently, black holes with strings have attracted a lot of attention [80][81][82]. Sood et al. proposed an RBH surrounded by the string cloud, but the string cloud makes this black hole solution no longer regular [83]. It would be very fascinating if string cloud would not insert singularities in the RBH. Simpson and Visser proposed a type of regular black hole known as black bounces [84]. The difference between this solution and the standard RBH is that it is achieved by modifying the black hole area, and it allows a nonzero radius throat in r = 0. Many studies have been done on black bounces including analysis of their properties [85][86][87][88][89][90][91][92]. Recently, Rodrigues et al. embedded the Simpson-Visser spacetime into a string cloud [93]. They demonstrate that the Simpson-Visser spacetime is still regular even if the string cloud exists. In this work, our goal is to study the effect of the presence of string cloud on the GW echoes of black bounces spacetime and explore what gravitational effects are caused by string cloud. Our work is organized as follows. In Sec. II, we briefly review the black bounces in a cloud of strings. In Sec. III, we discuss the scalar field and electromagnetic field perturbations for black bounces in a cloud of strings. In Sec. IV, we outline the time-domain integration method as well as the WKB method. In Sec. V, we present the quasinormal ringing and gravitational wave echoes of the scalar field and electromagnetic field perturbations to black bounces in a cloud of strings. Sec. VI is our main conclusion of the full text. In this work, we use the units G = = c = 1. II. A BRIEF REVIEW OF THE BLACK BOUNCES IN STRINGS SLOUD To gain black bounces in a cloud of string, Rodrigues et al. [93] considers the following Einstein equations R µν − 1 2 Rg µν = κ 2 T µν = κ 2 T M µν + κ 2 T CS µν ,(1) where T M µν = T SV µν + T N M C µν ,(2) where T SV µν denotes the stress-energy tensor related to the Simpson-Visser spacetime, and the information about the non-minimum coupling between the string cloud and the Simpson-Visser spacetime is included in the stress-energy tensor T N M C µν . Furthermore, T CS µν in Eq. (1) represents the stress-energy tensor of the string cloud, which can be written as T CS µν = ρΣ α µ Σ αν 8π √ −γ ,(3) where ρ represents the density of the string cloud. T CS µν must satisfy the following conservation laws ∇ µ T CS µν = ∇ µ ρΣ µa Σ α ν 8π √ −γ = ∇ µ (ρΣ µα ) Σ α ν 8π √ −γ + ρΣ µα ∇ µ Σ α ν 8π √ −γ = 0. (4) By solving the above Einstein field equations, Rodrigues et al. obtain the following black bounces with the string cloud [93] ds 2 = f (r)dt 2 − f (r) −1 dr 2 − R 2 dθ 2 + sin 2 θdφ 2 ,(5)where f (r) = 1 − L − 2M √ a 2 + r 2 , R = a 2 + r 2 . (6) If a = 0, this spacetime can be reduced to the Letelier spacetime, and this spacetime can be reduced to the Simpson-Visser spacetime when L = 0. If L = 1, this spacetime will have no event horizon, so the range of the string parameter L is 0 < L < 1. In addition, the value of the parameter a has a critical value a c = 2M √ 1 − 2L + L 2 .(7) The black bounce with the string cloud will correspond to a different spacetime for different a: i) regular black hole with string cloud for 0 < a < a c ; ii) one-way wormhole with string cloud for a = a c ; iii) traversable wormhole with string cloud for a > a c . III. MASTER WAVE EQUATION The covariant equations of scalar field perturbation can be written as 1 √ −g ∂ µ √ −gg µν ∂ ν Ψ = 0,(8) considering the black bounces surrounded by the string cloud we studied, we can get − 1 f (r) d 2 Ψ d 2 t + 1 (r 2 + a 2 ) 2rf (r) d dr Ψ + r 2 + a 2 df (r) dr d dr Ψ + r 2 + a 2 f (r) d 2 d 2 r Ψ + 1 (r 2 + a 2 ) 1 sin θ ∂ θ sin θ∂ θ Ψ + 1 sin 2 θ ∂ 2 φ Ψ = 0.(9) Since the spacetime we are studying is spherically symmetric, we can achieve separation of variables through the following ansatz Ψ(t, r, θ, φ) = l,m ψ(t, r)Y lm (θ, φ)/R,(10) where R is the function of radial coordinate r and the parameter a, which has been defined in equation (6), and Y lm (θ, φ) are the spherical harmonic function. After separating the variables and using the properties of spherical harmonics, we can simplify equations (9) to the following form d 2 ψ dt 2 − d 2 ψ dr 2 * + V (r)ψ = 0,(11) where tortoise coordinate r * can be defined by dr * = 1 f (r) dr = 1 1 − L − 2M √ r 2 +a 2 dr.(12) Moreover, the effective potentials for scalar field perturbation can be written as V (r) = 1 − L − 2M √ r 2 + a 2 ( + 1) r 2 + a 2 + 2M r 2 + a 2 −2M − (−1 + L) √ a 2 + r 2 (a 2 + r 2 ) 5/2 .(13) The motion equation of the electromagnetic field in the curved spacetime background can be written as 1 √ −g ∂ µ √ −gF γσ g γµ g σν = 0(14) where A µ being the four vector potential, and F γσ = ∂ γ A σ − ∂ σ A γ . Since spacetime has spherical symmetry, we have A µ (t, r, θ, φ) = l,m         0 0 p lm (t,r) sin θ ∂ φ Y lm −p lm (t, r) sin θ∂ θ Y lm     +     f lm (t, r)Y lm h lm (t, r)Y lm k lm (t, r)∂ θ Y lm k lm (t, r)∂ φ Y lm         ,(15) where the term on the left has odd parity (−1) l+1 , and the term on the right has even parity (−1) l . Substituting the above equation into (14), we can get ∂ 2 ψ elec ∂t 2 − ∂ 2 ψ elec ∂r 2 * + V elec (r)ψ elec = 0,(16) where V elec (r) denotes the effective potential of the electromagnetic field perturbation, V (r) = 1 − L − 2M √ r 2 + a 2 ( + 1) r 2 + a 2 .(17) In Fig. 1, we present the effective potential of the scalar field perturbation for different a with M = 0.5, l = 1, L = 0.1 and for different L with M = 0.5, l = 1, a = 0.1 as the function of the tortoise coordinate r * . Here we are studying l = 1 mode of scalar field perturbation mainly because the peak value of l = 0 mode is too small. From Fig. 1, we can see that the effective potential is the single peak, which indicates that the black bounce in a cloud of strings at this time is the black hole spacetime with the string cloud. These results show that the effective potential is very sensitive to changes in L, but not particularly sensitive to changes in a. In Fig. 2, we show the effective potential of black bounces surrounded by the string cloud under electromagnetic field perturbation. We can observe that when the value of parameter a is less than its threshold a c (when M = 0.5, L = 0.1, the value of the threshold a c is 1.11111), the effective potential is the single peak (blue solid line on the left panel for a = 1.1), and when the value of parameter a is greater than its threshold a c , the effective potential has two peaks. Moreover, one can see that the change of a has almost no effect on the peak value of the effective potential. But as a increases, the depth and width of the effective potential decrease, and it eventually becomes a single peak effective potential (green solid line on the right panel for a = 1.7). Although we have not given the effective potential image of a larger than the threshold under scalar field perturbation, we have verified that similar behavior can appear under scalar field perturbation. In Fig. 3, we give the effect of parameter L on the effective potential under electromagnetic field perturbation. We can see that the effective potential behaves similarly to the influence of the parameter a, but the depth of the potential well is very shallow. IV. THE METHODS In this section, we introduce numerically solving the wave equation for black bounces in a cloud of strings to obtain the time-domain profiles. We use the light cone coordinates u = t − r * , v = t + r * ,(18) then Eq. (11) can be written as ∂ 2 ∂u∂v ψ(u, v) + 1 4 V (r)ψ(u, v) = 0.(19) We adopt the discretization scheme suggested by Gundlach and Price et al. [94,95] ψ N = ψ E +ψ W −ψ S −∆u∆vV ψ W + ψ E 8 +O ∆ 4 .(20) where S = (u, v), W = (u + ∆u, v), E = (u, v + ∆v), N = (u + ∆u, v + ∆v). Moreover, we use the Gaussian initial pulse [96][97][98] for two null surface, i.e. u = u 0 and v = v 0 ψ (u = u 0 , v) = exp − (v−vc) 2 2σ 2 , ψ (u, v = v 0 ) = 0.(21) in our work, we take σ = 3, v c = 10. For the frequency domain, we use the WKB method to calculate the QNM frequencies. Schutz and Will first used the WKB method to calculate the quasi-normal scale of black holes in 1985 [99], and they subsequently extended it to the third-order WKB method with higher accuracy [100]. Moreover, Konoplya extended it to the sixth-order [101,102]. When using the Padé approximation [103,104], WKB method can even be generalized to the more accurate thirteenth order. The higher-order WKB method take the form [105] ω 2 = V 0 + A 2 K 2 + A 4 K 2 + A 6 K 2 + · · · − iK −2V 2 1 + A 3 K 2 + A 5 K 2 + A 7 K 2 . . . ,(22) where K denotes half-integer values. The correction term A k (K 2 ) depends on the derivative of the effective potential at its maximum value. V. QUASINORMAL MODES AND ECHOES OF BLACK BOUNCES SURROUNDED BY THE STRING CLOUD A. Quasinormal modes of the regular black hole surrounded by the string cloud Here we first study the case of 0 < a < a c , that is, the regular black hole with the string cloud. In TABLE I, we give the fundamental QNM frequencies (l = 1, n = 0) of scalar field perturbation for black bounces in a cloud of strings with M = 0.5. One can see that when L is fixed and a is changed, both the real and imaginary parts of the QNM frequencies decrease with the increase of a, which implies that its actual oscillation frequencies decrease with the increase of a, while a decrease in its damping rate means that its decay time becomes longer as a increases. When a is fixed and L is increased, both the real and imaginary parts of the QNM frequency are also decreased, which indicates that L and a have similar contributions to the QNM frequencies for the scalar field perturbation to black bounces in a cloud of strings. In TABLE II, we give the fundamental QNM frequencies (l = 1, n = 0) of electromagnetic field perturbation for black bounces in a cloud of strings with M = 0.5. Unlike the case of scalar field perturbations, when L is fixed and a is changed. The real part of the QNM frequency increases as the parameter a increases and the imaginary part decreases as the parameter a increases. These results show that its true oscillation frequency increases with the increase of a, and its decay time increases with the increase of a. When a is fixed and L is changed, the results show that the contribution of L is similar to that of the scalar field. Fig. 4, the blue solid line represents a = 0.1, the black solid line represents a = 0.6, and the red solid line represents a = 1.1. The corresponding effective potential is given in Fig. 1. One can see that the decay time of quasinormal ringing is the longest when a is larger, which indicates that its damping rate should be smaller for the larger a. In other words, the imaginary part of its quasinormal modes frequency is smaller. Such a result is a good validation of our results shown in TABLE I and II, i.e. the imaginary parts of the quasinormal mode frequencies decrease as a increases. In Fig. 5, we present the time-domain profiles of the scalar field perturbation (left panel) for different L with M = 0.5, l = 1, a = 0.1, and the timedomain profiles of the electromagnetic field perturbation (right panel) for different L with M = 0.5, l = 1, a = 0.1. We can clearly see the quasinormal ringing after the initial pulse, which represents the unique " fingerprint" of black bounce in a could of strings. Furthermore, we can find that the contribution of L to the quasinormal ringing for black bounces in a cloud of strings is similar to that of a, but the quasinormal ringing is more sensitive to L than a. As we expected, because the effective potential is a single peak and the existence of the black hole event horizon, there is no gravitational wave echoes signal here. In addition, late-time tails are also shown after quasinormal ringing. B. Echoes of the wormhole surrounded by the string cloud For black bounce in a could of strings, when the parameter a increases, it can change from a black hole to a wormhole. It should be noted that when a = a c , the space-time we are studying becomes a one-way wormhole with the string cloud, where the QNM has similar behavior to the regular black hole with the string cloud, and has no other distinctive characteristics in quasinormal ringing. Therefore we do not give the corresponding results. Now, let's study the case of a > a c , which is the traversable wormhole with the string cloud. In Fig. 6, we present the effective potential and corresponding GWs echoes of the scalar field perturbation to black bounces in a cloud of strings with M = 0.5, l = 1, L = 0.1, a = 1.112. One can see that when a is increased slightly above the threshold a c (a c = 1.11111 for M = 0.5, L = 0.1), there are two peaks in the effective potential. It implies that the black bounce in a could of string at this time has become a wormhole spacetime. Due to the large distance between the two peaks of the effective potential, the two peaks can scatter waves independently. Therefore, the gravitational wave will be reflected by both peaks and will be repetitively reflected in the potential well, while a fraction of the wave also passes through the potential barrier. This allows observers to see the gravitational wave echoes. Note that Ref. [107] shows the event horizon with quantum nature will also reflect gravitational waves so that the echoes appear. From the right panel of Fig. 6, one can see clear gravitational waves echoes after the initial quasinormal ringing. Since the potential well at this time is wider, the time for the gravitational wave to reach another peak from one peak will be relatively long. Therefore, we see a long time interval between the first gravitational wave echo signal and the initial quasinormal ringing. As a increasing, we can see from Fig. 7 and Fig. 8 that the peak value of the effective potential hardly changes, but its potential well width becomes smaller and smaller. This means that the time required for gravitational waves to reach another peak from one peak becomes shorter so that the gravitational wave echo signal appears sooner after the initial quasinormal ringing, and the time interval between gravitational wave echoes is smaller when a is larger. We only made a qualitative analysis of time delay between gravitational wave echoes, while Refs [42,76] conducted a quantitative study on it, which proved that time delay has the logarithmic dependence on the width of the cavity. In Figs. 9, 10, and 11, we show the GWs echoes of electromagnetic field perturbation to black bounces in a cloud of strings. Time-domain profiles for electromagnetic field perturbations near the threshold a c (a c = 1.11111 for M = 0.5, L = 0.1) show the distinct echoes signal. But the echoes signal seems to become less clear as a increases. If a continues to increase, the echoes will become characteristic quasinormal ringing with a power-law tail, as shown in Fig. 12 (right panel). Such similar characteristics also exist for scalar field perturbations to black bounces in a cloud of strings, as shown in Fig. 12 (left panel). We also study the effect of the string cloud parameter L on the quasinormal ringing of the wormhole surrounded by the string cloud in Fig. 13. We can observe that there are weak echoes signal for the quasinormal ringing corresponding to the red solid line, but not for other cases. Perhaps the answer can be found in the effective potential. From Fig. 3, we can see that the string cloud parameter L has a very significant impact on the effective potential. Its increase causes the effective potential to change from unimodal to bimodal and then unimodal again. It is the double-peak effective potential (red solid line) in Fig. 3 that causes the gravitational wave to be captured in the potential well, so that the echoes signal appear. VI. CONCLUSION In this work, we studied the gravitational wave echoes for the black bounces surrounded by the string cloud. The distinctive feature of the black bounces with a cloud of strings is that when parameter a reaches a certain threshold a c , it can transform from a black hole to a wormhole, which is characterized by the emission of gravitational wave echoes signals. For the regular black hole (0 < a < a c ) with a cloud of strings, due to the existence of the event horizon, we did not find the gravitational wave echoes. This is consistent with the fact that Schwarzschild black holes have no gravitational wave echoes in the framework of general relativity. For wormholes (a > a c ) with a cloud of string, we obtained clear gravitational wave echoes signals after initial ringing. We demonstrate that the two peaks of the effective potential are the necessary conditions for the generation of gravitational wave echoes, and the shape of the potential well plays a decisive role in the gravitational wave echoes. When the parameter a is closer to the threshold a c , the width of the potential well is wider, making it easier for us to observe the gravitational wave echoes signal after the perturbations. As the parameter a increases, the width of the potential well becomes smaller and smaller so that the time interval between gravitational wave echoes becomes smaller and smaller until the echoes disappear. This may cause great difficulties to detect the black bounces surrounded by the string cloud experimentally through the gravitational wave echoes. Furthermore, we find that a has a very small effect on the peak of the effective potential, but the increase in string cloud parameter L has a very significant effect on the peak of the effective potential. In the process of L increasing, the effective poten-tial also changes from the single peak to the double peaks, and then to the single peak again. Although the effective potential exhibits the double peaks, the potential well is so shallow that perturbations can easily escape from the potential well. Therefore, we can only observe the weak echoes signal. On the other hand, by comparing with the work of Churilova and Stuchlik [75], we find that the strings cloud has the following effects on the black bounces spacetime: (i) It extends the parameter range of black bounces spacetime keeping as a regular black hole; (ii) The presence of the string cloud depresses the peak of the effective potentials barrier; (iii) It reduces the real oscillation frequency of gravitational waves and reduces the damping rate of gravitational wave signals. It should be noted that the parameter L related to the strings will not affect the appearance of the echoes. As long as the appropriate parameter a is selected, we can observe the echoes, but the existence of the strings makes the threshold a c larger. We discussed the gravitational wave echoes of the black bounces surrounded by the string cloud under the scalar field and electromagnetic field perturbation. The behavior of the gravitational wave echoes under the two kinds of perturbations are very similar, as a result we believe that the similar behavior can also be continued in the Dirac field perturbation [108] or the gravitational perturbation. It might also be very interesting to examine it in future work. FIG. 1 . 1The effective potential of the scalar field perturbation for different a (left panel) with M = 0.5, l = 1, L = 0.1 and for different L (right panel) with M = 0.5, l = 1, a = 0.1. FIG. 2 .FIG. 3 . 23The effective potential of the electromagnetic field perturbation for different a with M = 0.5, l = 1, The effective potential of the electromagnetic field perturbation for different L with M = 0.5, l = 1, a = 1.6 (left) and a = 2 (right) FIG. 4 .FIG. 5 . 45Comparing the QNM of the black bounces surrounded by the string cloud with the QNM of Schwarzschild black holes (0.585867 − 0.195321i with s = 0, l = 1, 0.496527-. Fundamental QNM frequencies (l = 1, n = 0) of scalar field perturbations for black bounces in a cloud of strings with M = 0.5. . Fundamental QNM frequencies (l = 1, n = 0) of electromagnetic field perturbations for black bounces in a cloud of strings with M = 0The time-domain profiles of the scalar field perturbation (left panel) for different a with M = 0.5, l = 1, L = 0.1. The time-domain profiles of the electromagnetic field perturbation (right panel) for different a with M = 0.5, l = 1, The time-domain profiles of the scalar field perturbation (left panel) for different L with M = 0.5, l = 1, a = 0.1. The time-domain profiles of the electromagnetic field perturbation (right panel) for different L with M = 0.5, l = 1, a = 0.1. 0.184975 with s = −1, l = 1, which can be obtained on the website [106]), we find that the oscillation frequencies and damping rates of both the scalar field and the electromagnetic field perturbation are smaller than the results of the Schwarzschild black hole. In Fig. 4, we show the time-domain profiles of the scalar field perturbation (left panel) for different a with M = 0.5, l = 1, L = 0.1, and the timedomain profiles of the electromagnetic field perturbation (right panel) for different a with M = 0.5, l = 1, L = 0.1. In FIG. 6 . 6The effective potential and gravitational wave echoes of the scalar field perturbation to black bounces in a cloud of strings with M = 0.5, l = 1, L = 0.1, a = 1.12. FIG. 7 . 7The effective potential and gravitational wave echoes of the scalar field perturbation to black bounces in a cloud of strings with M = 0.5, l = 1, L = 0.1, a = 1.13. FIG. 8 .FIG. 9 .FIG. 10 . 8910The effective potential and gravitational wave echoes of the scalar field perturbation to black bounces in a cloud of strings with M = 0.5, l = 1, L = 0.1, a = 1.14. The GWs echoes of electromagnetic field perturbation (left panel) and semilogarithmic plots for the GWs echoes of electromagnetic field (right panel) to black bounces in a cloud of strings with M = 0.5, l = 1, L = 0.1, The GWs echoes of electromagnetic field perturbation (left panel) and semilogarithmic plots for the GWs echoes of electromagnetic field (right panel) to black bounces in a cloud of strings with M = 0.5, l = 1, L = 0.1, a = 1.13. FIG. 11 .FIG. 12 .FIG. 13 . 111213The GWs echoes of electromagnetic field perturbation (left panel) and semilogarithmic plots for the GWs echoes of electromagnetic field (right panel) to black bounces in a cloud of strings with M = 0.5, l = 1, L = 0.1, Semilogarithmic plots for the time-evolution of scalar field perturbation (left panel for l = 0) and electromagnetic field perturbation (right panel for l = 1) to black bounces in a cloud of strings with M = 0.5, Semilogarithmic plots for the time-evolution of electromagnetic field perturbation to black bounces in a cloud of strings with M = 0.5, l = 1, a = 1.6 (left panel) and a = 2 (right panel). ACKNOWLEDGMENTSWe are very grateful to Prof. Manuel E. Rodrigues for useful correspondences. Virgo), Astrophys. B P Abbott, LIGO Scientific10.3847/2041-8205/818/2/L22arXiv:1602.03846J. Lett. 81822astro-ph.HEB. P. Abbott et al. (LIGO Scientific, Virgo), As- trophys. J. Lett. 818, L22 (2016), arXiv:1602.03846 [astro-ph.HE]. . 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{'abstract': 'In the string theory, the fundamental blocks of nature are not particles but one-dimensional strings. Therefore, a generalization of this idea is to think of it as a cloud of strings. Rodrigues et al. embedded the black bounces spacetime into the string cloud, which demonstrates that the existence of the string cloud makes the Bardeen black hole singular, while the black bounces spacetime remains regular. On the other hand, the echoes are the correction to the late stage of the quasinormal ringing for a black hole, which is caused by the deviation of the spacetime relative to the initial black hole spacetime geometry in the near-horizon region. In this work, we study the gravitational wave echoes of black bounces spacetime surrounded by a cloud of strings under scalar field and electromagnetic field perturbation to explore the effects caused by a string cloud in the near-horizon region. The ringing of the regular black hole and traversable wormhole with string cloud are presented. Our results demonstrate that the black bounce spacetime with strings cloud is characterized by gravitational wave echoes as it transitions from regular black holes to wormholes, i.e. the echoes signal will facilitate us to distinguish between black holes and the wormholes in black bounces surrounded by the string cloud.', 'arxivid': '2210.12641', 'author': ['Yi Yang \nCollege of Physics\nGuizhou University\n550025GuiyangChina\n', 'Dong Liu \nCollege of Physics\nGuizhou University\n550025GuiyangChina\n', 'Zhaoyi Xu \nCollege of Physics\nGuizhou University\n550025GuiyangChina\n', 'Zheng-Wen Long \nCollege of Physics\nGuizhou University\n550025GuiyangChina\n'], 'authoraffiliation': ['College of Physics\nGuizhou University\n550025GuiyangChina', 'College of Physics\nGuizhou University\n550025GuiyangChina', 'College of Physics\nGuizhou University\n550025GuiyangChina', 'College of Physics\nGuizhou University\n550025GuiyangChina'], 'corpusid': 254564016, 'doi': '10.1140/epjc/s10052-023-11382-5', 'github_urls': [], 'n_tokens_mistral': 22141, 'n_tokens_neox': 16744, 'n_words': 7767, 'pdfsha': 'd452cec112f561832fbed635aa452abca6602dd0', 'pdfurls': ['https://export.arxiv.org/pdf/2210.12641v3.pdf'], 'title': ['Ringing and echoes from black bounces surrounded by the string cloud', 'Ringing and echoes from black bounces surrounded by the string cloud'], 'venue': []}

arxiv

Backward Pilot Strategy in Constrained Sampling Problems * 7 Jun 2017 Chencheng Cai Department of Statistics Rutgers University Rutgers University 08854PiscatawayNJUSA Rong Chen rongchen@stat.rutgers.edu. Department of Statistics Rutgers University Rutgers University 08854PiscatawayNJUSA Rong Chen Department of Statistics Rutgers University Rutgers University 08854PiscatawayNJUSA Backward Pilot Strategy in Constrained Sampling Problems * 7 Jun 2017* Chen's research was supported in part by National Science Foundation grants DMS-1503409 and DMS-1209085. Corresponding author: 0Backward pilotSequential Monte CarloSequential Importance ResamplingPriority scoreconstrained sampling problem Sequential Monte Carlo methods are a class of Monte Carlo methods that is used to obtain random samples of a high dimensional random variable in a sequential fashion. Many problems encountered involve different types of constraints. These constraints make the problem much more challenging. In this paper we formulate a general framework of using SMC for constrained problems, extending the backward pilot idea ofLin et al. (2010). Several new algorithms are developed and demonstrated. It is noted that all information observed or imposed on the underlying system can be viewed as constraints. Hence the approach outlined in this paper can be useful in many applications. Introduction Stochastic dynamic systems are widely used to model the dynamic behavior of random variables with a wide range of applications in physics, finance, engineering and other fields. One of the important problems of studying complex dynamic systems is to sample paths following the underlying stochastic process. Such paths can be used for statistical inferences under the Monte Carlo framework. In practice, a stochastic system often comes with observable information including direct/indirect measurements, external constraints and others. For example, in state-space model, noisy measurements of the underlying latent state are observed. In a diffusion bridge problem, the beginning and ending points of the diffusion process are fixed. Such information often creates challenges in sampling the paths. In this paper, we treat all such information as constraints. Noisy measurements are considered as weak constraints and fixed boundary conditions are strong constraints. The Sequential Monte Carlo (SMC) methods is a class of sampling methods that utilize the sequential nature of the underlying process with wide applications (Gordon et al., 1993;Kong et al., 1994;Avitzour, 1995;Liu and Chen, 1995;Kitagawa, 1996;Kim et al., 1998;Liu and Chen, 1998;Pitt and Shephard, 1999;Chen et al., 2000;Doucet et al., 2001;Liu, 2001;Fong et al., 2002;Godsill et al., 2004). The Sequential Importance Sampling (SIS) (Kong et al., 1994;Liu, 2001;Liu and Chen, 1998) and Sequential Importance Resampling (SIR) (Gordon et al., 1993;Liu and Chen, 1995;Kitagawa, 1996;Pitt and Shephard, 1999;Chopin, 2004;Del Moral, 2004) schemes embedded in SMC enable sampling from complex target distributions. However, the choice of the proposal distribution in SIS and the choice of priority score in SIR are critical in sample quality and inference efficiency. For example, in a diffusion bridge problem, in which the start and end points of a diffusion process are exactly enforced, Pedersen (1995) proposed to generate the samples through underlying diffusion process without considering the end point constraint and then force the samples to connect with the fixed endpoint. It may not be efficient due to the large deviation of the end of the forward path from the enforced end point. Durham and Gallant (2002) proposed a method based on SIS with linear interpolation as the proposal distribution. It may not be efficient for non-linear processes. Lin et al. (2010) generated bridge samples based on a backward pilot strategy. In their procedure, a pilot run is conducted backward from the fixed end point to determine the priority scores for resampling in a forward SIR procedure. The backward pilot strategy achieves good efficiency in diffusion bridge problems. This approach improves the forward sampling by bringing future information and constraints for effective sampling and with minimum additional computational costs. In this article, we extend the procedure of Lin et al. (2010) to more general settings. Specifi-cally, the problem of simulating a stochastic process under constraints is more formally stated in a general setting that contains many problems as its specially cases, including the standard state space models. The general setting also allows the discussion of a formal guidance of improving the efficiency in developing SMC implementations for such problems. Under the setting, we propose a general backward-pilot framework for constrained sampling problems with measure theory interpretation. Under such a framework, several types of constraints are discussed, with their corresponding implementations. Links to existing procedures are also discussed. The developed approaches are demonstrated with several examples. The rest of this article is organized as follows. In section 2, the constrained sampling problem is formally stated and a general principle of backward pilot is proposed. Section 3 discusses several special constrained problems in detail with their corresponding implementations of the proposed method. Section 4 presents several applications and Section 5 concludes. 2 Constrained Sampling and the Backward Pilots Approach Sampling from Forward Stochastic Systems A discrete-time forward stochastic system X is described as a sequence of random states at each integer time, i.e. X = {X t } t=0,1,2,... , whose dynamics is governed entirely by an initial state distribution and a forward propagation equation. Here we focus on simulating path from a complete known system, hence suppress all parameters in the system. A typical setting is shown as follows p(X 0 = x 0 ) = π 0 (x 0 ) initial state distribution p(X t = x t |x t−1 ) = π t (x t |x t−1 ), t = 1, 2, . . . forward propagation equation where x t = (x 0 , x 1 , . . . , x t ) is the cumulative state vector up to time t 0. The joint distribution of the partial sequence up to time T is p(x T ) = π 0 (x 0 ) T t=1 π t (x t |x t−1 ). The sequential nature results in the forward propagation sampling algorithm of the possible paths following the underlying system, shown in Figure 1. This forward propagation algorithm is also applicable for continuous-time stochastic processes when a discrete approximation is available. Suppose a continuous-time stochastic process X = {X t } t>0 is governed by a diffusion stochastic differential equation dX t = µ(X t , t)dt + σ(X t , t)dW t ,(1) where µ(X t , t) and σ(X t , t) are the corresponding drift and diffusion coefficients and {W t } t>0 is a Wiener process. Let 0 = τ 0 < τ 1 < · · · < τ L−1 < τ L = T be a sequence of intermediate points Forward propagation algorithm • Sample x (i) 0 from π 0 (x 0 ) for i = 1, 2, . . . , n. • for t from 1 to T : connecting time 0 and time T , such that τ k = kδ for k = 0, 1, . . . , L. By Euler-Maruyama method, system (1) can be approximated by -draw x (i) t from π t (x t |x (i) t−1 ) for i = 1, 2, . . . , n. -set x (i) t = (x (i) t−1 , x (i) t ) for i = 1, 2, . . . , n. • return {x (i) T } i=1,...,n .X τ k+1 = X τ k + µ(X τ k , τ k )δ + σ(X τ k , τ k )(W τ k+1 − W τ k ) + o p (δ),(2) where o p (δ) denotes the error in discretization. When π t (x t |x (i) t−1 ) is difficult to sample, importance sampling can be applied. The basic idea of importance sampling is to sample from a different distribution, called a trial distribution, and assign proper weights to account for the discrepancy between the target and trial distributions. Given W τ k+1 − W τ k ∼ N (0, δ), This gives a weighted sample {(x (i) T , w (i) T )} i=1,. ..,n . A sample is said to be properly weighted if for any measurable function h, the weighted average converges to its expectation under π(·), i.e. n i=1 w (i) T h(x (i) T ) n i=1 w (i) T −→ E π (h(x T )), as n → ∞. Importance sampling is often used as a variance reduction technique in Monte Carlo simulation. For general sampling purpose when w(x) = π(x)/q(x) is used, the χ 2 -divergence between target distribution π(x T ) and trial distribution q(x T ), which is defined by D χ 2 (π||q) = Ω π(x T ) q(x T ) 2 q(x T )dx T − 1 = Ω [π(x T )] 2 q(x T ) dx T − 1 = Var q (w),(3) is often used as a performance measure (Liu, 2001). Reduction of the χ 2 divergence is a general guideline in designing SMC algorithms. Constrained Sampling Problems Suppose a compounded constraint/information set I is imposed on the underlying process and this constraint consists of a sequence of atomic information at each time t, such that I = I 0 ∩ I 1 ∩ · · · ∩ I T = T t=0 I t ,(4) where I t is the available information on the state variable X t at time t. For example, if we observe Y t ∼ N (X t , σ 2 ), a direct measurement of X t with error, or more generally Y t = g t (X t , e t ) with noise e t , then I t = {Y t }. For a fixed point constraint on X t , we have I t = {X t = c}. When there is no additional information at time t, we assume I t = {X t ∈ Ω}, a trivial constraint, where Ω is the support. The joint distribution of the full process x T of this constrained problem is p(x T |I) = p(x 0 |I) T t=1 p(x t |x t−1 , I).(5) It induces a sequence of marginal posterior probability measures Q 0 , . . . , Q T , such that Q t (x t ) = p(x 0 |I) t s=1 p(x s |x s−1 , I), for t = 0, 1, . . . , T. The above measure corresponds to the true marginal posterior distribution and implies the recursion relationship Q t+1 (x t+1 ) = Q t (x t )p(x t+1 |x t , I), which reveals a way to update sample at time t + 1 given a sample at time t. However, under this measure, the term p(x t+1 |x t , I) is usually difficult to evaluate or sample from at time t + 1 since I involves the entire information set from t = 1 to T . Hence a sequential sampling procedure cannot be directly applied. To utilize the importance sampling approach (Kong et al., 1994;Liu, 2001;Liu and Chen, 1998), conventional SMC approach uses a sequence of forward propagation probability measures P 0 , . . . , P T such that P t (x t ) = p(x 0 |I [0,t] ) t s=1 p(x s |x s−1 , I [0,t] ), for t = 0, 1, . . . , T, where I [0,t] = t s=0 I s denotes all information/constraints up to time t (inclusive). Under these forward propagation measures, the recursion relationship becomes P t+1 (x t+1 ) = p(x 0 |I [0,t+1] ) t+1 s=1 p(x s |x s−1 , I [0,t+1] ) = p(I t+1 |x 0 , I [0,t] )p(x 0 |I [0,t] ) p(I t+1 |I [0,t] ) t+1 s=1 p(I t+1 |x s , I [0,t] )p(x s |x s−1 , I [0,t] ) p(I t+1 |x s−1 , I [0,t] ) ∝ P t (x t ) · p(x t+1 |x t , I [0,t] )p(I t+1 |x t+1 , I [0,t] ) = P t (x t ) · π t+1 (x t+1 |x t )p(I t+1 |x t+1 ). This setting makes it easy to update sample in a sequential way. Once X t is properly sampled with respect to P t , X t+1 can also be sampled by importance sampling without difficulty, provided that π t+1 (x t+1 |x t ) and p(I t+1 |x t+1 ) are relatively easy to work with. They are of low dimensions, hence a reasonable and efficient trial distribution can often be found. The marginal posterior measure Q t and the forward propagation measure P t agree at time T . Therefore, in the end, the samples drawn using the forward propagation measure P T are indeed samples from the joint distribution (5). However, since conditioned on different information sets, P t and Q t are generally different. The difference has a direct impact on the efficiency. For example, if all I t are trivial, with the exception that I 0 and I T are the only two non-trivial constraints, then we have, for t < T , P t (x t ) ∝ π 0 (x 0 |I 0 ) t s=1 π s (x s |x s−1 ) , and for t = T , P T (x T ) ∝ π 0 (x 0 |I 0 ) T s=1 π s (x s |x s−1 ) p(I T |x T ). A conventional SMC starts sampling at time 0, propagates T steps forward using the simple forward propagation algorithm to get x T ∼ π 0 (x 0 |I 0 ) T s=1 π s (x s |x s−1 ) , and finally adjusts weights by p(I T |x T ) at the last step. However, when I T is a rare event such that x T |I 0 and x T |I 0 , I T have a large χ 2 divergence, the samples from the forward propagation are not representative, resulting in a large weight variance Var q (w). Even though all paths are properly weighted, the large discrepancy between the sampling distribution and the target distribution gives rise to a poor performance and less accurate inference when the Monte Carlo sample size is finite. Without using any future information and constraints, sampling with respect to forward sampling measure P t fails to correct the sample proactively. It is also not efficient especially when I is sparse and the marginal likelihood p(I) := p(I|x T )π(x T )dx T is extremely small. To overcome this drawback, we propose another sequence of probability measures Q * t which has easy sequential implementation and uses part of future information to correct the Monte Carlo samples proactively. Define Q * t as Q * t (x t ) = p(x 0 |I [0,t + ] ) t s=1 p(x s |x s−1 , I [0,t + ] ), for t = 0, 1, . . . , T, where t + t is the next time when a non-trivial constraint is imposed after time t (inclusive). This sequence of measures is a compromise between the marginal posterior measure Q t and the forward propagation measure P t , where the former considers the whole information set and the latter ignores all future constraints. When I t is trivial, measure P t implies a blind propagation without any constraint, while measure Q * t seeks the next available non-trivial constraint I t+ for guidance, but not the entire future information set as the measure Q t would require. In most cases, the next available non-trivial constraint I t + plays an important role in shaping the path distribution. Hence the measure Q * t is expected to approximate the marginal posterior measure Q t reasonably well. Similar to P t , the measure Q * t gives the recursion rule Q * t+1 (x t+1 ) ∝ Q * t (x t ) · p(x t+1 |x t , I [0,t + ] ) · p(I (t + ,(t+1) + ] |x t+1 , I [0,t + ] ) =      Q * t (x t )π t+1 (x t+1 |x t ) p(I (t+1) + |x t+1 ) p(I t + |x t ) if t + > t, Q * t (x t )π t+1 (x t+1 |x t )p(I (t+1) + |x t+1 ) if t + = t. All three measures P, Q and Q * agree at time T . Notice that, Q * t (x t ) ∝ P t (x t )p(I t+ |x t ). A sample under measure P t can be easily changed to measure Q * by multiplying the weight by p(I t+ |x t ), or conducting a resampling step with priority score proportional to p(I t+ |x t ). Therefore, sampling under measure Q * has a similar procedure as measure P except for the weight adjustment steps. The challenge of using measure Q * is to estimate p(I t + |x t ). In Section 2.3, we will discuss how to evaluate (estimate) the probability by a backward pilot sample. The corresponding sequential Monte Carlo algorithm with importance resampling can be viewed as sampling under measure P and resampling under measure Q * . A detailed Constrained Sequential Monte Carlo (cSMC) algorithm is depicted in Figure 2, with the resampling step at time t carried out before sampling x t (see Liu and Chen (1998)). The key step in cSMC is the resampling priority score (6). The 'optimal' priority score is β (i) t−1 = w (i) t−1 p(I t + |x (i) t−1 ) under Q * t measure (see below). Since p(I t + |x (i) t−1 ) is difficult to obtain, approximation is often used. The better the approximation is, the more efficient the algorithm is. In general, resampling is done to preventing the samples from weight collapse (Kong et al., 1994;Liu, 2001;Liu and Chen, 1998). After a resampling step, the samples with a low priority score tend to be replaced by those with high priority Fearnhead, 2008; constrained Sequential Monte Carlo algorithm (cSMC) • sample x (i) 0 from π 0 (X 0 ) for i = 1, 2, . . . , n. • initialize the weights w (i) 0 = 1 for i = 1, 2, . . . , n. • for time t from 1 to T : resampling step: * assign a priority score β (i) t−1 = w (i) t−1p (I t + |x (i) t−1 ),(6)to each sample x (i) t−1 , i = 1, 2, . . . , n. * sample {J i } i=1,...,n from index set {1, 2, . . . , n} according to priority scores {β (1) t−1 , . . . , β (n) t−1 }. * set x (i) t−1 ← x (J i ) t−1 for i = 1, . . . , n. * update weights w * (i) t−1 ← w (J i ) t−1 /β (J i ) t−1 for i = 1, . . . , n. -propagation step: * draw x (i) t from x (i) t ∼ q t (X t |x (i) t−1 , I t ),(7) for i = 1, 2, . . . , n. * set x Wang et al., 2002;Doucet et al., 2006). Priority scores are the sampler's preferences over different paths. In our constrained Sequential Monte Carlo, priority scores take the probability p(I t + |x t ) into consideration. On one hand, paths with larger tendency to comply with the next non-trivial constraint I t + are more likely to be selected. On the other hand, paths with little probabilities to satisfy the constraint I t + are more likely to be eliminated by the resampling step. (i) t = (x (i) t−1 , x (i) t ) for i = 1, . . . , n. * update weights w (i) t ← w * (i) t−1 × π t (x (i) t |x (i) t−1 )p(I t |x (i) t ) q t (x (i) t |x (i) t−1 , I t ) (8) for i = 1, . . . , n. • return the weighted sample {(x (i) T , w (i) )} i=1,...,n . This choice of the priority scores properly guides the sampler using the future information I t + and eliminates the unlikely samples proactively. We point out that at each time t < T , the weighted samples {(x (i) nomial resampling (Gordon et al., 1993) draws indices independently from a multinomial distribution with probability proportional to the priority scores. Residual resampling (Liu and Chen, 1998) produces nβ (i) replicates of index i, where · is the integer part, and then draws the rest from a multinomial distribution. Stratified resampling (Kitagawa, 1996) splits the index set into several strata and applies multinomial resampling within each stratum. Both residual resampling and stratified resampling ensure that the difference between duplication number of particle i and its expectation is less than one. The choice of the priority score β t−1 (6) in Figure 2 can be seen from the perspective of qualify control. The optimal choice of priority score minimizes χ 2 divergence in (3). At time t, suppose x t−1 is drawn from the sampling distribution Q t−1 (x t−1 ) and all future sampling is perfect. Then we have Var Q (w T ) = E p(x T |I) Q t−1 (x t−1 )β (j) t−1 p(x t , . . . , x T |x t−1 , I) 2 − 1 = [p(x T |I)] 2 Q t−1 (x t−1 )β (j) t−1 p(x t , . . . , x T |x t−1 , I) dx 0 · · · dx T − 1 = T s=1 π s (x s |x s−1 ) T s=0 p(I s |x s ) 2 p(I [t,T ] |x t−1 ) Q t−1 (x t−1 )β (j) t−1 T s=t π s (x s |x s−1 ) T s=t p(I s |x s )[p(I)] 2 dx 0 · · · dx T − 1 = 1 [p(I)] 2 t−1 s=1 π s (x s |x s−1 ) t−1 s=0 p(I s |x s ) 2 p(I [t,T ] |x t−1 ) Q t−1 (x t−1 )β (j) t−1 × T s=t π s (x s |x s−1 )p(I s |x s )dx t · · · dx T · dx 0 · · · dx t−1 − 1 = 1 [p(I)] 2 t−1 s=1 π s (x s |x s−1 ) t−1 s=0 p(I s |x s ) 2 [p(I [t,T ] |x t−1 )] 2 Q t−1 (x t−1 )β (j) t−1 dx 0 · · · dx t−1 − 1. The optimal choice of priority scores is thus given by β t−1 ∝ t−1 s=1 π s (x s |x s−1 ) t−1 s=0 p(I s |x s )p(I [t,T ] |x t−1 ) Q t−1 (x t−1 ) = w t−1 p(I [t,T ] |x t−1 ). The optimal priority score satisfies β t−1 ∝ w t−1 p(I t + |x t−1 )p(I (t + ,T ] |x t−1 , I t + ) ∝ ∼ w t−1 p(I t + |x t−1 ). The last approximate proportionality holds if p(I (t + ,T ] |x t−1 , I t + ) ≈ c, for all x t−1 .(9) The condition is satisfied in many cases, including when I t + is a fixed point constraint or when t + is far away from t. This shows the approximate optimality of the choice of priority scores β t in cSMC algorithm. Backward Pilots Here we develop a procedure to estimate p(I t + |x t−1 ). Suppose the underlying process is Markovian. The close-form of p(I t + |x t−1 ) involves the integral p(I t + |x t−1 ) = · · · p(I t + |x t + ) t + s=t π s (x s |x s−1 )dx t · · · dx t + . It is usually difficult to obtain the integral directly. However, since the integrand has a sequential structure, it is possible to draw sample paths (x t , x t+1 , . . . , x t + ) from a distribution whose density is proportional to p(I t + |x t + ) t + s=t π s (x s |x s−1 ). In this case the marginal sampling distribution for X t−1 is proportional to p(I t + |x t−1 ) and the marginal samples {x (j) t−1 } j=1,...,m from the generated paths can be used to estimate p(I t + |x t−1 ) by a nonparametric density estimator. The sampling of (x t−1 , . . . , x t + ) can be done by following the conventional SMC algorithm, but in the opposite direction: starting from the constrained pointx t + and then propagating backwards to obtaiñ Lin et al. (2010) named it as the backward pilot method. Herer t + andr s are trial distributions in this backward sampling procedure. They ensure the backward chain (x t + , . . . ,x t−1 ) can be properly sampled even when the underlying process is irreversible. Once a marginal sample for X t−1 is available, the term p(I t + |x t−1 ) can be easily estimated by:p x t + −1 , . . . ,x t−1 . Backward Pilot • drawx (j) t + fromr t + (X t + ) for j = 1, . . . , m • set weightsw (j) t + = p(I t + |x (j) t + ) r t + (x (j) t + ) for j = 1, . . . , m • for time s from t + − 1 to t − 1 -(optional) resampling if necessary -drawx (j) s fromr s (X s |x (j) s+1 ) for j = 1, . . . , m -update weights:w (j) s ←−w (j) s+1 π s+1 (x (j) s+1 |x (j) s ) r s (x (j) s |x s+1 ) for j = 1, . . . , m • return the weighted sample {(x (j) t−1 ,w (j) )} j=1,...,m .(I t + |x t−1 ) ∝ m j=1w (j) t−1 K h (x (j) t−1 − x t−1 ),(10) where K h (·) is a kernel function with bandwidth h. This estimation will be used to obtain β (i) t−1 in cSMC algorithm. We abbreviate cSMC with backward pilot strategy as cSMC-bp. according to the resampling score β t−1 in (6). Special Cases In this section, we discuss several special cases of constrained sampling problems. All these problems can be solved under the cSMC-bp scheme. Frequent Constraint Problems -the State Space Model Consider a type of weak constraint problems where X = {X t } t=0,1,...,T is a discrete-time stochastic process governed by a forward propagation equation π t (x t |x t−1 ) and at each time t > 0, we observe Y t that is related to X t with uncertainty. Suppose that the distribution of Y t is entirely determined by X t through a conditional distribution p(Y t = y t |X t = x t ) = g t (y t |x t ).(11) The above is the standard state space model (SSM) where (11) This problem can be solved by sampling X T first using simple forward propagation in section 3.1 and re-weighting the paths by p(Y|X ) = T t=1 p(y t | x t ), though it is not efficient. SMC method recursively utilizes the information I t = {Y t } during the propagation by importance sampling. It has been shown to be extremely useful and efficient if implemented properly. A variety of SMC implementations can be viewed under the cSMC framework. For example, the standard particle filter (Gordon et al., 1993) ignores the information I t = y t in the resampling step and uses β (i) (6) and q t (x t |x t−1 , I t ) ∝ π t (x t |x t−1 ) as the proposal in (7). The weight update in (8) then becomes w (i) t = p(I t |x t ) = p(y t |x t ). There is no backward pilot needed as p(I t |x t ) = p(y t |x t ) is given precisely. This weight is used in the resampling of x (i) t at the beginning of the t + 1 step. The full information particle filter of Kong et al. (1994) and Liu and Chen (1998) (6) as well, but with a full information proposal distribution q t (x t |x t−1 , I t ) ∝ π(x t |x t−1 )p(y t | x t ) with the current information I t = y t for the prorogation in (7), and weight update in (8) being t−1 = w (i) t−1 inuses β (i) t−1 = w (i) t−1 inw (i) t = π(x t |x t−1 )p(y t | x t )dx t . The auxiliary particle filter (Pitt and Shephard, 1999) uses an approximation of the optimal priority score (6) in cSMC algorithm. Specifically, instead of using backward pilots, an approximation such asp (I t + | x (i) t−1 ) = p(y t | µ (i) t|(t−1) ), is used, where µ t|(t−1) is an estimate of x t (e.g. the mean, mode or other suitable statistics) under the conditional distribution p(x t | x (i) t−1 ). Such a resampling step incorporates the information I t+ = y t , hence provides a set of more efficient samples for propagation. The independent particle filter (Lin et al., 2005) uses a different approximation of the optimal priority score (6) in cSMC algorithm. By drawing a set of m backward pilots, x (j) t ∼r t (X t ) using either the information I t = y t , say x (j) t ∝ p(Y t | X t ) , or an approximation of it, and p(I t + | x (i) t−1 ) = 1 m m j=1 p(x (j) t | x (i) t−1 )w (j) t .(12) wherew (j) t = p(y t | x (j) t )/r t (x (j) t ) is the backward pilot weight. This approach has been shown to be effective when the observation noise is small, or I t is a relatively strong constraint. It incorporates the strong constraint in the resampling of x (i) t−1 , before sampling x t . Lin et al. (2005) showed that this approach is effective even if the summation in (12) is further approximated by a partial sum of randomly selected summands. A straightforward implementation of cSMC-bp of Section 2.3 would sample the backward pilot (x t , x t−1 ) based on I t = y t , and then estimatep(I t + | x (i) t−1 ) using (10). However, this may not be efficient as the backward pilots do not incorporate the available information y t−1 , and it incurs two steps of random samples. A more efficient approach in the frequent constraint case is to propagate first to get a set of x (ij) t ∼ p(x t | x (i) t−1 ), j = 1, . . . , m, then usê p(I t + | x (i) t−1 ) ∝ p(I t + | x t )p(x t | x (i) t−1 )dx t ≈ 1 m m j=1 p(y t | x (ij) t ). To save computational time, m = 1 can be used. Strong Constrained Problems An extremely strong constraint has a delta function as its density function p(I|X T ) = 0 a.e.. One example is the diffusion bridge problem mentioned before. Suppose X is governed by a state equation π t (x t |x t−1 ) and the constraint is I T = {X T = x * T }. This problem requires to sample from the constrained distribution p(x T |x T = x * T ). In diffusion bridge problems, the starting distribution is given by x 0 = x * 0 . For the fixed end-point constraint, notice that the constrained joint distribution p(x T |x t = x * T ) ∝ T −1 t=1 π t (x t |x t−1 ) π T (x * T |x T −1 ) has a sequential structure up to time T −1. Under the fixed point constraint, the suggested measure Q * t equals the marginal posterior measure Q t for all time t. A cSMC-bp algorithm for this problem has been described in sections 2.2 and 2.3. Particularly, the priority score uses β (i) t−1 = w (i) t−1 p(x * T |x (i) t−1 ). The term p(x * T |x (i) t−1 ) measures the likelihood that each path reaches the fixed end point. It guides the sample to move towards that endpoint. It can be evaluated by a backward pilot run, which samples from the endpoint x T = x * T and propagates backward as described in section 2.3. In the backward pilot run, the trial distributionr s (x s |x s+1 ) is used to propagate the backward sample. When the process is reversible and the backward conditional distribution π(x s |x s+1 ) is available, the backward trial distribution can be chosen as exact:r s (x s |x s+1 ) = π(x s |x s+1 ). In other cases when the backward conditional distribution π(x s |x s+1 ) is intractable, the backward trial distributionr s is chosen to be close to the exact distribution. In the above strong constrained problem, condition (9) is satisfied because I t + is the only constraint and p(Ω|I T , x t ) = 1 for all x t . When the trial distribution in propagation step is chosen as exact: q t (x t |x (i) t−1 ) = π t (x t |x (i) t−1 ), the cSMC algorithm sample follows the same dynamics as the original process, no matter linear or nonlinear. Systems with Intermediate Constraints The cSMC algorithm can also be applied to sparse constraint cases when most of the atomic information I t is trivial. Consider a forward propagation system with intermediate noisy observations. Suppose {X t } t=0,1,...,T is such a discretized process with fixed-point constrains X 0 = x * 0 and X T = x * T . And K noisy observations {Y k } k=1,...,K are captured for time {kM } k=1,...,K , where (K + 1)M = T . Those noisy observations are governed by the observation equation: Y k ∼ g k (Y k |X kM ) for k = 1, . . . , K. This model can also be viewed as a state space model of length (K+2), denoted as {X kM , Y k } k=0,1,...,K+1 , with M − 1 interpolated points between adjacent states. The standard SMC sampler propagates forward at each time t and resamples only at time M, 2M, . . . , KM . As discussed in section 2.2, the sequence of propagations between t = kM and t = (k + 1)M without future constraints could result in a large divergence between the sampling distribution and the target distribution at time t = (k + 1)M , and thus is less efficient. Here we propose a special procedure under the general cSMC framework. At each time t, the sampler resamples under probability measure Q * t with the next non-trivial constraint I t + = I t M taken into consideration. Notice that K intermediate observations split the whole chain into K + 1 segments as shown in Figure 5. The first segment, along with the first observation (X 0 , X 1 , . . . , X M , Y 1 ), can be viewed as a system with strong constrains in which Y 1 is part of the stochastic process. And the observation equation p(Y 1 | X M ) for Y 1 works as the state equation of Y 1 conditioned on X M , or the forward propagation mechanism p(Y 1 | X M ). In such a setting, {Y 1 = y 1 } is now the fixed-point constraint at the end t = M + 1. Based on the same procedure for strong constrained problem, a backward pilot run is firstly conducted, followed by a forward propagation with importance resampling using priority scores estimated from backward samples. In the end, the marginal sample of (X 0 , X 1 , . . . , X M ), which is obtained by discarding Y 1 , can be used as an initial sample for further segments. Consequently, the first segment, as a subproblem, can be properly sampled under cSMCbp framework. Once the first k segments are sampled and properly weighted, the next segment (X kM , X kM +1 , . . . , X (k+1)M , Y k+1 ) forms the same strong constrained problem with Y k+1 as the fixed endpoint. Since the samples of the chain (X 0 , . . . , X kM ) is already obtained from the previous cSMC-bp runs, it follows the same procedure as previous segments to extend samples from time kM to time (k +1)M . By solving all the (K + 1) segmental subproblems sequentially in the same way, the entire path can thus be sampled. This strategy involves (K + 1) runs of backward pilot, one for each segment. The total computational effort in generating backward pilot is linear in T . Forward propagation and importance resampling procedures are the same at each time t except that the backward pilot samples used in priority score estimation differ for different segments. The cost of forward procedure, therefore, is also linear in T . Hence the whole computational efforts of this constrained SMC is linear in T . In the meantime, efficiency is improved by more fully utilizing the dynamic nature of underlying process. When M = 1, this setting becomes the standard State Space Model as discussed in Section 3.1. Figure 5: Segmentation of a chain with K intermediate observations. X 0 · · · X M · · · X 2M · · · X KM · · · X T Y 1 Y 2 Y K Segment 1 Segment 2 Segment K+1 No backward pilot is needed as the future information p(I t + |x t ) = p(y t |x t ) is available. When M is small, a different approach using forward pilots has been proposed (Wang et al., 2002;Lin et al., 2013). Multi-level System: State Space Model with Routine Observations & Fixed Point Constraints The backward pilot strategy can also be used to solve the case with multi-level constraints, when the constraints have a hierarchical structure, with one level of weak constrained and another level of strong constraints. In this section, we demonstrate this strategy to solve a special multi-level case, in which two fixed-point constraints are imposed on a standard state space model. Suppose a state space model is governed by the following equations, for t = 1, . . . , T p(X t = x t |X t−1 = x t−1 ) = π t (x t |x t−1 ), p(Y t = y t |X t = x t ) = g t (y t |x t ). Two fixed-point constraints are imposed on this process, X 0 = x * 0 and X T = x * T . The routine observations are viewed as a layer of weak constraints and the fixed point constraint is viewed as a layer of strong constraints. To utilize our constrained SMC method, we suppress the weak constraints layer and define a new forward propagation system such that its forward propagation equation is π * t (x t |x t−1 , Y) := π t (x t |x t−1 , y t ) ∝ π t (x t |x t−1 )g t (y t |x t ).(13) The problem turns into a strong constrained problem with fixed point constraints. The corresponding underlying process is governed by (13). In the resampling step of cSMC-bp, the priority score can be chosen as β t = w t p * (x * T |x t , Y) = w t p(x * T |x t , y t+1 , . . . , y T −1 ). The term p(x * T |x t , y t+1 , . . . , y T −1 ) is the probability for a path to reach the fixed endpoint conditioned on the weak constraints. This value can also be evaluated by a kernel density estimator on the backward pilot sample. Note that in this case, backward pilot run is sampled from the distribution with the density p(x t , . . . , x T −1 |y t+1 , . . . , y T −1 , x T ) ∝ T s=t+1 π s (x s |x s−1 ) T −1 s=t+1 g t (y s |x s ) = 1 g t (y t |x t ) T −1 s=t π s+1 (x s+1 |x s )g s (y s |x s ). As in the standard SMC for state space models, the backward trial distribution can be set just as used in standard particle filter withr s (x s |x s+1 ) ∝ π s+1 (x s+1 |x s ). One can also user s (x s |x s+1 ) ∝ g s (y s |x s ) as in independent particle filter orr s (x s |x s+1 ) ∝ π s+1 (x s+1 |x s )g s (y s |x s ) using full information. In the propagation step, a standard SMC procedure for state space model is used. Compared with a pure state space model without fixed point constraints, the additional level of strong constraints only affects the priority scores in resampling, with the propagation step unchanged. Examples System with Intermediate Observations Suppose a diffusion process on {X t }, t ∈ [0, 90] is governed by a stochastic differential equation as follows (Beskos et al., 2006;Lin et al., 2010) dX t = sin(X t − π)dt + dW t , where W t is a Winer process. This nonlinear process shows a jumping behavior among stable levels at X = 2kπ, k ∈ Z. With Euler approximation, the continuous diffusion process can be discretized by inserting intermediate points with interval δ X t i+1 = X t i + δ sin(X t i − π) + ε t i+1 , where t i = iδ and ε t i ∼ N (0, √ δ). Two noisy observations are made at time t = 30 and at time t = 60: Y 30 ∼ N (X 30 , σ 2 ), Y 60 ∼ N (X 60 , σ 2 ), along with two fixed point constraints X 0 = x * 0 , X 90 = x * 90 . The discretized model is demonstrated in Figure 6. In this experiment, we use the following parameter setting: X 0 = x * 0 X δ · · · X 30 · · · X 60 · · · X 90−δ X 90 = x * 90 Y 30 = y 30 Y 60 = y 60 Figure 6: Discretized Model X 0 · · · X 30 · · · X 60 · · · X 90 Y 30 Y 60 X 0 · · · X 30 · · · X 60 · · · X 90 Y 30 Y 60 X 0 · · · X 30 · · · X 60 · · · X 90 Y 30 Y 60 x * 0 = 0, x * T = −1.17, y 30 = 1.49, y 60 = −5.91 and δ = 0.1. The two intermediate points split the time line into three segments. The segmental sampling procedure is demonstrated in Figure 7. The sequential sampling procedure consists of three periods with different settings of I t + . In the first period, the chain (X 0 , . . . , X 30 , Y 30 ) is viewed as a strong constrained problem and is sampled by the cSMC-bp algorithm. In this period, backward pilot sample from the fixed point Y 30 is used to estimate the priority scores in the forward propagation. For the second and third period, sample paths keep propagating and get resampled according to the priority scores estimated from the backward pilot sample generated backwards from Y 60 and X 90 = x * 90 . In this study, three levels of error are investigated: σ = 0.01 for very accurate observations, σ = 1 for moderate accurate observations and σ = 2 for untrusted observations. Note that this process has stable levels and the gap between levels is 2π. The four observations (0, 1.49, −5.91, −1.17) correspond to stable levels 0, 0, −2π and 0 accordingly. It implies that the process is likely to fluctuate around stable level 0 during the first period. Next, the process jumps to stable level −2π in the second period and then jumps back to stable level 0 in the third period. Figure 9 shows the histogram of the marginal sampling distribution for X 60 . When the observations are accurate (σ = 0.01), the two observations perform like fixed point constraints that forcing all sample paths to pass through the observations at time 30 and time 60. When the observation error is large (σ = 2), a high portion of sample paths remains at the original stable level while a smaller portion of paths is drawn towards the observations. The moderate error case (σ = 1) is a compromise between the observation and the tendency of the process with no constraints. The marginal samples for X 60 are distributed around the observation with a moderate variance. Samples from all three levels of error retain the jumping nature of underlying process and the cSMC-bp approach is capable of dealing with different levels of observational errors. Setting 2 Suppose all the settings are the same as the above section, except that y 30 = 6.49. We take this value as it is close to 2π. Since y 30 and y 60 differ by a gap of two stable levels, this is a much rare event. In this case, the sample size increases to 5000 in order to overcome the degeneracy. Sample paths and marginal histograms for different levels of errors are shown in Figure 10 and Figure 11. As the error level decreases, more sample paths are drawn towards the observations. And in the large error case, most of the samples are concentrated around 0. Those three plots provide the evidence that the priority scores we used is effective in its power for all the three error levels under this extreme setting. Sampling Constrained Trading Paths In asset portfolio management, the optimal trading path problem is a class of optimization problem which typically maximizes the utility function including the transaction cost. Due to the complicated structure, it is difficult to find a closed-form solution. Kolm and Ritter (2015) turn such an optimization problem into a state space model and explore Monte Carlo methods to numerically solve this problem. Let X = (X 0 , X 1 , . . . , X T ) be a trading path where X t represents the holding position in shares at time t. The optimization problem is to maximize u(X) = − T t=1 f t (|X t − X t−1 |) − T t=0 g t (Y t − X t ),(14) where (Y 0 , Y 1 , . . . , Y T ) is a predetermined optimal trading path in an ideal world without transaction costs, typically the optimal strategy under Markowitz mean variance optimality. Here f t (·) is the transaction cost function and g t (·) stands for the utility loss due to the departure of the realized path from the ideal path. In most cases, g t (·) is a quadratic function under exponential utility loss and the transaction cost is expressed as f t (δ) = c 0 |δ| + c 1 δ 2 with c 0 and c 1 both non-negative (Kolm and Ritter, 2015). An emulating state space model can be implemented such that the likelihood has the form p(X) ∝ exp{κu(X)} ∝ T t=1 exp{−κf t (|X t − X t−1 |)} T t=0 exp{−κg t (Y t − X t )}.(15) Then the process of finding the most likely path of p(X) is the same as the optimization problem in (14). One of such state space models can be represented by the following state equation and observation equation: X t = X t−1 + δ t , where p(δ t ) ∝ exp{− 1 2σ 2 x (δ 2 t + 2|αδ t |)},(16)Y t = X t + ε t , where ε t ∼ N (0, σ 2 y ).(17) The joint distribution of X t of the system is the one in (15). In practice, a starting position X 0 and the target end position X T are often imposed for optimal execution of a (large) order with minimum market impact. Here for simplicity we imposed two end points X 0 = X T = 0. Any other positions can be implemented similarly. This constrained trading path sampling problem now becomes a state space model problem with fixed point constraints. Suppose the term structure of the asset consists of two exponentially-decaying alpha models. The corresponding ideal trading path is given by Y t = 25 exp{−(t + 1)/8} − 40 exp{−(t + 1)/4}, for t = 1, . . . , 20. We set the variances in the system (16) and (17) as σ 2 x = 0.25 and σ 2 y = 1. In the example, we investigate two special cases of α in the state equation (16). When α = 0, the model is linear Gaussian, and an efficient solution is feasible by Kalman filter. We will compare the results from the proposed Monte Carlo algorithm with that from Kalman filter. When α = 0.5, the model involves a Laplace distribution and thus is nonlinear. The optimal path is obtained by numerical methods. In both cases, the cSMC-bp is compared to a standard SMC that naively forces all paths to connect to the end point. The samples from standard SMC have a much larger variance and most of them lie outsides the 95% confidence region, while most samples from cSMC-bp stay within the 95% confidence region. For samples from cSMC-bp, the backward pilot samples bring the information from the future and guide the paths by importance resampling. Without using any future information, standard SMC sampler propagates blindly and suffers a large divergency between the sampling distribution and the target distribution in the end. We firstly run a standard SMC sampling with 1,000,000 sample paths to obtain the most likely path estimation, together with an approximate 95% confidence band. The sample paths generated by both two methods, along with the most likely path estimate and the approximate 95% confidence band are plotted in Figure 15. Guided by priority scores with future constraint, most samples from our constrained SMC method propagate within the 95% confidence band. The optimal path remains constant between time 8 and time 13 as l1 penalty is involved. Figure 17 plots the sampling distribution and kernel density at four time points. The true target density is estimated from a run with 1,000,000 paths. At time 4 and time 19, the sampling distribution of cSMC-bp is much closer to the target one than the standard SMC. Suppose sample average is still used to estimate the optimal path. The mean squared error of estimating the optimal path using sample marginal mean plotted in Figure 13 suggests that cSMC-bp reduces MSE at most time points, especially in the periods 1 t 7 and 13 t 19. Summary In this article, we have proposed a general framework of cSMC-bp algorithm for constrained sampling problems. The key idea is to use the next available information I t + to adjust the sampling These constraints go beyond the scope of fixed points but can still be solved using cSMC-bp. Compared with the standard SMC algorithm, cSMC-bp reduces the divergence between sampling distribution and the true distribution by taking future information into consideration at each intermediate step. And the additional computational burden from the backward pilot run is limited. Consequently, cSMC-bp achieves a smaller estimation error, as illustrated in the trading path problem in section 4.2. Compared with other possible SMC algorithms under sequential importance sampling (SIR) scheme, cSMC-bp is approximately optimal in reducing χ 2 divergence given condition (9). Figure 1 : 1Forward propagation algorithm Figure 2 2Figure 2: cSMC algorithm Figure 3 : 3Algorithm for generating backward pilots Figure 4 Figure 4 : 44demonstrates the cSMC-bp procedure, where the marginal backward pilot sample is used to estimate the empirical distribution of p(I t + | x t−1 ), shown with the heatmap (in log scale) on the right side of the figure. The left side of the figure shows five forward paths, to be resampled Use backward pilot sample as the priority score at time t = 10 is called the observation equation. The observation sequence Y = {Y t } t=1,...,T is viewed as a set of frequent constraints in our constrained problem framework. Figure 7 : 7Segmental Figure 8 8plots 1000 sample paths for each level of error. We use 300 backward pilot samples in each segment. Figure 8 : 0 Figure 9 : 0 Figure 10 : 0 Figure 11 : 809010011Sampled Paths for (top) σ = 0.01, (mid) σ = 1.0 and (bottom) σ = 2.Histogram for X 60 for (top) σ = 0.01, (mid) σ = 1.0 and (bottom) σ = 2.Sampled Paths for (top) σ = 0.01, (mid) σ = 1.0 and (bottom) σ = 2.Histogram for X 60 for (top) σ = 0.01, (mid) σ = 1.0 and (bottom) σ = 2.0 linear and Gaussian case, 2000 sample paths are drawn from the cSMC-bp using 300 backward samples. For the purpose of comparison, standard SMC draws 2300 paths such that both methods have a similar computational burden. The sample paths from those two methods and the 95% confidence interval band obtained by the optimal Kalman filter are plotted inFigure 14. Figure 16 Figure 12 : 1612shows the marginal kernel density estimation of un-weighted samples(left column) and weighted samples (right column). Both methods produce properly weighted samples, as the marginal density for the weighted samples are both close to the theoretical one at each time.However, the marginal sampling distribution for X 19 under standard SMC method has a large divergence from the theoretical distribution. This divergence results in a low efficiency.The mean squared errors (MSE) of estimating the marginal mean at each time t, using 500 sample paths with 1000 replications, are reported for each time t in Figure 12 for both standard SMC and cSMC-bp. In the period 8 t 17 where the fixed points have limited effects, standard SMC and cSMC-bp have similar performance. In the period 1 t 7 where the observation y t changes over time dramatically, cSMC-bp results in a smaller error than the standard SMC as future information are incorporated in sampling of cSMC-bp. In the period 18 t 19 where the endpoint constraint takes effect, MSE is smaller using the cSMC-bp approach. Mean squared error curve for sample average under setting 1. 4.2.2 Case 2: α = 0.5 distribution at each intermediate point t. The backward pilot strategy proposed by Lin et al. (2010) enables the estimation of priority scores. This framework is compatible with previous studies on state space model sampling and diffusion bridge problems as they can be viewed as special cases of constrained problems. The sampling procedure of cSMC-bp coincide with the standard SMC in a state space model problem and Lin et al. (2010)'s algorithm in the diffusion bridge sampling problem. Compared with Lin et al. (2010)'s backward pilot algorithm on diffusion bridge problems, our framework does not require each constraint to be fixed point type and can deal with a wider range of constraints as long as (4) is satisfied. Examples for two special cases are demonstrated in Section 4: one with noisy intermediate observation constraints and the other with multilevel constraints. Figure 13 : 13Mean squared error curve for sample average under setting 2. Figure 14 :Figure 15 :Figure 16 : 141516Sampling paths from (left) standard Sequential Monte Carlo algorithm (right) constrained SMC with backward pilot (left) standard SMC (right) constrained SMC with backward pilot For model with α = 0, plots of (Left Column) marginal sampling distribution and (Right Column) marginal density at (Row 1) time 1, (Row 2) time 4, (Row 3) time 12 and (Row 4) time 19. Figure 17 : 17For model with α = 0.5, plots for (Left Column) marginal sampling distribution and (Right Column) marginal density at (Row 1) time 1, (Row 2) time 4, (Row 3) time 12 and (Row 4) time 19. the continuous-time stochastic process {X t } t∈[0,T ] now is approximated by the discrete version {X τ k } k=0,··· ,L with the forward propagation equation in (2). The discrete approximation for continuous-time stochastic process can thus be sampled by forward propagation algorithm. High accuracy can be achieved by increasing the number of intermediate points at the cost of more computation burden. In most applications, the number of intermediate points is chosen as a compromise between discretization error and computational efficiency. t , w(i) t } at the end of cSMC inFigure 2is properly weighted with respect to P t , not the desired Q * t , except when t + = t, the nontrivial constraint time. However,{(x (i) t , w (i) t p(I t+ | x (i)t )} is properly weighted with respect to Q * t when t + > t, though one would need to be able to calculatep(I t+ | x (i) t ) precisely.There are varies ways of generating index {J i } i=1,...,n in the resampling step inFigure 2. Multi- Stochastic simulation Bayesian approach to multitarget tracking. D Avitzour, IEE Proceedings-Radar, Sonar and Navigation. 142Avitzour, D. (1995), "Stochastic simulation Bayesian approach to multitarget tracking," IEE Proceedings-Radar, Sonar and Navigation, 142, 41-44. Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes (with discussion). A Beskos, O Papaspiliopoulos, G O Roberts, P Fearnhead, Journal of the Royal Statistical Society: Series B (Statistical Methodology). 68Beskos, A., Papaspiliopoulos, O., Roberts, G. O., and Fearnhead, P. 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{'abstract': 'Sequential Monte Carlo methods are a class of Monte Carlo methods that is used to obtain random samples of a high dimensional random variable in a sequential fashion. Many problems encountered involve different types of constraints. These constraints make the problem much more challenging. In this paper we formulate a general framework of using SMC for constrained problems, extending the backward pilot idea ofLin et al. (2010). Several new algorithms are developed and demonstrated. It is noted that all information observed or imposed on the underlying system can be viewed as constraints. Hence the approach outlined in this paper can be useful in many applications.', 'arxivid': '1706.02348', 'author': ['Chencheng Cai \nDepartment of Statistics\nRutgers University\nRutgers University\n08854PiscatawayNJUSA\n', 'Rong Chen rongchen@stat.rutgers.edu. \nDepartment of Statistics\nRutgers University\nRutgers University\n08854PiscatawayNJUSA\n', 'Rong Chen \nDepartment of Statistics\nRutgers University\nRutgers University\n08854PiscatawayNJUSA\n'], 'authoraffiliation': ['Department of Statistics\nRutgers University\nRutgers University\n08854PiscatawayNJUSA', 'Department of Statistics\nRutgers University\nRutgers University\n08854PiscatawayNJUSA', 'Department of Statistics\nRutgers University\nRutgers University\n08854PiscatawayNJUSA'], 'corpusid': 88515492, 'doi': '10.5705/ss.202022.0185', 'github_urls': [], 'n_tokens_mistral': 17455, 'n_tokens_neox': 15539, 'n_words': 9931, 'pdfsha': '0e57168108c27b48548c3f61a46743669a6d0b2e', 'pdfurls': ['https://arxiv.org/pdf/1706.02348v2.pdf'], 'title': ['Backward Pilot Strategy in Constrained Sampling Problems *', 'Backward Pilot Strategy in Constrained Sampling Problems *'], 'venue': []}

arxiv

CASE: Aligning Coarse-to-Fine Cognition and Affection for Empathetic Response Generation Jinfeng Zhou DCST Institute for Artificial Intelligence The CoAI Group State Key Lab of Intelligent Technology and Systems Beijing National Research Center for Information Science and Technology Tsinghua University 100084BeijingChina College of Intelligence and Computing Tianjin University TianjinChina Chujie Zheng DCST Institute for Artificial Intelligence The CoAI Group State Key Lab of Intelligent Technology and Systems Beijing National Research Center for Information Science and Technology Tsinghua University 100084BeijingChina Bo Wang bo_wang@tju.edu.cn College of Intelligence and Computing Tianjin University TianjinChina Zheng Zhang DCST Institute for Artificial Intelligence The CoAI Group State Key Lab of Intelligent Technology and Systems Beijing National Research Center for Information Science and Technology Tsinghua University 100084BeijingChina Minlie Huang aihuang@tsinghua.edu.cn DCST Institute for Artificial Intelligence The CoAI Group State Key Lab of Intelligent Technology and Systems Beijing National Research Center for Information Science and Technology Tsinghua University 100084BeijingChina CASE: Aligning Coarse-to-Fine Cognition and Affection for Empathetic Response Generation Empathetic conversation is psychologically supposed to be the result of conscious alignment and interaction between the cognition and affection of empathy. However, existing empathetic dialogue models usually consider only the affective aspect or treat cognition and affection in isolation, which limits the capability of empathetic response generation. In this work, we propose the CASE model for empathetic dialogue generation. It first builds upon a commonsense cognition graph and an emotional concept graph and then aligns the user's cognition and affection at both the coarsegrained and fine-grained levels. Through automatic and manual evaluation, we demonstrate that CASE outperforms state-of-the-art baselines of empathetic dialogues and can generate more empathetic and informative responses. Introduction Human empathetic conversations allow both parties to understand each other's experiences and feelings (Keskin, 2014), which is crucial for establishing seamless relationships (Zech and Rimé, 2005) and is also integral to building a trustful conversational AI (Huang et al., 2020;. In social psychology, empathy consists of two aspects: cognition and affection (Davis, 1983). The cognitive aspect corresponding to the understanding of the user's situation and experiences (Cuff et al., 2016). The affective aspect requires the comprehension of the user's emotional state and his/her potential emotional reaction (Elliott et al., 2018). Although existing work of empathetic dialogue involves both aspects of empathy, there are still issues that need to be addressed. First, most work Lin et al., 2019;Majumder Figure 1: Examples from the EMPATHETICDIA-LOGUES dataset. The alignment of cognition and affection (i.e., emotional state and emotional reaction) leads to highly empathetic and informative expression in responses. Li et al., 2020 considers only the affective aspect, like detecting the user's emotional state to enhance empathy expression. Second, although recent work explored both roles of cognition and affection in empathy expression (Zheng et al., 2021a;Sabour et al., 2022), they usually treat cognition and affection in isolation without considering their relationship. However, human empathetic responses often result from conscious alignment and interaction between cognition and affection of empathy (Westbrook et al., 2011). For one thing, the user's overall emotional state manifested in the context suggests the user's attitude toward current situation (i.e., cognition). Thus, for the listener, aligning the user's expressed cognition to the proper emotional state is essential for an appropriate empathetic response. As in case-1 of Figure 1, the alignment of cognition (i.e., intent "to go to the beach") with different emo-tional states (i.e., "excited" vs. "disappointed") produces different appropriate empathetic expressions (i.e., "love" and "which beach are you going to go to" vs. "hate" and "waiting for the beach"), respectively. For another, the user's situation drives the listener to infer the deeper specific cognitions and associate them with the underlying emotional reactions. In this way, the listener can produce a more actively empathetic response instead of only understanding and repeating the user's expressed cognition. As in case-2 of Figure 1, building association between inferred cognitions and emotional reactions, i.e., "to give up" and "frustrated" vs. "to try harder" and "hopeful", yields cognitively distinct but highly empathetic responses, i.e., response-2a vs. response-2b. The two cases highlight the necessity of aligning cognition and affection on both overall and specific (i.e., coarse and fine-grained) level for empathy modeling in response generation. To this end, we align Cognition and Affection for reSponding Empathetically (CASE) on coarse and fine-grained levels by fusing sentence-level commonsense knowledge from COMET (Bosselut et al., 2019) and word-level concept knowledge from ConceptNet (Speer et al., 2017). Commonsense knowledge infers the user's situation as cognition and infers emotional reactions to the situation, which are implied in the dialogue. Concept knowledge serves to extract the emotional state manifested in the dialogue. For encoding the two types of knowledge, we first construct commonsense cognition graph and emotional concept graph, where the initial independent representation of cognition and emotional concept is carefully adjusted by dialogue context adopting graph transformers. Then, we design a two-level strategy to align cognition and affection using mutual information maximization (MIM) (Appendix A) (Hjelm et al., 2019). The coarse-grained level considers overall cognition and affection manifested in the dialogue context to align contextual cognition and contextual emotional state, which are extracted with a knowledge discernment mechanism. The fine-grained level builds the fine-grained association between cognition and affection implied in the dialogue to align each specific cognition and corresponding emotional reaction. Further, an empathy-aware decoder is devised for generating empathetic expressions. Our contributions are summarized as follows: (1) We devise a unified framework to model the interaction between cognition and affection for in-tegrated empathetic response generation. (2) We construct two heterogeneous graphs involving commonsense and concept knowledge to aid in the modeling of cognition and affection. (3) We propose a two-level strategy to align coarse-grained and fine-grained cognition and affection adopting mutual information maximization. (4) Extensive experiments demonstrate the superior of CASE in automatic and manual evaluation. 2 Related Work Emotional & Empathetic Conversation Emotional conversation gives the manually specified label preset as the emotion generated in the response (Zhou et al., 2018;Wei et al., 2019;Peng et al., 2022). Instead of giving a predefined emotion label, empathetic conversation Kim et al., 2022) involves cognitive and affective empathy (Davis, 1983) and aims to fully understand the interlocutor's situation and feelings and respond empathically (Keskin, 2014;Zheng et al., 2021b). For one thing, most existing works only focus on the affective aspect of empathy and make efforts to detect contextual emotion Lin et al., 2019;Majumder et al., 2020;Li et al., 2020 while ignoring the cognitive aspect. For another, some research utilizes commonsense as cognition to refine empathetic considerations (Sabour et al., 2022). However, the relatively independent modeling between the two aspects (i.e., cognition and affection) violates their interrelated characteristics. Commonsense & Concept Knowledge As a commonsense knowledge base, ATOMIC focuses on inferential knowledge organized as typed if-then relations. Six commonsense reasoning relations are defined for the person involved in an event, four of which are used to reason commonsense cognitions of a given event, i.e., PersonX's intent before the event (xIntent), what PersonX need to do before the event (xNeed), what PersonX want after the event (xWant), and the effect of the event on PersonX (xEffect). Each commonsense cognition is aligned with user's emotional reaction to the situation implied in the dialogue inferred by xReact (i.e., PersonX's reaction to the event) in our approach. To obtain inferential commonsense knowledge, we use COMET (Bosselut et al., 2019), a pretrained generative model, to generate rich commonsense statements. Unlike commonsense knowledge that provides sentence-level commonsense expression, we adopt ConceptNet (Speer et al., 2017) as concept knowledge, which provides word-level human knowledge and is widely used in various NLP tasks Zhong et al., 2021;. Following , we use NRC_VAD (Mohammad, 2018) to assign emotion intensity to concepts in ConceptNet (processing details are in ) severed to extract the contextual emotional state manifested in the context, and align it with contextual cognition. Fig. 2. The dialogue context X = [x 1 , . . . , x N ] contains N utterances, where x i denotes the i-th utterance. CASE contains three stages: (1) The graph encoding stage constructs and encodes heterogeneous commonsense cognition graph G CS and emotional concept graph G EC from the dialogue context X. Approach CASE framework is in (2) The coarse-to-fine alignment aligns coarse-grained (between contextual cognition and contextual emotional state) and fine-grained (between each specific cognition and corresponding emotional reaction) cognition and affection adopting MIM. (3) The empathy-aware decoder integrates the aligned cognition and affection to generate the response Y = [y 1 , y 2 , . . . , y M ] with empathetic and informative expressions. Graph Encoding Commonsense Cognition Graph Construction Given the last utterance x N of the dialogue context X, we segment it into the sub-utterances U = [u 0 , u 1 , u 2 , . . . , u t ], where we prepend the whole x N as u 0 for maintaining the global information of x N . We use COMET to infer l commonsense cognition knowledge K r i = [k r i,1 , k r i,2 , . . . , k r i,l ] for each u i ∈ U , where r is one of the four commonsense relations R = {xIntent, xNeed, xWant, xEffect}, similar to Sabour et al. (2022). The idea is that human responses tend to inherit the above and transfer the topic. There are differences in the topic and connotation of different sub-utterances affecting the listeners' concerns when responding empathetically. For constructing the heterogeneous commonsense cognition graph G CS , we use the utterance set U and the commonsense cognition knowledge set K CS = t i=0 r∈R K r i as vertices, i.e., vertex set V CS = U ∪ K CS . There are seven relations of undirected edges that connect vertices. (1) The selfloop relation for each vertex v CS i ∈ V CS . (2) The global relation between the whole x N (i.e., u 0 ) and its sub-utterances u i (i ≥ 1). (3) The temporary relation between any two successive sub-utterances u j and u j+1 . (4) The four commonsense relations, i.e., xIntent, xNeed, xWant, xEffect, between the utterance u i ∈ U and the corresponding K r i . We use a Transformer-based sentence encoder (cognition encoder) to first encode the vertices V CS of the graph G CS . For each v CS i ∈ V CS , we prepend with a special token [CLS]. Following Devlin et al. (2019), we collect the [CLS] representation as the initial embedding matrix for G CS . Emotional Concept Graph Construction We concatenate the utterances in the dialogue context X to obtain the token set T , i.e., T = x 1 ⊕ . . . ⊕ x N = [w 1 , . . . , w n ], where n is the number of all the tokens in the utterances in X. Following , we use ConceptNet to infer the related concepts for each token w i ∈ T , among which only the the top N emotional concepts (according to the emotion intensity ω(c)) are used for constructing G EC . Subsequently, the vertices V EC in the heterogeneous emotional concept graph G EC contains a [CLS] token, the dialogue context tokens T , and the above obtained emotional concepts. There are four relations of undirected edges that connect vertices. (1) The self-loop relation for each vertex v EC i ∈ V EC . (2) The global relation between the [CLS] token and other ones. (3) The temporary relation between any two successive tokens. (4) The emotional concept relation among a token and its related emotional concepts. We initialize the vertex embedding for G EC by summing up the token embedding, the positional embedding, and the type embedding for each vertex (signaling whether it is a emotional concept or not). Graph Encoder Given the commonsense cognition graph G CS , to capture the semantic relationship between vertices, we adopt the Relation-Enhanced Graph Transformer for graph encoding. It employs a relation-enhanced multi-head attention mechanism (MHA) to encode vertex embeddingv v i for vertex v i (we omit the superscripts CS for simplicity) as: v v i = M HA v k ∈V CS q v i , k v k , v v k , (1) where the semantic relations between vertices are injected into the query and key vectors: (2) where l v i →v k and l v k →v i are learnable relation embeddings between vertices v i and v k . The selfattention is subsequently followed by a residual connection and a feed-forward layer, as done in the standard Transformer encoder (Vaswani et al., 2017). Finally, we obtain the commonsense cognition embedding cs i for each v CS i ∈ V CS . To encode the emotional concept graph G EC , we adopt a vanilla Graph Transformer (i.e., omitting the relation enhancement part in the above Graph Transformer). By superimposing the emotion intensity of each token, we obtain the emotional concept embedding ec i for each v EC i ∈ V EC . q v i = v v i + l v i →v k , k v k = v v k + l v k →v i , Coarse-to-Fine Alignment Context Encoding Following previous works (Majumder et al., 2020;Sabour et al., 2022), we concatenate all the utterances in the dialogue context X and prepend with a [CLS] token: [CLS] ⊕ x 1 ⊕ . . . ⊕ x N . This sequence is fed into a standard Transformer encoder (context encoder) to obtain the representation S X of the dialogue context. We denote the representation of [CLS] as s X . Coarse-grained Alignment To reproduce the interaction of cognition and affection manifested in the dialogue context, we align contextual cognition and contextual emotional state at an overall level. They are separately acquired by cognitive and emotional knowledge discernment mechanisms, which select golden-like knowledge guided by response. To obtain the contextual cognitive representation r cog , the knowledge discernment calculates the prior cognitive distribution P CS (cs i | X) over the commonsense cognition knowledge (that is, only K CS rather than all the vertices V CS in G CS , and we thus use 1 ≤ i ≤ |K CS | for simplicity): r cog = |K CS | i=1 P CS (cs i | X) · cs i ,(3)P CS (cs i | X) = softmax i (cs T i ϕ CS (s X )),(4) where ϕ CS (·) is a MLP layer activated by tanh. Similarly, we calculate the prior emotional distribution P EC (ec i | X) (1 ≤ i ≤ |V EC |) and obtain the contextual emotional representation r emo . During training, we use the ground truth response Y to guide the learning of knowledge discernment mechanisms. We feed Y into the cognition encoder (used for initializing the embeddings of G CS above) and the context encoder to get the hidden states S cog Y and S ctx Y , where the [CLS] representations are s ctx Y and s cog Y respectively. The posterior cognitive distribution P CS (cs i | Y ) and the emotional one P EC (ec i | Y ) are calculated as: P CS (cs i | Y ) = softmax i cs T i s cog Y ,(5)P EC (ec i | Y ) = softmax i ec T i s ctx Y .(6) We then optimize the KL divergence between the prior and posterior distributions during training: L KL = L CS KL + L EC KL ,(7)L CS KL = |K CS | i=1 P CS (cs i | Y ) · log P CS (cs i | Y ) P CS (cs i | X) , L EC KL = |V EC | i=1 P CS (ec i | Y ) · log P EC (ec i | Y ) P EC (ec i | X) . To further ensure the accuracy of discerned knowledge, similar to Bai et al. (2021), we employ the BOW loss to force the relevancy between cognitive / emotional knowledge and the target response. The BOW loss L BOW is defined as: L BOW = − 1 |B| yt∈B log η(y t | r cog , r emo ),(8) where η(·) is a MLP layer followed by softmax and the output dimension is the vocabulary size, B denotes the word bags of Y , r cog = |K CS | i=1 P CS (cs i | Y ) · cs i , and r emo = |V EC | i=1 P EC (ec i | Y ) · ec i . Finally, we align the coarse-grained representations of the contextual cognition r cog and the contextual emotional state r emo using mutual information maximization (MIM). Specifically, we adopt the binary cross-entropy (BCE) loss L coarse as the mutual information estimator that maximizes the mutual information between r cog and r emo : L coarse = 2f coarse (r cog , r emo ) − log remo exp(f coarse (r cog , r emo )) − log rcog exp(f coarse ( r cog , r emo )),(9) where r emo and r cog are the encoded negative samples. f coarse (·, ·) is a scoring function implemented with a bilinear layer activated by sigmoid function: f coarse (a, b) = σ a T W coarse b .(10) Fine-grained Alignment To simulate the interaction of fine-grained cognition and affection implied in the dialogue during human express empathy, the fine-grained alignment builds the finegrained association between each inferred specific cognition and corresponding emotional reaction. For each u i ∈ U , we infer the commonsense knowledge about emotional reaction K xReact i = k xReact i,1 , . . . , k xReact i,l using COMET, which is regarded as the user's possible emotional reaction to the current cognitive situation. Since k xReact i,j ∈ K xReact i is usually an emotion word (e.g., happy, sad), we concatenate K xReact i and feed it into the Transformer-based encoder (reaction encoder) to get the representation of the emotional reaction H er i . Similar to (Majumder et al., 2020) and (Sabour et al., 2022), we use the average pooling to represent the reaction sequence, i.e., h er i = Average (H er i ). To avoid over-alignment of outof-context emotional reaction with cognition, we inject contextual information into the representation of reaction. We first connect h er i with the context representation S X at the token level, i.e., S er i [j] = S X [j] ⊕ h er i . Then another Transformerbased encoder takes S er i as input and output the fused information S er i . We take the hidden representation of [CLS] in S er i as the emotional reaction representation er i of u i . Finally, we build the association between the inferred specific cognition { l j=1 cs r i,j } from u i for r ∈ R = {xIntent, xNeed, xWant, xEffect} and the emotional reaction er i using MIM. Recall that t i=0 r∈R l j=1 cs r i,j exactly correspond to the commonsense cognition knowlege set K CS . The fine-grained BCE Loss L f ine is defined as: L f ine = t i=0 r∈R l j=1 2f f ine cs r i,j , er i − log er i exp f f ine cs r i,j , er i − log cs r i,j exp f f ine cs r i,j , er i ,(11) where er i and cs r i,j are the encoded negative samples. f f ine (·, ·) is implemented as: f f ine (a, b) = σ a T W f ine b .(12) Altogether, the coarse-to-fine alignment module can be jointly optimized by L align loss: L align = L BOW +L KL +L coarse +αL f ine ,(13) where α is a hyper-parameter. Emotion Prediction We fuse the contextual emotional state and emotional reaction to distill the affective representation, where we use er 0 as the distillation signal of emotional reaction. This is because er 0 is derived from the speaker's last utterance and represents the overall emotional reaction. A gating mechanism is designed to capture affective representation r af f : r af f = µ · r emo + (1 − µ) · er 0 ,(14)µ = σ w T af f [r emo ; er 0 ] .(15) We project r af f to predict the user's emotion e: P emo (e) = softmax (W emo r af f ) , which is supervised by the ground truth emotion label e * using the cross-entropy loss: L emo = − log P emo (e * ) .(17) Empathy-aware Response Generation We employ a Transformer-based decoder to generate the response. To improve empathy perception in response generation, we devise two strategies to fuse the two parts of empathy (i.e., cognition and affection). First, we concatenate the cognitive and affective signals r cog and r af f with the dialogue context representation S X at the token level, which is then processed by a MLP layer activated by ReLU to integrate cognition and affection into the dialogue context: S X [i] = M LP (S X [i] ⊕ r cog ⊕ r af f ) .(18) Second, we modify the original Transformer decoder layer by adding two new cross-attention to integrate commonsense cognition knowledge K CS = {cs i } |K CS | i=1 and emotional concept knowl- edge K EC = {ec i } |V EC | i=1 , which are inserted between the self-attention and cross-attention for S X . The decoder then predicts the next token y m given the previously decoded tokens y <m , as done in the standard Transformer decoder. We use the negative log-likelihood loss L gen to optimize the decoder: L gen = − M m=1 log P (y m | X, G CS , G EC , y <m ) . (19) Finally, we jointly optimize the alignment loss, emotion prediction loss, generation loss, and diversity loss proposed by Sabour et al. (2022) as: L = γ 1 L align +γ 2 L emo +γ 3 L gen +γ 4 L div , where γ 1 , γ 2 , γ 3 and γ 4 are hyper-parameters. model that combines the output of multiple decoders for generating. (4) MIME (Majumder et al., 2020): An empathy dialogue model that mimics the user's emotion for responding. (5) EmpDG (Li et al., 2020): An empathy dialogue generator that utilizes multi-resolution user emotions and feedback. (6) KEMP : A knowledgeaware empathy dialogue model that only uses concept knowledge. (7) CEM (Sabour et al., 2022): A commonsense-aware empathetic chatting machine that only exploits commonsense knowledge. Implementation Details We implemented all models with Pytorch. We initialize the word embeddings with pretrained GloVE word vectors (Pennington et al., 2014). The dimensionality of embeddings is set to 300 for all corresponding modules. We set hyper-parameters l = 5, N = 10, α = 0.2, γ 1 = γ 2 = γ 3 = 1 and γ 4 = 1.5. We use Adam optimizer (Kingma and Ba, 2015) with β 1 = 0.9 and β 2 = 0.98. The batch size is 16 and early stopping is adopted. The initial learning rate is set to 0.0001 and we varied it during training following Vaswani et al. (2017). The maximum decoding step is set to 30 during inference. All models are trained on a GPU-P100 machine. The training process of CASE is split into two phases. We first minimize L BOW for pretraining knowledge discernment mechanisms, and then minimize L for training overall model. Automatic Evaluation In the model's generation evaluation, we adopt the widely used Perplexity (PPL) and Distinct-1/2 (Dist-1/2) (Li et al., 2016). Perplexity evaluates the general generation quality of a model. Distinct-1/2 evaluates the generated diversity by measuring the ratio of unique unigrams/bigrams in the response. In the model's emotion classification evaluation, we measure the accuracy (Acc) of emotion prediction. Following KEMP and CEM, we do not report word overlap-based automatic metrics (Liu et al., 2016), e.g., BLEU (Papineni et al., 2002). In Table 1, our model outperforms all baselines and achieves a significant improvement on all metrics. First, our model achieves about 4.0% reduction on PPL compared to the best baseline, which shows that CASE is more likely to generate ground truth responses. Second, our model achieves 15.6% and 41.2% improvement on Dist-1/2 compared to CEM, which indicates the superiority of CASE in generating informative responses at the unigrams and bigrams level. This is attributed to the coarse-to-fine alignment that allows CASE to inject more informative commonsense cognition on the premise of ensuring the perplexity of the generated response. Third, our model achieves about 17.9% and 7.8% improvement in prediction accuracy compared to KEMP and CEM, respectively. This verifies that CASE considers both aspects of affection (i.e., contextual emotional state and emotional reaction) more effectively than focusing only on a single aspect as KEMP and CEM. Overall-to-Part Ablation Study We conduct an overall-to-part ablation study in Table 2. In the overall ablation, first, we remove the commonsense cognition graph and emotional concept graph, called "w/o Graph". The emotion prediction accuracy decreases significantly, which indicates that the two heterogeneous graphs make remarkable contribution to detecting emotion. Second, we remove the coarse-to-fine alignment, called "w/o Align". The diversity of generation decreases significantly and emotion prediction accuracy drops distinctly. It supports our motivation that the alignment of cognition and affection leads to informative and highly empathetic expression. In the part ablation, first, we remove two graphs, called "w/o CSGraph" and "w/o ECGraph", respectively. From the results, we find that the contribution of the commonsense cognition graph is mainly to improve the diversity of generation (i.e., Dist-1/2), while the role of the emotional concept graph is mainly located in the recognition of emotion (i.e., Acc). This also supports our constructed motivation. Second, we remove coarse-grained and fine-grained alignments, called "w/o CGAlign" and "w/o FGAlign", respectively. We observe that the alignment at the fine-grained level is more significant than the coarse-grained level in terms of overall contribution. This also matches our intuition that building the fine-grained association between cognition and affection is closer to the conscious interaction process during human express empathy. Human Evaluation Human Evaluation of CASE and Baselines Here, 200 contexts are randomly sampled and each context is associated with two responses generated from our CASE and baseline. Following Sabour et al. (2022), three crowdsourcing workers are asked to choose the better one (Win) from two responses by considering three aspects, respectively, i.e., (1) Coherence (Coh.): which model's response is more fluent and context-related? (2) Empathy (Emp.): which model's response expresses a better understanding of the user's situation and feelings? (3) Informativeness (Inf.): which model's response incorporates more information related to the context? We use the Fleiss' kappa (κ) (Fleiss, 1971) to measure the inter-annotator agreement. As in Table 3, the results show that CASE outperforms three more competitive baselines on all three aspects. Especially, CASE outperforms baselines significantly in terms of empathy and informativeness, which shows the superior of modeling the interaction between cognition and affection of empathy, Human Evaluation on Variants of CASE To more intuitively verify the role of the key components of CASE in language expression, especially empathy ability, we conduct a scoring human evaluation for the variants of CASE. Besides the same settings as above, we require annotating the Overall preference score (1-5). As in Table 4, CASE achieves the highest scores in all aspects, indicating that all components contribute. The low empathy scores of "w/o Graph" and "w/o Align" as well as their variants further confirm the crucial role of graph structure and the effectiveness of alignment. Applicability Analysis To analyze the applicability of our method, we build it on the pre-trained model to explore whether it brings further benefits. We integrate Blender-Bot (Roller et al., 2021) into CASE by replacing the encoder and decoder, and take the vanilla Bart (Lewis et al., 2020) and BlenderBot as baselines. All pre-trained models are small versions. As in Table 5, we found that CASE-BlenderBot integrating our method significantly outperforms finetune-only baselines. Although the overall performance of simple finetuning has achieved stage success, it is limited by the quality and scale of the dataset and lacks a more fine-grained design for the trait of human conversation. This also demonstrates the highlevel applicability of our method for uncovering the underlying mechanisms of human conversation. Case Study Two cases from six models are selected in Table 6, among which CASE is more likely to express informative cognition in a highly empathetic tone. This is due to two main advantages: (1) Effective alignment between cognition and affection on two levels. For example, in the first case, on the fine-grained level, CASE associates the cognition "to be safe" with the affection "good" (i.e., emotional reaction) to appease the user's "Terrified" experience, i.e., "to stay safe" and "get a little better", in response. In the second case, on the coarse-grained level, in the user's "Embarrassed" emotional state, CASE expresses empathetic affection "it is not too bad" with an informative cognitive statement, i.e., "get it fixed", in response. (2) Accurate identification of the conversational emotion integrating emotional concepts and reactions, being consistent with "Acc". For instance, in the first case, the correct conversational emotion "Terrified" tends to be identified in the emotional concepts ("frighten, terrify, etc."), while in the second case, the one "Embarrassed" tends to be identified in the emotional reactions ("embarrassed, ashamed, etc."). Compared with baselines that cannot correctly perform two cases simultaneously, CASE identifies correct emotion in both cases by integrating emotional concepts and reactions. Conclusion and Future Work In this paper, for responding empathetically, we propose CASE to align cognition and affection by simulating their conscious interaction in human conversation. Extensive experiments verify the superiority of CASE on overall quality and empathy performance. Our work will also encourage future work to model the more complex interaction between cognition and affection in human conversation as well as other human language behaviors Zheng et al., 2023). Limitations We discuss two limitations of this work as follows: One limitation of our work is the lack of taskspecific automatic metrics to evaluate the empathy of generated responses. Therefore, the evaluation of empathy relies more on human evaluation. Although human evaluation is a golden standard, automatic metrics help to conduct large-scale investigations. This is also a common limitation in current works on empathetic dialogue. Emotion Terrified Context Speaker: As a kid, I was so frightened any time there was a tornado warning. Not sure if it was the wizard of oz effect or what i just remember being frozen with fear. Listener: Oh, wow! My son was terrified of tornadoes because of that darn movie, ...... Speaker: My mom said the best thing once when I was babysitting and a tornado warning went off and I did not want my own fear to affect the little boy I was with. She was on the phone and just said you have big plans for the future, right? Nothing is going to get in the way of that. For some reason, It worked a charm. MoEL I am sorry to hear that. I hope you can get better soon! MIME I am sure it was a great feeling. EmpDG I am glad you are ok! I am sure you will have a great time. KEMP I would be too! I would have been terrified! CEM I am sure you will be fine! Cognition To make a plan, To be safe, To be helpful, Have a good life Reaction Scared, Hopeful, Nervous, Good, Happy Concept frighten, surprising, terrify, aghast, good, charming, protect CASE That is good, I am glad you are able to stay safe and get a little better! Ground-Truth Your mom sounds awesome! Wow, what a cool thing to say! Emotion Embarrassed Context Speaker: I burned my hair with my hair drier I am so embarrased to go out in public. Listener: I am sorry to hear that, my wife has done the same thing. She wore a hat for almost a month when she left the house. Speaker: Yeah it is the worst, I look so weird with my hair like this it is so embarassing. MoEL That is a pretty funny story. MIME I am sure it was a good idea. EmpDG Oh, that is a shame. I hope you are not too! KEMP I am sure you will not have to be able to do it again. CEM I am sorry to hear that. Cognition To look bad, To get a haircut, To fix it, To fix the hair, To get a wig Reaction Embarrassed, Bad, Ashamed, Sad Concept bad, ill, evil, freak, supernatural, exhausted, hope, sorrow CASE I hope you are able to get it fixed, or just let her know it is not too bad. Ground-Truth Well, like I told her, it will grow out, it will just take time. Table 6: Generated responses from baselines and CASE. Commonsense cognition, emotional reactions, reasoned emotional concepts by contextual words, and corresponding informative expressions in responses are highlighted. The second limitation is the passive response to the user's cognition and affection. In many scenarios, empathy is used as a strategy for emotional support by responding to the user's cognition and affection. However, besides passive response, emotional support also requires active emotion elicitation, which can be studied in future work. Ethical Considerations In this paper, our experiments adopt the widely used EMPATHETICDIALOGUES benchmark, an open-source dataset collected from Amazon Mechanical Turk (MTurk) that does not contain personal information. We also ensure the anonymization of the human evaluation. We believe that this work honors the ethical code of ACL. Figure 2 : 2The architecture of the proposed CASE model. Dataset The experiments are conducted on the widely used EMPATHETICDIALOGUES (Rashkin et al., 2019) dataset, comprising 25k open domain conversations. In a conversation, the speaker confides personal experiences, and the listener infers the situation and emotion of the speaker and responds empathetically. Following Rashkin et al. (2019), we split the train/valid/test set by 8:1:1. Baselines (1) Transformer (Vaswani et al., 2017): A vanilla Transformer-based response generation model. (2) Multi-TRS (Rashkin et al., 2019): A multi-task Transformer model that jointly optimizes response generation and emotion prediction. (3) MoEL (Lin et al., 2019): An empathy dialogue Table 3 : 3Human evaluation results (%) of CASE and baselines. The agreement ratio kappa κ ∈ [0.41, 0.6] denotes the moderate agreement. †, ‡ represent signif- icant improvement with p-value < 0.1/0.05, respec- tively. Table 4 : 4Human evaluation results of CASE's variants.Models PPL Dist-1 Dist-2 Acc Bart 15.17 2.77 16.41 0.419 BlenderBot 15.22 2.70 16.20 0.470 CASE-BlenderBot 15.40 2.92 17.66 0.492 Table 5 : 5Analysis of integrating pre-trained model.and supports the observations from automatic eval- uation. AcknowledgementsThis work was also supported by the National Natural Science Foundation of China (with No. 62272340, 61876128, 62276187).A Mutual Information MaximizationMutual information maximization (MIM) aims to measure the dependence between two random variables X and Y , and the mutual information (MI) between them is defined as: M I(X, Y ) = D KL (P (X, Y ) P (X)P (Y )). However, maximizing MI directly is normally intractable. A successful practice to estimate MI with a lower bound is InfoNCE(Kong et al., 2020). Given two different views x and y of an input, InfoNCE is defined by:where f θ is a learnable function with parameter θ. The set Y draws samples from a proposal distribution Q( Y ), and it comprises | Y | − 1 negative samples and a positive sample y. 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{'abstract': "Empathetic conversation is psychologically supposed to be the result of conscious alignment and interaction between the cognition and affection of empathy. However, existing empathetic dialogue models usually consider only the affective aspect or treat cognition and affection in isolation, which limits the capability of empathetic response generation. In this work, we propose the CASE model for empathetic dialogue generation. It first builds upon a commonsense cognition graph and an emotional concept graph and then aligns the user's cognition and affection at both the coarsegrained and fine-grained levels. Through automatic and manual evaluation, we demonstrate that CASE outperforms state-of-the-art baselines of empathetic dialogues and can generate more empathetic and informative responses.", 'arxivid': '2208.08845', 'author': ['Jinfeng Zhou \nDCST\nInstitute for Artificial Intelligence\nThe CoAI Group\nState Key Lab of Intelligent Technology and Systems\n\n\nBeijing National Research Center for Information Science and Technology\nTsinghua University\n100084BeijingChina\n\nCollege of Intelligence and Computing\nTianjin University\nTianjinChina\n', 'Chujie Zheng \nDCST\nInstitute for Artificial Intelligence\nThe CoAI Group\nState Key Lab of Intelligent Technology and Systems\n\n\nBeijing National Research Center for Information Science and Technology\nTsinghua University\n100084BeijingChina\n', 'Bo Wang bo_wang@tju.edu.cn \nCollege of Intelligence and Computing\nTianjin University\nTianjinChina\n', 'Zheng Zhang \nDCST\nInstitute for Artificial Intelligence\nThe CoAI Group\nState Key Lab of Intelligent Technology and Systems\n\n\nBeijing National Research Center for Information Science and Technology\nTsinghua University\n100084BeijingChina\n', 'Minlie Huang aihuang@tsinghua.edu.cn \nDCST\nInstitute for Artificial Intelligence\nThe CoAI Group\nState Key Lab of Intelligent Technology and Systems\n\n\nBeijing National Research Center for Information Science and Technology\nTsinghua University\n100084BeijingChina\n'], 'authoraffiliation': ['DCST\nInstitute for Artificial Intelligence\nThe CoAI Group\nState Key Lab of Intelligent Technology and Systems\n', 'Beijing National Research Center for Information Science and Technology\nTsinghua University\n100084BeijingChina', 'College of Intelligence and Computing\nTianjin University\nTianjinChina', 'DCST\nInstitute for Artificial Intelligence\nThe CoAI Group\nState Key Lab of Intelligent Technology and Systems\n', 'Beijing National Research Center for Information Science and Technology\nTsinghua University\n100084BeijingChina', 'College of Intelligence and Computing\nTianjin University\nTianjinChina', 'DCST\nInstitute for Artificial Intelligence\nThe CoAI Group\nState Key Lab of Intelligent Technology and Systems\n', 'Beijing National Research Center for Information Science and Technology\nTsinghua University\n100084BeijingChina', 'DCST\nInstitute for Artificial Intelligence\nThe CoAI Group\nState Key Lab of Intelligent Technology and Systems\n', 'Beijing National Research Center for Information Science and Technology\nTsinghua University\n100084BeijingChina'], 'corpusid': 251643380, 'doi': '10.48550/arxiv.2208.08845', 'github_urls': [], 'n_tokens_mistral': 20590, 'n_tokens_neox': 17300, 'n_words': 10057, 'pdfsha': '75a106f87d0169d2c759eb1a158e0f55ffe85a31', 'pdfurls': ['https://export.arxiv.org/pdf/2208.08845v2.pdf'], 'title': ['CASE: Aligning Coarse-to-Fine Cognition and Affection for Empathetic Response Generation', 'CASE: Aligning Coarse-to-Fine Cognition and Affection for Empathetic Response Generation'], 'venue': []}

arxiv

Sequential Double Robustness in Right-Censored Longitudinal Models Alexander R Luedtke Vaccine and Infectious Disease Division Fred Hutchinson Cancer Research Center USA Public Health Sciences Division Fred Hutchinson Cancer Research Center USA Oleg Sofrygin Division of Biostatistics University of California BerkeleyUSA Mark J Van Der Laan Division of Biostatistics University of California BerkeleyUSA Marco Carone Vaccine and Infectious Disease Division Fred Hutchinson Cancer Research Center USA Department of Biostatistics University of Washington USA Sequential Double Robustness in Right-Censored Longitudinal Models double robustnessefficient estimationinverse probability weightinglongitu- dinal dataright-censoringsequential regressiontargeted minimum loss-based estimation Consider estimating the G-formula for the counterfactual mean outcome under a given treatment regime in a longitudinal study. Bang and Robins provided an estimator for this quantity that relies on a sequential regression formulation of this parameter. This approach is doubly robust in that it is consistent if either the outcome regressions or the treatment mechanisms are consistently estimated. We define a stronger notion of double robustness, termed sequential double robustness, for estimators of the longitudinal G-formula. The definition emerges naturally from a more general definition of sequential double robustness for the outcome regression estimators. An outcome regression estimator is sequentially doubly robust (SDR) if, at each subsequent time point, either the outcome regression or the treatment mechanism is consistently estimated. This form of robustness is exactly what one would anticipate is attainable by studying the remainder term of a first-order expansion of the G-formula parameter. We show that a particular implementation of an existing procedure is SDR. We also introduce a novel SDR estimator, whose development involves a novel translation of ideas used in targeted minimum loss-based estimation to the infinite-dimensional setting. A Rotnitzky, Q Lei, M Sued, and J Robins. Improved Double-Robust Estimation in missing data and causal inference models. Biometrika, 99(2):439-456, 2012. D Rubin and M J van der Laan. A doubly robust censoring unbiased transformation. Int J Biostat, 3(1): S Seaman and A Copas. Doubly robust generalized estimating equations for longitudinal data. Statistics in medicine, 28(6):937-955, 2009. O Sofrygin, M J van der Laan, and R Neugebauer. simcausal {R} Package: Conducting Transparent and Reproducible Simulation Studies of Causal Effect Estimation with Complex Longitudinal Data. Journal of Statistical Software, In Press. O Sofrygin, M J van der Laan, and R Neugebauer. {simcausal}: Simulating Longitudinal Data with Causal Inference Applications, 2015. URL http://cran.r-project.org/package=simcausal. O Sofrygin, M J van der Laan, and R Neugebauer. stremr: Streamlined Estimation of Survival for Static, Dynamic and Stochastic Treatment and Monitoring Regimes, 2016. URL https://github.com/osofr/ stremr. A A Tsiatis, M Davidian, and W Cao. Improved doubly robust estimation when data are monotonely coarsened, with application to longitudinal studies with dropout. Biometrics, 67(2):536-545, 2011. S van de Geer. Estimating a regression function. The Annals of Statistics, pages 907-924, 1990. M J van der Laan. Targeted estimation of nuisance parameters to obtain valid statistical inference. The international journal of biostatistics, 10(1):29-57, 2014. M J van der Laan and S Dudoit. Unified cross-validation methodology for selection among estimators and a general cross-validated adaptive epsilon-net estimator: finite sample oracle inequalities and examples. . M J van der Laan and S Gruber. Targeted minimum loss based estimation of causal effects of multiple time point interventions. Int J Biostat, 8(1):Article 9, 2012. M J van der Laan and S Gruber. One-Step Targeted Minimum Loss-based Estimation Based on Universal Least Favorable One-Dimensional Submodels. Introduction Consider a longitudinal study, where for each individual in the study we have observed timevarying covariates and treatment indicator, and we also observe an outcome at the end of the study. For simplicity, we suppose that the aim of the study is to estimate the G-formula that, under the consistency and sequential randomization assumptions (Robins, 1986;Pearl, 2009), is identified with the counterfactual mean outcome if each participant had received treatment at every time point. As is well known, estimating the end-of-study mean outcome in a study subject to dropout can, under standard assumptions, be equivalently described in this manner (we review this fact in Appendix B). Throughout we refer to the probability of receiving treatment at time k, conditional on the observed past, as the time k treatment mechanism and the G-formula identified with the mean outcome, conditional on the observed past before time k treatment, under receiving treatment at all treatments at or after time k as the time k outcome regression. The last several decades have seen extensive work on estimating parameters from our right-censored longitudinal data structure, including the G-formula. Reviews of the exist- an abbreviated version. Early methods include inverse probability weighted methods (e.g., Robins, 1993;Robins et al., 2000) and structural nested mean/G-estimation methods (e.g., Robins, 1989Robins, , 1994. These approaches are respectively consistent if the treatment mechanism or the outcome regressions correctly specified, and are asymptotically normal if one has correctly specified a parametric form for these components of the observed data distribution. More recently, there has been extensive development of so-called doubly robust (DR) methods, which are consistent if either the treatment mechanism at each time point or the outcome regression at each time point is consistently estimated. One can also establish asymptotic normality under some additional conditions. Robins (1999) introduced a sequential methodology for estimating this quantity, which represents an extension of the single time point methodology given in Scharfstein et al. (1999). Bang and Robins (2005) introduced a simplification of the approach that allows one to estimate sequential regressions conditional on the past history rather than the full distribution of the time-varying covariates. Van der Laan and Gruber (2012) extended this procedure to allow for data adaptive estimation, and Rotnitzky et al. (2012) extended it to allow for oracle-type model selection so that their estimator achieves the optimal efficiency for a (possibly multivariate) parameter within a prespecified class when the treatment mechanism is correctly specified by a parametric model. Tsiatis et al. (2011) also describe an approach to exploit a correctly specified parametric model for the treatment/missingness mechanism, though using estimating equations rather than a sequential regression procedure. Seaman and Copas (2009) describe an earlier DR generalized estimating equation methodology for longitudinal data structures. In this work, we describe a new form of robustness, which we term sequential double robustness. An estimator of the G-formula parameter is sequentially doubly robust (SDR) if it is consistent provided that, at each time point, either the outcome regression or the treatment mechanism is consistently estimated. This property is stronger than the traditional definition of double robustness, which either requires all of the outcome regression estimates or all of the treatment mechanism estimates to be consistent. We also define sequential double robustness for the outcome regression estimate (see Section 2.2). We show that an existing estimator achieves this property and present a new estimator that achieves this new form of double robustness and is guaranteed to respect known bounds on the outcome. Developing this latter estimator involves translating ideas from targeted minimum loss-based estimation (TMLE) to estimate non-pathwise differentiable infinite-dimensional parameters for which square root sample size convergence rates are not typically possible. We note that this extension is distinct from the recent work of van der Laan and Gruber (2016), which describes a TMLE for infinite-dimensional parameters for which each component is pathwise differentiable. We instead draw inspiration from the recent work of Kennedy et al. (2016), which gave an implementation of an infinite-dimensional targeting step in a continuous point treatment setting, where this step is implemented via locally linear regression. As is typical in our setting, no component of their infinite-dimensional parameter was pathwise differentiable. The paper is organized as follows. Section 2 outlines our objective, with the parameter(s) of interest defined in Section 2.1 and sequential double robustness in Section 2.2. A variationindependent, but more restrictive, formulation of sequential double robustness is presented in Appendix D. An analysis of the SDR properties of some existing data adaptive outcome regression estimators is given in Section 3. A general template for deriving an SDR procedure is given in Section 4. Our new SDR procedure, which represents what we term an infinitedimensional targeted minimum loss-based estimator (iTMLE), is described in Section 5. Formal properties of the empirical risk minimization (ERM) variant of our procedure are given in Section 6. In practice, ERMs will be prone to overfitting when used with our procedure. Hence, in Appendix E we present a variant of our procedure that relies on cross-validation. We recommend this variant be used in practice. Though more notationally burdensome, the proofs of the validity of the cross-validated procedure are nearly identical to the proofs of the validity of the ERM approach and so are omitted. Section 7 presents simulation results. Section 8 concludes with a discussion and directions for future work. Some of the proofs are given in the main text, and the others can be found in Appendix A. Appendix B demonstrates that general discretely right-censored data structures and timeto-event outcomes can be handled using our methodology. Objective Parameter(s) of interest Consider the longitudinal data structure that, at each time k = 1, . . . , K, consists of covariates L k and subsequent treatment (or right censoring indicator) A k . At time K + 1, the data structure contains a final outcome L K+1 . For each time k = 1, . . . , K + 1, letH k denote the observations made through the covariate observation at time point k. Thus,H 1 = L 1 and, for k = 1, . . . , K + 1,H k = (H k−1 , A k−1 , L k ). Note that our entire longitudinal data structure is contained inH K+1 . To focus ideas, we assume throughout that the outcome L K+1 is bounded in the unit interval. We will make repeated use of the logit link function Ψ −1 (x) = log x 1−x and the expit function Ψ(x) = 1/(1 + e −x ). Suppose we observe an i.i.d. sampleH K+1 [1], . . . ,H K+1 [n] drawn from a distribution P belonging to some model M. Throughout we will refer to an arbitrary element in M by P . We will use E to denote expectations under P , and E to denote expectations under P . For each k ∈ {1, . . . , K}, we define the treatment mechanism pointwise as π k (h k ) ≡ P (A k = 1|h k ), where here and throughout writing a conditioning statement on a lower case variable is equivalent to writing a conditioning statement on the upper case random variable equal to the lower case realization, e.g. here we are conditioning on the eventH k =h k . For ease of notation, we denote π k applied to observation i as π k [i], i.e. π k [i] ≡ π k (H k [i]). Similarly, for an estimateπ k of π k we letπ k [i] ≡π k (H k [i]) . Throughout we assume the strong positivity assumption that there exists a δ > 0 so that P {π k (H k ) > δ} = 1 for all k. For a history vectorh K+1 , define Q K+1 h K+1 ≡ K+1 . For k = K, . . . , 1, recursively define Q k h k ≡ E Q k+1 H k+1 h k , A k = 1 , and define Q 0 ≡ E[Q 1 (H 1 )]. For ease of notation, we write Q k [i] ≡ Q k H k [i] . For an estimateQ k , we similarly writeQ k [i]. For k ≥ 1, our objective will be to estimate Q k as well as possible in terms of some user-specified criterion. We focus on the mean-squared error criterion in this work. For k = 0, our objective will be to obtain a consistent estimate of Q k for which there exist reasonable conditions for its asymptotically linearity. We use many recursions over k = K, . . . , 0 in this work. When k = 0 this requires some conventions: 0 k =1 . . . = 0 (sums are zero); 0 k =1 . . . = 1 (products are one);H 0 = ∅ (the time 0 covariate is empty); A 0 = 1 and π k (h 0 ) = 1 (the time 0 intervention is always 1); and, for f :H 0 → R, f (H 0 ) = f (functions applied to the empty set can also be written as constants). We ignore measurability considerations in this work, with the understanding that modifications may be needed to make our arguments precise. (Sequential) double robustness Throughout we will make use of the following non-technical conditions, defined for each k = 0, . . . , K. OR.k) The functional form of the outcome regression at time k, i.e. Q k , is correctly specified by the estimation procedure, or at least arbitrarily well approximated asymptotically. TM.k) The treatment mechanism at time point k, i.e. π k , is consistently estimated. We use these conditions informally to discuss properties of existing estimator and our new estimator until Section 5, where we begin to present formal conditions for the validity of our estimator. Not that TM.k requires consistent estimation, and OR.k requires correct specification. This discrepancy occurs because we will use OR.k as a part of a sufficient condition for consistent estimation of Q k : to directly impose consistency of Q k would yield a tautology. Consider estimation of Q 0 . The current literature appears to define double robustness as follows (van der Laan and Robins, 2003;Bang and Robins, 2005;Tsiatis et al., 2011;Rotnitzky et al., 2012): Definition 1 (Double robustness for Q 0 ). A doubly robust estimator of Q 0 is an estimator that is consistent if either (i) OR.k holds for all k = 1, . . . , K or (ii) TM.k holds for all k = 1, . . . , K. These estimators are referred to as DR because there are two possibilities for obtaining consistent estimation. In this work, we define a more general form of robustness, which we refer to as sequential double robustness. Definition 2 (Sequential double robustness for Q 0 ). A sequentially doubly robust estimator of Q 0 is an estimator that is consistent if, for each k = 1, . . . , K, either OR.k or TM.k holds. Clearly a sequentially doubly robust estimator is double robust, but the converse need not hold. In fact, there are 2 K ways that the estimation can satisfy one and only one of OR.k and TM.k, k = 1, . . . , K, though of course certain of these possibilities may be less likely than others (see Appendix C for a comparison with a 2 K -robust estimator given by Vansteelandt et al. in a different longitudinal setting). Appendix D gives a variation-independent, but more restrictive, formulation of the OR.k conditions. We also define sequential double robustness for the estimation of Q k for general k. Definition 3 (Sequential double robustness for general Q k ). A sequentially doubly robust estimator of Q k is an estimator that is consistent if OR.k holds and, for each k > k, either OR.k or TM.k holds. Because OR.k is a triviality when k = 0 (the functional form is correctly specified by a constant), the definition of sequential double robustness given specifically for Q 0 is a special case of this definition, though having Definition 2 readily available to contrast against Definition 1 is useful. The objective of this work will be to present sequentially doubly robust estimators of Q k , k = 0, . . . , K. Our estimator of Q 0 will also be efficient among all regular and asymptotically linear estimators under some additional conditions. Detailed Overview of Existing Procedures for Estimating the Outcome Regressions In the introduction, we gave a broad overview of the existing literature for estimating mean outcomes from monotonely coarsened data structures. We now give a deeper review of the existing literature, where here we only focus on methods that allow semi-or nonparametric estimation of the outcome regressions. We note that one could also study the parametric methods from the introduction, incorporating basis function transformations of the covariates of increasing dimension to allow for increasingly flexible estimation of the outcome regressions. We do not consider such approaches here, though we note that, to our knowledge, none of the approaches in the introduction have been shown to be SDR even in the parametric case. First, we describe a method that uses DR unbiased transformations of the data, i.e. distribution dependent pseudo-outcomes with conditional expectation equal to the parameter of interest given correct specification of an outcome regression or treatment mechanism. Variants of these unbiased transformations were given in Rubin and van der Laan (2007), which represent a DR extension of the unbiased transformations presented earlier in the literature (see, e.g., Buckley and James, 1979;Koul et al., 1981;Fan and Gijbels, 1996). We describe an SDR implementation of this unbiased transform estimator, and we also discuss its shortcomings. We then describe inverse probability weighted (IPW) loss functions as presented in van der Laan and Dudoit (2003). We also briefly discuss the augmented IPW (AIPW) version of this loss function (van der Laan and Dudoit, 2003;Keles et al., 2004) and its double robustness property. Finally, we discuss sequential regression procedures in the vein of Bang and Robins (2005). Doubly robust unbiased transformations. An unbiased transform for Q k is a distribution-dependent mapping Γ P k :H K+1 → R such that Γ P k (H K+1 ) has mean Q k (H k ) whenH K+1 is drawn from the conditional distribution of P given that (H k , A k ) = (h k , 1). Early work on these transformations used imputationbased approaches (e.g., Buckley and James, 1979), whose consistency relies on consistently estimating Q k , k > k, or on IPW approaches (e.g., Koul et al., 1981), whose consistency relies on consistently estimating π k , k > k. Rubin and van der Laan (2007) presented an AIPW unbiased transformation. For our problem, one could estimate Q k using the DR transform Γ k [i] ≡ K k =k+1 k k =k+1 A k [i] π k [i] {Q k +1 [i] − Q k [i]} + Q k+1 [i], regressing Γ k [i] againstH k [i] for all subjects i with A k [i] = 1. In practice, the transformation Γ k [i] ≡ K k =k+1 k k =k+1 A k [i] π k [i] Q k +1 [i] −Q k [i] +Q k+1 [i], is used, where each instance of Q k and π k , k > k, has been replaced by an estimate. ifQ K is estimated via this procedure, thenQ K−1 using as initial estimateQ K , thenQ K−2 using as initial estimatesQ K−1 andQ K , etc., then this procedure is SDR in the sense of Definition 3. To our knowledge, the sequential double robustness property of this estimator has not been recognized in the literature. While one could theoretically constrain the estimated regression function to respect the [0, 1] bounds (and use, e.g., a cross-entropy loss), few existing regression software packages allow for such constraints on the model. The stability of such a procedure has also not been evaluated when there are near-positivity violations, i.e. K k=1π k (H k ) is near zero with non-negligible positivity overH k ∼ P . We evaluate this procedure in Section 7. Algorithm 1 SDR estimation of Q k via doubly robust transformations (particular implementation of Rubin and van der Laan, 2007) This function runs regressions using covariateH k for each k, where the regression algorithms may be k-dependent. 1: procedure SDR.Unbiased.Trans 2: LetQ K+1 (H K+1 ) ≡ L K+1 . 3: Obtain estimatesπ k , k = 1, . . . , K, via any desired technique. 4: for k = K, . . . , 0 do 5: Using observations i with A k [i] = 1, regress Γ k [i] againstH k [i], where Γ k [i] ≡ K k =k+1 k k =k+1 A k [i] π k [i] Q k +1 [i] −Q k [i] +Q k+1 [i]. Label the outputQ k . An alternative approach uses the IPW loss function of van der Laan and Dudoit (2003). Before describing this approach, we introduce the notion of a loss function. Suppose that one wishes to estimate a feature f 0 of a distribution ν. For example, ν may be the distribution of a predictor-outcome pair Z ≡ (X, Y ), and f 0 may be the conditional expectation function x → E ν [Y |x].z; f ) → −[y log f (x) + (1 − y) log{1 − f (x)}]. We now return the presentation of the IPW loss. For simplicity, we focus on the squarederror loss. The IPW loss is given by L k (h K+1 ; Q k ) ≡ K k =k+1 a k π k (h k ) { K+1 − Q k (h k )} 2 . The standard change of measure argument associated with IPW estimators shows that weight feature that is standard in most regression algorithms. In practice we may not know π k , so we replace them with estimates and denote the corresponding loss by L k . It is well known that are also efficiency gains that result from estimating π k even when the treatment mechanism is known (van der Laan and Robins, 2003). Under a Glivenko-Cantelli condition and a consistency condition on eachπ k , k > k, the empirical mean of the above L k is consistent for Risk(Q k ) for each Q k . This proves the consistency of an empirical risk minimizer Q k provided the class over whichQ k is selected is not too large (satisfies a Glivenko-Cantelli condition). To conclude, roughly speaking, this procedure yields a consistent estimate of Q k if eachπ k is consistently estimated and the regression is correctly specified (or approximates Q k arbitrarily well asymptotically). Q k = argmin Q k E L k (H K+1 ; Q k ) A k = 1 , There also exists an AIPW version of this loss function (van der Laan and Dudoit, 2003; Keles et al., 2004). Implementing this loss function involves treating the Q k -specific loss function applied to the dataH K+1 , i.e. L F (h K+1 ; Q k ) ≡ { K+1 − Q k (h k )} 2 , as the outcome and estimating the iterative conditional expectations as defined by the G-formula that sets A k = 1 for all k ≥ k. Roughly speaking, this estimation procedure is DR in the sense defined in this paper, though we note that its double robustness is now in terms of estimation of the iterative expectations of L F (H K+1 ; Q k ) rather than of the outcome regressions as in OR.k. However, it does not appear to satisfy an analogue of the sequential double robustness property unless the estimation of the G-formula for the L F (H K+1 ; Q k ) outcome regressions is estimated in an SDR fashion. If one estimates the outcome regressions in the G-formula for outcome L F (H K+1 ; Q k ) using AIPW loss functions, then clearly the argument is circular so that this procedure is not SDR unless yet another set of outcomes is estimated in an SDR fashion. Alternatively, one could estimate the L F (H K+1 ; Q k ) outcome regressions using the unbiased transformation of L F (H K+1 ; Q k ). One disadvantage of using the AIPW loss is that it does not appear to be easily implementable with existing software. Sequential regression. Bang and Robins (2005) proposed a procedure that takes advantage of the recursive definition of these Q k functions. They aimed to estimate Q 0 . An instance of their procedure for binary outcomes is displayed in Algorithm 2. In short, it first correctly specifiesQ K+1 (H K+1 ) ≡ L K+1 . Now, iteratively from k = K to k = 0, it uses observations i satisfying k k =1 A k [i] = 1 to regressQ k+1 [i] againstH k [i] , using a parametric fit and the logit link function. Each parametric fit includes a linear term with covariate 1 k k =1πk [i] . These linear terms were added to ensure thatQ 0 solves the efficient estimating equation, i.e. that n i=1 K k=0 k k =1 A k [i] π k [i] Q k+1 [i] −Q k [i] = 0. In particular, each k-specific term in the above sum is equal to zero thanks to the fitting of k . Van der Laan and Gruber (2012) extended this procedure to allowQ k to be estimated data adaptively. They refer to their estimator as a longitudinal targeted minimum lossbased estimator (LTMLE). There are several implementations of this procedure: we choose the implementation that most closely resembles the new procedure we will introduce later in this work. The LTMLE splits the estimation of each Q k into two steps. First, one obtains an initial estimateQ init k of Q k using a data adaptive estimation procedure by re- gressingQ k+1 (H k+1 [i]) againstH k [i], using all observations i with A k [i] = 1. Next, one fits the interceptˆ k in an intercept-only logistic regression using outcomeQ k+1 (H k+1 [i]), offset Ψ −1 [Q init k (H k [i])], and observation weights k k =1 A k [i] π k [i] . The estimateQ k is then given bȳ h k → Ψ Ψ −1 [Q init k (h k )] +ˆ k . Van der Laan and Gruber (2012) refer to the fitting ofˆ k as "targeting" the initial estimate of Q k towards Q 0 . While both the Bang and Robins (2005) and van der Laan and Gruber (2012) procedures are DR forQ 0 , neither is SDR forQ 0 . Furthermore, neither is SDR for Q k , k ≥ 1, in the sense of Definition 3. In particular, when k ≥ 1 consistent estimation of Q k from these procedures generally relies on OR.k , k ≥ k. Though not discussed in the literature, these procedures can be consistent when only some outcome regressions and some treatment mechanisms are consistently estimated. In particular if there exists some k such that OR.k holds for all k ≥ k and TM.k holds for all k < k, then these estimators will be consistent for Q 0 . If the order in which the outcome regressions and treatment mechanisms are consistently estimated is flipped (OR.k , k < k, and TM.k , k ≥ k), then these estimators will not yield a consistent estimate of Q 0 . A General Template for Achieving Sequential Double Robustness We now give a general template for achieving sequential double robustness. This template hinges on a straightforward induction argument, where we show that achieving an SDR estimator at time k + 1 yields an SDR estimator at time k. In this section, we letQ k andπ k Algorithm 2 DR estimation of Q 0 (variant of Bang and Robins, 2005) For each k ≥ 1, s β k k :H k → R is a parametric model for Q k indexed by β k ∈ R d k , where d k is finite. Let β 0 = ∅, and s β 0 0 ≡ 1/2. 1: procedure DR.Q 2: LetQ K+1 (H K+1 ) ≡ L K+1 . 3: Obtain estimatesπ k , k = 1, . . . , K, via any desired technique. 4: for k = K, . . . , 0 do 5: For (β k , k ) ∈ R d k +1 , defineQ β k , k k as h k → Ψ s β k k (h k ) + k k k =1π k (h k ) . 6: Define (β k ,ˆ k ) as the arguments (β k , k ) minimizing − i: k k =1 A k [i]=1 Q k+1 [i] logQ β k , k k [i] + {1 −Q k+1 [i]} log{1 −Q β k , k k [i]} . If s β k k (h k ) = R(h k ), β k for some transformation R, then the above can be optimized in the programming language R by running a linear-logistic regression of the [0, 1]-boundedQ k+1 [i] against R(H k [i]) and 1/ k k =1π k (h k ) among all individuals with k k =1 A k [i] = 1. 7: LetQ k ≡Qβ k ,ˆ k k . 8: return {Q k : k = 0, . . . , K} respectively denote generic estimates of the outcome regression and treatment mechanism at time k. We will often make use of the following strong positivity assumption on our treatment mechanism estimates. Though we introduced this condition earlier, we name it here for clarity. This condition can be enforced in the estimation procedure via truncation. SP.k) There exists a δ > 0 such that, for each k > k, P {π k (H k ) > δ} = 0. To ease notation, when Q k or π k , or an estimate thereof, fall within an expectation, we often omit the dependence onH k . For a real-valued functionh k → f (h k ), we denote the L 2 (P ) norm by f = E[f (H k ) 2 ] 1/2 . We define the following useful objects: D k k (Q k +1 ,Q k )(h k ) ≡ − E A k π k k k =k+1 A k π k Q k +1 −Q k h k , k ≥ k Rem k k (Q k )(h k ) ≡ E A k π k k −1 k =k+1 A k π k 1 − π k π k Q k − Q k h k , k > k.(1) Note that, for each (Q k +1 ,Q k ), D k k (Q k +1 ,Q k ) is a function mapping ah k to the real line, and similarly for Rem k k (Q k ). In the remainder of this section we study a particular estimator (Q k +1 ,Q k ), and so use the simpler notation D k k ≡ D k k (Q k +1 ,Q k ) and Rem k k ≡ Rem k k (Q k ). The more general definitions will be useful in the coming sections. We also define D k (h k ) ≡ K k =k D k k (h k ). The following first-order expansion will prove useful. Lemma 1 (First-order expansion of Q k ). If SP.k holds, then, for P -almost allh k ∈H k , Q k (h k ) − Q k (h k ) = D k (h k ) + K k =k Rem k k (h k ).(2) The proof of Lemma 1 is given in Appendix A. By the triangle inequality, Cauchy-Schwarz, and the positivity assumption on theπ k estimates, the above implies that Q k − Q k ≤ D k + K k =k+1 Rem k k ≤ D k + K k =k+1 O π k − π k Q k − Q k .(3) A simple induction argument shows that Q k − Q k ≤ D k + K k =k+1 O π k − π k D k .(4) The above teaches us how to obtain an SDR-type estimator. We say SDR-type because we replace the conditions OR.k, OR.k by FO.k, FO.k ("correct first-order behavior at time k, k "), where FO.k) D k converges to zero in probability. The following result is an immediate consequence of (4). Theorem 2 (Achieving an SDR-Type Estimator). Fix k and suppose that SP.k holds with probability approaching one. If FO.k and, at each time k > k, either TM.k or FO.k , then Q k → Q k , i.e. Q k − Q k = o P (1). In the remainder, we do not make any distinctions between SDR and SDR-type estimators. Like OR.k, FO.k requires correctly specifying a regression of dimensionH k . Depending on how a procedure makes D k small, this regression may only require consistent specification of the functional form of Q k , e.g. that Q k belongs to some (possibly infinite-dimensional) smooth class. In Remark 1, we sketch an argument showing that FO.k and OR.k are equivalent for the (S)DR unbiased transformation approach. This will not hold for our upcoming TMLE extension, because that extension runs separate regressions to minimize each D k k , k ≥ k. In practice this may not matter if a these regressions are sufficiently flexible. Remark 1. We now connect the (S)DR unbiased transformation approach to the first-order expansion (2) given in this section. We have the following algebraic identity for P almost allh k : E Γ k A k = 1,h k −Q k (h k ) = −D k (h k ). Thus, E Γ k A k = 1,H k = · −Q k (·) = o P (1) if and only if D k = o P (1). Note that the objective of a regression of Γ k [i] againstH k [i] among all individuals with A k [i] = 1 is to ensure E Γ k A k = 1,H k = · ≈Q k (·), where this approximation can be made precise by using "≈" to mean closeness in L 2 (P ) norm. One could alternatively ensure closeness with respect to a different criterion by choosing a loss function other than the squared-error loss. Often, though not always, closeness in one loss implies closeness in another loss (see Theorem 3 for a connection between the squared error and cross-entropy losses in our setting). 2 Remark 2. In (3), we applied Cauchy-Schwarz to show that each Rem k k , k > k, is big-oh of π k − π k Q k − Q k . One can also obtain the bound Rem k k E E Q k − Q k 2 H k E (π k − π k ) 2 H k 1/2 . The left-hand side will converge if, for eachh k , either Q k or π k is consistently estimated across allh k that have time-k history equal toh k . This is weaker than requiring that either π k or Q k is consistently estimated, since we only require that the union ofh k values on which each of these quantities are consistently estimated is equal to the support ofH k ∼ P . We will not use this bound in the remainder. 2 Remark 3. Let f ∞ denote the P essential supremum norm and · 1 denote the L 1 (P ) norm. In this case, we have that K k =k+1 Rem k k ∞ π k − π k 1 Q k k − Q k ∞ . The above seems likely to be useful for constructing confidence bands for Q k . In particular, the above suggests that, under some conditions, Q k − Q k ≈ D k . Therefore, it generally suffices to develop a confidence band for D k , which is a regression with the dimension of H k . If one uses the (S)DR unbiased transformation approach from Remark 1 and implements the time k regression using a kernel regression procedure, then one should be able to study a kernel-weighted empirical process to develop confidence bands. This will presumably give meaningful confidence bands when the dimension ofH k is not too large, e.g. when k = 1 and there is a single or a small handful of baseline covariates of interest. We will examine this in detail in future works. 2 D k, * k and Rem k, * k are functions mapping from the support ofH k ∼ P to the real line, so that it makes sense to take L 2 (P ) norms of these objects to quantify their magnitude. Algorithm 3 iTMLE for Q k This function runs regressions using covariateH k for each k ≥ k 0 . These regressions may be k-dependent. In particular, they should be intercept-only logistic regressions if k = 0. 1: procedure SDR.Q(k 0 ) 2: Obtain estimatesπ k , k = k 0 + 1, . . . , K, via any desired technique. 3: LetQ * K+1 ≡ L K+1 4: for k = K, . . . , k 0 do Estimate Q k 5: InitializeQ k +1 k ≡ 1/2 6: for k = k , k − 1, . . . , k 0 do Target estimate of Q k towards Q k . 7: For each function k k :H k → R, defineQ k+1, k k k as h k → Ψ Ψ −1 [Q k+1 k (h k )] + k k (h k ) . 8: Using all observations i = 1, . . . , n, fitˆ k k by running a regression using the cross-entropy loss and the logit link function with: • Outcome:Q * ¯ k k : H k → R satisfying E k k =k+1 A k π k Q k+1,¯ k k k (H k ) h k , A k = 1 = E k k =k+1 A k π k Q * k +1 h k , A k = 1 ,(5) where we note that a sufficient condition for this¯ k k to exist is that P {Q * k +1 (H k +1 ) = 0} < 1 and P {Q * k +1 (H k +1 ) = 1} < 1, i.e. thatQ * k +1 is not degenerate at zero or one. Define the conditional excess risk by E k k (h k ) ≡ E L k k (H k +1 ;ˆ k k ) − L k k (H k +1 ;¯ k k ) h k . The excess risk is defined as the average conditional excess risk, i.e. E[E k k (H k )]. The upcoming lemma bounds the term D k, * k from the upper bound in Lemma 1 by the excess risk of the procedure for estimatingˆ k k plus the deviation between the estimate of Q k targeted towards estimating Q k , k ≥ k, and the estimate of Q k that is targeted towards estimating Q k , k ≥ k 0 . By the triangle inequality, controlling D k, * k for all k ≥ k suffices to control k ≥k D k, * k and, by Lemma 1, plays an important role in controlling Q * k − Q k . Theorem 3 (Upper bounding D k, * k by an excess risk). Fix k , k satisfying k ≥ k ≥ k 0 . If P {Q * k +1 = 0} < 1, P {Q * k +1 = 1} < 1, and SP.k 0 holds, then, with probability one over drawsH k ∼ P , D k k (Q * k +1 ,Q k k )(H k ) 2 E k k (H k ). Furthermore, D k, * k E E k k (H k ) 1/2 + Q * k −Q k k . The proof of Theorem 3 is given in Appendix A. The above shows that D k, * k converges to zero in probability if the excess risk ofˆ k k for¯ k k converges to zero in probability and all targeting steps of the estimate of Q k that occur after the estimate is successfully targeted towards Q k (small excess risk) have little effect on the estimate. We will formally show that empirical risk minimizers satisfy this latter condition. Note also that, for k = k 0 , i.e. for the final targeting step for the estimate of each Q k ,Q * k = Q k 0 k by definition so that the latter term above is zero. Thus, D k 0 , * k 2 converges to zero at least as quickly as does the excess risk E E k 0 k (H k 0 ) . Explicit guarantees for empirical risk minimizers To establish concrete results about the iTMLE, it is easiest to analyze one particular class of estimators. Here we focus on estimators derived from empirical risk minimization (ERM; we also use ERM to denote "empirical risk minimizer") (see, e.g., van de Geer, 1990;Vapnik, 1991;Bartlett and Mendelson, 2006). In our simulation, we run our simulation with several regression approaches to demonstrate its greater generality. We first give a brief review of ERM. Again suppose Z ≡ (X, Y ) ∼ ν and that f 0 minimizes the risk corresponding to some loss L . An ERM attempts to estimate f 0 by lettingf = argmin f ∈F ν n L (·; f ), where ν n L (·; f ) is the empirical mean of L (Z; f ) from an i.i.d. sample of size n drawn from ν, and F is some user-specified index set. While in practice this index set may depend on sample size, in this work we focus our analysis on a sample-size-independent F. One could alternatively study a sieved estimator for which F grows with sample size. For ease of notation, we assume that, when estimating each Q k , the same class F k is used to estimate both k k , k > k, and also to estimate Q k . We assume that F k contains the trivial function mapping eachh k to zero. It is hard to imagine a useful class F k that would violate this condition. We assume that eachˆ k k is obtained via ERM, so that k k ∈ argmin k k ∈F k P n L k k (·; k k ). For each k, use the following correct specification assumptions in this section. CS.k) F k contains the data-dependent functions¯ k k defined in (5), k ≥ k, with probability approaching one. Remark 4 (Alternative to CS). If eachQ * k , k > k, has a (possibly misspecified) limit, then one can replace the condition that each¯ k k , which relies on the sample-dependent estimatê Q * k , falls in F k with the condition that the limit of¯ k k , k ≥ k, falls in F k . This is useful because, whenQ * k , k > k, is consistent, the limit of¯ k k is the constant function zero. Hence, for these k our assumption that F k contain this trivial function suffices. For k = k, replacing the above assumption by the assumption that the limit of¯ k k ∈ F k is like assuming OR.k provided at least one of one ofQ * k+1 orπ k+1 is consistent. 2 For ease of analysis, we also rely on the following assumption on each F k . BD) For each k ≥ k 0 , the elements in F k are uniformly bounded in some interval [−c, c], c < ∞. Remark 5 (Conditions on outcome regressions can be weakened). Combining the above with CS.k) shows that we are assuming that Q k is bounded away from zero or one. More precisely, we are assuming that E[Q * k+1 |H k = ·, A k = 1] is bounded away from zero or one, where, under our assumptions, this quantity is consistent for Q k (·). This could be weakened to a moment condition on elements of F k . One could also extend our analysis to timeto-event data (see Appendix B) in which an observed event in the past implies that the regression function is strictly equal to one at all subsequent time points. 2 Each result also uses an empirical process condition to ensure that the class F k is not too large, and also that the estimates of the propensity scores are well-behaved with probability approaching one (van der Vaart and Wellner, 1996). DC) For all k ≥ k 0 , F k is a Donsker class andπ k belongs to a fixed Donsker class D k with probability approaching one. Remark 6. Under the weaker condition that F k is a Glivenko-Cantelli class andπ k belongs to a fixed Glivenko-Cantelli class D k with probability approaching one, one can obtain the same results as those that we will present in this section, with the only change being that each o P (n −1/4 ) term is replaced by an o P (1) term. 2 We have the following result. Theorem 4 (ERMs achieve SDR estimation of Q k ). If k ≥ k 0 is such that CS.k, SP.k holds with probability approaching one, BD, and DC, then, with probability approaching one, Q * k − Q k k ≥k+1 Q * k − Q k π k − π k + k ≥k Q * k −Q k k + E[E k k (H k )] 1/2 . Furthermore, Q * k − Q k k ≥k+1 Q * k − Q k π k − π k + o P (n −1/4 ). Proof of Theorem 4. We start by using the bound in Lemma 1. Lemma A.1 in Appendix A.1 shows that each E[E k k (H k )] = o P (n −1/2 ). Cauchy-Schwarz and the fact that SP.k holds with probability approaching one show that each Rem k k is upper bounded by a constant times Q * k − Q k π k − π k with probability approaching one. The upcoming Theorem 5 shows that CS.k and the other conditions of this theorem imply that targeting the estimates of each Q k has little effect on the estimates, i.e. Q * k −Q k k = o P (n −1/4 ). We now make explicit the sense in which the above establishes the SDR property of our estimator. Suppose that, for each k > k 0 , at least one of CS.k and TM.k, i.e. π k − π k = o P (1), holds. A straightforward induction argument with inductive hypothesis "CS.k implies Q * k − Q k = o P (1)" from k = K, . . . , k 0 then shows that Q * k 0 − Q k 0 = o P (1). Thus, our approach is SDR once we replace OR.k by the related, but somewhat more technical, condition CS.k. If each π k − π k is o P (n −1/4 ), which is achievable ifπ k is an ERM from a correctly specified Donsker class, then the same induction argument holds, but now with the o P (1) replaced by o P (n −1/4 ) so that Q * k 0 − Q k 0 = o P (n −1/4 ). We now present Theorem 5, which we used in the proof of Theorem 4 to show that, for k ≥ k ≥ k 0 , targeting the estimate of Q k towards all k = k − 1, k − 2, . . . , k 0 has little effect if on the estimate if CS.k holds. Theorem 5 (Conditions under which the targeting step has little effect). Fix a k ≥ k 0 for which CS.k holds and let k ≥ k. If SP.k 0 holds with probability approaching one, BD, and DC, then Q * k −Q k k = o P (n −1/4 ). The proof of Theorem 5, which relies on an induction argument, is given in Appendix A. Remark 7 (Improved rate quantification). Note that, under entropy integral bounds on the Donsker classes in DC, one could tighten our analysis to improve our understanding of the o P (n −1/4 ) rates above using local properties of the empirical process. For example, one could use local maximal inequalities for bracketing entropy (Lemma 3.4.2 in van der Vaart and Wellner, 1996) or for uniform entropy (van der Vaart and Wellner, 2011). Alternatively, one could work with local Rademacher complexities (Bartlett et al., 2005). We do not focus on these refinements in this work. 2 Remark 8 (Conjectured rate optimality when DR terms small). Suppose that CS.k and TM.k hold at every time point k > k 0 and that CS.k 0 holds. Further suppose that π k − π k = o P (n −1/4 ) for each k > k 0 , which will be the case if each π k is estimated using an ERM over a correctly specified Donsker class. In this case a simple induction argument shows that Q * k 0 − Q k 0 k≥k 0 E[E k k (H k )] 1/2 + o P (n −1/2 ) ≤ k≥k 0 (P n − P ) L k 0 k (· ;¯ k 0 k ) − L k 0 k (· ;ˆ k 0 k ) 1/2 + o P (n −1/2 ). The first inequality holds whether or not CS.k 0 is true, but the second uses CS.k 0 . In particular, the second inequality holds because eachˆ k 0 k is an ERM over F k 0 and each¯ k 0 k ∈ F k 0 by CS.k 0 . Even in a correctly specified parametric model, the leading term is only O P (n −1/2 ), so the leading sum above is always expected to dominate. As the rate of convergence of the empirical processes can be controlled by the size of the class F k 0 , we see that Q * k 0 − Q k 0 converges to zero at the rate dictated by size of the class F k 0 , where the size of the class can, for example, be quantified using metric entropy (van der Vaart and Wellner, 1996). Compare this to earlier sequential regression procedures, whose rate of convergence is typically dominated by the size of the largest F k , k ≥ k 0 . AsH k 0 is necessarily lower dimensional thanH k , k > k 0 , we would typically expect that F k 0 has a smaller entropy integral than F k . Hence, we expect that traditional sequential regression procedures have rate dominated by the size of F K . It seems likely that this fact enables the construction of confidence sets for Q k 0 . We will examine this further in future works. k (h k+1 ) ≡ k k =1 a k π k (h k ) Q * k+1 (h k+1 ) −Q 1, 0 k k (h k ) , whereQ 1, 0 k k =Q k 0 +1, k 0 k k is defined in Algorithm 3. In particular, the fact that, for each k, {L 0 k (·, 0 k ) : 0 k ∈ F 0 = R} is a parametric class ensures that 0 = ∂ ∂ 0 k P n L 0 k (·, 0 k ) 0 k =ˆ 0 k = P n IFˆ 0 k k . Noting that D 0 k = −P IFˆ 0 k k , K k=0 D 0 k = (P n − P ) K k=0 IFˆ 0 k k = (P n − P ) IF −(P n − P ) IF − K k=0 IFˆ 0 k k . By IF − K k=0 IFˆ 0 k k = o P (1), DC, permanence properties of Donsker classes (van der Vaart and Wellner, 1996), BD), and SP.k), the latter term is o P (n −1/2 ). Thus, by Lemma 1, Q * 0 − Q 0 = (P n − P ) IF + K k=1 Rem 0 k +o P (n −1/2 ), and soQ * 0 is an asymptotically linear estimator of Q 0 with influence function IF if the remainder term K k=1 Rem 0 k is o P (n −1/2 ). If one has not used known values of each π k or correctly specified a parametric model for each π k , k ≥ 1, then often this IF is the canonical gradient in the nonparametric model (Pfanzagl, 1990). 2 Simulation Simulation methods We conduct a simulation study that evaluates the finite sample behavior of the two SDR methods presented. All simulations report (i) the mean-squared error (MSE) for the outcome regression that conditions on baseline covariates only, i.e. Q 1 ; (ii) the bias and the coverage of a two-sided 95% confidence interval given by the various estimators for the marginal parameter Q 0 . As most existing methods are designed to focus on estimating Q 0 , they will not necessarily perform well for Q 1 . We compare the performance of the following estimators: LTMLE, doubly robust unbiased transformation (DR Transform), and iTMLE. We also implement a naïve plug-in esti- is evaluated based on the following four regression specification scenarios of the remaining outcome regressions and propensity scores: Qc.gc, when all Q k and π k are based on correctly specified regressions; Qi.gc, all Q k for k > 1 are incorrect, and π k are correctly specified for all k; Qc.gi, when Q k are correctly specified for all k, while π k are incorrect for all k; Qi.gi, Q k are incorrectly specified for k > 1 and g k is incorrectly specified for all k. The second simulation scenario (Simulation 2) is based on a longitudinal data structure with 5 time-points, i.e, K = 5. The estimates are evaluated from a simulated sample of 5,000 i.i.d. units. The types of regressions considered in this simulation include the four scenarios from Simulation 1 (Qc.gc, Qi.gc, Qc.gi and Qi.gi), except that the estimation of Q k is based on non-parametric regression approaches (details below). We define the "correct" estimation scenario for Q k (i.e, Qc.gc and Qc.gi) by including all the relevant time-varying and baseline covariates, whereas the "incorrect" estimation scenario for Q k means that we exclude some of the key time-varying or baseline covariates. We estimate each π k via the main-terms logistic regression. The incorrect estimatorπ k of π k is then obtained by running an intercept-only logistic regression. We also consider an additional scenario (QSDR.gSDR), where Q 5 is incorrect, while Q k are correct for all k < 5, and, conversely, π 5 is correct, while π k are incorrect for all k < 5. This scenario mimics data for which the last outcome regression is a high-dimensional and biologically complex mechanism and is unlikely to be correctly specified, while the exposure mechanism at the last time-point is actually known. For Simulation 2, the non-parametric estimation of Q k is based on a discrete super-learner (van der Laan et al., 2007). The ensemble library of candidate learners includes 18 estimators from xgboost R package (Chen and Guestrin, 2016), which is a high-performance implementation of the gradient boosting machines (GBMs) (Friedman, 2001), as well as a main-terms logistic regression (GLM). The best performing model in the ensemble is selected via 5-fold cross-validation. We found that using the ensemble of highly data-adaptive xgboost learners for all Q k , k = 1, . . . , 5, was prone to overfitting. To mitigate this overfitting we employ xgboost-based learners only for estimating Q 5 and Q 4 , and we use the logistic regression for estimating Q 3 , Q 2 and Q 1 . In both simulation scenarios, the iTMLE targeting steps are based on super-learner ensembles that include 3 GBMs from xgboost R package, a main terms logistic regression, a univariate intercept-only logistic regression and an empty learner that does not updateQ k . The targeted iTMLE update is then defined by the convex combination of predictions from each learner in the super-learner ensemble, where this combination is fitted using the novel cross-validation scheme presented in Appendix E. The regression specification for DR Transform relies on exactly the same estimation approaches as described for Q k in Simulation 1 and 2. However, since the transformed estimatesΓ k [i] often result in some values being outside of (0, 1), the standard statistical R software, such as GLM, typically produces an error. To overcome this, we modified the R package xgboost to produce valid regression estimates with DR transformed outcomesΓ k [i], even if they happen to fall outside of (0, 1). The source code and the installation instructions for our modified version of xgboost are available at github.com/osofr/xgboost. Simulation results The simulation results for the relative MSE estimation of Q 1 for Simulation 1 and 2 are presented in Figure 1. These results clearly demonstrate that, depending on the scenario, the iTMLE and DR Transform either outperform or perform comparably to Direct Plugin and LTMLE. The Simulation 1 and 2 results for the relative absolute bias in estimation of Q 0 are presented in Figure 2. for the coverage and mean length of the two-sided 95% CIs for Q 0 in Simulation 1 and 2 are presented in Figure 3. The confidence interval coverage and width appear to be comparable between the two SDR methods and the LTMLE. The only exception is for the QSDR.gSDR scenario, where the LTMLE has roughly 10% coverage while the SDR approaches achieve nearly the nominal coverage level at roughly the same mean confidence interval width. observations. Simulation 2 is based on longitudinal data with 5 time-points and n=5,000 observations. The iTMLE and DR Transform typically outperform or perform comparably to both competitors. Figure 3: Coverage (left panels) and mean length (right panels) of the two-sided 95% CIs for Q 0 in simulation scenario 1 (top panels) and simulation scenario 2 (bottom panels). Confidence interval coverage and width appear to be comparable between the two SDR methods and the LTMLE. The only exception is for the QSDR.gSDR scenario, where the LTMLE has roughly 10% coverage, whereas the SDR approaches nearly achieve the nominal coverage level. Discussion We have introduced a new form of robustness, which we term sequential double robustness. This form of robustness allows the misspecification of either the censoring mechanism or outcome regression functional form at each time point. We studied the extent to which existing estimators for univariate G-formula parameter do or do not satisfy this property. We then presented a general SDR estimation strategy (iTMLE), which represents an exten-sion of existing targeted minimum loss-based estimation to the infinite-dimensional setting. The iTMLE is iterative, leveraging the SDR property from temporally subsequent outcome regressions to ensure the SDR property at the current outcome regression. We presented a high level argument supporting the SDR nature of a general iTMLE (Section 5), and formally established that a special case of our estimation scheme (based on empirical risk minimization) is SDR (Section 6). In practice we believe the ERM procedure is prone to overfitting, and so we suggest using the cross-validation selector presented in Appendix E. Beyond the added robustness property of our new estimator, we presented heuristic arguments for why the iTMLE more appropriately accounts for the dimension of the outcome regression problem than typical sequential regression procedures (Remark 8). We made certain decisions in this manuscript to improve readability. We focused on an outcome that is bounded in [0, 1], and as a consequence all regressions were run using a cross-entropy loss function. Like targeted minimum-loss based estimation, this method immediately extends beyond binary outcomes by choosing a different loss function. To expedite our analysis of the ERM special case, we also assumed that the outcome regressions were bounded away from zero or one in Section 6. This decision artificially disallows timeto-event outcomes (Appendix B). We note that the assumptions needed for our TMLE extension to be SDR are stronger than those needed for the implementation of the (S)DR unbiased transformation given in Section 3 in that we require correct specification of the functional form of subsequent time point residuals, rather than just of the outcome regressions. This additional assumption appears plausible provided we use a flexible regression framework to estimate these quantities. We note that the the DR unbiased transformation approach requires using pseudo-outcomes that may fall well outside of the outcome range [0, 1] if the number of time points is large or the treatment mechanism ever falls close to zero. As the bound on the outcome in a regression plays a major role on the practical performance of the procedure, we expect our method to outperform the DR unbiased transformation approach in these settings. Our iTMLE can be used to naturally extend recent estimators that yield doubly robust inference for the G-formula if either the outcome regression or treatment mechanism is correctly specified (van der Laan, 2014; Benkeser et al., 2016). In particular, we will replace their univariate fluctuation submodels, fit with a logistic regression, by infinite-dimensional fluctuation submodels, fit via a data adaptive regression procedure. Rather than obtaining DR inference, we will obtain what we refer to as an SDR rate, i.e. a rate of convergence dictated by the entropy of the class used to estimate the outcome regression k if, for each k > k, either the time k outcome regression or treatment mechanism is correctly specified. The SDR rate property is stronger than what we have established for the estimator in this paper, which we have only shown to achieve the rate dictated by the time k outcome regression class if, at each subsequent time point, both the outcome regression and treatment mechanism are correctly specified. Benkeser et al. (2016) recently showed that a one-step estimator fails to obtain valid doubly robust inference in the single time point setting, and Benkeser (2015) showed a similar result in the longitudinal setting. In our setting, the analogue will be to show that the natural modification to the DR unbiased transformation method fails to obtain an SDR rate. We expect our method to enable the construction of confidence sets and bands for time k outcome regressions that shrink at the rate dictated by the entropy of the class used to estimate outcome regression k, rather than at the rate dictated by the entropy of the largest class used to estimate the outcome regressions k ≥ k. We will explore this exciting development in a future work. Appendix A Proofs Proof of Lemma 1. Note that Q k k (h k ) − Q k (h k ) = − E A k π k Q k+1 −Q k h k + E Q k+1 − Q k+1 h k , A k = 1 . The leading term is D k k (h k ). To see that the latter term equals K k =k+1 [D k k (h k )+Rem k (h k )], recursively (from k = k to k = K − 1) apply the following relationship to the inner expectation in the final term of E Q k +1 − Q k +1 h k , a k = − E A k +1 π k +1 Q k +2 −Q k +1 h k , a k + E 1 − π k +1 π k +1 Q k +1 − Q k +1 h k , a k + E A k +1 π k +1 E Q k +2 − Q k +2 H k +1 , A k +1 h k , a k , where the recursion ends at k = K − 1 becauseQ K+1 = Q K+1 . Proof of Theorem 3. Fixh k . Define Gh k (ε) ≡ E L k k (H k +1 ;h k → ε) h k . LetĠh k (ε) ≡ ∂ ∂ε Gh k (ε). The chain rule shows that ∂[Ġh k (ε)] 2 ∂Gh k (ε) = 2Gh k (ε), whereGh k (ε) ≡ ∂ ∂εĠhk (ε) . By the mean value theorem, there exists a c in the range of 2Gh k (·) such thaṫ Gh k [ˆ k k (h k )] 2 −Ġh k [¯ k k (h k )] 2 = c Gh k [ˆ k k (h k )] − Gh k [¯ k k (h k )] As¯ k k is the risk minimizer over all functions¯ k k : H k → R,Ġh k [¯ k k (h k )] = 0. Straightforward calculations show that 2Gh k (ε) = 2a k k k =k+1 A k π k Q k+1,h k →ε (relying on δ only) times Q * k −Q k k .E[E k k (H k )] = o P (n −1/2 ) for each k ≥ k. Proof of Lemma A.1. We use empirical process notation so that, for a distribution ν and function f , νf = E ν [f (Z)]. Asˆ k k is an empirical risk minimizer, CS.k) implies that P n L k k (· ;ˆ k k ) ≤ P n L k k (· ;¯ k k ). Hence, E[E k k (H k )] = P L k k (·;ˆ k k ) − L k k (·;¯ k k ) ≤ (P n − P ) L k k (· ;¯ k k ) − L k k (· ;ˆ k k ) . The remainder of the proof shows that the right-hand side is o P (n −1/2 ). By BD), CS.k), and permanence properties of Donsker classes (e.g., Chapter 2.10 in van der Vaart and Wellner, 1996), the right-hand side is O P (n −1/2 ). Using the bounds onQ k+1 k , k > k, BD), and SP.k), standard arguments used to show that the cross-entropy loss is quadratic (see, e.g., Lemma 2 in van der Laan et al., 2004) show that P L k k (·;ˆ k k ) − L k k (·;¯ k k ) 2 P L k k (·;ˆ k k ) − L k k (·;¯ k k ) . (A.1) We have already shown that the left-hand side is O P (n −1/2 ). Combining this with DC, permanence properties of Donsker classes, and the asymptotic equicontinuity of Donsker classes (e.g., Lemma 19.24 in van der Vaart, 1998), (P n − P ) L k k (· ;¯ k k ) − L k k (· ;ˆ k k ) = o P (n −1/2 ). Proof of Theorem 5. Let k, k satisfy the conditions of the theorem. We give proof by induction on k = k, k − 1, . . . , k 0 . The inductive hypothesis at k = k 0 includes our desired result that Q * k −Q k k = o P (n −1/4 ). Induction Hypothesis: IH(k ). Q k k −Q k k = o P (n −1/4 ) and E[E k k (H k )] = o P (n −1/2 ). Base Case: k = k. Q k k −Q k k = 0, so is o P (n −1/4 ) with much to spare. Lemma A.1 shows that CS.k plus the other conditions of this theorem imply that E[E k k (H k )] = o P (n −1/2 ). Induction Step: Suppose IH(k + 1) holds. By the triangle inequality, Q k k −Q k k ≤ Q k k −Q k +1 k + Q k +1 k −Q k k . By IH(k + 1), the leading term above is o P (n −1/4 ). In the remainder we establish that We start by giving a useful upper bound for −¯ k k for a general function :H k → R that falls in L 2 (P ). Because¯ k k is a risk minimizer over all functions mapping fromH k to R, aH k -pointwise second-order Taylor expansion shows that, for some˜ :H k → R that falls in between and¯ k k , Q k +1 k −Q k k = o P (n −1/4 ),E L k k (H k +1 ; ) − L k k (H k +1 ;¯ k k ) = E E L k k (H k +1 ; ) − L k k (H k +1 ;¯ k k ) H k = 1 2 E { (H k ) −¯ k k (H k )} 2 E A k k k =k +1 A k π k Q k +1,˜ k (1 −Q k +1,˜ k ) H k ≥ c −¯ k k 2 for an appropriately specified constant c > 0, where we used BD. The triangle inequality and two applications of the preceding display (at = 0 and =ˆ k k ) show that ˆ k k ≤ ¯ k k + ˆ k k −¯ k k E L k k (H k +1 ; 0) − L k k (H k +1 ;¯ k k ) 1/2 + E[E k k (H k )] 1/2 . The square of the latter term upper bounds as follows: E[E k k (H k )] = P L k k (·;ˆ k k ) − L k k (·; 0) + P L k k (·; 0) − L k k (·;¯ k k ) ≤ P L k k (·;ˆ k k ) − L k k (·; 0) − (P n − P ) L k k (·; 0) − L k k (·;¯ k k ) = P L k k (·;ˆ k k ) − L k k (·; 0) + (P n − P ) L k k (·; 0) − L k k (·;¯ k k ) , (A.2) where the inequality uses that the constant function zero is in F k and the latter equality uses that¯ k k is the true risk minimizer andˆ k k is the empirical risk minimizer. The subadditivity of x → x 1/2 thus yields that ˆ k k E L k k (H k +1 ; 0) − L k k (H k +1 ;¯ k k ) 1/2 + (P n − P ) L k k (·; 0) − L k k (·;¯ k k ) 1/2 . The square of the first term above upper bounds as follows: E L k k (H k +1 ; 0) − L k k (H k +1 ;¯ k k ) = E A k π k +1 E L k +1 k (H k +1 ;ˆ k +1 k ) − L k +1 k (H k +1 ;ˆ k +1 k +¯ k k ) H k +1 E A k π k +1 E L k +1 k (H k +1 ;ˆ k +1 k ) − L k +1 k (H k +1 ;¯ k +1 k ) H k +1 E[E k +1 k (H k )]. (A.3) The equality is an algebraic identity, the first inequality uses that¯ k+1 k is the risk minimizer among all functions mapping fromH k +1 → R, and the second inequality uses SP.k 0 . By IH(k + 1), the right-hand side is o P (n −1/2 ). Returning to the preceding display, ˆ k k (P n − P ) L k k (·; 0) − L k k (·;¯ k k ) 1/2 + o P (n −1/4 ). By DC, BD, SP.k 0 , and Lemma 19.24 of van der Vaart (1998), the former term on the right satisfies (P n − P ) L k k (·;ˆ k k ) − L k k (·; 0) =        o P (n −1/2 ), if ˆ k k = o P (1), O P (n −1/2 ), otherwise. (A.4) A first application of the above result shows that ˆ k k = O P (n −1/4 ). A second application shows that ˆ k k = o P (n −1/4 ). Recall that Q k +1 k −Q k k ˆ k k , thereby establishing the first part of IH(k ). For the second part of IH(k ), note that it suffices to bound the two terms on the right-hand side of (A.2). The first term is controlled using that the right-hand side of (A.3) is o P (n −1/2 ) under IH(k + 1). The second term is o P (n −1/2 ) by the above since ˆ k k = o P (n −1/4 ). Hence, we have established the second part of IH(k ), namely that E[E k k (H k )] = o P (n −1/2 ). B Right-censored data structures and time-to-event outcomes Remark 10 (Right-censored data structures). General discretely right-censored data structures can be expressed using our notation. In what follows we mimic the introduction to discretely right-censored data structures given in Bang and Robins (2005). For each k, let L k ≡ (L 1 , . . . , L k ). Let C be a discrete censoring time taking value in 1, . . . , K + 1. The observation is censored after time C, so that we observe (C,L C ). Under the missing at random assumption, P (C = k|C ≥ k,L K+1 ) = P (C = k|C ≥ k,L k ). If one wishes to estimate E[L K+1 |L k ] for some k ≤ K, then one can use that, under the missing at random assumption, this estimand is equal toQ k (L k ), whereQ k , k ≥ k, is recursively defined as Q K+1 (L K+1 ) ≡L K+1 , andQ k (L k ) ≡ E[Q k +1 (L k +1 )|C > k ,L k ]. To see the equivalence with our data structure, let A k ≡ 1 {C>k} . Then E[L K+1 |¯ k ] =Q k (h k ), whereh k is the history vector with time k ≤ k covariates k and time k < k treatment A k = 1. 2 Remark 11 (Time-to-event outcomes). Let C be a censoring time taking values in 1, . . . , K, +∞, let T be a survival time taking values in 1, . . . , K + 1, +∞, andL T ≡ (L 1 , . . . , L T ) denote a vector of covariates up to the survival time, where each L k , k ≤ K, contains an indicator that T ≤ k and L K+1 = 1 {T ≤K+1} . We wish to estimate P (T ≤ K + 1|¯ k ). One way to express the observed data structure is to write (Y, ∆,L Y ), where Y ≡ min{T, C} and ∆ ≡ 1 {T ≤C} . Alternatively, the observed data structure can be expressed using ourH K+1 notation, where each A k ≡ 1 {Y >k}∪{∆=1} . Under the sequential randomization assumption that P (C = k|C ≥ k, T > k,L T ) = P (C = k|C ≥ k, T > k,L k ) with probability one, one can show that P (T ≤ K + 1|¯ k ) is equal to Q k (h k ), where each a k , k < k, inh k is equal to one. Working with time-to-event data requires one additional consideration compared to the longitudinal treatment setting that we consider in the main text. In particular, once the indicator that T ≤ k inL k is equal to one, it is automatically true that L K+1 = 1. Thus, one should deterministically set estimates of Q k (L k ) equal to one for all suchL k . 2 C Sequential double robustness and 2 K robustness Sequential double robustness is a special case of the general notion of 2 K robustness as introduced in Section 4.1 of Vansteelandt et al. (2007). Nevertheless, as we discuss below, the form of 2 K robustness considered in that work is very different from the one we have considered. We refer to our form of robustness as sequential double robustness because we believe it emphasizes the fact that the robustness in question applies at each specific time point in the longitudinal structure. Note, however, that this estimator cannot be evaluated when dropout is monotone. If one views our context as a monotone missing data problem (Remark 10 in Appendix B), then there is probability zero of observing an individual after their first time of censoring. This forces us to use a more intricate identifiability result for the outcome regressions than that used by Vansteelandt et al. (2007), including the marginal parameter Q 0 . To the best of our knowledge, this new form of robustness for the sequential G-formula is novel: we are not aware of another work that has described, let alone exhibited estimators proven to achieve, this property. D Variation-independent formulation of sequential double robustness We now present a variation independent formulation of sequential double robustness and establish its achievability. This formulation is more restrictive than that given in Definition 3, but makes clear that there are scenarios in which one could correctly specify OR.k but not OR.k, k > k. For each k ≥ 1, let P k denote the distribution of L k+1 conditional on A k = 1,H k that is implied by P . Consider an estimation procedure that estimates each P k separately, where we note that these P k are variation independent, both of one another and of the treatment mechanisms, in the sense that knowing P k places no restriction on the set of possible realizations of P k , k = k, or of π k , k arbitrary. Thus, so our procedure can estimate all of these conditional distributions a priori. Define the following alternative to OR.k: FD.k) The distribution P k is correctly specified by the estimation procedure, or at least arbitrarily well approximated asymptotically. Above "FD" refers to correct specification of a component of the Full Data distribution. Consider an alternative definition of sequential double robustness that replaces OR.k and OR.k in Definition 3 by FD.k and FD.k . First note that, by recursive applications of this definition, from k = K, . . . , 0, we see that a procedure satisfying this alternative SDR definition implicitly correctly specifies the functional form of the time k outcome regression (i.e., satisfies OR.k) at each time point for which FD.k holds and, for each k > k, either FD.k or TM.k holds. Secondly, an estimator achieving this alternative SDR property is achievable using the (S)DR unbiased transformation presented in Section 3, where the regressions are fitted via kernel regression. Here we used the fact that kernel regression represents a kernel density estimation based plug-in estimator for the regression function. The downside to this procedure is that it requires correctly estimating possibly high-dimensional conditional densities. We therefore prefer an alternative approach that allows us to incorporate modern regression techniques -see Section 5. Nonetheless, the variation independence of the procedure discussed in this appendix: FD.k holding does not logically imply that FD.k holds for k = k , just as TM.k does not logically imply that TM.k holds for k = k . E Mitigating overfitting via cross-validation In this section, we describe a variant of V -fold cross-validation that can be used to estimate Note thatθ v[i] depends on the data in v[i] only through the selectedm and, if the loss function depends on nuisance parameters, then also through the cross-validation-selected candidated algorithms from these nuisance parameters. As we will show in a future work, the fact thatθ v[i] only depends on validation set i through discrete quantities ensures that our cross-validation scheme for estimatingm satisfies oracle inequalities analogous to those presented in van der Laan and Dudoit (2003). While the above procedure is written as a discrete cross-validation selector, we note that the super-learner algorithm, which replaces the discrete choice of size M with all convex combinations of M algorithms, can be arbitrarily well approximated by forming an -net over the M − 1 simplex, where now each convex combination of these M algorithms is treated as a candidate (van der Laan et al., 2006). F Simulated data-generating distributions All simulations are carried out in R programming package using simcausal package (Sofrygin et al., 2015). F.1 Simulation 1 The simulation is implemented by sampling longitudinal data over 3 time-points and n = 500 i.i.d. subjects. Briefly, this simulation represents a simple data structure with binary exposure that can be assigned separately at time-points k = 1, . . . , 3 and a binary outcome of interest Y K evaluated at K = 3. Recall from the main text that Ψ(x) ≡ 1/(1 + e −x ). The longitudinal structure on each subject was sampled according to the following structural equation model for time-point t = 1: L t ∼ Normal(0, 1) A t ∼ Bernoulli (Ψ{L t }) Y t = 0. Followed by time-point t = 2: L t ∼ Normal(0, 1) A t ∼ Bernoulli (Ψ{L t + A t−1 }) Y t = 0. Followed by time-point t = 3: L t ∼ Normal(L t−2 A t−1 + A t−2 L t−1 + L t−1 A t−1 , 1) A t ∼ Bernoulli (Ψ{L t + A t−1 }) Y t = Bernoulli (Ψ{L t−1 A t + A t−1 L t + L t A t }) . F.2 Simulation 2 The simulation is implemented by sampling longitudinal data over 5 time-points and n = 5, 000 i.i.d. subjects. The data-generating distribution for Simulation 2 is more complex and higher dimensional than that in Simulation 1. Briefly, for Simulation 2 we let A k = 1 denote standard of care and A k = 0 denote the experimental new treatment at time-point k (note: our coding is the reverse of the more standard way to denote A k = 1 as the new treatment at k). The subject can switch from the standard of care at any time during the follow-up k = 1, . . . , K. However, once the subject switches he or she is forced to stay on the new treatment until the end of the follow-up. We assume that the outcome of interest, Y K , is a binary indicator of the adverse event at the final follow-up time-point K. Furthermore, switching to the experimental treatment A k = 0 at any k lowers the probability of the final adverse event at K. Finally, the probability of receiving the experimental treatment A k = 0 increases once the subject's risk for experiencing the adverse end-of-the study event becomes high, where subject's risk at each k is being assessed conditionally on the fact that he or she will remain on the standard of care. Thus, an incorrectly specified regression of π k in such a data-generating process would miss the informative switching to the experimental treatment, yielding a biased estimate of Q 0 for IPW-based estimator that requires consistent π k . The longitudinal structure on each subject is sampled according to the following struc-tural equation model at time-point t = 1: U L,t ∼ Normal(0, 1) W t ∼ Normal(0, 1) L t = |U L,t | Z t = L t A t ∼ Bernoulli (Ψ{L t }) Y t = 0. Followed by time-point t = 2: U L,t ∼ Normal(A t−2 L t−1 + L t−1 A t−1 , 1) L t = |U L,t | Z t = −2 + 0.5L t−1 + L t A t ∼ Bernoulli A t−1 Ψ{1.7 − 2.0 1 {Ψ(Zt)>0.85} } Y t ∼ Bernoulli (Ψ{−3Y t−1 + 0.5L t−1 A t + 0.5A t−1 L t + 0.5L t A t }) . U L,t ∼ Normal(0, 1) L t = |U L,t | Z t = −2 + 0.5L t−1 + L t A t ∼ Bernoulli A t−1 Ψ{1.7 − 2.0 1 {Ψ(Zt)>0.9} } Y t ∼ Bernoulli (Ψ{−3 + 0.5L t−1 A t + 0.5A t−1 L t + 0.5L t A t }) . Followed by time-point t = 4: U L,t ∼ Normal(L t−2 A t−1 + A t−2 L t−1 + L t−1 A t−1 , 1) L t = |U L,t | Z t = −2 + 0.5L t−1 + L t A t ∼ Bernoulli A t−1 Ψ{1.7 − 2.0 1 {Ψ(Zt)>0.80} } Y t ∼ Bernoulli (Ψ{−3Y t−1 + 0.5L t−1 A t + 0.5A t−1 L t + 0.5L t A t }) . Finally, followed by time-point t = 5: U L,t ∼ Normal(L t−3 A t−1 + A t−3 L t−1 + L t−1 A t−1 , 1) L t = |U L,t | Z t = −1 + 0.25L t−1 + 0.5L t − 0.1L t L t−1 + 1.5W 0 L t−1 ing literature are given in Tsiatis et al. (2011); Rotnitzky et al. (2012) -here we provide return {Q k : k = 0, . . . , K} (A)IPW loss functions. For each f , let z → L (z; f ) denote a real-valued function. We call L a loss function if f 0 = argmin f E ν [L (Z; f )] over an appropriate index set for f . The quantity E ν [L (Z; f )] is referred to as the risk of f . Examples of loss functions for the conditional mean f 0 include the squared-error loss (z; f ) → (y − f (x)) 2 and, for z bounded in [0, 1], the cross-entropy loss ( 2 Remark 9 ( 29Asymptotic linearity). A stronger result than that of Theorem 4 can be obtained if k 0 = 0 so that Q k 0 is a real number. Suppose that the conditions of Theorem 4 hold and there exists a function IF :H K+1 → R such that IF − mator (Direct Plugin), that letsQ K+1 (H K+1 ) ≡ L K+1 , and recursively from k = K, . . . , 0, regressesQ k+1 [i] againstH k [i] among all individuals i with A k [i] = 1. The naïve plug-in estimator yields estimates of Q 0 and Q 1 , though does not yield a confidence interval for Q 0 .Finally, for estimation of the marginal mean parameter Q 0 , we also evaluate the bias for the inverse probability weighted estimator (IPW). The performance of these estimators is evaluated under various model specification scenarios for the outcome regressionsQ k and the propensity scoresπ k , as described below. We do not evaluate the performance of the Bang & Robins (BR) estimator in its original form because LTMLE can be viewed as its robust extension. Some of the data generating distributions used in this simulation study would yield an inconsistent BR, thus not providing a fair comparison of this estimation approach.The performance of the estimators is evaluated over 1000 replicas of the observed data. All simulation and estimation is carried out in the R language (R Core Team, 2016) using the packages simcausal (Sofrygin et al.) and stremr(Sofrygin et al., 2016). The full R code for this simulation study is available in a github repository github.com/osofr/SDRsimstudy.Our simulation study consists of two scenarios. The data-generating distributions for Simulations 1 and 2 are described in detail in Appendix F. Here we give an overview of the simulation methods.Simulation 1 is a proof of concept with a simple longitudinal structure and 3 timepoint interventions, i.e., K = 3. For this simulation scenario, the estimates are evaluated from a sample of n = 500 i.i.d. units. The outcome regressions Q k and propensity scores π k are estimated using the main terms logistic regressions. The estimator of the outcome regression Q 1 is always correctly specified in all four of these scenarios. Each estimator Figure 1 : 1Relative MSE forQ 1 for simulation scenario 1 (top panel) and simulation scenario 2 (bottom panel). Simulation 1 is based on longitudinal data with 3 time-points and n=500 Figure 2 : 2Relative absolute bias forQ 0 for simulation scenario 1 (top panel) and simulation scenario 2 (bottom panel). Simulation 1 is based on longitudinal data with 3 time-points and n=500 observations. Simulation 2 is based on longitudinal data with 5 time-points and n=5,000 observations. The performance of LTMLE, iTMLE, and DR Transform is similar. and along the way we also establish that E[E k k (H k )] = o P (n −1/2 ). By the Lipschitz property of the expit function, the first part of IH(k ). Vansteelandt et al. (2007) constructed a 2 K multiply robust estimator of the vector of marginal means over K time points in a different context. In particular, they considered a setting where one observes repeated measures, with an outcome at each time point, but the missingness pattern is nonmonotone and there is always a positive probability of a participant being observed again after having missed a visit. Because participants always have a nonzero chance of returning, the mean of the (potentially missing) outcome at time k can be estimated using a single augmented inverse-probability weighted estimator -see the representation of the time point specific mean outcome parameter in their Theorem 1. To estimate the dimension-K vector of mean outcomes at all time points, they use K parallel doubly robust estimators, one for each time point, where by "parallel" we mean that the fit of the estimator at one time point has no impact on the fit of the estimator at another time point. Since the assumptions used for each of these K doubly robust estimators are variation independent, this yields 2 K ways to correctly estimate the vector of interest. Nevertheless, the resulting estimator of the mean outcome at the final time point is only doubly robust rather than 2 K robust. In the setting of Vansteelandt et al. (2007), where the missingness pattern is nonmonotone and there is always a possibility of a subject returning to the trial, their estimator may be preferred to ours because it only requires consistent estimation of a final time point propensity score or final time outcome regression. In particular, it does not require estimation of any of the other time point propensity scores or outcome regressions. L each Q k . Let S 1 , . . . , S V be mutually exclusive and exhaustive subsets of {1, . . . , n} of approximately equal size, determined independently of the observationsH K+1 [1], . . . ,H K+1 [n]. The set S v is referred to as validation set v, and its complement is referred to as training set v. For each i, let v[i] denote the validation set S v[i] to which i belongs, i.e. the validation set for observation i. Suppose we wish to estimate some parameter θ of an arbitrary distribution ν on Z ≡ (X, Y ), e.g. θ(x) = E ν [Y |x]. Suppose that we have M candidate regression algorithms for estimating θ. For each candidate algorithm m and subset indicator v, letθ v,m denote an estimate of θ by running algorithm m on observations in training set v. The cross-validation selector for m is defined (Z;θ v[i],m ) for some appropriately defined loss L . If the loss function depends on nuisance parameters that must be estimated from the data, then one can replace this loss by a loss with the nuisance parameter estimates obtained from training set v[i], up to the dependence of these parameters on their own cross-validation selectors of an algorithm m. We letθ v[i] ≡θ v[i],m . Followed by time-point t = 3: Consider the procedure that regresses Γ k [i] againstH k [i] for all i such that A k [i] = 1. This procedure is DR in the sense that it is consistent if OR.k holds and either (i) OR.k holdsfor all k > k or (ii) TM.k holds for all k > k. If applied iteratively as in Algorithm 1, i.e. While overall the performance the LTMLE, iTMLE, and DR Transform is similar across different scenarios, the notable exceptions are the scenario Qi.gc in Simulation 1, where DR Transform appears to outperform other methods, the scenario Qc.gi in Simulation 2, where DR Transform outperforms the rest, and the scenario QSDR.gSDR in Simulation 2, where both SDR methods outperform the LTMLE. The simulation results The only exception for Simulation 1 is under Qi.gc, where DR Transform outperforms other methods. The only exceptions for Simulation 2 are for Qc.gi, where DR Transform outperforms other methods, and QSDR.gSDR, where both SDR methods outperform LTMLE.q q q q q q q q q Simulation 1 Simulation 2 0.10 0.30 0.50 0.70 0.85 0.95 Qc.gc Qi.gc Qc.gi Qi.gi Qc.gc Qi.gc Qc.gi Qi.gi QSDR.gSDR Coverage Probability q q q q q q q q q Simulation 1 Simulation 2 0.1 1.0 Mean CI Length Estimator q iTMLE DR Transform LTMLE Direct Plugin IPW New Sequential Regression ProcedureWe now present a novel, SDR procedure for estimating Q k 0 , k 0 ≥ 0. We extend the univariate targeting step used in the procedure of van der Laan and Gruber (2012) to infinitedimensional targeting steps towards estimating Q k 0 . We refer to this new procedure as iTMLE. The iTMLE is presented in Algorithm 3. For each k, we denote the estimate of Q k that is targeted towards the outcome regressions at all outcome regressions k satisfying k ≤ k ≤ k − 1 byQ k k (intuition on the targeting steps will be developed in what follows), and we letQ * k ≡Q k 0 k .In this and the proceeding section, we abbreviate the following definitions from (1) for all k , k with k ≥ k ≥ k 0 : D k, * k ≡ D k k (Q * k +1 ,Q * k ) and Rem k, * k ≡ Rem k k (Q * k ). Recall that k +1 • Offset: logitQ k+1 k • Predictor:H k • Weight: A k k k =k+1 A k π k 9:Q k k ≡Q k+1,ˆ k k k 10: LetQ * k ≡Q k 0 k Targeted towards all Q k , k < k 11: returnQ * k 0We now analyze the procedure that targets the estimate of Q k towards Q k , k ≥ k.Define the data-dependent loss functionL k k (h k +1 ; k k ) = −a k k k =k+1 a k π k Q * k +1 logQ k+1, k k k + [1 −Q * k +1 ] log 1 −Q k+1, k k k ,where above we suppressed the dependence ofπ k ,Q k k +1 , andQ k+1, k k k onh k ,h k +1 , and h k , respectively. It is not difficult to verify that E[L k k (H k +1 ; k k )] is minimized at the k (1 −Q k+1,h k →ε k ),which is uniformly upper bounded by δ k −k /2. Furthermore, because¯ k k is a risk minimizer,Gh k [ˆ k k (h k )]−Gh k [¯ k k (h k )] ≥ 0. Further note thatĠh k [ˆ k k (h k )] 2 = π k (h k ) 2 D k k (Q * k +1 ,Q k k )(h k ) 2 , and also that D k k (Q * k +1 ,Q k k )(h k ) 2 D k k (Q * k +1 ,Q k k )(h k ) 2 . Hence, we have shown that D k k (Q * k +1 ,Q k k )(h k ) 2 Gh k [ˆ k k (h k )] − Gh k [¯ k k (h k )] = E k k (h k ).This yield the almost sure pointwise bound. Take an expectation overH k ∼ P on both sides, taking the square root, and applying the triangle inequality shows thatD k, * k ≤ D k k (Q * k +1 ,Q k k ) + D k k − D k k (Q * k +1 ,Q k k ) E E k k (h k ) 1/2 + D k k − D k k (Q * k +1 ,Q k k ) .The positivity assumption shows that the latter term upper bounds by a positive constant AcknowledgementsThis work was partially supported by the National Institute of Allergy and Infectious Disease at the NationalAlgorithm 4 Cross-validated iTMLE for Q kSelects between M k with covariateH k , k ≥ k 0 . When k 0 = 0, M k 0 = 1 and the only candidate regression is an intercept-only logistic regression. 1: procedure SDR.Q(k 0 ) 2:for v=1,. . . ,V do Estimate treatment mechanisms.3:Using only observations in training set v, obtain estimatesπ k,v of π k , k = k 0 , . . . , K, via any desired technique.4:LetQ * K+1 ≡ L K+1 5:Fit candidates on training set v.9:For each function k k : Doubly robust estimation in missing data and causal inference models. H Bang, J M Robins, Biometrics. 61H Bang and J M Robins. Doubly robust estimation in missing data and causal inference models. Biometrics, 61:962-972, 2005. Empirical minimization. Probability Theory and Related Fields. P L Bartlett, S Mendelson, 135P L Bartlett and S Mendelson. Empirical minimization. Probability Theory and Related Fields, 135(3): 311-334, 2006. Local rademacher complexities. P L Bartlett, O Bousquet, S Mendelson, The Annals of Statistics. 334P L Bartlett, O Bousquet, and S Mendelson. Local rademacher complexities. The Annals of Statistics, 33 (4):1497-1537, 2005. Data-adaptive Estimation in Longitudinal Data Structures with Applications in Vaccine Efficacy Trials. D Benkeser, University of WashingtonPhD thesisD Benkeser. Data-adaptive Estimation in Longitudinal Data Structures with Applications in Vaccine Efficacy Trials. PhD thesis, University of Washington, 2015. Doubly-Robust Nonparametric Inference on the Average Treatment Effect. D Benkeser, M J Carone, P Van Der Laan, Gilbert, 326Division of Biostatistics, University of California, BerkeleyTechnical reportD Benkeser, M Carone, M J van der Laan, and P Gilbert. Doubly-Robust Nonparametric Inference on the Average Treatment Effect. Technical report, 326, Division of Biostatistics, University of California, Berkeley, 2016. Linear regression with censored data. J Buckley, James, Biometrika. J Buckley and I James. Linear regression with censored data. Biometrika, pages 429-436, 1979. XGBoost: A Scalable Tree Boosting System. T Chen, C Guestrin, KDD 2016. T Chen and C Guestrin. XGBoost: A Scalable Tree Boosting System. In KDD 2016, 2016. Local polynomial modelling and its applications: monographs on statistics and applied probability 66. J Fan, Gijbels, CRC Press66J Fan and I Gijbels. Local polynomial modelling and its applications: monographs on statistics and applied probability 66, volume 66. CRC Press, 1996. Greedy function approximation: a gradient boosting machine. J H Friedman, Annals of Statistics. J H Friedman. Greedy function approximation: a gradient boosting machine. Annals of Statistics, pages 1189--1232, 2001. Asymptotically optimal model selection method with right censored outcomes. S Keles, S Van Der Laan, Dudoit, Bernoulli. S Keles, M van der Laan, and S Dudoit. Asymptotically optimal model selection method with right censored outcomes. Bernoulli, pages 1011-1037, 2004. Non-parametric methods for doubly robust estimation of continuous treatment effects. E H Kennedy, M D Ma, D S Mchugh, Small, Journal of the Royal Statistical Society: Series B (Statistical Methodology). E H Kennedy, Z Ma, M D McHugh, and D S Small. Non-parametric methods for doubly robust estimation of continuous treatment effects. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 2016. Regression analysis with randomly right censored data. H Koul, Susarla, Van Ryzin, Ann Stat. 9H Koul, V Susarla, and J van Ryzin. Regression analysis with randomly right censored data. Ann Stat, 9: 1276-1288, 1981. Causality: Models, Reasoning and Inference. J Pearl, Cambridge University PressNew York2nd editionJ Pearl. Causality: Models, Reasoning and Inference. Cambridge University Press, New York, 2nd edition, 2009. Estimation in semiparametric models. J Pfanzagl, SpringerBerlin Heidelberg New YorkJ Pfanzagl. Estimation in semiparametric models. Springer, Berlin Heidelberg New York, 1990. R Core Team, R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing. Vienna, AustriaR Core Team. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, 2016. URL https://www.r-project.org/. A new approach to causal inference in mortality studies with a sustained exposure periodapplication to control of the healthy worker survivor effect. J M Robins, Mathematical Modelling. 79J M Robins. A new approach to causal inference in mortality studies with a sustained exposure perio- dapplication to control of the healthy worker survivor effect. Mathematical Modelling, 7(9):1393-1512, 1986. J M Robins, The analysis of randomized and non-randomized AIDS treatment trials using a new approach in causal inference in longitudinal studies. L Sechrest, H Freeman, and A MulleyU.S; Washington D.C.National Center for Health SErvices ResearchHealth Service Methodology: A Focus on AIDS. A t ∼ Bernoulli A t−1 Ψ{2 − 2.0 1 {Ψ(Zt)>0.80} } Y t ∼ Bernoulli (Ψ{−Y t−1 + A t + Z t A t + 0.20A t−1 L t })J M Robins. The analysis of randomized and non-randomized AIDS treatment trials using a new approach in causal inference in longitudinal studies. In L Sechrest, H Freeman, and A Mulley, editors, Health Service Methodology: A Focus on AIDS, pages 113-159. U.S. Public Health Service, National Center for Health SErvices Research, Washington D.C., 1989. A t ∼ Bernoulli A t−1 Ψ{2 − 2.0 1 {Ψ(Zt)>0.80} } Y t ∼ Bernoulli (Ψ{−Y t−1 + A t + Z t A t + 0.20A t−1 L t }) .

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{'abstract': 'Consider estimating the G-formula for the counterfactual mean outcome under a given treatment regime in a longitudinal study. Bang and Robins provided an estimator for this quantity that relies on a sequential regression formulation of this parameter. This approach is doubly robust in that it is consistent if either the outcome regressions or the treatment mechanisms are consistently estimated. We define a stronger notion of double robustness, termed sequential double robustness, for estimators of the longitudinal G-formula. The definition emerges naturally from a more general definition of sequential double robustness for the outcome regression estimators. An outcome regression estimator is sequentially doubly robust (SDR) if, at each subsequent time point, either the outcome regression or the treatment mechanism is consistently estimated. This form of robustness is exactly what one would anticipate is attainable by studying the remainder term of a first-order expansion of the G-formula parameter. We show that a particular implementation of an existing procedure is SDR. We also introduce a novel SDR estimator, whose development involves a novel translation of ideas used in targeted minimum loss-based estimation to the infinite-dimensional setting. A Rotnitzky, Q Lei, M Sued, and J Robins. Improved Double-Robust Estimation in missing data and causal inference models. Biometrika, 99(2):439-456, 2012. D Rubin and M J van der Laan. A doubly robust censoring unbiased transformation. Int J Biostat, 3(1): S Seaman and A Copas. Doubly robust generalized estimating equations for longitudinal data. Statistics in medicine, 28(6):937-955, 2009. O Sofrygin, M J van der Laan, and R Neugebauer. simcausal {R} Package: Conducting Transparent and Reproducible Simulation Studies of Causal Effect Estimation with Complex Longitudinal Data. Journal of Statistical Software, In Press. O Sofrygin, M J van der Laan, and R Neugebauer. {simcausal}: Simulating Longitudinal Data with Causal Inference Applications, 2015. URL http://cran.r-project.org/package=simcausal. O Sofrygin, M J van der Laan, and R Neugebauer. stremr: Streamlined Estimation of Survival for Static, Dynamic and Stochastic Treatment and Monitoring Regimes, 2016. URL https://github.com/osofr/ stremr. A A Tsiatis, M Davidian, and W Cao. Improved doubly robust estimation when data are monotonely coarsened, with application to longitudinal studies with dropout. Biometrics, 67(2):536-545, 2011. S van de Geer. Estimating a regression function. The Annals of Statistics, pages 907-924, 1990. M J van der Laan. Targeted estimation of nuisance parameters to obtain valid statistical inference. The international journal of biostatistics, 10(1):29-57, 2014. M J van der Laan and S Dudoit. Unified cross-validation methodology for selection among estimators and a general cross-validated adaptive epsilon-net estimator: finite sample oracle inequalities and examples. . M J van der Laan and S Gruber. Targeted minimum loss based estimation of causal effects of multiple time point interventions. Int J Biostat, 8(1):Article 9, 2012. M J van der Laan and S Gruber. One-Step Targeted Minimum Loss-based Estimation Based on Universal Least Favorable One-Dimensional Submodels.', 'arxivid': '1705.02459', 'author': ['Alexander R Luedtke \nVaccine and Infectious Disease Division\nFred Hutchinson Cancer Research Center\nUSA\n\nPublic Health Sciences Division\nFred Hutchinson Cancer Research Center\nUSA\n', 'Oleg Sofrygin \nDivision of Biostatistics\nUniversity of California\nBerkeleyUSA\n', 'Mark J Van Der Laan \nDivision of Biostatistics\nUniversity of California\nBerkeleyUSA\n', 'Marco Carone \nVaccine and Infectious Disease Division\nFred Hutchinson Cancer Research Center\nUSA\n\nDepartment of Biostatistics\nUniversity of Washington\nUSA\n'], 'authoraffiliation': ['Vaccine and Infectious Disease Division\nFred Hutchinson Cancer Research Center\nUSA', 'Public Health Sciences Division\nFred Hutchinson Cancer Research Center\nUSA', 'Division of Biostatistics\nUniversity of California\nBerkeleyUSA', 'Division of Biostatistics\nUniversity of California\nBerkeleyUSA', 'Vaccine and Infectious Disease Division\nFred Hutchinson Cancer Research Center\nUSA', 'Department of Biostatistics\nUniversity of Washington\nUSA'], 'corpusid': 88517768, 'doi': None, 'github_urls': ['https://github.com/osofr/'], 'n_tokens_mistral': 26349, 'n_tokens_neox': 23756, 'n_words': 15928, 'pdfsha': '9d1297f833c31bc679c0577e4b186e349b100d6a', 'pdfurls': ['https://arxiv.org/pdf/1705.02459v2.pdf'], 'title': ['Sequential Double Robustness in Right-Censored Longitudinal Models', 'Sequential Double Robustness in Right-Censored Longitudinal Models'], 'venue': []}

arxiv

Epitaxial van der Waals heterostructures of Cr 2 Te 3 on 2D materials Quentin Guillet Université Grenoble Alpes CEA CNRS IRIG-SPINTEC 38000GrenobleFrance Libor Vojáček Université Grenoble Alpes CEA CNRS IRIG-SPINTEC 38000GrenobleFrance Djordje Dosenovic Université Grenoble Alpes CEA IRIG-MEM 38000GrenobleFrance Fatima Ibrahim Université Grenoble Alpes CEA CNRS IRIG-SPINTEC 38000GrenobleFrance Hervé Boukari Institut Neel Université Grenoble Alpes CNRS 38000GrenobleFrance Jing Li Université Grenoble Alpes CEA F-38000GrenobleLetiFrance Fadi Choueikani Synchrotron SOLEIL L'Orme des Merisiers91190Saint-AubinFrance Philippe Ohresser Synchrotron SOLEIL L'Orme des Merisiers91190Saint-AubinFrance Abdelkarim Ouerghi Université Paris-Saclay CNRS Centre de Nanosciences et de Nanotechnologies PalaiseauFrance Florie Mesple Université Grenoble Alpes CEA CNRS IRIG-PHELIQS 38000GrenobleFrance Vincent Renard Université Grenoble Alpes CEA CNRS IRIG-PHELIQS 38000GrenobleFrance Jean-François Jacquot Université Grenoble Alpes CEA CNRS IRIG-SYMMES 38000GrenobleFrance Denis Jalabert Université Grenoble Alpes CEA IRIG-MEM 38000GrenobleFrance Hanako Okuno Université Grenoble Alpes CEA IRIG-MEM 38000GrenobleFrance Mairbek Chshiev Université Grenoble Alpes CEA CNRS IRIG-SPINTEC 38000GrenobleFrance Institut Universitaire de France 75231ParisFrance Céline Vergnaud Université Grenoble Alpes CEA CNRS IRIG-SPINTEC 38000GrenobleFrance Frédéric Bonell Université Grenoble Alpes CEA CNRS IRIG-SPINTEC 38000GrenobleFrance Alain Marty Université Grenoble Alpes CEA CNRS IRIG-SPINTEC 38000GrenobleFrance Matthieu Jamet Université Grenoble Alpes CEA CNRS IRIG-SPINTEC 38000GrenobleFrance Epitaxial van der Waals heterostructures of Cr 2 Te 3 on 2D materials (Dated: March 7, 2023) Achieving large-scale growth of two-dimensional (2D) ferromagnetic materials with high Curie temperature (TC) and perpendicular magnetic anisotropy (PMA) is highly desirable for the development of ultra-compact magnetic sensors and magnetic memories. In this context, van der Waals (vdW) Cr2Te3 appears as a promising candidate. Bulk Cr2Te3 exhibits strong PMA and a TC of 180 K. Moreover, both PMA and TC might be adjusted in ultrathin films by engineering composition, strain, or applying an electric field. In this work, we demonstrate the molecular beam epitaxy (MBE) growth of vdW heterostructures of five-monolayer quasi-freestanding Cr2Te3 on three classes of 2D materials: graphene (semimetal), WSe2 (semiconductor) and Bi2Te3 (topological insulator). By combining structural and chemical analysis down to the atomic level with ab initio calculations, we confirm the single crystalline character of Cr2Te3 films on the 2D materials with sharp vdW interfaces. They all exhibit PMA and TC close to the bulk Cr2Te3 value of 180 K. Ab initio calculations confirm this PMA and show how its strength depends on strain. Finally, Hall measurements reveal a strong anomalous Hall effect, which changes sign at a given temperature. We theoretically explain this effect by a sign change of the Berry phase close to the Fermi level. This transition temperature depends on the 2D material in proximity, notably as a consequence of charge transfer. MBE-grown Cr2Te3/2D material bilayers constitute model systems for the further development of spintronic devices combining PMA, large spin-orbit coupling and sharp vdW interface. arXiv:2303.03076v1 [cond-mat.mtrl-sci] I. INTRODUCTION The discovery of ferromagnetic order in two-dimensional (2D) materials like Cr 2 Ge 2 Te 6 [1] and CrI 3 [2] has paved the way for the development of new van der Waals (vdW) heterostructures [3]. Combined with the large spin-orbit coupling and low crystal symmetries of 2D materials like transition metal dichalcogenides (TMD) [4], 2D ferromagnets represent a key ingredient to construct ultracompact devices for spintronic applications [5] such as spin transfer torque (STT) or spin-orbit torque (SOT) magnetic random access memories (MRAMs). These technologies based on 2D materials would allow for the miniaturization of today's devices as well as a sizeable reduction of energy consumption [6]. For this purpose, 2D ferromagnets with Curie temperatures (T C ) higher than room temperature and with perpendicular magnetic anisotropy (PMA) are required [7]. Fe x GeTe 2 (x=3, 4 or 5) [8] and Cr 1+δ Te 2 (0 ≤ δ ≤ 1) [6][7][8][9][10][11][12][13][14][15][16][17][18][19] have emerged recently as the two most promising families of materials to achieve such conditions. Cr 1+δ Te 2 * QG and LV contributed equally to this work materials are composed of 1T-CrTe 2 monolayers (ML) separated by a variable amount of intercalated chromium atoms (from empty to fully occupied). CrTe 2 is a vdW ferromagnet with room temperature ferromagnetic order (T C = 315 K) [9][10][11], whereas Cr 1+δ Te 2 (δ > 0) are quasi vdW ferromagnets with T C ranging from 160 K to 350 K and varying magnetic anisotropy from out-of-plane to inplane easy axis of magnetization [9][10][11][12][13][14]. Magnetic properties of Cr 1+δ Te 2 have been shown to depend on its stoichiometry [12,13], its thickness in case of thin films [14], the Cr-Te flux ratio during crystal growth [15] and strain in the layer [16,17]. The stoichiometry of the stack could be adjusted by post-growth annealing [12] or by changing elemental fluxes [18]. Highly efficient control of magnetic properties is required for spintronic applications [5] and it is, therefore, necessary to understand the growth mechanisms of these materials, especially for the development of functional vdW heterostructures. Exotic topological phenomena such as the topological Hall effect have also been reported in Cr 2 Te 3 /Bi 2 Te 3 bilayers [19,20] or Cr 2 Te 3 /Cr 2 Se 3 [21]. Moreover, non-collinear spin textures were shown in Cr 2 Te 3 as a consequence of antiferromagnetic coupling between neighboring chromium atoms [22] making it an interesting host for exotic, trivial or topological spin textures. In this work, we report the vdW epitaxy [23,24] of 5 ML of Cr 2 Te 3 on three different 2D materials, namely graphene (a semimetal with exceptional electronic properties), WSe 2 (a transition metal dichalcogenide semiconductor exhibiting strong photo-luminescence and spinvalley locking in its monolayer form), and Bi 2 Te 3 (a topological insulator with strong spin-orbit interaction). Particular care was given to their full structural and magnetic characterizations including the determination of the film stoichiometry. Those bilayers represent model systems to study proximity effects in vdW heterostructures, interface spin textures as well as spin-orbit torques. The Cr 2 Te 3 films were grown by molecular beam epitaxy (MBE) in ultrahigh vacuum (UHV) by depositing simultaneously Cr and Te atoms. They exhibit in-plane compression and out-of-plane expansion with respect to the bulk phase. This strain is shown to vary with the postgrowth annealing, but it is almost independent of the 2D layer underneath. Indeed, Cr 2 Te 3 films on graphene and WSe 2 annealed at 400°C show the same lattice parameters, which are equal to the ones of 5 ML free-standing Cr 2 Te 3 calculated by ab initio methods. This demonstrates that the vdW epitaxy of Cr 2 Te 3 on 2D materials leads, after annealing, to the formation of quasifreestanding films with negligible interaction with the substrate. We then correlate the PMA of Cr 2 Te 3 with strain and confirm our experimental findings using ab initio calculations. Finally, magnetotransport measurements reveal a change of sign of the anomalous Hall effect in Cr 2 Te 3 with temperature and point out a charge transfer from the substrate to the film changing the ptype doping level of Cr 2 Te 3 . This effect has already been observed in several vdW heterostructures [25,26]. The charge transfer is shown to govern the temperature at which the anomalous Hall effect changes sign. We reproduce theoretically this effect by showing a sign change of the Berry phase close to the Fermi level. Finally, our work demonstrates the ability of MBE to synthesize model vdW heterostructures incorporating 2D materials and quasi vdW ferromagnets which are highly promising for future 2D-based spintronic devices. II. METHODS A. Experimental methods All the films were grown by MBE using a home-designed UHV system. Metallic elements (Cr, W, Bi and Al) were evaporated using an electron gun and the growth rate was controlled using a quartz microbalance, whereas chalcogens (Te, Se) were evaporated from Knudsen cells. Their elemental fluxes were measured by a pressure gauge. The substrates were attached to a molybloc by wetting In underneath. The temperature of the samples during growth was controlled by a thermocouple touching the backside of the molybloc. Scanning transmission electron microscopy (STEM) mea-surements were performed using a Cs-corrected FEI Themis at 200 kV. HAADF-STEM (high-angle annular dark field) images were acquired using a convergence angle of 20 mrad and collecting electrons scattered at angles higher than 60 mrad. STEM specimens were prepared by the focused ion beam (FIB) lift-out technique using Zeiss Crossbeam 550. The sample was coated with protective carbon and platinum layers prior to the FIB cut. The out-of-plane x-ray diffraction (XRD) measurements were performed using a Panalytical Empyrean diffractometer operated at 35 kV and 50 mA, with a cobalt source, (Kα = 1.79Å). A PIXcel-3D detector allowed a resolution of 0.02°per pixel, in combination with a divergence slit of 0.125°on the source side. Grazing in-plane XRD measurements were performed with a SmartLab Rigaku diffractometer equipped with a copper rotating anode (Kα = 1.54Å) operating at 45 kV and 200 mA. Collimators with a resolution of 0.5°were used both on the source and the detector sides. The grazing incidence close to the critical angle of the substrate was optimized to maximize the intensity of the Cr 2 Te 3 Bragg peaks. Both diffractometers were equipped with multi-layer mirrors on the source side and Kβ filter on the detector side. Raman measurements were performed with a Horiba Raman setup with a 632 nm laser excitation source and a spot size of 0.5 µm. The signal was collected by using a 1800 grooves/mm grating. Rutherford backscattering (RBS) measurements were performed with a 4 He + beam delivered by the SAFIR Platform at Sorbonne University in Paris at beam energies ranging from 1.5 to 2.0 MeV. For all samples, the scattering angle was set to 160°and the resolution of the detector was 13.5 keV. To avoid channeling effects, the samples were tilted with respect to the normal of the sample in two perpendicular directions. The magnetic properties were measured by superconducting quantum interference device (SQUID) magnetometry with the magnetic field applied parallel or perpendicular to the film plane. The measurements were performed using a Quantum Design magnetic property measurement system. The diamagnetic contribution was subtracted using the data at high field (≥ 3T) and some parasitic contributions were corrected by subtracting signals measured well above the T C of the systems (at 350K). This method has already been used successfully by Ribeiro et al. [8] and confirmed by comparing it with magnetic moments extracted from x-ray magnetic circular dichroism (XMCD) measurements. The XMCD measurements were performed on the DEIMOS beamline [27] of synchrotron SOLEIL (Saint Aubin, France). The signals were recorded using the Total Electron Yield (TEY) method. Each XMCD spectrum was obtained from four measurements, where both the circular helicity and the direction of the applied magnetic field were flipped. The XAS data are then averaged (the signals of opposite helicity and field) and normalized to the absorption at the pre-edge of chromium (565 eV). The XMCD spectra are normalized to their maximum for comparison. In order to carry out magnetotransport measurements, we processed Hall bars out of Cr 2 Te 3 films by laser lithography and argon etching. Electrical contacts were made of e-beam evaporated Ti(10 nm)/Au(100 nm) bilayers. The length and width of Hall bars were approximately 100 µm and 10 µm respectively. All the electrical measurements were performed using an OXFORD Spectromag setup working in the 1.6-300 K temperature range with magnetic fields up to 7 Tesla. The anomalous Hall contribution was obtained by fitting the experimental data with a hyperbolic tangent function. B. Calculation methods The ab initio calculations were performed using density functional theory (DFT) as implemented in the Vienna ab initio simulation package (VASP) [28,29] with the generalized gradient approximation (GGA) pseudopotentials in the Perdew-Burke-Ernzerhof (PBE) parametrization [30]. The DFT+U approach using Dudarev's formulation [31] was applied with an effective Hubbard correction U eff = 2.1 eV to localize the Cr d-orbitals. A Cr pseudopotential with semicore p electrons was chosen and an energy cutoff of 330 eV was used for the plane-wave basis. The van der Waals interaction was approximated by the DFT-D3 method [32] with the Becke-Johnson damping [33]. To compute the relaxed heterostructures of Cr 2 Te 3 with 2D materials, the relative orientation of the two materials in the calculation was not taken from an experiment, but chosen in a systematic way [34] to minimize the lattice mismatch. This captures more realistically the weak epitaxy of the heterostructure. The magnetic anisotropy calculation procedure is described in [35]. A 9x9x5 k-point mesh was found to be sufficient. The volume was fixed at its calculated equilibrium bulk value, while the in-plane and out-of-plane lattice parameters (a and c) were varied. A demagnetizing energy contribution E demag = −µ 0 M 2 s /2 was added to the calculated magnetocrystalline energy, using the experimental M s value ≈ 300 kA/m. The anomalous Hall effect was computed [36] by constructing a tight-binding Hamiltonian based on maximally localized Wannier functions using the Wannier90 package [37]. We verified carefully that the model reproduces well the band structure of Cr 2 Te 3 from the original DFT calculation. Using the WannierBerri package [38,39], the Berry curvature was then calculated for a dense k-point mesh and integrated over the Brillouin zone to obtain the Berry phase, proportional to the anomalous Hall conductivity, at various Fermi level positions. III. SAMPLE PREPARATION In this study, we have grown samples of 5 layers of Cr 2 Te 3 corresponding to a thickness of 6.1 nm on three different vdW surfaces: 1 monolayer of WSe 2 deposited on GaAs, 10 layers of Bi 2 Te 3 on Al 2 O 3 , which were both grown in situ by MBE and monolayer graphene, which was obtained by the controlled graphitization of 4H-SiC(0001) [40] in another reactor. WSe 2 was grown epitaxially on Se-passivated GaAs(111)B as detailed in [41]. Bi 2 Te 3 was grown epitaxially on sapphire. For this purpose, sapphire substrates were first annealed in air for one hour at 1000°C with a heating ramp of 40 minutes starting from room temperature. They were additionally annealed in situ for 30 minutes at 800°C. 10 quintuple layers of Bi 2 Te 3 were then grown by co-evaporating Bi and Te from an electron gun and a cracker cell at deposition rates of 0.057 and 0.1Å/s, respectively. The substrate temperature was maintained at 250°C during the growth. Post-growth annealing at 300°C under Te flux was done for 10 minutes to improve the crystal quality. Finally, Gr/SiC layers were annealed in situ for 30 minutes at 650°C after their transfer. The Cr 2 Te 3 films were grown using a two-step process as sketched in Fig. 1(a). The growth temperature, Te:Cr ratio and deposition rate were set at 300°C, 10, and 0.25 L/min respectively. The Te cell shutter was opened 1 minute before chromium deposition to ensure that the surface of the substrate was saturated in Te at the first stage of growth [see Fig. 1(a)]. After the growth, the samples were annealed at 400°C for 10 minutes using the same Te flux as during the growth and a heating ramp of 40°C/min. The samples were then cooled down to 50°C and 3 nm of aluminum was deposited to prevent oxidation of the layers during transfers between experimental setups. The film morphology was monitored in situ by reflection high-energy electron diffraction (RHEED) as can be seen in Fig. 1(b) in the case of WSe 2 . A streaky diffraction pattern was observed indicating a flat and well-crystallized surface. The different diffraction patterns along the two high symmetry axes of the WSe 2 substrate [ Fig. 1(c)-(e)] indicate a good alignment of Cr 2 Te 3 grains with the underlying layer. After annealing, the width of the diffraction rods is approximately divided by 2 as a consequence of the larger grain size [ Fig. 1 (d)-(f)]. We made similar RHEED observations for the two other vdW substrates Bi 2 Te 3 and graphene (see the Supplemental Material Fig. S1), except an increased isotropic contribution on graphene attributed to a lower interaction with the substrate. IV. STRUCTURAL PROPERTIES We found similar structural characteristics for Cr 2 Te 3 grown on WSe 2 and graphene. Therefore, we present the results of the growth on WSe 2 and the ones on graphene are given in the Supplemental Material. The results on Bi 2 Te 3 are shown in Fig. 5. Figure 2(a) is a cross-section STEM image of the layers, revealing a sharp and well-defined interface between the vdW ferromagnet and the 2D layer as evidenced by the W-Te distance between W atoms of WSe 2 and the first Te atoms plane of Cr 2 Te 3 with a value of 5.3Å. This value is taken to obtain a better experimental determination of the gap (due to the large atomic number of W with respect to Se) as can be seen in Fig. 2(b) showing a line profile along the c-direction of the heterostructure. This corresponds to a vdW gap ∆c vdW of 3.5Å if we assume a relaxed WSe 2 layer, in agreement with our XRD data (see Fig. 3). It is worth noting that we resolve an intensity difference between fully and partially filled Cr planes. Indeed, every two Cr planes, only one third of the lattice sites are occupied by intercalated Cr atoms in the case of Cr 2 Te 3 . We also observe that the monolayer of WSe 2 remains intact after the growth of Cr 2 Te 3 on top. The experimental W-Te distance was compared with the one obtained by ab initio calculations performed on To accurately determine the lattice parameters of Cr 2 Te 3 layers and crystal orientation with respect to each of the substrates, systematic XRD analysis were performed to extract the in-plane and out-of-plane lattice parameters. These measurements allowed us to measure accurately the strain in each layer when compared with the bulk lattice parameters. Figure 3(a), 3(b) and 3(c) show XRD out-ofplane, in-plane radial and in-plane azimuthal scans of Cr 2 Te 3 /WSe 2 /GaAs (sample 2, see Table I) respectively. The diffraction patterns of Cr 2 Te 3 deposited on WSe 2 reveal the single crystalline character of the film and the clear epitaxial relationship with WSe 2 . The thin Bragg peaks in radial scans in Fig. 3 (b) (Full Widths at Half Maximum -FWHM-of 0.48°and 0.70°) indicate the large grain size and the uniformity of the lattice parameter. In Fig. 3(c), the mosaic spread (FWHM of 1.54°) is negligibly small which confirms the perfect orientation of Cr 2 Te 3 on WSe 2 . All the XRD data are summarized in Table I. Compared to bulk values of Cr 2 Te 3 with a = 6.812Å and c = 12.07Å [42], we systematically found an in-plane compressive strain and a resulting out-ofplane expansion. We found similar lattice parameters regardless of the substrate underneath although the mismatch between inter-atomic distances is very large (WSe 2 :+19.1%/Gr:+56.3%/Bi 2 Te 3 :-10.8%). An in-plane compressive strain would be expected for Cr 2 Te 3 deposited on WSe 2 and graphene, whereas an in-plane tensile strain would be expected for the growth on Bi 2 Te 3 . Moreover, we could not find any commensurable relationship between the in-plane lattice parameter of Cr 2 Te 3 and the one of the substrate. This was confirmed by ab initio calculations: the lattice parameter of the vdW heterostructure corresponds to the one of bulk Cr 2 Te 3 above 7 MLs of Cr 2 Te 3 , as demonstrated for Cr 2 Te 3 /Gr (see the Supplemental Material Fig. S2). We thus conclude about the pure vdW interaction between Cr 2 Te 3 and the substrate. The slight difference between lattice parameters might be due to the surface topography (presence of steps, terrace, etc.) and the microscopic structure of Cr 2 Te 3 (grain size, grain boundaries, etc.). However, the energy given to the system by annealing seems to be the driving force to control the final crystal structure since all the lattice parameters converge to the same values after annealing at 400°C. Besides, layers grown on graphene exhibit a much larger mosaic spread even after annealing, indicating even lower interaction between Cr 2 Te 3 and the substrate during growth. The measured lattice parameters match well with the ab initio calculations performed on ∼ 5 nm-thick freestanding films which corresponds to the experimental thickness (see Fig. 4). In particular, for the Cr 2 Te 3 films annealed on graphene and WSe 2 (samples 2 and 4), the experimental lattice parameters fall exactly on the theoretical curve confirming a weak interaction between the film and the substrate. The case of Cr 2 Te 3 grown on Bi 2 Te 3 is discussed later in Fig. 5. In Table I, we also show the measured composition of some selected samples by RBS (see the Supplemental Material Fig. S4) and found compositions very close to Cr 2 Te 3 . No measurement could be performed on GaAs substrates as Ga and As are heavier than chromium, causing the Cr signal peak to lie in the substrate background, preventing any determination of the Cr:Te ratio in these samples. Raman spectroscopy was also performed before and after the growth of Cr 2 Te 3 to control the quality and integrity of the 2D layers. Figure 3(d) depicts the Raman spectra of WSe 2 /GaAs and Cr 2 Te 3 /WSe 2 /GaAs. The width and position of WSe 2 peaks are preserved, indicating that the deposition of Cr 2 Te 3 did not alter the WSe 2 layer. The reference signal of WSe 2 /GaAs (green curve) was measured with a 532 nm laser instead of 633 nm as the other samples, explaining the intensity differences. Similar observations have been made on the Cr 2 Te 3 /graphene heterostructure as shown in the Supplemental Material Fig. S5. Figure 5(a) shows the Raman spectra of Cr 2 Te 3 /Bi 2 Te 3 /Al 2 O 3 at different stages of growth. We detected two characteristic peaks of Bi 2 Te 3 at 101.8 cm −1 and 133.5 cm −1 , which correspond to the E 2 g and A 2 1g vibrational modes and have also been reported in [43]. After the deposition of 5 layers of Cr 2 Te 3 at 300°C, those peaks remained unchanged (the amplitude drop is explained by the partial absorption of the laser fluence in the metallic Cr 2 Te 3 layer). However, when the sample was annealed at 400°C, the two characteristic peaks of Bi 2 Te 3 disappeared. Indeed, x-ray diffraction measurements performed before and after annealing in Fig. 5(b) clearly show the disappearance of Bi 2 Te 3 after thermal annealing. Finally, in Fig. 5(c), RBS measurements on the annealed sample show the absence of Bi in the heterostructure. This reveals that Bi 2 Te 3 was evaporated during annealing leaving the Cr 2 Te 3 film on the pristine sapphire substrate. We shaded the structural data of this sample (n°6) in Table I because Cr 2 Te 3 is standing directly on the sapphire substrate after annealing. Moreover, in this case, the substrate is no more vdW and defects might have been created in Cr 2 Te 3 by evaporation of the Bi 2 Te 3 layer underneath. V. MAGNETIC PROPERTIES Hysteresis loops were measured by SQUID magnetometry at 5 K and are displayed in Fig. 6. For all samples, the easy axis of magnetization was found along the caxis and by integrating the difference of area between the out-of-plane and the in-plane magnetization curves, the magnetic anisotropy energy (MAE) could be experimentally derived for all the samples. The origin of ferromagnetism in our layers was confirmed by XMCD performed at the SOLEIL synchrotron radiation source. The energy spectra are shown in Fig. 7 and a hysteresis loop is displayed in the Supplemental Material Fig. S6. A clear magnetic dichroism signal with a similar spectral shape was obtained for all the three different substrates. This proves that the chemical environment of Cr atoms in Cr 2 Te 3 films is essentially independent of the substrate. The lower magnetic moment for the sample on Bi 2 Te 3 /Al 2 O 3 [ Fig. 7(c)] is explained by a lower sample thickness (three monolayers instead of five). To better understand the magnetic properties, the magnetic anisotropy energy was calculated theoretically as a function of strain for bulk Cr 2 Te 3 and was compared to experimental values in Fig. 8. The results reveal that the MAE is correlated to the strain of the layers. Overall, the trend and magnitude correspond well with experimental data. In particular, there is no sharp discontinuous change from positive to negative anisotropy values, as reported in [14]. However, the experimental data show larger PMA values compared to the theory. Since our calculations were performed in bulk Cr 2 Te 3 , we can attribute this shift to the presence of interfacial PMA at the Cr 2 Te 3 /substrate or Cr 2 Te 3 /AlO x capping layer interfaces. To determine the T C of each annealed sample, we recorded the remanent magnetization after saturation at 5 T (with 5 K steps and no external field) as a function of temperature (Fig. 9). A value close to 180 K was found for the three substrates demonstrating again the very weak interaction between Cr 2 Te 3 and the vdW substrates. Here, we believe that the T C is fully determined by the 2:3 stoichiometry of the films. VI. MAGNETOTRANSPORT To study the magnetotransport properties, we performed four-probe resistance measurements and found an increasing longitudinal resistivity of Cr 2 Te 3 layers with temperature indicating a metallic character (see the Supplemental Material Fig. S8). The resistivity is of the order of 500 µΩ.cm at 4 K. Figure 10(a) shows the Hall resistivity of 5 ML of Cr 2 Te 3 deposited on WSe 2 (sample 2) as a function of the perpendicular magnetic field at different temperatures. For visibility, the ordinary Hall slope was subtracted and a carrier density of 1.6×10 15 holes/cm 2 was extracted at 50 K, compared to 7.0×10 15 holes/cm 2 for 5 ML of Cr 2 Te 3 directly deposited on sapphire (see the Supplemental Material Fig. S9), indicating a charge transfer from the WSe 2 layer. The clear anomalous Hall signal confirmed the strong PMA of the ferromagnet. The same measurements were performed for a sample grown on Bi 2 Te 3 (sample 6) and annealed at 400°C (resulting in Bi 2 Te 3 evaporation), as shown in Fig. 10(b). The ordinary hall slope was removed and a carrier density of 4.5×10 15 holes/cm 2 was extracted at 50 K. Since there is no charge transfer with sapphire, the difference in carrier density with Cr 2 Te 3 directly grown on sapphire could be explained by the presence of defects at the interface introduced during the evaporation of the Bi 2 Te 3 layer. In Fig. 11(a), the anomalous Hall resistivity is extracted for the two samples on sapphire as a function of temperature. We observe in both cases a sign change of the anomalous Hall resistivity below the Curie temperature of 180 K. Similar observations were reported in [44,45]. The possible origin of this effect is discussed in the following as a consequence of the energy-dependent Berry phase of Cr 2 Te 3 . In the temperature range of the AH resistivity sign reversal, a resonance of the Hall signal manifested as peaks at the coercive fields can be observed. Figure 11(b) shows the Hall resistivity after subtraction of the ordinary and anomalous Hall effect at two temperatures below the sign change and one above. The bumps are enhanced when the temperature is closer (but still lower) than the temperature of the sign change and disappear above it. The width of the bumps also decreases with temperature, which could be related to the shrinking of the coercive field. The physical origin of such an effect is still under debate. In a similar heterostructure, Chen et al. [19] interpreted it as the topological Hall effect, which would originate from the presence of magnetic skyrmions. Skyrmions nucleate during the magnetization reversal and give rise to an extra transverse transport channel inducing a peak in the Hall resistivity. Imaging such spin textures has been performed by Lorentz-TEM in Cr 3 Te 4 layers [46]. Nevertheless, another explanation has been put forward by other groups as two anomalous Hall contributions with opposite signs [47]. The origin could be thickness variations, inhomogeneities in the film or interface effects leading to the sign of the AHE being different [48,49]. In the case of Cr 2 Te 3 these peaks appear close to the anomalous Hall resistivity sign change. If the thickness of the layer is not strictly constant over the Hall bar, some areas could have slightly different temperatures at which the anomalous Hall signal changes sign. In this case, for intermediate temperatures, two AHE components with opposite signs would indeed add up and could explain the observed behavior. The sign reversal of the anomalous Hall effect observed experimentally can be elucidated by ab initio calculations. The longitudinal resistivity is in the range where the contribution to AHE from intrinsic and impurity scattering components coexist, while the intrinsic part stays significant [45]. We thus calculated the intrinsic contribution to AHE for bulk Cr 2 Te 3 (see Methods). As shown in Fig. 12, it exhibits a clear sign reversal very close to E F (∼ −10 meV). This is in contrast with previous calculations [21] where the sign reversal occurs 330 meV above E F . This difference is due to the inclusion of the vdW corrections in our DFT calculations (see the Supplemental Material Fig. S10). We consider three different mechanisms influencing the value and sign of the anomalous Hall conductivity: (a) thermal broadening around the Fermi level (of the order of k B T , i.e., 15 meV for ∆T = 180 K), (b) charge transfer with the substrate (which we calculated was greatest on graphene inducing a Fermi level shift of ≈ +50 meV), and (c) outof-plane strain (see the Supplemental Material Fig. S11). All these effects change the system energy in a range compatible with the calculations in Fig. 12. We believe that the strain dependence of σ int. AH is at the origin of the change of sign of the AHE reported in Fig. 11(a). Anisotropic lattice expansion with temperature was reported for Cr 1+δ Te 2 [50], which directly affects the AHE conductivity. To illustrate this qualitative argument, we chose in Fig. 12 two reasonable strain values in agreement with the structural data we obtained, keeping in mind that films in this work have c/a ranging between 1.797 and 1.868 (see Fig. 8). If the Fermi level of the sample lies in the red shaded area (between -8 and 0 meV), the evolution of c/a from 1.79 to 1.82 with temperature would lead to a sign change of σ int. AH . Another effect that could influence this picture is the thermal broadening of the Fermi-Dirac distribution upon heating. However, we obtained a mostly linear dependence of σ int. AH on energy close to the Fermi level. When considering contributions above and below E F , both would cancel out as the thermal broadening is symmetric. Finally, the role of the substrate has to be also accounted for. As shown experimentally, charge transfer with the 2D materials was observed and leads to a shift of the Fermi level. This explains why the effect is present for samples standing on sapphire and not for the one on WSe 2 . Indeed, for sapphire, the Fermi level is the lowest (carrier density of 4.5×10 15 holes/cm 2 and 7.0×10 15 holes/cm 2 ) so that we observe a sign change whereas the Fermi level is shifted up for WSe 2 (1.6×10 15 holes/cm 2 ) and the sign change is absent. This observation is in agreement with the fact that the sign change in ab initio calculations occurs for lower energies as shown in Fig. 12. Finally, in Fig. 13, we present magnetotransport measurements on 5 layers of Cr 2 Te 3 grown on graphene/SiC (sample 3). Both layers are metallic and contribute to conduction. The Hall resistivity is plotted as a function of the applied perpendicular magnetic field at different temperatures. No anomalous Hall resistivity sign change is measurable below the Curie temperature. This observation is in good agreement with ab initio calculations since the extracted carrier density at 50 K, which is the lowest: 1.4×10 14 holes/cm 2 , corresponds to a Fermi level position shifted towards higher values. On top of the anomalous Hall contribution following the magnetization reversal at 0.5 T (at 50 K), another step close to 0.4 T is also present. This two-step signal behavior (absent in SQUID and XMCD measurements) vanishes progressively when increasing the temperature and disappears around 100 K, well below the Curie temperature. The origin of this effect needs further investigation and is out of the scope of the present work. VII. CONCLUSION In conclusion, we reported the vdW epitaxy of Cr 2 Te 3 on three different 2D materials. We revealed the pristine interface and the preservation of the intrinsic properties of the underlying layers after the growth of the vdW ferromagnet. We demonstrated the free-standing character of Cr 2 Te 3 layers grown on these 2D materials after an annealing step at 400°C. Besides, the energy given to the system during the growth was identified as a way to control the crystal structure and tune the magnetic properties. We observed a correlation between the PMA energy of the system and the lattice parameters which was elucidated by ab initio calculations. Finally, we theoretically predicted a strain-sensitive sign change of the Berry phase very close to the Fermi level explaining the measured sign change of AHE with temperature. Charge transfer between the 2D layers and Cr 2 Te 3 was shown to directly affect the temperature at which the AHE changes sign by shifting the Fermi level. To summarize, this system outputs highly tunable structural, magnetic and electrical properties, which presents an important asset for future spintronic applications. ACKNOWLEDGMENTS This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 800945 -NUMER-ICS -H2020-MSCA-COFUND-2017 and grant agreement 881603 (Graphene Flagship). We also acknowledge the French National Research Agency through the MAG-ICVALLEY project (ANR-18-CE24-0007) and the ESR-Equipex+ project 2D-MAG on two-dimensional magnetic materials. We acknowledge the financial support from the ANR project ELMAX (ANR-20-CE24-0015) and from the LANEF framework (ANR-10-LABX-51-01) for its support with mutualized infrastructure. This work was partly supported by the French RENATECH network. XMCD experiments were performed on the DEIMOS beamline at SOLEIL Synchrotron, France (proposal number 20220542). We are grateful to the SOLEIL staff for smoothly running the facility. TABLE I: Growth/annealing temperature and structural parameters measured by x-ray diffraction and chemical composition from RBS, with a (c) the in-plane (out-of-plane) lattice parameter, the radial width (∆θ // ) of the (300) diffraction peak and the mosaic spread (∆φ) measured on the same Bragg peak. Films with E F within the red energy range will experience σ int. AH sign reversal upon this possible strain change with temperature. Similar to [S1] we define the formation energy by: F form = (F tot − n Cr µ Cr − n Te µ Te )/N atoms(1) where F tot is the total calculated free energy, µ i is the chemical potential and n i is the number of atoms of element i in the unit cell, and N atoms is the total number of atoms per unit cell. The chemical potential µ is taken either as the free energy of an isolated atom (Cr or Te) or the free energy per atom in the bulk of the given material. Both cases give the same qualitative result regarding the surface termination (see Table S1). From our calculations, we conclude that the Te-terminated surface is indeed the most stable (Table S1), although the formation energy of the intercalated Cr-terminated structure is very close -the difference is only 35 meV/atom (for µ from the Cr and Te bulk structures). S4. STOICHIOMETRY OF THE LAYERS To verify the stoichiometry of the grown samples, Rutherford Back Scattering (RBS) was performed (see Fig. S4). The sample on graphene (n°3) was capped with amorphous Se and the one on Bi 2 Te 3 (sample 5) was protected with Al. The ratio found is close to the expected 40% Cr and 60% Te. S5. RAMAN SCATTERING OF MONOLAYER GRAPHENE Raman scattering is used to control the graphene quality [S4]. We show in Fig. S5 that graphene is mostly unaffected by the deposition of Cr 2 Te 3 . FIG. S5: Two characteristic Raman scattering peaks of monolayer graphene before and after deposition of Cr 2 Te 3 . S6. HYSTERESIS LOOP MEASURED BY XMCD To confirm the origin of the magnetic signal in Cr 2 Te 3 layers, we measured at the Soleil synchrotron source an hysteresis loop at the chromium L 3 edge and obtained a consistent signal with the experimental observations by SQUID magnetometry. (when the Bi 2 Te 3 layer evaporated), Cr 2 Te 3 was directly deposited on sapphire and measured (see Fig. S9). The extracted carrier density was found higher (7.0*10 15 holes/cm 2 at 50 K) and a similar sign change of the anomalous Hall resistivity is noticeable below the Curie temperature. The temperature at which the resistivity switches sign occurs below the sample deposited on Bi 2 Te 3 . This difference is attributed to the presence of defects after removal of Bi 2 Te 3 and therefore to different Fermi levels in these two samples. Extracted anomalous Hall resistivity as a function of temperature. S10. ANOMALOUS HALL EFFECT SIGN REVERSAL The anomalous Hall effect sign reversal around the Fermi level is sensitive to strain [ Fig. S10(a)], but not to volume expansion [ Fig. S10(d)]. The strain can significantly change both the value of σ intr. AH at E F [ Fig. S10(b)] and the Fermi level position at which the sign reversal occurs [Fig. S10(c)]. In comparison, a realistic thermal volume expansion (0.5% per 300K [S5]) does not change the σ intr. AH curve characteristics significantly [ Fig. S10(e)-(f)]. Note that an equivalent calculation in Ref. [S6] does not show the reversal around E F . We could reproduce this result by neglecting the vdW correction. The vdW correction is, however, important, due to the presence of the pseudo vdW gap in Cr 2 Te 3 . Otherwise, the relaxed unit cell volume is overestimated (by ≈ 8%) compared to both experiment and the calculation with vdW correction included. A reversal is still present in Ref. [S6], but at E − E F = 300 meV. We performed a charge transfer calculation and show that a charge transfer from the 2D materials to Cr 2 Te 3 should indeed increase its Fermi level, but only by 25 meV at most. Therefore a sign reversal region around 300 meV seems unrealistic. We calculate heterostructures of Cr 2 Te 3 /2D materials to estimate the charge transfer between the studied 2D materials and Cr 2 Te 3 (see Fig. S11). We observe a positive electron transfer to Cr 2 Te 3 in all three cases. The resulting shift of the Cr 2 Te 3 Fermi level compared to its bulk value is estimated by dividing the number of transferred electrons/unit cell by the density of states of bulk Cr 2 Te 3 at E F , in this case, 8.05 electrons/eV for the considered unit cell. In the case of graphene, the charge transfer is one order of magnitude larger than for WSe 2 or Bi 2 Te 3 , leading to an estimated ∆E F ≈ 54 meV. Cr 2 2Te 3 /WSe 2 consisting of 1.5 unit cell thick Cr 2 Te 3 on top of a single layer of WSe 2 [see Fig. 2(c)]. The calculated distance is 5.08Å, in good agreement with the experimental one of 5.3Å. FIG. 1 : 1MBE growth of Cr 2 Te 3 on the 2D transition metal dichalcogenide WSe 2 . (a) Sketch of the deposition procedure. The growth temperature was 300°C and in situ annealing was performed at 400°C. (b) in situ RHEED images of 1 layer WSe 2 deposited on GaAs(111)B along two crystal directions (Time = 0 min). (c) RHEED pattern after the deposition of Cr 2 Te 3 (Time = 21 min). (d) RHEED pattern after annealing (Time = 35 min). (e,f) Intensity profiles of the RHEED diffraction pattern for images c (blue dashed box) and d (red dashed box) respectively. N°2D layer Temperature a (Å) FIG. 2 : 2(a) Low-pass filtered HAADF-STEM image of 5 layers Cr 2 Te 3 grown on one monolayer WSe 2 deposited on a GaAs(111)B surface. The van der Waals gap between the layers is shown to highlight the high quality of the interface. Arrows on the right side indicate the position of the atomic planes noted in the line profile. (b) Line profile along the c-direction of Cr 2 Te 3 layers (yellow arrow) with intensity distinction between partially and fully occupied Cr planes [see crystal structure in (c)]. (c) A unit cell of the calculated Cr 2 Te 3 /WSe 2 heterostructure -in the interstitial planes, only 1/3 of the lattice sites is occupied by the intercalated Cr atoms. The ab initio-calculated W-Te distance is 5.08Å. FIG. 3: Post-growth characterization of the crystal structure of Cr 2 Te 3 /WSe 2 /GaAs (sample 2). (a) Out-of-plane Θ/2Θ XRD scan shows, in addition to Cr 2 Te 3 (00l) peaks (red), GaAs substrate peaks (black) with weak additional peaks due to spurious radiations not completely eliminated by the mirror and the Kβ filter (grey). (b) In-plane radial XRD scans performed along the GaAs substrate R=(hh0 ) direction, and R+30°=(2hhh) direction. These scans show the substrate peaks (black), WSe 2 (green), and Cr 2 Te 3 peaks (red) labeled with their FWHM. (c) In-plane azimuthal XRD scan of the (300) peak measured within a range of 100°shows thin peaks with a FWHM of 1.54°separated by 60°corresponding to the 6-fold symmetry of the crystal. (d) Raman spectra of WSe 2 before and after deposition of Cr 2 Te 3 . FIG. 4: Top: Cr 2 Te 3 bulk crystal structure and a thin free-standing film constructed from it. The in-plane lattice parameter a was fixed at a range of values while the atomic positions were relaxed to obtain the out-of-plane lattice parameter c. Bottom: The calculated and experimental lattice parameters for free-standing slabs and for bulk structures. The measured values follow well the trend calculated for free-standing Cr 2 Te 3 films. FIG. 5: Structural properties of Cr 2 Te 3 on Bi 2 Te 3 /Al 2 O 3 . (a) Raman spectra of the sapphire substrate, Bi 2 Te 3 /sapphire, and Cr 2 Te 3 /Bi 2 Te 3 /sapphire with and without annealing. Positions and full widths at half maximum of Bi 2 Te 3 peaks are indicated. (b) Radial x-ray diffraction spectra for Cr 2 Te 3 /Bi 2 Te 3 /Al 2 O 3 without (top in blue) and after (bottom in red) annealing. (c) RBS of Cr 2 Te 3 grown on Bi 2 Te 3 / Al 2 O 3 without (top in blue) and with (bottom in red) annealing. No elemental Bi can be found after annealing. FIG. 6: SQUID hysteresis loops with out-of-plane (⊥) and in-plane (//) applied magnetic field measured at 5 K are plotted after the removal of the substrate diamagnetic contribution. (a) (b) (c) Measurements on samples 1, 3 and 5 (without annealing on WSe 2 /GaAs, graphene/SiC, and Bi 2 Te 3 /Al 2 O 3 ). (d) (e) (f) Same measurements on samples 2, 4 and 6 (annealed at 400°C). FIG. 7: Top: x-ray absorption spectroscopy (XAS) and bottom: x-ray magnetic circular dichroism (XMCD) measurements performed on Cr 2 Te 3 layers grown and annealed on (a) WSe 2 (sample 2), (b) graphene (sample 4) and (c) Bi 2 Te 3 (sample 6). FIG. 8: Magnetic anisotropy energy of bulk Cr 2 Te 3 as a function of strain compared for experiment and theory. It is a sum of the DFT-calculated magnetocrystalline energy with a demagnetizing energy contribution of -0.06 MJ/m 3 corresponding to the experimentally measured magnetization of ≈ 300 kA/m. FIG. 9: Remanent magnetization of Cr 2 Te 3 layers grown and annealed on WSe 2 (sample 2), graphene (sample 4), and Bi 2 Te 3 (sample 6) as a function of temperature with no external field. FIG. 10: (a) Temperature-dependent Hall resistivity of Cr 2 Te 3 /WSe 2 /GaAs (sample 2) after removal of the ordinary Hall slope with a magnetic field applied out-of-plane. Inset: an optical image of the Hall bar device processed by laser lithography with Ti(10nm)/Au(100nm) contacts. (b) Temperature-dependent Hall resistivity of Cr 2 Te 3 /Bi 2 Te 3 /Al 2 O 3 (sample 6) after annealing (evaporation of Bi 2 Te 3 ) and subtraction of the ordinary Hall slope. FIG. 11: (a) Anomalous Hall resistivity of Cr 2 Te 3 /Al 2 O 3 as a function of temperature for direct growth on sapphire, after thermal removal of the Bi 2 Te 3 layer and growth on WSe 2 . (b) Hall resistivity of Cr 2 Te 3 deposited on Bi 2 Te 3 and annealed after subtracting the ordinary and anomalous Hall contributions. The curves are vertically shifted for clarity. FIG. 12: Intrinsic Anomalous Hall conductivity in bulk Cr 2 Te 3 as a function of energy. A sign reversal occurs close to E F . Two different strains (c/a ratios) are considered, in agreement with the experimental ones. FIG. 13 : 2 132Temperature-dependent Hall resistivity of Cr 2 Te 3 /graphene/SiC. An arbitrary slope of 4 µΩ.cm/T has been subtracted for comparison with Fig. 10. CONTENTS S1. RHEED images of Cr 2 Te 3 layers on Graphene/SiC and Bi 2 Te 3 /Al 2 O 3 3 S2. Lattice parameter of heterostructures with increasing Cr Te Layer-resolved magnetic moments in 2D material/Cr 2 Te 3 heterostructures from ab initio calculations 8 S8. Metallic resistivity of Cr 2 Te 3 layers 9 S9. Hall resistivity of Cr 2 Te 3 on sapphire 10 S10. Anomalous Hall effect sign reversal 11 S11. Charge transfer between Cr 2 Te 3 and 2D materials 12 S1. RHEED IMAGES OF CR 2 TE 3 LAYERS ON GRAPHENE/SIC AND BI 2 TE 3 /AL 2 O 3 (1) On Graphene/SiC (2) On Bi 2 Te 3 /Al 2 O 3FIG. S1: (a) In-situ RHEED images of the 2D material along two main crystal directions. (b) RHEED pattern after the deposition of Cr 2 Te 3 at 300°C. (c) RHEED pattern after annealing at 400°C.S2. LATTICE PARAMETER OF HETEROSTRUCTURES WITH INCREASING CR 2 TE 3 THICKNESS We argue that the choice of a particular 2D material has a negligible effect on the Cr 2 Te 3 /2D material lattice parameters. We support this by performing DFT relaxation of Cr 2 Te 3 /graphene heterostructures with different Cr 2 Te 3 thicknesses (see Fig. S2). A linear trend is observed. The value interpolates to the one of Cr 2 Te 3 bulk at ≈ 7 MLs of Cr 2 Te 3 , while the experimentally grown films have a thickness of 5 MLs. Hence, the lattice parameter of Cr 2 Te 3 is expected to retain its bulk value. Even more when considering that in the calculation the two materials are bound to occupy the same unit cell. This makes their lattice parameters coupled more strongly than in reality. FIG. S2: The in-plane lattice parameter of Cr 2 Te 3 /graphene heterostructure with increasing Cr 2 Te 3 thickness. FIG. S4 : S4Stoichiometry of Cr x Te 3 (a) RBS of Cr 1.88 Te 3 deposited on Gr/SiC (sample 3). (b) RBS of Cr 1.97 Te 3 grown on Bi 2 Te 3 / Al 2 O 3 (sample 5). The exact stoichiometry of Bi 2 Te 3 was considered to deduce the Cr/Te ratio in Cr 2 Te 3 . FIG. S6 : S6Hysteresis loop of Cr 2 Te 3 /Graphene/SiC measured at 5 K with a perpendicular applied magnetic field by recording the difference of XMCD signals at the chromium L 3 edge (corresponding to the maximum dichroic signal) and at 565 eV (pre-edge).S7. LAYER-RESOLVED MAGNETIC MOMENTS IN 2D MATERIAL/CR 2 TE 3 HETEROSTRUCTURES FROM AB INITIO CALCULATIONS We simulate heterostructures of Cr 2 Te 3 on Bi 2 Te 3 , graphene, and WSe 2 , as well as Cr 2 Te 3 slabs suspended in vacuum. None of these show a significant change of the interfacial magnetic moments in Cr 2 Te 3 − all the Cr and Te atoms keep their magnetic moments of µ Cr ≈ 3.3 µ B and µ Te ≈ −0.3 µ B , see Fig. S7. 8 FIG. S7: Layer-resolved magnetic moments in ab initio-calculated Cr 2 Te 3 /2D material heterostructures. The magnetic moments of the 2D materials are negligible. Moreover, we do not observe any significant modification of the Cr 2 Te 3 magnetic moments at the interface.S8. METALLIC RESISTIVITY OF CR 2 TE 3 LAYERS The magnetoresistance of 5 layers Cr 2 Te 3 deposited on the insulating substrate Al 2 O 3 is shown in Fig. S8. The resistivity is growing with temperature indicating the metallic nature of Cr 2 Te 3 layers. A change in the slope is noticeable around the Curie temperature of the material. FIG. S8: Longitudinal resistivity of Cr 2 Te 3 deposited on sapphire versus temperature. S9. HALL RESISTIVITY OF CR 2 TE 3 ON SAPPHIRE To understand the Hall signal measured on Cr 2 Te 3 /Bi 2 Te 3 /Al 2 O 3 annealed at 400°C FIG. S9: (a) Hall resistivity of Cr 2 Te 3 deposited on sapphire at different temperatures. (b) FIG. S10: (a) The intrinsic AHC σ intr. AH as a function of the Fermi level position for bulk Cr 2 Te 3 under strain and (b) volume expansion. The strain can have a significant effect on AHC. (c) σ intr. AH at the Fermi level as a function of strain and (d) volume expansion. (e) Shift of the (right-most) sign reversal energy as a function of strain and (f) volume expansion. S11. CHARGE TRANSFER BETWEEN CR 2 TE 3 AND 2D MATERIALS S3. CR 2 TE 3 TERMINATION STABILITYTo build heterostructures made of Cr 2 Te 3 and 2D materials, it is necessary to determine the stable termination of Cr 2 Te 3 , as there are 4 distinct possible terminations(Fig. S3).It has been shown by Wen et al.[S1] that Cr 2 Te 3 should be terminated by Te, with a complete CrTe 2 layer at the boundary. The same conclusion was reached by Lasek et al.[S2] performing molecular dynamics of differently terminated Cr 2 Te 3 on MoS 2 . On the contrary, Bian et al.[S3] concluded that the intercalated Cr is the one that should lie at the interface, arguing that this leads to strain relaxation. Due to this discordance in the literature we performed our own surface stability calculations. TABLE S1 : S1Chemical potentials of Cr and Te atoms and bulk. The formation energy of 4 differently-terminated Cr 2 Te 3 thin films in vacuum. The most stable is structure III, see Fig. S3 for its definition. FIG. S3: Cr 2 Te 3 thin films in vacuum used for the calculation of the surface-terminationstability. Structure III (terminated by a complete CrTe 2 layer) is the most stable, followed by the intercalated Cr-terminated structure I. This corresponds to Ref.[S1].µ (eV) F tot (eV) n Cr n Te F atoms form (eV) F bulk form (eV) Cr atom -4.89 structure I -462.477 33 48 -3.318 -0.365 Te atom -0.68 structure II -474.406 35 48 -3.264 -0.305 Cr bulk -8.10 structure III -504.478 35 54 -3.336 -0.390 Te bulk -3.45 structure IV -483.289 33 54 -3.282 -0.342 Discovery of intrinsic ferromagnetism in two-dimensional van der Waals crystals. 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Zhang, Room-temperature magnetic skyrmions and large topological hall effect in chromium telluride engineered by self-intercalation, Advanced Materials , 2205967 (2022). Distinguishing the two-component anomalous hall effect from the topological hall effect. L Tai, B Dai, J Li, H Huang, S K Chong, K L Wong, H Zhang, P Zhang, P Deng, C Eckberg, G Qiu, H He, D Wu, S Xu, A Davydov, R Wu, K L Wang, ACS Nano. 1617336L. Tai, B. Dai, J. Li, H. Huang, S. K. Chong, K. L. Wong, H. Zhang, P. Zhang, P. Deng, C. Eckberg, G. Qiu, H. He, D. Wu, S. Xu, A. Davydov, R. Wu, and K. L. Wang, Distinguishing the two-component anomalous hall effect from the topological hall effect, ACS Nano 16, 17336 (2022). Controllable thickness inhomogeneity and berry curvature engineering of anomalous hall effect in srruo3 ultrathin films. L Wang, Q Feng, H G Lee, E K Ko, Q Lu, T W Noh, Nano Letters. 202468L. Wang, Q. Feng, H. G. Lee, E. K. Ko, Q. Lu, and T. W. Noh, Controllable thickness inhomogeneity and berry curvature engineering of anomalous hall effect in srruo3 ultrathin films, Nano Letters 20, 2468 (2020). Challenges in identifying chiral spin textures via the topological Hall effect. G Kimbell, C Kim, W Wu, M Cuoco, J W A Robinson, Communications Materials. 31G. Kimbell, C. Kim, W. Wu, M. Cuoco, and J. W. A. Robinson, Challenges in identifying chiral spin textures via the topological Hall effect, Communications Materials 3, 1 (2022). C Li, K Liu, D Jiang, C Jin, T Pei, T Wen, B Yue, Y Wang, Diverse Thermal Expansion Behaviors in Ferromagnetic Cr1-dTe with NiAs-Type, Defective Structures. 6114641C. Li, K. Liu, D. Jiang, C. Jin, T. Pei, T. Wen, B. Yue, and Y. Wang, Diverse Thermal Expansion Behaviors in Ferromagnetic Cr1-dTe with NiAs-Type, Defective Structures, Inorganic Chemistry 61, 14641 (2022). S11: Charge transfer calculation for Cr 2 Te 3 on top of (a) graphene, (b) Bi 2 Te 3 , and (c) WSe 2 . Yellow cloud signifies positive and blue signifies negative electron transfer. * Qg, Lv, In green is given the number of electrons (per unit cell) transferred from the 2D material to* QG and LV contributed equally to this work FIG. S11: Charge transfer calculation for Cr 2 Te 3 on top of (a) graphene, (b) Bi 2 Te 3 , and (c) WSe 2 . Yellow cloud signifies positive and blue signifies negative electron transfer. In green is given the number of electrons (per unit cell) transferred from the 2D material to Cr 2 Te 3 . An estimation of the Fermi level shift ∆E F compared to bulk Cr 2 Te 3 resulting from this charge transfer is also given. Cr 2 Te 3 . An estimation of the Fermi level shift ∆E F compared to bulk Cr 2 Te 3 resulting from this charge transfer is also given. Y S1, Z Wen, Y Liu, C Zhang, B Xia, X Zhai, G Zhang, C Zhai, P Shen, R He, L Cheng, Y Yin, M Yao, Z Sendeku, X Wang, C Ye, C Liu, C Jiang, Y Shan, J Long, He, Tunable Room-Temperature Ferromagnetism in Two-Dimensional Cr 2 Te 3. 203130S1. Y. Wen, Z. Liu, Y. Zhang, C. Xia, B. Zhai, X. Zhang, G. Zhai, C. Shen, P. He, R. Cheng, L. Yin, Y. Yao, M. Getaye Sendeku, Z. Wang, X. Ye, C. Liu, C. Jiang, C. Shan, Y. Long, and J. He, Tunable Room-Temperature Ferromagnetism in Two-Dimensional Cr 2 Te 3 , Nano Letters 20, 3130 (2020). K S2, P M Lasek, K Coelho, Y Zberecki, S K Xin, J Kolekar, M Li, Batzill, Cr-) Tellurides: From Monolayer Ditellurides to Multilayer Self-Intercalation Compounds. Ti148473Molecular Beam Epitaxy of Transition MetalS2. K. Lasek, P. M. Coelho, K. Zberecki, Y. Xin, S. K. Kolekar, J. Li, and M. Batzill, Molecular Beam Epitaxy of Transition Metal (Ti-, V-, and Cr-) Tellurides: From Monolayer Ditellurides to Multilayer Self-Intercalation Compounds, ACS Nano 14, 8473 (2020). M Bian, A N Kamenskii, M Han, W Li, S Wei, X Tian, D B Eason, F Sun, K He, H Hui, F Yao, R Sabirianov, J P Bird, C Yang, J Miao, J Lin, S A Crooker, Y Hou, H Zeng, Covalent 2D Cr2Te3 ferromagnet. 9205M. Bian, A. N. Kamenskii, M. Han, W. Li, S. Wei, X. Tian, D. B. Eason, F. Sun, K. He, H. Hui, F. Yao, R. Sabirianov, J. P. Bird, C. Yang, J. Miao, J. Lin, S. A. Crooker, Y. Hou, and H. Zeng, Covalent 2D Cr2Te3 ferromagnet, Materials Research Letters 9, 205 (2021). Raman Studies of Monolayer Graphene: The Substrate Effect. Y S4, Z H Wang, T Ni, Z X Yu, H M Shen, Y H Wang, W Wu, A T Chen, Shen Wee, The Journal of Physical Chemistry C. 11210637S4. Y. y. Wang, Z. h. Ni, T. Yu, Z. X. Shen, H. m. Wang, Y. h. Wu, W. Chen, and A. T. Shen Wee, Raman Studies of Monolayer Graphene: The Substrate Effect, The Journal of Physical Chemistry C 112, 10637 (2008). Y S5, Y Kubota, T Okamoto, T Kanematsu, D Yajima, K Hirai, Takenaka, arXiv:2211.13388Large magnetic-field-induced strains in sintered chromium tellurides. arXiv preprintS5. Y. Kubota, Y. Okamoto, T. Kanematsu, T. Yajima, D. Hirai, and K. Takenaka, Large magnetic-field-induced strains in sintered chromium tellurides, arXiv preprint arXiv:2211.13388 (2022). . J H S6, H R Jeon, H Na, S Kim, S Lee, J Song, S Kim, J Park, H Kim, G Noh, S.-K Kim, S.-H Jerng, Chun, Emergent Topological Hall Effect from Exchange Coupling inS6. J. H. Jeon, H. R. Na, H. Kim, S. Lee, S. Song, J. Kim, S. Park, J. Kim, H. Noh, G. Kim, S.-K. Jerng, and S.-H. Chun, Emergent Topological Hall Effect from Exchange Coupling in Ferromagnetic Cr2Te3/Noncoplanar Antiferromagnetic Cr2Se3 Bilayers. 9Ferromagnetic Cr2Te3/Noncoplanar Antiferromagnetic Cr2Se3 Bilayers, ACS Nano , 9 (2022).

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{'abstract': 'Achieving large-scale growth of two-dimensional (2D) ferromagnetic materials with high Curie temperature (TC) and perpendicular magnetic anisotropy (PMA) is highly desirable for the development of ultra-compact magnetic sensors and magnetic memories. In this context, van der Waals (vdW) Cr2Te3 appears as a promising candidate. Bulk Cr2Te3 exhibits strong PMA and a TC of 180 K. Moreover, both PMA and TC might be adjusted in ultrathin films by engineering composition, strain, or applying an electric field. In this work, we demonstrate the molecular beam epitaxy (MBE) growth of vdW heterostructures of five-monolayer quasi-freestanding Cr2Te3 on three classes of 2D materials: graphene (semimetal), WSe2 (semiconductor) and Bi2Te3 (topological insulator). By combining structural and chemical analysis down to the atomic level with ab initio calculations, we confirm the single crystalline character of Cr2Te3 films on the 2D materials with sharp vdW interfaces. They all exhibit PMA and TC close to the bulk Cr2Te3 value of 180 K. Ab initio calculations confirm this PMA and show how its strength depends on strain. Finally, Hall measurements reveal a strong anomalous Hall effect, which changes sign at a given temperature. We theoretically explain this effect by a sign change of the Berry phase close to the Fermi level. This transition temperature depends on the 2D material in proximity, notably as a consequence of charge transfer. MBE-grown Cr2Te3/2D material bilayers constitute model systems for the further development of spintronic devices combining PMA, large spin-orbit coupling and sharp vdW interface. arXiv:2303.03076v1 [cond-mat.mtrl-sci]', 'arxivid': '2303.03076', 'author': ['Quentin Guillet \nUniversité Grenoble Alpes\nCEA\nCNRS\nIRIG-SPINTEC\n38000GrenobleFrance\n', 'Libor Vojáček \nUniversité Grenoble Alpes\nCEA\nCNRS\nIRIG-SPINTEC\n38000GrenobleFrance\n', 'Djordje Dosenovic \nUniversité Grenoble Alpes\nCEA\nIRIG-MEM\n38000GrenobleFrance\n', 'Fatima Ibrahim \nUniversité Grenoble Alpes\nCEA\nCNRS\nIRIG-SPINTEC\n38000GrenobleFrance\n', 'Hervé Boukari \nInstitut Neel\nUniversité Grenoble Alpes\nCNRS\n38000GrenobleFrance\n', 'Jing Li \nUniversité Grenoble Alpes\nCEA\nF-38000GrenobleLetiFrance\n', "Fadi Choueikani \nSynchrotron SOLEIL\nL'Orme des Merisiers91190Saint-AubinFrance\n", "Philippe Ohresser \nSynchrotron SOLEIL\nL'Orme des Merisiers91190Saint-AubinFrance\n", 'Abdelkarim Ouerghi \nUniversité Paris-Saclay\nCNRS\nCentre de Nanosciences et de Nanotechnologies\nPalaiseauFrance\n', 'Florie Mesple \nUniversité Grenoble Alpes\nCEA\nCNRS\nIRIG-PHELIQS\n38000GrenobleFrance\n', 'Vincent Renard \nUniversité Grenoble Alpes\nCEA\nCNRS\nIRIG-PHELIQS\n38000GrenobleFrance\n', 'Jean-François Jacquot \nUniversité Grenoble Alpes\nCEA\nCNRS\nIRIG-SYMMES\n38000GrenobleFrance\n', 'Denis Jalabert \nUniversité Grenoble Alpes\nCEA\nIRIG-MEM\n38000GrenobleFrance\n', 'Hanako Okuno \nUniversité Grenoble Alpes\nCEA\nIRIG-MEM\n38000GrenobleFrance\n', 'Mairbek Chshiev \nUniversité Grenoble Alpes\nCEA\nCNRS\nIRIG-SPINTEC\n38000GrenobleFrance\n\nInstitut Universitaire de France\n75231ParisFrance\n', 'Céline Vergnaud \nUniversité Grenoble Alpes\nCEA\nCNRS\nIRIG-SPINTEC\n38000GrenobleFrance\n', 'Frédéric Bonell \nUniversité Grenoble Alpes\nCEA\nCNRS\nIRIG-SPINTEC\n38000GrenobleFrance\n', 'Alain Marty \nUniversité Grenoble Alpes\nCEA\nCNRS\nIRIG-SPINTEC\n38000GrenobleFrance\n', 'Matthieu Jamet \nUniversité Grenoble Alpes\nCEA\nCNRS\nIRIG-SPINTEC\n38000GrenobleFrance\n'], 'authoraffiliation': ['Université Grenoble Alpes\nCEA\nCNRS\nIRIG-SPINTEC\n38000GrenobleFrance', 'Université Grenoble Alpes\nCEA\nCNRS\nIRIG-SPINTEC\n38000GrenobleFrance', 'Université Grenoble Alpes\nCEA\nIRIG-MEM\n38000GrenobleFrance', 'Université Grenoble Alpes\nCEA\nCNRS\nIRIG-SPINTEC\n38000GrenobleFrance', 'Institut Neel\nUniversité Grenoble Alpes\nCNRS\n38000GrenobleFrance', 'Université Grenoble Alpes\nCEA\nF-38000GrenobleLetiFrance', "Synchrotron SOLEIL\nL'Orme des Merisiers91190Saint-AubinFrance", "Synchrotron SOLEIL\nL'Orme des Merisiers91190Saint-AubinFrance", 'Université Paris-Saclay\nCNRS\nCentre de Nanosciences et de Nanotechnologies\nPalaiseauFrance', 'Université Grenoble Alpes\nCEA\nCNRS\nIRIG-PHELIQS\n38000GrenobleFrance', 'Université Grenoble Alpes\nCEA\nCNRS\nIRIG-PHELIQS\n38000GrenobleFrance', 'Université Grenoble Alpes\nCEA\nCNRS\nIRIG-SYMMES\n38000GrenobleFrance', 'Université Grenoble Alpes\nCEA\nIRIG-MEM\n38000GrenobleFrance', 'Université Grenoble Alpes\nCEA\nIRIG-MEM\n38000GrenobleFrance', 'Université Grenoble Alpes\nCEA\nCNRS\nIRIG-SPINTEC\n38000GrenobleFrance', 'Institut Universitaire de France\n75231ParisFrance', 'Université Grenoble Alpes\nCEA\nCNRS\nIRIG-SPINTEC\n38000GrenobleFrance', 'Université Grenoble Alpes\nCEA\nCNRS\nIRIG-SPINTEC\n38000GrenobleFrance', 'Université Grenoble Alpes\nCEA\nCNRS\nIRIG-SPINTEC\n38000GrenobleFrance', 'Université Grenoble Alpes\nCEA\nCNRS\nIRIG-SPINTEC\n38000GrenobleFrance'], 'corpusid': 257365457, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 26844, 'n_tokens_neox': 22957, 'n_words': 13509, 'pdfsha': '61f12c0a1ce6200cbdc669332f563f955bbb2e2e', 'pdfurls': ['https://export.arxiv.org/pdf/2303.03076v1.pdf'], 'title': ['Epitaxial van der Waals heterostructures of Cr 2 Te 3 on 2D materials', 'Epitaxial van der Waals heterostructures of Cr 2 Te 3 on 2D materials'], 'venue': []}

arxiv

How predictions of cosmological models depend on Hubble parameter data sets 19 Jul 2018 G S Sharov V O Vasiliev How predictions of cosmological models depend on Hubble parameter data sets 19 Jul 2018 1 Tver state university 170002, Sadovyj per. 35, Tver, Russia * We explore recent estimations of the Hubble parameter H depending on redshift z, which include 31 H(z) data points measured from differential ages of galaxies and 26 data points, obtained with other methods. We describe these data together with Union 2.1 observations of Type Ia supernovae and observed parameters of baryon acoustic oscillations with 2 cosmological models: the standard cold dark matter model with the Λ term (ΛCDM) and the model with generalized Chaplygin gas (GCG). For these models with different sets of H(z) data we calculate two-parameter and one-parameter distributions of χ 2 functions for all observed effects, estimate optimal values of model parameters and their 1σ errors. For both considered models the results appeared to be strongly depending on a choice of Hubble parameter data sets if we use all 57 H(z) data points or only 31 data points from differential ages. This strong dependence can be explained in connection with 4 H(z) data points with high redshifts z > 2. We explore recent estimations of the Hubble parameter H depending on redshift z, which include 31 H(z) data points measured from differential ages of galaxies and 26 data points, obtained with other methods. We describe these data together with Union 2.1 observations of Type Ia supernovae and observed parameters of baryon acoustic oscillations with 2 cosmological models: the standard cold dark matter model with the Λ term (ΛCDM) and the model with generalized Chaplygin gas (GCG). For these models with different sets of H(z) data we calculate two-parameter and one-parameter distributions of χ 2 functions for all observed effects, estimate optimal values of model parameters and their 1σ errors. For both considered models the results appeared to be strongly depending on a choice of Hubble parameter data sets if we use all 57 H(z) data points or only 31 data points from differential ages. This strong dependence can be explained in connection with 4 H(z) data points with high redshifts z > 2. I. INTRODUCTION The latest astronomical observations and their astrophysical interpretation [1] let cosmologists conclude that our Universe demonstrates accelerated expansion and it contains ≃ 4% of visible baryonic matter, about 26% of cold dark matter and ≃ 70% of dark energy (DE). The visible and dark matter have properties of cold dust with close to zero pressure. However dark energy has another equation of state with large negative pressure p DE close to its energy density −ρ DE with minus sign. Such a form of matter is considered as a source of the current cosmological acceleration, it helps us to construct a model that can describe all available now observational data and restrictions [1][2][3][4]. The simplest way to modify the Einstein theory of gravitation and to include dark energy with the mentioned properties is to add the Λ term into the Einstein equations. In this case cosmological solutions can demonstrate accelerated expansion. The resulting dynamical equations may be also obtained, if we add the dark energy component with the equation of state p DE = −ρ DE to the usual visible matter and cold dark matter components. This cosmological model is called ΛCDM (the Λ term with cold dark matter), it is now the most popular and usually considered as the standard model in interpretation of observational data [1][2][3]. However, the ΛCDM model has some problems, in particular, vague nature of dark energy and dark matter, the fine tuning problem for the small observed value of Λ and the coincidence problem with surprising proximity of DE and matter contribution in total energy balance nowadays [5,6]. Due to these reasons cosmologists suggest a lot of alternative models (see reviews [5][6][7]), in particular, scenarios with nontrivial equations of state [8][9][10][11], with interaction between dark components [12][13][14][15], with F (R) Lagrangian [16][17][18], additional space dimensions [19] and many others. In particular, in this paper together with the ΛCDM model we consider the model with generalized Chaplygin gas (GCG) [8][9][10][11]. In this model two dark fluids -dark energy and dark matter are unified and represented as one dark component (generalized Chaplygin gas) with the following equation of state connecting energy density ρ g and pressure p g : p g = −B ρ −α g .(1) Here B and α are positive constants. This fluid generalizes the classical Chaplygin gas [8] with the equation of state p = const/ρ. For the models ΛCDM and GCG in this paper we calculate limitations on model parameters determined from available recent observations including the Type Ia supernovae data (SN Ia) from Union 2.1 satellite [4], observable parameters baryon acoustic oscillations (BAO) and we pay special attention to different data sets of the Hubble parameter estimations H(z). Type Ia supernovae are usually considered as standard candles in the Universe, because they give possibility for each event to determine its epoch and the distance (luminosity distance) to this object. Supernova is an exploding star with huge energy release, creating a shock wave on the expanding shell [20]. They are observed in rather far galaxies because of their giant luminosity. All supernovae are classified in correspondence with time dependence of the their brightness (the light curve) and their spectrum. In particular, stars of Type I have hydrogen-deficient optical spectrum and they belong to Type Ia subdivision, if they also have strong absorption near the silicon line 615 nm. For Type Ia supernovae astronomers can definitely determine their luminosity distances from light curves. In this paper Sect. III we use the Union 2.1 compilation [4] with 580 SN Ia. The observable effect of baryon acoustic oscillations (BAO) is generated by acoustic waves with ions (baryons), which propagated in the relativistic plasma before the recombination epoch and stopped after the drag era corresponding to z d ≃ 1059.3 [1]. This effect is observed as disturbances (a bump) in the correlation function of the galaxy distribution at the sound horizon scale r s (z d ) [1,21]. In Sect. III we analyze two types of observational manifestations the BAO effect from Refs. [22] - [39], in particular, estimations of the Hubble parameter H(z) for different redshifts z [28] - [39]. The Hubble parameter H is the logarithmic derivative of the scale factor a with respect to time t, redshift z is also expressed via a H =ȧ a , z = a 0 a − 1 = 1 a − 1,(2) if we choose here and below the value a nowadays: a 0 = a(t 0 ) = 1. The Hubble parameter H(z) as the function of z may be estimated with different methods: in addition to the mentioned BAO effects [28] - [39] (26 data points) we also have the H(z) data measured from differential ages of galaxies [40] - [46] (31 data points are tabulated Sect. III). In this paper we compare different approaches in choosing H(z) data, make calculations with all 57 H(z) data points or only 31 points from differential ages and demonstrate for 2 popular cosmological models ΛCDM and GCG that predictions of optimal model parameters strongly depend on a considered Hubble parameter data set. In Sect. II we make a brief review of the models ΛCDM and GCG and their dynamics, in Sect. III describe observational data and in Sect. IV we demonstrate and analyze the results of our calculations. II. MODELS For the ΛCDM model and the model with generalized Chaplygin gas (GCG) the dynamical equations are deduced from the Einstein equations for the Robertson-Walker metric with the curvature sign k ds 2 = −dt 2 + a 2 (t) (1 − kr 2 ) −1 dr 2 + r 2 dΩ and may be reduced to the system 3ȧ 2 + k a 2 = 8πGρ + Λ,(3)ρ = −3ȧ a (ρ + p).(4) Here the dot denotes the time derivative, ρ and p are correspondingly the energy density and pressure of all matter, G is the Newtonian gravitational constant, the constant Λ equals zero for the GCG model, the speed of light c = 1. Eq. (4) is the consequence of the continuity condition ∇ µ T µ ν = 0. For both considered models we can neglect the fraction of relativistic matter (radiation and neutrinos), because the radiation-matter ratio is rather small ρ r /ρ m ≃ 3 · 10 −4 [1] for observable values z ≤ 2.36. In the ΛCDM model baryons and dark matter may be considered as one component with density ρ = ρ b + ρ dm that behaves like dust because of zero pressure p = 0. In this case we use the solution ρ/ρ 0 = (a/a 0 ) −3 of Eq. (4) and rewrite the Friedmann equation (3) in the form H 2 H 2 0 = Ω m a −3 + Ω Λ + Ω k a −2 = Ω m (1 + z) 3 + Ω Λ + Ω k (1 + z) 2 .(5) We divided Eq. (3) by 3H 2 0 , used Eq. (2) and the following notations for the present time fractions of matter, dark energy (Λ term) and curvature correspondingly: Ω m = 8πGρ(t 0 ) 3H 2 0 , Ω Λ = Λ 3H 2 0 , Ω k = − k H 2 0 .(6) These values are connected by the equality Ω m + Ω Λ + Ω k = 1,(7) resulting from Eq. (5) if we fix t = t 0 . Thus, in description of the mentioned observational data the ΛCDM model has 3 independent parameters: H 0 , Ω m and Ω Λ (or Ω k ). The GCG model includes two matter components: baryons and the generalized Chaplygin gas, the common density is ρ = ρ b + ρ g . Unlike the ΛCDM in the GCG model one should separately consider baryonic matter (it may include some part of cold dark matter) and introduce the corresponding fraction Ω b = 8πGρ b (t 0 ) 3H 2 0 as an additional model parameter. However in Ref. [11] we demonstrated, that results of calculations very weakly depend on Ω b . So in this paper we consider the simplified model with one (gas) component and suppose Ω b = 0 or ρ = ρ g . In this case one can substitute the equation of state (1) into Eq. (4), integrate it and obtain the following consequence of the Friedmann equation (3) [9][10][11]: H 2 H 2 0 = Ω k a −2 + (1 − Ω k ) B s + (1 − B s ) a −3(1+α) 1/(1+α) .(8) Here the dimensionless parameter B s = Bρ −1−α 0 is used instead of B. If we exclude the mentioned above parameter Ω b , the GCG model will have 4 independent parameters: α, B s , Ω k and H 0 . III. OBSERVATIONAL DATA A. Supernovae Ia data In Sect. I we briefly mentioned the observational data under investigation and here we describe details. For Type Ia Supernovae (SN Ia) we use N SN = 580 data points from the table [4] after the Union 2.1 satellite investigation. This compilation provides observed (estimated) values of distance moduli µ i = µ obs i for redshifts z i in the interval 0 < z i ≤ 1.41. We fit free parameters of our models, when compare µ obs i with theoretical values µ th (z i ) of the distance moduli, which are logarithms µ th i = µ(D L ) = 5 log 10 (D L /10pc) of the luminosity distance [1,5]: D L (z) = c (1 + z) H 0 S k H 0 z 0 dz H(z) , S k (x) =      sinh (x √ Ω k )/ √ Ω k , Ω k > 0, x, Ω k = 0, sin (x |Ω k |)/ |Ω k |, Ω k < 0. (9) For a cosmological model with theoretical value H(z) (5) or (8) depending on model parameters p 1 , p 2 , . . . we calculate the distance D L (z) and the corresponding χ 2 function, that measures differences between the SN Ia observational data and predictions of a model: [4]. For the Union 2.1 data [4] the standard marginalization over the nuisance parameter H 0 is required [11], it is made as the minimum over H 0 in the expression (10). χ 2 SN (p 1 , p 2 , . . .) = min H 0 N SN i,j=1 ∆µ i (C −1 SN ) ij ∆µ j ,(10)where ∆µ i = µ th (z i , p 1 , . . .) − µ obs i , C SN is the 580 × 580 covariance matrix B. BAO data For baryon acoustic oscillations (BAO) we take into account the values d z (z i ) [21] d z (z) = r s (z d ) D V (z) , D V (z) = czD 2 L (z) (1 + z) 2 H(z) 1/3 .(11) They were extracted for redshifts (redshift ranges) z = z i from a peak in the correlation function of the galaxy distribution at the comoving sound horizon scale r s (z d ). The value z d corresponds to decoupling of photons, for the sound horizon scale r s (z d ) here we use the following fitting formula [11] r s (z d ) = (r d · h) f id h , (r d · h) f id = 104.57 Mpc, h = H 0 100 km/(s · Mpc) ,(12) providing true h dependence of r d . The value (r d · h) f id = 104.57 ± 1.44 Mpc is the best fit for the ΛCDM model [11]. In our calculations we use N BAO = 26 BAO data points for d z (z) (11) from Refs. [22] - [33], tabulated here in Table I. We add 9 new points from Ref. [33] to 17 ones, which were used earlier in Refs. [10,11,14,15,18]. We use the covariance matrix C d for correlated data from Refs. [22,25] described in detail in Ref. [11]. So the χ 2 function for the value (11) yields χ 2 BAO (p 1 , p 2 , . . .) = ∆d · C −1 d (∆d) T , ∆d i = d obs z (z i ) − d th z (z i ).(13)A(z) = H 0 √ Ω m cz D V (z), because it essentially depends on Ω m , however Ω m is not the model parameter for the GCG model (see Table III). C. H(z) data The Hubble parameter values H at certain redshifts z can be measured with two methods: (1) extraction H(z) from line-of-sight BAO data [28] - [39] including analysis of correlation functions of luminous red galaxies [28,37], and (2) H(z) estimations from differential ages ∆t of galaxies (DA method) [40] - [46] via Eq. (2) and the following relation: H(z) =ȧ a = − 1 1 + z dz dt ≃ − 1 1 + z ∆z ∆t . The maximal set with N H = 57 recent estimations of H(z) is shown in Fig. 1 and in Table II below, it includes 31 data points measured with DA method (the left side) and 26 data points (the right side), obtained with BAO and other methods. The χ 2 function for the H(z) data is χ 2 H (p 1 , p 2 , . . .) = N H i=1 [H i − H th (z i , p 1 , p 2 , . . .)] 2 σ 2 H,i .(14) In papers [14,18] we used only N H = 30 H(z) data points estimated from DA method to avoid additional correlation with the BAO data from Table I. This consideration should be taken into account in the present paper: in the next section we calculate separately the χ 2 function with N H = 31 DA data points from the left column of Table II (30 points from Refs. [14,18] and the recent point from Ref. [46]) and compare these results with the full H(z) data from Table II with N H = 57 data points. In Fig. 1 the H(z) data points from dependence with the optimal parameters from Table III for the ΛCDM and GCG models with 57 and 31 H(z) data points. IV. RESULTS OF ANALYSIS For any cosmological model we investigate the space of its model parameters p 1 , p 2 , . . . (they are Ω m , Ω Λ , H 0 for the ΛCDM and α, B s , Ω k , H 0 for the GCG model) and search the optimal values of these parameters, which yield the most successful description of the observational data from Sect. III. To achieve this purpose, for any set of parameters p 1 , p 2 , . . . we use the dependence H(z) (5) or (8), calculate the integral in Eq. (9), the distances D L = D th L (z) and D th V (z) (11), the values µ th , d th z , the χ 2 functions χ 2 SN (10), χ 2 BAO (13), χ 2 H (14) and the summarized function χ 2 tot = χ 2 SN + χ 2 BAO + χ 2 H .(15) We search minima of the functions χ 2 H and χ 2 tot in the parameter spaces of a model in the two mentioned variants of the H(z) data sets: with all N H = 57 data points from Table II and with only N H = 31 data points from Refs. [40] - [46], estimated via the DA method. For both considered models we calculate two-parameter distributions of min χ 2 tot in planes of [44] two model parameters, for example, We use this functions to determine one-parameter distributions and the corresponding likelihood functions: m χ tot (p j ) = min other p k χ 2 tot (p 1 , . . .), L tot (p j ) = exp − m χ tot (p j ) − m abs 2 .(17) Here m abs is the absolute minimum of χ 2 tot . The results of these calculations for the ΛCDM model with three independent parameters Ω m , Ω Λ and H 0 are presented in Figs. 2, 3 and in Table III. In the top-left panel of Fig. 2 we draw the contour plots at 1σ (68.27%), 2σ (95.45%) and 3σ (99.73%) confidence level for the two-parameter distributions (16) of χ 2 tot in the (Ω m , Ω Λ ) plane. The green filled contours describe the m χ tot (Ω m , Ω Λ ) function for all 57 H(z) data points, the magenta contours present the case with 31 DA H(z) data points. Here the function (16) is In the top-right panel of Fig. 2 we compare the mentioned contours for χ 2 tot (with the same colors) and the similar contours for the function χ 2 H (14), more correctly, m χ tot (Ω m , Ω Λ ) = min H 0 χ 2 tot (Ω m , Ω Λ , H 0 ).(18)m χ H (Ω m , Ω Λ ) = min H 0 χ 2 H (Ω m , Ω Λ , H 0 ). This distribution includes only H(z) data. The green circles and magenta stars in Fig. 2 denote the minimum points of m χ tot (Ω m , Ω Λ ) (and, naturally, for χ 2 tot ) correspondingly for 57 and 31 H(z) data points. Their coordinates (the optimal values of parameters) are tabulated in Table III. In the same way, the minimum points for χ 2 H are shown in the top-right panel as the deep green square and brown hexagram. In the bottom panels of Fig. 2 we compare the one-parameter distributions (17) Table III (for χ 2 tot ). In Fig. 2 we see the interesting phenomenon: the optimal values of parameters Ω m , Ω Λ (and positions of minimum points for χ 2 ) are essentially different for the two considered cases with 57 and 31 H(z) data points. This divergence takes place for χ 2 tot (the left panels in Fig. 2), for example, these estimations for Ω m are correspondingly Ω m = 0.282 ± 0.021 and Ω m = 0.349 ± 0.041 (see Table III): the last value 0.349 is beyond 2σ confidence level for the N H = 57 case. However for χ 2 H this divergence is stronger, the correspondent estimations are Ω m = 0.227 +0.036 −0.041 (for N H = 57) and Ω m = 0.359 +0.197 −0.232 (for N H = 31). This is natural, because the summands χ 2 SN + χ 2 BAO in χ 2 tot moderate this effect. Below we concentrate on the more relevant summarized function χ 2 tot . In Fig. 3 we present other two-and one-parameter distributions of χ 2 tot and the likelihood functions for the ΛCDM model. In particular, in the top-right panel the contour plots for m χ tot (Ω k , H 0 ) = min Ωm χ 2 tot are shown for the cases N H = 57 and N H = 31 in the same notations. In these calculation we consider the curvature fraction Ω k as an independent parameter (together with Ω m , H 0 ), the fraction Ω Λ is expressed via Eq. (7): Ω Λ = 1 − Ω m − Ω k . The two-parameter distributions (18) Table III for the cases N H = 57 and N H = 31. The 1σ errors are calculated from the correspondent likelihood functions (17) L tot (p i ). We should emphasize, that the number N p of model parameters is essential, when we comrade different models. So we also use the Akaike information criterion [11,47] AIC = min χ 2 tot + 2N p . Here N p = 3 for the ΛCDM model. The similar estimations for the ΛCDM model were made in many papers, in particular, in Refs. [1-3, 11, 47-49] both DA and BAO methods and calculated Ω m = 0.237 ± 0.051, Ω Λ = 0.66 ± 0.20. However, when they excluded 3 data points [30,31,36] with z ≥ 2.3, they obtained the enhanced values for both parameters Ω m = 0.40 +0. 18 −0.14 , Ω Λ = 0.92 +0.34 −0.23 (compare with our results for χ 2 H in Fig. 2). If we compare our results for the ΛCDM model with the latest Planck data [1] (Ω m = 0.308 ± 0.012, Ω Λ = 0.692 ± 0.012, Ω k = −0.005 +0.016 −0.017 , H 0 = 67.8 ± 0.9 km c −1 Mpc −1 ), we will find some tension for Ω Λ , Omega k in the case N H = 31 and for H 0 in both cases because of too low estimation of H 0 in Ref. [1]. The influence of a chosen H(z) data set takes place not only for the ΛCDM model. One can see in Fig. 4 and in Table III, that for the GCG model this influence is even more strong. In the top panels we demonstrate the contour plots for two-parameter distributions (16) of χ 2 tot in the (α, B s ) and (Ω k , B s ) planes for the cases N H = 57 (blue filled contours) and N H = 31 (red contours). In particular, the two-parameter distributions (16) in the top-left panel are m χ tot (α, B s ) = min Ω k ,H 0 χ 2 tot (α, B s , Ω k , H 0 ). The circles and stars show the points of minima for χ 2 tot . The similar two-parameter contour plots for the GCG model in the (Ω k , H 0 ) plane are drawn in Fig. 5. The one-parameter distributions m χ tot (α), m χ tot (B s ), m χ tot (Ω k ) and the corresponding likelihood functions (17) L tot (p i ) are shown in the middle and bottom panels of Fig. 4. Fig. 4 and Table III demonstrate, that for the GCG model the best fitted values of α, B s , Ω k strongly depend on a Hubble parameter data: N H = 57 (all data points) or N H = 31 (only from DA method). In particular, the best fitted values α ≃ −0.124, Ω k ≃ −0.192 for N H = 57 change their signs and become α ≃ +0.647, Ω k ≃ +0.019, if N H = 31. In Fig. 5 we compare the ΛCDM and GCG models in the plane (Ω k , H 0 ) of their common parameters. For both models we draw the one-parameter distributions m χ tot (Ω k ), m χ tot (H 0 ) (they help us to compare the best results min χ tot for these models) and the likelihood functions L tot (Ω k ), L tot (H 0 ). In the top-left panel of Fig. 5 the filled contours describe the GCG model with N H = 57, Fig. 5 is useful, when we want to compare predictions the ΛCDM and GCG models in the considered cases N H = 57 and N H = 31. The plots L tot (Ω k ) and L tot (H 0 ) show differences of the best fitted values, the plots m χ tot (Ω k ) and m χ tot (H 0 ) describe effectiveness of these models. Mere detailed information is tabulated in Table III. V. CONCLUSION In this paper we describe the observational data for Type Ia supernovae [4], BAO (Table I) and two data sets of the Hubble parameter data H(z) (all N H = 57 data points from Table II and only 31 data points from differential ages) with the ΛCDM model and the model with generalized Chaplygin gas (GCG). The results are demonstrated in Table III: for all models and variants of N H we calculated the minimal values of the function χ 2 tot (15), the results of Akaike information criterion (19) and the best fitted values of model parameters with 1σ errors. For the GCG model we achieve the best minimal values of min χ 2 tot , however the Akaike criterion gives advantage to the ΛCDM model, because it has the small number N p = 3 of model parameters (degrees of freedom) in comparison with with N p = 4 for GCG. But the most striking result of our calculations for both models is the large difference between the best fitted values of model parameters in the cases with N H = 57 H(z) data points from Table II and N H = 31 data points, obtained with DA method (the left hand side of Table II). For the case N H = 57 these results are close to the estimations for these models in Ref. [11], because in that paper we used H(z) data points from both DA and BAO methods (though there were N H = 38 points). This essential divergence between the predictions of the variants with all N H = 57 and N H = 31 DA data points is seen visually in Fig. 1. It may be explained and connected with 4 H(z) data points [30,31,36,39] with high redshifts z ≥ 2.3. These data points, obtained with BAO method (see the right hand side of Table II) have small errors σ H and strongly influence on a model predictions, when we take these points into account (in the case N H = 57). Otherwise, when we include only N H = 31 DA data points, this effect disappears. 2 tot (p 1 , p 2 , p 3 , . . .). FIG. 2 : 2The ΛCDM model: 1σ, 2σ and 3σ contour plots for two-parameter distributions m χ tot (Ω m , Ω Λ ) are drawn in (Ω m , Ω Λ ) plane for 57 and 31 H(z) data points in comparison with contours for min top-right panel). The corresponding one-parameter distributions m χ tot (Ω m ) and m χ H (Ω m ) are in the bottom panels. Ω m , Ω Λ ). These distributions and the corresponding likelihood functions(17) determine 1σ estimates in m χ tot (Ω m , Ω Λ ) for N H = 57 and 31 in the top-right panel of Figs. 2, 3 let us calculate the one-parameter distributions m χ tot (Ω m ), m χ tot (Ω Λ ) and the likelihood functions (17) L tot (Ω m ), L tot (Ω Λ ) shown in the middle and bottom panels of Fig. 3. The functions L tot (H 0 ) are deduces from the two-parameter distributions in the (Ω k , H 0 ) plane. The best fitted values of min χ 2 tot and the model parameters Ω m , Ω Λ , Ω k , H 0 for the ΛCDM model are presented in FIG. 3 : 3for describing the Type Ia supernovae, H(z), BAO and other data in various combinations. One can observe the following effect (connected with the described above): the estimations of Ω m , Ω Λ , Ω k and H 0 in different papers essentially depend on a chosen H(z) data set. For example, the authors of Refs. [49] used the χ 2 H function with N H = 41 data points from The ΛCDM model with 57 and 31 H(z) data points: contour plots in 2 planes, one-parameter distributions and likelihood functions. FIG. 4 :FIG. 5 : 45The GCG model with N H = 57 (blue) and N H = 31 (red): two-parameter, one-parameter distributions and likelihood functions for χ 2 tot . Comparison of the two-parameter distributions min χ 2 tot (Ω k , H 0 ) for the ΛCDM and GCG models in the plane (Ω k , H 0 ) of their common parameters for the cases with 57 and 31 H(z) data points (the top-left panel). The corresponding one-parameter distributions are in other panels. Notations correspond to the previous figures. other contours differ in their color. The points of minima are marked here as the circle (GCG, N H = 57), the pentagram (GCG, N H = 31), the square (ΛCDM, N H = 57) and the hexagrams (ΛCDM, N H = 31) of the corresponding color. TABLE I : IValues d z (z) = r s (z d )/D V (z)(11) with errors and references Unlike Refs.[11,14,15,18] we do not use in this paper the observational value[21] z d z (z) σ d Refs z d z (z) σ d Refs 0.106 0.336 0.015 [24] 0.44 0.0916 0.0071 [25] 0.15 0.2232 0.0084 [27] 0.44 0.0874 0.0010 [33] 0.20 0.1905 0.0061 [22] 0.48 0.0816 0.0009 [33] 0.275 0.1390 0.0037 [22] 0.52 0.0786 0.0009 [33] 0.278 0.1394 0.0049 [23] 0.56 0.0741 0.0008 [33] 0.31 0.1222 0.0021 [33] 0.57 0.0739 0.0043 [29] 0.314 0.1239 0.0033 [25] 0.57 0.0726 0.0014 [32] 0.32 0.1181 0.0026 [32] 0.59 0.0711 0.0010 [33] 0.35 0.1097 0.0036 [22] 0.60 0.0726 0.0034 [25] 0.35 0.1126 0.0022 [26] 0.64 0.0675 0.0011 [33] 0.35 0.1161 0.0146 [28] 0.73 0.0592 0.0032 [25] 0.36 0.1053 0.0018 [33] 2.34 0.0320 0.0021 [31] 0.40 0.0949 0.0014 [33] 2.36 0.0329 0.0017 [30] Table II estimated with DA and BAO methods are shown as correspondingly red stars and cyan diamonds. The lines demonstrate the best fitted H(z)0 0.5 1 1.5 2 50 100 150 200 z H(z) , km (s Mpc) −1 DA method BAO method GCG 57 GCG 31 ΛCDM 57 ΛCDM 31 FIG. 1: H(z) data from Table II, stars and diamonds denote data points correspondingly from DA and BAO methods. The lines are the best fitted for the ΛCDM and GCG models with 57 and 31 H(z) data points. TABLE II : IIHubble parameter values H(z) with errors σ H from DA and BAO methods.DA method BAO method z H(z) σ H Refs z H(z) σ H Refs 0.070 69 19.6 [42] 0.24 79.69 2.99 [34] 0.090 69 12 [40] 0.30 81.7 6.22 [37] 0.120 68.6 26.2 [42] 0.31 78.18 4.74 [33] 0.170 83 8 [40] 0.34 83.8 3.66 [34] 0.1791 75 4 [43] 0.35 82.7 9.1 [28] 0.1993 75 5 [43] 0.36 79.94 3.38 [33] 0.200 72.9 29.6 [42] 0.38 81.5 1.9 [38] 0.270 77 14 [40] 0.40 82.04 2.03 [33] 0.280 88.8 36.6 [42] 0.43 86.45 3.97 [34] 0.3519 83 14 [43] 0.44 82.6 7.8 [35] 0.3802 83 13.5 [45] 0.44 84.81 1.83 [33] 0.400 95 17 [40] 0.48 87.79 2.03 [33] 0.4004 77 10.2 [45] 0.51 90.4 1.9 [38] 0.4247 87.1 11.2 [45] 0.52 94.35 2.64 [33] 0.4497 92.8 12.9 [45] 0.56 93.34 2.3 [33] 0.470 89 34 [46] 0.57 87.6 7.8 [29] 0.4783 80.9 9 [45] 0.57 96.8 3.4 [32] 0.480 97 62 [41] 0.59 98.48 3.18 [33] 0.593 104 13 [43] 0.60 87.9 6.1 [35] 0.6797 92 8 [43] 0.61 97.3 2.1 [38] 0.7812 105 12 [43] 0.64 98.82 2.98 [33] 0.8754 125 17 [43] 0.73 97.3 7.0 [35] 0.880 90 40 [41] 2.30 224 8.6 [36] 0.900 117 23 [40] 2.33 224 8 [39] 1.037 154 20 [43] 2.34 222 8.5 [31] 1.300 168 17 [40] 2.36 226 9.3 [30] 1.363 160 33.6 [44] 1.430 177 18 [40] 1.530 140 14 [40] 1.750 202 40 [40] 1.965 186.5 50.4 TABLE III : IIIOptimal values and 1σ estimates of model parameters Model min χ 2 tot AIC H 0 Ω k other parameters ΛCDM 610.31 616.31 71.35 +0.63 −0.62 −0.085 ± 0.048 Ω m = 0.282 ± 0.021, 57 H(z) Ω Λ = 0.803 ± 0.028 ΛCDM 588.96 594.96 71.77 +1.70 −1.69 −0.224 +0.085 −0.084 Ω m = 0.349 ± 0.041, 31 H(z) Ω Λ = 0.875 ± 0.045 GCG 609.94 617.94 71.68 +0.82 −0.83 −0.192 +0.188 −0.170 α = −0.124 +0.235 −0.138 , 57 H(z) B s = 0.705 +0.065 −0.044 GCG 587.93 595.93 70.46 +2.16 −2.51 +0.019 +0.541 −0.255 α = 0.647 +3.25 −0.64 , 31 H(z) B s = 0.826 +0.284 −0.111 Planck 2015 results. 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{'abstract': '1 Tver state university 170002, Sadovyj per. 35, Tver, Russia * We explore recent estimations of the Hubble parameter H depending on redshift z, which include 31 H(z) data points measured from differential ages of galaxies and 26 data points, obtained with other methods. We describe these data together with Union 2.1 observations of Type Ia supernovae and observed parameters of baryon acoustic oscillations with 2 cosmological models: the standard cold dark matter model with the Λ term (ΛCDM) and the model with generalized Chaplygin gas (GCG). For these models with different sets of H(z) data we calculate two-parameter and one-parameter distributions of χ 2 functions for all observed effects, estimate optimal values of model parameters and their 1σ errors. For both considered models the results appeared to be strongly depending on a choice of Hubble parameter data sets if we use all 57 H(z) data points or only 31 data points from differential ages. This strong dependence can be explained in connection with 4 H(z) data points with high redshifts z > 2.', 'arxivid': '1807.07323', 'author': ['G S Sharov ', 'V O Vasiliev '], 'authoraffiliation': [], 'corpusid': 118995692, 'doi': '10.26456/mmg/2018-611', 'github_urls': [], 'n_tokens_mistral': 18092, 'n_tokens_neox': 14249, 'n_words': 7927, 'pdfsha': '21a878993dd0de368306ab1702999150c200ae47', 'pdfurls': ['https://arxiv.org/pdf/1807.07323v1.pdf'], 'title': ['How predictions of cosmological models depend on Hubble parameter data sets', 'How predictions of cosmological models depend on Hubble parameter data sets'], 'venue': []}

arxiv

Axion-sourced fireballs from supernovae Melissa Diamond Damiano F G Fiorillo Niels Bohr International Academy & DARK Niels Bohr Institute University of Copenhagen Blegdamsvej 172100CopenhagenDenmark Gustavo Marques-Tavares Maryland Center for Fundamental Physics Department of Physics University of Maryland 20742College ParkMDU.S.A Edoardo Vitagliano Racah Institute of Physics Hebrew University of Jerusalem 91904JerusalemIsrael Arthur B Mcdonald Canadian Astroparticle Physics Institute Queens University K7L 3N6KingstonOntarioCanada Axion-sourced fireballs from supernovae (Dated: March 22, 2023) New feebly interacting particles would emerge from a supernova core with 100-MeV-range energies and produce γ-rays by subsequent decays. These would contribute to the diffuse cosmic γ-ray background or would have shown up in the Solar Maximum Mission (SMM) satellite from SN 1987A. However, we show for the example of axion-like particles (ALPs) that, even at distances beyond the progenitor star, the decay photons may not escape, and can instead form a fireball, a plasma shell with T < ∼ 1 MeV. Thus, existing arguments do not exclude ALPs with few 10 MeV masses and a two-photon coupling of a few 10 −10 GeV −1 . However, the energy would have showed up in sub-MeV photons, which were not seen from SN 1987A in the Pioneer Venus Orbiter (PVO), closing again this new window. A careful re-assessment is required for other particles that were constrained in similar ways. I. INTRODUCTION The core of a collapsing star is one of the hottest and densest regions in the Universe. Therefore, it can be a factory of high-energy (around 100 MeV) particles beyond the standard model, in particular, feebly interacting particles (FIPs), such as sterile neutrinos, dark photons, new scalars, QCD axions and axionlike particles, and many others. It comes as no surprise, nowadays, that-despite the small number of neutrino events detected at several neutrino experimentssupernova 1987A (SN 1987A) represented a bonanza for bounds on FIPs. A renowned example is the SN 1987A energy-loss bound [1][2][3], that limits the luminosity of a novel particle φ to be smaller than the neutrino luminosity, L φ < ∼ L ν , evaluated at 1 second after the bounce [4,5]. While a large body of literature is dedicated to light particles, the last decade has seen an evergrowing interest in the high-mass part of FIP parameter space, above 1 keV, that could have an impact on cosmology [6][7][8][9], astrophysical transients [10,11], and play the role of dark matter mediator [12,13]. Heavy FIPs can be probed down to luminosities much smaller than L ν . Depending on their lifetime and decay channels, they can travel for distances that are either smaller or larger than the progenitor radius. If the mean-free path against decay is small enough, FIPs can decay in the mantle of the progenitor, lighting up the SN [14]. Since the explosion energy of SNe is much smaller than the energy released in neutrinos, the radiative decay of FIPs to charged leptons and photons must be suppressed [15,16]. This argument, applied to low energy SNe, limits the lifetime of radiatively decaying particles even further [17]. If the mean-free path against the decay is larger (or if the FIP decays to neutrinos), the decay product of FIPs produced by SN 1987A could r G g J s x m G 0 / Q q V P 6 K F b a P W n R T P 2 d y L A w Z i w i N y m w H Z p F b y r + 5 3 V S G 1 + F G Z N J a q k k 8 4 / i l C O r 0 P R 8 1 G e a E s v H j m C i m d s V k S H W m F h X U t G V E C y e v E y a 1 U p w X q n e X 5 R r 1 3 k d B T i G E z i D A C 6 h B r d Q h w Y Q k P A M r / D m G e / F e / c + 5 q M r X p 4 5 g j / w P n 8 A N 3 a Q n g = = < / l a t e x i t > o R g + e V V 0 q p V g 4 t q 7 f 6 y U r / J 4 y i i E 3 S K z l G A r l A d 3 a E G a i K C H t E z e k V v n v J e v H f v Y 9 F a 8 P K Z Y / Q H 3 u c P i D m P G Q = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " b d A S 1 D U 9 p L D w m R S u o G 8 e t V l J o I w = " > A A A B 7 X i c b V B N S 8 N A E N 3 4 W e t X 1 a O X x S J 4 K k k V 9 F j U g 8 c K 9 g P a U D b b S b t 2 k w 2 7 E 6 G E / g c v H h T x 6 v / x 5 r 9 x 2 + a g r Q 8 G H u / N M D M v S K Q w 6 L r f z s r q 2 v r G Z m G r u L 2 z u 7 d f O j h s G p V q D g 2 u p N L t g B m Q I o Y G C p T Q T j S w K J D Q C k Y 3 U 7 / 1 B N o I F T / g O A E / Y o N Y h I I z t F K z e w s S W a 9 U d i v u D H S Z e D k p k x z 1 X u m r 2 1 c 8 j S B G L p k x H c 9 N 0 M + Y R s E l T I r d 1 E D C + I g N o G N p z C I w f j a 7 d k J P r d K n o d K 2 Y q Q z 9 f d E x i J j x l F g O y O G Q 7 P o T c X / v E 6 K 4 Z W f i T h J E W I + X x S m k q K i 0 9 d p X 2 j g K M e W M K 6 F v Z X y I d O M o w 2 o a E P w F l 9 e J s 1 q x T u v V O 8 v y r X r P I 4 C O S Y n 5 I x 4 5 J L U y B 2 p k w b h 5 J E 8 k 1 f y 5 i j n x X l 3 P u a t K 0 4 + c 0 T + w P n 8 A W H a j w A = < / l a t e x i t > j Q w i V z N y K 6 Y B I Q r W J r G R C c O d f X i T N a s U 9 q 1 R v z 8 u 1 q z y O I j p E R + g E u e g C 1 d A N q q M G o u g R P a N X 9 G Y 9 W S / W u / U x a y 1 Y + c w + + g P r 8 w f 5 n 5 V u < / l a t e x i t > FIG. 1. (Out-of-scale) Schematic representation of the fireball sourced by axions from a supernova. Axions are produced in the proto-neutron star, travel outside of the progenitor star, and decay exterior to its surface. The produced γ-rays, rather than conserving their spectrum, form a fireball. b J L z W j i E v a K s k P I 1 u d h M h e 7 e 8 k = " > A A A B 6 n i c b V D L S g N B E O y N r x h f U Y 9 e B o M g C G E 3 C n o M e v E Y 0 T w g W c P s p J M M m Z 1 d Z m a F s O Q T v H h Q x K t f 5 M 2 / c Z L s Q R M L G o q q b r q 7 g l h w b V z 3 2 8 m t r K 6 t b + Q 3 C 1 v b O 7 t 7 x f 2 D h o 4 S x b D O I h G p V k A 1 C i 6 x b r g R 2 I o V 0 j A Q 2 A x G N 1 O / + Y R K 8 0 g + m H G M f k g H k v c 5 o 8 Z K 9 / h 4 1 i 2 W 3 L I 7 A 1 k m X k Z K k K H W L X 5 1 e h F L Q p S G C a p 1 2 3 N j 4 6 d U G c 4 E T g q d R G N M 2 Y g O s G 2 p p C F q P 5 2 d O i E n V u m R f q R s S U N m 6 u + J l I Z a j 8 P A d o b U D P W i N x X / 8 9 q J 6 V / 5 K Z d x Y l C y + a J + I o i J y P R v 0 u M K m R F j S y h T 3 N 5 K 2 J A q y o x N p 2 B D 8 B Z f X i a N S t k 7 L 1 f u L k r V 6 y y O P B z B M Z y C B 5 d Q h V u o Q R 0 Y D O A Z X u H N E c 6 L 8 + 5 8 z F t z T j Z z C H / g f P 4 A 5 W C N i g = = < / l a t e x i t > e < l a t e x i t s h a 1 _ b a s e 6 4 = " + P r g j 4 d J 2 c Q d 1 C M x h 1 e X p N Y s T W M = " > A A A B 6 n i c b V D L S g N B E O y N r x h f U Y 9 e B o P g x b A b B T 0 G v X i M a B 6 Q r G F 2 0 k m G z M 4 u M 7 N C W P I J X j w o 4 t U v 8 u b f O E n 2 o I k F D U V V N 9 1 d Q S y 4 N q 7 7 7 e R W V t f W N / K b h a 3 t n d 2 9 4 v 5 B Q 0 e J Y l h n k Y h U K 6 A a B Z d Y N 9 w I b M U K a R g I b A a j m 6 n f f E K l e S Q f z D h G P 6 Q D y f u c U W O l e 3 w 8 6 x Z L b t m d g S w T L y M l y F D r F r 8 6 v Y g l I U r D B N W 6 7 b m x 8 V O q D G c C J 4 V O o j G m b E Q H 2 L Z U 0 h C 1 n 8 5 O n Z A T q / R I P 1 K 2 p C E z 9 f d E S k O t x 2 F g O 0 N q h n r R m 4 r / e e 3 E 9 K / 8 l M s 4 M S j Z f F E / E c R E Z P o 3 6 X G F z I i x J Z Q p b m 8 l b E g V Z c a m U 7 A h e I s v L 5 N G p e y d l y t 3 F 6 X q d R Z H H o 7 g G E 7 B g 0 u o w i 3 U o A 4 M B v A M r / D m C O f F e X c + have been seen directly in the detectors on-line during the explosion [18][19][20][21]. For example, the decay to 100-MeV neutrinos would have been seen in Cherenkov detectors, so the lack of such events gives constraints superseding the energy-loss bound by one order of magnitude in coupling [22]. The focus of this paper will be on heavy axion-like particles (hereafter axions) that can decay to photons through the coupling − 1 4 g aγγ aFF . The daughter photons would have contributed to the diffuse cosmic γray background [16,24,25] and, for SN 1987A, would have showed up in the Gamma-Ray Spectrometer on the Solar Maximum Mission satellite [26]. The latter mea- . We also show bounds from low-energy supernovae (red) [17], γ-rays due to axion decays from SN 1987A (blue) [21,23], and diffuse supernova background of γ-rays from axion decays (green) [16,17]. The constraints from low-energy supernovae are the only ones unaffected by the fireball formation. Bounds from γ-ray decay at the Solar Maximum Mission and from the diffuse supernova background should be reevaluated in the fireball formation region. On the other hand, we find that the entire region, which we hatch in blue, is robustly excluded by the Pioneer Venus Orbiter observations. burst. Its observations have been used to constrain radiatively decaying particles by looking for γ-ray emission [16,21,23,27]. For such constraint to apply, the photons produced in the axion decay should have been in the energy interval detected by SMM [18]. If axions with a mass m a > ∼ 1 MeV were produced in SN 1987A, after escaping from the progenitor they would have decayed to γray photons with energy around 100 MeV. If the density of such photons was sufficiently large, they would rapidly produce a fireball, creating a plasma of electrons, positrons and photons, as sketched in Fig. 1. Several processes rapidly drive the plasma to a much lower temperature. After this rapid thermalization, the plasma would follow the evolution of a "standard" fireball [28,29], though with a much smaller initial temperature and, potentially, negligible baryon load. 1 The gas would first expand adiabatically, converting the temperature into bulk momentum, and then expand freely. The photons produced in axion decays from SN 1987A would have had an energy E γ < ∼ 1 MeV. Most of these photons would 1 The decay of heavy axions was proposed as a mechanism to produce the fireball sourcing gamma-ray bursts or as a SN catalyzer in Ref. [30], but the axion luminosity needed to source such kind of fireball requires couplings that are excluded. not have been energetic enough to be detected by SMM, though we are still able to put constraints using the data of the Pioneer Venus Orbiter Satellite (PVO) [31], which had an energy window 0.2 − 2 MeV. Our main results are collected in Fig. 2. Our new PVO bound covers the region of the parameter space carved by the fireball formation and previously thought to be excluded by SMM. The paper is structured as follows. In Section II we describe the first stages of the fireball, from its formation to thermalization. In Section III we analyze the period of expansion. Section IV and Section V are dedicated to the impact of the fireball formation on respectively the SN 1987A and diffuse γ-ray bounds, as well as to the new bounds we place on the axion parameter space. Finally, we devote Section VI to our conclusions. II. FORMATION OF THE FIREBALL Axions are produced in the protoneutron star (PNS) at the center of the supernova, mainly via Primakoff emission and photon-photon coalescence. Their spectrum is parameterized in Refs. [21,23]. Both papers account for Primakoff, and neglect coalescence, which is reasonable at light masses. Since we want to compare the region of fireball formation with the bounds drawn in Refs. [21,23], we self-consistently use their expres-sions; coalescence would increase the total number of photons injected and enhance the possibility of fireball formation. Further, we are mostly interested in masses below 60 MeV, as confirmed by our results, where coalescence should have little impact (see e.g. the supplemental material of Ref. [17]). The axion spectrum is extracted from Ref. [23] as dN a dE a = C 2 E 2 a exp(E a /T eff ) − 1 σ 0 (E a , g aγ , κ s , m a ). (1) The numerical values for the parameters C 2 , κ s , T eff are all taken from Ref. [23]; the expression for the Primakoff cross section for massive particles, σ 0 (E a , g aγ , κ s , m a ), is taken from Ref. [21]. Only those axions which decay outside of the progenitor, with a radius of approximately R = 3 × 10 12 cm, contribute to the photons which are detected at Earth. Therefore, the total energy injected by the axions is E = E a dN a dE a e −R/ (Ea) dE a ,(2) where the decay length is (E a ) = E 2 a − m 2 a 64π g 2 aγ m 4 a .(3) We can similarly determine the total number of axions injected N and the total radial momentum injected P. A. Geometry of the fireball The massive axions propagate with a speed close to the speed of light, approximately radially, and decay substantially far from the center of the supernova. The decay is in reality a continuous process; however, most of the energy from the axion decay is injected at a distance of the order of the decay length. The latter is larger than the typical thickness of the axion shell propagating from the supernova at small couplings and masses. Based on these facts, we can schematically model the propagation as a shell of axions, with a thickness of the order of ∆ 1 3 s, the typical timescale over which they are emitted from the supernova. The typical decay radius for axions that decay outside the progenitor is determined by averaging over both the energy distribution and the radius of decay distribution, e −(r dec −R)/ (Ea) dr dec / (E a ), where r dec > R; this leads to the average radius r = R + dE a (E a )e −R/ (Ea) dNa dEa dE a e −R/ (Ea) dNa dEa .(4) For simplicity, here and henceforth we will indicate the average of a quantity x over the energy fluence of the axions decaying outside the progenitor as x = dE a xe −R/ (Ea) dNa dEa dE a e −R/ (Ea) dNa dEa .(5) To estimate the thickness of the photon shell, we determine the quadratic dispersion in the radius of the photons evaluated at the timet = τ ( E a ) + R/v(E a ) = ( (E a ) + R)/v(E a ) , where v(E a ) = 1 − m 2 a /E 2 a ; this is the mean time at which the bulk of the axions outside the progenitor decay. If the axion is produced in the PNS at a time t i , and decays at time t dec , the photon position at time t is r γ = v(E a ) [t dec − t i ] + t − t dec .(6) The assumption that photons propagate radially is valid for ALPs with masses much belowĒ a 100 MeV, the typical ALP energy. We will be interested in typical masses below 60 MeV, since at larger masses we find no fireball formation. Therefore, accounting for the angle with which the photons are emitted in the decay, typically of the order m a /Ē a , would account for 10% corrections to the average radius of photon emission. Without loss of generality, we shift the origin of time and define t i to be uniformly distributed between −∆ 1 /2 and ∆ 1 /2, where the time window is given a representative value ∆ 1 = 3 s; with this choice the average value of t i vanishes. At a fixed energy, the decay time is averaged over the ex- ponential distribution e −[t dec −R/v(Ea)]/τ (Ea) dt dec /τ (E a ) for t dec > R/v(E a ). The average position of a photon at time t therefore is r γ = t + (v(E a ) − 1) τ (E a ) + R v(E a ) .(7) Evaluating this att, we find, as we should, our previous estimate for the fireball radius, r γ = v(E a ) τ (E a ) + R v(E a ) .(8) We now compute the average of the squared radius and define the thickness as ∆ 2 = r 2 γ − r γ 2 .(9) Expanding the averages over the exponential distribution, this is ∆ 2 = ∆ 2 1 v(E a ) 2 12 + (1 − v) 2 R 2 v(E a ) 2 + 2Rτ (E a ) v(E a ) + 2τ (E a ) 2 − τ (E a ) + R v(E a ) − v(E a )τ (E a ) + R 2 .(10) The first term corresponds to the average thickness of the axion shell, while the other terms correspond to the thickness induced by the delay between the slower axions yet to decay and the photons. Notice that for ultrarelativistic axions the last two terms become arbitrarily small, since axions and photons move with arbitrarily close velocities. Therefore, its order of magnitude for strongly boosted ALPs, as we can approximately consider for masses below 60 MeV that are of interest here, is rm 2 a /Ē 2 a . The thickness of the photon shell will therefore be determined by the largest among the thickness of the ALP shell and the approximate spread rm 2 a /Ē 2 a . Notice that in the thickness of the shell the corrections due to the non-collinear emission of photons in ALP decay might lead to corrections by factors of order 1, since these are suppressed by one factor of m 2 a /Ē 2 a and must be compared with terms of the same order of magnitude. On the other hand, we will find later that the thickness of the shell in the end never enters the final results presented here. Therefore, we stick to this simple estimate which might be corrected by factors of order unity, since none of our results are affected by it. B. Thermalization via pair production If the photon density is large enough, it can form a plasma by producing pairs of e ± . The pair production process (and subsequent Compton and Rutherford scattering) leads to kinetic equilibrium, with both γ and e ± acquiring thermal spectra. Since pair production is not able to change the total number of particles, chemical equilibrium cannot be fully reached, and both species develop a common temperature T i and (negative) chemical potential −µ. Furthermore, they behave as a tightly coupled fluid with a bulk Lorentz factor γ. In the following, we will refer with primes to lab-frame quantities, and without primes to plasma-frame quantities. If the number density of photons n 0 = 2N /4πr 2 ∆ is too low, they are not able to produce pairs of electrons and positrons. The condition for pair-production reactions to be efficient is n 0 σ γγ→e + e − ∆ 1,(11) where σ γγ→e + e − is the pair production cross section, to be evaluated at the typical photon energies in the plasma frame, where the photons are isotropic. The cross section is therefore [32] σ γγ→e + e − = πα 2 E 2 γ 2 + 2m 2 e E 2 γ − m 4 e E 4 γ × log E γ m e + E 2 γ m 2 e − 1 − 1 − m 2 e E 2 γ 1 + m 2 e E 2 γ   ,(12) where E γ is the center-of-mass energy of any of the two colliding photons. We average this cross section over a uniform distribution of the pitch angle between the two incoming photons, with all the photons having a constant energy m a /2 in the comoving frame. In reality, there will be photons from axions moving with different speeds with a relative angular distribution different from the isotropic one and with a relative boost with respect to one another; however, these small corrections typically change negligibly the shape or normalization of our curves corresponding to fireball formation, due to the large power of the coupling appearing in these curves (see below). If Eq. (11) is satisfied, a population of electrons and positrons is nearly immediately established in the plasma. Because the typical energies at this stage are much larger than the electron mass, the energy and number of particles are equally shared among the two populations of electrons and positrons and the population of photons. The corresponding fluid has equation of state p = ρ/3, where p is the fluid pressure and ρ is the fluid energy density. Therefore, we may deduce its bulk velocity taking the ratio of the conservation of energy density (see, e.g., Ref. [33], Chap. 2) 4γ 2 − 1 3 ρ = E 4πr 2 ∆(13) and momentum density 4 3 γ 2 ρv = P 4πr 2 ∆ .(14) Here, v is the bulk velocity and γ is the bulk Lorentz factor. After taking the ratio, we obtain v = 2E P − 4E 2 P 2 − 3.(15) We can also deduce the initial temperature of this plasma using the conservation of the total number of particles, which at this stage is valid and yields γn = 2N 4πr 2 ∆ ,(16) where n is the number density of the fluid. Since at this stage chemical equilibrium is not maintained, the chemical potential of photons, electrons, and positrons is equal to a common (negative) value µ γ = µ e + = µ e − = −|µ|, such that µ T . Therefore, we can use the equation of state of a gas of relativistic Boltzmann particles: taking the ratio of Eqs. (14) and (16) we obtain the initial temperature of the plasma as T i = P 8γvN . C. Thermalization via bremsstrahlung With a sufficiently large concentration of e ± , bremsstrahlung reactions can become fast enough to drive the system towards chemical equilibrium, increasing the particle number. The formation of the fireball requires pair production to equilibrate, while the particle number dilution also entails bremsstrahlung. For both reactions to be fast, axions should be produced in large quantities, with a large fraction decaying outside the progenitor. The condition for the initial population of electrons and positrons to be equilibrated by bremsstrahlung is 2 3 2N 4πr 2 ∆ σ ee→eeγ (T i )∆ 1,(18) where the factor 2/3 accounts for the fact that only the electron-positron part of the plasma interacts via bremsstrahlung and σ ee→eeγ (T i ) is the cross section for bremsstrahlung from ultrarelativistic electrons and positrons. The total bremsstrahlung cross section diverges, due to the emission of soft photons with negligible energies which, however, do not contribute to the thermalization of the bulk of the plasma. We use as an estimate of the bremsstrahlung cross section the averaged energy emission rate per electron from Eq. (16) of Ref. [10], defined as σ ee→eeγ = 1 ρ e n e dE γ dt ,(19) where n e is the electron density, ρ e is the electron energy density, and dE γ /dt is the total energy emitted by bremsstrahlung per unit volume per unit time. For bremsstrahlung, different prescriptions are sometimes used in the literature, such as using the total cross section for emission of a photon with an energy larger than a typical electron energy; but this again leads to very similar results to the criterion used here. Fig. 3 shows, for each axion mass, the minimum value of the coupling for which the energy density of photons is large enough to activate pair production and bremsstrahlung (the latter assuming, of course, that electrons and positrons are produced). For low axion masses, and correspondingly low average energies for the photons, pair production equilibrates at lower coupling than bremsstrahlung, as expected given that bremsstrahlung is suppressed by a higher power of the coupling. At large masses, as soon as electrons and positrons are produced, bremsstrahlung reactions become faster than pair production. The reason is that the average energy per particle, m a /2 in the rest frame, is so large that the pair annihilation cross section is suppressed as σ e + e − →γγ ∼ α 2 /(m a /2) 2 , whereas the bremsstrahlung cross section is σ ee→eeγ ∼ α 3 /m 2 e , up to logarithmic corrections. Therefore, for m a ∼ 2m e / √ α ∼ 10 MeV, bremsstrahlung is faster than pair production. The subsequent thermalization depends on whether pair production or bremsstrahlung is the faster process. The final state of the plasma is, however, independent of this. Nevertheless, we keep the discussion separate for these two cases for clarity. At large enough couplings and masses, the fireball does not form anymore; the reason is that the fraction of photons decaying outside the progenitor star becomes so low that pair production is not efficient. For this reason, we find no fireball formation for axions heavier than about 60 MeV. D. Final state of the fireball Since the total energy released outside of the progenitor, for a benchmark axion mass m a = 40 MeV and coupling g aγγ = 2 × 10 −10 GeV −1 , is E = 1.22 × 10 50 erg, and the estimated volume of the shell is of order 4πr 2 ∆ = 3.60 × 10 38 cm 3 , thermalization tries to drive the plasma towards significantly low temperatures, of the order of T 15E/4π 3 r 2 ∆ 1/4 192 eV. 2 However, this state is never really reached. Thermalization via bremsstrahlung is interrupted much sooner, since at temperatures smaller than m e = 0.51 MeV the electron population is depleted by the faster pair annihilation. For m a < 10 MeV, in the intersection of the red and blue regions of Fig. 3, pair production is immediately the faster process. The role of pair production is there- fore that of maintaining the chemical potential of electrons and photons equal to a common value µ γ = µ e + = µ e − = −|µ|. The slower bremsstrahlung reaction (still faster than the timescale associated with the expansion of course) gradually drives this chemical potential towards 0, with the electrons and positrons emitting photons to dilute the excess average energy per particle. As the chemical potential drops, so does the temperature, since the number of particles is increasing and therefore the average energy per particle decreases. This emission of particles is interrupted when the electron number density is so low that bremsstrahlung cannot proceed any further. Therefore, the final state of the thermalization is a plasma of electrons and positrons with an equal chemical potential µ, with a temperature T and bulk motion γ determined by the initial energy and momentum released by the axions, and by the condition that bremsstrahlung is out of equilibrium γn e (T, µ)v th σ ee→eeγ ∆ = 1. We use here the non-relativistic expression for the bremsstrahlung cross section, since the number density of electrons and positrons starts to be kinematically suppressed only at temperatures lower than the electron mass. For the same reason, we also account for a factor v th corresponding to the typical thermal velocities of the electrons in the plasma frame, of the order of m e /T . For the full rate of bremsstrahlung interaction, we use the expression [10] σ ee→eeγ v th = 64α 3 3 πT m 3 e .(21) For m a > 10 MeV, in the intersection of the red and blue regions of Fig. 3, after the population of relativistic electrons and positrons is established, bremsstrahlung becomes immediately the faster process. Therefore, in this intermediate stage, the chemical potentials of electrons and photons are not bound to be equal. On the contrary, electrons and positrons start to radiate photons by bremsstrahlung rapidly in order to drive the chemical potential of the photons to zero. In doing this, they dilute the average energy and reduce the temperature of the plasma. Therefore, the initially equal chemical potential of electrons and photons now are unbalanced, with |µ γ | < |µ e + | = |µ e − |. This state of affairs is interrupted once the temperature of the plasma drops below 5 MeV, at which point pair production becomes again the dominant reaction as we have seen above. Therefore, there is a second stage at which pair production rapidly drives the chemical potentials of electrons and photons, which had been unbalanced by bremsstrahlung, to a common value intermediate between the two, so now µ γ = µ e + = µ e − = −|µ|. At this point, the subsequent evolution is identical to the case m a < 10 MeV: pair production is so rapid that it maintains the chemical potentials equal at all time, and bremsstrahlung slowly dilutes the number density by trying to bring µ to 0, and the chemical equilibration is interrupted once the bremsstrahlung freezes out, namely when the condition in Eq. (20) is reached. Thus, the cases m a < 10 MeV and m a > 10 MeV differ only in their approach to thermalization, but the final state is determined by the same conditions: conservation of energy, conservation of radial momentum, and decoupling of bremsstrahlung. The first two conditions are enforced by simply determining the total energy and radial momentum density of the initial axion shell, and imposing it equal to the energy and radial momentum density of the relativistic fluid of photons and electrons. The three conditions determine the three quantities of the plasma immediately after thermalization, namely the Lorentz factor γ, the temperature in rest frame T , and the common chemical potential of photons and electrons µ/T . The determination of the final state of the plasma is simplified by making two assumptions, both of which are verified by the final result. First, since the thermaliza-tion is interrupted because the electron density is suppressed, we can already guess that the plasma at the end of thermalization will be dominantly composed of photons. Therefore, in its energy density and pressure, we may neglect the electron-positron contribution and use again the equation of state of relativistic Boltzmann particles with p = ρ/3. This means that Eqs. (13), (14) and (15) are still satisfied. The second assumption that we make is that in the final state µ T ; this implies that thermalization is interrupted much before chemical equilibrium is reached, so that µ does not lower significantly. With this assumption, photons behave as relativistic Boltzmann particles and electrons/positrons behave as non-relativistic Boltzmann particles. Taking the ratio of Eqs. (20) and (14), after replacing the cross section Eq. (21) and the corresponding energy density and pressure for photons and number density for electrons/positrons, we find e −me/T T 3 = 3πγvr 2 √ 2α 3 P .(22) One may obtain an approximate solution to this equation in the limit T m e , which in turn requires the condition 3πγvr 2 m 3 e √ 2α 3 P 1.(23) In this limit, the solution is approximately T ∼ m e − log 3πγvr 2 m 3 e √ 2α 3 P .(24) The conditions for applicability are not always verified in our region of interest, which is why for the results shown in the figures we numerically solve Eq. (22); however, Eq. (24) has some pedagogical value, since it shows that the final temperature is a small fraction of the electron mass of order 10% with weak dependence on the initial parameters of the fireball. One may compare this with the initial temperature of the plasma before bremsstrahlung sets it, in Eq. (17), which on the other hand is of the order of the initial average energy per particle in the comoving frame. Finally, the chemical potential can be obtained from any of the equations Eqs. (14), (13) and (20); for example, using Eq. (14) we find µ T = log 32γ 2 vr 2 ∆T 4 πP .(25) Numerically, we find that this quantity is indeed much larger than 1 in all the region of fireball formation. Fig. 4 shows the properties of the plasma formed after thermalization. The plasma moves with a bulk Lorentz factor which is larger at lighter masses, due to the relativistic beaming in the photons emitted. The temperature reached is a fraction of the electron mass, with little dependence on the amount of energy injected; the chemical potential is larger close to the boundaries of the region, where the final state is farthest from the chemical equilibrium state. Consistently with our approximation scheme, the chemical potential never reaches values below about 8T . Therefore, at the end of the thermalization phase, the plasma is composed of a dominant population of photons and a subdominant population of e ± , with a sub-MeV temperature. III. FIREBALL EXPANSION The subsequent evolution is driven by radial expansion. As long as pair annihilation remains fast, this expansion happens in the hydrodynamical regime. In the case of γ 1, this regime is well-known from the analogous fireball formed in γ-ray bursts [28]. In our parameter space, we find γ to be always larger than 2.5, which means that the assumption of ultra-relativistic motion is reasonably satisfied; this approach is all the more justified since the radial expansion increases the bulk kinetic energy at the expense of the internal energy, leading to a rapid increase of γ. Furthermore, since the number of particles cannot change in this phase of expansion, the average energy per particle (in the laboratory frame) is not significantly affected by the details of the expansion. For γ 1, the thickness of the shell remains constant, since the fluid moves radially with the speed of light. The evolution of the thermodynamic quantities is ruled by the conservation laws of the entropy per particle, σ γ (r)/n γ (r), where σ γ is the entropy density; the average energy per particle, γ(r)ρ γ (r)/n γ (r), with ρ γ being the rest-frame energy density; and the total number of photons, γn γ 4πr 2 ∆. Unprimed quantities refer to the frame comoving with the fluid. These conservation laws happen to be identical to the fireball equations when chemical equilibrium is maintained, as derived from the hydrodynamical equations, see, e.g., Ref. [28]. We are assuming the plasma to be dominated by photons, so that the e ± component can be neglected. The conservation of entropy per particle implies that the ratio µ/T remains constant, since for photons σ γ /n γ is only a function of µ/T . In turn, the conservation of the average energy per particle implies the constancy of the product γT . Finally, the conservation of the photon number implies the scaling laws γ ∝ r, and therefore T ∝ r −1 . The expansion ends when pair annihilation runs out of equilibrium. At this point, photons moving at an angle θ with the radial direction will have a thermal distribution with an effective temperature T = T f /γ f (1 − v f cos θ), where the subscript f means that the quantity must be evaluated at the moment of decoupling. Since the photons reaching Earth will mostly be forward, with an angle θ γ −1 , we may safely take the limit θ 0, and therefore photons reaching Earth will have an average temperature T 2T f γ f for γ f 1. Using the constancy of T γ during the hydrodynamic expansion phase, this can be evaluated at the beginning of the expansion phase, that is, immediately after thermalization. As it should be, this quantity is of the same order of the average energy per particle in the lab-frame. A. Remaining fraction of high-energy photons Fig. 5 shows the average energy of the thermal distribution seen at Earth, namely three times the effective temperature, in the parameter space of interest. At low axion mass, the average energy can reach up to the order of MeV, mostly because of the higher bulk Lorentz factor due to the relativistic motion of the light axions. However, already at about 10 MeV, the average energy becomes sub-MeV. To qualitatively understand the impact on the bounds from the non-observation of γ-rays in Refs. [21,23], we look at the fraction of photons that, after thermalization, remain at an energy higher than 25 MeV, which is the threshold energy above which upper bounds on the fluence have been set. Therefore, we show in the middle panel of Fig. 5 the fraction of energy remaining in photons above E min = 25 MeV, the minimum energy at which data were available from SN 1987A. Since the photon spectrum has a Boltzmann shape, namely proportional to E 2 e −E/2γT , this fraction is f spectrum = +∞ Emin e −E/2γT E 3 dE +∞ 0 e −E/2γT E 3 dE = e −Emin/2γT 1 + E min 2γT + 1 2 E min 2γT 2 + 1 6 E min 2γT 3 .(26) This fraction rapidly drops by more than 4 orders of magnitude at a mass larger than about 5 MeV. All the previous discussion relates to the bulk of the photons produced in the decay of the axion, at distances comparable with the mean decay length or the progenitor radius. However, at sufficiently large radii, the photons coming from the decay of the small fraction of surviving axions may not be able to thermalize with the dominant bath of lower energy photons, if the density of the photon shell has sufficiently rarefied. This small fraction of the total photon flux reaches Earth with the originally expected large energies. To quantify the impact of this contribution, we estimate up to what radius the high-energy photons from late axion decays can thermalize with the bulk photon fluid. The dominant channel for thermalization is pair annihilation. While the temperature of the photon fluid is significantly lower than m e , the photons from axion decay have a typical lab-frame energyĒ a /2, withĒ a ∼ 100 MeV. In the rest frame of the decaying axion, the high-energy photons are isotropically distributed. Boosting to the laboratory frame, and then into the frame comoving with the plasma, we find that the center-of-mass energy for the collision of the two photons is E 2 CM = 3γTĒ a (1 − V v) ,(27) where we have assumed an average energy 3T for the restframe photons in the plasma; here V = 1 − γ −2 is the bulk plasma velocity and v = 1 − m 2 a /Ē 2 a is the typical axion velocity. As long as E CM > m e , photons from axion decay are kinematically allowed to collide with the photons from the bulk of the plasma. Notice that the minimum value that E 2 CM can take as the plasma expands and γ becomes larger is simply obtained by writing V = 1 E 2 CM,min = 3γTĒ a (1 − v) .(28) A sufficient condition for the reaction to be kinematically allowed is E CM,min > m e . The reaction will remain in equilibrium provided that γn γ πα 2 ∆ E 2 CM 1,(29) where we estimate the pair production cross-section as σ γγ→e + e − πα 2 /s, with s the Mandelstam parameter of the collision. This estimate holds up to factors of order unity. Once the radius of the fireball has expanded by a factor x to a radius r pair = xr, the condition for freezeout of pair production reaction becomes n γ,0 πα 2 ∆ 3T 0Ēa x 2 1 − 1 − m 2 ā E 2 a 1 − 1 γ 2 x 2 = 1,(30) where we identify by the suffix 0 the quantities at the beginning of the expansion. Expanding 1 − 1 γ 2 x 2 1 − 1 2γ 2 x 2 , we can obtain the factor of expansion as x 2 = nγ,0π∆α 2 3T0Ēa − v 2γ 2 1 − v .(31) This defines the maximum radius r pair = xr out of which injected photons are not able to thermalize with the bulk of the plasma. At this radius, the fraction of energy injected is exponentially suppressed as f pair = dE a E a e −rpair/ (Ea) dNa dEa dE a E a e −R/ (Ea) dNa dEa .(32) We show this suppression factor in the right panel of Fig. 5: this should be interpreted as the fraction of energy produced by axions at a large enough radius that the decay photons are not able to thermalize with the remaining bath of low-energy photons. The red boundary identifies the region within which E CM,min > m e , and therefore our treatment above is applicable. We define the fraction of energy which is still injected above the photon energy of 4 MeV as f = max [f pair , f spectrum ]. . We also show the fraction of axion decaying far enough from the center that the produced high-energy photons cannot thermalize and reach the Earth with their original energies; the red line is the boundary of the region in which the pair production reaction is always kinematically allowed from the criterion ECM,min > me. IV. UPDATED BOUNDS FROM SN 1987A Since the current bounds from SN 1987A rely on the energy window above 25 MeV [21,23], fireball formation can impact them. Unless the axion is very light and the photon energy remains large due to the large Lorentz factor, fireball formation reduces the photon flux above 25 MeV by more than ten orders of magnitude. (We comment on SMM data in the interval 4 − 25 MeV in Appendix A.) This already indicates that a region of the parameter space, previously excluded by non-observation by SMM of the decay photons, may be ruled in-though, as we will see, this is not the case. Thus, our first result is the region identified by the black solid line in Fig. 2, in which the bounds from both γ-ray decay from SN 1987A and diffuse supernova background need to be reevaluated, since the spectrum adopted in previous literature for γ-rays should actually be replaced by a Boltzmann spectrum with a temperature 2γT . Notice that this region is highly complementary to bounds from energy deposition in low-energy supernovae. If axions produced in the PNS travel and decay in the mantle, the ejecta kinetic energy becomes too large. Assuming that less than 0.1 B (1 B (bethe) = 10 51 erg) energy is deposited, as observed in low-energy supernovae, the solid red bounds are obtained [17]. A particularly conservative limit of 1 B leads to the dotted red bound. Since these bounds are calorimetric, they are not strongly affected-if at all-by the formation of a fireball. While the limits from SMM do not apply straightforwardly anymore, we find that the entire region of fireball formation is directly excluded by the non-observation of γ-rays at the Pierre Venus Orbiter (PVO). Launched in 1978 (last contact in 1992), PVO was part of NASA'S Pioneer Venus project and featured a γ-ray burst detector (OGBD) with two NaI photomultiplier detector units [34,35]. The sensitivity window of this detector was in a much lower energy range, between 0.2 MeV and 2 MeV, which is just the region in which the average energy of the thermalized spectrum lies in the region of fireball formation. Moreover, PVO had a 4π acceptance and, being in orbit around Venus, it was obviously outside of the Earth's radiation belts, so the background was particularly low [36]. As reported by Ref. [36], at the time of SN 1987A no excess over the background was observed in any of the energy bins observed by the instrument. While the response functions of the experiment are not published in this paper, we may extract a coarse bound on the fluence from Ref. [37], which reports the detection of γ-ray bursts (GRB) in the same energy window with a fluence of the order of about 10 −4 erg cm −2 . At the distance of SN 1987A of about 50 kpc, this means that a minimum total energy E of about 3 × 10 43 erg should have been detected. In the whole region of fireball formation, we find that the total energy E is always larger than 5 × 10 48 erg, namely more than five orders of magnitude higher than the approximate exclusion. Therefore, even without refining the calculation, we can already claim that the entire region of fireball formation is excluded on the basis of the PVO non-observation of a γ-ray burst in the low energy range 0.2 − 2 MeV. In fact, the bounds from PVO may extend even outside the region of fireball formation, since the γ-ray spectrum from ALP decay is more or less flat so that a non-negligible fraction of energy would be released in the low-energy window even without fireball formation; however, since in these regions bounds would be only complementary to the ones from SMM, we do not investigate this question further. It is logical to ask if other detectors could probe the low-energy flux of the fireball from SN 1987A. 3 While the GRS onboard of the SMM could detect photons even down to tens of keV [39], no available public data exists of SN 1987A in this energy band. Other hard X-ray detectors which observed SN 1987A, such as a JPL balloon [40] and the Ginga satellite [41], did so only hundreds of days after the collapse. V. UPDATED BOUNDS FROM DIFFUSE GAMMA-RAY BACKGROUND Photons originating from the decay of heavy axions produced by all past SNe would contribute to the diffuse cosmic γ-ray background. The idea of constraining neutrino radiative decays in this way dates back to the 1970s [42], and the argument has been revisited recently with applications to axions (see e.g. [16,25]). We here follow closely Ref. [16] to compute the expected photon flux ignoring the possibility of a fireball. Assuming that all SNe occur at z = 1, the photon energy flux (energy per unit area, time, and solid angle) due to axion decays is ω 2 dΦ γ dω = 1 4π ζ a E SN E av n cc 2(T eff + 2ω)ω 2 T 2 eff e −2ω/T eff ,(33) where ζ a is a fudge factor proportional to g 2 aγγ , E SN = 3 × 10 53 erg is the typical energy released by a SN, T eff is the effective temperature akin to the one appearing in Eq. (1), E av = 3T eff is the axion average energy, and n cc 10 7 Mpc −3 is the core-collapse number per comoving volume per redshift interval. This spectrum has a maximum at ω max = T eff (1 + √ 3)/2 1.37 T eff . The diffuse γ flux produced by axions should be compared with the measurements of the extragalactic background radiation [24]. For axions with masses and couplings not allowing the formation of a fireball, we can use the range of the extragalactic background spectrum 2-200 MeV [24], ω 2 dΦ observed γ dω 2−200 MeV 2×10 −3 MeV cm −2 s −1 ster −1 . (34) Therefore, the bound is found comparing Eq. (33) evaluated at ω max , ω 2 dΦ γ dω max = ζ a E SN n cc 7 + 4 √ 3 12π e −(1+ √ 3) = ζ a n cc 10 7 Mpc −3 46.2 MeV cm −2 s −1 ster −1 , with Eq. (34). One obtains [16] ζ a < ∼ 0.43 × 10 −4 n cc , in Ref. [38]. independently of the assumed average energy of the emitted bosons. This corresponds to a constraint on axions with a photon coupling g aγγ < ∼ 10 −10 GeV −1 [16,17]. The result does not depend strongly on the assumed redshift dependence of the cosmic core-collapse rate n cc . The formation of the fireball impacts the diffuse γ-ray bounds: as the average energy of photons is driven below 1 MeV, the flux from axion decays needs to be compared to a much larger astrophysical background (see e.g. Figure 10 of Ref. [24]). A very conservative estimate can be obtained using the largest value of the extragalactic background light energy flux, ω 2 dΦ observed γ dω < ∼ 100 keV < ∼ 5 × 10 −2 MeV cm −2 s −1 ster −1 . (37) This is 25 times larger than the value that is observed in the 2-200 MeV energy range. Therefore, the diffuse γ-ray bound is relaxed at most by a factor of 5 (g aγγ < ∼ 5 × 10 −10 GeV −1 ) in the region corresponding to fireball formation. On the other hand, the bound rapidly degrades at m a 50 MeV [17], so we cannot commit to a precise evaluation of this region, and a dedicated analysis is needed to revisit this bound at large masses. VI. DISCUSSION AND OUTLOOK Heavy axions coupling to photons can be produced in the hot core of supernovae and decay back to photons with around 100 MeV. While it has always been assumed that such photons would travel freely, in part of the axion parameter space photons form a fireball, and their energy degrades to values below 1 MeV. Part of the parameter space previously thought to be excluded by observations of SN 1987A with SMM [21,23] is actually excluded by PVO data [31,36], as the energy of the photons at Earth would have been much lower than expected. PVO observations, previously applied only to neutrino radiative decays [36], are relevant since cosmological constraints apply in the region of interest only for a large reheating temperature [7,9]. Constraints arising from the extragalactic background light (see e.g. Ref. [16] and references therein) also get relaxed. Other complementary probes in the regions might be, for example, very low luminosity SNe and their light-curve shape and spectral line velocities, that can potentially probe unexplored parts of the parameter space [17]. Another interesting probe could rely on past [43] and future neutron star merger observations, which we plan to explore in forthcoming work. We stress that our discussion applies to different models, such as axions coupling to charged leptons [16,27] and heavy neutral leptons featuring a dipole portal [44]. Therefore, constraints arising from SN γ-ray observations should be updated to include PVO data. Fireball formation is even easier for axions with an electron coupling, since they can decay to electron-positron pairs and bremsstrahlung can drive the thermalization without the need of producing pairs through two-photon annihilation. ACKNOWLEDGMENTS We thank Hans-Thomas Janka, Georg Raffelt, and Irene Tamborra for comments on the first draft of this paper. MD acknowledges the support of the National Sciences and Engineering Research Council of Canada In the main text we have discussed what is the fraction of photons remaining above 25 MeV, since Refs. [21,23] both use γ-ray data above this energy. In reality, SMM did have additional observations down to about 4 MeV. Therefore, it makes sense to also look at the fraction of photons remaining above 4 MeV; this would provide a guide as to how powerful the bounds from SMM could be by accounting also for these additional data which were not used in past approaches. Fig. 6 shows this fraction of energy. We find this to be significantly higher than the fraction of energy injected above 25 MeV, which is expected since the tail of the spectrum decreases exponentially. Therefore, SMM may in principle still constrain a significant part of the region of fireball formation, though with different data and a different approach than what has been done in the past. However, we do not follow up on this question further, since the bounds from PVO robustly exclude the entirety of the region. FIG. 2 . 2sured the fluence in the interval 4.1 − 100 MeV during a 223.2 s interval coincident with the SN 1987A neutrino arXiv:2303.11395v1 [hep-ph] Region of fireball formation in the axion mass and coupling space (solid black line) FIG. 3 . 3Regions of the parameter space in which the relevant processes for chemical equilibrium are activated. In the blue region, the photon plasma from axion decay is dense enough that pair production equilibrates, leading to the formation of an electron-positron component in the plasma. In the red region, if electrons and positrons are produced, the number density is diluted by bremsstrahlung emission. Thus, in the intersection of the blue and red region, number density dilution via bremsstrahlung is effective. FIG. 4 . 4Final state of the plasma. From left to right panel, we show bulk Lorentz factor, rest-frame temperature, and rest frame chemical potential of the plasma, as a function of the axion mass and coupling. All quantities are shown only in the region in which the plasma forms. FIG. 5 . 5Spectral properties of the γ-ray emission from the fireball. We show the average energy of the thermalized photons (left), and the fraction of the thermalized photons integrated above 25 MeV (middle) FIG. 6 . 6Fraction of the thermalized photons integrated above 4 MeV. (NSERC). DF is supported by the Villum Fonden under project no.29388. This project has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No. 847523 'INTERACTIONS'. GMT acknowledges support by the National Science Foundation under Grant Number PHY-2210361 and by the US-Israeli BSF Grant 2018236. 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{'abstract': 'New feebly interacting particles would emerge from a supernova core with 100-MeV-range energies and produce γ-rays by subsequent decays. These would contribute to the diffuse cosmic γ-ray background or would have shown up in the Solar Maximum Mission (SMM) satellite from SN 1987A. However, we show for the example of axion-like particles (ALPs) that, even at distances beyond the progenitor star, the decay photons may not escape, and can instead form a fireball, a plasma shell with T < ∼ 1 MeV. Thus, existing arguments do not exclude ALPs with few 10 MeV masses and a two-photon coupling of a few 10 −10 GeV −1 . However, the energy would have showed up in sub-MeV photons, which were not seen from SN 1987A in the Pioneer Venus Orbiter (PVO), closing again this new window. A careful re-assessment is required for other particles that were constrained in similar ways.', 'arxivid': '2303.11395', 'author': ['Melissa Diamond ', 'Damiano F G Fiorillo \nNiels Bohr International Academy & DARK\nNiels Bohr Institute\nUniversity of Copenhagen\nBlegdamsvej 172100CopenhagenDenmark\n', 'Gustavo Marques-Tavares \nMaryland Center for Fundamental Physics\nDepartment of Physics\nUniversity of Maryland\n20742College ParkMDU.S.A\n', 'Edoardo Vitagliano \nRacah Institute of Physics\nHebrew University of Jerusalem\n91904JerusalemIsrael\n', 'Arthur B Mcdonald Canadian ', '\nAstroparticle Physics Institute\nQueens University\nK7L 3N6KingstonOntarioCanada\n'], 'authoraffiliation': ['Niels Bohr International Academy & DARK\nNiels Bohr Institute\nUniversity of Copenhagen\nBlegdamsvej 172100CopenhagenDenmark', 'Maryland Center for Fundamental Physics\nDepartment of Physics\nUniversity of Maryland\n20742College ParkMDU.S.A', 'Racah Institute of Physics\nHebrew University of Jerusalem\n91904JerusalemIsrael', 'Astroparticle Physics Institute\nQueens University\nK7L 3N6KingstonOntarioCanada'], 'corpusid': 257636649, 'doi': '10.1103/physrevd.107.103029', 'github_urls': [], 'n_tokens_mistral': 21242, 'n_tokens_neox': 17834, 'n_words': 11812, 'pdfsha': 'dc469b117955ec693171594ad413e7af1197bdaf', 'pdfurls': ['https://export.arxiv.org/pdf/2303.11395v1.pdf'], 'title': ['Axion-sourced fireballs from supernovae', 'Axion-sourced fireballs from supernovae'], 'venue': []}

arxiv

Transformation Electronics: Tailoring Electron's Effective Mass Mário G Silveirinha mario.silveirinha@co.it.pt Department of Electrical and Systems Engineering University of Pennsylvania PhiladelphiaPAU.S.A Department of Electrical Engineering -Instituto de Telecomunicações University of Coimbra Portugal Nader Engheta engheta@ee.upenn.edu Department of Electrical and Systems Engineering University of Pennsylvania PhiladelphiaPAU.S.A Department of Electrical Engineering -Instituto de Telecomunicações University of Coimbra Portugal Transformation Electronics: Tailoring Electron's Effective Mass 1 PACS: 42.70.Qs, 73.21.Cd, 73.23.-b 73.22.-f * To whom correspondence should be addressed: The speed of integrated circuits is ultimately limited by the mobility of electrons or holes, which depend on the effective mass in a semiconductor. Here, building on an analogy with electromagnetic metamaterials and transformation optics, we describe a new transport regime in a semiconductor superlattice characterized by extreme anisotropy of the effective mass and a low intrinsic resistance to movement -with zero effective mass -along some preferred direction of electron motion. We theoretically demonstrate that such regime may permit an ultra fast, extremely strong electron response, and significantly high conductivity, which, notably may be weakly dependent on the temperature at low temperatures. These ideas may pave the way for faster electronic devices and detectors and new functional materials with a strong electrical response in the infrared regime. In 1969, Esaki and Tsu suggested that by either periodically doping a monocrystalline semiconductor or by varying the composition of the alloy, quantum mechanical effects should be observed in a new physical scale 1 , so that the conduction and valence bands of such superlattices are structured in the form of many sub-bands 1,2 , and in particular they predicted the possibility of a negative differential conductance. 1 This pioneering work has set the stage for the dispersion engineering in semiconductor superlattices. This conceptual breakthrough and other prior key proposals (e.g. the idea of quasi-electric fields 3 ), are the foundation of many spectacular advances in semiconductor technology 4 , and has enabled among others the development of the quantum cascade laser 5 , and the realization of ultrahigh mobilities in semiconductor superlattices and quantum wells 6,7 . Following these advancements, more recently, there has been a huge activity in the study of a new class of mesoscopic materials -metamaterials -whose electromagnetic properties are determined mainly by the geometry and material of its constituents, rather from the chemical composition 8,9 . Such line of research has resulted in the development of double negative materials, which promise erasing diffraction effects and perfect lensing 8 . Until now, the obvious analogy between superlattices and electromagnetic metamaterials received little attention, apart from isolated studies 10,11 . Here, inspired by the exciting paradigm offered by electromagnetic metamaterials and transformation optics 8,9 , we develop the paradigm of "transformation electronics", wherein the electron wave packets are constrained to move along desired paths, and predict a totally new transport regime in a semiconductor superlattice based on the extreme anisotropy of the effective mass. In a semiconductor the effective mass determines the inertia of the electron to an external stimulus. The finite value of the mobility ultimately limits the speed of integrated circuits and other devices. In most electronic circuits the electron flow is supposed to occur along a predetermined path, e.g. down the passageway connecting two transistors. However, typically only a small portion of the available free carriers responds effectively to an external electric field, i.e. those whose velocity 1 g E − = ∇ k v h is parallel to the impressed field. Would it however be possible to engineer the electron mass in such a way that all the available electronic states contribute to the electron flow? Moreover, would it however be possible to reverse or "cancel" the effects of the intrinsic electron resistance to movement, along the preferred direction of motion? A superlattice with the properties implicit in the first question must be anisotropic. Indeed, in order that and in addition to have a weak resistance to movement ( * 0 zz m = ). A zero mass has been previously predicted to occur at contacts between semiconductors with normal and inverted band structures 14 , but not an extreme anisotropy regime. 1 g E − = ∇ k v h is To achieve this, we draw on an analogy with electromagnetic metamaterials. The intriguing tunneling phenomena observed in electromagnetic metamaterials are rooted in the fact that two materials such that 1 2 ε ε = − and 1 2 μ μ = − , with ε being the permittivity and μ the permeability, "electromagnetically annihilate" one another. 8,15 It is thus natural to wonder if in electronics it may be possible to identify complementary materials that when paired yield * 0 m ≈ . Since, the effective mass of the carriers is expected to be determined by some averaging of the values of * m in the superlattice constituents, this suggests that one should look for materials wherein * m has different signs. Even though unusual, the carriers can have a negative effective mass, notably in semiconductors and alloys with a negative energy band gap. 16 Examples of such materials are mercury-telluride (HgTe) [a group II-VI degenerate semiconductor] and some alloys of mercury-cadmium-telluride (HgCdTe), which have an inverted band structure 16,17 , so that the 8 Γ (P-type) valence bands lie above the conduction band 6 Γ (Stype), and the effective masses of both electrons and holes ( * , c h m ) are negative. In Refs. [18,19] we develop a formal analogy between the Helmholtz equation for the electromagnetic field and a Schrödinger-type equation for the envelope wavefunction consistent with the standard Kane model (k⋅p method) for semiconductors with a zincblende structrure. 20 Within this formalism, that is consistent with Bastard's theory 22 , the electron is described by a single component wavefunction, ψ , which may be regarded as the spatially averaged microscopic of wavefunction. This contrasts with the conventional k⋅p approach where the electron is described by a multi-component wavefunction. 20 For the case of Bloch waves, ψ may be identified with the zero-th order Fourier harmonic of the microscopic wavefunction. 18,19 Related averaging procedures have been considered previously in the context of electromagnetic metamaterials 21 . The wavefunction in the superlattice satisfies: ( ) ( ) ( ) 2 1 , 0 2 , V E E m E ψ ψ ⎛ ⎞ − ∇⋅ ∇ + − = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ r r h .(1) The effective potential ( ) ( ) ( ) 6 , c V E E E Γ = = r r r is determined( ) ( ) ( ) ( ) 2 , /2 v v P m E E E ≈ − ⎡ ⎤ ⎣ ⎦ r r r , where ( ) ( ) 8 v E E Γ = r r is the valence band energy level, 0 m is the free-electron mass, 2 2 0 2 / P E P m = h , P is Kane's parameter 20 , and ( ) 0 v / 3 P P E m = has dimensions of velocity. The dispersive mass, ( ) , m m E = r , should not be confused with the effective mass 1 * 2 2 / i j m E k k − ⎡ ⎤ = ∂ ∂ ∂ ⎣ ⎦ h determinedE E Λ = − is such that ,1 ,2 0 g g E E < Λ < + so that( ) ( ) ( ) k m k m k a k d k d k d k d k m k m ⎛ ⎞ = − + ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ (2) where 1 2 a d d = + is the lattice constant, ( ) ( ) ( ) 2 , | | 2 2 i z i i m E k E V E k = − − h , ||ẑ k = + k k z( ) ( )1 2 f f = = . ( For ternary alloys of Hg , one may estimate that the lattice constant should not be greater than ( ) 1/ 2 2 2 2 2 || v 1 4 v / z P P E k k − = ± + Λ h h .(4)( ) max~0 .1 / P B a v k T π h , which at room temperature gives max~8 a nm. Ideally, the energy dispersion should be independent of || k . This is achieved close to the plane 0 z k = , where the constant energy surfaces are flat, whereas for larger values of z k they become hyperbolic (Fig. 3b). Indeed, within the validity of Eq. (4), the ideal case requires Λ → ∞ . In Fig. 3d, it is shown that if the lattice constants of the materials are slightly mismatched, the energy dispersion is perturbed and a small band gap may appear. Even in this non-ideal scenario, the effective mass * zz M remains near zero, whereas * || M remains extremely large (not shown). The transport properties of the superlattice, and most notably the conductivity, may be radically different from those of the constituent semiconductors. A detailed calculation shows that within the validity of Eq. (4), the intraband conductivity is given by 19 ( ) 2 2 intra,xx intra,yy 2 1 1 1 2 v 6 B P ie k T D D ω ⎛ ⎞ = = + − ⎜ ⎟ ⎝ ⎠ σ σ h h (5a) ( ) 2 2 intra,zz 2 1 1 v 2 P ie D ω π Λ ⎛ ⎞ = − ⎜ ⎟ ⎝ ⎠ σ h h (5b) where ( ) 2 ||,max 2 v / 1 P D k = Λ + h and ||,max~/ s k a π is a cut-off parameter. 19 It is assumed that the Fermi level lies exactly at 0 E = . Moreover, we neglect scattering effects due to defects or interface mismatch, which may in any case be modeled phenomenologically by replacing ω by i ω + Γ in the above formulas, where Γ represents a collision frequency. It should be mentioned the effective medium model based on Eq. (1) is unable to predict the dispersion of (the hybridized) heavy-hole states, and thus their contribution to the conductivity was not considered. However, since the heavy-hole mini-bands are expected to be nearly flat they should not influence much the transport properties. Equation (5) 1 1 1 1 6 2 v v P P g E D π ⎛ ⎞ Λ = − ⎜ ⎟ ⎝ ⎠ h h( ) ( ) 2 3 2 , which is independent of the energy and does not vanish at the Fermi level (Fig. 4d). Hence, at the Fermi level the surfaces of constant energy are not reduced to a point as in a normal semiconductor (or graphene), but instead are collapsed into the 0 z k = plane (a square-shaped surface). The electrons occupying such states can respond effectively to an external field oriented along z, which explains the finite conductivity in the 0 T → limit. In conclusion, we have investigated the transport properties of a novel metamaterialinspired superlattice, characterized by linear energy dispersion along some preferred direction of carrier motion and extreme anisotropy. The condition * 0 zz M = results from pairing materials with band-gaps of different signs that effectively interact as "matterantimatter", in the same manner as ENG and MNG metamaterials electromagnetically annihilate one another. Our ideas may establish a new paradigm for an ultra-fast and extremely strong electronic response, which may be nearly independent of temperature in the limit 0 T → , and exciting new developments in electronics and photonics. As the concept of "Transformation Optics" enables tailoring the path of light, in our system we can have the same but for electrons, namely, the electron's path may be constrained so that the electrons are forced to move along a preferred direction. Since it may be possible to vary the parameters of semiconductors continuously either by doping or by controlling the material composition or -in case of a 2D electron gas -by tailoring externally the potential seen by the electrons with a top-gate, we envision that some of the ideas of Transformation Optics can be brought to the field of electronics. by the energy level of the conduction band in each component of the heterostructure. Provided the effect of the spin-orbit split-off bands is negligible, the dispersive (energy-dependent) mass there is no overlap between conduction and valence bands in the two materials (Fig. 1c). From the analogy between the Schrödinger [Eq. (1)] and Helmholtz equations outlined in 18,19 , we have the correspondences between the parameters ε and μ (permittivity and permeability) and V and m : to 10 . Hence, it follows that the material with positive band gap ( ,1 0 g E > ) is seen by an electron with energy E in the band gap as a (ENG material), whereas the material with negative band gap ( (MNG material) [see Fig. 1b]. We calculated analytically the dispersion of the superlattice Bloch modes, using Eq. (1) and imposing generalized Ben Daniel-Duke boundary conditions at the interfaces 22 . Our Kronig-Penney type model yields 19 , , 2 . 2the wave vector, and the effective parameters of the semiconductors are , The conduction mini-band resulting from the hybridization of the energy diagrams of the two semiconductors emerges at the energy level (for both electrons and holes) provided the spatially averaged band-gap energy ( ,av g E ) and the filling ratio of the materials satisfy 19 : M Thus, the energy dispersion along the z-direction varies linearly, = . Hence, even though our system is fully three-dimensional and the wavefunction is not a pseudo-spinor as in graphene, the electron transport along z may be somewhat analogous to that in graphene. On the other hand, to ẑ , and thus all the associated electronic states contribute to an electron flow along the z-direction, as expected from * || M = ∞ . These properties are confirmed byFig. 3a [obtained using Eq. (2)] which depicts the energy dispersion for the bulk semiconductors. 17 Fig. 3c shows that the dispersion calculated with Eq. (4) captures accurately the results of the Kronig-Penney model. The value of v P in the superlattice is similar to that of the Fermi velocity in graphene17 ], and hence, in the limit of low scattering the electron response in the superlattice can be extremely fast. Similar to photonic metamaterials a description of the superlattice in terms of effective parameters is possible when 1 ka << , where k represents the wave vector in a generic region. The spread of the wave vector is determined by the temperature, and thus at low temperatures the effective medium theory is expected to be quite accurate. Based on Eq. (4), imposing ~B E k predicts that the conductivity along the z-direction is independent of the be extremely large. Even though Eq. (5) was derived using the approximate dispersion (4),Fig. 4ashows that it describes fairly well the conductivity calculated using Eq. (2) (the case 0.4 g E Λ = models the superlattice Hg 0.65 Cd 0.35 Te-HgTe). Due to the extreme anisotropy, at low temperatures most of the states contribute to the electron flow along z, and thus the conductivity in the x-y plane vanishes in the limit 0 T → . The anisotropy ratio, as well as the absolute value of zz σ , are enhanced with larger values of Λ , because larger values of Λ yield an energy dispersion closer to the ideal case: .Fig. 4b and 4cshow that the superlattice conductivity can be made several orders of magnitude (~3 10 at 300 T K = ) larger than that of the constituent materials. To shed light on the intriguing independence of zz σ on T, using Eq. (4) we calculated the density of states in the superlattice, FiguresFig. 1 . 1Transformation Electronics and Electronic Metamaterials: Sketch of the geometry and electronic band diagram of the elements of the superlattice. (a) Geometry of a stratified superlattice formed by alternating layers of semiconductor alloys with band gaps with different signs. (b) Electromagnetic analogue of the superlattice for energy levels close to 0 eff E V − ≈ : in the band gap the semiconductor with positive (negative) band gap is the electronic analogue of . (c) Detailed energy band structure of each layer of the superlattice, showing the valence band-offset Λ between the two semiconductors. Fig. 2 . 2Effective parameters (mass and potential) of the semiconductor alloys. Left axis: dispersive mass as a function of the normalized electron energy E; Solid lines: exact result taking into account the effect of the split-off bands (see Ref. [18, 19]); Dashed lines: linear mass approximation described in the main text. As seen, the effect of the split-off bands in negligible in the energy range of interest. Right axis: effective potential ( E V − ) as a function of the normalized electron energy E. Fig. 3 .Fig. 4 .. 34Electronic band structure of the superlattice electronic metamaterials. Conductivity and density of states of the superlattice. (a) Conductivity of the semiconductor superlattice (SL) at 10THz as a function of the temperature for different values of the valence band offset: indicate the direction of increasing Λ . The solid lines were calculated using the "exact" energy dispersion of the superlattice, whereas the dashed lines were obtained from Eqs. (5a)-(5b). (b) Conductivity of the semiconductor superlattice as a function of frequency for different values of the valence band offset at 300K. (c) Similar to (a) but for the normalized intraband conductivity of Hg 0.65 Cd 0.35 Te, assuming that the Fermi level lies at the midpoint of the energy band gap. (d). Normalized density of states of the semiconductor superlattice for different values of the valence band offset. In all the calculations it was assumed that ||,max 0.Since ( ) g E was computed based on the approximate model (4), the results of Fig. 4d are meaningful only for E << Λ . parallel to the desired direction of flow (let us say z), .e. the resistance to a flow in the x-y plane must be extremely large. To satisfy the requirements implicit in the second question it isit is necessary that the energy dispersion ( ) E E = k depends exclusively on the wave vector component z k , and hence the effective mass tensor satisfies ( ) 1 * * 2 2 2 / xx yy y m m E k − = = ∂ ∂ =∞ h , inecessary that ( ) 1 * 2 2 2 / zz z m E k − = ∂ ∂ h be near zero. Thus ideally we should have * * xx yy m m = =∞ and * 0 zz m = , and thus an effective mass tensor characterized by extreme anisotropy. Notably, heterostructures with extreme anisotropy have received some attention in recent years due to their potentials in collimating both light 12 and electrons 13 . However, our findings are fundamentally different from previous studies: we deal with a bulk semiconductor superlattice, and show how by combining two different semiconductors it may be possible to super-collimate the electron flow ( * * xx yy m m = =∞) by the curvature of the energy diagram. For narrow gap semiconductors * m satisfies (for bothLet us consider a superlattice formed by slabs of two narrow gap semiconductors alternately stacked along the z-direction(Fig. 1a). Each semiconductor layer (i=1,2) haselectrons and holes): * * 2 2v g c h P E m m ≈ ≈ , where g c v E E E = − is the band-gap energy of the semiconductor. The sign of * m is the same as that of g E . thickness i d , and is described by parameters i V and ( ) i i m m E = , and the band gap energies of the semiconductors have different signs so that ,1 0 g E > (e.g. an alloy of HgCdTe) and ,2 0 g E < (e.g. HgTe). In addition, the valence band offset ,2 ,1 v v 1-x Cd x Te we have ,1,2 v v P P ≈ , because Kane's P parameter varies little with the mole fraction x. 17 , and hence Eq. (3) reduces to ,av 0 g E = . This can be realized taking the negative band gap material as HgTe ( ,2 0.3 g E e V = − 16,17 ), and the positive band gap material as Hg 0.65 Cd 0.35 Te, which has [ ] ,1 0.3 g E eV = + 23 . Fig. 2 shows the effective dispersive mass and effective potential calculated using our model (with 0.40 0.12 g E eV Λ = = 24 ) confirming that the effective parameters of each material have different signs. In the conditions of Eq. 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{'abstract': 'The speed of integrated circuits is ultimately limited by the mobility of electrons or holes, which depend on the effective mass in a semiconductor. Here, building on an analogy with electromagnetic metamaterials and transformation optics, we describe a new transport regime in a semiconductor superlattice characterized by extreme anisotropy of the effective mass and a low intrinsic resistance to movement -with zero effective mass -along some preferred direction of electron motion. We theoretically demonstrate that such regime may permit an ultra fast, extremely strong electron response, and significantly high conductivity, which, notably may be weakly dependent on the temperature at low temperatures. These ideas may pave the way for faster electronic devices and detectors and new functional materials with a strong electrical response in the infrared regime.', 'arxivid': '1205.6325', 'author': ['Mário G Silveirinha mario.silveirinha@co.it.pt \nDepartment of Electrical and Systems Engineering\nUniversity of Pennsylvania\nPhiladelphiaPAU.S.A\n\nDepartment of Electrical Engineering -Instituto de Telecomunicações\nUniversity of Coimbra\nPortugal\n', 'Nader Engheta engheta@ee.upenn.edu \nDepartment of Electrical and Systems Engineering\nUniversity of Pennsylvania\nPhiladelphiaPAU.S.A\n\nDepartment of Electrical Engineering -Instituto de Telecomunicações\nUniversity of Coimbra\nPortugal\n'], 'authoraffiliation': ['Department of Electrical and Systems Engineering\nUniversity of Pennsylvania\nPhiladelphiaPAU.S.A', 'Department of Electrical Engineering -Instituto de Telecomunicações\nUniversity of Coimbra\nPortugal', 'Department of Electrical and Systems Engineering\nUniversity of Pennsylvania\nPhiladelphiaPAU.S.A', 'Department of Electrical Engineering -Instituto de Telecomunicações\nUniversity of Coimbra\nPortugal'], 'corpusid': 118626721, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 8118, 'n_tokens_neox': 6815, 'n_words': 4020, 'pdfsha': '557c211a70c396731484fbf805c38aea24ff439a', 'pdfurls': ['https://export.arxiv.org/pdf/1205.6325v2.pdf'], 'title': ["Transformation Electronics: Tailoring Electron's Effective Mass", "Transformation Electronics: Tailoring Electron's Effective Mass"], 'venue': []}

arxiv

Beyond Starobinsky inflation July 2018 23 Jul 2018 Yermek Aldabergenov aldabergenov-yermek@ed.tmu.ac.jp Department of Physics Tokyo Metropolitan University Minami-ohsawa 1-1, Hachioji-shi192-0397TokyoJapan Research School of High-Energy Physics Tomsk Polytechnic University 2a Lenin Ave634050TomskRussian Federation Ryotaro Ishikawa ishikawa-ryotaro@ed.tmu.ac.jp Department of Physics Tokyo Metropolitan University Minami-ohsawa 1-1, Hachioji-shi192-0397TokyoJapan Sergei V Ketov ketov@tmu.ac.jp Department of Physics Tokyo Metropolitan University Minami-ohsawa 1-1, Hachioji-shi192-0397TokyoJapan Research School of High-Energy Physics Tomsk Polytechnic University 2a Lenin Ave634050TomskRussian Federation Kavli Institute for the Physics and Mathematics of the Universe (IPMU) The University of Tokyo 277-8568ChibaJapan Sergey I Kruglov serguei.krouglov@utoronto.ca Department of Chemical and Physical Sciences University of Toronto 3359 Mississauga RoadL5L 1C6North, MississaugaOntarioCanada Beyond Starobinsky inflation July 2018 23 Jul 2018 A supergravity extension of the (R + R 2 ) gravity with the additional (Born-Infeld) structure of a massive vector multiplet gives rise to the specific F (R) gravity, whose structure is investigated in detail. The massive vector multiplet has inflaton (scalaron), goldstino and a massive vector field as its field components. The model describes Starobinsky inflation and allows us to extrapolate the F (R) function beyond the inflationary scale (up to Planck scale). We observe some differences versus the (R + R 2 ) gravity, and several breaking patterns of the well known correspondence between the F (R) gravity and the scalar-tensor gravity. arXiv:1807.08394v1 [hep-th] Introduction An ultimate theory of cosmological inflation should be based on quantum gravity and is yet to be constructed. This is related to another open problem of finding an ultraviolet (UV) completion of any phenomenologically viable inflationary model. Amongst the most successful and popular inflationary models, Starobinsky inflationary model of (R + R 2 ) gravity [1] is special because it is entirely based on gravitational interactions. This model is, however, non-renormalizable and has the UV-cutoff given by Planck scale. In addition, when extrapolating the (R + R 2 ) gravity beyond the inflationary scale of about 10 13 GeV, i.e. when going to the very large curvature regime, we are left with the scale-invariant R 2 gravity. The original motivation in [1] was to get rid of the initial singularity of Einstein-Friedmann gravity, in addition to describing inflation in the early Universe. However, demanding the asymptotical scale invariance at very high energies is clearly not the only option. Hence, is still the open question: what should we expect beyond Starobinsky inflation? To address this question at least partially, one needs a motivated extension of the (R + R 2 ) gravity in a specific framework. In this paper, we address the issue in four-dimensional N = 1 supergravity. The importance of the inflationary model building in supergravity stems from the natural objective to unify gravity with particle physics beyond the Standard Model of elementary particles and beyond the Standard (ΛCDM) Model of cosmology, see e.g., [2,3] for a review. Though supergravity can be considered as the low-energy effective theory of (compactified) superstrings, and the latter can be viewed as a consistent theory of quantum gravity, we obviously need more specific assumptions. Our additional specific assumptions in this paper are the following: • Starobinsky inflationary model should be embedded into a four-dimensional N = 1 supergravity, with linearly realized (manifest) local supersymmetry, • inflaton (scalaron) should belong to a massive N = 1 vector supermultiplet, • the kinetic terms of the vector supermultiplet should have the Born-Infeld (or Dirac-Born-Infeld) structure, inspired by superstrings and D-branes. This leads to the specific (modified) F (R) gravity model, whose peculiar structure is in the focus of our investigation in this paper. Our paper is organized as follows. In Sec. 2 we outline Born-Infeld (BI) non-linear electrodynamics and the supergravity theory with the BI structure. In Sec. 3 we review the Starobinsky inflationary model. In Sec. 4 we study in detail the F (R) gravity extension of the (R +R 2 ) gravity, originating from the supergravity theory. In Sec. 5 we present the dual description of the same F (R) gravity in terms of the scalar-tensor gravity. Our Conclusion is Sec. 6. In Appendix we formulate the full supergravity theory in terms of superfields in curved superspace. Born-Infeld structure in gravity and supergravity Born-Infeld (BI) Lagrangian was originally introduced [4] as a non-linear generalization of the Lagrangian of Maxwell electrodynamics in terms of the abelian field strength F µν = ∂ µ A ν − ∂ ν A µ , L BI = −b −2 − det η µν + b e F µν − 1 = − 1 4e 2 F µν F µν + O(F 4 ) ,(1) where we have introduced the dimensional (BI) coupling constant b = M −2 BI and the gauge (dimensionless) coupling constant e. Being minimally coupled to gravity, the BI action reads S BI = b −2 d 4 x √ −g − − det g µν + b e F µν .(2) This BI structure also arises (i) in the bosonic part of the open superstring effective action [5], (ii) as part of Dirac-Born-Infeld effective action of a D3-brane [6], and (iii) as part of Maxwell-Goldstone action describing partial supersymmetry breaking of N = 2 supersymmetry to N = 1 supersymmetry [7,8]. In string theory, b = 2πα , while the BI scale M BI does not have to coincide with M Pl . 1 In N = 1 supersymmetry and supergravity, a vector field belongs to an N = 1 vector multiplet, whose supergravity couplings are naturally (off-shell) described in superconformal tensor calculus [11] and in curved superspace [12]. A massive N = 1 vector multiplet has a single (real) scalar field amongst its bosonic field components, in addition to a massive vector field. In this paper, we identify this real scalar with inflaton, and unify it with the massive vector field whose kinetic terms are assumed to have the BI structure in N = 1 supergravity (we do not assume any relation between our massive vector field and electromagnetic field). The full action of the self-interacting massive vector multiplet with the BI structure in supergravity is very complicated: it was found by using the superconformal tensor calculus in [13], and we present this action in Appendix, by using superfields in curved superspace. 2 In particular, local supersymmetry (SUSY) is spontaneously broken in this theory (after inflation also), while goldstino is identified with a massive "photino" in the same vector multiplet with inflaton. For our purposes in this paper, it is enough to notice that in the dual (modified supergravity) picture the BI structure just leads to the presence of the contribution 12R 2 /(e 2 M 4 BI ) under the square root of the BI term, in addition to the F µν -dependent terms there. When ignoring all other interactions besides the modified gravity itself (i.e. keeping only the R-dependent terms), it gives rise to the following F (R) gravity model (see Ref. [13] and Appendix): S = d 4 x √ −g M 2 Pl 2 R + M 4 BI 3 1 + 12R 2 e 2 M 4 BI − 1 .(3) It is this modified gravity theory that is the main subject of our investigation in this paper. It is worth noticing that it does not imply the upper bound on the values of R, unlike the original BI theory (1) that limits the maximal values of the gauge field strength components. It is worth noticing here that the idea of finding a "BI-extension" of Einstein gravity is old but still popular, although it lacks a good definition and guiding principles, see e.g., [17] for classification of many such extensions in gravitational theory, and [18] for other proposals to an F (R) gravity function of the BI-type. A "BI-extension" of N = 1 supergravity is more restrictive, but it suffers similar problems, see e.g., [19] for some specific proposals of BI supergravity in curved superspace. Equation (3) is just the specific extension of Starobinsky (R + R 2 ) gravity in the framework of F (R) gravity derived from supergravity and inspired by string theory. It is directly related to the BI action (1) that arises together with the F (R) gravity (3) in the same supergravity theory having the BI structure. It is also worth mentioning that Starobinsky inflation is equivalent to the so-called Higgs inflation in gravity and supergravity, because both lead to the same inflationary observables [20]. Starobinsky inflation and F (R) gravity Starobinsky model of inflation is defined by the action [1] S Star. = M 2 Pl 2 d 4 x √ −g R + 1 6m 2 R 2 ,(4) where we have introduced the reduced Planck mass M Pl = 1/ √ 8πG N ≈ 2.4 × 10 18 GeV, and the scalaron (inflaton) mass m as the only parameter. We use the spacetime signature (−, +, +, +, ). The (R + R 2 ) gravity model (4) can be considered as the simplest extension of the standard Einstein-Hilbert action in the context of (modified) F (R) gravity theories with an action S F = M 2 Pl 2 d 4 x √ −g F (R) ,(5) in terms of the function F (R) of the scalar curvature R. The F (R) gravity action (5) is classically equivalent to S[g µν , χ] = M 2 Pl 2 d 4 x √ −g [F (χ)(R − χ) + F (χ)](6) with the real scalar field χ, provided that F = 0 that we always assume. Here the primes denote the derivatives with respect to the argument. The equivalence is easy to verify because the χfield equation implies χ = R. In turn, the factor F in front of the R in (6) can be (generically) eliminated by a Weyl transformation of metric g µν , that transforms the action (6) into the action of the scalar field χ minimally coupled to Einstein gravity and having the scalar potential V = M 2 Pl 2 χF (χ) − F (χ) F (χ) 2 .(7) Differentiating this scalar potential yields dV dχ = M 2 Pl 2 F (χ) [2F (χ) − χF (χ)] (F (χ)) 3 .(8) The kinetic term of χ becomes canonically normalized after the field redefinition χ(ϕ) as F (χ) = exp 2 3 ϕ/M Pl , ϕ = √ 3M Pl √ 2 ln F (χ) ,(9) in terms of the canonical inflaton field ϕ, with the total acton S quintessence [g µν , ϕ] = M 2 Pl 2 d 4 x √ −gR − d 4 x √ −g 1 2 g µν ∂ µ ϕ∂ ν ϕ + V (ϕ) .(10) The classical and quantum stability conditions of F (R) gravity theory are given by [3] F (R) > 0 and F (R) > 0 , and they are obviously satisfied for Starobinsky model (4) for R > 0. Differentiating the scalar potential V in Eq. (7) with respect to ϕ yields dV dϕ = dV dχ dχ dϕ = M 2 Pl 2 χF + F − F F 2 − 2 χF − F F 3 F dχ dϕ ,(12) where we have dχ dϕ = dχ dF dF dϕ = dF dϕ dF dχ = √ 2 √ 3M Pl F F .(13) This implies dV dϕ = M Pl 2F − χF √ 6F 2 .(14) Combining Eqs. (7) and (14) yields R and F in terms of the scalar potential V , R = √ 6 M Pl dV dϕ + 4V M 2 Pl exp 2 3 ϕ/M Pl ,(15)F = √ 6 M Pl dV dϕ + 2V M 2 Pl exp 2 2 3 ϕ/M Pl .(16) These equations define the function F (R) in the parametric form, in terms of a scalar potential V (ϕ), i.e. the inverse transformation to (7). This is known as the classical equivalence (duality) between the F (R) gravity theories (5) and the scalar-tensor (quintessence) theories of gravity (10). In the case of Starobinsky model (4), one gets the famous potential V (ϕ) = 3 4 M 2 Pl m 2 1 − exp − 2 3 ϕ/M Pl 2 .(17) This scalar potential is bounded from below (non-negative and stable), and it has the absolute minimum at ϕ = 0 corresponding to a Minkowski vacuum. The scalar potential (17) also has a plateau of positive height (related to inflationary energy density), that gives rise to slow roll of inflaton in the inflationary era. The Starobinsky model (4) is the particular case of the so-called α-attractor inflationary models [21], and is also a member of the close family of viable inflationary models of F (R) gravity, originating from higher dimensions [22]. A duration of inflation is measured in the slow roll approximation by the e-foldings number N e ≈ 1 M 2 Pl ϕ * ϕ end V V dϕ ,(18) where ϕ * is the inflaton value at the reference scale (horizon crossing), and ϕ end is the inflaton value at the end of inflation when one of the slow roll parameters ε V (ϕ) = M 2 Pl 2 V V 2 and η V (ϕ) = M 2 Pl V V ,(19) is no longer small (close to 1). The amplitude of scalar perturbations at horizon crossing is given by [23] A = V 3 * 12π 2 M 6 Pl (V * ) 2 = 3m 2 8π 2 M 2 Pl sinh 4 ϕ * √ 6M Pl .(20) The Starobinsky model (4) is the excellent model of cosmological inflation, in very good agreement with the Planck data [24,25,26]. The Planck satellite mission measurements of the Cosmic Microwave Background (CMB) radiation [24,25,26] give the scalar perturbations tilt as n s ≈ 1 + 2η V − 6ε V ≈ 0.968 ± 0.006 and restrict the tensor-to-scalar ratio as r ≈ 16ε V < 0.08. The Starobinsky inflation yields r ≈ 12/N 2 e ≈ 0.004 and n s ≈ 1 − 2/N e , where N e is the e-foldings number between 50 and 60, with the best fit at N e ≈ 55 [27,28]. The Starobinsky model (4) is geometrical (based on gravity only), while its (mass) parameter m is fixed by the observed CMB amplitude (COBE, WMAP) as m ≈ 3 · 10 13 GeV or m M Pl ≈ 1.3 · 10 −5 .(21) A numerical analysis of (18) with the potential (17) (4 + 3 √ 3) ≈ 0.5 ,(22) where we have used N e ≈ 55. BI-modified Starobinsky model In accordance to (5), the modified gravity theory (3) has F (R) = R + 2g 2 3β 1 + 12βR 2 − 1 ,(23) where we have introduced the parameters g = 1/(eM Pl ) and β = 1/(e 2 M 4 BI ). In this parametrization, our F -function (23) exactly agrees with Eq. (37) of Ref. [13]. When assuming 12βR 2 1, the function (23) gives rise to the (R + R 2 ) gravity model of Starobinsky in (4), as it should. It allows us to identify g 2 = 1 24m 2 and e 2 = 24 m M Pl 2 ≈ 4 · 10 −9 ,(24) where we have used (21). In terms of the dimensionless quantitiesF = F/M 2 Pl andR = R/M 2 Pl , and the dimensionless parameters α = M BI M Pl andγ = eα 2 ,(25) we have the dimensionless functioñ F (R) =R + 2 3 α 4 1 + 12R 2 /γ 2 − 1(26) A global shape of this function is given in Fig. 1. The function (23) is well defined for any values of R, and implies three physical regimes: • the small curvature regime, where gravity is described by the standard Einstein-Hilbert action, • the inflationary regime, where gravity is described by Starobinsky (R + R 2 ) action (4), • the high curvature regime, where gravity is again described by the Einstein-Hilbert action, though with the different (larger) effective Planck scale M Pl,effective = M Pl 1 + 4g 2 / √ 3β 1/2 ≤ 189M Pl for large positive values of R. Static solutions to the F (R) gravity field equations with R = const. ≡ R 0 follow from our equations (8) and (14), and are given by solutions to the algebraic equation [29] RF (R) = 2F (R) , In our case (23), with F (R) = 1 + 8g 2 R 1 + 12βR 2 > 0 for R ≥ 0 ,(28) we find 8g 2 R 2 0 1 + 12βR 2 0 = R 0 + 4g 2 3β 1 + 12βR 2 0 − 1(29) that gives rise to the condition R 0 4(16g 4 − 3β)R 3 0 + 32g 2 R 2 0 − R 0 + 8g 2 3β = 0.(30) Besides the trivial solution R 0 = 0 corresponding to a stable Minkowski vacuum, any other real positive solution (R 0 > 0) must obey the cubic equation aR 3 0 + bR 2 0 + cR 0 + d = 0,(31) whose coefficients are a = 4(16g 4 −3β), b = 32g 2 , c = −1 and d = 8g 2 /(3β). By using the standard replacement y = R 0 + b 3a ,(32) we can bring (31) to the canonical form y 3 + 3py + 2q = 0,(33) where we have 2q = 2b 3 27a 3 − bc 3a 2 + d a = 4g 2 (1152g 8 − 104g 4 β + 27β 2 ) 27β(16g 4 − 3β) 3 ,(34) and 3p = 3ac − b 2 3a 2 = 9β − 304g 4 12(16g 4 − 3β) 2 .(35) The number of real solutions depends upon the sign of the cubic discriminant D = q 2 + p 3 that in our case reads D = (144g 4 + β)(32g 4 + 3β) 2 5184β 2 (16g 4 − 3β) 4 .(36) Since D > 0, there is only one real solution. Our numerical studies show that this root R 0 is negative (e.g., with α = 1 we find R 0 ≈ −10 −7 M 2 Pl ). The second derivative of the F (R) gravity function (23) F (R) = 8g 2 (1 + 12βR 2 ) 3/2 > 0(37) can be compared to the laboratory bound of Eöt-Wash experiment [30] : F (0) ≤ 2 × 10 −6 cm 2 or g < 0.5 × 10 −3 cm 2 ,(38) that is well satisfied because of (21) and (24). Scalar-tensor gravity and inflaton scalar potential It is instructive to study the same gravitational model (3) in the dual (scalar-tensor gravity) picture defined by (7), (9) and (10). The classical equivalence (duality) between the F (R) gravity theories and their scalar-tensor gravity (or quintessence) counterparts is well known, see e.g., [31]. Our equation (9) impliesR γ = 1 2γ 1 − e − √ 2/3φ 16α 2 − 3γ 2 1 − e − √ 2/3φ 2 ,(39) where we have introduced the dimensionless inflaton fieldφ = ϕ/M Pl . Actually, (9) determines R 2 as the function of ϕ, and our sign choice in (39) comes from demanding a plateau of the scalar potential at positive values of R. In turn, our equation (7) yields V = α 4 3 1 + 12R 2 /γ 2 1 + 12R 2 /γ 2 − 1 8α 4γ−1 (R/γ) + 1 + 12R 2 /γ 2 2 ,(40) where we have introduced the dimensionless scalar potentialṼ = V /M 4 Pl . The scalar potential V (φ) is obtained via a substitution of (39) into (40), while the value of the parameterγ, according to Sections 3 and 4, is given byγ ≈ 6.3 · 10 −5 α 2 . A profile of the scalar potential is given in Fig. 2. As expected, the scalar potential V (ϕ) has a plateau for positive values of ϕ and R, which corresponds to Starobinsky inflation (Sec. 3). As is clear from (39), the higher the values of ϕ and R are, the closer the potential V (ϕ) to the Starobinsky potential (17) with V max. = 3 4 m 2 M 2 Pl is. Hence, the BI structure does not play a significant role for positive values of ϕ and R. When formally sending ϕ → +∞ in (39), we getR max. =γ 2 2 √ 16α 2 −3γ 2 > 0. The scalar-tensor gravity description does not exist forR >R max. , whereas theF (R) gravity description (26) is well defined there. This is an explicit example of breaking the naive equivalence between the two dual descriptions. Though the scalar potential V (ϕ) cannot be trusted for large negative values of ϕ and R, because of intense particle production (reheating) starting near the absolute minimum of the scalar potential, it is instructive to illustrate two more breaking patterns of the naive equivalence between F (R) gravity theories and scalar-tensor gravity theories in our specific example. First, we observe the infinite maximum of the scalar potential in Fig. 2. It happens when the expression under the root in the denominator of (40) vanishes, that corresponds to zero of F (R) in (7) at a negative value of R. Since this occurs at a finite value of R, it represents an example of of the broken correspondence, when the F (R) gravity description is regular, but the scalar-tensor description is singular. Second, yet another example of the broken correspondence is given by the wall on the lefthand-side of Fig. 2. This wall appears when the expression under the root in the denominator of (39) vanishes at a finite value of ϕ that gives rise to the infinite values of R and the scalar potential V (ϕ), although the value of V (R) remains finite. Beyond the wall, the scalar-tensor gravity description does not exist in our case. Conclusion Our main results are given in Sections 4 and 5. They provide a viable extension of Starobinsky (R + R 2 ) inflationary model, motivated by the Born-Infeld structure in supergravity, in turn, motivated by string theory. Our physical motivation is to explore the range of energies beyond the Starobinsky inflationary scale of approximately 10 13 GeV up to the (reduced) Planck scale of approximately 10 18 GeV, by using the specific modified gravity function (3) derived from the supergravity model under our assumptions formulated in Sec. 1. The significant deviation between our modified F (R) gravity model and Starobinsky (R + R 2 ) gravity model takes place only for very large positive curvature, with the asymptotic R 2 gravity being replaced by the asymptotic Einstein-Hilbert gravity having a larger effective Planck scale. The corresponding values of the inflaton field are trans-Planckian, so that the asymptotic gravity is supposed to be considered with a grain of salt, because it may be affected by quantum gravity effects. On the other side, we found explicit examples of breaking the naive correspondence between the F (R) gravity theories and the scalar-tensor gravity theories in our model. They are, however, of academic interest in the inflationary physics context, because they occur at large negative values of the curvature. with ω ≡ D 2 W 2 /8 and the BI coupling κ = b −2 = M −4 BI . The Lagrangian (44) can be expanded as e −1 L = 1 2 R + 3 4 B m B m − 3 2 B m A m + M 4 BI 3   1 − 3 2M 4 BI e 2 F 2 − 8(R + 3 2 B m B m ) 2 + 3 4M 4 BI e 2 2 (FF ) 2 − 1   + . . . ,(46) where we have kept only the relevant terms. Using B m = F mn = 0 as a solution, we get (3). The physical conditions imply the rangeR ∈ [−1, 1] (i.e. up to the UV-cutoff) and α ∈ [0.01, 1] (i.e. between the Grand Unification scale and Planck scale), so thatγ ∈ 6.3 · [10 −7 , 10 −5 ]. The Starobinsky inflation takes place for 0 <R γ. Figure 1 : 1the profile of the F (R) gravity function (23) for α = 1 andγ −2 = 10 5 . This value of the parameterγ is only chosen to demonstrate the global shape of the function. Figure 2 : 2the profile of the V (ϕ) function (40) for α = 1 andγ = 6.3 · 10 −5 . This function is not well defined for all values ofφ. It reproduces the inflationary potential(17) for the relevant values ofφ (Sec. 3). The infinite maximum occurs atφ ≈ −0.6 that corresponds toR ≈ −5 · 10 −10 . The only anti-de-Sitter minimum occurs atφ ≈ −6.5 that corresponds to the rootR 0 ≈ −10 −7 found in Sec. 4. The wall on the left-hand-side, where V sharply goes up to infinity, appears atφ ≈ −9. See also[9,10] for more about special properties of the BI action and its supersymmetric extensions.2 See also[14,15,16] for related papers. AcknowledgementsYA and SVK are supported in part by the Competitiveness Enhancement Program of Tomsk Polytechnic University in Russia. SVK is also supported in part by a Grant-in-Aid of the Japanese Society for Promotion of Science (JSPS) under No. 26400252, and the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. One of the authors (SVK) is grateful to Ignatios Antoniadis for useful discussions.Appendix: supergravity with BI structure in superspaceThe supersymmetric extension of the (R+R 2 ) gravity (with Maxwell structure) in the new-minimal formulation of N = 1 supergravity is given by eq. (38) of Ref.[13]in the superconformal tensor calculus. In curved superspace, with M Pl = 1, the Lagrangian reads[32,33]where V R is the gauge multiplet of SUSY algebra, representing the new-minimal set of supergravity field components, W α is its superfield strength, and γ ∼ e −2 is the R 2 parameter. The superfield V R has the following bosonic components (in a Wess-Zumino gauge):where A m is the (dynamical) gauge field,is the gravitational D-term, and B m is the auxiliary vector field of supergravity multiplet. The old-minimal set of supergravity is also present via E and R that is hidden in the definition of the superfield strength W α ≡ − 1 4 (D 2 − 8R)D α V R . After identifying the "old" auxiliary field b m with the "new" auxiliary field B m as b m = − 3 2 B m , we can expand the Lagrangian (41) as follows:where we have kept only the relevant terms. When allowing the superfield V R to be massive (or not using a WZ gauge), the complex scalar M of the old-minimal set[12]also appears. The BI extension of the supergravity theory (41) can be written down as follows:where the BI structure function Λ is given by (see e.g., Ref.[9]) A New type of isotropic cosmological models without singularity. A A Starobinsky, Phys. Lett. 9199A. A. Starobinsky, "A New type of isotropic cosmological models without singularity," Phys. Lett. 91B (1980) 99. Supergravity based inflation models: a review. M Yamaguchi, arXiv:1101.2488Class. Quant. Grav. 28103001astro-ph.COM. Yamaguchi, "Supergravity based inflation models: a review," Class. Quant. Grav. 28 (2011) 103001 [arXiv:1101.2488 [astro-ph.CO]]. S V Ketov, arXiv:1201.2239Supergravity and Early Universe: the Meeting Point of Cosmology and High-Energy Physics. 281330021hep-thS. V. Ketov, "Supergravity and Early Universe: the Meeting Point of Cosmology and High-Energy Physics," Int. J. Mod. Phys. 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Higgs mechanism and cosmological constant in N = 1 supergravity with inflaton in a vector multiplet. Y Aldabergenov, S V Ketov, arXiv:1701.08240Eur. Phys. J. 77233hep-thY. Aldabergenov and S. V. Ketov, "Higgs mechanism and cosmological constant in N = 1 supergravity with inflaton in a vector multiplet," Eur. Phys. J. C77 (2017) 233 [arXiv:1701.08240 [hep-th]]. Gravitino and Polonyi production in supergravity. A Addazi, S V Ketov, M Yu, Khlopov, arXiv:1708.05393hep-thA. Addazi, S. V. Ketov and M. Yu. Khlopov, "Gravitino and Polonyi production in supergravity," [arXiv:1708.05393 [hep-th]]. Born-Infeld inspired modifications of gravity. J Beltran Jimenez, L Heisenberg, G J Olmo, D Rubiera-Garcia, arXiv:1704.03351Phys. Repts. 7271gr-qcJ. Beltran Jimenez, L. Heisenberg, G. J. Olmo and D. Rubiera-Garcia, "Born-Infeld inspired modifications of gravity", Phys. Repts. 727 (2018) 1 [arXiv:1704.03351[gr-qc]]. Notes on Born-Infeld-like modified gravity. S I Kruglov, arXiv:1403.0675Astrophys. Space Sci. 36173gr-qcS. I. Kruglov, "Notes on Born-Infeld-like modified gravity", Astrophys. Space Sci. 361 (2016) 73 [arXiv:1403.0675[gr-qc]]. 4D, N=1 Born-Infeld supergravity. S J Gates, Jr , S V Ketov, arXiv:hep-th/0104233Class. Quant. Grav. 183561hep-thS. J. Gates, Jr., and S. V. Ketov, "4D, N=1 Born-Infeld supergravity", Class. Quant. Grav. 18 (2001) 3561 [arXiv:hep-th/0104233[hep-th]]. Inflation and non-minimal scalar-curvature coupling in gravity and supergravity. S V Ketov, A A Starobinsky, arXiv:1203.0805JCAP. 120822hep-thS. V. Ketov and A. A. Starobinsky, "Inflation and non-minimal scalar-curvature coupling in gravity and supergravity", JCAP 1208 (2012) 022 [arXiv:1203.0805 [hep-th]]. Unity of Cosmological Inflation Attractors. M Galante, R Kallosh, A Linde, D Roest, arXiv:1412.3797Phys. Rev. Lett. 114141302hep-thM. Galante, R. Kallosh, A. Linde and D. Roest, "Unity of Cosmological Inflation Attractors", Phys. Rev. Lett. 114 (2015) 141302 [arXiv:1412.3797 [hep-th]]. Inflation from higher dimensions. H Nakada, S V Ketov, arXiv:1710.02259Phys. Rev. D. 96123530hep-thH. Nakada and S. V. Ketov, "Inflation from higher dimensions", Phys. Rev. D 96 (2017) 123530 [arXiv:1710.02259 [hep-th]]. Calculations of inflaton decays and reheating: with applications to no-scale inflation models. J Ellis, M A G Garcia, D V Nanopoulos, K A Olive, arXiv:1505.06986JCAP. 150750hep-phJ. Ellis, M. A. G. Garcia, D. V. Nanopoulos and K. A. Olive, "Calculations of inflaton decays and reheating: with applications to no-scale inflation models", JCAP 1507 (2015) 050 [arXiv:1505.06986 [hep-ph]]. P A R Ade, Planck CollaborationarXiv:1502.01589Planck 2015 results. XIII. Cosmological parameters. 59413astro-ph.COP. A. R. Ade et al. [Planck Collaboration], "Planck 2015 results. XIII. Cosmological parameters," Astron. Astrophys. 594 (2016) A13 [arXiv:1502.01589 [astro-ph.CO]]. P A R Ade, Planck CollaborationarXiv:1502.02114Planck 2015 results. XX. Constraints on inflation. 59420astro-ph.COP. A. R. Ade et al. [Planck Collaboration], "Planck 2015 results. XX. Constraints on inflation," Astron. Astrophys. 594 (2016) A20 [arXiv:1502.02114 [astro-ph.CO]]. Improved Constraints on Cosmology and Foregrounds from BICEP2 and Keck Array Cosmic Microwave Background Data with Inclusion of 95 GHz Band. P A R Ade, BICEP2 and Keck Array CollaborationsarXiv:1510.09217Phys. Rev. Lett. 11631302astro-ph.COP. A. R. Ade et al. [BICEP2 and Keck Array Collaborations], "Improved Constraints on Cosmology and Foregrounds from BICEP2 and Keck Array Cosmic Microwave Background Data with Inclusion of 95 GHz Band," Phys. Rev. Lett. 116 (2016) 031302 [arXiv:1510.09217 [astro-ph.CO]]. Quantum Fluctuations and a Nonsingular Universe. V F Mukhanov, G V Chibisov, Pisma Zh. Eksp. Teor. Fiz. 33532JETP Lett.V. F. Mukhanov and G. V. Chibisov, "Quantum Fluctuations and a Nonsingular Universe", JETP Lett. 33 (1981) 532 [Pisma Zh. Eksp. Teor. Fiz. 33 (1981) 549]. Fourth-order gravity as the inflationary model revisited. S Kaneda, S V Ketov, N Watanabe, arXiv:1001.5118Mod. Phys. Lett. 252753hep-thS. Kaneda, S.V. Ketov and N. Watanabe, "Fourth-order gravity as the inflationary model revisited", Mod. Phys. Lett. A25 (2010) 2753 [arXiv:1001.5118 [hep-th]]. The Stability of General Relativistic Cosmological Theory. J D Barrow, A C Ottewill, J. Phys. 162757J. D. Barrow and A. C. Ottewill, "The Stability of General Relativistic Cosmological Theory", J. Phys. A16 (1983) 2757. Tests of the gravitational inverse-square law below the dark-energy length scale. D J Kapner, T S Cook, E G Adelberger, J H Gundlach, B R Heckel, C D Hoyle, H E Swanson, arXiv:hep-ph/0611184Phys. Rev. Lett. 9821101D. J. Kapner, T. S. Cook, E. G. Adelberger, J. H. Gundlach, B. R. Heckel, C. D. Hoyle and H. E. Swanson, "Tests of the gravitational inverse-square law below the dark-energy length scale", Phys. Rev. Lett. 98 (2007) 021101 [arXiv:hep-ph/0611184]. The scalar-tensor theory of gravitation. Y Fujii, K-I Maeda, Cambridge Univ. PressY. Fujii and K-I. Maeda, "The scalar-tensor theory of gravitation", Cambridge Univ. Press, 2007. New minimal higher derivative supergravity coupled to matter. S Cecotti, S Ferrara, M Porrati, S Sabharwal, Nucl. Phys. 306160S. Cecotti, S. Ferrara, M. Porrati and S. Sabharwal, "New minimal higher derivative supergravity coupled to matter", Nucl. Phys. B306 (1988) 160. On the Starobinsky Model of Inflation from Supergravity. F Farakos, A Kehagias, A Riotto, arXiv:1307.1137Nucl. Phys. 876187hep-thF. Farakos, A. Kehagias and A. Riotto, "On the Starobinsky Model of Inflation from Supergravity", Nucl. Phys. B876 (2013) 187 [arXiv:1307.1137 [hep-th]].

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{'abstract': 'A supergravity extension of the (R + R 2 ) gravity with the additional (Born-Infeld) structure of a massive vector multiplet gives rise to the specific F (R) gravity, whose structure is investigated in detail. The massive vector multiplet has inflaton (scalaron), goldstino and a massive vector field as its field components. The model describes Starobinsky inflation and allows us to extrapolate the F (R) function beyond the inflationary scale (up to Planck scale). We observe some differences versus the (R + R 2 ) gravity, and several breaking patterns of the well known correspondence between the F (R) gravity and the scalar-tensor gravity. arXiv:1807.08394v1 [hep-th]', 'arxivid': '1807.08394', 'author': ['Yermek Aldabergenov aldabergenov-yermek@ed.tmu.ac.jp \nDepartment of Physics\nTokyo Metropolitan University\nMinami-ohsawa 1-1, Hachioji-shi192-0397TokyoJapan\n\nResearch School of High-Energy Physics\nTomsk Polytechnic University\n2a Lenin Ave634050TomskRussian Federation\n', 'Ryotaro Ishikawa ishikawa-ryotaro@ed.tmu.ac.jp \nDepartment of Physics\nTokyo Metropolitan University\nMinami-ohsawa 1-1, Hachioji-shi192-0397TokyoJapan\n', 'Sergei V Ketov ketov@tmu.ac.jp \nDepartment of Physics\nTokyo Metropolitan University\nMinami-ohsawa 1-1, Hachioji-shi192-0397TokyoJapan\n\nResearch School of High-Energy Physics\nTomsk Polytechnic University\n2a Lenin Ave634050TomskRussian Federation\n\nKavli Institute for the Physics and Mathematics of the Universe (IPMU)\nThe University of Tokyo\n277-8568ChibaJapan\n', 'Sergey I Kruglov serguei.krouglov@utoronto.ca \nDepartment of Chemical and Physical Sciences\nUniversity of Toronto\n3359 Mississauga RoadL5L 1C6North, MississaugaOntarioCanada\n'], 'authoraffiliation': ['Department of Physics\nTokyo Metropolitan University\nMinami-ohsawa 1-1, Hachioji-shi192-0397TokyoJapan', 'Research School of High-Energy Physics\nTomsk Polytechnic University\n2a Lenin Ave634050TomskRussian Federation', 'Department of Physics\nTokyo Metropolitan University\nMinami-ohsawa 1-1, Hachioji-shi192-0397TokyoJapan', 'Department of Physics\nTokyo Metropolitan University\nMinami-ohsawa 1-1, Hachioji-shi192-0397TokyoJapan', 'Research School of High-Energy Physics\nTomsk Polytechnic University\n2a Lenin Ave634050TomskRussian Federation', 'Kavli Institute for the Physics and Mathematics of the Universe (IPMU)\nThe University of Tokyo\n277-8568ChibaJapan', 'Department of Chemical and Physical Sciences\nUniversity of Toronto\n3359 Mississauga RoadL5L 1C6North, MississaugaOntarioCanada'], 'corpusid': 115134983, 'doi': '10.1103/physrevd.98.083511', 'github_urls': [], 'n_tokens_mistral': 11834, 'n_tokens_neox': 9733, 'n_words': 5524, 'pdfsha': 'f9ee2dc4c51eede5908c4facc8521bcee22fbeb6', 'pdfurls': ['https://arxiv.org/pdf/1807.08394v1.pdf'], 'title': ['Beyond Starobinsky inflation', 'Beyond Starobinsky inflation'], 'venue': []}

arxiv

Unified Analysis on L 1 over L 2 Minimization for signal recovery 22 Jan 2023 Min Tao taom@nju.edu.cn Xiao-Ping Zhang Min Tao Xiao-Ping Zhang Department of Mathematics Department of Electrical, Computer and Biomedical Engineering National Key Laboratory for Novel Software Technology Nan-jing University 210093NanjingChina Ryerson University M5B 2K3TorontoONCanada Unified Analysis on L 1 over L 2 Minimization for signal recovery 22 Jan 2023Received: date / Accepted: dateNoname manuscript No. (will be inserted by the editor)Sparse recovery · fractional programming · coherent dictionary · d-stationarity Mathematics Subject Classification (2010) MSC 90C26 · MSC 90C90 · 49N45 In this paper, we carry out a unified study for L 1 over L 2 sparsity promoting models, which are widely used in the regime of coherent dictionaries for recovering sparse nonnegative/arbitrary signals. First, we provide a unified theoretical analysis on the existence of the global solutions of the constrained and the unconstrained L 1 /L 2 models. Second, we analyze the sparse property of any local minimizer of these L 1 /L 2 models which serves as a certificate to rule out the nonlocal-minimizer stationary solutions. Third, we derive an analytical solution for the proximal operator of the L 1 /L 2 with nonnegative constraint. Equipped with this, we apply the alternating direction method of multipliers to the unconstrained model with nonnegative constraint in a particular splitting way, referred to as ADMM + p . We establish its global convergence to a d-stationary solution (sharpest stationary) without the Kurdyka-Lojasiewicz assumption. Extensive numerical simulations confirm the superior of ADMM + p over the state-of-the-art methods in sparse recovery. In particular, ADMM + p reduces computational time by about 95% ∼ 99% while achieving a much higher accuracy than the commonly used scaled gradient projection method for the wavelength misalignment problem. Compressive sensing (CS) is to seek the sparsest solution from a set of undersampled linear measurements. Mathematically, a fundamental problem in CS can be formulated as a constrained model, min x∈X x 0 , s.t. Ax = b,(1) where A ∈ R m×n (m ≪ n) is a sensing matrix and the observation of b ∈ R n , · 0 is the L 0 norm (the number of nonzero elements). We consider the recovery of nonnegative/arbitrary compressed signal, which corresponds to X = R n + and R n , respectively. Unfortunately, the optimization (1) is known to be NP-hard [28]. One common approach is to relax L 0 norm to L 1 norm, leading to basis pursuit model [7]. Theoretically, the exact recovery by the L 1 minimization is guaranteed under the restricted isometry property [5] or null space property [9]. Although the L 1 minimization technique has been widely used, it is not able to reconstruct the sparsest solutions when columns of A are highly coherent, such as those applications rising from discretization of continuum image problems (such as medical and radar) when the grid spacing is below the Rayleigh threshold [12]. As such, various nonconvex regularizers, such as the L p (quasi-)norm (0 < p < 1) [6], L 1 -L 2 [38], transform L 1 [29], the ratio of L 1 over L 2 norm (L 1 /L 2 ) [38] have been developed to enhance the recovery quality. Among these nonconvex regularizations, L 1 /L 2 can approximate L 0 norm very well when the domain is without origin, due to its being scale-invariant and parameter-free as well as L 0 norm. For one-sparse signal, the L 1 /L 2 is the same as the L 0 norm. The L 1 /L 2 arose as sparseness measure [17,18] and has attracted a considerable attention due to its wide applications, e.g., nonlinear matrix factorization [26] and blind deconvolution [19,32]. In this paper, we focus on two commonly-used L 1 /L 2 models for signal recovery, i.e., the constrained and the penalized/unconstrained models: min x x 1 x 2 s.t. x ∈ H := {x ∈ X | Ax = b},(2) and min x∈X F (x) := γ x 1 x 2 + 1 2 Ax − b 2 2 ,(3) where A and b are defined identically as in (1). The penalized/unconstrained model (3) can tackle both noisy and noiseless observations while (2) can only deal with unnoisy data, it is more meaningful to develop efficient and convergent algorithms to solve (3). First, to recover the signal, we need to solve L 1 over L 2 minimization models (2) or (3). The first question is whether these models are well-defined. Recently, Zeng et al. [39] analyze the existence of global optimal solutions of the constrained model (2) for the case of X = R n . However, the nonemptyness of the global solution set of the model of (2) with X = R n + and the model (3) with X = R n + /R n (i.e., R n + and R n ) have not been studied. Many early works focus on developing different optimization algorithms for solving L 1 /L 2 minimization. To name a few, the scaled gradient projection method (SGPM) [11,38] for (3) with X = R n + , as well as the alternating direction method of multiplier (ADMM) approach [31] for (2) with X = R n , accelerated schemes for solving the constrained model (2) with X = R n [36], all lack global convergence guarantees. There are two exceptions: Zeng et al. [39] apply moving-balls-approximation based algorithms to solve L 1 /L 2 minimization over an inequality constraint for arbitrary signal and prove its local linear convergence under some conditions. Our previous work [34] proposes an ADMM-based algorithm for solving (3) with X = R n with global convergence guarantee. Although there do exist a few different reformulations for the unconstrained model (3) with X = R n + , it turns out that most of these for implementing ADMM result in a violation of convergence guarantee [16,21]. It is still unknown how to solve the model (3) with convergence guarantee where X = R n + . In this paper, we carry out a unified theoretical study on (2) and (3). First, we provide a unified analysis on the existence of global solutions of (2) and (3). Inspired by [39], we introduce an auxiliary optimization problem and verify that the solution set of (2) is nonempty if the objective function value of (2) is strictly less than that of the auxiliary optimization problem. A similar result is also proved for the model (3) while the proof is much more complicated. Then, we illustrate that this sufficient condition can be guaranteed by the µspherical section property of N (A) (i.e., the null space of A) [35]. Second, we exploit the sparse property of any local minimizer of (2) or (3). In particular, we prove that any feasible vector cannot be a local minimizer if the columns of A restricted on the support set are linearly dependent. Third, we design an efficient and convergent algorithm for the penalized model (3) with X = R n + . To do so, we derive a closed-form solution for one global solution of the proximal operator of (L 1 /L 2 ) + (i.e., x 1 x 2 + ι R n + (x) and ι R n + (x) is the indicator function of R n + ) and accompanied by a practical solver. Equipped with this, we propose a specific variable-splitting scheme of ADMM for solving (3) with X = R n + . We referred to it as ADMM + p by incorporating the practical proximal solver. Although there already exist many seminal works on convergence analysis for the nonsmooth nonconvex problem, e.g., [1,16,21], all these convergence results focus on converging to a stationary point which is much weaker than to a dstationary point [10,23,30]. Furthermore, all these approaches achieve global convergence by assuming the introduced merit function with the Kurdyka-Lojasiewicz (KL) property. In contrast, we introduce a novel merit function T (see (31)) instead of the augmented Lagrange function or its variants [16,21] for aiding the global convergence analysis. Then, we establish the global convergence of ADMM + p converging to a d-stationary point by proving the merit function with the KL property. We conduct extensive experiments on algorithmic behaviors and various sparse recovery model comparisons, testing on two specific applications. All of these showcase the superior performance of the proposed approach over the state-of-the-art in sparse nonnegative signal recovery. In particular, ADMM + p always converges to a more accurate solution (d-stationary) and significantly reduces the computational time in comparison with SGPM and accelerated proximal gradient methods (monotone version with fixed stepsize/line search and its nonmonotone versions). In summary, our contributions are threefold: (1) We provide a unified theoretical analysis on the existence of global solutions of the constrained model (2) and unconstrained model (3). (2) We exploit the sparse property of any local minimizer of the constrained model (2) and unconstrained model (3). This property serves as a certificate to rule out the stationary solution that is not a local minimizer. (3) We derive an analytic solution of the proximal operator of (L 1 /L 2 ) + which allows us to design an efficient algorithm for the unconstrained model (3) with X = R n + , i.e., ADMM + p . We establish its global convergence to a d-stationary solution without KL assumption. Extensively numerical simulations further verify the computational efficiency of the ADMM + p over the state-of-the-art in sparse recovery. The rest of this paper proceeds as follows. We describe the notations and definitions in Section 1. In Section 2, we elaborate on the existence of optimal solutions of (2) and (3). We analyze the sparse property and provide the exact recovery theory for (2) in Section 3. In Section 4, the proximal operator for (L 1 /L 2 ) + is derived. Then, we solve the unconstrained model (3) with X = R n + via ADMM, where its global convergence is established. Section 5 devotes extensive experiments to showcase the superior performance of the proposed approach in sparse recovery. Conclusions are given in Section 6. Preliminary We use a bold letter to denote a vector, e.g., x ∈ R n , and x i , x 0 and |x| denote the i-th entry of x, its zero norm of x and the vector with the absolute value of x for each entry, respectively. x 2 and x p (0 < p < 1) denote its 2norm and p-norm ( x p = ( n i=1 x p i ) 1/p ), respectively. The subscript 2 in · 2 is omitted when there is no ambiguity. We use R n + , R n − and R n ++ to denote the set of nonnegative, nonpositive and positive vectors, respectively. The notation of 1 represents a vector with all entries equal to 1. I n is n × n identity matrix, and ⊙ presents the componentwise product. We define [n] := {1, 2, . . . , n}. Given an index set D ⊆ [n], we use ♯(D) and D c to present the cardinality of D and its complementary set. We specify that X is either R n + or R n throughout this paper. For a matrix A ∈ R m×n , we denote AX = {y | y = Ax, x ∈ X } and refer to the projection onto the closed set of AX as Proj (AX ) (·). For a closed set S ∈ R n , we use the notation ι S (x) to represent the indicator function of the set S. Given a matrix A ∈ R m×n or a vector x ∈ R n and an index set Λ ⊆ [n], we use A Λ , x| Λ to denote A[:, i] i∈Λ and a subvector of x with entries in Λ, respectively. N (A) denotes the null space of A and r(A) denotes the rank of A. Given a square matrix A, A ≻ 0 means that A is a positive definite matrix. For a vector x, we use the notation of x ≤ 0 to represent that not all the entries of x are nonpositive. Given ǫ > 0, we use B ǫ (x) andB ǫ (x) to denote the open ball of {x | x−x < ǫ} and the open ball without the center, respectively. Given two sets of A and B, we use the notation A\B to represent the intersection of A and the complement of B. The notation of R n + /R n means R n + or R n . An extended-real-valued function f : R n → (−∞, +∞] is said to be proper if its domain domf := {x | f (x) < ∞} is nonempty. A proper function f is said to be closed if it is lower semi-continuous. For a proper closed function f andx ∈ domf , the regular subdifferential∂f (x) and the limiting subdifferential ∂f (x) [24] are defined aŝ ∂f (x) = v lim x→x inf x =x f (x) − f (x) − v, x −x x −x ≥ 0 , ∂f (x) := v ∃ x k →x, f (x k ) → f (x), v k ∈∂f (x k ), v k → v , respectively. Note ∂ι X (x) = {0} if X = R n , and ∂ ιX (x) = {d | d Λ = 0, d Λ c ≤ 0, Λ = supp(x)} if X = R n + . Throughout the paper, we assume that b = 0 and H = φ. By defining 0 1 0 2 = 1, the objective functions of (2) and (3) are lower semi-continuous over X . Suppose f be a proper lower semicontinuous function, we define the proximal mapping [33,Definition 1.22] : Prox f (v) = arg min x f (x) + 1 2 x − v 2 . For nonconvex and nonsmooth programs, there are various stationary concepts and d-stationary is arguably the sharpest kind among them [10,23,30]. A pointx( = 0) ∈ X is called a d-stationary point to (3), if it satisfies F ′ (x; x −x) ≥ 0. ∀x ∈ X , where F ′ (x; x −x) is the directional derivative of F (·). According to [8,Definition 2.3.4] and [23,Fact 5],x( = 0) ∈ X is a d-stationary point of (3) if and only if x −x, γ 1 x 2 − x 1 x 3 2x + A ⊤ (Ax − b) ≥ 0, ∀x ∈ X ,(4) where 1 ∈ R n by noting that x 1 x 2 + ι R n + (x) = 1 ⊤ x x 2 + ι R n + (x). Next, we review the concepts of locally sparse set and the uniformity of a vector [38] and the KL property [3] which is widely used in convergence analysis. Definition 1 x ∈ H is called locally sparse if ∄ y ∈ H\{x} (H defined in (2)) such that supp(y) ⊆ supp(x). Denote by H L = {x ∈ H | x is locally sparse}. Definition 2 The uniformity of x, κ(x) is the ratio between the smallest nonzero absolute entry and the largest one in the sense of absolute value, i.e. 0 < κ(x) := min i∈supp(x) |x i | max i∈supp(x) |x i | ≤ 1. Definition 3 We say a proper closed function h : R n → (−∞, +∞] satisfies KL property at a pointx ∈ dom∂h if there exist a constant α ∈ (0, ∞], a neighborhood U ofx, and a continuous concave function φ : [0, ν) → [0, ∞) with φ(0) = 0 such that (i) φ is continuously differentiable on (0, ν) with φ ′ > 0 on (0, ν); (ii) for every x ∈ U with h(x) < h(x) < h(x) + ν, it holds that φ ′ (h(x) − h(x))dist(0, ∂h(x)) ≥ 1. Our analysis on the existence of globally optimal solutions is based on the spherical section property (SSP) [35,39]. Definition 4 Let m, n be two positive integers such that m < n. Let V be an (n − m)-dimensional subspace of R n and µ be a positive integer. We say that V has the µ-spherical section property if inf v∈V \{0} v 1 v 2 ≥ m µ . If A ∈ R m×n (m < n) is a random matrix with i.i.d. standard Gaussian entries, then its nullspace has the µ-spherical section property with high probability [39]. Consider the following problem: min x∈R n f (x) = h(x) + g(x),(5) where h : R n → R is L-smooth (possibly, nonconvex) and g : R n → R is proper lower semicontinuous function. The forward-backward algorithm generates the iterate as follows: x k+1 ∈ arg min x Prox αg (x k − α∇h(x k )),(6) where the step size α ∈ (0, 1/L) to guarantee that any accumulation point of the sequence generated above is a stationary point of (5) [4,21]. Existence of optimal solutions We explore the conditions to guarantee the existence of global solutions of (2) and (3). First, we verify that the solution set of (2) is nonempty if the optimal value of (2) is less than that of the auxiliary problem. Second, we prove that the solution set of (3) is nonempty when the objective function value of newly-introduced constrained model is less than the auxiliary problem. Finally, we further show that these sufficient conditions can be guaranteed by the µ-spherical section property of the null space of the measurement matrix. Our analysis is inspired by [39] and introduce the following auxiliary problem: f * dc := inf d∈F0 d 1 d 2 where F 0 := {d | Ad = 0, d ∈ X , d = 0}.(7) For analysis convenience, we denote the optimal value of (2) as f * pc and f * pc < +∞. We recall x k is a minimizing sequence of (2) if x k ∈ H for each k and lim k→∞ x k 1 x k 2 = f * pc . Therefore we only need to characterize the existence of unbounded minimizing sequence. Before that, we provide a sufficient condition to guarantee the solution set of (7) nonempty. Lemma 1 Let f * dc be defined in (7). Assume that N (A) ∩ X = {0}. Then, f * dc < +∞ and the solution set of (7) is nonempty. Proof First, the feasible set of (7) is nonempty due to N (A) ∩ X = {0}. The objective function value is lower bounded, i.e., d 1 d 2 ≥ 1. Suppose that there exists a minimizing sequence d k of (7) that is unbounded with d k 1 dk 2 → f * dc as k → +∞. Thus, f * dc < +∞. Consequently, by defining d k =d k dk 2 , the sequence d k is satisfying d k ∈ F 0 and lim k→+∞ d k 1 d k 2 = f * dc . Since the sequence of d k is bounded, it has one accumulation point d * ∈ F 0 . Therefore, the solution value of (7) is attainable by d * . The following proposition shows that the optimal value of (2) is upper bounded by that of (7) when the feasible sets of both (2) and (7) are nonempty. Proposition 1 Suppose that N (A) ∩ X = {0}. Then, f * pc ≤ f * dc . Proof Since N (A) ∩ X = {0}, it leads to F 0 = φ. For anyx ∈ H and anŷ d ∈ F 0 , we havex+τd ∈ H where τ > 0. Thus, it leads to f * pc = inf x∈H x 1 x 2 ≤ x+τd 1 x+τd 2 . Next, we have that lim τ →+∞ x+τd 1 x+τd 2 = lim τ →+∞ x/τ +d 1 x/τ +d 2 = d 1 d 2 . Consequently, for anyd ∈ F 0 , it yields that f * pc ≤ d 1 d 2 . By taking infimum on both sides of the above inequality with respect tod, we have the desired inequality. (2) and (7). Then, f * pc = f * dc if and only if there exists a minimizing sequence of (2) that is unbounded. Theorem 1 Assume that N (A) ∩ X = {0}. Consider Proof For the case of X = R n , it has been proved in [39,Lemma 3.3] that f * pc = f * dc if and only if there exists a minimizing sequence of (2) that is unbounded. The proof for the case of X = R n + is similar to the case of X = R n , and thus omitted here. (2) and (7). If f * pc < f * dc , the solution set of (2) is nonempty. Corollary 1 Suppose that N (A) ∩ X = {0}. Consider Proof It follows directly by combining Theorem 1 and Proposition 1. Next, we analyze the existence of the global solution of the penalized model (3). In doing so, we introduce a constrained problem parameterized by the vector c: min x x 1 x 2 s.t. Ax = c, x ∈ X ,(8) and denote the optimal value of (8) as f * pc (c) and f * pc (c) < +∞ for any c ∈ AX . (3) and (8). Theorem 2 Suppose that N (A) ∩ X = {0}. ConsiderIf f * pc (c) < f * dc where c = Proj AX (b) , the optimal value of (3) can be attainable. Proof First, define r(x) = γ x 1 x 2 . Assume that x k is a minimizing sequence of (3), i.e., lim k→∞ F x k = F * .(9) If the sequence x k is bounded, then it has a subsequence x kj converging to some x * . Hence, x * is an optimal solution of (3). Otherwise, the sequence x k is unbounded, i.e., x k → ∞ as k → ∞. Since the sequence of {r(x k )} is bounded below and F * is finite, it leads to the sequence of { 1 2 Ax k − b 2 2 } is bounded above. It implies that the subsequence of {Ax k } is bounded. Since {Ax k } is bounded, it has a sequence that converges to y * . Without loss of generality, we assume that Ax k → y * . Let I = {j : {x k j } is bounded}. Then, it follows that {A I x k I } is bounded. This together with boundedness of {Ax k } implies that {A I c x k I c } is also bounded. Next, for each k, we consider the following linear system, A I c y k = A I c x k I c , y k ∈ Y, where Y := R ♯(I c ) if X = R n and Y := R ♯(I c ) + if X = R n + . Obviously, the solution set of the above linear system is nonempty due to at least one solution y k = x k I c for each k. Using Hoffman's Error Bound [15], there exist a vector y k satisfying A I c y k = A I c x k I c and a constant ζ > 0 depending only on A I c such that y k ≤ ζ A I c x k I c . By settingx k = (x k I , y k ), it leads to Ax k = Ax k andx k → x * for convenience due to its boundedness of {x k }. Obviously, y * = Ax * thanks to Ax kj = Ax kj . In the following, we divide into two cases to verify. Case 1. If there exists two subsequences {x kj } and {x kj } such that x kj 1 x kj 2 ≥ x kj 1 x kj 2 , ∀ j.(10) Since Ax kj = Ax kj , we have that F (x kj ) ≥ F (x kj ). By usingx kj → x * and F lower semi-continuous, it yields that lim j→∞ F (x kj ) ≥ F (x * ). On the other hand, F * = lim k→∞ F (x k ) = lim j→∞ F (x kj ) ≥ lim j→∞ F (x kj ) ≥ F (x * ). Invoking the definition of F * , it leads to F (x * ) = F * and x * is an optimal solution. Case 2. If there does not exist such two sequences {x kj } and {x kj } satisfying (10), it implies there exists an index K such that x k 1 x k 2 < x k 1 x k 2 , ∀ k ≥ K. Next, we further divide into two cases to verify. (a) Suppose that y * ∈ Proj (AX ) (b). Then, the solution set of (8) with c = y * is nonempty due to f * pc (y * ) < f * dc . We assume thatx is an optimal solution of (8) with c = y * . Since (9) and Ax k → y * , then lim k→∞ x k 1 x k 2 = (F * − 1 2 y * − b 2 2 ) < +∞. Next, we verify that lim k→∞ x k 1 x k 2 ≤ x 1 x 2 = f * pc (y * ).(11) Suppose not, i.e., lim k→∞ x k 1 x k 2 > x 1 x 2 , it implies that F (x) < F * since Ax = Ax * = y * . It contradicts the definition of F * . Thus, (11) holds. We definex k := x k x k 2 . Taking k → ∞, Ax k → y * ⇒ Ax k → 0, since x k 2 → ∞. Sincex k is bounded, it has a subsequencex kj →x wherex satisfiesx ∈ F 0 . lim k→∞ x k 1 x k 2 = x 1 x 2 ≥ f * dc > f * pc (y * ),(12) where the first inequality is due tox ∈ F 0 and the last inequality follows from y * ∈ Proj AX (b). Note that the above inequality contradicts (11). Hence, this case cannot happen. (b) Suppose that y * ∈ Proj AX (b). Then, we choose one vectorŷ ∈ Proj AX (b). Consider the constrained problem (8) with c :=ŷ. Since f * pc (ŷ) < f * dc , the solution set of (8) with c :=ŷ is nonempty due to Corollary 1. We assume thatx is an optimal solution of (8) with c :=ŷ. Similar to case (a), one can derive a version of (12) as follows lim k→∞ γ x k 1 x k 2 = γ x 1 x 2 ≥ γf * dc ,(13) where the definition ofx is the same as Case (a). By notingŷ ∈ Proj AX (b) and y * ∈ AX since Ax k ∈ AX and Ax k → y * , it leads to 1 2 ŷ − b 2 ≤ 1 2 y * − b 2 . Thus, 1 2 Ax − b 2 ≤ 1 2 y * − b 2 . By noting r(x) = γf * pc (ŷ) < γf * dc , and combining with the above inequality, it yields that F (x) < γf * dc + 1 2 y * − b 2 .(14) On the other hand, combining (13) with the fact of Ax k → y * , we have γf * dc + 1 2 y * − b 2 ≤ lim k→∞ γ x k 1 x k 2 + 1 2 Ax k − b 2 2 = F * . In view of (14) and the above inequality, it leads to F (x) < F * which contradicts the definition of F * . Thus, the sequence {x k } is bounded, and thus it has an accumulation point x * which is an optimal solution of (3). Next, we present the theorem on the existence of global solutions of (2) and (3). The proof follows the line of arguments as in [39,Theorem 3.4], thus omitted here. Theorem 3 Consider (2) and (3). Suppose that N (A) has the µ-spherical section property for some µ > 0. Then, the following assertions hold: (i) If there existsx ∈ R n such that x 0 < m/µ,x ∈ X and Ax = b, then the set of optimal solution of (2) is nonempty. (ii) If there existsx ∈ R n such that x 0 < m/µ,x ∈ X , and Ax = c where c = Proj AX (b) , then the set of optimal solutions of (3) is nonempty. Next, we consider how to guarantee 0 not being a globally optimal solutions for (2) and (3). In view of b = 0, 0 cannot be a globally optimal solution of the constrained model (2). In Theorem 4, we provide sufficient conditions to guarantee that 0 cannot be a globally optimal solution of (3). Theorem 4 Suppose that one of the following assumptions holds: (i) b ∈ AX and 0 < γ < b 2 2 2( √ n−1) ; (ii) There exists a vectorx ∈ X such that Ax − b 2 ≤ ε (ε ≪ b 2 ) and 0 < γ < b 2 2 −ε 2 2( √ n−1) . Then, the optimal solution of (3) cannot be 0. Proof (i) We use contradiction to show it. Suppose that 0 is a global solution of (3). Since b ∈ AX , we choose a vectorx ∈ X such that Ax = b. Then, it leads to γ √ n < γ + 1 2 b 2 which implies that F (x) < F (0). It contradicts to 0 being a global solution of (3). (ii) The proof is similar to (i), thus omitted here. Remark 1 The assumptions of (i) and (ii) in Theorem 4 correspond to the cases of noiseless and noisy observations, respectively. Sparse property We demonstrate the sparsity of the local minimizers of (2) and (3) in the sense that minimizing L 1 /L 2 or (L 1 /L 2 ) + only extract linearly independent columns from the sensing matrix A. With this, we provide a much more easily checkable exact recovery condition than [38, Theorem III.2] for the constrained model both for arbitrary and nonnegative signals. Theorem 5 Let x * (x * = 0) be a local minimizer of the constrained problem (2) and Λ * = supp(x * ). Then, (A Λ * ) ⊤ (A Λ * ) ≻ 0. Proof We divide into two cases to verify. Case 1. X = R n . Let x * be a local minimizer of the constrained model (2). We use contradiction. Suppose the columns of A Λ * are linearly dependent; then there exists v = 0 and v ∈ N (A) such that supp(v) ⊆ Λ * . For any fixed neighborhood B r (x * ) of x * , we scale v so that v 2 < min{min i∈Λ * |x * i |, r}. Consider two feasible vectors in B r (x * ),x = x * + v andx = x * − v. Since supp(v) ⊆ Λ * , we have supp(x) ⊆ Λ * and supp(x) ⊆ Λ * . Since for any i ∈ Λ * , |x * i | ± sign(x * i )v i ≥ min i∈Λ * |x * i | − v 2 > 0, ∀ i ∈ Λ * . Thus, (x * ± v) i = sign(x * i )(|x * i | ± sign(x * i )v i ) for any i ∈ Λ * . It implies that x * ,x andx are located in the same octant. Consequently, x * 1 = 1 2 ( x 1 + x 1 ) , x * 2 < 1 2 ( x 2 + x 2 ) . Suppose not. Then, there exists a positive scalar κ( = 1) such thatx = κx which contradicts the facts of Ax = b and Ax = b. Finally, it yields that x * 1 x * 2 > min x 1 x 2 , x 1 x 2 , which contradicts the fact that x * is a local minimizer. Case 2. X = R n + . The proof is similar to Case 1, thus omitted here. Next, we show that the conclusion of Theorem 5 also holds for the unconstrained model. Theorem 6 Let x * be a local minimizer of the unconstrained problem (3) and Λ * = supp(x * ). Then, (A Λ * ) ⊤ (A Λ * ) ≻ 0. Proof First, we show that x * is also a local minimizer of the constrained problem (2) where b := Ax * . Suppose not. Then, for any r > 0, there exists x r ∈ B r (x * ) ∩ X such that Ax r = Ax * and xr 1 (3), which contradicts that x * is a local minimizer of (3). Therefore, x * is a local minimizer of (2) where b := Ax * . By invoking Theorem 5, the conclusion follows directly. xr 2 < x * 1 x * 2 . It further implies that F (x r ) < F (x * ) where F is defined in Remark 2 From Theorems 5 and 6, we see that if a computed solution x from the model (2) or (3) fails to extract linearly independent columns from the sensing matrix A. Then, x cannot be a local minimizer. The next lemma presents a sufficient and necessary condition for characterizing x ∈ H L which turns out to be checkable. Lemma 2 x ∈ H L if and only if A ⊤ Λ A Λ ≻ 0 where Λ = supp(x). Proof We use contradiction to show the direction of "only if". Let α = ♯(Λ). Suppose not. It implies that there exists a vector v ∈ R α (v = 0) such that v ∈ N (A Λ ). Thus, there exists a vectorṽ ∈ R n such that (ṽ)| Λ = v and (ṽ)| Λ c = 0. We define an index set: Λṽ := supp(ṽ). Note that supp(ṽ) ⊆ Λ. Let ζ := min i∈Λṽ | xĩ vi | > 0 andĩ ∈ arg min i∈Λṽ | xĩ vi |. Next, we define the vector: x = x − sign(xĩṽĩ)ζṽ. Then,x ∈ H, and it is much sparser than x and supp(x) ⊆ supp(x) which contradicts x ∈ H L . Second, for the direction of "if", we also use contradiction. Suppose there exists a feasible solution y( = x) such that supp(y) ⊆ supp(x), and Ay = b, y ∈ X . Then, define v : = y − x( = 0) ∈ N (A Λ ) which contradicts (A Λ ) ⊤ (A Λ ) ≻ 0. Combining Theorems 5, 6 and Lemma 2, we conclude the following corollary. Corollary 2 We have these facts hold: (i) Suppose that rank(A) = m and x * is a local minimizer of (2) and (3), the sparsity of x * is at most m. Equipped with Lemma 2, we provide an exact recovery condition of the constrained model (2) κ(x) > ( x 0 − x 0 − s) 2 s , ∀ x ∈F \{x 0 },(15)F := {x ∈ H | Λ = supp(x), A ⊤ Λ A Λ ≻ 0}, where H is defined in (2). Then, x 0 also uniquely solves (2). Computational approach We focus on solving (3) with X = R n + . Inspired by [34], we derive the closedform solution of the proximal operator of (L 1 /L 2 ) + , and accompanied by a practical solver for finding one global solution. Proximal operator Define a proximal operator of (L 1 /L 2 ) + (:= x 1 x 2 + ι R n + (x)) with a parameter ρ > 0 as Prox [(L1/L2) + /ρ] (q) := arg min x∈R n + x 1 x 2 + ρ 2 x − q 2 2 .(16) It follows from [33, Definition 1.23] and [33,Theorem 1.25], the solution set of (16) is nonempty. Obviously, if q ∈ R n − , the solution of (16) is 0. Next, Example 1 shows that the optimal solution of (16) may not be unique. Example 1. Let n = 2 and q 1 = q 2 = 2( √ 2 − 1). Consider an objective function min x∈R n + |x 1 | + |x 2 | x 2 1 + x 2 2 + 1 2 (x 1 − q 1 ) 2 + 1 2 (x 2 − q 2 ) 2 . Indeed, it has three globally optimal solutions: x 1 = ( 2( √ 2 − 1), 0) ⊤ , x 2 = (0, 2( √ 2 − 1)) ⊤ and x 3 = 2( √ 2 − 1), 2( √ 2 − 1) ⊤ . Next, we characterize one of globally optimal solutions of Prox [(L1/L2) + /ρ] (q) in a closed-form. Theorem 8 Given q ∈ R n and q ≤ 0 and ρ > 0. We can sort q in a descending order in a way of q π(1) ≥ · · · ≥ q π(ν) > 0 ≥ q π(ν+1) ≥ · · · ≥ q π(n) where π is a proper permutation of [n]. Then, the following assertions hold: (i) There exists an optimal solutionx of (16) such that it has the same descending order as q, i.e., x π(1) ≥ · · · ≥x π(ν) ≥ 0 =x π(ν+1) = · · · =x π(n) . (ii) We denote the multiplicity of the largest magnitude in q as µ, i.e., q π(1) = · · · = q π(µ) > q π(µ+1) . One of the following assertions holds: (a) If 0 < ρ ≤ 1/(q 2 π(1) ), then (16) has a one-sparse solution given bȳ x π(i) = q π(i) i = 1; 0 otherwise. (b) If ρ > 1 q 2 π(1) , there exist an integer t (t ≤ ν) and a scalar pair of (a, r) such that (Q t = t i=1 q π(i) )          a 2 r 3 − ρa + ρQ t − t r = 0, r 3 − t i=1 q 2 π(i) r + Q t − a ρ = 0, and the r is also satisfied with q π(t) − 1 ρr > 0 and q π(t+1) − 1 ρr ≤ 0, and the vectorx is characterized bȳ x π(i) =    ρq π(i) − 1 r ρ − a r 3 1 ≤ i ≤ t, 0 otherwise, is an optimal solution of (16), where t = x 0 , a = x 1 , r = x 2 . Proof (i) First, we verify that for any global solutionx, we have q i > q j ⇒x i ≥x j .(18) We use contradiction. Define the objective function of (16) by f (x) = x 1 x 2 + ι ≥0 (x) + ρ 2 x − q 2 2 . Suppose not. It means thatx i <x j . Then, we exchange these two entries inx to obtain a new vectorx. Then, f (x) < f (x) since (q i −x i ) 2 + (q j −x j ) 2 > (q i −x j ) 2 + (q j −x i ) 2 . It contradictsx being a global solution. Thus, (18) holds. Furthermore, ifx i ,x j > 0, we can strengthen the conclusion in (18) to "x i >x j ". Indeed, by invoking the optimality conditions of (16), it leads to (x −x) ⊤ 1 x 2 − x 1 x 3 2x + ρ(x − q) ≥ 0, ∀x ∈ R n + .(19)Define Υ := 1 x 2 − x 1 x 3 2x + ρ(x − q). Sincex i ,x j > 0, it leads to Υ i = Υ j = 0.(20) Suppose not. Then,x i =x j . It follows from the above equality that q i = q j which contradicts q i > q j . Therefore,x i >x j . If there exists several entries of q with the same value, the corresponding entries inx can be arranged in a descending order. Thus, there exists a global solutionx such that x π(1) ≥x π(2) ≥ · · · ≥x π(n) . Next, we show that q i ≤ 0 ⇒x i = 0.(22) We use contradiction. Suppose not. Then, there exists at least one indexî such that qî ≤ 0 andxî > 0. It follows from (19) that xî > 0 ⇒ Υî = 0.(23) In the following, we divide into two cases to verify. Case 1. ρ ≥ a/r 3 . Sincexî > 0, Υî = 1 x 2 + (ρ − a r 3 )xî − ρqî > 0 due to qî ≤ 0 and ρ − a/r 3 ≥ 0, and it contradicts (23). Case 2. ρ < a/r 3 . Invoking (23), it leads tō x i = ρq i − 1 r ρ − a r 3 , ∀ i ∈ supp(x).(24) We define two index sets: Λ + := {i | q i > 0} and Λ − := {i | q i ≤ 0}. We dividē x into two parts: x + =x | Λ + andx − =x | Λ − . By assumption, we know that x − = 0. Thus,x + = 0 due to (21). Picking up i ∈ Λ + and setting j =î ∈ Λ − (i.e.,x j > 0). Since q i > q j andx j > 0, it leads tox i > 0 due to (18). It implies that i, j ∈ supp(x). Consequently, it follows from (24) that x i <x j ,(25) due to ρ < a/r 3 . On the other hand, sincex is an optimal solution, we have proved that q i > q j ⇒x i >x j , which contradicts (25). Therefore, the assertion (22) holds. Thus, the assertion (i) follows immediately. (ii) In view of (22), it implies that the minimization problem (16) amounts to solving a low-dimension minimization problem: arg min y∈R ν y 1 y 2 + ρ 2 y − p 2 2 ,(26) where p = q | σ and σ := {π(1), · · · , π(ν)}. Denotingȳ as an optimal solution of (26), the vector x defined by x| σ =ȳ and x| σ c = 0 is an optimal solution of (16). By invoking [34, Theorem 3.3], the assertion of (ii) follows immediately. x 2 + ρ 2 x − q 2 2 . In [34,Theorem 3.3], one global solution of Prox [(L1/L2) + /ρ] (q) has been characterized in a closed-form. It includes two cases: (a) If 0 < ρ ≤ 1/(q 2 π(1) ), there is a onesparse solution; (b) If ρ > 1/q 2 π(1) , there is a t-sparse solution. Algorithm 3.1 [34] either returns a one-sparse solution for case (a) or produces a t-sparse solution for case (b). For the latter case, it adopts a bisection search to find the true sparsity t and incorporates the fixed-point iterative method to get the unique solution pair (a, r) [34, Lemma 3.6] of the two-dimension nonlinear system. With this (a, r), Algorithm 3.1 of [34] computes the t-sparse solution in a closed-form. More discussions can be found in [34,Section 3]. In summary, an overall algorithm for findingx ∈ Prox [(L1/L2) + /ρ] (q) with q ∈ R n and ρ > 0 is presented in Algorithm 1. Algorithm 1 Finding a solution of Prox [(L1/L2) + /ρ] (q) Require: ρ > 0, q ∈ R n , q π(1) ≥ · · · ≥ q π(ν) > 0 ≥ q π(ν+1) ≥ · · · ≥ q π(n) . Set p = q|σ where σ = {π(1), · · · , π(ν)}. 1: Using [34, Algorithm 3.1] to findȳ ∈ Prox [(L 1 /L 2 )/ρ] (p). 2: Definex|σ =ȳ andx| σ c = 0. 3: Outputx. (3) with X = R n + Although there exist a few different ways for reformulating the unconstrained model (3) with X = R n + , most of them result in a scheme of ADMM with violation of convergence guarantee [16,21,37]. Equipped with the newlyderived solution of the proximity of (L 1 /L 2 ) + , we apply ADMM to (3) in a particular splitting way: ADMM for solving min x,y∈X γ x 1 x 2 + 1 2 Ay − b 2 2 s.t. x = y, x ∈ X .(27) The augmented Lagrangian of (27) is defined by L A (x, y, z) = γ x 1 x 2 +ι X (x)+ 1 2 Ay − b 2 + z ⊤ (x − y)+ β 2 x − y 2 2 ,(28) where z is the Lagrangian multiplier and β > 0 is the penalty parameter. Given (y k , z k ), the ADMM scheme generates the iterative sequence {w k } (w k = (x k , y k , z k )) as follows,      x k+1 ∈ arg min x∈X L A (x, y k , z k ),(29a) y k+1 = arg min y L A (x k+1 , y, z k ), (29b) z k+1 = z k + β(x k+1 − y k+1 ).(29c) The x-subproblem (29a) amounts to x k+1 ∈ Prox [ γ β (L1/L2) + ] (y k − z k /β). By using Sherman-Morrison-Woodbury Theorem, the y-subproblem (29b) can be given by a more efficient scheme: y k+1 = M A ⊤ b β + z k β + x k+1 ,(30) where M = I n − 1 β A ⊤ (I m + 1 β AA ⊤ ) −1 A since m ≪ n. We summarize the overall scheme in Algorithm 2, and denote it by ADMM + p . Global Convergence In contrast to the existing literature on the convergence analysis of ADMM or its variants [16,21,34,37], we proves it converges to a d-stationary point Algorithm 2 ADMM + p Require: A ∈ R m×n , b ∈ R m , β, ε > 0. 1: Initialize: y 0 = z 0 . 2: while k < kmax or x k−1 − x k / x k > ε do 3: Solving the x-subproblem (29a) via Algorithm 1. 4: Computing y k+1 via (30). 5: Updating z k+1 via (29c). 6: end while without the KL assumption. We define the merit function: T (x, y) = γ x 1 x 2 +ι X (x)+ 1 2 Ax − b 2 + β 2 x − y 2 2 ,(31) and denote T k := T (x k , y k ) for succinctness. Lemma 3 Let {w k } be the sequence generated by ADMM + p . If β > 2L, then there exists a constant c 1 > 0 such that T k+1 ≤ T k − c 1 y k − y k+1 2 2 . Proof First, it follows from the optimality condition of (29b) that z k+1 = A ⊤ (Ay k+1 − b). Then, it further implies that z k − z k+1 2 ≤ L y k − y k+1 2 ,(32) where L = σ max (A ⊤ A) where σ max (·) represents the largest eigenvalue. Next, invoking (29a), it leads to L A (x k+1 , y k , z k ) ≤ L A (x k , y k , z k ). Then, using (29b), it yields that L A (x k+1 , y k+1 , z k ) ≤ L A (x k+1 , y k , z k ) − β 2 y k − y k+1 2 , which is due to L A (x k+1 , y, z k ) is strongly convex with respect to y with strongly convex coefficient of β 2 . In view of (29c), we obtain that L A (x k+1 , y k+1 , z k+1 ) = L A (x k+1 , y k+1 , z k ) + 1 β z k − z k+1 2 . Combining above three inequalities with (29c), we have that L A (x k+1 , y k+1 , z k+1 ) ≤ L A (x k , y k , z k ) − (β/2 − L 2 /β) y k − y k+1 2 . (33) Next, we show that T k+1 ≤ L A (x k+1 , y k+1 , z k+1 ) + L 2 x k+1 − y k+1 2 .(34) Recall the definition T k+1 in (31) and L A (x k+1 , y k+1 , z k+1 ) in (28). To show (34), we only need to prove that 1 2 Ax k+1 − b 2 ≤ 1 2 Ay k+1 − b 2 + (z k+1 ) ⊤ (x k+1 − y k+1 ) + L 2 x k+1 − y k+1 2 . Invoking the optimality condition of (29b), it leads to z k+1 = ∇( 1 2 Ay k+1 − b 2 ). By using this fact and L = σ max (A ⊤ A), the above inequality follows directly. Consequently, T k+1 ≤ L A (x k+1 , y k+1 , z k+1 ) + L 2 x k+1 − y k+1 2 ≤ L A (x k , y k , z k ) − 3L 8 y k − y k+1 2 ,(35) where the first inequality is due to (34), the second is due to (29c), (33), (32) and β > 2L. Next, we have that T k ≥ L A (x k , y k , z k ). Combining (35) with the above inequality, the assertion holds with c 1 = 3L 8 . Lemma 4 Let {w k } be the sequence generated by ADMM + p . Then there exists a constant c 2 > 0 such that dist(0, ∂T (x k+1 , y k+1 )) ≤ c 2 y k+1 − y k 2 . Proof The proof is similar to [34,Lemma 5.7] and thus omitted. Next, we present the subsequential convergence of ADMM + p under the boundedness of {x k } which is a standard assumption to ensure existence of accumulation point [1,39]. The boundedness of {x k } can be guaranteed by the boundedness of the set of {x ∈ X |F (x) ≤ F (x 0 )} which can be further ensured by no nonnegative vectors in N (A). The proof of the following theorem is standard [21,34] and thus omitted. Theorem 9 Let {w k } be the sequence generated by ADMM + p . If {x k } is bounded and β > 2L, we have the following statements: (i) lim k→∞ x k − x k+1 = 0, lim k→∞ y k − y k+1 = 0, and lim k→∞ z k − z k+1 = 0; (ii) The sequence {w k } has at least one accumulation point w ∞ . Next, we show the global convergence of ADMM + p to a d-stationary point by assuming A ⊤ b ≤ 0, the boundedness of {x k } and β sufficiently large. The first assumption is to guarantee 0 not being a accumulation point. The latter two assumptions are usually imposed for the convergence [16,21,34]. Theorem 10 Let {w k } be the sequence generated by ADMM + p . If A ⊤ b ≤ 0, β > 2L, and {x k } is bounded, then (i) any accumulation point of {x k } is a dstationary point of (3), (ii) {w k } has finite length, i.e. ∞ k=1 w k+1 − w k < ∞, and hence {w k } converges to a stationary point w ∞ := (x ∞ , y ∞ , z ∞ ) satisfying      (x − x ∞ ) ⊤ γ( 1 x ∞ 2 − x ∞ 1 x ∞ 2 x ∞ ) + z ∞ ≥ 0 ∀x ∈ X , A ⊤ (Ay ∞ − b) − z ∞ = 0, x ∞ = y ∞ .(36) Proof (i) We first show that any accumulation point x ∞ of the sequence {x k } generated by (29) cannot be 0. Suppose not. Then, there exists a subsequence of {w kj } converging to w ∞ where x kj → x ∞ = 0. Thus, x kj +1 → x ∞ = 0 due to Theorem 9. Also, one has y kj − 1 β z kj → ξ ∞ := y ∞ − 1 β z ∞ . Next, we show that ξ ∞ ∈ R n − . In what follows, we show the solution of (16) is 0 if and only if q ∈ R n − in (16). For the "if" part, it is obviously true. For the "only if" part, i.e., if 0 is a solution of (16), then q ∈ R n − . Suppose not. Then, q ∈ R n − . Thus, there exists at least one index (without loss of generality) q 1 > 0 and q 1 ≥ q 2 ≥ · · · ≥ q n . We define q + = max(q, 0). Thus, q + = 0. According to [34,Theorem 3.2], we see that the solution of arg min x∈R n x 1 x 2 + ρ 2 x − q + 2 2 ,(37) cannot be 0 since the solution of (37) is at least one-sparse. It contradicts 0 being a solution of (16). It follows from (29a) that x kj +1 ∈ Prox [(L1/L2) + /ρ] (y kj − 1 β z kj ). Taking j → ∞ and invoking [33,Theorem 1.25 ], we have x ∞ ∈ Prox [(L1/L2) + /ρ] (y ∞ − 1 β z ∞ ). Consequently, ξ ∞ ≤ 0 due to x ∞ = 0. Since y kj → y ∞ = 0 due to x ∞ −y ∞ = 0, z kj → −βξ ∞ . Invoking z kj = A ⊤ (Ay kj −b) and letting j → ∞, it leads to A ⊤ b = βξ ∞ which contradicts to A ⊤ b ≤ 0. Thus, x ∞ = 0. Next, we show any accumulation point x ∞ of {x k } is a d-stationary point of (3) with X = R n + . The sequence of {w k } is bounded and hence it has a subsequence {w kj } such that w kj → w ∞ as j → +∞. From the optimality condition of (29), we have        (x − x k+1 ) ⊤ γ( 1 x k+1 2 − x k+1 1 x k+1 2 x k+1 ) + z k + β(x k+1 − y k ) ≥ 0 ∀x ∈ X , A ⊤ (Ay k+1 − b) − z k+1 = 0, β(x k+1 − y k+1 ) + z k − z k+1 = 0.(38) The above system is also true when k := k j . Note that w kj +1 → w ∞ as j → +∞ due to Theorem 9 and w kj → w ∞ . Then, taking limit on both sides of the system (38) with k := k j , we have that w ∞ is satisfying (36) due to Theorem 9 and x ∞ = 0. By eliminating y ∞ and z ∞ from (36), we have (4) holds withx = x ∞ , which implies that x ∞ is a d-stationary point of (3) with X = R n + . (ii) According to [2], if at least one of the two subanalytic functions maps bounded sets to bounded sets, then their sum is subanalytic. Since 1 2 Ax−b 2 2 is real analytic and maps bounded sets to bounded sets, and the function γ x 1 x 2 + ι R n + (x)+ι {x| x ≥ε} (x) is semianalytic (for any sufficiently small ε > 0) [39], then their sum is also subanalytic. Similarly, the function T (x, y) defined in (31) is also subanalytic. Furthermore, invoking Lemma 4, any accumulation point (x ∞ , y ∞ ) of {x k , y k } generated from (29) satisfies 0 ∈ (∂ x T (x ∞ , y ∞ ), ∂ y T (x ∞ , y ∞ )) with x ∞ = y ∞ . Thus, (x ∞ , y ∞ ) can not be (0, 0). DefineŨ = {u := (x, y) ∈ R n × R n | u 2 ≥ ǫ} with 0 < ǫ < 1 2 u ∞ 2 . Invoking [3, Theorem 3.1], the merit function T (x, y)|Ũ satisfies the KL property since T (x, y)|Ũ is continuous and its domain is closed. Therefore, T (x, y) satisfies the KL property at the point (x ∞ , x ∞ ). The remaining proof is standard and similar to [21,Theorem 4], thus omitted here. Numerical results In this section, we compare ADMM + p with state-of-the-art methods in sparse recovery. We focus on the sparse recovery problem with the compressive matrix is highly coherent, on which L 1 minimization fails. All these algorithms are implemented on MATLAB R2016a, and performed on a desktop with Windows 10 and an Intel Core i7-7600U CPU processor (2.80GH) with 16GB memory. The stopping criterion is as follows: RelChg := x k − x k−1 2 max { x k−1 2 , 0.1} < Tol or k max > 5n.(39) We set Tol as Tol = 10 −6 if σ = 0, 0.01 * σ if σ > 0, where σ is the variance of the noise (σ = 0 means the noiseless case). Two types of sensing matrices are considered: (I) Oversampled DCT. A = [a 1 , a 2 , . . . , a n ] ∈ R m×n with each column a j := 1 √ m cos 2πwj F (j = 1, . . . , n), where w ∈ R m is an uniformly distribution on [0, 1] random vector and F ∈ R + controls the coherence. (II) Gaussian matrix. A is subject to N (0, Σ) with the covariance matrix given by Σ = {(1 − r)I n (i = j) + r} i,j with 1 > r > 0. We generate an s-sparse ground truth signal x * = |x| ∈ R n + with each nonzero entry ofx following a Gaussian normal distribution. Algorithmic behaviors In the literature, there are some efficient methods applicable to the model [32]. For a fair comparison, we incorporate Algorithm 1 in each algorithm for computing the proximal operator of (L 1 /L 2 ) + . We test on two types of matrices (Gaussian matrix, oversampled DCT) with ground-truth (3), and β = 0.025 in ADMM + p . According to the theoretical results in [14,22,27], each of GIST, APG1, APG2, APG3 and APG4 clusters at a critical point. In Table 1, we test on two types of matrices with three different choices of initial points (the first two in the MATLAB scripts): (1) rand(n,1); (2) abs(randn(n,1)); (3) The solution of L 1 minimization. For each instance, we run 20 trials for all of these algorithms and record the average results. We report the computing time in seconds (Time), the objective function value (Obj) and the relative error (RErr:= x k −x * 2 x * 2 ) when the stopping criterion (39) is satisfied. Data in this table show that ADMM + p converges faster than the other comparing algorithms except the cases of L 1 solution respectively under the Gaussian matrix and the oversampled DCT. For each scenario, ADMM + p always achieves the lowest quantity of RErr, and its performance is very robust to the choices of initial points. This advantage represents another advantage of the proposed ADMM + p over the other comparing algorithms, such as GIST, various versions of APG and SOOT whose numerical performances are sensitive to the initial points. In Figure 1, we depict RErr with respect to iteration number from ADMM + p with other comparing algorithms. Each plot in Figure 1 corresponds to the two types of initial points under two types of compressive matrices: the left is from rand(n,1) under oversampled DCT matrix and the right is from abs(randn(n,1)) with Gaussian matrix. Clearly, ADMM + p converges much faster than the others and always achieves the lowest quantity of RErr among these comparing algorithms for both cases. 10 0 Fig. 1 The evolution of RErr with respect to the iteration number (It.): Initialized from rand(n,1) under oversampled DCT matrix (left), and from abs(randn(n,1)) under Gaussian matrix (right). x k − x * 2/ x * 2 GIST ADMM + p APG1 APG2 APG3 APG4 SOOT Comparison on various models We show the efficiency of the proposed ADMM + p for (L 1 /L 2 ) + minimization under the noiseless observation. We compare with other sparse recovery unconstrained models: L 1 , L 1/2 [6], and L 1 -L 2 [25], all in an unconstrained formulation without nonnegative constraint. We use the default setting for each algorithm and unify their stopping criteria as (39) and set γ = 10 −6 in all these models due to the noisefree. We consider over-sampled DCT matrix with F = 10 and Gaussian matrix with r = 0.8 of size 64 × 1024, and the sparsity ranging from 2 to 24 with an increment of 2. The fidelity of sparse signal recovery is evaluated in terms of success rate, model failure and algorithm failure rates [31,34]. If the relative error of the reconstructed solutionx to the ground truth x * is less than 10 −3 , we refer to it as a success. Success rate is defined as the number of successes over the number of trials. Furthermore, we classify the failure of not recovery as model/algorithm failures by comparing the objective function F (·) at the ground truth x * and the reconstructed solutionx. If F (x * ) < F (x), we refer it to as algorithm failure. Otherwise, we have model failure. In Figure 2, we present success rate and model/algorithm failure rates for (L 1 /L 2 ) + , L 1 , L 1/2 and L 1 -L 2 by randomly simulating 50 trials for each scenario and computing the average results. For the oversampled DCT case, (L 1 /L 2 ) + achieves the highest success rate. For the Gaussian matrix case, (L 1 /L 2 ) + exhibits a slightly better than L 1/2 when the sparsity is less than 22 and otherwise comparable to L 1/2 in terms of success rate. Nevertheless, (L 1 /L 2 ) + performs much better than L 1 and L 1 -L 2 for Gaussian case regarding success rate. Based on model/algorithm failure rates in Figure 2, we observe that the algorithm failure rates of (L 1 /L 2 ) + are much lower than that of L 1/2 and L 1 -L 2 for both cases and achieves the lowest (as well as the L 1 ) for the oversampled DCT case. The model failure rates of (L 1 /L 2 ) + rank second for both cases and are always worse than L 1 -L 2 and L 1/2 and better than L 1 . These results illustrate the efficiency of the ADMM + p for both types of the sensing matrices and prompt us to further work on the model improvement of (L 1 /L 2 ) + when the sparsity level is increasing. Recovery of nonnegative signal from coherent dictionaries We illustrate the efficiency of the ADMM + p for solving Examples 1-3 of [38] by comparing with the scaled gradient projection method (SGPM) which is a state-of-the-art algorithm in (L 1 /L 2 ) area [11,38]. These examples are constructed to show the superiority of (L 1 /L 2 ) minimization over L 1 or L p (0 < p < 1) minimization. The SGPM is forward-backward algorithm applied to (5) by setting g(x) = ι R n + (x) and h(x) = γ x 1 x 2 + 1 2 Ax − b 2 2 with line research. All these three examples are linear systems, i.e., Ax = b and denote the corresponding matrix as A (i) , b (i) for each i ∈ {1, 2, 3}. We test on Examples 1-3, all these matrices A (i) (i = 1, 2, 3) are defined with values of n = 50, 100 and p = 0.9, 0.95, and the vectors of b (i) (i = 1, 2, 3) are of n random numbers subjected to uniform distribution on [0, 1]. The model parameter of γ in (3) is set to 0.01. For the SGPM, we use the defaulted setting as in [11,38], i.e., δ = 1, c 0 = 10 −9 , ξ 1 = 2, ξ 2 = 10 and σ = 0.01. Set β = 0.8 in ADMM + p . For these three examples, we set the initial point as x 0 = 0.05(100 + 0.01η i ) and η i ∼ N (0, 1). In order to measure the extent of satisfying optimality condition of (4), we define the Karush-Kuhn-Tucker (KKT) residual on the support set (KKT R ) of the last iteratex as: KKT R = γ sign(xΛ) x 2 − x 1 x 3 2xΛ + (AΛ) ⊤ (Ax − b) 2 , whereΛ = supp(x). In Table 2, we record the results of ADMM + p and SGPM in terms of final objective function value of (3) (Obj), the KKT R and computational time in seconds (Time). Table 2 clearly shows that ADMM + p performs much better than SGPM in terms of achieving much lower objective function values, ending up with higher accuracy while taking less time. For the scenario of n = 50 and p = 0.9 of Example 2, we find that ADMM + p recovers the one-sparse solution x (2) = [2, 0, · · · , 0] ⊤ while SGPM does not. DOAS We consider the wavelength misalignment problem in different optical absorption spectroscopy analysis (DOAS). More specifically, J(λ) = M j=1 a j y j (λ + v j (λ)) + η(λ). J(λ) presents the data and y j (λ + v j (λ)) denotes the reference spectra at the deformed wavelength λ + v j (λ) where v j (·) denotes the deformations. The noise η(λ) are given at the wavelength λ and {a j } M j=1 are coefficients. In our experiments, we generate a dictionary for three reference gases (M = 3): HONO, NO2 and O3, and then deform each with a set of linear functions, i.e., v j (λ) = p j λ + q j . We use B j (j = 1, · · · , M ) to denote a matrix with each column being deformed basis, i.e.. y j (λ + p k λ + q ℓ ) (k = 1, · · · , K; ℓ = 1, · · · , L) and y j ∈ R 1024 ; p k = −1.01 + 0.01k, q ℓ = −1.1 + 0.1ℓ. By setting K = L = 21, there is a total of 441 linearly deformed references for each of the three groups. We generate the dictionary by imitating the relative magnitudes of a real DOAS dataset [13] with normalization to the dictionary. Then, to generate the data a j , we randomly pick up one entry with random magnitudes whose mean values are 1, 0.1, 2 for HONO, NO2 and O3, respectively. Finally, the synthetic data J (λ) is generated by adding zero-mean Gaussian noise. We test five different noise levels: std = 0, 1e − 3, 5e − 3, 1e − 2, 5e − 2. We solve the wavelength misalignment by considering the following model: min {xj}j 1 2 J − [B 1 , · · · , B M ]    x 1 . . . x M    2 +γ M j=1 R (x j ) ,(40) where R(·) represents the regularization function, and x j ∈ R 441 (j = 1, 2, 3). We test (40) on different regularization functions to enforce sparsity. In particular, we set R(x) = ι R n + (x), x 1 + ι R n + (x), x 1 − x 2 , x 1/2 , x 1 x 2 + ι R n + (x) in (40), respectively. We refer to these models as non-negative least square (NNLS), non-negative unconstrained L 1 (NNL1), L 1 -L 2 , L 1/2 , (L 1 /L 2 ) + . For (40) with (L 1 /L 2 ) + regularizer, we adopt ADMM + p and SGPM to solve it. For L 1 -L 2 , we use Algorithm 1 in [24]. For NNLS, we use MATLAB's lsqnonneg function. As for NNL1, we solve it by ADMM and for L 1/2 , we solve it by [20]. For all these methods, we use the default setting. Tables 3 and 4 show the errors (err = x−x * 2 ) between the reconstructed vectors and the ground-truth, and computational time (Time (s)) under different amounts of noise, respectively. Each recorded value is the average of 20 random realizations. In Fig. 2, the ground truth and the error vectors of these comparing algorithms defined by the constructed signals minus the true signal are presented in Plots (a) and (b), for the scenario of std = 0.05. The horizontal heavy yellow line surrounded by cyan is caused by the full-dimension error vectors from L 1 -L 2 , SGPM, NNL1, L 1/2 . The deviation of ADMM + p is much smaller than the others. ADMM + p achieves the best recovery quality in the sense of highest accuracy and sparsity. All the results demonstrate that ADMM + p is comparable to NNLS in terms of accuracy for noiseless data, and even more accurate than NNLS for noisy cases. In comparison with NNL1, L 1 -L 2 , ADMM + p also ends up with much higher accuracy and takes less time. In contrast with L 1/2 , ADMM + p converges to a much more accurate solution while consuming a bit more time. Besides, for solving the same (L 1 /L 2 ) + model, ADMM + p costs significantly less time than SGPM while still achieving a much more accurate solution. More specificially, ADMM + p reduces computational time by about 95% ∼ 99% compared to SGPM. Conclusions We carry out a unified theoretical study on both L 1 /L 2 minimization models, including the constrained and the unconstrained. First, we prove that the existence of the globally optimal solution can be guaranteed by the µ-spherical section property of the null space of the matrix A. Second, we analyze the sparsity property of the constrained and the unconstrained models. Third, we derive a closed-form solution of the proximal operator of (L 1 /L 2 ) + . Equipped with this, we propose a specific splitting scheme (ADMM + p ) to solve the unconstrained (L 1 /L 2 ) + model. We establish its global convergence to a d-stationary solution by verifying the KL property of the merit function. Numerical simulations validate our analyses and demonstrate that ADMM + p outperforms other state-of-the-art methods in sparse recovery. current study are available from the corresponding author on reasonable request. Declarations Conflict of interests The authors have no relevant financial or non-financial interests to disclose. (ii) The model(2) has a finite number of local minimizers. (iii) If x * is a local minimizer of (2), then x * ∈ H L where H L is defined in Definition 1.Proof The proof for (i) and (ii) are elementary, thus omitted. (iii) It follows from Theorems 5, 6 and Lemma 2 directly. Theorem 7 7If x 0 uniquely solves (1) and x 0 0 = s and if For finding one global solution of (26), a fast solver has been developed in [34, Algorithm 3.1]. Indeed, Algorithm 3.1 of [34] aims to find one global solution of Prox [(L1/L2) + /ρ] (q) := arg min x∈R n x 1 ( 3 ) 3with X = R n + , including General Iterative Shrinkage Thresholding (GIST) ([27, Algorithm 2], [14, Algorithm 1]) and monotone accelerated proximal gradient method (APG) with fixed stepsize (APG1) [22, Algorithm 1], monotone APG with line search (APG2) [22, Algorithm 2], nonmonotone APG with fixed stepsize (APG3) [22, Algorithm 3], nonmonotone APG with line search (APG4) [22, Algorithm 4] and the smoothed L 1 /L 2 approach (SOOT) proposed in Fig. 2 2Comparison results in the noisefree case based on the oversampled DCT matrix with F = 10 (left) and and Gaussian matrix with r = 0.8 (right). From top to bottom: success rates, algorithm failures and model failures. FundingFig. 3 3Min Tao was partially supported by National Key Research and Development Program of China (2018AAA0101100), the Natural Science Foundation of China (No. 11971228) and Jiangsu University QingLan Project. The work of Xiao-Ping Zhang is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), Grant No. RGPIN-2020-04661. Data Availibility The datasets generated during and/or analysed during the Comparison results for NNLS, NNL1, L 1 -L 2 , L 1/2 , SGPM and ADMM + p on DOAS data with additive noise (std = 5e − 2). (a) The ground truth with sparsity 3 (top); (b) The error vectors (ERR=x − x * ) from these comparing algorithms, and its nonzero numbers of these error vectors are 27, 1323, 1323, 1323, 1323, 18, respectively (bottom). Table 1 1Average computation results generated from different initial points.ADMM + p GIST APG1 APG2 APG3 APG4 SOOT Gaussian matrix, initial point: rand(n,1) Obj 2.38e-03 1.44e-01 2.72e-01 3.14e-02 2.83e-01 8.18e-03 7.99e+01 Time 0.23 7.17 3.64 31.92 2.25 14.89 10.25 RErr 4.91e-05 1.45e-02 5.53e-03 6.18e-03 5.76e-03 2.64e-03 2.83e-01 Gaussian matrix, initial point: abs(randn(n,1)) Obj 2.60e-03 1.27e-01 3.03e-01 1.04e-01 3.17e-01 8.26e-03 7.89e+01 Time 0.22 6.89 3.86 55.48 2.35 15.83 10.39 RErr 5.17e-05 1.36e-02 6.12e-03 1.23e-02 6.26e-03 2.18e-03 2.83e-01 Gaussian matrix, initial point: L1 solution Obj 3.52e-04 3.94e-04 3.94e-04 3.94e-04 3.94e-04 3.93e-04 3.82e-04 Time 0.15 0.071 0.031 0.050 0.029 0.033 0.63 RErr 7.19e-06 8.51e-05 8.51e-05 8.51e-05 8.51e-05 8.51e-05 8.43e-05 Oversampled DCT, initial point: rand(n,1) Obj 1.64e-04 6.80e-04 1.68e-04 3.84e-04 1.67e-04 2.96e-04 2.10e-03 Time 0.68 6.30 1.54 24.67 1.02 52.81 5.43 RErr 1.45e-05 3.73e-02 1.76e-03 2.37e-02 1.56e-03 1.81e-02 6.97e-02 Oversampled DCT, initial point: abs(randn(n,1)) Obj 1.64e-04 6.87e-04 1.70e-04 3.94e-04 1.71e-04 3.25e-04 2.14e-03 Time 0.68 6.40 1.44 24.53 0.82 53.20 5.09 RErr 1.45e-05 3.77e-02 2.40e-03 2.43e-02 2.56e-03 2.08e-02 7.05e-02 Oversampled DCT, initial point: L1 solution Obj 1.64e-04 1.64e-04 1.64e-04 1.64e-04 1.64e-04 1.64e-04 1.64e-04 Time 0.34 0.016 0.056 0.053 0.038 0.030 0.28 RErr 1.01e-05 2.28e-05 1.29e-05 1.57e-05 1.29e-05 2.06e-05 2.17e-05 signals of sparsity 15. The size of the sensing matrix is 128 × 1024. We set γ = 0.001 in Table 2 2Comparison between ADMM + p and SGPM on Examples 1, 2 and 3 via solving (3).(Ex. ,n, p) Obj KKT R Time SGPM ADMM + p SGPM ADMM + p SGPM ADMM + p (1,50,0.95) 5.64 0.011 3.35 2.18 × 10 −4 0.63 0.33 (1,50,0.9) 5.63 0.010 3.35 4.38 × 10 −4 0.64 0.25 (1,100,0.95) 11.2 0.010 4.73 1.22 × 10 −4 2.11 0.97 (1,100,0.9) 11.2 0.010 4.73 2.69 × 10 −4 2.02 0.81 (2,50,0.95) 5.64 0.010 3.35 5.69 × 10 −6 0.61 0.45 (2,50,0.9) 5.63 0.010 3.35 1.05 × 10 −5 0.59 0.16 (2,100,0.95) 11.2 0.010 4.73 9.92 × 10 −6 1.72 0.92 (2,100,0.9) 11.2 0.010 4.72 1.18 × 10 −5 1.98 0.27 (3,50,0.95) 3.00 0.084 2.42 2.56 × 10 −3 0.61 0.34 (3,50,0.9) 2.98 0.083 2.41 1.97 × 10 −3 0.66 0.38 (3,100,0.95) 5.81 0.11 3.38 1.24 × 10 −3 1.81 0.95 (3,100,0.9) 5.77 0.11 3.37 7.00 × 10 −4 1.88 0.95 Table 3 3Reconstructed error (err = x − x * 2 ) for DOAS.std NNLS NNL1 L 1 -L 2 L 1/2 (L 1 /L 2 ) + ADMM + p SGPM 0 7.72e-16 2.70e-03 5.21e-05 3.60e-03 2.58e-05 4.80e-02 0.001 4.91e-03 7.95e-03 8.76e-04 2.41e-02 4.37e-04 7.60e-02 0.005 3.26e-02 1.92e-02 3.05e-03 6.95e-02 2.03e-03 3.75e-01 0.01 1.46e-01 1.61e-01 4.90e-03 1.04e-01 4.25e-03 3.84e-01 0.05 1.73e-01 1.75e-01 2.39e-02 1.30e-01 2.00e-02 5.67e-01 Table 4 4Computational time (s) for DOAS under different noisy level.std NNLS NNL1 L 1 -L 2 L 1/2 (L 1 /L 2 ) + ADMM + p SGPM 0 0.021 8.05 13.30 0.10 5.66 3110.00 0.001 0.047 8.19 35.60 0.11 6.59 3030.00 0.005 0.016 7.77 28.10 0.13 7.17 437.00 0.01 0.057 8.50 33.90 0.13 8.34 167.00 0.05 0.052 8.82 70.90 0.13 4.00 276.00 Optim., 31 (2021), pp. 1576-1603. 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{'abstract': 'In this paper, we carry out a unified study for L 1 over L 2 sparsity promoting models, which are widely used in the regime of coherent dictionaries for recovering sparse nonnegative/arbitrary signals. First, we provide a unified theoretical analysis on the existence of the global solutions of the constrained and the unconstrained L 1 /L 2 models. Second, we analyze the sparse property of any local minimizer of these L 1 /L 2 models which serves as a certificate to rule out the nonlocal-minimizer stationary solutions. Third, we derive an analytical solution for the proximal operator of the L 1 /L 2 with nonnegative constraint. Equipped with this, we apply the alternating direction method of multipliers to the unconstrained model with nonnegative constraint in a particular splitting way, referred to as ADMM + p . We establish its global convergence to a d-stationary solution (sharpest stationary) without the Kurdyka-Lojasiewicz assumption. Extensive numerical simulations confirm the superior of ADMM + p over the state-of-the-art methods in sparse recovery. In particular, ADMM + p reduces computational time by about 95% ∼ 99% while achieving a much higher accuracy than the commonly used scaled gradient projection method for the wavelength misalignment problem.', 'arxivid': '2108.01269', 'author': ['Min Tao taom@nju.edu.cn ', 'Xiao-Ping Zhang ', 'Min Tao ', 'Xiao-Ping Zhang ', '\nDepartment of Mathematics\nDepartment of Electrical, Computer and Biomedical Engineering\nNational Key Laboratory for Novel Software Technology\nNan-jing University\n210093NanjingChina\n', '\nRyerson University\nM5B 2K3TorontoONCanada\n'], 'authoraffiliation': ['Department of Mathematics\nDepartment of Electrical, Computer and Biomedical Engineering\nNational Key Laboratory for Novel Software Technology\nNan-jing University\n210093NanjingChina', 'Ryerson University\nM5B 2K3TorontoONCanada'], 'corpusid': 256105797, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 26922, 'n_tokens_neox': 23411, 'n_words': 13515, 'pdfsha': 'b010a70e22aa413bee91d5b995dba49b7d432770', 'pdfurls': ['https://export.arxiv.org/pdf/2108.01269v2.pdf'], 'title': ['Unified Analysis on L 1 over L 2 Minimization for signal recovery', 'Unified Analysis on L 1 over L 2 Minimization for signal recovery'], 'venue': []}

arxiv

Random Matrix Spectral Form Factor of Dual-Unitary Quantum Circuits Jul 2021 B Bertini · P Kos · T Prosen Bruno Bertini bruno.bertini@physics.ox.ac.uk Pavel Kos pavel.kos@fmf.uni-lj.si Tomaž Prosen tomaz.prosen@fmf.uni-lj.si Bruno Bertini Pavel Kos Tomaž Prosen Theoretical Physics Faculty of Mathematics and Physics Oxford University Parks RoadOX1 3PUOxfordUK Faculty of Mathematics and Physics University of Ljubljana Jadranska 19, SI1000 Ljubl-janaSlovenia University of Ljubljana Jadranska 19, SI1000 Ljubl-janaSlovenia Random Matrix Spectral Form Factor of Dual-Unitary Quantum Circuits 16Jul 2021Received: date / Accepted: dateNoname manuscript No. (will be inserted by the editor)Quantum chaos · Spectral form factor · Random matrix theory · Floquet quantum circuits · Dual unitarity We investigate a class of local quantum circuits on chains of d−level systems (qudits) that share the so-called 'dual unitarity' property. In essence, the latter property implies that these systems generate unitary dynamics not only when propagating in time, but also when propagating in space. We consider space-time homogeneous (Floquet) circuits and perturb them with a quenched single-site disorder, i.e. by applying independent single site random unitaries drawn from arbitrary non-singular distribution over SU(d), e.g. one concentrated around the identity, after each layer of the circuit. We identify the spectral form factor at time t in the limit of long chains as the dimension of the commutant of a finite set of operators on a qudit ring of t sites. For general dual unitary circuits of qubits (d = 2) and a family of their extensions to higher d > 2, we provide an explicit construction of the commutant and prove that spectral form factor exactly matches the prediction of circular unitary ensemble for all t, if only the local 2-qubit gates are different from a SWAP (non-interacting gate). We discuss and partly prove possible extensions of our results to weaker (more singular) forms of disorder averaging, as well as to quantum circuits with time-reversal symmetry, and for computing higher moments of the spectral form factor. Introduction: Quantum chaos conjecture and many-body systems The ubiquitousness of random matrix theory (RMT) [1,2,3] descriptions for a diverse range of phenomena in Nature and Society is the example par excellence of effectiveness of mathematics. In quantum dynamical systems, the fact that the fluctuations in the spectra of unitary evolution operators can be described in terms of structureless ensembles of RMT, characterised solely by unitary and anti-unitary symmetries, has been identified as a defining property of quantum chaos. The presence of RMT spectral correlations has been related to Hamiltonian chaos of the corresponding limiting classical dynamical system via the so-called quantum chaos conjecture 1 (QCC) [4,5,6], while the absence thereof is linked to integrability or regularity of the corresponding classical motion via the Berry-Tabor conjecture [7,8]. A heuristic proof of QCC in terms of semiclassical periodic orbit theory has been a decades long tour de force [9,10,11], while a rigorous proof has so far been possible only in a rather restricted setting of completely connected quantum graphs [12]. In systems which lack a small parameter (e.g. an effective Planck's constant), such as extended (many-body) systems of locally interacting quantum spins or fermions, the mechanisms for the validity of QCC have remained obscure despite a plethora of empirical evidence, see e.g. Refs. [13,14,15,16,17]. For such systems, lacking any meaningful limiting classical chaotic dynamics, the agreement of spectral fluctuations with RMT can be considered as the most versatile definition of quantum chaos and as a robust empirical method for detection of quantum (non)integrability. Recently we proposed a rigorous methodology which lead to the first proof of the emergence of RMT spectral 2-point correlation functions in the thermodynamic limit, for a particular locally interacting chain of quantum spins 1/2 [18]. In this paper, we present a generalisation of such methodology and show that it can be extended to a much broader class of systems. In particular, we reformulate our approach in the general language of local quantum circuits -which are the standard minimal model of quantum many-body (extended) systems with local interactions [19,20,21] -and show that it leads to exact results (in the thermodynamic limit) whenever the "local gates" (the unitary matrices encoding the nearest-neighbour interactions of local quantum circuits) are dual-unitary, i.e. they generate unitary evolution in both time and space. The method described here applies to quantum circuits of qudits (or arbitrary spins) and can easily account for spatially inhomogeneous interactions. Moreover, in contrast to Ref. [18], here we focus on the generic case of systems without anti-unitary symmetries (like time-reversal) while only sketch the extension of the results to generic time-reversal invariant case. In summary, in this paper we identify the key mathematical steps for approaching the problem of characterizing quantum ergodicity [22,23] and quantum chaos [24,20,25] through spectral correlations in extended quantum spin lattice systems. Although our results hold in a specific setting (and in the thermodynamic limit only), they represent the first rigorous proof of the emergence of RMT behaviour in a class of extended quantum spin systems to the best of our knowledge. They pertain to both the case of quenched disorder and the clean limit, and provide the first proof of the QCC in the many-body realm. The rest of the paper is laid out as follows. In Sec. 2 we introduce the basic concepts and provide their definitions. In Sec. 3 we state and interpret our main results, while in Sec. 4 we elaborate the proofs. In Sec. 5 we discuss some straightforward extensions and generalisations of our results. While the treatment of spatially inhomogeneous dual-unitary interactions in Sec. 5.1 is rigorous, the extensions of the techniques to study fluctuations (5.2) and singular disorder distributions (5.3) are speculative at this point. Basic concepts Floquet quantum circuits In this work we consider a class of quantum many-body systems known as Floquet local quantum circuits. They consist of a set of 2L, L ∈ N, quantum variables with d internal states ("qudits"), that can be thought of as arranged on a 1-dimensional periodic lattice Λ L = 1 2 Z 2L . The Hilbert space of the system is H 2L = (H 1 ) ⊗2L = C N ,(1) where the "local Hilbert space" H 1 = C d is the Hilbert space of a single qudit and N = d 2L is the dimension of H 2L . In these systems the time evolution is discrete and generated by integer powers of the unitary operator U L := x∈ZL η x,L (U x,1 ) x∈ZL+ 1 2 η x,L (U x, 1 2 ) ,(2) conventionally called the "Floquet operator". In writing Eq. (2) we introduced the following definitions. (i) We indicated by η x,n : End(H 2 ) → End(H 2n ), with n ∈ N and x ∈ Λ n , the linear map defined by η x,n (O) := Π 2x−1 2n (O ⊗ ½ 2(n−1) )Π −(2x−1) 2n .(3) Here H n = (H 1 ) ⊗n denotes the Hilbert space of a periodic qudit chain of n sites, while Π n and ½ n designate respectively the periodic shift operator and the identity operator over H n . Explicitly we have Π n |j 1 ⊗ |j 2 ⊗ · · · |j n = |j n ⊗ |j 1 ⊗ · · · |j n−1 , with Π n n = ½ n , where R = {|j ; j = 0, . . . , d − 1},(5) is the canonical orthonormal basis of H 1 = C d . (ii) We introduced the function U ·,· : Λ L × 1 2 Z 2 → U(d 2 ), where U(N ) denotes the group of N × N unitary matrices. The operators U x, 1 2 , U x,1 ≡ U x,0 ∈ U(d 2 ) define the interaction among neighbouring qudits at sites x − 1 2 , x for half-odd integer and integer times respectively. These operators encode all physical information about the dynamics and will be referred to as the "local gates". This setting can be regarded as the simplest possible modelling for extended quantum many-body systems [20,21]. Indeed, it captures what can be considered the primary and most essential feature of an extended system, i.e., the locality of the interactions. It is precisely this feature that distinguishes an extended system from a single quantum variable with arbitrary many internal states, for example, a single (arbitrary high) spin. Note that even though local quantum circuits describe time-depended dynamics, they can also be thought of as approximations of time-independent local interactions obtained through the use of the Suzuki-Trotter decomposition [26,27]. Finally, we stress that this precise setting emerges naturally in the context of quantum simulation, for instance, through the use of the recently developed Google's Sycamore processor [28]. The time evolution generated by (2) admits the following convenient graphical representation U t L = 1 2 1 2 3 2 1 2 3 2 5 2 1 2 3 · · · L ≡ 0 x t τ . . . ,(6) where each local gate is represented by U x,τ ≡ U x,mod(τ,1) = , x ∈ Λ L , τ ∈ 1 2 Z,(7) and different shades illustrate distinct matrices. The function mod(x, n) indicates the remainder upon division by n. Note that leftmost and rightmost gates are connected because of periodic boundary conditions. Finally, we point out that the dynamics generated by (2) are time-reversal invariant if there exist a unitary operator K ∈ U(d 2L ) such that [1] KU L K † = U T L and K T = ±K ,(8) where (·) T denotes transposition in the canonical basis (5) and (·) † Hermitian conjugation. Symmetric and antisymmetric matrices correspond respectively to cases where the anti-unitary operator implementing time reversal on the Hilbert space squares to plus or minus one [1]. The first, "regular", kind of time-reversal symmetry emerges in physical systems with integer total angular momentum and is associated with orthogonal ensembles of RMT, while the second characterises systems with half-odd integer spin and is associated with symplectic ensembles [1]. Our Setting Here we consider local gates of the form U x+ 1 2 , 1 2 = (u x ⊗ u x+ 1 2 ) U = ,(9a)U x,1 = (w mod(x− 1 2 ,L) ⊗ w x ) W = , x ∈ Z L ,(9b) where U, W ∈ U(d 2 ) act non-trivially on a pair of neighbouring qudits and u x , w x ∈ U(d) on a single one (we hence represented them graphically as balls acting on a single wire). Therefore we have U L = 0 1 1 2 1 2 3 2 5 2 1 2 3 · · · .(10) In particular, it is immediate to see that, choosing local gates (9a)-(9b) with U = U T , W = W T , w x = u T x , ∀ x ∈ Λ L ,(11) the condition (8) is fulfilled with K = x∈ZL+ 1 2 η x,L (W ) = K T .(12) Namely, the dynamics generated by (10) are time-reversal invariant. Here we consider both the time-reversal-invariant and the non-time-reversal-invariant cases. We remark that in (9a)-(9b) we assumed the 2-site gates U, W to be the same for all x. In physical terms this means that we consider interactions that are homogeneous in space, while we allow for some position-dependent 'external fields' (encoded in the single-site gates u x , w x ). The extension of our results to fully inhomogeneous systems is discussed in Sec. 5.1. Spectral form factor The objective of this paper is the study of spectral statistics of the Floquet operator. Namely, we consider the distribution of the elements of the spectrum of the unitary Floquet matrix spect[U L ] = {e iϕj ; j = 1, 2 . . . , N } ,(13) where ϕ j -conventionally referred to as quasienergies -can be taken to be in [0, 2π). Considering spect[U L ] as a one-dimensional gas on the circle S 1 , we analyse its 2-point correlation functions, specifically, the spectral form factor (SFF) defined as K(t, L) := E |tr U t L | 2 = E   N j,j ′ =1 e i(ϕj−ϕ j ′ )t   , t, L ∈ N .(14) Here E[·] is an average over an ensemble of similar systems. The average is necessary to smear out the fluctuations of |tr U t L | 2 , which do not die out even in the limit of large L, and extract the universal behaviour. We shall see later that very mild forms of averaging are sufficient (we remind the reader that the results are most interesting in the limit of clean systems), specifically we will consider cases where u x and w x are i.i.d. for x ∈ Λ L densely covering an arbitrary small ball around the identity in SU(d). The SFF is directly connected to the Fourier transform of the quasi-energy 2-point function. Indeed, introducing the n-point function as ρ n (ϑ 1 , . . . , ϑ n ) := E   N j1 =... =jn=1 n k=1 δ(ϑ k − ϕ j k )   ,(15) the SFF can be expressed as K(t, L) = [0,2π] 2 dϑ 1 dϑ 2 e i(ϑ1−ϑ2)t ρ 2 (ϑ 1 , ϑ 2 ) + N .(16) Since this quantity measures correlations between quasienergy levels at arbitrary distance, it is very convenient to analyse extended systems for large volume L where neighbouring quasienergy levels become exponentially close in L and one has to look at correlations on larger scales. Spectral form factor for random unitary matrices Before moving to the analysis of (14) for local quantum circuits let us briefly recall our point of reference: the SFF of random unitary matrices. In this case the average E [·] in (14) is replaced by the integration over an ensemble of random unitary matrices of dimension N , i.e. K ens (t, N ) := |tr U t | 2 dµ ens (U),(17) and the result depends on the precise form of the measure dµ ens (U) of the ensemble considered. Specifically, in this paper we are interested in the two most common cases: (i) systems without anti-unitary symmetries, and (ii) systems with regular time-reversal symmetry (squaring to the identity). In these two cases the relevant ensembles of random matrices are two of Dyson's circular ensembles: the Circular Unitary Ensemble (CUE) and the Circular Orthogonal Ensemble (COE). The CUE measure is the invariant Haar measure over U(N ), while the COE measure is defined for symmetric unitary matrices and is uniquely specified by the property of being invariant under orthogonal transformations [1]. In these two cases the result reads as [29] K CUE (t, N ) = min(t, N ) , K COE (t, N ) = 2min(t, N )   1 − min(t,N ) m=1 1 2m + 2max(t, N ) − N − 1   . (19)(18) In particular, in the thermodynamic limit L → ∞ (N → ∞) they simplify to lim N →∞ K CUE (t, N ) = t, lim N →∞ K COE (t, N ) = 2t .(20) The main result of our paper is the proof that one recovers the r.h.s. of (20) by computing exactly the expression (14) for a broad class of Floquet quantum circuits. Spectral form factor of Floquet quantum circuits For local quantum circuits the SFF (14) can be represented diagrammatically as follows K(t, L) = E tr(U L ) t tr(U † L ) t = E ,(21) where we represented the trace in the forward time sheet (tr U t L ) using the diagram (10) and that in the backward time sheet (tr (U † L ) t ) by introducing U † = , W † = , u † x , w † x = .(22) Once again shades of the same colour denote different matrices. Note that top and bottom lines at the same positions within both sheets are connected because of the traces. Folding the backward sheet (blue) underneath the forward one (red) we write the folded circuit representation of the SFF K(t, L) = E ,(23) where we introduced "doubled" or thickened wires = ,(24) and "doubled" gates = = U ⊗ U * , = = W ⊗ W * ,(25)= = u x ⊗ u * x , w x ⊗ w * x . Here and in the following (·) * denotes complex conjugation in the canonical basis (5). Local disorder averaging As mentioned in Sec. 2.3 the definition of SFF requires an average. Since our interest is mainly on clean systems, we consider averages over onsite disorder that can be made arbitrary weak. This kind of disorder is arguably the most harmless form of disorder that one can introduce in the system because it does not couple different sites. In particular, we focus on the following generic model of on-site disorder where the local gates (9a)-(9b) are specified by fixed unitary interactions U, W ∈ U(d 2 ), and site-dependent local gates u x , w x ∈ SU(d) of the general form u x = e iθ0,x·σ , w x = e iθ1,x·σ T , x ∈ Λ L , θ ι,x ∈ R d 2 −1 .(26) The vector σ = (σ 1 , σ 2 , . . . , σ d 2 −1 ) is formed by Generalised Gell-Mann matrices σ a [30] (Pauli matrices for d = 2, Gell-Mann matrices for d = 3, etc.), the Hermitian generators of su(d), and σ T = (σ T 1 , σ T 2 , . . . , σ T d 2 −1 ) is the vector of the corresponding transposed generators. The expectation can be explicitly written in terms of a factorised measure as: E[f ] = f (θ) L−1 x=0 1 ι,ι ′ =0 g ιι ′ (θ ι,x+ ι ′ 2 )d d 2 −1 θ ι,x+ ι ′ 2 , θ ≡ (θ ι,x ) ι=0,1 x∈ΛL . (27) where g ιι ′ ∈ L 1 [R d 2 −1 ] are arbitrary probability densities of i.i.d. random variables θ ι,x . Note that distributions on integer (ι ′ = 0) and half-odd-integer (ι ′ = 1) sublattices are generally different. Space-time duality The key property of E[·] (27) is the factorization with respect to a spatial coordinate x. This means that, even though the diagram (23) cannot be thought of as the trace of the product of t transfer matrices in the time direction (because the average couples different time layers), it can be thought of as the trace of the product of L transfer matrices in the space direction. Specifically, K(t, L) = E E E E E . (28) In equations this is expressed as K(t, L) = E tr U t L tr U t L * = E tr (U L ⊗ U * L ) t = E tr L x=1Ũ t (x) ⊗Ũ t (x) * = tr E Ũ t ⊗Ũ * t L = tr T L ,(29) where the tensor product operates between the two different time sheets, and we introduced the following definitions: (i) "Dual" Floquet operator propagating in the space-direction over the Hilbert space H 2t of 2t qudits, explicitly depending on the position x ∈ Z L : U t (x) := τ ∈Zt+ 1 2 η τ,t (Ũ (u x− 1 2 ⊗ w T x− 1 2 )) τ ∈Zt η τ,t (W (w x ⊗ u T x )) .(30) HereŨ ,W ∈ End(H 2 ) are the "dual" 2-body interaction gates defined via the space-time duality mapping˜: End(H 2 ) → End(H 2 ). Specifically, for any O ∈ End(H 2 ) with matrix elements O i1i2,j1j2 = i 1 | ⊗ i 2 | O |j 1 ⊗ |j 2 ,(31) we defineÕ jl,ik := O ij,kl , i, j, k, l ∈ {0, 1, . . . , d − 1} .(32) We see thatŨ ij,kl andW ij,kl correspond to a particular reshuffling of the indices of U ij,kl and W ij,kl . (ii) SFF-transfer matrix: T := E Ũ t ⊗Ũ * t ∈ End(H 2t ⊗ H 2t ).(33) Note that T does not depend on position x due to the identical distribution of (θ ι, x− 1 2 , θ ι,x ) for all x ∈ Z L . More specifically, performing explicitly the average via (27), we find T = (Ũ ⊗Ũ * )O † 1 (W ⊗W * )O 0 ,(34) where we introduced U := τ ∈Zt+ 1 2 η τ,t (Ũ ) ,(35)W := τ ∈Zt η τ,t (W ) ,(36)O ι ′ := O 0ι ′ O 1ι ′ = O 1ι ′ O 0ι ′ ,(37)O ιι ′ := d d 2 −1 θ g ιι ′ (θ) exp (iθ · (M ι ⊗ ½ 2t − ½ 2t ⊗ M * ι )) .(38) Here M ι = (M 1,ι , M 2,ι , . . . , M d 2 −1,ι ) with M a,M a,ι := τ ∈Zt+ 1 2 ι σ a,τ , σ a,τ := Π 2τ 2t (σ a ⊗ ½ 2t−1 )Π −2τ 2t , ι ∈ {0, 1} .(39) Normalisability and non-negativity of the probability densities g ιι ′ imply the following important properties of the operators O ιι ′ : (a) O ιι ′ is a non-expansive mapping: O ιι ′ ≤ d d 2 −1 θ |g ιι ′ (θ)| exp (iθ · (M ι ⊗ ½ 2t − ½ 2t ⊗ M * ι )) = 1 . (40) (b) Let O ιι ′ |B = e iφ |B for some |B ∈ H 2t ⊗ H 2t and φ ∈ R. Then: φ = 0, and (M a,ι ⊗½ 2t −½ 2t ⊗M * a,ι ) |B = 0 , ∀a ∈ {1, 2, . . . , d 2 −1} . (41) Indeed, the assumptions imply e iθ·(M ι ⊗½2t−½2t⊗M * ι ) |B = e iφ |B ,(42) for a dense set S ∋ θ with positive measure, e.g. the support of g ιι ′ or its dense subset. Without loss of generality we can assume that S contains the origin 0 ∈ S. Taking a partial derivative ∂ ∂θa | θ=0 of (42) we obtain (41). As convenient examples we can consider a Gaussian measure g ιι ′ (θ) = d 2 −1 a=1 1 √ 2πν aιι ′ exp − 1 2 θ 2 a ν 2 aιι ′ ,(43) or a box-measure g ιι ′ (θ) = d 2 −1 a=1 1 2ν aιι ′ Θ(ν aιι ′ − |θ a |) ,(44) where choosing sufficiently small nonvanishing variabilities ν aιι ′ > 0 allows for arbitrary concentration of measure around the identity in SU(d), and hence description of an "almost clean system". Relevant limits For the class of dual-unitary circuits, which is the main focus of this paper and will be elaborated in the next section, the expression (29) in terms of the map T [which is, in fact, a vectorised form of a completely positive trace preserving and unital mapping over End(H 2t )] allows us to explicitly compute the SFF at any fixed t in the thermodynamic limit L → ∞. In particular, as we show below, we find that lim L→∞ K(t, L) = lim N →∞ K RMT (t, N ) = t, ∀t.(45) This fact is quite remarkable and signals a special property of the dualunitary systems considered here. Indeed, in generic quantum chaotic systems one typically observes that there exists a timescale t * (L) such that the universal behaviour described by RMT emerges only for times t > t * (L). This timescale, usually referred to as the Thouless (or Ehrenfest) time, is typically observed to grow monotonically with L [25,20,36]. The fact that for us, instead, t * (L) is strictly equal to zero can be interpreted as a sort of "critical chaotic" (scale-free) property of dual-unitary systems. It would certainly be desirable to address spectral correlations on other scales. For instance, those on the scale of mean level spacing (which becomes exponentially small in L for a many-body system) and correspond to times t of the order of Heisenberg time N = 2 L . This would require studying the scaling limit K(τ ) = lim L→∞ 2 −L K ⌊2 L τ ⌋, L .(46) At the moment, however, we do not foresee any method to rigorously attack this challenging issue. 3 Statement of the main results Exact SFF at large L for dual-unitary circuits Using the representation (29) we see that to compute the SFF one has to determine the spectrum of T. In particular, to obtain K(t, L) in the large size limit L → ∞, it is sufficient to find all the eigenvalues with maximal magnitude. To achieve this goal we consider a special class of local quantum circuits called dual-unitary circuits [31]. These systems are characterised by the property that their dual local gates (cf. (32)) are unitary. Specifically, we consider local gates U and W (cf. (9a, 9b)) that simultaneously fulfil U † U = U U † = ½,Ũ †Ũ =ŨŨ † = ½ ,(47)W † W = W W † = ½,W †W =WW † = ½ .(48) The above conditions admit non-trivial solutions for any local dimension d ≥ 2 [32,33]. A complete classification of solutions, however, has been achieved only for d = 2 [31]. An immediate consequence of (47) and (48) is that bothŨ (35) andW (36) are unitary. This allows us to prove the following Lemma: Lemma 1 For dual-unitary circuits the matrix T (33) fulfils the properties: (i) |λ| ≤ 1 for all λ ∈ spect(T). (ii) If T|A = e iφ |A , φ ∈ R, then (Ũ ⊗Ũ * ) · (W ⊗W * ) |A = e iφ |A , (M a,ι ⊗ ½ 2t − ½ 2t ⊗ M * a,ι )(W ⊗W * ) |A = 0 ,(49)(M a,ι ⊗ ½ 2t − ½ 2t ⊗ M * a,ι ) |A = 0, ι ∈ {0, 1}, a ∈ {1, 2, . . . , d 2 − 1} . (iii) For any unimodular eigenvalue λ, |λ| = 1, its algebraic and geometric multiplicities coincide (i.e. its Jordan blocks are trivial). In essence, (i) and (iii) mean that T is a linear non-expansive mapping over H 2t ⊗ H 2t , while (ii) suggests that computation of invariant subspaces can be reduced to simpler algebraic problems. Indeed, it ensures that for dual-unitary circuits N (φ) := 1 L L ℓ=1 e −iφℓ K(t, ℓ)(50) approaches a finite value when L → ∞ which is given by the number of linearly independent solutions |A of the system of equations (49). The number of solutions for φ = 0 give the SFF averaged over the system size, while showing that φ = 0 is the only phase for which there are nontrivial solutions gives the thermodynamic limit lim L→∞ K(t, L). In order to further simplify the conditions (49) we introduce a vectoroperator isomorphism H⊗H vo ←→ End(H). Specifically, we define in the canonical basis (5): |j ⊗ |j ′ vo ←→ |j j ′ | .(51) This implies that the problem of finding all linearly independent states |A solving (49) is mapped to the one of finding all linearly independent operators A satisfying, for all a ∈ {1, 2, . . . , d 2 − 1} and ι ∈ {0, 1}: UWAW †Ũ † = e iφ A, [M a,ι , A] = 0, [W † M a,ιW , A] = 0 .(52) The above conditions can be simplified further by making use of an explicit parametrisation of a dual-unitary matrices. Specifically, we parametrise the matrix D ∈ End(C d ⊗ C d ) fulfilling DD † = D † D = ½ andDD † =D †D = ½ as follows D = (u 1 ⊗ u 2 )Se iJs3⊗s3 (u 3 ⊗ u 4 ), J ∈ [0, π],(53) where u j ∈ U(d) are arbitrary unitary matrices ('local gates') and S ∈ End(C d ⊗ C d ) is the SWAP operator defined as S |j 1 ⊗ |j 2 = |j 2 ⊗ |j 1 ∀j 1 , j 2 ∈ {0, 1, . . . d − 1} .(54) Finally, here and in the following s 1 , s 2 , s 3 designate the 'spin matrices' car- rying d−dimensional irreducible representation of SU (2) over H 1 , satisfying [s a , s b ] = i 3 c=1 ǫ abc s c ,(55) where ǫ abc is the three-dimensional Levi-Civita tensor, and we choose s 3 to be diagonal in the canonical basis (5) s 3 = diag − d − 1 2 , − d − 3 2 , . . . , d − 1 2 .(56) Local embeddings into End(H 2t ) are, like in (39), defined as s a,τ := Π 2τ 2t (s a ⊗ ½ 2t−1 )Π −2τ 2t .(57) Note that for d = 2 the parametrisation (53) exhausts all dual-unitary circuits [31] while for d > 2 it characterises a physically interesting sub-class [33]. Plugging (53) in the definitions (35)-(36) we find U = e iθ e iα0·M 0 e iα1·M 1Ṽ e iβ 0 ·M0 e iβ 1 ·M 1 , W = e iθ ′ e iγ 0 ·M 0 e iγ 1 ·M 1Ṽ′ e iδ0·M 0 e iδ1·M1 ,(58) where α ι , β ι , γ ι , δ ι ∈ R d 2 −1 , θ, θ ′ ∈ R, and we introduced V := (Se iJs3⊗s3 ) ⊗t ,(59)V ′ := Π 2t (Se iJ ′ s3⊗s3 ) ⊗t Π † 2t . We are then able to simplify the conditions for the existence of unimodular eigenvalues and write their invariant eigenoperator spaces in terms of a simple algebraic commutant: Lemma 2 For J, J ′ = 0 the conditions (52) cannot be met unless φ = 0. In this case, they are equivalent to [A, M a,ι ] = 0, [A, M ab,ι ] = 0 , a, b ∈ {1, 2, . . . , d 2 − 1}, ι ∈ {0, 1} . (60) Here we introduced the 2-site magnetization operators of the even and odd spin sub-lattices M ab,ι := τ ∈Zt+ 1 2 ι σ a,τ σ b,τ + 1 2 , ι ∈ {0, 1} .(61) As a corollary of Lemma 1 and Lemma 2, we can express the SFF in the limit L → ∞ in terms of the dimension of the eigenspace of eigenvalue 1, which in turn (Lemma 2) equals the dimension (in End(H 2t )) of the commutant M ′ of the set M := {M a,ι } a,ι ∪ {M ab,ι } a,b,ι .(62) Namely, lim L→∞ K(t, L) = dim M ′ .(63) In fact, M ′ can be completely characterised: Theorem 1 The commutant M ′ is the span of the representation of the cyclic group C t of even-site translations on a periodic chain of 2t spins: M ′ = span{Π 2τ 2t ; τ = 0, 1, . . . t − 1} .(64) Hence, we arrive at the following corollary, which summarises our first main result Corollary 1 For local quantum circuits (2) with local gates of the form (9a,9b) and U = (u 1 ⊗ u 2 )Se iJs3⊗s3 (u 3 ⊗ u 4 ), W = (u ′ 1 ⊗ u ′ 2 )Se iJ ′ s3⊗s3 (u ′ 3 ⊗ u ′ 4 ) (65) where u j , u ′ j ∈ U(d) and J, J ′ = 0, the SFF (14) averaged according to the measure (27) fulfils lim L→∞ K(t, L) = t .(66) This is precisely the CUE result for all times. It is remarkable that 2-point spectral correlations of dual-unitary circuits agree with RMT at all scales. Note that the restriction J, J ′ = 0 for the validity of the statement is not surprising. Indeed, for J = 0 the gate U does not encode interactions among the qudits, they are evolved in an entirely independent fashion (an analogous conclusion holds concerning W for J ′ = 0). This means that if one of J and J ′ is equal to zero not all the qudits are coupled by the dynamics and one cannot expect U L in Eq. (2) to behave like a random matrix on the whole Hilbert space. Results on SFF at large L for T-symmetric dual-unitary circuits To obtain circuits with time-reversal symmetry one has to choose local gates U, W ∈ U(d 2 ) and on-site disorder u x , w x ∈ U(d) which are compatible with the conditions (11). A generic choice is U = U T , W = W T ,(67) and gates u x , w x of the form u x = e iθx·σ , w x = e iθx·σ T = u T x , x ∈ Λ L , θ x ∈ R d 2 −1 .(68) where now θ x is the same in both u x and w x . The expectation can again be explicitly written in terms of a factorised measure as: E T [f ] = f (θ) L−1 x=0 1 ι ′ =0 g ι ′ (θ x+ ι ′ 2 )d d 2 −1 θ x+ ι ′ 2 , θ ≡ (θ x ) x∈ΛL .(69) where g ι ′ ∈ L 1 [R d 2 −1 ] is a pair of arbitrary probability densities of i.i.d. random variables θ x on integer (ι ′ = 0) and half-odd-integer (ι ′ = 1) sublattices. Considering local gates fulfilling the conditions (67, 68) and averaging according to the measure E T [·], one can repeat the reasoning of Sec. 2.7 and conclude K T (t, L) = E T tr U t L tr U t L * = tr T L T ,(70) with T T := (Ũ ⊗Ũ * )O † T,1 (W ⊗W * )O T,0 .(71) HereŨ, andW are defined as in Eqs. (35,36) while time-reversal symmetric averaging operator O T,ι ′ reads as O T,ι ′ := d d 2 −1 θ g ι ′ (θ) exp (iθ · (M ⊗ ½ 2t − ½ 2t ⊗ M * )) .(72) Here Assuming that, together with the conditions (67, 68), the local gates also fulfil (47, 48) (i.e. they are dual-unitary), and noting that the averaging operator also satisifies the properties (40,41), one can immediately write an analogue of the Lemma 1 for the transfer matrix T T (its proof 4.1 carries over). Namely, one can prove that T T is again a linear non-expansive mapping over H 2t ⊗H 2t , and its eigenvectors corresponding to unimodular eigenvalues are determined by the conditions (49) with M a,ι replaced by M a . Following the logic of Sec. 3.1, one can map the problem of finding all linear independent eigenvectors of T T corresponding to unimodular eigenvalues to a simpler algebraic problem. In this case the equivalent algebraic problem is to find all linearly independent operators A fulfilling UWAW †Ũ † = e iφ A, [M a , A] = 0, [W † M aW , A] = 0 .(74) for some φ ∈ R and all a ∈ {1, 2, . . . , d 2 − 1}. Using the explicit from (53) of the 2-site dual-unitary gates we can again simplify the conditions (74) and write their invariant eigenoperator spaces in terms of simple algebraic commutants. In this case, to lift some technical complications, we specialise the treatment to the case of qubits (d = 2): Lemma 3 For J, J ′ = 0 and d = 2 the conditions (74) cannot be met unless φ = 0. In this case, they are equivalent to [A, M a ] = 0, [A, M ab,ι + R 2t M ab,ι R 2t ] = 0 , a, b ∈ {1, 2, 3} , ι ∈ {0, 1} , (75) where the 2-site magnetization operators of the even and odd spin sub-lattices are defined in (61) and R 2t is the reflection of the time lattice Λ t around the centre, i.e. R 2t |j 1 ⊗ |j 2 ⊗ · · · |j 2t = |j 2t ⊗ · · · |j 2 ⊗ |j 1 . As before, this lemma allows us to express the SFF in the limit L → ∞ in terms of the dimension (in End(H 2t )) of the commutant M ′ T of the set M T := {M a } a ∪ {M ab,ι + R 2t M ab,ι R 2t } a,b,ι .(77) Namely, lim L→∞ K T (t, L) = dim M ′ T .(78) The commutant M ′ T can again be completely characterised by proving a statement analogous to Theorem 1, which we present here without proof. Conjecture 1 The commutant M ′ T is the linear span of the representation of the dihedral group D t , i.e. the symmetry group of a polygon of t vertices, on a periodic chain of 2t spins: M ′ T = span{R n 2t Π 2τ 2t ; τ = 0, 1, . . . t − 1, n = 0, 1} .(79) The number of independent elements of the dihedral group is established by: Lemma 4 The number of linearly independent elements in the representation of D t in H t formed by {R n 2t Π 2τ 2t } n=0,1 τ =0,1,...t−1 is 2t. Hence, we arrive at the following corollary: Corollary 2 For local quantum circuits (2) with d = 2, local gates of the form (9a,9b), and U = (u 1 ⊗u 2 )Se iJs3⊗s3 (u T 1 ⊗u T 2 ), W = (u ′ 1 ⊗u ′ 2 )Se iJ ′ s3⊗s3 (u ′ T 1 ⊗u ′ T 2 ),(80) where u j , u ′ j ∈ U(2) and J, J ′ = 0, the SFF (14) averaged according to the measure (69) fulfils lim L→∞ K T (t, L) = 2t .(81) This is precisely the COE result for all times. Once again we see that 2-point spectral correlations of dual-unitary circuits agree with RMT at all scales. The proof of Conjecture 1 follows the same ideas as that of Theorem 1 but is technically more involved and will be presented elsewhere [34]. A similar proof, for the special case of the time-reversal symmetric self-dual kicked Ising model, has been presented in the supplemental material of Ref. [18]. Proofs Proof of Lemma 1 As a consequence of properties (40,41) the spectra of O ιι ′ belong to the open unit disk with 1 attached to it, D 1 = {z ∈ C; |z| < 1}∪{1} ⊃ spect(O ιι ′ ). Since O 0ι ′ and O 1ι ′ commute, spect(O ι ′ = O 0ι ′ O 1ι ′ ) ⊂ D 1 , and all eigenvectors |R of O ι ′ with unique unimodular eigenvalue 1 are characterised by (M a,ι ⊗ ½ 2t − ½ 2t ⊗ M * a,ι ) |R = 0, ι ∈ {0, 1}, a ∈ {1, 2, . . . , d 2 − 1} . (82) Since M a,ι ⊗ ½ 2t − ½ 2t ⊗ M * a,ι are Hermitian, exactly the same conditions uniquely characterise the eigenvalue 1 eigenvectors of O † ι ′ . We now turn to SFF transfer matrix (34) and write T † T = O † 0 (W ⊗W * ) † O 1 O † 1 (W ⊗W * )O 0 .(83) Let |A be a normalised eigenvector of T associated to the eigenvalue λ. Considering the expectation value of (83) we have, since Finally, we prove the last point by contradiction: assuming that the eigenvalue λ corresponds to a non-trivial Jordan block, there must exist a normalised vector |B such that B| O † ι ′ O ι ′ |B ≤ B|B , B| O ι ′ O † ι ′ |B ≤ B|B , for any |B : |λ| 2 = A|T † T|A = A|O † 0 (W ⊗W * ) † O 1 O † 1 (W ⊗W * )O 0 |A ≤ A|O † 0 (W ⊗W * ) † (W ⊗W * )O 0 |A = A|O † 0 O 0 |A ≤ 1 ,(84)T |B = λ |B + α |A , α = 0 ,(85) where |A is the eigenvector corresponding to the eigenvalue λ (where we can choose A|B = 0). Reasoning as above we have B|T † T|B = 1 + |α| 2 ≥ 1 ,(86) which is a contradiction. Proof of Lemma 2 Plugging (58) into the first condition (52) and using the second condition to commute α · M ι around A we bring the conditions (52) to the following form VṼ ′ AṼ ′ †Ṽ † = e iφ A, [M a,ι , A] = 0, [Ṽ ′ † M a,ιṼ ′ , A] = 0,(87) where a ∈ {1, 2, 3}, ι ∈ {0, 1}. Let us now consider more closely the operator in the last relation. Considering for example ι = 0 sublattice and a combination of generators which yields the first spin matrix s 1 = α · σ, V ′ † (α·M 0 )Ṽ ′ = τ ∈Zt exp −iJ ′ s 3,τ − 1 2 s 3,τ s 1,τ − 1 2 exp iJ ′ s 3,τ − 1 2 s 3,τ ,(88) where we used S † (½ ⊗ s a )S = s a ⊗ ½. Resolving the identity in an eigenbasis of s 3 we find V ′ † (α·M 0 )Ṽ ′ = τ ∈Zt s 1,τ − 1 2 cos(J ′ s 3,τ ) + s 2,τ− 1 2 sin(J ′ s 3,τ ).(89) Next, we consider V ′ † (α·M 0 )Ṽ ′ − e i π 2 α·M0Ṽ′ † (α·M 0 )Ṽ ′ e −i π 2 α·M0 = = 2 τ ∈Zt s 2,τ − 1 2 sin(J ′ s 3,τ ).(90) Furthermore, since the adjoint representation of SU(d) is irreducible, we can for any non-vanishing vector β ∈ R d 2 −1 , and b ∈ {1, . . . , d 2 − 1} find a vector γ ∈ R d 2 −1 , such that e iγ·M ι (β · σ τ + 1 2 ι )e −iγ·M ι = σ b,τ + 1 2 ι , τ ∈ Z t .(92) This means that, conjugating the operators on r.h.s. of (90) with appropriate γ · M ι on integer (ι = 0) and half-odd integer (ι = 1) spin sub-lattices independently, we can produce any operator of the form M ab,1 (cf. (61)). Since A commutes with M a,ι , we have for J ′ = 0: [A, M ab,1 ] = 0, a, b ∈ {1, 2, . . . , d 2 − 1}.(93) To obtain an analogous statement for M ab,0 we first note that combining the first and last relation of (87) yields [ṼM a,0Ṽ † , A] = 0 , a ∈ {1, 2, . . . , d 2 − 1} .(94) Proceeding as before we find V(α·M 1 )Ṽ † = τ ∈Zt cos(Js 3,τ )s 1,τ + 1 2 − sin(Js 3,τ )s 2,τ + 1 2 ,(95) and e i π 2 α·M 1Ṽ (α·M 1 )Ṽ † e −i π 2 α·M 1 −Ṽ(α·M 1 )Ṽ † = 2 τ ∈Zt sin(Js 3,τ )s 2,τ + 1 2 . (96) Assuming J = 0, we can repeat the reasoning after (90) and find: [A, M ab,0 ] = 0, a, b ∈ {1, 2, . . . , d 2 − 1}.(97) Now we note that V and V ′ can be written in terms of double magnetisations (61). This can be seen by observing that d 2 −1 a=1 σ a ⊗ σ a(98) is the quadratic Casimir operator of the representation of SU(d) over C d ⊗ C d . Therefore, we must have d 2 −1 a=1 σ a ⊗ σ a = c + ½ + S 2 + c − ½ − S 2 ,(99) for some c ± ∈ R. Indeed, the symmetric and antisymmetric subspaces of C d ⊗ C d contain irreducible representations. Fixing the constants using the explicit form of {σ a } (see, e.g., Ref. [30]) we find c ± = ±2 − 2/d. Using S = e −i π 2 e i π 2 S ,(100) and writing s 3 = a γ a σ a we finally find V = e −i π 2 t exp i π 2 τ ∈Zt S τ exp iJ ab γ a γ b M ab,0 = e −i π 2d t(1−d) exp i π 4 d 2 −1 a=1 M aa,0 exp iJ ab γ a γ b M ab,0 ,(101)V ′ = e −i π 2 t Π 2t exp i π 2 τ ∈Zt S τ exp iJ ′ a,b γ a γ b M ab,0 Π † 2t = e −i π 2d t(1−d) exp i π 4 d 2 −1 a=1 M aa,1 exp iJ ′ ab γ a γ b M ab,1 ,(102) where we introduced S τ := ½ 2τ −1 ⊗ S ⊗ ½ 2t−2τ −1 .(103) Equations (93), (97), (101) and (102) imply that if J, J ′ = 0 VṼ ′ AṼ ′ †Ṽ † = A ,(104) so that the first of conditions (87) can be fulfilled only for φ = 0 and in that case it follows from (93), (97), and the second of (87). This proves the Lemma. Proof of Theorem 1 To prove the Theorem we use of the following Lemma. Our statement (Theorem 1) follows from a simple combination of Lemma 5 (which is proven later below) and the Schur's Lemma. If some A fulfils the conditions (60), it commutes with all elements of the algebra K generated by {M a,ι } and {M ab,ι }. Since the representations of the algebra K in the eigenspaces W k are irreducible and inequivalent, Schur's Lemma implies A = t−1 k=0 c k Q k ,(105) where c k ∈ C are arbitrary coefficients and Q k := 1 t t−1 τ =0 e 2πiτ k/t Π 2τ 2t , k ∈ {0, . . . , t − 1},(106)are orthogonal projectors on {W k } t−1 k=0 , i.e. Q k Q k ′ = δ k,k ′ Q k . This proves that A is a linear combination of t cyclic translations Π 2τ 2t , i.e. K ′ = M ′ = span{Π 2τ 2t ; τ = 0, 1 . . . , t − 1}, concluding the proof of Theorem 1. Proof of Lemma 5 We begin by introducing the following shorthand notation S ± := τ ∈Λt s ±,τ s 3,τ + d − 1 2 ,(107) R ±,n := τ ∈Λt s 3,τ + d − 1 2 s n ±,τ + 1 2 ,(108) T ±,n := τ ∈Λt s 3,τ − d − 1 2 s n ±,τ − 1 2 ,(109)M ±∓ := τ ∈Λt s ±,τ s ∓,τ + 1 2 ,(110)Z ι := t∈Zt s 3,τ + 1 2 ι ,(111) where n ∈ {1, . . . , d − 1} and (61)). s ±,τ = s 1,τ ± is 2,τ √ 2 , τ ∈ Λ t ,(112) In addition, we also introduce the set of vectors (states) in H t S k :=                |0 ⊗2t ∪ {|n 0 ; n ∈ I} ∪ {|n, ν, m ℓ,0 ; n, m ∈ I, ν ∈ J ℓ−2 , 2 ≤ ℓ ≤ 2t} k = 0 {|n k ; n ∈ I} ∪ {|n, ν, m ℓ,k ; n, m ∈ I, ν ∈ J ℓ−2 , 2 ≤ ℓ ≤ 2t} k ∈ {1, . . . ,2t−1}(113) where we defined the sets I := {1, . . . , d − 1}, (114) J := {0, 1, . . . , d − 1},(115) and the vectors |n k := 1 √ 2t 2t−1 j=0 e i πk t j Π j 2t s n +,0 |0 ⊗2t ,(116)|n 1 , 0 · · · 0 ℓ1−1 , n 2 , · · · , n a , 0 · · · 0 ℓa−1 , n a+1 ℓ,k := 1 √ 2t 2t−1 j=0 e i πk t j Π j 2t s n1 +,0 s n2 +, ℓ 1 2 · · · s na +, ℓ−ℓa 2 s na+1 +, ℓ−1 2 |0 ⊗2t ,(117)with {ℓ j } a j=1 ⊂ {1, . . . , 2t − 1} fulfilling N ∋ ℓ := 1 + a j=1 ℓ j ≤ 2t .(118) We note that the 'empty' state |0 ∈ R (cf. (5)) satisfies s 3 |0 = − d − 1 2 |0 ,(119) (cf. (56)). The integer ℓ shall be referred to as the length of the states (117) (one can verify that ℓ ≥ 2), while the states (116) have conventionally a unit length. For each value of k the set (113) contains (d − 1) 2 d ℓ−2 states for every length ℓ ≥ 2 and d − 1 states for ℓ = 1. Note that for each k the states in (113) have momentum πk/t, i.e. Π 2t |n, ν, m ℓ,k = e −i πk t |n, ν, m ℓ,k .(120) The set S k is complete in V k -the eigenspace of single-site shift Π 2t associated with momentum πk/t -but are not all linearly independent: While for ℓ < t the states are clearly orthonormal, some of the states with ℓ ≥ t can be represented by a string with a shorter length or they have multiple representations with the same length. One can then construct a basis B k of V k by extracting from S k the maximal subset of linearly independent vectors. Here we want to prove the following (Lemma 5): The representation of the algebra K generated by {M a,ι } and {M ab,ι } is irreducible in W k ≡ span(B k ∪ B k+t ), k ∈ {0, 1, . . . , t − 1} ,(121) specifically in the eigenspace of (Π 2t ) 2 corresponding to the eigenvalue e −2πik/t . Moreover, the irreducible representations in different W k are inequivalent. Noting that W k are closed under the action of K (all generators commute with (Π 2t ) 2 ) we have that the following three requirements imply the statement of the Lemma: (1) All vectors in B k are mapped into one another by elements of the algebra K. (2) There is an element of K mapping |1 k and |1 k+t one into another. (3) There is no unitary matrix C such that (Z 1 ) k = C(Z 1 ) p C † ,(122)(Z 0 ) k = C(Z 0 ) p C † ,(123)(M 2 +− ) k = C(M 2 +− ) p C † ,(124) where (·) p denotes the projection to W p , if p = k. Proof of (1). We prove the statement by showing the validity of a sufficient condition: all states (113) are mapped into one-another by elements of the algebra K. This condition is sufficient because the elements of B k are a subset of the states (113). We begin by proving that one can map |1 k into every state (113). First, we note that using S ± |n k = n |n ± 1 k (125) we can map |1 k to all states |n k with n ∈ {2, . . . , d − 1} and, for k = 0, also to |0 ⊗2t . Next, we observe that using R +,1 |n k = n |n, 1 2,k and then repeatedly applying S + |n, m k = n |n + 1, m 2,k + m |n, m + 1 2,k we can map |1 k into every state |n, m 2,k with n, m ∈ {1, . . . , d − 1}. We proceed using an inductive argument. Assuming that we can access every state |m, ν, m ℓ ′ ,k of length ℓ ′ < ℓ we shall prove that we can access every state of length ℓ, for ℓ ≥ 3. This follows straightforwardly from the relations R +,1 |n, ν, m ℓ−1,k ≃ m |n, ν, m, 1 ℓ,k ,(128)M −+ |n, ν, 1 ℓ−1,k ≃ |n, ν, 0, 1 ℓ,k ,(129) where ≃ denotes equality up to states of length < ℓ which can be accessed by assumption, and the repeated application of S + |n, ν, m, m 2 ℓ,k ≃ m 2 |n, ν, m, m 2 + 1 ℓ,k , S + |n, ν, 0, m 2 ℓ,k ≃ m 2 |n, ν, 0, m 2 + 1 ℓ,k . In Eq. (130), ≃ denotes equality up to states of the form |n ′ , ν ′ , m ′ , m 2 ℓ,k that are accessed at the previous step (n, ν, and m are arbitrary). Analogously in Eq. (131), ≃ denotes equality up to states of the form |n ′ , ν ′ , 0, m 2 ℓ,k . This means that for every state |n, ν, m ℓ,k there exist an operator B n,ν,m ∈ K such that B n,ν,m |1 k = |n, ν, m ℓ,k . Then we can construct an operator mapping the arbitrary vector |n ′ , ν ′ , m ′ ℓ ′ ,k into the arbitrary vector |n, ν, m ℓ,k A n,ν,m;n ′ ,ν ′ ,m ′ = B n,ν,m |1 k k 1| B † n ′ ,ν ′ ,m ′ .(133) To prove that this operator is in K we fist note that, since the generators are Hermitian, we have that if B n,ν,m ∈ K also B † n,ν,m ∈ K. We then just need to prove that |1 k k 1| ∈ K. This is explicitly done by observing |1 k k 1| = (S − ) d−2 (S + ) d−2 T 2t−1 −,d−1 R 2t−1 +,d−1 k 1|(S − ) d−2 (S + ) d−2 T 2t−1 −,d−1 R 2t−1 +,d−1 |1 k , k = 0 ∨ d = 2 , (134) |1 k k 1| = T 2t−2 −,1 R 2t−2 +,1 k 1|T 2t−2 −,1 R 2t−2 +,1 |1 k , k = 0 ∧ d = 2 . (135) Proof of (2). This point is immediate. Indeed, one can directly verify that (Z 0 − Z 1 ) |1 k = |1 k+t .(136) Proof of (3). The statement is trivial whenever k or p are 0. Indeed, |0 ⊗2t is the only eigenstate of Z 0 + Z 1 corresponding to eigenvalue 2t and does not appear for p = 0. This means that the sum of (122) and (123) can never be satisfied. To prove (3) for k, p = 0 we note that |1 (0) k := |1 k + |1 k+t √ 2 ,(137) is the only eigenstate of Z 1 and Z 0 with eigenvalues t and t − 2 respectively. This means that (122) and (123) can be fulfilled only if |1 (0) k is an eigenstate of C. In turn, this implies that 1|(M +− ) 2 |1 (0) (0) k k = e −2iπk/t ,(138) is invariant under the mapping implemented by C. Since e −2iπk/t = e −2iπp/t for k = p ∈ {1, . . . , t − 1} we conclude that there can be no transformation C fulfilling (123). Proof of Lemma 3 We consider d = 2, where s a = 1 2 σ a , a ∈ {1, 2, 3}, and begin by writing the analogue of (87). To this aim we plug the form (53) in the definitions (35,36) and use the constraints (67, 68) to find U = e iθ e iα·MṼ e iβ·M ,W = e iθ ′ e iγ·MṼ′ e iδ·M ,(139) where α, β, γ, δ ∈ R 3 , θ, θ ′ ∈ R, M = (M 1 , M 2 , M 3 ), whileṼ andṼ ′ are defined in (59). Substituting now (139) into the first condition (74) and using the second condition to commute α·M around A we find the desired analogue of (87) VṼ ′ AṼ ′ †Ṽ † = e iφ A, [M a , A] = 0, [Ṽ ′ † M aṼ ′ , A] = 0, a ∈ {1, 2, 3} . (140) Next we consider V ′ † M 1Ṽ ′ =Ṽ ′ † M 1,1Ṽ ′ +Ṽ ′ † M 1,0Ṽ ′ = sin(2J ′ )(M 32,1 + M 23,1 ) + cos(2J ′ )M 1 ,(141)VM 1Ṽ † =ṼM 1,1Ṽ † +ṼM 1,0Ṽ † = − sin(2J)(M 32,0 + M 23,0 ) + cos(2J)M 1 .(142) where we used (89) and (95) specialised to the case d = 2. From these relations we see that, for J, J ′ = 0, A commutes with {M 32,ι + M 23,ι } ι=0,1 . Using the second of (140) to make generic SU(2) rotations we find that actually A commutes with the following 10 operators In fact, A commutes also with {M aa,ι }. To show that we consider the following objects Pῑ(P ι Pῑ) t 2 −1 (M 11,ι − M 33,ι )(P ι Pῑ) 1− t 2 P −1 ι = −S 3 M 11,ι − M 33,ι , t even (144) (P ι Pῑ) t−1 2 (M 11,ι − M 33,ι )(P ι Pῑ) − t−1 2 = −S 3 M 11,ῑ + M 33,ῑ , t odd(145) for ι ∈ {0, 1},ῑ := 1 − ι, and we defined S a = σ ⊗2t a = i 2t e i π 2 Ma , P ι = e i π 4 (M11,ι−M22,ι) .(146) The left hand sides of (144) and (145) commute with A by construction, therefore we find [A, S 3 M 11,ι + M 33,ι ] = 0, t even (147) [A, S 3 M 11,ι − M 33,ι ] = 0, t odd.(148) Using that A commutes with {S a } a=1,2,3 and with the operators (143) we then have [A, (S a + ½ t )M bb,ι ] = (S a + ½ t )[A, M bb,ι ] = 0, t even (149) [A, (S a − ½ t )M bb,ι ] = (S a − ½ t )[A, M bb,ι ] = 0, t odd .(150) At this point we observe spect(S 1 + S 2 + S 3 ) = {(−1) t 3, (−1) t+1 } ,(151) where the multiplicities (both algebraic and geometrical) of the two eigenvalues are 2 2t−2 and 3 · 2 2t−2 respectively. Thus, by summing (149, 150) over a we obtain that the commutators are multiplied by invertible operators. Therefore, we finally arrive at [A, M bb,ι ] = 0 , b ∈ {1, 2, 3} .(152) Putting it all together we find [A, M ab,ι + R 2t M ab,ι R 2t ] = 0 .(153) At this point, consideringṼṼ ′ and using ((101), (102)) we havẽ VṼ ′ AṼ ′ †Ṽ † = A ,(154) so that the first of conditions (140) can be fulfilled only for φ = 0, and in that case it follows from (152), and the second of (140). This proves the Lemma. Proof of Lemma 4 To prove the statement we note that the set of operators {R n 2t Π 2τ 2t } n=0,1 τ =0,1,...t−1 can be written as Π 2τ 2t = t−1 k=0 e −2πiτ k/t Q k , R 2t Π 2τ 2t = t−1 k=0 e −2πiτ k/t Q ′ k ,(155) where Q k are the orthogonal projectors defined in Eq. (106) and we introduced Q ′ k := 1 t t−1 τ =0 e 2πiτ k/t R 2t Π 2τ 2t , k ∈ {0, . . . , t − 1}.(156) Since the mapping between {R n 2t Π 2τ 2t } n=0,1 τ =0,1,...t−1 and {Q k , Q ′ k } k=0,1,...t−1 is invertible it is sufficient to prove that the latter operators are linearly independent. To this aim we note that {Q k , Q ′ k } k=1,...t−1 are obviously linearly independent. This can be seen by writing them in a basis of eigenstates of Π 2t and noting distinct operators are non zero on distinct, non-overlapping, blocks. Moreover, noting that all {Q k , Q ′ k } k=1,...t−1 are zero when reduced to the block of zero two-momentum (i.e. the one composed by eigenstates of Π 2t with eigenvalues 1 and −1), we have that the only two operators which can be linearly dependent are Q 0 and Q ′ 0 . To prove that such operators are independent we note that Q 0 |n 0 = Q ′ 0 |n 0 = |n 0 , Q 0 |n t = −Q ′ 0 |n t = |n t , n ∈ I ,(157) where |n 0 , |n t are defined in Eq. (113) and the set I in (114). This implies that αQ 0 + βQ ′ 0 = 0(158) only if α = β = 0 and concludes the proof. Discussion of the results and their possible extensions In this section we discuss some generalisations and extensions of our results. While the extension to inhomogeneous interactions in Sec. 5.1 is rigorous, the other two subsections discussing fluctuations of SFF (5.2) and singular disorder distributions (5.3) are currently of speculative nature. Spatially inhomogeneous interactions The space-time duality approach adopted in this paper treats separately each point in space and it is therefore convenient to study general inhomogeneous interactions. However, our results as elaborated in Sec. 3 are not directly applicable to this case. Here we explain how to extend them focussing for simplicity on the case of no time reversal symmetry. We begin by observing that for position-dependent local gates, Eq. (29) is substituted with K(t, L) = tr (T 1 T 2 · · · T L ) , where T x is defined as in Eq. (34) with O given in Eq. (38) whileŨ andW are replaced byW x := τ ∈Zt η τ,t (W x+ 1 2 ),(160)U x := τ ∈Zt+ 1 2 η τ,t (Ũ x ).(161) Then we choose dual-unitary gates U x , W x of the form (53), assuming J x , J ′ x = 0 for all x. We use Lemma 1, Lemma 2, and Theorem 1 to find K(t, L) = t + tr (R 1 R 2 · · · R L ) ,(162) where R x = (½ − P)T x (½ − P) and P is the projector onto the eigenspace of T x associated to the eigenvalue 1. In writing (162) we used that, due to Lemma 1, Lemma 2, and Theorem 1, such eigenspace is the same for all x. From (162) we see that to recover the result (66) we need to show that lim L→∞ tr (R 1 R 2 · · · R L ) = 0 . For example this would hold if R x < 1, where · denotes the operator norm. The results of Sec. 3, however, are not sufficient to infer (163). To overcome this problem, we focus on the case of L even and make the following different replacement T x−1 T x = P +R x−1,x ⇒ K(t, L) = t + tr(R 1,2R3,4 · · ·R L−1,L ) . (164) whereR x−1,x = (½ − P)T x−1 T x (½ − P).(165) For the new operatorR x−1,x we are able to prove the following theorem (the proof is provided at the end of the subsection): Theorem 2 For J x , J ′ x = 0, R x−1,x < 1.(166) This means that if there is a finite density of points x with nonzero couplings (J x , J ′ x = 0), i.e. finite density of non-SWAP gates, then lim L→∞ tr R 1,2 · · ·R L−1,L = 0 ,(167) and lim L→∞ K(t, L) = tr P = t . A completely analogous treatment holds for L odd, e.g., considering K(t, L) = t + tr(R 12 · · ·R L−2,L−1 R L ) . Proof of Theorem 2 To prove the statement it is sufficient to show that if A|(T x−1 T x ) † T x−1 T x |A = 1(170) for some state |A , then P |A = |A .(171) This follows immediately from Theorem 1 and the following two lemmas A|(T 0 T) † T 0 T|A = 1 (172) where T 0 and T are transfer matrices of the form (34) (with unitary matrices U 0 andW 0 andŨ andW respectively), then (M a,ι ⊗ ½ 2t − ½ 2t ⊗ M * a,ι )(W 0 ⊗W * 0 )(Ũ ⊗Ũ * )(W ⊗W * ) |A = 0 ,(173)(M a,ι ⊗ ½ 2t − ½ 2t ⊗ M * a,ι )(Ũ ⊗Ũ * )(W ⊗W * ) |A = 0 ,(174)(M a,ι ⊗ ½ 2t − ½ 2t ⊗ M * a,ι )(W ⊗W * ) |A = 0 ,(175)(M a,ι ⊗ ½ 2t − ½ 2t ⊗ M * a,ι ) |A = 0 ,(176) where ι ∈ {0, 1}, a ∈ {1, 2, . . . , d 2 − 1}. Proof of Lemma 6 Considering the expectation value (172), and using that B| O † ι ′ O ι ′ |B ≤ B|B , B| O ι ′ O † ι ′ |B ≤ B|B ,(178) for any |B , we have 1 = A|(T 0 T) † T 0 T|A = O † 1 (W 0 ⊗W * 0 )O 0 (Ũ ⊗Ũ * )O † 1 (W ⊗W * )O 0 |A 2 (179) ≤ O † 1 (W 0 ⊗W * 0 )O 0 (Ũ ⊗Ũ * )O † 1 (W ⊗W * )|A 2 ≤ O † 1 (W 0 ⊗W * 0 )O 0 (Ũ ⊗Ũ * )(W ⊗W * )|A 2 ≤ O † 1 (W 0 ⊗W * 0 )(Ũ ⊗Ũ * )(W ⊗W * )|A 2 ≤ (W 0 ⊗W * 0 )(Ũ ⊗Ũ * )(W ⊗W * )|A 2 = |A 2 = 1 . This can hold only if all four inequalities in (179) are saturated. Using (82) we see that this happens only if the conditions (173 -176) are satisfied. Proof of Lemma 7 Plugging the forms (58) we bring (174 -176) in the form [M a,ι , A] = 0, [Ṽ ′ † M a,ιṼ ′ , A] = 0, [Ṽ ′ †Ṽ † M a,ιṼṼ ′ , A] = 0,(180) where a ∈ {1, 2, . . . , d 2 − 1}, ι ∈ {0, 1}. As shown in the proof of Lemma 2 we have that the first two relations (180) imply that for J ′ = 0: [A, M ab,1 ] = 0, a, b ∈ {1, 2, . . . , d 2 − 1}.(181) Using the above relation and Eq. (102) we have that the third of (180) is equivalent to [Ṽ † M a,ιṼ , A] = 0.(182) Proceeding as in (95, 97) we then find that for J = 0: [A, M ab,0 ] = 0, a, b ∈ {1, 2, . . . , d 2 − 1}.(183) Fluctuations of the spectral form factor The approach presented in the current manuscript can also be applied to study the fluctuations of the SFF (14) (this idea has recently been exploited in Ref. [35] in the special case of the self-dual kicked Ising model and in Ref. [36] for the large d asymptotics of Floquet chains with Haar random interactions). Specifically, it can be employed to compute the higher moments of |tr U t L | 2 with respect to the i.i.d. on-site disorder distribution K n (t, L) := E |tr U t L | 2n = E     N i,j=1 e i(ϕi−ϕj )t   n   , t, L ∈ N, n ≥ 1. (184) Indeed, exploiting the space-time duality described in Sec. 2.7 we can express the above quantities as K n (t, L) = tr T L n , with T n = (Ũ ⊗n ⊗ (Ũ * ) ⊗n )O † 1;n (W ⊗n ⊗ (W * ) ⊗n )O 0;n ,(186) whereŨ andW are defined in (35,36) while we introduced where the generalised sublattice magnetisation M a,ι is defined in Eq. (39). Note that, for definiteness, here we considered systems without time-reversal symmetry. O ι ′ ;n := O 0ι ′ ;n O 1ι ′ ;n = O 1ι ′ ;n O 0ι ′ ;n ,(187)O ιι ′ ;n := d d 2 −1 θ g ιι ′ (θ) exp iθ · (M ι;n ⊗ ½ 2t − ½ 2t ⊗ M * ι;n ) .(188) Using that the matrix T n has the same structure as T in (34) we can directly repeat the treatment described in Sec. 3.1. In particular, for local gates of the form (53) we find lim L→∞ K n (t, L) = dim M ′ n ,(190) where we introduced the set M n := {M a,ι;n } a,ι ∪ {M ab,ι;n } a,b,ι , (192) All elements of (191) are invariant under permutations of the n copies of H t in H ⊗n t and under 2-site translations within each copy, i.e., they commute with A p;τ1,...τn = Γ (p)(Π 2τ1 2t ⊗ · · · ⊗ Π 2τn 2t ), τ 1 , . . . , τ n ∈ {0, . . . , t − 1} , (193) where Γ (·) is a representation of S n , the symmetric group of n letters, on H tn ≡ H ⊗n t and p ∈ S n . Specifically, Γ (p) |A 1 ⊗ |A 2 ⊗ · · · ⊗ |A n = |A p(1) ⊗ |A p(2) ⊗ · · · ⊗ |A p(n) . This means that A n = span{A p;τ1,τ2...τn ; τ 1 , τ 2 . . . , τ n ∈ {0, . . . , t − 1}, p ∈ S n } is a vector subspace of M ′ n and hence lim L→∞ K n (t, L) = dim M ′ n ≥ n! · t n = lim N →∞ |tr U t | 2n dµ CUE (U),(196) where dµ CUE (U) is the CUE measure and N is the dimension of the matrix U. In analogy with what happens for n = 1 (c.f. Theorem 1) we expect A n and M ′ n to actually coincide, leading to an equality sign in (196). However, we leave the formal proof of this statement to future work. Similar conclusions (with CUE replaced by COE) hold in the time-reversal invariant case. Singular on-site disorder distributions As discussed in Sec. 2.3 we expect our results to be stable under modifications of the averaging procedure as long as such modifications do not introduce spatial correlations. In our setting this can be verified explicitly by considering singular disorder distributions of local gates u x , w x , Eq. (26), supported on lower-dimensional submanifolds of SU(d) that include the identity. For instance, one can imagine having some of the components of θ ι,x in (26) (or θ x in (68)) set to zero for all ι and x. Physically, this choice describes a weaker external noise where, for example, the random magnetic fields are imposed only along certain specific directions rather than isotropically. In this case we expect that away from certain "resonances", namely for almost all 2-site dual-unitary gates U, W , the treatment described in the previous sections is still applicable. In particular we anticipate that the SFF will still be characterised by the commutants of (64) or (79) depending on whether or not the problem is time-reversal symmetric. Let us illustrate the main steps that can be used to prove this idea. We consider for simplicity the case of qubits (d = 2) and absence of time-reversal symmetry. Denoting by I the subset of indices I ⊂ {1, 2, 3} such that {θ a,ι,x } a∈I are not set to zero, and repeating the steps of Sec. 2.7 and 3.1, one readily finds the following analogue of the conditions (52) UWAW †Ũ † = e iφ A , Here we used that {M a,ι } generate the su(2) algebra. A completely analogous reasoning applies for the set of equivalent matrices {W † M a,ιW }. The only possible non-trivial choice is then to take the set I composed by single element, which, without loss of generality, can be set to 3. This corresponds to u x , w x being restricted to some U(1) subgroup of SU (2). To show that the our treatment applies also in this case we need to prove the analogue of Lemma 2. Namely, we need to show that if A fulfils (197) then it commutes with {M a,ι } a=1,2,3;ι=0,1 , {M ab,ι } a,b=1,2,3;ι=0,1 . This can be done by considering the following set of 15 operators where we introduced the short-hand notation N ι :=W † M 3,ιW , ι = 0, 1. Since the operators (199) are constructed by taking commutators of M 3,ι , andW † M 3,ιW , they commute with A. Moreover, they can be written as linear combinations of {M a,ι } a=1,2,3;ι=0,1 and {M ab,1 } a,b=1,2,3 . This means that if we can prove that the operators in S 1 are linearly independent we immediately have We could not prove explicitly the linear independence of the set (199) but we verified numerically that it holds almost always (away from special, measurezero set of U , W -the so-called "resonances"). ι denoting the representation of the Hermitian generators of su(d) in the lattice made of integer and half-oddinteger time indices M = (M 1 , M 2 , . . . , M d 2 −1 ) with M a denoting the representation of the a-th Hermitian generator of su(d) in the full time lattice M a := τ ∈Λt σ a,τ = M a,0 + M a,1 . which proves point (i). The eigenvalue λ is unimodular only if both inequalities in (84) are saturated. The second one implies (82) for |R = |A , i.e. the second line of (49), while the first one implies (82) for |R = (W ⊗W * ) |A , i.e. the third line of (49). Since O 0 |A = |A , O † 1 (W ⊗W * ) |A = (W ⊗W * ) |A we have the first line of (49). This proves point (ii). Since sin(J ′ s 3 ) is Hermitian and traceless it can be expanded in terms of the generators {σ a }, i.e. sin(J ′ s 3 ) = c(J ′ ) · σ , where c(J ′ ) = 0 for J ′ = 0 . {M 12,ι + M 21,ι ,M 13,ι + M 31,ι , M 23,ι + M 32,ι , M 11,ι − M 22,ι , M 11,ι − M 33,ι } ι=0,1 . Lemma 7 7ForŨ andW of the form (58) with J = 0 and J ′ = 0 the conditions (174), (175), and (176) are equivalent to [A, M a,ι ] = 0 , [A, M ab,ι ] = 0 , a, b ∈ {1, 2, . . . , d 2 − 1}, ι ∈ {0, 1} . (177) and finally M ι;n = (M 1,ι;n , M 2,ι;n , . . . , M d 2 −1,ι;n ) with t ⊗M a,ι ⊗ ½ ⊗(n−1−j) t , a ∈ {1, . . . , d 2 − 1}, ι ∈ {0, 1}, (189) b ∈ {1, . . . , d 2 − 1}, ι ∈ {0, 1} . [M a,ι , A] = 0 ,[W † M a,ιW , A] = 0 , ι ∈ {0, 1} , a ∈ I .(197)At this point we note that if I has at least two elements these conditions are equivalent to (52). This follows by observing that if A commutes with M a,ι and M b,ι it also commutes with their commutator[M a,ι , M b,ι ] = i 3 c=1ǫ abc M c,ι . S 1 1={{M 3,ι } ι=0,1 , {[N ι , M 3,ι ′ ]} ι,ι ′ =0,1 , {[[N ι , M 3,ι ′ ], M 3,ι ′ ]} ι,ι ′ =0,1 , {N ι } ι=0,1 , [[N 0 , M 3,0 ], N 0 ], [[N 0 , M 3,1 ], N 0 ], [[N 1 , M 3,0 ], N 1 ]} , [ A, M a,ι ] = 0, [A, M ab,1 ] = 0 , a, b ∈ {1, 2, 3}, ι ∈ {0, 1} . (201) Lemma 5 Let K ⊆ End(H 2t ) be a multiplicative algebra of operators overH 2t , generated by elements of M = {M a,ι } a,ι ∪ {M ab,ι } a,b,ι . The representations of K over the eigenspaces {W k ⊂ H 2t } t−1k=0 of the 2-site shift operator (Π 2t ) 2 W k = e −2πik/t W k are all irreducible and inequivalent. are local spin raising/lowering operators. Using the commutation relations among {σ a } it is straightforward to show that all operators (107)-(111) can be expressed as linear combinations of M a,ι , M ab,ι (cf. (39) and Lemma 6 For dual-unitary circuits, if a state |A fulfils Sometimes referred to also as Bohigas-Giannoni-Schmit conjecture. 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{'abstract': "We investigate a class of local quantum circuits on chains of d−level systems (qudits) that share the so-called 'dual unitarity' property. In essence, the latter property implies that these systems generate unitary dynamics not only when propagating in time, but also when propagating in space. We consider space-time homogeneous (Floquet) circuits and perturb them with a quenched single-site disorder, i.e. by applying independent single site random unitaries drawn from arbitrary non-singular distribution over SU(d), e.g. one concentrated around the identity, after each layer of the circuit. We identify the spectral form factor at time t in the limit of long chains as the dimension of the commutant of a finite set of operators on a qudit ring of t sites. For general dual unitary circuits of qubits (d = 2) and a family of their extensions to higher d > 2, we provide an explicit construction of the commutant and prove that spectral form factor exactly matches the prediction of circular unitary ensemble for all t, if only the local 2-qubit gates are different from a SWAP (non-interacting gate). We discuss and partly prove possible extensions of our results to weaker (more singular) forms of disorder averaging, as well as to quantum circuits with time-reversal symmetry, and for computing higher moments of the spectral form factor.", 'arxivid': '2012.12254', 'author': ['B Bertini ', '· P Kos ', '· T Prosen ', 'Bruno Bertini bruno.bertini@physics.ox.ac.uk ', 'Pavel Kos pavel.kos@fmf.uni-lj.si ', 'Tomaž Prosen tomaz.prosen@fmf.uni-lj.si ', 'Bruno Bertini ', 'Pavel Kos ', 'Tomaž Prosen ', '\nTheoretical Physics\nFaculty of Mathematics and Physics\nOxford University\nParks RoadOX1 3PUOxfordUK\n', '\nFaculty of Mathematics and Physics\nUniversity of Ljubljana\nJadranska 19, SI1000 Ljubl-janaSlovenia\n', '\nUniversity of Ljubljana\nJadranska 19, SI1000 Ljubl-janaSlovenia\n'], 'authoraffiliation': ['Theoretical Physics\nFaculty of Mathematics and Physics\nOxford University\nParks RoadOX1 3PUOxfordUK', 'Faculty of Mathematics and Physics\nUniversity of Ljubljana\nJadranska 19, SI1000 Ljubl-janaSlovenia', 'University of Ljubljana\nJadranska 19, SI1000 Ljubl-janaSlovenia'], 'corpusid': 229348964, 'doi': '10.1007/s00220-021-04139-2', 'github_urls': [], 'n_tokens_mistral': 27539, 'n_tokens_neox': 23830, 'n_words': 13312, 'pdfsha': 'e8e22d4f2f99c0ee273c8d3529bf986e77a793bd', 'pdfurls': ['https://arxiv.org/pdf/2012.12254v3.pdf'], 'title': ['Random Matrix Spectral Form Factor of Dual-Unitary Quantum Circuits', 'Random Matrix Spectral Form Factor of Dual-Unitary Quantum Circuits'], 'venue': []}

arxiv

Computing coset leaders of binary codes 28 Jan 2010 M Borges-Quintana M A Borges-Trenard · E Martínez-Moro Computing coset leaders of binary codes 28 Jan 2010Received: date / Revised: dateDes. Codes Cryptogr. manuscript No. (will be inserted by the editor)Binary codes · Cosets leaders · Gröbner representations We present an algorithm for computing the set of all coset leaders of a binary code C ⊂ F n 2 . The method is adapted from some of the techniques related to the computation of Gröbner representations associated with codes. The algorithm provides a Gröbner representation of the binary code and the set of coset leaders CL(C ). Its efficiency stands of the fact that its complexity is linear on the number of elements of CL(C ), which is smaller than exhaustive search in F n 2 . Introduction The error-correction problem in coding theory addresses given a received word recovering the codeword closest to it with respect to the Hamming distance. This previous statement is the usual formulation of the Complete Decoding Problem (CDP). The t-bounded distance decoding (t-BDD) algorithms determine a codeword (if such a word exists) which is at distance less or equal to t to the received word. If t is the covering radius of the code then the bounded distance decoding problem is the same as CDP. In the CDP of a linear code of length n , C ⊂ F n q those errors that can be corrected are just the coset leaders, which are vectors of smallest weight in the cosets F n q /C . When there is more than one leader in a coset there is more than one choice for the error. Therefore the following problem naturally follows known as the coset weights problem(CWP): Input: A binary r × n matrix H, a vector s ∈ F r 2 and a non-negative integer t. Problem: Does a binary vector e ∈ F r 2 of Hamming weight at most t exist such that He = s? Recall that H can be seen as the parity check matrix of a binary code and s the syndrome of a received word, thus the knowledge of e would solve the t-BDD problem. Unfortunatelly the hope of finding an efficient t-BDD algorithm is very bleak since it was proven that the CWP is NP-complete [4]. Thus the computation of all the coset leaders is also NP-complete. The study of the set of coset leaders is also related to the study of the set of minimal codewords, which have been used in the Maximum Likelihood Decoding Analysis [2] and which are also related to the minimal access structure of secret sharing schemes [13]. Furthermore, the computation of all coset leaders of a code allows to know more about its internal structure [7]. All problems mentioned before are considered to be hard computational problems (see for example [2,4]) even if preprocesing is allowed [10]. However, taking into account the nature of the problem, to develop an algorithm for computing the set of all coset leaders of a binary code, in the vector space of 2 n vectors, by generating a number of vectors close to the cardinality of this set may be quite efficient. This is our purpose extending some results on Gröbner representations for binary codes. For previous results and applications of Gröbner representations for linear codes see [6,7,5], for a summary of the whole material we refer the reader to [8]. We extend some settings of previous work in order to obtain the set of all coset leaders keeping record of the additive structure in the cosets. The presentation of the paper is done in a "Gröbner bases"-free context. The outline of the paper is as follows. In Section 2 we review some of the standard facts on binary linear codes and their Gröbner representation. Section 3 gives a concise presentation of the results in this paper. Definition 3 corresponds to the construction of List, the main object that is used in the algorithm proposed, while Theorem 2 guarantees that all coset leaders will belong to List. In this section it is presented the algorithm CLBC for computing the set of all coset leaders. At the end of the section we show an example to illustrate a computational approach applied to a binary linear code with 64 cosets and 118 coset leaders. In Section 4 we discuss some complexity issues. Finally some conclusion are given which include further research. Preliminaries Let F 2 be the finite field with 2 elements and F n 2 be the F 2 -vector space of dimension n. We will call the vectors in F n 2 words. A linear code C of dimension k and length n is the image of an injective linear mapping L : F k 2 → F n 2 , where k ≤ n, i.e. C = L(F k 2 ) . From now on, we will use the term code to mean binary linear code. The elements in C are called codewords. For a word y ∈ F n 2 the support of y is supp(y) = {i ∈ {1, . . . , n} | y i = 0} and the Hamming weight of y is given by weight(y) the cardinal of supp(y). The Hamming distance between two words c 1 , c 2 is d(c 1 , c 2 ) = weight(c 1 − c 2 ) and the minimum distance d of a code is the minimum weight among all the non-zero codewords. It is well known that CDP has a unique solution for those vectors in B(C ,t) = {y ∈ F n 2 | ∃ c ∈ C s.t. d(c, y) ≤ d−1 2 } where [·] is the greatest integer function. Definition 1 The words of minimum Hamming weight in the cosets of F n 2 /C are called coset leaders. Cosets corresponding to B(C ,t) have a unique leader however in general outside B(C ,t) there may be also cosets with a unique leader, i.e. those cosets with only one leader could be more than |B(C ,t)|. Let CL(C ) denote the set of coset leaders of the code C and CL(y) the subset of coset leaders corresponding to the coset C + y. Let e i i = 1, . . . , n be the i-th vector of the canonical basis X = {e 1 , . . . , e n } ⊂ F n 2 and 0 the zero vector of F n 2 . The following theorem gives some relations between cosets [12, Corollary 11.7.7]. Theorem 1 Let w ∈ CL(C ), such that w = w 1 +e i for some w 1 ∈ F n 2 and i ∈ {1, . . . , n}. Then, w 1 ∈ CL(w 1 ). Definition 2 A Gröbner representation of F n 2 /C [6,8] is a pair N, φ where N is a transversal of F n 2 /C such that 0 ∈ N and for each n ∈ N \ {0} there exists e i , i ∈ 1, . . . , n, such that n = n ′ + e i and n ′ ∈ N, and φ : N × {e i } n i=1 → N be the function that maps each pair (n, e i ) to the element of N which belongs to the coset of n + e i . Computing the set of coset leaders A key point of the algorithms for computing Gröbner representations is the construction of an object we called List which is an ordered set of elements of F n 2 w.r.t. a linear order ≺ defined as follows: w ≺ v if weight(w) < weight(v) or weight(w) = weight(v) and weight(w) ≺ 1 weight(v), where ≺ 1 is any admissible order on F n 2 in the sense given in [3, p. 167]. We will call such kind of orders as weight compatible orderings. Definition 3 (Construction of List) . Let List be the ordered structure given by the following axioms 1. 0 ∈ List. 2. If v ∈ List, let N(v) = min ≺ {w | w ∈ List ∩ (C + v)}. 3. If v ∈ List is such that weight(v) = weight(N(v)), then {v + e i : i / ∈ supp(v), i ∈ {1, . . ., n}} ⊂ List. Remark 1 The set N is the subset of List such that N(v) is the least element in List ∩ (C + v), i.e. N(v) ∈ C + v and N(v) = min ≺ {w | w ∈ List ∩ (C + v)}. Remark 2 We start List with 0 and we will see that, whenever condition 3. holds for v, the vector v will be a coset leader of C + v. Then, we insert v + e i to List, for i = 1, . . . , n and i / ∈ supp(v) (see Theorem 1). It is not necessary to introduce v + e i , for i ∈ supp(v), because in this case v + e i ≺ v and then v + e i has been already considered in List since it is an ordered structure. Next theorem states that List in Definition 3 includes the set of coset leaders. Theorem 2 Let w ∈ CL(C ), then w ∈ List. Proof We will proceed by Noetherian Induction on the words of F n 2 with the ordering ≺, since ≺ has the property of being a well-founded ordering or Noetherian ordering (i.e., any descending chain of words is finite) [3, Theorem 4.62, p. 168]. The word 0 ∈ CL(C ) and by definition it belongs to List, let w ∈ CL(C ) \ {0}. Assume the property valid for any word less than w with respect to ≺, i.e. u ∈ List provided u ∈ CL(C ) and u ≺ w. Taking i ∈ supp(w), we write w = u + e i where u ∈ F n 2 , then by Theorem 1, u ∈ CL(C ). In addition, supp(u) = supp(w) − 1, thus u ≺ w. Therefore, by applying the induction principle we have u ∈ List. If u is a coset leader belonging to List it is clear that weight(u) = weight(N(u)). As a consequence, by 3. in Definition 3, w = u + e i ∈ List. ⊓ ⊔ Algorithm 1 (CLBC) Input: A parity check matrix of a binary code C . Output: CL(C ), φ : The set of all coset leaders and the function Matphi. 1: List ← [0], N ← / 0, r ← 0, CL(C ) ← [] 2: while List = / 0 do 3: τ ← NextTerm[List], s ← τH 4: j ← Member[s, {s 1 , . . . , s r }] 5: if j = false then 6: for k such that τ = τ ′ + e k with τ ′ ∈ N do φ (τ ′ , e k ) ← τ j 7: if weight(τ) = weight(τ j ) then 8: CL(C )[τ j ] ← CL(C )[τ j ] ∪ {τ} Theorem 3 CLBC computes the set of coset leaders of a given binary code and its corresponding Matphi. Proof Note that when an element τ is taken out from List by NextTerm the elements to be inserted in List by InsertNext are of the form τ + e k , where k / ∈ supp(τ). As a consequence, τ + e k ≻ τ. Then all elements generated by CLBC in List, after τ is taken out, are greater than τ. Therefore, when τ is the first element in List in Step 3, all elements of List that shall be analyzed by CLBC are greater than τ. Let us prove that the procedure generates List according to Definition 3. First, by Step 1, 0 ∈ List. Let τ = NextTerm[List] in Step 3, in this step the syndrome s of τ is computed. Thus we have two cases regarding the result of Step 4, namely 1. Assume j = "false", thus Steps 5 and 11 guaranty us to find N(τ) as the least element in List having the same syndrome as τ. When we are in this case (τ = N(τ)), in Step 14 it is performed (ii) of Definition 3. 2. On the other hand, assume j = "false" (an element N(τ) = τ j have been already computed), if we have weight(τ) = weight(τ j ), in Step 9 it is performed (ii) of Definition 3. Therefore CLBC constructs the object List following Definition 3. By Theorem 2, the set of coset leaders is a subset of List. Then Steps 8 and 13 assure the computation of this set. The procedure computes the Matphi structure as a direct consequence of Steps 6, 16, 17. The reason for including Step 17 is that in this case τ r + e k = τ ′ ≺ τ r and those elements τ r + e k , where k ∈ supp(τ r ), are not inserted in List since τ ′ has been already considered when τ r is computed as a new element of N. In addition, since ≺ is admissible, all subwords of τ r are also least elements of their cosets according to ≺, so τ ′ = N(τ ′ ) (note τ ′ is a coset leader and by Theorem 2 it belongs to List). We have proved that CLBC guarantees the outputs we are claiming. Termination is a consequence of the fact that the cardinal of the set of elements belonging to List is least than n|CL(C )|. Then, to a certain extent (when the set of coset leaders has been computed) no more elements are inserted in List in Steps 9 and 14. Therefore, the list get empty, and by Step 2, CLBC terminates. ⊓ ⊔ Remark 3 1. We introduce in this paper steps 7, 8 and 9. In our predecessor algorithms where it was not necessary to introduce them [6,7,8]. These steps, as well as a modified definition of list, are forced by the fact that all coset leaders are incorporated. 2. In Steps from 14 to 17, the pairs (τ r , e k ), where k ∈ supp(τ r ), are not necessary to compute the coset leaders but for computing the structure Matphi and could be removed if we are only interested in the coset leaders.         . We use GAP 4.12 [11] and the GAP's package GUAVA 3.10 for Coding Theory. We have built in this framework a collection of programs we call GBLA LC "Gröbner Bases by Linear Algebra and Linear Codes" [9]. In particular, we have run the function "CLBC" of GBLA LC (Coset Leaders of Binary Codes), it gives a list of three objects as an output, the first one is the set of coset leaders, the second one the function Matphi, and the third one the error correcting capability of the code. The complete set of coset leaders is the list below with 64 components and each component corresponds to a coset with its cosets leaders, the set N is composed by the first elements of each component. We have indicated with arrows some places in the list below, which we are going to use during the example. The elements in the list CL(C ) are Note that y = e 4 + e 5 + e 6 in CL(C ) 46 = [e 1 + e 2 + e 4 , e 1 + e 6 + e 8 , e 2 + e 5 + e 8 , e 4 + e 5 + e 6 ] (pointed by arrows) is a coset leader such that none subword of y is in N (see the previous elements pointed by arrows). This shows the importance of considering all coset leaders in 3. of Definition 3 and not only the coset leaders belonging to N like in previous works. The algorithm could be adapted without incrementing the complexity to get more information like the Covering radius and the Newton radius. In this case by analyzing the last element CL(C ) 64 = [e 5 + e 9 + e 10 ] we have a coset of highest weight which also contains only one leader; therefore, the Covering radius and the Newton radius are equal to 3. Moreover, for computing these parameters the algorithm do not need to run until the very end. As we state before, the set N would be enough to compute the Weight Distribution of the Coset Leaders WDCL = (α 0 , . . . , α n ) where α i is the number of cosets with coset leaders of weight i, i = 1, . . . , n of the code. We can provide also an object which gives more information about the structure of the code, that would be the numbers of coset leaders in each coset (#(CL)). 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 4, 1, 1, 1, 1, 2 It is also interesting to note that there are more cosets with one leader (19) out the cosets corresponding to B(C ,t) than cosets corresponding to B(C ,t) |{CL(y) | y ∈ B(C ,t)}| = 11. Therefore, there are 30 of the 64 cosets where the CDP has a unique solution. Complexity Analysis For a detailed complexity analysis and some useful considerations from the computational point of view we refer the reader to [7]. Section 6 of that paper is devoted to discuss in details about the computational complexity and space complexity of the setting for computing Gröbner basis representation for binary codes and general linear codes. The method is also compared with other existent methods for similar purposes. In the case of this paper, the difference is that we work out the set of all coset leaders and not only a set of canonical forms. Next theorem shows an upper bound for the number of iterations that will perform CLBC. Theorem 4 CLBC computes the set of coset leaders of a given binary code C of length n after at most n|CL(C )| iterations. Proof Note that the number of iterations is exactly the size of List. In the proof of Theorem 3 was shown that the algorithm follows this definition to construct the object List. It is clear that the size of List is bounded by n|CL(C )|, note that we can write List, as a set as follows List = {w + e i | w ∈ CL(C ) and ∈ {1, . . . , n}}. ⊓ ⊔ Remark 4 1. By the proof above we require a memory space of O(n|CL(C )|). We assume that for computing the set of coset leaders it is required at least O(|CL(C )|); therefore, CLBC is near the optimal case of memory requirements. 2. CLBC generates at most n|CL(C )| words from F n 2 to compute the set of coset leaders. An algorithm for computing this set needs to generate from F n 2 at least a subset formed by all coset leaders, i.e. |CL(C )|; therefore, the algorithm is near the optimal case of computational complexity. Conclusion The Algorithm CLBC is formulated in this paper, which turns out to be quite efficient for computing all coset leaders of a binary code from memory requirements and computational complexity view. Although, as it is expected, the complexity of the algorithm is exponential (in the number of check positions). The difference of CLBC with its predecessors rely also in the computation of all coset leaders instead of a set of representative leaders for the cosets, however, it supports the computation of the function Matphi. We remark that the computation of Matphi is not necessary at all for the main goal of computing the coset leaders. We have kept this resource in the algorithm because this structure provides some computational advantages (see [6,7,8]) and, although the algorithm will be clearly faster without computing Matphi, the nature of the computational complexity and space complexity will remain the same. Unfortunately, a generalization to non-binary linear codes is not trivial from this work. The main reason seems to be that the solution based on Hamming weight compatible orderings will not continue being possible; the error vector ordering we have used in the general approach [7] is not a total ordering, although it allowed to set up the computational environment in order to compute Gröbner representations for linear codes. r ← r + 1, s r ← s, τ r ← τ, N ← N ∪ {τ r } 13: CL(C )[τ r ] ← CL(C )[τ r ] ∪ {τ} 14: List ← InsertNext[τ r , List] 15:for k such that τ r = τ ′ + e k with τ ′ ∈ N do 16:φ (τ ′ , e k ) ← τ r 17: φ (τ r , e k ) 20: end while 21: return CL(C ), φ Where 1. InsertNext[τ, List] Inserts all the sums τ +e k in List, where k / ∈ supp(τ), and keeps List in increasing order w.r.t. the ordering ≺. 2. NextTerm[List] returns the first element from List and deletes it from this set. 3. Member[ob j, G] returns the position j of ob j in G if ob j ∈ G and false otherwise. [ [ 1 ] 1, [e 1 ], [e 2 ], [e 3 ], [e 4 ], [e 5 ], [e 6 ], [e 7 ], [e 8 ], [e 9 ], [e 10 ], [e 1 + e 2 , → e 5 + e 6 ←], [e 1 + e 3 , e 5 + e 7 ], [e 1 + e 4 , e 5 + e 8 ], [e 1 + e 5 , e 2 + e 6 , e 3 + e 7 , e 4 + e 8 ], [e 1 + e 6 , e 2 + e 5 ], [e 1 + e 7 , e 3 + e 5 ], [e 1 + e 8 , → e 4 + e 5 ←], [e 1 + e 9 ], [e 1 + e 10 ], [e 2 + e 3 , e 6 + e 7 ], [e 2 + e 4 , e 6 + e 8 ], [e 2 + e 7 , e 3 + e 6 ], [e 2 + e 8 , → e 4 + e 6 ←], [e 2 + e 9 ], [e 2 + e 10 ], [e 3 + e 4 , e 7 + e 8 ], [e 3 + e 8 , e 4 + e 7 ], [e 3 + e 9 ], [e 3 + e 10 ], [e 4 + e 9 ], [e 4 + e 10 ], [e 5 + e 9 ], [e 5 + e 10 ], [e 6 + e 9 ], [e 6 + e 10 ], [e 7 + e 9 ], [e 7 + e 10 ], [e 8 + e 9 ], [e 8 + e 10 ], [e 9 + e 10 ], [e 1 + e 2 + e 3 , e 1 + e 6 + e 7 , e 2 + e 5 + e 7 , e 3 + e 5 + e 6 ], → [e 1 + e 2 + e 4 , e 1 + e 6 + e 8 , e 2 + e 5 + e 8 , e 4 + e 5 + e 6 ] ←, [e 1 + e 2 + e 7 , e 1 + e 3 + e 6 , e 2 + e 3 + e 5 , e 5 + e 6 + e 7 ], [e 1 + e 2 + e 8 , e 1 + e 4 + e 6 , e 2 + e 4 + e 5 , e 5 + e 6 + e 8 ], [e 1 + e 2 + e 9 , e 5 + e 6 + e 9 ], [e 1 + e 2 + e 10 , e 5 + e 6 + e 10 ], [e 1 + e 3 + e 4 , e 1 + e 7 + e 8 , e 3 + e 5 + e 8 , e 4 + e 5 + e 7 ], [e 1 + e 3 + e 8 , e 1 + e 4 + e 7 , e 3 + e 4 + e 5 , e 5 + e 7 + e 8 ], [e 1 + e 3 + e 9 , e 5 + e 7 + e 9 ], [e 1 + e 3 + e 10 , e 5 + e 7 + e 10 ], [e 1 + e 4 + e 9 , e 5 + e 8 + e 9 ], [e 1 + e 4 + e 10 , e 5 + e 8 + e 10 ], [e 1 + e 5 + e 9 , e 2 + e 6 + e 9 , e 3 + e 7 + e 9 , e 4 + e 8 + e 9 ], [e 1 + e 5 + e 10 , e 2 + e 6 + e 10 , e 3 + e 7 + e 10 , e 4 + e 8 + e 10 ], [e 1 + e 6 + e 9 , e 2 + e 5 + e 9 ], [e 1 + e 6 + e 10 , e 2 + e 5 + e 10 ], [e 1 + e 7 + e 9 , e 3 + e 5 + e 9 ], [e 1 + e 7 + e 10 , e 3 + e 5 + e 10 ], [e 1 + e 8 + e 9 , e 4 + e 5 + e 9 ], [e 1 + e 8 + e 10 , e 4 + e 5 + e 10 ], [e 1 + e 9 + e 10 ], [e 2 + e 3 + e 8 , e 2 + e 4 + e 7 , e 3 + e 4 + e 6 , e 6 + e 7 + e 8 ], [e 5 + e 9 + e 10 ] ] An introduction to Gröbner bases. 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{'abstract': 'We present an algorithm for computing the set of all coset leaders of a binary code C ⊂ F n 2 . The method is adapted from some of the techniques related to the computation of Gröbner representations associated with codes. The algorithm provides a Gröbner representation of the binary code and the set of coset leaders CL(C ). Its efficiency stands of the fact that its complexity is linear on the number of elements of CL(C ), which is smaller than exhaustive search in F n 2 .', 'arxivid': '1001.5074', 'author': ['M Borges-Quintana ', 'M A Borges-Trenard ', '· E Martínez-Moro '], 'authoraffiliation': [], 'corpusid': 17743716, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 7970, 'n_tokens_neox': 6985, 'n_words': 4475, 'pdfsha': 'c46da426f908f1fe5c80af94630339531219d711', 'pdfurls': ['https://arxiv.org/pdf/1001.5074v1.pdf'], 'title': ['Computing coset leaders of binary codes', 'Computing coset leaders of binary codes'], 'venue': []}

arxiv

A New Definition of Exoplanet Habitability: Introducing the Photosynthetic Habitable Zone April 14, 2023 C Hall Department of Physics and Astronomy The University of Georgia 30602AthensGAUSA Center for Simulational Physics The University of Georgia 30602AthensGAUSA P C Stancil Department of Physics and Astronomy The University of Georgia 30602AthensGAUSA Center for Simulational Physics The University of Georgia 30602AthensGAUSA J P Terry Department of Physics and Astronomy The University of Georgia 30602AthensGAUSA Center for Simulational Physics The University of Georgia 30602AthensGAUSA C K Ellison Department of Microbiology The University of Georgia 30602AthensGAUSA A New Definition of Exoplanet Habitability: Introducing the Photosynthetic Habitable Zone April 14, 2023Submitted to ApJLDraft version Typeset using L A T E X twocolumn style in AASTeX631 It may be possible to detect biosignatures of photosynthesis in an exoplanet's atmosphere. However, such a detection would likely require a dedicated study, occupying a large amount of telescope time. It is therefore prudent, while searching for signs of life that we may recognise, to pick the best target possible. In this work, we present a new region, the "photosynthetic habitable zone" -the distance from a star where both liquid water and oxygenic photosynthesis can occur. It is therefore the region where detectable biosignatures of oxygenic photosynthesis are most likely to occur. Our analysis indicates that in the most ideal conditions for life and no atmospheric effects, the photosynthetic habitable zone is almost as broad as the habitable zone. On the other hand, if conditions for life are anything less than excellent and atmospheric effects are even moderate, the photosynthetic habitable zone is concentrated at larger separations around more massive stars. Such cases are also not tidally locked to their host star, which could result in planetary rotation periods similar to the Earth's. We identify five planets, Kepler-452 b, Kepler-1638 b, Kepler-1544 b and Kepler-62 e and Kepler-62 f, that are consistently in the photosynthetic habitable zone for a variety of conditions, and we predict their day lengths to be between 9 and 11 hours. We conclude that the parameter space in which we should search for signs of life is much narrower than the standard habitable zone. INTRODUCTION Since the first exoplanet atmospheric detection around a 1.35 R J planet (Charbonneau et al. 2002), astronomers have been pushing the limits to smaller and smaller planets, such as the detection of water vapour around the 8M E planet K12-18b (Tsiaras et al. 2019). The continued discovery of exoplanet atmospheres, and increasing technological capability, has raised the prospect of finding a planet that may be inhabited by life. Subsequently, it has been determined that the characterization and detection of biosignatures -atmospheric spectral features that could indicate signs of life on a planet -should be an area of focus for astrobiology (see, e.g. Seager et al. 2012;Kaltenegger 2017;Lammer et al. 2019). To date, over 5000 1 exoplanets have been discovered using a mix of ground-based and space-based methods. With the successful launch of JWST and future observatories such as the European Extremely Large Telescope (E-ELT) and the Thirty Meter Telescope (TMT), we are moving from the era of exoplanet discovery to exoplanet atmospheric characterization. However, characterizing these worlds remains an enormous challenge. For example, the most promising O 2 feature for JWST appears to be the O 2 → X collisional induced adsorption band at 6.4 µm (Fauchez et al. 2020), but even for a target such as TRAPPIST 1-e (Gillon et al. 2016(Gillon et al. , 2017 this would require more than 700 transits, longer than the anticipated lifetime of JWST given that TRAPPIST 1-e has a 6 day orbital period and is only visible to JWST for less than a third of the year (Gillon et al. 2020). Even the more favourable strong O 3 band at 10µm would require more than 100 transit observations on a planet such as TRAPPIST 1-e (Lin et al. 2021) to detect it at just 3σ. Fortunately, O 2 and O 3 are not the only biosignatures. More generally, atmospheric chemical disequilibrium, characterised by the coexistence of two or more longterm incompatible gases (Lovelock 1965;Sagan et al. 1993;Cockell et al. 2009), can be considered a sign of ongoing life. The Archean Earth had a biogenic disequilibrium caused by the coexistence of N 2 , CH 4 , CO 2 , and liquid water, which could be possible to remotely detect on an Earth-sized planet (Krissansen-Totton et al. 2018). Simultaneous detection of abundant CH 4 and CO 2 is therefore considered a biosignature. Happily, detecting this CH 4 -CO 2 pair is feasible, requiring ∼ 5 − 30 co-added JWST transits depending on if the stratosphere is dry or has a cloud or haze layer (Mikal-Evans 2022). Observing resources are valuable and finite, so choosing the best targets to search for biosignatures of any kind is imperative. The main criterium is whether the planet can sustain liquid water on its surface by residing an appropriate semi-major axis from its host star, referred to as the habitable zone (Huang 1959;Kasting et al. 1993). However, liquid water alone is not enough for life. Life requires energy to remain out of equilibrium with its environment. For almost all the biomass on Earth, this energy source is oxygenic photosynthesis ( Bar-On et al. 2018). We therefore suggest that a new criterium be used to determine where biosignatures may be found. In this work, we demonstrate that, like the habitable zone, the photosynthetic habitable zone is a bounded strip on a plot of stellar mass against semimajor axis. It occurs where both liquid water and photosynthesis is simultaneously possible. It is where the search for life in the Universe should be concentrated under the assumption of biosignatures similar to those generated by past or present Earth. We detail our calculations of the habitable zone in Section 2.1, our calculations of photosynthesis rate curves in Section 2.2, and the photosynthetic habitable zone in Section 2.3. We present our results in Section 3 and discuss assumptions and limitations in Section 4. We summarise and present our conclusion in Section 5. METHODS The Habitable Zone We used pre-main sequence (PMS) evolutionary models (Baraffe et al. 2015) 2 to obtain stellar effective temperature, T eff , as a function of stellar mass, M * . We assumed an age of 1 Gyr, corresponding to the approximate time primitive life first appeared on Earth (e.g. Dodd et al. 2017;Cavalazzi et al. 2021). The habitable zone (HZ) for each stellar mass was calculated using the method described in Kopparapu et al. (2013b). We use their derived relationships between HZ effective temper- ature, T eff and stellar fluxes, S eff , in the range 2600 K T eff 7200K: S eff = S eff + aT + bT 2 + cT 3 + dT 4 ,(1) where T = T eff − 5780 K, and S eff = L 4πR 2 σT 4 ,(2) where L is stellar luminosity and R is stellar radius. The coefficients a, b, c and d are determined by scenario, for example runaway greenhouse (Inner HZ) and maximum greenhouse (outer HZ). We used updated coefficient values published online 3 as per Kopparapu et al. (2013a), and detail them in Table 1. The corresponding HZ distance for a given star is then d = L/L S eff 0.5 AU,(3) where L/L is the luminosity of the star compared to the Sun. The Photosynthesis zone Photosynthesis is the chemical reaction by which organisms use energy from sunlight to synthesis sugar from carbon dioxiode and water: 6CO 2 + 6H 2 O hν −→ C 6 H 12 O 6 + 6O 2 .(4) Three variables directly impact the rate of photosynthesis (Gaastra 1959): light intensity (I), temperature (T ), and carbon dioxide (CO 2 ). Water availability indirectly impacts the rate of photosynthesis, as water stress causes plant structures to wilt, reducing CO 2 availability (Muller et al. 2011). In this work, we make two assumptions: 1) that the photosynthetic life we consider is ocean based, and so has an unlimited reservoir of water and 2) the global average of CO 2 concentration is sufficiently high that photosynthesis is not rate-limited by CO 2 concentration. Today's CO 2 levels are ∼ 0.04% by volume, and the early Archaen Earth had an estimated atmospheric CO 2 concentration of 70% by volume (see, e.g., Lehmer et al. 2020), so this assumption is probably reasonable. The chlorophyll a-normalised net photosynthetic rate, P , as a function of irradiance intensity, I, is given by (Eilers & Peeters 1988): P (I) = I αI 2 + βI + γ − R rate ,(5) where α and β are dimensionless parameters, γ is defined as the reciprocal of the light-limited initial slope of the P-I curve, and R rate is the dark respiration rate, the minimum rate at which glucose is combined enzymatically with oxygen to release energy and CO 2 . Net photosynthetic rate is the total output of molecular oxygen per unit biomass per unit time [µmol O 2 mg −1 h −1 ]. The maximum photosynthetic rate is given by P max rate = 1 β + 2 √ αγ − R rate .(6) The parameters α, β and γ in Equations 5 and 6 are determined by performing best-fit analysis to empirically determined photosynthesis rate curves of phytoplankton in the laboratory setting, the values used in this work are α = 1.0 × 10 −5 , β = 1.0 × 10 −3 , and γ = 2.0 (Yang et al. 2020). Three values of the dark respiration rate are explored, which determine the quality of conditions for life. As conditions for life become less favourable, R rate becomes a greater fraction of P max rate (Geider & Osborne 1989). We consider excellent conditions to be Earth-like (R rate = 0.3 P max rate ), and optimistic and pessimistic conditions to be R rate = 0.6 P max rate and R rate = 0.8 P max rate respectively. The P-I curve from Equation 5 is shown in Figure 1. The green dotted line in Figure 1 shows the line where the net photosynthesis rate is P (I) = 0, due to rate of primary production equalling the rate of respiration. Any irradiance intensity that causes a total net negative photosynthesis rate will result in no net oxygen being produced. Therefore, for photosynthesis to increase atmospheric content of O 2 , P > 0 is the absolute lowest limit required. We term the region of parameter space where P (I) > 0 (Equation 5) the "photosynthesis zone" (PZ) for convenience, and it is shown in pink in Figure. 2. Photosynthesis cannot proceed where P ≤ 0, since the organism's respiration rate is exceeding that of its primary production. Intensity of PAR at Earth P(I) = Rrate Figure 1. Net photosynthetic rate versus irradiance intensity. Dotted green line indicates where rate of photosynthetic production of O2 is equal to consumption of O2 during respiration, such that net photosynthetic production of O2 is zero. Rate of respiration is Rrate = 20 µmol O2 mg −1 h −1 . Orange vertical line is intensity of PAR at Earth. The Photosynthetic Habitable Zone We suggest the existence of a photosynthetic habitable zone (PHZ), a region of parameter space where the habitable zone overlaps the photosynthesis zone. It is in the photosynthetic habitable zone, rather than the habitable zone, that humanity should concentrate its search for spectral signs of life, since they can only be present where both liquid water and P (I) > 0 occur. To obtain the region where P (I) > 0, we calculate the irradiance intensity as a function of stellar mass and planet semimajor axis. We begin with the Planck function: B(λ, T ) = 2hc 2 λ 5 e hc λk B T − 1 −1 ,(7) and obtain the intensity of photosynthetically active radiation, I PAR , by integrating between λ min =400 nm and λ max =700 nm I PAR = λmax λmin 2hc 2 /λ 5 e hc/λkBT − 1 dλ.(8) The number of photons emitted by the star per unit time,Ṅ , in this wavelength range is then obtained by multiplying Equation 8 by the surface area of the star, and dividing by the energy of each photon, so that we obtaiṅ N = 4πR 2 λmax λmin 2c λ 4 exp hc λk B T − 1 −1 dλ. (9) Assuming a circular orbit, the photon flux at the top of a planetary atmosphere, Φ, at a distance a from the star is Φ =Ṅ 4πa 2 .(10) We account for atmospheric attenuation such that the light intensity at the bottom of the atmosphere is I = f a ·Ṅ 4πa 2(11) where f a ≤ 1.0 is the fractional attenuation due to atmospheric effects. We consider three attenuation efficiencies: no attenuation, f a = 1.0, moderate attenuation f a = 0.6, and Earth-like attenuation, f a = 0.2 (Sarmiento & Gruber. 2006;Lingam & Loeb 2021a). The intensity in Equation 11 is used in Equation 5 to obtain P (I) that would be achieved by a phytoplanktonlike species as a function of stellar mass and planet semimajor axis. RESULTS We find the existence of the photosynthetic habitable zone (PHZ), shown in green in Figure 2), where photosynthesizing life could exist and therefore leave behind atmospheric biosignatures. It occurs where the P (I, T ) > 0 region (pink in Figure 2) overlaps with the region where liquid water can exist (the Habitable Zone, blue in Figure 2). Figure 2 shows nine scenarios. On the y axis of the whole plot, atmospheric attenuation of photons is increasing. On the x-axis of the whole plot, the quality of the conditions for life are increasing (i.e., the maximum photosynthetic rate is much higher than the baseline respiration rate). The "Excellent" column corresponds to conditions for photosynthesizing lifeforms on Earth with respiration rates at 30% of maximum photosynthesis rates, typical for marine phytoplankton such as Isochrysis galbana, Platymonas subcordiformis, etc (Ippoliti et al. 2016;Yang et al. 2020). As the quality of conditions for life increases, the rate of respiration as a fraction of the maximum photosynthetic rate attainable decreases, resulting in a larger PZ. As atmospheric attenuation increases to Earth-like levels, interestingly, this decreases the size of the PZ, reducing the parameter space over which biosignatures could be found. This suggests that it may be easier for largescale photosynthesizing organisms, such as cyanobacterial mats, to occur on planets with more tenuous atmospheres. Overplotted as yellow symbols are planets that spend at least 10% of their orbit in the HZ and have radii R p < 1.8 R E , so could have a solid surface, since planets with R p 1.8 R E are likely to be gas-dominated (Lehmer & Catling 2017;Fulton et al. 2017). Additionally, we also plot 3 recently identified water world candidates, Kepler-138 c and d (Piaulet et al. 2022) and TOI-1452 b (Cadieux et al. 2022). While these fall outside the HZ, Kepler-138 d and TOI-1452 b fall inside the PZ. Although liquid water has not been directly detected on these planets, measurements imply that these planets may be similar to the water-rich icy moons of the solar system, such as Europa or Enceladus. These criteria reduce the ∼5000 known exoplanets down to 29 planets of interest. Overplotted on Figure 2 is the tidal lock radius (Peale 1977;Kasting et al. 1993): r lock = 0.027 P 0 t Q 1/6 M 1/3 * ,(12) where P 0 is the original rotation period of the planet, t is the age of the system (1 Gyr), M * is the stellar mass and Q −1 is the solid body plus ocean specific dissipation function. We use Q = 100, for the solid line, and Q = 10 and Q = 1000 for the upper and lower limits of the tidal lock radius. We assume P 0 = 13.5 hours, i.e. the day length of Earth when it was 1 Gyr old. There are several planets that are in or near the PHZ largely regardless of the quality of conditions for life or atmospheric effects, Kepler-452 b, Kepler-1638 b, Kepler-1544 b and Kepler-62 e and f. These planets should therefore be the most promising for detecting biosignatures. It is also possible that some of these planets with R 1.5 R E are water worlds (Luque & Pallé 2022). The least promising candidates are those around the lowest mass host stars, M * 0.4 M , since their position in Figure 2 coincides with the PHZ for less than half of our considered scenarios. As a final result, we posit that meaningful discussions of the habitable zone should account for where photosynthesis is also possible, since almost all life on Earth depends on photosynthesis either directly or indirectly. We therefore suggest replacing the use of the phrase "Habitable Zone" with "Photosynthetic Habitable Zone", when searching for biosignatures. DISCUSSION Conventional photosynthesis, as experienced on Earth, takes place during the day via photosynthetically active radiation (PAR) received directly from the Sun. We do not consider here any other scenario, such as starlight PAR, moonlight PAR, planetlight PAR, or speculative biological adaptations. We work under the limiting assumption that life would exist approximately "as we know it". While other scenarios are possible in principle, they require increasingly complex caveats, such as an older universe or photosynthesis only occurring at full moon (see, e.g. Raven & Cockell 2006). Furthermore, we focus our consideration on atmospheric biosignatures, Removed temperature dependence -bz-baraffe-loop-2023-02-16.py Figure 2. The Photosynthetic Habitable Zone. Region where positive net photosynthesis is possible shown in pink, and habitable zone shown in blue. The overlapping region is where photosynthesis can actually occur, and is named "The photosynthetic habitable zone', since it is possible that oxygenic photosynthesis, and therefore biosignatures, could exist on planets in this location. Overplotted in yellow markers are planets of interest, i.e. planets in the habitable zone expected to have a solid surface (R 1.8 RE). Blue markers indicate water world candidates. Earth is shown by white Earth symbol. The tidal lock radius with upper and lower limits is also shown. rather than biological surface features, since we expect atmospheric signatures to be detectable even when the disk-averaged spectrum features, such as the vegetation red edge (Seager et al. 2005), are not (Cockell et al. 2009). Photon attenuation Quality of conditions for life Our analysis here intends to show a general trend rather than demarcate absolute boundaries, since the fit parameters (α, β and γ) in the P − I relation can take a variety of values as long as the curve retains the same functional form, that is, an initial slope, a peak, and steady decline due to photoinhibition of photosynthesis at high intensity (see, e.g., Platt & Jassby 1976;Platt et al. 1981;Eilers & Peeters 1988;Ye et al. 2013). Furthermore, atmospheric attenuation is a function of column density, which is a function of both planet mass and planet size. While a super Earth may have a more massive atmosphere than Earth, a super Earth also has a larger surface area which results in atmospheric mass scaling more slowly than the increase in planet mass (Elkins-Tanton & Seager 2008). Additionally, super Earth outgassing rates may be lower than the Earth's (Stamenković et al. 2012), so a more rigorous consideration of atmospheric attenuation must determine the role that planet mass plays. It is unclear what effect tidal locking would have on the development of photosynthesizing life on another planet, since much of life on earth depends on 24hour circadian cycles to regulate physiological function (Dvornyk et al. 2003). An absence of periodicity on tidally locked planets would likely impact the evolution of biological regulation in those systems. The night side of the planet could not support photosynthesis since it does not receive PAR. This immediately discounts half of the surface area of the planet. On the other hand, always receiving PAR could potentially increase the rate of net O 2 production, since intensity does not wax and wane during the course of the day. In either case, it seems clear that there is a link between Earth's rotation rate and oxygenation, with longer days associated with higher oxygenation rates (Klatt et al. 2021). Even if this is not the case, and the circadian rhythm for phototrophs simply originates in environmental behaviour rather than any advantage in photosynthetic production, our analysis indicates that the PHZ predominantly exists outside the tidal locking radius for all cases, suggesting that the search for life elsewhere in the Universe should be focused around non-tidally locked planets. An important limitation of our work is that we do not consider the impact of temperature on photosynthesis rates. The planet equilibrium temperature depends on Bond albedo, which is a function of both stellar type (e.g. Kopparapu et al. 2013b) and planet properties (Shields et al. 2013;Rushby et al. 2019), with variations up to an order of magnitude around an F type star depending on planet composition. Correctly calculating planet surface temperature therefore requires 1D energy-balance climate models, which will be explored in a future work (Hall et al. in prep). However, the HZ models that we use here have already placed bounds on surface temperature, such that there would be liquid water and therefore temperatures in the range 0 − 100 deg C. Photosynthesis takes place on Earth in a variety of temperatures. In plants, it is generally constrained to lower temperature ranges (10 • C -40 • C) before suffering irreversible damage (Berry & Bjorkman 1980), while in cyanobacteria the preferred range is somewhat higher, with the upper limit for non-thermophiles ∼ 73 • C (Ward et al. 2012). A few points are worth noting -first, at low temperatures, photosynthesis is both enzymelimited and phosphate limited due to a reduction in the availability of phosphate in chloroplasts, while at higher temperatures proteins become denatured. High or low temperatures could also affect the stability and fluidity of the cellular membranes in which photosynthesis machinery components localize, resulting in more or less efficient biochemical reactions at the membrane interface. At moderate temperatures (∼10-35 • C), photosynthesis is mostly limited by the rate of CO 2 diffusion. This is another limitation of our work -cyanobacteria exist in mat-like colonies that have a z-depth, and we have assumed instead that any colony is essentially infinitesimally thin and we therefore do not need to consider diffusion equations. Another limitation of not considering the z-depth is that attenuation of photon flux by water occurs (Lingam & Loeb 2021b), affecting light availability to organisms found at different water column depths. Chloroplast-based photosynthesis in terrestrial plants largely depends on chlorophyll a for optimal absorbance of violet and orange light, while cyanobacteria possess additional light-harvesting phytochromes which allow light capture at wavelengths outside of optimal chlorophyll a absorbance (Kehoe 2010). If these phytoplankton exist solely underwater, the PHZ could therefore move closer to the central star, which may increase or decrease the size of the PHZ. Advanced modelling of microbial benthic ecology would be best suited to this problem, and we leave this to future work. We have also assumed that any extant life in the Universe shares a biochemistry similar enough to photosynthesizing lifeforms on Earth that we would recognise its signatures. Even on Earth, so-called "exotic photosynthesis" exists, such as infrared photosynthesis in anoxygenic photosynthetic organisms (Heath et al. 1999). This anoxygenic photosynthesis uses hydrogen sulfide instead of water as the reductant, and produces sulphur instead of oxygen as a byproduct, e.g: 6CO 2 + 12H 2 S hν −→ C 6 H 12 O 6 + 12 S + 6H 2 O. (13) The pigments used to carry out anaerobic photosynthesis are similar to chlorophyll, but have peak absorption in the near-IR due to molecular differences. While significant atmospheric sulphuric acid in this scenario could be considered a biosignature, there is a large risk of false-positive results due to its occurrence in many nonbiological processes (Domagal-Goldman et al. 2011), which is why it is not targeted as a biosignature. In addition to this, anoxygenic photosynthesis is likely an evolutionary precursor to oxygenic photosynthesis, with biogeochemical changes on a terrestrial planet forcing a switch to an oxygen producing version (Raven 2007(Raven , 2009b. One thing that we do not consider here is the effect of planet mass on atmospheric composition and density. For example, a super-Earth that is outside the HZ, that retains its primordial H-He dominated atmosphere could have surface temperatures that are warm enough to host liquid water (Mol Lous et al. 2022). The same could therefore also be true of regions that we have determined too cold for net positive photosynthesis. On the other hand, we also determine in this work having less atmospheric attenuation results in a broader PHZ, which counteracts the positive effect of the retained H-He atmosphere. A further interesting avenue of exploration is regarding the impact of CO 2 on planetary temperature through the greenhouse effect, along with the impact on photosynthetic rates. An inherent assumption of our work is that photosynthesis is not limited by reduced CO 2 availability, because the fluctuations on Earth are small. The global average concentration of CO 2 today is 400 ppm, and was significantly higher when life first emerged due to Earth's secondary atmosphere. Spacebased observatories show the CO 2 concentration varies only at ∼few ppm levels (Hakkarainen et al. 2016(Hakkarainen et al. , 2019, whereas rate limitation requires a decrease of ∼tens of ppm below this 400ppm level (e.g. Moss 1962;Gabrielsen 1948). However, the flip side of this is that we have not explored what this means for atmospheres rich in CO 2 , which would be likely to increase the expected surface temperature as a function of instellation. A selfconsistent CO 2 -instellation habitable zone should be calculated for this. Finally, it could be possible that most habitable worlds in the Universe simply have no detectable signs of life (Cockell 2014), either because they are uninhabited, are too young to have evolved life yet (< 1 Gyr based on the Earth's fossil record), have biotic chemistry at concentrations too low to detect, or the biotic atmospheric chemistry is indistinguishable from the abiotic. Planet rotation periods Cyanobacteria appeared in the fossil record ∼3.5 Gyr ago, just ∼1 Gyr after the Earth's formation. The concentration of O 2 remained at primordial values of 10 −3 of present atmospheric levels (PALs) until ∼2.1 Gyr, when a dramatic increase in atmospheric O 2 occurred -the so-called Great Oxidation Event (GOE). Earth's rotation period is currently 24 hours having been slowed by tidal interaction with the moon, but is likely to have been as low as 6 hours 4 Gyr ago (Lambeck 1980;Ćuk & Stewart 2012;. Earth's rotation period may therefore have increased by more than a factor of two since the evolution of photosynthesis. Recently, it has been postulated that the GOE occurred when Earth's daylength increased to ∼16 hours (Klatt et al. 2021). To test this hypothesis, Klatt et al. (2021) performed numerical simulations of movement of O 2 in cyanobacterial mats for daylengths between 12 and 52 hours, using simulated diel light cycle illumination. They found that longer days resulted in higher net O 2 flux through the cyanobacteria mat, and verified these results by taking measurements from real cyanobacterial colonies in controlled conditions. This lead them to conclude that increases in daylength could plausibly have influenced Earth's oxygenation, particularly around key oxidation events, and thus helped to pave the way for the evolution of plants and animals as we know them. At 1 Gyr, Earth's daylength was ∼ 13 hours with gravitational modelling predicting a spin-down to 16 hour days by ∼1.9 Gyr (late Archean). Within the next 0.3 Gyr the atmospheric O 2 concentration increased to ∼0.1 PAL. Assuming Earthlike biology, the work of Klatt et al. (2021) suggests that a planetary period of 16 hours could be an important factor in producing an oxygen-rich atmosphere to support life as we know it. Unfortunately, the rotation period, or daylength, is measured only for a handful of exoplanets (2M1207 b, PSO J318.5, GQ Lup b and β Pic b), all of which are massive, fast rotators (for a summary see Scholz et al. 2018). An empirical spin-mass relationship, where equatorial velocity is given by v eq ∝ √ M , is known to fit the solar system planets. However, both Mercury and Venus do not fit this trend due to tidal interactions with the Sun, and the Earth's tidal interactions with the moon also result in deviation from this relationship. The solar system trend is: v eq = A(M/M J ) 1 2 ,(14) with v eq in units of km s −1 and A = 13.1. The rotation period of the planet in seconds is then T = 2πR A M M J 1 2(15) where R is the planet radius. If the exoplanet radius is known (or estimated), then the period of the planet in 24-hour days is T = 0.632 M E M 1 2 R R E ,(16) and can therefore be predicted directly. However, for small radii planets (as we consider here), they generally do not also have associated mass measurements, but estimates can be made using a mass-radius relationship. We take the mass and radius values for low mass planets with an Earth-like composition from Table 1 of Fortney et al. (2007), which gives us a best-fit mass-radius relation of R p = M 0.27 p(17) with a high coefficient of determination R 2 = 0.98. We use this to estimate planet mass, given in the fifth column of Table 2. For Kepler-452 b, we calculate the rotation rate using the observed mass and retrieve 11 hours. For all other planets in our sample, we use the mass estimate from Eq. 17. The relationship is shown in Figure 3. If longer daylengths are required for atmospheric oxygenation, then our sample of planets may not yet be rotating slowly enough. If, however, their rotation has been slowed due to moons, then these planets may have the right conditions for their own GOE. It is useful to note that alternatively, if the radius is unknown and the mass is known, then a density can be assumed, and we can instead write: T = 0.632 M E M 1 6 ρ E ρ 1 3 ,(18) A tidally-locked planet will experience constant daylight on one half of its surface, and constant darkness (or reflected moonlight Lingam & Loeb 2020) on the other, and will therefore not experience the diurnal variation in light intensity of non tidally-locked planets. It is unclear whether this could be helpful or harmful to oxygenic photosynthesis. On the one hand, constant illumination means photosynthesis is always possible as long all other conditions allow, and on the other hand, dark respiration is not. A key question is therefore: is there an advantage to the light-dark cycle for life ? Tang & Vincent (2000) Figure 3. Mass-period relationship. Day lengths of planets either measured directly (blue points) or estimated through empirically determined relationship in Eq. 16 (green and yellow points). Yellow points indicate that mass was also estimated using Eq. 17. Grey solid lines are plots of Equation 18 for a plausible range of planet densities. Horizontal dashed lines mark the range of Earth's daylength predicted for the GOE (Klatt et al. 2021). the effects of daylight length and temperature on arctic cyanobacteria. The total daylength was held constant at 24 hours, and they varied the length of daytime, L, between 8 and 24 hours for three fixed temperatures of 5, 15, and 25 • C. The cyanobacteria growth rates increased with increasing L for 5 • C, but they plateaued with L for 10 • C and 15 • C, resulting in a reduction in net photosynthesis that was largest for the 24 hour daylight case. Unfortunately, the experimental errors on the measured respiration rates were too large to discriminate between the different conditions. However, other work has shown that dark respiration rates are at their peak shortly after the transition from lightness to darkness, and steeply decline as the period of darkness increases (Markager et al. 1992). In a similar vein, peak O 2 production (rather than respiration) was found to occur at different times in the 24 hour and 52 hour daylengths of Klatt et al. (2021). Peak O 2 production occurred before noon in the 52 hour daylength and after noon in the 24 hour daylength. More work is therefore required to determine how tidal locking may impact the photosynthesis-respiration cycle for the tidally locked planets in this work. CONCLUSION We have demonstrated the existence of a photosynthetic habitable zone (PHZ). It is the distance from the host star where the habitable zone overlaps with where photosynthesis is possible. We argue that the search for biosignatures of oxygenic photosynthesizing life forms should be concentrated in the PHZ if we expect photosynthesis in the Universe to proceed in a similar manner to photosynthesis on Earth. The PHZ becomes smaller with increasing atmospheric attenuation (i.e., more dense atmospheres), and so may make life less likely on super-Earths, since their larger gravitational field can hold onto more atmosphere. The PHZ also becomes smaller as the conditions for life become less favourable, which we describe as respiration rate relative to maximum possible photosynthetic rate, increasing. We therefore conclude that the parameter space for signs of life is far narrower than the standard HZ. Out of the nine scenarios we considered, we found TRAPPIST 1-e to be in the photosynthetic habitable zone for one scenario -little atmospheric affects and excellent conditions for life. However, it is almost certainly tidally locked, and it is not clear how or if photosynthetic life can proceed on tidally locked planets. Furthermore, the global circulation models of tidally-locked planets by Lobo et al. (2022) find that the HZ is limited to a narrow strip along the terminator for water-limited rocky planets. This reduces both the fraction of planet surface area for liquid water and cyanobacteria mats, and potentially the amount of water for photosynthesis. It may therefore not be the best place to focus the search for signs of life. We identify five planets, Kepler-452 b, Kepler-1638 b, Kepler-1544 b and Kepler-62 e and Kepler-62 f, that are consistently in the PHZ in a variety of environments. For Kepler-452 b, we calculate that it should have a rotation period of 11 hours. The other four planets are estimated to have rotation periods between 9 and 11 hours. We suggest the search for signs of life elsewhere in the Universe should begin in earnest on the candidate planets we have identified. ACKNOWLEDGEMENTS With special thanks from CH to Duncan H. Forgan (Forgan 2019). We thank the referee for their work reviewing this manuscript. CH also thanks Ken Rice, Nancy Kiang and Ethan Van Woerkom for insightful discussion. This study was supported in part by resources and technical expertise from the Georgia Advanced Computing Resource Center, a partnership between the University of Georgia's Office of the Vice President for Research and Office of the Vice President for Information Technology. This work has made use of the NASA Exoplanet Catalogue https://exoplanets.nasa. gov/discovery/exoplanet-catalog/, and the exoplanet archive https://exoplanetarchive.ipac.caltech.edu/ 2 http://perso.ens-lyon.fr/isabelle.baraffe/BHAC15dir/ BHAC15 iso.2massConstant Inner HZ Outer HZ (Runaway Greenhouse) (Max. Grenhouse ) S eff 1.107 0.356 a 1.332 × 10 −4 6.171 × 10 −5 b 1.580 × 10 −8 1.698 × 10 −9 c −8.308 × 10 −12 −3.198 × 10 −12 d −1.931 × 10 −15 −5.575 × 10 −16 Table 1. Coefficients used in Eq. 1 to calculate habitable stellar fluxes, and corresponding habitable zones. Table 2 . 2Properties of candidate exoplanets in the photosynthetic habitable zone assuming excellent conditions. Daylengths are estimated using the empirical relation of Equation 16, which is not valid if the planet is tidally-locked (TL). If the mass is not known, it is estimated using Eq. 17. Values from the NASA exoplanet archive.Planet a [au] M * [M ] M [ME] Mest. [ME] R [RE] P[days] day [hrs] day [hrs] (R = M 0.27 ) Calculated Estimated GJ-1061 d 0.054 0.125 1.64 1.73 1.16 13.00 TL TL GJ-667 Cc 0.125 0.330 3.81 4.95 1.54 28.10 TL TL K2-72e 0.106 0.270 2.21 2.57 1.29 24.20 TL TL Kepler-1229 b 0.290 0.430 - 3.48 1.40 86.80 - 11.39 Kepler-138 c 0.091 0.535 2.3 4.60 1.51 13.78 TL TL Kepler-138 d 0.129 0.525 2.1 2.03 1.21 23.09 TL TL Kepler-1544 b 0.542 0.810 - 8.46 1.78 168.80 - 9.28 Kepler-1638 b 0.745 0.970 - 10.16 1.87 259.30 - 8.90 Kepler-1649 c 0.083 0.198 - 1.24 1.06 19.50 - 14.43 Kepler-1652 b 0.165 0.404 - 5.70 1.60 38.10 - 10.16 Kepler-186 f 0.432 0.544 - 1.79 1.17 129.90 - 13.27 Kepler-283 c 0.341 0.664 - 9.19 1.82 92.70 - 9.11 Kepler-296 f 0.255 0.498 - 8.82 1.80 63.30 - 9.19 Kepler-442 b 0.409 0.610 - 3.04 1.35 112.30 - 11.75 Kepler-452 b 1.048 1.037 5 6.11 1.63 384.80 11.057 10.00 Kepler-62 e 0.427 0.690 - 5.83 1.61 122.40 - 10.11 Kepler-62 f 0.718 0.690 - 3.57 1.41 267.30 - 11.32 LHS-1140 b 0.096 0.179 6.38 6.25 1.64 24.70 TL TL LP 890-9 c 0.040 0.118 - 3.21 1.37 8.46 - 11.60 Luyten b 0.091 0.260 2.89 8.82 1.80 18.65 TL TL Proxima Centauri b 0.049 0.122 1.27 2.64 1.30 11.19 TL TL Teegarden's b 0.026 0.093 1.05 1.08 1.02 4.91 TL TL Teegarden's c 0.044 0.093 1.11 1.16 1.04 11.40 TL TL TOI-1452 b 0.061 0.249 4.82 6.71 1.67 11.06 TL TL TOI-700 d 0.163 0.416 1.72 1.62 1.14 37.40 TL TL TRAPPIST-1 d 0.022 0.090 0.39 0.40 0.78 4.05 TL TL TRAPPIST-1 e 0.029 0.090 0.69 0.73 0.92 6.10 TL TL TRAPPIST-1 f 0.038 0.090 1.04 1.16 1.04 9.21 TL TL TRAPPIST-1 g 0.047 0.090 1.32 1.57 1.13 12.40 TL TL explored this in an experiment probing /Users/cass/work/photosynth/python_files_used_in_paper/day_vs_mass.py Great oxidation event = . = . = .10 −4 10 −3 10 −2 10 −1 10 0 10 1 3lanet Pass [0 J ] 0.2 0.5 1.0 5RtatiRn periRd [days] ρ / ρ E 2 ρ / ρ E 1 ρ / ρ E 1 / 4 ρ / ρ E 1 / 8 ρ / ρ E 1 / 1 6 EstiPated rRt. rates 0easured rRt. rates Calculated rRt. rates Earth Mars Jupiter Neptune Pict b β PSO J318.5 2M1207 b Uranus Kepler- 452 b http://depts.washington.edu/naivpl/sites/default/files/hz 0. shtml#overlay-context=content/hz-calculator Y M Bar-On, R Phillips, R Milo, Proceedings of the National Academy of Sciences. the National Academy of Sciences1156506Bar-On, Y. 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{'abstract': 'It may be possible to detect biosignatures of photosynthesis in an exoplanet\'s atmosphere. However, such a detection would likely require a dedicated study, occupying a large amount of telescope time. It is therefore prudent, while searching for signs of life that we may recognise, to pick the best target possible. In this work, we present a new region, the "photosynthetic habitable zone" -the distance from a star where both liquid water and oxygenic photosynthesis can occur. It is therefore the region where detectable biosignatures of oxygenic photosynthesis are most likely to occur. Our analysis indicates that in the most ideal conditions for life and no atmospheric effects, the photosynthetic habitable zone is almost as broad as the habitable zone. On the other hand, if conditions for life are anything less than excellent and atmospheric effects are even moderate, the photosynthetic habitable zone is concentrated at larger separations around more massive stars. Such cases are also not tidally locked to their host star, which could result in planetary rotation periods similar to the Earth\'s. We identify five planets, Kepler-452 b, Kepler-1638 b, Kepler-1544 b and Kepler-62 e and Kepler-62 f, that are consistently in the photosynthetic habitable zone for a variety of conditions, and we predict their day lengths to be between 9 and 11 hours. We conclude that the parameter space in which we should search for signs of life is much narrower than the standard habitable zone.', 'arxivid': '2301.13836', 'author': ['C Hall \nDepartment of Physics and Astronomy\nThe University of Georgia\n30602AthensGAUSA\n\nCenter for Simulational Physics\nThe University of Georgia\n30602AthensGAUSA\n', 'P C Stancil \nDepartment of Physics and Astronomy\nThe University of Georgia\n30602AthensGAUSA\n\nCenter for Simulational Physics\nThe University of Georgia\n30602AthensGAUSA\n', 'J P Terry \nDepartment of Physics and Astronomy\nThe University of Georgia\n30602AthensGAUSA\n\nCenter for Simulational Physics\nThe University of Georgia\n30602AthensGAUSA\n', 'C K Ellison \nDepartment of Microbiology\nThe University of Georgia\n30602AthensGAUSA\n'], 'authoraffiliation': ['Department of Physics and Astronomy\nThe University of Georgia\n30602AthensGAUSA', 'Center for Simulational Physics\nThe University of Georgia\n30602AthensGAUSA', 'Department of Physics and Astronomy\nThe University of Georgia\n30602AthensGAUSA', 'Center for Simulational Physics\nThe University of Georgia\n30602AthensGAUSA', 'Department of Physics and Astronomy\nThe University of Georgia\n30602AthensGAUSA', 'Center for Simulational Physics\nThe University of Georgia\n30602AthensGAUSA', 'Department of Microbiology\nThe University of Georgia\n30602AthensGAUSA'], 'corpusid': 256415889, 'doi': '10.3847/2041-8213/acccfb', 'github_urls': [], 'n_tokens_mistral': 16957, 'n_tokens_neox': 13731, 'n_words': 7864, 'pdfsha': '8744888c07e697a6772ab2339998743a034310c9', 'pdfurls': ['https://export.arxiv.org/pdf/2301.13836v2.pdf'], 'title': ['A New Definition of Exoplanet Habitability: Introducing the Photosynthetic Habitable Zone', 'A New Definition of Exoplanet Habitability: Introducing the Photosynthetic Habitable Zone'], 'venue': []}

arxiv

Holomorphic Yang-Mills Theory and Variation of the Donaldson Invariants arXiv:hep-th/9503036v2 17 Mar 1995 Seungjoon Hyun hyun@phya.yonsei.ac.kr††e-mail:j.park@swansea.ac.uk Jae-Suk Park Institute for Mathematical Sciences Department of Physics Theoretical Physics Group Yonsei University 120-749SeoulKorea University of Wales SA2 8PPSwansea SwanseaUK Holomorphic Yang-Mills Theory and Variation of the Donaldson Invariants arXiv:hep-th/9503036v2 17 Mar 1995 We study the path integrals of the holomorphic Yang-Mills theory on compact Kähler surface with b + 2 = 1. Based on the results, we examine the correlation functions of the topological Yang-Mills theory and the corresponding Donaldson invariants as well as their transition formulas.Feb, 1995 † Introduction The Donaldson polynomial invariants are powerful tools for classifying the smooth structures of four-manifolds [1] [2]. For a Riemann manifold with b + 2 ≥ 3, these polynomials are well-defined and metric independent, as long as it is generic. On the other hand, for a manifold with b + 2 = 1, the polynomials depend on metric in a very interesting way. This is due to the reducible anti-self-dual (ASD) connections which appear at finite points of the generic smooth path connecting two generic metrics. The first example was studied by Donaldson in his seminal paper on the failure of the h-cobordism conjecture [3]. His formula was further studied in detail by Friedman-Morgan [4], and generalized by Kotschick [5], Mong [6], and Kotschick-Morgan [7]. The topological Yang-Mills (TYM) theory proposed by Witten is a field theoretic interpretation of the Donaldson invariants [8]. The relation with the theory to the BRST quantization were much studied in [9][10] [11] [12][13] [14]. 1 It is also a new mathematical viewpoint on the Donaldson invariants due to the Atiyah-Jeffrey re-interpretation [16] of the Witten's approach as an infinite dimensional generalization of the Mathai-Quillen representative [17] of the equivariant Thom class. The TYM theory on compact Kähler surface was studied in some detail by the second author [18]. It was shown that the theory has two global fermionic symmetries. 2 The theory can be interpreted as an infinite dimensional version of a Dolbeault equivariant cohomological analogue of the Mathai-Quillen formalism [20] [21]. Witten studied the Kähler case by twisting the N = 2 super-Yang-Mills theory and determined the Donaldson Polynomial invariants for Kähler surfaces with b + 2 ≥ 3 almost completely [22]. His approach gives yet another new perspective on the Donaldson theory since he related the theory to the infrared behavior of the N = 1 super-Yang-Mills theory. On the other hand, to the authors' knowledge, no serious attempt has been made for the field theoretic study of the Donaldson polynomial invariants for manifolds with b + 2 = 1. 3 The TYM theory may not be well-defined in the presence of the reducible 1 We refer to ref. [15] as a review and for further references. 2 The first formulation of the TYM theory on Kähler surface was given by [19]. 3 In ref. [23], it was shown that the variation of the Donaldson invariants is related to the holomorphic anomaly. anti-self-dual (ASD) connections. The reducible connections also contribute to the noncompactness of the space where the path integral is localized, due to the flat directions of the scalar potential. This makes the topological interpretation of the theory unclear [24]. Furthermore, the crucial perturbation of Witten [22], inducing the mass gap to the theory, is not applicable to this case 4 . In this paper, we study the SU (2) Donaldson polynomial invariants of a simply connected compact Kähler surface X with the vanishing geometric genus p g (X) = 0, i.e., b + 2 (X) = 1, based on the holomorphic Yang-Mills (HYM) theory [25]. The HYM theory was proposed by the second author to provide yet another field theoretic interpretation of the Donaldson invariants of Kähler surface, adopting the two dimensional construction of Witten [26]. It was shown that there exists a simple mapping to TYM theory on Kähler surface analoguos to the mapping from two-dimensional TYM theory to the physical YM theory. It turns out that the HYM theory is more useful for manifold with p g = 0 5 . A la HYM theory we will show that the path integral approach to the Donaldson theory is well-defined whatever properties the moduli space of ASD connections has [8]. The basic idea of the HYM theory is that the moduli space of ASD connections over Kähler surface is the symplectic quotients in the space A 1,1 of all holomorphic connections. Classically, the action functional of the HYM theory is the norm squared of moment map. The basic method of our calculation is the fixed point theorem of Witten [27]. Unfortunately, we could calculate only one of the two branches of the fixed points. The branch we calculate does not contain any information on the genuine diffeomorphism invariants. 4 In ref. [20], we have exploited the origin of the mass gap in terms of the Dolbeault model of the equivariant cohomology. 5 The HYM theory was also used to determine the non-algebraic part of the Donaldson invariants of manifold with H 2,0 = 0 [20]. While we were struggling with this problem, the Donaldson theory evolved into the new dimension by the fundamental works of Seiberg and Witten on the strong coupling properties of the N = 2 super-Yang-Mills theory [28] [29]. By exploiting a dual Donaldson theory (Seiberg-Witten theory), Witten determined the invariants completely for Kähler surface with b + 2 ≥ 3 [30]. This new method resolves all of the difficulties of the Donaldson theory as well as the TYM theory, such as the non-compactness of the moduli space of ASD connections. For a manifold with b + 2 = 1, Witten's new approach has an additional complication comparing to the case with b + 2 ≥ 3 [30]. The calculation we have done in this paper can be viewed as the path integral contributed from the generic points of the quantum moduli space (the complex u-plane in [30]) except the two singular points. We will argue that the remaining branch of the partition function of HYM theory may be obtained by the Seiberg-Witten invariants which corresponds to the path integral contributed from the two singular points in the quantum moduli space. As a matter of fact, the expectation values of TYM theory are not identical with the corresponding Donaldson invariants. The difference is originated from the compactification of the moduli space ASD connections in the mathematical definition of the Donaldson invariants. In the field theoretic approach, both TYM and HYM theories do not refer to any compactification procedure. In any case, the path integrals are turned out to be well-defined though the moduli space is rarely compact. To recover the original Donaldson invariants some explicit prescription for incorporation of the compactification may be required. There are many reasons that the only difference between the two approaches is that the Donaldson theory has additional contributions due to reducible connections with lower instanton numbers. However, the additional part does not have any relevant information on the diffeomorphism class of the underlying manifolds. We will interpret our results as the contributions of the original moduli space of ASD connections which is the top stratum of the compactified moduli space. Then, our results can be used to determine the transition formula of the Donaldson invariants. In some respects our results are more general than the known mathematical results. We also determine the non-analytic (non-polynomial) property of the invariants when the moduli space is singular. We believe that our formula gives a very clear and an intuitive understanding of the mechanism of the variation of the Donaldson invariants. This allows us to predict a rather detailed formula of the Donaldson invariants if the transition formula is known. The invariants and the explicit transition formulas for some special cases were calculated in [31] [32]. While we are preparing this paper the general transition formula appeared in term of some enumerative geometry of the Hilbert schemes [33][34] [35]. The paper of Friedman and Qin [33] contains some explicit results on the transition formula. The paper of Ellingsrud and Göttsche [34]contains more general explicit results. 6 . Their mathematical picture on the variation of the moduli space is quite similar to our physical picture. At the moment, the invariants are calculated only for some special cases and no general explicit transition formula is known [31] [32]. This paper is organized as follows. In Sect. 2, we review the Donaldson polynomial invariants, the TYM theory and the HYM theory. We show that the HYM theory is a natural field theoretic model for the Donaldson invariants of manifolds with p g = 0. In Sect. 3, we give the path integral calculation of the HYM theory and obtain an effective theory. In Sect. 4, we examine the small coupling behavior of the partition function and expectation values of topological observables. Then, we study the variation of the path integral according to the variation of the metric. In Sect. 5, we obtain the expectation values in the TYM theory from the result of HYM theory. We argue that the unknown branch of the path integral corresponds to the Seiberg-Witten monopole invariants. After a brief discussion of some properties of the monopole invariants, we show that the expectation values of TYM theory depends only on certain chamber structures in the space of metric. In Sect. 6, we discuss some natural implications of our results on the Donaldson invariant and its transition formula, as well as some conjectures and speculations. The Donaldson Polynomials and Holomorphic Yang-Mills Theory In this section, we review the Donaldson theory, TYM theory and HYM theory on a manifold with b + 2 (X) = 1. Throughout this paper, for simplicity, we consider a simply connected compact algebraic surface X and the SU (2) polynomial invariants only. There is no loss of generality to consider the algebraic surface only since any Kähler surface deforms to an algebraic one. We hope we can treat more general gauge group in the future. 6 We would like to thank L. Göttsche for pointing us out, after our first version, the expanded version of [34] which contains the explicit results. Let X be a simply connected projective algebraic surface with Kähler-Hodge metric g and associated Kähler form ω of type (1,1). Since X is a projective space, we have an ample line bundle H over X whose first Chern Class c 1 (H) ∈ H 2 (X; Z) is given by the Kähler form ω. We will occasionally confuse a line bundle with its first Chern class as well as with the the corresponding divisor. By the Kodaria projective embedding theorem, sections of some positive tensor power H ⊗m of the ample line bundle define projective embedding of X to some complex projective space whose hyperplane section class is Poincaré dual to m[ω]. A line bundle L is ample if and only if L · L > 0 and L · c > 0 for every complex curve c in X, where ' · ' denotes the intersection pairings, i.e., L · L ≡ X c 1 (L) ∧ c 1 (L). They defines a positive cone called ample cone (or Kähler cone) Cone X in the space of H 2 (X; R). For a given metric we can decompose the space H 2 (X) of harmonic forms on X into the space H 2 + (X) of self-dual and the space H 2 − (X) of anti-self-dual harmonic forms. Similarly, we have b 2 = b + 2 + b − 2 where b 2 = dim H 2 (X) and b ± 2 = dim H 2 ± (X). From the Hodge index theorem, we have b + 2 = 1 + 2p g , (2.1) where the one-dimensional piece is spanned by the Kähler form ω and the geometric genus p g is the number of (harmonic) holomorphic two-forms. For an arbitrary vector bundle valued two-form we have the similar decompositions, i.e. the self-dual part of the curvature two-form F + (A) can be decomposed as F + (A) = F 2,0 (A) + f (A)ω + F 0,2 (A), (2.2) where f (A) = 1 2 ΛF 1,1 (A). Here Λ denotes the algebraic trace operator which is adjoint to the wedge multiplication of the Kähler form. The intersection form q X between harmonic two-forms is an unimodular (due to the Poincaré duality) and symmetric b 2 × b 2 matrix with signature (b + 2 − b − 2 ) . That is, upon diagonalization of q X it has b + 2 positive entries and b − 2 negative entries. It is also called of type (b + 2 , b − 2 ). This comes from the simple fact that q X (α, α) = α · α = X α ∧ α = X α + ∧ α + + X α − ∧ α − = X α + ∧ * α + − X α − ∧ * α − = X |α + | 2 dµ − X |α − | 2 dµ, (2.3) where * is the Hodge star operator and dµ = ω 2 /2! is the volume form. Thus X has b + 2 = 1 if and only if p g = 0 and its intersection form is of type (1, b − 2 ). The Donaldson polynomial invariants Let E be a complex vector bundle over X with the reduction of the structure group to SU (2). The bundle E is classified by the instanton number k, k =< c 2 (E), X >= 1 8π 2 X Tr (F ∧ F ) ∈ Z + , (2.4) where Tr is the trace in the 2-dimensional representation and Tr (ξ) 2 = −|ξ| 2 on su (2). Let A k be the space of all connections of E and A 1,1 k ∈ A k be the subspace consisting of the connections whose curvatures are of type (1, 1), i.e. A ∈ A 1,1 k iff F 0,2 (A) = 0. We will call a connection A holomorphic if A ∈ A 1,1 k . Let G be the group of the gauge transformations. We denote M k (g) to the moduli space ASD connections with respect to a (Riemann) metric g. We denote A * k the space of irreducible connections, and thus, A * 1,1 k ≡ A * k ∩ A 1,1 k and M * k (g) ≡ A * k ∩ M k (g). The virtual complex dimension of M k (g) is d = 4k − 3. We briefly review a definition of the Donaldson polynomial invariants 7 . For a manifold with b + 2 > 0, there are no reducible ASD connections for a generic choice of the metric. Let g be such a generic metric. Then M k (g) = M * k (g) is a smooth manifold with actual complex dimension d. Since the moduli space is rarely compact one should compactify M * µ : H 2 (X; Z) −→ H 2 (M * k (g); Z) (2.6) 7 There are several other more conceptually elaborated and powerful definitions. For details, the reader can consult the excellent book [2]. The definition of the invariants in this introduction is to emphasize that one should also take care of the reducible ASD connections with lower instanton numbers. 8 For manifold with p g = 0, it was recently shown that the compactified moduli space M k can be identified with the total space itself [36]. For k > 1 (the stable range), the compactified moduli space carries the fundamental homology class. The Donaldson polynomial q X,g,k , q X,g,k (Σ 1 , ..., Σ 4k−3 ) =< µ(Σ 1 ) ⌣ ... ⌣ µ(Σ 4k−3 ), [M k (g)] > (2.8) defines the map; H 2 (X, Z) × · · · × H 2 (X, Z) → Z, which is well-defined for the generic metric g. The above definition of the Donaldson polynomial is basically the same as those for manifolds with b + 2 > 1. The special feature of the manifold with b + 2 = 1 is that the polynomial actually depends on the metric. To show that the polynomial (2.8) is metric independent, one should consider a smooth generic path g t of metrics joining the two generic metrics and show that its value does not change according to the variation of the metric. This amounts to show that the one parameter family of the moduli space M k (g t ) or rather its compactification M k (g t ) does not depend on g t at the level of homology. This can be ensured if there is no point in g t which admits the reducible ASD connections. A SU (2) connection is reducible if the SU (2)-bundle reduces to direct sum of line bundles E = ζ ⊕ ζ −1 . Since the bundle E is classified by the second Chern-Class, from the Whitney formula, we have the following condition for the bundle reduction k =< c 2 (E), X >= −c 1 (ζ) · c 1 (ζ −1 ) = −c 1 (ζ) · c 1 (ζ) = −ζ · ζ (2.9) where c 1 (ζ) = H 2 (X, Z). Since k should be positive definite to admit ASD connection, the self-intersection of line bundle should be negative definite to solve above equation. Let ω g be the family of the self-dual two-forms associated with a metric g. The metric admits reducible ASD connection if and only if X c 1 (ζ) ∧ ω g = 0. (2.10) If the number of self-dual harmonic form, whose self-intersection is positive definite, is greater than 0, we can avoid the reducible connection by perturbing metric such that there are no reducible connection for generic choice of metric. Now we consider a smooth generic path g t of the metric joining two generic metrics. For manifold with b + 2 > 1 such a path can always avoid metric admitting the reducible ASD connection. However, for manifold with b + 2 = 1 there can be at least finite number of the points in g t which admit reducible ASD connections since the subspace of ASD two-forms in the space H 2 (X; R) has the codimension one. Since we are considering the compactified moduli space, we should also consider all the bundle reductions given by −k ≤ ζ · ζ ≤ −1, (2.11) where ζ ∈ H 2 (X; Z). Then, the compactified moduli space M k (g) does not contain any reducible ASD connection if and only if X c 1 (ζ) ∧ ω g = 0 for all ζ satisfying (2.11). A proper definition of the Donaldson polynomials requires the systematic understanding of the appearance of the reducible ASD connection as one varies the metric. The chamber structure Let Ω X be the positive cone in H 2 (X; R) defined by Ω X = {θ ∈ H 2 (X; R)|θ · θ > 0}. (2.12) Since the intersection form q X is of type (1, n) the positive cone has two connected components. For each element ζ satisfying (2.11) one defines the wall W ζ = W −ζ by the intersection of the hyperplane ζ ⊥ ∈ H 2 (X; R) orthogonal to ζ, i.e., ζ · ζ ⊥ = 0, with Ω X . We denote W ℓ by the collection of walls defined by all ζ ∈ H 2 (X; Z) satisfying ζ · ζ = −ℓ. We also denote the system W k of walls by W k = 1≤ℓ≤k W ℓ . (2.13) The set C k X of chambers 9 is the set of the connected components of Ω X after removing W k . 9 The physicist reader may find it easy to understand the chamber structure by an analogy with (1 + n) dimensional Minkowski space with metric diag(1, −1, . . . , −1) . This analogy is rigorous if the intersection form is odd. The vector space H 2 (X; R) corresponds to the Minkowski space. The intersection form is just the metric form and the positive cone corresponds to the future lightcone. We consider the future cone which contains the ample cone. An integral class ζ ∈ H 2 (X; Z) corresponds to a vector and its intersection number to the norm squared of the vector in the Minkowski space. An integral class with negative self-intersection number corresponds to a spacelike vector. Then the wall is defined by a space-like hyperplane. Definitely, the hyperplane orthogonal to a vector intersect with the time-like space if and only if the vector is space-like. Without loss of generality one can only consider one of the two connected components of Ω X which contains the Kähler cone. It is convenient to choose a level set H(q) ∈ Ω X defined by the n-dimensional hyperbolic space satisfying θ · θ = 1. A metric g determines a line 10 in Ω X made up of the cohomology classes represented by g-self-dual harmonic two Fig. 1. A typical chamber structure [3] for a manifold X of type (1, 2) and k = 1 i.e., ζ ·ζ = −1. forms ω g . Let [ω g ]H(q) W g C [ω ] W ζ g The right-hand side is the the Poincaré model for H(q) and C denote the chamber containing ω g . The pattern repeats to infinity. For a smooth path g t of metrics we have the corresponding path [ω g t ] in H(q). Although the moduli space M k (g t ) and its compactification certainly depends on t, it may not be changed at the level of homology. If [ω g t ] is contained in a chamber, the homology class of [M k (g t )] does not depend on g t . On the other hand, if ω g t crosses the walls, special things happen such that the moduli space is changed even at the level of homology. The variation of the homology class of the moduli space is essentially due to the appearance of the reducible ASD connection. 10 Note that the space of self-dual harmonic two-forms is one-dimensional. After picking a generic metric g such that [ω g ] lies in one of the chambers C ∈ C k X and consider q X,g,k (Σ 4k−3 ) (2.14) an element of Sym 4k−3 (H 2 (X; Z)), we can define the map Γ k X : C → Sym 4k−3 (H 2 (X; Z)),(2.15) which depends only on the chamber structure C k X in Ω X . The SU (2)-invariants of X introduced by Donaldson [3] and extended by Mong [6] and by Kotschick-Morgan [5] [7] are the assignments Γ k X : C k X → Sym 4k−3 (H 2 (X, Z)), (2.16) The polynomial Γ k X (C) depends only on the chamber with following properties: i) Γ k X (−C) = −Γ k X (C) ii) If f : X 1 → X 2 is an orientation preserving diffeomorphism between two such mani- folds, then Γ k X 1 (f * (C)) = f * (Γ k X 2 (C)). The computation of the Donaldson invariant amounts to determine the invariant in a certain chamber and to find a general transition formula its variations when [ω g t ] cross the walls. The Donaldson polynomial invariants q X,g,k (Σ 4k−3 ) may not be well-defined if [ω g ] lies one of the walls due to the singularity in the moduli space. One can also extend the definition of the invariants including the four-dimensional class such that q X,g,k (Σ 4k−3−2r (pt) r ). We will denote Γ k,r X (C) for the corresponding assignments which also depend only on the chamber structure. This extension becomes more problematic when the moduli space has singularity. The topological Yang-Mills theory To begin with, we recall the N = 2 TYM theory on compact Kähler surfaces [18][22] [20]. The theory has N = 2 global supersymmetry whose conserved charges s ands can be identified with the operators of G-equivariant Dolbeault cohomology of A. The algebra for the basic multiplet (A ′ , A ′′ , ψ,ψ, ϕ) [18] is sA ′ = −ψ, sA ′ = 0, sA ′′ = 0, sA ′′ = −ψ, sψ = 0, sψ = −i∂ A ϕ, sψ = −i∂ A ϕ, sψ = 0,s ϕ = 0, sϕ = 0, (2.17) where A ′ and A ′′ denote the holomorphic and anti-holomorphic parts of the connection one-form A = A ′ + A ′′ , ψ ∈ Ω 1,0 (g E ),ψ ∈ Ω 0,1 (g E ) and ϕ ∈ Ω 0 (g E ). Note that ψ can be identified with holomorphic (co)tangent vectors on A. The fields have additional quantum numbers characterized by the degree ( * , * ). The operator s carries the degree (1, 0) and s carries the degree (0, 1). Assigning the degree (0, 0) to the connection A, ϕ is of degree (1,1). In terms of the equivariant cohomology the above algebra can be represented as follows. We let Ω * , * (A) be the Dolbeault complex on A. Now we interpret Fun(Lie(G)) to the algebra of polynomial functions generated by ϕ a . Then the desired Dolbeault model of the G-equivariant complex is Ω * , * G = (Ω * , * (A) ⊗ Fun(G)) G . The associated differential operators with the degrees (1, 0) and (0, 1) are s ands, represented by s = − i ψ i ∂ ∂A ′i + i ī ,a ϕ a Vī a ∂ ∂ψī , s = − īψī ∂ ∂A ′′ī + i i,a ϕ a V i a ∂ ∂ψ i , (2.18) where i,ī are the local holomorphic and anti-holomorphic indices tangent to A, respectively. We have s 2 = 0, ss +ss = −iϕ a L a ,s 2 = 0, (2.19) Thus, {s,s} = 0 on the G-invariant subspace Ω * , * G of Ω * , * (A) ⊗ Fun(Lie(G)). We define the G-equivariant Dolbeault cohomology H * , * G (A) by the pairs (Ω * , * G (A),s). It was shown that for manifold with p g = 0 the s-cohomology is isomorphic to thes cohomology [20]. The action functional of N = 2 TYM theory can be viewed as the Dolbeault equivariant cohomological version of the Mathai-Quillen representative of the universal Thom class 11 of the infinite dimensional bundle A → A/G. We consider the vector space V of g E -valued self-dual two-forms with a linear G action on it. Then we can form a homology quotient (2.20) where F + (A) denotes the self-dual part of the curvature two form F (A) and defines a section s of E G . Now, the moduli space of ASD connections is the zero set of the section s. In the Kähler geometry, we can decompose the vector space V into the vector spaces E G = A × G V . Then there exists a equivariant map F + : A −→ V by A −→ F + (A),V = V 2,0 ⊕ V 1,1 ω ⊕ V 0,2 of (2, 0)-forms, (1, 1)-forms proportional to the Kähler form and (0, 2)-forms. Using the natural complex structure on A induced from X, we can also decompose the section s (the equivariant map) into F 2,0 : A ′ −→ V 2,0 by A ′ −→ F 2,0 (A ′ ), F 1,1 ω : A −→ V 1,1 ω by A −→ f (A ′ , A ′′ )ω, F 0,2 : A ′′ −→ V 0,2 by A ′′ −→ F 0,2 (A ′′ ), (2.21) where f (A) = 1 2 ΛF 1,1 (A). To write the Mathai-Quillen representative of the Universal Thom class, we should introduce the set of anti-ghost multiplets for each components of the above map. Geometrically, the anti-ghost multiplets are various equivariant differential forms living in the dual vector space of V and their spectrum and algebra can be uniquely determined in terms of the Dolbeault model of the equivariant cohomology [21](see also [18] [20]). This leads to a commuting anti-ghostφ ∈ Ω 0 (g E ) living in the dual vector space V * 1,1 ω with degree (−1, −1). Then we have multiplet (φ, iχ 0 , −iχ 0 , H 0 ) with transformation laws sφ = −iχ 0 , sφ = iχ 0 , sχ 0 = H 0 − 1 2 [ϕ,φ], sχ 0 = H 0 + 1 2 [ϕ,φ], sχ 0 = 0, sχ 0 = 0, sH 0 = − i 2 [ϕ, χ 0 ], sH 0 = − i 2 [ϕ,χ 0 ]. (2.22) We also have an anti-commuting anti-ghost χ 2,0 in the dual vector space V 2,0 with degree (−1, 0) and an anti-commuting anti-ghostχ 0,2 living in the dual vector space V * 0,2 with degree (0, −1) with transformation laws sχ 2,0 = 0, sχ 2,0 = H 2,0 , sχ 0,2 = H 0,2 , sχ 0,2 = 0, sH 2,0 = −i[ϕ, χ 2,0 ],sH 2,0 = 0, sH 0,2 = 0, sH 0,2 = −i[ϕ,χ 0,2 ]. (2.23) The action functional (the universal Thom class) is given by S = − is 1 h 2 Trχ 0,2 ∧ * F 2,0 − is 1 h 2 Tr χ 2,0 ∧ * F 0,2 − (ss −ss) 1 h 2 Tr χ 2,0 ∧ * χ 0,2 − (ss −ss) 2 1 h 2 Tr φf + χ 0χ0 ω 2 . (2.24) A small calculation gives S = 1 h 2 X Tr − 1 2 F 2,0 ∧ * F 0,2 + iχ 2,0 ∧ * ∂ Aψ + iχ 0,2 ∧ * ∂ A ψ − 2i[ϕ, χ 2,0 ] ∧ * χ 0,2 − 1 2 f 2 − 2i[ϕ, χ 0 ]χ 0 −χ 0 ∂ * A ψ + χ 0∂ * Aψ − 1 2 [ϕ,φ] 2 + 1 2φ d * A d A ϕ − 2iΛ[ψ,ψ] ω 2 2! . (2.25) where we have integrated out auxiliary fields H 2,0 , H 0 , H 0,2 and used the Kähler identities, ∂ * A = i[∂ A , Λ], ∂ * A = −i[∂ A , Λ], (2.26) The bosonic kinetic terms of the action is S = − 1 h 2 Tr 1 2 |F + | 2 + 1 2 |d A ϕ| 2 dµ + . . . . (2.27) The first term is the norm square of the equivariant section of E G . In the h 2 → 0, the dominant contribution of the path integral comes from the instantons F + (A) = 0 and the configuration satisfying d A ϕ = 0. (2.28) If there is a non-zero solution ϕ of the equation (2.28) , it means the instanton A is reducible (abelian). Then the SU (2) group reduces to the U (1) subgroup, ϕ = ϕ c T 3 = − i 2 ϕ c 0 0 −ϕ c ∈ su(2). (2.29) Note also that, for reducible connection, the superpotential term + 1 h 2 Tr 1 2 [ϕ,φ] 2 dµ,(2.30) vanishes, which corresponds to the flat direction in the physical terms. If the reducible instantons appear, the path integral has additional contribution from the vector space of the ϕ-zero-modes. Of course the moduli space of ASD connections becomes singular. Then, the semi-classical description breaks down due to the singularity and the topological interpretation of the theory can be invalidated due to the non-compactness of the vector space of the ϕ-zero-modes [24]. One can also view the localization of the path integral by the fixed point locus;    sψ = −i∂ A ϕ = 0 sψ = −i∂ A ϕ = 0 =⇒ d A ϕ = 0                          sχ 0,2 = − i 2 F 2,0 (A) = 0 sχ 2,0 = − i 2 F 0,2 (A) = 0 sχ 0 = − i 2 f (A) − 1 2 [ϕ,φ] = 0 sχ 0 = − i 2 f (A) + 1 2 [ϕ,φ] = 0 =⇒    F + A = 0 [ϕ,φ] 2 = 0 (2.31) Picking a two-dimensional class Σ ∈ H 2 (X; Z), one can define a topological observable µ(Σ) ≡ 1 4π 2 Σ Tr iϕF + ψ ∧ψ ≡ 1 4π 2 X Tr iϕF + ψ ∧ψ ∧ α Σ , (2.32) where α Σ ∈ H 2 (X; Z) is Poincaré dual to Σ. This observable is the field theoretic representation of the Donaldson's µ-map (2.6). One also defines the observable Θ corresponding to the four-dimensional class, µ(pt), Θ = 1 8π 2 X ω 2 2! Tr φ 2 . (2.33) Assuming that there is no reducible instanton, the correlation function µ(Σ 1 ) · · · µ(S s )Θ r = 1 vol(G) DX e −S µ(Σ 1 ) · · · µ(Σ s )Θ r ,(2.34) is the path integral representation of the Donaldson invariant q k,X (Σ 1 , . . . , Σ s , (pt) r ). Due to the ghost number anomaly, the correlation function (2.34) always vanish unless dim C M k = s + 2r. It is not entirely clear how the path integral (2.34) takes care of the non-compactness of the instanton moduli space. However, at least for manifolds with b + 2 ≥ 3, Witten's explicit results show that the path integral correctly leads to the Donaldson invariants. Clearly the path integral localizes to the moduli space of ASD connections rather than the compactified one. We do not know any clear reasoning why the path integral correctly reproduce the Donaldson invariant. One may argue that the additional space added for the compactification does not contribute to the Donaldson invariants. In fact more elaborated definitions of the invariants, compared to that of (2.8), clearly indicate such a property under certain conditions [1] [2]. For a manifold with b + 2 = 1, on the other hand, the path integral does not exactly represent the Donaldson invariants. This can be easily seen if one considers the reducible connections. Clearly the path integral localizes to the moduli space of ASD connections rather than the compactified one. In the path integral approach, then, we only need to worry about the reducible connections corresponding to the bundle reduction (2.9). In any case, one can insist that the path integral approach is well-defined whatever properties the instanton moduli space has. We believe that Witten's explicit results on the b + 2 ≥ 3 cases and our partial result in this paper for b + 2 = 1 support such a viewpoint. We would like to add the following remarks. i) The form (2.24) of the action functional has been uniquely determined. We could correctly recover every term in the action functional of the N = 2 super-Yang-Mills theory. The global supersymmetry transformation laws and the action functional are rigorously identical to those of the twisted theory [22]. On the other hand, The usual approach based on the de Rham model of the equivariant cohomology or the N = 1 global supersymmetry does not leads to the complete determination of the anti-ghost multiplets and the correct action functional. One should add, so called, projection term and non-minimal term [41]. ii) The usual approach to TYM theory on Kähler surface based on the N = 1 global supersymmetry can not explain the perturbation of Witten, breaking the N = 2 symmetry down to N = 1 symmetry [22] [20]. Unfortunately, the perturbation is not applicable for manifolds with p g = 0. The holomorphic Yang-Mills theory An obvious way out of the difficulty of the TYM theory with the reducible connections is to eliminate the zero-mode of ϕ from the theory as originally suggested by Witten in the two-dimensional model of the TYM theory [26]. The remarkable fact is that his method eventually leads to a non-abelian localization theorem of the theory of the equivariant (de Rham) cohomology 12 . The HYM theory is an analogous prescription for the Donaldson theory and it is related to a Dolbeault equivariant cohomological version of the non-Abelian localization theorem [18] [20]. Consequently the HYM theory is a suitable model for the Donaldson invariants on Kähler surface with b + 2 = 1. The basic observation is that the reduction of the path integral of the TYM theory to the instanton moduli space is achieved by the following two steps; i) restriction of A to A 1,1 , ii) restriction of A 1,1 to the solution space of ASD connections and reduction to the instanton moduli space by dividing by the gauge group G. Then the second step can be replaced with the symplectic reduction. Now we we can deform the (1, 1) part of the action by the one-parameter family of the action, S(t) = S + t(ss −ss) − 1 h 2 M ω 2 2! Trφ 2 , (2.35) where t is a real positive deformation parameter. After some Gaussian integrals we are left with 13 S(t) = − is 1 h 2 Trχ 0,2 ∧ * F 2,0 − is 1 h 2 Tr χ 2,0 ∧ * F 0,2 + ss − ss 2 1 2h 2 t M ω 2 2! Tr f 2 . (2.36) The action functional of the HYM theory is defined by S(t) H = 1 h 2 X Tr −iH 2,0 ∧ * F 0,2 − iH 0,2 ∧ * F 2,0 + iχ 2,0 ∧ * ∂ Aψ + iχ 0,2 ∧ * ∂ A ψ − 1 4π 2 X Tr iϕF + ψ ∧ψ ∧ ω − ε 8π 2 X ω 2 2! Tr ϕ 2 + 1 4π 2 ε X Tr (F 2,0 ∧ F 0,2 + 1 2 F 1,1 ∧ F 1,1 ) + ss − ss 2 1 2h 2 t M ω 2 2! Tr f 2 ,(2.37) where ε is positive number. Several remarks are in order. 12 The equivariant localization was studied in the mathematical literatures [ [50] 13 Here, we choose the delta-function gauge for simplicity. i) The first line of the action is identical to the part of the TYM action (Thom class) which leads to a clear cut reduction of T * A to T * A 1,1 without any quantum correction. Thus we can regard the theory as the one defined on T * A 1,1 . Similar hybrid model of the de Rahm model was suggested in [51] independently to [25]. ii) The termω ≡ 1 4π 2 X Tr iϕF + ψ ∧ψ ∧ ω (2.38) is the equivariant extension of the Kähler form 1 4π 2 X Tr (ψ ∧ψ) ∧ ω on A 1,1 . It also define a two dimensional class µ(H) of the Donaldson invariants associated to the ample class H. iii) The term Θ ≡ 1 8π 2 X ω 2 2! Tr ϕ 2 (2.39) is the four dimensional class of the Donaldson invariant. iv) The term 1 2h 2 t M ω 2 2! Tr f 2 is proportional to the norm squared < ,> of the moment map m : A 1,1 → Ω 0 (g E ) * , m(A) = − 1 4π 2 F 1,1 A ∧ ω = − 1 4π 2 f ω 2 , (2.40) where Ω 0 (g E ) * = Ω 4 (g E ) denotes dual of Ω 0 (g E ) = Lie(G). Since the path integral is independent of t, we can set t → 0, and hence the path integral gets contributions only from the critical set of the function < m, m >, i.e., < f, d A f >= 0. We set t → ∞ so that we can omit the equivariantly exact form. The resulting action, S H = − 1 4π 2 X Tr iϕF 1,1 + ψ ∧ψ ∧ ω − ε 8π 2 X ω 2 2! Tr ϕ 2 + k ε + · · · ,(2.Z(ε, k) = e − k ε × 1 vol(G) T * A 1,1 DA ′ DA ′′ Dψ Dψ Dϕ × exp 1 4π 2 M Tr iϕF 1,1 + ψ ∧ψ ∧ ω + ε 8π 2 M ω 2 2! Tr ϕ 2 . (2.43) In the limit ε → 0, the partition function is related to certain expectation value of the original TYM theory, up to the exponentially small terms, Z(ε, k) = e − k ε exp(ω + εΘ) + exponentially small terms = e − k ε [(4k−3)/2] r=0 ε r (d − 2r)!r! ω d−2r Θ r +exponentially small terms,(2.44) We assumed that the path integrals are defined with respect to the metric whose Kähler form lies in one of the chambers. Otherwise, as we will show in Sect. 4, the partition function contains a non-analytic term proportional to ε 2k−3/2 , which is the contribution due to the reducible instantons. There is another way of justifying that the path integral can be expressed as the sum We can further apply the fixed point theorem to calculate the partition function. The path integral can be done by evaluating exactly at the fixed point locus and by evaluating one-loop contribution of the normal modes to the fixed point locus [27]. This is the basic method of our calculation. The partition function of the two-dimensional physical Yang-Mills theory, which is the similar low dimensional cousin of HYM theory, has been calculated also by adopting the fixed point theorem [52] [53]. Before moving to the next section, we should add a cautionary remark on the dual roles ofω (eq.(2.38)). We are interested in the variation of < µ(Σ) d−2r (pt) r > where Σ is an arbitrary fixed element of H 2 (X; Z) according to the changes of metric. On the other hand, the Kähler form ω and its Poincaré dual H in the action and inω vary as we change the metric. This amounts to using the different two dimensional classes rather than fixed one. Thus, we should read the relation (2.44) very carefully. We will return this in Sect. 5. The Path Integral We use the action functional of the HYM theory in the delta function gauge S H = 1 h 2 X Tr −iH 2,0 ∧ * F 0,2 − iH 0,2 ∧ * F 2,0 + iχ 2,0 ∧ * ∂ Aψ + iχ 0,2 ∧ * ∂ A ψ − 1 4π 2 X Tr iϕF 1,1 + ψ ∧ψ ∧ ω − ε 8π 2 X ω 2 2! Tr ϕ 2 (3.1) The partition function of the HYM theory has contributions from the two branches. A. The non-abelian branch : ϕ f = 0 where the full SU (2) symmetry is restored, i.e., the irreducible ASD connections. B. The abelian branch : ϕ f = 0 where the non-abelian symmetry breaks down to abelian one, i.e., reducible holomorphic connections including reducible ASD connections, if any. Then, we can divide the partition function Z(ε, k) of N = 2 HYM theory as the sum of contributions of the two branches, Z(ε, k) = Z A (ε, k) + Z B (ε, k). (3.2) The zero coupling limit of Z(ε, k) can be identified with the symplectic volume of the instanton moduli space M k (g) with respect to a Kähler metric g [25]. Of course M k (g) is rarely compact. We always assume there are no χ 2,0 andχ 0,2 zero-modes. We calculate the partition function Z B (ε, k) contributed from the branch B as a simple application of Witten's fixed point theorem [27]. 14 The fixed points locus The HYM theory has the same global supersymmetry transformation laws as the N = 2 TYM theory. We only deal with the branch B, ϕ f = 0. In this branch the BRST fixed point is 15 ϕ f = ϕ c T 3 = constant.A 1,1 k \A * 1,1 k = A k \A * 1,1 k . A holomorphic connection A ∈ A 1,1 k endows E with a holomorphic structure E A . The connection A is reducible if and only if E A splits into the sum of holomorphic line bundles E A = L A ⊕ L −1 A (3.6) satisfying c 1 (L A ) · c 1 (L A ) = −k,(3.7) H 1,1 (X; Z) × H 1,1 (X; Z) → Z : 14 On the manifold b + 2 > 1 the branch B is absent generically. This means that the deformation to HYM theory is not particulary useful. We do not how to evaluate the crucial branch A. A careful application of the abelianization techniques [47][53] [54] may be used to deal with the non-Abelian branch. 15 According to our convention of the Tr , the Lie algebra generators are anti-hermitian given by T a = σ a 2i , T ± = T 1 ± iT 2 ,(3.3) with Pauli matrices σ a and Tr T a T b = − 1 2 δ ab .(3. 4) Since we consider the case p g = 0, the above parings become the intersection parings q X : H 2 (X; Z) × H 2 (X; Z) → Z: ζ · ζ = −k, (3.8) where ζ ∈ H 2 (X; Z). Obviously, the solutions of the above equation are independent of the metrics on X. Furthermore we always obtain in pairs ±ζ. For the simply connected case, that we are considering, it is known that the reducible connection corresponding to the pairs ±ζ is unique up to gauge equivalence class [2]. Thus, it is sufficient to solve the above intersection pairings to determine the gauge equivalence class of the abelian critical points of HYM theory. It also follows that the abelian fixed point locus is nondegenerate and isolated set of points. Then, the path integral calculation based on the fixed point theorem gives an exact answer. We will confuse the abelian critical point with its associated line bundle as well as its first Chern class. The value of the curvature two-form F f ∈ Ω 1,1 (End 0 (E A f ) at the fixed point locus F f = −2πi ζ 0 0 −ζ ,(3.S H ≈ S f + S (2) ,(3.11) where the action S f in the fixed points is given by the intersection number S f = i 2π ϕ c (ζ · H) + ε 32π 2 (H · H)ϕ 2 c ,(3.12) for every integral cohomology classes ζ ∈ H 2 (X; Z) satisfying ζ · ζ = −k. The gauge fixing We choose unitary gauge in which ϕ ± = 0, (3.13) where ϕ = (ϕ c + ϕ 3 )T 3 + ϕ + T + + ϕ − T − . (3.14) In this gauge ϕ has values on the maximal torus(Cartan sub-algebra). By following the standard Faddev-Povov gauge fixing, we introduce a new nilpotent BRST operator δ with the algebra δϕ ± = ±c ± (ϕ c + ϕ 3 ), δc ± = 0 δϕ 3 = δϕ c = 0, δc ± = b ± , δb ± = 0,(3.15) where c ± andc ± are anti-commuting ghosts and anti-ghosts respectively, and b ± are commuting auxiliary fields. The action for gauge fixing terms reads S gauge =δ i 4π 2 X i c − ϕ + +c + ϕ − = 1 4π 2 X − b − ϕ + + b + ϕ − +c − (ϕ c + ϕ 3 )c + −c + (ϕ c + ϕ 3 )c − ω 2 2! . The transverse part (I) The transverse part corresponding the ± part of the Lie algebra can be easily calculated by simple Gaussian integrals. Bosonic Sector (doublet i. e. ± part) The quadratic action relevant to this sector is given by S (2) ± (bose) = i h 2 X H 2,0 + ∧ * ∂ a A ′′ − + H 2,0 − ∧ * ∂ a A ′′ + + H 0,2 + ∧ * ∂ a A ′ − + H 0,2 − ∧ * ∂ a A ′ + + i 4π 2 X iϕ c A ′ + ∧ A ′′ − +A ′′ + ∧ A ′ − ∧ ω (3.18) where ∂ a α ± = ∂α ± ± iA f α ± (3.19) for any doublet α ± . This is a Gaussian integral which can be evaluated by shifting the variables A ′ ± → A ′ ± ± 4π 2 h 2 ϕ c ∂ * a H 2,0 ± A ′′ ± → A ′′ ± ∓ 4π 2 h 2 ϕ c∂ * a H 0,2 ± (3.20) In terms of new variables, we have S (2) ± (bose) = − 4π 2 i h 4 X 1 ϕ c ∂ * a H 2,0 + ∧ * ∂ * a H 0,2 − +∂ * a H 0,2 + ∧ * ∂ * a H 2,0 − − i 4π 2 X ϕ c A ′ + ∧ * A ′′ − +A ′′ + ∧ * A ′ − ,(3.21) where we have used the Kähler identities (2.26) as well as the relation 16 X α 1,0 ∧ α 0,1 ∧ ω = i X α 1,0 ∧ * α 0,1 ,(3.22) and self-duality of H. By the Gaussian integrals over A and H's, we have det ± (ϕ c ) −1/2 Ω 1,0 ⊕Ω 0,1 · det ± ( ∂ a ∂ * a ϕ c ) −1/2 Ω 2,0 · det ± (∂ a∂ * a ϕ c ) −1/2 Ω 0,2 . (3.23) The fermionic sector (doublet i. e. ± part) We can compute, in a similar fashion, the contribution from the transverse fermion doublets S (2) ± (f ermi) = − i h 2 X χ 2,0 + ∧ * ∂ aψ− + χ 2,0 − ∧ * ∂ aψ+ +χ 0,2 + ∧ * ∂ a ψ − +χ 0,2 − ∧ * ∂ a ψ + + i 4π 2 X ψ + ∧ * ψ − + ψ − ∧ * ψ + (3.24) where we have used (3.22) . After changing variables by ψ ± → ψ ± − 4π 2 i h 2 ∂ * a χ 2,0 ± ψ ± →ψ ± − 4π 2 i h 2∂ * aχ 2,0 ± (3.25) we have S (2) ± (f ermi) = − 4π 2 i h 4 X ∂ * a χ 2,0 + ∧ * ∂ * aχ 0,2 − + ∂ * a χ 2,0 − ∧ * ∂ * aχ 0,2 + + i 4π 2 X ψ + ∧ * ψ − + ψ − ∧ * ψ + (3.26) The Gaussian integrals over ψ and χ's give det ± (∂ a ∂ * a ) 1/2 Ω 2,0 · det ± (∂ a∂ * a ) 1/2 Ω 0,2 . (3.27) The transverse part (II) : U (1) singlets The remaining quadratic action is given by S (t) 3 = i h 2 X 1 2 H 2,0 3 ∧ * ∂A ′′ 3 + 1 2 H 0,2 3 ∧ * ∂A ′ 3 − 1 2 χ 2,0 3 ∧ * ∂ψ 3 − 1 2χ 0,2 3 ∧ * ∂ψ 3 + 1 4π 2 X i 2 ϕ 3 ∂A ′′ 3 +∂A ′ 3 + ψ 3ψ3 ∧ ω + ε 8π 2 X ω 2 2! 1 2 ϕ 2 3 . (3.28) We will show that the bosonic and fermionic contributions in this part cancel exactly each other and so give trivial result. Note that every field in here doesn't contain any zero mode for ∂ and∂ operator. We explicitly decomposed T 3 component of ϕ as zero mode ϕ c and non-zero mode ϕ 3 from the beginning and similarly for the U (1) connections as the topologically non-trivial part, which is contained in the action S f , and trivial one. integrations, we get the delta function constraints ∂A ′′ 3 = 0, ∂A ′ 3 = 0. (3.29) Similarly the χ 2,0 3 andχ 0,2 3 integrations give the constraints ∂ψ 3 = 0, ∂ψ 3 = 0. (3.30) From the Kähler identities (2.26), we have ∂A ′ 3 = ∂ * A ′ 3 = 0, ∂A ′′ 3 =∂ * A ′′ 3 = 0, ∂ψ 3 = ∂ * ψ 3 = 0, ∂ψ 3 =∂ * ψ 3 = 0. (3.31) This shows that all of them are harmonic one-forms which implies they actually vanishes. Thus the delta function constraints (3.31) and (3.30) are equivalent to the delta function constraint, x∈X δ(A ′ 3 (x))δ(A ′′ 3 (x))δ(ψ 3 (x))δ(ψ 3 (x)). Thus the integrations over A ′ 3 , A ′′ 3 , ψ 3 ,ψ 3 just lead to an unity. Finally, the integration of non-constant mode ϕ 3 is trivial as only the quadratic term left in the action. det ± (ϕ c ) Ω 0 · det ± (ϕ c ) −1/2 Ω 1,0 ⊕Ω 0,1 · det ± (ϕ c ) −1/2 Ω 2,0 ⊕Ω 0,2 = det ± (ϕ c ) 1/2 (Ω 0 ⊖Ω 1,0 ⊕Ω 2,0 )⊕(Ω 0 ⊖Ω 0,1 ⊕Ω 0,2 ) = det(ϕ c ) index∂ (+) +index∂ (−) = ϕ 2(1−h 0,1 +h 0,2 )−4k c = ϕ 2−4k c , (3.32) where we used the index theorem of the twisted Dolbeault operators. The Partition Function The path integral of the previous section shows that partition function essentially reduces to the following expression Z B (ε, k) = ζ i ·ζ i =−k ∞ −∞ dϕ 1 ϕ 4k−2 exp − i 2π ϕ(ζ i · H) − ε 32π 2 ϕ 2 (H · H) , (4.1) where we omitted the superscript c from ϕ. In this section we examine the small coupling behavior of the above partition function 18 and its variations according to the metric. The Small Coupling Behavior The above equation has pole at ϕ = 0, which can be removed by deforming the contour from R to P ± = {Im ϕ = ±κ} and taking the limit κ → 0. Of course the integral should be independent to the contours. For ϕ ∈ P ± , we can write Z B (ε, k) = − ζ i ·ζ i =−k 1 (4k − 3)! ∞ 0 ds s 4k−3 P ± dϕ exp ±iϕs − i 2π ϕ(ζ i · H) − ε 32π 2 ϕ 2 (H · H) = − ζ i ·ζ i =−k 1 (4k − 3)! 32π 3 ε(H · H) ∞ 0 ds s 4k−3 exp − 8π 2 ε(H · H) s ± ζ i · H 2π 2 . (4.2) We choose the + sign and decompose the above equation as Z B (ε, k) = − ζ i ·H<0 1 (4k − 3)! 2π ε ′ ∞ 0 ds s 4k−3 e − 1 2ε ′ s+ ζ i ·H 2π 2 − ζ i ·H=0 1 (4k − 3)! 2π ε ′ ∞ 0 ds s 4k−3 e − 1 2ε ′ s 2 − ζ j ·H>0 1 (4k − 3)! 2π ε ′ ∞ 0 ds s 4k−3 e − 1 2ε ′ s+ ζ j ·H 2π 2 , (4.3) where we set ε ′ = ε(H · H)/16π 2 . One can easily see the last term vanishes in the limit 18 The similar asymtotic estimation of the finite dimensional integral (4.1) was discussed by Wu [46] in his study of the abelian localization as a special case of the non-abelian localization. Note that the branch B is governed by abelian localization rather than the sophiscated non-abelian localization. ε ′ → 0 up to exponentially small term as follows; ζ i ·H>0 1 (4k − 3)! 2π ε ′ ∞ 0 ds s 4k−3 e − 1 2ε ′ s+ ζ i ·H 2π 2 ≤ ζ j ·H>0 1 (4k − 3)! 2π ε ′ ∞ 0 ds s 4k−3 e − 1 2ε ′ s 2 + ζ j ·H 2π 2 = ζ j ·H>0 1 (4k − 3)! 2π ε ′ ∞ 0 ds s 4k−3 e − 1 2ε ′ s 2 · e − 1 ε · 2(ζ j ·H) 2 H·H =O e − δ 2 ε , (4.4) where δ 2 = M in j ( 2(ζ j ·H) 2 H·H ). Similarly we can extract the polynomial dependency on ε, apart from the exponentially small terms, from the first term; − ζ i ·H<0 1 (4k − 3)! 2π ε ′ ∞ 0 ds s 4k−3 e − 1 2ε ′ s+ ζ i ·H 2π 2 = − ζ j ·H<0 1 (4k − 3)! 2π ε ′ ∞ −∞ ds s 4k−3 e − 1 2ε ′ s+ ζ j ·H 2π 2 + O e − δ 2 ε = − ζ j ·H<0 1 (4k − 3)! 2π ε ′ ∞ −∞ ds ′ s ′ − ζ j · H 2π 4k−3 e − 1 2ε ′ s ′2 + O e − δ 2 ε = ζ i ·H<0 [d/2] r=0 2π (d − 2r)! r! 2 r ζ i · H 2π d−2r (ε ′ ) r + O e − δ 2 ε = 1 (2π) d−1 ζ i ·H<0 [d/2] r=0 ε r (d − 2r)! r! 2 3r (ζ i · H) d−2r (H · H) r + O e − δ 2 ε , (4.5) where d = 4k −3. Assume that there are no reducible ASD connections, i.e., no line bundle ζ i with ζ i · H = 0, such that the second term absent. Then, our result for Z ′ (ε, k) is Z B (ε, k) = 1 (2π) d−1 ζ i ·H<0 [d/2] r=0 ε r (d − 2r)! r! 2 3r (ζ i · H) 4k−3−2r (H · H) r +O e − δ 2 ε , (4.6) where the summation runs over every divisor satisfying ζ i · ζ i = −k and ζ i · H < 0. If we choose the − sign in the beginning we have the same pattern of the asymptotic behaviors and the divisors satisfying ζ i · H > 0 only contributes. Since the solutions of ζ i · ζ i = −k always arise in pairs ±ζ i , we have the same result. The Variation of the Partition Function We have seen that the Abelian critical points of HYM theory are the reducible con- Before going on, we should remark that the structure of the abelian critical points is isomorphic to the chamber structure determined by the system W k ⊂ W k of walls in the positive cone. In particular, for each line bundle associated with an abelian critical point ζ, there is an associated wall W ζ in the positive cone. Let H + and H − be ample divisor lying in the two chambers C + and C − separated by the single wall W ζ . We assume that ζ has negative degree with respect to ample divisor H + lying in the chamber C + , i.e., deg(ζ) = X c 1 (ζ) ∧ ω = ζ · H = i 2π X F c ∧ ω = if c (A) 2π X ω ∧ ω. ζ · H + < 0. Let H 0 denote ample divisor lying on the wall. If we change the metric, such that the ample divisor H starts from C + , crosses the wall and goes to the chamber C − , we have 19 ζ · H + < 0, ζ · H 0 = 0, ζ · H − > 0. (4.8) Now it is clear what we have actually done by deforming TYM theory to HYM theory. We mapped all the possible reducible instantons, which appear as we vary the Kähler metric in every possible way, to the abelian critical points of the HYM theory. In the limit ε → 0, the partition function of HYM theory is such that we sum up only critical points with negative degree. As one can easily see, we lost a critical point ζ with negative degree by passing through the wall W ζ . On the other hand, there is also another abelian critical point −ζ which defines the same wall W ζ with ζ. Thus, we get a new critical point −ζ of negative degree instead. We will collectively denote the ample classes lying in a C ± by H ± . We change the metric such that our ample class H crosses just one wall W ζ , defined by a certain divisor ζ satisfying ζ · H + < 0 and ζ · ζ = −k, and moves to another chamber C − . Then we have ζ · H − > 0 by definition. This also amounts that no other line bundles change the sign of their degrees. The contribution of ζ to the partition function Z B (ε, k)(C + ) is given by 1 (2π) 4k−4 [d/2] r=0 ε r (d − 2r)! r! 2 3r (ζ · H + ) 4k−3−2r (H + · H + ) r + O e − δ 2 ε , (4.9) while Z B (ε, k)(C − ) no longer receives contributions from ζ. On the other hand, Z B (ε, k)(C − ) receives contributions from −ζ, since −ζ · H − < 0, given by 1 (2π) 4k−4 [d/2] r=0 ε r (d − 2r)! r! 2 3r (−ζ · H − ) 4k−3−2r (H − · H − ) r + O e − δ 2 ε ,(4.10) Without loss of generality, we will use a normalization H ·H = 1, i.e., H + ·H + = H − ·H − = 1. The non-analytic part Up to now, we have considered the case that the metric does not admit reducible ASD connections. Now we allow the Kähler metric to lie on one of the walls W k such that there is a reducible ASD connection. We define the multiplicity m of the reducible instanton by the number of the walls intersect at the point where the Kähler metric lies on. Then the partition function (3.2) contains a non-analytic term which is not a polynomial of ε. Now the partition function (4.6) has additional contribution given by − ζ i ·H=0 1 (4k − 3)! 2π ε ′ ∞ 0 ds s 4k−3 e − 1 2ε ′ s 2 = − m (4k − 3)! 2π ε ′ ∞ 0 ds s 4k−3 e − 1 2ε ′ s 2 = − (2π) 1/2 m (4k − 3)!! (ε ′ ) 2k−3/2 ,(4.11) leading to the modified partition function Z ′ (ε, k) = 1 (2π) 4k−4 ζ i ·H<0 [d/2] r=0 ε r (d − 2r)! r! 2 3r (ζ i · H) 4k−3−2r − 1 (2π) 4k−4 m (2π) 1/2 (4k − 3)!! (ε/2) 2k−3/2 + O e − δ 2 ε . (4.12) Our calculation precisely shows that the non-analytic term is the contribution of the re- d i=1 µ(Σ i ) ′ = d i=1 µ(Σ i ) + O e − δ 2 ε . (5.1) This can be further generalized to include the four-dimensional class Θ d−2r i=1 µ(Σ i ) ′ = 2m+n=2r ε m m!n! d−2r i=1 µ(Σ i )ω n Θ m + O e − δ 2 ε . (5.2) Thus it is sufficient to calculate (5.2) to determine the general expectation values of TYM theory for a simply connected manifold. 21 20 We refer the readers to the remarks in Sect. 4.4 of ref. [26]. 21 Actually, it is sufficient to determine the term of power ε r in the RHS of (5.2). The expectation value of the observables in HYM theory can be also written as the sum of the contributions of the two branches A and B; d−2r i=1 µ(Σ i ) ′ = d−2r i=1 µ(Σ i ) ′ A + d−2r i=1 µ(Σ i ) ′ B . (5.3) We should emphasize here that, in the HYM theory, all the terms that do not vanish exponentially must be interpreted as the contributions of the ASD connections. 22 That is, < · · · > ′ B for the branch B is also the contributions of ASD connections up to the exponentially small terms for ε → 0. Clearly, < · · · > ′ A for the branch A does not contain any exponentially small term. Consequently, we can divided any expectation value of TYM theory as d−2r i=1 µ(Σ i ) Θ r = d−2r i=1 µ(Σ i ) Θ r A + d−2r i=1 µ(Σ i ) Θ r B (5.4) where d−2r i=1 µ(Σ i ) Θ r ′ A = d−2r i=1 µ(Σ i ) Θ r A , d−2r i=1 µ(Σ i ) Θ r ′ B = d−2r i=1 µ(Σ i ) Θ r B + O e − δ 2 ε . (5.5) If we consider manifold with b + 2 ≥ 3, the contribution of the branch B is generically absent for both HYM and TYM theories. In Sect. 5.1, the contribution of the branch B to the expectation values is evaluated using the similar method as for the partition function Z B (ε, k). However, we do not know how to calculate the contribution of the branch A. It is quite natural to believe that the Seiberg-Witten monopole invariants correspond to the branch A. In Sect. 5.2, we briefly review some known properties of the Seiberg-Witten invariants for manifolds with b + 2 = 1. Then, we study the variation of the expectation values of TYM theory in Sect. 5.3. The branch B As usual, we choose a generic metric g which does not admit the zero-modes ofχ 0,2 . That is, the second instanton cohomology group is trivial. Then the only source of the singularities in the moduli space of ASD connection M k (g) is the reducible instantons d−2r l=1 µ(Σ l ) ′ B = ζ i ·ζ i =−k d−2r l=1 − i(ζ i · Σ l ) 2π ∞ −∞ dϕ 1 ϕ 2r+1 exp − i 2π ϕ(ζ i · H) − ε 32π 2 ϕ 2 (5.6) We can study the ε → 0 limit using the methods in Sect. 4.1., d−2r l=1 µ(Σ l ) ′ B = ζ i ·H<0 d−2r l=1 ζ i · Σ l 2π 1 (2r)! 2π ε ′ ∞ 0 ds s 2r e − 1 2ε ′ s+ ζ i ·H 2π 2 + ζ i ·H=0 d−2r l=1 ζ i · Σ l 2π 1 (2r)! 2π ε ′ ∞ 0 ds s 2r e − 1 2ε ′ s 2 + ζ j ·H>0 d−2r l=1 ζ i · Σ l 2π 1 (2r)! 2π ε ′ ∞ 0 ds s 2r e − 1 2ε ′ s+ ζ j ·H 2π 2 , (5.7) The third term above vanishes exponentially fast for ε → 0. It is amusing to note that the second term in ( µ(Σ l ) Θ r B = 1 (2π) d−1 ζ i ·H<0, ζ i ·ζ i =−k 1 2 3r d−2r l=1 (ζ i · Σ l ). (5.9) Then, the general form of the expectation values of the topological observables of TYM theory is given by d−2r l=1 µ(Σ l ) Θ r = d−2r l=1 µ(Σ l ) Θ r A + 1 (2π) d−1 ζ i ·H<0, ζ i ·ζ i =−k 1 2 3r d−2r l=1 (ζ i · Σ l ). (5.10) The Seiberg-Witten invariants One of the open problem in our approach is to uncover the genuine diffeomorphism invariants. This is because we computed the path integral only for one of the two branches of the fixed points. Clearly, the branch B, which we have calculated, is generically absent for manifold with p g > 1. For the fixed point ϕ = 0 (the branch A), we have the full SU (2) symmetry and the path integral has contribution from the irreducible instantons. There is no reason that the contribution would vanish. Unfortunately, the path integral for this branch reduces to a formal expression such as one integral over the moduli space of irreducible instantons. Thus, we need an alternative approach to deal with this branch. Note that the limit t → 0 in eq.(2.29) can also be viewed as the limit h 2 → ∞ with t For manifold with p g = 0, no such perturbation is possible and one should integrate over the whole complex plane [30]. 24 The extra contributions from the generic points dim R M x SW = W x = x · x − (2χ + 3σ) 4 . (5.11) The amazing fact is that the moduli space is compact. If W x = 0 such that x 2 = 2χ + 3σ, one simply counts the number of the points with sign according to a suitable orientation. The integer n x is that algebraic sum of the points. If W x = 0 and W x = 2a, one defines iii) The precise relation between the Seiberg-Witten invariants and the Donaldson invariants for manifold of simple type with b + 2 ≥ 3 has been established by Witten [30]. The Seiberg-Witten class x corresponds to the basic class of Kronheimer-Mrowka and n x to the coefficient associated to the basic class [57]. The only other informations in the Donaldson invariant are all homotopy invariants, such as the intersection form. Now we come back to our original problem for a simply connected Kähler surface with p g = 0. Similarly to the Donaldson invariants, the Seiberg-Witten invariants (x, n x ) is a topological invariants if n x does not change for a smooth path of metric joining two generic metrics. It turns out n x jumps by ±1 if there is a metric such that the monopole equation reduces to the equation for the abelian instanton [30][58] [59]. This amounts to X x ∧ ω g = 0. n x = M x SW ν(Σ) a ,(5. (5.13) Since ω g belongs to the positive cone, the above can happen if and only if x · x < 0, as we discussed before. Conversely, n x is metric independent if x · x ≥ 0. A necessary condition that n x can be non-vanishing is the non-negativity of the dimen- sion W x , x · x ≥ 2χ + 3σ. (5.14) Using 2χ + 3σ = 9 − n for manifold of type (1, n) with H 1 (X; R) = 0, one can conclude that all the Seiberg-Witten invariants are metric independent for manifold with n ≤ 9 (or equivalently b − 2 ≤ 9). Now consider the manifold of type (1, 9 + N ) with N > 0. Any possibly non-vanishing, at least for certain metric, Seiberg-Witten class x with x ≡ w 2 (X) mod 2 should satisfy x · x ≥ −N. x into {x inv } ∪ {x ′ } where x inv · x inv ≥ 0. Clearly n x inv is metric independent. One can easily show that all the n x inv are identically vanish. Recall that all n x vanish for any x with a metric which admits positive scalar curvature. From Consider one parameter family of g t joining two generic metrics g −1 and g 1 with g o such that [ω g 0 ] ∈ x ⊥ and g t cross the wall x ⊥ transversely. Witten showed that c 1 (X) ∧ ω g = −K X ∧ ω g = 1 2 R g ω g ∧ ω g where ω g ∧ ω g isn x (g −1 ) = n x (g 1 ) ± 1. (5.18) The expectation values in TYM theory At present we do not know the precise relation between the Seiberg-Witten invariants and the expectation values d−2r l=1 µ(Σ l ) Θ r A . In any case, d−2r l=1 µ(Σ l ) Θ r A should be, in general, polynomials of degree d in H 2 (X; Z) depending on (x, n x ) and q X . Since the values at the both two branches are well-defined, we can confirm Witten's original claim that the path integral approach is well-defined whatever properties the moduli space of ASD connections has [8]. If we consider manifold of type (1, n) with n ≤ 9, the variation of the expectation value (5.10) of TYM theory comes only from the branch B. The relevant chamber structure in this case is the set C k X of the connected components in the positive cone after removing the system W k of walls defined by all the two-dimensional integral classes ζ i satisfying ζ i · ζ i = −k. The expectation values d−2r l=1 µ(Σ l ) Θ r depend on metric only by the chamber structure C k X . Let C + and C − be the two chambers separated by the single wall W ζ with the same condition as (4.8). From (5.10), we have d−2r l=1 µ(Σ l ) Θ r C + − d−2r l=1 µ(Σ l ) Θ r C − = 2 (2π) d−1 2 3r d−2r l=1 (ζ · Σ l ). (5.19) For a manifold of type (1, 9 + N ) with N > 0, the general transition formula can be more complicated. We should consider the both chamber structures for the branches A and B. Assuming that the metric crosses a single wall W ζ which does not intersect with the walls x ⊥ defined by x satisfying (5.17), the transition formula is given by (5.19). The Relations with the Donaldson Invariants In this section, we discuss some conjectural relations between the expectation values of TYM theory and the corresponding Donaldson invariants. Clearly, the expectation value (5.10) does not coincide to the corresponding Donaldson invariants q k,X (Σ 1 , . . . , Σ d−2r , (pt) r ). Throughout this paper, we have tried to convince the (especially the mathematician) readers that the path integral approach to the gauge theoretic invariants are well-defined whatever properties the moduli space of ASD connections has. It is clear, due to Witten, that the TYM theory correctly determines the Donaldson invariants at least for manifold with b + 2 ≥ 3. For a manifold with b + 2 = 1, the expectation values of TYM theory in general have additional contributions essentially due to the reducible connections. We have shown that the expectation values are well-defined and those additional contributions can be determined exactly. In any case, the path integral approaches do not refer to the compactification procedure to get well-defined results. The equivalence between the TYM theory and Donaldson theory for the manifolds with b + 2 ≥ 3 implies that q k,X (Σ 1 , .., Σ d−2r , (pt) r ) A = d−2r l=1 µ(Σ l ) Θ r A . (6.2) Then the remaining part q k,X (Σ 1 , .., Σ d−2r , (pt) r ) B should be expressed as the sum of contributions of reducible holomorphic connections with the instanton numbers 1, 2, ...., k. In particular, the expression d−2r l=1 µ(Σ l ) Θ r B can be regarded as the contribution of the reducible holomorphic connections with the instanton number k; q k,X (Σ 1 , .., Σ d−2r , (pt) r ) B = 1 2 ζ i ·H<0, ζ i ·ζ i =−k 1 2 3r d−2r l=1 (ζ i · Σ l ) + . . . ,(6.3) where we normalized the expectation value by multiplying an universal factor (−1) k (2π) 4k−4 2 . Throughout this section, we always assume that the path of metric does not cross the walls responsible for the changes of the Seiberg-Witten invariants. We change the metric such that our ample class H cross just one wall W ζ , defined by a certain divisor ζ satisfying ζ · H + < 0 and ζ · ζ = −k, and move to another chamber C − with ζ · H − > 0. Provided that the path of metric does not cross another walls defined by divisors ζ ′ satisfying ζ ′ · ζ ′ = −1, ..., −k + 1, we immediately have the transition formula Γ k,r X (C + ) − Γ k,r X (C − ) = (−1) k 1 2 3r ζ d−2r . (6.4) or q X,k,C + (Σ d−2r , (pt) r ) − q X,k,C − (Σ d−2r , (pt) r ) = (−1) k 1 2 3r (ζ · Σ) d−2r . (6.5) The transition formula (6.4) is a generalization of the formulas obtained by Mong [6] and Kotschick [5] for r = 0 by Friedman and Qin [33] for the case r = 1. It also agrees with the results of Ellingsrud and Göttsche [34]for general values of r up to certain normalization difference. The problem associated with the compactification Now the question is to determine the contribution of the reducible connections from the lower stratas in the compactified moduli space. If we compactify the moduli space, the path integral will get additional contributions from the reducible critical points whose associated divisors satisfying −k + 1 ≤ ζ · ζ < 0. Thus, we will simply view our result as the contribution from the top strata. To determine the extra contributions, at least partially, we will use the recent result of Hu and Li [36]which shows that M k = k−1 ℓ=0 M k−ℓ × Sym ℓ (X),(6.6) if k is sufficiently large (the condition that the Kähler metrics on X behave as generic metrics). We consider a d-dimensional subspace N k−ℓ = N k−ℓ × Sym ℓ (X) ⊂ M k such that M k−ℓ ⊂ N k−ℓ , dim C (N ℓ ) = d − 2ℓ. We also assume that N k−ℓ \M k−ℓ does not contains any reducible connections, for arbitrary metric, other than M k−ℓ \M * k−ℓ . We consider k ℓ=0 N ℓ ⊂ M k . The restriction of µ(Σ) to N k−ℓ × Sym ℓ (X) will be in the form µ(Σ) ℓ = µ(Σ) ℓ + 2H ′ ,(6.7) where H ′ ∈ H 2 (Sym ℓ (X)). We can consider the part of the Donaldson invariants contributed from N k−ℓ , µ(Σ) d , N k−ℓ = 2 2ℓ d! (d − 2ℓ)!(2ℓ)! µ(Σ) d−2ℓ ℓ , N k−ℓ (H ′ ) 2ℓ , Sym ℓ (X) = 2 ℓ d! (d − 2ℓ)!ℓ! µ(Σ) d−2ℓ ℓ , N k−ℓ q ℓ ,(6.8) where we have used (H ′ ) 2ℓ , Sym ℓ (X) = (2ℓ)! 2 ℓ ℓ! q ℓ , where q ∈ Sym 2 (H 2 (X; Z)). Let a smooth path of Kähler metric meet only one wall W ζ defined by ζ such that ζ · ζ = −k + ℓ ζ , ζ · H + < 0 < ζ · H − . (6.13) We have Γ k X (C + ) − Γ k X (C − ) = (−1) k−ℓ ζ 2 ℓ ζ d! (d − 2ℓ ζ )! ℓ ζ ! (ζ) d−2ℓ ζ q ℓ ζ + . . . (6.14) Now we can easily generalize the result to the polynomials including the fourdimensional class This formula has been obtained by Kotschick and Morgan [7] for r = 0, by Friedman and q X,k (Σ d−2r (pt) r ) B = 1 2 k ℓ=0 (−1) k−ℓ 2 ℓ−3r (d − 2r)! r! (d − 2ℓ − 2r)! ℓ! ζ i ·H<0, ζ i ·ζ i =−k+ℓ (ζ i · Σ) d−2ℓ−2r q ℓ + · · · ,(6. Qin [33] for r = 1 and by Ellingsrud and Göttsche [34] for general r. The paper [33] has some explicit results beyond the leading term. In the paper [34], up to the leading 3-terms were calculated for general r. The variation of the moduli space It is well-known that the image of moment map is a convex cone in a positive Weyl chamber of the Lie algebra and the critical values of the moment map define a system of walls in the convex cone [60]. It is also known that the symplectic quotients undergo a Hu [63]. The variation of GIT quotients was also studied independently by Witten in the context of two-dimensional supersymmetric theory and the quantum cohomology rings [67]. We note that Witten's picture is quite similar to our approach. It is not yet clear if a natural compactification of the moduli space can be obtained using some path integral methods. However, our result is sufficient to predict the general form of the Donaldson invariants as (6.1)(6.2)(6.3). If we consider when the variation of the Donaldson invariants get contribution only from q X,k,C(X) (Σ d−2r (pt) r ) B , the transition formula of the Donaldson invariants is sufficient to determine q X,k,C(X) (Σ d−2r (pt) r ) B as one can see from the relation between (6.17) to (6.16) and (6.15). The partition function of the theory can be expressed by the values of the moment map at the critical points of the action functional. The spectrum of the critical points, as one varies the metric, depends on a certain chamber structure of the positive cone. By studying the partition function and some correlation functions in the small coupling limit, one can obtain the expectation values of certain topological observables of TYM theory. It turns out those expectation values are well-defined and depend only on the chamber structure of the metric. k to get a fundamental homology class. The Donaldson-Uhlenbeck compactification M k of M * k is the closed subset of the embedding of M k to the disjoint union 8 , k ℓ=0 M ℓ × Sym k−ℓ (X). (2.5) The compactified space includes the ASD moduli spaces with lower instanton numbers in the lower stratas which can contain reducible ASD connections. To get a well-defined invariants we should choose the metric such that it does not admit any reducible ASD connections with instanton numbers 1, ..., k. The genericity of the metric always means to satisfy this additional requirement. Let be the Donaldson µ-map. Then we have a natural extension of µ µ : H 2 (X; Z) −→ H 2 (M k (g); Z).(2.7) 41) after the Gaussian integral over ϕ, is identical to the physical Yang-Mills theory restricted to the space A 1,1 of holomorphic connection. It is proportional to the normed-square of moment map up to topological terms. The real number ε corresponds to the coupling constant. The classical equation of motion is given by F 2, 0 0(A) = F 0,2 (A) = 0, d A f (A) = 0, (2.42) which is the Yang-Mills equation of motion on A 1,1 . The partition function of the HYM theory exactly reduces to an infinite dimensional non-abelian equivariant integration formula, of contributions of the critical points. The action functional has the global N = 2 supersymmetry, thus, we can use the fixed point theorem of Witten. The important fixed point equation issϕ = −i∂ A ϕ = 0, sϕ = −i∂ A ϕ = 0. (2.45)This equation shows that non-zero solutions for ϕ (the zero-modes of ϕ) appear for reducible connections. By eliminating ϕ using Gaussian integral gives2if + εϕ = 0. (2.46)Combining the above two equations we are led to the fixed point equation d A f = 0. Thus, the zero-modes of ϕ are no longer associated with the reducible instanton, rather they are mapped into higher critical points. determine the fixed point solution for the gauge connections. They are given by the reducible (holomorphic) connections which, sometimes, will be called the abelian critical points. Consider the space A k of all connections of a SU (2)-vector-bundle E over X with a given instanton number k. We denote A 1,1 k be the subspace which consists of holomorphic connections of A k and A * k be the space of irreducible connections. The space of reducible connections is then by the first Chern class of the line bundles ζ satisfying (3.8). The values of the other fields in the fixed point locus can be read from (2.17) and (2.23) ψ f =ψ f = H , the values of χ 2,0 andχ 0,2 in the fixed point locus need not to be zero. It is sufficient to take values in the Cartan (abelian) sub-algebra to satisfy sH 2,0 = −i[ϕ, χ 2,0 ] = 0 orsH 2,0 = −i[ϕ,χ 0,2 ] = 0. Since we are dealing with the manifold satisfying H 2,0 (X) = H 0,2 (X) = 0, we can set the values zero without loss of generality. We expand the action functional about the fixed point locus up to the quadratic terms in the action action has the δ-BRST symmetry and, hence we can use the fixed point theorem. The fixed point locus for δ-BRST is c ± = b ± = 0. The Gaussian integrations overc and c give det ± (ϕ c ) Furthermore as h 1 , 0 10= h 0,1 = h 2,0 = h 0,2 = 0, where h p,q denotes the Hodge number, in our considerations all other fields do not have any zero-mode 17 .17 Actually we can remove the condition h 1,0 = h 0,1 = 0. We leave this as an exercise. From H 2 2 By combining (3.17) , (3.23) and (3.27) , we have the contributions of the transverse degrees of freedom as follows: nections which can be uniquely determined by the 2-dimensional integral classes (or line bundles) ζ i satisfying ζ i · ζ i = −k. Clearly, the notion of the reducible connections is metric independent. On the other hand, the notion of the reducible ASD connections is evidently metric dependent. An Abelian critical point ζ is ASD if and only if ζ · H = 0 where H is Poincaré dual to the Kähler form associated with the given metric. Equivalently, a reducible connection A is ASD if and only if the degree of the associated holomorphic line bundle ζ is zero, depends only on the cohomology classes of the first Chern class and of the Kähler form. The value f c (A f ) of the critical points d A f c (A) = 0 is also determined by the degree and the self-intersection number of the Kähler form. Note that the critical points of the moment map m(A) are also determined by the same data. We call a reducible connection as a higher critical point if its degree is non-zero. ducible instanton. The similar phenomenon has been observed in the two-dimensional version of the TYM theory by Witten. 20 5. The Relations with the TYM Theory Now we are going to extract the expectation values of topological observables in the TYM theory. To obtain the precise relations with the TYM theory when we vary the metric, it is conceptually more clear and convenient to use the expectation values of the topological observables rather than the partition function itself. The HYM theory has the same set of topological observables (2.32)(2.33) with the TYM theory. The relation (2.44)between certain correlation function of TYM and the partition of HYM theory can be generalized[26][25]. Picking two-dimensional classes Σ i ∈ H 2 (X; Z), the expectation value d i=1 µ(Σ i ) ′ evaluated in the HYM in the limit ε → 0 is related to that of TYM theory by the formula 5.7), corresponding to the contributions of the reducible ASD connections, always vanish. If we have a solution ζ for ζ · ζ = −k and ζ · H = 0, we always have another solution −ζ. Since d − 2r = 4k − 3 − 2r is always an odd integer, their contributions cancel with each other. Thus, the presence of the reducible ASD connections.The first term can be calculated in the same way as (4.5), which gives i ·Σ l )+lower orders in ε+O e − fixed. In this limit, the semi-classical analysis is invalidated. At this point we can utilize the fundamental results of Seiberg-Witten on the strong coupling behaviour of the untwisted N = 2 super-Yang-Mills theory[28][29][30]. Their result can be essentially summarized by the quantum moduli space parametrized by a complex variable u, which corresponds to the observable Θ in the twisted theory. Classically, there is a singularity at the origin u = 0 where the full SU (2) symmetry is restored. Quantum mechanically, the complex u-plane has two singularities at u = ±1. The singularities in that plane represent the appearance of new massless particles. For manifolds with p g > 1 one can introduce a perturbation utilizing holomorphic two-forms such that the only contribution comes from the two singular points. Such an effective low energy theory turns out to be an N = 2 super-Maxwell theory coupled with hyper-multiplet. One can twist this theory and the resulting theory gives the dual description of the Donaldson invariants[30].23 23 The Seiberg-Witten monopole equation is the close cousin of the equation appeared in the Vafa and Witten's paper on a twisted N = 4 super-Yang-Mills theory[23]. The simlarity has an obvious origin since the N = 4 theory can be viewed as an N = 2 theory coupled with a N = 2 matter multiplet in the adjoint represenation. The similar equation also appears in[55], although we do not know the origin of the similarity. We would like to mention that the general N = 2 super-Yang-Mills theory with hypermultiplets can be twisted to define a set of topological field theories which lead to certain non-abelian version of the Seiberg-Witten monopole equation[56].24 It was noticed that the Donaldson invariants and Seiberg-Witten invariants are very different for manifold with b + 2 = 1. There is no mystery in it since Seiberg-Witten monopole invariant is one of two parts of the Donaldson invariant. of the quantum moduli space are the contributions of the abelian instantons. What we have calculated in this paper is essentially such contributions. We can view the original Seiberg-Witten monopole invariants as the contribution of the branch A. The Seiberg-Witten invariants can be viewed as the pairs (x, n x ) where x ∈ H 2 (X; Z) with x ≡ w 2 (X) mod 2 and n x is an integer associated with x. For each x we have an associated holomorphic line bundle L such that x = −c 1 (L 2 ) = −2c 1 (L). For each L we have the Seiberg-Witten monopole equation. The dimension of the moduli space M SW of that monopoles is given by 12) where ν : H 2 (X; Z) → H 2 (M SW ; Z) analogous to the Donaldson µ-map. We would like to remark the followings.i) The Seiberg-Witten invariants are metric independent for manifold with b + 2 > 1. Witten completely determined the invariants for Kähler surface with b + 2 > 1. For Kähler surface with p g > 1, all the higher dimensional Seiberg-Witten invariants vanish (the simple condition). On the other hand, it is not known if such a property still hold for manifold with b + 2 = 1.ii) The Seiberg-Witten invariants vanish if a metric admits positive scalar curvature. It is easy to see that the Kähler surface with p g > 1 does not admit a metric of positive curvature except for the hyperKähler case. So there are at least two Seiberg-Witten invariants including (K X , 1) and (−K X , −1). divide the set of all such Now the underlying reason for the discrepancy between the expectation values of TYM theory and the Donaldson invariants is clear. If we incorporated compactification of the moduli space, the path integral will receive additional contributions from the reducible connections with lower instanton numbers. Similarly to the expectation value of the TYM theory, one can divide the Donaldson invariant into the sum of two branches q k,X (Σ 1 , .., Σ d−2r , (pt) r ) = q k,X (Σ 1 , .., Σ d−2r , (pt) r ) A + q k,X (Σ 1 , .., Σ d−2r , (pt) r ) B . (6.1) can use our previous result to determine µ(H) summation runs over every divisor satisfying ζ i · ζ i = −k + ℓ. By summing up, we can writeq X,k,C(X) (Σ d ) − 2ℓ)! ℓ! (ζ i · Σ) d−2ℓ q ℓ + . . . − 2ℓ)! ℓ! (ζ i ) d−2ℓ q ℓ + . . .(6.12) X k−ℓ 2 ℓ−3r (d − 2r)! r! (d − 2ℓ − 2r)! ℓ! ζ i ·H<0, ζ i ·ζ i =−k+ℓ (ζ i ) d−2ℓ−2r q ℓ + · · · . (C − ) = (−1) k−ℓ ζ 2 ℓ ζ −3r (d − 2r)! r! (d − 2ℓ ζ − 2r)! ℓ ζ ! (ζ) d−2ℓ ζ −2r q ℓ ζ + · · · . (6.17) specific birational transformations closely related to the variation of GIT quotients as the values of moment map cross the wall[61][62][63]. The walls are determined by the critical points of the moment map. Of course, all these are rigorous for finite dimensional compact manifold with compact group actions.Formally speaking, our problem is an infinite dimensional analogue of the variation of the symplectic quotients. The moduli space of ASD connection can be identified with the symplectic quotients m(0) −1 /G. An interesting fact is that the chamber structures in the positive cone of the manifold and those in the convex cone sitting on image of the moment map are determined by the same data. Our results imply that the two chamber structures are isomorphic! Naively speaking, this suggests that the moduli space of ASD connections undergoes certain birational transformations if the metric cross that walls. This picture also coincides with our general strategy for deriving the transition formula. Losing a higher critical point ζ and getting another higher critical point −ζ is analogous to the blown-down and successive blown-up. In the algebro-geometrical method, one constructs the moduli space of H-stable bundles over algebraic surface and studies the variation of the moduli space under the changes of the polarizations [64][65]. One of the advantage of the algebro-geometrical approach is that it has a natural way of the compactification of the moduli space. Since the moduli space of H-semi-stable bundles contains the moduli space of the H-stable bundles as a Zariski open subset, it gives a natural compactification. It is well-known that the diffeomorphism class of the moduli space depends on the chamber structure in the ample (Kähler) cone. Recently several papers on the variation of the moduli space of H-semi-stable bundle under the changes of the polarization of the ample class appeared [34][35][33]. They show that the moduli space undergoes specific birational transformation similar to the variation of geometrical invariant theory (GIT) quotients studied by Thaddeus [66][62] and Dolgachev- positive definite, we have n x inv is metric independent it should vanish identically. Consequently, every possibly non-vanishing x at least for certain metric satisfies−N ≤ x · x ≤ −1Now we can conclude that for manifold of type(1, 9 + N ) there are no Seiberg-Witten class which are independent of the metric.25 Similarly to the Donaldson invariants, the Seiberg-Witten invariant depends only on the chamber structure defined by x satisfying(5.17) and x ≡ w 2 (X) mod 2.and x ≡ w 2 (X) mod 2. (5.17) The standard reference on the de Rham equivariant cohomology is the book[37]. 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{'abstract': 'We study the path integrals of the holomorphic Yang-Mills theory on compact Kähler surface with b + 2 = 1. Based on the results, we examine the correlation functions of the topological Yang-Mills theory and the corresponding Donaldson invariants as well as their transition formulas.Feb, 1995 †', 'arxivid': 'hep-th/9503036', 'author': ['Seungjoon Hyun hyun@phya.yonsei.ac.kr††e-mail:j.park@swansea.ac.uk ', 'Jae-Suk Park ', '\nInstitute for Mathematical Sciences\nDepartment of Physics\nTheoretical Physics Group\nYonsei University\n120-749SeoulKorea\n', '\nUniversity of Wales\nSA2 8PPSwansea SwanseaUK\n'], 'authoraffiliation': ['Institute for Mathematical Sciences\nDepartment of Physics\nTheoretical Physics Group\nYonsei University\n120-749SeoulKorea', 'University of Wales\nSA2 8PPSwansea SwanseaUK'], 'corpusid': 17404128, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 32466, 'n_tokens_neox': 28325, 'n_words': 17304, 'pdfsha': 'e7060a3968cd0b7df149b5e28203e9217f9643c0', 'pdfurls': ['https://export.arxiv.org/pdf/hep-th/9503036v2.pdf'], 'title': ['Holomorphic Yang-Mills Theory and Variation of the Donaldson Invariants', 'Holomorphic Yang-Mills Theory and Variation of the Donaldson Invariants'], 'venue': []}

arxiv

Computing square roots in quaternion algebras 4 Jan 2023 Przemysław Koprowski przemyslaw.koprowski@us.edu.pl Institute of Mathematics University of Silesia in Katowice ul Bankowa 1440-007KatowicePoland Computing square roots in quaternion algebras 4 Jan 20231square root computationquaternion algebranumber fieldsglobal fields We present an explicit algorithmic method for computing square roots in quaternion algebras over global fields of characteristic different from 2. Introduction The computation of square roots is one of the most basic operations in mathematics. Effective methods for computing square roots are among the oldest algorithms in the realm of computational mathematics. In fact, Heron's method for a numerical approximation of a square root of a real number is two thousand years old and preceded by the Euclidean algorithm (wildly believed to be the oldest mathematical algorithm) by only about three to four centuries (for an in-depth discussion on the chronology see [1]). Although numerous methods for computing square roots in various algebraic structures are known nowadays, some important omissions prevail. Among them are general quaternion algebras. Computation of square roots in the algebra of Hamilton quaternions H = −1,−1 R is well-known (see [2]) and very simple as for every quaternion ∈ H there is a subfield K ∼ = C of H containing , and so the computation the square root in H can be reduced to the computation of the square root in C. It is no longer so in a general quaternion algebra Q = α,β K for an arbitrary field K and two elements α, β ∈ K × . To the best of our knowledge, no algorithm for computing quaternionic square roots exists in the literature. One possible explanation for this (quite surprising) fact is that in the commutative case when one considers a field extension L/K, a typical way to compute a square root of an element a ∈ L is to factor the polynomial x 2 − a in L[x]. However, for quaternion algebras, there are no known polynomial factorization algorithms. The sole purpose of this paper is to correct this evident omission and present an explicit algorithm for computing square roots in quaternion algebras over arbitrary global fields of characteristic different from 2. Notation Throughout this paper, K will denote an arbitrary global field of characteristic char K = 2. Hence, K is either a number field, i.e. a finite extension of Q (then its characteristic is just 0) or a global function field, that is, a finite extension of a rational function field over a finite field F q , where q is a power of an odd prime. The set of nonzero elements of K is denoted K × . Recall that a quaternion algebra Q = α,β K over K is a 4-dimensional K-algebra with a basis {1, i, j, k} and a multiplication gathered by the rules: i 2 = α, j 2 = β, ij = k = −ji. As usual, we shall identify the field K with the subfield K · 1 of Q, which is known to coincide with the center Z(Q) of Q. We refer the reader to [3,4] for a comprehensive presentation of the theory of quaternion algebras. A quaternion is called pure (see e.g., [4, Definition 5.2.1]) if ∈ span K {i, j, k}. Every quaternion ∈ Q can be uniquely expressed as a sum = a + 0 of a scalar a ∈ K and a pure quaternion 0 . We write := a − 0 for the conjugate of . The map that sends a quaternion to its conjugate is an involution. If x is an element of either a quadratic field extension L = K √ α of K or a quaternion algebra Q = α,β K over K, we write N (x) := xx and call it the norm of x. If the domain is not clear from the context, we write N L/K or N Q/K . Remark 2.1. When Q is a quaternion algebra, the norm of in the above sense should not be confused with the determinant of the endomorphism of Q defined by the multiplication by , which is often also called the norm. For this reason, in [3,4] the map → is called the reduced norm and denoted nrd. In that manner, our terminology in the present paper agrees with the one used by Lam in [5] but not with the one used by Vigneras in [3] and Voight in [4]. Equivalency classes of valuations on K are called places. Throughout this paper, places are denoted using fraktur letters p, q, r. Every place of a global field is either archimedean, when it extends the standard absolute value on Q (then the field K is necessarily a number field) or non-archimedean. Over a global function field, every place is non-archimedean. To avoid monotonous repetitions, nonarchimedean places will also be called primes (or finite primes when we want to emphasize the fact that they are non-archimedean). The completion of K with respect to a place p is denoted K p . If p is a finite prime, we write ord p : K → Z to denote the corresponding (normalized) discrete valuation on K. The prime p is called dyadic if ord p 2 = 0. The map ord p induces a natural map K × /K ×2 → Z /2Z on the group of square classes of K that is again denoted ord p . If p is an archimedean place, then the completion K p is isomorphic either to C or to R. The places of the second kind are called real. We write sgn r a for the sign of a ∈ K with respect to for the unique ordering of K induced by a real place r. Given some nonzero elements a 1 , . . . , a n ∈ K we denote by a 1 , . . . , a n the quadratic form a 1 x 2 1 + · · · + a n x 2 n . Further, if p is a place and a, b ∈ K × we write (a, b) p for the Hilber symbol of a and b at p, that is (a, b) p := 1 if a,b K ∼ = M 2 K p , −1 otherwise. For a quadratic form ξ = a 1 , . . . , a n we define its Hasse invariant s p ξ at p by the formula (see e.g., [5,Definition V.3.17]): s p ξ := i<j (a i , a j ) p . Finally, abusing the notation harmlessly, by log -1 we will denote the (unique) isomorphism from the multiplicative group {±1} to the additive group {0, 1} with addition modulo 2. Square roots of non-central elements Let us begin by writing down the explicit formula for a square in quaternion algebra so that we can easily reference it in the discussion that follows. Observation 3.1. If = q 0 + q 1 i + q 2 j + q 3 k ∈ Q is a quaternion, then 2 = (q 2 0 + q 2 1 α + q 2 2 β − q 2 3 αβ) + 2q 0 q 1 i + 2q 0 q 2 j + 2q 0 q 3 k = (2q 2 0 − N ( )) + 2q 0 · (q 1 i + q 2 j + q 3 k). (1) An immediate consequence of the previous observation is the following rather well-known fact. Corollary 3.2. If ∈ Q is a pure quaternion, then 2 ∈ Z(Q) = K. Another direct consequence of Eq. (1) is the following observation that may be treated as a partial converse of Corollary 3.2. Observation 3.3. Let ∈ Q be a square root of some element a ∈ K. Then is either pure or ∈ K. Proof: Let = q 0 + q 1 i + q 2 j + q 3 k. If 2 = a ∈ K then by Eq. (1) we have 2q 0 q 1 = 2q 0 q 2 = 2q 0 q 3 = 0. Therefore, if is not pure, that is if q 0 = 0, then q 1 = q 2 = q 3 = 0, hence ∈ K. ⊓ ⊔ Combining Corollary 3.2 with Observation 3.3 we see that for computing the square roots in quaternion algebras it is crucial to distinguish between the case when one computes a quaternionic square root of an element in K (i.e., in the center of Q) and the case when the argument comes from Q \ Z(Q). It turns out that the latter case is, in fact, trivial and requires nothing more than high-school mathematics. Algorithm 1. Let Q = α,β K be a quaternion algebra over a field K of characteristic char K = 2. Given a quaternion = q 0 + q 1 i + q 2 j + q 3 k ∈ Q \ Z(Q), this algorithm outputs its square root or reports a failure when is not a square. 1. Check if the norm N ( ) of is a square in K. (a) If it is not, then report a failure and quit. (b) If it is, let d be an element of K such that d 2 = N ( ). 2. Check if any of the following two elements is a square in K: a + := q 0 + d 2 , a − := q 0 − d 2 . 3. If neither of them is a square, then report a failure and quit. 4. Otherwise, fix r 0 such that either r 2 0 = a + or r 2 0 = a − . Set r 1 := q 1 2r 0 , r 2 := q 2 2r 0 , r 3 := q 3 2r 0 . 6. Output r = r 0 + r 1 i + r 2 j + r 3 k. Proof of correctness: Since the norm N : Q → K is multiplicative, it is obvious that if N ( ) / ∈ K 2 , then cannot be a square in Q. This fact justifies the early exit in step (1a) of the algorithm. Assume that N ( ) = d 2 and let r = r 0 + r 1 i + r 2 j + r 3 k be the sought square root of , if it exists. By Eq. (1) we have q 1 = 2r 0 r 1 , q 2 = 2r 0 r 2 , q 3 = 2r 0 r 3 . It is, thus, clear that it suffices to find r 0 . Again by Eq. (1) we may write q 0 = r 2 0 + r 2 1 α + r 2 2 β − r 2 3 αβ = r 2 0 + q 1 2r 0 2 α + q 2 2r 0 2 β − q 3 2r 0 2 αβ. The above formula can be rewritten in the form of a bi-quadratic equation: 4r 4 0 − 4q 0 r 2 0 + q 2 1 α + q 2 2 β − q 2 3 αβ = 0. If we treat the left-hand-side as a quadratic equation in r 2 0 , then its discriminant equals 16 · N ( ) = (4d) 2 , hence r 2 0 = q 0 ± d 2 = a ± . It follows that the sought quaternion r exists if and only if either a + or a − is a square in K. This proves the correctness of the algorithm. ⊓ ⊔ Remark 3.4. In the above proof, we constructed the square root r of a quaternion ∈ Q \ Z(Q) by solving a bi-quadratic equation. Such equations in general, may have four roots. Hence, one may suspect that there are four distinct quaternions r such that r 2 = . It is not the case. It is clear from the above proof that ∈ Q \ Z(Q) has only finitely moan square roots in Q. Now, if r 2 = Q, then r is a root of a quaternionic polynomial x 2 − . But [6,Theorem 5] asserts that a quadratic polynomial over Q which has more than two zeros must have infinitely many of them. This way, we conclude that has just two square roots. Notice that for hamiltonian quaternions this fact has been observed already 80 years ago by Niven in [2]. Square roots of central elements. Split case It is evident from the preceding section that the only non-trivial case that must be considered is how to compute a quaternionic square root of an element of the base field K, which is not a square in K. In contrast to the previous case (cf. Remark 3.4), in general, an element a ∈ K = Z(Q) may have infinitely many square roots in Q. Once again, for hamiltonian quaternions it has been observed already by Niven. First, we need, however, to introduce an auxiliary algorithm that is not specific to quaternions, as it deals with an arbitrary quadratic form. Recall that a quadratic form is called isotropic (see e.g., [5,Definition I.3.1]) if it represents zero non-trivially. It is well known (see, e.g., [5,Theorem I.3.4]) that every isotropic form represents all elements of K. Algorithm 2. Let ξ be an isotropic quadratic form of dimension n over a field K of characteristic char K = 2. Given an element a ∈ K and a vector V ∈ K n such that ξ(V ) = 0, this algorithm outputs a vector W ∈ K n satisfying the condition ξ(W ) = a. 1. Find a vector U ∈ K n such that U and V are linearly independent. 2. Set b := ξ(U ) and c : = 1 /2 · ξ(U + V ) − ξ(U ) . Output W := U + a − b 2c · V. Proof of correctness: Just compute: ξ(W ) = ξ U + a − b 2c · V = ξ(U ) + a − b 2c · ξ(U + V ) − ξ(U ) − ξ(V ) + (a − b) 2 4c 2 ξ(V ) = b + a − b 2c · 2c + 0 = a ⊓ ⊔ Recall that a quaternion algebra Q = α,β K is said to split (see e.g., [4,Definition 5.4.5]) if Q is isomorphic to the matrix ring M 2 K. It is well known (see e.g., [4,Theorem 5.4.4] or [5, Theorem III.2.7]) that Q is split if and only if the quadratic form −α, −β, αβ is isotropic. If it is the case, the preceding algorithm combined with Eq. (1) lets us compute the quaternionic square root of any element of the base field. In particular, when K is a global field, char K = 2, then the computation of the square root of a ∈ K in a split quaternion algebra boils down to solving a norm equation in a quadratic extension of K. Algorithms for the latter task are well known. They can be found in [7,8,9,10,11]. Algorithm 3. Let Q = α,β K be a split quaternion algebra over a global field K of characteristic char K = 2. Given a nonzero element a ∈ K, this algorithm outputs a pure quaternion ∈ Q such that 2 = a. 1. Check if α is a square in K. If there is c ∈ K × such that c 2 = α, then set V := (0, c, 1). Otherwise, if α / ∈ K ×2 , then: (a) Construct a quadratic field extension L = K √ α of K. 4. Output = 0 + w 1 i + w 2 j + w 3 k. Proof of correctness: We claim that the vector V constructed either in step (1) or in step (2) of the algorithm is an isotropic vector for ξ. First, suppose that α is a square in K. Say α = c 2 for some c ∈ K × . Then −α · 0 2 − β · c 2 + αβ · 1 2 = 0. Conversely, assume that α / ∈ K ×2 and so L = K √ α is a proper extension of K. Let λ = b + c √ α be an element of L such that N (λ) = − α /β. Then − α β = λλ = b 2 − αc 2 . It follows that −α · 1 2 − β · b 2 + αβ · c 2 = 0. Hence, in both cases V is an isotropic vector of ξ, as claimed. Consequently, executing Algorithm 2 in step (3) we obtain a vector W satisfying the condition ξ(W ) = −a. Now, by Eq. (1) the square of the quaternion outputted by the algorithm equals 2 = −N ( ) = −ξ(W ) = a. Thus, to conclude the proof, we only need to show that the norm equation in step (2b) is solvable. But this follows immediately from the fact that Q is split. Hence ξ is isotropic. Indeed, if V = (v 1 , v 2 , v 3 ) is an isotropic vector of ξ, then −α · v 2 1 − β · v 2 2 + αβ · v 2 3 = 0. Observe that v 1 must be nonzero since otherwise, α would be a square. It follows that − α β = v 2 v 1 2 − α v 3 v 1 2 = N L/K v 2 v 1 + v 3 v 1 √ α . Therefore, the norm equation is solvable, as claimed. Square roots of central elements. Non-split case Now the only case left to be dealt with is when a ∈ K × but Q is not split. Here we have to solve not one but two norm equations (see Algorithm 5 below). First, however, we need to introduce the following auxiliary algorithm that constructs an element simultaneously represented by two binary forms. Recall (see e.g., [5, Definition I.2.1]) that for a given quadratic form ξ of dimension d, we denote the set of nonzero elements of K represented by ξ by the symbol D K (ξ) := ξ(V ) | V ∈ K d and ξ(V ) = 0 . Let P be any finite set of primes of K. Recall that an element a ∈ K × is called P-singular if ord p a ≡ 0 (mod 2) for all finite primes p / ∈ P. The set of all P-singular elements forms a subgroup of the group K × containing K ×2 . Thus, the notion of P-singularity generalizes naturally to the square classes. Define the set E P := aK ×2 | a is P-singular of P-singular square classes. It is a subgroup of the group K × /K ×2 of square classes of K, hence a vector space over F 2 . It is known that the dimension of this vector space is finite. In fact it equals (see e.g., [12, p. 607 ]) dim F 2 E P = |P| + dim F 2 CP /C 2 P , where C P is the P-class group of K. There is a number of know algorithms to construct a basis of this vector space. For details see e.g., [13,14,15]. Algorithm 4. Let K be a global field of characteristic char K = 2. Given two binary quadratic forms ξ = x 0 , x 1 and ζ = z 0 , z 1 over K with x 0 , x 1 , z 0 , z 1 = 0, this algorithm outputs a nonzero element d ∈ K × such that d ∈ D K (ξ) ∩ D K (ζ) or reports a failure if there is no such d. (c) Construct a vector W = (w 1 , . . . , w r ) ∈ {0, 1} r setting w i = log -1 sgn r i x 0 if sgn r i x 0 x 1 = 1, log -1 sgn r i z 0 if sgn r i x 0 x 1 = −1. Otherwise, if the field K is non-real, set R := ∅, r = 0 and W := (). 6. Repeat the following steps until the sought element d is found: u i = log -1 (x 0 , x 1 ) p i and v i = log -1 (z 0 , z 1 ) p i . (d) Construct matrices A = (a ij ) and B = (b ij ), with k = |B| columns and s = |P| rows, setting a ij = log -1 (−x 0 x 1 , β j ) p i and b ij = log -1 (−z 0 z 1 , β j ) p i . (e) If R = ∅ construct a matrix C = (c ij ) with k columns and r = |R| rows, setting c ij = log -1 sgn r i β j . Otherwise, when R = ∅, set C = (). (f) Check if the following system of F 2 -linear equations has a solution    A B C    ·     x 1 . . . x k     =    U V W    (Ϙ) (g) If it does, denote the solution by (ε 1 , . . . , ε k ) ∈ {0, 1} k . Output d = β ε 1 1 · · · β ε k k and quit. (h) If the system (Ϙ) has no solution, then append a new prime p to P (see Remark 5.1 below) and reiterate the loop. Proof of correctness: First, suppose that −x 0 x 1 is a square in K. This means that the form ξ is isotropic (see, e.g., [5,Theorem I.3.2]). Hence, by [5,Theorem I.3.4] it represents every element of K. In particular, it represents z 0 . Since ζ also represents z 0 (trivially), step (1) of the algorithm outputs the correct result. The same argument also applies to step (2), when it is the form ζ that is isotropic. It is also clear that the sets D K (ξ) and D K (ζ) of elements represented by ξ and ζ, intersect if and only if ξ ⊥ (−ζ) is isotropic. This justifies the test in step (3). Therefore, without loss of generality, for the remainder of the proof, we may assume that ξ ⊥ (−ζ) is isotropic while both forms ξ and ζ are anisotropic. We will first show that the algorithm terminates. Let W = (w 0 , w 1 , w 2 , w 3 ) ∈ K 4 be an isotropic vector of ξ ⊥ (−ζ). Denote e := ξ(w 0 , w 1 ) = ζ(w 2 , w 3 ). Further, let R and P be the sets of places (real and non-archimedean, respectively) constructed in steps (4-5) of the algorithm. Now, [17, Lemma 2.1] asserts that there exists a finite prime p 0 of K and an element d ∈ K × such that: i. ord p d = 0 for every finite prime p / ∈ P ∪ {p 0 }; ii. d ≡ e (mod p 1+ordp 4 ) for every p ∈ P; iii. ord p 0 d = 1; iv. sgn r d = sgn r e for every real places r of K. Let B = {β 1 , . . . , β k } be a basis of the group E P∪{p 0 } of P ∪ {p 0 } -singular square classes. The element d is P ∪ {p 0 } -singular, hence it can be expressed in the form d = β ε 1 1 · · · β ε k k , where ε 1 , . . . , ε k ∈ F 2 are the coordinates of d with respect to B. Fix a real place r i ∈ R. First, suppose that sgn r i x 0 x 1 = 1, so the form ξ ⊗ K p is definite. Then sgn r i (−d) = sgn r i (−e) = sgn r i x 0 since −e, x 0 , x 1 is isotropic. But this implies that k j=1 (−1) c ij ε j = k j=1 sgn r i β ε j j = sgn r i d = sgn r i x 0 = (−1) w i . Consequently c i1 ε 1 + · · · + c ik ε k = w i .(2) Conversely, assume that ξ ⊗ K r i is indefinete, hence ζ ⊗ K r i must be definete. Applying the same arguments to the form ζ instead of ξ, we show that Eq. (2) also holds in this case. Now fix a finite prime p i ∈ P. Observe that by the local square theorem (see, e.g., [5, Theorem VI.2.19]) condition (ii) implies that the local squares classes dK ×2 p i and eK ×2 p i coincide. It follows that the form −d, [5,Proposition V.3.22] asserts that the Hasse invariant of −d, x 0 , x 1 ⊗ K p i equals x 0 , x 1 ⊗ K p i ∼ = −e, x 0 , x 1 ⊗ K p i is isotropic. Now,s p i −d, x 0 , x 1 = (−1, x 0 x 1 · d) p i . This can be rewritten as (−x 0 x 1 , d) p i = (x 0 , x 1 ) p i . Substituting β ε 1 1 · · · β ε k k for d we obtain k j=1 (−x 0 x 1 , β j ) ε j p i = (x 0 , x 1 ) p i . Now, (x 0 , x 1 ) p i = (−1) u i and (−x 0 x 1 , β j ) p i = (−1) a ij , where u i , a ij ∈ {0, 1} are the elements constructed in steps (6c-6d). Therefore, the last condition can be expressed as a linear equation over F 2 : a i1 ε 1 + · · · + a ik ε k = u i .(3) Finally, we will show that the above equation also holds for the index i = 0, that is for the prime p 0 In that case, p is certainly non-dyadic, and all three elements x 0 , x 1 , and d have even valuations at p. Hence, [5,Corollary VI.2.5] asserts that −d, x 0 , x 1 ⊗ K p is isotropic. On the other hand, we know from the first part of the proof that if p = p i ∈ P, then d satisfies the condition (−x 0 x 1 , d) p = (x 0 , x 1 ) p , which is equivalent to s p −d, x 0 , x 1 = (−1, x 0 x 1 · d) p . The later condition implies that −d, x 0 , x 1 ⊗ K p is isotropic, again by [5,Proposition V.3.22]. The very same arguments may be applied to the form −d, z 0 , z 1 ⊗ K p . All in all, we have shown that the forms −d, x 0 , x 1 and −d, z 0 , z 1 are locally isotropic in every completion of K. Thus, they are isotropic over K by the Hasse-Minkowski principle (see e.g., [5,Theorem VI.3.1]). This means that the forms ξ and ζ represent d over K by [5,Corollary I.3.5]. ⊓ ⊔ Remark 5.1. To rigorously prove that Algorithm 4 terminates, we must show that the new primes may be appended to the set P in such a way that after finitely many iterations the set will contain the prime p 0 specified in the proof of correctness. If K is a number field, let p 1 , p 2 , p 3 , . . . = 2, 3, 5, . . . be the (strictly increasing) sequence of all prime numbers. On the other hand, if K is a global function field, i.e., a finite extension of a rational function field F q (x), let p 1 , p 2 , p 3 , . . . be a sequence of all the irreducible polynomials from F q [x] ordered in such a way that deg p j ≤ deg p j+1 for every j. Now, appending new primes to P, we can first use the places of K that extend p 1 , then the ones that extend p 2 , then p 3 , and so on. It is clear that this is an exhaustive method, so eventually, we will append p 0 and consequently terminate the algorithm. This proves that the algorithm stops but does not present a complete picture. The existence of the prime p 0 follows from [17, Lemma 2.1], which in turn uses Chebotarev's density theorem. In particular, if P 0 denotes the set of primes of K such that appending any of them makes the algorithm stop, then the set P 0 has positive density. This means that in practice, one can just append primes to P at random with exponentially diminishing probability that the system (Ϙ) fails to be solvable. We are now in a position to present an algorithm that computes a square root of a scalar in a non-split quaternion algebra. Algorithm 5. Let Q = α,β K be a non-split quaternion algebra over a global field of characteristic char K = 2. Given a nonzero element a ∈ K this algorithm outputs a quaternion ∈ Q such that 2 = a or reports a failure if a is not a square in Q. 1. Check if a is a square in K. If there is c ∈ K × such that a = c 2 , then output = c+0i +0j +0k and quit. 2. Check if aα is a square in K. If there is c ∈ K × such that aα = c 2 , then output = 0+ ( c /α)i + 0j + 0k and quit. 3. Check if aβ is a square in K. If there is c ∈ K × such that aβ = c 2 , then output = 0 + 0i + ( c /β)j + 0k and quit. 4. Execute Algorithm 4 with input ξ = a, −α and ζ = β, −αβ . If it fails, then report a failure and quit. Otherwise, let d ∈ K × denote the outputted element represented by these two binary forms. Construct two quadratic extensions of K: L := K √ α and M := K √ aα . 6. Solve the following two norm equations: 7. Output = 0 + a · m 1 m 0 i + l 0 m 0 j + l 1 m 0 k. Proof of correctness: The correctness of the results outputted in step (1) is obvious as is the correctness of output of steps (2)(3). Indeed, if aα = c 2 for some c ∈ K × and = ( c /α)i, then 2 = α · c 2 /α 2 = a. In the remainder of the proof, we can, thus, assume that neither a nor aα is a square in K. Likewise, α is not a square, either, since otherwise, the quaternion algebra Q would split. Therefore, L and M are proper quadratic extensions of K. It follows from Eq. (1) that a is a square of some pure quaternion = q 1 i +q 2 j +q 3 k if and only if a · 1 2 − α · q 2 1 = β · q 2 2 − αβ · q 2 3 . This equality is equivalent to the condition that the sets of elements of K represented by the binary forms ξ = a, −α and ζ = β, −αβ have a non-empty intersection. This proves the correctness of step (4). Now, assume that Algorithm 4 returned some element d ∈ D K (ξ) ∩ D K (ζ). Then there are l 0 , l 1 , m 0 , m 1 ∈ K such that d = am 2 0 − α(am 1 ) 2 = a · N M/K (m 0 + m 1 √ aα) d = βl 2 0 − αβl 2 1 = β · N L/K (l 0 + l 1 √ α) . Rearranging the terms we have . Now, the right-hand-side is nothing else but the square of the quaternion constructed in step (7). This proves that the algorithm is correct. ⊓ ⊔ (b ) )Solve the norm equation N L/K (x) = − α β and denote the solution byλ = b + c √ α. (c) Set V := (1, b, c).3. Let ξ := −α, −β, αβ be the pure subform of the norm form of Q. Execute Algorithm 2 with the input (−a, V, ξ) to construct a vector W = (w 1 , w 2 , w 3 ) such that ξ(W ) = −a. . 1 . 1The construction of the isotropic vector V in steps (1-2) of Algorithm 3 is equivalent to establishing an explicit isomorphism Q ∼ = M 2 K. For details, see[5, Chapter III]. Of course, if the quaternion algebra Q is fixed, the vector V should be computed only once and cached between successive computations of square roots. Remark 4. 2 . 2If the isomorphism Q ∼ = M 2 K is a priori known explicitly, then the computation of the quaternionic square root of any a ∈ K × trivializes, as we have the identity 1 . 1If −x 0 x 1 is a square in K, then output z 0 and quit.2. Likewise, if −z 0 z 1 is a square in K, then output x 0 and quit.3. Check (using e.g.,[16, Algorithm 5]) whether the formξ ⊥ (−ζ) = x 0 , x 1 , −z 0 , −z 1is isotropic. If it is not, then report a failure and quit.4. Construct a set P consisting of all dyadic places of K (if there are any) and of all these nondyadic primes of K where at least one of the elements x 0 , x 1 , z 0 , z 1 has an odd valuation.5. IfK is a formally real number field, then:(a) Construct the set R of all the real places of K, where either ξ or ζ is definite and denote its cardinality by r, i.e. R = r | sgn r x 0 x 1 = 1 or sgn r z 0 z 1 = 1 , r = |R|.(b) [Notation only] Let r 1 , . . . , r r be all the elements of R. (a ) [ )Notation only] Let p 1 , . . . , p s be all the elements of P.(b) Construct a basis B = {β 1 , . . . , β k } of the group E P of P-singular square classes. (c) Construct vectors U = (u 1 , . . . , u s ) and V = (v 1 , . . . , v s ) setting appended to P. This fact follows from Hilbert reciprocity law (see, e.g.,[5,Theorem VI.5.5]). We already know that for every i ∈ {1, . . . , s} we haveThe same also holds for primes not in P. Indeed, if q / ∈ P ∪ {p 0 } then q is non-dyadin and all three elements x 0 , x 1 and d have even valuations at q. Consequently, by [5, Corollary VI.2.5] one obtainsNow, by Hilbert reciprocity law, we can writeHence, in the same way as above, we show that Eq. (3) also holds for i = 0. Applying the same arguments to the form ζ, we obtainfor all i ∈ {0, 1, . . . , s}. All in all, we have proved that Eq. (Ϙ) has a solution in E P∪{p 0 } . Now, for every P ′ ⊇ P ∪ {p 0 } we have E P∪{p 0 } ⊆ E P ′ , hence once the prime p 0 is appended to P the algorithm terminates (see also Remark 5.1 w below). Now, when we have proved that the algorithm stops, we must show that it outputs a correct result. To this end, we will show that the forms −d, x 0 , x 1 and −d, z 0 , z 1 are locally isotropic in every completion of K. The assumptions are symmetric with respect to both forms, except in real places. Hence it generally suffices to prove the isotropy of one of them.Both forms are trivially isotropic in all complex completions of K (provided that there are any) and in all real completions K r for r / ∈ R. Fix now a real place r i ∈ R. First, assume that the form x 0 , x 1 ⊗ K r i is definite. From the preceding part we know that the element d = β ε 1 1 · · · β ε k k , constructed by the algorithm, satisfies the condition sgn r i d = sgn r i x 0 . Therefore the form −d, x 0 , x 1 ⊗ K r i is isotropic. Now, the form ξ ⊥ (−ζ) is isotropic because otherwise, the execution of the algorithm would have been interrupted already in step (3). Thus, either sgn r i z 0 = sgn r i x 0 = sgn r i d or sgn r i z 1 = sgn r i x 0 = sgn r i d. In both cases, we have that the form −d, z 0 , z 1 ⊗ K r i is isotropic, as well. Conversely, assume that ξ ⊗ K r i is indefinite, and so it is ζ ⊗ K r i that must be definite. 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{'abstract': 'We present an explicit algorithmic method for computing square roots in quaternion algebras over global fields of characteristic different from 2.', 'arxivid': '2301.00743', 'author': ['Przemysław Koprowski przemyslaw.koprowski@us.edu.pl \nInstitute of Mathematics\nUniversity of Silesia in Katowice ul\nBankowa 1440-007KatowicePoland\n'], 'authoraffiliation': ['Institute of Mathematics\nUniversity of Silesia in Katowice ul\nBankowa 1440-007KatowicePoland'], 'corpusid': 255372627, 'doi': '10.48550/arxiv.2301.00743', 'github_urls': [], 'n_tokens_mistral': 12615, 'n_tokens_neox': 10619, 'n_words': 6515, 'pdfsha': '24c367e37591d97e7d249f1a524808aa59566696', 'pdfurls': ['https://export.arxiv.org/pdf/2301.00743v2.pdf'], 'title': ['Computing square roots in quaternion algebras', 'Computing square roots in quaternion algebras'], 'venue': []}

arxiv

A SYMMETRIC FRACTIONAL-ORDER REDUCTION METHOD FOR DIRECT NONUNIFORM APPROXIMATIONS OF SEMILINEAR DIFFUSION-WAVE EQUATIONS * Pin Lyu Seakweng Vong A SYMMETRIC FRACTIONAL-ORDER REDUCTION METHOD FOR DIRECT NONUNIFORM APPROXIMATIONS OF SEMILINEAR DIFFUSION-WAVE EQUATIONS * diffusion-wave equationweak singularitynonuniform meshadaptive mesh AMS subject classifications 65M0665M1235B6535R11 We introduce a symmetric fractional-order reduction (SFOR) method to construct numerical algorithms on general nonuniform temporal meshes for semilinear fractional diffusion-wave equations. By using the novel order reduction method, the governing problem is transformed to an equivalent coupled system, where the explicit orders of time-fractional derivatives involved are all α/2 (1 < α < 2). The linearized L1 scheme and Alikhanov scheme are then proposed on general time meshes. Under some reasonable regularity assumptions and weak restrictions on meshes, the optimal convergence is derived for the two kinds of difference schemes by H 2 energy method. An adaptive time stepping strategy which based on the (fast linearized) L1 and Alikhanov algorithms is designed for the semilinear diffusion-wave equations. Numerical examples are provided to confirm the accuracy and efficiency of proposed algorithms. 1. Introduction. In this paper, we consider numerical methods of the semilinear diffusion-wave equation: D α t u = ν 2 ∆u + f (u, x, t), x ∈ Ω, t ∈ (0, T ], (1.1) subject to the initial conditions u(x, 0) = ϕ(x) and u t (x, 0) =φ(x) for x ∈ Ω, and the homogeneous boundary condition u(x, t) = 0 for x ∈ ∂Ω; where Ω = (x l , x r ) × (y l , y r ), 1 < α < 2, ν is a constant, and D δ t denotes the Caputo derivative of order δ: The diffusion-wave equation, which is also called the time-fractional wave equation, can be applied to describe evolution processes intermediate between diffusion and wave propagation. For example, it governs the propagation of mechanical waves in viscoelastic media [24,25]. The practical applications of equation (1.1) span diversely many disciplines, such as the image processing [3,39], the universal electromagnetic, acoustic and mechanical response [31]. It is well known that the solutions of the sub-diffusion equations (also called the time-fractional diffusion equations) typically exhibit weak initial singularities [9,34,35], and it causes that the traditional time-stepping methods fail to preserve their desired convergence rate [9]. The same phenomenon occurs for the diffusion-wave equations. For example, Jin, Lazarov and Zhou [10,Theorem A.4] show that the solution of the linear diffusion-wave equation (f = f (x, t)) satisfies that ∂ m t u L 2 (Ω) ≤ C T t α−m f W m−1,∞ (0,T ;L 2 (Ω)) , m = 1, 2, if f ∈ W 1,∞ (0, T ; L 2 (Ω)) and ϕ =φ = 0. Other studies on regularities can be found in [10,26,34]. Recently some excellent works have been done on the numerical approximation of linear diffusion-wave equations taking the weak initial singularities into account. The convolution quadrature methods generated by backward difference formulas are rigorously discussed in [10], where the first-and second-order temporal convergence rates are obtained under proper assumptions of the given data, and their discrete maximal regularities are further studied by Jin, Li and Zhou [11]. Lately, for the problem with nonsmooth data, a Petrov-Galerkin method and a time-stepping discontinuous Galerkin method are proposed in [22] (Luo, Li and Xie) and [14] (Li, Wang and Xie), where the temporal convergence rate is (3 − α)/2-order and about first-order respectively. Numerical schemes with classical L1 approximation in time and the standard P1-element in space are also implemented in [13] to have the temporal accuracies of O(τ 3−α ) and O(τ 2 ) provided the ratio τ α /h 2 min is uniformly bounded. We note that the numerical methods in the above works [10,11,13,14,22] are implemented on uniform temporal steps. On the other hand, Mustapha & McLean [29] and Mustapha & Schötzau [30] considered the time-stepping discontinuous Galerkin methods on nonuniform temporal meshes to solve the following kind of fractional wave equation: u t + I β Au(t) = f (t), for β ∈ (0, 1) and t ∈ (0, T ], (1.2) where A is a self-adjoint linear elliptic spatial operator. It can be observed that the above integrodifferential problem is (mathematically) equivalent to the linear case of (1.1) under suitable assumptions on f and initial data. Their methods are illuminating and efficient with good temporal accuracies. Laplace transform methods and convolution quadrature methods on uniform temporal steps are also discussed respectively by McLean & Thomée [27,28] and Cuesta et al. [4][5][6] for the above integrodifferential problem, where the reference [4] is for the semilinear case f (t) = f (u, ∇u, x, t). However, the above numerical methods for solving (1.2) may not be easily extended to the semilinear problem (1.1) due to the nonlinearity D β t f (u, t). To the best of our knowledge, there are still challenges for numerical methods of the diffusion-wave equation. In this paper, we will address the following issues: (i) establishing and analyzing difference schemes by the classical L1 [32] and Alikhanov [1] approximations on nonuniform temporal meshes (especially on more general meshes) for the semilinear diffusion-wave equation with typical weak singular solutions; (ii) studying efficient numerical algorithms, such as the adaptive time-stepping algorithm, for the semilinear diffusion-wave equation in order to deal with the highly oscillatory variations in time since the problem (1.1) leads to a mixed behavior of diffusion and wave propagation. Before introducing our main approach, we review two classical and popular algorithms. The first one is the L1 algorithm [32], which was generated by Lagrange linear interpolation formula, it is a direct and convenient approximation formula in constructing numerical methods for sub-diffusion problems, e.g., [21,36,38] where it was employed on uniform temporal grids. Recently, the L1 method on graded temporal meshes, with monotonically increasing step sizes, was analyzed in Stynes, O'Riordan & Gracia [35] and Kopteva [12] to resolve the sub-diffusion equations with weakly singular solutions. The other one is the Alikhanov algorithm, which was firstly proposed by Alikhanov [1] by combining linear and quadratic interpolations skillfully at an off-set time point on uniform mesh for the subdiffusion problem with sufficiently smooth solution. Implementation of this algorithm on graded mesh was discussed by Chen and Stynes [2] and the second-order convergence concerning with the weak initial singularities was established. Particularly, Liao, Li and Zhang [15] presented a novel and technical framework to derive the optimal convergence result of the nonuniform L1 scheme. The techniques were then generalized in Liao, McLean and Zhang [17], which were further extended to a linearized scheme for the semilinear sub-diffusion equations [20] and the Alikhanov scheme on more general nonuniform meshes [16]. We remark that the above methods [2,12,[15][16][17]20,35] on nonuniform meshes are all for sub-diffusion problems. In view of the high efficiency and broad potential applications of the L1 and the Alikhanov algorithms, it is of high scientific value to consider their nonuniform versions for resolving (at least) the weak initial singularities of the diffusion-wave problem. In [36], Sun and Wu utilized a fixed-order reduction method, i.e., by taking an auxiliary function v = u t , to rewrite a linear case of equation (1.1) to the following coupled equations: D α−1 t v = ν 2 ∆u + f (x, t), (1.3) v = u t , (1.4) for x ∈ Ω, t ∈ (0, T ]. We note that the time-fractional derivative on the auxiliary function v in equation (1.3) is of order α − 1 which belongs to (0, 1), so the system (1.3)-(1.4) is not structure consistency in the time derivative order point of view. The diffusion-wave equation can be solved following the standard framework of the L1 method on uniform temporal meshes. Although it is easy to extend the above order reduction method to the corresponding nonuniform L2 scheme (see [36] for its uniform version), we find that it may be difficult to establish its stability and convergence on more general time meshes. Therefore, for the first time, we present a new order reduction method by introducing a novel auxiliary function v = D α 2 t u, where u = u−tφ, which is a non-fixed-order reduction technique, and we call the symmetric fractionalorder reduction (SFOR) method. The semilinear diffusion-wave equation (1.1) is then skillfully rewritten as coupled equations with nice structure or having the feature of structure consistency, i.e. (2.3)-(2.4), see Section 2 for more details. Basing on this equivalent formulation, we can construct the implicit and linearized L1 and Alikhanov algorithms on possible nonuniform time partitions 0 = t 0 < t 1 < · · · < t N = T for a given positive integer N , and discuss their unconditional convergence by utilizing the framework of [15][16][17]20]. Throughout this paper we assume that the solution satisfies the following regularity: ∂ (k) t u H 4 (Ω) ≤ C u (1 + t σ1−k ) and ∂ (k) t v H 4 (Ω) ≤ C u (1 + t σ2−k ), k = 0, 1, 2, 3, (1.5) for t ∈ (0, T ], where σ 1 ∈ (1, 2) ∪ (2, 3) and σ 2 ∈ (α/2, 1) ∪ (1, 2) are two regularity parameters. Our analysis are under the weak mesh assumption: MA. There is a constant C γ > 0 such that τ k ≤ C γ τ min{1, t 1−1/γ k } for 1 ≤ k ≤ N , with t k ≤ C γ t k−1 and τ k /t k ≤ C γ τ k−1 /t k−1 for 2 ≤ k ≤ N , where γ ≥ 1 is the mesh parameter, τ k := t k − t k−1 denotes the k-th time step size for 1 ≤ k ≤ N and τ := max 1≤k≤N {τ k }. We prove that our method can achieve the desired optimal temporal convergence orders (see Theorem 3.5), that is O(τ min{2− α 2 ,γσ1,γσ2} ) for the nonuniform L1 algorithm and O(τ min{2,γσ1,γσ2} ) for the nonuniform Alikhanov algorithm. We note that the sum-of-exponentials approximations [8,19] can also be directly adopted to the proposed nonuniform L1 and Alikhanov algorithms to reduce the memory storage and computational costs. We further design an adaptive time-stepping strategy according to the two kinds of algorithms, which is robust and accurate for dealing with not only the weak initial singularities but also the rapid temporal oscillations of the semilinear diffusion-wave problem. The main contributions of this paper are summarized below: • We propose a novel order reduction method (SFOR) which enables the nonuniform L1 and Alikhanov algorithms for the semilinear diffusion-wave equation can be constructed and analyzed. • Based on some reasonable regularity assumptions and weak mesh restrictions, we obtain the optimal convergence orders: the temporal convergence rate is up to (2 − α/2)-order for the L1 algorithm and second-order for the Alikhanov algorithm. • An adaptive time-stepping strategy is designed for the semilinear diffusion-wave equation to efficiently resolve possible oscillations of the solution. The rest of the paper is organized as follows. In section 2, we present the novel SFOR method and equivalently rewrite the semilinear diffusion-wave equation into coupled equations. In Section 3, we construct and analyze the linearized nonuniform L1 and the nonuniform Alikhanov algorithms, and obtain their optimal convergences unconditionally by H 2 energy method. Furthermore, we design an adaptive time-stepping method by combining the proposed nonuniform (fast linearized) L1 and Alikhanov algorithms. Numerical examples are provided in Section 4 to demonstrate the accuracy and efficiency. A brief conclusion is followed in Section 5, and the analysis of truncation errors is given in Section 7. Throughout the paper, we use C to denote a generic constant which may depends on the data of the governing problem but is independent of time and space step sizes (or nodes). The SFOR method. In this section, we propose a symmetric fractional-order reduction (SFOR) method such that the technical framework proposed in [15][16][17]20] can be adopted to analyze implicit numerical schemes for solving the diffusion-wave equation (1.1) on temporal nonuniform mesh. The basic idea of the SFOR method is presented in the following lemma. Lemma 2.1. For α ∈ (1, 2) and u(t) ∈ C 1 ([0, T ]) ∩ C 2 ((0, T ]), it holds that D α t u(t) = D α 2 t D α 2 t u(t) − u (0)ω 2−α (t). (2.1) Moreover, if we take u(t) := u(t) − tu (0), then D α t u(t) = D α t u(t) = D α 2 t D α 2 t u(t) . (2.2) Proof. Taking v(t) := D α 2 t u(t), one has v = D α 2 t u(t) = (I α 2 u )(t) = t 0 ω 1− α 2 (t − s)u (s) ds = − u (s)ω 2− α 2 (t − s)| t 0 + t 0 ω 2− α 2 (t − s)u (s) ds =u (0)ω 2− α 2 (t) + t 0 ω 2− α 2 (t − s)u (s) ds, where the integration by parts has been utilized. Then v t = d dt D α 2 t u(t) = u (0)ω 1− α 2 (t) + t 0 ω 1− α 2 (t − s)u (s) ds = u (0)ω 1− α 2 (t) + (I 1− α 2 u )(t). Hence, using the composition property I p I q g(t) = I p+q g(t) (p, q > 0) [33, pp. 59], we get D α 2 t D α 2 t u(t) = D α 2 t v(t) = (I 1− α 2 v )(t) =u (0)(I 1− α 2 ω 1− α 2 )(t) + I 1− α 2 I 1− α 2 u (t) =u (0)ω 2−α (t) + I 2−α u (t) =u (0)ω 2−α (t) + D α t u(t), implying (2.1) is true. The equality (2.2) can be obtained directly by taking v := D α 2 t u(t) in the above derivations. Now we take u(x, t) := u(x, t) − tφ(x) and v(x, t) := D α 2 t u(x, t) . From (1.5) and using the Sobolev embedding theorem, we have u t (x, t) ∞ ≤ C(1 + t σ1−1 ) for t ∈ (0, T ]. Then utilizing the Comparison theorem for integrals (see pp. 400-401 in [40]), one has |v(x, 0)| = lim t→0 D α 2 t u(x, t) ≤ 1 Γ(1 − α 2 ) lim t→0 t 0 (t − s) − α 2 |u t (x, s)| ds ≤C lim t→0 t 0 (t − s) − α 2 (1 + s σ1−1 ) ds ≤ C lim t→0 (t 1− α 2 + t σ1− α 2 ) = 0, which gives v(x, 0) = 0. Thus, by Lemma 2.1, the equation (1.1) can be equivalently solved by the following coupled equations: D α 2 t v = ν 2 ∆u + f (u, x, t) + t∆φ, x ∈ Ω, t ∈ (0, T ], (2.3) v = D α 2 t u, x ∈ Ω, t ∈ (0, T ], (2.4) provided u = u + tφ, the initial conditions u(x, 0) = ϕ(x), v(x, 0) = 0 for x ∈ Ω, and boundary conditions u(x, t) = v(x, t) = 0 for x ∈ ∂Ω. One can observe that, by utilizing the proposed SFOR method, the explicit orders of the timefractional derivatives in the resulting coupled equations (2.3) and (2.4) are all α/2. Therefore, they can be discretized by the same strategy (e.g., the L1 or Alikhanov approximations). Remark 2.2. We observe from our numerical experiments that, by extracting the singular term u (0)ω 2−α (t) in (2.1), the proposed algorithms will have more regular accuracy due to the regularity of the remaining part. This is the reason why we define the auxiliary function v = D α 2 t u with u = u − tφ, instead of v = D α 2 t u. 3. Numerical algorithms. Preliminary. Our main concern is the time approximation of (1.1). Here and hereafter, g k and g k h denotes the numerical approximations of g(t k ) and g(x h , t k ), respectively. Define the off-set time points and grid functions t n−θ := θt n−1 + (1 − θ)t n and g n−θ := θg n−1 + (1 − θ)g n , 1 ≤ n ≤ N. Denote β := α/2. The Caputo derivative D β t g(t n−θ ) can be formally approximated by the following discrete Caputo derivative with convolution structure: (D β τ g) n−θ := n k=1 A (n) n−k ∇ τ g k , where ∇ τ g k = g k − g k−1 . (3.1) The general discretization (3.1) includes two practical ones. It leads to the L1 formula while θ = 0 (see also (3.2)) and yields the Alikhanov formula while θ = β/2 (see also (3.4)). To efficiently solve the semilinear diffusion-wave equation with possible weak singular or more complicated solutions, we next give more explicit formulations of these two classical approximations on possible nonuniform meshes, which have also been rigorously studied in [15][16][17]. Nonuniform L1 formula. The L1 formula on general mesh for the approximation of the Caputo derivative D β t g(t n ) is given as: (D β τ g) n := n k=1 t k t k−1 ω 1−β (t n − s)(Π 1,k g(s)) ds = n k=1 A (n) n−k ∇ τ g k , (3.2) where Π 1,k represents the linear interpolation operator, and A (n) n−k := t k t k−1 ω 1−β (t n − s) τ k ds. (3.3) Nonuniform Alikhanov formula. Denote θ := β/2 = α/4, and define the discrete coefficients a (n) n−k := 1 τ k min{t k ,t n−θ } t k−1 ω 1−β (t n−θ − s) ds, 1 ≤ k ≤ n; b (n) n−k := 2 τ k (τ k + τ k+1 ) t k t k−1 ω 1−β (t n−θ − s)(s − t k− 1 2 ) ds, 1 ≤ k ≤ n − 1. Referring to [16], the Alikhanov formula on general mesh for the approximation of the Caputo deriv- ative D β t g(t n−θ ) is (D β τ g) n−θ := n−1 k=1 t k t k−1 ω 1−β (t n−θ − s)(Π 2,k g(s)) ds + t n−θ tn−1 ω 1−β (t n−θ − s)(Π 1,n g(s)) ds = n k=1 A (n) n−k ∇ τ g k , (3.4) where Π 2,k denotes the quadratic interpolation operator, and the discrete convolution kernels A A (n) n−k :=      a (n) 0 + ρ n−1 b (n) 1 , k = n, a (n) n−k + ρ k−1 b (n) n−k+1 − b (n) n−k , 2 ≤ k ≤ n − 1, a (n) n−1 − b (n) n−1 , k = 1, for n ≥ 2, (3.5) with ρ k := τ k /τ k+1 and ρ := max k {ρ k } being the local time step-size ratios and the maximum ratio, respectively. Remark 3.1. In the rest of this paper, we will use the general form (3.1) to represent the nonuniform L1 formula and Alikhanov formula. The discrete coefficients A (n) n−k and the related properties studied later correspondingly refer to those of the nonuniform L1 formula and the Alikhanov formula while θ = 0 and θ = β/2, respectively. The following two basic properties have been verified in [16,17] for the discrete coefficients of the nonuniform L1 formula (with π A = 1) and the nonuniform Alikhanov formula (with π A = 11/4 and ρ = 7/4), which are required in the numerical analysis of corresponding algorithms: A1. The discrete kernels are positive and monotone: A (n) 0 ≥ A (n) 1 ≥ · · · ≥ A (n) n−1 > 0; A2. There is a constant π A > 0 such that A (n) n−k ≥ 1 π A t k t k−1 ω 1−β (tn−s) τ k ds for 1 ≤ k ≤ n ≤ N . With A1-A2,(D β τ g) n−θ , g n−θ ≥ 1 2 n k=1 A (n) n−k ∇ τ ( g k 2 ) for 1 ≤ n ≤ N. (3.6) A discrete fractional Grönwall inequality proposed in [17, Theorem 3.1] is a crucial tool in the numerical analysis of fractional problems. As required in the analysis later, we present a slightly modified version in the following. It is easy to trace the proof of [17, Theorem 3.1] to justify the modification, here we skip its trivial derivations. Lemma 3.2. Let (g n ) N n=1 and (λ l ) N −1 l=0 be given nonnegative sequences. Assume that there exists a constant Λ (independent of the step sizes) such that Λ ≥ N −1 l=0 λ l , and that the maximum step size satisfies max 1≤n≤N τ n ≤ 1 β 4π A Γ(2 − β)Λ . Then, for any nonnegative sequence (v k ) N k=0 and (w k ) N k=0 satisfying n k=1 A (n) n−k ∇ τ (v k ) 2 + (w k ) 2 ≤ n k=1 λ n−k v k−θ + w k−θ 2 + (v n−θ + w n−θ )g n , 1 ≤ n ≤ N, it holds that v n + w n ≤ 4E β (4 max(1, ρ)π A Λt β n )   v 0 + w 0 + max 1≤k≤n k j=1 P (k) k−j g j   for 1 ≤ n ≤ N, (3.7) where E β (z) = ∞ k=0 z k Γ(1+kβ) is the Mittag-Leffler function. The coefficients P k−j−1 − A (k) k−j P (n) n−k , 1 ≤ j ≤ n − 1. (3.8) It has been shown in [17, Lemmma 2.1] that the kernels satisfy 0 ≤ P (n) n−j ≤ π A Γ(2 − β)τ β j , n j=1 P (n) n−j ω 1−β (t j ) ≤ π A , 1 ≤ j ≤ n ≤ N.= {x h = (x l + ih x , y l + jh y )|0 ≤ i ≤ M x , 0 ≤ j ≤ M y }. For any grid functions u h := {u i,j = u(x i , t j )|(x i , t j ) ∈Ω h }, we employ standard five-point finite difference operator ∆ h := δ 2 x +δ 2 y onΩ h to discretize the Laplacian operator ∆, where δ 2 x u i,j := (u i+1,j −2u i,j +u i−1,j )/h 2 x and δ 2 y u i,j is defined similarly. Denote F (u n−θ h ) := f (u n−1 h , x h , t n−θ ) + (1 − θ)∂ u f (u n−1 h , x h , t n−θ )(u n h − u n−1 h ), 1 ≤ n ≤ N. The linearized and implicit difference schemes which based on the L1 and the Alikhanov approximations on general nonuniform temporal meshes for the problem (2.3)-(2.4) or the problem (1.1) are constructed as follows: (D β τ v h ) n−θ = ν 2 ∆ h u n−θ h + F (u n−θ h ) + t n−θ ∆φ h , x ∈ Ω h , 1 ≤ n ≤ N ; (3.10) v n−θ h = (D β τ u h ) n−θ , x ∈ Ω h , 1 ≤ n ≤ N ; (3.11) u n h = u n h + t nφh , x ∈ Ω h , 0 ≤ n ≤ N, (3.12) equipped with the initial conditions u 0 h = ϕ(x h ) and v 0 h = 0 for x ∈ Ω h , and the boundary conditions u n h = v n h = 0 for x ∈ ∂Ω h , 1 ≤ n ≤ N .w = D β t v − f (u, x, t) + t∆φ, x ∈ Ω, t ∈ (0, T ]; w = ν 2 ∆u, x ∈ Ω, t ∈ (0, T ]; v = D β t u, x ∈ Ω, t ∈ (0, T ]. Utilizing the nonuniform L1 and Alikhanov formulas and the linearized technique to approximate the above equations, we obtain an auxiliary system of (3.10)-(3.12): Take Ω h =Ω h ∩ Ω and ∂Ω h =Ω h ∩ ∂Ω. For u h , v h belonging to the space of grid functions which vanish on ∂Ω h , we introduce the discrete inner product u, v := h x h y x h ∈Ω h u h v h , the discrete L 2 -norm u := u, u , the discrete L ∞ -norm u ∞ := max{|u h |}, the discrete H 1 seminorms δ x u and δ y u , and ∇ h u := δ x u 2 + δ y u 2 , where δ x u i− 1 2 ,j := (u i,j − u i−1,j )/h x and similar definition works for δ y u i,j− 1 2 . One can easily check that ∆ h u, u = − ∇ h u 2 , and, for some positive constantsC Ω ,Ĉ Ω , the embedding inequalities are valid ( [18]): u ≤C Ω ∇ h u and max{ ∇ h u , u ∞ } ≤Ĉ Ω ∆ h u . For simplicity of presentation, we take h := max{h x , h y } and C Ω = max{C Ω ,Ĉ Ω }. w n−θ h = (D β τ v h ) n−θ − F (u n−θ h ) + t n−θ ∆φ h , x h ∈ Ω h , 1 ≤ n ≤ N ; (3.13) w n h = ν 2 ∆ h u n h , x h ∈ Ω h , 0 ≤ n ≤ N ; (3.14) v n−θ h = (D β τ u h ) n−θ , x h ∈ Ω h , 1 ≤ n ≤ N ; (3.15) u n h = u n h + t nφh , x ∈ Ω h , 0 ≤ n ≤ N. For ϑ ∈ (0, 1], let U n−ϑ h := u(x h , t n−ϑ ) and denote the solution errors u n h := U n h − u n h = u(x h , t n ) − u n h ,ṽ n h := v(x h , t n ) − v n h , andw n h := w(x h , t n ) − w n h . One can obtain the error system of (3.13)-(3.16): w n−θ h = (D β τṽh ) n−θ − N n−θ h − (T f ) n−θ h + (T v1 ) n−θ h − (T w ) n−θ h , x h ∈ Ω h , 1 ≤ n ≤ N ; (3.17)w n h = ν 2 ∆ hũ n h + ν 2 S n h , x h ∈ Ω h , 1 ≤ n ≤ N ; (3.18)ṽ n−θ h = (D β τũh ) n−θ + (T u ) n−θ h − (T v2 ) n−θ h , x h ∈ Ω h , 1 ≤ n ≤ N ; (3.19)ũ 0 h =ṽ 0 h =w 0 h = 0, x h ∈Ω h ;ũ n h =ṽ n h = 0, x h ∈ ∂Ω h , 1 ≤ n ≤ N. where (T f ) n−θ h , (T v1 ) n−θ h , (T w ) n−θ h , (T u ) n−θ h , (T v2 ) n−θ h and S n h are the temporal and spatial truncation errors, see more details in Appendix (Section 7); and N n−θ h :=F (U n−θ h ) − F (u n−θ h ) =(1 − θ) ∂ u f (u n−1 h , x h , t n−θ )∇ τũ n h +ũ n−1 h ∇ τ U n h 1 0 ∂ 2 u f (sU n−1 h + (1 − s)u n−1 h , x h , t n−θ ) ds +ũ n−1 h 1 0 ∂ u f (sU n−1 h + (1 − s)u n−1 h , x h , t n−θ ) ds.= (D β τ ∆ hṽh ) n−θ + ∆ h N n−θ h − ∆ h (T f ) n−θ h + ∆ h (T v1 ) n−θ h − ∆ h (T w ) n−θ h , x h ∈ Ω h , 1 ≤ n ≤ N ; (D β τwh ) n−θ = ν 2 (D β τ ∆ hũh ) n−θ + ν 2 (D β τ S h ) n−θ , x h ∈ Ω h , 1 ≤ n ≤ N ; (3.20) ∆ hṽ n−θ h = (D β τ ∆ hũh ) n−θ + ∆ h (T u ) n−θ h − ∆ h (T v2 ) n−θ h , x h ∈ Ω h , 1 ≤ n ≤ N ; (3.21)ũ 0 h =ṽ 0 h =w 0 h = 0, x h ∈Ω h ;ũ n h =ṽ n h = 0, x h ∈ ∂Ω h , 1 ≤ n ≤ N. By eliminating the term (D β τ ∆ hũh ) n−θ in (3.20) and (3.21), we get ∆ hw n−θ h = (D β τ ∆ hṽh ) n−θ + ∆ h N n−θ h − ∆ h (T f ) n−θ h + ∆ h (T v1 ) n−θ h − ∆ h (T w ) n−θ h , x h ∈ Ω h , 1 ≤ n ≤ N ; (3.22) 1 ν 2 (D β τwh ) n−θ = ∆ hṽ n−θ h − ∆ h (T u ) n−θ h + ∆ h (T v2 ) n−θ h + (D β τ S h ) n−θ , x h ∈ Ω h , 1 ≤ n ≤ N ; (3.23)ũ 0 h =ṽ 0 h =w 0 h = 0, x h ∈Ω h ;ũ n h =ṽ n h = 0, x h ∈ ∂Ω h , 1 ≤ n ≤ N. Lemma 3.4. Let F(ψ(x), x) ∈ C 2 (R × Ω), and {ψ h } be a grid function which satisfy max{ ψ ∞ , ∇ h ψ , ∆ h ψ } ≤ C ψ . Then there is a constant C F > 0 dependent on C ψ and C Ω such that ∆ h [F(ψ, x)v] ≤ C F ∆ h v . Proof. The proof can be worked out following that of [20, Lemma 4.1] just by routine computations on δ x F(ψ i− 1 2 ,j , (x i− 1 2 , y j )), δ y F(ψ i,j− 1 2 , (x i , y j− 1 2 )), δ 2 x F(ψ i,j , (x i , y j )) and δ 2 y F(ψ i,j , (x i , y j )) using the Taylor formula with integral remainder. We next show the unconditional convergence of the proposed linearized scheme (3.10)-(3.12) based on the H 2 energy method ( [18,20]). Theorem 3.5. Let f ∈ C (4,2,0) (R × Ω × [0, T ]) . If the assumptions in (1.5) and the mesh assumption MA hold, the linearized schemes (3.10)-(3.12) are unconditional convergent with (3.24) ∆ hũ n + ∇ hṽ n ≤ C(τ min{2−β,γσ1,γσ2} + h 2 ), if θ = 0; C(τ min{2,γσ1,γσ2} + h 2 ), if θ = β 2 ; for 1 ≤ n ≤ N. Proof. Taking inner product of equations (3.22) and (3.23) withṽ n−θ h andw n−θ h respectively, we have ∆ hw n−θ ,ṽ n−θ = (D β τ ∆ hṽ ) n−θ ,ṽ n−θ + ∆ h N n−θ − ∆ h (T f ) n−θ + ∆ h (T v1 ) n−θ − ∆ h (T w ) n−θ ,ṽ n−θ (3.25) and 1 ν 2 (D β τw ) n−θ ,w n−θ = ∆ hṽ n−θ ,w n−θ + −∆ h (T u ) n−θ + ∆ h (T v2 ) n−θ + (D β τ S) n−θ ,w n−θ . (3.26) With the identity ∆ hw n−θ ,ṽ n−θ = ∆ hṽ n−θ ,w n−θ and the zero boundary conditions ofṽ n−θ h , it follows form (3.25)-(3.26) that 1 ν 2 (D β τw ) n−θ ,w n−θ + (D β τ ∇ hṽ ) n−θ , ∇ hṽ n−θ = ∆ h N n−θ − ∆ h (T f ) n−θ + ∆ h (T v1 ) n−θ − ∆ h (T w ) n−θ ,ṽ n−θ + −∆ h (T u ) n−θ + ∆ h (T v2 ) n−θ + (D β τ S) n−θ ,w n−θ . Utilizing (3.6) and the Cauchy-Schwarz inequality, the above equation leads to n k=1 A (n) n−k ∇ τ ( w n 2 + ν 2 ∇ hṽ n 2 ) ≤2ν 2 ∆ h N n−θ + ∆ h (T f ) n−θ + ∆ h (T v1 ) n−θ + ∆ h (T w ) n−θ ṽ n−θ + 2ν 2 ∆ h (T u ) n−θ + ∆ h (T v2 ) n−θ + (D β τ S) n−θ w n−θ ≤2ν(1 + ν)(1 + C Ω ) ∆ h N n−θ + T n−θ w n−θ + ν ∇ hṽ n−θ , (3.27) where T n−θ := ∆ h (T f ) n−θ + ∆ h (T v1 ) n−θ + ∆ h (T w ) n−θ + ∆ h (T u ) n−θ + ∆ h (T v2 ) n−θ + (D β τ S) n−θ . From (7.1) and (7.3)-(7.8), there exist positive constant C r such that (3.28) n j=1 P (n) n−j T j−θ ≤ C r (τ min{2−β,γσ1,γσ2} + h 2 ), for θ = 0; C r (τ min{2,γσ1,γσ2} + h 2 ), for θ = β 2 . From the regularity assumptions in (1.5), we introduce the following constant C 0 = max 0≤n≤N { U n ∞ , ∇ h U n , ∆ h U n }. (3.29) The mathematical induction method will be applied to show that w n + ν ∇ hṽ n ≤ E nT n−θ , 1 ≤ n ≤ N, (3.30) where E n := 4E β (2 max(1, ρ)π A Λt β n ) with Λ = 2ν(1 + ν)C f (1 + C Ω ) and T n−θ := 2ν(1 + ν)(1 + C Ω ) × C r (τ min{2−β,γσ1,γσ2} + h 2 ) + π A Γ(1 − β)C f C u t β n h 2 , for θ = 0; C r (τ min{2,γσ1,γσ2} + h 2 ) + π A Γ(1 − β)C f C u t β n h 2 , for θ = β 2 , in which C f = max{(1 − θ)C 1 , C 2 [1 + ((1 − θ)(C 2 + 1) + 1)/θ]} with C 1 and C 2 being two proper positive constants which depend on C 0 and C Ω . While n = 1, it holds thatũ 0 h = 0 and u 0 h = U 0 h ≤ C 0 . Suppose f ∈ C (3,2,0) (R × Ω × [0, T ]),∆ h N 1−θ = (1 − θ) ∆ h f u (u 0 h , x, t 1−θ )ũ 1 h ≤ (1 − θ)C 1 ∆ hũ 1 h ≤ (1 − θ)C 1 ( w 1 + C u h 2 ). (3.31) For simplicity, denote w (n−θ) := (1 − θ) w n + θ w n−1 . Similarly, we define ∇ hṽ (n−θ) . The triangle inequality gives w n−θ ≤ w (n−θ) and ∇ hṽ n−θ ≤ ∇ hṽ (n−θ) . Then, it follows from (3.27) and (3.31) that A (1) 0 ∇ τ ( w 1 2 + ν 2 ∇ hṽ 1 2 ) ≤ 2ν(1 + ν)C 1 (1 + C Ω ) w (1−θ) + ν ∇ hṽ (1−θ) 2 +2ν(1 + ν)(1 + C Ω ) T 1−θ + (1 − θ)C 1 C u h 2 w (1−θ) + ν ∇ hṽ (1−θ) . Thus, applying Lemma 3.2 on the above inequality, and utilizing (3.28), we get w 1 + ν ∇ hṽ 1 ≤ E 1 2ν(1 + ν)(1 + C Ω )P (1) 0 T 1−θ + (1 − θ)C 1 C u h 2 ≤ E 1T 1−θ , which means that (3.30) holds for n = 1. Assume that (3.30) is valid for 1 ≤ k ≤ n − 1 (n ≥ 2). The eq. (3.18) and discrete embedding inequalities imply that max{ ũ k ∞ , ∇ hũ k , ∆ hũ k } ≤ max{1, C Ω } 1 ν 2 w k + S k ≤ max{1, C Ω } 1 ν 2 E kT k−θ + C u h 2 ≤ 1, for 1 ≤ k ≤ n − 1 and small step sizes. So according to (3.29), the numerical solutions satisfy max{ u k ∞ , ∇ h u k , ∆ h u k } ≤ C 0 + 1. Now for k = n, suppose f ∈ C (4,2,0) (R × Ω × [0, T ]). By Lemma 3.4 there exists a positive constant C 2 such that ∆ h N n−θ ≤(1 − θ) ∆ h [f u (u n−1 , x, t n−θ )∇ τũ n ] + 1 0 ∆ h [f u (sU n−1 + (1 − s)u n−1 , x, t n−θ )ũ n−1 ∇ τ U n ] ds + 1 0 ∆ h [f u (sU n−1 + (1 − s)u n−1 , x, t n−θ )ũ n−1 ] ds ≤(1 − θ)C 2 ∆ h (∇ τũ n ) + C 2 ∆ hũ n−1 + C 2 ∆ hũ n−1 ≤C f (1 − θ) ∆ hũ n + θ ∆ hũ n−1 ≤C f w (n−θ) + C f C u h 2 .n−k ∇ τ ( w n 2 + ν 2 ∇ hṽ n 2 ) ≤ 2ν(1 + ν)C f (1 + C Ω ) w (n−θ) + ν ∇ hṽ (n−θ) 2 +2ν(1 + ν)(1 + C Ω ) T n−θ + C f C u h 2 w (n−θ) + ν ∇ hṽ (n−θ) . Applying Lemma 3.2 and utilizing (3.28) again, it yields w n + ν ∇ hṽ n ≤ E n   2ν(1 + ν)(1 + C Ω ) max 1≤k≤n k j=1 P (k) k−j T j−θ + C f C u h 2   ≤ E nT n−θ . Therefore (3.30) is verified. Finally, the desired result (3.24) is reached by (3.18) and unifying the constants. Remark 3.6. A memory and computational storage saving technique (called SOE approximation) investigated in [8] (see also [8,Theorem 2.5] or [19,Lemma 5.1]) to compute the discrete Caputo derivative can be directly employed to the nonuniform L1 and Alikhanov formulas, the corresponding coefficients of fast L1 and fast Alikhanov formulas preserve the properties A1-A2 [17,19] which further ensure the theoretical analysis of the associated fast schemes. Therefore, in our later implementation of adaptive time stepping strategy and numerical tests, we will always utilize the fast L1 formula [17, Example 2] and the fast Alikhanov formula [19, eq. (5.3)] while applying the proposed algorithms (3.10)-(3.12) with θ = 0 and θ = β/2, respectively. 3.4. Adaptive time-stepping strategy. The time mesh assumption in Theorems 3.5 permits us to establish adaptive time-stepping strategy based on the fast L1 and fast Alikhanov algorithms to reduce the computational costs while solving the semilinear diffusion-wave equation, especially when the solution of the governing problem may possess highly oscillatory feature in time. In the following, we refer to [7,19] for designing an adaptive time-stepping algorithm of the semilinear diffusion-wave equation (1.1), the strategy is presented in Algorithm 3.1. The adaptive time step size in Algorithm 3.1 is updated by τ ada (e, τ ) = S tol e 1 2 τ, where S, tol denote the safety coefficient and the tolerance, respectively. Update time step size τ n+2 ← min{max{τ min , τ ada }, τ max }; 6: else 7: Reformulate the time-step size τ n+1 ← max{min{max{τ min , τ ada }, τ max }, 2 3 τ n+1 }; f (u, x, t) = −u 3 + [sin(πx) sin(πy)(1 + t + t α )] 3 + sin(πx) sin(πy) Γ(α + 1) + 2π 2 (1 + t + t α ) . In this situation, the exact solution is u = sin(πx) sin(πy)(1 + t + t α ). One may notice that the regularity parameters in (1.5) are σ 1 = α and σ 2 = α/2 for Example 1. Therefore, according to Theorem 3.5, the optimal mesh parameter is γ opt = (4 − α)/α for the nonuniform L1 scheme and takes the value γ opt = 4/α for the nonuniform Alikhanov scheme. They are all bounded for α ∈ (1, 2). These bounded mesh parameters keep the robustness of the algorithms in practical implementation if the graded mesh t k = T (k/N ) γ is imposed to deal with the weak initial singularity. On the other hand, the optimal grading parameters of the nonuniform schemes in [12,15,16,19,20,35] will grow without bound while the fractional order becomes small as they are all possessing the form γ opt = r/α where α ∈ (0, 1) for the sub-diffusion problems and r should be the optimal time rate, this generally lead to practical limitations. Since the spatial error O(h 2 ) is standard, we only display the temporal accuracy of the fast L1 and fast Alikhanov schemes. For Example 1, we fixed a fine spatial grid mesh with M = 1000 such that the temporal errors dominate the spatial errors. In each tests, the time interval [0, T ] is divided into two parts [0, T 0 ] and (T 0 , T ] with total N time nodes. A graded mesh with t k = T 0 (k/N 0 ) γ in the first interval [0, T 0 ] is utilized to resolve the weak initial singularity, where T 0 = min{1/γ, T }. The discrete H 2 -norm errors e H 2 (N ) = max 1≤n≤N U n − u n H 2 are recorded in each run, and the temporal convergence order is given by Order = log 2 e H 2 (N/2) e H 2 (N ) . Tables 1-3 record the numerical results of the proposed fast L1 scheme with different grading parameters when solving the example for different α. One can observe that the L1 scheme works accurately with the optimal temporal convergence of O(τ min{2− α Tables 4-6. The temporal convergence of O(τ min{2,γ α 2 } ) is well reflected and the optimal second-order convergence is apparent while γ ≥ γ opt = 4/α. In order to show the efficiency of the adaptive algorithm, the fast linearized Alikhanov scheme is applied at the same time to find the solution in the interval (T 0 , T ]. Its temporal mesh is graded (with γ = α/4) in [0, T 0 ] and is uniform in (T 0 , T ]. In the following, we use 'Graded-Uniform' to represent this scheme. Figure 1 displays the numerical solution in maximum-norm of the Algorithm 3.1 and the Graded-Uniform scheme for α = 1.5. It implies that the adaptive mesh suits well with a dense uniform mesh in (T 0 , T ], provided that the adaptive mesh requires 277 time nodes in the remain interval (T 0 , T ] whereas the uniform mesh needs 970 time nodes. Figure 2 gives the solution contour plots of the solutions by the adaptive strategy, which simply shows the wave interactions of the example at different time. The variation of the temporal step sizes of the adaptive strategy with its comparison to those of the Graded-Uniform scheme are presented in Figure 3. The results indicate that the adaptive time-stepping strategy should be efficient and robust in the long time simulation of the semilinear diffusion-wave equations especially when the solution may exhibit high oscillations in time. Then we define a function T n f (x) by (T f ) n h = T n f (x h ). If the assumptions in (1.5) and MA are satisfied, by Lemma 7.2 and the differential formula of composite function, we can obtain (7.1) n j=1 P (n) n−j ∆ h (T f ) n−θ ≤ Cτ min{2γσ1,2} , θ = 0; Cτ min{γσ1,2} , θ = β 2 ; 1 ≤ n ≤ N. For the spatial error, based on the regularity condition, it is easy to know that S n ≤ C u h 2 , 1 ≤ n ≤ N. • For L1 approximation (θ = 0): We have (T w ) n h = (T v2 ) n h = 0 in this situation. According to [15,Lemma 3.3] and [20,Lemma 3.3], the global consistency error of the L1 approximation can be presented in the following lemma. Lemma 7.3. Assume that g ∈ C 2 ((0, T ]) and there exists a constant C g > 0 such that |g (t)| ≤ C g (1 + t σ−2 ), 0 < t ≤ T, where σ ∈ (0, 1) ∪ (1, 2) is a regularity parameter. If the assumption MA holds, it follows that n j=1 P (n) n−j |R j | ≤ C g τ σ 1 /σ + 1 1 − β max 2≤k≤n (t k − t 1 ) β t σ−2 k−1 τ 2−β k ≤ Cτ min{2−β,γσ} . Define the functions T n v1 (x) and T n u (x) by (T v1 ) n h := T n v1 (x h ) and (T u ) n h := T n u (x h ) respectively. Using similar techniques for (7.1), and Lemma 7.3 with the in assumptions (1.5) and MA, we have The global consistency error estimate of the Alikhanov approximation is estimated in the next lemma. n−j |R j−θ | ≤ C g τ σ 1 /σ + t σ−3 1 τ 3 2 + 1 1 − β max 2≤k≤n t β k t σ−3 k−1 τ 3 k /τ β k−1 . By Lemma 7.4, Lemma 7.1, the assumptions in (1.5) and MA, it is easy to get that n j=1 P (n) n−j ∆ h (T v1 ) j−θ h ≤ C τ σ2 1 + τ 3 2 τ σ2−3 1 + max 2≤k≤n (t k − t 1 ) β t σ2−3 k−1 τ 3−β k ≤ Cτ min{3−β,γσ2} , (7.5) n j=1 P (n) n−j ∆ h (T u ) j−θ h ≤ C τ σ1 1 + τ 3 2 τ σ1−3 1 + max 2≤k≤n (t k − t 1 ) β t σ1−3 k−1 τ 3−β k ≤ Cτ min{3−β,γσ1} , (7.6) n j=1 P (n) n−j ∆ h (T w ) j−θ h ≤ C τ σ1+β 1 + max 2≤k≤n t σ1−2 k−1 τ 2 k ≤ Cτ min{2,γσ1} , (7.7) n j=1 P (n) n−j ∆ h (T v2 ) j−θ h ≤ C τ σ2+β 1 + max 2≤k≤n t σ2−2 k−1 τ 2 k ≤ Cτ min{2,γσ2} . (7.8) (t) := (I n−δ u (n) )(t) for t > 0 and n − 1 < δ < n, in which I β represents the Riemann-Liouville fractional integral of order β:I β u(t) := t 0 ω β (t − s)u(s) ds with ω β (t) = t β−1 Γ(β). in (3.7) are called the complementary discrete kernels ([17]) which are defined on the convolution coefficients A L1 and Alikhanov algorithms. We now implement linearized algorithms on temporal nonuniform meshes to solve the coupled equations (2.3)-(2.4) based on the nonuniform L1 and Alikhanov formulas. Some basic notations in the spatial direction are needed. The uniform spatial step sizes are denoted by h x := (x r − x l )/M x and h y := (y r − y l )/M y respectively, where M x , M y are positive integers. The mesh space is given byΩ h : Remark 3.3. The equations (3.10)-(3.12) represent two different numerical algorithms for solving the semilinear diffusion-wave equation. It is the nonuniform L1 algorithm while θ = 0 and is the nonuniform Alikhanov algorithm while θ = β/2 = α/4. In order to analyze the two proposed algorithms, we consider an equivalent form of (3.10)-(3.12). Firstly, denote w := D β t v − f (u, x, t) + t∆φ with the initial condition w(x, 0) := ν 2 ∆ϕ and the boundary w(x, t) := −f (0, x, t). Then (2.3)-(2.4) can be rewritten as can see that equations (3.10)-(3.12) are equivalent to (3.13)-(3.16) by eliminating the functions w n−θ h and w n h . 3.3. Unconditional convergence. In this subsection, we show the unconditional convergence of the proposed nonuniform L1 and Alikhanov algorithms (3.10)-(3.12) according to their auxiliary system (3.13)-(3.16). by Lemma 3.4 and (7.2), there exists a positive constant C 1 such that Algorithm 3. 1 1 : 11Adaptive time-stepping strategy Given: u n , v n and time step τ n+1 Compute (fast) Alikhanov scheme ((3.10)-(3.12) for θ = β/2) with time step τ n+1 ;3: Calculate e n+1 = u experiments. Numerical examples are carried out in this section to show the accuracy and efficiency of proposed algorithms. The absolute tolerance error and the cut-off time ∆t of fast L1 formula [17, Example 2] and the fast Alikhanov formula [19, Lemma 5.1] are set as = 10 −12 and ∆t = τ 1 in all of the following tests. Example 1. We first consider the problem (1.1) with Ω = (0, 1) 2 , T = 1, ν = 1 and For the second interval (T 0 , T ], we use random time-step sizes τ N0+k = (T − T 0 ) k / N1 k=1 k for N 1 = N − N 0 , where k take values in (0, 1) randomly. For this example, we take N 0 = N T +1−γ −1 . Example 2 . 2Consider the semilinear problem (1.1) with Ω = (−1, 1) 2 , ν = 1 and f (u, x, t) = −u 3 . The initial data are given as ϕ(x) = (x 2 − 1)(y 2 − 1){exp{−10((x + 0.4) 2 + y 2 )] + exp[−10((x − 0.4) 2 + y 2 )]},φ(x) = 0. For Example 2, we choose the spatial node M = 100 and also divide the time interval [0, T ] into two parts [0, T 0 ] and (T 0 , T ] with T 0 = 0.02. The Alikhanov algorithm on graded mesh with t k = T 0 (k/N 0 ) γ (γ = α/4) in the first interval [0, T 0 ] is utilized to resolve the possible weak initial singularity, where N 0 = 30. For the remaining interval (T 0 , T ], we employ the proposed adaptive time stepping strategy (Algorithm 3.1) to compute the numerical solution until T = 10. The parameters of the adaptive algorithm for solving this example are tol = 10 −3 , S = 0.9, τ min = 10 −3 , τ max = 10 −1 , τ N0+1 = τ N0 . Fig. 1 . 1The numerical solution in maximum-norm of the Algorithm 3.1 and the Graded-Uniform scheme for Example 2 with α = 1.5. Fig. 2 . 2Contour plots of the solutions of Algorithm 3.1 for Example 2 at different time with α = 1.5. Fig. 3 . 3The variation of time step sizes of the Algorithm 3.1 and the Graded-Uniform scheme for Example 2 with α = 1.5. situations: θ = 0 and θ = β/2. For a function g(t), define the global errorR n−θ := (D β t g)(t n−θ ) − (D β τ g) n−θ , 1 ≤ n ≤ N. • For Alikhanov approximation (θ = β/2): Lemma 7.4. ( [16, Lemma 3.6]) Assume that g ∈ C 3 ((0, T ]) and there exists a constant C g > 0 such that |g (t)| ≤ C g (1 + t σ−3 ), 0 < t ≤ T,where σ ∈ (0, 1) ∪ (1, 2) is a regularity parameter. a natural and important property is valid for the nonuniform L1 formula [15, proof of Theorem 2.1] and the nonuniform Alikhanov formula [16, Corollary 2.3]: Table 1 1Numerical accuracy in temporal direction of fast L1 scheme for Example 1, where α = 1.1.γ = 1 γ opt = (4 − α)/α ≈ 2.64 γ = 9 8 γ opt ≈ 2.97 N e H 2 (N ) Order e H 2 (N ) Order e H 2 (N ) Order 16 4.2668e-02 * 1.4285e-01 * 2.4493e-01 * 32 3.3723e-02 0.34 3.5519e-02 2.01 5.1210e-02 2.26 64 2.2386e-02 0.59 1.0731e-02 1.73 1.2611e-02 2.02 128 1.3688e-02 0.71 2.4621e-03 2.12 3.8861e-03 1.70 Theoretical Order 0.55 1.45 1.45 Table 2 2Numerical accuracy in temporal direction of fast L1 scheme for Example 1, where α = 1.5.γ = 1 γ opt = (4 − α)/α ≈ 1.67 γ = 9 8 γ opt ≈ 1.88 N e H 2 (N ) Order e H 2 (N ) Order e H 2 (N ) Order 16 3.4875e-02 * 2.2507e-01 * 2.5657e-01 * 32 1.2196e-02 1.52 3.6991e-02 2.61 3.4006e-02 2.92 64 8.7566e-03 0.48 1.2921e-02 1.52 1.3962e-02 1.28 128 5.5637e-03 0.65 4.2362e-03 1.61 3.8805e-03 1.85 Theoretical Order 0.75 1.25 1.25 Table 3 3Numerical accuracy in temporal direction of fast L1 scheme for Example 1, where α = 1.9.γ = 1 γ opt = (4 − α)/α ≈ 1.11 γ = 9 8 γ opt ≈ 1.24 N e H 2 (N ) Order e H 2 (N ) Order e H 2 (N ) Order 16 7.4641e-02 * 7.4027e-02 * 1.3225e-01 * 32 3.7415e-02 1.00 3.4234e-02 1.11 3.4250e-02 1.95 64 1.7990e-02 1.06 1.6886e-02 1.02 1.6737e-02 1.03 128 8.4156e-03 1.10 8.0910e-03 1.06 8.0567e-03 1.05 Theoretical Order 0.95 1.05 1.05 Table 4 4Numerical accuracy in temporal direction of fast Alikhanov scheme for Example 1, where α = 1.2.γ = 1 γ opt = 4/α ≈ 3.33 γ = 9 8 γ opt ≈ 3.75 N e H 2 (N ) Order e H 2 (N ) Order e H 2 (N ) Order 16 5.2656e-02 * 1.2494e-01 * .5496e-01 * 32 3.2671e-02 0.69 3.3236e-02 1.91 4.1575e-02 1.90 64 2.0683e-02 0.66 8.5962e-03 1.95 1.0801e-02 1.94 128 1.1645e-02 0.83 2.1990e-03 1.97 2.8352e-03 1.93 Theoretical Order 0.60 2.00 2.00 Table 5 5Numerical accuracy in temporal direction of fast Alikhanov scheme for Example 1, where α = 1.5.γ = 1 γ opt = 4/α ≈ 2.67 γ = 9 8 γ opt = 3 N e H 2 (N ) Order e H 2 (N ) Order e H 2 (N ) Order 16 3.0823e-02 * 7.0440e-02 * 8.7416e-02 * 32 1.3857e-02 1.15 1.8560e-02 1.92 2.3212e-02 1.91 64 6.2024e-03 1.16 4.7736e-03 1.96 5.9919e-03 1.95 128 2.6236e-03 1.24 1.2150e-03 1.97 1.5269e-03 1.97 Theoretical Order 0.75 2.00 2.00 Table 6 6Numerical accuracy in temporal direction of fast Alikhanov scheme for Example 1, where α = 1.8. fast Alikhanov scheme are carried out for the example, and the results are listed inγ = 1 γ opt = 4/α ≈ 2.22 γ = 9 8 γ opt = 2.50 N e H 2 (N ) Order e H 2 (N ) Order e H 2 (N ) Order 16 1.9521e-02 * 3.5560e-02 * 4.3938e-02 * 32 6.7203e-03 1.54 9.3089e-03 1.93 1.1590e-02 1.92 64 2.6309e-03 1.35 2.3828e-03 1.97 2.9755e-03 1.96 128 1.1487e-03 1.20 6.0470e-04 1.98 7.5559e-04 1.98 Theoretical Order 0.90 2.00 2.00 ,γ α 2 } ). Similar numerical tests of the 5. Concluding remarks. We proposed a novel order reduction method to equivalently rewrite the semilinear diffusion-wave equation into coupled equations, where the explicit time-fractional derivative orders are all α/2. The L1 and Alikhanov schemes combining with linearized approximations have been constructed for the equivalent problem. By using H 2 energy method, unconditional convergences (Theorem 3.5) were obtained for the two proposed algorithms under reasonable regularity assumptions and weak mesh restrictions. An adaptive time-stepping strategy was then designed for the semilinear problem to deal with possible temporal oscillations of the solution. The theoretical results were well demonstrated by our numerical experiments.We finally point out several relevant issues that deserve for further study: (i) deriving the regularity of the linear and semilinear diffusion-wave equations for the difference schemes; (ii) studying the energy properties of the nonlinear diffusion-wave equations for both the continuous and discrete versions, noting that corresponding properties were investigated recently for the nonlinear sub-diffusion problems[37]; (iii) extending the proposed methods to some related problems, such as the multi-term time-fractional wave equation[23].6. Acknowledgement. The authors are very grateful to Prof. Hong-lin Liao for his great help on the design of the SFOR method and valuable suggestions on other parts of the whole paper.According to [16, Lemma 3.8 and Theorem 3.9], we have the follow lemma on estimating the time weighted approximation.Lemma 7.1. Assume that g ∈ C 2 ((0, T ]) and there exists a constant C g > 0 such thatwhere σ ∈ (0, 1) ∪ (1, 2) is a regularity parameter. Denote the local truncation error of g n−ϑ (hereIf the mesh assumption MA holds, thenThe following lemma is provided to analyze (T f ) n−θ h , which is analogous to Lemma 3.4 in[20].Lemma 7.2. Assume that η ∈ C([0, T ]) ∩ C 2 ((0, T ]) and there exists a constant C u > 0 such thatwhere σ ∈ (0, 1) ∪ (1, 2) is a regularity parameter. Assume further that the nonlinear function f (u, x, t) ∈ C 4 (R) with respect to u. Denote η n = η(t n ) and the local truncation errorIf the assumption MA holds, then n j=1 P (n)Proof. Denote R n−θ η := η(t n−θ )−η n−θ . We have R n−θ η = 0 while θ = 0. By the Taylor expansion, we haveFollowing the proof of [20, Lemma 3.4] and using Lemma 7.1, the desired result holds immediately.For x ∈ Ω, let ξ n (x) be a spatially continues function and denote ξ n h := ξ n (x h ). 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{'abstract': 'We introduce a symmetric fractional-order reduction (SFOR) method to construct numerical algorithms on general nonuniform temporal meshes for semilinear fractional diffusion-wave equations. By using the novel order reduction method, the governing problem is transformed to an equivalent coupled system, where the explicit orders of time-fractional derivatives involved are all α/2 (1 < α < 2). The linearized L1 scheme and Alikhanov scheme are then proposed on general time meshes. Under some reasonable regularity assumptions and weak restrictions on meshes, the optimal convergence is derived for the two kinds of difference schemes by H 2 energy method. An adaptive time stepping strategy which based on the (fast linearized) L1 and Alikhanov algorithms is designed for the semilinear diffusion-wave equations. Numerical examples are provided to confirm the accuracy and efficiency of proposed algorithms.', 'arxivid': '2101.09678', 'author': ['Pin Lyu ', 'Seakweng Vong '], 'authoraffiliation': [], 'corpusid': 231698462, 'doi': '10.1007/s10915-022-02000-9', 'github_urls': [], 'n_tokens_mistral': 24566, 'n_tokens_neox': 20729, 'n_words': 10732, 'pdfsha': 'b9626c42611c706685509a21d35a84658ce0149b', 'pdfurls': ['https://arxiv.org/pdf/2101.09678v3.pdf'], 'title': ['A SYMMETRIC FRACTIONAL-ORDER REDUCTION METHOD FOR DIRECT NONUNIFORM APPROXIMATIONS OF SEMILINEAR DIFFUSION-WAVE EQUATIONS *', 'A SYMMETRIC FRACTIONAL-ORDER REDUCTION METHOD FOR DIRECT NONUNIFORM APPROXIMATIONS OF SEMILINEAR DIFFUSION-WAVE EQUATIONS *'], 'venue': []}

arxiv

Constraining Co-Varying Coupling Constants from Globular Cluster Age Rajendra P Gupta rgupta4@uottawa.ca Department of Physics University of Ottawa K1N 6N5OttawaONCanada Constraining Co-Varying Coupling Constants from Globular Cluster Age 1starsmain-sequenceglobular clustersstellar agescosmologyDirac cosmologyvarying coupling constants Equations governing the evolution of a star involve multiple coupling constants. Thus, the time it spends as a main-sequence star can be expected to depend on whether or not such constants vary over the time scale of stellar evolution. When the star belongs to a globular cluster, the star's age cannot exceed that of the globular cluster, and the latter cannot exceed the age of the Universe. This fact can be used to constrain or verify the variation of the coupling constants, i.e., the speed of light , the gravitational constant , the Planck constant ℎ, and the Boltzmann constant . We have estimated the age of the main-sequence star analytically from the time it takes to synthesize all its hydrogen into helium under fixed and varying coupling constants scenarios. When we permitted the interrelated variation of the four constants (~3~ℎ 3~3/2 ) and differentiated between the cosmological energy and local energy conservation laws, we could show that the variation of the constants established in our earlier studies, i.e., ̇⁄ = 3⁄ = 3ℎ̇ℎ ⁄ = 1.5̇⁄ = 3.90(±0.04) × 10 −10 yr −1 at the current cosmic time is consistent with the present work. Nevertheless, the challenge remains to come up with an experiment, astrometric or terrestrial, that can unequivocally prove or falsify the predicted variation. Introduction The most studied fundamental constants for their potential variations are the dimensionless fine structure constant and the dimensionful gravitational constant , especially since Dirac [1] predicted their variations by analyzing his large number hypothesis. They have been studied copiously, theoretically as well as observationally. Uzan [2,3] has thoroughly reviewed variations of these constants, among others, including the speed of light , the Planck constant ℎ, and the Boltzmann constant . Our interest in this study is in the variations of , , ℎ, and . We will, therefore, briefly state their current status without attempting to be comprehensive. Based on the work of Jordan [4], Brans and Dicke [5] developed a scalar-tensor theory of gravitation wherein 1/ was raised to the status of a scalar field potential that could vary spatially and temporally. It was Teller [6] who first attempted to determine a constraint on the variation of from the stellar scaling laws applied to the evolution of Solar luminosity and the environment required for the existence of life on Earth in the past. Many methods have been developed since then to determine the variation of , which have all resulted in its variation well below Dirac's prediction [1]. These include methods based on solar evolution [6][7][8], lunar occultation and eclipses [9], paleontological evidence [10], white dwarf cooling and pulsation [11][12][13], galaxy cluster and globular cluster dimensions [14], age of globular clusters [15], neutron star masses and ages [16], cosmic microwave background temperature anisotropies [17,18], bigbang nucleosynthesis abundances [19,20], asteroseismology [21], lunar laser ranging [22,23], the evolution of planetary orbits [24][25][26], binary pulsars [27][28][29], supernovae type Ia luminosity evolution [30][31][32], and gravitational waves [33][34][35]. Despite Einstein developing his ground-breaking theory of special relativity based on the constancy of the speed of light, he did consider its potential variation [36]. It was followed by the varying speed of light theories by Dicke [37], Petit [38], and Moffatt [39,40]. Albrecht and Magueijo [41] and Barrow [42] developed such a theory in which Lorentz invariance is broken as there is a preferred frame in which scalar field is minimally coupled to gravity. Other proposals include locally invariant theories [43,44] and vector field theories that cause spontaneous violation of Lorentz invariance [45]. Efforts for constraining the variation of the speed of light include (1) Qui et al. [46]-using supernovae type Ia observations, powerlaw variation, Hubble parameter ( ), BAO (baryonic acoustic oscillations), and CMB (cosmic microwave background); (2) Salzano et al. [47]-using angular diameter distance ( ) maximum, ( ), simulated data, and BAO; (3) Cai et al. [48]-using ( ) and luminosity distance ( ) independently determined by Suzuki et al. [49]; (4) Cao et al. [50]-using ( ) measurement radio quasars at redshift = 1.7; (5) Cao et al. [51]-suggesting the use of gravitational lensing with background sources, such as supernovae type Ia and quasars; (6) Lee [52]-with statistical analysis of gravitational lensing data on velocity dispersion measurements of 161 systems; and (7) Mendonca et al. [53]-using galaxy clusters mass-fraction measurements. The Planck constant and the Boltzmann constants are two more constants that play essential roles in our research. It is worth mentioning the efforts of Mangano et al. [54], de Gosson [55], and Dannenberg [56] in considering the variation of the Planck constant ℎ. The time-dependent stochastic fluctuations of ℎ were studied by Mangano et al. [54]. The effect of varying ℎ on mixed quantum states was considered by de Gosson [55]. Dannenberg [56] reviewed the relevant literature and elevated ℎ to the status of a dynamical field, which through the Lagrangian density derivatives, couples to other fields and to itself. The Doppler broadening of absorption lines in thermal equilibrium can provide a direct measurement of the Boltzmann constant e.g., [57,58]. The spectral line profile analysis of quasars and interstellar media should be able to constrain the variation. Studying the possible variation of one coupling constant, while keeping fixed other coupling constants shown to be correlated through one common dimensionless function [59], may not be prudent. It likely yields erroneous results when several constants are present in the equations used for the analysis of data. Therefore, we permit in our studies the simultaneous variation of all the interrelated constants , , ℎ, and . As a result, we were able to (i) resolve the primordial lithium problem [60], (ii) find a reasonable solution to the faint young Sun problem [61], (iii) show that orbital timing studies do not constrain the variation of [62], (iv) prove that gravitational lensing cannot determine the variation of [63], (v) establish that SNe Ia data are consistent with the co-varying coupling constants (CCC) model [64], (vi) attest the consistency of the CCC model with bright and extreme quasar Hubble diagrams [65] as well as gamma-ray burst data [66], and (vii) show from local energy conservation the interrelationship ~3~ℎ 3~3/2 among the constant [59]. Cuzinatto et al. [67] have theoretically confirmed ~3. They considered a scalar-tensor theory of gravity wherein the scalar field includes the gravitational coupling and the speed of light , both of which are allowed to be functions of the spacetime coordinates. Our work on the resolution of the faint young Sun problem [61] has special relevance to this work as it still remains amicably unresolved using standard stellar evolution models. These models suggest that the solar luminosity several billion years ago was significantly lower than today, and thus the Earth's temperature would be too low for life to evolve as it is today. It is a standard practice to apply local energy conservation laws to the problems involving evolutions at a cosmological time scale when it is well known that energy is not conserved in general relativity and cosmology [68][69][70][71][72][73][74]. Any conclusion reached due to this practice should thus be treated with caution and not considered proof against Dirac's prediction. As in our recent study towards resolving the faint young Sun problem [61], we will deviate from this practice in this work as well, i.e., we will distinguish between cosmological energy and local energy (discussed in Section 4). Our focus in this paper is to explore how the variation of physical constants affects the stellar ages in globular clusters. For this, we will closely follow the analytical approach of Degl'Innocenti et al. [15] and Gupta [61] summarized in Section 2. The results from this approach are presented in Section 3. Section 4 comprises discussion, and Section 5 states our conclusion. Analytical Background Degl'Innocenti et al. [15] expressed in the context of the main-sequence (MS) stellar evolution on the H-R diagram: 'If gravity varies in time, the modification of the MS evolutionary time-scale can be estimated by a surprisingly accurate analytical method.' Accordingly, the MS luminosity of a star of a given mass and initial elemental composition is determined by the gravitational constant and the age of the star. Such a star brightens as it burns hydrogen to helium in its core. Denoting the helium abundance in the central region of the star where hydrogen burns as , we could approximately write ∝ ( ) ( )(1) where and are some functions of and , respectively. Since helium is produced at a rate proportional to , ∝ ∝ ( ) ( ) ⇒ ( ) ∝ ( ( ))(2) Thus, for a star that is born at = with helium abundance (= 0.25) and turns off the MS today ( = 0 ) with ≈ 1 at its center, we may write ∫ ( ) ∝ ∫ ( ( )) 0 1(3) Degl'innocenti et al. [15] assumed to be a power-law function given by = ( ( ) 0 ⁄ ) and determined = 5.6. Therefore, they could write the relation between the apparent turnoff age of a star * corresponding to the constant scenario, and the turnoff age of the star with time-dependent , as follows: * = ∫ ( ( ) 0 ) 0 0 − (4) Assuming a linear dependence of on time given by ( ) = 0 (1 + Γ 0 ( − 0 )), writing = + 1 (= 6.6), integrating Equation (4), and rearranging, they could write * = Γ 0 1 − (1 − Γ 0 ) = 1 − (1 − Γ 0 * ) 1⁄ Γ 0 *(5) They also considered the power-law variation of , i.e., ( ) = 0 ( 0 ⁄ ) and obtained * = (1 + ) 0 ⁄ 1 − (1 − 0 ⁄ ) 1+(6) The authors then analyzed Equations (5) and (6) to establish the bounds on ̇/ , i.e., on the variation, that would yield reasonable main-sequence stellar ages. We will modify the above approach in line with our recent work related to resolving the faint young Sun problem [61]. The luminosity in that work for a star of mass and radius , following Newman and Rood [75], is expressed as ~0 5.5 −0.5 ( ) 7.5(7) Here = 2 5 4 /15 2 ℎ 3 is the Stefan-Boltzmann constant, is the mass of a hydrogen atom, is the mean particle mass measured as a fraction of , and the opacity is defined by Kramer's opacity law = 0 −3.5 , with being the density and the temperature. The stellar energy equation may be written as [76] (p. 23) ( + + + ) + + = 0 (8) The net stellar energy loss due to the kinetic energy , the gravitational energy , the internal energy , and the nuclear energy , results in a stellar luminosity that comprises the bolometric luminosity and the neutrino luminosity . All these energies and luminosities are assumed to be local. However, in the CCC approach, we must also consider the change in cosmological energy and the corresponding luminosity (Section 4 below and [61]). Taking the logarithmic time derivative of Equation (7) yields: ̇=̇−̇0 0 + 5.5̇− 0.5̇+ 7.5̇+ 7.5̇− 7.5̇ ≡̇+̇+̇. (9) Here variations of all the physical constants , , ℎ, and , which are explicitly or implicitly contained in equation (9) relate to changes in the unobservable cosmological luminosity : ̇=⁄ −̇0 0 ⁄ + 7.5̇⁄ ⁄ − 7. 5̇⁄ . For example, changes in the unobservable cosmological luminosity include changes in the rest-mass energy of the Sun due to changes in the speed of light, but it is not reflected in Equation (7) and thus not considered. Since this energy is cosmological, its inclusion will not affect our findings. Let us now examine the other terms in Equation (9), which are relevant to the observable parameters: ̇+̇= 5.5̇− 0.5̇+ 7.5̇(10) 1. The solar mass-loss rate due to nuclear fusion is ≈ 7 × 10 −14 y −1 , and that due to solar winds is ≈ 2 × 10 −14 yr −1 [77], giving a total mass-loss rate of ̇/ ≈ −9 × 10 −14 yr −1 . Similar mass loss is expected from main-sequence stars of interest in this work. It was shown to be negligible. 2. In the CCC approach, we have ~3~ℎ 3~3/2 . Additionally, is measured in units of in relativity and in the CCC model, i.e., → / 0 . We may therefore write ̇/ →̇/ +/ . Here, corresponds to in the standard model. Since ~ as per the stellar scaling laws [78], we could expect ̇⁄ ≈≈ −9 × 10 −14 , i.e., the change in for main-sequence stars in the standard model can also be considered negligible see also [75]. We may, therefore, write ̇/ =/ . 3. We have to now focus on the last term / . Taking as the mean mass fraction of hydrogen (assumed to be 0.75 initially when the star became a main-sequence star) and Z as the mean mass fraction of elements heavier than helium, may be written [79] (p. 54); [78] (p. 42): 1 = 2 + 3 4 + 1 2(11) where = 1 − − is the mean mass fraction of helium, and electrons are included in determining the mean particle mass. Since ≪ , we have: ≅ (2 − 1.25 ) −1(12) We may now write the luminosity function that includes only the factors in Equation (7) that are relevant in determining the stellar evolution, i.e., and , ∝ −0.5 (2 − 1.25 ) −7.5 ∝ /(13) Here the last proportionality is from Equation (2). Upon writing ∝ ( ) from above, we have The reason we have used the Newman and Rood's exponent of 7.5 on rather than the more recent exponent of 5.6 determined by Degl'innocenti is that the former yields even stronger constraints on the constants' variation than the latter. So, if our constraint is good for the Newman and Rood's exponent, it would be good for Degl'innocenti equations as well. Another reason to consider Newman and Rood's equations is that it explicitly includes other relevant constants also, and their treatment cannot be ignored in an analysis where these constants are also varying. We are now ready to compare the findings of Degl'Innocenti et al. [15] with ours. Figure 1. Variation of , , ℎ, and with cosmic time. Only c and h vary similarly. G and k vary differently. However, variations of all these constants are interrelated through a common function [59] and not arbitrarily forced. Results In order for the results of the two findings to be comparable, we have to present them in the same format. Thus, even though the CCC results relate to the variation of the speed of light rather than the gravitational constant, we will use the relationship ~3 to convert the variation to the variation: ̇⁄ = 3⁄ . Let us first consider the calculations of Degl'Innocenti et al. [15]. For the linear dependence of on time ( ) = 0 (1 + Γ 0 ( − 0 )), we have ̇0 0 ⁄ = Γ 0 , and for the power-law variation of , i.e., ( ) = 0 ( 0 ⁄ ) , we obtain ̇0 0 ⁄ = 0 ⁄ . They assumed that the star in a GC was born possibly about a Gyr or so after the Big Bang ( 0 = 14 Gyr), and therefore they set / 0 ≈ 1. We may thus write Equations (5) (1 − ℛ ) 6.6 = 1 − 6.6 (21) If we consider 0.5 ≤ ℛ ≤ 1.5, i.e., could be 50% lower or 50% higher than * , equation (21) yields real values = −0.487 for ℛ = 0.5 and = 0.099 for ℛ = 1.5. Taking * = 14 Gyr, and since ≡ * 0 0 ⁄ , we determine −35 × 10 −12 yr −1 ≤̇0 0 ⁄ ≤ 7 × 10 −12 yr −1 when variation is linear. Solution of Equation (20) for the same bounds on ℛ, i.e., on / * , and 0 = 14 Gyr, yields −6 × 10 −12 yr −1 ≤̇0 0 ⁄ ≤ 6 × 10 −12 yr −1 when variation follows a power-law. One might ask, why consider the age ratio ℛ corresponding to varying versus constant as 0.5 ≤ ℛ ≤ 1.5? We could have chosen other limits, but these are more restrictive than those used by Degl'Innocenti et al. in their work [15] and thus would provide more conservative constraints than their work. Consider now the CCC model. For the linear variation of ( ) = 0 (1 + 1 ( − 0 )) ⇒̇0 0 ⁄ = 1 , and since Following the same steps as above with ≡ * 0 3 0 ⁄ , we determine for this case −429 × 10 −12 yr −1 ≤ 0 0 ⁄ ≤ 429 × 10 −12 yr −1 for 0.5 ≤ ℛ ≤ 1.5. The constraint for the CCC case is substantially more lax compared to the other case. We would like to consider now what ratio of ℛ, i.e., / * we obtain when we use the previously determined value of ̇0 0 ⁄ = 390(±4) × 10 −12 yr −1 contained in equation (18) [60,61,64,65]. This equation yields / * = 0.73 to 1, well within the bound considered above. It is also depicted in Figure 2 for different values of the main-sequence stellar ages. The findings of this study are summarized in Table 1. The ̇0 0 ⁄ values correspond to the ratios ℛ ≡ / * for the models we have considered. The last row has only one ratio calculated for the ̇0 0 ⁄ value predicted by the CCC model. ⁄ values corresponding to the ratios ℛ ≡ / * for the models considered in this study: Degl'Innocenti's-linear variation model with ( ) = 0 (1 + Γ 0 ( − 0 )), Degl'Innocenti's-powerlaw variation model with ( ) = 0 ( 0 ⁄ ) , the CCC linear model with ( ) = 0 (1 + 1 ( − 0 )), and the CCC prediction ( ( ) 0 ⁄ ) −1/2 = 1 + 0.0651( 0 − ) − 0.0013( 0 − ) 2 , as described in Section 3. The last row has only one ratio that has been calculated for the ̇0 0 ⁄ value predicted by the CCC model. Model Discussion We have used the approach of Degl'Innocenti et al. [15] and modified it to be compliant with the CCC approach to explore if the variation of the gravitational constant predicted by the latter is consistent with the age of the globular cluster stars it determines. We applied the same assumptions for the cosmological and local energy conservation that we recently used to resolve the faint young Sun problem [61]. The interrelationship among the speed of light , the gravitational constant , the Planck constant ℎ, and the Boltzmann constant of the CCC model, i.e., ~3~ℎ 3~3/2 , was used to relate the variation with the variation for comparing our findings with those of Degl'Innocenti et al. [15]. We determined that the CCC constraint on ̇0 0 ⁄ is over fifty times laxer than that estimated by Degl'Innocenti et al. [15]. It is worth noticing that in addition to power law and linear variation of the constant consideres by Degl'Innocenti et al., we considered the variation of constants related through the function exp( 1.8 − 1). The main reason for this discrepancy can be traced back to the application of local energy conservation laws to the non-local effects in general relativity and cosmology [61]. Energy conservation is not generally possible in general relativity and cosmology [68][69][70][71][72][73][74], which is also applicable to quantum mechanical systems [80]. Harrison [69,81] observed that the energy lost from one spatial region of a thermodynamically expanding Universe could not show up in another spatial region as work because, cosmologically, all regions are equivalent. Thus, all regions are expanding at the same rate. This led him to conclude energy is not conserved in the Universe. For a commoving ball of photon gas in the Universe, Peebles [68] considered energy loss inside it. He observed, "while energy conservation is a good local concept, …, and can be defined more generally in the special case of an isolated system in asymptotically flat space, there is not a general global energy conservation law in general relativity theory". An analysis of the deep conceptual problem encountered with the standard cosmological model in the context of the violation of energy conservation for commoving volumes was delineated by Baryshev [70]. His work also contains several vital references. Several non-conservation theories of gravity do not require null divergence of the stress-energy tensor [74]. The long and complex story relating to the debate on energy conservation in general relativity among some of the most brilliant minds-Felix Klein, Emmy Noether, David Hilbert, Albert Einstein, and others-was narrated by Brading [82]. In the work of Degl'Innocenti et al. [15] and other studies involving the evolution of stellar bodies to constrain the variation of gravitational constant e.g., [6,8,12,13,16,32], the potential energy change resulting from evolving contributes to the luminosity function. However, this energy change is cosmological and not available to affect the observable luminosity. Similarly, the energy change resulting from variations of , ℎ, or does not contribute to the observable luminosity. For example, even a meager 1% increase in the speed of light would cause the solar mass-energy to increase by 2%. If released as observable luminosity over the life of the Sun, the solar luminosity would increase by about an unacceptable two orders of magnitude over 5 billion years. The energy non-conservation is evident when we allow energy density associated with the cosmological constant Λ to remain constant while the Universe expands. As a result, the total energy of the Universe increases without a concomitant decrease of any known source of energy. Similarly, one does not know where exactly the energy of a photon goes when it is redshifted in an expanding Universe. Thus, the energy conservation in cosmology must involve the sources and sinks of energy that are not directly observable, such as the vacuum energy of the quantum field theory. When one ignores this requirement, it naturally leads to unrealistic constraints on the variation of the constants. Just as a side note in the context of globular clusters, it is worth mentioning briefly the work of Dearborn and Schramm [14]. They estimated the effect of variation on the dimensions of these clusters for constraining . They considered in their study how the self-binding energy of a cluster would evolve when is not constant and consequently affect the dimensions of a cluster. Once the variation of and the effect of variation on the unit of length are considered, the kinetic energy is affected precisely with the same proportionality as the potential energy. The net result is that the dimensions of the clusters are not affected by the variation of the constants, and this fact cannot be used for constraining ̇⁄ . The challenge remains to come up with a controlled experiment, astrometric or terrestrial, that could unequivocally prove the variation of any of the four constants without constraining others to their fixed currently known values. Since the Planck constant is already possible to measure with a precision of about 10 parts in a billion using the Kibble balance e.g., [83][84][85][86], it is obvious to consider if it can be improved so that when measured over say a one-to-ten-year period, it could constrain the variation of ℎ. We have explored this possibility [87] as ℎ is used now to define the unit of mass and weighing of a mass in the Kibble balance involves , , and ℎ through mass which is proportional to ℎ/( 2 ) [88]. XRCD (x-ray crystal density) method can also determine the Planck constant [89]) with precision similar to the Kibble balance. Another high precision method of measuring ℎ is based on photoemission spectroscopy [90,91]. The precision of this method is currently lower than the Kibble balance and XRCD method but may have the potential to increase significantly by improving the design of the electron spectrometer and other components of the apparatus. Another possibility is a high precision determination of the Boltzmann constant [92,93] which can already be measured with an accuracy approaching 1 part in a million. In these experimental methods and any others that may be conceived, it will need to be ensured that one takes into account the variation of other constants that might directly or indirectly influence the outcome of the measurements. Conclusions The ages of the main-sequence stars in globular clusters are not in conflict with those determined using the co-varying coupling constant (CCC) approach, provided: 1. Energy conservation includes the cosmological energy along with the local energy. 2. Variations of gravitational constant , the speed of light , the Planck constant ℎ, and the Boltzmann constant are considered interrelated as ~3~ℎ 3~3/2 . The variations of and , i.e., ̇⁄ = 3⁄ = 390(±4) × 10 −10 yr −1 , determined in earlier publications are consistent with the present study. However, the challenge is to design an experiment, astrometric or terrestrial, which could, without doubt, establish the variation of any of the four constants without fixing remaining to their known laboratory-measured values. 1 . 1The left-hand side of the equation yields a fixed value (= 8.03) and thus is irrelevant for our comparative analysis. Thus, instead of Equation (4), we may write * If we assume ( ) evolves linearly, i.e., in the CCC theory, the constants do not evolve linearly with time. Their evolution is typically expressed in terms of the scale factor or redshift . For the speed of light, = 0 exp( 1.8 − 1). Following our earlier work e.g.,[61,64], when translated into time variation with the time unit of Gyr, it can be graphically depicted as inFigure We and / * = ℛ, Equation(19)may be written as Figure 2 . 2Variation of the ratio / * with stellar main-sequence age for the CCC model value of ̇0 0 ⁄ = 390 × 10 −12 yr −1 . Funding: this research was partially funded by Macronix Research Corporation research grant2020-23.Institutional Review Board Statement: Not applicableInformed Consent Statement: Not applicableData Availability Statement: No external data was used in this research. Table 1 . 1The ̇0 0 Acknowledgments:The author is grateful to Barry Wood of NRC (National Research Council of Canada) for an informed discussion on the limitation of the Kibble balance for measuring the Planck constant at a precision better than ten parts in a billion. 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The Boltzmann project. Metrologia 2018, 55, R1-R20. The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods. Disclaimer/Publisher&apos;s Note, instructions or products referred to in the contentDisclaimer/Publisher's Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

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{'abstract': "Equations governing the evolution of a star involve multiple coupling constants. Thus, the time it spends as a main-sequence star can be expected to depend on whether or not such constants vary over the time scale of stellar evolution. When the star belongs to a globular cluster, the star's age cannot exceed that of the globular cluster, and the latter cannot exceed the age of the Universe. This fact can be used to constrain or verify the variation of the coupling constants, i.e., the speed of light , the gravitational constant , the Planck constant ℎ, and the Boltzmann constant . We have estimated the age of the main-sequence star analytically from the time it takes to synthesize all its hydrogen into helium under fixed and varying coupling constants scenarios. When we permitted the interrelated variation of the four constants (~3~ℎ 3~3/2 ) and differentiated between the cosmological energy and local energy conservation laws, we could show that the variation of the constants established in our earlier studies, i.e., ̇⁄ = 3⁄ = 3ℎ̇ℎ ⁄ = 1.5̇⁄ = 3.90(±0.04) × 10 −10 yr −1 at the current cosmic time is consistent with the present work. Nevertheless, the challenge remains to come up with an experiment, astrometric or terrestrial, that can unequivocally prove or falsify the predicted variation.", 'arxivid': '2302.00552', 'author': ['Rajendra P Gupta rgupta4@uottawa.ca \nDepartment of Physics\nUniversity of Ottawa\nK1N 6N5OttawaONCanada\n'], 'authoraffiliation': ['Department of Physics\nUniversity of Ottawa\nK1N 6N5OttawaONCanada'], 'corpusid': 256445732, 'doi': '10.3390/universe9020070', 'github_urls': [], 'n_tokens_mistral': 17039, 'n_tokens_neox': 14240, 'n_words': 8109, 'pdfsha': 'bf804784407aa636c004180aac861b5c992d7022', 'pdfurls': ['https://export.arxiv.org/pdf/2302.00552v1.pdf'], 'title': ['Constraining Co-Varying Coupling Constants from Globular Cluster Age', 'Constraining Co-Varying Coupling Constants from Globular Cluster Age'], 'venue': []}

arxiv

Approximate Solutions, Thermal Properties and Superstatistics Solutions to Schrödinger Equation I B Okon Department of Physics Theoretical Physics Group University of Uyo Nigeria C A Onate Department of Physical Sciences Landmark University Omu-AranNigeria E Omugbe Department of Physics Federal University of Petroleum Resources EffurunNigeria U S Okorie Department of Physics Akwa Ibom State University Ikot Akpaden, UyoNigeria A D Antia Department of Physics Theoretical Physics Group University of Uyo Nigeria M C Onyeaju Department of Physics Theoretical Physics Group University of Port Harcourt Nigeria Chen Wen-Li School of Intelligent Science and Information Engineering Xi'an Peihua University Xi'anChina J P Araujo Department of Mathematics Federal Institute of the Southeast of Minas Gerais Juiz de ForaBrazil Approximate Solutions, Thermal Properties and Superstatistics Solutions to Schrödinger Equation Thermal propertiesSuperstatisticsCPSEHPParametric Nikiforov-Uvarov methodSchrodinger Wave equation In this work, we apply the parametric Nikiforov-Uvarov method to obtain eigen solutions and total normalized wave function of Schrödinger equation express in terms of Jacobi polynomial using Coulomb plus Screened Exponential Hyperbolic potential (CPSEHP), where we obtained the probability density plots for the proposed potential for various orbital angular quantum number, as well as some special cases (Hellmann and Yukawa potential).The proposed potential is best suitable for smaller values of the screening parameter .The resulting energy eigen equation is presented in a close form and extended to study thermal properties and superstatistics express in terms of partition function () Z and other thermodynamic properties such as; vibrational mean energy () U , vibrational specific heat capacity () C ,vibrational entropy () S and vibrational free energy () F . Using the resulting energy equation and with the help of Matlab software, the numerical bound state solutions were obtained for various values of the screening parameter ( ) as well as different expectation values via Hellmann-Feynman Theorem (HFT). The trend of the partition function and other thermodynamic properties obtained for both thermal properties and superstatistics were in excellent agreement with the existing literatures. Due to the analytical mathematical complexities, the superstatistics and thermal properties were evaluated using Mathematica 10.0 version software. The proposed potential model reduces to Hellmann potential, Yukawa potential, Screened Hyperbolic potential and Coulomb potential as special cases. Introduction The approximate analytical solutions of one-dimensional radial Schrödinger wave equation with a multiple potential function has been studied using a suitable approximation scheme to the centrifugal term within the frame work of parametric Nikiforov-Uvarov method [1]. The solutions to the wave equations in quantum mechanics and applied physics play crucial role in understanding the importance of physical systems [2]. The two most important parts in studying Schrödinger wave equations are the total wave function and energy eigenvalues [3]. The analytic solutions of wave equations for some physical potentials are possible for 0 l  . For 0 l  , special approximation scheme like the Greene-Aldrich and Pekeris approximations are employ to deal with the centrifugal barrier in order to obtain an approximate bound state solutions [4][5][6]. The Greene-Aldrich approximation scheme is mostly applicable for short range potentials [7]. Eigen solutions for both relativistic and nonrelativistic wave equations have been studied with different methods which include: Exact quantisation, WKB, Nikiforov-Uvarov method (NU), Laplace transform technique, asymptotic iteration method, proper quantisation, supersymmetric quantum mechanics approach, vibrational approach, formula method, factorisation method, and the Shifted 1/N-expansion method [8][9][10][11][12][13]. Bound state solutions obtained from Schrodinger wave equation has practical applications in investigating tunnelling rate of quantum mechanical systems [14] and mass spectra of quarkonia systems [15][16][17][18][19]. Among other goals achieve in this research article is to apply the Hellmann-Feynman Theorem (HFT) to eigen equation of the Schrödinger wave equations to obtain expectation values of [20][21][22][23][24][25]. To engage HFT in calculating the expectation values, one needs to promote the fixed parameter which appears in the Hamiltonian to be continuous variable in order to ease the mathematical purpose of taking the derivative [26]. Similarly, application of Hellmann-Feynman Theorem provides a less mathematical approach of obtaining expectation values of a quantum mechanical systems [27][28]. Some of the potential models considered within the framework of relativistic and nonrelativistic wave equations are: Hulthen-Yukawa Inversely quadratic potential [29], noncentral Inversely quadratic potential [30], Modified Hylleraas potential [31], Yukawa, Hulthen, Eckart, Deng-Fan, Pseudoharmonic , Kratzer, Woods-saxon, double ring shape , Coulomb, Tietz -wei , Tietz-Hua , Deng-Fan, Manning-Rosen , trigonometric Rosen-Morse, hyperbolic scalar and vector potential and exponential type potentials among others [32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50]. Coulomb, hyperbolic and screened exponential type potentials have been of interest to researchers in recent times because of their enormous applications in both chemical and physical sciences. In view of this, Parmar [51] studied ultrageneralized exponential hyperbolic potential where he obtained energy eigenvalues, un-normalised wave function and the partition function. This potential reduces to Yukawa potential, Screened cosine kratzer potential, Manning-Rosen potential, Hulthen plus Inversely quadratic exponential Mie-type potential and many others. Okon et al. [52], in their studies, obtained eigen solutions to Schrödinger equation with trigonometric Inversely quadratic plus Coulombic hyperbolic potential where they obtained energy eigen equation and normalised wave function using Nikiforov-Uvarov method. Onate [53] examined bound state solutions of the Schrödinger equation with second Poschl-Teller-like potential where he obtained vibrational partition function, mean energy, vibrational specific heat capacity and mean free energy. In that work, the Poschl-Teller like potential was expressed in (hyperbolic form). The practical application of energy eigen equation of Schrödinger equation in investigating the partition function, thermodynamic properties and superstatistics arouses the interest of many researchers. Recently, Okon et al. [54], obtained the thermodynamic properties and bound state solutions of the Schrödinger equation using Mobius square plus screened Kratzer potential for two diatomic systems (Carbon(II) oxide and Scandium Flouride) within the framework of Nikiforov-Uvarov method. Their results were in agreement to semi-classical WKB among others. They presented energy eigen equation in a close form in order to obtain partition function and other thermodynamic properties. Omugbe et al. [55], recently studied the unified treatment of the nonrelativistic bound state solutions, thermodynamic properties and expectation values of exponentialtype potentials where they obtained the thermodynamic properties within the framework of semiclassical WKB approach. The authors studied the special cases of the potential as Eckart, Manning-Rosen and Hulthén potentials. Besides, Oyewumi et al. [56], studied the thermodynamic properties and the approximate solutions of the Schrodinger equation with shifted Deng-Fan potential model within the framework of asymptotic Iteration method where they apply Perkeris-type approximation to centrifugal term to obtain rotational-vibrational energy eigenvalues for selected diatomic systems A lot of research have been carried out by A.N. Ikot, U. S Okorie and co-authors. These can be seen in Refs. [57][58][59][60]. Also, Boumali and Hassanabadi [61] studied thermal properties of a two dimensional Dirac oscillator under an external magnetic field where they obtained relativistic spin-1\2 fermions subject to Dirac oscillator coupling and a constant magnetic field in both commutative and noncommutative space. In this work, we propose a novel potential called Coulomb plus screened Hyperbolic potential to study bound state solutions, expectation values, superstatistics and thermal properties within the framework of parametric Nikiforov-Uvarov method. This article is divided into 9 sections. The introduction is given in section 1. The parametric Nikiforov-Uvarov method is presented in section 2. The solutions of the radial Schrodinger wave equation is presented in section 3. The application of Hellmann-Feynman Theorem to obtain expectation values is presented in section 4. The thermodynamic properties and superstatistics formulations are presented in sections 5 and 6 respectively. Numerical results and discussion are presented sections 7 and 8 respectively, and the article is concluded in section 9 The propose Coulomb plus screened hyperbolic exponential potential (CPSHEP) is given as 12 The graph of Pekeris approximation to centrifugal term is given below Parametric Nikiforov-Uvarov (NU) Method The NU method is based on reducing second order linear differential equation to a generalized equation of hyper-geometric type and provides exact solutions in terms of special orthogonal functions like Jacobi and Laguerre as well as corresponding energy eigenvalues [62][63][64][65][66][67][68][69]. The reference equation for parametric NU method according to Tezcan and Sever [70] is given as 2 12 1 2 3 22 33 1 ( ) ( ) ( ) 0. (1 ) (1 ) ( ) 2 ( 1) ( ) ( ) 0. 2 d R r l l E V r R r dr r           (7) Equation (7) can only be solved analytically to obtain exact solution if the angular orbital quantum number 0 l  . However, for 0 l  equation (7) can only be solve by using the approximations in (2) to the centrifugal term. Substituting equation (1) into (7) gives 22 12 2 2 2 2 cosh ( ) 2 ( 1) ( ) 0. 2 r r nl v v e d R r Be l l E R r dr r r r r                 (8) By substituting equation (2)                                                         (14)                                        (15) To obtain the normalization constant of equation (15) The wave function is assumed to be in bound at ∈ (0, ∞) and = − ∈ (1, 0) Equation (15) to ∈ (−1, 1). Then equation (19) reduces to   2 2 1 2 1 2 2 ,2 1 1 11 ( ) 1 2 2 2 nl n N zz P z dz                        .(20) Using the standard integral           2 1 1 ,1 1 2 1 1 11 () 2 2 ! 1 2 1 xy xy xy n x n y n ww P w dw n x y n x y n                                   .(21)Let , 2 1, 2 z w x y      . Then using equation (20) the normalization constant can be obtained as           22 2 ( !) 2 2 2 2 2 2 2 2 1 nl n n n N nn                     .(22) Hence the total normalized wave function is given as             2 ,2 1 , 22 2 ( !) 2 2 2 2 2 ( ) (1 ) (1 2 ) 2 2 2 1 n l n n n n R s s s P s nn                          .(23) Expectation Values Using Hellmann-Feynman Theorem In this section, some expectation values are obtain using Hellmann-Feynman Theorem (HFT). According to Hellmann-Feynman Theorem, the Hamiltonian H for a particular quantum mechanical system is express as a function of some parameters q . Let () Eq and () q  be the eigenvalues and eigen function of the Hamiltonian. Then () ( ) ( ) nl E Hq qq qq       .(24) For the purpose of clarity, the Hamiltonian for the propose potential using HFT is   21 2 cosh 11 2 1 . 2 2 2 nl l v r l l Q Q                         (26) where   1 2 2 2 2 1 2 2 2 2 2 6 2 2 2 2 2 2 2 7 2 2 cosh 2 2 cosh 1 1 2 1 1 (2 1) ( 1) 2 1 2 2 2 2 2 cosh 1 1 2 2 2 2 2 cosh 1 1 2 2 v v v B l n l l l l l Q v n l v B v Q n l                                                                                                       1 2 2 2 2 ( 1) 2 cosh 1 1 2 2 ll v n l                                              (27) Expectation value for r v nl Qe v nl                                                 .(28)nl Q ll T v B v l vv B ll Q v v v l nl                                                                                     .(29) Expectation value for 2 nl p  The relationship between T and 2 p is given as 2 2 p T   , therefore   2 2 2 2 2 2 7 2 2 1 2 2 2 1 22 2 2 2 7 2 2 2 2 2 2 2 2 2 ( 1) 4 2 cosh 2 1 2 cosh 2 2 ( 1) cosh 2 2 cosh 1 2 cosh 11 2 22 nl Q p l l v B v l v v B ll Q v v v l nl                                                                                                       .(30) Thermodynamic Properties In this section, we present the thermodynamic properties for the potential model. The thermodynamic properties of quantum systems can be obtained from the exact partition function given by   where k and T are Boltzmann constant and absolute temperature respectively. In the classical limit, the summation in equation (31) can be replaced with an integral: 0 () n E Z e dn       .(32) In order to obtain the partition function, the energy equation (14) can be presented in a close and compact form as   2 1 2 22 22 2 2 2 1 2 2 2 2 2 2 ( 1) 1 2 cosh 1 1 2 8 2 2 2 cosh 1 1 2 2 nl v B ll ll v E v n l v n l                                                                                  .(33) Equation (33) can further be simplified to     2 3 12 nl Q E Q Q n n                  ,(34) where   2 22 22 1 2 1 1 2 3 2 2 2 1 2 2 cosh 2 1 1 , ,( 1), 2 8 2 2 ll v v B Q v Q Q l l l                                (35) and equation (34) can be represented in the form   2 2 2 3 2 3 1 2 2 2 3 1 2 2 nl Q Q Q E Q Q Q Q Q Q                (36) where, n  .(37)                                                                                                                                                            .(40) (v) Vibrational specific heat capacity is given as   2 2 2 2 2 2 ln ( ) Q Z C k k Q e                                                                                                                                .                       (43) where                                                                                                              ) Q Q Q erf Q Q e erf Q erf Q Q Q erf e erf e erf                                                                              ) , ( ) 2 2 , ( ) ( ) 4 ( Superstatistics Formulation Superstatistics is the superposition of two different statistics which is applicable to driven non equilibrium systems to statistical intensive parameter (  ) fluctuation [71]. This intensive parameter which undergoes spatio-temporal fluctuations include: chemical potential and energy fluctuation which is basically describe in terms of effective Boltzmann factor [72]. According to Edet et.al [73], the effective Boltzmann factor is given as   0 ( , ) E B E e f d           ,(45) where   ( , ) f        is the Dirac delta function. However, the generalized Boltzmann factor express in terms of deformation parameter q is given as 22 ( ) 1 2 E q B E e E        .(46) The partition function for superstatistics formalism is then given as 0 () s Z B E dn    .(47) Substituting equation (34) into equation (46) 8 1 2 4 Q Q Q Q Q Q s q Q Q q q Q q Q Q Q Q Z e q Q Q Q Q Q Q Q Q q Q Q Q Q Q Q                                                                               .(50) We use the same procedure of thermodynamic section 5 to obtain superstatistics vibrational mean energy ( Numerical Results Using Matlab 10.0 version, the numerical bound state solutions for the proposed potential was calculated using (14) for different quantum state. Also, using equations (18), (20), (21) and (22) r   . 12 0.2, 0.1 , 0.2 , 1 B v V v V       n l 2 nl r   0.01,     2 A  2 nl r   0.03,     2 A  2 nl r   0.03,     2 A  2 nl r   0.04,     2 A  0 0 -0.1 B v V v V       n l 1 nl r   0.01,     1 A  1 nl r   0.02,     1 A  1 nl r   0.03,     1 A  1 nl r   0.04,     1 A  0 0 -0 The required energy equation is         2 22 1 22 2 1 2 ( 1) 1 1 2 8 1 nl B v l l ll E v n l nl                                       .(52)                                                           .(56 8 Discussion Figure 1 is the graph of Pekeries approximation against the screening parameter  . This graph shows that the approximation is suitable for the proposed potential. Variation of the probability density against the internuclear separation at various quantum state for 0 l  and 1 l  respectively are shown in figure 2. The variation of the probability are similar but for 0, l  there is a more concentration of the electron density at the origin for all the quantum state studied. The concentration is higher for 0. l  At every value of the internuclear distance, the probability density for 0 l  is higher than the probability density for 1 l  . Figure 3: Variation of the probability density against the internuclear separation at various quantum state for 0 l  and 1 l  respectively for Hellmann potential is presented. A more concentration of the electron density is observed at the origin in both cases. It is also seen that the probability density obtained for 0 l  are lower than the probability density obtained for 1. l  In figure 4, we presented the variation of the probability density against the internuclear separation at various quantum state for 0 l  and 1 l  respectively for Yukawa potential. There is more concentration of the electron density at the origin for 0 l  , but this situation is not the same for 1. l  However, the probability density at the second excited state for 1, l  remains constant for all values of the internuclear separation while for 0, l  the revise is the case. In Figure 5, the variation partition function for non-superstatistics and superstatistics with the temperature parameter were observed. The partition function increases non-linearly with β. In both cases, the partition function diverged as β increases, but later converges for the superstatistics. The partition function converges as β becomes positive in the superstatistics. In Figure 6, we presented the variation of vibrational mean energy against  for thermodynamic properties and superstatistics respectively. For the non-superstatistics, the mean energy increases as β goes up for all values of λ. However, at higher values of β, the mean energy for various λ tends to converge. The superstatistics mean energy rises as the temperature of the system decreases. However, at a certain absolute temperature, the superstatistics mean energy increases vertically for various values of the deformed parameter. The variation are opposite. In Figure 7, the variation of the heat capacity against β for four values of λ and q are shown. In each case, the heat capacity decreases monotonically with an increasing β for the non-superstatistics. At zero value of β, the heat capacity for various λ converged and diverge as β increases gradually. For the superstatistics, the heat capacity for various deformed parameter rises while the temperature cools down. The specific heat capacity has a turning point when β equals -150, the specific heat capacity for the superstatistics has a turning point and then converged at absolute zero. In figure 8, the vibrational entropy decreases and diverged while the temperature of the system decreases (β increases) in the non-superstatistics. This decrease is sharper for negative values of the entropy when β is almost constant. The superstatistics entropy varies inversely with β (directly with temperature). This means that when the temperature of the system is raised, the disorderliness of the system also increases for every value of the deformed parameter. The entropy for the superstatistics converged as the temperature parameter tends to zero. In Figure 10, the variation of free energy against the β is seen to be two different steps in the case of a non-superstatistics. Between 0 and 60 values of β, the free energy increases steadily for the various values of λ but beyond this range, the free energy increases sharply at constant β. For the superstatistics, the free energy increases monotonically as the temperature of the system reduces gradually. The free energy is always higher when the deformed parameter is increased. p  respectively. Here, the numerical values in all cases decreases with an increase in the screening parameter. Conclusion In this work, we apply the parametric Nikiforov-Uvarov method to obtain the bound state solutions of Coulomb plus screened-exponential hyperbolic potential. The resulting energy eigen equation were presented in a close and compact form. The research work was extended to study thermal properties, superstatistics and various expectation values. The propose potential also reduce to Hellmann potential, Yukawa potential, Screened Hyperbolic potential and Coulomb potential as special cases. The normalized wave function for the mother potential and that of the Hellmann potential are similar but the normalized wave function of the Yukawa potential seams different. The trend of the thermodynamic and superstatistics curves are in agreement to results of an existing literature. The results of the thermodynamic properties and superstatistics revealed that the effect of the temperature on the thermodynamic properties and the superstatistics are similar. Finally, this research work has practical applications in Physical and chemical sciences. Data Availability: The data used for this work are generated using Matlab programme  analytically. The Hellmann-Feynman Theorem gives an insight about chemical bonding and other forces existing among atoms of molecules Figure 1 : 1The graph of Pekeries approximation for various values of  Figure 2 :Figure 3 : 23Variation of the probability density against the internuclear separation at various at various quantum states Variation of the probability density against the internuclear separation at various quantum state for 0 l  and 1 l  respectively for Hellmann potential. Figure 4 : 4Variation of the probability density against the internuclear separation at various quantum state for 0 l  and 1 l  333respectively for Yukawa potential.  is an upper bound of the vibrational quantum number obtained from the numerical F ) from the partition equation(41). However, this solution is not included in the article because of the lengthy and bulky analytical equations. Figure 5 : 5Variation of partition function with  and q for thermodynamic properties and superstatistics respectively. Figure 6 : 6Variation of Vibrational mean energy with  and q for thermodynamic properties and superstatistics respectively. Figure 7 : 7Variation of Vibrational specific heat capacity with  and q for thermodynamic properties and superstatistics respectively. Figure 8 : 8Variation of Vibrational entropy with  and q for thermodynamic properties and superstatistics respectively. Figure 11 : 11Variation of Vibrational free energy with  and q for thermodynamic properties and superstatistics respectively. The Pekeries -like approximation to the centrifugal term is given as2 cosh ( ) , r vv B V r e r r r           (1) where 1 v and 2 v are the potential depth, B is a real constant parameter,  is the adjustable screening parameter.     2 2 2 11 . 1 1 r r rr e e           into(8) gives the following equation.        2 2 2 2 12 22 22 cosh ( ) 2 ( 1) ( ) 0. 11 1 2 1 r r nl rr rr v v e d R r B e l l E R r dr ee ee                             (9) By defining r se    , and with some simple algebraic simplification, equation (9) can be presented in the form             22 2 1 2 2 2 2 2 2 2 12 1 ( ) 1 ( ) 0 1 1 2 ( 1) s s d R s dR Rs ds s s ds ss s l l                            2 22 22 2 1 2 2 8 cosh 2 1 1 2 1 4 ( 1) 22 2 cosh 21 1 22 8 cosh 2 1 1 4 ( 1) nl vv B n n n l l v ll ll Ev v n l l ,QQ Q Q Q e e Q Q Q Q Q Q e erf        gives the generalized Boltzmann factor equation as    2 2 23 2 2 3 1 2 2 2 2 22 23 2 2 3 1 2 ( ) 1 2 2 QQ Q Q Q Q QQ q B E Q Q Q Q e                                   . (48) Using equation (47), the superstatistics partition function equation is given as     2 2 23 2 2 2 3 1 2 2 2 22 23 2 2 3 1 2 0 12 2 QQ Q Q Q Q s QQ q Z e Q Q Q Q e d                                 . (49) Using Mathematica 10.0 version, the partition obtain from equation (47) is        2 1 2 3 2 2 3 333 23 2 2 2 1 1 2 2 3 2 2 3 2 2 2 1 2 3 2 3 2 3 2 2 2 2 2 2 2 3 1 2 2 3 32 8 3 4 4 1 8 1 2 16 Table 2 . 2Expectation values for2 nl Table 3 . 3Expectation values for1 nl r   . 12 0.2, 0.1 , 0.2 , Table 4 . 4Expectation values fornl Bv  , into equation (1), the potential reduces to screened-hyperbolic inversely quadratic potential(b) Yukawa potential: If 12 0 vv  , then equation (1) reduces to Yukawa potential () r Be vr r    . (53) The corresponding energy eigen equation is       2 22 22 2 2 ( 1) 1 1 2 8 1 nl B ll ll E n l nl                        . 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{'abstract': 'In this work, we apply the parametric Nikiforov-Uvarov method to obtain eigen solutions and total normalized wave function of Schrödinger equation express in terms of Jacobi polynomial using Coulomb plus Screened Exponential Hyperbolic potential (CPSEHP), where we obtained the probability density plots for the proposed potential for various orbital angular quantum number, as well as some special cases (Hellmann and Yukawa potential).The proposed potential is best suitable for smaller values of the screening parameter\uf061 .The resulting energy eigen equation is presented in a close form and extended to study thermal properties and superstatistics express in terms of partition function () Z and other thermodynamic properties such as; vibrational mean energy () U , vibrational specific heat capacity () C ,vibrational entropy () S and vibrational free energy () F . Using the resulting energy equation and with the help of Matlab software, the numerical bound state solutions were obtained for various values of the screening parameter (\uf061 ) as well as different expectation values via Hellmann-Feynman Theorem (HFT). The trend of the partition function and other thermodynamic properties obtained for both thermal properties and superstatistics were in excellent agreement with the existing literatures. Due to the analytical mathematical complexities, the superstatistics and thermal properties were evaluated using Mathematica 10.0 version software. The proposed potential model reduces to Hellmann potential, Yukawa potential, Screened Hyperbolic potential and Coulomb potential as special cases.', 'arxivid': '2110.09896', 'author': ['I B Okon \nDepartment of Physics\nTheoretical Physics Group\nUniversity of Uyo\nNigeria\n', 'C A Onate \nDepartment of Physical Sciences\nLandmark University\nOmu-AranNigeria\n', 'E Omugbe \nDepartment of Physics\nFederal University of Petroleum Resources\nEffurunNigeria\n', 'U S Okorie \nDepartment of Physics\nAkwa Ibom State University\nIkot Akpaden, UyoNigeria\n', 'A D Antia \nDepartment of Physics\nTheoretical Physics Group\nUniversity of Uyo\nNigeria\n', 'M C Onyeaju \nDepartment of Physics\nTheoretical Physics Group\nUniversity of Port Harcourt\nNigeria\n', "Chen Wen-Li \nSchool of Intelligent Science and Information Engineering\nXi'an Peihua University\nXi'anChina\n", 'J P Araujo \nDepartment of Mathematics\nFederal Institute of the Southeast of Minas Gerais\nJuiz de ForaBrazil\n'], 'authoraffiliation': ['Department of Physics\nTheoretical Physics Group\nUniversity of Uyo\nNigeria', 'Department of Physical Sciences\nLandmark University\nOmu-AranNigeria', 'Department of Physics\nFederal University of Petroleum Resources\nEffurunNigeria', 'Department of Physics\nAkwa Ibom State University\nIkot Akpaden, UyoNigeria', 'Department of Physics\nTheoretical Physics Group\nUniversity of Uyo\nNigeria', 'Department of Physics\nTheoretical Physics Group\nUniversity of Port Harcourt\nNigeria', "School of Intelligent Science and Information Engineering\nXi'an Peihua University\nXi'anChina", 'Department of Mathematics\nFederal Institute of the Southeast of Minas Gerais\nJuiz de ForaBrazil'], 'corpusid': 239024475, 'doi': '10.1155/2022/5178247', 'github_urls': [], 'n_tokens_mistral': 28979, 'n_tokens_neox': 22970, 'n_words': 9788, 'pdfsha': '1d5d3024360a080e01fd4eaca676eec101f8ba40', 'pdfurls': ['https://arxiv.org/pdf/2110.09896v1.pdf'], 'title': ['Approximate Solutions, Thermal Properties and Superstatistics Solutions to Schrödinger Equation', 'Approximate Solutions, Thermal Properties and Superstatistics Solutions to Schrödinger Equation'], 'venue': []}

arxiv

Canonical Logic Programs are Succinctly Incomparable with Propositional Formulas * 24 Jan 2015 January 27, 2015 Yuping Shen Institute of Logic and Cognition Department of Philosophy Sun Yat-sen University 510275GuangzhouP.R. China Xishun Zhao Institute of Logic and Cognition Department of Philosophy Sun Yat-sen University 510275GuangzhouP.R. China Canonical Logic Programs are Succinctly Incomparable with Propositional Formulas * 24 Jan 2015 January 27, 2015 Canonical (logic) programs (CP) refer to normal logic programs augmented with connective not not. In this paper we address the question of whether CP are succinctly incomparable with propositional formulas (PF). Our main result shows that the PARITY problem, which can be polynomially represented in PF but only has exponential representations in CP. In other words, PARITY separates PF from CP. Simply speaking, this means that exponential size blowup is generally inevitable when translating a set of formulas in PF into an equivalent program in CP (without introducing new variables). Furthermore, since it has been shown by Lifschitz and Razborov that there is also a problem that separates CP from PF (assuming P NC 1 /poly), it follows that CP and PF are indeed succinctly incomparable. From the view of the theory of computation, the above result may also be considered as the separation of two models of computation, i.e., we identify a language in NC 1 /poly which is not in the set of languages computable by polynomial size CP programs. * Extended version of a paper with the same name in KR2014. † Corresponding Author. Introduction The study of logic programs under answer set semantics, i.e., answer set programming (ASP) [15,26,5], has been an active area in artificial intelligence since the past decades. As a competing approach to SAT [4], ASP has been successfully applied in many fields like Planning, Commonsense Reasoning, Scheduling, etc. The relationship between logic programs and propositional formulas (PF) gains a lot of attention in the literature. A well-known theorem shown by Lin & Zhao [29] gives a method for translating a normal (logic) program (LP) to a (logically) equivalent set of formulas in PF, without introducing additional variables. However, it has been observed that the translation may result in an exponential number of socalled loop formulas in the worst case. In 2006, Lifschitz and Razborov proved that such exponential blowup is generally inevitable, more precisely, they showed that (a variant of) the P-complete problem PATH has polynomial size representations in LP, however, it cannot be polynomially represented in PF (assuming P NC 1 /poly) [28]. In other words, we say PATH separates LP from PF. As noted in [28], PF can be considered as a special case of (nondisjunctive) nested programs (NLP) [25], which is a general form of programs that subsumes LP and some other kinds of programs. Therefore, NLP is stronger than PF in terms of the succinctness criterion (or the "comparative linguistics" approach) proposed in [17]: That is, we consider formalism A to be stronger than formalism B if and only if any knowledge base (KB) in B has an equivalent KB in A that is only polynomially longer, while there is a KB in A that can be translated to B only with an exponential blowup. So the following footnote in [26] seems convincing at first glance: ...ASP appears to be stronger than SAT in the sense of the "comparative linguistics" approach to knowledge representation... However, since ASP involves many kinds of programs, the above statement probably needs further clarification. Particularly, the so-called (nondisjunctive) canonical programs (CP) 1 [22,25,24], is a "minimal" form of ASP that is equally expressive as PF, but looks more likely not succinctly stronger. So a question naturally arises: Does there exist a problem that separates CP from PF? If there is such a problem, then CP and PF are succinctly incomparable (assuming P NC 1 /poly). In this paper we address the question and give a positive answer. Our main result shows that the problem PARITY separates PF from CP. Simply speaking, this means an exponential size blowup is generally inevitable when translating a set of formulas in PF into an equivalent program in CP (without introducing new variables). The PARITY problem asks whether a binary string contains an odd number of 1's, and it is well-known that (i) PARITY∈ NC 1 /poly, i.e., it has polynomial representations in PF 2 [3,21], (ii) PARITY / ∈ AC 0 , i.e., it cannot be represented by polynomial size boolean circuits with constant depth and unbounded fan-in [14,20]. To show PARITY separates PF from CP, we provide a procedure that simplifies every PARITY program Π into a shorter, loop-free program Π ′ . By Lin-Zhao Theorem (or the (generalized) Fages Theorem [11,10,33]), Π ′ is equivalent to its completion Comp(Π ′ ), the latter is essentially a constant depth, unbounded fan-in circuit whose size is polynomially bounded by |Π ′ |. According to PARITY / ∈ AC 0 , these circuits must be of exponential size, consequently, there are no polynomial size CP programs for PARITY. From the view of the theory of computation, the above result may also be considered as the separation of two models of computation [31], i.e., we identify a language in NC 1 /poly which is not in the set of languages computable by polynomial size CP programs. Based on the observation, we point out more separation results on some classes of logic programs, e.g., PARITY separates logic programs with cardinality constraints and choice rules (CC) [32] from CP; assuming P NC 1 /poly, CP and definite causal theories (DT) [30,16] are succinctly incomparable; two-valued programs (TV) [27] are strictly more succinct than CP and DT, etc. The rest of the paper is organized as follows: Section 2 gives preliminaries to the semantics of canonical programs, the concepts of succinctness and the PARITY problem. In Section 3 we briefly review the notation of boolean circuit, the completion semantics and the Lin-Zhao theorem. Section 4 illustrates how to simply an arbitrary PARITY program to be loop-free and presents the main theorem. In Section 5 we discuss the importance of succinctness research and point out more results on a family of logic program classes. Conclusions are drawn in the last Section. Background Canonical Programs The following notations are adopted from [25,22]. A rule element e is defined as e := ⊤ | ⊥ | x | not x | not not x in which ⊤, ⊥ are 0-ary connectives, x is a (boolean) variable (or an atom) and not is a unary connective 3 . A (nondisjunctive canonical) rule is an expression of the form H ← B(1) where the head H is either a variable or the connective ⊥, and the body B is a finite set of rule elements. A canonical program (CP) Π is a finite set of rules, Π is normal if it contains no connectives not not. A normal program Π is basic if it contains no connectives not. The following is a canonical program: x 1 ← not not x 1 , x 2 ← not not x 2 , x 3 ← not x 1 , not x 2 , x 3 ← x 1 , x 2 .(2) The satisfaction relation |= between a set of variables I and a rule element is defined as follows: • I |= ⊤ and I ⊥, • I |= x iff I |= not not x iff x ∈ I, • I |= not x iff x / ∈ I. Say I satisfies a set of rule elements B if I satisfies each rule elements in B. We say I is closed under a program Π, if I is closed under every rule in Π, i.e., for each rule H ← B ∈ Π, I |= H whenever I |= B. Let Π be a basic program, Cn(Π) denotes the minimal set (in terms of inclusion) closed under Π, we say I is an answer set of Π if I = Cn(Π). Note that a basic program has exactly one answer set. The reduct Π I of a canonical program Π w.r.t. I is a set of rules obtained from Π via: (i) Replacing each not not x with ⊤ if I |= x, and with ⊥ otherwise; (ii) Replacing each not x with ⊤ if I x, and with ⊥ otherwise. Observe that Π I must be a basic program. We say I is an answer set of Π if I = Cn(Π I ), i.e., I is an answer set of Π I . The following single rule canonical program Π: x ← not not x(3) has two answer sets {x} and ∅. To see this, check that Π {x} is {x ← ⊤}, whose only answer set is {x}. Similarly, Π ∅ is {x ← ⊥}, whose only answer set is ∅. For convenience, ⊤ in the body is often omitted. For a set of rule elements B, define var(B) = {e ∈ B : e is a variable}. E.g., var({x 1 , not x 2 , not not x 3 }) = {x 1 }. Let Π be a program, by var(Π) we denote the set of all variables involved in Π and by Ans(Π) we denote the set of all answer sets of Π. E.g., let Π be program (2), then var(Π) = {x 1 , x 2 , x 3 } and Ans(Π) = {{x 1 , x 2 , x 3 }, {x 1 }, {x 2 }, {x 3 }}. As a convention, by Π n we refer to a program with n variables {x 1 , . . . , x n }, i.e., var(Π n ) = {x 1 , . . . , x n }. The size |Π n | of a program Π n , is the number of rules in it. Problem Representation and Succinctness A string is a finite sequence of bits from {0, 1}. A string w of length n (i.e., w ∈ {0, 1} n ) can be written as w 1 w 2 . . . w n , in which each bit w i ∈ {0, 1}. Note that a string w ∈ {0, 1} n defines a subset of variables {x 1 , . . . , x n }, e.g., 1010 stands for {x 1 , x 3 }. So a set of variables and a string is regarded as the same. A problem (or language ) L is a set of strings. Definition 2.1 (Problem Representation). A problem L can be represented in a class of programs C (i.e., L ∈ C), if there exists a sequence of programs {Π n } (n = 1, 2, . . .) in C that computes L, i.e., for every string w ∈ {0, 1} n , w ∈ L ⇔ w ∈ Ans(Π n ). Moreover, say L has polynomial representations in C (i.e., L ∈ Poly-C), if L ∈ C and |Π n | is bounded by a polynomial p(n). The following concept is adopted from [17,13]. Definition 2.2 (Succinctness). Let C, C ′ be two classes of programs and for every problem L, L ∈ C ⇔ L ∈ C ′ . Say C is at least as succinct as C ′ (i.e., C ′ C), if for every problem L, L ∈ Poly-C ′ ⇒ L ∈ Poly-C. If L ∈ Poly-C but L ∈ Poly-C ′ (i.e., C C ′ ), then L separates C from C ′ . If C ′ C and C C ′ , then C is strictly more succinct than C ′ (i.e., C ′ ≺ C). Moreover, C, C ′ are succinctly incomparable if there is a problem L separates C from C ′ , and vice versa ( i.e.,C C ′ and C ′ C). Please note that the above notions also apply to formalisms like PF or boolean circuits, etc. The PARITY Problem The PARITY problem is defined as: PARITY = {Binary strings with an odd number of 1's}. We may simply call a string in PARITY an odd string, and PARITY n denotes the set of odd strings of length n. Observe that PARITY n contains 2 n−1 strings. It is not hard to see that PARITY n for n = 1, 2 can be computed by normal programs Π 1 = {x 1 ←} and Π 2 = {x 1 ← not x 2 , x 2 ← not x 1 } respectively. Since Ans(Π 1 ) = {1} (i.e., {x 1 }), and Ans(Π 2 ) = {10, 01} (i.e., {x 1 }, {x 2 }). However, as stated below, PARITY n for n ≥ 3 have no representations in normal programs. Proof. Suppose there is a normal program Π n that computes PARITY n for a fixed n ≥ 3. Then {x 1 } and {x 1 , x 2 , x 3 }, which are two odd strings, belong to Ans(Π n ). However, this is impossible since it contradicts the anti-chain property of Π n [28]: if strings I, I ′ ∈ Ans(Π n ) and I ⊆ I ′ then I = I ′ . On the other hand, the anti-chain property is suppressed in CP. E.g., the answer set 111 of program (2) is a superset of the other three answer sets 100, 010, 001. Clearly, program (2) represents PARITY 3 , moreover, it suggests a "pattern" for representing PARITY n : The first part of the program (e.g., the first two rules in (2)) generates all possible strings of n − 1 bits, the second part identifies the last bit to produce an odd string. Therefore, it is straightforward to give a sequence of canonical programs {Π n } for PARITY n . The following is a PARITY 4 program generated from the pattern: x 1 ← not not x 1 , x 2 ← not not x 2 , x 3 ← not not x 3 , x 4 ← x 1 , x 2 , not x 3 , x 4 ← x 1 , x 3 , not x 2 , x 4 ← x 2 , x 3 , not x 1 , x 4 ← not x 1 , not x 2 , not x 3 .(4) Please note that the number of rules involved in the second part of the pattern grows exponentially, since the number of odd strings with the last bit 1 grows exponentially. Theorem 2.2 (PARITY∈CP). PARITY can be represented by exponential size canonical programs. By PF we denote propositional formulas built on classical connectives {∧, ∨, ¬} with boolean variables. Related concepts like satisfaction, model etc., are defined as usual. By M (φ) we denoted the set of models of φ. The size |φ| of a formula φ is the number of connectives occur in it. PARITY n for n = 1, 2 can be represented by formulas x 1 and (x 1 ∧ ¬x 2 ) ∨ (¬x 2 ∧ x 1 ). Furthermore, it is a textbook result that PARITY n for n ≥ 3 has polynomial size formulas in PF, i.e., PARITY∈ NC 1 /poly (or Poly-PF) [3,21]. Boolean Circuits, Completion and PARITY n Programs for n ≤ 2 Boolean Circuits A (boolean) circuit is a directed, cycle-free graph where each node is either a gate marked with one of {∧, ∨, ¬} or a boolean variable. The in-degree (resp. out-degree) of a node is called its fan-in (resp. fan-out). A node marked with a variable always has fan-in 0 and is called an input. The output of the circuit is one gate designated with fan-out 0. The value of a circuit C n under inputs x 1 , . . . , x n , denoted by C n (x 1 , . . . , x n ), is the value of the output obtained from an iterative calculation through the inputs and the intermediate gates in the usual way. The size |C n | of a circuit C n is the number of gates occur in it. The depth of a circuit is the length of the longest path from an input to the output. We say a circuit computes (or represents) a problem L ⊆ {0, 1} n , if w ∈ L ⇔ C n (w) = 1. E.g., a circuit C 2 that computes PARITY 2 is shown in Fig. 1. If L consists of strings of arbitrary lengths, then we introduce a sequence of circuits {C n }(n = 1, 2, . . .) to represent L, as indicated in Definition 2.1. A circuit is said with bounded fan-in if each gate has at most fain-in 2. If we do not have such restriction then the circuit is with unbounded fan-in. The class AC 0 exactly contains all problems that can be computed by a sequence of circuits {C n } in which the circuits C n have constant depth and polynomial size p(n). E.g., a sequence of polynomial size CNFs {ψ n } computes an AC 0 language, in which a CNF is a conjunction of clause of the form (L 1 ∨ . . . ∨ L m ), where each L i is either a variable x or a negated variable ¬x. Observe that CNF has constant depth 2 (¬ is usually not counted in the depth), and each clause can be regarded as an unbounded fan-in gate ∨ with m inputs. Note that {ψ n } cannot represent PARITY since PARITY / ∈ AC 0 . For more details about circuits, please see [3]. Completion and Related Theorems The completion Comp(Π) [6, 10] of a canonical program Π, consists of a set (or conjunction) of formulas 4 : • x ≡B 1 ∨B 2 ∨ · · · ∨B m , where x ← B 1 , . . . , x ← B m are all rules in Π with head x, and eachB i is the conjunction of rule elements in B i with connective not replaced by ¬, • x ≡ ⊥, if x is not a head of any rule in Π, • ¬B, if a rule ⊥ ← B is in Π. Proposition 3.1. Let Π be an arbitrary canonical program. Then Comp(Π) is a constant depth, unbounded fan-in circuit whose size is polynomially bounded by |Π|. Proof. All propositional formulas are circuits of fan-out 1, so Comp(Π) is definitely a circuit. Clearly, its size is polynomially bounded by |Π|, and its depth is a constant for arbitrary program Π. Moreover, there are no restrictions on the number of rule elements in a body or the number of rules in Π, therefore the corresponding gates in Comp(Π) are with unbounded fan-in. It is well-known that every answer set of a canonical program Π is a model of Comp(Π), but the inverse is generally not hold. E.g., the completion of the program x ← x has two models {x} and ∅, while it has a unique answer set ∅. It turns out that x ← x gives rise to a so-called loop, which leads to an inappropriate model. It is shown in [29,23] that the so-called loop formulas LF (Π) nicely eliminate inappropriate models of Comp(Π), s.t. the models of the union (or conjunction) of LF (Π) and Comp(Π) are coincided with Ans(Π). The (positive) dependency graph [2] of a canonical program Π is a pair (N, E) in which the set of nodes N = var(Π), and E contains a directed edge (x, x ′ ) iff there is a rule H ← B in Π s.t. H = x and x ′ ∈ B. Note that rule elements of the form not x ′ or not not x ′ in B do not contribute to the edges. A non-empty set of variables U ⊆ var(Π) is called a loop of Π, if i) U is a singleton {x} and (x, x) ∈ E, or ii) U is not a singleton and the restriction of the graph on U is strongly connected. Let U be a loop of Π, define R − (U, Π) = {H ← B ∈ Π : H ∈ U, ¬∃ variable x ∈ B s.t. x ∈ U }. Let {B 1 , . . . , B m } be all the bodies of the rules in R − (U, Π), then the loop formula LF (U, Π) is the following: ¬[B 1 ∨ . . . ∨B m ] ⊃ x∈U ¬x.(5) LF (Π) denotes the conjunction of all loop formulas of Π. Theorem 3.1 (Lin-Zhao Theorem[29, 23]). Let Π be a canonical program. Then Π is equivalent to Comp(Π) ∪ LF (Π), i.e., Ans(Π) = M (Comp(Π) ∪ LF (Π)). By Theorem 3.1 (or the (generalized) Fages theorem [11,10,33]), if Π has no loops, then LF (Π) is a tautology ⊤ and Π is equivalent to Comp(Π) (i.e., completion-equivalent). PARITY n Programs for n ≤ 2 Proposition 3.2. Let Π 1 be a PARITY 1 canonical program. Then Π 1 is equivalent to Comp(Π 1 ). Proof. By Theorem 3.1, the unique answer set {x 1 } of Π 1 is a model of Comp(Π 1 )∪ LF (Π 1 ), which also is a model of LF (Π 1 ). There are two cases about the loops in Π 1 : (i) Π 1 has no loops. LF (Π 1 ) is simply ⊤; (ii) Π 1 has a singleton loop {x 1 }. Recall that LF (Π 1 ) is a formula of the form ¬[B 1 ∨ . . . ∨B m ] ⊃ ¬x 1 , in which B 1 . . . B m are all the bodies of rules in R − ({x 1 }, Π 1 ). In both cases, ∅ is a model of LF (Π 1 ), so LF (Π 1 ) is a tautology. Therefore, Π 1 is equivalent to Comp(Π 1 ). Observe that Proposition 3.2 does not hold for PARITY 2 programs. Consider the following PARITY 2 program: x 1 ← not x 2 , x 2 ← not x 1 , x 1 ← x 1 , x 2 ← x 2 .(6) Clearly, {x 1 , x 2 } (i.e., 11) is not an answer set of (6), but a model of its completion {x 1 ≡ x 1 ∨ ¬x 2 , x 2 ≡ x 2 ∨ ¬x 1 }. Note that the rules {x 1 ← x 1 , x 2 ← x 2 } contribute to so-called singleton loops. We may check that without the above two rules, program (6) is a completionequivalent PARITY 2 program. In fact, such "singleton loop" rules can be always safely removed, as stated in Proposition 3.3. Let Π be a basic program and I be a set of variables, define the Knaster-Tarski operator [2] as T Π (I) = {H : H ← B ∈ Π and I |= B}. The operator T is monotone w.r.t. I therefore has a least fixed point T ∞ Π (∅), which can be computed by : (i) T 0 Π (∅) = ∅; (ii) T i+1 Π (∅) = T Π (T i Π (∅)) and (iii) T ∞ Π (∅) = i≥0 (T i Π (∅)). Moreover, T is also monotone w.r.t. Π for a given I, i.e., T Π (I) ⊆ T Π ′ (I) if Π ⊆ Π ′ . It is pointed out in [15,33] that I ∈ Ans(Π) iff I = T ∞ Π I (∅) for a canonical program Π. Proposition 3.3. Let Π be a canonical program. Then removing each rule x ← B ∈ Π with x ∈ var(B) results in a program Π ′ s.t. Ans(Π) = Ans(Π ′ ). Proof. It is sufficient to show T ∞ Π I (∅) = T ∞ Π ′I (∅) for any set I of variables. Suppose H ∈ T ∞ Π I (∅) for some I, then ∃i > 0, H ∈ T i Π I (∅) and H / ∈ T i−1 Π I (∅). It is not hard to see that H must be obtained from a rule H ← B in Π s.t. H / ∈ var(B), H ← var(B) ∈ Π I and T i−1 Π I (∅) |= var(B). Note that H ← B ∈ Π ′ and H ← var(B) is in Π ′I as well. Now we show H ∈ T ∞ Π ′I (∅). Suppose H ∈ T 1 Π I (∅). So a rule H ← is in Π I and Π ′I , clearly H ∈ T ∞ Π ′I (∅). Let k > 1 and assume for all i < k, T i Π I (∅) ⊆ T ∞ Π ′I (∅). Suppose H ′ ∈ T k Π I (∅), then ∃H ′ ← var(B ′ ) ∈ Π I s.t. T k−1 Π I (∅) |= var(B ′ ). Obviously H ′ ∈ T ∞ Π ′I (∅) since H ′ ← var(B ′ ) ∈ Π ′I and T ∞ Π ′I (∅) |= var(B ′ ) by induction hypothesis. Therefore T ∞ Π I (∅) ⊆ T ∞ Π ′I (∅). Note that Π ′I ⊆ Π I since Π ′ ⊆ Π. It follows that T ∞ Π ′I (∅) ⊆ T ∞ Π I (∅) due to the monotonicity of operator T . Hence T ∞ Π I (∅) = T ∞ Π ′I (∅) . It turns out that we have a more general observation: deleting all rules with variables in the body (thus removing all loops) does not affect the answer sets of a PARITY 2 program! Proposition 3.4. Let Π 2 be a PARITY 2 canonical program. Then there is a PARITY 2 program Π ′ 2 which is equivalent to Comp(Π ′ 2 ) and |Π ′ 2 | ≤ |Π 2 |. Proof. W.l.o.g., assume Π 2 has no singleton loops. Let Π ′ 2 = {H ← B ∈ Π 2 : var(B) = ∅}, clearly Π ′ 2 ⊆ Π 2 and thus |Π ′ 2 | ≤ |Π 2 |. To see Π ′ 2 is also a PARITY 2 program, it is sufficient to show for any I ∈ Ans(Π 2 ), Cn(Π I 2 ) = Cn(Π ′I 2 ). Suppose H ∈ I, i.e., H ∈ Cn(Π I 2 ). We claim that H must be obtained from a rule H ← B in Π 2 s.t. (i) I |= B, and (ii) var(B) = ∅. Clearly (i) holds. To see (ii), note that Π 2 has exactly two answer sets {x 1 } and {x 2 }. W.l.o.g., let I = {x 1 } thus H = x 1 . Since Π 2 has no singleton loops, x 1 / ∈ var(B), and x 2 / ∈ var(B) since I |= B. Hence var(B) = ∅. Now it is easy to see H ← B ∈ Π ′ 2 and H ←∈ Π ′I 2 since I |= B and var(B) = ∅. Thus H ∈ Cn(Π ′I 2 ). Therefore Cn(Π I 2 ) ⊆ Cn(Π ′I 2 ). Since Π ′ 2 ⊆ Π 2 , we have Cn(Π ′I 2 ) ⊆ Cn(Π I 2 ) due to the monotonicity of operator Cn(·). Consequently, Cn(Π I 2 ) = Cn(Π ′I 2 ). Observe that Π ′ 2 has no loops, so Π ′ 2 is equivalent to Comp(Π ′ 2 ) . Consider the following PARITY 2 program (7), which has a non-singleton loop {x 1 , x 2 } but not completion-equivalent. One may see that removing the two rules in the second line makes it completion-equivalent, without affecting its answer sets. x 1 ← not x 2 , x 1 ← x 2 , not not x 1 , x 2 ← not x 1 , x 2 ← x 1 , not not x 2 .(7) In the following, we shall introduce a general approach to simply an arbitrary PARITY program to be completion-equivalent. General Simplification of PARITY n Programs Let B be a set of rule elements built on associated variables V = {x 1 , . . . , x n }. We say B is consistent if there is a set of variables I s.t. I |= B. Define S(B) to be the set {I ⊆ V : I |= B}. E.g., let V = {x 1 , x 2 , x 3 , x 4 } and B = {x 2 , not x 3 , not not x 4 }, then B is consistent and S(B) = {{x 1 , x 2 , x 4 }, {x 2 , x 4 }} = {1101, 0101}. Clearly, if B is not consistent then S(B) = ∅. Note that if a rule has an inconsistent body, then it is redundant and can be safely removed. We say B covers a variable x ∈ V iff x ∈ B or not x ∈ B or not not x ∈ B. If B covers every variable in V then B fully covers V . E.g., B = {x 1 , not x 2 , not not x 3 } fully covers V = {x 1 , x 2 , x 3 }. Obviously, B is consistent and fully covers V iff S(B) contains a unique string. In the next section, we stipulate that the set of associated variables is var(Π n ) whenever Π n is the program under discussion, we also assume that a PARITY program has no singleton-loops and contains no inconsistent bodies. Simplifying Full Coverage Rules A rule H ← B ∈ Π n is a full coverage rule if B fully covers var(Π n ). Proof. We show for any set I of variables, I = Cn(Π I n ) iff I = Cn(Π ′I n ). Observe that Π ′ n ⊆ Π n , then Cn(Π ′I n ) ⊆ Cn(Π I n ) for any I. So it is sufficient to show Cn(Π I n ) ⊆ Cn(Π ′I n ). Assume Cn(Π I n ) Cn(Π ′I n ) for some I. It must be the case that x ∈ Cn(Π I n ) and x / ∈ Cn(Π ′I n ) since Π ′ n ∪ {x ← B} = Π n . Moreover, we have x ← var(B) ∈ Π I n and Cn(Π I n ) |= var(B). The former implies that I |= B \ var(B). Since not not x ∈ B \ var(B), we have I |= not not x (i.e., x ∈ I). Now suppose I = Cn(Π I n ), then I is an odd string. However, recall that I |= B \ var(B) and I = Cn(Π I n ) |= var(B). Hence we have I |= B, i.e., I ∈ S(B). This contradicts the fact that I ∈ S(B) is an even string. So Cn(Π I n ) ⊆ Cn(Π ′I n ). Suppose I = Cn(Π ′I n ). As mentioned above, Cn(Π I n ) Cn(Π ′I n ) implies that x / ∈ Cn(Π ′I n ) and x ∈ I. However, recall that I = Cn(Π ′I n ), hence x ∈ Cn(Π ′I n ), a contradiction. So Cn(Π I n ) ⊆ Cn(Π ′I n ). Note that Lemma 4.1 also justifies our simplification for (7). Proof. We show that I = Cn(Π I n ) iff I = Cn(Π ′I n ) for any set I of variables. Suppose I = Cn(Π I n ), we shall prove Cn(Π I n ) = Cn(Π ′I n ). Consider the following cases: Suppose I = Cn(Π ′I n ), we shall prove Cn(Π I n ) = Cn(Π ′I n ). Consider the following cases: • I |= B \ var(B). Since B ′ = B \ {not not x}, clearly, B ′ \ var(B ′ ) ⊆ B \ var(B), hence I |= B ′ \ var(B ′ ). It follows that x ← var(B) ∈ Π I n and x ← var(B ′ ) ∈ Π ′I n . Furthermore, note that Π n \ {x ← B} = Π ′ n \ {x ← B ′ }• x ← var(B ′ ) / ∈ Π ′I n . Clearly, I B ′ \ var(B ′ ). So I B \ var(B) since B ′ \ var(B ′ ) ⊆ B \ var(B) . Therefore x ← var(B) / ∈ Π I n and we have Π ′I n = Π I n . Hence Cn(Π ′I n ) = Cn(Π I n ). • x ← var(B ′ ) ∈ Π ′I n . There are two subcases: -x ← var(B) ∈ Π I n . Similarly, Π ′I n = Π I n and then Cn(Π ′I n ) = Cn(Π I n ) . -x ← var(B) / ∈ Π I n . Clearly, I not not x, i.e., x / ∈ I. Furthermore, recall that Π ′I n = Π I n ∪ {x ← var(B ′ )}, We shall show Cn(Π ′I n ) = Cn(Π I n ). Obviously Cn(Π I n ) ⊆ Cn(Π ′I n ). Now assume Cn(Π ′I n ) Cn(Π I n ). It must be the case that x ∈ Cn(Π ′I n ) but x / ∈ Cn(Π I n ). However, since I = Cn(Π ′I n ), we have x ∈ I, a contradiction. So Cn(Π ′I n ) ⊆ Cn(Π I n ), therefore Cn(Π ′I n ) = Cn(Π I n ). Standard PARITY n Programs A PARITY n program Π n is standard if for each rule x ← B ∈ Π n , not not x / ∈ B whenever S(B ∪ {x}) contains a unique string. E.g., the PARITY program (2) is standard, while (7) is not. Note that if Π n is standard, then for any rule x ← B ∈ Π n , B does not cover x, i.e., x / ∈ B, not x / ∈ B and not not x / ∈ B, since Π n has no singleton loops and S(B ∪ {x}) is consistent. The proof idea of Proposition 4.2 is that every standard PARITY n program Π n can be equivalently rewritten to a loop-free program Π ′ n by replacing each x ∈ var(B) with not not x for every rule body B in Π n . By the Lin-Zhao Theorem or the (generalized) Fages Theorem, Π ′ n is equivalent to its completion Comp(Π ′ n ). And then the proposition follows from the fact that Comp(Π ′ n ) = Comp(Π n ), since not is treated as classical negation ¬ in the completion. The detailed proof is presented in subsection 4.3. Proof of Proposition 4.2 For technical reasons, we divide the rewriting procedure into two steps, in the first step a standard PARITY program is converted to so-called almost pure program and in the second step the program is converted to a pure one, i.e., a PARITY program that does not have any loops. Before doing so we show some lemmas. Proof. Note that for any rule x ← B in a PARITY 1 program, B ∪ {x} must fully cover var(Π 1 ) since Π 1 involves only one variable. So in the following we consider n ≥ 2. (i) Equivalently, we show that if B is consistent and not not x / ∈ B, then B ∪ {x} fully covers var(Π n ). Assume B ∪ {x} does not fully cover var(Π n ). It follows that B covers 0 ≤ i < n variables in var(Π n ) (i.e., B does not fully cover var(Π n )). Consider the following cases: • not x ∈ B. Note that B is consistent and n ≥ 2. It is not hard to see S(B) has exactly 2 n−i−1 ≥ 1 odd strings. It means there is at least one odd string I, I |= B and I x. Therefore I is not close under x ← B. However, since Π n is a PARITY n program, every odd string must be closed under x ← B. A contradiction. • not x / ∈ B. B does not cover x, since not not x / ∈ B and x / ∈ B for Π n has no singleton loops. Recall that B is consistent and n ≥ 2, thus S(B) has exactly 2 n−i−1 odd strings. Obviously, half of these strings do not satisfy x. To be more precise, there are 2 n−i−2 odd strings I, I |= B and I x. We have 2 n−i−2 ≥ 1 since i is at most n − 2. In other words, there is at least one odd string I which is not close under x ← B. Again a contradiction. Consequently, B ∪ {x} must fully cover var(Π n ). (ii) There are two cases about H: • H is ⊥. Assume B does not fully cover var(Π n ), i.e., B covers i variables in var(Π n ) with 0 ≤ i < n. Since B is consistent, it is easy to see S(B) has exactly 2 n−i−1 ≥ 1 odd strings. So there exists at least one odd string I is not closed under ⊥ ← B. A contradiction. • H is a variable x ∈ var(Π n ). Since B ∪{x} is inconsistent, we have not x ∈ B. It is not hard to see in this case x ← B can be rewritten as ⊥ ← B. By an argument similar to the above, B must fully cover var(Π n ). Almost Pure PARITY n Programs Let Π n be a standard PARITY n program in CP, by F − (Π n ) we denote the set of rules H ← B ∈ Π n s.t. B ∪ {H} does not fully cover var(Π n ), by F + (Π n ) we denote Π n \ F − (Π n ). If for each rule H ← B ∈ F + (Π n ) we have var(B) = ∅, then Π n is called almost pure. By Lemma 4.4, it is not hard to see that every rule of the form x ← B, not not x is in F − (Π n ), and every rule of the form ⊥ ← B or x ← B, not x is in F + (Π n ). ← B ∈ F + (Π n ) with H ← B ′ . Clearly Π ′ n is almost pure and |Π ′ n | ≤ |Π n |. It remains to prove that Π ′ n is also a PARITY n program, i.e., I = T ∞ Π I n (∅) iff I = T ∞ Π ′I n (∅). (⇒) Suppose I is an answer set of Π n , i.e., I = T ∞ Π I n (∅), we shall show T ∞ Π I n (∅) = T ∞ Π ′I n (∅): • T ∞ Π I n (∅) ⊆ T ∞ Π ′I n (∅). Note that ⊥ / ∈ T ∞ Π I n (∅) since T ∞ Π I n (∅) isΠ ′I n , trivially, x ∈ T ∞ Π ′I n (∅). Therefore, T ∞ Π I n (∅) ⊆ T ∞ Π ′I n (∅). • T ∞ Π ′I n (∅) ⊆ T ∞ Π I n (∅). We first show ⊥ / ∈ T ∞ Π ′I n (∅). Assume ⊥ ∈ T ∞ Π ′I n (∅), then ∃⊥ ← B 1 ∈ Π ′ n s.t. I |= B 1 \ var(B 1 ) . Consider its source ⊥ ← B in Π n . Recall that Π n has no singleton loops and B is consistent since Π n is standard. Furthermore, B ∪ {⊥} is inconsistent, then ⊥ ← B ∈ F + (Π n ) by Lemma 4.4 (ii). So var(B 1 ) = ∅, I |= B 1 and thus I |= B. The latter means that I is not closed under ⊥ ← B ∈ Π n , which contradicts the fact that I is an answer set of Π n . So ⊥ / ∈ T ∞ Π ′I n (∅). Now suppose x ∈ T 1 Π ′I n (∅), then ∃x ← B 1 ∈ Π ′ n s.t. x ←∈ Π ′I n and I |= B 1 . Consider the source of x ← B 1 : (i) x ← B 1 ∈ Π n , var(B 1 ) = ∅. Since I |= B 1 , x ←∈ Π I n , we have x ∈ T ∞ Π I n (∅). (ii) x ← B ∈ F + (Π n ), var(B) = ∅ and B 1 = B ′ . Note that I |= B since I |= B 1 . Furthermore, I is closed under x ← B since I is an answer set of Π n . So x ∈ I, i.e, x ∈ T ∞ Π I n (∅). Suppose x ∈ T k Π ′I n (∅) but x / ∈ T k−1 Π ′I n (∅) for some k > 1. It means that ∃x ← B 1 ∈ Π ′ n s.t. var(B 1 ) = ∅, x ← var(B 1 ) ∈ Π ′I n , I |= B 1 \ var(B 1 ) and T k−1 Π ′I n (∅) |= var(B 1 ). Note that var(B 1 ) = ∅ implies that x ← B 1 ∈ F − (Π n ), B 1 ∪ {x} does not fully cover var(Π n ). By Lemma 4.4 (i), we have not not x ∈ B 1 . Recall that I |= B 1 \ var(B 1 ), so I |= not not x, i.e., x ∈ T ∞ Π I n (∅). Therefore, T ∞ Π ′I n (∅) ⊆ T ∞ Π I n (∅). (⇐) Suppose I is an answer set of Π ′ n , i.e., I = T ∞ Π ′I n (∅), we shall show T ∞ Π ′I n (∅) = T ∞ Π I n (∅): • T ∞ Π ′I n (∅) ⊆ T ∞ Π I n (∅). Note that ⊥ / ∈ T ∞ Π ′I n (∅). Suppose x ∈ T 1 Π ′I n (∅), then ∃x ← B 1 ∈ Π ′ n s.t. var(B 1 ) = ∅, x ←∈ Π ′I n and I |= B 1 . Now consider the source of x ← B 1 : Let k > 1 and assume for all i < k, T i (i) x ← B 1 ∈ Π n . So x ←∈ Π I n and clearly x ∈ T ∞ Π I n (∅). (ii) x ← B ∈ F + (Π n ),Π ′I n (∅) ⊆ T ∞ Π I n (∅). Suppose x ∈ T k Π ′I n (∅) but x / ∈ T k−1 Π ′I n (∅). Then ∃x ← B 1 ∈ Π ′ n s.t. var(B 1 ) = ∅, x ← var(B 1 ) ∈ Π ′I n , I |= B 1 \ var(B 1 ) and T k−1 Π ′I n (∅) |= var(B 1 ). Note that var(B 1 ) = ∅ implies x ← B 1 ∈ F − (Π n ). Moreover, x ← var(B 1 ) ∈ Π I n . By inductive hypothesis, T ∞ Π I n (∅) |= var(B 1 ), therefore x ∈ T ∞ Π I n (∅). Consequently, T ∞ Π ′I n (∅) ⊆ T ∞ Π I n (∅). • T ∞ Π I n (∅) ⊆ T ∞ Π ′I n (∅). We first show x ∈ T ∞ Π I n (∅) implies x ∈ T ∞ Π ′I n (∅). Suppose x ∈ T 1 Π I n (∅), then ∃x ← B ∈ Π n , var(B) = ∅ and I |= B. It follows that x ← B ∈ Π ′ n and x ←∈ Π ′I n . Clearly, x ∈ T ∞ Π ′I n (∅) . Let k > 1 and assume x ← B ∈ Π ′ n , in both cases I |= x since I is an answer set of Π ′ n and must be closed under every rule of Π ′ n .Consequently, for all i < k, x ∈ T i Π I n (∅) implies x ∈ T ∞ Π ′I n (∅). Suppose x ∈ T k Π I n (∅) but x / ∈ T k−1 Π I n (∅). Then ∃x ← B ∈ Π n s.t. x ← var(B) ∈ Π I n ,x ∈ T ∞ Π I n (∅) implies x ∈ T ∞ Π ′I n (∅). It remains to show ⊥ / ∈ T ∞ Π I n (∅). Assume ⊥ ∈ T ∞ Π I n (∅), then ∃⊥ ← B in F + (Π n ) s.t. I |= B \ Pure PARITY n Programs Let Π n be an almost pure PARITY n program. If for every rule H ← B ∈ Π n we have var(B) = ∅, then Π n is called pure. Clearly, a pure program has no loops and is hence completion-equivalent. Proposition 4.4. Let Π n be an almost pure PARITY n program. Then there is a pure PARITY n program Π ′ n s.t. |Π ′ n | ≤ |Π n |. Proof. A rule H ← B is non-pure if var(B) = ∅. We show by induction on the number m of non-pure rules in Π n . Base step m = 0, the claim trivially holds. Let m > 0 and assume the claim holds for all almost pure PARITY n programs containing j < m non-pure rules. Suppose Π n is an almost pure PARITY n program with m non-pure rules, and let H ← B be a non-pure rule in Π n . Note that H ← B must be in F − (Π n ) and H is a variable x, since Π n is almost pure. Let B ′ be obtained from B by replacing each variable x ∈ var(B) with not not x. Note that I |= B iff I |= B ′ for any set of variables I. Let Π be Π n \ {H ← B} and let Π ′′ n = Π ∪ {H ← B ′ }. Clearly, Π ′′ n is almost pure and |Π ′′ n | ≤ |Π n |. We shall show that Π ′′ n is also a PARITY n program, i.e., I = T ∞ Π I n (∅) iff I = T ∞ Π ′′I n (∅). (⇒) Suppose I = T ∞ Π I n (∅), we prove T ∞ Π I n (∅) = T ∞ Π ′′I n (∅): • T ∞ Π I n (∅) ⊆ T ∞ Π ′′I n (∅). Note that ⊥ / ∈ T ∞ Π I n (∅) since T ∞ Π I n (∅) is an answer set. Suppose x ∈ T 1 Π I n (∅), then ∃x ← B 1 ∈ Π n s.t. var(B 1 ) = ∅ and I |= B 1 . Clearly, x ← B 1 ∈ Π ′′ n and x ←∈ Π ′′I n , thus x ∈ T ∞ Π ′′I n (∅). Let k > 1 and assume for all i < k, T i Π I n (∅) ⊆ T ∞ Π ′′I n (∅). Suppose x ∈ T k Π I n (∅) but x / ∈ T k−1 Π I n (∅). Then ∃x ← B 1 ∈ Π n s.t. var(B 1 ) = ∅, x ← var(B 1 ) ∈ Π I n and T k−1 Π I n (∅) |= var(B 1 ). Hence if x = H, B 1 = B then x ← B ′ ∈ Π ′′ n , otherwise x ← B 1 ∈ Π ′′ n . The former implies x ←∈ Π ′′I n since var(B ′ ) = ∅, and I |= B ′ due to I |= B 1 , trivially, x ∈ T ∞ Π ′′I n (∅). The latter implies x ← var(B 1 ) ∈ Π ′′I n , clearly, T ∞ Π ′′I n (∅) |= var(B 1 ) by induction hypothesis, and thus x ∈ T ∞ Π ′′I n (∅). Therefore, T ∞ Π I n (∅) ⊆ T ∞ Π ′′I n (∅). • T ∞ Π ′′I n (∅) ⊆ T ∞ Π I n (∅). We first show that ⊥ / ∈ T ∞ Π ′′I n (∅). Assume ⊥ ∈ T ∞ Π ′′I n (∅). So ∃⊥ ← B 1 ∈ Π ′′ n s.t. ⊥ ← var(B 1 ) ∈ Π ′′I n and I |= B 1 \ var(B 1 ). Note that ⊥ ← B 1 must be in F + (Π n ) since Π n is almost pure. Hence var(B 1 ) = ∅, I |= B 1 thus I |= B. The latter means that I is not closed under ⊥ ← B 1 ∈ Π n , which contradicts the fact that I is an answer set of Π n . So ⊥ / ∈ T ∞ Π ′′I n (∅). Suppose x ∈ T 1 Π ′′I n (∅), then ∃x ← B 1 ∈ Π ′′ n s.t. x ←∈ Π ′′I n and I |= B 1 . There are two cases about the source of x ← B 1 : . Note that Π ′′I n = Π I ∪ {x ←}, and x / ∈ var(B) since Π n is almost pure, it has no singleton loops. We prove by induction that (i) x ← B 1 ∈ Π n , var(B 1 ) = ∅. Since I |= B 1 , so x ←∈ Π I n , x ∈ T ∞ Π I n (∅). (ii)x ← B 1 is obtained from H ← B ∈ F − (Π n ), i.e., H = x, var(B) = ∅ and B 1 = B ′ . Note that I |= B since I |= B 1 . Recall that not not x ∈ B, so x ∈ I, i.e, x ∈ T ∞ Π I n (∅). Now suppose x ∈ T k Π ′′I n (∅) but x / ∈ T k−1 Π ′′I n (∅) for some k > 1. It means that ∃x ← B 1 ∈ Π ′′ n s.t. var(B 1 ) = ∅, x ← var(B 1 ) ∈ Π ′′I n , I |= B 1 \ var(B 1 ) and T k−1 Π ′′I n (∅) |= var(B 1 ). Since Π n is almost pure, var(B 1 ) = ∅ implies that x ← B 1 ∈ F − (Π n ). By Lemma 4.4 (i), not not x ∈ B 1 . Recall that I |= B 1 \ var(B 1 ), so I |= not not x, i.e., x ∈ T ∞ Π I n (∅). Therefore, T ∞ Π ′′I n (∅) ⊆ T ∞ Π I n (∅). (⇐) Suppose I = T ∞ Π ′′I n (∅), we show T ∞ Π ′′I n (∅) = T ∞ Π I n (∅). • T ∞ Π ′′I n (∅) ⊆ T ∞ Π I n (∅). Note that ⊥ / ∈ T ∞ Π ′′I n (∅). Suppose x ∈ T 1 Π ′′I n (∅), then ∃x ← B 1 ∈ Π ′′T ∞ Π I (∅) |= var(B). Suppose x ′ ∈ var(B) and x ′ ∈ T 1 Π ′′I n (∅), then ∃x ′ ← B 2 ∈ Π ′′ , var(B 2 ) = ∅ and I |= B 2 . Clearly, x ′ ←∈ Π I , thus x ′ ∈ T ∞ Π I (∅) . Let s > 1 and assume for all t < s, x ′ ∈ var(B) and x ′ ∈ T t Π ′′I n (∅) implies x ′ ∈ T ∞ Π I n (∅). Suppose x ′ ∈ var(B) and x ′ ∈ T s Π ′′I n (∅). Then ∃x ′ ← B 2 ∈ Π ′′ , x ′ ← var(B 2 ) ∈ Π ′′I and T s−1 Π ′′I n (∅) |= var(B 2 ). Recall that x = x ′ since x / ∈ var(B), so x ′ ← var(B 2 ) ∈ Π I . By induction hypothesis, T ∞ Π I n (∅) |= var(B 2 ), x ′ ∈ T ∞ Π I n (∅). Hence T ∞ Π I (∅) |= var(B). Furthermore, Π I n = Π I ∪ {x ← var(B)}, so T ∞ Π I (∅) ⊆ T ∞ Π I n (∅), T ∞ Π I n (∅) |= var(B). Therefore T ∞ Π I n (∅) |= x, i.e., x ∈ T ∞ Π I n (∅). Discussion and Some More Results Interestingly, our main result may at first appear counter-intuitive: the P-complete problem PATH has Poly-CP representations, while this does not hold for an "easy" problem PARITY. Actually, there is no contradiction. As noted in [1,8], a complete problem in a complexity class can be represented in a formalism C, does not imply that all problems in that class can be represented in C. Generally speaking, the research of succinctness [17,7,19,13] gives us a deeper understanding about KR formalisms, for it reveals their (in)abilities of concisely representing different problems under the condition that the encoded models are the same. In terms of the theory of computation, succinctness essentially concerns with the computational power of different formalisms (i.e., models of computation). This is particularly interesting if the formalisms are equally expressive and share the same reasoning complexity. E.g., logic programs with cardinality constraints and choice rules (CC, without classic negation ¬) [32], (simple) definite causal theories (S/DT) [16] and two-valued programs (TV) [27] are as expressive as PF and NP-complete for consistency checking. But they have a non-trivial succinctness picture, see Fig. 2. Besides the theoretical interests, succinctness also tells us something like "which for what is the best" in choosing KR formalisms for a given application. E.g., one should choose ASP instead of SAT if the application involves reasoning about PATH or Transitive Closure 5 , because the former provides compact representations to avoid unnecessary overload. Recall that from the complexity viewpoint, even one extra variable may double the search space for intractable problems. In the following we shall briefly discuss some succinctness results illustrated in Fig. 2, note that all mentioned formalisms have the same expressive power and same reasoning complexity. Logic Programs with Cardinality Constraints (CC) Simply speaking, CC extends normal programs (LP) with so-called cardinality constraints and choice rules [32]. A choice rule {x} ←(8) has two answer sets {x} and ∅, i.e., same as x ← not not x. Moreover, a choice rule {x 1 , . . . , x n } ← produces 2 n answer sets, i.e., all subsets of {x 1 , . . . , x n }. A cardinality constraint is an expression of the form l ≤ B ≤ u(9) in which B is a finite set of rule elements of the form x or not x, and integer l (resp. u) is the lower (resp. upper) bound on B. In this paper we assume the magnitude of l (and u) is polynomially bounded by n. Intuitively, a set of variables I satisfies (9), if the number of satisfied rule elements in B fulfills the related bounds. E.g., {x 1 } satisfies 1 ≤ {x 1 , x 2 , x 3 } ≤ 1 but not 2 ≤ {x 1 , x 2 , x 3 } ≤ 3, while {x 2 , x 3 } satisfies the latter. Informally, we may think of (9) as a special kind of rule element, and the answer set semantics is defined accordingly. The following is a PARITY 3 program in CC: {x 1 , x 2 , x 3 } ←, ⊥ ← 0 ≤ {x 1 , x 2 , x 3 } ≤ 0, ⊥ ← 2 ≤ {x 1 , x 2 , x 3 } ≤ 2,(10) Clearly, the pattern applies to all PARITY n and the program grows linearly. We define the size of a CC program to be the number of cardinality constraints occur in it. Theorem 5.1 (PARITY∈Poly-CC). PARITY has polynomial size programs in CC. An equivalent translation from CC to NLP was presented in [12], however, the translation may involve exponential size blowup, since every cardinality constraint is simply converted to a formula via a brute force enumeration. In fact, such a translation can be reduced to be polynomial by adopting a non-trivial, sophisticated encoding for so-called threshold functions 6 . Therefore, we have: Definite Causal Theories (DT) A variable x or negated variable ¬x is called a literal. A definite (causal) theory D n on signature {x 1 , . . . , x n } is a finite set of (causal) rules of the form H ⇐ G(11) in which H is either a literal or ⊥, and G is a propositional formula. If every G is a conjunction of variables or negated variables, then D n is called simple (SDT) 7 . The reduct D I n of D n w.r.t. a set of variables I, is the set of the heads H of all rules in D n whose bodies G are satisfied by I. Say I is a model of D n if I is the unique model of D I n . The following theory: x ⇐ x, ¬x ⇐ ¬x (12) has two models {x} and ∅, which is equivalent to program x ← not not x or {x} ←. If a definite theory D n is simple, then its size |D n | is defined as the number of rules in it, otherwise |D n | is the number of connectives in it. It is well-known that D n is equivalent to its (literal) completion Comp(D n ), in which Comp(D n ) is similarly defined as for logic programs [30,16]. It means that definite theories are fragments of PF, i.e., DT PF. Therefore, the problems that can be represented by Poly-DT are in NC 1 /poly as well. Moreover, the completion of a simple definite theory is also a constant depth, unbounded fan-in circuit whose size is polynomially bounded. By a proof similar to that of Theorem 4.1, we have the following theorem: Consider the (non-simple) causal theory (13) for PARITY 2 , where the body of the last rule is the negation of a PARITY 2 formula: x 1 ⇐ x 1 , ¬x 1 ⇐ ¬x 1 , x 2 ⇐ x 2 , ¬x 2 ⇐ ¬x 2 , ⊥ ⇐ ¬((x 1 ∧ ¬x 2 ) ∨ (¬x 2 ∧ x 1 )). (13) Recall that PARITY have polynomial formulas in PF, therefore it is not hard to see we can have polynomial DT theory for PARITY by the above pattern. Theorem 5.4 (PARITY∈Poly-DT). PARITY has polynomial size theories in DT. Since PATH is P-complete [28], therefore if PATH has polynomial representations in Poly-DT, then P ⊆ NC 1 /poly, which is believed impossible. Theorem 5.5 (PATH / ∈Poly-DT). Suppose P NC 1 /poly. Then PATH has no polynomial size definite theories. By the fact that PATH has polynomial size CP programs, we have: Corollary 5.1. Suppose P NC 1 /poly. Then CP and DT are succinctly incomparable. It is worth to point out that some difficulties observed in the literature could be nicely explained by the above succinctness results. E.g., DT has been observed hard to concisely encode Transitive Closure (TC) [16,9]. Recall that Poly-DT represents problems in NC 1 /Poly, and TC is a problem in NC 2 /poly [18], a class widely believed strictly contains NC 1 /poly. So unless the two classes coincide, TC has no polynomial size definite theories. Two-Valued Logic Programs (TV) A (two-valued) program [27] Π n on signature {x 1 , . . . , x n } is a finite set of (twovalued) rules of the form: H ← B : G(14) in which B ∪ {H} is a finite set of literals and G is a formula. The reduct Π I n of Π n w.r.t. a set of variables I, is the set of rules H ← B(15) from Π n s.t. I satisfies G. A set of literals J is closed under rule (15) if H ∈ J whenever B ⊆ J. We say I is a model of Π n if I is the unique model of the minimal closure J under every rule of Π I n . The following program Π 2 in TV x ←: x, ¬x ←: ¬x (16) has two models {x} and ∅, which is equivalent to (12). The following observations were pointed out in [27]. A formula φ n can be rewritten in TV (i ∈ {1, . . . , n}) 8 : x i ←: x i , ¬x i ←: ¬x i , ⊥ ←: ¬φ n . A causal rule H ⇐ G can be equivalently rewritten as H ←: G. Moreover, to equivalently rewrite a CP program Π n , each rule: H ← u 1 , . . . , u j , not y j+1 , . . . , not y m , not not z m+1 , . . . , not not z k can be translated as: H ← u 1 , . . . , u j : ¬y j+1 ∧, . . . ¬y m ∧ z m+1 ∧ . . . z k (19) and add ¬x ←: ¬x for every x ∈ var(Π n ). All together, we have: Theorem 5.6. Two valued programs are strictly more succinct than: (i) propositional formulas and definite theories, if P NC 1 /poly; (ii) canonical programs. Conclusions The main result of the paper is that the PARITY problem separates PF from CP, i.e., PARITY has no polynomial size CP programs, but has polynomial size PF formulas. Together with Lifschitz and Razborov's separation result, i.e., there exists a problem separates CP from PF (assuming P NC 1 /poly), we conclude that the two well-known KR formalisms are succinctly incomparable. In other words, if we consider CP and PF as two different models of computation, the above result just states that they are incomparable in terms of computational power. We also give a non-trivial succinctness picture on a family of logic program classes which posses the same expressive power and same reasoning complexity as PF. In future work, we plan to investigate some missing connections in Fig. 2, e.g., we conjecture that there is a problem separates NLP from CP, SDT and CP are succinctly incomparable. Theorem 2.1 (PARITY / ∈LP). PARITY cannot be represented by normal programs. Figure 1: A Parity 2 Circuit Lemma 4. 1 . 1Let Π n be a PARITY n program. Suppose there is a rule x ← B in Π n s.t.not not x ∈ B and S(B) contains a unique even string. Then removing x ← B from Π n results in a PARITY n program Π ′ n . Lemma 4. 2 . 2Let Π n be a PARITY n program. Suppose there is a rule x ← B in Π n s.t. not not x ∈ B and S(B) contains a unique odd string. Then replacing its body B with B ′ = B \ {not not x} results in a PARITY n program Π ′ n . and var(B) = var(B ′ ), thus Π I n = Π ′I n . So Cn(Π I n ) = Cn(Π ′I n ). • I B \ var(B). Consider the following subcases: -I B ′ \ var(B ′ ). Similarly, we have Π ′I n = Π I n , thus Cn(Π ′I n ) = Cn(Π I n ). -I |= B ′ \ var(B ′ ). Clearly, in this case I not not x. Now suppose I |= var(B ′ ), so we have I |= B ′ . Recall that (i) not not x ∈ B, (ii) x / ∈ B since Π n has no singleton loops, (iii) B ′ = B \ {not not x} and (iv) S(B) contains a unique odd string, say I ′ . It follows that S(B ′ ) = {I ′ , I ′ \ {x}}. Obviously I must be I ′ \ {x} since I ′ |= not not x. However, this is a contradiction since I ′ \ {x} is an even string and I is an odd string since I is an answer set of Π n . So suppose I var(B). Note that in this case Π ′I n = Π I n ∪ {x ← var(B)}, we show Cn(Π I n ) = Cn(Π I n ∪ {x ← var(B)}), i.e., Cn(Π I n ) = Cn(Π ′I n ). Firstly, Cn(Π I n ) ⊆ Cn(Π I n ∪ {x ← var(B)}) due to the monotonicity of operator Cn(·). Assume Cn(Π I n ∪ {x ← var(B)}) Cn(Π I n ), it must be Cn(Π I n ) |= var(B) and x ∈ Cn(Π I n ∪ {x ← var(B)}), x / ∈ Cn(Π I n ). However this is impossible since I = Cn(Π I n ) and I var(B). Therefore Cn(Π I n ∪ {x ← var(B)}) ⊆ Cn(Π I n ). Proposition 4. 1 . 1Let Π n be a PARITY n program. Then there is a standard PARITY n program Π ′ n s.t. |Π ′ n | ≤ |Π n |. Proof. For each rule x ← B ∈ Π n in which not not x ∈ B and S(B) contains a unique string: (i) Delete x ← B from Π n if S(B) contains an even string; (ii) Remove not not x from B if S(B) contains an odd string. By Lemma 4.1 and 4.2, the above procedure results in a standard PARITY n program Π ′ n and |Π ′ n | ≤ |Π n |. Proposition 4. 2 . 2Let Π n be a standard PARITY n program. Then Π n is equivalent to its completion Comp(Π n ). Lemma 4. 3 . 3Let Π n be a standard PARITY n program. For each rule x ← B ∈ Π n , if S(B ∪ {x}) contains a unique string, then the string must be odd.Proof. Since Π n is standard, B does not cover x. So we have S(B) = {I, I \{x}}.Assume I is an even string, then I \ {x} must be an odd string. It follows that I \ {x} is not closed under x ← B, since I \ {x} |= B but I \ {x} x. However, Π n is a PARITY n program, every odd string must be closed under x ← B. A contradiction. Lemma 4. 4 . 4Let Π n be a PARITY n program. (i) If there is a rule x ← B ∈ Π n s.t. B is consistent and B ∪ {x} does not fully cover var(Π n ), then not not x ∈ B. (ii) If there is a rule H ← B ∈ Π n s.t. B is consistent and B ∪ {H} is inconsistent, then B fully covers var(Π n ). Proposition 4. 3 . 3Let Π n be a standard PARITY n program. Then there is an almost pure PARITY n program Π ′ n s.t. |Π ′ n | ≤ |Π n |. Proof. Let B ′ be the set obtained from B by replacing every x ∈ var(B) with not not x. Note that I |= B iff I |= B ′ for any set of variables I. Let Π ′ n be the program obtained from Π n by replacing every rule H ), then ∃x ← B ∈ Π n s.t. var(B) = ∅ and I |= B.Clearly, we have x ← B ∈ Π ′ n . It follows that x ←∈ Π ′I n and then x ∈ T ∞ Π ′I n (∅). Let k > 1 and assume for all i < k, Then ∃x ← B ∈ Π n s.t. x ← var(B) |= var(B). Observe that either x ← B ∈ Π ′ n or x ← B ′ ∈ Π ′ n . The former implies that x ← var(B) ∈ Π ′I n , clearly, T ∞ Π ′I n (∅) |= var(B) by induction hypothesis, and thus x ∈ T ∞ Π ′I n (∅). The latter implies that x ←∈ var(B) = ∅ and B 1 = B. Note that I |= B 1 ∪ {x} since x ∈ I and I |= B 1 , it follows that I |= B ∪ {x}. Clearly, B ∪ {x} is consistent and fully covers var(Π n ). By Lemma 4.3, I is exactly the unique odd string in S(B ∪ {x}).Recall that Π n is a PARITY n program, so I must be an answer set of Π n , i.e., I = T ∞ var(B) = ∅, I |= B \ var(B) and T |= var(B), i.e., I |= var(B). Now I |= B since I |= B \ var(B) and I |= var(B), hence I |= B ′ . Observe that either x ← B ′ ∈ Π ′ n or |= var(B) for some k ≥ 1. Notice that the latter means I |= var(B), |= var(B) by the previous result and I = T ∞ Π ′I n (∅). So I |= B, i.e., I |= B ′ . However, note that ⊥ ← B ′ in Π ′ n and I is not closed under ⊥ ← B ′ . This contradicts the fact that I is an answer set of Π ′ n . Therefore ⊥ / n s.t. x ←∈ Π ′′I n and I |= B 1 . Now consider the source of x ← B 1 : (i) x ← B 1 ∈ Π n and var(B 1 ) = ∅. It follows that x ←∈ Π I n and clearly x ∈ T ∞ ) x ← B 1 is obtained from H ← B ∈ F − (Π n ), i.e., H = x and B 1 = B ′ . So x ← var(B) ∈ Π I n and I |= B. The latter implies T ∞ |= var(B) . 2 . 2Suppose P NC 1 /poly. Then CP and PF are succinctly incomparable. Theorem 5.3 (PARITY /∈Poly-SDT). PARITY has no polynomial size theories in SDT. Theorem 5.2. NLP is at least as succinct as CC.6 E.g., see Chapter 2 of[31]. Extends LP with connective not not. 2 NC 1 /poly (or non-uniform NC 1 ) exactly contains languages computable (i.e., representable) by polynomial size propositional formulas. According to[25], not not not x can be replaced by not x. For convenience, we slightly abuse the connective ≡ here. An NL-complete problem. It is believed that NL NC 1 /Poly. SDT is originally named as Objective Programs in[30]. ⊥ ←: ¬φn is a shorthand of x1 ←: ¬φn, ¬x1 ←: ¬φn. AcknowledgementWe are grateful to the anonymous reviewers for their valuable comments. Thanks to Shiguang Feng, Yan Zhang, Jiankun He, Guangrui Dang and Xiaolong Liang for their helpful discussions. The research was partially supported by NSFC Grant 61272059, MOE Grant 11JJD720020, NSSFC Grant 13&ZD186, 14CZX058 and the Fundamental Research Funds for the Central Universities Grant 1409025.Let k > 1 and assume for all.In both cases I |= x since I is an answer set of Π ′′ n and it must be closed under every rule of Π ′′ n . Consequently,⊥ ←∈ Π I n and I |= B 1 . Notice that var(B 1 ) = ∅ since Π n is almost pure. Furthermore, ⊥ ← B 1 must be in Π ′′ n , therefore I |= ⊥ since I |= B 1 .However, this contradicts the fact that I is an answer set of Π ′′ n . Therefore ⊥ / ∈ T ∞ Π I n (∅), and hence T ∞Consequently, Π ′′ n is an almost pure PARITY n program with m − 1 non-pure rules. By induction hypothesis, there is a pure PARITY n program Π ′ n with |Π ′ n | ≤ |Π ′′ n | ≤ |Π n |.The Main ResultsThe main lemma below easily follows from Proposition 4.1 and 4.2. Proof. 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{'abstract': 'Canonical (logic) programs (CP) refer to normal logic programs augmented with connective not not. In this paper we address the question of whether CP are succinctly incomparable with propositional formulas (PF). Our main result shows that the PARITY problem, which can be polynomially represented in PF but only has exponential representations in CP. In other words, PARITY separates PF from CP. Simply speaking, this means that exponential size blowup is generally inevitable when translating a set of formulas in PF into an equivalent program in CP (without introducing new variables). Furthermore, since it has been shown by Lifschitz and Razborov that there is also a problem that separates CP from PF (assuming P NC 1 /poly), it follows that CP and PF are indeed succinctly incomparable. From the view of the theory of computation, the above result may also be considered as the separation of two models of computation, i.e., we identify a language in NC 1 /poly which is not in the set of languages computable by polynomial size CP programs. * Extended version of a paper with the same name in KR2014. † Corresponding Author.', 'arxivid': '1412.0320', 'author': ['Yuping Shen \nInstitute of Logic and Cognition\nDepartment of Philosophy\nSun Yat-sen University\n510275GuangzhouP.R. China\n', 'Xishun Zhao \nInstitute of Logic and Cognition\nDepartment of Philosophy\nSun Yat-sen University\n510275GuangzhouP.R. China\n'], 'authoraffiliation': ['Institute of Logic and Cognition\nDepartment of Philosophy\nSun Yat-sen University\n510275GuangzhouP.R. China', 'Institute of Logic and Cognition\nDepartment of Philosophy\nSun Yat-sen University\n510275GuangzhouP.R. China'], 'corpusid': 2707306, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 23809, 'n_tokens_neox': 21272, 'n_words': 12832, 'pdfsha': '2a212e1156d8bc54e90ee00dfbb32f4dc60d6232', 'pdfurls': ['https://arxiv.org/pdf/1412.0320v2.pdf'], 'title': ['Canonical Logic Programs are Succinctly Incomparable with Propositional Formulas *', 'Canonical Logic Programs are Succinctly Incomparable with Propositional Formulas *'], 'venue': []}

arxiv

XVoxel-Based Parametric Design Optimization of Feature Models Ming Li State Key Laboratory of CAD&CG Zhejiang University 310027HangzhouChina Chengfeng Lin State Key Laboratory of CAD&CG Zhejiang University 310027HangzhouChina Wei Chen State Key Laboratory of CAD&CG Zhejiang University 310027HangzhouChina Yusheng Liu State Key Laboratory of CAD&CG Zhejiang University 310027HangzhouChina Shuming Gao State Key Laboratory of CAD&CG Zhejiang University 310027HangzhouChina Qiang Zou State Key Laboratory of CAD&CG Zhejiang University 310027HangzhouChina XVoxel-Based Parametric Design Optimization of Feature Models Design optimizationParametric modelingFeature modelsExtended voxels (XVoxels)Semantic voxelsFinite cell method (FCM)CAD/CAE integration Parametric optimization is an important product design technique, especially in the context of the modern parametric feature-based CAD paradigm. Realizing its full potential, however, requires a closed loop between CAD and CAE (i.e., CAD/CAE integration) with automatic design modifications and simulation updates. Conventionally the approach of model conversion is often employed to form the loop, but this way of working is hard to automate and requires manual inputs. As a result, the overall optimization process is too laborious to be acceptable. To address this issue, a new method for parametric optimization is introduced in this paper, based on a unified model representation scheme called eXtended Voxels (XVoxels). This scheme hybridizes feature models and voxel models into a new concept of semantic voxels, where the voxel part is responsible for FEM solving, and the semantic part responsible for high-level information to capture both design and simulation intents. As such, it can establish a direct mapping between design models and analysis models, which in turn enables automatic updates on simulation results for design modifications, and vice versa-effectively a closed loop between CAD and CAE. In addition, robust and efficient geometric algorithms for manipulating XVoxel models and efficient numerical methods (based on the recent finite cell method) for simulating XVoxel models are provided. The presented method has been validated by a series of case studies of increasing complexity to demonstrate its effectiveness. In particular, a computational efficiency improvement of up to 55.8 times the existing FCM method has been seen. Introduction Design optimization has been recognized as one of the dominant industrial practices for product design due to improved product quality, reduced cost, and shorter time to market [1]. The optimization may be done in various ways, and parametric optimization is among the primary [2]. It optimizes engineering meaningful parameters that are embedded in feature-based CAD models with externally defined objective functions [3]. Design optimization of this sort has seen applications in many fields, including automotive, shipbuilding, and aerospace industries. While parametric optimization is very relevant and beneficial in the context of modern feature-based CAD [4], its full realization is not trivial. Its working relies on a closed loop between CAD and CAE with automatic design modifications and simulation updates [5,6]. Forming such a loop is difficult because of the different information contents stored in CAD models and CAE models [1]. Specifically, a CAD model is designated to have an accurate description of the design in order to automate any queries from manufacturing and assembling. It usually consists of feature history, geometric constraints, and parameter definitions (which, altogether, encompass design intent) [4]. A CAE model contains data on the boundary conditions, material distribution, and volumetric meshes that are suitable for conducting the finite element method or the like, which encompass simulation intent [5,7]. To solve the discrepancy between CAD models and CAE models, the approach of model conversion is often employed. As illustrated in Fig. 1, a typical conversion begins with a feature model, then goes through steps of boundary representation (B-rep) generation, model simplification, volumetric mesh generation, and boundary condition specification, cumulatively into a simulation model ready for FEM solving. The solving results will then be used to generate parametric modifications on the feature model for the next optimization iteration. Repeating these procedures will lead to an optimized design. Despite its conceptual simplicity, there are several technical difficulties in the conversion. Typically the model simplification and mesh generation steps are hard to automate and require manual inputs [8][9][10][11], the design optimization and adjustment cannot be directly fed back to CAD modeling operations [12], and the important design intent could be lost after conversion [5,7]. In the context of iterative design optimization, such inefficiency will be much amplified and consequently, the overall process is too laborious to be acceptable. It has been reported that manual intervention accounts for about 80% of the overall design time in the conversion-based process described above [5,13]. In view of the above issues, a unified representation scheme that can completely, compactly, and associatively represent the contents of both CAD and CAE models has been recognized as a much-desired method for parametric design optimization [5]. This paper follows this direction and proposes a new representation scheme called Extended Voxel (XVoxel) to address the problem. It essentially makes use of semantic voxels (as will be detailed in Section 3), where the voxel part is responsible for FEM solving, and the semantic part responsible for high-level information to capture both design and simulation intents. In a nutshell, XVoxel models provide the following advantages: • Design and simulation intents can be preserved in the loop of design, simulation, and optimization, which otherwise are lost in the conversion-based approach and have to be reconstructed. The relevant details can be found in Sections 3.1 and 4.3. • Generation of analysis geometries and volumetric meshes can be done virtually. As such, labor-intensive and nonrobust model simplification and mesh generation can be possibly avoided, and boundary conditions can be wellretained over the course of optimization iterations. The relevant details can be found in Sections 3.3 and 3.4. • Modifications on design parameters and updates on simulation results can be associated automatically and locally, allowing automatic and efficient looping among design, simulation, and optimization. The relevant details can be found in Sections 3.2 and 4.3. • Simulation of the feature models can be done efficiently on a coarse XVoxel model while attaining high accuracy through the combination of the fictitious domain technique and material-aware shape functions. The relevant details can be found in Section 4.1 and 4.2. The following sections begin with a review on existing parametric optimization methods in Section 2. A detailed description on XVoxel models is given in Section 3. The XVoxel-based simulation and design optimization are presented in Sections 4 and 5, respectively. Application examples and comparisons with existing methods are provided in Section 6, followed by conclusions in Section 7. Related work Parametric optimization aims to find the optimal design parameters regarding certain performance metrics. Related approaches include conversion-based optimization, unified model-based optimization, and parameter-driven topology optimization. Conversion-based parametric optimization The conversion-based parametric optimization is the de facto standard in practice, but it may require significant manual effort for complex CAD models or boundary conditions to ensure that all conversion steps can be carried out successively [1,6]. A typical conversion procedure involves model simplification, volumetric mesh generation, boundary condition specification, and design modification. Despite the progress on simplifying geometries using methods like feature suppression [8], direct modeling [14], virtual topology [15,16], etc., current methods either have restricted applicability or have robustness issues, thereby requiring considerable manual intervention. Generation of unstructured meshes, e.g., tetrahedron meshes, is an almost solved problem [17,18]. However, automatically generating structured meshes, which are preferable in applications requiring high computational accuracy and efficiency, still remains an open issue [19]. Boundary conditions are largely specified manually in practice, which thus requires huge human efforts in design optimization that loops even thousands of times between the feature model and its simulation. Clearly, this is unacceptable. A common way to address this issue is via assigning fixed boundary conditions or imposing simple varying loads, e.g., in topology optimization [20]. These approaches however would restrict the range of the problem under study. The issue was addressed in a broad sense by defining simulation intent by incorporating concepts of cellular modeling and equivalencing [5,7]. It shares similar spirits with the present work but does not involve standard voxels for performing the simulation. The involvement of manual intervention clearly decreases design efficiency and makes it hard, if not impossible, to automate design optimization. Note also that in the conversion-based parametric optimization, the underlying FE mesh is varied during each step of the design update, resulting in a varied design space. As a consequence, it usually tends to result in an unstable optimization convergence. Unified model-based parametric optimization Existing unified model-based parametric optimization approaches mainly include isogeometric analysis (IGA), embedded domain, or their combinations. IGA, initialized by Hughes et al. [21], uses a unified geometric representation scheme, i.e., NURBS (non-uniform rational B-spline), for both design and analysis. Basically, it discards the use of explicit meshes but employs the knot vector and spline basis of a NURBS surface to directly generate the elements and shape functions for FEM solving [22]. As the mesh generation step is eliminated (in principle), IGA provides a tighter integration between CAD and CAE for automatic design optimization, and the benefits extend beyond integration to higher simulation accuracy and efficiency [2,[23][24][25]. However, IGA only works well on the surface model having a regular parametric domain. For general shapes composed of trimmed NURBS surfaces or 3D volumetric models, quadrilateral meshing of its boundary, or hexahedral meshing of its volume are inevitable, which are challenging research topics in their own right. The XVoxel method to be presented does not have this issue because there are no B-rep models or meshing processes involved. This advantage manifests itself through situations where the B-rep model given to IGA is complex and introduces robustness issues in model simplification and difficulties in quad/hex meshing. It should, however, be noted that IGA has higher simulation accuracy and can directly take B-rep models as input, while XVoxel is not able to do so. IGA is thus preferred in such situations. Unlike IGA which revolves around the design model (i.e., NURBS), the embedded domain approach such as finite cell method (FCM) [26] focuses on the other side, i.e., meshes. It uses the same regular background mesh (e.g., a grid) to carry out FEM solving regardless of the design model's variations. As such, no mesh generation is needed when the design model is modified during optimization. This is essentially achieved through high-order finite elements and weak enforcement of unfitted essential boundary conditions. FCM was also used together with IGA to utilize both of their advantages [27,28], most of which did not discuss its work on feature models. Recently, Wassermann et al. [29] studied the problem of conducting FCM on CSG model, which mainly studied the point membership classification problem for different primitives while the present study focuses on the overall integration flowchart for parametric optimization of feature models. The embedded domain approach is to be combined with the feature-based approach in this work to enable the embedding of design and simulation semantics within the background mesh (which is otherwise purely geometric), where the background mesh serves as a common data structure, and the embedded semantics provide automatic links between design modifications and simulation updates. Parameter-driven topology optimization Research efforts have been devoted toward parametric optimization of CAD models, which mainly focus on finding the optimal shape parameters in describing a specific CAD part but seldom addressed the issue of integrating design semantics into the optimization process. Chen et al. considered using R-functions for design optimization with topological changes [3,30]. Zhu et al. proposed a direct simulation approach for CAD models undergoing parametric modifications [31] using a model reduction technique called PGD (Proper Generalized Decomposition) [32]. Schulz et al. developed an exploration tool for interactive exploration and optimization of parametric CAD models [33] via precomputations. More recently, Hafner et al. proposed a generic shape optimization method, called X-CAD, for CAD models based on the eXtended Finite Element Method (XFEM) [34]. These approaches did not involve a complex model-conversion process but worked on an embedded background mesh so as to automate the overall process. To keep the design intent, the adjustment should be made on feature parameters or the feature history of CAD models [35]. Conducting topology optimization under constraints of specific CAD features has attracted research interests. Zhang and his colleagues have studied extensively the topic [36][37][38][39] for practical engineering design. Recently, Guo introduced a novel topology optimization method of MMC (Moving Morphable Components) [40][41][42], which uses deforming bars as primitive features in topology optimization process for ease of geometric control. However, most of the approaches only studied abstract and single parametric features without design history. Recently, Liu and To [12] first included the feature modeling history of CAD models in the design optimization process. This work follows this direction but employs a more automatic and efficient method, i.e., XVoxel, to carry out the optimization of feature parameters by embedding design intent in the overall optimization process. XVoxel models This section introduces features, voxels, and their combination into XVoxels, as well as the data structure and algorithms for constructing and manipulating XVoxel models. From features and voxels to XVoxels There is no widely accepted definition of features. The one this work employs is given by Shah [4]: a feature is a generic portion of a model's shape that has certain engineering significance. Roughly speaking, features are clusters of geometric entities in a CAD model, which can be used as information containers to carry domain-specific attributes, e.g., materials and boundary conditions. A feature model is a set of features, combined in a way similar to traditional constructive solid geometry, as shown in Fig. 2. Practically almost all of today's commercial CAD systems use features as an internal representation for constructing and/or editing their CAD models [43]. The user designs a feature by first defining a topology of geometric entities then specifying geometric constraints relating them. A feature can be positioned anywhere in space, or relatively to existing features (through, again, geometric constraints). As such, geometric entities of a CAD model are stored associatively and hierarchically. Changes to the parameters of those features can then be propagated automatically in a pre-defined fashion [44]. This is the basis upon which parametric design optimization becomes possible. A voxel is a cube-like element in space, and a voxel model is a collection of voxels comprising a three-dimensional geometry of interest. A voxel model can be stored as an array of voxels occupied by the geometry or a grid with binary labels indicating the occupancy relationship between each voxel and the geometry; see also Fig. 3. The former storage scheme is often used to represent static geometries, and the latter used to represent dynamic geometries (and therefore the chosen one in this work). This work proposes to combine features with voxels, i.e., embedding features into voxels. Traditional CAD/CAE integration methods consistently use features as information containers to store design intent (e.g., shape parameterization) and simulation intent (e.g., meshing procedures and boundary conditions) [45]. In this work, voxels are information containers where design intent and simulation intent reside. This shift leads to the notion of semantic voxels (named XVoxels in this work). The primary benefit of doing so is that explicit generation of analysis geometry and meshes can be mostly avoided, and then an automatic, closed loop between CAD and CAE can be achieved. This will be demonstrated in the next few subsections. We begin with the specific data structure used to represent XVoxel models and some primitive operations used to manipulate them. XVoxel representation and operations An XVoxel model consists of two components: a list of features and an array of voxel attributes, as shown in Fig. 3. The feature list is nothing but an unordered set of features (with boundary conditions, material properties, etc. already associated). The voxel attributes associate each voxel with the features occupying it. Three feature attributes are stored: feature occupancy, feature nature, and feature history. For a voxel, feature occupancy describes whether it is completely or partially occupied by a feature; feature nature indicates whether an occupying feature is adding material or subtracting material; feature history refers to the precedence of all occupying features of the voxel. Consider, for example, the model in Fig. 3, and focus on feature F1. It occupies the voxels colored blue. Voxels at its boundary have partial occupancy, while those in its interior have complete occupancy. The plus signs in Fig. 3b indicate that the occupied voxels are positive (the same as F1's nature). Following the same principle, two additional arrays of voxel attributes can be generated for features F2 and F3, as shown in Fig. 3b. Combining these three arrays of voxel attributes in their chronological order (i.e., F1 → F2 → F3) results in an XVoxel model, where each voxel maintains an ordered list of 3-tuples ( f eature index, f eature nature, occupancy completeness), as shown by the rightmost four lists in Fig. 3b. In XVoxel models, determining a model's actual shape relies merely on XVoxel nature, which refers to the nature of the last feature in the attributes list of individual XVoxels. This is because whether the last feature adds or subtracts material, it will override any preceding operations. One exceptional situation is when the last feature partially occupies an XVoxel; this XVoxel's nature is a compound result of the last few features in the attribute list, from the last feature with a complete occupancy to the end. In the following, an XVoxel of this kind is referred to as compound nature. Special algorithms will be developed in the next subsection to handle this situation when using XVoxel to conduct its property simulation. The above statements seemingly imply that there is no need for storing all historical feature natures of an XVoxel, but only the last one (or ones). They are actually saved for providing easy ways to carry out XVoxel operations, as detailed below. In particular, the novel idea of constantly storing negative feature nature, rather than immediately discarding it after feature Booleans as in conventional feature modeling approaches, allows all operations to work locally, efficiently, and robustly. Feature Addition This operation creates a new feature by instantiating a chosen feature class with user-specified feature parameter values. After instantiation, the feature's shape extent is used to determine which voxels it occupies, then append the feature's attributes (i.e., the 3-tuple described above) to the end of those voxels' attribute lists. This addition operation is the basis of constructing an XVoxel model from a given feature model. We simply repeat this operation over all features of the model in their chronological order. Feature Deletion The selected feature is simply removed from the XVoxel model's feature list, with feature dependencies updated accordingly and its attributes removed from relevant XVoxels' attribute lists. To facilitate the retrieval of relevant XVoxels, we further associate each feature with a list of XVoxel indices it occupies in the XVoxel data structure (which can be easily recorded during feature addition). For every single relevant XVoxel, we linearly search the corresponding feature entry in its attribute list and, once found, simply remove it from its current position. (Note that in practice, because attributes in each XVoxel are stored as a linked list, a postprocessing step to correct the linking pointers of remaining entries in the list is needed.) If parallel computing is enabled, we can search and do the removal for all XVoxels simultaneously, without the need for the associativity from features to relevant voxels. Multiple features can also be deleted in parallel. It should, however, be noted that to avoid race conditions when deleting feature attributes at the same XVoxel, we lock the list when the entry removal operation is being carried out for a feature. Parameter Editing This operation modifies features' parameter values. In the background, we first delete it from the XVoxel model, then re-add its modified version to the XVoxel model according to its original precedence in the feature history. As such, no additional algorithms are needed. Considering that features are often interdependent [46], the above two procedures are modified to include the dependent features of the feature being edited. Feature Rearrangement This operation modifies the order of features (under the condition that feature dependencies will not be broken). What we need to do is simply updating the orders in individual XVoxels' attribute lists to accommodate the rearrangement. --- - - (a) (b)+ + + + + --- --- --- 1 2 … F1 + P 1 2 … F1 + c F2 - c 1 2 … F1 + c F3 As can be seen, there is no time-consuming and non-robust geometric computing involved in the above operations, except for the determination of voxels occupied by a feature to be added in the addition operation. All operations boil down to manipulating entries in a certain linked list, which is easy to implement, robust, and efficient. For the determination of occupied voxels, the essential task involved is to voxelize the shape of a given feature, using the same resolution as the XVoxel model. Note that voxelization is done on individual feature shapes here, which are usually primitives like cuboids or spheres, not on the overall combined shape of all features, which is otherwise complex. Many algorithms exist to voxelize a feature's B-rep model, and the method developed by Young and Krishnamurthy [47] is employed in this work due to its high efficiency. The B-rep model of a feature is often made readily available during feature instantiation, a function provided by almost all modern commercial CAD modelers. Virtual model simplification Model simplification 1 is to remove some design features (e.g., small drilled holes) that are of little significance to simulation. This task becomes straightforward if XVoxel models are used. What we need to do is applying the delete operation described in the previous subsection. The only issue is that, similar to traditional feature-based model simplification approaches, directly removing a feature may cause the persistent naming problem, ultimately breaking the design-analysis cycle. To be more specific, features are made interdependent in feature modeling to enable automatic propagation of parameter changes [44]. Removing a feature makes any inter-dependencies related to it undefined, and then the whole model becomes invalid, which is the so-called persistent naming problem [50]. To solve this issue, we customize the delete operation slightly. The delete operation in Section 3.2 directly removes a feature from the XVoxel model's feature list. Instead, we retain it but make it transparent to voxel attributes by associating the feature list with a bitmask whose 0-elements indicate that features at their positions have been removed, virtually. As such, the difficult persistent naming problem is avoided and meanwhile, there are no real geometric operations involved in model simplification. Another benefit of doing so is that boundary conditions, once associated with certain features, can retain over the course of optimization iterations regardless of design modifications because those features are completely stored in XVoxel models. Point membership classification Traditionally, what comes next after model simplification is generating boundary-conformed meshes for downstream task of simulations. This is, however, a field not all major questions have been answered [21]. This work reformulates the problem as an underlying problem of point membership classification (PMC) for Gaussian integral point selection. It is to be further combined with the recently developed method of FCM for physical simulation, which embeds the physical domain of computation (i.e., the geometry of the feature model) in a larger, regular mesh like a grid, and then transforms the FE computation onto the embedding meshes [26]. (A detailed introduction to FCM will be given in the next section.) This way of working is a perfect match for XVoxel models. If an XVoxel has positive nature and complete occupancy, it is completely within the model shape. Then the stiffness matrix for this XVoxel can be computed in the exact same way as conventional FEM does. If an XVoxel has compound nature, the XVoxel crosses the boundary of the model shape. According to FCM, voxel subdivision is needed to generate stiffness matrices for such boundary XVoxels, which in turn relies on the operator of point membership classification (PMC) to determine if a sample integration point within a boundary XVoxel is IN/ON/OUT the model shape. Because XVoxel models have prepared the history of feature occupancy for every XVoxel, the problem of PMC against the overall model shape can be converted to a sequence of much simpler PMCs against individual features [51]. The conversion consists of three major steps: (1) screening relevant features; (2) evaluating PMC against each screened feature; and (3) compiling evaluation results to the final IN/ON/OUT decision. Clearly, not all features occupying an XVoxel contribute to its final shape (i.e., which portion of the XVoxel is solid or void). According to the XVoxel's attribute list, candidate features include those ranging from the last feature with a complete occupancy to the end feature (see Section 3.2). For this reason, the screening step can be simply done by tracing from the back of the attribute list up to the first entry having the complete occupancy attribute. Having relevant features in place, we next determine the IN/ON/OUT relationship between a query integration point and each of the features. Let the relevant features be denoted by f 1 , f 2 , · · · , f n , and their corresponding implicit representation denoted by φ 1 , φ 2 , · · · , φ n . In this work, feature implicitization is done by first triangulating its B-rep model with a sufficient high accuracy, then building a KD-tree for the triangles to allow fast query of the (approximated) signed distance between a given point and the feature, similar to the method presented in [52]. Note that alternative methods surely exist [53], and we choose this one for its simplicity and efficiency. Whether a given point x is IN/ON/OUT feature f i is determined by the sign of φ i (x):          φ i (x) > 0 → x IN f i , φ i (x) = 0 → x ON f i , φ i (x) < 0 → x OUT f i .(1) To compile individual classification results to the final IN/ON/OUT decision, we again make use of the feature history stored in each XVoxel. First, the features classified as OUT are filtered out from the relevant feature set because they contribute nothing to the process of adding/removing material. Then, the final IN/ON/OUT decision is the same as the nature of the last remaining relevant features: if the nature is positive, the material is added to the query point, and the final decision is IN/ON; otherwise, the final decision is OUT. This is because the last material removing/adding operation overrides all the preceding operations. Altogether, they yield a method to generate a "mesh" suitable for FCM solving from an XVoxel model. The mesh is not explicitly generated but through combining the fixed grid carrying the XVoxel model and an implicit PMC operator developed specifically for XVoxel models. The method is thus easy to implement. It should, however, be noted that, the use of triangulation in feature implicitization will introduce errors, and therefore possible misclassifications in PMC. In fact, this is generally acceptable since we can triangulate at a high accuracy. Also, due to the integral nature of FCM, it is not very sensitive to such misclassifications. XVoxel-based simulation In this work, simulation is to be carried out using a fictitious domain approach following a FCM-like framework [26], which can work directly on voxel models. This approach's low computational efficiency is improved in two aspects: (1) by introducing material-aware piecewise matrix-valued shape functions, called CBN (Curved Bridge Node) shape functions following the previous study in [54]; (2) by utilizing the local computation of XVoxel models. Finite cell method (FCM) for XVoxel-based simulation The basic idea of FCM is to use a simple regular structured mesh to approximate the solution fields. This is achieved by combining the fictitious domain idea with the benefits of highorder finite elements, thus avoiding the costly and even labourintensive meshing process. The FCM concept is interpreted by a 2D linear elasticity problem in Fig. 4. Let Ω p ∈ R 2 be the physical domain, Γ D the Dirichlet boundary, and Γ N the Neumman boundary under external loading τ. A linear elasticity analysis problem on Ω p is studied to find the displacement u satisfying a(u, v) = l(v), ∀ v ∈ H 1 0 (Ω),(2) where a(u, v) = Ω p ε(u) T Dε(v) dV = Ω H(x)ε(u) T Dε(v) dV,(3) and l(v) = Ω p f ·v dV+ Γ N τ·v dΓ = Ω H(x)f ·v dV+ Γ N τ·v dΓ,(4) where H 1 (Ω) and H 1 0 (Ω) are the usual Sobolev vector spaces, f is the body force, σ(u) is the second-order stress tensor defined via Hooke's law, σ(u) = D : ε(u), ε(u) = 1 2 (∇u + ∇u T )(5) for a fourth-order elasticity tensor D. Here in Eqs. (3) and (4), the computation domain is converted from Ω p to the embedded domain Ω by incorporating the fictitious domain material which is defined via a Heaviside function H(Φ(x)) , H(Φ(x)) =        1, if Φ(x) > 0, α, otherwise ,(6) where Φ is the SDF (Signed Distance Function) of the feature model Ω p and α is a small positive coefficient, say 10 −8 , to avoid ill-conditionedness on the stiffness matrix. The Nitsche's method [55] was usually adopted to weakly impose Dirchlet boundary conditions in FCM; we are not going into details here. Following a classical Galerkin FE method, the solution u(x) to Eq. (2) is approximated as a linear combination of higher-order shape (base) functions N α (x) for each regular grid (or voxel) Ω α ⊂ Ω. Specifically, the overall displacement on any point of x ∈ Ω p can be interpolated from an assembly sum u(x) ≈ N(x)Q = M α=1 N α (x) Q α , x ∈ Ω,(7) where N(x) is the collection of bases N α (x), Q is the collection of Q α , a displacement vector per voxel Ω α . Accordingly, the displacement Q to Eq. (2) is computed as the solution to a linear system KQ = F,(8) where the stiffness matrix and load vector are assembly from their element stiffness matrix K α and element load vector on a regular grid (or XVoxels) K = α K α , F = α F α .(9) FCM transfers the challenges of mesh generation to the numerical integration of discontinuous integrands in Equation (3) and (4). An adaptive Gauss integration is usually applied to improve its accuracy; see Fig. 5. The high-order shape functions N α (x) in FCM requires a huge number of Gaussian points, which involves huge computational costs as compared with FEA and occupies the dominant computational costs of FCM. CBN shape functions for efficient FCM computation Following previous study [54], material-aware CBN shape functions are introduced in this work to replace the higher-order shape functions in FCM to accelerate the computations. In our adopted version of CBN, a 4×4 grid is formed by introducing twelve additional virtual nodes on each face of a voxel element besides its original 4 nodes (all together 80 nodes for a 3D XVoxel). Displacement on these CBN nodes are collected into a vector Q and taken as DOFs for solution computation. The higher order shape function N α in Eq. (7) is replaced by the following one composed of linear shape functions on fine mesh via a transformation matrixΦ α : N α (x) = N α,h (x)Φ α , x ∈ Ω α(10) where N α,h (x) is an assembly of the nodal shape functions on the fine mesh of Ω α . The CBN transformation matrixΦ α aims to map the CBN nodal values to the interior values in Ω α . It is derived as a product of boundary interpolation matrix Ψ and boundary-interior transformation matrixM α , as follows, Φ α =M α Ψ,(11) where Ψ andM α maps the displacements from the CBNs to the boundary nodes and then to the full fine nodes in Ω α . The boundary interpolation matrix Ψ maps the CBN nodal values to the fine mesh boundary nodal values of Ω α . It is derived by constructing a bi-cubic Bézier interpolation surface over the face of interest, taking the CBN as control points. The matrix Ψ is derived by evaluating the surfaces at fine mesh nodes within the face, and collecting them in a matrix form all the values row by row for the six faces of the mesh α. The transformation matrixM α maps the boundary node values to those of the fine mesh in Ω α . It is derived from the local simulation on the fine mesh of Ω α with the equilibrium equation k b k bi k ib k i q b q i = f b 0 ,(12) where k b , k i , k bi , k ib are the sub-matrices of the local stiffness matrix k α on Ω α , q b , q i is respectively vector of the boundary, interior nodes, and f b is vector of exposed forces on the boundary nodes formed by harmonic analysis [54]. We have from the second-row the relation of q i = M α q b , for M α = −k −1 i k ib . Accordingly, assembling q i and q b as q = [q b , q i ] T , we have the form ofM α , M α = [I 2b , −k −1 i k ib ] T ,(13) where I 2b is the 2b × 2b identity matrix. Once the CBN shape functions are derived, the solution to the linear elasticity problem in Eq. (2) can be similarly attained, following a classical Galerkin FE method. More technical details are referred to [54]. XVoxel-based local simulation of feature models Based on the approach of FCM for simulation, in combination with CBN, the XVoxel-based approach for simulation of modified feature model is developed below. The feature model is generally modified via updating feature parameters, which may change the topology and geometry of the final B-rep model. As long as the feature model are updated, element stiffness matrix K α of each voxel need to be re-computed, which accounts most for the computation costs. FCM equipped with local voxel updates can accelerate the computations in two ways: (a) the element stiffness matrix of each full-voxel or void-voxel is identical; the voxels along the boundary are called cut voxels; (b) the element stiffness matrices of voxels not affected by updated features remain unchanged. Case (a) can be easily resolved via a pre-computation strategy. For case (b), the element stiffness matrix can be incrementally updated by updating and querying voxel-feature membership table via the PMC algorithm described in Section 3.4, where voxels affected by the updated features can be quickly located, called active voxels, and consequently only their stiffness matrices are re-computed. This can significantly reduce computation costs. XVoxel-based parametric design optimization Using XVoxels, the parametric design optimization works over a fixed regular grid under controlled simulation accuracy, and on direct updates of feature parameters. During the process, the sensitivities with respect to the design parameters is derived for parameter updates. The locality information of XVoxel provides an efficient sensitivity computation either via finite difference or via a derived analytical expressions. Let Ω p be a CAD model with features f 1 , f 2 , · · · , f n . For ease of explanation, each feature f i is assumed to take only one parameter p i . The classical compliance minimization problem is studied to find the optimized design parameters p = (p 1 , p 2 , · · · , p n ): min p C(u, p) = u T Ku, s.t.            Ku = F, V = Ω H(Φ(x, p))dΩ ≤V, p i ≤ p i ≤p i , i = 1, 2 . . . , n,(14) in which V andV are the total structural volume and maximum volume constraint, the Heaviside function H(·) is used to indicate structural boundary, p i andp i are lower and upper bounds of the design variable p i . The optimization problem Eq. (14) is to be solved following a numerical gradient-based approach Globally Convergent Method of Moving Asymptotes (GCMMA) [56] for its robust convergence in design optimization. It approximates the original nonconvex problem through a set of convex sub-problems by using the gradients of the optimization objective and constraints with respect to the design variables p derived below. The gradient computation follows the chain rule. First consider the sensitivities of stiffness matrix K with respect to design parameter p i . Rewritting ψ(x) = B T DB for conciseness, we have ∂K ∂p i = ∂ ∂p i Ω B T DBH(Φ(x, p))dΩ = Ω B T DB ∂H(Φ) ∂Φ ∂Φ ∂p i dΩ = Ω ψ(x) ∂H(Φ) ∂Φ ∂Φ ∂p i dΩ.(15) The key point of above equation is to compute derivative of Heaviside function. We bring in Dirac delta functionδ(Φ) δ(Φ) = ∇H(Φ) · ∇Φ ∇Φ = dH(Φ) dΦ ∇Φ · ∇Φ ∇Φ = dH(Φ) dΦ ∇Φ ,(16) Consequently, Eq. (15) is rewritten as ∂K ∂p i = Ω ψ(x) ∂H(Φ) ∂Φ ∂Φ ∂p i dΩ = Ω ψ(x) ∂Φ ∂p i 1 ∇Φ ∂H(Φ) ∂Φ ∇Φ dΩ = Ω ψ(x) ∂Φ ∂p i 1 ∇Φ δ (Φ)dΩ = ∂Ω p ψ(x) ∂Φ ∂p i 1 ∇Φ dΓ,(18) where ∂Ω p denotes boundary of feature model Ω p . This way, the volume integral of sensitivities is transformed into a boundary integral. According to the expression of Φ(x, p) in Eq. (6), we further have for Eq. (18), ∂Φ(x, p) ∂p i = n j=1 ∂Φ(x, p) ∂φ j · ∂φ j ∂p i ,(19)∇Φ(x, p) = n j=1 ∂Φ(x, p) ∂φ j ∇φ j .(20) Noting that design variable p i is only associated to one feature f i , we have ∂φ j ∂p i = 0, j i. Let S i = ∂Φ(x,p) ∂φ i be the logical operationde defined by parent bifurcation nodes of f i in CSG tree, and S i ∈ {−1, 1}. Accordingly, the integral domain of Eq. (18) can be reduced from boundary ∂Ω p of whole feature model Ω p to boundary ∂ f i of a feature f i , that is, computations. ∂Φ(x, p) ∂p i = ∂Φ(x, p) ∂φ i · ∂φ i ∂p i = S i ∂φ i ∂p i ,(22) which avoids redundant integration. Similarly, we have ∇Φ(x, p) = S i ∇φ i = ∇φ i(23) for quadrature points along boundary of feature f i . Accordingly, the sensitivities in Eq. (18) is reduced to ∂K ∂p i = ∂Ω p ψ(x) ∂Φ ∂p i 1 ∇Φ dΓ = ∂ f i S i ψ(x) ∂φ i ∂p i 1 ∇φ i dΓ.(24) Afterwards, we consider sensitivities of structural compliance C, ∂C ∂p i = ∂F T ∂p i u + F T ∂u ∂p i = ∂F T ∂p i u + F T K −1 ∂F ∂p i − ∂K ∂p i u = 2 ∂F T ∂p i u − u T ∂K ∂p i u.(25) Assuming for simplicity the independence of load F and feature design variables, the first term in the above equation is zero. Based on sensitivities of stiffness matrix in Eq. (24), we have ∂C ∂p i = −u T ∂K ∂p i u = −u T ( ∂ f i S i ψ(x) ∂φ i ∂p i 1 ∇φ i dΓ)u = −u T ( ∂ f i S i B T DB ∂φ i ∂p i 1 ∇φ i dΓ)u.(26) In our numerical implementation, we use triangles in 3D (lines in 2D) to approximate structural boundary of model for adaptive integration; see Fig. 6 for an illustration. Numerical examples and discussions The proposed XVoxel-based method for parametric design optimization of feature models has been implemented in Matlab on a computer with Intel Core i7-12700 3.6 GHz CPU, 64GB RAM. Five different examples are shown to demonstrate its effectiveness: the first three on simulation during interactive editing to test its computational accuracy and efficiency, and the last two on its usage for feature-based design optimization. The CAD model sizes are all measured in micrometer, and the material has a Young's modulus E = 2e 11 Pa and Poisson's ratio ν = 0.3. In FCM computing, all examples have the octree refinement depth d = 3 and the shape function order p = 2, except for the first example where d = 4 and p = 3. The locality characteristics of XVoxel are measured in terms of the number of active voxels (i.e. voxels affected by local feature updates) against that of FCM. The fidelity of XVoxel or FCM is measured via its displacement residual (in terms of top 10%) against the benchmark: r u = u 1 − u 0 u 0 ,(27) where u 1 , u 0 are the computed and the benchmark displacements, respectively. The experimental settings and results are summarized in Table 1, including mesh size, DOFs, timing (per step/iter), the number of active voxels and relative error r u . In all these examples, FEA simulation results on tetrahedral meshes (in Ansys Workbench 22R1) were taken as the benchmark. Three other approaches were tested to show the method's simulation accuracy and efficiency: standard FCM approach [26], XVoxel-FM combining FCM with XVoxel, XVoxel-CBN combining CBNbased FCM [54]; the last two are our approaches. The DOFs of FEA and XVoxel were set approximately same for the comparisons to be fair. As can be seen from Table 1, the error of FCM (XVoxel) is as low as 0.005%, demonstrating its high accuracy. FCM and XVoxel-FCM always have the same simulation accuracy, and CBN-FCM is very close to them. Other examples may have higher error due to the need for balancing accuracy and effiency. Note that FCM (XVoxel) can reach a prescribed accuracy voxel refinement or degree elevation [26]. In all examples, FEA is much more efficient than FCM while XVoxel-FCM improves the efficiency, resulting in a similar computation time to FEA, due to its local computations. XVoxel-CBN greatly improves the efficiency of XVoxel-FCM due to its usage of piecewise linear shape functions. We have to mention again that in comparison with FEA FCM or XVoxel (either FCM or CBN version) has a prominent advantage in its much easier and more robust voxolization than FEA's tetrahedral meshing. Example #1: an L-shaped model for simulation accuracy testing The accuracy of the proposed method was first tested on a classic L-shaped model as shown in Fig. 7, constructed by combining two cubes and one rounded corner. The model is fixed on its upper face and subject to a downward traction of τ = 100N/mm 2 on its right face. The tetrahedral mesh of FEA has 6.3K elements, and the FCM has 768 voxels. The rounded corner radii was varied from 6mm to 2mm at a step of −1mm. The simulation error, the number of active voxels, and timings for each step were respectively plotted in Figs. 8(a),(b),(c). FEA and XVoxel-FCM has a very close approximation at an error r u = 0.03%, as can also be observed from distributions of their displacement norm and von Mises stress in Figs. 9(a),(b). We also notice from Figs. 9(c) and 8(b),(c) that FCM and XVoxel-FCM have exactly the same number of active voxels and computational timings in this example. This is because the cut cells in FCM are just the active voxels due to the regular shape of the L-shaped model. Example #2: a connector model for simulation efficiency testing The second test was conducted on the engine connector as shown in Fig. 10 to test XVoxel's ability in handling more complex models. The model is fixed on its left hole, subject to horizontal and vertical tractions on its right hole. The FEA tetrahedral mesh has 23K elements while the XVoxel has 7.9K voxels. The connector model was modified in Fig. 12(a) by the following six steps of feature operations: 1. Add a pair of inner groove made of two cylinders and their tangents. Note that the two sides of connector are symmetrical, and we only consider design parameters on one side. 2. Translate the two cylinders by changing parameters d 1 from 25 to 30, d 2 from 55 to 40. 3. Modify the right cylinder's radius r 2 from 5 to 7.5. 4. Modify the inner groove's depth h from 1.5 to 2.5. 5. Modify design parameter h from 2.5 to 3.5 and remove the round corner so that the inner groove goes through the whole model. 6. Modify design parameters r 1 from 5 to 3, r 2 from 7.5 to 5. XVoxel has a close approximation to FEA with a maximal r u = 0.33% as observed from plot in Fig. 11(a), or the comparison of simulation results in Figs. 12(b) and (c). We also noticed from Figs. 12(d) and 11(b) and (c) that the XVoxel (XVoxel-FCM and XVoxel-CBN) has much less computational time than FCM during the model modifications (after the first step) as its local voxel update immensely decreases the number of active voxels. Example #3: a pump model under drastic topology variation and varying loads The proposed method's potentiality in handling drastic topology variations and varying loads was further tested on the complex pump model in Fig. 13. The pump's bottom is fixed and its top and outer side are exerted by forces of 200N, 100N respectively. The FEA has 216K tetrahedral elements while FCM (XVoxel) has 40K voxels. The model was edited by the following steps, during which both the model's topology and external loadings are varied: 1. Input an initial model consisting of different cylinders. 2. Add a round corner feature f 1 . 3. Add feature f 2 which consists a cube and a cylinder. 4. Add a negative feature f 3 as a union of four cylinders. 5. Add a negative feature f 4 . 6. Add a feature f 5 . 7. Add a feature f 6 consisting of four spheres. 8. Add a negative feature f 7 . 9. Add a feature f 8 consisting of a cube and a cylinder. The resulting models during the modification were shown in Figs. 14(a)-(c), with the associated active voxels given in Fig. 14(b). Statistics of the relative errors, numbers of active voxels, timings were compared in Fig. 15, which indicates XVoxel's high simulation accuracy and efficiency, as already confirmed in Examples #1 and #2. We in particular observed from Fig. 15(b) that the number of active voxels of XVoxel is only around 1/4 of FCM's, demonstrating XVoxel's strong ability in properly selecting active voxels regardless of the complex feature shape and feature operations. The nice property in turn resulted in a 4-time efficiency improvement of XVoxel-FCM in comparison to FCM; XVoxel-CBN even achieved a 50 times efficiency improvement. Such efficiency is very useful in obtaining interactive simulation feedback in modifying designs. Example #4: a bracket model for parametric design optimization The proposed method's ability in feature-based parametric design optimization was first tested on an asymmetric bracket model in Fig. 16, which has a negative groove feature composed of several cylinders and prisms. The model was fixed on its left hole, and subject to an axial load of 100 √ 5N on its right hole. The model is discretized into 9.1K voxels for FCM (XVoxel) based simulation. The design goal is to minimize the bracket's compliance under a volume ratio of 0.9 by varying the locations and sizes of the groove. The design variables are: circle centers x i , y i and radii r i (i=1, 2, 3, 4) of the four corner circles, their inscribed circle radius r 5 and circumscribed circles' radii r 6 , r 7 , r 8 (in 2D plane). The associated geometric constraints are formulated as follows so as to produce a valid geometry:                                          Ku = F, V ≤ 0.9V 0 , ||(x i , y i ) − O 5 || + r i = r 5 , i = 1, 2 ||(x i , y i ) − O 6 || − r i = r 6 , i = 3, 4 ||(x i , y i ) − O 7 || − r i = r 7 , i = 1, 4 ||(x i , y i ) − O 8 || − r i = r 8 , i = 2, 3 x min i ≤ x i ≤ x max i , i = 1, ..., 4 y min i ≤ y i ≤ y max i , i = 1, ..., 4 r min j ≤ r j ≤ r max j , j = 1, ..., 8 where V 0 is the volume of the original model, O i 's are the 2D circle centers' coordinates, and ranges of x i , y i , r j were set so that the features would not move out of the bracket. Some intermediate structures during optimization were shown in Fig. 17(a), where the groove gradually enlarged its size to meet the volume constraint while moving to the left side for performance improvement. Stress distributions were also plotted in Fig. 17(b), and the active voxels were shown in gray in Fig. 17(c). The optimization was stopped after 100 iterations, where the features and their relative constraints were all maintained during the optimization; the convergence curve was plotted in Fig. 18(a). During the optimization, XVoxel only had approximately 1/8 active voxels of FCM, with a much-improved efficiency; see also Figs. 18(b) and (c). XVoxel-FCM and XVoxel-CBN respectively achieved around 13× and 50× efficiency improvements compared to FCM. By maintaining the feature lists during optimization, the simulation was greatly accelerated by local recomputations for active voxels. As can be seen from Fig. 18(a), the compliance curve did not go steadily, but first went up quickly to the peak, then went down. This is because compliance is highly sensitive to volume changes, and the volume factor dominates the optimization before reaching the specified volume limit. That is, the optimization will quickly reduce the volume toward the specified volume limit at the beginning (see the volume curve in the same figure). After getting the peak (at the 10th iteration), the optimization algorithm (i.e., the globally convergent method of moving asymptotes) reduces compliance effectively while keeping the volume above the limit. This process corresponds to the going down phase in Fig. 18(a). (a) (b) (c) (d) (e) In conducting the optimization, the sensitivities for this example were derived using finite differences. Without proper handling, the computation would be very expensive as it requires a complete FE recomputation for all 16 design parameters. Instead, XVoxel requires much fewer active voxels for the finite difference computations, as only one feature parameter was varied in each finite difference process; see also Fig. 17(d) for an illustration. 6.5. Example #5: a bearing bracket model for parametric design optimization with varied topology The proposed method's robustness in handling complex parametric design optimization with varied structural topology was tested on a bearing bracket model in Fig. 19, where the design model, CSG tree, boundary conditions, discrete voxels were respectively shown in Figs. 19(a), (b), (c), (d). The FCM (XVoxel) has 33K voxels, and the volume ratio constraint was set to be 0.6. The model has 11 CBS (Closed B-Spline) negative features, each with 24 control radii and 2 position coordinates, where the first and last radii of each feature are driven parameters for higher-order geometric continuity. In order to maintain the features to produce a valid structure of complex topology, features f 1 and f 2 were restricted to move only along the y-axis, features f 3 , f 4 , f 5 , f 7 , f 8 , f 9 were fixed, and features f 6 , f 10 The XVoxel model has around half active voxels of FCM, and XVoxel-FCM is about 1.3× faster than FCM in each iteration. The speedup is much smaller in comparison with Example #4, as the CBS features were distributed more broadly within the bracket model, and therefore resulted in more active voxels. Nevertheless, XVoxel-CBN has gained about 5× efficiency improvement. Again, the proposed method remains robust in the feature-based design optimization framework in handling such a large-scale model with a large number of design variables and drastic topology changes. In the end, we further test the approach's ability in handing features of varied locations, sizes or orientations using the example in Fig. 22. The approach's potentiality is also tested in removing unnecessary features in the construction history during the optimization. The model has two plates supported by 9 cylinders in the middle. Each cylindrical feature contains 5 design variables, including position coordinates x i , y i , directions α i , β i and a radius r i for i = 1, . . . , 9. The upper surface of the top plate is exerted by a radiant force F defined as F(x, y) = F c cos ( (x − x c ) 2 + (y − y c ) 2 /(9 √ 2) · π 2 ) 8 , where F c = 50N, x c = x c = 9mm, 0 ≤ x, y ≤ 12 while the lower surface of the bottom plate is fixed, as shown in Fig. 22(c). Three different tests are conducted at a volume fraction of 1.2 times the original volume of the cylinders by optimizing x i , y i and r i ;α i , β i and r i ; all the variables. The optimized structures and convergence curves are shown in Fig. 23. All the cases lead to reliable convergence and produce structures respectively of compliance 39.3, 47.1, and 36.0. It can be observed that the full variable optimization has the best convergence rate. Meanwhile, note in Fig. 24(c) that three design features are able to run out of the design area during optimization, demonstrating the approach's ability in removing unnecessary features automatically. In this case, we can conveniently remove the features from the feature list. The variations of the features are also shown in Fig. 24. Discussion and limitations As one may have noticed from the above examples, although the final results of our XVoxel method can admit topology changes, their overall shapes still follow a similar structural pattern to those of the initial designs before optimization. This is because the method is designated for feature-based CAD and parametric optimization, where feature semantics (and therefore the overall shapes) often need to be respected. For example, the boundary representation of the final results is consistently composed of smooth parametric surfaces; there is no way the boundary takes discrete, free shapes (which is the case for SIMP or the voxel density method [20]). For this reason, our method works better for situations like fine or semi-fine design tuning. Dimensional variations and topology changes can be large (as demonstrated by the example in Fig. 20) but cannot be radically different. If design changes of this sort are desired, other methods, e.g., the voxel density method, should be used. For the same reason, our method needs to take as input an initial design that does not deviate too much from the final result. This is different from methods like the voxel density method. Their input can even be a block without bearing any similarity to the final result. Conclusions An XVoxel-based parametric design optimization method for feature models has been presented in this paper. The proposed method combines the local regularity of voxel models and the global semantic information of feature models to facilitate the automatic linking between CAD and CAE. By further integrating XVoxels, FCM and CBN, design modifications and simulation updates can be looped in an efficient and robust manner, without involving labor-intensive conversion between CAD models and CAE models. The effectiveness of the proposed method has been validated by various numerical examples with complex topology variations and varying loads. And a computational efficiency improvement of up to 55.8 times the existing FCM method has been achieved, see Table 2. We consider the proposed XVoxel method as an alternative attempt toward the long-standing research objective of a unified model representation scheme that can completely, compactly, and associatively represent the contents of both CAD and CAE models. The method builds itself upon a new concept called semantic voxels to provide the advantage of avoiding B-rep model simplification and mesh generation. These two procedures could present a particular challenge for existing methods; for example, the quad/hex meshing required by IGA is never easy if the geometry is complex. For this reason, a typical use case where XVoxel is preferable over the others is when the design to be optimized is given as a feature model and its overall shape is complex. A couple of interesting improvement directions for the XVoxel method are noted here. As already noted in Sections 3.3 and 4.3, XVoxel models can much reduce the dependence of simulation on model simplification and can be directly used to guide the simulation-suitable model simplification process. Nevertheless, the method, in its current form, is still not able to handle dimension reduction, which is the other important step in model idealization [10,48,49]. Extending the method to including dimension reduction is among the research studies to be carried out in our group. Another interesting improvement direction is that the proposed method has only been implemented in Matlab for proof of concept. Its further implementation on basis of commercial/opensource feature modelers, e.g., Open CASCADE, and then release as an open-source plugin is of great interest to our future research work. In industrial design optimization, innovative designs often require heavy optimization within a large design space but time resources are limited. This entails the use of dimensionality reduction techniques on the design space during optimization. The proposed XVoxel method has a good potential to integrate with dimensionality reduction methods, e.g., parametric model embedding [57], due to its generality on the input model. Such an interesting integration is among our future work. Another improvement direction lies in the efficiency of the proposed method. Currently, our use of octrees in finding Gaussian points leads to a time-consuming simulation. If augmented with some adaptive meshing method (e.g., [58]), the proposed method can be much accelerated. It is also worth noting that a model may correspond to multiple construction ways using Boolean operations. For different construction ways, their design variation spaces may be different, so do the corresponding optimization processes. Therefore, the construction way needs to be carefully thought out before using our method. The proposed method, in its current form, only focuses on parametric optimization on given models, and it cannot automatically find an appropriate construction way. Such an automatic selection mechanism is of great interest to future work. In addition, If a B-Rep model is provided, a Boundary-to-CSG conversion procedure (e.g., the method presented in [59]) is necessary for the proposed method to work. Another limitation of the proposed method is that the optimiza-tion result is affected by the size of the cells used in the simulation, a consequence of using FCM for simulation (see [60] for a detailed discussion). Currently, there is no principled way to choose the best cell size, and the usual solution is using empirical tuning to find a good cell size. Fig. 1 . 1Traditional conversion-based process of parametric design optimization. Fig. 2 . 2A CSG example. Fig. 3 . 3Illustration of the XVoxel data structure: (a) a feature model (the plus sign means Boolean addition, and the minus sign Boolean subtraction); and (b) its corresponding XVoxel model. Fig. 4 . 4The embedded domain Ω consists of the physical domain Ω p and the fictitious domain Ω f , and the influence of Ω f is penalized by material parameter 10 −q . Fig. 5 . 5In 2D, adaptive Gaussian integration is recursively refined towards boundary of physical domain (gray domain), yielding a quadtree structure (thin blue grids) of finite cells (bold black grids); Quadtrees depth ranging from 0 to 5 are shown here. The voxels along the boundary are called cut voxels. Fig. 6 . 6Sensitivity computation (in 2D) as adaptive boundary integration of finite cell (bold black grid): In each sub-cell (thin blue grid), the boundary is approximated with line segments (green), along which Gauss quadrature points (red) are taken. Fig. 7 . 7Example #1. (a) Parameters (mm) of the L-shaped model, where the radius R gradually varies from 6 to 2 with a step size of −1; (b) The CSG of the model; (c) Boundary conditions, where γ D is fixed and τ = 100N/mm 2 in γ N ; (d) FEA mesh with 6,325 tetrahedral elements; (e) FCM (XVoxel) mesh with 3 × 15 × 15 voxels. Fig. 8 . 8Performance statistics of Example #1 in Fig. 7 : (a) Displacement residual r u between XVoxel and FCM; (b) The number of active voxels by FCM and XVoxel; (c) Timing of four methods FEA, FCM, XVoxel-FCM (XVoxel based on FCM) and XVoxel-CBN (XVoxel based on CBN). Fig. 9 .Fig. 10 . 910Results for an L-shaped model in editing step 1, 3 and 5. (a): Displacement norm (mm) from FEA (left) and FCM/XVoxel (right); (b) Von Mise stress (MPa) from FEA (left) and FCM/XVoxel (right); (c) Active voxels (in grey) of XVoxel. Example #2. (a) Parameters of the piston model, where modified variables are d 1 , d 2 , r 1 , r 2 and h; (b) The CSG of the model; (c) Boundary conditions where Γ D is fixed and τ x = 100N/m 2 and τ y = 200N/m 2 were applied to Γ N as (sinusoidal) bearing loads; (d) FEA mesh of 22,789 tetrahedral elements in step 1; (e) FCM (XVoxel) mesh of 55 × 16 × 9 voxels. Fig. 11 . 11Performance statistics of Example #2 in Fig. 10 : (a) Displacement residual r u between XVoxel and FCM; (b) The number of active voxels by FCM and XVoxel; (c) Timing of four methods FEA, FCM, XVoxel-FCM and XVoxel-CBN. Fig. 12 . 12(a)The models for steps 1 to 6: Initial model with a pair of symmetrical grooves; Modify d1 and d2 to translate grooves; Increase the right cylinder's radius r2; Increase grooves' depth h; Increase groove' depth h to run through the model; Decrease radii r 1 and r 2 to narrow the groove ; Results for a connector model in editing steps 1, 3, 5 and 6 (b) Displacement norm (mm) by FEA (left) and FCM/XVoxel (right), (c) Von Mise stress (MPa) from FEA (left) and FCM/XVoxel (right), (d) Active voxels (in grey) of XVoxel. 10 . 10Add a negative feature f 9 . 11-15. Modify design parameter d 1 from 76 to 56 at a step of -4. 16-20. Modify design parameter d 1 from 50 to 70 at a step of 4. Fig. 13 . 13Example #3. (a) Parameters (mm) of the pump model at the 10th step, where d 1 and d 2 are the variables to modify in step 11 − 20; (b) The CSG of the model at the 10th step, where f i are the features to add in step 1 − 10, i = 1, ..., 9; (c) Boundary condition, where γ D is fixed and τ z = 200N and τ y = 100N; (d) FEA mesh with 216,411 tetrahedron at the 10th step; (e) FCM (XVoxel) mesh with 25 × 43 × 37 voxels. Fig. 14 . 14(a)The models in editing steps 1-10, 15 and 20, (b)Active voxels (in grey) of XVoxel in editing steps 1-10, 15 and 20, (c)The simulation results for a pump model in editing steps 1, 5, 10, 15 and 20. Displacement norm (mm) of FEA (up) and of FCM/XVoxel (down). Fig. 15 .Fig. 16 . 1516Performance statistics of Example #3 in Fig.13 : (a) Displacement residual r u between XVoxel and FCM; (b) The number of active voxels by FCM and XVoxel; (c) Timing of four methods FEA, FCM, XVoxel-FCM and XVoxel-CBN. Example #4. (a) Parameters (mm) of the model. The position and size of a groove are optimized by changing the position and radius of the tangential cylinders that form it, where the design variables are x i , y i and r j (i=1, 2, ..., 4; j=1, 2, ..., 8); (b) The CSG of the model; (c) Boundary conditions, where Γ D is fixed and τ x = 100N, τ y = 200N are applied on Γ N as (sinusoidal) bearing loads; (d) FCM (XVoxel) mesh with 52 × 25 × 7 voxels. and f 11 only moved along a prescribed plane; see also Fig. 19(a). Altogether, there are in total (24 − 2 + 2) × 11 − 14 = 250 design variables. The intermediate results of the examples are given in Fig. 20 and performance statistics of the example in Fig. 21. As can be seen from Fig. 20, during the optimization, the CBS features gradually expanded to meet the volume constraint, resulting in drastic topology variations in the arm and base part of the bearing bracket, before reaching convergence after 35 iterations. The XVoxel-based optimization successfully handled the topological changes. Fig. 17 . 17Results for a bracket model in optimization iterations of steps 1, 10, 30 and 100 (a) The model, (b) Von Mise stress (MPa) of FCM/XVoxel, (c) Active voxels (in grey) of XVoxel, (d) Active voxels (in grey) for calculating the sensitivities with respect to x 1 , y 1 , r 1 (left) and r 5 (right) in the first iteration. Fig. 18 .Fig. 19 . 1819Performance statistics of Example #4 in Fig. 16. : (a) Convergence curves of compliance and volume by XVoxel and FCM; (b) The number of active voxels by FCM and XVoxel; (c) Timing of three methods FCM, XVoxel-FCM and XVoxel-CBN. Example #5. (a) The CBS (Closed B-Spline) design features f i , i = 1, ..., 11 and restricted area R 1 (in yellow) for f j , j = 3, 4, 5, 7, 8, 9, R 2 (in yellow) for f 6 , R 3 (in green) for f 10 and R 4 (in red) for f 11 . The x-coordinate of f i , i = 1, 2 and x-, y-coordinates of f j , i = 3, 4, 5, 7, 8, 9 are fixed. There are 250 design variables in total, and each feature contains 22 control radii and unfixed x-, y-coordinates; (b) The CSG of the model; (c) Boundary conditions, where Γ D is fixed and τ x = 100N and τ y = 100N are exerted on Γ N as (sinusoidal) bearing loads; (d) FCM (XVoxel) mesh with 50 × 17 × 39 voxles. Fig. 20 . 20Results for a bearing bracket model in optimization iterations of steps 1, 10, 30 and 100: (a) the models, (b) Von Mise stress (MPa) of FCM/XVoxel, (c) active voxels (in grey) of XVoxel. Fig. 21 .Fig. 22 . 2122Performance statistics of Example #5 in Fig. 19 : (a) Convergence curves of compliance and volume by XVoxel and FCM; (b) The number of active voxels by FCM and XVoxel; (c) Timing of three methods FCM, XVoxel-FCM and XVoxel-CBN. Example #6. (a) Parameters (mm) of the model, where the design variables are x i , y i , α i , β i and r i for i = 1, 2, ..., 9; (b) The CSG of the model; (c)Boundary conditions, where Γ D is fixed and Γ N is exerted by a radially decayed force field F(x, y) = F c cos ( (x − x c ) 2 + (y − y c ) 2 F c = 50N, x c = x c = 9mm, 0 ≤ x, y ≤ 12. Fig. 23 .Fig. 24 . 2324Optimized models and convergence curves of structural compliance and cylindrical volumes: (a) x i , y i and r i as design variables; (b) α i , β i and r i as design variables; (c) all 5 design variables; in Fig. 22. The disappearance of features during optimization in Fig. 20(c). where∇Φ = ∂Φ ∂x 2 + ∂Φ ∂y 2 + ∂Φ ∂z 2 . Table 1 : 1Summary of the performance of XVoxel on the tested numerical examples in comparison with those using FEA and FCM (standard FCM, XVoxel-FCM combining FCM with XVoxel, XVoxel-CBN combining CBN-based FCM and XVoxel). Here, d is the depth of octree refinement for element integration and p is the order of shape functions.Example Step (Iter) Mesh Size DOFs Timings (per Step/Iter) Active Voxel Number r u (%) FEA FCM(XVoxel) FEA FCM (XVoxel-FCM) XVoxel-CBN FEA FCM XVoxel-FCM XVoxel-CBN FCM XVoxel #1 1 13,801 675 63,372 63,480 47,280 5.3 35.1 35.1 6.0 33 33 0.0061 2 14,118 64,833 5.2 28.8 28.9 2.9 21 21 0.0169 3 13,807 63,600 5.2 23.2 23.7 1.8 21 21 0.0263 4 13,844 63,714 6.2 17.3 17.7 1.3 15 15 0.0049 5 13,908 64,029 5.2 11.6 11.8 0.8 9 9 0.0047 #2 1 22,789 4,004 109,632 108,135 255,024 8.5 44.9 44.9 13.8 1,182 1,182 0.0328 2 22,579 108,777 7.3 44.2 6.3 1.1 1,182 121 0.0825 3 22,326 108,006 7.2 42.6 9.4 1.8 1,182 183 0.0812 4 22,465 109,350 7.3 44.3 9.8 2.1 1,185 189 0.1555 5 22,611 109,569 7.3 43.7 5.6 1.4 1,191 128 0.3221 6 22,326 108,336 8.5 40.2 8.5 1.4 1,141 227 0.0263 #3 1 222,816 39,775 1,019,364 998,325 2,365,080 22.1 140.1 143.1 80.9 6,040 6,040 0.0928 2 217,244 1,001,976 19.7 151.5 33.9 5.2 6,048 397 1.0381 3 218,659 1,006,104 21.6 152.0 35.0 6.0 6,156 491 1.1859 4 215,383 996,989 23.1 166.2 35.3 5.4 6,356 412 1.3995 5 215,185 996,513 20.8 161.3 28.0 1.1 6,369 97 1.4215 6 214,206 991,353 23.5 167.1 35.8 4.5 6,545 458 1.4720 7 216,983 1,002,681 21.0 168.0 31.0 3.3 6,561 216 1.4834 8 215,132 997,344 20.9 174.4 34.8 3.9 6,681 325 2.5318 9 214,288 993,975 18.0 181.3 30.3 2.0 6,789 190 2.3235 10 216,411 1,004,442 21.3 210.0 29.5 2.2 6,633 102 2.4980 15 214,690 992,787 23.7 201.5 48.2 9.0 5,837 677 2.7177 20 213,749 992,295 23.5 223.4 42.6 5.5 6,485 459 3.0647 #4 1 9,100 240,975 250,638 1,490.8 130.3 55.2 2,256 2,256 10 1,411.0 109.6 25.2 2,174 308 30 1,401.8 101.3 24.8 2,166 274 100 1,467.5 102.4 25.3 2,172 255 #5 1 33,150 279,265 282,006 146.0 148.0 60.4 4,592 4,592 10 141.5 85.8 30.7 3,904 1,644 30 156.0 92.7 33.2 3,952 1,611 100 166.6 97.2 32.6 3,962 1,486 (a) (b) (c) (d) (e) Table 2 : 2Performance comparison of FCM, XVoxel-FCM and XVoxel-CBN on the tested numerical examples. The table shows the size of mesh, the number of DOFs, the total time and the total acceleration ratio.Example Mesh Size DOFs Timings (total Steps/Iterations) Total Acceleration Ratio ( Based on FCM ) FCM(XVoxel-FCM) XVoxel-CBN FCM XVoxel-FCM XVoxel-CBN XVoxel-FCM XVoxel-CBN #1 675 63,480 42,280 116.0 117.2 12.8 0.990 9.06 #2 4,004 108,135 255,024 259.9 84.5 21.6 3.08 12.0 #3 39,775 998,325 2,365,080 3,801.4 883.6 174.6 4.30 21.8 #4 9,100 240,975 250,638 145,022.8 10,321.3 2,597.2 14.1 55.8 #5 33,150 279,265 282,006 15,849.0 9,358.5 3,278.5 1.69 4.83 6.6. Example #6: bearing plates containing varying cylindri- cal supports It should be noted that a more general concept than model simplification is model idealization, which includes an additional dimension reduction task[10,48,49]. As XVoxels models are three-dimensional, they are not able to handle dimension-reduced geometries in their current form. 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{'abstract': 'Parametric optimization is an important product design technique, especially in the context of the modern parametric feature-based CAD paradigm. Realizing its full potential, however, requires a closed loop between CAD and CAE (i.e., CAD/CAE integration) with automatic design modifications and simulation updates. Conventionally the approach of model conversion is often employed to form the loop, but this way of working is hard to automate and requires manual inputs. As a result, the overall optimization process is too laborious to be acceptable. To address this issue, a new method for parametric optimization is introduced in this paper, based on a unified model representation scheme called eXtended Voxels (XVoxels). This scheme hybridizes feature models and voxel models into a new concept of semantic voxels, where the voxel part is responsible for FEM solving, and the semantic part responsible for high-level information to capture both design and simulation intents. As such, it can establish a direct mapping between design models and analysis models, which in turn enables automatic updates on simulation results for design modifications, and vice versa-effectively a closed loop between CAD and CAE. In addition, robust and efficient geometric algorithms for manipulating XVoxel models and efficient numerical methods (based on the recent finite cell method) for simulating XVoxel models are provided. The presented method has been validated by a series of case studies of increasing complexity to demonstrate its effectiveness. In particular, a computational efficiency improvement of up to 55.8 times the existing FCM method has been seen.', 'arxivid': '2303.15316', 'author': ['Ming Li \nState Key Laboratory of CAD&CG\nZhejiang University\n310027HangzhouChina\n', 'Chengfeng Lin \nState Key Laboratory of CAD&CG\nZhejiang University\n310027HangzhouChina\n', 'Wei Chen \nState Key Laboratory of CAD&CG\nZhejiang University\n310027HangzhouChina\n', 'Yusheng Liu \nState Key Laboratory of CAD&CG\nZhejiang University\n310027HangzhouChina\n', 'Shuming Gao \nState Key Laboratory of CAD&CG\nZhejiang University\n310027HangzhouChina\n', 'Qiang Zou \nState Key Laboratory of CAD&CG\nZhejiang University\n310027HangzhouChina\n', 'Ming Li \nState Key Laboratory of CAD&CG\nZhejiang University\n310027HangzhouChina\n', 'Chengfeng Lin \nState Key Laboratory of CAD&CG\nZhejiang University\n310027HangzhouChina\n', 'Wei Chen \nState Key Laboratory of CAD&CG\nZhejiang University\n310027HangzhouChina\n', 'Yusheng Liu \nState Key Laboratory of CAD&CG\nZhejiang University\n310027HangzhouChina\n', 'Shuming Gao \nState Key Laboratory of CAD&CG\nZhejiang University\n310027HangzhouChina\n', 'Qiang Zou \nState Key Laboratory of CAD&CG\nZhejiang University\n310027HangzhouChina\n'], 'authoraffiliation': ['State Key Laboratory of CAD&CG\nZhejiang University\n310027HangzhouChina', 'State Key Laboratory of CAD&CG\nZhejiang University\n310027HangzhouChina', 'State Key Laboratory of CAD&CG\nZhejiang University\n310027HangzhouChina', 'State Key Laboratory of CAD&CG\nZhejiang University\n310027HangzhouChina', 'State Key Laboratory of CAD&CG\nZhejiang University\n310027HangzhouChina', 'State Key Laboratory of CAD&CG\nZhejiang University\n310027HangzhouChina', 'State Key Laboratory of CAD&CG\nZhejiang University\n310027HangzhouChina', 'State Key Laboratory of CAD&CG\nZhejiang University\n310027HangzhouChina', 'State Key Laboratory of CAD&CG\nZhejiang University\n310027HangzhouChina', 'State Key Laboratory of CAD&CG\nZhejiang University\n310027HangzhouChina', 'State Key Laboratory of CAD&CG\nZhejiang University\n310027HangzhouChina', 'State Key Laboratory of CAD&CG\nZhejiang University\n310027HangzhouChina'], 'corpusid': 257766558, 'doi': '10.1016/j.cad.2023.103528', 'github_urls': [], 'n_tokens_mistral': 27259, 'n_tokens_neox': 23479, 'n_words': 14441, 'pdfsha': 'f2e57602c6e6d07d86559554c02da09237f96654', 'pdfurls': ['https://export.arxiv.org/pdf/2303.15316v1.pdf'], 'title': ['XVoxel-Based Parametric Design Optimization of Feature Models', 'XVoxel-Based Parametric Design Optimization of Feature Models', 'XVoxel-Based Parametric Design Optimization of Feature Models', 'XVoxel-Based Parametric Design Optimization of Feature Models'], 'venue': []}

arxiv

Generalized q-deformed oscillators, q-Hermite polyno- mials, generalized coherent states 20 Jul 2006 I M Burban Institute for Theoretical Physics Ukrainian National Academy of Sciences Metrologichna st. 14b03143KyivUkraine Generalized q-deformed oscillators, q-Hermite polyno- mials, generalized coherent states 20 Jul 2006arXiv:math-ph/0607045v1 The aim of this paper is to study generalized q-analogs of the well-known q-deformed harmonic oscillators and to connect them with q-Hermite polynomials. We give a construction of the appropriate oscillatorlike algebras and show that corresponding Hermite polynomials are generalization of the discrete q-Hermite I and the discrete q-Hermite II polynomials. We also construct generalized coherent states of Barut-Girardello type for oscillator-like systems connected with these polynomials. Introduction The simplest deformation of the canonical commutation relations has been emerged in context of the study of the dual resonance models of the strong interaction theories [1]. More general deformaion of these relations was considered in connection with description of representations of quantum groups [2], [3]. It was introduced in order to extend the method of realization of generators of Lie algebras by creation and annihilation operators (the Jordan-Wigner construction) to the quantum case. Since then various generalized q-analogs of these deformations became the subject of mathematical and theoretical physics. Their relation to the noncommutative geometry, special functions of the q-analysis and other subjects of the mathematics became evident. A connection between harmonic oscillators and Hermite polynomials in the quantum mechanics is well-known. This connection was generalized to the q-deformed cases as well. The generalized q-deformed oscillators [4], [5], [6], [7] are related to the q-deformed Hermite polynomials in the same way as the standard quantum oscillator is connected with the classical Hermite polynomials. In the work [8] the spectra of the position Q and the momentum P operators for various qdeformed oscillators in the Fock representation has been described. There the spectral measures and the generalized eigenfunctions of these operators has been found. They are expressed in terms of certain q-Hermitian polynomials. Various orthogonal deformed q-Hermite polynomials can appear depending on the type of a deformation of the oscillator algebra. For example, in the case of Arik-Coon deformation with parameter q > 1 one gets Hermite polynomials for which orthogonal measure is known. The same problems has been studied for more general form of the operators Q and the P [9]. In this case the spectral measure and the eigenfunctions has been expressed in terms of the discrete q-Hermite polynomials. The spectral measure of the position operator of Biedenharn-McFarlane oscillator has been calculated in the case indetermine Hamburger moment problem [10]. Coherent states of the oscillator-like systems, connected with some q-Hermite polynomials have been constructed in [6], [15]. Naturally, a problem of generalization of these results to the case of other q-deformations of the harmonic oscillator algebra arises. In this paper we consider oscillator-like systems giving a description of the generalized q-deformed oscillators and connect them with the generalized discrete q-Hermite I and q-Hermite II polynomials. These systems involve, as particular cases, the known one-parameter deformations of oscillator algebras. In Section 2, we give a structure function, defining relations and the position and momentum operators Q and P of the corresponding deformed oscillator algebra. In the Fock representation the operators Q and P have a Jacobi matrix form. We investigate the self-adjointness properties of these operators. In Section 3 we give examples of the oscillator-like systems connected with discrete q-Hermite I and the generalized discrete q-Hermite I polynomials. We establish the orthogonality relations for these polynomials and, as a consequence, obtain a spectrum of the position operators of these systems. The same problem is solved in Section 4 for discrete Hermite II and generalized discrete Hermite II polynomials of the corresponding oscillator algebras. In Section 5 we study generalized coherent states of oscillator algebras corresponding to discrete q-Hermite I, generalize discrete q-Hermite I and II polynomials on the basis of the method suggested in [6], [15]. The generalized q-deformed oscillator and its Heisenberg-Weyl algebra is defined by the structure function f (n) = f n (a positive function satisfying f (0) = 0), which fix an associative algebra generated by the elements {1, a, a + , N}, satisfying the defining relations [N, a] = −a, [N, a + ] = a + ,(1) a + a = f (N), aa + = f (N + 1). (2) The structure functions f (n) characterize the deformation scheme. The Fock realization of these relations and the number of particles operator have the form a|n = f n−1 |n − 1 , a + |n = f n |n + 1 ,(3)N|n = n|n .(4) As an example we consider the special case f n = (n + 1) 1/2 to get the Heisenberg-Weyl algebra of the quantum oscillator of quantum mechanics. It is generated by the generators a, a + , N and its defining relations are [N, a] = −a, [N, a + ] = a + ,(5) [a, a + ] = 1, [a, a] = [a + , a + ] = 0. Recall that the Fock realization of this Heisenberg-Weyl algebra is a|n = √ n|n − 1 , a + |n = √ n + 1|n + 1 ,(7) N|n = a + a|n = n|n . The position Q and the momentum P operators are unbounded operators defined on a dense domain in the Hilbert space H and satisfy the famous commutation relation [Q, P ] = iI. These operators are related to the creation and annihilation operators a + and a from (7) by the formulae Q = a + + a, P = 1 i (a + − a).(9) Each of these operators have a Jacobi matrix form and due to the condition ∞ n=0 1 √ n = ∞(10) is a self-adjoint operator (Carleman's Lemma) [11]. There is a well-known the connection between the quantum-mechanical position and momentum operators Q and P and the Hermite polynomials H n (x). The spectrum of the operator Q is continuous and in order to find its generalized eigenvectors we have to describe solutions of the equation Q|x = x|x . To do this, ones consider the expansion |x = ∞ n=0 P n (x)|n of the vector |x in the Fock space H. Using the equations (7) and Q|x = x|x we obtain the following recurrence relation (n + 1) 1/2 P n+1 (x) + n 1/2 P n−1 (x) = xP n (x), n = 0, 1, . . . (11) for coefficients P n (x). This relation has the solutions P n (x) = 1 (2 n πn!) 1/2 H n (x),(12) with the initial conditions P −1 (x) = 0, P 0 (x) = 1. The polynomials H n (x) satisfy the equation xH n (x) = 1 2 H n+1 (x) + nH n−1 (x)(13) and can be written as H n (x) = [n/2] k=0 n!(−1) k k!(n − 2k)! x n−2k .(14) They can be represented by means of hypergeometric function as H n (x) = (2x) n 2 F 0 −n/2, −(n − 1)/2 − − 1 x 2 .(15) The polynomials ψ n (x) = 1 √ π2 n n! 1/2 H n (x)(16) are orthonormal with respect to the measure dω(x) = e −x 2 dx and give wave functions of the Hamiltonian H = a + a + aa + of the harmonic oscillator corresponding to the eigenvalues λ n = 2n + 1, n ≥ 0. Generalized oscillator algebras, position and momentum operators and spectrum of Hamiltonian We consider oscillator-like systems defined by the structure function f n = q α(n+1)+β/2 1 − q (l−1)(n+1) 1 − q l−1 1/2(17) of oscillator algebra (see (Sec. 3) and (Sec. 4)). For the special values of the parameters α, β, l, q this function reproduce known versions of deformations of the oscillator Heisenberg-Weyl algebra: the Arik-Coon, the Biedenharn-Macfarlane and the other ones. The defining relations for these algebras can be written as aa + − q 2α a + a = q 2α(N +1)+β q (l−1)N ,(18)aa + − q 2α+l−1 a + a = q 2α(N +1)+β ,(19) or in a more compact form aa + − q 2α a + a = q 2α(N +1)+β q ′N ,(20)aa + − q 2α q ′ a + a = q 2α(N +1)+β ,(21) where q ′ = q l−1 . The Fock realization of the operators a, a + in the Fock space H is a|n = q αn+β/2 1 − q (l−1)n 1 − q l−1 1/2 |n − 1 ,(22)a + |n = q α(n+1)+β/2 1 − q (l−1)(n+1) 1 − q l−1 1/2 |n + 1 .(23) A spectrum of the Hamiltonian H = a + a + aa + of these oscillator-like systems is discrete and is given by the expression λ n = q 2αn+β (1−q l−1 ) −1 1 − q (l−1)n +q 2α(n+1)+β (1−q l−1 ) −1 1 − q (l−1)(n+1) ,(24) where n ≥ 0. However, wave functions corresponding to these eigenvalues are defined in a more complicated way. In the orthonormal basis |n , n = 1, 2, . . . of the Hilbert space H the position and momentum operators Q and P of these oscillators systems are given by by the Jacobi matrices Q|n = f n |n + 1 + f n−1 |n − 1 ,(25)P |n = 1 i (f n |n + 1 − f n−1 |n − 1 ).(26) Unlike to the case of standard quantum oscillator, in this one the selfadjointness of the operator Q depends on the values of parameters q, α, l. Due to Theorem 1.1, Chapter VII in [11] the deficiency indices of the operator Q are (0, 0) and then its closureQ is a self-adjoint operator, or they are (1, 1) and then the operatorQ allows self-adjoint extensions. According to Theorem 1.5, Chapter VII in [11], if the function f (n) from (17) satisfy the conditions f n−1 f n+1 ≤ f 2 n , ∞ n=0 1 f n < ∞,(27) then the deficiency indices of Q are (1, 1). The first condition of (27) is reduced to the inequality q −(l−1) − q l−1 ≥ 2 what is satisfied for all positive q. The convergence of the series (27) depends on the values of parameters of the deformation. Namely, q < 1,          α < 0, l − 1 > 0, convergent, α > 0, l − 1 > 0, divergent, α + l − 1 < 0, l − 1 < 0, convergent, α + l − 1 > 0, l − 1 < 0, divergent,(28) and q > 1,          α < 0, l − 1 > 0, divergent, α > 0, l − 1 > 0, convergent, α + l − 1 < 0, l − 1 < 0, divergent, α + l − 1 > 0, l − 1 < 0, convergent.(29) In particular, this choice of the structure function unify the following cases of the q-deformed of the oscillator algebras: the Biedenharn-Macfarlane de- formation (α = 1/2, β = −1, l = −1, q < 1), and (α = −1/2, β = 1, l − 1 = 2, q > 1) [N, a] = −a, [N, a + ] = a + ,(30)aa + − qa + a = q −N , aa + − q −1 a + a = q N ,(31) and its symmetric generalization (α = 1/2, β = −1, l ∈ R), [N, a] = −a, [N, a + ] = a + ,(32)aa + − qa + a = q lN , aa + − q l a + a = q N ,(33) the deformation associated with the discrete q-Hermite I polynomials (α = 1/2, β = −1, l = 2, q < 1) [N, a] = −a, [N, a + ] = a + ,(34)aa + − qa + a = q 2N , aa + − q 2 a + a = q N(35) and the deformation associated with the discrete q-Hermite II polynomials (α = −1, β = 2, l = 2, q < 1) [N, a] = −a, [N, a + ] = a + ,(36)aa + − q −1 a + a = q −2N , aa + − q −2 a + a = q −N .(37) 3. Generalized q-deformed oscillator-like systems and generalized discrete q-Hermite I polynomials First of all we consider a deformed oscillator in the case when the parameters in (17) take the values α = 1/2, β = −1, l = 2, 0 < q < 1. The structure function (17) in this case is written as f n = q (n+1)/2−1/2 (1 − q) −1/2 (1 − q n+1 ) 1/2 .(38) The Fock representation of the a and a + operators for (34), (35) are a|n = ( 1 1 − q ) 1/2 q (n−1)/2 (1 − q n ) 1/2 |n − 1 ,(39)a + |n = ( 1 1 − q ) 1/2 q n/2 (1 − q n+1 ) 1/2 |n + 1 .(40) The deformed canonical commutation relations take the form (34), (35) or [a, a + ] = q N 1 − q N +1 1 − q − q N −1 1 − q N 1 − q .(41) Recall that the Hamiltonian of this oscillator H = aa + +a + a has a discrete spectrum H|n = λ n |n , where λ n = q n (1 − q) −1 (1 − q n+1 ) + q n−1 (1 − q) −1 (1 − q n ), n ≥ 0.(42) To find the wave functions corresponding to these eigenvalues we proceed as in the case of the standard quantum oscillator (see Introduction). The position operator Q and the momentum operator P are given in the basis |n , n = 0, 1, . . . of the Hilbert space H by Jacobi matrices. The generalized eigenvectors {|x } of the operator Q, Q|x = x|x , form a continuous basis of the Hilbert space H and coefficients P n (x; q)|n , satisfy the recurrence relation xP (0) n (x; q) = 1 1 − q 1/2 q n/2 ×(1 − q n+1 ) 1/2 P (0) n+1 (x; q) + 1 1 − q 1/2 q (n−1)/2 (1 − q n ) 1/2 P (0) n−1 (x; q). (43) If we do the rescaling of variables y = (1 − q) 1/2 x and denote ψ (0) n (x; q) = P (0) n ((1 − q) −1/2 x; q) , then the previous relation is reduced to xψ (0) n (x; q) = q n/2 (1 − q n+1 ) 1/2 ψ (0) n+1 (x; q) + q (n−1)/2 (1 − q n ) 1/2 ψ (0) n−1 (x; q).(44) After replacement ψ (0) n (x; q) = q −n(n−1)/4 (q; q) 1/2 n h (0) n (x; q),(45) we obtain the recurrence relation for the discrete q-Hermite I polynomials [12] x h (0) n (x; q) = h (0) n+1 (x; q) + q n−1 (1 − q n )h (0) n−1 (x; q).(46) Together with the initial condition h (0) 0 (x; q) = 1 it defines the discrete q-Hermitian I polynomials [12], [13] represented as h (0) n (x; q) = [n/2] k=0 (q; q) n (q 2 ; q 2 ) k (q; q) n−2k (−1) k q k(k−1) x n−2k .(47) They can be written by means of the basic hypergeometric function as h (0) n (x; q) = x n 2 φ 0 q −n , q −n+1 − q 2 ; q 2n−1 x 2 .(48) Now, the solution of the equation (43) can be represented by the expression P (0) n (x; q) = q −n(n−1)/4 (q; q) 1/2 n h (0) n ( 1 − qx; q).(49) It follows from [12], [13] that these polynomials are orthogonal with respect to the discrete measure d ω (0) (x) = 1 2 (q; q 2 ) ∞ δ(x − q 0 √ 1 − q ) dx + k>0 √ 1 − q|x| 2 (q 2 (1 − q)x 2 , q; q 2 ) ∞ (q; q) ∞ δ(x − q k √ 1 − q ) dx + k>0 √ 1 − q|x| 2 (q 2 (1 − q)x 2 , q; q 2 ) ∞ (q; q) ∞ δ(x + q k √ 1 − q ) dx.(50) and the orthogonality relation is δ mn (q; q) n = 1 2 (q; q 2 ) ∞ P (0) m (1; q)P (0) n (1; q) + ∞ k=0 {P (0) m (q k ; q)P (0) n (q k ; q)+P (0) m (−q k ; q)P (0) n (−q k ; q)} q k 2 (q 2k+2 , q; q 2 ) ∞ (q; q 2 ) ∞ (q 2 ; q 2 ) ∞ . (51) It follows that spectrum of the position operator Q is Sp Q = ±1 √ 1 − q , ±q √ 1 − q , . . . , ±q k √ 1 − q , . . . ; k ≥ 0 .(52) The extension of this method for the generalized oscillator (17), (20), (21), determined by the formulas (22) and (23) for q < 1 gives the recurrence relations xP n (x; q) = 1 1 − q ′ 1/2 q α(n+1)+β/2 (1 − q ′(n+1) ) 1/2 P n+1 (x; q) + 1 1 − q ′ ) 1/2 q αn+β/2 (1 − q ′n ) 1/2 P n−1 (x; q).(53) If we rescale the variables y = (1−q ′ ) 1/2 x, then P n (x; q) = ψ n ((1−q ′ ) 1/2 x; q) yields xψ n (x; q) = q α(n+1)+β/2 (1 − q ′n ) 1/2 ψ n+1 (x; q) + q αn+β/2 (1 − q ′n ) 1/2 ψ n−1 (x; q). (54) Representing the function ψ n (x; q) as ψ n (x; q) = q −αn 2 /2 q (α+β)n/2 (q ′ ; q ′ ) 1/2 n h n (x; q)(55) we obtain from (54) the recurrent relation for the generalized q-Hermite polynomials h n (x; q) : xh n (x; q) = h n+1 (x; q) + q 2αn+β (1 − q ′n )h n−1 (x; q).(56) This equation can be solved by means of the anzatz h n (x; q) = [n/2] k=0 (q ′ ; q ′ ) n ((a n , c n ); (1, q ′d )) k (q ′ ; q ′ ) n−2k (−1) k q ′k(k−1) x n−2k ,(57) where we use the notation [14] ((a, c); (p, q)) k = 1, if k = 0; (a − c)(ap − cq) . . . (ap k−1 − cq k−1 ), otherwise,(58) q ′ = q l−1 , and a n , c n , d are unknown quantities. It is easy to see that this anzatz leads the relation (56) to the identity 1 − q ′n+1 − q 2αn+β (a n − c n q ′d(k−1) )q ′−2(k−1) = 1 − q ′n−2k+1 .(59) This identity admits the solutions a n = q −2αn−β q ′n−1 , c n = q −2αn−β q ′n+1 , d = 2 and an easy calculation gives ((a n , c n ); (1, q ′d )) k = q −k(2αn+β) q ′k(n−1) (q ′2 ; q ′2 ) k . The resulting expressions for the generalized q-Hermite polynomials can be written as the polynomial of degree n in x h n (x; q) = [n/2] k=0 (q ′ ; q ′ ) n (q ′2 ; q ′2 )) k (q ′ ; q ′ ) n−2k (−1) k q (2αn+β)k q ′k(k−n) x n−2k .(61) They can be represented in terms of the basic hypergeometric function as h n (x; q) = x n 2 φ 0 q ′−n , q ′−n+1 − q ′2 ; q 2αn+β q ′n x 2 .(62) It is easy to see that for α = 1 2 , β = −1, l = 2 the solution (61) of (56) reduces to the solution (47) of (46). At last, the solutions P n (x; q) of the equations (53) with the initial conditions P −1 (x; q) = 0, P 0 (x; q) = 1 can be written as polynomials of degree n in x : P n (x; q) = q −αn 2 /2 q α+β 2 n (q ′ ; q ′ ) 1/2 n h n ( 1 − q ′ x; q).(63) Now we restrict ourselves by the condition α = (l − 1)/2 in (63). Then P n (x; q) = q −αn(n−1)/2 (q ′ ; q ′ ) 1/2 n h 0 n (q −(2α+β)/2 1 − q ′ x; q ′ ).(64) These polynomials are orthogonal with respect to the discrete measure dω(x) = q −(2α+β) √ 1 − q ′ 2 (q ′ ; q ′2 ) ∞ δ(x − q ′0 q −(2α+β) √ 1 − q ′ )dx + k>0 q −(2α+β)/2 √ 1 − q ′ |x| 2 (q −(2α+β) (q ′2 (1 − q ′ )x 2 , q ′ ; q 2 ) ∞ (q ′ ; q ′ ) ∞ δ(x− q ′k q −(2α+β)/2 √ 1 − q ′ )dx + k>0 q −(2α+β)/2 √ 1 − q ′ |x| 2 (q −(2α+β) (q ′2 (1 − q ′ )x 2 , q ′ ; q 2 ) ∞ (q ′ ; q ′ ) ∞ δ(x+ q ′k q −(2α+β)/2 √ 1 − q ′ )dx. (65) The orthogonality relation has the form δ mn (q ′ ; q ′ ) n = 1 2 (q ′ ; q ′ ) ∞ (q ′2 ; q ′2 ) ∞ P m (1; q)P n (1; q) + k>0 {P m (q ′k ; q)P n (q ′k ; q)+P m (−q ′k ; q)P n (−q ′k ; q)} q ′k 2 (q ′2k+2 , q ′ ; q ′2 ) ∞ (q ′ ; q ′2 ) ∞ (q ′2 ; q ′2 ) ∞ . (66) From this it follows that spectrum of the position operator Q is Sp Q = ±q (2α+β)/2 √ 1 − q ′ , ±q (2α+β)/2 q ′ √ 1 − q ′ , . . . , ±q (2α+β)/2 q ′k √ 1 − q ′ , . . . ; k ≥ 0 . (67) Generalized q-deformed oscillator-like systems and generalized discrete q-Hermite II polynomials If we fix in (17) the values of the parameters α = −1, β = 2, l = 2, 0 < q < 1, the structure function f n is reduced to the form f n = q −(n+1)+1 (1 − q) −1/2 (1 − q n+1 ) 1/2 .(68) The Fock representation of the creation and the annihilation operators of the relations (36), (37) is given by a|n = q 1 − q 1/2 q −n+1/2 (1 − q n ) 1/2 |n − 1 , a + |n = q 1 − q 1/2 q −n−1/2 (1 − q n+1 ) 1/2 |n + 1 .(69) It follows that aa + |n = q −2n 1 − q n+1 1 − q |n , a + a|n = q −2n+2 1 − q n 1 − q |n ,(70) and commutation relation (37) can be written in the symbolic form [a, a + ] = q −2N 1 − q N +1 1 − q − q −2N +2 1 − q N 1 − q .(71) The Hamiltonian H of this oscillator-like system has the discrete spectrum H|n = λ n |n , where λ n = q −2n (1 − q) −1 (1 − q n+1 ) + q 2−2n (1 − q) −1 (1 − q n ),q 1 − q 1/2 q −(n+1)+1/2 ×(1 − q n+1 ) 1/2P 0 n+1 (x; q) + q 1 − q 1/2 q −n+1/2 (1 − q n )P 0 n−1 (x; q). (73) Introducing the rescaling y = q −1/2 (1 −q) 1/2 x and the functionψ n (x; q) = P (0) n (q 1/2 (1 − q) −1/2 x; q) we obtain the equation xψ (0) n (x; q) = q −(n+1)+1/2 ×(1 − q n+1 ) 1/2ψ (0) n+1 (x; q) + q −n+1/2 (1 − q n ) 1/2ψ (0) n−1 (x; q).(74) After the replacementψ (0) n (x; q) = q n 2 /2 (q; q) 1/2 nh (0) n (x; q),(75) we obtain the recurrence relation for the discrete q-Hermite II polynomials [12] xh (0) n (x; q) =h (0) n+1 (x; q) + q −2n+1 (1 − q n )h (0) n−1 (x; q)(76) which together with the initial conditionh (0) 0 (x; q) = 1 define the discrete q-Hermite II polynomials h (0) n (x; q) = [n/2] k=0 (q; q) n (q 2 ; q 2 ) k (q; q) n−2k (−1) k q 2k(k−n)+k x n−2k .(77) These polynomials can be represented in terms of the basic hypergeometric function:h (0) n (x; q) = x n 2 φ 1 q −n q −n+1 0 q 2 ; − q 2 x 2 .(78) The solution of the equations (73) with the initial conditionsP (0) −1 (x; q) = 0,P(0) 0 (x; q) = 1 can be given in the form P (0) n (x; q) = q n 2 /2 (q; q) 1/2 nh (0) n (q −1/2 1 − qx; q).(79) It follows from [13] that these polynomials are orthogonal with respect to the discrete measure dω (0) (x) = ∞ k=−∞ c −1 q −1/2 1 − qw(q −1/2 1 − qx; q) x δ(x − cq k q −1/2 √ 1 − q ) dx − ∞ k=−∞ c −1 q −1/2 1 − qw(−q −1/2 1 − qx; q) x δ(x + cq k q −1/2 √ 1 − q ) dx. (80) The orthogonality relation for these polynomials has the form ∞ k=−∞ {P (0) m (cq k ; q)P (0) n (cq k ; q) +P (0) m (−cq k ; q)P (0) n (−cq k ; q)}w(cq k ; q)q k = 2 (q 2 , −c 2 q, −c −2 q; q 2 ) ∞ (q, −c 2 , −c −2 q 2 ; q 2 ) ∞ δ mn , c > 0,(81) where w(x; q) = 1/(−x 2 ; q 2 ) ∞ . It follows that spectrum of the position operator Q is Sp Q = ±c q −1/2 √ 1 − q , ±cq q −1/2 √ 1 − q , . . . , ±cq k q −1/2 √ 1 − q , . . . ; k ≥ 0 . (82) A connection of the discrete q-Hermite I polynomials and the discrete q-Hermite II polynomials are given by q → 1/q. Indeed, we have the relations (1/q ′ ; 1/q ′ ) n q ′−k(k−1) (1/q ′ ; 1/q ′ ) n−2k = (q ′ ; q ′ ) n (q ′ ; q ′ ) n−2k q ′k 2 −2kn ,(83)(1/q 2 ; 1/q 2 ) k = (−1) k (q 2 ; q 2 ) k q −k(k+1)(84) leading to the identity [13] h (0) n (ix; q −1 ) = i nh(0) n (x; q).(85) This identity reflects the transition q → q −1 from the oscillator (34), (35) to the oscillator (36), (37). Now we consider the generalized oscillator (17), (20), (21) represented by operators (22), (23), (29) for q < 1. Then instead (53) we have the equality xP n (x; q) = q β/2 1 − q ′ 1/2 q α(n+1)+β/4 ×(1 − q ′n+1 ) 1/2P n+1 (x; q) + q β/2 1 − q ′ 1/2 q αn+β/4 (1 − q ′n ) 1/2P n−1 (x; q), (86) or xψ n = q α(n+1)+β/4 (1 − q ′n+1 ) 1/2ψ n+1 (x; q) + q αn+β/4 (1 − q ′n ) 1/2ψ n−1 (x; q),(87) whereψ n (x; q) =P n ((q β/4 (1−q ′ )) −1/2 x; q). Representing the functionψ n (x; q) asψ n (x; q) = q −αn 2 /2 q (2α+β)n/4 (q ′ ; q ′ ) 1/2 nh n (x; q)(88) we obtain the recurrence relatioñ h n+1 (x; q) + q 2αn+β/2 (1 − q ′n )h n−1 (x; q) = xh n (x; q). The solution of this equation can be obtained by means of the anzatz h n (x; q) = [n/2] k=0 (q ′ ; q ′ ) n ((a n , c n ); (1, q ′d )) k (q ′ ; q ′ ) n−2k (−1) k q ′k(2k−2n+1) x n−2k ,(90) which generalizes (77). It is easy to see that it reduces the relations (89) to the identity (1 − q ′n−2k+1 )q ′2k = 1 − q ′n+1 − q 2αn+β/2 (a n − c n q ′d(k−1) )q −(2k−2n−1) q ′2(n−1) .(91) We obtain the solution a n = q −2αn−β/2 q ′−2n+1 , c n = q −2αn−β/2 q ′−2n+3 , d = 2.(92) An easy calculation gives ((a n , c n ); (1, q ′d )) k = q −k(2αn+β/2) q ′−k(2n−1) (q ′2 ; q ′2 ) k . The resulting expressioñ h n (x; q) = [n/2] k=0 (q ′ ; q ′ ) n (q ′2 ; q ′2 ) k (q ′ ; q ′ ) n−2k (−1) k q (2αn+β/2)k q ′2k 2 x n−2k(93) defines a generalized of the q-Hermite polynomials which can be written in terms of the basic hypergeometric function, h n (x; q) = x n 2 φ 1 q ′−n , q ′−n+1 0 q ′2 ; − q 2αn+β/2 q ′2n+1 x 2 .(94) It is easy to see that for the special values α = −1, β = 2, l = 2 the solution (93) is reduced to the solution (77) of the relation (76). Finally, the solutionP n (x; q) with the initial conditionsP −1 (x; q) = 0,P 0 (x; q) = 1 of the equation (86) are given by the formulã P n (x; q) = q −αn 2 /2 q (2α+β)n/4 (q ′ ; q ′ ) 1/2 nh n (q −β/4 1 − q ′ x; q).(95) From now on we restrict ourselves by the condition α = −(l − 1) in (95). ThenP n (x; q) = q −αn 2 /2 (q ′ ; q ′ ) 1/2 nh 0 n (q −(α+β)/2 1 − q ′ x; q ′ ).(96) These polynomials are orthogonal with respect to the discrete measure dω(x) = ∞ k=−∞ c −1 q −(α+β)/2 1 − q ′ w(q −(α+β)/2 1 − q ′ x; q ′ ) x δ(x− cq ′k q −(α+β)/2 √ 1 − q ′ ) dx − ∞ k=−∞ c −1 q −(α+β)/2 1 − q ′ w(q −(α+β)/2) 1 − q ′ x; q ′ ) x δ(x+ cq ′k q −(α+β)/2 √ 1 − q ′ ) dx (97) and the orthogonality relation is ∞ k=−∞ {P m (cq ′k ; q)P n (cq ′k ; q) +P m (−cq ′k ; q)P n (−cq ′k ; q)}w(cq ′k ; q)q ′k = 2 (q ′2 , −c 2 q ′ , −c −2 q ′ ; q ′2 ) ∞ (q ′ ; q ′ ) n q −αn 2 (q ′ , −c 2 , −c −2 q ′2 ; q ′2 ) ∞ q ′n 2 δ mn , c > 0,(98) where w(x; q) = 1/(−x 2 ; q 2 ) ∞ . It follows that spectrum of the position operator Q is Sp Q = ±c q −(α+β)/2 √ 1 − q ′ , ±cq ′ q −(α+β)/2 √ 1 − q ′ , . . . , ±cq ′k q −(α+β)/2 √ 1 − q ′ , . . . ; k ≥ 0 . (99) A connection of the generalized q-Hermitian I and the generalized q-Hermitian II polynomials is not evident at all as in non -generalized case. Unfortunately the change q → 1/q does not lead to the analogous of (85) for the generalized q-Hermitian I and the generalized q-Hermitian II polynomials (61) and (93). It is evident from (22) and (23). Instead of this in this case we have h n (x; 1/q) = [n/2] k=0 (q ′ ; q ′ ) n (q ′2 ; q ′2 ) k (q ′ ; q ′ ) n−2k (−1) k q −k(2αn+β) q ′k(2k−n) x n−2k . (100) It is easy to see that for α = 1 2 , β = −1, l − 1 = 1 this relation gives identity (85). Barut -Girardello coherent states of oscillators associated with generalized discrete q-Hermite I and II polynomials It is known that the coherent states in the ordinary Lie algebras are very useful for studying the representation theory. The generalized coherent states are very useful in the study of representation of quantum group and physics, in particular, in quantum optics. Barut-Girardello type coherent states of oscillator algebras have been studied for oscillator-like system connected with some orthogonal polynomials. The family coherent states associated with discrete q-Hermite polynomials of type II have been described in [6] and [15]. In this section we give solution the same problem for discrete q-Hermite I and generalized discrete q-Hermite I and II polynomials. First of all we prove the formula for a generating function of q-Hermite I polynomials connected with the appropriate q-oscillator ∞ n=0 (−1) n q −n(n−1)/2 (q; q) n h (0) n (x; q)t n = (qt; 1/q) ∞ 1 φ 1 x tq 1/q; −tq . (101) Let us denote the left hand side of this identity by Φ(x, q, t) = ∞ n=0 (−1) n q −n(n−1)/2 (q; q) n h (0) n (x, q)t n .(102) Then Φ(ix, 1/q, t) = ∞ n=0 (−1) n q n(n−1) (q; q) nh (0) (x; q)( it q ) n(103) Using formula (3.29.12) of ref. [13] ∞ n=0 (−1) n q n(n−1) (q; q) nh (0) (x; q)t n = (−it; q) ∞ 1 φ 1 ix −it q; it(104) we obtain Φ(ix, 1/q, t) = (−t/q; q) ∞ 1 φ 1 ix t/q q; −t/q(105) from which easy follows (101). The Barut-Girardello coherent states of a oscillator (1), (2) in the Fock representation space H are defined as eigenvectors of annihilation operator a : a |z = z |z , z ∈ C, given by the formula |z = N −1 ∞ n=0 z n f n−1 ! |n ,(107) where N is normalized factor. By definition the basis vectors |n of H are taken as polynomials ψ (0) n (x; q) of (45). In the case of the oscillator corresponding q-Hermite I polynomials we have f n−1 = 1/(1 − q))q n−1 (1 − q n ). It follows f n−1 ! = 1/(1 − q) n q n(n−1)/2 (q; q) n (108) and coherent state (107) can be written as |z = N −1 (|z| 2 ) ∞ n=0 ( √ 1 − qz) n q n(n−1)/2 (q; q) n q −n(n−1)/4 (q; q) 1/2 n h (0) n (x; q) (109) = N −1 (|z| 2 ) ∞ n=0 (−1) n q −n(n−1)/2 (q; q) n h (0) n (x, q)(− 1 − qz) n(110) (take into account of (101)) = N −1 (|z| 2 )(q(− 1 − qz); 1/q) ∞ 1 φ 1 x (− √ 1 − qz)q 1/q; q 1 − qz . (111) Easy calculation gives the normalized factor N 2 (|z| 2 ) = 2 φ 0 0, 0 − q; (1 − q)z .(112) The overlapping of two coherent states is z 1 |z 2 = 2 φ 0 0, 0 − q; (1 − q)z 1 z 2 .(113) The terminal expression for coherent state |z of the q-oscillator corresponding discrete q-Hermite I polynomials has the form |z = 2 φ 0 0, 0 − q; (1 − q)z 1/2 ×(− 1 − qz); 1/q) ∞ 1 φ 1 x −q √ 1 − qz 1/q; q 1 − qz .(114) The family of coherent states associated with oscillator-like system corresponding to generalized discrete q-Hermite I polynomials (61) can be obtained the same method. In this case f n−1 ! = 1/(1 − q ′ ) n q αn 2 q (α+β)n (q ′ ; q ′ ) n(115) and basis vectors |n are taken as polynomials ψ n (x; q) of (55). This leads (107) to |z = N −1 (|z| 2 ) ∞ n=0 ( √ 1 − q ′ z) n q αn 2 q (α+β)n (q ′ ; q ′ ) n q −αn 2 /2 h n (x; q) q (α+β)n/2 (q ′ ; q ′ ) 1/2 n , or |z = N −1 (|z| 2 ) ∞ n=0 (−1) n q −αn(n−1) (q; q) n h n (x; q) − √ 1 − q ′ q 2α+β z n(116) The generating function for generalized discrete q-Hermite I polynomials (the extension of (101)) is given by the formula ∞ n=0 (−1) n q −(2α+β)n q −αn(n−1) (q; q) n h n (x, q)t n = (tq ′ ; 1/q ′ ) ∞ 1 φ 1 q −(2αn+β)/2 q ′(n−1)/2 x q ′ t 1/q ′ ; −tq ′ . Comparing the expressions (116) and (117) we obtain |z = N −1 (|z| 2 )(−q −(2α+β) q ′ 1 − q ′ z; 1/q ′ ) ∞ × 1 φ 1 q −(2αn+β)/2 q ′(n−1)/2 x −q −(2α+β) q ′ √ 1 − q ′ z 1/q ′ ; q −2α+β q ′ 1 − q ′ z .(118) The short calculation of a normalizing factor of the coherent state (118) gives N 2 (|z| 2 ) = ∞ n=0 √ 1 − q ′ q 2α+β |z| 2 n q −αn(n−1) (q ′ ; q ′ ) n . The description of the Barut-Girardello type coherent states for the oscillatorlike systems, connected with discrete q-Hermite II polynomials has been done in [6]. Therefore in the further we consider the coherent states, connected only with oscillator-like systems connected with generalized discrete q-Hermite II. They are given by the expression (107), where f n−1 ! = q βn /(1 − q ′ ) n q αn(n+1) (q ′ ; q ′ ) n (120) and basis vectors |n are given byψ n (x, q) of (75), namely, |z = N −1 z n q βn /(1 − q ′ ) n q αn(n+1) (q ′ ; q ′ ) n q −αn 2 /2 q (2α+β)n/4) (q ′ ; q ′ ) 1/2 nh n (x; q), or |z = N −1 (|z| 2 ) ∞ n=0 (−1) n q −αn 2 (q; q) n h n (x; q) −q −β/4 q −(2α+β)/2 1 − q ′ z n . (122) The generation function for polynomialsh n (x; q) is defined by ∞ n=0 (−1) n q −αn 2 (q; q) n h n (x; q) t n = (−iq β/4 q ′1/2 t; q ′ ) ∞ 1 φ 1 iq −(2αn+β)/2 q ′(2n−1)/2 x −iq β/4 q ′β/4 t q ′ ; itq β/4 q ′1/2 (123) (the extension of the formula (3.29.12) of ref. [13]). Using this identity the coherent state |z of (122) can represented by expression |z = N −1 (|z| 2 ) ×(i q ′ (1 − q ′ ) q (2α+β)/2 z; q ′ ) ∞ 1 φ 1 iq −(2αn+β)/2 q ′(2n−1)/2 x i q ′ (1−q ′ ) q (2α+β)/2 z q ′ ; −i q ′ (1 − q ′ ) q (2α+β)/2 z . (124) At last, the normalizing factor of the generalized coherent state (124) can be written as N 2 (|z| 2 ) = ∞ n=0 q −αn(n+1) (q ′ ; q ′ ) n 1 − q ′ q β |z| 2 n .(125) We have not established completeness (over-completeness) of the given set of generalized coherent states. It will be done in a forthcoming paper. n ≥ 0. (72) As in the previous section the position and momentum operators Q and P in the basis |n of the Hilbert space H are represented by Jacobi matrices. The coefficientsP 0 n (x; q) of the transition |x = ∞ n=0P 0 n (x; q)|x from the basis {|n } to the basis {|x }, , Q|x = x|x , satisfy the relations xP 0 n (x; q) = Acknowledgements I would like to thank A.U. Klimyk for many useful discussions, valuable suggestions and observations. This research was partially supported by Grant 10.01/015 of the State Foundation of Fundamental Research of Ukraine. . A Arik, D D Coon, Lam, J. Math. Phys. 161776Arik A., Coon D. D. and Lam, J. Math. Phys. 16 (1975), 1776. The quantum group SU q (2) and a q analog of the boson operator. L C Biedenharn, J. Phys. A. 22837Biedenharn, L. C., The quantum group SU q (2) and a q analog of the boson operator, J. Phys. A 22 (1989), L837. On q analogs of the quantum harmonic oscillator and the quantum group SU q (2). A J Macfarlane, J. Phys. A. 224581Macfarlane A. J., On q analogs of the quantum harmonic oscillator and the quantum group SU q (2), J. Phys. A 22 (1989) L4581. Deformed oscillators and their applications. E V Damaskinsky, P P Kulish, Zap. Nauchn. Sem. LOMI. 18937Damaskinsky E. V., Kulish P. P., Deformed oscillators and their ap- plications, Zap. Nauchn. Sem. LOMI 189, (1991) 37 . E V Damaskinsky, P P Kulish, q-Hermite polynomials and qoscillators. 81Damaskinsky E. V., Kulish P. P., q-Hermite polynomials and q- oscillators, Zap. Nauch. Sem. POMI 199 1992, 81 . V V Borzov, E V Damaskinsky, arXiv:math.QA/0307356Generalized coherent states for q-oscillator connected with q-Hermite polynomials. Borzov V. V., Damaskinsky E. V., Generalized coherent states for q-oscillator connected with q-Hermite polynomials, arXiv:math. QA/0307356. V V Borzov, E V Damaskinsky, S B Yegorov, arXiv:q-alg/9509022Some remarks on the representations of the generalized deformed oscillator algebra. Borzov V. V., Damaskinsky E. V., Yegorov S. B., Some remarks on the representations of the generalized deformed oscillator algebra, arXiv:q-alg/9509022. On spectral properties of q-oscillator operators. I M Burban, A U Klimyk, Lett. Math. Phys. 2913Burban I. M., Klimyk A. U., On spectral properties of q-oscillator operators, Lett. Math. Phys. 29 (1991), 13. On position and momentum operators in the q-oscillator algebra. W.-Sang Chung, A U Klimyk, J. Math. Phys. 372917Chung W.-Sang, Klimyk A. U., On position and momentum operators in the q-oscillator algebra, J. Math. Phys. 37(2) (1992) 917. On position operator spectral measure for deformedin the case of indetermined Hamburger moment problem. V V Borzov, E V Damaskinsky, P P Kulish, Reviews in Math. Phys. 12691Borzov V.V., Damaskinsky E.V., Kulish P.P., On position operator spectral measure for deformedin the case of indetermined Hamburger moment problem, Reviews in Math. Phys. 12 (2000), 691. Expansionin eigenfunctions of self-adjoint operators. Ju M Berezanskii, American Mathematical SocietyProvidence, R. IBerezanskii, Ju. M., Expansionin eigenfunctions of self-adjoint oper- ators, (American Mathematical Society, Providence, R. I. 1968). . G Gasper, M Rahman, Basic Hypergeometric Series. Cambridge University PressGasper G., Rahman M., Basic Hypergeometric Series, (Cambridge University Press, Cambridge, 1996). The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue. R Koekoek, R F Swarttouw, arXiv:math.CA/9602214Delft University of TechnologyReportKoekoek R., Swarttouw R. F., The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, (Report no.94 -05, Delft University of Technology, 1994), [arXiv: math. CA/9602214]. Two-parameter quantum algebras, twinbasic numbers, and associated generalized hypergeometric series. R Jagannathan, K Rao, arXiv:math.NT/0602613Jagannathan, R., Rao K., Two-parameter quantum algebras, twin- basic numbers, and associated generalized hypergeometric series, arXiv: math. NT/0602613. Generalized coherent states for q-oscillator connected with discrete q-Hermite polynomials. V V Borzov, E V Damaskinsky, arXiv:math.quant-ph/0407252Borzov V. V., Damaskinsky E.V., Generalized coherent states for q-oscillator connected with discrete q-Hermite polynomials, arXiv: math. quant-ph/0407252.

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{'abstract': 'The aim of this paper is to study generalized q-analogs of the well-known q-deformed harmonic oscillators and to connect them with q-Hermite polynomials. We give a construction of the appropriate oscillatorlike algebras and show that corresponding Hermite polynomials are generalization of the discrete q-Hermite I and the discrete q-Hermite II polynomials. We also construct generalized coherent states of Barut-Girardello type for oscillator-like systems connected with these polynomials.', 'arxivid': 'math-ph/0607045', 'author': ['I M Burban \nInstitute for Theoretical Physics\nUkrainian National Academy of Sciences\nMetrologichna st. 14b03143KyivUkraine\n'], 'authoraffiliation': ['Institute for Theoretical Physics\nUkrainian National Academy of Sciences\nMetrologichna st. 14b03143KyivUkraine'], 'corpusid': 15472375, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 14432, 'n_tokens_neox': 12965, 'n_words': 6675, 'pdfsha': '90d716987d98aa47780c8b1ddb21ad3f0feccf57', 'pdfurls': ['https://export.arxiv.org/pdf/math-ph/0607045v1.pdf'], 'title': ['Generalized q-deformed oscillators, q-Hermite polyno- mials, generalized coherent states', 'Generalized q-deformed oscillators, q-Hermite polyno- mials, generalized coherent states'], 'venue': []}

arxiv

Relational Context Learning for Human-Object Interaction Detection Sanghyun Kim sanghyun.kim@postech.ac.kr Pohang University of Science and Technology (POSTECH) South Korea Deunsol Jung deunsol.jung@postech.ac.kr Pohang University of Science and Technology (POSTECH) South Korea Minsu Cho mscho@postech.ac.kr Pohang University of Science and Technology (POSTECH) South Korea Relational Context Learning for Human-Object Interaction Detection Recent state-of-the-art methods for HOI detection typically build on transformer architectures with two decoder branches, one for human-object pair detection and the other for interaction classification. Such disentangled transformers, however, may suffer from insufficient context exchange between the branches and lead to a lack of context information for relational reasoning, which is critical in discovering HOI instances. In this work, we propose the multiplex relation network (MUREN) that performs rich context exchange between three decoder branches using unary, pairwise, and ternary relations of human, object, and interaction tokens. The proposed method learns comprehensive relational contexts for discovering HOI instances, achieving state-of-the-art performance on two standard benchmarks for HOI detection, HICO-DET and V-COCO. Introduction The task of Human-Object Interaction (HOI) detection is to discover the instances of ⟨human, object, interaction⟩ from a given image, which reveal semantic structures of human activities in the image. The results can be useful for a wide range of computer vision problems such as human action recognition [1,25,42], image retrieval [9,33,37], and image captioning [12,34,36] where a comprehensive visual understanding of the relationships between humans and objects is required for high-level reasoning. With the recent success of transformer networks [31] in object detection [2,45], transformer-based HOI detection methods [4,15,16,29,38,44,46] have been actively developed to become a dominant base architecture for the task. Existing transformer-based methods for HOI detection can be roughly divided into two types: single-branch and twobranch. The single-branch methods [16,29,46] update a token set through a single transformer decoder and detect HOI instances using the subsequent FFNs directly. As a single transformer decoder is responsible for all sub-tasks (i.e., < human, bicycle, riding > Ternary Unary Pairwise Figure 1. The illustration of relation context information in an HOI instance. We define three types of relation context information in an HOI instance: unary, pairwise, and ternary relation contexts. Each relation context provides useful information for detecting an HOI instance. For example, in our method, the unary context about an interaction (green) helps to infer that a human (yellow) and an object (red) are associated with the interaction, and vice versa. Our method utilizes the multiplex relation context consisting of the three relation contexts to perform context exchange for relational reasoning. human detection, object detection, and interaction classification), they are limited in adapting to the different subtasks with multi-task learning, simultaneously [38]. To resolve the issue, the two-branch methods [4,15,38,40,44] adopt two separated transformer decoder branches where one detects human-object pairs from a human-object token set while the other classifies interaction classes between human-object pairs from an interaction token set. However, the insufficient context exchange between the branches prevents the two-branch methods [15,38,40] from learning relational contexts, which plays a crucial role in identifying HOI instances. Although some methods [4,44] tackle this issue with additional context exchange, they are limited to propagating human-object context to interaction context. To address the problem, we introduce the MUtiplex RElation Network (MUREN) that performs rich context exchange using unary, pairwise, and ternary relations of human, object, and interaction tokens for relational reasoning. As illustrated in Figure 1, we define three types of relation context information in an HOI instance: unary, pairwise, and ternary, each of which provides useful information to discover HOI instances. The ternary relation context gives holistic information about the HOI instance while the unary and pairwise relation contexts provide more fine-grained information about the HOI instance. For example, as shown in Figure 1, the unary context about an interaction (e.g., 'riding') helps to infer which pair of a human and an object is associated with the interaction in a given image, and the pairwise context between a human and an interaction (e.g., 'human' and 'riding') helps to detect an object (e.g., 'bicycle'). Motivated by this, our multiplex relation embedding module constructs the context information that consists of the three relation contexts, thus effectively exploiting their benefits for relational reasoning. Since each sub-task requires different context information for relational reasoning, our attentive fusion module selects requisite context information for each sub-task from multiplex relation context and propagates the selected context information for context exchange between the branches. Unlike previous methods [4,15,38,44], we adopt three decoder branches which are responsible for human detection, object detection, and interaction classification, respectively. Therefore, the proposed method learns discriminative representation for each sub-task. We evaluate MUREN on two public benchmarks, HICO-DET [3] and V-COCO [10], showing that MUREN achieves state-of-the-art performance on two benchmarks. The ablation study demonstrates the effectiveness of the multiplex relation embedding module and the attentive fusion module. Our contribution can be summarized as follows: • We propose multiplex relation embedding module for HOI detection, which generates context information using unary, pairwise, and ternary relations in an HOI instance. • We propose the attentive fusion module that effectively propagates requisite context information for context exchange. • We design a three-branch architecture to learn more discriminative features for sub-tasks, i.e., human detection, object detection, and interaction classification. • Our proposed method, dubbed MUREN, outperforms state-of-the-art methods on HICO-DET and V-COCO benchmarks. Related Work CNN-based HOI Methods. Previous CNN-based HOI methods can be categorized into two groups: two-stage methods and one-stage methods. Two-stage HOI methods [7,8,13,18,19,26,30,32,39] first detect the human and the object instances using an offthe-shelf detector (e.g., Faster R-CNN [27]) and predict the interaction between all possible pairs of a human and an object. To create discriminative instance features for HOI detection, they additionally utilize spatial features [8,19,35], linguistic features [7,23], and human pose features [11,19] with visual features. Some approaches [7,26,30,32,39] utilize the graph structure and exchange the context information of the instance features for relational reasoning between the nodes. DRG [7] proposes human-centric and object-centric graphs to perform context exchange focused on relevant context information. SCG [39] transforms and propagates the context information to the nodes in a graph conditioned on spatial relation. On the other hand, previous one-stage HOI methods [6,14,20] detect human-object pairs and classify the interactions between human-object pairs in an end-to-end manner. These methods utilize the interaction region to match the interaction and a pair of a human box and an object box. UnionDet [14] proposes a unionlevel detector to find the union box of human and object for matching a human-object pair. PPDM [20] detects interaction centers and points to the center point of the human and object box to predict HOI instances. Transformer-based HOI Methods. Inspired by DETR [2], a number of work [4,15,16,29,40,44,46] have adopted the transformer-based object detector to solve HOI detection. They can be divided into two folds: single-branch and two-branch methods. The singlebranch methods [16,29,46] predict the HOI instances with a single transformer decoder. MSTR [29] utilizes multi-scale features to extract discriminative features for the HOI instances. In contrast, two-branch methods [4,15,38,40,44] adopt two transformer decoder branches, one is responsible for human-object pair detection and the other for interaction classification. HOTR [15] detects the instances in an image in detection branch and predicts the interaction with additional offsets to associate humans and objects in interaction branch. Although they extract discriminative features for each sub-task, there is no context exchange for relational reasoning, bringing performance degradation in HOI detection. To alleviate this, AS-NET [4] and DisTR [44] perform the message passing for relational reasoning between two branches. However, they only propagate human-object context information for interaction classification. In this paper, we exchange the context among branches with the multiplex relation context. The multiplex relation context, Figure 2. The overall architecture of MUREN. The proposed method adopts three-branch architecture: human branch, object branch, and interaction branch. Each branch is responsible for human detection, object detection, interaction classification. The input image is fed into the CNN backbone followed by the transformer encoder to extract the image tokens. A transformer decoder layer in each branch layer extracts the task-specific tokens for predicting the sub-task. The MURE takes the task-specific tokens as input and generates the multiplex relation context for relational reasoning. The attentive fusion module propagates the multiplex relation context to each sub-task for context exchange. The outputs at the last layer of each branch are fed into to predict the HOI instances. Positional Encoding F F N ⋯ ! " # " $ " ⋯ F F N ⋯ ( ! % , ! % ) ( # % , # % ) ( $ % , $ % ) ⋯ <human,Flatten ⋯ M U R E ! & ! % ! " # & # % # " $ & $ % $ " ! # $ F F N ⋯ ! & # & $ & ⋯ & % " dsfasldkfjadsklfjasdklfjaskldfgjaskldf which considers all relation contexts in an HOI instance, gives relational semantics for relational reasoning. We also extract more discriminative features for each sub-task via three-branch. Problem Definition Given an input image, the goal of HOI detection is to predict a visually-grounded set of HOI instances for object classes O and interaction classes I. An HOI instance consists of four components: a bounding box of human b H i ∈ R 4 , a bounding box of object b O i ∈ R 4 , a one-hot vector of object label c O i ∈ {0, 1} |O| , and a one-hot vector of interaction label c I i ∈ {0, 1} |I| , where | · | denotes the size of a set. The output of HOI detection is thus expressed by a set of HOI instances {(b H i , b O i , c O i , c I i )}. Method The proposed network, MUREN, is illustrated in Figure 2. Given an input image, it extracts image tokens via a CNN backbone followed by a transformer encoder. The image tokens are fed to three independent branches to perform three sub-task: human detection, object detection, and interaction classification. In each branch, a transformer decoder layer refines N learnable tokens using the image tokens as keys and values to extract task-specific tokens. Using the task-specific tokens of each branch, our multiplex relation embedding module (MURE) generates the context information for relational reasoning. The attentive fusion module then integrates the context information across the taskspecific tokens for human, object, and interaction branches, propagating the results to the next layer. After repeating this process for L times, FFNs predict the set of HOI instances. In the remainder of this section, we explain the details of each component in MUREN. Image Encoding Following the previous work [2,29,47], we use a transformer encoder with a CNN backbone to extract image tokens. The CNN backbone takes an input image to extract an image feature map. The image feature map is fed into 1 × 1 convolution layer to reduce the channel dimension to D, and the positional encoding [2] is added to the image feature map to reflect the spatial configuration of the feature map. The feature map is then tokenized by flattening and fed into the transformer encoder to produce image tokens X ∈ R T ×D for the subsequent networks, where T and D are the number of the image tokens and the channel dimension, respectively. HOI Token Decoding Different from previous two-branch methods [4,15,44], we design an architecture consisting of three branches which is responsible for human detection, object detection, and interaction classification, respectively. Each branch τ , consisting of L layers, takes the learnable tokens Q τ = {q τ i } N i=1 and the image tokens X as inputs , where τ ∈ {H, O, I} indicates human, object, and interaction respectively. At each layer, Q τ is refined through a transformer decoder layer followed by a MURE module and an attentive fusion module. Specifically, the three branches take learnable tokens Q H , Q O , Q I ∈ R N ×D for human, object, and interaction branches, respectively. In l-th layer of the branch τ , a transformer decoder layer Dec τ (l) updates Q τ (l−1) , the output of previous layer of the branch τ , by attending X to generate task-specific tokens F τ (l) = {f τ (l),i } N i=1 which contain the context information for predicting a sub-task which the branch τ is responsible for: F τ (l) = Dec τ (l) (Q τ (l−1) , X),(1) where Dec(q, kv) denotes a transformer decoder layer. Relational Contextualization As mentioned above, relational reasoning is crucial to identify HOI instances. However, since the task-specific tokens are generated from the separated branches, the tokens suffer from a lack of relational context information. To mitigate this issue, we propose multiplex relation embedding module (MURE) which generates multiplex relation context for relational reasoning. The multiplex relation context contains the unary, pairwise, and ternary relation contexts to exploit useful information in each relation context, as shown in Figure 3. Specifically, the MURE first constructs the ternary relation context f HOI i ∈ R D for i-th HOI instance by concatenating each f τ i followed by an MLP layer. f HOI i = MLP([f H i ; f O i ; f I i ]),(2) where [·; ·] is a concatenation operation. We omit the subscript l for the sake of simplicity. Since the ternary relation takes the overall understanding of each sub-task into account, it gives holistic context information about the HOI instance. On the other hand, since the unary and the pairwise relations take a fine-grained level understanding of each sub-task into account, they give the fine-grained context information about the HOI instance. To exploit both holistic and fine-grained context information, we embed the unary and the pairwise relation contexts within the ternary relation context with a sequential manner. In detail, we apply a self-attention on a set of i-th task- specific tokens {f H i , f O i , f I i } to consider the unary relation for i-th HOI instance as Eq. 3. Then, the unary-relation context U i is embedded into ternary relation context using a cross-attention as Eq. 4: U i = SelfAttn({f H i , f O i , f I i }),(3)f HOI i = CrossAttn(f HOI i , U i ),(4) where we denote SelfAttn(·) as a self-attention operation and CrossAttn(q, kv) as a cross-attention operation for simplicity. To embed the pairwise relation context within the ternary relation context, we extract the pairwise features of f HO , f HI , f OI ∈ R D for respective human-object, humaninteraction, object-interaction relation as follows: f HO i = MLP([f H i ; f O i ]),(5)f HI i = MLP([f H i ; f I i ]),(6)f OI i = MLP([f O i ; f I i ]).(7) Similar to the above, we apply the self attention on a set of pairwise features to consider the pairwise relation for i-th HOI instance, and the cross attention to embed the pairwise relation contexts within ternary relation context: P i = SelfAttn({f HO i , f HI i , f OI i }),(8)f HOI i = CrossAttn(f HOI i , P i ).(9) Finally, thef HOI i is transformed to generate the multiplex relation context m i as follows by attending the image tokens X: m i = CrossAttn(f HOI i , X).(10) It is noteworthy that our high-order (ternary and pairwise) feature functions have a form of non-linear function, i.e., MLP, on top of a tuple of multiple inputs, which is not reducible to a sum of multiple functions of individual lowerorder inputs in general. Such a high-order feature function thus can learn the structural relations of the inputs in the tuple, considering all the inputs jointly. For example, a ternary function of three coordinates f (a, b, c) can compute the angle feature between ab and ac, which cannot be computed by an individual unary function, g(a), g(b), or g(c) as well as their linear combination. In a similar vein, our ternary feature functions, i.e., Eq. 2, can effectively learn to capture structural relations which are not easily composable from unary and pairwise feature functions. Attentive Fusion Our attentive fusion module aims to propagate the multiplex relation context to the task-specific tokens for context exchange. Since each sub-task requires different context information for relational reasoning, the multiplex relation context is transformed using MLP with each task-specific token to propagate the context information conditioned on each sub-task. We further utilize the channel attention to select the requisite context information for each sub-task. Then, the refined tokens Q τ (l) , the output of l-th layer of branch τ , is generated by propagating the requisite context information to the task-specific tokens F τ (l) . Formally, the channel attention α and the refined tokens Q τ (l) are formulated as follows: α = σ(MLP([f τ (l),i ; m (l),i ]))(11)q τ (l),i = f τ (l),i + α ⊙ MLP([f τ (l),i ; m (l),i ]),(12) where we denote ⊙ and σ as element-wise multiplication, and sigmoid function, respectively. As the refined tokens Q τ (l) is generated via context exchange with the multiplex relation context, it deduces the comprehensive relational understanding to discover HOI instances. The Q τ (L) , the output of last layer of branch τ , is fed into FFNs to predict a set of the HOI predictions. Formally, given the Q τ (L) , the MUREN predicts a set of HOI predic- tions {(b H i , b O i , p O i , p I i )} N i=1 using FFNs as follows: b H i = FFN hbox (q H (L),i ) ∈ R 4 ,(13)b O i = FFN obox (q O (L),i ) ∈ R 4 ,(14)p O i = δ(FFN oc (q O (L),i )) ∈ R |O| ,(15)p I i = σ(FFN ic (q I (L),i )) ∈ R |I| ,(16) where δ is a softmax operation, and p O i , p I i are class probability of object and interaction, respectively. Training Objective For training our proposed method, we follow previous transformer-based methods [29,38,44]. We adopt the Hungarian Matching [17] to assign the ground-truth HOI instances to the predictions. MUREN is trained with multitask loss composed of four losses: L1 loss [27] L L1 and GIoU loss [28] L GIoU for the bounding box regression, cross-entropy loss L oc for the object classification, and focal loss [21] L ic for the interaction classification. The total loss L is formulated as: L = λ L1 L L1 + λ GIoU L GIoU + λ oc L oc + λ ic L ic ,(17) where λ L1 , λ GIoU , λ oc , and λ ic are the hyper-parameters for weighting each loss. Additionally, we apply intermediate supervision for better representation learning. Specifically, we attach the same FFNs to each decoding branch layer to calculate the intermediate loss. This auxiliary loss is computed the same as L. Inference Given the set of HOI predictions, we generate a set of HOI instances {(b H i , b O i , c O i,j ′ , c I i,t )| i ∈ N, k ∈ R |I| , j ′ = argmax j p O i,j }, where c O i,j ′ ∈ R |O| , c I i,t ∈ R |I| are one-hot vectors with the j-th and t-th index set to 1, respectively. Following [38], we then select top-k score HOI instances, where the score is given by p O i,j ′ · p I i,t . Experiments Datasets and Metrics We evaluate our model on the two public benchmark datasets: HICO-DET [3] and V-COCO [10]. HICO-DET has 38,118 images for training and 9,658 images for testing. It contains 80 object classes, 117 interaction classes and 600 HOI classes, which are a pair of an object class and an interaction class (e.g., 'riding bicycle'). We evaluate the proposed method on Default and Known Object settings. In the Default setting, the AP is calculated across all testing images for each HOI class. The Known Object setting calculates the AP of an HOI class over the images containing the object in the HOI class (e.g., the AP of an HOI class 'riding bicycle' is only calculated on the images which contain the object 'bicycle'). Following the previous work [38], we report the mAP under three splits (Full, Rare, and Non-Rare) for each setting. The Full, Rare, and Non-Rare splits contain all 600 HOI classes, 138 HOI classes, which have less than 10 training samples for each class, and 462 HOI classes, which have more than 10 training samples for each class, respectively. V-COCO is a subset of the MS-COCO [22] dataset. It consists of 5400 and 4,946 images for training, and testing. It has 80 object classes and 29 action classes. Following the evaluation settings in [15], we evaluate the proposed method on scenario 1 and scenario 2, and report role average precision under two scenarios (AP #1 role for scenario 1 and AP #2 role for scenario 2). In scenario 1, the model should predict the bounding box of the occluded object as [0,0,0,0]. In contrast, the predicted bounding box of the occluded object is ignored on calculating the AP role in scenario 2. Implementation Details The encoder in MUREN adopts ResNet-50 as a CNN backbone followed by a 6-layer transformer encoder. We set the number of branch layers L to 6. For the training, we set the number of queries N to 64 for HICO-DET and 100 for V-COCO following [38]. The weight of loss λ L1 , λ GIoU , λ oc , λ ic is set to 2.5, 1, 1, 1, respectively. The network is initialized with the parameters of DETR [2] pretrained on MS-COCO [22]. We optimize our network by AdamW [24] with the weight decay 1e−4. We set the initial learning rate of the CNN backbone to 1e−5 and the other component to 1e−4. The model is trained with 100 epoch. For the V-COCO, we freeze the CNN backbone to prevent overfitting, and set the learning rate to 4e−5. All experiments are conducted with a batch size of 16 on 4 RTX 3090 GPUs. Table 1 and Table 2 show the performance comparison of the proposed method with the previous HOI methods. As posed method achieves state-of-the-art performance on Default and Known Object settings against existing CNN-and transformer-based methods. Compared with the previous CNN-based methods [7,26,30,32,39], which utilize the graph structure for context exchange, MUREN shows significant improvements. We also surpass the previous singlebranch methods [16,29,46]. It illustrates that it is crucial extracting the task-specific tokens for each sub-task with different branches. In particular, we outperform the previous two-branch methods [4,15,38,40,44]. DisTR [44] and AS-NET [4] perform context exchange for relational reasoning, but they only propagate the context information of the human and the object to the interaction branch for interaction classification. Instead, we exchange the context information among the three branches, selecting requisite context information from the multiplex relation context for each sub-task. These results illustrate the advantage of context exchange between each branch using the multiplex relation context for relational reasoning. Moreover, MUREN shows better performance without using any additional information (e.g., spatial and linguistic information) compared with [16,[39][40][41]. We also outperform [29,38,46] which utilize a deeper backbone to extract discriminative features for each sub-task. These results illustrate that three-branch architecture and context exchange with multiplex relation context for relational reasoning provide more discriminative features to predict each sub-task. We further evaluate MUREN on the V-COCO dataset and observe similar results as in the HICO-DET dataset. As shown in Table 2, MUREN achieves state-of-the-art performances across all the metrics compared with existing methods. Comparison with State-of-the-Art Ablation Study We conduct various ablation studies on the V-COCO dataset to validate the effectiveness of MUREN. Impact of each relation context information on relational reasoning. We utilize the multiplex relation context, which contains the unary, pairwise, and ternary relation context, for relational reasoning. To investigate the impact of each relation context information on relational reasoning, we gradually add each relation context information to the baseline, which predicts the HOI instances without context exchange among each branch for relational reasoning. As shown in Table 3, we observe that context exchange using the ternary relation context gives 4.55%p, 4.22%p improvement with a large margin in AP #1 role and AP #2 role , respectively. This result indicates that context exchange for relational reasoning is essential for discovering the HOI instance and ternary relation context promotes relational reasoning providing holistic information about the HOI instances. Besides, when the model exploits ternary and unary relation contexts, the model shows an additional performance improvement. We observe similar results on the model which utilizes both ternary and pairwise relation contexts. It indicates that the fine-grained relation contexts provide useful information for relational reasoning to predict HOI instances. When we use all the relation context information in HOI instance, the model shows a significant performance increase of 6.23%p and 5.86%p in AP #1 role and AP #2 role , compared with the baseline. It demonstrates that each relation context information complements the others, and thus the multiplex relation context provides rich information for relational reasoning and brings performance gain in HOI detection. Impact of the multiplex relation context on each subtask. For investigating the propagation impact of the multiplex relation context on the sub-tasks, we gradually add the propagation the multiplex relation context to each branch. When we propagate the multiplex relation context to one of the detection branches (i.e., human branch and object branch), we observe that the model consistently shows performance improvement compared with the baseline, as shown in Table 4. We also observe the performance gains when the model propagates the multiplex relation context to both human and object branch. It indicates that relational context information is required to detect the human and the object in the HOI detection. In particular, when the model propagates the multiplex relation context to the interaction branch, MUREN shows the notable performance gains of 3.19%p 2. It indicates that the multiplex relation context is essential to interaction classification which requires a comprehensive relational understanding between the human and the object. The entire model of MUREN, which propagates the relation context information to all sub-tasks, achieves the highest performance with a significant margin compared with the other model variants. The results demonstrate that context exchange among the three branches is essential to identify HOI instances and plays a crucial role in the comprehensive relational understanding. Impact of attentive fusion module on context exchange. MUREN exchanges relational context information between each branch via the attentive fusion module. To investigate the impact of the attentive fusion module, we remove the attentive fusion module and fuse both the task-specific tokens and the multiplex relation context with an element-wise addition operation for the baseline. As shown in Table 5 Impact of the context information selection for each subtask. In the attentive fusion module, we select requisite context information for each sub-task from the multiplex relation context. We further analyze the impact of the context information selection as shown in Table 5. To select the requisite context information for each sub-task, we utilize 1) transforming multiplex relation context conditioned on a task-specific token ('conditioning' in Table 5) and 2) channel attention mechanism ('channel' in Table 5). We observe that the model, which utilizes one of 'conditioning' and 'channel', gains performance improvement. We also observe that the model with both 'conditioning' and 'channel' shows better performance than the other model variants. The results demonstrate that each sub-task requires different context information for relational reasoning, and thus it is important to propagate the requisite context for each sub-task. Our attentive fusion module effectively selects requisite context information for each sub-task. Impact of disentangling human and object branches. Human plays a central and an active role for HOI, which is distinctive from a relatively passive role of object, and thus requires a dedicated module to capture relevant attributes and semantics such as pose and clothing. We evaluated in Table 6 the effect of sharing parameters between human and object branches; we gradually increased the number of layers that share parameters between the two branches. The results show that increasing the number of shared layers drops the performance and the full-sharing model, MUREN-(6), results in 2.2%p and 1.9%p decrease in performance at two scenarios, respectively, compared with non-sharing model, MUREN-(0). This is a significant drop also compared to MUREN † , which has a similar number of parameters with MUREN-(6) by adjusting the number of layer L of MUREN, indicating that separating human and object branches is important indeed for HOI detection. Qualitative Results We visualize HOI detection results and the cross attention map of each branch and the multiplex relation embedding module (MURE) in Fig. 4. As shown in Fig. 4b, c, the human and the object branches focus on the instance extremities to detect the human and the object. In the Fig. 4d, we observe that the interaction branch attends to the regions where the interaction exists between the human and the object. These results indicate that the task-specific tokens contain context information for predicting each sub-task. We also observe that the cross-attention map in MURE highlights the overall region that contains the relational semantics about the HOI instance as shown in Fig. 4e. It demonstrates that MURE captures the context information about HOI instance for relational reasoning. Conclusion We have proposed MUREN, a one-stage method that effectively performs the context exchange between the three branches for HOI detection. By leveraging relation contexts for relational reasoning in MURE and using the attention fusion module to select requisite context information for each sub-task, MUREN can learn discriminative features to predict each sub-task. Our extensive experiments demonstrate the importance of context exchange between the branches and the effectiveness of MUREN, which achieves stateof-the-art performance on both HICO-DET and V-COCO benchmarks and its components. Figure 3 . 3The architecture of the multiplex relation embedding module (MURE). MURE takes i-th task-specific tokens and the image tokens as input, and embed the unary and pairwise relation contexts into the ternary relation context. The multiplex relation context, the output of MURE, is fed into subsequent attentive fusion module for context exchange. Figure 4 . 4The visualization of the HOI detection results and the cross-attention map in each branch and the multiplex relation embedding module (MURE). Best viewed in color. horse, ride>Attentive Fusion Module ⋯ % Transformer Decoder Layer ⋯ Object branch Attentive Fusion Module ⋯ & Transformer Decoder Layer ⋯ Human branch Attentive Fusion Module ⋯ " Transformer Decoder Layer ⋯ Interaction branch × × × CNN Backbone Input Image Transformer Encoder shown inTable 1, on the HICO-DET dataset, the pro-Table 1. Performance comparison on the HICO-DET[3] dataset. The letters in Feature column stand for A: Appearance/Visual features, S: Spatial features, L: Linguistic features, P: Human pose features, M: Multi-scale features. The best score is highlighted in bold, and the second-best score is underscored.Method Backbone Feature Default Known Object Full Rare Non-Rare Full Rare Non-Rare CNN-based methods iCAN [8] R50 A+S 14.84 10.45 16.15 16.26 11.33 17.73 TIN [19] R50 A+S+P 22.90 14.97 25.26 - - - GPNN [26] R101 A 13.11 9.34 14.23 - - - DRG [7] R50-FPN A+S+L+M 24.53 19.47 26.04 27.98 23.11 29.43 VSGNet [30] R152 A+S 19.80 16.05 20.91 - - - wang et al. [32] R50-FPN A+S+M 17.57 16.85 17.78 21.00 20.74 21.08 IDN [18] R50 A+S 26.29 22.61 27.39 28.24 24.47 29.37 VCL [13] R50 A 23.63 17.21 25.55 25.98 19.12 28.03 UnionDet [14] R50 A 17.58 11.72 19.33 19.76 14.68 21.27 GGNet [43] HG104 A 28.83 22.13 30.84 27.36 20.23 29.48 SCG [39] R50-FPN A+S+M 31.33 24.72 33.31 34.37 27.18 36.52 Transformer-based methods PST [5] R50 A 23.93 14.98 26.60 26.42 17.61 29.05 HoiTrans [46] R101 A 26.61 19.15 28.84 29.13 20.98 31.57 HOTR [15] R50 A 25.10 17.34 27.42 - - - AS-Net [4] R50 A 28.87 24.25 30.25 31.74 27.07 33.14 QPIC [29] R101 A 29.90 23.92 31.69 32.38 26.06 34.27 MSTR [16] R50 A+M 31.17 25.31 32.92 34.02 28.83 35.57 CDN [38] R101 A 32.07 27.19 33.53 34.79 29.48 36.38 UPT [40] R50 A+S 31.66 25.94 33.36 35.05 29.27 36.77 DisTR [44] R50 A 31.75 27.45 33.03 34.50 30.13 35.81 STIP [41] R50 A+S+L 32.22 28.15 33.43 35.29 31.43 36.45 Ours R50 A 32.87 28.67 34.12 35.52 30.88 36.91 Table 4. The impact of the multiplex relation context on each subtask. The 'human', 'object', and 'interaction' columns indicate the propagation of the multiplex relation context to human, object, and interaction branch, respectively.Table 5. Ablations studies on each component in the attentive fusion module. 'conditioning' and 'channel' indicate transforming multiplex relation context conditioned on a task-specific token and channel attention mechanism.and 2.77%p on scenario 1 and scenario ternary unary pairwise AP #1 role AP #2 role - - - 62.52 65.14 ✓ - - 67.07 69.36 ✓ ✓ - 68.12 70.31 ✓ - ✓ 67.67 70.02 ✓ ✓ ✓ 68.75 71.00 Table 3. The impact of each relation context information on rela- tional reasoning. The 'ternary', 'unary', and 'pairwise' columns indicate the ternary, unary and pairwise relation context. human object interaction AP #1 role AP #2 role - - - 62.52 65.14 ✓ - - 64.44 66.62 - ✓ - 63.66 66.00 ✓ ✓ - 65.29 67.5 - - ✓ 65.71 67.91 ✓ ✓ ✓ 68.75 71.00 conditioning channel AP #1 role AP #2 role - - 66.50 68.96 ✓ - 66.95 69.23 - ✓ 67.10 69.49 ✓ ✓ 68.75 71.00 , the performance drops by 2.25%p and 2.04%p in the two scenarios. It shows the effectiveness of our attentive fusion module for context exchange between the branches.Table 6. The Impact of disentangling human and object branches. MUREN-(k) denotes the sharing of parameters between the human and object branches across k layers. The parameters are shared only between corresponding layers. MUREN † is variant of MUREN by adjusting the number of layer L.Method AP #1 role AP #2 role Params (M) MUREN-(0) 68.8 71.0 69.3 MUREN-(3) 67.1 69.3 64.3 MUREN-(6) 66.6 69.1 59.6 MUREN † 68.3 70.6 59.6 Acknowledgements. This work was supported by the IITP grants (2021-0-00537: Visual common sense through selfsupervised learning for restoration of invisible parts in images (50%), 2021-0-02068: AI Innovation Hub (40%), and 2019-0-01906: AI graduate school program at POSTECH (10%)) funded by the Korea government (MSIT). Zero-shot action recognition from diverse object-scene compositions. Carlo Bretti, Pascal Mettes, arXiv:2110.13479arXiv preprintCarlo Bretti and Pascal Mettes. Zero-shot action recogni- tion from diverse object-scene compositions. arXiv preprint arXiv:2110.13479, 2021. 1 End-toend object detection with transformers. Nicolas Carion, Francisco Massa, Gabriel Synnaeve, Nicolas Usunier, Alexander Kirillov, Sergey Zagoruyko, European conference on computer vision. SpringerNicolas Carion, Francisco Massa, Gabriel Synnaeve, Nicolas Usunier, Alexander Kirillov, and Sergey Zagoruyko. End-to- end object detection with transformers. In European confer- ence on computer vision, pages 213-229. 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{'abstract': 'Recent state-of-the-art methods for HOI detection typically build on transformer architectures with two decoder branches, one for human-object pair detection and the other for interaction classification. Such disentangled transformers, however, may suffer from insufficient context exchange between the branches and lead to a lack of context information for relational reasoning, which is critical in discovering HOI instances. In this work, we propose the multiplex relation network (MUREN) that performs rich context exchange between three decoder branches using unary, pairwise, and ternary relations of human, object, and interaction tokens. The proposed method learns comprehensive relational contexts for discovering HOI instances, achieving state-of-the-art performance on two standard benchmarks for HOI detection, HICO-DET and V-COCO.', 'arxivid': '2304.04997', 'author': ['Sanghyun Kim sanghyun.kim@postech.ac.kr \nPohang University of Science and Technology (POSTECH)\nSouth Korea\n', 'Deunsol Jung deunsol.jung@postech.ac.kr \nPohang University of Science and Technology (POSTECH)\nSouth Korea\n', 'Minsu Cho mscho@postech.ac.kr \nPohang University of Science and Technology (POSTECH)\nSouth Korea\n'], 'authoraffiliation': ['Pohang University of Science and Technology (POSTECH)\nSouth Korea', 'Pohang University of Science and Technology (POSTECH)\nSouth Korea', 'Pohang University of Science and Technology (POSTECH)\nSouth Korea'], 'corpusid': 258060116, 'doi': '10.48550/arxiv.2304.04997', 'github_urls': [], 'n_tokens_mistral': 17813, 'n_tokens_neox': 15311, 'n_words': 8620, 'pdfsha': 'e850cadb3df2a6594dc55cabc7883d1fba7fea1f', 'pdfurls': ['https://export.arxiv.org/pdf/2304.04997v1.pdf'], 'title': ['Relational Context Learning for Human-Object Interaction Detection', 'Relational Context Learning for Human-Object Interaction Detection'], 'venue': []}

arxiv

Azimuthal fermionic current in the cosmic string spacetime induced by a magnetic tube November 9, 2018 M S Maior De Sousa Departamento de Física-CCEN Universidade Federal da Paraíba 58.059-970, J. PessoaPB C. Postal 5.008Brazil R F Ribeiro Departamento de Física-CCEN Universidade Federal da Paraíba 58.059-970, J. PessoaPB C. Postal 5.008Brazil E R Bezerra De Mello Departamento de Física-CCEN Universidade Federal da Paraíba 58.059-970, J. PessoaPB C. Postal 5.008Brazil Azimuthal fermionic current in the cosmic string spacetime induced by a magnetic tube November 9, 2018numbers:1127 + d0462 + v9880Cq In this paper, we analyze the vacuum azimuthal fermionic current induced by a magnetic field confined in a cylindrical tube of finite radius a, in the cosmic string spacetime. Three distinct configurations for the magnetic field are taken into account: (i) a cylindrical shell of radius a, (ii) a magnetic field proportional to 1/r and (iii) a constant magnetic field. In these three cases, the axis of the infinitely long tube of radius a coincides with the cosmic string; moreover, we only develop this analysis for the region outside the tube. In order to do that, we explicitly construct the corresponding complete set of normalized wave-functions. We show that in the region outside the tube, the induced current is decomposed into a part corresponding to a zero-thickness magnetic flux in addition to a core-induced contribution. The latter presents specific form depending on the magnetic field configuration considered. The zero-thickness contribution depends only on the fractional part of the ration of the magnetic flux inside the tube by the quantum one. As to the core-induced contribution, it depends on the total magnetic flux inside the tube, and consequently, in general, it is not a periodic function of the flux. Introduction The existence of a magnetic flux tube penetrating a type II superconductor, named vortex, was first demonstrated by Abrikosov [1], by using the Ginzburg-Landau theory of superconductivity. Some years later, Nielsen and Olesen [2] have shown, by using a classical relativistic field theory, composed by Higgs fields interacting with Abelian one, that presents spontaneously gauge symmetry broken, contains static cylindrically symmetric solution carrying a magnetic flux. This configuration corresponds to the vortex solution. The equations of motion associated for this system form a set of coupled non-linear differential equation, that, in general, has no closed solutions. The analysis of the influence of this system on the geometry of the spacetime was analyzed numerically by Garfinkle [3] and Laguna [4] many years ago. In these papers the authors showed that, the vortex has a inner structure characterized by a non-vanishing core carrying a magnetic flux, whose extension is determined by the energy scale where the symmetry is broken. Two length scales naturally appear, the one related with the extension of the magnetic flux proportional to the inverse of vector field mass, m v , and the other associated with the inverse of the scalar field mass, m s . The latter being the radius that the scalar field reaches its vacuum value. Moreover the authors also verify that asymptotically the surface perpendicular to the vortex corresponds to a Minkowski one minus a wedge. According to the Big Bang theory, during its expansion the universe underwent to a series of phase transition. In most interesting model of high-energy physics, the formation of topological defects such as domain walls, monopoles, cosmic string, among others are predicted to occur [5]. The cosmic string, a linear topological defect, is the most studied. Though the recent observations of the cosmic microwave background radiation have ruled out them as the primary source for primordial density perturbations, cosmic strings give rise to a number of interesting physical effects such as gamma rays bursts [6], the emission of gravitational waves [7] and the generation of high-energy cosmic rays [8]. String-like defects also appear in a number of condensed matter systems, including liquid crystals and graphene made structures. The complete analysis of the behavior of a quantum charged field in the neighborhood a Nielsen and Olesen (NO) string must take into account the influence of the geometry of the spacetime and also the presence of the magnetic field. Two distinct approach for this system are: (i) To consider the string as an idealized linear topological defect, having a magnetic field running along it. This case can be treated analytically. (ii) To consider the non-zero thickness for the string. Unfortunately this problem is analytically intractable. In the idealized model, the conical structure of the spacetime modifies the zero-point vacuum fluctuation of quantized fields inducing non-vanishing vacuum expectation values (VEV) for important physical observable, such as the energy-momentum tensor. See for instance references given in [9], for the cases of uncharged scalar, fermionic and vector fields. Considering the presence magnetic flux, additional polarization effects associated with charged quantum fields take place [10]- [13]. In particular this magnetic flux induces non-vanishing current densities, j µ . This phenomenon has been investigated for scalar fields in Ref. [14,15]. The analysis of induced fermionic currents in higherdimensional cosmic string spacetime in the presence of a magnetic flux have been developed in Ref. [16]. In all these analysis, the authors have shown that induced azimuthal vacuum current densities take place if the ratio of the magnetic flux by the quantum one has a nonzero fractional part. 1 Because the analysis of a quantum system in a realistic model for the NO vortex cannot be exactly solvable, an intermediate approach can be adopted. Here we assume an approximated model that consists to consider the spacetime produced by the string as being conical everywhere, but having a non-zero thickness magnetic field surrounding it. In this way, some improvements can be obtained. This approach was used in [19,20] to calculate the VEV of massless charged scalar and fermionic energy-momentum tensors, T µν , respectively. More recently the scalar vacuum current induced by a magnetic flux in a cosmic string considering a non-vanishing core has been developed in [21]. In these calculations the vacuum polarization effects were developed for the region outside the tube, and shown to present two contributions. The first one associated with a zero-thickness magnetic flux, and the second induced by the non-vanishing core, named core-induced contributions. The first contribution is a periodic function of the magnetic flux inside the tube, with the period equal to the quantum flux, Φ 0 = 2π/e; as to the second, in general, are not periodic function of the quantum flux. In fact the core-induced contributions depends on the total magnetic flux inside the core. The later is new one, and its corresponding result may shed light upon features of finite core effects in more realistic models. Three different configurations of magnetic flux will be considered: (i) A magnetic field on a cylindrical shell, (ii) a magnetic field proportional to 1/r, and finally (iii) a homogeneous magnetic field inside the tube. This paper is organized as follow. In section 2 we describe the background geometry of the spacetime and the configurations of the magnetic fields. We provide the general structure of the complete set of normalized positive-and negative-energy fermionic mode functions in the region outside the tube, which can be used for each magnetic field configuration. In section 3, by using the mode-summation method, we calculate the induced azimuthal current density. We show that this current can be decomposed in two parts: The first one corresponds to the induced current by a zero-thickness magnetic flux in the geometry of an idealized cosmic string, and the second one is induced by the non-zero core of the magnetic tube. The latter is calculated, separately, for the three kinds of magnetic flux considered. The behavior of them are discussed in various asymptotic regions of the parameters. We also present some plots associated with the core-induced azimuthal current exhibiting its behavior as function of the most relevant physical variables. Our most relevant conclusions are summarized in section 4. The geometry and the fermionic wave-functions The background geometry associated with an idealized cosmic string along the z-axis, can be given, by using cylindrical coordinates, through the line element below: ds 2 = dt 2 − dr 2 − r 2 dφ 2 − dz 2 , (2.1) where the coordinates take values in range r ≥ 0, 0 ≤ φ ≤ φ 0 = 2π/q and −∞ ≤ (t, z) ≤ +∞. The parameter q associated with the planar angle deficit is related to the mass per unit length of the string, µ 0 , by q −1 = 1 − 4µ 0 . The quantum dynamic of a massive charged spinor field in curved space-time and in the presence of an electromagnetic four-vector potential, A µ , is described by the Dirac equation, iγ µ (∇ µ + ieA µ )ψ − mψ = 0, ∇ µ = ∂ µ + Γ µ ,(2.2) where γ µ are the Dirac matrices in curved space-time and Γ µ is the spin connection. For the geometry in consideration the gamma matrices can be written in the form, γ 0 = γ (0) = 1 0 0 −1 , γ l = 0 σ l −σ l 0 , (2.3) where we have introduced the 2 × 2 matrices for l = (r, φ, z): σ r = 0 e −iqφ e iqφ 0 , σ φ = − i r 0 e −iqφ −e iqφ 0 , σ z = 1 0 0 −1 . (2.4) As to the four vector potential we consider A µ = (0, 0, A φ (r), 0) ,(2.5) with A φ (r) = − qΦ 2π a(r) . For the second and third models, representing a magnetic field proportional to 1/r and an homogeneous field, respectively, the radial function a(r) reads, In the expressions above Θ(z) represents the Heaviside function, and Φ the total magnetic flux. For positive-energy solutions, assuming the time-dependence of the eigenfunctions in the form e −iEt and decomposing the spinor ψ into the upper (ψ + ) and lower (ψ − ) components, we find the following equations a(r) = f (r)Θ(a − r) + Θ(r − a) ,(2.(E − m)ψ + + i σ l (∂ l + ieA l ) + 1 − q 2r σ r ψ − = 0 , (2.10) (E + m)ψ − + i σ l (∂ l + ieA l ) + 1 − q 2r σ r ψ + = 0 . (2.11) Substituting the function ψ − from the second equation into the first one, we obtain the second order differential equation for the spinor ψ + : r 2 ∂ 2 r + r∂ r + ∂ φ + ieA φ − i 1 − q 2 σ z 2 + r 2 (∂ 2 z + E 2 − m 2 ) − e r σ z ∂ r A φ ψ + = 0 . (2.12) The same equation is obtained for ψ − . So, we may say that the general solutions to ψ + and ψ − can be express in terms of the ansatz below, compatible with the cylindrical symmetry of the physical system, ψ + = e −ip.x e iqnφ R 1 (r) R 2 (r)e iqφ , (2.13) ψ − = e −ip.x e iqnφ R 3 (r) R 4 (r)e iqφ ,(2.14) where p.x ≡ Et − kz. This function is eigenfunction of the total angular momentum along the string, J z ,Ĵ z ψ = −i∂ φ + i q 2 γ (1) γ (2) ψ = qjψ ,(2.15) where j = n + 1/2, n = 0, ±1, ±2, ... . In order to construct the complete set of the wave-functions we shall consider, separately, the equation (2.12) in the regions r < a and r > a. For the inner region, three different configurations of magnetic field have been already specified by the four-vector potential (2.5)-(2.9). For the outer region, r > a, there is no magnetic field and the vector potential given by, A φ = − qΦ 2π , (2.16) being Φ the magnetic flux. So, we find that the external positive-energy solution of the Dirac equation is given in terms of Bessel, J µ (z), and Neumann, Y µ (z), functions. A similar result is verified for the lower components of the Dirac spinor. Consequently we can write: ψ (+) = e −ip.x e iqnφ     C 1 J β j (λr) + D 1 Y β j (λr) C 2 J β j + j (λr) + D 2 Y β j + j (λr) e iqφ A 1 J β j (λr) + B 1 Y β j (λr) A 2 J β j + j (λr) + B 2 Y β j + j (λr) e iqφ     , (2.17) with n = j − 1/2 being an integer number. We have defined β j = q|j + α| − j 2 , (2.18) with j = 1 for j ≥ −α and j = −1 for j < −α, being α = eA φ /q = −Φ/Φ 0 , Here Φ 0 = 2π/e is the quantum flux. As we can see, we have introduced a set of eight constants C i , D i , A i and B i with i = 1, 2 in the general solution above. The energy is expressed in terms of λ, k and m by the relation E = λ 2 + k 2 + m 2 . (2.19) We can find a relation between the constants of the the upper and lower solutions in (2.17) by the use of (2.10) and (2.11). The relations are given by A 1 = kC 1 − i j λC 2 E + m , A 2 = − kC 2 − i j λC 1 E + m (2.20) B 1 = kD 1 − i j λD 2 E + m , B 2 = − kD 2 − i j λD 1 E + m . (2.21) In addition, for the further specification of the eigenfunctions, we can impose extra conditions relating the above constants. As such a condition, following [22], we will require the following relations between the upper and lower components: R 3 (r) = ρ s R 1 (r), R 4 (r) = − R 2 (r) ρ s , (2.22) with ρ s = E + s √ λ 2 + m 2 k , s = ±1 . (2.23) Doing this we obtain the following relations: A 1 = ρ s C 1 , A 2 = −C 2 /ρ s , (2.24) B 1 = ρ s D 1 , B 2 = −D 2 /ρ s . (2.25) Hence, the positive frequency exterior solutions to the Dirac equation, specified by the set of quantum numbers σ = (λ, j, k, s), has the form ψ (+) σ(out) (x) = e −ip.x e iqnφ      C 1 J β j (λr) + D 1 Y β j (λr) i j ρ s b (+) s C 1 J β j + j (λr) + D 1 Y β j + j (λr) e iqφ ρ s C 1 J β j (λr) + D 1 Y β j (λr) −i j b (+) s C 1 J β j + j (λr) + D 1 Y β j + j (λr) e iqφ      , (2.26) where we have introduced b (±) s = ±m + s √ λ 2 + m 2 λ . (2.27) For the region inside, r < a, we have three different configurations of magnetic field, as we have already mentioned. In this way we have three different solutions for (2.12). Let us represent each radial function by R (i) l , where i = 1, 2, 3, is the index associated with the model and l = 1, 2 the index specifying the the spinor components. For the inner region we can write the positive-energy spinor field in the general form below: ψ (+) i(in) (x) = C (i) e −ip.x e iqnφ      R (i) 1 (λ, r) iρ s b (+) s R (i) 2 (λ, r)e iqφ ρ s R (i) 1 (λ, r) −ib (+) s R (i) 2 (λ, r)e iqφ      . (2.28) The coefficients C 1 and D 1 in (2.26) and C (i) in (2.28) can be determined by the continuity condition of the fermionic wave function at r = a. After some intermediate steps we can write, C 1 = − π 2 (λa)C (i) R (i) 1 (λ, a)Ỹ β j (λa) , (2.29) D 1 = π 2 (λa)C (i) R (i) 1 (λ, a)J β j (λa) , (2.30) whereZ β j (z) = j Z β j + j (z) − V (i) j (λ, a)Z β j (z) , with V (i) j (λ, a) = R (i) 2 (λ, a) R (i) 1 (λ, a) . (2.31) In (2.31) Z µ represents the Bessel functions J µ or Y µ . With this notation all the informations about the inner structure of the magnetic field is contained in the coefficient V (i) j . Finally the constant C (i) can be obtained form the normalization condition, d 3 x g (3) ψ (+) σ † ψ (+) σ = δ σ,σ , (2.32) where delta symbol on the right-hand side is understood as the Dirac delta function for continuous quantum numbers λ and k, and the Kronecker delta for discrete ones n and s, and g (3) is the determinant of the spatial metric tensor. The integral over the radial coordinate should be done in the interval [0, ∞). In this case two different expressions for the wavefunction must be used. The function (2.28) for r ∈ [0, a] and (2.26) for r ∈ [0, ∞). The integral over the interior region is finite, consequently the dominant contribution for λ = λ comes from the integration in the exterior region. By using the standard integrals involving the cylindrical Bessel functions, we find (2π) 2 [|C 1 | 2 + |D 1 | 2 ] = qλ (1 + ρ 2 s )(1 + (b (+) s ) 2 ) . (2.33) Substituting (2.29) and (2.30) into the above equation we find: C (i) R (i) 1 (λ, a) = Ξ(λ, a) , (2.34) with Ξ(λ, a) = 1 aπ 2 q λ 1 (1 + ρ 2 s ) 1 (1 + (b (+) s ) 2 ) 1/2 1 (Ỹ β j (λa)) 2 + (J β j (λa)) 2 . (2.35) This relation determines the normalization constant for the interior wave function. The out-side negative-energy fermionic wave-function can be obtained in a similar procedure. So, the positive-and negative-energy wavefunctions are specified by the complete set of quantum numbers σ = (λ, k, j, s). These functions can be written as show below: ψ (±) σ(out) (x) = C (±) (out) e ∓i(Et−kz) e iq(j−1/2)φ      g β j (λa, λr) , ±i j ρ s b (±) s g β j + j (λa, λr)e iqφ ρ s g β j (λa, λr) ∓i j b (±) s g β j + j (λa, λr)e iqφ      ,(2.36) where we have introduced the notations, g β j (λa, λr) =Ỹ β j (λa)J β j (λr) −J β j (λa)Y β j (λr) (Ỹ β j (λa)) 2 + (J β j (λa)) 2 , (2.37) g β j + j (λa, λr) =Ỹ β j (λa)J β j + j (λr) −J β j (λa)Y β j + j (λr) (Ỹ β j (λa)) 2 + (J β j (λa)) 2 , (2.38) ρ s = E + s √ λ 2 + m 2 k , s = ±1 , (2.39) b (±) s = ±m + s √ λ 2 + m 2 λ (2.40) and C (±) (out) = 1 2π   qλ (1 + ρ 2 s ) 1 + (b (±) s ) 2   1/2 ,(2.41) Having the negative-energy wave-function for the region outside the magnetic flux, i.e., for r > a, we can use it for the investigation of the vacuum azimuthal fermionic current density. Induced Azimuthal Fermionic current The vacuum expectation value (VEV) of the fermionic current density operator, j µ = eψγ µ ψ, can be obtained by using the mode sum formula, j µ (x) = e σψ (−) σ (x)γ µ ψ (−) σ (x) ,(3.1) where we are using the compact notation defined by σ = ∞ 0 dλ +∞ −∞ dk j=±1/2,... s=±1 . (3.2) Specifically the VEV of the azimuthal current density is given by, j φ (x) = e σ (ψ (−) σ (x)) † γ 0 γ φ ψ (−) σ (x) .j φ = − eq 2π 2 r ∞ −∞ dk ∞ 0 λ 2 dλ √ λ 2 + k 2 + m 2 j j g β j (λa, λr)g β j + j (λa, λr) . (3.4) Developing the product of g β j (λa, λr)g β j + j (λa, λr) in a convenient form, i.e., separating the contributions that does not depend on the inner structure of the magnetic field from the other that does, we can written the above result as the sum of two terms as shown below: j φ (x) = j φ (x) s + j φ (x) c . (3.5) The first term, j φ (x) s , corresponds to the azimuthal current density in the geometry of a straight cosmic having a magnetic flux running along its core, and the second, j φ (x) c , is induced by the magnetic tube of radius a. At this point we would like to analyze separately both contributions. Azimuthal current induced by a zero-thickness magnetic flux The azimuthal current induced by a magnetic flux running along the idealized cosmic string is given by, j φ (x) s = − eq 2π 2 r ∞ −∞ dk ∞ 0 dλλ 2 √ λ 2 + k 2 + m 2 j j J β j (λr)J β j + j (λr) . (3.6) The explicit calculation of this contribution was given in [23]. Here we briefly review its more important results. Using the identity bellow, 1 √ m 2 + k 2 + λ 2 = 2 √ π ∞ 0 dt e −(m 2 +k 2 +λ 2 )t 2 ,(3.7) into (3.6), it is possible to integrate over the variable λ by using the results form [24]: where I(q, α 0 , y) is defined by ∞ 0 dλλ 2 e −λ 2 t 2 J β j (λr)J β j + j (λr) = e −r 2 /(2t 2 ) 4t 4 r j I β j (r 2 /(2t 2 )) − I β j + j (r 2 /(2t 2 )) .I(q, α 0 , z) = j I β j (z) = ∞ n=0 I q(n+α 0 +1/2)−1/2 (z) + I q(n−α 0 +1/2)+1/2 (z) ,(3.10) and j I β j + j (z) = I(q, −α 0 , z) . (3.11) In the above development, we have used used the notation α = eA φ /q = −Φ/Φ 0 = n 0 + α 0 ,(3.12) with n 0 being an integer number. So we conclude that (3.9) is an odd function of α 0 . In [25] we have presented an integral representation for I(q, α 0 , y): I(q, α 0 , z) = e z q − 1 π ∞ 0 dy e −z cosh y f (q, α 0 , y) cosh(qy) − cos(qπ) + 2 q p k=1 (−1) k cos[2πk(α 0 − 1/2q)]e z cos(2πk/q) ,(3.13) with 2p < q < 2p + 2 and with the notation For 1 q < 2, the last term on the right-hand side of Eq. (3.13) is absent. By using the result (3.13), and after the integration over y, the expression (3.9) is presented in the form j φ s = − em 2 π 2 r 2 p k=1 (−1) k sin(πk/q) sin(2πkα 0 )K 2 (2mr sin(πk/q)) + q π ∞ 0 dy g(q, α 0 , 2y)K 2 (2mr cosh y) [cosh(2qy) − cos(qπ)] cosh y . (3.15) In the above expression K ν (x) is the Macdonald function, and g(q, α 0 , y) = cos [qπ (1/2 + α 0 )] cosh [q (1/2 − α 0 ) y] − cos [qπ (1/2 − α 0 )] cosh [q (1/2 + α 0 ) y] . (3.16) As we can see j φ s depends only on the fractional part of the ration of the total flux by the quantum one, α 0 , and vanishes when this parameter is zero. We can say that this current is a manifestation of the Aharonov-Bohm effect. Core-induced azimuthal current The core-induced azimuthal current, j φ (x) c , can be written in a compact for by, j φ (x) c = eq (2π) 2 r ∞ −∞ dk ∞ 0 dλλ 2 √ λ 2 + k 2 + m 2 × j jJβ j (λa) 2 l=1 H (l) β j (λr)H (l) β j + j (λr) H (l) β j (λa) ,(3.17) where H l ν (x) with l = 1, 2 represents the Hankel functions. In order to develop this calculation, we rotate the integrals contour in the complex plane λ as follows: by the angle π/2 for l = 1 and −π/2 for l = 2. By using the property below, V (i) j (±iλ, a) = ±iIm{V (i) j (iλ, a)} ,(3.18) one can see that the integral over the segments (0, i √ m 2 + k 2 ) and (0, −i √ m 2 + k 2 ) are canceled. In the remaining integral over the imaginary axis we introduce the modified Bessel functions. Moreover, writing imaginary integral variable by λ = ±iz, the core-induced azimuthal current reads, j φ (x) c = − eq π 3 r ∞ 0 dk ∞ √ k 2 +m 2 dzz 2 √ z 2 − k 2 − m 2 j K β j (zr)K β j + j (zr)F (i) j (za) , (3.19) where we use the notation F (i) j (y) = I β j + j (y) − Im[V (i) j (iy/a, a)]I β j (y) K β j + j (y) + Im[V (i) j (iy/a, a)]K β j (y) . (3.20) After a convenient coordinate transformations we write (3.19) as follow: j φ (x) c = − eq π 2 r 4 ∞ mr z 2 dz j K β j (z)K β j + j (z)F (i) j (z(a/r)) . (3.21) Before to start explicit numerical analysis related to the core-induced azimuthal current, let us now evaluate its behavior at large distance from the core. First we consider massive fields and in the limit mr >> ma. In order to develop this analysis, we assume that the product K β j (z)K β j + j (z) can be expressed in terms of their corresponding asymptotic forms. So we have the induced current density given below j φ c ≈ − eq 2πr 4 ∞ mr dz ze −2z j F (i) j (z(a/r)) . (3.22) The dominant contribution for this integral is given from the region near the lower limit of integration. Then the leading order contribution is, j φ (r) c ≈ − eqm 4 4π(mr) 3 e −2mr j F (i) j (ma) . (3.23) The behavior of the zero-thickness azimuthal current, j φ c , decays as e −2mr sin(π/q) /(mr) 5/2 for q > 2; consequently the latter dominates the total azimuthal current at large distances. Our next analysis is to study the behavior of azimuthal current for massless fields for r >> a. In order to do that, we shall use explicitly the radial functions in the region inside the tube. In [26,27] we provided exact solutions for R 1 (r) and R 2 (r) for the three different configurations of magnetic field. They are: 1. For the cylindrical shell: (3.24) where ν j = q|j| −˜ j 2 , with˜ j = 1 for j > 0 and˜ j = −1 for j < 0. R (1) 1 (r) = J ν j (λr) R (1) 2 (r) =ˆ j J ν j +ˆ j (λr) , 2. For the magnetic field proportional to 1/r: 3. For the homogeneous magnetic field: R (2) 1 (r) = M κ,ν j (ξr) √ r R (2) 2 (r) = C (2) j M κ,ν j +ˆ j (ξr) √ r , (3.25) where ξ = 2 a q 2 α 2 − λ 2 a 2 , κ = − 2q 2 jα ξa (3.26) and C (2) j = λ ξ 1 (2q|j|+1) , j > 0 . − ξ λ (2q|j| + 1), j < 0 .R (3) 1 (r) = M κ− 1 4 , ν j 2 (τ r 2 ) r R (3) 2 (r) = C (3) j M κ+ 1 4 , ν j +ˆ j 2 (τ r 2 ) r , (3.28) where τ and κ are given by where τ and κ are given by τ = qα/a 2 , κ = λ 2 4τ − qj 2 , (3.29) with C (3) j = λ √ τ 1 2q|j|+1 j > 0 . − √ τ λ (2q|j| + 1), j < 0 . (3.30) For the second and third models, the radial functions are given in terms of Whittaker functions, M κ,ν (z) [24]. Now we would like to discuss the behavior of the core-induced azimuthal current at large distance of the core, considering massless fields. Instead to use the summation on the angular moment j in (3.21), we use n = j − 1/2. In this way, we shall use a new notation, F (i) n ≡ F (i) j . Moreover, we change n by n − n 0 , being n 0 given in (3.12). So from (3.21) we can write, j φ (x) c = − eq π 2 r 4 ∞ 0 dzz 2 ∞ n=−∞ K β (z)Kβ(z)F (i) n−n 0 (z(a/r)) . (3.31) In the above expression we are using the notation: F (i) n−n 0 (z(a/r)) = Iβ(z(a/r)) − Im[V (i) n−n 0 (iz, (a/r))]I β (z(a/r)) Kβ(z(a/r)) + Im[V (i) n−n 0 (iz, (a/r))]K β (z(a/r)) . (3.32) In (3.32) the orders of Bessel functions are given by β = q|n + 1/2 + α 0 | − 1 2 |n + 1/2 + n 0 | n + 1/2 + α 0 , β = q|n + 1/2 + α 0 | + 1 2 |n + 1/2 + n 0 | n + 1/2 + α 0 . (3.33) Expanding the integrand of (3.31) in powers of a/r, we keep only the dominant term that is given by the smaller power of this ratio. For this analysis we have two possibilities: for α 0 > 0 (0 ≤ α 0 < 1/2) this term is given by n = −1, and for α 0 < 0 (−1/2 < α 0 ≤ 0) this term is given for n = 0. Moreover, using the expansions for the modified Bessel functions for small arguments [28], the leader contributions are: • For α 0 > 0: F (i) −1−n 0 z a r ≈ 2 Γ 2 (β) 1 + iV (i) −1−n 0 (iz, a/r) z a r az 2rβ Γ(1−β) Γ(β) − iV (i) −1−n 0 (iz, a/r) z R r az 2r 2r az 2β . (3.34) • For α 0 < 0, and F (i) −n 0 z a r ≈ − 2 βΓ 2 (β) 1 + iV (i) −n 0 (iz, a/r) z R r 2rβ az 2r az 2β − i Γ(1−β) Γ(β) V (i) −n 0 (iz, a/r) z R r az 2r . (3.35) The next steps are the calculations of the dominants contribution for the coefficient that contains all the information about the core, V (i) −1−n 0 (iz, a/r) and V (i) −n 0 (iz, a/r), for the three models. This can be done by explicit substitution of the radial functions, R (i) 1 (iz, a/r) and R (i) 2 (iz, a/r), into (2.31). So, after some intermediate steps, we find: j φ (r) c ≈ 2 |α 0 | α 0 eq π 2 r 4 β − χ (l) 2r a 2β χ (l) β 2β + 1 , (3.36) where β = q 1 2 − |α 0 | + 1 2 (3.37) and χ (l) is a parameter depending on the specific model adopted for the magnetic field, given bellow by: χ (l) =                ν = q|n 0 − 1 2 |α 0 | α 0 | − 1 2 |α 0 | α 0 , for the model (i) qα(q + 1) M q 2 |α 0 | α 0 ,ν (2qα) M q 2 |α 0 | α 0 ,ν+1 (2qα) , for the model (ii) √ qα 2 (q + 1) M − 1 2 |α 0 | α 0 q+1 2 , ν 2 (τ R 2 ) M − 1 2 |α 0 | α 0 q+1 2 , ν 2 (τ R 2 ) , for the model (iii) . (3.38) On basis of the above results we can say that for the three models considered, the core-induced azimuthal current density decays with, 1 r 4 (r/a) 2β , for large distance from the tube. In [23] we have shown that for massless fields the zero-thickness azimuthal current decays with 1/r 4 . So, we conclude that for large distance from the core, the total azimuthal current, (3.5), is dominated by the zero-thickness contribution. The next investigation is the behavior of j φ (r) c near the core, for the three models. In general the current diverges in this region. The dominant. To find the leading term it is convenient to introduce a new variable z = β j x, and use the uniform expansion for large order for the modified Bessel functions [28]. However, before to do that, we would like to notice that, changing n → −n − 1 the summation over j keeps unchanged, but the parameter ν j change as ν j →ν j andν j → ν j . If, in addition, we also change α → −α, then β j →β j andβ j → β j . It means that, when we make n → −n − 1 and α → −α we have F (i) j (y) −→ −F (i) j (y). On basis of this analysis, and considering α > 0, the behavior of the core-induced azimuthal current near the boundary is given by, j φ (x) c ≈ 2 eq π 2 r 4 n>0 β 3 j ∞ mr β j dx x 2 F (i) j (β j x(a/r))K β j (β j x)K β j + j (β j x) . (3.39) Because we are considering n >> 1, from now on we use the approximation, β j ≈ ν j ≈ qn. For the three models, it is necessary to find the leading term of F (i) j for large value of j. We have shown in [26], that for the three models the leading terms have the same structure given below, F (1,2) j (qnx(a/r)) ≈ 1 4q 2 π e 2qnη n 2 (1 + e 2η ) , (3.40) whereη = 1 + x 2 (a/r) 2 . Finally for the three models, the core-induced azimuthal current density is given by j φ (x) c ≈ eq 4π 2 r 4 n>0 ∞ mr qn dz z 2 e −2qn(η−η) (1 + e 2η ) √ 1 + z 2 ,(3.41) where η = √ 1 + x 2 . Using the approximation η −η ≈ z(1 − a/r), and observing that the denominator of the integrand in (3.41) can be approximate to unity, for the three models, we have: j φ (x) c ≈ eq 4π 2 r 4 n>0 ∞ mr qn dz z 2 e −2qn(1−a/r) . (3.42) Solving the above integral we have j φ (r) c ≈ e (4πq) 2 1 r 1 (r − a) 3 . (3.43) Here, we must notice that the current density diverges near the boundary. Because the zero-thickness azimuthal current, j φ (r) s , presents a finite value near the boundary, we can conclude that, in this region, the total azimuthal current, (3.5) is dominated by the core-induced contribution. Let us now present some additional informations which are not provided by the analytical expressions. In Fig. 1 we exhibit the dependence of the core-induced azimuthal current density with mr considering q = 1.5 and ma = 1. On the left plot, we present the behavior for the current induced by the cylindrical shell of magnetic field, taking into account positive and negative value for α. On the right plot we exhibit its behavior as function of mr, for the three different models of magnetic fields considering α = 2.1. By this plot we can infer that for a given point outside the tube, the intensity of the current induced by the first model is the biggest one. In Fig 2, we exhibit, for the magnetic field concentrated in a cylindrical shell, the behavior of j φ (r) c as function of mr considering q = 1.5, 2.0, 2.5. By this plot we can infer that the intensity of the current increases with q. Here we adopted α = 1.2 and ma = 1. In Fig 3 we exhibit the behavior of the core-induced current as function of α, considering ma = 1 and mr = 2. On the left plot we display the current induced by the first model of magnetic field for q = 1.5, 2.0, 3.5. On the right plot we consider the currents induced by the three different models, adopting q = 1.5. For both plots, we assume that α varies in the interval [−7.0, 7.0]. By them we can infer once more that, the intensity of the current increases with q (left plot) and the first model provide the current with biggest intensity (right plot). . Figure 2: The core-induced azimuthal current density in units of m 4 e, due to a magnetic field concentrated on a shell, as function of mr considering q = 1.5, 2.0, 2.5. In this plot we adopted α = 1.2. Conclusions In this paper we have analyzed the influence of the conical topology of the spacetime and the finite core effect, on the vacuum expectation value of fermionic azimuthal current, induced by magnetic fluxes of finite extension. In the model considered we have adopted that the geometry of the spacetime corresponds to an idealized cosmic string everywhere, surrounded by a magnetic tube of radius a. In this tube three different configurations of magnetic fields have been taken into account: a cylindrical shell, a field decaying with 1/r and finally a homogeneous magnetic field. In order to develop these analysis we had to construct the normalized fermionic wavefunctions for the region outside the tube, and calculated the fermionic azimuthal current density by using the mode summation method. Although the magnetic field vanishes outside the tube, the magnetic field inside induces a non-vanishing azimuthal current density in the exterior region. This phenomenon is explicitly manifested in the structure of the induced current. The latter has been decomposed in two distinct contributions: The first one corresponds to the idealized cosmic string with a magnetic flux running along its core. It is given in (3.15) and is a periodic function of the total flux with the period equal to quantum flux, Φ 0 = 2π/e. The second contribution named core-induced given by (3.21), takes into account a specific configuration for the magnetic field inside the tube, and in general, is not a periodic function of the magnetic flux and depends on the total magnetic flux inside the core. By general analysis we could observe that the core-induced azimuthal current decays with e −2mr /(mr) 3 for r >> a (see (3.23)). Because the corresponding zero-thickness azimuthal cur-rent decays as e −2mr sin(π/q) /(mr) 5/2 , the latter dominates the total azimuthal current. For massless field and at large distance from the core, the result was given by Eq. (3.36), and explicitly we see that this current decays with 1 r 4 (r/a) 2β . Comparing this result with the corresponding one for the zero-thickness azimuthal current which decays with 1/r 4 , we conclude that for large distance the total azimuthal current is dominated by the zero-thickness contribution. Finally for point near the tube core, the core-induced current diverges with 1 r 4 (1−a/r) 3 , as shown by (3.43). So this contribution is dominant in this region. Finally we have also provide, by using numerical evaluation, the behavior of the core-induced current as function of several physical quantities relevant in our analysis. In Fig. 1 we have two plots. In the left plot an expected result is presented. The current changes its sign when we change the sign of α. In the right plot, it is exhibited the behavior of the current for the three models of magnetic field as function of mr. It is shown that the intensity of the current induced by the first model is the largest one. In Fig. 2, we exhibit the behavior of current density for the first model as function of mr, for fixed value of α and different values of q. By this plot we can infer that the intensity of the current increasing with q. Finally Fig. 3, we have displayed current as function of the intensity of the magnetic flux. In the left plot we have considered only the first model fixing mr and varying q. This plot reinforces the fact that the intensity of the current increases with q; moreover, it shows that the current is not a periodic function of the flux. The right plot exhibits the behavior of the core-induced current, for the three models, for fixed value of q. Also this plot reinforce that the first model provides the current with biggest intensity. ( 2 . 6 ) 26For the first model, representing a cylindrical magnetic shell, a(r) = Θ(r − a) .(2.7) (2.36),(2.41) and the explicit form of the Dirac matrices given by (2.3) and (2.4) the following expression for the induced azimuthal current density in the region outside the tube is obtained: e −y−m 2 r 2 /(2y) [I(q, α 0 , y) − I(q, −α 0 , y)] , (3.9) f (q, α 0 , y) = cos [qπ (1/2 − α 0 )] cosh [(qα 0 + q/2 − 1/2) y] − cos [qπ (1/2 + α 0 )] cosh [(qα 0 − q/2 − 1/2) y] .(3.14) Figure 1 : 1The core-induced azimuthal current density is plotted, in units of m 4 e as function of mr for q = 1.5 and ma = 1. On the left plot we consider the current induced by the magnetic field configuration of the first model, and taking α = 2.1 and α = −2.1. On the right plot we display the intensity of the core-induced current for the three different models of magnetic fields considering α = 2.1. Figure 3 : 3The core-induced azimuthal current density is plotted in units of m 4 e, as function of α for ma = 1 and mr = 2. On the left plot, we exhibit only the current induced by the first model considering different values of q, and on the right plot, we exhibit the current induced by the three different models of magnetic fields, considering q = 1.5. Moreover, induced fermionic and scalar current densities in compactified cosmic string spacetimes have been considered in[17] and[18]. . A A Abrikosov, Zh. Eksp. Teor. Fiz. 321442A. A. Abrikosov, Zh. Eksp. Teor. Fiz. 32, 1442 (1957). . N B Nielsen, P Olesen, Nucl. Phys. B. 6145N.B. Nielsen, P. Olesen, Nucl. Phys. B 61, 45 (1973). . D Garfinkle, Phys. Rev. D. 311323D. Garfinkle, Phys. Rev. D 31 1323 (1985). . P Laguna-Castillo, R A Matzner, Phys. Rev. D. 352933P. Laguna-Castillo and R. A. Matzner, Phys. Rev. D 35 2933 (1987). A Vilenkin, E P S Shellard, Cosmic String and Other Topological Defects. CambridgeCambridge University PressA. 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{'abstract': 'In this paper, we analyze the vacuum azimuthal fermionic current induced by a magnetic field confined in a cylindrical tube of finite radius a, in the cosmic string spacetime. Three distinct configurations for the magnetic field are taken into account: (i) a cylindrical shell of radius a, (ii) a magnetic field proportional to 1/r and (iii) a constant magnetic field. In these three cases, the axis of the infinitely long tube of radius a coincides with the cosmic string; moreover, we only develop this analysis for the region outside the tube. In order to do that, we explicitly construct the corresponding complete set of normalized wave-functions. We show that in the region outside the tube, the induced current is decomposed into a part corresponding to a zero-thickness magnetic flux in addition to a core-induced contribution. The latter presents specific form depending on the magnetic field configuration considered. The zero-thickness contribution depends only on the fractional part of the ration of the magnetic flux inside the tube by the quantum one. As to the core-induced contribution, it depends on the total magnetic flux inside the tube, and consequently, in general, it is not a periodic function of the flux.', 'arxivid': '1801.01344', 'author': ['M S Maior De Sousa \nDepartamento de Física-CCEN\nUniversidade Federal da Paraíba\n58.059-970, J. PessoaPB C. Postal 5.008Brazil\n', 'R F Ribeiro \nDepartamento de Física-CCEN\nUniversidade Federal da Paraíba\n58.059-970, J. PessoaPB C. Postal 5.008Brazil\n', 'E R Bezerra De Mello \nDepartamento de Física-CCEN\nUniversidade Federal da Paraíba\n58.059-970, J. PessoaPB C. Postal 5.008Brazil\n'], 'authoraffiliation': ['Departamento de Física-CCEN\nUniversidade Federal da Paraíba\n58.059-970, J. PessoaPB C. Postal 5.008Brazil', 'Departamento de Física-CCEN\nUniversidade Federal da Paraíba\n58.059-970, J. PessoaPB C. Postal 5.008Brazil', 'Departamento de Física-CCEN\nUniversidade Federal da Paraíba\n58.059-970, J. PessoaPB C. Postal 5.008Brazil'], 'corpusid': 55408501, 'doi': '10.3390/particles1010006', 'github_urls': [], 'n_tokens_mistral': 14791, 'n_tokens_neox': 12798, 'n_words': 7643, 'pdfsha': '618b82e5b517845b82d01c4eed1de5129e4e0613', 'pdfurls': ['https://arxiv.org/pdf/1801.01344v1.pdf'], 'title': ['Azimuthal fermionic current in the cosmic string spacetime induced by a magnetic tube', 'Azimuthal fermionic current in the cosmic string spacetime induced by a magnetic tube'], 'venue': []}

arxiv

arXiv:math-ph/0105043v4 5 Nov 2001 Finite Energy Superluminal Solutions of Maxwell Equations March 28, 2022 E Capelas De Oliveira Institute of Mathematics, Statistics and Scientific Computation IMECC-UNICAMP CP 6065 13083-970CampinasSPBrazil W A Rodrigues Jr Department of Mathematics ‡ On leave in absence: Institute of Mathematics, Statistics and Scientific Computation, UNICAMP and Wernher von Braun Advanced Research Center, UNISAL University of Liverpool L69 3BXLiverpoolUK † arXiv:math-ph/0105043v4 5 Nov 2001 Finite Energy Superluminal Solutions of Maxwell Equations March 28, 2022 We exhibit exact finite energy superluminal solutions of Maxwell equations in vacuum and discuss the physical meaning of these solu- Recently, some papers [1,2] have appeared in the literature showing that in some hypothetical media there is the possibility of the existence of superluminal electromagnetic pulses (solutions of Maxwell equations) such their fronts travel in the media with superluminal velocities. Now, the solutions discovered in [1,2], despite their theoretical interest have infinite energy and as such cannot be produced in the physical world. Only finite aperture approximations to these waves can eventually be produced (supposing the existence of the special media). The objective of this letter is to show that in contrast to the solutions discovered in [1,2] (that, as already said have infinite energy), there exist vacuum solutions of Maxwell equations which are finite energy superluminal solutions . These new solutions, as we shall see, appear when we solve some Sommerfeld like problems 3,4 to be reported below. We discuss if the new solutions can be realized in the physical world. Moreover, we emphasize that the new solutions correspond to phenomenon distinct to already observed wave motion with superluminal [5][6][7][8] (or even negative [9,10]) group velocities. In the case,e.g., of experiments [5][6][7][8] only the peaks of the waves travel (for a while) with superluminal velocity whereas their fronts always travel at the velocity of light. We start by recalling how to write electromagnetic field configurations in terms of Hertz potentials [11,12]. Suppose we have a Hertz potential Π m of magnetic type. In what follows we use units such that the velocity of light is c = 1. Then, the associated electromagnetic field is given by E = − ∂ ∂t (∇ × Π m ), B = ∇ × ∇ × Π m .(1) Let us take Π m = Φê z . Then, since the Hertz potential (in vacuum) satisfies a homogeneous wave equation, we have that Φ = 0.(2) The Sommerfeld problem (not to be confused with a Cauchy problem) to be considered here is the following. In a given inertial frame (the laboratory 1 ) find a solution Φ X : (t, x) → C (where C is the field of complex numbers) for eq.(2) satisfying the following boundary conditions 2 at the z = 0 plane,        Φ X (t, ρ, 0) = T(t) ∞ −∞ dωB(ω)J 0 (ωρ sin η)e −iωt , ∂Φ X (t,ρ,z) ∂z z=0 = iT(t) cos η ∞ −∞ dωB(ω)J 0 (ωρ sin η)k(ω)e −iωt ,(3) where T(t) = [Θ(t + T ) − Θ(t − T )] , Θ is the Heaviside function, k(ω) = ω, and η is a constant called the axicon angle [13][14][15][16][17][18] and B(k) is an appropriate frequency distribution. As showed in [14] the solution of eq.(2) (for z > 0, t > T ) which satisfies the Sommerfeld conditions is Φ X (t, ρ, z) =    ∞ −∞ dωB(ω)J 0 (ωρ sin η)e −iω(t−z cos η) for |t − z cos η| < T 0 for |t − z cos η| > T.(4) We call Φ X a scalar superluminal X-pulse. Now, as is well known, the energy density for a complex field configuration, like the Φ X , is (5) and the energy of the field configuration can be calculated by the volume integral of u on a constant time hyperplane, say t = T ′ > T . The calculation is easy when done in cylindrical coordinates. Recalling that from eq.(4) it follows that the support of the pulse at t = T ′ is △z = 2T / cos η, we have u = (∂ t Φ X )(∂ t Φ * X ) + (∂ x Φ X )(∂ x Φ * X ) + (∂ y Φ X )(∂ y Φ * X ) + (∂ z Φ X )(∂ z Φ * X ),E = 8πT sin 2 η cos η ∞ −∞ |B(k)| 2 kdk,(6) where the kinetic and potential energy terms give equal contributions. Eq. (6) gives finite energy for the scalar X-pulse for an infinity of frequency distribution functions B(k), such that |B(k)| 2 be null for k < 0. A trivial example is B(k) = [ Θ(k) − Θ(k − k 0 )] , with k 0 a constant. Now, we study the electromagnetic case. The non null components of the electromagnetic field 3 corresponding to a magnetic Hertz potential Π m = Φ Xêz are (for z > 0, t > T )                    E θ = i sin η ∞ −∞ dkB(k)k 2 J 1 (kρ sin η)e −ik(t−z cos η) , B ρ = −i 2 sin 2η ∞ −∞ dkB(k)k 2 J 1 (kρ sin η)e −ik(t−z cos η) , B z = sin 2 η ∞ −∞ dkB(k)k 2 J 0 (kρ sin η)e −ik(t−z cos η) , for |t − z cos η| < T, E θ = B ρ = B z = 0, for |t − z cos η| > T.(7) Now, using the standard energy density of the electromagnetic field [11,12], the energy of the superluminal electromagnetic X pulse results, E X = 1 2 2π 0 zmax z min ∞ 0 E θ E * θ + B ρ B * ρ + B z B * z ρdρdzdθ = 4πT cos η ∞ −∞ |B(k)| 2 k 3 dk.(8) Eq. (8) gives finite energy for superluminal solutions of Maxwell equations satisfying Sommerfeld boundary conditions (here expressed through conditions for the associated Hertz potential) for an infinity of possible frequency distributions B(k), as in the scalar case. We have four comments before ending this letter: (i) What does our finite energy solution (for the scalar wave equation) look like for an observer in a Lorentz frame Z ∈ sec T M, Z = 1 √ 1 − V 2 (∂ t + V ∂ z ),(9) which is moving with velocity V = cos η relative to the laboratory (the frame L = ∂ t ∈ sec T M )? As can be easily verified the transformed solution is: Φ ′ X (t ′ , ρ, z ′ ) =    ∞ −∞ dωB(ω)J 0 (ωρ sin η)e −iω sin ηt´f or |t ′ | < T / sin η 0 for |t ′ | > T / sin η.(10) The solution is independent of the spatial coordinate z and corresponds to a standing wave occupying all the rest space of the Z frame and that exists only for the time interval △t ′ = 2T / sin η. Is this result non physical? If not, what is the meaning of such a wave for the observers of the Z frame? As a Minkowski diagram can show, the wave stands for a finite period of time according to the time order of the Z frame because it is going to the past of the Z's observers. This must be a normal phenomenon if relativity theory is true and genuine superluminal motion exists. The observers at the Z frame will compute an infinite energy for that wave, but since they know relativity theory they will interpret the whole phenomena as follows: the wave that stands for a finite period of time at our frame is a superluminal finite energy wave produced in a laboratory ( the L frame) that is moving with velocity −1/ cos η relative to our frame (i.e., Z frame). Of course, the Z frames physicists cannot produce such a wave in their frame, due to two reasons. The first is that the wave according to them has infinite energy and the second, which is the crucial one, is simply because the device which produced it is at rest in another frame (the L frame). According to the Principle of Relativity the Z frame physicists can duplicate in their frame the device used in the L frame and launch a wave like the one given by eq.(4) (with boundary conditions like in eq.(3)) with the (t, ρ, z) substituted by (t ′ , ρ, z ′ ). Of course, if that would be possible, we would arrive at well known paradoxical situations 4 , that fortunately need not to be discussed here (see (iii) below). Note also that the Z frame mathematicians aware of the interpretation given by their fellow physicists can obtain directly the solution given by eq.(10) by solving a generalized mixed boundary value problem, where the boundary conditions are: Φ ′ X (t ′ , ρ, z ′ )| z ′ =− cos ηt ′ = [Θ(sin ηt ′ + T ) − Θ(sin ηt ′ − T )] ∞ −∞ dωB(ω)J 0 (ωρ sin η)e −iω sin ηt ′ γ ∂ ∂z ′ − γV ∂ ∂t ′ Φ ′ X (t ′ , ρ, z ′ )| z ′ =− cos ηt ′ = i cos η [Θ(sin ηt ′ + T ) − Θ(sin ηt ′ − T )] ∞ −∞ dωB(ω)ωJ 0 (ωρ sin η)e −iω sin ηt ′ . (ii) Of course, an analogous analysis holds for the finite energy superluminal solutions of Maxwell equations that we have just found. It is worth saying here that the existence of such solutions does not conflict with the famous result on the Cauchy problem concerning the Maxwell equations. That result says: any electromagnetic field configuration with compact support at t = 0, let us say for | x| ≤ R, is such that the field is null for t > 0 for all | x| ≥ R + t. 5 4 More details on this issue can be found in [18]. 5 A proof of an anlagous theorem for the homogeneous wave equation can be found in [13]. For Maxwell equations see [14]. (iii) Is it possible to build a physical device to launch a finite energy superluminal electromagnetic X pulse? Our answer is no. Indeed, finite aperture approximations (FAA) to exact superluminal X-like solutions of Maxwell equations (which, of course have finite energy) have already been produced [7,8]. However, these FAA are such that their peaks move with velocity v > 1 but their front always moves with the speed of light. This result has been predicted in [16,18] and is endorsed by the experimental results of [7,8] as proved in [13]. Now, concerning the solutions we just found, in order for them to be produced (by an antenna) as real physical waves it is necessary to produce waves that extend in all the z = 0 plane where the antenna is located for the time interval −T < t < T . Of course, this is physically impossible because it would require that the antenna should be an infinite one. (iv) Besides the superluminal solutions just found, there are also finite energy subluminal solutions (to be reported elsewhere). We must say that even if the new superluminal solutions cannot be produced by physical devices the only possible reason for their non existence in our universe is that of a possible violation of the principle of relativity. Eventually these new superluminal solutions may also find applications in the understanding of some fundamental issues concerning the nonlocality problem in quantum mechanics [21]. The laboratory is modeled by time like vector field L = ∂ ∂t ∈ sec T M .2 The necessity for these boundary conditions is proved in[13]. Called a superluminal electromagnetic X pulse[13,14]. Lorentz Invariant Superluminal Tunneling. in publ. P Ghose, M K Samal, Phys. Rev. E. 6436620P. Ghose and M.K. Samal, Lorentz Invariant Superluminal Tunneling. in publ. Phys. Rev. E 64, # 036620 (2001).(http://www.arXiv:quant- ph/0011033 v3). Possibility of a light pulse with speed greater than c. X Zhou, Phys. Lett. A. 278X. Zhou, Possibility of a light pulse with speed greater than c, Phys. Lett. A 278, 1-5 (2001) L Brillouin, Wave Propagation and Group Velocity. New YorkAcademic PressL. Brillouin, Wave Propagation and Group Velocity, Academic Press, New York, 1960. Transient Tunnel Effect and Sommerfeld Problem. F A Mehmeti, Akademie VerlagBerlinF.A. Mehmeti,Transient Tunnel Effect and Sommerfeld Problem, Akademie Verlag, Berlin, 1996. On superluminal barrier traversal. A Enders, G Nimtz, J. Phys. I (France). 2A. Enders, G.Nimtz, On superluminal barrier traversal, J. Phys. I (France) 2, 1693-1698 (1992). Measurement of the single-photon tunneling time. A M Steinberg, P G Kwiat, R Y Chiao, Phys. Rev. Lett. 71A. M. Steinberg, P.G. Kwiat, and R.Y. Chiao, Measurement of the single-photon tunneling time, Phys. Rev. Lett.71, 708-711 (1993) Evidence of X-shaped propagation-invariant localized light waves. P Saari, K Reivelt, Phys. Rev. Lett. 21P. Saari and K. Reivelt, Evidence of X-shaped propagation-invariant localized light waves, Phys. Rev. Lett. 21, 4135-4138 (1997). Observation of superluminal behaviors in wave propagation. D Mugnai, A Ranfagni, R Ruggeri, Phys. Rev. Lett. 80D. Mugnai , A. Ranfagni, and R. Ruggeri, Observation of superluminal behaviors in wave propagation, Phys. Rev. Lett. 80, 4830-4833 (2000). Optical pulse propagation at negative group velocities due to nearby gain line. E L Bolda, J C Garrison, R Y Chiao, Phys.Rev. A. 49E. L. Bolda, J. C. Garrison, and R.Y. Chiao, Optical pulse propagation at negative group velocities due to nearby gain line, Phys.Rev. A 49, 2938-2947 (1994) Gain-assisted superluminal light propagation. L J Wang, A Kumzmich, A Dogariu, Nature. 406L.J. Wang, A. Kumzmich, and A. Dogariu, Gain-assisted superluminal light propagation, Nature 406, 277-279 (2000). J A Stratton, Electromagnetic Theory. New YorkMcGraw-HillJ. A. Stratton, Electromagnetic Theory, McGraw-Hill, New York, 1941. W K H Panofski, M Phillips, Classical Electricity and Magnetism. Reading, MAAddison-Wesley2nd ed.W.K.H. Panofski, and M. Phillips, Classical Electricity and Magnetism, 2nd ed., Addison-Wesley, Reading, MA, 1962. Causal explanation of superluminal behavior of microwave propagation in free space. W A RodriguesJr, D S Thober, A L XavierJr, Phys. Lett. A. 284W.A. Rodrigues Jr., D.S. Thober, and A.L. Xavier Jr., Causal explana- tion of superluminal behavior of microwave propagation in free space, Phys. Lett. A 284, 217-224 (2001). Thoughtful comments on 'Bessel beams and signal propagation. E Capelas De Oliveira, W A RodriguesJr, D S Thober, A L XavierJr, Phys. Lett. A. 284E. Capelas de Oliveira, W.A. Rodrigues Jr., D.S. Thober and A.L. Xavier Jr., Thoughtful comments on 'Bessel beams and signal propa- gation', Phys. Lett. A 284, 296-303 (2001). Exact solutions for nondiffracting beams.1.The scalar theory. J Durnin, J. Opt. Soc. Am. A. 4J. Durnin, Exact solutions for nondiffracting beams.1.The scalar theory, J. Opt. Soc. Am. A 4, 651-654 (1987). On the existence of undistorted progressive waves (UPWs) of arbitrary speeds 0 ≤ v < ∞ in nature. W A RodriguesJr, J Y Lu, Found. Phys. 27W.A. Rodrigues Jr., and J.Y. Lu, On the existence of undistorted pro- gressive waves (UPWs) of arbitrary speeds 0 ≤ v < ∞ in nature. Found. Phys. 27, 435-503 (1997). Superluminal electromagentic waves in free space. E , Capelas Oliveira, W A RodriguesJr, Ann. der Physik. 7E. Capelas Oliveira, and W A. Rodrigues Jr., Superluminal electroma- gentic waves in free space, Ann. der Physik 7, 654-659 (1998). What is Superluminal Wave Motion. J E Maiorino, W A RodriguesJr, Sci. and Tech. Mag. 42J. E. Maiorino, and W.A. Rodrigues Jr., What is Superluminal Wave Motion? electronic book at http://www.cptec.br/stm, Sci. and Tech. Mag. 4(2) (1999). M E Taylor, Pseudo Differential Operators. PrincetonPrinceton Univ. PressM.E. Taylor, Pseudo Differential Operators. Princeton Univ. Press, Princeton, 1981. . R Courant, D Hilbert, Methods of Mathematical Physics. 2John Wiley and SonsR. Courant, and D. Hilbert, Methods of Mathematical Physics, vol. 2. John Wiley and Sons, New York, 1966. A A Grib, W A RodriguesJr, Nonlocality in Quantum Physics. New YorkKluwer Acad./Plenum PublA.A. Grib, and W.A. Rodrigues Jr., Nonlocality in Quantum Physics. Kluwer Acad./Plenum Publ., New York, 1999.

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{'abstract': 'We exhibit exact finite energy superluminal solutions of Maxwell equations in vacuum and discuss the physical meaning of these solu-', 'arxivid': 'math-ph/0105043', 'author': ['E Capelas De Oliveira \nInstitute of Mathematics, Statistics and Scientific Computation\nIMECC-UNICAMP CP 6065\n13083-970CampinasSPBrazil\n', 'W A Rodrigues Jr\nDepartment of Mathematics\n‡ On leave in absence: Institute of Mathematics, Statistics and Scientific Computation, UNICAMP and Wernher von Braun Advanced Research Center, UNISAL\nUniversity of Liverpool\nL69 3BXLiverpoolUK\n', '† '], 'authoraffiliation': ['Institute of Mathematics, Statistics and Scientific Computation\nIMECC-UNICAMP CP 6065\n13083-970CampinasSPBrazil', 'Department of Mathematics\n‡ On leave in absence: Institute of Mathematics, Statistics and Scientific Computation, UNICAMP and Wernher von Braun Advanced Research Center, UNISAL\nUniversity of Liverpool\nL69 3BXLiverpoolUK'], 'corpusid': 119742178, 'doi': '10.1016/s0375-9601(01)00748-4', 'github_urls': [], 'n_tokens_mistral': 5182, 'n_tokens_neox': 4524, 'n_words': 2586, 'pdfsha': '0aa9c9d5b041bde260a60d3b9c1c93bd14301af9', 'pdfurls': ['https://export.arxiv.org/pdf/math-ph/0105043v4.pdf'], 'title': ['arXiv:math-ph/0105043v4 5 Nov 2001 Finite Energy Superluminal Solutions of Maxwell Equations', 'arXiv:math-ph/0105043v4 5 Nov 2001 Finite Energy Superluminal Solutions of Maxwell Equations'], 'venue': []}

arxiv

LARGE DEVIATIONS FOR A MEAN FIELD MODEL OF SYSTEMIC RISK 29 Aug 2012 Josselin Garnier George Papanicolaou ANDTzu-Wei Yang LARGE DEVIATIONS FOR A MEAN FIELD MODEL OF SYSTEMIC RISK 29 Aug 2012arXiv:1204.3536v2 [q-fin.RM]mean fieldlarge deviationssystemic riskdynamic phase transitions AMS subject classifications 60F1060K3591B3082C26 We consider a system of diffusion processes that interact through their empirical mean and have a stabilizing force acting on each of them, corresponding to a bistable potential. There are three parameters that characterize the system: the strength of the intrinsic stabilization, the strength of the external random perturbations, and the degree of cooperation or interaction between them. The latter is the rate of mean reversion of each component to the empirical mean of the system. We interpret this model in the context of systemic risk and analyze in detail the effect of cooperation between the components, that is, the rate of mean reversion. We show that in a certain regime of parameters increasing cooperation tends to increase the stability of the individual agents but it also increases the overall or systemic risk. We use the theory of large deviations of diffusions interacting through their mean field. Introduction. Systemic risk is the risk that in an interconnected system of agents that can fail individually, a large number of them fails simultaneously or nearly so, leading to the overall failure of the system. It is a property of the interconnected system as a whole, and not only of the individual components, in the sense that assessment of the risk of individual failure alone cannot provide an assessment of the systemic risk. The interconnectivity of the agents, its form and evolution, play an essential role in systemic risk assessment [6]. In this paper we consider a simple model of interacting agents for which systemic risk can be assessed analytically in some interesting cases. Each agent can be in one of two states, a normal and a failed one, and it can undergo transitions between them. We assume that the dynamic evolution of each agent has the following features. First, there is an intrinsic stabilization mechanism that tends to keep the agents near the normal state. Second, there are external destabilizing forces that tend to push away from the normal state and are modeled by stochastic processes. Third, there is cooperation among the agents that acts as individual stabilizer by diversification. This means that in such a system we expect that there is a decrease in the risk of destabilization or "failure" for each agent because of the cooperation or diversification. What is less obvious is the effect of cooperation on the overall or system's risk, which can be defined in a precise way for the model considered here. We show in this paper that for the models under consideration and in a certain regime of parameters, the systemic risk increases with increasing cooperation. The aim of this paper is to analyze this tradeoff between individual risk and systemic risk for a class of interacting systems subject to failure. Perhaps a simple mathematical model of interacting agents having the features we want is a system of stochastic differential equations with mean-field interaction. Let x j (t) be the state of risk of agent or component j, taking real values. For j = 1, . . . , N , the x j (t)'s are modeled as continuous-time stochastic processes satisfying the system of Itô stochastic differential equations: dx j (t) = −hU (x j (t))dt + θ(x(t) − x j (t))dt + σdw j (t), (1.1) with given initial conditions. Here −hU (y) = −hV ′ (y) is the restoring force, V is a potential which we assume has two stable states, and {w j (t)} N j=1 are independent, standard Brownian motions. The parameter h controls the level of intrinsic stabilization and σ is the strength of the destabilizing random forces. The interaction or cooperation is the mean reversion term with rate of mean reversion θ and with x(t) := 1 N N i=1 x i (t) denoting the empirical mean of the processes, that is, the empirical mean of the individual risks. For θ > 0 the individual risk processes tend to mean-revert to their empirical mean, which is a simple but non-trivial form of cooperation. We take the empirical meanx(t) to be a measure of the systemic risk. The bi-stable-state structure of V (y) determines the normal and failed states of the agents. We will assume in this paper that U (y) = y 3 − y, so that V (y) = 1 4 y 4 − 1 2 y 2 + c and we take c = 0 since it is inessential. The two stable states are then ±1 and we let −1 be the normal state and +1 to be the failed state. The potential V (y) ensures that each risk variable x j (t) stays around −1 (normal) or +1 (failed). The evolution of the system is characterized by the initial conditions, the three parameters (h, θ, σ) and by the system size N . We have chosen a mean-field interaction because it is a simple form of cooperative behavior. More elaborate models are considered in Section 3, where some heterogeneity is introduced between the components of the system. For mean-field models a natural measure of systemic risk is the transition probability of the empirical mean x(t) from the normal state to the failed state. More precisely, the mathematical problem we address here is this: For N large we calculate approximately such transition probabilities and analyze how they depend on h, σ and θ, the three parameters of the system. We are interested in a regime of these parameters for which there are two collective, that is, large N , equilibria centered around the normal and failed states. These two equilibria can be identified through the mean-field limit of the system, that is, the weak limit in probability of the empirical density of the agents risk x j . Mean field models with multiple stable points, not only bistable ones, could be considered but their analysis is more involved while the main result about systemic risk, and dependence on the parameters (h, θ, σ) and by the system size N , is clearly seen in the bistable model that we consider here. The mathematical analysis of bistable mean field models like (1.1) was initiated by Dawson [9,18], including the mean field limit, the existence of multiple equilibria, and a fluctuation theory. Non-equilibrium statistical mechanics and phase transitions have been studied extensively in the sciences [19]. The large deviation theory that we use here was developed by Dawson and Gärtner [10,11]. In particular, they introduced and analyzed the rate function for large deviations associated with (1.1) when N is large and with more general potentials [11]. Their theory may be considered as an infinite dimensional extension of the Freidlin-Wentzell theory of large deviations for stochastic differential equations with small noise [16,14]. The main result in this paper is the analysis of this rate function for small h. That is, for a shallow twowell potential, where transitions from one well (quasi-equilibrium) to the other are exponentially small in N , the "constant" in the exponent is small when h is small. Other mean field models have been studied in [33,18,27,2,28,30,15], and large deviations results for various models can be found in [12,1,29,13,22,8,7]. In [7] a general large deviations theory is developed for a model with both drift and volatility interactions, as well as with degenerate noise, using weak convergence and optimal control methods. The main contribution of the paper as far as systemic risk theory is concerned is the demonstration that, within the range of the bistable mean field model (1.1), while cooperation between agents decreases the individual risk of each agent, the systemic or overall risk is increased. This is discussed in detail in Section 6.4, in terms of the three parameters (h, θ, σ), with h small. The fact that reducing individual risk by cooperation or diversification can lead to increased systemic risk has been anticipated in macroeconomics and elsewhere and it has been extensively discussed, modeled, and analyzed in [31,4,20,17,26,32,5,3,21,24]. However, the dynamic phase transitions formulation and the large deviations theory exploited in this paper have not been used in the economics literature, to our knowledge. The use of coupled stochastic equations for modeling evolution of individual risk and the effects of interactions among agents is also considered in [4,23] where there is some discussion regarding the economic interpretation of the variables {x j (t)}. They could, for example, represent some form of equity ratio in a very simple model in insurance or banking. The paper is organized as follows. In Section 2, we briefly review the classical mean-field limit in [9], and we discuss the intrinsic stability of equilibria [9] when h is small. Section 3 generalizes (1.1) by replacing the rate of mean reversion θ by an agent-dependent θ j . The mean-field limit and the explicit conditions are also studied. In Section 4, we carry out numerical simulations of both the homogeneous and the heterogeneous model in various parameter ranges. Section 5 uses the large deviation principle in [10] to formulate the dynamic phase transition of interest here, that is, the system transition from the normal state to the failed state. In Section 6, we specialize the large deviations theory when h is small so as to obtain a result from which the systemic risk as a function the basic parameters (h, θ, σ) can be assessed and interpreted. In Section 7 we introduce a formal expansion of the rate function for small h and obtain a reduced variational principle for the systemic risk that appears to come from a large deviations principle for an one-dimensional dynamical system. It gives, of course, the same results about systemic risk as described in Section 6. In Section 8 we discuss the case where there is diversity in mean reversion and it is shown that under some natural conditions the heterogeneous model is systemically more unstable than the homogeneous one. The technical details of the proofs are in the appendices. 2. The Mean-Field Limit. We briefly review the mean field limit in [9,18] and carry out a small h analysis of results since they will be used in calculating large deviation probabilities. We want to analyze the systemic behavior of the interacting diffusion processes (1.1), through their empirical meanx(t), but this is not possible in a direct way since (1.1) is nonlinear. We consider instead the empirical density of x j (t), which is a measure valued process that has a limit as N → ∞. Let M 1 (R) be the space of probability measures endowed with the weak (Prohorov) topology and let C([0, T ], M 1 (R)) be the space of continuous M 1 (R)-valued processes on [0, T ] endowed with the corresponding weak topology. Define the empirical probability measure process X N (t, dy) := 1 N N j=1 δ xj(t) (dy) and note that X N ∈ C([0, T ], M 1 (R)). The mean field limit theorem for X N , proved in [9,18], is as follows: Theorem 2.1. (Dawson, 1983) Assume that the force is U (y) = y 3 − y and that X N (0) converges weakly to a probability measure ν 0 . Then the measure valued process X N converges weakly in law as N → ∞ to a deterministic process with density u(t, y)dy ∈ C([0, T ], M 1 (R)), which is the unique weak solution of the Fokker-Planck equation: ∂ ∂t u = h ∂ ∂y [U (y)u] − θ ∂ ∂y yu(t, y)dy − y u + 1 2 σ 2 ∂ 2 ∂y 2 u, (2.1) with initial condition ν 0 . By Theorem 2.1, we can analyze u and view X N as a perturbation of u for N large. We may considerx(t) in the same way becausex(t) = yX N (t, dy). However, the limit problem is infinitely dimensional, as is expected. Explicit solutions of (2.1) are not available in general, but we can find equilibrium solutions. Assuming that ξ = lim t→∞ yu(t, y)dy, then an equilibrium solution u e ξ satisfies h d dy [(y 3 − y)u e ξ ] − θ d dy [(ξ − y)u e ξ ] + 1 2 σ 2 d 2 dy 2 u e ξ = 0, and has the form u e ξ (y) = 1 Z ξ 2π σ 2 2θ exp − (y − ξ) 2 2 σ 2 2θ − h 2 σ 2 V (y) ,(2.2) with Z ξ the normalization constant: Z ξ = 1 2π σ 2 2θ exp − (y − ξ) 2 2 σ 2 2θ − h 2 σ 2 V (y) dy. Now ξ must satisfy the compatibility or consistency condition: ξ = m(ξ) := yu e ξ (y)dy. (2.3) Finding equilibrium solutions has thus been reduced to finding solutions of this equation. For U (y) = y 3 − y, ξ = 0 is a solution for (2.3). With the same U (y), it can be shown (see also [9, Theorem 3.3.1 and 3.3.2]) that there are two additional non-zero solutions ±ξ b if and only if d dξ m(0) > 1, and for given h and θ, there exists a critical σ c (h, θ) > 0 such that d dξ m(0) > 1 if and only if σ < σ c (h, θ). An explanation for this bifurcation at equilibrium is that when σ ≥ σ c , randomness dominates the interaction among the components, i.e., θ(x(t) − x j (t))dt is negligible. In this case, the system behaves like N independent diffusions and hence, by the symmetry of V (y), at any given time roughly half of them stay around −1 and half around +1 so the average is 0. When, however, σ < σ c , then the interactive force is significantly larger (now σdw j (t) is less important). Therefore all agents stay around the same place (either −ξ b or +ξ b ) and the zero average equilibrium is unstable. Since we want to model systemic risk phenomena, we assume that σ < σ c throughout this paper, and we regard −ξ b as the normal state of the system and +ξ b as the failed state. The calculation of transitions probabilities between these two states is our objective. For small h we can approximate the solution of (2.3) to order O(h) as follows. Proposition 2.2. For small h, the critical value σ c can be expanded as σ c = 2θ 3 + O(h). (2.4) In addition, the non-zero solutions ±ξ b are ± ξ b = ± 1 − 3 σ 2 2θ 1 + h 6 σ 2 σ 2 2θ 2 1 − 2(σ 2 /2θ) 1 − 3(σ 2 /2θ) + O(h 2 ). (2.5) Proof. See Appendix A. From Proposition 2.2, we see the relation between the existence of the bi-stable states and the ratio σ 2 /2θ: For a given θ, and for small h, (2.3) has non-zero solutions if and only if 3σ 2 /2θ < 1. Moreover, these non-zero solutions ±ξ b are generally not ±1 since the magnitude |ξ b | is less than 1. Note that the coefficient of order h in the expansion (2.5) depends significantly on θ and σ. Thus, when 3σ 2 /2θ tends to 1, ξ b in (2.5) will not go to +∞ while, in fact, ξ b goes to 0. From the O(1) term in (2.5), we also see that ξ b is roughly decreasing as σ 2 /2θ is increasing. 3. Diversity of Sensitivities. We can generalize (1.1) by allowing for agent dependent coefficients. We consider a particular case in which each agent can have a different rate of mean reversion to the empirical mean, that is, for j = 1, . . . , N , dx j = −h ∂ ∂x j V (x j )dt + σdw j + θ j (x − x j )dt,(3.1) and as before V (y) = 1 4 y 4 − 1 2 y 2 . We consider the case where θ 1 , . . . , θ N take K distinct positive numbers, Θ 1 , . . . , Θ K . We define I l = {j : θ j = Θ l }, ρ l = |I l |/N and X l N = 1 ρ l N j∈I l δ xj . Assuming that lim N →∞ ρ l exists and is positive for all l, the limit of (X 1 N , . . . , X K N ) as N → ∞ are the weak solutions (u 1 , . . . , u K ) of the set of K coupled Fokker-Planck equations. Theorem 3.1. Assume that U (y) = y 3 − y and that (X 1 N (0), . . . , X K N (0)) converge weakly in probability to the probability measures (ν 1 , . . . , ν K ). Then the measure valued vector process (X 1 N , . . . , X K N ) converges weakly as N → ∞ to the weak solution (u 1 , . . . , u K ) of the system of the Fokker-Planck equations: ∂ ∂t u 1 = 1 2 σ 2 ∂ 2 ∂y 2 u 1 − Θ 1 ∂ ∂y y K l=1 ρ l u l (t, y)dy − y u 1 + h ∂ ∂y [U (y)u 1 ] (3.2) . . . ∂ ∂t u K = 1 2 σ 2 ∂ 2 ∂y 2 u K − Θ K ∂ ∂y y K l=1 ρ l u l (t, y)dy − y u K + h ∂ ∂y [U (y)u K ], with initial condition (ν 1 , . . . , ν K ). Proof. See Appendix B.1 for the outline of the proof following [18]. The equilibrium solutions {u e l,ξ } K l=1 have the form and {u e l,−ξ b } K l=1 if and only if d dξ m(0) > 1. The numerical simulations presented in the next section show that diversity in the rate of mean reversion can have significant impact on the stability of the mean-field model. u e l,ξ (y) = 1 Z l,ξ 2π σ 2 2Θ l exp − (y − ξ) 2 2 σ 2 2Θ l − h 2 σ 2 V (y) (3.3) Z l,ξ = 1 2π σ 2 2Θ l exp − (y − ξ) 2 2 σ 2 2Θ l − h 2 σ 2 V (y) dy, As in the homogeneous case, we can get an approximate condition for equilibrium bifurcation for small h. σ div c = K l=1 ρ l Θ l / K l=1 3ρ l 2Θ 2 l + O(h). Proof. See Appendix B.2. We note that diversity does affect the threshold condition and makes the analysis more difficult. The non-zero solutions ±ξ b can be computed approximately when h is small: ± ξ b = ± K l=1 ρ l Θ l 1 − 3 σ 2 2Θ l / K l=1 ρ l Θ l + O(h). (3.5) Higher order terms in the expansion of (3.5) can also be obtained but we will omit them in this paper. In the following Proposition we show that σ div This result shows that when there is diversity the parameter region of existence of equilibria ±ξ b is smaller than in the homogeneous case . From this observation we can anticipate that these equilibria are less stable in the presence of diversity, and this is confirmed next by numerical simulations and analytically. By noting that ξ homo b = 1 − (σ 2 /σ homo c ) 2 + O(h) and ξ div b = 1 − (σ 2 /σ div c ) 2 + O(h), we have the following corollary: Corollary 3.4. With θ = K l=1 ρ l Θ l , we have 1 > ξ homo b ≥ ξ div b for small h. 4. Numerical Simulations. Before going into a detailed analysis of the models, we carry out numerical simulations of (1.1) and (3.1) so as to get a quick impression of their behavior. We discretize with a uniform time grid, and let X n j denote the simulated X j at time n∆t. Homogeneous Model. We simulate (1.1) using the Euler scheme Simulations for different σ. The system has two (statistically) stable equilibria when σ is below the critical value or otherwise has single stable state 0. For small h, 3σ 2 /2θ < 1 is the approximate criterion. Simulations for different θ. The system has two stable equilibria if θ is above the critical value or otherwise has single stable state 0. For small h, 3σ 2 /2θ < 1 is the approximate criterion. X n+1 j = X n j − hU (X n j )∆t + σ∆W n+1 j + θ( 1 N N k=1 X n k − X n j )∆t. We take U (y) = y 3 − y, = 1, X 0 j = −1, ∆t = 0.02, and let {∆W n j } j,n be independent Gaussian random variables with mean zero and variance ∆t. In the figures presented, the dashed lines show the numerical solutions of the compatibility equation (2.3), ξ = m(ξ). As noted earlier, if d dξ m(0) ≤ 1, then 0 = m(0) is the unique solution and 0 is a stable state. Therefore we should observe that the systemic risk fluctuates around 0. If d dξ m(0) > 1, there are two additional non-zero solutions ±ξ b = m(±ξ b ) and ±ξ b are stable while 0 is unstable. We also know that when h is small, the condition d dξ m(0) > 1 can be simplified to be 3σ 2 /2θ < 1. Figure 4.2 illustrate the behavior of the empirical mean as the system transitions from having two equilibria to having a single one, which is controlled by the value of d dξ m(0). This is an instance of a bifurcation of equilibria. From Proposition 2.2, we know that when h is small, the existence condition of two equilibria, d dξ m(0) > 1, can be approximated by the condition 3σ 2 /2θ < 1. In the simulations, we let h = 0.1 so the approximate condition 3σ 2 /2θ < 1 can be applied. In Figure 4.1 we change σ but fix the other parameters, and consider the three cases Figure 4.2 we change θ. We can see that even though the parameters varied in the numerical simulations are not the same, the bifurcation behavior is similar. Figure 4.3 shows the effect of increasing h on the system stability. By stability we mean resistance to the transition of the empirical mean of the system from one state to the other (because the model is symmetric). The parameter h is proportional to the height of the potential barrier of each agent. Thus we increase the overall system stability if we increase the component's stability. This observation is analogous to comments in [31,25,26]. It is clear that h influences system stability substantially. system is more stable. These stability phenomena will be quantified with the large deviations analysis of Section 5. d dξ m(0) < 1 (3σ 2 /2θ > 1), d dξ m(0) ≈ 1 (3σ 2 /2θ = 1) and d dξ m(0) > 1 (3σ 2 /2θ < 1). In Heterogeneous Model. For the heterogeneous model, θ is replaced by θ j , and the discretization is X n+1 j = X n j − hU (X n j )∆t + σ∆W n+1 j + θ j ( 1 N N k=1 X n k − X n j )∆t, (4.2) with the same parameter settings. The different values of θ j are controlled by the parameters Θ l and ρ l . In the simulation, we take K = 3 and {Θ l } K l=1 = {Θ L , Θ M , Θ H } for a system a low, medium and high rates of mean reversion to the empirical mean, that is, the systemic risk. We also take {ρ l } K l=1 = {ρ L , ρ M , ρ H } for the corresponding fractions. We use the normalized standard deviation of the distribution of θ j values in order to quantify diversity. We find that the heterogeneous model behaves like the homogeneous one when h, σ and N change. But, diversity on the rates of mean reversion has significant impact on system stability. As in the homogeneous case, in Figure 4.5 we consider cases with σ below, close to and above the critical value. The results are similar to the homogeneous case as expected. For σ below the critical value we have two equlibria and for σ above the critical value one equilibrium. The condition d dξ m(0) > 1 is still necessary and sufficient for the existence two equlibria. The condition K l=1 (ρ l /Θ l )(3σ 2 /2Θ l −1) < 1 is also a good approximation to the exact one when h is small. The parameter h and the system size N are closely associated with system stability. We note that in Figure 4.6 and Figure 4.7 when h or N are increased, the system becomes visibly more stable. Another observation is that with h, σ and N fixed, and with the mean of θ j of (4.2) equal to θ of (4.1), the heterogeneous system is consistently more unstable than the corresponding homogeneous model (see Fig. 4.5. Effect of changes in σ. The system has two stable equllibria when σ is below the critical value and has single one otherwise. For small h, We also change the diversity of θ j by changing Θ l and ρ l . To compare with the homogeneous case, in Figure 4.8 and Figure 4.9 we change the standard deviation of θ j while the mean of θ j is fixed. In this most interesting part of the simulations we see that when we increase the standard deviation of diversity values, the number of transitions is notably larger than that in the homogeneous case. K l=1 (ρ l /Θ l )(3σ 2 /2Θ l − 1) < 1 is the approximate criterion. Large Deviations. In the previous two sections we saw both analytically and numerically that for large N , the empirical density X N (t, dy) is close (weakly, in probability) to the solution of the Fokker-Planck equation (2.1), and so the meanx(t) in (1.1) stays around the first order moment of the deterministic limit, ∞ −∞ yu(t, y)dy. If the condition of existence of two equilibria is satisfied, thenx(t) will remain close to either −ξ b or +ξ b for relatively long time intervals, depending in particular on the parameter h. However, as long as N < ∞, as we have seen in the simulations the random forcing by the Brownian motions {w j (t)} N j=1 will cause transitions with non-zero probability. A systemic transition is the event thatx(t) is displaced from ±ξ b to ∓ξ b within a finite time horizon. Thus, systemic transition means that a large number of agents transition in a finite time. In this paper, we are interested in computing the probability of such a systemic transition. Mathematically, given a finite time horizon [0, T ] and the conditions for existence of two equilibria, we want to compute the probability P(x(0) = −ξ b ,x(T ) = ξ b ) (5.1) when N is large and as a function of the parameters (h, θ, σ) in (1.1). Large Deviations of Mean-fields. According to [10], we can calculate this probability asymptotically for large N using large deviations. To state the large deviations theory that we will use, we will review briefly some notation and terminology from [10]. (y)φ n (t, dy) < ∞. • Given ν ∈ M ∞ (R), we let E ν = {φ ∈ C([0, T ], M ∞ (R)) : φ(0) = ν}, endowed with the relative topology. To simplify the notation, we rewrite (2.1) as u t = L * u u + hM * u, where Fig. 4.9. The effect of changes in ρ l , with Θ l and the mean of θ j fixed. Increasing diversity tends to destabilize the system. L * ψ φ = 1 2 σ 2 φ yy + θ ∂ ∂y y − yψ(t, y)dy φ , M * φ = ∂ ∂y [U (y)φ] .function I h : − inf φ∈Å I h (φ) ≤ lim inf N →∞ 1 N log P(X N ∈ A) ≤ lim sup N →∞ 1 N log P(X N ∈ A) ≤ − inf φ∈Ā I h (φ), whereÅ andĀ are the interior and closure of A in E ν , respectively, and I h (φ) = 1 2σ 2 T 0 sup f : φ,f 2 y =0 J h (φ, f )dt, (5.2) J h (φ, f ) = φ t − L * φ φ − hM * φ, f 2 / φ, f 2 y , φ, f = ∞ −∞ f (y)φ(dy), if φ(t) is absolutely continuous in t ∈ [0, T ] and I h (φ) = ∞ otherwise. Remark. Here for φ ∈ E ν and t ∈ [0, T ], φ(t) is viewed as a real Schwartz distribution on R, L * ψ and M * φ are differential operators in the distribution sense, and f in (5.2) is a real Schwartz test function. The definition of absolute continuity for the path of measures (φ(t)) t∈[0,T ] is in the sense of Definition 4.1 in [10], that is to say: for each compact set K ⊂ R there exists a neighborhood U K of the null function in the set of test functions with compact support in K and an absolutely continuous function H K from [0, T ] to R such that | φ(t), f − φ(s), f | ≤ |H K (t) − H K (s)| for all s, t ∈ [0, T ] and f ∈ U K . Note that by Lemma 4.2 in [10], if φ(t) is absolutely continuous in t ∈ [0, T ], φ t (t) exists in the distribution sense almost everywhere on t ∈ [0, T ]. In order to use Theorem 5.1, we let ν = u e −ξ b in (2.2) and define the rare event A of systemic transition by A = φ ∈ E ν : φ(T ) = u e ξ b .(5.3) However, sinceÅ is an empty set, Theorem 5.1 give a trivial lower bound for the probability in question. Therefore we consider instead the closed rare event A δ : A δ = φ ∈ E ν : ρ(φ(T ), u e ξ b ) ≤ δ . Then Theorem 5.1 implies that − inf φ∈Å δ I h (φ) ≤ lim inf N →∞ 1 N log P(X N ∈ A δ ) ≤ lim sup N →∞ 1 N log P(X N ∈ A δ ) ≤ − inf φ∈A δ I h (φ). In addition, we show that inf φ∈A δ I h (φ) can be bounded from below by inf φ∈A I h (φ) as δ → 0. Lemma 5.2. By definition inf φ∈A δ I h (φ) is decreasing with δ > 0 and bounded from above by inf φ∈A I h (φ). In addition, lim δ→0 inf φ∈A δ I h (φ) ≥ inf φ∈A I h (φ). Proof. See Appendix C. Combining Lemma 5.2 and the fact that inf φ∈Å δ I h (φ) ≤ inf φ∈A I h (φ), for any ǫ > 0, we have for sufficiently small δ > 0 − inf φ∈A I h (φ) ≤ lim inf N →∞ 1 N log P(X N ∈ A δ ) ≤ lim sup N →∞ 1 N log P(X N ∈ A δ ) ≤ − inf φ∈A I h (φ) + ǫ. Therefore for large N and sufficiently small δ, P(X N ∈ A δ ) ≈ exp −N inf φ∈A I h (φ) . (5.4) This tells us that a larger system has a more stable empirical mean trajectory, which is consistent with what we have seen in the numerical simulation. Now the main step is finding inf φ∈A I h (φ), which is a min-max variational problem inf φ∈A I h (φ) = inf φ∈A 1 2σ 2 T 0 sup f : φ,f 2 y =0 φ t − L * φ φ − hM * φ, f 2 / φ, f 2 y dt,(5.5) where the f in the sup is a real Schwartz test function. An Alternative Expression for the Rate Function. The representation of the rate function (5.2) is somewhat complicated, but we can simplify it if φ has the density with some additional properties. If φ is a density function such that φ(t, y) is smooth, rapidly decreasing in y ∈ R for each t ∈ [0, T ] and is absolutely continuous in t ∈ [0, T ] for each y ∈ R, then let g(t, y) satisfy φ t − L * φ φ − hM * φ = (φg) y . (5.6) Note that because of the properties of φ, the left hand side of (5.6) is well-defined in y ∈ R and almost everywhere in t ∈ [0, T ]. In addition, because φ is positive valued, g exists and is unique except on a measure zero set in [0, T ]. Note that for the pair (φ, g) satisfying (5.6) sup f : φ(t),f 2 y =0 J h (φ(t), f ) = sup f : φ(t),f 2 y =0 φ(t), f y g 2 / φ(t), f 2 y = φ(t), g 2 , and therefore we have the following proposition. Proposition 5.3. If φ is a density function such that φ(t) is a Schwartz function for each t ∈ [0, T ] and is absolutely continuous in t ∈ [0, T ] for each y ∈ R, and g(t, y) satisfies (5.6), the rate function I h (φ) in (5.2) can be written in the form I h (φ) = 1 2σ 2 T 0 φ, g 2 dt. (5.7) We interpret (5.6) and (5.7) as follows. The function g is regarded as the driving force making φ deviate from the solution of the Fokker-Planck equation (2.1), and I h (φ) is the L 2 (φ) norm of g, which measures how difficult it is to have this deviation φ. 6. Small h Analysis. The goal of this section is to analyze the min-max problem (5.5) which controls the asymptotic systemic transition probability. This problem is nonlinear and infinitely dimensional and is difficult to analyze. To get some useful information about it we will assume that h is small and analyze it in this regime. We will first solve (5.5) when h is exactly 0, and then we will get rigorous upper and lower bounds for (5.5) when h is nonzero but small. We will then compare the large deviations result with the local fluctuation theory of a single agent so as to explain why interconnectedness destabilizes the system. 6.1. The h = 0 and the Small h Analysis. We note that when h = 0, u e ±ξ b = u e ±ξ0 , where u e ±ξ0 (y) = 1 2π σ 2 2θ exp − (y − (±ξ 0 )) 2 2 σ 2 2θ , ξ 0 = 1 − 3 σ 2 2θ . (6.1) In this case, (5.5) is solvable and the optimal path is a Gaussian, starting from u e −ξ0 and ending in u e +ξ0 . Theorem 6.1. Let h = 0 and define p e (t, y) = 1 2π σ 2 2θ exp − (y − a e (t)) 2 2 σ 2 2θ , a e (t) = 2ξ 0 T t − ξ 0 . (6.2) Then p e ∈ A is the unique minimizer for (5.5) and inf φ∈A I 0 (φ) = I 0 (p e ) = 2ξ 2 0 σ 2 T . Proof. See Appendix D.1. We show next that (5.5) is continuous at h = 0. Theorem 6.2. There exists γ(h) such that γ(h) → 0 as h → 0 and inf φ∈A I h (φ) − 2ξ 2 b σ 2 T ≤ γ(h). (6.3) We recall here that ξ b = ξ 0 + hξ 1 + O(h 2 ), ξ 1 = 1 − 3 σ 2 2θ 6 σ 2 σ 2 2θ 2 1 − 2(σ 2 /2θ) 1 − 3(σ 2 /2θ) . (6.4) Proof. See Appendix D.2 and D.3. As it is stated we could replace ξ b by ξ 0 in Theorem 6.2, since ξ b = ξ 0 + o(1) as h → 0. We will see in the next section (in Proposition 7.5) that γ(h) = O(h 2 ). In fact we show this rigorously for the upper bound but only formally for the lower bound. Since ξ b = ξ 0 + hξ 1 + O(h 2 ) we see that the term 2ξ 2 b /(σ 2 T ) contains the leading-order term and the first-order correction in the h-expansion of inf φ∈A I h (φ). Large Deviations for the First Exit Time. In this subsection, we consider the rare event B of systemic transition at some time before T : B δ = {φ ∈ E ν : ∃t ∈ (0, T ], ρ(φ(t), u e ξ b ) ≤ δ}. In other words, B δ = ∪ t∈(0,T ] A δ (t), where A δ (t) = {φ ∈ E ν : ρ(φ(t), u e ξ b ) ≤ δ}. We let B := B 0 . We then have that Lemma 6.3. By definition inf φ∈B δ I h (φ) is decreasing with δ > 0 and bounded from above by inf φ∈B I h (φ). In addition, lim δ→0 inf φ∈B δ I h (φ) = inf φ∈∪ t∈(0,T ] A(t) I h (φ) = inf φ∈B I h (φ), where A(t) := A 0 (t). Proof. See Appendix D.4. From Theorem 6.2, we see that in the sense of large deviations the probability of system failure at some time before time T is essentially the same as the probability of system failure at time T . Corollary 6.4. For any t 1 < t 2 , there exists a sufficiently small h such that inf φ∈A(t1) I h (φ) > inf φ∈A(t2) I h (φ). Consequently, inf φ∈B I h (φ) ≈ inf φ∈A(T ) I h (φ) for small h. Comparison with the Fluctuation Theory of a Single Agent. To get a better understanding of the large deviations results we need to carry out a standard fluctuation theory for a single agent. We assume that x j (0) = −1 for all j and that the x j (t)'s are in the vicinity of −1 so that we can linearize (1.1): x j (t) = −1 + z j (t),x(t) = −1 +z(t),z(t) = 1 N N j=1 z j (t). For V (y) = 1 4 y 4 − 1 2 y 2 , z j (t) andz(t) satisfy the linear stochastic differential equations dz j = −(θ + 2h)z j dt + θzdt + σdw j , dz = −2hzdt + σ N N j=1 dw j , with z j (0) =z(0) = 0. The processes z j (t) andz(t) are Gaussian and the mean and variance functions are easily calculated. We are especially interested in their behavior for large N . Lemma 6.5. For all t ≥ 0, Ez j (t) = Ez(t) = 0 and Varz(t) = σ 2 N (1 − e −4ht ). In addition, Varz j (t) → σ 2 2(θ+2h) (1 − e −2(θ+2h)t ) as N → ∞, uniformly in t ≥ 0. From Lemma 6.5, we see that σ 2 /N and σ 2 /2(θ + 2h) should be sufficiently small so that linearization is consistent with the results it produces. Increased Probability of Large Deviations for Increased θ and Its Systemic Risk Interpretation. We have now the analytical results with which we may conclude that individual risk diversification may increase the systemic risk. Assume that σ 2 /N and σ 2 /2(θ + 2h) are sufficiently small and N is large. From Lemma 6.5, the risk x j (t) of the agent j is approximately a Gaussian process with the stationary distribution N (−1, σ 2 /2(θ + 2h)). If the external risk, σ is high, then in order to keep the risk x j (t) at an acceptable level, the agent may increase the intrinsic stability, h, or share the risk with other agents, that is, increase θ. Increasing h is in general more costly (cuts into profits) than increasing θ, and at the individual agent level there is no difference in risk assessment between increasing h and increasing θ. Therefore the agents are likely to increase θ and reduce individual risk by diversifying it. Note that σ 2 /2(θ + 2h) σ 2 /2θ when σ 2 and θ are significantly larger than h. Thus, individual agents can maintain low locally assessed risk by diversification, even in a very uncertain environment. What is not perceived by the individual agents, however, is that risk diversification may increases the systemic risk while it reduces their individual risk. Because σ 2 and θ are significantly larger than h, the small h analysis can be applied and from (5.4) and Theorem 6.2, the systemic risk (the probability of the system failure) is P(X N ∈ B δ ) ≈ exp −N 2ξ 2 b σ 2 T , for small δ and h, ξ b = 1 − 3 σ 2 2θ 1 + h 6 σ 2 σ 2 2θ 2 1 − 2(σ 2 /2θ) 1 − 3(σ 2 /2θ) + O(h 2 ). We see that there are additional systemic-level σ 2 terms in the exponent and ξ b , which can not be observed by the agents, increasing the systemic risk, even if the individual risk σ 2 /2θ is fixed. In other words, the individual agents may believe that they are able to withstand larger external fluctuations as long as their risk can be diversified, but a higher σ tends to destabilize the system. A Reduced Large Deviations Principle for Small h. In Section 6.1, we show that the large deviation problem inf φ∈A I h (φ) is continuous in h so that we have the upper and lower bounds for inf φ∈A I h (φ) when h is small. In this section, we analyze with a formal expansion the optimal path for inf φ∈A I h (φ) by assuming that it is of the form p e + O(h), motivated by the fact that the optimal path is p e for h = 0. In this way, we can obtain a reduced large deviations principle (a reduced Freidlin-Wentzell theory) for the systemic risk. That is, we obtain a reduced rate function corresponding to a finite dimensional system after ignoring higher order terms. The reduced rate function has all relevant information up to O(h 2 ) terms, and we also need to expand φ to O(h 2 ). We assume that the optimal φ = p + hq (1) + h 2 q (2) + . . ., where p(t, y) = 1 2π σ 2 2θ exp − (y − a(t)) 2 2 σ 2 2θ , a(t) = φ, y . In other words, we let the first moment of φ be determined by a(t), and from the zero h case we know that a(t) = a e (t) + O(h). From the form of p and (5.6), a natural parameterization for q (1) and q (2) is the Hermite expansion q (1) (t, y) = ∞ n=2 b n (t) ∂ n ∂y n p(t, y), q (2) (t, y) = ∞ n=2 c n (t) ∂ n ∂y n p(t, y). Note that by the properties of p and a(t), q (1) , y n = q (2) , y n = 0 for n = 0, 1 so we can start the Hermite expansion from n = 2. The formal expansion result of this section is that if the optimal φ = p + hq (1) + h 2 q (2) , then inf φ∈A I h (φ) ≈ inf a(t):0≤t≤T a(0)=−ξ b a(T )=ξ b 1 2σ 2 T 0 d dt a + h(a 3 + 3 σ 2 2θ a − a) 2 dt, (7.1) for small h. Note that a(t) = φ, y =x(t). The right hand side of (7.1) is an onedimensional variational problem that has the form of a rate function of the Freidlin-Wentzell theory. In fact, the right side of (7.1) is the large deviations variational problem for the rate function of the small-noise stochastic differential equation dx(t) = −h x 3 (t) − 1 − 3σ 2 2θ x(t) dt + ǫσdw(t) (7.2) where here ǫ = 1/ √ N is small. Note that 3σ 2 /2θ < 1, as assumed above, and therefore (7.2) also represents a bi-stable structure. In the remainder of this section we describe how this result is obtained by formal expansions and then in Section 7.3 we show how we recover from (7.1) the main result of the paper stated in the previous section. An important remark about the expansion is that the Hermite functions are a basis of the L 2 space and thus p+hq (1) +h 2 q (2) is generally a signed measure. However, if q (1) and q (2) can be expressed as the linear combinations of finite Hermite functions, then we can see that for any ǫ > 0, there exists a sufficiently small h such that the negative part of p + hq (1) + h 2 q (2) is less than ǫ. 7.1. Optimization over g. The first step in finding the optimal φ = p + hq (1) + h 2 q (2) is determining the optimal g by using (5.6) for φ. Once we obtain g, we can compute I h (φ) by using (5.7). It is also natural to assume that g = g (0) +hg (1) +h 2 g (2) along with the Hermite expansion: g (0) = p −1 ∞ n=0 α n (t) ∂ n ∂y n p, g (1) = p −1 ∞ n=0 β n (t) ∂ n ∂y n p, g (2) = p −1 ∞ n=0 γ n (t) ∂ n ∂y n p. In addition, since q (1) , y = q (2) , y = 0, we can see that φ = p + hq (1) + h 2 q (2) satisfies L * φ φ = L * p p + hL * p q (1) + h 2 L * p q (2) , M * φ = M * p + hM * q (1) + h 2 M * q (2) . The force U (y) = y 3 − y can also be expanded in Hermite polynomials: U (y) = p −1 3 n=0 δ n (t) ∂ n ∂y n p. Now everything is expanded in the orthogonal basis and we can find the optimal g (0) and g (1) by putting everything into (5.6) and comparing coefficients. Lemma 7.1. With the expansions mentioned above, the optimal g (0) is − d dt a, and the optimal β n for g (1) are β n =      −δ 0 = − p, U (y) , n = 0, d dt b n+1 + θ(n + 1)b n+1 − δ n , 1 ≤ n ≤ 3, d dt b n+1 + θ(n + 1)b n+1 , n ≥ 4. (7.3) Proof. See Appendix E.1. It remains to determine g (2) . From (5.7) we see that the only contribution of g (2) to I h up to O(h 2 ) is p, 2g (0) g (2) = −2γ 0 d dt a. Thus it suffices to determine γ 0 , which can also be obtained from (5.6). Lemma 7.2. With the expansions mentioned above, the optimal γ 0 is γ 0 = − q (1) , U (y) + g (1) . Proof. See Appendix E.2. 7.2. Optimization over φ. We are now ready to find the optimal φ. For given φ = p + hq (1) + h 2 q (2) and the corresponding optimal g = g (0) + hg (1) + h 2 g (2) , (5.7) gives (1) , and therefore I h (φ) = 1 2σ 2 T 0 p + hq (1) + h 2 q (2) , (g (0) + hg (1) + h 2 g (2) ) 2 dt = 1 2σ 2 T 0 p, (g (0) ) 2 dt + h 2σ 2 T 0 p, 2g (0) g (1) dt + h 2 2σ 2 T 0 p, (g (1) ) 2 + 2g (0) g (2) + q (1) , 2g (0) g (1) dt + O(h 3 ). From Lemma 7.2, p, 2g (0) g (2) = −2g (0) q (1) , U (y) + gp, 2g (0) g (2) + q (1) , 2g (0) g (1) = −2g (0) q (1) , U (y) = −2g (0) 3 n=2 H n δ n b n , where H n (t) := p −1 , (∂ n p/∂y n ) 2 . We note that p, 2g (0) g (1) = −2g (0) δ 0 , p, (g (1) ) 2 = δ 2 0 + ∞ n=1 H n β 2 n , p, (g (0) ) 2 = (g (0) ) 2 . Then I h (φ) can be written as I h (φ) = 1 2σ 2 T 0 (g (0) − hδ 0 ) 2 dt + h 2 2σ 2 T 0 (H 1 β 2 1 − 2H 2 g (0) δ 2 b 2 )dt (7.4) + h 2 2σ 2 T 0 (H 2 β 2 2 − 2H 3 g (0) δ 3 b 3 )dt + h 2 2σ 2 ∞ n=3 T 0 H n β 2 n dt + O(h 3 ). We see that a and b n are coupled at the O(h 2 ) level of (7.4). However, from the results of the zero h case, a = a e + O(h) and p = p e + O(h) so we can decouple a and b n and express the expanded I h (φ) up to O(h 2 ) as the sum of independent terms. Proposition 7.3. To order O(h 2 ), the rate function I h (φ) can be written as the sum of independent terms: I h (φ) = 1 2σ 2 T 0 (g (0) − hδ 0 ) 2 dt + h 2 2σ 2 T 0 (H 1β 2 1 + 2 d dt a eH 2δ2 b 2 )dt (7.5) + h 2 2σ 2 T 0 (H 2β 2 2 + 2 d dt a eH 3δ3 b 3 )dt + h 2 2σ 2 ∞ n=3 T 0H nβ 2 n dt + O(h 3 ), whereH n (t) = (p e ) −1 , (∂ n p e /∂y n ) 2 , U (y) = (p e ) −1 3 n=0δ n (t) ∂ n ∂y n p e , and β n =      −δ 0 = − p e , U (y) , n = 0, d dt b n+1 + θ(n + 1)b n+1 −δ n , 1 ≤ n ≤ 3, d dt b n+1 + θ(n + 1)b n+1 , n ≥ 4. (7.6) We can see from (7.5) that q (2) does not appear in terms up to O(h 2 ). From the h expansion of u e ±ξ b in (2.2), and the fact that V (y) is a polynomial of degree four, we have b n+1 (0) = b n+1 (T ) = 0 for n ≥ 4. The variational problem for b n+1 is to minimize T 0H nβ 2 n dt whereβ n is given in terms of b n+1 by (7.6). The obvious solution of this problem is b n+1 = 0 andβ n = 0 for n ≥ 4. Consequently, in order to find the optimal φ for I h (φ) in (7.5), we may solve separately the variational problems for a, b 1 , b 2 and b 3 . Probability of Systemic Transitions for Small h. We consider the small probability of systemic transitions for large N and small h through the large deviation inf φ∈A I h (φ). Here we consider the solution up to O(h) terms. That is, using (7.5), we solve the variational problem for a(t): inf a(t):0≤t≤T a(0)=−ξ b a(T )=ξ b T 0 (g (0) − hδ 0 ) 2 dt = inf a(t):0≤t≤T a(0)=−ξ b a(T )=ξ b T 0 ( d dt a + h(a 3 + 3 σ 2 2θ a − a)) 2 dt. (7.7) By simple calculus of variations methods we find the optimal a. Lemma 7.4. The optimal a(t) for (7.7) satisfies the second order ordinary differential equation d 2 dt 2 a = h 2 (a 3 + (3 σ 2 2θ − 1)a)(3a 2 + (3 σ 2 2θ − 1)) with a(0) = −ξ b and a(T ) = ξ b . Consequently, the optimal path is a(t) = 2ξ b T t − ξ b + O(h 2 ). (7.8) By inserting (7.8) into (7.7) we obtain inf φ∈A I h (φ) up to O(h). Proposition 7.5. For small h, the large deviations problem, inf φ∈A I h (φ), up to O(h), is inf φ∈A I h (φ) = 2ξ 0 σ 2 T (ξ 0 + 2hξ 1 ) + O(h 2 ), (7.9) where ξ b = ξ 0 + hξ 1 + O(h 2 ) from (2.5). Note that ξ 1 is positive because 2θ > 3σ 2 . Proof. See Appendix E.3. The asymptotic probability of systemic transition for large N and sufficiently small δ and h has the form P(X N ∈ A δ ) ≈ exp −N inf φ∈A I h (φ) = exp −N 2ξ 0 σ 2 T (ξ 0 + 2hξ 1 ) + O(h 2 ) . Effect of Diversity of Sensitivities on the Transition Probability. We consider the situation introduced in Section 3 and analyze it when h = 0. We aim at computing the transition probability in this situation. The K partial empirical averagesx k (t) := 1 |I k | j∈I k x j (t), k = 1, . . . , K (8.1) then satisfy a closed system of stochastic differential equations dx k = σ √ ρ k N dw k (t) − θ k (x k −x)dt (8.2) wherew k are independent Brownian motions and the empirical meanx(t) can be expressed in terms of the partial averages as x(t) = K k=1 ρ kxk (t) Proposition 8.1. Ifx k (0) = −ξ b for all k = 1, . . . , K, thenx(T ) is a Gaussian random variable with mean −ξ b and variance σ 2 T := Var(x(T )) given by σ 2 T = σ 2 N T 0 ̺ T e Ms R −1 (e Ms ) T ̺ds (8.3) where ̺ is the K-dimensional column vector (ρ k ) k=1,...,K , M and R are the K × K matrices defined by M ij = −θ i (δ ij − ρ j ), R ij = ρ i δ ij , i, j = 1, . . . , K, and T stands for the transpose. Proof. See Appendix F.1. We can then deduce that the transition probability is p T ≈ exp − 2ξ 2 b σ 2 T (8.4) Our next goal is to study the impact of the diversity on the transition probability. Proposition 8.2. Let us assume that the diversity is small: θ k =θ(1 + δα k ), δ ≪ 1 where k ρ k α k = 0 so thatθ is the mean value of the θ k 's. The equilibrium position ξ b , the variance σ 2 T and the transition probability p T can be expanded as powers of δ as ξ 2 b = 1 − 3σ 2 2θ − δ 2 k ρ k α 2 k 3σ 2 2θ + O(δ 3 ), σ 2 T = σ 2 T N 1 + δ 2 k ρ k α 2 k 1 T T 0 (1 − e −θs ) 2 ds + O(δ 3 ) , p T ≈ exp − 2N σ 2 T 1 − 3σ 2 2θ − δ 2 k ρ k α 2 k 3σ 2 2θ + 1 T T 0 (1 − e −θs ) 2 ds . Proof. See Appendix F.2. This proposition shows that the diversity reduces the gap between the two equilibrium states and enhances the fluctuations of the empirical mean. Both effects contribute to the increase of the systemic transition probability. 9. Summary and Conclusions. The aim of this paper is to introduce and analyze a mathematical model for the evolution of risk in a system of interacting agents where cooperation between them can reduce their individual risk of failure but increase the systemic or overall risk. The model we use is a system of bistable diffusion processes that interact through their empirical mean, a mean field model. We take the rate of mean reversion to the empirical mean θ as a measure of cooperation, the depth of the bistable potential h as a measure of intrinsic stability of each agent, and the strength of the external random perturbations σ as the level of uncertainty in which the agents function. Using the theory of large deviations we calculate the probability that the system will transition from one of the two bistable states to the other during a time interval of length T , when the number of agents N is large and when h is small. In this regime of parameters we find that systemic risk increases with cooperation. The formula from which we draw this conclusion is given is Section 6.4. We also show that when the rate of mean reversion to the empirical mean varies among the different agents, that is, when there is diversity in the cooperative behavior then the probability of transitions increases, which means that the systemic risk increases. Acknowledgement. This work is partly supported by the Department of Energy [National Nuclear Security Administration] under Award Number NA28614, and partly by AFOSR grant FA9550-11-1-0266. Appendix A. Proof of Proposition 2.2. For small h, we view u e ξ as a perturbed Gaussian density function. Let p ξ (y) be the Gaussian density function with mean ξ and variance σ 2 /2θ, Y be the Gaussian random variable with the density p ξ , and η = 2/σ 2 . By using the expansion exp(−hηV ) = 1 − hηV + h 2 η 2 V 2 /2 + O(h 3 ), we have Z ξ = 1 − hηEV (Y ) + 1 2 h 2 η 2 EV 2 (Y ) + O(h 3 ) Z −1 ξ = 1 + hηEV (Y ) − 1 2 h 2 η 2 EV 2 (Y ) + h 2 η 2 (EV (Y )) 2 + O(h 3 ). Then we calculate m(ξ) as follows: m(ξ) = Z −1 ξ y 1 − hηV + 1 2 h 2 η 2 V 2 + O(h 3 ) p ξ (y)dy = Z −1 ξ ξ − hηE[Y V (Y )] + 1 2 h 2 η 2 E[Y V 2 (Y )] + O(h 3 ) = ξ + hη{ξEV (Y ) − E[Y V (Y )]} + h 2 η 2 {− 1 2 ξEV 2 (Y ) + ξ(EV (Y )) 2 − EV (Y )E[Y V (Y )] + 1 2 E[Y V 2 (Y )]} + O(h 3 ) = ξ − hη σ 2 2θ EV y (Y ) + h 2 η 2 σ 2 2θ {E[V (Y )V y (Y )] − EV (Y )EV y (Y )} + O(h 3 ) = ξ − hη σ 2 2θ EV y (Y ) + h 2 η 2 σ 2 2θ Cov(V y (Y ), V (Y )) + O(h 3 ). The compatibility condition ξ b = m(ξ b ) gives EV y (Y ) − hηCov(V y (Y ), V (Y )) + O(h 2 ) = 0. (A.1) Assuming that ξ b = ξ 0 + hξ 1 + O(h 2 ), the O(1) terms in (A.1) give ξ 3 0 + 3 σ 2 2θ ξ 0 − ξ 0 = ξ 0 (ξ 2 0 + 3 σ 2 2θ − 1) = 0. Then ξ 0 = 0, ± 1 − 3σ 2 /2θ if 3σ 2 < 2θ, or otherwise ξ 0 = 0. In order to obtain the nontrivial result, we suppose that 3σ 2 < 2θ and ξ 0 takes ± 1 − 3σ 2 /2θ in the later calculations. Note that EV y (Y ) = ξ 3 + (3σ 2 /2θ − 1)ξ = 2hξ 2 0 ξ 1 + O(h 2 ), and Cov(V y (Y ), V (Y )) = E[V (Y )V y (Y )] + O(h) = E[( 1 4 Y 4 − 1 2 Y 2 )(Y 3 − Y )] + O(h) = E[ 1 4 Y 7 − 3 4 Y 5 + 1 2 Y 3 ] + O(h). Along with the identity ξ 2 0 + 3σ 2 /2θ = 1, we have For a test function f ∈ S(R), we define X f,l N (t) = f (y), X l N (t, y) = j∈I l f (x j (t))/|I l |. By Itô's formula, EY 3 = ξ 0 + O(h), EY 5 = 1 + 4 σ 2 2θ − 6 σ 2 2θ 2 ξ 0 + O(h), EY 7 = 1 + 12 σ 2 2θ + 6 σ 2 2θ 2 − 48 σ 2 2θ 3 ξ 0 + O(h). Then Cov(V y (Y ), V (Y )) = 6(σ 2 /2θ) 2 (1 − 2σ 2 /2θ)ξ 0 + O(h). The O(h) terms in (A.1) imply ξ 1 = 3η(σ 2 /2θ) 2 (1 − 2σ 2 /2θ)/ξ 0 .dX f,l N = 1 |I l | j∈I l [−hU (x j )dt + σdw j + Θ l (x − x j )dt]f y (x j ) + 1 2 σ 2 f yy (x j )dt = −hU f y + Θ l ( y, K l=1 ρ l X l N − y)f y + σ 2 2 f yy , X l N dt + f y , σ |I l | j∈I l δ xj dw j . Then by the integration by parts, we write dX l N = {(hU X l N ) y − [Θ l ( y, K l=1 ρ l X l N − y)X l N ] y + σ 2 2 (X l N ) yy }dt − σ |I l | j∈I l (δ xj ) y dw j . For simplicity, we prove the case that K = 2 and the general case is similar. We let X 1,×n N × X 2,×n N denote the product measure on R 2n : X 1,×n N × X 2,×n N (y 1 , . . . , y 2n ) = X 1 N (t, y 1 ) · · · X 1 N (t, y n )X 2 N (t, y n+1 ) · · · X 2 N (t, y 2n ). For a test function f ∈ S(R 2n ), we have d f, X 1,×n N × X 2,×n N = d f, X 1,×n N × X 2,×n N (1) + d f, X 1,×n N × X 2,×n N (2) , where (1) and (2) denote the first and the second order terms of d f, X 1,×n N × X 2,×n N , respectively: d f, X 1,×n N × X 2,×n N (1) = n j=1 f, dX 1 N (t, y j ) × X 1,×(n−1),j N × X 2,×n N + 2n j=n+1 f, dX 2 N (t, y j ) × X 1,×n,j N × X 2,×(n−1),j N d f, X 1,×n N ×X 2,×n N (2) = 1 2 n j,k=1 j =k f, dX 1 N (t, y j )×dX 1 N (t, y k )×X 1,×(n−2),j,k N ×X 2,×n N + 1 2 2n j,k=n+1 j =k f, dX 2 N (t, y j ) × dX 2 N (t, y k ) × X 1,×n N × X 2,×(n−2),j,k N + 1 2 n j=1 2n k=n+1 f, dX 1 N (t, y j ) × dX 2 N (t, y k ) × X 1,×(n−1),j N × X 2,×(n−1),k N . Note that for j = k, dX l N (t, y j ) × dX l N (t, y k ) = σ 2 |I l | 2 i∈I l (δ xi (y j )) j (δ xi (y k )) k dt = σ 2 ρ 2 l N (δ(y k − y j )X l N (t, y j )) jk dt, and dX 1 N (t, y j ) × dX 2 N (t, y k ) = 0. If we analogously represent the generator G (X 1,×n , where (u 1 , u 2 ) satisfying (3.2). Then the limit of (X 1 N , X 2 N ) is a solution of the martingale problem associated to (3.2). In addition, by [18,Corollary 2.10], the solution is unique and therefore (X 1 N , X 2 N ) → (u 1 , u 2 ) weakly as N → ∞. yu e l,ξ (y)dy. Note that d dξ Z l,ξ = (2Θ l /σ 2 )( yu e l,ξ dy − ξ)Z l,ξ and N ,X 2,×n N ) f of f, X 1,×n N × X 2,×n N as G (X 1,×n N ,X 2,×n N ) f = G (1) (X 1,×n N ,X 2,×n N ) f + G (2) (X 1,×n N ,X 2,×n N ) f,d 2 dξ 2 Z l,ξ = 2Θ l σ 2 Z l,ξ d dξ yu e l,ξ dy − 1 + 2Θ l σ 2 yu e l,ξ dy − ξ d dξ Z l,ξ (B.1) = 2Θ l σ 2 Z l,ξ d dξ yu e l,ξ dy − 1 + 2Θ l σ 2 2 Z l,ξ yu e l,ξ dy − ξ 2 . On the other hand, we can also compute d 2 dξ 2 Z l,ξ by directly taking the twice derivatives of Z l,ξ : Note that yu e l,0 dy = 0, so d dξ m(0) = K l=1 ρ l (2Θ l /σ 2 ) y 2 u e l,0 dy. By using the same trick in the proof of Proposition 2.2, let p l (y) be the Gaussian density function with mean 0 and variance σ 2 /2Θ l , Y l be the Gaussian random variable with the density p l , and η = 2/σ 2 . Then for small h, Z −1 l,0 = 1 + hηEV (Y l ) + O(h 2 ), and y 2 u e l,0 dy = Z −1 l,0 d 2 dξ 2 Z l,ξ = − 2Θ l σ 2 Z l,ξ + 2Θ l σ 2 2 Z l,ξ (y − ξ) 2 u e l,y 2 (1 − hηV + O(h 2 ))p l (y)dy = Z −1 l,0 (EY 2 l − hηE[Y 2 l V (Y l )] + O(h 2 )) = EY 2 l + hη(EY 2 l EV (Y l ) − E[Y 2 l V (Y l )]) + O(h 2 ). Therefore d dξ m(0) > 1 if and only if K l=1 ρ l (2Θ l /σ 2 )(EY 2 l EV (Y l ) − E[Y 2 l V (Y l )]) > 0. Note that EY 2 l = σ 2 /2Θ l , EV (Y l ) = (3/4)(EY 2 l ) 2 − (1/2)EY 2 l , and E[Y 2 l V (Y l )] = (15/4)(EY 2 l ) 3 − (3/2)(EY 2 l ) 2 . Then the sufficient and necessary condition becomes K l=1 ρ l Θ l 1 − 3 σ 2 2Θ l > 0. B.3. Proof of Proposition 3.3. It is equivalent to show that K l=1 ρ l /Θ l ≤ K l=1 ρ l Θ l K l=1 ρ l /Θ 2 l . First note that by the Cauchy-Schwarz inequality, K l=1 ρ l Θ l 2 = K l=1 √ ρ l Θ l × √ ρ l 2 ≤ K l=1 ρ l Θ 2 l K l=1 ρ l = K l=1 ρ l Θ 2 l . Then it suffices to show that 1 ≤ K l=1 ρ l Θ l K l=1 ρ l /Θ l . Again by the Cauchy-Schwarz inequality, K l=1 ρ l Θ l K l=1 ρ l Θ l ≥ K l=1 ρ l Θ l ρ l Θ l = K l=1 ρ l = 1. for sufficiently small h. For t ∈ [0, δT ], because φ u is simply the convex combination of u e −ξ b and p u , φ u can be bounded by a δT -independent constant. To compute g u from (5.6), it is also easy to see that L * φ u φ u and M * φ u can be bounded by δT -independent constants. The only term we need to worry is (p u (t)−u e −ξ b )/δT from computing φ u t (t). However, p u (t) is differentiable at t = 0 and p u (0) → u e −ξ b as h → 0 so we can bound (p u (t)−u e −ξ b )/δT by a δT -independent constant with suitable h. Thus g u is bounded independently of δT and so we can find a δT -independent constant c u > 0 such that φ u , (g u ) 2 < c u . The same argument works for t ∈ [T − δT, T ] and we have the desired result. D.3. Proof of Theorem 6.2 (Lower Bounds). From (D.1), there exists some constant C such that inf φ∈A I h (φ) ≤ C for all h ≤ h 0 . Then we can assume that I h (φ) ≤ C for all φ ∈ A and all h ≤ h 0 without loss of generality. The following lemma shows that the first and second moments of all φ ∈ A are uniformly bounded. Lemma D.5. Given C > 0, there exists R > 0 such that for any φ ∈ A with I h (φ) ≤ C for some h ≥ 0, then sup t∈[0,T ] φ(t), y 2 ≤ sup t∈[0,T ] φ(t), y 2 ≤ R. Proof. Recall that M R (R) = {φ ∈ M 1 (R), ϕ(y)φ(dy) ≤ R} and M ∞ (R) = ∪ R>0 M R (R) with the inductive topology. Here we focus on the case that ϕ = 1 + y 2 in order to obtain the uniform result, and let M 2 R (R) and M 2 ∞ (R) denote the spaces with the quadratic Lyapunov function ϕ. The proof is an application of Theorem 5.1(c), Theorem 5.3 and Lemma 5.5 of [10]. By Theorem 5. µ, L µ ϕ + hMϕ + 1 2 ϕ 2 y / µ, ϕ , with ϕ(y) = 1 + y 2 . Obviously we can find the uniform r and λ for all h ≥ 0 and also the uniform R. Then any φ of interest are in C([0, T ], M 2 R (R)) and thus have the uniform bounded first and second order moments. Now we derive that lower bound. The key idea is that because we have the universal upper bound for the first and second moments of all φ ∈ A and for all h ≤ h 0 , Chebyshev's inequality implies the uniform convergence. 1(c), if φ ∈ C([0, T ], M 2 ∞ (R)) with φ(0) = u e −ξ b and I h (φ) ≤ C for some h ≥ 0, Proposition D.6. For any ǫ > 0, then for all sufficiently small h, inf φ∈A I h (φ) ≥ 1 2σ 2 T 0 p u , ( d dt a u − h(y 3 − y)) 2 dt − ǫ. (D.2) Proof. Define f M = ι * f M , wheref M is a piecewise linear function and ι is the standard mollifier: i=−2 (iM − 1, iM + 1). Because for all φ ∈ A, φ(t), (f M y ) 2 ≤ 1, we can estimate the rate function: I h (φ) ≥ 1 2σ 2 T 0 φ t − L * φ φ − hM * φ, f M 2 dt ≥ 1 2σ 2 T T 0 φ t − L * φ φ − hM * φ, f M dt 2 . Then we estimate the integrand term by term. By Lemma D.5, the following convergences are all uniform in φ ∈ A and h ≤ h 0 . First we have Again by Chebyshev's inequality, the right hand side vanishes as M → ∞. Finally we estimate M * φ, f M . Since f M is compactly supported, | M * φ, f M | = | φ, (y 3 − y)f M y | ≤ (2M + 1) 3 + (2M + 1). For a fixed M , we can choose a sufficiently small h such that h| M * φ, f M | is small. Consequently, for any ǫ > 0, we can first choose a sufficiently large M and then there exists a sufficiently small h such that inf φ∈A I h (φ) ≥ 2ξ 2 b σ 2 T − ǫ. E.3. Proof of Proposition 7.5. We write a(t) = a 0 (t) + ha 1 (t) + O(h 2 ) with a 0 (t) = 2ξ 0 t/T − ξ 0 and a 1 (t) = 2ξ 1 t/T − ξ 1 . Then we put a(t) into (7.7) and we have inf φ∈A I h (φ) = 1 2σ 2 T 0 ( d dt a 0 ) 2 + 2h( d dt a 0 )(a 3 0 + (3 σ 2 2θ − 1)a 0 + d dt a 1 ) dt + O(h 2 ). We note that d dt a 0 is a constant, and a 0 (t) and a 3 0 (t) are odd functions with respect to t = T /2. Then inf φ∈A I h (φ) = 1 2σ 2 T 0 d dt a 0 2 + 2h d dt a 0 d dt a 1 dt + O(h 2 ) = 1 2σ 2 T 0 2ξ 0 T 2 + 2h 2ξ 0 T 2ξ 1 T dt + O(h 2 ) = 2ξ 0 σ 2 T (ξ 0 + 2hξ 1 ) + O(h 2 ). Appendix F. Proofs in Section 8. F.1. Proof of Proposition 8.1. The system of SDEs (8.1) for the vectorX(t) = (x k (t)) k=1,...,K has the form dX(t) = MX(t) + σ √ N R −1/2 dW (t) whereW (t) = (w k (t)) k=1,...,K is a column vector. This system can be solved: Using the fact thatM T ̺ = 0 and N R −1 ̺ = N u = 0, we obtain X(t) = e MtX (0) + σ √ N̺ T e Mt R −1 (e Mt ) T ̺ = ̺ T R −1 ̺ + δ 2 (1 − e −θt ) 2 ̺ T N R −1 N T ̺ + O(δ 3 ) We have ̺ T R −1 ̺ = 1 and ̺ T N R −1 N T ̺ = k ρ k α 2 k which gives the expansion of the variance σ 2 T . Finally the expansion of the transition probability can be obtained by substituting the expansions of ξ 2 b and σ 2 T into (8.4). For U (y) = y 3 − y, ξ = 0 is the trivial solution of (3.4), and a simple extension of Theorem 3.3.1 in[9], shows that there are two sets of non-trivial solutions {u e l,ξ b } K l=1 Proposition 3 . 2 . 32The compatibility condition (3.4) has non-zero solutions if and only if σ < σ div c . For small h, σ div c has the expansion c , the critical value (2.4) of the homogeneous case. Proposition 3.3. With θ = K l=1 ρ l Θ l , we have σ homo c ≥ σ div c for small h. Proof. See Appendix B.3. Fig. 4.1. Simulations for different σ. The system has two (statistically) stable equilibria when σ is below the critical value or otherwise has single stable state 0. For small h, 3σ 2 /2θ < 1 is the approximate criterion. Fig. 4.2. Simulations for different θ. The system has two stable equilibria if θ is above the critical value or otherwise has single stable state 0. For small h, 3σ 2 /2θ < 1 is the approximate criterion. Figure 4 4 Figure 4 .Fig. 4 . 3 .Fig. 4 . 4 . 443444 illustrates the effect of system size on its stability. The effect of changing h. Increasing it stabilizes the system. Influence of the system size N . A larger system tends to have a more stable behavior. Figure 4 . 43 and Figure 4.4). Clearly diversity tends to destabilize the system. Fig. 4 . 6 . 46Effect of changing h. Increasing it stabilizes the system. Fig. 4 . 8 . 48Effect of changing the system size N . Larger system have a more stable behavior. The effect of changes in Θ l . The median of the diversity values is fixed but the low and high sensitivities are changed to adjust the level of diversity of θ j while ρ l and the mean of θ j are the same. Increasing diversity tends to destabilize the system.• M 1 (R) is the space of probability measures on R with the Prohorov metric ρ, associated with weak convergence. • C([0, T ], M 1 (R)) is the space of continuous functions from [0, T ] to M 1 (R) with the metric sup 0≤t≤T ρ(φ 1 (t), φ 2 (t)). • M R (R) = {µ ∈ M 1 (R), ϕ(y)µ(dy) ≤ R}, where ϕ ∈ C 2 (R) is a nonnegative function with lim |x|→∞ ϕ(x) = ∞. From [10], if U (y) = y 3 − y, we can choose ϕ(y) = 1 + y 2 + γy 4 , 0 ≤ γ ≤ h/2. • M ∞ (R) = ∪ R>0 M R (R) = {µ ∈ M 1 (R), ϕ(y)µ(dy) < ∞} endowed with the inductive topology: µ n → µ in M ∞ (R) if and only if µ n → µ in M 1 (R) and sup n ϕ(y)µ n (dy) < ∞. • C([0, T ], M ∞ (R)) is the space of continuous functions from [0, T ] to M ∞ (R) endowed with the topology: φ n (·) → φ(·) in C([0, T ], M ∞ (R)) if and only if φ n (·) → φ(·) in C([0, T ], M 1 (R)) and sup 0≤t≤T sup n ϕ Theorem 5. 1 . 1(Dawson and Gärtner, 1987) Given a finite horizon [0, T ], ν ∈ M ∞ (R) and A ⊆ E ν , if X N (0) = 1 N N j=1 δ xj (0) → ν in M ∞ (R) as N → ∞, then the law of X N (t) = 1 N N j=1 δ xj(t) satisfies the large deviation principle with the good rate B. 1 . 1Proof of Theorem 3.1. The proof contains three steps. B.1.1. Existence and Uniqueness of the Weak Solution of the McKean-Vlasov Equation. The existence and uniqueness of a probability measure valued process (u 1 (t), . . . , u K (t)) that is a weak solution of the McKean-Vlasov equation (3.2) is guaranteed by [18, Theorem 2.11]. B.1.2. Weak Compactness of the Empirical Process. By Prohorov's theorem, it suffices to prove that the sequence {(X 1 N , . . . , X K N )} ∞ N =1 is weakly compact by t, dy), |y| ] < ∞, which can be done by using the calculations similar to (B1) and (B2) in [9]. B.1.3. Identification of the Limit. → 0 as N → ∞ and G 2 ) f , the generator of f, u ×n 1 ×u ×n 2 ξ dy − ( yu e l,ξ dy) 2 . then φ is in an h-dependent compact set K. By Theorem 5.3 the compact set K is contained in C([0, T ], M 2 R (R)) for an h-dependent R > 0. Finally, by Lemma 5.5 and Theorem 5.1(c), it suffices to let R ≥ e λT (C + r), where r and λ satisfy r ≥ 2 ϕ(y)u e −ξ b (y)dy, λ ≥ sup µ∈M1(R) (−M, M ) −y + 2M, y ∈ [M, 2M ] −y − 2M, y ∈ [−2M, −M ] 0, otherwise , ι(y) = Z exp( 1 y 2 −1 ), y 2 < 1 0, otherwise. Then f M is a smooth function with the compact support [−2M − 1, 2M + 1]. In addition, f M (y) ≡ y on (−M + 1, M − 1), |f M x | ≤ 1, and |f M xx | is uniformly bounded for all M and is nonzero only on ∪ 2 , f M dt = u e ξ b , f M − u e −ξ b , f M . u e ±ξ b are exponentially decaying functions so u e ±ξ b , f M converges to ±ξ b rapidly as M → ∞. We note that L * φ φ, f M = σ 2 φ, f M yy /2 − θ φ, (y − a)fM y . By reading the properties of f M yy and Chebyshev's inequality, we have φ, f M yy → 0 as M → ∞. We write φ, (y − a)Since a is bounded and φ, f M y → 1 as M → ∞, a(1 − φ, f M y ) → 0 as M → ∞. t 0 e̺( 0M(t−s) R −1/2 dW (s) Ifx k (0) = −ξ b , then, using the fact that the uniform vector is in the null space of M , we have e MtX (0) =X(0). As a corollary we get the explicit representation of the empirical mean:T e M(t−s) R −1/2 dW (s) This shows the desired result.F.2. Proof of Proposition 8.2. The expansion of ξ 2 b follows from the explicit expression (3.5). The expansion of σ 2 T follows from the expansion of (8.3) and uses the properties of the matrix M . We have M = −θM − δθN , withM = I − u̺ T , where u = (1, . . . , 1) is the K-dimensional column vector, N ij = α i (δ ij − ρ j ), i, j = 1, . . . , K.The matrixM satisfiesM n =M for all n ≥ 1 and therefore −θt) n n! (M + δN ) n Using the fact thatM T ̺ = 0 (and again thatM n =M for n ≥ 1), we can expand̺ T e Mt = ̺ T + δ̺ T (−θt)N + (e −θt − 1 +θt)NM + δ 2 ̺ T (θt) 2 2 N 2 + e −θt − 1 +θt − (θt) 2 2 N 2M − 3(NM ) 2 + NM N −θt e −θt − 1 +θt (NM ) 2 ̺ + O(δ 3 ).Using the fact thatM T N T ̺ = N T ̺ andM T (N T ) 2 ̺ = (N T ) 2 ̺, this can be simplified into̺ T e Mt = ̺ T + δ̺ T (e −θt − 1)N + δ 2 ̺ T (θt) 2 − (1 +θt)(e −θt − 1 +θt) N 2 + O(δ 3 ).Consequently̺ T e Mt R −1 (e Mt ) T ̺ = ̺ T (I + (e −θt − 1)M )R −1 (I + (e −θt − 1)M T )̺ + 2δ̺ T (e −θt − 1)N R −1 (I + (e −θt − 1)M T )̺ + 2δ 2 ̺ T (θt) 2 − (1 +θt)(e −θt − 1 +θt) N 2 R −1 (I + (e −θt − 1)M T )̺ + δ 2 ̺ T (e −θt − 1)N R −1 (e −θt − 1)N T ̺ + O(δ 3 ). . 5 .N=100, h=0.1, σ=1, Θ=[5;10;15] ρ=[0.33;0.34;0.33], stdev(θ)/mean(θ)=0.40825 dm(0)/dξ=1.0096, Σ i (ρ i /Θ i )(3σ N=100, h=0.1, σ=2.1218, Θ=[5;10;15] ρ=[0.33;0.34;0.33], stdev(θ)/mean(θ)=0.40825 dm(0)/dξ=1.0002, Σ i (ρ i /Θ i )(3σ 2 /2Θ i −1)=1.0408e−017 N=100, h=0.1, σ=3, Θ=[5;10;15] ρ=[0.33;0.34;0.33], stdev(θ)/mean(θ)=0.40825 dm(0)/dξ=0.98922, Σ i (ρ i /Θ i )(3σ 2 /2Θ i −1)=0.12192 /2Θ i −1)=−0.0949 t Systemic Risk 0 2000 4000 6000 8000 10000 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 t Systemic Risk 0 2000 4000 6000 8000 10000 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 t Systemic Risk B.2. Proof of Proposition 3.2. All we need to show is that for small h, d dξ m(0) > 1 if and only if σ < σ div c , where m(ξ) is defined by(3.4). We obtaind dξ m by calculate d dξ Appendix C. Proof of Lemma 5.2. It suffices to show the case that δ = 1/n. For each n, let φ n ∈ A 1/n , such that inf φ∈A 1/n I h (φ) ≤ I h (φ n ) < inf φ∈A 1/n I h (φ) + 1/n; {I h (φ n )} are bounded from above by inf φ∈A I h (φ) + 1 < ∞. Because I h is a good rate function, and by Proposition B.13 of[18], compactness is equivalent to sequentially compactness in C([0, T ], M ∞ (R)), {φ n } has a convergent subsequence {φ n k } whose limit φ * is in A. As I h is lower semicontinuous, thenAppendix D. Proofs in Section 6. D.1. Proof of Theorem 6.1. We prove it in three steps. The first step is to show that there exists a uniform lower bound of I 0 (φ), for all φ ∈ A.Lemma D.1. If h = 0, then inf φ∈A I 0 (φ) ≥ 2ξ 2 0 /(σ 2 T ). Proof. For any φ ∈ A, a(t) denotes yφ(t, dy). We observe thatThen after taking the infimum over φ ∈ A, we haveThe last equality is obtained by a simple calculus of variation with the optimal path a(t) = 2ξ 0 t/T − ξ 0 . The second step is to show that I 0 (p e ) = 2ξ 2 0 /(σ 2 T ). Then inf φ∈A I 0 (φ) = 2ξ 2 0 /(σ 2 T ) and therefore p e is a minimizer for (5.5). Lemma D.2. If h = 0, andthen p e ∈ A and I 0 (p e ) = 2ξ 2 0 /(σ 2 T ). Proof. By reading (5.6) with φ = p e and h = 0, we have p e t = L * p e p e + (p e g) y . One can easily check that L * p e p e = 0 and p e t = −p e y d dt a e (t). Then we have g = − d dt a e (t) and by (5.7),Finally we prove that for h = 0, the minimizer p e is unique.Lemma D.3. For h = 0, p e is the unique minimizer for (5.5).Proof. From the previous lemmas, we find that if φ is a minimizer then a(t) = yφ(t, dy) must be a e (t), and f = − d dt a e (t)y is a global maximizer of J 0 (φ, ·). Then for any test functionf , d dǫ J 0 (φ, − d dt a e (t)y + ǫf ) = 0 at ǫ = 0. By a simple calculus of variations, φ satisfies the linear parabolic PDE:with the initial condition φ(0) = u e −ξ0 , and that implies the uniqueness of the minimizer, which is p e . D.2. Proof of Theorem 6.2 (Upper Bounds). Define the test function:We recall that from(2.3)It is not difficult to see that the first term of the right hand side of (D.1) is equal to 2ξ 2 b /(σ 2 T ) up to a term of order h as h → 0. Proof. We construct the test function φ u ∈ A as follows:where δT will be determined later. Note that inf φ∈A I h (φ) ≤ I h (φ u ) so we just need to compute I h (φ u ). Let g u satisfy (5.6) for φ = φ u . For t ∈ (δT, T − δT ), φ u (t) = p u (t), and it is easy to see that p u t = − d dt a u p u y and L * p u p u = 0. Therefore for t ∈ (δT, T −δT ), g u = − d dt a u − h(y 3 − y) by (5.6). From (5.7), we haveThe rest is to show that for any ǫ > 0, there exists a sufficiently small h such that the last term in the last equation is bounded by ǫ. It suffices to show that for any δT > 0, we can choose a sufficiently small h such that φ u , (g u ) 2 is bounded by a δTindependent constant c u > 0 for t ∈ [0, δT ] ∪ [T − δT, T ]. If so, then let δT < ǫσ 2 /c u andAppendix E. Proofs in Section 7.E.1. Proof of Lemma 7.1. We note that p t = −p y d dt a and thereforeAfter collecting O(1) terms in (5.6) and integrating over y, we haveThen g (0) = − d dt a. Now we collect O(h) terms in (5.6) and integrating over y. We getβ n ∂ n ∂y n p.Using the fact thatβ n ∂ n ∂y n p, and the optimal β n are obtained by comparing the coefficients.E.2. Proof of Lemma 7.2. Let ψ(2)denote the anti-derivative of q (2) that vanishes at −∞. After collecting O(h 2 ) terms in (5.6) and integrating over y. We have ψ (2) t = 1 2 σ 2 q (2) y + θ(y − a)q (2) + U (y)q (1) + q (2) g (0) + q (1) g (1) + pg(2). (E.1)Note that pg (2) = ∞ n=0 γ n ∂ n ∂y n p, so γ 0 is obtained by integrating (E.1) from y = −∞ to y = ∞. Then we have γ 0 = − q (1) , U (y) + g(1). < inf φ∈B 1/n I h (φ) + 1/n; {I h (φ n )} are bounded from above by inf φ∈B I h (φ) +. I h (φ nI h (φ n ) < inf φ∈B 1/n I h (φ) + 1/n; {I h (φ n )} are bounded from above by inf φ∈B I h (φ) + Let {t n k } be a convergent subsequence of {t n }. Because I h is a good rate function, and by Proposition B.13 of [18], compactness is equivalent to sequentially compactness in C. &lt; ∞ , {φ n k } has a convergent subsequence {φ n k ′ } whose limit φ * is in A(t * ) where t * = lim t n k . As I h is lower semicontinuous, then REFERENCES< ∞. Let {t n k } be a convergent subsequence of {t n }. Because I h is a good rate function, and by Proposition B.13 of [18], compactness is equivalent to sequentially compactness in C([0, T ], M ∞ (R)), {φ n k } has a convergent subsequence {φ n k ′ } whose limit φ * is in A(t * ) where t * = lim t n k . As I h is lower semicontinuous, then REFERENCES Large deviations for Langevin spin glass dynamics. 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{'abstract': 'We consider a system of diffusion processes that interact through their empirical mean and have a stabilizing force acting on each of them, corresponding to a bistable potential. There are three parameters that characterize the system: the strength of the intrinsic stabilization, the strength of the external random perturbations, and the degree of cooperation or interaction between them. The latter is the rate of mean reversion of each component to the empirical mean of the system. We interpret this model in the context of systemic risk and analyze in detail the effect of cooperation between the components, that is, the rate of mean reversion. We show that in a certain regime of parameters increasing cooperation tends to increase the stability of the individual agents but it also increases the overall or systemic risk. We use the theory of large deviations of diffusions interacting through their mean field.', 'arxivid': '1204.3536', 'author': ['Josselin Garnier ', 'George Papanicolaou ', 'ANDTzu-Wei Yang '], 'authoraffiliation': [], 'corpusid': 15204579, 'doi': '10.1137/12087387x', 'github_urls': [], 'n_tokens_mistral': 29110, 'n_tokens_neox': 25744, 'n_words': 15814, 'pdfsha': '0d02d33e87102675c3dc3cf4fc33c65a894b9226', 'pdfurls': ['https://arxiv.org/pdf/1204.3536v2.pdf'], 'title': ['LARGE DEVIATIONS FOR A MEAN FIELD MODEL OF SYSTEMIC RISK', 'LARGE DEVIATIONS FOR A MEAN FIELD MODEL OF SYSTEMIC RISK'], 'venue': []}

arxiv

On the geometry of the f -invariant 22 Sep 2009 Hanno Von Bodecker Fakultät für Mathematik Ruhr-Universität Bochum 44780BochumGermany On the geometry of the f -invariant 22 Sep 20091 The f -invariant is a higher version of the e-invariant that takes values in the divided congruences between modular forms; it can be formulated as an elliptic genus of manifolds with corners of codimension two.In this thesis, we develop a geometrical interpretation of the finvariant in terms of index theory, thereby providing an analytical link between the stable homotopy groups of the spheres and the arithmetic of modular forms. In particular, we are able to establish a formula that allows us to compute the f -invariant from a single face. Furthermore, we apply our results to the situation of cartesian products and principal circle bundles, performing explicit calculations. * Contents The computation of the stable homotopy groups of the sphere, which, by the Pontrjagin-Thom construction, can be interpreted as the bordism groups of framed manifolds, is one of the most fundamental problems in pure mathematics, and the Adams-Novikov spectral sequence (ANSS) serves as a powerful tool to attack this problem, see e.g. [Rav04]. In [Lau00], the ANSS is interpreted in terms of manifolds with corners, their codimension corresponding to the AN filtration degree; furthermore, an invariant for elements of second AN filtration is proposed. As a follow-up to the degree and the e-invariant, this so-called f -invariant arises as an elliptic genus of manifolds with corners of codimension two and takes values in the divided congruences between modular forms. It is well-known that the classical genera of closed manifolds can be understood in terms of index theory, and the seminal work of Atiyah, Patodi, and Singer on index theory on manifolds with smooth boundary [APS75a] can be used to relate the spectral asymmetry of Dirac operators to the e-invariant [APS75b]. The purpose of this thesis is to show that these powerful ideas also enable us to provide geometrical insight into the f -invariant. The outline is as follows: Section 1 consists of a brief recollection of the relevant background material, namely tangential structures, k -manifolds, characteristic forms, some index theory results, and the Hirzebruch elliptic genus of level N, followed by an (admittedly biased) exposition of the index theoretical aspects of the e-invariant; lastly, we recall the definition of the f -invariant in terms of the relative classes of a suitable 2 -manifold. In section 2, we set out to explore the geometry behind the occurrence of the divided congruences in the definition of the f -invariant. A somewhat surprising result is the following: Theorem. Let X be a compact manifold of dimension 2n, and let E 1 , E 2 , and F be hermitian vector bundles over X such that E 1 ⊕ E 2 ∼ = T X st , and that there is a given trivialization ψ : E 1 | ∂X ∼ = ∂X ×C k . Then, for compatible connections, the inhomogeneous combination of modular forms given by X Ell(∇ E 1 )Ell 0 (∇ E 2 )ch(∇ F ) has an integral q-expansion. Having this result at our disposal, we write down a similar expression on a (U, f r) 2 -manifold and determine which reductions have to be made in order to retrieve the information that depends only on its corner; we call the resulting geometrical invariantf , and it will serve as a tool for an index theoretical computation of the topological f -invariant later on. Section 3 is devoted to establishing a method of calculating the f -invariant in a more analytical fashion, based onf . To this end, we are going to introduce the notion of a (U, f r) 2 f -manifold; roughly speaking, these manifolds are families of (U, f r)-manifolds parameterized by (U, f r)-manifolds. This construction enables us to reformulate the f -invariant in the following form: Theorem. Let Z be a (U, f r) 2 f -manifold, and let ∇ 1 = π * ∇ E and ∇ 2 be compatible connections on E 1 = π * E and E 2 , respectively. Then the finvariant of its corner M is given by f (M) ≡ Bê Γ Ell(∇ E ), where we defined (a de Rham representative of ) the e Γ -invariant of a family: This result furnishes a close analogy to the e-invariant, and makes possible the use of 'classical' techniques from index theory to compute the f -invariant from a single face: Corollary. Let Z be as above. Assume that the kernel of the twisted Dirac family ð Γ ∂X is of constant rank along the fibers. Then the f -invariant of the corner is given by B η ð Γ ∂X + 1 2 ch ∇ ker ð Γ ∂X + Z ′′ /B cs Ell(∇ E ) . Furthermore, we establish a vanishing result: Theorem. Let M be the codimension-three corner of a (U, f r) 3 f -manifold Y . Then the f -invariant of M is trivial. The following sections focus on sample calculations and further simplifications: In section 4, we treat cartesian products, in which case we can actually reduce the computations to the corner itself: Theorem. Let Y 1 , Y 2 be odd-dimensional framed manifolds, and let m(Y i ) be any modular form of weight (dim Y i + 1)/2 w.r.t. Γ = Γ 1 (N) such that m(Y i ) ≡ e C (Y i ) mod Z Γ [[q]]. Then we have f (Y 1 × Y 2 ) ≡ m(Y 1 )e C (Y 2 ) ≡ −m(Y 2 )e C (Y 1 ). Thus, the f -invariant of a product is determined by the e C -invariants of the factors, and the latter can be calculated by various means. We illustrate this result by performing explicit calculations at the level N = 3, covering a broad variety of products. In section 5, we turn our attention to principal circle bundles, and our index theoretical approach yields the following: Corollary. Let L be a hermitian line with unitary connection ∇ L over a (U, f r)-manifold B of dimension 2n + 2, and let S(L) | be the framed circle bundle over ∂B. Then we have f (S(L) | ) ≡ n k=0 B k+1 (k + 1)! B iF L 2π k Ell(∇ E ) . As an application, we treat the case of principal torus bundles over a framed base. In particular, we perform explicit calculations for the generic situation up to (total) dimension 14, enabling us to determine the necessary and sufficient conditions for non-triviality (at the level N = 3) in this range. Finally, we decided to provide a rather extensive appendix: Besides some useful formulae, we remind the reader of the theory of modular forms of level N, deriving explicit expressions for N = 3. Furthermore we expand the Hirzebruch genus at this level, thereby correcting some errors in the appendix of [HBJ92]. Moreover, we compile a list of the congruences relevant to the computations carried out in the main part. For the sake of completeness, we also included a derivation of theη-form in the situation of a principal circle bundle (following [Zha94]). who has always been willing to share his insight and interest in a wide variety of mathematical topics. Furthermore, I am grateful to all the members of the topology chair for creating such a pleasant atmosphere. Last but not least, I acknowledge financial support from the DFG within the Graduiertenkolleg 1150 "Homotopy and Cohomology". Preliminaries Tangential structures We start by reviewing some basic definitions appearing in bordism theories, see e.g. [CF66], [Sto68]: Let X be a smooth compact manifold of dimension n and consider 'the' stable tangent bundle T X st = T X ⊕ X × R 2k−n , 2k ≥ n + 2; we suppress the dependence on k, as it does not matter for our purposes (as long as 2k ≥ n + 2). A stably almost complex structure on X is a complex structure on T X st , i.e. a linear bundle map J covering the identity and squaring to minus the identity on the fiber. Thus, T X st may be identified with a complex vector bundle E (of rank k); consequently, we can define its Chern classes, and there is a preferred orientation on X. We denote the underlying homotopy class of the stably almost complex structure by φ, which we will refer to as a U-structure; phrased differently, a U-structure is a lift (up to homotopy) of the classfying map of the tangent bundle: BU X T X st / / φ 7 7 o o o o o o o o o o o o o BO A pair (X, φ) is called a U-manifold; it will simply be denoted by X if confusion is unlikely. Any U-structure admits a 'negative', −φ, and we set −(X, φ) = (X, −φ); furthermore, U-structures are compatible w.r.t. taking boundaries, i.e. ∂(X, φ) = (∂X, ∂φ). We define an equivalence relation called U-bordism, X 1 ∼ U X 2 , if there is a U-manifold W such that ∂W ∼ = X 1 ⊔−X 2 . Disjoint union and cartesian product turn the set of U-bordism classes into a graded ring, the complex bordism ring Ω U * . Similarly, a framing of X is trivialization of the stable tangent bundle, ψ : T X st ∼ = X × R 2k ; thus, up to homotopy, we have a lift of the classifying map to EO: EO X T X st / / 7 7 o o o o o o o o o o o o o BO A framed manifold is a manifold with a homotopy class of trivializations of the stable tangent bundle. Take note that, by pulling back the canonical complex structure on R 2k , any framed manifold becomes a U-manifold. Again, framings admit negatives, are compatible w.r.t. taking boundaries, and we deem two framed manifolds to be equivalent, X 1 ∼ f r X 2 , if there is a framed manifold W such that ∂W ∼ = X 1 ⊔ −X 2 ; the resulting graded ring, the framed bordism ring, will be denoted by Ω f r * . Let us sketch the relation to homotopy theory: We may embed the closed framed manifold X into R N , for N sufficiently large; up to homotopy, the framing is equivalent to a trivialization of the stable normal bundle νX, i.e. we obtain a map ϕ : νX → R N −n . (1) The normal bundle may be identified with a tubular neighborhood of X, and the map (1) extends to a map of spheres, ϕ : S N → S N −n , by sending the complement of the tubular neighborhood to the point at infinity; this is the so-called Pontrjagin-Thom construction. By Freudenthal's theorem, π n+k S n is independent of n provided that n > k + 1; we call it the k th stable homotopy group of the sphere (or the k th stable stem) and denote it by π st k . The Pontrjagin-Thom construction depends only on the framed bordism class of X, and it is well-known that the map Ω f r k → π st k , [X] → [φ], is an isomorphism. k -manifolds Later on, we want to allow manifolds to have corners: Recall from [Jän68] that we can define a smooth n-dimensional manifold Z with corners as being differentiably modeled on the open sets of {x ∈ R n |x 1 ≥ 0, . . . , x n ≥ 0}. If x ∈ Z is represented by (x 1 , . . . x n ) in a local coordinate system, we denote by c(x) the number of zeros in this n-tuple; this number is independent of the choice of coordinate system. Note that x belongs to the closure of at most c(x) different connectedness components of {p ∈ Z|c(p) = 1}. We call Z a manifold with faces, if each x ∈ Z does belong to the closure of c(x) different components of {p ∈ Z|c(p) = 1}. For a manifold with faces, the closure of a connectedness component of {p ∈ Z|c(p) = 1} has the structure of an (n − 1)-dimensional manifold with corners and is called a connected face of Z; any union of pairwise disjoint connected faces is called a face of Z. A k -manifold is an n-dimensional manifold with faces Z together with a k-tuple (∂ 1 Z, . . . , ∂ k Z) of faces such that (i) ∂ 1 Z ∪ · · · ∪ ∂ k Z = ∂Z and (ii) ∂ i Z ∩ ∂ j Z is a face of ∂ i Z and of ∂ j Z for i = j. In particular, a 0 -manifold is a manifold without boundary; in the situation of a 1 -manifold, we recover the usual concept of a manifold with boundary. Connections, curvature, and Chern forms For the most part, we are adopting the notational conventions of [BGV04]: Let E be a complex vector bundle over a compact manifold X. We denote the space of all (smooth) sections by Γ(X, E), or, if confusion is unlikely, simply by Γ(E). A connection (or covariant derivative) on E is a differential operator ∇ : Γ(E) → Γ(T * X ⊗ E) satisfying the Leibniz rule; usually we restrict our attention to covariant derivatives preserving a given hermitian metric on E, in which case we call the connection unitary. A covariant derivative extends to E-valued differential forms, which we denote by Ω i (X, E). The curvature of a covariant derivative is the End(E)-valued two-form on X given by F (u, v) = ∇ u ∇ v − ∇ v ∇ u − ∇ [u,v] , where u and v are vector fields on X. Of course, these defintions carry over to the case of real vector bundles as well; in the situation of the Levi-Civita connection, we will denote the so(T X)-valued Riemannian curvature form by R. These concepts may be generalized to Z/2-graded bundles E, which will be referred to as superbundles; for such a superbundle, there is a total Z/2grading on the space of E-valued forms, Ω ± = Ω 2i (X, E ± ) ⊕ Ω 2i+1 (X, E ∓ ). A superconnection on E is an odd-parity first-order differential operator A : Ω ± → Ω ∓ satisfying the Leibniz rule in the Z/2-graded sense, and its action extends naturally to Ω(X, End(E)). The curvature of a superconnection A is the operator A 2 , which is given by the action of an End(E)-valued differential form F of even total degree; it satisfies the Bianchi identity, AF = 0. Due to the supercommutativity of ΛT * X, we obtain a canonical bundle map Str : ΛT * X ⊗ End(E) → ΛT * X; applied to sections, this yields the so-called supertrace on End(E)-valued forms. Now, if A 2 ∈ Ω + (X, End(E)) is the curvature of a superconnection, we may apply the supertrace to any analytical function f of A 2 to obtain an even differential form on X, the Chern-Weil form of A corresponding to f ; it is closed due to the Bianchi identity. Furthermore, the transgression formula, d dt Str(f (A 2 t )) = d Str dA t dt f ′ (A 2 t ) , applied to the family A t = (1 − t)A 0 + tA 1 = A 0 + tω and integrated w.r.t. t, yields the relation Str(f (A 2 1 )) − Str(f (A 2 0 )) = d 1 0 Str(ωf ′ (A 2 t ))dt.(2) In particular, we may apply these constructions to an ordinary bundle with connection, and, contrary to [BGV04], we normalize the characteristic forms such that they represent rational characteristic classes. Thus, if E is a hermitian vector bundle with unitary connection ∇ E and curvature F E , we denote the Chern character form by ch ∇ E = tr exp iF E 2π , and the Todd genus form by T d ∇ E = det iF/2π 1 − exp (−iF/2π) = exp tr ln iF/2π 1 − exp (−iF/2π) . Similarly, for a real vector bundle with connection ∇ and curvature R we denote the genus form bŷ A(∇) = det 1/2 R/4πi sinh(R/4πi) = exp 1 2 tr ln R/4πi sinh(R/4πi) . Obviously, a unitary connection ∇ E on a hermitian vector bundle E induces a connection ∇ E R on the underlying real bundle and, via the trace, a connection ∇ det E on the hermitian line det E = Λ max E, which enables us to express T d ∇ E = ∇ E R exp tr iF E 4π . Later on, it will be convenient to be able to change the connection on the underlying real bundle; by means of the transgression formula (2), we can construct a differential form cs (T d, ∇ 0 , ∇ 1 ) satisfying d cs (T d, ∇ 0 , ∇ 1 ) =  (∇ R 1 ) −Â(∇ R 0 ) exp tr iF E 4π ,(3) which we call the Chern-Simons form (associated to the Todd genus). This construction extends to twisted versions of the Todd genus as well, and we will often denote the corresponding forms simply by cs. Some classical index theory Recall that the Atiyah-Singer index theorem [AS68a] identifies the analytical index of an elliptic (i.e. Fredholm) operator D on a closed oriented manifold X with the topological index; more precisely, we obtain a map that sends the class of the symbol of D -viewed as an element of the compactly supported K-theory of the tangent bundle -to the formal difference of the kernel and cokernel of D (an element in the K-theory of the point), and the numerical value of the index can be computed using a simple formula by passing to ordinary cohomology [AS68b]. An alternative route to obtaining the index formula makes use of the heat kernel approach (our main reference is [BGV04]); its main advantage is the fact that it is local in the sense that it yields a differential form which is constructed canonically from the metrical connections on the bundles involved and equates to the index upon integration. For our purposes, it will suffice to restrict our attention to the index theory of twisted Spin C Dirac operators, although the theorems stated below hold for more general Clifford modules. We define the group Spin C (n) as (see e.g. [LM89]) Spin C (n) = Spin(n) × Z/2 U(1), and a Spin C structure on an oriented Riemannian manifold X of dimension n is a choice of a hermitian line L such that w 2 (T X) ≡ c 1 (L) mod 2; clearly, the existence of a Spin C structure is equivalent to the condition βw 2 (T X) = 0, where β is the Bockstein. A unitary connection ∇ L on L, together with the Levi-Civita connection ∇ LC on T X, determines a Spin C connection ∇; conversely, a Spin C connections projects (under the canonical twofold covering) to a connection on the SO(n) × U(1) principal bundle and its associated vector bundles. Take note that, for a hermitian line L with curvature F L , the characteristic form given by ch ∇ L 1/2 = exp iF L 4π is well-defined even if L does not admit a global square root. Obviously, the complex representations of Spin C (n) are the same as those of Spin(n), so we get an associated complex spinor bundle S over X; furthermore, if n is even, S is Z/2-graded by means of the chirality operator γ, which is defined by Clifford multiplication with i n/2 e 1 . . . e n . We define the Spin C Dirac operator by composing covariant differentiation with Clifford multiplication: ð = e i · ∇ e i : Γ(X, S) → Γ(X, S), where the e i constitute a local orthonormal frame for T X. Now let dim X = n be even; then ð anticommutes with γ, so we may decompose it into ð = 0 ð − ð + 0 , where ð ± are the restrictions to sections of S ± , and ð − is the adjoint of ð + , due to the unitarity of ∇. We define the index of the Spin C Dirac operator to be Ind(ð) = dim ker ð + − dim ker ð − . Given a hermitian vector bundle E over X with unitary connection ∇ E , we may form the twisted Spin C Dirac operator, ð ⊗ E : Γ(X, S ⊗ E) → Γ(X, S ⊗ E), by using the tensor product connection on S ⊗ E, and all of our discussion above holds verbatim. Finally, we can state Theorem 1.1 (Atiyah-Singer). Let X be a closed, oriented, even-dimensional Riemannian manifold with Spin C structure defined by a hermitian line L and let E be a hermitian vector bundle with unitary connection. Then the index of the twisted Spin C Dirac operator is given by the formula Ind(ð ⊗ E) = X (∇ T X,LC )ch(∇ L 1/2 )ch(∇ E ). Remark 1.2. If X is a U-manifold, we get a canonical Spin C structure by setting L = Λ max T X st . Take note that in this situation A(∇ T X,LC )ch(∇ L 1/2 )(4) represents the Todd class, but we warn the reader that, unless X is Kähler, the Levi-Civita connection will not be compatible with a complex structure on T X. On the other hand, if X is spin, i.e. w 2 (T X) = 0, we may choose L to be trivial, in which case we essentially recover the situation of 'the' Dirac operator, usually denoted / D (the precise definition of / D, which would take into account the choice of Spin structure and avoid additional complexification, shall not be needed in this thesis). Imposing global boundary conditions and requiring product type structures near the boundary, Atiyah, Patodi, and Singer were able to generalize the formula in Theorem 1.1 to the situation where X has smooth non-empty boundary [APS75a]: Restriction to the boundary ∂X induces an operator ð ∂X ⊗ E : Γ (S + ⊗ E)| ∂X → Γ (S + ⊗ E)| ∂X which is formally self-adjoint and elliptic; we may decompose the L 2 completion of Γ((S + ⊗ E)| ∂X ) into eigenspaces, and, letting P ≥0 denote the orthogonal projection onto the non-negative part, we define Γ(S + ⊗ E, P ≥0 ) = s ∈ Γ(S + ⊗ E) | P ≥0 (s| ∂X ) = 0 . Then, according to [APS75a], ð + ⊗ E : Γ(S + ⊗ E, P ≥0 ) → Γ(S − ⊗ E)(5) defines an elliptic problem with finite index. Defining the function η(ð ∂X ⊗ E, s) = λ∈spec\{0} λ|λ| −s−1 , Re(s) >> 1, which extends meromorphically and is holomorphic at s = 0, we may state: Theorem 1.3 (Atiyah-Patodi-Singer) . Let E be a hermitian vector bundle with unitary connection over an oriented, even-dimensional, compact Riemannian manifold X with Spin C structure defined by a hermitian line L. Assuming product type structures near the boundary ∂X, the index of the twisted Spin C Dirac operator w.r.t. the condition (5) is given by the formula Ind AP S (ð ⊗ E) = X (∇ T X,LC )ch(∇ L 1/2 )ch(∇ E ) − ξ(ð ∂X ⊗ E), where ξ(ð ∂X ⊗ E) = 1 2 η(ð ∂X ⊗ E, 0) + 1 2 dim ker(ð ∂X ⊗ E). Next, we are interested in generalizations of the results above to the case of families: Let π : Z → B be a submersion that defines a fiber bundle with typical fiber X and let the vertical tangent bundle T (Z/B) be equipped with a metric g T (Z/B) ; furthermore, we make a choice of splitting T Z ∼ = T (Z/B) ⊕ π * T B. Using an auxiliary metric on T B, we form the metric g = g T (Z/B) ⊕ π * g T B ; (6) let ∇ T Z be its Levi-Civita connection, and let P : T Z → T (Z/B) denote the orthogonal projection; by setting ∇ T (Z/B) = P • ∇ T Z • P,(7) we obtain a connection ∇ T (Z/B) on T (Z/B); in particular, it does not depend on the metric g T B and restricts to the Levi-Civita connection on each fiber. Take note that the connection on T Z given by ∇ ⊕ = ∇ T (Z/B) ⊕ π * ∇ T B preserves the metric (6), but it is not torsion-free. We define the curvature of the fibration T (u, v) = −P [u, v], where u and v are horizontal vectors. Fixing a Spin C structure on the vertical tangent, we may form the associated complex spinor bundle S; given a hermitian vector bundle E over Z with unitary connection ∇ E , we get a family of twisted Dirac operators {ð b ⊗ E} b∈B , where ð b ⊗ E : Γ ((S ⊗ E)| π −1 b ) → Γ ((S ⊗ E)| π −1 b ) .(8) Let us briefly comment on the situation where the fiber X is even-dimensional and closed [AS71]: If we assume that the kernel and cokernel of ð + ⊗ E form vector bundles over B, we can define the index bundle as the formal difference class, Ind(ð ⊗ E) = ker ð + ⊗ E ⊖ ker ð − ⊗ E ∈ K(B).(9) The caveat is that, generically, the kernels do not form vector bundles. However, this situation may be remedied by modifying the family of operators using a compact perturbation; as such a perturbation does not affect the index, we obtain a well-defined K-theory class, which, by abuse of notation, will still be denoted as in (9). Furthermore, the K-theoretical construction of the index map generalizes to the families situation, and, upon applying the Chern character, one obtains the following formula in H * (B, Q) [AS71]: ch(Ind(ð ⊗ E)) = π * [Â(T (Z/B)) exp( 1 2 c 1 (L))ch(E)].(10) There is a heat kernel proof of this result, due to Bismut [Bis85], which we shall omit; however, it introduces a concept that will be needed later on: Given a family of twisted Dirac operators D constructed in the situation of a closed (but not necessarily even-dimensional) typical fiber, we define the Bismut superconnection to be A t = √ tD +∇ − c(T ) 4 √ t ,(11) where∇ is the natural lift of ∇ to the infinite dimensional bundle of sections of the vertical spinor bundle and c(T ) denotes Clifford multiplication with the curvature of the fibration; we refer the reader to [Bis85], [BGV04] for further details. With this in mind, let us consider the family (8) in the situation of an even-dimensional typical fiber X with smooth, non-empty boundary, where g T (Z/B) is assumed to be of product type near the boundary. Then the induced family of Dirac operators on the boundary gives rise to an associated Bismut superconnection A t , and, if the kernel of D forms a vector bundle over B, then the following differential form, η = 1 √ π ∞ 0 Tr ev dA t dt exp(−A 2 t ) dt, where Tr ev denotes the even form part of the trace, is well-defined, see e.g. [BGV04]. This defines an even form on B, and its rescaled version η| deg 2k = (2πi) −kη | deg 2k , will be referred to as theη-form. Finally, a generalization of Theorem 1.3 to the situation of families, which is due to Bismut and Cheeger [BC90a], [BC90b], [BC91], reads: Theorem 1.4 (Bismut-Cheeger). Let Z → B be a fiber bundle, the evendimensional typical fiber X having smooth, non-empty boundary, and let ð⊗E be a twisted Dirac family constructed using a metric that is of product type near the boundary. If the kernel of the twisted Dirac operator induced on the fiberwise boundary is of constant rank, then the index bundle w.r.t. the APS boundary condition is well-defined; furthermore, a representative in cohomology of the Chern character of the index bundle is given by the following smooth differential form on B: Z/B  (∇ T (Z/B) )ch(∇ L 1/2 )ch(∇ E ) −η(ð ∂ ⊗ E) − 1 2 ch ∇ ker(ð ∂ ⊗E) . Hirzebruch elliptic genera Let us recall the definition of the Hirzebruch elliptic genus of level N associated to the congruence subgroup Γ = Γ 1 (N) [HBJ92]: For fixed N > 1, let ζ be a primitive N th root of unity and let q = exp(2πiτ ), τ ∈ h. We consider the function Ell Γ (x) = x 1 − e −x 1 − ζe −x 1 − ζ ∞ n=1 (1 − q n ) 2 (1 − q n e x )(1 − q n e −x ) 1 − q n e x /ζ 1 − q n /ζ 1 − ζq n e −x 1 − ζq n as a power series in the indeterminate x of degree two. Making use of standard results on elliptic functions, it can be shown that the coefficient of x m is a modular form of weight m w.r.t. Γ (confer e.g. [HBJ92]); an alternative route is to rewrite the power series such that modularity becomes manifest, see appendix C. For a hermitian vector bundle E with unitary connection ∇ E , we define the elliptic genus form to be Ell Γ ∇ E = exp tr ln Ell Γ iF E 2π .(12) If we introduce the power operations with respect to a formal parameter t, S t (V ) = k≥0 t k S k (V ), Λ t (V ) = k≥0 t k Λ k (V ), which extend to K-theory classes in the obvious way [Ati89], we see that the underlying cohomology class of (12) is given by Ell Γ (E) = χ −ζ (E)ch ∞ n=1 S q n (E ⊗ C) ⊗ Λ −q n /ζ E ⊗ Λ −ζq n E * ,(13) where the bar denotes virtual reduction of the complex bundles involved, and χ y is the stable χ y -genus, i.e. χ y (E) = (1 + y) −rkE χ y (E) = (1 + y) −rkE T d(E)ch(Λ y E * ). We have the following well-known result: Proposition 1.5. Let X be a closed U-manifold. Then the elliptic genus of X has an integral q-expansion, i.e. Ell Γ (T X), [X] ∈ Z[ζ, 1/N][[q]]. Proof. Multiplying the LHS by (1 − ζ) rk(T X st ) , expanding the formal power series, and grouping powers of ζ, we see that, by Theorem 1.1, every coefficient admits an interpretation as the index of a suitably twisted Spin C Dirac operator. Since (1 − ζ) −1 ∈ Z[ζ, 1/N], the claim is proven. For convenience, we are going to delete the fixed group Γ from the notation (after all, we suppressed the dependence on ζ from the very beginning); furthermore, we introduce the abbreviations Ell 0 = Ell| q=0 , Ell = Ell − Ell 0 . The e-invariant The original formulation of the e-invariant, e : π st 2k+1 → Q/Z, is due to Adams [Ada66]; for our purposes however, it will be more convenient to use the cobordism description given by Conner and Floyd [CF66]: Definition 1. A (U, f r)-manifold is a compact U-manifold X with smooth boundary and a trivialization of E ∼ = T X st over the boundary, i.e. a bundle map ψ : E| ∂X ∼ = ∂X × C k . In particular, ψ provides a framing for ∂X; using the relative characteristic classes of the complex vector bundle E ∼ = T X st , the complex e-invariant of the framed bordism class of ∂X is defined to be e C (∂X) ≡ T d(E), [X, ∂X] mod Z.(14) As mentioned in the introduction, the e-invariant admits an interpretation in terms of index theory, and it can be related to (and computed from) the spectral asymmetry encoded into η(ð) [APS75b]; we begin by rephrasing the RHS of (14): The framing induces a hermitian metric on E| ∂X ; we extend it to E such that it is of product type near the boundary. Furthermore, we endow E with a unitary connection ∇ E that restricts to the canonical flat connection specified by the trivialization, i.e. the one w.r.t. which the frame is parallel; this enables us to rewrite T d(E), [X, ∂X] = X T d(∇ E ). Now we can show that e C is well-defined: Let X ′ be another (U, f r)manifold having the same framed boundary, i.e. ∂X ′ ∼ = ∂X, and let W be the closed U-manifold obtained by gluing X and −X ′ along the boundary. Then, by the linearity of the integral, the relative Todd genera of X and X ′ differ by the Todd genus of W , which is an integer by Theorem 1.1. Noting that the integrand is trivial on any framed bordism W ′ shows that e C depends only on the framed bordism class of ∂X. In order to compute the e C -invariant analytically, we consider the canonical Spin C Dirac operator on the (U, f r)-manifold X and apply Theorem 1.3, X T d ∇ T X,LC ≡ ξ (ð ∂X ) mod Z, where we used T d(∇ T X,LC ) as the shorthand notation for (4), i.e. the local index form associated to the Spin C Dirac operator built from the Levi-Civita connection ∇ LC on T X and the connection on det E (induced by ∇ E ). This implies e C (∂X) ≡ ξ (ð ∂X ) + X T d ∇ E − T d ∇ T X,LC mod Z, but with the help of (3) and Stokes' theorem, the integral can be reduced to an integral over ∂X. Thus, the e-invariant is computable from geometrical data on M = ∂X itself: e C (M) ≡ ξ (ð M ) + M cs mod Z.(15) Remark 1.6. In [APS75b], Atiyah, Patodi, and Singer actually treat the real e-invariant, e R : π st 4k−1 → Q/Z: Since MSpin 4k−1 = 0, the framed manifold M is the boundary of a Spin-manifold N. Considering the Dirac operator / D, the quaternionic structure of the spinors in dimensions 4k, 4k − 1, for 2 ∤ k implies that the kernels are even-dimensional, so one obtains the refined result ǫ(k) M cs + ξ(/ D M ) ≡ ǫ(k)  (T N), [N, M] ≡ e R (M) mod Z, where ǫ(k) = 1 if 2|k and 1/2 otherwise. We would like to point out that we have MSU 4k−1 = 0 as well [CF66]; furthermore, the first Chern class of an SU-manifold is trivial, in which case the Todd genus coincides withÂ, showing that e R /ǫ ≡ e C mod Z. Admittedly, the formula (15) seems of little practical use, as one rarely is in the situation to compute the spectrum of ð explicitly. There are, however, some notable exceptions: In particular, the analytical computation of the (real) e-invariant for nilmanifolds covered by Heisenberg groups has been carried out by Deninger and Singhof, thus exhibiting a family representing (twice) the generator of Im(J) in dimension 8k + 3 (8k + 7) [DS84]. On the other hand, index theory considerations yield a vanishing theorem for compact Lie groups of higher rank [AS74]; strictly speaking, this result is formulated for e R and holds under the additional assumption of semi-simplicity. The vanishing of the complex e-invariant for higher rank Lie groups (not necessarily semi-simple) can also be deduced from the algebraic-topological results of [Kna78], see [Lau00] for a geometrical interpretation. The topological f -invariant Recall that the Adams-Novikov spectral sequence gives rise to a filtration of the stable stems. Geometrically, the AN filtration can be understood in terms of manifolds with corners [Lau00]: A framed manifold is in k th filtration if it occurs as the codimension-k corner of a so-called (U, f r) kmanifold; in particular, we already defined the e C -invariant for boundaries of (U, f r)-manifolds, i.e. for manifolds in first filtration. Definition 2. A (U, f r) 2 -manifold is a compact 2 -manifold Z together with two complex vector bundles E 1 , E 2 , with trivializations over the faces ∂ 1 Z, ∂ 2 Z, respectively, i.e. a choice of bundle maps ψ i : E i | ∂ i Z ∼ = ∂ i Z × C k i , and an isomorphism T Z st ∼ = E 1 ⊕ E 2 (in the stable sense). Fixing Γ = Γ 1 (N), we set Z Γ = Z[ζ, 1/N] and denote by M Γ * the graded ring of modular forms w.r.t. Γ which expand integrally, i.e. which lie in Z Γ [[q]]. We define the ring of divided congruences D Γ to consist of those rational combinations of modular forms which expand integrally; this ring can be filtered by setting D Γ k = f = k i=0 f i f i ∈ M Γ i ⊗ Q, f ∈ Z Γ [[q]] . Finally, we introduce D Γ k = D Γ k + M Γ 0 ⊗ Q + M Γ k ⊗ Q. Let M 2n be the codimension-two corner of a (U, f r) 2 -manifold Z. Using the relative Chern classes of the split tangent bundle, the f -invariant of the framed bordism class of M is defined to be f (M) ≡ (Ell(E 1 ) − 1)(Ell 0 (E 2 ) − 1), [Z, ∂Z] mod D Γ n+1 ,(16) hence f takes values in D Γ n+1 ⊗ Q/Z. We refer to [Lau99] and [Lau00] for details concerning the homotopy theoretical construction and its interpretation as a genus arising from an (MU 2 -) orientation of a suitable 2 -spectrum E; what we are aiming at in this thesis, however, is to provide a geometrical interpretation of (16), in a fashion similar to the last section. ξ ∇ E ≡ ξ(ð ⊗ E) − rk(E)ξ(ð) mod Z,(17) where ð denotes the canonical Spin C Dirac operator on M, is independent of the metric, hence an R/Z-valued invariant of the flat bundle E. This can be seen as follows: Since Ω U odd (BU) = 0, we can find a U-manifold X with boundary M and a vector bundleÊ over X (not necessarily flat) that extends E; then Theorem 1.3 yields ξ ∇ E ≡ X ch ∇Ê − rkÊ T d(∇ T X ) mod Z,(18) which is the mod Z reduction of the evaluation of a real relative cohomology class and easily seen to be independent of all choices. Obviously, this invariance property persists if we couple E to twisted versions of ð, i.e. we may considerξ ∇ E ⊗ F ≡ ξ(ð ⊗ F ⊗ E) − rk(E)ξ(ð ⊗ F ) mod Z, for some hermitian vector bundle F with unitary connection. Take note that even if E is trivial, the invariant can be non-zero; however, a choice of trivialization induces a canonical flat connection ∇ p.g. , namely the one with respect to which the global section trivializing the principal U-bundle is parallel (strictly speaking, this yields is a connection on the principal bundle, but since it canonically induces connections on any associated vector bundle, we do not bother to distinguish). In physics terminology, this is a so-called globally pure gauge connection, and it has vanishing holonomy along all closed paths in M. Lemma 2.1. Let E be a trivialized hermitian vector bundle over an odddimensional closed U-manifold M, and let ∇ p.g. be the unitary connection preserving the trivialization. Then we havẽ ξ (∇ p.g. ) ≡ 0 mod Z; this remains true if we twist the Spin C Dirac operator with an auxiliary hermitian vector bundle F with unitary connection. Proof. Since we do not need the full generality of [APS75b, Theorem 3.3], we may argue as follows: Clearly, we have −ξ(ð M ) ≡ ξ(ð −M ) mod Z, so we can interpret each summand of (17) separately in terms of index theory on manifolds X, X ′ with opposite boundary; in particular, we may represent (17) by X T d(∇ T X )ch(∇Ê) + X ′ T d(∇ T X ′ ) rkE. Furthermore, using the trivialization of E,Ê and the trivial bundle rkE patch together to form a hermitian bundle over the closed U-manifold X ∪X ′ ; by Theorem 1.1, the sum of the integrals yields an integer. For the twisted case, we notice that F also extends to X and X ′ (since Ω U odd (BU × BU) = 0) and that the multiplication of the integrands by ch(∇ F ) does not change the validity of our argument above. This result might seem a little bit dull, but it enables us to establish a surprising relation between trivialized vector bundles and divided congruences: Theorem 2.2. Let X be a compact manifold of dimension 2n, and let E 1 , E 2 , and F be hermitian vector bundles over X such that E 1 ⊕ E 2 ∼ = T X st , and that there is a given trivialization ψ : E 1 | ∂X ∼ = ∂X × C k . Equip E 1 with any unitary connection ∇ E 1 that restricts to the pure gauge connection on the boundary. Then, for arbitrary unitary connections ∇ E 2 and ∇ F , X Ell(∇ E 1 )Ell 0 (∇ E 2 )ch(∇ F ) ∈ D Γ n . Proof. By the multiplicativity of Ell 0 , the integrand may be rewritten as Ell 0 (∇ 1 ⊕ ∇ 2 )ch ∞ n=1 S q n E 1 ⊗ C ⊗ Λ −q n /ζ E 1 ⊗ Λ −ζq n E * 1 − 1 ch(∇ F ), where we dropped the symbol ∇ inside the curly brackets in favor of notational simplicity. Furthermore, Ell 0 itself is a twisted version of the Todd genus, so Ell 0 (∇ 1 ⊕ ∇ 2 ) = (1 − ζ) −rk(E 1 ⊕E 2 ) T d(∇ 1 ⊕ ∇ 2 )ch ∇ Λ −ζ (E 1 ⊕E 2 ) * , but T d(∇ 1 ⊕ ∇ 2 ) agrees with T d(∇ T X ) up to the differential of a Chern-Simons term; the latter does not contribute to the integral, since it gets multiplied by a relative characteristic form vanishing on ∂X. Then we observe that the formal vector bundle inside the curly brackets expands such that each summand contains at least one factor that is a virtually reduced bundle, so we may apply Theorem 1.3 and Lemma 2.1 to establish integrality. On the other hand, the integral takes values in inhomogeneous modular forms, due to the factor Ell(∇ 1 ); in fact, we may decompose the latter into the difference of Ell(∇ 1 )−1 and Ell 0 (∇ 1 )−1, which, due to the pure gauge condition, represent classes in k H 2k (X, ∂X; Q) ⊗ M Γ k and H even (X, ∂X; Q[ζ]), respectively. Example 2.3. We may consider the following special case: If X is a (U, f r)manifold of dimension 2n, we may choose E 1 ∼ = T X st , E 2 = 0 and F = 1. Then we see that the a priori rational modular form of weight n given by the elliptic genus of X actually admits an integral q-expansion once we remove its constant term: X Ell(∇ E 1 ) − X Ell 0 (∇ E 1 ) = X Ell(∇ E 1 ) ∈ D Γ n . Remark 2.4. The preceding example seems to be well-known; for instance, it may also be deduced from the results of [Lau99], albeit with considerably more effort, at least compared to the simple geometrical statement of Theorem 2.2. Construction off Having established an integrality result which may serve as a substitute for the Atiyah-Singer index theorem, we may now leave the realm of manifolds with smooth boundary and turn our attention to manifolds with corners of codimension two. In the following, we consider (U, f r) 2 -manifolds Z of (positive) even dimension. We also want additional geometrical structures: From now on, we endow all our bundles with hermitian metrics and unitary connections which are of product type near the respective faces; furthermore, we want these connections to preserve the respective trivializations on the faces, i.e. we require that they restrict to the pure gauge ones. Let us call these connections compatible. Definition 3. For a (U, f r) 2 -manifold Z of real dimension 2n + 2 and any compatible connections ∇ i on the E i , we seť F (Z, ∇ 1 , ∇ 2 ) = Z Ell(∇ 1 )Ell 0 (∇ 2 ) ∈ n+1 k=0 M Γ k ⊗ R. If we allow (U, f r) 2 -manifolds with empty corner, we obtain the following integrality results forF : Proposition 2.5. Let E 1 and E 2 be hermitian vector bundles over a closed manifold X of dimension 2n + 2 such that E 1 ⊕ E 2 ∼ = T X st (in the stable sense). For any unitary connections ∇ i we havě F (X, ∇ 1 , ∇ 2 ) ≡ 0 mod D Γ n+1 . Proof. Integrality is established by either making use of Theorem 1.1, or by applying Theorem 2.2 to the situation of ∂Z = ∅. Proposition 2.6. Let X be a compact manifold of dimension 2n + 2 and let E and F be hermitian vector bundles over X such that E ⊕ F ∼ = T X st (in the stable sense) and that there is a given trivialization ψ : F | ∂X ∼ = ∂X × C k . Regard X as a (U, f r) 2 -manifold with empty corner, and choose any unitary connection ∇ F that restricts to the pure gauge one on the boundary. Then we have: (i)F (X, ∇ F , ∇ E ) ≡ 0 mod D Γ n+1 . (ii)F (X, ∇ E , ∇ F ) ≡ 0 mod D Γ n+1 + M Γ 0 ⊗ R + M Γ n+1 ⊗ R. Proof. The first statement is clear by Theorem 2.2. For the second statement, we make use of the identity Ell(∇ 1 )Ell 0 (∇ 2 ) = Ell(∇ ⊕ ) − Ell(∇ 1 ) Ell(∇ 2 ) − Ell(∇ 2 )Ell 0 (∇ 1 ),(19) where ∇ ⊕ = ∇ 1 ⊕ ∇ 2 . Thus, up to modular terms of top and zero weight, i.e. up to Ell(∇ ⊕ ), we can expressF (X, ∇ E , ∇ F ) in terms to which Theorem 2.2 applies. Furthermore, we have a stability result concerning the splitting, namely: Proposition 2.7. Let (Z, E 1 , E 2 ) be a (U, f r) 2 -manifold of dimension 2n+2. Suppose that we have a different splitting of the tangent bundle of the same manifold, T Z ∼ = E ′ 1 ⊕ E ′ 2 , such that E 1 | ∂Z = E ′ 1 | ∂Z ⊕ F | ∂Z , where F is trivialized over all of ∂Z, and let ∇ 0 be any unitary connection on F that restricts to the pure gauge one on both faces. Then we havě F (Z, ∇ ′ 1 ⊕ ∇ 0 , ∇ 2 ) ≡F (Z, ∇ ′ 1 , ∇ 0 ⊕ ∇ 2 ) mod D Γ n+1 . Proof. From (19) we deduce Ell (∇ ′ 1 ⊕ ∇ 0 ) − Ell (∇ ′ 1 ) Ell 0 (∇ 0 ) = Ell (∇ ′ 1 ) Ell (∇ 0 ) ; then we multiply by Ell 0 (∇ 2 ), integrate, and apply Theorem 2.2 making use of the fact that, since the integrand vanishes near the corner, the integral over Z will yield the same result as the integral over a manifold Z s obtained from smoothing the corner. The preceding results suggest the following: Definition 4. Let M be a closed manifold of positive even dimension 2n, which is the corner of a (U, f r) 2 -manifold Z, and therefore inherits a splitting of its framing. Then, using compatible connections, we seť f(M, ∇ 1 | M , ∇ 2 | M ) ≡F (Z, ∇ 1 , ∇ 2 ) mod D Γ n+1 + M Γ 0 ⊗ R + M Γ n+1 ⊗ R. (20) We callf the geometrical f-invariant. In fact, this is well-defined. Proof. First of all, two (U, f r) 2 -manifolds Z 1 , Z 2 , having in common one face (and therefore having the same corner), in the sense that, say, ∂ 1 Z 1 ∼ = ∂ 1 Z 2 together with identifications of the respective E i thereon, give rise to congruentf -invariants, for we may glue −Z 1 and Z 2 along this face. By assumption, the metric and connections near the boundary are of product type, so everything fits together to yield a manifold Y with smooth boundary to which we may apply Proposition 2.6. Similarly, any other manifold Z 3 coinciding with Z 2 on the other face will have the samef as well. Finally, given Z 1 and Z 3 having in common the corner M (together with an identification of the trivialized vector bundles E i | thereon), there always exists a suitable (U, f r) 2 -manifold Z 2 : The faces ∂ 1 Z 3 and ∂ 2 Z 1 fit together along M to form a topological U-manifold N of odd dimension; furthermore the restrictions of the vector bundles (and the connections thereon) fit together to form vector bundlesẼ i . Since Ω U odd (BU) = 0, there exists a U-manifold P such that ∂P ∼ = N, and a complex vector bundle F 1 over P extendingẼ 1 . Specifying a vector bundle representative of the K-theory class [T P st ⊖ F 1 ], P is turned into the desired (U, f r) 2 -manifold Z 2 . Thus, we have succeeded in constructing a geometrical invariant of the corner of a (U, f r) 2 -manifold; furthermore,f bears a striking resemblance to the topological f -invariant, and, in fact, the former will serve as a tool for the index theoretical computation of the latter in the following section. Calculability It is a natural question to ask whether the (geometrical) f -invariant is computable using index theory. In order to establish a formula that is similar to (15), we have to address the following problems: (i) Analysis: A good starting point would be an index theorem on manifolds with corners of codimension two -alas, there are no theorems comparable to the generality of [APS75a]; however, we would like to mention [Mül96], where an index formula is proved under the assumption that the induced Dirac operators are invertible. The results of [HMM97] show that, without this assumption, it is still possible to obtain an 'index formula', but the latter holds only modulo the integers. (ii) Modularity: We want our formula to yield a result that is still recognizable as a combination of modular forms, but this property would inadvertently be spoiled by reducing modulo the index (which takes values in Z Γ [[q]]); furthermore, working one operator at a time, we obtain just a finite amount of coefficients of a q-expansion. Unfortunately, it is unclear under which conditions a finite amount of reduced coefficients can be lifted to an inhomogeneous combination of modular forms, and this task is complicated by the fact thatf is defined only up to real modular forms of top degree. (iii) Geometry: Lastly, we have to keep in mind that the definition of the f -invariant makes use of a (U, f r) 2 -manifold, the construction of which is also a non-trivial task. Our approach is to simplify matters by making some assumptions on the underlying geometry, i.e. we seek out (U, f r) 2 -manifolds that are sufficiently 'nice', in the sense that they allow the problems (i) and (ii) to be resolved. Corners via fiber bundles As a first step, we restrict our attention to manifolds of the following form: Definition 5. We define a 2 f -manifold to be a compact 2 -manifold that is a fiber bundle π : Z −→ B, where both the fiber X and the base B are even-dimensional compact 1manifolds, and the faces are given by Z ′ = ∂ 1 Z, X → Z ′ → ∂B, Z ′′ = ∂ 2 Z, ∂X → Z ′′ → B, which are fiber bundles themselves. Expecting such manifolds to be accessible to index theory considerations, we proceed along the lines of section 1.4, i.e. we introduce metrics g T (Z/B) and g T B (which are assumed to be of product type near the respective faces), make a choice of splitting T Z ∼ = T (Z/B) ⊕ π * T B, and construct the connection ∇ ⊕ = ∇ T (Z/B) ⊕ π * ∇ T B . As a model situation, we consider Z to be equipped with fixed Spin structures on the bundles; these induce natural orientations, and we may compute the integral of the genus form using integration over the fiber: finally, we apply Theorem 1.3 to the first summand and Stokes' theorem to the second and obtain: Z (∇ ⊕ ) = Z (π * ∇ T B )Â(∇ T (Z/B) ) = B  (∇ T B ) Z/B (∇ T (Z/B) ) .Z (∇ ⊕ ) = Ind(/ D ⊗ E) + ξ(/ D ∂B ⊗ E) + ∂B ωÂ(∇ T B ) + Bη (∇ T B ). Thus, we have an interpretation of the integral of the genus form in terms of an index and contributions from each of the two faces; the caveat is that this is just a formal result, in the sense that we had to introduce the form ω to store information we usually do not have access to. Rest assured however, that this information will not be needed for the computation of the f -invariant. Application to thef -invariant Let us extend the construction of the previous section to incorporate complex structures by making the following Definition 6. A (U, f r) 2 f -manifold consists of • a 2 f -manifold Z, • a vector bundle E over B that turns B into a (U, f r)-manifold, • a complex vector bundle E 2 stably isomorphic to T (Z/B) st • a trivialization of the restriction of E 2 to the face Z ′′ → B, • and an isomorphism E 2 ⊕ π * E ∼ = T Z st (in the stable sense). Again, we equip the complex bundles with hermitian metrics that restrict to the ones induced by the trivializations, and call unitary connections thereon compatible, if they restrict to the pure gauge ones. Then we have: Theorem 3.1. Let Z be a (U, f r) 2 f -manifold of dimension 2n + 2, and let ∇ 1 = π * ∇ E and ∇ 2 be compatible connections on E 1 = π * E and E 2 , respectively. Then thef-invariant of its corner M is given by f (M, π * ∇ E | M , ∇ 2 | M ) ≡ Bê Γ Ell(∇ E ), where we defined (a de Rham representative of ) the e Γ -invariant of a family: e Γ ≡ Z/B Ell 0 (∇ 2 ) mod im(ch : K(B) ⊗ Z Γ → H even (B, Q[ζ])) dR . Proof. First of all, we notice that T (Z/B) inherits a natural orientation from E 2 , so we may integrate over the fiber, Z Ell 0 (∇ 2 ) Ell π * ∇ E = B Ell ∇ E Z/B Ell 0 (∇ 2 ) . Then we observe that, by Theorem 2.2, B ch(∇ F ) Ell(∇ E ) yields a divided congruence for arbitrary hermitian vector bundles F with unitary connection; finally, for arbitrary ω ∈ Ω odd (B), we have Remark 3.2. The rationale behind the definition ofê Γ is to exhibit an analogy as close as possible to the e-invariant, thus paving the way for Corollary 3.3. However, the results of [BS07] 1 (which I was unaware of until recently) show that one can actually define an e-invariant of a family of framed manifolds in terms of smooth K-theory. Take note that E 2 also induces a canonical Spin C structure on the vertical tangent. Thus, given a metric and connection on T (Z/B) (which we assume to be of product type near the face Z ′′ ), we can construct a family of Dirac operators coupled to the formal vector bundle (1 − ζ) −rkE 2 Λ −ζ E * 2 ; let us denote this family by ð Γ . Corollary 3.3. Let Z be as in Theorem 3.1, and let ð Γ be constructed as above. If the kernel of ð Γ ∂X is of constant rank along the fibers, then thě f -invariant of the corner M is given by B η ð Γ ∂X + 1 2 ch ∇ ker ð Γ ∂X + Z ′′ /B cs Ell(∇ E ) . In particular, this expression is calculable from the face Z ′′ . Proof. We replace ∇ R 2 by ∇ T (Z/B) in the genus form underlying Ell 0 , hence giving rise to a Chern-Simons term of the form (3), but clearly, Z/B d cs ≡ Z ′′ /B cs mod im d : Ω odd (B) → Ω even (B) ; thus, taking into account the indeterminacy ofê Γ , the application of Theorem 1.4 yields the claim. Remark 3.4. We would like to point out that we can actually obtain a formula that does not require the kernel of ð Γ ∂X to form a vector bundle: In [MP97], Melrose and Piazza prove a general index theorem for families with boundary by making use of b-calculus techniques and introducing the notion of a spectral section P . The formal application of their theorem to our situation is straightforward, but since their result reduces to Theorem 1.4 under the aforementioned assumption (which will indeed be satisfied in the examples considered in this thesis), we are not going to elaborate on this. In fact, Corollary 3.3 gives an index theoretical formula for the topological f -invariant, for we have: Proof. The Chern-Weil forms constructed from the curvature of a compatible connection ∇ i represent characteristic classes of E i relative to ∂ i Z. Since ∇ 1 = π * ∇ E comes from the base, we have Z Ell 0 (∇ 2 ) Ell(∇ 1 ) = Z (Ell 0 (∇ 2 ) − 1) Ell(∇ 1 ); integrating over the fiber, we observe that the closed differential form on B given by Z/B (Ell 0 (∇ 2 ) − 1) has rational periods, i.e. it represents a class in H * (B, Q[ζ]). On the other hand, the differential form Ell(∇ E ) can be considered as a representative of a class in H even (B, ∂B; Q) ⊗ D Γ . Thus, the integral yields a rational combination of divided congruences, whereas the indeterminacy inê Γ manifests itself in true (i.e. integral) divided congruences (by Theorem 2.2). Finally, we observe that Z (Ell 0 (∇ 2 ) − 1) Ell(∇ 1 ) = Z (Ell 0 (∇ 2 ) − 1)(Ell(∇ 1 ) − 1) − Z (Ell 0 (∇ 2 ) − 1)(Ell 0 (∇ 1 ) − 1), and the same reasoning as above shows that the second expression takes values in Q[ζ], whereas the first expression can be identified with (16). Remark 3.6. We would like to stress that it is the fact that Ell(∇ 1 ) comes from the base that allows us to show that the integrals above take rational values instead of real ones. Summarizing, we see that the distinction betweenf and f becomes negligible in the context of (U, f r) 2 f -manifolds: The former comes with a natural lift to the latter, and this property is respected by our formulae (Theorem 3.1 and Corollary 3.3). A vanishing theorem It can be shown algebraically that the topological f -invariant vanishes on framed manifolds which are in third filtration, i.e. which lift to (U, f r) 3manifolds [Lau00]. Here we shall provide geometrical insight by considering the following situation: Definition 7. A (U, f r) 3 f -manifold is a 3 -manifold Y that is a fiber bundle over a (U, f r) 2 f -manifold B ′ where the typical fiber is a compact 1 -manifold; this time, we do not require B ′ to be of even dimension. In addition, there is a complex vector bundle E 3 ∼ = T (Y /B ′ ) st that is trivialized over the face ∂ 3 Y → B ′ , and there is an isomorphism E 3 ⊕ π * T Z st ∼ = T Y st . Proof. First of all, we know thatf depends only on the corner and its split framing; furthermore, by Proposition 2.7, all the possible splittings induced by the (U, f r) 3 f -structure yield congruent results. Thus, we may computě f (M) from Z = π −1 (∂ 1 B ′ ), which we endow with its obvious (U, f r) 2 fstructure. Successive integration along the fibers yieldš f (M, π * (∇ 1 ⊕ ∇ 2 )| M , ∇ 3 | M ) ≡ Z Ell 0 (∇ 3 ) Ell(π * (∇ 2 ⊕ ∇ 1 )) = ∂ 1 B ′ Ell(∇ 1 ⊕ ∇ 2 ) Ell 0 (∇ 3 ) = ∂ 1 B ′ Ell(∇ 2 ) Ell 0 (∇ 3 ) = ∂B Ell(∇ 2 ) Ell 0 (∇ 3 ) , where we made use of the fact that ∇ 1 is flat on Z. By assumption, the integrand, a closed form, extends over B, so the integral vanishes by Stokes' theorem. The f -invariant of cartesian products Now it is the time to illustrate the preceding ideas: Let X, B be (U, f r)manifolds of even dimension. Then the cartesian product Z = B×X becomes a (U, f r) 2 f -manifold in the obvious way; furthermore, since Z is a trivial fiber bundle,ê Γ is concentrated in degree zero, hence a constant function e Γ on B, so the formula of Theorem 3.1 simplifies tǒ f (∂B × ∂X) ≡ e Γ B Ell(∇ E ).(21) Of course, the kernel of the operator ð Γ ∂X forms a trivial vector bundle on B, so Corollary 3.3 applies; we can do better though: Lemma 4.1. Let X be a (U, f r)-manifold, and let e Γ (∂X) ≡ Ell Γ 0 (T X), [X, ∂X] mod Z Γ . Then we have e Γ (∂X) ≡ e C (∂X) mod Z Γ . Proof. We equip the hermitian vector bundle E ∼ = T X st with a unitary connection ∇ E preserving the trivialization of E| ∂X ; then the bundle Λ −ζ E * inherits a connection ∇ Λ −ζ E * preserving the induced trivialization, and by Lemma 2.1 we have X T d(∇ E )ch ∇ Λ −ζ E * ≡ (1 − ζ) rkE X T d(∇ E ) mod Z[ζ]. Stabilizing, i.e. multiplying by (1 − ζ) −rkE ∈ Z Γ , the claim follows. This enables us to establish the following remarkable result: Theorem 4.2. Let Y 1 , Y 2 be odd-dimensional framed manifolds, and let m(Y i ) be any modular form of weight (dim Y i + 1)/2 w.r.t. the fixed con- gruence subgroup Γ = Γ 1 (N) such thatm(Y i ) = m(Y i ) − e C (Y i ) ∈ Z Γ [[q]]. Then we havef (Y 1 × Y 2 ) ≡m(Y 1 )e C (Y 2 ) ≡ −m(Y 2 )e C (Y 1 ). In particular, thef -invariant of a product is antisymmetric under exchange of the factors. Proof. Combining (21) and Lemma 4.1, we know thať f(Y 1 × Y 2 ) ≡m ′ (Y 1 )e C (Y 2 ), where, by Example 2.3,m ′ (Y 1 ) is a divided congruence of the form m ′ −e Γ (Y 1 ) for a rational homogeneous modular form m ′ of weight (dim Y 1 + 1)/2. If we choose any divided congruencem(Y 2 ) = m(Y 2 ) − e C (Y 2 ) for a suitable homogeneous modular form m(Y 2 ) of level N and weight (dim Y 2 + 1)/2, we obtain:f (Y 1 × Y 2 ) ≡m ′ (Y 1 )e C (Y 2 ) ≡m ′ (Y 1 )m(Y 2 ) ≡ −e Γ (Y 1 )m(Y 2 ) ≡ −e Γ (Y 1 )m(Y 2 ) ≡ −e C (Y 1 )m(Y 2 ) ≡ −m(Y 1 )m(Y 2 ) ≡m(Y 1 )e C (Y 2 ) . Take note that Lemma 4.1 and Theorem 2.2 ensure the existence ofm(Y i ). This immediately implies: Sample calculations at level three Due to Theorem 4.2, it is quite simple to determine a representative of the finvariant of a product, but it still has to be checked whether f is non-trivial, which requires explicit calculations using divided congruences. Throughout this section, we fix Γ = Γ 1 (3), as N = 3 is the smallest level at which two is not inverted. The ring of modular forms for Γ 1 (3) is generated by E 1 = 1 + 6 n d|n ( d 3 )q n , E 3 = 1 − 9 n d|n d 2 ( d 3 )q n , which are of weight one and three, respectively; we refer to the appendix for more details. A word on notation: Although by Proposition 3.5 the distinction is unnecessary, we are still going to use the notationf in this section, and indicate the framed manifold it is computed from; if we write f , its argument will be given as the underlying element in the stable stems, which we denote by names prevalent in the literature, see e.g. [Rav04]. Proposition 4.5. Let Y and Y ′ be framed manifolds that represent a generator ν of π st 3 ∼ = Z/24. Then we havě f(Y × Y ′ ) ≡ 1 2 E 2 1 − 1 12 2 , and this is non-trivial in D Γ 4 ⊗ Q/Z. Proof. We consider the sphere S 3 as the sphere bundle S(L) of the Hopf line L over S 2 ; framing the base and the vertical tangent in a straightforward manner, we obtain a framing for the total space. By evaluating the (relative) Todd genus on the associated disk bundle, we compute (see also Remark 5.5) that e C (S(L)) is given by − 1 12 ; in view of Remark 1.6, the e R -invariant must be either − 1 24 or 11 24 , thus we conclude that S(L) represents ν. We apply Theorem 4.2, choose −m(ν) = E 2 1 − 1 12 = ∞ n=1   3∤d|n d   q n ∈ Z[[q]], and compute 1 12 E 2 1 − 1 12 ≡ − 1 2 E 2 1 − 1 12 2 ≡ 1 2 E 2 1 − 1 12 2 . To see that this is non-trivial in D Γ 4 ⊗ Q/Z, we have to compare the qexpansion of f (ν 2 ) ≡ 1 2 E 2 1 − 1 12 2 = 1 2 q 2 + 3q 3 + 11 2 q 4 + O(q 5 ) to those of a suitable basis of M Γ 4 ⊗ Q. A convenient choice consists of G * 4 and G 4 , since the q-expansions of the latter two agree on powers of q not divisible by three; but looking at G 4 − 1 240 = q + 9q 2 + O(q 3 ), we immediately deduce that f (ν 2 ) is non-trivial. Proposition 4.6. Let Y and Y ′ be framed manifolds that represent a generator σ of π st 7 ∼ = Z/240. Then we havě f(Y × Y ′ ) ≡ 1 2 E 4 − 1 240 2 , and this is non-trivial in D Γ 8 ⊗ Q/Z. Proof. Similar to the preceding case, the sphere S 7 , considered as the sphere of the quaternion line over S 4 , represents σ, see e.g. [CF66]. Then (at least up to sign conventions) e C (σ) = 1 240 , so we choosem(σ) = 1 240 (E 4 − 1) and 'complete the square'. Thus, f (σ 2 ) ≡ 1 2 E 4 − 1 240 2 = 1 2 q 2 + 9q 3 + 137 2 q 4 + 325q 5 + 1175q 6 + O(q 7 ), and it is straightforward to show that this is non-trivial by making use of the first, second, third, and sixth coefficients of G * 8 + 1093 240 = q + 129q 2 + q 3 + 16513q 4 + 78126q 5 + 129q 6 + O(q 7 ) E 8 − 1 480 = q + 129q 2 + 2188q 3 + 16513q 4 + 78126q 5 + 282252q 6 + O(q 7 ) E 8 1 − 1 48 = q + 21q 2 + 253q 3 + 1933q 4 + 9870q 5 + 35553q 6 + O(q 7 ). Recall from [Ada66] that there is an 8-periodic family in the stable stems generalizing η ∈ π st 1 . Although we are not going to give an explicit representative, we denote by µ k any framed manifold such that [µ k ] ∈ π st 8k+1 represents a member of this family (take note that our indexing differs from Adams'). Proposition 4.7. For any representatives µ of this family, we havě f (µ k × µ l ) ≡ 1 2 E 1 − 1 2 . Proof. According to [Ada66], we have e C (µ k ) = 1/2, and this is the only possible non-trivial value of e C in dimension 8k + 1. Thus, we judiciously choosef (µ k × µ l ) ≡ 1 2 E 4k+1 1 − 1 2 , but 1 2 E 4k+1 1 − 1 2 − 1 2 E 1 − 1 2 = E 4k 1 − 1 4 E 1 ∈ Z[[q]]. Remark 4.8. Take note that Proposition 4.7 just shows that the f -invariant of all products of the form µ k × µ l admits a universal representative; in low dimensions, non-triviality of this representative is readily verified by hand (cf. Example 4.9), but we do not know how to establish this in full generality. On the other hand, it is known that these products represent non-trivial elements in π st 8(k+l)+2 : As shown in [Ada66], e C (µ k ) = 1/2 is equivalent to d R (µ k ) = 0, and, by the properties of the degree, then also d R (µ k × µ l ) = 0. Example 4.9. It is easy to check that [µ 0 ] = η ∈ π st 1 ∼ = Z/2 can be represented by the circle with its non-bounding framing: Using a Fourier decomposition, we see that the Spin C Dirac operator has symmetric spectrum and a single zero mode, so we conclude e C (η) = 1 2 . We also see that f (η 2 ) ≡ 1 2 E 1 − 1 2 ≡ 1 2 n d|n ( d 3 )q n = 1 2 q + 1 2 q 3 + O(q 4 ) is non-trivial in D Γ 2 ⊗ Q/Z, since its q-expansion obviously cannot be congruent to a rational multiple of (E 2 1 − 1)/12 = q + 3q 2 + O(q 3 ). The images of e C in dimensions 8k + 3, 8k + 7 are known to be isomorphic to cyclic groups of order d 4k+2 , 2d 4k+4 , respectively, where d 2k denotes the denominator of B 2k /2k ([Ada66], see also [CF66]). The elements of π st 8k+3 , π st 8k+7 on which these values are attained lie in the image of the so-called J-homomorphism, J : π r SO → π st r , r ≥ 1. Proposition 4.10. Let M 8k+3 be a representative of the generator of Im(J) in dimension 8k + 3. Thenf (M 8k+3 × µ t ) ≡ 0. Proof. Without loss of generality we may assume that e C (M 8k+3 ) is represented by B 4k+2 /(4k + 2), so, for k > 0, we havě f(M 8k+3 × µ t ) ≡ 1 2 B 4k+2 (1 − E 4k+2 ) 4k + 2 = σ 4k+1 (n)q n , whereas for k = 0 we havě f (M 3 × µ t ) ≡ 1 2 E 2 1 − 1 12 ≡ 1 2 E 2 1 − 1 12 + E 4t 1 E 3 − 1 ≡ 1 2 E 2 1 − 1 12 + E 3 − 1 , which expands integrally by Proposition D.2. Proposition 4.11. Let M 8l−1 be a representative of the generator of Im(J) in dimension 8l − 1. Theň f (M 8l−1 × µ t ) ≡ 1 2 E 4 − 1 16 . Proof. The theorem of von Staudt allows the computation of the denominator of the Bernoulli numbers (see e.g. [Apo76]). More precisely, let d 2k denote the denominator of B 2k /2k; if 2 i |2k, then 2 i+1 |d 2k , and this result is sharp. In particular, this implies that for l = (2n + 1)2 m , 2 m+4 |2d 4l (and 2d 4l is precisely the order of Im(J) in dimension 8l − 1). Writing G 4l = B 4l 8l (1 − E 4l ) , B 4l 8l = n 4l 2d 4l = n 4l 2 m+4 u 4l , we have 1 2G 4l ≡ 1 2G 4l + u 4l − 1 2G 4l = 1 2 n 4l (1 − E 4l ) 2 m+4 . With the help of Lemma D.1 we compute that, modulo D 4(l+t)+1 , − n 4l 2 m+5 E 4l ≡ n 4l 2 m+5 E 4t+1 1 ≡ − n 4l 2 m+5 E l 4 ≡ 1 2 E l 4 − 1 2 m+4 , from which it follows thať f (M 8l−1 × µ t ) ≡ 1 2 E l 4 − 1 2 4+m . Now let l = (2n + 1)2 m , l ′ = (2n ′ + 1)2 m , n ≤ n ′ . Then 1 2 E l ′ 4 − E l 4 2 4+m = E l ′ −l 4 − 1 2 4+m+1 E l 4 ∈ Z[[q]], which means we can always reduce to the situation l = 2 m ; but we also have 1 2 (E 2 m 4 − 1) − 2 m (E 4 − 1) 2 4+m ∈ Z[[q]], thus proving the claim. Remark 4.12. This situation is akin to Proposition 4.7/Remark 4.8: From [Ada66] (see also [Rav04]), we know that ηx 2l , where x 2l denotes the generator of Im(J) in dimension 8l−1, is the generator of Im(J) in dimension 8l, hence non-trivial in π st 8l . Example 4.13. The lowest-dimensional example can be realized geometrically by considering the spheres S 7 and S 1 with framings that represent σ and η, respectively. In D Γ 5 ⊗ Q/Z, we may modify the q-expansion to read f (ησ) ≡ 1 2 E 4 − 1 240 ≡ 1 2 E 4 − 1 240 + 1 2 E Γ 5 − 1 3 = q − 3q 2 + 29 2 q 3 + 157q 4 + O(q 5 ), and now a quick comparison with the first three coefficients of E Γ 5 − 1 3 = q − 15q 2 + q 3 + 241q 4 + O(q 5 ), E 5 1 − 1 6 = 5q + 60q 2 + 365q 3 + 1205q 4 + O(q 5 ), shows that f (ησ) cannot be completed to an integral q-expansion. Proposition 4.14. Let Y 4k−1 represent a generator of Im(J) in dimension 4k − 1, k > 1, and let Y ′ represent ν. Then we havě f (Y 4k−1 × Y ′ ) ≡ 0. Proof. For k = 2l we may proceed as in the previous proof: f (Y 8l−1 × Y ′ ) ≡ 1 4G 4l ≡ 1 4G 4l ± u 4l ∓ 1 4G 4l = ± 1 4 n 4l (1 − E 4l ) 2 m+4 ≡ ∓ n 4l 2 m+6 E 2 1 ≡ ± n 4l 4 E l 4 − 1 2 m+4 ≡ ± n 4l 4 (E 4 − 1) · l 2 m+4 , where the sign is chosen according to whether u 4l ≡ ±1 mod 4, and the last step follows from (26) and (27). Making use of Proposition D.3, we see that in D Γ 4l+2 ⊗ Q/Z: E 4 − 1 2 6 ≡ E 2 1 − 1 2 5 − 1 4 (E 2 3 − 1) − 1 8 (E 3 1 E 3 − 1) ≡ E 2 1 − 1 2 5 − 1 4 (E l−1 4 E 2 3 − 1) − 1 8 (E l−1 4 E 3 1 E 3 − 1) ≡ E 2 1 − 1 2 5 ≡ − E l 4 − 1 2 5 ≡ −l E 4 − 1 2 5 ; if l is even, we are done, otherwise we have to iterate once. Now let k = 2l − 1 > 1. Then we havě f (Y 8l−5 × Y ′ ) ≡ 1 4 2G 4l−2 ≡ 1 2 n 4l−2 (1 − E 4l−2 ) 8 ≡ n 4l−2 4 E 2 1 − 1 4 , but the latter is seen to be congruent to a form of top weight, E 2 1 − 1 2 4 ≡ E 4 − 1 2 5 + 1 2 (E 2 3 − 1) + 1 4 (E 3 1 E 3 − 1) ≡ E 4 − 1 2 5 + 1 2 (E 4l−6 1 E 2 3 − 1) + 1 4 (E 4l−6+3 1 E 3 − 1) ≡ E 4 − 1 2 5 ≡ l E 4 − 1 2 5+m ≡ E l 4 − 1 2 5+m , thus vanishes in D Γ 4l ⊗ Q/Z. The f -invariant of principal circle bundles Let L be a hermitian line bundle with unitary connection ∇ L over a compact manifold B; restricting to the unit disk in the fibers, we get a bundle Z = D(L). Furthermore, the connection ∇ L induces a splitting of the tangent bundle. Clearly, the vertical tangent bundle is Proof. In terms of the iR-valued curvature two-form F ∇ 2 , we have T (Z/B) ∼ = π * L,T d (∇ 2 ) − Ell 0 (∇ 2 ) = − ζ 1 − ζ iF ∇ 2 2π ; upon integration along the fiber, the RHS takes values in Z Γ , and, when computing f , this integer will get multiplied by the divided congruence B Ell(∇ E ). In order to make contact with index theory, we endow the vertical tangent bundle with a metric g T (Z/B) such that it is of product type near the boundary and that the circle acts isometrically. Then we construct the connection ∇ T (Z/B) as in section 1.4; if we denote by e the vertical unit tangent of the circle bundle, we see that ∇ T (Z/B) e, e = 0, which implies that ∇ T (Z/B) e = 0, i.e. this connection is already trivializing (cf. [Zha94]). Now we can prove: Proposition 5.2. In the situation of the disk bundle D(L), the cohomology class of theê Γ -form is given by [ê Γ ] ≡ ∞ k=0 B k+1 (c 1 (L)) k (k + 1)! mod im(ch : K(B) ⊗ Z Γ → H even (B, Q[ζ])). Proof. By Lemma 5.1, it is sufficient to consider Z/B T d (∇ 2 ) = Z/B T d ∇ T (Z/B) + Z/B d cs T d, ∇ T (Z/B) , ∇ 2 ; analogously to Corollary 3.3, the last summand can be reduced to an integral over S(L)/B, at least up to exact forms on the base. Clearly, we have cs T d, ∇ T (Z/B) , ∇ 2 = cs(Â, ∇ T (Z/B) , ∇ R 2 ) exp iF ∇ 2 4π , and since theÂ-genus form comes from an even power series, formula (2) yields cs(Â) = ∞ k=0 c k 1 0 tr ωF 2k+1 t dt, for explicitly calculable coefficients c k ; but the so(2)-valued (i.e. abelian) curvature two-forms of ∇ T (Z/B) and ∇ R 2 vanish upon restriction to S(L), hence so do F t and cs. Now, by assumption, the circle acts isometrically on the fibers, which implies that the kernel of the boundary family ð ∂ is induced by the (trivial) S 1representation ker ð S 1 , hence it is of constant rank. Therefore, we may apply Theorem 1.4, and since ch(ker ð ∂ ) = 1, we have [ê Γ ] ≡ [η(ð)] + 1 2 mod im(ch : K(B) ⊗ Z Γ → H even (B, Q[ζ])). Luckily, theη-form of the Dirac operator for principal circle bundles has been computed in [Zha94] and [Goe00] (see also appendix E); its underlying cohomology class is [η(ð)] = ∞ k=1 B k+1 (c 1 (L)) k (k + 1)! ,(22) and the addition of 1 2 = B 1 (1) completes the proof. Corollary 5.3. Let L be a hermitian line with unitary connection ∇ L over a (U, f r)-manifold B of dimension 2n + 2. If we denote by S(L) | the framed circle bundle over ∂B, we have f (S(L) | ) ≡ n k=0 B k+1 (k + 1)! B iF L 2π k Ell(∇ E ) . Proof. The first Chern class of L may be represented by the normalized curvature two-form, so the result follows from Proposition 3.5, Theorem 3.1, Proposition 5.2, and the fact that powers k > n cannot contribute (take note that Ell is concentrated in positive degree). Remark 5.4. Modulo the integers, [ê Γ ] is an odd function of c 1 (L). Thus, if we replace L by L * in Corollary 5.3, we see that the f -invariant just changes sign. Remark 5.5. We may also use Proposition 5.2 to compute the e C -invariant of a principal circle bundle S(L) over a closed framed base B of dimension 2k: Take note that if E is a hermitian vector bundle with unitary connection, then the fact that B is framed implies that B ch(∇ E ) is an integer, the index of ð⊗E. Thus, modulo the integers, the evaluation of the relative Todd genus of the (U, f r)-manifold D(L) reduces to the evaluation of [η(ð)] + 1 2 on B, which implies that e C (S(L)) ≡ B k+1 k + 1 B ch ∇ L mod Z. In particular, this formula applies to the nilmanifolds of [DS84], cf. [Goe00]. Calculations for torus bundles As an application of Corollary 5.3, we consider the situation where the (U, f r)-manifold B is a disk bundle itself: Let L, L ′ be hermitian lines over a closed, framed manifold B ′ of positive even dimension; we choose a connection on L ′ and turn D(L ′ ) into a (U, f r)-manifold B. Pulling back the line L, we proceed similarly to obtain a (U, f r) 2 f -manifold D(π * L). In particular, the corner M is a principal torus bundle, which we refer to as the double transfer of B ′ (with respect to the lines L, L ′ ). Remark 5.6. The iterated transfers appear in Knapp's work investigating the Adams filtration of Lie groups [Kna78], treating a (compact) Lie group G as principal bundle over G/T , where T is a maximal torus. In particular, the tangent bundle of G/T is stably trivial, and there is a convenient description of the cohomology of G/T in terms of the roots [BH58]. Lemma 5.7. Let Γ = Γ 1 (3), let L and L ′ be hermitian lines over a closed framed manifold B ′ of dimension 2n > 0, and let M be the double transfer constructed above. Then, denoting the first Chern classes of L and L ′ by x and y, respectively, the f -invariant of M is given by f (M) ≡ n−1 k=1 B k+1 (k + 1)! x k Ell(y)/y, [B ′ ] . Proof. Clearly, the pair (D(L ′ ), S(L ′ )) may be identified with the Thom space of L ′ ; furthermore, the tangent of the (U, f r)-manifold B is stably isomorphic to π * L ′ , so the Thom isomorphism yields n k=0 B k+1 (k+1)! x k Ell(y)/y, [B ′ ] = n k=0 B k+1 (k+1)! (π * x) k Ell(T B), [D(L ′ ), S(L ′ )] , and the RHS is precisely the formula of Corollary 5.3. If we interpret the summand for k = 0 as 1 2 Ell(y)/y, [B ′ ] = e C (S 1 ) Ell(y)/y, [B ′ ] ≡f(S(L ′ )) × S 1 ), the latter is easily seen to be congruent to zero by Propositions 4.10, 4.11, and the fact that e C (S(L ′ )) is represented by an integer multiple of B n+1 /(n + 1), cf. Remark 5.5. Similarly, the contribution proportional to x n , [B ′ ] may be identified withf (S 1 × S(L)) ≡ 0. Remark 5.8. It should be noted that the explicit choice of framing for B ′ plays only a minor rôle in the formula above; in particular, orientationpreserving reframings of B ′ lead to a double transfer having the same finvariant. For the remainder of this section, we are going to fix Γ = Γ 1 (3) and use the notations of Lemma 5.7; we are going to use the latter to compute the f -invariant of the general double transfer in a given dimension, thus enabling us to determine the precise conditions for non-triviality. Clearly, the double transfer on a two-dimensional base will have vanishing f -invariant; increasing dimensions, things become more interesting, starting with: Proof. By Lemma 5.7, we just have to consider the coefficient of xy 2 ; by (24) and (25), it is given by 1 12 iE 3 1 − iE 3 18 √ 3 ≡ − i √ 3 3 1 2 E 2 1 − 1 12 1 − E 3 9 + E 2 1 − 1 12 E 3 1 − 1 18 ≡ 1 2 E 2 1 − 1 12 E 3 − 1 9 ≡ 1 2 E 2 1 − 1 12 2 = 1 2 q 2 + 3q 3 + 11 2 q 4 + O(q 5 ), which is seen to be non-trivial by checking the first and fourth coefficients of 5 E Γ 5 − 1 3 = 5q − 75q 2 + 5q 3 + 1205q 4 + O(q 5 ), E 5 1 − 1 6 = 5q + 60q 2 + 365q 3 + 1205q 4 + O(q 5 ); obviously, this argument still holds true if we add f (ησ) ≡ 1 2 E 4 − 1 240 + 1 2 E Γ 5 − 1 3 = q − 3q 2 + 29 2 q 3 + 157q 4 + O(q 5 ). Example 5.12. We borrow an example from [Lau00], choosing B ′ = SU(3)/T and x and y to be the simple roots. A straightforward computation yields xy 2 , [B ′ ] = 3, hence establishing non-triviality. Proposition 5.13. Let B ′ be a closed framed manifold of dimension eight. Then the double transfer w.r.t. any lines L, L ′ has trivial f -invariant in D Γ 6 ⊗ Q/Z. Proof. By Lemma 5.7, we know that f is represented by the evaluation of 1 12 13(E 4 1 − 1) − 16(E 1 E 3 − 1) 2160 y 3 x − 1 720 E 2 1 − 1 12 yx 3 ,but 1 12 13(E 4 1 − 1) − 16(E 1 E 3 − 1) 2160 = 1 12 5(E 4 1 − 1) − 2(E 4 − 1) 2160 ≡ 1 8 E 4 1 − 1 8 − 1 2 E 4 − 1 16 ≡ 1 8 E 4 1 − 1 8 + 9 E 2 1 − 1 12 3 − 1 2 E 4 − 1 16 ≡ E 2 1 − 1 2 6 − 1 2 E 4 − 1 16 ≡ − E 4 − 1 2 6 − 1 2 E 4 − 1 16 = − 3 4 E 4 − 1 16 and − 1 720 E 2 1 − 1 12 ≡ 1 36 E 4 − 1 240 ≡ 1 4 E 4 − 1 16 may be identified with (multiples of) f (νσ), but the latter is trivial by Proposition 4.14. Remark 5.14. Revisiting the above at level two, an admittedly tedious calculation along the lines of this section shows that the corresponding finvariant will be non-trivial if and only if 3 ∤ x 3 y, [B ′ ] , and Spin(5)/T fits the bill; since this result also follows from [Lau00], we do not bother with details. Proposition 5.15. Let B ′ be a closed framed manifold of dimension ten. Then the double transfer w.r.t. any lines L, L ′ has trivial f -invariant in D Γ 7 ⊗ Q/Z. Proof. By Lemma 5.7, it is sufficient to evaluate i √ 3 xy 4 1 12 E 2 1 (E 3 1 − E 3 ) 216 − x 3 y 2 1 720 E 3 1 − E 3 18 on B ′ . In D Γ 1 (3) 7 ⊗ Q/Z, we may rewrite 1 12 E 2 1 (E 3 1 − E 3 ) 216 ≡ − 1 4 · 9 E 2 1 − 1 4 2 + E 4 − 1 16 E 3 1 − E 3 9 , 1 720 E 3 1 − E 3 18 ≡ − 1 2 · 3 E 4 − 1 240 E 3 1 − E 3 9 . By coupling the Dirac operator on B ′ to L ′ ⊗ L ⊕ L ′ ⊗ L * ⊖ 2L ′ , we obtain the divisibility result 5!| (20x 3 y 2 + 10xy 4 ), [B ′ ] , i.e. 4| (2x 3 y 2 + xy 4 ), [B ′ ] ; interchanging L and L ′ , we get 4| (2y 3 x 2 +yx 4 ), [B ′ ] . But a short calculation with Steenrod squares (see e.g. [Ste62], [MS74]) shows that x 3 y 2 is already even: Recall that, on a framed manifold, Sq k vanishes on classes of codimension k; furthermore, Sq 1 (i.e. the Bockstein) vanishes on integral classes, so we compute 0 ≡ Sq 2 (x 3 y) ≡ Sq 2 (x 3 )y + x 3 Sq 2 (y) ≡ x 4 y + x 3 y 2 ≡ x 3 y 2 mod 2. Thus, 4| xy 4 , [B ′ ] , and consequently, f admits an integral q-expansion. Proof. Let L, L ′ be hermitian lines such that their first Chern classes are x, y, respectively; we endow the lines with unitary connections and compute the index of several twisted Dirac operators to obtain the following divisibility results: 6!| (x + y) 6 + (x − y) 6 − 2x 6 − 2y 6 , [B ′ ] ⇒ 8| x 4 y 2 + x 2 y 4 , [B ′ ] , 6!| (x + 2y) 6 + (2x + y) 6 − x 6 − y 6 , [B ′ ] ⇒ 16| 12(x 5 y + xy 5 ) + 60(x 4 y 2 + x 2 y 4 ), [B ′ ] . Thus, we also have 8| 6(x 5 y + xy 5 ), [B ′ ] , and from 6!| 6(x 5 y + xy 5 ) + 20x 3 y 3 + 15(x 4 y 2 + x 2 y 4 ), [B ′ ] we may now deduce 2| x 3 y 3 , [B ′ ] . Finally, making use of Steenrod squares, we have 0 ≡ Sq 4 (x 3 y) ≡ x 5 y + x 4 y 2 ≡ x 5 y + Sq 6 (x 2 y) ≡ x 5 y mod 2, so x 5 y and, analogously, xy 5 are even. 1 2 E 2 1 − 1 12 3 ∈ D Γ 8 ⊗ Q/Z; furthermore, this differs from the situation of Proposition 4.6. Note added for v2: The attempt to streamline the proof of Proposition 5.17 introduced an error in the previous version (v1); the present version (v2) remedies this by reverting to the original, correct form of Lemma D.4 and the subsequent calculations used in this proof. Proof. We apply Lemma 5.7, and by Lemma 5.16, we gain an extra factor of two; in particular, the coefficient of 1 2 x 3 y 3 , [B ′ ] reads 2 · 1 720 16(E 1 E 3 − 1) − 13(E 4 1 − 1) 2160 = − 2 3 3 · 240 2(E 4 − 1) − 5(E 4 1 − 1) 240 ≡ 2 3 3 E 4 − 1 240 2 + 1 3 4 · 240 E 4 1 − 1 8 ≡ − 1 3 4 · 8 E 4 − 1 240 ≡ 5 3 3 E 4 − 1 240 2 ; thus, we are left with the contributions 2 · 121(E 6 1 − 1) − 152(E 3 1 E 3 − 1) + 40(E 2 3 − 1) 2 7 · 3 6 · 5 · 7 , 2 · 1 2 5 · 3 3 · 5 · 7 E 2 1 − 1 12 , corresponding to 1 2 x 5 y, [B ′ ] and 1 2 xy 5 , [B ′ ] , respectively. Furthermore, the divisibility results of Lemma 5.16 imply that we may write x 5 y, [B ′ ] = 60k − xy 5 , [B ′ ] − 20 3 1 2 x 3 y 3 , [B ′ ] ; combined with 20 E 2 1 − 1 2 7 · 5 · 7 ≡ − 1 4 E 6 − 1 7 · 8 ≡ − E 2 1 − 1 2 5 ≡ 0, where triviality follows from Lemma D.4, we end up with an expression for f determined by xy 5 , [B ′ ] alone. We now set xy 5 , [B ′ ] = 2 (or, more generally, an odd multiple thereof), write −2 = 18 − 20, and simplify to arrive at f ≡ 121E 6 1 − 152E 3 1 E 3 + 40E 2 3 2 7 · 3 6 · 5 · 7 · 2 + E 2 1 2 7 · 3 4 · 5 · 7 · 18, which is congruent to the desired result by Proposition D.5. Non-triviality is established by comparing the first, second and fourth coefficient of the q-expansion of 1 2 E 2 1 − 1 12 3 = 1 2 q 3 + 9 2 q 4 + O(q 5 ) to those of Example 5.18. We choose B ′ to be the framed manifold G 2 /T . Let us recall some facts about its cohomology ring [BS55]: Rationally, it is generated by classes α, β ∈ H 2 (G 2 /T, Z) subject to the relations α 2 + 3β 2 + 3αβ = 0, α 6 = 0 = β 6 ; furthermore, we have αβ 5 , [G 2 /T ] = 2, which is precisely what we need, i.e. we take x = α, y = β. Remark 5.19. It is of course possible to continue the program initiated above, i.e. to calculate the f -invariant of the generic double transfer systematically; however, as the computations become increasingly more involved, one should look for results complementing our approach. In fact, such results exist (at least partially): It is possible to compute the f -invariant algebraically, never leaving the context of the ANSS, and this has been done for several beta-elements in [HN07]. In particular, a straightforward comparison of their results to Proposition 5.17 (making use of Lemma D.4) shows that the example above represents β 3 at the prime two. (Of course, this does not come as a surprise, since, by our Proposition 5.17, it cannot be σ 2 = β 4/4 .) A Useful formulae . . . from analytic number theory (See e.g. [Apo76]) The Hurwitz zeta function is defined by analytic continuation of the series ζ(s, x) = ∞ n=0 (n + x) −s , x > 0 and s > 1. Take note that ζ(s, 1) = ζ(s) is the usual Riemann zeta function. It satisfies the functional equation ζ 1 − s, m n = 2Γ(s) (2πn) s n k=1 cos πs 2 − 2πkm n ζ s, k n for integers 1 ≤ m ≤ n. We also have the relation ζ(−n, x) = − B n+1 (x) n + 1 , where the Bernoulli polynomials are given by If no argument is indicated, it is understood to be one, i.e. B n = B n (1). te xt e t − 1 = ∞ k=0 B k (x) t k k! ; . . . involving modular forms Let Γ ⊂ SL(2, Z) be a subgroup of finite index, e.g. the following congruence subgroups of level N > 1 [HBJ92], [Sch74]: (i) f is holomorphic on h, Γ 0 (N) = γ ∈ SL(2, Z) | γ ≡ * * 0 * mod N , Γ 1 (N) = γ ∈ SL(2, Z) | γ ≡ 1 * 0 1 mod N ;(ii) for all γ = a b c d ∈ Γ, we have: (cτ + d) k f (τ ) = f aτ +b cτ +d , (iii) and for every S = a b c d ∈ SL(2, Z), (cτ + d) −k f aτ +b cτ +d admits a Fourier expansion of the form n≥0 a n q n/N . The vector space of modular forms of a given weight k is finite-dimensional; the valence formula implies the following upper bound: dim M k (Γ) ≤ 1 + k 12 [P SL(2, Z) : P Γ].(23) Take note that if f (τ ) is a modular form w.r.t. SL(2, Z), then g(τ ) = f (Nτ ) is a modular form w.r.t. Γ 0 (N): Proof. Let N|b and a b c d ∈ SL(2, Z). Then we use 1/N 0 0 1 a b c d N 0 0 1 = a b/N Nc d to obtain any element in Γ 0 (N). Assuming f to be of weight k, we have (cNτ + d) k f (Nτ ) = f aNτ + b cNτ + d = f N aτ + b/N Ncτ + d . B Eisenstein series Level 1 Taking the logarithmic derivative of the product formula of the sine, we obtain π cot πz = iπ(1 − 2 ∞ k=0 e 2πikz ) = 1 z + ∞ n=1 1 z + n + 1 z − n , assuming that z ∈ h. Successive differentiation w.r.t. z yields (for r > 1) 1 z r + ∞ n=1 1 (z + n) r + 1 (z − n) r = (−2πi) r (r − 1)! ∞ k=1 k r−1 e 2πikz . We may now define the Eisenstein series: Let k > 2 be an even integer; then E k (τ ) = ζ(k) −1 m,n ′ (mτ + n) −k = 1 + 2 (−2πi) k ζ(k)(k − 1)! ∞ m=1 ∞ d=1 d k−1 q dm , where the prime denotes the omission of the term m = n = 0. This in turn implies E k (τ ) = 1 − 2k B k ∞ n=1 σ k−1 (n)q n , where σ k (n) = d|n d k . It is straightforward to see that E k (τ + 1) = E k (τ ) and τ k E k (τ ) = E k (−1/τ ). For k = 2, we define E 2 = 1 − 24 ∞ n=1 σ 1 (n)q n , which is holomorphic, but not modular anymore; instead, we have τ −2 E 2 (−1/τ ) = E 2 + 6 iπτ , which means thatÊ 2 = E 2 − 3 πℑ(τ ) behaves well w.r.t. Γ = SL(2, Z). Sometimes it will be convenient to use another normalization of the Eisenstein series, namely G k (τ ) = − B k 2k E k (τ ). Level N Fixing a level N > 1, we may choose to sum over a sublattice (m 1 , m 2 ) ≡ (a 1 , a 2 ) mod N, so, for any integer k ≥ 3, we define G (a 1 ,a 2 ) k (τ ) = m≡a (N ) (m 1 τ + m 2 ) −k . Obviously, we have (cτ + d) −k G (a 1 ,a 2 ) k aτ + b cτ + d = m≡a (N ) ((m 1 a + m 2 c)τ + (m 1 b + m 2 d)) −k , and we may change the summation to run over the lattice m ′ = mγ ≡ aγ; take note that, modulo N, Γ 1 (N) preserves a = (0, a 2 ). In order to obtain a q-expansion, we recast the expression for G by splitting the sums Thus, the normalized odd Eisenstein series for Γ 1 (3) is given by E Γ 1 (3) 2n+1 (τ ) = 1 − 2n + 1 3 2n B 2n+1 (1/3) ∞ l=1   d|l d 2n ( d 3 )   q l . Similarly, for even k = 2n + 2 we compute b k c k = (−1) n+1 (2n + 1)! (2π) 2n+2 (ζ(2n + 2, 1/3) + ζ(2n + 2, 2/3)) = ζ(−(2n + 1), 1) − 3 2n+2 ζ(−(2n + 1), 1/3) (σ 1 (n) − 3σ 1 (n/3))q n ; this is proportional to E 2 (τ ) − 3E 2 (3τ ) =Ê 2 (τ ) − 3Ê 2 (3τ ), hence even modular for Γ 0 (3). It is also possible to define a first Eisenstein series by introducing a regularization scheme that preserves modularity (cf. e.g. [Sch74]); we still focus on the level N = 3 and define 3 . Putting everything together, we see that the regularized (and normalized) Eisenstein series has the same q-expansion that would have been expected from the naïve formula: E Γ 1 (3) 1 = 1 + 6 ∞ n=1   d|n ( d 3 )   q n . Finally, from (23), we know that dim M k (Γ 1 (3)) ≤ 1 + k 3 ; thus, it is easy to see that the ring of modular forms w.r.t. Γ 1 (3) is generated by the first and third Eisenstein series; as confusion is unlikely, we denote them by E 1 and E 3 , respectively. C Expanding the Hirzebruch genus Following [HBJ92], we introduce the Φ-function, Φ(τ, x) = (ξ 1/2 − ξ −1/2 ) ∞ n=1 (1 − ξq n )(1 − ξ −1 q n ) (1 − q n ) 2 = x exp − ∞ k=1 2 (2k)! G 2k (τ )x 2k , where ξ = exp x and q = exp(2πiτ ); for We set e ω = ζ = 1, and, at level three, ζ 3 = 1. Then, e x + ζ e x − ζ = (e 3x − 1) −1 (ζ − ζ 2 )(e 2x − e x ) + e 3x − e 2x − e x + 1 = (ζ − ζ 2 ) ∞ k=−1 (B k+1 (2/3) − B k+1 (1/3)) (3x) k (k + 1)! + ∞ k=−1 (B k+1 (1) + B k+1 (0) − B k+1 (2/3) − B k+1 (1/3)) (3x) k (k + 1)! = −2(ζ − ζ 2 ) ∞ k=0 3 2k B 2k+1 (1/3) x 2k (2k + 1)! + ∞ n=0 (3 2n+2 − 1)B 2n+2 x 2n+1 (2n + 2)! . The last step follows from the fact that the minus first summands cancel, and we used Thus, we may express the elliptic genus of level three as x Φ(τ, x − ω) Φ(τ, x)Φ(τ, −ω) = exp 3 ∞ n=1 x 2n (2n)! G * 2n (τ ) − 2 ∞ k=0 x 2k+1 (2k + 1)! G (−ω) 2k+1 (τ ) , where G * 2n (τ ) = G 2n (τ ) − 3 2n−1 G 2n (3τ ), G (−ω) 2k+1 (τ ) = e ω − e −ω 2 3 2k B 2k+1 (1/3) 2k + 1 E Γ 1 (3) 2k+1 (τ ), hence modularity is manifest. Choosing ω = 2πi/3, the first few terms of the genus, when expressed in terms of E 1 and E 3 , read Ell Γ 1 (3) (x) = 1 + iE 1 2 √ 3 x + E 2 1 12 x 2 + iE 3 1 − iE 3 18 √ 3 x 3 + 13E 4 1 − 16E 1 E 3 2160 x 4 + iE 2 1 (E 3 1 − E 3 ) 216 √ 3 x 5 + 121E 6 1 − 152E 3 1 E 3 + 40E 2 3 272160 x 6 + O(x 7 ).(24) We also list the first few terms of the expansion of the Todd genus: x 1 − e −x = 1 + 1 2 x + 1 12 x 2 − 1 720 x 4 + 1 30240 x 6 + O(x 8 ). (25) D Useful congruences The ring of modular forms w.r.t. Γ = Γ 1 (3) is generated by E 1 = 1 + 6 ∞ n=1   d|n ( d 3 )   q n , E 3 = 1 − 9 ∞ n=1   d|n ( d 3 )d 2   q n , so it is straightforward to check that E 4 = 1 + 240 ∞ n=1   d|n d 3   q n = 9E 4 1 − 8E 1 E 3 , E 6 = 1 − 504 ∞ n=1   d|n d 5   q n = −27E 6 1 + 36E 3 1 E 3 − 8E 2 3 , E 8 = 1 + 480 ∞ n=1   d|n d 7   q n = E 2 4 , Since exp(aT ) is to be understood as formal power series, we may integrate for k = 0 k ∞ 0 1 t 2 ∞ l=0 (πkT /(2t)) l l! exp − k 2 π 2 t dt = k ∞ 0 ∞ l=0 (πkT x/2) l l! exp −k 2 π 2 x dx = 1 kπ 2 ∞ l=0 T 2πk l = 1 π 2 1 k − T 2π . Thus, putting everything together, we have 2η = i π k =0 1 T 2π − k = i π π cot T 2 − 2π T = coth T 2i − 2i T . Obviously, we may identify the basic two-form with minus i times the curvature two-form of the hermitian line, i.e. T = −iF ; therefore, the normalized η-form is given byη (ð) = ∞ k=1 B k+1 (k + 1)! iF 2π k . structures . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 k -manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Connections, curvature, and Chern forms . . . . . . . . . . . . 8 1.4 Some classical index theory . . . . . . . . . . . . . . . . . . . 10 1.5 Hirzebruch elliptic genera . . . . . . . . . . . . . . . . . . . . 15 1.6 The e-invariant . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.7 The topological f -invariant . . . . . . . . . . . . . . . . . . . . 19 2 The geometricalf -invariant 20 2.1 Divided congruences from trivialized vector bundles . . . . . . 20 2.2 Construction off . . . . . . . . . . . . . . . . . . . . . . . . . 22 3 Calculability 25 3.1 Corners via fiber bundles . . . . . . . . . . . . . . . . . . . . . 26 3.2 Application to thef -invariant . . . . . . . . . . . . . . . . . . 27 3.3 A vanishing theorem . . . . . . . . . . . . . . . . . . . . . . . 30 4 The f -invariant of cartesian products 31 4.1 Sample calculations at level three . . . . . . . . . . . . . . . . 33 5 The f -invariant of principal circle bundles 39 5.1 Calculations for torus bundles . . . . . . . . . . . . . . . . . Ell 0 (∇ 2 ) mod im(ch : K Γ (B) → H even (B, Q[ζ])) dR . For simplicity, we assume the Dirac operator of the boundary family to be invertible; thus, the application of Theorem 1.4 yields Z/B (∇ T (Z/B) ) = (ch(Ind)) dR +η, where the subscript indicates that we are dealing with a de Rham representative of the Chern character of the index bundle. Let us choose a virtual vector bundle E with unitary connection ∇ E , such that [E] = [E 1 ⊖ E 2 ] = [Ind]; then we may write (ch(Ind)) dR = ch(∇ E ) + dω, for some ω ∈ Ω odd (B), i.e. Z (∇ ⊕ ) = B  (∇ T B ) ch(∇ E ) + dω +η ; ' theorem and the flatness of ∇ E restricted to the boundary. Proposition 3 . 5 . 35Let Z be as in Theorem 3.1. Then we haveBê Γ Ell(∇ E ) ∈ D Γ n+1 ⊗ Q/Z,and this gives a representative of the topological f -invariant of M. Theorem 3 . 7 . 37Let M be the codimension-three corner of a (U, f r) 3 f -manifold Y . Thenf (M) ≡ 0. Corollary 4. 3 . 3Let M be the cartesian product of two odd-dimensional framed factors, one of which has vanishing e C -invariant. Thenf (M) ≡ 0. Remark 4. 4 . 4While this result is similar to Theorem 3.7, it does not require any geometrical assumptions concerning (U, f r) 3 -structures. Remark 4 . 15 . 415We would like to point out that Propositions 4.5 through 4.14 may be thought of as an elliptic analogue of [Rav04, Theorem 5.5.8. (b)]. therefore it inherits a natural complex structure. Restricting to the sphere bundle S(L), we have(π * L)| S(L) ∼ = T (Z/B)| S(L) ∼ = T (S(L)/B) ⊕ S(L) × R,and we can trivialize the vertical tangent of the principal circle bundle S(L), hence also (π * L)| S(L) . Thus, if L is a hermitian line with connection over an even-dimensional (U, f r)-manifold B, then D(L) can be turned into a (U, f r) 2 f -manifold. Let us compute the f -invariant in this situation: Given any compatible connections we may invoke Theorem 3.1; further simplification can be achieved by a result similar to Lemma 4.1:Lemma 5.1. In the situation of a disk bundle Z = D(L), we may replace Ell 0 (∇ 2 ) by T d(∇ 2 ) for the computation of theê Γ -form in Theorem 3.1. Proposition 5 . 9 . 59Let B ′ be a closed framed manifold of dimension four. The double transfer w.r.t. the lines L, L ′ has non-trivial f -invariant if and only if xy, [B ′ ] is odd, in which case it is , [B ′ ] ; thus, we can proceed as in Proposition 4.5.Example 5.10. An obvious choice is to take B ′ to be S 2 × S 2 , which we may think of as Spin(4)/T . Taking x and y to be (minus) the generators of the cohomology of the respective factors, we essentially recover the situation of Proposition 4.5.Proposition 5.11. Let B ′ be a closed framed manifold of dimension six. The double transfer w.r.t. the lines L, L ′ has non-trivial f -invariant if and only if xy 2 , [B ′ ] is odd, in which case it is given by this differs from the situation of Example 4.13. Lemma 5 . 16 . 516Let B ′ be a closed, framed manifold of dimension twelve and let x, y ∈ H 2 (B ′ , Z). Then x 3 y 3 , [B ′ ] and xy 5 , [B ′ ] are even. k ≥ 0 be an integer. Recall that a function f : h → C is called a modular form of weight k w.r.t. Γ if: 2πid(m 1 τ −a 2 )/N , where c k = (−2πi/N) k /((k − 1)!) and b k = n≡a 2 n −k .Level 3 Let us focus on N = 3 and a ≡ (0, 1). For k = 2n + 1, we have b 2n+1 = 3 −k ζ(2n + 1, 1/3) − 3 −k ζ(2n + 1 0 0(e 2πid/3 + e −2πid/3 )q dl is a modular form w.r.t. Γ = Γ 1 (3). In order to treat the situation n = 1 τ + m 2 ) −1 |m 1 τ + m 2 | −s , which converges for s sufficiently large, so we rearrange the sums asG ≡a 2 m −1 2 |m 2 | −s if a 1 ≡ 0 mod 3 0 otherwise .Next, we observe that for fixed zΨ(u, s) = k∈Z (z + k + u)[(z + k + u)(z + k + u)] k + u) −1 |z + k + u| −s e u) −1 |z + u| −s e −2πimu du , so we have c m m 1 τ + a 2 3 , 0 = −2πisgn(m)e 2πim(m 1 τ +a 2 )/3 for mm 1 > 0 0 for mm 1 u) −1 |τ + u| −s du,which is holomorphic at s = 0, and, in particular, does not contribute for a 1 ≡ 0. Finally, we evaluate b by consideringlim s→1 [ζ(s, x) − 1/(s − 1)] = −Γ ′ (x)/Γ(x) = −ψ(x),where the digamma function ψ satisfiesψ(1 − x) − ψ(x) = π cot( B n ( 1 ) 1= 3 n−1 (B n (1/3) + B n (2/3) + B n (1)), B n (1 − x) = (−1) n B n ( 2 The geometricalf -invariant 2.1 Divided congruences from trivialized vector bundles Recall from [APS75b] that the spectral information encoded in ξ can be used to formulate an invariant for flat vector bundles: Let M be a closed U-manifold of odd dimension and let E be a hermitian vector bundle with flat unitary connection ∇ E ; then the expressioñ Proposition 5.17. Let B ′ be a closed framed manifold of dimension twelve. The double transfer w.r.t. lines L, L ′ has non-trivial f -invariant if and only if 1 2 xy 5 , [B ′ ] is odd, in which case it is given by Thanks to U. Bunke for pointing out this reference AcknowledgementsFirst of all, I would like to thank my advisor, Professor Gerd Laures, for introducing me to the fascinating subject of elliptic cohomology and its diverse applications, and for proposing the topic of and supervising the work on this thesis. At the same time, I want to thank Professor Uwe Abresch,Integrating and exponentiatingTherefore we have:Deriving the c k at level threeBy definition,E Derivation of theη-formWe use the notation from the proof of Proposition 5.2 and follow[Zha94]. The superconnection associated to the family of Spin C Dirac operators on the circle bundle S(L) is given byIntroducing a Grassmann variable z, we may write down a Weitzenböck-type formulawhere we identified T on the RHS with a basic two-form (by contraction with the dual of the vertical unit tangent e). Now we have to computewhere Tr z denotes the trace restricted to the coefficient of z. 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Differential Geom. 441Werner Müller, On the L 2 -index of Dirac operators on mani- folds with corners of codimension two. I, J. Differential Geom. 44 (1996), no. 1, 97-177. Complex cobordism and stable homotopy groups of spheres. C Douglas, Ravenel, AMS Chelsea PublishingProvidence, RI2nd ed.Douglas C. Ravenel, Complex cobordism and stable homotopy groups of spheres. 2nd ed., Providence, RI: AMS Chelsea Pub- lishing., 2004. Die Grundlehren der mathematischen Wissenschaften. B Schoeneberg, J. R. Smart and E. A. SchwandtSpringer-Verlag203New YorkElliptic modular functions: an introductionB. Schoeneberg, Elliptic modular functions: an introduction, Springer-Verlag, New York, 1974, Translated from the German by J. R. Smart and E. A. Schwandt, Die Grundlehren der mathe- matischen Wissenschaften, Band 203. N E Steenrod, Cohomology operations, Lectures by N. E. Steenrod written and revised by D. B. A. Epstein. Annals of Mathematics Studies. Princeton, N.J.Princeton University PressN. E. Steenrod, Cohomology operations, Lectures by N. E. Steen- rod written and revised by D. B. A. Epstein. Annals of Mathemat- ics Studies, No. 50, Princeton University Press, Princeton, N.J., 1962. Robert E Stong, Notes on cobordism theory, Mathematical notes. Princeton, N.J.Princeton University PressRobert E. Stong, Notes on cobordism theory, Mathematical notes, Princeton University Press, Princeton, N.J., 1968. Circle bundles, adiabatic limits of η-invariants and Rokhlin congruences. Weiping Zhang, Ann. Inst. Fourier. 441Weiping Zhang, Circle bundles, adiabatic limits of η-invariants and Rokhlin congruences., Ann. Inst. Fourier 44 (1994), no. 1, 249-270.

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{'abstract': 'The f -invariant is a higher version of the e-invariant that takes values in the divided congruences between modular forms; it can be formulated as an elliptic genus of manifolds with corners of codimension two.In this thesis, we develop a geometrical interpretation of the finvariant in terms of index theory, thereby providing an analytical link between the stable homotopy groups of the spheres and the arithmetic of modular forms. In particular, we are able to establish a formula that allows us to compute the f -invariant from a single face. Furthermore, we apply our results to the situation of cartesian products and principal circle bundles, performing explicit calculations. *', 'arxivid': '0808.0428', 'author': ['Hanno Von Bodecker \nFakultät für Mathematik\nRuhr-Universität Bochum\n44780BochumGermany\n'], 'authoraffiliation': ['Fakultät für Mathematik\nRuhr-Universität Bochum\n44780BochumGermany'], 'corpusid': 18706222, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 37334, 'n_tokens_neox': 31843, 'n_words': 19713, 'pdfsha': 'ebb99004e47f869edea5a73db5250de3d465d1b1', 'pdfurls': ['https://arxiv.org/pdf/0808.0428v2.pdf'], 'title': ['On the geometry of the f -invariant', 'On the geometry of the f -invariant'], 'venue': []}

arxiv

Solutions of a Dirac hydrogenlike meson and scalar confinement at low orbital angular momentum states 28 Jan 1998 M A Avila Facultad de Ciencias UAEM 62210MorelosCuernavacaMexico Solutions of a Dirac hydrogenlike meson and scalar confinement at low orbital angular momentum states 28 Jan 1998PACS number(s) : 1239Pn; 1238Aw * Electronic address: manuel@servmfcuaemmx Wavefunctions of a heavy-light quark (Q,q) system described by a covariant Dirac hamiltonian are analyzed. By assuming that the confinement potential is a Lorentz scalar (S), the slope of the Isgur-Wise function is calculated at zero recoil point. The result obtained is ξ ′ (1) = −0.93 ± 0.05. This means that the solutions are perfectly consistent. If relativistic corrections in the light quark wave functions are included the result is ξ ′ (1) = −1.01 ± 0.04. From heavy-light data this suggests that if relativistic effects are considered, scalar confinement is reliable in low orbital angular momentum states. A covariant Dirac-like hamiltonian describing a heavy quark-light quark (Q,q) system has been introduced in Ref. [1]. This hamiltonian contains all of the relativistic corrections. These effects come mainly from the recoil of the heavy quark. The model has been successful in applications involving both hydrogenlike (B−, D−) meson spectroscopy and also in calculations of weak mixing angles [2] from exclusive semileptonic B-decays. The hamiltonian of Ref. [1] contains two potentials, one of them is a central linear Lorentz scalar potential (S) which describes dynamically the confinement, and the other is a Coulomb potential (U) describing the short range color interquark interaction. To be strong enough, the non-perturbative scalar potential S does not feel any relativistic corrections but the Coulomb potential U still is susceptible to them. In fact, in the model of Ref. [1] the potential U incorporates relativistic corrections to first order in m/M Q , where m is the light quark mass and M Q the heavy quark mass. One question about the above description still remains to be studied and it is the concerning to the consistency of its solutions. For this reason, the purpose of this work is to examine the solutions of [1] when the system (Q,q) is in low orbital angular momentum states. In order to do this, we subject the wavefunctions to several phenomenological tests. For instance, we check that they are normalizable. The way in which this is done is through an approach introduced in Ref. [3]. Another thing which is checked in the present work is that the solutions account for the so called Klein paradox. This effect consists in the mixing of positive-negative energy states [4]. The third and last test employed on the wavefunctions is to calculate the slope of the Isgur -Wise (IW) function at zero recoil point, ξ ′ (1) [5]. This quantity is calculated by assuming a scalar confinement potential (S) and also in the heavy quark symmetry limit (M Q → ∞). The value obtained for the slope is compared to other values of this quantity previously found using different approaches. After we check that the wavefunctions of Ref. [1] are consistent, then we include relativistic corrections. These corrected solutions are employed to make the corrections to ξ ′ (1) that result from a finite M Q . Finally, the present work is concluded by discussing the nature of confinement in a (Q,q) system in low angular momentum states. Let us first check the normalizability of the solutions of Ref. [1]. To do this, we follow an approach similar to that of Ref. [3]. In [3] it was pointed out that a relativistic quarkonia system (q, q) with a vector confining potential V v = β 1 β 2 κ v r ( κ v > 0 ) could have unphysical non-normalizable solutions. On the other hand, if the confining potential is of a Lorentz scalar nature, V s = κ s r ( κ s > 0 ), this problem would not arise. If one uses both kind of potentials, it is necessary that scalar confinement be stronger than vector confinement (e.g. k s > k v ) in order to avoid singularities in the wave function. In [3] a possible singularity in the (q, q) wave function is avoided through the use of a Salpeter (no-pair) equation. In order to illustrate the procedure, let us consider a simple scheme where the (Q,q) bound system is described by a Dirac equation. We assume that the only interaction existing between the quarks is that of confinement and that the confinement potential is composed of two parts. One of them is scalar, S = k s r (k s > 0), and the other vector, V = k v r (k s > 0). In this way, the respective equation in the C.M. system is α · p + mβ + βS + V ψ = Eψ,(1) where p is the light quark momentum. Let us decompose the wave function as ψ = ψ + + ψ − , ψ ± ≡ P ± ψ,(2) where P ± ≡ (1 ± β)/2 are the standard projection operators into the upper and lower components of the wave function. Then, by using (1) and (2), we obtain ψ − = α · p E + m + (k s − k v ) r ψ + .(3) From the last expression it is evident that if E > 0 and k v > k s , the lower sector of the wave function has a singularity at r = (E + m)/(k v − k s ). Consequently, the norm of the wave function < ψ | ψ >=< ψ + | ψ + > + < ψ − | ψ − >,(4) is not finite. In particular, we can observe from Eq. (3) that with a vector confinement (S = 0), the wavefunctions would not be finite. But, if the confinement is scalar (V = 0), the solutions are finite. Let us turn now to examine the normalization of the wavefunctions of the covariant hamiltonian which was introduced in Ref. [1] for describing a (Q,q) system. This hamiltonian is α · p + m β + p 2 2M Q + M Q + U(r) + U(r) 2 M Q α · p + α ·rr · p + βS(r) ψ = E ψ,(5) where p is the light quark momentum, V s (r) ≡ βS(r) = βκ s r (κ s > 0) is the scalar confining linear potential and U(r) = −ξ/r (ξ > 0) is a color Coulomb-like potential. Note that terms proportional to U (r) 2 M Q and to p 2 2 M Q in this hamiltonian arise from the recoil of the heavy quark. Using Eqs. (2) and (5) we obtain ψ − = α · p + U (r) 2 M Q α · p + α ·rr · p E + m − p 2 2 M Q − M Q − U(r) + S(r) ψ + ,(6) where |E| ≥ M Q + m. Since − U(r) + V (r) > 0 then for ǫ ≡ E − M Q − m ≥ 0 the wave function ψ − could have an unphysical singularity, which would depend strictly on the size of the term p 2 2 M Q 1 . According to Heavy Quark Effective Theory (HQET) [5] this term represents the exchanged momentum between the bound quarks. It acquires a maximum value in the nonperturbative regime (quarks and gluons inside the meson) [6]. Thus, if we denote by R the radius of the hadron, then from HQET it follows that p 2 2 M Q ∼ 1/R f ms −1 = 0.2/R GeV.(7) Thus, if we define the denominator of Eq. (6) as h(r) ≡ ǫ + 2m − U(r) + S(r) − p 2 2 M Q ,(8) and use (7) together with U = − ξ/r f ms −1 = − 0.2 ξ/ | r | GeV, S = κ s r f ms 1 = 5 κ s | r | GeV −1 , and x ≡ r/R , we obtain h(r) = ǫ + 2m + (κ s R 2 x 2 − x + ξ)/r.(9) In order to show that h(r) is positive for any value of r, let us note that the quantity between parenthesis in the last equation is positive at x = 0 and it does not take negative values for any value of r providing that 4 R 2 κ s ξ > 1. For κ = 0.2 GeV −2 and ξ = 0.445 [1], this last condition means R > 1.6 GeV −1 = 0.32 f ms. Since this condition is very reasonable for a hadronic radius, we can conclude that h(r) > 0 for any value of r. In particular, we have shown that the denominator h(r) of Eq. (6) is not zero for any value of r. Therefore, the solutions of Eq. (5) are finite and normalizable. Let us proceed now with the second test of the solutions of the hamiltonian introduced in Ref. [1]. For this purpose we impose the s-wave classical turning points condition p = 0 in Eq. (5). Once we do this, we obtain β ( m + S ) + M Q + U = E which becomes E + − M Q = m + S + U,(10)E − − M Q = −m − S + U,(11) for the positive (β = 1) and negative (β = −1) energy states respectively 2 . At this stage there are two possibilities, either U = 0 or U = 0. If we turn off the Coulomb interaction ( U = 0 ) in the above equations we are 2 The condition p = 0 can be seen from the MIT bag model point of view as the boundary condition that prevents light quark q current flux leaving through the meson bag. In this way, we expect typical values for the returning point (r r.p. ) of order r r.p. ≥ R ∼ 0.78. f m [7]- [8] reproducing the same analysis as Ref. [4]. In this work the confinement in a Dirac equation was analyzed by assuming that the confinement potentials were of both kinds, scalar (S) and vector (V ). It was found that the structure of the confinement must be scalar (V = 0) in order to avoid the Klein paradox (mixing of positive with negative energy states). For the above reason, in the present work we are just considering the second and more realistic situation where U = 0. Thus, if we turn on the color Coulomb interaction and plot Eqs. (10) and (11) with S(r) = κ s r and U(r) = −ξ / r for two different values of the light quark masses m = 0 , and m = 0.5 GeV, we obtain the plot of Fig. 1. As can be seen from this figure, the singularity in the Coulomb potential at r = 0 distorts and mixes the positive and negative energy states inside the domain of perturbative physics i. e. r ≪ R 3 . But (as should be expected of a good and consistent confinement potential) this effect dissappears in the confinement region r ≃ R. In other words, Fig. 1 indicates that the scalar confining potential S of Eq. (5) accounts for the Klein paradox in the confining region r ≃ R independent of the value of the light quark mass. Furthermore, we note also from Fig. 1 that the critical values of the radial coordinate (r c ) for which the physical condition E + − M Q − m ≥ 0 begins to be valid, independently of the value of m, are those such that r > r c where r c ∼ 0.3 f ms. If we think of r c as the critical value dividing the perturbative and the non perturbative region, it follows that the critical energy necessary to reach the non perturbative region (r > r c ) would be ǫ c ∼ 1/r c ∼ 0.67 GeV . Another sign of consistency is that precisely M D(Ds) − M c ∼ M B − M b ∼ 0.66 GeV ∼ ǫ c [1]. It is easy to see that a confining linear vector potential (V = κ v r) without a scalar potential (S = 0) in Eq. (5) is automatically discarded. In fact, in this case the s-wave returning point condition p = 0 would yield E + − M Q = m + V + U,(12)E − − M Q = −m + V + U.(13) For large values of r say r ∼ R where the Coulomb potential vanishes (U ∼ 0), then Eqs. (12) and (13) become E − ≃ E + ≃ M Q + V > 0 ; r ∼ R.(14) This result is unphysical since it means that the negative and positive energy states are mixed in the non-perturbative region. In this way, we can conclude that only the scalar potential of Eq. (5) can account for the Klein paradox. This result makes the solutions of (5) consistent. In order to continue the analysis of the solutions of Eq. (5) let us turn now to the calculation of the slope of the IW function at zero recoil point. By solving Eq. (5) in the heavy quark symmetry limit (M Q → ∞), we find the light quark wavefunctions. With these solutions the IW function is calculated through the expression [8] ξ(ω) = 2 ω + 1 j 0 (2 E q ω − 1 ω + 1 r) ,(15) where E q is the light quark energy. The average is defined as f ≡ Ω dr r 2 ψ(r) † f (r) ψ(r),(16) where Ω is the spatial region explored by the wave function. In this work Ω is taken as a sphere of radius R . Likewise, for the calculation of the slope of the IW function at zero recoil point, we use both the solutions of (5) in the heavy quark symmetry limit and the expression [8] ξ ′ (1) = − ( 1 2 + 1 3 E 2 q < r 2 >).(17) Once we do the above we obtain ξ ′ (1) ≃ −0.93 ± 0.05.(18) We have allowed the light quark mass to run from 0 to 0.25 GeV and the hadron radius as R = 0.78 f m = 3.9 GeV −1 [7]. The above value for ξ ′ (1) is consistent with values of this quantity previously obtained in other works. For instance, in Ref. [4] ξ ′ (1) ≃ − 0.90 was obtained by solving the Dirac equation with scalar confinement (S) through a variational method. While in Ref. [9] by analyzing sum rules it was found ξ ′ (1) ≃ − 0.65. In [10] a relativistic flux tube mesonic model was employed and gave the result ξ ′ (1) ≃ − 0.93, which is exactly the same value found in the present work. The coincidences between (18) with values for ξ ′ (1) obtained with such different approaches allow us to assert that solutions of (5) can go through this last test. Since the solutions of (5) have passed satisfactorily the three different tests above cited we can conclude that the model introduced in Ref. [1] is perfectly consistent to describe a (Q,q) system in low orbital angular momentum states. Now that we have checked the consistency of Eq. (5), let us employ its solutions to discuss the nature of confinement in a (Q,q) system. As was pointed out in Ref. [4], the value of the IW function constitutes a sensitive test for the kind of confinement potential. According to this work, there is an apparent conflict at the moment between the values for ξ ′ (1) obtained from a Dirac equation with confining scalar on the one hand and that extracted from heavy-light 'data' on the other. The values of ξ ′ (1) from the data are higher than that calculated with a Dirac equation. To quote just a few values obtained from different Lattice QCD (LQCD) calculations, ξ ′ (1) LQCD = −1.0 [11], ξ ′ (1) LQCD = −1.2 [12], and ξ ′ (1) LQCD = −1.16. Such a large discrepancy between ξ ′ (1) and ξ ′ (1) LQCD in [4] questions the scheme of a scalar confining potential in a Dirac equation to describe a (Q,q) system. In fact, the authors of [4] propose that if instead of a Dirac equation with scalar confining potential a Salpeter (no-pair) equation with a vector confining potential is used to calculate ξ ′ (1), the values obtained for ξ ′ (1) would be in better agreement with heavy-light 'data'. The respective values for ξ ′ (1) found in [4] with these two different models are ξ ′ (1) Dirac ≃ −.90 and ξ ′ (1) no−pair ≃ −1.2. From these values they then argue in favor of a no-pair equation with vector confinement. We want to stress at this point that the values obtained in [4] for ξ ′ (1) do not contain in them corrections coming from assuming M Q finite. Furthermore, it is claimed in this work that if one includes these corrections in the light quark wave functions, the scheme of a Dirac equation with vector confining potential would be the correct one for the description of a (Q,q) system. That these corrections are necessary is evident from the following three facts: 1. The limit M Q → ∞ where both the IW function and its slope are properly defined are just a good approximation to reality. 2. Even in the limit M Q → ∞, the IW function and its slope are of a non-perturbative nature. Consequently their values rely on parameters (quark masses, Bag radius, tensions, etc) independently of the approach employed. 3. A good model for the description of a (Q,q) is that which retains its validity in the limit M Q → ∞. Furthermore, this model can be seen as a good heavy quark picture modified by finite relativistic corrections. As we stated above, we consider it important to include relativistic corrections to the slope of the IW form factor at zero recoil point. The way we incorporate these relativistic corrections to ξ ′ (1) is as follows. First, we obtain the light quark wavefunction from Eq. (5) with finite M Q using for this purpose the same parameters as in Ref. [1]. These solutions are then substituted in Eq. (17). Once the above procedure is implemented, we obtain ξ ′ (1) corr = −1.01 ± 0.04, where the light quark mass has taken the same range of values as in Eq. (18). As can be seen in Eq. (19) the value for ξ ′ (1) with relativistic corrections is in considerably better agreement with heavy-light data than those without relativistic corrections in Eq. (18) and Ref. [4]. The corrected value for ξ ′ (1) of Eq. (19) modifies sustantially any conclusion about the nature of confinement. In fact, this result induces us to conclude that a Dirac equation with scalar confinement constitutes a good way of describing a (Q,q) system in low orbital angular momentum states providing relativistic corrections are taken into account. In the infinitly heavy quark mass limit this term does not exist. Consequently if ǫ > 0, the solutions of Eq. (5) are perfectly well defined for any value of r. In fact, as is well known, this mixing inside the confinement region r ≪ R is responsible for the bound state spectra in a hydrogenlike system (Q,q). http://arxiv.org/ps/hep-ph/9710241v3 http://arxiv.org/ps/hep-ph/9710241v3 Acknowledgement I thank to N. A. and to N. H. with whom part of this work was done. I thank also to M. S. for helping with the preparation of the manuscript. I acknowledge support from CONACyT grant 3135. . M Avila, Phys. Rev. D. 49309M. Avila, Phys. Rev. D 49, (1994) 309. . M Avila, J. Phys. G. 21615M. Avila, J. Phys. G 21, (1995) 615. . J Sucher, Phys. Rev. A. 22348J. Sucher, Phys. Rev. A 22 (1980) 348; . Phys. Rev. D. 515965Phys. Rev. D 51, (1995) 5965. . M G Olsson, S Veseli, K Williams, Phys. Rev. D. 515079M. G. Olsson, S. Veseli, and K. Williams, Phys. Rev. D 51, (1995) 5079. . N Isgur, M B Wise, Phys. Lett. 69527ibid 237N. Isgur and M. B. Wise, Phys. Lett. 69 (1989) 111 ; ibid 237 (1990) 527. . M Neubert, Phys. Rept. 245259M. Neubert, Phys. Rept. 245 (1994) 259. . D Izat, C De Tar, M Stephenson, Nucl. Phys. B. 199269D. Izat, C. De Tar and M. Stephenson, Nucl. Phys. B 199 (1982) 269. . M Sadzikowski, K Zalewski, Z. Phys. C. 59677M. Sadzikowski and K. Zalewski, Z. Phys. C 59 (1993) 677; . H Ogaasen, M Sadzikowski, Z. Phys. C. 64427H. H ogaasen and M. Sadzikowski, Z. Phys. C 64 (1994) 427. . M Blok, M Shifman, Phys. Rev. D. 472949M. Blok and M. Shifman, Phys. Rev. D 47 (1993) 2949. . M G Olsson, S Veseli, Phys. Rev. D. 512224M. G. Olsson and S. Veseli, Phys. Rev. D 51, (1995) 2224. . C W Bernard, Y Shen, A Soni, Nucl. Phys. (Proc. Suppl.) B. 30473C. W. Bernard, Y. Shen and A. Soni, Nucl. Phys. (Proc. Suppl.) B 30, (1993) 473. . S P Booth, Phys. Rev. Lett. 72S. P. Booth et al, Phys. Rev. Lett. 72, (1994) 462. FIGURES Figure 1. Behavior of the right hand side of Eqs. (10) and (11) with U(r) = −ξ/r and S(r) = κ s r for two different light quark masses: 1a). m = 0 and 1b). m = 0.5 GeV . The values employed for ξ and κ s are the same of Ref. Figure 1. Behavior of the right hand side of Eqs. (10) and (11) with U(r) = −ξ/r and S(r) = κ s r for two different light quark masses: 1a). m = 0 and 1b). m = 0.5 GeV . The values employed for ξ and κ s are the same of Ref. . − Gev, GeV −2 .

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{'abstract': 'Wavefunctions of a heavy-light quark (Q,q) system described by a covariant Dirac hamiltonian are analyzed. By assuming that the confinement potential is a Lorentz scalar (S), the slope of the Isgur-Wise function is calculated at zero recoil point. The result obtained is ξ ′ (1) = −0.93 ± 0.05. This means that the solutions are perfectly consistent. If relativistic corrections in the light quark wave functions are included the result is ξ ′ (1) = −1.01 ± 0.04. From heavy-light data this suggests that if relativistic effects are considered, scalar confinement is reliable in low orbital angular momentum states.', 'arxivid': 'hep-ph/9710241', 'author': ['M A Avila \nFacultad de Ciencias\nUAEM\n62210MorelosCuernavacaMexico\n'], 'authoraffiliation': ['Facultad de Ciencias\nUAEM\n62210MorelosCuernavacaMexico'], 'corpusid': 1098123, 'doi': '10.1142/s0217751x99000853', 'github_urls': [], 'n_tokens_mistral': 6011, 'n_tokens_neox': 5304, 'n_words': 3504, 'pdfsha': 'eab732741aaedf4dde93eefba20122b680bdb307', 'pdfurls': ['https://export.arxiv.org/pdf/hep-ph/9710241v3.pdf'], 'title': ['Solutions of a Dirac hydrogenlike meson and scalar confinement at low orbital angular momentum states', 'Solutions of a Dirac hydrogenlike meson and scalar confinement at low orbital angular momentum states'], 'venue': []}

arxiv

ConvGenVisMo: Evaluation of Conversational Generative Vision Models Narjes Nikzad Khasmakhi Meysam Asgari-Chenaghlu Nabiha Asghar Philipp Schaer Dietlind Zühlke ConvGenVisMo: Evaluation of Conversational Generative Vision Models Conversational generative vision models (CGVMs) like Visual ChatGPT (Wu et al., 2023) have recently emerged from the synthesis of computer vision and natural language processing techniques. These models enable more natural and interactive communication between humans and machines, because they can understand verbal inputs from users and generate responses in natural language along with visual outputs. To make informed decisions about the usage and deployment of these models, it is important to analyze their performance through a suitable evaluation framework on realistic datasets. In this paper, we present ConvGenVisMo, a framework for the novel task of evaluating CGVMs. ConvGenVisMo introduces a new benchmark evaluation dataset for this task, and also provides a suite of existing and new automated evaluation metrics to evaluate the outputs. All ConvGenVisMo assets, including the dataset and the evaluation code, will be made available publicly on Github 1 . 2 https://chat.openai.com/ 3 https://blog.google/technology/ai/ bard-google-ai-search-updates/ 4 We will increase the number of samples in future work. Introduction Conversational models are AI-powered systems that use natural language processing (NLP) and machine learning algorithms to carry out human-like conversations with users. The state of the art has rapidly advanced in this area with the advent of large language models (LLMs) like GPT-4 (Ope-nAI, 2023), LaMDA (et al., 2022b), PaLM (et al., 2022a and Llama (Touvron et al., 2023), as well as training methods like Reinforcement Learning from Human Feedback (RLHF) (Christiano et al., 2017). Two prominent examples of LLM-based conversational models are OpenAI's 1 TH Köln -University of Applied Sciences, Cologne, Germany 2 Ultimate, Berlin, Germany 3 Microsoft, Redmond WA, USA (This work is not sponsored by or done at Microsoft). Correspondence to: Narjes Nikzad Khasmakhi <narjes.nikzad khasmakhi@thkoeln.de>. 1 https://github.com/nabihach/ConvGenVisMo ChatGPT 2 and Google's Bard 3 . They demonstrate excellent command over language syntax and semantics, can carry out fluent multi-hop conversations with humans without losing long-term context, and are able to perform few-shot question answering with surprisingly high accuracy. An important question about LLMs is, can they turn conversations with humans into actions? For instance, state of the art text-to-image models like Stable Diffusion (Rombach et al., 2021) and DALL-E can understand textual prompts to generate highly complex images, but this is a single-round process; they are not able to generate and/or refine images iteratively over the course of an entire conversation. Training a multi-modal conversational model is one feasible solution for turning conversations with humans into actions. However, this approach is prohibitive because it requires a lot of training data to tune the large number of parameters needed in such multi-modal models. The next best alternative is to synthesize a trained conversation model with a trained vision model to obtain a conversational generative vision model (CGVM). A user can converse with a CGVM's language component, which outputs a summary of the conversation and passes it to the CGVM's visual component, which in turn outputs a realistic image relevant to the conversation. Visual ChatGPT (Wu et al., 2023) is the first effort to synthesize LLMs with image generation models. It is reasonable to expect CGVMs to become ubiquitous in the near future, which leads to the need for an appropriate performance evaluation framework for them. However, due to the novelty of the task CGVMs perform, there is no benchmark dataset available to assess their output quality. Moreover, it is not clear how existing image quality assessment metrics should be used to evaluate these models. In this study, we develop two ideas to bridge the aforementioned gaps in CGVM evaluation. First, we curate a new benchmark dataset for CGVM evaluation. It contains 180 manually collected and carefully labelled image-conversation pairs 4 . Each sample in the dataset mimics the scenario where a human is conversation- Where is the fox? It is sitting in middle of a jungle. Can you describe more the scence? There is snow on the ground, Fox is in middle sitting a very tiny green grass. I see, is there any other details about it? There are some dead trees in the background and foreground as well. ally instructing a model to generate and refine an image. Concretely, each sample contains an image, and a conversation between two real users about that image. Only one user has access to the image, and is explaining its contents to the second user who does not have access to the image. The second user listens to the first user's description of the image and may ask questions about it to get a better understanding of it. The dataset provides six categories of images: product, nature, human, animal, cartoon, and paintings. These categories allow broad coverage of conversational topics. Note that we curate this new dataset of images from scratch and purposefully do not use existing image datasets like MS COCO (Lin et al., 2014), PASCAL VOC (Everingham et al., 2010), or ImageNet (Deng et al., 2009), because pre-trained language and vision models may have already seen them and may perform unrealistically well on them. Second, we collect two groups of new and existing evaluation metrics that are appropriate for CGVM evaluation. The first group consists of semantic and computation metrics that assess the quality of the CGVM generated images with and without reference images. The second group uses a novel Element Presence Score to assess element-based overlap between the generated and ground-truth images. Figure 1 shows an overview of the CGVM evaluation process. Thus, our main contribution in this work is an evaluation framework for CGVMs, called ConvGenVisMo: • ConvGenVisMo introduces the CGVM evaluation task, and the first benchmark dataset for it. The dataset consists of unseen and carefully curated imageconversation pairs spanning several image categories. • ConvGenVisMo provides a suite of several existing and new evaluation metrics to assess the performance of CGVMs. • We demonstrate ConvGenVisMo's usage and results on a CGVM that is obtained from the synthesis of ChatGPT and Dreamstudio 5 . ConvGenVisMo: Task and Dataset The task at hand is to evaluate the quality of a conversational generative vision model (CGVM). By definition, a CGVM is a model that carries out natural language conversations with humans and produces visual outputs if needed. For example, it can respond to questions about an image it generates, and can handle image editing instructions to further refine the image. For the CGVM evaluation task described above, we introduce the ConvGenVisMo benchmark dataset which contains 180 samples, each containing an image and a multi-hop conversation between two humans about that image. The image is revealed to only one user, whose goal is to describe it to the other user through a back and forth conversation. The person who does not have access to the image commences the conversation. This individual is referred to as "Joe" in the dataset. The second individual, named "Jill", provides descriptions of the image, pointing out the presence or absence of specific elements and explaining their significance. When Joe does not understand certain aspects of the image, he seeks clarification. For the sake of simplicity, we only use text-based conversations in the dataset. Inclusion of speech-based conversations is set aside as future work. Note that this dataset can also be used for evaluating visual question answering models, because it contains contextual questions and answers with respect to an image. We define the dataset mathematically as follows. Each image Y i , where 1 ≤ i ≤ 180, is associated with a conversation C i . Each conversation C i is a time-series given by C i = {(M Joe , M Jill ) t : t ∈ N} i ,(1) where the tuple (M Joe , M Jill ) t is the t'th conversation hop consisting of the message sent by Joe to Jill and the response sent by Jill to Joe at step t. Since all images are inaccessible to Joe, every conversation starts with him asking questions to Jill about the image. The full dataset is given by D = {(C i , Y i , e i ) : i ∈ {1, 2, ..., 180}}(2) where e i is the metadata of the image Y i and contains valuable information useful for evaluation purposes. Data Collection and Statistics We collected the images from three different sources: unique (not publicly available), Pinterest, and Flickr. We have tried to minimize the number of images from the two latter sources, because it is reasonable to assume that existing pretrained models have seen them during training. The images are pre-classified into six categories: cartoon, nature, painting, product, animal, and human, with each category containing 30 samples. A sample image and its corresponding conversation are shown in Figure 2 and Figure 14 respectively. Figures 3a-3b show the distribution statistics of conversations in the dataset. Images categorized as 'Human' are the most common, and are widely spread across different conversation lengths. This is because most images in the dataset have been captured from mobile phones. In contrast, the 'Product' category is the most uncommon, and has the smallest spread across conversation lengths. Figures 4a-4b shows the distribution statistics of images in the dataset. The most common image source is 'unique', which represents private, unseen images taken from personal mobile phones (and donated to us voluntarily). The wordcloud shown here is generated from visual element annotation of the images in each category. Figure 5 illustrates the input and output of ChatGPT for each hop of an example conversation. Each element of the chats array shows a hop. Each element of the llm desc array shows the generated summary of the conversation upto the corresponding hop. ConvGenVisMo Evaluation Metrics Automatic performance evaluation of CGVMs is an open research question, with no clear metrics proposed to date. In this paper, we explore several existing metrics from the literature on conversational models and visual question answering. We also propose a novel element presence score. Image Quality Assessment (IQA) Metrics Assessing the quality of generated images remains a crucial challenge in the field of generative vision models. To this end, numerous metrics have been proposed, including image quality assessment (IQA) metrics, which are categorized into 'No Reference' (NR) or 'Full Reference' (FR) metrics. NR-IQA metrics are generally useful when reference images are not available, but we can still use them in our study to capture the intrinsic quality of the CGVM-generated images. We use Blind/Referenceless Image Spatial Quality Evaluator (BRISQUE) (Mittal et al., 2012), which is a popular NR metric. FR-IQA metrics are used when reference images to are available for comparison, and can be categorized into two types: semantic and computational. Computational FR-IQA metrics use algorithmic approaches to compare the generated images with reference images. Peak Signal-to-Noise Ratio (PSNR) (Wang et al., 2004) is a well-known FR-IQA metric that calculates the difference between a signal's maximal strength and the power of corrupting noise that degrades the accuracy of its representation. In our case, reference image can be treated as signals and the CGVM generated image as the noise. For a given reference image Y and a generated imageŶ , PSNR is given by: PSNR(Y,Ŷ ) = log 10 R 2 MSE(Y,Ŷ )(3) where R is the highest possible pixel value for an image (e.g., 255 for an 8-bit image) and the MSE is the Mean Squared Error given by: MSE(Y,Ŷ ) = 1 M N M i=1 N j=1 (Y [i, j] −Ŷ [i, j]) (4) Here, M and N refer to the number of rows and columns of an image, respectively. The indices i and j represent the row and column indices, respectively. A higher PSNR value indicates that the CGVM's output is more similar to the reference image, because their pixel values are closer to each other. Another common FR-IQA metric is Universal Quality Index (UQI) (Wang & Bovik, 2002). It compares the correlation (spatial arrangement of pixels), luminance (average pixel intensity), and contrast values of a generated image with those of the reference image. UQI is given by whereȲ andȲ are defined asȲ = 1 UQI(Y,Ŷ ) = σ YŶ σ Y σŶ · 2ȲȲ (Ȳ ) 2 + (Ȳ ) 2 · 2σ Y σŶ σ 2 Y + σ 2 Y(5)N N i=1 Y i andȲ = 1 N N i=1Ŷ i respectively, σ 2 Y and σ 2 Y denote the variances of Y andŶ respectively, and σ YŶ is the covariance of Y andŶ . Structural Similarity Index (SSIM) (Wang et al., 2004) is another popular FR-IQA metric, which is predicated on the notion that changes in structural information are more perceptible to the human visual system than changes in pixel values alone. SSIM allows us to take into account the structural similarity of two images. It computes luminance (l), contrast (c) and structure (s) similarities as follows: l(Y,Ŷ ) = 2µ Y µŶ + c 1 µ 2 Y + µ 2 Y + c 1 (6a) c(Y,Ŷ ) = 2σ Y σŶ + c 2 σ 2 Y + σ 2 Y + c 2 (6b) s(Y,Ŷ ) = σ YŶ + c 3 σ Y σŶ + c 3 (6c) where µ Y and µŶ are the pixel sample means of Y and Y , respectively. To stabilize the division with a weak denominator, three variables c 1 = (k 1 L) 2 , c 2 = (k 2 L) 2 , and c 3 = c 2 /2 are used. k 1 , k 2 are hyper-parameters and L is the dynamic range for pixel values. Based on the Eq. 6, SSIM is defined as SSIM(Y,Ŷ ) = l(Y,Ŷ ) α · c(Y,Ŷ ) β · s(Y,Ŷ ) γ (7) where α, β and γ are the weights to combine the three components. Note that both SSIM and UQI use luminance and contrast in their evaluation of image quality, however they do so in different ways and for different purposes. In SSIM, the contrast similarity signifies the standard deviation of the pixel values, and the luminance comparison quantifies the average brightness of the images. The similarity of the edges and textures of the reference and generated images is measured by the structural comparison. It is important to note that each aforementioned metric has advantages and limitations, thus we use multiple such metrics in our evaluation framework. Computational IQA metrics have a serious limitation: they are often divorced from human perception. For example, rotated images are identical for a human viewer, but computational IQA techniques will give them low similarity scores. Therefore, it is essential to develop alternative metrics which factor in human perception and judgement. Semantic FR-IQA metrics provide a solution to the aforementioned issue by taking into account the human perception of image quality. Semantic FR-IQA metrics incorporate features such as sharpness, color accuracy, and contrast to evaluate the image quality. CLIP similarity score is an important metric in this category, which can be used to measure the semantic alignment between two images in a high dimensional vector space. A higher CLIP score indicates higher similarity between two images. CLIP score is given by CLIP score (Y,Ŷ ) = cos(CLIP(Y ), CLIP(Ŷ ))(8) where CLIP() is a transformation function that maps an image to a semantic vector space. While CLIP score offers advantages in evaluating image Ground Truth Elements Generated Image Elements Fox Leaves Trees Snow Grass Figure 6. EPPr and EPRe for hop-2 for example in Fig. 1; Also known as: EPPr@2 and EPRe@2. quality by capturing semantic information, it may have limitations in accurately considering the presence of specific objects in the images. Therefore, there is a need for evaluation metrics that specifically measure the presence of elements or objects in images. Element Presence Scores For two images to be considered similar, they must have common elements (e.g. people or objects). We capture this notion with Element Presence Scores. Element Presence Precision (EPPr) is defined as number of overlapping elements present in the generated image and ground-truth, divided by number of elements in the generated image. Element Presence Recall (EPRe), on the other hand, is defined as number of overlapping elements present in the generated image and ground-truth, divided by number of elements in the ground-truth image. Intuitively, these scores combine the notion of element presence with the regular definitions of precision and recall. EPPr(Y,Ŷ ) = Elements(Ŷ ) ∩ Elements(Y ) Elements(Ŷ )(9)EPRe(Y,Ŷ ) = Elements(Ŷ ) ∩ Elements(Y ) Elements(Y )(10) We can similarly define Element Presence F1 (EPF1) score. For element detection in images, we can use out-of-the-box automated object detection methods, or have humans annotate harder examples. Figure 6 shows the EP computations on an example image. Intersection over Union (IoU) is another metric that can be used to calculate the presence of objects and their respective position using an object detection method. It is given by , Figure 7. IoU metric for ground truth image (left) and generated image (right). IoU(Y,Ŷ ) = 1 N N e=1 B e (Y ) ∩ B e (Ŷ ) B e (Y ) ∪ B e (Ŷ )(11) where N is the number of objects in the ground truth image and e represents each one of these objects. B is a function that gives the bounding box of each object e in the image. Figure 7 shows the intuition of IoU computation on example images. IoU is mainly useful in cases where the position of objects in the generated image is important, however it is possible that objects don't not appear in either of the images. Given this fact, we propose three variants of: Common-IoU, Precision-IoU, and Recall-IoU. Common-IoU is calculated in the same way but only for objects that are present in both of the images. Precision-IoU calculates IoU for objects present in the generated image and when an object only appears in the generated but not in the original image, the score for that specific object is zero. Similarly, Recall-IoU takes only the original image objects into account and gives a zero score for that specific object if it is not present in the generated image. For each object, we compute IoU for all of the instances of that object in the image and then average the result over all objects. Top Conversation Hops We further introduce the concept of @K for each evaluation metric described previously, which denotes the value of that metric upto the K'th conversation hop. An example is EPPr@K, read as Element Presence Precision at K. In the same vein, we can define the notion of image generated at K, given bŷ Y K = f ({(M Joe , M Jill ) t : 1 ≤ t ≤ K})(12) Experimental evaluation Here we provide details of the CGVM evaluation process. Evaluation process and settings We demonstrate the ConvGenVisMo evaluation framework on a CGVM which is a combination of ChatGPT ("text-davinci-003" conversation model) and DreamStudio ("stable-diffusion-xl-beta-v2-2-2" vision model). The overall evaluation process consists of three steps: 1. Generate conversation summaries: For each conversation sample in the ConvGenVisMo dataset, we generate a textual summary of the conversation upto each hop, using ChatGPT. For example, if a conversation sample has 2 hops, we generate two summaries for it. The first summary is for the first hop, and the second summary is of both hops. Figure 5 illustrates the input and output of ChatGPT for each hop of an example conversation. We use the following prompt for ChatGPT: "Below is a conversation between Joe and Jill, about an image. Use this conversation to generate a description of the image, such that it can be given as input to a text-to-image model as a prompt." 2. Generate images for summaries: For each summary of each conversation obtained in step 1, we generate an image using DreamStudio. Figure 1 shows an example conversation and the respective image generated at each conversation hop. Comparison of generated images with ground truth: For each conversation, we compare its associated generated images with the ground truth image using the automated evaluation metrics. We then aggregate the statistics of this analysis. To ensure accurate computation of metrics, we perform a number of preprocessing steps on all the images. We standardize the size and format of each image, perform object detection on them using the DETR-100 algorithm (Carion et al., 2020), and use linear interpolation to normalize the number conversation hops, so that they fall within the range 0 to 1. A hop value of 0 represents the first conversation hop, while a value of 1 corresponds to the last conversation hop. This allows meaningful comparisons between conversations that have varying numbers of hops. Evaluation Results Here we present the results for the two groups of CGVM evaluation metrics. CGVM Performance on IQA Metrics We organize our IQA results by the types of IQA metrics: NR-IQA, computational FR-IQA, and semantic FR-IQA. NR-IQA metric -BRISQUE: Figure 8 illustrates the BRISQUE score corresponding to the number of hops. We see a somewhat linear trend, with a slight decrease after reaching a peak. A lower BRISQUE score corresponds to better perceptual quality, so the fact that the scores remain within a narrow range (20-26) suggests that the created images have consistently high quality overall. This in turn suggests that the model is good at preserving the quality of the generated images. Computational FR-IQA metrics -PSNR, SSIM, UQI: Figures 9a-9d show the PSNR, SSIM, and UQI scores. We observe that these scores yield low values with minimal variation. As new details are introduced in each hop, they do not accurately capture the perceptual impact of the added details, especially if the changes are significant. This observation aligns with the fact that PSNR, SSIM, and UQI primarily focus on measuring pixel-wise differences, structural similarities, and universal quality aspects, respectively, rather than being concerned with higher-level image semantic characteristics or the presence of specific objects. Semantic FR-IQA metric -CLIP Score: Figure 10a demonstrates that as the number of hops increases, the CLIP score values also increase. This observation indicates that with each additional hop, more details are incorporated into the image, resulting in a higher similarity to the original image. In other words, the CLIP score accurately reflects the changes and improvements in the image as each step progresses. Figure 10b provides a visual representation of the CLIP scores for different image categories. The animal category obtains the highest CLIP score, indicating a strong correlation between the generated images and the original images in this category. On the other hand, the human category exhibits the lowest CLIP score, suggesting a relatively weaker resemblance between the generated images and the original images in this category. The variations in CLIP scores across different categories highlight the varying performance of the CGVM in generating images related to different object categories. CGVM Performance on Element Presence Scores Recall that the Element Presence Scores provide insights into the model's ability to accurately generate objects and their spatial alignment with the ground truth objects in the images. On the other hand, the IoU score assesses the model's ability to accurately place the generated objects within the images. As shown in Figure 11a, the EPRe score improves as the number of hops increases in the CGVM. The EPRe score specifically evaluates the model's ability to generate images that correctly consider the presence of specific objects discussed in the conversation. In other words, as each hop progresses in the conversation, the model could add more objects to the generated images. Figure 11b illustrates a decrease in the EPPr score as the number of hops increases in the CGVM. This observation suggests while the model has the ability to add more objects to the generated images, there is a potential for including irrelevant or incorrect objects that do not align with the ground truth or user expectations. Since there are conflicting trends between the EPRe and EPPr scores, the EPF1 score serves as a useful metric to provide a balanced evaluation of the model's performance. Figure 11c demonstrates that the EPF1 score increases as the number of hops increases in the CGVM. This indicates that, overall, the model improves in terms of generating images that accurately incorporate the discussed objects while minimizing irrelevant or incorrect inclusions. Figures 11d-11f illustrate the element presence scores for each image category. From the figures, it can be inferred that the animal category exhibits the highest element presence score, indicating that the generated images in this category accurately consider the presence of specific objects related to animals discussed in the conversation. This aligns with the high CLIP score observed in the animal category, suggesting a strong semantic alignment with the intended objects. Furthermore, the animal category exhibits a lower precision score. This implies that while the model incorporates more objects, some of them may not be entirely relevant or contextually appropriate based on the ground truth or user expectations. This discrepancy between EPRe and EPPr highlights the challenge of achieving a balance between including relevant objects and avoiding the addition of irrelevant or incorrect ones. Similarly, the human category shows a high EPRe score, indicating that the model successfully includes human-related objects discussed in the conversation. However, it also exhibits a lower precision score, suggesting that the generated images may still contain some irrelevant or contextually inappropriate elements. Results for IoU scores are presented in Table 1. The benchmark model achieves a very high score on a sample but overall for all of the scores in this category, it achieves around 0.2 and 0.3. Common-IoU score achieves a higher value and it is expected because of its nature, common objects are more likely to have similar bounding boxes and get less zero values for the objects that do not exist. Discussion Some key challenges we observed in the evaluation of CGVMs were: 1) the tendency of the LLMs to hallucinate and introduce additional details without fully considering the context of the conversation, 2) CGVM's failure to pay sufficient attention to details, and 3) ensuring that the generated images contain all significant elements discussion in the input conversation summary. One of the key challenges we observed in the evaluation of CGVMs is the tendency of the LLMs to introduce additional details without fully considering the context of the conversation. For example, Figure 12a shows sample #54 where "Jill" answers that "I can see a girl" , while the LLM model adds descriptions such as "the girl is standing in the center of the image" or "the girl is wearing a white dress and a pink flower in her hair." These details may not be explicitly mentioned or requested in the conversation, indicating that the LLM component tends to be proactive in generating additional information. In the subsequent hop, the LLM tends to remove the previously added details and the vision model generates the new image based on the updated conversation context. This is the main reason that can help explain the observed increase in the EPRe and decrease in the EPPr in CGVMs. The inclusion of more details may also introduce a higher likelihood of including irrelevant or incorrect objects, leading to a decrease in precision. The second challenge we encountered was, the CGVM fails to pay sufficient attention to details. For example in the final hop of sample #54, as shown in Figure 12a, it is noticed that the vision model neglects information about the girl lying on the snow and primarily focuses on incorporating the newly provided details about the girl's clothes. Additionally, in another example as shown in Figure 12b, the model initially generates an image of a pink panther in the first step, and the output indicates that the model does not know the pink panther as a specific cartoon figure. However, as the conversation progresses and reaches its conclusion, the final output disregards the color aspect and includes a black and white panther instead. This demonstrates how the model can neglect certain details, such as color information, throughout the course of the dialogue. Third, it is a challenge to coordinate the LLM and vision model such that the generated images encompass all significant elements discussed in the conversation. For instance, in Figure 13a, the input (original conversation) and output of ChatGPT are provided. When the term "package" is mentioned in the conversation within the product category, although the LLM can recognize the term and include it in the summary, the vision model completely disregards it, as shown in Figure 13b. Consequently, the output generated by the model does not include any package. This observation highlights a challenge in maintaining consistency and comprehensiveness throughout the conversation. While the LLM may introduce new information, it is crucial for the vision model to consider and retain important details mentioned earlier in the conversation. Neglecting relevant information can result in an incomplete representation of the scene. Introducing a memory mechanism in CGVMs can address the challenge of retaining and effectively utilizing both previously mentioned details and new information. By incorporating a memory component, the model can store relevant information from previous hops and access it when generating subsequent images. The evaluation results shed light on the appropriate metrics for investigating the performance of CGVMs. The findings indicate that computational FR-IQA scores may not be reliable options for assessing the step-by-step improvements in generated images for each hop. These metrics are highly sensitive to even slight changes. On the other hand, semantic FR-IQA measurements, such as the CLIP score, prove to be valuable in evaluating the progression and enhancement of generated images as the number of hops increases. They provide a quantitative measure of the increasing similarity between the generated images and the original image. Additionally, the presence of specific objects is crucial, and metrics like Element Presence Scores play a critical role in assessing the generated images in terms of the presence of desired elements or objects. Semantic similarity and element-wise precision and recall are not always directly correlated. It is possible for a generated image to contain the same objects as the ground truth image, yet lack semantic similarity. Differences in background, context, or depicted actions can contribute to the semantic disparity between the two images. Conversely, two images can have a close semantic meaning while exhibiting fewer interactions on the element level. These nuances highlight the importance of employing multiple evaluation scores. By considering different metrics, we can capture various aspects of image quality and better understand the strengths and limitations of the generative model. Furthermore, it is crucial to acknowledge that metrics like CLIP score and element-wise scores do not consider the size and location of the elements in the images. For instance, if an image is mirrored, it may receive high scores in terms of CLIP and element presence. Also, if the generated objects are smaller in size compared to the ground truth objects, the same issue will occur. While IOU scores do not yield identical values in these two scenarios. IOU scores take into account both the size and location of the objects, providing a more comprehensive assessment. We emphasize here that the primary objective of this paper was to identify suitable evaluation methodologies that can be applied to a wide range of CGVMs. The intention was not to solely assess the performance of specific models including ChatGPT or DreamStudio. Rather, these models were utilized as means to explore solutions for evaluating CGVMs. Conclusion This paper introduces ConvGenVisMo, a novel evaluation task for conversational generative vision models (CGVMs), along with the first benchmark dataset specifically designed for this task. The dataset comprises diverse image-conversation pairs across various image categories. Con-vGenVisMo also presents a comprehensive suite of evaluation metrics, combining existing and newly developed measures, to effectively assess the performance of CGVMs. The results obtained from the evaluation emphasize the importance of employing a combination of metrics to comprehensively evaluate the performance of CGVMs. (a) Input and output of ChatGPT for sample #47 (b) Output of DreamStudio for sample #47 Figure 13. An example that illustrates how the CGVM visual component disregarded the term "package" when generating the output. Figure 1 . 1Overview of CGVM evaluation process. Two users converse about a ground truth image. Then the CGVM summarizes conversations upto each hop, and generates an image corresponding to each summary. Finally, the generated images are compared with the ground truth image using the evaluation metrics. Figure 2 . 2A sample image from the ConvGenVisMo dataset. Figure 3 . 3Conversation-level statistics of theConvGenVisMo dataset. Figure 4 . 4Image-level statistics of the ConvGenVisMo dataset. Figure 5 . 5An example of ChatGPT input and output. Figure 8 . 8Model performance based on NR-IQA metric: BRISQUE score. Figure 9 .Figure 10 . 910Model performance based on computational FR-IQA metrics. Model performance based on semantic FR-IQA metric: CLIP score. Figure 11 . 11Model performance based on Element Presence Scores: (a-c) based on hops, and (d-f) based on each image category. and output of ChatGPT for sample 54. Figure 12 . 12Some data samples that highlight how the CGVM model's visual component forgets important details from previous hops. Table 1. Model performance based on IoU scores.Scores Mean Std Maximum Common-IoU 0.298925 0.199805 0.789078 Precision-IoU 0.223916 0.200435 0.789078 Recall-IoU 0.273368 0.203606 0.789078 https://beta.dreamstudio.ai/generate AcknowledgementThe authors express their gratitude to 1) Shervin Minaee for reviewing this work and providing very insightful comments, and 2) all individuals who voluntarily donated their images for this study. . Openai, Chatgpt, Openai. ChatGPT. https://chat.openai.com/. End-to-end object detection with transformers. N Carion, F Massa, G Synnaeve, N Usunier, A Kirillov, S Zagoruyko, Computer Vision-ECCV 2020: 16th European Conference. Glasgow, UKSpringerProceedings, Part I 16Carion, N., Massa, F., Synnaeve, G., Usunier, N., Kirillov, A., and Zagoruyko, S. End-to-end object detection with transformers. In Computer Vision-ECCV 2020: 16th European Conference, Glasgow, UK, August 23-28, 2020, Proceedings, Part I 16, pp. 213-229. Springer, 2020. 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{'abstract': 'Conversational generative vision models (CGVMs) like Visual ChatGPT (Wu et al., 2023) have recently emerged from the synthesis of computer vision and natural language processing techniques. These models enable more natural and interactive communication between humans and machines, because they can understand verbal inputs from users and generate responses in natural language along with visual outputs. To make informed decisions about the usage and deployment of these models, it is important to analyze their performance through a suitable evaluation framework on realistic datasets. In this paper, we present ConvGenVisMo, a framework for the novel task of evaluating CGVMs. ConvGenVisMo introduces a new benchmark evaluation dataset for this task, and also provides a suite of existing and new automated evaluation metrics to evaluate the outputs. All ConvGenVisMo assets, including the dataset and the evaluation code, will be made available publicly on Github 1 . 2 https://chat.openai.com/ 3 https://blog.google/technology/ai/ bard-google-ai-search-updates/ 4 We will increase the number of samples in future work.', 'arxivid': '2305.17784', 'author': ['Narjes Nikzad Khasmakhi ', 'Meysam Asgari-Chenaghlu ', 'Nabiha Asghar ', 'Philipp Schaer ', 'Dietlind Zühlke '], 'authoraffiliation': [], 'corpusid': 258959337, 'doi': '10.48550/arxiv.2305.17784', 'github_urls': ['https://github.com/nabihach/ConvGenVisMo'], 'n_tokens_mistral': 10405, 'n_tokens_neox': 9318, 'n_words': 6225, 'pdfsha': '7e72eb196b7c90b3a5d6385af536fe8e5934fb82', 'pdfurls': ['https://export.arxiv.org/pdf/2305.17784v1.pdf'], 'title': ['ConvGenVisMo: Evaluation of Conversational Generative Vision Models', 'ConvGenVisMo: Evaluation of Conversational Generative Vision Models'], 'venue': []}

arxiv

USING AUXILIARY TASKS IN MULTIMODAL FUSION OF WAV2VEC 2.0 AND BERT FOR MULTIMODAL EMOTION RECOGNITION Dekai Sun Harbin Institute of Technology HarbinChina Yancheng He Harbin Institute of Technology HarbinChina Jiqing Han Harbin Institute of Technology HarbinChina USING AUXILIARY TASKS IN MULTIMODAL FUSION OF WAV2VEC 2.0 AND BERT FOR MULTIMODAL EMOTION RECOGNITION Index Terms-Multimodal emotion recognitionBERTWav2vec20Cross-attentionAuxiliary task The lack of data and the difficulty of multimodal fusion have always been challenges for multimodal emotion recognition (MER). In this paper, we propose to use pretrained models as upstream network, wav2vec 2.0 for audio modality and BERT for text modality, and finetune them in downstream task of MER to cope with the lack of data. For the difficulty of multimodal fusion, we use a K-layer multi-head attention mechanism as a downstream fusion module. Starting from the MER task itself, we design two auxiliary tasks to alleviate the insufficient fusion between modalities and guide the network to capture and align emotion-related features. Compared to the previous state-of-the-art models, we achieve a better performance by 78.42% Weighted Accuracy (WA) and 79.71% Unweighted Accuracy (UA) on the IEMOCAP dataset. INTRODUCTION Multimodal emotion recognition is a significant capability in human-machine interaction and has attracted widespread attention in industry and academia. As we all know that emotions are expressed in extremely complex and ambiguous ways, perhaps through linguistic content, speech intonation, facial expression and body actions. There have been many related studies on text emotion recognition [1,2], and also on audio emotion recognition [3,4,5]. However, by observing these results, the research on single modality has reached a certain bottleneck which leads to increasing attention devoted to the use of multimodal approach. Some studies propose that the information of different modalities is often complementary and verified, and the full use of the information of different modalities can help the model to better learn the key content [6,7]. In recent years, pretrained self-supervised learning has performed prominently in several research fields such as natural language processing (NLP) [8] and automatic speech recognition (ASR) [9]. For the multimodal emotion recognition (MER) task, there are also studies that have done a lot of exploration on the basis of pretrained models. For the * Corresponding authors: jqhan@hit.edu.cn first time, Siriwardhana et al. [5] jointly finetuned modalityspecific "BERT-like" pretrained Self Supervised Learning (SSL) architectures to represent both audio and text modalities for the task of MER. Similarly, Yang et al. [10] also proposed to finetune two pretrained self-supervised learning models (Text-RoBERTa and Speech-RoBERTa) for MER. Based on pretrained models, Zhao et al. [11] explored Multilevel fusion approaches, including coattention-based early fusion and late fusion with the models trained on both embeddings. Compared with the MCSAN [12] using traditional features (MFCC & GloVe) for modal fusion, the works mentioned above have greatly improved performance. From the perspective of making full use of contextual data, Wu et al. [13] took advantage of contextual information and proposed a two-branch neural network structure including time synchronous branch and time asynchronous branch. By modifying the structure of network, SMCN [14] realize multi-modal alignment which can capture the global connections without interfering with unimodal learning. However, these previous works focused more on sophisticated fusion structure design and the use of larger and stronger pretrained models, or the use of contextual information that breaks data constraints. They did not start from the MER task itself to explore the bottleneck of insufficient fusion, or capture the feature of emotion itself and the alignment of emotion in different modalities. We believe that the parameters of the network are already sufficient, and the complex fusion module design has not brought enough benefits. Thus, we hope to guide the model to fully exploit the potential of the fusion module by designing just the right auxiliary tasks. In this work, we propose a modular end-to-end approach for the MER task. The general framework is shown in figure 1. First, we learn the semantic information of the respective modalities through the pretrained models, wav2vec 2.0 [9] for audio modality and BERT [8] for text modality. Then, we map text and audio modal feature information into a unified semantic vector space through a k-layer cross-attention mechanism for more adequate modal fusion. Furthermore, we design two auxiliary tasks to help fully fuse the features of the two modalities and learn the alignment information of the emotion itself between different modalities. In the first one, we randomly recombine text and audio modalities and let the model to predict the combination of the two modali- ties through the vector obtained by fusion. This decoupling of multimodal data enables the model to see more complex input combinations, and the constraint of this auxiliary task forces the network to not ignore the role of any modality in the task of MER. In the second one, we randomly replace one of the modalities with other data of the same emotion category, and hope that the model can capture the feature related to emotion and the alignment information beyond the content itself. We comprehensively evaluated the performance of the model proposed on the IEMOCAP dataset in terms of average weighted accuracy (WA) and unweighted accuracy (UA). In additional, we compared it with the SOTA methods and presented relevant ablation experiments that illustrate the effectiveness of each module. METHOD The framework of our proposed model is showned in Figure 1, which consists of three modules, i.e., text encoder, audio encoder, and fusion module. Text Encoder The emergence of BERT has brought NLP into a new era, and gradually refreshed the effect of multiple NLP domain tasks. And "Pretrain + Finetune" has gradually become a new paradigm. Pre-training models such as BERT can be used to transform text into word vectors with contextual semantic in-formation. In this paper, we choose bert-base-uncased 1 as the text modal encoder, which consists 12 layers of transformer encoder. It converts the text into 768-dimensional vectors, which are fed into the fusion module. During training, we also finetune its weights to make it more suitable for our multimodal emotion recognition task. Audio Encoder We choose wav2vec2-base 2 as the audio modality encoder, which consists of feature encoder, contextualized representations with Transformers, and quantization module. The base model contains 12 transformer blocks, and it is pretrained in Librispeech corpus containing 960 hours of 16kHz speech. It is able to learn 768-dimensional latent representation directly from raw audio every 20ms (16Khz sampling rate). We also finetune its parameters during training similar to BERT. Fusion Module The fusion module is based on the multi-head cross attention mechanism [15]. In addition, two auxiliary tasks (Section 2.4) help the model to better handle the feature relationship between the two modalities. Figure 3 shows the specific details of the fusion module, and each layer of the fusion module consists of two branches, which have the same structure but different Q, K, and V. In addition, we use residual linking to reduce the loss of information of the original modalities. The calculation process of multi-head cross attention is as follows: F a = F a + Attention at (Q a , K t , V t )(1)Attention at (Q a , K t , V t ) = sof tmax( Q a K T t d Kt )V t (2) F t = F t + Attention ta (Q t , K a , V a )(3)Attention ta (Q t , K a , V a ) = sof tmax( Q t K T a d Ka )V a(4) where subscript a represents audio modality and subscript t represents text modality. d Ka and d Kt represent dimension of the embeddings. F t : (B, T t , C) is the text feature outputed by BERT, and F a : (B, T a , C) is the audio feature outputed by Wav2vec 2.0. Q a , K a , V a are given here (same of Q t , K t , V t ): Q a = W Q F a + b Q a (5) K a = W K F a + b K a (6) V a = W V F a + b V a(7) Finally, we average pooling F a and F t in the time dimension, and concatenate them in the feature dimension to obtain the fusion embedding (B, 2C), which is sent to the classifier to get the emotion category. Auxiliary Tasks In order to help the model fully fuse the features of the two modalities and learn the alignment information of the emotion itself between different modalities, we design two auxiliary modal interaction tasks. Auxiliary Task1 In MER tasks, audio and text have the same semantics. In the modal fusion of the downstream network, we analyze that the reason for insufficient fusion comes from the fact that the overall emotional orientation can be obtained just from the information of one modality. In some cases, this approach leads to the right results. But for complex cases, we want the network to be more "humble", making full use of the information of the two modalities. As shown in Figure 2, we decouple the pairs of {Audio, Text} in a batch of data, and then randomly scramble and recombine them to get Aux batch1. During the training process, we not only let the model predict the emotion category of the original data pair, but also predict the combined category of this reorganized data pair {Audio, Text} (a total of emotion num × emotion num kinds), and its label (label new ) is defined as follows: label original = label a = label t (8) label new = label a × emotion nums + label t(9) The main task MER requires the downstream network to receive the features from the two modalities and output the emotion category, while the auxiliary task 1 requires the downstream network to predict not only the emotion but also the combination of the two modalities according to the fusion embedding. It forces the downstream network to not ignore any modal information during the feature fusion process of the two modalities, that is, both modal information contributes to the final fusion embedding. Auxiliary Task2 In order to guide the fusion network to learn the alignment information of emotion itself between different modalities, we break the strong semantic correlation between modalities. As shown in Figure 2, for the pairs of {Audio, Text} in a batch of data, we randomly replace one of the modalities (Audio or Text) with other data of the same emotion category. In Aux batch2, different modalities have same emotional label but different semantics. We hope that the fusion network can focus on the features of emotion itself in different modalities and align them. At the same time, the model can better learn common features of the same emotion category. EXPERIMENTAL SETUP Dataset The dataset used in the experiment is the Interactive Emotional Dyadic Motion Cap-ture (IEMOCAP) [16], which is a dialogues dataset and performs improvised and scripts by 10 actors. The 10 actors are divided into 5 sessions, and every session consists of 1 male and 1 female. There are a total of 7529 utterances in IEMOCAP (happy 595, excited 1,041, angry 1,103, sad 1,084, neutral 1,708, frustra-tion 1,849, fear 40, surprise 107, disgust 2). To be consistent and compare with previous studies [17], only utterances with ground truth labels belonging to "angry", "happy", "excited", "sad", and "neutral" were used. The "excited" class was merged with "happy" to better balance the size of each emotion class, which results in a total of 5,531 utterances (happy 1,636, angry 1,103, sad 1,084, neutral 1,708). Implementation Details In order to fully evaluate our proposed model and maintain the same test conditions as previous studies [13], a leave-onesession-out 5-fold cross-validation (CV) configuration was implemented to evaluate our model. We divide IEMOCAP into five folds according to sessions in our experiments. At each fold we keep one session for testing, and other sessions are used for training. Therefore, for each fold we can get one result, and we take the average of the results as the final result of our experiments. We implement our model within the PyTorch framework and select the AdamW [18] optimizer for model optimization with a learning rate of 1 × 10 −5 , where cross attention had 8 heads. Table 1 shows the performance of our method on audioonly, text-only, and multimodal (audio and text) emotion recognition tasks. Compared with a single modality, we simply concatenate the features of the two modalities and feed them into a downstream network constructed with a fully connected layer (FC), which improves the performance by about 6%. Further, we use the single-layer (K=1) multi-head cross-attention downstream network in Figure 3 for modality fusion, which achieves WA : 77.19%, UA : 78.47%. In the current state, we also verify the gains of Auxiliary Task 1 and Auxiliary Task 2, of which Auxiliary Task 2 has the better performance. We also try to use both auxiliary tasks with performance WA : 78.34%, UA : 79.59%. Table 2 shows that when both auxiliary tasks are used simultaneously, the effect of multi-head cross-attention layer K on the performance of emotion recognition task. When K is 2, we get the best performance WA : 78.42%, UA : 79.71%. We found that with the introduction of auxiliary tasks, the overall training objective of the model became difficult to achieve. By appropriately increasing the number of layers in the downstream network, we could obtain better performance. However, due to the limited size of the IEMOCAP dataset, continuously increasing the number of network layers will make it difficult to fully train the network parameters, resulting in performance degradation. The performance of previous stateof-the-art multimodal models is mentioned in Table 3, and our proposed method has better performance than previous works. Table 3. Comparison of the 5-fold CV results of previous state-of-the-art multimodal models and our model on the IEMOCAP. Methods WA(%) UA(%) BERT + Wav2vec2 [11] − 76.31 RoBERTa-text&audio [10] 77.70 78.50 BERT + FBK [13] 77.57 78.41 SMCN [14] 75.60 77.60 BERT + FBK [19] 70.56 71.46 MCSAN [12] 61.20 56.00 Our proposed (best) 78.42 79.71 RESULTS CONCLUSION In this paper, we propose to use wav2vec 2.0 and BERT as upstream network and K-layer downstream network based on multi-head cross-attention mechanism for multimodal emotion recognition task. In addition, we design two auxiliary tasks for the model to help the audio and text be fully integrated, and capture and align the features of emotion itself in different modalities. Finally our method outperforms the previous work on the 5-fold CV result of IEMOCAP, achieved the state-of-the-art, WA : 78.42%, UA : 79.71%. Fig. 1 . 1Our framework Fig. 2 . 2Auxiliary Batch Fig. 3 . 3Proposed model structure Table 1 . 1Weighted Accuracy (WA) and Unweighted Accuracy (UA) of the 5-fold CV results using single modality and multi modality.(FC -Fully Connected; CA -Multi-Head Cross Attention (K=1); Aux1 -Auxiliary Task1; Aux2 -Auxiliary Task2.)Table 2. Performance with different K (the number of layers of Multi-Head Cross Attention (CA)).Methods WA(%) UA(%) Text-only BERT 70.53 71.79 Audio-only Wav2vec2 69.92 70.68 Audio and Text BERT+Wav2vec2+FC 76.24 77.20 BERT+Wav2vec2+CA 77.19 78.47 BERT+Wav2vec2+CA+Aux1 77.67 79.16 BERT+Wav2vec2+CA+Aux2 78.11 79.47 BERT+Wav2vec2+CA+Aux1&2 78.34 79.59 Methods K WA(%) UA(%) BERT+Wav2vec2+CA+Aux1&2 1 78.34 79.59 BERT+Wav2vec2+CA+Aux1&2 2 78.42 79.71 BERT+Wav2vec2+CA+Aux1&2 3 77.68 79.41 https://huggingface.com/bert-base-uncased 2 https://huggingface.co/facebook/wav2vec2-base Comparative analyses of bert, roberta, distilbert, and xlnet for text-based emotion recognition. 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IEEE, 2022, pp. 6922-6926.

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{'abstract': 'The lack of data and the difficulty of multimodal fusion have always been challenges for multimodal emotion recognition (MER). In this paper, we propose to use pretrained models as upstream network, wav2vec 2.0 for audio modality and BERT for text modality, and finetune them in downstream task of MER to cope with the lack of data. For the difficulty of multimodal fusion, we use a K-layer multi-head attention mechanism as a downstream fusion module. Starting from the MER task itself, we design two auxiliary tasks to alleviate the insufficient fusion between modalities and guide the network to capture and align emotion-related features. Compared to the previous state-of-the-art models, we achieve a better performance by 78.42% Weighted Accuracy (WA) and 79.71% Unweighted Accuracy (UA) on the IEMOCAP dataset.', 'arxivid': '2302.13661', 'author': ['Dekai Sun \nHarbin Institute of Technology\nHarbinChina\n', 'Yancheng He \nHarbin Institute of Technology\nHarbinChina\n', 'Jiqing Han \nHarbin Institute of Technology\nHarbinChina\n'], 'authoraffiliation': ['Harbin Institute of Technology\nHarbinChina', 'Harbin Institute of Technology\nHarbinChina', 'Harbin Institute of Technology\nHarbinChina'], 'corpusid': 257219615, 'doi': '10.1109/icassp49357.2023.10096586', 'github_urls': [], 'n_tokens_mistral': 7238, 'n_tokens_neox': 6203, 'n_words': 3538, 'pdfsha': 'c4738cb67a019ec97936e736146aff6094c48308', 'pdfurls': ['https://export.arxiv.org/pdf/2302.13661v1.pdf'], 'title': ['USING AUXILIARY TASKS IN MULTIMODAL FUSION OF WAV2VEC 2.0 AND BERT FOR MULTIMODAL EMOTION RECOGNITION', 'USING AUXILIARY TASKS IN MULTIMODAL FUSION OF WAV2VEC 2.0 AND BERT FOR MULTIMODAL EMOTION RECOGNITION'], 'venue': []}

arxiv

Diagram of High-Energy Nuclear Collisions * 13 Mar 2023 Evgeny Andronov St. Petersburg State University St. PetersburgRussia Magdalena Kuich University of Warsaw WarsawPoland Marek Gazdzicki Geothe-University Frankfurt am Main Germany Jan Kochanowski University KielcePoland Diagram of High-Energy Nuclear Collisions * 13 Mar 2023* This paper is based on work performed before February 24th 2022. 1numbers: 2575q2575Nq2460Ky Keywords: high-energy collisionsstrongly interacting matterquark-gluon plasmastringsresonances Many new particles, mostly hadrons, are produced in high-energy collisions between atomic nuclei.The most popular models describing the hadron-production process are based on the creation, evolution and decay of resonances, strings or quark-gluon plasma. The validity of these models is under vivid discussion, and it seems that a common framework for this discussion is missing. Here, for the first time, we explicitly introduce the diagram of high-energy nuclear collisions, where domains of the dominance of different hadron-production processes in the space of laboratory-controlled parameters, the collision energy and nuclear-mass number of colliding nuclei are indicated. We argue that the recent experimental results suggest the location of boundaries between the domains, allowing for the first time to sketch an example diagram. Finally, we discuss the immediate implications for experimental measurements and model development following the proposed sketch of the diagram. I. INTRODUCTION One of the crucial issues of contemporary physics is understanding strong interactionsthe interactions defining properties of atomic nuclei and collisions between them. Nuclear collisions at high energies lead to the production of many new particles, predominately strongly interacting hadrons. With the advent of the quark model of hadrons and the development of the commonly accepted theory of strong interactions, quantum chromodynamics (QCD) naturally led to expectations that matter at very high densities may exist in a state of quasi-free quarks and gluons, the quark-gluon plasma (QGP) [1]. There are numerous indications that QGP is created in heavy-ion collisions at high energies; for review, see Refs. [2][3][4][5]. The theoretical description of high-energy nuclear collision is not an easy task. This may be attributed to the difficulty of obtaining unique and quantitative predictions from QCD. In particular, even the formation of QGP in heavy-ion collisions is beyond the predictability of QCD. Consequently, the bulk properties of high-energy nuclear collisions are described by phenomenological models. Over time, three classes of them gained in popularity: (i) One postulates that the final hadronic state emerges from the quark-gluon plasma's creation, evolution, and hadronisation [6]. A key input -the QGP equation of state -can be estimated using lattice-QCD calculations [7]. This process will be labelled as QGP ; (ii) One assumes hadrons originate from the formation, evolution and fragmentation of strings -the gluon fields between a pair of colour charges forming a narrow flux tube [8]. Strings are typically oriented along the collision axis, and they have a continuous masses spectrum. Symmetries and experimental results are used to determine model parameters. This process will be labelled as strings; (iii) One describes the production of final state hadrons by creation, evolution and decay of hadronic resonances [9] -excited states of stable hadrons. Resonances do not have a preferred elongation direction and have a discrete mass spectrum. Experimental results are used to determine model parameters. This hadron-production process will be labelled as resonances. Do the processes reflect reality? If yes, what are the domains of their applicability? Answer-2 ing these questions is directly related to understanding intriguing changes of hadron-production properties observed experimentally by varying collision energy and the mass number of colliding nuclei. This task goes hand in hand with selecting measurable quantities sensitive to a transition between the processes. In this paper, we focus the discussion on the ratio of positively charged kaons and pions measured at mid-rapidity, the K + /π + ratio. This measure can be interpreted as a good approximation of the strange to non-strange quarks ratio. Due to mass and number differences between strange and non-strange particles (quarks and gluons or hadrons), the ratio is expected to be sensitive to the hadron-production process [10,11] -it is expected to be sensitive to a changeover between different processes. With the above and the availability of the rich experimental data, the choice of the K + /π + ratio as the subject of this paper was most suitable. For a quantitative comparison of the experimental results with model predictions, we selected PHSD [12,13] and SMASH [14,15] models. This is motivated by their important features. Both models give predictions in the full range collision energy and masses of the colliding nuclei covered by the experimental data. The SMASH model includes resonances and strings, whereas the PHSD model also includes QGP. We review the experimental results and suggest the first answers to the questions asked in Section II. Section III introduces the diagram of high-energy nuclear collisions, and we summarise our findings in a diagram sketch. Finally, we discuss the implications following the sketch for experimental measurements and developing models. II. GUIDING IDEAS AND EXPERIMENTAL RESULTS Heavy-ion collisions. The richest experimental results on the collision energy dependence of hadron-production properties concern collisions between two heavy atomic nuclei, Pb+Pb and Au+Au collisions. Over the last 40 years, they were recorded in the hunt for QGP and the energy threshold Refs. [2][3][4][5]). The most popular plot illustrating this assessment is presented in Figure 1 (left). It shows the collision energy dependence of the K + /π + ratio in central heavy-ion collisions. The ratio shows the so-called horn structure. Following a fast rise, the ratio passes through a maximum in the CERN SPS energy range, at approximately 8 GeV, then decreases and settles to a plateau which continues up to the CERN LHC energies. Kaons are the lightest strange hadrons, and due to approximate isospin symmetry, the K + yield counts about half of the strange quarks produced in the collisions and contained in the reaction products [11]. Thus, Figure 1 (left) demonstrates that the fraction of strangeness carrying particles in the produced matter passes through a sharp maximum at the SPS energy range in central heavy-ion collisions; for a detailed explanation; see Ref. [4]. The standard modelling of heavy-ion collisions [6] includes the formation of high-density matter (be it QGP or hadronic matter) at the early stage of a collision, its expansion and the decoupling of hadrons that freely stream to particle detectors. A statistical description of the early stage [11] led to predictions of the collision energy dependence of bulk hadron production properties. In particular, the horn structure was predicted as the signal of the onset of deconfinement. In the model, it reflects the decrease in the ratio of strange to nonstrange degrees of freedom when deconfinement sets in. Experimental data are compared with calculations of the PHSD model [12,13] that incorporates the QGP creation at sufficiently high densities and chiral-symmetry restoration in the dense hadronic matter. The model catches the basic properties of the data; see Figure 1 (left). This further supports interpreting the horn maximum at √ s N N ≈ 8 GeV as the beginning of the QGP creation. Moreover, the SMASH model [14,15], which does not include the QGP creation qualitatively, fails to reproduce the results; see Figure 1 (left). One should, however, note that there are significant uncertainties in modelling both production processes; see below for an example. Proton-proton interactions. Measurements of proton-proton interactions started long before the first experiments studying heavy-ion collisions. The primary goal of the study of p+p interactions was understanding (iii) At the LHC energies, the p+p ratio is about 20% lower than the heavy-ion one. The most popular modelling of proton-proton interactions at high energies includes strings' formation, evolution, and fragmentation. The widely used approaches are the Lund [34], EPOS [8] and Dual Parton [35,36] models. At low collision energies, the validity of the string approach breaks, and one replaces it with the creation of resonances and their decay; for a detailed explanation, see Ref. [15]. These two processes are implemented in the PHSD [12,13] and SMASH [14,15] models. Their predictions for the collision energy dependence of the K + /π + ratio in p+p interactions are shown in Figure 1 (right). Significant differences between them shed light on the uncertainty of the predictions. Taking into account this uncertainty, one concludes that the models reproduce the bulk properties of the data. The effect of the changeover from resonances to strings (onset of strings) was studied in detail within the UrQMD model [37,38]. Within SMASH [14,15], the changeover causes a wiggle in the collision energy dependence of the K + /π + ratio, which can be seen in Figure 1 (right) by enlarging the plot. In PHSD [12,13], a sharp transition is located at √ s N N ≈ 2.6 GeV -close to the threshold for kaon production-and thus its effect on the ratio is hard to observe. The open question discussed in Ref. [33] is whether the break (ii) in the collision energy dependence of the experimental ratio at √ s N N ≈ 8 GeV is due to the onset of strings or is related to the onset of deconfinement. One notes the following regarding the similarity of the ratio in p+p and Pb+Pb collisions at LHC (iii). It was reported that relative strange hadron yields in p+p interactions at LHC smoothly increase with increasing charged-particle multiplicity and for high multiplicity interactions are close to those in Pb+Pb collisions [39]. Moreover, recent LHC data on the azimuthal angle distribution of charged particles in high multiplicity p+p interactions [40][41][42] show anisotropies up to the recently observed only in heavy-ion collisions and attributed to the hydrodynamical expansion of matter [43]. This suggests that QGP may also be produced in p+p 6 interactions at the LHC energies, at least in collisions with sufficiently high hadron multiplicity. Collisions of intermediate-mass nuclei. The collision-energy dependence of hadron-production properties in collisions of intermediate-mass nuclei is the least established one. The only systematic measurements have been performed at the CERN SPS by NA61/SHINE [44]. They were motivated by a search for the critical point of strongly interacting matter and a need to establish the nuclear mass dependence of the horn structure [45]. [16][17][18][19][20][21][22][23][24][25][26] and inelastic p+p [19,[27][28][29][30][31] interactions, results on central Be+Be [46], C+C [47,48], Si+Al [49], Si+Si [47,48], Ar+Sc (preliminary) [50,51], Ni+Ni [52,53] and Xe+Xe [54] collisions are shown. Points presenting Pb+Pb/Au+Au and p+p are plotted in pale colours to emphasize intermediate-mass nuclei results. A different system-size dependence is predicted within statistical models of nucleus-nucleus collisions. The strangeness conservation imposed on the whole system leads to a fast increase of the ratio with increasing system size to its upper limit given by the grand-canonical-ensemble approximation. The effect is referred to as canonical strangeness suppression and has been extensively studied since 1980; see, e.g., Refs. [55][56][57]. The PHSD model predictions shown in Figure 3 show a gradual ratio increase with W . However, in this model, the change is likely to be also caused by smoothly increasing contributions from QGP and chiral symmetry restoration. The PHSD model describes the main properties of the data significantly better. tions [27] and central Be+Be [46], C+C [47], Si+Si [47], Ar+Sc (preliminary) [50,51], Au+Au [22], Pb+Pb [16,17] collisions. Experimental results were compared with the PHSD [12,13] (open crosses) and SMASH [14,15] (open circles) predictions. Lines are plotted to guide the eye. However, it fails to reproduce the jump between the results for p+p and Be+Be collisions and the results for heavier nuclei at √ s N N ≈ 17 GeV; see Figure 3 (right). With increasing collision energy and nuclear mass number of colliding nuclei, the number of produced strings and their density is expected to increase. The idea that, at sufficiently high densities, the strings would be close enough to interact and change their properties has been developing over the last 40 years. Many approaches have been proposed, in particular, colour ropes [58], string fusion [59][60][61][62][63], core formation [64], string melting [65] and percolation [66,67]. A model that explicitly involves the rapid string-QGP changeover was proposed recently. It is a string collapse pictured as the black hole formation using the AdS/CFT duality [68][69][70]. Thus, it is natural to interpret the jump as due to a rapid changeover from strings to QGP. This changeover is called the onset of QGP fireball. The gradual increase of the ratio at low collision energies (see Figure 3 (left)) is also not reproduced by the models. This can be due to the (i) Approaching equilibrium with increasing system size and evolution time; (ii) Weakening of the canonical strangeness suppression with increasing system size; (iii) Increasing role of chiral symmetry restoration in dense hadronic matter. III. DIAGRAM OF HIGH-ENERGY NUCLEAR COLLISIONS Here, for the first time, we explicitly introduce a concept of the diagram of high-energy nuclear collisions and then, based on the experimental data and ideas discussed above, sketch its example version. To sketch the example diagram, the hadron-production processes discussed above are selected: (i) Creation, evolution and decay of resonances; (ii) Formation, evolution and fragmentation of strings; (iii) Creation, evolution and hadronisation of QGP. In addition, based on the discussion of the experimental results presented in the previous section, we assume that (i) The Pb+Pb horn locates the resonances-QGP changeover at √ s N N ≈ 8 GeV; (ii) The p+p break locates the resonances-strings changeover at √ s N N ≈ 8 GeV; (iii) The jump between p+p/Be+Be and Ar+Sc/Pb+Pb plateaus locates the strings-QGP changeover at √ s N N ≈ 17 GeV; 10 (iv) The LHC p+p data imply QGP creation in (high multiplicity) p+p interactions at sufficiently high (order of 1 TeV) energies. The diagram of high-energy nuclear collisions following these assumptions is sketched in Thus the string domain disappears, and one observes direct resonances-QGP changeover. This locates the resonances-QGP changeover at the energy of the resonances-strings one. (ii) It is interesting to consider other diagrams of high-energy collisions. Here, we discuss a simple example of the hadron-resonance gas diagram. Hagedorn's early papers postulated that hadrons in high-energy collisions are produced according to statistical thermodynamics [71]. Thus, following Hagedorn's postulate, the diagram would include only one production process -the statistical-thermodynamical production, with Hagedorn's temperature T H ≈ 150 MeV. This model is clearly in contradiction with the experimental results, as it predicts the K + /π + ratio to be independent of energy and nuclear mass number of colliding nuclei. Over the years, the simple Hagedorn approach evolved into many models that are much more flexible in fitting the data; for a recent review, see Ref. [72]. In particular, it has been popular to fit mean hadron multiplicities, which include multiplicities of kaons and pions, assuming that a hadron gas in equilibrium is created at high-energy collisions. The temperature, the baryon chemical potential, and the gas volume are free parameters of the model and are fitted to the data from each reaction separately. The model cannot predict the energy and nuclear mass dependence of hadron production in this formulation. Thus, it is unsuitable for the diagram construction. To verify the assumptions and the diagram sketched on Figure 4, further analysis of the existing data and new experimental measurements as well as the development of models is needed. Concerning modelling, there is a need for the development of dynamical models that include all three production processes. In this paper, these models are represented by PHSD [12,13] which reproduces experimental results significantly better than the SMASH model [14,15]. The latter includes only two approaches to hadron production, resonances and strings. Still, the PHSD model misses important features of the experimental data shown in Figures 1 and 3. One must reconsider the nature of the changeover between different processes to improve predictions. Concerning the further analysis of the existing data, one should extend the presented studies to other quantities which characterise hadron production in high-energy nuclear collisions. In particular, quantities sensitive to the collective flow of matter, radial and anisotropic should be sensitive to the production mechanisms discussed. This important study goes beyond the scope of this introductory paper. Finally, concerning the new experimental measurements, data on light and medium mass nuclei collisions are needed-in particular, a precision system-size dependence to locate the strings-QGP changeover. Such a study was launched by NA61/SHINE at the CERN SPS, and its continuation is considered in the following years [73,74]. It would be important to perform the corresponding measurements in the full range of available energies, from the FAIR SIS-100 through NICA and SPS to CERN LHC energies. In 2024, a beam of oxygen ions is considered at the SPS and LHC in CERN [74,75], making a good start for further study. Prospects of studies with the intermediate-mass nuclear beams (e.g., Ar+Ar or Kr+Kr) at LHC energies are also vividly discussed [76]. of its creation -the onset of deconfinement. Many fixed-target and collider experiments in the US (Lawrence Berkeley Laboratory, LBL, and Brookhaven National Laboratory, BNL) and European (European Organization of Nuclear Research, CERN and Helmholtz Centre for 3 Heavy Ion Research, GSI) laboratories have been conducting the measurements. The results are consistent with the onset of deconfinement being located at ( √ s N N ≈ 8 GeV) and the QGP being created at the early stage of heavy-ion collisions at higher collision energies (for review, see Figure 1 : 1Collision energy dependence of the K + /π + multiplicity ratio at mid-rapidity in central heavy-ion collisions (Pb+Pb[16][17][18][19] and Au+Au[20][21][22][23][24][25][26]) (left) and in inelastic p+p interactions[19,[27][28][29][30][31] (right). Open cross points present the PHSD[12, 13] predictions, while open circles -the SMASH[14,15] predictions. Lines connecting the points are plotted to guide the eye. strong interactions. With increasing collision energy, more and heavier hadrons have been produced. To understand the early results, the string model was invented[32]. The experimental results also contributed to the formulation of QCD -the nowadays commonly accepted theory of strong interactions. While there is no first-principles derivation of strings from QCD, some properties of a string can be derived from QCD.Paradoxically, QCD demotivated studies of bulk properties of p+p interactions. This is because of difficulties in obtaining unique and quantitative predictions from QCD. Consequently, the world data on p+p interactions are not as rich as the corresponding results on heavy-ion collisions. The compiled data on the K + /π + ratio at mid-rapidity in inelastic p+p interactions is shown inFigure 1(right). Precise measurements are available at the CERN SPS and LHC energies. The p+p results still allow for important observations:(i) At the SPS energies, the ratio in p+p interactions is about a factor of two lower N N ≈ 8 GeV, a break in the collision energy dependence of the ratio is observed in p+p interactions instead of the horn seen in heavy-ion collisions. For a more detailed analysis of the p+p break, see Ref.[33]; The data on collisions of intermediate-mass nuclei are summarised in Figure 2. The results of K + /π + ratio for central Pb+Pb/Au+Au and inelastic p+p are also plotted for comparison in a light colour. The main observations are: (i) The ratio in Be+Be collisions is similar to the one in p+p interactions in the whole SPS energy range;(ii) There is no horn structure in Ar+Sc collisions;(iii) The ratio in Ar+Sc collisions at the top SPS energy is similar to the one in Pb+Pb collisions. Figure 3 3shows results on the K + /π + ratio measured at √ s N N ≈ 7.7 GeV (the left plot) and √ s N N ≈ 17 GeV (the right plot) as a function of the mean number of nucleons that participated in inelastic interactions, the so-called number of wounded nucleons W . Since,, at high collision energies, the ratio is weakly dependent on the collision energy, results from central Au+Au collisions at √ s N N = 19.6 GeV were also included in the plot.Let us start a discussion of model predictions concerning the system-size dependence of the ratio from the string models. For simplicity of the arguments, we assume that the string formation, evolution and fragmentation is independent of W . The yields of K + and π + mesons are proportional to the mean number of strings, and consequently, their ratio is independent of the mean number of strings, and thus it is independent of W . The predictions of the SMASH model shown inFigure 3approximately follow this naive expectation. Figure 2 : 2Status of experimental results on energy dependence of the K + /π + ratio at mid-rapidity in high-energy nuclear collisions. In addition to the previously shown results on central heavyion collisions Figure 3 : 3The K + /π + ratio at mid-rapidity measured at √ s N N ≈ 7.7 GeV (left) and √ s N N ≈ 17 GeV (right) as a function of a mean number of wounded nucleons, W , in inelastic p+p interac- The diagram of high-energy nuclear collisions is defined as a plot showing domains of the dominance of different hadron-production processes in high-energy nuclear collisions. The domains are indicated in the space of laboratory-controlled parameters, the collision energy and the nuclear-mass number of colliding nuclei. For simplicity, we consider only central nucleusnucleus collisions -collisions in which a large fraction of nucleons participated in inelastic interactions ( W /A ≈ 1). Figure 4 Figure 4 : 44Schematic diagram of high-energy nuclear collisions outlined in colliding nuclei mass number, A, and collision energy, √ s N N variables. Domains in which hadron production is dominated by the creation, evolution and decay of resonances, strings and quark-gluon plasma are indicated as resonances, strings and QGP, respectively, while thick coloured lines show the changeover regions between the domains. Two comments are in order here. (i) The changeover resonances-strings and resonances-QGP are located at similar collision energies (≈8 GeV/c). This suggests that the resonances-QGP changeover is driven by the resonances-strings one. At high masses of colliding nuclei, strings produced above at the resonances-strings changeover would have density exceeding the strings-QGP changeover. AcknowledgmentsThe authors thank Mark Gorenstein and Edward Shuryak as well as members of the NA61/ Quantum Chromodynamics and the Theory of Superdense Matter. E V Shuryak, Phys. Rept. 61E. V. Shuryak, "Quantum Chromodynamics and the Theory of Superdense Matter," Phys. Rept., vol. 61, pp. 71-158, 1980. 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{'abstract': 'Many new particles, mostly hadrons, are produced in high-energy collisions between atomic nuclei.The most popular models describing the hadron-production process are based on the creation, evolution and decay of resonances, strings or quark-gluon plasma. The validity of these models is under vivid discussion, and it seems that a common framework for this discussion is missing. Here, for the first time, we explicitly introduce the diagram of high-energy nuclear collisions, where domains of the dominance of different hadron-production processes in the space of laboratory-controlled parameters, the collision energy and nuclear-mass number of colliding nuclei are indicated. We argue that the recent experimental results suggest the location of boundaries between the domains, allowing for the first time to sketch an example diagram. Finally, we discuss the immediate implications for experimental measurements and model development following the proposed sketch of the diagram.', 'arxivid': '2205.06726', 'author': ['Evgeny Andronov \nSt. Petersburg State University\nSt. PetersburgRussia\n', 'Magdalena Kuich \nUniversity of Warsaw\nWarsawPoland\n', 'Marek Gazdzicki \nGeothe-University Frankfurt am Main\nGermany\n\nJan Kochanowski University\nKielcePoland\n'], 'authoraffiliation': ['St. Petersburg State University\nSt. PetersburgRussia', 'University of Warsaw\nWarsawPoland', 'Geothe-University Frankfurt am Main\nGermany', 'Jan Kochanowski University\nKielcePoland'], 'corpusid': 248798716, 'doi': '10.3390/universe9020106', 'github_urls': [], 'n_tokens_mistral': 14727, 'n_tokens_neox': 12256, 'n_words': 7015, 'pdfsha': 'f080537d65535346c5e5bb9638c42315e2c382a2', 'pdfurls': ['https://export.arxiv.org/pdf/2205.06726v5.pdf'], 'title': ['Diagram of High-Energy Nuclear Collisions *', 'Diagram of High-Energy Nuclear Collisions *'], 'venue': []}

arxiv

Bad Habits: Policy Confounding and Out-of-Trajectory Generalization in RL Miguel Suau Delft University of Technology Matthijs T J Spaan Delft University of Technology Frans A Oliehoek f.a.oliehoek@tudelft.nl Delft University of Technology Bad Habits: Policy Confounding and Out-of-Trajectory Generalization in RL Reinforcement learning agents may sometimes develop habits that are effective only when specific policies are followed. After an initial exploration phase in which agents try out different actions, they eventually converge toward a particular policy. When this occurs, the distribution of state-action trajectories becomes narrower, and agents start experiencing the same transitions again and again. At this point, spurious correlations may arise. Agents may then pick up on these correlations and learn state representations that do not generalize beyond the agent's trajectory distribution. In this paper, we provide a mathematical characterization of this phenomenon, which we refer to as policy confounding, and show, through a series of examples, when and how it occurs in practice.Preprint. Under review. Introduction This morning, I went to the kitchen for a coffee. When I arrived, I forgot why I was there, so I got myself a coffee-How often do you do something without paying close attention to your actions? Have you ever caught yourself thinking about something else while washing the dishes, making coffee, or cycling? Acting out of habit is a vital human skill as it allows us to concentrate on more important matters while carrying out routine tasks. You can commute to work while thinking about how to persuade your boss to give you a salary raise or prepare dinner while imagining your next holidays in the Alps. However, unlike in the above example, habits can also lead to undesired outcomes when we fail to recognize that the context has changed. You may hop in your car and start driving towards work even though it is a Sunday and you actually want to go to the grocery store, or you may flip the light switch when leaving a room even though the lights are already off. Here we show how reinforcement learning (RL) agents may also suffer from this phenomenon. Agents can exploit spurious correlations (Pearl et al., 2016) between observed variables and rewards to build simple habits that require little effort to carry out. Such correlations are induced by the agent's policy and hence can be relied upon so long as said policy is followed consistently. However, as we shall see, even minor trajectory deviations can result in catastrophic outcomes. Ideally, the agent should only pick up on correlations that are stable across policies. That is, independently of the trajectories being followed. We refer to this objective as out-of-trajectory (OOT) generalization. Contributions This paper characterizes policy confounding, a term we use to name the abovedescribed phenomenon. To do so, we introduce a mathematical framework that helps us investigate different types of state representations. Moreover, we provide a series of clarifying examples that illustrate how, as a result of policy confounding, the agent may learn representations based on spurious correlations that do not guarantee OOT generalization. Unfortunately, we do not have a complete answer for how to prevent policy confounding. However, we suggest a few off-the-shelf solutions that may help mitigate its effects. We hope this paper will create awareness among the RL community about the risks of policy confounding and inspire further research on this topic. Example: Frozen T-Maze We now provide an example to illustrate the phenomenon of policy confounding and motivate the need for careful analysis. The environment shown in Figure 1 is a variant of the popular T-Maze environment (Bakker, 2001). The agent receives a binary signal, green or purple, at the start location. Then, it needs to move to the right and reach the correct goal at the end of the maze (ignore the blue cells and the black vertical arrow in the middle of the maze for now). The agent obtains a reward of +1 for moving to the green (purple) goal when having received the green (purple) signal and a reward of −1 otherwise. At first sight, one may think that the only way the agent can solve the task is if, at every cell along its trajectory, it can recall the initial signal. However, once the agent figures out the shortest path to each of the two goals (depicted by the green and purple arrows), the agent may safely forget the initial signal. The agent knows that whenever it is at any of the cells along the green (purple) path, it must have received the green (purple) signal. Hence, it can simply move toward the right goal on the basis of its own location. Sticking to this habit is optimal so long as the agent commits to always taking these two paths. 1 It is also essential that the environment's dynamics remain the same since even the slightest change in the agent's trajectories may erase the spurious correlation induced by the agent's policy between the agent's location and the correct goal. To show that this actually occurs in practice, we train agents in the original environment (train env) and evaluate them on a variant of the same (eval env), where some ice (blue) has appeared in the middle of the maze. The ice makes the agent slip from the upper cell to the bottom cell and vice versa. The plot on the right of Figure 1 shows the return averaged over 10 trials. The performance drop in the evaluation environment (blue curve) suggests that the agents' policies do not generalize. The ice confuses the agents, who, after being pushed away from their preferred trajectories, can no longer select the right goal. More details about this experiment are provided in Section 7. Related Work The presence of spurious correlations in the training data is a well-studied problem in machine learning. These correlations often provide convenient shortcuts that a model can exploit to make predictions (Beery et al., 2018). However, the performance of a model that relies on them may significantly deteriorate under different data distributions (Quionero-Candela et al., 2009;Arjovsky, 2021). Langosco et al. (2022) show that RL agents may use certain environment features as proxies for choosing their actions. These features, which show only in the training environments, happen to be spuriously correlated with the agent's objectives. In contrast, we demonstrate that, as a result of policy confounding, agents may directly take part in the formation of spurious correlations. A few prior works have already reported empirical evidence of particular forms of policy confounding, showing that in deterministic environments, agents can rely on information that correlates with the agent's progress in an episode to determine the optimal actions. This strategy is effective because under fixed policies, features such as timers (Song et al., 2020), agent's postures (Lan et al., 2023), or previous action sequences (Machado et al., 2018) can be directly mapped to the agent's state. These works provide various hypotheses to justify their experimental observations. Here, we contribute an overarching theory that explains the underlying causes and mechanisms behind these results, along with a series of examples illustrating other types of policy confounding. Please refer to Appendix C for more details on related work. Preliminaries Although, as we shall see in the experiments, policy confounding can occur even when states are fully observable, in order to understand the idea, it is useful to formulate the setting as partially observable (Kaelbling et al., 1996). Moreover, since we model values and policies using (parametric) functions rather than tables, we use state variables or state factors to represent the different states of the environment (Boutilier et al., 1999). Definition 1 (FPOMDP). A factored partially observable Markov decision process (FPOMDP) is a tuple ⟨S, F, A, T, R, O, X, Y ⟩ where S is the set of states, F is the set of state variables (or state factors) F = {f 1 , ..., f l } so that every state s t ∈ S = × l i=1 f i is represented as a vector s = ⟨f 1 , ..., f l ⟩, A is the set of actions a t , T (s t+1 | s t , a t ) is the transition probability function, R(s t , a t ) is the reward function which determines the immediate reward r t , and O(s t ) is the observation or emission function, which selects a subset of observed variables X t ⊆ F (which may be different depending on the state s t ), and discards the hidden variables Y t = F \ X t , such that the agent's observations o t ∈ × mt i=1 X t are represented as vectors o t = ⟨x 1 t , ..., x mt t ⟩ with m t ≤ l. In this setting, the agent must keep track of past actions and observations to make the right action choices (Singh et al., 1994). The optimal policy is a mapping from the past action-observation history, h t = ⟨o 1 , a 1 , ..., a t−1 , o t ⟩, to a probability distribution ∆(A) over actions A, π : H → ∆(A), where H is the set of all possible histories of any length. We use the random variable τ = ⟨o 1 , a 1 , ..., a T −1 , o T ⟩ to denote the agent's trajectory in an episode, with T being the episode's horizon. Knowing that the full history constitutes a Markov representation, we can reformulate the FPOMDP into a factored history MDP (FHMDP). Definition 2 (FHMPD). A factored history Markov decision process (FHMDP) is a tuple ⟨H, Θ, A, T h , R h ⟩, where H is the set of all possible histories of any length, Θ denotes the set of variables in the history, with Θ t denoting the set of actions A and observation variables X in a history of length t, Θ t = {x 1 1 , ..., x m1 1 , a 1 , ..., x 1 t , ..., x mt t , a t }, such that we write their Cartesian product, H t = {x 1 1 × ... × x m1 1 × a 1 × ... × x 1 t × ... × x mt t × a t }, simply as H t = ×Θ t , T h (ht+1 = ⟨ht, at, ot+1⟩ | ht, at) ≜ s t+1 ,s t ∈S O(st+1)T (st+1 | st, at) Pr(st | ht) is the history transition function, 2 and R h (ht, at) ≜ s t ∈S R(st, at) Pr(st | ht) is the history reward function. This formulation is convenient because it allows solving the POMDP using MDP methods. Yet, due to combinatorial explosion, learning a policy that conditions on the full history is generally infeasible. Fortunately, in many problems, not all the information is strictly relevant; the agent can usually find compact representations of the history, that are sufficient for solving the task (McCallum, 1995). History representations Factored representations are useful because they readily define relationships between (states) histories. Histories can be compared to one another by looking at the individual values the different variables take. Removing some of the variables in Θ t has the effect of grouping together those histories that share the same values for the remaining ones. Thus, in contrast with most of the theoretical work in RL, which treats histories (states) as independent entities, we can define history (state) abstractions at the variable level instead of doing so at the history (state) level (Li et al., 2006). Definition 3 (History representation). A history representation is a function Φ : H t →H t , with H t = ×Θ t ,H t = ×Θ t , andΘ t ⊆ Θ t . Intuitively a history representation Φ(h t ) is a context-specific projection of a history h t ∈ H t = ×Θ t onto a lower dimensional spaceH t = ×Θ t defined by a subset of its variables,Θ t ⊆ Θ t . We use {h t } Φ = {h ′ t ∈ H t : Φ(h ′ t ) = Φ(h t )} to denote h t 's equivalence class under Φ. Markov history representations As noted in Section 4, the agent should strive for history representations with few variables. Yet, not all history representations will be sufficient to learn the optimal policy; some may exclude variables that contain useful information for the task at hand. Definition 4 (Markov history representation). A history representation Φ(h t ) is said to be Markov if, for all h t , h t+1 ∈ H, a t ∈ A, R h (ht, at) = R h (Φ(ht), at) and h ′ t+1 ∈{h t+1 } Φ T h (h ′ t+1 | ht, at) = Pr(Φ(ht+1) | Φ(ht), at), where R h (Φ(h t ), a t ) = {R(h ′ t , a t )} h ′ t ∈{ht} Φ is the reward at any h ′ t ∈ {h t } Φ . The above definition is equivalent to the notion of bisimulation (Dean and Givan, 1997;Givan et al., 2003) or model-irrelevance state abstraction (Li et al., 2006). Representations satisfying these conditions are guaranteed to be equivalent to the original representation. That is, for any given policy and initial history, the expected return (i.e., cumulative reward; Sutton and Barto, 2018) is the same when conditioning on the full history or on the Markov history representation. Note that a history representation Φ such that Φ(h t ) = h t , for all h t ∈ H, is, in itself, Markov. Definition 5 (Minimal history representation). A history representation Φ * : H t →H * t with H * t = ×Θ * t is said to be minimal, if all other history representations Φ : H t →H t withH t = ×Θ t and |Θ t | ⊂ |Θ * t |, for at least one h t ∈ H, are not Markov. In other words, Φ * t (h t ) is minimal when none of the remaining variables can be removed while the representation remains Markov. Hence, we say that a minimal history representation Φ * t (h t ) is a sufficient statistic of the full history. Definition 6 (Superfluous variable). Let {Θ * t } ∪Φ * be the union of variables in all possible minimal history representations. A variable Θ i t ∈ Θ t is said to be superfluous, if Θ i t / ∈ {Θ * t } ∪Φ * . π-Markov history representations Considering that the agent's policy will rarely visit all possible histories, the notion of Markov history representation seems excessively strict. We now define a relaxed version that guarantees the representation to be Markov when a specific policy π is followed. Definition 7 (π-Markov history representation). A history representation Φ π (h t ) is said to be π-Markov if, for all h t , h t+1 ∈ H π , a t ∈ supp(π(· | h t )), Policy Confounding We are now ready to describe how and when policy confounding occurs, as well as why we should care, and how we should go about preventing it. The proofs for all theoretical results are deferred to Appendix A. Policy confounding arises naturally as the agent improves its policy. Normally, at the beginning of training, the agent takes exploratory actions to determine which ones yield high rewards. It is only after the agent has committed to a particular policy that we start seeing how some of the variables in its history become irrelevant for predicting future states and rewards. The agent may then choose to ignore these variables and exclude them from its representation if keeping them takes extra 'effort'. The next result demonstrates that a π-Markov history representation Φ π requires at most the same variables, and in some cases fewer, than a minimal history representation Φ * , while still satisfying the Markov conditions for those histories visited under π, h t ∈ H π . Proposition 1. Let Φ * be the set of all possible minimal history representations, where every Φ * ∈ Φ * is defined as Φ * : H t →H * t withH * t = ×Θ * t . For all π and all Φ * ∈ Φ * , there exists a π-Markov history representation Φ π : H π t →H π t withH π t = ×Θ π t such that for all h t ∈ H π , Θ π t ⊆Θ * t . Moreover, there exist cases for whichΘ π t is a proper subset,Θ π t ̸ =Θ * t . Although the result above seems intuitive, its truth may appear incidental. While it is clear that Φ π will never require more variables than the corresponding minimal history representation Φ * , whether or not Φ π will require fewer, seems just an arbitrary consequence of the policy being followed. Moreover, since the variables inΘ * t are all strictly relevant for predicting transitions and rewards, one may think that a policy π inducing representations such thatΘ π t ⊂Θ * t can never be optimal. However, as shown by the following example, it turns out that the histories visited by a particular policy, especially if it is the optimal policy, tend to contain a lot of redundant information. This is particularly true in environments where future observations are heavily influenced by past actions and observations. In such cases, the current observation often reveals a lot about the agent's trajectory. Example 1. (Frozen T-Maze) Let us consider the Frozen T-Maze again (Section 2). Figure 3 shows a dynamic Bayesian network (DBN; Murphy, 2002) describing the dynamics of the environment. Observation variables are denoted by x, while hidden variables are denoted by y. The nodes labeled as x 2 represent the agent's location from t = 0 to t = 8. All intermediate nodes between t = 0 and t = 7 are omitted for simplicity. The nodes labeled as y indicate whether the goal is to go to the green or the purple cell (see Figure 1). Note that y always takes the same value at all timesteps within an episode (either green or purple). The information in y is hidden and only passed to the agent at the start location through the node x 1 0 . On the one hand, if actions are not specified by any particular policy, but simply sampled at random (left diagram), to determine the reward r 8 at t = 8, one needs to know the signal x 1 0 received at t = 0 and the agent's current location x 2 8 . These are highlighted by the green circles in the left DBN. This is because the actions ⟨a 0 , ..., a 7 ⟩ appear as exogenous variables and can take any possible value. Hence, the reward could be either −0.1, (per timestep penalty), −1 (wrong goal), or +1 (correct goal) depending on the actual values of x 1 1 and x 2 8 . On the other hand, when actions are sampled from the optimal policy π * (right DBN), knowing x 2 8 (green circle) is sufficient to determine r 8 . In this second case, π * makes the action a 0 , and thus all future agent locations, dependent on the initial signal x 1 0 . This occurs because, under the optimal policy (green and purple paths in Figure 1), the agent always takes the action 'move up' when receiving the green signal or 'move down' when receiving the purple signal, and then follows the shortest path towards each of the goals. As such, we have that, from t = 1 onward, Φ π * (h t ) = x 2 t is a π-Markov history representation since it constitutes a sufficient statistic of the history h t under π * . Finally, note that, for the same reason, from t = 1, actions may also condition only on x 2 . The phenomenon highlighted by the previous example is the result of a spurious correlation induced by the optimal policy between the agent's locations ⟨x 2 0 , ..., x 2 8 ⟩ and the reward r 8 . Generally speaking, this occurs because policies act as confounders, opening backdoor paths between future histories/rewards and the variables in the current history h t (Pearl, 2000). This is shown by the DBN depicted in Figure 9, where we see that the policy influences both the current history and also future histories/rewards, hence potentially affecting the conditional relationships between some of their variables. For instance, in the above example, R π * (x 2 8 = 'agent at green goal') = +1 when following π * , while for an arbitrary π, R(x 2 8 = 'agent at green goal') = ±1. Definition 9 (Policy Confounding). A history representation Φ : H t →H t is said to be confounded by a policy π if, for some h t , h t+1 ∈ H, a t ∈ A, R π (Φ(ht), at) ̸ = R π (do(Φ(ht)), at) or Pr π (Φ(ht+1) | Φ(ht), at) ̸ = Pr π (Φ(ht+1) | do(Φ(ht)), at) The operator do(·) is known as the do-operator, and it is used to represent physical interventions in a system (Pearl, 2000). These interventions are meant to distinguish cause-effect relations from mere statistical associations. In our case, do(Φ(h t )) means setting the variables forming the history representation Φ(h t ) to a particular value and considering all possible histories in the equivalence class, h ′ t ∈ {h t } Φ . That is, independently of what policy is being followed. It is easy to show that the underlying reason why a π-Markov history representation may require fewer variables than the minimal history representation (as in Example 1) is indeed policy confounding. Theorem 1. Let Φ * : H t →H * Leveraging spurious correlations to develop simple habits can be advantageous when resources such as memory, computing power, or data are limited. Agents can disregard and exclude from their representation those variables that are redundant under their policies. However, the challenge is that some of these variables may be crucial to ensure that the agent behaves correctly when the context changes. In the Frozen T-Maze example from Section 2, we observed how the agent could no longer find the correct goal when the ice pushed it away from the optimal trajectory. This is a specific case of a well-researched issue known as out-of-distribution (OOD) generalization (Quionero-Candela et al., 2009;Arjovsky, 2021). We refer to it as out-of-trajectory (OOT) generalization to highlight that the problem arises due to repeatedly sampling from the same policy and thus following the same trajectories. In contrast to previous works (Kirk et al., 2023) that address generalization to environments that differ from the training environment, our objective here is to generalize to trajectories the agent never (or only rarely) takes. 3 Ideally, the agent should aim to learn representations that enable it to predict future rewards and transitions even when experiencing slight variations in its trajectory. Based on Definition 4, we know that, in general, only a Markov history representation satisfies these requirements. However, computing such representations is typically intractable (Ferns et al., 2006), and thus most standard RL methods usually learn representations by maximizing an objective function that depends on the distribution of trajectories P b (τ ) visited under a behavior policy b (e.g., expected return, E τ ∼P b (τ ) [G(τ )]; Sutton and Barto, 2018). The problem is that b may favor certain trajectories over others, which may lead to the exploitation of spurious correlations in the learned representation. When should we worry about OOT generalization in practice? The previous section highlighted the generalization failures of representations that depend on spurious correlations. Now, let us delve into the circumstances in which policy confounding is most prone to cause problems. Function approximation Function approximation has enabled traditional RL methods to scale to high-dimensional problems with long-term memory dependencies, where storing values in lookup tables is infeasible. Using parametric functions (e.g., neural networks) to model policies and value functions, agents can learn abstractions by grouping together histories if these yield the same transitions and rewards. As mentioned before, abstractions occur naturally when histories are represented by a set of variables since the functions simply need to ignore some of these variables. However, this also implies that value functions and policies are exposed to spurious correlations. If a particular variable becomes irrelevant due to policy confounding, the function may learn to ignore it and remove it from its representation (Example 1). This is in contrast to tabular representations, where, every history takes a separate entry, and even though there exist algorithms that perform history (state) abstractions in tabular settings (Andre and Russell, 2002;Givan et al., 2003), these abstractions are normally formed offline before learning (computing) the policy, hence avoiding the risk of policy confounding. Narrow trajectory distributions In practice, agents are less prone to policy confounding when the trajectory distribution P b (τ ) is broad (i.e., when b encompasses a wide set of trajectories) than when it is narrow. This is because the spurious correlations present in certain trajectories are less likely to have an effect on the learned representations. On-policy methods (e.g., SARSA, Actor-Critic; Sutton and Barto, 2018) are particularly troublesome for this reason since the same policy being updated must also be used to collect the samples. Yet, even when the trajectory distribution is narrow, there is no reason why the agent should pick up on spurious correlations while its policy is still being updated. Only when the agent commits to a particular policy should we start worrying about policy confounding. At this point, lots of the same trajectories are being used for training, and the agent may 'forget' (French, 1999) that, even though certain variables may no longer be needed to represent the current policy, they were important under previous policies. This generally occurs at the end of training when the agent has converged to a particular policy. However, if policy confounding occurs earlier during training, it may prevent the agent from further improving its policy (Nikishin et al., 2022; please refer to Section C for more details). What can we do to improve OOT generalization? As mentioned in the introduction, we do not have a complete answer to the problem of policy confounding. Yet, here we offer a few off-the-shelf solutions that, while perhaps limited in scope, can help mitigate the problem in some situations. These solutions revolve around the idea of broadening the distribution of trajectories so as to dilute the spurious correlations introduced by certain policies. Off-policy methods We already explained in Section 6.2 that on-policy methods are particularly prone to policy confounding since they are restricted to using samples coming from the same policy. A rather obvious solution is to instead use off-policy methods, which allow using data generated from previous policies. Because the samples belong to a mixture of policies it is less likely that the model will pick up the spurious correlations present on specific trajectories. However, as we shall see in the experiments, this alternative works only when replay buffers are large enough. This is because standard replay buffers are implemented as queues, and hence the first experiences coming in are the first being removed. This implies that a replay buffer that is too small will contain samples coming from few and very similar policies. Since there is a limit on how large replay buffers are allowed to be, future research could explore other, more sophisticated, ways of deciding what samples to store and which ones to remove (Schaul et al., 2016). Exploration and domain randomization When allowed, exploration may mitigate the effects of policy confounding and prevent agents from overfitting their preferred trajectories. Exploration strategies have already been used for the purpose of generalization; to guarantee robustness to perturbations in the environment dynamics (Eysenbach and Levine, 2022), or to boost generalization to unseen environments (Jiang et al., 2022). The goal for us is to remove, to the extent possible, the spurious correlations introduced by the current policy. Unfortunately, though, exploration is not always without cost. Safety-critical applications require the agent to stay within certain boundaries (Altman, 1999;García and Fernández, 2015). When training on a simulator, an alternative to exploration is domain randomization (Tobin et al., 2017;Peng et al., 2018;Machado et al., 2018). The empirical results reported in the next section suggest that agents become less susceptible to policy confounding when adding enough stochasticity to the environment or to the policy. Yet, there is a limit on how much noise can be added to the environment or the policy without altering the optimal policy ( Sutton and Barto, 2018, Example 6.6: Cliff Walking). Experiments The goal of the experiments is to: (1) demonstrate that the phenomenon of policy confounding described by the theory does occur in practice, (2) uncover the circumstances under which agents are most likely to suffer the effects of policy confounding and fail to generalize, and (3) evaluate how effective the strategies proposed in the previous section are in mitigating these effects. Experimental setup Agents are trained with an off-policy method, DQN (Mnih et al., 2015) and an on-policy method, PPO (Schulman et al., 2017). To be able to analyze the learned representations more easily, we represent policies and value functions as feedforward neural networks and use a stack of past observations as input in the environments that require memory. We report the mean return as a function of the number of training steps. Training is interleaved with periodic evaluations on the original environments and variants thereof used for validation. The results are averaged over 10 random seeds. Please refer to Appendix F for more details about the experimental setup. Environments We ran our experiments on three grid-world environments: the Frozen T-Maze from Section 2, and the below described Key2Door, and Diversion environments. We use these as pedagogical examples to clarify the ideas introduced by the theory. Nonetheless, in Appendix C, we refer to previous works showing evidence of particular forms of policy confounding in high dimensional domains. Example 2. Key2Door. Here, the agent needs to collect a key placed at the beginning of the corridor in Figure 4 (left) and then open the door at the end. The observations do not show whether the key has already been collected. Thus, to solve the task in the minimum number of steps, the agent must remember that it already got the key when going to the door. Yet, since during training, the agent always starts the episode at the first cell from the left, when moving towards the door, the agent can forget about the key once it has reached the third cell. As in the Frozen T-Maze example, the agent can build the habit of using its own location to tell whether it has or has not got the key yet. This, can only occur when the agent consistently follows the optimal policy, depicted by the purple arrow. Otherwise, if the agent moves randomly through the corridor, it is impossible to tell whether the key has or has not been collected. In contrast, in the evaluation environment, the agent always starts at the second to last cell, this confuses the agent, which is used to already having the key by the time it reaches said cell. A DBN describing the dynamics of the environment is provided in Appendix D. Example 3. Diversion. Here, the agent must move from the start state to the goal state in Figure 4 (right). The observations are length-8 binary vectors. The first 7 elements indicate the column where the agent is located. The last element indicates the row. This environment aims to show that policy confounding can occur not only when the environment is partially observable, as was the case in the previous examples, but also in fully observable scenarios. After the agent learns the optimal trajectory depicted by the green arrow, it can disregard the last element in the observation vector. This is because, if the agent does not deviate, the bottom row is never visited. Rather than forgetting past information, the agent ignores the last element in the current observation vector for being irrelevant when following the optimal trajectory. We train the agent in the original environment and evaluate it in a version with a yellow diversion sign in the middle of the maze that forces the agent to move to the bottom row. A DBN describing the dynamics of the environment is provided in Appendix D. Results On-policy vs. off-policy The results in Figure 5 reveal the same pattern in all three environments. PPO fails to generalize outside the agent's preferred trajectories. After an initial phase where the average returns on the training and evaluation environments increase ('PPO train' and 'PPO eval'), the return on the evaluation environments ('PPO eval') starts decreasing when the agent commits to a particular trajectory, as a result of policy confounding. In contrast, since the training samples come from a mixture of policies, DQN performs optimally in both variants of the environments ('DQN train' and 'DQN eval') long after converging to the optimal policy. 4 A visualization of the history representations learned with PPO, showing that the policy does ignore variables that are necessary for generalization, is provided in Appendix E.1. Large vs. small replay buffers We mentioned in Section 6.3 that the effectiveness of off-policy methods against policy confounding depends on the size of the replay buffer. The results in Figure 6 (left) confirm this claim. The plot shows the performance of DQN in the Frozen T-Maze environment when the size of the replay buffer contains 100K experiences and when it only contains the last 10K experiences. We see that in the second case, the agents performance in the evaluation environment decreases (red curve left plot). This is because, after the initial exploration phase, the distribution of trajectories becomes too narrow, and the spurious correlations induced by the latest policies dominate the replay buffer. Similar results for the other two environments are provided in Appendix E.2. Exploration and domain randomization The last experiment shows that if sufficient exploration is allowed, DQN may still generalize to different trajectories, even when using small replay buffers (blue curve right plot Figure 6). In the original configuration, the exploration rate ϵ for DQN starts at ϵ = 1 and decays linearly to ϵ = 0.0 after 20K steps. For this experiment, we set the final exploration rate ϵ = 0.1. In contrast, since exploration in PPO is normally controlled by the entropy bonus, which makes it hard to ensure fixed exploration rates, we add noise to the environment instead. The red curve in Figure 6 (right) shows that when we train in an environment where the agent's actions are overridden by a random action with 20% probability, the performance of PPO in the evaluation environment does not degrade after the agent has converged to the optimal policy. This suggests that the added noise prevents the samples containing spurious correlations from dominating the training batches. However, it may also happen that random noise is not sufficient to remove the spurious correlations. As shown in Figure 13 (Appendix E.2), in the Key2Door environment, neither forcing the agent to take random actions 20% of the time nor setting ϵ = 0.1, solves the OOT generalization problem. Similar results for Diversion are provided in Appendix E.2. Conclusion This paper described the phenomenon of policy confounding. We showed both theoretically and empirically how as a result of following certain trajectories, agents may pick up on spurious correlations, and build habits that are not robust to trajectory deviations. We also uncovered the circumstances under which policy confounding is most likely to occur in practice and suggested a few ad hoc solutions that may mitigate its effects. We conceive this paper as a stepping stone to explore more sophisticated solutions. An interesting avenue for future research is the integration of tools from the field of causal inference (Pearl et al., 2016;Peters et al., 2017) to aid the agent in forming history representations that are grounded on causal relationships rather than mere statistical associations (Lu et al., 2018;Zhang et al., 2020;Sontakke et al., 2021;Saengkyongam et al., 2023). which is precisely the first condition in Definition 4, R h (Φ π (h t ), a t ) = R h (h t , a t ),(4) for all h t ∈ H and a t ∈ A. Analogously, we have that, Pr π (Φ π (h t+1 ) | Φ π (h t ), a t ) = Pr π (Φ π (h t+1 ) | do(Φ π (h t )), a t ) = Pr(Φ π (h t+1 ) | Φ π (h t ), a t )(5) where the second equality reflects that the above must hold independently of π. Hence, we have that for all h t , h t+1 ∈ H and h ′ t ∈ {h t } Φ , Pr(Φ π (h t+1 ) | Φ π (h t ), a t ) = Pr(Φ π (h t+1 ) | Φ π (h ′ t ), a t ),(6) which means that, for all h t , h t+1 ∈ H and a t ∈ A, Pr(Φ π (h t+1 ) | Φ π (h t ), a t ) = Pr(Φ π (h t+1 ) | h t , a t ) = h ′ t+1 ∈{ht+1} Φ π T h (h ′ t+1 | h t , a t ),(7) which is the second condition in Definition 4. Equations (4) and (7) reveal that if the assumption is true (i.e., Φ π is not confounded by the policy), then Φ π is not just π-Markov but actually strictly Markov (Definition 4). However, we know that Φ * (h t ) is the minimal history representation, which contradicts the above statement, since, according to Definition 5, there is no proper subset ofΘ * t , for all h t ∈ H, such that the representation remains Markov. Hence,Θ π t ⊂Θ * t implies policy confounding. Proposition 2. Let {Θ * t } ∪Φ * be the union of variables in all possible minimal history representations. There exist cases where, for some π, there is a π-minimal history representation Φ π * : H π t →H π * t withH π * t = ×Θ π * t such thatΘ π * t \ {Θ * t } ∪Φ * ̸ = ∅. Proof (sketch). Consider a deterministic MDP with a deterministic policy. Imagine there exists a variable X 1 that is perfectly correlated with the episode's timestep t, but that is generally irrelevant to the task. The variable X 1 would constitute in itself a valid π-Markov history representation since it can be used to determine transitions and rewards so long as a deterministic policy is followed. At the same time, X 1 would not enter the minimal Markov history representation because it is useless under stochastic policies. Example 4 below illustrates this situation. Song et al. (2020). Figure 7 shows a grid world environment., The agent must go from the start cell to the goal cell. The agent must avoid the yellow cells; stepping on those yields a −0.1 penalty. There is a is +1 reward for reaching the goal. The agent can observe its own location within the maze x and the current timestep t. The two diagrams in Figure 8 are DBNs describing the environment dynamics. When actions are considered exogenous random variables (left diagram), the only way to estimate the reward at t = 10 is by looking at the agent's location. In contrast, when actions are determined by the policy (right diagram), the time variable becomes a proxy for the agent's location. This is because the start location and the sequence of actions are fixed. This implies that t is a perfectly valid π-Markov history representation under π * . Moreover, as shown by the DBN on the right, the optimal policy may simply rely on t to determine the optimal action. C Further Related Work Early evidence of policy confounding Although to the best of our knowledge, we are the first to bring forward and describe mathematically the idea of policy confounding, a few prior works have reported evidence of particular forms of policy confounding. In their review of the Arcade Learning Environment (ALE; Bellemare et al., 2013), Machado et al. (2018) explain that because the games are fully deterministic (i.e., initial states are fixed and transitions are deterministic), open-loop policies that memorize good action sequences can achieve high scores in ALE. Clearly, this can only occur if the policies themselves are also deterministic. In such cases, policies, acting as confounders, induce a spurious correlation between the past action sequences and the environment states. Similarly, Song et al. (2020) showed, by means of saliency maps, how agents may learn to use irrelevant features of the environment that happen to be correlated with the agent's progress, such as background clouds or the game timer, as clues for outputting optimal actions. In this case, the policy is again a confounder for all these, a priori irrelevant, features. Zhang et al. (2018b) provide empirical results showing how large neural networks may overfit their training environments and, even when trained on a collection of procedurally generated environments, memorize the optimal action for each observation. Zhang et al. (2018a) shows how, when trained on a small subset of trajectories, agents fail to generalize to a set of test trajectories generated by the same simulator. Lan et al. (2023) report evidence of well-trained agents failing to perform well on Mujoco environments when starting from trajectories (states) that are out of the distribution induced by the agent's policy. We conceive this as a simple form of policy confounding. Since the Mujoco environments are also deterministic, agents following a fixed policy can memorize the best actions to take for each state instantiation, potentially relying on superfluous features. Hence, they can overfit to unnatural postures that would not occur under different policies. Finally, Nikishin et al. (2022) describe a phenomenon named 'primacy bias', which prevents agents trained on poor trajectories from further improving their policies. The authors show that this issue is particularly relevant when training relies heavily on early data coming from a fixed random policy. We hypothesize that one of the causes for this is also policy confounding. The random policy may induce spurious correlations that lead to the formation of rigid history (state) representations that are hard to recover from. Generalization Generalization is a hot topic in machine learning. The promise of a model performing well in contexts other than those encountered during training is undoubtedly appealing. In the realm of reinforcement learning, the majority of research focuses on generalization to environments that, despite sharing a similar structure, differ somewhat from the training environment (Kirk et al., 2023). These differences range from small variations in the transition dynamics (e.g., sim-to-real transfer; Higgins et al., 2017;Tobin et al., 2017;Peng et al., 2018;Zhao et al., 2020), changes in the observations (i.e., modifying irrelevant information, such as noise: Mandlekar et al., 2017;Ornia et al., 2022, or background variables: Zhang et al., 2020Stone et al., 2021), to alterations in the reward function, resulting in different goals or tasks (Taylor and Stone, 2009;Lazaric, 2012;Muller-Brockhausen et al., 2021). Instead, we focus on the problem of OOT generalization. Keeping the environment unchanged, we aim to ensure that agents perform effectively when confronted with situations that differ from those encountered along their preferred trajectories. State abstraction State abstraction is concerned with removing from the representation all that state information that is irrelevant to the task. In contrast, we are worried about learning representations containing too little information, which can lead to state aliasing. Nonetheless, as argued by McCallum (1995), state abstraction and state aliasing are two sides of the same coin. That is why we borrowed the mathematical frameworks of state abstraction to describe the phenomenon of policy confounding. Li et al. (2006) provide a taxonomy of the types of state abstraction and how they relate to one another. Givan et al. (2003) introduce the concept of bisimulation, which is equivalent to our definition of Markov history representation (Definition 4) but for states instead of histories. Ferns et al. (2006) proposes a method for measuring the similarity between two states. Castro (2020) notes that this metric is prohibitively expensive and suggests using a relaxed version that computes state similarity relative to a given policy. This is similar to our notion of π-Markov history representation (Definition 7). While the end goal of this metric is to group together states that are similar under a given policy, here we argue that this may lead to poor OOT generalization. Figure 9: Two DBNs representing the dynamics of the Key2Door environment, when actions are sampled at random (left), and when they are determined by the optimal policy (right). The nodes labeled as x represent the agent's location, while the nodes labeled as y represent whether or not the key has been collected. The agent can only see x. Hence, when actions that are sampled are random (left), the agent must remember its past locations to determine the reward r 7 . Note that only x 1 and x 7 are highlighted in the left DBN. However, other variables in ⟨x 2 , ..., x 6 ⟩ might be needed, depending on when the key is collected. In contrast, when following the optimal policy, only x 7 is needed. In this second case, knowing the current location is sufficient to determine whether the key has been collected. , and when they are determined by the optimal policy (right). The nodes labeled as x 1 indicate the row where the agent is located; the nodes labeled as x 2 indicate the column. We see that when actions are sampled at random, both x 1 6 and x 2 6 are necessary to determine r 6 . However, when actions are determined by the optimal policy, x 2 6 is sufficient, as the agent always stays at the top row. E Experimental Results E.1 Learned history representations The results reported in Section 7 show that the OOT generalization problem exists. However, some may still wonder if the underlying reason is truly policy confounding. To confirm this, we compare the outputs of the policy at every state in the Frozen T-Maze when being fed the same histories (observation stack) but two different signals. That is, we permute the variable containing the signal (x 1 in the diagram of Figure 2) and leave the rest of the variables in the observation stack unchanged. We then feed the two versions to the policy network and measure the KL divergence between the two output probabilities. This metric is a proxy for how much the agent attends to the signal in every state. The heatmaps in Figure 11 show the KL divergences at various points during training (0, 10K, 30K, and 100K timesteps) when the true signal is 'green' and we replace it with 'purple'. We omit the two goal states since no actions are taken there. We see that initially (top left heatmap), the signal has very little influence on the policy (note the scale of the colormap is 10 × −6), after 10K steps, the agent learns that the signal is very important when at the top right state (top right heatmap). After this, we start seeing how the influence of the signal at the top right state becomes less strong (bottom left heatmap) until it eventually disappears (bottom right heatmap). In contrast, the influence of the signal at the initial state becomes more and more important, indicating that after taking the first action, the agent ignores the signal and only attends to its own location. The results for the alternative case, purple signal being replaced by green signal, are shown in Figure 12. Figure 11: A visualization of the learned history representations. The heatmaps show the KL divergence between the action probabilities when feeding the policy network a stack of the past 10 observations and when feeding the same stack but with the value of the signal being switched from green to purple, after 0 (top left), 10K (top right), 30K (bottom left), and 100K (bottom right) timesteps of training. Figure 12: A visualization of the learned history representations. The heatmaps show the KL divergence between the action probabilities when feeding the policy network a stack of the past 10 observations and when feeding the same stack but with the value of the signal being switched from purple to green, after 0 (top left), 10K (top right), 30K (bottom left), and 100K (bottom right) timesteps of training. Figures 13 and 14 report the results of the experiments described in Section 7 (paragraphs 2 and 3) for Key2Door and Diversion. We see how the buffer size also affects the performance of DQN in the two environments (left plots). We also see that exploration/domain randomization does improve OOT generalization in Diversion but not in Key2Door. E.2 Buffer size and exploration/domain randomization F Further Experimental Details We ran our experiments on an Intel i7-8650U CPU with 8 cores. Agents were trained with Stable Baselines3 (Raffin et al., 2021). Most hyperparameters were set to their default values except for the ones reported in Tables 1 (PPO) and 2 (DQN), which seemed to work better than the default values. Figure 1 : 1Left: An illustration of the Frozen T-Maze environment. Right: Learning curves when evaluated in the Frozen T-Maze environment with (blue curve) and without (red curve) ice. Figure 2 : 2Two DBNs representing the dynamics of the Frozen T-Maze environment, when actions are sampled at random (left), and when they are determined by the optimal policy (right). Figure 3 : 3A DBN illustrating the phenomenon of policy confounding. The policy opens backdoor path that can affect conditional relations between the variables in h t and h t+1 Figure 4 : 4Illustrations of the Key2Door (left) and Diversion (right) environments. Figure 5 :Figure 6 : 56DQN vs. PPO in the train and evaluation variants of Frozen T-Maze (left), Key2Door (middle), and Diversion (right). Frozen T-Maze. Left: DQN small vs. large buffer sizes. Right: PPO and DQN when adding stochasticity. Figure 7 : 7An illustration of the watch-the-time environment. ... ... ... ... Figure 8 : 8Two DBNs representing the dynamics of the watch-the-time environment, when actions are sampled at random (left), and when they are determined by the optimal policy (right). Example 4 . 4(Watch the Time) This example is inspired by the empirical results of Figure 10 : 10Two DBNs representing the dynamics of the Diversion environment, when actions are sampled at random (left) Figure 13 :Figure 14 : 1314Key2Door. Left: DQN small vs. large buffer sizes. Right: PPO and DQN when adding stochasticity. Diversion. Left: DQN small vs. large buffer sizes. Right: PPO and DQN when adding stochasticity. Table 1 : 1PPO hyperparameters.Rollout steps 128 Batch size 32 Learning rate 2.5e-4 Number epoch 3 Entropy coefficient 1.0e-2 Clip range 0.1 Value coefficient 1 Number Neurons 1st layer 128 Number Neurons 2nd layer 128 Table 2 : 2DQN hyperparameters.Buffer size 1.0e5 Learning starts 1.0e3 Learning rate 2.5e-4 Batch size 256 Initial exploration bonus 1.0 Final exploration bonus 0.0 Exploration fraction 0.2 Train frequency 5 Number Neurons 1st layer 128 Number Neurons 2nd layer 128 Note that the two paths highlighted inFigure 1are not the only optimal paths. However, for the agent to be able to ignore the initial signal, it is important that the paths do not overlap. Note that we sum over st+1 because multiple states may emit the same observation ot+1. R h (ht, at) = R π h (Φ π (ht), at) andh ′ t+1 ∈{h t+1 } Φ π T h (h ′ t+1 | ht, at) = Pr π (Φ π (ht+1) | Φ π (ht), at), where H π ⊆ H denotes the histories visited under π, R π h (Φ π (h t ), a t ) = {R h (h ′ t , a t )} h ′ t ∈{ht} Φ π , {h t } Φ π = {h ′ t ∈ H π t : Φ π (h ′ t ) = Φ π (h t )},and Pr π is probability under π. Definition 8 (π-minimal history representation). A history representation Φ π * : H π t →H π * t with H π * t = ×Θ π * t is said to be π-minimal, if all other history representations Φ : H π t →H π t with H π t = ×Θ t and |Θ t | ⊂ |Θ π * t |, for at least one h t ∈ H π , are not π-Markov. t withH * t = ×Θ * t be a minimal history representation. If, for some π, there is a π-Markov history representation Φ π : H π t →H π t withH π t = ×Θ π t , such thatΘ π t ⊂Θ * t for some h t ∈ H, then Φ π is confounded by policy π.Finally, to conclude this section, we demonstrate that even though, in Example 1, the variables included in the π-minimal history representation are a subset of the variables in the minimal history representation,Θ π * t ⊂Θ * t , this is not always the case, asΘ π * t may contain superfluous variables (Definition 6). An example illustrating this situation is provided in Appendix B (Example 4).Proposition 2. Let {Θ * t } ∪Φ * be the union of variables in all possible minimal history representations. There exist cases where, for some π, there is a π-minimal history representation Φ π * : H π t →H π * t withH π * t = ×Θ π * t such thatΘ π * t \ {Θ * t } ∪Φ * ̸ = ∅.6.1 Why should we care about policy confounding? Note that in the Frozen T-Maze environment, the ice does change the environment dynamics. However, its purpose is to compel the agent to take trajectories different from the optimal ones. The way we implemented it, the effect of the ice would be equivalent to forcing the agent to move down twice when in the top cell or move up twice when in the bottom cell. These trajectories are feasible in the original environment. The small gap between 'DQN train' and 'DQN eval' is due to the −0.1 penalty per timestep. In all three environments, the shortest path is longer in the evaluation environment than in the training environment. Peters, J., Janzing, D., and Schölkopf, B. (2017). Elements of causal inference: foundations and learning algorithms. The MIT Press. Quionero-Candela, J., Sugiyama, M., Schwaighofer, A., and Lawrence, N. D. (2009). Dataset shift in machine learning. The MIT Press. Raffin, A., Hill, A., Gleave, A., Kanervisto, A., Ernestus, M., and Dormann, N. (2021). Stable-baselines3: Reliable reinforcement learning implementations. Journal of Machine Learning Research, 22(268):1-8. AcknowledgementsThis project received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programA Proofs Lemma 1. Let Φ π1 * be the set of all possible π-minimal history representations under π 1 , where every Φ π1 * ∈ Φ π1 * is defined as Φ π1 * : H π1 t →H π1 * t andH π1 * t = ×Θ π1 * t , and let π 2 be a second policy such that for all h t ∈ H π1 t ∩ H π2 t , supp (π 2 (· | h t )) ⊆ supp (π 1 (· | h t )) .For all Φ π1 * ∈ Φ π1 * , there exists a π-Markov history representation under policy π 2 , Φ π2 :Proof. First, it is easy to show thatIn particular,In such cases, we know that there is at least one action a ′ for whichfrom H π 2 but possibly also subsequent histories that can only be reached from h ′ t+1 . Further, since H π2 ⊂ H π1 , we know that, for every Φ π1 * ∈ Φ π1 * , there must be a Φ π2 * that requires, at most, the same number of variables,Θ π2 t ⊆Θ π1 * t and, in some cases, fewer,Θ π1 * t ̸ =Θ π2 * t (e.g., Frozen T-Maze example).Proposition 1. Let Φ * be the set of all possible minimal history representations, where every Φ * ∈ Φ * is defined as Φ * : H t →H * t withH * t = ×Θ * t . For all π and all Φ * ∈ Φ * , there exists a π-Markov history representation Φ π : H π t →H π t withH π t = ×Θ π t such that for all h t ∈ H π , Θ π t ⊆Θ * t . Moreover, there exist cases for whichΘ π t is a proper subset,Θ π t ̸ =Θ * t .Proof. The proof follows from Lemma 1. We know that, in general, H π ⊆ H, and if π(a ′ t |h ′ t ) = 0 for at least one pair a ′ t ∈ A, h ′ t ∈ H, then H π ⊂ H. Hence, for every Φ * there is a Φ π such thatΘ π t ⊆Θ * t , and in some cases, when H π ⊂ H, we may haveΘ π t ̸ =Θ * t (e.g., Frozen T-Maze example).Theorem 1. 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{'abstract': "Reinforcement learning agents may sometimes develop habits that are effective only when specific policies are followed. After an initial exploration phase in which agents try out different actions, they eventually converge toward a particular policy. When this occurs, the distribution of state-action trajectories becomes narrower, and agents start experiencing the same transitions again and again. At this point, spurious correlations may arise. Agents may then pick up on these correlations and learn state representations that do not generalize beyond the agent's trajectory distribution. In this paper, we provide a mathematical characterization of this phenomenon, which we refer to as policy confounding, and show, through a series of examples, when and how it occurs in practice.Preprint. Under review.", 'arxivid': '2306.02419', 'author': ['Miguel Suau \nDelft University of Technology\n\n', 'Matthijs T J Spaan \nDelft University of Technology\n\n', 'Frans A Oliehoek f.a.oliehoek@tudelft.nl \nDelft University of Technology\n\n'], 'authoraffiliation': ['Delft University of Technology\n', 'Delft University of Technology\n', 'Delft University of Technology\n'], 'corpusid': 259076389, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 19728, 'n_tokens_neox': 17511, 'n_words': 11588, 'pdfsha': '061eed292b71f07c6d0cec2cf89f6144a9e33730', 'pdfurls': ['https://export.arxiv.org/pdf/2306.02419v1.pdf'], 'title': ['Bad Habits: Policy Confounding and Out-of-Trajectory Generalization in RL', 'Bad Habits: Policy Confounding and Out-of-Trajectory Generalization in RL'], 'venue': []}

arxiv

Artificial intelligence for Sustainability in Energy Industry: A Contextual Topic Modeling and Content Analys Tahereh Saheb t.saheb@modares.ac.ir Mohammad Dehghani mohamad.dehqani@modares.ac.ir Management Studies Center Industrial and Systems Engineering Tarbiat Modares University TehranIran Tarbiat Modares University Tehran Iran Artificial intelligence for Sustainability in Energy Industry: A Contextual Topic Modeling and Content Analys Research Assistant Professor, Science & Technology Studies Group, 1 Corresponding Author 2Artificial intelligencesustainabilityenergytopic modelingcontent analysissustainable energy Parallel to the rising debates over sustainable energy and artificial intelligence solutions, the world is currently discussing the ethics of artificial intelligence and its possible negative effects on society and the environment. In these arguments, sustainable AI is proposed, which aims at advancing the pathway toward sustainability, such as sustainable energy. In this paper, we offered a novel contextual topic modeling combining LDA, BERT and Clustering. We then combined these computational analyses with content analysis of related scientific publications to identify the main scholarly topics, sub-themes and cross-topic themes within scientific research on sustainable AI in energy. Our research identified eight dominant topics including sustainable buildings, AI-based DSSs for urban water management, climate artificial intelligence, Agriculture 4, convergence of AI with IoT, AI-based evaluation of renewable technologies, smart campus and engineering education and AI-based optimization. We then recommended 14 potential future research strands based on the observed theoretical gaps. Theoretically, this analysis contributes to the existing literature on sustainable AI and sustainable energy, and practically, it intends to act as a general guide for energy engineers and scientists, AI scientists, and social scientists to widen their knowledge of sustainability in AI and energy convergence research.On May 29th 2021, we searched the following keywords inside the title, keyword, and abstract: "artificial intelligence" OR "AI" AND "sustainable" OR "sustainability" AND "energy". This search resulted in the retrieval of 981 documents. Following that, we restricted the document type to Articles and the language toEnglish. This exclusion resulted in 296 articles. Following that, we manually evaluated the titles and abstracts of the articles to identify the most pertinent ones that examined the role of artificial intelligence in ensuring the energy sector's sustainability. This screening yielded 182 publications spanning the years 2004 to 2022.Given that abstracts of research articles are the most succinct summary of key ideas [22], we included abstracts of the final publications in the study's corpus.Preprocessing and Post-Processing StagesPython 3.7.9 was utilized for pre-and post-processing, as well as for topic modeling analysis. We preprocessed our corpus using the NLTK and Scikit-learn packages, as well as Regular Expressions or RegEX. We import the word tokenize from the NLTK to begin the tokenization process. After removing punctuation, we lowercased our characters and deleted all numeric characters, punctuation, and whitespace.Additionally, we eliminated no-word repetitions and anything enclosed in parenthesis. Additionally, we eliminated the NLTK library's stopwords.We reviewed the first findings and created a manual exclusion list for more relevant topic identification during the postprocessing step. We added the core keywords (i.e. artificial intelligence, AI, energy, sustainable, sustainability) in the exclusion list to enhance the coherence of the findings. We used stemming throughout the preprocessing step; however, after observing the first results, we decided to remove the stemming to make the words displayed in the word clouds more understandable. We next used the lemmatization procedure, which we abandoned following the findings of the word clouds in order to make our topic labeling approach more comprehensible. Additionally, we estimated the TF-IDF score for each word in the corpus. We eliminated words with scores that were lower than the median of all TF-IDF values. We calculated the TF-IDF scores using the Scikit-learn package. The maximum TF-IDF score was set to 0.8 and the minimum value at 0.11. Additionally, we incorporated unigrams and bigrams.Topic ModelingWe applied the following libraries to conduct the topic modeling: Pandas to read the dataset, Gensim to perform LDA, Transformers to perform BERT, Keras to perform auto-encoding, and Seaborn and Matplotlib to visualize the results. We imported the TFID vectorizer from the Scikit-learn feature extraction and KMeans from the Scikit-learn cluster. The probabilistic topic assignment vector was constructed using LDA, while the sentence embedding vector was constructed using BERT. To begin, we used the TF-IDF, Introduction The rise of unsustainable practices and procedures co-occurred with the rising urbanization and civilization have driven the emergence of AI-based solutions to assist the path toward sustainability [1][2][3]. Excessive consumption and unsustainable energy sources, which have increased at an unprecedented rate due to factors such as urbanization, improper building construction, transportation, environmental changes, and population growth, have pressured the energy industry to pursue clean energy sources and smart solutions [4]. The deployment of alternative energy sources and access to sustainable energy are pillars of global economic growth [5] and fight against environmental hazards, in particular climate change [6]. Thus, the energy sector has focused its efforts not only on developing new sources of energy, but also on inventing novel technical solutions that increase the efficiency of existing mitigation measures [7]. AI-based interventions, which are available in the form of both hard and soft solutions, such as robots and algorithms and models, are one of these solutions that have come to assist humanity [8]. Artificial intelligence can provide a wide range of intelligent solutions, from predictive and prescriptive energy consumption insights to intelligent energy generation and distribution. Parallel to the escalating discussions over sustainable energy and artificial intelligence solutions, the world is now debating the ethics of artificial intelligence and its potentially negative effects on society and the environment. Ethical AI considers not just AI's moral dimensions, but also its epistemic perspectives [9]. While prior studies have urged scholars to focus on the epistemological aspects of sustainable AI and to open the black box of algorithms to develop sustainable models and algorithms [10], other researches have concentrated on AI for social good and its favorable societal and environmental circumstances [11,12]; such as the development of sustainable AI. In this article, we define sustainable AI as AI that is designed to achieve sustainability and is called AI for sustainability, as differed from AI that is designed to be sustainable and is called sustainability of AI [10]. In this paper, the term "sustainable AI" refers to the extent to which artificial intelligence can help society accomplish their sustainability goals [13,14]. The energy industry is one of the core industries that will benefit from sustainable AI, which will aid in the development of energy sustainability [15]. Sustainable energy strives to fulfill today's energy demand without depleting energy supplies or harming the environment. Sustainable energy systems are regarded as a requirement for achieving all the Sustainable Development Goals (SDGs) [16]. Sustainable artificial intelligence can help to expedite the development of sustainable energy [14]. To advance sustainable energy, the industry has supplied a wide variety of choices, including wind energy, fossil fuels, solar energy, and bioenergy. It's also vital to recognize how academics have dealt with the confluence of sustainability, artificial intelligence, and energy. This research is novel from various perspectives. First, this study intends to foster discussions on sustainable AI by identifying the most important research issues in the area, highlighting intellectual gaps, and proposing potential research streams. It is obvious that the energy sector and scientific research and innovation are inextricably linked. Scientific research is seen to be the cornerstone of technological advancements [17]. Identifying the intellectual frameworks of scientific research across time and the historical progression of its themes can have a huge influence on the effectiveness or failure of new technological solutions. To our knowledge, scientific research on sustainable energy is lacking a coherent understanding of how artificial intelligence has been integrated into this domain and how it should be conducted in the future. It is therefore imperative to perform a mixed-method literature review to have a deeper understanding of the deployment of AI to achieve sustainable energy in order to identify existing research gaps and potential future research streams. The second aspect of this research that distinguishes it from prior research is its novel methodology. Extensive literature reviews are conducted by scholars using bibliometric methodologies [18][19][20] or topic modeling techniques such as Latent Dirichlet Allocation (LDA) [21,22] or qualitative content analysis [23]. As a result, we incorporated all the aforementioned review methodologies to ensure that their findings were complementary. Furthermore, because both bibliometric and LDA topic modeling are based on keyword cooccurrence analysis, we included a contextual embedding-based topic modeling analysis that incorporates use of sentences as fundamental units of analysis. This method which is the latest development in natural language processing (NLP) is offered by Google under the name of Bidirectional Encoder Representations for Transformers (BERT) [24] . BERT makes use of the Transformer library, which uses machine learning to discover contextual relationships between words in a text. Our integrated adoption of computational and advanced topic modeling tools, as well as qualitative analysis, enables us to gain highly objective, coherent, superior, and meta-analytical insight into present research on sustainable artificial intelligence in energy and to forecast its future. The final contribution of this research is that we offer a thorough list of research gaps and potential research agendas that may be used to increase the depth of research on sustainable artificial intelligence in the energy industry In sum, the theoretical contribution of this research is to extent the literatures on sustainable AI and sustainable energy by determining the key academic themes, sub-themes and cross-topic common themes addressed by scientists working on sustainable AI in energy, as well as how these subjects have evolved over time. Practically, this research attempts to enlighten policymakers, the energy sector, and engineers and developers of artificial intelligence about the productivity of science while emphasizing the challenges that require more AI-based responses. Additionally, it encourages policymakers to design artificial intelligence regulations that promote the development of sustainable AI in the energy sector while mitigating the unintended consequences of unsustainable energy sources and AI solutions. 4 The study is structured as follows: we begin with an explanation of our methodology and then go on to the findings, which include our topic modeling and content analysis of topics. We conclude the study by discussing our findings, theoretical research gaps, and potential future research directions. We also discussed the theoretical and practical contribution of the study. We conclude the paper with a conclusion. Methodology It is a widely held belief among researchers that each quantitative and qualitative research technique has inherent strengths and weaknesses; hence, combining both methods is advised to ensure that their results complement one another. We drew on and included four complimentary sets of research methodologies in our study. Three of these, BERT, LDA topic modeling and clustering are connected with text mining techniques. Additionally, we supplemented these quantitative findings with a qualitative topic-based content analysis. Our mixed-methods approach is new in three ways. First, we employed computational approaches such as BERT, LDA, and clustering to discover the thematic content of research on sustainable AI in energy. Second, we conducted a comprehensive analysis of the retrieved topics using content analysis as a qualitative approach. Third, we integrated LDA and BERT topic modeling approaches in this study to achieve the highest level of topic identification accuracy. Our suggested mixed-method methodology may be used by researchers from a variety of disciplines to improve our understanding of quantitative and computational analyses through the use of topic-based content analysis. LDA is predicated on the premise that documents are made of topics and that some words are more likely to occur in certain topics than others (Xie et al., 2020). While LDA has been regularly used by academics to identify topics, it does have some limitations due to the fact that it is a word co-occurrence analysis and so cannot incorporate the entire content of the sentence. Additionally, it does not do well on short texts [26]. Additionally, the outcomes of LDA may be challenging for humans to comprehend and consume [27]. By contrast, BERT topic modeling is focused on detecting semantic similarity and integrating topics with pretrained contextual representations [28] It substantially enhances the coherence of neural topic models by including contextual information into the topic modeling process [29]. BERT makes use of the Transformer library, which has an Autoencoder technique: an encoder that scans the text input. We combined the LDA and BERT vectors in this study to improve topic recognition and clustering. Moreover, because one of the most difficult aspects of word-sentence embedding is dealing with high dimensions, we applied the Uniform Manifold Approximation and Projection (UMAP) approach. In comparison to other approaches, UMAP is one of the most efficient implementations of manifold learning [30]. order to balance the information content of each vectors. We incorporated the Keras package to process the auto-encoder in order to learn a lower-dimensional latent space representation for the concatenated vector. To ensure the clusters were of good quality, we calculated the Silhouette Score, which was 0.566 and near to one for LDA+BERT+ Clustering. TFIDF+clustering received a score of 0.048, while BERT+clustering received a score of 0.095 ( Figure 2). The Silhouette Score is used for cluster quality [31]. The score ranges from -1 to 1. If the score is near to one, the cluster is dense and well isolated from neighboring clusters. In comparison to other topic modeling techniques, LDA BERT Clustering is closer to 1, indicating that the clusters are of excellent quality. The final topic identification obtained by LDA+BERT+Clustering Algorithms is depicted in Figure 3. We utilized the UMAP package to do dimension reductions and set the topic count to eight. We also evaluated several topic clustering, including 10, 4, and 6. The authors determined that eight topics were better separated from one another and had a greater density within each topic; this demonstrates the excellent quality of clustering. As indicated by the percentage of documents contained inside each topic, approximately 11% of documents belong to topic 0 and approximately 16% to topic 1. Clustering resulted in a balanced distribution of documents within each topic, confirming the clustering's excellent quality. TF-IDF Clustering BERT LDA Figure 3 The global view of the topic model on sustainable AI in energy research area. We integrated LDA, BERT and clusetering for topic modeling detection. uncovered eight different topics. These topics will be described, and then a content analysis of the papers that are associated with each one will be carried out throughout this part of the article. Results Descriptive Analysis These articles were organized according to their relative likelihood of belonging to each topic. As seen in The word cloud visualization ( Figure 6.0) shows the identified topics after labeling based on the topic three keywords. The Figure 6 shows that the first three most-used terms in each subject are as follows: Topic 1(building, consumption, environment); topic 2 (design, water, decision); topic 3 (building, climate, fuel); topic 4 (decision, agriculture, improve); topic 5 (IoT, devices, consumption); topic 6 (urban, technology, industrial); topic 7 (engineering, efficiency, students); topic 8 (optimization, efficient, building). Figure 4 The distribution of documents across topics The evolution of topics over time Once we scoured the corpus for hidden topics, we determined how often they appear throughout time. however, topic reached its apex in 2019 and 2020. The topic of AI for energy efficiency has shown a reasonably steady increase from 2013, with its greatest growth occurring between 2020 and 2021. In 2020, significant academic focus was given to AI-based DSSs for urban water management. Content analysis to detect topics, sub-themes and cross-topic common themes In this part of the paper, we conducted content analysis of detected topics for three purposes: First, to detect the general topics from articles; second, to identify the sub-themes from each topic, and third to find the cross-topic common themes. Topic 1: Sustainable Buildings and Energy Consumption The primary concerns of topic 1 are related to the design of automated and intelligent systems and the incorporation of cutting-edge technologies, particularly IoT and AI-based DSSs, in order to construct sustainable buildings. These buildings will be part of the sustainable cities initiative, which aims to promote sustainable energy consumption and smart grids. One of the primary scholarly interests is the creation of sustainable buildings and smart grids for the purpose of reducing energy consumption. One way to accomplish this aim is to redefine the design and architecture of buildings, whether residential, public, commercial, industrial, or manufacturing. According to studies, the application of automation and intelligent systems in the construction of sustainable buildings will result in sustainable energy usage [32,33]. Several AI-based approaches are proposed to achieve a more sustainable building, including building management systems, knowledge-based engineering (KBE), fuzzy logic, neural [34]. From a broad standpoint, sustainable building development falls under the umbrella of sustainable smart cities and reducing building energy consumption [35]. Additionally, scholars have drawn inspiration from nature and advocated regenerative design influenced by nature for pattern detection, prediction, optimization, and planning of buildings [36]. Additionally, scholars discuss the potential of AI in reducing CO2 emissions in buildings, suggesting that AI may be used to construct smart multi-energy systems, such as those found in industrial districts, resulting in significant energy savings and CO2 emission reductions (Simeoni, Nardin and Ciotti, 2018 ). As a result, sustainable building design would be a way to combat climate change. Several additional studies integrate AI solutions with other cutting-edge technologies, most notably the Internet of Things and big data, to improve not only the design and optimization of sustainable buildings, but also the efficiency of their power usage (Chui, Lytras and Visvizi, 2018). For instance, one project focused on the application of IoT in public buildings in order to discover and anticipate energy usage trends [39]. A preceding study, for illustration, outlines the obstacles involved in understanding the semantics of IoT devices using machine learning models. Image Encoded Time Series has been identified as an alternate method to other statistical feature-based inference [35]. Sustainability analysts from [40] and [41] studies have also advocated for continual monitoring of sustainability metrics by integrating AI with DSSs or ambient intelligence. Both residential buildings and plants and commercial buildings and offices have the same issue in regard to energy usage. Previous studies incorporated multi-objective and multi-attribute decision making modeling as well as impact evaluation of the emission outputs to help designers and manufacturers to make environmentally sustainable decisions about the designs and production of facilities [42]. Researchers also believe that in order to provide bulk energy consumption forecast, control, and management, simulation techniques could be utilized [15], for instance in public buildings, offices and factories. Due to new modes of consumption and distributed intelligence, the electrical power grids have been also influenced, and as a result, smart energy grids have been generated to achieve sustainability [43]. Topic 2: AI-based DSSs for Sustainable Urban Water Management The second topic is sustainable water management, which includes utilizing AI to create DSSs for consumption and water usage. Forecasting, real-time monitoring, and customized and adjustable pricing and tariffs are the primary strategies. AI is used with other sophisticated technologies to assist in the development of a smart city. The previous studies have postulated several approaches, such as optimization and AI-based decision support systems, for water infrastructure management [44], better delivery of public services of smart cities such as water treatment and supply [45], AI-based water pricing and tariff options [46] and sustainable water consumption [47]. For this goal, AI is integrated with recent technological advances in urban life. This includes using open source data, employing deep learning algorithms, and developing smart street lighting systems. Such decisions about social impacts of smartphone applications or smart travel behavior are also examined [48]. AI techniques are utilized to anticipate water resource management [49], such as water quality by adopting algorithms such as neuro-fuzzy inference system [50]. Real-time optimization of water resources and cloud technologies are integrated with visual recognition techniques and created to improve efficiency with irrigation systems [51]. A study conducted on ecological water governance implementation using AI found that including algorithms into the system yields higher-quality information and better prediction models for accurate evaluation of water quality [52]. AI may be used for tracking water use and demand as well as forecasting water quality, but it can also be used for estimating water infrastructure maintenance, monitoring dam conditions, water-related diseases and disasters [53] and water reuse [54]. By critiquing conventional decision support systems, research offer alternatives based on artificial intelligence, such as a systematic decision process [55], sustainability ranking framework based on Mamdani Fuzzy Logic Inference Systems to develop a sustainable desalination plant [56] or an comprehensive and flexible decision-making process fueled by social learning and engagement aimed at ensuring the urban water system's environmental and energy sustainability [57]. One research offers a unique DSS for analyzing the energy effect of each of the urban water cycle's macro-sectors, including assessing the system's energy balance and proposing potential energy-efficient solutions ( Puleo et al., 2016). Topic 3: Climate Artificial Intelligence (Climate Informatics) Climate informatics, specially climate artificial intelligence as a new field of study is concerned with issues such as AI-based DSSs to reduce greenhouse gas emissions, optimizing grid assets, enhancing climate resiliency and reliability, increasing energy efficiency, forecasting energy consumption and modeling earth systems. Moreover, within this topic, scholars have addressed the issue of explainable and trustworthy AL models due to the controversial nature of climate change. Climate change has compelled societies to seek alternate energy sources and fuels [59]. Climate informatics [60], such as several AI-based solutions, including novel algorithms and DSSs, have been hugely beneficial in lowering greenhouse gas emissions in the energy sector. By improving grid assets, and strengthening climate adaptability these innovations have greatly contributed to this ultimate goal [15]. Reliable and explainable artificial intelligence models, as advocated in prior studies, might help stakeholders and decision-makers achieve climate-resilient and sustainable development goals [61]. By integrating advanced machine learing techniques, AI can propose fresh insights in complex climate simulations in the field of climate modeling [62]. Energy consumption patterns might undergo considerable changes due to climatic change, which means AI forecasts can aid in estimating future energy use for various climate scenarios [63]. It's not only businesses and other organizations that are using AI algorithms these days-AI algorithms are also being utilized to foster sustainable urban growth and mitigate climate change by examining how future urban expansion will affect material and energy flows [64]. Fossil fuel, used as the primary energy source, is the primary contributor to human greenhouse gases that influence the climate. AI is extensively utilized for decreasing carbon footprints and for avoiding fossil fuel combustion [65] as prior studies show that AI can act as an automated carbon tracker [66]. Artificial intelligence-powered technologies may help investors in analyzing a company's climate effect while making investment choices [67]. By drawing attention to climate change through visualization techniques, they help to educate the public on the effects of climate change [68] Ultimately, AI algorithms may provide great resources for climate change conflicts, including in the field of modeling earth systems [69], teleconnections [70], weather forecasting ( McGovern and Elmore, 2017), future climate scenarios [72], climate impacts [73] and climate extremes [74]. Topic 4: Agriculture 4.0 and Sustainable Sources of Energy The fourth area that academics in the field of sustainable AI for energy extensively address is the development of smart agriculture and sustainable energy sources. The primary issue in this subject is how to combine advanced technologies like IoT, drones, and renewable energy with AI in order to create automated and real-time systems. According to some researchers, the agriculture industry is suffering from an insufficient application of responsible innovation [75]. As a result, the researchers are calling for a system referred to as Responsible Agriculture 4.0, which incorporates drones, IoT, robotics, vertical farms, AI, and solar and wind power linked to microgrids [76][77][78]. When it comes to the productivity of agriculture, factors such as the cost of energy for cultivation are equally significant [79]. Based on the premise that most agricultural machinery operates on fossil fuels, it may potentially contribute to climate change. Thus, new energy solutions, and AI-based approaches are provided. One way in which bioproduction and renewable energy may positively influence sustainable agriculture and farming is via the development of bioproduction and renewable energy [80]. Proposing new AI methods to forecast agricultural energy use has also been researched [79]. biomass may also be used to provide sustainable energy in agriculture, and care should be taken to avoid any injuries [81]. Real-time alerting systems, AI-based DSSs, real-time DSS forecasting models, and alternative energy sources such as solar and wind play a vital role in sustainable agriculture [82]. Maximizing agricultural production and economic stabilization while minimizing the use of natural resources and their harmful environmental consequences may be accomplished using renewable energy and AI [82]. Artificial intelligence enables academics to provide accurate forecasts of agricultural energy use [83]. Especially, a drastic shift toward sustainability in agricultural practices has occurred because of its confluence with other cutting-edge 14 technology, including sensors, DSSs, greenhouse monitoring, intelligent farm equipment, and drone-based crop imaging. [84]. 15 AI is used in tandem with a number of cutting-edge technologies for sustainable energy development, such as improved energy conservation [85] and building intelligent energy management [86] such as building management systems [35]. Internet of Things (IoT) is one of the most promising and pervasive technologies [85]; whose integration with AI has generated a revolution in the energy sector. There are many functions in creating sustainable energy in the IoT-enabled smart city dubbed City 4.0 [87] such as simulation and optimization of power plant energy sustainability [86]. City systems such as water and electricity, as well as other infrastructures, such as data analytics, will be driven by sensor and data collection in the smart city [87]. A significant use of IoT is in the design of intelligent buildings, which with AI included may support a goal of energy or water conservation [39,88], for instance, by educating the citizens on how to use energy more effectively and giving them warnings if they are using excessive amounts of energy. [89]. IoT is integral to modern grid development as well. In particular, it seeks to transform the traditional, fossil-fuel-based power grids with distributed energy resources and integrate it with cutting-edge technology such as artificial intelligence for improved grid management [90]. In the same manner, Blockchain has also been considered to be a viable alternative for smart cities. Fusing blockchain with AI may be leveraged for smart services, including energy load forecasting, categorizing customers, and evaluating energy load [91]. Smart connected devices such as IoT devices have successfully employed blockchain in time to retain these devices safe and secure in a blockchain network [92]. The effect of IoT and AI on agriculture and food sectors is also substantial [93,94]. Manufacturing facilities such as food factories and plants may be transformed more intelligent and more environmentally friendly via the use of IoT and AI, which merge with nonthermal and advanced thermal technologies [94]. Sustainable and green IoT are other topics covered in this subject. The two main objectives of the literature on green IoT are to increase the recyclability and usefulness of IoT devices, as well as to minimize the carbon footprints of such devices. The second objective is to incorporate more effective life cycle assessment (LCA) methods integrating artificial intelligence (AI) in order to cut costs and time [95]. Another of the many topics that apply to IoT is with developing smart campuses, which are carbon neutral, energy efficient, use less water, and are laced with various high-quality green energy tools [96] and smart teaching and learning platforms [97]. Researchers have identified the positive traits of IoT devices, but they've also forewarned about the possible risks of the devices and proposed various techniques for detecting weaknesses [93] or challenges regarding the heterogeneity of smart devices and their associated meta-data [35]. Topic 6: AI-based Evaluation of Renewable Energy Technologies Scholarly interest has been generated by the discussion of leveraging AI for DSSs to enhance the efficiency of conventional system evaluations for renewable energy technologies. To a great extent, a sustainable future will depend on maximizing the use of energy sources that cannot be depleted [98]. Artificial intelligence is important for the survival of the future by leveraging a wide range of renewable energy technologies such as biomass energy, wind energy, solar energy, geothermal energy, hydro energy, marine energy, bioenergy, hydrogen energy, and hybrid energy [99]. AI is used to evaluate renewable energy solutions based on their cost of energy production, carbon footprint, affordability of renewable resources, and energy conversion efficiency [100]. Artificial intelligence will ensure the most effective use of these resources while also pushing for improved management and distribution systems [14]. Distributed energy management, generating, forecasting, grid health monitoring, and fault detection are also made more efficient by using automated AI systems [101]. AI can help disperse the supply and demand of energy in real-time and improve energy consumption and storage allocation (Sun, Dong and Liang, 2016). To mitigate against the barrier of utilizing renewable energy technology, the following measures are taken: Renewable energy sustainability is evaluated [103]; in addition, the turbulent and sporadic character of renewable energy data is addressed [104]. One research group claims that standard techniques such as LCA and EIA (Environmental Impact Assessment) may be improved by developing more advanced digital intelligent decision-making systems, or DSSs. It is feasible that improved assessments of renewable energy sources may be achieved via intelligent and automated technologies [105]. With the smart mechanisms in place, long-term detrimental consequences can be calculated, as well as visible and invisible factors [106]. Artificial intelligence (AI) increases the adaptability of power systems, providing DSSs for energy storage applications [107]. For instance, to ensure more use of battery-electric buses, and minimize the effect on the power grids, the researchers developed an AI-powered DSS [108]. Another research leveraged AI to create a DSS for forecasting future energy consumption patterns, and to provide a solution for utilizing renewable energy alternatives [109]. Topic 7: Smart Campus & Engineering Education It is possible to break down the discussions inside this topic into two distinct types: those about engineering education and those which deal with using AI and IoT to construct intelligent campuses to help maintain sustainability objectives. The two themes represent two elements of education: one dealing with the learning contents, and the other with behavioral outcomes of developing smart campuses.To build a model of smart campuses, we should focus on incorporating IoT into the infrastructure, with subsequent implementations of smart apps and services, with smart educational tools and pedagogies and smart analysis as well [97]. A smart campus is in charge of energy consumption scheduling, while its telecommunications infrastructure serves as the place where data transfers are conducted [110]. Integrating cutting-edge technology, a smart campus captures real-time data on energy usage, renewable energy power generation , air quality, and more [111]. Another point of view is that higher education should equip itself with relevant skills and competences to help in realizing long-term sustainable objectives [112]. The energy sustainability in this respect may be addressed via engineering education and engineering assistance for high-level strategic decision-making [113]. This objective can be achieved by using innovative instructional programs, alongside cutting-edge technology such as artificial intelligence and the Internet of Things. A living lab campus equipped with technology, as well as a deep well of talent and competency, may serve as a digital platform for education and sustainable growth [114]. For illustration, to support ongoing research, teaching, and learning on sustainable development, the University of British Columbia (UBC) implemented the Campus as a Living Laboratory project, which included AI and IoT and other cutting-edge technologies [115]. Furthermore, there have been several research done to help AI seamlessly integrate with current educational institutions in order to aid in sustainable development learning [116]. Topic 8: AI for Energy Optimization Conventional optimization methods may be a roadblock for making progress toward sustainability, and AIbased solutions can help eliminate such roadblocks. Whilst renewable energy sources, like solar and wind, have many merits, there are some downsides to consider. They are usually not always available and often rely on the climate, which renders employing them complicated [117]. A proper optimization of energy may be utilized to minimize greenhouse gas emissions and cut energy usage. Efforts to reduce costs and side effects of energy consumption are facilitated using optimization models [118]. Computational and intelligent resources have enabled academics to progress with optimization problems by employing advanced AI methods. Manufacturers have developed numerous energy-efficient appliances for this reason. Even if the deployment of digital technologies in buildings will likely lead to improved energy efficiency, that is not the sole solution. Studies recommend implementing energy-saving measures that don't just target environmental variables, but also include building inhabitants' comfort and preferences, which is achievable via the integration of AI-augmented algorithms [119]. For illustration, AI algorithms that not only monitor current actions but also give real-time alerts and warnings to users and providers allow optimization to be significantly accelerated. Some approaches, such as algorithms that use energy consumption data to lower energy costs in buildings that use advanced AI, are only one example of how AI and advanced technology may be used to benefit society [120]. Weather has a direct effect on energy consumption, which is indisputable. To ensure the winter heating demand of non-residential buildings was calculated correctly, researchers used an optimized artificial neural network method to determine and forecast this need [121]. By utilizing AI along with the use of smart metering and non-intrusive load monitoring, one may improve energy efficiency by evaluating the electricity use of appliances [38]. Using a new approach, researchers found that the GP model was capable of making accurate predictions and a multi-objective genetic algorithm, NSGA-II, was also capable of optimizing sustainable building design [32]. The use of a fuzzy-enhanced energy system model to represent a route to a sustainable energy system has also been presented in another research [122]. The views of other researchers in the field include techniques based on artificial neural networks, evolutionary algorithms, swarm intelligence, and their hybrids, all of which rely on biological inspiration. These findings imply that sustainable energy development is computationally challenging conventional optimization, demanding advanced techniques [123]. Discussion, Theoretical Gaps, and Future Strands of Research To For topic 1, the key problems are the importance of sustainable buildings for smart city development and smart grid services. The issue of AI and its application in decision-making, pricing, forecasting, and sustainable consumption are all addressed in this topic. To reach sustainability, various cutting-edge technologies are tied to AI. One problem which may be especially neglected is the use of AI technology to make buildings eco-friendlier and enhance their inhabitants' feeling of accountability toward sustainability. One approach might be to design real-time warning systems to ensure people are prohibited from excessive energy use, while also ensuring that they benefit from the AI-based solutions. Convergence research may also explore how green architecture is uniquely enabled to deal with complex issues, including environmental efficiency, such as using eco-lighting, natural ventilation, shading, green roofs, and artificial intelligence. Most of prior research focuses on eco-design and overlooks other factors of green architecture. However, there is a dearth of distributed energy resource optimization models, particularly due to the emergence of blockchain. Figure 6 Identified cross-topic common themes As shown in Figure 8, we discovered six core problems that were prevalent throughout the majority of the topics. For example, tariff and price models based on artificial intelligence are prevalent in topics 1 and 2; while economic issues in general are a concern in topics 4, 6, and 8. The dilemma of sustainable consumption is prevalent in all of these topics, demonstrating the critical role of AI in attaining sustainable energy use. Forecasting is inextricably connected to sustainable consumption, since more than half of the topics cover both; demonstrating the progress of AI forecasting algorithms for sustainable consumption. Forecasting, on the other hand, is not restricted to anticipating consumption patterns. The topic's second significant recurring theme is the development of AI-based DSSs. The majority of research have contested traditional DSSs and devised decision-making systems based on artificial intelligence. Sustainable building, urban water management, climate change, and renewable energy evaluation have all been substantially influenced by AI-based DSSs. Automated and real-time systems enabled by artificial intelligence are also discussed in relation to buildings, agriculture, the Internet of Things, and renewable energy technologies. Scholars have combined various digital technologies to promote sustainability in the energy sector via the management of buildings, water, agriculture, IoT, and smart campuses. Theoretical and Practical Contribution Theoretical Contribution Our results supplement existing work on sustainable AI and sustainable energy by delivering the following results. Results from this study provide and highlight a thematic map of the sustainable AI research topics existing in several fields, such as energy, ethics, and management. We developed a novel mixed-method approach, the contextual topic modeling and content analysis, to visualize the latent knowledge structures pertaining to AI and sustainability and energy. This yielded in a conceptual framework representing the main topics, subtopics and common terms in each topic pertaining to sustainable AI in energy. Using LDA and BERT, eight themes related to AI in the sustainability and energy sectors were discovered. We provided the most likely terms for each topic, as well as the distribution of articles and topics throughout time. Finally, by using a thematic analysis method, we identified and qualitatively analyzed the hidden themes. Second, we examined and analyzed hidden sub-themes within each topic, as well as common themes between topics, using a content analysis method. Figure 8 illustrates the sub-domain themes within each topic, whereas Figure 9 depicts the common cross-topic themes. Our content analysis of each topic reveals six recurring themes: sustainable consumption, AI-based DSSs, forecasting models, economic and pricing problems, automated and real-time systems, and convergence with digital technology. To further our knowledge, we highlighted how these themes intersect across topics in order to articulate the commonalities across topics. These six separate but related topics demonstrate that sustainable AI solutions can be observed at a range of behavioral, decision-making, economic, operational, and technical dimensions. At the behavioral level, shifts in consumption patterns are illustrated; at the decision-making level, decision automation is outlined; at the economic level, personalized tariffing is demonstrated; at the operational level, automation and real-time operations are addressed; and at the technological level, convergence with other technologies is studied. Practical Implications This research provides energy engineers, social scientists, scientists, and policymakers with a variety of insights. Engineers may develop sustainable energy products and services. Energy scientists can also integrate sustainability considerations into their research and development of new energy sources such as renewable energy. In their discussions on AI and energy, social scientists may also emphasize ethical problems, including sustainability. Additionally, policymakers may create and construct new laws and policy initiatives aimed at mitigating the harmful effects of unsustainable energy on society and the environment. Conclusion To discover heavily discussed scholarly topics, our study utilized a new topic modeling technique. While this illustration depicts the trajectory of previous efforts, it also prompted us to propose a number of possible future research strands targeted at increasing energy sector sustainability via the application of artificial intelligence technology. The aim of this study is to further the conversation on sustainable AI and energy, as well as their intersection, in order to get a deeper understanding of how AI may be incorporated to achieve sustainability in the energy sector. Figure 1 Figure 2 12The concatenating and encoding LDA and BERT vectors to extract contextual topics The separate and independent results of topic modeling of research on sustainable AI in energy by using TF-IDF, BERT and LDA algorithms Figure 3 . 30 shows a representation of the topic model on sustainable AI in energy research field with respect to the overall global view. This visualization represents the topic modeling results, where topics are illustrated as clusters on a two-dimensional plane. Also shown in Figure 4 is the word cloud visualization of the topics with the most frequently used terms in each topic. Topics 1, 2, and 3 represent the greatest research interest in the model based on 8 topics and including 21.67%, 17.22%, and 15.0% of the corpus. Our research Figure 4 . 40, the three most-covered topics by academia are topic 1: Sustainable buildings (22.5%), Topic 2: AIbased DSSs for urban water management (16.5%) and Topic 3: Climate Artificial Intelligence (14.8%). About 54% of the articles in the corpus are concerned with these three themes. Figure 5 5depicts the ratios of all the eight topics (beginning in 2004 and extending into 2021). Since 2018 forward, topics have garnered a substantial amount of academic interest. Specifically, the first topic, which is about the design of sustainable buildings and minimizing energy usage via the application of artificial intelligence. This subject gained considerable attention between 2012 and 2014, but then slipped off the spotlight between 2015 and 2018. The discussions about AI-based evaluation of renewable energy solutions peaked around 2008 but then became less prominent until 2019. Climate artificial intelligence experienced two distinct phases, with the second one peaking in 2015 and 2016 and the first between 2009 and 2012; Figure 5 5The evolution of topics over time Figure 6 6Topics detected by the combination of LDA+BERT+Clustering algorithms on sustainable AI in energy sector identify the relevant research topics in the literature on artificial intelligence for sustainability in the energy industry, we performed a contextual topic modeling combined with qualitative cluster analysis. We went beyond previous approaches in developing this novel analysis by combining three algorithms of topic modeling (LDA, BERT, and clustering) with content analysis. In this research, eight academic topics were discovered including sustainable buildings and energy consumption, AI based DSSs for sustainable urban water management, climate artificial intelligence, agriculture 4.0 and sustainable sources of energy, convergence of IoT and AI for sustainable smart cities, AI-based evaluation of renewable energy technologies, smart campus and engineering education and AI for energy optimization. Concerns and problems addressed in each topic are summarized in Figure 7. The Figure illustrates that each topic addresses a number of specific issues, which some of them overlap. Figure 7 7Possible future streams of research pertaining to each topic networks, genetic algorithms, and Monte-Carlo simulationTopic 1: Sustainable Buildings and Energy Consumption Topic 2: AI-based DSSs for Sustainable Urban Water Management Topic 3: Climate Artificial Intelligence Topic 4: Agriculture 4.0 and Sustainable Sources of Energy Topic 5: Convergence of IoT & AI for Sustainable Smart Cities Topic 6: AI-based Evaluation of Renewable Energy Technologies topic 7: Smart Campus & Engineering Education Topic 8: AI for Energy Optimization Convergence of IoT & AI for Sustainable Smart CitiesA significant step in the implementation of sustainable energy solutions is to implement smart cities and services using internet of things technology. This topic exhibits how AI and IoT operate together to drive environmental progress. Much of this topic focuses on measure such as smart buildings, smart grid systems, green IoT, and smart campuses.Topic 1: Sustainable Buildings and Energy Consumption Topic 2: AI-based DSSs for Sustainable Urban Water Management Topic 3: Climate Artificial Intelligence (Climate Informatics) Topic 4: Agriculture 4.0 and Sustainable Sources of Energy Topic 5: Convergence of IoT & AI for Sustainable Smart Cities Topic 6: AI-based Evaluation of Renewable Energy Technologies Topic 7: Engineering Education & Smart Campus Topic 8: AI for Energy Optimization 16 Topic 5: future study area is the confluence of smart grids, renewable energy, and 5G technology, since these technologies have the potential to generate enormous volumes of big data. Furthermore, the use of AI in transportation seems worthy of analysis, for example, with regard to traffic predictions, public transit planning, and so on.The agricultural 4.0 and sustainable energy sources are examined in Topic 4. Many problems relevant to the subject of "prosperity, sustainable consumption, forecasting, and convergence with other automated and realtime technologies" are covered in this topic. There is only a limited body of studies dedicated to precision farming and digital mapping, but both developments promise to lead to better knowledge of the environment and to improved energy management. Precision farming by assessing soil nutrients, detecting humidity in the air, and monitoring crops allows farmers to leverage digital maps for better energy management and fight against climate change. Other related areas of study include developing automated working environments. It is worthwhile to investigate the effect that artificial intelligence and other green technologies will have onthe working conditions of farmers and farm operators, since AI may help with deeper speculations of working conditions in farms.In Topic 5, convergent IoT and AI technologies for smart city development were addressed. The primary goal of this topic was to discuss issues around sustainable consumption, LCA analysis, and the development of intelligent energy grids. Pervasive Wi-Fi connection, due to its ability to save energy, is critical in this subject. Additionally, a significant problem is open data sharing in energy management. AI-based assessment of renewable energy technologies, such as DSSs, financial problems, sustainable consumption, and automated and real-time systems are all issues in this topic that focus on renewable energy. One potential study path in this topic involves the challenges that AI algorithms and models face when attempting to evaluate renewable energy solutions. Other sophisticated AI systems, such as deep learning, make use of supervised learning using human-annotated data, and thus they are limited when it comes to complicated situations.The subject of smart campus and engineering education is examined in the seventh topic. Labs that facilitate continuous innovation are discussed in this article, as well as the idea of sustainable consumption, AI skills, and convergence with other technologies. There is an imperative requirement for further research to clarify how AI might be leveraged for practical learning and training for a range of stakeholders across businesses, farmers, residents, and employees in relation to energy management. AI is discussed in relation to energy optimization in Topic 8 of the study. This subject covers many elements of sustainable optimization, including forecasting, consumption, affordable pricing, and societal and financial impacts.Topic 2 addresses sustainable urban water management via the use of AI-based DSSs. Conventional DSSs were under criticism from academics who suggested alternatives, and innovative approaches to DSSs were revealed, particularly with regard to water utilities in a smart city. The second discussion point, focused on sustainable consumption and real-time and predictive modeling, is also addressed in topic 2. Mitigating urban problems, notably air pollution, waste management, and wastewater management, are applicable here to exemplify how smart energy management leveraging AI improves environmental sustainability. Topic 3 deals with the connection between climate change and artificial intelligence, and the emergence of the climate informatics field. This topic highlights the role of trustworthy of explainable AI algorithms, an issue which is marginalized in other topics. As a result, a future potential study direction may be the development of ethical artificial intelligence in other topics to help with the sustainable management of energy. One prospective Figure 5 Sub-themes extracted from each topic Taxonomy research of artificial intelligence for deterministic solar power forecasting. Energy Convers. 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{'abstract': 'Parallel to the rising debates over sustainable energy and artificial intelligence solutions, the world is currently discussing the ethics of artificial intelligence and its possible negative effects on society and the environment. In these arguments, sustainable AI is proposed, which aims at advancing the pathway toward sustainability, such as sustainable energy. In this paper, we offered a novel contextual topic modeling combining LDA, BERT and Clustering. We then combined these computational analyses with content analysis of related scientific publications to identify the main scholarly topics, sub-themes and cross-topic themes within scientific research on sustainable AI in energy. Our research identified eight dominant topics including sustainable buildings, AI-based DSSs for urban water management, climate artificial intelligence, Agriculture 4, convergence of AI with IoT, AI-based evaluation of renewable technologies, smart campus and engineering education and AI-based optimization. We then recommended 14 potential future research strands based on the observed theoretical gaps. Theoretically, this analysis contributes to the existing literature on sustainable AI and sustainable energy, and practically, it intends to act as a general guide for energy engineers and scientists, AI scientists, and social scientists to widen their knowledge of sustainability in AI and energy convergence research.On May 29th 2021, we searched the following keywords inside the title, keyword, and abstract: "artificial intelligence" OR "AI" AND "sustainable" OR "sustainability" AND "energy". This search resulted in the retrieval of 981 documents. Following that, we restricted the document type to Articles and the language toEnglish. This exclusion resulted in 296 articles. Following that, we manually evaluated the titles and abstracts of the articles to identify the most pertinent ones that examined the role of artificial intelligence in ensuring the energy sector\'s sustainability. This screening yielded 182 publications spanning the years 2004 to 2022.Given that abstracts of research articles are the most succinct summary of key ideas [22], we included abstracts of the final publications in the study\'s corpus.Preprocessing and Post-Processing StagesPython 3.7.9 was utilized for pre-and post-processing, as well as for topic modeling analysis. We preprocessed our corpus using the NLTK and Scikit-learn packages, as well as Regular Expressions or RegEX. We import the word tokenize from the NLTK to begin the tokenization process. After removing punctuation, we lowercased our characters and deleted all numeric characters, punctuation, and whitespace.Additionally, we eliminated no-word repetitions and anything enclosed in parenthesis. Additionally, we eliminated the NLTK library\'s stopwords.We reviewed the first findings and created a manual exclusion list for more relevant topic identification during the postprocessing step. We added the core keywords (i.e. artificial intelligence, AI, energy, sustainable, sustainability) in the exclusion list to enhance the coherence of the findings. We used stemming throughout the preprocessing step; however, after observing the first results, we decided to remove the stemming to make the words displayed in the word clouds more understandable. We next used the lemmatization procedure, which we abandoned following the findings of the word clouds in order to make our topic labeling approach more comprehensible. Additionally, we estimated the TF-IDF score for each word in the corpus. We eliminated words with scores that were lower than the median of all TF-IDF values. We calculated the TF-IDF scores using the Scikit-learn package. The maximum TF-IDF score was set to 0.8 and the minimum value at 0.11. Additionally, we incorporated unigrams and bigrams.Topic ModelingWe applied the following libraries to conduct the topic modeling: Pandas to read the dataset, Gensim to perform LDA, Transformers to perform BERT, Keras to perform auto-encoding, and Seaborn and Matplotlib to visualize the results. We imported the TFID vectorizer from the Scikit-learn feature extraction and KMeans from the Scikit-learn cluster. The probabilistic topic assignment vector was constructed using LDA, while the sentence embedding vector was constructed using BERT. To begin, we used the TF-IDF,', 'arxivid': '2110.00828', 'author': ['Tahereh Saheb t.saheb@modares.ac.ir ', 'Mohammad Dehghani mohamad.dehqani@modares.ac.ir ', '\nManagement Studies Center\nIndustrial and Systems Engineering\nTarbiat Modares University\nTehranIran\n', '\nTarbiat Modares University Tehran\nIran\n'], 'authoraffiliation': ['Management Studies Center\nIndustrial and Systems Engineering\nTarbiat Modares University\nTehranIran', 'Tarbiat Modares University Tehran\nIran'], 'corpusid': 238259404, 'doi': '10.1016/j.suscom.2022.100699', 'github_urls': [], 'n_tokens_mistral': 30366, 'n_tokens_neox': 25132, 'n_words': 14845, 'pdfsha': '391453413761a841adfd1239bb30ccc86f909bda', 'pdfurls': ['https://arxiv.org/pdf/2110.00828v1.pdf'], 'title': ['Artificial intelligence for Sustainability in Energy Industry: A Contextual Topic Modeling and Content Analys', 'Artificial intelligence for Sustainability in Energy Industry: A Contextual Topic Modeling and Content Analys'], 'venue': []}

arxiv

Precision cosmology with primordial GW backgrounds in presence of astrophysical foregrounds 13 Dec 2022 D Racco dracco@stanford.edu Stanford Institute for Theoretical Physics Stanford University 94305StanfordCAUSA D Poletti University of Milano-Bicocca Piazza della Scienza3 -20126Milano (MI)Italy Precision cosmology with primordial GW backgrounds in presence of astrophysical foregrounds 13 Dec 2022Prepared for submission to JCAP The era of Gravitational-Wave (GW) astronomy will grant the detection of the astrophysical GW background from unresolved mergers of binary black holes, and the prospect of probing the presence of primordial GW backgrounds. In particular, the lowfrequency tail of the GW spectrum for causally-generated primordial signals (like a phase transition) offers an excellent opportunity to measure unambiguously cosmological parameters as the equation of state of the universe, or free-streaming particles at epochs well before recombination. We discuss whether this programme is jeopardised by the uncertainties onIntroductionThe first direct observation [1] of Gravitational Waves (GWs) performed in 2015 by the interferometers of LIGO, whose measurements were joined shortly after by the companion VIRGO and later by KAGRA, marked the onset of the GW astronomy era. After that first detection of the merger of a black hole binary (BBH), the catalogues released by the LVK consortium after the run O3 of observation [2, 3] collect almost 90 compact binary coalescences, predominantly BBH mergers together with two binary neutron star (BNS) mergers (including the first multi-messenger detection of a merger [4-6]), and a few candidate BH-NS merger. Direct observations of GWs offer a completely new window to explore the Universe, which has spurred many directions to interpret the incoming wealth of data for a variety of scientific programs in astrophysics, cosmology and fundamental physics[7][8][9].Besides the observation of individually resolved mergers, another frontier lies ahead in GW astronomy: the detection of the stochastic GW background (SGWB) filling our Universe (see[10,11]for recent reviews). This background is generated by the stochastic superposition of multiple GWs travelling cosmological distances and originates from various contributions. A class of GW backgrounds is of astrophysical nature, and consists of the incoherent superposition of the GWs generated by all the binaries that have merged throughout the cosmic history, including BBH, BNS (which are subleading in terms of SGWB), and White Dwarfs binaries (BWD).Along with this background, that will be measured [12] by the end of the mission of LVK[13]or at the latest by future ground-based (CE, ET) [14-17] and space-based (LISA, BBO, MAGIS, DECIGO)[18][19][20][21]experiments, there could lie a primordial signal generated in the early cosmological history[22][23][24][25][26][27][28]. Such a discovery would offer to us the farthest signal that we have ever detected, way beyond what is presently visible in the Cosmic Microwave Background (CMB). GWs can travel to us (almost [29]) unperturbed, offering us with a pristine glance into the earliest cosmological epochs. For these reasons, a primordial GW background can be a powerful tool to probe the cosmological evolution and the particle content of the Universe at primordial times, since shortly after the generation of the GWs until today. The power spectrum of the SGWB offers the tantalising prospect of performing precision measurements for cosmology [30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49]. To put this program in practice, we need to know what was the original spectral shape of the SGWB at formation, to detect the subsequent modifications throughout the cosmological history. This is not the typical case, on general grounds. A remarkable exception comes from the class of GW backgrounds that are generated from a phenomenon happening within a finite time and locally in each spatial patch, the prototypical example being a phase transition (PT). In that case, the low-frequency tail of the spectrum, corresponding to wavelengths larger than the Hubble radius (or horizon) at the time of GW generation, is constrained by causality arguments to have a power-law dependence, whose tilt informs us of the expansion history of the Universe and the particle content of the Universe [50][51][52]. The physical origin of this result can be explained in simple terms [50] by considering the evolution of modes on scales larger than the correlation length of the source of GWs. The result is that the spectrum at low frequencies (or causalitylimited tail) is independent of the generation mechanism, and can only be affected by (i ) the expansion history, which affects the redshift of waves inside the horizon and the evolution of super-horizon modes, and (ii ) the propagation of GWs, which do not travel as free waves if there is anisotropic stress in the cosmic fluid. The latter can notably be sourced by freestreaming particles, providing us with the unique opportunity of detecting their presence before the epoch of recombination. The question to which we wish to provide an answer in this paper is the following one: In the simultaneous presence of astrophysical GW backgrounds and of a primordial GW background of causal origin (e.g. from a PT), how well can we perform precision cosmology through the low-frequency part of the primordial signal? In order to discriminate these backgrounds (we refer to the astrophysical ones as foregrounds) we consider the analysis of the global average signal in frequency space, analogously to the measurement by COBE of the black-body spectrum of the CMB [53]. This path, that has been mostly explored so far in the literature [54][55][56][57][58][59][60][61], is not the only option: more refined measurements of the SGWB will detect angular anisotropies to some degree of accuracy, enabling us to use this information to further discriminate primordial and astrophysical SGWBs [29,62,63]. In this paper, we focus on the range of frequencies (10 −4 − 10 −1 Hz) that will be measured by the space-based experiment LISA. Among the reasons for this, this experiment is at an advanced stage among various proposals (after the successful performance of the LISA pathfinder), and its GW-frequency range will provide an interesting window on primordial signals from PTs at energies not far from the present collider reach. Finally, this range is particularly illustrative for our method thanks to the simultaneous presence of various astrophysical foregrounds: GWs from BWD mergers in our Galaxy will be detectable and, although a significant fraction will be resolvable, there will be an irreducible unresolved component in the LISA frequency band. Ground-based experiments, like ET or CE, will measure the peak of the BBH foreground, which has more significant modelling uncertainties; in the frequency range of LISA, instead, it is a good approximation to consider its spectral shape to be known (as we detail in the following together with all the caveats and assumptions). We perform a Fisher matrix forecast of the sensitivity of LISA to features in the causality-limited tail (i.e. low-frequency range) of a primordial SGWB, while marginalising over astrophysical foregrounds. Previous studies have performed related analyses, looking at the sensitivity of LISA to a primordial SGWB that follows a power-law spectrum (in particular, flat for inflation or cosmic strings) [54,56,57,59,60] or from a PT (with a focus on the model-dependent high-frequency part of the spectrum) [58,61], in presence of astrophysical foregrounds from WD mergers and a power-law SGWB from unresolved BBHs. An interesting study [68] shows the potential for BBH foreground cleaning at LISA by exploiting the measurements at higher frequencies in ground-based experiments. We differ here from previous studies in two aspects. First, we assess more carefully the spectral shape expected for astrophysical foregrounds, discussing in particular the validity of the extrapolation to low frequencies of the slope of the SGWB from BBH (Sec. 2.1, 2.2). Secondly, we make advantage of the statistical method introduced in [64] to optimally filter the primordial signal in presence of foregrounds with known shape and unknown normalisation. This method is particularly suitable for our case, because the spectral shape of the SGWB from BH binaries can be treated as fixed up to a good accuracy in the frequency range of interest, and deviations (due in particular to eccentric binaries) can be shown to have a moderate impact under the assumptions that we discuss. Also the SGWB from WD binaries, after the subtraction of resolvable mergers, will be measured by LISA and we will be able to treat it as a known source in the frequency range where LISA is most sensitive. We begin in Sec. 2 with the discussion of the various astrophysical foregrounds which are relevant to the range of LISA, while in Sec. 3 we review the class of primordial signals (i.e. whose production is causality-limited, like PT) that allows us to perform precision cosmology. We discuss in Sec. 4 the statistical treatment that we use to perform the analysis. In Sec. 5 we show our results for the sensitivity to cosmological observables in a primordial SGWB in presence of astrophysical foregrounds, and in Sec. 6 we summarise our conclusions. Astrophysical backgrounds of gravitational waves In this section, we discuss the most relevant astrophysical foregrounds for the detection of a primordial SGWB, that are the unresolved mergers of binary compact objects, considering in particular the frequency range of LISA. We begin by reviewing the contribution from BBHs in Sec. 2.1 to understand the spectrum of the associated SGWB, and we elaborate on the contribution from eccentric binaries (which is a novelty of this paper) in Sec. 2.2. Then we discuss BWDs, which are another major contribution for the LISA frequency range, in Sec. 2.3. The SGWBs expected from BNS and NS-BH binaries are subleading to the SGWB from BBHs and display a completely analogous spectrum (see e.g. [65,66]), so we do not discuss them further. Black hole binaries (BBH) As mentioned in the introduction, the detection of the SGWB from unresolved BBHs is one of the scientific goals of the LVK collaboration at their design sensitivity. At present, the collaboration has not detected a signal, neither in searches for an isotropic [13] nor anisotropic [67] SGWB. The present upper limit after O3 for the isotropic SGWB, parameterised as a power law, is around Ω (bbh) gw (25 Hz) 10 −8 , with small variations as a function of the powerlaw tilt and the prior on Ω gw . The expected signal lies one order of magnitude below the present sensitivity: a recent assessment by the LVK collaboration, based on the merger rates measured through O3, infers [65] Ω (bbh) gw (25 Hz) = 5.0 +1.4 −1.8 · 10 −10 , (2.1) and we use this central value for our analysis. The calculation of the SGWB from unresolved BBHs is affected by many astrophysical uncertainties, including the star formation rate and the average metallicity as a function of redshift, the BH formation rate, and subsequently the BH binary formation rate, which in turn depends on a variety of possible formation channels [69]. Remarkably, many of these astrophysical effects mostly manifest themselves as an uncertainty on the normalisation of the signal. This is particularly important for us with respect to the knowledge of the spectral shape of the SGWB. The measurement of Ω (bbh) gw (f ) in the final stages of LVK will provide us with observational input for the calibration of the astrophysical uncertainties behind the BBH merger rate. Ground-based experiments are limited by seismic noise to measure this SGWB at frequencies f ∼ 10 − 10 2 Hz, just below its peak around 100-500 Hz [65]. In order to make use of this measurement to deduce the SGWB expected at lower frequencies, we need to extrapolate from the measured amplitude of the BBH foreground with the knowledge of the spectral shape [70]. The prediction from General Relativity (GR) of the spectral shape of the superposition of GWs from BBH mergers is a power law with spectral tilt of 2/3, with some specifications that we discuss below. This scaling can be justified as follows. By highlighting the main frequency dependencies, and remembering that the energy density of GWs is proportional to f 2 , Ω gw (f ) ≡ 1 ρ cr dρ gw (f ) d ln f ∝ f dρ gw (f ) df ∝ f · f 2 A 2 (f ),(2.2) where ρ cr = 3H 2 0 M 2 p is the critical energy density, A(f ) is the amplitude of the emitted GW in Fourier space, and for a circular orbit with non-spinning, point-like sources is (see e.g. [71,Problem 4.1] for a derivation) A(f ) ∝ f − 7 6 ⇒ dΩ gw (f ) d ln f ∝ f 2/3 . (2.3) A few qualifications to this equation are in order. 3) is obtained in flat background, for non-spinning BHs, and neglecting the secondary mass. It is possible to take into account all of these effects systematically by means of a perturbative expansion of GR corrections, known as Post-Newtonian (PN) expansion (see [72] and [71,Chapter 5] for reviews on the topic). The expansion parameter is x ≡ G(m 1 + m 2 )2πf s c 3 = R S 2r ≈ v 2 c 2 , (2.5) where f s is the source rotation frequency, R S = 2G(m 1 + m 2 )/c 2 is the Schwarzschild radius of the source, and v is the velocity of the inspiralling bodies. In order to detect the merger signal out of the large noise, we need to track the number of GW cycles at least up to O(1). This requires a very accurate determination of the inspiralling phase, including corrections up to the order x 7/2 or 3.5PN (see [72] for state-of-the-art results in the field). For the sake of discussing the leading corrections to the GW amplitude in Eq. (2.3), it is enough the consider the first leading correction. We can understand from Eq. (2.5) that the leading order in the PN expansion amounts to neglect GR corrections up to when the inspiralling bodies approach each other to distances close to R S , and reach relativistic speeds. If, for the frequency range that we are interested in (in our case, down to f ∼ mHz), the leading contributions to Ω (bbh) gw (f ) come from light BBHs far from merging (see the discussion at point 1), that is f f s,isco , we see that x ∼ 0 to a good accuracy, and the PN corrections to A(f ) are negligible. In this regime (also called "restricted" PN [71]) where the PN corrections to the A(f ) are neglected, the corrections to Eq. (2.3) due to m 2 > 0, m 1 = m 2 and spins S 1 , S 2 = 0 (which can be found in [72]) are irrelevant. There is only one orbital parameter which is still relevant to Ω (bbh) gw , the eccentricity e (see point 3). 3. Eq. (2.3) is valid for circular orbits with e = 0. This is a good assumption for all the BBHs that are born with small eccentricity, and also for any BBH after some time through the inspiral phase, because the GW emission tends to quickly reduce the eccentricity [73,74]. If a sizable fraction of BBHs arises from a formation channel which leads to large e > 0 at formation, then its GW emissions (for about a decade in frequency through the inspiral) are affected by the eccentricity. We dedicate Sec. 2.2 to discuss how we take this effect into account. In summary, we have justified why it is a good approximation to treat the SGWB from unresolved BBHs as a power law Ω (bbh) gw (f ) ∝ f 2/3 below its peak around 10 − 100 Hz. The class of eccentric binaries, whose amplitude evolution differs from non-eccentric binaries, deserves a more careful assessment, that we discuss in Sec. 2.2, but seems not to affect significantly this prediction down to the frequency range of LISA. A separate effect that could be relevant for this discussion arises when we consider the subtraction from Ω (bbh) gw (f ) of single binaries whose waveform can be individually identified, as studied e.g. by [54,75]. The reconstruction of the BBH parameters for these resolvable contributions is inevitably imperfect, so that there is a residual difference between the actual waveform and the reconstructed one. The superposition of these residuals should still follow the f 2/3 power law, because if both the actual GW amplitude and the reconstructed one follow Eq. (2.3), then their difference should too 1 . For this reason, the subtraction procedure should not affect our discussion. Eccentric black hole binaries As illustrated in the previous section, the most significant contribution that could imply deviations of Ω (bbh) gw (f ) ∝ f 2/3 at f ∼ mHz are the GWs emitted from BBHs with large e at formation. There are known astrophysical environments leading to the formation of BBHs with large eccentricity (see [69] for a review of BBH formation channels). Quieter environments in the gravitational field of the galaxy allow binary stellar systems to evolve via the formation of a common envelope of stellar material towards the end of their life, whose friction brings them close enough that they form individual black holes that merge within a Hubble time. These isolated BBHs typically display e ≈ 0 at formation. Different formation channels lead to the generation of a BBH as a result of dynamical interactions with other stars or BHs, and the eccentricity of the merging binary can be of order 1. Such dynamical BBHs can arise in dense stellar systems, such as star clusters, where 3-body interactions are frequent. 2 One of the scientific goals of the GW community is to infer the fractions contributing to the population of merging BBHs for each formation channel, on the basis of the LVK data. These fractions cannot be discriminated from e at merging, given that the orbits have already circularised, but from the study of of mass, spin and redshift distributions and the comparison with simulated catalogues. Future data and developments of semi-analytical modelling of BBH-forming environments will allow to refine these analyses, and a growing body of studies in the literature (with very few exceptions [79]) finds that a fraction O(0.1 − 1) of the BBHs measured by LVK is of dynamical origin [80][81][82][83][84]. The effect of eccentric binaries on the SGWB from unresolved binaries is the following (see [73,74] for the original analysis). The Fourier spectrum of the GWs emitted by an eccentric binary is continuous, rather than discrete as a circular binary. The peak frequency f p , corresponding to the separation of the mergers at periastron, scales with time asḟ p ∼ f 11/3 p √ 1 − e 2 e→1 → 0. A BBH spends more time emitting GWs at f p if the orbit is eccentric, and more BBHs accumulate in that frequency bin, potentially distorting the power-law shape of the SGWB (see [85] for a recent analysis). In order to assess the net impact of eccentric binaries on the spectral shape of Ω (bbh) gw , the way forward is a comprehensive modelling of many astrophysical variables determining the BBH formation rate. These include in particular metallicity, star formation rate, stellar binary formation rate, efficiency in the ejection of stellar material in the common envelope, and 3-body dynamics in the environment. Many groups are facing this programme with semi-numerical approaches (see e.g. [66,70,86] and references therein), and we can make use of two recent studies to get a concrete expectation of the size of the residual SGWB of eccentric BBH with respect to the power-law f 2/3 : Ω (bbh,e>0) gw (f ) ≡ Ω (bbh) gw (f ) − Ω (bbh) gw (f ) e=0 , (2.6) 2 A known phenomenon leading to eccentric BBHs out of 3-body systems are Lidov-Kozai oscillations [76][77][78]. In these hierarchical systems, a light secondary mass orbits around the heavy primary (inner orbit), and a far heavy perturber orbits around the centre-of-mass of primary and secondary (outer orbit). Inner and outer orbits are inclined by an inclination angle i, and the eccentricity of the inner orbit is e. The conservation of the angular momentum L secondary in the direction of L perturber implies that √ 1 − e 2 cos i is constant. On very long (Kozai) timescales, e and i oscillate: when the inner orbit becomes more inclined (compared to the outer orbit plane), its eccentricity increases. If dissipative processes (like GW emission) reduce the inner orbit radius faster than the Kozai timescale, then the inner system decouples from the perturber and undergoes merging. where Ω (bbh) gw (f )| e=0 ∝ f 2/3 is defined as the SGWB computed by fictitiously ignoring the eccentricity effects on the inspiral. A first ingredient to assess this is an estimate of the fraction of observed BBH mergers coming from dynamic environments that are known to produce eccentric BBHs. The recent state-of-the-art analysis of [86] quantifies, for a few benchmarks choices for the astrophysical inputs, the fractions of mergers from isolated binaries, and from binaries in nuclear star clusters (NSC), globular star clusters (GSC) and young star clusters (YSC). Starting from their population synthesis pipeline, they infer the expected distributions for the BBH merger parameters, which are then compared to data in order to select the more realistic astrophysical benchmarks. We make use of the average fraction of mergers from YSCs from [86] to properly weigh the residual SGWB Ω (bbh,e>0) gw obtained in the followup study [66], which simulates a catalogue of compact objects 3 in YSCs and computes the SGWB from the binaries merging within a Hubble time. The result of our estimate is (for a wide frequency range f µHz) In the remainder of our analysis, we conservatively assume that future refinements of population synthesis studies (informed by new data from resolved mergers at LVK), and the inclusion of additional galactic environments in these analyses, will at most increase the estimate of this SGWB by a factor of 10: Ω (bbh,e>0) gw (f ) 10 −14 (for BBHs in YSC [66]) ,(2.Ω (bbh,e>0) gw (f ) 10 −12 (conservative; this work) . (2.9) We understand that this procedure, admittedly approximate, has the goal of providing us with an educated guess of the final impact of eccentric binaries on the f 2/3 power-law shape, in order to justify our treatment of Ω (bbh) gw in the following (see Sec. 4). As we elaborate further later, we actually believe that progress in the direction of estimating the impact of eccentric binaries on the SGWB expected from BBHs will be very valuable to improve our sensitivity to primordial SGWBs. Before concluding this section, we would like to comment on another possible source of uncertainty. A class of BBHs which could affect the assumption (1) that we discussed after Eq. (2.3) are Extreme Mass-Ratio Inspirals (EMRI), composed by an intermediate-mass and a stellar-mass BHs. The EMRIs that are visible with LISA have a heavy progenitor around 10 5−7 M , which falls in the mass range of the BHs typically hosted at galactic centres. If the merger rates of EMRIs turned out to be comparable to the ones of stellar-mass BHs, their f s,isco would lie around the peak sensitivity of LISA. From the viewpoint of the spectral shape of Ω (bbh) gw (f ), this could imprint distortions on the power-law behaviour around the typical merging frequencies of EMRIs. Recent analyses like [87] find that the EMRI contribution to the SWGB is still smaller than the one from stellar-mass BHs, but the uncertainty in the modelling of their formation rate is significant, and further studies will be relevant. For what concerns the eccentricity of EMRIs, the possible presence of an AGN disk (so-called wet EMRIs) increases the friction and quickly dissipates e, as compared to EMRI without disk (dry EMRIs) where dynamical formation is at play and e > 0. Recent studies [88] find the rate from wet EMRIs to dominate, thus reducing the impact of eccentric EMRIs on deviations from the f 2/3 power law. White dwarf binaries (BWD) White dwarf binaries are an important class of astrophysical foregrounds for GW observations around mHz and below. White dwarfs have a mass typically around a solar mass and are about a thousand times larger than a BH of 1M (and accordingly less compact), so that their binary merger emits much less GW power and the peak frequency is a thousand times smaller. Although less powerful, the large number of such binaries in our own Milky way make them an important foreground for any GW experiment at f mHz. The foreground from extragalactic BWD mergers will be a further source of astrophysical uncertainty. It is expected to be subdominant to the background from galactic BWDs, but could be not negligible as compared to a weakly visible primordial signal, and it is important to improve its modelling [68]. This superposition of GW signals can be controlled and reduced by identifying the loudest binary signals and removing them from the time-domain data stream. This procedure was first exemplified by [89], to which we refer (together with more recent analyses as e.g. [90][91][92]) for a detailed discussion. Starting from the BWD catalogue of [93], Ref. [89] computed the expected SGWB from their mergers. This background, from an operational point of view, adds up on top of the detector noise. It is possible to identify and remove the loudest BWDs contributing to this background: for each of the BWD of the catalogue, they computed their Signal-to-Noise Ratio (SNR) where the noise is understood as the sum of instrumental noise and BWD background. The loudest mergers that pass a threshold SNR are subtracted from the noise, reducing its size. The procedure can be iterated, and converges to an irreducible background of unresolved BWDs, after the successful removal of ∼ O(10 4 ) of the loudest binaries. The key feature that allows the removal of the loudest binaries is that they are approaching the merger phase (when the amplitude of the emitted GWs increases) and their GW frequency is sweeping more and more rapidly the frequency range. Therefore, if we consider the number of binaries from a sample population that is emitting GWs in a given frequency bin, it gets smaller as we consider bins at higher frequencies. When this number reduces to a few, then the corresponding BWD signals are resolvable and can be subtracted; at lower-frequency bins, the superposition of multiple signals is instead not resolvable, originating an irreducible foreground [89]. For this high-frequency bins where the expected number of sources is of order one and decreasing, the number of detected BWDs should follow the tail of a Poissonian distribution, motivating an exponential decay in the number of unresolved sources. This is the reason why a common parametrisation [90][91][92]94] for the SGWB from BWDs has an exponential cut-off for f > f knee (last term in this equation): Ω (bwd) gw (f ) 4π 2 f 3 3H 2 0 A (bwd) Hz f Hz −7/3 e − f f 1 α 1 + tanh ((f knee − f )/f 2 ) 2 (2.10) where H 0 = h · 100 km/s/Mpc is the present Hubble rate, and f 1 , α, f knee , f 2 are fit parameters. 4 The knee-frequency where the confusion noise drops slowly decreases with the time duration of the mission, as more BWDs can be identified and subtracted. For our 4 Refs. [95,96] (with a sign correction in [97]) modify the second-to-last term in Eq. (2.10) into exp − (f / Hz) α − β sin(κf ) , where the sin term does not have a physical interpretation and ends up being numerically almost irrelevant for the best-fit parameters. Also, we consider physically more appropriate to include a free-parameter f1 ∼ mHz in the fit, rather than fixing it to 1 Hz where Ω (bwd) gw is totally negligible. We decide then to follow the more recent convention in Eq. (2.10). At any rate, this factor has the only purpose of accommodating a better fit to the residual confusion noise around the knee frequency. analysis, we adopt from [91] (which is based on the BWD catalogue from [98]) the best-fit values A (bwd) = 2.7 · 10 −44 , f 1 = 0.64 mHz, α = 1.26, f knee = 2.0 mHz, f 2 = 0.28 mHz. In summary, for our purposes the relevant point about the confusion noise of galactic BWDs is that, after removal of the resolvable loudest binaries, the residual foreground falls exponentially at f ∼ mHz, around the peak sensitivity of LISA. It is therefore safe to assume that the frequency range above this value will not be significantly contaminated by the foreground from BWDs [68]. The exponential drop in Ω (bwd) gw is the main spectral feature that is relevant in the following. Primordial backgrounds of gravitational waves for precision cosmology Among the possible primordial GW backgrounds that could populate our Universe, which include backgrounds generated during inflation [22-24, 27, 28, 99], the subsequent reheating phase [100][101][102][103][104][105], phase transitions (PT) [44] and by topological defects (such as cosmic strings) [10], or finally by scalar perturbations at 2nd order in perturbation theory [106][107][108][109][110][111], in this paper we are interested in the exciting prospect of precision measurements of cosmological parameters from the GW spectrum. Many studies discussed related effects and their observability in various scenarios [30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49]. Similar effects, like variations in the number of relativistic degrees of freedom g , or in the equation-of-state parameter w of the universe, manifest themselves as modifications to the spectral shape of a primordial SGWB. To this end, we should know the primordial shape of the signal, before the cosmological history affected it. This can be confidently assumed for the class of GW backgrounds generated by phenomena that occur locally in space and time, independently in each Hubble patch. In this regime, the principle of causality fixes the spectral shape of the SGWB at low frequencies, corresponding to wavelengths larger than the correlation length of the GW source [50]. In a radiation-dominated Universe, and if the GW propagate as free waves, the spectrum at low frequencies must go as f 3 [112]. As illustrated in [50,51], this behaviour can only be affected by the evolution of the universe (i.e. by modifications to w, or g and hence to the Hubble rate) or by the propagation of the GWs, which can be damped by anisotropic stress sourced e.g. by free-streaming relativistic particles. The primary example of a causality-limited source of a SGWB is a first-order PT. In this paper, we consider for concreteness a primordial GW signal sourced by a PT, and we assess the sensitivity to cosmological parameters from the causality-limited tail of the spectrum, similarly to [51] but accounting also for the presence of astrophysical GW foregrounds. We consider a PT occurring when the Universe has a temperature T pt , whose duration is β −1 and releasing an energy fraction α of the total energy density of the Universe. The GW energy spectrum consists of the contributions from various sources (bubble collisions, sound waves and turbulence in the plasma; for recent reviews, see [44,113]); for simplicity, we consider the contribution that is typically the largest, i.e. from sound waves in the plasma. The results of numerical simulations can be approximated by the analytical formula [114] h 2 Ω (pt) gw (f ) = 1.2·10 −6 H * β κ v α 1 + α 2 106.75 g 1 3 f f * 3 7 4 + 3(f /f * ) 2 7 2 ·F(f, θ) (3.1) where the peak frequency f * is roughly of the order of the Hubble rate at T pt , and corresponds to a present-day frequency f * = 9 · 10 −3 mHz 1 v w β H * T pt 100 GeV g 106.75 1 6 . (3. 2) The energy fraction α mostly affects the normalisation of Ω (pt) gw , and the inverse duration of the PT β shifts Ω (pt) gw both vertically and horizontally. Other parameters of the PT are the latent heat fraction that gets converted into bulk motion of the fluid κ v , and the bubble wall terminal velocity v w . For definiteness, we fix the following values for the PT parameters: β = 4H * , α = 0.1 , v w = 1 , g = 106.75 , κ v = α 0.73 + 0.083 √ α + α . (3.3) We consider the following possible modifications to the low-frequency tail of the GW spectrum. In the case when the equation-of-state parameter of the Universe is w = 1 3 at the time when the Hubble rate is equal to a frequency f w ≤ f * shortly after the PT, the prefactor is [50] 5 F(f, w) = f f w 2 j 1−3w 1+3w f f w 2 + y 1−3w 1+3w f f w 2 (3.4) where j n (z), y n (z) are spherical Bessel functions (notice that F(f, w = 1 3 ) = 1). The impact of this factor is shown in Fig. 1 for a few values of w: for a standard radiation-dominated cosmology, w = 1 3 and the scaling of the SGWB at low frequencies is f 3 , while for different values of w the spectral tilt is 1+15w 1+3w . Another unique probe of the early Universe that is offered by the causality-limited tail of the GW spectrum is the measurement of the fraction f fs of energy density of relativistic free-streaming particles around T pt . In presence of free-streaming particles that modify the propagation of GWs in the universe, the modification to the primordial background in Eq. (3.1) is F(f, f fs ) = f f * 2 j αfs f f * 2 + y αfs f f * 2 , α fs = − 1 2 + 1 4 − 8 5 f fs (3.5) where f * is the peak frequency given in Eq. (3.2). In the absence of free-streaming particles, F(f, f fs = 0) = 1, whereas for f fs > 0 the tilt at low frequencies becomes steeper, from 3 to 3 + 16 5 f fs (and for f fs > 5 32 saturates to 4, with an additional small oscillating pattern imprinted on the power-law spectrum [50]) 6 . Optimal estimator for a primordial SGWB in presence of astrophysical foregrounds As we discussed in Sec. 2, the two main astrophysical foregrounds in the frequency range mHz − Hz are the unresolvable superpositions of GWs from mergers of BBHs and BWDs, and we justified why both their spectral shapes have two distinguishing features. The BBH background can be approximated to a good accuracy as a f 2/3 power law down to Ω (f ) can be expected (see Eq. (2.9)). The BWD background (arising from the confusion noise of binaries that cannot be isolated in a frequency bin and subtracted), is supposed to fall off exponentially around f knee (see Eq. (2.10)). Therefore, we argue that the leading uncertainties in these astrophysical foregrounds are encapsulated in one free parameter each, that is their overall normalisation. This will be our working assumption in the remainder of the paper, and we have provided justifications for this hypothesis in the discussion of Sec. 2. We then assess the sensitivity to a primordial signal by performing a simple Fisher matrix analysis. The optimal filter, that enhances the SNR for a given signal by accounting for the presence of astrophysical foregrounds, is derived in [64]. We summarise here those results and we refer to [64] for a complete and detailed discussion. Let us denote the signal d I (t) measured in the channel I of a GW detector (I can run e.g. over the two independent channels of the LISA proposal, or over the interferometers of LVK) as the sum of a GW signal s I (t) and the noise n I (t) in the detector: d I (t) = s I (t) + n I (t) , (4.1) where the the signal s I (t) is related to the GW in frequency space h P (f,n) (P being the polarisation +, ×, andn the unit vector in the direction of propagation k) through s I (t) = P =+,׈+ ∞ −∞ dfˆd 2n F (P ) I (n, f ) h P (f,n) e i2πf (t−n· x I ) , (4.2) where the response function F (P ) I (n, f ) of a channel I depends on the properties of the detector. We define the spectral densities of signal and noise, h * P (f,n)h P (f,n ) = 1 8π δ (2) (n −n ) δ P,P δ(f − f ) S h (f ) , (4.3) n I (f )n * J (f ) = 1 2 δ(f − f ) N IJ (f ) . (4.4) The signal can be extracted from the correlation estimator x IJ =ˆT /2 −T /2 dtˆT /2 −T /2 dt d I (t)d J (t ) − n I (t)n J (t ) Q IJ (t, t ) , (4.5) where T is the time duration of the experiment (we take T = 4y for LISA), and Q is an arbitrary filter function that we choose in order to maximise the sensitivity to signal. We now define the signal spectral density as the sum of a primordial and an astrophysical component (we discuss here for simplicity the case of one astrophysical foreground, and we refer to App. A in [64] for the formulae in the case of multiple foregrounds) S h (f ) = S prim (f ) + A astro S astro (f ) , (4.6) where A astro is an unknown normalisation parameter for which some information is available. In particular, we denote by A astro its expected value and by σ 2 astro its variance. We can introduce a modified estimator in presence of astrophysical foreground y IJ = x IJ − A astro Tˆ∞ 0 df S astro (f ) R IJ (f ) Q IJ (f ) , (4.7) where R is the response function defined by R IJ (f ) ≡ 1 4πˆd 2n P =+,× F (P ) * I (n, f ) F (P ) J (n, f )e −i2πfn·( x I − x J ) ,(4.8) depending on the properties of the detectors I, J and their relative orientation and distance. The expectation value of y IJ gives the primordial signal in frequency space: y IJ = Tˆ∞ 0 df S prim (f ) R IJ (f ) Q IJ (f ) . (4.9) We now define the signal-to-noise ratio SNR as snr = y IJ y 2 IJ − y IJ 2 , (4.10) and we want to choose an optimal filter function Q IJ (f ) in frequency space to maximise the SNR for a given primordial signal. It is possible to show that this is equal to the Fisher information on the amplitude of the signal, under the assumption of Gaussianity [55,115,116]. We can express the final result of [64] in the following compact form, which allows a geometrical interpretation of the result. First, we define the following scalar product between integrable functions in frequency space, u · v ≡ 2Tˆ∞ 0 df N 2 IJ (f ) u * (f ) v(f ) ,(4.11) where the noise spectral density N 2 IJ (f ) is given by N 2 IJ (f ) ≡ N II (f ) N JJ (f ) + N 2 IJ (f ) . (4.12) We also denote s ≡ R IJ (f ) N 2 IJ (f ) S prim (f ) , a ≡ R IJ (f ) N 2 IJ (f ) S astro (f ) . We can then write the optimal filter function in presence of astrophysical foregrounds simply as Q IJ (f ) = s − a s · a σ −2 astro + a · a , (4.14) as opposed to the standard optimal filter in the absence of foregrounds, which is just Q IJ (f ) = s. The optimal filter ranges from s in the limit σ astro → 0 (perfectly known foreground) to a function becoming increasingly orthogonal to a for larger σ astro . In other words, in presence of foregrounds, the optimal filter counterweights the astrophysical components by subtracting a component proportional to a while computing the SNR. This can be nicely seen in the illustrative example of [64, Fig. 3] where, for growing values of σ astro , Q IJ (f ) becomes negative across some frequency ranges, so that the filter does not pick up power from foreground-like spectral shapes. Finally, the SNR in presence of an astrophysical foreground can be simply written as snr 2 = s · s − (s · a) 2 σ −2 astro + a · a ,(4.15) which highlights the reduction of the SNR when the foreground a has a "parallel" component to the signal s, i.e. when the spectral shapes of signal and foreground are similar. Two limits are instructive to consider. The SNR approaches its value s · s without foregrounds when s · a s · s, that is when s and a are dominated by different frequency ranges and their shapes do not significantly overlap. Another limit in which the SNR recovers the value without foregrounds is when σ astro → 0: in that case, the foreground is perfectly known and our sensitivity to any signal sitting above it is only mildly affected. We are interested in the case in which the unknown parameter θ that we want to measure in the signal is not the amplitude of such signal (that we assume to be measured), but a cosmological parameter θ = w or f fs distorting the low-frequency tail of the signal (see Sec. 3), that affects the signal non-linearly as in Eqs. (3.4) and (3.5). We can apply the previous results to the estimate of the sensitivity to the parameter θ by including s among the templates (in the generalisation of Eq. (4.15) to multi-component subtraction, see [64]), and inserting ∂ θ s (rather than s) in Eq. (4.15). Under the same assumptions mentioned above, the snr 2 = 1/σ 2 θ can be shown to be the Fisher information on θ. We have now introduced all the ingredients to compute the sensitivity to cosmological parameters in a primordial GW signal, in presence of astrophysical foregrounds. Results We now apply the procedure illustrated in detail in [64] and outlined in Sec. 4 to the scenario that we have been considering throughout this paper. We would like to quantify the sensitivity to the spectral features in the low-frequency (causality-limited) range of the primordial SGWB generated by a local phenomenon like a PT, in presence of astrophysical GW foregrounds. We have discussed in Sec. 2 why it is a good assumption to treat the SGWB spectra from BWD and from BBH mergers as known up to a normalisation factor, and up to a contribution from eccentric BBHs that we have quantified in Eq. (2.9) and that we separately keep into account as Ω (bbh,e>0) gw in the following. Therefore we model Ω For what concerns the contribution Ω (bbh,e>0) gw from eccentric BBH mergers, we model it as a constant foreground in frequency space, with a normalisation given in Eq. (2.9) with an uncertainty σ (bbh,e>0) . This assumption is partly motivated by expectation that, in the frequency range of LISA, its spectrum should not vary by orders of magnitude, and partly by the results of [66] 7 , which find a featureless spectrum. We do not expect this assumption to impact much on our results. The reason is that, as long as the actual component Ω (bbh) gw (f ) is not similar in spectrum to the primordial signal, and is limited from above not to exceed a value of the order of Eq. (2.9), its precise shape does not modify significantly the optimal filter function. This statement can be reinforced in the future by means of further progress from the joint effort of astrophysical modelling and GW observations to corner down the uncertainties on the contribution from eccentric binaries. We show the results of our analysis for two significant cases of primordial signals, as we discussed in Sec. 3. We consider the potential to perform precision cosmology after the discovery of a primordial signal from a PT, for which we assume a spectral shape as in Eq. without including the uncertainty on astrophysical foregrounds (red curve), accounting for BWDs and BBHs (blue dot-dashed), and by also adding the background from eccentric BBHs (green dashed). The relative uncertainties on the astrophysical foregrounds are taken to be 1; their exact value does not matter, as the optimal filter removes all the signal component parallel to the foreground (see Eq. (4.15)). Bottom right plot: the same as the left plot, but with σ (bbh) = σ (bwd) = 2%, to show the typical fractional uncertainty below which astrophysical foregrounds are known well enough not to degrade the sensitivity to the signal. In each figure, the top panel shows: • the LISA noise sensitivity curve (for the AA and EE channels; we refer to [117] for the calculation of LISA noise curves and response functions). We remind the reader that these strain sensitivity curves should not be confused with the power-law integrated sensitivity curves (that stretch down to Ω gw ∼ 10 −13 for LISA, see e.g. [116, Fig. 2]), which represent the envelope of the upper limits at each frequency on the GW power-law signal that is detectable with a given SNR. While the latter allows for a quick graphical check about whether a GW background is detectable, the original noise sensitivity curve is required to compute the SNR; Presence of free-streaming particles f FS = 0 f FS = 5% f FS = 15% • the primordial signal for a few values of w, f fs and for the PT parameters listed in the box on the upper right. The dotted curve, for reference, shows the input for the calculation of the sensitivity to θ = w, f fs , that is ∂ θ Ω The lower panels of Figs. 1 and 2 show the main results of this paper, that is the sensitivity to variations in w (or f fs ) around a reference value. For simplicity, in these plots we vary only this parameter, and keep the other PT parameters fixed: the reason is that the impact of w, f fs (which affects the low-frequency range of the spectrum) is quite distinct from the other parameters, which do not alter the causal part of the primordial SGWB, so there aren't significant degeneracies. The three curves show σ w (or σ ffs ) for the following assumptions: • red curve: without accounting for the uncertainty on any astrophysical foreground (the results are in agreement with [51]). This curve is peaked for f * around the maximum sensitivity of LISA (around a few mHz) where the signal is highest above the noise, and is weaker elsewhere, roughly mirroring the noise curve of LISA; • blue dot-dashed curve: accounting for the unknown normalisation of Ω (bwd) gw (f ) and Ω (bbh) gw (f ) with the fractional uncertainties σ (bwd) , σ (bbh) listed in the legends. The bottom left plot assumes these astrophysical uncertainties to be ∼ O(1): their actual value does not matter quantitatively, because for this size of σ, the component of the signal parallel to the foregrounds is discarded entirely (see Eq. (4.15)), which makes the forecast largely insensitive to the actual amplitude of the foregrounds. The impact of a foreground on the deterioration of the SNR depends on its degeneracy in frequency shape with the primordial signal. The bottom right plot shows the values of σ (bwd) , σ (bbh) (below ∼ 5% of fractional uncertainty) at which the astrophysical foregrounds are constrained enough that the degradation of the SNR starts to be less significant. • green dashed curve: the same as the previous curve, with the inclusion of Ω (bbh,e>0) gw from the eccentric binaries. We keep σ (bbh,e>0) = 1 in both the bottom plots, as this background suffers from larger uncertainties. We can draw the following conclusions from the results of Figs. 1 and 2. As we can see from the bottom right plots, in order to accurately detect a peaked primordial signal we need to know the normalisation of the astrophysical signal with an accuracy around a few percent, for its impact on the SNR to be negligible. Such a precision is unrealistic, given the wide set of astrophysical uncertainties that are encapsulated in the normalisation of these foregrounds, so we cannot neglect these uncertainties and they have an impact on our analysis. As we quantitatively assess their impact, though, we can see from the bottom left plots that, even when their normalisation is highly uncertain, the knowledge of their spectral shape strongly reduces the degradation of the SNR. The increase in σ w , σ ffs is by a factor of a few, and only matters where the primordial and astrophysical signals peak around the same frequencies. As long as our modelling of the astrophysical foregrounds allows us to exclude "non-standard" contributions with a shape (and amplitude) comparable to the primordial signal we are looking for, those uncertainties do not hinder our discovery potential with GW detectors. Conclusions As we advance in the era of GW astronomy, ongoing measurements of the BBH merger rate from LVK improve the estimate of the expected GW background from unresolved BBH mergers, that should be eventually measured by the collaboration at the achievement of their design sensitivity. This measurement will corner down part of the uncertainties on the astrophysical foreground from BBH mergers that are expected (together with BWDs) to populate the lower GW frequencies that will be probed by incoming space-based missions, such as LISA. Keeping these astrophysical foregrounds under control is essential to our power of discovering a primordial GW background, which would enable us to test the physics describing primordial epochs which are complementary to what is testable through current probes In the case of a causality-limited GW generation (such as a phase transition), this would open a window for precision cosmology through the accurate measurement of the low-frequency range of the spectrum [50]. The knowledge of the astrophysical foregrounds is essential to this goal. In this paper, we consider them in detail in the frequency range of LISA, and we show that it is possible (with some assumptions that we detail in the following) to describe them with a linear model, where the unknown parameter is a prefactor multiplying a known spectral shape. We then use the simple and numerically powerful formalism described in [64] to estimate the SNR by marginalising over the astrophysical foregrounds. We find that the sensitivity to a primordial signal that is not degenerate in spectrum with the astrophysical foregrounds remains promising. There are three main assumptions that underlie this treatment for LISA, as illustrated in Sec. 2. Two of them concern the extrapolation of the BBH foreground as a f 2/3 power law from the 10 − 100 Hz range of LVK to the mHz range. First, we need to confirm (as currently supported by e.g. [87]) that the largest contribution to Ω (bbh) gw (f ) at f ∼ mHz − Hz comes from the inspiral phase of lighter BHs merging in the LVK band, and not from heavier BHs (such as EMRIs) merging at lower frequencies. A second point of concern is the impact of eccentric BBHs on Ω (bbh) gw , as e is the only binary parameter that significantly modifies the amplitude |h(f )| at f f isco . Recent population synthesis studies [66,70,82,86] support the hypothesis that eccentric binaries do not alter Ω (bbh) gw ∝ f 2/3 above Ω gw ∼ 10 −13 − 10 −12 . Future developments, regarding in particular the astrophysical modelling of dynamical channels for BBH formation, refinements on BH and BBH population synthesis, and observational input from LVK, will reinforce these findings. A third relevant astrophysical ingredient is the confusion foreground from BWDs which cannot be individually resolved. Recent studies for the galactic Ω (bwd) gw , as [90][91][92], confirm that this foreground will fall exponentially at f > mHz above the peak sensitivity of LISA, and future developments of population synthesis studies will refine our modelling of the residual foreground at lower frequencies. Improvement in the estimation of the foreground from extragalactic BWDs is also required (see e.g. [68] for work in this direction), to assess its potential degeneracy with a weak primordial signal. After having justified our modelling of the astrophysical uncertainties, we show how the sensitivity to cosmological parameters for precision cosmology is not dramatically impacted by the uncertainty on the foreground normalisation. As visible in Figs. 1 and 2, the sensitivity to the equation-of-state parameter of the Universe w or to the fraction of free-streaming particles f fs after a phase transition, worsen only by a factor of a few, and only where the signal is closer in shape and frequency to an astrophysical foreground. This conclusion is guaranteed as long as the signal s is not degenerate in frequency space with an unknown astrophysical foreground a, in the mathematical sense of s · a s · s, and as long as the signal is the larger than the variance on the astrophysical normalisation, σ 2 astro s · s (see the definition of this scalar product as an integral over frequencies in Eq. (4.11), (4.15)). Our findings offer bright prospects for the detection of a primordial GW signal even in presence of astrophysical foregrounds, and at the same time highlight the importance of an accurate modelling of the astrophysical foregrounds from BBH and BWD mergers, since the knowledge of their spectral shape is essential to reduce their impact on the signal sensitivity. A major reward of this programme is an unhindered potential to perform precision cosmology upon detection of a primordial GW signal. Contributions. D.R. lead the project, defined the analysis for all aspects related to the astrophysical and primordial signals, and wrote the manuscript. D.P. curated the foreground marginalization, the SNR methodology and their implementation, and contributed to the discussion of the results. and (2.10) up to the normalisation factors not being perfectly known, and we assume them to lie within fractional uncertainties σ (bbh) , σ (bwd) from their central values. (3.1). Two key observables are the equation-of-state parameter w of the Universe for a few e-folds after the PT, and the energy fraction f fs of relativistic free-streaming particles in the Universe at the time of the PT. The two cases are shown respectively in Figs. 1 and 2. σσFigure 1 . 1: β = 4H * , α = 0.1, f w = 0.5 f * Variation of w after PT BWD = σ BBH,e=0 = σ BBH,e>0 = σ PT = 1 w/ astro fg (BWD, BBHe=0, BBHe>0) w/ some astro fg (BWD, BWD = σ BBH,e=0 = 0.02, σ BBH,e>0 = σ PT = 1 w/ astro fg (BWD, BBHe=0, BBHe>0) w/ some astro fg (BWD, Impact of the uncertainty of astrophysical foregrounds on the measurement of a primordial GW background signal. Top plot: comparison of the noise sensitivity of LISA to Ω gw (f ) (green shaded curve), main astrophysical foregrounds in this frequency range (binary White Dwarfs in purple, Binary Black Holes extrapolating Ω (bbh) gw ∝ f 2/3 in blue, and contribution from eccentric binaries in green; see Sec. 2), a primordial Phase transition followed by a phase with w = 1 3 (red-orange curves, and ∂ w Ω(pt) gw w=1/3 is shown by a dotted line; see Sec. 3). Bottom left plot: sensitivity to σ w around w = 1 3 β = 4H * , α = 0.1 σσFigure 2 . 2BWD = σ BBH,e=0 = σ BBH,e>0 = σ PT = 1 w/ astro fg (BWD, BBH e=0 , BBH e>0 ) w/ some astro fg (BWD, BBH e=0 ) BWD = σ BBH,e=0 = 0.02, σ BBH,e>0 = σ PT = 1 w/ astro fg (BWD, BBH e=0 , BBH e>0 ) w/ some astro fg (BWD, BBH e=0 ) The same asFig. 2, but looking for the presence of an energy fraction f fs of free-streaming particles in the Universe at the time of the phase transition. The top plot shows three reference values for f fs , and we consider in the bottom plots the sensitivity to σ ffs around f fs = 1%. f ) (see Sec. 4);• the three astrophysical foregrounds discussed in Sec. 2 (BWDs, BBHs, eccentric BBHs). We notice that in[54] the residual Ω (bbh) gw after subtraction deviates from f 2/3 , but this effect disappears when accounting for the reconstruction error on more parameters than just chirp mass Mz, coalescence phase φc and coalescence time tc, as apparent from Figs. 6 and 7 of[75]. In the findings of[66], only BBHs end up being relevant in terms of SGWB from unresolved mergers, while BNS and NS-BH binaries give a sub-leading contribution. Strictly speaking, also the physical frequency f is affected if we modify w with respect to the radiationdominated case, as the total expansion factor of the universe changes. 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{'abstract': 'The era of Gravitational-Wave (GW) astronomy will grant the detection of the astrophysical GW background from unresolved mergers of binary black holes, and the prospect of probing the presence of primordial GW backgrounds. In particular, the lowfrequency tail of the GW spectrum for causally-generated primordial signals (like a phase transition) offers an excellent opportunity to measure unambiguously cosmological parameters as the equation of state of the universe, or free-streaming particles at epochs well before recombination. We discuss whether this programme is jeopardised by the uncertainties onIntroductionThe first direct observation [1] of Gravitational Waves (GWs) performed in 2015 by the interferometers of LIGO, whose measurements were joined shortly after by the companion VIRGO and later by KAGRA, marked the onset of the GW astronomy era. After that first detection of the merger of a black hole binary (BBH), the catalogues released by the LVK consortium after the run O3 of observation [2, 3] collect almost 90 compact binary coalescences, predominantly BBH mergers together with two binary neutron star (BNS) mergers (including the first multi-messenger detection of a merger [4-6]), and a few candidate BH-NS merger. Direct observations of GWs offer a completely new window to explore the Universe, which has spurred many directions to interpret the incoming wealth of data for a variety of scientific programs in astrophysics, cosmology and fundamental physics[7][8][9].Besides the observation of individually resolved mergers, another frontier lies ahead in GW astronomy: the detection of the stochastic GW background (SGWB) filling our Universe (see[10,11]for recent reviews). This background is generated by the stochastic superposition of multiple GWs travelling cosmological distances and originates from various contributions. A class of GW backgrounds is of astrophysical nature, and consists of the incoherent superposition of the GWs generated by all the binaries that have merged throughout the cosmic history, including BBH, BNS (which are subleading in terms of SGWB), and White Dwarfs binaries (BWD).Along with this background, that will be measured [12] by the end of the mission of LVK[13]or at the latest by future ground-based (CE, ET) [14-17] and space-based (LISA, BBO, MAGIS, DECIGO)[18][19][20][21]experiments, there could lie a primordial signal generated in the early cosmological history[22][23][24][25][26][27][28]. Such a discovery would offer to us the farthest signal that we have ever detected, way beyond what is presently visible in the Cosmic Microwave Background (CMB). GWs can travel to us (almost [29]) unperturbed, offering us with a pristine glance into the earliest cosmological epochs. For these reasons, a primordial GW background can be a powerful tool to probe the cosmological evolution and the particle', 'arxivid': '2212.06602', 'author': ['D Racco dracco@stanford.edu \nStanford Institute for Theoretical Physics\nStanford University\n94305StanfordCAUSA\n', 'D Poletti \nUniversity of Milano-Bicocca\nPiazza della Scienza3 -20126Milano (MI)Italy\n'], 'authoraffiliation': ['Stanford Institute for Theoretical Physics\nStanford University\n94305StanfordCAUSA', 'University of Milano-Bicocca\nPiazza della Scienza3 -20126Milano (MI)Italy'], 'corpusid': 254591508, 'doi': '10.1088/1475-7516/2023/04/054', 'github_urls': [], 'n_tokens_mistral': 32808, 'n_tokens_neox': 26163, 'n_words': 14794, 'pdfsha': '782fc58c147a53e433f99991c801e28a29bb5f4b', 'pdfurls': ['https://export.arxiv.org/pdf/2212.06602v1.pdf'], 'title': ['Precision cosmology with primordial GW backgrounds in presence of astrophysical foregrounds', 'Precision cosmology with primordial GW backgrounds in presence of astrophysical foregrounds'], 'venue': []}

arxiv

The Taylor Interpolation through FFT Algorithm for Electromagnetic Wave Propagation and Scattering 10 Oct 2006 March 31, 2022 Shaolin Liao sliao@wisc.edu Department of Electric and Computer Engineering University of Wisconsin 1415 Engineering Drive53706Madison, MadisonWIUSA The Taylor Interpolation through FFT Algorithm for Electromagnetic Wave Propagation and Scattering 10 Oct 2006 March 31, 2022arXiv:physics/0610057v1 [physics.comp-ph]numbers: 4120Jb8440-x9430Tz MSC numbers: 41A5841A6065D1565Dxx68W2583C50 The Taylor Interpolation through FFT (TI-FFT) algorithm for the computation of the electromagnetic wave propagation in the quasi-planar geometry within the half-space is proposed in this article. There are two types of TI-FFT algorithm, i.e., the spatial TI-FFT and the spectral TI-FFT. The former works in the spatial domain and the latter works in the spectral domain. It has been shown that the optimized computational complexity is the same for both types of TI-FFT algorithm, which is N optopt r is the optimized number of slicing reference planes and N opt o is the optimized order of Taylor series.Detailed analysis shows that N opt o is closely related to the algorithm's computational accuracy γ TI , which is given as N opt o ∼ − ln γ TI and the optimized spatial slicing spacing between two adjacent spatial reference planes δ opt z only depends on the characteristic wavelength λ c of the electromagnetic wave, which is given as δ opt z ∼ 1 17 λ c . The planar TI-FFT algorithm allows a large sampling spacing required by the sampling theorem. What's more, the algorithm is free of singularities and it works particularly well for the narrow-band beam and the quasi-planar geometry. I. INTRODUCTION The computation of electromagnetic wave propagation using the direct integration method is not efficient for the large-scale computation because the direct integration method has a daunting computational complexity of O (N 2 ) for an N = N x × N y computational grid, e.g., in the beam-shaping mirror system design for the Quasi-Optical (QO) gyrotron application, days of computation is required [1,2,3,4,5]. Fortunately, when the computational geometry is a plane, the FFT has been shown to be efficient in the electromagnetic wave computation [6,7,8], which has a computational complexity of O(N log 2 N) and a low sampling rate only limited by the Nyquist rate. For the quasi-planar geometry, it will be shown in this article that the FFT can still be used with the help of the Taylor Interpolation (TI) technique. The rest of this article is organized as follows. Section II gives the 2-Dimensional (2D) Fourier spectrum of the electromagnetic wave in its closed-form expression. Section III presents the optimized spatial and spectral types of TI-FFT algorithm. In Section IV, one numerical example is used to show the performance of the planar TI-FFT algorithm. Section V discusses the advantages and problems of the planar TI-FFT algorithm; some helpful suggestions are given. Finally, Section VI summarizes the planar TI-FFT algorithm. The scheme used to illustrate the planar TI-FFT algorithm is shown in Fig. 1 and the time dependence e jωt has been assumed in this article. II. ELECTROMAGNETIC WAVE IN THE SPECTRAL DOMAIN In this section, the 2D Fourier spectrum and far-field of the electromagnetic wave for the radiation integral are shown to be closely related to each other. A. The radiation integral For given electric and magnetic surface currents (J s , J ms ), the radiating electric field E can be obtained under the Lorenz condition [9,10], which is given as where, ∇ ′ is the gradient operator on the source coordinate r ′ and the scalar Green's function is given as G(R) = e −jk|R| 4π|R| , R ≡ r − r ′ .(2) B. The 2D Fourier spectrum of the scalar Green's function Now apply the 2D Fourier transform on the scalar Green's function G(R) in (2), G(k x , k y , r ′ ) ≡ FT 2D    G(R)    = 1 2π ∞ x=−∞ ∞ y=−∞ e −jk|R| 4π|R| e jkxx e jkyy dxdy,(3) where k z and the 2D Fourier transform has been defined as k z =      k 2 − k 2 x − k 2 y , k 2 x + k 2 y < k 2 −j k 2 x + k 2 y − k 2 , k 2 x + k 2 y ≥ k 2 ,(4)FT 2D · = 1 2π ∞ x=−∞ e jkxx ∞ y=−∞ · e jkyy dy dx,(5) From (3), G(k x , k y , r ′ ) = 1 2π e jkxx ′ e jkyy ′ ∞ x=−∞      e jkx(x−x ′ ) × ∞ y=−∞    e −jk|R| 4π|R| e jky(y−y ′ )    dy      dx,(6)Changing variables u = x − x ′ , v = y − y ′ and w = z − z ′ , (6) becomes, FT 2D    G(R)    = 1 2π e jkxx ′ e jkyy ′ ∞ u=−∞      e jkxu × ∞ v=−∞    e −jk|R| 4π|R| e jkvv    dv      du(7) In the cylindrical coordinate, |R| = (r ⊥ ) 2 + w 2(8) where r ⊥ = u 2 + v 2 and the following relation can be obtained from (8), dr ⊥ = |R| r ⊥ d|R|(9) Now, express (7) in the cylindrical coordinate with the help of (9), G(k x , k y , r ′ ) = 1 4π e jkxx ′ e jkyy ′ ∞ |R|=|w|      e −jk|R| 1 2π 2π φ=0    e −jk ⊥ r ⊥ cos(ψ−φ)    dφ      d|R|(10) where ψ = arctan ky kx and φ = arctan v u . The integration over φ is the Bessel function of the first kind of order 0 and (10) reduces to G(k x , k y , r ′ ) = 1 4π e jkxx ′ e jkyy ′ ∞ |R|=|w|    e −jk|R| × J 0    k ⊥ |R| 2 − w 2       d|R| = −j 4πk z e jkxx ′ e jkyy ′ e −jkz|z−z ′ |(11) Because only half-space z > z ′ is of interest, only the 2D Fourier spectrum for half-space z > z ′ will be considered in the rest of this article, which is obtained from (11) as G > (k x , k y , r ′ ) = −j 4πk z e jk·r ′ e −jkzz(12) C. 2D Fourier spectra of Green's function related expressions The 2D Fourier spectra of the derivatives (order n) of the scalar Green's function can be obtained from the property of the Fourier transform [7], ∂ (n) G(R) ∂τ (n) =⇒ (−jk τ ) n G > (k x , k y , r ′ ), τ = x, y, z.(13) Particularly, for the first-order and second-order derivatives, ∂G(R) ∂τ =⇒ −k τ 4πk z e jk·r ′ e −jkzz , τ = x, y, z.(14)∂ 2 G(R) ∂τ 2 =⇒ jk 2 τ 4πk z e jk·r ′ e −jkzz , τ = x, y, z.(15) Similarly, the 2D Fourier spectra of the following expressions can be obtained for half- space z > z ′ , FT 2D ∇G(R) =⇒ −jkG > (k x , k y , r ′ ). (16) FT 2D ∇ 2 G(R) =⇒ −k 2 G > (k x , k y , r ′ ).(17) FT 2D ∇∇G(R) =⇒ −kkG > (k x , k y , r ′ ). (18) FT 2D G e (R) =⇒ G > (k x , k y , r ′ ) I − kk k 2 .(19)FT 2D G m (R) =⇒ −jG > (k x , k y , r ′ ) k × I .(20) where the dyadic Green's functions of the electric type (G e ) and the magnetic type (G m ) are given as G e (R) = I + 1 k 2 ∇∇ G(R).(21)G m (R) = ∇G(R) × I.(22) In the far-field limit, R ≃ r → ∞, G(r) = e −jk|r| 4π|r| , R = r in the far-field limit. Similarly, the first-order derivative of the Green's function in the far-field limit can be obtained as ∂G(r) ∂τ = τ |r| −jk − 1 |r| 2 e −jk|r| 4π|r| ≃ −j τ |r| k e −jk|r| 4π|r| = −jk τ e −jk|r| 4π|r| , τ = x, y, z (24) where only 1 |r| term is kept and the other terms ( 1 |r| 2 , 1 |r| 3 , · · ·) are ignored. In derivation of (24), the following relation has been used in the far-field limit, k τ k = τ |r| , τ = x, y, z.(25) Following the similar procedure given in (24), the far-fields of derivatives (order n) of the scalar Green's function are obtained as ∂ (n) G(r) ∂τ (n) = (−jk τ ) n e −jk|r| 4π|r| , τ = x, y, z.(26) It is not difficult to see that the far-fields and the 2D Fourier spectra are closely related to each other. E. The 2D Fourier spectra of the 3D spatial convolutions It is not difficult to see that, the radiation integral in (1) can be expressed as the sum of the 3D spatial convolutions of some source terms with the Green's function related expressions. For simplicity, let's consider the 3D spatial convolution of an arbitrary source term s with the scalar Green's function G, s(r) 3D G(r) = S s(r ′ )G(R) dS ′ ,(27) Now, apply the 2D Fourier transform on (27) and express the scalar Green's function G(R) in the spectral domain, S(k x , k y ) ≡ FT 2D s(r) 3D G(r) (28) = 1 2π ∞ x=−∞ ∞ y=−∞      e jkxx e jkyy S    s(r ′ ) ×    1 2π ∞ k ′ x =−∞ ∞ k ′ y =−∞ e −jk ′ x (x−x ′ ) e −jk ′ y (y−y ′ ) × −je −jk ′ z (z−z ′ ) 4πk ′ z dk ′ x dk ′ y       dS ′      dxdy, First, do the integral over (x, y), (28) reduces to S(k x , k y ) = S      s(r ′ ) ∞ k ′ x =−∞ ∞ k ′ y =−∞ e jk ′ x x ′ e jk ′ y y ′ (29) × −je −jk ′ z (z−z ′ ) 4πk ′ z δ(k ′ x − k x )δ(k ′ y − k y )dk ′ x dk ′ y         dS ′ , Next, do the integral over (k ′ x , k ′ y ) and (29) reduces to FT 2D s(r) 3D G(r) = L    s (r)    G > (k x , k y , 0),(30) where L in (30) is the radiation vector [10] for source term s, which is defined as L    s (r)    = S s(r ′ )e jk·r ′ dS ′ ,(31) It is not difficult to see that the radiation vector L in (31) reduces to the regular 2D Fourier spectrum when surface S is a plane located at z ′ = 0. L    s (r)    z ′ =0 = 2πFT 2D    s(x, y)    ,(32) where the dummy primed (x ′ , y ′ ) have been replaced with (x, y). Substitute (31) into (30), S(k x , k y ) = −je −jkzz 4πk z L    s (r)    = G(k x , k y , 0)L    s (r)    .(33) It is not difficult to see that the radiation vector L in (31) is closely related to the far-field by letting r → ∞, s(r) 3D G(r) r→∞ = e −jk|r| 4π|r| L    s (r)    .(34) From (34), if e −jk|r| 4π|r| can be ignored, the radiation vector L can be considered as the far-field pattern, which means that when the far-field is obtained, the radiation vector and the 2D Fourier spectrum of the 3D convolution are also obtained, from (34) and (33) respectively. F. 2D Fourier spectrum of the radiation integral From (33) and (1), the 2D Fourier spectrum of the radiation integral (denoted as F ) is obtained, which is F = −j ωǫ G > (k x , k y , 0)            k 2 L    J(r)    − k τ =x,y,z    k τ L    J τ (r)       + ωǫL    J m (r)    × k            .(35) G. Electromagnetic field on a plane After the 2D Fourier spectrum F has been obtained, the electric field E can be expressed in the PWS form [11,12], which is given as, E(r) = IFT 2D    F 0 (k x , k y )e −jkzz   (36) where F 0 and the 2D Inverse Fourier Transform have been defined as F 0 (k x , k y ) = F 0xx + F 0yŷ + F 0zẑ = F (k x , k y )e jkzz , F 0z = − k x F 0x + k y F 0y k z , IFT 2D · = 1 2π ∞ kx=−∞ e −jkxx ∞ ky=−∞ · e −jkyy dk y dk x . III. THE PLANAR TI-FFT ALGORITHM In this section, the optimized spatial and spectral TI-FFTs are presented. It will be shown that both of them have the same computational complexity for the same quasi-planar surface. A. The spatial TI-FFT algorithm It has been shown in (36) that the electric field E on a plane can be evaluated through the 2D inverse Fourier transform. For a quasi-planar surface, the TI technique can be used, which leads to the spatial TI-FFT algorithm (where the quasi-planar surface is sliced into many small spatial subdomains, as shown Fig. 1). Rewrite the electric field E in (36) as follows, (37) where z min denotes the minimum value of z. Now, express e −j△kz△z into a Taylor series on the spatial reference plane located at z = z r , E(r) = e −jkz IFT 2D    F 0 (k x , k y ) e j△kz△z    , F 0 (k x , k y ) = F 0 (k x , k y )e j△kzz mine j△kz△z = e j△kz△zr No n=0 1 n! (j△k z ) n z − z r n ,(38) where △z r = z r − z min and N o is the order of Taylor series. Substitute (38) into (37), the spatial TI-FFT algorithm for the electric field E is obtained, E(r) = e −jkz No n=0      1 n!   j (z − z r )    n IFT 2D F 0 (k x , k y )e j△kz△zr △k z n      ,(39) The number of spatial reference planes N r required in the computation depends on the spatial slicing spacing δ z ≡ max z − z r = z r+1 − z r and the characteristic surface variation △z c (within which the electromagnetic field is of interest): N r ∝ △z c /δ z . The readers should note that the actual maximum interpolation distance is δz 2 , which is located at the middle of two adjacent spatial reference planes, but δ z is used in this article to simplify the notation. Apparently, to achieve the desired computational accuracy (denoted as γ TI ), the choice of the spatial slicing spacing δ z between two adjacent spatial reference planes depends on △k z,c , which is defined as △k z,c ≡ k − k z,c = k − k 2 − k 2 ⊥,c = kα, α = 1 − 1 − k ⊥,c k 2 ,(40) where, k ⊥,c is the characteristic bandwidth (beyond which the 2D Fourier spectrum F is negligible) and is defined on x-y plane. It is clear that the smaller the bandwidth k ⊥,c , the larger the δ z could be, which also means a smaller N r . So, a narrow-band beam and a small surface variation △z c (quasi-planar geometry) are in favor of the planar TI-FFT algorithm. In view of the importance of the spatial slicing spacing δ z , it is helpful to define the characteristic wave length λ c for a narrow-band beam. From (40), λ c ≡ 2π △k z,c = 2π k − k 2 − k 2 ⊥,c = λ α ,(41) For a narrow-band beam (k ⊥,c ≪ k), λ c ∼ 2 k k ⊥,c 2 λ.(42)k ⊥,c = k. It can be seen from (38) and (39) that, for the given computational accuracy γ TI , the spatial slicing spacing δ z should satisfy the following relation, γ TI ∼ O (△k z,c δ z ) No(43)→ δ z ∼ 1 kα 1 γ TI − 1 No = λ 2πα 1 γ TI − 1 No ,(44) For a narrow-band beam (k ⊥,c ≪ k), δ z ∼ 1 π k k ⊥,c 2 1 γ TI − 1 No λ.(45) Now consider a quasi-planar surface with a characteristic surface variation of △z c = N z λ, from (44) the number of spatial reference planes N r is given as N r = △z c δ z ∼ 2πα 1 γ TI 1 No N z ,(46) For a narrow-band beam (k ⊥,c ≪ k), N r ∼ π k ⊥,c k 1 γ TI 1 No N z .(47) The number of FFT operations N FFT and the computational complexity CPU are obtained as N FFT = N o × N r = 2πα 1 γ TI 1 No N o N z ,(48)CPU = N FFT O N log 2 N = 2πα 1 γ TI 1 No N o N z O N log 2 N ,(49) For a narrow-band beam (k ⊥,c ≪ k), N FFT ∼ π k ⊥,c k 2 1 γ TI 1 No N o N z .(50)CPU ∼ π k ⊥,c k 2 1 γ TI 1 No N o N z O N log 2 N .(51) For a narrow-band beam, the computational complexity CPU has a square law dependence on the characteristic bandwidth k ⊥,c of the electromagnetic wave and have a linear dependence on the surface variation (△z c = N z λ). The computational complexity CPU also has an inverse N o th -root dependence on the computational accuracy γ TI . So the characteristic bandwidth k ⊥,c has the most significant effect on the computational complexity of the planar TI-FFT algorithm. It can be seen from (48) where "round" means to round the value to its nearest integer (actually, to achieve a higher computational accuracy γ TI , the upper-bound could be used but the computational complexity CPU is a little higher). The optimized spatial slicing spacing δ opt z can be obtained from (44) and (53), which is ∂N FFT ∂N o N opt o = 0 → ∂ ln [N o ] − ln [γ TI ] 1 No ∂N o N opt o = 0,(52)N opt o ∼ round    ln 1 γ TI    = round    − 0.1151γ TI (dB)    .(53)δ opt z ∼ λ 2πα 1 γ TI − 1 ln [ 1 γ TI ] = λ 2πeα ∼ 1 17 λ c ,(54) For a narrow-band beam (k ⊥,c ≪ k), δ opt z ∼ 1 eπ k k ⊥,c 2 λ,(55) where e ∼ 2.718 is the natural logarithmic base. It is interesting to note that the optimized spatial slicing spacing δ z doesn't depend on the computational accuracy γ TI and strongly The optimized number of spatial reference planes N opt r is given as N opt r = △z c δ z = 2πeαN z ∼ 17αN z ,(56) For a narrow-band beam (k ⊥,c ≪ k), N opt r ∼ πe k ⊥,c k 2 N z .(57) Substitute (53) into (48) and (49), the optimized number of FFT operations N opt FFT and the optimized computational complexity CPU opt can also be obtained, N opt FFT = 2πeα ln 1 γ TI N z ,(58) For a narrow-band beam (k ⊥,c ≪ k), N opt FFT ∼ πe k ⊥,c k 2 ln 1 γ TI N z .(59) The optimized computational complexity CPU opt is given as CPU opt ∼ N opt FFT O N log 2 N .(60) B. The spectral TI-FFT algorithm It has been shown in (35) that the computation of the 2D Fourier spectrum F is equivalent to evaluate the radiation vector L. For the quasi-planar geometry, the FFT can still be used with the help of the TI technique, which leads to the spectral TI-FFT algorithm (where the spherical spectral surface is sliced into many small spectral subdomains, as shown Fig. 5). From (31), the radiation vector L can be rewritten as L   f (r)    = 2πe jkzz min FT 2D    f(r)e jkz△z    ,(61) where f(r) = s(r) n·ẑ andn is the normal to surface S. Now the Taylor expansion of L in (61) over k z is given as L   f (r)    = 2πe jkzz min No n=0      1 n! j [k z − k z,r ] n FT 2D f (r) △z n      ,(62) where k z,r denotes the spectral reference plane. For the given computational accuracy γ TI , the spectral slicing spacing δk z ≡ max k z − k z,r = k z,r+1 − k z,r should satisfy the following relation, is used for half-space z > z ′ computation in this article. k z,r and k z,r+1 denote the r th and (r + 1) th spectral reference planes respectively. δ kz is the spectral slicing spacing. γ TI ∼ O (δ kz △z c ) No ,(63)→ δ kz ∼ 1 △z c 1 γ TI − 1 No ∼ 1 N z λ 1 γ TI − 1 No ,(64) The number of spectral reference planes N r is given as N r = △k z,c δ kz ∼ 2πα 1 γ TI 1 No N z ,(65) The number of FFT operations is given as N FFT ∼ 2πα 1 γ TI 1 No N o N z .(66) It is obvious that N r in (65) and N FFT in (66) are the same as those given in (46) and (48), which also means that the spatial and spectral TI-FFTs have the same optimized computational complexity. IV. COMPUTATIONAL RESULTS To show the efficiency of the planar TI-FFT algorithm, the direct integration of the radiation integral in (1) has been used to make comparison with the planar TI-FFT algorithm. The numerical example used for such purpose is a 110 GHz Fundamental Gaussian Beam (FGB) scattered by a PEC quasi-planar surface with a sin wave perturbation. The 110 GHz FGB has a wavelength of λ ∼ 2.7 mm. A. The numerical results The incident 110 GHz FGB propagates at −ẑ direction and has a beam waist radius of w = 1 cm. The quasi-planar PEC surface with a sine wave perturbation is described as z(x, y) = −2.5λ + 0.5λ cos 2π x 15λ cos 2π y 15λ . In the numerical implementation of the planar TI-FFT algorithm, the computational where the quasi-planar surface described in (67) has a characteristic surface variation △z c ∼ 1λ. The scattered output field E s are evaluated on plane z = 0 (where the incident 110 GHz FGB starts to propagate). Fig. 6, Fig. 7 and Fig. 8 show the magnitude patterns of x-, y-, and z-components of the scattered output field E s . The comparison of the result obtained from the planar TI-FFT algorithm and that from the direct integration method is given in Fig. 9, for both the magnitudes and the real parts, which shows that the planar TI-FFT algorithm has the desired −80 dB computational accuracy. B. The CPU time and the accuracy The CPU time t TI for the planar TI-FFT algorithm and t DI for the direct integration method have been summarized in Table I, together with the coupling coefficient defined as C τ ≡ E s TI,τ [E s DI,τ ] * dxdy |E s TI,τ | 2 dxdy |E s DI,τ | 2 dxdy z=0,(69) where, E s TI,τ and E s DI,τ (τ = x, y, z) denote the scattered output field components obtained from the planar TI-FFT algorithm and the direct integration method respectively. From TABLE I, it can be seen that, even though at a large sampling spacing δ = 0.46λ (N x = N y ∼ 128), the coupling coefficients are still well above 90.00%. At this sampling rate, the direct integration method using Simpson's 1/3 rule is not accurate enough [1,2,3]. Also note that the coupling coefficients C τ (τ = x, y, z) reach their maximum values of 99.99% at N x = N y ∼ 256 (δ x = δ y = 0.23λ), after which the accuracies remain constant and thus the Nyquist rate can be estimated roughly as N Nyquist ∼ 256. The reason for this phenomenon is, that after the sampling rate increases above the Nyquist rate, further increasing the sampling rate will not give more information or computational accuracy. The CPU time for the planar TI-FFT algorithm t TI and for the direct integration method t DI are shown in Fig. 10. The ratio t DI /t TI is shown in Fig. 11. N x,y δ(λ) t TI (sec.) t DI (sec.) t DI /t TI C x (%) C y (%) C z (% All work was done in Matlab 7.0.1, on a 1.66 GHz PC (Intel Core Duo), with 512 MB Memory. V. DISCUSSION: PROBLEMS AND POSSIBLE SOLUTIONS Although the planar TI-FFT algorithm has so many advantages given above, some problems do exist in the practical applications. Complicate geometry As an example, consider surface S shown in Fig. 12, where the surface itself is not a quasi-planar surface and the direct implementation of the planar TI-FFT algorithm requires a large number of FFT operations, which can be seen from the spatial reference planes with a spatial slicing spacing δ z . The problem can be solved by dividing surface S into two surface patches △S 1 and △S 2 , which can be considered as quasi-planar surfaces and the planar TI-FFT can be used on them independently, with coordinate systems selected based on the spatial reference planes. At the extreme limit where surface patches △S 1 and △S 2 are planes, the number of FFT operations reduces to N FFT = 2. Observation points not on the computational grid It is well-known that the FFT requires an even grid spacing (but δ x and δ y need not to be equal), which raises the question of how to calculate the electric field at points that are not exactly on the computational grid, e.g., the red filled circles in Fig. 13. One solution for this problem is to zero-pad the computational grid in the spectral domain, which corresponds to the interpolation of the computational grid in the spatial domain, as shown in Fig. 14. In the above example, it has been assumed that the observation points are evenly distributed coordinate system whose z-coordinate is perpendicular to the slicing spatial reference planes. and the interpolation results are exact provided that the sampling rate is above the Nyquist rate [7]. For complicate observation point configurations (e.g., unevenly distributed points), the approximate techniques like the Gauss's forward/backward interpolations can be used. The translation in spatial domain In the real situation, the source field surface and the observation surface are separate far away from each other (see Fig. 15). It is not practical nor necessary to use a large computational grid that covers both the source field surface and the observation surface. This kind of problem can be solved by using two computational grids, one for the source field surface and the other for the observation surface, with the same grid spacings (δ x , δ y ). Then the translation of the observation coordinate system in the spatial domain, which is denoted as (x 0 , y 0 ), corresponds to the phase shift in the spectral domain. Suppose the electric field in the source coordinate system is expressed as E(x ′ − x 0 , y ′ − y 0 ), according to the property of the Fourier transform [7], the electric field E(x, y) in the observation coordinate system is given as Computational redundancy In the numerical implementation of the planar TI-FFT algorithm, the spatial domain or the spectral domain are divided into many small subdomains where the FFT can be used to interpolate the electromagnetic field (see Fig. 1 and Fig. 5). However, the FFT operation is done on the whole spatial or spectral domain even though the interpolation is only necessary on the relatively small subdomain, which causes the computational redundancy in the planar TI-FFT algorithm. Fortunately, the computational redundancy is small for a quasi-planar surface and a narrow-band beam. In this article, the optimized planar TI-FFT algorithm for the computation of electromagnetic wave propagation has been introduced for the narrow-band beam and the quasiplanar geometry. Two types of TI-FFT algorithm are available, i.e., the spatial TI-FFT and the spectral TI-FFT. The former is for computation of electromagnetic wave on the quasi-planar surface and the latter is for computation of the 2D Fourier spectrum of the electromagnetic wave. The optimized order of Taylor series used in the planar TI-FFT algorithm is found to be closely related to the algorithm's computational accuracy γ TI , which is given as N opt o ∼ − ln γ TI and the optimized spatial slicing spacing between two adjacent spatial reference planes only depends on the characteristic wavelength λ c of the electromagnetic wave, which is δ opt z ∼ 1 17 λ c . The optimized computational complexity is given as N opt r N opt o O (N log 2 N) for an N = N x × N y computational grid. The planar TI-FFT algorithm allows a low sampling rate (large sampling spacing) required by the sampling theorem. Also, the algorithm doesn't have the problem of singularity. The planar TI-FFT algorithm has applications in near-field and far-field computations, beam-shaping mirror system designs, diffraction and scattering phenomena, millimeter wave propagation, and microwave imaging in the half-space scenario. FIG. 1 : 12 J s (r ′ )G(R) +    J s (r ′ ) · ∇ ′    ∇ ′ G(R) − jωǫJ ms (r ′ ) × ∇ ′ G(R)Electromagnetic wave propagation and scattering: the computation of the electromagnetic wave propagation (the incident file E i ) onto the PEC surface S is implemented through the spatial TI-FFT and the computation of the scattered electromagnetic field E s from the PEC surface S is implemented through the spectral TI-FFTs and the inverse Fourier transform. δz is the spatial slicing spacing in the spatial TI-FFT. Fig. 2 2plots the exact value in (41) and approximation in (42) of the characteristic wavelength λ c for different characteristic bandwidth k ⊥,c , from which it can be seen that the maximum deviation of the approximation from the exact value is 1λ, which occurs at FIG. 2 : 2The plots of the characteristic wavelength λ c for different k z,c . The exact value (line) is given in (41) and the approximation (dots) is given in (42). The plots show that λ c ≫ λ for a narrow-band beam. The maximum deviation of the approximation from the exact value is 1λ, which occurs at k z,c = 0 (k ⊥,c = k). FIG. 3 : 3The linear dependence of the optimized order of Taylor series N opt o on the computational accuracy γ TI (dB). It can be seen that N opt o = (2, 5, 7, 9) for γ TI = (−20, −40, −60, −80) dB respectively. Fig. 3 3also shows the linear dependence of the optimized order of Taylor series N opt o on the computational accuracy γ TI (dB), which has been shown in (53). FIG. 4 : 4Plots of the exact value (line) of δ opt z given in (54) and approximation (dots) given in (55) for different characteristic bandwidth k ⊥,c .depends on the characteristic bandwidth k ⊥,c (inverse square law).Fig. 4shows δ opt z for different characteristic bandwidth k ⊥,c , from which it can be seen that δ opt z > 0.5λ for k z,c > 0.9k (k ⊥,c < 0.436k). FIG. 5 : 5The spectral domain division for the spectral and TI-FFT: Only k z > 0 half sphere surface FIG. 6 : 6The x-component (magnitude) of the scattered output field. FIG. 7 :∼ 2 , 72The y-component (magnitude) of the scattered output field. accuracy γ TI = 0.0001 (−80 dB) has been used and the following optimized quantities are obtained from (53)-N opt FFT ∼ 18, CPU opt ∼ 18O N log 2 N . (68) FIG. 8 : 8The z-component (magnitude) of the scattered output field. FIG. 9 : 9The comparison of the scattered output field on plane z = 0, across the maximum value point of |E s x | and atx direction: a) is for E s x ; and b) is for E s z ; solid lines (TI-FFT) and circles (direct integration method) are magnitudes; dashed lines (TI-FFT) and dots (direct integration method) are real parts. 10: The CPU time (t TI , t DI ) comparison: bars in gray color are for the planar TI-FFT algorithm and bars in black color are for the direct integration method. Note that the CPU time is in logarithmic scale (10-base). FIG. 11 : 11The efficiency of the planar TI-FFT algorithm: the ratio of t DI /t TI for N x = N y = (128, 256, 512, 1024). . 12: An example of complicate surface S that can be divided into two quasi-planar surface patches △S 1 and △S 2 . The computations of each surface patch is done in its corresponding FIG. 13 : 13The problem of computation of electromagnetic field on the observation points that are not on the computational grid (4 × 4), which are denoted as red filled circles in the spatial domain (assume that they are evenly distributed). (δ kx , δ ky ) are grid spacings in the spectral domain. (δ x , δ y ) are grid spacings in the space domain. FIG. 14 : 14The zero-padding in the spectral domain (4×4 → 8×8) corresponding to the interpolation in the spatial domain (4 × 4 → 8 × 8). (δ kx , δ ky ) are still the same after zero-padding. But grid spacings in the spatial domain become (δ x /2, δ y /2) after interpolation. FIG. 15 : 15The translation of the source coordinate system o ′ (0,0) to the observation coordinate system o(x 0 , y 0 ) in the spatial domain. Both the source and observation coordinate systems should have the same grid spacings (δ x , δ y ). TABLE I : ICPU time (t TI , t DI ) and coupling coefficient C τ A new fast algorithm for field propagation between arbitrary smooth surfaces. Shaolin Liao, R J Vernon, 10.1109/ICIMW.2005.1572687the joint 30 th Infrared and Millimeter Waves and 13 th International Conference on Terahertz Electronics. Williamsburg, Virginia, USA2Shaolin Liao and R. J. Vernon, "A new fast algorithm for field propagation between arbitrary smooth surfaces", In: the joint 30 th Infrared and Millimeter Waves and 13 th International Conference on Terahertz Electronics, Williamsburg, Virginia, USA, 2005, ISBN: 0-7803-9348- 1, INSPEC number: 8788764, DOI: 10.1109/ICIMW.2005.1572687, Vol. 2, pp. 606-607. Improved performance of three-mirror beam-shaping systems and application to step-tunable converters. R Cao, R J Vernon, 10.1109/ICIMW.2005.1572692the joint 30 th Infrared and Millimeter Waves and 13 th International Conference on Terahertz Electronics. Williamsburg, Virginia, USA2R. Cao and R. J. Vernon, "Improved performance of three-mirror beam-shaping systems and application to step-tunable converters", In: the joint 30 th Infrared and Millimeter Waves and 13 th International Conference on Terahertz Electronics, Williamsburg, Virginia, USA, 2005, ISBN: 0-7803-9348-1, INSPEC number: 8788768, DOI: 10.1109/ICIMW.2005.1572692, Vol. 2, pp. 616-617. Iterative design of a cylinder-based beam-shaping mirror pair for use in a gyrotron internal quasi-optical mode converter. P Michael, R J Perkins, Vernon, the 29 th Infrared and Millimeter Waves Conference. Karlsruhe, GermanyMichael P. Perkins and R. J. Vernon, "Iterative design of a cylinder-based beam-shaping mirror pair for use in a gyrotron internal quasi-optical mode converter", In: the 29 th Infrared and Millimeter Waves Conference, Karlsruhe, Germany, Sep. 27-Oct. 1, 2004. Sub-THz beam-shaping mirror designs for quasi-optical mode converter in high-power gyrotrons. Shaolin Liao, R J Vernon, J. Electromagn. Waves and Appl. 214Shaolin Liao and R. J. Vernon, "Sub-THz beam-shaping mirror designs for quasi-optical mode converter in high-power gyrotrons", J. Electromagn. Waves and Appl., scheduled for volume 21, number 4, page 425-439, 2007. On fast computation of electromagnetic wave propagation through FFT. Shaolin Liao, the 7 th International Symposium on Antennas, Propagation, and EM Theory (ISAPE2006). Guilin, ChinaShaolin Liao et. al., "On fast computation of electromagnetic wave propagation through FFT", the 7 th International Symposium on Antennas, Propagation, and EM Theory (ISAPE2006), Guilin, China, Oct. 26-Oct. 29, 2006. An algorithm for the machine caculation of complex Fourier series. J W Cooley, J W Tukey, Math. Comput. 19297301J. W. Cooley, J. W. Tukey, "An algorithm for the machine caculation of complex Fourier series", Math. Comput., 19, 297301, 1965. . A V Oppenheim, R W Schaffer, Ditital Signal Processing. Prentice-HallA. V. Oppenheim, R. W. Schaffer, Ditital Signal Processing, Prentice-Hall, Englewood, Cliffs, NJ, 1975. An examination of the theory and practices of planar near-field measurement. J H Johnson, Wang, IEEE Trans. on Antennas and Propagat. 366Johnson J. H. Wang, "An examination of the theory and practices of planar near-field mea- surement", IEEE Trans. on Antennas and Propagat., Vol. 36, No. 6, Jun., 1988. R E Collin, Field Theory of Guided Waves. IEEE presssecond editionR. E. Collin, Field Theory of Guided Waves, second edition, IEEE press, 1991. Advanced Engineering Electromagnetic. C A Balanis, John Wiley & Son's IncC. A. Balanis, Advanced Engineering Electromagnetic, John Wiley & Son's Inc., 1989. G T Whittaker, G N Watson, Mordern Analysis, ch. XVIII, 4. LondonCambridge Univ. Pressth ed.G. T. Whittaker, G. N. Watson, Mordern Analysis, ch. XVIII, 4 th ed., London: Cambridge Univ. Press, 1927. The concept of an angular spectrum of a plane wave, and its relations to that of polar diagram and aperture distribution. H G Booker, P C Clemmow, Proc. Inst. Elec. Engr. 97H. G. Booker, P. C. Clemmow, "The concept of an angular spectrum of a plane wave, and its relations to that of polar diagram and aperture distribution", Proc. Inst. Elec. Engr., 97, 1950, pp. 11-17.

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{'abstract': "The Taylor Interpolation through FFT (TI-FFT) algorithm for the computation of the electromagnetic wave propagation in the quasi-planar geometry within the half-space is proposed in this article. There are two types of TI-FFT algorithm, i.e., the spatial TI-FFT and the spectral TI-FFT. The former works in the spatial domain and the latter works in the spectral domain. It has been shown that the optimized computational complexity is the same for both types of TI-FFT algorithm, which is N optopt r is the optimized number of slicing reference planes and N opt o is the optimized order of Taylor series.Detailed analysis shows that N opt o is closely related to the algorithm's computational accuracy γ TI , which is given as N opt o ∼ − ln γ TI and the optimized spatial slicing spacing between two adjacent spatial reference planes δ opt z only depends on the characteristic wavelength λ c of the electromagnetic wave, which is given as δ opt z ∼ 1 17 λ c . The planar TI-FFT algorithm allows a large sampling spacing required by the sampling theorem. What's more, the algorithm is free of singularities and it works particularly well for the narrow-band beam and the quasi-planar geometry.", 'arxivid': 'physics/0610057', 'author': ['Shaolin Liao sliao@wisc.edu \nDepartment of Electric and Computer Engineering\nUniversity of Wisconsin\n1415 Engineering Drive53706Madison, MadisonWIUSA\n'], 'authoraffiliation': ['Department of Electric and Computer Engineering\nUniversity of Wisconsin\n1415 Engineering Drive53706Madison, MadisonWIUSA'], 'corpusid': 118162558, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 12620, 'n_tokens_neox': 10771, 'n_words': 6355, 'pdfsha': 'b316e8bb2e215fcdd53e21705619dc07c5c69eb9', 'pdfurls': ['https://export.arxiv.org/pdf/physics/0610057v1.pdf'], 'title': ['The Taylor Interpolation through FFT Algorithm for Electromagnetic Wave Propagation and Scattering', 'The Taylor Interpolation through FFT Algorithm for Electromagnetic Wave Propagation and Scattering'], 'venue': []}

arxiv

Sparse Sums of Positive Semidefinite Matrices 18 Oct 2011 Marcel K De Carli Silva mksilva@uwaterloo.ca. Department of Combinatorics and Optimization University of Waterloo Nicholas J A Harvey Department of Computer Science University of British Columbia Cristiane M Sato Sparse Sums of Positive Semidefinite Matrices 18 Oct 2011 Recently there has been much interest in "sparsifying" sums of rank one matrices: modifying the coefficients such that only a few are nonzero, while approximately preserving the matrix that results from the sum. Results of this sort have found applications in many different areas, including sparsifying graphs. In this paper we consider the more general problem of sparsifying sums of positive semidefinite matrices that have arbitrary rank.We give several algorithms for solving this problem. The first algorithm is based on the method of Batson, Spielman and Srivastava (2009). The second algorithm is based on the matrix multiplicative weights update method of Arora and Kale(2007). We also highlight an interesting connection between these two algorithms.Our algorithms have numerous applications. We show how they can be used to construct graph sparsifiers with auxiliary constraints, sparsifiers of hypergraphs, and sparse solutions to semidefinite programs. Introduction A sparsifier of a graph is a subgraph that approximately preserves some structural properties of the graph. The original work in this area studied cut sparsifiers, which are weighted subgraphs that approximate every cut arbitrarily well. The celebrated work of Benczúr and Karger [5,6] proved that every undirected graph with n vertices and m edges (and potentially non-negative weights on its edges) has a subgraph with only O(n log n/ε 2 ) edges (and new weights on those edges) such that, for every cut, the weight of the cut in the original graph and its subgraph agree up to a multiplicative factor of (1 ± ε). Benczúr and Karger also gave a randomized algorithm to construct a cut sparsifier inÕ(m/ε 2 ) time. Recent work has extended and improved their algorithm in various ways [10,11,12,14,15]. Spielman and Teng [39] introduced spectral sparsifiers, which are weighted subgraphs such that the quadratic forms defined by the Laplacians of the graph and the sparsifier agree up to a multiplicative factor of (1 ± ε). Spectral sparsifiers are also cut sparsifiers, as can be seen by evaluating these quadratic forms at {0, 1}-vectors. They proved that every undirected graph with n vertices and m edges (and potentially nonnegative weights on its edges) has a spectral sparsifier with only n polylog(n)/ε 2 edges (and new weights on those edges). Spielman and Srivastava [38] reduce the graph sparsification problem to the following abstract problem in matrix theory. Problem 1. Let v 1 , . . . , v m ∈ R n be vectors and let B = i v i v T i . Given ε ∈ (0, 1), find a vector y ∈ R m with small support such that y ≥ 0 and B i y i v i v T i (1 + ε)B. (1) (Here the notation X Y means that the matrix Y − X is positive semidefinite.) Spielman and Srivastava [38] observe that Problem 1 can be solved using known concentration bounds on operator-valued random variables, specifically Rudelson's sampling lemma [32,33]. This approach yields a vector y with support size O(n log n/ε 2 ), and therefore yields a construction of spectral sparsifiers with O(n log n/ε 2 ) edges. Their algorithm relies on the linear system solver of Spielman and Teng [39], which was significantly simplified by Koutis, Miller and Peng [24]. Recent work [23] has improved the space usage of Spielman and Srivastava's algorithm. In subsequent work, Batson, Spielman and Srivastava [4] give a deterministic algorithm that solves Problem 1 and produces a vector y with support size O(n/ε 2 ). Consequently they obtain improved spectral sparsifiers with O(n/ε 2 ) edges. This work led to important progress in metric embeddings [29,34], convex geometry [40] and Banach space theory [37]. In this paper, we focus on a more general problem. Problem 2. Let B 1 , . . . , B m be symmetric, positive semidefinite matrices of size n × n and let B = i B i . Given ε ∈ (0, 1), find a vector y ∈ R m with small support such that y ≥ 0 and B i y i B i (1 + ε)B.(2) This problem can also be solved by known concentration bounds: Ahlswede and Winter [1] give a method for generalizing Chernoff-like bounds to operator-valued random variables, and one of their theorems [1,Theorem 19] directly yields a solution to Problem 2. (Other expositions of these results also exist [41,16].) This approach yields a vector y with support size O(n log n/ε 2 ). See Section 3 for more details. This paper gives two improved solutions to Problem 2. Our interest in this topic is motivated by several applications, such as constructing sparsifiers with certain auxiliary properties and sparsifiers for hypergraphs. We discuss these applications in Section 1.2. Our Results We give several efficient algorithms for solving Problem 2. Our strongest solution is: Theorem 3. Let B 1 , . . . , B m be symmetric, positive semidefinite matrices of size n × n and arbitrary rank. Set B := i B i . For any ε ∈ (0, 1), there is a deterministic algorithm to construct a vector y ∈ R m with O(n/ε 2 ) nonzero entries such that y ≥ 0 and B i y i B i (1 + ε)B. The algorithm runs in O(mn 3 /ε 2 ) time. Moreover, the result continues to hold if the input matrices B 1 , . . . , B m are Hermitian and positive semidefinite. Our proof of Theorem 3 is quite simple and builds on results of Batson, Spielman and Srivastava [4]. We remark that the assumption that the B i 's are positive semidefinite cannot be removed; see Appendix D. We also give a second solution to Problem 2 which is quantitatively weaker, although it is based on very general machinery which might prove useful in further applications or generalizations of Problem 2. This second solution is based on the matrix multiplicative weights update method (MMWUM) of Arora and Kale [3,22]. By a black-box application of their theorems we obtain a deterministic algorithm to construct a vector y with O(n log n/ε 3 ) nonzero entries. By slightly refining their analysis we can improve the number of nonzero entries to O(n log n/ε 2 ). We remark that Orecchia and Vishnoi [30] have used MMWUM for solving the balanced separator problem; this can be used as a subroutine in Spielman and Teng's algorithm for constructing spectral sparsifiers. Another virtue of our second solution is that it illustrates that the surprising Batson-Spielman-Srivastava (BSS) algorithm is actually closely related to MMWUM. In particular, the algorithms underlying our two solutions are identical, except for the use of slightly different potential functions. We explain this connection in Section 8. Applications In this section, we present several applications of Problem 2. Proofs are given in Appendix A. Sparsifiers with costs. Corollary 4. Let G = (V, E) be a graph, let w : E → R + be a weight function, and let c 1 , . . . , c k : E → R + be cost functions, with k = O(n). Let L G (w) denote the Laplacian matrix for graph G with weight function w. For any real ε ∈ (0, 1), there is a deterministic polynomial-time algorithm to find a subgraph H of G and a weight function w H : E(H) → R + such that The inequalities L G (w) L H (w H ) (1 + ε)L G (w) are equivalent to the condition that the subgraph H (with weights w H ) is a spectral sparsifier of G (with weights w). We remark that existing methods for producing sparsifiers have low probability of approximately satisfying even a single cost function (i.e., the case k = 1). L G (w) L H (w H ) (1 + ε)L G (w), One potentially interesting application of sparsifiers with costs is as follows. L G (w) L H (w H ) (1 + ε)L G (w), (1 − ε) e∈E i w e ≤ e∈E(H)∩E i w H,e ≤ (1 + ε) e∈E i w e for all i, and |E(H)| = O((n + k)/ε 2 ). Hypergraph sparsifiers. Let H = (V, E) be a hypergraph, and let w : E → R + . We follow the definition of Laplacian for hypergraphs as in [31]. For each hyperedge E ∈ E, define its Laplacian L E as the graph Laplacian of a graph on V whose edge set forms a clique on E. Define the Laplacian for the hypergraph H with weight function w as the matrix L H (w) := E∈E w E L E . Corollary 6 (Spectral sparsifiers for hypergraphs). For any real ε ∈ (0, 1), there is a deterministic polynomialtime algorithm to find a sub-hypergraph G of H and a weight function w G : E(G) → R + such that L H (w) L G (w G ) (1 + ε)L H (w), and |E(G)| = O(n/ε 2 ). This corollary concerns spectral sparsifiers. It is also interesting to study sparsifiers that approximately preserve all cuts. There are several ways to extend the definition of "the weight of a cut" from ordinary graphs to hypergraphs. We consider the following two definitions, where S is any set of vertices in a hypergraph H with edge weights w. • w(δ H (S)): This is the sum of the weights of all hyperedges that contain at least one vertex in S and at least one vertex in S := V \ S. • w * (δ H (S)): This is defined to be E∈E w E · |S ∩ E| · |S ∩ E|. Obviously these definitions agree in ordinary graphs. Corollary 7 (Cut sparsifiers for hypergraphs, second definition). For any real ε ∈ (0, 1), there is a deterministic polynomial-time algorithm to find a sub-hypergraph G of H and a weight function w G : E(G) → R + such that w * (δ H (S)) ≤ w * G (δ G (S)) ≤ (1 + ε)w * (δ H (S)) for every S ⊆ V , and |E(G)| = O(n/ε 2 ). Corollary 8 (Cut sparsifiers for hypergraphs, first definition). Assume that H is an r-uniform hypergraph. For any real ε ∈ (0, 1), there is a deterministic polynomial-time algorithm to find a sub-hypergraph G of H and a weight function w G : E(G) → R + such that (r − 1) r 2 /4 w(δ H (S)) ≤ w G (δ G (S)) ≤ (1 + ε)r 2 4(r − 1) w(δ H (S)) ∀S ⊆ V, and |E(G)| = O(n/ε 2 ). In other words, the sparsified hypergraph G approximates the weight of the cuts in the hypergraph H to within a factor Θ(r 2 ). For the special case r = 3, we can achieve (1 + ε)-approximate sparsification for all cuts, even under the first definition. Corollary 9 (Cut sparsifiers for 3-uniform hypergraphs). Assume that H is a 3-uniform hypergraph. For any ε ∈ (0, 1), there is a deterministic polynomial-time algorithm to find a sub-hypergraph G of H and a weight function w G : E(G) → R + such that w(δ H (S)) ≤ w G (δ G (S)) ≤ (1 + ε)w(δ H (S)) ∀S ⊆ V,min c T z : i z i A i B, z ∈ R m , z ≥ 0 has a feasible solution z * . Then, for any real ε ∈ (0, 1), it has a feasible solutionz with at most O(n/ε 2 ) nonzero entries and c Tz ≤ (1 + ε)c T z * . Several important SDPs can be cast as in Corollary 10; see, e.g., [19,20]. Recently, Jain and Yao [21] gave a parallel approximation algorithm for SDPs in this form with B positive semidefinite. Lovász theta number. For a graph G = (V, E) on n nodes, let t ′ (G) denote the square of the minimum radius of an Euclidean ball in R n such that there is a map from V to points in the ball such that adjacent vertices are mapped to points at distance at least 1. Also, let ϑ ′ (G) denote the variant of the Lovász theta number introduced in [27] and [35]. Corollary 11. Let G = (V, E) be a graph. For any real ε ∈ (0, 1), there is a deterministic polynomial-time algorithm to find a subgraph H of G such that (1 − ε)t ′ (G) ≤ t ′ (H) ≤ t ′ (G) and |E(H)| = O(n/ε 2 ). Corollary 12. Let G = (V, E) be a graph. For any real ε ∈ (0, 1), there is a deterministic polynomial-time algorithm to find a supergraph H of G such that Approximate Carathéodory theorems. One immediate application for Theorem 3 is an approximate Carathéodory-type theorem. A classic result of this sort is: ϑ ′ (G) 1 − ε + εϑ ′ (G) ≤ ϑ ′ (H) ≤ ϑ ′ (G) and |E(H)| = n 2 − O(n/ε 2 ). Corollary 13. Let G be a graph such that ϑ ′ (G) = o( √ n). For any real γ > 0, there is a supergraph H of G such that ϑ ′ (G) 1 + γ ≤ ϑ ′ (H) ≤ ϑ ′ (G) and |E(H)| = n 2 − O(nϑ(G) 2 /γ 2 ). Theorem 15 (Althöfer [2], Lipton-Young [25]). Let v 1 , . . . , v m ∈ [0, 1] n and let λ ∈ R m satisfy λ ≥ 0 and i λ i = 1. Then there exists µ ∈ R m with µ ≥ 0, i µ i = 1 and only O(log n/ε 2 ) nonzero entries such that i λ i v i − i µ i v i ∞ ≤ ε. This theorem follows from simple random sampling arguments, but it has several interesting consequences, including the existence of sparse, low-regret solutions to zero-sum games. The following corollary of Theorem 3 can be viewed as a matrix generalization of Theorem 15. (1 − ε)B i µ i B i (1 + ε)B. Although the support size in Theorem 15 is much smaller than in Corollary 16, the latter provides a multiplicative error bound whereas the former only provides an additive error bound. Theorem 15 can be modified to give multiplicative error bounds if we allow µ to have O(n log n/ε 2 ) non-zero entries. However such a result is not interesting as Carathéodory's theorem provides a µ with only n + 1 non-zero entries and no error (i.e., ǫ = 0). In contrast, Carathéodory's theorem is very weak in the scenario of Corollary 16 as it only provides a µ with n(n + 1)/2 + 1 nonzero entries. Sparsifiers on subgraphs. Corollary 17. Let G = (V, E) be a graph, let w : E → R + be a weight function, and let F be a collection of subgraphs of G such that F ∈F |V (F )| = O(n). For any real ε ∈ (0, 1), there is a deterministic polynomial-time algorithm to find a subgraph H of G and a weight function w H : E(H) → R + such that |E(H)| = O(n/ε 2 ) and L G (w) L H (w H ) (1 + ε)L G (w), L F (w F ) L H∩F (w H ↾ E(H∩F ) ) (1 + ε)L F (w F ) for all F ∈ F , where w F := w↾ E(F ) is the restriction of w to the coordinates E(F ) and H ∩ F = V (F ), E(F ) ∩ E(H) . Preliminaries For a non-negative integer n, we denote [n] := {1, . . . , n}. The non-negative reals are denoted by R + . The set of n × n symmetric matrices is denoted by S n . The set of symmetric, n × n positive semidefinite (resp., positive definite) matrices is denoted by S n + (resp., S n ++ ). Recall that X ∈ S n is positive semidefinite if v T Xv ≥ 0 for all v ∈ R n , and X is positive definite if X is positive semidefinite and v T Xv = 0 implies v = 0. Sometimes we denote X ∈ S n + by X 0 and the notation X Y means that X − Y 0. For X ∈ S n and a, b ∈ R, the notation X ∈ [a, b] means that aI X bI, where I is the identity matrix. For X ∈ S n , its trace is Tr X := n i=1 X ii , its largest (resp., smallest) eigenvalue is denoted by λ max (X) (resp., λ min (X)). The vector space S n can be endowed with the trace inner product ·, · defined by X, Y := Tr(XY ) = i,j X ij Y ij for every X, Y ∈ S n . We shall repeatedly use that Tr(XY ) = Tr(Y X) for any matrices X, Y for which the products XY and Y X make sense. Let G = (V, E) be a graph. The canonical basis vectors of R V are { e i : i ∈ V }, and the canonical basis vectors of R E are { e {i,j} : {i, j} ∈ E}. The Laplacian of G is the linear transformation L G (·) : R E → S V defined by L G (w) = {i,j}∈E w {i,j} (e i − e j )(e i − e j ) T . When dealing with Problem 2, we may assume that B = I. See [4, Proof of Theorem 1.1] for the details of the reduction. Solving Problem 2 by Ahlswede-Winter As mentioned earlier, Spielman and Srivastava [38] explain how Problem 1 can be solved by Rudelson's sampling lemma. This lemma can be easily generalized to handle matrices of arbitrary rank using the Ahlswede-Winter inequality, yielding a solution to Problem 2. Let X be a random matrix such that X = B i / Tr B i with probability p i := Tr B i / Tr I. Since B i 0 and i B i = I, the p i 's define a probability distribution. Theorem 18 ([1, Theorem 19]). Let X, X 1 , . . . , X T be i.i.d. random variables with values in S n such that X i ∈ [0, 1] for every i and E(X) = µI with µ ∈ [0, 1]. Let ε ∈ (0, 1/2). Then P 1 µT T i=1 X i ∈ [1 − ε, 1 + ε] ≤ 2n · exp −T ε 2 µ 2 ln 2 . In our case, E(X) = (1/n)I and X ∈ [0, 1]. So µ = 1/n. Thus, if T > (2 ln 2) · ln n+2 ln 2 ε 2 µ = O(n log n/ε 2 ), then P 1 µT T i=1 X i ∈ [1 − ε, 1 + ε] < 1/2. Thus, with constant probability, we obtain a solution y to Problem 2 where y has only O(n log n/ε 2 ) non-zero entries. Solving Problem 2 by BSS In our modification of the BSS algorithm [4], we keep a matrix A of the form A = i y i B i with y ≥ 0, starting with A = 0, and at each iteration we add another term αB j to A. We enforce the invariant that the eigenvalues of A lie in [ℓ, u], where u and ℓ are parameters given by u = u 0 + tδ U and ℓ = ℓ 0 + tδ L after t iterations. This procedure is presented in Algorithm 1. The step of the algorithm which finds B j and α can be done by exhaustive search on j and binary search on α. Instead of the binary search, one could also compare the quantities U A(t−1) (B j ) and L A(t−1) (B j ) defined below. In the original BSS algorithm, the matrices are rank one: B j = v j v T j for some vector v j . Their Lemmas 3.3 and 3.4 give sufficient conditions on the new term αv j v T j so that the invariant on the eigenvalues is maintained; Lemma 3.5 gives sufficient conditions on the remaining parameters so that a suitable new term αv j v T j exists with α > 0. In this section we generalize those lemmas to allow B i matrices of arbitrary rank. Let A ∈ S n . If u ∈ R with λ max (A) < u, define Φ u (A) := Tr(uI − A) −1 . If ℓ ∈ R with λ min (A) > ℓ, define Φ ℓ (A) := Tr(A − ℓI) −1 . Note that Φ ℓ (A) = i 1/(λ i − ℓ) and Φ u (A) = i 1/(u − λ i ), where λ 1 , . . . , λ n are the eigenvalues of A. Lemma 19 (Analog of Lemma 3.3 in [4]). Let A ∈ S n and X ∈ S n + with X = 0. Let u ∈ R and δ U > 0. Suppose λ max (A) < u. Let u ′ := u + δ U and M := u ′ I − A. If 1 α ≥ M −2 , X Φ u (A) − Φ u ′ (A) + M −1 , X =: U A (X),then λ max (A + αX) < u ′ and Φ u ′ (A + αX) ≤ Φ u (A). Algorithm 1 A procedure for solving Problem 2 based on the BSS method. procedure SparsifySumOfMatricesByBSS(B 1 , . . . , B m , ε) input: Matrices B 1 , . . . , B m ∈ S n + such that i B i = I, and a parameter ε ∈ (0, 1). output: A vector y with O(n/ε 2 ) nonzero entries such that I i y i B i (1 + O(ε))I. Initially A(0) := 0 and y(0) := 0. Set parameters u 0 , ℓ 0 , δ L , δ U as in (5) and T := 4n/ε 2 . Define the potential functions Φ u (A) := Tr(uI − A) −1 and Φ ℓ (A) := Tr(A − ℓI) −1 . For t = 1, . . . , T Set u t := u t−1 + δ U and ℓ t := ℓ t−1 + δ L . Find a matrix B j and a value α > 0 such that A(t − 1) + αB j ∈ [ℓ t , u t ], and Φ ut (A(t − 1) + αB j ) ≤ Φ u t−1 (A(t − 1)) and Φ ℓt (A(t − 1) + αB j ) ≤ Φ ℓ t−1 (A(t − 1)). Set A(t) := A(t − 1) + αB j and y(t) := y(t − 1) + αe j . Return y(T )/λ min (A(T )). Proof. Clearly M ≻ 0. Let V := X 1/2 . By the Sherman-Morrison-Woodbury formula [13], Φ u ′ (A + αX) = Tr(M − αV V T ) −1 = Tr M −1 + αM −1 V (I − αV T M −1 V ) −1 V T M −1 = Φ u ′ (A) + Tr αM −1 V (I − αV T M −1 V ) −1 V T M −1 . Since M −1 ≻ 0, X = 0 and Φ u (A) > Φ u ′ (A), our hypotheses imply 1/α > M −1 , X = Tr(V T M −1 V ) ≥ λ max (V T M −1 V ) ≥ 0, so β := λ min (I − αV T M −1 V ) = 1 − αλ max (V T M −1 V ) > 0 and by, e.g., [18, Corollary 7.7.4], 0 ≺ βI I − αV T M −1 V =⇒ 0 ≺ (I − αV T M −1 V ) −1 β −1 I. Thus, Φ u ′ (A + αX) ≤ Φ u ′ (A) + αβ −1 Tr(V T M −2 V ) = Φ u (A) − (Φ u (A) − Φ u ′ (A)) + αβ −1 M −2 , X To prove that Φ u ′ (A + αX) ≤ Φ u (A), it suffices to show that αβ −1 M −2 , X ≤ Φ u (A) − Φ u ′ (A). This is equivalent to M −2 , X 1/α − λ max (V T M −1 V ) ≤ Φ u (A) − Φ u ′ (A), which follows from 1/α ≥ U A (X) since λ max (V T M −1 V ) ≤ Tr(V T M −1 V ) = M −1 , X . It remains to show that λ max (A + αX) < u ′ . Suppose not. Choose ε ∈ (0, δ U ) such that 1/ε > Φ u (A). By continuity, for some α ′ ∈ (0, α) we have λ max (A + α ′ X) = u ′ − ε. Since 1/α ′ ≥ 1/α ≥ U A (X), we get Φ u ′ (A + α ′ X) ≥ 1/ε > Φ u (A) ≥ Φ u ′ (A + α ′ X), a contradiction. Lemma 20 (Analog of Lemma 3.4 in [4]). Let A ∈ S n and X ∈ S n + , with n ≥ 2. Let ℓ ∈ R and δ L > 0. Suppose λ min (A) > ℓ and Φ ℓ (A) ≤ 1/δ L . Let ℓ ′ := ℓ + δ L and N := A − ℓ ′ I. If 0 < 1 α ≤ N −2 , X Φ ℓ ′ (A) − Φ ℓ (A) − N −1 , X =: L A (X),then λ min (A + αX) > ℓ ′ and Φ ℓ ′ (A + αX) ≤ Φ ℓ (A). Moreover, N ≻ 0. Proof. Note that λ min (A) > ℓ and Φ ℓ (A) ≤ 1/δ L imply that N ≻ 0, and therefore λ min (A + αX) > ℓ ′ . Let V := X 1/2 . By the Sherman-Morrison-Woodbury formula, Φ ℓ ′ (A + αX) = Tr(N + αV V T ) −1 = Tr N −1 − αN −1 V (I + αV T N −1 V ) −1 V T N −1 = Φ ℓ ′ (A) − Tr αN −1 V (I + αV T N −1 V ) −1 V T N −1 . For β := λ max (I + αV T N −1 V ), we have 0 ≺ I + αV T N −1 V βI =⇒ 0 ≺ β −1 I (I + αV T N −1 V ) −1 . Thus, Φ ℓ ′ (A + αX) ≤ Φ ℓ ′ (A) − αβ −1 Tr(V T N −2 V ) = Φ ℓ (A) + (Φ ℓ ′ (A) − Φ ℓ (A)) − αβ −1 N −2 , X We will be done if we show that αβ −1 N −2 , X ≥ Φ ℓ ′ (A) − Φ ℓ (A) . This is equivalent to N −2 , X 1/α + λ max (V T N −1 V ) ≥ Φ ℓ ′ (A) − Φ ℓ (A) which follows from 0 < 1/α ≤ L A (X), since Φ ℓ ′ (A) > Φ ℓ (A), N ≻ 0, and λ max (V T N −1 V ) ≤ Tr(V T N −1 V ) = N −1 , X . The next lemma can be proved by a syntactic modification of the proof of Lemma 3.5 in [4]. Lemma 21 (Analog of Lemma 3.5 in [4]). Let A ∈ S n with n ≥ 2, and let u, ℓ ∈ R and ε U , δ U , ε L , δ L > 0 such that λ max (A) < u, λ min (A) > ℓ, Φ u (A) ≤ ε U , and Φ ℓ (A) ≤ ε L . Let B 1 , . . . , B m ∈ S n such that i B i = I. If 0 ≤ 1 δ U + ε U ≤ 1 δ L − ε L (3) then there exists j ∈ [m] and α > 0 for which L A (B j ) ≥ 1/α ≥ U A (B j ). Proof. As in [4, Lemma 3.5], it suffices to show that i L A (B i ) ≥ i U A (B i ). Let u ′ := u + δ U , M := u ′ I − A, ℓ ′ := ℓ + δ L , and N := A − ℓ ′ I. It follows from the bilinearity of ·, · and the assumption i B i = I that i U A (B i ) = Tr M −2 Φ u (A) − Φ u ′ (A) + Tr M −1 (4a) i L A (B i ) = Tr N −2 Φ ℓ ′ (A) − Φ ℓ (A) − Tr N −1 (4b) It is shown in [4, Lemma 3.5] that (4a) is at most (4b), completing the proof. Now we set the parameters of Lemma 21 similarly as in [4]: δ L := 1 ε L := ε 2 ℓ 0 := − n ε L δ U := 2 + ε 2 − ε ε U := ε 2δ U u 0 := n ε U .(5) So (3) holds with equality. If A is the matrix obtained after T = 4n/ε 2 iterations, then λ max (A) λ min (A) ≤ u 0 + T δ U ℓ 0 + T δ L = 2 + ε 2 − ε 2 ≤ 1 + ε 1 − ε so A ′ := A/λ min (A) satisfies I A ′ (1+ε)I/(1−ε) and A ′ is a positive linear combination of O(n/ε 2 ) of the matrices B i . It is easy to check that the previous lemmas also hold if we replace the set S n of symmetric matrices of size n × n by the set H n of Hermitian matrices of size n × n. Running Time At each iteration, we must compute U A (B j ) and L A (B j ) for each j ∈ [m]. The functions U A (X) and L A (X) are the inner products of X with certain matrices that can be obtained from A in time O(n 3 ). Thus, each iteration runs in time O(n 3 +mn 2 ) = O(mn 2 ), and the total running time after T = 4n/ε 2 iterations is O(mn 3 /ε 2 ). We remark that the reduction to the case B = I can be made in time O(mn 3 ). This concludes the proof of Theorem 3. If the matrices B i have O(1) nonzero entries, as in the graph sparsification problem, the algorithm can be made to run in time O(n 4 /ε 2 + mn/ε 2 ). We briefly sketch the details. To reduce the problem to the case that B = I, we first compute (B + ) 1/2 , where B + is the Moore-Penrose pseudoinverse of B. Define the function f (X) := (B + ) 1/2 X(B + ) 1/2 on S n . The reduction now calls for replacing each input matrix B i by f (B i ) and the matrix B by f (B). But we shall not do this. Instead, we do some preprocessing at each iteration as follows. The function U A (X) (as well as L A (X)) is the inner product of X with a certain matrix V . Hence, U A (f (B j )) = V, f (B j ) = f (V ), B j for every j, since f is self-adjoint. Thus, to compute U A (f (B j )) for each j, Solving Problem 2 by MMWUM Observe that the set of all vectors y that are feasible for (2) is the feasible region of a semidefinite program (SDP). So solving Problem 2 amounts to finding a sparse solution to this SDP. Here "sparse" means that there are few non-zero entries in the solution y; this differs from other notions of "low-complexity" SDP solutions, such as the low-rank solutions studied by So, Ye and Zhang [36]. It has long been known known that the multiplicative weight update method can be used to construct sparse solutions for some linear programs. A prominent example is the construction of sparse, low-regret solutions to zero-sum games [9,43,44]. (Another example is the work of Charikar et al. [7] on approximating metrics by few tree metrics.) Building on that idea, one might imagine that Arora and Kale's matrix multiplicative update method (MMWUM) [3] can construct sparse solutions to (2). In this section, we show that this is indeed possible: we obtain a solution y to Problem 2 with O(n log n/ε 3 ) nonzero entries. Overview of MMWUM The MMWUM is an algorithm that helps us approximately solve an SDP feasibility problem. The gist of (a slight modification of) the method is contained in the following result (its proof can be found in Appendix B): Theorem 22. Let T, K, n 1 , . . . , n K be positive integers. Let C k , A 1,k , . . . , A m,k ∈ S n k for k ∈ [K]. For each k ∈ [K], let η k > 0 and 0 < β k ≤ 1/2. Given X 1 , . . . , X K ∈ S n , consider the system m i=1 y i A i,k , X k ≥ C k , X k − η k Tr X k , ∀k ∈ [K], and y ∈ R m + .(6)For each k ∈ [K], let {P k , N k } be a partition of [T ], let 0 < ℓ k ≤ ρ k , and let W (t) k ∈ S n and ℓ (t) k ∈ R for t ∈ [T + 1]. Let y (t) ∈ R m for t ∈ [T ] . Suppose the following properties hold: W (t+1) k = exp − β k ℓ k + ρ k t τ =1 m i=1 y (τ ) i A i,k − C k + ℓ (τ ) k I , ∀t ∈ {0, . . . , T }, ∀k ∈ [K], y = y (t) is a solution for (6) with X k = W (t) k , ∀k ∈ [K], ∀t ∈ [T ], m i=1 y (t) i A i,k − C k ∈ [−ℓ k , ρ k ], if t ∈ P k , [−ρ k , ℓ k ], if t ∈ N k , ∀t ∈ [T ], k ∈ [K], ℓ (t) k = ℓ k , ∀t ∈ P k , ∀k ∈ [K], and ℓ (t) k = −ℓ k , ∀t ∈ N k , ∀k ∈ [K] . Defineȳ := 1 T T t=1 y (t) . Then, m i=1ȳ i A i,k − C k − β k ℓ k + (ρ k + ℓ k ) ln n T β k + (1 + β k )η k I, ∀k ∈ [K].(7) Take K = 2, set C 1 := I and C 2 := −I, and put A i,1 := B i and A i,2 := −B i for each i ∈ [m]. Then Theorem 22 shows that finding a solution for (2) reduces to constructing an oracle that solves linear systems of the form (6) with a few extra technical properties involving the parameters ℓ k and ρ k , and adjusting the other parameters so that the error term on the right-hand side of (7) is ≤ ε. To obtain a feasible solution for (2) that is also sparse, the idea is to design an implementation of the oracle that returns a vector y (t) with only one nonzero entry at each iteration t of MMWUM, and to adjust the parameters so that, after T = O(n log n/ε 3 ) iterations, the smallest and largest eigenvalues of m i=1ȳ i B i are ε-close to 1. Sinceȳ is the average of the y (t) 's, the resultingȳ will have at most T nonzero entries. We set the remaining parameters as follows: Then the error term on the right-hand side of (7) is βℓ + (ρ + ℓ) ln n T β + (1 + β)η = ε 4 + ε 2 + 1 + ε 4 ε 8 = 7ε 8 + ε 2 32 ≤ ε.(8) Thus, (2) follows from (7) and (8). Moreover, T = O(n log n/ε 3 ), as desired. The Oracle It remains to implement the oracle. Consider an iteration t, and let X 1 := W (t) 1 and X 2 := W (t) 2 be given. We must find y := y (t) ∈ R m + with at most one nonzero entry such that m i=1 y i X 1 , B i ≥ (1 − η) Tr X 1 , m i=1 y i X 2 , B i ≤ (1 + η) Tr X 2 , and m i=1 y i B i ∈ [0, ρ]. Since y should have only one nonzero entry, it suffices to find j ∈ [m] and α ∈ R + such that α X 1 , B j ≥ (1 − η) Tr X 1 , α X 2 , B j ≤ (1 + η) Tr X 2 , α Tr B j ≤ ρ.(9) Here we are using the fact that λ max (B j ) ≤ Tr B j since B j 0. We will show that such j and α exist. Due to the definition of W 1 and W 2 , the oracle can assume that X 1 is a scalar multiple of X −1 2 , although we will not make use of that fact. Proposition 23. Let B 1 , . . . , B m ∈ S n + such that m i=1 B i = I. Let η > 0 and X 1 , X 2 ∈ S n ++ . Then, for ρ := (1 + η)n/η, there exist j ∈ [m] and α ≥ 0 such that (9) holds. Proof. By possibly dropping some B i 's, we may assume that B i = 0 for every i ∈ [m]. Define p i := X 1 , B i / Tr X 1 > 0 for every i ∈ [m]. Consider the probability space on [m] where j is sampled from [m] with probability p j . The fact that m j=1 p j = 1 follows from m i=1 B i = I. Then E j [p −1 j Tr B j ] = m i=1 Tr B i = Tr I = n. By Markov's inequality, P p −1 j Tr B j ≤ (1 + η) η n = 1 − P p −1 j Tr B j > (1 + η) η n > 1 − η 1 + η = 1 1 + η .(10) Next note that E j [p −1 j X 2 , B j ] = m i=1 X 2 , B i = X 2 , I = Tr X 2 . Together with Markov's inequality, this yields P p −1 j X 2 , B j ≤ (1 + η) Tr X 2 = 1 − P p −1 j X 2 , B j > (1 + η) Tr X 2 > 1 − 1 1 + η .(11) It follows from (10) and (11) that there exists j ∈ [m] satisfying p −1 j X 2 , B j ≤ (1 + η) Tr X 2 , and p −1 j Tr B j ≤ 1 + η η n = ρ. Set α := p −1 j and note that α X 1 , B j = p −1 j X 1 , B j = Tr X 1 ≥ (1 − η) Tr X 1 . Hence, j and α satisfy (9). The following proposition, proven in Appendix C, shows that the parameters achieved by Proposition 23 is essentially optimal. Proposition 24. Any oracle for satisfying (9) must have ρ = Ω(n/η), even if the B i matrices have rank one, and even if X 1 is a scalar multiple of X −1 2 . We also point out that a naive application of MMWUM as stated by Kale in [22] does not work. In his description of MMWUM, the parameter K is fixed as 1. So we must correspondingly adjust our input matrices to be block-diagonal, e.g., C has two blocks: I and −I. However, applying Theorem 22 in this manner would lead to a sparsifier with Ω(n 2 ) edges. The reason is that the parameter ρ needs to be Ω(n), and we must choose ℓ = ρ since the spectrum of m i=1 y i A i − C is symmetric around zero for any y. Thus, to get the error term on the right-hand side of (7) to be ≤ ε, we would need to take T = Ω(n 2 ). Solving Problem 2 by a Width-Free MMWUM The algorithm of Section 5 solves Problem 2 with only O(n log n/ε 3 ) nonzero entries, which is slightly worse than the O(n log n/ε 2 ) nonzero entries achieved by the Ahlswede-Winter method discussed in Section 3. The main reason for this discrepancy is that MMWUM requires us to bound the "width" of the oracle using the parameter ρ; formally, the oracle must the inequality α Tr B j ≤ ρ in (9). In order to satisfy this width constraint, the oracle loses an extra factor of O(1/ε), and this is necessary as shown in Proposition 24. In this section, we slightly refine MMWUM to avoid its dependence on the width. This allows us to simplify our oracle and avoid losing the extra factor of O(1/ε). We obtain a solution to Problem 2 with only only O(n log n/ε 2 ) nonzero entries, matching the sparsity of the solutions obtained by the Ahlswede-Winter inequality. The following theorem is our width-free variant of MMWUM. We remark that the method described in this theorem is geared towards solving Problem 2 and is not necessarily useful for all applications of MMWUM. T be a positive integer. Let B 1 , . . . , B m ∈ S n + be nonzero. Let γ, η, δ L , δ U > 0. For any given X L , X U ∈ S n , consider the system Theorem 25. Let δ U ≥ exp(γα Tr B j ) − 1 Tr B j X U , B j , δ L ≤ 1 − exp(−γα Tr B j ) Tr B j X L , B j , α ∈ R + , j ∈ [m].(12) For each t ∈ {0, . . . , T + 1}, let A(t), W L (t), W U (t) ∈ S n , let α(t) ∈ R + , and let j(t) ∈ [m]. Suppose the following properties hold: A(t) = t τ =1 α(τ )B j(τ ) , ∀t ∈ {0, . . . , T }, W U (t + 1) = exp(γA(t)) and W L (t + 1) = exp(−γA(t)), ∀t ∈ {0, . . . , T }, (α, B j ) = (α(t), B j(t) ) is a solution for (12) with (X U , X L ) = W U (t) Tr W U (t) , W L (t) Tr W L (t) , ∀t ∈ [T ]. Then A(T ) T ∈ log(1 − δ L ) −1 γ − log n T γ , log(1 + δ U ) γ + log n T γ .(13) Proof. We will use Golden-Thompson inequality: Tr(exp(A + B)) ≤ Tr(exp(A) exp(B)), ∀A, B ∈ S n .(14) We will also make use of the following facts. First, exp(cx) ≤ 1 + exp(c · b) − 1 b x ∀c ∈ R, b > 0, x ∈ [0, b]. For X ∈ S n + , we have λ max (X) ≤ Tr X, so X ∈ [0, Tr X], and exp(cX) I + exp(c · Tr X) − 1 Tr X X. For each t ∈ [T + 1], define Φ L (t) := Tr W L (t) and Φ U (t) = Tr W U (t). For each t ∈ [T ], Φ U (t + 1) = Tr exp(γA(t)) = Tr exp(γA(t − 1) + γαB j ) (14) ≤ Tr exp(γA(t − 1)) exp(γαB j ) (15) ≤ Tr exp(γA(t − 1)) exp(γα Tr B j ) − 1 Tr B j B j + I = exp(γα Tr B j ) − 1 Tr B j Tr(exp(γA(t − 1))B j ) + Tr(exp(γA(t − 1))) = exp(γα Tr B j ) − 1 Tr B j W U (t), B j + Φ U (t)(12)≤ (1 + δ U )Φ U (t),(16) where we abbreviated j := j(t) and α := α(t). Since A(0) = 0, we have that Φ U (1) = Tr I = n. Using (16), after T iterations, Φ U (T + 1) ≤ (1 + δ U ) T n. Thus, exp(γλ max (A(T ))) ≤ n i=1 exp(γλ i ) = Tr W U (T + 1) = Φ U (T + 1) ≤ (1 + δ U ) T n, where λ 1 , . . . , λ n are the eigenvalues of A(T ). And so γλ max (A(T )) ≤ T log(1 + δ U ) + log n, which implies the upper bound in (13). The proof of the lower bound is analogous. Next we establish conditions under which we can construct an oracle for solving the system (12). The proof consists of algebraic manipulations and an averaging argument analogous to the proof of Lemma 3.5 in [4]. Theorem 26. Let B 1 , . . . , B m ∈ S n + be nonzero such that m i=1 B i = I. Let δ U , δ L > 0 be such that 1 δ L − n ≥ 1 δ U .(17) Then, for any X L , X U ∈ S n ++ with trace one, the system (12) has a solution. Proof. The first inequality in (12) is equivalent to Tr B j exp(γα Tr B j ) − 1 ≥ X U , B j δ U .(18) Using the identity 1 1−1/x = 1 + 1 x−1 , the second inequality in (12) is equivalent to Tr B j exp(γα Tr B j ) − 1 ≤ X L , B j δ L − Tr B j .(19) We will choose j ∈ [m] so that X L , B j δ L − Tr B j ≥ X U , B j δ U(20) and set α so that (18) holds with equality. Then both (18) and (19) will hold. Note that α ≥ 0 since e γα Tr B j = 1 + δ U Tr B j / X U , B j > 1 and γ Tr B j > 0. To see that there exists j ∈ [m] satisfying (20), note that, by (17) and m i=1 B i = I, m i=1 X L , B i δ L − Tr B i = Tr X L δ L − n = 1 δ L − n ≥ 1 δ U = Tr X U δ U = m i=1 X U , B i δ U . This concludes the proof. Finally, let us show how to set the parameters to get a sparsifier. Given ε ∈ (0, 1), set η := ε/2, δ U := η n , δ L := η (1 + η)n , T := n log n η 2 . By our choice of δ L and δ U , we have 1/δ L − n = (1 + η)n/η − n = n/η = 1/δ U , so (17) holds with equality. After we run the modified version of MMWUM given by Theorem 25, we obtain a matrix A(T ). SetĀ := A(T )/T . By Theorem 25, λ max (Ā) ≤ log(1 + δ U ) γ + log n T γ ≤ δ U + η 2 n /γ = 1 + η nγ/η . We will use that − log(1 − x) ≥ x for x < 1. Thus, λ min (Ā) ≥ log(1 − δ L ) −1 γ − log n T γ ≥ δ L − η 2 n /γ = 1/(1 + η) − η nγ/η ≥ 1 − 2η nγ/η . So if we choose γ = η/n then (1 − ε)I Ā (1 + ε)I andĀ is of the form i y i B i with y ≥ 0 and has at most T = O(n log n/ε 2 ) nonzero entries. Remark. The choice of γ is actually irrelevant here. We could choose γ > 0 arbitrarily, then definē A = A(T ) · (nγ/ηT ) and the desired conclusion would hold. Solving Problem 2 by Pessimistic Estimators An anonymous reviewer for a preliminary draft of this paper raised the possibility of designing another deterministic solution to Problem 2. The proposal was to use the pessimistic estimators of Wigderson and Xiao [42] to derandomize the random sampling approach of Section 3. In this section we show that this proposal indeed works. We remark that pessimistic estimators were also used by Hofmeister and Lefmann [17] to derandomize the proof of Theorem 15. It is known that there is a close relationship between pessimistic estimators and multiplicative weight update methods. (See, for example, the work of Young [44].) However, the two methods are not identical, and in particular the algorithm presented in this section is not identical to either of our algorithms based on MMWUM. To illustrate one difference, notice that the algorithm in Section 3 has the property that its output vector y has every component y i equal to an integer multiple of n/(T · Tr B i ). The algorithm of this section also has that property as it is a derandomization of the algorithm in Section 3. However, the algorithms in Sections 4, 5 and 6 do not have that property. i and any fixed x 1 , ..., x i ∈ [n]: P X i+1 ,...,X T (x 1 , . . . , x i , X i+1 , . . . , X T ) ∈ S ≤ φ i (x 1 , . . . , x i ). For any E X i+1 (φ i+1 (x 1 , . . . , x i , X i+1 )) ≤ φ i (x 1 , . . . , x i ). Note that the function φ 0 depends on no variables and is therefore just a scalar in [0, 1]. A nice property of this definition is that it allows compositions very easily. That is, if we have pessimistic estimators φ 0 , . . . , φ T and ψ 0 , . . . , ψ T for events S and S ′ , resp., then φ 0 + ψ 0 , . . . , φ T + ψ T are pessimistic estimators for the event S ∩ S ′ (see Lemma 3.3 in [42]). The key point of this method is that, if there are pessimistic estimators φ 0 , . . . , φ T , such that φ 0 < 1 and each φ i can be computed efficiently, then one can find (x 1 , . . . , x T ) ∈ S efficiently. Let X 1 , . . . , X T be be i.i.d. random variables with same distribution as the random variable X as defined in Section 3. Wigderson and Xiao [42] considered the event S ≥ = { X : 1 T T i=1 X i (1 − ε)µI} and obtained 1 the following pessimistic estimators: φ 0 = ne tT (1−ε)µ E X exp(−tX) T ≤ n exp(−T ε 2 µ/(2 ln 2)); φ i (x 1 , . . . , x i ) := e tT (1−ε)µ Tr exp(− j i=1 tx i ) · E X exp(−tX) T −i , where t = log 1−(1−ε)µ (1−µ)(1−ε) . Similarly, for the event S ≤ = { X : 1 T T i=1 X i (1 + ε) µI}, one can find the following pessimistic estimators ψ 0 = ne −t ′ T (1+ε)µ E X exp(t ′ X) T ≤ n exp(−T ε 2 µ/(2 ln 2)); ψ i (x 1 , . . . , x i ) := e −t ′ T (1+ε)µ Tr exp( i j=1 t ′ x j ) · E X exp(t ′ X) T −i , where t ′ = log (1+ε)(1−µ) 1−(1+ε)µ . If we choose T > (2 ln 2) ln(2n)/(ε 2 µ) = (2 ln 2)n ln(2n)/ε 2 , then φ 0 + ψ 0 < 1. Each φ i , ψ i can be computed efficiently and so one can find in polynomial time (x 1 , . . . , x T ) ∈ S ≥ ∩ S ≤ . Comparing BSS and MMWUM In this section we show a striking similarity between the algorithms presented in Sections 4 and 6. The proof of Theorem 25 defines two potential functions for each iteration t. The proof shows that, for the algorithm of Section 6, the potentials must change as follows: Φ U (t + 1) ≤ (1 + δ U )Φ U (t) ∀t ∈ {0, . . . , T − 1} Φ L (t + 1) ≤ (1 − δ L )Φ L (t) ∀t ∈ {0, . . . , T − 1}.(22) Instead of requiring these potentials to grow and shrink in this way, we could instead parameterize the potential functions by the iteration number t and then simply require that the potential do not grow from iteration to iteration. To formalize this alternative approach, let us define the new potential functions For t = 1, . . . , T Set u t := u t−1 + ∆ U and ℓ t := ℓ t−1 + ∆ L . Find a matrix B j and a value α > 0 such that Ψ ut (A(t − 1) + αB j ) ≤ Ψ u t−1 (A(t − 1)) and Ψ ℓt (A(t − 1) + αB j ) ≤ Ψ ℓ t−1 (A(t − 1)). Set A(t) := A(t − 1) + αB j and y(t) := y(t − 1) + αe j . Return y(T )/λ min (A(T )). (22) governing the algorithm's change in potentials are equivalent to inequalities in (23). Proposition 28. The inequalities in Ψ (t+1)∆ U (A(t) + αB j ) ≤ Ψ t∆ U (A(t)) Ψ (t+1)∆ L (A(t) + αB j ) ≤ Ψ t∆ L (A(t))(23) Proof. Obviously (22) is equivalent to (1 + δ U ) −(t+1) · Φ U (t + 1) ≤ (1 + δ U ) −t · Φ U (t) ∀t ∈ {0, . . . , T − 1}, (1 − δ L ) −(t+1) · Φ L (t + 1) ≤ (1 − δ L ) −t · Φ L (t) ∀t ∈ {0, . . . , T − 1}. By the definition of Φ U and Φ L , and by properties of the exponential function, these inequalities are equivalent to Tr exp(−(t + 1)∆ U I + γA(t + 1)) ≤ Tr exp(−t∆ U I + γA(t)), Tr exp((t + 1)∆ L I − γA(t + 1)) ≤ Tr exp(t∆ L I − γA(t)). Writing A(t + 1) = A(t) + αB j , these inequalities in (24) are equivalent to (23). Algorithm 2 gives pseudocode for the algorithm of Section 6, using the functions Ψ u and Ψ ℓ to control the change in potentials. The main point of this section is to observe that Algorithms 1 and 2 are identical with the exception of different parameters and different potential functions. We believe that this similarity between these two algorithms is intriguing, especially since the BSS algorithm has been called "highly original" by Naor [28]. In retrospect, it would have been perhaps more natural to develop the BSS algorithm by the following logical progression of ideas: first observe that MMWUM is useful for giving sparse solutions to SDPs, then design Algorithm 2, then later realize that a clever refinement of it leads to Algorithm 1 and its improved analysis. It is remarkable that Batson, Spielman and Srivastava developed their algorithm from first principles, apparently without knowing this connection to established algorithmic techniques. With the advantage of hindsight (i.e., the knowledge that the BSS algorithm exists), we now explain how one might be tempted to refine Algorithm 2. It is quite tempting to modify the potential functions to more strongly penalize eigenvalues which deviate from the desired range. The natural approach to do this would be to increase the derivatives of the potential function by increasing the parameter γ. However, as remarked at the end of Section 6, the algorithm is actually unaffected by varying γ! Thus, to improve Algorithm 2, one must seek a more substantially different potential function. Focusing on the upper potential, we consider the question: is there a function f : R → R with steeper derivatives than exp(u − x) and such that, for any matrices Proof. For every edge e = ij ∈ E, let B e be the direct sum w ij (e i − e j )(e i − e j ) T ⊕ c 1,e ⊕ · · · ⊕ c k,e . Let B := L G (w) ⊕ w T c 1 ⊕ · · · ⊕ w T c k . The result follows immediately by applying Theorem 3 to these matrices. Proof. For each i, let c i : E → R be the characteristic vector of E i . Now apply Corollary 4. L G (w) L H (w H ) (1 + ε)L G (w), (1 − ε) e∈E i w e ≤ e∈E(H)∩E i w H,e ≤ (1 + ε) Corollary 6 (Spectral sparsifiers for hypergraphs). For any real ε ∈ (0, 1), there is a deterministic polynomial-time algorithm to find a sub-hypergraph G of H and a weight function w G : E(G) → R + such that L H (w) L G (w G ) (1 + ε)L H (w), and |E(G)| = O(n/ε 2 ). Proof. The result follows directly by applying Theorem 3 to the matrices w E L E . Corollary 7 (Cut sparsifiers for hypergraphs, second definition). For any real ε ∈ (0, 1), there is a deterministic polynomial-time algorithm to find a sub-hypergraph G of H and a weight function w G : E(G) → R + such that w * (δ H (S)) ≤ w * G (δ G (S)) ≤ (1 + ε)w * (δ H (S)) for every S ⊆ V , and |E(G)| = O(n/ε 2 ). Proof. Note that w * (δ H (S)) is obtained by evaluating the quadratic form x T L H (w)x, where x is the characteristic vector of S. Thus the sparsifier produced by Corollary 6 satisfies the desired inequalities. Corollary 8 (Cut sparsifiers for hypergraphs, first definition). Assume that H is an r-uniform hypergraph. For any real ε ∈ (0, 1), there is a deterministic polynomial-time algorithm to find a sub-hypergraph G of H and a weight function w G : E(G) → R + such that (r − 1) r 2 /4 w(δ H (S)) ≤ w G (δ G (S)) ≤ (1 + ε)r 2 4(r − 1) w(δ H (S)) ∀S ⊆ V, and |E(G)| = O(n/ε 2 ). In other words, the sparsified hypergraph G approximates the weight of the cuts in the hypergraph H to within a factor Θ(r 2 ). Proof. For any r-uniform hypergraph H, it is easy to see that (r − 1)w(δ H (S)) ≤ w * (δ H (S)) ≤ ⌊r/2⌋⌈r/2⌉w(δ H (S)) ∀S ⊆ V.(25) Thus the sparsifier produced by Corollary 6 satisfies the desired inequalities. Corollary 9 (Cut sparsifiers for 3-uniform hypergraphs). Assume that H is a 3-uniform hypergraph. For any ε ∈ (0, 1), there is a deterministic polynomial-time algorithm to find a sub-hypergraph G of H and a weight function w G : E(G) → R + such that w(δ H (S)) ≤ w G (δ G (S)) ≤ (1 + ε)w(δ H (S)) ∀S ⊆ V, and |E(G)| = O(n/ε 2 ). Proof. Since r = 3, a consequence of (25) is that w * (δ H (S)) = 2w(δ H (S)) for every S. Thus the sparsifier produced by Corollary 6 satisfies the desired inequalities. min c T z : i z i A i B, z ∈ R m , z ≥ 0 has a feasible solution z * . Then, for any real ε ∈ (0, 1), it has a feasible solutionz with at most O(n/ε 2 ) nonzero entries and c Tz ≤ (1 + ε)c T z * . Proof. Let B ′ i := z * i A i 0 0 c i z * i for every i ∈ [m] and B ′ := D 0 0 c T z * , where D := i z * i A i B. Then B ′ i 0 and B ′ = i B ′ i . By applying Theorem 3, we obtain y ∈ R m with y ≥ 0 and O(n/ε 2 ) nonzero entries such that i y i z * i A i D B and i c i y i z * i ≤ (1 + ε)c T z * . Thus, we can takez i = y i z * i for every i ∈ [m]. Corollary 11. Let G = (V, E) be a graph. For any real ε ∈ (0, 1), there is a deterministic polynomial-time algorithm to find a subgraph H of G such that (1 − ε)t ′ (G) ≤ t ′ (H) ≤ t ′ (G) and |E(H)| = O(n/ε 2 ). Proof. It is straightforward to formulate t ′ (G) as an SDP (see, e.g., [26]) so that its dual has an optimal solution and there is no duality gap. The dual can be written as: max e∈E z e : Diag(y) L G (z), v∈V y v = 1, z ≥ 0(26) The proof is now almost identical to the proof of Corollary 10. Let (z * , y * ) be an optimal solution. Using Theorem 3, we obtainz ∈ R E withz ≥ 0 and O(n/ε 2 ) nonzero entries such that (y * ,z) is feasible in (26) and has objective value e∈E(H)z e ≥ (1 − ε)t(G), where H = (V, E(H)) and E(H) is the support ofz. Thenz is also feasible for the SDP defined using H instead of G, which shows that t ′ (H) ≥ (1 − ε)t ′ (G). Corollary 12. Let G = (V, E) be a graph. For any real ε ∈ (0, 1), there is a deterministic polynomial-time algorithm to find a supergraph H of G such that ϑ ′ (G) 1 − ε + εϑ ′ (G) ≤ ϑ ′ (H) ≤ ϑ ′ (G) and |E(H)| = n 2 − O(n/ε 2 ). Proof. For a graph G = (V, E), define t(G) as the square of the minimum radius of a hypersphere on R n such that there is a map from V to the hypersphere such that adjacent vertices are mapped to points at distance exactly 1. Lovász [26] noted that t(G) is related to the Lovász theta number ϑ(G) of the complement G of G by the formula 2t(G) + 1/ϑ(G) = 1; see [8] for a proof. By repeating the same proof for t ′ (G), one finds that 2t ′ (G) + 1/ϑ ′ (G) = 1. The result now follows from Corollary 11 via this formula. Corollary 13. Let G be a graph such that ϑ ′ (G) = o( √ n). For any real γ > 0, there is a supergraph H of G such that ϑ ′ (G) 1 + γ ≤ ϑ ′ (H) ≤ ϑ ′ (G) and |E(H)| = n 2 − O(nϑ(G) 2 /γ 2 ). Proof. Apply Corollary 12 with ε := γ/ϑ ′ (G). Corollary 14. Let G be a graph such that ϑ ′ (G) = Ω( √ n). For any real γ ≥ 1, there is a supergraph H of G such that ϑ ′ (H) = Ω( √ n/γ) and |E(H)| = n 2 − O(n 2 /γ 2 ). Proof. Apply Corollary 12 with ε := γ/ √ n. (1 − ε)B i µ i B i (1 + ε)B. Proof. Let B ′ i := λ i B i 0 0 λ i for every i ∈ [m] and B ′ := B 0 0 1 , so that B ′ i 0 and B ′ = i B ′ i . By applying Theorem 3, we obtain y ∈ R m with y ≥ 0 and O(n/ε 2 ) nonzero entries such that B ′ i y i B ′ i (1 + ε)B ′ or, equivalently, B y y i λ i B i (1 + ε)B and 1 ≤ i y i λ i ≤ 1 + ε. Let µ ∈ R m be defined by µ i := y i λ i /( i y i λ i ). Then µ ≥ 0 and i µ i = 1, and (1 − ε)B B 1 + ε B i y i λ i i µ i B i 1 + ε i y i λ i B (1 + ε)B. This completes the proof. L G (w) L H (w H ) (1 + ε)L G (w), L F (w F ) L H∩F (w H ↾ E(H∩F ) ) (1 + ε)L F (w F ) for all F ∈ F , where w F := w↾ E(F ) is the restriction of w to the coordinates E(F ) and H ∩F = V (F ), E(F ) ∩ E(H) . Proof. For each edge e ∈ E, define B e := w e L G (χ e ) ⊕ F ∈F L F (χ e ↾ E(F ) ) , where χ e denotes the characteristic vector of {e} as a subset of E. Now apply Theorem 3. B The MMWUM In this section we provide some proofs about the MMWUM. These proofs are due to Kale [22]. Our set up and conclusions are slightly different and we modified the proofs accordingly. We reproduce the proofs here for the sake of completeness. Theorem 22 can be viewed as a block-friendly version of MMWUM. First we show the version with only one block. It is basically the same as [22,Theorem 13 in Chapter 4]. Theorem 29. Let T be a positive integer. Let C, A 1 , . . . , A m ∈ S n . Let η > 0 and 0 < β ≤ 1/2. For any given X ∈ S n , consider the system m i=1 y i A i , X ≥ C, X − η Tr X, and y ∈ R m + . Let {P, N } be a partition of [T ], let 0 < ℓ ≤ ρ, and let W (t) ∈ S n and ℓ (t) ∈ R for t ∈ [T + 1]. Let y (t) ∈ R m for t ∈ [T ]. Suppose the following properties hold: where we have used Golden-Thompson's inequality (14). Using the fact that e x is convex, one can prove that W (t+1) = exp − β ℓ + ρ t τ =1 m i=1 y (τ ) i A i − C + ℓ (τ ) I ,0 A I =⇒ exp(−βA) I − β 1 A, −I A 0 =⇒ exp(−βA) I − β 2 A. Suppose that t ∈ P. Then exp(−βM (t) ) I − β 1 M (t) , and since W (t) 0, we get Φ (t+1) ≤ W (t) , exp(−βM (t) ) ≤ W (t) , I − β 1 M (t) = Tr(W (t) ) − β 1 W (t) , M (t) = Tr(W (t) ) − Tr(W (t) )β 1 P (t) , M (t) = Tr(W (t) ) 1 − β 1 P (t) , M (t) = Φ (t) 1 − β 1 P (t) , M (t) ≤ Φ (t) exp(−β 1 P (t) , M (t) ). and e x − 1 ≤ x(1 + x), ∀x ∈ [0, 1 2 ] So our choice of β 1 and β 2 ensures that 1 − β ≤ β 1 /β and 1 + β ≥ β 2 /β. We can now show the proof of Theorem 29. = 1 (ℓ + ρ) Tr W (t) m i=1 y (t) i A i , W (t) − C, W (t) + ℓ (t) ℓ + ρ ≥ − η ℓ + ρ + ℓ (t) ℓ + ρ , since y (t) is a solution for (27) with X := W (t) . Thus, by (29), t∈P (1 − β)(ℓ (t) − η) ℓ + ρ + t∈N (1 + β)(ℓ (t) − η) ℓ + ρ ≤ 1 ρ + ℓ λ min T t=1 m i=1 y (t) i A i − C + ℓ (t) I + ln n β . Multiply through by ℓ + ρ and move ℓ (t) I out of λ min (·): t∈P (1 − β)ℓ (t) + t∈N (1 + β)ℓ (t) − T (1 + β)η ≤ λ min T t=1 m i=1 y (t) i A i − C + T t=1 ℓ (t) + (ρ + ℓ) ln n β . Thus, t∈P −βℓ (t) + t∈N βℓ (t) ≤ λ min T t=1 m i=1 y (t) i A i − C + (ρ + ℓ) ln n β + T (1 + β)η. Next note that t∈P −ℓ (t) + t∈N ℓ (t) = t∈P −ℓ + t∈N −ℓ = −T ℓ, so Theorem 22 can be easily proved from Theorem 29. First, we apply Theorem 29 separately for each block. In each iteration, y (t) is a solution for (27) for all blocks simultaneously, and so the conclusion in (28) holds for all blocks with sameȳ. This new algorithm can be seen as equivalent to running K copies of MMWUM, each with different input data, with the caveat that all copies run for the same number of iterations and the vector y (t) returned from the oracle is the same for all copies at each iteration t. 0 ≤ λ min T t=1 m i=1 y (t) i A i − C + βT ℓ + (ρ + ℓ) ln n β + T (1 + β)η. C Optimality of MMWUM Oracle Proposition 24. Any oracle for satisfying (9) must have ρ = Ω(n/η), even if the B i matrices have rank one, and even if X 1 is a scalar multiple of X −1 2 . w H,e c i,e ≤ (1 + ε) e∈E w e c i,e for all i and |E(H)| = O(n/ε 2 ). and |E(G)| = O(n/ε 2 ). Sparse solutions to semidefinite programs. Corollary 10. Let A 1 , . . . , A m be symmetric, positive semidefinite matrices of size n × n, and let B be a symmetric matrix of size n × n. Let c ∈ R m with c ≥ 0. Suppose that the semidefinite program (SDP) Corollary 14 . 14Let G be a graph such that ϑ ′ (G) = Ω( √ n). For any real γ ≥ 1, there is a supergraph H of G such that ϑ ′ (H) = Ω( √ n/γ) and |E(H)| = n 2 − O(n 2 /γ 2 ). Corollary 16 . 16Let B 1 , . . . , B m be symmetric, positive semidefinite matrices of size n × n and let λ ∈ R m satisfy λ ≥ 0 and i λ i = 1. Let B = i λ i B i . For any ε ∈ (0, 1), there exists µ ≥ 0 with i µ i = 1 such that µ has O(n/ε 2 ) nonzero entries and we first compute the matrix f (V ) in time O(n 3 ), and now the inner product U A (f (B j )) = f (V ), B j can be computed in constant time for each j, since B j has O(1) nonzero entries. Thus, each iteration runs in time O(n 3 + m) and the total running time is O(n 4 /ε 2 + mn/ε 2 ). β := β 1 := β 2 12= ℓ 1 := ℓ 2 := 1, ρ := ρ 1 := ρ 2 := 1 + η η n, P 1 := N 2 := [T ], N 1 := P 2 := ∅. Definition 27 ( 27Definition 3.1 in[42]). Let X = (X 1 , . . . , X T ) be random variables distributed over[m]. Let S be an event with P( X ∈ S) > 0. We say that φ 0 , . . . , φ T , φ i : [m] i → [0, 1], are pessimistic estimators for S if the following hold.1. For any i and any fixed x 1 , . . . , x i ∈ [m], we have that Φ U (t) := Tr W U (t) = Tr exp(γA(t)) Φ L (t) := Tr W L (t) = Tr exp(−γA(t)) Ψ u (A) := Tr exp(−uI + γA), Ψ ℓ (A) := Tr exp(ℓI − γA) and define the parameters ∆ U = ln(1 + δ U ) and ∆ L = ln (1 − δ L ) −1 .Algorithm 2 A procedure for solving Problem 2 based on the Width-Free MMWUM method. procedure SparsifySumOfMatricesByMMWUM(B 1 , . . . , B m , ε) input: Matrices B 1 , . . . , B m ∈ S n + such that i B i = I, and a parameter ε ∈ (0, 1). output: A vector y with O(n log n/ε 2 ) nonzero entries such that I i y i B i (1 + O(ε))I. Initially A(0) := 0, and y(0) := 0. Set parameters u 0 := 0, ℓ 0 := 0, ∆ U := ln(1 + δ U ), ∆ L := ln (1 − δ L ) −1 , where δ U , δ L and T are as defined in (21). Define the potential functions Ψ u (A) := Tr exp(−uI + γA) and Ψ ℓ (A) := Tr exp(ℓI − γA). A and B, Tr f (A + B) can be easily related to Tr f (A)? The natural candidates to try are f (x) = − log(u − x) and f (x) = (u − x) −1 since, in both cases, Tr f (A + B) can be related to Tr f (A) by the Sherman-Morrison-Woodbury formula. We do not know whether the choice f (x) = − log(u − x) can be made to work. However, choosing f (x) = (u − x) −1 , one arrives at Algorithm 1, our generalization of the BSS algorithm. Of course, even after arriving at this algorithm, one must also analyze it, and this requires the delicate calculations that were accomplished by Batson, Spielman and Srivastava. A Proofs of the Applications Corollary 4. Let G = (V, E) be a graph, let w : E → R + bea weight function, and let c 1 , .. . , c k : E → R + be cost functions, with k = O(n). Let L G (w) denote the Laplacian matrix for graph G with weight function w. For any real ε ∈ (0, 1), there is a deterministic polynomial-time algorithm to find a subgraph H of G and a weight function w H : E(H) → R + such thatL G (w) L H (w H ) (1 + ε)L G (w),e∈E w e c i,e ≤ e∈E(H) w H,e c i,e ≤ (1 + ε) e∈E w e c i,e for all i and |E(H)| = O(n/ε 2 ). Corollary 5 . 5Let G = (V, E) be a graph and let w : E → R + be a weight function. Let E 1 , . . . , E k be a partition of the edges, i.e., each edge is colored with one of k colors. For any real ε ∈ (0, 1), there is a deterministic polynomial-time algorithm to find a subgraph H of G and a weight function w H : E(H) → R + such that for all i, and |E(H)| = O((n + k)/ε 2 ). Corollary 10 . 10Let A 1 , . . . , A m be symmetric, positive semidefinite matrices of size n × n, and let B be a symmetric matrix of size n × n. Let c ∈ R m with c ≥ 0. Suppose that the semidefinite program (SDP) Corollary 16 . 16Let B 1 , . . . , B m be symmetric, positive semidefinite matrices of size n × n and let λ ∈ R m satisfy λ ≥ 0 and i λ i = 1. Let B = i λ i B i . For any ε ∈ (0, 1), there exists µ ≥ 0 with i µ i = 1 such that µ has O(n/ε 2 ) nonzero entries and Corollary 17 . 17Let G = (V, E) be a graph, let w : E → R + be a weight function, and let F be a collection of subgraphs of G such that F ∈F |V (F )| = O(n). For any real ε ∈ (0, 1), there is a deterministic polynomial-time algorithm to find a subgraph H of G and a weight function w H : E(H) → R + such that |E(H)| = O(n/ε 2 ) and iM ∀t ∈ {0, . . . , T }, y = y (t) is a solution for(27) with X = W (t) , ∀t ∈ [A i − C ∈ [−ℓ, ρ], if t ∈ P, [−ρ, ℓ], if t ∈ N , ∀t ∈ [T ],ℓ (t) = ℓ, ∀t ∈ P, and ℓ (t) = −ℓ, ∀t ∈ N . tool for the proof of Theorem 29 is the following result:Theorem 30 (Kale [22,Corollary 3 in Chapter 3]). Let 0 < β ≤ 1/2. Let T be a positive integer. Let {P, N } be a partition of [T ], and let M (t) ∈ S n for t ∈ [T ] and W (t) ∈ S n for t ∈ [T + 1] with the following properties: Proof. Set Φ (t) := Tr(W (t) ) for t ∈ [T + 1]. Put β 1 := 1 − e −β and β 2 := e β − 1. Then, for any t ∈ [T ], Φ (t+1) = Tr(W (t+1) ) = Tr exp −β (τ ) exp −βM (t) = Tr W (t) exp(−βM (t) )= W (t) , exp(−βM (t) ) , Proof of Theorem 29 .ii 29Let M (t) A i − C + ℓ (t) I and P (t) := W (t) / Tr W (t) for every t.For every t ≤ T , using(27),M (t) , P (t) A i , P (t) − C, P (t) + ℓ (t) I, P (t) Corollary 5 (Rainbow Sparsifiers). Let G = (V, E) be a graph and let w : E → R + be a weight function. Let E 1 , . . . , E k be a partition of the edges, i.e., each edge is colored with one of k colors. For any real ε ∈ (0, 1), there is a deterministic polynomial-time algorithm to find a subgraph H of G and a weight function w H : E(H) → R + such that There was an factor of n in the φi that can be removed. AcknowledgementsWe thank Satyen Kale for helpful discussions.Proof. Let k = n/3, let I k be the identity of size k × k, and let e j ∈ R k be the jth standard basis vector. Let ζ = 3η and definewhere ⊗ denotes tensor product. For j = 1, . . . , k, define, since satisfying (9) would lead to a contradiction:for sufficiently small η. So the oracle must choose a matrix B i,j with i = 3. In this case,This shows that ρ = Ω(n/η). Proof. Let P := { (i, j) : i, j ∈ [n], i < j}. For (i, j) ∈ P, let E ij := e i e T j + e j e T i . Let J denote the matrix of all ones. Then 2I + (i,j)∈P E ij = I + J =: B ≻ 0. Let ε ∈ (0, 1) and suppose that (1−ε)B 2tI + (i,j)∈P z ij E ij for some t ∈ R and z ∈ R P . By taking the inner product with E ab on both sides, we see that 0 < 2(1− ε) ≤ z ab for every (a, b) ∈ P. Similarly, we find that 0 < 2n(1− ε) ≤ 2nt. Strong converse for identification via quantum channels. 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Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems. In Proceedings of the 36th Annual ACM Symposium on Theory of Computing (STOC), pages 81-90, 2004. On contact points of convex bodies. Nikhil Srivastava, Nikhil Srivastava. On contact points of convex bodies, 2009. http://www.cs.yale.edu/homes/srivastava/papers/contact.pdf. A note on sums of independent random matrices after Ahlswede-Winter. Roman Vershynin, Roman Vershynin. A note on sums of independent random matrices after Ahlswede-Winter, 2008. http://www-personal.umich.edu/˜romanv/teaching/reading-group/ahlswede-winter.pdf. Derandomizing the Ahlswede-Winter matrix-valued Chernoff bound using pessimistic estimators and applications. Avi Wigderson, David Xiao, Theory of Computing. 4Avi Wigderson and David Xiao. Derandomizing the Ahlswede-Winter matrix-valued Chernoff bound using pessimistic estimators and applications. Theory of Computing, 4(3), 2008. Greedy algorithms by derandomizing unknown distributions. Neal Young, 1087Department of ORIE, Cornell UniversityTechnical ReportNeal Young. Greedy algorithms by derandomizing unknown distributions. Technical Report 1087, Department of ORIE, Cornell University, March 1994. http://hdl.handle.net/1813/8971. Randomized rounding without solving the linear program. Neal Young, Proceedings of the 6th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). the 6th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Similarly, if t ∈ N , then Φ (t+1) ≤ Φ (t) exp(−β 2 P (tNeal Young. Randomized rounding without solving the linear program. In Proceedings of the 6th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 170-178, 1995. Similarly, if t ∈ N , then Φ (t+1) ≤ Φ (t) exp(−β 2 P (t) , M (t) ). By induction on t, and using Φ (1) = Tr(I) = n, we get. By induction on t, and using Φ (1) = Tr(I) = n, we get

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{'abstract': 'Recently there has been much interest in "sparsifying" sums of rank one matrices: modifying the coefficients such that only a few are nonzero, while approximately preserving the matrix that results from the sum. Results of this sort have found applications in many different areas, including sparsifying graphs. In this paper we consider the more general problem of sparsifying sums of positive semidefinite matrices that have arbitrary rank.We give several algorithms for solving this problem. The first algorithm is based on the method of Batson, Spielman and Srivastava (2009). The second algorithm is based on the matrix multiplicative weights update method of Arora and Kale(2007). We also highlight an interesting connection between these two algorithms.Our algorithms have numerous applications. We show how they can be used to construct graph sparsifiers with auxiliary constraints, sparsifiers of hypergraphs, and sparse solutions to semidefinite programs.', 'arxivid': '1107.0088', 'author': ['Marcel K De ', 'Carli Silva mksilva@uwaterloo.ca. \nDepartment of Combinatorics and Optimization\nUniversity of Waterloo\n\n', 'Nicholas J A Harvey \nDepartment of Computer Science\nUniversity of British Columbia\n\n', 'Cristiane M Sato '], 'authoraffiliation': ['Department of Combinatorics and Optimization\nUniversity of Waterloo\n', 'Department of Computer Science\nUniversity of British Columbia\n'], 'corpusid': 12980437, 'doi': '10.1145/2746241', 'github_urls': [], 'n_tokens_mistral': 27015, 'n_tokens_neox': 24038, 'n_words': 14277, 'pdfsha': '95eb8a3b7e5e6b0a6570b010205bf0fd7cc573b1', 'pdfurls': ['https://arxiv.org/pdf/1107.0088v2.pdf'], 'title': ['Sparse Sums of Positive Semidefinite Matrices', 'Sparse Sums of Positive Semidefinite Matrices'], 'venue': []}

arxiv

All the Groups of Signal Analysis from the (1 + 1)-affine Galilei Group 19 Jun 2011 June 21, 2011 S Hasibul Department of Mathematics and Statistics Concordia University H3G 1M8MontréalQuébecCanada Hassan Chowdhury †e-mail:schowdhury@mathstat.concordia.ca††e-mail:stali@mathstat.concordia.ca Department of Mathematics and Statistics Concordia University H3G 1M8MontréalQuébecCanada S Twareque Department of Mathematics and Statistics Concordia University H3G 1M8MontréalQuébecCanada Ali † Department of Mathematics and Statistics Concordia University H3G 1M8MontréalQuébecCanada All the Groups of Signal Analysis from the (1 + 1)-affine Galilei Group 19 Jun 2011 June 21, 2011 We study the relationship between the (1 + 1)-affine Galilei group and four groups of interest in signal analysis and image processing, viz., the wavelet or the affine group of the line, the Weyl-Heisenberg, the shearlet and the Stockwell groups. We show how all these groups can be obtained either directly as subgroups, or as subgroups of central extensions of the affine Galilei group. We also study this at the level of unitary representations of the groups on Hilbert spaces.Extension to the affine Galilei groupWe start with the (1+1)-Galilei group G 0 which, as we said, is the kinematical group of a non-relativistic space-time of (1 + 1)-dimensions. This is a three parameter group, an element of which we shall denote by (b, a, v). The parameters b, a, and v stand for time translation, space translation and the Galilean or velocity boost, respectively. Under the action of this group, a space-time point (x, t) transforms in the following mannerx → x + vt + a t → t + b Introduction There are a number of groups that are used in the current literature, on signal analysis and image processing, to construct signal transforms, as functions representing the signals over convenient parameter spaces. Of these, the most commonly used are the wavelet group, i.e., the affine group of the real line R, the Heisenberg and the Weyl-Heisenberg groups and the more recently introduced Stockwell and shearlet groups. Another set of groups, which are extensions of the Heisenberg group by one-parameter dilations, were introduced in [12]. These include the shearlet group as a special case and hence are also relevant for constructing signal transforms. As the name suggests, the wavelet group [1,7,14] is used to build the well-known continuous wavelet transform while the shearlet transform, using the shearlet group [6], is applicable to situations where the signal to be analyzed has undergone shearing transformations. The Weyl or Weyl-Heisenberg group leads to the windowed Fourier transform, useful in time-frequency analysis [1,3,7], while the Stockwell transform [5,11,13] combines features of both the wavelet and time-frequency transforms. As an interesting result, we show that the Stockwell group is just a trivial central extension of the wavelet group. (Of course, the wavelet group has no non-trivial central extensions.) This also has the implication that the unitary irreducible representation of the Stockwell group is square-integrable over a homogeneous space (the space consisting of the affine group parameters), a fact studied in [11]. The matrix representations of these various groups are as follows. A generic element of the Heisenberg group is given by a 3 × 3 matrix, g =   1 x y 0 1 z 0 0 1   , x, y, z ∈ R ,(1) while its one-parameter family of extensions obtained in [12] have the form g =   e σ ve σ p+1 a 0 e σ p+1 b 0 0 1   , −1 < p ≤ 1 , a, b, v, σ ∈ R .(2) with the shearlet group, which is a special case (p = 1), being of the type, g =   µ ν √ µ α 0 √ µ β 0 0 1   , µ > 0, ν, α, β ∈ R .(3) The connected affine or wavelet group is given by 2 × 2 matrices of the form g = d t 0 1 , d > 0 , t ∈ R ,(4) and finally, the Stockwell group can be represented by a 4 × 4 matrix, g =     1 γδ 0 θ 0 γ 0 1 − γ 0 0 1 γ 0 0 0 0 1     , γ > 0, δ, θ ∈ R .(5) The question naturally arises as to whether there exists a matrix group which contains all the above groups as subgroups. It is also noteworthy that all these groups consist of upper triangular matrices. The purpose of this paper is firstly, to answer the above question., i.e., we show how all these groups can be obtained as subgroups of various extensions of the Galilei group in (1 + 1)-dimensions. This group is a physical kinematical group, which incorporates the symmetry of non-relativistic motion in a (1 + 1)-dimensional space-time. More precisely, we shall first extend this group by space and time dilations to obtain the (1 + 1)-affine Galilei group, which will then be shown to contain all the above groups as subgroups, except the Stockwell group. This last group which, as we mentioned earlier, is a trivial central extension of the wavelet group, will be obtained as a subgroup of a trivial central extension of the Galilei-Schrödinger group, which itself is a subgroup of the affine Galilei group. As a second and related problem we study how unitary irreducible representations of the affine Galilei and the various centrally extended Galilei-Schrödinger group decompose when restricted to the above subgroups. This would shed light on how signal transforms related to the bigger groups decompose into linear combinations of transforms based on the smaller subgroups. Physically this could correspond to situations where certain parameters of a more detailed transform are averaged over or ignored. The group element g = (b, a, v) can be faithfully represented by a 3 × 3 upper triangular matrix, g =   1 b a 0 1 v 0 0 1   ,(6) so that matrix multiplication captures the group composition law. This group, also known as the Heisenberg group in the mathematical and signal analysis literature, is a central extension of the group of translations of R 2 , (translations in time and velocity.) The exponent giving this extension is ξ H (x, x ′ ) = bv ′ ,(7) where, x = (b, v), x ′ = (b ′ , v ′ ). In the physical literature one usually works with another extension of R 2 , the resulting group being referred to as the Weyl-Heisenberg group. This latter group is constructed using an exponent which is projectively equivalent to (7). We shall come back to this point later. Next we construct a different kind of an extension of the Galilei group G 0 , by forming its semidirect product with D 2 , the two-dimensional dilation group, i.e., we introduce two dilations (of space and time). The resulting group G 0 ⋊ D 2 will be denoted G aff . If the space and time dilations are given by σ and τ , respectively, and a generic group element of G aff is written (b, a, v, σ, τ ), then the corresponding group composition law reads (b, a, v, σ, τ )(b ′ , a ′ , v ′ , σ ′ , τ ′ ) = (b + e τ b ′ , a + e τ b ′ v + e σ a ′ , v + e σ−τ v ′ , σ + σ ′ , τ + τ ′ ) .(8) We shall refer to G aff as the affine Galilei group. It has the matrix representation (b, a, v, σ, τ ) aff =   e σ ve τ a 0 e τ b 0 0 1  (9) 3 From affine Galilei to extended Heisenberg, shearlet and wavelet groups In this section, starting from the affine Galilei group G aff , we first derive the family of extensions G p H of the Heisenberg group, originally obtained in [12]. Following this, we shall show how the reduced shearlet group, constructed in [6] is in fact one of the above groups. Finally, we shall obtain the wavelet group as another subgroup of the affine Galilei group. In subsequent sections, using the matrix representations of two central extensions (one of them being a trivial extension) of the Galilei-Schrödinger group G s , we shall demonstrate that the Weyl-Heisenberg group and the connected Stockwell group are subgroups of these centrally extended groups. In other words, we shall have shown that all the groups of interest in timefrequency analysis and signal processing are obtainable from a single group, the affine Galilei G aff . Extended Heisenberg group G p H as subgroup of affine Galilei group G aff Let us construct a family of subgroups of the the affine Galilei group G aff = G 0 ⋊ D 2 by restricting the two dilations σ and τ to lie on a line τ = mσ, where m is a constant. The special case where m = 2 is called the Galilei-Schrödinger group [2]. We shall come back to this group later. Consider first the the family of (non-isomorphic) extensions G p H of the Heisenberg group, worked out in [12]. This family of groups is parametrized by a real number p, where −1 < p ≤ 1. The corresponding group law reads (b, a, v, σ)(b ′ , a ′ , v ′ , σ ′ ) = (b+e σ p+1 b ′ , a+e σ a ′ +e σ p+1 vb ′ , e pσ p+1 v ′ +v, σ+σ ′ ). (10) The matrix representation of the above family of Lie groups, referred to in ( [12]) as the extended Heisenberg groups, is easily seen to be (b, a, v, σ) p H =   e σ ve σ p+1 a 0 e σ p+1 b 0 0 1   , −1 < p ≤ 1.(11) Comparing with (9), we immediately see that the groups G p H are subgroups of the (1+1) affine Galilei group G aff of the type where the two dilations are restricted to the line τ = mσ, with m = 1 p+1 . Reduced shearlet group as subgroup of the affine Galilei group G aff The reduced shearlet group S, as described in [6], has a generic element, s = (µ, ν, α, β), µ ∈ R + , ν ∈ R and (α, β) ∈ R 2 , with the multiplication law (µ 1 , ν 1 , α 1 , β 1 )(µ 2 , ν 2 , α 2 , β 2 ) = (µ 1 µ 2 , ν 1 + ν 2 √ µ 1 , α 1 + µ 1 α 2 + ν 1 √ µ 1 β 2 , β 1 + √ µ 1 β 2 ) . (12) The matrix representation for the group S is as follows (µ, ν, α, β) =   µ ν √ µ α 0 √ µ β 0 0 1  (13) Comparing with (11), we see that this group corresponds to the special case p = 1, i.e., m = 1 2 , (b, a, v, σ) S := (b, a, v, σ) p=1 H =   e σ ve σ 2 a 0 e σ 2 b 0 0 1   ,(14) and the explicit identification e σ −→ µ v −→ ν a −→ α b −→ β . Thus, the reduced shearlet group S is a member of the family of extensions G p H of Heisenberg group (with p = 1) and hence also a subgroup of the (1 + 1)-affine Galilei group. G aff . Wavelet group as subgroup of the affine Galilei group G aff The connected affine group or the wavelet group is a two-parameter group G aff + which consists of transformations on R given by x → dx + t,(15) where x ∈ R, d > 0 and t ∈ R. Here d and t can be regarded as the dilation and translation parameters, respectively. The group law for this group is given by (d 1 , t 1 )(d 2 , t 2 ) = (d 1 d 2 , d 1 t 2 + t 1 )(16) The matrix representation of G aff + , compatible with the above group law, is given by (d, t) = d t 0 1(17) In the matrix (14) of the reduced shearlet group if we set b = v = 0, we are left with s | Wavelet =   e σ 0 a 0 e σ 2 0 0 0 1   ,(18) which is a 3 × 3 faithful matrix representation of G aff + with the following identification d −→ e σ t −→ a , i.e., we ave obtained the wavelet group as a subgroup of the reduced shearlet group S and hence of the affine Galilei group G aff . Thus, so far we have obtained all the groups mentioned in Section 1, except for the Stockwell group, as subgroups of the affine Galilei group. Although we shall later obtain the Stockwell group as a subgroup of a trivial central extension of the Galilei-Schrödinger group, which is itself a subgroup of the affine Galilei group, we might mention already here that we could obtain the Stockwell group also as a trivial central extension of the wavelet group. In this sense, we could have started with a trivial extension of the affine Galilei group and obtained all the groups mentioned in Section 1 essentially as subgroups of it. Extensions of the affine Galilei and related groups The Galilei group G 0 , has a non-trivial central extension [8], and in fact, there is only one such extension, up to projective equivalence. This extension, which we describe below, incorporates the quantum kinematics of a physical system in a space-time of (1 + 1)-dimensions. Let M be a non-zero, positive real number; the local exponent ξ : G 0 × G 0 → R, giving the extension in question is: ξ(g, g ′ ) = M[va ′ + 1 2 b ′ v 2 ] ,(19)where g ≡ (b, a, v) and g ′ ≡ (b ′ , a ′ , v ′ ) are elements of G 0 . We denote this extended group by G M ; writing a generic element of G M as (θ, b, a, v), the group multiplication law reads, (θ, b, a, v)(θ ′ , b ′ , a ′ , v ′ ) = (θ + θ ′ + M[va ′ + 1 2 b ′ v 2 ], b + b ′ , a + a ′ + vb ′ , v + v ′ )(20) We shall refer to G M as the quantum Galilei group. Non-central extension of affine Galilei group The group G aff does not have non-trivial central extensions. Consequently, it cannot be used in quantum mechanics, since a trivial extension fails to generate mass [2]. From a physical point of view, it is therefore more meaningful to take the quantum Galilei group G M and to form its semidirect product with D 2 . This way, we arrive at G M aff = G M ⋊ D 2 , which is a non-central extension of the affine Galilei group. For simplicity we will call this group the extended affine Galilei group. Denoting a generic group element of this group by (θ, b, a, v, σ, τ ), the group multiplication law reads (θ, b, a, v, σ, τ )(θ ′ , b ′ , a ′ , v ′ , σ ′ , τ ′ ) = (θ + e 2σ−τ θ ′ + M[e σ va ′ + 1 2 e τ v 2 b ′ ], b + e τ b ′ , a + e τ b ′ v + e σ a ′ , v + e σ−τ v ′ , σ + σ ′ , τ + τ ′ )(21) The matrix representation of an element of G M aff , consistent with the above multiplication rule is (θ, b, a, v, σ, τ ) M aff =     e σ ve τ 0 a 0 e τ 0 b Mve σ 1 2 Mv 2 e τ e 2σ−τ θ 0 0 0 1    (22) As mentioned earlier (see [8]), all the multipliers for the (1+1) dimensional quantum Galilei group G M are equivalent, i.e., there is only one equivalence class in the multiplier group of the (1 + 1)-dimensional Galilei group G 0 . In other words H 2 (G 0 , U(1)) is just one dimensional. It is noteworthy in this context that equation (22) is a matrix representation of G M aff provided that the multiplier we choose, from the one dimensional group H 2 (G 0 , U(1)) to obtain G M during the two step construction of G M aff , has the form e iξ(g 1 ,g 2 ) , with ξ given by equation (19). Choosing another, though equivalent, multiplier will alter the form of the matrix (22). Galilei-Schrödinger group: central extensions Let us consider the particular case of the subgroup of G aff when τ = 2σ, i.e., m = 2 (or p = − 1 2 in (11)). We denote the resulting one-dimensional dilation group by D s and the corresponding subgroup of G aff by G s , so that G s = G 0 ⋊D s . In the literature, this group is known as the Galilei-Schrödinger group [2]. It is easy to construct a central extension, denoted G M s , of G s by U(1), using a local exponent ξ : G s × G s → R, or equivalently, using the multiplier exp iξ : G s × G s → U(1). We mention in this context that since we prefer working with addition rather than multiplication, we shall henceforth talk in terms of exponents rather than multipliers. We proceed to construct two extensions of the Galilei-Schrödinger group, using two equivalent multipliers, and a third extension using a trivial or exact multiplier. To do that we first note that the group multiplication law for G s is given by (b, a, v, σ)(b ′ , a ′ , v ′ , σ ′ ) = (b + e 2σ b ′ , a + e σ a ′ + e 2σ vb ′ , v + e −σ v ′ , σ + σ ′ ) (23) where a generic element of the group is denoted as (b, a, v, σ). Now using the exponent ξ((b, a, v, σ); (b ′ , a ′ , v ′ , σ ′ )) = M[ve σ a ′ + 1 2 v 2 e 2σ b ′ ] ,(24) we obtain a central extension G M s of G s by U(1). The group law for the centrally extended group G M s therefore reads (θ, b, a, v, σ)(θ ′ , b ′ , a ′ , v ′ , σ ′ ) = (θ + θ ′ + M[ve σ a ′ + 1 2 v 2 e 2σ b ′ ], b + e 2σ b ′ , a + e 2σ vb ′ + e σ a ′ , v + e −σ v ′ , σ + σ ′ ) ,(25) which is consistent with the matrix representation, (θ, b, a, v, σ) M s =     e σ ve 2σ 0 a 0 e 2σ 0 b Mve σ 1 2 Mv 2 e 2σ 1 θ 0 0 0 1     .(26) Comparing (22) and (26) we easily see that G M s ⊂ G M aff , which is clear since we have just set τ = 2σ. It ought to be noted here, that in going from G 0 to G M s , two extensions were involved: first we extended G 0 to the Galilei-Schrödinger group G s , by taking the semidirect product of the former with the dilation group D s , and then doing a central extension of this enlarged group. We could equivalently have reversed the process, i.e., first done a central extension of G 0 to obtain the quantum Galilei group G M and then taken a semi-direct of this group with D s to again arrive at G M s . In other words, in this case the two procedures commute. Next consider a second local exponent, ξ 1 : G s × G s → R given by ξ 1 ((b, a, v, σ); (b ′ , a ′ , v ′ , σ ′ )) = M 2 [−vv ′ b ′ e σ + va ′ e σ − av ′ e −σ ] .(27) This exponent is easily seen to be equivalent equivalent to ξ, given in (24). Indeed, the difference of the above two exponents, ξ − ξ 1 = M 2 [v 2 e 2σ b ′ + vv ′ b ′ e σ + va ′ e σ + v ′ ae −σ ] = M 2 (a + e 2σ vb ′ + e σ a ′ )(v + e −σ v ′ ) − M 2 av − M 2 a ′ v ′ (28) is a trivial exponent. In other words (28) can be rewritten in terms of the continuous function ζ M : G s → R, ξ − ξ 1 = ζ M ((b, a, v, σ)(b ′ , a ′ , v ′ , σ ′ )) − ζ M (b, a, v, σ) − ζ M (b ′ , a ′ , v ′ , σ ′ ), (29) where ζ M (b, a, v, σ) = M 2 av. Let G M ′ s denote the central extension of G s by U(1) with respect to the exponent ξ 1 given by equation (27). The group multiplication law for G M ′ s reads (θ, b, a, v, σ)(θ ′ , b ′ , a ′ , v ′ , σ ′ ) = (θ + θ ′ + M 2 [−vv ′ b ′ e σ + va ′ e σ − av ′ e −σ ], b + e 2σ b ′ , a + e σ a ′ + e 2σ vb ′ , v + e −σ v ′ , σ + σ ′ )(30) The matrix representation for G M ′ s , compatible with the group law, (30) is (θ, b, a, v, σ) M ′ s =     e σ −e −σ b 0 a − vb 0 e −σ 0 −v 1 2 Mve σ 1 2 Mae −σ 1 θ 0 0 0 1     .(31) Finally, we extend the Galilei-Schrodinger group G s centrally by U(1) with respect to the trivial exponent ξ 2 : G s × G s → R given by ξ 2 ((b, a, v, σ); (b ′ , a ′ , v ′ , σ ′ )) = ae −σ (1 − e −σ ′ ) − e σ−σ ′ vb ′ .(32) We call this extension G T s . Again, it is straight forward to verify the fact that the exponent given in (32) is indeed trivial, since it can be rewritten in terms of the continuous function ζ T : G s → R, ξ 2 ((b, a, v, σ); (b ′ , a ′ , v ′ , σ ′ )) = ζ T (b, a, v, σ) + ζ T (b ′ , a ′ , v ′ , σ ′ ) − ζ T ((b, a, v, σ)(b ′ , a ′ , v ′ , σ ′ )) , where ζ T (b, a, v, σ) = ae −σ . Thus, the group law for the trivially extended Galilei-Schrodinger group G T s reads (θ, b, a, v, σ)(θ ′ , b ′ , a ′ , v ′ , σ ′ ) = (θ + θ ′ + [ae −σ (1 − e −σ ′ ) − e σ−σ ′ vb ′ ], b + e 2σ b ′ , a + e σ a ′ + e 2σ vb ′ , v + e −σ v ′ , σ + σ ′ )(33) The matrix representation of G T s compatible with the above group law is given by (θ, b, a, v, σ) T s =     1 ae −σ −e σ v θ 0 e −σ 0 1 − e −σ 0 −e −σ b e σ e −σ b 0 0 0 1    (34) From Galilei-Schrödinger to Weyl-Heisenberg and Stockwell groups In this section we obtain the Weyl-Heisenberg and Stockwell groups as subgroups of the centrally extended Galilei-Schrödinger groups. We shall also re-derive the Heisenberg group, which by construction was a subgroup of the affine Galilei group G aff , this time as a subgroup of one of the central extensions of the Galilei-Schrödinger group. Heisenberg and Weyl-Heisenberg groups as subgroups of centrally extended Galilei-Schrödinger groups As mentioned in Section 2, the Heisenberg group is identical to the (1 + 1)-Galilei group G 0 , which means that it is trivially a subgroup of the affine Galilei group G aff . Moreover, the Heisenberg group is a central extension of the two-dimensional translation group of the plane, via the local exponent ξ H in (7). As also indicated earlier, in the physical literature one uses a different, but projectively equivalent, exponent ξ WH (see (41) below) to do this extension, the resulting group being called the Weyl-Heisenberg group. Thus, although the Heisenberg and the Weyl-Heisenberg groups are projectively equivalent, we shall continue to differentiate between them in this paper. We now proceed to obtain these groups as subgroups of central extensions of the Galilei-Schrödinger group. Changing notations a bit let (q, p) denote a point in the plane R 2 . In constructing the Heisenberg group G H one uses the local exponent, ξ H ((q, p); (q ′ , p ′ )) = pq ′ .(35) Writing a general element of this group as g = (θ, q, p), θ ∈ R, (q, p) ∈ R 2 , the group multiplication law reads (θ, q, p)(θ ′ , q ′ , p ′ ) = (θ + θ ′ + pq ′ , q + q ′ , p + p ′ ),(36) with the matrix representation being (θ, q, p) H =   1 p θ 0 1 q 0 0 1   .(37) Now we form the subgroup G M s | H of the centrally extended Galilei-Schrodinger group G M s by setting b = σ = 0, θ ∈ R and (a, v) ∈ R 2 . The matrix representation of G M s | H then has the form (see (26)): (θ, 0, a, v, 0) M s := (θ, a, v) M s | H =     1 v 0 a 0 1 0 0 Mv 1 2 Mv 2 1 θ 0 0 0 1     ,(38) which under the identification Mv −→ p a −→ q θ −→ θ(39) reduces to (θ, q, p) M s | H =     1 p M 0 q 0 1 0 0 p p 2 2M 1 θ 0 0 0 1     .(40) Here we assume that the mass term M is never zero. The above 4 × 4 matrix is a faithful representation of the Heisenberg group G H , compatible with the group law (36). Thus, the Heisenberg group constructed using the ξ H in (35), can also be obtained as a subgroup of the nontrivial central extension G M s of the Galilei-Schrödinger group. To obtain the Weyl-Heisenberg group in a similar manner, consider the local exponent ξ WH ((q, p); (q ′ , p ′ )) = 1 2 (pq ′ − p ′ q) .(41) It is straightforward to verify that this exponent is equivalent to ξ H in (35). Indeed, ξ H − ξ WH = pq ′ − 1 2 (pq ′ − p ′ q) = 1 2 pq ′ + 1 2 p ′ q = 1 2 (p + p ′ )(q + q ′ ) − 1 2 pq − 1 2 p ′ q ′ = ζ((q, p); (q ′ , p ′ )) − ζ(q, p) − ζ(q ′ , p ′ ) , where ζ is a real valued continuous function defined on the group of translations of R 2 , and hence ξ H − ξ WH is a trivial exponent. Using the exponent ξ WH we extend the group of translations of R 2 to form the Weyl-Heisenberg group G WH , which then obeys the following group law: (θ, q, p)(θ ′ , q ′ , p ′ ) = (θ + θ ′ + 1 2 (pq ′ − p ′ q), q + q ′ , p + p ′ )(42) The matrix representation compatible with the above group law can be written as (θ, q, p) WH =     1 0 0 q 0 1 0 −p 1 2 p 1 2 q 1 θ 0 0 0 1     .(43) Forming now the subgroup G M ′ s | WH of the centrally extended Galilei-Schrödinger group G M ′ s , obtained by setting b = σ = 0, θ ∈ R and (a, v) ∈ R 2 (see (31)), we get for its matrix representation (θ, 0, a, v, 0) M ′ s := (θ, a, v) M ′ s | WH =     1 0 0 a 0 1 0 −v 1 2 Mv 1 2 Ma 1 θ 0 0 0 1     .(44) Making again the identification (39), this becomes (θ, q, p) M ′ s | WH =     1 0 0 q 0 1 0 − p M 1 2 p 1 2 Mq 1 θ 0 0 0 1     .(45) Here we assume once more that the mass term M is not zero. While the above matrix is not exactly of the same form as the one given in (43) Connected Stockwell group as subgroup of the trivial central extension G T s of the Galilei-Schródinger group The connected Stockwell group G SW (see [5,11] for definition and properties) can be seen as a trivial central extension of a group G ′ aff , isomorphic to the connected affine group G aff + (see (17)). Given a group element (γ, δ) ∈ R >0 ×R, we define the group law for G ′ aff by (γ 1 , δ 1 )(γ 2 , δ 2 ) = (γ 1 γ 2 , δ 1 + 1 γ 1 δ 2 )(46) Comparing with (16), we identify the group homomorphism f : G aff + −→ G ′ aff f (γ, δ) = ( 1 γ , δ) .(47) Let us extend the group G ′ aff centrally using the exponent ξ s ((γ 1 , δ 1 ); (γ 2 , δ 2 )) = γ 1 δ 1 (1 − γ 2 ) = γ 1 δ 1 + γ 2 δ 2 − (γ 1 γ 2 )(δ 1 + δ 2 γ 1 ) .(48) This is in fact a trivial exponent since it can be written in terms of the continuous function ζ s : G ′ aff → R: ξ s ((γ 1 , δ 1 ); (γ 2 , δ 2 )) = ζ s (γ 1 , δ 1 ) + ζ s (γ 2 , δ 2 ) − ζ s ((γ 1 , δ 1 )(γ 2 , δ 2 )) , where ζ s (γ, δ) = γδ. The group so extended obeys the multiplication rule (θ 1 , γ 1 , δ 1 )(θ 2 , γ 2 , δ 2 ) = (θ 1 + θ 2 + [γ 1 δ 1 (1 − γ 2 )], γ 1 γ 2 , δ 1 + 1 γ 1 δ 2 ) ,(50) which is the product rule for elements of the Stockwell group G SW . This proves that the Stockwell group is a trivial central extension of the wavelet or affine group. The matrix representation of a group element of G SW is seen to be (θ, γ, δ) SW =   1 γδ θ 0 γ 1 − γ 0 0 1   .(51) We now show that this group can also be obtained as a subgroup of the trivially extended Galilei-Schrödinger group G T s (see ((32) -(34)). Indeed, comparing (32) to (48) it is clear that the former exponent reduces to he latter if v is set equal to zero. Next, setting v = b = 0 in G T s we see that (34) reduces to (θ, 0, a, 0, σ) T s := (θ, a, σ) T s | SW =     1 ae −σ 0 θ 0 e −σ 0 1 − e −σ 0 0 e σ 0 0 0 0 1     .(52) The identification e −σ −→ γ a −→ δ θ −→ θ and subsequent elimination of the redundant third row and column is then seen to yield the matrix (51). We can conveniently depict all these various extensions and reductions to subgroups by means of a diagram. Galilei group G 0 ≃Heisenberg group G H . (b, a, v) Affine Galilei group G aff . (b, a, v, σ, τ ) aff Galilei- Schrödinger group G s . (b, a, v, σ) s Nontrivial central extension of Galilei- Schrödinger group G M s . (θ, b, a, v, σ) M s Nontrivial central extension of Galilei- Schrödinger group G M ′ s . (θ, b, a, v, σ) M ′ s Weyl-Heisenberg group G WH . (θ, a, v) WH Trivial Central extension of Galilei- Schrödinger group G T s . (θ, b, a, v, σ) T s Heisenberg group G H . (θ, a, v) H Connected Stockwell group G SW . (θ, a, σ) SW Extended Heisenberg groups G p H . (b, a, v, σ) p H Reduced Shearlet group S. (b, a, v, σ) S Wavelet group G aff + . (a, σ) Wavelet Connected Stockwell group G SW . (θ, a, σ) SW Extension τ = 2σ Extension Extension Extension b = σ = 0 b = σ = 0 τ = 1 p+1 σ p = 1 p = − 1 2 v = b = 0 b = v = 0 Trivial central extension 6 Decomposition of UIRs of the affine Galilei group and central extensions of the Galilei-Schrödinger group restricted to various subgroups The general procedure for building signal transforms, starting from a group G is first to define functions over the group using matrix elements of unitary irreducible representations. Provided these functions possess certain desirable properties which, among others, enable one to reconstruct the signal, they can be used as transforms describing the signal. In other words, the signal transforms are functions which encode the properties of the signal in terms of the group parameters. It is therefore of interest to construct unitary irreducible representations of the various groups discussed in the previous sections and to see how representations of the smaller subgroups, relevant to signal analysis, sit inside representations of the bigger groups. The affine Galilei group G aff was defined in Section 2, following which in Section 3 we studied its restriction to various subgroups of interest. In this section we shall first construct unitary irreducible representations of the affine Galilei group and then study their restrictions to the reduced shearlet and wavelet subgroups. In later subsections we will find the UIRs of the two central extensions of the Galilei-Schrödinger and look at their restrictions to the Heisenberg group G H and the connected Stockwell group G SW . UIRs of affine Galilei group restricted to the reduced shearlet group The group law and matrix representation of the affine Galilei group G aff was given in (8) and (9). From the matrix representation, we easily infer the semidirect product structure, G aff = T ⋊ V, where T is an abelian subgroup, with generic element (b, a) and V is the subgroup generated by the elements (v, σ, τ ). Now, the action of (v, σ, τ ) on the element (b, a) as determined by (8) is seen to be (v, σ, τ )(b, a) = (e τ b, e τ vb + e σ a)(53) We also have (v, σ, τ ) −1 (b, a) = (e −τ b, e −σ (a − vb)) . Now let (E, p) denote a generic element of T * , the dual of T and the corresponding character by < (E, p) | (b, a) >= e i(Eb+pa) The action of (v, σ, τ ) ∈ V on (E, p) ∈ T * is then defined by < (v, σ, τ )(E, p) | (b, a) > =< (E, p) | (v, σ, τ ) −1 (b, a) > =< (E, p) | (e −τ b, e −σ (a − vb)) > = e i[(e −τ E−e −σ pv)b+e −σ pa] ,(55) from which we easily find the dual action (E, p) −→ (Ē,p), E = e −τ E − e −σ pv p = e −σ p(56) which we can now use to compute the dual orbits. We see that the sign of p is an invariant for the same orbit while E takes on all real values independently. In other words, the orbits are (i) the two open half planes R × R ≷ 0 , one corresponding to all the positive values of p and the other corresponding to the negative ones, (ii) the two half lines R ≷ 0 , with p = 0, E ≷ 0, and (iii) the degenerate orbit E = p = 0. Note that none of these orbits are open-free (in the sense of [4]). Now using (54) and (56) we obtain (v, σ, τ ) −1 (E, p) = (E ′ , p ′ ) = (e τ (E + pv), e σ p)(57) From this it follows that dE ′ dp ′ = e σ+τ dE dp , on R × R ≷0 ,(58) and dE ′ = e τ dE , on R ≷0 .(59) Using the Mackey's theory of induced representations [9,10], we obtain four unitary irreducible representations of G aff , corresponding to the above four orbits. We denote the representations corresponding to the two halfplanar orbits R × R ≷ 0 by U ± aff , defined on L 2 (R × R ± , dE dp), and the representations on the half lines R ≷ 0 , on L 2 (R ± , dE), by V ± aff . The representations are easily computed to be and (U ± aff (b, a, v, σ, τ )ψ)(E, p) = e(V ± aff (b, a, v, σ, τ )ψ)(E) = e τ 2 e iEbψ (e τ E) , E ≷ 0 .(61) Note that the last two representations are trivial on the subgroup of G aff with a = v = σ = 0, i.e., the affine or wavelet group defined by the two remaining parameters b, τ , and in fact, constitute the two unitary irreducible representations of that group. As is well known, these two representations of the affine group are square integrable and give rise to wavelet transforms. We saw in Section 3.2 that the (reduced) shearlet group S is the subgroup of G aff corresponding to τ = 1 2 σ. Restricting U ± aff in (60) to this subgroup we get A quick examination of (56) shows that R×R ≷ 0 are both open free orbits under the action of S. Also, as representations of the (reduced) shearlet group the two representations (60) are irreducible and hence square-integrable. Indeed, these are the representations used to build the shearlet transforms. (U ± aff | S (b, a, v, σ)ψ)(E, p) = e UIRs of affine Galilei group G aff restricted to the wavelet group We saw in Section 3.3 that the wavelet or affine group G aff + could be obtained from the shearlet group as the subgroup with b = v = 0, or directly from the affine galilei group G aff as the subgroup with b = v = τ = 0. Setting b = v = τ = 0 in the representations U ± aff in (60) we obtain (U ± aff | Wavelet (0, a, 0, σ, 0)ψ)(E, p) = e σ 2 e ipaψ (E, e σ p)(63) as representations of the wavelet group G aff + on L 2 (R × R ± , dE dp). However, these representations are not irreducible. Indeed, noting that L 2 (R × R ± , dE dp) ≃ L 2 (R, dE) ⊗ L 2 (R ± , dp), the representations (63) are immediately seen to be of the form U ± aff | Wavelet = I ⊗ U ± Wavelet ,(64) where I is the identity operator on L 2 (R, dE) and U ± Wavelet are the two unitary irreducible representations of G aff + on L 2 (R ± , dp), given by (U ± Wavelet (a, σ)ψ)(p) = e A decomposition of (64) into irreducibles is easily done. Indeed, let {φ n } ∞ n=0 be an orthonormal basis of L 2 (R, dE) and H n the one-dimensional subspaces spanned byφ n , n = 0, 1, 2, . . . , ∞, so that L 2 (R, dE) = ⊕ ∞ n=0 H n . It is then immediately clear that U ± aff | Wavelet (0, a, 0, σ, 0) = ⊕ ∞ n=0 U ±, n Wavelet (a, σ) ,(66) where U ±, n Wavelet is an irreducible representation of G aff + which is simply a direct product of the trivial representation of the wavelet group on H n with the irreducible representation U ± Wavelet on L 2 (R ± , dp) given in (65). This decomposition also implies, that the shearlet transform, when restricted to the parameters of the wavelet group, decomposes into an infinite sum of wavelet transforms. UIRs of centrally extended Galilei-Schrödinger group G M s restricted to the Heisenberg group G H The group law for the centrally extended Galilei-Schrödinger group G M s , formed using the exponent ξ in (24), is given by (25) and the corresponding matrix representation by (26). From the matrix representation one can deduce the semidirect product structure G M s = T ⋊ V where T is an abelian subgroup with generic element (θ, b, a) and V a semi-simple group consisting of the elements (v, σ). Note that that V is just the affine or wavelet group which also has a semidirect product structure, since (v 1 , σ 1 )(v 2 , σ 2 ) = (v 1 + e −σ 1 v 2 , σ 1 + σ 2 ). Now let (q, E, p) denote a generic element of T * , the dual of T and consider the character < (q, E, p) | (θ, b, a) >= e i(qθ+Eb+pa) . The action of the subgroup V on the abelian subgroup T follows from (25) (v, σ)(θ, b, a) = (θ + M[ve σ a + 1 2 e 2σ v 2 b], be 2σ , e σ a + e 2σ vb) .(67) Now the action of (v, σ) ∈ V on (q, E, p) ∈ T * is defined by < (v, σ)(q, E, p) | (θ, b, a) > =< (q, E, p) | (v, σ) −1 (θ, b, a) > =< (q, E, p) | (θ + M[−va + 1 2 v 2 b], e −2σ b, e −σ (a − vb)) > = e i[qθ+(e −2σ E−e −σ pv+ 1 2 qM v 2 )b+(e −σ p−qM v)a](68) Thus dual orbit elements (q,Ē,p) corresponding to a fixed value of (q, E, p) are given byq = q E = e −2σ E − e −σ pv + 1 2 qMv 2 p = e −σ p − qMv ,(69)so that,Ē −p 2 2qM = e −2σ (E − p 2 2qM )(70) where we assume that q = 0. Since q remains invariant under the transformation (69), we takeq = q = κ. We thus get two dual orbits, the interior and exterior of the parabola given by E − p 2 2κM = 0, lying on the two-dimensional plane determined by q = κ in theq-Ē-p space. The parabola E − p 2 2κM = 0 itself determines an orbit and there are additional orbits when q = 0. Here we shall only consider the first two orbits, i.e., the interior and exterior of the parabola, for each non-zero κ ∈ R. Let us introduce the new variables p = k 1 E − p 2 2κM = k 2(71) Then, for fixed value of q = κ, the coordinates (k 1 , k 2 ) are easily seen to transform ask 1 = e −σ k 1 − κMv k 2 = e −2σ k 2(72) In these new coordinates, (v, σ)(q, k 1 , k 2 ) = (q, e −σ k 1 − qMv, e −2σ k 2 ) , and (v, σ) −1 (k 1 , k 2 ) = (e σ (k 1 + κMv), e 2σ k 2 ) := (k ′ 1 , k ′ 2 ) ,(73) so that, k ′ 1 = e σ (k 1 + κMv) k ′ 2 = e 2σ k 2 . Therefore we obtain dk ′ 1 dk ′ 2 = e 3σ dk 1 dk 2(74) Using again the method of induced representations, we arrive at the two UIRs of G M s defined on either L 2 (R × R ± , dk 1 dk 2 ), for each non-zero value of q = κ, (U κ ± (θ, b, a, v, σ)ψ)(k 1 , k 2 ) = e 3σ 2 e i(κθ+k 1 a+{k 2 + (k 1 ) 2 2κM }b)ψ (e σ (k 1 + κMv), e 2σ k 2 ) . (75) Let us now go back to the Heisenberg group G H , as discussed in Section 5.1 and construct its unitary irreducible representations, following similar techniques. From the matrix representation in (37) we infer the semidirect product structure, G H = T ⋊ A where (θ, q) constitute elements of the abelian subgroup T and p is an element of the subgroup A. Now p ∈ A acts on (θ, q) ∈ T in the following manner p(θ, q) = (θ + pq, q)(76) We now denote by (s, t) a geneirc element of T * , the dual of the abelian subgroup T . Let us take the character < (s, t) | (θ, q) >= e i(sθ+tq) ; then < p(s, t) | (θ, q) > = < (s,t) | (θ, q) > = e i(sθ+tq) = < (s, t) | p −1 (θ, q) > = < (s, t) | (θ − pq, q) > = e i[sθ+(t−sp)q](77) For fixed (s, t) the coordinates of its orbit orbits under the action of A arē s = s t = t − sp(78) Thus, the dual orbits are a family of parallel straight lines, one for each value of s and dt is the invariant measure on the orbit. Once again, using Mackey's theory of induced representation we obtain the UIR, corresponding to each dual orbit, i.e., for each fixed value of s: (U s H (θ, q, p)ψ)(t) = e isθ e itqψ (t + sp) ,(79) on the Hilbert space L 2 (R, dt). Now the restriction of the UIR (75) of the centrally extended Galilei-Schrödinger group G M s to the Heisenberg group G H is seen to be (U κ ± | H (θ, 0, a, v, 0)ψ)(k 1 , k 2 ) = e i(κθ+k 1 a)ψ (k 1 + κMv, k 2 )(80) Thus, U κ ± | H = U κ H ⊗ I ±(81) where U κ H is the unitary irreducible representation of the Heisenberg group on L 2 (R, dk 1 ) and I ± are the identity operators on L 2 (R ± , dk 2 ). Once again we can decompose this representation as an infinite direct sum of irreducibles, U κ ± | H = ⊕ ∞ n=0 U ±, n κ . just as in (66). Here each U ±, n κ is a copy of the UIR (79) with s = κ on the Hilbert space L 2 (R, dk 1 ) times a trivial representation on a one dimensional subspace of L 2 (R ± , dk 2 ). We also recall that in Section 5.1 we obtained the Weyl-Heisenberg group G WH as a subgroup of the centrally extended Galilei-Schrödinger group G M ′ s . We could just as well have obtained similar representations of G WH and their decomposition into irreducibles from the UIR's of G M ′ s . UIRs of cenrally extended (trivial) Galilei-Schrödinger group G T s restricted to connected Stockwell group In Section 4.2 we had introduced the Galilei-Schrödinger group G s , by setting τ = 2σ in the affine Galilei group (see (9)). Later we obtained a central extension of it using the trivial exponent ξ 2 in (32). Here we shall obtain UIRs of this centrally extended group by first finding unitary irreducible representations of G s itself. The matrix representation of G s is found by substituting τ = 2σ in (9): (b, a, v, σ) s =   e σ ve 2σ a 0 e 2σ b 0 0 1   .(82) From this follows the semi-direct product structure, G s = T ⋊ V where the abelian subgroup T consists of elements (b, a) and the subgroup V consists of the elements (v, σ). Now let (E, p) denote a generic element of T * , the dual to T and consider the corresponding character < (E, p); (b, a) >= e i(Eb+pa) . The action of the subgroup V on the abelian subgroup T can be immediately read off. We find, We shall only consider orbits for which p = 0. Making a change of variables (E, p) → (t = E p 2 , p), the orbit equations become t ′ = t + v p p ′ = pe σ(84) Thus we get two orbits in the t-p space, namely, the two disjoint open half planes (p ≷ 0). Also, dt ′ dp ′ = e σ dt dp Again, following the standard Mackey construction we get the following two unitary irreducible representations of the ordinary Galilei-Schrödinger group, corresponding to these two orbits R × R ± in the t-p space: (U ± (b, a, v, σ)ψ)(t, p) = e i(tp 2 b+pa) e σ 2ψ (t + v p , e σ p) The representations are carried by the Hilbert spaces L 2 (R × R ± , dt dp), respectively. In Section 4.2 the trivial exponent ξ 2 was shown to arise from the continuous function ζ T : G s → R given by ζ T (g) = ae −σ .(87) where g ≡ (b, a, v, σ) is a generic element of G s . In terms of this continuous function it follows immediately thatŨ ± (g) = e iζ T (g) U ± (g) are projective representations of the Galilei-Schrodinger group G s . In other words, where I is the identity operator on L 2 (R, dt) and U ± SW are UIRs of the connected Stockwell group on L 2 (R ± , dt). The representation (89) again decomposes in the usual manner into an infinite direct sum of irreducibles. We remark here that the UIRs of the Stockwell group G SW are not squareintegrable (over the whole group). However, since taking θ = 0 in (88) yields a projective representation of the affine group, the two non-trivial representaions of which are both square-integrable, this fact can be exploited to arrive at square-integrability over the homogeneous space G SW /Θ, where Θ is the phase subgroup. This is exactly the sense in which square-integrability for representations of the Stockwell group has been defined in [11] and is in accordance with the theory of square-integrability modulo subgroups (see, for example [1]). Conclusion The fact that the various groups of signal analysis enumerated in Section 1 are all obtainable from the affine Galilei group shows a remarkable unity in their structures and consequently of their unitary irreducible representations. In later publications we propose to make a comparative study of the structures of their co-adjoint orbits and Wigner functions built on them. From the point of view of signal transforms, all this could lead to a deeper understanding of how signal transforms, defined over a larger set of parameters, reduce when a smaller set of parameters is used, with the original signal still being reconstructible from the smaller set. , it does reproduce the group multiplication rule (42). Moreover, the two matrix representations are equivalent, via the intertwining matrix e., we have S (θ, q, p) WH S −1 = (θ, q, p) M ′ s | WH . In this way we have shown that the Weyl-Heisenberg group G WH is a subgroup of the nontrivial central extension G M ′ s of Galilei-Schrödinger group. Figure 1 : 1Flowchart showing the passage from the (1+1)-affine Galilei group to the various groups of signal analysis. 2 e i(Eb+pa)ψ (e τ (E + pv), e σ p) , p ≷ 0 , (60) E + pv), e σ p) , p ≷ 0 . (62) σ 2 e 2ipaψ (e σ p) . (v, σ) −1 (b, a) = (e −2σ b, e −σ (a + vb)) , and the action of (v, σ) ∈ V on (E, p) ∈ T * :< (v, σ)(E, p); (b, a) >= e i[(e −2σ E+e −σ pv)b+e −σ pa] Thus, writing (v, σ) −1 (E, p) = (E ′ , p ′ )we get the equations for the dual orbit, corresponding to (E, p) E ′ = e 2σ (E + pv) p ′ = pe σ , b, a, v, σ) := e iθŨ ± (b, a, v, σ) are unitary irreducible representations of the trivial central extension G T s of the Galilei-Schrödinger group. Next the UIRs U T,± s restricted to the connected Stockwell group have the form (U T,± s | SW (θ, 0, a, 0, σ))(t, p) = e i(θ+ae −σ ) e ipa e σ 2ψ (t, e σ p) Coherent States, Wavelets and Their Generalizations. S T Ali, J-P Antoine, J.-P Gazeau, Springer-VerlagNew YorkS.T. Ali, J-P. Antoine, and J.-P. Gazeau. Coherent States, Wavelets and Their Generalizations. Springer-Verlag, New York, 2000. Galilean wavelets: Coherent states of the affine galilei group. J-P Antoine, I Mahara, Journal of Mathematical Physics. 4011J-P. Antoine and I. Mahara. Galilean wavelets: Coherent states of the affine galilei group. Journal of Mathematical Physics, 40(11):5956-5971, 1999. J.-P Antoine, R Murenzi, P Vandergheynst, S T Ali, Twodimensional Wavelets and their Relatives. Cambridge University PressJ.-P. Antoine, R. Murenzi, P. Vandergheynst, and S.T. Ali. Two- dimensional Wavelets and their Relatives. Cambridge University Press, 2004. Wavelets from square-integrable representations. D Bernier, K F Taylor, SIAM J. Math. Anal. 27D. Bernier and K.F. Taylor. Wavelets from square-integrable represen- tations. SIAM J. Math. Anal., 27:594-608, 1996. A group representation related to the stockwell transform. Indiana University mathematics journal. P Boggiatto, C Fernandez, A Galbis, 58P. Boggiatto, C. Fernandez., and A. Galbis. A group representation re- lated to the stockwell transform. Indiana University mathematics jour- nal, 58(5):2277-2304, 2009. Shearlet coorbit spaces and associated banach frames. S Dahlke, G Kutyniok, G Steidl, Gerd Teschke, Applied and Computational Harmonic Analysis. 272S. Dahlke, G. Kutyniok, G. Steidl, and Gerd Teschke. Shearlet coor- bit spaces and associated banach frames. Applied and Computational Harmonic Analysis, 27(2):195-214, 2009. . I Daubechies, Ten Lectures on Wavelets. SIAM. I. Daubechies. Ten Lectures on Wavelets. SIAM, Philadelphia, 1992. Galilei group and galilean invariance. J-M Levy-Leblond, Group Theory and Its Applications. E.M. LoeblNew YorkAcademic PressIIJ-M. Levy-Leblond. Galilei group and galilean invariance. In E.M. Loebl, editor, Group Theory and Its Applications, volume II, pages 221- 299. Academic Press, New York, 1971. Imprimitivity for representations of locally compact groups. I. G W Mackey, Proc. Nat. Acad. Sci. Nat. Acad. Sci35G.W. Mackey. Imprimitivity for representations of locally compact groups. I. Proc. Nat. Acad. Sci., 35:537-545, 1949. Unitary representations of group extensions. G W Mackey, I. Acta Mathematica. 991G.W. Mackey. Unitary representations of group extensions. I. Acta Mathematica, 99(1):265-311, 1958. Square integrable group representations and localization operators for modified stockwell transform. S Molahajloo, M W Wong, Rend. Sem. Mat. Univ. Pol. Torino. 672S. Molahajloo and M.W. Wong. Square integrable group representations and localization operators for modified stockwell transform. Rend. Sem. Mat. Univ. Pol. Torino, 67(2):215-227, 2009. Extensions of the heisenberg group and wavelet analysis in the plane. E Schulz, K F Taylor, CRM Proceedings and Lecture Notes. Serge Dubuc and Gilles Deslauriers18E. Schulz and K.F. Taylor. Extensions of the heisenberg group and wavelet analysis in the plane. In Serge Dubuc and Gilles Deslauriers, editors, CRM Proceedings and Lecture Notes, volume 18, pages 99-107, 1999. Localization of the complex spectrum: the s transform. R G Stockwell, L Mansinha, R P Lowe, IEEE Trans. Signal Processing. 44R.G. Stockwell, L. Mansinha, and R.P. Lowe. Localization of the com- plex spectrum: the s transform. IEEE Trans. Signal Processing, 44:998- 1001, 1996. Analyse continue par ondelettes. InterÉditions/CNRS Éditions. B Torrésani, ParisB. Torrésani. Analyse continue par ondelettes. InterÉditions/CNRS Éditions, Paris, 1995.

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{'abstract': 'We study the relationship between the (1 + 1)-affine Galilei group and four groups of interest in signal analysis and image processing, viz., the wavelet or the affine group of the line, the Weyl-Heisenberg, the shearlet and the Stockwell groups. We show how all these groups can be obtained either directly as subgroups, or as subgroups of central extensions of the affine Galilei group. We also study this at the level of unitary representations of the groups on Hilbert spaces.Extension to the affine Galilei groupWe start with the (1+1)-Galilei group G 0 which, as we said, is the kinematical group of a non-relativistic space-time of (1 + 1)-dimensions. This is a three parameter group, an element of which we shall denote by (b, a, v). The parameters b, a, and v stand for time translation, space translation and the Galilean or velocity boost, respectively. Under the action of this group, a space-time point (x, t) transforms in the following mannerx → x + vt + a t → t + b', 'arxivid': '1106.3768', 'author': ['S Hasibul \nDepartment of Mathematics and Statistics\nConcordia University\nH3G 1M8MontréalQuébecCanada\n', 'Hassan Chowdhury †e-mail:schowdhury@mathstat.concordia.ca††e-mail:stali@mathstat.concordia.ca \nDepartment of Mathematics and Statistics\nConcordia University\nH3G 1M8MontréalQuébecCanada\n', 'S Twareque \nDepartment of Mathematics and Statistics\nConcordia University\nH3G 1M8MontréalQuébecCanada\n', 'Ali † \nDepartment of Mathematics and Statistics\nConcordia University\nH3G 1M8MontréalQuébecCanada\n'], 'authoraffiliation': ['Department of Mathematics and Statistics\nConcordia University\nH3G 1M8MontréalQuébecCanada', 'Department of Mathematics and Statistics\nConcordia University\nH3G 1M8MontréalQuébecCanada', 'Department of Mathematics and Statistics\nConcordia University\nH3G 1M8MontréalQuébecCanada', 'Department of Mathematics and Statistics\nConcordia University\nH3G 1M8MontréalQuébecCanada'], 'corpusid': 56566721, 'doi': '10.1063/1.3652697', 'github_urls': [], 'n_tokens_mistral': 16664, 'n_tokens_neox': 14403, 'n_words': 9004, 'pdfsha': '2b3a6eb01eede4bd46eca323ae36a9ae692b6fa1', 'pdfurls': ['https://arxiv.org/pdf/1106.3768v1.pdf'], 'title': ['All the Groups of Signal Analysis from the (1 + 1)-affine Galilei Group', 'All the Groups of Signal Analysis from the (1 + 1)-affine Galilei Group'], 'venue': []}

arxiv

On Robust Numerical Solver for ODE via Self-Attention Mechanism Zhongzhan Huang Mingfu Liang Liang Lin On Robust Numerical Solver for ODE via Self-Attention Mechanism With the development of deep learning techniques, AI-enhanced numerical solvers are expected to become a new paradigm for solving differential equations due to their versatility and effectiveness in alleviating the accuracy-speed trade-off in traditional numerical solvers. However, this paradigm still inevitably requires a large amount of high-quality data, whose acquisition is often very expensive in natural science and engineering problems. Therefore, in this paper, we explore training efficient and robust AI-enhanced numerical solvers with a small data size by mitigating intrinsic noise disturbances. We first analyze the ability of the self-attention mechanism to regulate noise in supervised learning and then propose a simple-yet-effective numerical solver, AttSolver, which introduces an additive self-attention mechanism to the numerical solution of differential equations based on the dynamical system perspective of the residual neural network. Our results on benchmarks, ranging from high-dimensional problems to chaotic systems, demonstrate the effectiveness of AttSolver in generally improving the performance of existing traditional numerical solvers without any elaborated model crafting. Finally, we analyze the convergence, generalization, and robustness of the proposed method experimentally and theoretically. Introduction Mathematical models, represented by differential equations, are widely used in tackling various natural science and engineering problems, such as weather forecasting (Touma et al., 2021), transportation networks (Saberi et al., 2020), power grid management (Gholami & Sun, 2022), drug discovery (Aulin et al., 2021), etc. Effective solutions to these mathematical models can help researchers investigate and analyze Figure 1: The comparison of (a) ResNet and (c) traditional forward numerical solver. (d) is a numerical solver with the self-attention mechanism (ours) inspired by the structure of (b) ResNet with self-attention. denotes the elementwise multiplication and ⊕ denotes the addition operator. Different from the common practice that multiple the output of the attention module to f (x t | θ t ) in (b), we instead add the output of the attention module to S (f , u n , ∆t) in (d), based on the optimization and error analysis in Section 6. the evolution process of the research objects to predict their future states. Those solutions are mainly obtained by specific numerical solvers, such as the Euler method (Shampine, 2018), Runge-Kutta method (Butcher, 2016), etc., which discretizes the solution space with a sufficiently fine step size and then performs numerical integration to achieve the solution. However, these classic numerical solvers encounter a trade-off between computation speed and solution accuracy. As the step size becomes smaller (finer), the accuracy is higher while the solution speed slows down; when the step size becomes larger (coarser), the solution speed increases, but the accuracy decreases. Confronted with this predicament, many recent works Li et al., 2020;Liang et al., 2022;Choudhary et al., 2020) on solving differential equations have introduced AI techniques, i.e., using data-driven neural networks to fit and replace differential operators or functions in the equations. Under the acceleration of GPU and the excellent approximation ability of neural networks, these works start to balance the speed and accuracy of solving differential equations. Although AI technology has achieved excellent results in solving differential equations, the versatility of these purely data-driven methods (PDDM) is still limited due to the inherent characteristics (Wang et al., 2022b) of differential equations and neural networks. Specifically, some differential equations are chaotic (Greydanus et al., 2019), i.e. sensitive to the initial values of inputs, which is hard for PDDM to fundamentally model the ubiquitous and elusive randomness of chaos with only finite data, leading to poor generalization (Abu-Mostafa et al., 2012); some differential equations are stiff (nonsmooth) (Liang et al., 2022), and their solutions may undergo drastic changes in a very short time interval, which pose a great challenge for PDDM to handle stiff problems since neural networks tend to approximate smooth functions (Xu et al., 2020;Cao et al., 2019) intrinsically. To reduce this interference, a paradigm that shifts from "neural network replacement" to "neural network enhancement" has been recently proposed in (Huang et al., 2022b) called NeurVec, which uses neural networks to compensate for the errors generated by numerical solvers in the coarse step size ∆t c . Specifically, for the differential equation du/dt = f (u), u(0) = c 0 , the forward numerical method of such a paradigm is u n+1 = u n + S(f , u n , ∆t c )∆t c Integration term + Net(u n |φ, D f ) Compensation term ,(1) where Net(·) is a neural network, S(·) is a numerical integration scheme, φ is the learnable parameters, D f denotes the high-quality data, e.g. high-precision observation data or synthetic data generated by fine step size ∆t f . With suitable coarser step size ∆t c , the computation speed increases while the Eq.(1) still achieves high accuracy through the compensation term trained by D f , which alleviates the accuracy-speed trade-off in traditional numerical solvers. More details are introduced in Section 2.1. Compared to PDDM, this new paradigm can largely exploit the stability advantages of pure mathematical methods, given their physical correctness, to alleviate instability problems from chaos and stiffness. At the same time, it can enjoy the computation speedup as an AI-enhanced numerical solver. However, since the prediction of the compensation term is achieved through deep learning, this new paradigm still inevitably requires numerous high-quality training data to ensure the prediction accuracy of the compensation term. Moreover, since most differential equations that require numerical solutions do not have analytical solutions, those high-quality data can only be observed or generated by investing a lot of labor and equipment costs. These nonnegligible costs hinder the implementation of the new paradigm. To this end, we ask a critical question: Is it possible to train an efficient and robust AI-enhanced numerical solver with a small amount of data? To answer this question, we explore the feasibility of obtaining AI-enhanced numerical solvers using a small amount of data, both empirically and theoretically in this paper. For this more realistic setting, due to the intrinsic differences between the obtained (observed or generated) training data and their corresponding ground truth, the inherent noise in the obtained data will readily hurt model training, especially when the data size is small (see Section 2.3 for details). This points out the main challenge of this critical setting, i.e., alleviating the adverse impact of the naturally existing noise in the training of the predictor of the compensation term. To this end, we first analyze the noise issue faced by AIenhanced numerical solvers training with a small amount of data. Then we correspondingly analyze the role of selfattention mechanisms in noise regulation (Liang et al., 2020;Zuo et al., 2022) in general supervised deep learning. Based on the dynamic system view (Weinan, 2017;Lu et al., 2018;Chang et al., 2017) of residual neural networks, we introduce self-attention mechanisms into the numerical solver, adapt the self-attention mechanisms based on the analysis and insights of numerical errors, and propose a simpleyet-effective novel numerical solver, AttSolver. Extensive experiments conducted on three benchmarks, i.e., a highdimensional problem and chaotic systems demonstrate the effectiveness of AttSolver in generally improving the performance of existing traditional numerical solvers without any elaborated model crafting. Finally, we analyze the convergence, generalization, and robustness of the proposed method experimentally or theoretically. We summarize our contributions as follows: • To the best of our knowledge, our proposed Att-Solver is the first to introduce self-attention mechanisms into numerical solvers for differential equations. With a small amount of data, experiments on highdimensional and chaotic systems show that our method can still effectively enhance multiple traditional numerical solvers. • We also analyze the limitations and provide experimental or theoretical evidence for the convergence, generalization, and robustness of the proposed AttSolver. Preliminaries AI-enhanced numerical solver As mentioned in Section 1, for ordinary differential equation du/dt = f (u), u(0) = c 0 , we can obtain a numerical solution by a forward numerical solver (Butcher, 2016;Ames, 2014) given by the iterative formula u n+1 = u n + S(f , u n , ∆t)∆t, u 0 = c 0 .(2) e.g., for the Euler method (Shampine, 2018), we have S(f , u n , ∆t)∆t = f (u n )∆t, where ∆t is a given step size, and u n ∈ R d is an approximated solution at time n i=0 ∆t. As shown in Eq.(1), the main idea of the AI-enhanced numerical solver is to use a neural network Net(u n |φ, D f ) as a corrector to compensate for the error of the integration term S(f , u n , ∆t c )∆t c with the coarse step size ∆t c . Compared to PDDM, this method has good acceleration while maintaining numerical accuracy. For example, if D f is synthetic data generated by a fine step size ∆t f , then theoretically its evaluation speed is O( ∆tc (1+ )∆t f ) times (Huang et al., 2022b) that of the traditional method Eq.(2), where > 0 is related to the inference speed of the neural network. The dynamical system view of residual block The residual block structure has been successfully applied to many well-known neural network architectures, e.g., ResNet (He et al., 2016), UNet (Ronneberger et al., 2015, Transformer (Liu et al., 2021). As shown in Fig.1(a), the residual blocks in one stage can be written as x t+1 = x t + f (x t ; θ t ),(3) where x t ∈ R d is the input of neural network f (·; θ t ) with the learnable parameters θ t in t th block . Several recent studies (Weinan, 2017;Queiruga et al., 2020;Zhu et al., 2022;Meunier et al., 2022) have uncovered valuable connections between residual blocks and dynamic systems. i.e., the residual blocks can be interpreted as one step of forward numerical methods in Eq. (2) and Fig.1(c). The initial condition u 0 = c 0 corresponds to the initial input x 0 of the network, and u t corresponds to the input feature x t in t th block. The output of neural network f (·; θ t ) in t th block can be regarded as an integration S(u t ; f , ∆t) with step size ∆t and numerical integration scheme S. Given the above connection, some dynamical system theories (Chang et al., 2017;Chen et al., 2018) can be transferred to the analysis of residual neural networks, and more efficient neural network structures or deep learning applications (Huang et al., 2022a;Lu et al., 2018) can be designed. The inherent noise in data The high-quality data required by AI-enhanced numerical solvers inevitably have inherent noise in their acquisition process. On the one hand, the error in the observed data can come from the error of the measuring instruments. Theories like the uncertainty principle (Busch et al., 2007) also reveals the difficulty of collecting high-precision observation data. On the other hand, synthetic data, the other primary data acquisition method, inherently contains noise during their generation. Concretely, the choice of the numerical integration scheme S in Eq. (2) and the data storage strategy both cause unavoidable losses on the accuracy of the synthesized data and hence introduce noise naturally. Last but not least, in many specific problems, we can only obtain the data by experiments. In this process, the objective noise may exhibit passively interfering with the data, such as interference from some unknown electromagnetic waves (Yao et al., 2021) or intermittent vibrations from the nearby subway (Wang et al., 2022a), etc. Method Inherent noise in the data tends to have an adverse effect on training neural networks, and learning with noisy data generally increases the training loss (Song et al., 2022). Such an effect will be more pronounced when the data size is limited (Abu-Mostafa et al., 2012). Additionally, for many complex differential equations, small noise can easily be accumulated and amplified in the iterative solving process of Eq.(1) or Eq. (2) due to chaotic issues (Huang et al., 2022b), leading to numerical explosions. As mentioned in Section 2.3, since these noises are almost inevitable, we need to regulate the noise to relieve their adverse impact. It should be noted that for the network used in general supervised learning tasks, such as a residual neural network in Fig.1(a), the noise in the training data, e.g., the batch noise introduced by batch training (Liang et al., 2020), can be regulated by adding a self-attention module, e.g., Fig.1(b). Now we provide the Theorem 3.1 as the theoretical evidence to justify the effect of the self-attention mechanism as a noise regulator. Theorem 3.1. Consider a L layers residual neural network with self-attention module Q(·), i.e., x t+1 = x t + f (x t ; W t ) Q[f (x t ; W t )] , t = 0, 1, ..., L − 1. And there exists a constant r s.t. max{|x t ∇ x Q|} ≤ r. Let be the perturbation from the noise and satisfies x 0 − x 0 = , and t = x t − x t , we have t+1 ≤ t (1 + (max{Q[f (x t ; W t )]} + r) W t 2 ) ,(4) where max refers to the largest element in a vector. Proof. (See Appendix B). Theorem 3.1 reveals the principle of the self-attention mechanism on regulating the noise. Given an input x 0 and a perturbation , we consider the noise input x 0 satisfied x 0 ∈ {x| x − x 0 = }. According to Eq.(4), for a L layers ResNet, the noise impact is L = x L − x L and L = x L − x L ≤ L−1 1 +Q(x L−1 ) W L−1 2 ≤ L−1 t=1 1 + W t 2Q (x t )(5) whereQ (x t ) = max{Q[f (x t ; W t )]} + r depends on the self-attention term Q[f (x t ; W t )]. Since W t 2Q (x t ) ≥ 0, the upper bound of the noise impact L in Eq.(5) will be Algorithm 1 The processing of AttSolver. Input: A coarse step size ∆tc; The number of step N which satisfied the evaluation time T = N ∆tc; A given equation du/dt = f (u), u(0) = c0; A given numerical integration scheme S; The high-quality dataset D(traj(u)). large if the number of layers L is large. Thus the noise will have a pivotal impact on the network's final output, especially in a deep network. In this case, the self-attention mechanism can changeQ(x t ) to adaptively control the upper bound L−1 t=1 1 + W t 2Q (x t ) . Note that generally, Q(x t ) will not converge to 0 to obtain the minimum upper bound, i.e. , since appropriate noise can enhance the robustness of the network and improve the generalization (Zhang et al., 2017;Cao et al., 2020). Moreover, ifQ(x t ) → 0, the forward process will degenerate to x t+1 = x t , which may affect the representation learning of the network. AttSolver: AI-enhanced Numerical Solver with Additive Self-Attention Following the dynamical system view of residual block, in this section, we consider introducing self-attention mechanisms into numerical solvers to regulate the inherent noise when training with limited data, and propose a simple-yeteffective AI-enhanced numerical solver AttSolver, which is shown in Algorithm 1 and Fig.1(d). The main iterative formula of AttSolver is defined as Eq. (6), and we highlight the additive self-attention with red color. u n+1 =û n +Ŝ∆t c +Q[Ŝ|φ],(6) whereŜ = S(f ,û n , ∆t c ) and φ is the learnable parameters of self-attention module. For estimated trajectoryû = [û 1 , ...,û N ] and the ground truth trajectory from the highquality dataset D(traj(u)), the loss R e can be defined as R e = c n · 1 N û − u 2 2 ,(7) where c n is a constant which can alleviate the problem that the value of R e is too small due to the magnitude of u in some specific differential equations being too small. At this time, the gradient will be small enough and affect the optimization of the learnable parameter φ. Compared with Eq. (1), Net(u n |φ, D f ) is replaced by a self-attention term Q [Ŝ|φ]. Note that according to the self-attention mechanism shown in Theorem 3.1, the self-attention term should be multiplicative, i.e., Eq.(6) can be rewritten aŝ u n+1 =û n +Ŝ∆t c Q[Ŝ|φ].(8) In Section 6. we will discuss the reasons why we should use additive rather than multiplicative self-attention mechanisms for AI-enhanced numerical solvers in two-fold: (1) The multiplicative attention will interfere with the step size, leading to an unstable solution; (2) Eq.(8) will bring negative impacts on the optimization of the self-attention module. ForŜ ∈ R d , the architecture of self-attention module is Q[Ŝ|φ] = W h • a • · · · • W 2 • a • W 1 [Ŝ],(9) where a is rational activation function (Boullé et al., 2020); W i , i=2, · · · , h− 1 are d 1 × d 1 matrices, W h ∈ R d×d1 and W 1 ∈ R d1×d . We set d 1 = 1024 and h = 2 by default. Next, we take AttSolver with step size k∆t and the Euler method as an example, and provide the convergence analysis for AttSolver in Theorem 3.2. Theorem 3.2. We consider ODE du/dt = f (u), u(0) = c 0 and Euler method u n+1 = u n + ∆tf (u n ) . We assume that (1) f is Lipschitz continuous with Lipschitz constant L and (2) the second derivative of the true solution u is uniformly bounded by M > 0, i.e., u ∞ ≤ M on [0, T ]. Moreover, we assume that the attention module in AttSolver is Lipschitz continuous with Lipschitz constant k∆tL att . For the solution of AttSolverû with step size k∆t, we have |û N − u(T )| ≤ α∆t + β √ δ,(10)where α = 1 2L M exp(2T L), β = √ T exp(T L(1+Latt) √ L(1+Latt) and δ is a error term about the training loss R e . If the AttSlover can fit the training data well, i.e., R e → 0, the error δ → 0. Proof. (See Appendix C). Previous works (Du et al., 2019;Jacot et al., 2018) based on neural tangent kernel theory reveal that under certain mild conditions, gradient descent allows a neural network to converge to a globally optimal solution, where the loss R e in Eq.(7) tends to 0. Then according to Theorem 3.2, we have δ → 0, and thus |û N − u(T )| = O(∆t). In this case, the AttSolver with step size k∆t can achieve the same accuracy as the Euler method with the step size ∆t, whose global truncation error is also O(∆t) (Butcher, 2016). However, the evaluation speed of AttSolver is approximately O(k) times greater than that of the Euler method in this situation. Table 1: AttSolver for four forward numerical solvers on the spring-mass system with different dimensions. NeurVec is the state-of-the-art AI-enhanced Numerical solver. "data↓ 50%" denotes that we reduce the amount of training data by 50%. Red color means the accuracy of AttSolver can be improved with the SOTA method, and conversely, we use Green color. Experiment In this section, we consider two perspectives to verify the effectiveness of AttSolver: (1) on different numerical solvers and (2) different differential equation benchmarks. Specifically, since AttSolver is an AI-enhanced method for numerical solvers, we use many commonly used forward numerical solvers as backbones to evaluate the enhancement achieved by AttSolver, including the Euler method, Improved Euler method, 3rd and 4th order Runge-Kutta methods (see Appendix A for details.). On the other hand, to demonstrate that AttSolver is competent for complex differential equations, we further experiment on two chaotic dynamical systems on the 4th order Runge-Kutta method, i.e., k-link pendulum and elastic pendulum. For all experiments in this section, for fair comparisons, we follow the settings in (Huang et al., 2022b;Chen et al., 2020) for all generations of initial conditions and metrics. We elaborate on the rationale of choosing comparison methods in Appendix A. AttSolver for different numerical solvers In this section, we consider a high-dimensional linear system, namely the spring-mass system, and four forward numerical solvers (the details of these solvers can be found in Appendix A). In a spring-mass system, there are d masses and d + 1 springs connecting in sequence, and they are placed horizontally with two ends connected to two fixed blocks. The corresponding ODE of this system is d dt qi pi = pi/m ki (qi−1 − qi) + ki+1 (qi+1 − qi) ,(11) i = 1, 2, · · · , d, q 0 = q d+1 = 0, where m i and k i are the mass of the i th mass and force coefficient of the i th spring, respectively. The momentum and the position of i th mass are denoted as p i and q i . We adopt the coarse step size ∆t = 2e − 1 for the numerical solver and the fine step size 1e − 3 for training the enhanced method. The experiment results at evaluation time T = 20 are shown in Table 1. For the Euler method with low simulation accuracy, our AttSolver achieves consistent performance with the stateof-the-art (SOTA) enhanced method, i.e., NeurVec (Huang et al., 2022b), for the spring-mass system in different dimensions. For other numerical solvers with higher accuracy, AttSolver can better enhance the solver than NeurVec. Even for the spring-mass system with increasing dimensions, although the difficulty of the simulation increases, AttSolver can still maintain the performance. In addition, we can observe that our AttSolver can achieve similar performance as the SOTA method with less data size , showing that we are capable of training an efficient and robust AI-enhanced numerical solver with a small amount of data. In fact, such performance improvements as in Table 1 and significant, and as mentioned in Section 3, small errors can easily be accumulated and amplified in the iterative solving process. We will give a corresponding example of the numerical explosion in Section 6 ( Fig.3). AttSolver for different systems In this section, we reduce the amount of training data by 50% and use two chaotic dynamical systems to verify the effectiveness of our AttSolver, i.e., the Elastic and K-link pendulum. The Elastic pendulum considers a ball without volume connected to an elastic rod. Under the effect of gravity and force of spring (Breitenberger & Mueller, 1981), the motion of the ball will be chaotic, and its ODE is d dt    θ rθ r    =    θ r 1 r (−g sin θ −θṙ) rθ 2 − k m (r − l0) + g cos θ     ,(12) where k, m, l 0 , and g are related constants. There are two variables θ and r in Eq.(12). Specifically, r is the length of the spring, and θ is the angle between the spring and the vertical axis. For K-link pendulum, it considers K balls connected end to end with K rods under the effect of gravity (Lopes & Tenreiro Machado, 2017), and its ODE is d(θ,θ)/dt = (θ, A −1 b).(13) where θ = (θ 1 , θ 2 , · · · , θ K ) and θ i is the angle between the i th rod and the vertical axis. Let b = (b 1 , b 2 , · · · , b K ) and b i = − K j=1 c(i, j)θ 2 j sin (θ i − θ j ) −(K −i+1)g sin θ i . A is a K × K matrix and the element in A i,j is [K − max(i, j) + 1] cos (θ i − θ j ). The simulation results are shown in Fig.2. We first analyze the train/validation loss curves. For the K-link pendulum, AttSolver's loss is significantly smaller than that of the SOTA method in both the training and validation phase. For the elastic pendulum, at the beginning of the training phase, AttSolver's curve is smoother, and the loss is about 3∼4 orders of magnitude better than NeurVec. As the epoch increases, the loss curves of AttSolver are similar to those of the SOTA method in both training and validation phases until about 400 epochs, and we can observe that AttSolver can converge in the local optimal point with a smaller loss. Next, for Fig.2b and 2d, AttSolver can significantly reduce the error of the traditional method, i.e., 4th order Runge-Kutta methods, under the same step size 1e − 1 in the test phase, indicating that it is feasible to compensate the error of the numerical solver by the self-attention mechanism. Moreover, at the significance level α = 0.05, our method has a smaller MSE loss and can still significantly outperform the SOTA method. Ablation study In this section, we perform several ablation studies on Att-Solver. In Table 2, we analyze the impact of the depth h and width d 1 for the self-attention module in Eq.(9). We Stable Simulation Train loss (log) Figure 3: The noise attack experiments for the Elastic pendulum under different data sizes. AttSolver, with the self-attention mechanism, can better mitigate the adverse effects of noise than the SOTA method. observe that the depth h has little effect on the MSE loss in all benchmarks, but the model's performance is positively correlated with the width d 1 . Therefore, to increase the inference speed, we choose a sufficiently shallow depth h = 2 and appropriate width d 1 = 1024 for AttSolver. Then we Spring-mass 4.57e-4 (↓ 99.44%) 3.67e-6 (↓ 29.78%) 2.58e-6 K-link pendulum 6.36e-6 (↓ 99.93%) 4.18e-9 (↑ 1.91%) 4.26e-9 Elastic pendulum 1.39e-4 (↓ 99.61%) 1.65e-6 (↓ 67.21%) 5.41e-7 study popular residual structures for AttSolver. From Table 3, the skip connection will bring significant negative impacts on simulation for both high-dimension linear systems and chaotic dynamical systems. This motivates us to adopt a simple stacking structure like Eq.(9) for our Att-Solver. Moreover, we analyze the form of the input of the self-attention module, where we takeŜ as input instead of the complete integration termŜ∆t c in Eq.(9). In Table 3, we show that the inputŜ has better performance than input S∆t c , except for K-link pendulum (similar performance). Discussion The robustness of AttSolver. Our proposed method Att-Solver is inspired by the self-attention mechanism in ResNet, whose robustness is guaranteed by Theorem 3.1. In Fig.3, we empirically explore the robustness of AttSolver by noise attack experiments under different data sizes. Specifically, in the training process, we can interfere with the training phase of AttSolver by adding constant noise σ = 1e − 5 tô u n , i.e.,û n ←û n + σ for all n in Eq.(6). Experimental results show that, compared with the SOTA method, our proposed method can significantly regulate the noise, leading to smaller training loss and stable simulations. In contrast, the SOTA method explodes numerically at the beginning of the simulation due to the effect of noise in all settings. Why not multiplicative attention in AttSolver? Now we analyze the multiplicative attention in Eq.(8). In mathematics, multiplicative attention may incorrectly estimate the integration termŜ∆t c Q[Ŝ|φ] and cause an unstable solution. The reason is that the step size ∆t c inŜ used for discretization is not equal to the step size ∆t c := ∆t c Q[Ŝ|φ] used for integration under the multiplicative attention, unless Q[Ŝ|φ] is a constant vector I whose elements are all 1. In fact, Q[Ŝ|φ] does converge around I, as shown in Fig.4b. From an optimization viewpoint, this is because, in Eq.(1), the AI-enhanced method aims to compensate for the errors of the traditional numerical solver, and hence the magnitude of the compensation term c will be small. Then if multiplicative attention is used, we havê S∆t c + c =Ŝ∆t c Q[Ŝ|φ],(14) i.e., (Q[Ŝ|φ] − I)Ŝ∆t c = c . SinceŜ∆t c c , we have Q[Ŝ|φ] ≈ I. This means the self-attention module will tend On Robust Numerical Solver for ODE via Self-Attention Mechanism where the termŜ∆t c Q[Ŝ|φ] can be regarded as the additive attention term forû n +Ŝ∆t c and Eq.(15) has the same form as Eq.(6). Therefore, for AttSolver, we adopt additive attention instead of multiplicative attention. Limitations of AttSolver. From the experiment results in Section 4 and Section 5, the proposed AttSolver can achieve good enough simulation performance with less training data. However, AttSolver does not completely prevent the solution from being disturbed under the inherent noise in the dataset. So we aim to alleviate the noise issue rather than solve it completely. If we want to further mitigate this issue, we may need more elaborate network structures and training settings to enhance the effectiveness of our AttSolver. Theorem 6.1. We consider the SOTA method and AttSolver under the view of Vapnik-Chervonenkis theory (Vapnik, 1999). For > 0, when the data size is more than N , the empirical error of two methods satisfy R e (φ|Neur.) ≤ and R e (φ|Att.) ≤ . For small enough 0 and Euler method, we have N (Att.) N (Neur.),(16) where N ( * ) is the lower bound of the data size that the generalization error of method * can reach (1 − 0 ) −1 . Proof. (See Appendix D). Now we extend our discussion of the data size, where some analyses have been given by Theorem 6.1 under the Euler method and some mild assumptions. Compared to the SOTA method, AttSolver requires a smaller data size to achieve the same generalization error, which is consistent with the experimental results in Table 1. Although our proposed method does reduce the data size required by the SOTA method, the accuracy may not be sufficient for all the problems in the natural sciences and engineering. Therefore, if we need to solve a problem that requires high accuracy, it would be better to increase the training data size as much as possible to further improve the performance of the AIenhance solver than only relying on the algorithmic design. This is because we can observe when the complete training data is used, the improvement brought by AttSolver may be less than 10%, which is reasonable since the AttSolver is designed tailored to the data-insufficient scenario and we have not explicitly maximized the performance of AttSolver when the data is sufficient. In the future, we will explore improving the AttSolver in the data-sufficient scenario. Conclusion This paper discusses how to train an effective and robust AIenhanced numerical solver for differential equations with a small amount of data to alleviate the high data acquisition costs. Using the dynamical system view of ResNet, we introduce the self-attention mechanism into the numerical solver and propose AttSolver to mitigate the adverse effects of the inherent noise when the data is limited. Experimental results show the effectiveness of AttSolver in improving traditional numerical solvers with limited data, where we also analyze the convergence, generalization, and robustness. A. The details of the proposed algorithm. On the rationale of choosing comparison methods. Existing frontier works on solving differential equations based on neural networks can be divided into "neural network replacement" and "neural network enhancement" methods. The core idea of "neural network replacement" methods is to replace the role of the numerical solver in a data-driven manner. Different kinds of "neural network replacement" methods are differentiated by their replacement approaches. First, if a large amount of high-quality and uniformly sampled data is available, some specially structured neural networks (Wang et al., 2022b;Geneva & Zabaras, 2022;Liang et al., 2021) can directly approximate the integral curve such that it can replace the whole numerical solver without considering the physical properties of the equation. On the other hand, the PINN methods (Choudhary et al., 2020;Raissi et al., 2019;Ji et al., 2021;Cai et al., 2021) consider using sampling to replace the differential process while incorporating the physical information of the equations by constructing suitable loss functions. Last but not least, the operator-based methods focus on replacing the differential operators (Li et al., 2021;Lu et al., 2021) in a data-driven manner, where its basic idea is to approximate the solution operator by a neural network and replace the traditional solver during numerical simulations to improve the efficiency of the solution process. Although the "neural network replacement" method has an excellent performance in solving many differential equations, these methods rely heavily on neural networks, so there are still some problems that cannot be easily overcome. For instance, The operator-based methods prefer certain solutions with clean boundary regions (Ji et al., 2021), and the fitting of operators is relatively sensitive to model initialization and structural design (Meng et al., 2022); The method of directly fitting the integral curve may ignore the effect of chaos or stiffness in the differential equation (Liang et al., 2022). Moreover, the PINN methods do not have versatility for the initial values of the equation and we need to be retrained (Zhong et al., 2022;Psaros et al., 2022; the network for different initial values, etc. In contrast, the "neural network enhancement" method does not completely replace the numerical solver. Instead, it only leverages the neural network to augment the existing numerical solver (Huang et al., 2022b). Such a paradigm reduces the dependence on data-driven methods and ensures physical correctness since the traditional numerical solver still plays the main role in solving differential equations. Moreover, as most practical applications use the traditional solver paradigm as a primary choice, the "neural network enhancement" method has another advantage compared to the "neural network replacement" method in that we can directly integrate the enhancement method into the existing numerical solver system without changing any user's habits or interrupting the deployed solver system. Therefore, due to the difference in applicable scenarios and the scope of application between the paradigms of "neural network replacement" and "neural network enhancement", in this paper, we primarily compare the method belonging to the "neural network enhancement" category, e.g., NeurVec (Huang et al., 2022b), to ensure an apple-to-apple comparison to avoid the unreasonable and even impossible apple-to-orange comparison. The details of dataset. We summarize the information on training, validation, and test data for all benchmarks in this paper in Table 4. We obtain the discrete solutions every step size up to the model time T and the evaluation time for all experimental results of MSE loss in this paper is T . The Generative method of Spring-mass during training depends on the numerical solver used in Table 1. Moreover, the dimension of the Spring-mass system also depends on d in Table 1, and if d = 20, the dimension is 2d = 40. "RK4" denotes 4th order Runge-Kutta method. Numerical solvers. In this paper, we consider four forward numerical solvers to validate the effectiveness of our proposed AttSolver, including the Euler method, Improved Euler method, 3rd and 4th order Runge-Kutta methods. In this section, we introduce these solvers. As mentioned in Eq. (2), these solvers have different integration terms S(f, u n , ∆t). [f (u n ) + f (u n + ∆tf (u n ))] O(∆t 2 ) 3rd order Runge-Kutta ∆t(λ 1 K 1 + λ 2 K 2 + λ 3 K 3 ) O(∆t 3 ) 4th order Runge-Kutta ∆t(β 1 J 1 + β 2 J 2 + β 3 J 3 + β 4 J 4 ) O(∆t 4 ) For 3rd order Runge-Kutta method ,the coefficients λ 1 = λ 3 = 1 6 and λ 2 = 2 3 . Besides, K 1 = f (u n ), K 2 = f (u n + ∆t 2 K 1 ) and K 3 = f (u n − ∆tK 1 + 2∆tK 2 ). For the 4th order Runge-Kutta method, β 1 = β 4 = 1 6 and β 2 = β 3 = 1 3 . Moreover, J 1 = f (u n ), J 2 = f (u n + ∆t 2 J 1 ), J 3 = f (u n + ∆t 2 J 2 ) and J 4 = f (u n + ∆tJ 3 ). Generally speaking, 4th order Runge-Kutta method has the smallest Global truncation error O(∆t 4 ), i.e., 4th order Runge-Kutta method has the highest accuracy. However, due to its most complex integration term ∆t(β 1 J 1 + β 2 J 2 + β 3 J 3 + β 4 J 4 ), its simulation speed is the slowest among the four numerical solvers in Table 6. B. The proof of Theorem 3.1 Theorem 3.1. Consider a L layers residual neural network with self-attention module Q(·), i.e., x t+1 = x t + f (x t ; W t ) Q[f (x t ; W t )] , t = 0, 1, ..., L − 1. And there exists a constant r s.t. max{|x t ∇ x Q|} ≤ r. Let be the perturbation coming from noise and satisfies x 0 − x 0 = , and t = x t − x t , we have t+1 ≤ t (1 + (max{Q[f (x t ; W t )]} + r) W t 2 ) ,(17) where max refers to the largest element in a vector. Proof. Let H(x) = f (x t ; W t ) Q[f (x t ; W t )] , according to Taylor expansion, we have H(x t ) = H(x t ) + (x t − x t ) T ∇ x H(x)| x=xt(18)For t+1 = x t+1 − x t+1 , we have t+1 = x t+1 − x t+1 = x t + f (x t ; W t ) Q[f (x t ; W t )] − x t − f (x t ; W t ) Q[f (x t ; W t )] = (x t − x t ) + f (x t ; W t ) Q[f (x t ; W t )] − f (x t ; W t ) Q[f (x t ; W t )] = (x t − x t ) + (x t − x t ) T ∇ x H(x)| x=xt Since Eq.(18) ≤ (x t − x t ) · (1 + ∇ x H(x)| x=xt ) Since ab ≤ a b = t (1 + ∇ x H(x)| x=xt ) Next, we analyze ∇ x H(x)| x=xt . We also follow and consider the assumption (Yang et al., 2020) that f (x t ; W t ) is consists of linear transformation and ReLU non-linear activation A, i.e., f ( x t ; W t ) = AW t x t . Note that A is a diagonal matrix A =      a 1 0 0 0 0 a 2 0 0 0 0 . . . 0 0 0 0 a d      , a 1 , a 2 , .., a d ∈ {0, 1},(19) where d is dimension of x t and and each element a i , i = 1, 2, ..., d in A is 0 or 1. If the i th element in x t is positive, the value of a i = 1, otherwise equals to zero. Therefore the 2-norm, i.e., the largest singular value, of A is less than 1, i.e., A 2 = λ max A T A = λ max A ≤ 1(20) Hence, we can further estimate ∇ x H(x)| x=xt that ∇ x H(x)| x=xt = ∇ x f (x; W t )| x=xt Q[f (x t ; W t )] + f (x t ; W t ) ∇ x Q| x=xt ≤ AW t I Q[f (x t ; W t )] + AW t x t ∇ x Q| x=xt ≤ max{Q[f (x t ; W t )]} · A W t + r · A W t Since max{|x t ∇ x Q|} ≤ r ≤ (max{Q[f (x t ; W t )]} + r) W t Since Eq.(20) Therefore, we have t+1 ≤ t (1 + (max{Q[f (x t ; W t )]} + r) W t 2 ). C. The proof of Theorem 3.2 Theorem 3.2. We consider ODE du/dt = f (u), u(0) = c 0 and Euler method u n+1 = u n + ∆tf (u n ) . We assume that (1) f is Lipschitz continuous with Lipschitz constant L and (2) the second derivative of the true solution u is uniformly bounded by M > 0, i.e., u ∞ ≤ M on [0, T ]. Moreover, we assume that the attention module in AttSolver is Lipschitz continuous with Lipschitz constant k∆tL att . For the solution of AttSolverû with step size k∆t, we have |û N − u(T )| ≤ α∆t + β √ δ,(21)where α = 1 2L M exp(2T L), β = √ T exp(T L(1+Latt) √ L(1+Latt) and δ is a error term about the training of AttSlover. If the AttSlover can fit the training data well, the error δ → 0. Lemma C.1. If the assumptions (1) and (2) in Theorem 3.2 hold, for Euler method u n+1 = u n + ∆tf (u n ), we have |u N k − u(T )| ≤ M exp(2T L) 2L ∆t,(22) where 0 = t 0 < t 1 < · · · < t N k = T be uniform points on [0, T ] and ∆t = T N k . Lemma C.2. For any n ∈ N + and x > 0, we have (1 + x) n ≤ exp(nx),(23) Proof. (For Lemma C.2). According to the Taylor expansion, we have exp(nx) = ∞ i=0 (nx) i i! ≥ ∞ i=0 n i x i i! ≥ ∞ i=0 i−1 j=0 (n − j) x i i! = ∞ i=0 C i n x i = (1 + x) n(24) Proof. (For Theorem 3.2.) Let's consider the discretization for time interval [0, T ] as shown in Fig.5, and E n :=û n − u nk Original Euler method AttSolver(ours) Figure 5: The discretization of the original Euler method (Left) and our AttSolver (Right). |û N k − u(T )| ≤ |u N k − u(T )| + |û N − u N k | Since |a − b| ≤ |a| + |b| = M exp(2T L) 2L ∆t + E N Since Lemma C.1 Next we estimate the upper bound of E N . For any n ≥ 0 we havê u n+1 − u k(n+1) =û n + f (û n )(k∆t) + Q(f (û n )|φ) − u k(n+1) =û n − u kn + f (û n ) − f (u kn ) (k∆t) + Q(f (û n )|φ) − Q(f (u kn )|φ) − (k∆t)V n . where V n is an error term about training, i.e., V n = 1 k∆t (u k(n+1) − u kn − f (u kn )k∆t − Q(f (u nk )|φ)). Next, from Lipschitz conditions and the triangle inequality, we have |û n+1 − u k(n+1) | ≤ |û n − u kn | + L|û n − u kn |(k∆t) + k∆tL att |f (û n ) − f (u kn )| + (k∆t)|V n | ≤ |û n − u kn | + L|û n − u kn |(k∆t) + k∆tL att · L|û n − u kn | + (k∆t)|V n | = (1 + k∆tL + k∆tL att L)|û n − u kn | + (k∆t)|V n |. Let w = (1 + k∆tL + k∆tL N V L). Eq. (27) can be rewritten as |E n+1 | ≤ w|E n | + (k∆t)|V n | ≤ w(w|E n−1 | + (k∆t)|V n−1 |) + (k∆t)|V n | = w 2 |E n−1 | + w(k∆t)|V n−1 | + (k∆t)|V n | ≤ w n+1 |E 0 | + (k∆t) n i=0 w i |V n−i | = (k∆t) n i=0 w i |V n−i |,(28) where E 0 = 0 as E 0 =û 0 − u 0 = c 0 − c 0 = 0. Let δ = 1 N ( V 0 2 2 + V 1 2 2 + ... + V N −1 2 2 )(29) By the Cauchy inequality and Eq. (28), |E N | = (k∆t) N −1 i=0 w i |V N −1−i | ≤ (k∆t)( N −1 i=0 w 2i ) 1 2 ( N −1 i=0 |V N −1−i | 2 ) 1 2 Since Cauchy inequality = (k∆t) (w 2N − 1)/(w 2 − 1) √ N δ. Since Eq. (29) Next, we simplify the term (w 2N − 1)/(w 2 − 1). Note that w = (1 + k∆tL + k∆tL att L) ≥ 1, hence w 2 − 1 ≥ w − 1.(30) Therefore, Hence, we have . Since Eq.(26), if the AttSlover can fit the training data well, V n 2 → 0, and we have δ → 0. |û N − u(T )| ≤ α∆t + β √ δ,(31) On Robust Numerical Solver for ODE via Self-Attention Mechanism D. The proof of Theorem 6.1. Theorem 6.1. We consider the SOTA method and AttSolver under the view of Vapnik-Chervonenkis theory (Vapnik, 1999). For > 0, when the data size is more than N , the empirical error of two methods satisfy R e (φ|Neur.) ≤ and R e (φ|Att.) ≤ . For small enough 0 and Euler method, we have N (Att.) N (Neur.),(32) where N ( * ) is the lower bound of the data size that the generalization error of method * can reach (1 − 0 ) −1 . Lemma D.1. Consider the set of the models S k = {f (·|w), w ∈ ω k }. If S 1 ⊂ S 2 ⊂ · · · ⊂ S k ⊂ · · · , h 1 ≤ h 2 ≤ · · · ≤ h k ≤ · · · , where h i is the Vapnik-Chervonenkis dimension of the model in set S i , i = 1, 2, · · · . Lemma D.2. Let f (x) = (ln x + α)/x, x ∈ (0, +∞) and α ∈ R. f (x) achieves its maximum value at x = e 1+α , when x ∈ (0, e 1+α ), f (x) rises monotonically, and when x ∈ (e 1+α , +∞), f (x) decreases monotonically. Lemma D.3. Let f (x) = x exp( −a x ) where a > 0 and x ∈ R. Then f (x) rises monotonically. Lemma D.4. Let W (x) be Lambert W function, if |x| < 1/e, the Taylor expansion of W (x) is W (x) = ∞ n=1 (−n) n−1 n! x n = x − x 2 + 3 2 x 3 − 8 3 x 4 + · · · .(34) Assumption D.5. For x ∈ R and a general real function f (x), we consider the maps f 1 : x → ∇ x f (x) T f (x) and f 2 : f (x) → ∇ x f (x) T f (x) . For a small enough > 0, if S 1 and S 2 are the sets that ∀g 1 (φ) ∈ S 1 and g 2 (φ) ∈ S 2 , g 1 (φ) − f 1 ≤ , g 2 (φ) − f 2 ≤ ,(35) we assume that S 1 ⊂ S 2 . This assumption means that the complexity of the model for fitting f 1 is higher than that for f 2 . Proof. (For Theorem 6.1). According to Vapnik-Chervonenkis theory (Vapnik, 1999) for regression, for any data distribution P (x, y), model A (φ), the generalization error R(φ) and empirical error R e (φ), the inequality R(φ) ≤ R e (φ) 1 − c δ(h A (φ) , N ) −1 + ,(36) Figure 2 : 2The simulation on different chaotic systems with step size 1e − 1. a and c are the train/validation loss curves; b and d is the Mean Squared Error (MSE) loss on the test set. The significance level α is 0.05. Notation: "*": p < 0.05. Figure 4 : 4a. The loss curves for two kinds of attention. The loss minimization of multiplicative attention is fast at first and then slow during the last epochs, which have a local minimum with a large loss; b. The mean of attention value Q[Ŝ|φ] (blue) while using Eq.(8), which quickly converges to the vicinity of constant 1 during training. to fit a constant vector, which will bring negative impacts on its optimization (Wang et al., 2022b), like Fig.4a. 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(w 2N − 1)/(w 2 − 1) ≤ [(1 + k∆tL + k∆tL att L) 2N ]/[k∆tL + k∆tL att L] Since Eq.(30) ≤ exp(2N k∆tL(1 + L att )) k∆tL(1 + L att ) where α = 1 2L M exp(2T L), β = √ T exp(T L(1+Latt) √ L(1+Latt) holds with probability 1 − η. h A (φ) is Vapnik-Chervonenkis dimension of model A (φ), N is the data size, c usually is set as 1, and δ(h, N ) = (h ln N h + 1 − ln η)/N.(37)Therefore, from Eq.(36), for the generalization error R(φ|Neur.) and R(φ|Att.), we have R(φ|Neur.) ≤ R e (φ|Neur.) 1 − c δ(h Neur.(φ) , N ) and R(φ|Att.) ≤ R e (φ|Att.) 1 − c δ(h Att.(φ) , N ) are meaningful0.5 2.5 4.5 6.5 time T 8.5 1e-4 1e-6 1e-8 1e-10 1e-12 ) g o l ( s s o l E S M 4th Runge-Kutta Method NeurVec (SOTA) AttSolver 0 100 200 300 Epoch 400 500 1e1 1e-1 1e-3 1e-5 1e-7 Train/Validation loss (log) a b Train: NeurVec (SOTA) Train: AttSolver Test: NeurVec (SOTA) Test: AttSolver 0.5 2.5 4.5 6.5 time T 8.5 1e-2 1e-4 1e-6 1e-8 1e-10 ) g o l ( s s o l E S M 4th Runge-Kutta Method NeurVec (SOTA) AttSolver d 0 100 200 300 Epoch 400 500 c 1e9 1e5 1e1 1e-3 1e-7 50 100 1e4 1e0 Train: NeurVec (SOTA) Train: AttSolver Test: NeurVec (SOTA) Test: AttSolver Elastic pendulum Elastic pendulum K-link pendulum K-link pendulum Train/Validation loss (log) Table 2 : 2The impact of the depth h and width d 1 of the self-attention module on the simulation performance. Rel. Infe. Speed is the relative inference speed of the network based on our proposed AttSolver setting, i.e., h = 2 and d 1 = 1024.Spring-mass K-link pendulum Elastic pendulum MSE loss Rel. Infe. Speed MSE loss Rel. Infe. Speed MSE loss Rel. Infe. Speed h = 2 (ours) 2.58e-6 - 4.26e-9 - 5.41e-7 - h = 3 2.49e-6 (↑ 3.61%) ↓ 88.49% 4.39e-9 (↓ 2.52%) ↓ 91.60% 5.28e-7 (↑ 2.46%) ↓ 90.89% h = 4 2.55e-6 (↑ 1.18%) ↓ 93.95% 4.42e-9 (↓ 3.62%) ↓ 95.64% 5.58e-7 (↓ 3.05%) ↓ 95.33% d 1 = 512 3.76e-5 (↓ 83.36%) ↑ 27.59% 2.56e-8 (↓ 93.14%) ↑ 15.36% 2.56e-6 (↓ 78.87%) ↑ 17.25% d 1 = 1024 (ours) 2.58e-6 - 4.26e-9 - 5.41e-7 - d 1 = 2048 2.41e-6 (↑ 7.05%) ↓ 26.48% 3.98e-9 (↑ 7.03%) ↓ 23.12% 4.83e-7 (↑ 12.01%) ↓ 24.63% Data size ↓50% + Noise Data size ↓75% + Noise Data size ↓90% + Noise AttSolver NeurVec (SOTA) AttSolver NeurVec (SOTA) AttSolver NeurVec (SOTA) 0 100 200 300 Epoch 400 500 0 100 200 300 Epoch 400 500 0 100 200 300 Epoch 400 500 0 2 4 6 Time 8 10 1e1 1e-1 1e-3 1e-5 1e2 1e-0 1e-2 1e-4 1e3 1e1 1e-1 1e-3 ) g o l ( s s o l E S M 1e24 1e14 1e4 1e-6 NeurVec (↓50%) NeurVec (↓75%) NeurVec (↓90%) AttSolver (↓50%) AttSolver (↓75%) AttSolver (↓90%) Numerical Explosion Table 3 : 3The impact of skip connection and considering ∆t c as a part of the input.Benchmark w/ skip w/o ∆tc w/o ∆tc & w/o skip (ours) Table 4 : 4Summary of the datasets mentioned in this paper.Benchmark Type Dimension Data size Step size Generative Method Evaluation time T Spring-mass Train * 5k 1e-3 * 20 Spring-mass Validation * 0.1k 1e-5 RK4 20 Spring-mass Test * 5k 1e-5 RK4 20 2-link pendulum Train 4 20k 1e-3 RK4 10 2-link pendulum Validation 4 1k 1e-5 RK4 10 2-link pendulum Test 4 10k 1e-5 RK4 10 Elastic pendulum Train 4 20k 1e-3 RK4 50 Elastic pendulum Validation 4 1k 1e-5 RK4 50 Elastic pendulum Test 4 10k 1e-5 RK4 50 Table 5 : 5The initialization of different benchmarks. "Uniform random" means that the variables are sampled with uniform distribution of a given range. "Constant" means the variable is initialized as a constant. The dimension of p and q in Spring-mass system depends on d inTable 1, and if d = 20, their dimension are d = 20.Task Variable Dimension Type Range of initialization Model input? Spring-mass p * Uniform random [−2.5, 2.5] 20 Spring-mass q * Uniform random [−2.5, 2.5] 20 Elastic pendulum θ 1 Uniform random [0, π/8] Elastic pendulum r 1 Constant 10 Elastic pendulumθ 1 Constant 0 Elastic pendulumṙ 1 Constant 0 Elastic pendulum l 0 1 Constant 10 Elastic pendulum g 1 Constant 9.8 Elastic pendulum k 1 Constant 40 Elastic pendulum m 1 Constant 1 2-link pendulum θ 2 Uniform random [0, π/8] 2 2-link pendulumθ 2 Constant 0 2-link pendulum m 1 Constant 1 2-link pendulum g 1 Constant 9.8 Table 6 : 6The integration terms for different kinds of numerical solvers with step size ∆t.Numerical solver Integration term S(f, u n , ∆t) Global truncation error Euler ∆tf (u n ) O(∆t) Improved Euler ∆t 2 Next, we consider how many data size can the term 1 − δ(h A (φ) , N ) −1 reach the accuracy (1 − 0 ) −1 . Let δ := (lnN + α)/N , whereN = N/h A (φ) and α = 1 − ln η/h A (φ) .(Lehtonen, 2016), andFrom Lemma D.2, we know that when the data sizeholds with probability 1 − η. 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{'abstract': 'With the development of deep learning techniques, AI-enhanced numerical solvers are expected to become a new paradigm for solving differential equations due to their versatility and effectiveness in alleviating the accuracy-speed trade-off in traditional numerical solvers. However, this paradigm still inevitably requires a large amount of high-quality data, whose acquisition is often very expensive in natural science and engineering problems. Therefore, in this paper, we explore training efficient and robust AI-enhanced numerical solvers with a small data size by mitigating intrinsic noise disturbances. We first analyze the ability of the self-attention mechanism to regulate noise in supervised learning and then propose a simple-yet-effective numerical solver, AttSolver, which introduces an additive self-attention mechanism to the numerical solution of differential equations based on the dynamical system perspective of the residual neural network. Our results on benchmarks, ranging from high-dimensional problems to chaotic systems, demonstrate the effectiveness of AttSolver in generally improving the performance of existing traditional numerical solvers without any elaborated model crafting. Finally, we analyze the convergence, generalization, and robustness of the proposed method experimentally and theoretically.', 'arxivid': '2302.10184', 'author': ['Zhongzhan Huang ', 'Mingfu Liang ', 'Liang Lin '], 'authoraffiliation': [], 'corpusid': 257050822, 'doi': '10.48550/arxiv.2302.10184', 'github_urls': [], 'n_tokens_mistral': 21294, 'n_tokens_neox': 18528, 'n_words': 10492, 'pdfsha': '0b90f5c635131a6d918dbc6943540ee0691b6817', 'pdfurls': ['https://export.arxiv.org/pdf/2302.10184v1.pdf'], 'title': ['On Robust Numerical Solver for ODE via Self-Attention Mechanism', 'On Robust Numerical Solver for ODE via Self-Attention Mechanism'], 'venue': []}

arxiv

TESS Asteroseismic Analysis of HD 76920: The Giant Star Hosting An Extremely Eccentric Exoplanet February 7, 2023 6 Feb 2023 Chen Jiang jiangc@mps.mpg.de Max-Planck-Institut für Sonnensystemforschung Justus-von-Liebig-Weg 337077GöttingenGermany Tao Wu wutao@ynao.ac.cn Yunnan Observatories Chinese Academy of Sciences 396 Yangfangwang Guandu District 650216KunmingPeople's Republic of China Key Laboratory for the Structure and Evolution of Celestial Objects Chinese Academy of Sciences 396 Yangfangwang Guandu District 650216KunmingPeople's Republic of China Center for Astronomical Mega-Science Chinese Academy of Sciences 20A Datun Road100012Chaoyang District, BeijingPeople's Republic of China University of Chinese Academy of Sciences 100049BeijingPeople's Republic of China Institute of Theoretical Physics Shanxi University 030006TaiyuanChina Adina D Feinstein Department of Astronomy and Astrophysics University of Chicago 5640 S. Ellis Ave60637ChicagoILUSA Keivan G Stassun Department of Physics and Astronomy Vanderbilt University 37235NashvilleTNUSA Sylvain N Breton Département d'Astrophysique/AIM CEA/IRFU CNRS/INSU Univ. Paris-Saclay & Univ. de Paris 91191Gif-sur-YvetteFrance Mia S Lundkvist Department of Physics and Astronomy Stellar Astrophysics Centre (SAC) Aarhus University Ny Munkegade 120DK-8000Aarhus CDenmark Przemys Law J Miko Astronomical Observatory University of Warsaw Al. Ujazdowskie 400-478WarsawPoland Astronomical Institute University of Wroc law Miko laja Kopernika 11, 51-622 Wroc lawPoland Charlotte Gehan Max-Planck-Institut für Sonnensystemforschung Justus-von-Liebig-Weg 337077GöttingenGermany Instituto de Astrofísica e Ciências do Espaço Universidade do Porto Rua das Estrelas4150-762PortoPortugal Tiago L Campante Instituto de Astrofísica e Ciências do Espaço Universidade do Porto Rua das Estrelas4150-762PortoPortugal Departamento de Física e Astronomia Faculdade de Ciências da Universidade do Porto Rua do Campo Alegre s/n4169-007PortoPortugal Diego Bossini Instituto de Astrofísica e Ciências do Espaço Universidade do Porto Rua das Estrelas4150-762PortoPortugal Stephen R Kane Department of Earth and Planetary Sciences University of California 92521RiversideCAUSA Jia Mian Joel Ong Department of Astronomy Yale University P.O. Box 20810106520-8101New HavenCTUSA Mutlu Yıldız Department of Astronomy and Space Sciences Science Faculty Ege University 35100İzmirBornovaTürkiye Cenk Kayhan Department of Astronomy and Space Sciences Science Faculty Erciyes University 38030KayseriMelikgaziTürkiye Zeynep Ç Elik Orhan Department of Astronomy and Space Sciences Science Faculty Ege University 35100İzmirBornovaTürkiye Sibelörtel Xinyi Zhang Department of Astronomy and Space Sciences Science Faculty Ege University 35100İzmirBornovaTürkiye State Key Laboratory of Lunar and Planetary Sciences Macau University of Science and Technology MacauPeople's Republic of China Margarida S Cunha Instituto de Astrofísica e Ciências do Espaço Universidade do Porto Rua das Estrelas4150-762PortoPortugal Departamento de Física e Astronomia Faculdade de Ciências da Universidade do Porto Rua do Campo Alegre s/n4169-007PortoPortugal Bruno Lustosa De Moura Departamento de Fisica Universidade Federal do Rio Grande do Norte 59072-970NatalBrazil Grande do Norte-IFRN Instituto Federal do Rio Brazil Jie Yu Max-Planck-Institut für Sonnensystemforschung Justus-von-Liebig-Weg 337077GöttingenGermany Daniel Huber Institute for Astronomy University of Hawai'i 2680 Woodlawn Drive96822HonoluluHIUSA Ou (欧建文)Jian-Wen Robert A Wittenmyer School of Physics and Electromechanical Engineering Guangdong Province Shaoguan University 512005ShaoguanChina Centre for Astrophysics University of Southern Queensland USQ Toowoomba 4350QLDAustralia Laurent Gizon Max-Planck-Institut für Sonnensystemforschung Justus-von-Liebig-Weg 337077GöttingenGermany Institut für Astrophysik Georg-August-Universität Göttingen Friedrich-Hund-Platz 137077GöttingenGermany William J Chaplin Department of Physics and Astronomy Stellar Astrophysics Centre (SAC) Aarhus University Ny Munkegade 1208000Aarhus CDenmark School of Physics and Astronomy University of Birmingham B15 2TTBirminghamUK Chen Jiang School of Physics Sydney Institute for Astronomy (SIfA) University of Sydney 2006NSWAustralia Centre for Exoplanets and Habitability University of Warwick CV4 7ALCoventryUK Centre for Space Domain Awareness University of Warwick CV4 7ALCoventryUK Department of Physics University of Warwick CV4 7ALCoventryUK INAF -Osservatorio Astrofisico di Catania via S. Sofia 7895123CataniaItaly Department of Chemistry & Physics FGCU Boulevard S Florida Gulf Coast University 10501, 33965Fort MyersFLUSA School of Physics University of New South Wales 2052NSWAustralia Instituto de Astrofísica de Canarias (IAC) E-38205La Laguna, TenerifeSpain Departamento de Astrofísica Universidad de La Laguna (ULL) E-38206La Laguna, TenerifeSpain Université Paris-Saclay Université Paris Cité CEA CNRS 91191Gif-sur-YvetteAIMFrance TESS Asteroseismic Analysis of HD 76920: The Giant Star Hosting An Extremely Eccentric Exoplanet February 7, 2023 6 Feb 2023Draft version Typeset using L A T E X default style in AASTeX631 Timothy R. Bedding , 9, 10 Dimitri Veras , 11, 12, 13 Enrico Corsaro , 14 Derek L. Buzasi , 15 Dennis Stello , 16, 9, 10 Yaguang Li(李亚光) , 9, 10 Savita Mathur , 17, 18 Rafael A. García , 19 Corresponding author: Corresponding author: Tao Wu 2 Jiang et al. 37 Center for Space Science, NYUAD Institute, New York University Abu Dhabi, Abu Dhabi, UAEasteroseismology -stars: individual (HD 76920) -planets and satellites: dynamical evolution and stability The Transiting Exoplanet Survey Satellite (TESS) mission searches for new exoplanets. The observing strategy of TESS results in high-precision photometry of millions of stars across the sky, allowing for detailed asteroseismic studies of individual systems. In this work, we present a detailed asteroseismic analysis of the giant star HD 76920 hosting a highly eccentric giant planet (e = 0.878) with an orbital period of 415 days, using 5 sectors of TESS light curve that cover around 140 days of data. Solar-like oscillations in HD 76920 are detected around 52 µHz by TESS for the first time. By utilizing asteroseismic modeling that takes classical observational parameters and stellar oscillation frequencies as constraints, we determine improved measurements of the stellar mass (1.22 ± 0.11M ), radius (8.68 ± 0.34 R ), and age (5.2 ± 1.4 Gyr). With the updated parameters of the host star, we update the semi-major axis and mass of the planet as a = 1.165 ± 0.035 au and M p sin i = 3.57 ± 0.22 M Jup . With an orbital pericenter of 0.142 ± 0.005 au, we confirm that the planet is currently far away enough from the star to experience negligible tidal decay until being engulfed in the stellar envelope. We also confirm that this event will occur within about 100 Myr, depending on the stellar model used. INTRODUCTION The excellent quality of photometric data from space observation missions, such as CoRoT (Baglin et al. 2006) and Kepler (Borucki et al. 2010), allow for major advancements in the understanding of stellar interior physics and evolution using asteroseismology. Asteroseismology is the study of internal structure of stars by the interpretation of their oscillation frequencies. In particular, the detection of oscillations in solar-type and red giant stars has led to breakthroughs such as the discovery of fast core rotation (Beck et al. 2012) and a way to distinguish between hydrogen-shell-burning stars and stars that are also burning helium in their cores (Bedding et al. 2011). The advent of space photometry has also brought in advancements of data analysis techniques (e.g., Davies & Miglio 2016;Lund et al. 2017;Corsaro & De Ridder 2014;Corsaro et al. 2015) and stellar modeling strategies (e.g., Wu & Li 2016;Serenelli et al. 2017;Silva Aguirre et al. 2017;Wu & Li 2017). Furthermore, by using individual oscillation frequencies as constraints in the model optimization process (Metcalfe et al. 2010;Jiang et al. 2011;Mathur et al. 2012;Paxton et al. 2013;Rendle et al. 2019), asteroseismic modeling has proven to be a robust tool to determine fundamental stellar properties, including stellar distances (Silva Aguirre et al. 2012;Rodrigues et al. 2014), radii and masses (Casagrande et al. 2014;Pinsonneault et al. 2014;Sharma et al. 2016), and most importantly ages and core size for red giants and clump stars (Casagrande et al. 2016;Anders et al. 2017;Pinsonneault et al. 2018;Zhang et al. 2018Zhang et al. , 2020. Consequently, this enables us to characterize systematically the properties of the exoplanet-host stars through asteroseismology, which in turn provides an unprecedented level of precision in the parameters estimated for the planets (Ballard et al. 2014;Campante et al. 2015;Lundkvist et al. 2016;Kayhan et al. 2019). Furthermore, the synergy between asteroseismology and exoplanetary research also enables us to set constraints on the spin-orbit alignment of exoplanet systems (Huber et al. 2013;Benomar et al. 2014;Chaplin et al. 2014;Lund et al. 2014;Campante et al. 2016a;Kamiaka et al. 2019) and to perform statistical inferences on the planetary orbital eccentricities, by making use of asterodensity profiling (Kane et al. 2012;Sliski & Kipping 2014;Van Eylen & Albrecht 2015;Van Eylen et al. 2019). The Transiting Exoplanet Survey Satellite (TESS) Mission (Ricker et al. 2015) is NASA's near all-sky survey for exoplanets, which launched in 2018. The large sample of monitored systems guarantees the synergy between asterosismology and exoplanetary science to continue to expand (Campante et al. 2018;Huber 2018;Hatt et al. 2022). TESS searches for exoplanets using the transit method in an area 400 times larger than that covered by the Kepler mission. Although the exploration of new exoplanetary systems is the main scientific goal of the mission, TESS has also provided aids toward the characterization of previously known systems . Thanks to the high-quality of TESS photometry and large sky coverage, oscillations are expected to be detected in hundreds of thousands of solar-like oscillators (Campante et al. 2018;Huber 2018;Schofield et al. 2019), including several hundred asteroseismic exoplanet hosts (Campante et al. 2016b). Detections of oscillations by TESS in previously known exoplanet-host stars were reported by several works (e.g. Campante et al. 2019;Jiang et al. 2020b;Nielsen et al. 2020;Hill et al. 2021;Huber et al. 2022), following on the discovery of the first planet transiting a star in which oscillations could be measured by TESS . These extraordinary synergies between asteroseismology and exoplanetary science significantly improve our understanding of planet systems outside of the solar system and provide insight into the occurrence rates of exoplanets as a function of their host stars' property and evolutionary state, as well as the planets' mass, size and orbital architecture. One of the most interesting architectures is the planets having very eccentric orbits that are considered as a possible origin of hot Jupiter (see Dawson & Johnson 2018, for a review). Planets in very eccentric orbits are a prime example and testbed of how planetary systems form and evolve. The first planet discovered to orbit around a evolved giant star, ι Draconis b, is on a highly eccentric orbit with e = 0.71 (Frink et al. 2002;Kane et al. 2010). Even larger orbital eccentricity is found in a planet around the giant star HIP 126844 with e = 0.76 (Adamów et al. 2012(Adamów et al. , 2018. In this work, using TESS data, we present an asteroseismic analysis of the K giant star HD 76920 (TIC 302372658) known to host a planet. At the time that HD 76920 b was first detected through the radial-velocity (RV) survey of Pan-Pacific Planet Search (Wittenmyer et al. 2017), it was the most eccentric exoplanet known to orbit an evolved star, with an orbital eccentricity of 0.856 ± 0.009. Later, with the help of new multi-site RV measurements, Bergmann et al. (2021) refined the planetary properties, finding an even higher eccentricity of 0.8783 ± 0.0025 and an orbital period of 415.891 +0.043 −0.039 days, a minimum planet mass of 3.13 +0.41 −0.43 M Jup , and a semi-major axis of 1.091 +0.068 −0.077 au. There is no evidence of any unseen binary companion, suggesting a scattering event rather than Kozai oscillations as the probable explanation for the observed eccentricity, and making the system valuable to the study the evolution and occurrence of planets around evolved stars. TESS detected solar-like oscillations in HD 76920 for the first time. Bergmann et al. (2021) analyzed three sectors (9, 10 and 11) of TESS data and estimated the stellar mass and radius through the scaling relations (Brown et al. 1991;Kjeldsen & Bedding 1995;Stello et al. 2008;Kallinger et al. 2010) using the measurements of global seismic parameters. While the scaling relations provide decent estimates of the stellar mass and radius for stars showing solar-like oscillations, improvements, in terms of accuracy and precision, of the estimates can be achieved by using asteroseismic modeling and individual oscillation modes. In this paper, we aim to analyze the solar-like oscillations from 5 sectors of TESS photometric light curve. Including these oscillations in detailed stellar modeling can help us derive precise fundamental stellar properties of the host star. The improved stellar parameters can further be used to update the properties of the orbiting planet, which is of great importance in characterizing the planetary system. The determined stellar age from asteroseismology and modeling provide information about the evolution of the system, which assists at predicting the final fate of the planet. OBSERVATIONS TESS Observations HD 76920 has been observed by TESS in Sectors 9, 10, 11 at 30-min cadence during its Cycle 1 observations, and Sectors 36 and 37 at 10-min cadence 1 during its Cycle 3 observations. To extract these light curves, we used the open-source Python package eleanor (v2.0.3; Feinstein et al. 2019) 2 . eleanor performs background subtraction, systematics corrections, and aperture selection per each sector of data. We extracted 13×13 pixels postage stamps and applied the eleanor default apertures for light curve extraction (top panel of Figure 1). Although the aperture selection a Two SED pipelines are used to obtain A V , F bol , R and L (Section 2.2). The second value of each parameter is from the SEDEX pipeline (Yu et al. 2021(Yu et al. , 2022. process is optimized for exoplanet searches, we found also these default apertures to perform well for asteroseismic measurements. We used the eleanor corrected flux; this flux option corrects for systematic issues by regressing against a linear model of time, background, and position and removing trends by the co-trending basis vectors provided by the Science Process Operations Center pipeline (Jenkins et al. 2016). We additionally applied quality masks, provided by eleanor, to remove any bad cadences and removed outliers ≥ 7σ from the median of each light curve. The resulting light curves are shown in the bottom panel of Figure 1. No transit of the planet was detected in existing TESS data due to the short observation coverage compared to its long orbital period. TESS is expected to observe HD 76920 again for 5 months in Cycle 5. Broadband Photometry and Spectral Energy Distribution As an independent determination of the basic stellar parameters, we performed an analysis of the broadband spectral energy distribution (SED) of the star together with the Gaia DR3 parallax (with no systematic offset applied; see, e.g., Stassun & Torres 2021), in order to determine an empirical measurement of the stellar radius, following the procedures described in Stassun & Torres (2016) and Stassun et al. (2017Stassun et al. ( , 2018. We pulled the B T V T magnitudes from Tycho-2, the JHK S magnitudes from the Two Micron All Sky Survey, the W1-W4 magnitudes from the Wide-field Infrared Survey Explorer, and the G BP G RP magnitudes from Gaia DR3. Together, the available photometry spans the full stellar SED over the wavelength range 0.4-22 µm (see Figure 2). We performed a fit using Kurucz stellar atmosphere models, with the effective temperature (T eff ), surface gravity (log g), and metallicity ([Fe/H]) adopted from the spectroscopic analysis of Wittenmyer et al. (2017). The remaining free parameter is the extinction A V , which we limited to the maximum line-of-sight extinction from the Galactic dust maps of Schlegel et al. (1998). The resulting fit ( Figure 2) has a reduced χ 2 of 2.2, and best-fit A V = 0.20 ± 0.05. Integrating the (unreddened) model SED gives the bolometric flux at Earth, F bol = 3.12 ± 0.11 × 10 −8 erg s −1 cm −2 . Taking the F bol and T eff together with the Gaia parallax, gives the stellar radius, R = 8.64 ± 0.37 R , revealing the star to be clearly evolved. In addition, we can estimate the stellar mass directly from R together with the spectroscopic log g from (Wittenmyer et al. 2017, see Table 1), which gives M = 2.37 ± 0.41 M ; this is higher than the value obtained from the empirical relations of Torres et al. (2010), giving M = 1.68 ± 0.10 M , indicating an overestimate of spectroscopic log g (see Section 4). Combining F bol and Gaia DR3 parallax allows us to derive a luminosity L = 32.54 ± 1.17 L . We also performed an independent SED fitting using the SEDEX pipeline (Yu et al. 2021(Yu et al. , 2022 with MARCS atmosphere models (Gustafsson et al. 2008) and same input atmospheric parameters. The resulting parameters from both SED fittings are well consistent, as listed in Table 1. Global Oscillation Parameters For the seismic analysis, the TESS light curve prepared by eleanor was distributed to several groups using different pipelines (e.g., Huber et al. 2009;Mathur et al. 2010;Jiang et al. 2011;Lundkvist 2015;Campante et al. 2017;Yu et al. 2018;Corsaro et al. 2020;De Moura et al. 2020;Li et al. 2020) to extract the seismic parameters. Figure 3 shows the power density spectrum of HD 76920 computed based on the eleanor light curve, combining both Cycle 1 and Cycle 3 data. The 10-min cadence sectors were rebinned to 30-min cadence, with a simple average over three measurements. The power spectrum shows a frequency-dependent background signal due to stellar activity, granulation, and faculae that can be modeled by a superposition of several Lorentzian-like functions (i.e., Harvey-like models, Harvey 1985;Karoff 2008;Jiang et al. 2011;Kallinger et al. 2014;Corsaro et al. 2017), and a white noise term. The background shown as the cyan dotted curve in Figure 3 was obtained by fitting the background model with two Harvey-like components and one white noise to the smoothed power spectrum. The target pixel time series of the Cycle 3 sectors have been detrended with a moving median to remove long timescale stellar variations. As a result, the background noise at low frequency is greatly suppressed during this process. Therefore, the background fit displayed in Figure 3 disregarded frequencies below ∼10 µHz. The global seismic parameters such as the frequency of maximum power (ν max ) and the mean large frequency separation (∆ν) were based on the analysis of the background-corrected power spectrum generated by each group. In general, ν max was measured by fitting a Gaussian distribution profile to the power excess hump of the smoothed power spectrum (e.g., Hekker et al. 2010) or through the 2D autocorrelation function (e.g., Viani et al. 2019). To measure ∆ν, techniques like autocorrelation of the amplitude spectrum (e.g., Huber et al. 2009;Mosser & Appourchaux 2009), power spectrum of the power spectrum (e.g., Kjeldsen & Bedding 1995;Mathur et al. 2010;Jiang 2015), matched filter response function (e.g., Gilliland et al. 2011), and asymptotic or linear fit to the frequencies of the radial modes (individual mode extraction given in Section 3.2) were used. However, a clear shift of the oscillation power excess region to lower frequencies is observed in the power spectrum generated with Cycle 3 data (Sectors 36 and 37), compared with the Cycle 1 one (Sectors 9 to 11), as illustrated in the upper panel of Figure 4. This is principally due to the stochastic nature of the oscillations so that the change in mode amplitudes impacts the value of ν max . This shift of the power excess region may lead to a larger uncertainty in ν max when compared to the corresponding formal uncertainty originating from different pipelines. With this in mind, we computed consolidated results from the 8 independent determinations of the global seismic parameters. In particular, we adopted the mean values of the parameter estimates returned by all methods and recalculated the uncertainties by adding in quadrature the corresponding formal uncertainty and the standard deviation. The consolidated results are ν max = 53.2 ± 2.3 µHz and ∆ν = 5.53 ± 0.15 µHz. We note that due to the intrinsic change of the distribution of mode amplitudes observed between Cycle 1 and Cycle 3 data, this uncertainty of ∆ν resulting from our statistical consolidation approach, can also be overestimated and can be decreased by examining the different ∆ν values against theéchelle diagrams (Stello et al. 2011). As an alternative, instead of taking into account the scatter across results, we also quote here the pipeline results that are of the smallest deviation from the mean values of all pipeline results as ν max = 52.4 ± 0.3 µHz and ∆ν = 5.52 ± 0.02 µHz, considering both parameters simultaneously (i.e. both parameters returned by the same pipeline). This level of uncertainty of ∆ν is of comparable magnitude to those extracted from 2 sectors of TESS observations of red giants (Silva Aguirre et al. 2020). The combined power spectrum corrected from the background model is shown in the bottom panel of Figure 4, where signals with ν < 50 µHz are largely enhanced by the Cycle 3 data, and those with ν > 50 µHz are due to the Cycle 1 data. The asteroseismic analysis discussed in Section 3 were performed based on the combined power spectrum (lower panel of Figure 4). Individual Mode Frequencies In the bottom panel of Figure 4 the combined power spectrum shows a regular series of peaks corresponding to solarlike oscillations within the frequency range between 30 to 80 µHz. To extract individual oscillation modes from the power spectrum several independent methods ranging from traditional iterative fitting of sine waves, i.e., pre-whitening (e.g. Kjeldsen et al. 2005;Lenz & Breger 2005;Bedding et al. 2007;Jiang et al. 2011), to fitting of Lorentzian mode profiles individually or in a global power spectrum model (e.g. Handberg & Campante 2011;Appourchaux et al. 2012;Mosser et al. 2012;Corsaro et al. 2015;Vrard et al. 2015;Davies & Miglio 2016;Handberg et al. 2017;Roxburgh 2017;Kallinger et al. 2018;Corsaro et al. 2020;Li et al. 2020;Breton et al. 2022) were used by different pipelines. The extracted mode frequencies and corresponding uncertainties are listed in Table 2. The extracted radial modes also allowed us to measure ∆ν by fitting a straight line to the radial-mode frequencies. Thus, the slope of the line is ∆ν that is 5.62 ± 0.03 µHz using this method. Theéchelle diagram (Bedding & Kjeldsen 2010) generated using this ∆ν is depicted in Figure 5. The two vertical ridges located near the right edge of the figure correspond to = 0 (red circles) and 2 (blue triangles) modes. However, due to the relatively short duration of our TESS data for HD 76920, clear mixed-mode pattern is not visible in the power spectrum or in theéchelle diagram, though there are a few peaks corresponding to = 1 mixed modes (green diamonds in Figure 5) appearing in the spectrum. In Table 2 we also list the = 1 modes that are extracted by at least two independent sources, including a pair of mixed modes around 53 µHz that were simultaneously extracted by 3 independent sources. ASTEROSEISMIC MODELLING Asteroseismic modelling is a powerful tool to estimate fundamental stellar properties. Five independent teams performed modeling efforts to search for the stellar models that best match the classical and asteroseismic constraints from observations for HD 76920. These teams employ different stellar evolution codes (ASTEC, MESA; Christensen-Dalsgaard 2008a; Paxton et al. 2011Paxton et al. , 2013Paxton et al. , 2015, Zhang et al. 2022). The input physics adopted by each team is detailed in Table 3. Diffusion and overshoot were turned off except for one team. For the modeling, all five teams used the luminosity L of 32.54 ± 1.17 L derived with Gaia DR3 parallax (Section 2.2) and the spectroscopic measurements, i.e., the effective temperature T eff and metallicity [Fe/H] from Wittenmyer et al. (2017), as constraints. As for the seismic constraints, the large frequency separation ∆ν = 5.52 ± 0.02 µHz and = 0 and 2 mode frequencies in Table 2 were used 3 . Generally, the most p-dominated = 2 mixed modes of each model are selected to be compared with the observations, except for team Ong who computed the pure quadrupole p-modes of stellar models using the π-mode isolation condition of Ong & Basu (2020) as implemented in GYRE 4 . The constraint on ν max were not considered in the modeling due to the discrepancy of the oscillation power excess region between the sectors (Section 3.1). We measured the phase shift c of the central radial mode, which is the linear offset in the asymptotic fit to the acoustic modes, as ∼ 0.94, indicating that the star is still on the red giant branch (RGB) burning only hydrogen in a shell, because the more evolved horizontal branch stars (e.g. red clump and secondary clump stars) that have already ignited helium in the core would have a systematically smaller c than their counterparts on the RGB . Moreover, horizontal branch tracks hardly cross the observational constraints in the HR diagram, thus, we limited the modeling only to the RGB. The outputs from the five teams are generally in agreement with each other (Table 4), though differences are inevitably seen due to the diversity of modeling codes, procedures and input physics adopted by different teams. We note that the amount of convective overmixing (f ov = 0.006) adopted by team Ong is mainly for numerical softening and is too small to induce an appreciable physical difference to the spectroscopic parameters (see discussions in, e.g., Claret & Torres 2017Guo & Li 2019;Zhang et al. 2022). Furthermore, since we have only used even-degree ( = 2) p-dominated modes in this work, the effects of overshoot on the g-mode cavity (e.g. Lindsay et al. 2022) are not relevant. For the even-degree p-modes, envelope overshoot will reduce the amplitude of the convective-glitch signature induced into the p-modes, which is already very small on the main sequence, and decreases in amplitude as the convective boundary retreats closer to the core as the star ascends the RGB. The consolidated values (the mean from all sources) for stellar mass M ,seis , radius R ,seis , age t, surface gravity log g and density ρ are summarized in Table 1, constraining the corresponding parameters to a precision level of 9%, 4%, 27%, 1%, 15%, respectively, which are the most precise results for HD 76920 so far. The final uncertainties on these stellar parameters were recalculated by adding in quadrature the corresponding formal uncertainty for a given parameter to the standard deviation of the parameter estimates returned by all teams. Therefore, both random and systematic errors arising from the diversity of modeling methods from different teams were taken into account. The estimated log g from modeling is distinctly smaller than the spectroscopic results reported by Wittenmyer et al. (2017), but matches the one measured by Bergmann et al. (2021) (who used updated spectroscopy data) within 1σ, confirming the SED results (see Section 2.2). Figure 6 shows the locations of the sampling points by BESTP in the HR diagram, along with a series of evolutionary tracks with different initial masses generated by ASTEC. The sampling points are color-coded according to the normalized Note-Each mode is labeled according to its mode degree , radial order n (or radial order of acoustic-component np for non-radial modes) from the best-fitting model. ν and σν are the mode cyclic frequency and uncertainty. Modes identified simultaneously by at least two independent methods/sources are selected and used as modeling constraints. All these modes have a height-to-background ratio (SNR) larger than 3. The noise at each frequency is calculated as the average amplitude in the residual periodogram in a frequency range (20 µHz boxsize) that encloses the mode peak after the frequency is pre-whitened . likelihood, thus, redder samples have higher possibilities as the representation of the real star. According to the figure, HD 76920 is most likely approaching closely to the RGB luminosity bump, where the properties of mixed-modes are significantly impacted by the buoyancy glitch (Cunha et al. 2015(Cunha et al. , 2019Jiang et al. 2020a), a signature that can help us inspect the stellar mid-layer structures (Pinçon et al. 2020;Jiang et al. 2022). However, the short observation duration for HD 76920 limits the detection of mixed-modes from the TESS power spectrum, thus, such investigation of stellar interior structure with the help of mixed modes is not feasible with current TESS data. Nevertheless, by matching the theoretical oscillation frequencies calculated for the best-fitting model with the observed TESS power spectrum (Figure 7), we could confirm the mode degree and identify the mode order for the oscillation modes that were extracted by different groups ( Table 2). The theoretical modes used to identify observed modes are corrected for the surface effect (e.g., Houdek et al. 2017) that yields a systematic offset between the calculated and the observed oscillation frequencies. For theoretical dipolar mixed modes, correction for the surface effect was not performed due to the lack of enough observed dipolar frequencies, thus, we did not show them in Figure 7. CHARACTERIZATION OF HD 76920 b According to the prediction for the transit ephemeris made by Bergmann et al. (2021), if HD 76920 b were to transit, it would have done so during TESS Sector 9. However, no clear transit signal was found by Bergmann et al. (2021) or in our data. Thus, we could not measure the planetary orbit or eccentricity through the TESS photometric data. However, we can combine our newly computed asteroseismic stellar mass (M ,seis = 1.22 ± 0.11M ) with the orbital period, RV semi-amplitude and eccentricity values found by Bergmann et al. (2021) to compute updated values for the planet's semi-major axis and minimum mass. Using values from Bergmann et al. (2021), we find a = 1.165 ± 0.035 au and M p sin i = 3.57±0.22 M Jup . These updated parameters place the orbital pericenter of the planet at 0.142±0.005 au, revealing that the planet's orbit is sufficiently far from the star such that tidal decay is currently negligible until the (Table 2) frequencies are marked with red circles ( = 0), green diamonds ( = 1) and blue triangles ( = 2). star expands its radius by at least 50-100% (Villaver & Livio 2009) and subsequently engulfs the planet, in line with the findings of Bergmann et al. (2021). The time at which the star will engulf the planet will depend on when the star began ascending the RGB. During this phase of stellar evolution, the star's radius will expand to a distance approximately one order of magnitude greater than the planet's orbital pericenter, allowing the star to easily and quickly engulf the planet. For instance, further evolving the best-fitting model ( Figure 6) along the evolutionary track, we predict that the expanding star will have a radius one order of magnitude greater than the planet's orbital pericenter and engulf the planet in about 130 Myr. As an independent estimation of the planetary engulfment time, we also calculated models using the SSE prescription from Hurley et al. (2000). According to this prescription, a 1.22 M star with a metallicity of Z = 0.02 will begin the red giant phase at about 5.58 Gyr and leave it at 6.01 Gyr. During this interval, the planet will be engulfed. Asteroseismology has allowed us to estimate the current age of the star to be 5.2 ± 1.4 Gyr. This value clearly demonstrates that the star is on the verge of engulfing the planet. However, the uncertainty on the age is about triple the value of the duration of the red giant phase. Nevertheless, asteroseismology has also allowed us to constrain the stellar radius to a value of 8.68 ± 0.34 R . This value corresponds to an age from the SSE model of 5.939 ± 0.004 Gyr, meaning that the planet will be engulfed in less than about 150 Myr. However, this approximate upper bound is likely an overestimate given the fact that the star's metallicity is sub-Solar. Step overshoot e None None None None Diffusion Thoul et al. (1994) None Other stellar models produce similar results, just renormalized to other absolute ages within the measured asteroseismic uncertainty. For example, a 1.22 M star with a sub-Solar metallicity of Z = 0.01 will begin the red giant phase at about 4.72 Gyr and leave it at 5.08 Gyr. In this case, the asteroseismically measured radius corresponds to an age from the SSE model of 5.007 ± 0.004 Gyr. This value illustrates that the planet will be engulfed within about 70 Myr. This result is in agreement with that of Bergmann et al. (2021). CONCLUSIONS In this work, we have analyzed the TESS photometric data for HD 76920 to determine the star's fundamental parameters using asteroseismology, and to characterize the exoplanet system consisting of a planet with an extremely large orbital eccentricity. In total, 5 sectors of TESS light curves are used for the extraction of asteroseismic parameters for the host star, including 17 individual oscillation frequencies. Modeling by various pipelines that utilize the extracted asteroseismic parameters, classical spectroscopic observables, as well as luminosity from Gaia DR3 parallax, places strong constraints on the stellar parameters. Through the asteroseismic analysis, we obtain a value for the stellar mass of 1.22 ± 0.11M , a stellar radius of 8.68 ± 0.34 R and an age of 5.2 ± 1.4 Gyr, which provide the most precise estimations for HD 76920 to date. The stellar models reveal that the star is ascending the red giant branch and most likely approaching closely to the luminosity bump where the properties of mixed-modes are significantly impacted by Normalized likelihood Figure 6. Evolutionary tracks for a series of ASTEC models with different initial masses but same chemical abundance (X = 0.714 and Z = 0.0142, corresponding to [Fe/H] = -0.09) and mixing length parameter (α = 1.927) that are the optimization outputs of the BESTP pipeline. The initial masses of the models increase from 0.8 to 2.0 M with a step of 0.1 M . The evolutionary track indicated by orange is with an initial mass of 1.2 M that is closest to the adopted value (Table 1). Sampling points by BESTP are drawn in the diagram and also in the inset, with color-coded normalized likelihood values. The green star marks the location of our best-fitting model (M = 1.21M , X = 0.713 and Z = 0.0145) from BESTP. the buoyancy glitch. However, the current power spectrum of HD 76920 from TESS does not allow the extraction of a sufficient number of mixed modes as would be required for the investigation of the buoyancy glitch. The updated stellar parameters of the host star from our asteroseismic analysis have enabled improved estimations for the semi-major axis and mass of the planet as a = 1.165±0.035 au and M p sin i = 3.57±0.22 M Jup . With an orbital pericenter of 0.142 ± 0.005 au, we confirm that the planet is currently far away enough from the star to experience negligible tidal decay before being engulfed in the stellar envelope. However, we predict that this event will occur within about 100 Myr, depending on the stellar model used. HD 76920 will be observed in 5 more sectors by TESS in Cycle 5. The prolonged data will possibly enable the detection of mixed-modes, and thus, the investigation of stellar interior through these modes. Moreover, our asteroseismic analysis emphasizes the potential of TESS for characterizing exoplanet systems through the synergy between exoplanet research and asteroseismology. . Background-corrected PSD (dark grey) plotted in theéchelle diagram format that divides the spectrum into bins each ∆ν wide. The blue and red peaks are the = 0 and 2 modes, respectively, of the best-fitting model returned by BESTP ( Figure 6). The line width and amplitude of the theoretical modes are derived using the formulae introduced in Lund et al. (2017) and Ball et al. (2018). The corresponding observed frequencies and uncertainties (Table 2) are indicated by the horizontal spans, with additional = 1 modes in green. b Based on extrapolated relations of Torres et al. (2010). References-(1) Stassun et al. (2018b), (2) van Leeuwen (2007), (3) Houk & Cowley (1975), (4) Wittenmyer et al. (2017), (5) Bergmann et al.(2021), (6) Gaia Collaboration et al. (2021), (7) this work. Figure 1 . 1eleanor best-fit apertures (top) overlaid on the TESS target pixel files (TPFs) extracted per each sector. The TPFs are all scaled from 0 to 30000 e −1 s −1 . We use these apertures to extract the flux within eleanor, which are then corrected via the default eleanor corrected flux routine (bottom). Apertures and light curves are colored by TESS sector. Figure 2 . 2Spectral energy distribution of HD 76920. Red symbols represent the observed photometric measurements, where the horizontal bars represent the effective width of the passband. Blue symbols are the model fluxes from the best-fit Kurucz atmosphere model (black). oscillation codes (ADIPLS, GYRE; Christensen-Dalsgaard 2008b; Townsend & Teitler 2013), and optimization methods (Mier 2017; Kayhan et al. 2019; Wu & Li 2019; Yıldız et al. 2019; Jiang & Gizon 2021; Figure 3 . 3Power spectral density (PSD) of HD 76920 and corresponding global background model fit (cyan dotted curve). The background model consist of two Harvey-like profiles (green and blue dashed curves) and white noise (orange dashed line). The solid red curve depicts the summation of the background and a Gaussian fit (purple dashed) to the oscillation power excess envelope. The PSD is generated using 5 sectors of TESS data(Figure 1). The 10-min cadence sectors are rebinned to 30-min cadence, with simple average over three measurements. The original PSD is shown in gray and a heavily smoothed (Gaussian with an FWHM of ∆ν) version in black. The fit displayed here disregarded frequencies below ∼10 µHz (Section 3.1). Figure 4 . 4Background corrected power spectra of HD 76920 depicted in the power excess region. Top panel: power spectra of the Cycle 1 (Sectors 9 to 11) and Cycle 3 (Sectors 36 and 37) light curve. The Cycle 1 data shows a larger value of νmax, compared with the Cycle 3 data. Bottom panel: power spectra of the combined light curve that is used for the asteroseismic analysis. Figure 5 . 5Échelle diagram of the background-corrected PSD, folded on an large frequency separation of ∆ν = 5.62 µHz estimated by fitting the radial-mode frequencies (c.f. Section 3.2). Identified individual mode a The MESA/OPAL tables are based on the 2005 update of the OPAL EoS tables(Rogers & Nayfonov 2002). b The adopted methods for surface correction areKjeldsen et al. (2008) (KB) andBall & Gizon (2014) two-term correction c Solar composition given inGrevesse & Sauval (1998) (GS98)and Asplund et al. (2009) (AGSS09) are used for initial chemical composition. d The atmosphere choices used in MESA are introduced in Paxton et al. (2011), and ASTEC uses the Kurucz model (Kurucz 1991) for atmosphere. e The overshoot parameters fov and f0,ov are set as 0.006 and 0.003. The overshooting is applied at all convective boundaries. References-Ong: Mier (2017); Kayhan: Kayhan et al. (2019); Izmir: Yıldız et al. (2019); Zhang: Zhang et al. (2022); BESTP: Jiang & Gizon (2021) Figure 7 7project leading to this publication has received funding from the B-type Strategic Priority Program of the Chinese Academy of Sciences (Grant No. XDB41000000). ADF acknowledges the support from the National Science Table 1 . 1Stellar Parameters for HD 76920Parameter Value References Basic Properties TIC 302372658 1 Hipparcos ID 43803 2 TESS Mag. 6.87 1 Sp. Type K1 III 3 Spectroscopy T eff (K) 4698 ± 100 4 4664 ± 53 5 [Fe/H] (dex) −0.11 ± 0.10 4 −0.19 ± 0.06 5 log g (cgs) 2.94 ± 0.15 4 2.71 ± 0.04 5 SED & Gaia DR3 Parallax a π (mas) 5.4618 ± 0.0187 6 AV 0.20 ± 0.05 7 0.15 ± 0.07 7 F bol (erg s −1 cm −2 )(3.12 ± 0.11) × 10 −8 7 (3.08 ± 0.11) × 10 −8 7 R (R ) 8.64 ± 0.37 7 8.70 ± 0.25 7 L (L ) 32.54 ± 1.17 7 32.06 ± 1.16 7 M (M ) 1.68 ± 0.10 b 7 Asteroseismology ∆ν (µHz) 5.53 ± 0.15 7 νmax (µHz) 53.2 ± 2.3 7 M ,seis (M ) 1.22 ± 0.11 7 R ,seis (R ) 8.68 ± 0.34 7 ρ (gcc) 0.0026 ± 0.0004 7 log g (cgs) 2.648 ± 0.037 7 t (Gyr) 5.2 ± 1.4 7 Table 2 . 2Extracted Oscillation Frequencies from TESS light curve and Mode Identification based on the best-fitting model for HD 76920n(np) ν σν SNR (µHz) (µHz) 0 7 44.80 0.13 6.63 0 8 50.32 0.10 8.08 0 9 55.85 0.08 12.50 0 10 61.48 0.08 6.71 0 11 67.18 0.13 3.83 0 12 72.73 0.15 4.0 1 6 42.41 0.08 12.05 1 8 53.12 0.03 9.54 1 8 53.27 0.03 6.73 1 9 58.90 0.07 6.95 1 10 64.13 0.08 3.61 2 5 38.61 0.29 3.71 2 6 43.93 0.24 5.88 2 7 49.53 0.12 7.13 2 8 55.10 0.07 5.42 2 9 60.59 0.20 4.48 2 10 66.32 0.07 3.89 Table 3 . 3Modeling configurations from different sources. One entry is used where all five teams used the same input physics.Team Ong Kayhan Izmir Zhang BESTP Evolution code MESA (r12778) MESA (r12778) MESA (r15140) MESA (r10398) ASTEC Oscillation code GYRE ADIPLS ADIPLS ADIPLS ADIPLS EoS a MESA/OPAL MESA/OPAL MESA/OPAL MESA/OPAL OPAL Surface Correction b BG-2term KB KB None BG-2term Nuclear reactions NACRE (Angulo et al. 1999) High-T opacities OPAL (Iglesias & Rogers 1993, 1996) Low-T opacities Ferguson et al. (2005) Solar mixture c GS98 AGSS09 AGSS09 GS98 AGSS09 Atmosphere d Eddington gray simple photosphere simple photosphere Eddington gray Kurucz αMLT 1.83 2.175 1.828 2.0 1.7 -2.1 Overshoot Table 4 . 4Modeling results from different sources. log g (cgs) 2.650 ± 0.008 2.709 ± 0.002 2.630 ± 0.019 2.630 ± 0.001 2.620 ± 0.010Team Ong Kayhan Izmir Zhang BESTP M (M ) 1.31 ± 0.06 1.28 ± 0.16 1.15 ± 0.14 1.17 ± 0.01 1.20 ± 0.10 R (R ) 8.96 ± 0.14 8.29 ± 0.14 8.62 ± 0.39 8.67 ± 0.02 8.87 ± 0.26 t (Gyr) 4.1 ± 0.7 4.3 ± 0.5 5.8 ± 2.1 6.3 ± 0.06 5.6 ± 1.4 * NSF Graduate Research Fellow HD 76920 has also been observed in Sector 38 at 10-min cadence, but the data was not released at the time of this analysis. 2 For sectors 36 and 37, eleanor uses the TessCut tool(Brasseur et al. 2019). . 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{'abstract': 'The Transiting Exoplanet Survey Satellite (TESS) mission searches for new exoplanets. The observing strategy of TESS results in high-precision photometry of millions of stars across the sky, allowing for detailed asteroseismic studies of individual systems. In this work, we present a detailed asteroseismic analysis of the giant star HD 76920 hosting a highly eccentric giant planet (e = 0.878) with an orbital period of 415 days, using 5 sectors of TESS light curve that cover around 140 days of data. Solar-like oscillations in HD 76920 are detected around 52 µHz by TESS for the first time. By utilizing asteroseismic modeling that takes classical observational parameters and stellar oscillation frequencies as constraints, we determine improved measurements of the stellar mass (1.22 ± 0.11M ), radius (8.68 ± 0.34 R ), and age (5.2 ± 1.4 Gyr). With the updated parameters of the host star, we update the semi-major axis and mass of the planet as a = 1.165 ± 0.035 au and M p sin i = 3.57 ± 0.22 M Jup . With an orbital pericenter of 0.142 ± 0.005 au, we confirm that the planet is currently far away enough from the star to experience negligible tidal decay until being engulfed in the stellar envelope. We also confirm that this event will occur within about 100 Myr, depending on the stellar model used.', 'arxivid': '2302.01102', 'author': ['Chen Jiang jiangc@mps.mpg.de \nMax-Planck-Institut für Sonnensystemforschung\nJustus-von-Liebig-Weg 337077GöttingenGermany\n', "Tao Wu wutao@ynao.ac.cn \nYunnan Observatories\nChinese Academy of Sciences\n396 Yangfangwang\n\nGuandu District\n650216KunmingPeople's Republic of China\n\nKey Laboratory for the Structure and Evolution of Celestial Objects\nChinese Academy of Sciences\n396 Yangfangwang\n\nGuandu District\n650216KunmingPeople's Republic of China\n\nCenter for Astronomical Mega-Science\nChinese Academy of Sciences\n20A Datun Road100012Chaoyang District, BeijingPeople's Republic of China\n\nUniversity of Chinese Academy of Sciences\n100049BeijingPeople's Republic of China\n\nInstitute of Theoretical Physics\nShanxi University\n030006TaiyuanChina\n", 'Adina D Feinstein \nDepartment of Astronomy and Astrophysics\nUniversity of Chicago\n5640 S. Ellis Ave60637ChicagoILUSA\n', 'Keivan G Stassun \nDepartment of Physics and Astronomy\nVanderbilt University\n37235NashvilleTNUSA\n', "Sylvain N Breton \nDépartement d'Astrophysique/AIM\nCEA/IRFU\nCNRS/INSU\nUniv. Paris-Saclay & Univ. de Paris\n91191Gif-sur-YvetteFrance\n", 'Mia S Lundkvist \nDepartment of Physics and Astronomy\nStellar Astrophysics Centre (SAC)\nAarhus University\nNy Munkegade 120DK-8000Aarhus CDenmark\n', 'Przemys Law ', 'J Miko \nAstronomical Observatory\nUniversity of Warsaw\nAl. Ujazdowskie 400-478WarsawPoland\n\nAstronomical Institute\nUniversity of Wroc law\nMiko laja Kopernika 11, 51-622 Wroc lawPoland\n', 'Charlotte Gehan \nMax-Planck-Institut für Sonnensystemforschung\nJustus-von-Liebig-Weg 337077GöttingenGermany\n\nInstituto de Astrofísica e Ciências do Espaço\nUniversidade do Porto\nRua das Estrelas4150-762PortoPortugal\n', 'Tiago L Campante \nInstituto de Astrofísica e Ciências do Espaço\nUniversidade do Porto\nRua das Estrelas4150-762PortoPortugal\n\nDepartamento de Física e Astronomia\nFaculdade de Ciências da Universidade do Porto\nRua do Campo Alegre\ns/n4169-007PortoPortugal\n', 'Diego Bossini \nInstituto de Astrofísica e Ciências do Espaço\nUniversidade do Porto\nRua das Estrelas4150-762PortoPortugal\n', 'Stephen R Kane \nDepartment of Earth and Planetary Sciences\nUniversity of California\n92521RiversideCAUSA\n', 'Jia Mian ', 'Joel Ong \nDepartment of Astronomy\nYale University\nP.O. Box 20810106520-8101New HavenCTUSA\n', 'Mutlu Yıldız \nDepartment of Astronomy and Space Sciences\nScience Faculty\nEge University\n35100İzmirBornovaTürkiye\n', 'Cenk Kayhan \nDepartment of Astronomy and Space Sciences\nScience Faculty\nErciyes University\n38030KayseriMelikgaziTürkiye\n', 'Zeynep Ç Elik Orhan \nDepartment of Astronomy and Space Sciences\nScience Faculty\nEge University\n35100İzmirBornovaTürkiye\n', 'Sibelörtel ', "Xinyi Zhang \nDepartment of Astronomy and Space Sciences\nScience Faculty\nEge University\n35100İzmirBornovaTürkiye\n\nState Key Laboratory of Lunar and Planetary Sciences\nMacau University of Science and Technology\nMacauPeople's Republic of China\n", 'Margarida S Cunha \nInstituto de Astrofísica e Ciências do Espaço\nUniversidade do Porto\nRua das Estrelas4150-762PortoPortugal\n\nDepartamento de Física e Astronomia\nFaculdade de Ciências da Universidade do Porto\nRua do Campo Alegre\ns/n4169-007PortoPortugal\n', 'Bruno Lustosa De Moura \nDepartamento de Fisica\nUniversidade Federal do Rio Grande do Norte\n59072-970NatalBrazil\n\nGrande do Norte-IFRN\nInstituto Federal do Rio\nBrazil\n', 'Jie Yu \nMax-Planck-Institut für Sonnensystemforschung\nJustus-von-Liebig-Weg 337077GöttingenGermany\n', "Daniel Huber \nInstitute for Astronomy\nUniversity of Hawai'i\n2680 Woodlawn Drive96822HonoluluHIUSA\n", 'Ou (欧建文)Jian-Wen ', 'Robert A Wittenmyer \nSchool of Physics and Electromechanical Engineering\nGuangdong Province\nShaoguan University\n512005ShaoguanChina\n\nCentre for Astrophysics\nUniversity of Southern Queensland\nUSQ Toowoomba\n4350QLDAustralia\n', 'Laurent Gizon \nMax-Planck-Institut für Sonnensystemforschung\nJustus-von-Liebig-Weg 337077GöttingenGermany\n\nInstitut für Astrophysik\nGeorg-August-Universität Göttingen\nFriedrich-Hund-Platz 137077GöttingenGermany\n', 'William J Chaplin \nDepartment of Physics and Astronomy\nStellar Astrophysics Centre (SAC)\nAarhus University\nNy Munkegade 1208000Aarhus CDenmark\n\nSchool of Physics and Astronomy\nUniversity of Birmingham\nB15 2TTBirminghamUK\n', 'Chen Jiang ', '\nSchool of Physics\nSydney Institute for Astronomy (SIfA)\nUniversity of Sydney\n2006NSWAustralia\n', '\nCentre for Exoplanets and Habitability\nUniversity of Warwick\nCV4 7ALCoventryUK\n', '\nCentre for Space Domain Awareness\nUniversity of Warwick\nCV4 7ALCoventryUK\n', '\nDepartment of Physics\nUniversity of Warwick\nCV4 7ALCoventryUK\n', '\nINAF -Osservatorio Astrofisico di Catania\nvia S. Sofia 7895123CataniaItaly\n', '\nDepartment of Chemistry & Physics\nFGCU Boulevard S\nFlorida Gulf Coast University\n10501, 33965Fort MyersFLUSA\n', '\nSchool of Physics\nUniversity of New South Wales\n2052NSWAustralia\n', '\nInstituto de Astrofísica de Canarias (IAC)\nE-38205La Laguna, TenerifeSpain\n', '\nDepartamento de Astrofísica\nUniversidad de La Laguna (ULL)\nE-38206La Laguna, TenerifeSpain\n', '\nUniversité Paris-Saclay\nUniversité Paris Cité\nCEA\nCNRS\n91191Gif-sur-YvetteAIMFrance\n'], 'authoraffiliation': ['Max-Planck-Institut für Sonnensystemforschung\nJustus-von-Liebig-Weg 337077GöttingenGermany', 'Yunnan Observatories\nChinese Academy of Sciences\n396 Yangfangwang', "Guandu District\n650216KunmingPeople's Republic of China", 'Key Laboratory for the Structure and Evolution of Celestial Objects\nChinese Academy of Sciences\n396 Yangfangwang', "Guandu District\n650216KunmingPeople's Republic of China", "Center for Astronomical Mega-Science\nChinese Academy of Sciences\n20A Datun Road100012Chaoyang District, BeijingPeople's Republic of China", "University of Chinese Academy of Sciences\n100049BeijingPeople's Republic of China", 'Institute of Theoretical Physics\nShanxi University\n030006TaiyuanChina', 'Department of Astronomy and Astrophysics\nUniversity of Chicago\n5640 S. Ellis Ave60637ChicagoILUSA', 'Department of Physics and Astronomy\nVanderbilt University\n37235NashvilleTNUSA', "Département d'Astrophysique/AIM\nCEA/IRFU\nCNRS/INSU\nUniv. Paris-Saclay & Univ. de Paris\n91191Gif-sur-YvetteFrance", 'Department of Physics and Astronomy\nStellar Astrophysics Centre (SAC)\nAarhus University\nNy Munkegade 120DK-8000Aarhus CDenmark', 'Astronomical Observatory\nUniversity of Warsaw\nAl. Ujazdowskie 400-478WarsawPoland', 'Astronomical Institute\nUniversity of Wroc law\nMiko laja Kopernika 11, 51-622 Wroc lawPoland', 'Max-Planck-Institut für Sonnensystemforschung\nJustus-von-Liebig-Weg 337077GöttingenGermany', 'Instituto de Astrofísica e Ciências do Espaço\nUniversidade do Porto\nRua das Estrelas4150-762PortoPortugal', 'Instituto de Astrofísica e Ciências do Espaço\nUniversidade do Porto\nRua das Estrelas4150-762PortoPortugal', 'Departamento de Física e Astronomia\nFaculdade de Ciências da Universidade do Porto\nRua do Campo Alegre\ns/n4169-007PortoPortugal', 'Instituto de Astrofísica e Ciências do Espaço\nUniversidade do Porto\nRua das Estrelas4150-762PortoPortugal', 'Department of Earth and Planetary Sciences\nUniversity of California\n92521RiversideCAUSA', 'Department of Astronomy\nYale University\nP.O. Box 20810106520-8101New HavenCTUSA', 'Department of Astronomy and Space Sciences\nScience Faculty\nEge University\n35100İzmirBornovaTürkiye', 'Department of Astronomy and Space Sciences\nScience Faculty\nErciyes University\n38030KayseriMelikgaziTürkiye', 'Department of Astronomy and Space Sciences\nScience Faculty\nEge University\n35100İzmirBornovaTürkiye', 'Department of Astronomy and Space Sciences\nScience Faculty\nEge University\n35100İzmirBornovaTürkiye', "State Key Laboratory of Lunar and Planetary Sciences\nMacau University of Science and Technology\nMacauPeople's Republic of China", 'Instituto de Astrofísica e Ciências do Espaço\nUniversidade do Porto\nRua das Estrelas4150-762PortoPortugal', 'Departamento de Física e Astronomia\nFaculdade de Ciências da Universidade do Porto\nRua do Campo Alegre\ns/n4169-007PortoPortugal', 'Departamento de Fisica\nUniversidade Federal do Rio Grande do Norte\n59072-970NatalBrazil', 'Grande do Norte-IFRN\nInstituto Federal do Rio\nBrazil', 'Max-Planck-Institut für Sonnensystemforschung\nJustus-von-Liebig-Weg 337077GöttingenGermany', "Institute for Astronomy\nUniversity of Hawai'i\n2680 Woodlawn Drive96822HonoluluHIUSA", 'School of Physics and Electromechanical Engineering\nGuangdong Province\nShaoguan University\n512005ShaoguanChina', 'Centre for Astrophysics\nUniversity of Southern Queensland\nUSQ Toowoomba\n4350QLDAustralia', 'Max-Planck-Institut für Sonnensystemforschung\nJustus-von-Liebig-Weg 337077GöttingenGermany', 'Institut für Astrophysik\nGeorg-August-Universität Göttingen\nFriedrich-Hund-Platz 137077GöttingenGermany', 'Department of Physics and Astronomy\nStellar Astrophysics Centre (SAC)\nAarhus University\nNy Munkegade 1208000Aarhus CDenmark', 'School of Physics and Astronomy\nUniversity of Birmingham\nB15 2TTBirminghamUK', 'School of Physics\nSydney Institute for Astronomy (SIfA)\nUniversity of Sydney\n2006NSWAustralia', 'Centre for Exoplanets and Habitability\nUniversity of Warwick\nCV4 7ALCoventryUK', 'Centre for Space Domain Awareness\nUniversity of Warwick\nCV4 7ALCoventryUK', 'Department of Physics\nUniversity of Warwick\nCV4 7ALCoventryUK', 'INAF -Osservatorio Astrofisico di Catania\nvia S. Sofia 7895123CataniaItaly', 'Department of Chemistry & Physics\nFGCU Boulevard S\nFlorida Gulf Coast University\n10501, 33965Fort MyersFLUSA', 'School of Physics\nUniversity of New South Wales\n2052NSWAustralia', 'Instituto de Astrofísica de Canarias (IAC)\nE-38205La Laguna, TenerifeSpain', 'Departamento de Astrofísica\nUniversidad de La Laguna (ULL)\nE-38206La Laguna, TenerifeSpain', 'Université Paris-Saclay\nUniversité Paris Cité\nCEA\nCNRS\n91191Gif-sur-YvetteAIMFrance'], 'corpusid': 256503778, 'doi': '10.3847/1538-4357/acb8ac', 'github_urls': [], 'n_tokens_mistral': 31927, 'n_tokens_neox': 23966, 'n_words': 10430, 'pdfsha': '27a74dfd43d888a95e82db196abd974d3309813c', 'pdfurls': ['https://export.arxiv.org/pdf/2302.01102v2.pdf'], 'title': ['TESS Asteroseismic Analysis of HD 76920: The Giant Star Hosting An Extremely Eccentric Exoplanet', 'TESS Asteroseismic Analysis of HD 76920: The Giant Star Hosting An Extremely Eccentric Exoplanet'], 'venue': []}

arxiv

A simplified multiphase multiscale model for tissue growth 25 Jun 2018 June 26, 2018 E C Holden Centre for Mathematical Medicine and Biology School of Mathematical Sciences University of Nottingham NG7 2RDUniversity Park, NottinghamUK B S Brook Centre for Mathematical Medicine and Biology School of Mathematical Sciences University of Nottingham NG7 2RDUniversity Park, NottinghamUK S J Chapman Mathematical Institute University of Oxford Radcliffe Observatory Quarter Woodstock RoadOX2 6GGOxfordUK R D O&apos;dea Centre for Mathematical Medicine and Biology School of Mathematical Sciences University of Nottingham NG7 2RDUniversity Park, NottinghamUK A simplified multiphase multiscale model for tissue growth 25 Jun 2018 June 26, 201810.1017/S1446181118000044) In this paper, we derive an effective macroscale description suitable to describe the growth of biological tissue within a porous tissue-engineering scaffold. As in our recent work (Holden et al. "A multiphase multiscale model for nutrient limited tissue growth", The ANZIAM Journal, 2018, doi:10.1017/S1446181118000044) the underlying tissue dynamics is described as a multiphase mixture, thereby naturally accommodating features such as interstitial growth and active cell motion. Via a linearisation of the underlying multiphase model (whose nonlinearity poses significant challenge for such analyses), we obtain, by means of multiple-scales homogenisation, a simplified macroscale model that nevertheless retains explicit dependence on both the microscale scaffold structure and the tissue dynamics. The model we obtain comprises Darcy flow, and differential equations for the volume fraction of cells within the scaffold and the concentration of nutrient, required for growth. These are coupled to underlying Stokes-type cell problems that provide permeability tensors to parameterise the macroscale description. In Holden et al., the cell problems retain macroscale dependence, posing significant computational challenges; here, we obtain a decoupled system whereby the quasi-steady cell-problems may be solved separately from the macroscale description, thereby greatly reducing the complexity associated with fully-coupled multiscale descriptions. Moreover, we indicate how the formulation is influenced by a set of alternative microscale boundary conditions. Introduction Tissue growth is a complex and inherently multiscale phenomenon, whose unified description requires the integration of insight obtained at one scale with observations at another. For example growth processes (or disease manifestation) at the organ scale are driven by microscopic events at the (sub-)cellular scale that themselves are influenced by macroscopic dynamics. Such complexity necessitates investigation by theoretical means, to provide supporting insight not available by experimental investigation alone. However, the complex multiscale interactions (from intra-and inter-cell signalling pathways, cell biomechanics and migration to tissue-level patterning and mechanics) leads inevitably to formulations that are analytically and computationally intractable, or are otherwise highly idealised. For this reason, a significant area of research is dedicated to developing various mathematical and computational techniques that enable efficient coupling between dynamics occurring on multiple scales (see, e.g. [1,16,23] and references therein). This article is concerned with the method of multiple-scale asymptotic homogenisation that provides a coarse-scale description of the tissue dynamics, while still incorporating aspects of the microscale physics. Such methods have long theoretical history (e.g. [2,13]), and have been widely used to describe descriptions of porous and poroelastic materials in applied/industrial settings, such as the study of soil and reservoirs [26,27]; see [8,24] for reviews. The key feature of this approach is to derive suitable macroscale equations from an underlying microscale description, rather than stating them ab initio. Coupling to the microscale physics is effected by suitable problems defined on a prototypical 'unit cell'. These so-called 'cell problems', determine microscale behaviour that is subsequently employed to specify effective coefficients in the macroscale description. More contemporary studies have employed these methods in a biological setting. Of particular relevance to the current work is a series of studies that seek to describe growing tissues. In [22] a simple solid-accretion-based model of nutrient-limited tissue growth within a porous scaffold, was considered. A macroscale description of growth and transport was obtained using a multiple-scales technique, to accommodate explicit dependence on microscale dynamics and structure. A similar analysis by Penta et al. [25] described accretion in a poroelastic setting. To permit analysis, the authors of [22,25] (and other similar studies) exploit asymptotic restrictions on the underlying model, considering slow (quasi-static) growth and linearised deformation. In Collis et al. [7], such assumptions are relaxed to consider a macroscale representation of finite volumetric nutrient-limited growth of a hyperelastic solid, employing the Arbitrary-Lagrangian-Eulerian approach [4]. Collis et al. [5,6] sought to address the highly idealised representation of growth in the aforementioned studies, by employing a multiphase description for the underlying tissue dynamics that naturally accommodates the complexity associated with tissue growth dynamics, such as interstitial growth and active cell motion. However, the simplifying adoption of a large-drag limit employed therein constrains growth to a thin boundary layer and the resulting model is effectively equivalent to accretion. This deficiency was addressed in Holden et al. [10] to obtain an effective macroscale description of nutrient limited tissue growth on an artificial scaffold, in which the assumption of large interphase drag is relaxed so that active cell motion is permitted, caused by the cells' tendency to aggregate or repel. Analytical progress was effected by a linearisation that ameliorates problems associated with complex mass-transfer considered in the multiphase model (see [7] for a discussion), and allows one to obtain a more tractable description that permits coupling between micro-and macro-scale processes. The derived model comprises a Darcy flow, a partial differential equation for the volume fraction of cells within the scaffold, and an advection-reaction equation for the nutrient concentration, coupled to the underlying microscale dynamics via suitable cell problems. Importantly, and in contrast to other similar studies, these unit cell problems are themselves parameterised by the macroscale dynamics, so that the micro-and macro-scale descriptions are fully coupled. Here, we show how through a modification to the analysis of [10], the more standard de-coupling between microscale and macroscale can be effected, leading to a system whereby the quasi-steady cell-problems may be solved separately from the macroscale description, thereby greatly simplifying the computational difficulty associated with fully-coupled multiscale descriptions. In addition, we indicate how this formulation is modified under alternative boundary conditions to those employed in [10]. Model formulation We consider a model of broad relevance to tissue engineering applications where tissue growth occurs on a structured periodic scaffold, such as can be achieved through the use of 3D printing [12,29]. We emphasise that the model set-up considered herein is identical to that of [10] (itself following closely [15], that builds on the general theory of multiphase porous flow developed in [3,9,17]), and so only a very brief summary is included here. The microscale domain is denoted Ω, with boundary ∂Ω, and has characteristic lengthscale l * . This domain comprises scaffold, Ω S , tissue Ω T and interstitial fluid Ω I . The scaffold boundary is denoted by ∂Ω S and the tissue-interstitial boundary by Γ; see Figure 1 for a schematic diagram. The macroscopic lengthscale (associated with the full extent of the scaffold) is denoted L. The lengthscales in question are well-separated such that ε = l L ≪ 1. We model the porous scaffold material as a rigid solid, and the tissue as a two phase mixture of cells and interstitial fluid which covers the scaffold, whilst the interstitial space contains only fluid. Henceforth, we refer to the interstitial fluid as water, for concision. Both cells and water are modelled as viscous fluids, described by a Stokes flow. Increase in the cell volume fraction of the mixture depends on the concentration of a generic diffusible nutrient, as well as the availability of water. Tissue growth is represented by movement of the boundary Γ, occuring as a consequence of nutrient limited phase transition or cell aggregation/repulsion. Model equations The equations governing the multiphase mixture in the tissue domain are as follows: θ n + θ w = 1,(1)ρ i ∂θ i ∂t + ∇ · (θ i v i ) = S i ; i = n, w,(2)∇ · (θ n v n + θ w v w ) = 1 ρ n − 1 ρ w S n ,(3)∇ · (θ i σ i ) + f ij = 0; i = n, w,(4) wherein the subscript 'n' or 'w' denotes variables associated with the cell or water phases, respectively. The volume fraction of the i th phase is denoted θ i , with associated density ρ i , velocity v i , mass source S i (obeying S n = −S w ). The stress tensor of phase i is denoted σ I and the interphase force associated with the action of phase j on phase i by f ij , these being defined as follows: σ i = −p i I + µ i ∇v i + (∇v i ) ⊤ − 2 3 ∇ · v i I ,(5)f ij = p∇θ i + βθ i θ j (v j − v I ) .(6) In (5) p i denotes the pressure of the i th phase, whose constant viscosity is µ i . Consistent with the two-phase representation we adopt, in (6) we have assumed that the only interphase interaction that exists is passive viscous drag, with coefficient β. The common 'mixture pressure' is denoted by p, which is related to the individual phase pressures by p w = p and p n = p + φ n , where φ n represents an additional intraphase pressure, generated by cell-cell interactions, that can lead to active cell motion. This is specified as: φ n = θ n −ν + κθ n 1 − θ n ,(7) in which the first term represents aggregation with strength ν > 0, while the second term curtails this, with strength κ > 0. Note in particular the singularity that occurs as θ n → 1, representing high repulsion occurring when all available space is occupied by cells. Lastly, we note that it is at times notationally convenient to refer to weighted mixture variables, represented by the subscript T ; for example, we define: v T = θ n v n + θ w v w . We reiterate that the above described model is identical to that presented in [10] and follows closely that developed in [15]. As noted above, in the interstitium (Ω I ), a corresponding viscous flow model is adopted; however, since there are no cells (θ n = 0, θ w = 1) equations (3)-(6) reduce to a standard incompressible Stokes flow. Variables in this domain are denoted with a subscript I. A generic nutrient of concentration c, on which mitosis depends, is advected by the flows, diffuses (with diffusivity D I ), and in the tissue domain Ω T is taken up by the cell phase at a rate Λ according to: ∂c I ∂t + ∇ · (c I v I ) = ∇ · (D I ∇c I ) , in Ω I ,(8)∂c T ∂t + ∇ · (c T v T ) = ∇ · (D T ∇c T ) − Λ, in Ω T .(9) Equation (9) arises from the sum of the nutrient transport equations in each phase, under the assumption that due to rapid equilibriation across cell membranes, the concentration in the cell and water phases is equal (as employed in [14]). We assume that the diffusivity in the interstitium, D I is constant, but specify D T = D T (θ n ), and Λ = Λ(θ n , c). Boundary conditions At the scaffold-tissue boundary, we impose no-slip and no-penetration conditions, so that: v i = 0, ∇c · n S = 0 on Ω S ,(10) wherein n S denotes the outward normal to the scaffold surface. The tissue and interstitial domains are separated by the free interface Γ, whose evolution is given by ∂F ∂t + ∇F · v Γ = 0,(11) where v Γ is the boundary velocity and F denotes the position of the moving interface through the level set equation F (x, t) = 0. On this interface, the dynamics are coupled via the following flux and stress continuity conditions: ρ n θ n (v n − v Γ ) · n = 0, (12) ρ w θ w (v w − v Γ ) · n = ρ w θ w (v I − v Γ ) · n,(13)[c i (v i − v Γ ) · n − D i ∇c i · n] + − = 0 (14) [v i · t] + − = 0, [σ i · n] + − = 0, [c i ] + − = 0,(15) where i = T, I, n is the outward normal to Γ and [ ] + − denotes the jump across the interface. We remark that our model describes a complex free-boundary problem in which the interface position Γ is not known, and should be determined as part of the solution. However, in the multiscale analysis that follows, the boundary velocity remains undetermined. In order to close the model, we are therefore required to specify constitutively this motion; this issue is considered in more detail in [10]. Nondimensionalisation and linearisation We non-dimensionalise our model equations by using the following scalings x = lx, v i = Vv i , c i = Cĉ i , p = µ n V lp , t = l Vt ,(16)S n = ρ n V lŜ n , β = µ n l 2β , D i V l = 1 Pe i , Λ = V lΛ(17) in which circumflexes denote dimensionless variables, i = n, w, T, I, and V and C are a characteristic microscale velocity and nutrient concentration. Henceforth, we drop the circumflex notation for simplicity. We reduce the degree of nonlinearity of the microscale model to enable a more straightforward multiscale analysis by linearising the equations about a uniform steady state, across Ω T , as follows: θ n = θ * n + δθ n,1 + · · · ,(18) with corresponding expansions for the other model variables, and where 0 < δ ≪ 1 and asterisks denote steady-state values. For concision we do not state the linearised model here (the reader is referred to equations (2.27)-(2.32) in [10]) but we highlight in particular that the steady state volume fraction, nutrient concentration and velocity are defined by S n (θ * n , θ * w , c * ) = 0, Λ(θ * n , c * ) = 0,(19) and v * i = 0. Moreover, the source, uptake and intraphase interaction terms that appear in the following are the first order linear corrections, defined by: S n,1 = ∂S n ∂θ n (θ * n , c * ) θ n,1 + ∂S n ∂c (θ * n , c * ) c T ,1 , φ n,1 = 1 θ * n ∂ (θ n φ n ) ∂θ n (θ * n ) θ n,1 ,(20)Λ 1 = ∂Λ ∂θ n (θ * n , c * ) θ n,1 + ∂Λ ∂c (θ * n , c * ) c T ,1 .(21) Multiple scales analysis We now work with the linearised version of the model described in §2 and, for the sake of clarity, drop the associated subscripts. To derive a suitable macroscale description incorporating the microscale growth, dynamics and structure, we follow (e.g.) [5,22,28] in using the method of multiple scales. Correspondingly we rescale such that the timescale under consideration is that of macroscale advection and the pressure scaling results in the appropriate leading order problem: t = εt, p = 1 εp(22) in which tildes denote the rescaled variables. We drop the tilde notation for convenience as we work exclusively with the rescaled variables in subsequent sections. This choice of time rescaling simplifies the analysis by resulting in a quasi-steady problem at leading order. Next we introduce a macroscale coordinate X where X = εx (x being the microscale coordinate) and expand in multiple-scales form as follows: ψ(x, X, t; ε) = ψ (0) (x, X, t) + εψ (1) (x, X, t) + . . .(23)∇ = ∇ x + ε∇ X , ∇ 2 = ∇ 2 x + 2ε∇ x · ∇ X + ε 2 ∇ 2 X .(24) Moreover, in addition to the boundary conditions (10), (12)-(15) we require that ψ (i) for i = 0, 1, . . . are periodic in x. We now analyse the equations at each order in ε, with the aim of obtaining a description of the macroscale growth and transport Microscale governing equations at each order in ε At O(1), the equations and boundary conditions in the tissue domain Ω T are as follows: θ * n ∇ x · v (0) n = S (0) n ,(25)θ * w ∇ x · v (0) w = −ρS (0) n ,(26)θ (0) n + θ (0) w = 0,(27)∇ x · v (0) T = (1 −ρ) S (0) n ,(28)θ * n ∇ x p (0) w + φ (0) n = 0,(29)θ * w ∇ x p (0) w = 0,(30)∇ x · c * v (0) T = 1 P e T ∇ 2 x c (0) T − Λ (0) .(31) In the interstitial domain, Ω I : ∇ x · v (0) I = 0,(32)∇ x p (0) I = 0,(33)∇ x · c * v (0) I = 1 P e I ∇ 2 x c (0) I .(34) On the interface, Γ, the boundary conditions are: θ * n v (0) n − v (0) Γ · n = 0,(35)θ * w v (0) w − v (0) Γ · n = v (0) I − v (0) Γ · n,(36)v (0) T · t = v (0) I · t,(37)− p (0) w + θ * n φ (0) n I = −p (0) I I,(38)∇ x F * · v (0) Γ = 0,(39)c * v (0) T − v (0) Γ · n − 1 P e T ∇ x c (0) T · n = c * v (0) I − v (0) Γ · n − 1 P e I ∇ x c (0) I · n,(40)c (0) T = c (0) I ,(41) where v (0) T = θ * n v (0) n + θ * w v (0) w .(42) On the scaffold surface, ∂Ω S , we impose: v (0) n = v (0) w = 0, ∇ x c (0) T · n S = 0.(43) Lastly, we note that, in view of (20) and (21), the phase transfer, intraphase pressure and nutrient uptake functions depend only on θ (0) n , c(0) T (and the relevant steady states), so that S (0) n = S n,1 θ (0) n , c (0) T , φ (0) n = φ n,1 θ (0) n , Λ (0) = Λ 1 θ (0) n , c (0) T .(44) Equation (39) tells us that ∇ x F * = 0 or v (0) Γ · n = 0, but the latter holds most generally, so we take the boundary to be stationary at this order and, for consistency, we rescale S (0) n and Λ (0) to O(ε). Following the arguments in [6,10,22] we find that pressures, nutrient concentrations and cell volume fraction are independent of the microscale variable x, i.e. p (0) (X, t) = p (0) I (X, t) = p (0) w (X, t) + θ * n φ (0) n (X, t),(45)c (0) (X, t) = c (0) T (X, t) = c (0) I (X, t),(46)θ (0) n = θ (0) n (X, t).(47) At O(ε) the governing equations in Ω T are: ∂θ (0) n ∂t + θ * n ∇ x · v (1) n + ∇ X · v (0) n = S (0) n ,(48)∇ x · v (1) T + ∇ X · v (0) T = (1 −ρ) S (0) n ,(49)θ * n ∇ x p (1) w + φ (1) n + ∇ X p (0) w + φ (0) n + βθ * w v (0) n − v (0) w − ∇ 2 x v (0) n = 0,(50)θ * w ∇ x p (1) w + ∇ X p (0) w + βθ * n v (0) w − v (0) n − µ∇ 2 x v (0) w = 0,(51)∂c (0) ∂t + c * ∇ x · v (1) T + ∇ X · v (0) T = 1 P eT ∇ 2 x c (1) T − Λ (0) ,(52) where v (1) T = θ * n v (1) n + θ * w v(1) w . In Ω I : ∇ x · v (1) I + ∇ X · v (0) I = 0,(53)∇ x p (1) I + ∇ X p (0) − µ∇ 2 x v (0) I = 0,(54)∂c (0) ∂t + c * ∇ x · v (1) I + ∇ X · v (0) I = 1 P e I ∇ 2 x c (1) I .(55) On the tissue-intertitium interface, Γ, we obtain: θ * n v (1) n − v (1) Γ · n = 0,(56)θ * w v (1) w − v (1) Γ · n = v (1) I − v (1) Γ · n,(57)v (1) T · t = v (1) I · t,(58)− p (1) w + θ * n φ (1) n I + θ * n ∇ x v (0) n + ∇ x v (0) n T + µθ * w ∇ x v (0) w + ∇ x v (0) w T = −p (1) I I + µ ∇ x v (0) I + ∇ x v (0) I T ,(59)∂F (0) ∂t + ∇ x F * · v (1) Γ = 0,(60)c * v (1) T − v (1) Γ · n − 1 P e T ∇ x c (1) T + ∇ X c (0) · n = c * v (1) I − v (1) Γ · n − 1 P e I ∇ x c (1) I + ∇ X c (0) · n, (61) c (1) T = c (1) I .(62) Finally, on ∂Ω S we apply: v (1) n = v (1) w = 0, ∇ x c (1) T · n + ∇ X c (0) T · n S = 0.(63) Macroscale description In this subsection, we seek a macroscale representation of the flow, growth and transport described by the above equations. To effect this, we require a method of averaging variables across the various domains of the periodic cell. We therefore define the following integral average for some variable, g, over domain Ω I by g = 1 |Ω| ΩI g dV,(64) where Ω = Ω I ∪ Ω T ∪ Ω S . Velocity and pressure ansatz To determine the flow in Ω T and Ω I specified by the system of equations given in §3.1, we follow, e.g., [6,8,18,22,28] in exploiting the linearity of the momentum equations (50), (51) and (54) by taking an appropriate form for the macroscale velocities and microscale pressures to be given by the following ansatz: v (0) i = −K i ∇ X p (0) and p (1) i = −a i · ∇ X p (0) −p i ,(65) where p (0) is the overall macroscale pressure, K i are tensors describing the permeability, a I are first order tensors imparting microscale pressure variation, andp i are the mean (microscale-invariant) values of the first order pressures in Ω I . In the linearised model of [10], this choice of ansatz results in unit cell problem that are parameterised by the macroscale pressure and cell volume fraction (through φ (0) ) so that the micro-and macro-scale descriptions are fully-coupled (see equations (3.12)-(3.14) in that paper). This provides a significant challenge from a computational point of view. Here, we seek to remove this complexity; since both the macroscale pressure and active cell behaviour terms appear linearly in the momentum equations (50), (51) and (54), a more appropriate for the macroscale velocities and microscale pressures takes the following form: v (0) i = −K i ∇ X p (0) w − M i ∇ X φ (0) n ,(66)p (1) i = −a i · ∇ X p (0) w − b i · ∇ X φ (0) n −p i .(67) Note that rather than an ansatz in terms the overall macroscale pressure p (0) , in equations (66) n . In interstitial domain, the original ansatz (65) remains suitable. Microscale cell problems Substituting (66), (67) into the conservation of mass equations (25) and (26), and the momentum equations (50) and (51), we obtain the following modified Stokes-type cell problems in Ω T ∇ x · K T n = 0, ∇ x · M T n = 0,(68)∇ x · K T w = 0, ∇ x · M T w = 0,(69)∇ x a T n − I − ∇ 2 x K n − βθ * w (K w − K n ) = 0,(70)∇ x a T w − I − µ∇ 2 x K w − βθ * n (K n − K w ) = 0,(71)∇ x b T n − I − ∇ 2 x M n − βθ * w (M w − M n ) = 0,(72)∇ x b T w − µ∇ 2 x M w − βθ * n (M n − M w ) = 0.(73) Similarly, in Ω I , standard Stokes problems are obtained, as follows: ∇ x · K T I = 0,(74)∇ x a T I − I − µ∇ 2 x K I = 0.(75) These cell problems are coupled together through the conditions (35)-(37) and (59) specified on the interface, Γ, which supply K T I n = 0, K T n n = 0, K T w n = 0, M T n n = 0, M T w n = 0,(76) − a T ⊗ n + ∇K T + (∇K T ) T n = −a I ⊗ n + µ ∇K I + (∇K I ) T n, − b T ⊗ n + ∇M T + (∇M T ) T n = θ * n −a I ⊗ n + µ ∇K I + (∇K I ) T n ,(77) in which (45) has been employed to replace p (0) , and where K T = θ * n K n + µθ * w K w , a T = θ * n a n + θ * w a w ,(79)M T = θ * n M n + µθ * w M w , b T = θ * n b n + θ * w b w .(80) Lastly, on ∂Ω S , (63) provides K n = 0, K w = 0, M n = 0, M w = 0.(81) For uniqueness in the above cell problems, we use a standard approach see, e.g. [19,22,25,28]) and impose that in the relevant domain a i = 0, b i = 0.(82) We note that, while a standard Stokes-type cell problem is obtained in Ω I , the multiphase dynamics in Ω T leads to signifcantly increased complexity. In particular, we obtain a set of coupled modified Stokes problems, determining the permeability tensors K i , M i and extra pressures a i , b i for each phase, which are further coupled to the flow in Ω I via stress and velocity continuity boundary conditions. Furthermore, we highlight that whilst the number of cell problems has increased as a result of the change in ansatz from that employed in [10], we find that the permeability tensors are no longer dependent on macroscale pressures. The system we obtain therefore represents a significant simplification, taking the more familiar de-coupled form, whereby the quasi-steady cell problems can be solved separately from the macroscale description, that we obtain below. Averaging The macroscale flow is obtained by averaging (67) via the definition (64) to obtain v (0) i = − K i ∇ X p (0) w − M i ∇ X φ (0) n ,(83) wherein p w (0) and φ n , and equations (7), (20)) are obtained from the following system, derived from the average (exploiting the divergence theorem) of equations (48), (49), (52) and (55): ∂ ∂t θ (0) n T + θ * n ∇ X · v (0) n T + v (1) Γ · n Γ = S (0) n T ,(84)∇ X · K ∇ X p (0) w +M ∇ X φ (0) n = − (1 −ρ) S (0) n T ,(85)Φ T ∪I ∂c (0) ∂t + c * (1 −ρ) S (0) n T = − Λ (0) T .(86) Equation (86) arises from the sum of the averaged form of (52) and (55) and the tensorsK andM are given byK = θ * n K n + θ * w K w T + K I I ,M = θ * n M n + θ * w M w T + θ * n K I I ,(87) where the individual permeability tensors K i and M i are determined from the set of coupled Stokes problems (68)-(78). We remark that while the modification to the unit cell problems outlined above is significant, the impact of our modification to the approach of [10] on the macroscale description is less significant, being restricted to the redefinition of the relevant permeability tensors, and the associated velocities and pressures (in particular in the the explicit appearance of ∇ X φ (0) n terms associated with active cell motion). The governing system itself is of identical structure, and comprises a macroscale Darcy flow PDE, coupled to reaction equations describing tissue component volume fractions and nutrient concentration. Lastly, we note that as is common in analyses of this type, the macroscale model we obtain is not closed: we are required to specify constitutively the O(ε) boundary velocity v (1) Γ · n (cf. [6,11]). This is explored in [10] by means of detailed investigation of the travelling wave properties of the microscale multiphase model, but we do not pursue this here. Lastly, we note in passing that in the limit case of inviscid water (that employed in [10] for illustrative numerical simulations), the overall pressure p (0) is zero and consequently so is p (0) w + θ * n φ (0) n . This means that the new ansatz (66), (67) can be rewritten as v (0) i = −K i + 1 θ * n M i ∇ X p (0) w , p (1) i = −a i + 1 θ * n b i ∇ X p (0) w −p i ,(88) which is equivalent to the form used in [10] (see equation (3.17) therein) where the terms in square brackets are given by single tensors. Alternative boundary conditions 4.1 Cell motion on the scaffold surface In the model described above, we impose no-slip and no-penetration conditions on the scaffold boundary Ω S . While these are a sensible choice, reflecting the solid nature of the scaffold material, in some cases, a less restrictive choice may be of interest. For example, as well as the active motion embodied by the intraphase pressure φ n , cells may exhibit significant haptotactic motion on the scaffold surface itself. This is especially pertinent to the tissue engineering application under study, in which scaffolds may be produced to include substrate-bound chemoattractants thereby promoting cell ingress (see, e.g., [20,21,30] and references therein). We do not consider a haptotactic model here, but consider the following simple alternative choice of boundary condition permitting cell motion v n = b ∂v n ∂n(89) where b is a constant of proportionality and ∂/∂n denotes the normal derivative. We retain the no-penetration condition v i · n = 0 on ∂Ω S since the scaffold remains solid. The effective macroscale equations remain the same in each case and the only change to the Stokes problem is in the tissue-scaffold boundary conditions. The above equations give K n = b ∂K n ∂n , M n = b ∂M n ∂n , K T n n = 0, M T n n = 0(90) as the set of alternative boundary conditions to be applied on K n and M n at ∂Ω S in the Stokes problem. (Note that the slip condition is of similar form to that obtained by Irons et al. [11] in a similar cell problem, for a porous medium growth model.) Nutrient flux For completeness, we also indicate the influence of alternative concentration flux boundary conditions applied on the tissue-interstitium interface Γ. In the model developed above, we impose continuity of flux and concentration. As discussed in [28] two further options are suggested, which we now consider: Option 1 -Membrane law The flux of nutrient concentration across the boundary is proportional to the concentration jump. This widely-used approach demands: (c I v I − D I ∇c I ) · n = (c T v T − D T ∇c T ) · n = r (c T − c I ) ,(91) where r is a constant reflecting the permeability of the tissue boundary to nutrient flux. Option 2 -Concentration jump due to species solubility Alternatively, a concentration jump may be permitted, as a consequence of reduced solvability in the tissue compared to the interstitium (cf. Henry's law for gases in which the concentration c and partial pressure P of a gas in solution are related through the c = γP , where γ denotes the solvability): αc I = c T ,(92) where we assume for simplicity that α is a constant, although in a more general formulation it may be suitable to specify α = α(θ n ). In the following, we investigate the choice of boundary condition, and scaling of associated constant, on the effective macroscale description. Note that the choice of condition has no direct impact on the Stokes problem on the periodic cell. Option 1 -Membrane law Firstly we linearise the boundary condition (91); assuming that at steady state the nutrient concentration is equal and uniform across both domains, Ω I and Ω T we obtain: c * v I − v Γ − 1 P e I ∇c I · n = c * v T − v Γ − 1 P e T ∇c T · n = r (c T − c I ) .(93) In the subsequent multiple scales analysis we consider two further scaling subcases on the membrane permeability; namely r = O(1) or r = εr, withr = O(1). At leading order the boundary condition reads in each case: − 1 P e I ∇ x c (0) I · n = − 1 P e T ∇ x c (0) T · n = r c (0) T − c (0) I 0(94) We recall that the leading order problem is quasi-steady, so there is no growth of Ω T , flux of fluid across the interface or nutrient uptake (see §3.1); correspondingly, and in line with the linearised model set-up, it is sensible to assume that there is no induced diffusive transport of nutrient across Γ either. In the first sub-case this implies that, since r = 0, c Following through the rest of the analysis as described above and in [10] for O(1) membrane permeability, we find that the effective macroscale equation is unchanged, Φ T ∪I ∂c (0) ∂t + c * (1 −ρ) S (0) n T = − Λ (0) T .(95) In the second sub-case, there are minor differences imbued by the fact that a leading-order concentration jump may be permitted and we obtain the following macroscale representation in each domain Φ T ∂c (0) T ∂t + c * (1 −ρ) S (0) n T = − r c (0) T − c (0) I Γ − Λ (0) T (96) Φ I ∂c (0) I ∂t = r c (0) T − c (0) I Γ ,(97) which are identical to those presented in [28], except that advective transport is linearised in our description. Macroscale nutrient concentration in this case is given by two coupled equations, one for each of c Option 2 -Concentration jump due to species solubility We remark that when linearising the model equations we can no longer assume that at steady state nutrient concentration c * is uniform across the entire unit cell (unless α = 1, which returns us our original representation). We instead suppose that nutrient concentration is uniform in each domain, connected by the boundary condition, i.e. c T = c * T + δc T 1 + ... (98) c I = c * I + δc I 1 + ...(99) where αc * I = c * T .(100) As previously, c * T is defined by (19). The linearised and rescaled equations for the nutrient concentration are given by: ε ∂c T ,1 ∂t + ∇ · (c * T (θ * n v n,1 + θ * w v w,1 )) = 1 P e T ∇ 2 c T ,1 − Λ 1 in Ω T ,(101)ε ∂c I ,1 ∂t + ∇ · (c * I v I ,1 ) = 1 P e I ∇ 2 c I ,1 in Ω I ,(102)c * i (v i,1 − v Γ,1 ) · n − 1 P e i ∇c i,1 · n + − = 0,(103)αc I ,1 = c T ,1 on Γ.(104) All other equations remain unchanged from the original analysis, except that in cell proliferation and nutrient uptake terms c * is replaced by c * T . In the folllowing, the subscripts associated with the linearisation are omitted for clarity. At leading order, we find, via standard arguments, that both c At O(ε) the relevant equations are: ∂c (0) T ∂t + c * T ∇ x · θ * n v (1) n + θ * w v (1) w + ∇ X · θ * n v (0) n + θ * w v (0) w = 1 P e T ∇ 2 x c (1) T − Λ (0) in Ω T ,(105)∂c (0) I ∂t + c * I ∇ x · v (1) I + ∇ X · v (0) I = 1 P e I ∇ 2 x c (1) I in Ω I ,(106)c * i v (1) i − v (1) Γ · n − 1 P e i ∇ x c (1) i + ∇ X c (0) i · n + − = 0,(107)αc (1) I = c (1) T on Γ.(108) On averaging (105) and (106) over their domains, we obtain: Φ T ∂c (0) T ∂t + c * T (1 −ρ) S (0) n T = 1 P e T ∇ x c (1) T · n Γ − Λ (0) T ,(109)Φ I ∂c (0) I ∂t = − 1 P e I ∇ x c (1) I · n Γ .(110) Averaging boundary condition (107) over Γ and rearranging, we find that 1 P e T ∇ x c (1) T · n Γ − 1 P e I ∇ x c(1)I · n Γ = (c * T − c * I ) Q (0) Γ ,(111) where Q (0) = v(1)I − v (1) Γ · n = v (1) T − v (1) Γ · n = θ * w v (1) w − v (1) Γ · n(112) describes the leading order flux of material across the boundary of the tissue domain. Summing (109) and (110), and exploiting (100) and (104) to eliminate c * I and c (0) I , we obtain Φ T + 1 α Φ I ∂c (0) T ∂t + c * T (1 −ρ) S (0) n T = c * T 1 − 1 α Q (0) Γ − Λ (0) T ,(113) and Q (0) must be determined. Note that when α = 1, i.e. we have continuity of concentration on the boundary, we obtain (86) as in the original model, and Q (0) no longer appears. Discussion In this paper, we have derived an effective description for a growing tissue, by means of two-scale asymptotics. We considered a rigid periodic lattice-like structure covered by a layer of growing tissue. The model is therefore applicable to problems in regenerative medicine, such as tissue growth within a tissue engineering scaffold (our primary motivation), or biofilm growth, for example in the subsurface or the fouling of filters. Multiscale homogenisation techniques are increasing in popularity in biologically-inspired models, with a recent series of studies seeking to incorporate growth [6,7,10,22,25]. As in [10], here, we seek to accommodate a more complex description of tissue growth than one comprising a solid undergoing accretion [22,25] or volumetric growth [7], by employing a multiphase fluid tissue model that naturally accommodates aspects such as interstitial growth and active cell motion, while still obtaining a tractable macroscale description. (A multiphase approach was used in [6]; however, exploiting the limit of large interphase drag reduces the dynamics to effectively an accretion-type process.) In [10], this deficiency was addressed to obtain an effective description of tissue growth that retains active cell motion is permitted, caused by their tendency to aggregate or repel. Analytical progress was effected by a linearisation that ameliorates problems associated with complex masstransfer considered in the multiphase model; however, the macroscale description obtained was fully coupled to the microscale unit cell problems, thereby providing a significant computational challenge in the general case (decoupling is obtained in the inviscid limit case). Here, we address this feature by adopting a more suitable solution ansatz to describe the velocities and pressures in the system, that respects the linear structure of the relevant momentum equations. This analysis provides a macroscale model of very similar structure to that presented in [10], parameterised by permeability tensors, provided by a set of modified Stokes-type cell problems. The contribution of this work is that, unlike that presented in [10], the cell problems are independent of the macroscale description, leading to a system whereby the quasi-steady cell problems may be solved separately from the macroscale description, thereby greatly simplifying the computational difficulty associated with fully-coupled multiscale descriptions. Moreover, we also demonstrate how the model formulation is changed under a set of alternative microscale boundary conditions associated with, for example, cell motion over the scaffold surface, alternative nutrient flux dynamics across the tissue-interstitium boundary. Figure 1 : 1Schematic diagram of the microscale domain Ω illustrating a periodic scaffold covered with a layer of tissue, indicating the scaffold, Ω S , tissue Ω T and interstitial fluid Ω I domains. 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Microstructural influences on growth and transport in biological tissue-a multiscale description. L Irons, J Collis, R D O&apos;dea, Modeling of microscale transport in biological processes. ElsevierL. Irons, J. Collis, and R.D. O'Dea. Microstructural influences on growth and transport in biological tissue-a multiscale description. In Modeling of microscale transport in biological processes, pages 311-334. Elsevier, 2017. Three-dimensional printing of high-content graphene scaffolds for electronic and biomedical applications. Ethan B Adam E Jakus, Alexandra L Secor, Rutz, W Sumanas, Jordan, C Mark, Ramille N Hersam, Shah, ACS Nano. 94Adam E Jakus, Ethan B Secor, Alexandra L Rutz, Sumanas W Jordan, Mark C Hersam, and Ramille N Shah. Three-dimensional printing of high-content graphene scaffolds for electronic and biomedical applications. ACS Nano, 9(4):4636-4648, 2015. Darcy's law for flow in porous media and the two-space method. 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Ptashnyk and T. Roose. Derivation of a macroscopic model for transport of strongly sorbed solutes in the soil using homogenization theory. SIAM Journal on Applied Mathematics, 70(7): 2097-2118, 2010. Multiscale modelling of fluid and drug transport in vascular tumours. R J Shipley, S J Chapman, Bull. Math. Biol. 726R.J. Shipley and S.J. Chapman. Multiscale modelling of fluid and drug transport in vascular tumours. Bull. Math. Biol., 72(6):1464-1491, 2010. Reinforcement of hydrogels using threedimensionally printed microfibres. J Visser, F P W Melchels, J E Jeon, E M Van Bussel, L S Kimpton, H M Byrne, W J A Dhert, P D Dalton, D W Hutmacher, J Malda, Nature communications. 6J. Visser, F.P.W. Melchels, J.E. Jeon, E.M. van Bussel, L.S. Kimpton, H.M. Byrne, W.J.A. Dhert, P.D. Dalton, D.W. Hutmacher, and J. Malda. Reinforcement of hydrogels using three- dimensionally printed microfibres. Nature communications, 6, 2015. Haptotaxis is cell type specific and limited by substrate adhesiveness. 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{'abstract': 'In this paper, we derive an effective macroscale description suitable to describe the growth of biological tissue within a porous tissue-engineering scaffold. As in our recent work (Holden et al. "A multiphase multiscale model for nutrient limited tissue growth", The ANZIAM Journal, 2018, doi:10.1017/S1446181118000044) the underlying tissue dynamics is described as a multiphase mixture, thereby naturally accommodating features such as interstitial growth and active cell motion. Via a linearisation of the underlying multiphase model (whose nonlinearity poses significant challenge for such analyses), we obtain, by means of multiple-scales homogenisation, a simplified macroscale model that nevertheless retains explicit dependence on both the microscale scaffold structure and the tissue dynamics. The model we obtain comprises Darcy flow, and differential equations for the volume fraction of cells within the scaffold and the concentration of nutrient, required for growth. These are coupled to underlying Stokes-type cell problems that provide permeability tensors to parameterise the macroscale description. In Holden et al., the cell problems retain macroscale dependence, posing significant computational challenges; here, we obtain a decoupled system whereby the quasi-steady cell-problems may be solved separately from the macroscale description, thereby greatly reducing the complexity associated with fully-coupled multiscale descriptions. Moreover, we indicate how the formulation is influenced by a set of alternative microscale boundary conditions.', 'arxivid': '1806.09388', 'author': ['E C Holden \nCentre for Mathematical Medicine and Biology\nSchool of Mathematical Sciences\nUniversity of Nottingham\nNG7 2RDUniversity Park, NottinghamUK\n', 'B S Brook \nCentre for Mathematical Medicine and Biology\nSchool of Mathematical Sciences\nUniversity of Nottingham\nNG7 2RDUniversity Park, NottinghamUK\n', 'S J Chapman \nMathematical Institute\nUniversity of Oxford\nRadcliffe Observatory Quarter\nWoodstock RoadOX2 6GGOxfordUK\n', 'R D O&apos;dea \nCentre for Mathematical Medicine and Biology\nSchool of Mathematical Sciences\nUniversity of Nottingham\nNG7 2RDUniversity Park, NottinghamUK\n'], 'authoraffiliation': ['Centre for Mathematical Medicine and Biology\nSchool of Mathematical Sciences\nUniversity of Nottingham\nNG7 2RDUniversity Park, NottinghamUK', 'Centre for Mathematical Medicine and Biology\nSchool of Mathematical Sciences\nUniversity of Nottingham\nNG7 2RDUniversity Park, NottinghamUK', 'Mathematical Institute\nUniversity of Oxford\nRadcliffe Observatory Quarter\nWoodstock RoadOX2 6GGOxfordUK', 'Centre for Mathematical Medicine and Biology\nSchool of Mathematical Sciences\nUniversity of Nottingham\nNG7 2RDUniversity Park, NottinghamUK'], 'corpusid': 49409242, 'doi': '10.1017/s1446181119000130', 'github_urls': [], 'n_tokens_mistral': 15570, 'n_tokens_neox': 13512, 'n_words': 8266, 'pdfsha': '6a44e6d32796001a4d18daff41124e2ee9178f2a', 'pdfurls': ['https://arxiv.org/pdf/1806.09388v1.pdf'], 'title': ['A simplified multiphase multiscale model for tissue growth', 'A simplified multiphase multiscale model for tissue growth'], 'venue': []}

arxiv

The structure of AK 2 -manifolds. * 21 Jan 2003 November 21, 2018 Paul-Andi Nagy The structure of AK 2 -manifolds. * 21 Jan 2003 November 21, 2018 We study special almost Kähler manifolds whose curvature tensor satisfies the second curvature condition of Gray. It is shown that for such manifolds, the torsion of the first canonical Hermitian is parallel. This enables us to show that every AK 2 -manifold has parallel torsion. Some applications of this result, concerning the existence of orthogonal almost Kähler structures on spaces of constant curvature, are given. * MSC 2000 : 53B20, 53C25 Keywords : Almost Kähler manifold, parallel torsion non-compact type of any complex dimension n ≥ 3 admitting almost Kähler structures commuting with the invariant Kähler one[4]. Not that, at the opposite, the real hyperbolic space of dimension at least 4 do not admits, even locally, orthogonal almost Kähler structures[22,8]. In dimension 4, examples of local Ricci flat almost Kähler metrics are constructed in[3,8,18] . In the same paper, a potential source of compact almost Kähler, Einstein manifolds is considered, namely those compact Kähler manifolds whose Ricci tensor admits two distinct, constant eigenvalues; integrability is proven under certain positivity conditions. The rest of known results, most of them enforcing or replacing the Einstein condition with some other natural curvature assumption are mainly in dimension 4. To cite only a few of them, we mention the beautiful series of papers [5, 2, 6] giving a complete local and global classfication of almost Kähler manifolds of 4 dimensions satisfying the second and third Gray condition on the Riemannian curvature tensor. Other recent results, again in 4-dimensions, are concerned with the study of local obstructions to the existence of Einstein metrics [8], ⋆-Einstein metrics [19], etc.The aim of this paper is to close the circle of ideas from our previous paper [17] and to show coincidence between the class of AK 2 -manifolds-that is almost Kähler manifolds whose curvature tensor satisfies the second Gray's condition-and almost Kähler manifolds with parallel torsion. Here, both torsion and parallelism are with respect ot the first canonical Hermitian connection. More precisely, let us recall that it was proven in[17]that in an open dense set every AK 2 locally splits a the Riemannian product of an almost Kähler manifolds with parallel torsion and a special AK 2 -manifold (see section 3 for the definition). Our main present result is to give a classification of special AK 2 -manifolds. Theorem 1.1 Let (M 2n , g, J) be a special AK 2 -manifold. Then M has parallel torsion and (M, g, J) is in fact a locally 3-symmetric space. Therefore we get the precise description, in terms of the torsion of almost Kähler manifolds whose curvature tensor satisfies the second Gray condition. Theorem 1.2 Let (M 2n , g, J) be an almost Kähler manifold in the class AK 2 . Then the torsion of (g, J) is parallel with respect to the first canonical Hermitian connection. As a corollary we obtain that there are no Einstein manifolds in the class AK 2 . This follows from [17] as a simple application of Sekigawa's formula combined with the paralleism of the torsion. Let us note that our actula classification of AK 2 -manifolds is of course not a complete one. What it remains to be done is to classify almost Kähler manifolds with parallel torsion with respect to the first canonical Hermitian connection.Our present classification enables some applications relating to the non-existence of almost Kähler structures in hyperbolic geometry. Theorem 1.3 (i) Let (M, g) be a manifold of constant sectional curvature. If there exists an almost complex structure J such that (g, J) is almost Kähler, then g is a flat metric. (ii) Let (M 2n , g, J) be a Kähler manifold of constant negative holomorphic sectional curvature. If I is an almost complex structure commuting with I such that (g, I) is almost Kähler, then I has to be in fact a Kähler structure. Introduction An almost Kähler manifold (shortly AK) is a Riemannian manifold (M 2n , g), together with a compatible almost complex structure J, such that the Kähler form ω = g(J·, ·) is closed. Hence, almost Kähler geometry is nothing else that symplectic geometry with a prefered metric and complex structure. Since symplectic manifolds often arise in this way, it is rather natural to ask under which conditions on the metric we get integrability of the almost complex J. In this direction, a famous conjecture of S. I. Golberg asserts that every compact, Einstein, almost Kähler manifold is, in fact, Kähler. At our present knowledge, this conjecture is still open. Nevertheless, they are a certain number of partial results, supporting this conjecture. First of all, K. Sekigawa proved [23] that the Goldberg conjecture is true when the scalar curvature is positive. We have to note that the Golberg conjecture is definitively not true with the compacity assumption removed. In fact, there are Hermitian symmetric spaces of The result in (i) is not new, with exception of its proof. In dimension beyond 8 it was proven in [22], and in dimension 4 and 6 in [8]. Note that in dimension 4 the result of (ii) was already obtained by J. Armstrong, as a consequence of his classification of almost Kähler, Einstein, 4-manifold of class AK 3 . In dimension 6 and beyond, the result is, at our knowledge, new. Also, note that, by results in [5], that they are Hermitian symmetric spaces of non-compact type, admitting a reversing strictly almost Kähler structure. Other symmetric spaces can also support strictly almost Kähler structures [21]. The paper is organised as follows. In section 2 we review some elementary facts from almost Kähler geometry and also a few results from our previous paper [17], to be used in the subsequent. Section 3 is devoted to the investigation of the curvature tensor of special AK 2 manifolds. Some important technical tools are developed and at the end of the section it is proved that any special AK 2 -manifold with parallel torsion has to be locally 3-symmetric. The fourth section contains the proof of the theorem 1.1, whose basic ingredient is the observation that a partial holonomy reduction (with respect to the canonical Hermitian connection) extends in a canonical way to a global one. Preliminaries Let us consider an almost Hermitian manifold (M 2n , g, J), that is a Riemannian manifold endowed with a compatible complex structure. We denote by ∇ the Levi-Civita connection of the Riemannian metric g. Consider now the tensor ∇J, the first derivative of the almost complex structure and recall that for all X in T M we have that ∇ X J is a skew-symmetric (with respect to g) endomorphism of T M, which anticommutes with J. The tensor ∇J can be used to distinguish various classes of almost Hermitian manifolds. For example, (M 2n , g, J) is quasi-Kähler iff ∇ JX J = −J∇ X J for all X in T M. If ω = g(J·, ·) denotes the Kähler form of the almost Hermitian structure (g, J), we have an almost Kähler structure (AK for short), iff dω = 0. We also recall the well known fact that almost Kähler manifolds are always quasi-Kähler. The almost complex structure J defines a Hermitian structure if it is integrable, that is the Nijenhuis tensor N J defined by N J (X, Y ) = [JX, JY ] − [X, Y ] − J[X, JY ] − J[JX, Y ] for all vector fields X and Y on M identically vanishes. This is also equivalent to ∇ JX J = J∇ X J whenever X is in T M. Therefore, an almost Kähler manifold which is also Hermitian must be Kähler. In the rest of this section (M 2n , g, J) will be an almost Kähler manifold. We begin to recall some basic facts about the various notions of Ricci tensors. Let Ric be the Ricci tensor of the Riemannian metric g. We denote by Ric ′ and Ric ′′ the J-invariant resp. the J-anti-invariant part of the tensor Ric. Then the Ricci form is defined by ρ =< Ric ′ J·, · > . We define the ⋆-Ricci form by ρ ⋆ = 1 2 2n i=1 R(e i , Je i ) where {e i , 1 ≤ i ≤ 2n} is any local orthonormal basis in T M. Note that ρ ⋆ is not, in general, J-invariant. The ⋆-Ricci form is related to the Ricci form by 2.1 ρ ⋆ − ρ = 1 2 ∇ ⋆ ∇ω. The (classical) proof of this fact consists in using the Weitzenböck formula for the harmonic 2-form ω. Taking the scalar product with ω we obtain : s ⋆ − s = 1 2 |∇J| 2 where the ⋆-scalar curvature is defined by s ⋆ = 2 < R(ω), ω >. A basic object in almost Kähler geometry is the first canonical Hermitian connection, defined by : ∇ X Y = ∇ X Y + η X Y for all vector fields X and Y on M. Here, the tensor η is given by η X Y = 1 2 (∇ X J)JY . Then ∇ is a metric Hermitian connection, that is it respects the metric and the almost complex structure. The torsion of ∇ is defined by T X Y = η X Y − η Y X. Now, it is worthly to note that the almost Kähler condition (i.e. dω = 0) is equivalent to < T X Y, Z >= − < η Z X, Y > for all X, Y and Z in T M. In the subsequent we will refer simply to the tensor T as the torsion of the almost Kähler manifold (M 2n , g, J). Our main object of study in this paper is the class of almost Kähler manifolds introduced by the following definition. AK 2 iff (∇ X η)(Y, Z) = (∇ Y η)(X, Z) for all vector fields X, Y and Z on M. A number of usefull properties can be derived from the previous caracterization. At first let us define the Kähler nullity of the almost Kähler structure (g, J) to be H = {v : η v = 0}. The orthogonal complement of H in T M will be denoted by V. Since our study is purely local we can assume without loss of generality that H (and hence V) have constant rank (this happens anyway in any connected component of a dense open subset of M). Then every closed property proved locally will extend to the whole manifold. Proposition 2.2 [17] We have : (i) both distributions V and H are integrable. (ii) ∇ V η = 0 for all V in V. (iii) for any V, W in V and X in H we have that ∇ V W and ∇ V X belong to V and H respectively. Let us denote now by R the curvature tensor of connection ∇. It has the following symmetry property : 2.2 R(X, Y, Z, U) − R(Z, U, X, Y ) =< [η X , η Y ]Z, U > − < [η Z , η U ]X, Y > whever X, Y, Z, U are in T M. We end this section by recalling a technical consequence of proposition 2.1. Lemma 2.1 [17] Let V, W belong to V and X, Y be in H. We have : (i) R(V, W )η = 0 (ii) [R(X, Y ), η V ] = η β V (X,Y ) where β V (X, Y ) = η η V Y X − η η V X Y . Special AK 2 -manifolds This section will be devoted to devellop a number of preliminary results to be used in the proof of theorem 1.1. We begin by recalling the definition of special AK 2manifolds which is of esentially algebraic nature. Definition 3.1 Let (M 2n , g, J) be in the class AK 2 . It is said to be special if and only if η V V = H where H is the Kähler nullity of (g, J) and V its orthogonal complement in T M. Algebraically speaking, the special condition ensures the vanishing of the torsion on V, and also the symmetry of the restriction to V of the tensor η. We equally note that for a special AK 2 -manifold we also have : (∇ V J)H = V. From a geometric viewpoint, the special condition tells us that the integral manifolds of the integrable distribution V inherits from (M, g, J) the structure of Kähler manifolds. In the rest of this section we will work on a given AK 2 -manifold (M 2n , g, J). The notations of the previous definition are to used without further comment. We are going to investigate the action of the curvature tensor R on the decomposition T M = V ⊕ H. Our starting point is the following intermediary result. Lemma 3.1 Let (M 2m , g, J) be a special AK 2 manifold. Then the following holds : 3.1 2R(V 3 , V 4 , V 2 , η V 1 X) = R(V 3 , V 4 , X, η V 1 V 2 ) for all V i , 1 ≤ i ≤ 4 in V and X in H. Proof : We will make use of the following formula from [17], a consequence of the second Bianchi identity for the connection ∇: 3.2 R(η V 2 X, V 1 , V 2 , V 3 , V 4 ) − R(η V 1 X, V 2 , V 3 , V 4 ) = − < [η V 3 , η V 4 ]X, T V 1 V 2 > whenever V i , 1 ≤ i ≤ 4 are in V and X is in H. First, we note that under the special condition the right hand side of (3.2) vanishes so that we have R(η V 2 X, V 1 , V 2 , V 3 , V 4 ) = R(η V 1 X, V 2 , V 3 , V 4 ) . Now, using the symmetry property (2.2) we obtain : R(V 3 , V 4 , η V 2 X, V 1 )+ < [η η V 2 X , η V 1 ]V 3 , V 4 > − < [η V 3 , η V 4 ]η V 2 X, V 1 >= R(V 3 , V 4 , η V 1 X, V 2 )+ < [η η V 1 X , η V 2 ]V 3 , V 4 > − < [η V 3 , η V 4 ]η V 1 X, V 2 > Using the vanishing of the torsion T on V, a standard verification leads to < [η η V 2 X , η V 1 ]V 3 , V 4 > − < [η V 3 , η V 4 ]η V 2 X, V 1 >= 0 hence 3.3 R(V 3 , V 4 , η V 2 X, V 1 ) = R(V 3 , V 4 , η V 1 X, V 2 ). But (R(V 3 , V 4 ).η)(V 2 , X) = 0 for i = 1, 2 (see lemma 2.1, (i)) . Plugging this in the previous equation we obtain : < η R(V 3 ,V 4 )V 2 X, V 1 > + < η V 2 R(V 3 , V 4 )X, V 1 >= R(V 3 , V 4 , η V 1 X, V 2 ). Invoking proposition 2.2, (iii), one finds that the operator R(V 3 , V 3 ) preserves V and H. Using again the vanishing of the torsion on V we have < η R(V 3 ,V 4 )V 2 X, V 1 >= − < X, η R(V 3 ,V 4 )V 2 V 1 >= − < X, η V 1 R(V 3 , V 4 )V 2 >= R(V 3 , V 4 , V 2 , η V 1 X) and the conclusion is now immediate The relation in lemma 3.1 shows that the restriction of R to V is completely determined by the mixed curvature terms of type R(V, W, X, Y ) with V, W in V and X, Y in H. To investigate these terms we introduce now the configuration tensor A : H × H → V by setting : ∇ X Y =∇ X Y + A X Y for all X, Y in H. In a similar way, we define B : H × V → V by ∇ X V =∇ X V + B X V. Since H is integrable A is a symmetric tensor, that is A X Y = A Y X for all X, Y in H. It is immediate to establish that [A X , J] = 0 for all X in H. Now, the parallelism of η in the direction of V together with the caracterization in proposition 2.1 translates into additional algebraic properties of the tensor A, as follows. Lemma 3.2 Let X, Y be in H and V, W in V respectively. We have : (i) B X (η V Y ) = η V (A X Y ). (ii) A X (η V W ) = η V (B X W ). Proof : Having in mind proposition 2.2, (iii) it suffices to project on H and V respectively the identities (∇ X η)(V, Y ) = 0 and (∇ X η)(V, W ) = 0. The configuration tensor A can be used to compute parts of the curvature tensor R in the following way. Lemma 3.3 Let V, W and X, Y be vector fields in V and H respectively. We have : (i) R(V, X, W, Y ) =< W, (∇ V A)(X, Y ) > − < B X V, B Y W > (ii) < W, (∇ V A)(X, Y ) >=< V, (∇ W A)(X, Y ) > . (iii) R(V, W, X, Y ) = − < B X V, B Y W > + < B X W, B Y V > for all V, W in V and X, Y in H. Proof : The proof of (i) will be omitted since a standard computation using only the proposition 2.2, (iii). Now, (ii) comes from (i) by means of the symmetry property (2.2). To prove (iii) one uses the first Bianchi identity for R when noticing that the latter do not contains derivatives of the torsion, in virtue of proposition 2.1 and proposition 2.2, (ii). We will start now to compute parts of the curvature tensor R. To begin with, let us define the symmetric, J-invariant, partial Ricci tensors r 1 : V → V and r 2 : H → H by setting : v k ∈V R(v k , Jv k )V = r 1 (JV ) v k ∈V R(v k , Jv k )X = r 2 (JX) Then we have : Proposition 3.1 The partial Ricci tensors r 1 , r 2 can be computed by the following formulas : (i) < r 1 V, W >= e i ∈H < B e i V, B e i W > (ii) < r 2 X, Y >= −2 v k ∈V < B X v k , B Y v k > where V, W are in V and X, Y belong to H and {v k }, {e i } are arbitrary orthonomal basis in V and H respectively. Proof : (ii) follows by a simple computation involving lemma 3.3, (iii). Let us prove (i). Using R(v k , Jv k , V, η W X) = v k ∈V R(v k , Jv k , W, η V X) = 1 2 v k ∈V R(v k , Jv k , X, η V W ) = − v k ∈V < B X v k , B η V W (Jv k ) > . Now, we have : v k ∈V < B X v k , B η V W (Jv k ) >= − v k ∈V e i ∈H < e i , B η V W (Jv k ) >< v k , A e i X > = e i ∈H < A e i η V W, JA e i X > . Or using in an apropriate way lemma 3.2 we have < A e i η V W, JA e i X >=< η V B e i W, A e i JX >= − < B e i W, B e i η V (JX) >= < B e i W, B e i Jη V X > and the conclusion is now straightforward. We are now able to give the main technical result of this section. Proposition 3.2 Let (M 2n , g, J), n ≥ 2 be a special AK 2 -manifold. Then the following holds : 3.4 ∆ V |A| 2 = −5|r 1 | 2 − 2|∇ V A| 2 . Here, ∇ V denotes the restriction of ∇ to V and ∆ V is the corresponding partial Laplacian, acting on functions. Proof : From lemma 3.3, (ii) we deduce that 3.5 (∇ JV A)(JX, Y ) = (∇ V A)(X, Y ) for all V in V and X, Y in H respectively. We consider now the partial Laplacian D V , acting on A by : (D V A)(X, Y ) = − v k ∈V (∇ 2 v k ,v k A)(X, Y ) for all X, Y belonging to H, where {v k } is an arbitrary local orthonormal basis of V. Derivating (3.5) it follows that (D V A)(X, Y ) = 1 2 J v k ∈V (R(v k , Jv k ).A)(X, Y ). Using proposition 3.1, we obtain further : (D V A)(X, Y ) = − 1 2 (r 1 A X Y + 2A r 2 X Y + 2A X r 2 Y ). Taking the scalar product with A gives now < D V A, A >= − 1 2 (|r 1 | 2 + 4|r 2 | 2 ), or further < D V A, A >= − 5 2 |r 1 | 2 , after noticing that |r 1 | 2 = |r 2 | 2 . Now , the standard Weitzenböck formula gives 1 2 ∆ V |A| 2 =< D V A, A > −|∇ V A| 2 and the claimed formula follows now easily. Then it can be shown that (M 2n , g, I) is almost Kähler, and one can even show after some calculation that (g, I) belongs to the class AK 3 . Then the use of Sekigawa's formula gives exactly (3.4). Of course, one has to use (3.1) and furthermore compute all the remaining curvature terms. Since the calculations are of more length we prefered the direct approach. (ii) If the manifold (M, g, I) belongs to the class AK 2 then the function |∇I| 2 is known to be constant and by the previous proposition we get that (g, I) is in fact a Kähler structure. Proof of theorem 1.1 In this section we will give the proof of the theorem 1.1. This will be done by showing that it is always possible to restrict, at least locally and in dimension at least 6, the study of special AK 2 -manifolds, to the case when the function |A| 2 is constant. This, together with proposition 3.2 of the previous section, will enable us to prove theorem 1.1. As in the section 3 let (M 2n , g, J) be a special AK 2 -manifold, with Kähler nullity H and let V be the distribution orthogonal to H. We will first study the integral manifolds of H. For every X in H define a linear map : γ X : V → V by γ X V = η V X. The vanishing of the torsion on V implies that γ X is symmetric for all X in H. Let us denote byR the curvature tensor of the connection∇, where we recall that∇ is the orthogonal projection of ∇ onto the decomposition T M = V ⊕ H (see section 3). The maps γ X are in relation with the curvature of H (with respect to the connection∇), as the following lemma shows. R(X, Y, η V W, Z) =< [[γ X , γ Y ], γ Z ]V, W > . Proof : Using the definition of∇ we obtain after a short computation that 4.1 R(X, Y, Z ′ , Z) =R(X, Y, Z ′ , Z)+ < A Y Z ′ , A X Z > − < A X Z ′ , A Y Z > for all X,R(X, Y, W, η V Z) − R(W, η V Z, X, Y ) = − < [η W , η η V Z ]X, Y > since H is the Kähler nullity of (g, J). The use of lemma 3.3, (iii) gives then R(W, η V Z, X, Y ) = − < B X W, B Y (η V Z) > + < B X (η V Z), B Y W > = − < B X W, η V (A Y Z) > + < η V (A X Z), B Y W > =< A X (η V W ), A Y Z > − < A X Z, A Y (η V W ) > . where we used succesivelly lemma 3.2, (i) and (ii). It follows that R(X, Y, W, η V Z) =< A X (η V W ), A Y Z > − < A X Z, A Y (η V W ) > − < [η W , η η V Z ]X, Y > . Using this in (4.2) and taking Z ′ = η V W in (4.1) we get R(X, Y, η V W, Z) =< η β V (X,Y ) W, Z > + < [η W , η η V Z ]X, Y > . It remains to take into account, in the last equation, the definition of the maps γ U , U in H and our lemma follows routineously. Our basic tool in the study of special AK 2 -manifolds will be the following intermediary result showing that a partial holonomy reduction of H extends in a canonical way to a holonomy reduction over T M. V 1 ⊕ V 2 = V. Moreover we have η V 1 V 2 = 0 and η V i V i = H i , i = 1, 2. (ii) The decomposition T M = (V 1 ⊕ H 1 ) ⊕ (V 2 ⊕ H 2 ) defines a local splitting of M into the Riemannian product of two special AK 2 -manifolds, with corresponding Kähler nullities H 1 and H 2 . Proof : (i) Let X 1 and X 2 be in H 1 and H 2 respectively. Then the partial parallelism of H 1 , together with the symmetry property ofR (a consequence of (4.1) and (2.2)) ensures thatR(X 1 , X 2 , η V W, Z) = for all V, W in V and Z in H. Then, by the previous lemma we obtain [[γ X 1 , γ X 2 ], γ Z ] = 0 for all Z in H. Taking Z = X 1 we find that γ 2 X 1 γ X 2 + γ X 2 γ 2 X 1 = 2γ X 1 γ X 2 γ X 1 . We change now X 2 in JX 2 in the previous equation and take into account that γ JX = γ X J = −Jγ X . It follows that γ 2 X 1 γ X 2 + γ X 2 γ 2 X 1 = −2γ X 1 γ X 2 γ X 1 hence we must have γ 2 X 1 γ X 2 + γ X 2 γ 2 X 1 = γ X 1 γ X 2 γ X 1 = 0. This yields to γ 3 X 1 γ X 2 = 0 and since γ X is a symmetric operator for all X in H we get that γ X 1 γ X 2 = 0. But this fact is easily seen to be equivalent to the orthogonality of the spaces V 1 = η V H 1 and V 2 = η V H 2 . The remaining afirmations of (i) are direct consequences of this fact. (ii) We are going to prove first that the distribution V 1 ⊕ H 1 is ∇-parallel. Let U be in T M and V, W in V 1 . As (∇ U η)(V, W ) = (∇ V η)(U, W ) = 0 we get that ∇ U (η V W ) = η ∇ U V W + η V (∇ U W ) belongs to η T M V 1 + η V 1 T M = V 1 ⊕ H 1 . Since H 1 = η V 1 V 1 we conclude that ∇ U X belongs to V 1 ⊕ H 1 for all X in H 1 . Take now V in V 1 and X in H 1 . As before, we have : ∇ U (η V X) = η ∇ U V X + η V (∇ U X) belongs to η T M H 1 + η V 1 T M = V 1 ⊕ H 1 and using the fact that η V 1 H 1 = V 1 we con- clude that V 1 ⊕ H 1 is ∇-parallel. In the same way it can be proven that V 2 ⊕ H 2 is ∇-parallel. Now, if E i = V i ⊕ H i for i = 1, 2 it follows from (i) that η E i E i ⊆ E i and η E i E j = 0 if i = j. This shows that E 1 and E 2 are in fact ∇-parallel and the proof is finished. We will now study properties of the tensor r 2 : H → H defined in the previous section. Lemma 4.2 Let (M 2n , g, J) be a special AK 2 -manifold. Then : (∇ X r 2 )Y = 0 for all X, Y in H. Proof : We will make use of the following formula which has been proven in [17] : 4.3 (∇ X R)(V 1 , V 2 , V 3 , V 4 ) = 0 for all X in H and V i in V, 1 ≤ i ≤ 4. It follows easily that (∇ X R)(V 1 , V 2 , V 3 , V 4 ) = 0. Now, we recall that lemma 3.1 states that 2R(V 3 , V 4 , V 2 , η V 1 Y ) = R(V 3 , V 4 , Y, η V 1 V 2 ) whenever Y belongs to H and V i in V, 1 ≤ i ≤ 4. To conclude, it suffices to derive the last equation in the direction of X in H, with respect to the connection∇, and next take (4.3) into account. We will give now the proof of the theorem 1.1 stated in the introduction. Proof of theorem 1.1 Let U the open dense set of M where we have an orthogonal splitting : H = H 1 ⊕ . . . H p where H i , 1 ≤ i ≤ p are the eigenbundles of r 2 with corresponding eigenfunctions λ i , 1 ≤ i ≤ p. Note that by lemma 4.2 we have in the standard way X.λ i = 0 for all X in H and 1 ≤ i ≤ p hence the distributions H i are∇-parallel, inside H. We set V i = η V H i , 1 ≤ i ≤ p and use proposition 4.1 to obtain an orthogonal, J-invariant and ∇-parallel decomposition : 4.4 T M = p i=1 (V i ⊕ H i ). Of course, each factor corresponds to a special AK 2 -manifold such the corresponding tensor r 2 has the Kähler nullity as eigenspace. Therefore it remains us to consider this situation. Suppose that (M 2n , g, J) is special AK 2 -manifold, with decomposition T M = V ⊕ H and such that r 2 = λ · 1 H . Then using (3.1) and the fact that η V H = V we obtain that v k ∈V R(v k , Jv k , V, W ) = − λ 2 < JV, W > for all V, W in V. If dim R V = 2 the special condition implies that dim R H = 2 and we know by the work in [5] that the torsion has to be parallel. If the dimension is greater, using the second Bianchi identity for ∇ on V, exactly in the way one shows that a manifold of dimension greater than 3, with Ricci tensor proportional to the metric tensor is Einstein, one obtains that V.λ = 0 for all V in V. But using proposition 3.2 it follows immediately that the configuration tensor A vanishes, ensuring the ∇parallelism of the decomposition T M = V ⊕ H. Moreover, by propositions 2.1 and 2.2, (ii) we obtain that the torsion is also ∇-parallel. Now it clear that each factor of the decomposition (4.4) has parallel torsion and it follows that the torsion of M is parallel over U and by continuity over M, hence the first part of theorem 1.1 is proved. It remains us to show that (M 2n , g, J) is locally 3-symmetric. The vanishing of A implies by lemmas 3.1 and 3.3 that curvature terms of the form R(V 1 , V 2 , V 3 , V 4 ), R(V 1 , V 2 , X, Y ) and R(V 1 , X, V 2 , Y ) where V i , 1 ≤ i ≤ 4 are in V and X, Y in H must be all equal to 0. Furthermore, the restriction of R to H is computed by lemma 4.1 and using the parallelism of the torsion it is an easy exercise to see that ∇R = 0, in other words ∇ is an Ambrose-Singer connection. We conclude now by [15]. It is a good place now to introduce, in view of future use, the following definition. Definition 4.1 A locally 3-symmetric space of type I is a special AK 2 with parallel torsion. Remark 4.1 From the proof of theorem 1.1 we get the explicit dependence on the torsion of the curvature tensor R of a space of type I. This can be used to get algebraic caracterizations as a homogeoneous space of such a manifold. Since this discussion is beyond the scope of the present paper it will be omitted. Proof of theorem 1.3 : It is easy to see that an almost Kähler structure satisfying the conditions in (i) or (ii) has to satisfy the second Gray condition and therefore must have parallel torsion. We conclude by recalling (cf. [17]) that any Einstein manifold supporting a compatible almost Kähler structure with parallel torsion is Kähler. Definition 2. 1 1An almost Kähler manifold (M 2n , g, J) belongs to the class AK 2 iff its Riemannian curvature tensor satisfies the identity : R(X, Y, Z, U) − R(JX, JY, Z, U) = R(JX, Y, JZ, U) + R(JX, Y, Z, JU) for all X, Y, Z, U in T M. Manifolds within this class have a simple caracterization in terms of the torsion of the canonical Hermitian connection. Proposition 2.1 [17] An almost Kähler manifold (M 2n , g, J) belongs to the class (3.3), lemma 3.1 and (i) we obtain that v k ∈V way of proving formula (3.4) is the following. Define an almost complex structure I on M by setting I = J on V and I = −J on H. Lemma 4. 1 1Let X, Y, Z be in H and V, W in V. We have : Y, Z, Z ′ in H. Now, by lemma 2.1, (ii) one obtains 4.2 R(X, Y, η V W, Z) + R(X, Y, W, η V Z) =< η β V (X,Y ) W, Z > . But the symmetry property (2.2) yields to Proposition 4. 1 1Let (M, g, J) be a special AK 2 -manifold. Assume that we have an orthogonal, J-invariant decomposition H 1 ⊕ H 2 which is also∇-parallel (inside H). Then : (i) If we put V i = η V H i for i = 1, 2 then we have an orthogonal and J-invariant decomposition Curvature properties of twistor spaces of quaternionic Kähler manifolds. Grantcharov B G Alexandrov, Ivanov , J. 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Uni. Parma, IV. Ser. 13LEDGER, A.J., L. VANHECKE, On a theorem of Kiricenko relating to 3-symmetric spaces, Riv. Mat. Uni. Parma, IV. Ser.13, 367-372 (1987). Nearly Kähler geometry and Riemannian foliations. P-A Nagy, Asian J. Math. 63to appearNAGY, P-A., Nearly Kähler geometry and Riemannian foliations , Asian J. Math. 6, no.3 (2002), 481-504, to appear. P-A Nagy, arXiv:math.DG/0301069Torsion in almost Kähler geometry, preprint. NAGY, P-A., Torsion in almost Kähler geometry, preprint, arXiv:math.DG/0301069. A four-dimensional example of Ricci flat metric admitting Kähler non-Kähler structure. P Nurowski, M Przanowski, Classical Quantum Gravity. 163NUROWSKI, P., PRZANOWSKI, M, A four-dimensional example of Ricci flat metric admitting Kähler non-Kähler structure , Classical Quantum Gravity 16 (1999), no.3, L9-L13. Four dimensional almost Kähler Einstein and ⋆-Einstein manifolds. T Oguro, K Sekigawa, Geom. Dedicata. 691OGURO. T, SEKIGAWA, K. Four dimensional almost Kähler Einstein and ⋆-Einstein mani- folds, Geom. Dedicata 69 (1998), no.1, 91-112. F Tricerri, L Vanhecke, Curvature tensors on almost Hermitian manifolds. 267F. TRICERRI, L. VANHECKE, Curvature tensors on almost Hermitian manifolds, Trans. Amer. math. Soc. 267 (1981), 365-398. Kähler structures on the Riemannian product of a 3-dimensional hyperbolic space and a real line. T Oguro, Sekigawa, Almost, Tsukuba J. Math. 20OGURO, T, SEKIGAWA, K Almost Kähler structures on the Riemannian product of a 3- dimensional hyperbolic space and a real line, Tsukuba J. Math. 20 (1986), 151-161. A note on almost Kähler manifolds. Z Olszak, Bull. Acad. Polon. Sci. XXVI. Z. OLSZAK, A note on almost Kähler manifolds, Bull. Acad. Polon. Sci. XXVI (1978), 199-206. on some compact Einstein almost Kähler manifolds. K Sekigawa, J. Math. Soc. Japan. 36SEKIGAWA, K, on some compact Einstein almost Kähler manifolds, J. Math. Soc. Japan 36 (1987), 677-684. . Paul-Andi, Nagy Institut de Mathématiques rue E. Argand. 112007Neuchâtel email : Paul.Nagy@unine.chPaul-Andi Nagy Institut de Mathématiques rue E. Argand 11, CH-2007, Neuchâtel email : Paul.Nagy@unine.ch

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{'abstract': "We study special almost Kähler manifolds whose curvature tensor satisfies the second curvature condition of Gray. It is shown that for such manifolds, the torsion of the first canonical Hermitian is parallel. This enables us to show that every AK 2 -manifold has parallel torsion. Some applications of this result, concerning the existence of orthogonal almost Kähler structures on spaces of constant curvature, are given. * MSC 2000 : 53B20, 53C25 Keywords : Almost Kähler manifold, parallel torsion non-compact type of any complex dimension n ≥ 3 admitting almost Kähler structures commuting with the invariant Kähler one[4]. Not that, at the opposite, the real hyperbolic space of dimension at least 4 do not admits, even locally, orthogonal almost Kähler structures[22,8]. In dimension 4, examples of local Ricci flat almost Kähler metrics are constructed in[3,8,18] . In the same paper, a potential source of compact almost Kähler, Einstein manifolds is considered, namely those compact Kähler manifolds whose Ricci tensor admits two distinct, constant eigenvalues; integrability is proven under certain positivity conditions. The rest of known results, most of them enforcing or replacing the Einstein condition with some other natural curvature assumption are mainly in dimension 4. To cite only a few of them, we mention the beautiful series of papers [5, 2, 6] giving a complete local and global classfication of almost Kähler manifolds of 4 dimensions satisfying the second and third Gray condition on the Riemannian curvature tensor. Other recent results, again in 4-dimensions, are concerned with the study of local obstructions to the existence of Einstein metrics [8], ⋆-Einstein metrics [19], etc.The aim of this paper is to close the circle of ideas from our previous paper [17] and to show coincidence between the class of AK 2 -manifolds-that is almost Kähler manifolds whose curvature tensor satisfies the second Gray's condition-and almost Kähler manifolds with parallel torsion. Here, both torsion and parallelism are with respect ot the first canonical Hermitian connection. More precisely, let us recall that it was proven in[17]that in an open dense set every AK 2 locally splits a the Riemannian product of an almost Kähler manifolds with parallel torsion and a special AK 2 -manifold (see section 3 for the definition). Our main present result is to give a classification of special AK 2 -manifolds. Theorem 1.1 Let (M 2n , g, J) be a special AK 2 -manifold. Then M has parallel torsion and (M, g, J) is in fact a locally 3-symmetric space. Therefore we get the precise description, in terms of the torsion of almost Kähler manifolds whose curvature tensor satisfies the second Gray condition. Theorem 1.2 Let (M 2n , g, J) be an almost Kähler manifold in the class AK 2 . Then the torsion of (g, J) is parallel with respect to the first canonical Hermitian connection. As a corollary we obtain that there are no Einstein manifolds in the class AK 2 . This follows from [17] as a simple application of Sekigawa's formula combined with the paralleism of the torsion. Let us note that our actula classification of AK 2 -manifolds is of course not a complete one. What it remains to be done is to classify almost Kähler manifolds with parallel torsion with respect to the first canonical Hermitian connection.Our present classification enables some applications relating to the non-existence of almost Kähler structures in hyperbolic geometry. Theorem 1.3 (i) Let (M, g) be a manifold of constant sectional curvature. If there exists an almost complex structure J such that (g, J) is almost Kähler, then g is a flat metric. (ii) Let (M 2n , g, J) be a Kähler manifold of constant negative holomorphic sectional curvature. If I is an almost complex structure commuting with I such that (g, I) is almost Kähler, then I has to be in fact a Kähler structure.", 'arxivid': 'math/0301228', 'author': ['Paul-Andi Nagy '], 'authoraffiliation': [], 'corpusid': 17118334, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 11710, 'n_tokens_neox': 10015, 'n_words': 6429, 'pdfsha': 'fe4040f0b4c67ac388fc36ba6bc5a5896f691667', 'pdfurls': ['https://arxiv.org/pdf/math/0301228v1.pdf'], 'title': ['The structure of AK 2 -manifolds. *', 'The structure of AK 2 -manifolds. *'], 'venue': []}

arxiv

Estimating entanglement monotones with a generalization of the Wootters formula 18 Nov 2012 (Dated: May 2, 2014) Zhi-Hua Chen Department of Science Zhijiang College Zhejiang University of Technology 310024HangzhouChina Zhi-Hao Ma Department of Mathematics Shanghai Jiaotong University 200240ShanghaiChina Otfried Gühne Naturwissenschaftlich-Technische Fakultät Universität Siegen Walter-Flex-Straße 357068SiegenGermany Simone Severini Department of Computer Science Department of Physics & Astronomy University College London Gower StWC1E 6BTLondonUnited Kingdom Estimating entanglement monotones with a generalization of the Wootters formula 18 Nov 2012 (Dated: May 2, 2014)arXiv:1207.2889v2 [quant-ph] Entanglement monotones, such as the concurrence, are useful tools to characterize quantum correlations in various physical systems. The computation of the concurrence involves, however, difficult optimizations and only for the simplest case of two qubits a closed formula was found by Wootters [Phys. Rev. Lett. 80, 2245(1998]. We show how this approach can be generalized, resulting in lower bounds on the concurrence for higher dimensional systems as well as for multipartite systems. We demonstrate that for certain families of states our results constitute the strongest bipartite entanglement criterion so far; moreover, they allow to recognize novel families of multiparticle bound entangled states. Introduction -Entanglement proved itself to be a fundamental concept in physics, with applications spanning virtually all areas of quantum science: these include antipodal topics such as the black hole information paradox and industrial realizations of quantum cryptographic devices. By definition, entanglement between two or more particles is given by those quantum correlations, which cannot be created by local operations or classical communication (LOCC). For the case of more than two particles, also different classes of entanglement can be distinguished. For the quantification of entanglement and also for the discrimination between entanglement classes one can use so-called entanglement measures or entanglement monotones -parameters that indeed are non-increasing under LOCC. The concurrence and the entanglement of formation are important parameters of this kind [1]. A central problem for the quantification of entanglement is the fact that nearly all entanglement monotones are extremely difficult to compute. Indeed, most definitions of entanglement monotones contain nontrivial optimizations, such as the optimization over all possible LOCC protocols or the minimization over all possible decompositions of a given density matrix. This difficulty is an important issue for the application of monotones to real world problems or experiments. A milestone in the theory of entanglement measures was the derivation of a closed formula for the concurrence of two qubits by Wootters in 1998 [2]. In this work, it was shown how the minimization over all state decompositions can be done for such a special case. Consequently, the Wootters' formula has lead to many applications of the concurrence, e.g. for characterizing phase transitions in spin models [3]. In the following years, the formula has been shown to work also for a special type of multipartite measures by Uhlmann [4]. Furthermore, the concurrence can also be computed for some special states with high symmetries [5]. In the present Letter, we generalize the idea of Wootters to compute lower bounds on the concurrence. Our methods work for higher dimensional bipartite systems as well as for multipartite systems. Compared with the large amount of research about lower bounds on entanglement measures [6][7][8] our approach has substantial advantages: for the bipartite case we discuss a family of bound entangled states and show that our result gives the strongest separability criterion so far; for the multipartite examples, our estimates give the precise value of the multipartite concurrence, and allow to identify a novel and simple family of bound entangled states. Finally, our approach can also be used to estimate other quantities besides the concurrence, which might be useful to deal with entanglement monotones based on antilinear operators and combs [9]. It should be noted that lower bounds on the concurrence based on Wootters' idea have appeared in the literature before [8]; as we will see, however, the existing approaches are fundamentally limited. Setting the stage -To start, let us recall the main definitions. For a general m × n-dimensional bipartite pure quantum state ̺ AB = |ψ ψ| on H A ⊗ H B , the concurrence [1,10] can be defined as C(|ψ ) = 2 (1 − Tr̺ 2 A ),(1) where ̺ A = Tr B (|ψ ψ|) is the reduced density matrix of the first particle [11]. A pure state is separable if and only if its concurrence is zero. The above definitions are extended to mixed states via the so-called convex roof construction, C(̺) = min {pi,|ψi } i p i C(|ψ i ),(2) where the minimization is meant as an optimization over all possible ensemble realizations ̺ = i p i |ψ i ψ i |, where p i ≥ 0 and i p i = 1. The decomposition attaining the minimum is said to be the optimal decomposition. Clearly, this is a difficult optimization problem, and different estimates have been obtained [6][7][8]. The bipartite bound -For our approach, we first need to reformulate the definition of the concurrence. The pure state |ψ can be expressed in a product basis as |ψ = m i=1 n j=1 ψ ij |ij . Furthermore, we can define on H A the generators of the group SO(m) as L α = |i j| − |j i|. There are m(m − 1)/2 generators of this type, similarly, there are n(n − 1)/2 generators S β of SO(n) on H B . Then, a direct calculation for the ψ ij shows that one can express the concurrence as (see also Ref. [12]) C 2 (|ψ ) = 2 1 − Tr̺ 2 A = α,β | ψ|L α ⊗ S β |ψ * | 2 ,(3) where |ψ * denotes the complex conjugation. In the following, it is convenient to use a single index for the matrices L α ⊗ S β and we define J t = L α ⊗ S β , where the index t runs from 1 to N = [mn(m − 1)(n − 1)]/4. In order to formulate our bound, we first fix an integer k. We then choose a subset of indices t = {t 1 , ..., t k } ⊂ {1, ..., N }, where we use the ordering t i < t i+1 . Moreover, we can choose k complex numbers u = {u s } for which the absolute values are bounded via |u s | ≤ 1. Then, we consider the quantity ∆ k (̺, t, u) = max 0, λ (1) mn − i>1 λ (i) mn ;(4) here the numbers λ X = ̺ k s=1 u s J ts ̺ * k s=1 u * s J ts(5) in non-increasing order. Alternatively, one can say that the λ (j) mn are the eigenvalues of the hermitean matrix Y = √ ̺( s u s J ts )̺ * ( s u * s J ts ) √ ̺.(6) For our given k, we consider the set of all possible t and choose for any of them a different vector u and compute the corresponding ∆ k (̺, t, u). This leads to N k numbers and for these we can state our first main result: Observation 1. Let ̺ be a density matrix acting on an m× n-dimensional bipartite quantum system and consider for fixed k all the possible t and a possible choice of u as discussed above. Then, a lower bound on the concurrence is given by C(̺) 2 ≥ N k 2 N k t [∆ k (̺, t, u)] 2 .(7) Especially, if ̺ is separable then ∆ k (̺, t, u) = 0 for any choice of k, t and u. Before proving this theorem, let us discuss some of its implications. Eq. (7) is a lower bound for the concurrence for any given choice of the u. In order to obtain a good bound, the set of the u has to be optimized for the given state ̺. Often this has to be done numerically, but we will also present examples, where a good choice of the J ts is given analytically. Second, for the case of two qubits there is only one possible generator, namely L α = S β = |0 1| − |1 0| = iσ y . This implies that the only possibility in Observation 1 is k = N = 1, and then Eq. (7) reduces to the well known formula for the concurrence of mixed states. Of course, obtaining a closed formula for the concurrence is a significantly more advanced result as one has to prove in addition that equality holds. In Refs. [2,4] this has been achieved by writing down an explicit decomposition. This is, however, beyond the scope of the present Letter, we focus on the problem of deriving lower bounds. Finally, one should add that other researchers have obtained lower bounds on the concurrence by using the formulation of Eq. (3) and ideas similar to the original construction [8]. In these works, the terms | ψ|L α ⊗ S β |ψ * | 2 are estimated separately. A single observable L α ⊗ S β , however, acts on a 2 × 2 subspace only, and for these subspaces the criterion of the positivity of the partial transpose (PPT) is a necessary and sufficient criterion for entanglement [1]. This implies that the approaches in Refs. [8] can never detect weak forms of entanglement, such as bound entanglement which is not detected by the PPT criterion [13]. On the other side, Observation 1, represents a strong criterion for bound entanglement, as we will see below. Proof of Observation 1. First we prove that for a fixed k, and fixed vector t we have that min {pi,|ψi } i p i | ψ i | k s=1 u s J ts |ψ * i | ≥ ∆ k (̺, t, u),(8) where the minimum is taken over all decompositions ̺ = i p i |ψ i ψ i |. Let λ i and |χ i be the eigenvalues and the eigenvectors of ̺. It is known that any decomposition of ̺ is connected to the eigenvalue decomposition via a unitary matrix U ij , namely one has [14]. Therefore, we have √ p i |ψ i = mn j=1 U * ij ( λ j |χ j )√ p i p j ψ i | k s=1 u s J ts |ψ * j = (U Y U T ) ij , where the ma- trix Y is defined by Y αβ = λ α λ β χ α | k s=1 u s J ts |χ * β . Since the J k are symmetric, the matrix Y = Y T is complex and symmetric and we can use Takagi's factorization [15] to write Y = V DV T with a real diagonal matrix D. The entries of D are nonnegative and given by the square roots of the eigenvalues of Y Y † . Then, following directly the argumentation of Ref. [2] we have: min {pi,|ψi } i p i | ψ i | k s=1 u s J ts |ψ * i | = min W =UV i |[W DW T ] ii | ≥ λ (1) mn − i>1 λ (i) mn , = ∆ k (̺, t, u)(9) where λ (j) mn are the entries of D in decreasing order. These quantities are, however, nothing but the eigenvalues of X in Eq. (5). Therefore, if a state ̺ is separable then a decomposition into pure states without concurrence exists. Due to Eq. (3) all the mean values of J k vanish, which implies already that ∆ k (̺, t, u) = 0. It remains to show that ∆ k (̺, t, u) can give a lower bound on the concurrence also for entangled states. Suppose that ̺ = i p i |ψ i ψ i | is an optimal decomposition of ̺. Then C(̺) = i p i C(|ψ i ) = i p i N t=1 | ψ i |J t |ψ * i | 2 . From the argumentation above, we know that for fixed k and t and fixed t 1 , ..., t k the estimates ∆ k (̺, t, u) ≤ i p i k s=1 | ψ i |u s J ts |ψ * i | ≤ i p i k s=1 | ψ i |J ts |ψ * i | hold. Finally, using the rule ( k j=1 x j ) 2 ≤ k j x 2 j and the Cauchy-Schwartz inequality we can directly estimate the right-hand side of Eq. (7) as: t [∆ k (̺, t, u)] 2 ≤ k 2 N N k C(̺) 2 .(10) The details of this calculation are given in the Appendix A1 [16]. This concludes the proof of Observation 1. Before proceeding to the examples, let us discuss the properties of the concurrence that were used in the proof. The starting point was Eq. (3) and the only further requirement needed was that the fact that the J t = J T t were symmetric [17]. Moreover, if A t = −A T t were antisymmetric, then one has for any state | ψ|A t |ψ * | 2 = 0, so restricting to symmetric J t can be done without loosing generality. In summary, the convex roof of any quantity E(|ψ ), which can be written as E 2 (|ψ ) = t ±m t | ψ|M t |ψ * | 2 ,(11) can be estimated with our methods: one can split each M t in a symmetric and an antisymmetric part and estimate the contributions from the symmetric part. The fact that some of the coefficients m t can be negative does not matter: using the relation t | ψ|G t |ψ * | 2 = 1 (where the G t form an orthonormal basis of the operator space) one can rewrite E 2 (|ψ ) as a sum with only positive coefficients minus a constant term. Bound entangled states as an example -In order to show that Observation 1 results in a stronger separability criterion than best methods that are currently known, we consider the family of 3 × 3 bound entangled states introduced by P. Horodecki [18]. This family of states ̺ P H a is not detected by PPT criterion, but is nevertheless entangled for any 0 < a < 1. The detailed form of these states is given in Appendix A2 [16]. We consider a mixture of these states with white noise, ̺ a (p) = p̺ P H a + (1 − p)1 1/9 and ask for the minimal p, so that the entanglement in ̺ a (p) is still detected. First, we use Observation 1 with the purpose of detecting entanglement and find the optimal J t via numerical optimization. We finally compare our values with the values obtained via different known criteria: the Zhang-Zhang-Zhang-Guo (ZZZG) criterion [19], the Ma and Bao (MB) criterion [20], and the method based on symmetric extensions and semidefinite programming (SDP) [21,22]. We also used the algorithm proposed in Ref. [23] to prove separability of quantum states. This allows to compute values of p, for which ̺ a (p) is provably separable. The results are given in Fig. 1. One clearly sees that Observation 1 provides the best criterion, but the comparison with the separability algorithm also suggests that Observation 1 does not detect all states. Estimating the multipartite concurrence -For simplicity, we only discuss the three particle case, but our results can be directly generalized to arbitrary N -partite states. Let us consider a pure state |ψ in a d × d × d-system. Its concurrence is given by C τ (|ψ ) = [3 − (Tr̺ 2 1 + Tr̺ 2 2 + Tr̺ 2 3 )],(12) where ̺ 1 = Tr 23 (̺), etc. are the reduced one-particle states. For this definition, it directly follows that for pure states C τ (|ψ ) 2 = 1 2 ([C (1|23) (|ψ )] 2 + [C (2|13) (|ψ )] 2 + [C (3|12) (|ψ )] 2 ), where C (1|23) (|ψ ), etc. are the corresponding bipartite concurrences. This definition is extended to mixed states via the convex roof construction. Clearly, C τ (̺) = 0 if and only if ̺ is a fully separable state. A first possibility to estimate the multipartite concurrence is to start with an estimate of the bipartite concurrence for each bipartition (as in Observation 1), and then estimate the total concurrence C τ from it. This is indeed a viable way, in Appendix A3 [16] we present and discuss a corresponding theorem. The disadvantage of this approach is that there are states which are separable for any bipartition, but not fully separable [24]. For them, this method will not succeed, since all the bipartite concurrences vanish. To overcome this limitation, one should note that C τ (|ψ ) 2 is of the same structure as Eq. (11): we define the operators J t as before, but separately for any bipartition and write J 1|23 t = L 1 α ⊗S 23 β and similarly for the other bipartitions. Then we have the expression C τ (|ψ ) 2 := 1 2 t [| ψ|J 1|23 t |ψ * | 2 + | ψ|J 2|13 t |ψ * | 2 + | ψ|J 3|12 t |ψ * | 2 ]. So we have to consider ∆ tot k (̺, t, x) = max (0, λ (1) mn − i>1 λ (i) mn ),(13) where the λ (j) mn are the square roots of eigenvalues of X tot =̺ k s=1 (u s J 1|23 ts + v s J 2|13 ts + w s J 3|12 ts )̺ * × k s=1 (u * s J 1|23 ts + v * s J 2|13 ts + w * s J 3|12 ts )(14) in decreasing order. Here, x = ( u, v, w) denotes a triple of complex vectors which are normalized as in Observation 1 and t = {t 1 , ..., t k }. For this quantity we can state the following: Observation 2. For any arbitrary mixed state on H⊗ H ⊗ H and for every fixed k and for arbitrary x we have: N 6k 2 N k t [(∆ tot k (̺, t, x)) 2 ] ≤ C τ (̺) 2 .(15) A proof is given in the Appendix A4 [16]. Multipartite Examples -We will consider two simple examples for three qubits, but these already demonstrate two interesting points: first, they give an idea how the observables J t and the coefficients x can be chosen; second, it turns out that the entanglement criterion in Observation 2 is strong and allows to identify a novel family of bound entangled states. As the first example, we consider the three-qubit Greenberger-Horne-Zeilinger (GHZ) state |GHZ = (|000 + |111 )/ √ 2 and mix it with white noise, ̺ G (p) = p|GHZ GHZ| + (1 − p)1 1/8. Then we take the singlequbit operator S (a) = |0 1| − |1 0| and the two-qubit operator L (bc) = |00 11| − |11 00| and from them we form the operators J i|jk = S (i) ⊗ L (jk) for all three bipartitions. Applying Observation 2 for the choice k = 1 and u 1 = v 1 = w 1 = 1, one finds already from a single term in the sum of Eq. (15) that the three-qubit concurrence is bounded by (C τ [̺ G (p)]) 2 ≥ 1 6 3 4 [5p − 1] 2 .(16) For p = 1, this reproduces exactly concurrence of the pure GHZ state. Moreover, this bound shows that the state ̺ G (p) is entangled for p > 1/5. This means that Observation 2 provides a necessary and sufficient criterion for entanglement for the family of states ̺ G (p), since it is known that for p ≤ 1/5 these states are separable [25]. In fact, Eq. (16) gives a linear lower bound on the convex function C τ [̺ G (p)] and this bound coincides with the exact value on the points p = 1/5 and p = 1. This means that the bound equals the exact value on the whole interval p ∈ [1/5; 1] and for them we have C τ [̺ G (p)] = ( 3 4 [5p − 1])/ √ 6. As the second example, we consider the three-qubit W state |W = (|001 + |010 + |100 )/ √ 3 mixed with white noise, ̺ W (p) = p|W W | + (1 − p)1 1/8. In this case, we use again the operator S (a) = |0 1| − |1 0|, but for two qubits we use the L (bc) = |00 10|−|10 00| and from them we form the operators J i|jk = S (i) ⊗L (jk) . Applying Observation 2 for k = 1 and u 1 = v 1 = w 1 = 1, we find that (C τ [̺ W (p)]) 2 ≥ (1/96)[p(8 + √ 3) − √ 3] 2 , especially, the state ̺ W (p) is entangled for p > p s = √ 3/(8 + √ 3) ≈ 0.178. This is a remarkable value for several reasons. First, using the separability algorithm from Ref. [23], one can prove that the states ̺ W (p) are fully separable for p ≤ 0.177, giving strong evidence that Observation 2 provides a necessary and sufficient criterion for the family of states ̺ W (p). Second, these calculations show that the states ̺ W (p) exhibit quite peculiar entanglement properties: one can directly check that for p ≤ 3(8 √ 2 − 3)/119 ≈ 0.2096 the states have a positive partial transpose for any bipartition, and using the separability algorithm [23] one finds that for p ≤ 0.2095 the states are indeed separable for any bipartition. Hence, for p ∈ [p s ; 0.2095] the states ̺ W (p) are separable for any bipartition, but not fully separable. This implies that they are bound entangled: no entanglement can be distilled from them, even if two of the three parties join. It was known that such states exist [24], however, the existing examples required a sophisticated construction. It is surprising that the simple family ̺ W (p) includes bound entangled states and it underlines the power of our approach that these states can be detected with Observation 2. Finally, the bound entanglement in the family ̺ W (p) can easily be generated experimentally (contrary to other known examples of bound entangled states) since adding noise to a pure state is easy in practically any experimental implementation. Conclusion -We have provided a general method to bound entanglement monotones by extending in a nontrivial way the original construction of Wootters [2], an approach that works for both bipartite and multipartite concurrence. We leave open the problem of determining for which states our method gives the exact value of the concurrence. It would also be interesting to broaden our approach to the general classification of invariants of quantum states [9], since this may help to understand the different entanglement classes for multiparticle systems. We thank M. Hofmann for discussions. Z. C. is supported by NSF of China (11201427), Z. M. is supported by the NSF of China (10901103), O. G. is supported by the EU (Marie Curie CIG 293993/ENFOQI) and the BMBF (Chist-Era Project QUASAR), and S. S. is supported by the Royal Society. Let us write Γ (i) k = | ψ i |J t k |ψ * i |. By making use of the rule ( k j=1 x j ) 2 ≤ k k j=1 x 2 j and the Cauchy-Schwartz inequality we can estimate the following: t [∆ k (̺, t, u)] 2 ≤ t i p i k s=1 Γ (i) s 2 = i p 2 i t k s=1 Γ (i) s 2 + 2 i<j p i p j t k h=1 Γ (i) h k m=1 Γ (j) m ≤ i p 2 i t k k s=1 (Γ (i) s ) 2 + 2 i<j p i p j t k h=1 Γ (i) h 2 t k m=1 Γ (j) m 2 ≤ k i p 2 i t k s=1 (Γ (i) s ) 2 + 2k i<j p i p j t k h=1 (Γ (i) h ) 2 t k m=1 (Γ (j) m ) 2 = k 2 N N k i p 2 i N t=1 (Γ (i) t ) 2 + 2k 2 N N k i<j p i p j N t=1 (Γ (i) t ) 2 N t=1 (Γ (j) t ) 2 = k 2 N N k i p i N t=1 (Γ i t ) 2 2 = k 2 N N k C(̺) 2 .(17) We would like to add that for the case k = N also a different bound can be proved: Consider ∆ tot (̺, u) = ∆ N (̺, t, u) where u is now normalized as a vector, that is N s=1 u * s u s = 1. Note that since k = N , t is fixed and denotes all matrices J t . Then, one has that C(̺) ≥ ∆ tot (̺, u). This can be seen as follows. First, Eq. (8) in the main text can be proven just as before. Then, we have for the optimal decomposition ∆ tot (̺, u) ≤ i p i N t=1 | ψ i |u t J t |ψ * i | = i p i N t=1 |u t || ψ i |J t |ψ * i | ≤ i p i N t=1 | ψ i |J t |ψ * i | 2 = C(̺), where also the Cauchy-Schwartz inequality has been used. A2: The family of bound entangled states The family of bound entangled states from Ref. [18] are explicitly given by ̺ P H a = 1 8a + 1              0 1+a 2               .(19) These states have a non-negative partial transpose and are not distillable, but they are entangled for any 0 < a < 1. This concludes the proof. are the square roots of the eigenvalues of online) Detecting entanglement in the Horodecki 3×3 bound entangled state mixed with white noise. The criterion of Observation 1 (points denoted by OBS1) is stronger than previously known criteria. For values of p smaller than the values given by SEP the states ̺a(p) are proven to be separable. See text for further details. A3: Estimates on the multipartite concurrence from bipartite bounds We define the operators J k as before, but separately for any bipartition. That is, we write J 1|23 k = L 1 α ⊗S 23 β and similarly for the other bipartitions. Then, we obtain the expression C τ (|ψ )2where the numbers λ in non-increasing order and similarly for the other bipartitions. We can then state:Observation 3. Let ̺ be a density matrix on a tripartite d × d × d-system and consider for fixed k all the possible t and a possible choice of u as in Observation 1. Then, a lower bound on the multipartite concurrence is given byHere, the coefficients u can be chosen separately for any t and any bipartition. Especially, if ̺ is fully separable then ∆ a|bc k (̺, t, u) = 0 for any choice of k, t and u.Proof. First, one finds in the same way as in the bipartite case:and analogous bounds for the other bipartitions. It remains to bound C τ (̺) from the values of C 1|23 (̺), C 2|13 (̺) and C 3|12 (̺). For that, let us assume that ̺ = i p i |ψ i ψ i | is an optimal decomposition when computing the convex roof of C τ (̺). Let us denote ΘUsing the Cauchy-Schwartz inequality we have thatwhich proves the claim.A4: Proof of Observation 2First, as in Observation 1, we can prove thatBy denoting Θ . Quant. Inf. Comp. M. B. Plenio and S. Virmani71For reviews see: M. B. Plenio and S. Virmani, Quant. Inf. Comp. 7, 1 (2007); . R Horodecki, Rev. Mod. Phys. 81865R. Horodecki et al., Rev. Mod. Phys. 81, 865 (2009); . F Mintert, Phys. Rep. 4154F. Mintert et al., Phys. Rep. 415, 4 (2005); . O Gühne, G Tóth, Phys. Rep. 4741O. Gühne and G. Tóth, Phys. Rep. 474, 1 (2009). . W K Wootters, Phys. Rev. Lett. 802245W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998). . A Osterloh, Nature. 416608A. Osterloh et al., Nature 416, 608 (2002); . T J Osborne, M A Nielsen, Phys. Rev. A. 6632110T. J. Osborne and M. A. Nielsen, Phys. Rev. A 66, 032110 (2002). . A Uhlmann, Phys. Rev. A. 6232307A. Uhlmann, Phys. Rev. A 62 032307 (2000); . Entropy. 121799Entropy 12, 1799 (2010). . B M Terhal, K G H Vollbrecht, Phys. Rev. Lett. 852625B. M. Terhal and K. G. H. Vollbrecht, Phys. Rev. Lett. 85, 2625 (2000); . K G H Vollbrecht, R F Werner, Phys. Rev. A. 6462307K. G. H. Vollbrecht and R. F. Werner, Phys. Rev. A 64, 062307 (2001); . P Rungta, C M Caves, Phys. Rev. A. 6712307P. Rungta and C. M. Caves, Phys. Rev. A 67, 012307 (2003). . F Mintert, M Kus, A Buchleitner, Phys. Rev. Lett. 95260502F. Mintert, M. Kus, and A. Buchleitner, Phys. Rev. Lett. 95, 260502 (2005); . F Mintert, A Buchleitner, Phys. Rev. A. 7212336F. Mintert and A. Buchleitner, Phys. Rev. A 72, 012336 (2005); . K Chen, S Albeverio, S.-M Fei, Phys. Rev. Lett. 9540504K. Chen, S. Albeverio, and S.-M. Fei, Phys. Rev. Lett. 95, 040504 (2005); . J I De Vicente, Phys. Rev. A. 75E39903J. I. de Vicente, Phys. Rev. A 75, 052320 (2007); 77, 039903(E) (2008); . O Gittsovich, O Gühne, Phys. Rev. A. 8132333O. Gittsovich and O. Gühne, Phys. Rev. A 81, 032333 (2010). . E Gerjuoy, Phys. Rev. A. 6752308E. Gerjuoy, Phys. Rev. A 67, 052308 (2003); . Y.-C Ou, H Fan, S.-M Fei, Phys. Rev. A. 7812311Y.-C. Ou, H. Fan, and S.-M. Fei, Phys. Rev. A 78, 012311 (2008); . M Li, S.-M Fei, Z.-X Wang, J. Phys. A: Math. Theor. 42145303M. Li, S.-M. Fei, and Z.-X. Wang, J. Phys. A: Math. Theor. 42, 145303 (2009). . A Osterloh, J Siewert, Phys. Rev. A. 7212337A. Osterloh and J. Siewert, Phys. Rev. A 72, 012337 (2005). . S Hill, W K Wootters, Phys. Rev. Lett. 785022S. Hill and W. K. Wootters Phys. Rev. Lett. 78, 5022 (1997); . P Rungta, Phys. Rev. A. 6442315P. Rungta et al., Phys. Rev. A 64, 042315 (2001). Note that originally the concurrence was expressed differently via a spin-flip operation. 2, 10Note that originally the concurrence was expressed dif- ferently via a spin-flip operation [2, 10]. . S J Akhtarshenas, J. Phys. A: Math. Gen. 386777S. J. Akhtarshenas, J. Phys. A: Math. Gen. 38, 6777 (2005). This is also explicitely stated in Ref. This is also explicitely stated in Ref. [8] (b). Nielsen and I. Chuang, Quantum information and computation Cambridge University Press. See Theorem 2.6 on page 103 of M. Nielsen and I. Chuang, Quantum information and computation Cam- bridge University Press, 2011. Matrix analysis. Cambridge University PressSee Corollary 4.4.4 on page 204 of R. Horn and C. JohnsonSee Corollary 4.4.4 on page 204 of R. Horn and C. John- son, Matrix analysis, Cambridge University Press, 1990. See the Supplemental Material for the Appendices. See the Supplemental Material for the Appendices. Of course, for any observable one can always find a basis where it is symmetric. but in the present case this basis has to coincide with the basis where the complex conjugation |ψ * is takenOf course, for any observable one can always find a ba- sis where it is symmetric, but in the present case this basis has to coincide with the basis where the complex conjugation |ψ * is taken. . P Horodecki, Phys. Lett. A. 232333P. Horodecki, Phys. Lett. A 232, 333 (1997). . C J Zhang, Phys. Rev. A. 7760301C. J. Zhang et al., Phys. Rev. A 77, 060301(R) (2008); . O Gittsovich, Phys. Rev. A. 7852319O. Gittsovich et al., Phys. Rev. A 78, 052319 (2008). . Z Ma, M Bao, Phys. Rev. A. 8234305Z. Ma and M. Bao, Phys. Rev. A 82, 034305 (2010). We used the program available from www. A C Doherty, P A Parrilo, F M Spedalieri, Phys. Rev. Lett. 88187904A. C. Doherty, P. A. Parrilo, and F. M. Spedalieri, Phys. Rev. Lett. 88, 187904 (2002). We used the program available from www.iqi.caltech.edu/ documents/spedalieri/pptsetest1.m. ](b), the computable cross-norm or realignment (CCNR) criterion and the covariance matrix criterion. The bounds from Ref. We also considered the algebraic bound from Ref. [6](a) the quasipure approximation from Ref. 6] are of a similar strength as the SDP criterion, e.g. the algebraic bound from Ref. [6](a) results for a = 0.2 in p = 0.968 and the bound from Ref. [6](b) in p = 0.971. The other criteria are weak in comparison with the SDPWe also considered the algebraic bound from Ref. [6](a) the quasipure approximation from Ref. [6](b), the com- putable cross-norm or realignment (CCNR) criterion and the covariance matrix criterion. The bounds from Ref. [6] are of a similar strength as the SDP criterion, e.g. the algebraic bound from Ref. [6](a) results for a = 0.2 in p = 0.968 and the bound from Ref. [6](b) in p = 0.971. The other criteria are weak in comparison with the SDP. . 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{'abstract': 'Entanglement monotones, such as the concurrence, are useful tools to characterize quantum correlations in various physical systems. The computation of the concurrence involves, however, difficult optimizations and only for the simplest case of two qubits a closed formula was found by Wootters [Phys. Rev. Lett. 80, 2245(1998]. We show how this approach can be generalized, resulting in lower bounds on the concurrence for higher dimensional systems as well as for multipartite systems. We demonstrate that for certain families of states our results constitute the strongest bipartite entanglement criterion so far; moreover, they allow to recognize novel families of multiparticle bound entangled states.', 'arxivid': '1207.2889', 'author': ['Zhi-Hua Chen \nDepartment of Science\nZhijiang College\nZhejiang University of Technology\n310024HangzhouChina\n', 'Zhi-Hao Ma \nDepartment of Mathematics\nShanghai Jiaotong University\n200240ShanghaiChina\n', 'Otfried Gühne \nNaturwissenschaftlich-Technische Fakultät\nUniversität Siegen\nWalter-Flex-Straße 357068SiegenGermany\n', 'Simone Severini \nDepartment of Computer Science\nDepartment of Physics & Astronomy\nUniversity College London\nGower StWC1E 6BTLondonUnited Kingdom\n'], 'authoraffiliation': ['Department of Science\nZhijiang College\nZhejiang University of Technology\n310024HangzhouChina', 'Department of Mathematics\nShanghai Jiaotong University\n200240ShanghaiChina', 'Naturwissenschaftlich-Technische Fakultät\nUniversität Siegen\nWalter-Flex-Straße 357068SiegenGermany', 'Department of Computer Science\nDepartment of Physics & Astronomy\nUniversity College London\nGower StWC1E 6BTLondonUnited Kingdom'], 'corpusid': 17180291, 'doi': '10.1103/physrevlett.109.200503', 'github_urls': [], 'n_tokens_mistral': 10402, 'n_tokens_neox': 8951, 'n_words': 5474, 'pdfsha': '72c457582a80c95abe35e813f3ec4a11094c93be', 'pdfurls': ['https://arxiv.org/pdf/1207.2889v2.pdf'], 'title': ['Estimating entanglement monotones with a generalization of the Wootters formula', 'Estimating entanglement monotones with a generalization of the Wootters formula'], 'venue': []}

arxiv

Spectral Approach to the Relativistic Inverse Stellar Structure Problem II 1 Mar 2014 Lee Lindblom Theoretical Astrophysics California Institute of Technology 350-1791125PasadenaCA Nathaniel M Indik Theoretical Astrophysics California Institute of Technology 350-1791125PasadenaCA Spectral Approach to the Relativistic Inverse Stellar Structure Problem II 1 Mar 2014(Dated: March 4, 2014)arXiv:1310.0803v2 [astro-ph.HE]numbers: 0440Dg9760Jd2660Kp2660Dd The inverse stellar structure problem determines the equation of state of the matter in stars from a knowledge of their macroscopic observables (e.g. their masses and radii). This problem was solved in a previous paper by constructing a spectral representation of the equation of state whose stellar models match a prescribed set of macroscopic observables. This paper improves and extends that work in two significant ways: i) The method is made more robust by accounting for an unexpected feature of the enthalpy based representations of the equations of state used in this work. After making the appropriate modifications, accurate initial guesses for the spectral parameters are no longer needed so Monte-Carlo techniques can now be used to ensure the best fit to the observables.ii) The method is extended here to use masses and tidal deformabilities (which will be measured by gravitational wave observations of neutron-star mergers) as the macroscopic observables instead of masses and radii. The accuracy and reliability of this extended and more robust spectral method is evaluated in this paper using mock data for observables from stars based on 34 different theoretical models of the high density neutron-star equation of state. In qualitative agreement with earlier work, these tests suggest the high density part of the neutron-star equation of state could be determined at the few-percent accuracy level using high quality measurements of the masses and radii (or masses and tidal deformabilities) of just two or three neutron stars. I. INTRODUCTION The purpose of this paper is to improve and extend the spectral approach to solving the relativistic inverse stellar structure problem developed in our earlier paper, Lindblom and Indik [1]. In that approach the density ǫ and pressure p of the matter in a particular class of stars (e.g. neutron stars) are represented as faithful parametric expressions of the form: ǫ(h, γ k ) and p(h, γ k ), where h is the enthalpy of the material, and γ k are parameters that specify the particular equation of state. Faithful in this context means that any physical equation of state has such a representation while every choice of γ k represents a physically possible equation of state (cf. Lindblom [2]). Given a specific equation of state in this form, it is straightforward to solve the relativistic stellar structure equations to construct stellar models and their macroscopic observables, e.g. their masses M (h c , γ k ) and radii R(h c , γ k ). These macroscopic observables depend on the equation of state through the parameters γ k , as well as the central enthalpy h c (or equivalently the central pressure or density) of the particular stellar model. Our approach to the inverse stellar structure problem [1] determines the equation of state by adjusting the parameters γ k (and h i c ) in the model observables, e.g. M (h i c , γ k ) and R(h i c , γ k ), to match a set of prescribed values of those observables, e.g. M i and R i . The spectral approach to the relativistic inverse stellar structure problem (summarized above) was tested in our first paper, Lindblom and Indik [1], using mock observational data, M i and R i , constructed from 34 different theoretical models of the highest density part of the neutron-star equation of state. Sequences of approximate solutions to this problem were constructed by de-termining the spectral parameters γ k that minimize the quantity χ 2 defined by, χ 2 (γ k , h i c ) = 1 N stars Nstars i=1 log M (h i c , γ k ) M i 2 + log R(h i c , γ k ) R i 2 .(1) The accuracies of the resulting spectral equations of state were then evaluated by comparing with the exact equations of state. Those tests showed that the spectral equations of state provide good approximate solutions to the relativistic inverse stellar structure problem, with (average) error levels of just a few percent using (mock) observational data from only two or three stars. These tests also showed that the accuracy of the approximations got better (on average) when more data were used and more spectral parameters were fixed by the data. Unfortunately, our implementation of the spectral approach described above had a serious flaw. The method worked very well if the search for the minimum of χ 2 (γ k , h i c ) in Eq. (1) began with a reasonably accurate initial estimate for the spectral parameters γ k . Without an accurate initial guess, however, the code used to solve this non-linear least squares problem often crashed. This flaw made it impossible to perform searches for the true global minimum of χ 2 (γ k , h i c ), or to investigate the structure of that minimum (in γ k parameter space). One of the main objectives of this paper is to understand the cause of this problem, and to use this understanding to develop a more robust implementation of the spectral approach. The root problem turned out to be a subtle and unexpected feature of the enthalpy based representations of the equations of state. This feature is described in some detail in Sec. II, along with the changes in our initial implementation of the spectral approach to the inverse stellar structure problem needed to accommodate it. Using the resulting more robust approach, Sec. II contains a more thorough and systematic study of the mathematical convergence of the sequence of approximate spectral equations of state produced by this method. Our analysis of the relativistic inverse stellar structure problem up to this point has assumed that the masses and radii of neutron stars would be the first observables measured accurately. This may turn out to be the case, but the spectral approach for solving this problem does not (in principle) depend very strongly on exactly which observables are used. Recent work [3][4][5][6][7][8][9][10][11][12][13][14] has shown that gravitational-wave observations of binary neutron-star mergers should provide accurate measurements of the masses and tidal deformabilities of neutron stars once the advanced LIGO-VIRGO network of detectors becomes operational (within the next few years). The possibility of using this type of observational data to solve the inverse stellar structure problem is explored in Sec. III of this paper. A new and more efficient method for evaluating the tidal deformabilities Λ(h c , γ k ) of neutron-star models is presented in Appendix C, along with an efficient method for evaluating its derivatives with respect to the parameters h c and γ k . The inverse stellar structure problem is tested in Sec. III with masses and tidal deformabilities computed from the same catalog of 34 theoretical neutron-star equations of state used in our previous studies. These tests show that the high density part of the neutron-star equation of state could be determined using precision measurements of the masses and tidal deformabilities of just two or three neutron stars at about the same level of accuracy that could be achieved using mass and radius data. Our analysis of the relativistic inverse stellar structure problem (begun in our first paper [1] and continued here) focuses on understanding some of the fundamental mathematical aspects of this problem. Is it possible to determine the neutron-star equation of state exactly from a complete exact knowledge of the macroscopic observable properties of these stars, i.e., does this problem have a unique solution? Can numerical methods can be devised whose approximate solutions converge to the exact equation of state when a complete exact knowledge of the macroscopic observables of these stars is available? What level of numerical approximation and how many macroscopic observable data points are needed to achieve reasonable levels of accuracy for "realistic" neutron-star equations of state? A number of researchers have studied various observational and data-analysis questions associated with the inverse stellar structure problem, both in the context of using mass and radius observations [15][16][17][18][19], and in the context of using mass and tidal deformability measurements from gravitational-wave observations [3][4][5][6][7][8][9][10][11][12][13][14]. To our knowledge, our studies of the more fundamental questions about solving the inverse stellar structure problem described in our papers are unique. We discuss in more detail some of the basic differences between our results and those reported by others in Sec. IV. II. IMPROVING THE METHOD The spectral approach to the relativistic inverse stellar structure problem outlined above requires the use of a faithful parametric representation of the equation of state. There are a variety of ways to construct such representations (cf. Lindblom [2]), but the most useful for solving the relativistic stellar structure problem (and its inverse) represent the energy density ǫ and pressure p of the stellar matter as functions of the relativistic enthalpy: ǫ(h, γ k ) and p(h, γ k ). The parameters γ k specify the particular equation of state, and the relativistic specific enthalpy h is defined by the integral, h(p) = p 0 dp ′ ǫ(p ′ ) + p ′ .(2) Representing the equation of state in this way makes it possible to transform the stellar structure equations into a form that can be solved numerically more accurately and efficiently than the standard Oppenheimer-Volkoff form of the equations [1,20]. An important feature of the enthalpy (from the perspective of the inverse stellar structure problem) is the unexpected diversity of its high pressure limit: h ∞ ≡ lim p→∞ h(p). This limit is infinite in some equations of state, while it is finite in others. For example, an equation of state of the form ǫ = ǫ 0 + p, has an enthalpy given by h(p) = log 1 + 2p/ǫ 0 with h ∞ = ∞. However the equation of state ǫ = ǫ 0 e p/p0 − p, has an enthalpy given by h(p) = p 0 (1 − e −p/p0 )/ǫ 0 , with h ∞ = p 0 /ǫ 0 . This diversity in h ∞ complicates the problem of writing a robust code to find the minimum of χ 2 (γ k , h i c ). For any given equation of state, the parameters h i c that specify the central enthalpy of each stellar model must satisfy h i c ≤ h ∞ . Since h ∞ depends on the equation of state, these conditions on h i c also depend on the parameters γ k used to specify the particular equation of state: h i c ≤ h ∞ (γ k ) . Any algorithm that explores the structure of the function χ 2 (γ k , h i c ) to find its minimum, must therefore ensure that the inequalities h i c ≤ h ∞ (γ k ) are satisfied at every step of the process. We assumed (implicitly) in our original implementation of the spectral approach that h ∞ = ∞, so it seemed unnecessary to check the conditions h i c ≤ h ∞ (γ k ). This error is benign whenever the initial choices for the parameters γ k and h i c are close to a minimum where the conditions are satisfied. However, this limitation prevented us from exploring the structure of χ 2 (γ k , h i c ) except in the immediate neighborhood of a good initial estimate. Whenever the condition h i c ≤ h ∞ (γ k ) was violated for some reason, our original code produced unpredictable results: sometimes generating unphysical (e.g. negative) densities, and sometimes simply crashing. This limitation therefore prevented us from using Monte Carlo methods to explore the γ k and h i c parameter space more widely, and made it impossible to determine whether any particular local minimum of χ 2 (γ k , h i c , ) was also its global minimum. The minima of complicated non-linear functions like χ 2 (γ k , h i c ) are generally found numerically using iterative methods. At an abstract level these methods begin with some choice of the parameters which are then refined in some way to produce an estimate that is closer to a minimum. This process is repeated until an appropriate convergence criterion is satisfied. At each step in this process the parameters must satisfy h i c ≤ h ∞ (γ k ), or the code which evaluates ǫ(h, γ k , ) and p(h, γ k ) will fail whenever h enters the range h ∞ ≤ h ≤ h i c . The upper limit on the range of physical enthalpies h ∞ (γ k ) must therefore be re-evaluated at each step that changes the values of the spectral parameters γ k . Appendix A describes in detail how the value of a good estimate h max ≤ h ∞ (γ k ) can be determined for the spectral equations of state used in our approach. The conditions h i c ≤ h max are then checked at each step of the iterative process that finds the minimum of χ 2 (γ k , h i c ). If any of the h i c violate this condition at any step, then all the h i c at this step are scaled (down) so the conditions h i c ≤ h max are satisfied before proceeding. Testing and re-scaling the h i c (if necessary) at each step is the biggest improvement in our new more robust implementation of the spectral approach to the inverse stellar structure problem. With this change it becomes possible to use Monte Carlo methods to explore the global minimum of χ 2 (γ k , h i c ). This new improved implementation of the spectral approach to the relativistic inverse stellar structure problem has been tested using mock observational data for the masses and radii based on the 34 theoretical high-density neutron-star equations of state. These mock data sets consist of N stars [M i , R i ] data pairs, with the masses uniformly spaced between 1.2M ⊙ (a typical minimum mass for astrophysical neutron-stars) and the maximum mass M max for each theoretical equation of state. See Read, et al. [21] for descriptions of these 34 theoretical equations of state used in our tests, along with citations to the original nuclear physics papers on which they are based. The mock data used here differ in only two minor ways from those used in our original work [1]. First, the method of extrapolating above and below the highest and lowest entries in those tabulated theoretical equations of state was changed slightly for these new tests. The new versions of our interpolation and extrapolation formulas are given in detail in Appendix B, while the old version is described in Appendix B of Ref. [1]. The second change made some (minor) corrections to some of the theoretical equation of state tables. In particular we found that some of the tabulated equations of state were non-monotonic (and therefore non-physical) at a density of about 1.67 × 10 12 g/cm 3 . The effected equations of state were: APR1, APR2, APR3, APR4, ENG, H1, H2, H3, MPA1, MS1B, MS1, PCL2, PS, WFF1, WFF2, and WFF3. We corrected these problems simply by removing the one row in each table at the density where this nonmonotonicity occurred. The resulting interpolated equations of state are then monotonic. The result of these two minor changes made it possible to compute stellar models and their observational properties based on these tabulated equations of state more accurately and reliably. In these tests of our new improved implementation of the spectral approach to the inverse stellar structure problem, we begin the calculation of the minimum of χ(γ k , h i c ) by choosing a good initial estimate for the parameters γ k and h i c . We refine this initial estimate using the Levenberg-Marquardt algorithm [22] to find a local minimum of χ(γ k , h i c ). Once completed, we explore the neighborhood of this minimum by adding small random changes to each of the parameters γ k and h i c . The minimum of χ(γ k , h i c ) is then recomputed using Levenberg-Marquardt with these randomized initial parameter values. This process is repeated until a minimum is found with χ(γ k , h i c ) < 10 −10 , or until 100 subsequent randomized steps fail to reduce the smallest minimum further. The results of these tests are summarized in Table I. For each equation of state the inverse stellar structure problem has been solved by fitting N γ k different spectral parameters to mock data sets containing N = N stars = N γ k pairs of mass M i and radius R i data. The minimum value of the fitting function χ N is given for each of these solutions in Table I. Two additional quantities, ∆ MR N and Υ MR N are also included in Table I that measure how accurately the N parameter spectral equation of state agrees with the original used to compute the mock mass-radius observables. The function ∆ MR N is defined by: ∆ MR N 2 = 1 N eos Neos i=1 log ǫ(h i , γ k ) ǫ i 2 .(3) The sum in Eq. (3) therefore measures the average error in the spectral part of the equation of state [i.e., the part with densities above ǫ(h 0 )] that occur within neutron stars. 1 The best possible spectral fit to each of these theoretical neutron-star equations of state was determined in Ref. [2], and the average errors ∆ EOS N of those best N γ k parameter spectral fits are given in Table II of that reference. The quantity Υ MR N measures the relative accuracy between the N parameter spectral equation of state determined by solving the inverse stellar structure problem, and the best possible spectral fit: Υ MR N = ∆ MR N ∆ EOS N .(4) Except for the improvements described above, the tests performed here are identical to those performed in our original implementation of the spectral approach. The new results given in Table I are therefore directly comparable to those given in Table I of Lindblom and Indik [1]. The most obvious differences between the two tables are the values of χ N . All of the new χ N (except one) are less than our convergence criterion, χ N < 10 −10 , while in contrast very few of the original χ N were able to meet this condition. These improvements in the values of χ N are due (mostly) to the use of Monte Carlo methods to ensure that a global rather than just a local minimum of Table I = 0.017, The errors in the fits with N γ k = 2 and N γ k = 3 are almost identical to those from the original tests. But the errors in the fits with N γ k = 4 and N γ k = 5 are slightly larger. The basic reason for these differences comes from the simple fact that the original method used good initial estimates of the parameters γ k and h i c , followed by Levenberg-Marquardt minimization to find the nearest minimum. This local minimum was not always the global minimum of χ 2 (γ k , h i c ), and in some cases (especially for larger values of N γ k ) the real global minimum has somewhat larger equation of state errors than the local minimum. Despite these increases, however, the improved method still provides very good approximations to the neutron-star equation of state: i.e., average accuracy levels of just a few percent are achieved using using high precision (mock) observational data from just two or three stars. χ 2 (γ k , h i c ) is obtained. The parameters ∆ MR N in In a few cases, the equation of state errors ∆ MR N and Υ MR N in Table I are much larger than the values found using our original methods in Ref. [1]. In these cases the error quantities appear non-convergent as the number of parameters N γ k is increased. We now believe that the least squares method itself may be responsible for some of these failures. It is well known, for example, that interpolating polynomials constructed by least squares fits to data at equally spaced points are unreliable when N 2 > 4K, where N is the order of the polynomial fit and K the number of data points, cf §4.3.4 of Dahlquist and Björck [23]. When N exceeds this amount, the least squares method tends to produce fits that accurately pass through the K fixed data points, but oscillate wildly about the true solution between these points. This is referred to in the literature as the Runge phenomenon. While the particular non-linear least squares minimization used in our spectral method is not strictly equivalent to polynomial interpolation, our expectation is that our method probably exhibits some form of Runge phenomenon unless appropriate restrictions are made on the number of spectral parameters, i.e., some condition of the form N γ k < F (N stars ). At present we do not know an analytical expression for the function F (N stars ) that determines this stability criterion, but we can explore this question by examining the numerical convergence of our spectral equations of state. To do that we have examined in more detail the spectral solutions using mock observational data constructed from the PAL6 and the BGN1H1 equations of state. These cases represent the best (PAL6) and the worst (BGN1H1) spectral representations of the 34 equations of state used in our tests [1,2]. Figures 1 and 2 show the dependence of the error quantities ∆ MR Nγ k for these cases as functions of the number of data points N stars used in the solution. The results in the best case, Fig. 1, show the exponential spectral convergence that is expected in the high N limit. There are no significant changes in ∆ MR Nγ k (N stars ) as N stars is increased above the minimum N stars = N γ k , and ∆ MR Nγ k decreases exponentially as N γ k increases. The worst case, Fig. 2, shows definite signs of the Runge phenomenon. The error functions ∆ MR Nγ k (N stars ) for fixed N γ k in this case decrease significantly as N stars increases. The BGN1H1 equation of state has a strong phase transition in the density range where the spectral methods are used, so it is not really surprising that even in the large N stars limit the spectral equations of state in this case have yet to enter the convergent range for the relatively small values of N γ k used in these tests. The good news is that even in this terrible case, the errors in the inferred spectral equations of state are never worse than about 20%, and it appears that results in the 5-10% range can be obtained using high quality observational data from about six stars. EOS ∆ M R 2 ∆ M R 3 ∆ M R 4 ∆ M R 5 Υ M R 2 Υ M R 3 Υ M R 4 Υ M R 5 χ2 χ3 χ4 χ5 PAL6 0. We have also examined the numerical convergence of our spectral fits in more detail for several additional cases that show significant deviations from ideal convergence: PS, GS2, ALF1, and ALF3. The sequences of error measures ∆ MR Nγ k given in Table I Table I for those cases are in fact anomalous. The other cases, ALF1 and ALF3, that we have studied in more detail are more problematic. The results for the ALF3 case are shown in Fig. 5 . We do not know exactly what is causing this problem in these cases. One possibility is that our method for finding the minimum of χ 2 (γ k , h i c ) fails for some reason in these cases for larger values of N γ k . Another possibility is that these equations of state require more terms in their spectral expansions before they become truly convergent. All we can say at this point is that the spectral representations for these anomalous cases appear to be more reliable for solutions with smaller numbers of spectral parameters, i.e., the N γ k = 2 and N γ k = 3 cases, than they do for the solutions with larger numbers of parameters. III. TIDAL DEFORMABILITY When a star in a binary system interacts with the tidal field of its companion, it is deformed by an amount that depends on the internal structure of that star and hence the equation of state of the material from which it is made. These tidal deformations can significantly effect the phase evolutions of the last parts of the orbits of compact binary systems, so the gravitational waves emitted by such systems will contain the imprints of those tidal interactions [24][25][26][27][28][29]. Accurate observations of the gravitational waves from neutron-star binary systems will make it possible therefore to measure the tidal properties of these stars. A number of studies [3][4][5][6][7][8][9][10][11][12][13][14] have shown that the macroscopic neutron-star observable best determined by such gravitational-wave measurements are the masses M and the tidal deformabilities λ. This section explores the question, How well can the neutron-star equation of state be determined from accurate measurements of M and λ? The tidal deformability λ of a star is defined as the proportionality factor in the relationship between the tidal field from a star's companion, E ij , and the star's quadrupole moment, Q ij , induced by that tidal interaction: Q ij = −λE ij . This tidal deformability λ is related to the tidal Love number k 2 by λ = 2k 2 R 5 /3, and to the dimensionless tidal deformability Λ: Λ = λ/M 5 = (2k 2 /3)(R/M ) 5 . Some studies [8,12] suggest that the dimensionless tidal deformability Λ can be determined somewhat more accurately by gravitational wave observations than λ, so we use Λ in our analysis of this version of the inverse stellar structure problem. The equations needed to compute Λ (or equivalently λ or k 2 ) for relativistic neutron stars were first derived by Hinderer [4,5]. Appendix C presents a more efficient way to compute Λ(h c , γ k ), as well as its derivatives with respect to the parameters γ k and h c for the enthalpy based representations of the parametric equations of state used in our solution of the inverse stellar structure problem: ∂Λ/∂γ k and ∂Λ/∂h c . These derivatives are used by the Levenberg-Marquardt algorithm as part of our method of finding the global minimum of χ 2 (γ k , h i c ). The spectral approach to the solution of the inverse stellar structure problem described in Sec. I does not depend very strongly on which macroscopic observables are used. It is straightforward to replace the data for ob-served masses M i and radii R i , with those for observed masses M i and tidal deformabilities Λ i . The corresponding model observables, M (h i c , γ k ) and Λ(h i c , γ k ), are evaluated using our parametrized equations of state, ǫ(h, γ k ) and p(h, γ k ) with the methods described in Appendix C. The equation of state parameters γ k (and the central enthalpy parameters h i c ) are then fixed by minimizing the quantity χ(γ k , h i c ) that measures the differences between the observed data and the model observables: χ 2 (γ k , h i c ) = 1 N stars Nstars i=1 log M (h i c , γ k ) M i 2 + log Λ(h i c , γ k ) Λ i 2 .(5) We have tested the spectral approach to the relativistic inverse stellar structure problem (with the improvements described in Sec. II) using the masses and tidal deformabilities as observables. The mock observational data for the masses and tidal deformabilities used in these tests are based on the same selections of stellar models computed with the same 34 theoretical high-density neutron-star equations of state used in the tests described in Sec. II. The results of these tests are summarized in Table II. For each equation of state the inverse stellar structure problem has been solved by fitting N γ k different spectral parameters to mock data sets containing N = N stars = N γ k pairs of mass M i and tidal deformability Λ i data. The minimum value of the fitting function χ N is given for each of these solutions in Table II. Two additional quantities, ∆ MΛ N and Υ MΛ N are also included in Table II that The results for the M Λ case shown in Table II are very similar, both quantitatively and qualitatively, to those from the M R case shown in Table I. All of the χ N in Table II meet our convergence criterion χ N < 10 −10 , except the N γ k = 5 case of the PS equation of state. This is the same exceptional case as in Table I, suggesting there is some pathology in this particular equation of state that keeps our code from finding accurate reproducible solutions to the standard stellar structure problem. Similar problems were eliminated when we corrected the nonmonotonicity problems in some of the equations of state, as described in Sec. II. Unfortunately, we have not been able to identify any similar problem with the PS equation of state. The parameters ∆ MΛ N in Table II that quantify the errors in the spectral equations of state for the M Λ case are very similar to those found using using M R data in Table I . The averages of these quantities (over the 34 different theoretical equations of state) in these tests are = 0.029. The errors in the M Λ cases with N γ k = 2 and N γ k = 3 are almost identical to those from the analogous M R cases. But the errors in the cases with N γ k = 4 and N γ k = 5 are slightly larger. We don't know exactly why. We note that the M Λ cases with poorest convergence properties are the same ones that show poor convergence using M R data. This suggests that this anomalous behavior may be caused by some pathological feature of these particular equations of state, rather than some general failure of the method itself. IV. DISCUSSION In summary, we have improved our method of solving the relativistic inverse stellar structure problem using faithful spectral expansions of the unknown high density part of the equation of state. This method is based on minimizing a function χ that measures the differences between a given set of observables, e.g. [M i , R i ], and model values of these observables, e.g. M (h i c , γ k ) and R(h i c , γ k ). Our improved methods described in Sec. II are much better at finding the global minimum of this complicated non-linear function χ of the model parameters γ k and h i c . The numerical tests of our improved method, described in Sec. II, consistently give much smaller values of χ than those in the tests of our original method [1]. We have also expanded our new method in Sec. III to solve the inverse stellar structure problem using the mass and tidal deformability of a star as the observables: [M i , Λ i ]. To do this we have developed (in Appendix C) more efficient and accurate ways to evaluate the tidal deformability Λ(h c , γ k ) and its derivatives with respect to h c and γ k . Our analysis of the relativistic inverse stellar structure problem, introduced in Refs. [1,20] and continued here in Secs. II and III, has focused on understanding some of the fundamental mathematical aspects of this problem. Is it possible to determine the neutron-star equation of state exactly from a complete knowledge of the macroscopic observable properties of these stars, i.e., does this problem have a unique solution? Can numerical methods be devised whose approximate solutions converge to the exact equation of state when a complete exact knowledge of the macroscopic observables of these stars is available? What level of numerical approximation and how many macroscopic observable data points are needed to achieve reasonable levels of accuracy for "realistic" neutron-star equations of state? While various observational and data-analysis questions related to this problem have been studied previously by a number of researchers, our studies of these fundamental questions are unique (to our knowledge). An essential element of any practical robust solution to the inverse stellar structure problem (in our opinion) is the use of faithful parametric representations of the equation of state. These faithful representations must not exclude any physically possible equation of state, and conversely no choice of parameters may correspond to a physically impossible equation of state. To our knowledge the only faithful parametric representations of the high density equation of state discussed in the literature are the piecewise-polytropic representations of Read, et al. [21], and our spectral representations [2] (which in general are somewhat more accurate for a given number of parameters than the piecewise-polytropes). Ozel and collaborators [15,17,18] and Steiner and collaborators [16,19] have used low-order piecewisepolytropic models of the equation of state to solve the inverse stellar structure problem using presently available mass and radius measurements of neutron stars. Both groups have studied the accuracy with which the presently available [M i , R i ] data have been determined observationally. Both groups have done careful studies of the effects of these measurement errors on the accuracy with which the parameters in their high-density equation of state models are determined in this way. However, neither group has considered some of the more fundamental questions like those studied here, e.g., how accurately their solutions to the inverse stellar structure problem represent the actual neutron-star equation of state, or whether their method converges when higher-order parametric equation of state models are used in the solution. A number of researchers have shown that tidal effects in compact binary systems can influence the gravitational waveforms they emit in an equation of state dependent way [24][25][26][27][28][29]. Flanagan and Hinderer showed that a neutron-star's tidal deformability was the particular stellar characteristic that determines the leading-order effect on these gravitational waveforms [3]. Hinderer was the first to derive the equations that determine the tidal deformability from the structure of a relativistic stellar model [4,5]. Hinderer and collaborators were the first to explore how the tidal deformability depends on the equation of state by evaluating it numerically for a number of theoretical neutron-star equations of state [7]. We have extended this basic formalism for evaluating the tidal deformability in this paper in two important ways. First, we derive (in Appendix C) an expression for the tidal deformability in terms of a solution to a first-order, rather than a second-order, differential equation. Our expression can therefore be evaluated numerically more accurately and efficiently. Second, we derive a set of differential equations whose solutions determine the variations of the tidal deformability with respect to the equation of state parameters. These expressions make it possible to determine these equation of state parameters from tidal deformability data more accurately and efficiently. A number of researchers have studied how the tidal deformability of neutron stars can be measured from observations of the gravitational waves emitted by compact binary systems [6][7][8][9][10][11][12][13][14]. These researchers have constructed post-Newtonian [7,13,14], effective one body [9], and numerical relativity models [6,8,[10][11][12] of the waveforms produced by these systems. They have also explored in great detail (using a variety of data-analysis methods) the expected accuracy with which the tidal deformability should be measured by the next generation of gravitational wave detectors (advanced LIGO, etc.). These researchers have shown, for example, that such measure-ments are likely to be accurate enough to distinguish between some of the published theoretical neutron-star equations of state. None of these studies, however, has considered any of the more fundamental questions about the relativistic inverse stellar structure problem that we consider here. They have not proposed a method for determining the equation of state itself from these gravitational wave measurements, nor have they estimated how accurately it could be determined. Our study presented in Sec. III of this paper is therefore unique (to our knowledge) in its exploration of some of the fundamental questions associated with the mass and tidal deformability version of the inverse stellar structure problem. The spectral approach to the solution of the inverse stellar structure problem introduced in Ref. [1] and improved and extended in Secs. II and III of this paper has been shown to be quite effective in determining the high-density neutron-star equation of state using highaccuracy measurements of the mass and radius (or the mass and tidal deformability) of just two or three neutron stars. However, many basic questions remain unanswered. The equations of state produced by our current implementation of the spectral approach do not converge to the exact equation of state in a few cases as the number of observational data points is increased. At the present time we do not understand the reason for this. More study of the mathematical properties of the inverse stellar structure problem is therefore needed to resolve these remaining questions. Our studies of the inverse stellar structure problem have also assumed that the observational data were ideal. Additional research is therefore needed to explore the robustness of our approach before it can be used as a practical tool for analyzing observational data. How do the errors in the approximate spectral equations of state change when more realistic [M i , R i ] or [M i , Λ i ] data are used? The data used in our tests were idealized in two important ways. First, the mock [M i , R i ] or [M i , Λ i ] data were supplied with very high precision. Real astrophysical measurements of these quantities will have significant errors. How will measurement errors influence the accuracy of the equation of state that is constructed by these techniques? Second, the mock [M i , R i ] or [M i , Λ i ] data used in our tests were chosen to cover uniformly the astrophysically relevant range of neutron-star masses. Real astrophysical measurements will not be distributed in such a complete and orderly way. How will the accuracy of the implied equation of state be affected by different, presumably less ideal, data distributions? In particular, how does the accuracy of the highest-density part of the equation of state depend on the mass of the most massive neutron-star for which observational data are available? , where h 0 is the lower bound on the enthalpy, h 0 ≤ h, in the domain where the spectral expansion is to be used. This is a standard spectral expansion of the function log Γ(h) in which the [log(h/h 0 )] k are the spectral basis functions and the γ k are the spectral expansion coefficients (or parameters). The equation of state functions p(h, γ k ) and ǫ(h, γ k ) are obtained from Γ(h, γ k ) by integrating the system of ordinary differential equations, dp dh = ǫ + p,(A3)dǫ dh = (ǫ + p) 2 pΓ(h) ,(A4) that follow from the definitions of h and Γ in Eqs. (2) and (A1). The general solution to these equations can be reduced to quadratures: p(h) = p 0 exp h h0 e h ′ dh ′ µ(h ′ ) ,(A5)ǫ(h) = p(h) e h − µ(h) µ(h) ,(A6) where µ(h) is defined as. µ(h) = p 0 e h0 ǫ 0 + p 0 + h h0 Γ(h ′ ) − 1 Γ(h ′ ) e h ′ dh ′ .(A7) The constants p 0 and ǫ 0 are defined by p 0 = p(h 0 ) and ǫ 0 = ǫ(h 0 ) respectively. Equations (A5)-(A7) show that ǫ(h) and p(h) are finite (for h 0 ≤ h < ∞) unless there exists an h = h ∞ where µ(h ∞ ) = 0. The problem of finding h ∞ is reduced therefore to the problem of finding the first zero of µ(h) above h 0 . It is not necessary for our purposes to know the exact value of h ∞ . Rather a firm estimate h max < h ∞ that is beyond the range of h occurring in neutron stars is all that is needed. Equation (A7) shows that µ(h 0 ) > 0 and that µ(h) is monotonically increasing unless Γ(h) < 1. The first step in finding a useful estimate h max is to evaluate Γ(h) (which can be done very efficiently) on a mesh of points covering the range h 0 ≤ h ≤ h 0 e 5 . If Γ(h) ≥ 1 throughout this range, then we simply set h max = h 0 e 5 . The upper limit of this range needs to be larger than any value of h that is likely to occur within a neutron star. For the cases we have studied the value h 0 e 5 is a factor of 4 or 5 larger than any h we have seen in a neutron-star model, but its value could (and should) be adjusted upward as needed. If one of the mesh points, h n , is found where Γ(h n ) < 1, then we evaluate µ(h) on a second mesh of points that covers the range h n ≤ h ≤ h 0 e 5 . If µ(h) is positive throughout this range, then we again set h max = h 0 e 5 . If µ(h) is found to become negative somewhere in this range then we use standard numerical root finding methods to determine the location of h ∞ where µ(h ∞ ) = 0. In this case we set h max = h ∞ . Appendix B: Interpolating and Extrapolating Equation of State Tables This appendix describes the method for interpolating between table entries for the exact equation of states used in the tests described here. This change was motivated by our need to find the tidal deformabilities Λ of stellar models with these equations of state. The equations that determine Λ depend on the adiabatic index of the material. In our original work the equation of state below the first table entry was assumed to have uniform density, and therefore infinite adiabatic index. This choice made it difficult therefore to evaluate Λ. Consequently the method used here to extrapolate below the lowest table entries has been changed. For clarity, this appendix provides a complete description of the interpolation methods used in this paper. We assume that the exact equation of state is represented as a table of energy densities ǫ i and corresponding pressures p i for i = 1, ..., N . For our purposes here we will convert these to an equation of state of the form ǫ = ǫ(h) and p = p(h) in the following way. We do this by assuming that the exact equation of state is obtained for values intermediate between those given in the table, ǫ i ≤ ǫ ≤ ǫ i+1 , by the interpolation formula: p p i = ǫ ǫ i ci+1 ,(B1)c i+1 = log(p i+1 /p i ) log(ǫ i+1 /ǫ i ) .(B2) For smaller values of the density than the lowest entry in the table, ǫ ≤ ǫ 1 , we assume, p p 1 = ǫ ǫ 1 5/3 ,(B3) and for larger values of the density than the highest entry, ǫ ≥ ǫ N , we assume, p p N = ǫ ǫ N cN . (B4) The low density extrapolation given in Eq. (B3) assumes that the equation of state is that of a low temperature non-relativistic Fermi gas with adiabatic index 5/3, while the high density extrapolation given in Eq. (B4) just extends the tabulated portion of the equation of state smoothly. Given this prescription for interpolation, it is straightforward to show that the values of the enthalpy h(p) = p 0 dp ′ ǫ(p ′ ) + p ′ ,(B5) are given at the table entry values h i = h(p i ), by h 1 = 5 2 log ǫ 1 + p 1 ǫ 1 ,(B6)h i+1 = h i + c i+1 c i+1 − 1 log ǫ i (ǫ i+1 + p i+1 ) ǫ i+1 (ǫ i + p i ) . (B7) The pressure is determined as a function of the enthalpy, by performing the integral in Eq. (B5) to give h(p), and then inverting. It is slightly more convenient to perform this inversion to give ǫ(h), from which it is straightforward to determine p(h) through Eqs. (B3) and (B1): ǫ(h) = ǫ 1 ǫ 1 p 1 exp 2h 5 − 1 3/2 (B8) for h ≤ h 1 , ǫ(h) = ǫ i ǫ i + p i p i exp c i+1 − 1 c i+1 (h − h i ) − ǫ i p i 1/(ci+1−1) (B9) for h i ≤ h ≤ h i+1 , and ǫ(h) = ǫ N ǫ N + p N p N exp c N − 1 c N (h − h N ) − ǫ N p N 1/(cN −1) ,(B10)for h ≥ h N . Appendix C: Computing Λ and its Derivatives A number of studies [3][4][5][6][7][8][9][10][11][12][13][14] have shown that the mass M and the tidal deformability λ are the neutron-star observables best measured by gravitational wave observations of neutron-star binary systems, while some studies [8,12] suggest that the dimensionless tidal deformability Λ = λ/M 5 can be determined somewhat more accurately than λ itself. Hinderer [4,5] derived the expressions for the tidal deformability λ, or equivalently the dimensionless tidal deformability Λ of a relativistic stellar model, in terms of the gravitational compactness C = M/R and a quantity Y that measures the relativistic quadrupole gravitational potential induced by the tidal deformation. Using those expressions, the dimensionless tidal deformability Λ can be expressed in terms of C and Y in the following way, Λ(C, Y ) = 16 15Ξ (1 − 2C) 2 [2 + 2C(Y − 1) − Y ], (C1) where Ξ is given by Ξ(C, Y ) = 4C 3 [13 − 11Y + C(3Y − 2) + 2C 2 (1 + Y )] +3(1 − 2C) 2 [2 − Y + 2C(Y − 1)] log(1 − 2C) +2C[6 − 3Y + 3C(5Y − 8)].(C2) This dimensionless tidal deformability Λ is the observable we use in our study of the inverse stellar structure problem in Sec. III of this paper. The gravitational compactness C = M/R of a relativistic stellar model is computed by solving the Oppenheimer-Volkoff equations: dm dr = 4πr 2 ǫ,(C3)dp dr = −(ǫ + p) m + 4πr 3 p r(r − 2m) . (C4) The radius of the star, R, is the surface where the pressure vanishes, p(R) = 0, while the star's total mass, M , is M = m(R). 2 The relativistic quadrupole gravitational potential, H, induced by the tidal interaction is determined by solving the Regge-Wheeler equation (cf Hinderer [4,5]): 0 = d 2 H dr 2 + 2 r + 2m + 4πr 3 (p − ǫ) r(r − 2m) dH dr + 4πr 5ǫ + 9p + (ǫ + p) 2 pΓ − 6 r − 4(m + 4πr 3 p) 2 r 2 (r − 2m) H r − 2m . (C5) 2 In this paper we use geometrical units in which the gravitational constant G and the speed of light c are one: G = c = 1. The potential Y that appears in the expression for the tidal deformability Λ, Eq. (C1), is defined as Y = (R/H)(dH/dr) evaluated at the surface of the star. Since H itself does not enter the expression for Λ, it is more efficient to transform Eq. (C5) into a form that determines only the part of the potential that is needed: y = r H dH dr . (C6) The resulting first-order equation for y is given by, 0 = dy dr − y 2 r − r + 4πr 3 (p − ǫ) r(r − 2m) y + 4(m + 4πr 3 p) 2 r(r − 2m) 2 + 6 r − 2m − 4πr 2 r − 2m 5ǫ + 9p + (ǫ + p) 2 pΓ . (C7) The potential Y that appears in the expression for Λ is just the surface value of the potential y determined by solving Eq. (C7): Y = y(R). The solutions to Eqs. (C3), (C4), and (C7) therefore determine the mass M , the radius R and the quadrupole deformation Y of a relativistic stellar model. The tidal deformability Λ is then determined algebraically from Eq. (C1) with C = M/R. This third-order system of ordinary differential equations to determine M and Λ is therefore more efficient to solve numerically than the original fourth-order system, Eqs. (C3), (C4), and (C5), derived by Hinderer [4,5]. In our previous work on the inverse stellar structure problem [1], we found that the Oppenheimer-Volkoff equations could be solved more accurately and efficiently by transforming them into a form that determines the mass m(h) and radius r(h) as functions of the relativistic enthalpy h. We use this same transformation in this work to change Eq. (C7) for the relativistic quadrupole deformation y(r) into an equation for y(h). The resulting transformed stellar structure equations are, p) − 16πr 3 r − 2m ,(C33) ∂Y ∂Γ = − 4πr 3 (ǫ + p) 2 pΓ 2 (m + 4πr 3 p) . (C34) For the case when η = γ k , the derivatives ∂ǫ/∂γ k , ∂p/∂γ k and ∂Γ/∂γ k are determined from the equations that determine the spectral representation of the equation of state. The needed expressions are given by, The integrals needed to determine these quantities can be performed accurately and efficiently using Gaussian quadrature. The equation of state does not depend on the parameter h c , and so ∂ǫ/∂h c = ∂p/∂h c = ∂Γ/∂h c = 0. Consequently the equations that determine ∂m/∂h c , ∂r/∂h c and ∂y/∂h c in Eqs. (C19)-(C21) are somewhat simpler than those for ∂m/∂γ k , ∂r/∂γ k and ∂y/∂γ k . ∂μ(h) ∂γ k = h h0 log h ′ h 0 k e h ′ dh ′ Γ(h ′ ) , (C35) ∂p(h) ∂γ k = −p(h) h h0 ∂μ(h ′ ) ∂γ k e h ′ dh ′ [μ(h ′ )] 2 ,(C36) The functions ∂m/∂η, ∂r/∂η and ∂y/∂η are determined by solving Eqs. (C19)-(C21) numerically. This can be done by integrating them from the center of the star where h = h c out to the surface of the star where h = 0. To do this we need to impose the appropriate boundary conditions for these functions at h = h c . The needed boundary conditions can be found by differentiating the power series solutions, Eqs. (C11)-(C13) with respect to the parameters η. The quantities r 1 , r 3 , m 3 , m 5 , and y 2 which appear in these power series solutions, depend on the central values of the thermodynamic quantities ǫ c = ǫ(h c ), p c = p(h c ), and Γ c = Γ(h c ), and through them the parameters η = {h c , γ k }. For the case where η = γ k these derivatives can be written as ∂r(h) ∂γ k = ∂r 1 ∂ǫ c ∂ǫ c ∂γ k + that quantify the errors in the spectral equations of state are slightly larger (on average) than those obtained using our original implementation of the method. The averages of these quantities (over the 34 different theoretical equations of state) in the new tests are ∆ of the number of mass-radii data points, Nstars, used to fix the spectral parameters γ k in an Nγ k parameter spectral expansion. These results use mass-radius data computed with the PAL6 equation of state. 35 3.2 × 10 −11 6.9 × 10 −11 6.7 × 10 −11 8.0 × 10 −11 BGN1H1 0.1352 0.1702 0.1356 0.1382 1.54 3.40 3.06 3.94 3.4 × 10 −11 5.2 × 10 −11 6.4 × 10 −11 9.2 × 10 −11 GNH3 0.0174 0.0183 0.0389 0.0171 1.27 1.92 4.72 2.93 8.5 × 10 −12 3.0 × 10 −11 5.7 × 10 −11 9.5 × 10 −11 H1 0.0294 0.0161 0.0127 0.0105 1.44 1.29 1.47 1.45 4.5 × 10 −11 4.1 × 10 −11 7.2 × 10 −11 9.3 × 10 −11 H2 0.0211 0.0279 0.0146 0.0221 1.19 2.01 2.12 3.22 1.7 × 10 −11 6.4 × 10 −11 4.3 × 10 −11 8.5 × 10 −11 H3 0.0139 0.0201 0.0176 0.0097 1.09 1.79 2.08 1.39 3.1 × 10 −11 3.8 × 10 −11 9.7 × 10 −11 9.9 × 10 −11 H4 0.0132 0.0259 0.0187 0.0105 1.28 2.60 2.76 1.56 8.3 × 10 −11 6.1 × 10 −11 5.7 × 10 −11 8.5 × 10 −11 H5 0.0140 0.0296 0.0118 0.0160 1.02 2.21 2.00 3.25 3.6 × 10 −11 1.0 × 10 −10 6.4 × 10 −11 8.7 × 10 −11 H6 0.0150 0.0141 0.0205 0.0157 1.09 1.03 1.57 1.38 9.1 × 10 −11 9.8 × 10 −11 7.9 × 10 −11 1.0 × 10 −10 H7 0.0134 0.0212 0.0124 0.0129 1.09 1.88 2.17 2.28 1.9 × 10 −11 8.3 × 10 −11 9.4 × 10 −11 8.5 × 10 −11 PCL2 0.0374 0.0152 0.0101 0.0250 1.35 1.16 1.04 3.06 4.8 × 10 −11 6.4 × 10 −11 2.6 × 10 −11 9.3 × 10 −11 ALF1 0.0796 0.0664 0.1040 0.0768 1.08 1.39 2.59 2.70 6.8 × 10 −11 5.6 × 10 −11 9.3 × 10 −11 9.0 × 10 −11 ALF2 0.0723 0.0598 0.0485 0.0218 1.04 1.21 1.75 1.22 5.8 × 10 −11 8.1 × 10 −11 8.6 × 10 −11 9.3 × 10 −11 ALF3 0.0404 0.0178 0.0202 0.1229 1.04 1.19 1.43 9.13 2.2 × 10 −11 5.2 × 10 −11 8.8 × 10 −11 3.1 × 10 −11 ALF4 0.0839 0.0182 0.0218 0.0394 1.18 1.35 2.19 4.15 7.6 × 10 −11 5.2 × 10 −11 9.1 × 10 −11 9.2 × 10 −11 Averages 0.0396 0.0289 0.0276 0.0239 1.22 1.65 2.14 2.77 clearly appear to be nonconvergent for those cases. The PS equation of state is also anomalous because it is the only case where our method fails to find a minimum of χ 2 (γ k , h i c ) satisfying our convergence criterion:χ(γ k , h i c ) ≤ 10 −10 . Figures 3 and 4 show the error quantities ∆ MR Nγ k for the PS and the of the number of mass-radii data points, Nstars, used to fix the spectral parameters γ k in an Nγ k parameter spectral expansion. These results use mass-radius data computed with the BGN1H1 equation of state. GS2 cases as functions of the number of data points N stars used to construct the solutions. These cases both show definite signs of the Runge phenomenon: the error functions ∆ MR Nγ k (N stars ) for fixed N γ k decrease significantly as N stars increases. So the unexpectedly large values of ∆ MR Nγ k (N stars ) seen in the N γ k = N stars solutions reported in of the number of mass-radii data points, Nstars, used to fix the spectral parameters γ k in an Nγ k parameter spectral expansion. These results use mass-radius data computed with the PS equation of state. FIG. 4 : 4Equation of state errors ∆ M RNγ k as functions of the number of mass-radii data points, Nstars, used to fix the spectral parameters γ k in an Nγ k parameter spectral expansion. These results use mass-radius data computed with the GS2 equation of state. of the number of mass-radii data points, Nstars, used to fix the spectral parameters γ k in an Nγ k parameter spectral expansion. These results use mass-radius data computed with the ALF3 equation of state. measure how accurately the N parameter spectral equation of state agrees with the original used to compute the mock mass and tidal deformability observables. These equation of state error measures, ∆ MΛ N and Υ MΛ N , are defined exactly as they were for the spectral equations of state computed from mass-radius data in Eqs. (3) and (4). II: Accuracies of the neutron-star equations of state obtained by solving the inverse stellar structure problem. ∆ M Λ N measures the average fractional error of the equation of state obtained by fitting to N different [Mi, Λi] data pairs. The parameter Υ M Λ N measures the ratio of ∆ M Λ N to the accuracy of the optimal N -parameter spectral fit to each equation of state. The parameter χN measures the accuracy with which the model masses M (h i c , γ k ) and tidal deformability Λ(h i c , γ k ) produced by the approximate spectral equation of state match the exact Mi and Λi data. 10 −12 7.6 × 10 −12 3.7 × 10 −12 8.0 × 10 −11 BBB2 0.0344 0.0368 0.0357 0.0143 1.04 1.28 1.64 1.55 6.8 × 10 −12 6.0 × 10 −12 1.9 × 10 −11 2.5 × 10 −11 BPAL12 0.0184 0.0118 0.0076 0.0090 1.07 1.19 1.54 4.04 9.3 × 10 −12 2.2 × 10 −11 1.3 × 10 −11 9.8 × 10 −11 ENG 0.0219 0.0243 0.0207 0.0520 1.08 1.31 1.40 4.62 4.0 × 10 −12 4.1 × 10 −12 2.2 × 10 −11 1.7 × 10 −11 MPA1 0.0301 0.0043 0.0061 0.0081 1.17 1.33 1.98 3.58 1.4 × 10 −11 1.3 × 10 −11 1.5 × 10 −11 1.6 × 10 −11 MS1 0.0465 0.0141 0.0129 0.0008 1.62 2.49 3.56 2.44 1.7 × 10 −11 1.7 × 10 −11 9.8 × 10 −12 1.9 × 10 −11 MS2 0.0155 0.0042 0.0009 0.0005 1.32 1.80 2.18 3.20 1.6 × 10 −13 4.2 × 10 −13 5.6 × 10 −13 5.7 × 10 −13 MS1B 0.0304 0.0135 0.0084 0.0014 1.52 2.10 2.82 5.08 8.0 × 10 −12 5.1 × 10 −12 1.4 × 10 −11 1.5 × 10 −11 PS 0.1044 0.0740 0.1120 0.0439 1.66 2.46 3.73 2.62 3.7 × 10 −12 1.4 × 10 −11 2.9 × 10 −11 8.0 × 10 −5 GS1 0.1018 0.0648 0.0386 0.0493 1.14 1.68 1.02 1.97 3.0 × 10 −12 1.3 × 10 −12 2.6 × 10 −12 9.5 × 10 −12 GS2 0.0909 0.0855 0.1164 0.0537 1.50 1.95 2.67 1.70 2.7 × 10 −12 4.3 × 10 −12 1.6 × 10 −11 1.9 × 10 −11 BGN1H1 0.1356 0.1652 0.1445 0.1363 1.55 3.30 3.26 3.89 1.6 × 10 −11 1.7 × 10 −11 3.5 × 10 −11 4.6 × 10 −11 GNH3 0.0182 0.0171 0.0397 0.0216 1.32 1.80 4.82 3.70 7.2 × 10 −12 5.7 × 10 −12 6.3 × 10 −12 4.3 × 10 −11 H1 0.0309 0.0154 0.0124 0.0107 1.51 1.23 1.45 1.49 2.5 × 10 −11 3.6 × 10 −11 2.1 × 10 −11 3.6 × 10 −11 H2 0.0226 0.0265 0.0153 0.0263 1.27 1.90 2.22 3.83 1.6 × 10 −11 1.1 × 10 −11 1.5 × 10 −11 1.5 × 10 −11 H3 0.0151 0.0186 0.0177 0.0118 1.18 1.66 2.09 1.70 2.0 × 10 −11 1.4 × 10 −11 1.2 × 10 −11 4.2 × 10 −11 H4 0.0119 0.0256 0.0211 0.0141 1.15 2.57 3.11 2.09 2.1 × 10 −11 1.2 × 10 −11 1.6 × 10 −11 2.1 × 10 −11 H5 0.0141 0.0293 0.0145 0.0221 1.03 2.19 2.46 4.48 7.5 × 10 −12 1.9 × 10 −11 9.7 × 10 −12 1.4 × 10 −11 H6 0.0160 0.0144 0.0204 0.0160 1.16 1.05 1.56 1.40 1.1 × 10 −11 1.2 × 10 −11 1.2 × 10 −11 1.3 × 10 −11 H7 0.0142 0.0205 0.0136 0.0170 1.16 1.83 2.38 3.00 8.7 × 10 −12 1.1 × 10 −11 2.0 × 10 −11 2.3 × 10 −11 PCL2 0.0378 0.0154 0.0103 0.0288 1.37 1.18 1.07 3.52 1.2 × 10 −11 2.4 × 10 −11 2.2 × 10 −11 4.3 × 10 −11 ALF1 0.0795 0.0704 0.1427 0.1225 1.08 1.47 3.55 4.31 3.8 × 10 −11 8.9 × 10 −12 3.8 × 10 −11 3.3 × 10 −11 ALF2 0.0725 0.0630 0.0479 0.0225 1.04 1.28 1.73 1.26 1.3 × 10 −11 1.1 × 10 −11 1.6 × 10 −11 2.3 × 10 −11 ALF3 0.0408 0.0203 0.0200 0.1566 1.05 1.36 1.42 11.64 1.1 × 10 −11 1.2 × 10 −11 2.8 × 10 −11 9.2 × 10 −11 ALF40.0793 0.0193 0.0213 0.0600 1.12 1.43 2.14 6.33 7.9 × 10 −12 1.3 × 10 −11 1.5 × 10 −11 3.0 × 10 quantities M(m, r, ǫ, p), R(m, r, p) and Y(y, m, r, ǫ, p, Γ) merely represent the expressions on the right sides.The enthalpy based representation of the stellar structure Eqs. (C8)-(C10) are solved numerically by specifying conditions, m(h c ) = r(h c ) = 0 and y(h c ) = 2, at the center of the star where h = h c and then integrating these equations out to the surface of the star where is over the points, [ǫ i , h i ] from the tabulated theoretical equation of state table. Only the N eos points that lie in the range h 0 ≤ h i ≤ max h c are included in this sum, where h 0 is the lower limit of the spectral domain, and max h c is the central value of hin the maximum mass neutron star for this equation of state. The quantity ∆ MR N TABLE I : IAccuracies of the neutron-star equations of state obtained by solving the inverse stellar structure problem using mass-radius data. ∆ M R N measures the average fractional error of the equation of state obtained by fitting to N different [Mi, Ri] data pairs. The parameter Υ M R N measures the ratio of ∆ M R N to the errors in the optimal N -parameter spectral fit to each equation of state. The parameter χN measures the accuracy with which the model masses M (h i c , γ k ) and radii R(h i c , γ k ) produced by the approximate spectral equation of state match the exact Mi and Ri data. × 10 −11 4.4 × 10 −11 9.6 × 10 −11 6.8 × 10 −11 BPAL12 0.0181 0.0107 0.0068 0.0075 1.06 1.08 1.37 3.36 4.6 × 10 −12 1.3 × 10 −11 4.6 × 10 −11 5.6 × 10 −11 × 10 −11 5.6 × 10 −11 6.6 × 10 −11 8.9 × 10 −11 MS2 0.0159 0.0044 0.0009 0.0006 1.35 1.86 2.17 3.41 1.3 × 10 −15 8.6 × 10 −16 1.3 × 10 −15 1.1 × 10 −15 MS1B 0.0305 0.0149 0.0084 0.0017 1.53 2.32 2.85 6.08 6.6 × 10 −11 6.3 × 10 −11 9.4 × 10 −11 9.3 × 10 −11 PS 0.1047 0.0779 0.1125 0.0432 1.67 2.59 3.74 2.58 6.9 × 10 −11 7.8 × 10 −11 5.7 × 10 −11 1.5 × 10 −5 GS1 0.0965 0.0604 0.0388 0.0445 1.08 1.56 1.03 1.78 5.1 × 10 −12 2.0 × 10 −12 6.1 × 10 −12 1.8 × 10 −11 GS2 0.0885 0.0888 0.1144 0.0426 1.46 2.02 2.63 1.0034 0.0018 0.0007 0.0003 1.06 1.09 1.33 1.91 9.2 × 10 −12 4.1 × 10 −11 5.1 × 10 −11 5.2 × 10 −11 SLy 0.0107 0.0040 0.0022 0.0011 1.17 1.13 1.30 1.68 4.2 × 10 −11 5.3 × 10 −11 7.9 × 10 −11 8.3 × 10 −11 APR1 0.0746 0.0422 0.0314 0.0172 1.05 1.27 1.68 2.10 4.1 × 10 −11 2.2 × 10 −11 8.8 × 10 −11 9.4 × 10 −11 APR2 0.0313 0.0165 0.0094 0.0068 1.01 1.18 1.49 2.02 3.9 × 10 −11 8.8 × 10 −11 3.2 × 10 −11 7.2 × 10 −11 APR3 0.0266 0.0061 0.0030 0.0022 1.06 1.13 1.24 1.49 3.2 × 10 −11 2.4 × 10 −11 9.4 × 10 −11 9.8 × 10 −11 APR4 0.0258 0.0037 0.0017 0.0016 1.03 1.23 1.26 1.16 5.6 × 10 −11 3.4 × 10 −11 7.7 × 10 −11 8.1 × 10 −11 FPS 0.0047 0.0061 0.0096 0.0049 1.06 1.44 2.53 2.69 2.6 × 10 −11 3.7 × 10 −11 7.5 × 10 −11 8.3 × 10 −11 WFF1 0.0552 0.0169 0.0220 0.0158 1.04 1.59 3.19 2.41 9.6 × 10 −11 6.0 × 10 −11 6.6 × 10 −11 9.6 × 10 −11 WFF2 0.0277 0.0146 0.0084 0.0055 1.01 1.21 1.18 1.46 3.4 × 10 −11 6.4 × 10 −11 7.8 × 10 −11 9.5 × 10 −11 WFF3 0.0127 0.0147 0.0124 0.0110 1.14 1.43 2.09 1.98 3.0 × 10 −11 4.0 × 10 −11 7.1 × 10 −11 8.8 × 10 −11 BBB2 0.0332 0.0328 0.0303 0.0131 1.01 1.14 1.39 1.42 1.2 ENG 0.0204 0.0247 0.0201 0.0478 1.01 1.33 1.36 4.25 3.6 × 10 −11 5.6 × 10 −11 7.9 × 10 −11 9.7 × 10 −11 MPA1 0.0328 0.0040 0.0049 0.0057 1.27 1.23 1.60 2.50 7.8 × 10 −11 3.6 × 10 −11 2.3 × 10 −11 7.4 × 10 −11 MS1 0.0474 0.0157 0.0132 0.0009 1.65 2.77 3.63 2.49 5.6 , while those for the ALF1 case (not shown) are similar. These cases show no sign of the Runge phenomenon, yet the higher order errors ∆ MR in the ALF1 case) are much larger than the lower order errors ∆ MR5 (and ∆ MR 4 2 and ∆ MR 3 TABLE The tests of our solution to the [M i , Λ i ] version of the inverse stellar structure problem show that accurate measurements of [M i , Λ i ] data can determine the neutron-star equation of state about as accurately as it could using the same number of accurate [M i , R i ] data. Using only two [M i , R i ] or [M i , Λ i ] data points, this new method can determine the high density part of the neutron-star equation of state that is present in these stars with errors (on average) of just a few percent. Eqs. (C8)-(C10) with respect to η:The various derivatives ∂M/∂m, etc. are determined directly from the stellar structure equations, Eqs.d dh ∂m ∂η = ∂M ∂m ∂m ∂η + ∂M ∂r ∂r ∂η + ∂M ∂ǫ ∂ǫ ∂η + ∂M ∂p ∂p ∂η , (C19) d dh ∂r ∂η = ∂R ∂m ∂m ∂η + ∂R ∂r ∂r ∂η + ∂R ∂p ∂p ∂η , (C20) d dh ∂y ∂η = ∂Y ∂y ∂y ∂η + ∂Y ∂m ∂m ∂η + ∂Y ∂r ∂r ∂η + ∂Y ∂ǫ ∂ǫ ∂η + ∂Y ∂p ∂p ∂η + ∂Y ∂Γ ∂Γ ∂η . (C21) (C8)- (C10): ∂M ∂m = 8πr 3 ǫ − M m + 4πr 3 p , (C22) ∂M ∂r = −4πr 2 3pM + 2ǫ(2r − 3m) m + 4πr 3 p , (C23) ∂M ∂p = − 4πr 3 M m + 4πr 3 p , (C24) ∂M ∂ǫ = − 4πr 3 (r − 2m) m + 4πr 3 p , (C25) ∂R ∂m = 2r − R m + 4πr 3 p , (C26) ∂R ∂r = − 12πr 2 pR + 2(r − m) m + 4πr 3 p , (C27) ∂R ∂p = − 4πr 3 R m + 4πr 3 p , (C28) ∂Y ∂y = 1 + (r − 2m)2y + r − m − 4πr 3 ǫ m + 4πr 3 p , (C29) ∂Y ∂m = − (r − 2m)(y + 1)y (m + 4πr 3 p) 2 − (2y + 1)y m + 4πr 3 p − 4 r − 2m − (m − 4πr 3 ǫ)y (m + 4πr 3 p) 2 − 4πr 3 (5ǫ + 9p) − 6r (m + 4πr 3 p) 2 − 4πr 3 (ǫ + p) 2 pΓ(m + 4πr 3 p) 2 − 8(m + 4πr 3 p) (r − 2m) 2 , (C30) ∂Y ∂r = (y + 1)y m + 4πr 3 p − 12πr 2 p(r − 2m)(y + 1)y (m + 4πr 3 p) 2 − 12πr 2 p(m − 4πr 3 ǫ)y (m + 4πr 3 p) 2 − 12πr 2 ǫy m + 4πr 3 p + 12πr 2 (5ǫ + 9p) − 6 m + 4πr 3 p + 12πr 2 (ǫ + p) 2 pΓ(m + 4πr 3 p) − 12πr 2 p[4πr 3 (5ǫ + 9p) − 6r] (m + 4πr 3 p) 2 − 48πr 2 p r − 2m − 48π 2 r 5 (ǫ + p) 2 Γ(m + 4πr 3 p) 2 + 4(m + 4πr 3 p) (r − 2m) 2 , (C31) ∂Y ∂ǫ = 4πr 3 (5 − y) m + 4πr 3 p + 8πr 3 (ǫ + p) pΓ(m + 4πr 3 p) , (C32) ∂Y ∂p = − 4πr 3 y[(r − 2m)(y + 1) + m − 4πr 3 ǫ] (m + 4πr 3 p) 2 − 4πr 3 [4πr 3 (5ǫ + 9p) − 6r] (m + 4πr 3 p) 2 + 36πr 3 m + 4πr 3 p − 16π 2 r 6 (ǫ + p) 2 pΓ(m + 4πr 3 p) 2 + 8πr 3 (ǫ + p) pΓ(m + 4πr 3 p) − 4πr 3 (ǫ + p) 2 p 2 Γ(m + 4πr 3 We follow the convention used in Read, et al.[21] and in Lindblom and Indik[1] and choose the density ǫ(h 0 ) at the lower end of the spectral domain to be about half nuclear density. We note that the power series expansion given in Eq. (16) of Hinderer[4,5] for H(r) near r = 0 contains a typographical error, which has been corrected in our derivation of Eqs. (C13) and (C18). AcknowledgmentsWe thank John Friedman, Tanja Hinderer, Benjamin Lackey, and Manuel Tiglio for helpful discussions concerning this research. A portion of this research was carried out during the time L.L. was a visitor at the Leonard E. Parker Center for Gravitation, Cosmology and Astrophysics, University of Wisconsin at Milwaukee. This research was supported in part by a grant from the Sherman Fairchild Foundation and by NSF grants PHY1005655 and DMS1065438.Appendix A: Estimating h∞(γ k )The parametric representation of the equation of state used in our analysis, ǫ = ǫ(h, γ k ) and p = p(h, γ k ), is constructed from a spectral expansion of the adiabatic index Γ(h) of the material[2]:= exp k γ k log h h 0 k h = 0. The right sides of Eqs. (C8)-(C10), i.e., the functions M(m, r, ǫ, p), R(m, r, p) and Y(y, m, r, ǫ, p, Γ) are singular at the center of the star h = h c . Consequently it is necessary to start any numerical integration of these equations slightly away from that singular point. The needed starting conditions can be obtained using a power series solution to the equations. The needed power series can be written in the form, 3where r 1 , r 3 , m 3 , m 5 and y 2 are given:The quantities ǫ c , p c and Γ c in these expressions are the energy density, pressure and the adiabatic index evaluated at the center of the star where h = h c : ǫ c = ǫ(h c ), p c = p(h c ), and Γ c = Γ(h c ). We obtain the total mass M (h c , γ k ) and dimensionless tidal deformation Λ(h c , γ k ) by solving Eqs. (C8)-(C10) numerically starting at h = h c using an equation of state with spectral parameters γ k . The total mass is simply the surface value M (h c , γ k ) = m(0) of this solution, while Λ(h c , γ k ) is determined from Eq. (C1) using the surface values C = m(0)/r(0) and Y = y(0). It will be useful for our least-squares minimization problem to know how the solutions to Eqs. (C8)-(C10) change as the parameters h c and γ k are varied. Let η denote any one of the parameters: η = {h c , γ k }. We wish to derive equations for the derivatives of the solutions to these equations with respect to these parameters: ∂m/∂η, ∂r/∂η and ∂h/∂η. It is straightforward to determine the needed auxiliary equations by differentiating,The derivatives of r 1 , r 3 , m 3 , m 5 and y 2 with respect to the parameters ǫ c , p c and Γ c which appear in Eqs. (C39)-(C41) are given by,The values of the derivatives ∂p c /∂γ k , ∂ǫ c /∂γ k and ∂Γ c /∂γ k are obtained by evaluating Eqs.For the case where η = h c the expressions for the derivatives ∂r/∂η, ∂m/∂η and ∂y/∂η have somewhat different forms because h c appears explicitly in the expansions in Eqs. (C11)-(C13). Differentiating these series with respect to h c , keeping only the leading terms, givesThe discussion to this point has shown how to evaluate the derivatives of M , R and Y with respect to the parameters η = {h c , γ k }. The quantity of primary interest in the discussion of the inverse stellar structure problem in Sec. III is the tidal deformability Λ. Its derivatives are determined by those of M , R and Y :The derivatives ∂Λ/∂C and ∂Λ/∂Y are given by,whereIn summary, the macroscopic stellar properties M , R and Y are determined by solving the stellar structure Eqs. (C8)-(C10). The dimensionless tidal deformability Λ is then determined algebraically from them using Eq. (C1). The derivatives of these properties ∂M/∂η, ∂R/∂η, and ∂Y /∂η with respect to the parameters η = {h c , γ k } are determined by solving the perturbed stellar structure Eqs. (C19)-(C21). The derivatives of the dimensionless tidal deformability ∂Λ/∂η are then determined algebraically from them using Eq. (C60). . 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{'abstract': 'The inverse stellar structure problem determines the equation of state of the matter in stars from a knowledge of their macroscopic observables (e.g. their masses and radii). This problem was solved in a previous paper by constructing a spectral representation of the equation of state whose stellar models match a prescribed set of macroscopic observables. This paper improves and extends that work in two significant ways: i) The method is made more robust by accounting for an unexpected feature of the enthalpy based representations of the equations of state used in this work. After making the appropriate modifications, accurate initial guesses for the spectral parameters are no longer needed so Monte-Carlo techniques can now be used to ensure the best fit to the observables.ii) The method is extended here to use masses and tidal deformabilities (which will be measured by gravitational wave observations of neutron-star mergers) as the macroscopic observables instead of masses and radii. The accuracy and reliability of this extended and more robust spectral method is evaluated in this paper using mock data for observables from stars based on 34 different theoretical models of the high density neutron-star equation of state. In qualitative agreement with earlier work, these tests suggest the high density part of the neutron-star equation of state could be determined at the few-percent accuracy level using high quality measurements of the masses and radii (or masses and tidal deformabilities) of just two or three neutron stars.', 'arxivid': '1207.3744', 'author': ['Lee Lindblom \nTheoretical Astrophysics\nCalifornia Institute of Technology\n350-1791125PasadenaCA\n', 'Nathaniel M Indik \nTheoretical Astrophysics\nCalifornia Institute of Technology\n350-1791125PasadenaCA\n'], 'authoraffiliation': ['Theoretical Astrophysics\nCalifornia Institute of Technology\n350-1791125PasadenaCA', 'Theoretical Astrophysics\nCalifornia Institute of Technology\n350-1791125PasadenaCA'], 'corpusid': 1081118, 'doi': '10.1103/physrevd.89.064003', 'github_urls': [], 'n_tokens_mistral': 24476, 'n_tokens_neox': 20571, 'n_words': 12665, 'pdfsha': 'f4b26a41266f4676a124aad09b9c7f37136f7301', 'pdfurls': ['https://arxiv.org/pdf/1310.0803v3.pdf'], 'title': ['Spectral Approach to the Relativistic Inverse Stellar Structure Problem II', 'Spectral Approach to the Relativistic Inverse Stellar Structure Problem II'], 'venue': []}

arxiv

Symmetric Morse potential is exactly solvable 28 Nov 2016 Ryu Sasaki ryu@yukawa.kyoto-u.ac.jp Faculty of Science Shinshu University 390-8621MatsumotoJapan Symmetric Morse potential is exactly solvable 28 Nov 2016piecewise analytic potentialsbound statesWhittaker functionpure imaginary zerosor- thogonality theoremsHilbert-Pólya conjectureassociated Hamiltonians Morse potential V M (x) = g 2 exp(2x) − g(2h + 1) exp(x) is defined on the full line, −∞ < x < ∞ and it defines an exactly solvable 1-d quantum mechanical system with finitely many discrete eigenstates. By taking its right half 0 ≤ x < ∞ and glueing it with the left half of its mirror(|x|)is obtained. The quantum mechanical system of this piecewise analytic potential has infinitely many discrete eigenstates with the corresponding eigenfunctions given by the Whittaker W function. The eigenvalues are the square of the zeros of the Whittaker function W k,ν (x) and its linear combination with W ′ k,ν (x) as a function of ν with fixed k and x. This quantum mechanical system seems to offer an interesting example for discussing the Hilbert-Pólya conjecture on the pure imaginary zeros of Riemann zeta function ζ(s) on Re(s) = 1 2 . Introduction This is a third paper discussing exact solvability of one dimensional quantum mechanical systems having piecewise analytic potentials which are mirror symmetric V (−x) = V (x) with respect to the origin. In previous works, a weak attractive piecewise analytic exponential potential V (x) = −g 2 exp(−|x|) [1] and a confining piecewise analytic exponential potential V (x) = g 2 exp(2|x|) [2,3] have been discussed. 1 The present work could be understood as a one parameter generalisation of these results [5]. The eigensystem of the right half of the Morse potential was discussed in detail by Lagarias [6]. The present paper is a modest supplement of this seminal work. The spectra or the eigenvalues of this type of exactly solvable quantum mechanical systems are very different from those of the 'ordinary' exactly solvable systems [7]- [12] based on shape invariance [13], [14]. In the latter, the n-th eigenvalue, E n counted from the ground state, is a simple elementary function of n; linear, quadratic, inverse quadratic or q-quadratic for those belonging to the 'discrete' quantum mechanics [15]- [17]. In contrast, the eigenvalues of the Hamiltonian with the symmetric Morse potential are the square of the zeros of the Whittaker function W k,ν (2g) [18] and its derivative regarded as a function of ν with fixed k and g (28)-(37). Correspondingly, the eigenvalues of the piecewise symmetric exponential potential [2,3] are the square of the zeros of the modified Bessel function of the second kind K ν (g) and its derivative regarded as a function of ν with fixed g. As is well known K ν (x) is related to the Whittaker W function (38). With this feature of the spectra, the Hamiltonian system with the symmetric Morse potential offers an interesting example for Hilbert-Pólya conjecture [19,20,6] on the pure imaginary zeros of Riemann zeta function ζ(s) on Re(s) = 1 2 . As for the asymptotic distribution of the zeros (eigenvalues), one can apply WKB approximation or Bohr-Sommerfeld quantum condition as demonstrated for the symmetric piecewise analytic exponential potential [2]. In this connection we would like to point out the usefulness of certain deformation procedures applicable to any 1-d quantum mechanical system including the discrete quantum mechanics, in which the Hamiltonian is a self-adjoint second order difference operator [15]- [17]. By multiple application of Crum's transformations [21], one can delete as many lowest lying eigenstates as wanted. By Krein-Adler transformations [22,23], finitely many eigenstates specified by the set D = {d 1 , . . . , d L } can be deleted so long as the labels satisfy conditions for the positivity of the norm. Here d j ∈ Z ≥0 is the label of the eigenfunction corresponding to the number of nodes. The present paper is organised as follows. In §2 the essence of the original Morse potential on the full line is summarised. Section three is the main part of the paper deriving the eigensystems of the symmetric Morse by imposing matching and finite norm conditions. In §3.2 the formulas of the Crum and Krein-Adler transformations are briefly recapitulated. The corresponding orthogonality relations of the deformed Hamiltonian systems are also presented. The final section is for a summary and comments. Original Morse Potential Let us recapitulate the main results of the 1-d quantum mechanical system with the original Morse potential defined on the full line: H = − d 2 dx 2 + g 2 e 2x − g(2h + 1)e x , −∞ < x < ∞, g > 0, h ∈ R.(1) For positive h > 0, the eigenvalue problem Hφ n (x) = E n φ n (x), n = 0, 1, . . . ,(2) has finitely many discrete eigenstates E n = −(h − n) 2 , n = 0, 1, . . . , [h] ′ ,(3)φ n (x) = φ 0 (x)P n ρ(x) , φ 0 (x) = e hx− 1 2 ρ(x) , ρ(x) def = 2g exp(x),(4)P n ρ(x) = ρ(x) −n L (2h−2n) n ρ(x) ,(5) in which [n] ′ means the greatest integer not exceeding n and L (α) n (x) is the Laguerre polynomial of degree n in x. Although the minimum of the potential exists for − 1 2 < h, minV M (x) = −(h + 1 2 ) 2 < 0 = V M (−∞), the system has continuous spectrum only for h ≤ 0. This is explained by the 'zero point energy'. The system is shape invariant as the potential of the first associated Hamiltonian H [1] (see (46) in §3.2) V [1] M (x) = V M (x) − 2∂ 2 x log φ 0 (x) = g 2 e 2x − g(2h − 1)e x ,(6) has the same form as V M (x) with h replaced by h − 1 and g remains unchanged. Symmetric Morse Potential Now let us discuss the Schrödinger equation (2), with the symmetric Morse potential V (x) = g 2 exp(2|x|) − g(2h + 1) exp(|x|) = 1 4 ρ(x) 2 − (h + 1 2 )ρ(x) = 1 4 ρ(x) 2 − kρ(x), k def = h + 1 2 , ρ(x) def = 2g exp(|x|).(7) Here we have introduced parameter k instead of h for convenience and the definition of ρ(x) is now mirror symmetric ρ(−x) = ρ(x). Now the potential grows indefinitely at the boundaries x = ±∞ and the system has infinitely many bound-states with positive eigenvalues E m > 0, on top of the finitely many negative eigenvalues E m < 0 which could exist when k > 0. The corresponding eigenfunctions must be normalizable, ψ m (x) ∈ L 2 (R). Since the potential is parity invariant, V (−x) = V (x) , the eigenfunctions are also parity invariant, ψ m (−x) = (−1) m ψ m (x) .(8) According to the conventional oscillation theorems [24] the subscript m counts the nodes in −∞ < x < ∞. Moreover, we may only consider the positive half-line x ≥ 0, even parity : ψ ′ 2n (0) = 0, odd parity : ψ 2n+1 (0) = 0 ,(9) i.e., with the eigenfunctions constrained by the parity-dependent boundary condition at the origin. One could say that 1-d quantum mechanical systems with mirror symmetric poten- tial V (−x) = V (x) Eigenfunctions Let us look for the solutions of Schrödinger equation (2) with the symmetric Morse potential (7) with positive energy E = ν 2 , ν > 0, in the following form: ψ(x) = ρ(x) − 1 2 φ ρ(x) .(10) It is now rewritten as that for the Whittaker function [18]: positive energy: d 2 φ(ρ) dρ 2 + − 1 4 + k ρ + 1 4 + ν 2 ρ 2 φ(ρ) = 0.(11) In the same ansatz (10) the solution with negative (non-positive) energy E = −µ 2 , µ ≥ 0 is rewritten as negative energy: d 2 φ(ρ) dρ 2 + − 1 4 + k ρ + 1 4 − µ 2 ρ 2 φ(ρ) = 0.(12) Among possible sets of general solutions, we choose the following Whittaker W functions. For the positive energy solutions even: ψ (e) (x) = ρ(x) − 1 2 A W k,iν ρ(x) + B W −k,iν −ρ(x) ,(13)odd: ψ (o) (x) = ρ(x) − 1 2 C W k,iν ρ(x) + D W −k,iν −ρ(x) ,(14) and for the negative energy solutions even: ψ (e) (x) = ρ(x) − 1 2 A W k,µ ρ(x) + B W −k,µ −ρ(x) ,(15)odd: ψ (o) (x) = ρ(x) − 1 2 C W k,µ ρ(x) + D W −k,µ −ρ(x) .(16) The matching condition at the origin (9) can be easily met by considering the derivative positive energy: dψ (e) (x) dx x=0 = − 1 2 ρ − 1 2 0 A −W k,iν (ρ 0 ) + 2ρ 0 W ′ k,iν (ρ 0 ) −B W −k,iν (−ρ 0 ) + 2ρ 0 W ′ −k,iν (−ρ 0 ) ,(17) negative energy: dψ (e) (x) dx x=0 = − 1 2 ρ − 1 2 0 A −W k,µ (ρ 0 ) + 2ρ 0 W ′ k,µ (ρ 0 ) −B W −k,µ (−ρ 0 ) + 2ρ 0 W ′ −k,µ (−ρ 0 ) ,(18) in which ρ 0 def = ρ(0) = 2g. It is a regular point of Whittaker W functions (13)- (16). As is clear from the equations (11), (12), ρ = 0 is a regular singular point with the characteristic exponents 1 2 ± iν and 1 2 ± µ, respectively. Since ρ = 0 is not included in the domain of the present problem, another set of solutions including the Whittaker M functions having these characteristic exponents is irrelevant. Thus, wave functions satisfying the matching conditions (9) at the origin can be easily found. For positive eigenvalues, they are ψ (e) (x) = ρ(x) − 1 2 A(k, ν, ρ 0 ) W k,iν ρ(x) + B(k, ν, ρ 0 ) W −k,iν −ρ(x) , A(k, ν, ρ 0 ) def = W −k,iν (−ρ 0 ) + 2ρ 0 W ′ −k,iν (−ρ 0 ),(19)B(k, ν, ρ 0 ) def = −W k,iν (ρ 0 ) + 2ρ 0 W ′ k,iν (ρ 0 ),(20)ψ (o) (x) = ρ(x) − 1 2 C(k, ν, ρ 0 ) W k,iν ρ(x) + D(k, ν, ρ 0 ) W −k,iν −ρ(x) ,(21)C(k, ν, ρ 0 ) def = −W −k,iν (−ρ 0 ), D(k, ν, ρ 0 ) def = W k,iν (ρ 0 ).(22) For negative eigenvalues, they are ψ (e) (x) = ρ(x) − 1 2 A(k, µ, ρ 0 ) W k,µ ρ(x) + B(k, µ, ρ 0 ) W −k,µ −ρ(x) , A(k, µ, ρ 0 ) def = W −k,µ (−ρ 0 ) + 2ρ 0 W ′ −k,µ (−ρ 0 ),(23)B(k, µ, ρ 0 ) def = −W k,µ (ρ 0 ) + 2ρ 0 W ′ k,µ (ρ 0 ),(24)ψ (o) (x) = ρ(x) − 1 2 C(k, µ, ρ 0 ) W k,µ ρ(x) + D(k, µ, ρ 0 ) W −k,µ −ρ(x) ,(25)C(k, µ, ρ 0 ) def = −W −k,µ (−ρ 0 ), D(k, µ, ρ 0 ) def = W k,µ (ρ 0 ).(26) The asymptotic condition at x → +∞, (ρ → +∞) is easily imposed. The Whittaker W function has the following asymptotic behaviour [18] W k,µ (x) ∼ e − 1 2 x x k 1 + O( 1 x ) ∼ W k,iν (x),(27) as |x| → ∞. For k < 0, the system has positive energy eigenstates only and they are numbered by the conditions even: − W k,iν 2n (ρ 0 ) + 2ρ 0 W ′ k,iν 2n (ρ 0 ) = 0, n = 0, 1, . . . ,(28) odd: W k,iν 2n+1 (ρ 0 ) = 0, n = 0, 1, . . . ,(29) with the corresponding eigenfunctions: ψ 2n (x) = ρ(x) − 1 2 W k,iν 2n ρ(x) , E 2n = ν 2 2n , n = 0, 1, . . . ,(30)ψ 2n+1 (x) = sign(x)ρ(x) − 1 2 W k,iν 2n+1 ρ(x) , E 2n+1 = ν 2 2n+1 , n = 0, 1, . . . ,(31)0 < g(g − k) < E 0 < E 1 < E 2 < · · · ⇔ g(g − k) < ν 0 < ν 1 < ν 2 < · · · .(32) For k > 0, there are approximately k − 1 eigenstates with negative energy. These eigenvalues are determined by the conditions even: − W k,µ 2n (ρ 0 ) + 2ρ 0 W ′ k,µ 2n (ρ 0 ) = 0, n = 0, 1, . . . ,(33) odd: W k,µ 2n+1 (ρ 0 ) = 0, n = 0, 1, . . . . The corresponding eigenfunctions are ψ 2n (x) = ρ(x) − 1 2 W k,µ 2n ρ(x) , E 2n = −µ 2 2n , n = 0, 1, . . . , ψ 2n+1 (x) = sign(x)ρ(x) − 1 2 W k,µ 2n+1 ρ(x) , E 2n+1 = −µ 2 2n+1 , n = 0, 1, . . . , .(35)− k 2 < E 0 < E 1 < E 2 < · · · < 0 ⇔ k > µ 0 > µ 1 > µ 2 > · · · > 0.(36) The eigenstates with positive eigenvalues are numbered after the negative ones. The eigenfunctions have the same form as (30), (31) and the eigenvalues are determined by the same conditions (28) and (29) but the numbering follows that of the negative energy ones. For k = 0 the symmetric Morse potential (7) reduces to the confining piecewise analytic exponential potential V (x) = g 2 exp(2|x|) discussed in a previous paper [2]. It has positive energy eigenvalues only and its eigenfunctions are the modified Bessel function of the second kind K iν (x), which is related to Whittaker W function ( [25] Vol. 1, §6.9 formula (14)) K α (x) = π 2x W 0,α (2x).(38) The factor 2 among the arguments is reflected by the factor two in the definitions of ρ(x) in [2] and in this paper. This also explains the extra factor ρ(x) − 1 2 in the wavefunction ψ(x) formula (10) compared to the counterpart in [2]. By using (38) one can deduce the condition (28) gives K ′ iν 2n (g) = 0 when k = 0. In this manner the exact solvability of the symmetric Morse potential (7) is established. The orthogonality relations among the eigenfunctions have the following forms: ∞ 0 e −x W k,iν 2n (2g e x )W k,iν 2m (2g e x ) dx ∝ δ n m ,(39)∞ 0 e −x W k,iν 2n+1 (2g e x )W k,iν 2m+1 (2g e x ) dx ∝ δ n m ,(40)∞ 0 e −x W k,µ 2n (2g e x )W k,µ 2m (2g e x ) dx ∝ δ n m ,(41)∞ 0 e −x W k,µ 2n+1 (2g e x )W k,µ 2m+1 (2g e x ) dx ∝ δ n m ,(42)∞ 0 e −x W k,µ 2n (2g e x )W k,iν 2m (2g e x ) dx ∝ δ n m ,(43)∞ 0 e −x W k,µ 2n+1 (2g e x )W k,iν 2m+1 (2g e x ) dx ∝ δ n m .(44) Deformed Hamiltonians When a 1-d Hamiltonian (Sturm-Liouville) system {H, E n , ψ n } is given, it is possible to construct deformed systems in which finitely many eigenvalues and corresponding eigenfunctions are deleted. The simplest one due to Crum [21] is to delete the lowest lying L levels are essentially iso-spectral, that is, the remaining eigenvalues are unchanged: {E j , ψ j (x)}, j = 0,H [L] ψ [L] n (x) = E n ψ [L] n (x), n = L, L + 1, . . . ,(45)H [L] def = H [0] − 2∂ 2 x log |W[ψ 0 , ψ 1 , . . . , ψ L−1 ](x)| ,(46)ψ [L] n (x) def = W[ψ 0 , ψ 1 , . . . , ψ L−1 , ψ n ](x) W[ψ 0 , ψ 1 , . . . , ψ L−1 ](x) , (ψ [L] n , ψ [L] m ) = L−1 j=0 (E n − E j )(ψ n , ψ m ),(47) in which the Wronskian of n-functions {f 1 , . . . , f n } is defined by formula W [f 1 , . . . , f n ](x) def = det d j−1 f k (x) dx j−1 1≤j,k≤n .(48) This result is obtained from a multiple application of the Darboux transformations. Another deformation method is due to Krein [22] and Adler [23]. It deletes finitely many eigenlevels specified by the set D = {d 1 , d 2 , . . . , d L }, d j ∈ Z ≥0 satisfying the conditions L j=1 (m − d j ) ≥ 0, ∀m ∈ Z ≥0 .(49) The deformed Hamiltonian system {H D , E n , ψ D;n (x)}, is given by Parallel expressions for the Krein-Adler deformations can be obtained easily. It is easy to see that the systems are parity invariant: H D ψ D;n (x) = E n ψ D;n (x), n ∈ Z ≥0 \D,(50)H D def = H [0] − 2∂ 2 x log |W[ψ d 1 , ψ d 2 , . . . , ψ d L ](x)| ,(51)ψ D;n (x) def = W[ψ d 1 , ψ d 2 , . . . , ψ d L , ψ n ](x) W[ψ d 1 , ψ d 2 , . . . , ψ d L ](x) , (ψ D;n , ψ D;m ) = L j=1 (E n − E d j )(ψ n ,V [L] (x) def = V (x) − 2∂ 2 x log |W[ψ 0 , ψ 1 , . . . , ψ L−1 ](x)| , V [L] (−x) = V [L] (x),(53)ψ [L] n (−x) = (−1) L+n ψ [L] n (x).(54) Because of the parity, the orthogonality relations among the even and odd eigenfunctions are trivial and those even-even and odd-odd δ n m ∝ (ψ [L] n , ψ [L] m ) = ∞ −∞ ψ [L] n (x)ψ [L] m (x)dx(55) can be rewritten as those on the positive x-axis δ n m ∝ ∞ 0 ψ [L] 2n (x)ψ [L] 2m (x)dx,(56)δ n m ∝ ∞ 0 ψ [L] 2n+1 (x)ψ [L] 2m+1 (x)dx.(57) In the following we consider the case of k ≤ 0 so that all the eigenvalues are positive. For the case k > 0, similar expressions can be obtained with relatively more notational com- W[ψ 0 , ψ n ](x) = W[W k,iν 0 (ρ), W k,iνn (ρ)](ρ), · · · · · · W[ψ 0 , ψ 1 , . . . , ψ L−1 , ψ n ](x) = ρ (L−1)(L+1)/2 · × W[W k,iν 0 (ρ), . . . , W k,iν L−1 (ρ), W k,iνn (ρ)](ρ).(58) It is straightforward to evaluate V [L] (x) asymptotically by using that of Whittaker W function W k,ν (x) (27): It has the form V [L] (x) = 1 4 ρ(x) 2 − (k − L)ρ(x) + O( 1 x ), |x| → ∞, which is not shape invariant but the parameter k(h) retains the property of the original Morse potential (6). The results obtained in the previous subsection can be stated as various Theorems on Whittaker W functions: Theorem 3.1 Pure imaginary zeros When Whittaker W functions W k,ν (x), d dx W k,ν (x) are regarded as functions of the parameter α for fixed k and x > 0, they have infinitely many pure imaginary zeros: −W k,iλ j (x) + 2x dW k,iλ j (x) dx = 0, 0 < x 2 < λ 0 < λ 1 < λ 2 < · · · ,(59)W k,iη j (x) = 0, 0 < x 2 < η 0 < η 1 < η 2 < · · · .(60) They are interlaced by the oscillation theorem: 0 < x 2 < λ 0 < η 0 < λ 1 < η 1 < λ 2 < η 2 < · · · .(61) Since the discrete eigenvalues of one dimensional quantum mechanics are always simple, all these zeros are also simple. ∞ x W k,iλ j (ρ)W k,iλ k (ρ) dρ ρ 2 = 0, j = k,(62) odd : ∞ x W k,iη j (ρ)W k,iη j (ρ) dρ ρ 2 = 0, j = k.(63) Let us denote these two types of zeros by one consecutive sequence ({ν j }): ν 0 ≡ λ 0 , ν 1 ≡ η 0 , ν 2 ≡ λ 1 , ν 3 ≡ η 1 , . . . , . The orthogonality relations of the eigenfunctions (56)-(57) of the L-th associated Hamiltonian system can be stated as odd : ∞ x W[W k,iν 0 , . . . ,W k,iν L−1 ,W k,iν 2n+1 ](ρ)W[W k,iν 0 , . . . ,W k,iν L−1 ,W k,iν 2m+1 ](ρ) W[W k,iν 0 , . . . , W k,iν L−1 ](ρ) 2 ρ 2(L−1) dρ = 0, n = m.(65) Theorem 3.2 is the special case (L = 0) of Theorem 3.3. As for the asymptotic distribution of the pure imaginary zeros {ν n }, n ≫ 1 [6], we can make a conjecture based on the WKB approximation or the so-called Bohr-Sommerfeld quantum condition p(x)dx = 2π(n + 1 2 ). Here p(x) is the momentum at x determined by the energy conservation p(x) 2 + g 2 e 2x − 2gke x = E n = ν 2 n . In terms of the elementary integral 4 log[(k+ √ k 2 +ν 2 n )/g] 0 ν 2 n − g 2 e 2x + 2gke x dx = 2π(n + 1 2 ),(66) the asymptotic dependence of ν n on n is obtained. Summary and Comments Following the examples of the weak attractive piecewise analytic exponential potential V (x) = −g 2 exp(−|x|) [1], the confining non-analytic exponential potential V (x) = g 2 exp(2|x|) [2,3] and the half line Morse potential [6], the exact solvability of the quantum system with the symmetric Morse potential (7) is demonstrated. Certain similarity to the original Morse potential is observed. Depending on the sign of the parameter k, the system has positive energy eigenstates only (k ≤ 0) and positive and finitely many negative eigenvalues (k > 0). The mirror symmetric potential imposes the Neumann boundary condition for the even level eigenfunctions and the Dirichlet for the odd level eigenfunctions. The eigenvalues are determined as the zeros of the Whittaker W function W k,ν (x) and its linear combination with W ′ k,ν (x) regarded as the function of ν with fixed k and x. Resulting orthogonality relations among the eigenfunctions are explored in some detail. Two types of deformed Hamiltonian systems are mentioned for possible relevance to Hilbert-Pólya conjecture. These deformations could be used to enhance the precision of numerical fitting of the eigenvalues of any model by allowing to delete finitely many eigenvalues subject to certain conditions. For possible relevance to Hilbert-Pólya conjecture, it would be desirable to generate many more quantum mechanical Hamiltonian systems having similar features to the present example, hopefully with more parameters. One direction would be to look for systems having (confluent) basic hypergeometric (q-hypergeometric) functions as eigenfunctions. The 'discrete' quantum mechanics [15,16,17] have many such examples. are equipped two types of eigenfunctions, one satisfying the Neumann boundary condition at x = 0 and the other the Dirichlet condition. The eigenvalues are selected by requiring the coefficients B(k, ν, ρ 0 ) (20) and D(k, ν, ρ 0 ) (22) of the divergent term W −k,iν −ρ(x) should vanish for the positive energy eigenstates and the coefficients B(k, µ, ρ 0 ) (24) D(k, µ, ρ 0 ) (26) of the divergent term W −k,µ −ρ(x) should vanish for the negative energy eigenstates. 1, . . . , L − 1 from the the original one-dimensional Hamiltonian system H = H [0] , {E n , ψ n (x)}, n = 0, 1, . . .. The deformed Hamiltonian systems H [L] L = 1, 2, . . ., ψ m ). (52) The above conditions on the set D (49) are necessary and sufficient for the positivity of norms and self-adjointness of the deformed Hamiltonian H D . Let us apply Crum's sequence to the present Hamiltonian (2), (7), (30)-(32), (35)-(37). W k,iν 0 , . . . , W k,iν L−1 , W k,iν 2n ](ρ)W[W k,iν 0 , . . . , W k,iν L−1 , W k,iν 2m ](ρ) W[W k,iν 0 , . . . , W k,iν L−1 ] Theorem 3.2 Orthogonality relation I The Whittaker W function with the above pure imaginary parameters {iλ j } (59), {iη j } (60) satisfy the following orthogonality relations(x > 0): even : The piecewise linear potential V (x) = |x|, exactly solvable by Airy functions[4], is probably the first example of this type. AcknowledgementsThe author thanks Jeffrey Lagarias for enlightening communication. He also thanks Milosh Znojil for many interesting discussions on exact solvability. One-dimensional Schrödinger equation with non-analytic potential V (x) = −g 2 exp(−|x|) and its exact Bessel-function solvability. R Sasaki, M Znojil, arXiv:1605.07310J. Phys. 4912pp445303Nr.. math-phR. Sasaki and M. Znojil, "One-dimensional Schrödinger equation with non-analytic po- tential V (x) = −g 2 exp(−|x|) and its exact Bessel-function solvability," J. Phys. A49 (2016) Nr.44 445303 (12pp), arXiv:1605.07310[math-ph]. Confining non-analytic exponential potential V (x) = g 2 exp (2|x|) and its exact Bessel-function solvability. R Sasaki, arXiv:1611.02467math-phR. Sasaki, Confining non-analytic exponential potential V (x) = g 2 exp (2|x|) and its exact Bessel-function solvability," arXiv:1611.02467[math-ph]. Symmetrized exponential oscillator. M , arXiv:1609.00166Modern Phys. Lett. 3111pp1650195quant-phM. Znojil, "Symmetrized exponential oscillator," Modern Phys. Lett. A31 (2016) 1650195 (11pp), arXiv:1609.00166[quant-ph]. O Vallée, M Soares, Airy Functions and Applications to Physics. LondonImperial College Press2nd ed.O. Vallée and M. Soares, Airy Functions and Applications to Physics, 2nd ed., Imperial College Press, London (2010). Morse potential, symmetric Morse potential and bracketed bound-state energies. M , arXiv:1603.09483Mod. Phys. Lett. 311650088quant-phM. Znojil, "Morse potential, symmetric Morse potential and bracketed bound-state energies," Mod. Phys. Lett. A31 (2016) 1650088, arXiv:1603.09483[quant-ph]. The Schrödinger operator with Morse potential on the right half-line. J C Lagarias, arXiv:0712.3238Commun. Number Theor. Phys. 2math.SPJ. C. Lagarias, "The Schrödinger operator with Morse potential on the right half-line," Commun. Number Theor. Phys. 2 (2009) 323-361, arXiv:0712.3238[math.SP]. The factorization method. L Infeld, T E Hull, Rev. Mod. Phys. 23L. Infeld and T. E. Hull, "The factorization method," Rev. Mod. Phys. 23 (1951) 21-68. Supersymmetry and quantum mechanics. F Cooper, A Khare, U Sukhatme, Phys. Rep. 251F. Cooper, A. Khare and U. Sukhatme, "Supersymmetry and quantum mechanics," Phys. Rep. 251 (1995) 267-388. An extended class of orthogonal polynomials defined by a Sturm-Liouville problem. D Gómez-Ullate, N Kamran, R Milson, arXiv:0805.3376arXiv:0807.3939J. Math. Anal. Appl. 162J. Approx Theory. math-phD. Gómez-Ullate, N. Kamran and R. Milson, "An extension of Bochner's problem: ex- ceptional invariant subspaces," J. Approx Theory 162 (2010) 987-1006, arXiv:0805. 3376[math-ph]; "An extended class of orthogonal polynomials defined by a Sturm- Liouville problem," J. Math. Anal. Appl. 359 (2009) 352-367, arXiv:0807.3939 [math-ph]. Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry. C Quesne, arXiv:0807.4087J. Phys. 41392001quant-phC. Quesne, "Exceptional orthogonal polynomials, exactly solvable potentials and super- symmetry," J. Phys. A41 (2008) 392001, arXiv:0807.4087[quant-ph]. Infinitely many shape invariant potentials and new orthogonal polynomials. S Odake, R Sasaki, arXiv:0906.0142Phys. Lett. 679math-phS. Odake and R. Sasaki, "Infinitely many shape invariant potentials and new orthogonal polynomials," Phys. Lett. B679 (2009) 414-417, arXiv:0906.0142[math-ph]. Exactly Solvable Quantum Mechanics and Infinite Families of Multi-indexed Orthogonal Polynomials. S Odake, R Sasaki, arXiv:1105.0508Phys. Lett. 702math-phS. Odake and R. Sasaki, "Exactly Solvable Quantum Mechanics and Infinite Families of Multi-indexed Orthogonal Polynomials," Phys. Lett. B702 (2011) 164-170, arXiv: 1105.0508[math-ph]. Derivation of exact spectra of the Schroedinger equation by means of supersymmetry. L E Gendenshtein, JETP Lett. 38L. E. Gendenshtein, "Derivation of exact spectra of the Schroedinger equation by means of supersymmetry," JETP Lett. 38 (1983) 356-359. Explicit wavefunctions for shapeinvariant potentials by operator technique. J W Dabrowska, A Khare, U P Sukhatme, J. Phys. A. 21J. W. Dabrowska, A. Khare and U.P. Sukhatme, "Explicit wavefunctions for shapein- variant potentials by operator technique," J. Phys. A 21 (1988) L195-L200. Discrete quantum mechanics," (Topical Review). S Odake, R Sasaki, arXiv:1104.0473J. Phys. 4447ppmath-phS. Odake and R. Sasaki, "Discrete quantum mechanics," (Topical Review) J. Phys. A44 (2011) 353001 (47pp), arXiv:1104.0473[math-ph]. Orthogonal Polynomials from Hermitian Matrices. S Odake, R Sasaki, arXiv:0712.4106J. Math. Phys. 4943pp53503math.CAS. Odake and R. Sasaki, "Orthogonal Polynomials from Hermitian Matrices," J. Math. Phys. 49 (2008) 053503 (43pp), arXiv:0712.4106[math.CA]. Exactly solvable 'discrete' quantum mechanics; shape invariance, Heisenberg solutions, annihilation-creation operators and coherent states. S Odake, R Sasaki, ProgS. Odake and R. Sasaki, "Exactly solvable 'discrete' quantum mechanics; shape invari- ance, Heisenberg solutions, annihilation-creation operators and coherent states," Prog. . arXiv:0802.1075Theor. Phys. 119quant-phTheor. Phys. 119 (2008) 663-700, arXiv:0802.1075[quant-ph]. E T Whittaker, G N Watson, A course of modern analysis. Cambridge Univ. Press Cambridge4th edE. T. Whittaker and G. N. Watson, A course of modern analysis, 4th ed. Cambridge Univ. Press Cambridge, (1927). Bemerkungüber die Integraldarstellung der Riemannschen ξ-Function. G Pólya, J. Reine Angew. Math. 305Acta Math.G. Pólya, "Bemerkungüber die Integraldarstellung der Riemannschen ξ-Function," Acta Math. 305 (1926) 305-317; "Über trigonometrische Integrale mit nur rellen Nullstellen," J. Reine Angew. Math. 158 (1927) 6-18. Using integrals of squares of certain real-valued special functions to prove that the Pólya Ξ * (z) function, the functions K iz (a), a > 0 and some other entire functions have only real zeros. G Gasper, arXiv:0801.2996World Sci. Publ. Topics in classical analysis and applications in honor of Daniel Waterman. math.CVG. Gasper, "Using integrals of squares of certain real-valued special functions to prove that the Pólya Ξ * (z) function, the functions K iz (a), a > 0 and some other en- tire functions have only real zeros." Topics in classical analysis and applications in honor of Daniel Waterman, 102-109, World Sci. Publ., Hackensack, NJ, (2008), arXiv:0801.2996[math.CV]. Associated Sturm-Liouville systems. M M Crum, arXiv:physics/9908019Quart. J. Math. Oxford Ser. 2M. M. Crum, "Associated Sturm-Liouville systems," Quart. J. Math. Oxford Ser. (2) 6 (1955) 121-127, arXiv:physics/9908019. On continuous analogue of Christoffel's formula in orthogonal polynomial theory. M G Krein, Doklady Acad. Nauk. CCCP. 113M. G. Krein, "On continuous analogue of Christoffel's formula in orthogonal polynomial theory," Doklady Acad. Nauk. CCCP, 113 (1957) 970-973. A modification of Crum's method. V É Adler, Theor. Math. Phys. 101V.É. Adler, "A modification of Crum's method," Theor. Math. Phys. 101 (1994) 1381- 1386. E Hille, Ordinary Differential Equations in the Complex Domain. New YorkWileyE. Hille, Ordinary Differential Equations in the Complex Domain, Wiley, New York, (1976). A Erdélyi, Higher Transcendental Functions. New YorkMcGraw-Hill Book Company1A. Erdélyi, ed. Higher Transcendental Functions, Vol. 1, 2, McGraw-Hill Book Company, New York (1953).

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{'abstract': 'Morse potential V M (x) = g 2 exp(2x) − g(2h + 1) exp(x) is defined on the full line, −∞ < x < ∞ and it defines an exactly solvable 1-d quantum mechanical system with finitely many discrete eigenstates. By taking its right half 0 ≤ x < ∞ and glueing it with the left half of its mirror(|x|)is obtained. The quantum mechanical system of this piecewise analytic potential has infinitely many discrete eigenstates with the corresponding eigenfunctions given by the Whittaker W function. The eigenvalues are the square of the zeros of the Whittaker function W k,ν (x) and its linear combination with W ′ k,ν (x) as a function of ν with fixed k and x. This quantum mechanical system seems to offer an interesting example for discussing the Hilbert-Pólya conjecture on the pure imaginary zeros of Riemann zeta function ζ(s) on Re(s) = 1 2 .', 'arxivid': '1611.05952', 'author': ['Ryu Sasaki ryu@yukawa.kyoto-u.ac.jp \nFaculty of Science\nShinshu University\n390-8621MatsumotoJapan\n'], 'authoraffiliation': ['Faculty of Science\nShinshu University\n390-8621MatsumotoJapan'], 'corpusid': 119158553, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 10903, 'n_tokens_neox': 9238, 'n_words': 4729, 'pdfsha': '7f2f689c14bfa0c219792f653cf33589126c74f0', 'pdfurls': ['https://arxiv.org/pdf/1611.05952v2.pdf'], 'title': ['Symmetric Morse potential is exactly solvable', 'Symmetric Morse potential is exactly solvable'], 'venue': []}

arxiv

Extended Addressing Machines for PCF, with Explicit Substitutions Benedetto Intrigila Giulio Manzonetto Nicolas Münnich Dipartimento di Ingegneria dell'Impresa Univ. USPN, Sorbonne Paris Cité, LIPN, UMR 7030 University of Rome "Tor Vergata" Italy CNRS F-93430VilletaneuseFrance Extended Addressing Machines for PCF, with Explicit Substitutions Theoretical Informatics ENTICS Proceedings of And Computer Science 1Addressing machinesPCFexplicit substitutionscomputational model Addressing machines have been introduced as a formalism to construct models of the pure, untyped λ-calculus. We extend the syntax of their programs by adding instructions for executing arithmetic operations on natural numbers, and introduce a reflection principle allowing certain machines to access their own address and perform recursive calls. We prove that the resulting extended addressing machines naturally model a weak call-by-name PCF with explicit substitutions. Finally, we show that they are also well-suited for representing regular PCF programs (closed terms) computing natural numbers.PreliminariesThe paradigmatic programming language PCF [23] is a simply typed λ-calculus enriched with constants representing natural numbers, the fundamental arithmetical operations, an if-then-else conditional instruction, and a fixed-point operator. We give PCF for granted and rather present EPCF, an extension of PCF with explicit substitutions[19]. We draw conclusions for the standard PCF by exploiting the fact that they are equivalent on programs (closed terms) of type int. Definition 1.1 Consider fixed a countably infinite set Var of variables. EPCF terms and explicit substitutions are defined by (for n ≥ 0 and x ∈ Var): L, M, N ::= x | M · N | λx.M σ | 0 | pred M | succ M | ifz(L, M, N ) | fix M σ, ρ ::= [x 1 ← (σ 1 , M 1 ), . . . , x n ← (σ n , M n )] As is customary, M · N stands for the application of a term M to its argument N , 0 represents the 6 The reader interested in a comparison with other abstract machines or formalisms is invited to consult [14].This technical appendix is devoted to provide the proofs that have been partially given, or completely omitted, in the body of the paper. As an abbreviation, we write IH for "induction hypothesis".A.1 Proofs of Section 1Proof.[Proof of Lemma 1.8] Items (i) and (ii) are straightforward. We prove (iii). Given an EPCF term M , an EPCF value V , an explicit substitution σ, a type α, and a context Γ such that σ ⊲ M ⇓ d V, σ |= Γ, Γ ⊢ M : α, we prove by induction on a derivation of σ ⊲ M ⇓ d V that ⊢ V : α.Case (nat): In this case M = V = n, for n ∈ N, and α = int. By the typing rules of EPCF, ⊢ n : int. Case (fun): In this case M = λx.M ′ ρ , V = λx.M ′ σ + ρ , α = β → γ. As Γ ⊢ λx. Introduction Turing machines (TM) and λ-calculus constitute two fundamental formalisms in theoretical computer science. Because of the difficulty in emulating higher-order calculations on a TM, their equivalence on partial numeric functions is not obtained directly, but rather composing different encodings. As a consequence, no model of λ-calculus (λ-model) based on TM's has arisen in the literature so far. Recently, Della Penna et al. have successfully built a λ-model based on so-called addressing machines (AM) [11]. The intent is to propose a model of computation, alternative to von Neumann architecture, where computation is based on communication between machines rather than performing local operations. In fact, these machines are solely capable of manipulating the addresses of other machines-this opens the way for modelling higherorder computations since functions can be passed via their addresses. An AM can read an address from its input-tape, store in a register the result of applying an address to another and, finally, pass the execution 8-2 Exxtended Addresing Machines for PCF,... to another machine by calling its address (possibly extending its input-tape). The set of instructions is deliberately small, to identify the minimal setting needed to represent λ-terms. The downside is that performing calculations on natural numbers is as awkward as using Church numerals in λ-calculus. Contents. In this paper we extend the formalism of AM's with a set of instructions representing basic arithmetic operations and conditional tests on natural numbers. As we are entering a world of machines and addresses, we need specific machines to represent numerals and assign them recognizable addresses. Finally, in order to model recursion, we rely on the existence of machines representing fixed point combinators. These machines can be programmed in the original formalism but we can avoid any dependency on self-application by manipulating the addressing mechanism so that they have access to their own address. This can be seen as a very basic version of the reflection principle which is present in some programming languages. We call the resulting formalism extended addressing machines (EAMs). Considering these features, one might expect EAMs to be well-suited for simulating Plotkin's Programming Computable Functions (PCF) [23], a simply typed λ-calculus with constants, arithmetical operations, conditional testing and a fixed point combinator. A PCF term of the form (λx.M )N can indeed be translated into a machine M reading as input (x) from its tape the address of N. As M has control over the computation, it naturally models a weak leftmost call-by-name evaluation. However, while in the contractum M [N/x] of the redex the substitution is instantaneous, M needs to pass the address of N to the machines representing the subterms of M , with the substitution only being performed if N gains control of the computation. As a result, rather than PCF, EAMs naturally emulate the behavior of EPCF-a weak call-by-name PCF with explicit substitutions that are only performed "on demand", as in [19]. We endow EAMs with a typing mechanism based on simple types and define a type-preserving translation from well-typed EPCF terms to EAMs. Subsequently, we prove that also the operational behavior of EPCF is faithfully represented by the translation. Finally, by showing the equivalence between PCF and EPCF on terminating programs of type int, we are capable of drawing conclusions for the original language PCF. In this paper we mainly focus on the properties of the translation, but our long-term goal is to construct a sequential model of higher-order computations. The problem of finding a fully abstract model of PCF was originally proposed by Robin Milner in [21] and is a difficult one. A model is called fully abstract (FA) whenever two programs sharing the type α get the same denotation in the model if and only if they are observationally indistinguishable when plugged in the same context C[] of type α → int. Therefore, a FA model provides a semantic characterization of the observational equivalence of PCF. Quoting from [3]: "the problem is to understand what should be meant by a semantic characterization [. . . ] Our view is that the essential content of the problem, what makes it important, is that it calls for a semantic characterization of sequential, functional computation at higher-types". A celebrated result is that FA models of PCF can be obtained by defining suitable categories of games [4,3,13]. Preliminary investigations show that EAMs open the way to construct a more 'computational' FA model. E.g., in [21], the model construction starts with first-order definable functions and requires-to cope with fixed point operator-the addition of extra 'limit points' to ensure that the resulting partial order is direct complete. In the game semantics approach the fixed point operator is treated similarly, namely via its canonical interpretation in a cpo-enriched Cartesian closed category [4]. On the contrary, in our approach no limit construction is required to give the fixed point operator a meaning. The fact that EAMs possess a given recursor having its own address stored inside is easily obtained from a mathematical point of view and, as argued above, can be seen as an abstract view of the usual implementation of recursion. We believe this new point of view may increase our understanding of PCF observational equivalence. Outline. The paper is organized as follows. In Section 1 we introduce the language EPCF along with its syntax, simply typed assignment system and associated (call-by-name) big-step operational semantics. In Section 2 we define EAMs (no familiarity with [11] is assumed) and introduce their operational semantics. In Section 3 we describe a type-checking algorithm for determining whether an EAM is well-typed. In Section 4 we present our main results, namely: (i) the translation of a well-typed EPCF term is an EAM typable with the same type (Theorem 4.6); (ii) if an EPCF term reduces to a value, then their translations as machines are interconvertible (Thm. 4.7); Intrigila, Manzonetto, Münnich (iii) the operational semantics of PCF and EPCF coincide on terminating programs of type int (Thm. 4.13); (iv) the translation of a PCF program computing a number is an EAM evaluating the corresponding numeral (Theorem 4.14). 8-3 n ∈ N σ ⊲ n ⇓ d n (nat) σ ⊲ λx.M ρ ⇓ d λx.M σ + ρ (fun) σ(x) = (ρ, N ) ρ ⊲ N ⇓ d V σ ⊲ x ⇓ d V (var) σ ⊲ M · (fix M ) ⇓ d V σ ⊲ fix M ⇓ d V (fix) σ ⊲ M ⇓ d 0 σ ⊲ N 1 ⇓ d V 1 σ ⊲ ifz(M, N 1 , N 2 ) ⇓ d V 1 (ifz 0 ) σ ⊲ M ⇓ d n + 1 σ ⊲ N 2 ⇓ d V 2 σ ⊲ ifz(M, N 1 , N 2 ) ⇓ d V 2 (ifz >0 ) σ ⊲ M ⇓ d n + 1 σ ⊲ pred M ⇓ d n (pr) σ ⊲ M ⇓ d 0 σ ⊲ pred M ⇓ d 0 (pr 0 ) σ ⊲ M ⇓ d n σ ⊲ succ M ⇓ d n + 1 (sc) σ ⊲ M ⇓ d λx.M ′ ρ ρ + [x ← (σ, N )] ⊲ M ′ ⇓ d V σ ⊲ M · N ⇓ d V (β v ) Related works. A preliminary version of AMs was introduced in Della Penna's MSc thesis [10] in order to model computation as communication between distinguished processes by means of their addresses. They were subsequently refined in [11] with the theoretical purpose of constructing a model of λ-calculus. Similarly, our paper should be seen as a first step towards the construction of a denotational model of PCF. Thus, the natural comparison is 6 with other models rather than other machine-based formalisms that have been proposed in the literature (e.g., call-by-name: SECD [16], KAM [15], call-by-need: TIM [12], Lazy KAM [6,17]); call-by-value: ZINC [18]) from which they differ at an implementational level. Compared with models of PCF based on Scott-continuous functions [21,5,7], EAMs provide a more operational interpretation of a program and naturally avoid parallel features that would lead to the failure of FA as in the continuous semantics. Compared with Curien's sequential algorithms [8] and categories of games [4,13] they share the intensionality of programs' denotations, while presenting an original way of modelling sequential computation. The model based on AMs also bares some similarities with the categories of assembly used to model PCF [20], mostly on a philosophical level, in the sense that these models are based on the 'codes' (rather than addresses) of recursive functions realizing a formula ( ∼ = type). Concerning explicit substitutions we refer to the pioneering articles [1,2,9,19]. Explicit substitutions have been barely considered in the context of PCF-with the notable exception of [24]. 8-4 Exxtended Addresing Machines for PCF,... natural number 0, pred and succ indicate the predecessor and successor respectively, ifz is the conditional test on zero, and finally, fix is a fixed-point operator. We assume that application -often denoted as juxtaposition -associates to the left and has higher precedence than abstraction. Concerning λx.M σ , it represents an abstraction where σ is a list of assignments from variables to closures (terms with the associated substitutions), where each variable can only have one closure assigned to it. Hereafter terms are considered up to renaming of bound variables. Therefore the symbol = will denote syntactic equality up to α-conversion. Notation 1 (i) For every n ∈ N, we let n = succ n (0). In particular, 0 is an alternative notation for 0. (ii) As a syntactic sugar, we write λx.M for λx.M . With this notation in place, PCF terms are simply EPCF terms containing empty explicit substitutions. (iv) succ2 = (λsn.s · (s · n)) · succ1, representing the function f (x) = x + 2. (v) add aux = λf xy.ifz(y, x, (f · (succ x)) · (pred y)), i.e. the functional Φ f (x, y) = x, if y = 0, f (x + 1, y − 1), if y > 0. (vi) add = fix (add aux), i.e., the recursive definition of addition f (x, y) = x + y. The operational semantics of EPCF is defined via a call-by-name big-step (leftmost) weak reduction. Otherwise, we say that M is a non-terminating, or looping, term. Example 1. 5 We show some of the terms from Example 1.3, at work. Intrigila, Manzonetto, Münnich 8-5 Γ, x : α ⊢ E x : α (ax) Γ ⊢ E 0 : int (0) Γ ⊢ E M : α → α Γ ⊢ E fix M : α (Y) Γ ⊢ E M : int Γ ⊢ E succ M : int (+) Γ ⊢ E M : int Γ ⊢ E pred M : int (−) σ |= ∆ Γ, ∆, x : α ⊢ E M : β Γ ⊢ E λx.M σ : α → β (→ I ) Γ ⊢ E M : α → β Γ ⊢ E N : α Γ ⊢ E M · N : β (→ E ) Γ ⊢ E L : int Γ ⊢ E M : α Γ ⊢ E N : α Γ ⊢ E ifz(L, M, N ) : α (ifz) [] |= ∅ (σ 0 ) σ |= Γ ρ |= ∆ ∆ ⊢ E M : α σ + [x ← (ρ, M )] |= Γ, x : α (σ) Fig. 2. EPCF type assignment system. (i) We have [ ] ⊲ succ1 · 0 ⇓ d 1. To get the reader familiar with the operational semantics, we give the details: [ ] ⊲ λx.succ (x) ⇓ d λx.succ (x) (fun) [ ] ⊲ 0 ⇓ d 0 (nat) [x ← ([ ], 0)] ⊲ x ⇓ d 0 (var) [x ← ([ ], 0)] ⊲ succ (x) ⇓ d 1 (sc) [ ] ⊲ (λx.succ (x)) · 0 ⇓ d 1 (β v ) (ii) Similarly, [ ] ⊲ I · 4 ⇓ d 4, [ ] ⊲ I · I ⇓ d I, [ ] ⊲ succ2 · 1 ⇓ d 3 and [ ] ⊲ add · 5 · 1 ⇓ d 6. (iii) Since Ω is looping, there is no V ∈ Val such that [ ] ⊲ Ω ⇓ d V is derivable. We now endow EPCF terms with a type system based on simple types. Definition 1.6 (i) The set T of (simple) types over a ground type int is inductively defined by the grammar: α, β ::= int | α → β (T) The arrow associates to the right, in other words we write α 1 → · · · → α n → β for α 1 → (· · · → (α n → β) · · · ) (= α → β, for short). (ii) A typing context Γ is given by a set of associations between variables and types, written x 1 : α 1 , . . . , x n : α n . In this case, we let dom(Γ) = {x 1 , . . . , x n }. When writing Γ, x : α, we silently assume that x / ∈ dom(Γ). (iii) Typing judgements are triples, denoted Γ ⊢ E M : α, where Γ is a typing context, M is an EPCF term and α ∈ T. (iv) Typing derivations are finite trees built bottom-up in such a way that the root has shape Γ ⊢ E M : α and every node is an instance of a rule from Figure 2. In the rule (→ I ) we assume wlog that x / ∈ Γ, by α-conversion. We also use an auxiliary predicate σ |= Γ whose intuitive meaning is that Γ is a typing context constructed from an explicit substitution σ. (v) When writing Γ ⊢ E M : α, we mean that this typing judgement is derivable. (vi) We say that M is typable if Γ ⊢ E M : α is derivable for some Γ, α.(i) [ ] |= ∅ (σ 0 ) [ ] |= ∅ (σ 0 ) ⊢ E 0 : int (0) [y ← ([ ], 0)] |= y : int (σ) y : int, x : α ⊢ E y : int (ax) y : int, x : α ⊢ E succ (y) : int (+) ⊢ E λx.succ (y) [y ← ([ ], 0)] : α → int (→ I ) (ii) ⊢ E (λx.succ (x)) · 0 : int. 8-6 Exxtended Addresing Machines for PCF,... (iii) ⊢ E (λsn.s · (s · n)) · (λx.succ (x)) : int → int. (iv) ⊢ E fix (λf xy.ifz(y, x, f · (succ x) · (pred y))) : int → int → int. (v) ⊢ E Ω : α, for all α ∈ T. The following lemma summarizes the main (rather standard) properties of the language EPCF. Lemma 1.8 Let M be an EPCF term, V ∈ Val, α, β ∈ T and Γ be a context. (i) (Syntax directedness) Every derivable judgement Γ ⊢ E M : α admits a unique derivation. (ii) (Strengthening) If Γ, x : β ⊢ E M : α and x / ∈ FV(M ) then Γ ⊢ E M : α. (iii) (Subject reduction) For M closed, ⊢ E M : α and [ ] ⊲ M ⇓ d V entail ⊢ E V : α. It follows that, if an EPCF program M is typable, then it is also typable in the empty context. Extended Addressing Machines We extend the addressing machines from [11] with instructions for performing arithmetic operations and conditional testing. Natural numbers are represented by particular machines playing the role of numerals. Main definitions We consider fixed a countably infinite set A of addresses together with a distinguished countable subset X ⊂ A, such that A − X remains infinite. Intuitively, X is the set of addresses that we reserve for the numerals, therefore hereafter we work under the hypothesis that X = N, an assumption that we can make without loss of generality. Let ∅ / ∈ A be a "null" constant corresponding to an uninitialised register. Set A ∅ = A ∪ {∅}. Definition 2.1 (i) An A-valued tape T is a finite ordered list of addresses T = [a 1 , . . . , a n ] with a i ∈ A for all i (1 ≤ i ≤ n). When A is clear from the context, we simply call T a tape. We denote by T A the set of all A-valued tapes. (ii) Let a ∈ A and T, T ′ ∈ T A . We denote by a :: T the tape having a as first element and T as tail. We write T @ T ′ for the concatenation of T and T ′ , which is an A-valued tape itself. (iii) Given an index i ≥ 0, an A ∅ -valued register R i is a memory-cell capable of storing either ∅ or an address a ∈ A. We write !R i to represent the value stored in the register R i . (The notation !R i is borrowed from ML, where ! represents an explicit dereferencing operator.) (iv) Given A ∅ -valued registers R 0 , . . . , R n for n ≥ 0, an address a ∈ A and an index i ≥ 0, we write R[R i := a] for the list of registers R where the value of R i has been updated by setting !R i = a. Notice that, whenever i > n, we assume that the contents of R remains unchanged, i.e. R[R i := a] = R. Intuitively, the contents of the registers R 0 , . . . , R n constitutes the state of a machine, while the tape correspond to the list of its inputs. The addressing machines from [11] are endowed with only three instructions (i, j, k, l range over indices of registers): 1. Load i : reads an address a from the input tape, assuming it is non-empty, and stores a in the register R i . If the tape is empty then the machine suspends its execution without raising an error. 2. k App(i, j) : reads the addresses a 1 , a 2 from R i and R j respectively, and stores in R k the address of the machine obtained by extending the tape of the machine of address a 1 with the address a 2 . The resulting address is not calculated internally but rather obtained calling an external application map. 3. Call i : transfers the computation to the machine having as address the value stored in R i , whose tape is extended with the remainder of the current machine's tape. As a general principle, writing on a non-existing register does not cause issues as the value is simply discarded-this is in fact the way one can erase an argument. The attempt of reading an uninitialized Intrigila, Manzonetto, Münnich 8-7 register would raise an error-we however show that these kind of errors can be avoided statically (see Lemma 2.4). We enrich the above set of instructions with arithmetic operations mimicking the ones present in PCF: 4. l Test(i, j, k): implements the "is zero? " test on !R i . Assuming that the value of R i is an address n ∈ N, the instruction stores in R l the value of R j or R k , depending on whether n = 0. 5. j Pred(i): if !R i ∈ N, the value of R j becomes !R i ⊖ 1 = max(!R i − 1, 0). 6. j Succ(i): if !R i ∈ N, then the value of R j becomes !R i + 1. Notice that the instructions above need R i to contain a natural number to perform the corresponding operation. However, they are also supposed to work on addresses of machines that compute a numeral. For this reason, the machine whose address is stored in R i must first be executed, and only if the computation terminates with a numeral is the arithmetic operation performed. Clearly, if the computation terminates in an address not representing a numeral, then an error should be raised at execution time. We will see that these kind of errors can be avoided using a type inference algorithm (see Proposition 3.5, below). Definition 2.2 (i) A program P is a finite list of instructions generated by the following grammar, where ε represents the empty string and i, j, k, l are indices of registers: P ::= Load i; P | A A ::= k App(i, j); A | l Test(i, j, k); A | j Pred(i); A | j Succ(i); A | C C ::= Call i | ε Thus, a program starts with a list of Load's, continues with a list of App, Test, Pred, Succ, and possibly ends with a Call. Each of these lists may be empty, in particular the empty program ε can be generated. (ii) In a program, we write Load (i 1 , . . . , i n ) as an abbreviation for the instructions Load i 1 ; · · · ; Load i n . (iii) Let P be a program, r ≥ 0, and I ⊆ {0, . . . , r − 1} be a set of indices corresponding to the indices of initialized registers. Define the relation I |= r P , whose intent is to specify that P does not read uninitialized registers, as the least relation closed under the rules: I |= r ε i ∈ I I |= r Call i I ∪ {j} |= r A i ∈ I j < r I |= r j Pred(i); A I ∪ {i} |= r P i < r I |= r Load i; P I |= r P i ≥ r I |= r Load i; P I ∪ {j} |= r A i ∈ I j < r I |= r j Succ(i); A I ∪ {k} |= r A i, j ∈ I k < r I |= r k App(i, j); A I ∪ {l} |= r A i, j, k ∈ I l < r I |= r l Test(i, j, k); A (iv) A program P is valid with respect to R 0 , . . . , R r−1 if R |= r P holds for R = {i | R i = ∅ ∧ 0 ≤ i < r}. Example 2.3 For each of these programs, we specify its validity with respect to R 0 = 7, R 1 = a, R 2 = ∅ (i.e., r = 3). where P = Load (1, . . . , n + 1); 0 App(0, 1); · · · ; 0 App(0, n + 1); 1 App(1, 2); · · · ; 1 App(1, n + 1); 1 App(1, 0); Call 1 We now enter into the details of the addressing mechanism which constitutes the core of this formalism. Definition 2.6 Recall that N stands for an infinite subset of A, here identified with the set of natural numbers, and Y a n has been introduced in Definition 2.5(vi). (i) Since M A is countable, we can fix a bijective function # : M A → A satisfying the following conditions: (a) (Numerals) ∀n ∈ N . #n = n, where n is the n-th numeral machine; (b) (Fixed point combinator) for all n ≥ 0, there exists an address a ∈ A − N such that #(Y a n ) = a. We say that the bijection #(·) is an address table map and call the element #M the address of the EAM M. We simply write Y n for the machine satisfying the equation above and a Yn for its address, i.e. #(Y n ) = a Yn . (ii) For a ∈ A, we write # −1 (a) for the unique machine having address a, i.e., # −1 (a) = M ⇐⇒ #M = a. and a non-computable function # : M K c → K c as a map. In an implementation of EAMs the address table map should be computable-one can choose a fresh address from A whenever a new machine is constructed, save the correspondence in some table and retrieve it in constant time. The results we present are independent from the choice of #. Operational semantics The operational semantics of extended addressing machines is given through a small-step rewriting system. The reduction strategy is deterministic, since the only applicable rule at every step is univocally determined by the first instruction of the internal program, the contents of the registers and the head of the tape. Notice that since the redexes in Figure 3 are not overlapping, the confluence of ։ c follows easily (cf. 8-10 Exxtended Addresing Machines for PCF,... Unconditional rewriting rules R, Call i; P, T → c # −1 (!R i ) @ T R, Load i; P, a :: T → c R[R i := a], P, T R, k App(i, j); P, T → c R[R k := !R i · !R j ], P, T Under the assumption that # −1 (!R i ) → c (i.e., it is in final state). R, j Pred(i); P, T → c R[R j := !R i ⊖ 1, P, T , if !R i ∈ N, err, otherwise. R, j Succ(i); P, T → c R[R j := !R i ⊕ 1, P, T , if !R i ∈ N, err, otherwise. R, l Test(i, j, k); P, T → c      R[R l := !R j ], P, T , if !R i = 0, R[R l := !R k ], P, T , if !R i ∈ N + , err, otherwise. Under the assumption that # −1 (!R i ) → c A (i.e., it is not in final state). Typing Algorithm Recall that the set T of (simple) types has been introduced in Definition 1.6(i). We now show that certain EAMs can be typed, and that typable machines do not raise error during their execution. (iv) For R i 1 , . . . , R in ∈ R, write R i 1 : β i 1 , . . . , R in : β in |= R if # −1 (!R j ) : β j , for all j ∈ {i 1 , . . . , i n }. The algorithm in Figure 4 deserves some discussion. As it is presented as a set of inference rules, one should reason bottom-up. To give a machine M a type α, one needs to derive the judgement ⊢ M : α. The machines n and Y n are recognizable from their addresses and the rules (nat) and (fix) can thus be given higher precedence. Otherwise, the rule (R T ) allows to check whether the value in a register is typable and only retain its type, the rule (R ∅ ) allows to get rid of uninitialized registers. Once this initial step is performed, one needs to derive a judgement of the form R i 1 : β i 1 , . . . , R in : β in (P, T ) : α, where P and T are the program and the input tape of the original machine respectively. This is done by verifying the coherence of the instructions in the program with the types of the registers and of the values in the input tape. As a final consideration, notice that the rules in Figure 4 can only be considered as an algorithm when the address table map is effectively given. Otherwise, the algorithm would depend on an oracle deciding a = #M. Intrigila, Manzonetto, Münnich 8-11 #M ∈ N ⊢ M : int nat #M = a Yn δ = δ 1 → · · · → δ n ⊢ M : ( δ → α → α) → δ → α fix n ∆ (P, T ) : α ∆ ⊢ (), P, T : α R () ∆ ⊢ R 0 , . . . , R r−1 , P, T : α !R r = ∅ ∆ ⊢ (R 0 , . . . , R r ), P, T : α R ∅ R r : β, ∆ ⊢ R 0 , . . . , R r−1 , P, T : α ⊢ # −1 (!R r ) : β ∆ ⊢ (R 0 , . . . , R r ), P, T : α R T ∆[R i : β] (P, []) : α ∆ (Load i; P, []) : β → α load ∅ ∆[R i : β] (P, T ) : α ⊢ # −1 (a) : β ∆ (Load i; P, a :: T ) : α load T (∆, R i : int)[R j : int] (P, T ) : α ∆, R i : int (j Pred(i); P, T ) : α pred (∆, R i : int)[R j : int] (P, T ) : α ∆, R i : int (j Succ(i); P, T ) : α succ (∆, R i : int, R j : β, R k : β)[R l : β] (P, T ) : α ∆, R i : int, R j : β, R k : β (l Test(i, j, k); P, T ) : α test (∆, R i : α → β, R j : α)[R k : β] (P, T ) : δ ∆, R i : α → β, R j : α (k App(i, j); P, T ) : δ app ⊢ M 1 : α 1 · · · ⊢ M n : α n ∆, R i : α 1 → · · · → α n → α (Call i, [#M 1 , . . . , #M n ]) : α call(i) ⊢ Succ2 : int → int (ii) ⊢ Add : int → int → int, where Add = Y 0 @ [#Add aux] (iii) For a smaller example, like ⊢ Succ1 : int → int, we can provide the whole derivation tree: Exxtended Addresing Machines for PCF,... derivable. (ii) In the rules (R T ) and (load T ), one needs to show that a type for the premises exists. As the set of types is countable, and effectively given, one can easily design an algorithm constructing a derivation tree (by dovetailing). However, the algorithm cannot terminate when executed on M 0 from Remark 2.9. ✷ The machine M 0 in Remark 2.9 cannot be typable because it would require an infinite derivation tree. Notice that, in this case, β = α → α. Using # −1 (a Y 0 ) = Y 0 , we derive: Case P = Call i. Then R i : α 1 → · · · → α n → α, T = [#M 1 , . . . , #M n ] and ⊢ M j : α j , for all j ≤ n. In this case, N = # −1 (!(M.R i )) @ T with ⊢ # −1 (!(M.R i )) : α 1 → · · · → α n → α, so we conclude by (i). R 0 : α, R 1 : α (Call 0, []) : α call R 0 : α → α, R 1 : (α → α) → α (1 App(1, 0); · · · , []) : α app; app ⊢ N : α → α R 1 : (α → α) → α (Load 0; · · · , [#N]) : α load T R 1 : (α → α) → α R 0 = ∅, Load 0; · · · , [#N] : α R () ; R ∅ fix 0 ⊢ Y 0 : (α → α) → α ⊢ (R 0 = ∅, R 1 = a Y0 ), All other cases follows easily from the IH. (iii) Assume that ⊢ M : int and M ։ c N for some N in final state. By (ii), we obtain that ⊢ N : int holds, therefore N = n since numerals are the only machines in final state typable with int. (iv) The three cases from Figure 3 where a machine can raise an error are ruled out by the typing rules (pred), (succ) and (test), respectively. Therefore, no error can be raised during the execution. ✷ Translation and Simulation We define a type-preserving translation from EPCF terms to extended addressing machines. More precisely, we show that if Γ ⊢ E M : α is derivable then M is transformed into a machine M which is typable with the same α. By Proposition 3.5, M never raises a runtime error and well-typedness is preserved during its execution. We then show that if a well-typed EPCF program M computes a value n, then its translation for all a, b, c, d 1 , . . . , d n ∈ A): (i) Pr n i @ [d 1 , . . . , d n ] ։ c d i , for 1 ≤ i ≤ n; (ii) Apply n @ [a, b, d 1 , . . . , d n ] ։ c # −1 (a) @ [d 1 , . . . , d n , b · d 1 · · · d n ] ; (iii) Pred n @ [a, d 1 , . . . , d n ] ։ c R 0 = a · d 1 · · · d n , R, ; 0 Pred(0); Call 0, [] ; (iv) Succ n @ [a, d 1 , . . . , d n ] ։ c R 0 = a · d 1 · · · d n , R, ; 0 Succ(0); Call 0, [] ; (v) Ifz n @ [a, b, c, d 1 , . . . , d n ] ։ c R 0 = a · d, R 1 = b · d, R 2 = c · d, R, 0 Test(0, 1, 2); Call 0, [] . Proof. Easy. As an example, we give a possible definition of the predecessor: (i) ⊢ Pr n i : δ → δ i , with δ = δ 1 → · · · → δ n ; (ii) ⊢ Apply n : ( δ → β → α) → ( δ → β) → δ → α; (iii) ⊢ Pred n : ( δ → int) → δ → int; (iv) ⊢ Succ n : ( δ → int) → δ → int; (v) ⊢ Ifz n : ( δ → int) → ( δ → α) → ( δ → α) → δ → α; Proof. The naive implementations are, in fact, typable. ✷ We will show that, using the auxiliary EAMs given in Lemma 4.2, we can translate any EPCF term into an EAM. In order to proceed by induction, we first need to define the size of an EPCF term. Intuitively, an EPCF term M having x 1 , . . . , x n as free variables is translated as an EAM M loading n arguments as input. M · N x = Apply n @ [# M x , # N x ] ; k x = Pr n+1 1 @ [k] , where k ∈ N; pred M x = Pred n @ [# M x ] ; succ M x = Succ n @ [# M x ] ; ifz(L, M, N ) x = Ifz n @ [# L x , # M x , # N x ] ; fix M x = Y n @ [# M x ] . We show the extended abstract machines associated by this translation to some of our running examples. Example 4.5 1. (λx.succ (x) ) · 0 = Succ 1 @ [#Pr 1 1 , 0] . 2. (λsn.s(sn))(λx.succ (x)) = Apply 2 @ [#Pr 2 1 , #(Apply 2 @ [#Pr 2 1 , #Pr 2 2 ] ), #(Succ 1 @ [#Pr 1 1 ] )] . 3. add = Y 0 @ # λf xy.ifz(y, x, (f · (succ x) · (pred y))) = Y 0 @ # ifz(y, x, (f · (succ x) · (pred y))) f,x,y , = Y 0 @ #(Ifz 3 @ [#Pr 3 3 , #Pr 3 2 , # f · (succ x) · (pred y) f,x,y ] ) , where f ·(succ x)·(pred y) f,x,y = Apply 3 @ [#(Apply 3 @ [#Pr 3 1 , #(Succ 3 @ [#Pr 3 2 ] )] ), #(Pred 3 @ [#Pr 3 3 ] )] . Theorem 4.6 Let M be an EPCF term, α ∈ T, Γ = x 1 : β 1 , . . . , x n : β n . Then Γ ⊢ E M : α ⇒ ⊢ M x 1 ,...,xn : β 1 → · · · → β n → α. Proof. By induction on a derivation of Γ ⊢ E M : α. As an induction loading, one needs to prove simultaneously that for all explicit substitutions σ with dom(σ) = {x 1 , . . . , x n }, if σ |= x 1 : β 1 , . . . , x n : β n then ⊢ σ(x i ) : β i , for all i ≤ n. ✷ Theorem 4.7 Let M be an EPCF term and V ∈ Val. Then Γ, x : α ⊢ x : α σ ⊲ M ⇓ d V ⇒ σ, M ↔ c V Proof. ByU ∈ Val U ⇓ U (val) P ⇓ 0 pred P ⇓ 0 (pr 0 ) P ⇓ n + 1 pred P ⇓ n (pr) P ⇓ 0 Q ⇓ U 1 ifz(P, Q, Q ′ ) ⇓ U 1 (ifz 0 ) P ⇓ n + 1 Q ′ ⇓ U 2 ifz(P, Q, Q ′ ) ⇓ U 2 (ifz >0 ) P ⇓ n succ P ⇓ n + 1 (sc) P · (fix P ) ⇓ U fix P ⇓ U (fix) P ⇓ λx.P ′ P ′ [Q/x] ⇓ U P · Q ⇓ U (β v )Γ, x : α ⊢ P : β Γ ⊢ λx.P : α → β Γ ⊢ 0 : int Γ ⊢ P : α → β Γ ⊢ Q : α Γ ⊢ P Q : β Γ ⊢ P : int Γ ⊢ pred P : int. Γ ⊢ P : α → α Γ ⊢ fix P : α Γ ⊢ P : int Γ ⊢ succ P : int. Γ ⊢ P : int Γ ⊢ Q : α Γ ⊢ Q ′ : α Γ ⊢ ifz(P, Q, Q ′ ) : α= x | P · Q | λx.P | 0 | pred P | succ P | ifz(P, Q, Q ′ ) | fix P (ii) A closed PCF term P is called a PCF program. (iii) A PCF value U is a term of the form λx.P or n, for some n ≥ 0. (iv) Given a PCF term P and a value U , we write P ⇓ U if this judgement can be obtained by applying the rules from Figure 5. (v) The set T of simple types and typing contexts have already been defined in items (i) and (ii) of Definition 1.6, respectively. (vi) Given a PCF term P , a typing context Γ and α ∈ T, we write Γ ⊢ PCF P : α if this typing judgement is derivable from the rules of Figure 6. Recall that any PCF program P can be seen as an EPCF term, thanks to the notation λx.N := λx.N . However, the hypotheses ⊢ PCF P : int and P ⇓ n are a priori not sufficient for applying Corollary 4.8, since one needs to show that also the corresponding EPCF judgments ⊢ E P : int and P ⇓ d n hold. The former is established by the following lemma. An EPCF term is easily translated into PCF by performing all its explicit substitutions. The converse is trickier as the representation is not unique: for every PCF term P there are several decompositions P = P ′ [Q 1 /x 1 , . . . , Q n /x n ]. Recall that the size |(σ, M )| has been defined in Definition 4.3. To show the equivalence between PCF and EPCF, we need yet another auxiliary lemma. All other cases derive straightforwardly from the IH. (ii) By an easy induction on P , using (i). ✷ Proof. (Proof sketch) For the full proof, we refer to the technical Appendix A. (i) Proceed by induction on a derivation of σ ⊲ M ⇓ d V , using Lemma 4.12(i) in the (β v )-case. (ii) By induction on the lexicographically ordered pairs, whose first component is the length of a derivation of P ⇓ U and second component is |(σ, M )|, using Lemma 4.12(ii) in the (β v )-case. ✷ As promised, we now draw conclusions for the regular PCF. As customary in PCF, we are interested on the properties of closed terms having ground type. Theorem 4.14 For a PCF program P of type int, P ⇓ n entails P ։ c n. Proof. Note that P is also an EPCF term such that ([], P ) ∈ P † , and that ⊢ E P : int by Lemma 4.10. Thus [ ] ⊲ P ⇓ d n by Theorem 4.13(ii). Conclude by Corollary 4.8. ✷ Fig. 1 . 1The big-step operational semantics of EPCF. Definition 1.2 (i) In an explicit substitutionσ = [x 1 ← (σ 1 , M 1 ), . . . , x n ← (σ n , M n )]the x i 's are assumed to be fresh and distinguished.(ii) By (i), we can define σ(x i ) = (σ i , M i ). (iii) The domain of σ is given by dom(σ) = {x 1 , . . . , x n }.(iv) We write σ + ρ for the concatenation of σ and ρ, and in this case we assume dom(σ) ∩ dom(ρ) = ∅.The set FV(M ) of free variables of an EPCF term M is defined as usual, except for the abstraction case FV(λx.M σ ) = FV(M ) − ({x} ∪ dom(σ)). The term M is closed if FV(M ) = ∅, and in that case it is called an EPCF program. (iii) For n ∈ N, we often write λx 1 . . . λx n .M as λx 1 . . . x n .M , or even λ x.M when n is clear from the context. Summing up, and recalling that · is left associative, λx 1 x 2 x 3 .L · M · N stands for λx 1 .(λx 2 .(λx 3 .((L · M ) · N ) ) ) . (iv) M [N/x] denotes the capture-free substitution of N for all free occurrences of x in M . Example 1. 3 3We introduce some notations for the following (E)PCF programs, that will be used as running examples.(i) I = λx.x, representing the identity. (ii) Ω = fix (I) representing the paradigmatic looping program. (iii) succ1 = λx.succ (x), representing the successor function. Definition 1.4 (i) We let Val = {n | n ∈ N} ∪ {λx.M σ | M is an EPCF term} be the set of EPCF values. (ii)The big-step weak reduction is the least relation ⇓ d from EPCF terms to Val, closed under the rules ofFigure 1. (iii) We say that an EPCF program M is terminating whenever M ⇓ d V holds, for some V ∈ Val. Example 1. 7 7The following are examples of derivable typing judgments. P 3 = 3Load (0, 2, 8); Call 8 (calling the uninitialized register R 8 , thus not valid) Lemma 2.4 Given A ∅ -valued registers R and a program P it is decidable whether P is valid w.r.t. R. Decidability follows from the syntax directedness of Definition 2.2(iii), and the preservation of the invariant I ⊆ {0, . . . , r − 1}, since I is only extended with k < r. ✷ Definition 2.5 (i) An extended addressing machine (EAM ) M with r registers over A is given by a tuple: M = R 0 , . . . , R r−1 , P, T where R are A ∅ -valued registers, P is a program valid w.r.t. R and T ∈ T A is an (input) tape.(ii) We write M.r for the number of registers of M, M.R i for its i-th register, M.P for the associated program and M.T for its input tape. When writing "R i = a" in a tuple we indicate that R i is present and !R i = a. (iii) We say that an extended addressing machine M as above is stuck, written stuck(M), whenever its program has shape M.P = Load i; P but its input-tape is empty M.T = []. Otherwise M is ready, written ¬stuck(M). (iv) The set of all extended addressing machines over A will be denoted by M A .(v) For n ≥ 0, the n-th numeral machine is defined n = R 0 , ε, [] with !R 0 = n. (vi) For n ≥ 0 and a ∈ A, define Y a n = (R 0 = a, R 1 = ∅, . . . , R n+1 = ∅, P, [] ( iii) Given M ∈ M A and T ′ ∈ T A , we write M @ T ′ for the machine M. R, M.P, M.T @ T ′ . (iv) Define the application map (·) : A × A → A by setting a · b = #(# −1 (a) @ [b] ), i.e., the application of a to b is the unique address c of the EAM obtained by adding b at the end of the input tape of the EAM # −1 (a). Example 2.7 The following are examples of EAMs (whose registers are assumed uninitialized, i.e. R = ∅). (i) Succ1 := R 0 , 0Load 0; 0 Succ(0); Call 0, [] . (ii) Succ2 := R 0 , R 1 , Load 0; Load 1; 1 App(0, 1); 1 App(0, 1); Call 1, [a S ] , where a S = #Succ1. (iii) Add aux := R, P, [] with Add aux.r = 5 and P = Load ( Remark 2. 8 8In general, there are uncountably many possible address table maps of arbitrary computational complexity. A natural example of such maps is given by Gödelization, which can be performed effectively. The framework is however more general and allows to consider non-r.e. sets of addresses like the complement K c of the halting set K = {(i, x) | the program i terminates when run on input x} Remark 2. 9 9Depending on the chosen address table map, it might be possible to construct infinite (static) chains of EAMs (M) n∈N , e.g., M n = R 0 = #M n+1 , ε, [] . Definition 2 . 210 We introduce a fresh constant err / ∈ M A to represent a machine raising an error.(i) Define a reduction strategy → c on EAMs, representing one step of computation, as the least relation → c ⊆ M A × (M A ∪ {err}) closed under the rules in Figure 3. (ii) The multistep reduction ։ c is defined as the transitive-reflexive closure of → c . (iii) Given M, N, M ։ c N, we write |M ։ c N| ∈ N for the length of the (unique) reduction path from M to N. (iv) For M, N ∈ M A , we write M ↔ c N if they have a common reduct Z ∈ M A ∪ {err}, i.e. M ։ c Z c և N. (v) An extended address machine M: is in final state if it cannot reduce, written M → c ; reaches a final state if M ։ c M ′ for some M ′ ∈ M A in final state; raises an error if M ։ c err; does not terminate, otherwise. [ 11 , 11Lemma 2.11(2)]). Lemma 2 . 211 If M ։ c M ′ , then M @ #N ։ c M ′ @ #N . Proof. By induction on the length of M ։ c M ′ . ✷ Example 2.12 See Example 2.7 for the definition of Succ1, Succ2, Add aux.(i) We have Succ1 @ [0] ։ c 1 and Succ2 @ [1] ։ c 3. (ii) Define Add = Y 0 @ [#Add aux], an EAM performing the addition. We show:Add @ [1, 3] → c (R 0 = a Y 0 , R 1 = #Add aux), 0 App(0, 1); 1 App(1, 0); Call 1, [1, 3] ։ c R, Load (0, 1, 2); 3 Pred(1); 4 Succ(2); 0 App(0, 3); 0 App(0, 4); 0 Test(1, 2, 0); Call 0, [#Add, 1, 3] ։ c R 0 = #Add, R 1 = 1, R 2 = 3, R 3 , R 4 , 3 Pred(1); 4 Succ(2); 0 App(0, 3); 0 App(0, 4); 0 Test(1, 2, 0); Call 0, [] ։ c R 0 = #(Add @ [0, 4] ), R 1 = 1, R 2 = 3, R 3 = 0, R 4 = 4, 0 Test(1, 2, 0); Call 0, [] ։ c R 0 = #(Add @ [0, 5] ), R 1 = 0, R 2 = 4, R 3 = 0, R 4 = 5, 0 Test(1, 2, 0); Call 0, [] ։ c 4 (iii) For I = R 0 = ∅, Load 0; Call 0, [] , Y 0 @ [#I] ։ c I @ [#(Y 0 @ [#I] )] . (iv) Y n @ [#M, d 1 , . . . , d n ] ։ c M @ [d 1 , . . . , d n , #(Y n @ [#M, d 1 , . . . , d n ] )] , for all n ≥ 0, M ∈ M A , d ∈ A. R, j Fig. 3 . j3Pred(i); P, T → c R[R i := #A], j Pred(i); P, T R, j Succ(i); P, T → c R[R i := #A], j Succ(i); P, T R, l Test(i, j, k); P, T → c R[R i := #A], l Test(i, j, k); P, T Small-step operational semantics for extended addressing machines. Definition 3. 1 1(i) A typing context ∆ is a finite set of associations between registers and types, represented as a list R i 1 : α 1 , . . . , R in : α n . The indices i 1 , . . . , i n are not necessarily consecutive.(ii) We denote by ∆[R i : α] the typing context ∆ where the type associated with R i becomes α. If R i is not present in ∆, then ∆[R i : α] = ∆, R i : α. (iii) Let ∆ be a typing context, M ∈ M A , P be a program, T ∈ T A and α ∈ T. We define the typing judgements ∆ ⊢ M : α ∆ (P, T ) : α by mutual induction as the least relations closed under the rules ofFigure 4. The rules (nat) and (fix) are the base cases and take precedence over (R ∅ ) and (R T ). Fig. 4 . 4Typing rules for extended addressing machines. Remark 3 . 2 32(i) For all M ∈ M A and α ∈ T, we have ⊢ M : α if and only if there exists a ∈ A such that both # −1 (a) : α and #M = a hold. (ii) If #M / ∈ N ∪ {a Yn | n ≥ 0}, then ⊢ M : α ⇐⇒ ∃∆ . [∆ |= M. R ∧ ∆ (M.P, M.T ) : α] (iii) The higher priority assigned to the rules (nat) and (fix) does not modify the set of typable machines, rather guarantees the syntax-directedness of the system. Example 3. 3 3The following typing judgements are derivable. R 0 0: int Call 0, [] : int call R 0 : int 0 Succ(0); Call 0, [] : int succ Load 0; 0 Succ(0); Call 0, [] : int → int load ∅ ⊢ (), Load 0; 0 Succ(0); Call 0, [] : int → int R () !R 0 = ∅ ⊢ (R 0 = ∅), Load 0; 0 Succ(0); Call 0, [] : int → int R ∅ Lemma 3.4 Let M ∈ M A , α ∈ T. Assume that # : M → A is effectively given. ( i ) iIf M = R = ∅, P, [] then the typing algorithm is capable of deciding whether ⊢ M : α holds. (ii) In general, the typing algorithm semi-decides whether ⊢ M : α holds. Proof. (Sketch) (i) In this case, ⊢ M : α holds if and only if (M.P, []) does. By induction on the length of M.P , one verifies if it is possible to construct a derivation. Otherwise, conclude that ⊢ M : α is not 8-12 Proposition 3. 5 5Let M, M ′ , N, ∈ M A and α, β ∈ T. (i) If ⊢ M : β → α and ⊢ N : β then ⊢ M @ [#N] : α. (ii) If ⊢ M : α and M → c N then ⊢ N : α. (iii) If ⊢ M : int then either M does not terminate or M ։ c n, for some n ≥ 0. (iv) If ⊢ M : α then M does not raise an error. Proof. (i) Simultaneously, one proves that ∆ (P, T ) : β → α and ⊢ N : β imply ∆ (P, T @ [#N] ) : α. Proceed by induction on a derivation of ⊢ M : β → α (resp. ∆ (P, T ) : β → α). Case (nat) is vacuous. Case (fix n ). We show the case for n = 0, the others being similar. By definition of Y 0 , we have: Y 0 @ [#N] = (∅, a Y 0 ), Load 0; 1 App(1, 0); 0 App(0, 1); Call 0, [#N] . Load 0; 1 App(1, 0); 0 App(0, 1); Call 0, [#N] : α R T Case load ∅ . Then P = Load i; P ′ , T = [] and ∆[R i : β] (P ′ , []) : α. By assumption ⊢ N : β, so we conclude ∆ (Load i; P ′ , []) : α by applying load T . All other cases derive straightforwardly from the IH. (ii) The cases M = Y n or M = n for some n ∈ N are vacuous, as these machines are in final state. Otherwise, by Remark 3.2(ii), ∆ (M.P, M.T ) : α for some ∆ |= M. R. By cases on the shape of M.P . Case P = Load i; P ′ . Then M.T = a :: T ′ otherwise M would be in final state, and N = R[R i := a], P ′ , T ′ . From (Load T ) we get ∆[R i : β] (P ′ , T ′ ) : α for some β ∈ T satisfying # −1 (a) : β. As ∆ |= R we derive ∆[R i : β] |= R[R i := a], so as N = R[R i := a], P ′ , T ′ , by Remark 3.2(ii), ⊢ N : α. Pred n = R 0 , . . . , R n , Load (0, . . . , n); 0 App(0, 1); · · · ; 0 App(0, n); 0 Pred(0); Call 0, [] The others are similar. ✷ Lemma 4.2 The EAMs in the previous lemma can be defined in order to ensure their typability (for all n ≥ 0, α, β, γ, δ i ∈ T): Definition 4. 3 3Let M be an EPCF term and σ be an explicit substitution. The sizes |−| of σ, M and (σ, M ) are defined by mutual induction, e.g.|[]| = 0, |(σ, M )| = |σ| + |M |, |ρ + [x ← (ρ ′ , N )]| = |ρ| + |(ρ ′ , N )|, |λx.M σ | = |(σ, M )| + 1,and the other cases of |M | are standard whence they are omitted. Definition 4 . 4 ( 44Translation) Let M be an EPCF term and σ be an explicit substitution such that FV(M ) ⊆ dom(σ) ∪ { x}, where x = x 1 , . . . , x n . The translation of the pair (σ, M ) (w.r.t x) is a machine denoted σ, M x ∈ M A , or simply M x when σ is empty. 7 The machine σ, M x is defined by = σ, M x,y , where wlog y / ∈ x; Fig. 5 . 5The big-step operational semantics of PCF. Fig. 6 . 6The type inference rules of PCF. Definition 4. 9 9(i) PCF terms are defined by the grammar (for n ≥ 0, x ∈ Var): P, Q, Q ′ :: Lemma 4.10 Let M be a PCF term, α ∈ T and Γ be a context. ThenΓ ⊢ PCF M : α ⇒ Γ ⊢ E M : αProof. By a straightforward induction on a derivation of Γ ⊢ PCF M . ✷ Definition 4.11Let M be an EPCF term and σ be an explicit substitution. Define a PCF term (σ, M ) λx.M ρ ) * = λx.(σ + ρ, M ) * , (σ, M · N ) * = (σ, M ) * · (σ, N ) * , (σ, fix M ) * = fix ((σ, M ) * ), (σ, 0) * = 0, (σ, pred M ) * = pred (σ, M ) * , (σ, succ M ) * = succ (σ, M ) * ,(σ, ifz(L, M, N )) * = ifz((σ, L) * , (σ, M ) * , (σ, N ) * ). For a PCF term P , define P † = {(σ, M ) | (σ, M ) * = P }. Let M, N be EPCF terms, σ, ρ be explicit substitutions and x be a variable.(σ + [x ← (ρ, N )], M ) * = (σ, M ) * [(ρ, N ) * /x](ii) Let P, Q be PCF terms with FV(P ) ⊆ {x} and Q closed. For all EPCF terms M, N and explicit substitutions σ, ρ, we have:(σ, M ) ∈ P † ∧ (ρ, N ) ∈ Q † ⇒ (σ + [x ← (ρ, N )], M ) ∈ (P [Q/x]) †We rely on the freshness hypothesis on the variables in dom(σ).Proof.(i) By structural induction on M . Case M = y, with y = x: There are two subcases. • If y ∈ dom(σ), then (σ + [x ← (ρ, N )], y) * = σ(y) = σ(y)[(ρ, N ) * /x] = (σ, y) * [(ρ, N ) * /x], since x / ∈ FV(σ(y)); • if y / ∈ dom(σ), then (σ + [x ← (ρ, N )], y) * = y = y[(ρ, N ) * /x] = (σ, y) * [(ρ, N ) * /x]. Case M = x: Then (σ + [x ← (ρ, N )], x) * = (ρ, N ) * = x[(ρ, N ) * /x] = (σ, x) * [(ρ, N ) * /x]. Case M = λy.M ′ : Wlog, we may assume y = x. We have (σ + [x ← (ρ, N )], λy.M ′ ) * = λy.(σ + [x ← (ρ, N )], M ′ ) * = λy.((σ, M ′ ) * [(ρ, N ) * /x]) = (λy.(σ, M ′ ) * )[(ρ, N ) * /x] = ((σ, λy.M ′ ) * )[(ρ, N ) * /x]. Theorem 4. 13 13The big-step weak reduction of EPCF is equivalent to the usual big-step operational semantics of PCF. Formally:(i) Given an EPCF program M , a value V and an explicit substitution σ, we have:σ ⊲ M ⇓ d V ⇒ (σ, M ) * ⇓ ([ ], V ) *(ii) Given a PCF program P and PCF value U . If P ⇓ U then ∀(σ, M ) ∈ P † , ∃V ∈ Val . ( σ ⊲ M ⇓ d V and ([ ], V ) ∈ U † ) M reduces to the corresponding EAM n. Finally, this result is transported to PCF using their equivalence on programs of type int.We start by showing that EAMs implementing the main PCF instructions are definable. We do not need any machinery for representing explicit substitutions because they are naturally modelled by the evaluation strategy of EAMs.Lemma 4.1 Let n ≥ 0. There are EAMs satisfying (Intrigila, Manzonetto, Münnich 8-13 induction on a derivation of σ ⊲ M ⇓ d V . ✷Corollary 4.8 For an EPCF program M of type ⊢ E M : int we have [ ] ⊲ M ⇓ d n ⇒ M ։ c n Proof. Assume that [ ] ⊲ M ⇓ d n. By Theorem 4.7, we have [ ], M ↔ c n . Since n ։ c n and the numeral machine n is in final state we conclude M ։ c n.✷4.1 Applying the translation to regular PCFLet us show how to apply our machinery to the usual (call-by-name) PCF. Our presentation follows[22].Intrigila, Manzonetto, Münnich 8-15 In other words, we set M x = [], M x . Intrigila, Manzonetto, Münnich8-19Case (β v ): In this case M = N · L. By the operational semantics of EPCF, σ ⊲ N ⇓ d λx.N ′ ρ and ρ + [x ← (σ, L)] ⊲ N ′ ⇓ d V . By the type system of EPCF, Γ ⊢ N : β → α, Γ ⊢ L : β. By IH, ⊢ λx.N ′ ρ : β → α, and then by the type system of EPCF ρ |= ∆, ∆, x : β ⊢ N ′ : α. By the type system we also have [x ← (σ, L)] |= x : β, so ρ + [x ← (σ, L)] |= ∆, x : β, and thus by IH we conclude ⊢ V : α.All other cases derive straightforwardly from applying the rules of the type system and the IH. ✷A.2 Proofs of Section 4Proof.[Proof ofTheorem 4.6]We prove the following statements by mutual induction and call the respective inductive hypotheses IH1 and IH2.(i) Let M be an EPCF term, Γ = x 1 : β 1 , . . . , x n : β n and α ∈ T. Then Γ ⊢ M : α ⇒ ⊢ M x 1 ,...,xn : β 1 → · · · → β n → α. (ii) For all Γ = x 1 : β 1 , . . . , x n : β n and σ, with dom(σ) = {x 1 , . . . , x n }, we have σ |= Γ ⇒ ⊢ σ(x 1 ) : β 1 , . . . , ⊢ σ(x n ) : β n .We start with the cases concerning (i).Case (ax): Then, x 1 : β 1 , . . . , x n : β n ⊢ x i : β i , and x i x 1 ,...,xn = Pr n i . By Lemma 4.2(i) we conclude ⊢ Pr n i : β 1 → · · · → β n → β i .Case (0): In this case, we have Γ ⊢ 0 : int and 0 x 1 ,...,xn = Pr n+1From the hypothesis IH1, we obtain ⊢ M ′ x 1 ,...,xn : β → α → α. By Proposition 3.5(i), we conclude that ⊢ Y n @ [# M ′ x 1 ,...,xn ] : β 1 → · · · → β n → α. Case (ifz): Assume Γ ⊢ ifz(L, N 1 , N 2 ) : α since Γ ⊢ L : int and Γ ⊢ N i : α, for i ∈ {1, 2}. By definition of the translation, we have ifz(L,, 2}, and thus by Proposition 3.5(i), weCase (→ I ): Assume that Γ ⊢ λz.M ′ σ : α 1 → α 2 , for α = α 1 → α 2 , because there is ∆ = y 1 : δ 1 , . . . , y m : δ m such that σ |= ∆ and Γ, ∆, z : α 1 ⊢ M ′ : α 2 . Then σ |= ∆ entails σ = [y 1 ← (ρ 1 , N 1 ), . . . , y m ← (ρ m , N m )] for appropriate ρ, N . By definition, we have λz.M ′ σ x 1 ,...,xn = σ, M ′ x,z = M ′ y 1 ,...,ym, x,z @ [# ρ 1 , N 1 , . . . , # ρ m , N m ] . By applying IH1, we obtain ⊢ M ′ y, x,z : δ → β → α 1 → α 2 . From IH2, we get ⊢ σ 1 , N 1 : δ 1 · · · ⊢ σ m , N m : δ m . Finally, by Proposition 3.5(i), we derive ⊢ M ′ y, x,z @ [# σ 1 , N 1 , . . . , # σ m , N m ] : β → α 1 → α 2 . Case (→ E ): In this case Γ ⊢ M 1 · M 2 : α since, for some δ ∈ T, Γ ⊢ M 1 : δ → α and Γ ⊢ M 2 : δ. By definition,8-20Exxtended Addresing Machines for PCF,...We now consider the cases concerning (ii).Case (σ 0 ) : In this case [ ] |= ∅, so we have nothing to prove.Case (σ): In this case Γ = Γ ′ , x n : β n and. , x n } and sometimes use the convenient notation # σ(Case (nat): Then M = V = k, for some k ≥ 0. Recall that we assume k = #k. Then using Lemma 4.1(i),The case follows by reflexivity of ↔ c .Easy calculations give: Conclude as follows:Case (pr 0 ): Analogous to the previous case, using the fact that Pred(0) is 0.Case 7: (sc) M = succ M ′ , V = n + 1 and σ ⊲ M ′ ⇓ d n, for some n ≥ 0. By IH we get σ, M ′ ↔ c n = # −1 (n) and, since the n-th numeral machine is in final state, we derive σ, M ′ ։ c # −1 (n). Conclude as follows:We conclude since, by IH, σ, N 2 ↔ c V .By IH we get σ, L ↔ c 0 , and since the (n + 1)-th numeral machine is in final state, we derive σ, L ։ c # −1 (0). Thenby Lemma 4.1(v),We conclude since, by IH, σ, N 1 ↔ c V .8-22Exxtended Addresing Machines for PCF,...Case (fix): ThenThis concludes the proof. Case (nat): In this case M = V = n for some n ≥ 0. By definition, (σ, n) * = n, so we apply PCF's rule (val) and get n ⇓ n.Case (fix): In this case M = fix N and σ ⊲N ·(fix N ) ⇓ d V . From the IH we get (σ, N ·(fix N )) * ⇓ ([ ], V ) * . By definition, we have (σ, N · (fix N )) * = (σ, N ) * · (fix (σ, N ) * ) and fix (σ, N ) * = (σ, fix N ) * . By applying PCF's rule (fix), we obtainAll other cases derive straightforwardly from the IH. ✷Proof.[Proof of Theorem 4.13(ii)] For a PCF program P and PCF value U , we show that P ⇓ U entails:By induction on the lexicographically ordered pairs, whose first component is the length of a derivation of P ⇓ U and second component is |(σ, M )|. First, consider the case M = y and σ(y) = (ρ, N ), with (ρ, N ) ∈ P † . In this case, the length of the derivation (ρ, N ) * ⇓ U remained unchanged, while |(ρ, N )| < |(σ, M )|. Thus, we may use the IH and conclude by applying (var). Therefore, in the following we assume that M is not a variable.Intrigila, Manzonetto, Münnich8-23Case (val): In this case P = U . Given (σ, M ) ∈ P † , we distinguish several cases:• Case P = U = λx.P 0 . Then, M = λx.M 0 ρ with P 0 = (σ + ρ, M 0 ) * . Setting V = λx.M 0 σ + ρ we obtain M ⇓ d V by (fun) with ([], V ) * = λx.(σ + ρ, M 0 ) = λx.P 0 = U. • P = U = 0. It follows M = V = 0 and σ ⊲ 0 ⇓ d 0 by (nat). • P = U = succ (n), for some n ∈ N, and M is not a variable.There are two possibilities: · M = succ (n) in which case we are done, since σ ⊲ n + 1 ⇓ d n + 1. · M = succ (y) with σ(y) = (ρ, N ) and (ρ, N ) ∈ n † . Again, the length of the derivation (ρ, N ) * ⇓ n is unchanged, while |(ρ, N )| < |(σ, M )|. Once applied the IH, we conclude by (var) + (sc).Case (β v ): P = P 1 · P 2 with P 1 ⇓ λx.Q 1 and Q 1 [P 2 /x] ⇓ U for some Q 1 . Notice that, since P is closed so are P 1 , P 2 and hence FV(Q 1 ) ⊆ {x}. Now, (σ, M ) * = P entails M = M 1 · M 2 with (σ, M 1 ) ∈ P † 1 and (σ, M 2 ) ∈ P † 2 . By ind. hyp., there is V ′ ∈ Val such that σ ⊲ M 1 ⇓ d V 1 with ([], V 1 ) ∈ (λx.Q 1 ) † . This implies V 1 = λx.N 1 ρ for some (ρ, N 1 ) ∈ Q † 1 . By Lemma 4.12(ii), we get (ρ + [x ← (σ, M 2 )], N 1 ) ∈ (Q 1 [P 2 /x]) † . By ind. hyp., there is V ∈ Val such that ρ + [x ← (σ, M 2 )] ⊲ N 1 ⇓ d V and V ∈ U † . Conclude by EPCF's (β v ).Case (fix): P = fix Q and Q · (fix Q) ⇓ V . Then (σ, M ) ∈ P † entails M = fix N with (σ, N ) ∈ Q † . It follows that (σ, N · (fix N )) ∈ (Q · (fix Q)) † , therefore by IH we get σ ⊲ N · (fix N ) ⇓ d V for some V ∈ U † . We conclude by applying EPCF's rule (fix).All other cases derive straightforwardly from the IH. ✷ Explicit substitutions. M Abadi, L Cardelli, P.-L Curien, J.-J Lévy, Conference Record of the Seventeenth Annual ACM Symposium on Principles of Programming Languages. F. E. AllenSan Francisco, California, USAACM PressAbadi, M., L. Cardelli, P.-L. Curien and J.-J. Lévy, Explicit substitutions, in: F. E. Allen, editor, Conference Record of the Seventeenth Annual ACM Symposium on Principles of Programming Languages, San Francisco, California, USA, January 1990, pages 31-46, ACM Press (1990). . 10.1145/96709.96712https://doi.org/10.1145/96709.96712 Explicit substitutions. M Abadi, L Cardelli, P.-L Curien, J.-J Lévy, 10.1017/S0956796800000186J. Funct. Program. 1Abadi, M., L. Cardelli, P.-L. Curien and J.-J. Lévy, Explicit substitutions, J. Funct. Program. 1, pages 375-416 (1991). https://doi.org/10.1017/S0956796800000186 Full abstraction for PCF. S Abramsky, R Jagadeesan, P Malacaria, 10.1006/inco.2000.2930Inf. Comput. 163Abramsky, S., R. Jagadeesan and P. Malacaria, Full abstraction for PCF, Inf. Comput. 163, pages 409-470 (2000). https://doi.org/10.1006/inco.2000.2930 Full abstraction for PCF. S Abramsky, P Malacaria, R Jagadeesan, 10.1007/3-540-57887-0_87Theoretical Aspects of Computer Software, International Conference TACS '94. M. Hagiya and J. C. MitchellSendai, JapanSpringer789Abramsky, S., P. Malacaria and R. Jagadeesan, Full abstraction for PCF, in: M. Hagiya and J. C. Mitchell, editors, Theoretical Aspects of Computer Software, International Conference TACS '94, Sendai, Japan, April 19-22, 1994, Proceedings, volume 789 of Lecture Notes in Computer Science, pages 1-15, Springer (1994). https://doi.org/10.1007/3-540-57887-0_87 Full abstraction for sequential languages: state of the art. G Berry, J.-J P.-L. Curien, Lévy, Algebraic methods in semantics. M. Nivat and J. ReynoldsCambridge University PressBerry, G., P.-L. Curien and J.-J. Lévy, Full abstraction for sequential languages: state of the art, in: M. Nivat and J. Reynolds, editors, Algebraic methods in semantics, pages 89-132, Cambridge University Press (1985). Available online at https://hal.inria.fr/inria-00076361/document. Machines a environnement pour la reduction symbolique et l'evaluation partielle. P Cregut, French. Paris VIIUniversité Paris-DiderotPh.D. thesisCregut, P., Machines a environnement pour la reduction symbolique et l'evaluation partielle, Ph.D. thesis, Université Paris-Diderot (Paris VII) (1991). In French. http://www.theses.fr/1991PA077152 Definability and full abstraction. P Curien, 10.1016/j.entcs.2007.02.011Electron. Notes Theor. Comput. Sci. 172Curien, P., Definability and full abstraction, Electron. Notes Theor. Comput. Sci. 172, pages 301-310 (2007). https://doi.org/10.1016/j.entcs.2007.02.011 Observable algorithms on concrete data structures. P.-L Curien, Proceedings of the Seventh Annual Symposium on Logic in Computer Science (LICS '92). the Seventh Annual Symposium on Logic in Computer Science (LICS '92)Santa Cruz, California, USAIEEE Computer SocietyCurien, P.-L., Observable algorithms on concrete data structures, in: Proceedings of the Seventh Annual Symposium on Logic in Computer Science (LICS '92), Santa Cruz, California, USA, June 22-25, 1992, pages 432-443, IEEE Computer Society (1992). . 10.1109/LICS.1992.185554https://doi.org/10.1109/LICS.1992.185554 Confluence properties of weak and strong calculi of explicit substitutions. P.-L Curien, T Hardin, J.-J Lévy, J. ACM. 43Curien, P.-L., T. Hardin and J.-J. Lévy, Confluence properties of weak and strong calculi of explicit substitutions, J. ACM 43, pages 362-397 (1996). . 10.1145/226643.226675https://doi.org/10.1145/226643.226675 Una semantica operazionale per il network computing: le macchine di Turing virtuali, Master's thesis, Università degli Studi di L'Aquila (1996-97). G Della Penna, In ItalianDella Penna, G., Una semantica operazionale per il network computing: le macchine di Turing virtuali, Master's thesis, Università degli Studi di L'Aquila (1996-97). In Italian. Addressing machines as models of lambda-calculus. G Della Penna, B Intrigila, G Manzonetto, 10.48550/arXiv.2107.00319Log. Methods Comput. Sci. 18Della Penna, G., B. Intrigila and G. Manzonetto, Addressing machines as models of lambda-calculus, Log. Methods Comput. Sci. 18 (2022). https://doi.org/10.48550/arXiv.2107.00319 Tim: A simple, lazy abstract machine to execute supercombinators. J Fairbairn, S Wray, 10.1007/3-540-18317-5_3978-3-540-47879-9Functional Programming Languages and Computer Architecture. G. KahnSpringerFairbairn, J. and S. Wray, Tim: A simple, lazy abstract machine to execute supercombinators, in: G. Kahn, editor, Functional Programming Languages and Computer Architecture, pages 34-45, Springer Berlin Heidelberg, Berlin, Heidelberg (1987), ISBN 978-3-540-47879-9. https://doi.org/10.1007/3-540-18317-5_3 Exxtended Addresing Machines for PCF. Exxtended Addresing Machines for PCF,... On full abstraction for PCF: I, II, and III. J M E Hyland, C L Ong, 10.1006/inco.2000.2917Inf. Comput. 163Hyland, J. M. E. and C. L. Ong, On full abstraction for PCF: I, II, and III, Inf. Comput. 163, pages 285-408 (2000). https://doi.org/10.1006/inco.2000.2917 Extended addressing machines a comparative view. B Intrigila, G Manzonetto, N Münnich, Intrigila, B., G. Manzonetto and N. Münnich, Extended addressing machines a comparative view (2022). https://lipn.univ-paris13.fr/~gmanzonetto/papers/NotesIMM22 A call-by-name lambda-calculus machine. J.-L Krivine, 10.1007/s10990-007-9018-91388-3690Higher Order Symbol. Comput. 20Krivine, J.-L., A call-by-name lambda-calculus machine, Higher Order Symbol. Comput. 20, page 199-207 (2007), ISSN 1388-3690. https://doi.org/10.1007/s10990-007-9018-9 The Mechanical Evaluation of Expressions. P J Landin, 10.1093/comjnl/6.4.3080010- 4620The Computer Journal. 6Landin, P. J., The Mechanical Evaluation of Expressions, The Computer Journal 6, pages 308-320 (1964), ISSN 0010- 4620. https://academic.oup.com/comjnl/article-pdf/6/4/308/1067901/6-4-308.pdf. https://doi.org/10.1093/comjnl/6.4.308 Explaining the lazy Krivine machine using explicit substitution and addresses, Higher-Order and Symbolic Computation. F Lang, Lang, F., Explaining the lazy Krivine machine using explicit substitution and addresses, Higher-Order and Symbolic Computation (2007). The ZINC experiment: an economical implementation of the ML language. X Leroy, Technical report. 117INRIALeroy, X., The ZINC experiment: an economical implementation of the ML language, Technical report 117, INRIA (1990). Explicit substitutions and programming languages. J.-J Lévy, L Maranget, 10.1007/3-540-46691-6_14Foundations of Software Technology and Theoretical Computer Science, 19th Conference. C. P. Rangan, V. Raman and R. RamanujamChennai, IndiaSpringer1738Lévy, J.-J. and L. Maranget, Explicit substitutions and programming languages, in: C. P. Rangan, V. Raman and R. Ramanujam, editors, Foundations of Software Technology and Theoretical Computer Science, 19th Conference, Chennai, India, December 13-15, 1999, Proceedings, volume 1738 of Lecture Notes in Computer Science, pages 181- 200, Springer (1999). https://doi.org/10.1007/3-540-46691-6_14 Realizability toposes and language semantics. J Longley, University of EdinburghPh.D. thesisLongley, J., Realizability toposes and language semantics, Ph.D. thesis, University of Edinburgh (1995). https://www.lfcs.inf.ed.ac.uk/reports/95/ECS-LFCS-95-332/ Fully abstract models of typed λ-calculi. R Milner, 10.1016/0304-3975(77)90053-6Theor. Comput. Sci. 4Milner, R., Fully abstract models of typed λ-calculi, Theor. Comput. Sci. 4, pages 1-22 (1977). https://doi.org/10.1016/0304-3975(77)90053-6 C.-H L Ong, Correspondence between operational and denotational semantics of PCF. S. Abramsky, D. Gabbay and T. S. E. MaibaumOxford University Press4Semantic ModellingOng, C.-H. L., Correspondence between operational and denotational semantics of PCF, in: S. Abramsky, D. Gabbay and T. S. E. Maibaum, editors, Semantic Modelling, volume 4 of Handbook of Logic in Computer Science, pages 269-356, Oxford University Press (1995). LCF considered as a programming language. G D Plotkin, 10.1016/0304-3975(77)90044-5Theor. Comput. Sci. 5Plotkin, G. D., LCF considered as a programming language, Theor. Comput. Sci. 5, pages 223-255 (1977). https://doi.org/10.1016/0304-3975(77)90044-5 An operational semantics of sharing in lazy evaluation. J Seaman, S P Iyer, Sci. Comput. Program. 27Seaman, J. and S. P. Iyer, An operational semantics of sharing in lazy evaluation, Sci. Comput. Program. 27, pages 289-322 (1996). . 10.1016/0167-6423(96)00012-3https://doi.org/10.1016/0167-6423(96)00012-3

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{'abstract': 'Addressing machines have been introduced as a formalism to construct models of the pure, untyped λ-calculus. We extend the syntax of their programs by adding instructions for executing arithmetic operations on natural numbers, and introduce a reflection principle allowing certain machines to access their own address and perform recursive calls. We prove that the resulting extended addressing machines naturally model a weak call-by-name PCF with explicit substitutions. Finally, we show that they are also well-suited for representing regular PCF programs (closed terms) computing natural numbers.PreliminariesThe paradigmatic programming language PCF [23] is a simply typed λ-calculus enriched with constants representing natural numbers, the fundamental arithmetical operations, an if-then-else conditional instruction, and a fixed-point operator. We give PCF for granted and rather present EPCF, an extension of PCF with explicit substitutions[19]. We draw conclusions for the standard PCF by exploiting the fact that they are equivalent on programs (closed terms) of type int. Definition 1.1 Consider fixed a countably infinite set Var of variables. EPCF terms and explicit substitutions are defined by (for n ≥ 0 and x ∈ Var): L, M, N ::= x | M · N | λx.M σ | 0 | pred M | succ M | ifz(L, M, N ) | fix M σ, ρ ::= [x 1 ← (σ 1 , M 1 ), . . . , x n ← (σ n , M n )] As is customary, M · N stands for the application of a term M to its argument N , 0 represents the 6 The reader interested in a comparison with other abstract machines or formalisms is invited to consult [14].This technical appendix is devoted to provide the proofs that have been partially given, or completely omitted, in the body of the paper. As an abbreviation, we write IH for "induction hypothesis".A.1 Proofs of Section 1Proof.[Proof of Lemma 1.8] Items (i) and (ii) are straightforward. We prove (iii). Given an EPCF term M , an EPCF value V , an explicit substitution σ, a type α, and a context Γ such that σ ⊲ M ⇓ d V, σ |= Γ, Γ ⊢ M : α, we prove by induction on a derivation of σ ⊲ M ⇓ d V that ⊢ V : α.Case (nat): In this case M = V = n, for n ∈ N, and α = int. By the typing rules of EPCF, ⊢ n : int. Case (fun): In this case M = λx.M ′ ρ , V = λx.M ′ σ + ρ , α = β → γ. As Γ ⊢ λx.', 'arxivid': '2212.11147', 'author': ['Benedetto Intrigila ', 'Giulio Manzonetto ', 'Nicolas Münnich ', '\nDipartimento di Ingegneria dell\'Impresa\nUniv. USPN, Sorbonne Paris Cité, LIPN, UMR 7030\nUniversity of Rome "Tor Vergata"\nItaly\n', '\nCNRS\nF-93430VilletaneuseFrance\n'], 'authoraffiliation': ['Dipartimento di Ingegneria dell\'Impresa\nUniv. USPN, Sorbonne Paris Cité, LIPN, UMR 7030\nUniversity of Rome "Tor Vergata"\nItaly', 'CNRS\nF-93430VilletaneuseFrance'], 'corpusid': 245739928, 'doi': '10.46298/entics.10533', 'github_urls': [], 'n_tokens_mistral': 24164, 'n_tokens_neox': 21186, 'n_words': 11964, 'pdfsha': '88307460e05d8a9463914bb910871d76e8c46e6c', 'pdfurls': ['https://export.arxiv.org/pdf/2212.11147v4.pdf'], 'title': ['Extended Addressing Machines for PCF, with Explicit Substitutions', 'Extended Addressing Machines for PCF, with Explicit Substitutions'], 'venue': ['Theoretical Informatics ENTICS Proceedings of And Computer Science']}

arxiv

Obtaining Hydrogen energy wave functions using the Runge-Lenz vector 4 May 2018 May 7, 2018 Chun-Khiang Chua Department of Physics and Chung Yuan Center for High Energy Physics Chung Yuan Christian University Chung-Li32023TaoyuanTaiwan, Republic of China Obtaining Hydrogen energy wave functions using the Runge-Lenz vector 4 May 2018 May 7, 2018 The Pauli method of quantizing the Hydrogen system using the Runge-Lenz vector is ingenious. It is well known that the energy spectrum is identical with the one obtained from the Schrödinger equation and the consistency contributed significantly to the development of Quantum Mechanics in the early days. Since the Runge-Lenz vector is a vector and it commutes with the Hamiltonian, it is natural to use it to connect energy eigenstate |n, l, m with other degenerate states |n, l ± 1, m ′ . Recursive relations can be obtained and the wave functions of the whole spectrum can be obtained easily. Note that the recursive relations are consistent with those used in factorizing the Schrödinger equation. Nevertheless, the present analysis provide a better reasoning originated from the conserved vector, the Runge-Lenz vector. As in the Pauli analysis, group theory or symmetry plays a prominent role in the present analysis, while the rest of the derivations are mostly elementary. I. RUNGE-LENZ VECTOR AND THE HYDROGEN ENERGY SPECTRUM The Pauli method of quantizing the Hydrogen system using the Runge-Lenz vector is ingenious. [1] The energy spectrum is identical with the the one obtained from the Schrödinger equation [2] and the consistency contributed greatly to the development of Quantum Mechanics in the early days. Some early development along this line can be found in [3]. Since the Runge-Lenz vector is a vector and it commutes with the Hamiltonian, it is natural to use it to connect energy eigenstate |n, l, m with other degenerate states |n, l ± 1, m ′ . It will be interesting to use it obtain the corresponding wave functions and to show explicitly that they are identical to the results obtained from the Schödinger equation. As we shall see they recursive relations of radial wave functions can be obtained. They are consistent with the results obtained by factorized the Schrödinger equation [4][5][6][7] (see also [8]). Nevertheless we believe that the present approach is more natural as it makes good use of the Runge-Lenz vector, a conserved vector of the system. In other words, we provide a reasoning for the factorization results. The wave functions of the whole spectrum can be obtained easily. We will briefly discuss the E > 0 case and see that the corresponding wave functions can also be verified. As in the Pauli analysis, group theory or symmetry plays a prominent role in the present analysis. The lay our of this paper is as following. In the first section we briefly go through the derivations of the Hydrogen spectrum 1 and will concentrate on obtaining wave function via the Rune-Lenz vector in the next section, which is followed by a conclusion. An appendix is added for the derivation of some relevant matrix elements using group theory. A. Runge-Lenz vector The Hamiltonian of the Hydrogen atom is given by H = p 2 2µ − Ze 2 4πǫ 0 1 r = p 2 2µ − κ 1 r ,(1) with κ ≡ Ze 2 /4πǫ 0 . The Runge-Lenz vector is defined as: A ≡ 1 2µ ( p × L − L × p) − κ r r .(2) The Runge-Lenz vector satisfies the following relations: [1] L · A = A · L = 0,(3)[H, A] = 0, [H, L] = 0,(4)[L i , L j ] = ihǫ ijk L k , [L i , A j ] = ihǫ ijk A k , [A i , A j ] = ih(− 2 µ H)ǫ ijk L k ,(5) 1 Our derivations follow closely to those in Ref. [10]. A · A = 2 µ H( L 2 +h 2 ) + κ 2 .(6) Note that the Runge-Lenz vector is a conserved operator. The above relations will be useful in obtaining the Hydrogen energy spectrum. [1] B. Hydrogen energy spectrum The eigenvalue equation of the Hydrogen Hamiltonian is given by H|E, α = E|E, α = −|E||E, α ,(7) where we only consider the E < 0 case here and α is a possible quantum number. The set of the eigenstates {|E, α , |E, β . . . } with E fixed spans the degenerate space of eigenstates all having the same energy. From Eq. (4), we know that the Hamiltonian commutes with L and A. Hence, it is useful to define the following matrices: (L i ) αβ ≡ E, α|L i |E, β , (A i ) αβ ≡ E, α|A i |E, β ,(8) and the relations in Eqs. (3), (5) and (6) correspond to the following relations of matrices: (L · A) αβ = (A · L) αβ = 0,(9)A 2 αβ = − 2|E| µ (L 2 +h 2 1) αβ + κ 2 1 αβ ,(10) and [L i , L j ] αβ = ihǫ ijk (L k ) αβ , [L i , A j ] αβ = ihǫ ijk (A k ) αβ , [A i , A j ] αβ = ihǫ ijk (− 2E µ )(L k ) αβ = ihǫ ijk ( 2|E| µ )(L k ) αβ .(11) With the following definition (A ′ i ) αβ ≡ µ 2|E| (A i ) αβ ,(12) the above equations become (L · A ′ ) αβ = (A ′ · L) αβ = 0,(13)(A ′ 2 + L 2 ) αβ = −h 2 + µκ 2 2|E| 1 αβ ,(14) and [L i , L j ] αβ = ihǫ ijk (L k ) αβ , [L i , A ′ j ] αβ = ihǫ ijk (A ′ k ) αβ , [A ′ i , A ′ j ] αβ = ihǫ ijk (L k ) αβ .(15) The above equation, Eq. (15), implies that L i and A ′ j are the generators of the O(4) group. [1] The quantization of the Hydrogen system can be achieved by using group theory. [1] A representation of O(4) can be expressed as a direct product of two SO(3) representations as following. 2 Defining two new sets of operators B (+) i and B (−) i , B (±) i ≡ 1 2 (L i ± A ′ i ),(16) Eq. (15) becomes [B (+) i , B (+) j ] αβ = ihǫ ijk (B (+) k ) αβ , [B (−) i , B (−) j ] αβ = ihǫ ijk (B (−) k ) αβ , [B (+) i , B (−) j ] αβ = 0.(17) It is clear that B ( B (+) ) 2 , B (+) z , ( B (−) ) 2 , B (−) z .(18) We can now return to the usual Dirac notation. The corresponding eigenvalue equations are H|E, b (+) , m (+) , b (−) , m (−) = E|E, b (+) , m (+) , b (−) , m (−) , ( B (±) ) 2 |E, b (+) , m (+) , b (−) , m (−) =h 2 b (±) (b (±) + 1)|E, b (+) , m (+) , b (−) , m (−) , B (±) z |E, b (+) , m (+) , b (−) , m (−) = m (±)h |E, b (+) , m (+) , b (−) , m (−) ,(19)with b (+) , b (−) = 0, 1/2, 1, 3/2, ... and −b (±) ≤ m (±) ≤ b (±) . Note that we have ( B (±) ) 2 = 1 4 ( L 2 ± L · A ′ ± A ′ · L + A ′2 ) = 1 4 ( L 2 + A ′2 ) = 1 4 (−h 2 − µ 2H κ 2 ).(20) Hence, we have ( B (+) ) 2 = ( B (−) ) 2 . Eq. (20) implies the following relation: ( B (±) ) 2 |E, b, m (+) , m (−) =h 2 b(b + 1)|E, b, m (+) , m (−) , = 1 4 (−h 2 − µ 2E κ 2 )|E, b, m (+) , m (−) ,(21)with b ≡ b (+) = b (−) = 0, 1/2, 1, 3/2, ... and −b ≤ m (±) ≤ b. The energy eigenvalue E = E n can now be obtained as [1] E n = − κ 2 µ 2h 2 (2b + 1) 2 = − κ 2 µ 2h 2 n 2 = − Z 2 e 4 µ 32π 2 ǫ 2 0h 2 n 2 ,(22) with n defined as n ≡ 2b + 1 = 1, 2, 3, .... It will be useful to define the (reduced) Bohr radius, a 0 ≡ 4πǫ 0h 2 /e 2 µ = (h/µc)/α = Zh 2 /κµ, where α ≡ e 2 /4πǫhc ≃ 1/137 is the fine structure constant. The energy spectrum obtained by Pauli [1] is consistent with the one obtained from the Schrödinger equation [2] and the consistency contributed significantly to the development of Quantum Mechanics in the early days. II. HYDROGEN ATOM ENERGY EIGENSTATE A. Connecting degenerate states using the Runge-Lenz vector Since the Runge-Lenz vector is a vector and it commutes with the Hamiltonian, it is natural to use it to connect energy eigenstate |n, l, m with other degenerate states |n, l ± 1, m ′ , |n, l, m ′ . As shown in the previous section the energy eigenstates of the Hydrogen system are |n, b = (n − 1)/2, m (+) , m (−) , which satisfy H n, n − 1 2 , m (+) , m (−) = E n n, n − 1 2 , m (+) , m (−) , ( B (±) ) 2 n, n − 1 2 , m (+) , m (−) =h 2 n 2 − 1 4 n, n − 1 2 , m (+) , m (−) , B (±) z n, n − 1 2 , m (+) , m (−) = m (±)h n, n − 1 2 , m (+) , m (−) ,(23) with −(n − 1)/2 ≤ m (±) ≤ (n − 1)/2. In general these eigenstates do not have specified angular momentum quantum numbers and are different from the energy eigenstates in a more familiar basis: H|n, l, m = E n |n, l, m , L 2 |n, l, m = l(l + 1)h 2 |n, l, m , L z |n, l, m = m|n, l, m ,(24) with −l ≤ m ≤ l. Since the angular momentum can be obtained through the following equation, see Eq. (16), L = B (+) + B (−) ,(25) with B (+) and B (−) viewed as two independent spin operators, the |n, l, m state can be constructed as in the analysis of the addition of angular momentum: |n, l, m = m (+) ,m (−) n, n − 1 2 , m (+) , m (−) n − 1 2 , m (+) , n − 1 2 , m (−) l, m ,(26)where n−1 2 , m (+) , n−1 2 , m (−) |l, m is the ClebschGordan coefficient. Note that the minimum of l is |(n − 1)/2 − (n − 1)/2| = 0, while the maximum is (n − 1)/2 + (n − 1)/2 = n − 1. These are consistent with the result obtained by using the Schödinger equation. The Runge-Lenz vector A commutes with the Hamiltonian H. It can connect eigenstate |n, lm with other degenerate energy eigenstates |n, l ± 1, m ′ , |n, l, m ′ . To proceed we define A ± ≡ A x ± iA y , B (±) ± ≡ B (±) x ± iB (±) y .(27) where we apply A − on the m = l state, which is found to be useful in obtaining the radial wave function in later discussion. Using the familiar formula of non-vanishing matrix elements of lowering operator L − = L x −iL y , we have n, l ′ , l − 1|B (+) − |n, l, l = l ′ , l − 1|b, m (+) − 1, b, m (−) (b + m (+) )(b − m (+) + 1)h b, m (+) , b, m (−) |l, l , n, l ′ , l − 1|B (−) − |n, l, l = l ′ , l − 1|b, m (+) , b, m (−) − 1 (b + m (−) )(b − m (−) + 1)h b, m (+) , b, m (−) |l, l ,(29) with m (±) summed. Possible non-vanishing matrix elements are for l ′ = l, l ± 1 as B (±) are vector operators. Using group theory the above matrix elements (for l ′ = l, l ± 1) are found to be n, l − 1, l − 1|B (±) − |n, l, l = ∓ l(n 2 − l 2 ) 2(2l + 1)h , n, l + 1, l − 1|B (±) − |n, l, l = ± n 2 − (l + 1) 2 2(2l + 1)(2l + 3)h , n, l, l − 1|B (±) − |n, l, l = l 2h ,(30) where the derivation are shown in Appendix A. Note that the above results also hold for the l = 0 case, where the equation implies that n, l − 1, l − 1|B =        Z na 0 2(n 2 −(l+1) 2 ) (2l+1)(2l+3) , l ′ = l + 1, − Z na 0 2l(n 2 −l 2 ) (2l+1) , l ′ = l − 1 0, l ′ = l ± 1,(31) where we have made use of −2µE n /h 2 = κ 2 µ 2 /h 4 n 2 = (Z/na 0 ) 2 . Substitute them into Eq. (28), we finally obtain our master formula: μ h 2 A − |n, l, l = Z na 0 2(n 2 − (l + 1) 2 ) (2l + 1)(2l + 3) |n, l + 1, l − 1 − 2l(n 2 − l 2 ) (2l + 1) |n, l − 1, l − 1 . (32) Note that as in the Pauli analysis, the above master formula follows from group theory or symmetry. As we shall see the above equation can provides recursive relations on degenerate states, and the relations are powerful enough to determine the wave functions. B. Obtaining the radial wave function R nl (r) Recursive relations of R nl (r) We will obtain the recursive relations of the radial wave functions from the master formula here. Note that most of the derivations are elementary. Using r|n, l, m = R nl (r)Y lm (θ, φ) and r|A − |n, l, l = (A − ) op r|n, l, l ,(33) the master formula Eq. (32) can be expressed as: μ h 2 (A − ) op R n,l (r)Y l,l (θ, φ) = Z na 0 2(n 2 − (l + 1) 2 ) (2l + 1)(2l + 3) R n,l+1 (r)Y l+1,l−1 (θ, φ) − 2l(n 2 − l 2 ) (2l + 1) R n,l−1 (r)Y l−1,l−1 (θ, φ) .(34) To proceed we need to work out the left-hand-side of the above equation. Using p× L+ L× p = 2ih p, it is convenient to express the ± components of the Runge-Lenz vector as A ± = 1 µ ( p × L − ih p) ± − κ r ± r = 1 µ (±ip z L ± ) + 1 µ (∓ip ± L z − ihp ± ) − κ r ± r ,(35) where we have defined r ± ≡ x ± iy. Consequently, the operator (A − ) op in Eq. (34) is given by (A − ) op = − 1 µ (ip z L − ) op + 1 µ [ip − (L z −h)] op − κ r − r ,(36) with (p ± ) op =h i (∂ x ± i∂ y ) = 2h i ∂ ∂r ∓ , (L z ) op =h i (x∂ y − y∂ x ) =h r + ∂ ∂r + − r − ∂ ∂r − , (L ± ) op = ∓ir ± (p z ) op ± iz(p ± ) op = ∓h r ± ∂ ∂z − 2z ∂ ∂r ∓ .(37) Using r 2 = r + r − + z 2 and ∂ ∂r ± f (r) = r ∓ 2r df dr , ∂ ∂z f (r) = z r df dr ,(38) it can be shown from Eq. (37) that we must have (L z ) op f (r) = (L ± ) op f (r) = 0, which are expected from the familiar forms of ( L) op in wave mechanics as they only involve derivatives with respect to θ and φ. Note that it is convenient to use the (r + , r − , z) coordinate, as Y l,±l (θ, φ) in the left-hand-side of Eq. (34) have simple forms in terms of r ± and r: Y l,±l = (∓) l 2 l l! (2l + 1)! 4π (r ± ) l r l ∝ (r ± ) l r l .(39) From Eqs. (37), (38) and the fact that (L z ) op and (L − ) op do not act on any function of r, it can be easily shown that we must have (ip − ) op [(L z ) op −h] (r + ) l r l f (r) = 2h 2 (l − 1) ∂ ∂r + (r + ) l r l f (r), (ip z ) op (L − ) op (r + ) l r l f (r) = −lh 2 ∂ ∂z 2z r + (r + ) l r l f (r) ,(40) and, consequently, the left-hand-side of Eq. (34) becomes µ(A − ) op h 2 (r + ) l r l R nl (r) = l ∂ ∂z 2z r + + 2(l − 1) ∂ ∂r + − κμ h 2 r − r (r + ) l R nl (r) r l .(41) It can be further expressed as µ(A − ) op h 2 (r + ) l r l R nl (r) = (r + ) l 2l r + + 2l z r + z r d dr + 2l(l − 1) r + + (l − 1) r − r d dr − Z a 0 r − r R nl (r) r l ,(42) where we have made use of Eq. (38) and κµ/h 2 = Z/a 0 . Use again z 2 = r 2 − r + r − , we obtain µ(A − ) op h 2 (r + ) l r l R nl (r) = (r + ) l 2l r + l + r d dr − r − r (l + 1) d dr + Z a 0 R nl (r) r l ,(43) or, equivalently, [see Eq. (39)] µ(A − ) op h 2 Y l,l R nl (r) = r l−1 r r + Y l,l 2l 1 + r d dr − r l r − r Y l,l (l + 1) d dr + Z a 0 R nl (r) r l . (44) It is useful to note that the spherical harmonics Y l,l has the following properties: 3 r r + Y l,l = − (2l + 1) 2l Y l−1,l−1 , r − r Y l,l = 8π 3 Y 1,−1 Y l,l = − 2l 2l + 1 Y l−1,l−1 + 2 (2l + 1)(2l + 3) Y l+1,l−1 ,(45) and the above equation can be expressed as µ(A − ) op h 2 Y l,l R nl (r) = − 2 (2l + 1)(2l + 3) r l Y l+1,l−1 (l + 1) d dr + Z a 0 R nl (r) r l − 2l 2l + 1 r l−1 Y l−1,l−1 (2l + 1)l + lr d dr − Zr a 0 R nl (r) r l .(46) Compare the above equation to the master formula Eq. (34), we finally obtain − Z n 2 − (l + 1) 2 na 0 R n,l+1 (r) r l+1 = (l + 1) r d dr + Z a 0 r R nl (r) r l , Z √ n 2 − l 2 na 0 R n,l−1 (r) r l−1 = (2l + 1)l + lr d dr − r Z a 0 R nl (r) r l .(47) These are the recursive relations of the radial wave function R nl (r) and they are important results of this section. The above relations are consistent to the results found in [4][5][6] (see also [8]) using the factorization method. Nevertheless we believe that according to the properties of the Runge-Lenz vector, which is a conserved vector, the above derivation is the most natural way to obtain them. In other words, we provide a reasoning for the factorization results. As we shall see shortly they are powerful enough to determine the radial wave functions. 2. Solve for R n,n−1 (r) using the recursive relation with the normalization constant, c n = 2 n n n−1/2 (2n)! Z na 0 3/2 ,(50) obtained from the usual normalization condition ∞ 0 dr r 2 R 2 n.n−1 (r) = 1. In fact, the above result can be quickly obtained by noting |n, l = n − 1, m = ±(n − 1) = n, b (±) = n − 1 2 , m (+) = ± n − 1 2 , m (−) = ± n − 1 2 ,(51) and A ± ∝ B (+) ± − B (−) ± , which imply: A ± |n, n − 1, ±(n − 1) = 0.(52) From A ± = 1 µ (±ip z L ± ) + 1 µ (∓ip ± L z − ihp ± ) − κ r ± r ,(53) we have 0 = A ± |n, n − 1, ±(n − 1) = − i nh µ p ± − κ r ± r |n, n − 1, ±(n − 1) .(54) Using r|n, l, m = R nl (r)Y lm (θ, φ) the above equation takes the following form: nih µ (p ± ) op + κ r ± r R n,n−1 (r)Y n−1,±(n−1) = 0. With the help of Y l,±l ∝ (r ± ) l /r l and (p ± ) op (r ± ) l f (r) = 2h i ∂ ∂r ∓ (r ± ) l f (r) = (r ± ) lh i r ± r df (r) dr ,(56) we clearly see that the it is equivalent to Eq. (48). Obtaining other R nl (r) Once R n,n−1 (r) is known, other R n,l (r) can be obtained readily by applying the second relation of the recursive relations given in Eq. (47): R n,l−1 (r) r l−1 = n √ n 2 − l 2 (2l + 1)la 0 Z + la 0 r Z d dr − r R nl (r) r l .(57) For illustration, we apply the above equation on the l = n − 1 radial wave function and obtain R n,n−2 (r) = c n n(n − 1) √ 2n − 1 Zr a 0 n−2 1 − Z n(n − 1)a 0 r e − Z na 0 r ,(58) and, sequentially, applying the relation on l = n − 2, we have R n,n−3 (r) = 1 2 c n Zr a 0 n−3 (2n − 3)(n − 2)n 2 (2n − 1)(n − 1) 1 − 2Zr n(n − 2)a 0 + 2Z 2 r 2 n 2 (6 − 7n + 2n 2 )a 2 0 e − Zr na 0 .(59) In principle, the procedure can be carried out to obtain all R nl (r). It will be useful to show explicitly some of the radial wave functions obtained: R 10 (r) = 2 Z a 0 3/2 e −Zr/a 0 , R 21 (r) = 1 √ 3 Z 2a 0 3/2 Zr a 0 e −Zr/2a 0 , R 20 (r) = 2 Z 2a 0 3/2 1 − Zr 2a 0 e −Zr/2a 0 , R 32 (r) = 2 √ 2 27 √ 5 Z and compare them to those obtained by solving the Schrödinger equation directly, see for example, [12]. Indeed, it is clear that they are consistent with the results obtained in the two approaches. 4 It is interesting that even the phase conventions match. It can be compared to the radial equation from the Schrödinger equation, see for example, [12] d 2 dr 2 + 2(l + 1) r d dr + 2Z a 0 r + 2μ h 2 E n R nl r l = 0.(64) It is clear that Eq. (63) is same as the above equation with E n = −Z 2h2 /2µa 2 0 n 2 . D. Wave functions in the E > 0 case We briefly discuss the E > 0 case here. For the E > 0 case, as in the E < 0 case, one can still define B (±) i ≡ 1 2 (L i ± iA ′ ) ≡ 1 2 L i ± i µ 2E A ,(65) which satisfy [B (+) i , B (+) j ] αβ = ihǫ ijk (B (+) k ) αβ , [B (−) i , B (−) j ] αβ = ihǫ ijk (B (−) k ) αβ , [B (+) i , B (−) j ] αβ = 0,(66) and ( B (±) ) 2 = 1 4 ( L 2 ± i L · A ′ ± i A ′ · L − A ′2 ) = 1 4 ( L 2 − A ′2 ) = 1 4 (−h 2 − µ 2E κ 2 )1.(67) However, now the situation is different. The above relations cannot be used to obtain energy eigenvalue as in the E < 0 case. The reason is that B (±) are no longer hermitian. The usual procedure of obtaining quantum numbers in L 2 and L z cannot be used in ( B (±) ) 2 and B satisfy the usual rotation group [SO(3)] algebra. Conventionally the simultaneous eigenstates in the |α space are choosen to be the eigenstates of the following mutual commuting matrices: From Eq. (16), we have μ h 2 A − |n, l, l = l ′ =l,l±1 |n, l ′ , l − 1 n, l ′ , l − 1|(B (+) − − B (−) − )|n, l, l h 2µE n −h 2 , − |n, l, l and n, l, l − 1|B (±) − |n, l, l are vanishing as they should. Using the above equation, the corresponding matrix elements of A − are given by μ h 2 n, l ′ , l − 1|A − |n, l, l = n, l ′ , l − 1|(B For3 l = n − 1, the first relation of the recursive relations in Eq. The first relation follows from Eq. (39), while the second relation follows from the familiar relationY l,m Y l ′ ,m ′ = L,M (2l+1)(2l ′ +1) 4π(2L+1)l, 0, l ′ , 0|L, 0 l, m, l ′ , m ′ |L, M Y L,M , see for example[11]. It is probable that a modern reader is more familiar with the case of the SO(3, 1) Lorentz group, which can be analyzed using a similar manipulation, see, for example, ref.[9]. Note that there is a typo in the normalization factor of R 31 (r) in[12].5 The additional factors 1/(l + 1) 2 and 1/l 2 are designed to remove the coefficients of d 2 /dr 2 .6 The equation is said to be factorizable[4][5][6][7]. AcknowledgmentsThe authors are grateful to Chung-Wen Kao for discussions. This research was supported in part by the Ministry of Science and Technology of R.O.C. under Grant Nos. 103-2112-M-033-002-MY3 and 106-2112-M-033-004-MY3.C. The radial wave function R nl satisfies the radial Schrödinger equationApplying the recursive relations, Eq. (47), on R n,l /r l can bring it to R n,l±1 /r l±1 and suitably apply the relations again can bring them back to R n,l /r l . These procedures produce the following identities on R nl /r l ,5 61 (l + 1) 2 (2l + 3)(l + 1) + (l + 1)r d dr − r Z a 0 Z a 0 r + (l + 1) r d dr R nl (r) r l = − Z 2 (n 2 − (l + 1) 2 ) n 2 a 2 0 (l + 1) 2 R nl (r) r l ,giving,or, equivalently,z . For example, the above equation show that ( B (±) ) 2 are negative matrices and the usual steps in quantizing angular momentum break down in quantizing this system.Nevertheless some results obtained in the previous section can still be used. In particular the recursive relations similar to Eq. (47) can be used to find the wave function in the E > 0 case. Replacing n by iν in Eq. (47), we have the following recursive relationswhich lead toor, simply,The above equation can match to the Schrödinger equation (with E > 0):by takingwith k ≡ √ 2µE/h. Solving the Schrödinger equation now reduce to finding functions that satisfy the recursive relations in Eq. (69). From[13], we find that the Coulomb functions, u l (η, ρ) = F l (η, ρ) and G l (η, ρ), have the following recursive relations:or, equivalently,Compare the above relations with those in Eq. (68), we see that by takingthe following R l /r l ,with constants a, b, satisfies the recursive relations in Eq. (68). Indeed, the above result on R l is consistent with those obtained by solving the Schrödinger equation directly, see for example,[14].III. CONCLUSIONSIn this work we follow the Pauli method of quantizing the Hydrogen system using the Runge-Lenz vector. Since the Runge-Lenz vector is a vector and it commutes with the Hamiltonian, it is natural to use it to connect energy eigenstate |n, l, m with other degenerate states |n, l ± 1, m ′ . In this work, we found the recursive relations for the radial wave functions. They are consistent with the results found in[4][5][6][7]using the factorization method. Nevertheless we believe that as the present approach makes good use of the properties of the Runge-Lenz vector, which is a conserved vector, it is the most natural way to obtain the recursive relations. The wave functions of the whole spectrum can be obtained easily. These radial wave functions are shown to satisfy the Schrödinger equation. In addition, using the recursive relations the wave functions in the E > 0 case can also be verified. As in the Pauli analysis, group theory plays a prominent role in this analysis, while the rest of the derivations are mostly elementary.with m (±) summed. First we note that from the known symmetrical properties of Clebsch-Gordan coefficients,for integers l, l ′ , which is certainly the case here.Using the known formula in the study of addition of angular momenta[15](see, also[16]),where { } is the Wigner 6-j symbol and j ′ 1 ||T k (1)||j 1 is the reduced matrix element, we must haveand, consequently, we obtain− |n, l, l = ± n 2 − (l + 1) 2 2(2l + 1)(2l + 3)h , n, l, l − 1|B − |n, l, l are vanishing as they should. Furthermore, in the above derivation we encounter 1/(n 2 − 1) 1/2 factors in Eq. (A6), and, hence, n needs to be greater than 1. Nevertheless, using Eq. (A1) and by direct computation, it is easy to check that the above equation also holds for the n = 1 case. W Pauli, 10.1007/BF01450175ber das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik. B.L. van der WaardenNorth-Holland, Amsterdam36translated in Sources of Quantum MechanicsW. Pauli, "ber das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik," Z. Phys. 36, no. 5, 336 (1926) doi:10.1007/BF01450175; translated in Sources of Quantum Me- chanics, edited by B.L. van der Waarden, North-Holland, Amsterdam, 1967, pp. 387-415. Quantisierung als Eigenwertproblem. E Schrdinger, 10.1002/andp.19263840404Annalen Phys. 3844361Annalen Phys. 79, no. Ser.IV, 361 (1926)E. Schrdinger, "Quantisierung als Eigenwertproblem," Annalen Phys. 384, no. 4, 361 (1926) [Annalen Phys. 79, no. Ser.IV, 361 (1926)]. doi:10.1002/andp.19263840404 On the Theory of the hydrogen atoms. V Fock ; /Bf01336904 M, C Bander, Itzykson, 10.1103/RevModPhys.38.330Rev. Mod. Phys. 98330Z. Phys.V. Fock, "On the Theory of the hydrogen atoms," Z. Phys. 98, 145 (1935); doi:10.1007/BF01336904 M. Bander and C. Itzykson, Rev. Mod. Phys. 38, 330 (1966). doi:10.1103/RevModPhys.38.330 A method of determining quantum-mechanical eigenvalues and eigenfunctions. E Schrödinger, Proc. Roy. Irish Acad. (Sect. A). Roy. Irish Acad. (Sect. A)469E. Schrödinger, "A method of determining quantum-mechanical eigenvalues and eigenfunc- tions," Proc. Roy. Irish Acad. (Sect. A) 46, 9 (1940). The factorization method. L Infeld, T E Hull, 10.1103/RevModPhys.23.21Rev. Mod. Phys. 2321L. Infeld and T. E. Hull, "The factorization method," Rev. Mod. Phys. 23, 21 (1951). doi:10.1103/RevModPhys.23.21 Factorization of the radial Schrödinger equation and four kinds of raising and lowering operators of 3-D and 2-D hydrogen atoms. Y F Liu, Y A Lei, J Y Zeng, AS-ITP-96-19Y. F. Liu, Y. A. Lei and J. Y. Zeng, "Factorization of the radial Schrödinger equation and four kinds of raising and lowering operators of 3-D and 2-D hydrogen atoms," AS-ITP-96-19; Factorization of the radial Schrödinger equation of the hydrogen atom. B-W Xu, F-M Kong, Phys. Lett. A. 259212B-W. Xu and F-M. Kong, "Factorization of the radial Schrödinger equation of the hydrogen atom", Phys. Lett. A 259, 212 (1999). On the determination of radial matrix elements for high-n transitions in hydrogenic atoms and ions. J D Hey, J. Phys. B. 392641J.D. Hey, "On the determination of radial matrix elements for high-n transitions in hydrogenic atoms and ions", J. Phys. B, 39, 2641 (2006). Heisenberg's quantum mechanics. M Razavy, 10.1142/7702World ScientificHackensack, USAM. Razavy, "Heisenberg's quantum mechanics," Hackensack, USA: World Scientific (2011) 657 p doi:10.1142/7702 S Coleman, arXiv:1110.5013Notes from Sidney Coleman's Physics 253a: Quantum Field Theory. physics.ed-phS. Coleman, "Notes from Sidney Coleman's Physics 253a: Quantum Field Theory," arXiv:1110.5013 [physics.ed-ph]. Quantum Mechanics. L Schiff, J Bandhyopadhyay, Mc Graw Hill India4th. editionL. Schiff and J Bandhyopadhyay, "Quantum Mechanics", 4th. edition, Mc Graw Hill India, 2014. Mathematical Methods for Physicists. G B Arfken, H J Weber, F E Harris, Academic Press Massachusetts7th editionG. B. Arfken, H. J. Weber, F. E. Harris, "Mathematical Methods for Physicists", 7th edition, Academic Press Massachusetts, 2012. Quantum Physics. S Gasiorowicz, John Wiley & Son3rd EdS. Gasiorowicz, "Quantum Physics", 3rd Ed., John Wiley & Son, 2003. Handbook of Mathematical Functions. Stegun Abramowitz, 14Abramowitz and Stegun, "Handbook of Mathematical Functions," chapter 14. L D Landau, E M Lifshitz, Quantum Mechanics : Non-Relativistic Theory. Pergamon PressCourse of theoretical physics III. 3rd ed.L. D. Landau and E. M. Lifshitz, "Quantum Mechanics : Non-Relativistic Theory," (Course of theoretical physics III, 3rd ed.), Pergamon Press (1977). Angular Momentum Techniques in Quantum Mechanics. V Devanathan, Kluwer AcademicV. Devanathan, " Angular Momentum Techniques in Quantum Mechanics," Kluwer Academic, (2002), chapter 7. Elementary Theory of Angular Momentum. M E Rose, DoverM. E. Rose, "Elementary Theory of Angular Momentum", Dover (2011), chapter 6.

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{'abstract': 'The Pauli method of quantizing the Hydrogen system using the Runge-Lenz vector is ingenious. It is well known that the energy spectrum is identical with the one obtained from the Schrödinger equation and the consistency contributed significantly to the development of Quantum Mechanics in the early days. Since the Runge-Lenz vector is a vector and it commutes with the Hamiltonian, it is natural to use it to connect energy eigenstate |n, l, m with other degenerate states |n, l ± 1, m ′ . Recursive relations can be obtained and the wave functions of the whole spectrum can be obtained easily. Note that the recursive relations are consistent with those used in factorizing the Schrödinger equation. Nevertheless, the present analysis provide a better reasoning originated from the conserved vector, the Runge-Lenz vector. As in the Pauli analysis, group theory or symmetry plays a prominent role in the present analysis, while the rest of the derivations are mostly elementary.', 'arxivid': '1805.01610', 'author': ['Chun-Khiang Chua \nDepartment of Physics and Chung Yuan Center for High Energy Physics\nChung Yuan Christian University\nChung-Li32023TaoyuanTaiwan, Republic of China\n'], 'authoraffiliation': ['Department of Physics and Chung Yuan Center for High Energy Physics\nChung Yuan Christian University\nChung-Li32023TaoyuanTaiwan, Republic of China'], 'corpusid': 119464972, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 10775, 'n_tokens_neox': 9294, 'n_words': 5375, 'pdfsha': '3f913c2647261090c248d8a3194ec2d8bf6c455b', 'pdfurls': ['https://arxiv.org/pdf/1805.01610v1.pdf'], 'title': ['Obtaining Hydrogen energy wave functions using the Runge-Lenz vector', 'Obtaining Hydrogen energy wave functions using the Runge-Lenz vector'], 'venue': []}

arxiv

Do language models have coherent mental models of everyday things? Yuling Gu yulingg@allenai.org Allen Institute for AI SeattleWA Bhavana Dalvi bhavanad@allenai.org Allen Institute for AI SeattleWA Mishra Allen Institute for AI SeattleWA Peter Clark peterc@allenai.org Allen Institute for AI SeattleWA Do language models have coherent mental models of everyday things? When people think of everyday things like an egg, they typically have a mental image associated with it. This allows them to correctly judge, for example, that "the yolk surrounds the shell" is a false statement. Do language models similarly have a coherent picture of such everyday things? To investigate this, we propose a benchmark dataset consisting of 100 everyday things, their parts, and the relationships between these parts, expressed as 11,720 "X relation Y?" true/false questions. Using these questions as probes, we observe that state-ofthe-art pre-trained language models (LMs) like GPT-3 and Macaw have fragments of knowledge about these everyday things, but do not have fully coherent "parts mental models" (54-59% accurate, 19-43% conditional constraint violation). We propose an extension where we add a constraint satisfaction layer on top of the LM's raw predictions to apply commonsense constraints. As well as removing inconsistencies, we find that this also significantly improves accuracy (by 16-20%), suggesting how the incoherence of the LM's pictures of everyday things can be significantly reduced. 1 Introduction Psychologists and cognitive scientists hypothesize that humans develop mental models of the world, namely internal, conceptual representations of the environment which we base our decisions and actions on (Ha and Schmidhuber, 2018;Jonassen and Henning, 1996). Hespos and Spelke (2004) observed that 5-month-old human infants exhibit understanding of mechanical properties of objects in terms of arrangements and motions of surfaces, well before they can understand language. Drawing loosely on this idea, but without making any claims about how LMs reason internally (Shanahan,Figure 1: While humans appear to have coherent mental pictures of everyday things (e.g., an egg, A), our question-asking probes suggest that LMs do not (e.g., one LM answered that the egg white both surrounds and is surrounded by the shell, B). This model incoherence can be reduced by applying commonsense constraints (e.g., surrounds is asymmetric), resulting in a more coherent parts model (C). 2022; Andreas, 2022), we investigate if pre-trained language models show evidence of coherent internal representations of everyday things, analogous to human mental models, via probing. We focus on mental models in the context of ordinary objects that we encounter in our everyday lives. Such commonsense knowledge helps us understand how these everyday things work and how to interact with them. For example, when someone tries to make a fried egg, they know that it has a shell and that it can be cracked open to reveal the egg white and yolk inside. However, if a system does not have a coherent picture of such everyday things, thinking that the egg yolk surrounds the shell, then it might have to resort to ridiculous approaches such as trying to scrape the egg yolk off the shell into the pan. We explore a first version of this, in which we consider only knowledge about an object's parts and their relationships. We refer to this knowledge as a parts mental model. We first create a benchmark dataset of 100 everyday things, by asking human annotators to draw a graph representing their parts mental model (e.g., Figure 2) depicting the parts of an everyday thing, spatial relationships, connections between its parts and functional dependencies (if any). Then we probe two representative state-of-the-art LMs with questions about these everyday things. We find that the LMs' parts mental models are generally of poor quality. Further, model predictions can violate basic consistency constraints e.g. transitivity. To alleviate this, we apply constraint reasoning to derive more accurate and consistent mental models of everyday things, correcting some of the LMs' original inconsistencies. This is illustrated in Figure 1. Our contributions are: 1. We present a benchmark dataset of parts mental models consisting of 100 everyday things, 2.2K parts and 11.7K relationships. 2. We show that SOTA LMs like Macaw are poor at answering relationship queries between parts of everyday things. The parts mental models derived using their predictions are only 54-59% accurate, and significantly inconsistent (19-43% conditional violation τ ). 3. We propose a neuro-symbolic method that applies constraint reasoning on top of raw LM predictions as a way of obtaining more consistent (0% conditional violation τ ) and more accurate mental models (16-20% improvement). This suggests a broader cognitive architecture (LM + reasoner) for future systems, to better construct mental models than the LM alone. Related work Mental models: The idea of mental models (Johnson-Laird, 1983) is not new. Many years ago, Craik (1943) proposed that thinking itself is the manipulation of internal representations of the world. Craik (1943) described mental models as a 'small-scale model' of external reality and of its own possible actions within someone's head. Such a mental model is useful in many ways, including allowing one to try out various alternatives, make conclusions, react to future situations, learn from past events, and in general, improve competency. Years later, when Johnson-Laird (2006) outlined the mental processes that underlie human reasoning, he based his discussion on the fundamental assumption that human beings can construct internal representations of spatial layouts, and specified mental models to be iconic. In his words, a mental model's "parts and the relations among them correspond to the parts of the layout and the relations among them." While coherent internal representations of spatial layouts are crucial for human reasoning, their role, coherence, and even existence in LMs have not been systematically explored. In this work, we try to bridge this gap by proposing a benchmark dataset and methodology to compare human internal representations of spatial layouts of everyday things with those of LMs. Prior datasets: Prior works on reasoning about object/body parts include Li et al. (2019b) which focused on human body parts and human interaction with other objects. The PTR benchmark (Hong et al., 2021) is a QA dataset about objects and their parts, combining 5 everyday things: chair, table, bed, refrigerator, and cart, to create questions across 70K different scenes. Ji et al. (2022) used tangram puzzles to analyze shape naming, part naming and segmentation divergence across participants when they see a certain shape. Contributing to this existing body of datasets, the dataset we introduce serves as a resource for researchers to study canonical parts mental models for a wide variety of everyday things, focusing on relationships between parts of objects, which is fundamental to how humans think and interact with these things. Large language models: Despite recent advances in LMs, studies suggest that they still struggle at reasoning with real-world entities and concepts. Bisk et al. (2020) found that when LMs answer questions involving physical commonsense reasoning, their performance at that time was near chance level for questions involving spatial relations like "top" and "bottom." Sahu et al. (2022) demonstrated the lack of conceptual consistency in LMs by correlating models' answers on commonsense reasoning questions (CSQA dataset) and their answers on associated conceptual questions from ConceptNet knowledge base. To improve existing systems, progress has been made such as by imposing constraints with neuro-symbolic approaches (Nye et al., 2021;Mitchell et al., 2022) and incorporating both textual and visual information (Dan et al., 2020). Inspired by recent progress, we propose a constraint reasoning method that applies hard commonsense constraints (e.g., if 'A above B' is True then 'A below B' cannot be True) on top of raw LM predictions to produce more accurate and consistent mental models of everyday things. Parts mental models and Task We define "parts mental model" for everyday things in this section. Then in the rest of the paper, we describe how we collect a dataset for them, measure LMs' coherence on them, and finally apply external reasoning to improve the accuracy and consistency of LMs' parts mental model. Here, we use parts mental model to mean a partsfocused subset of a complete mental model of an entity. We represent a parts mental model as a directed graph where parts of the everyday thing form the nodes of this graph and these nodes are connected with edges indicating how these parts are related to each other. Based on prior works such as Renz (2002) and Gunning et al. (2010), we selected 11 spatial orientation relations to focus on. In addition, we augmented these with relations describing connectivity and functional dependency. In total, we consider 14 relationships (across these 3 categories) between parts, listed in Table 2. Note that the notion of a single "parts mental model" for an everyday thing is somewhat uncon-strained (e.g., which parts to pick? what version of the entity are we talking about?). To make this task more well-defined, we also provide a predefined list of parts as a guide (details in Section 4.1), and the task for annotators or a model is to specify relationships between them as they see appropriate, using our ontology of relationships. This is important so that we can do meaningful comparisons between language models and humans' notion of parts mental models of everyday things. Figure 2 shows two examples of parts mental models in our dataset, where edges encode relationships between parts. E.g., in a tree, "trunk is above the roots"; in a flashlight, "bulb requires the batteries," etc. Inspired by previous literature, we envision that such parts mental models would play a key role when one carries out daily activities involving these everyday things. Task Here we define our task: "Construct a parts mental model for everyday things" with the following input/output specifications: • Input: Everyday thing, Parts list, Relation vocabulary (14 relations). • Output: List of tuples (x, r, y) where relation r holds between parts x and y. In Section 4 we describe how we acquire a benchmark dataset by asking human annotators to carry out this task. Once we have collected gold-standard parts mental models for everyday things based on the human annotations, we prompt LMs for their Given Type Relations Spatial orientation part of, has part, inside, contains, in front of, behind, above, below, surrounds, surrounded by, next to * Connectivity directly connected to * Functional dependency requires 2 , required by parts mental models and evaluate how well they do on this task. Our proposed method to measure this is described in Section 5. In particular, we are interested in (1) how accurate are LM-generated parts mental models when compared to gold-standard models in our dataset and (2) ignoring accuracy, how consistent are these generated parts mental models with respect to basic commonsense constraints? I.e., Do they at least conform to the 4 types of commonsense constraints laid out in Section 5.2 e.g., 'above' and 'below' are inverse relations, so if the LM predicts that in a tree, (trunk is above the roots) then it should also predict (roots are below the trunk). Everyday Things Dataset: ParRoT (Parts and Relations of Things) We created a dataset of common entities that one would encounter in their daily life. For each everyday thing, our dataset (ParRoT) contains a "parts mental model" in the form of a graph, which depicts parts of the entity and relational information about the parts. Such a graph encodes a partsfocused mental model of that everyday thing, potentially useful for reasoning about how the entity works and how to interact with it. Everyday entities We first compiled a list of entities from children's books, vocabulary lists (Grades 1-8), and online web search. 3 For the unique entities in this list, the authors manually filtered out those entities that are not common in everyday setting or have too few (i.e. only 1 or 2 parts) or too many parts (composite scenes). Specifically, we kept 100 entities that are common everyday things that a child would be familiar with, with a mix of natural and man-made things. This annotation task involves answering the following question for each item in the list: "Do you imagine this is something that most people would have seen in their everyday lives?" We recognize there could be many variants of a single everyday entity e.g. different types of coffee makers. To narrow down the possibilities, the authors picked a diagram for each everyday thing via web search and carefully annotated a parts list for each of them to guide the level of granularity we are looking for. In some cases, the entity name was qualified to disambiguate further e.g. "digital clinical thermometer" instead of just "thermometer." Mental model annotations We ask crowdworkers to draw sketches of everyday things covering spatial relations, connectivity, and functional dependencies between parts (Table 2). To encourage the format of the mental model graphs to be more standardized across annotators, we ask that the nodes (in circles) mainly contain labels from the "Parts list" provided. However, to collect mental models that are most natural to the workers, they were also told that they can ignore parts in the "Parts list" if they seem unimportant, or add extra parts that seem important. We also specified for edges to be labeled with the relations shown in Table 2. 4 Given the name of an everyday thing, list of parts, and example diagram, 3 crowdworkers were recruited to sketch mental models for each everyday thing. 5 Figure 2 shows examples of such sketches. According to Norman (2013), mapping that takes advantage of spatial analogies leads to immediate understanding and is more natural. Sketching out such a graph allows workers more flexibility in taking advantage of spatial analogies between the actual entity and the sketch (see flashlight example in Figure 2). Therefore, we hypothesize that drawing a graph would be easier or more natural for crowdworkers than typing a list of relations. 6 Statistics ParRoT consists of 100 everyday things ranging from devices like coffee maker, space heater to natural entities like tree and butterfly with number of parts (provided as a seed list to crowdworkers) ranging from 3-14. We collected 3 mental models per everyday thing. We take the parts mental models annotated by crowdworkers to be correct but not complete. I.e., they may include only those relations that they think are salient for the everyday thing, and also omit the ones that can be easily inferred from what they have annotated e.g., when (trunk is above the roots) is annotated, (roots are below the trunk) can be omitted (Figure 2, tree example). For each everyday thing's mental model annotation, with the relation tuples annotated, we automatically add relations that are implied via enrichment based on 4 types of constraints (symmetric, asymmetric, inverse, and transitive). The inferred relations include both relations that are labeled True (e.g. A above B being True implies that B below A is True) and relations that are labeled False (e.g. A above B being True implies B above A is False). This gives a total of 11.7K gold relation tuples (6894 with "True" as gold labels and 4826 with "False" as gold labels). Table 1 provides additional dataset statistics. Appendix C discusses the unanimity and diversity of mental models for these everyday things. Measuring and Improving Parts Mental Models Our proposed approach, ParRoT-Con, 7 comprises two main components. 8 The first component "Probing a Pre-trained Language Model" sends an exhaustive list of relation queries to a LM querying for every relation between each pair of parts (e.g. all relationships between egg white, yolk, shell, shell membrane and air cell). This gives us a large set of candidate relation tuples along with the model's confidence in each of them. Incorrect relation predictions can result in inconsistencies in the mental model. E.g, "egg white both surrounds and is surrounded by the egg shell." The second component "constraint reasoning" then applies a constraint satisfaction layer on top of these raw predictions to choose a subset of these relation tuples that are maximally probable and minimally conflicting with each other. Note that ParRoT-Con is a zero-shot approach, where both probing LMs and constraint reasoning steps do not require any task-specific fine-tuning or re-training. Probing a Pre-trained Language Model We use the following pre-trained language models for our study: GPT-3 (Brown et al., 2020) and Macaw 9 . We probe them using True/False questions of type: "Judge whether this statement is true or false: In an <everyday thing>, <part1 relation part2>." For each query q, we record an answer a ∈ {T rue, F alse}, and the model's beliefs about the likelihood of the relation being "True" as p(T rue|q) p(T rue|q) + p(F alse|q) . Constraint Reasoning We observed a significant amount of inconsistency in raw predictions from these LMs by considering the following constraints: • Symmetric relations: This constraint ensures symmetric relations like "directly connected to" and "next to" hold both ways. i.e. x rln y ↔ y rln x • Asymmetric relations: For asymmetric relations like part of, has part, inside, contains, in front of, behind, above, below, surrounds, surrounded by, requires, required by, this constraint makes sure that both "x rln y" and "y rln x" cannot be true at the same time. i.e. ¬(x rln y) ∨ ¬(y rln x) • Inverse relations: For a set of inverse relations e.g. above vs below, this constraint makes sure that (x above y) and (y below x) have the same truth value. i.e. x rln y ↔ y inverse(rln) x • Transitive relations: For relations like inside, contains, in front of, behind, above, below, surrounds, surrounded by, this constraint will impose transitivity. i.e. x rln y ∧ y rln z → x rln z In this step, we try to resolve inconsistencies in LMs' raw predictions by solving a MaxSAT constraint satisfaction problem where each (x, relation, y) tuple is represented as a variable with confidence value from the LM used as its weight (soft clause). We introduce 4 types of hard constraints (listed above) between these variables as hard clauses and any constraint violation results in an extremely high penalty. Given a WCNF formula with these, a weighted MaxSAT solver tries to find an optimal assignment of truth values to relation tuples that maximizes the sum of weights of satisfied soft clauses and satisfies all the formula's hard clauses. We use the RC2 MaxSAT solver (Ignatiev et al., 2018b) in PySAT (Ignatiev et al., 2018a). Results and Analysis Evaluation Metrics We evaluate the parts mental models produced by the two LMs in terms of accuracy and consistency: Accuracy: We compute the True/False accuracy of parts mental models based on the 11.7K gold relation tuples present in ParRoT. Results Q1: How consistent are LMs when they answer questions about everyday things? We measure the consistency of parts mental models constructed by LMs based on 4 types of constraints described in Section 5.2. This measurement is purely based on LMs' predictions and is independent of relations in the gold mental models acquired for the everyday things. Table 3 shows that LMs contradict themselves (19-43% conditional violation) when we ask them multiple questions about parts of the same everyday thing to probe for their parts mental model. E.g., in Appendix D, the LM believes that in an egg, "yolk surrounds the shell" and "shell surrounds the yolk" are both True. Table 3 also breaks down the LMs' inconsistency across 4 types of constraints. We observe that GPT-3 struggles with maintaining consistency for symmetric and inverse relations, whereas Macaw-11B finds it most challenging to satisfy constraints for asymmetric relations. Q2: Do language models have accurate mental models of everyday things? Next, we investigate how accurate are these parts mental models when compared to gold mental models in our ParRoT dataset. Table 4 shows that such queries pertaining to parts of everyday things are challenging for even SOTA models, with an average accuracy of 54-59%. This is barely better than the majority class baseline at 59% and random chance at 50%. The LMs' low performance shows that ParRoT is a challenging dataset, which is expected given the fact that this dataset queries for commonsense knowledge about everyday things (e.g. spatial relationship between parts of a device) that are often omitted in text, and hence less likely seen during pre-training. Further, by construction, our queries minimally differ e.g. for relations between parts of a tree, the edit distance between a statement with true relation "the leaves are above the roots" and false relation "the leaves are below the roots" is just 1 word. This makes our task even more challenging as the models need to understand the semantics of relational phrases to give the correct answer. Q3: Does ParRoT-Con, our proposed constraint reasoning approach, help create more accurate mental models? Our proposed approach, ParRoT-Con, utilizes the inherent inconsistency in LMs' raw predictions to self-correct their own parts mental models. It finds an optimal assignment of truth values to relation tuples that accounts for both the model's original beliefs (about the likelihood of each relation statement being True or False), and the 4 types of commonsense constraints imposed. By imposing the commonsense constraints as hard constraints, our proposed method produces perfectly consistent mental models for all LMs with respect to the imposed constraints i.e. % conditional violation becomes 0 for all columns in Table 3. Using these basic commonsense constraints, ParRoT-Con improves parts mental model accuracy significantly by 16-20% on ParRoT (Table 4). Further analysis Most effective range We analyze what is the quality range of mental models that ParRoT-Con is most effective on. We quantify the quality of parts mental models by defining accuracy@s, a metric that says a mental model is correct if the proportion of correct relations is at least s%. We then plot the percentage of mental models (out of 300) that are correct vs accuracy@s for different values of s, where s ∈ {50, 60, 70, 80, 90, 100}. Figure 3 shows that ParRoT-Con not only effectively increases the percentage of mental models that are approximately correct (s = 50, 60) but also the percentage of mental models that are (almost) totally correct (s = 90, 100). The improvements with constraint reasoning are even more prominent when it comes to increasing the percentage of mental models that are at least 60-80% accurate. This is likely attributed to the improvement in mental models that have enough signals from LMs' raw predictions and also enough margin to improve. Figure 4 shows that the base LMs are more accurate in predictions for queries containing relationships like 'part of' which is more likely to be stated in text than spatial relations like 'above', 'below', and 'behind' which are lower-level physical details often not mentioned in text. Different models also differ in which relationships they perform better on: e.g. GPT-3 performs poorly on bi-directional relations like 'connects' and 'next to', with accuracy way below chance level, while Macaw-11B achieves around 70% accuracy for queries involving these relations. Accuracy of parts mental models per relation Success and failure across models per everyday thing LMs show both similarities and differences in what everyday things they have better mental models of. For each model, Figure 5 shows the top 20 everyday things that the models performed best on in terms of base LM accuracy. Both GPT-3 and Macaw-11B perform well on the following everyday things: sandwich, kayak, dog, kite, bird, rat, cat, pencil sharpener, tree, cable car, and butterfly. It is interesting to see that both models perform well on several natural living things like animals (e.g. dog, bird, rat, cat), insect (e.g. butterfly), and plant (e.g. tree). Figure 6 shows the top 20 everyday things that the models performed worst on in terms of base LM accuracy. We observe that (a) GPT-3 (b) Macaw-11B Figure 3: Percentage of correct mental models vs accuracy@s shows that for both GPT-3 and Macaw-11B, there is a higher percentage of correct mental models after constraint reasoning (orange) as compared to raw LM predictions (blue), no matter the threshold for considering a mental model to be correct is lower or higher. For improvements from constraint reasoning (black), we observe the highest increase in percentage of mental models that are at least 60-80% accurate. entities like typewriter, bed, air conditional, and computer are challenging for both models to form accurate mental models of. Although the models share some similarities in what everyday things they have better/worse mental models of, they also show differences, especially for man-made devices: e.g. GPT-3 does well but Macaw-11B performs poorly on forming an accurate parts mental model of piano; Macaw-11B does well, but GPT-3 performs poorly on devices like doorbell, digital clinical thermometer, and binoculars. Conclusion Do language models have coherent mental models of everyday things? To systematically study this question, we present a benchmark dataset, ParRoT, consisting of 300 human-constructed mental models for 100 everyday objects, including over 2K parts and 11.7K relationships between these parts. Our experiments reveal that even SOTA LMs generally have poor mental models (inaccurate and violating basic commonsense constraints) of everyday things, thus providing insight into their apparent knowledge and behavior not previously explored. We apply constraint reasoning on top of base LM predictions to construct more coherent mental models. Our method, ParRoT-Con, improves both accuracy (up to 20% improvement) and consistency (up to 43% improvement) of such parts mental models. This suggests a broader cognitive architecture (LM + reasoner) for future systems, to construct more coherent mental models than using the LM alone. : 20 everyday things that each model achieved worst performance on, based on models' raw predictions (i.e. Base LM). In many of these cases, the accuracy of the parts mental models produced by the base LM is at around or below chance level and constraint reasoning boosts accuracy to beyond 50%. Limitations Common everyday things change over the years. While we try to choose ones that are in children's vocabulary, over decades, devices evolve and humans change in which things they interact with more frequently, affecting which relationships would be more prominent in an average person's mental model. So the parts mental models in such a dataset may not stay constant over time (e.g. some entities may be less familiar and certain relations may be less salient to annotators of the future). It would be interesting to use our ParRoT dataset as a point of comparison when studying mental models of everyday things in the future to reveal interesting insights on how humans' mental models of everyday things evolve over time. Other important future directions include to explore how more coherent mental models can help in complex reasoning tasks about everyday things, combine these parts mental models with mental models along other dimensions e.g. Gu et al. (2022a,b), as well as using our dataset of commonsense queries about everyday things as a source of follow-up questions for existing QA tasks e.g., PIQA (Bisk et al., 2020) and CSQA (Talmor et al., 2019). This paper only focuses on relationships (spatial orientation, connectivity, and functional dependency) between parts of everyday things. However, our approach ParRoT-Con is easily extensible to other applications such as: • spatial relations in other domains e.g. for geographical distances, we can similarly impose constraints on inverse relations like closer and further • temporal relations e.g. on a timeline, if event A occurred before event B, then event B cannot have occurred before event A (before is asymmetric) We leave the demonstration of the generalizability of our approach to future works. Ethics Statement All annotators that participated in the data collection process have been anonymized. The only personal information we collect is the worker IDs from Amazon Mechanical Turk, which we will not release. No personally identifiable information is contained in our dataset or otherwise released. We took great care to pay fair wages, and were responsive to feedback and questions throughout the data collection process. This study involves the use of large-scale language models. We only use them to generate True/False answers to questions about parts of everyday things, therefore we do not foresee any substantial ethical issues with their use for research presented in this submission. Our participants were recruited on the Amazon Mechanical Turk platform. The workers met minimum qualification in AMT: 95% approval rate. They were from US locations and rated at Amazon's Masters Level. Workers were paid at a rate of ≈$15/hr. C Unanimity and diversity in parts mental models People vary greatly in how they construct mental models, but the underlying reasoning is often structurally similar i.e. in accordance with commonsense constraints (Halford, 1993;Jonassen and Henning, 1996). In our ParRoT dataset, similarly, contradictions amongst crowdworkers (e.g., for guitar, one worker annotated that the neck is part of the fingerboard, while another annotated that the fingerboard is part of the neck) are extremely rare. There are only 80 instances out of 11720 in total in our entire dataset (0.68%) -less than 1%. We also looked at relations overlapped across workers in our dataset to analyze if workers pay attention to similar or different aspects of everyday things. To do so, we gathered a set of (p1, rln, p2) relations that are common across all 3 annotators for each everyday thing. These relationships are ones that achieved full agreement across all the 3 assigned annotators for that everyday thing in terms of the spatial/connectivity/functional relationship annotated and the parts involved. Together, we refer to this set as the ParRoT++ dataset. Table 5 summarizes the number of such high-agreement relationships for each everyday thing. Everyday things with few or no high-agreement relationships (refer Figure 7 for an example) imply higher diversity among annotators in terms of which spatial/connectivity/functional relationship and what parts they decided to include in their annotations. There are a total of 508 overlapped relations in ParRoT++, out of the 11720 in ParRoT, suggesting that attention is often paid to different aspects of everyday things. In Table 6, we present accuracy on ParRoT++, revealing similar results for relationships that achieved full agreement across all assigned annotators. Using basic commonsense constraints, ParRoT-Con improves parts mental model accuracy significantly by 16-22% on ParRoT++. These trends are similar to that obtained for ParRoT, illustrating that the results hold across all gold-standard parts relations, regardless of whether they are more unanimous or diverse across annotators. Table 6: Comparing the accuracy of parts mental models before and after constraint reasoning on ParRoT++ dataset. Note that all 3 models are accurate but there is some divergence in terms of (1) part names: e.g., 'head' vs 'forehead' and (2) which relation tuples they consider salient. Similar forms of diversity have been reported in Ji et al. (2022), for instance, as part naming divergence and segmentation divergence. D Pictorial illustration of ParRoT-Con Our proposed approach, ParRoT-Con, is illustrated in Figure 8 with an example everyday entity "egg". Figure 8: When asked about relationships between parts of an everyday thing, LMs can produce inconsistent relations. E.g., GPT-3 believes that in an egg, "yolk surrounds the shell" and "shell surrounds the yolk" are both True. Our proposed neuro-symbolic method, ParRoT-Con, applies constraint reasoning over raw LM predictions to produce more accurate and consistent mental models of everyday things. E Accuracy on different everyday things Table 7 gives example prompts and GPT-3's responses (includes both correct and incorrect) for entity "tree". Top 20 and bottom 20 everyday things that each model achieved best and worst performance on are shown in Figures 5 and 6 respectively. Further, Figure 11 demonstrates everyday things with 21st to 80th ranking in terms of the base LM accuracy. Model Prompt Model's Answer GPT-3 Judge whether this statement is true or false: In a tree, twig is directly connected to the branches. True (correct) GPT-3 Judge whether this statement is true or false: In a tree, trunk is above the roots. False (incorrect) GPT-3 Judge whether this statement is true or false: In a tree, roots are surrounded by the trunk. True (incorrect) GPT-3 Judge whether this statement is true or false: In a tree, trunk is below the roots. False (correct) Table 7: Example prompts and GPT-3's responses for an everyday entity "tree". F Use of models for inference For all experiments in this paper we used existing models/toolkits without any re-training or fine-tuning. We used GPT-3 text-davinci-003 and Macaw (T5-11B based) as representative LMs for our experiments. To probe GPT-3 text-davinci-003, we used their web API which took around 30 to 60 msec per relation tuple (one T/F question). To probe Macaw, we used two 48GB GPUs and it takes around 10.4 msec per relation tuple. We also run a MaxSAT solver for each everyday entity's parts mental model. To solve a constraint satisfaction problem per parts mental model takes a few msec up to around 3 minutes depending on the WCNF formula involved. G On the use of our dataset and code We have made all data and code used in this paper publicly available. Our dataset and code are released for research purposes only. Consistency: Following Kassner et al. (2021); Mitchell et al. (2022), we adapt the Conditional Violation (τ ) (Li et al., 2019a) metric to measure inconsistency across the 4 types of constraints defined in Section 5.2. For constraints L(x) → R(x) imposed on samples x ∈ D, where D is the dataset, Figure 4 : 4Accuracy of base LM and improvement achieved through constraint reasoning on different relations in ParRoT dataset. Figure 5 :Figure 6 5620 everyday things that each model achieved best performance on, based on models' raw predictions (i.e. Base LM). In almost all cases, constraint reasoning boosts the accuracy of the parts mental models produced by the base LM, pushing it even closer to 100%. Figure 7 : 7Example parts mental model annotations from ParRoT: (a) we provide the crowdworkers a diagram of cow retrieved from the Web. (b), (c), (d) are parts mental model sketches by 3 different crowdworkers. Table 2 : 2Relationships encoded in "parts mental models" of everyday things. Among these relations, 'next to' and 'directly connected to' relations are bi-directional, whereas the other 12 relations are uni-directional. Table 3 : 3Parts mental models constructed by LMs are significantly inconsistent with respect to their own predictions, violating basic commonsense constraints. In brackets, we indicate (# violations) / (# constraints fired).# params Base LM (%) ParRoT-Con (%) Improve (%) GPT-3 (text- davinci-003) 175B 53.83 70.26 16.42 Macaw-11B 11B 59.45 79.28 19.84 Table 4 : 4Comparing the accuracy of parts mental models before and after constraint reasoning on ParRoT dataset. pillow, truck, washing machine, door, hair dryer, rocket, screw, toaster, butterfly, chair, knife, photo frame, shoe, baby bottle, bed, bird cage, car, chainsaw, electric tea kettle, humidifier, piano 2 binoculars, digital camera, zipper, apple, digital clinical thermometer, earphone, flower, windmill, backpack, dog, doorbell, lightbulb, bat, cat, umbrella, stethoscope, tent 0 air conditioner, bicycle, blender, boat, glider, guitar, house, pencil sharpener, table fan, dryer, pencil, suitcase, telephone, microscope, refrigerator, space heater, typewriter, violin, wall clock, window, bookcase, bus, cable car, calculator, saucepan, train, cow, rat, table lamp# full-agreem. relations Everyday thing(s) 36 coffee maker, fish 28 rabbit 18 deer 16 egg, electric stove, tree 14 ink pen 12 laptop, sandwich, rice cooker, airplane, table 10 fire extinguisher, bird 8 elevator, flashlight, stroller, dishwasher, kayak, ship, teapot, telescope, corn, hot air balloon, microwave 6 wheelchair, barbeque grill, kite, microphone, computer, duck, helicopter 4 Table 5 : 5Number of relationships that achieved full agreement across all the 3 assigned annotators for each everyday thing. Higher number of such relations indicates more unanimous parts mental model annotations, whereas lower number reflects more diversity.# params Base LM (%) ParRoT-Con (%) Improve (%) GPT-3 (text- davinci-003) 175B 55.51 71.13 15.62 Macaw-11B 11B 60.04 82.41 22.38 We make our data and code publicly available at https: //github.com/allenai/everyday-things. A requires B denotes A cannot perform its primary function without B. Appendix A provides more details on the source of the list of everyday things. For ease of annotation, they do not need to repeat annotations that mean the same thing. e.g. if they annotated (x, above, y), they do not need to annotate (y, below, x) again. We automatically generate these in our data post-processing.5 More details can be found in Appendix B. 6 Later these sketches are transcribed into (x, r, y) tuples. First obtain the output of "stochastic parrots,"(Bender et al., 2021) then apply constraints to reason on top of the output. 8 See Appendix DFigure 8for an illustration. 9 A SOTA T5-11B based question-answering system that outperforms GPT-3 on some QA tasks. AcknowledgementsWe thank the anonymous ACL reviewers, as well as Ernest Davis, Chris Callison-Burch and members of the Aristo team at AI2 for their valuable feedback on an earlier draft.A Source of everyday thingsWe compiled a list of 100 everyday things from:1. Children's books (b) Select from all the nouns from an 8th-grade vocabulary list that were also under either "artifact" or "device" in WordNet(Miller, 1994)3. Online web search B Details on mental model annotation taskMechanical Turk task instructions: Language models as agent models. Jacob Andreas , Findings of the Association for Computational Linguistics: EMNLP 2022. Abu Dhabi, United Arab EmiratesAssociation for Computational LinguisticsJacob Andreas. 2022. Language models as agent mod- els. In Findings of the Association for Computational Linguistics: EMNLP 2022, pages 5769-5779, Abu Dhabi, United Arab Emirates. Association for Com- putational Linguistics. On the dangers of stochastic parrots: Can language models be too big?. Emily M Bender, Timnit Gebru, Angelina Mcmillan-Major, Shmargaret Shmitchell, 10.1145/3442188.3445922Proceedings of the 2021 ACM Conference on Fairness, Accountability, and Transparency, FAccT '21. the 2021 ACM Conference on Fairness, Accountability, and Transparency, FAccT '21New York, NY, USAAssociation for Computing MachineryEmily M. Bender, Timnit Gebru, Angelina McMillan- Major, and Shmargaret Shmitchell. 2021. On the dangers of stochastic parrots: Can language mod- els be too big? In Proceedings of the 2021 ACM Conference on Fairness, Accountability, and Trans- parency, FAccT '21, page 610-623, New York, NY, USA. Association for Computing Machinery. 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We provide some examples to illustrate how the lack of coherent mental models of everyday things may also appear for other models of the GPT-3.5 family, like ChatGPT in Figure 9. Others have also found ChatGPT responses that convey ridiculous interactions with everyday things e.g. it generates that. H Faqs, Q , Does ChatGPT do better? From informal tests, we find that ChatGPT is not devoid of mistakes either. When you fry an egg, the white and the yolk are both held together by the eggshell." (See Figure 10H FAQs Q: Does ChatGPT do better? From informal tests, we find that ChatGPT is not devoid of mistakes either. We provide some examples to illustrate how the lack of coherent mental models of everyday things may also appear for other models of the GPT-3.5 family, like ChatGPT in Figure 9. Others have also found ChatGPT responses that convey ridiculous interactions with everyday things e.g. it generates that "When you fry an egg, the white and the yolk are both held together by the eggshell." (See Figure 10) GPT-3 and ChatGPT models are often updated, when were the models accessed for your experiments? In our experiments with GPT-3, we used the text-davinci-003 model and queried the API on December 16, 2022 (during the period of time between 12 PM to 3.30 PM PST). ChatGPT as in Figure. 9at around 9.30 PM PST. It would be interesting for researchers to investigate if future versions of the systems can construct better parts mental models of everyday thingsQ: GPT-3 and ChatGPT models are often updated, when were the models accessed for your experiments? In our experiments with GPT-3, we used the text-davinci-003 model and queried the API on Decem- ber 16, 2022 (during the period of time between 12 PM to 3.30 PM PST). ChatGPT as in Figure 9 was accessed on December 17, 2022 (at around 9.30 PM PST). It would be interesting for researchers to in- vestigate if future versions of the systems can construct better parts mental models of everyday things. We enforced a set of manual and automated checks during data acquisition which includes collecting mental model sketches and transcribing them into relation tuples. Manual checks: We randomly sampled 15 mental model sketches and made sure that the transcription of relation tuples was accurate i.e. all the relations tuples in mental model sketches drawn by crowdworkers were precisely added to our dataset. We also checked the quality and format of sketches ('.png' files) which will be released with our dataset. Automated checks: After enriching with implied relations, we also programatically checked that all individual mental models (total of 11. Q: How do you ensure high-quality mental models are acquired via crowdsourcing. 7K relations) in ParRoT are fully consistent (based on the 4 commonsense constraints described in Section 5.2)Q: How do you ensure high-quality mental models are acquired via crowdsourcing? We enforced a set of manual and automated checks during data acquisition which includes collecting mental model sketches and transcribing them into relation tuples. Manual checks: We randomly sampled 15 mental model sketches and made sure that the tran- scription of relation tuples was accurate i.e. all the relations tuples in mental model sketches drawn by crowdworkers were precisely added to our dataset. We also checked the quality and format of sketches ('.png' files) which will be released with our dataset. Automated checks: After enriching with implied relations, we also programatically checked that all individual mental models (total of 11.7K relations) in ParRoT are fully consistent (based on the 4 commonsense constraints described in Section 5.2). UnifiedQA-large pointed towards the same trends. Do similar trends apply to smaller models? Experiments on Macaw-3B, Macaw-large. interested researchers to experiment with other models of interest to themQ: Do similar trends apply to smaller models? Experiments on Macaw-3B, Macaw-large, UnifiedQA-large pointed towards the same trends. We also make our code and data fully accessible at https://github.com/allenai/everyday-things for interested researchers to experiment with other models of interest to them. ChatGPT also seems to have incoherent mental pictures of everyday things. ChatGPT Figure 10: ChatGPT provides ridiculous responses regarding daily life activities such as frying an egg, illustrating poor mental models of everyday things and interactions with them. (Example by @bio_bootloader. Q: Can ParRoT-Con be applied to other languages? While our dataset is in English, relationships between parts of everyday things could indeed be authored for/ translated into other languages. 9We made our code and data publicly available, so others could use the infrastructure to apply the technique to other languages. at 11:59 AM Dec 3, 2022.) (a) GPT-3 (b) Macaw-11BQ: Can ParRoT-Con be applied to other languages? While our dataset is in English, relationships between parts of everyday things could indeed be authored for/ translated into other languages. We made our code and data publicly available, so others could use the infrastructure to apply the technique to other languages. ChatGPT Figure 9: Like GPT-3 (text-davinci-003), ChatGPT also seems to have incoherent mental pictures of everyday things. ChatGPT Figure 10: ChatGPT provides ridiculous responses regarding daily life activities such as frying an egg, illustrating poor mental models of everyday things and interactions with them. (Example by @bio_bootloader, posted on Twitter https://twitter.com/bio_bootloader/status/1599131249553330176/photo/1 at 11:59 AM Dec 3, 2022.) (a) GPT-3 (b) Macaw-11B Performance on other everyday things. Accuracy of base LM and improvement achieved through constraint reasoning on different everyday things in our dataset. Figure. 11Figure 11: Performance on other everyday things. Accuracy of base LM and improvement achieved through constraint reasoning on different everyday things in our dataset.

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{'abstract': 'When people think of everyday things like an egg, they typically have a mental image associated with it. This allows them to correctly judge, for example, that "the yolk surrounds the shell" is a false statement. Do language models similarly have a coherent picture of such everyday things? To investigate this, we propose a benchmark dataset consisting of 100 everyday things, their parts, and the relationships between these parts, expressed as 11,720 "X relation Y?" true/false questions. Using these questions as probes, we observe that state-ofthe-art pre-trained language models (LMs) like GPT-3 and Macaw have fragments of knowledge about these everyday things, but do not have fully coherent "parts mental models" (54-59% accurate, 19-43% conditional constraint violation). We propose an extension where we add a constraint satisfaction layer on top of the LM\'s raw predictions to apply commonsense constraints. As well as removing inconsistencies, we find that this also significantly improves accuracy (by 16-20%), suggesting how the incoherence of the LM\'s pictures of everyday things can be significantly reduced. 1', 'arxivid': '2212.10029', 'author': ['Yuling Gu yulingg@allenai.org \nAllen Institute for AI\nSeattleWA\n', 'Bhavana Dalvi bhavanad@allenai.org \nAllen Institute for AI\nSeattleWA\n', 'Mishra \nAllen Institute for AI\nSeattleWA\n', 'Peter Clark peterc@allenai.org \nAllen Institute for AI\nSeattleWA\n'], 'authoraffiliation': ['Allen Institute for AI\nSeattleWA', 'Allen Institute for AI\nSeattleWA', 'Allen Institute for AI\nSeattleWA', 'Allen Institute for AI\nSeattleWA'], 'corpusid': 254877558, 'doi': '10.48550/arxiv.2212.10029', 'github_urls': ['https://github.com/allenai/everyday-things'], 'n_tokens_mistral': 15798, 'n_tokens_neox': 13870, 'n_words': 8916, 'pdfsha': '74c4d59076d6af78d3399662c4b9dc7ff08a76cc', 'pdfurls': ['https://export.arxiv.org/pdf/2212.10029v3.pdf'], 'title': ['Do language models have coherent mental models of everyday things?', 'Do language models have coherent mental models of everyday things?'], 'venue': []}

arxiv

ON THE EFFECTIVENESS OF OUT-OF-DISTRIBUTION DATA IN SELF-SUPERVISED LONG-TAIL LEARNING Jianhong Bai Zhejiang University Zuozhu Liu Zhejiang University Hualiang Wang The Hong Kong University of Science and Technology Jin Hao Harvard University Yang Feng Angelalign Technology Huanpeng Chu Zhejiang University Haoji Hu Zhejiang University ON THE EFFECTIVENESS OF OUT-OF-DISTRIBUTION DATA IN SELF-SUPERVISED LONG-TAIL LEARNING Published as a conference paper at ICLR 2023 Though Self-supervised learning (SSL) has been widely studied as a promising technique for representation learning, it doesn't generalize well on long-tailed datasets due to the majority classes dominating the feature space. Recent work shows that the long-tailed learning performance could be boosted by sampling extra in-domain (ID) data for self-supervised training, however, large-scale ID data which can rebalance the minority classes are expensive to collect. In this paper, we propose an alternative but easy-to-use and effective solution, Contrastive with Out-of-distribution (OOD) data for Long-Tail learning (COLT), which can effectively exploit OOD data to dynamically re-balance the feature space. We empirically identify the counter-intuitive usefulness of OOD samples in SSL long-tailed learning and principally design a novel SSL method. Concretely, we first localize the 'head' and 'tail' samples by assigning a tailness score to each OOD sample based on its neighborhoods in the feature space. Then, we propose an online OOD sampling strategy to dynamically re-balance the feature space. Finally, we enforce the model to be capable of distinguishing ID and OOD samples by a distributionlevel supervised contrastive loss. Extensive experiments are conducted on various datasets and several state-of-the-art SSL frameworks to verify the effectiveness of the proposed method. The results show that our method significantly improves the performance of SSL on long-tailed datasets by a large margin, and even outperforms previous work which uses external ID data. Our code is available at INTRODUCTION Self-supervised learning (SSL) methods He et al., 2020;Grill et al., 2020) provide distinctive and transferable representations in an unsupervised manner. However, most SSL methods are performed on well-curated and balanced datasets (e.g., ImageNet), while many real-world datasets in practical applications, such as medical imaging and self-driving cars, usually follow a long-tailed distribution (Spain & Perona, 2007). Recent research indicates that existing SSL methods exhibit severe performance degradation when exposed to imbalanced datasets. To enhance the robustness of SSL methods under long-tailed data, several pioneering methods (Jiang et al., 2021b;Zhou et al., 2022) are proposed for a feasible migration of cost-sensitive learning, which is widely studied in supervised long-tail learning (Elkan, 2001;Sun et al., 2007;Cui et al., 2019b;Wang et al., 2022). The high-level intuition of these methods is to re-balance classes by adjusting loss values for different classes, i.e., forcing the model to pay more attention to tail samples. Another promising line of work explores the probability of improving the SSL methods with external data. (Jiang et al., 2021a) suggests re-balancing the class distributions by sampling external in-distribution (ID) tail instances in the wild. Nevertheless, they still require available ID samples in the sampling pool, which is hard to collect in many real-world scenarios, e.g., medical image diagnosis (Ju et al., 2021) or species classification (Miao et al., 2021). The aforementioned findings and challenges motivate us to investigate another more practical and challenging setting: when the ID data is not available, can we leverage the out-of-distribution (OOD) data to improve the performance of SSL in long-tailed learning? Compare to MAK Jiang et al. (2021a) that assumes external ID samples are available, while we consider a more practical scenario where we only have access to OOD data that can be easily collected (e.g., downloaded from the internet). A very recent work (Wei et al., 2022) proposes to re-balance the class priors by assigning labels to OOD images following a pre-defined distribution. However, it is performed in a supervised manner while not directly applicable to the SSL frameworks. In this paper, we proposed a novel and principal method to exploit the unlabeled OOD data to improve SSL performance on long-tailed learning. As suggested in previous research, the standard contrastive learning would naturally put more weight on the loss of majority classes and less weight on that of minority classes, resulting in imbalanced feature spaces and poor linear separability on tail samples (Kang et al., 2020;Li et al., 2022). However, rebalancing minorities with ID samples, no matter labeled or unlabeled, is quite expensive. To alleviate these issues, we devise a framework, Contrastive Learning with OOD data for Long-Tailed learning (COLT), to dynamically augment the minorities with unlabeled OOD samples which are close to tail classes in the feature space. As illustrated in Fig. 1, our COLT can significantly improve SSL baselines in terms of the Alignment and Uniformity (Wang & Isola, 2020), two widely-used metrics to evaluate the performance of contrastive learning methods, demonstrating the effectiveness of our method. The pipeline of our method is illustrated in Fig. 2. To augment the long-tail ID dataset, we define a tailness score to localize the head and tail samples in an unsupervised manner. Afterward, we design an online sampling strategy to dynamically re-balance the long-tail distribution by selecting OOD samples close (with a large cosine similarity in the feature space) to the head or tail classes based on a predefined budget allocation function. We follow the intuition to allocate more OOD samples to the tail classes for rebalancing. Those selected OOD samples are augmented with the ID dataset for contrastive training, where an additional distribution-level supervised contrastive loss makes the model aware of the samples from different distributions. Experimental results on four long-tail datasets demonstrate that COLT can greatly improve the performance of various SSL methods and even surpass the state-of-the-art baselines with auxiliary ID data. We also conduct comprehensive analyses to understand the effectiveness of COLT. Our contributions can be summarized as: • We raise the question of whether we can and how to improve SSL on long-tailed datasets effectively with external unlabeled OOD data, which is better aligned with the practical scenarios but counter-intuitive to most existing work and rarely investigated before. • We design a novel yet easy-to-use SSL method, which is composed of tailness score estimation, dynamic sampling strategies, and additional contrastive losses for long-tail learning with external OOD samples, to alleviate the imbalance issues during contrastive learning. • We conducted extensive experiments on various datasets and SSL frameworks to verify and understand the effectiveness of the proposed method. Our method consistently outperforms baselines by a large margin with the consistent agreement between the superior performance and various feature quality evaluation metrics of contrastive learning. RELATED WORKS Supervised learning with imbalanced datasets Early attempts aim to highlight the minority samples by re-balancing strategy. These methods fall into two categories: re-sampling at the data level (Shen et al., 2016;Zou et al., 2018;Geifman & El-Yaniv, 2017), or re-weighting at the loss (gradient) level (Cao et al., 2019;Jamal et al., 2020). Due to the usage of label-related information, the above methods can not be generalized to the unsupervised field. (Kang et al., 2019) suggests that the scheme of decoupling learning representations and classifiers benefits long-tail learning. The feasibility of two-stage training promotes the exploration in unsupervised scenarios. Self-supervised long tail learning (Yang & Xu, 2020) is, to our best, the first to analyze the performance of SSL methods in long-tail learning and verify the effectiveness of self-supervised pretraining theoretically and experimentally. However, shows that SSL methodsalthough more robust than the supervised methods -are not immune to the imbalanced datasets. Follow-up studies improve the ability of SSL methods on long-tailed datasets. Motivated by the observation that deep neural networks would easily forget hard samples after pruning (Hooker et al., 2019), (Jiang et al., 2021b) proposed a self-competitor to pay more attention to the hard (tail) samples. BCL (Zhou et al., 2022) involved the memorization effect of deep neural networks (Zhang et al., 2021b) into contrastive learning, i.e., they emphasize samples from tail by assigning more powerful augmentation based on the memorization clue. We show that our method is non-conflict with existing methods and can further improve the balancedness and accuracy (Section 4.2). Learning with auxiliary data Auxiliary data is widely used in the field of deep learning for different purposes, e.g., improving model robustness (Lee et al., 2020), combating label noise (Wei et al., 2021), OOD detection (Liang et al., 2018;Hendrycks et al., 2018a), domain generalization (Li et al., 2021;Long et al., 2015), neural network compression (Fang et al., 2021), training large models (Alayrac et al., 2022;. In long-tail learning, MAK (Jiang et al., 2021a) suggests tackling the dataset imbalance problem by sampling in-distribution tail classes' data from an open-world sampling pool. On the contrary, we explore the probability of helping long-tail learning with OOD samples, i.e., none of the ID samples are included in the sampling pool. Open-Sampling (Wei et al., 2022) utilizes the OOD samples by assigning a label to each sample following a pre-defined label distribution. Their work is performed under supervised scenarios, and the OOD data is not filtered, which results in a massive computation overhead. METHOD PRELIMINARIES Unsupervised visual representation learning methods aim to find an optimal embedding function f , which projects input image X ∈ R CHW to the feature space Z ∈ R d with z = f (x), such that z retains the discriminative semantic information of the input image. SimCLR is one of the state-of-the-art unsupervised learning frameworks, and its training objective is defined as: L CL = 1 N N i=1 − log exp(z i · z + i /τ ) exp(z i · z + i /τ ) + z − i ∈Z − exp(z i · z − i /τ ) ,(1) where (z i , z + i ) is the positive pair of instance i, z − i indicates the negative samples from the negative set Z − , and τ is the temperature hyper-parameter. In practice, a batch of images is augmented twice in different augmentations, the positive pair is formulated as the two views of the same image, and the negative samples are the views of other images. LOCALIZE TAIL SAMPLES IN SELF-SUPERVISED TRAINING Due to the label-agnostic assumption in the pre-training state, the first step of the proposed method is to localize tail samples. As mentioned earlier, the majority classes dominate the feature space, and tail instances turn out to be outliers. Moreover, the minority classes have lower intra-class consistency (Li et al., 2022). Hence, a sparse neighborhood could be a reliable proxy to identify the tail samples (More analysis can be found in Section 4.4). Specifically, we use top-k% (k = 2 Figure 2: Overview of Contrastive with Out-of-distribution data for Long-Tail learning (COLT). COLT can be easily plugged into most SSL frameworks. Proposed components are denoted as red. in practice) largest negative logits of each sample to depict the feature space neighborhood during training. Given a training sample x i , its negative logits p − i is the following: p − i = exp(z i · z − i /τ ) exp(z i · z + i /τ ) + z − i ∈Z − exp(z i · z − i /τ ) .(2) Considering implementing SimCLR with batch size B, each image has 2(B − 1) negative samples. Then, we define s i t = − top−k% p − i as the tailness score for each ID instance x i . During training, we perform a momentum update to the tailness score, i.e., s i,0 t = s i t , s i,n t = ms i,n−1 t + (1 − m)s i,n t where m ∈ [0, 1) is the momentum coefficient. The momentum update makes the tailness score more robust and discriminative to the tail samples. A higher value of s i t indicates sample x i has a more sparse neighborhood in the feature space and implies that it belongs to the tail classes with a larger probability. Experiments in Fig 3e empirically demonstrated that tail samples could be effectively discovered by our proposed tailness score. DYNAMICALLY RE-BALANCE THE FEATURE SPACE WITH ONLINE SAMPLING The core of our approach is to sample OOD images from the sampling pool S ood and further rebalance the original long-tail ID dataset and the feature space. First, we obtain C feature prototypes z ci from ID training set S id via K-means clustering. Note that we use the features at the last projection layer since the contrastive process is performed on this layer. The cluster-wise tailness score s ci t is defined as the mean of tailness score in cluster c i , i.e., s ci t = zj ∈ci s j t /|c i |, here |c i | is the number of instances in cluster c i . Then, we obtain each cluster's sampling budget K ′ as follows: K ′ = K · sof tmax( s c t /τ c ), s c t = s c t − mean(s c t ) std(s c t ) ,(3) where K refers to the total sampling budget, K ′ ∈ R C is the sampling budget assigned to each cluster, s c t is the normalized cluster tailness score. Empirically, we assign more sampling budget to the tailness clusters to be consistent with the idea of re-balancing the feature space. We sample OOD images whose feature is close to (higher cosine similarity) the ID prototypes z ci . To fully exploit the OOD data, we re-sample from the S ood every T epoch. The motivation behind this is: i) the sampled OOD data can be well-separated from S id after a few epochs, therefore becoming less effective to re-balance the feature space; ii) over-fitting to the OOD data can be toxic to the ID performance (Wei et al., 2022). From another perspective, this online sampling strategy lets the ID training set (especially the tail samples) continuously be exposed to the more newly sampled effective negative samples, forcing the model gives more distinctive embeddings and better fit the ID distribution. The online sampling process is summarized in Algorithm 2. Algorithm 1 The overall pipeline of COLT. Algorithm 2 our online sampling strategy. Input: ID train set S id , OOD dataset S ood , model θ, sample budget K, cluster number C, similarity metric sim(·), hyper-parameter τ c . Output: new train set S train . Input: ID train set S id , OOD dataset S ood , sam- ple budget K, train epoch T , momentum coeffi- cient m, warm-up epochs w, sample interval r, cluster number C, hyper-parameter k, τ c . Output: pre-trained model parameter θ T . Initialize: model parameter θ 0 , the original train set S train = S id . if epoch = 0 then Train model θ 0 with Eq. 1 and compute s 0 t ; end if for epoch = 1, · · · , T − 1 do if epoch ≥ w then if (epoch − w) %r = 0 Calculate both ID features z id and OOD features z ood through model θ; Obtain C ID prototypes z ci via K-means clustering in the projected feature space; Calculate cluster-wise tailness score by s ci t = zj ∈ci s j t /|c i |; Assign each cluster a sample budget K ′ ci with Eq. 3; Initialize the sample set S sample = ∅; for i = 0, · · · , C − 1 do Initialize subset S i sample = ∅; while |S sample | < K ′ ci do u = arg max xj ∈S ood sim(z j , z ci ); S i sample = S i sample ∪ {u}; end while S sample = S sample ∪ S i sample ; end for S train = S train ∪ S sample . AWARENESS OF THE OUT-OF-DISTRIBUTION DATA Section 3.2 and Section 3.3 introduce our sampling strategy toward OOD images. To involve the sampled OOD subset S sample in training, a feasible way is directly using the augmented training set (containing both ID and OOD samples) to train the model with Eq. 1. However, we would argue that giving equal treatment to all samples may not be the optimal choice (details in Section 4). One natural idea is to let the model be aware of that there are two kinds of samples from different domains. Hence, we define an indicator ϕ to provide weakly supervised (distribution only) information: ϕ(x i ) = +1, x i ∈ S id ; −1, x i ∈ S ood .(4) Afterward, we add a supervised contrastive loss (Khosla et al., 2020) to both ID and OOD samples: L SCL = 1 N N i=1 1 |P (i)| p∈P (i) − log exp(z i · z p /τ ) exp(z i · z p /τ ) + n∈N (i) exp(z i · z n /τ ) ,(5) where P (i) ≡ {p : ϕ(x p ) = ϕ(x i )} is the set of indices of the same domain within the mini-batch, |P (i)| is its cardinality and the negative index set N (i) ≡ {n : ϕ(x n ) ̸ = ϕ(x i )} contains index from different distribution. Fig 3c illustrates that the proposed distribution-awareness loss improves not only the overall performance but also facilitates a more balanced feature space. It's worth noting that the proposed loss only utilizes the distribution information as the supervised term, while the labels for both ID and OOD samples are unavailable during the self-supervised training stage. Finally, we scale the supervised loss with α and add it to the contrastive loss in Eq 1: L COLT = L CL + αL SCL .(6) EXPERIMENTS In this section, we first introduce the datasets and experimental settings (Section 4.1) and evaluate the proposed COLT in three aspects: accuracy and balancedness(Section 4.2), versatility and complexity (Section 4.3). Then, we verify whether our method can 1), localize tail samples, 2), re-balance the feature space. Finally, we provide a comprehensive analysis of COLT (Section 4.4). DATASETS AND SETTINGS We conduct experiments on four popular datasets. CIFAR-10-LT/CIFAR-100-LT are long-tail subsets sampled from the original CIFAR10/CIFAR100 (Cui et al., 2019a). We set the imbalance ratio to 100 in default. Following (Wei et al., 2022), we use 300K Random Images (Hendrycks et al., 2018b) as the OOD dataset. ImageNet-100-LT is proposed by (Jiang et al., 2021b) with 12K images sampled from ImageNet-100 with Pareto distribution. We use ImageNet-R (Hendrycks et al., 2021) Evaluation protocols To verify the balancedness and separability of the feature space, we report performance under two widely-used evaluation protocols in SSL: linear-probing and few-shot. For both protocols, we first perform self-supervised training on the encoder model to get the optimized visual representation. Then, we fine-tune a linear classifier on top of the fixed encoder. The only difference between linear-probing and few-shot learning is that we use the full dataset for linear probing and 1% samples of the full dataset for few-shot learning during fine-tuning. Measurement metrics As a common practice in long tail learning, we divide each dataset into three disjoint groups in terms of the instance number of each class: {Many, Median, Few}. By calculating the standard deviation of the accuracy of the three groups, we can quantitatively analyze the balancedness of a feature space (Jiang et al., 2021b). The linear separability of the feature space is evaluated by the overall accuracy. Training settings We evaluate our method with SimCLR framework in default. We also conduct experiments on several state-of-the-art methods in self-supervised long tail learning (Jiang et al., 2021b;Zhou et al., 2022). We adopt Resnet-18 (He et al., 2016) for small datasets (CIFAR-10-LT/CIFAR-100-LT), and Resnet-50 for large datasets (ImageNet-100-LT/Places-LT), respectively. More details can be found in Appendix. COLT'S ACCURACY, BALANCEDNESS AND VERSATILITY The main results of the proposed approach in various datasets and settings are presented in Table 1 and Table 2. We sample K = 10, 000 OOD images on every r = 25 epoch for CIFAR-10-LT/CIFAR-100-LT, Places-LT, and r = 50 for ImageNet-100-LT. COLT significantly outperforms the baseline (vanilla SimCLR) by a large margin (about 10% for long-tail CIFAR, 5% for ImageNet-100-LT, 1.6% for Places-LT). Besides, the performance gain of the minority classes (Median & Few) is more notable (e.g., about 12% for long-tailed CIFAR-100). Meanwhile, COLT yields a balanced feature space. Following previous works (Jiang et al., 2021b) (Zhou et al., 2022, we measure the balancedness of a feature space through the accuracy's standard deviation from Many, Median and Few. COLT significantly narrows the performance gap between the three groups (much lower Std), which indicates we learn a more balanced feature space. To evaluate the versatility of COLT, we carry out experiments on top of several improved SSL frameworks for long-tail learning, i.e., SDCLR (Jiang et al., 2021b) and BCL (Zhou et al., 2022). Table 1 and Table 2 also summarized COLT performance on these two methods. We can observe that incorporating our method into existing state-of-the-art methods can consistently improve their performance, which indicates that our method is robust to the underlying SSL frameworks. COLT VS BASELINES WITH AUXILIARY DATA We also compare COLT with methods that make use of external data. MAK (Jiang et al., 2021a) is the state-of-the-art method that proposes a sampling strategy to re-balance the training set by sample in-distribution tail class instances from an open-world sampling pool. We compare the proposed COLT with MAK in Table 3, noting that "random" refers to random sampling from the external dataset according to the budget. We observe both higher accuracy and balancedness under different sample budgets on ImageNet-100 and Places. It indicates COLT leverages OOD data in a more efficient way. Furthermore, we ask the question that whether OOD samples can replace ID samples to help long-tail learning. We obtain a positive answer from empirical results in Table 4. We compare the result of COLT and MAK on auxiliary data which involve ID samples. COLT achieves better performance on most of the metrics, even compared with sampling in an entirely ID dataset. On the other, COLT has less computational overhead. MAK applies a three-stage pipeline: pre-train the model with ID samples, sample from the sampling pool, and re-train a model from the beginning. In contrast, COLT samples during the training process, resulted in a single-stage pipeline. The online sampling strategy not only fully utilized the external datasets but also reduced the computation overhead significantly 1 . Open-Sampling (Wei et al., 2022) also uses OOD data to help long-tail learning. Different from ours, they use a large data budget (300K for CIFAR), while COLT improves the baselines with a much smaller budget (10K for CIFAR). ANALYSIS AND ABLATION STUDY The choice of OOD dataset We conduct experiments on CIFAR-100-LT and replace the OOD dataset while maintaining other settings unchanged. As shown in Fig 3a, our method improves the ID accuracy when using 300K Random Images (Hendrycks et al., 2018b), STL (Coates et al., 2011), ImageNet-100, Places with 9.81%, 5.19%, 9.43%, 5.74% respectively. Besides, sampling on Gaussian Noise provides limited help (less than 1%) or degradation to ID accuracy. The effectiveness of distribution-awareness loss COLT introduces a supervised contrastive loss to explicitly separate samples from different distributions. We conduct an ablation study on the proposed loss in Fig 3c, and the results show that the proposed loss not only improves the overall accuracy but also significantly alleviates the imbalance (i.e., much lower std). The effect of sampling budget In Fig 3b, we compare the performance gains of COLT under different budgets. COLT consistently outperforms the random sampling strategy, i.e., leveraging OOD samples more effectively. Moreover, though a larger budget will give better performance, the performance gain almost plateaus with a budget of 10-15k in COLT, indicating better data efficiency. Comparison with semi-supervised methods Performing semi-supervised learning is also a natural choice for utilizing external unlabeled data. We implement FixMatch (Sohn et al., 2020), FlexMatch (Zhang et al., 2021a), ABC (Lee et al., 2021), and DARP (Kim et al., 2020) on long-tailed CIFAR-100, the first two are general semi-supervised methods, and the last two are elaborately designed for long-tail learning. In Table 5, we compare the results in such semi-supervised learning scenarios (labeled: CIFAR-100-LT, unlabeled: 300K Random Images) to supervised, self-supervised, and COLT. It can be observed that 1), external unlabeled OOD data can also be helpful when performing semi-supervised learning 2), the performance gains of COLT (about 10%) are more significant than incorporating OOD data via semi-supervised training. This could be attributed to most semisupervised methods considering unlabeled data is also ID. It may need some special design for unlabeled OOD data, e.g., resist some "toxic" samples or redesign the pseudo labels for OOD data. Changing of hyper-parameters We also show the effect of hyper-parameters involved in COLT. Fig 3d shows the empirical results of changing resample interval r. We can observe that reasonably small intervals lead to higher accuracy. Fig 3f shows the classification accuracy when changing k. The limited fluctuations in performance prove that COLT is robust to hyper-parameters. The ability of tail sample mining Recall in Section 3.2, we localize tail samples by assigning a predefined "tailness score" to each sample. In order to verify the effectiveness of tailness score, we select the top 10% samples with the highest tailness score as a subset and calculate the ratio of the percentage of {Major / Minor} samples in this subset to the percentage of whole dataset:ϕ = T ∩S sub id T ∩S id where T denotes the target group, S id is the whole in-distribution dataset, S sub id is the subset of samples which have top γ% highest tailness score, γ is set to 10. ϕ reflects the ability to identify tail samples: when the target group is Minor/Major, higher/lower ϕ indicates a method localizes tail samples well. As illustrated in Fig 3e, COLT discovers more samples from the tail than BCL. CONCLUSION AND LIMITATIONS In this paper, we propose a novel SSL pipeline COLT, which is, to our best, the first attempt to extend additional training samples from OOD datasets for improved SSL long-tailed learning. COLT includes three steps, unsupervised localizing head/tail samples, re-balancing the feature space by online sampling, and SSL with additional distribution-level supervised contrastive loss. Extensive experiments show that our method significantly and consistently improves the performance of SSL on various long-tailed datasets. There are nevertheless some limitations. First, more theoretical analyses are needed to better understand the effectiveness of OOD samples. Besides, for a given long-tail ID dataset, how to specify the best OOD dataset that gives the largest improvements is also worth exploring. We hope our work can promote the exploration of OOD data in long-tail scenarios. REPRODUCIBILITY We uploaded the code to ensure the reproducibility of the proposed method, which can be found at: https://github.com/JianhongBai/COLT. APPENDIX A DATASETS AND TRAINING DETAILS CIFAR-10-LT/CIFAR-100-LT are first introduced by (Cui et al., 2019a), which are long-tail subsets sampled from the original CIFAR10/CIFAR100. The imbalance ratio is defined as the instance number of the largest class divided by the smallest class. To better reflect the performance difference, we set the imbalance ratio to 100 in default. Following (Wei et al., 2022), we use 300K Random Images (Hendrycks et al., 2018b) 2 as the OOD dataset. In addition, we also conduct experiments with OOD datasets as STL-10 (Coates et al., 2011), which contains 5,000 labeled images and 100,000 unlabeled images in 10 classes with a resolution of 96x96. We use all the unlabeled images as external OOD data. ImageNet-100-LT is proposed by (Jiang et al., 2021b). it contains about 12K images sampled from ImageNet-100 with Pareto distribution. The instance number of each class ranges from 1,280 to 5. We use ImageNet-R (Hendrycks et al., 2021) as the OOD dataset. The dataset contains 30K images with several renditions (e.g., art, cartoons, deviantart) of ImageNet classes. Training details We implement all our techniques using PyTorch (Paszke et al., 2017) and conduct the experiments using RTX3090 GPUs. We evaluate our method with SimCLR framework with batch size 512 for small datasets (CIFAR-10-LT/CIFAR-100-LT) and 256 for large datasets (ImageNet-100-LT/Places-LT) in default. We adopt Resnet-18 (He et al., 2016) for small datasets and Resnet-50 for large datasets, respectively. In our paper, we evaluate COLT's performance under two evaluation protocols in self-supervised learning: linear-probing and few-shot. For both protocols, we first perform self-supervised training on the encoder model to get the optimized visual representation. Then, we fine-tune a linear classifier on top of the encoder (fixed during training the classifier). The only difference between linear probing and few-shot learning is we use the full dataset for linear probing and 1% samples of the full dataset for few-shot learning during finetuning. We keep all settings in the fine-tuning stage (e.g., optimizer, learning rate, batch size) the same as (Jiang et al., 2021b). Places-LT Mainly Following (Jiang et al., 2021b;Zhou et al., 2022;Jiang et al., 2021a), we pre-train all the baselines and COLT with 2000 epochs on CIFAR10/100, 1000 epochs on ImageNet-100, 500 epochs on Places. As for the fine-tuning stage, the "linear-probing" and "few-shot" results are produced by fine-tuning the classifier for 30 epochs and 100 epochs, respectively. To make a fair comparison, we implement COLT and all baselines with the same data augmentation strategies. We sample K = 10, 000 OOD images on every r = 25 epoch for CIFAR-10-LT/CIFAR-100-LT, Places-LT, and r = 50 for ImageNet-100-LT. APPENDIX B MORE EMPIRICAL RESULTS We present the experiment results on ImageNet-100 with ImageNet-R or Places69 as the external OOD dataset in Table 6. We compare COLT with methods that make use of external data. MAK (Jiang et al., 2021a) is the state-of-the-art method that proposes a sampling strategy to re-balance the training set by sample in-distribution tail class instances. Note that "random" refers to randomly sampling from the external dataset according to the sampling budget. We observe both higher accuracy and balancedness under different sample budgets on different OOD datasets. It indicates COLT leverages OOD data in a more efficient way. Besides, the performance gain of the minority classes (Median & Few) is more notable. Meanwhile, COLT yields balancedness feature space. Following previous works (Jiang et al., 2021b) (Zhou et al., 2022, we measure the balancedness of a feature space through the accuracy's standard deviation (Std) from Many, Median and Few. To further demonstrate that our proposed COLT is also effective on the non-curated open-world datasets, we conduct experiments on ImageNet-100-LT with a 50K subset of Open Images (Krasin et al., 2017) (a dataset of about 9 million images belonging to over 6000 categories.) as the OOD dataset. We can also observe a significant improvement in both the accuracy (especially for {Median, Few}) and the balancedness of the feature space. As suggested in previous research, the standard contrastive learning would naturally put more weight on the loss of majority classes and less weight on that of minority classes, resulting in imbalanced feature spaces and poor linear separability on tail samples (Kang et al., 2020;Li et al., 2022), i.e., the majority classes dominate the feature space and tail instances turn out to be outliers. To quantitatively analyze the imbalance of the feature space, we define a metric called Normalized Misclassification Matrix (NMM): NMM ij = m ij / n k=1 m ik |T j |/ n k=1 |T k | ,(7) where T k is the k-th split of the long-tailed train set, satisfying S id = ∪ n k=1 T k and T i ∩ T j = ∅, ∀i ̸ = j. In this paper, we follow the common practice in long-tail learning that split the dataset to S id = T = {T Few , T Median , T Many} according to the instance number in each class, and |T j | denote the class number in split T j . m ij represents the number of (misclassified) instances belonging to split T i but are classified to split T j . Note that m ii indicates both the ground truth label and the (wrong) prediction fall into split T i . Intuitively, if the feature space is perfectly balanced (i.e., equal margin between different classes), each element in NMM is approximately 1.0, i.e., the misclassified samples nearly randomly fall into each split. On the contrary, higher (lower) mean margins between classes in T i and T j result in lower (higher) m ij . The results are shown in Fig 4b and Fig 4c. Fig 4b reflects that majority classes have a higher margin to other classes than minority classes. In other words, the model is more likely to confuse samples from a minority class with other minority classes, implying an imbalanced feature space. Fig 4c exhibits the result of COLT based on SimCLR with 10K OOD data. It's observed that COLT alleviates the imbalance issue since COLT augments minority classes with more instances which could be interpreted as an implicit loss re-weighting strategy. OOD DATA BRIDGE INSTANCES FROM MINORITY CLASSES In this section, we demonstrate the effect of OOD data on the perspective of contrastive learning. A recent work gives a theoretical understanding of contrastive learning based on augmentation overlap. Concretely, they suggest that the samples of the same class could be very alike after aggressive data augmentations. Thus, the pretext task of aligning positive samples can facilitate the model to learn class-separable representations. They define the augmentation graph G = (V, E) as: N samples are the vertices of the graph, and there exists an edge e ij when sample i and sample j has overlapped views. According to their theory, intra-class augmentation overlap is a sufficient condition for gathering features from the same class. In this case, we compute the ratio of connected nodes (degree is not zero) to measure the extent of intra-class augmentation overlap in class k, denoted as score s k c . Ideally, each class should have s k c = 1, which indicates all samples from the same class will be clustered together during the contrastive training process. A smaller s k c indicates lower intra-class consistency and vice versa. , We choose a random subset of test images and randomly augment them 20 times. Then, we calculate the instance distance in the representation space and draw edges for image pairs whose smallest view distance is below a small threshold. We visualize the samples with t-SNE and denote edges between ID instances in black and edges between ID and OOD samples, forming new connections in red. : Linear regression results between the minority proportion in a cluster and the cluster's tailness score on long-tailed CIFAR, ImageNet-100, and Places. We set cluster number C = 10. We visualize the augmentation graph of CIFAR-10 in Fig 5 and compute the connectivity score for each class. We observe a smaller s c for minority classes than for majority classes, which is also consistent with the theoretical analysis in . Nevertheless, since we dynamically add OOD samples around ID (especially minority classes) samples, more instances from the ID minority class can be reachable on the augmentation graph via OOD samples (majority/minority connectivity improvement: 0.03/0.31). The red edges in Fig 5b illustrate that OOD samples help long-tail learning by bridging samples from minority classes, further improving intra-class consistency. APPENDIX D MORE ANALYSIS ABOUT COLT Different tail estimation strategies In our paper, we defined a tailness score based on top-k% (k = 2 in practice) largest negative logits to localize the head and tail samples in an unsupervised manner. We also make a comparison to other alternative strategies. In Tab. 8, we provide results on locating tail samples by a radius-based definition (with radius as the sum of negative logits and select different radius thresholds) of tailness score or simply using the method in BCL (Zhou et al., 2022). The sampling budget is set to 5K, and the ID and OOD dataset is CIFAR-100-LT and 300K Random Images respectively. We can notice that there is no significant difference between COLT with Top-k% and radius-based methods, while both of them surpass BCL. Is the unsupervised clustering reliable? In the paper, we propose to perform a K-means clustering to the samples from S id , then calculate the cluster-wise sampling budget via the tailness score. Since our ultimate goal is to sample more (less) OOD data similar to the minority (majority) samples, it's natural to ask how well does this clustering work, whether we assigned more budget to the tail samples. To this end, we propose to measure the clustering and sampling quality by the ratio of minority samples in a cluster and its corresponding cluster-wise tailness score. Results are visualized in Fig 6. It's observed that the minority proportion of some clusters is close to 1, while others are composed of samples from majority classes. Besides, the cluster-wise tailness score shows a linear correlation of the minority proportion, implying we do allocate more sampling budget to tail classes according to Eq. 3. Impact of the cluster number Recall in Sec 3.3 we first perform a K-means clustering to the ID samples, then select OOD samples close (with a large cosine similarity in the feature space) to the head or tail classes based on the budget allocation function. To verify 1), whether the cluster number will dramatically affect the performance 2), how the performance is when clustering based on the ground-truth label information rather than a self-supervised manner. We conduct experiments with various cluster number C in Tab. 9, the OOD dataset is 300K Random Images (Hendrycks et al., 2018b), and the sampling budget is set to 5K. Note that the supervised clustering method is referred to as Oracle. The results show the performance of COLT will not be significantly affected by the number of clusters. Furthermore, cluster according to the label information (Oracle) achieves a similar performance compared to the unsupervised clustering, implying the K-means clustering can be used as a feasible alternative to roughly separates the minority from the majority classes in selfsupervised scenarios. The scale / balancedness of the OOD dataset In our work, we evaluate the proposed COLT on both balanced datasets, which are used as OOD datasets in previous works (Kumar et al., 2021;Wei et al., 2022) and non-curated open-world datasets such as 300K Random Images (Hendrycks et al., 2018b). Although COLT performs well on the aforementioned datasets, we intend to further ask will COLT perform well when the OOD dataset is also long-tailed? Tab. 10 shows the performance of COLT on CIFAR-100 when the OOD dataset (ImageNet-100) has various imbalance ratios. The similar accuracy suggests COLT is robust to the imbalancedness of the OOD dataset to some extent. This could be attributed to the dynamic sampling procedure filtering out most of the unhelpful OOD samples so that the performance will not be dramatically affected by changes in the external OOD dataset. Another observation can be found from Tab. 11, where we measured how the OOD dataset's scale influences COLT. We form the external OOD dataset of different scales by gradually increasing the number of samples in Random 300K Images (Hendrycks et al., 2018b), and implementing COLT on those subsets of Random 300K Images. Conform to intuition, the scale of the OOD dataset is positively correlated with the performance of COLT since we may select more desired samples in a larger candidate set. However, the problem can be tricky when the dataset is also changed. For example, we observe a similar performance gain on ImageNet-100-LT when utilizing Places-69 (about 98K images) and ImageNet-R (30K images) as the auxiliary OOD dataset, which implies the scale of the OOD dataset is not the only factor affecting the performance. Published as a conference paper at ICLR 2023 Figure 1 : 1(1a): Feature space uniformity of different SSL frameworks. (1b): Visualization of the alignment property of samples in minority classes and majority classes w/ or w/o COLT. The experiment is conducted with ResNet-18 on CIFAR-100-LT. as the OOD dataset. Places-LT (Liu et al., 2019) contains about 62.5K images sampled from the large-scale scene-centric Places dataset (Zhou et al., 2017) with Pareto distribution. Places-Extra69 (Zhou et al., 2017) is utilized as the OOD dataset. Ablation on hyper-parameter k. Figure 3 : 3Analytical experiments of COLT on CIFAR-100-LT. (3a): accuracy when changing the external OOD dataset. (3b): accuracy when sampling different numbers of OOD samples on 300K Random Images. (3c): Top-1 accuracy and standard derivation (Std) of COLT with or without the proposed distribution loss. (3d): accuracy with various sampling intervals r. (3e): A higher ϕ tail and a lower ϕ head implies mining tail samples more precisely. (3f): accuracy with various k. The original Places (Zhou et al., 2017) is a large-scale scene-centric dataset. Places-LT (Liu et al., 2019) contains about 62.5K images sampled from Places with Pareto distribution. The instance number of each class ranges from 4,980 to 5. Places-Extra69 (Zhou et al., 2017) is utilized as the OOD dataset. It includes 98,721 images for 69 scene categories besides the 365 scene categories in Places. Fig 4 . 4Fig 4a indicates when the train set is balanced, the mean margin of different classes is approximately equal (NMM ij ≈ 1), note that the split {T Few , T Median , T Many } in Fig 4a is consistent with Figure 4 :Figure 5 : 45Normalized Misclassification Matrix (NMM) on the test set of CIFAR-100 with different frameworks and train sets. (4a): SimCLR trained on balanced CIFAR-100. (4b): SimCLR trained on long-tailed CIFAR-100. (4c): implement COLT on top of SimCLR trained on long-tailed CIFAR-100. The augmentation graph of CIFAR-10. Similar to Figure 6 6Figure 6: Linear regression results between the minority proportion in a cluster and the cluster's tailness score on long-tailed CIFAR, ImageNet-100, and Places. We set cluster number C = 10. then Sample by performing Algorithm 2. end if Calculate the supervised contrastive loss with Eq. 5 Use S train to train θ epoch with Eq. 6;else Use S train to train θ epoch with Eq. 1; end if Compute s i t , and update s i,epoch t . end for Table 1 : 1Test accuracy (%) and balancedness (Std↓) on CIFAR-10-LT and CIFAR-100-LT.Metric Many↑ Median ↑ Few ↑ Std ↓ All ↑ Many ↑ Median ↑ Few ↑ Std ↓ All ↑Method CIFAR-10-LT CIFAR-100-LT SimCLR 82.40 73.91 70.19 5.11 75.34 51.50 45.58 45.96 2.71 47.65 +COLT 87.50 81.65 80.80 2.98 83.15 57.94 56.74 57.72 0.52 57.46 SDCLR 86.69 82.15 76.23 4.28 81.74 58.54 55.70 52.10 2.64 55.48 +COLT 90.87 84.28 81.45 3.95 85.41 63.28 60.85 59.42 1.59 61.18 BCL-I 86.97 82.40 76.45 4.31 81.99 58.92 54.63 53.58 2.31 55.70 +COLT 89.03 85.10 80.36 3.55 84.86 61.12 57.03 55.82 2.27 57.98 Table 2 : 2Test accuracy (%) and balancedness (Std↓) on ImageNet-100-LT and Places-LT.Metric Many ↑ Median ↑ Few ↑ Std ↓ All ↑ Many ↑ Median ↑ Few ↑ Std ↓ All ↑Method ImageNet-100-LT Places-LT SimCLR 70.96 65.33 61.89 3.74 67.08 40.02 46.61 49.38 3.93 44.78 +COLT 75.13 71.38 66.62 3.48 72.22 41.55 48.40 50.54 3.83 46.36 SDCLR 71.13 66.04 62.31 3.61 67.54 40.13 46.61 48.90 3.71 44.73 +COLT 75.13 70.25 67.69 3.08 71.82 41.72 48.42 50.78 3.84 46.47 Table 3 : 3Compare the proposed COLT with random sample and MAK under the same sampling pool and sampling budget. The best performance under each setting is marked as bold. Median ↑ Few ↑ Std ↓ All ↑ID dataset Sampling pool Budget Method Protocol Many ↑ Image Net-100 Image Net-R 5K random few-shot 51.76 41.45 38.12 5.81 45.04 linear-probing 72.22 66.49 63.45 3.64 68.33 MAK few-shot 52.49 43.54 39.00 5.65 46.48 linear-probing 73.24 67.62 64.14 3.75 69.36 COLT few-shot 54.10 46.01 42.32 4.92 48.69 linear-probing 74.52 69.67 67.53 2.92 71.28 10K random few-shot 52.82 42.88 40.27 5.41 46.42 linear-probing 74.33 68.52 62.65 4.77 70.02 MAK few-shot 54.33 45.01 40.12 5.90 48.01 linear-probing 75.57 68.20 66.29 4.07 70.83 COLT few-shot 54.26 46.54 43.38 4.57 49.14 linear-probing 75.13 71.38 66.62 3.48 72.22 Places 365 Places 69 10K random few-shot 30.61 33.94 37.55 2.83 33.45 linear-probing 40.21 47.59 50.51 4.33 45.51 MAK few-shot 30.41 34.47 37.59 2.94 33.62 linear-probing 40.83 47.78 50.72 4.15 45.86 COLT few-shot 31.04 34.65 37.49 2.64 33.91 linear-probing 41.55 48.40 50.54 3.83 46.36 Table 4 : 4Compare the test accuracy (%) on ImageNet-100-LT of the proposed COLT with MAK which use ID data. The best performance is marked as bold.Method Extra type Sample set Many ↑ Median ↑ Few ↑ Std ↓ All ↑ MAK ID IN-900 75.7±0.5 70.4±0.6 66.9±0.6 3.0±0.4 72.0±0.5 ID & OOD IPM 74.7±0.2 69.2±0.7 66.6±0.7 3.3±0.3 71.1±0.5 OOD ImageNet-R 75.6±0.4 68.2±0.8 66.3±0.8 4.1±0.6 70.8±0.5 COLT OOD ImageNet-R 75.3±0.3 70.9±0.8 69.5±0.3 2.4±0.7 72.4±0.3 Table 5 : 5Comparison of semi-supervised and self-supervised methods when leveraging OOD data.Method ID- Supervised FixMatch FlexMatch ABC DARP SimCLR SimCLR +COLT(10K) Accuracy 44.13 47.38 50.40 51.22 50.94 47.65 57.46 10 50 100 Imbalance ratio 45 50 55 60 65 Accuracy (%) baseline 300K Images STL ImageNet-100 Places Gaussian (a) External dataset 5K 10K 15K 20K Sample budget 45 50 55 60 Accuracy (%) Random sample Ours Tongzhou Wang and Phillip Isola. Understanding contrastive representation learning through alignment and uniformity on the hypersphere. 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In Proceedings of the European conference on computer vision (ECCV), pp. 289-305, 2018. Table 6 : 6Accuracy (%) and balancedness (Std) on ImageNet-100 with different OOD datasets. Sampling pool Budget Method Protocol Many ↑ Median ↑ Few ↑ Std ↓ All ↑ None - - few-shot 48.92 39.54 34.15 6.10 42.51 linear-probing 69.53 63.74 59.88 3.97 65.50 Places69 10K random few-shot 52.97 44.03 39.98 5.43 46.99 linear-probing 73.49 67.57 63.05 4.28 69.29 MAK few-shot 54.16 45.17 41.43 5.34 48.19 linear-probing 74.69 68.60 64.63 4.14 70.46 COLT few-shot 55.82 48.32 43.49 4.83 49.24 linear-probing 75.23 71.42 66.46 3.59 72.26 Image Net-R 5K random few-shot 51.76 41.45 38.12 5.81 45.04 linear-probing 72.22 66.49 63.45 3.64 68.33 MAK few-shot 52.49 43.54 39.00 5.65 46.48 linear-probing 73.24 67.62 64.14 3.75 69.36 COLT few-shot 54.10 46.01 42.32 4.92 48.69 linear-probing 74.52 69.67 67.53 2.92 71.28 10K random few-shot 52.82 42.88 40.27 5.41 46.42 linear-probing 74.33 68.52 62.65 4.77 70.02 MAK few-shot 54.33 45.01 40.12 5.90 48.01 linear-probing 75.57 68.20 66.29 4.07 70.83 COLT few-shot 54.26 46.54 43.38 4.57 49.14 linear-probing 75.13 71.38 66.62 3.48 72.22 Table 7 : 7Accuracy (%) and balancedness (Std) on ImageNet-100 with external 50K Open-Images (Krasin et al., 2017) as OOD dataset. Method Budget Protocol Many ↑ Median ↑ Few ↑ Std ↓ All ↑ SimCLR - few-shot 49.74 41.00 36.62 5.45 43.84 linear-probing 70.82 65.33 62.31 3.52 67.08 SimCLR+COLT 5K few-shot 54.46 47.12 43.23 4.66 49.48 linear-probing 75.62 70.87 67.29 3.41 72.26 10K few-shot 57.54 48.12 44.00 5.67 51.26 linear-probing 76.87 71.00 69.23 3.27 73.06 APPENDIX C HOW DO OOD DATA CONTRIBUTE TO LONG-TAIL LEARNING? OOD DATA RE-BALANCE THE FEATURE SPACE Table 8 : 8Comparison on different tail estimation strategies.Accuracy 54.20 ± 0.35 53.19 ± 0.47 53.33 ± 0.28 54.00 ± 0.21 53.87 ± 0.22 53.0 ± 0.34 52.97 ± 0.30Method COLT Radius-based BCL Threshold - 0.08 0.09 0.10 0.11 0.12 - Table 9 : 9Ablation study on the cluster number C on CIFAR-100-LT.Cluster Number C 10 20 50 100 100 (Oracle) Accuracy 54.20 ± 0.35 53.61 ± 0.18 54.06 ± 0.22 53.81 ± 0.29 54.16 ± 0.10 Table 10 : 10COLT's accuracy when OOD dataset is also imbalanced.Imbalance ratio 1 10 50 100 Accuracy 52.98 ± 0.33 52.63 ± 0.19 52.74 ± 0.25 52.66 ± 0.41 Table 11 : 11COLT's accuracy with different scale OOD dataset. Accuracy 53.43 ± 0.26 53.99 ± 0.18 54.20 ± 0.35OOD Images Number (K) 100 200 300 Although we sample for multiple times (every r epochs) in COLT, the time of performing sampling once is less than training for one epoch; therefore can be ignored compared to 1.7x training epochs brought by MAK. 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Zhongqi Miao, Ziwei Liu, Kaitlyn M Gaynor, S Meredith, Stella X Palmer, Wayne M Yu, Getz, Nature Machine Intelligence. 310Zhongqi Miao, Ziwei Liu, Kaitlyn M Gaynor, Meredith S Palmer, Stella X Yu, and Wayne M Getz. Iterative human and automated identification of wildlife images. Nature Machine Intelligence, 3 (10):885-895, 2021. Automatic differentiation in pytorch. Adam Paszke, Sam Gross, Soumith Chintala, Gregory Chanan, Edward Yang, Zachary Devito, Zeming Lin, Alban Desmaison, Luca Antiga, Adam Lerer, Adam Paszke, Sam Gross, Soumith Chintala, Gregory Chanan, Edward Yang, Zachary DeVito, Zeming Lin, Alban Desmaison, Luca Antiga, and Adam Lerer. Automatic differentiation in pytorch. 2017. Relay backpropagation for effective learning of deep convolutional neural networks. Li Shen, Zhouchen Lin, Qingming Huang, European conference on computer vision. SpringerLi Shen, Zhouchen Lin, and Qingming Huang. Relay backpropagation for effective learning of deep convolutional neural networks. In European conference on computer vision, pp. 467-482. Springer, 2016. Fixmatch: Simplifying semi-supervised learning with consistency and confidence. Kihyuk Sohn, David Berthelot, Nicholas Carlini, Zizhao Zhang, Han Zhang, Colin A Raffel, Ekin Dogus Cubuk, Alexey Kurakin, Chun-Liang Li, Advances in neural information processing systems. 33Kihyuk Sohn, David Berthelot, Nicholas Carlini, Zizhao Zhang, Han Zhang, Colin A Raffel, Ekin Dogus Cubuk, Alexey Kurakin, and Chun-Liang Li. Fixmatch: Simplifying semi-supervised learning with consistency and confidence. Advances in neural information processing systems, 33:596-608, 2020. Measuring and predicting importance of objects in our visual world. Merrielle Spain, Pietro Perona, Merrielle Spain and Pietro Perona. Measuring and predicting importance of objects in our visual world. 2007. Cost-sensitive boosting for classification of imbalanced data. Yanmin Sun, S Mohamed, Kamel, K C Andrew, Yang Wong, Wang, Pattern recognition. 4012Yanmin Sun, Mohamed S Kamel, Andrew KC Wong, and Yang Wang. Cost-sensitive boosting for classification of imbalanced data. Pattern recognition, 40(12):3358-3378, 2007. Contrastive multiview coding. Yonglong Tian, Dilip Krishnan, Phillip Isola, European conference on computer vision. SpringerYonglong Tian, Dilip Krishnan, and Phillip Isola. Contrastive multiview coding. In European conference on computer vision, pp. 776-794. Springer, 2020. 80 million tiny images: A large data set for nonparametric object and scene recognition. IEEE transactions on pattern analysis and machine intelligence. Antonio Torralba, Rob Fergus, William T Freeman, 30Antonio Torralba, Rob Fergus, and William T Freeman. 80 million tiny images: A large data set for nonparametric object and scene recognition. IEEE transactions on pattern analysis and machine intelligence, 30(11):1958-1970, 2008. Towards calibrated hyper-sphere representation via distribution overlap coefficient for long-tailed learning. Hualiang Wang, Siming Fu, Xiaoxuan He, Hangxiang Fang, Zuozhu Liu, Haoji Hu, Computer Vision-ECCV 2022: 17th European Conference. Tel Aviv, IsraelSpringerProceedings, Part XXIVHualiang Wang, Siming Fu, Xiaoxuan He, Hangxiang Fang, Zuozhu Liu, and Haoji Hu. Towards calibrated hyper-sphere representation via distribution overlap coefficient for long-tailed learning. In Computer Vision-ECCV 2022: 17th European Conference, Tel Aviv, Israel, October 23-27, 2022, Proceedings, Part XXIV, pp. 179-196. Springer, 2022.

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{'abstract': "Though Self-supervised learning (SSL) has been widely studied as a promising technique for representation learning, it doesn't generalize well on long-tailed datasets due to the majority classes dominating the feature space. Recent work shows that the long-tailed learning performance could be boosted by sampling extra in-domain (ID) data for self-supervised training, however, large-scale ID data which can rebalance the minority classes are expensive to collect. In this paper, we propose an alternative but easy-to-use and effective solution, Contrastive with Out-of-distribution (OOD) data for Long-Tail learning (COLT), which can effectively exploit OOD data to dynamically re-balance the feature space. We empirically identify the counter-intuitive usefulness of OOD samples in SSL long-tailed learning and principally design a novel SSL method. Concretely, we first localize the 'head' and 'tail' samples by assigning a tailness score to each OOD sample based on its neighborhoods in the feature space. Then, we propose an online OOD sampling strategy to dynamically re-balance the feature space. Finally, we enforce the model to be capable of distinguishing ID and OOD samples by a distributionlevel supervised contrastive loss. Extensive experiments are conducted on various datasets and several state-of-the-art SSL frameworks to verify the effectiveness of the proposed method. The results show that our method significantly improves the performance of SSL on long-tailed datasets by a large margin, and even outperforms previous work which uses external ID data. Our code is available at", 'arxivid': '2306.04934', 'author': ['Jianhong Bai \nZhejiang University\n\n', 'Zuozhu Liu \nZhejiang University\n\n', 'Hualiang Wang \nThe Hong Kong University of Science and Technology\n\n', 'Jin Hao \nHarvard University\n\n', 'Yang Feng \nAngelalign Technology\n\n', 'Huanpeng Chu \nZhejiang University\n\n', 'Haoji Hu \nZhejiang University\n\n'], 'authoraffiliation': ['Zhejiang University\n', 'Zhejiang University\n', 'The Hong Kong University of Science and Technology\n', 'Harvard University\n', 'Angelalign Technology\n', 'Zhejiang University\n', 'Zhejiang University\n'], 'corpusid': 259108315, 'doi': None, 'github_urls': ['https://github.com/JianhongBai/COLT.', 'https://github.com/openimages,'], 'n_tokens_mistral': 22966, 'n_tokens_neox': 19211, 'n_words': 10781, 'pdfsha': '87fc362bb981e3de271a2338b015818c6acc9794', 'pdfurls': ['https://export.arxiv.org/pdf/2306.04934v1.pdf'], 'title': ['ON THE EFFECTIVENESS OF OUT-OF-DISTRIBUTION DATA IN SELF-SUPERVISED LONG-TAIL LEARNING', 'ON THE EFFECTIVENESS OF OUT-OF-DISTRIBUTION DATA IN SELF-SUPERVISED LONG-TAIL LEARNING'], 'venue': []}

arxiv

Modeling radio communication blackout and blackout mitigation in hypersonic vehicles 24 Jul 2014 Madhusudhan Kundrapu John Loverich Kristian Beckwith Peter Stoltz Alexey Shashurin Michael Keidar Tech-X Corporation 80303BoulderCOUSA The George Washington University 20052WashingtonDCUSA Modeling radio communication blackout and blackout mitigation in hypersonic vehicles 24 Jul 2014Submitted to AIAA Journal of Spacecraft and Rockets NomenclatureSubmitted to AIAA Journal of Spacecraft and Rockets 1 Associate Research Scientist 2 Research Scientist 3 Research Scientist 4 Vice President 5 Research Scientist, Mechanical and Aerospace Engineering 6 Professor, Mechanical and Aerospace Engineering A procedure for the modeling and analysis of radio communication blackout of hypersonic vehicles is presented. A weakly ionized plasma generated around the surface of a hypersonic reentry vehicle traveling at Mach 23 was simulated using full Navier-Stokes equations in multi-species single fluid form. A seven species air chemistry model is used to compute the individual species densities in air including ionization -plasma densities are compared with experiment. The electromagnetic wave's interaction with the plasma layer is modeled using multi-fluid equations for fluid transport and full Maxwell's equations for the electromagnetic fields. The multi-fluid solver is verified for a whistler wave propagating through a slab. First principles radio communication blackout over a hypersonic vehicle is demonstrated along with a simple blackout mitigation scheme using a magnetic window. R = momentum exchange rate per unit volume, N/m 3 t = time, s T = temperature, K u = velocity, m/s Hypersonic vehicles are subjected to severe aerothermal heating due to the formation of shock waves in front of the vehicle. Flow Mach numbers exceeding four are classified as hypersonic [1] and in this regime the kinetic energy of the flow, when converted to internal energy through the shock provides a significant increase in the fluid temperature. The temperatures quite often exceeds the dissociation and ionization limits of the flow species and results in the formation of a weakly ionized plasma layer around the vehicle. The electrons in the plasma layer may interrupt the propagation of radio frequency electromagnetic waves if the plasma electron oscillation frequency exceeds that of the electromagnetic wave frequency. This phenomenon is commonly called radio communication blackout. For instance, a 1.6 GHz radio wave will be interrupted by a plasma layer of density of 3.5 × 10 16 . Blackout mitigation is an important requirement for the design of hypersonic vehicles, especially for those vehicles in steady state hypersonic flight such as those envisioned by NASA and the US Air Force. A few mitigation mechanisms described in the literature are the magnetic window [13,24,32], electrophilic fluid injection [2], wave frequency modification, aerodynamic shape modification, E × B drift [21,22], resonant transmission [33], time varying magnetic field [3] and electron acoustic wave transmission [34]. The magnetic window uses a static magnetic field to convert the free space radio wave to a whistler wave in the plasma. [23] Electrophilic injection uses an electrophilic substance injected into the fluid to decrease the electron density. Wave frequency and aerodynamic shape modification have design limitations so may be impractical in many cases. The E ×B drift accelerates the ions in the layer there by decreasing the plasma density near the antenna. Resonant transmission uses surface wave resonance to enhance transmission through the plasma layer. The time varying magnetic field approach uses the hall effect to expel ions. Electron acoustic wave transmission works by converting the wave into an electron acoustic wave in the plasma layer. Numerical simulations of blackout mitigation techniques are valuable during the design phase of hypersonic vehicles. Radio communication blackout modeling of aerospace vehicles with full wave electromagnetics has been investigated by several groups with many different codes. Takahashi [14,15] uses a CFD tool to compute the plasma distribution and then a FDTD solver with a modified permittivity to account for the presence of a plasma. Thoma[32] used the high density FDTD PIC code LSP to investigate the magnetic window with a horn antenna surrounded by an assumed plasma distribution. Visbal [4] uses a multi-fluid electromagnetic approach to modeling radio communication blackout on an overset mesh, without an investigation of steady magnetic field effects. The scope of this paper is to show the modeling approach that works for realistic vehicles in complex geometries and can be used to simulate the feasibility of blackout mitigation devices for hypersonic vehicles. The RAM C reentry vehicle is used for this demonstration as there is significant experimental data for comparison. USim [28,30,31], a commercial code developed by Tech-X Cor-poration for general fluid plasma modeling on unstructured grids, is used for all simulations in this paper. This paper is organized as follows: (1) Modeling and simulation of the multi-species hypersonic flow over the RAM C reentry vehicle to obtain the plasma density distribution (2) is the sum of internal energy, kinetic energy and the chemical energy of the fluid. The fluid was assumed to be Newtonian and obeys the Stoke's hypothesis of zero bulk viscosity. Further, the fluid obeys the ideal gas law for equation of state. ∂ρ ∂t + ∇ · (ρ u) = 0 (1) ∂ (ρ u) ∂t + ∇ · (ρ u u + pI) = ∇ · τ (2) ∂ (e) ∂t + ∇ · ( u (e + p)) = ∇ (τ · u) + ∇ · (k T ∇T )(3) where, ρ = i n i m i (4) p = ρRT (5) τ = − 2 3 µ (∇ · u) I + µ ∇ u + (∇ u) T (6) e = p γ − 1 + 1 2 ρ u · u + i n i H i(7) and γ = c p c p − R (8) B. Species transport The mass conservation of the individual species in the bulk fluid is satisfied separately for each of the species using Eq. 9. The velocity u is same as that of the bulk fluid. The right hand side of Eq. 9 represents the rate of change of species density due to the chemical reactions. ∂n i ∂t + ∇ · ( un i ) = s i(9) C. Material properties The properties viscosity, thermal conductivity and the specific heat of the individual species were obtained from the kinetic theory of gases as given by the Eqs. (11)- (13). The fluid thermal conductivity k and viscosity µ in Eqs. (1)-(3) were obtained using mole fraction averaging while the specific heat c p was obtained using the mass fraction averaging. The gas constant R was computed using the mole fraction averaged molecular weight. c p i = f 2 + 1 R i (10) µ i = 5 16 √ πm i k B T (πσ 2 Ω)(11)k i = 5 2 c v i µ i(12)c v i = c p i − R i(13) D. Electromagnetic multi-fluid A multi-fluid model was used for the interaction of radio wave with the plasma. Maxwell's equations were used to solve evolution of electric and magnetic fields. Ampere's law and Faraday's law are given by Eqs. (14) and (15) respectively. The right hand side of Eq. (14) is the sum of the current densities of the conducting species. The divergence equations (16) and (17) should be satisfied along with the Ampere's and Faraday's laws. ∂ E ∂t − c 2 ∇ × B = − 1 ǫ 0 α q α ρ α u α m α (14) ∂ B ∂t + ∇ × E = 0 (15) ∇ · E = 1 ǫ 0 α q α ρ α m α (16) ∇ · B = 0(17) The transport of the multi-fluid system was modeled using the system of equations Eqs. (18)- (20). Index α is for any fluid. The first terms on the RHS of Eq.(19) the electric and magnetic Lorentz forces. The third term is the net momentum exchange with the remaining fluids in the system. [25] The first term on the RHS of Eq.(20) is for Joule heating and the fourth and fifth terms are kinetic energy and internal energy exchange terms respectively. [25] ∂ρ α ∂t + ∇ · (ρ α u α ) = 0 (18) ∂ (ρ α u α ) ∂t + ∇ · (ρ α u α u α + p α I) = ρ α m α q α E + u α × B + ∇ · τ α + R α (19) ∂ (e α ) ∂t + ∇ · ( u α (e + p α )) = ρ α m α q α u α · E + ∇ (τ α · u α ) + ∇ · (k α ∇T α ) + V α · R α + Q α(20) where, V α = i ρ i u i / i ρ i(21)R α = − i ρ α m α µ αi ζ −1 αi ( u α − u i )(22) and Q α = − i 3k B ρ α m α [µ αi / (m α + m i )] ζ −1 αi (T α − T i )(23) importantly, this model describes electromagnetic wave propagation in free space (when the charged species densities are zero) and in a conducting fluid (when the charged species densities are non-zero). In particular it describes reflection of electromagnetic waves off of an over-dense plasma as well as electromagnetic wave propagation in a plasma including the changes caused by external magnetic fields. The model is more complete to that of magnetohydrodynamics (MHD) and can be thought of as MHD without the assumption of quasi-neutrality, without the assumption that the light wave is infinitely fast and by using the full Ohm's law, (including electron inertia) in the MHD system. Restricting ourselves to two-fluids for the moment (electrons and ions only), 4 parameters that can be derived from this model, will be important in determining the electromagnetic wave propagation characteristics in the plasma. The first is the electron plasma frequency Ω p e = n e q 2 e m e ǫ 0(24) the second is the electron cyclotron frequency Ω c e = q e B 0 m e(25) followed by the ion plasma frequency Ω p i = n i q 2 i m i ǫ 0(26) and the ion cyclotron frequency Ω c i = q i B 0 m i(27) These parameters will be used in the discussion of the whistler wave. E. Solution methodology The equation systems given in Sec. II were solved using a generalized unstructured grid finite volume solver, USim [28,30,31]. Though multi-fluid electromagnetic solvers have been developed throughout the years by several researchers [11, 12, 18-20, 29, 35, 36], the present solver is the first solver using an unstructured formulation and running on an unstructured grid [30] as prior codes were based on multi-block logically Cartesian grids. The flux reconstruction on the cell faces was carried out using second order accurate Monotonic Upstream-Centered Scheme for Conservation Laws (MUSCL) [8]. The right and left fluxes on the cell faces were obtained by extrapolating the cell centered gradient of the conserved variable. The spurious oscillations that may arise due to the flux reconstruction are limited using flux limiters such as Van-Leer limiter. [8] The cell centered gradient was computed using the weighted least squares method. A second degree polynomial was considered in this work. The actual flux on the cell face was then obtained using an approximate Riemann flux. In this work HLLE approximate Riemann flux [9] was considered for the fluid equations while full were considered from the Ref. [5]. The unstructured grid used for the simulation was shown in Electron density (m −3 ) USim peak density USim wall density Flight test peak density curves are peak density and surface density respectively. The middle curve is the averaged peak density measurements from Ref. [6] of the trend and in the design point view, the higher values of the simulation to this extent are acceptable in terms of the factor of safety. The wall densities are well below the peak values, for the simple reason that the wall temperature is much below the boundary layer temperature. B. Dispersion relation for waves parallel to the magnetic field The magnetic window approach to radio communication blackout mitigation takes advantages of special properties of electromagnetic wave propagation in plasmas in the presence of a magnetic field. A derivation of the whistler wave (as well as many other plasma waves) can be found in many plasma physics text books including [37]. In the case below, the dispersion relation is derived from the twofluid electromagnetic plasma model and written in terms of plasma parameters, Ω c e ,Ω c i ,Ω p e ,Ω p i and the speed of light c, the results are given as follows. The R-Mode dispersion relation is given by k 2 = w 2 w 2 + w(Ω ci − Ω ce ) − Ω ce Ω ci − Ω 2 pe − Ω 2 pi c 2 (w − Ω ce )(w + Ω ci )(28) ignoring ion motion we get k 2 = w w 2 − wΩ ce − Ω 2 pe c 2 (w − Ω ce )(29) The L-Mode dispersion relation is given by k 2 = w 2 w 2 + w(Ω ce − Ω ci ) − Ω ce Ω ci − Ω 2 pe − Ω 2 pi c 2 (w + Ω ce )(w − Ω ci )(30) ignoring ion motion we get fields. E y and B z are shown in Fig. 8(a) and 8(b) respectively. The amplitudes of the electric and magnetic fields doubled since a standing wave is formed upon reflection from the plasma slab. The figure also shows the simulation accuracy of the two-fluid solver. [10,17] A constant magnetic field of B 0x 1 T was applied in the domain to create a magnetic window 20) is neglected too as the collision frequency [26,27] of the electrons and neutrals is less than an order of magnitude the plasma frequency. Figure 10 shows the comparison. The contour flood in Fig.10(a)represents the plasma electron frequency and the contour lines show the electron neutral collision frequency. In the present simulation, the electron neutral collision frequency is highest among the collisions of the remaining species. The highest values of the frequencies are seen near the stagnation region, where the densities are high. The contours clearly show that plasma frequency is higher than the collision frequency everywhere within the plasma layer. Also, a line plot comparison along the stagnation line is shown in Fig.10(b) to get a better picture of the comparison of magnitudes. k 2 = w w 2 + wΩ ce − Ω ce Ω 2 pe c 2 (w + Ω ce )(31) The wave reflection by the plasma layer of the RAM C is shown in Fig.11. The frequency of the plane wave originating at the top boundary is 1.6 GHz. The wave components at the top boundary are E x = ca 0 sin(2πf t), B z = E y /c and the remaining components are equal to zero. of wave occurs. These magnetic field strengths are slightly large, however, many hypersonic vehicles of interest will actually have a considerably smaller plasma density and thus require a much weaker field to allow whistler wave propagation -this situation described can be considered an extreme case. In addition, it's possible that using a weaker field will still allow whistler wave propagation as the evanescent waves may propagate through the outer edge of the plasma where the field is weak, then propagate as whistler waves closer to the vehicle surface. Figure 12 shows wave, it also focuses the wave through the converging magnetic field, which is in agreement with the observations made in Ref. [38]. The ability to recover the original signal on the surface can be checked to verify that a useable signal can be obtained. The best way to do this is to find the frequency of the signal and its energy density at the surface. The frequency of the whistler wave is obtained by taking Fourier transform of the wave history recorded on the surface of RAM C. Figure 13 shows the frequency of E y at (0.25555, -0.15829). The highest peak corresponds to a frequency of 1.6 GHz verifying that the original signal can be recovered at the vehicle surface. In addition to the frequency match, the signal strength can also be calculated. The comparison of the energy density of the signal in the free space and that on the surface gives an idea of the signal strength. The wave's energy density is computed using Eq.32. A comparison of the electromagnetic wave at t = 4.5 ns is due to the added reflected components. It can be seen from Fig.14(b) that the whistler wave's energy is sufficient to be received by the antenna. The whistler wave's energy density is about 40% that of the original wave. Q EM = 1 2 ǫ 0 E · E + 1 µ 0 B · B(32) Another interesting observation made during the simulations is the amplification of the whistler wave energy density with the increase of magnetic field. For instance, increasing the coil current by 4 times generates a maximum magnetic field of 3.1 T on the surface. The strength of the magnetic field at the plasma layer's edge where whistler wave propagation occurs in this case is around 0.8 T. The wave energy density history in a magnetic widow created with the increased magnetic field is shown in Fig.15. The energy density in this case is 400% of the source wave. In fact, the energy is amplified by about four times the wave's energy in the free space. As explained previously, the amplification is due to the focusing of wave along the converging field lines. This amplification could be useful in the cases where, the original signal itself is weak. densities ranging from 40% to 400% using magnetic fields of 0.15T and 0.8 T respectively at the edge of the plasma layer. However, the configuration can be further optimized by changing the orientation of the magnetic field lines after obtaining the plasma distributions for all the critical flight conditions in terms of angle of attack, speed and altitude. The designer can also test other mitigation methods using the same model described in this paper. For instance, electrophilic fluid injection method can be tested by adding the additional reactions to the existing reactions set of the multi-species transport equations to establish the reduced plasma density. The electron acoustic wave transmission can be tested by including the multi-fluid advection terms in the analysis [34] so that the electron acoustic wave is simulated. Similarly, resonant transmission can be modeled with the equations described. V. Summary Validation of the plasma density distribution with the results from literature (3) Modeling of electromagnetic wave propagation into the plasma and validation with the dispersion relation and (4) Finally, the propagation of plane EM wave on to the vehicle's surface through the plasma layer using a magnetic window and the whistler wave conversion. II. Mathematical formulation A. Bulk fluid transport A generalized model for simulating the compressible flow with reacting multi-species is given in this section. The Navier-Stokes equations in conservative form Eqs. (1)-(3) were used for the conservation of fluid mass, momentum, and total energy respectively. The total energy e in Eq. (3) wave flux was chosen for the Maxwell's equations. The diffusion fluxes were evaluated by computing the least squares gradient on the cell faces and then performing a surface integral according to the Gauss divergence theorem. Reaction rates were integrated using the Boost ODE integrator[40]. The left and right hand side of the Eqs. (1)-(9), (14), and (15) were evaluated separately and added together and then integrated in time using Runge-Kutta method. A third order RK integration scheme was used. Operator splitting was used for the reaction terms and super time stepping for the diffusion terms. III. Validation of the results A. Reactive flow simulation The simulation was performed for a velocity of 7650 m/s, density 2.816×10 −4 kg/m 3 and temperature 244.3 K. These values correspond to an altitude of 61 km. The air species N 2 , O 2 , N , O, N O, N O + , and e (electron) were considered for the reaction chemistry and the reaction rates Fig. ( 1 1). Cubit[39] grid generation software was used for the grid generation. The contour flood represents the area of cells in m 2 . The average edge lengths vary from around 0.5 mm at the nose cap region to 1 mm on the lateral surface.Flow enters the domain from the top boundary ofFig. 1. An axisymmetric boundary condition Fig. 1 Fig. 2 12The unstructured grid used for the simulation. The contour flood represents the cell area.was imposed on the axis (left boundary). On that wall, a standard no slip and radiation equilibrium temperature were imposed. Outflow boundary conditions were used on the remaining boundaries.The temperature and electron density distributions are shown inFig. 2. The peak values of average temperature and the electron density are 21860 K and 1.18 × 10 20 m −3 respectively existing in the stagnation region.Figure 3shows the comparison of the surface and peak electron densities in the plasma layer with the reflectometer measurements presented in Ref.[6]. The measurementsrepresent the time averaged values of the electron density measured using 15 reflectometers of four different frequencies placed at four stations on the wall of RAM C. The cut-off densities associated with the four frequencies are 1.52 × 10 19 ,1.25 × 10 18 ,1.37 × 10 17 , and 1.54 × 10 16 m −3 respectively.[7] The first station was located at 0.0457 m from the nosecap tip. The remaining three stations locations along with the measured peak densities are shown by squares in the figure. The dashed curve represents the curve fit of the measured data points. The bottom most curve is the surface density distribution while the top most curve is the peak density of electrons in the plasma layer ofFig.2(b). The peak values of the simulation are about three times the values from the measurements. Inclusion of radiation losses from the plasma and the diffusion of electrons in the simulation could decrease the density to some extent. Moreover, the comparison shows a good agreement in terms Flow parameters on RAM C. (a) Temperature distribution and ( Fig. 3 3Validation of the electron density in the plasma layer of RAM C. The top and bottom Fig. 4 4shows the frequency vs wave number plotted for the electromagnetic wave in a plasma without a background magnetic field. At all frequencies below the plasma frequency the wave is evanescent.By adding a magnetic field the dispersion relation changes and the wave can propagate through the plasma at frequencies below the electron cyclotron frequency.Fig.5 shows the frequency vs wave number plotted for the R-Mode and L-Mode waves in non-dimensional units in a plasma with a background magnetic field. The background field can be adjusted so that the signal can propagate through the plasma as a whistler wave which is the magnetic window. The Fig.6 shows the EM wave propagation in a neutral fluid. A plane wave was excited from the left boundary with the components E y = c a 0 sin(2πf t) and B z = E y /c. The frequency of the wave was 1.6 GHz. Uninterrupted propagation of the wave can be clearly seen. A uniform plasma slab of thickness 0.3 m was then added in the domain at x = -0.15 m. The plasma density was 10 19m −3 . Since the frequency of the plasma, 28.4 GHz is much greater than the wave frequency, the wave was completely reflected by the plasma.Figure 7shows the reflection of electric and magnetic Submitted to AIAA Journal of Spacecraft and Rockets Fig. 4 4Dispersion graph for electromagnetic waves traveling in a plasma with no background magnetic field, frequency vs wave number in normalized units. The vacuum electromagnetic wave is provided for comparison. Notice that below the plasma frequency the electromagnetic wave in the plasma does not propagate. This is the reason for radio communication blackout. Fig. 5 5Dispersion graph for electromagnetic waves traveling parallel to the magnetic field, frequency vs wave number in normalized units. The R-mode wave has two branches, the lower frequency branch which propagates below the plasma frequency is known as the whistler wave. The whistler wave has a cutoff at the electron cyclotron frequency. In the presence of a magnetic field then it is possible for the electromagnetic wave to penetrate the over-dense plasma. The electron cyclotron frequency must be greater than the signal frequency and this puts a lower bound on the magnetic field strength that should be used. Fig. 6 Fig. 7 Fig. 8 Fig. 9 6789The (a) y component of electric field and the (b) z component of the magnetic field in the EM wave of frequency 1.6 GHz traveling in neutral fluid. The (a)y component of electric field and the (b) z component of the magnetic field of the EM wave of frequency 1.6 GHz traveling in a domain with plasma slab of thickness 0.3 m. The slab starts from x = -0.15 m.The wave is completely reflected by the plasma hence no propagation in the plasma and beyond the slab. for the propagation of the wave in whistler mode through the plasma. Figure 8 shows the wave components E y . The whistler wave's accuracy is compared with the analytical solution obtained from the dispersion theory. The Dispersion relation shows that the wave number of the whistler wave in the plasma slab is 23.89. The Fourier transform of the wave in the spatial domain gives the wavenumber spectrum. The Fourier transform E x in Fig.8 is shown in Fig.9. The first peak is located around k = 5.33 and the second peak around k=23.89. The first peak represents the The (a)y component of electric field and the (b) z component of the magnetic field of the EM wave of frequency 1.6 GHz traveling in a domain with plasma slab of thickness 0.3 m. The slab starts from x = -0.15 m. A background magnetic field of B0x = 1 T is applied in the domain. The wave propagates through the plasma as a Whistler wave. The wavenumber of the plane wave in neutral fluid and plasma. wave in the neutral zone and the second peak corresponds to the wave in the uniform plasma. Note that the amplitude does not match with the values shown in Fig.8 since the exact wave numbers 5.33 and 23.89 were not resolved by the grid. A more refined grid is required to represent the wave numbers of interest, in which case, the spectral amplitude will match the amplitude of the waves in the simulation. Eqs.(18)-(20), the advection and viscous diffusion occur on much larger time scales when compared to the time scale of the plasma oscillations and EM wave. For instance, the smallest advection and diffusion time scales in the stagnation region are 4.8 × 10 −6 and 8.9 × 10 −6 s respectively. Whereas the plasma oscillations occur on the time scale of 1.1 × 10 −11 s. In the aft region (y=-1.25m), the minimum advection and diffusion times are 2.73 × 10 −7 and 1.78 × 10 −6 s respectively. The plasma frequency time scale is 1.01 × 10 −10 s. Overall, the timescales of the advection and diffusion are more than three orders of magnitude the plasma oscillation time scale. Hence, the advection and the viscous diffusion terms were neglected. Note that the length scale used in the estimation of time scales was the average edge length of the local cell. The collision term in Eqs.(18)-( Figures 11 (Fig. 10 FigFig 1110a) and11(b) represent the x,y components of the electric field.Figure 11(c) represents the z-component of the magnetic field. The contour flood shows the amplitudes of the wave. The dashed contour lines represent the plasma frequency. It can be clearly observed from the figures that the wave is completely reflected by the plasma layer once the plasma frequency is 1.6 GHz. Note that the flood contour levels are limited between peak positive and negative amplitudes of the Comparison of the plasma electron frequency ωpe and the collision frequency of electrons with neutrals ζe−n. (a) contour lines of ζe−n and the contour flood of ωpe. (b) Line plot of the frequencies along the stagnation line. original wave, in order to make the wave visible. The wave's amplitude increases by about 10 times at the edge of the plasma layer due to the resonance of the evanescent wave.[16] The amplified wave propagates along the plasma layer's edge. B. Magnetic window whistler mode The magnetic field was applied on the surface near to the nosecap using a current carrying coil of radius 0.1 m centered at (0.15, -0.15). The current was 1.5 × 10 5 A. In practice a permanent magnet would be used to generate the field. The magnetic field lines colored in magnitude can be seen in all of the subplots of Fig.12. The maximum field strength available on the RAM C surface is 0.77 T while the value is around 0.125T at the the edge of the plasma layer where significant propagation . 11 EM wave reflection in the plasma layer of RAM C. (a) x component of the electric field, (b) y component of the electric field, and (c) z component of the magnetic field. The contour lines represent the plasma electron frequency. . 12 EM wave propagation in whistler mode in to the plasma layer of RAM C. The imposed magnetic field is shown by the streamlines. The three components of the electric and magnetic field are shown by the contour floods of (a), (b), (c), (d), (e), (f ) respectively. The black dashed contour line corresponds to the plasma electron frequency of 1.6 GHz. the flood contours of electric and magnetic field components of the EM wave. The cutoff frequency of plasma f pe = 1.6 GHz is depicted by the dashed contour line. The three components of the electric and magnetic field are shown in Figs.12(a)-12(c) and Figs.12(d)-12(f) respectively. It has to be noted that, the electric field contours are limited between -100 and 100 V/m, in order to improve the visibility of the whistler wave. It is clear now that the wave signal propagates through the plasma layer in the whistler mode. The additional components arising in Fig.12 when compared to the Fig.11 are due to the circular polarization of the wave around the magnetic field lines. The circularly polarized wave propagates parallel to the magnetic field lines. Hence the angle between the wave vector and the magnetic field lines at the edge of the plasma layer plays an important role. The wave does not propagate along the field lines perpendicular to the direction of propagation. It can also be observed from the figure that the magnetic window not only allows the passage of the original electromagnetic Fig. 13 13Whistler wave frequency on the surface of RAM C. energy density of the wave in the free space and that of the whistler wave on the surface of RAM C at (0.25555, -0.15829) is shown in Fig. 14. The Fig.14(a) corresponds to the wave in the free space and bottom subplot is for the Whistler wave. The slight rise in the energy of the free space FromFig. 14 Fig. 15 1415the above feasiblity analysis, it can be said that the magnetic window subjected to a realistic flight condition is capable of propagating the signal on to the vehicle's surface with energy The energy density recorded in the free space and on the surface of RAM C. The maximum magnetic field on the RAM C surface is 0.77 T and the magnetic field at the plasma layer's edge where the wave propagates is 0.125 T. (a) Recorded in free space and (b) on the RAM C surface at (0.25555, -0.15829). The recorded energy density on the surface of RAM C at (0.258043, -0.172272). The maximum magnetic field on the RAM C surface is 3.1 T and the magnetic field at the plasma layer's edge where the wave propagates is 0.8 T. A procedure to model and simulate the hypersonic flow and the vehicle's communication blackout is shown. The plasma density on the RAM C vehicle showed a good agreement with the reflectometer measurements from the literature. Addition of radiation losses to the reactive flow could further improve the accuracy the simulation results. The results of Maxwell equation solver of USim are validated with the analytical solution of whistler wave propagation in one dimensional plasma layer. The Whistler mode propagation of the wave on the RAM C surface is demonstrated successfully. The frequency and the energy density of the wave signal recorded on the surface of RAM C showed a good possibility of recovering the signal propagated in whistler mode. Although only the magnetic window was investigated, the same plasma model together with the solver can be used to investigate many radio blackout mitigation schemes including electron acoustic wave transmission and resonant transmission. AcknowledgmentsThe authors are thankful to the financial support from AFOSR (grant numbers FA9550-12-C-0039 and FA9550-14-C-0004). The authors thank Dr. Thomas Jenkins, Dr. Ming-Chieh Lin, and Dr. David Smithe of Tech-X Corporation, for their helpful comments. Hypersonic aerothermodynamics. J J Bertin, AIAABertin, J. J.,"Hypersonic aerothermodynamics," AIAA, 1994, pp.4-5. System and method for reducing plasma induced communication disruption utilizing electrophilic injection and sharp reentry vehicle nose shaping. J W Meyer, Lockheed Martin Corp, Palo Alto, U S Ca, Patent, Publication No. US7237752 B1. 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{'abstract': "A procedure for the modeling and analysis of radio communication blackout of hypersonic vehicles is presented. A weakly ionized plasma generated around the surface of a hypersonic reentry vehicle traveling at Mach 23 was simulated using full Navier-Stokes equations in multi-species single fluid form. A seven species air chemistry model is used to compute the individual species densities in air including ionization -plasma densities are compared with experiment. The electromagnetic wave's interaction with the plasma layer is modeled using multi-fluid equations for fluid transport and full Maxwell's equations for the electromagnetic fields. The multi-fluid solver is verified for a whistler wave propagating through a slab. First principles radio communication blackout over a hypersonic vehicle is demonstrated along with a simple blackout mitigation scheme using a magnetic window. R = momentum exchange rate per unit volume, N/m 3 t = time, s T = temperature, K u = velocity, m/s", 'arxivid': '1407.6635', 'author': ['Madhusudhan Kundrapu ', 'John Loverich ', 'Kristian Beckwith ', 'Peter Stoltz ', 'Alexey Shashurin ', 'Michael Keidar ', '\nTech-X Corporation\n80303BoulderCOUSA\n', '\nThe George Washington University\n20052WashingtonDCUSA\n'], 'authoraffiliation': ['Tech-X Corporation\n80303BoulderCOUSA', 'The George Washington University\n20052WashingtonDCUSA'], 'corpusid': 118694347, 'doi': '10.2514/1.a33122', 'github_urls': [], 'n_tokens_mistral': 12959, 'n_tokens_neox': 11064, 'n_words': 7345, 'pdfsha': '9025ef7f54a8bb1bf3db803fd310038d7d3e10f2', 'pdfurls': ['https://arxiv.org/pdf/1407.6635v2.pdf'], 'title': ['Modeling radio communication blackout and blackout mitigation in hypersonic vehicles', 'Modeling radio communication blackout and blackout mitigation in hypersonic vehicles'], 'venue': []}

arxiv

Field evidence for the initiation of isolated aeolian sand patches 20 Feb 2023 P Delorme School of Geography and Environmental Science University of Southampton SouthamptonUK now at: Energy and Environment Institute University of Hull HullUK J M Nield School of Geography and Environmental Science University of Southampton SouthamptonUK G F S Wiggs School of Geography and the Environment University of Oxford OxfordUK M C Baddock Geography and Environment Loughborough University LoughboroughUK N R Bristow Mechanical Engineering St Anthony Falls Laboratory University of Minnesota MinneapolisUSA J L Best K T Christensen P Claudin Physique et Mécanique des Milieux Hétérogènes, CNRS, ESPCI Paris PSL Research University Université Paris Cité Sorbonne UniversitéParisFrance Field evidence for the initiation of isolated aeolian sand patches 20 Feb 2023manuscript submitted to Geophysical Research Letters manuscript submitted to Geophysical Research Letters6 Departments of Geology, Geography and GIS, Mechanical Science and Engineering and Ven Te Chow Hydrosystems Laboratory, University of Illinois at Urbana-Champaign, USA 7 Departments of Mechanical, Materials and Aerospace Engineering and Civil, Architectural and Environmental Engineering, Illinois Institute of Technology, USA Corresponding author: Pauline Delorme, p.m.delorme@hull.ac.uk -1- Key Points:• Sand patches can emerge on non-erodible surfaces. • Differing surface characteristics control particle behaviour. • Field measurements demonstrate the key role of sand transport in bedform initiation.AbstractSand patches are one of the precursors to early-stage protodunes and occur widely in both desert and coastal aeolian environments. Here we show field evidence of a mechanism to explain the initiation of sand patches on non-erodible surfaces, such as desert gravels and moist beaches. Changes in sand transport dynamics, directly associated with the height of the saltation layer and variable transport law, observed at the boundary between non-erodible and erodible surfaces lead to sand deposition on the erodible surface. This explains how sand patches can form on surfaces with limited sand availability where linear stability of dune theory does not apply. This new mechanism is supported by field observations that evidence both the change in transport rate over different surfaces and in-situ patch formation that leads to modification of transport dynamics at the surface boundary.Plain Language SummarySand patches can be observed in various environments such as beaches and gravel plains in deserts. Expected to be precursors of dunes when sediment supply is limited, these bedforms are typically a few centimeters high and present a reverse longitudinal elevation profile, with a sharp upwind edge and a smooth downwind tail. Based on field measurements, we propose a formation mechanism for these patches associated with the sensitive nature of wind-blown sand transport to changing bed conditions: sand saltation is reduced at the transition from a solid to an erodible surface, hence favouring deposition on the patches. This allows us to explain their typical meter-scale length as well as their asymmetric shapes. Introduction Isolated low-angle sand patches are commonly observed in desert and coastal regions on non-erodible surfaces, such as gravel plains or moist beaches (Figure 1, e.g. Lancaster, 1996;Hesp & Arens, 1997;Nield, 2011). These bedforms are typically several centimeters high, exhibit reverse longitudinal asymmetry compared to mature dunes, and can develop rapidly over several hours. Extensive research has explored the physical dynamics and morphology of mature desert sand dunes (Bagnold, 1937(Bagnold, , 1941Lancaster, 1982;Werner, 1990;Andreotti et al., 2002a;Charru et al., 2013;du Pont, 2015;Wiggs, 2021). We also have some evidence of the dynamics by which emerging dunes might grow into early-stage protodunes and more mature dune forms (Kocurek et al., 1992;Elbelrhiti, 2012;Hage et al., 2018;Montreuil et al., 2020), where the subtle coupling of topography, wind flow, and sediment transport acts to reinforce their growth (Baddock et al., 2018;Delorme et al., 2020;Gadal, Narteau, Ewing, et al., 2020;Lü et al., 2021;Bristow et al., 2022). However, our knowledge of the processes resulting in, and the relevant time and length scales associated with, the initial deposition of sand on a non-erodible surface remains incomplete and unquantified, although such processes possibly represent a fundamental stage in the origin of aeolian dunes. There are two clear sets of processes by which aeolian dunes are thought to be established (Courrech du Pont et al., 2014;du Pont, 2015). The first is associated with the hydrodynamic instability of an erodible granular flat bed with unlimited sand availability (Warren, 1979;Andreotti et al., 2002a;Claudin et al., 2013;Charru et al., 2013). This instability results from the combination of the response of wind stress to the modulated topographic profile, and the response of sand transport to the spatial variation in that wind stress (Charru et al., 2013). The former drives the instability where the wind stress maximum is shifted upwind of a dune crest Lü et al., 2021); the latter controls the emerging dune size with a relaxation process over a (saturation) length, L sat (Sauermann et al., 2001;Andreotti et al., 2010;Pähtz et al., 2013;Selmani et al., 2018). The resulting dune pattern consists of straight-crested bedforms growing in am- plitude with an orientation controlled by the wind regime Delorme et al., 2020). The second set of processes is associated with the growth of finger-like dunes developing across a non-erodible surface from isolated sand sources (Courrech du Pont et al., 2014;Rozier et al., 2019;Gadal, Narteau, Du Pont, et al., 2020). In this case, the dunes, well separated by interdunes where sand is scarce, present a finger-like shape and grow in length in a direction between those of the dominant winds . Experiments in wind tunnels have also highlighted the critical role of boundary conditions in determining saltation dynamics and sand transport rates (e.g. Ho et al., 2012;Kamath et al., 2022) and this offers a potential further means by which dunes may establish. These experiments have provided evidence for the existence of distinctly different transport rates on erodible and non-erodible or moist surfaces (Neuman & Scott, 1998;Ho et al., 2011). Larger sediment fluxes on non-erodible beds have been interpreted as a consequence of a negligible feedback between the mobile grains on the flow. This is in contrast to the wind velocity 'focal point' that exists when saltation takes place over an erodible granular bed where the saltating grains comprise a momentum sink on the overlying flow (Bagnold, 1937;Ungar & Haff, 1987;Creyssels et al., 2009;Durán et al., 2011;Ho et al., 2014;Valance et al., 2015). Here, we propose a new mode for sand patch and protodune initiation associated with the sensitive nature of the transport law in response to changing bed conditions. We find that sand transport rates responding to non-erodible to erodible bed conditions can explain the emergence of isolated, meter-scale sand patches on gravelly interdune areas or moist beaches ( Figure 1). Our field data in support of this process, quantitatively capturing the emergence of a sand patch and the change in saltation this produces, allows us to propose a conceptual model for early-stage protodune growth from a flat bed. Methods Sediment transport measurements were undertaken in the Skeleton Coast National Park, Namibia on sand and gravel surfaces between the 13th and 15th September 2019. Here, wind speed was measured simultaneously on both surfaces using hotwire anemometers (DANTEC 54T35 probes) at a height of 0.085 m and a frequency of 0.1 Hz. Co-located sediment transport was measured via laser particle counters (Wenglor YH03PCT8, following the methods of Barchyn et al. (2014)), Sensit contact particle counters and modified Bagnold sand traps. Saltation height was measured, using a Leica P20 terrestrial laser scanner (TLS) following the methods of , in a 1 m 2 area immediately upwind of the wind and sand transport instrument arrays, alternating between each of the gravel and sand sites. Additional measurements were undertaken to quantify both saltation height and surface topographic change during the initial formation of a sand patch using Leica P20 and P50 TLS instruments placed downwind of an emerging patch at Great Sand Dunes National Park, Colorado, USA on the 15th April 2019. Details on the data processing methods can be found in the Supplementary Information. Evidence for Differing Sand Transport Processes on Surfaces with Different Erodibility Our measurements show evidence of different particle behavior over the erodible and non-erodible beds. We find that the saltation height on the erodible surface is invariant with wind velocity whereas it increases with wind velocity on the non-erodible surface, as has been noted by other researchers (Bagnold, 1937(Bagnold, , 1941Creyssels et al., 2009;Ho et al., 2012;Martin & Kok, 2017, Figure 2a). This field measured saltation height behavior then drives a change in sediment transport law on the erodible and non-erodible surface, as confirmed by our three independent measures of sand transport: a vertical array of Wenglor laser counters (Figure 2b Figures 2 b, c, and d show that for a given wind velocity, the amount of sand transported over the non-erodible surface is greater than that transported over the erodible surface. According to Bagnold (1937), the velocity of saltating grains over the erodible bed is independent of the wind velocity, and consequently the sand flux over an erodible surface scales quadratically with the wind speed (Ungar & Haff, 1987;Werner, 1990, orange dashed lines Figure 2b and d). However, over the non-erodible bed, the particle velocity increases with wind velocity, thereby establishing a cubic dependence of sand transport on wind velocity (Ho et al., 2011, black dashed lines Figure 2b and d). Two equations can thus be proposed to fit our datasets: Q sat = p Q ref u 2 − u 2 t u 2 t ,(1) for the erodible surface datasets, and, Q sat = p Q ref u 2 − u 2 t u 2 t u u t ,(2) for the non-erodible surface datasets, with Q ref as the reference flux that is dependent on the sand characteristics, u t , the threshold velocity, and p, a fitting parameter (see Supplementary Information for details on values for each measurement method). Because of this change in transport law, to respect mass balance, the transition from nonerodible to erodible bed should thus generate sand deposition. Based on our field measurements, we propose a conceptual model to explain the emergence of an isolated sand patch on a flat, non-erodible bed with limited sand availability. We consider a flat, non-erodible surface (represented in black on Figure 3a) adjacent to an erodible zone (in orange). Due to this change in surface characteristics, and according to equations 1 and 2, a drop in the saturated sand flux at the transition from the non-erodible to erodible surface should occur (blue line on Figure 3a). However, the flux does not adjust instantaneously to its new saturated value, but responds with a characteristic relaxation length, called the saturation length L sat , to reach Q sat (Sauermann et al., 2001;Andreotti et al., 2010;Pähtz et al., 2013;Selmani et al., 2018). The red line represents this decrease in sand flux downwind of the non-erodible/erodible bed boundary ( Figure 3b). To respect mass balance, the excess sand transported on the non-erodible surface must deposit at the non-erodible/erodible transition following the decrease in sand flux over L sat , which thereby leads to the formation of a sand deposit (Figure 3b). The rapid decrease in sand flux at the transition from a non-erodible to erodible surface (red line) thus generates a sand patch with an asymmetric shape, possessing a sharp upwind edge with a smooth downwind tail (Figure 3b). This simple conceptual model assumes a constant wind velocity above threshold, and a sharp transition from a non-erodible to erodible surface. In the next section, we compare qualitatively the topography of an incipient bedform in the field to the idealized patch presented in Figure 3b. Field Evidence Sand transport measurements over a centimeter-high initial sand patch are challenging in the field as the placement of instruments can modify or destroy the emerging bedform by disrupting the windflow. Consequently, we measure concurrently the topography of an emerging sand patch and the saltation layer height with a non-invasive TLS. According to the measurements presented in Figure 2a, we can use the dependence of the saltation layer height upon the wind velocity as a proxy for the appropriate transport law. To confirm that the change in sand flux acts as a driver for sand patch initiation, we measured the topography and saltation layer height pre-(black) and post-(orange) emergence of a sand patch on a sediment availability-limited, non-erodible surface (Figure 4; field site and method are described in Supplementary Information). Figure 4 shows the height of the saltation layer is constant above the non-erodible surface, whereas it decreases over the developing patch due to its erodible sand surface. When sand particles start to travel over the erodible surface, each grain impact with the bed generates a particle ejection (splash effect), so that this process is consumes energy. Consequently, saltating particles lose energy and experience a lower jump height, causing a decrease in the height of the saltation layer (Bagnold, 1937;Ho et al., 2012Ho et al., , 2014Valance et al., 2015). As predicted by our conceptual model (Figure 3), the observed initial sand patch exhibits a reverse asymmetry, with the steepest slope at the upwind edge. Our field measurements ( Figure 4) show a rapid decrease in saltation height from the upwind edge of the patch to a distance 1.4±0.3 m downwind of the patch toe. According to our conceptual model, sand deposition occurs over the saturation length. Although the relationship between L sat and the grain diameter is still a matter of debate (Pähtz et al., 2013;Pähtz & Durán, 2017;Selmani et al., 2018), here we follow Andreotti et al. (2010) to estimate L sat as L sat ≈ 2.2 ρ s ρ d(3) At the Great Sand Dunes field site, the grain size is d=350 ± 50 µm, mass density is ρ s =2650 kg m −3 , and the air density ρ = 1.2 kg m −3 that yields a saturation length of 1.7 ± 0.25 m, in good agreement with our field measurements. This therefore suggests that the saturation length sets the length of the incipient sand patch. Discussion and Conclusions Combining field measurements and a simple physically-based model, we propose a mechanism to explain the initiation of aeolian sand patches where there is limited sand availability. A change in surface characteristics (erodible/non-erodible or dry/moist) is critical, and leads to a modification of the sand transport dynamics. In agreement with previous studies, we show that the quantity of transported sand, and height of particle saltation, drops when encountering an erodible surface. The corresponding decrease in sand flux generates deposition in order to satisfy mass balance, thus adding sediment to the patch. Moreover, our field measurements demonstrate that the saturation length controls the size of the emerging deposit associated with the spatial relaxation of flux. Besides a change in surface mobility, the second critical parameter controlling sand patch emergence is the incoming sand flux. In our conceptual model, we assume the incoming sand flux equals the saturated sand flux associated with the non-erodible surface. However, the value of incoming flux depends largely on the sand source availability upwind of the initial patch. Without appropriate sand supply, such incipient bedforms are likely to degrade rapidly (Lancaster, 1996;Nield, 2011). The majority of sand patches develop in interdune areas (Lancaster, 1996) and beaches (Hesp & Arens, 1997;Baddock et al., 2018;Hage et al., 2018;Montreuil et al., 2020), and in these cases sand sources are provided by the surrounding dry sandy surfaces. However, in the case of a succession or field of patches, if all the excess sand is deposited on the upwind erodible surfaces (as in the case of our conceptual schematics), then sediment supply would be further reduced to downwind patches. This condition likely creates a control on sand feeding of downwind patches and suggests there is a role for temporal wind fluctuations, both in strength and direction, in maintaining a broad field of multiple sand patches. As sand starts to be deposited, the initial bedform will interact with the wind flow and consequently the downwind variation of the sand flux will depend not only on the nature of the substrate (erodible/ non-erodible) but also on the underlying and developing topography du Pont, 2015;Bristow et al., 2022). Consequently, to develop the conceptual arguments presented herein and investigate the conditions under which the aeolian sand patch is most likely to evolve, the present model needs further development to include full coupling between wind, transport and topography. In order to examine propagative solutions in a simplified dune model that accounted for these couplings, Andreotti et al. (2002b) identified flat bedform profiles without slipfaces (patches), but these solutions did not account for the change of transport law when bed conditions varied. However, these results did show the necessity of an incoming flux for these solutions to exist. The present study shows, for the first time, that it is possible to develop a sand patch on a non-erodible surface without any additional perturbation from the topography of the bed, and opens the way for study of the evolution of isolated sand patches towards larger bedforms and fully developed dunes (Kocurek et al., 1992;Bristow et al., 2022). Data Availability The data used in this manuscript can be found in the NERC National Geological Data Center: Huab river valley dataset (https://doi.org/10.5285/99e4446f-c43a-492d-83c9-e896206649c0, Nield et al., 2022a) and Great Sand Dunes National Park dataset (https://doi.org/10.5285/46e9ff95-27ca-4d3b-b587-fc9ce22c5781, Nield et al., 2022b). Supplementary figures and text can be found in the supporting information. MET andNCRST (permits 1913/2014;2051/20152168/2016. Data processing used the IRIDIS Southampton Computing Facility. J. M. Nield was supported by a Department of Geology and Geophysics, Texas A&M University, Michel T Halbouty Visiting Chair during the GSD field campaign. We thank B. Andreotti, C. Gadal, C. Narteau and TOAD project partners for useful discussions. We also thank Patrick Hesp and an anonymous reviewer for their careful reading of our manuscript and their insightful comments and suggestions. Acknowledgments Supporting Information for "Field evidence for the initiation of isolated aeolian sand patches" Introduction Our study involved two field campaigns. Data collected in the Skeleton Coast National Park, Namibia, was used to confirm the differences in sand transport dynamics on erodible and non-erodible surfaces. The saltation dynamics that occurred during the emergence of a sand patch on a non-erodible surface were measured at Great Sand Dunes National Park, Colorado, USA. The following section contain the description of the field sites and the methods used to collect and analyze the data. Text S1-Field sites The study area in the Huab Valley dune field of the Skeleton Coast National Park, northwest Namibia, consisted of a flat surface covered with gravel and an ephemeral sand patch aligned with the predominant south-southwest winds (Lancaster, 1982). The gravel acts as an armor layer, which makes the surface non-erodible, in contrast to sand patches that are erodible. We measured sand transport, saltation height and wind velocity over both surfaces during periods of high sand transport. The measurements were performed simultaneously over both surfaces, a co-located sand patch and gravel surface from the 13 th to the 15 th September 2019. The Medano Creek site, close to the Visitor Center at Great Sand Dunes National Park, Colorado, USA, consisted of a moist sand surface, adjacent to an ephemeral creek bed, with sand being supplied upwind from a field of protodunes. The moisture was large enough to prevent sand being eroded from the surface (i.e. it was a non-erodible surface), but not moist enough for adhesion to occur. We measured saltation height and surface change over this surface during an hour-long period on the 15 th April 2019. 8 Text S2-Wind Velocity and Sand Transport (Skeleton Coast National Pak, Namibia) We measured the wind velocity at a height of 0.085 m with a hotwire anemometer (DANTEC 54T35 probes), at a frequency of 0.1 Hz. We used three different methods to quantify sand transport on both erodible and non-erodible surfaces. i) Five laser particle counters (Wenglor YH03PCT8) were positioned in a vertical array at height of (Figure 5a). ii) A Sensit contact particle counter, which counts sand particles within 0.03 m from the surface recorded total counts every 10 seconds. iii) A modified Bagnold sand trap sampled moving sand between 0 to 0.5 m above the surface over a time period varying between 10 and 30 minutes (Figure 6a). The sand traps allowed us to quantify the physical characteristics of the transported sand. We measured the grain size using a laser granulometer (Malvern Mastersizer 3000), and the density of the sand using a pycnometer. We then used this particle size distribution to convert particle counts from the Wenglors into sand flux following the method of Barchyn et al. (2014). All the datasets are subdivided into 30-sec intervals. The methods used to estimate the sand flux are described below: • Wenglor array: The particle count recorded by each Wenglor instrument is converted into sand flux using the method proposed by Barchyn et al. (2014). For each time period, we integrated the sand flux density at each elevation, q, using the corresponding exponential law to obtain the sand flux, Q (equation 4 and Figure 5b). Q = z=∞ z=0 q 0 exp −z z 0 dz,(4) where q 0 and z 0 are the parameters of the exponential fit. • Bagnold sand traps: We positioned four Bagnold traps approximately 50 cm from each other and spanwise to incoming sand transport. Each trap comprised 25 slots of 2 cm-height. By measuring the amount of sand trapped in each slot, we could estimate the sand flux density at each elevation, q. We finally fit the exponential law (equation 4) to the averaged sand flux (averaged over the four Bagnold traps), to calculate the sand flux, Q ( Figure 6). • Sensit-piezoelectric counter: The Sensit piezoelectric counter detects particle impacts from all directions between 0 and 3 cm above the surface. We converted the a b Sand flux density (kg m -2 s -1 ) Figure 6. a) Frontal view of Bagnold traps in the field. b) Sand flux measured at each elevation (red dots), the hatched area is the total flux Q, and the blue area is the error calculated from the exponential fit. particle counts into a flux using: Q = nV ρ A h,(5) where n is the counts per second, V the volume of each sand particle, A the area of the Sensit and h the measurement height. To compare transport at similar wind speeds on different surfaces, we concatenated all the data to generate datasets for erodible and non-erodible surfaces that we then binned according to the wind velocity, with an increment of 0.1 m s −1 . Finally, we fitted the transport law, Equation (1) and (2), to the two datasets, erodible and non-erodible, respectively. The reference flux Q ref is defined as, Q ref = ρ s d ρ s ρ gd,(6) equals 1.8 kg m −1 s −1 . Using the Wenglor counter dataset, we find a threshold velocity u t = 6.2 m s −1 (at 0.085 m), and the fitting parameter p equals 0.004 and 0.009 respectively for the erodible and non-erodible surfaces. For the Sensit datasets, we obtain u t = 5.5 m s −1 and p equals 0.0007, and 0.0014 respectively for the erodible and nonerodible surfaces. Due to the intermittency of the measurement with the Bagnold traps, there was insufficient data to fit a transport law. However, the Bagnold trap data show qualitatively the difference in transport capacity over each surface. Text S3-Maximum Saltation Height and Surface Change (Skeleton Coast National Park and Great Sand Dunes National Park) We measured the maximum saltation height over each surface using terrestrial laser scanning (TLS), following the methods of . An area of approximately 1 m 2 was scanned using a Leica P20 TLS over a 3-minute period, and a filter of 35 was applied to separate laser returns from above, and on, the surface. Data was gridded in 0.01 m 2 to minimize the impact of mixed pixels or large-scale topography. Within each scan, saltation heights were obtained for each grid square based on the maximum height of the laser returns above the surface and the minimum height of the surface. The maximum value for each scan was assigned a mean velocity from the hotwire anemometer. Data were then binned for each surface using 0.3 m s −1 velocity brackets, with a minimum of four points in each bracket and points where the standard deviation was greater than the mean value within the bracket were excluded. Additionally, at Great Sand Dunes National Park we used two TLS instruments, a Leica P20 and a Leica P50. As the sand patch that developed was the same order of magnitude in height as the saltating grains, we first concatenated four consecutive scans of the surface and applied the saltation filter to maximize the chance that scanning of the surface was not occluded by saltation. We then use each individual scan to plot the surface (gridded at 0.0001 m 2 ) and saltation (gridded at 0.01 m 2 ) at the start and end of the measurement period (1 hour). 10 Text S4-Estimation of the Saturation Length from the Evolution of the Saltation Layer Height The transition appears as a decreasing sigmoidal curve (Figures 4 and 7). To quantify this observation, we fit a hyperbolic tangent to the height profile, defined as, H = (H ne − H e ) tanh(−(x − x 0 )/l + 1 2 + H e , where H e and H ne are the normalized saltation height over the erodible and nonerodible surface, x 0 is the location of the transition and l is the exponential length. We find that the normalized saltation height plateaus at about 0.67 on the nonerodible surface (H ne ), and at about 0.08 on the erodible surface (H e ). The transition between these two areas is a smooth transition with a characteristic length 2l = 1.4± 0.3 m. Figure 1 . 1downwind of crest (m) Streamwise distance downwind of crest (m) Streamwise distance downwind of crest (m) Sand patches formed on different surfaces. Brancaster beach Norfolk, UK (a, d and g), Helga's dunefield, Namib Desert, Namibia (b, e and h), and Medano Creek, Great Sand Dunes National Park, Colorado, USA (c, f and i). ), Bagnold type sand traps (Figure 2c), and Sensit piezoelectric counters (Figure 2d). Figure 2 . 2Saltation height (a) and sediment flux (Q) as a function of wind velocity on both surfaces, as measured from Wenglor vertical array (b), Bagnold trap (c), and Sensit counters (d Figure 3 . 3Conceptual model for emergence of a sand patch driven by change in sand transport in the case of limited sand availability surface. (a) Pre-deposition state with the associated potential saturated sand flux (blue line). (b) Post-deposition state, with red line representing the actual sand flux. Figure 4 . 4TLS measured surface over an hour during the initial development of a sand patch and the corresponding relative saltation height over the same surface. Measurements were undertaken in the Great Sand Dunes National Park. The average wind speed measured at 0.1m above the surface during the experiment was 6.35 m s −1 . Relative saltation height is normalized by the maximum saltation height within each x-minute measurement period (the methods are detailed in the Supplementary Information). This work was funded by the TOAD (The Origin of Aeolian Dunes) project (funded by the Natural Environment Research Council, UK and National Science Foundation, USA; NE/R010196NSFGEO-NERC, NSF-GEO-1829541 and NSF-GEO-1829513). Research was undertaken at GSD under a Scientific Research and Collection permit GRSA-2018-SCI-004, and we are very grateful for support from A. Valdez and F. Bunch. For the Huab fieldwork, we acknowledge Gobabeb Namib Research Institute, J. Kazeurua, I. Matheus, L. 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Particle velocity dis- tribution in saltation transport. Physical Review E , 85 (5), 052301. Retrieved from https://doi.org/10.1103/PhysRevE.85.052301 Scaling laws in aeolian sand transport. T D Ho, A Valance, P Dupont, A O Moctar, 10.1103/PhysRevLett.106.094501Physical Review Letters. 106994501Ho, T. D., Valance, A., Dupont, P., & El Moctar, A. O. (2011). Scaling laws in aeo- lian sand transport. Physical Review Letters, 106 (9), 094501. doi: https://doi .org/10.1103/PhysRevLett.106.094501 Aeolian sand transport: Length and height distributions of saltation trajectories. T D Ho, A Valance, P Dupont, A O Moctar, 10.1016/j.aeolia.2013.11.004Aeolian Research. 12Ho, T. D., Valance, A., Dupont, P., & El Moctar, A. O. (2014). Aeolian sand transport: Length and height distributions of saltation trajectories. Aeolian Research, 12 , 65-74. doi: https://doi.org/10.1016/j.aeolia.2013.11.004 Scaling laws in aeolian sand transport under low sand availability. S Kamath, Y Shao, E J R Parteli, 10.1029/2022GL097767Geophysical Research Letters. 4911Kamath, S., Shao, Y., & Parteli, E. J. R. (2022). Scaling laws in aeolian sand transport under low sand availability. Geophysical Research Letters, 49 (11), e2022GL097767. doi: https://doi.org/10.1029/2022GL097767 . G Kocurek, M Townsley, E Yeh, K G Havholm, M L Sweet, Kocurek, G., Townsley, M., Yeh, E., Havholm, K. G., & Sweet, M. L. (1992). Dune and dune-field development on padre island, texas, with implications for interdune deposition and water-table-controlled accumulation. 10.1306/D4267974-2B26-11D7-8648000102C1865DJournal of Sedimentary Research. 624Dune and dune-field development on padre island, texas, with implications for interdune deposition and water-table-controlled accumulation. Journal of Sedimentary Research, 62 (4), 622-635. doi: https://doi.org/10.1306/ D4267974-2B26-11D7-8648000102C1865D Dunes on the skeleton coast, namibia (south west africa): geomorphology and grain size relationships. Earth surface processes and landforms. N Lancaster, 10.1002/esp.32900706067Lancaster, N. (1982). Dunes on the skeleton coast, namibia (south west africa): geomorphology and grain size relationships. Earth surface processes and land- forms, 7 (6), 575-587. doi: https://doi.org/10.1002/esp.3290070606 Field studies of sand patch initiation processes on the northern margin of the namib sand sea. N Lancaster, 10.1002/(SICI)1096-9837Earth Surface Processes and Landforms. 2110 947::AID-ESP634 3.0.CO;2-7Lancaster, N. (1996). Field studies of sand patch initiation processes on the north- ern margin of the namib sand sea. Earth Surface Processes and Landforms, 21 (10), 947-954. doi: https://doi.org/10.1002/(SICI)1096-9837(199610)21: 10 947::AID-ESP634 3.0.CO;2-7. Direct validation of dune instability theory. Proceedings of the National Academy of Sciences. P Lü, C Narteau, Z Dong, P Claudin, S Rodriguez, Z An, 10.1073/pnas.20241051181182024105118Courrech du PontLü, P., Narteau, C., Dong, Z., Claudin, P., Rodriguez, S., An, Z., . . . Courrech du Pont, S. (2021). Direct validation of dune instability theory. Proceed- ings of the National Academy of Sciences, 118 (17), e2024105118. doi: https://doi.org/10.1073/pnas.2024105118. . R L Martin, J F Kok, Martin, R. L., & Kok, J. F. (2017). Wind-invariant saltation heights imply linear scaling of aeolian saltation flux with shear stress. 10.1126/sciadv.1602569Science advances. 361602569Wind-invariant saltation heights imply lin- ear scaling of aeolian saltation flux with shear stress. Science advances, 3 (6), e1602569. doi: https://doi.org/10.1126/sciadv.1602569 . A.-L Montreuil, M Chen, E Brand, A De Wulf, L De Sloover, S Dan, T Verwaest, Montreuil, A.-L., Chen, M., Brand, E., De Wulf, A., De Sloover, L., Dan, S., & Verwaest, T. (2020). Early-stage aeolian dune development and dynamics on the upper-beach. 336-340.doi:Doi:10.2112/SI95-065.1Journal of Coastal Research. 95SIEarly-stage aeolian dune development and dynam- ics on the upper-beach. Journal of Coastal Research, 95 (SI), 336-340. doi: Doi:10.2112/SI95-065.1 A wind tunnel study of the influence of pore water on aeolian sediment transport. C M Neuman, M M Scott, 10.1006/jare.1997.0371Journal of Arid Environments. 393Neuman, C. M., & Scott, M. M. (1998). A wind tunnel study of the influence of pore water on aeolian sediment transport. Journal of Arid Environments, 39 (3), 403-419. doi: https://doi.org/10.1006/jare.1997.0371 Surface moisture-induced feedback in aeolian environments. J M Nield, 10.1130/G32151.1Geology. 3910Nield, J. M. (2011). Surface moisture-induced feedback in aeolian environments. Geology, 39 (10), 915-918. doi: https://doi.org/10.1130/G32151.1 The application of terrestrial laser scanning to aeolian saltation cloud measurement and its response to changing surface moisture. J M Nield, G F S Wiggs, 10.1002/esp.2102Earth Surface Processes and Landforms. 36Nield, J. M., & Wiggs, G. F. S. (2011). The application of terrestrial laser scanning to aeolian saltation cloud measurement and its response to changing surface moisture. Earth Surface Processes and Landforms, 36 (2), 273-278. doi: https://doi.org/10.1002/esp.2102 Surface and meteorological data at huab river valley, skeleton coast national park. J M Nield, G F S Wiggs, M C Baddock, P Delorme, Nield, J. M., Wiggs, G. F. S., Baddock, M. C., & Delorme, P. (2022a). Surface and meteorological data at huab river valley, skeleton coast national park, namibia in september 2019. . 10.5285/99e4446f-c43a-492d-83c9-e896206649c0NERC EDS National Geoscience Data Centre. (DatasetNERC EDS National Geoscience Data Centre. (Dataset). doi: https://doi.org/10.5285/99e4446f-c43a-492d-83c9-e896206649c0 Surface and meteorological data at medano creek, great sand dunes national park, colorado, usa on 15th. J M Nield, G F S Wiggs, M C Baddock, P Delorme, 10.5285/46e9ff95-27ca-4d3b-b587-fc9ce22c5781NERC EDS National Geoscience Data Centre. (Dataset). doi. Nield, J. M., Wiggs, G. F. S., Baddock, M. C., & Delorme, P. (2022b). Surface and meteorological data at medano creek, great sand dunes national park, colorado, usa on 15th april 2019. NERC EDS National Geoscience Data Centre. (Dataset). doi: https://doi.org/10.5285/46e9ff95-27ca-4d3b-b587 -fc9ce22c5781 Aeolian sand strip mobility and protodune development on a drying beach: examining surface moisture and surface roughness patterns measured by terrestrial laser scanning. J M Nield, G F S Wiggs, R S Squirrell, Doi:10.1002/esp.2071Earth Surface Processes and Landforms. 36Nield, J. M., Wiggs, G. F. S., & Squirrell, R. S. (2011). Aeolian sand strip mobility and protodune development on a drying beach: examining surface moisture and surface roughness patterns measured by terrestrial laser scanning. Earth Surface Processes and Landforms, 36 (4), 513-522. doi: Doi:10.1002/esp.2071 Fluid forces or impacts: What governs the entrainment of soil particles in sediment transport mediated by a newtonian fluid?. T Pähtz, O Durán, 10.1103/PhysRevFluids.2.074303Physical Review Fluids. 2774303Pähtz, T., & Durán, O. (2017). Fluid forces or impacts: What governs the en- trainment of soil particles in sediment transport mediated by a newtonian fluid? Physical Review Fluids, 2 (7), 074303. doi: https://doi.org/10.1103/ PhysRevFluids.2.074303 Flux saturation length of sediment transport. T Pähtz, J F Kok, E J R Parteli, H J Herrmann, 10.1103/PhysRevLett.111.218002Physical review letters. 11121218002Pähtz, T., Kok, J. F., Parteli, E. J. R., & Herrmann, H. J. (2013). Flux saturation length of sediment transport. Physical review letters, 111 (21), 218002. doi: https://doi.org/10.1103/PhysRevLett.111.218002 . O Rozier, C Narteau, C Gadal, P Claudin, S Courrech Du Pont, Rozier, O., Narteau, C., Gadal, C., Claudin, P., & Courrech du Pont, S. (2019). Elongation and stability of a linear dune. 10.1029/2019GL085147Geophysical Research Letters. 4624Elongation and stability of a linear dune. Geophysical Research Letters, 46 (24), 14521-14530. doi: https://doi.org/10.1029/2019GL085147 Continuum saltation model for sand dunes. G Sauermann, K Kroy, H J Herrmann, 10.1103/PhysRevE.64.031305Physical Review E. 64331305Sauermann, G., Kroy, K., & Herrmann, H. J. (2001). Continuum saltation model for sand dunes. Physical Review E , 64 (3), 031305. doi: https://doi.org/10.1103/ PhysRevE.64.031305 Aeolian sand transport in out-of-equilibrium regimes. H Selmani, A Valance, A Ould El Moctar, P Dupont, R Zegadi, 10.1002/2017GL076937Geophysical Research Letters. 454Selmani, H., Valance, A., Ould El Moctar, A., Dupont, P., & Zegadi, R. (2018). Ae- olian sand transport in out-of-equilibrium regimes. Geophysical Research Let- ters, 45 (4), 1838-1844. doi: https://doi.org/10.1002/2017GL076937 Steady state saltation in air. J E Ungar, P K Haff, 10.1111/j.1365-3091.1987.tb00778.xValanceThe physics of aeolian sand transport. A., Rasmussen, K. R., El Moctar, A. O., & Dupont, P.34Ungar, J. E., & Haff, P. K. (1987). Steady state saltation in air. Sedimentology, 34 (2), 289-299. doi: https://doi.org/10.1111/j.1365-3091.1987.tb00778.x Valance, A., Rasmussen, K. R., El Moctar, A. O., & Dupont, P. (2015). The physics of aeolian sand transport. . 10.1016/j.crhy.2015.01.006Comptes Rendus Physique. 161Comptes Rendus Physique, 16 (1), 105-117. doi: https://doi.org/10.1016/j.crhy.2015.01.006 A steady-state model of wind-blown sand transport. A Warren, S1, B T Werner, 10.1086/629371The Journal of Geology. 3251Process in geomorphologyWarren, A. (1979). Aeolian processes. Process in geomorphology, 325 , S1. Werner, B. T. (1990). A steady-state model of wind-blown sand transport. The Journal of Geology, 98 (1), 1-17. doi: https://doi.org/10.1086/629371 7.17 dune morphology and dynamics. G Wiggs, 10.1016/B978-0-12-818234-5.00073-Academic PressTreatise on GeomorphologySecond EditionWiggs, G. (2021). 7.17 dune morphology and dynamics. Treatise on Geomorphol- ogy (Second Edition), Academic Press. doi: https://doi.org/10.1016/B978-0-12 -818234-5.00073-

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{'abstract': 'Key Points:• Sand patches can emerge on non-erodible surfaces. • Differing surface characteristics control particle behaviour. • Field measurements demonstrate the key role of sand transport in bedform initiation.AbstractSand patches are one of the precursors to early-stage protodunes and occur widely in both desert and coastal aeolian environments. Here we show field evidence of a mechanism to explain the initiation of sand patches on non-erodible surfaces, such as desert gravels and moist beaches. Changes in sand transport dynamics, directly associated with the height of the saltation layer and variable transport law, observed at the boundary between non-erodible and erodible surfaces lead to sand deposition on the erodible surface. This explains how sand patches can form on surfaces with limited sand availability where linear stability of dune theory does not apply. This new mechanism is supported by field observations that evidence both the change in transport rate over different surfaces and in-situ patch formation that leads to modification of transport dynamics at the surface boundary.Plain Language SummarySand patches can be observed in various environments such as beaches and gravel plains in deserts. Expected to be precursors of dunes when sediment supply is limited, these bedforms are typically a few centimeters high and present a reverse longitudinal elevation profile, with a sharp upwind edge and a smooth downwind tail. Based on field measurements, we propose a formation mechanism for these patches associated with the sensitive nature of wind-blown sand transport to changing bed conditions: sand saltation is reduced at the transition from a solid to an erodible surface, hence favouring deposition on the patches. This allows us to explain their typical meter-scale length as well as their asymmetric shapes.', 'arxivid': '2302.09895', 'author': ['P Delorme \nSchool of Geography and Environmental Science\nUniversity of Southampton\nSouthamptonUK\n\nnow at: Energy and Environment Institute\nUniversity of Hull\nHullUK\n', 'J M Nield \nSchool of Geography and Environmental Science\nUniversity of Southampton\nSouthamptonUK\n', 'G F S Wiggs \nSchool of Geography and the Environment\nUniversity of Oxford\nOxfordUK\n', 'M C Baddock \nGeography and Environment\nLoughborough University\nLoughboroughUK\n', 'N R Bristow \nMechanical Engineering\nSt Anthony Falls Laboratory\nUniversity of Minnesota\nMinneapolisUSA\n', 'J L Best ', 'K T Christensen ', 'P Claudin \nPhysique et Mécanique des Milieux Hétérogènes, CNRS, ESPCI Paris\nPSL Research University\nUniversité Paris Cité\nSorbonne UniversitéParisFrance\n'], 'authoraffiliation': ['School of Geography and Environmental Science\nUniversity of Southampton\nSouthamptonUK', 'now at: Energy and Environment Institute\nUniversity of Hull\nHullUK', 'School of Geography and Environmental Science\nUniversity of Southampton\nSouthamptonUK', 'School of Geography and the Environment\nUniversity of Oxford\nOxfordUK', 'Geography and Environment\nLoughborough University\nLoughboroughUK', 'Mechanical Engineering\nSt Anthony Falls Laboratory\nUniversity of Minnesota\nMinneapolisUSA', 'Physique et Mécanique des Milieux Hétérogènes, CNRS, ESPCI Paris\nPSL Research University\nUniversité Paris Cité\nSorbonne UniversitéParisFrance'], 'corpusid': 255742459, 'doi': '10.1029/2022gl101553', 'github_urls': [], 'n_tokens_mistral': 15851, 'n_tokens_neox': 13023, 'n_words': 7002, 'pdfsha': 'd28cabef81bead980a3e6cd06065351a2993e699', 'pdfurls': ['https://export.arxiv.org/pdf/2302.09895v1.pdf'], 'title': ['Field evidence for the initiation of isolated aeolian sand patches', 'Field evidence for the initiation of isolated aeolian sand patches'], 'venue': []}

arxiv

Exact-exchange Kohn-Sham potential, surface energy, and work function of jellium slabs C M Horowitz Donostia International Physics Center (DIPC) E-20018San SebastianSpain Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas, (INIFTA) UNLP CCT La Plata-CONICET Sucursal 4, Casilla de Correo 16, 1900) La PlataArgentina C R Proetto Institüt für Teoretische Physik Freie Universitat Berlin Arnimallee 14D-14195BerlinGermany J M Pitarke CIC nanoGUNE Consolider Basque Country Mikeletegi Pasealekua 56, E20009DonostiaSpain Materia Kondentsatuaren Fisika Saila and Centro Física Materiales, CSIC-UPV/EHU 644 Posta KutxatilaE-48080Bilbo, Basque CountrySpain Exact-exchange Kohn-Sham potential, surface energy, and work function of jellium slabs 10.1103/PhysRevB.78.085126͑Received 22 May 2008; revised manuscript received 22 July 2008; published 25 August 2008͒and European Theoretical Spectroscopy Facility (ETSF) Exact-exchange self-consistent calculations of the Kohn-Sham potential, surface energy, and work function of jellium slabs are reported in the framework of the optimized effective potential ͑OEP͒ scheme of density functional theory. In the vacuum side of the jellium surface and at a distance z that is larger than the slab thickness, the exchange-only Kohn-Sham potential is found to be imagelike ͑ϳ−e 2 / z͒ but with a coefficient that differs from that of the classical image potential V im ͑z͒ =−e 2 / 4z. The three OEP contributions to the surface energy ͑kinetic, electrostatic, and exchange͒ are found to oscillate as a function of the slab thickness, as occurs in the case of the corresponding calculations based on the use of single-particle orbitals and energies obtained in the local-density approximation ͑LDA͒. The OEP work function presents large quantum size effects that are absent in the LDA and which reflect the intrinsic derivative discontinuity of the exact Kohn-Sham potential. I. INTRODUCTION The analysis of the electronic structure of metal surfaces poses a big theoretical challenge; a suitable calculational tool is needed for large, interacting, and strongly inhomogeneous many-electron systems. More than 30 years since its first application by Lang and Kohn 1,2 to the surface problem, little doubt exists that one method of choice for fulfilling this goal is density functional theory ͑DFT͒. 3,4 DFT aims to a microscopic understanding of atoms, molecules, clusters, surfaces, and bulk solids starting from the fundamental laws of quantum mechanics. In the Kohn-Sham 5 ͑KS͒ implementation of DFT, the complicated many-body problem is mapped to an effective single-particle problem, with particles subjected to an effective single-particle potential ͑the KS potential͒. Although this mapping is exact, it gives no clue as to how to calculate in practice the so-called exchange-correlation ͑xc͒ contribution to the KS potential. Lang and Kohn 1,2 solved this problem by using the local-density approximation ͑LDA͒ for the surface problem. In LDA, the xc potential at each point is taken to be that of a homogeneous interacting electron gas with the local density. Since then, many authors have calculated the electronic properties of metal surfaces by using either the LDA ͑Ref. 6͒ or further elaborations that incorporate nonlocal ingredients to the unknown xc functional. 7,8 Other schemes of the computational electronicstructure tool kit available for the investigation of solid surfaces are the Fermi hypernetted-chain ͑FHNC͒ method, 9,10 the GW approximation, 11 quantum Monte Carlo ͑QMC͒, [12][13][14] and the inhomogeneous Singwi-Tosi-Land-Sjölander ͑ISTLS͒ approach. 15 In the framework of the optimized effective potential ͑OEP͒ scheme of DFT, 16,17 which had been first used in the context of atomic physics, 18 correlation is ignored altogether and the exact-exchange KS potential is obtained. Several advantages are associated with the use of the exact-exchange energy functional of DFT: ͑i͒ it corrects the self-interaction problem inherent in approximate treatments of the exchange energy 19 ͑this problem is particularly acute for localized systems such as atoms and molecules, although it is not relevant for extended systems like bulk solids and solid surfaces͒; ͑ii͒ it yields great improvements in the study of the KS eigenvalue spectrum, 20 semiconductor band structures and excitations, 21 and nonlinear optical properties; 22 ͑iii͒ it yields the correct asymptotics; 23 ͑iv͒ it reproduces the derivative discontinuity which should be present in the KS exchange potential each time the number of particles crosses through an integer value; [24][25][26][27][28] and ͑v͒ it yields the correct twodimensional ͑2D͒ exchange energy per particle in the case of a quasi-2D electron gas. 29 It is the aim of this paper to provide benchmark exact-exchange OEP calculations for jellium slabs, with the expectation that more accurate DFT schemes that include correlation be developed by starting from a well founded exchange analysis and tested once reduced to their exchange-only ͑x-only͒ counterparts. The rest of the paper is organized as follows. We give in Sec. II the general theoretical background which will be used in the following sections. Section III is devoted to a discussion of the asymptotic behavior of the exact-exchange KS potential of jellium slabs; in Secs. IV and V we give the results that we have obtained for the OEP surface energy and work-function, respectively, and in Sec. VI we present the conclusions. II. OEP APPROACH Our calculations are restricted to a jellium-slab model of metal surfaces, where the discrete character of the positive ions inside the metal is replaced by a uniform distribution of positive charge ͑the jellium͒. The positive jellium density is defined as n + ͑z͒ = nͩ d 2 − ͯz+ d 2 ͯͪ, ͑1͒ which describes a slab of width d, number density n, 30 and jellium edges at z =−d and z =0. ͑x͒ represents the Heaviside step function: ͑x͒ =1 if x Ͼ 0 and ͑x͒ =0 if x Ͻ 0. A schematic view of our jellium slab is given in Fig. 1. Besides, and for convenience for the numerical calculations, infinite barriers are located far from the jellium edges, well inside the left and right evanescent vacuum regions. We have checked that these infinite barriers are located far enough for all the numerical calculations presented here to be independent of their precise location. 31 The jellium-slab model is invariant under translations in the x-y plane, so the KS eigenfunctions can be factorized as follows: i,k ͑r͒ = e ik· ͱ A i ͑z͒, ͑2͒ where and k are the in-plane coordinate and wave vector, respectively, and A represents a normalization area. i ͑z͒ are the normalized spin-degenerate eigenfunctions for electrons in slab discrete levels ͑SDLs͒ i ͑i =1,2,...͒ with energy i . They are the solutions of the effective one-dimensional KS equation, ĥ KS i ͑z͒ i ͑z͒ = ͫ − ប 2 2m e ‫ץ‬ 2 ‫ץ‬z 2 + V KS ͑z͒ − iͬ i ͑z͒ = 0, ͑3͒ with m e as the bare electron mass. The KS potential V KS entering Eq. ͑3͒ is the sum of two distinct contributions, V KS ͑z͒ = V H ͑z͒ + V xc ͑z͒, ͑4͒ where V H ͑z͒ is the classical ͑electrostatic͒ Hartree potential, given by 32 V H ͑z͒ = − 2e 2 ͵ −ϱ ϱ dzЈ͉z − zЈ͉͓n͑zЈ͒ − n + ͑zЈ͔͒. ͑5͒ Here, n͑z͒ is the electron number density, 33 n͑z͒ = 1 2 ͚ i occ. ͑k F i ͒ 2 ͉ i ͑z͉͒ 2 , ͑6͒ where k F i = ͱ 2m e ͑ − i ͒ / ប, and = ͑n , d͒ is the chemical potential, which in turn is determined from the neutrality condition for the whole system by the condition ͚ i occ. ͑k F i ͒ 2 =2dn. V xc ͑z͒ is the nonclassical xc potential, which is obtained as the functional derivative of the so-called xc energy functional E xc ͓n͑z͔͒, 34 V xc ͑z͒ ϵ 1 A ␦E xc ͓n͑z͔͒ ␦n͑z͒ . ͑7͒ Applications of DFT typically proceed from explicit density-dependent forms of E xc , as obtained using a variety of local or semilocal approximations. However, in the last few years increasing attention has been devoted to orbitaldependent forms of E xc : E xc = E xc ͓͕ i ͖ , ͕ i ͖͔, which are only implicit functionals of the electron density n͑z͒. In this case, one resorts to the OEP method 16 or, equivalently, uses repeatedly the chain rule for functional derivatives to obtain the following expression for the xc potential of Eq. ͑7͒ ͑Ref. 35͒: V xc ͑z͒ = 1 A ͚ i occ. ͵ −ϱ ϱ dzЈ ͵ −ϱ ϱ dzЉ ϫ ͫ ␦E xc ␦ i ͑zЉ͒ ␦ i ͑zЉ͒ ␦V KS ͑zЈ͒ + c.c. ͬ ␦V KS ͑zЈ͒ ␦n͑z͒ .␦ i ͑zЈ͒ ␦V KS ͑z͒ = i ͑z͒ ͚ j͑ i͒ j ͑zЈ͒ ‫ء‬ j ͑z͒ ͑ i − j ͒ ϵ i ͑z͒G i KS ͑zЈ,z͒, ͑11͒ and KS ͑z,zЈ͒ = ͚ i occ. ͵ −ϱ ϱ dzЉ ͫ ␦n͑z͒ ␦ i ͑zЉ͒ ␦ i ͑zЉ͒ ␦V KS ͑zЈ͒ + c.c. ͬ , ͑12͒ = 1 4 ͚ i occ. ͓͑k F i ͒ 2 i ͑z͒ ‫ء‬ i ͑zЈ͒G i KS ͑zЈ,z͒ + c.c.͔, ͑13͒ where G i KS ͑zЈ , z͒ is Green's function of noninteracting KS electrons. In the calculation of KS ͑z , zЈ͒, the chain rule for functional derivatives has been used; now we are considering the density itself as a functional of the occupied SDL. In obtaining Eq. ͑13͒ from Eq. ͑12͒, we have used Eq. ͑11͒ and also that ␦n͑z͒ ␦ i ͑zЈ͒ = ␦͑z − zЈ͒ ͑k F i ͒ 2 2 i ͑z͒ ‫ء‬ , which follows from Eq. ͑6͒. Introducing Eqs. ͑11͒ and ͑13͒ into the central Eq. ͑10͒, we obtain the final and compact version of the OEP integral equation for V xc ͑z͒, ͚ i occ. S i ͑z͒ = 0, ͑14͒ where S i ͑z͒ = ͑k F i ͒ 2 ⌿ i ͑z͒ ‫ء‬ i ͑z͒ + c.c. and ⌿ i ͑z͒ = ͚ j͑ i͒ j ͑z͒ ͑ i − j ͒ ͵ −ϱ ϱ j ͑zЈ͒ ‫ء‬ ⌬V xc i ͑zЈ͒ i ͑zЈ͒dzЈ. ͑15͒ Here, ⌬V xc i ͑z͒ = V xc ͑z͒ − u xc i ͑z͒, where u xc i ͑z͒ are SDLdependent xc potentials of the form 36 u xc i ͑z͒ ϵ ͓4/A͑k F i ͒ 2 i ͑z͒ ‫ء‬ ͔␦E xc /␦ i ͑z͒. The magnitudes ⌿ i ͑z͒ are called the "shifts," as they can be physically interpreted as the first-order corrections of the KS eigenfunctions i ͑z͒ under the perturbation ⌬V xc i ͑z͒. These shifts also provide a useful and practical tool for the numerical solution of the OEP equation. 37,38 From Eq. ͑15͒, we find the orthogonality constraint between the KS eigenfunctions and the shifts: ͐ i ͑z͒ ‫ء‬ ⌿ i ͑z͒dz = 0. It is also immediate that the shifts are invariant under the replacement V xc ͑z͒ → V xc ͑z͒ + ␣, with ␣ being an arbitrary constant. This means that the above set of equations determines V xc ͑z͒ up to an additive constant, which should be fixed by imposing a suitable boundary condition. Moreover, the shifts ⌿ i ͑z͒ are easily found to satisfy the following inhomogeneous differential equation: 20 ĥ KS i ͑z͒⌿ i ͑z͒ = − ͓⌬V xc i ͑z͒ − ⌬V xc i ͔ i ͑z͒. ͑16͒ Here, mean values are defined as Ō i = ͐ i ͑z͒ ‫ء‬ O i ͑z͒ i ͑z͒dz. Equations ͑3͒-͑5͒ and ͑14͒, which determine the local V xc ͑z͒ corresponding to a given SDL-dependent E xc , form a closed system of equations ͑the OEP equations͒, which should be solved in a self-consistent way. In order to accomplish some contact with other useful versions of the OEP equations for the present problem, a few additional steps are required. First of all, we write i ͑z͒ ‫ء‬ ĥ KS i ͑z͒⌿ i ͑z͒ = − ប 2 2m e ͫ i ͑z͒ ‫ء‬ ‫ץ‬ 2 ⌿ i ͑z͒ ‫ץ‬z 2 − ⌿ i ͑z͒ ‫ץ‬ 2 i ͑z͒ ‫ء‬ ‫ץ‬z 2 ͬ , which is easily obtained from Eq. ͑3͒. Second, we multiply the left-hand side of Eq. ͑16͒ by i ͑z͒ ‫ء‬ to obtain ប 2 2m e ͫ i ͑z͒ ‫ء‬ ‫ץ‬ 2 ⌿ i ͑z͒ ‫ץ‬z 2 − ⌿ i ͑z͒ ‫ץ‬ 2 i ͑z͒ ‫ء‬ ‫ץ‬z 2 ͬ = ͓⌬V xc i ͑z͒ − ⌬V xc i ͔͉ i ͑z͉͒ 2 . ͑17͒ Then, we start from the self-evident identity, V xc ͑z͒ = ͚ i occ. ͑k F i ͒ 2 ͉ i ͑z͉͒ 2 4n͑z͒ ϫ͓u xc i ͑z͒ + ⌬V xc i + ⌬V xc i ͑z͒ − ⌬V xc i + c.c.͔, ͑18͒ we eliminate the factor ͓⌬V xc i ͑z͒ − ⌬V xc i ͔͉ i ͑z͉͒ 2 by using Eq. ͑17͒, and we obtain V xc ͑z͒ = ͚ i occ. ͑k F i ͒ 2 4n͑z͒ ͭ ͉ i ͑z͉͒ 2 ͓u xc i ͑z͒ + ⌬V xc i ͔ + ប 2 2m e ͫ i ͑z͒ ‫ء‬ ‫ץ‬ 2 ⌿ i ͑z͒ ‫ץ‬z 2 − ⌿ i ͑z͒ ‫ץ‬ 2 i ͑z͒ ‫ء‬ ‫ץ‬z 2 ͬ + c.c. ͮ . ͑19͒ Finally, we proceed with the elimination from Eq. ͑19͒ of the term proportional to ‫ץ‬ 2 ⌿ i ͑z͒ / ‫ץ‬z 2 , the subsequent elimination of ‫ץ‬ 2 i ͑z͒ ‫ء‬ / ‫ץ‬z 2 proceeds via the KS equations, and as a result of all these manipulations we obtain the following expression for the DFT xc potential: 39 V xc ͑z͒ = V xc,1 ͑z͒ + V xc,2 ͑z͒, ͑20͒ where V xc,1 ͑z͒ = ͚ i occ. ͑k F i ͒ 2 ͉ i ͑z͉͒ 2 4n͑z͒ ͕u xc i ͑z͒ + ⌬V xc i + c.c.͖ and V xc,2 ͑z͒ = − 1 2n͑z͒ ͚ i occ. ͑ − i ͓͒͑k F i ͒ 2 ⌿ i ͑z͒ i ͑z͒ ‫ء‬ + ⌿ i Ј͑z͒ i Ј͑z͒ ‫ء‬ + c.c.͔, with primes denoting derivatives with respect to the z coordinate. It is important to note that Eqs. ͑14͒ and ͑20͒ are just two different, but fully equivalent, ways to obtain the OEP xc potential for the present problem. If the shifts ⌿ i ͑z͒ are ͑arbitrarily͒ forced to be identically equal to zero, the only term that survives is V xc,1 ͑z͒. This is exactly the Krieger-Li-Iafrate 24 ͑KLI͒ approximation, which brings the identification V xc,1 ͑z͒ϵV xc KLI ͑z͒. 40 As before, Eqs. ͑3͒-͑5͒ and ͑20͒ form a closed set of equations, which should be solved self-consistently. Both exchange and correlation have been included so far. Unless stated otherwise, we will now focus on the x-only case, where E xc , V xc ͑z͒, and u xc ͑z͒ are replaced by E x , V x ͑z͒, and u x ͑z͒, respectively. We have achieved the self-consistent numerical solution of the x-only version of the OEP equations by two different methods: ͑i͒ direct calculation of the shifts of Eq. ͑15͒, by solving Eq. ͑16͒, 37 and ͑ii͒ direct solution of the OEP integral equation for V x ͑z͒, as given by the x-only version of Eq. ͑14͒. 38 Both methods yield results that agree within numerical accuracy, although the first approach is found to be computationally more efficient than the second. Both methods face numerical instabilities beyond a critical coordinate z in the vacuum region. Finally, we note that the exact-exchange energy of a jellium slab is given by the following expression: E x ͑d͒ = A 4 ͚ i occ. ͑k F i ͒ 2 ͵ −ϱ ϱ dz͉ i ͑z͉͒ 2 u x i ͑z͒, ͑21͒ where u x i ͑z͒ represent the SDL-dependent exchange potentials, u x i ͑z͒ = − 2e 2 ͑k F i ͒ 2 ͚ j occ. j ͑z͒ ‫ء‬ i ͑z͒ ‫ء‬ ͵ −ϱ ϱ dzЈ i ͑zЈ͒ ‫ء‬ g͑⌬zk F i ,⌬zk F j ͒ j ͑zЈ͒ ͑⌬z͒ 3 , ͑22͒ with ⌬z = ͉z − zЈ͉, g͑s,sЈ͒ = ssЈ ͵ 0 ϱ J 1 ͑st͒J 1 ͑sЈt͒ ͱ 1 + t 2 dt t , ͑23͒ being the "universal" ͑that is, independent of V KS ͒ function introduced by Kohn and Mattsson 41 and J 1 ͑x͒ being the firstorder cylindrical Bessel function. 42 III. ASYMPTOTICS OF THE EXACT-EXCHANGE KS POTENTIAL The long-range behavior of V xc ͑z͒ in the vacuum region is an important and open issue in DFT studies of metal surfaces. 43 The aim of this section is to present a detailed derivation of the analytical asymptotic limit of V x ͑z͒ reported in Ref. 39 for a slab geometry. First of all, we note that by making the choice that V KS ͑z → ϱ͒ → 0, Eq. ͑3͒ leads us to the conclusion that i ͑z → ϱ͒ → e −z ͱ −2m e i /ប for all occupied i ͑disregarding a factor involving powers of z͒. We also remark the following points: ͑i͒ due to the exponential decay of V H ͑z → ϱ͒, the assumption V KS ͑z → ϱ͒ → 0 implies that V x ͑z → ϱ͒ → 0; ͑ii͒ for this choice of the zero of energy, one finds i Ͻ 0 for all occupied states; ͑iii͒ the slowest decaying of all the occupied SDL corresponds to i = m, where m is the highest occupied SDL. Now we look at the asymptotic behavior of the shifts ⌿ i ͑z͒. Turning to the x-only version of Eq. ͑16͒, ͫ − ប 2 2m e ‫ץ‬ 2 ‫ץ‬z 2 + V H ͑z͒ + V x ͑z͒ − iͬ ⌿ i ͑z͒ = − V x ͑z͒ i ͑z͒ + u x i ͑z͒ i ͑z͒ + ⌬V x i i ͑z͒, ͑24͒ we focus on the asymptotic behavior of the three terms on the right-hand side ͑rhs͒ of this equation, 20 V x ͑z → ϱ͒ i ͑z → ϱ͒ → V x ͑z → ϱ͒e −z␤ i , ͑25͒ u x i ͑z → ϱ͒ i ͑z → ϱ͒ → e −z␤ m , ͑26͒ ⌬V x i i ͑z → ϱ͒ → e −z␤ i , ͑27͒ with ␤ i = ͱ −2m e i / ប. Equation ͑26͒ follows from an inspection of Eq. ͑22͒ in the limit z → ϱ; in this limit, the sum over j is exponentially dominated by the term j = m, and the result of Eq. ͑26͒ follows at once. Hence, for i m Eq. ͑24͒ yields ͫ − ប 2 2m e ‫ץ‬ 2 ‫ץ‬z 2 − iͬ ⌿ i ͑z → ϱ͒ → e −z␤ m , ͑28͒ i.e., ⌿ i ͑z → ϱ͒ → e −z␤ m . For i = m, all three terms in the rhs of Eq. ͑24͒ decay equally ͑to exponential accuracy͒, and further analysis is necessary. Equation ͑14͒ can be rewritten as follows, ͑k F m ͒ 2 ⌿ m ͑z͒ ‫ء‬ m ͑z͒ + c.c. = − ͚ i=1 m−1 ͑k F i ͒ 2 ⌿ i ͑z͒ ‫ء‬ i ͑z͒ − c.c., ͑29͒ and by studying its asymptotic limit, it is clear that its rhs can be approximated by the term i = m −1 ͑with exponential ac-curacy͒. Given that both m ͑z → ϱ͒ and ⌿ m−1 ͑z → ϱ͒ decay as e −z␤ m−1 , it follows that ⌿ m ͑z → ϱ͒ decays as m−1 ͑z → ϱ͒, that is, ⌿ m ͑z → ϱ͒ → e −z␤ m−1 . Armed with these results, the asymptotic limit of V x ͑z͒ is immediate from Eq. ͑20͒: V x,2 ͑z → ϱ͒ tends exponentially to zero, while V x,1 ͑z → ϱ͒ → u x m ͑z → ϱ͒ + ⌬V x m . ͑30͒ The leading contribution to u x m ͑z → ϱ͒ is easily obtained from Eq. ͑22͒ by considering once again that in this regime the sum over j is exponentially dominated by the term j = m. For this case, the integral over the coordinate t can be evaluated analytically, yielding u x m ͑z → ϱ͒ → − e 2 ͵ −ϱ ϱ ͉ m ͑zЈ͉͒ 2 ͉z − zЈ͉ dzЈ ϫ ͫ 1 − I 1 ͑2k F m ͉z − zЈ͉͒ k F m ͉z − zЈ͉ + L 1 ͑2k F m ͉z − zЈ͉͒ k F m ͉z − zЈ͉ ͬ , ͑31͒ where I 1 and L 1 are the modified Bessel and Struve functions, respectively. 42 Noting now that in this regime k F m ͉z − zЈ͉Ӎk F m z ӷ 1, 44 it is permissible to expand the integrand of Eq. ͑31͒ as follows: u x m ͑z → ϱ͒ → − e 2 z ͵ −ϱ ϱ ͉ m ͑zЈ͉͒ 2 dzЈ ϫ ͫ 1 + zЈ z − 2 k F m z + O ͩ 1 z 2 ͪͬ . ͑32͒ Using the normalization of the orbitals m ͑z͒, we obtain 45 Since the exchange potential V x ͑z͒ has been chosen to vanish at large distances from the surface into the vacuum ͓V x ͑ϱ͒ =0͔, Eq. ͑30͒ leads us to the important constraint, u x m ͑z → ϱ͒ → − e 2 z ͩ1+ ␤ z +¯ͪ , ͑33͒ with ␤ = z m −2/ ͑k F m ͒.⌬V x m = V x m − ū x m = 0, ͑34͒ which fixes the undetermined constant in V x ͑z͒ discussed above. All numerical results presented here have been obtained by using this constraint. From Eqs. ͑30͒, ͑33͒, and ͑34͒, we conclude that V x ͑z → ϱ͒ → V x,1 ͑z → ϱ͒ → u x m ͑z → ϱ͒ → − e 2 z ͩ1+ ␤ z +¯ͪ , ͑35͒ which is the main result of this section. At this point, we emphasize that the asymptotics dictated by Eq. ͑35͒ hold only at z coordinates that are larger than 1 / k F m . As k F m is of the order of 1 / d ͑or smaller, depending on the actual value of d͒, Eq. ͑35͒ shows that the x-only KS potential happens to be four times larger than the classical image potential ͓V im ͑z͒ =−e 2 / 4z͔ only at a distance z that is considerably larger than the slab thickness. Furthermore, the arguments leading to Eqs. ͑26͒ and ͑31͒ are only valid for a discrete slab spectrum, such that there is a finite-energy-gap between m and the remaining occupied energy levels i ͑i Ͻ m͒. An extension of the present OEP framework to treat the case of a semi-infinite jellium surface 46 is now in progress. 47 Finally, we note that under the condition V KS ͑ϱ͒ = 0 Eq. ͑35͒ for the asymptotics of V x ͑z͒ remains valid when correlation is included in the evaluation of the shifts ⌿ i ͑z͒. The point here is that the shifts are separable in their exchange and correlation components, and they also satisfy separated differential equations ͓like Eq. ͑24͒ for exchange͔. Once exchange and correlation contributions are split, the analysis of the asymptotic behavior of V x ͑z͒ follows the same lines as above, and the asymptotic limit of Eq. ͑35͒ remains the same. IV. SURFACE ENERGY In this section, surface-energy calculations are presented, as obtained at the x-only level. The surface energy is the work required, per unit area of the new surface formed, to split the crystal in two along a plane. 1 For our slab geometry, ͑d͒ = 2E͑d͒ − E͑2d͒ 2A , ͑36͒ where E͑d͒ is the total ground-state energy for each half of the slab after it is split ͑width d͒, and E͑2d͒ is the total ground-state energy of the unsplit slab ͑width 2d͒, both the split and unsplit systems with the same jellium density. Following the standard DFT energy-functional partitioning, the surface energy ͑without correlation contribution͒ can be written as the sum of three terms, 6 ͑d͒ = K ͑d͒ + el ͑d͒ + x ͑d͒, ͑37͒ where K ͑d͒ is the noninteracting kinetic contribution to the surface energy, el ͑d͒ is the electrostatic surface energy due to all noncompensated positive and negative charge distributions in the slab, and x ͑d͒ is the exchange contribution to the surface energy. From elementary physical arguments, it follows that K ͑d͒ Ͻ 0, while el ͑d͒ and x ͑d͒ are both positive. 2 Also, the stability of the slab against spontaneous fragmentation is accomplished if ͑d͒ Ͼ 0. From Eqs. ͑36͒ and ͑37͒, one writes l ͑d͒ = ͓2E l ͑d͒ − E l ͑2d͔͒/͑2A͒, ͑38͒ with l = K ,el,x, and E K ͑d͒ = Aប 2 4m e ͚ i occ. ͑k F i ͒ 2 ͫ ͑k F i ͒ 2 2 − ͵ −ϱ ϱ i ͑z͒ ‫ץ‬ 2 i ͑z͒ ‫ץ‬z 2 dz ͬ , ͑39͒ E el ͑d͒ = A 2 ͵ −ϱ ϱ V H ͑z͓͒n͑z͒ − n + ͑z͔͒dz, ͑40͒ and ͓see Eqs. ͑21͒ and ͑22͔͒ E x ͑d͒ = − e 2 A 2 ͚ i,j occ. ͵ −ϱ ϱ dz ͵ −ϱ ϱ dzЈ ϫ i ͑z͒ ‫ء‬ j ͑zЈ͒ ‫ء‬ g͑k F i ⌬z,k F j ⌬z͒ j ͑z͒ i ͑zЈ͒ ͑⌬z͒ 3 . ͑41͒ The dependence on the slab width d in Eqs. ͑39͒-͑41͒ enters through the self-consistent KS eigenvalues ͑ i ͒ and eigenfunctions ͓ i ͑z͔͒. Alternatively, one can define the effective single-slab surface energies, 48 ͑d͒ = E͑d͒ − E unif ͑d͒ 2A ͑42͒ and l ͑d͒ = ͓E l ͑d͒ − E l unif ͑d͔͒/͑2A͒, ͑43͒ where E l unif ͑d͒ is the ground-state energy of a uniform slab of electron gas of size d and l = K ,el,x. 49 Note that Eq. ͑42͒ only reproduces the surface-energy definition of Eq. ͑36͒ as d → ϱ. However, for a correct extrapolation of finite-slab calculations to the infinite-width limit, 48 here we calculate numerically the three components of the surface energy from the single-slab ͓Eq. ͑43͔͒. We have checked that the differ-ences between surface energies obtained from Eqs. ͑36͒ and ͑42͒ are quite small even for the narrowest slabs studied and that both agree in the extrapolation toward the semi-infinite limit. Being the ground-state density the basic ingredient of DFT, we found interesting to compare the differences between the different density profiles that we have obtained. We exhibit in Fig. 2 the self-consistent electron-density profiles that we have obtained within the x-only LDA and OEP schemes for r s = 2.07 and d =8 F . 50 It is expected that the amplitude of the difference between both densities diminishes as z approaches the slab center, where both n LDA ͑z → −d / 2͒ and n OEP ͑z → −d / 2͒ should approach n as d → ϱ. Figure 2 shows that there are noticeable differences between both densities: n LDA ͑z͒ extends further into the vacuum region than n OEP ͑z͒, which is a result of the LDA orbitals being more extended or "diffuse" than their OEP counterparts, and the amplitude of the Friedel oscillations near the surface is larger for n OEP ͑z͒ than for n LDA ͑z͒. We have found the same behavior for other values of r s . Figure 3 shows the results that we have obtained for the slab kinetic surface energy, as a function of the slab width d, for r s = 2.07. As in the case of the electron density, we have performed these calculations within the x-only LDA and OEP schemes. In the LDA, the kinetic surface energy K ͑d͒ ͑LDA͒ is obtained by introducing the x-only self-consistent LDA eigenfunctions i LDA ͑z͒ and eigenvalues i LDA into the formally exact Eq. ͑39͒. In the OEP, the kinetic surface energy K ͑d͒ ͑OEP͒ is obtained by using the same equation ͓Eq. ͑39͔͒ but with the LDA eigenfunctions and eigenvalues replaced by their x-only OEP counterparts i OEP ͑z͒ and i OEP . The strong oscillations in both K ͑d͒ ͑LDA͒ and K ͑d͒ ͑OEP͒ are the result of the sequential filling of empty slab discrete levels as d increases. Maxima in K ͑d͒ correspond to the onset for the filling of a new slab discrete level. For this particular case, and following the extrapolation procedure of Ref. 48, we have obtained the infinite-width extrapolated surface energies K ͑LDA͒ = −4832 erg/ cm 2 ͑as reported in Ref. 48͒ and K ͑OEP͒ = −4720 erg/ cm 2 . Figure 4 displays the results that we have obtained for the electrostatic contribution to the surface energy, as a function of the slab width d and for r s = 2.07, again within the x-only LDA and OEP schemes. The electrostatic surface energies el ͑d͒ ͑LDA͒ and el ͑d͒ ͑OEP͒ are obtained from Eq. ͑40͒ by using either the x-only LDA electron density n LDA ͑z͒ or the x-only OEP electron density n OEP ͑z͒, respectively. In this case, the onset for the filling of a new slab discrete level is always associated with a minimum. Following the extrapolation procedure of Ref. 48, we have obtained the infinitewidth surface energies indicated by arrows in Fig. 4: el ͑LDA͒ = 1172 erg/ cm 2 ͑as reported in Ref. 48͒ and el ͑OEP͒ = 1103 erg/ cm 2 . In Fig. 5, we show the results that we have obtained for the exact-exchange contribution to the slab surface energy, as a function of the slab width d and for r s = 2.07, again within the x-only LDA and OEP schemes. As in the case of the kinetic and electrostatic surface energies, exact-exchange surface energies x ͑d͒ ͑LDA͒ and x ͑d͒ ͑OEP͒ ͓both derived LDA and OEP self-consistent electron densities and its difference for d =8 F and r s = 2.07. Note that n LDA ͑z͒ is slightly more diffuse than n OEP ͑z͒, as n LDA ͑z͒ − n OEP ͑z͒ Ͼ 0 for z outside the jellium edge ͑in the vacuum͒. from the formally exact Eq. ͑41͔͒ are obtained by using either the x-only self-consistent LDA eigenfunctions i LDA ͑z͒ and eigenvalues i LDA or their x-only OEP counterparts i OEP ͑z͒ and i OEP . For comparison, we have also calculated standard LDA-exchange surface energies, 1 x LDA = 1 2 ͵ −ϱ ϱ dzn LDA ͑z͕͒ x unif ͓n LDA ͑z͔͒ − x unif ͑n͖͒, ͑44͒ where x unif ͑n͒ is the exchange energy per particle of a uniform electron gas of density n, x unif ͑n͒ =−3e 2 ͑3 2 n͒ 1/3 / ͑4͒, and n LDA ͑z͒ represents the x-only LDA electron density. All x ͑d͒ ͑LDA͒, x ͑d͒ ͑OEP͒, and x LDA ͑d͒ exhibit the characteristic oscillatory behavior also shown by the other components of the surface energy. As in the case of the electrostatic surface energy, the onset for the filling of a new slab discrete level is associated with a minimum. Figure 5 shows that while the LDA ͓see Eq. ͑44͔͒ considerably overestimates the exchange surface energy, which is a known result, the exact-exchange surface energy is not very sensitive to the actual shape of the single-particle orbitals and energies, i.e., to whether LDA or OEP orbitals are used. Following the extrapolation procedure of Ref. 48, we have obtained the infinite-width surface energies indicated by arrows in Fig. 5: x LDA = 2767 erg/ cm 2 , x ͑LDA͒ = 2390 erg/ cm 2 ͑both as reported in Ref. 48͒, and x ͑OEP͒ = 2316 erg/ cm 2 . We have also computed kinetic, electrostatic, and exchange surface energies for other values of the electrondensity parameter r s , and we have obtained the infinite-width extrapolated results shown in Table I. A comparison of the LDA and OEP calculations presented in Table I shows that ͑i͒ LDA orbitals being more delocalized than the more realistic OEP orbitals, surface energies that are based on the use of LDA orbitals are too large relative to those obtained with the use of OEP orbitals, and ͑ii͒ the sum of kinetic, electrostatic, and exchange surface energies are not very sensitive to whether LDA or OEP is used in the evaluation of the singleparticle KS eigenfunctions and eigenvalues. V. WORK FUNCTION The work function W is the minimum work that must be done to remove an electron from the metal at zero temperature. In the context of DFT, the rigorous expression for the work function for a slab of thickness d is 51 W͑d͒ = V KS ͑ϱ͒ − , ͑45͒ where is the chemical potential. We note that as we are considering an electron system that is infinite in the x-y plane, electronic relaxation effects after removal of one electron are infinitesimal. For a slab geometry, the work function becomes size dependent through the chemical potential ͑n , d͒. We are imposing the boundary condition V KS ͑ϱ͒ = 0; accordingly, W͑d͒ =− Ͼ 0. Besides, the only energy of the full KS spectrum which has a physical significance is precisely the energy of the highest occupied level, which can be identified with . 52 The work function for a slab with r s = 2.07 and d =4 F is shown schematically in Fig. 1. For this particular case, nine SDLs are occupied and is between the ninth and tenth SDLs. Now we focus on the slab-width dependence of the work function. Figure 6 shows the result of the x-only calculations that we have performed within LDA and OEP ͓W LDA and W OEP ͔ for r s = 2.07. The weakly oscillating x-only W LDA ͑d͒ is equivalent to the slab-width dependent work-function reported by Schulte a long-time ago. 53 As discussed by Schulte, the oscillations in W LDA ͑d͒ are the result of a combination of the shift of the bottom of the slab potential well and an effective film thickness shift, both effects suffering from an abrupt change each time the number of occupied SDL changes by one. The important point here, however, is the much stronger oscillations found in our W OEP ͑d͒ calculations, whose explanation is provided now with some detail. First of all, we note that, strictly speaking, the OEP work function W OEP ͑d͒ exhibits discontinuities of large size each time a new SDL becomes infinitesimally occupied. The first discontinuity in Fig. 6 appears at the 1 SDL→ 2 SDL transition ͑for d Շ F / 2͒, the second discontinuity appears at the 2 SDL→ 3 SDL transition ͑for d Շ F ͒, and so on. In order of clarify the source of such a discontinuous behavior, we have plotted in Fig. 7 the OEP exchange potential for slightly increasing values of the slab width d, around the 6 SDL → 7 SDL transition. Each slab width d is characterized by a "filling factor" of the last occupied SDL, which is defined as follows: ␣ m ϵ − m m+1 − m . ͑46͒ Hence, ␣ m → 0 + ͑implying → m + ͒ corresponds to an infinitesimally small filling of the last occupied SDL ͑i = m͒, while ␣ m → 1 − corresponds to the threshold of occupancy of the next SDL ͑i = m +1͒. The key point here is the dramatic change in V x ͑z͒ when passing from the slab thickness corresponding to ␣ 6 =1 − to the infinitesimally thicker slab corresponding to ␣ 7 =0 + ͑ϳ10 −5 ͒. The remaining curves have been obtained for slab widths corresponding to the seventh SDL being progressively occupied; as ␣ 7 increases from 0 + to 1 − , V x ͑z͒ approaches the form it had at ␣ 6 =1 − , both in depth and asymptotic behavior, the only difference being a lateral shift of V x ͑z͒ to the right that is simply due to the larger value of d. Second, we note that the potential barrier that forms at the interface, right after the jellium edge on the vacuum side of the surface, exhibits both V x,1 ͑z͒ and V x,2 ͑z͒ contributions ͓see Eq. ͑20͔͒, so the KLI approximation ͓which sets V x,2 ͑z͒ϵ0͔ cannot be used for the analysis of the characteristic discontinuous behavior of the work function. In all cases in Fig. 7, V x ͑z → ϱ͒ → 0. While this is clearly seen in the figure for the curves corresponding to ␣ 6 =1 − and ␣ 7 =1 − ͓in which case k F m ϳ 1 / d; see the asymptotics of Eq. ͑35͔͒, it is not evident at all for the set of potentials with small occupancies of the last occupied level, i.e., ␣ 7 Ӷ 1. In this case, k F m Ӷ 1 / d and the asymptotic regime only takes place at z coordinates that go to infinity ͑as ␣ 7 → 0 + ͒ far beyond the z coordinates considered in Fig. 7. This is the situation for ␣ 7 Ӎ 10 −5 , 10 −4 , and 10 −3 . As a final remark on this figure, it is important to realize that in the bulk and near the interface the exchange potentials V x ͑z͒ corresponding to ␣ 6 =1 − and ␣ 7 =0 + are simply related through a single vertical ͑constant͒ shift. This property, which can be verified numerically from Fig. 7, may also be derived analytically ͑see below͒. Finally, we note that although we have restricted our discussion to the case of a particular SDL transition, the same happens at every highest occupied→ lowest unoccupied SDL transition. With the aim of understanding how this discontinuous behavior of V x ͑z͒ versus the slab width explains the results of Fig. 6 for the work function W OEP ͑d͒, we show in Fig. 8 the slab OEP electronic structure just before occupation of the SDL 7 ͑left panel͒, that is, at the slab width corresponding to ␣ 6 → 1 − , and just after occupation of the SDL 7 ͑right panel͒, i.e., at the slab width corresponding to ␣ 7 → 0 + . We note that while the Hartree potential approaches zero outside the surface exponentially and remains essentially unaffected by the infinitesimal population of the SDL 7 ͑compare left and right panels of Fig. 8͒, the OEP exchange potential ͓and therefore V KS ͑z͒ as well͔ suffers the abrupt jump explained in Fig. 7 which induces in turn the corresponding abrupt jump in the Fermi level. The net result in going from the left to the right panels of Fig. 8 is that the work function W OEP ͑d͒ suffers an abrupt ͑discontinuous͒ decrease, as the boundary condition V KS ͑ϱ͒ = 0 is rigorously valid in both cases. This discontinuous behavior of W OEP ͑d͒, shown schematically in Fig. 8, represents precisely the origin of the jumps that are visible in Fig. 6 at every threshold for SDL occupation. It is evident from Fig. 6 that the size of the discontinuity decreases as d increases. Finally, we investigate the size of the discontinuities that are visible in Fig. 6. For this, we rewrite the central OEP equation ͓as given by Eq. ͑14͔͒ in the following way: ͚ i=1 m−1 ͑k F i ͒ 2 ͵ −ϱ ϱ ͓V x ͑zЈ;m͒ − u x i ͑zЈ;m͔͒G i KS ͑z,zЈ͒ i ͑zЈ,z͒dzЈ + ͑k F m ͒ 2 ͵ −ϱ ϱ ͓V x ͑zЈ;m͒ − u x i ͑zЈ;m͔͒G m KS ͑z,zЈ͒ m ͑zЈ,z͒dzЈ + c.c. = 0, ͑47͒ where i ͑z , zЈ͒ = i ͑z͒ ‫ء‬ i ͑zЈ͒. In writing Eq. ͑47͒ the contribution of all the m − 1 occupied SDLs has been split from the contribution of the last occupied ͑m͒ SDL. The label m in V x ͑z ; m͒ and u x i ͑z ; m͒ has been introduced in order to emphasize that they are solutions of a system with m occupied SDLs. Let us now define a distance Z, such that for z Ͼ Z the electron density is dominated by the contribution of the last occupied ͑m͒ SDL, which is the one with the slowest decay. Equation ͑6͒ clearly shows that Z → ϱ when k F m → 0, which is the case whenever ␣ m → 0 + , i.e., whenever the filling of the last occupied SDL is infinitesimally small. We consider the following trial solution of Eq. ͑47͒: V x ͑z;m͒ = V x ͑z;m − 1͒ + C x ͑m͒ ͑ 48͒ for z Ͻ Z and k F m → 0, with C x ͑m͒ being a constant which depends on the last occupied SDL. Introducing this trial solution into Eq. ͑47͒, we obtain ͚ i=1 m−1 ͑k F i ͒ 2 ͵ −ϱ ϱ ͓V x ͑zЈ;m − 1͒ + C x ͑m͒ − u x i ͑zЈ;m͔͒G i KS ͑z,zЈ͒ i ͑zЈ,z͒dzЈ + ͑k F m ͒ 2 ϫ ͵ −ϱ ϱ ͓V x ͑zЈ;m − 1͒ + C x ͑m͒ − u x i ͑zЈ;m͔͒G m KS ͑z,zЈ͒ m ͑zЈ,z͒dzЈ + c.c. = 0. ͑49͒ In the limit k F m → 0, the second-line contribution of Eq. ͑49͒ is arbitrarily small; also, the KS wave functions i ͑z͒ and eigenvalue differences ͑denominators͒ entering G i KS ͑z , zЈ͒ should be extremely similar for the slab width corresponding to m − 1 occupied levels and ␣ m−1 → 1 − and the slab width corresponding to m occupied levels and ␣ m → 0 + . Therefore, an inspection of Eq. ͑22͒ leads us, using similar arguments, to the conclusion that u x i ͑z ; m͒ → u x i ͑z ; m −1͒ for all i Ͻ m, z Ͻ Z, and k F m → 0. Under these conditions, the first line of Eq. ͑49͒ reverts to the OEP equation for a slab width corresponding to m − 1 occupied states, and the proposal of Eq. ͑48͒ is proved. Considering now that C x ͑m͒ = V x ͑z ; m͒ − V x ͑z ; m −1͒ and taking the expectation value at the last occupied state ͑m −1͒ of m − 1 system, we find C x ͑m͒ = V x m−1 ͑m͒ − V x m−1 ͑m − 1͒. ͑50͒ Now, for the m − 1 system we can use the boundary condition V x m−1 ͑m −1͒ = ū x m−1 ͑m −1͒ and, once again, approximate ū x m−1 ͑m −1͒Ӎū x m−1 ͑m͒, yielding C x ͑m͒ = V x m−1 ͑m͒ − ū x m−1 ͑m͒, ͑51͒ which has the nice feature that both the exchange potential V x and the orbital-dependent exchange potential u x are referred to the m system. For the m system V x m ͑m͒ = ū x m ͑m͒, which does not prevent the constant C x ͑m͒ from being nonzero ͑as shown in Fig. 7͒ since the KS orbitals m−1 ͑z͒ and m ͑z͒ are different. As the slab width increases, m also increases and the difference between m−1 ͑z͒ and m ͑z͒ decreases, thereby leading to the expectation that C x ͑m͒ → 0 as d → ϱ. This is explicitly shown in Fig. 9. While this analysis explains why C x ͑m͒ 0 for any finite m, it does not give a hint about its sign; Fig. 9 shows, however, that C x ͑m͒ is positive for all m. This positive jump in V x ͑z͒ is exchange driven: at each threshold width for the occupation of a new level, a barrier appears against the occupancy of an empty SDL. This is due to the fact that intra-SDL exchange is stronger than inter-SDL exchange. As a consequence, the slab gains exchange energy by restricting new SDL occupancies. On the other hand, correlation induces in general a negative jump in V c ͑z͒, so the net jump in V xc ͑z͒ depends on the relative weight of exchange and correlation for each particular system. 54 Finally, we have observed numerically that the average of the OEP work functions for slab widths corresponding to ␣ m−1 → 1 − and ␣ m → 0 + remains the same ͑within error bars͒ for all the m values that we have considered. Hence, we have taken the infinite-width extrapolated work function to be simply that average. Table II exhibits the infinite-width x-only LDA and OEP work functions that we have obtained in this way for various values of the electron-density parameter r s . OEP work functions are slightly and systematically smaller than their LDA counterparts. VI. CONCLUSIONS We have reported benchmark exact-exchange selfconsistent calculations of the KS potential, surface energy, and work function of jellium slabs in the framework of the OEP scheme. Special emphasis has been put into the asymptotical behavior of the exact-exchange KS potential far into the vacuum and the large quantum size effects that are present in the slab-width dependence of the surface energy and work function. We have performed a detailed analysis of the asymptotics of the exact-exchange KS potential far into the vacuum, 39 showing that at a distance z that is larger than the slab thickness the exact-exchange potential takes an imagelike form, V x ͑z → ϱ͒ → −e 2 / z, but with a coefficient that differs from that of the classical image potential V im ͑z͒ =−e 2 / 4z. Although this result has been obtained in the x-only approximation, it is also true in the presence of correlation due to the separability of the basic OEP equations in their basic exchange and correlation components. The OEP kinetic, electrostatic, and exchange contributions to the surface energy of jellium slabs have been obtained as a function of the slab width d and for a set of electron densities characterized by the parameter r s . We have shown that these components of the surface energy are all oscillating functions of d, with the oscillating period being Ϸ F / 2. By a suitable extrapolation procedure, we have found the values of the different components of the surface energy of a semi-infinite jellium. We have compared our OEP surface energies with those obtained from the same formally exact expressions ͓see Eqs. ͑39͒-͑41͔͒ but using single-particle LDA wave functions and energies; we have found small differences between these OEP and LDA surface energies, which appear as a consequence of the LDA orbitals being slightly more delocalized ͑diffuse͒ than their more realistic OEP counterparts. Finally, we have performed x-only OEP calculations of the work function of jellium slabs, again as a function of the slab width d. We have found that the OEP work-function exhibits large quantum size effects that are absent in the LDA and which reflect the intrinsic derivative discontinuity of the exact KS potential. The amplitude of this discontinuity diminishes as the slab width increases and becomes arbitrarily small as d → ϱ, i.e., in the case of a semi-infinite system. This has been proved both analytically and numerically. We also note that although the precise value of the x-only OEP work functions reported here would change with the inclusion of correlation, the exact slab work function is expected to exhibit the large quantum size effects and discontinuities observed in the present work, barring possible accidental cancellations of exchange-driven and correlationdriven contributions to the total discontinuity. The presence of these large discontinuities in the x-only OEP slab work function ͑and presumably also in the actual work function that includes correlation͒ highlights the potential danger in which can be incurred by performing elaborated calculations for a restricted set of slab sizes without performing a suitable and reliable extrapolation toward the semi-infinite case. In summary, we expect that the benchmark exact-exchange OEP calculations reported here for jellium slabs will serve as motivation and as a starting point for the development of more realistic approximations for the exchange-correlation energy functional of jellium and real surfaces. FIG. 2. LDA and OEP self-consistent electron densities and its difference for d =8 F and r s = 2.07. Note that n LDA ͑z͒ is slightly more diffuse than n OEP ͑z͒, as n LDA ͑z͒ − n OEP ͑z͒ Ͼ 0 for z outside the jellium edge ͑in the vacuum͒. FIG. 3 . 3Kinetic surface energy, as a function of slab width d, for r s = 2.07, from Eq. ͑43͒, with l = K. Full line, OEP results; dotted line, LDA results. The two arrows on the right denote the extrapolated asymptotic values K ͑OEP͒ → −4720 erg/ cm 2 , K ͑LDA͒ → −4832 erg/ cm 2 . FIG. 4 . 4Electrostatic surface energy, as a function of slab width d, for r s = 2.07, from Eq. ͑43͒, with l = el. Full line, OEP results; dotted line, LDA results. The two arrows on the right denote the extrapolated asymptotic values el ͑OEP͒ → 1103 erg/ cm 2 , el ͑LDA͒ → 1172 erg/ cm 2 . FIG. 5 . 5Exchange surface energy, as a function of slab width d, for r s = 2.07, from Eq. ͑43͒, with l = x. Full line, OEP results; dotted line, LDA results; dash-dotted line, standard LDA-exchange results. The three arrows on the right denote the extrapolated asymptotic values x ͑OEP͒ → 2316 erg/ cm 2 , x ͑LDA͒ → 2390 erg/ cm 2 , and x LDA → 2767 erg/ cm 2 . FIG. 6 . 6Slab work function versus slab width d for r s = 2.07. Full line, OEP result; dotted line, LDA result. Occupation events corresponding to transitions from a slab with m occupied SDL toward m + 1 occupied SDL are denoted as m → m +1. FIG. 7 . 7Self-consistent OEP exchange potential, around the 6 → 7 SDL transition, for r s = 2.07. The origin of coordinate z for each slab has been taken at the slab center. The position of the right slab edge has been indicated by a vertical dashed line for each case. ACKNOWLEDGMENTS C.M.H. wishes to acknowledge the financial support received from CONICET of Argentina. J.M.P. acknowledges partial support by the University of the Basque Country, the Basque Unibertsitate eta Ikerketa Saila, the Spanish Ministerio de Educación y Ciencia ͑Grants No. FIS2006-01343 and No. CSD2006-53͒, and the EC Sixth Framework Network of Excellence NANOQUANTA ͑Grant No. NMP4-CT-2004-500198͒. C.R.P. was supported by the European Community through a Marie Curie IIF ͑Grant No. MIF1-CT-2006-040222͒. ͑8͒ Multiplying Eq. ͑8͒ by the KS density-response function KS ͑z , zЈ͒ϵ␦n͑z͒ / ␦V KS ͑zЈ͒, using the identity FIG. 1. Main features of the jellium-slab model of metal surfaces. Top panel: normalized jellium density ͓n + ͑z͔͒ and the selfconsistent OEP electron density n͑z͒ for two different values of the electron-density parameter r s . Lower panel: OEP Hartree, exchange, and Kohn-Sham potentials for r s = 2.07. Dotted lines denote KS eigenvalues, F is the Fermi energy, and W is the work function. d =4 F .The nice feature of Eq. ͑10͒ is that ␦ i ͑zЈ͒ / ␦V KS ͑z͒ and KS ͑z , zЈ͒ are simply obtained from the solutions of Eq. ͑3͒ as follows:͵ −ϱ ϱ KS ͑z,zЈ͒ KS −1 ͑zЈ,zЉ͒dz = ␦͑z − zЉ͒, ͑9͒ comparing Eqs. ͑7͒ and ͑8͒, and integrating over the coordi- nate z, one finds ͵ −ϱ ϱ ␦E xc ␦n͑zЈ͒ KS ͑z,zЈ͒dzЈ = ͚ i occ. ͵ −ϱ ϱ ͫ ␦E xc ␦ i ͑zЈ͒ ␦ i ͑zЈ͒ ␦V KS ͑z͒ + c.c. ͬ dzЈ. ͑10͒ -15 -10 -5 0 Energy [eV] V H (z) V x (z) V KS (z) Density [ n ] n + (z) n (z) r s = 2.07 n(z) r s = 6.00 0 -d / 2 -d i = 1 µ 2 3 4 5 6 7 8 9 W 2/3 1/3 1 r s = 2.07 10 z TABLE I . IInfinite-width extrapolated results for exchange-only kinetic ͓ K ͑LDA͒ and K ͑OEP͔͒, electrostatic ͓ el ͑LDA͒ and el ͑OEP͔͒, and exchange ͓ x LDA , x ͑LDA͒, and x ͑OEP͔͒ surface energies for different values or r s . ͑LDA͒ and ͑OEP͒ represent the sum of the corresponding exchange-only kinetic, electrostatic, and exchange surface energies. Empty entries in el ͑OEP͒ for the two largest r s studied are due to the fact that the corresponding magnitudes are so small that it is not possible to obtain a reliable extrapolated value. Units are erg/ cm 2 .r s K ͑LDA͒ K ͑OEP͒ el ͑LDA͒ el ͑OEP͒ x LDA x ͑LDA͒ x ͑OEP͒ ͑LDA͒ ͑OEP͒ 2.00 −5707 −5579 1390 1317 3131 2726 2649 −1591 −1613 2.07 −4832 −4720 1172 1103 2767 2390 2316 −1270 −1301 3.00 −770 −733 189 177 707 568 535 −13 −21 4.00 −169 −155 49 48 243 180 161 60 54 5.00 −46 −39 19 105 71 59 44 6.00 −13 −9 9 52 32 23 28 EXACT-EXCHANGE KOHN-SHAM POTENTIAL, SURFACE… PHYSICAL REVIEW B 78, 085126 ͑2008͒ 085126-7 FIG. 8. Left: electronic structure of the slab for ␣ 6 =1 − . Right: electronic structure of the slab for ␣ 7 =0 + . The work-function W jumps discontinuously from its left large value toward the smaller right value. Slab edge is at z =0.FIG. 9. Exchange-driven discontinuity C x ͑m͒ for increasing number of occupied slab levels, as follows from Eq. ͑51͒.-1 0 1 2 z [ λ F ] -15 -10 -5 0 Energy [eV] W µ V H V x V KS r s = 2.07 -1 0 1 2 z [ λ F ] -15 -10 -5 0 W µ V H V x V KS r s = 2.07 4 8 1 2 16 20 m 1 2 3 4 5 6 C x [ eV] r s = 2.07 TABLE II . IIInfinite-width extrapolated x-only LDA and OEP work functions for various values of r s . Units are eV.r s 2.00 2.07 3.00 4.00 5.00 6.00 W LDA 2.82 2.80 2.50 2.15 1.86 1.62 W OEP 2.64 2.63 2.49 2.11 1.84 1.61 *Permanent address: Centro Atómico Bariloche and Instituto Balseiro, 8400 S. C. de Bariloche. Río Negro, Argentina*Permanent address: Centro Atómico Bariloche and Instituto Balseiro, 8400 S. C. de Bariloche, Río Negro, Argentina. . N D Lang, W Kohn, Phys. Rev. B. 1N. D. Lang and W. Kohn, Phys. Rev. B 1, 4555 ͑1970͒. N D Lang, Solid State Physics. H. Eirenreich, F. Seitz, and D. Turnball ͑AcademicNew York28225N. D. Lang, in Solid State Physics, edited by H. Eirenreich, F. 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Dobson, A. García-Lekue, and J. P. Perdew, Phys. Rev. Lett. 100, 036401 ͑2008͒. T Grabo, J Kreibich, S Kurth, E K U Gross, Strong Coulomb Interactions in Electronic Structure Calculations: Beyond the Local Density Approximation. AmsterdamT. Grabo, J. Kreibich, S. Kurth, and E. K. U. Gross, in Strong Coulomb Interactions in Electronic Structure Calculations: Be- yond the Local Density Approximation, edited by V. I. Anisimov ͑Gordon and Breach, Amsterdam, 2000͒. E Engel, A Primer in Density Functional Theory. C. Fiolhais, F. Nogueira, and M. A. L. Marques ͑SpringerBerlin62056E. Engel, in A Primer in Density Functional Theory, Lecture Notes in Physics Vol. 620, edited by C. Fiolhais, F. Nogueira, and M. A. L. Marques ͑Springer, Berlin, 2003͒, p. 56. . J D Talman, W F Shadwick, Phys. Rev. A. 141976J. D. Talman and W. F. Shadwick, Phys. Rev. A 14, 36 ͑1976͒. . J P Perdew, A Zunger, Phys. Rev. B. 23J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 ͑1981͒. . F , Della Sala, A Görling, Phys. Rev. Lett. 892002J. Chem. Phys.F. Della Sala and A. Görling, Phys. Rev. Lett. 89, 033003 ͑2002͒; J. Chem. Phys. 116, 5374 ͑2002͒. . D M Bylander, L Kleinman, Phys. Rev. Lett. 741995D. M. Bylander and L. Kleinman, Phys. Rev. Lett. 74, 3660 ͑1995͒; . M Städele, J A Majewski, P Vogl, A Görling, 0964022002ibid. 79, 2089 ͑1997͒; Y.-H. Kim and A. Gorling, ibid. 89M. Städele, J. A. Majewski, P. Vogl, and A. Görling, ibid. 79, 2089 ͑1997͒; Y.-H. Kim and A. Gorling, ibid. 89, 096402 ͑2002͒. . S J A Van Gisbergen, P R T Schipper, O V Gritsenko, E J Baerends, J G Snijders, B Champagne, B Kirtman, Phys. Rev. Lett. 831999S. J. A. van Gisbergen, P. R. T. Schipper, O. V. Gritsenko, E. J. Baerends, J. G. Snijders, B. Champagne, and B. Kirtman, Phys. Rev. Lett. 83, 694 ͑1999͒. . T Kreibich, S Kurth, T Grabo, E K U Gross, Adv. Quantum Chem. 331998T. Kreibich, S. Kurth, T. Grabo, and E. K. U. Gross, Adv. Quan- tum Chem. 33, 31 ͑1998͒. . J B Krieger, Y Li, G J Iafrate, Phys. Rev. A. 451992J. B. Krieger, Y. Li, and G. J. Iafrate, Phys. Rev. A 45, 101 ͑1992͒. . J P Perdew, R G Parr, M Levy, J L Balduz, Phys. Rev. Lett. 49J. P. Perdew, R. G. Parr, M. Levy, and J. L. Balduz, Phys. Rev. Lett. 49, 1691 ͑1982͒. . J P Perdew, M Levy, Phys. Rev. Lett. 511983J. P. Perdew and M. Levy, Phys. Rev. Lett. 51, 1884 ͑1983͒. . J P Perdew, NATO ASI Ser., Ser. B. 123J. P. Perdew, NATO ASI Ser., Ser. B 123, 265 ͑1985͒. . D J Tozer, N C Handy, J. Chem. Phys. 1081998D. J. Tozer and N. C. Handy, J. Chem. Phys. 108, 2545 ͑1998͒. . Y.-H Kim, I.-H Lee, S Nagaraja, J P Leburton, R Q Hood, R M Martin, Phys. Rev. B. 612000Y.-H. Kim, I.-H. Lee, S. Nagaraja, J. P. Leburton, R. Q. Hood, and R. M. Martin, Phys. Rev. B 61, 5202 ͑2000͒; . P García-González, ibid. 6223212000P. García- González, ibid. 62, 2321 ͑2000͒; . L Pollack, J P Perdew ; ͑2000͒; P. García-González, R W Godby, J. Phys.: Condens. Matter. 12Phys. Rev. Lett.L. Pollack and J. P. Perdew, J. Phys.: Condens. Matter 12, 1239 ͑2000͒; P. García-González and R. W. Godby, Phys. Rev. Lett. 88, 056406 ͑2002͒. The electron-density parameter r s is defined as the radius of a sphere containing on average one electron, i.e., r s = ͑3 / 4na 0. The electron-density parameter r s is defined as the radius of a sphere containing on average one electron, i.e., r s = ͑3 / 4na 0 ͒ 1/3 . A convenient length unit for the present system. ͒ 1/3 . A convenient length unit for the present system We have found that two infinite barriers located at 2 F from each jellium edge are enough for this purpose in our system. We have found that two infinite barriers located at 2 F from each jellium edge are enough for this purpose in our system. Note that we have included in our Hartree potential the contribution which comes from the uniform jellium background ͑pro-portional to n + ͒. Alternatively, this contribution may be denoted separately as the "external potential. Note that we have included in our Hartree potential the contri- bution which comes from the uniform jellium background ͑pro- portional to n + ͒. Alternatively, this contribution may be denoted separately as the "external potential." Note that dimensions of Eq. ͑6͒ are ͑length͒ −3 , as correspond to our three-dimensional ͑3D͒ system. However, for the particular slab geometry, the number density only depends on one spatial coordinate ͑z͒. Note that dimensions of Eq. ͑6͒ are ͑length͒ −3 , as correspond to our three-dimensional ͑3D͒ system. However, for the particular slab geometry, the number density only depends on one spatial coordinate ͑z͒. Due to the imposed translational invariance in the x , y plane, functional derivatives in this work are conveniently defined as ␦f = ͓͐␦f͓g͔ / ␦g͑z͔͒␦g͑z͒dz, where ␦g͑z͒ represents a uniform variation of the function g͑r͒ in the plane r = z. This is the origin of the factor A −1 in Eq. ͑7͒.Due to the imposed translational invariance in the x , y plane, functional derivatives in this work are conveniently defined as ␦f = ͓͐␦f͓g͔ / ␦g͑z͔͒␦g͑z͒dz, where ␦g͑z͒ represents a uniform variation of the function g͑r͒ in the plane r = z. This is the origin of the factor A −1 in Eq. ͑7͒. We assume that the xc energy functional does not depend on the unoccupied eigenfunctions and eigenvalues of the KS equation, which is only known to be true in the case of the x-only energy functional. In general, the sum over the index i should run over the whole KS spectrum ͑Refs. 54 and 55͒. Also, for simplicity. we do not consider the possibility that E xc depends on the Kohn-Sham eigenvalues. Once again, this is only known to be true in the case of the x-only energy functional for fixed particle number. This is however, the only case to which all the numerical calculations presented below applyWe assume that the xc energy functional does not depend on the unoccupied eigenfunctions and eigenvalues of the KS equation, which is only known to be true in the case of the x-only energy functional. In general, the sum over the index i should run over the whole KS spectrum ͑Refs. 54 and 55͒. Also, for simplicity, we do not consider the possibility that E xc depends on the Kohn- Sham eigenvalues. Once again, this is only known to be true in the case of the x-only energy functional for fixed particle num- ber. This is however, the only case to which all the numerical calculations presented below apply. In the x-only version it can be written as Eq. ͑22͒. In the x-only version it can be written as Eq. ͑22͒. . S Kummel, J P Perdew, Phys. Rev. Lett. 90Phys. Rev. BS. Kummel and J. P. Perdew, Phys. Rev. Lett. 90, 043004 ͑2003͒; Phys. Rev. B 68, 035103 ͑2003͒. . S Rigamonti, C R Proetto, F A Reboredo, Europhys. Lett. 702005S. Rigamonti, C. R. Proetto, and F. A. Reboredo, Europhys. Lett. 70, 116 ͑2005͒. . C M Horowitz, C R Proetto, S Rigamonti, Phys. Rev. Lett. 97Note that there is a sign of difference between the definition of the shifts in this and the present workC. M. Horowitz, C. R. Proetto, and S. Rigamonti, Phys. Rev. Lett. 97, 026802 ͑2006͒. Note that there is a sign of difference between the definition of the shifts in this and the present work. Note that the equivalence is only valid for this specific case ͑shifts identically zero͒. As soon as the shifts are different from zero, V xc,1 ͑z͒ and V xc KLI ͑z͒ will be not equal due to the presence of V xc. 2 ͑z͒ in this more general situationNote that the equivalence is only valid for this specific case ͑shifts identically zero͒. As soon as the shifts are different from zero, V xc,1 ͑z͒ and V xc KLI ͑z͒ will be not equal due to the presence of V xc,2 ͑z͒ in this more general situation. . W Kohn, A E Mattsson, Phys. Rev. Lett. 811998W. Kohn and A. E. Mattsson, Phys. Rev. Lett. 81, 3487 ͑1998͒. Handbook of Mathematical Functions ͑Dover. M Abramowitz, I A Stegun, New YorkM. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions ͑Dover, New York, 1964͒. . J Jung, J E Alvarellos, E Chacon, P Garcia-Gonzalez, J. Phys.: Condens. Matter. 19266008 ͑2007͒J. Jung, J. E. Alvarellos, E. Chacon, and P. Garcia-Gonzalez, J. Phys.: Condens. Matter 19, 266008 ͑2007͒. The first approximation k F m ͉z − zЈ͉Ӎk F m z can be justified for z → ϱ, even considering that the zЈ integral covers the range from −ϱ to +ϱ. This is due to the fact that the main contribution to this integral comes from values of zЈ inside the slab, so the range of zЈ could be effectively restricted to the region −d Շ zЈ Շ 0The first approximation k F m ͉z − zЈ͉Ӎk F m z can be justified for z → ϱ, even considering that the zЈ integral covers the range from −ϱ to +ϱ. This is due to the fact that the main contribution to this integral comes from values of zЈ inside the slab, so the range of zЈ could be effectively restricted to the region −d Շ zЈ Շ 0. Note that z m =−d / 2. So ␤ is always negative as a sum of two negative terms. Note that z m =−d / 2. So ␤ is always negative as a sum of two negative terms. . Fred Nastos, Queen's UniversityPh.D. thesisFred Nastos, Ph.D. thesis, Queen's University, 2000. . C M Horowitz, C R Proetto, J M Pitarke, C. M. Horowitz, C. R. Proetto, and J. M. Pitarke ͑unpublished͒. . J M Pitarke, A G Eguiluz, Phys. Rev. B. 57J. M. Pitarke and A. G. Eguiluz, Phys. Rev. B 57, 6329 ͑1998͒; 63, 045116 ͑2001͒. By x-only LDA we mean that in the KS equations the actual exchange potential V x ͑z͒ is replaced at this point by the exchange potential of a uniform electron gas with the local density. i.e., V x LDA ͑z͒ =−͓6n͑z͒ / ͔ 1/3By x-only LDA we mean that in the KS equations the actual exchange potential V x ͑z͒ is replaced at this point by the ex- change potential of a uniform electron gas with the local density, i.e., V x LDA ͑z͒ =−͓6n͑z͒ / ͔ 1/3 . Note that as the work function is defined as the difference between two energies, it is independent of the choice for the zero energy factor ͑ef͒ energy. See Ref. 7. Note that as the work function is defined as the difference between two energies, it is independent of the choice for the zero energy factor ͑ef͒ energy. . F K Schulte, J. Phys. C. 7F. K. Schulte, J. Phys. C 7, L370 ͑1974͒. . F K Schulte, Surf. Sci. 551976F. K. Schulte, Surf. Sci. 55, 427 ͑1976͒. . S Rigamonti, C R Proetto, Phys. Rev. Lett. 98S. Rigamonti and C. R. Proetto, Phys. Rev. Lett. 98, 066806 ͑2007͒. . S Rigamonti, C R Proetto, Phys. Rev. B. 73S. Rigamonti and C. R. Proetto, Phys. Rev. B 73, 235319 ͑2006͒.

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{'abstract': 'Exact-exchange self-consistent calculations of the Kohn-Sham potential, surface energy, and work function of jellium slabs are reported in the framework of the optimized effective potential ͑OEP͒ scheme of density functional theory. In the vacuum side of the jellium surface and at a distance z that is larger than the slab thickness, the exchange-only Kohn-Sham potential is found to be imagelike ͑ϳ−e 2 / z͒ but with a coefficient that differs from that of the classical image potential V im ͑z͒ =−e 2 / 4z. The three OEP contributions to the surface energy ͑kinetic, electrostatic, and exchange͒ are found to oscillate as a function of the slab thickness, as occurs in the case of the corresponding calculations based on the use of single-particle orbitals and energies obtained in the local-density approximation ͑LDA͒. The OEP work function presents large quantum size effects that are absent in the LDA and which reflect the intrinsic derivative discontinuity of the exact Kohn-Sham potential.', 'arxivid': '0808.0580', 'author': ['C M Horowitz \nDonostia International Physics Center (DIPC)\nE-20018San SebastianSpain\n\nInstituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas, (INIFTA)\nUNLP\nCCT La Plata-CONICET\nSucursal 4, Casilla de Correo 16, 1900) La PlataArgentina\n', 'C R Proetto \nInstitüt für Teoretische Physik\nFreie Universitat Berlin\nArnimallee 14D-14195BerlinGermany\n', 'J M Pitarke \nCIC nanoGUNE Consolider\nBasque Country\nMikeletegi Pasealekua 56, E20009DonostiaSpain\n\nMateria Kondentsatuaren Fisika Saila and Centro Física Materiales, CSIC-UPV/EHU\n644 Posta KutxatilaE-48080Bilbo, Basque CountrySpain\n'], 'authoraffiliation': ['Donostia International Physics Center (DIPC)\nE-20018San SebastianSpain', 'Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas, (INIFTA)\nUNLP\nCCT La Plata-CONICET\nSucursal 4, Casilla de Correo 16, 1900) La PlataArgentina', 'Institüt für Teoretische Physik\nFreie Universitat Berlin\nArnimallee 14D-14195BerlinGermany', 'CIC nanoGUNE Consolider\nBasque Country\nMikeletegi Pasealekua 56, E20009DonostiaSpain', 'Materia Kondentsatuaren Fisika Saila and Centro Física Materiales, CSIC-UPV/EHU\n644 Posta KutxatilaE-48080Bilbo, Basque CountrySpain'], 'corpusid': 118516894, 'doi': '10.1103/physrevb.78.085126', 'github_urls': [], 'n_tokens_mistral': 23843, 'n_tokens_neox': 21534, 'n_words': 11000, 'pdfsha': '1326883a6203dc3814d7af6f871a6bb96ab27ef8', 'pdfurls': None, 'title': ['Exact-exchange Kohn-Sham potential, surface energy, and work function of jellium slabs', 'Exact-exchange Kohn-Sham potential, surface energy, and work function of jellium slabs'], 'venue': []}

arxiv

The Vacuum Energy from a New Perspective 0103161v3 23 Oct 2001 Antonio R Mondragon Center for Theoretical Physics Texas A&M University 77843College StationTexasUSA Roland E Allen Center for Theoretical Physics Texas A&M University 77843College StationTexasUSA The Vacuum Energy from a New Perspective 0103161v3 23 Oct 2001arXiv:astro-ph/ It is commonly believed that the vacuum energy problem points to the need for (1) a radically new formulation of gravitational physics and (2) a new principle which forces the vacuum stress-energy tensor (as measured by gravity) to be nearly zero. Here we point out that a new fundamental theory contains both features: (1) In this theory the vierbein is interpreted as the "superfluid velocity" associated with the order parameter Ψ s for a GUT-scale Higgs condensate.(2)The vacuum stress-energy tensor T vac µν is exactly zero in the vacuum state, because the action is extremalized with respect to variations in Ψ s . With inhomogeneouslydistributed matter present, T vac µν is shifted away from zero. The vacuum energy is one of the deepest issues in theoretical physics, and no conventional theory -including superstring/M theory -has offered a convincing solution to the problem of why the vacuum stress-energy tensor (as measured by gravity) is vastly smaller than expected but still nonzero [1][2][3]. In this paper we consider an unconventional theory which contains a radically new formulation of gravitational physics [4][5][6]. The gravitational vierbein is interpreted as the "superfluid velocity" of a GUT-scale condensate Ψ s which forms in the very early universe: g µν = η αβ e µ α e ν β , e µ α = v µ α , v µ = v µ α σ α with µ, α = 0, 1, 2, 3 (1) v µ = η µν v ν , mv µ = − iU −1 ∂ µ U , Ψ s = n 1/2 s Uη s , η † s η s = 1.(2) Here η µν = diag (-1,1,1,1) is the Minkowski metric tensor, the σ α are the identity matrix and three Pauli matrices, U is a 2 × 2 unitary matrix, η s is a constant 2-component vector, and n s is the condensate density. (After a Kaluza-Klein reduction from a higher-dimensional theory, the initial group of this order parameter is SO(10) × SU(2) × U(1). For the purposes of this paper, however, the gauge group SO(10) can be ignored, leaving the simpler description of (1) and (2).) In the present theory, Ψ s is not static but instead exhibits SU(2) × U(1) rotations as a function of position and time. This condensate also supports Planck-scale SU(2) instantons (in a Euclidean picture), which are analogous to the U(1) vortices in an ordinary superfluid. In the present theory, the Einstein-Hilbert action and the curvature of spacetime result from these instantons [4]. Quantum gravity has a natural cutoff at the energy scale m ∼ the Planck energy m P , but at lower energies one recovers the Einstein field equations δS total δg µν = δS vac δg µν + δS f ields δg µν + δS EH δg µν = 0 (3) or R µν − 1 2 g µν (4) R = 1 2 ℓ 2 P T total µν = 1 2 ℓ 2 P T vac µν + T f ields µν (4) where T vac µν = − 2 √ −g δS vac δg µν , T f ields µν = − 2 √ −g δS f ields δg µν (5) and ℓ P is the Planck length defined in Ref. 4. In the vacuum state, there is no stress-energy tensor for matter and radiation: T f ields µν = 0. We additionally assume that there there is no contribution from topological defects in the vacuum state: δS EH /δg µν = 0. This assumption will be discussed in more detail elsewhere [6], but it will be seen below that it leads to a consistent solution (whereas such a solution could not be obtained in conventional physics). We then have δS vac = δS total = 0 in the vacuum state (6) for arbitrary variations of the order parameter Ψ s . Variations in v µ α , however, are a special case of functional variations in Ψ s . It follows that the vacuum stress-energy tensor is exactly zero: T vac µν = 0 in the vacuum state. It is interesting to see in more detail how (7) 1 2 mη † s η µν v µ v ν η s + V + P + V vac = µ (8) V = bn s , P = − 1 2m n −1/2 s η µν ∂ µ ∂ ν n 1/2 s (9) where µ is a fundamental energy which plays the role of a chemical potential here and which is comparable to m P . We have added a term V vac which represents the contribution of all other vacuum fields to δS vac when Ψ s is varied. Let us rewrite (8) as − 1 2m n −1/2 s η µν ∂ µ ∂ ν n 1/2 s + bn s = µ − V vac − 1 2 mη µν e µ α e ν β η † s σ α σ β η s .(10) After V vac and e µ α are specified, the condensate density n s adjusts itself to satisfy (10), (6), and (7). One might express this result as follows: In the vacuum state, the vacuum stress-energy tensor is tuned to exactly zero through adjustments of the condensate density. Notice that the extremalization (3) in conventional physics requires a contribution from the Einstein-Hilbert action S EH even when δS f ields /δg µν = 0, but in the present theory this extremalization in the vacuum state can be accomplished with S vac alone. In a more general state with matter, radiation, and topological defects present, it is the total action which is extremalized in (3). Then δS vac /δg µν is shifted away from zero: T vac µν = 0 with matter and radiation present. There are two primary aspects of the vacuum energy problem [1,2]: (i) Why is the vacuum stress-energy tensor many orders of magnitude smaller than predicted by conventional physics? This question is addressed in (7). (ii) Why is the vacuum stress-energy tensor not exactly zero? This is addressed in (11). Although these are the "big" questions, one can add two more in the present context: (iii) Why was the vacuum energy density small compared to the density of matter and radiation during the period of big-bang nucleosynthesis? (iv) Why is it comparable to the density of matter now? To fully answer these questions will require a detailed treatment of how the vacuum energy is affected by the presence of matter and radiation. Suppose, however, that the dominant mechanism is a Casimir-like effect, in which the vacuum energy is modified by the boundary conditions imposed on the vacuum fields when there is an inhomogeneous distribution of matter. In a radiation-dominated universe, the energy density will be relatively homogeneous, and such an effect should be relatively small. In the present epoch, on the other hand, there is an extremely inhomogeneous distribution of matter. This plausibility argument indicates that the vacuum energy should play an important role only in the present epoch, and that the vacuum energy density (as measured by the stress-energy tensor) will be comparable to the density of inhomogeneously-distributed matter. can be achieved. According to the quantum Bernoulli equation (3.20) of Ref. 4, we have AcknowledgementThis work was supported by the Robert A. Welch Foundation. . S Weinberg, Rev. Mod. Phys. 611S. Weinberg, Rev. Mod. Phys. 61, 1 (1989). S Weinberg, astro-ph/0005265talk at the 4th International Symposium on Sources and Detection of Dark Matter in the Universe (DM2000) and. S. Weinberg, talk at the 4th International Symposium on Sources and Detection of Dark Matter in the Universe (DM2000) and astro-ph/0005265. E Witten, hep-ph/0002297talk at DM2000 and. E. Witten, talk at DM2000 and hep-ph/0002297. . R E Allen, hep-th/9612041Int. J. Mod. Phys. A. 122385R. E. Allen, Int. J. Mod. Phys. A 12, 2385 (1997) and hep-th/9612041. . R E Allen, hep-th/0008032R. E. Allen, to be published and hep-th/0008032. R E Allen, to be published and talk given at the conference on Problems with Vacuum Energy. CopenhagenR. E. Allen, to be published and talk given at the conference on Problems with Vacuum Energy (Copenhagen, August 24-26, 2000).

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{'abstract': 'It is commonly believed that the vacuum energy problem points to the need for (1) a radically new formulation of gravitational physics and (2) a new principle which forces the vacuum stress-energy tensor (as measured by gravity) to be nearly zero. Here we point out that a new fundamental theory contains both features: (1) In this theory the vierbein is interpreted as the "superfluid velocity" associated with the order parameter Ψ s for a GUT-scale Higgs condensate.(2)The vacuum stress-energy tensor T vac µν is exactly zero in the vacuum state, because the action is extremalized with respect to variations in Ψ s . With inhomogeneouslydistributed matter present, T vac µν is shifted away from zero.', 'arxivid': 'astro-ph/0103161', 'author': ['Antonio R Mondragon \nCenter for Theoretical Physics\nTexas A&M University\n77843College StationTexasUSA\n', 'Roland E Allen \nCenter for Theoretical Physics\nTexas A&M University\n77843College StationTexasUSA\n'], 'authoraffiliation': ['Center for Theoretical Physics\nTexas A&M University\n77843College StationTexasUSA', 'Center for Theoretical Physics\nTexas A&M University\n77843College StationTexasUSA'], 'corpusid': 506800, 'doi': '10.1063/1.1419576', 'github_urls': [], 'n_tokens_mistral': 2417, 'n_tokens_neox': 2048, 'n_words': 1360, 'pdfsha': 'd6726c6252b64ee252024daf3d59dc8612b79689', 'pdfurls': ['https://arxiv.org/pdf/astro-ph/0103161v3.pdf'], 'title': ['The Vacuum Energy from a New Perspective', 'The Vacuum Energy from a New Perspective'], 'venue': []}

arxiv

TempT: Temporal consistency for Test-time adaptation Onur Cezmi Mutlu Stanford University Mohammadmahdi Honarmand Stanford University Saimourya Surabhi Stanford University Dennis P Wall dpwall@stanford.edu Stanford University TempT: Temporal consistency for Test-time adaptation We introduce Temporal consistency for Test-time adaptation (TempT), a novel method for test-time adaptation on videos through the use of temporal coherence of predictions across sequential frames as a self-supervision signal.TempT is an approach with broad potential applications in computer vision tasks, including facial expression recognition (FER) in videos. We evaluate TempT's performance on the AffWild2 dataset. Our approach focuses solely on the unimodal visual aspect of the data and utilizes a popular 2D CNN backbone, in contrast to larger sequential or attention-based models used in other approaches. Our preliminary experimental results demonstrate that TempT has competitive performance compared to the previous years' reported performances, and its efficacy provides a compelling proof-of-concept for its use in various real-world applications. Introduction Affective computing aims to develop technologies with the capabilities like recognizing, interpreting, and simulating human affects. Expressions being one of the primary means of conveying emotion, facial expression recognition (FER) often constitutes an important part of human affective behavior analysis. There is an increasing number of use cases from driver safety applications to diagnosis and therapy of developmental problems of children [11]. With the continuous improvement in the computer vision field through extensive adoption of deep learning approaches, the real-world use of such algorithms is becoming easier and universal. However, the robustness and reliability of aforementioned algorithms tend to suffer from the domain shift phenomena which is still a prominent problem for computer vision models with limited generalization capability. The domain shift problem becomes even more pronounced in the "real world" scenarios due to uncontrollable environmental conditions. In the computer vision setting some examples to these conditions could be lighting, camera quality, motion, and resolution. Invariance and robust-ness against these variations is the main focus of domain adaptation and domain generalization research, with many successful algorithms already developed. In our work, we explore a specific subdomain of this field called Test-Time Adaptation (TTA), also referred to as Unsupervised Source-Free Domain Adaptation. In this setting, we assume no access to the target domain during training-time and no access to target domain labels in test-time. We treat each video as a new domain and our method adapts the trained model to a given video during test-time to improve its performance. We investigate the performance of our approach on the Facial Expression Recognition (FER) task, where the goal is to classify each frame in a video for Ekman emotions. The task of video assessment at the frame level is a natural environment for machine learning models with spatiotemporal inductive biases since the ability to model inter-frame relations could potentially be useful. Examples of such models are 3D convolutional neural networks (CNN) [10], attention-based models [1], or hybrid approaches combining 2D CNNs with recurrent neural networks (RNN) [37]. The first two of these approaches usually suffer from greater computational requirements than 2D CNNs, whereas the last method has unstable training time behavior under inputs with longer duration. There are numerous solutions to these problems including more efficient architectures as well as well-studied training paradigms, but in our work, we focus on exploring an adaptive approach where a simple 2D CNN model, which lacks useful biases for the setting, uses temporal predictive consistency as a self-supervision signal to adapt at test-time. For benchmarking purposes, we use Affwild2 [14][15][16][17][18][19][20][21][22]40] which is an invaluable FER dataset that contains over 500 videos and covers a wide variety of aforementioned variations. These qualities make it a suitable candidate for testing our algorithm. Related Work Facial Expression Recognition (FER) is a challenging task, especially in real-world scenarios. The difficulty arises from the fact that there is a significant amount of variation within each expression category, making it difficult to distinguish between different expressions. Additionally, there can be similarities between different expression categories, which further complicates the task of FER. This challenge is even more pronounced in real-world settings, where the lighting conditions, poses, and identities of the individuals can vary significantly. In such scenarios, even individuals with the same identity, pose, and lighting conditions can exhibit different expressions, while individuals with different identities, ages, gender, and pose can express the same emotion. Thus, FER is a task that requires robust algorithms that can effectively handle these intraclass variances and interclass similarities. In the past few years many Convolution Neural networks (CNN) based [4,23,27] and transformerbased architectures [39] have been proposed and significantly improve the performance of FER. As far as we are aware, there has been no prior research on test-time adaptation (TTA) for facial expression recognition (FER). Our work is an attempt to explore the use of TTA on FER tasks. It represents a novel approach to FER that has the potential to improve accuracy and opens up new avenues for research into TTA. Test-time Adaptation Early attempts for unsupervised domain adaptation were mainly based on updating running statistics of the batch normalization layers [26,33] with the new information from test data. [34] was one of the early works to propose using an auxiliary self-supervised task to be used in the test-time with the purpose of adapting the backbone parameters. [35] proposed using entropy minimization as the main adaptation goal and limiting the set of parameters to be updated to the weights of batch normalization layers (as opposed to updating statistics as before) which are shown to be highly expressive in [5]. Originating from the close ties of domain adaptation with fewshot learning [42] introduces a meta-learning-based solution where the loss to be used for adaptation is meta-learned. Finally, [41] and [28] report impressive adaptation results by combining image augmentation and entropy minimization to overcome the shortcomings of the latter in scenarios with large domain shifts. All of these works operate on static data that does not necessarily bear temporal correlations. Among them, only [35] explores continual adaptation to online data streams. [36] is a novel work that proposes a continual adaptation algorithm based on augmentation consistency. Yet, their al-gorithm makes an assumption of i.i.d. samples during test time, which may not always be correct. [6] addresses this issue and coins a new normalization layer that handles selective adaptation under non-iid data streams. To our knowledge, none of the works in the field exploits the temporal correlations in a given stream, and in our work, we aim to explore a possible direction for that. Our Approach Datasets and Preprocessing Focusing on training a computer vision model that operates on images (rather than videos), we have numerous data sources that are popular in the FER literature. We combine Affwild2 with Affectnet [30] and Real-world Affective Faces Database (RAF-DB) [24,25] to create a larger and more diverse training dataset. In our task, target classes are 7 basic emotions (also known as Ekman emotions [3]) plus an "other" class for expressions that do not fit into any category. Affwild2 is significantly larger in comparison to the others and has a label imbalance as can be seen in Fig. 2 and Fig. 3. In order to overcome this, we perform a random sampling on it by limiting the number of frames to 300 per video per expression class basis. Detailed label distribution of the resultant dataset is given in Tab. 1. We use provided cropped and aligned images in Af-fwild2, and others are only available in cropped versions, so we do not require any additional spatial preprocessing for any of the datasets. We then resize images to 112px×112px with antialiasing. For training purposes, we use common image augmentation methods such as color jitter, brightness and contrast shift, histogram equalization, channel dropout, blur, and random horizontal flip. Modeling Our approach is based on individual predictions on video frames, which allows us to use popular image-processing architectures in the literature. Due to their proven performance and stability of training, we use models from Resnet [7] family, with variations such as aggregated residual transformations [38] and squeeze-and-excitation blocks [9]. Generated embeddings are processed by two fullyconnected layers where the second, i.e. output, layer is subject to weight and input normalization [32] to prevent overconfidence, improve smoothness, and generalization. Significant class imbalance is a problem in this setting that needs to be addressed for successful supervised training. Label weighting, class up-sampling, and class down-sampling are classic methods to alleviate this issue, yet there are numerous scenarios where they fail to do so. We, therefore, adopt another approach namely Label-Distribution-Aware Margin Loss (LDAM) that was introduced in [2]. LDAM is similar to sample weighting in the sense that it modifies the loss depending on the class frequency but instead of using a multiplicative scaling, it intercepts with the class margins. The exact formulation is given in Eq. (1) where z is the unnormalized prediction vector,y is the ground truth class label, n j is the number of samples in class j and C is a temperature-like hyperparameter that tunes the effect of margins. LDAM enforces larger margins on minority classes which in return increases the model robustness and prevents overfitting. For more details, we refer the reader to the original paper. Supervised training of the model is then performed with back-propagation algorithm using defined LDAM loss to account for the skewed label distribution. Adam [13] optimizer with weight decay [29] is used for optimization where learning rates were subject to a step-decay schedule. Modeling and training were performed using PyTorch [31] framework on NVIDIA V100 GPUs. TempT: Temporal consistency for Test-time adaptation Being trained on static images as opposed to videos, 2D CNN models do not carry the implicit bias for the smoothness and/or consistency in their predictions across frames. We found empirically that such models contain stronger high-frequency components at the output, and when they are subject to a low-pass filter, the results look more desirable. We propose using this fact to generate a supervision signal to tune the network and improve classification performance. In particular, we temporally smooth the model predictions using a low-pass filter and set it as the desired signal. Purpose of setting this filtered signal as the target is to enforce the model to make temporally consistent predictions. We then calculate the mean-squared error between the original and target signals and use back-propagation to update a subset of model parameters. More formally, let x (t) ∈ R 112×112×3 be the t th frame of video and f (.) : R 112×112×3 → R 8 be the trained neural network of interest. We hypothesize that predictive coherence between consecutive samples can be used as an im- plicit Jacobian regularizer. In [8], it has been shown that regularization on the Frobenius norm of input-output Jacobian of a neural network can help the network attain flatter minima with higher robustness against input variations. Now, consider the case when the frame rate of a video is high enough. We can then approximate the Jacobian as in Eq. (2). J i,j (x (t) ) = ∂f i (x (t) ) ∂x (t) j ≈ f i (x (t) ) − f i (x (t−1) ) x (t) j − x (t−1) j(2) Then minimizing the Frobenius norm of the Jacobian becomes equivalent to minimizing inter-frame prediction differences as in Eq. (3) min J(x (t) ) F ≡ min i,j J 2 i,j (x (t) ) ≡ min f i (x (t) ) − f i (x (t−1) )(3) We empirically found that the initial distribution of prediction differences is heavy-tailed, with the tail being caused by momentary jumps in predictions due to problems at input cropping and/or sharp changes in activations due to model imperfections. When we used the target in Eq. (3) these outliers made the training process unstable for a significant portion of the experiments. We, therefore, chose to use another equivalent formulation to minimize the target. We first pass all frames from the pipeline to obtain an initial set of unnormalized scores y (t) ∈ R 8 . We then use the error signal in Eq. (4) as a self-supervision loss function to finetune the model. LP F (.) can be any low pass filter; in our experiments, we use a median filter, due to its robustness to outliers. L(y) = t y (t) − LP F (y) (t)(4) Using the entire video for adaptation may not be computationally feasible when the video duration is long. To alleviate this, we count the number of changes in model predictions using a sliding window and select the regions with the most changes to be the training regions that will compose the training batch. The updated version of the loss signal can be examined in Eq. (5) where R is the set of se-lected regions, and r indicated the range of frames to be considered. L(y) = r∈R t∈r y (t) − LP F (y) (t)(5) Being differentiable, this loss allows the use of backpropagation to update model parameters. The choice of parameters has an important effect on the performance of the adapted model since the selection defines the expressivity of the model and therefore the power of adaptive interventions. Following the analysis in [12], we select this subset to be the weight and bias terms in batch normalization layers while freezing the running statistics. This has been shown to yield enough expressivity while preventing overfitting. We then use AdamW optimizer with learning rate set to 0.0001, for the adaptation process and take 10 gradient steps, a number that has proven empirically optimal in our hyperparameter searches. Experiments We test TempT on the AffWild2 dataset and compare the results against the baseline model as well as another test time domain adaptation method, namely TENT [35]. We performed an extensive hyperparameter search on the adaptation parameters of TempT, such as the number of steps, learning rate, optimizer, etc., and report the performance of the best configuration in Tab. 2. Static models' performances are deterministic whereas for adaptation cases we report an average F1 score over 20 experiments to account for stochasticity arising from a random sampling of adaptation frames. We clearly see the positive effect of TempT on classification performance. One important observation of these results is the ability of adaptation to help a less complex model reach the performance of a much larger one. In this experiment, SE-ResNext-101 has 8 times the number of parameters of Resnet-18. Another observation of the results is the performance disruption that TENT introduces. It consistently hurt the performance of the baseline model and we argue that this is due to the highly correlated inputs that we have during test time, which is predicted in [6]. To further observe the changes that TempT induces, we also investigate time series generated by the model before and after adaptation. In Fig. 4 we provide such an example taken from 100 frame portion of a validation set video. On the top figure we provide unnormalized model outputs before and after adaptation, whereas bottom figure shows 'argmax' predictions. To create a cleaner top plot we omitted the classes that do not become the dominant prediction during this interval. From this visualization we can see, in a qualitative manner, that adaptation reduces the flickering behavior at the output and provides more coherent predictions over time, while increasing the F1 score for this particular video from 0. 39 understanding of this effect, we computed average number of decision changes before and after the adaptation on entire AffWild validation dataset. TempT reduces normalized number of changes (i.e. number of changes per frame) for a given video from 0.15 to 0.043. Conclusion and Future Work In our work, we explored a novel model-agnostic algorithm that can have real-life applications for similar tasks and showed that this adaptive method could enhance model performance without any additional means of supervision. On the other hand, performance variance due to stochasticity in the frame sampling process is a problem that needs to be addressed to obtain a more deterministic understanding of the limits and behavior of the algorithm. With such increased stability and the performance boost it brings, TempT can potentially enable more reliable use of models on edge devices while protecting user privacy. Figure 1 .Figure 2 . 12TempT Label distribution of Affwild2 train set Figure 3 . 3Label distribution of Affwild2 validation set L (z, y) = − log e zy−∆y e zy−∆y + j =y e zj j ∈ {1, . . . , k} Figure 4 . 4(top) Model outputs before (dashed) and after (solid) adaptation. (bottom) Model predictions before and after adaptation to 0.47. To have a quantitative Supervised TENT [35] TempT Table 2. Average F1 Score performances on validation setResnet-18 0.307 0.277 0.323 SE-ResNext-101 0.325 0.269 0.345 AcknowledgementsWe would like to thank all members of the Wall Lab for providing valuable feedback. The work was supported in part by funds to DPW from the National Institutes of Health (1R01LM013364-01, 1R01LM013083), the National Science Foundation (Award 2014232), Lucile Packard Foundation (Auxiliaries Endowment) the ISDB Transform Fund, and program grants from Stanford's Human-Centered Artificial Intelligence Program, and from the Wu Tsai Neurosciences Institute's Neuroscience:Translate Program. Vivit: A video vision transformer. Anurag Arnab, Mostafa Dehghani, Georg Heigold, Chen Sun, Mario Lučić, Cordelia Schmid, Proceedings of the IEEE/CVF international conference on computer vision. the IEEE/CVF international conference on computer visionAnurag Arnab, Mostafa Dehghani, Georg Heigold, Chen Sun, Mario Lučić, and Cordelia Schmid. Vivit: A video vision transformer. In Proceedings of the IEEE/CVF inter- national conference on computer vision, pages 6836-6846, 2021. 1 Learning imbalanced datasets with labeldistribution-aware margin loss. 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Zuxuan Wu, Xi Wang, Yu-Gang Jiang, Hao Ye, Xiangyang Xue, Proceedings of the 23rd ACM international conference on Multimedia. the 23rd ACM international conference on MultimediaZuxuan Wu, Xi Wang, Yu-Gang Jiang, Hao Ye, and Xi- angyang Xue. Modeling spatial-temporal clues in a hybrid deep learning framework for video classification. In Pro- ceedings of the 23rd ACM international conference on Mul- timedia, pages 461-470, 2015. 1 Aggregated residual transformations for deep neural networks. Saining Xie, Ross Girshick, Piotr Dollár, Zhuowen Tu, Kaiming He, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionSaining Xie, Ross Girshick, Piotr Dollár, Zhuowen Tu, and Kaiming He. Aggregated residual transformations for deep neural networks. 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{'abstract': "We introduce Temporal consistency for Test-time adaptation (TempT), a novel method for test-time adaptation on videos through the use of temporal coherence of predictions across sequential frames as a self-supervision signal.TempT is an approach with broad potential applications in computer vision tasks, including facial expression recognition (FER) in videos. We evaluate TempT's performance on the AffWild2 dataset. Our approach focuses solely on the unimodal visual aspect of the data and utilizes a popular 2D CNN backbone, in contrast to larger sequential or attention-based models used in other approaches. Our preliminary experimental results demonstrate that TempT has competitive performance compared to the previous years' reported performances, and its efficacy provides a compelling proof-of-concept for its use in various real-world applications.", 'arxivid': '2303.10536', 'author': ['Onur Cezmi Mutlu \nStanford University\n\n', 'Mohammadmahdi Honarmand \nStanford University\n\n', 'Saimourya Surabhi \nStanford University\n\n', 'Dennis P Wall dpwall@stanford.edu \nStanford University\n\n'], 'authoraffiliation': ['Stanford University\n', 'Stanford University\n', 'Stanford University\n', 'Stanford University\n'], 'corpusid': 257631932, 'doi': '10.48550/arxiv.2303.10536', 'github_urls': [], 'n_tokens_mistral': 9827, 'n_tokens_neox': 8378, 'n_words': 5029, 'pdfsha': 'adb47ff038ddff96c306ead838d820bc5e1e63fb', 'pdfurls': ['https://export.arxiv.org/pdf/2303.10536v2.pdf'], 'title': ['TempT: Temporal consistency for Test-time adaptation', 'TempT: Temporal consistency for Test-time adaptation'], 'venue': []}

arxiv

On Fully Dynamic Graph Sparsifiers 7 Apr 2016 Ittai Abraham David Durfee Georgia Institute of Technology University of Puerto Rico Rio Piedras Ioannis Koutis Sebastian Krinninger Max Planck Institute for Informatics Georgia Institute of Technology Richard Peng On Fully Dynamic Graph Sparsifiers 7 Apr 2016* VMware Research i We initiate the study of dynamic algorithms for graph sparsification problems and obtain fully dynamic algorithms, allowing both edge insertions and edge deletions, that take polylogarithmic time after each update in the graph. Our three main results are as follows. First, we give a fully dynamic algorithm for maintaining a (1 ± )-spectral sparsifier with amortized update time poly(log n, −1 ). Second, we give a fully dynamic algorithm for maintaining a (1 ± )-cut sparsifier with worst-case update time poly(log n, −1 ). Both sparsifiers have size n · poly(log n, −1 ). Third, we apply our dynamic sparsifier algorithm to obtain a fully dynamic algorithm for maintaining a (1 − )-approximate minimum cut in an unweighted, undirected, bipartite graph with amortized update time poly(log n, −1 ). Introduction Problems motivated by graph cuts are well studied in theory and practice. The prevalence of large graphs motivated sublinear time algorithms for cut based problems such as clustering [ST13, BBC + 12, ACL06, AP09, OV11,GT12]. In many cases such as social networks or road networks, these algorithms need to run on dynamically evolving graphs. In this paper, we study an approach for obtaining sublinear time algorithms for these problems based on dynamically maintaining graph sparsifiers. Recent years have seen a surge of interest in dynamic graph algorithms. On the one hand, very efficient algorithms, with polylogarithmic running time per update in the graph, could be found for some key problems in the field [HK99, HLT01, KKM13, OR10, NS13, BGS15, BHI15, BKS12, ACD + 16]. On the other hand, there are polynomial conditional lower bounds for many basic graph problems [AVW14,HKN + 15]. This leads to the question which problems can be solved with polylogarithmic update time. Another relatively recent trend in graph algorithmics is graph sparsification where we reduce the size of graphs while approximately preserving key properties such as the sizes of cuts [BK15]. These routines and their extensions to the spectral setting [ST11, BSS + 13] play central roles in a number of recent algorithmic advances [Mad10, She13, KLO + 14, PS14, ST14, KLP + 15, Pen16], often leading to graph algorithms that run in almost-linear time. In this paper, we study problems at the intersection of dynamic algorithms and graph sparsification, leveraging ideas from both fields. At the core of our approach are data structures that dynamically maintain graph sparsifiers in polylog n time per edge insertion or deletion. They are motivated by the spanner based constructions of spectral sparsifiers of Koutis [Kou14]. By modifying dynamic algorithms for spanners [BKS12], we obtain data structures that spend amortized polylog n per update. Our main result for spectral sparsifiers is: Theorem 1.1. Given a graph with polynomially bounded edge weights, we can dynamically maintain a (1 ± )-spectral sparsifier in poly(log n, −1 ) amortized time per edge insertion / deletion. When used as a black box, this routine allows us to run cut algorithms on sparse graphs instead of the original, denser network. Its guarantees interact well with most routines that compute minimum cuts or solve linear systems in the graph Laplacian. Some of them include: 1. min-cuts, sparsest cuts, and separators [She09], 2. eigenvector and heat kernel computations [OSV12] In many applications the full power of spectral sparsifiers is not needed, and it suffices to work with a cut sparsifier. As spectral approximations imply cut approximations, research in recent years has focused spectral sparsification algorithms [KL13, KLP12, KLM + 14, ZLO15,LS15,JK15]. In the dynamic setting however we get a strictly stronger result for cut sparsifiers than for spectral sparsifiers: we can dynamically maintain cut sparsifiers with polylogarithmic worst-case update time after each insertion / deletion. We achieve this by generalizing Koutis' sparsification paradigm [Kou14] and replacing spanners with approximate maximum spanning trees in the construction. While there are no non-trivial results for maintaining spanners with worst-case update time, spanning trees can be maintained with polylogarithmic worst-case update time by a recent breakthrough result [KKM13]. This allows us to obtain the following result for cut sparsifiers: Theorem 1.2. Given a graph with polynomially bounded edge weights, we can dynamically maintain a (1 ± )-cut sparsifier in poly(log n, −1 ) time per edge insertion / deletion in the worst case. We then explore more sophisticated applications of dynamic graph sparsifiers. A key property of these sparsifiers is that they have arboricity polylog n. This means the sparsifier is locally sparse, and can be represented as a union of spanning trees. This property is becoming increasingly important in recent works [NS13,PS16]: Peleg and Solomon [PS16] gave data structures for maintaining approximate maximum matchings on fully dynamic graphs with amortized cost parameterized by the arboricity of the graphs. We demonstrate the applicability of our data structures for designing better data structures on the undirected variant of the problem. Through a two-stage application of graph sparsifiers, we obtain the first non-separator based approach for dynamically maintaining (1 + )-approximate minimum cuts on fully dynamic graphs: Theorem 1.3. We can maintain an (1 − ) to the value of the maximum flow on a dynamically changing unweighted, undirected, bipartite graph, as well as query access to the associated minimum cut in poly(log n, −1 ) time per update. To obtain this result we give stronger guarantees for vertex sparsification in bipartite graphs, identical to the terminal cut sparsifier question addressed by Andoni, Gupta, and Krauthgamer [AGK14]. Our new analysis profits from the ideas we develop by going back and forth between combinatorial reductions and spectral sparsification. This allows us to analyze a vertex sampling process via a mirror edge sampling process, which is in turn much better understood. Overall, our algorithms bring together a wide range of tools from data structures, spanners, and randomized algorithms. We will provide more details on our routines, as well as how they relate to existing combinatorial and probabilistic tools in Section 3. Background Dynamic Graph Algorithms In this paper we consider undirected graphs G = (V, E) with n nodes and m edges that are either unweighted or have non-negative edge weights. We denote the weight of an edge e = (u, v) in a graph G by w G (e) or w G (u, v) and the ratio between the largest and the smallest edge weight by W . The weight w G (F ) of a set of edges F ⊆ E is the sum of the individual edge weights. We will assume that all weights are polynomially bounded because there are standard reductions from the general case using minimum spanning trees (e.g. [SS11] Section 10.2., [EES + 08] Theorem 5.2). Also, these contraction schemes in the data structure setting introduces another layer of complexity akin to dynamic connectivity, which we believe is best studied separately. A dynamic algorithm is a data structure for dynamically maintaining the result of a computation while the underlying input graph is updated periodically. We consider two types of updates: edge insertions and edge deletions. An incremental algorithm can handle only edge insertions, a decremental algorithm can handle only edge deletions, and a fully dynamic algorithm can handle both edge insertions and deletions. After every update in the graph, the dynamic algorithm is allowed to process the update to compute the new result. For the problem of maintaining a sparsifier, we want the algorithm to output the changes to the sparsifier (i.e., the edges to add to or remove from the sparsifier) after every update in the graph. Running Times and Success Probabilities The running time spent by the algorithm after every update is called update time. We distinguish between amortized and worst-case update time. A dynamic algorithm has amortized update time T (m, n, W ), if the total time spent after q updates in the graph is at most qT (m, n, W ). A dynamic algorithm has worst-case update time T (m, n, W ), if the total time spent after each update in the graph is at most T (m, n, W ). Here m refers to the maximum number of edges ever contained in the graph. All our algorithms are randomized. The guarantees we report in this paper (quality and size of sparsifier, and update time) will hold with high probability (w.h.p.), i.e. with probability at least 1 − 1/n c for some arbitrarily chosen constant c ≥ 1. These bounds are against an oblivious adversary who chooses its sequence of updates independently from the random choices made by the algorithm. In particular, this means that the adversary is not allowed to see the current edges of the sparsifier. As our composition of routines involve poly(n) calls, we will assume the composability of these w.h.p. bounds. Most of our update costs have the form O(log O(1) n −O(1) ), where is the approximation error. We will often state these as poly(log n, −1 ) when the exponents exceed 3, and explicitly otherwise. Cuts and Laplacians A cut U ⊆ V of G is a subset of nodes whose removal makes G disconnected. We denote by ∂ G (U ) the edges crossing the cut U , i.e., the set of edges with one endpoint in U and one endpoint in V \ U . The weight of the cut U is w G (∂ G (U )). An edge cut F ⊆ E of G is a a subset of edges whose removal makes G disconnected and the weight of the edge cut F is w G (F ). For every pair of nodes u and v, the local edge connectivity λ G (u, v) is the weight of the minimum edge cut separating u and v. If G is unweighted, then λ G (u, v) amounts to the number of edges that have to be removed from G to make u and v disconnected. For studying the spectral properties of G we treat the graph as a resistor network. For every edge e ∈ E we define the resistance of e as r G (e) = 1/w G (e). Assuming some arbitrary order v 1 , . . . v n on the nodes, the Laplacian matrix L G of an undirected graph G is the n × n matrix that in row i and column j contains the negated resistance −r G (v i , v j ) of the edge (v i , v j ) and in the i-th diagonal entry contains the weighted degree n j=1 r G (v i , v j ) of node v i . Note that Laplacian matrices are symmetric. The matrix L e of an edge e of G is the n × n Laplacian matrix of the subgraph of G containing only the edge e. It is 0 everywhere except for a 2 × 2 submatrix. The effective resistance R G (e) of an edge e = (v, u) is defined as the potential difference that has to be applied to u and v to drive one unit of current through the network. A closed form expression of the effective resistance is R G (e) = b u,v L † G b u,v , where L † G is the Moore-Penrose pseudo-inverse of the Laplacian matrix of G and b u,v is the n-dimensional vector that is 1 at position u, −1 at position v, and 0 otherwise. Graph Approximations The goal of graph sparsification is to find sparse subgraphs, or similar small objects, that approximately preserve certain metrics of the graph. We first define spectral sparsifiers where we require that Laplacian quadratic form of the graph is preserved approximately. Spectral sparsifiers play a pivotal role in fast algorithms for solving Laplacian systems, a special case of linear systems. Definition 2.1. A (1 ± )-spectral sparsifier H of a graph G is a subgraph of G with weights w H such that for every vector x ∈ R n (1 − )x L H x ≤ x L G x ≤ (1 + )x L H x . Using the Loewner ordering on matrices this condition can also be written as (1 − )L H L G (1 + )L H . An n × n matrix A is positive semi-definite, written as A 0, if x Ax ≥ 0 for all x ∈ R n . For two n × n matrices A and B we write A B as an abbreviation for A − B 0. Note that x L G x = (u,v)∈E w(u, v)(x(u) − x(v)) 2 where the vector x is treated as a function on the nodes and x(v) is the value of x for node v. A special case of such a function on the nodes is given by the binary indicator vector x U associated with a cut U , where x U (v) = 1 is v ∈ U and 0 otherwise. If limited to such indicator vectors, the sparsifier approximately preserves the value of every cut. Definition 2.2. A (1 ± )-cut sparsifier H of a graph G is a subgraph of G with weights w H such that for every subset U ⊆ V (1 − )w H (∂ H (U )) ≤ w G (∂ G (U )) ≤ (1 + )w H (∂ H (U )) . Sampling Schemes for Constructing Sparsifiers Most efficient constructions of sparsifiers are randomized, partly because when G is the complete graph, the resulting sparsifier needs to be an expander. These randomized schemes rely on importance sampling, which for each edge: 1. Keeps it with probability p e , 2. If the edge is kept, its weight is rescaled to we pe . A crucial property of this process is that the edge's expectation is preserved. As both cut and spectral sparsifiers can be viewed as preserving sums over linear combinations of edge weights, each of these terms have correct expectation. The concentration of such processes can then be bounded using either matrix concentration bounds in the spectral case [Tro12,SS11], or a variety of combinatorial arguments [BK15]. Our algorithms in this paper will use an even simpler version of this importance sampling scheme: all of our p e 's will be set to either 1 or 1/2. This scheme has a direct combinatorial interpretation: 1. Keep some of the edges. 2. Take a random half of the other edges, and double the weights of the edges kept. Note that composing such a routine O(log n) times gives a sparsifier, as long as the part we keep is small. So the main issue is to figure out how to get a small part to keep. Spanning Trees and Spanners A spanning forest F of G is a forest (i.e., acyclic graph) on a subset of the edges of G such that every pair of nodes that is connected in G is also connected in F . A minimum/maximum spanning forest is a spanning forest of minimum/maximum total weight. For every pair of nodes u and v we denote by d G (u, v) the distance between u and v (i.e., the length of the shortest path connecting u and v) in G with respect to the resistances. The graph sparsification concept also exists with respect to distances in the graph. Such sparse subgraphs that preserves distances approximately are called spanners. Definition 2.3. A spanner of stretch α, or short α-spanner, (where α ≥ 1) of an undirected (possibly weighted) graph G is a subgraph H of G such that, for every pair of nodes u and v, d H (u, v) ≤ αd G (u, v). Overview and Related Work Dynamic Spectral Sparsifier We first develop a fully dynamic algorithm for maintaining a spectral sparsifier of a graph with polylogarithmic amortized update time. Related Work. Spectral sparsifiers play important roles in fast numerical algorithms [BSS + 13]. Spielman and Teng were the first to study these objects [ST11]. Their algorithm constructs a (1 ± )-spectral sparsifier of size O(n · poly(log n, −1 )) in nearly linear time. This result has seen several improvements in recent years [SS11,dHS16,Zou12,ZLO15]. The state of the art in the sequential model is an algorithm by Lee and Sun [LS15] that computes a (1 ± )-spectral sparsifier of size O(n −2 ) in nearly linear time. Most closely related to the data structural question are streaming routines, both in one pass incremental [KL13], and turnstile [KLM + 14]. A survey of sparsifier constructions is given in [BSS + 13]. Many of these methods rely on solving linear systems built on the graph, for which there approaches with a combinatorial flavor using low-stretch spanning trees [KOS + 13, LS13] and purely numerical solvers relying on sparsifiers [PS14] or recursive constructions [KLP + 15]. We build on the spectral sparsifier obtained by a simple, combinatorial construction of Koutis [Kou14], which initially was geared towards parallel and distributed implementations. Sparsification Framework. In our framework we determine 'sampleable' edges by using spanners to compute a set of edges of bounded effective resistance. From these edges we then sample by coin flipping to obtain a (moderately sparser) spectral sparsifier in which the number of edges has been reduced by a constant fraction. This step can then be iterated a small number of times in order to compute the final sparsifier. Concretely, we define a t-bundle spanner B = T 1 ∪ · · · ∪ T t (for a suitable, polylogarithmic, value of t) as a sequence of spanners T 1 , . . . , T t where the edges of each spanner are removed from the graph before computing the next spanner, i.e., T 1 is a spanner of G, T 2 is a spanner of G \ T 1 , etc; here each spanner has stretch O(log n). We then sample each non-bundle edge in G \ B with some constant probability p and scale the edge weights of the sampled edges proportionally. The t-bundle spanner serves as a certificate for small resistance of the non-bundle edges in G \ B as it guarantees the presence of t disjoint paths of length at most the stretch of the spanner. Using this property one can apply matrix concentration bounds to show the t-bundle together with the sampled edges is a moderately sparse spectral sparsifier. We repeat this process of 'peeling off' a t-bundle from the graph and sampling from the remaining edges until the graph is sparse enough (which happens after a logarithmic number of iterations). Our final sparsifier consists of all t-bundles together with the sampled edges of the last stage. Towards a Dynamic Algorithm. To implement the spectral sparsification algorithm in the dynamic setting we need to dynamically maintain a t-bundle spanner. Our approach to this problem is to run t different instances of a dynamic spanner algorithm, in order to separately maintain a spanner T i for each graph G i = G \ i−1 j=1 T j , for 1 ≤ i ≤ t. A natural first idea would be to use the existing fully dynamic spanner algorithm of Baswana, Khurana, and Sarkar [BKS12] in a black-box fashion in order to separately maintain each spanner of a t-bundle. However, we do not know how to do this because of the following obstacle. A single update in G might lead to several changes of edges in the spanner T 1 , an average of Ω(log n) according to the amortized upper bound. This means that the next instance of the fully dynamic spanner algorithm which is used for maintaining T 2 , not only has to deal with the deletion in G but also the artificially created updates in G 2 = G \ T 1 . This of course propagates to more updates in all graphs G i . Observe also that any given update in G t caused by an update in G, can be requested repeatedly, as a result of subsequent updates in G. Without further guarantees, it seems that with this approach we can only hope for an upper bound of O(log t−1 n) (on average) on the number of changes to be processed for updating G t after a single update in G. That is too high because the sparsification algorithm requires us to take t = Ω(log n).Our solution to this problem lies in a substantial modification of the dynamic spanner algorithm in [BKS12] outlined below. Dynamic Spanners with Monotonicity. The spanner algorithm of Baswana et al. [BKS12] is at its core a decremental algorithm (i.e., allowing only edge deletions in G), which is subsequently leveraged into a fully dynamic algorithm by a black-box reduction. We follow the same approach by first designing a decremental algorithm for maintaining a t-bundle spanner. This is achieved by modifying the decremental spanner algorithm so so that, additional to its original guarantees, it has the following monotonicity property: Every time an edge is added to the spanner T , it stays in T until it is deleted from G. Recall that we initially want to maintain a t-bundle spanner T 1 , . . . , T t under edge deletions only. In general, whenever an edge is added to T 1 , it will cause its deletion from the graph G \ T 1 for which the spanner T 2 is maintained. Similarly, removing an edge from T 1 causes its insertion into G \ T 1 , unless the edge is deleted from G. This is precisely what the monotonicity property guarantees: that an edge will not be removed from T 1 unless deleted from G. The consequence is that no edge insertion can occur for G 2 = G \ T 1 . Inductively, no edge is ever inserted into G i , for each i. Therefore the algorithm for maintaining the spanner T i only has to deal with edge deletions from the graph G i , thus it becomes possible to run a different instance of the same decremental spanner algorithm for each G i . A single deletion from G can still generate many updates in the bundle. But for each i the instance of the dynamic spanner algorithm working on G i can only delete each edge once. Furthermore, we only run a small number t of instances. So the total number of updates remains bounded, allowing us to claim the upper bound on the amortized update time. In addition to the modification of the dynamic spanner algorithm, we have also deviated from Koutis' original scheme [Kou14] in that we explicitly 'peel off' each iteration's bundle from the graph. In this way we avoid that the t-bundles from different iterations share any edges, which seems hard to handle in the decremental setting we ultimately want to restrict ourselves to. The modified spanner algorithm now allows us to maintain t-bundles in polylogarithmic update time, which is the main building block of the sparsifier algorithm. The remaining parts of the algorithm, like sampling of the non-bundle edges by coin-flipping, can now be carried out in the straightforward way in polylogarithmic amortized update time. At any time, our modified spanner algorithm can work in a purely decremental setting. As mentioned above, the fully dynamic sparsifier algorithm is then obtained by a reduction from the decremental sparsifier algorithm. Dynamic Cut Sparsifier We then give dynamic algorithms for maintaining a (1 ± )-cut sparsifier. We obtain a fully dynamic algorithm with polylogarithmic worst-case update time by leveraging a recent worst-case update time algorithm for dynamically maintaining a spanning tree of a graph [KKM13]. As mentioned above, spectral sparsifiers are more general than cut sparsifiers. The big advantage of studying cut sparsification as a separate problem is that we can achieve polylogarithmic worst-case update time, where the update time guarantee holds for each individual update and is not amortized over a sequence of updates. Related Work. In the static setting, Benczúr and Karger [BK15] developed an algorithm for computing a (1 ± )-cut sparsifier of size O(n · poly(log n, −1 )) in nearly linear time. Their approach is to first compute a value called strength for each edge and then sampling each edge with probability proportional to its strength. Their proof uses a cut-counting argument that shows that the majority of cuts are large, and therefore less likely to deviate from their expectation. A union bound over these (highly skewed) probabilities then gives the overall w.h.p. success bound. This approach was refined by Fung et al. [FHH + 11] who show that a cut sparsifier can also be obtained by sampling each edge with probability inversely proportional to its (approximate) local edge connectivity, giving slightly better guarantees on the sparsifier. Our Framework. The algorithm is based on the observation that the spectral sparsification scheme outlined above in Section 3.1. becomes a cut sparsification algorithm if we simply replace spanners by maximum weight spanning trees (MSTs). This is inspired by sampling according to edge connectivities; the role of the MSTs is to certify lower bounds on the edge connectivities. We observe that the framework does not require us to use exact MSTs. For our t-bundles we can use a relaxed, approximate concept that we call α-MST that. Roughly speaking, an α-MST guarantees a 'stretch' of α in the infinity norm and, as long as it is sparse, does not necessarily have to be a tree. Similarly to before, we define a t-bundle α-MST B as the union of a sequence of α-MSTs T 1 , . . . T t where the edges of each tree are removed from the graph before computing the next α-MST. The role of α-MST is to certify uniform lower bounds on the connectivity of edges; these bounds are sufficiently large to allow uniform sampling with a fixed probability. This process of peeling and sampling is repeated sufficiently often and our cut sparsifier then is the union of all the t-bundle α-MSTs and the non-bundle edges remaining after taking out the last bundle. Thus, the cut sparsifier consists of a polylogarithmic number of α-MSTs and a few (polylogarithmic) additional edges. This means that for α-MSTs based on spanning trees, our cut sparsifiers are not only sparse, but also have polylogarithmic arboricity, which is the minimum number of forests into which a graph can be partitioned. Simple Fully Dynamic Algorithm. Our approach immediately yields a fully dynamic algorithm by using a fully dynamic algorithm for maintaining a spanning forest. Here we basically have two choices. Either we use the randomized algorithm of Kapron, King, and Mountjoy [KKM13] with polylogarithmic worst-case update time. Or we use the deterministic algorithm of Holm, de Lichtenberg, and Thorup [HLT01] with polylogarithmic amortized update time. The latter algorithm is slightly faster, at the cost of providing only amortized update-time guarantees. A t-bundle 2-MST can be maintained fully dynamically by running, for each of the log W weight classes of the graph, t instances of the dynamic spanning tree algorithm in a 'chain'. An important observation about the spanning forest algorithm is that with every update in the graph, at most one edge is changed in the spanning forest: If for example an edge is deleted from the spanning forest, it is replaced by another edge, but no other changes are added to the tree. Therefore a single update in G can only cause one update for each graph G i = G \ i−1 j=1 T j and T i . This means that each instance of the spanning forest algorithm creates at most one 'artificial' update that the next instance has to deal with. In this way, each dynamic spanning forest instance used for the t-bundle has polylogarithmic update time. As t = poly log n, the update time for maintaining a t-bundle is also polylogarithmic. The remaining steps of the algorithm can be carried out dynamically in the straightforward way and overall give us polylogarithmic worst-case or amortized update time. A technical detail of our algorithm is that the high-probability correctness achieved by the Chernoff bounds only holds for a polynomial number of updates in the graph. We thus have to restart the algorithm periodically. This is trivial when we are shooting for an amortized update time. For a worst-case guarantee we can neither completely restart the algorithm nor change all edges of the sparsifier in one time step. We therefore keep two instances of our algorithm that maintain two sparsifiers of two alternately growing and shrinking subgraphs that at any time partition the graph. This allows us to take a blend of these two subgraph sparsifiers as our end result and take turns in periodically restarting the two instances of the algorithm. (1 + )-Approximate Undirected Bipartite Flow We then study ways of utilizing our sparsifier constructions to give routines with truly sublinear update times. The problem that we work with, approximate maximum flow on bipartite graphs. Formally, we want to maintain an approximate maximum flow problem on a bipartite graph G A,B = (A, B, E) with demand −1 and 1 on each vertex in A and B, respectively. All edges are unit weight and we dynamically insert and delete edges. The maximum flow minimum cut theorem states that the objective here equals to the minimum s − t cut or maximum s − t flow in G, which will be G A,B where we add vertices s and t, and connect each vertex in A to s and each vertex in B to t. The only dynamic changes in this graph will be in edges between A and B. As our algorithms builds upon cut sparsifiers, and flow sparsifiers [KLO + 14] is a much more difficult construction, we will focus on the cut version throughout this paper. This problem is motivated by the dynamic approximate maximum matching problem, which differs in that the edges are directed, and oriented from A to B. This problem has received much attention recently [OR10,BGS15,NS13,GP13,PS16,BS16], and led to the key definition of low arboricity graphs [NS13,PS16]. On the other hand, bipartite graphs are known to be difficult to sparsify: the directed reachability matrix from A to B can encode Θ(n 2 ) bits of information. As a result, we study the undirected variant of this problem instead, with the hope that this framework can motivate other definitions of sparsification suitable for wider classes of graphs. Another related line of work are are fully dynamic algorithm for maintaining the global minimum cut [Tho07,TK00] with update time O( √ n polylog n). As there is significant differences between approximating global minimum cuts and st-minimum cuts in the static setting [Kar00], we believe that there are some challenges to adapting these techniques for this problem. The data structure by Thorup [Tho07] can either maintain up to polylogarithmic global edge connectivity exactly or, with high probability, arbitrary global edge connectivity with an approximation of 1 + o(1). The algorithms also maintain concrete (approximate) minimum cuts, where in the latter algorithm the update time increases to O( √ m polylog n) (and cut edges can be listed in time O(log n) per edge). Thorup's result was preceded by a randomized algorithm with worse approximation ratio for the global edge connectivity by Thorup and Karger [TK00] with update time O( √ n polylog n). At the start of Section 6 we will show that this formulation is in fact different than matchings. On the other hand, our incorporation of sparsifiers for maintaining solutions to this problem relies on three properties that hold in a variety of other settings: 1. The static version can be efficiently approximated. 2. The objective can be sparsified. 3. A small answer (for which the algorithm's current approximation may quickly become suboptimal) means the graph also has a small vertex cover. 4. The objective does not change much per each edge update. As with algorithms for maintain high quality matchings [GP13,PS16], our approach aims to get a small amortized cost by keeping the same minimum s − t cut for many consecutive dynamic steps. Specifically, if we have a minimum s − t cut of size (2 + 2 )OP T , then we know this cut will remain (2 + ) approximately optimal for 2 OP T dynamic steps. This allows us to only compute a new minimum s − t cut every 2 OP T dynamic steps. As checking for no edges would be an easy boundary case, we will assume throughout all the analysis that OP T > 0. To obtain an amortized O(poly(log n, −1 )) update cost, it suffices for this computation to take O(OP T · poly(log n, −1 )) time. In other words, we need to solve approximate maximum flow on a graph of size O(OP T · poly(log n, −1 )). Here we incorporate sparsifiers using the other crucial property used in matching data structures [OR10,GP13,PS16]: if OP T is small, G also has a small vertex cover. Lemma 3.1. The minimum vertex cover MVC in G has size at most OP T + 2 where OP T is the size of the minimum s − t cut in G. We utilize the low arboricity of our sparsifiers to find a small vertex cover with the additional property that all non-cover vertices have small degree. Then directly reducing the graph onto this (much) smaller vertex cover gives a (2 + ) approximation. Maintaining a sparsifier of this routine again leads to an overall routine that maintains a (2 + )-approximation in polylog n per update time, which we show in Section 6. The large approximation ratio motivated us to reexamine the sparsification routines, namely the one of reducing the graph to one whose size is proportional to the vertex cover. This is directly related to the terminal cut sparsifiers studied in [AGK14,KK15]. However, for an update time of poly(log n, −1), it is crucial for the vertex sparsifier to have size O(|VC| poly(log n, −1 )). As a result, instead of doing a direct union bound over all 2 |VC| cuts to get a size of poly(|VC|) as in [AGK14], we need to invoke cut counting as with cut sparsifier constructions. This necessitates the use of objects similar to t-bundles to identify edges with small connectivity. This leads to a sampling process motivated by the (2 + )-approximate routine, but works on vertices instead of edges. By relating the processes, we are able to absorb the factor 2 error into sparsifier size. In Section 7, we formalize this process, as well as its guarantees on graphs with bounded weights. However, the natural scheme of bucketing by edge weight is difficult to analyze because a sampled vertex could have non-zero degree in multiple buckets. Instead, we work around this issue via a pre-processing scheme on G that creates an approximation so that all vertices outside of VC have degree polylog n. This scheme is motivated in part by the weighted expanders constructions from [KLP + 15]. Bucketing after this processing step ensures that each vertex belongs to a unique bucket. In terms of a static sparsifier on terminals, the result that is most comparable to results from previous works is: Corollary 3.2. Given any graph G = (V, E) with weights within a O(poly(n)) factor, a vertex cover VC of G, where X = V \ VC, and some error , we can build a terminal-cut-sparsifier H with O(|VC| poly(log n, −1 )) vertices in O(m · poly(log n, −1 )) work. The concept of terminal-cut-sparsifier will be equivalent to that in [AGK14], and will be given formal treatment in Section 8 and Corollary 3.2 will be proven in Section 8.1.3. The idea behind this sparsification will be to maintain an -approximation on all minimally extended cuts of the terminals, which will be the vertex cover in our construction. These algorithmic extensions, as well as their incorporation into data structures are discussed in Section 8. Turning this into a dynamic routine leads to the result described in Theorem 1.3: a (1 + )-approximate solution that can be maintained in time polylog(n) per update. Discussion Graph Sparsification. We use a sparsification framework in which we 'peel off' bundles of sparse subgraphs to determine 'sampleable' edges, from which we then sample by coin flipping. This leads to combinatorial and surprisingly straightforward algorithms for maintaining graph sparsifiers. Additionally, this gives us low-arboricity sparsifiers; a property that we exploit for our main application. Although spectral sparsification is more general than cut sparsification. Our treatment of cut sparsification has two motivations. First, we can obtain stronger running time guarantees. Second, our sparsifier for the (1 + ) approximate algorithm hinges upon improved routines for vertex sparsification, a concept which leads to different objects in the spectral setting. Dynamic Graph Algorithms. In our sparsification framework we sequentially remove bundles of sparse subgraphs to determine 'sampleable' edges. This leads to 'chains' of dynamic algorithms where the output performed by one algorithm might result in updates to the input of the next algorithm. This motivates a more fine-grained view on of dynamic algorithms with the goal of obtaining strong bounds on the number of changes to the output. Future Work. The problem whether spectral sparsifiers can be maintained with polylogarithmic worst-case update time remains open. Our construction goes via spanners and therefore a natural question is whether spanners can be maintained with worst-case update time. Maybe there are also other more direct ways of maintaining the sparsifier. A more general question is whether we can find more dynamic algorithms for numerical problems. Our sparsification framework for peeling off subgraphs and uniformly sampling from the remaining edges is very general. Are there other sparse subgraphs we could start with in the peeling process? Which properties do the sparsifiers obtained in this way have? In particular, it would be interesting to see whether our techniques can be generalized to flow sparsifiers [KLO + 14, AGK14]. The combination of sparsifiers with density-sensitive approaches for dynamic graph data structures [NS13,PS16] provides an approach for obtaining poly(log, −1 ) update times. We believe this approach can be generalized to other graph cut problems. In particular, the flow networks solved for balanced cuts and graph partitioning are also bipartite and undirected, and therefore natural directions for future work. Dynamic Spectral Sparsifier In this section we give an algorithm for maintaining a spectral sparsifier under edge deletions and insertions with polylogarithmic amortized update time. The main result of this section is as follows. After giving an overview of our algorithm, we first explain our spectral sparsification scheme in a static setting and prove its properties. Subsequently, we show how we can dynamically maintain the edges of such a sparsifier by making this scheme dynamic. Algorithm Overview Sparsification Framework. In our framework we determine 'sampleable' edges by using spanners to compute a set of edges of bounded effective resistance. From these edges we then sample by coin flipping to obtain a (moderately sparser) spectral sparsifier in which the number of edges has been reduced by a constant fraction. This step can then be iterated a small number of times in order to compute the final sparsifier. Concretely, we define a t-bundle spanner B = T 1 ∪ · · · ∪ T t (for a suitable, polylogarithmic, value of t) as a sequence of spanners T 1 , . . . , T t where the edges of each spanner are removed from the graph before computing the next spanner, i.e., T 1 is a spanner of G, T 2 is a spanner of G \ T 1 , etc; here each spanner has stretch O(log n). We then sample each non-bundle edge in G \ B with some constant probability p and scale the edge weights of the sampled edges proportionally. The t-bundle spanner serves as a certificate for small resistance of the non-bundle edges in G \ B as it guarantees the presence of t disjoint paths of length at most the stretch of the spanner. Using this property one can apply matrix concentration bounds to show the t-bundle together with the sampled edges is a moderately sparse spectral sparsifier. We repeat this process of 'peeling off' a t-bundle from the graph and sampling from the remaining edges until the graph is sparse enough (which happens after a logarithmic number of iterations). Our final sparsifier consists of all t-bundles together with the sampled edges of the last stage. Towards a Dynamic Algorithm. To implement the spectral sparsification algorithm in the dynamic setting we need to dynamically maintain a t-bundle spanner. Our approach to this problem is to run t different instances of a dynamic spanner algorithm, in order to separately maintain a spanner T i for each graph G i = G \ i−1 j=1 T j , for 1 ≤ i ≤ t. A natural first idea would be to use the existing fully dynamic spanner algorithm of Baswana, Khurana, and Sarkar [BKS12] in a black-box fashion in order to separately maintain each spanner of a t-bundle. However, we do not know how to do this because of the following obstacle. A single update in G might lead to several changes of edges in the spanner T 1 , an average of Ω(log n) according to the amortized upper bound. This means that the next instance of the fully dynamic spanner algorithm which is used for maintaining T 2 , not only has to deal with the deletion in G but also the artificially created updates in G 2 = G \ T 1 . This of course propagates to more updates in all graphs G i . Observe also that any given update in G t caused by an update in G, can be requested repeatedly, as a result of subsequent updates in G. Without further guarantees, it seems that with this approach we can only hope for an upper bound of O(log t−1 n) (on average) on the number of changes to be processed for updating G t after a single update in G. That is too high because the sparsification algorithm requires us to take t = Ω(log n).Our solution to this problem lies in a substantial modification of the dynamic spanner algorithm in [BKS12] outlined below. Dynamic Spanners with Monotonicity. The spanner algorithm of Baswana et al. [BKS12] is at its core a decremental algorithm (i.e., allowing only edge deletions in G), which is subsequently leveraged into a fully dynamic algorithm by a black-box reduction. We follow the same approach by first designing a decremental algorithm for maintaining a t-bundle spanner. This is achieved by modifying the decremental spanner algorithm so so that, additional to its original guarantees, it has the following monotonicity property: Every time an edge is added to the spanner T , it stays in T until it is deleted from G. Recall that we initially want to maintain a t-bundle spanner T 1 , . . . , T t under edge deletions only. In general, whenever an edge is added to T 1 , it will cause its deletion from the graph G \ T 1 for which the spanner T 2 is maintained. Similarly, removing an edge from T 1 causes its insertion into G \ T 1 , unless the edge is deleted from G. This is precisely what the monotonicity property guarantees: that an edge will not be removed from T 1 unless deleted from G. The consequence is that no edge insertion can occur for G 2 = G \ T 1 . Inductively, no edge is ever inserted into G i , for each i. Therefore the algorithm for maintaining the spanner T i only has to deal with edge deletions from the graph G i , thus it becomes possible to run a different instance of the same decremental spanner algorithm for each G i . A single deletion from G can still generate many updates in the bundle. But for each i the instance of the dynamic spanner algorithm working on G i can only delete each edge once. Furthermore, we only run a small number t of instances. So the total number of updates remains bounded, allowing us to claim the upper bound on the amortized update time. In addition to the modification of the dynamic spanner algorithm, we have also deviated from Koutis' original scheme [Kou14] in that we explicitly 'peel off' each iteration's bundle from the graph. In this way we avoid that the t-bundles from different iterations share any edges, which seems hard to handle in the decremental setting we ultimately want to restrict ourselves to. The modified spanner algorithm now allows us to maintain t-bundles in polylogarithmic update time, which is the main building block of the sparsifier algorithm. The remaining parts of the algorithm, like sampling of the non-bundle edges by coin-flipping, can now be carried out in the straightforward way in polylogarithmic amortized update time. At any time, our modified spanner algorithm can work in a purely decremental setting. As mentioned above, the fully dynamic sparsifier algorithm is then obtained by a reduction from the decremental sparsifier algorithm. Spectral Sparsification As outlined above, iteratively 'peels off' bundles of spanners from the graph. Definition 4.2. A t-bundle α-spanner (where t ≥ 1, α ≥ 1) of an undirected graph G is the union T = k i=1 T i of a sequence of graphs T 1 , . . . , T k such that, for every 1 ≤ i ≤ k, T i is an α-spanner of G \ i−1 j=1 T j . The algorithm for spectral sparsification is presented in Figures 1 and 2. Algorithm Light-Spectral-Sparsify computes a moderately sparser (1± )-spectral sparsifier. Algorithm Spectral-Sparsify takes a parameter ρ and computes the sparsifier in k = log ρ iterations of Light-Spectral-Sparsify. We will now prove the properties of these algorithms. We first need the following lemma that shows how t-bundle spanners can be used to bound effective resistances. We highlight the main intuition of this crucial observation in our proof sketch. Lemma 4.3 ([Kou14]). Let G be a graph and B be a t-bundle α-spanner of G. For every edge e of G \ B, we have w G (e) · R G (e) ≤ α t which implies that w G (e) · L e α t · L G where L e is the n × n Laplacian of the unweighted edge e. Sketch. Fix some edge e = (u, v) of G \ B and let T 1 , . . . T t denote the (pairwise disjoint) α-spanners contained in B. For every 1 ≤ i ≤ t, let π i denote the shortest path from u to v in T i . The length of the path π in T i exceeds the distance from u to v in G \ i−1 j=1 T j by at most a factor of α (property of the spanner T i ). Since e is contained in G \ B, the latter distance is at most the resistance of the edge e as we have defined distances as the length of shortest paths with respect to the resistances of the edges. Consider each path π i as a subgraph of G and let Π be the subgraph consisting of all paths π i . Observe that Π consists of a parallel composition of paths, which in turn consists of a serial composition of edges, the we can view as resistors. We can now apply the well-known rules for serial and parallel composition for computing effective resistances and get the desired bounds. Our second tool in the analysis the following variant [Har12] of a matrix concentration inequality by Tropp [Tro12]. Theorem 4.4. Let Y 1 , . . . , Y k be independent positive semi-definite matrices of size n × n. Let Y = k i=1 Y i and Z = E [Y ]. Suppose Y i RZ, where R is a scalar, for every 1 ≤ i ≤ k. Then for all ∈ [0, 1] P k i=1 Y i (1 − )Z ≤ n · exp(− 2 /2R) P k i=1 Y i (1 + )Z ≤ n · exp(− 2 /3R) Given these facts we can now prove the following Lemma which is a slight generalization of a Lemma in [Kou14]. As the proof is quite standard we have moved it to Appendix A (together with the proofs of the subsequent two lemmas). For applying the lemma in our dynamic algorithm it is crucial that the input graph (which might be generated by another randomized algorithm) is independent of the random choices of algorithm Light-Spectral-Sparsify. Lemma 4.5. The output H of Light-Spectral-Sparsify is a (1 ± )-spectral sparsifier with probability at least 1 − n −(c+1) for any input graph G that is independent of the random choices of the algorithm. (G, ) 1. t ← 12(c + 1)α −2 ln n for some absolute constant c. Light-Spectral-Sparsify 2. let B = t j=1 T j be a t-bundle α-spanner of G 3. H := B 4. for each edge e ∈ G \ B1. k ← log ρ 2. G 0 ← G 3. B 0 ← (V, ∅) 4. for i = 1 to k (a) (H i , B i ) ← Light-Spectral-Sparsify(G i−1 , c, /(2k)) (b) G i ← H i \ B i (c) if G i has less than (c + 1) ln n edges then break By iteratively applying the sparsification of Light-Spectral-Sparsify as done in Spectral-Sparsify we obtain sparser and sparser cut sparsifiers. (* break loop *) 5. H ← 1≤j≤i B j ∪ G i 6. return (H, {B j } i j=1 , G i ) Lemma 4.6. The output H of algorithm Spectral-Sparsify is a (1 ± )-spectral sparsifier with probability at least 1 − 1/n c+1 for any input graph G that is independent of the random choices of the algorithm. Lemma 4.7. With probability at least 1−2n −c , the number of iterations before algorithm Spectral-Sparsify terminates is min{ log ρ , log m/((c + 1) log n) }. Moreover the size of H is O   1≤j≤i |B i | + m/ρ + c log n   , and the size of the third output of the graph is at most max{O(c log n), O(m/ρ)}. We conclude that with probability at least 1 − n −c our construction yields a (1 ± )-spectral sparsifier that also has the properties of Lemma 4.7. Typically, the t-bundle spanners will consist of a polylogarithmic number of spanners of size O(n poly log n) and thus the resulting spectral sparsifier will have size O(n poly log n, −1 + m/ρ). In each of the at most log n iterations the weight of the sampled edges is increased by a factor of 4. Thus, the ratio between the largest and the smallest edge weight in H is at most by a factor of O(n) more than in G, i.e., O(nW ). Decremental Spanner with Monotonicity Property We first develop the decremental spanner algorithm, which will give us a (log n)-spanner of size O(n poly (log n)) with a total update time of O(m poly (log n)). Our algorithm is a careful modification of the dynamic spanner algorithm of Baswana et al. [BKS12] having the following additional monotonicity property: Every time an edge is added to H, it stays in H until it is deleted from G by the adversary. Formally, we will prove the following theorem. Lemma 4.8. For every k ≥ 2 and every 0 < ≤ 1, there is a decremental algorithm for maintaining a (1 + )(2k − 1)-spanner H of expected size O(k 2 n 1+1/k log n log 1+ W ) for an undirected graph G with non-negative edge weights that has an expected total update time of O(k 2 m log n), where W is the ratio between the largest and the smallest edge weight in G. Additional H has the following property: Every time an edge is added to H, it stays in H until it is deleted from G. The bound on the expected size and the expected running time hold against an oblivious adversary. It would be possible to enforce the monotonicity property for any dynamic spanner algorithm by simply overriding the algorithms' decision for removing edges from the spanner before they are deleted from G. Without additional arguments however, the algorithm's bound on the size of the spanner might then not hold anymore. In particular, we do not know how obtain a version of the spanner of Baswana et al. that has the monotonicity property without modifying the internals of the algorithm. Similar to Baswana et al. [BKS12] we actually develop an algorithm for unweighted graphs and then extend it to weighted graphs as follows. Let W be the ratio of the largest to the smallest edge weight in G. Partition the edges into log 1+ W subgraphs based on their weights and maintain a (2k − 1)-spanner ignoring the weights. The union of these spanners will be a (1 + )(2k − 1)-spanner of G and the size increases by a factor of log 1+ W compared to the unweighted version. The update time stays the same as each update in the graph is performed only in one of the log 1+ W subgraphs. Therefore we assume in the following that G is an unweighted graph. Algorithm and Running Time We follow the approach of Baswana et al. and first explain how to maintain a clustering of the nodes and then define our spanner using this clustering. Clustering. Consider an unweighted undirected graph G = (V, E) undergoing edge deletions. Let S ⊆ V be a subset of the nodes used as cluster centers. Furthermore, consider a permutation σ on the set of nodes V and an integer i ≥ 0. The goal is to maintain a clustering C S,σ,i consisting of disjoint clusters with one cluster C S,σ,i [s] ⊆ V for every s ∈ S. Every node within distance i to the nodes in S is assigned to the cluster of its closest node in S, where ties are broken according to the permutation σ. More formally, v ∈ C S,σ,i [s] if and only if • d G (v, s) ≤ i and • for every s ∈ S \ {s} either -d G (v, s) < d G (v, s ) or -d G (v, s) = d G (v, s ) and σ(s) < σ (s). Observe that each cluster C S,σ,i [s] of a node s ∈ S can be organized as a tree consisting of shortest paths to s. We demand that in this tree every node v chooses the parent that comes first in the permutation σ among all candidates (i.e., among the nodes that are in the same cluster C i [s] as v and that are at distance d(v, s) − 1 from s). 1 These trees of the clusters define a forest F S,σ,i that we wish to maintain together with the clustering C S,σ,i . Using a modification of the Even-Shiloach algorithm [ES81] all the cluster trees of the clustering C i together can be maintained in total time O(im log n). For our version of the spanner that has the monotonicity property we additionally need the following observation whose proof is similar to the one of the lemma above. Lemma 4.11. For every node v the expected number of times v changes its parent in F S,σ,i is at most O(i log n). Proof. Remember that we assume the adversary to be oblivious, which means that the sequence of deletions is independent of the random choices of our algorithm. We divide the sequence of deletions into phases. For every 1 ≤ l ≤ i the l-th phase consists of the (possibly empty) subsequence of deletions during which the distance from v to S is exactly l, i.e., d G (v, S) = l. Consider first the case l ≥ 2. We will argue about possible 'configurations' (s, u) such that v is in the cluster of s and u is the parent of v that might occur in phase l. Let (s 1 , u 1 ), (s 2 , u 2 ), . . . , (s t (l) , u t (l) ) (where t (l) ≤ n 2 ) be the sequence of all pairs of nodes such that, at the beginning of phase l, for every 1 ≤ j ≤ t (l) , s j is at distance l from v and u j is a neighbor of v. The pairs (s i , u i ) in this sequence are ordered according to the point in phase l at which they cease to be possible configurations, i.e., at which either the distance of s i to v increases to more than l or u is not a neighbor of v anymore. Let A (l) j denote the event that, at some point during phase l, v is in the cluster of s j and u j is the parent of v. The expected number of times v changes its parent in F S,σ,i during phase l is equal to the expected number of j's such that event A (l) j takes place. Let B (l) j denote the event that (s j , u j ) is lexicographically first among all pairs (s j , u j ), . . . , (s t , u t (l) ) under the permutation σ, i.e., for all j ≤ j ≤ t (l) either σ(s j ) ≤ σ(s j ) or σ(s j ) = σ(s j ) and σ(u j ), σ(u j ). Observe that P A (l) j ≤ P B (l) j because the event A (l) j can only take place if the event B (l) j takes place. Furthermore, P B (l) j = 1/(t (l) − j + 1) as every pair of (distinct) nodes has the same probability of being first in the lexicographic order induced by σ. Thus, by linearity of expectation, the number of times v changes its parent in F S,σ,i during phase l is at most t (l) j=1 P A (l) j ≤ t (l) j=1 P B (l) j = t (l) j=1 1 t (l) − j + 1 = t (l) j=1 1 j = O(log t (l) ) = O(log n) . In the second case l = 1, a slightly simpler argument bounds the number of times v changes its parent (which is equal to the number of times v changes its cluster) by ordering the neighbors of v in the order of deleting their edge to v. This is the original argument of Baswana et al. [BKS12] of Lemma 4.10. We therefore also get that the number of times v changes its parent in F S,σ,i in phase 1 is at most O(log n). We now sum up the expected number of changes during all phases, and, by linearity of expectation, get that the number of times v changes its parent in F S,σ,i is at most O(i log n). Spanner. Let 2 ≤ k ≤ log n be a parameter of the algorithm. At the initialization, we first create a sequence of sets V = S 0 ⊇ S 1 ⊇ . . . ⊇ S k = ∅ by obtaining S i+1 from sampling each node of S i with probability n −1/k . Furthermore, we pick a random permutation σ of the nodes in V . We use the algorithm of Theorem 4.9 to maintain, for every 1 ≤ i ≤ k, the clustering C i := C S i ,σ together with the forest F i := F S i ,σ . Define the set V i as V i = {v ∈ V | d G (v, S i ) ≤ i}, i.e. , the set of nodes that are at distance at most i to some node of S i . Observe that the nodes in V i are exactly those nodes that are contained in some cluster C i [s] of the clustering C i . For every node v ∈ V i (where C i [s] is the cluster of v) we say that a cluster C i [s ] (for some s ∈ S i \ {s}) is neighboring to v if G contains an edge (v, v ) such that v ∈ C i [s ]. Our spanner H consists of the following two types of edges: 1. For every 1 ≤ i ≤ k, H contains all edges of the forest F i consisting of partial shortest path trees from the cluster centers. 2. For every 1 ≤ i ≤ k, every node node v ∈ V i \ V i+1 (contained in some cluster C i [s ]), and every neighboring cluster C i [s ], H contains one edge to C i [s ], i.e., one edge (v, v ) such that v ∈ C i [s ]. The first type of edges can be maintained together with the spanning forests of the clustering algorithm of Theorem 4.9. The second type of edges can be maintained with the following update rule: Every time the clustering of a node v ∈ V i \ V i+1 changes, we add to H one edge to each neighboring cluster. Every time such a 'selected' edge is deleted from G, we replace it with another edge to this neighboring cluster until all of them are used up. We now enforce the monotonicity property mentioned above in the straightforward way. Whenever we have added an edge to H, we only remove it again from H when it is also deleted from G. We argue below that this makes the size of the spanner only slightly worse than in the original construction of Baswana et al. Stretch and Size We now prove the guarantees on the stretch and size of H. The stretch argument is very similar to the ones of Baswana et al. We include it here for completeness. In the stretch argument we need stronger guarantees than Baswana et al. as we never remove edges from H, unless they are deleted from G as well. Lemma 4.12 ([BKS12]). H is a (2k − 1)-spanner of G. Proof. Consider any edge (u, v) of the current graph G and the first j such that u and v are both contained in V j and at least one of u or v is not contained in V j+1 . Without loss of generality assume that u / ∈ V j+1 . Since v ∈ V j , we know that v is contained in some cluster C j [s] and because of the edge (u, v) this cluster is neighboring to u. Similarly, the cluster of u is neighboring to v. Consider the node out of u and v that has changed its cluster within C i most recently (or take any of the two if both of them haven't changed their cluster since the initialization). Assume without loss of generality that this node was u. Then C i [s] has been a neighboring cluster of u at the time the cluster of u changed, and thus, the spanner H contains some edge (u, v ) such that v ∈ C j [s]. Using the cluster tree of C j [s] we find a path from v to v via s of length at most 2i in H. Thus, H contains a path from u to v of length at most 2i + 1 ≤ 2k − 1 as desired. Proof. Consider the first type of edges which are the ones stemming from the partial shortest path trees from the cluster centers. We charge to each node v a total of O(k 2 log n) edges given by all of v's parents in the partial shortest path trees from the cluster centers over the course of the algorithm. For every 1 ≤ i ≤ k, we know by Lemma 4.11 that the parent of v in F i changes at most O(i log n) times in expectation, which gives an overall bound of O(k 2 log n). We get the bound on the second type of edges by charging to each node v a total of O(k 2 n 1/k log n) edges. Consider a node v ∈ V i \ V i+1 for some 0 ≤ i ≤ k − 1. The number of neighboring clusters of v is equal to the number of nodes of S i that are at distance exactly i + 1 from v. Since v / ∈ V i+1 the number of such nodes is n 1/k in expectation. Thus, whenever a node v ∈ V i \ V i+1 changes its cluster in C i we can charge n 1/k to v i to pay for the n 1/k edges to neighboring clusters. As v changes its cluster in C i O(i log n) times by Lemma 4.10 and there are k clusterings, the total number of edges of the second type contained in H is O(k 2 n 1+1/k log n). Note that are allowed to multiply the two expectations because the random variables in question are independent. The overall bound of O(k 2 n 1+1/k log n) on the expected number of edges follows from the linearity of expectation. Decremental Spectral Sparsifier In the following we explain how to obtain a decremental algorithm for maintaining a spectral sparsifier using the template of Section 4.2. Internally we use our decremental spanner algorithm of Section 4.3. It is conceptually important for our approach to first develop a decremental algorithm, that is turned into a fully dynamic algorithm in Section 4.5. We follow the template of Section 4.2 by first showing how to maintain t-bundle spanners under edge deletions, and then giving decremental implementations of Light-Cut-Sparsify and Cut-Sparsify. The overall algorithm will use multiple instances of the dynamic spanner algorithm, where outputs of one instance will be used as the input of the next instance. We will do so in a strictly hierarchical manner which means that we can order the instances in a way such that the output of instance i only affects instances i + 1 and above. In this way it is guaranteed that the updates made to instance i are independent of the internal random choices of instance i, which means that each instance i is running in the oblivious-adversary setting required for Section 4.3. Decremental t-Bundle Spanners We first show how to maintain a t-bundle log n-spanner under edge deletions for some parameter t. Using the decremental spanner algorithm of Lemma 4.8 with k = (log n)/4 and = 1 we maintain a sequence H 1 , . . . H t of log n-spanners by maintaining H i as the spanner of G \ 1≤j≤i−1 H j . Here we have to argue that this is legal in the sense that every instance of the algorithm of Lemma 4.8 is run on a graph that only undergoes edge deletions. Lemma 4.14. If no edges are ever inserted into G after the initialization, then this also holds for G \ 1≤j≤i−1 H j for every 1 ≤ i ≤ t + 1. Proof. The proof is by induction on i. The claim is trivially true for i = 1 by the assumption that there are only deletions in G. For i ≥ 2 we the argument uses the monotonicity property of the dynamic algorithm for maintaining the spanner H i−1 . By the induction hypothesis we already know that no edges are ever added to the graph G \ 1≤j≤i−2 H j . Therefore the only possibility of an edge being added to G \ 1≤j≤i−1 H j would be to remove an edge e from H i−1 . However, by the monotonicity property, when e is removed from H i−1 , it is also deleted from G. Thus, e will not be inserted into G \ 1≤j≤i−2 H j . Our resulting t-bundle log n-spanner then is B = 1≤i≤t H i , the union of all these spanners. Since the H i s are disjoint the edges of B can be maintained in the obvious way by observing all changes to the H i s. By our choice of parameters, n 1/k = O(1) and thus the expected size of B is O(tn log 2 n log W ). Observe that Lemma 4.14 implies that no edges will ever be inserted into the complement G \ B, which will be relevant for our application in the spectral sparsifier algorithm. We can summarize the guarantees of our decremental t-bundle spanner algorithm as follows. Lemma 4.15. For every t ≥ 1, there is a decremental algorithm for maintaining a t-bundle log nspanner B of expected size O(tn log 2 n log W ) for an undirected graph G with non-negative edge weights that has an expected total update time of O(tm log 3 n), where W is the ratio between the largest and the smallest edge weight in G. Additional B has the following property: After the initialization, no edges are ever inserted into the graph G \ B. The bound on the expected size and the expected running time hold against an oblivious adversary. Dynamic Implementation of Light-Spectral-Sparsify We now show how to implement the algorithm Light-Spectral-Sparsify decrementally for a graph G undergoing edge deletions. For this algorithm we set t = 12(c + 3)α −2 ln n . Note that this value is slightly larger than the one proposed in the static pseudocode of Figure 1. For the sparsification proof in Section 4.2 we have to argue that by our choice of t certain events happen with high probability. In the dynamic algorithm we need ensure the correctness for up to n 2 versions of the graph, one version for each deletion in the graph. By increasing the multiplicative constant in t by 2 (as compared to the static proof of Section 4.2) all desired events happen with high probability for all, up to n 2 , versions of the graph by a union bound. The first ingredient of the algorithm is to maintain a t-bundle log n-spanner B of G under edge deletions using the algorithm of Lemma 4.15. We now explain how to maintain a graph Hwith the intention that H contains the sampled non-bundle edges of G \ B -as follows: At the initialization, we determine the graph H by sampling each edge of G \ B with probability 1/4 and adding it to H with weight 4w G (e). We then maintain H under the edge deletions in G using the following update rules: After every deletion in G we first propagate the update to the algorithm for maintaining the t-bundle spanner B, possibly changing B to react to the deletion. We then check whether the deletion in G and the change in B cause an deletion in the complement graph G \ B. Whenever an edge e is deleted from G \ B, it is removed from H . Note that by Lemma 4.15 no edge is ever inserted into G \ B. We now simply maintain the graph H as the union of B and H and make it the first output of our algorithm; the second output is B. By the update rules above (and the increased value of t to accommodate for the increased number of events), this decremental algorithm imitates the static algorithm of Figure 1 and for the resulting graph H we get the same guarantees as in Lemma 4.5. The total update time of our decremental version of Light-Spectral-Sparsify is O(tm log 3 n), as it is dominated by the time for maintaining the t-bundle log n-spanner B. As an additional property we get that no edge is ever added to the graph H = H \B. Furthermore, for all edges added to H weights are always increased by the same factor. Therefore the ratio between the largest and the smallest edge weight in H will always be bounded by W , which is the value of this quantity in G (before the first deletion). Dynamic Implementation of Spectral-Sparsify Finally, we show how to implement the algorithm Spectral-Sparsify decrementally for a graph G undergoing edge deletions. We set k = log ρ as in the pseudocode of Figure 2 and maintain k instances of the dynamic version of Light-Spectral-Sparsify above . We maintain the k graphs G 0 , . . . , G k , B 1 , . . . , B k , and H 1 , . . . , H k as in the pseudocode. For every 1 ≤ i ≤ k we maintain H i and B i as the two results of running the decremental version of Light-Spectral-Sparsify on G i−1 and maintain G i as the graph H i \ B i . As argued above (for H in Section 4.4.3), no edge is ever added to G i = H i \ B i for every 1 ≤ i ≤ k and we can thus use our purely decremental implementation of Light-Spectral-Sparsify. At the initialization, we additionally count the number of edges of every graph G i and ignore every graph G i with less than (c + 1) ln n edges. Formally we set k maximal such that G k has at least (c + 1) ln n edges. The output of our algorithm is the graph H = k i=1 B i ∪ G k . Now by the same arguments as for the static case, H gives the same guarantees as in Lemmas 4.6 and 4.7. Thus, by our choices of k and t, H is a (1 ± )-spectral sparsifier of size O(c −2 log 3 ρ log 4 n log W + mρ −1 ). As the total running time is dominated by the running time of the k instances of the decremental algorithm for Light-Spectral-Sparsify, the total update time is O(cm −2 log 3 ρ log 5 n). The guarantees of our decremental sparsifier algorithm can be summarized as follows. Lemma 4.16. For every 0 < ≤ 1, every 1 ≤ ρ ≤ m, and every c ≥ 1, there is a decremental algorithm for maintaining, with probability at least 1 − 1/n c against an oblivious adversary, a (1 ± )-spectral sparsifier H of size O(c −2 log 3 ρ log 4 n log W + mρ −1 ) for an undirected graph G with non-negative edge weights that has a total update time of O(cm −2 log 3 ρ log 5 n), where W is the ratio between the largest and the smallest edge weight in G. Turning Decremental Spectral Sparsifier into Fully Dynamic Spectral Sparsifier We use a well-known reduction to turn our decremental algorithm into a fully dynamic algorithm. Together with Lemma 4.16 this immediately implies Theorem 4.1. A similar reduction has been used by Baswana et al. [BKS12] to turn their decremental spanner algorithm into a fully dynamic one. The only additional aspect we need is the lemma below on the decomposability of spectral sparsifiers. We prove this property first and then give the reduction, which carries over almost literally from [BKS12]. ≤ i ≤ k, H i be a (1 ± )-spectral sparsifier of G i = (V, E i ). Then H = k i=1 H i is a (1 ± )-spectral sparsifier of G. Proof. Because H i is a spectral sparsifier of G i , for any vector x and i = 1, . . . , k we have (1 − )x T L H i x ≤ x T L G i x ≤ (1 + )x T L H i x Summing these k inequalities, we get that (1 − )x T L H x ≤ x T L G x ≤ (1 + )x T L H x, which by definition means that H is a (1 ± )-spectral sparsifier of H. Proof of Lemma 4.17. Set k = log (n 2 ) . For each 1 ≤ i ≤ k, we maintain a set E i ⊆ E of edges and an instance A i of the decremental algorithm running on the graph G i = (V, E i ). We also keep a binary counter C that counts the number of insertions modulo n 2 with the least significant bit in C being the right-most one. A deletion of some edge e is carried out by simply deleting e from the set E i it is contained in and propagating the deletion to instance A i of the decremental algorithm. An insertion of some edge e is carried out as follows. Let j be the highest (i.e., left-most) bit that gets flipped in the counter when increasing the number of insertions. Thus, in the updated counter the j-th bit is 1 and all lower bits (i.e., bits to the right of j) are 0. We first add the edge e as well as all edges in j−1 i=1 E i to E j . Then we set E i = ∅ for all 1 ≤ i ≤ j − 1. Finally, we re-initialize the instance A j on the new graph G j = (V, E j ). We know bound the total update time for each instance A i of the decremental algorithm. First, observe that the i-th bit of the binary counter is reset after every 2 i edge insertions. A simple induction then shows that at any time E i ≤ 2 i for all 1 ≤ i ≤ k. Now consider an arbitrary sequence of updates of length . The instance A i is re-initialized after every 2 i insertions. It will therefore be re-initialized at most /2 i times. For every re-initialization we pay a total update time of |E i | · T (|E i |, n, W ) ≤ 2 i T (m, n, W ). For the entire sequence of updates, the total time spent for instance A i is therefore ( /2 i ) · 2 i T (m, n, W ) = · T (m, n, W ). Thus we spend total time O( · T (m, n, W ) log n) for the whole algorithm, which amounts to an amortized update time of O(T (m, n, W ) log n). Dynamic Cut Sparsifier In this section we give an algorithm for maintaining a cut sparsifier under edge deletions and insertions with polylogarithmic worst-case update time. The main result of this section is as follows. By running the algorithm with basically ρ = m we additionally get that H has low arboricity, i.e., it can be partitioned into a polylogarithmic number of trees. We will algorithmically exploit the low arboricity property in Sections 6 and 8. Here, W is the ratio between the largest and the smallest edge weight in G. The ratio between the largest and the smallest edge weight in H is at most O(nW ). We can maintain a partition of H into disjoint forests T 1 , . . . , T k such that every node keeps a list of its neighbors together with its degree in each forest T i . After every update in G at most one edge is added to and at most one edge is removed from each forest T i . After giving an overview of our algorithm, we first explain our cut sparsification scheme in a static setting and prove its properties. Subsequently, we show how we can dynamically maintain the edges of such a sparsifier with both amortized and worst-case update times by making this scheme dynamic. Algorithm Overview Our Framework. The algorithm is based on the observation that the spectral sparsification scheme outlined above in Section 3.1. becomes a cut sparsification algorithm if we simply replace spanners by maximum weight spanning trees (MSTs). This is inspired by sampling according to edge connectivities; the role of the MSTs is to certify lower bounds on the edge connectivities. We observe that the framework does not require us to use exact MSTs. For our t-bundles we can use a relaxed, approximate concept that we call α-MST that. Roughly speaking, an α-MST guarantees a 'stretch' of α in the infinity norm and, as long as it is sparse, does not necessarily have to be a tree. Similarly to before, we define a t-bundle α-MST B as the union of a sequence of α-MSTs T 1 , . . . T t where the edges of each tree are removed from the graph before computing the next α-MST. The role of α-MST is to certify uniform lower bounds on the connectivity of edges; these bounds are sufficiently large to allow uniform sampling with a fixed probability. This process of peeling and sampling is repeated sufficiently often and our cut sparsifier then is the union of all the t-bundle α-MSTs and the non-bundle edges remaining after taking out the last bundle. Thus, the cut sparsifier consists of a polylogarithmic number of α-MSTs and a few (polylogarithmic) additional edges. This means that for α-MSTs based on spanning trees, our cut sparsifiers are not only sparse, but also have polylogarithmic arboricity, which is the minimum number of forests into which a graph can be partitioned. Simple Fully Dynamic Algorithm. Our approach immediately yields a fully dynamic algorithm by using a fully dynamic algorithm for maintaining a spanning forest. Here we basically have two choices. Either we use the randomized algorithm of Kapron, King, and Mountjoy [KKM13] with polylogarithmic worst-case update time. Or we use the deterministic algorithm of Holm, de Lichtenberg, and Thorup [HLT01] with polylogarithmic amortized update time. The latter algorithm is slightly faster, at the cost of providing only amortized update-time guarantees. A t-bundle 2-MST can be maintained fully dynamically by running, for each of the log W weight classes of the graph, t instances of the dynamic spanning tree algorithm in a 'chain'. An important observation about the spanning forest algorithm is that with every update in the graph, at most one edge is changed in the spanning forest: If for example an edge is deleted from the spanning forest, it is replaced by another edge, but no other changes are added to the tree. Therefore a single update in G can only cause one update for each graph G i = G \ i−1 j=1 T j and T i . This means that each instance of the spanning forest algorithm creates at most one 'artificial' update that the next instance has to deal with. In this way, each dynamic spanning forest instance used for the t-bundle has polylogarithmic update time. As t = poly log n, the update time for maintaining a t-bundle is also polylogarithmic. The remaining steps of the algorithm can be carried out dynamically in the straightforward way and overall give us polylogarithmic worst-case or amortized update time. A technical detail of our algorithm is that the high-probability correctness achieved by the Chernoff bounds only holds for a polynomial number of updates in the graph. We thus have to restart the algorithm periodically. This is trivial when we are shooting for an amortized update time. For a worst-case guarantee we can neither completely restart the algorithm nor change all edges of the sparsifier in one time step. We therefore keep two instances of our algorithm that maintain two sparsifiers of two alternately growing and shrinking subgraphs that at any time partition the graph. This allows us to take a blend of these two subgraph sparsifiers as our end result and take turns in periodically restarting the two instances of the algorithm. Definitions We will work with a relaxed notion of an MST, which will be useful when maintaining an exact maximum spanning tree is hard (as is the case for worst-case update time guarantees). Definition 5.3. A subgraph T of an undirected graph G is an α-MST (α ≥ 1) if for every edge e = (u, v) of G there is a path π from u to v such that w G (e) ≤ αw G (f ) for every edge f on π. Note that in this definition we do not demand that T is a tree; any subgraph with these properties will be fine. A maximum spanning tree in this terminology is a 1-MST. Definition 5.4. A t-bundle α-MST (t, α ≥ 1) of an undirected graph G is the union B = k i=1 T i of a sequence of graphs T 1 , . . . , T t such that, for every 1 ≤ i ≤ t, T i is an α-MST of G \ i−1 j=1 T j . We can imagine such a t-bundle being obtained by iteratively peeling-off α-MSTs from G. A Simple Cut Sparsification Algorithm We begin with algorithm Light-Cut-Sparsify in Figure 3; this is the core iteration used to compute a sparser cut approximation with approximately half the edges. Algorithm Cut-Sparsify in Figure 3 is the full sparsification routine. The properties of these algorithm are given in the following lemmas. Lemma 5.5. The output H of algorithm Light-Cut-Sparsify is a (1 ± )-cut approximation of the input G, with probability 1 − n −c . We will need a slight generalization of a Theorem in [FHH + 11]. 1. k ← log ρ 2. G 0 ← G 3. B 0 ← (V, ∅) 4. for i = 1 to k (a) (H i , B i ) ← Light-Cut-Sparsify(G, c + 1, /(2k)) (b) G i+1 ← H i \ B i (c) if G i+1 has less than (c + 2) ln n edges then break Proof. Suppose without loss of generality that the maximum weight in G is 1. We decompose G into log W edge-disjoint graphs, where G i consists of the edges with weights in (2 −(i+1) , 2 −i ] plus B i = B/2 −(i+1) , where B is the bundle returned by the algorithm. By definition of the α-MST t-bundle, the connectivity of each edge of G i \ B i in G i is at least 4ρc, for c = d log W where ρ is as defined in Lemma 5.6. Assume for a moment that all edges in B i are also in (2 −(i+1) , 2 −i ]. Then we can set p e = 1 for each e ∈ B i and p e = 1/4 for all other edges, and apply Lemma 5.6. In this way we get that H i is (1 ± )-cut sparsifier with probability at least 1 − n d log W . (* break loop *) 5. H ← 1≤j≤i B j ∪ G i+1 6. return (H, {B j } i j=1 , G i+1 ) The assumption about B i can be removed as follows. We observe that one can find a subgraph B i of B i (by splitting weights when needed, and dropping smaller weights), such that B i is a t-bundle α-MST of G i . This follows by the definition of the t-bundle α-MST . We can thus apply the lemma on G i = (G i \ B i ) ∪ B i , and get that the sampled graph H i is a (1 ± )-cut sparsifier. We then observe that G i = G i ∪ (B i \ B i ) and H i = G i ∪ (B i \ B i ), from which it follows that H i is a (1 ± )-cut sparsifier of G i . Note: The number of logarithms in Light-Cut-Sparsify is not optimal. One can argue that the lower bounds we compute can be used in place of the strong connectivities used in [BK15] and reduce by one the number of logarithms. It is also possible to replace log W with log n by carefully re-working some of the details in [BK15]. We finally have the following Lemmas. The proofs are identical to those for the corresponding Lemmas in Section 4, so we omit them. Lemma 5.7. The output H of algorithm Cut-Sparsify is a (1 ± )-spectral sparsifier of the input G, with probability at least 1 − 1/n c+1 . Lemma 5.8. With probability at least 1 − 2n −c , the number of iterations before algorithm Cut-Sparsify terminates is min{ log ρ , log m/((c + 1) log n) }. Moreover the size of H is O   1≤j≤i |B i | + m/ρ + c log n   , and the size of the third output of the graph is at most max{O(c log n), O(m/ρ)}. Dynamic Cut Sparsifier We now explain how to implement the cut sparsifier algorithm of Section 5.3 dynamically. The main building block of our algorithm is a fully dynamic algorithm for maintaining a spanning forest with polylogarithmic update time. We either use an algorithm with worst-case update time, or a slightly faster algorithm with amortized update time. In both algorithms, an insertion might join two subtrees of the forest and after a deletion the forest is repaired by trying to find a single replacement edge. This strongly bounds the number of changes in the forest after each update. Theorem 5.10 ([HLT01]). There is a fully dynamic deterministic algorithm for maintaining a minimum spanning forest T of a weighted undirected graph G with amortized update time O(log 2 n). Every time an edge e is inserted into G, the only potential change to T is the insertion of e Every time an edge e is deleted from G, the only potential change to T is the removal of e and possibly the addition of at most one other edge to T . We first explain how to use these algorithms in a straightforward way to maintain a 2-MST. Subsequently we show how to dynamically implement the procedures Light-Cut-Sparsify and Cut-Sparsify. The overall algorithm will use multiple instances of a dynamic spanning forest algorithm, where outputs of one instance will be used as the input of the next instance. We will do so in a strictly hierarchical manner which means that we can order the instances in a way such that the output of instance i only affects instances i + 1 and above. In this way it is guaranteed that the updates made to instance i are independent of the internal random choices of instance i, which means that each instance i is running in the oblivious-adversary setting required for Theorem 5.9. Dynamic Maintenance of 2-MST For every 0 ≤ i ≤ log W , let E i be the set of edges of weight between 2 i and 2 i+1 , i.e., E i = {e ∈ E | 2 i ≤ w G (e) < 2 i+1 }, and run a separate instance of the dynamic spanning forest algorithm for the edges in E i . For every 0 ≤ i ≤ log W , let F i be the spanning forest of the edges in E i maintained by the i-th instance. We claim that the union of all these trees is a 2-MST of G. Lemma 5.11. T = log W i=0 F i is a 2-MST of G. Proof. Consider some edge e = (u, v) of G and let i be the (unique) index such that 2 i ≤ w G (e) < 2 i+1 . Since F i is spanning tree of G, there is a path π from u to v in F i (and thus also in T ). Every edge f of π is in the same weight class as e, i.e., 2 i ≤ w G (f ) < 2 i+1 . Thus, w G (e) < 2 i+1 ≤ 2w G (f ) as desired. Every time an edge e is inserted or deleted, we determine the weight class i of e and perform the update in the i-th instance of the spanning forest algorithm. This 2-MST of size O(n log W ) can thus be maintained with the same asymptotic update time as the dynamic spanning forest algorithm. We now show how to maintain a t-bundle 2-MST and consequently a (1 ± )-cut sparsifier H according to the construction presented in Section 5.3. For the t-bundle 2-MST B = 1≤i≤k T i we maintain, for every 1 ≤ i ≤ t, a 2-MST of G \ i−1 j=1 T j . We now analyze how changes to G \ i−1 j=1 T j affect G \ i j=1 T j (for every 1 ≤ i ≤ k): • Whenever an edge e is inserted into G \ i−1 j=1 T j , the 2-MST algorithm either adds e to T i or not. -If e is added to T i , then G \ i j=1 T j does not change. -If e is not added to T i , then e is added to G \ i j=1 T j . • Whenever an edge e is deleted from G \ i−1 j=1 T j , either e is contained in T i or not. -If e is contained in T i , then e is removed from T i and some other edge f is added to T i . This edge f is removed from G \ i j=1 T j . 2 2 The edge e will not be added to G \ -If e is not contained in T i , then e is removed from G \ i j=1 T j . Thus, every change to G \ i−1 j=1 T j results in at most one change to G \ i j=1 T j . Consequently, a single update to G results to at most one update in each instance of the dynamic MST algorithm. For every update in G we therefore incur an amortized update time of O(t log 4 n). Thus, we can summarize the guarantees for maintaining a t-bundle 2-MST as follows. Dynamic Implementation of Light-Cut-Sparsify For this algorithm we set t = (C ξ + 2)dα −2 log W log 2 n. Note that this value is slightly larger than the one proposed in Figure 3. For the sparsification proof in Section 5.3 we have to argue that by our choice of t certain events happen with high probability. In the dynamic algorithm we need ensure the correctness for a polynomial number of versions of the graph, one version for each update made to the graph. We show in Section 5.5 that it is sufficient to be correct for up to 4n 2 updates to the graph, as then we can extend the algorithm to an arbitrarily long sequence of updates. By making t slightly large than in the static proof of Section 5.3 all the desired events happen with high probability for all 4n 2 versions of the graph by a union bound. The first ingredient of the algorithm is to dynamically maintain a t-bundle 2-MST B using the algorithm of Corollary 5.12 above. We now explain how to maintain a graph H -with the intention that H contains the sampled non-bundle edges of G \ B -as follows: After every update in G we first propagate the update to the algorithm for maintaining the t-bundle 2-MST B, possibly changing B to react to the update. We then check whether the update in G and the change in B cause an update in the complement graph G \ B. • Whenever an edge is inserted into G \ B, it is added to H with probability 1/4 and weight 4w G (e). • Whenever an edge e is deleted from G \ B, it is removed from H . We now simply maintain the graph H as the union of B and H and make it the first output of our algorithm; the second output is B. By the update rules above (and the increased value of t to accommodate for the increased number of events), this dynamic algorithm imitates the static algorithm of Figure 3 and for the resulting graph H we get the same guarantees as in Lemma 5.5. The update time of our dynamic version of Light-Spectral-Sparsify is O(t log 4 n) worst-case and O(t log 2 n) worst-case, as it is dominated by the time for maintaining the t-bundle 2-MST B. As an additional property we get that with every update in G at most one change is performed to H = H \ B. Furthermore, for all edges added to H weights are always increased by the same factor. Therefore the ratio between the largest and the smallest edge weight in H will always be bounded by W , which is the value of this quantity in G (before the first deletion). Dynamic Implementation of Cut-Sparsify We set k = log min(ρ, m/((c + 2) log n)) and maintain k instances of the dynamic version of Light-Cut-Sparsify above, using the other parameters just like in the pseudo-code of Figure 4. By this choice of k we ensure that we do not have to check the breaking condition in the pseudo-code explicitly, which is more suited for a dynamic setting where the number of edges in the maintained subgraphs might grow and shrink. We maintain the k graphs G 0 , . . . , G k , B 1 , . . . , B k , and H 1 , . . . , H k as in the pseudocode. For every 1 ≤ i ≤ k we maintain H i and B i as the two results of running the dynamic version of Light-Cut-Sparsify on G i−1 and maintain G i as the graph H i \ B i . The output of our algorithm is the graph H = k i=1 B i ∪ G k . Note that, by our choice of k, G k has at most max(m/ρ, (c + 2) log n) edges. Now by the same arguments as for the static case, H gives the same guarantees as in Lemmas 5.7 and 5.8 for up to a polynomial number of updates (here at most 4n 2 ) in the graph. As argued above (for H in Section 5.4.2), every update in G i−1 results in at most one change to G i = H i \ B i for every 1 ≤ i ≤ k. By an inductive argument this means that every update in G results in at most one change to G i for every 1 ≤ i ≤ k. As each instance of the dynamic Light-Cut-Sparsify algorithm has update time O(t log 4 n) worst-case or O(t log 2 n) amortized, this implies that our overall algorithm has update time O(kt log 4 n) or O(kt log 2 n), respectively. Together with Lemma 5.14 in Section 5.3, we have proved Theorem 5.1 stated at the beginning of this section. In Corollary 5.2 we additionally claim that for ρ = m we obtain a sparsifier with polylogarithmic arboricity. This is true because the cut sparsifier H mainly consists of a collection of bundles, which in turn consists of a collection of trees. In total, H consists of O(tk log W ) trees and O(c log n) remaining edges in G k , each of which can be seen as a separate tree. Furthermore we can maintain the collection of trees explicitly with appropriate data structures for storing them. Handling Arbitrarily Long Sequences of Updates The high-probability guarantees of the algorithm above only holds for a polynomially bounded number of updates. We now show how to extend it to an arbitrarily long sequence of updates providing the same asymptotic update time and size of the sparsifier. We do this by concurrently running two instances of the dynamic algorithm that periodically take turns in being restarted, which is a fairly standard approach for such situations. The only new aspect necessary for our purposes is that both instances explicitly maintain a sparsifier and when taking turns we cannot simply replace all the edges of one sparsifier with the edges of the other sparsifier as processing all these edges would violate the worst-case update time guarantee. For this reason we exploit the decomposability of graph sparsifiers and maintain a 'blend' of the two sparsifiers computed by the concurrent instances of the dynamic algorithm. This step is not necessary for other dynamic problems such as connectivity where we only have to make sure that the query is delegated to the currently active instance. ≤ i ≤ k, H i be a (1 ± )-cut sparsifier of G i = (V, E i ). Then H = k i=1 H i is a (1 ± )-cut sparsifier of G. Proof. Let U be a cut in G. First observe that w G (∂ G (U )) = w G ( k i=1 ∂ G i (U )) = k i=1 w G (∂ G i (U )) = k i=1 w G i (∂ G i (U )) and similarly w H (∂ H (U )) = k i=1 w H i (∂ H i (U )). Now since (1 − )w H i (∂ H i (U )) ≤ w G i (∂ G i (U )) ≤ (1 + )w H i (∂ H i (U )) for every 1 ≤ i ≤ k, we have (1 − )w H (∂ H (U )) = (1 − ) k i=1 w H i (∂ H i (U )) ≤ k i=1 w G i (∂ G i (U )) = w G (∂ G (U )) = · · · ≤ (1 + )w H (∂ H (U )) . Lemma 5.14. Assume there is a fully dynamic algorithm for maintaining a (1 ± )-cut (spectral) sparsifier of size at most S(m, n, W ) with worst-case update time T (m, n, W ) for up to 4n 2 updates in G. Then there also is a fully dynamic algorithm for maintaining a (1 ± )-cut (spectral) sparsifier of size at most O (S(m, n, W )) with worst-case update time O (T (m, n, W )) for an arbitrary number of updates. Proof. We exploit the decomposability of cut sparsifiers. We maintain a partition of G into two disjoint subgraphs G 1 and G 2 and run two instances A 1 and A 2 of the dynamic algorithm on G 1 and G 2 , respectively. These two algorithms maintain a (1 ± )-sparsifier of H 1 of G 1 and a (1 ± )-sparsifier H 2 of G 2 . By the decomposability stated in Lemmas 5.13 and 4.18, the union H := H 1 ∪ H 2 is a (1 ± )-sparsifier of G = G 1 ∪ G 2 . We divide the sequence of updates into phases of length n 2 each. In each phase of updates one of the two instances A 1 , A 2 is in the state growing and the other one is in the state shrinking. A 1 and A 2 switch their states at the end of each phase. In the following we describe the algorithm's actions during one phase. Assume without loss of generality that, in the phase we are fixing, A 1 is growing and A 2 is shrinking. At the beginning of the phase we restart the growing instance A 1 . We will orchestrate the algorithm in such a way that at the beginning of the phase G 1 is the empty graph and G 2 = G. After every update in G we execute the following steps: 1. If the update was the insertion of some edge e, then e is added to the graph G 1 and this insertion is propagated to the growing instance A 1 . 2. If the update was the deletion of some edge e, then e is removed from the graph G i it is contained in and this deletion is propagated to the corresponding instance A i . 3. In addition to processing the update in G, if G 2 is non-empty, then one arbitrary edge e is first removed from G 2 and deleted from instance A 2 and then added to G 1 and inserted into instance A 1 . Observe that these rules indeed guarantee that G 1 and G 2 are disjoint and together contain all edges of G. Furthermore, since the graph G 2 of the shrinking instance has at most n 2 edges at the beginning of the phase, the length of n 2 updates per phase guarantees that G 2 is empty at the end of the phase. Thus, the growing instance always starts with an empty graph G 1 . As both H 1 and H 2 have size at most S(n, m, W ), the size of H = H 1 ∪ H 2 is O (S(n, m, W )). With every update in G we perform at most 2 updates in each of A 1 and A 2 . It follows that the worst-case update time of our overall algorithm is O (T (m, n, W )). Furthermore since each of the instances A 1 and A 2 is restarted every other phase, each instance of the dynamic algorithm sees at most 4n 2 updates before it is restarted. Application of Dynamic Cut Sparsifier: Undirected Bipartite Min-Cut We now utilize our sparsifier data structure to maintain a (2+ )-approximate st-min-cut in amortized O(poly(log n, −1 )) time per update. In this section, we will define several tools that are crucial for the better analyses in Sections 7 and 8. This result is a weaker form of Theorem 1.2 with an approximation factor of 2 + instead of 1 + . The main result that we will show in this section is: Theorem 6.1. For every 0 < ≤ 1, there is a fully dynamic algorithm for maintaining a (2 + )approximate minimum cut in an unweighted undirected graph that's a bipartite graph with source/sink s and t attached to each of the partitions with amortized update time O(poly(log n, −1 )). To add motivation for solving this problem, we would like to point out that there are examples in which the maximum s − t flow is much larger than the minimum vertex cover, and we cannot simply consider the problem as finding a maximum matching in G A,B . Specifically, let A = A k 2 ∪ A k and B = B k 2 ∪ B k , where |A k |, |B k | = k and |A k 2 |, |B k 2 | = k 2 , then construct a complete bipartite graph on (A k 2 , B k ), (A k , B k ), (A k , B k 2 ) , while having no edges between A k 2 and B k 2 . A vertex cover would be A k ∪ B k ∪ {s, t}, but we can achieve a max-flow in G of Ω(k 2 ). Accordingly, the objective cannot be approximated using matching routines even in the static case. However, the solution can still be approximated using recent developments in flow algorithms [She13,KLO + 14,Pen16]. Below we will show that these routines can be sped up on dynamic graphs using multiple layers of sparsification. Specifically, the cut sparsifiers from Section 5.4 allow us to dynamically maintain a (1 + )-approximation of the solution value, as well as some form of query access to the minimum cut, in O(poly(log n, −1 )) per update. The section is organized as follows. Section 6.1 will give some of the high level ideas and critical observations on which our dynamic algorithm will hinge. Section 6.2 will present the dynamic algorithm for maintaining a (2 + )-approximate minimum s − t cut, prove that the approximation factor is correct, and show that the dynamic update time is O(poly(log n, −1 )) if we can dynamically update all data structures necessary for the algorithm in O(poly(log n, −1 )) time. Finally, Section 6.3 will present all of the necessary data structures and show how we can dynamically maintain them in O(poly(log n, −1 )) time. Key Observations and Definitions Our starting point is the observation that a small solution value implies a small vertex cover. E(A s , B t ), so |V A (A s , B t )| ≤ |E(A s , B t )|. We know G A,B is bipartite, so A t ∪ B s ∪ V A (A s , B t ) must be a vertex cover in G A,B , which implies |MVC| ≤ OP T + 2 by adding s and t to the cover. Our goal, for the rest of this section, is to show ways of reducing the graph onto a small vertex cover, while preserving the flow value. The first issue that we encounter is that the minimum vertex cover can also change during the updates. However, in our case, the low arboricity property of the sparsifier given in Corollary 5.2 gives a more direct way of obtaining a small cover: Lemma 6.2. For any tree T , the vertex cover of all vertices other than the leaves is within a 2-approximation of the minimum vertex cover. This is proven in Appendix B. We suspect that this is a folklore result, but it was difficult to find a citation of it, as there exist far better algorithms for maintaining vertex covers on dynamic trees [GS09]. Since there are at most O(poly(log n, −1 )) trees, and the overall vertex cover needs to be at least the size of any cover in one of the trees, we can set the cover as the set of all non-leaf vertices in the trees. = F 1 ∪ · . . . ∪ · F K . If VC is a branch vertex cover ofG, then, VC is a 2K-approximate vertex cover ofG. Furthermore, any x ∈ V \ VC has degree at most K inG Proof. Since the size of a minimum vertex cover in subgraph can only be smaller, we have |MVC F i | ≤ |MVC G |. Coupling this the choice of |VC| gives |VC i | ≤ 2|MVC F i |, and summing over all K trees gives the bound. The bound on the degree of x follows from all leaves having degree 1. We will ensure that s and t are placed in the cover, and use X to denote the non-cover vertices. If we let the neighborhood of x be N (x), its interaction with various partitions of S can be described as: Definition 6.5. For a cut on VC, S ⊆ VC, and a non-cover vertex x ∈ X with neighborhood N (x), let 1. w(x, S) := u∈S∩N (x) w(x, u), 2. w (x) (S) := min{w(x, S), w(x, VC \ S)}. Definition 6.6. Given a graph G = (V, E) and some V ⊆ V such that V is a vertex cover of G, and X = V \ V 1. For any S ⊂ V , let ∆ G (S) be the weight of cut S on G 2. For any S V ⊂ V , let the weight of a cut that is minimally extended from S V then be given by ∆ G (S V ) := ∆ G\X (S V ) + x∈X w (x) (S V ), Definition 6.7. Given G = (V G , E G ) and H = (V H , E H ) such that V H ⊆ V G and V is a vertex cover of both graphs 1. If V H = V G , then we say H ≈ G if for any S ⊂ V G (1 − )∆ H (S) ≤ ∆ G (S) ≤ (1 + )∆ H (S) 2. If V H ⊂ V G , then we say H ≈ V G, if for any S V ⊂ V (1 − )∆ H (S V ) ≤ ∆ G (S V ) ≤ (1 + )∆ H (S V ) Note that if some x ∈ X has degree 1, it will always belong to the same side as its neighbor in a minimum s − t cut; while if x is incident to two neighbors u and v, it will always go with the neighbor with smaller weight . Aka if w(x, u) ≤ w(x, v), then this is equivalent to an edge of weight w(x, u) between u and v. This suggests that we can reduce the star out of x, N x , to a set of edges on its neighborhood. We formalize the construction of this graph, K x , as well as the resulting graph by removing all of X below: Definition 6.8. Given a weighted graph G = (V, E) and w(u, v) → R + , and any S, let: K x be the clique generated by running VertexElimination: for any two neighbors u and v of x, the edge weight of (u, v) x is w(x, v)w(x, u) i∈N (x) w(x, i) . For some vertex cover VC and independent set X = V \ VC, we let G VC = (G \ X) ∪ x∈X K x Note that we're using a subscript x to denote the origin of the edge. Specifically, an edge e x ∈ G VC implies that e x ∈ K x , and an edge e ∅ ∈ G VC means it's from VC, aka e ∅ ∈ G \ X. Note that G VC also defines a weight for each cut S VC ⊂ VC, where ∆ G VC (S VC ). The crucial property of Definition 6.8 is that it preserves the values all cuts within a factor of 2. We prove the following in Appendix B. Theorem 6.9. Given a weighted graph G = (V, E) and w(u, v) → R + , with some vertex cover VC and independent set X = V \ VC. For any S V C ⊂ VC 1 2 ∆ G (S V C ) ≤ ∆ G VC (S V C ) ≤ ∆ G (S V C ) Lemma 6.10. Given G = (V, E) with all weights in [γ, γU ], along with vertex cover VC and independent set X, such that any x ∈ X has degree at most d. Then the weight of any edge in G VC is in [γ(dU ) −1 , γU ] Dynamic Algorithm for Maintaining a Minimum s − t Cut on Bipartite Graphs Our algorithm can then be viewed as dynamically maintaining this cover using two layers of dynamic graph sparsifiers intermixed with elimination routines. Its main steps are shown in Figure 5. One issue with maintaining a cut is that its two sides could have size O(n), which cannot be returned in amortized O(poly(log n, −1 )) time. Instead, we will maintain the cut S VC ⊂ VC with s ∈ S VC , and allow querying of any vertex. For a vertex v ∈ VC, return v is with s iff v ∈ S VC , which takes O(1) time. For a vertex x / ∈ VC, return that x is with s iff w(x, S VC ) = w (x) ( S VC ) in S = S V C ∪ {x ∈ V \ VC : w(x, S V C ) = w (x) ( S V C )}, the extension of S VC onG which allows for the O(poly(log n, −1 )) query computation by Corollary 5.2 and Corollary 6.4. We first establish the quality of this cut on H that we maintain: Further, each K x ofG VC has at most K 2 = O(poly(log n, −1 )) edges, soG VC has O(n·poly(log n, −1 )) edges. Corollary 5.2 then tells us that H has O(|VC| · poly(log n, −1 )) = O(OP T · poly(log n, −1 )) edges, and that we can find a (1 + ) approximate minimum s − t cut in H, S V C in O(OP T · poly(log n, −1 )) time. From Corollary 5.2, we assume that H ≈ G VC and G ≈ G with high probability. Suppose S V C ⊂ VC is returned as a (1 + )-approximate minimum s − t cut in H, and let S = S V C ∪ {x ∈ V \ VC : w(x, S V C ) = w (x) ( S V C )} be its extension ontoG. The left-hand side of Theorem 6.9 implies ∆G( S V C ) ≤ 2∆G V C ( S V C ), which along with the approximations G ≈ G andG V C ≈ H gives ∆ G ( S) ≤ (1 + )∆G( S) ≤ 2(1 + )∆G VC ( S V C ) ≤ 2(1 + ) 2 ∆ H ( S V C ) . On the other hand, let S ⊂ V be the minimum s − t cut in G, and S V C ⊂ VC be its restriction to VC. Since right-hand side of Theorem 6.9 is over optimum choices of V \ S V C , we have ∆G(S) ≥ ∆G(S V C ) ≥ ∆G V C (S V C ), which when combined with the approximations G ≈ G andG V C ≈ H gives ∆ G (S) ≥ (1 − )∆G(S) ≥ (1 − )∆G VC (S V C ) ≥ (1 − ) 2 ∆ H (S V C ). The result then follows from the near-optimality of S V C on H, ∆ H (S V C ) ≥ (1 − )∆ H ( S V C ). Corollary 6.12. The dynamic algorithm maintains a (2 + )-approximate minimum s − t cut in G, and will only compute an approximate minimum s − t cut on H every O( OP T ) dynamic steps. Proof. Choosing = O(1) in Theorem 6.11 can give a (2 + 2 )-approximate minimum s − t cut in G. Borrowing notation from the proof of Theorem 6.11, an approximate minimum s − t cut on H will be re-computed in 2 ∆ H ( S VC ) dynamic steps. OP T = ∆ G (S) ≤ ∆ G ( S) ≤ 2(1 + ) 2 ∆ H ( S VC ), so ∆ H ( S VC ) = O(OP T ) Dynamically Updating Data Structures As was shown in Corollary 6.12, the dynamic algorithm maintains a (2 + )-approximate minimum s − t cut of G, an approximate minimum s − t cut of H is computed every O( OP T ), and that computation takes O(OP T · poly(log n, −1 )) time from Theorem 6.11. Therefore, in order to establish that the amortized dynamic update time is O(poly(log n, −1 )), it suffices to show that all data structures can be maintained in O(poly(log n, −1 )) time per dynamic update, thereby finishing the proof of Theorem 6.1. As a result of Corollary 5.2, it suffices to show the following Theorem 6.13. For each addition/deletion of an edge inG, data structures forG, VC,G VC , and H can be maintained in O(poly(log n, −1 )) time. Bounds on the dynamic update time of each data structure will all ultimately follow from the O(poly(log n, −1 )) degree bound forG of all vertices not in the branch vertex cover, VC. This is a direct result of the O(poly(log n, −1 )) arboricity ofG from Corollary 5.2, and the properties of a branch vertex cover ofG in Corollary 6.4. Data structure forG: A list of O(poly(log n, −1 )) spanning forests, which we will denote SPANNERS G . Data structure for adjacency lists ofG:, We will denote it as ADJ-LISTG, and it will have, for each vertex v, two lists LEAF v and BRANCH v : • The list LEAF v will have the adjacency list of v for each spanning forest in SPANNERS G in which v is a leaf. • Similarly, the list BRANCH v will have the adjacency list of v edge for each spanning forest in SPANNERS G in which v is not a leaf. Data structure for VC: We will denote it as VCG, which will be a list of all vertices v whose list BRANCH v is non-empty. InsertVC(G, VC, v) 1. Delete all edges e v ∈ K v in GRAPH VC . 2. For all edges e adjacent to v in ADJ-LISTG, insert e ∅ into GRAPH VC . 2. Use all incident edges to compute K v and insert all e v ∈ K v into GRAPH VC Figure 7: Removing a Vertex from VC Data structure forG VC : We will denote it as GRAPH VC , and it will contain an adjacency list, ADJ v , for each vertex v ∈ VC. Assume that each ADJ v has a data structure such that deletion and insertion of any edge takes O(log n) time. Data structure for H: We will denote it as SPARSE VC , and it will be the sparsified multi-graph. We first show that moving a vertex in / out of the vertex cover can be done in O(poly(log n, −1 )) time, assuming that the degree of the vertex added/removed is small. Note that the small number of forests inG and the choice of VC allow us to meet this requirement. Lemma 6.14. If v is not in VCG, then running INSERT VC (v) on GRAPH VC , using ADJ-LISTG, will output GRAPH VC equivalent toG VC∪v of ADJ-LISTG in O(poly(log n, −1 )) time. Proof. Costs of the two steps are: 1. Delete all edges e v ∈ K v in GRAPH VC . This requires finding all incident vertices to v in LEAF v and BRANCH v , which is at most O(poly(log n, −1 )) because BRANCH v is empty due to v not in VCG. Every pair of vertices has a corresponding edge e v in GRAPH VC , so this takes O(poly(log n, −1 )) time. 2. There are at most O(poly(log n, −1 )) edges adjacent to v in ADJ-LISTG, so adding all these edges into GRAPH VC takes O(poly(log n, −1 )) time. If v is not in VC G , then v must only be incident to VC in ADJ-LISTG. Therefore inG VC∪v , v will only be incident to edges e ∅ for each e incident to v in ADJ-LISTG, and no edges e v will be iñ G VC∪v . INSERT VC (v) will perform exactly these operations on GRAPH VC . UpdateADJ(G, VC, e) 1. If e has been added/deleted, then add/delete e from the adjacency list of u and v for F i in ADJ-LISTG, which will be denoted L u,i and L v,i , respectively. 2. For u and v, if L v,i has at most one adjacent vertex, place it in LEAF v , otherwise place it in BRANCH v . 3. If the degree of u and v in F i is zero before adding e, then place L v,i in BRANCH v and L u,i in BRANCH u 4. For u and v, if degree of v is two before deleting e, check the other vertex incident to v, say it is w, and if w has degree one in F i then move L v,i to BRANCH v and L w,i to BRANCH w . 5. For u and v, if degree of v is one before adding e, check the other vertex incident to v, say it is w, and if w has degree one in F i then move L w,i to LEAF w . Proof. Costs of the two steps are: 1. At most O(poly(log n, −1 )) edges are adjacent to v in ADJ-LISTG, so deleting all these edges from GRAPH VC takes O(poly(log n, −1 )) time. 2. v / ∈ VC, so v has O(poly(log n, −1 )) neighbors, and using all incident edges to compute each e v ∈ K v and insert e v into GRAPH VC takes O(poly(log n, −1 )) time. If BRANCH v is empty, then v must only be incident to VC in ADJ-LISTG. Therefore iñ G VC\v , v will never be incident to any edges e ∅ , and for any of its neighbors w and z, (w, z) v will be in ADJ-LISTG. INSERT VC (v) will perform exactly these operations on GRAPH VC We now consider updating ADJ-LISTG given the addition/deletion of some edge. This process is simple in terms of time complexity, but has a small wrinkle in maintaining the correct LEAF and BRANCH structure. Specifically, for each forest, we can consider all of the degree one vertices to be leaves, except for when there is a disjoint edge in the forest. Accordingly, steps 3, 4, and 5 of the algorithm in Figure 8 will take care of this edge case. For all trees, other than a single edge, it suffices to put all vertices with degree ≥ 2 in the vertex cover, and 2-approx tree theorem tells us that this is a 2-approximate vertex cover. Step 3 and 4 of Update ADJ-LISTG ensure that in the single edge case, e = (u, v) that L v,i is in BRANCH v and L u,i is in BRANCH u , which is still a 2-approximate vertex cover. Further, step 5 ensures that anytime an edge is added to a tree that just contains a single edge, all vertices of degree one have their adjacency list moved to the LEAF list. Full Dynamic Update Process Finally, we consider the addition/deletion of an edge in SPANNERS G . Specifically, let the edge e = (u, v) be added/deleted from forest F i . The above two operations allow us to reduce it to the simpler case of both u and v being in VC. The update process will occur as follows: Proof of Theorem 6.13 : The full update process for ADJ-LISTG, VCG, and GRAPH VC only calls InsertVC, RemoveVC, and UpdateADJ a constant number of times. Therefore, by Lemma 6.14, Lemma 6.15, and Lemma 6.16 this process takes O(poly(log n, −1 )) time. This also implies that at most O(poly(log n, −1 )) edges can be added/deleted fromG V C , and by Corollary 5.2 maintaining H will take at most O(poly(log n, −1 )) time. Vertex Sampling in Bipartite Graphs We now design an improved method for reducing a graph onto one whose vertex size is O(|VC| poly(log n, −1 ))+ |X|/2. Instead of sampling edges of G VC , it samples vertices in X = V \ VC using G VC as a guide. This question that we're addressing, and the vertex sampling scheme, is identical to the terminal cut sparsifier question addressed in [AGK14]. In the next section we will apply this sampling scheme to obtain a vertex sparsification routine that will reduce onto a graph of size proportional to O(|VC| poly(log n, −1 )) without losing a factor of 2 approximation. We will reuse the notation from Section 6.1, and we encourage the reader to revisit the definitions in that subsection. For this section, we will exclusively be dealing with subsets of VC, and we will drop the VC subscript from each S VC . So, formally our goal is to find H so that for all S ⊂ VC, (1 − )∆ G (S) ≤ ∆ H (S) ≤ (1 − )∆ G (S). This sampling scheme allows us to keep expectation of the cuts on VC to be exactly the same, instead of having a factor 2 error from the conversion from G to G VC . The connection to G VC on the other hand allows us to bound the variance of this sampling process as before. In our application of this sampling routine to vertex sparsification, we will consider sparsifying G \ X separately, so for simplicity, we assume here that (V C, X) is a bipartition and G = x∈X N x Further, we first focus on the case where all vertices in X have degree d, and all edge weights in X are within a factor of U from each other. We will show reductions from general cases to ones meeting these assumptions in Subsection 8.1.1. As before, let G VC be the multigraph generated by the clique edges from Theorem 6.9: G VC = x∈X K x . Lemma 6.10 implies that the weights of every (multi) edge e x ∈ G VC are within a factor of O(U 2 d) from each other. As mentioned, we ultimately want to obtain a vertex sparsification scheme that reduces to size O(|VC| poly(log n, −1 )) for further application. As a result, instead of doing a direct union bound over all 2 |VC| cuts to get a size of poly(|VC|) as in [AGK14], we need to invoke cut counting as with cut sparsifier constructions. This necessitates the use of objects similar to t-bundles to identify edges with small connectivity. Our proof will use a similar structure to that of Fung et al. [FHH + 11], particularly the cutcounting based analysis of cut sparsifiers. We will follow their definitions, which are in turn based on the definition of edge strength by Benczur and Karger [BK15]. Definition 7.1. In a graph G, an edge is e k − heavy if the connectivity of its endpoints is at least k in G. Furthermore, for a cut S, its k − projection is the set of k − heavy edges in the edges cut, ∂(S). We will refer to edges that we cannot certify to be heavy as light. These edges are analogous to the bundle edges from the cut sparsifier routine from Section 5.4. Before we continue, we remark that these definitions of heavy/strong edges in [FHH + 11, BK15] is almost the opposite of definitions in spectral sparsification. In spectral sparsification, the edges with high leverage scores are kept, and the low leverage score ones are sampled. This issue can also be reflected in the robustness of this definition in the presence of weights: a natural way of generalizing heaviness is to divide the connectivity of uv by the weight w(u, v). This leads to a situation where halving the weight of an edge actually makes it heavier. In fact, these definitions of heaviness / strength are measuring the connectivity in the graph between the endpoints of e, instead of the strength of e itself. As our routines are in the cut-sparsification setting, we will use these definitions in this version in order to be consistent with previous works [FHH + 11, BK15], but may switch to a different set of notations in a future edit. The main result of [FHH + 11], when restricted to graphs with bounded edge weights, states that we can sample the O(log n −2 )-heavy edges by a factor of 2. Our goal is to prove the analogous statement for sampling heavy vertices, which we define as follows: Sample(G, VC, X heavy ) Input: Bipartite graph G with one bipartition VC, heavy subset X heavy of the other bipartition. Output: Bipartite graph H with bipartition (VC, XH). 1. Initialize H ← ∅, XH = ∅. 2. For every x ∈ X heavy , flip fair coin with probability 1/2, if returns heads: (a) H ← H + 2N x . (b) XH ← XH ∪ {x}N x for some x ∈ X heavy is k-connected in the graph G light VC = ∪ x / ∈X heavy K x . We will show in Section 8, these heavy/light subsets can be found by taking pre-images of more restricted versions of t-bundles on G VC . Our main structural result is that a heavy subset can be sampled uniformly while incurring -distortion. Lemma 7.3. Given a bipartite graph G between VC and X such that X has maximum degree d and all edge weights are in some range [γ, U γ], with U = O(poly(n)) and any non-negative γ. For any , there is a parameter t min = O(dU log n −2 ) such that if we're given a subset X light of X so that X heavy = X \ X light is γ(dU )t-heavy with t ≥ t min then the graph consisting of the light vertices and sampled heavy vertices, H = N (X light ) ∪ Sample(G, VC, X heavy ) meets the condition: |∆ G (S) − ∆ H (S)| ≤ ∆ G (S) for all subsets S ⊆ VC w.h.p. Here the constants in t min depends on the failure probability in the w.h.p. The cut-counting proof of cut-sparsifiers from [FHH + 11] essentially performs a union bound over distinct sets of k-heavy projections over all cuts. We will perform the same here, but over distinct partitions of N x over all x in X heavy . We can first define the partition of a single vertex by a cut S ⊆ VC as: N x (S) = {S ∩ N (x), N (x) \ S} . Then we can define an equivalence relation on cuts as: Definition 7.4. S 1 ≡ G S 2 if for any x ∈ X heavy , N x (S 1 ) = N x (S 2 ) Note that this equivalence ignores the presence of edges in X light . So we need to further take representatives of each equivalence class: Definition 7.5. Define S rep to be the set of subsets S ⊂ VC such that 1. For every S ∈ S rep , there is some x ∈ X heavy s.t. N x (S) = {N x , ∅}, aka N x is not entirely on one side of the cut. 2. For any S 1 , S 2 ∈ S rep , S 1 ≡ G S 2 3. For any S ⊂ VC such that S / ∈ S rep , there exists S ∈ S rep such that • S ≡ G S, and • ∆ G (S) ≤ ∆ G (S). An immediate consequence of condition 1 is that for any S ∈ S rep we have ∆ G (S) > γt(dU ) −1 . This set plays the same role as the unique k-projections in cut sparsifiers. Lemma 7.6. Let H be obtained from G by sampling on X heavy , then for any element of S rep , S we have: P H   S,S≡ G S |∆ G (S) − ∆ H (S)| > ∆ G (S)   = P H ∆ G (S) − ∆ H (S) > ∆ G (S) . Proof. Let G sample = G \ x∈X light N x and H sample = H \ x∈X light N x be the graphs being sampled. By construction of H, for any S ⊂ VC, |∆ G (S) − ∆ H (S)| = ∆ G sample (S) − ∆ H sample (S) . By construction of our equivalence relation, if S ≡ G S, ∆ G sample (S) − ∆ H sample (S) = ∆ G sample (S) − ∆ H sample (S) . due to them having the same part that's not in H. Therefore, the failure probability is limited by the element in the equivalence class with the smallest ∆ G (S), aka. S. Corollary 7.7. P H S⊂VC |∆ G (S) − ∆ H (S)| > ∆ G (S) = P H S∈S rep |∆ G (S) − ∆ H (S)| > ∆ G (S) The key observation is that the sizes of subsets of S rep of certain sizes can be bounded using cut-counting on G VC . For any S ⊂ VC, define K x (S) = E Kx (S ∩ N (x), N (x) \ S), which are the edges in K x crossing S. Similar to S 1 ≡ G S 2 , we can define S 1 ≡ G VC S 2 if for any x ∈ X heavy , K x (S 1 ) = K x (S 2 ) Lemma 7.8. For any S 1 , S 2 ⊆ VC, N x (S 1 ) = N x (S 2 ) iff K x (S 1 ) = K x (S 2 ). Therefore S 1 , S 2 ⊆ VC, S 1 ≡ G S 2 iff S 1 ≡ G VC S 2 . Proof. We construct K x as a clique, so K x (S 1 ) = K x (S 2 ) iff S 1 ∩ N (x) = S 2 ∩ N (x) or S 1 ∩ N (x) = N (x) \ S 2 Lemma 7.9. |{S ∈ S rep |∆ G VC (S) ≤ K}| is less than or equal to the number of distinct γ(dU ) −1 tprojections in cuts of weight at most K Proof. Lemma 7.8 gives that S rep has the following properties for G VC 1. For every S ∈ S rep , there is some x ∈ X heavy s.t. K x (S) = ∅. For any S 1 , S 2 ∈ S rep , S 1 ≡ G VC S 2 For any S ∈ S rep , let E heavy (S) denote all the γ(dU ) −1 t-heavy edges crossing S in G VC . The property above gives: x∈X heavy K x (S) is a non-empty subset of E heavy (S), and x∈X heavy K x (S 1 ) = x∈X heavy K x (S 2 ) ∀S 1 , S 2 ∈ S rep . Therefore, each S ∈ S rep such that ∆ G VC (S) ≤ K, must be a distinct γ(dU ) −1 t − projection of weight at most K It remains to combine this correspondence with cut counting to show the overall success probability of the vertex sampling routine. Proving this requires using Chernoff bounds. The bound that we will use is below, it can be viewed as a scalar version of Theorem 1 of [Tro12]. Lemma 7.10. Let Y 1 . . . Y n be random variables s.t. 1. 0 ≤ Y i ≤ 1. 2. µ i = E Y i [Y i ] 3. µ = i µ i Then for any ≥ 0 P Y 1 ...Yn i Y i > (1 + )µ ≤ exp − 2 µ 2 . P Y 1 ...Yn i Y i < (1 − )µ ≤ exp − 2 µ 2 . This bound can be invoked in our setting on a single cut S as follows: Lemma 7.11. For each cut S, we have P H [|∆ H (S) − ∆ G (S)| > ∆ G (S)] ≤ 2 exp − 2 ∆ G (S) 4γ . Proof. Let w max (S) = max x∈X {w (x) (S)}. We will only consider S ∈ S rep , so we know w max (S) > 0, which implies w max (S) ≥ γ. For each S ⊆ S rep and for all x ∈ X, let Y x (S) be the random variable such that either 1. Y x (S) = w (x) (S) 2w max (S) if x ∈ X light 2. Y x (S) equals w (x) (S) w max (S) w.p. 1/2, and 0 w.p. 1/2. Accordingly, we have x∈X E Yx(S) [Y x (S)] = 1 2w max (S) x∈X w (x) (S) = 1 2w max (S) ∆ G (S). The bound then follows from invoking Lemma 7.10. Proof. (Of Lemma 7.3) Let ∆ G VC (S) be the weight of cutting S ⊂ VC in G VC . From Theorem 6.9 for any S ∈ S rep that ∆ G (S) ≥ ∆ G VC (S). Therefore, S⊆S rep 2 exp − 2 ∆ G (S) 4γ ≤ S⊆S rep 2 exp − 2 ∆ G VC (S) 4γ The main cut-counting bound follows from Theorem 1.6 [FHH + 11] on multi-graphs, and by our construction of S rep gives: |{S ∈ S rep |∆ G VC (S) ≤ K}| ≤ n 2KdU (γt) −1 if K ≥ γ(dU ) −1 t, 0 otherwise. Each vertex adds weight at most γdU for any cut, so we can upper bound K by n 2 γU because d ≤ n. Invoking cut counting for intervals of length γ from K ≥ γ(dU ) −1 t to K ≤ n 2 γU allows us to bound the overall failure probability by: ≤ n 2 U i=(dU ) −1 t      S∈S rep γi≤∆ G VC (S)≤γ(i+1) 2 exp − 2 ∆ G VC (S) 4γ      ≤ n 2 U i=(dU ) −1 t 2n 2(i+1)dU t −1 exp − 2 i 4 = n 2 U i=(dU ) −1 t 2n 2(i+1)dU t −1 − 2 i 4 log n . (1) Note that we're free to choose t, and it can be checked that for U ≤ n c 1 , setting t ≥ (28 + 4c 1 )c 2 dU log n −2 bounds this by n −c 2 for any c 2 ≥ 1. Note that if U is larger than O(poly(n)), we could set t = O(dU 2 log n −2 ) and still achieve w.h.p., but for our practical purposes assuming U = O(poly(n)) is more than sufficient because U will always be O(poly(log n, −1)). Maintaining (1− )-Approximate Undirected Bipartite Min-Cut In this section, we will again consider the bipartite minimum s − t cut problem of Section 6, and will improve the approximation guarantee to (1 + ). This improvement will require many of the techniques from Section 6, but we will bypass the loss of a factor 2 approximation by utilizing the vertex sampling scheme presented in Section 7. A high level overview of these techniques is in Section 3.3. The dynamic algorithm given in this section will rely heavily on the definitions and observations of Subsection 6.1, which we encourage the reader to revisit. Lemma 7.3, along with the framework from Section 6 allow us sample a large set of vertices if the optimal minimum s − t cut is small, and will guarantee that the sampled vertices have O(poly(log n, −1 )) degree. However, Lemma 7.3 as stated require incident edges of all sampled vertices to have weight within factor O(poly(log n, −1 )) of one another. In this section, we integrate this subroutine into the data structure framework, leading to our main result for approximating undirected bipartite maximum flows: Theorem 1.3. We can maintain an (1 − ) to the value of the maximum flow on a dynamically changing unweighted, undirected, bipartite graph, as well as query access to the associated minimum cut in poly(log n, −1 ) time per update. Section 8.1 will show how the vertex sampling scheme given in Section 7 can be iteratively applied, reducing to a graph with O(|V C| poly(log n, −1 )) vertices and O(|V C| poly(log n, −1 )) edges. This section will first present the full vertex sparsification scheme, and then examine the two primary components of this scheme. Section 8.1.1 will show how we can pre-process a graph to ensure that all edge weights of each sampled vertex are close to each other, which will be necessary for bucketing sampled vertices. Section 8.1.2 will utilize these bounded properties and the vertex sampling of Section 7 to give a vertex sparsification scheme for each bucket, culminating in a proof of correctness for the full scheme in terms of approximation guarantees and bounds on the number of edges and vertices. Section 8.1.3 will extend vertex sparsification to general graphs without bounds on degree for the static case, proving Corollary 3.2. Section 8.2 will then use this vertex sparsification scheme along with many of the components from Section 6 to give a fully dynamic algorithm for maintaining a minimum s − t cut on a bipartite graph. The correctness of this algorithm will follow from the correctness of the dynamic algorithm in Section 6 and the correctness of vertex sparsification. Accordingly, it will then only be necessary to establish that we can dynamically update all necessary data structures in O(poly(log n, −1 )) time. Vertex Sparsification in Quasi-Bipartite Graphs The general framework of the routine is shown in Figure 10. Here the constant in front of t depends on W as well as in the w.h.p. condition. A proof of Theorem 8.1 will be given at the end of Section 8.1.2. Reduction to Bounded Weight Case The idea here will be to look at each N x and move the low weight edges into G \ X , thereby ensuring that the remaining edges in N x have weight within a O(poly(log n, −1 )) factor. This will create a multi-graph in G \ X, where will use the normal notation (u, v) x to denote an edge added by N x . Theorem 8.2. Given G with bipartition (V C, XG) with weights in [γ, γW ], such that the degree of each vertex in XG is bounded by d, for any , VertexBucketing(G, V C, XG, d, ) will return G = G \ X ∪ · G 1 ∪ · . . . ∪ · G L such that 1. G ≈ G 2. For each G i , the weights of G i are in [γ, 2γd −1 ] for some γ 3. Any edge e ∅ ∈ G must be in G \ X 4. If x ∈ X has non-zero degree in G i , then x has zero degree in G \ G i ,1. Initialize G \ X = G \ X, and G i = (VC, ∅) for i = 1 . . . L with L = O(log W ) 2. For each x ∈ XG (a) Let (x, u) be the edge with maximum weight in N x , where w(x, u) ∈ [γ2 i−1 , γ2 i ] (b) For each (x, v) ∈ N x , if w(x, v) < d w(x, u), then put (u, v) x in G \ X. Otherwise, put (x, v) in G i 3. Return the multi-graph G \ X, and graphs G 1 . . . G L Figure 11: Vertex Bucketing in G Proof. The only interesting cuts are singletons: 1. Removing v has w(x, v) before and after. 2. Removing x has w(x, u) + w(x, v) before, and w(x, u) after, a factor of difference since w(x, u) + w(x, v) ≤ (1 + )w(x, u). 3. Removing u has w(x, u) before, and w(x, u) + w(x, v) after, same as above. Invoking this repeatedly on small stars gives: Lemma 8.4. A star x with degree d can be reduced to one whose maximum and minimum weights is within a factor of O(d −1 ) while only distorting cuts by a factor of 1 + . Proof. Let the neighbors of x be v 1 . . . v d s.t. w(x, v 1 ) ≥ w(x, v 2 ) ≥ . . . ≥ w(x, v d ). Suppose w(x, v i ) < /dw(x, v 1 ) , then applying Lemma 8.3 gives a multiplicative error of 1 + /d. Applying this at most d times gives the approximation ratio, and moves all the light edges onto v 1 . Bounded Weight Vertex Sparsification The bucketing of vertices in the independent set ensures that all the weights in each bucket are within a factor O(d/ ), which will allow us to iteratively reduce the number of vertices by applying the Sample algorithm given in Section 7 O(log n) times. Note that our Sample algorithm doubles the weights of each sampled star, so N i x will denote the star x in ith iteration graph G i with updated weights for that graph. Assuming that this is the case, from Lemma 7.3, if we set = l , then with high probability ⊆ XG i of G i (b) (G i+1 , XG i+1 ) ← Sample(G i , VC, XG i , XG light i ) (c) Add x∈XG light i N i x to H 3. Return H = H ∪ G lG i ≈ VC G i+1 ∪ x∈XG light i N i x By construction, for all j < i, XG light j ∩ XG i = ∅, so adding each x∈XG light j N j x to both sides will still preserve the relation above. Applying this argument inductively and using = l gives H ≈ VC G with high probability. In order to complete the proof of Theorem 8.5, it is now necessary to show that for each G i and XG i , we can construct XG light i such that XG i \ XG light i is a 2 i γ(dU ) −1 t − heavy subset of XG i . The idea will simply be to construct XG light i from t disjoint spanning forests in G i VC with some additional properties that will allow O(poly(log n, −1 )) dynamic maintenance in the following subsection. Definition 8.6. Given G with vertex bipartition (VC, X), we say that F = F 1 ∪ · . . . ∪ · F t is a t-clique forest if Figure 13: Light Vertex Set of XG 2. For any x ∈ X, at most one edge e x ∈ K x is in F . For all x ∈ X such that F ∩ K x = ∅, for any e x = (u, v) x ∈ K x , u and v are connected in all F i Lemma 8.7. Given G with vertex bipartition (VC, X) such that all x ∈ X have maximum degree d, weights in [γ, γU ] and a t-clique forest F , if X light = {x ∈ X|F ∩ K x = ∅}, then X heavy = X \ X light is an γ(dU ) −1 t − heavy subset of X Proof. For some (u, v) x ∈ K x with x ∈ X heavy , suppose (u, v) x is in a cut S VC ⊂ VC such that ∆ G VC (S VC ) < γ(dU ) −1 t. From Lemma 6.10, all edges in G VC have weight at least γ(dU ) −1 . Therefore, there must exist some F j such that u and v are not connected, giving a contradiction. Note that after the algorithm terminates G i VC = x∈XG heavy i K x , which will be necessary for the dynamic maintenance. The following lemma follows by construction and the fact that each forest has at most |V C| − 1 edges. Lemma 8.8. F i is a t-clique forest of G i VC , |XG light i | ≤ t|V C|, and XG heavy i is a 2 i γ(dU ) −1 t−heavy subset of XG i Proof of Theorem 8.1 (1) The first part follows from Theorem 5.1 and the second part follows from Theorem 8.5 (2) Property (1) of Theorem 8.2 gives us G ≈ /2 G with U = 4d −1 for each G i from property (2). Then, property (3) implies that each G i is bipartite, and property (4) implies that each vertex in X G i is bounded by d. We can then apply Theorem 8.5 to each G i , with U = 4d −1 to get G i ≈ V C /2 H i with high probability. Note that we are implicitly assuming U = O(poly(n)), aka −1 = O(poly(n)). As was discussed at the end of Section 7, we could avoid this assumption by adding an extra −1 factor to the t-bundle, but any −1 = ω(poly(n)) loses any practical value. L = O(log W ) = O(poly(log n)) by assumption, and property (4) of Theorem 8.2 ensures that a vertex is only sampled in one G i , so taking the union over O(poly(log n)) buckets preserves G ≈ V C /2 H w.h.p. for sufficient constants in t. G ≈ /2 G is a stronger statement than G ≈ V C /2 G, implying G ≈ V C H (3) Edge weights are only changed in Sample where they are either doubled or left alone. VertexSparsify calls Sample at most O(log n) times for each bucket of G, giving the appropriate bound. Improved Static Algorithm for General Graphs Composing VertexSparsify O(log n) times, along with spectral sparsifiers, leads to a static routine: Corollary 3.2. Given any graph G = (V, E) with weights within a O(poly(n)) factor, a vertex cover VC of G, where X = V \ VC, and some error , we can build a terminal-cut-sparsifier H with O(|VC| poly(log n, −1 )) vertices in O(m · poly(log n, −1 )) work. Now that we have sufficient notation in place, by terminal − cut − sparsif ier, we mean that G ≈ VC H with high probability. Note that this is almost equivalent to Theorem 8.1, but we make no assumptions on the degree of vertices in X. Also, we will now specify poly(log n , −1 ) for the number of vertices as log 18 n −7 . Proof. Consider running the following routine iteratively: 1. Sparsify G with error = O(log n) and outputG 2. Find the bipartite subgraph G containing VC and vertices X G ⊆ X whose degree are less than O(log 2 n −2 ). Run VertexSparsify on G, VC, X G, with d = O(log 2 n −2 ) and with error = O(log n) , returning H G ←G \ G and H ← H ∪ H If at any point, we have |X| < |VC| log 17 n −7 , then return H ∪ G. From [SS11], and the number of edges inG is O(n log n −2 ) with high probability. Therefore, at least half of |X| have degree less than O(log 2 n −2 ) because otherwise the number of edges inG would be O(|X| log 2 n −2 ) = O(n log 2 n −2 ) by the assumption |X| ≥ |VC|. This eliminates half the vertices in X with high probability for every run of the routine, so the process can continue at most O(log n) times. From Theorem 8.1 each bucket of H will have at most O(|VC|t log n) vertices with t = O(d 2 log 3 n −3 ) and d = O(log 2 n −2 ), giving O(|VC| log 15 n −7 ). We run sparsification on G and VertexSparsify on G O(log n) times, so from the guarantees of Corollary 5.2 and property (3) of Theorem 8.1, the weights are within a factor O(n O(log n) ). Therefore, there are at most O(log 2 n) buckets of H, and at most O(|VC| log 17 n −7 ) vertices which has the appropriate size requirement. Sparsification gives G ≈ G with high probability, which is a stronger statement than G ≈ VC G . Theorem 8.1, which is still applicable for weight within a factor O(n O(log n) ), gives G ≈ VC H with high probability. Therefore (G \ G) ∪ H ≈ VC 2 G with high probability. Applying this inductively for O(log n) steps gives the desired relation by setting = O(log n) as was done in the iterative routine above. Sparsifying G requires O(m · poly(log n, −1 )) work [SS11]. Furthermore, in Section 8.2 we will show that VertexSparsify can be maintained dynamically in worst-case update time of O(poly(log n, −1 )), so it's static runtime must be O(m · poly(log n, −1 )). Dynamic Minimum Cut of Bipartite Graphs Now that we have the full process of VertexSparsify, we will give the dynamic algorithm for maintaining a (1+ )-approximate minimum cut in amortized O(poly(log n, −1 )) time. The algorithm in Figure 14 will be analogous to the one given in Section 6, but will replace sparsification of G VC with VertexSparsify, improving the approximation by a factor of 2. In this algorithm we run into the same issue of returning a cut of size O(n) in amortized O(poly(log n, −1 )) time, and will allow a similar querying scheme. Let V H be the non-zero degree vertex set of H. Our vertex sparsification process ensures that VC ⊆ V H , so for the computed S V H ⊂ V H , we will maintain the cut S V H ∩ VC ⊂ VC with s ∈ S V H . For a vertex v ∈ VC, return v is with s iff v ∈ S V H ∩ VC, which takes O(1) time. For a vertex x / ∈ VC, note that all of N (x) must be in VC, and return that x is with s iff w(x, S V H ∩ VC) = w (x) ( S V H ∩ VC) inG, taking O(poly(log n, −1 )) time to compute w(x, S V H ∩ VC) and w (x) ( S V H ∩ VC), by Corollary 5.2 and Corollary 6.4. Note that by restricting to VC we will be able take advantage of the approximation guarantees of vertex sparsification in the corollary below. Corollary 8.9. The dynamic algorithm maintains a (1 + )-approximate minimum s − t cut in G, and will only compute an approximate minimum s − t cut on H every O( OP T ) dynamic steps, taking O(OP T · poly(log n, −1 )) time each computation Proof.G = F 1 ∪ · . . . ∪ · F K for some K = O(poly(log n, −1 )) by Corollary 5.2, so from Lemma 3.1 and Corollary 6.4, we know |VC| = O(OP T · poly(log n, −1 )) and the degree of all vertices in XG is O(poly(log n, −1 )). From Corollary 5.2, the weights ofG are in [1, O(n)], and so property (1) of Theorem 8.1 implies that H has O(OP T · poly(log n, −1 )) edges. Therefore, we can find a (1 + ) approximate minimum s − t cut in H, in O(OP T · poly(log n, −1 )) time. Assume S V H ⊂ V H is returned as a (1 + )-approximate minimum s − t cut in H, with = O(1) . Let S VC = S V H ∩ VC be its restriction to VC, and let S = S V C ∪ {x ∈ XG : w(x, S V C ) = w (x) ( S V C )} be the extension of S VC ontoG, which is the cut returned by our vertex querying scheme. From Corollary 5.2 and Theorem 8.1, we have G ≈ G andG ≈ VC H, respectively, which gives ∆ G ( S) ≤ (1 + )∆G( S) = (1 + )∆G( S V C ) ≤ (1 + ) 2 ∆ H ( S V C ). On the other hand, let S ⊂ V be the minimum s − t cut in G, and S V C ⊂ VC be its restriction to VC. Using the fact that ∆G(S VC ) is the weight of the minimal extension of S VC inG, along with the approximations G ≈ G andG ≈ VC H gives ∆ G (S) ≥ (1 − )∆G(S) ≥ (1 − )∆G(S V C ) ≥ (1 − ) 2 ∆ H (S V C ). The near-optimality of S V H on H and setting S VC = S V H ∩ VC, gives, ∆ H (S V C ) ≥ (1 − )∆ H ( S V H ) ≥ (1 − )∆ H ( S VC ) Therefore, ∆ G ( S) ≤ (1 + ) 5 ∆ G (S), and by choosing = O(1) we maintain a (1 + 2 )-approximate minimum s − t cut in G. An approximate minimum s − t cut on H will be re-computed in 2 ∆ H ( S V H ) dynamic steps. OP T = ∆ G (S) ≤ (1 + )∆ H ( S V H ), so ∆ H ( S V H ) = O(OP T ) All that is left to be shown is that data structures can be maintained in O(poly(log n, −1 )) time per dynamic update. As a result of Corollary 5.2, it suffices to show the following Theorem 8.10. For each addition/deletion of an edge inG, maintaining G, H, and VC takes O(poly(log n, −1 )) time. As in Section 6.3, most of the necessary analysis for Theorem 8.10 will follow from the fact that all x ∈ XG have degree O(poly(log n, −1 )), and the only substantial changes made to the data structures in one dynamic step, are done within the neighborhood of some x ∈ XG. We will also assume all of the dynamic data structure analysis of Section 6.3 with regards to maintaining a corresponding G VC of some G. In the rest of this section, we will first examine dynamically maintaining the pre-processing routine, particularly when vertices are moved in and out of the vertex cover. Then we will consider dynamically maintaining our vertex sparsification routine. Most of the time complexity analysis will follow from Section 6.3, and the only tricky part will be ensuring that dynamic changes do not multiply along iterations of the sparsification routine. Maintaining G As with the multi-graph G VC , for G \ XG, an edge e ∅ denotes an edge originally inG and e x denotes an edge that was moved into G \ XG from N x . For each x ∈ XG, let x max denote the vertex such that (x, x max ) has the maximum weight in N x . Let bucket(x) be the i ∈ [L] such that w(x, x max ) ∈ [2 i−1 , 2 i ]. We can use 1 as our scalar here because all weights of G are 1, so from Corollary 5.2, all weights ofG are in [1, O(n)]. In order to maintain each x max , we will assume that the data structure ofG is such that the adjacency list of each x is sorted by edge weight. Consequently, edge insertions/deletions inG will require O(log n) time. Maintaining each bucket for an edge insertion/deletion inG will be analogous to maintaining G VC in Section 6.3. We will first show that moving a vertex in and out of XG can be done in O(poly(log n, −1 )) time, then give the overall update process, which will primarily just be composed of these two operations. If v is not in VC, then v cannot be incident to any vertices in XG. Therefore, placing v in VC implies that v cannot be incident to any edges in all G k and no edges e v exist in G \ XG. RemoveXG( G, XG, v) performs exactly these removals and inserts all necessary e ∅ incident to v into G \ XG Lemma 8.12. If v is not in VC, but was placed in VC for G, then running InsertXG( G, XG, v) If v is not in VC, but was placed in VC for G, then only edges e ∅ are incident to v in G. Removing v from VC requires deleting all of these edges. Further, all edges e in N v of sufficiently small weight must be moved to G \ XG as e v , and the rest of N v must be placed in the appropriate G i . InsertXG( G, XG, v) performs exactly these operations. will output G with v / ∈ VC in O(deg v log n) time, The full dynamic update process of G for each e = (u, v) insertion/deletion inG will then be as follows. 1 . For u and v, RemoveXG( G, XG, v) if v / ∈ VC 2. Update VC andG as done in section 5 3. Add/delete (u, v) ∅ from G \ XG 4. Update u max and v max , which will simply require looking at the first edge incident to u and v inG, as the list is sorted by weight 5. For u and v, InsertXG( G, XG, v) if v / ∈ VC Lemma 8.13. For each edge addition/deletion inG, maintaining G = G \ XG ∪ · G 1 ∪ · . . . ∪ · G L takes O(poly(log n, −1 )) time. Proof. Note that InsertXG( G, XG, v) and RemoveXG( G, XG, v) are only performed if v / ∈ VC, which implies that the degree of v inG is O(poly(log n, −1 )). Updating VC andG is known to take O(poly(log n, −1 )) time. Steps 3 and 4 clearly take O(log n) time. Therefore, the full runtime of this update process is O(poly(log n, −1 )). Maintaining BoundedVertexSparsify We will dynamically sparsify the multi-graph G \ XG as per usual, so each edge insertion/deletion requires O(poly(log n, −1 )) update time for G\XG. Accordingly, we will only consider maintaining the necessary data structures for BoundedVertexSparsify of each G k , which we will simply denote as G with bipartition (VC, XG). Alterations to G are made by the dynamic update process in the previous section, which implies that we only need to consider the following changes to G. Add/Delete a vertex x from X, and add/delete N x from G. Add/Delete an edge within N x for some x ∈ X. If an edge is added/deleted from N x , we will simply delete N x from G, and then add N x with the edge added/deleted to G. Accordingly, in order to establish that our data structures can be maintained in O(poly(log n, −1 )) update time, we just need to show that adding/deleting any N x from G can be done in O(poly(log n, −1 )) update time. For each level i of computing a light vertex set and running Sample, we need to maintain G i , G i VC , XG light i and all F i,j in F i . The data structures for G i and G i VC will be as in Subsection 6.3. Assume that the data structure of each F i,j is such that we can search for edges in O(log n)-time, either by search trees or linked lists with back pointers (see e.g. [CLR + 09], Chapters 10.2, 10.3, and InsertStar(G i , XG i , XG light i , N x ) 1. Update G i ← G i ∪ N x , XG i ← XG i ∪ x, and insert K x into G i VC 2. For the first e x ∈ K x that can be added to some F i,j : Update F i,j ← F i,j ∪ e x , XG light i ← XG light i ∪ x, and remove K x from G i VC 3. If no e x ∈ K x can be added to any F i,j , with probability 1 2 : run Figure 17: Add N x to G i 13). The data structure each XG light i will just be a list of vertices with insertion/deletion taking O(log n) time. InsertStar(G i+1 , XG i+1 , XG light i+1 , 2N x ) We will still assume edge additions/deletions in G i , G i VC can be maintained in O(poly(log n, −1 )), as was shown in Subsection 6.3. Most of the time complexity analysis will then follow from this, and we just need to establish that the additions/deletions will not multiply as we move down the pipeline. This will ultimately follow from our construction of the t-clique forests. Adding some N x to G i The algorithm in Figure 17 will add a vertex x to G i , along with the corresponding N x . Lemma 8.14. InsertStar(G i , XG i , XG light i , N x ) adds N x to G i while maintaining t-clique forest F i Proof. If some e x ∈ K x can be added to some F i,j , then by construction, F i ∩ K x = e x and x ∈ XG light i . Therefore, F i is still a t-clique forest, and x ∈ XG light i implies x / ∈ XG i+1 , so it is only necessary to add e x to F i,j and x to XG light i . If no e x ∈ K x can be added to any F i,j , then F i ∩ K x = ∅ and x ∈ XG heavy i . Therefore, F i is still a t-clique forest, and x ∈ XG heavy i implies a coin must be flipped to determine whether x is added to XG i+1 and 2N x is added to G i+1 . Furthermore, we still maintain G i VC = x∈XG heavy i K x Deleting some N x from G i The algorithm in Figure 18 will delete a vertex x from G i , along with the corresponding N x . 3. If no e x ∈ K x is in any F i,j , run RemoveStar(G i+1 , XG i+1 , XG light i+1 , 2N x ) if x ∈ XG i+1 Figure 18: Remove N x from G i to F i,j then y is added to XG light i and F i ∩ K y = f y . Therefore, F i is still a t-clique forest, and because y ∈ XG light i , it is now necessary to remove 2N y from G i+1 if y ∈ XG i+1 . If we have F i ∩ K x = ∅, then x ∈ XG heavy i and F i is still a t-clique forest. Further XG i+1 ⊆ XG i , so it is necessary to remove 2N x from G i+1 if x ∈ XG i+1 . Furthermore, we still maintain G i VC = x∈XG heavy i K x Lemma 8.16. For any addition/deletion of some x from XG 0 and N x from G 0 , maintaining H takes O(t · poly(log n, −1 )) time Proof. Checking each forest for an edge insertion/deletion takes O(t log n) time. It follows almost immediately from the analysis in Subsection 6.3 that the rest of the computation in one iteration of InsertStar and RemoveStar takes O(poly(log n, −1 )) time. Furthermore, both can make at most one recursive call to themselves, so adding/deleting N x from G 0 takes O(l · t · poly(log n, −1 )) time where l = O(log n). A Omitted Proofs of Section 4.2 In the following we give the omitted proofs of section Section 4.2, which mainly use standard arguments. Lemma 4.5. The output H of Light-Spectral-Sparsify is a (1 ± )-spectral sparsifier with probability at least 1 − n −(c+1) for any input graph G that is independent of the random choices of the algorithm. Proof. Let R = 2 3(c + 1) ln n . For every edge e ∈ G \ B, let X e be the random variable that is 4w G (e) · L e with probability 1/4 and 0 with probability 3/4. We further set L B (j) for every 1 ≤ j ≤ 1/R as follows: L B (j) i = R · L B i if 1 ≤ j ≤ 1/R L B i − 1/R R · L B i if j = 1/R Note that this definition simply guarantees that For every edge e ∈ G \ B, using Lemma 4.3, we have X e 4w G (e) · L e α t · L G ≤ R · L G . Furthermore, using B G, we have L B (j) i ≤ R · L B i R · L G i−1 for every 1 ≤ j ≤ 1/R . Thus, the preconditions of Theorem 4.4 are satisfied. We conclude that we have L G H (1 + )L G with probability at least n · exp(− 2 /2R) ≥ n · exp((c + 1) ln n) = 1/n c+1 . A symmetric argument can be used for (1 − )L G L H . Lemma 4.6. The output H of algorithm Spectral-Sparsify is a (1 ± )-spectral sparsifier with probability at least 1 − 1/n c+1 for any input graph G that is independent of the random choices of the algorithm. Proof. Note that since H = k i=1 B i ∪ G k we have L H = L G k + k i=1 L B i . We now prove by induction on j that L G k + k i=k−j+1 L B i (1 + /(2k)) j L G k−j . This claim is trivially true for j = 0. For 1 ≤ j ≤ k, we use the induction hypothesis and Lemma 4.5, which both hold with high probability, to get L G k + k i=k−j+1 L B i = L G k + k i=k−j+2 L B i + L B k−j+1 (1 + /(2k)) j−1 L G k−j+1 + L B k−j+1 (1 + /(2k)) j−1 (L G k−j+1 + L B k−j+1 ) (1 + /(2k)) j L G k−j . We now have L H (1 + /(2k)) k L G with high probability by setting j = k. Using symmetric arguments we can prove (1 − /(2k)) k L G L H . Since (1 − /(2k)) k ≥ 1 − and (1 + /(2k)) k ≤ 1 + , the claim follows. Proof. We will show that, with probability 1 − 2n −c+1 , every iteration j computes a graph G j+1 with half the number of edges in G j . By a union bound, the probability that this fails to be true for any j < n is at most 2n −c . This implies all claims. We use the following standard Chernoff bound: Let X = N k=1 X k , where X k = 1 with probability p k and X k = 0 with probability 1 − p k , and all X k are independent. Let µ = E [X] = N k=1 p k . Then P [X ≥ (1 + δ)µ] ≤ exp(− δ 2 2+δ µ) for all δ > 0. We apply this bound on the output of Light-Spectral-Sparsify for every j. Concretely, we assign a random variable to each edge e of G j , with X e = 1 if and only if e is added to G j+1 . Then E [X] = N/4. By construction, the number of edges in G j is N ≥ (c + 1) log n. Applying the Chernoff bound with δ = 2 we get because then it would be in p(T i ). Consequently, it must be connected to some vertex in p(T i ), and if |p(T i )| < |l(T i+1 )|, then two vertices in l(T i+1 ) must be connected to the same vertex in p(T i ), creating a cycle in T i , giving a contradiction. Thus |VC| = p(T d ) + d−1 i=0 (|p(T i )| + |l(T i+1 )|) ≤ p(T d ) + d−1 i=0 2|p(T i )| ≤ 2 d i=0 |p(T i )| = 2|MVC| , 3 . 3approximate Lipschitz learning on graphs [KRS + 15] and a variety of matrix polynomials in the graph Laplacian [CCL + 15]. Theorem 4. 1 . 1There exists a fully dynamic randomized algorithm with polylogarithmic update time for maintaining a (1 ± )-spectral sparsifier H of a graph G, with probability at least 1 − 1/n c for any 0 < ≤ 1 and c ≥ 1. Specifically, the amortized update time of the algorithm is O(c −2 log 3 ρ log 6 n) and the size of H is O(cn −2 log 3 ρ log 5 n log W + mρ −1 ) , where 1 ≤ ρ ≤ m is a parameter of choice. Here, W is the ratio between the largest and the smallest edge weight in G. The ratio between the largest and the smallest edge weight in H is at most O(nW ). Figure 1 : 1add e to H with w H (e) ← 4w G (e) 5. return (H, B) Light-Spectral-Sparsify (G, c, ). We give a dynamic implementation of this algorithm in Section 4.4.2. In particular we dynamically maintain the t-bundle α-spanner B which results in a dynamically changing graph G \ B. Figure 2 : 2Spectral-Sparsify (G, c, ). We give a dynamic implementation of this algorithm in Section 4.4.3. In particular we dynamically maintain each H i and B i as the result of a dynamic implementation of Light-Spectral-Sparsify which results in dynamically changing graphs G i . Theorem 4.9 ([BKS12]). Given a graph G = (V, E), a set S ⊆ V , a random permutation σ of V , and an integer i ≥ 0, there is a decremental algorithm for maintaining the clustering C S,σ,i and the corresponding forest F S,σ,i of partial shortest path trees from the cluster centers in expected total time O(mi log n).Note that we deviate from the original algorithm of Baswana et al. by choosing the parent in the tree of each cluster according to the random permutation. In the algorithm of Baswana et al. the parents in these trees were chosen arbitrarily. However, it can easily be checked that running time guarantee of Theorem 4.9 also holds for our modification.The running time analysis of Baswana et al. hinges on the fact that the expected number of times a node changes its cluster is O(i log n). Lemma 4.10 ([BKS12]). For every node v the expected number of times v changes its cluster in C S,σ,i is at most O(i log n). By charging time O(deg(v)) to every change of the cluster of v and every increase of the distance from v to S (which happens at most i times), Baswana et al. get a total update time of O(im log n) over all deletions in G. Lemma 4. 13 . 13The number of edges of H is O(k 2 n 1+1/k log n) in expectation. Lemma 4. 17 . 17Given a decremental algorithm for maintaining a (1 ± )-spectral (cut) sparsifier of size S(m, n, W ) for an undirected graph with total update time m · T (m, n, W ), there is a fully dynamic algorithm for maintaining a (1 ± )-spectral (cut) sparsifier of size O(S(m, n, W ) log n) with amortized update time O(T (m, n, W ) log n). Lemma 4. 18 ( 18Decomposability). Let G = (V, E) be an undirected weighted graph, let E 1 , . . . , E k be a partition of the set of edges E, and let, for every 1 Theorem 5. 1 . 1There exists a fully dynamic randomized algorithm with polylogarithmic update time for maintaining a (1 ± )-cut sparsifier H of a graph G, with probability at least 1 − n −c for any 0 < ≤ 1 and c ≥ 1. Specifically, the algorithm either has worst-case update time O(c −2 log 2 ρ log 5 n log W ) or amortized update time O(c −2 log 2 ρ log 3 n log W ) and the size of H is O(cn −2 log 2 ρ log n log W + mρ −1 ) , where 1 ≤ ρ ≤ m is a parameter of choice. Here, W is the ratio between the largest and the smallest edge weight in G. The ratio between the largest and the smallest edge weight in H is at most O(nW ). Corollary 5. 2 . 2There exists a fully dynamic randomized algorithm with polylogarithmic update time for maintaining a (1 ± )-cut sparsifier H of a graph G, with probability at least 1 − n −c for any 0 < ≤ 1 and c ≥ 1. Specifically, the algorithm either has worst-case update time O(c −2 log 7 n log W ) or amortized update time O(c −2 log 5 n log W ). The arboricity of H is k = O(c −2 log 3 n log W ). Lemma 5. 6 .Figure 3 : 63(generalization of Theorem 1.1[FHH + 11]) Let H be obtained from a graph G with weights in (1/2, 1] by independently sampling edge edge e with probability p e ≥ ρ/λ G (e), where ρ = C ξ c log 2 n/4 2 , and λ G (e) is the local edge connectivity of edge e, C ξ is an explicitly known constant. Then H is a (1 ± )-cut sparsifier, with probability at least 1 − n −c .Proof. (Sketch) The generalization lies in introducing the parameter c to control the probability of failure. This reflects the standard behavior of Chernoff bounds: increasing the number of samples by a factor of c drives down the failure probability by a factor of n −c . Also, the original theorem assumes that all edges are unweighted, but a standard variant of the Chernoff bound can absorb constant ranges, with a corresponding constant factor increase in the number of samples. Finally, the original theorem is stated with p e = ρ/λ G (e), but all arguments remain identical if this is relaxed to an inequality.1. t ← C ξ cα log W log 2 n/ 2 2. Let B be a t-bundle α-MST of G 3. H := B 4. For each edge e ∈ G \ B (a) With probability 1/4 add e to H with 4w H (e) ← w G (e) 5. Return (H, B) Light-Cut-Sparsify (G, c, ). We give a dynamic implementation of this algorithm in Section 5.4.2. In particular we dynamically maintain the t-bundle α-MST B which results in a dynamically changing graph G \ B. Figure 4 : 4Cut-Sparsify (G, c, ) We give a dynamic implementation of this algorithm in Section 5.4.3. In particular we dynamically maintain each H i and B i as the result of a dynamic implementation of Light-Cut-Sparsify which results in dynamically changing graphs G i . Theorem 5. 9 9([KKM13, GKK + 15]). There is a fully dynamic deterministic algorithm for maintaining a spanning forest T of an undirected graph G with worst-case update time O(log 4 n). Every time an edge e is inserted into G, the only potential change to T is the insertion of e Every time an edge e is deleted from G, the only potential change to T is the removal of e and possibly the addition of at most one other edge to T . The algorithm is correct with high probability against an oblivious adversary. j=1 Tj because it is removed from both G \ i−1 j=1 Tj and Ti. Corollary 5. 12 . 12There are fully dynamic algorithms for maintaining a t-bundle 2-MST B (where t ≥ 1 is an integer) of size O(tn log W ) with worst-case update time O(t log 4 n) or amortized update time O(t log 2 n), respectively. After every update in G, the graph G \ B changes by at most one edge. Lemma 5. 13 ( 13Decomposability). Let G = (V, E) be an undirected weighted graph, let E 1 , . . . , E k be a partition of the set of edges E, and let, for every 1 Lemma 3. 1 . 1The minimum vertex cover MVC in G has size at most OP T + 2 where OP T is the size of the minimum s − t cut in G.Proof. Denote the minimum s − t cut in G as (S,S) where S = {s} ∪ A s ∪ B s andS = {t} ∪ A t ∪ B t . Hence, we must have OP T ≥ |A t | + |B s | + |E(A s , B t )| whereE(A s , B t )are all of the edges between A s and B t .Let V A (A s , B t ) denote all of the vertices in A that are incident to an edge in Definition 6. 3 . 3Given a set of disjoint spanning forestsF = F 1 ∪ · . . . ∪ · F K , we say that VC = i∈[K] VC i is a branch vertex cover of F , if each VC i isthe set of all vertices other than the leaves in F i Corollary 6.4. For any graph G = (V, E) and corresponding sparsified graphG Figure 5 : 5Dynamic (2 + )-approximate Minimum s − t Cut G, taking O(poly(log n, −1 )) time to compute w(x, S VC ) and w(x, VC \ S VC ). Specifically, the cut will be Theorem 6. 11 . 11Computing a (1 + )-approximate minimum s − t cut in H as in Step 5 of Figure 5 takes O(OP T · poly(log n, −1 )) time for = O(1) , and cut S VC ⊂ VC can be extended to S a 2(1 + ) 5 -approximate minimum s − t cut in G with high probability Proof.G = F 1 ∪ · . . . ∪ · F K for some K = O(poly(log n, −1 )) by Corollary 5.2, so from Lemma 3.1 and Corollary 6.4, we know |VC| = O(OP T · poly(log n, −1 )). From Corollary 5.2, the weights ofG are in [1, O(n)], and Lemma 6.10 implies that the weights ofG VC are in [O(n −1 poly(log n, −1 )) −1 , O(n)]. Figure 6 : 6Moving a Vertex into VC RemoveVC(G, VC, v) 1. For all edges e adjacent to v in ADJ-LISTG, delete e ∅ from GRAPH VC . Figure 8 : 8Update ADJ-LISTG Lemma 6.15. If BRANCH v is empty, then running REMOVE VC (v) on GRAPH VC , using ADJ-LISTG, will output GRAPH VC equivalent toG VC\v of ADJ-LISTG in O(poly(log n, −1 )) time. Lemma 6 . 16 . 616UpdateADJ(G, VC, e) takes O(log n) time and all vertices v such that L v,i are in BRANCH v , maintain a 2-approximate vertex cover of F i . Proof. Finding the adjacency list of u and v for F i in ADJ-LISTG takes O(log n) time. The rest of the steps all take O(1) time, as they are just there to ensure we maintain the 2-approximate vertex cover of F i . 1 . 1For u and v, if v / ∈ VCG, then run InsertVC on GRAPH VC , VCG, and v 2. Update ADJ-LISTG 3. If e was added/deleted fromG, insert/delete edge e ∅ from GRAPH VC and insert u and v into VCG 4. For u and v, if BRANCH v is empty, then run RemoveVC on GRAPH VC , VCG, and v, and delete v from VCG By Lemma 6.14, GRAPH VC is equivalent toG VC∪{u,v} on updated ADJ-LISTG after step 3 because u and v are in VC. Similarly, the moving of u and v outside of VC ensures our final state is good. 3 . 3Return (H, XH). Figure 9 : 9Sampling Heavy Vertices Definition 7.2. A subset of X, X heavy is a k-heavy subset if every pair of vertices u, v in some Theorem 8. 1 . 1Given any graph G, vertex cover VC and XG = V \ VC, such that the degree of each vertex in XG is bounded by d, with weights in [γ, γW ] where log W = O(poly(log n)), and error . Then there is a t = O(d 2 log 3 n −3 ) whereby VertexSparsify(G, VC, XG, d, ) returns H s.t. w.h.p. 1. H \XG is a multi-graph on VC with O(|V C| poly(log n, −1 )) edges, and each H i is a bipartition with VC on one side, and at most O(|VC|t log n) vertices of XG on the other. 2. H ≈ VC G. VertexSparsify(G, VC, XG, d, ) Input: Graph G with vertex cover VC and XG = V \ VC, such that the degree of each vertex in XG is bounded by d. 1. Build G on the same vertex set as G s.t. G ≈ /2 G and for each x in X G, the weights are within a factor of O(d/ ) of each other.2. Bucket G by maximum edge weights in each N x into G 1 . . . G L , along with G \ XG 3. Set t = O(d 2 log 3 n −3 ), initialize H = (VC, ∅). 4. With error /2, sparsify G \ XG and BoundedVertexSparsify each G i , giving H i 5. Return the union of each sparsified graph, H = H \ XG ∪ · H 1 ∪ · . . . ∪ · H L . Figure 10 : 10Vertex Sampling in G 3. All edge weights of H are in [γ, O(γnW )] Theorem 8. 5 . 5Given a bipartite graph G with bipartition (V C, XG), and weights in [γ, U γ] where U = O(poly(n)), with degree of x ∈ XG bounded by d, and error . Then there is a t = O(dU log 3 n −2 ) whereby BoundedVertexSparsify(G, V C, XG, t) returns H, s.t. w.h.p. 1. Initialize G 0 ← G, XG 0 ← XG, and H ← ∅ 2. For each i = 0 to l − 1 (a) Compute a t-bundle vertex set XG light i Figure 12 : 12Bounded Weight Vertex Sparsification in G 1. H is a bipartition with VC on one side and at most O(|V C|t log n) vertices on the other 2. H ≈ V C G Proof. (1): Set l = O(log n) and note that |XG| ≤ n, so G l is unlikely to have many remaining vertices after sampling O(log n) times by a standard argument using concentration bounds. Then, Lemma 8.8 will show |XG light i | ≤ t|V C| for all i, giving the desired size. (2): By construction of BoundedVertexSparsify(G, V C, XG), the weights of each G i are in [2 i γ, 2 i γU ]. We will show in Lemma 8.8 that for each G i and XG i , we can find a t-bundle vertex set XG light i of XG i , such that XG heavy i = XG i \ XG light i is a 2 i γ(dU ) −1 t − heavy vertex subset. Figure 14 : 14Dynamic (1 + )-approximate Minimum s − t Cut Lemma 8 . 11 .Figure 15 :Figure 16 : 8111516If v is not in VC, then running RemoveXG( G,XG, v) will output G with v ∈ VC in O(deg v log n) time, where deg v is the degree of v inGProof. Costs of the three steps are:RemoveXG(G, XG, v) 1. Delete all edges e v incident to v max from G \ XG 2. Delete all edges incident to v from G bucket(v)3. For all edges e incident to v inG, add e ∅ into G \ XG Removing a Vertex from XGInsertXG( G, XG, v) 1. Delete all edges e ∅ incident to v in G \ XG 2. For all edges e = (v, w) ∈G incident to v (a) If w(v, w) < d w(v, v max ): insert (w, v max ) v into G \ XG (b) Otherwise: insert (v, w) into G bucket(v) Inserting a Vertex into XG 1. Deleting all edges e v incident to v max from G \ XG takes O(log n) time per deletion and O(deg v ) deletions. 2. Deleting all edges incident to v from G bucket(v) takes O(log n) time per deletion and O(deg v ) deletions.3. Adding e ∅ into G \ XG takes O(log n) time and is done for all edges e incident to v inG, so O(deg v ) times L B (j) = L B i and L B (j) i ≤ R · L B i for every 1 ≤ j ≤ 1/R . Wenow want to apply Theorem 4.4 with the random variables Y = e∈G\B X e + 1/R j=1 L B (j) and Z = L G . Observe that L e + L B = L G = Z . Lemma 4. 7 . 7With probability at least 1−2n −c , the number of iterations before algorithm Spectral-Sparsify terminates is min{ log ρ , log m/((c + 1) log n) }.Moreover the size of H isO   1≤j≤i |B i | + m/ρ + c log n   ,and the size of the third output of the graph is at most max{O(c log n), O(m/ρ)}. 1 . 1Dynamically maintain a sparsified G, which we will denoteG 2. Dynamically maintain a branch vertex cover, VC, ofG, where we ensure s, t ∈ VC 3. Dynamically maintain multi-graphG VC 4. Dynamically maintain a sparsifiedG VC , which we will denote as H with vertex set V 5. Every 2 ∆ H ( S VC ) dynamic steps, recompute S VC ⊂ VC, an approximate minimum s − t cut on H, ignoring all degree zero vertices 1 . 1Each F i is a forest of G VC and all are disjoint.LightVertices(G i , VC, XG i ) Input: Bipartite graph G i with bipartition (VC, XG i ) 1. Initialize XG light i ← ∅ and F i = j∈[t] F i,j with F i,j ← ∅ for all j 2. For each j = 1 to t (a) While some edge e x ∈ G i VC can be added to forest F i,j (b) Place e x in F i,j , place x in XG light i , and remove K x from G i VC 3. Return XG light i 1 . 1Dynamically maintain a sparsified G, which we will denoteG 2. Dynamically maintain a branch vertex cover, VC, onG, where we ensure s, t ∈ VC 3. Dynamically maintain a vertex sparsifiedG using VC and XG = V \ VC which we will denote H 4. Every 2 ∆ H ( S V H ) dynamic steps, recompute S V H ⊂ V H , an approximate minimum s − t cut on H, ignoring all degree zero vertices where deg v is the degree of v inG Proof. Costs of the two steps are: 1. Deleting all edges e ∅ incident to v in G \ XG takes O(log n) time per deletion and O(deg v ) deletions. 2. Checking if w(v, w) < d w(v, v max ) and inserting (w, v max ) v into G \ X or inserting (v, w) into G bucket(v) takes O(log n) time. This is done for all edges e = (v, w) ∈G incident to v, so O(deg v ) times Lemma 8.15. RemoveStar(G i , XG i , XG light i , N x ) removes N x from G i while maintaining t-clique forest F i Proof. If we had F i ∩ K x = e x , then x was in XG light i, so e x must be removed from some F i,j and x must be removed from XG light i . F i was a t-clique forest and G i VC = x∈XG heavy i K x (as was noted),implying that multiple edges in G i VC cannot be added to F i,j without creating a cycle. If f y is addedRemoveStar(G i , XG i , XG light i , N x ) 1. Update G i ← G i \ N x , XG i ← XG i \ x, and remove K x from G i VC 2. If some e x is in some F i,j (a) Update F i,j ← F i,j \ e x , XG light (b) If some edge f y ∈ G i VC can be added to F i,j • Update F i,j ← F i,j ∪ f y , XG light i ← XG light i ∪ y, and remove K y from G i VC • run RemoveStar(G i+1 , XG i+1 , XG light i+1 , 2N y ) if y ∈ XG i+1i ← XG light i \ x Using the permutation to choose a random parent is not part of the original construction of Baswana et al. Proof of Theorem 8.10 : Any edge insertion/deletion inG requires O(poly(log n, −1 )) update time for G and VC from Lemma 8.13. Therefore, there are at most O(poly(log n, −1 )) additions/deletions of some N x to some G i , which will require O(t · poly(log n, −1 )) update time from Lemma 8.16, where t = O(poly(log n, −1 )). Thus, the full dynamic update process of all data structures takes O(poly(log n, −1 )) time per dynamic update ofG.B Guarantees of Combinatorial ReductionsWe show some of the structural results necessary for the reductions in Sections 6, 7, and 8. We first show the guarantees of K x :Proof. (of Theorem 6.9) For any x ∈ X and S VC ⊂ VC, let w Kx (S VC ) denote the weight of cuttingand so by assumptionNext we bound the size of the vertex cover formed by removing all leaves, compared to the optimum.Proof. 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{'abstract': 'We initiate the study of dynamic algorithms for graph sparsification problems and obtain fully dynamic algorithms, allowing both edge insertions and edge deletions, that take polylogarithmic time after each update in the graph. Our three main results are as follows. First, we give a fully dynamic algorithm for maintaining a (1 ± )-spectral sparsifier with amortized update time poly(log n, −1 ). Second, we give a fully dynamic algorithm for maintaining a (1 ± )-cut sparsifier with worst-case update time poly(log n, −1 ). Both sparsifiers have size n · poly(log n, −1 ). Third, we apply our dynamic sparsifier algorithm to obtain a fully dynamic algorithm for maintaining a (1 − )-approximate minimum cut in an unweighted, undirected, bipartite graph with amortized update time poly(log n, −1 ).', 'arxivid': '1604.02094', 'author': ['Ittai Abraham ', 'David Durfee \nGeorgia Institute of Technology\nUniversity of Puerto Rico\nRio Piedras\n', 'Ioannis Koutis ', 'Sebastian Krinninger \nMax Planck Institute for Informatics\nGeorgia Institute of Technology\n\n', 'Richard Peng '], 'authoraffiliation': ['Georgia Institute of Technology\nUniversity of Puerto Rico\nRio Piedras', 'Max Planck Institute for Informatics\nGeorgia Institute of Technology\n'], 'corpusid': 3643068, 'doi': '10.1109/focs.2016.44', 'github_urls': [], 'n_tokens_mistral': 60325, 'n_tokens_neox': 54217, 'n_words': 35399, 'pdfsha': 'fcd98ff7503207622ee97dd9fef76ef4fe746d6c', 'pdfurls': ['https://arxiv.org/pdf/1604.02094v1.pdf'], 'title': ['On Fully Dynamic Graph Sparsifiers', 'On Fully Dynamic Graph Sparsifiers'], 'venue': []}

arxiv

Superembedding Formalism and Supertwistors 19 Dec 2012 Zuhair U Khandker Department of Physics Yale University 06520New HavenCT Daliang Li Department of Physics Yale University 06520New HavenCT Superembedding Formalism and Supertwistors 19 Dec 2012 We establish a correspondence between superembedding and supertwistor methods for constructing 4D N = 1 SCFT correlation functions by deriving a simple relation between tensors used in the two methods. Our discussion applies equally to 4D CFTs by simply reducing all formulas to the N = 0 case. I. INTRODUCTION The dynamics of four-dimensional (4D) conformal field theories (CFTs) and superconformal field theories (SCFTs) are tightly constrained by the underlying conformal symmetry. However, these symmetries are obscured when correlators are expressed in terms of standard Minkowski-space coordinates. Both the embedding method and the twistor method allow for the construction of manifestly covariant correlators. 1 Both methods exploit the identification of 4D N = 1 compactified Minkowski superspace M with G, the manifold of null two-dimensional subspaces of supertwistor space [14][15][16][17][18] (and references therein). However, the prescription for constructing correlators in the two approaches appear to be different: 4D superfields are mapped to embedding-or twistorspace fields with different transformation properties under the superconformal group. Consequently, the two methods utilize different tensors to build correlators. In this paper, we show that despite these differences, the two methods are equivalent due to a simple relation between the tensors appearing in each method. In Section II, we review the identification M ∼ = G and the description of G in both methods. In Section III, we review superembedding/supertwistor superfields. In Section IV we show the equivalence between the two methods. We work out an example correlator in Section V and conclude in Section VI. II. MINKOWSKI SUPERSPACE Both the superembedding and the supertwistor method for constructing correlators exploit the following fact: 4D compactified Minkowski superspace M, transforming under the N = 1 superconformal group SU(2, 2|1), can be identified with the manifold G consisting of null two-dimensional subspaces of supertwistor space C 4|1 [14][15][16][17][18] (and references therein). We briefly review this identification (for a more thorough treatment, see Ref. [18]). Supertwistor space C 4|1 [15] consists of vectors V A =   V a V˙a V 5   ,(1) 1 Embedding methods for CFTs go back to Refs. [1][2][3][4][5]. Recent developments for constructing correlators and conformal blocks appear in Refs. [6][7][8][9]. The generalization to 4D N = 1 SCFTs is given in Refs. [10,11]. Twistor methods for (S)CFT correlators are detailed in Refs. [12,13]. where V a and V˙a are bosonic while V 5 is fermionic. The 4D N = 1 superconformal group SU(2, 2|1) (SU for short) consists of 5 × 5 supermatrices U B A satisfying unitarity and unimodularity. 2 V A transforms in the fundamental representation of SU δ SU V A = iT B A V B ,(2) where T B A are the generators of SU. Supertwistor space is equipped with an SU-invariant inner product between vectors V A , W A given by V A W A ≡ VȦAȦ A W A , where VȦ = (V A ) † and AȦ A is the metric given in Eq. (28) of Ref. [10]. The conjugate supertwistor V A ≡ VȦAȦ A is an antifundamental, δ SU V A = −iV B T A B . We define G to be the manifold of null two-dimensional subspaces of C 4|1 . We review two ways of realizing G, which underpin the supertwistor and superembedding methods. A. 'Supertwistor' description of G A two-dimensional subspace of C 4|1 can be spanned with two linearly-independent 3 supertwistors V 1,2 A . We write this pair of supertwistors as Vc A , wherec = 1, 2. The conjugate supertwistor pair is given by V˙c A ≡ Vc A † AȦ A . We define V to be the space spanned by Vc A , V˙c A . To generate a two-dimensional null subspace, Vc A must satisfy the null condition V˙c A Vc A = 0,c,ċ = 1, 2.(3) The choice of Vc A , however, is not unique. One could perform a rotation V ′c A = Vd A gc d , where gc d ∈ GL(2, C) (GL for short), and V ′c A would describe the same subspace. We thus view Vc A as an object having both an SU index and a GL index, transforming as Eq. (2) under SU, but subject to the equivalence relation Vc A ∼ Vd A gc d , gc d ∈ GL.(4) Henceforth, indices with a tilde c,d,ċ,ḋ, ... = 1, 2 will denote GL indices. They are local indices, transforming under changes of basis localized on a particular two-dimensional subspace, and do not transform under SU. We consider the open domain of G where the 2 × 2 submatrix Vc a is non-singular, so it can be rotated into δc a by a GL tranformation, and Vc A can be written as Vc A = Pb A (g V )c b , where Pb A =    δb a iy˙ab 2iθb    and (g V )c b ∈ GL. (5) 2 Our notation for SU follows Ref. [10]. Indices A, B, . . . run over A = {a,ȧ, 5}, where a,ȧ = 1, 2 are SL(2, C) indices, and our conventions for two-component spinors are those of Wess and Bagger [19]. 3 In the terminology of Ref. [20], the bodies of V 1,2 A must be linearly independent. The Pb A , called the Poincaré section, label the equivalence classes of Vc A . An SU transformation induces a transformation on the Poincaré section, and one finds that (y, θ) transform precisely as coordinates on chiral superspace. Similarly, the conjugate supertwistor pair, V˙c A = (ḡ V )˙c˙b P˙b A , where P˙b A = −iȳ˙b a δ˙b˙a −2iθ˙b and (ḡ V )˙c˙b ∈ GL,(6) yields antichiral superspace. The null condition, Eq. (3), fixesȳ˙aã = y˙aã − 4iθ˙aθã, giving rise to the real coordinate x = y+ȳ 2 and standard 4D Minkowski superspace M ≡ R 4|4 . This construction is easily generalized to N -extended Minkowski superspace (including N = 0) as reviewed in [18] (see also Ref. [21]). To keep track of minus signs that arise when permuting objects like Vc A , we define σ(AB) ≡ −1 if A ∈ {a,ȧ} and B ∈ b,ḃ +1 otherwise,(7)σ(A) ≡ σ(AA).(8) For example, Vc A Vd B = −σ(A)σ(B)σ(AB)Vd B Vc A . B. 'Superembedding' description of G The SU-invariant inner product identifies the supertwistor V A as a one-form on the space of conjugate supertwistors. Therefore, another natural way to describe G is to use graded two-forms (bi-supertwistors) X AB [18]. In particular, one uses V, V to construct X AB ≡ − i 2 σ(B)Vc A Vd B ǫcd = X +    i 2 ǫ ab 1 2 (yǫ)˙b a θ a − 1 2 (yǫ)˙a b − i 2 y 2 ǫ˙a˙b i(yθ)˙a θ b i(yθ)˙b 2iθ 2   (9) and its conjugate X AB ≡ − i 2 σ(A)V˙c A V˙d B ǫ˙c˙d = X +   − i 2ȳ 2 ǫ ab − 1 2 (ǫȳ) aḃ −i(θȳ) a 1 2 (ǫȳ) ḃ a i 2 ǫ˙a˙bθȧ −i(θȳ) bθḃ −2iθ 2  (10) where X + = g V ≡ det (g V )c b , X + =ḡ V ≡ det (ḡ V )˙c˙d .(11) These definitions, together with the properties of V in Eq. (2) and Eqs. (3)(4) imply [10] X AB = σ(AB)X BA (12) δ SU X AB = iT A ′ A X A ′ B + σ(AB)iT B ′ B X B ′ A .(13)X AB = AȦ A AḂ B XȦḂ, where XȦḂ = (X BA ) † .(14) [ X AB X CD ] 16 = 0, X AB X CD 16 = 0,(15)X AB X BC 24 = 0,(16)X, X ∼ λX,λX , λ ∈ C − {0} .(17) where in Eqs. (15) and (16), the boldface subscripts denote the dimension of the SU irreducible representation being projected onto (see Ref. [10] for more details). The null condition, Eq. (3), impliesX AB X BC = 0 (which also follows from Eqs. (15,16)). In Ref. [10], superembedding space E was defined to be the space spanned by X, X satisfying Eqs. (12)(13)(14). The constraints in Eqs. (15)(16) and the identification in Eq. (17) reduce E to G [10,18]. In summary, both supertwistors, V, V , and superembedding coordinates, X, X , can be used to describe G. The advantage is that their SU transformation rules, Eqs. (2) and (13), respectively, are linear. In the rest of this paper, we will discuss how this can be used to simplify the construction of SCFT correlators. III. SUPERFIELDS Manifestly covariant correlators can be constructed for fields on E or V. To map results back to M requires a correspondence between these fields and standard superfields on M [11,13], which we now review. A generic primary superfield φ on M is specified by the quantum numbers (j,j, q,q) [23], where (j,j) are its SL(2, C) Lorentz quantum numbers and q,q are related to the scaling dimension ∆ and U(1) R charge of its lowest component field by q ≡ 1 2 ∆ + 3 2 R ,q ≡ 1 2 ∆ − 3 2 R .(18) We illustrate the procedure for uplifting φ c ∼ 1 2 , 0, q,q to a superfield Φ on either E or V. Superembedding Method Φ A X, X = X + −(q+ 1 2 ) X + −q · 2iX c A φ c x, θ,θ(19)Φ A λX,λX = λ −(q− 1 2 )λ−q Φ A X, X(20) Here, Φ A is an SU fundamental, this time with first four components fermionic and fifth component bosonic. Note that essentially X c A is being used as a vierbein in Eq. (19) to convert the Lorentz spinor index on φ c to an SU index on Φ A . To recover φ c , we plug the relation 2iX c a = X + δ c a into Eq. (19) and obtain, φ a = Φ A=a | X + ,X + =1 .(21) Given a correlator of Φ A , one simply uses Eq. (21) to recover the correlator of φ a . Supertwistor Method Φc V, V = (g V ) −(q+ 1 2 ) (ḡ V ) −q φ a x, θ,θ Vc a(22) Φc λV,λV = λ −2qλ−2q Φc V, V Here, Φc is an SU singlet; it only has a GL index. On the right-hand-side of Eq. (22) we see that Vc a is being used as a vierbein to convert the Lorentz spinor index on φ a to the GL index on Φc [13]. Given a correlator of Φc, one 'peels off' factors of Vc a to recover the correlator of φ a [13]. The verification of the above correspondence and the generalization to arbitrary (j,j, q,q) is given in Refs. [11,13]. IV. TENSORS AND INVARIANTS In the superembedding method, correlation functions of superfields on E are built from X and X. Any tensor constructed from these coordinates has at least two indices, and we utilize the notation of Ref. [11] for two-index tensors, 123 · · ·N B A ≡ X 1 σX 2 X 3 σ · · ·X N B A ,(24) with the obvious analogs for (12 · · · N) AB and 1 2 · · ·N AB . In the supertwistor method, superfields on V only have GL indices, so the building block for correlators is obtained by contracting the SU indices of a V˙c A i and a Vc jA to form [13] Y˙cc ij ≡ 1 2 V˙c A i Vc jA .(25) with the obvious analogs for Y˙ã˙b 123···N and Yãb 123···N . The equivalence of the superembedding and supertwistor methods stems from the following simple correspondence between the tensors in Eqs. (24) and (26): Y˙ãã 1···N = 2 g 1 g Nḡȧ 1ċ (1 · · · N)˙c c gã N c ,(27)1 · · ·N B A = 1 2 Vc 1A Y˙dd N ···1 V˙c B N ǫcdǫ˙d˙c(28) This is simply because when GL indices on two V 's are contracted as in Eq. (26), they form an X according to Eq. (9), similiarly forV andX. To derive Eq. (27) explicitly, one first observes that P (c=c) A = 2i X + X c A , P (ċ=ċ)A = − 2i X + X˙c A ,(29) which imply X AB = 2i X + σ(B)X c A X d B ǫ cd , X AB = 2i X + σ(A)X˙c A Xḋ B ǫ˙cḋ.(30) Eqs. (29) and (30) imply Eq. (27). Meanwhile, Eq. (28) follows from Eqs. (9,10). It follows that scalar invariants in both approaches are identical, 1 · · · N ≡ −1 A · · · N A = −Y˙ãã 1···N Y˙bb 1N ǫ˙ã˙bǫãb.(31) Both the superembedding and supertwistor methods reduce the construction of correlators to the task of enumerating tensors. Because of the correspondence in Eqs. (27,28) this task becomes equivalent in the two methods. The relation between the two methods can also be seen from Eqs. (19) and (22), which along with Eq. (9) imply that projections of superfields defined on E are related to superfields defined on V by Φ A=a = (g V )c a Φc. V. EXAMPLE Four-dimensional N = 1 SCFT two-and three-point functions were worked out previously [22][23][24]. Manifestly covariant two-point functions involving superfields in arbitrary SU representations were worked out using the superembedding and supertwistor approaches in Ref. [11] and Ref. [13], respectively. 4 Additionally, Ref. [11] worked out manifestly covariant expressions for three-point functions involving conserved current superfields. We consider here another correlator, the three-point function involving a superfield T cċ (x 1 , θ 1 ,θ 1 ) ∼ ( 1 2 , 1 2 , 3 2 , 3 2 ) 5 , a chiral scalar superfield φ(y 2 , θ 2 ) ∼ (0, 0, q 2, 0), and an antichiral scalar superfieldφ(ȳ 3 ,θ 3 ) ∼ (0, 0, 0,q 3 ), T cċ (x 1 , θ 1 ,θ 1 )φ(y 2 , θ 2 )φ(ȳ 3 ,θ 3 ) .(32) We start with the superembedding approach. First, we introduce superfields on E related 4 Ref. [11] made use of a variant of the index-free formalism introduced in Ref. [8]. 5 T cċ could be the supercurrent superfield [25], which contains the energy-momentum tensor, if it additionally satisfies D c T cċ =DċT cċ = 0, where D c is the N = 1 Poincaré super-covariant derivative. to the superfields on M by T B A (X 1 ,X 1 ) = X + 1 −2 X + 1 −2 X c 1A T cċXċ B 1 , Φ(X 2 ) = X + 2 −q 2 φ,Φ(X 3 ) = X + 3 −q 3φ ,(33) satisfying T B A (λX 1 ,λX 1 ) = λ −1λ−1 T B A (X 1 ,X 1 ), Φ(λX 2 ) = λ −q 2 Φ(X 2 ),Φ(λX 3 ) =λ −q 3Φ (X 3 ),(34) and consider the correlator T B A (X 1 ,X 1 )Φ(X 2 )Φ(X 3 ) = X + 1 −2 X + 1 −2 (X + 2 ) −q 2 (X + 3 ) −q 3 X c ′ 1A T c ′ċ′Xċ ′ B 1 φφ .(35) The RHS dictates that the index A attach to X 1 and the index B attach toX 1 . The only possible such tensor is (1321) B A . Then by homogeneity, up to an overall constant C, T B A (X 1 ,X 1 )Φ(X 2 )Φ(X 3 ) = C (1321) B A 13 2 21 2 23 q 2 −1 δ q 2 ,q 3(36) To recover the 4D correlator, one simply uses Eq. (21), T cċ (x 1 , θ 1 ,θ 1 )φ(y 2 , θ 2 )φ(ȳ 3 ,θ 3 ) = C (1321) B=ċ A=c 13 2 21 2 23 q 2 −1 (X + i =1) δ q 2 ,q 3 = C ′ y 13 + 4iθ 1θ3 cḋ y3 2 + 4iθ 3 θ 2 ḋ d y 21 + 4iθ 2θ1 dċ y3 1 + 2iθ 1 σθ 3 2 2 y1 2 + 2iθ 2 σθ 1 2 2 y3 2 + 2iθ 2 σθ 3 2 q 2 −1 δ q 2 ,q 3 ,(37) in agreement with Ref. [23]. We now consider the supertwistor method. The superfields on V are given by T˙cc(V 1 ,V 1 ) = g −2 1ḡ −2 1ḡċ 1ȧ T˙a a gc 1a , Φ(V 2 ) = g −q 2 2 φ,Φ(V 3 ) =ḡ −q 3 3φ (38) T˙cc(λV 1 ,λV 1 ) = λ −3λ−3 T˙cc(V 1 ,V 1 ), Φ(λV 2 ) = λ −2q 2 Φ(V 2 ),Φ(λV 3 ) =λ −2q 3Φ (V 3 ) (39) and we consider the correlator T˙cc(V 1 ,V 1 )Φ(V 2 )Φ(V 3 ) = g −2 1ḡ −2 1 g −q 2 2ḡ −q 3 3ḡċ 1ȧ T˙a a φφ gc 1a(40) The building blocks for this correlator are Y˙ãã 12 , Y˙ãã 31 , and Y˙ãã 32 . Just as in the superembedding approach, the RHS dictates that the indexċ attach to V 1 whilec attach to V 1 . Thus the tensor representing this correlator must start with Y˙cb 12 and end with Y˙bc 31 . The only possibility is Y˙cc 1231 , which is related to (1321) More complicated correlators involve the enumeration of more than one tensor. Eqs. (27,28) imply that this procedure is equivalent in both methods. VI. CONCLUSION We have shown an equivalence between the superembedding and supertwistor methods for constructing 4D N = 1 SCFT (and 4D CFT) correlators. For other applications, one of the methods may be more natural or efficient, so it seems worthwhile to have both in one's toolbag. Overall, we hope that the simplifications afforded by these approaches will eventually lead to new results in SCFTs. Thus far, correlators constructed using the superembedding and supertwistor approaches have been relatively simple. To construct more complicated correlators, it is imperative to have a better understanding of tensors. When constructing a correlation function, any tensor with the right index structure and homogeneity properties can appear but not all of them are linearly independent. We need a systematic way to identify such a linearly independent subset. We hope to explore this question further. .(27,28). Then by homogeneityT˙cc(V 1 ,V 1 )Φ(V 2 )Φ(V 3 ) = C ′′ 1 13 2 21 2 23 q 2 −1 δ q 2 ,q 3 Y˙cc 1231 .(41)Using Eq. (27) and comparing with Eq. (40) gives exactly Eq. (37). Note that Y˙cc ii = 0 because of Eq. (3). Also note that Y˙cc ij does not transform under SU and is a superconformal invariant. Since a GL index is a local index attached to a particular V i , it can only be contracted with another GL index attached to the same V i . Thus, two-index tensors built from Y˙cc ij can be written asY˙cc 123···N ≡ Y˙cã 1 12 ǫã 1ã2 Y˙ã 2ã2 32 ǫ˙ã 2ȧ3 Y˙ã 3ã3 34 · · · Y˙ã N−1c N −1N AcknowledgementsWe thank Walter Goldberger and Witold Skiba for comments on the manuscript. This work is supported in part by DOE grant DE-FG-02-92ER40704. . P A M Dirac, Annals Math. 37429P. A. M. Dirac, Annals Math. 37, 429 (1936). . H A Kastrup, Phys. Rev. 1501183H. A. Kastrup, Phys. 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{'abstract': 'We establish a correspondence between superembedding and supertwistor methods for constructing 4D N = 1 SCFT correlation functions by deriving a simple relation between tensors used in the two methods. Our discussion applies equally to 4D CFTs by simply reducing all formulas to the N = 0 case.', 'arxivid': '1212.0242', 'author': ['Zuhair U Khandker \nDepartment of Physics\nYale University\n06520New HavenCT\n', 'Daliang Li \nDepartment of Physics\nYale University\n06520New HavenCT\n'], 'authoraffiliation': ['Department of Physics\nYale University\n06520New HavenCT', 'Department of Physics\nYale University\n06520New HavenCT'], 'corpusid': 118565688, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 8886, 'n_tokens_neox': 7474, 'n_words': 3789, 'pdfsha': 'd7bfb6da341d6ae81fe1e512540905e0d94c4d42', 'pdfurls': ['https://arxiv.org/pdf/1212.0242v2.pdf'], 'title': ['Superembedding Formalism and Supertwistors', 'Superembedding Formalism and Supertwistors'], 'venue': []}

arxiv

Empirical Coordination with Two-Sided State Information and Correlated Source and State 16 Jun 2015 Maël Le Treust mael.le-treust@ensea.fr UMR 8051 ETIS ENSEA Université Cergy-Pontoise CNRS 6, avenue du Ponceau95014CERGY-PONTOISE CEDEXFRANCE Empirical Coordination with Two-Sided State Information and Correlated Source and State 16 Jun 2015Index Terms-Shannon TheoryState-dependent ChannelJoint Source-Channel CodingEmpirical CoordinationEmpiri- cal Distribution of SymbolsNon-Causal Encoding and DecodingCausal EncodingCausal Decoding The coordination of autonomous agents is a critical issue for decentralized communication networks. Instead of transmitting information, the agents interact in a coordinated manner in order to optimize a general objective function. A target joint probability distribution is achievable if there exists a code such that the sequences of symbols are jointly typical. The empirical coordination is strongly related to the joint source-channel coding with two-sided state information and correlated source and state. This problem is also connected to state communication and is open for non-causal encoder and decoder. We characterize the optimal solutions for perfect channel, for lossless decoding, for independent source and channel, for causal encoding and for causal decoding. I. INTRODUCTION The problem of the coordination of autonomous agents is the cornerstone of decentralized communication networks. Communication devices are considered as agents that interact in a coordinated manner in order to achieve a common objective, for example, the transmission of information. This analysis is based on a two step approach [1], [2]. The first step is the characterization of the set of achievable joint probability distributions over the symbols of source and channel. The second step is the maximization or the minimization of an objective function (source distortion or channel cost) by considering the set of achievable joint probability distributions. Empirical coordination has been investigated in [3], [4] with a rate-distortion perspective. In the literature of game theory, the agents coordinate their actions by implementing a coding scheme that satisfies an information constraint [5]. Empirical coordination was under investigation in [6] for perfect channel, in [7] for causal encoding, in [8] with channel feedback, in [9] for a multi-user network with an eavesdropper, in [10] for polar codes, in [2] for causal decoding, in [11], [12] with feedback from the source, and in [13] for lossless decoding and correlated source and state. The characterization of the set of achievable joint probability distributions is equivalent to the joint source-channel coding with two-sided state information [14], [15] and corre-lated source and state (Fig. 1). It is also related to the problem of state communication [16]. In this paper, we investigate the empirical coordination with non-causal encoding and decoding and we characterize the optimal solutions for three particular cases: perfect channel, lossless decoding and independent source and channel. We also characterize the optimal solutions for causal encoding and causal decoding with two-sided state information. U n S n Z n X n Y n V n P usz C T D Fig. 1. Non-causal encoding f : U n × S n → X n and decoding g : Y n × Z n → V n functions for i.i.d. source Pusz and channel T y|xs . Channel model and definitions are presented in Sec. II. In Sec. III, we provide achievability and converse results for noncausal encoding and decoding. In Sec. IV, we characterize optimal solutions for perfect channel, for lossless decoding and for independent source and channel. Causal encoding and decoding are presented in Sec. V. Conclusion is in Sec. VI and sketches of proof are provided in App. A, B, C, D, E. II. SYSTEM MODEL The problem under investigation is depicted in Fig. 1. Capital letter U denotes the random variable, lowercase letter u ∈ U denotes the realization and U n denotes the n-time cartesian product. U n , S n , Z n , X n , Y n , V n denote the sequences of random variables of source symbols u n = (u 1 , . . . , u n ) ∈ U n , of channel states s n ∈ S n , of state informations at the decoder z n ∈ Z n , of channel inputs x n ∈ X n , of channel outputs y n ∈ Y n and of outputs of the decoder v n ∈ V n . Sets U, S, Z, X , Y, V are discrete. ∆(X ) stands for the set of probability distributions P(X) over X . The total variation distance between the probability distributions Q and P is denoted by ||Q− P|| tv = 1/2 · x∈X |Q(x)− P(x)|. Notation 1(y|x) denotes the indicator function, that is equal to 1 if y = x and 0 otherwise. Markov chain property is denoted by Y − − X − − U and holds if for all (u, x, y) we have P(y|x, u) = P(y|x). Information source and channel states are correlated and i.i.d. distributed with P usz . Channel is memoryless with transition probability T y|xs . Statistics of P usz and T y|xs are known by both encoder C and decoder D. Definition II.1 Non-causal code c = (f, g) ∈ C(n) is defined by: f : U n × S n −→ X n , (1) g : Y n × Z n −→ V n .(2) N(u|u n ) denotes the occurrence number of symbol u ∈ U in the sequence u n . The empirical distribution Q n ∈ ∆(U × S × Z × X × Y × V) of (u n , s n , z n , x n , y n , v n ) is defined by: Q n (u, s, z, x, y, v) = N(u, s, z, x, y, v|u n , s n , z n , x n , y n , v n ) n , ∀(u, s, z, x, y, v) ∈ U × S × Z × X × Y × V.(3) Fix a target proba. distribution Q ∈ ∆(U ×S×Z×X ×Y×V), the error probability of the code c ∈ C(n) is defined by: P e (c) = P c Q n − Q tv ≥ ε ,(4) where Q n ∈ ∆(U ×S ×Z ×X ×Y ×V) is the random variable of the empirical distribution induced by the code c ∈ C(n) and the probability distributions P usz , T y|xs . Definition II.2 Probability distribution Q ∈ ∆(U × S × Z × X × Y × V) is achievable if for all ε > 0, there exists an ∈ N s.t. for all n ≥n, there exists a code c ∈ C(n) that satisfies: P e (c) = P c Q n − Q tv ≥ ε ≤ ε.(5) If the error probability P e (c) is small, the empirical frequency of symbols (u, s, z, x, y, v) ∈ U × S × Z × X × Y × V is close to the target probability distribution Q(u, s, z, x, y, v), i.e. the sequences (U n , S n , Z n , X n , Y n , V n ) ∈ A ⋆n ε (Q) are jointly typical, with large probability. In that case, the sequences of symbols are coordinated empirically. The performance of the coordination is evaluated by an objective function Φ : U × S × Z × X × Y × V → R, as stated in [2] and [13]. This approach is powerful and since the expectation E[d(u, v)] over the set of achievable distributions, provides directly the minimal distortion level d(u, v). This is valid for any function, as the channel cost c(x) or the utility Φ(u, s, z, x, y, v) of a decentralized network [5]. III. ACHIEVABILITY AND CONVERSE RESULTS We provide necessary and sufficient conditions for noncausal encoding and decoding. This problem is open. Theorem III.1 (Non-Causal Encoding and Decoding) 1) If the joint probability distribution Q(u, s, z, x, y, v) is achievable, then it satisfies the marginal and Markov chains: Q(u, s, z) = Pusz(u, s, z), Q(y|x, s) = T (y|x, s), Y − − (X, S) − − (U, Z), Z − − (U, S) − − (X, Y ).(6) 2) The joint probability distribution P usz (u, s, z)⊗Q(x|u, s)⊗ T (y|x, s) ⊗ Q(v|u, s, z, x, y) is achievable if: max Q∈Q I(W 1 , W 2 ; Y, Z) − I(W 1 , W 2 ; U, S) > 0,(7) 3) The joint probability distribution P usz (u, s, z)⊗Q(x|u, s)⊗ T (y|x, s) ⊗ Q(v|u, s, z, x, y) is not achievable if: max Q∈Q I(W 1 ; Y, Z|W 2 ) − I(W 2 ; U, S|W 1 ) < 0, (8) Q is the set of distributions Q ∈ ∆(U × S × Z × W 1 × W 2 × X × Y × V) with auxiliary random variables (W 1 , W 2 ) s.t.:              (w 1 ,w 2 )∈W 1 ×W 2 Q(u, s, z, w1, w2, x, y, v) = Pusz(u, s, z) ⊗ Q(x|u, s) ⊗ T (y|x, s) ⊗ Q(v|u, s, z, x, y), Y − − (X, S) − − (U, Z, W1, W2), Z − − (U, S) − − (X, Y, W1, W2), V − − (Y, Z, W1, W2) − − (U, S, X). The probability distribution Q ∈ Q decomposes as follows: Pusz(u, s, z) ⊗ Q(x, w1, w2|u, s) ⊗ T (y|x, s) ⊗ Q(v|y, z, w1, w2). The supports of the auxiliary random variables (W 1 , W 2 ) are bounded by max(|W 1 |, |W 2 |) ≤ (|B| + 1) · (|B| + 2) with B = U × S × Z × X × Y × V. Remark III.2 It is possible to send an additional message m ∈ M with rate R corresponding to left-hand side of equation (7), while still satisfying the coordination requirement. This remark also extends to the other results of this article. Sketches of proof are available in App. A and B. Achievability result of Theorem III.1 is based on hybrid coding [17], [18] and is stated in [7], with a unique auxiliary random variable W = (W 1 , W 2 ), without state informations S and Z. Empirical coordination is obtained by considering (U, S) as state information at the encoder and (Y, Z) as a state information at the decoder. This open problem is also related to "state communication" [16] which is solved for Gaussian channels [19], but remains open for non-causal encoding and decoding. In the following, we connect the duality result of [15] and the separation result of [14] to empirical coordination. IV. OPTIMAL SOLUTIONS FOR PARTICULAR CASES In this section, we characterize the joint probability distributions that are achievable for three particular cases. A. Perfect Channel is defined by T y|xs = 1(y|x) as in Fig. 2 U n S n Z n X n V n P usz C D Fig. 2. The perfect channel is defined by T y|xs = 1(y|x). Q(u, s, z) = Pusz(u, s, z), Z − − (U, S) − − X.(9) 2) The probability distribution P usz (u, s, z) ⊗ Q(x|u, s) ⊗ Q(v|u, s, z, x) is achievable if (10). The converse holds. Achievability comes from replacing Y and W 1 by X in Theorem III.1 and converse is in App. C. This results extends the coding theorem of Wyner-Ziv [20] with distortion d(u, v), to the framework of coordination. The message of rate R corresponds to the channel inputs X n of rate log 2 |X | and the optimal distortion level D ⋆ is obtained by taking the expectation E[d(u, v)], as in Corollary V.2. The main difference is that the symbols (U, S, V ) are coordinated with X. B. Lossless Decoding defined by Q(v|u, s, z, x, y) = 1(û|u) The characterization for lossless decoding with correlated source and state is in [13]. Achievability is also a particular case of Theorem III.1 by replacing random variables V and W 2 by U . Joint distribution P usz (u, s, z) ⊗ Q(x|u, s) ⊗ T (y|x, s) ⊗ 1(û|u) is achievable if (11). The converse holds. . This result extends the coding theorem of Gel'fand-Pinker [21] to the framework of coordination. The message is a sequence of source symbols U correlated with the states (S, Z) and channel inputs X. Duality between equations (10) and (11) recalls the duality between channel capacity and rate distortion, as mentioned in [15]. 2) Probability distribution P uz (u, z) ⊗ Q(v|u, z) ⊗ P s (s) ⊗ Q(x|s) ⊗ T (y|x, s) is achievable if (13). The converse holds. Q s is the set of product distributions Q uzw2v ⊗ Q sxw1y with auxiliary random variables (W 1 , W 2 ) satisfying conditions: Z − − U − − W 2 , V − − (Z, W 2 ) − − U, Y − − (X, S) − − W 1 . Supports of (W 1 , W 2 ) are bounded by (|B| + 1) · (|B| + 2). The product distribution Q ∈ Q s decomposes as follows: Puz(u, z) ⊗ Q(w 2 |u) ⊗ Q(v|z, w 2 ) ⊗ Ps(s) ⊗ Q(x, w 1 |s) ⊗ T (y|x, s). As mentioned in [2], the independence between the probability distributions of the source and channel induces the separation of the source coding and the channel coding. This result is related to Theorem 1 in [14], with expected distortion d(u, v) and channel cost c(x) functions. By considering (U, Z, W 2 , V ) independent of (S, X, W 1 , Y ) and introducing three indexes (m, l, j) with rates (R m , R l , R j ) satisfying, Rm + R l ≥ I(W2; U ), R l ≤ I(W2; Z), Rj ≥ I(W1; S), Rm + Rj ≤ I(W1; Y ),X i = f i (U i , S i ), (resp. V i = g i (Y i , Z i )). We characterize of the set of achievable joint probability distributions for causal encoding and for causal decoding. U i S i Z n X i Y n V n P usz C T D 2) The distribution P usz (u, s, z) ⊗ Q(x|u, s) ⊗ T (y|x, s) ⊗ Q(v|u, s, z, x, y) is achievable if (15). The converse holds. max Q∈Qe I(W 1 , W 2 ; Y, Z) − I(W 2 ; U, S|W 1 ) > 0,(15) Q e is the set of joint distributions Q uszw1w2xyv with auxiliary random variables (W 1 , W 2 ) satisfying marginal cond. and:          (U, S) independent of W1, X − − (U, S, W1) − − W2, Y − − (X, S) − − (U, Z, W1, W2), Z − − (U, S) − − (X, Y, W1, W2), V − − (Y, Z, W1, W2) − − (U, S, X). Supports of (W 1 , W 2 ) are bounded by (|B| + 1) · (|B| + 2). The probability distribution Q ∈ Q e decomposes as follows: P usz ⊗ Q w1 ⊗ Q w2|usw1 ⊗ Q x|usw1 ⊗ T y|xs ⊗ Q v|yzw1w2 . Proofs are available in App. A and B. This result is a particular case of Theorem 2 in [7], by considering (U, S) as information source and (Y, Z) as channel output. Empirical coordination is equivalent to joint source channel coding with two-sided state information and correlated source and state. Theorem V.1 also reduces to Theorem 3 in [16], by considering (U, S) as channel state and (Y, Z) as channel output. When considering strictly causal encoding instead of causal encoding, the optimal solution is obtained by replacing W 1 by X in equation (15). Corollary V.2 (Causal Encoding without Coordination) Consider a distortion function d : U × V → R. The distortion level D ≥ 0 is achievable if and only if: max Qw 1 ,Q w 2 |usw 1 ,Q x|usw 1 , Q v|yzw 1 w 2 ,E[d(U,V )]≤D I(W1, W2; Y, Z) − I(W2; U, S|W1) ≥ 0, Corollary V.2 is a direct consequence of Theorem V.1. The distortion level D ≥ 0 is achievable if and only if there exists a joint distribution P usz ⊗ Q w1 ⊗ Q w2|usw1 ⊗ Q x|usw1 ⊗ T y|xs ⊗ Q v|yzw1w2 that satisfies (16), such that E[d(U, V )] ≤ D. Similar analysis is mentioned in [2] for the proof of Theorem V.3. Causal decoding and non-causal encoding is considered in [2] without S and Z. Joint probability distribution P usz ⊗ Q x|us ⊗ T y|xs ⊗ Q v|uszxy is achievable if (16). Converse holds. (16) Remark that V depends on (Y, Z, W 2 ) but not on W 1 . When considering strictly causal decoding instead of causal decoding, the optimal solution is obtained by replacing W 2 by V in equation (16). More details are provided in [2]. max Q xw 1 w 2 |us , Q v|yzw 2 I(W 1 ; Y, Z|W 2 ) − I(W 1 , W 2 ; U, S) > 0, VI. CONCLUSION The problem of empirical coordination is closely related to the joint source-channel coding with two-sided state information with correlation between source and state. These two problems are also related to state communication. We provide achievability and converse results for the non-causal case, that is open. We characterize the optimal solutions for perfect channel, for lossless decoding, for independent source and channel, for causal encoding and for causal decoding. APPENDIX The full versions of the proofs are stated in [22]. A. Sketch of Achievability of Theorem III.1 Consider Q(u, s, z, w 1 , w 2 , x, y, v) ∈ Q that achieves the maximum in (7). There exists δ > 0 and rate R ≥ 0 such that: R ≥ I(W 1 , W 2 ; U, S) + δ,(17)R ≤ I(W 1 , W 2 ; Y, Z) − δ.(18) • Random codebook. We generate |M| = 2 nR pairs of sequences (W n 1 (m), W n 2 (m)) with index m ∈ M drawn from the i.i.d. marginal probability distribution Q ⊗n w1w2 . • Encoding function. Encoder finds the index m ∈ M such that (U n , S n , W n 1 (m), W n 2 (m)) ∈ A ⋆n ε (Q) are jointly typical. It sends X n drawn from Q ⊗n x|usw1w2 depending on (U n , S n , W n 1 (m), W n 2 (m)). • Decoding function. Decoder finds the index m ∈ M such that (Y n , Z n , W n 1 (m), W n 2 (m)) ∈ A ⋆n ε (Q) are jointly typical. Decoder returns V n drawn from Q ⊗n v|yzw1w2 depending on (Y n , Z n , W n 1 (m), W n 2 (m)). The pair (W 1 , W 2 ) can be replaced by a single W . From properties of typical sequences, packing and covering Lemmas stated in [23] pp. 27, 46 and 208, equations (17), (18) imply that the expected probability of error is bounded for all n ≥n: Ec P (U n , S n ) / ∈ A ⋆n ε (Q) ≤ ε,(19) Ec P ∀m ∈ M, (U n , S n , W n 1 (m), W n 2 (m)) / ∈ A ⋆n ε (Q) ≤ ε,(20)Ec P ∃m ′ = m, s.t. (Y n , Z n , W n 1 (m ′ ), W n 2 (m ′ )) ∈ A ⋆n ε (Q) ≤ ε. (21) There exists a code c ⋆ ∈ C(n) such that sequences are jointly typical for distribution P usz ⊗Q xw1w2|us ⊗T y|xs ⊗Q v|yzw1w2 with probability more than 1 − 3ε. B. Sketch of Converse of Theorem III.1 Consider code c(n) ∈ C with small error probability P e (c). 0 ≤ n i=1 I(U n i+1 , S n i+1 , Y n i+1 , Z n i+1 ; Yi, Zi|Y i−1 , Z i−1 ) − n i=1 I(Y i−1 , Z i−1 ; Ui, Si|U n i+1 , S n i+1 , Y n i+1 , Z n i+1 )(22) Eq. (22) is due to Csiszár Sum Identity and properties of MI. Eq. (23) introduces auxiliary random variables W 1,i = (U n i+1 , S n i+1 , Y n i+1 , Z n i+1 ) and W 2,i = (Y i−1 , Z i−1 ) satisfying: Y i − − (X i , S i ) − − (U i , Z i , W 1,i , W 2,i ),(25)Z i − − (U i , S i ) − − (X i , Y i , W 1,i , W 2,i ),(26)V i − − (Y i , Z i , W 1,i , W 2,i ) − − (U i , S i , X i ). (27) This is due to memoryless channel, i.i.d. source and non-causal decoding since (Y n , Z n ) is included in (Y i , Z i , W 1,i , W 2,i ). Eq. (24) comes from taking the maximum over the set Q. C. Sketch of Converse of Theorem IV.1 Consider code c(n) ∈ C with small error probability P e (c). 0 = H(X n , Z n ) − I(X n , Z n ; U n , S n ) − H(X n , Z n |U n , S n ) (28) ≤ n i=1 H(Xi, Zi) − n i=1 I(X n , Z n ; Ui, Si|U n i+1 , S n i+1 ) − H(X n , Z n |U n , S n ) I(W 2 ; 2Z|X) + H(X) − I(X, W 2 ; U, S) > 0,(10) Q p is the set of distributions Q uszw2xv with W 2 satisfying marginals and Z− −(U, S)− −(X, W 2 ) and V − −(X, Z, W 2 )− −(U, S). Support satisfies |W 2 | ≤ |B| + 1 and Q ∈ Q p decomposes: P usz (u, s, z) ⊗ Q(x|u, s) ⊗ Q(w 2 |u, s, x) ⊗ Q(v|x, z, w 2 ). 1 ; Y, Z) − I(W 1 ; S|U ) − H(U ) > 0, (11) Q l is the set of distributions Q uszw1xyû satisfying marginal conditions and Markov chains Y − − (X, S) − − (U, Z, W 1 ) and Z − − (U, S) − − (X, Y, W 1 ) Fig. 3 . 3C. Independent source (U, Z, V ), channel (S, X, Y ), Fig. The random variables of the source (U, Z, V ) are independent of the random variables of the channel (S, X, Y ). Theorem IV.2 (Independent Source and Channel) 1) If the product Q(u, z, v) ⊗ Q(s, x, y) of probability distribution is achievable, then it satisfies: Q(u, z) = Puz(u, z), Q(s) = Ps(s), Q(y|x, s) = T (y|x, s). ; Y ) + I(W2; Z) − I(W1; S) − I(W2; U ) > 0,(13) Fig. 4 . 4Causal encoding function f i : U i × S i → X , for all i ∈ {1, . . . , n}. Theorem V.1 (Causal Encoding and Non-Causal Decoding) 1) If Q(u, s, z, x, y, v) is achievable, then it satisfies: Q(u, s, z) = Pusz(u, s, z), Q(y|x, s) = T (y|x, s), Y − − (X, S) − − (U, Z), Z − − (U, S) − − (X, Y ). W1; Y, Z|W2) − I(W2; U, S|W1) . Xi, W2,i ; Xi, Zi) − I(Xi, W2,i; Ui, Si) (32) ≤ n · max Q∈Qp I(X, W2; X, Z) − I(X, W2; U, S) . (33) Eq. (28), (29) are due to properties of mutual information. Eq. (30) is due to the i.i.d. source (U, S). Eq. (31) is due to the i.i.d. source (U, S, Z): H(Z n |U n , S n ) = H(Z i |U i , S i ) = H(Z i |X n , Z −i , U n i+1 , S n i+1 , U i , S i ). Eq. (32) introduces auxiliary random variable W 2,i = (X −i , Z −i , U n i+1 , S n i+1 ) that satisfies Markov chains:This is due to i.i.d. source (U, S, Z) and non-causal decoding. Eq. (33) comes from taking the maximum over the set Q p .D. Sketch of Achievability of Theorem V.1Consider Q ∈ Q e that maximizes(15)and block-Markov code c ∈ C(n) over B ∈ N blocs of length n ∈ N with:• Random codebook. We generate |M| = 2 nR sequences W n 1 (m) drawn from Q ⊗n w1 with index m ∈ M. For each m ∈ M, we generate the same numberEquations (34), (35) imply for all n ≥n and for a large number of blocks B ∈ N, the sequences are jointly typical with large probability:E. Sketch of Converse of Theorem V.1Consider code c(n) ∈ C with small error probability P e (c).Eq. (36) is due to Csiszár Sum Identity and properties of MI. Eq. (37) introduces auxiliary random variables W 1,i = (U i−1 , S i−1 ) and W 2,i = (Y n i+1 , Z n i+1 ) satisfying:Eq. (38) comes from taking the maximum over the set Q e . 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{'abstract': 'The coordination of autonomous agents is a critical issue for decentralized communication networks. Instead of transmitting information, the agents interact in a coordinated manner in order to optimize a general objective function. A target joint probability distribution is achievable if there exists a code such that the sequences of symbols are jointly typical. The empirical coordination is strongly related to the joint source-channel coding with two-sided state information and correlated source and state. This problem is also connected to state communication and is open for non-causal encoder and decoder. We characterize the optimal solutions for perfect channel, for lossless decoding, for independent source and channel, for causal encoding and for causal decoding.', 'arxivid': '1506.04812', 'author': ['Maël Le Treust mael.le-treust@ensea.fr \nUMR 8051\nETIS\n\n\nENSEA\nUniversité Cergy-Pontoise\nCNRS\n6, avenue du Ponceau95014CERGY-PONTOISE CEDEXFRANCE\n'], 'authoraffiliation': ['UMR 8051\nETIS\n', 'ENSEA\nUniversité Cergy-Pontoise\nCNRS\n6, avenue du Ponceau95014CERGY-PONTOISE CEDEXFRANCE'], 'corpusid': 1434979, 'doi': '10.1109/isit.2015.7282498', 'github_urls': [], 'n_tokens_mistral': 9718, 'n_tokens_neox': 8703, 'n_words': 4944, 'pdfsha': 'e4e990d43904d31e72174520c3cfc88739716f61', 'pdfurls': ['https://arxiv.org/pdf/1506.04812v1.pdf'], 'title': ['Empirical Coordination with Two-Sided State Information and Correlated Source and State', 'Empirical Coordination with Two-Sided State Information and Correlated Source and State'], 'venue': []}

arxiv

Functional Sequential Treatment Allocation 21 Dec 2018 December 27, 2018 Anders Bredahl Kock anders.kock@economics.ox.ac.uk ECARES, SBS University of Oxford CREATES Aarhus University Université libre de Bruxelles Aarhus University EM David Preinerstorfer david.preinerstorfer@ulb.ac.be ECARES, SBS University of Oxford CREATES Aarhus University Université libre de Bruxelles Aarhus University EM Bezirgen Veliyev bveliyev@econ.au.dk ECARES, SBS University of Oxford CREATES Aarhus University Université libre de Bruxelles Aarhus University EM Functional Sequential Treatment Allocation 21 Dec 2018 December 27, 2018Sequential Treatment AllocationDistributional Policy EffectsStatistical Decision TheoryMinimax Optimal RegretMultiple Treatments In this paper we study a treatment allocation problem with multiple treatments, in which the individuals to be treated arrive sequentially. The goal of the policy maker is to treat every individual as well as possible. Which treatment is "best" is allowed to depend on various characteristics (functionals) of the individual-specific outcome distribution of each treatment. For example measures of welfare, inequality, or poverty. We propose the Functional Sequential Allocation policy, and provide upper bounds on the regret it incurs compared to the oracle policy that knows the best treatment for each individual. These upper bounds increase sublinearly in the number of treatment assignments and we show that this regret guarantee is minimax optimal. In addition, the assumptions under which this regret guarantee is established are as weak as possible -even a minimal weakening of them will imply non-existence of policies with regret increasing sub-linearly in the number of assignments. Furthermore, we provide an upper bound on the number of suboptimal assignments made by the FSA policy and show that every policy must make a number of suboptimal assignments at least of this order.JEL Classification: C18, C22, J68. Introduction In many treatment programs the individuals to be treated arrive sequentially. For example, workers become unemployed throughout the year or patients to be treated in a hospital fall at different points in time. Therefore, one must sequentially treat the individuals and thus sequentially learn about the treatment-specific outcome distribution as treatment outcomes are observed. In this paper, we study a setting in which a policy maker's objective is to treat as many individuals as well as possible in the course of the treatment period. We measure the attractiveness of a treatment by a (combination of) functionals of the distribution of treatment outcomes. Thus, the goal of the policy maker can be, for example, to assign as often as possible the treatment with the highest welfare (according to some welfare measure) or quantile, lowest uncertainty of outcome (according to a measure of dispersion), lowest inequality or poverty, or best tradeoff between several distributional characteristics of the treatments. In order to treat as many individuals as well as possible the policy maker must sequentially learn the distribution of treatment outcomes for the K available treatments, yet assign as few as possible individuals to suboptimal treatments. We propose the Functional Sequential Allocation (FSA) policy for the above problems and establish its properties. We first analyze a homogeneous treatment distribution setting, i.e.,the outcome distribution for each of the treatments is assumed to be the same for all individuals. To begin, we provide an upper bound on the maximal expected regret of the FSA policy compared to what could have been obtained had the population outcome distributions of the K treatments been known from the outset and one had always assigned the treatment maximizing the (combination of) functional characteristics of interest, cf. Theorem 3.1. Next, we show that FSA policy is near minimax optimal. More precisely, we show in Theorem 3.6 that every policy must incur of at least almost the same order as FSA policy. In the setting of heterogeneous treatment effects we assume that the policy maker observes a vector of characteristics (covariates) of the individual to be treated prior to treatment allocation. For each individual the best is now the one that maximizes the (combination of) functionals of the conditional distribution given the vector of characteristics. In Theorem 4.4 (cf. also Corollary 4.5) we show how the FSA policy can be adjusted in such a way that its maximal regret vis-à-vis the infeasible oracle policy (the policy that knows the population conditional distributions of treatment outcomes given characteristics and always assigns the best one) can be bounded from above and shown to increase sublinearly in the number of assignments. Theorem 5.2 shows that the assumptions under which the upper bounds on regret are established are essentially minimal as every policy must incur a regret which is linear in the number of assignments without such assumptions. Furthermore, Theorem 5.3 establishes the minimax optimality of the FSA policy up to logarithmic factors. Next, if for a large proportion of vectors of characteristics the best and second best treatment are not too similar, we show that the upper bound on the maximal regret compared to the infeasible oracle can be further sharpened, cf. Theorem 4.7. Furthermore, we show in Theorem 4.8 that the expected number of suboptimal assignments made increases sublinearly in the number of assignments. The latter can be interpreted as an ethical guarantee on the FSA policy; only few persons will receive a treatment which is not optimal for them. Furthermore, Theorem 5.1 (and its proof) shows that i) no policy can achieve much lower regret than the FSA policy, ii) every policy must make at least as many suboptimal assignments as the FSA policy. In this sense, the FSA policy is (minimax) optimal. Finally, in Theorem 5.4, we show that even though our sublinear regret bounds are increasing in the dimension d of the vector of characteristics, one should never ignore these, as maximal regret of any policy must increase even linearly in the number of assignments made if one ignores the characteristics. We note that the sequential setting considered in this paper differs from the classic treatment setting in which one often presupposes the existence of a data set that has already been sampled. Based on this data set, one is then interested in estimating the effect of a treatment. This effect is most often the difference in a suitable (conditional) expectation, i.e.,the focus is on the first moment of the distribution of treatment outcomes -a special case of the functionals studied in this paper. We also stress that the goal of the policy maker in this paper is to treat as many individuals as well as possible. This is not equivalent to assigning treatments in such a way that "information" about the treatments is maximized after the last treatment. Assigning treatments sequentially in a way that maximizes "information" at the end of the treatment period is also an interesting goal, which warrants further study. The latter objective, however, can sometimes be problematic as an algorithm designed for this purpose knowingly treats individuals in a suboptimal way in order to obtain information. Related literature Our paper is related to several strands of literature. First, our work relates to the literature on statistical treatment allocation rules. Here Manski (2004) did seminal work in proposing conditional empirical success rules which take a finite partition of the covariate space and on each set of this partition dictate to assign the treatment with the highest sample average. In choosing this partition, one faces a tradeoff between individualization and having enough observations for each group to estimate treatment effects precisely. In particular, Manski (2004) studies when full individualization is optimal. Stoye (2009) shed further light on this by showing that if one does not restrict how outcomes vary with covariates then full individualization is alway minimax optimal. This result relies on the fact that without any restrictions on how the outcome distribution varies with covariates, this relationship could be arbitrarily wiggly such that even seemingly similar individuals may carry no information about how treatments affect the other person. Furthermore, our work is related to the recent paper by Kitagawa and Tetenov (2018) who consider treatment allocation through an empirical welfare maximization lens. The authors take the view that realistic policies are often constrained to be simple due to ethical, legislative, or political reasons. Using techniques from empirical risk minimization they show how their procedure is minimax optimal within the considered class of realistic policies. Furthermore, Athey and Wager (2017) have used concepts from semiparametric efficiency theory to establish regret bounds that scale with the semiparametrically efficient variance. Finally, Kitagawa and Tetenov (2017) have considered treatment allocation in a setting in which one targets "equality-minded" social welfare functions. Other papers on statistical treatment rules in econometrics focusing on the case where the sample is given include Chamberlain (2000); Dehejia (2005); Hirano and Porter (2009); Bhattacharya and Dupas (2012); Stoye (2012); Tetenov (2012). Furthermore, Rothe (2010) has done work on inference on policy effects that need not be restricted to the mean of the distribution, cf. also Rothe (2012). The most important distinguishing features of our work compared to the classic literature on statistical treatment rules above is that we are working in a sequential setting where the individuals to be treated arrive gradually. Thus, we do not have a data set of size N at our disposal from the outset based on which the best treatment must be found. The sequential setting poses new challenges such as not maltreating too many individuals in the search for the best treatment. Furthermore, in contrast to most work, we focus on the problem of a policy maker who targets a general (combination of) functionals of the outcome distribution of the treatments. Second, the sequential setting adopted in this paper is related to the literature on multi-armed bandit problem which study sequential decision making under uncertainty. While this literature generally focuses on problem without covariates in which one is interested in the mean of the distribution only, we give an overview of the most related papers here. Robbins (1952) was the first to introduce an algorithm with provable performance guarantees. In particular, he showed that the average reward will converge to the mean of the best arm. The algorithm most relevant for our work is the Upper Confidence Bound (UCB) strategy of Lai and Robbins (1985), for targeting unconditional means, which was further refined in Auer et al. (2002). The underlying idea of our FSA policy is similar to the one of UCB. However, as will be seen, the analysis of the FSA policy is very different from the one of UCB since the FSA policy is designed to target (combinations of) general functionals of the distribution of treatment outcomes instead of the mean only. Furthermore, we allow for the presence of covariates on which the distribution of treatment outcomes can depend. In this sense, the works of Rigollet and Zeevi (2010) and Perchet and Rigollet (2013) are related to our paper -both consider a setting targeting the distribution with the highest conditional mean in the presence of covariates. The former paper studies a UCB type policy while the latter studies the successive elimination algorithm and verifies its minimax optimality. Kock and Thyrsgaard (2017) considered a setting in which the policy maker is also interested in how risky a treatment is and takes this into account by targeting a tradeoff between expected outcome and variance of the treatments. Finally, Cassel et al. (2018) consider bandit problems where the target can be a general (risk) criterion defined on the empirical distribution functions of the path of assignments. For a good general overview of multi-armed bandit problems for targeting the unconditional mean of the distributions to be sampled from we refer to Bubeck and Cesa-Bianchi (2012). Sequential treatment allocation without covariates To set the stage for the general setting, we begin by considering a treatment allocation problem where no covariates are observed prior to assigning the treatment. This can also be interpreted as (correctly) assuming that the outcome of the treatment is independent of the (observed) covariates. In most settings this is clearly not a realistic assumption. However, this stripped case clearly illustrates the main ideas of our policy . Notation We here present some notation used throughout the paper. For any x ∈ R d , ||x|| denotes the ℓ 2 -norm. Furthermore, for any a < b and d ∈ N, B([a, b] d ) denotes the Borel σ-field on [a, b] d equipped with the usual topology. Subsequently, let D cdf (R) denote the set of cumulative distribution functions (cdfs) on R, and for real numbers a < b let D cdf ([a, b]) denote the set of elements F ∈ D cdf (R) such that F (a−) = 0 and F (b) = 1. Furthermore, we shall denote the supremum metric on D cdf (R) by . ∞ . Setup Consider a setting where at each point in time t = 1, ..., N a policy maker must assign one of K treatments to an individual. Thus, t can also be thought of as indexing individuals instead of time. We shall allow the total number of treatments to be made, N, to be a random variable. This reflects that in many treatment problems the policy maker does not know a priori how many treatments will be made. For example, one does not know at the beginning of the year how many individuals will become unemployed. The aim of the policy maker is to assign as many of the N individuals as well as possible. However, the policy maker does not know which of the K treatments is the best. The observational structure is as follows: after assigning a treatment, its outcome is observed but the policy maker does not observe the counterfactuals, i.e., what would have happened if another treatment had been assigned. Upon observing the outcome of treatment t ∈ 1, ..., N − 1, individual t arrives and must be assigned to a treatment based on the information gathered from all previous assignments. Thus, the data set is gradually constructed in the course of the treatment program, and the policy maker seeks to sample in such a way as to maximize cumulative welfare by assigning the unknown best treatment as often as possible. To be precise, let a < b and let Y i,t ∈ [a, b] be the outcome of treatment i ∈ I := {1, ..., K} at time/individual t ∈ {1, ..., N}. A policy is a (recursively) defined sequence of (Borel measurable) functions π = {π t } ∞ t=1 which can depend on observed variables only: the first treatment is some element in I not depending on any non-existing previous treatment outcomes. The second treatment can depend on Z 1 := Y π 1 ,1 such that π 2 : [a, b] → I. In general, π t : [a, b] (t−1) → I and we write π t (Z t−1 ) where Z t−1 := (Y π t−1 (Z t−1 ),t−1 , ...., Y π 1 ,1 ). Thus, Z t−1 is the information available after the (t − 1)-th treatment outcome was observed. For convenience, the dependence of π t on Z t−1 is often suppressed. We stress that the policy maker can assign only one treatment π t ∈ I to each individual and observes the outcome of that treatment, i.e., Y πt,t , only. This is in accordance with most real life situations: one does generally not observe the counterfactuals. Note also that restricting attention to problems where only one of the K treatments can be assigned does not exclude that a treatment consists of a combination of several treatments (for example a combination of several drugs) -one simply defines this combined treatment as a separate treatment at the expense of increasing the set of potential treatments. While we assume that Y t = (Y 1,t , ..., Y K,t ) are distributed identically and independently across t, the joint distribution of the treatment outcomes is left unspecified. In particular, given t, the dependence structure of Y i,t and Y j,t is not restricted. Let P i denote the outcome distribution on B([a, b]) of treatment i with corresponding cdf F i . Ideally, the policy maker would like to assign every individual to the "best" treatment, in the sense that the outcome distribution for this treatment maximizes a functional T : D cdf ([a, b]) → R, where we recall that D cdf ([a, b]) denotes the set of cdfs F such that F (a−) = 0 and F (b) = 1, i.e., the set of cdfs with support [a, b]. The specific functional used depends on the application, and encodes the particular characteristic of the distribution the policy maker is interested in. To give specific examples, the functional could be a (combination of) welfare-, inequality-, or povertymeasures, see Appendix A. It could also be a quantile, a (trimmed) moment, a U-functional, or an L-functional see Appendix C. Given T, the goal is to find a policy π that minimizes the "regret" R N (π) = N t=1 max i∈I T(F i ) − T(F πt ) . (1) The loss for assigning treatment π t instead of a best treatment is max i∈I T(F i ) − T(F πt ). Thus, we consider a policy maker whose goal is to incur as little loss as possible for as many individuals as possible. Of course there are situations in which this is not the goal of the policy maker. For example, one may only be interested in using the treatments to gather as much information as possible about a characteristic of the treatment outcome distributions by the end of N treatments without regard to the loss each individual incurs. While this exception is definitely also an interesting problem, we shall focus on finding policies that minimize (1) since treating as many individuals as possible as well as possible is a common objective for policy makers. For every treatment i define ∆ i := max 1≤k≤K T(F k ) − T(F i ) as the loss due to assigning treatment i instead of an optimal one. Then, the regret can also be written as R N (π) = i:∆ i >0 ∆ i N t=1 1 {πt=i} = i:∆ i >0 ∆ i S i (N)(2) where S i (N) = N t=1 1 {πt=i} is the number of times treatment i is assigned in the course of N treatments. Throughout the paper, we shall assume that the functional T of interest satisfies the following assumption: Assumption 2.1. The functional T is well defined on D cdf ([a, b]) for a < b, and for D ⊆ D cdf ([a, b]) there exists a real number C such that: |T(F ) − T(G)| ≤ C F − G ∞ for every F ∈ D and every G ∈ D cdf ([a, b]). (3) Remark 2.2. The set D appearing in Assumption 2.1 encodes the assumptions imposed on the cdfs of each treatment outcome, i.e., on F 1 , . . . , F K . In particular, the larger D, the less restrictive are the assumptions imposed on F 1 , . . . , F K . Ideally, one would thus like D = D cdf ([a, b]), which, however, is too much to ask for many functionals. Furthermore, there is a trade-off between C and D, in the sense that a "larger" class D leads to a larger constant C. Note also that Assumption 2.1 implies that the restriction of T to D is Lipschitz continuous w.r.t. . ∞ . But, Assumption 2.1 does not require T to be Lipschitz continuous on all of D cdf ([a, b]). Remark 2.3. Assumption 2.1 is satisfied for many popular functionals arising in applied economics. We provide a detailed discussion together with formal results in Appendix A, where we consider many important inequality-, welfare-, and poverty measures. The inequality measures we discuss in Appendix A.1 include the Schutz-coefficient, the Gini-index, the class of linear inequality measures of Mehran (1976), the generalized entropy family (which includes Theil's index), the Atkinson family of inequality indices (Atkinson (1970)), and the family of Kolm-indices (Kolm (1976a)). In most cases, we discuss both relative and absolute versions of these measures. In Appendix A.2 we provide results for welfare measures based on inequality measures discussed in Appendix A.1. The poverty measures we discuss in Appendix A.3 are the headcount ratio, the family of poverty measures of Sen (1976) (in the generalized form of Kakwani (1980)), and the family of inequality measures suggested by Foster et al. (1984). We emphasize that the results in Appendices A.1, A.2 and A.3 are obtained from a set of more abstract results of independent interest that we establish in Appendix C. These results establish Assumption 2.1 for U-functionals (i.e., population versions of U-statistics), quantiles, L-functionals (population versions of L-statistics), and for truncated versions of U-functionals. We also need an assumption that guarantees that the functional T evaluated at empirical cdfs is measurable. Assumption 2.4. For every m ∈ N the function that maps x ∈ [a, b] m to T evaluated at m −1 m j=1 1 x j ≤· , the empirical cdf corresponding to x = (x 1 , . . . , x m ), is Borel measurable. Assumption 2.4 is typically satisfied and poses no practical restrictions. Functional sequential allocation policy and regret bounds We now turn to describing our treatment policy, the Functional Sequential Allocation (FSA) policy, and its properties. The policy is inspired by the UCB strategy of Lai and Robbins (1985) for multi-armed bandit problems. While the UCB policy was designed for targeting the mean of a distribution, the FSA policy can target any functional. Furthermore, Section 4 allows covariates to influence treatment outcomes. We need some more notation to introduce the FSA policy. Given a policy π, a natural number t and treatment i, we shall define the random set S i,t (π) = {s ∈ {1, . . . , t} : π s = i} ⊆ {1, . . . , t}. That is, S i,t (π) contains all those individuals s ∈ {1, . . . , t} the policy π has assigned to treatment i. We shall often just write S i,t instead of S i,t (π). Furthermore, we shall denote the cardinality of S i,t (π) by S i (t). Note that S i (t) = t s=1 1 {πs=i} holds, and that S i (t) is the number of times treatment i has been assigned in the first t treatments. To formulate our algorithm, we also define the empirical cdf F i,t (.) := S i (t) −1 s∈S i,t 1 {Y i,s ≤.} ,(4) Observe that this just denotes the empirical cdf based on the individuals assigned to treatment i up to time t. The policy we analyze is defined as follows: Functional Sequential Allocation: Let C be the Lipschitz coefficient in Assumption 2.1. Then, the FSA policyπ with parameter β > 0 proceeds as: 1. If t ∈ {1, . . . , K}, assign treatment t, i.e.π t = t. 2. If t ≥ K + 1, assignπ t ∈ arg max i∈I T(F i,t−1 ) + C β log(t)/2S i (t − 1) After the K initialization rounds, the FSA policy assigns a treatment that i) is promising in the sense that T(F i,t−1 ) is large or ii) has not been well explored in the sense that S i (t − 1) is small. The parameter β is chosen by the researcher and indicates the weight put on assigning scarcely explored treatments, i.e treatments with low S i (t − 1). A regret minimizing choice of β is given after Theorem 3.1 below. Note also that the FSA policy does not require knowledge of the, often unknown, number of treatments to be made (N). In this sense, the policy falls in the class of "anytime strategies", in the sense that it does not use the number of assignments to be made when making assignments and its regret guarantees hold for any termination point. Below, we use the notation log(x) = max(log(x), 1). Note that log(x) = log(x) if x ≥ e. Theorem 3.1. Suppose that the number of treatments, N, has expectation n and is independent of treatment outcomes. Under Assumptions 2.1 and 2.4, the cumulative regret of the FSA policyπ with parameter β > 2 satisfies sup E[R N (π)] ≤ c Knlog(n) (5) where the supremum is over all K-tuples of D and c = c(β, C) is a constant, defined in the proof of the theorem, that depends on β and C. Theorem 3.1 provides an upper bound on the maximal regret incurred by the FSA policy in the absence of covariates. As seen in Theorem 3.6 below, this bound is minimax optimal in n up to the factor log(n). Note that the "expected per person regret" E(R N (π))/n tend to zero as n tends to infinity. The choice parameter β can be chosen optimally as a function of C to minimize c. In particular, inspection of the proof shows that β = 2 + √ 2 minimizes c(β, C) and implies c ≤ √ 11C. Finally, we remark that it is sensible that the upper bound on regret is increasing in the number of available treatments K as it becomes harder to find the best treatment as the number of available treatments increases. We now turn to showing the near-minimax optimality of the upper bound on maximal regret in Theorem 3.1. It suffices to show that for any policy the maximal regret must be large against a certain family of K-tuples of distributions of treatment outcomes (we shall consider K = 2). To this end, consider the following family of distributions on B([0, 1]), but cf. also Remark 3.5 below. Definition 3.2. Let H = {P ha : a ∈ (−1, ∞)} where P ha is the distribution on B([0, 1]) with density h a (y) = (1 + a)y a 1 {y>0} , and corresponding cdf H a with H a (y) = y a+1 for y ∈ [0, 1], H a (y) = 0 for y ∈ (−∞, 0) and H a (y) = 1 else. Assumption 3.3. Assume that there existsā > −1, c > 0 and δ > 0 such that H a ∈ D for all a ∈ [ā − δ,ā + δ] ⊆ (−1, ∞). Furthermore, either T(H a 2 ) − T(H a 1 ) ≥ c(a 2 − a 1 )(6) holds for all a 1 , a 2 ∈ [ā − δ,ā + δ] ⊆ (−1, ∞) such that a 1 ≤ a 2 , or T(H a 2 ) − T(H a 1 ) ≤ −c(a 2 − a 1 ) holds for all a 1 , a 2 ∈ [ā − δ,ā + δ] ⊆ (−1, ∞) such that a 1 ≤ a 2 . Remark 3.4. The requirement on T in Assumption 3.3 is a local uniform monotonicity condition on a → T(H a ). It is rather mild and satisfied, for example, if a → T(H a ) is continuously differentiable on (ā − δ,ā + δ) with a derivative bounded away from zero (this can be seen by the mean value theorem). This requirement is, in turn, easily seen to be satisfied when T is any moment or quantile. Intuitively, Assumption 3.3 rules out that T is too flat. For example, all policies would incur zero regret if T is constant and it is thus sensible that some strict monotonicity is needed in order to prove non-trivial lower bounds on regret. We stress that the local uniform monotonicity is needed only at one pointā ∈ (−1, ∞). Remark 3.5. The only property of H that is used in the proof of Theorem 3.6 below is that the Kullback-Leibler divergence between any two members of H is sub-quadratic in a sense made precise in Lemma D.3 and surrounding discussion. Thus, the family H can be replaced by any other family of distributions with this sub-quadratic property as well as satisfying Assumption 3.3. Out next result shows that the maximal/uniform regret in Theorem 3.1 is optimal as a function of n up to a multiplicative factor log(n). It suffices to consider N non-random, [a, b] = [0, 1] and K = 2. Theorem 3.6. Let Assumptions 2.1 and 3.3 be satisfied and consider a treatment problem with N = n non-random and K = 2 treatments. Then, for any policy π, there exists a c l > 0 such that sup ER n (π) ≥ c l √ n, where the supremum is over all two-tuples of distributions on B([0, 1]). Remark 3.7. While the FSA policy does not need to know N, inspection of the proof of Theorem 3.6 shows that even when N is non-random (such that N = E[N] = n) and known, even a policy that requires knowledge of n must incur a maximal regret of order √ n. The FSA policy is guaranteed to incur a maximal regret not much more than this even without knowing n, cf. Theorem 3.1 above. Treatment allocation with covariates The results up to this point have been for treatment allocation problems without any covariates being observed on an individual prior to treatment allocation. While the results for this problem will be useful in the present section, it is often too restrictive to assume that the treatment outcomes do not depend on the characteristics of the person to be treated. For example, one medicine may work very well (in terms of the functional of interest) for one person while it may be outright dangerous to another person if he is allergic to some of the substances. We now suppose that prior to assigning each treatment the policy maker observes a vector of covariates. More precisely, let X t ∈ [0, 1] d , d ∈ N be the vector of covariates observed on individual t prior to the treatment assignment and assume that the random vector (Y 1,t , . . . , Y K,t , X t ) is iid across t. For each x ∈ [0, 1] d , let F i (·, x) be the distribution function of Y i,t conditional on X t = x. The corresponding probability measure on B([a, b]) is denoted P i (·, x). A policy is now a (recursively) defined sequence of Borel measurable functions π = {π t } ∞ t=1 which can depend on observed variables only: the first treatment can depend on X 1 only, so π 1 : [0, 1] d → I. The second treatment can depend on X 2 and Z 1 := (Y π 1 (X 1 ) , X 1 ) such that π 2 : [0, 1] d × [a, b] × [0, 1] d → I. In general π t : [0, 1] d × [[0, 1] d × [a, b]] t−1 → I and we write π t (X t , Z t−1 ) where Z t−1 := (Y π t−1 (Xt,Z t−1 ) , X t−1 , ...., Y π 1 (X 1 ) , X 1 ). We shall often use that it is convenient to suppress the dependence of π t on Z t−1 and write π t (X t ). In this case, π t is, of course, a random function as it implicitly depends on Z t−1 . The benchmark for our policy will be the infeasible (oracle) policy which knows the true conditional treatment outcome distributions F i (·, x) and for an individual with characteristics x assigns the treatment with the optimal conditional distribution, i.e assigns 1 π ⋆ (x) = arg max i∈I T(F i (·, x)) where ties are broken arbitrarily such that T(F π ⋆ (x) (·, x)) = max i=1,...,K T(F i (·, x)). This way of defining the oracle reflects the general difference to the setting without covariates: we now attempt to get as close as possible to the welfare we could have obtained had we known for each individual which treatment is best. The best treatment now depends on the characteristics of the individual and thus conditional distributions replace the unconditional ones in Section 2. One can still target all functionals T of the conditional distributions F i (·, x) as long as Assumption 2.1 is satisfied, cf. the examples given in Appendix A. In the presence of covariates, the regret of a policy π is defined as R N (π) = N t=1 T F π ⋆ (Xt) (·, X t ) − N t=1 T F πt(Xt) (·, X t )(7) Our goal is to provide sharp upper bounds on the expected value of the regret. Thus, we strive to get as close as possible to the welfare we could have attained had we assigned the optimal treatment π ⋆ (x) for each individual. We stress again that the optimal treatment now depends on the individual's characteristics through x. Remark 4.1. Without any assumptions on the map x → F i (y, x), the problem does not have any interesting solution in the sense that any policy has maximal regret that increases linearly in n (since Assumption 2.1 implies that T is bounded, such that no policy has regret of larger order than n, this also implies that any policy is minimax optimal). In fact we show in Theorem 5.2 below that even if x → F i (y, x) is continuous on (and thus also uniformly continuous on [0, 1] d ), the maximal regret of any policy must increase linearly in n. Hence, we impose the following, minimally stronger, Hölder continuity condition on F i (y, x) which will be just enough to ensure existence of (near) minimax optimal policies with sub-linear regret. Assumption 4.2. There exist γ ∈ (0, 1] and L > 0 such that for all i ∈ I |F i (y, x 1 ) − F i (y, x 2 )| ≤ L||x 1 − x 2 || γ for all y ∈ R and all x 1 , x 2 ∈ [0, 1] d . In addition, for each i ∈ I and x ∈ [0, 1] d , F i (·, x) belongs to D of Assumption 2.1. Assumption 4.2 requires that the distribution functions F i (·, x 1 ) and F i (·, x 2 ) are close to each other whenever x 1 and x 2 are close. The assumption essentially requires that individuals with similar characteristics have similar outcome distributions for each treatment. Since any policy must generically incur linear in n maximal regret without Assumption 4.2, Hölder continuity is in this sense the weakest possible form of continuity ensuring existence of policies with non-trivial upper bounds on regret. Assumption 4.3. The distribution P X of covariate X t has a density with respect to the Lebesgue measure on B([0, 1] d ) that is bounded above and below byc and c > 0, respectively. Assumption 4.3 restricts the covariates to be continuous but finitely discrete covariates can also be allowed for by simply running a separate policy for each of the values of the discrete covariates. The functional sequential allocation policy in the presence of covariates Heuristically, the idea of the FSA policy in thepresence of covariates is to group together individuals with similar values of the covariates, and then implement the FSA policy without covariates from Section 3 on each group separately. This amounts to targeting the treatment that is best on average in each group instead of fully individualizing the treatments. It thus strikes a middle ground between individualization and and having enough observations to estimate the treatment outcome distributions in each group. In other words, the grouping amounts to choosing a rectangular kernel in a sequential nonparametric estimation problem of x → F i (y, x). More precisely, let {B 1 , ..., B F } ⊆ B([0, 1] d ) be a partition of the space of covariates [0, 1] d , i.e. ∪ F i=1 B i = [0, 1] d and B i ∩B j = ∅ for i = j. In addition, let V j = sup x,y∈B j x − y be the maximal distance between any two points in B j andB j = λ d (B j ) > 0 be the Lebesgue measure of B j . In order to define the FSA policyπ in the presence of covariates precisely, let N j (t) = t s=1 1 {Xs∈B j } be the number of individuals with covariates in B j in the course of t treatment assignments and letπ B j ,r be the assignment made by the FSA policy without covariates to r-th individual in group B j with the parameter β > 2. 2 We then defineπ t : [0, 1] d → I as π t (x) =π B j ,N j (t) if x ∈ B j(8) whereπ B j ,N j (t) of course depends on the N j (t) previous treatment outcomes observed for individuals with covariates in B j . Note also that the covariates are only used to assign group membership to an individual. Since the FSA policy without covariates is used separately for each group, we are effectively targeting a treatment that is best for the average individual in each group. For group j, this means targeting a treatment that attains max i∈I T(F i j ) where F i j is the cumulative distribution function defined as F i j (y) := 1 P X (B j ) B j F i (y, x)P X (dx).(9) 2 We recommend running the FSA policy for each group with β = 2 + √ 2 as this minimizes c in Theorem 3.1. An upper bound on the maximal regret of the FSA policy with covariates Let S 0 = S 0 (γ, L, c,c) be the set of distributions of (Y 1,t , ..., Y K,t , X t ) that satisfy Assumptions 4.2 and 4.3. Theorem 4.4. Suppose that the number of treatments, N, has expectation n and is independent of treatment outcomes and covariates. Consider a partition characterized by {V 1 , ..., V F } and {B 1 , ...,B F }. Under Assumptions 2.1, 2.4, 4.2 and 4.3, there exists a c > 0 such that sup S 0 E[R N (π)] ≤ F j=1 c KcB j nlog(cB j n) + 2CLV γ j ncB j .(10) Theorem 4.4 gives an upper bound on the regret of the FSA policy in the presence of covariates for any choice of grouping individuals. This flexibility may be useful since a policy maker is sometimes constrained by ethical or legislative reasons in the way he groups individuals such that he can not choose the partition {B 1 , ..., B F } that minimizes the upper bound on regret in (10). The upper bound on regret consists of two parts. The first part is very similar to the uniform part of Theorem 3.1; the difference being that the total number of individuals expected to be treated, n, has now been replaced by an upper bound on the number of individuals expected to fall in group B j ,cnB j . Thus, the first part of the upper bound on regret is the regret we expect to accumulate on each group compared to always assigning the treatment that is best for the average individual in that group. The second part of the upper bound on expected regret is the approximation error incurred due to targeting the treatment that is best for the average individual in group B j , i.e.,targeting max i∈I T(F i j ), instead of targeting T F π ⋆ (x) (·, x) . It is sensible that this approximation error is increasing in the size of the groups as measured by V j andB j . A specific type of partition that achieves near-minimax optimal regret over the class of distributions S 0 is "quadratic groups". It uses hard thresholds for each entry of X t to create hypercubes that partition [0, 1] d . The groups thus created do not only attain low regret but are also relevant in practice due to their simplicity and resemblance to real ways of grouping people. More precisely, fix P ∈ N and define B k = x ∈ [0, 1] d : k l − 1 P ≤ x l ≤ ( * ) k l P , l = 1, ..., d(11) for k = (k 1 , ..., k d ) ∈ {1, ..., P } d where ≤ ( * ) is weak for k l = P and strict otherwise. Thus, P is the number of splits along each dimension of X t creating a partition of [0, 1] d consisting of P d smaller hypercubes B 1 , ..., B P d with side lengths 1/P . Corollary 4.5. Suppose that the horizon N has expectation n and is independent of treatment outcomes and covariates. Use the partition in (11) with P = ⌈n 1/(2γ+d) ⌉. Under Assumptions 2.1, 2.4, 4.2 and 4.3, we obtain that sup S 0 E R N (π) ≤ c Klog(n)n 1− γ 2γ+d .(12) for some c > 0. Corollary 4.5 reveals that it is possible to achieve sublinear (in n) regret under the smoothness on the conditional distributions guaranteed by Assumption 4.2. Note that a curse of dimensionality is present in the sense that the upper bound on regret gets close to linear in n as d, the number of covariates, increases. The presence of this effect is due to the fact that as part of the regret minimization, we sequentially estimate the conditional distributions of the treatment outcomes and each F i (y, ·) is a function of a d-dimensional variable. Finally, it is also to be expected that the regret is increasing in the number of available treatments K as more observations must be used for experimentation when more treatments are available. The upper bound on the maximal regret in Corollary 4.5 is near-minimax optimal as we shall make precise in Theorem 5.3 below. Thus, if there is nothing prohibiting the choice of groups in (11), not much can be gained from a maximal regret point-of-view in searching for "better" partitions under the given set of assumptions. Finally, we remark that the grouping in (11) with P = ⌈n 1/(2γ+d) ⌉ requires knowledge of the expected number of treatments n. If the exact number of treatments is known a priori (as it is in many classical treatment problems) then n is trivially known. If, however, n is unknown one can instead use the doubling trick to attain upper bound on the maximal regret that are of the same order as in Corollary 4.5, but with slightly higher multiplicative constants. In essence, the doubling trick works by resetting the policy at time 2 m , m ∈ N. The name "doubling trick" comes from the fact that the length between subsequent resets of the policy doubles between subsequent resets. Importantly, the length of each treatment period is known. The doubling trick is a general tool in games of unknown horizon and we refer to Shalev-Shwartz (2012) for more details. Stronger regret guarantees and number of suboptimal assignments So far, our results in the case where covariates are present have only assumed that the conditional distribution of the treatment outcomes is Hölder continuous, cf. Assumption 4.2. If, furthermore, it is also the case that the best and second best treatment are "well-separated", the upper bound on maximal regret in Section 4.2 can be lowered slightly. Formally, introduce the second best treatment π ♯ (x) that is, for any x ∈ [0, 1] d , if min i∈I T(F i (·, x)) < T F π ⋆ (x) (·, x) , then π ♯ (x) satisfies T F π ♯ (x) (·, x) = max i∈I {T(F i (·, x)) : T(F i (·, x)) < T F π ⋆ (x) (·, x) }, and π ♯ (x) = 1 if min i∈I T(F i (·, x)) = T F π ⋆ (x) (·, x) . We can now introduce the margin condition. Assumption 4.6. There exists α ∈ (0, 1) and C 0 > 0 such that P X 0 < T F π ⋆ (X) (·, X) − T F π ♯ (X) (·, X) ≤ δ ≤ C 0 δ α for all δ ∈ [0, 1]. The margin condition restricts how likely it is that the best and second best treatment are close to each other. In particular, it limits the probability of these treatments being almost equally good, i.e., it limits how likely it is that the best and second-best treatment are within a δ-margin. Assumptions of the margin condition type have previously been used in the works of Mammen and Tsybakov (1999), Tsybakov (2004), Audibert and Tsybakov (2007) in the statistics literature. In the context of statistical treatment rules, the margin condition has recently been used in the work of Kitagawa and Tetenov (2018) who considered a static (non-sequential arrival of individuals to be treated/information) treatment allocation problem. Finally, the margin condition was also used in the work of Perchet and Rigollet (2013) in the context of a multi-armed bandit problem. The margin condition does not only allow us to prove sharper upper bounds on maximal regret than in Section 4.2, it also allows us to provide an upper bound on the number of suboptimal assignments made by the FSA policy. In particular, we shall define the total number of suboptimal assignments for a policy π over the course of a total of N assignments as S N (π) = N t=1 1 πt(Xt,Z t−1 ) ∈arg max{T(F i (·,Xt)),i=1,...,K} . In Theorem 4.8 we establish an upper bound on E(S N (π)) which is near minimax optimal. Let S = S(γ, L, c,c, α, C 0 ) be the set of K-tuples of conditional distributions of (Y 1,t , . . . , Y K,t ) given X t that satisfy Assumptions 4.2, 4.3 and 4.6. The maximal regret over S of the FSA policy over can be bounded as follows. Theorem 4.7. Suppose that the horizon N has expectation n and is independent of treatment outcomes and covariates. Consider the partition in (11) and set P = ⌈n 1/(2γ+d) ⌉. Under Assumptions 2.1, 2.4, 4.2, 4.3 and 4.6, we obtain that sup S E[R N (π)] ≤ cKlog(n)n 1− γ(1+α) 2γ+d (13) for some constant c > 0. Compared to Corollary 4.5 the exponent on n in the upper bound on regret is now smaller. Thus, in the presence of the margin condition (Assumption 4.6), the regret guarantee on the FSA policy is stronger. Of course, since S ⊂ S 0 , this is not altogether surprising. We shall see in Theorem 5.1 below that the upper bound on maximal regret is minimax optimal in n up to logarithmic factors. Our next result shows that the upper bound on maximal regret does not come at the price of excessive experimentation leading to many suboptimal assignments. In fact, Lemma D.5 in the appendix shows that under the margin condition, an upper bound on E[S N (π)] for any policy π follows as a consequence of an upper bound on regret. Theorem 4.8. Suppose that the horizon N has expectation n and is independent of treatment outcomes and covariates. Consider the partition in (11) and set P = ⌈n 1/(2γ+d) ⌉. Under Assumptions 2.1, 2.4, 4.2, 4.3 and 4.6, we obtain that sup S E[S N (π)] ≤ c[Klog(n)] α 1+α n 1− αγ 2γ+d for some constant c > 0. The upper bound in Theorem 4.8 on the maximal number of suboptimal assignments made is a useful theoretical guarantee since it limits the number of individuals who receive suboptimal treatments. A policy which only ensures high total welfare (low regret), may not be ethically viable if too many individuals are maltreated. Finally we note that no policy can be expected to make substantially fewer suboptimal assignments than the FSA policy since Step 4 of the proof of Theorem 5.1 below actually shows that for any policy there exist distributions in S for which the number of suboptimal assignments must be at least of order n 1− αγ 2γ+d . Lower bounds on maximal regret and minimax optimality of the FSA policy In this section we prove formally the impossibility results mentioned in Remark 4.1 and establish (near) minimax optimality of the FSA policy in several settings. As was the case in the setting without covariates in Section 3, only a local uniform monotonicity of T over H suffices to establish tight lower bounds on maximal regret. We stress, however, that while the assumptions imposed in the present section are the same as in the case without covariates, the proofs are more involved as one must carefully construct conditional distributions of treatment outcomes satisfying Assumptions 4.2 and 4.6 against which large regret must be incurred by any policy. For all lower bounds, we consider the case of K = 2 available treatments. Fix a functional T and let Π denote the set of all policies π. For any 2-tuple of conditional distributions (F 1 , F 2 ) of Y 1,t and Y 2,t given X t in S and policy π, we make the dependence of regret on (F 1 , F 2 ) explicit by R n (π) = R n (π, F 1 , F 2 ) = n t=1 T F 1 (·, X t ) − T F 2 (·, X t ) 1 {π ⋆ (Xt) =πt(Xt,Z t−1 )} . (14) Theorem 5.1. Suppose that X t is uniformly distributed on [0, 1] d (thus c =c = 1 in Assumption 4.3). Then, under Assumptions 2.1 and 3.3, there exists a c l > 0 such that inf π∈Π sup (F 1 ,F 2 )∈S E[R n (π, F 1 , F 2 )] ≥ c l n 1− γ(1+α) 2γ+d .(15) Theorem 5.1 shows that the upper bound on maximal regret of the FSA policy obtained in Theorem 4.7 is only improvable by logarithmic factors. Put differently, the FSA policy is minimax optimal up to logarithmic factors. Maximal regret is linear without Assumption 4.2 Let C[0, 1] d denote the set of (uniformly) continuous functions on [0, 1] d and let S C denote the 2-tuples of distributions (F 1 , F 2 ) of Y 1,t and Y 2,t given X t such that F 1 , F 2 ∈ C[0, 1] d . The following theorem, which is a consequence of Theorem 5.1, shows that without the Hölder continuity imposed in Assumption 4.2, no policy exists that has sub-linear maximal regret in n. Furthermore, since every policy is guaranteed to incur no more than linear (in n) regret, this shows that the problem is not well posed without Assumption 4.2. More precisely, even when restricting attention to (uniformly) continuous F 1 and F 2 , maximal regret of any policy is linear in n. Thus, the following Theorem makes precise Remark 4.1 prior to Assumption 4.2. Theorem 5.2. Suppose that X t is uniformly distributed on [0, 1] d (thus c =c = 1 in Assumption 4.3). Then, under Assumptions 2.1 and 3.3, there exists a c l > 0 such that inf π∈Π sup (F 1 ,F 2 )∈S C E[R n (π, F 1 , F 2 )] ≥ c l n. Lower bound on maximal regret without margin condition The next result, which gives a lower bound on maximal regret in the absence of the margin condition in Assumption 4.6, is again a consequence of Theorem 5.1. Theorem 5.3. Suppose that X t is uniformly distributed on [0, 1] d (thus, c =c = 1 in Assumption 4.3). Then, under Assumptions 2.1 and 3.3, for all ε > 0 there exists a c l (ε) > 0 such that inf π∈Π sup (F 1 ,F 2 )∈S 0 E[R n (π, F 1 , F 2 )] ≥ c l (ε)n 1− γ 2γ+d n −ε .(16) Comparing the lower bound on maximal regret in Theorem 5.3 to the upper bound on maximal regret of the FSA policy established in Corollary 4.5 reveals that the FSA policy is near-optimal also in this setting. If a policy with strictly smaller maximal regret exists, the order of this improvement must be o(n ε ) for all ε > 0, e.g., logarithmic. Ignoring covariates Theorem 3.1 shows that the maximal regret for the FSA policy is guaranteed to increase not much faster than rate √ n in the absence of covariates. On the other hand, if γ(1+α) 2γ+d < 1 2 (which occurs in case 2γα < d), the maximal regret must increase faster than √ n for any policy in the presence of covariates, cf. Theorem 5.1. At first sight, one could be led to believe that it may sometimes be advantageous to ignore the covariates in order to achieve a lower maximal regret. Note, however, that the oracle targets are defined differently in the context of Theorems 3.1 and 5.1. More precisely, if covariates are available one targets the "best" conditional distribution. Our next result shows, that it is in fact a very bad idea to ignore the covariates unless one knows that these are irrelevant (which one rarely does). To be precise, consider polices that are a (recursively) defined sequence of Borel measurable functions π = {π t } ∞ t=1 which can depend on observed treatment outcomes (but not covariates): the first treatment is some element in I not depending on any non-existing previous treatment outcomes. The second treatment can depend on Z 1 := Y π 1 ,1 such that π 2 : [0, 1] → I. In general, π t : [0, 1] (t−1) → I and we write π t (Z t−1 ) where Z t−1 := (Y π t−1 (Z t−1 ) , ...., Y π 1 ). Thus, Z t−1 is the allowed to be used after the previous treatment outcome is observed. LetΠ denote the collection of such policies. Note the similarity to the definition of a policy in the setting without covariates, cf. Section 2.2. Theorem 5.4. Suppose that X t is uniformly distributed on [0, 1] d (thus c =c = 1 in Assumption 4.3). Then, under Assumptions 2.1 and 3.3, there exists a c l > 0 such that inf π∈Π sup (F 1 ,F 2 )∈S E[R n (π, F 1 , F 2 )] ≥ c l n.(17) Thus, the maximal regret of any policy ignoring covariates must increase at the worst-case linear rate in n. To illustrate the connections to the FSA policy note that for this policy ignoring the covariates amounts to assigning all individuals to the same group, i.e. F = 1 and thus V 1 = √ d,B 1 = 1. Using these quantities in Theorem 4.4 results in a upper bound on regret which is linear in n. A Verification of Assumption 2.1 for some inequality-, welfareand poverty-measures To keep the statements in the subsequent examples simple, and to give some concrete meaning to the discussion in Remark 2.2, we shall now define some sets of cdfs D that will show up frequently in the following discussion: Given real numbers a < b, we shall denote (i) the subset of all cdfs F in D cdf ([a, b]) that are continuous on [a, b] and right-differentiable on (a, b) with right-sided derivative F + , say, satisfying F + (x) ≤ s for all x ∈ (a, b) by D s ([a, b]); (ii) we shall denote the subset of all cdfs F in D cdf ([a, b]) that are continuous on [a, b] and right-differentiable on (a, b) with right-sided derivative F + , say, satisfying F + (x) ≥ r for all x ∈ (a, b) by D r ([a, b]); and (iii) we shall denote the subset of all cdfs F in D cdf ([a, b]) that are continuous on [a, b] and right-differentiable on (a, b) with right-sided derivative F + , say, satisfying r ≤ F + (x) ≤ s for all x ∈ (a, b) by D s r ([a, b]). Furthermore, we shall denote the subset of D s ([a, b]) consisting of all cdfs F ∈ D s ([a, b]) that are continuous on all of R (and not only on [a, b]) by C s ([a, b]), and correspondingly define C r ([a, b]) and C s r ([a, b]). To illustrate the scope of our results, and to facilitate their implementation in practice, we shall now discuss several functionals of interest in applied economics that satisfy Assumption 2.1. We emphasize that Appendix C contains a catalog of general methods for verifying Assumption 2.1, and that the results in the following three sections are established using these techniques. Therefore, in addition to their intrinsic importance, the following results, and in particular their proofs, also provide a pattern as to how Assumption 2.1 can be verified for functionals that are not explicitly discussed. A.1 Inequality measures In this section we verify Assumption 2.1 for functionals that measure the inequality inherent to an, e.g., "income", "wealth" or "productivity", distribution F . Such "inequality measures" are obviously relevant in situations where one intends to select, among various candidates, that treatment (e.g., the introduction of a certain tax) which leads to the most "equal" outcome distribution. To avoid possible misunderstanding, we emphasize that it is neither our goal to discuss theoretical foundations of inequality measures nor to point out their relative advantages and disadvantages. The functional must be chosen by the applied economist, who can-in making such a choice-rely on excellent book-length treatments, e.g., Lambert (2001), Chakravarty (2009) or Cowell (2011, as well as the original sources some of which we shall point out further below. We first discuss inequality measures that are derived from the Lorenz curve. The first such inequality measure we consider is the Schutz-coefficient S rel (F ), say, which is also known as the Hoover-index or Robin Hood-index. In plain words, this coefficient measures the maximal vertical distance between the 45 • line and the Lorenz curve corresponding to F (cf. Gastwirth (1971) or Equation (22) below for a formal definition of the Lorenz curve). It can be shown (e.g., Lambert (2001)) that S rel (F ) coincides with half the relative mean deviation index, i.e., S rel (F ) = 1 2µ(F ) |x − µ(F )|dF (x),(18) provided the expression is well defined. Here, the index "rel" signifies that this index is defined "relative" to the mean (as a consequence, multiplying each income by the same amount does not change the inequality index). A corresponding "absolute" variant (i.e., a measure which remains unchanged if one adds to every income the same amount) is obtained by multiplying the relative measure by the mean functional, and is denoted by S abs (F ) = 1 2 |x−µ(F )|dF (x). We refer to Kolm (1976a,b) for a discussion concerning relative and absolute inequality measures. The following lemma provides conditions under which the relative and absolute Schutz-coefficient satisfies Assumption 2.1. Lemma A.1. Let a < b be real numbers. Then, the absolute Schutz-coefficient T = S abs satisfies Assumption 2.1 with D = D cdf ([a, b]) and C = b − a. Next, assume that a ≥ 0, and define for every δ ∈ (a, b) and every s > 0 the set D(s, δ) := {F ∈ C s ([a, b]) : µ(F ) ≥ δ}.(19) Then, for every δ ∈ (a, b) and every s > 0 the relative Schutz-coefficient T = S rel ( defined as 0 for the cdf corresponding to point mass 1 at 0) satisfies Assumption 2.1 with D = D(s, δ) and constant C = (b − a)(2s + δ −1 ) + 5. The next index we consider is the Gini-index, which for a cdf F is defined as the area between the Lorenz curve and the 45 • line, and which can be written as (cf. again Lambert (2001)) G rel (F ) = 1 2µ(F ) |x 1 − x 2 |dF (x 1 )dF (x 2 ),(20) provided that the expression is well defined. A corresponding absolute inequality measure is .5 |x 1 − x 2 |dF (x 1 )dF (x 2 ), which we denote by G abs (F ). The following lemma provides conditions under which Assumption 2.1 is satisfied for these two Gini-indices: Lemma A.2. Let a < b be real numbers, and let D = D cdf ([a, b]). For this choice of a, b and D, the functional T = G abs satisfies Assumption 2.1 with constant C = b − a. Next, assume that a ≥ 0, and define for every δ ∈ (a, b) the set D(δ) := {F ∈ D cdf ([a, b]) : µ(F ) ≥ δ}.(21) Then, for every δ ∈ (a, b) the relative Gini-index T = G rel ( defined as 0 for the cdf corresponding to point mass 1 at 0) satisfies Assumption 2.1 with constant C = 4δ −1 (b − a). It can be verified that the Gini-index belongs to the class of linear inequality measures (cf. Mehran (1976)). Linear inequality measures are functionals of the form F → [0,1] (u − L(F, u))dW (u), where L(F, u) := µ(F ) −1 [0,u] q α (F )dα,(22) where W denotes a function on [0, 1] (independent of F ) with finite total variation. Note that L(F ; u) is the Lorenz curve corresponding to F evaluated at u (cf. Gastwirth (1971) and our Equation (94)). The following lemma provides conditions under which a linear inequality measure as defined in Equation (22) satisfies Assumption 2.1. The result relies on Lipschitz-type properties of the Lorenz curve established in Lemma C.14 in the appendix. The generality is bought at the price of comparably strong regularity conditions. This becomes apparent by comparing the regularity conditions to the ones in Lemma A.2. Lemma A.3. Let a < b be positive real numbers and let r > 0. Assume that W : [0, 1] → R has finite total variation κ, say. Then, the functional defined in Equation (22) satisfies Assumption 2.1 with D = C r ([a, b]), and constant C = κa −1 (r −1 + (b − a)a −1 ). An absolute version of the linear inequality measure in Equation (22) is obtained after multiplication with µ(F ), and is given by F → [0,1] (µ(F )u − U(F, u))dW (u), where U(F, u) := [0,u] q α (F )dα.(23) The following result provides conditions under which such absolute linear inequality measures satisfy Assumption 2.1. As usual, the regularity conditions required for absolute versions of an inequality measure are weaker than the ones needed for the relative version: Lemma A.4. Let a < b be real numbers and let r > 0. Assume that W : [0, 1] → R has finite total variation κ, say. Furthermore, denote | [0,1] udW (u)| =: c. Then, the functional defined in Equation (23) satisfies Assumption 2. 1 with D = C r ([a, b]), and constant C = c(b − a) + r −1 κ. Another important family of inequality indices is the so-called generalized entropy family, cf. Cowell (1980): Given a parameter α ∈ R, an inequality measure is obtained via (if the involved expressions are well defined) E c (F ) =        1 c(c−1) x/µ(F ) c − 1 dF (x) if c / ∈ {0, 1} x/µ(F ) log x/µ(F ) dF (x) if c = 1 log x/µ(F ) dF (x) if c = 0.(24) The inequality measures corresponding to c = 1 is known as Theil's entropy index (cf. also Theil (1967)), and the measure corresponding to c = 0 is known as the mean logarithmic deviation (cf. Lambert (2001), p.112). A formal result providing conditions under which generalized entropy measures in the previous display satisfy Assumption 2.1 is presented next. The regularity conditions we need to impose depend on c. In particular, support assumptions inherent in the definition of D are somewhat weaker in case c ∈ (0, 1). Lemma A.5. Let 0 ≤ a < b be real numbers, and let c ∈ R. 1. If c ∈ (0, 1), then, for every δ ∈ (a, b) the functional T = E c ( defined as 0 for the cdf corresponding to point mass 1 at 0) satisfies Assumption 2. 1 with D = D(δ) (cf. Equation 21) and constant C = |c(c − 1)| −1 δ −c (b c − a c ) + δ −1 (b − a) . 2. If c / ∈ [0, 1] and a > 0, then the functional T = E c satisfies Assumption 2.1 with D = D cdf ([a, b]) and C = |c(c − 1)| −1 [a −c |b c − a c | + |c| max (a/b) 2c−1 , (b/a) 2c−1 a −1 (b − a)]. 3. If c ∈ {0, 1} and a > 0, then the functional T = E c satisfies Assumption 2. 1 with D = D cdf ([a, b]) and with constant C = a −1 + log(b/a) if c = 0, and with constant C = a −1 [a,b] |1 + log(x)|dx + ba −1 | log(b/a)|(b − a) + a −1 if c = 0. We continue with a family of (relative) inequality indices introduced in Atkinson (1970). This family depends on an "inequality aversion" parameter ε ∈ (0, 1) ∪ (1, ∞). For a fixed ε in that range, the index obtained equals (if the involved quantities are well defined) A ε (F ) = 1 − 1 µ(F ) x 1−ε dF (x) 1/(1−ε) .(25) It is well known (cf., e.g., Lambert (2001) p.112) that A ε can be written as A ε (F ) = 1 − [ε(ε − 1)E 1−ε (F ) + 1] 1/(1−ε) .(26) Together with Lemma A.5 this relation can be used to obtain the following result: Lemma A.6. Let 0 ≤ a < b be real numbers, let ε ∈ (0, 1) ∪ (1, ∞) and set c(ε) = 1 − ε. 1. If ε ∈ (0, 1), then, for every δ ∈ (a, b) the functional T = A ε ( defined as 0 for the cdf corresponding to point mass 1 at 0) satisfies Assumption 2. 1 with D = D(δ) (cf. Equation 21) and constant C = c(ε) −1 δ −c(ε) (b c(ε) − a c(ε) ) + δ −1 (b − a) . 2. If ε ∈ (1, ∞) and a > 0, then the functional T = A ε satisfies Assumption 2.1 with D = D cdf ([a, b]) and constant C = c(ε) −1 (b/a) ε [a −c(ε) |b c(ε) − a c(ε) | + |ε||c(ε)| 2 (a/b) 2c(ε)−1 a −1 (b − a)] . As the last example in this section, we proceed to an important family of (absolute) inequality indices, the Kolm-indices (Kolm (1976a), cf. also the discussion in Section 1.8.1 of Chakravarty (2009)). Given a parameter κ > 0 the corresponding index is defined as K κ (F ) = κ −1 log e κ[µ(F )−x] dF (x) .(27) The following lemma verifies Assumption 2.1 for this class of inequality indices: Lemma A.7. Let a < b and let κ > 0. Then, the functional T = K κ satisfies Assumption 2.1 with D = D cdf ([a, b]) and constant C = κe κb (b − a) + [e −κa − e −κb ]. A.2 Welfare measures The most elementary class of social welfare functions are of the form (cf. Atkinson (1970)) F → u(x)dF (x),(28) for a utility function u. Such functionals can directly be dealt with the theory developed in Kock and Thyrsgaard (2017). We therefore refer to this article for corresponding results. There are many important social welfare functions, however, that are not of the simple form (28), and can thus not be treated with the results in Kock and Thyrsgaard (2017). Many such exceptional measures are related to a relative inequality measure F → I rel (F ), say, via the transformation W(F ) = µ(F )(1 − I rel (F ));(29) or (correspondingly) to an absolute inequality measure F → I abs (F ), say, via the transformation W(F ) = µ(F ) − I abs (F ).(30) The Gini social welfare function (obtained via the previous display upon choosing I abs = G abs , see the discussion after Equation (20) for a definition of G abs ) is one of the most important examples. The following result allows one to use the results from the preceding section in establishing Assumption 2.1 for the two types of social welfare functions in (29) and (30). Lemma A.8. Let a < b be real numbers. Then, the following holds: 1. Let the relative inequality measure I rel satisfy Assumption 2.1 with D rel and C. Suppose further that |1 − I rel | ≤ γ < ∞ holds. 3 Then, the social welfare function W derived via Equation (29) satisfies Assumption 2.1 with D = D rel and constant γ(b − a) + max(|a|, |b|)C. 2. Let the absolute inequality measure I abs satisfy Assumption 2.1 with D abs and C. Then, the social welfare function W derived via Equation (30) satisfies Assumption 2.1 with D = D abs and with constant (b − a) + C. The preceding lemma together with Lemma A.2 implies that the Gini social welfare function (defined directly after Equation (30) above) satisfies Assumption 2.1 with a < b real numbers, D = D cdf ([a, b]), and constant C = 2(b − a). Similar statements can easily be obtained for social welfare functions corresponding to the class of linear inequality measures via Lemma A.4 (which then covers the class of social welfare functions considered recently in Kitagawa and Tetenov (2017)), or via the other inequality measures discussed in the preceding section. A.3 Poverty measures Poverty indices are typically based on a poverty line, i.e., a threshold z below which an, e.g., income is classified as "poor". There are two basic approaches to defining z: the absolute approach considers z as fixed (i.e., independent of the underlying income distribution F ), whereas the relative approach views z = z(F ) as a function of the "income distribution" F . In particular, in the relative approach the poverty line adapts to growth/decline of the economy. To make this formal and to give an example, the following poverty line functional combines both approaches (cf. Kakwani (1986) and Lambert (2001), p.139) in taking a convex combination of a fixed amount z 0 and a centrality measure of the underlying income distribution: z m,z 0 ,δ (F ) = z 0 + δ(m(F ) − z 0 )(31) where z 0 > 0, 0 ≤ δ ≤ 1, and m is a location functional that either coincides with the mean functional µ, or the median functional q 1/2 . Note in particular that z m,z 0 ,0 = z 0 , i.e., this definition nests both an absolute and a relative approach. For concreteness, Lemma B.1 in Appendix B summarizes conditions under which the poverty line functionals in the family (31) satisfy Assumption 2.1. The first poverty measure we shall consider is the so-called headcount ratio, which is the proportion in a population F that qualifies as poor according to a given poverty line z: H z (F ) = F (z(F )).(32) For the sake of generality, the following lemma establishes conditions under which the headcount ratio satisfies Assumption 2.1 under high-level conditions concerning the poverty line functional z. Specific constants and domains for the concrete family of poverty lines defined in Equation (31) can immediately be obtained with Lemma B.1 in Appendix B. An analogous statement applies to the poverty measures introduced further below, and will not be restated. Lemma A.9. Let a = 0 < b, and let z : D cdf ([a, b]) → R denote a poverty line functional that satisfies Assumption 2.1 with D z and constant C z , say. Let s > 0. Then, H z satisfies Assumption 2.1 with D = D z ∩ D s ([a, b]) and C = C z s + 1. Certain disadvantages of the headcount ratio motivated Sen (1976) to introduce a different family of poverty measures, using an axiomatic approach. We shall now discuss this family in the generalized form of Kakwani (1980). Given a poverty line z and a "sensitivity parameter" κ ≥ 1, say, each element of this family of poverty indices is written as P SK (F ; z, κ) = (κ + 1) [0,z(F )] 1 − x z(F ) 1 − F (x) F (z(F )) κ dF (x),(33) with the convention that 0/0 := 0. A result discussing conditions under which P SK (F ; z, κ) satisfies Assumption 2.1, and which is again established under high-level assumptions on the poverty line z, is provided next. Lemma A.10. Let a = 0 < b, κ ≥ 1, and let z : D cdf ([a, b]) → R denote a poverty line functional that satisfies Assumption 2.1 with D z and constant C z , say. Suppose further that z ≥ z * > 0 holds for some real number z * . Let s > 0. Then T = P SK (.; z, κ) satisfies Assumption 2. 1 with D = D z ∩ D s ([a, b]) and C = (κ + 1)[1 + bz −2 * C z + κ[(2 + s)C z + 4] ]. Second, we consider a family, each element of which can be written as P F GT (F ; z, Λ) = [0,z(F )] Λ 1 − [x/z(F )] dF (x),(34) where Λ : [0, 1] → [0, 1] is non-decreasing, surjective and convex. This class contains (at least after monotonic transformations), e.g., the measures of Foster et al. (1984) or Chakravarty (1983) as special cases (cf. Lambert (2001) Chapter 6.3, and also the more recent review in Foster et al. (2010)). The following result provides conditions under which P F GT satisfies Assumption 2.1. Again the result is established under high-level assumptions on the poverty line z (note in particular that the poverty line in Equation (31) is greater or equal to (1 − δ)z 0 > 0 in case F is supported on [0, ∞)). Lemma A.11. Let a = 0 < b and let z : D cdf ([a, b]) → R denote a poverty line functional that satisfies Assumption 2.1 with D z and constant C z , say. Suppose further that z ≥ z * > 0 holds for some real number z * , that Λ : [0, 1] → [0, 1] is non-decreasing and surjective, and that Λ is the restriction of a convex real-valued function Λ * defined on an open interval in R containing [0, 1]. Denote the Lipschitz constant of Λ by C Λ . Then, T = P F GT (.; z, Λ) satisfies Assumption 2.1 with D = D z and C = bz −2 * C Λ C z + 1. As a direct application of Lemma A.11, we note that given a poverty line z the poverty measure of Foster et al. (1984) is obtained upon setting Λ(x) = x α in Equation (34). The conditions in the preceding lemma are satisfied for α ≥ 1 (in which case C Λ = α). The preceding lemma does not cover the range α ∈ [0, 1). Note that the functional corresponding to Λ(x) = x α with α = 0 coincides with the headcount ratio, which is already covered via Lemma A.9. Furthermore, the range α > 1 might be considered most important, as only such values of α guarantee that in addition to the "Monotonicity Axiom" ("Given other things, a reduction in the income of a poor household must increase the poverty measure"), which would be satisfied for all α ≥ 0, the inequality measure obtained also satisfies the "Transfer Axiom" ("Given other things, a pure transfer of income from a poor household to any other household that is richer must increase the poverty measure") of Sen (1976), cf. Proposition 1 in Foster et al. (2010). Both axioms are plausible requirements (albeit not undisputed, an early criticism being Kundu and Smith (1983)), but are not satisfied by the headcount ratio, cf. Sen (1976). B Proofs for Section A Proof of Lemma A.1: Given F, G ∈ D cdf ([a, b]) it holds that |S abs (F ) − S abs (G)| is not greater than 1/2- times [a,b] |x − µ(F )| − |x − µ(G)| dF (x) + [a,b] |x − µ(G)|dF (x) − [a,b] |x − µ(G)|dG(x) . Using the reverse triangle inequality, the first integral in the previous display can be bounded from above by |µ(F ) − µ(G)| ≤ (b − a) F − G ∞ (cf. Example C.3 for the inequality). Using Lemma C.2, the remaining expression to the right in the previous display is seen not to be greater than (b − a) F − G ∞ . Hence, the first statement follows (noting that S abs is obviously well defined on all of D cdf ([a, b])). Concerning the second claim, we first observe that for every F ∈ D cdf ([a, b]) it holds that 1 2 [a,b] |x − µ(F )|dF (x) = [a,µ(F )] (µ(F ) − x)dF (x).(35) Next, let s > 0, δ ∈ (a, b), F ∈ D(s, δ) and G ∈ D cdf ([a, b]). We consider two cases, and start with the case where µ(G) = 0 (implying that a = 0 and that G is the cdf corresponding to point mass at 0). Then, by convention, S rel (G) = 0, and it follows from Equation (35) (recalling that µ(F ) ≥ δ > 0) that |S rel (F ) − S rel (G)| ≤ [a,µ(F )] |1 − x/µ(F )|dF (x) ≤ F (µ(F )).(36) Since F is continuous 0 = F (0) = F (µ(G)) holds. It follows that F (µ(F )) = |F (µ(F ))−F (µ(G))|. Using the mean value theorem of Minassian (2007) and Example C. 3 we conclude that |F (µ(F )) − F (µ(G))| ≤ s(b − a) F − G ∞ . Next, we turn to the case where µ(G) > 0. First, we note that |S rel (F ) − S rel (G)| ≤ |F (µ(F )) − G(µ(G))| + [a,µ(F )] x µ(F ) dF (x) − [a,µ(G)] x µ(G) dG(x) . Consider the first term in absolute values in the previous display: by the triangle inequality: |F (µ(F )) − G(µ(G))| ≤ |F (µ(F )) − F (µ(G))| + F − G ∞ . From the mean value theorem for right-differentiable functions as in Minassian (2007), and the definition of D s ([a, b]), we obtain |F (µ(F ))−F (µ(G))| ≤ s|µ(F )−µ(G)| ≤ s(b−a) F −G ∞ , the second inequality following from Example C.3. Now, note that (incorporating the considerations in case µ(G) = 0) it remains to show that [a,µ(F )] x µ(F ) dF (x) − [a,µ(G)] x µ(G) dG(x) ≤ ((s + δ −1 )(b − a) + 4) F − G ∞ .(37) To this end, denote m := min(µ(F ), µ(G)), M := max(µ(F ), µ(G)), letF denote a cdf in {F, G} which realizes the latter maximum, and rewrite the difference of integrals inside the absolute value to the left in the preceding display as [a,m] x µ(F ) dF (x) − [a,m] x µ(F ) dG(x) ± (m,M ] x µ(F ) dF (x) + [a,µ(G)] x µ(F ) − x µ(G) dG(x), where the ± is "+" in caseF = F and "−" in caseF = G. Next, denote the difference of the first two integrals in the previous display by A, the third integral by B and the fourth by C, respectively. First, Lemma C.2 (applied with k = 1, c = a, d = m and ϕ(x) = x/µ(F )) implies (working with the upper bounds |M * | ≤ 1 and C ≤ 1 in Lemma C.2 for the special case under consideration) that |A| ≤ 2 F − G ∞ . Second, note that the integrand in B is smaller than 1, hence |B| ≤F (M) −F (m) ≤ F (M) − F (m) + 2 F − G ∞ ≤ s|µ(F ) − µ(G)| + 2 F − G ∞(38) where we used F − F ≤ F − G ∞ for the first inequality, and the mean value theorem of Minassian (2007) for the second. To obtain an upper bound for |B| we now use Example C.3 to see that the right hand side in the previous display is not greater than [s(b − a) + 2] F − G ∞ . Concerning |C| note that |C| ≤ [a,µ(G)] µ(G) µ(F ) − 1 dG(x) ≤ µ(G) µ(F ) − 1 . If µ(G)/µ(F ) ≥ 1, then the upper bound in the previous display is not greater (cf. Example C.3) than µ(F ) + (b − a) F − G ∞ /µ(F ) − 1 ≤ δ −1 (b − a) F − G ∞ , and the same bound holds if µ(G)/µ(F ) < 1. Hence, |C| ≤ δ −1 (b − a) F − G ∞ . Summarizing, |A| + |B| + |C| ≤ ((s + δ −1 )(b − a) + 4) F − G ∞ , which proves the statement in Equation (37). Proof of Lemma A.2: The first statement follows from Example C.6. To prove the statement concerning G rel , we first note that G rel is well defined on D cdf ([a, b]) (note that µ(F ) ≤ 0 implies that a = 0 and that µ F is point mass at 0, implying that G rel (F ) = 0). Let F, G ∈ D cdf ([a, b]). We now consider two cases, and start with the case where µ(G) = 0. Then, G rel (G) = 0 and |G rel (F ) − G rel (G)| = G rel (F ) ≤ δ −1 [µ(F ) − µ(G)] ≤ δ −1 (b − a) F − G ∞ ,(39) where we used Example C.3 in the last inequality. Next, in case µ(G) = 0, we have µ(G) > 0 (recall that a ≥ 0), and we set ϕ(x 1 , x 2 ) = |x 1 − x 2 |. Write |G rel (F ) − G rel (G)| ≤ A + B,(40) where A := δ −1 [a,b] [a,b] ϕ(x 1 , x 2 )dF (x 1 )dF (x 2 ) − [a,b] [a,b] ϕ(x 1 , x 2 )dG(x 1 )dG(x 2 ) ,(41) which, by Example C.6 is not greater than δ −1 2(b − a) F − G ∞ , and B := [a,b] [a,b] |(µ(F ) −1 − µ(G) −1 )ϕ(x 1 , x 2 )|dG(x 1 )dG(x 2 ).(42) By the reverse triangle inequality (using that µ(F ) > 0 and µ(G) > 0), we see that B ≤ |µ(F ) −1 − µ(G) −1 | [a,b] [a,b] |x 1 − x 2 |dG(x 1 )dG(x 2 ) ≤ 2 [µ(G)/µ(F )] − 1 .(43) Using Example C.3, we see that µ(F ) − (b − a) F − G ∞ ≤ µ(G) ≤ µ(F ) + (b − a) F − G ∞ , from which it is easy to conclude that − (b − a) F − G ∞ /µ(F ) ≤ [µ(G)/µ(F )] − 1 ≤ (b − a) F − G ∞ /µ(F ),(44) from which it follows that [µ(G)/µ(F )] − 1 ≤ δ −1 (b − a) F − G ∞ . Hence, in case µ(G) = 0, we obtain that |G rel (F ) − G rel (G)| ≤ 4δ −1 (b − a) F − G ∞ .(45) Together with the first case, this proves the result. Proof of Lemma A.3: First of all, the functional under consideration is trivially well defined on D cdf ([a, b]) (because a > 0 holds). Next, we apply Lemma C.12 together with Lemma C.14 to obtain that for every u ∈ [0, 1] the functional F → L(F, u) satisfies Assumption 2.1 with a, b and D (as in the statement of the present lemma) as in the statement of the theorem and with constant a −1 (r −1 + (b − a)a −1 ). The statement immediately follows. Proof of Lemma A.4: The triangle inequality, together with Example C.3 and Lemma C.13 (which is applicable due to Lemma C.12) immediately yield the claimed result. Proof of Lemma A.5: We start with the first statement. The functional T is obviously everywhere defined on D cdf ([a, b]) (in case µ(F ) = 0 it follows that a = 0 and that F corresponds to point mass 1 at 0 in which case T (F ) = 0, by definition). Next, let δ ∈ (a, b), let F ∈ D(δ) and let G ∈ D cdf ([a, b]). We consider first the case where µ(G) = 0 (implying that T(G) = 0). Then, noting that [a,b] x c dG(x) = 0, we conclude that |T(F ) − T(G)| = T(F ), the latter being not greater than 1 c|c − 1|δ c [a,b] x c dF (x) − [a,b] x c dG(x) ≤ b c − a c c|c − 1|δ c F − G ∞ ,(46) where we used Lemma C.2 for the last inequality. Next, consider the case where µ(G) > 0. We note that [a,b] (x/µ(F )) c dF (x) − [a,b] (x/µ(F )) c dG(x)(47) can be upper bounded by A + B with A := [a,b] (x/µ(F )) c dF (x) − [a,b] (x/µ(F )) c dG(x) ≤ b c − a c δ c F − G ∞(48) the inequality following from Lemma C.2, and B := |(1/µ(F )) c − (1/µ(G)) c | [a,b] x c dG(x) ≤ |(µ(G)/µ(F )) c − 1|,(49) the inequality following from Jensen's inequality (recalling that c ∈ (0, 1)). It remains to observe that the simple inequality |z c − 1| ≤ |x − 1| for z > 0 it follows that |(µ(G)/µ(F )) c − 1| ≤ |µ(G)/µ(F ) − 1| ≤ δ −1 (b − a) F − G ∞ ,(50) where the second inequality follows from Example C.3 together with µ(F ) ≥ δ. Hence, in case µ(G) > 0 we see that |T(F ) − T(G)| ≤ (c|c − 1|) −1 b c − a c δ c + δ −1 (b − a) F − G ∞ , which proves the first claim. We now prove the second claim. Since a > 0 holds in this case, µ(G) and µ(F ) can not be smaller than a. Hence the functional is well defined on all of D cdf ([a, b]). Furthermore, the expression in Equation (47) is greater than A + B, where A and B have been defined above. By Lemma C.2 it holds that A is not greater than a −c |b c − a c F − G ∞ . Furthermore, B is not greater than max (a/b) c , (b/a) c |(µ(F )/µ(G)) c − 1| ≤ |c| max (a/b) 2c−1 , (b/a) 2c−1 |µ(F )/µ(G) − 1| ≤ |c| max (a/b) 2c−1 , (b/a) 2c−1 a −1 (b − a) F − G ∞ ,(51) the first inequality following from |z c − 1| ≤ |c| max((a/b) c−1 , (b/a) c−1 )|z − 1| for z ∈ [a/b, b/a] (noting that this interval contains 1 and recalling that c / ∈ [0, 1]), and the second inequality following from Exercise C.3. We now turn to the last case where c ∈ {0, 1} (and a > 0 guaranteeing that the functional is then well defined on all of D cdf ([a, b])). We consider first the case where c = 0. The statement follows after noting that |T(F ) − T(G)| is not greater than C + D with C := [a,b] log(x)dF (x) − log(x)dG(x) ≤ log(b/a) F − G ∞(52) the inequality following from Lemma C.2, and D := [a,b] log(x/µ(F )) − log(x/µ(G)) dG(x) = | log(µ(G)/µ(F ))| ≤ log(1 + a −1 F − G ∞ ) ≤ a −1 F − G ∞ .(53) In case c = 1, let f (x) = (x/µ(F )) log(x/µ(F )) and g(x) = (x/µ(G)) log(x/µ(G)). Write |T(F ) − T(G)| ≤ [a,b] f (x)dF (x) − [a,b] f (x)dG(x) + [a,b] |f (x) − g(x)|dG(x).(54) From Lemma C.2 it follows that the first absolute value in the upper bound is not greater than a −1 [a,b] |1+ log(x)|dx F − G ∞ . Finally, noting that for every x ∈ [a, b] we have |f (x) − g(x)| = |x| |µ −1 (F ) − µ −1 (G)|| log(x/µ(F ))| + µ −1 (G) log(µ(F )/µ(G)) ≤ ba −1 | log(b/a)|(b − a) + a −1 F − G ∞ ,(55) where (in addition to a > 0) we used Exercise C.3 and the inequality for log(µ(G)/µ(F )) established above. The final claim follows. Proof of Lemma A.6: We start with Part 1: Let δ ∈ (a, b). From the first part of Lemma A.5 we obtain that in case ε ∈ (0, 1) the functional E c(ε) satisfies Assumption 2.1 with D = D(δ) (cf. Equation 21) and constant |εc(ε)| −1 δ −c(ε) (b c(ε) − a c(ε) ) + δ −1 (b − a) . It remains to observe that the function z → 1 − z 1/c(ε)(56) is Lipschitz continuous on [0, 1] with constant c(ε) −1 . The claim then follows from Lemma C.1 (with m = 1), the representation in Equation (26) together with the observation that 0 ≤ ε(ε − 1)E c(ε) (F ) + 1 ≤ 1 holds for every F ∈ D cdf ([a, b]) as a consequence of Jensen's inequality. Concerning Part 2: From the second part of Lemma A.5 we obtain that in case ε ∈ (1, ∞) the functional E c(ε) satisfies Assumption 2.1 with D = D cdf ([a, b]) and constant (note that 2c ( ε) − 1 < 0) equal to |c(ε)ε| −1 [a −c(ε) |b c(ε) − a c(ε) | + |c(ε)|(a/b) 2c(ε)−1 a −1 (b − a)]. We shall now argue similarly as in Part 1. The function in Equation (56) Proof of Lemma A.7: Clearly T is well defined on all of D cdf ([a, b]). Let F ∈ D cdf ([a, b]). Then, by Jensen's inequality: [a,b] e κ[µ(F )−x] dF (x) ≥ 1.(57) Since log restricted to [1, ∞) is Lipschitz continuous with constant 1, we obtain for any G ∈ D cdf ([a, b]) that |T(F ) − T(G)| ≤ κ −1 [a,b] e κ[µ(F )−x] dF (x) − [a,b] e κ[µ(G)−x] dG(x) .(58) Since the term in absolute values in the previous display is not greater than e κµ(F ) − e κµ(G) + [a,b] e −κx dF (x) − [a,b] e −κx dG(x)(59) which can be bounded from above by κe κb (b − a) F − G ∞ + [e −κa − e −κb ] F − G ∞ ,(60) where we used Example C.3 and Lemma C.2 Proof of Lemma A.8: Obviously, the welfare function W is well defined on D cdf ([a, b]) in both parts of the lemma. The first statement in the present lemma follows from the assumptions and Example C.3, noting that x 1 x 2 − y 1 y 2 = (x 1 − y 1 )x 2 − y 1 (y 2 − x 2 ) holds for real numbers x i , y i , i = 1, 2. The second statement follows directly from the assumptions and Example C.3. Thus H z is well defined on D cdf ([a, b]) as well. Finally, given F ∈ D and G ∈ D cdf ([a, b]), note that by definition and the triangle inequality: |H z (F ) − H z (G)| ≤ |F (z(F )) − F (z(G))| + F − G ∞ ≤ (C z s + 1) F − G ∞ ,(61) where we used that z satisfies Assumption 2.1 together with a mean-value theorem as in Minassian (2007) for the last inequality. Proof of Lemma A.10: Obviously, P SK (.; z, κ) is well defined on D cdf ([a, b]) because z ≥ z * > 0 holds by assumption, and due to our convention that 0/0 := 0 (noting also that F (x) = 0 for every x ∈ [0, z(F )] in case F (z(F )) = 0). Next, fix F ∈ D and G ∈ D cdf ([a, b]). Define for all x ∈ R f (x) := max(1 − [x/z(F )], 0)[1 − [F (x)/F (z(F ))]] κ , and analogously g(x) := max(1 − [x/z(G)], 0)[1 − [G(x)/G(z(G))]] κ . Define m := min(z(F ), z(G)) and M := max(z(F ), z(G)), and the following partition of [a, M] (using our convention 0/0 := 0): A := x ∈ [a, m] : F (x) F (z(F )) > G(x) G(z(G)) , where C = ∅ in case m = M. Next, write P SK (F ; z, κ) − P SK (G; z, κ) κ + 1 = [a,M ] [f (x) − g(x)]dF (x) + [a,b] g(x)F (x) − [a,b] g(x)G(x) , noting that f and g vanish for x > M; and denote the right-hand side by S 1 + S 2 , S 2 denoting the term in brackets to the far right. Since g([a, b]) ⊆ [0, 1] and because g is right-continuous (G is a cdf) and non-increasing, it hence follows from Lemma C.7 that |S 2 | ≤ F − G ∞ . Concerning S 1 , note that for every x ∈ [a, M] it holds that |f (x) − g(x)| is not greater than the sum of | max([1 − x z(F ) ], 0) − max([1 − x z(G) ], 0)| ≤ x|z(F ) −1 + z(G) −1 | ≤ xz −2 * |z(F ) + z(G)| ≤ bz −2 * C z F − G ∞ ,(62) (where we used that z ≥ z 0 to obtain the second inequality, and that z satisfies Assumption 2.1 to obtain the third) and |1 − [F (x)/F (z(F ))]| κ − |1 − [G(x)/G(z(G))]| κ ≤ κ|[F (x)/F (z(F ))] − [G(x)/G(z(G))]|(63) (where we used κ ≥ 1, the reverse triangle inequality, and Lemma A.9 to obtain the upper bound). It hence follows that |S 1 | is bounded from above by the sum of bz −2 * C z F − G ∞ and κ times A [F (x)/F (z(F ))] − [G(x)/G(z(G))]dF (x) + B [G(x)/G(z(G))] − [F (x)/F (z(F ))]dF (x) + C h(x)dF (x),(64) where for every x ∈ C we define h(x) :=        |F (x)/F (z(F ))| if m < M = z(F ) |G(x)/G(z(G))| if m < M = z(G) 0 if m = M.(65) Using 0 ≤ h ≤ 1, the mean value theorem in Minassian (2007), and that z satisfies Assumption 2.1, we see that the third integral in Equation (64) satisfies C h(x)dF (x) ≤ F (M) − F (m) ≤ sC z F − G ∞ .(66) To bound the expression in Equation (64), we also recall that Lemma A.9 shows that G(z(G)) − (C z + 1) F − G ∞ ≤ F (z(F )) ≤ G(z(G)) + (C z + 1) F − G ∞ .(67) We consider different cases: Consider first the case where F (z(F )) = 0: Then, the convention 0/0 := 0 implies A = ∅ and B = [a, m]. Furthermore, the integral over B in Equation (64) vanishes in this case, because m ≤ z(F ) implies F (m) = 0. By Equation (66), in this case it holds that the expression in Equation (64) does not exceed sC z F − G ∞ . Next, consider the case where G(z(G)) = 0 and F (z(F )) > 0. It follows from our convention that then A = {x ∈ [a, m] : F (x)/F (z(F )) > 0}, and that the integral over B in (64) vanishes. The integral over A is not greater than F (m) = F (m) − G(z(G)) ≤ F (z(F )) − G(z(G)) ≤ (C z + 1) F − G ∞ , where we used Equation (67) to obtain the last inequality. Together with Equation (66) we thus see that in this case the expression in Equation (64) does not exceed (C z (1 + s) + 1) F − G ∞ . Finally, consider the case where G(z(G)) and F (z(F )) are both positive. Then, we can write the integral over A in Equation (64) as F (z(F )) −1 A F (x) − F (z(F )) G(x) G(z(G)) dF (x) ≤ F (z(F )) −1 A [F (x) − G(x)] + (C z + 1) F − G ∞ G(x) G(z(G)) dF (x) ≤ F − G ∞ (C z + 2), where we used Equation (67) to obtain the first inequality. Similarly, the integral over B in Equation (64) can be shown not to be greater than F − G ∞ (C z + 2). Summarizing, in this last case the expression in Equation (64) does not exceed [(2 + s)C z + 4] F − G ∞ . In particular, this bound is bigger than the two bounds in the other two cases. Hence, we conclude that the expression in Equation (64) is not greater than [(2 + s)C z + 4] F − G ∞ in general. It follows that |S 1 | is bounded from above by [bz −2 * C z + κ[(2 + s)C z + 4]] F − G ∞ .(68) Since |S 2 | ≤ F − G ∞ , it follows that P SK (F ; z, κ) − P SK (G; z, κ) κ + 1 ≤ |S 1 | + |S 2 | ≤ [1 + bz −2 * C z + κ[(2 + s)C z + 4]] F − G ∞ . Proof of Lemma A.11: Obviously, P F GT (.; z, Λ) is well defined on D cdf ([a, b]) because z ≥ z * > 0 holds by assumption. Next, fix F ∈ D and G ∈ D cdf ([a, b]). Note that we can write P F GT (F ; z, Λ) = [a,b] Λ(max(1 − [x/z(F )], 0))dF (x).(69) Abbreviating f (x) = Λ(max(1 − [x/z(F )], 0)) and g(x) = Λ(max(1 − [x/z(G)], 0)) we obtain P F GT (F ; z, Λ) − P F GT (G; z, Λ) = [a,b] [f (x) − g(x)]dF (x) + [a,b] g(x)dF (x) − [a,b] g(x)dG(x) . Denote the first integral on the right by A, and the term in brackets to the far right by B. Noting that g : [a, b] → [0, 1] is continuous (Λ is the restriction of a convex function) and non-increasing, we obtain from Lemma C.7 that |B| ≤ F − G ∞ . Concerning A, we note that since Λ is the restriction of a convex function defined on an open interval containing [0, 1], Λ is Lipschitz continuous. Hence the constant C Λ as claimed in the statement of the lemma indeed exists. In particular, using this property and the inequality | max(1 − z 1 , 0) − max(1 − z 2 , 0)| ≤ |z 1 − z 2 | for nonnegative real numbers z 1 , z 2 , we can bound |A| ≤ C Λ b|[1/z(F )] − [1/z(G)]| ≤ C Λ bz −2 * |z(F ) − z(G)| ≤ C Λ bz −2 * C z F − G ∞ ,(70) where we used Lipschitz continuity of the map x → x −1 on [z * , ∞) (with constant z −2 * ), and the assumption that z satisfies Assumption 2.1 for obtaining the second inequality. Together with the upper bound on |B| we obtain the claimed statement. C General results for establishing Assumption 2.1 We here summarize in a self-contained way some results that turn out to be useful for establishing Assumption 2.1 for empirically relevant functionals T. Specific examples were discussed in Appendix A and include inequality measures (cf. Appendix A.1), welfare measures (cf. Appendix A.2), and poverty measures (cf. Appendix A.3). The techniques we describe are based on decomposability-properties of the functional, its specific structural (e.g., linearity) properties, and on properties of quantiles and quantile functions, or related quantities such as Lorenz curves. We emphasize that all of the results in the present section are fairly elementary, but are difficult to pinpoint in the literature in the form needed. We first start with a short section concerning notation. C.1 Notation We denote by D(R) the Banach space of real-valued bounded càdlàg functions equipped with the supremum norm G ∞ = sup{|G(x)| : x ∈ R}. The closed convex subset of D(R) consisting of all cumulative distribution functions (cdfs) shall be denoted by D cdf (R). Furthermore, given two real numbers a < b, we define the subset D cdf ((a, b]) of D cdf (R) as follows: Here F (a−) denotes the left-sided limit of F at a. Given a cdf F we denote by µ F the (uniquely defined) probability measure on the Borel sets of R that satisfies µ F ((−∞, x]) = F (x) for every x ∈ R, and, as usual, we denote the Lebesgue-Stieltjes integral of a µ F -integrable function f : R → R by R f (x)dF (x) := R f (x)dµ F (x). In the following subsections we shall repeatedly encounter functionals T with domain T ⊆ D cdf (R) and co-domain R, which are Lipschitz continuous (T being equipped with the metric induced by the supremum norm on D(R)): recall that a functional T : T → R is called Lipschitz continuous if there exists a nonnegative real number C so that for every F and G ∈ T it holds that |T(F ) − T(G)| ≤ C F − G ∞ .(71) We then call C a Lipschitz constant of T. When we say that a functional T is Lipschitz continuous with constant C, we do not imply that this is the smallest such constant. Recall from Remark 2.2 that if a functional T is Lipschitz continuous on T = D cdf ([a, b]) for real numbers a < b, then T satisfies Assumption 2.1 with D = D cdf ([a, b]). C.2 Decomposability Oftentimes functionals can be decomposed into a function of several "simpler" functionals. It is a basic, but useful, fact that if a functional can be written as a composition of functionals that satisfy Assumption 2.1 with a Lipschitz continuous function on the intermediating Euclidean space, this composition satisfies Assumption 2.1 as well. The result is as follows, its proof is trivial and omitted. C.3 U-functionals We here consider U-functionals (the corresponding sample plug-in variants being traditionally referred to as U-statistics, hence the name). The following result covers examples such as moments, certain concentration measures or dependence measures, cf. Chapter 5 in Serfling (2009), and see also the subsequent discussion for examples. Lemma C.2. Let a < b be real numbers and let ϕ : [a, b] k → R for some k ∈ N. Suppose that ϕ is bounded, and is symmetric in the sense that for all x i ∈ [a, b] for 1, . . . , k it holds that ϕ(x 1 , . . . , x k ) = ϕ(x π 1 , . . . , x π k ) for every permutation x π 1 , . . . , x π k of x 1 , . . . . [c,d] ϕ(x 1 , . . . , x k )dF (x 1 ) . . . dF (x k ), where we write m ϕ in case c = a and d = b. Then, m ϕ;c,d is Lipschitz continuous on D cdf ([a, b]) with constant kC * , where C * =            C if a = c, b = d C + m * if b = d C + M * if a = c C + m * + M * else,(73) and where m * := sup{|ϕ(c, x * 2 , . . . , x * k )| : x * 2 , . . . , x * k ∈ [c, d] k−1 } (74) M * := sup{|ϕ(d, x * 2 , . . . , x * k )| : x * 2 , . . . , x * k ∈ [c, d] k−1 }.(75) Proof. Note first that m ϕ;c,d (F ) is well defined (i.e., ϕ is integrable w.r.t. the k-fold product mea- sure k i=1 µ F ) on D cdf ([a, b] ) because ϕ is bounded. Next, we reduce the statement to the case k = 1: let F and G be elements of D cdf ([a, b]), let µ be a measure that dominates both µ F and µ G , let f and g denote µ-densities of µ F and µ G , respectively, w.r.t. µ. Then, by Fubini's theorem, m ϕ;c,d (F ) = [c,d] . . . [c,d] ϕ(x 1 , . . . , x k ) k j=1 f (x j )dµ(x 1 ) . . . dµ(x k ),(76) and an analogous expression (replacing the density f by the density g) corresponds to m ϕ;c,d (G). Recall also that for arbitrary real numbers a j , b j for j = 1, . . . , k we may write k j=1 a j − k j=1 b j = k j=1      j−1 i=1 a i   (a j − b j ) k i=j+1 b i    ,(77) where empty products are to be interpreted as 1. Equipped with (77), using Equation (76) . . . [c,d] ϕ (x 1 , . . . , x k )[f (x j ) − g(x j )]dµ(x j )dF (x 1 ) . . . dF (x j−1 )dµ(x 1 ) . . . dG(x j−1 ) . . . dG(x k ). Using the triangle inequality to upper bound |m ϕ;c,d (F ) − m ϕ;c,d (G)|, an application of the symmetry condition shows that it suffices to verify that for x * 2 , . . . , x * k in [c, d] k−1 arbitrary [c,d] ϕ(x, x * 2 . . . , x * k )dF (x) − [c,d] ϕ(x, x * 2 . . . , x * k )dG(x) ≤ C * F − G ∞ .(78) Let f : R → R be a continuous function (possibly depending on x * 2 , . . . , x * k ) such that f (x) = ϕ(x, x * 2 . . . , x * k ) holds for every x ∈ [c, d], and such that f (x) → 0 as x → −∞. An application of the integration-by-parts formula (as in, e.g., Exercise 34.b on p.108 in Folland (2013)) gives [c,d] ϕ (x, x * 2 . . . , x * k )dF (x) = [c,d] f (x)dF (x) = f (d)F (d) − f (c−)F (c−) − [c,d] F (x)dµ f (x),(79) an analogous statement holding for F replaced by G. Hence, the quantity to the left in the inequality in (78) is seen to be not greater than xdF (x). |f (d)||F (d) − G(d)| + |f (c)||F (c−) − G(c−)| + [c,d] F (x) − G(x)dµ f (x) .(80)Noting that |f (d)| ≤ M * , that |f (c)| ≤ m * , that |F (d) − G(d)| = 0 if d = b, that |F (c−) − G(c−)| = 0 if a = c, and furthermore noting that |F (d) − G(d)| ≤ F − G ∞ and |F (c−) − G(c−)| ≤ F − G ∞always Note that ϕ is bounded on [a, b], is trivially symmetric, and ϕ satisfies the continuity condition in Lemma C.2. Furthermore, the total variation of ϕ is x p dF (x). Note that ϕ is bounded on [a, b], is trivially symmetric, and ϕ satisfies the continuity condition in Lemma C.2. Furthermore, the total variation of ϕ is [0,b] |ϕ ′ (x)|dx = ϕ(b) = b p . As a consequence of Lemma C.2 the functional m ϕ is thus Lipschitz continuous on D cdf ([0, b]) with constant b p . Example C.5 (Variance). Let a < b be real numbers. Let k = 2 and set ϕ(x 1 , x 2 ) = .5(x 1 − x 2 ) 2 , i.e., we consider the variance F → .5 [a,b] [a,b] (x 1 − x 2 ) 2 dF (x 1 )dF (x 2 ) = [a,b] x 1 − [a,b] x 2 dF (x 2 ) 2 dF (x 1 ).(83) Note that ϕ is bounded on [a, b] 2 , is symmetric, and ϕ satisfies the continuity condition in Lemma C.2. For every x 2 ∈ [a, b] the total variation of x → .5(x−x 2 ) 2 is [a,b] |x−x 2 |dx ≤ 2 max(a 2 , b 2 )+.5(b 2 −a 2 ). It follows from Lemma C.2 that the variance functional is Lipschitz continuous with constant 2 max(a 2 , b 2 )+ .5(b 2 − a 2 ). Example C.6 (Gini-mean difference). Let a < b be real numbers, and let ϕ(x 1 , x 2 ) = |x 1 − x 2 |. This corresponds to the functional F → [a,b] [a,b] |x 1 − x 2 |dF (x 1 )dF (x 2 ),(84) which constitutes the numerator of the Gini-index and is sometimes called the Gini-mean difference. Clearly, ϕ is bounded and symmetric, and satisfies the continuity condition in Lemma C.2. Furthermore, for every x 2 ∈ [a, b] it holds that the total variation of x → |x − x 2 | equals (b − a). It follows from Lemma C.2 that m ϕ is Lipschitz continuous on D cdf ([a, b]) with constant 2(b − a). The following lemma is sometimes useful, because it avoids the continuity condition of the integrand in Lemma C.2 by working with a monotonicity condition. Lemma C.7. Let a < b be real numbers and let ϕ : [a, b] → R be right-continuous, and be non-decreasing or non-increasing. Then, the functional F → [a,b] ϕ(x)dF (x) (85) is Lipschitz continuous on D cdf ([a, b]) with constant |ϕ(b) − ϕ(a)|. Proof. Note first that the functional under consideration is well defined on D cdf ([a, b]); and that we only need to consider the case where ϕ is non-decreasing. To this end let F, G ∈ D cdf ([a, b]) and note that, by the transformation theorem, we have [a,b] ϕ (x)dF (x) − [a,b] ϕ(x)dG(x) = [ϕ(a),ϕ(b)] xdF ϕ (x) − [ϕ(a),ϕ(b)] xdG ϕ (x),(86) where F ϕ ∈ D cdf ([ϕ(a), ϕ(b)]) denotes the cdf corresponding to the image measure µ F • ϕ, and G ϕ ∈ D cdf ([ϕ(a), ϕ(b)]) is defined analogously. An application of Example C.3 thus shows that [a,b] ϕ (x)dF (x) − [a,b] ϕ(x)dG(x) ≤ [ϕ(b) − ϕ(a)] F ϕ − G ϕ ∞ .(87) It remains to observe that F ϕ − G ϕ ∞ ≤ F − G ∞ , by Lemma C.8. Lemma C.8. Let F and G be cdfs, and let ϕ : R → R be right-continuous, and be non-decreasing or non-increasing. Then, F ϕ − G ϕ ∞ ≤ F − G ∞ holds, where F ϕ denotes the cdf corresponding to the image measure µ F • ϕ, and G ϕ is defined analogously. Proof. First of all, note that F ϕ −G ϕ ∞ = sup z∈C(F,G) |F ϕ (z) −G ϕ (z)|, where C(F, G) ⊆ R is defined as the subset of points at which both F ϕ and G ϕ are continuous. Next, let z ∈ C(F, G) and define ϕ − (x) := inf{z ∈ R : ϕ(z) ≥ x}, i.e., a generalized inverse of ϕ. Part (5) of Proposition 1 in Embrechts and Hofert (2013) shows that for every z ∈ R we have A(z) := {x ∈ R : ϕ(x) < z} = {x ∈ R : x < ϕ − (z)}.(88) Using this expression for A(z), we can for every z ∈ C(F, G) rewrite |F ϕ (z) − G ϕ (z)| as |µ Fϕ ((−∞, z)) − µ Gϕ ((−∞, z))| = |µ F (A(z)) − µ G (A(z))| = |µ F ({x ∈ R : x < ϕ − (z)}) − µ G ({x ∈ R : x < ϕ − (z)})|. On the one hand, the expression to the far right in the previous display equals 0 ≤ F − G ∞ in case ϕ − (z) ∈ {−∞, +∞}. On the other hand, if ϕ − (z) ∈ R, the same expression is seen to equal |F (ϕ − (z)−) − G(ϕ − (z)−)| ≤ F − G ∞ . Since this argument goes through for every z ∈ C(F, G), we are done. C.4 Quantiles, quantile functions, L-functionals, Lorenz curve, and truncation In the present subsection we provide some results concerning Lipschitz continuity of quantiles-based functionals. For α ∈ [0, 1] we define the α-quantile of a cdf F as usual via q α (F ) = inf{x ∈ R : F (x) ≥ α}. Note that for α = 0 we have q α (F ) = −∞, and that (by monotonicity) the quantile function α → q α (F ) is B([0, 1]) − B(R) measurable. The first result is as follows: Lemma C.9. Let α ∈ (0, 1] and let F ∈ D cdf ([a, b]) for real numbers a < b. Suppose F (q α (F )) = α and that there exists a positive real number r so that F (q α (F ) − x) − α ≤ −rx if x > 0 and q α (F ) − x ≥ a. F (q α (F ) + x) − α ≥ rx if x > 0 and q α (F ) + x ≤ b.(89) Then, for every G ∈ D cdf ([a, b]) it holds that |q α (F ) − q α (G)| ≤ r −1 F − G ∞ . Consequently, denoting by D the set of all cdfs that satisfy the conditions imposed on F above, it follows that q α satisfies Assumption 2.1 with a, b, D and constant C = r −1 . Proof. Let G be an element of D cdf ([a, b]). The claim is trivial if F = G. Thus, we assume that F = G. Note that F (x) = 0 ≤ α for every x < a, and F (x) = 1 > α for every x ≥ b implies q α (F ) ∈ [a, b]; and that, by the same reasoning, q α (G) ∈ [a, b]. Now, on the one hand, if q α (F ) − r −1 G − F ∞ < a, then q α (G) ≥ q α (F ) − r −1 G − F ∞ . If, on the other hand, q α (F ) − r −1 G − F ∞ ≥ a, then from the first line in (89) with x = r −1 G − F ∞ one obtains α ≥ F (q α (F ) − r −1 G − F ∞ ) + G − F ∞ , thus α ≥ G(q α (F ) − r −1 G − F ∞ ) and hence, again, q α (G) ≥ q α (F ) − r −1 G − F ∞ . Similarly, on the one hand, if q α (F ) + r −1 G − F ∞ > b, then q α (G) ≤ q α (F ) + r −1 G − F ∞ . If, on the other hand q α (F ) + r −1 G − F ∞ ≤ b, then the second line in (89) with x = r −1 G − F ∞ shows that F (q α (F ) + r −1 G − F ∞ ) − G − F ∞ ≥ α, thus G(q α (F ) + r −1 G − F ∞ ) ≥ α, and hence, again, q α (G) ≤ q α (F ) + r −1 G − F ∞ . Summarizing yields |q α (F ) − q α (G)| ≤ r −1 F − G ∞ . The last statement is trivial. Example C.10 (Median). The median of a distribution F is defined as its α = 1/2 quantile q 1/2 (F ). Let a < b and r > 0 be real numbers, and denote by D the set of cdfs F so that F (q 1/2 (F )) = 1/2, and so that Equation (89) is satisfied for α = 1/2 (Lemma C.12 provides a sufficient condition for F ∈ D). Then, the functional F → q 1/2 (F ) satisfies Assumption 2.1 with a, b and D with constant C = r −1 . The second result is auxiliary, and concerns not a single quantile, but the Lipschitz continuity of the quantile function F → q . (F ) on certain subsets of [0, 1]. Lemma C.11. Let F ∈ D cdf ([a, b]) for real numbers a < b, and let α * < α * for α * and α * in (0, 1]. Suppose F (q α (F )) = α holds for every α ∈ [α * , α * ], and that there exists a positive real number r so that Equation (89) is satisfied for every α ∈ [α * , α * ]. Then, for every G ∈ D cdf ([a, b]) it holds that sup α∈[α * ,α * ] |q α (F ) − q α (G)| ≤ r −1 F − G ∞ . Proof. The statement follows immediately from Lemma C.9. A simple sufficient condition for the assumption on F in Lemma C.11 (and hence also for the assumption on F in Lemma C.9) is that F admits a density that is bounded from below (on the support of F ): Lemma C.12. Let a < b be real numbers and let F ∈ D cdf ([a, b]). Suppose F is continuous, and is right-sided differentiable on (a, b) with right-sided derivative F + , which furthermore satisfies F + (x) ≥ r for every x ∈ (a, b) for some r > 0. Then, F (q α (F )) = α and Equation (89) holds for every α ∈ (0, 1]. Proof. The condition F + (x) ≥ r for every x ∈ (a, b) for a r > 0 implies that F is strictly increasing on [a, b], which (together with continuity of F ) implies F (q α (F )) = α for every α ∈ (0, 1]. The second claim follows from the mean-value theorem for right-differentiable functions in Minassian (2007) T(F ) = [0,1] q α (F )J(α)dν(α) + d j=1 v i q p i (F ),(90) the sum to the right being interpreted as 0 if d = 0. Let F ∈ D cdf ([a, b]) satisfy F (q α (F )) = α for every α ∈ (0, 1], and suppose there is a positive real number r so that Equation (89) holds for every α ∈ (0, 1]. Then, for every G ∈ D cdf ([a, b]) it holds that |T(F ) − T(G)| ≤ r −1   c + d i=1 |v i |   F − G ∞ . Consequently, denoting by D the set of all cdfs that satisfy the conditions imposed on F above, it follows that T defined in Equation (90) satisfies Assumption 2.1 with a, b, D and constant C = r −1 [c+ d i=1 |v i |]. Proof. We first verify that T(F ) is well defined for every F ∈ D cdf ([a, b] g * (α) := g(α) if α ∈ (0, 1] 0 if α = 0.(91) Note further that the function α → q α (F ) is well defined on (0, 1] and its range is contained in [a, b] (cf. the argument in the beginning of the proof of Lemma C.9). It thus follows that |g * (α)| ≤ max(|a|, |b|)|J(α)|, and the integrability condition on J shows that g * (and thus g) is integrable. Now, let F and G be as in the statement of the lemma. Consider |T(F ) − T(G)|. Clearly, by the triangle inequality and Lemma C.11, it suffices to verify the statement for the case d = 0. Then, |T(F ) − T(G)| is not greater than (0,1] |q α (F ) − q α (G)||J(α)|dν(α).(92) Note that the function α → |q α (F )−q α (G)| is bounded on (0, 1]. By the monotonic convergence theorem for ε ց 0 the integral [ε,1] |q α (F )−q α (G)||J(α)|dν(α) converges to the integral in (92). But [ε,1] |q α (F )− q α (G)||J(α)|dν(α) ≤ r −1 c F − G ∞ by Lemma C.11. The last statement is trivial. One particularly important application concerns the so-called Lorenz curve associated with a cdf F which is defined (cf. Gastwirth (1971)) below in Equation (94). Let F ∈ D cdf ([a, b]) satisfy F (q α (F )) = α for every α ∈ (0, 1], and suppose there is a positive real number r so that Equation (89) holds for every α ∈ (0, 1]. Then, for every G ∈ D cdf ([a, b]) it holds that |Q(F, u) − Q(G, u)| ≤ r −1 u F − G ∞ ≤ r −1 F − G ∞ . Consequently, denoting by D the set of all cdfs that satisfy the conditions imposed on F above, it follows that T = Q(., u) satisfies Assumption 2.1 with a, b, D and constant C = r −1 u. Furthermore, if a > 0, then |L(F, u) − L(G, u)| ≤ a −1 (r −1 u + (b − a)a −1 ) F − G ∞ , and it follows that T = L(., u) satisfies Assumption 2.1 with a, b, D and constant C = a −1 (r −1 + (b − a)a −1 ). Proof. For the first claim, we just apply Lemma C.13 with ν equal to Lebesgue measure, J = 1 [0,u] , which satisfies the integrability condition with c = u ≤ 1. For the second claim, note that L(., u) is well defined on D cdf ([a, b]) because a > 0. Next, observe that for F and G as in the statement of the lemma we can bound |L(F, u) − L(G, u)| from above by µ(F ) −1 Q(F, u) − Q(G, u) + |1 − µ(F )/µ(G)| [0,u] q α (G)dα .(95) Since µ(F ) ≥ a, since q α (G) ≤ b for α ∈ (0, u], and because we already know that Q(F, u) − Q(G, u) ≤ r −1 u F − G ∞ , it remains to observe that |1 − (µ(F )/µ(G))| ≤ (b − a) F − G ∞ /µ(G) ≤ (b − a)a −1 F − G ∞(96) to conclude that the expression in (95) is not greater than a −1 r −1 u + (b − a)a −1 F − G ∞ . The final result in this section concerns trimmed generalized-mean functionals. We consider onesidedly trimmed functionals, the trimming affecting the lower or upper tail. Two-sided trimming can be dealt with similarly. We abstain from spelling out the details. ϕ(x)dF (x). Let F ∈ D cdf ([a, b]), assume that F is continuous, and right-sided differentiable on (a, b), with right-sided derivative F + satisfying r ≤ F + (x) ≤ κ for every x ∈ (a, b), and for positive real numbers κ and r. Then, for every G ∈ D cdf ([a, b]) it holds that |m t− ϕ;α (F ) − m t− ϕ;α (G)| ≤ [C + u(1 + κr −1 )] F − G ∞ , and |m t+ ϕ;α (F ) − m t+ ϕ;α (G)| ≤ [C + u(1 + κr −1 )] F − G ∞ ,(98) Consequently, denoting by D the set of all cdfs that satisfy the conditions imposed on F above, it follows that m t− ϕ;α and m t+ ϕ;α satisfy Assumption 2.1 with a, b, D and constant C + u(1 + κr −1 ). Proof. We only provide an argument for the first claimed inequality, the second is obtained analogously. Furthermore, throughout the proof we write m t ϕ;α instead of m t− ϕ;α . First, note that the functional m t ϕ;α (F ) is indeed well defined for every F ∈ D cdf ([a, b]). This follows from q α (F ) ∈ [a, b], and since ϕ is bounded on [a, b]. Next, let F be as in the statement of the lemma and satisfy the conditions imposed, and let G ∈ D cdf ([a, b]). Note first that q α (F ), q α (G) ∈ [a, b] (cf. the argument in the beginning of the proof of Lemma C.9). By the triangle inequality, |m t ϕ;α (F ) − m t ϕ;α (G)| ≤ A + B, where (using the notation introduced in Equation (72)) A := m ϕ;a,qα(G) (F ) − m ϕ;a,qα(G) (G) ≤ (C + u) F − G ∞ , the upper bound following from Lemma C.2, and B := g(x)|ϕ(x)|dF (x) ≤ u g(x)dF (x),(99) where g(x) = 1 [a,qα(F )] (x) − 1 [a,qα(G)] (x) . By continuity of F : g(x)dF (x) ≤ |F (q α (G)) − α|.(100) which, by the assumed behavior of the right-derivative of F and a mean-value theorem for rightdifferentiable functions (for example the one by Minassian (2007)), is not greater than κ|q α (G) − q α (F )| ≤ κr −1 F − G ∞(101) the last inequality following from Lemma C.9 together with Lemma C.12. This proves the claim. The last statement is trivial. D Proofs of results in Sections 3, 4 and 5 Throughout the appendix, KL(·, ·) denotes the Kullback-Leibler (KL) divergence between two probability measures (on a Borel σ-algebra clear from the context) or, if applicable, a version of their densities. D.1 Proofs of results in Section 3 D.1.1 Proof of Theorem 3.1 To establish Equation (5) we need to show that for every K-tuple (F 1 , . . . , F K ) with F i ∈ D, i = 1, . . . , K, we have E(R N (π)) ≤ c Knlog(n). Note that this inequality trivially holds in case T(F 1 ) = . . . = T(F K ). In particular, if T(F 1 ) = . . . = T(F K ) holds for every F 1 , . . . , F K such that F i ∈ D for i = 1, . . . , K there is nothing to prove. We thus assume without loss of generality that the K-tuple F 1 , . . . , F K is such that T(F i ) is not constant in i ∈ {1, . . . , K}; also implying that the constant C from Assumption 2.1 must satisfy C > 0. We now claim that for every i with ∆ i > 0 we have E[S i (N)] ≤ 2C 2 β log(n) ∆ 2 i + β + 2 β − 2 ,(102) where we recall that n = E(N). Before proving this claim, note that Assumption 2.1 implies ∆ i ≤ C, and that Equation (2) shows that E[R N (π)] = i:∆ i >0 ∆ i E[S i (N)],(103) which together with the claim in Equation (102) yield E[R N (π)] = i:∆ i >0 ∆ 2 i E[S i (N)] E[S i (N)] ≤ 2C 2 β log(n) + C 2 (β + 2)/(β − 2) i:∆ i >0 E[S i (N)] ≤ 2C 2 β log(n) + C 2 (β + 2)/(β − 2) √ Kn, where the last line follows from the Cauchy-Schwarz inequality and i: ∆ i >0 E[S i (N)] ≤ E[N] = n. Upon choosing c = C 2β + (β + 2)/(β − 2) we would thus obtain Equation (5). Therefore, it remains to prove the statement in Equation (102). Before doing that, we note for later use that Equations (102) and (103) also give the regret bound E[R N (π)] ≤ i:∆ i >0 2C 2 β log(n) ∆ i + β + 2 β − 2 ∆ i .(104) Now, to prove Equation (102), note that by Tonelli's theorem E(S i (N)) = E ∞ t=1 1 {N =t} S i (N) = ∞ t=1 E 1 {N =t} S i (t) ,(105) where we used that P(N ∈ N) = 1, a consequence of E(N) = n ∈ N. Denote the σ-algebra generated by Y 1 , . . . , Y t by A t . By assumption, σ(N) and A t are independent for every t ∈ N. Note furthermore that S i (t) is A t measurable for every t ∈ N. Hence, for every t ∈ N we have E 1 {N =t} S i (t) = E S i (t)E 1 {N =t} |A t = P (N = t) E S i (t) ,(106) from which it follows that E(S i (N)) = ∞ t=1 P (N = t) E S i (t) .(107) If Equation (102) were already known to be true for any N that coincides with its expectation with probability one, we could apply this to any N ≡ t (implying that the corresponding expectation equals t), which together with the previous display would deliver that E(S i (N)) ≤ 2C 2 β ∞ t=1 P (N = t) log(t) ∆ 2 i + β + 2 β − 2 ≤ 2C 2 log(n) ∆ 2 i + β + 2 β − 2 ,(108) where we used Jensen's inequality to obtain the last inequality. Therefore, it remains to establish Equation (102) for N such that P(N = E(N)) = 1. Let N satisfy P(N = E(N)) = 1. Note that without loss of generality we can assume that N = n holds everywhere. If n ≤ K, we have S i (n) ≤ 1 and hence (102) is obviously satisfied. From now on, we therefore let n ≥ K + 1. Furthermore, we fix an i such that ∆ i > 0. Now, for every t ∈ {K + 1, . . . , n}, we note that {π t = i} ⊆ A t ∪ B i,t ∪ C i,t , where A t := T(F i * ,t−1 ) + C β log(t)/2S i * (t − 1) ≤ T(F i * ) , B i,t := T(F i,t−1 ) > T(F i ) + C β log(t)/2S i (t − 1) , C i,t := S i (t − 1) < 2βC 2 log(n) ∆ 2 i , and where we define i * as the smallest element of arg max i=1,...,K T(F i ). Indeed, on the complement of A t ∪ B i,t ∪ C i,t we have T(F i * ,t−1 ) + C β log(t)/2S i * (t − 1) > T(F i * ) ≥ T(F i ) + 2C β log(n)/2S i (t − 1) ≥ T(F i ) + 2C β log(t)/2S i (t − 1) ≥ T(F i,t−1 ) + C β log(t)/2S i (t − 1), implyingπ t = i * , which contradictsπ t = i, because i = i * as ∆ i > 0. Using {π t = i} ⊆ A t ∪ B i,t ∪ C i,t and setting u := 2C 2 β log(n) ∆ 2 i , we now obtain (recall that n ≥ K + 1) that S i (n) = K t=1 1 {πt=i} + n t=K+1 1 {πt=i} = 1 + n t=K+1 1 {πt=i} = 1 + n t=K+1 1 {πt=i}∩C i,t + n t=K+1 1 {πt=i}∩C c i,t ≤ u + n t=K+1 1 At∪B i,t , where we also used 1 + n t=K+1 1 {πt=i}∩C i,t ≤ u. From the upper bound in the previous display we get E[S i (n)] ≤ u + n t=K+1 P(A t ) + P(B i,t ). We will show further below that: P(A t ) ≤ t s=1 P T(F i * ,s ) + C β log(t)/2s ≤ T(F i * )) P(B i,t ) ≤ t s=1 P T(F i,s ) > T(F i ) + C β log(t)/2s),(109) where for every s ∈ {1, . . . , t} and every l ∈ {i, i * } we define F l,s := s −1 s j=1 1 {Y l,j ≤.} . From Equation (109), Assumption 2.1 and the Dvoretzky-Kiefer-Wolfowitz (DKW) inequality in the form established in Corollary 1 in Massart (1990) (note that Equation (1.5) in Massart (1990) obviously remains valid if ">" is replaced by "≥") we then obtain P(A t ) ≤ t s=1 P(||F i * ,s − F i * || ∞ ≥ β log(t)/2s) ≤ 2 t s=1 1 t β = 2 t β−1 P(B i,t ) ≤ t s=1 P(||F i,s − F i || ∞ > β log(t)/2s) ≤ 2 t s=1 1 t β = 2 t β−1 . The identity n t=K+1 1 t β−1 ≤ ∞ K 1 x β−1 dx = 1 (β − 2)K β−2 ≤ 1 β − 2(110) combined with u ≤ 1 + 2C 2 β log(n)/∆ 2 i now establishes (102). It remains to verify the two inequalities in Equation (109). To this end we need some more notation: For every l ∈ {1, . . . , K}, every r ∈ N and every ω ∈ Ω let t l,r (ω) := inf{s ∈ N : s j=1 1 {π j (Z j−1 (ω))=l} = r}.(111) Lemma D.1. For every l ∈ {1, . . . , K}, every r ∈ N and every ω ∈ Ω it holds that t l,r (ω) ∈ N. Proof. Suppose there exists a triple l, r, ω such that s j=1 1 {π j (Zt(ω))=l} < r holds for every s ∈ N, implying in particular that 1 ≤ ∞ j=1 1 {π j (Z j−1 (ω))=l} =: κ(ω) < r,(112) where we used t l,1 = l. Let t ≥ K + 1. From the definition ofπ it follows that π t (Z t−1 (ω)) ∈ arg max j∈I T(F j,t−1 (.)(ω)) + C β log(t)/2S j (t − 1)(ω) ,(113) where we now evaluate all random variables at ω which we emphasize in the preceding display. In particular, T(F j,t−1 (.)(ω)) and S j (t − 1)(ω) are sequences of real numbers. For convenience, we shall writeF j,t−1 (ω) instead ofF j,t−1 (.)(ω) in what follows. From Equation (112) and the previous display it follows that eventually T(Fπ t(Zt−1(ω)),t−1 (ω)) + C β log(t)/2Sπ t(Zt−1(ω)) (t − 1)(ω) ≥ T(F l,t−1 (ω)) + C β log(t)/2S l (t − 1)(ω),(114) which is equivalent to (recall that C > 0 from the discussion in the first paragraph of the present subsection) a t T(Fπ t(Zt−1(ω)),t−1 (ω)) − T(F l,t−1 (ω)) ≥ [S l (t − 1)(ω)] −1/2 − [Sπ t(Zt−1(ω)) (t − 1)(ω)] −1/2 , (115) where a t := [C β log(t)/2] −1 → 0 as t → ∞. Consequently, the sequence on the left hand side of the previous inequality converges to 0 as t → ∞. To see this, let F ∈ D and note that |T(Fπ t(Zt−1(ω)),t−1 (ω))−T(F l,t−1 (ω))| ≤ |T(Fπ t(Zt−1(ω)),t−1 (ω))−T(F )|+|T(F )−T(F l,t−1 (ω))| ≤ 2C. (116) It thus follows that lim sup t→∞ [S l (t − 1)(ω)] −1/2 − [Sπ t(Zt−1(ω)) (t − 1)(ω)] −1/2 ≤ 0,(117) or equivalently, noting that Equation (112) implies lim t→∞ S l (t − 1)(ω) = κ(ω), that lim sup t→∞ Sπ t(Zt−1(ω)) (t − 1)(ω) ≤ κ(ω).(118) This, however, implies the contradiction that for every j = 1, . . . , K it must hold that lim t→∞ S j (t − 1)(ω) < ∞ (the limit existing due to monotonicity). To see the latter, suppose lim t→∞ S j (t − 1)(ω) = ∞ holds for treatment j. Define the subsequence t ′ := {t ∈ N :π t (Z t−1 (ω)) = j} of N, and note that t ′ → ∞ due to our assumption lim t→∞ S j (t−1)(ω) = ∞. Next, observe that S j (t ′ −1)(ω) = Sπ t ′ (Z t ′ −1 (ω)) (t ′ −1)(ω), a contradiction to the previous display. Before we proceed, let l ∈ {1, . . . , K}. Note that t l,r < t l,s holds for all pairs of natural numbers r < s. Note also that for all pairs of natural numbers r and s the event {t l,s = r} is measurable w.r.t. the σalgebra generated by Y 1 , . . . , Y r−1 , i.e., w.r.t. A r := σ(Y 1 , . . . , Y r−1 ). Lemma D.2. For every l ∈ {1, . . . , K} and every r ∈ N the joint distribution of Y l,1 , . . . , Y l,r coincides with the joint distribution of Y l,t l,1 , . . . , Y l,t l,r . Proof. Let l ∈ {1, . . . , K}. Note that for any r ∈ N the random variables Y l,t l,1 , . . . , Y l,t l,r are well defined as a consequence of Lemma D.1. To prove the statement, we use induction on r, and start with r = 1. In this case the statement is trivial, because t l,1 = l implies Y l,t l,1 = Y l,l which has the same distribution as Y l,1 . Next, assume that r > 1. By the induction hypothesis, we need to show that for A j ∈ B(R) for j = 1, . . . , r we have P Y l,t l,1 ∈ A 1 , . . . , Y l,t l,r ∈ A r = P Y l,t l,1 ∈ A 1 , . . . , Y l,t l,r−1 ∈ A r−1 P(Y l,r ∈ A r ). Let I r := A ⊆ N : |A| = r , and for any I ∈ I r , let I j , j = 1, ..., r denote the j-th element of I (I being ordered from smallest to largest). Observe that the random set {t l,1 , . . . , t l,r } takes its values in I for some I ⊆ I r , implying that I∈Ir r k=1 1 {t l,k =I k } = 1. We can thus write P Y l,t l,1 ∈ A 1 , . . . , Y l,t l,r ∈ A r = E   r j=1 1 A j (Y l,t l,j )   = E r j=1 1 A j (Y l,t l,j ) I∈Ir r k=1 1 {t l,k =I k } ,(120) which can further be rewritten as I∈Ir E   r j=1 1 A j (Y l,t l,j )1 {t l,j =I j }   = I∈Ir E   r j=1 1 A j (Y l,I j )1 {t l,j =I j }   .(121) Next, using that {t l,j = I j } ∈ A Ir−1 holds for every j = 1, . . . , r, we write the expectation to the far right in the previous display as E    E   r j=1 1 A j (Y l,I j )1 {t l,j =I j } A Ir−1      = E   r−1 j=1 1 A j (Y l,I j )1 {t l,j =I j } 1 {t l,r =Ir} E 1 Ar (Y l,Ir )|A Ir−1   . (122) Clearly Y l,Ir is independent of Y 1 , . . . , Y Ir−1 , and thus the inner conditional expectation coincides with E 1 Ar (Y l,Ir ) = P(Y l,Ir ∈ A r ) = P(Y l,r ∈ A r ). To prove (119) it hence remains to verify that I∈Ir E   r−1 j=1 1 A j (Y l,I j )1 {t l,j =I j } 1 {t l,r =Ir}   = P(Y l,t l,1 ∈ A 1 , . . . , Y l,t l,r−1 ∈ A r−1 ).(123) To see this, write I∈Ir E   r−1 j=1 1 A j (Y l,I j )1 {t l,j =I j } 1 {t l,r =Ir}   = I∈I r−1 k>I r−1 E   r−1 j=1 1 A j (Y l,I j )1 {t l,j =I j } 1 {t l,r =k}   = I∈I r−1 E   r−1 j=1 1 {t l,j =I j } 1 A j (Y l,I j ) k>I r−1 1 {t l,r =k}   , and note that t l,r > t l,r−1 implies that r−1 j=1 1 {t l,j =I j } k>I r−1 1 {t l,r =k} = r−1 j=1 1 {t l,j =I j } , which together with I∈I r−1 r−1 k=1 1 {t l,k =I k } = 1 shows that the expression to the right in the previous display equals I∈I r−1 E   r−1 j=1 1 {t l,j =I j } 1 A j (Y l,I j )   = I∈I r−1 E   r−1 j=1 1 A j (Y l,t l,j ) r−1 k=1 1 {t l,k =I k }   = E   r−1 j=1 1 A j (Y l,t l,j )   , the latter being equal to P(Y l,t l,1 ∈ A 1 , . . . , Y l,t l,r−1 ∈ A r−1 ). Finally, to obtain the upper bounds claimed in Equation (109), note first that P(B i,t ) = P(T(F i,t−1 ) > T(F i ) + C β log(t)/2S i (t − 1)) = t s=1 P(T(F i,t−1 ) > T(F i ) + C β log(t)/2s, S i (t − 1) = s). Note further that for every index s, on the event {S i (t − 1) = s} the empirical cdfF i,t−1 coincides by definition with s −1 s j=1 1 {Y i,t i,j ≤.} . Hence, the sum in the second line of the previous display is not greater than t s=1 P   T   s −1 s j=1 1 {Y i,t i,j ≤. }   > T(F i ) + C β log(t)/2s    . By Lemma D.2 the joint distribution of Y i,t i,1 , . . . , Y i,t i,s coincides with the joint distribution of Y i,1 , . . . , Y i,s . It thus follows that we can replace Y i,t i,1 , . . . , Y i,t i,s by Y i,1 , . . . , Y i,s in the previous display without changing the probabilities. In other words, we can replace s −1 s j=1 1 {Y i,t i,j ≤. } by F i,s in the previous display, from which the upper bound on P(B i,t ) in Equation (109) follows. The upper bound on P(A t ) is obtained analogously. D.1.2 Proof of Theorem 3.6 We begin with a lemma that provides an upper bound on the KL divergence between members of H as defined in Definition 3.2. In particular, the KL-divergence between two elements P ha and P h b of H is sub-quadratic in the distance between a and b. Lemma D.3. The KL divergence between any two elements of H (cf. Definition 3.2) satisfies: KL(h a , h b ) = 1 0 log h a (y) h b (y) h a (y)dy ≤ 1 (1 + b)(1 + a) (b − a) 2 Proof. By simple calculus we obtain that 1 0 log h a (y) h b (y) h a (y)dy = 1 0 log (1 + a)y a (1 + b)y b (1 + a)y a dy = 1 0 log 1 + a 1 + b + (a − b) log(y) (1 + a)y a dy ≤ a − b 1 + b + (a − b)(1 + a) 1 0 log(y)y a dy = a − b 1 + b − (a − b)(1 + a) (1 + a) 2 = a − b 1 + b − (a − b) (1 + a) = (a − b) 2 (1 + b)(1 + a) . Proof of Theorem 3.6. Throughout the proof we fix a policy π and assume without loss of generality that T(H a 2 ) − T(H a 2 ) ≥ c(a 2 − a 1 ) for all a 1 , a 2 ∈ [ā − δ,ā + δ] ⊆ (−1, ∞) and a 2 ≥ a 1 . The case where a → T(H a ) is locally uniformly decreasing follows analogously. Let treatment 1 have distribution Pā with cdf Hā and treatment 2 have distribution Pā −ε with cdf Hā −ε or Pā +ε with cdf Hā +ε for some ε > 0. It suffices to show that the maximal regret incurred over the two two-tuples (Hā, Hā −ε ) and (Hā, Hā +ε ) is greater than c √ n for some c > 0. Denote by P t π,−ε and P t π,ε , respectively, the distribution of (Y πt(Z t−1 ),t , ..., Y π 1 ,1 ) under the relevant tuple. Since sup j∈{−ε,ε} E n π,j R n (π) ≥ 1 2 E n π,−ε R n (π) + E n π,ε R n (π) = 1 2 n t=1 2 i=1 ∆ i E n π,−ε 1 {πt(Z t−1 )=i} + n t=1 2 i=1 ∆ i E n π,ε 1 {πt(Z t−1 )=i} ≥ cε 2 n t=1 E n π,−ε 1 {πt(Z t−1 )=2} + n t=1 E n π,ε 1 {πt(Z t−1 )=1} . where the third estimate used that T(Hā +ε )−T(Hā) and T(Hā)−T(Hā −ε ) are bounded from below by cε for ε ≤ δ. Next, note that E n π,−ε 1 {πt(Z t−1 )=2} + E n π,ε 1 {πt(Z t−1 )=1} = E n π,−ε 1 {πt(Z t−1 )=2} + 1 − E n π,ε 1 {πt(Z t−1 )=2} is the sum of type 1 and 2 errors for the testing problem H 0 : P = P n π,−ε vs H a : P = P n π,ε for the test 1 {πt(Z t−1 )=2} . Thus, using Theorem 2.2(iii) of Tsybakov (2009), we get for t = 2, ..., n E n π,−ε 1 {πt(Z t−1 )=2} + E n π,ε 1 {πt(Z t−1 )=1} ≥ 1 4 exp −KL(P n π,−ε , P n π,ε ) . Using the chain rule for Kullback-Leibler divergence, cf. Theorem 2.5.3 of Cover and Thomas (2012), KL(P n π,−ε , P n π,ε ) = KL(P n−1 π,−ε , P n−1 π,ε ) + E n−1 π,−ε KL(P π,−ε,n , P π,ε,n ) where P π,j,n , j ∈ {−ε, ε} is the conditional distribution of Y πn(Z n−1 ),n given Z n−1 under the policy π and distribution P n π,j . Since Y 1,n and Y 2,n are independent of Z n−1 , we observe P π,j,n = Pā1 {πn(Z n−1 )=1} + Pā +j 1 {πn(Z n−1 )=2} . Hence, by Lemma D.3, KL(P π,−ε,n , P π,ε,n ) ≤ KL(Hā −ε , Hā +ε )1 {πn(Z n−1 )=2} ≤ 4ε 2 (1 +ā − δ) 2 1 {πn(Z n−1 )=2} . Thus, by induction, we observe that KL(P n π,−ε , P n π,ε ) ≤ 4ε 2 (1 +ā − δ) 2 N π =cε 2 N π , withc = 4 (1+ā−δ) 2 and N π = E n π,−ε n t=1 1 {πt(Z t−1 )=2} . But since we also have sup j∈{−ε,ε} E n π,j R n (π) ≥ cε 2 N π we conclude that sup j∈{−ε,ε} E n π,j R n (π) ≥ cε 2 max n 4 exp(−cε 2 N π ), N π ≥ cε 4 n 4 exp(−cε 2 N π ) + N π ≥ cε 4 inf z≥0 n 4 exp(−cε 2 z) + z . Note that z * := argmin n 4 exp(−cε 2 z) + z = log cε 2 n 4 /(cε 2 ), is positive if ε 2 > 4 nc . Thus, choosing ε = 8 nc (which is less than δ for n ≥ n 0 = ⌈ 8 δ 2c ⌉) shows sup j∈{−ε,ε} E n π,j R n (π) ≥ c √c log(2) √ 128 · √ n. The first n 0 terms are handled by using sup j∈{−ε,ε} E n π,j R n (π) ≥ cε 4 inf z≥0 n 4 exp(−cε 2 z) + z with ε = δ/2 and by choosing the constant in the statement of the theorem small enough. D.2 Proofs of results in Section 4 First, we provide an auxiliary result that will be useful in the proofs of Theorems 4.4 and 4.7. In the local treatment problem for individuals with covariates in B j , * denotes the index of a treatment in the set arg max T(F i j ). That is, T(F * j ) = max i∈I T(F i j ). To save on notation, we shall write π t (X t ) instead of π t (X t , Z t−1 ) throughout this section. Lemma D.4. Suppose that Assumption 2.1 and 4.2 are satisfied and a grouping is characterised by {V 1 , ..., V F } and {B 1 , ...,B F }. Then, for any i ∈ I, j ∈ {1, . . . , F } and x,x ∈ B j , we obtain that |T(F i (·, x)) − T(F i (·,x))| ≤ CLV γ j , |T F π ⋆ (x) (·, x) − T F π ⋆ (x) (·,x) | ≤ CLV γ j , |T(F i j ) − T(F i (·, x))| ≤ CLV γ j , |T(F ⋆ (·, x)) − T(F * j )| ≤ CLV γ j . Proof. Fix i, j and x,x ∈ B j . Assumption 4.2 implies that ||F i (·, x) − F i (·,x)|| ∞ ≤ L||x −x|| γ ≤ LV γ j . Then, the first statement follows immediately from Assumption 2.1. Using this result, we also obtain the second statement via |T F π ⋆ (x) (·, x) − T F π ⋆ (x) (·,x) | = | max i∈I T(F i (·, x)) − max i∈I T(F i (·,x))| ≤ max i∈I |T(F i (·, x)) − T(F i (·,x))| ≤ CLV γ j . Now, we move to the third part. For any s and x, we have that F i j (y) − F i (y, x) = 1 P X (B j ) B j (F i (y, s) − F i (y, x))P X (ds). As x, s ∈ B j , Assumption 4.2 leads to |F i (y, s) − F i (y, x)| ≤ LV γ j and hence ||F i j − F i (·, x)|| ∞ ≤ LV γ j for each x ∈ B j . Then, the third claim is the direct consequence of Assumption 2.1 on T. Concerning the last statement, we observe that |T(F ⋆ (·, x)) − T(F * j )| = max i∈I T(F i (·, x)) − max i∈I T(F i j ) ≤ max i∈I |T(F i (·, x)) − T(F i j )| ≤ CLV γ j , which finishes the proof. Now, we are ready to deal with the proofs of main results. Proof of Theorem 4.4. First, we write R N (π) = F j=1R j (π) with R j (π) := N t=1 T F π ⋆ (Xt) (·, X t ) − T Fπ t(Xt ) (·, X t ) 1 {Xt∈B j } . Fix j ∈ {1, . . . , F }. Recalling the definition of F i j in (9), each summand in the previous display can be written as T F π ⋆ (Xt) (·, X t ) − T(F * j ) + T(F * j ) − T(Fπ t(Xt ) j ) + T(Fπ t(Xt) j ) − T Fπ t(Xt ) (·, X t ) 1 {Xt∈B j } ,(124) which by Lemma D.4 is not greater than T(F * j ) − T(Fπ t(Xt) j ) + 2CLV γ j . Therefore, we obtaiñ R j (π) ≤ N t=1 T F * j − T Fπ t(Xt) j 1 {Xt∈B j } + 2CLV γ j N t=1 1 {Xt∈B j } .(125) By Wald's identity E( N t=1 1 {Xt∈B j } ) ≤ ncB j . Hence, to prove the theorem, it remains to show that for some c (which in fact will be the same c(β, C) as in Theorem 3.1) it holds that E   N t=1 T F * j − T Fπ t(Xt ) j 1 {Xt∈B j }   ≤ c KcB j nlog(cB j n).(126) To this end, for every m ∈ N and every v = (v 1 , . . . , v m ) ∈ {1, . . . , F } m , we define the event Ω(m, v) := {N = m, X 1 ∈ B v 1 , . . . , X m ∈ B vm } ⊆ Ω.(127)E(f ) = m∈N v∈{1,...,F } m E(1 Ω(m,v) f ) = m∈N v∈{1,...,F } m P(Ω(m, v))E(f |Ω(m, v)),(128) where we define E(f |Ω(m, v)) := P −1 (Ω(m, v))E(1 Ω(m,v) f ) in case P(Ω(m, v)) > 0 and E(f |Ω(m, v)) := 0 else. Fix m and v, and assume that {v s : s ∈ {1, . . . , m}, v s = j} is not empty. Denote the elements of the latter set by t 1 , . . . , tm, ordered from smallest to largest. On the event Ω(m, v) (i.e., for every ω ∈ Ω(m, v)) we can use the definition ofπ to rewrite (recall the definition of the FSA policyπ in the no-covariate case) f =m s=1 T(F * j ) − T Fπ s(Zs−1) j ,(129) where for every s > 1 we define Z s−1 = (Yπ s−1 ,t s−1 , . . . , Yπ 1 ,t 1 ), and for s = 1 we recall from the definition of the FSA policy thatπ 1 := 1, which is deterministic. The previous display shows that on the event Ω(m, v) we can write f as a function of (Y t 1 , . . . , Y tm ), i.e., as H(Y t 1 , . . . , Y tm ), say. We conclude that E(f |Ω(m, v)) = E m s=1 T(F * j ) − T Fπ s(Zs−1) j Ω(m, v) = E H(Y t 1 , . . . , Y tm )|Ω(m, v) .(130) The quantity to the right equals E * (H(Y t 1 , . . . , Y tm )), where the probability measure P * corresponding to E * is defined as the P-measure with density P −1 (Ω(m, v))1 Ω(m,v) . Note that for A i ∈ B(R K ) for i = 1, . . . , m we have that P * (Y t 1 ∈ A 1 , . . . , Y tm ∈ Am) equals (using various independence properties of the observations and N) P −1 (Ω(m, v))P Y t 1 ∈ A 1 , . . . , Y tm ∈ Am, Ω(m, v) =m s=1 P(Y ts ∈ A s , X ts ∈ B j ) P(X ts ∈ B j ) (131) =m s=1 P(Y ts ∈ A s |{X ts ∈ B j }).(132) We thus see that P * is them-fold product of Q(.) := P(Y 1 ∈ .|{X 1 ∈ B j }). For i.i.d. random Kvectors Y * 1 , . . . , Y * m , say, each with distribution Q (which exist possibly after enlarging the underlying probability space), it hence follows from the definition of H that E(H(Y t 1 , . . . , Y tm )|Ω(m, v)) = E(H(Y * 1 , . . . , Y * m )) = E m s=1 T(F * j ) − T Fπ s(Z * s−1 ) j(133) where Z * s−1 = (Y * π s−1 ,s−1 , . . . , Y * π 1 ,1 ) for s > 1. Noting that the r-th marginal of Q has cdf F r j , it now follows from Theorem 3.1 applied with marginal distribution Q and (constant) N =m that the quantity in the previous display, and thus E(f |Ω(m, v)) is not greater than c Kmlog(m). It thus follows from (128) (noting that f vanishes on those exceptional sets Ω(m, v) for which the set {v s : s ∈ {1, . . . , m}, v s = j} is empty, and to which the just derived upper bound does not apply) that E(f ) ≤ c m∈N v∈{1,...,F } m P(Ω(m, v)) Kmlog(m).(134) Recall, thatm = |{v s : s ∈ {1, . . . , m}, v s = j}|. Hence, we can interpretm as a random variable on the set of all tuples (m, v), over which the sum in the previous display extends, equipped with the probability measure P(Ω(m, v)). It remains to observe that the function h defined via x → Kxlog(x) is concave on [0, ∞), which allows us to apply Jensen's inequality to upper bound the right hand side in the previous display by c Kxlog(x) with x = m∈N v∈{1,...,F } m P(Ω(m, v))m = E m∈N v∈{1,...,F } m 1 Ω(m,v)m = E m∈N v∈{1,...,F } m 1 Ω(m,v) N s=1 1 Xs∈B j (135) = E N s=1 m∈N v∈{1,...,F } m 1 Ω(m,v) 1 Xs∈B j = E N s=1 1 Xs∈B j m∈N v∈{1,...,F } m 1 Ω(m,v) = E N s=1 1 Xs∈B j .(136) We used Tonelli's theorem in the second equality. We know already that x ≤cB j n. Since the function h is also monotonically increasing, it follows that E(f ) ≤ ch(cB j n), which is the statement in Equation (126). Proof of Corollary 4.5. The given choice of groups results in F = P d ,B j = P −d and V j = √ dP −1 . Hence, Theorem 4.4 and the choice P = ⌈n 1 2γ+d ⌉ yields E[R N (π)] ≤ cP d KnP −d log(nP −d ) + nP −γ−d ≤ c Klog(n)P d √ nP −d + nP −γ−d ≤ c Klog(n)n 1− γ 2γ+d which is the claimed result. Proof of Theorem 4.7. Define c 1 := 4CLd γ/2 + 1. Recalling P = ⌈n 1/(2γ+d) ⌉, we shall assume without loss of generality that n is large enough (n ≥ n 0 , say) such that c 1 P −γ ≤ 1 holds (this will allow us to apply Assumption 4.6 with δ = c 1 P −γ in the arguments below). Note that by Assumption 2.1 for n < n 0 it holds that E[R N (π)] ≤ Cn 0 . Hence, c in the statement of Theorem 4.7 can be chosen large enough to deal with the initial terms smaller than n 0 . Throughout the proof the bins are sorted in the lexicographic order and we shall write B 1 , ..., B P d for the P d bins. The proof is divided into several steps: The bins corresponding to indices in J , J s , and J w will be referred to as "well-behaved", "strongly ill-behaved" and "weakly ill-behaved" bins, respectively. Note that J w and J ∪ J s are clearly disjoint. That J and J s are disjoint is shown in Step 2 below. Hence, the sets of bins corresponding to indices in J , J s , J w constitute a partition of the set of all P d bins B j , and we can thus write E(R N (π)) = j∈Js E(R j (π)) + j∈Jw E(R j (π)) + j∈J E(R j (π)),(138) whereR j (π) := N t=1 T F π ⋆ (Xt) (·, X t ) − T Fπ t(Xt ) (·, X t ) 1 {Xt∈B j } .(139) Step 2: Strongly ill-behaved bins. For every j ∈ J s , by definition, there exists ax ∈ B j such that T F π ⋆ (x) (·,x) = T F π ♯ (x) (·,x) . From the definition of π ♯ it thus follows that T F π ⋆ (x) (·,x) = T F i (·,x) for every i ∈ I. Therefore, for every x ∈ B j and every i ∈ I, Lemma D.4 yields T(F π ⋆ (x) (·, x)) − T(F i (·, x)) = T(F π ⋆ (x) (·, x)) − T(F i (·, x)) − [T(F π ⋆ (x) (·,x)) − T(F i (·,x))] (140) ≤ 2CLd γ/2 P −γ ≤ c 1 P −γ .(141) First of all, this shows that J and J s are disjoint. Furthermore, from Equations (139) and (140), we obtain j∈JsR j (π) ≤ c 1 P −γ j∈Js N t=1 1 {Xt∈B j } 1 {0<T(F π ⋆ (X t ) (·,Xt))−T(F π ♯ (X t ) (·,Xt))} (142) ≤ c 1 P −γ N t=1 1 {0<T(F π ⋆ (X t ) (·,Xt))−T(F π ♯ (X t ) (·,Xt))≤c 1 P −γ } .(143) From Condition 4.6 we hence obtain: j∈Js E[R j (π)] ≤ c 1 nP −γ P X 0 < T F π ⋆ (X) (·, X) − T F π ♯ (X) (·, X) ≤ c 1 P −γ ≤ C 0 c 1+α 1 nP −γ(1+α) . (144) Step 3: Weakly ill-behaved bins. Since {X t ∈ B j } for j ∈ J w are disjoint subsets of {0 < T(F π ⋆ (Xt) (·, X t )) − T(F π ♯ (Xt) (·, X t )) ≤ c 1 P −γ }, we obtain from Condition 4.6, recall that P(X t ∈ B j ) ≥ c P d , that |J w | c P d ≤ j∈Jw P(X t ∈ B j ) ≤ P X 0 < T F π ⋆ (X) (·, X) − T F π ♯ (X) (·, X) ≤ c 1 P −γ ≤ C 0 c α 1 P −γα ,(145) which yields |J w | ≤ (C 0 c α 1 /c)P d−γα . Using (125) and (126) with V j = √ dP −1 andB j = P −d we obtain E[R j (π)] ≤ c ′ Knlog(n)P −d/2 + nP −γ−d ,(146) where c ′ depends on d, L, γ,c, C, β, but not on n. Combining this with (146) leads to j∈Jw E[R j (π)] ≤ c ′′ Knlog(n)P d/2−γα + nP −γ(1+α) ,(147) where c ′′ depends on d, L, γ, c,c, C, C 0 , α, β, but not on n. Step 4: Well-behaved bins. For every j ∈ J let x j ∈ B j be such that T(F π ⋆ (x j ) (·, x j )) − T(F π ♯ (x j ) (·, x j )) > c 1 P −γ .(148) Next, define the following sets of indices ("corresponding to the optimal and suboptimal arms given x j "): I ⋆ j := {i ∈ I : T F π ⋆ (x j ) (·, x j ) = T(F i (·, x j ))}, I 0 j := {i ∈ I : T F π ⋆ (x j ) (·, x j ) − T(F i (·, x j )) > c 1 P −γ }. Clearly π ⋆ (x j ) ∈ I ⋆ j and π ♯ (x j ) ∈ I 0 j (cf. (148)). Hence I ⋆ j and I 0 j define a nontrivial partition of I. For every j ∈ J we can thus decomposeR j (π) defined in Equation (139) as the sum of R j,I ⋆ j (π) := i∈I ⋆ j N t=1 T F π ⋆ (Xt) (·, X t ) − T F i (·, X t ) 1 {Xt∈B j } 1 {πt(Xt)=i} , R j,I 0 j (π) := i∈I 0 j N t=1 T F π ⋆ (Xt) (·, X t ) − T F i (·, X t ) 1 {Xt∈B j } 1 {πt(Xt)=i} .(149) Step 4a: A bound for E(R j,I ⋆ j (π)). For any i ∈ I ⋆ j and every x ∈ B j satisfying T(F π ⋆ (x) (·, x)) = T(F i (·, x)), the triangle inequality, the definition of π ♯ , and Lemma D.4 yield 0 < T(F π ⋆ (x) (·, x)) − T(F π ♯ (x) (·, x)) ≤ T(F π ⋆ (x) (·, x)) − T(F i (·, x)) = T(F π ⋆ (x) (·, x)) − T(F π ⋆ (x j ) (·, x j )) + T(F i (·, x j )) − T(F i (·, x)) ≤ 2CLd γ/2 P −γ ≤ c 1 P −γ , the last inequality following from c 1 = 4CLd γ/2 + 1. But this means that for any i ∈ I ⋆ j and every x ∈ B j T(F π ⋆ (x) (·, x)) − T(F i (·, x)) ≤ c 1 P −γ 1 {0<T(F π ⋆ (x) (·,x))−T(F π ♯ (x) (·,x))≤c 1 P −γ } .(150) We deduce E[R j,I ⋆ j (π)] ≤ E N t=1 c 1 P −γ 1 {0<T(F π ⋆ (X t ) (·,Xt))−T(F π ♯ (X t ) (·,Xt))≤c 1 P −γ } 1 {Xt∈B j } ≤ nc 1 P −γ q j ,(151) where q j := P(0 < T(F π ⋆ (Xt) (·, X t )) − T(F π ♯ (Xt) (·, X t )) ≤ c 1 P −γ , X t ∈ B j ), which is independent of t due to the X t being identically distributed. Step 4b: A bound for E(R j,I 0 j (π)). By Lemma D.4 for every x ∈ B j and every i ∈ I 0 j we have T(F π ⋆ (x) (·, x)) − T(F i (·, x)) ≤ T(F * j ) − T(F i j ) + c 1 P −γ ,(152) from which it follows that E[R j,I 0 j (π)] ≤ E i∈I 0 j N t=1 T F * j − T F i j 1 {Xt∈B j } 1 {πt=i} + c 1 P −γ E i∈I 0 j N t=1 1 {Xt∈B j } 1 {πt=i} , = i∈I 0 j ∆ i j ES(i, N, j) + c 1 P −γ i∈I 0 j ES(i, N, j),(153) where for every i ∈ I 0 j the sum S(i, N, j) : = N t=1 1 {Xt∈B j } 1 {πt=i} , and where ∆ i j := T(F * j ) − T(F i j ) . We now claim that (this claim will be verified before moving to Step 4c below) ES(i, N, j) ≤ 2C 2 β log(cnP −d ) [∆ i j ] 2 + β + 2 β − 2 .(154) Defining ∆ j := min i∈I 0 j ∆ i j , noting that max i∈I 0 j ∆ i j ≤ 2C by Assumption 2.1, and combining Equations (153) and (154) we obtain the bound E[R j,I 0 j (π)] ≤ K 2C 2 β log(cnP −d ) ∆ j 1 + c 1 P −γ ∆ j + (c 1 + 2C)K β + 2 β − 2 .(155) Now, it remains to prove the claim in Equation (154). To this end we apply a conditioning argument as in the proof of Theorem 4.4. We shall now use some quantities (in particular the sets Ω(m, v)) that were defined in that proof: note that ES(i, N, j) = m∈N v∈{1,...,F } m P(Ω(m, v))E(S(i, N, j)|Ω(m, v)).(156) Arguing as in the proof of Theorem 4.4, it is now easy to see that E(S(i, N, j)|Ω(m, v)) can be written as the expected number of times arm i is selected in running the FSA policyπ (without covariates) in a problem withm (fixed) i.i.d. inputs with distribution Q. We can hence apply the bound in Equation (102), to the just mentioned problem, to obtain that E(S(i, N, j)|Ω(m, v)) ≤ 2C 2 β log(m) [∆ i j ] 2 + β + 2 β − 2 .(157) We can now combine the obtained inequality with Equation (156) to see that ES(i, N, j) ≤ m∈N v∈{1,...,F } m P(Ω(m, v)) 2C 2 β log(m) [∆ i j ] 2 + β + 2 β − 2 .(158) The claim in (154) now follows from Jensen's inequality, and (cf. the end of the proof of Theorem 4.4) m∈N v∈{1,...,F } m P(Ω(m, v))m ≤cB j n =cnP −d . (159) Step 4c: A bound for E(R j (π)) with j ∈ J . For all i ∈ I 0 j and all x ∈ B j the triangle inequality and Lemma D.4 with V j = √ dP −1 yield c 1 P −γ <|T(F π ⋆ (x j ) (·, x j )) − T(F i (·, x j ))| ≤|T(F π ⋆ (x j ) (·, x j )) − T(F π ⋆ (x) (·, x))| + |T(F π ⋆ (x) (·, x)) − T(F i (·, x))| + |T(F i (·, x)) − T(F i (·, x j ))| ≤2CLd γ/2 P −γ + |T(F π ⋆ (x) (·, x)) − T(F i (·, x))|. Recalling that c 1 = 4CLd γ/2 + 1, we obtain T F π ⋆ (x) (·, x) − T(F i (·, x)) > (1 + 2CLd γ/2 )P −γ .(160) [In particular, since I 0 j = ∅ holds, 0 < T(F π ⋆ (x) (·, x)) − T(F π ♯ (x) (·, x)) for all x ∈ B j if j ∈ J , an observation we shall need later in Step 4d.] For each i ∈ I 0 j and every x ∈ B j , (160) and Lemma D.4 imply ∆ i j = T(F * j ) − T(F i j ) ≥ T(F π ⋆ (x) (·, x)) − T(F i (·, x)) − 2CLd γ/2 P −γ > P −γ ;(161) in particular for any j ∈ J and any i ∈ I 0 j we have ∆ j = min i∈I 0 j ∆ i j > P −γ . Recalling thatR j (π) = R j,I * j (π) +R j,I 0 j (π), we combine (151) and (155) (with the just observed ∆ j > P −γ ) to see that for any j ∈ J E[R j (π)] ≤ nc 1 P −γ q j + 2C 2 (c 1 + 1)Kβ log(cnP −d ) ∆ j + (c 1 + 2C)K β + 2 β − 2 .(162) Step 4d: A bound for j∈J E[R j (π)]. Using Equation (162) and |J | ≤ P d we obtain j∈J E[R j (π)] ≤ (c 1 + 2C)K β + 2 β − 2 P d + nc 1 P −γ j∈J q j + j∈J 2C 2 (c 1 + 1)Kβ log(cnP −d ) ∆ j .(163) Since the B j are disjoint, recalling the definition of q j after Equation (151) we obtain nc 1 P −γ j∈J q j ≤ c 1 nP −γ P X 0 < T(F π ⋆ (X) (·, X)) − T(F π ♯ (X) (·, X)) < c 1 P −γ ≤ C 0 c 1+α 1 nP −γ(1+α) ,(164) where we used Assumption 4.6 to obtain the last inequality. To deal with the last term in (163), we need a better lower bound on the ∆ j -s than the already available P −γ . For notational simplicity, let's suppose that the well-behaved bins are indexed as J = {1, 2, . . . , j 1 } such that 0 < ∆ 1 ≤ ∆ 2 ≤ . . . ≤ ∆ j 1 (cf. Equation (161) and the ensuing discussion for 0 < ∆ 1 ). Fix j ∈ J . Then, for any k = 1, . . . , j, we claim that: B k ⊆ x : 0 < T(F π ⋆ (x) (·, x)) − T(F π ♯ (x) (·, x)) < ∆ j + 2CLd γ/2 P −γ .(165) To see (165), note that, by definition, there exists an i ∈ I 0 k such that ∆ k = T(F * k ) − T(F i k ). For x ∈ B k Lemma D.4 yields (the first inequality following from the observation after Equation (160)) 0 < T(F π ⋆ (x) (·, x)) − T(F π ♯ (x) (·, x)) ≤ T(F π ⋆ (x) (·, x)) − T(F i (·, x)) ≤ ∆ k + 2CLd γ/2 P −γ ≤ ∆ j + 2CLd γ/2 P −γ , and thus x is an element of the set on the right-hand-side of (165). Since all bins B k are disjoint and ∆ j + 2CLd γ/2 P −γ ≤ c 1 ∆ j (obtained by recalling c 1 = 4CLd γ/2 + 1, and using the observation ∆ j > P −γ made directly after Equation (161)), the inclusion (165) yields that for any j ∈ J : P X 0 < T(F π ⋆ (X) (·, X)) − T(F π ♯ (X) (·, X)) < c 1 ∆ j ≥ j k=1 P X (B k ) ≥ cj P d .(166) Let's denote j 2 := max{j ∈ J : ∆ j ≤ 1/c 1 } (here interpreting the maximum of an empty set as 0). Then, for each j ∈ {1, . . . , j 2 } by Assumption 4.6 : P X 0 < T(F π ⋆ (X) (·, X)) − T(F π ♯ (X) (·, X)) < c 1 ∆ j ≤ C 0 (c 1 ∆ j ) α .(167) Combining (166), (167), and ∆ j > P −γ , for any j ∈ {1, . . . , j 2 } we get ∆ j ≥ max c * jP −d 1/α , P −γ , with constant c * := c −1 1 c 1/α C −1/α . Combining this with the identity ∆ j > 1/c 1 for j > j 2 , we obtain that j∈J 1 ∆ j ≤ j 2 j=1 min c −1 * P d /j 1/α , P γ + j 1 j=j 2 +1 c 1 ≤ P d j=1 min c −1 * P d /j 1/α , P γ + c 1 P d . ForP := ⌈P d−αγ ⌉ (in fact for anyP ∈ {1, . . . , P d }, and thus in particular for our particular choice) it holds that (110)). Hence, combining Equations (163), (164) with the bounds in the previous two displays we obtain P d j=1 min c −1 * P d /j 1/α , P γ ≤P j=1 P γ + c −1 * P d/α ∞ j=P +1 j −1/α ≤ c * * P d+γ(1−α) , for c * * := [2 + c −1 * (α −1 − 1) −1 ], where we used ∞ j=P +1 j −1/α ≤ (α −1 − 1) −1P 1−α −1 (cf.j∈J E[R j (π)] ≤ c ′′′ nP −γ(1+α) + Klog(nP −d )P d + Klog(nP −d )P d+γ(1−α) ,(168) for a constant c ′′′ , say, that depends on d, L, γ, c, C, C 0 , α and β, but not on n. Step 5: Combining. From Equations (138), (144), (147) and (168) we obtain E[R N (π)] ≤ c ′′′′ 4 nP −γ(1+α) + Knlog(n)P d/2−γα + Klog(nP −d )P d + Klog(nP −d )P d+γ(1−α)(169) for a constant c ′′′′ that depends on d, L, γ, c, C, C 0 , α and β, but not on n. From P = ⌈n 1/(2γ+d) ⌉ we get n ≤ P 2γ+d , and obtain E[R N (π)] ≤ c ′′′′ 4 Klog(n) nP −γ(1+α) + n 1/2 P d/2−γα + 2P d+γ(1−α) ≤ c ′′′′ Klog(n)P d+γ(1−α) ,(170) from which the conclusion follows. The following lemma allows to upper bound the number of suboptimal assignments made by the FSA policy. It will also play a crucial role in providing a lower bound on regret in Section D.3 since it establishes that if the margin condition (Assumption 4.6) is in place, such a lower bound can be obtained by lower bounding the number of suboptimal assignments made. Lemma D.5. Let a functional T be given and assume that Assumption 4.6 is satisfied. Furthermore, N is independent of all covariates and has expectation n. Then, there exists aC > 0 such that for any policy π E[R N (π)] ≥Cn −1/α E[S N (π)] 1+1/α .(171) Proof. Choose D 0 ≥ 2 such that 1/(C 0 D 0 ) 1/α < 1. We show that E[R N (π)] ≥Cn −1/α E[S N (π)] 1+1/α .(172) forC =C(α) = (1 − 1/D 0 )/(C 0 D 0 ) 1/α . If E[S N (π)] = 0, (172) is trivially valid. Thus, suppose that E[S N (π)] > 0. Note that for any δ > 0, R N (π) ≥ δ N t=1 1 {T(F π ⋆ (X t ) (·,Xt))−T(F ♯(X t ) (·,Xt))>δ} 1 πt(Xt,Z t−1 ) ∈arg max{T(F i (·,Xt)), i=1,...,K} = δS N (π) − δ N t=1 1 {T(F π ⋆ (X t ) (·,Xt))−T(F ♯(X t ) (·,Xt))≤δ} 1 πt(Xt,Z t−1 ) ∈arg max{T(F i (·,Xt)), i=1,...,K} = δS N (π) − δ N t=1 1 {0<T(F π ⋆ (X t ) (·,Xt))−T(F ♯(X t ) (·,Xt))≤δ} 1 πt(Xt,Z t−1 ) ∈arg max{T(F i (·,Xt)), i=1,...,K} ≥ δS N (π) − δ N t=1 1 {0<T(F π ⋆ (X t ) (·,Xt))−T(F ♯(X t ) (·,Xt))≤δ} , where the second equality used that if π t (X t , Z t−1 ) ∈ arg max {T(F i (·, X t )), i = 1, ..., K}, then 0 < T(F π ⋆ (Xt) (·, X t )) − T(F ♯(Xt) (·, X t )). Choosing δ := (E[S N (π)]/(nC 0 D 0 )) 1/α ≤ 1/(C 0 D 0 ) 1/α < 1 (the inequality following from E[S N (π)] ≤ E(N) ≤ n), Assumption 4.6 yields E[R N (π)] ≥ δ(E[S N (π)] − C 0 nδ α ) = δ(1 − 1/D 0 )E[S N (π)] =Cn −1/α E[S N (π)] 1+1/α ,(173) which proves (172). Proof of Theorem 4.8. Combine Theorem 4.7 and Lemma D.5. D.3 Proofs for Section 5 We begin with an auxiliary lemma bounding the Kolmogorov distance between any two members of H as defined in Definition 3.2. Lemma D.6. For all a 1 < a 2 in (−1, ∞) it holds that H a 1 − H a 2 ∞ = a 1 + 1 a 2 + 1 (a 1 +1)/(a 2 −a 1 ) a 2 − a 1 a 2 + 1 ≤ a 2 − a 1 a 2 + 1 .(174) Proof. Let a 1 < a 2 be elements of (−1, ∞). By definition of the . ∞ -norm and the cdf H a it holds that H a 1 − H a 2 ∞ = sup x∈[0,1] |x a 1 +1 − x a 2 +1 |.(175) For every x ∈ [0, 1] the function a → x a+1 is strictly decreasing on (−1, ∞). Hence, using a 1 < a 2 , it follows that the supremum to the right in the previous display equals sup x∈[0,1] x a 1 +1 − x a 2 +1 = max x∈(0,1) x a 1 +1 − x a 2 +1 ,(176) the equality being trivial. It is elementary to verify (e.g., by checking the first and second order conditions for a maximum) that the maximum in the previous display is attained at x * := a 1 + 1 a 2 + 1 1/(a 2 −a 1 ) ,(177) from which it follows that H a 1 − H a 2 ∞ = x a 1 +1 * − x a 2 +1 * = x a 1 +1 * 1 − x a 2 −a 1 * = x a 1 +1 * a 2 − a 1 a 2 + 1 ,(178) which proves the claimed equality, the inequality being a trivial consequence of x * ∈ (0, 1). Lemma D.7. Suppose the functional T satisfies Assumptions 2.1 and 3.3. Withā and δ as in Assumption 3.3, denote the image of T over A(δ) := {H a : a ∈ [ā − δ,ā + δ]} by T(A(δ)). Then, T(A(δ)) is a non-empty compact interval and 1. the function from [ā − δ,ā + δ] to T(A(δ)) defined via a → T(H a ) is Lipschitz continuous and possesses an inverse that is Lipschitz continuous. 2. for any non-empty compact interval I ⊆ T(A(δ)), there exists a Lipschitz continuous function A : T(A(δ)) → [ā − δ,ā + δ] such that for any function f : [0, 1] d → I it holds that T(H A(f (x)) ) = f (x) for every x ∈ [0, 1] d .(179) Proof. Consider part 1. first. It suffices to verify the statement under the condition in Equation (6) (the other statement can be obtained from this one upon passing from T to −T). Under Equation (6), one has T(A(δ)) = [T(Hā −δ ), T(Hā +δ )] which is a non-empty compact interval. Furthermore, it suffices to verify that there exists a constant L > 0, say, such that for every pair a 1 = a 2 in [ā − δ,ā + δ] it holds that L −1 |a 1 − a 2 | ≤ |T(H a 1 ) − T(H a 2 )| ≤ L|a 1 − a 2 |.(180) For the upper bound in the previous display, we use Assumption 2.1 together with Lemma D.6 to obtain T(H a 1 ) − T(H a 2 ) ∞ ≤ C H a 1 − H a 2 ∞ ≤ C(min(a 1 , a 2 ) + 1) −1 |a 2 − a 1 |. Since min(a 1 , a 2 ) ≥ a − δ > −1 the second inequality in the previous display follows with L = (ā − δ + 1) −1 C. Next, observe that the assumption in Equation (6) implies the first inequality in the previous display with constant c (instead of L −1 ). Increasing L, if necessary, proves the first part of the lemma. Denoting the inverse of a → T(H a ) by A one has in particular that T(H A(z) ) = z for every z ∈ T(A(δ)) which yields the second part of the lemma upon using that f (x) ∈ I ⊆ T(A(δ)) for any x ∈ [0, 1] d . In proving Theorem 5.1 it will be useful to make the dependence of regret R n (π) = R n (π, F 1 , F 2 ) = n t=1 T F 1 (·, X t ) − T F 2 (·, X t ) 1 {π ⋆ (Xt) =πt(Xt,Z t−1 )} .(181) and and the number of suboptimal assignments S n (π) = S n (π, F 1 , F 2 ) = n t=1 1 {T(F 1 (·,Xt)) =T(F 2 (·,Xt)), π ⋆ (Xt) =πt(Xt,Z t−1 )} . on the conditional distributions F 1 and F 1 explicit for any policy π. We make the following remark on notation prior to proving Theorem 5.1 Proof of Theorem 5.1. Throughout the proof, fix a functional T satisfying Assumptions 2.1 and 3.3 as well as a policy π. It will be notationally convenient to label treatment 2 as -1. The idea of proving a lower bound on regret follows the general pattern of reducing the problem to a testing problem as in Chapter 2 of Tsybakov (2009). The proof consists of four steps. First, we construct a certain set of Hölder continuous functions C. Second, based on C, we construct a set S of 2-tuples of (conditional) distributions on the Borel sets of [0, 1]. Third, these distributions are shown to satisfy Assumptions 4.2 and 4.6, i.e S ⊆ S such that the treatment problem falls under the assumptions of Theorem 4.7 Fourth, we show that for any policy π sup (F 1 ,F −1 )∈S E[R n (π, F 1 , F −1 )] ≥ c 2 n 1− γ(1+α) 2γ+d . for some c 2 > 0 independent of π. To establish the above display, we use Lemma D.5 to conclude that it suffices to provide a lower bound on the number of suboptimal treatments made by π. This, in turn, is achieved by using standard techniques for obtaining minimax lower bounds (e.g., Chapter 2 of Tsybakov (2009)). In particular, by lower bounding the sum of Type 1 and Type 2 errors in a certain binary testing problem. Step 1: Construction of the Hölder class C. For P ≥ 2, let B 1 , B 2 , . . . , B P d be the hypercubes defined in (11) sorted in the lexicographic order. Let q i , i = 1, ..., P d be the center of B i . Furthermore, set m = ⌈0.5P d−αγ ⌉ and note that m ≤ P d for P ≥ 2. Next, set Ω m = {−1, 1} m and observe that |Ω m | = 2 m . For each ω ∈ Ω m , we define the function f ω on [0, 1] d as f ω (x) = 1 2 + m j=1 ω j ϕ j (x), where ϕ j (x) = 4 −1 P −γ φ(2P (x − q j ))1 B j (x) and φ(x) = (1 − ||x|| ∞ ) γ , and we let x ∞ = max 1≤i≤d |x i | for x ∈ R d . With this notation in place, we can define the class of functions C m := C = {f ω : ω ∈ Ω m }. We now show that each element of C is Hölder continuous. More precisely, for each f ω ∈ C, we show that for any pair x 1 , x 2 ∈ [0, 1] d |f ω (x 1 ) − f ω (x 2 )| ≤ 2 −1 ||x 1 − x 2 || γ , where . denotes the Euclidean norm. Observe that for any pair x 1 , x 2 ∈ [0, 1] d one has |φ(x 1 )−φ(x 2 )| ≤ ||x 1 − x 2 || γ ∞ ≤ ||x 1 − x 2 || γ where the first inequality uses i) for a, b ≥ 0 one has a γ ≤ b γ + |b − a| γ for γ ≤ 1 and ii) the reverse triangle inequality for · ∞ . If x 1 , x 2 ∈ B j for some j ∈ {1, ..., P d }, the definition of f ω and |φ( x 1 ) − φ(x 2 )| ≤ ||x 1 − x 2 || γ lead to |f ω (x 1 ) − f ω (x 2 )| ≤ |ϕ j (x 1 ) − ϕ j (x 2 )| ≤ 4 −1 P −γ (2P ) γ ||x 1 − q j − (x 2 − q j )|| γ ≤ 2 −1 ||x 1 − x 2 || γ .(182) Suppose instead that x 1 ∈ B j , x 2 ∈ B k for some j = k. Let S = {θx 1 + (1 − θ)x 2 : θ ∈ [0, 1]} be the line connecting x 1 and x 2 . Define y 1 = argmin z∈S∩B j ||z − x 2 || and y 2 = argmin z∈S∩B k ||z − x 1 ||. Noting that ϕ j (y 1 ) = ϕ k (y 2 ) = 0 we obtain from (182) that |f ω (x 1 ) − f ω (x 2 )| = |ω j ϕ j (x 1 ) − ω k ϕ k (x 2 )| ≤ |ω j ϕ j (x 1 ) − ω j ϕ j (y 1 )| + |ω j ϕ j (y 1 ) − ω k ϕ k (y 2 )| + |ω k ϕ k (y 2 ) − ω k ϕ k (x 2 )| ≤ 4 −1 2 γ ||x 1 − y 1 || γ + 4 −1 2 γ ||y 2 − x 2 || γ ≤ 2 −1 ||x 1 − x 2 || γ where we exploited ||x 1 − x 2 || = ||x 1 − y 1 || + ||y 1 − y 2 || + ||y 2 − x 2 || ≥ ||x 1 − y 1 || + ||y 2 − x 2 || combined with the inequality (a γ + b γ ) ≤ 2 1−γ (a + b) γ for γ ≤ 1. Step 2: Construction of S. Observe that the range of each f ∈ C is contained in [1/4, 3/4]. If necessary, a linear transformation of T ensures that [1/4, 3/4] ⊆ T(A(δ)) (with T(A(δ)) as in Lemma D.7) without affecting the rates of the lower bound on maximal regret. Part 2. of Lemma D.7 now implies for each f ∈ C there exists a g f : [0, 1] d → [ā − δ,ā + δ] with g f (x) = A(f (x)) for all x ∈ [0, 1] d and A is Lipschitz continuous such that T(H g f (x) ) = f (x) . Let P f,1 (·, x) be the distribution on the Borel sets of [0, 1] with cdf H g f (x) (y) = y g f (x)+1 . Borel measurability of x → P f,1 (A, x) for each A ∈ B([0, 1]) follows from continuity of g f and Scheffé's Lemma. Let the joint distribution P f of (Y 1,t , X t ) on B([0, 1] 1+d ) be defined as P f (A × B) = B P f,1 (A, x)P X (dx) for A ∈ B([0, 1]) and B ∈ B([0, 1] d ). To finish the construction of the distribution of (Y −1,t , Y 1,t , X t ) on B([0, 1] 2+d ), denoted P f,1/2 , let P 1/2,−1 (·) be the distribution on the Borel sets of [0, 1] with cdf H g 1/2 (y) = y g 1/2 +1 where g 1/2 = A(1/2) is a constant (function constant in x). Finally, let P f,1/2 (A 1 × A 2 × B) = A 2 ×B P 1/2,−1 (A 1 )dP f (y, x) = P 1/2,−1 (A 1 ) · P f (A 2 × B) for A 1 , A 2 ∈ B([0, 1]) and B ∈ B([0, 1] d ). Thus, Y −1,t is independent of (Y 1,t , X t ). This independence is merely chosen for concreteness as the important ingredients in Steps 3 and 4 below are the conditional distributions of P f,1 (of Y 1,t given X t ) and P 1/2,−1 (of Y −1,t given X t ), respectively 4 . With these definitions in place, for each f ∈ C, let P t π,f be the distribution of Z t on the Borel sets of R (d+1)t with corresponding expectation E t π,f . Define P π,f,1,t (A, (x, z)) := P f,1 (A, x) for every A ∈ B([0, 1]) and (x, z) ∈ R d × R (d+1)(t−1) . By independence of (Y 1,t , X t ) and Z t−1 the Markov kernel P π,f,1,t defines a regular conditional probability of Y 1,t given (X t , Z t−1 ). Similarly, P π,f,−1,t (A, (x, z)) := P 1/2,−1 (A) defines a regular conditional distribution of Y −1,t given (X t , Z t−1 ). To exhibit P t π,f explicitly, let u 1 = x 1 ∈ R d and z 1 = (y 1 , x 1 ) ∈ R × R d . For s = 2, ..., t let z s = (y s , x s , z s−1 ) ∈ R (d+1)s . Then, for any A ∈ B([0, 1]), u s = (x s , z s−1 ) ∈ R d × R (s−1)(d+1) and f ∈ C define (1 + g f (x s ))y g f (xs) s 1 {πs(us)=1} + (1 + g 1/2 )y g 1/2 s 1 {πs(us)=−1} dy s . Note that P π,f,s defines a regular conditional distribution of Y πt(Xt,Z t−1 ),t given (X t , Z t−1 ). Thus, setting d s (z s ) = (1 + g f (x s ))y g f (xs) s 1 {πs(us)=1} + (1 + g 1/2 )y g 1/2 s 1 {πs(us)=−1} , we observe by (184) for A s ∈ B([0, 1]), B s ∈ B([0, 1] d ), s = 1, ..., t P t π,f (× t s=1 (A s × B s )) = × t−1 s=1 (As×Bs) Bt P π,f,t (A t , (x t , z t−1 ))P X (dx t )P t−1 π,f (dz t−1 ) = × t−1 s=1 (As×Bs) Bt At d t (z t )dy t dx t P t−1 π,f (dz t−1 ) which can be used as the induction step to show (the induction start is trivial) that P t π,f is absolutely continuous with density d(z t ) = t s=1 d s (z s ) with respect to the t(d + 1)-dimensional Lebesgue measure (restricted to [0, 1] t(d+1) ). Now, set S = (H g f (x) , H g 1/2 ) : f ∈ C . Step 3: Verifying that S ⊆ S. To verify that S ⊆ S we show that for every f ∈ C: i) one has ||H g f (x 1 ) − H g f (x 2 ) || ∞ ≤ L||x 1 − x 2 || γ for some L > 0 (Assumption 4.2), and ii) the margin condition (Assumption 4.6) is satisfied. Verifying Assumption 4.2: We begin by verifying that for each f ∈ C one has ||H g f (x 1 ) − H g f (x 2 ) || ∞ ≤ L||x 1 − x 2 || γ for some L > 0. Note that by Lemma D.6 ||H g f (x 1 ) (y) − H g f (x 2 ) (y)|| ∞ ≤ |g f (x 1 ) − g f (x 2 )| a − δ + 1 ,ā − δ > −1 such that the conclusion follows upon recalling that g f (x) = A(f (x)) with Lipschitz continuous A and Hölder continuous f . Denoting by c 1 the Lipschitz constant of A, we can choose L = c 1 2(ā−δ+1) . Since H g 1/2 (y) does not depend on x it is Hölder continuous as well. Verifying Assumption 4.6: We next verify that each tuple in S satisfies the margin condition. To be precise, we shall show that for every f ∈ C P X 0 < |T(H g f (X) ) − T(H g 1/2 (X) )| ≤ δ ≤ 8dδ α for all δ ∈ [0, 1], which will verify the margin condition with C 0 = 8d since there are only two treatments. To this end, we note that T(H g f (X) ) − T(H g 1/2 ) = f ω (X) − 1/2 for some ω ∈ Ω m . Since P X is the uniform distribution on [0, 1] d and recalling φ(x) = (1 − ||x|| ∞ ) γ , for any ω ∈ Ω m , the substitution u = 2P x − 2P q 1 yields P X (0 < |f ω (X) − 1/2| ≤ δ) = m j=1 P X (0 < |f ω (X) − 1/2| ≤ δ, X ∈ B j ) = mP X (0 < φ(2P (X − q 1 )) ≤ 4P γ δ, X ∈ B 1 ) = m(2P ) −d 2P B 1 −2P q 1 1 {φ(x)≤4P γ δ} dx = m(2P ) −d [−1,1] d 1 {φ(x)≤4P γ δ} dx = mP −d [0,1] d 1 {φ(x)≤4P γ δ} dx, where the last equality follows from φ(x) being invariant to changing the signs of the coordinates of x. To bound the last line of the above display consider two cases. If 4P γ δ > 1, then P X (0 < |f ω (X) − 1/2| ≤ δ) = mP −d [0,1] d 1 {φ(x)≤4P γ δ} dx = mP −d ≤ 2P −γα ≤ 8δ α , where we used m = ⌈0.5P d−γα ⌉ ≤ 0.5P d−γα + 1 ≤ 2P d−γα and α ∈ (0, 1). On the other hand, if 4P γ δ ≤ 1, we obtain that P X (0 < |f ω (X) − 1/2| ≤ δ) = mP −d [0,1] d 1 {φ(x)≤4P γ δ} dx = mP −d − mP −d [0,1] d 1 {||x||∞<1−4 1/γ δ 1/γ P } dx = mP −d [1 − (1 − 4 1/γ δ 1/γ P ) d ] ≤ mP −d d4 1/γ δ 1/γ P ≤ 2dP 1−αγ 4 1/γ δ 1/γ ≤ 2d(4δ) α ≤ 8dδ α , which establishes (185). Step 4: Lower bounding sup (F 1 ,F −1 )∈S E(R n (π, F 1 , F −1 )). By Lemma D.5 it suffices to show that sup (F 1 ,F −1 )∈S E(S n (π, F 1 , F −1 )) ≥ c 3 n 1− αγ d+2γ for some c 3 > 0 independent of π. Note also that the left hand side of (186) is equal to sup f ∈C E n π,f S n (π) which we shall now lower bound. Since X t is independent of Z t−1 and π ⋆ (x) = sign(f ω (x) − 1/2), sup f ∈C E n π,f [S n (π)] = sup ω∈Ωm n t=1 E t−1 π,fω P X π t (X t , Z t−1 ) = sign(f ω (X t ) − 1/2), f ω (X t ) = 1/2) ≥ sup ω∈Ωm m j=1 n t=1 E t−1 π,fω [P X (π t (X t , Z t−1 ) = ω j , X t ∈ B j )] ≥ 1 2 m m j=1 n t=1 ω∈Ωm E t−1 π,fω [P X (π t (X t , Z t−1 ) = ω j , X t ∈ B j )]. Note that for every j ∈ {1, ..., m} and t ∈ {1, ..., n}, Q j t := ω∈Ωm E t−1 π,fω [P X (π t (X t , Z t−1 ) = ω j , X t ∈ B j )] = ω −j ∈Ω m−1 i∈{−1,1} E t−1 π,f ω i −j [P X (π t (X t , Z t−1 ) = i, X t ∈ B j )] where ω −j = (ω 1 , ..., ω j−1 , ω j+1 , ..., ω m ) and ω i −j = (ω 1 , ..., ω j−1 , i, ω j+1 , ..., ω m ) for i ∈ {−1, 1}. Note that for any u ∈ R (t−1)(d+1) , P X (π t (X t , u) = i, X t ∈ B j ) = P j X (π t (X t , u) = i)/P d with P j X (A) = P X (A|X t ∈ B j ) for any A ∈ B([0, 1] d ). Expectations with respect to P j X are denoted by E j X . Hence, for every ω −j ∈ Ω m−1 , i∈{−1,1} E t−1 π,f ω i −j [P X (π t (X t , Z t−1 ) = i, X t ∈ B j )] = 1 P d i∈{−1,1} E t−1 π,f ω i −j [P j X (π t (X t , Z t−1 ) = i)]. Here i∈{−1,1} E t−1 π,f ω i −j [P j X (π t (X t , Z t−1 ) = i)] = E t−1 π,f ω −1 −j E j X 1 {πt(Xt,Z t−1 )=1} + 1 − E t−1 π,f ω 1 −j E j X 1 {πt(Xt,Z t−1 )=1} is the sum of Type 1 and Type 2 errors for the testing problem H 0 : P t−1 π,f ⊗ P j X = P t−1 π,f ω −1 −j ⊗ P j X vs H a : P t−1 π,f ⊗ P j X = P t−1 π,f ω 1 −j ⊗ P j X for the test 1 {πt(Xt,Z t−1 )=1} . For any test π t this sum can be bounded from below, using Theorem 2.2(iii) of Tsybakov (2009) , by i∈{−1,1} E t−1 π,f ω i −j [P j X (π t (X t , Z t−1 ) = i)] ≥ 1 4 exp −KL P t−1 π,f ω −1 −j ⊗ P j X , P t−1 π,f ω 1 −j ⊗ P j X = 1 4 exp −KL P t−1 π,f ω −1 −j , P t−1 π,f ω 1 −j Thus, for every ω −j ∈ Ω m−1 , i∈{−1,1} E t−1 π,f ω i −j [P X (π t (X t , Z t−1 ) = i, X t ∈ B j )] ≥ 1 4P d exp −KL P t−1 π,f ω −1 −j , P t−1 π,f ω 1 −j and we next bound KL P t−1 π,f ω −1 −j , P t−1 π,f ω 1 −j from above. Using the chain rule for Kullback-Leibler divergence, cf. Theorem 2.5.3 of Cover and Thomas (2012) 5 , it follows that KL P t−1 π,f ω −1 −j , P t−1 π,f ω 1 −j = KL P t−2 π,f ω −1 −j ⊗ P X , P t−2 π,f ω 1 −j ⊗ P X + E t−2 π,f ω −1 −j E X KL P π,f ω −1 −j ,t−1 , P π,f ω 1 −j ,t−1 = KL P t−2 π,f ω −1 −j , P t−2 π,f ω 1 −j + E t−2 π,f ω −1 −j E X KL P π,f ω −1 −j ,t−1 , P π,f ω 1 −j ,t−1 . To proceed, note that by (183) for any s = 1, ..., t − 1, u = (x, z) ∈ R d × R (s−1)(d+1) (where u = x ∈ R d for s = 1) and f ∈ C KL P π,f ω −1 −j ,s , P π,f ω 1 Next, observe that the function f ω −1 −j − f ω 1 −j is γ-Hölder continuous with constant 2. Furthermore, it vanishes on the boundary of B j . Using these observations along with (188) and g f (x) = A(f (x)) for any f ∈ C, one obtains for any s = 1, ..., t − 1: KL P π,f ω −1 −j ,s , P π,f ω 1 −j ,s = KL P f ω −1 −j ,1 , P f ω 1 −j ,1 1 {πs(u)=1} ≤ 1 (1 +ā − δ) 2 g f ω −1 −j (x) − g f ω 1 −j (x) 2 1 {πs(u)=1,x∈B j } ≤ c 2 1 (1 +ā − δ) 2 f ω −1 −j (x) − f ω 1 −j (x) 2 1 {πs(z)=1,x∈B j } ≤ 4c 2 P 2γ 1 {πs(u)=1,x∈B j } , for a constantc = c 1 d γ/2 /(1 +ā − δ) (with c 1 being the Lipschitz constant of A). It thus follows by induction that for any t = 1, ..., n, j = 1, ..., m and policy π KL P t−1 π,f ω −1 −j , P t−1 π,f ω 1 −j ≤ 4c 2 P 2γ N j,π , where N j,π := E n−1 π,f ω −1 −j E X n s=1 1 {πs(Xs,Z s−1 )=1,Xs∈B j } . Thus, n t=1 Q j t ≥ n 2 m−1 4P d exp − 4c 2 P 2γ N j,π On the other hand, one also has n t=1 Q j t = n t=1 ω −j ∈Ω m−1 i∈{−1,1} E t−1 π,f ω i −j [P X (π t (X t , Z t−1 ) = i, X t ∈ B j )] ≥ n t=1 ω −j ∈Ω m−1 E t−1 π,f ω −1 −j [P X (π t (X t , Z t−1 ) = 1, X t ∈ B j )] = 2 m−1 N j,π . Using the above two displays in (187) for a constant c 3 = 0.5 d+2 d ln (2) 32 c 1 d 1+ā−δ ∨ 1 − 2 d+2 depending on neither α nor γ. Proof of Theorem 5.2. The proof of this theorem relies on Theorem 5.1 and the notation used is as in that theorem. Let F (γ) = f : [0, 1] d → [1/4, 3/4] such that |f (x 1 ) − f (x 2 )| ≤ 1/2||x 1 − x 2 || γ for all x 1 , x 2 ∈ [0, 1] d . Note that γ>0 F (γ) ⊆ C[0, 1] d . Thus, since c l in Theorem 5.1 does not depend on γ, we get sup f ∈C [0,1] d E n π,f [R n (π)] ≥ sup f ∈ γ>0 F (γ) E n π,f [R n (π)] ≥ c l n. Proof of Theorem 5.3. Fix a policy π and choose α ∈ (0, 1) such that γα/(2γ +d) < ε. Observe that S ⊆ S 0 for all α > 0. Furthermore, since c l = c l (α) in Theorem 5.1 equalsC(α)c 1+1/α 3 withC(α) as in Lemma D.5 (the dependence on α is suppressed there) and c 3 as in the last line of the proof of Theorem 5.1 one has that sup (F 1 ,F 2 )∈S 0 E[R n (π, F 1 , F 2 )] ≥ sup (F 1 ,F 2 )∈S E[R n (π, F 1 , F 2 )] ≥ c l (α)n 1− γ(1+α) 2γ+d ≥ c l (α)n 1− γ 2γ+d n −ε . Since α depends on ε, we write c l (ε) instead of c l (α). Proof of Theorem 5.4. Throughout this proof we shall use notation defined in the proof of Theorem 5.1. Fix a policy π ∈Π and let m 1 = 2m with m = ⌈0.5P d−αγ ⌉. Note that m 1 ≤ P d for P ≥ P 0 for P 0 sufficiently large. Set P = P 0 . Define ω (1) := (ι m , −ι m ), where ι m is a row vector of ones of length m, and let ω (−1) := −ω (1) . Set f i = f ω (i) for i ∈ {−1, 1}. Given f ∈ {f 1 , f −1 } ⊆ C m 1 , define P f,1 , P 1/2,1 , P 1/2,−1 and P t π,f as in Step 2 of the proof of Theorem 5.1. From the argument given in Step 3 of that proof it follows that C 1 := {(H g f (x) , H g 1/2 ) : f ∈ {f 1 , f −1 }} ⊆ S. Next, we defineZ t = (Y πt(Z t−1 ),t , ..., Y π 1 ,1 ), and we denote the distribution ofZ t byP t π,f . We claim thatP t π,f 1 =P t π,f −1 for every t ≥ 1. To this end, note first that from π ∈Π it follows thatZ t = F t (Y t , . . . , Y 1 ) for some measurable function F t . Hence, since the Y t are i.i.d., in order to prove the claim it is enough to verify that the distribution of the random vector Y 1 does not depend on f ∈ C 1 . Using the notation in Step 2 of the proof of Theorem 5.1 this is equivalent to: P f 1 ,1/2 (A 1 × A 2 × [0, 1] d ) = P f −1 ,1/2 (A 1 × A 2 × [0, 1] d ) for all Borel sets A 1 , A 2 in [0, 1]. To verify this equivalent condition, we write P f 1 ,1/2 (A 1 × A 2 × [0, 1] d ) = P 1/2,−1 (A 1 ) P f 1 ,1 (A 2 , x)P X (dx) as P 1/2,−1 (A 1 ) P d j=1 B j P f 1 ,1 (A 2 , x)P X (dx) = P 1/2,−1 (A 1 ) P d j=1 B j P f −1 ,1 (A 2 , x)P X (dx),(190) the latter coinciding with P f −1 ,1/2 (A 1 × A 2 × [0, 1] d ), which proves the claim. Here we have used that ω (1) = −ω (−1) implies B j P f 1 ,1 (A 2 , x)P X (dx) = B 2m+1−j P f −1 ,1 (A 2 , x)P X (dx) for j = 1, . . . , 2m, and that B j P f 1 ,1 (A 2 , x)P X (dx) = B j P f −1 ,1 (A 2 , x)P X (dx) for j = 2m + 1, . . . , P d . Now, to prove the theorem, by Lemma D.5 it suffices to show that sup S E[S n (π)] increases linearly in n. Using Equation (189) we see that sup S E[S n (π)] ≥ sup C 1 E[S n (π)] = sup f 1 ,f −1 E n π,f [S n (π)]. Next, arguing as in Equation (187) and exploiting π ∈Π we obtain sup f 1 ,f −1 E n π,f [S n (π)] ≥ sup i∈{−1,1} m j=1 n t=1 E t−1 π,f i P X π t (Z t−1 ) = i, X t ∈ B j = sup i∈{−1,1} m j=1 n t=1 P t−1 π,f i π t (Z t−1 ) = i P X (B j ) = P −d sup i∈{−1,1} m j=1 n t=1P t−1 π,f i π t (Z t−1 ) = i ≥ (2P d ) −1 m j=1 n t=1 P t−1 π,f 1 π t (Z t−1 ) = −1 +P t−1 π,f −1 π t (Z t−1 ) = 1 = mn 2P d ≥ 1 4P αγ 0 n, where we used independence of X t andZ t−1 to obtain the first equality, that each summand in the last double sum equals one (recall thatP t−1 π,f 1 =P t−1 π,f −1 ) to obtain the last equality, and the definition of m to obtain the final lower bound. is Lipschitz continuous on [(b/a) c(ε) , (a/b) c(ε) ] with constant c(ε) −1 (b/a) ε . From Equation(26)and because (b/a) c(ε) ≤ ε(ε − 1)E 1−ε (F ) + 1 ≤ (a/b) c(ε) holds for every F ∈ D cdf ([a, b]) (a > 0 and c(ε) < 0) the claim follows from Lemma C.1 (with m = 1). Lemma B. 1 . 1Let a < b be real numbers, z 0 > 0 and 0 ≤ δ ≤ 1. Then, the following holds:1. If δ = 0, then z m,z 0 ,δ satisfies Assumption 2.1 with D = D cdf ([a, b]), and C = 0.2. If δ > 0 and m = µ(.), then z m,z 0 ,δ satisfies Assumption 2.1 with D = D cdf ([a, b]) and C = δ(b − a).3. If δ > 0 and m = q 1/2 (.), then, for every r > 0 the poverty line z m,z 0 ,δ satisfies Assumption 2.1 with D = C r ([a, b]), and C = r −1 δ.Proof of Lemma B.1: Recall from Equation 31 that by definition z m,z 0 ,δ (F ) = z 0 + δ(m(F ) − z 0 ). The first statement is trivial; the second follows directly from Example C.3; and the third follows from Lemma C.12 and Example C.10.Proof of Lemma A.9: Since z satisfies Assumption 2.1 the functional z is well defined on D cdf([a, b]). B := [a, m]\A, and C := (m, M], F ∈ D cdf ((a, b]) if and only if F ∈ D cdf (R), F (a) = 0 and F (b) = 1. Likewise, we define the subset D cdf ([a, b]) of D cdf (R) as follows: F ∈ D cdf ([a, b]) if and only if F ∈ D cdf (R), F (a−) = 0 and F (b) = 1. Lemma C. 1 . 1Let a < b be real numbers, and let D ⊆ D cdf ([a, b]). Suppose that T i for i = 1, . . . , m satisfies Assumption 2.1 with a, b and D and with constant C i , respectively. Let I i := T i (D cdf ([a, b])) and set I = × m i=1 I i . Assume G : I → R is Lipschitz continuous with constant C, when I ⊆ R m is equipped with the metric induced by the norm . 1 on R m . Then, T = G • (T 1 , . . . , T m ) satisfies Assumption 2.1 with a, b and D and with constant C m i=1 C i . , and Fubini's theorem, we write m ϕ;c,d (F ) − m ϕ;c,d (G) as k j=1[c,d] hold, the statement in (78) follows from the total variation of µ f on [c, d] being not greater than C.Example C.3 (Mean). Let a < b be real numbers. Let k = 1 and set ϕ(x) = x, i.e., we consider the mean functional F → µ(F ), say, defined via F →[a,b] [a,b] |ϕ ′ (x)|dx = (b − a). As a consequence of Lemma C.2 the functional m ϕ is thus Lipschitz continuous on D cdf ([a, b]) with constant (b − a).Example C.4 (Moments). For simplicity, let a = 0 and b > 0. Let k = 1 and set ϕ(x) = x p for some p > 0, i.e., we consider the p-mean functional F →[0,b] (noting that q α (F ) ∈ [a, b] for every α ∈ (0, 1], cf. the proof of Lemma C.9).The next result in this section concerns population versions of L-statistics.Lemma C.13. Let ν be a measure on the Borel sets of [0, 1], and let J : [0, 1] → R be such that [0,1] |J(α)|dν(α) = c < ∞. Assume further that ν(0) = 0. Let d ∈ N ∪ {0}, let 0 < p 1 < . . . < p d ≤ 1, and let v 1 , . . . , v d be real numbers. Let a < b be real numbers and define on D cdf ([a, b]) the functional Lemma C. 14 . 14Let a < b be real numbers and define on D cdf ([a, b]) the family of functionals indexed by u ∈ [0, 1] and defined by if a > 0, define the family of functionals indexed by u ∈ [0, 1] via L(F, u) := µ(F ) Lemma C. 15 . 15Let a < b be real numbers, let ϕ : R → R, let ϕ restricted to [a, b] be continuous, let the total variation of ϕ on [a, b] be not greater than C, and let |ϕ(x)| ≤ u hold for all x ∈ [a, b]. Furthermore, let α ∈ (0, 1). For F ∈ D cdf ([a, b]) define m t− ϕ;α (F ) := [a,qα(F )] ϕ(x)dF (x) and m t+ ϕ;α (F ) := [qα(F ),b] P π,f,s (A, u s ) := P π,f,1,s (A, u s )1 {πs(us)=1} + P π,f,−1,s (A, u s )1 {πs(us)=−1} = P f,1 (A, x s )1 {πs(us)=1} + P 1/2,−1 (A)1 {πs(us) . . , x k . Let a ≤ c < d ≤ b.total variation not greater than C ∈ R. For F ∈ D cdf ([a, b]) define the functional m ϕ;c,d as the iterated Lebesgue-Stieltjes integralSuppose that for every x * 2 , . . . , x * k ∈ [c, d] k−1 the function x → ϕ(x, x * 2 , . . . , x * k ) defined on [c, d] is continuous and has m ϕ;c,d (F ) := [c,d] yields sup f sup∈C E n π,f [S n (π)] ≥ P 2γ N j,π , N j,π P 2γ z + z . P 2γ z + z = P 2γ 4c 2 log c 2 nP −d−2γis strictly positive if and only if P < nc 2 1/(d+2γ) in which case, upon choosing P = ⌈0.5 nc 2 1/(d+2γ) ⌉ and recalling m = ⌈0.5P d−αγ ⌉, we get1 2 m j=1 max n 4P d exp − 4c 2 ≥ 1 4 m j=1 n 4P d exp − 4c 2 P 2γ N j,π + N j,π ≥ m 4 inf z≥0 n 4P d exp − 4c 2 The unique z * = argmin z≥0 n 4P d exp − 4c 2 sup f ∈C E n π,f [S n (π)] ≥ m 4 P 2γ 4c 2 ln c 2 nP −d−2γ ≥ P d+γ(2−α) 32c 2 ln(2 d+2γ ) ≥ 0.5 d+γ(2−α) 32c 2 (nc 2 ) d+γ(2−α) d+2γ ln(2 d+2γ ) ≥ 0.5 d+2 d ln(2) 32c − 2αγ d+2γ n 1− αγ d+2γ ≥ 0.5 d+2 d ln(2) 32 c 1 d 1 +ā − δ ∨ 1 − 2αγ d+2γ n 1− αγ d+2γ ≥ 0.5 d+2 d ln(2) 32 c 1 d 1 +ā − δ ∨ 1 − 2 d+2 n 1− αγ d+2γ = c 3 n 1− αγ d+2γ Note that we allow π t , the assignment of a policy π for individual t, to be a function of X t and Z t−1 . Thus, one may argue that the oracle should assign the treatment i ∈ I for which the conditional distribution of Y i,t given X t and Z t−1 maximizes T. However, by independence of (Y i,t , X t ) and Z t−1 , this distribution is P i (·, x), i.e.,the conditional distribution of Y i,t given X t . Note that for relative inequality measures I rel defined on D cdf ([a, b]) for 0 ≤ a < b it is typically the case that I rel (F ) ∈ [0, 1], and hence 1 − I rel (F ) ∈ [0, 1] as well. Recall that the marginal distribution of Y −1,t , P 1/2,−1 , is also the conditional distribution of Y −1,t given X t by the independence of Y −1,t and X t . While the proof inCover and Thomas (2012) is for discrete measures, the same proof technique applies equally well to measures equivalent to a product of Lebesgue measures which is the case in our setting as observed at the end of Step 2. S Athey, S Wager, arXiv:1702.02896Efficient policy learning. arXiv preprintAthey, S. and S. Wager (2017). Efficient policy learning. arXiv preprint arXiv:1702.02896 . On the measurement of inequality. A B Atkinson, Journal of Economic Theory. 23Atkinson, A. B. (1970). On the measurement of inequality. Journal of Economic Theory 2 (3), 244-263. Fast learning rates for plug-in classifiers. J.-Y Audibert, A B Tsybakov, Annals of Statistics. 352Audibert, J.-Y. and A. B. Tsybakov (2007). Fast learning rates for plug-in classifiers. Annals of Statistics 35 (2), 608-633. Finite-time analysis of the multiarmed bandit problem. P Auer, N Cesa-Bianchi, P Fischer, Machine Learning. 472-3Auer, P., N. Cesa-Bianchi, and P. Fischer (2002). Finite-time analysis of the multiarmed bandit problem. Machine Learning 47 (2-3), 235-256. Inferring welfare maximizing treatment assignment under budget constraints. D Bhattacharya, P Dupas, Journal of Econometrics. 1671Bhattacharya, D. and P. Dupas (2012). Inferring welfare maximizing treatment assignment under budget constraints. Journal of Econometrics 167 (1), 168-196. Regret analysis of stochastic and nonstochastic multi-armed bandit problems. S Bubeck, N Cesa-Bianchi, Foundations and Trends R in Machine Learning. 51Bubeck, S. and N. Cesa-Bianchi (2012). Regret analysis of stochastic and nonstochastic multi-armed bandit problems. Foundations and Trends R in Machine Learning 5 (1), 1-122. A general approach to multi-armed bandits under risk criteria. A Cassel, S Mannor, A Zeevi, arXiv:1806.01380arXiv preprintCassel, A., S. Mannor, and A. Zeevi (2018). A general approach to multi-armed bandits under risk criteria. arXiv preprint arXiv:1806.01380 . A new index of poverty. S R Chakravarty, Mathematical Social Sciences. 63Chakravarty, S. R. (1983). A new index of poverty. Mathematical Social Sciences 6 (3), 307-313. Inequality, polarization and poverty. S R Chakravarty, SpringerChakravarty, S. R. (2009). Inequality, polarization and poverty. Springer. Econometrics and decision theory. G Chamberlain, Journal of Econometrics. 952Chamberlain, G. (2000). Econometrics and decision theory. Journal of Econometrics 95 (2), 255-283. Elements of information theory. T M Cover, J A Thomas, John Wiley & SonsCover, T. M. and J. A. Thomas (2012). Elements of information theory. John Wiley & Sons. Measuring inequality. F Cowell, Oxford University PressCowell, F. (2011). Measuring inequality. Oxford University Press. Generalized entropy and the measurement of distributional change. F A Cowell, European Economic Review. 131Cowell, F. A. (1980). Generalized entropy and the measurement of distributional change. European Economic Review 13 (1), 147-159. Program evaluation as a decision problem. R H Dehejia, Journal of Econometrics. 1251Dehejia, R. H. (2005). Program evaluation as a decision problem. Journal of Econometrics 125 (1), 141-173. A note on generalized inverses. P Embrechts, M Hofert, Mathematical Methods of Operations Research. 773Embrechts, P. and M. Hofert (2013). A note on generalized inverses. Mathematical Methods of Operations Research 77 (3), 423-432. Real Analysis: Modern Techniques and their Applications. G B Folland, WileyFolland, G. B. (2013). Real Analysis: Modern Techniques and their Applications. Wiley. A class of decomposable poverty measures. J Foster, J Greer, E Thorbecke, Econometrica. 523Foster, J., J. Greer, and E. Thorbecke (1984). A class of decomposable poverty measures. Economet- rica 52 (3), 761-766. The Foster-Greer-Thorbecke (FGT) poverty measures: 25 years later. J Foster, J Greer, E Thorbecke, The Journal of Economic Inequality. 84Foster, J., J. Greer, and E. Thorbecke (2010). The Foster-Greer-Thorbecke (FGT) poverty measures: 25 years later. The Journal of Economic Inequality 8 (4), 491-524. A general definition of the Lorenz curve. J L Gastwirth, Econometrica. 396Gastwirth, J. L. (1971). A general definition of the Lorenz curve. Econometrica 39 (6), 1037-1039. Asymptotics for statistical treatment rules. K Hirano, J R Porter, Econometrica. 775Hirano, K. and J. R. Porter (2009). Asymptotics for statistical treatment rules. Econometrica 77 (5), 1683-1701. On a class of poverty measures. N Kakwani, Econometrica. Kakwani, N. (1980). On a class of poverty measures. Econometrica, 437-446. Analyzing redistribution policies: a study using Australian data. N Kakwani, Cambridge University PressKakwani, N. (1986). Analyzing redistribution policies: a study using Australian data. Cambridge Uni- versity Press. Equality-minded treatment choice. T Kitagawa, A Tetenov, Working paper .Kitagawa, T. and A. Tetenov (2017). Equality-minded treatment choice. Working paper . Who should be treated? Empirical welfare maximization methods for treatment choice. T Kitagawa, A Tetenov, Econometrica. 862Kitagawa, T. and A. Tetenov (2018). Who should be treated? Empirical welfare maximization methods for treatment choice. Econometrica 86 (2), 591-616. Optimal sequential treatment allocation. A B Kock, M Thyrsgaard, arXiv:1705.09952arXiv preprintKock, A. B. and M. Thyrsgaard (2017). Optimal sequential treatment allocation. arXiv preprint arXiv:1705.09952 . 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V Perchet, P Rigollet, Annals of Statistics. Perchet, V. and P. Rigollet (2013). The multi-armed bandit problem with covariates. Annals of Statistics, 693-721. Nonparametric bandits with covariates. P Rigollet, A Zeevi, Proceedings of COLT. COLTRigollet, P. and A. Zeevi (2010). Nonparametric bandits with covariates. Proceedings of COLT . Some aspects of the sequential design of experiments. H Robbins, Bulletin of the Americal Mathematical Society. 585Robbins, H. (1952). Some aspects of the sequential design of experiments. Bulletin of the Americal Mathematical Society 58 (5), 527-535. Nonparametric estimation of distributional policy effects. C Rothe, Journal of Econometrics. 1551Rothe, C. (2010). Nonparametric estimation of distributional policy effects. Journal of Economet- rics 155 (1), 56-70. Partial distributional policy effects. C Rothe, Econometrica. 805Rothe, C. (2012). Partial distributional policy effects. Econometrica 80 (5), 2269-2301. 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{'abstract': 'In this paper we study a treatment allocation problem with multiple treatments, in which the individuals to be treated arrive sequentially. The goal of the policy maker is to treat every individual as well as possible. Which treatment is "best" is allowed to depend on various characteristics (functionals) of the individual-specific outcome distribution of each treatment. For example measures of welfare, inequality, or poverty. We propose the Functional Sequential Allocation policy, and provide upper bounds on the regret it incurs compared to the oracle policy that knows the best treatment for each individual. These upper bounds increase sublinearly in the number of treatment assignments and we show that this regret guarantee is minimax optimal. In addition, the assumptions under which this regret guarantee is established are as weak as possible -even a minimal weakening of them will imply non-existence of policies with regret increasing sub-linearly in the number of assignments. Furthermore, we provide an upper bound on the number of suboptimal assignments made by the FSA policy and show that every policy must make a number of suboptimal assignments at least of this order.JEL Classification: C18, C22, J68.', 'arxivid': '1812.09408', 'author': ['Anders Bredahl Kock anders.kock@economics.ox.ac.uk \nECARES, SBS\nUniversity of Oxford CREATES\nAarhus University\nUniversité libre de Bruxelles\nAarhus University\nEM\n', 'David Preinerstorfer david.preinerstorfer@ulb.ac.be \nECARES, SBS\nUniversity of Oxford CREATES\nAarhus University\nUniversité libre de Bruxelles\nAarhus University\nEM\n', 'Bezirgen Veliyev bveliyev@econ.au.dk \nECARES, SBS\nUniversity of Oxford CREATES\nAarhus University\nUniversité libre de Bruxelles\nAarhus University\nEM\n'], 'authoraffiliation': ['ECARES, SBS\nUniversity of Oxford CREATES\nAarhus University\nUniversité libre de Bruxelles\nAarhus University\nEM', 'ECARES, SBS\nUniversity of Oxford CREATES\nAarhus University\nUniversité libre de Bruxelles\nAarhus University\nEM', 'ECARES, SBS\nUniversity of Oxford CREATES\nAarhus University\nUniversité libre de Bruxelles\nAarhus University\nEM'], 'corpusid': 69517956, 'doi': '10.1080/01621459.2020.1851236', 'github_urls': [], 'n_tokens_mistral': 67852, 'n_tokens_neox': 60819, 'n_words': 35227, 'pdfsha': '591c554d1bfddb722e293dd5bba45e6ec47b932e', 'pdfurls': ['https://arxiv.org/pdf/1812.09408v1.pdf'], 'title': ['Functional Sequential Treatment Allocation', 'Functional Sequential Treatment Allocation'], 'venue': []}

arxiv

Identifying network topologies via quantum walk distributions Claudia Benedetti Dipartimento di Fisica "Aldo Pontremoli" Università degli Studi di Milano 20133MilanItaly Ilaria Gianani Dipartimento di Scienze Università degli Studi Roma Tre Via della Vasca Navale 8400146RomeItaly Identifying network topologies via quantum walk distributions Control and characterization of networks is a paramount step for the development of many quantum technologies. Even for moderate-sized networks, this amounts to explore an extremely vast parameters space in search for the couplings defining the network topology. Here we explore the use of a genetic algorithm to retrieve the topology of a network from the measured probability distribution obtained from the evolution of a continuous-time quantum walk on the network. Our result shows that the algorithm is capable of efficiently retrieving the required information even in the presence of noise.Networks are a fundamental model to understand the underlying properties of complex systems. They are invaluable tools to describe phenomena happening at different scales ranging from social interactions[1,2], to biological processes[3][4][5][6][7], from the configurations of molecules[8,9], to the structure of internet [10-13] and physical systems alike[14][15][16][17][18]. In the context of quantum technologies, networks constitute the prime structure of communication and computation protocols[19][20][21][22][23]. Understanding how quantum information can be reliably transmitted between distant nodes of a network, or routed among different computational units, is a key step and requires a full characterization of the network's structure. While a direct control may not be attainable with the required accuracy and precision, a straightforward strategy to provide such characterization is that of probing the network with a walker that gathers information on its topology by undergoing an evolution which depends on the network's structure. This is the case of continuous-time quantum walks (CTQWs)[24][25][26][27][28][29][30][31][32][33], which thus emerge as a natural paradigm for tackling this task.Two different scenarios may present: the topology of the network may be known, but an accurate estimation of the coupling strengths between each node may be required. This is tantamount to estimating multiple parameters, and can be address in quantum metrological terms[34][35][36][37]. It might otherwise be the case that the topology of the network is not known in advance. Whether one is interested in characterizing a physical network or a simulated one, this will be relying on an experimental platform controlled with set of experimental parameters Λ exp . These need to be mapped to the associated set of parameters describing the CTQW happening on the network, Λ QW , i.e. the Hamiltonian parameters of the quantum walk which, assuming all coupling strengths are fixed to unity and on-site energies to zero, coincide with the adjacency matrix identifying the topology of the network. In order to asses the evolution of the probe, one has to address an observable, such as * ilaria.gianani@uniroma3.it the spatial probability distribution on the network, which will strongly depend on the network's topology. However, since an analytical description of this distribution for CTQWs is often unattainable, and furthermore the relation between the QW Hamiltonian and its probability distribution is highly non-linear, performing a direct inversion can be involved. At the same time, the parameter space in this instance becomes exceedingly large for this to be treated as an estimation problem. An alternative solution is to cast the issue in terms of a search problem. Having access solely to the initial state of the probe and to the measured experimental distributions at fixed times, the task becomes that of finding an adjacency matrix that matches the evolution. Here we tackle this matter by using a genetic algorithm. We use the algorithm to successfully retrieve different topologies in the ideal case as well as when the measured probabilities are affected by noise.We consider a CTQW with zero on-site energies, defined by the couplings Λ QW = {J xy } between two nodes of the network x and y, such that its Hamiltoian is:(1)We assume that the couplings J xy can take only two values: J xy = 0 if the link between two nodes is off, or J xy = 1 if the link is on, so that each edge is bound to have the same strength. The Hamiltonian thus coincides with the adjacency matrix of the network, hence, determining its parameters amounts to determining the network's topology. The evolution of a walker in the initial state |ψ 0 is described by the unitary operator e −iHt . The probability of occupying a site x at a time t is then:Given an undirected graph of n sites, our objective is that of retrieving the couplings Λ QW = {J 12 , . . . , J (n−1)n }, i.e. a binary string of length n c =n(n−1)/2, having access only to the initial state of the network and to the probabilities p x (t k , Λ QW ) measured at times t k . We tackle this challenge by means of a genetic algorithm (GA). GAs are versatile iterative search algorithms inspired by natural selection and have been extensively arXiv:2301.13842v1 [quant-ph] 31 Jan 2023 Control and characterization of networks is a paramount step for the development of many quantum technologies. Even for moderate-sized networks, this amounts to explore an extremely vast parameters space in search for the couplings defining the network topology. Here we explore the use of a genetic algorithm to retrieve the topology of a network from the measured probability distribution obtained from the evolution of a continuous-time quantum walk on the network. Our result shows that the algorithm is capable of efficiently retrieving the required information even in the presence of noise. Networks are a fundamental model to understand the underlying properties of complex systems. They are invaluable tools to describe phenomena happening at different scales ranging from social interactions [1,2], to biological processes [3][4][5][6][7], from the configurations of molecules [8,9], to the structure of internet [10][11][12][13] and physical systems alike [14][15][16][17][18]. In the context of quantum technologies, networks constitute the prime structure of communication and computation protocols [19][20][21][22][23]. Understanding how quantum information can be reliably transmitted between distant nodes of a network, or routed among different computational units, is a key step and requires a full characterization of the network's structure. While a direct control may not be attainable with the required accuracy and precision, a straightforward strategy to provide such characterization is that of probing the network with a walker that gathers information on its topology by undergoing an evolution which depends on the network's structure. This is the case of continuous-time quantum walks (CTQWs) [24][25][26][27][28][29][30][31][32][33], which thus emerge as a natural paradigm for tackling this task. Two different scenarios may present: the topology of the network may be known, but an accurate estimation of the coupling strengths between each node may be required. This is tantamount to estimating multiple parameters, and can be address in quantum metrological terms [34][35][36][37]. It might otherwise be the case that the topology of the network is not known in advance. Whether one is interested in characterizing a physical network or a simulated one, this will be relying on an experimental platform controlled with set of experimental parameters Λ exp . These need to be mapped to the associated set of parameters describing the CTQW happening on the network, Λ QW , i.e. the Hamiltonian parameters of the quantum walk which, assuming all coupling strengths are fixed to unity and on-site energies to zero, coincide with the adjacency matrix identifying the topology of the network. In order to asses the evolution of the probe, one has to address an observable, such as * ilaria.gianani@uniroma3.it the spatial probability distribution on the network, which will strongly depend on the network's topology. However, since an analytical description of this distribution for CTQWs is often unattainable, and furthermore the relation between the QW Hamiltonian and its probability distribution is highly non-linear, performing a direct inversion can be involved. At the same time, the parameter space in this instance becomes exceedingly large for this to be treated as an estimation problem. An alternative solution is to cast the issue in terms of a search problem. Having access solely to the initial state of the probe and to the measured experimental distributions at fixed times, the task becomes that of finding an adjacency matrix that matches the evolution. Here we tackle this matter by using a genetic algorithm. We use the algorithm to successfully retrieve different topologies in the ideal case as well as when the measured probabilities are affected by noise. We consider a CTQW with zero on-site energies, defined by the couplings Λ QW = {J xy } between two nodes of the network x and y, such that its Hamiltoian is: H(Λ QW ) = xy J xy |x y| .(1) We assume that the couplings J xy can take only two values: J xy = 0 if the link between two nodes is off, or J xy = 1 if the link is on, so that each edge is bound to have the same strength. The Hamiltonian thus coincides with the adjacency matrix of the network, hence, determining its parameters amounts to determining the network's topology. The evolution of a walker in the initial state |ψ 0 is described by the unitary operator e −iHt . The probability of occupying a site x at a time t is then: p x (t, Λ QW ) = | x| e −iH(Λ QW )t |ψ 0 | 2 .(2) Given an undirected graph of n sites, our objective is that of retrieving the couplings Λ QW = {J 12 , . . . , J (n−1)n }, i.e. a binary string of length n c =n(n−1)/2, having access only to the initial state of the network and to the probabilities p x (t k , Λ QW ) measured at times t k . We tackle this challenge by means of a genetic algorithm (GA). GAs are versatile iterative search algorithms inspired by natural selection and have been extensively Given an initial probe state |ψ0 and a network with unknown topology controlled by a set of experimental parameters, we aim at retrieving the topology of the network measuring the probability distributions of the the probe evolved with a CTQW. This is achievede through a genetic algorithm in which the probability distributions are employed to evaluate the fitness score, as described in the main text. employed for quantum tasks [38][39][40][41]. They rely on the evolution of a population of individuals, each defined by a chromosome string and a fitness score, which breed new individuals replacing the previous population at each iteration. By promoting the reproduction of the fittest individuals while introducing various mechanisms to ensure enough genetic variability, GAs allow to efficiently retrieve the optimal solution [42,43]. We encode the chromosomes as binary strings Λ i of length n c , so that each gene constituting the chromosome is a coupling J xy . The fitness of each individual is evaluated as follows: Λ i is used to evolve the initial state of the probe up to selected times t k obtaining the probability distributions p x (t k , Λ i ). For practical purposes, we concatenate the probabilities at different times in a single array that we call π x ({t k }, Λ i ). Using multiple times allows to remove eventual ambiguities and to mitigate the effects of local minima, thus improving the performance of the algorithm. We then check the distance between these probabilities and the measured ones π x ({t k }, Λ QW ) e.g. by using the Kullback-Leibler divergence. When the distance is null, Λ i = Λ QW . The value of the distance will be the fitness score of each individual. Thus, in our case, the more fit an individual, the smaller its fitness score. The correct couplings will be those having a fitness score equal to 0. The algorithm scheme is shown in Fig. 1 and operates as follows: An initial random population of size n p is generated, and its fitness is evaluated as described above. An elitist function selects a small percentage p e of individuals with the best fitness scores to constitute the hall of fame, which will be cloned in the next generation. The whole population is then entered in a tournament where k in-dividuals at the time compete to be selected for breeding the next generation. This is achieved through a crossover strategy in which the chromosomes of the selected parents are mixed with a probability p c . The size of the population is kept constant through each generation, so that each selected pair of individuals will produce two children. In order to ensure genetic diversity, with a small probability p m , children can undergo mutations, consisting in bit flips. The new born children together with the cloned hall of fame individuals become the next generation, and the algorithm continues iteratively, stopping either when the optimal individual is found (i.e. fitness score equals to zero), or when a maximum number of generations n g is reached. A full depiction of the algorithm and of the implementation of the genetic operations are reported in the Supplementary Information. The initial state of the probe as well as the evolution times at which the probabilities are measured play a fundamental role towards the success of the algorithm: for instance, choosing a localized state may result in discarding part of the network, if composed by two or more disjoints subnetworks; evolving the state for too short a time in a large network, may preclude the state to reach the whole network. While we do not perform a full optimization of the initial state, we choose one that allows to explore a large variety of different topologies and network sizes. Also all the hyperparameters defining the algorithm (population size n p , elitist population p e , individuals involved in each tournament k, crossover probability p c , mutation probability p m , max number of generations n g ) can be optimized in accordance with the task at hand and specifically with the network size. In our analysis we vary the network size to explore how the algorithm scales with an increasing number of couplings, but for the sake of simplicity we have chosen to keep all hyperparameters fixed aside from the population size n p . Our results hence are but a lower bound to the achievable performance attainable by fine-tuning for a fixed network size. Here we report the results obtained with a star graph, a complete graph and a graph with an arbitrary topology. This last network is a simplified version of the graph in Ref [44] describing the relations between the characters in the novel Les Misérables [45]. Results for additional topologies (line and circle) and further details on the generation of the Les Misérables graph can be found in the Supplementary Information. In order to test the algorithm, we inspect networks with nodes from n=5 to n=10, thus we search for binary strings with length n c = 10 to n c = 45. We measure the probability distributions at two distinct times, t 1 = 0.5 and t 2 = 0.6. As mentioned above, all hyperparameters are kept fixed (see Supplementary Information), aside from the population size n p which we scale as n p = 2 · n 2 c . This ensures a trade-off between computation time and performance, and allows us to provide a controlled comparison for the performance at different sizes. We fix the maximum number of generations to n g = 100, and, for each configuration, we run the algorithm N=100 times. We first consider the ideal case in which the probability distributions are noiseless. The results are reported in Fig 2, which shows the couplings values (green = 1, fuchsia=0) obtained for each run of N for the star (a-f), complete (h-m) and Les Misérables graph (o-t), as well as the success rate in each instance (g, n, u). Fig. 2 highlights how most of the times when the algorithm fails it returns the same (wrong) couplings. This effect is due to the algorithm getting stuck in the same local minima because for the chosen evolution times there are multiple configurations leading to probabilities which are very close to the true one. The most affected network is the complete, whose success rate, for n=10, drops to 31%. However, it is sufficient to run the algorithm including also a third probability measured at time t 3 = 1, and a success rate of 73% is recovered (see the Supplementary Information). In Fig. 3 we report the number of generations needed for convergence as a function of n, which predictably increases with the number of network sites, as does the search space. Our results show a remarkable efficiency of the search algorithm employed: in fact, the number of possible combinations Λ i scales with 2 nc , while we are inspecting, at most, 2 · n 2 c · n g combinations, assuming the worst case scenario in which we run the algorithm for the maximum number of generations and completely replace the population each time. For n=10, when n c = 45 the combinations hence amount to ∼ 3.5 · 10 13 , and we are exploring less than ∼ 4 · 10 5 configurations. In a real-life scenario, the probabilities p(t k , Λ QW ) used to evaluate the fitness score would be affected by noise. This needs to be accounted for when evaluating the distance by setting a threshold value T below which two probabilities are considered equal. The algorithm thus needs to be modified to halt whether the distance between the measured and evaluated probabilities is smaller than T , which counts as a success, or when it reaches the maximum number of generations, in which case the algorithm has failed. Depending on the value of T, there can be four outcomes: 1) True negative: The algorithm fails to reach T and the couplings are not found. 2) False negative: The algorithm fails to reach T, but the exact string of couplings has been found. This happens if T is set too low compared to the noise affecting the probabilities. 3) True positive: the alogirhtm successfully finds a fitness below T, and that corresponds to the exact couplings. 4) False positive: the algorithm successfully finds a fitness below T, but the couplings are not correct. This happens when the threshold is set too high compared to the noise, and hence the algorithm stops be-fore it can converge. In order to test this behaviour, we simulate the measured probabilities for a network with n=5 for a star topology and a fully connected topology, using the same hyperparameters as before aside from the max number of generations which we set to n g = 5. We know from the ideal case (Fig. 3) that for these topologies the algorithm converges in more than 5 generations, so we do expect to have some true negative outcomes. We simulate the probability measurements with a total of N r resources ranging from N r = 500 to N r = 5000, and through a Montecarlo (MC) routine we add Possionian noise to the simulated probabilities. For each MC run, we average the successes/fails over N=10 runs of the algorithm. We record the results for threshold values ranging from T= 4 · 10 −4 to T= 0.2. In Fig. 4 we report the results of the success/fail rates over 100 MC runs for the star network (a-b) and complete network (c-d) with N r = 500 (a-c) and N r = 5000 (b-d) as a function of the threshold value (for other N r see Supplementary Information). As expected, we can observe the four behaviours described early: when T is too low, the outcomes are dominated by false negatives (light red), with a small percentage of true negatives, due to the fact that the algorithm would take more than 5 generations to converge. As the threshold increases so do the number of true positives, while the true negatives remain constant. For larger thresholds both the true positive and true negative drop. The algorithm always satisfies the threshold condition before it can converge to the actual solution. In conclusion, we have employed a genetic algorithm to retrieve the topology of a network having access solely to the initial state of a probe undergoing a CTWQ and to the measured probability distributions at given times. We have explored the performance of the algorithm for different network sizes and topologies, as well as when the measured probabilities are affected by Poissonian noise. The algorithm maintains high performance levels for all the configurations explored, which could be further optimized by fine-tuning the hyperparameters for a specific network size. The genetic algorithm is particularly suited to address large parameter spaces, however increasing the network size by order of magnitudes or remove the constraint on the coupling strength would make it challenging in terms of computational times and resources. In order to achieve such scalability, a perspective is that of extending these results to incorporated machine learning techniques. By relying solely on measured probabilities, our technique provides a simple but yet effective strategy for the routine characterization of networks, and as such constitutes an enabling step towards most developing quantum technologies based on complex networks [46][47][48][49][50][51], as well as a tool for exploring new involved simulation regimes which have non-trivial mapping between the experimental control and the CTQW parameters [52][53][54]. The algorithm begins with an initial population initialized by generating n p random binary arrays Λ i of size n c , containing the couplings J xy , which in this representation correspond to the genes of each individual. These n p chromosomes correspond to the zeroth generation. While the number of generations is lower than n g , we proceed as follows: We evaluate the score S i of each string Λ i using the Fitness function described in details in the next section. The best fitness score, corresponding to the lowest value, and the relative couplings are stored. If the score is equal to zero, the optimal solution has been found, the algorithm stops and returns it. If the condition is not met, the algorithm continues by selecting the fittest p e · n p chromosomes Λ i and places them in a hall of fame, to be cloned in the following generation. Since the population size has to stay constant, we need to create the remaining n p (1 − p e ) individuals which will populate the next generation together with those cloned from the hall of fame. In order to do so we select the best parents from the whole population (including the hall of fame). This is achieved with the Tournament selection function, which randomly selects k individuals at a time and returns the best among them (lowest fitness score). The random selection of the k competitors ensures that the chosen individuals are not necessarily the best in the population. In this way, genetic diversity is ensured to mitigate the presence of local minima. Once the parents are selected, they are mixed through the Crossover function, which returns two children which, with probability p c , are composed by a mixture of the parents chromosomes. To further ensure genetic diversity, the genes of the children can undergo mutations with mutation probability p m . When a mutation happens, the gene is flipped. The generated children together with the hall of fame constitute the new generation. The algorithm repeats until either a chromosome with fitness score equal to zero is found, or the maximum number of generations is reached. The pseudocode of the algorithm reported in Algorithm 1. We define the genetic functions which are used in the algorithm: Fitness evaluation. The algorithm evaluates the fitness of each individual in the population Λ i by evolving the initial state using the couplings in Λ i and measuring the distance between the generated and measured probabilities concatenated at different times t k , i.e. π x ({t k }, Λ i ) and π x ({t k }, Λ QW ) respectively. The distance is measured with the Kullback-Leibler divergence (KLD), defined as: KLD(Λ i ) = x π x ({t k }, Λ i ) log π x ({t k }, Λ i ) π x ({t k }, Λ QW ) .(3) We note we have also tried metrics such as the Kolmogorov distance, obtaining analogous results. Torunament selection. We select the individuals for reproduction among the whole population running repeated tournaments between k individuals at a time. We need to select n p (1 − p e ) individuals so that, since every couple will produce two children with probability p c , the size of the population remains unchanged at each iteration. During each tournament, k individuals at random are selected among the whole population. The fittest one among the k (i.e. that with the smallest KLD) is chosen as a parent. Crossover. Children are created two at a time. Both are initialized with the chromosome of one of their parents each. With a probability p c , their chromosomes are crossed over. If the crossover happens, a random integer number smaller than n c is selected, and serves as the splitting point for the chromosome of the two parents: one child's chromosome will be comprised of the chromosome of the first parent up to the splitting point, and of the second parent thereafter -and viceversa for the other child. Mutation For each child, each gene can undergo a mutation with a probability p m . This is achieved by selecting a random number between 0 and 1. If the number is smaller than p m , then the gene will be flipped. The pseudocode for each function is reported in Algorithm 2. Algorithm 2 Genetic functions 1: function Fitness(Λi, π(t k , Λ QW )): 2: Evaluate π(t k , Λi) 3: Evaluate KLD(π(t k , Λi), π(t k , Λ QW )) 4 In order to test the algorithm on a graph with a random topology we adopt a simplified version of the graph describing the connections between the characters in the novel Les Misérables by V. Hugo. We use only the main characters, and we fix all the coupling strengths to 1. We start from n = 5 characters, and then increase n by introducing new characters. The resulting graphs are reported in Fig. 5 D. Results for additional topologies In Fig. 6 we report the results obtained without noise for the line and circle topologies. Panels (a-f) and (h-m) show the couplings for the N=100 runs of the algorithm, while panels (g,n) show the success rate. In Fig. 7 we report the generations needed for convergence. E. Complete Network with n=10 As shown in the main text, the complete network for n=10 is the most affected by local minima, which pre-vent the algorithm to converge to the correct solution dropping the success rate to 31%. This is because at the chosen times, there are configurations leading to similar probabilities than the complete one. However, it is sufficient to repeat the algorithm adding a third probability measured at t 3 = 1, to drastically increase the success rate up to 73%. The retrieved couplings are reported in Fig. 8. F. Results with noise for additional resources We report additional results with noise for N r = 1000, and N r = 2500 for the star and the complete networks with n=5. The success/fail rates are shown in Fig. 9 FIG. 1 . 1Conceptual scheme. FIG. 2 . 2Results without noise Retrieved couplings for N runs of the algorithm for star (a-f), complete (h-m), and random (o-t) networks for varying network sizes: (a,h,o) n=5, (b,i,p) n=6, (c,j,q) n=7, (d,k,r) n=8, (e,l,s) n=9 (f,m,t) n=10. (g,n,u) success rates as a function of network size. Green indicates a coupling equal to 1, fuchsia a coupling equal to 0. FIG. 3 . 3Algorithm convergence. Average numbers of generations required for convergence over N runs of the algorithm for a star (a), complete (b), and random (c) network as described in the main text. The shaded region is the standard deviation error over the N runs. FIG. 4 . 4Results with noise. Success/fail rates over the MC runs for the algorithm performed on a star network (a-b) and a complete network (c-d) for Nr = 500 resources (a-c) and Nr = 5000 resources (b-d). Light green: true positives; Dark green: false positives; Light red: false negatives; Dark red: true negatives. Randomly generate np binary arrays {Λi} 3: Pgen ← {Λi} Initialize population 4: while gen < ng do 5:for i = 0 → np − 1 do 6:Si= Fitness(Λi, π ({t k }, Λ QW ) FIG. 7 . 7Generations required for convergence for a line network (upper panel) and a circle network (lower panel) FIG. 8. Retrieved couplings for a n=10 complete network using probability distribution measured at three evolution times: t1 = 0.5, t2 = 0.6, and t3 = 1 TABLE I . IGenetic algorithm parameters B. Genetic Operations : 27 : 27return KLD 5: end function 6: function Tournament(Pgen, S): id← random integer in [0, np] for j = 0 → k − 2 do aux← random integer in [0, np) if S[aux] < S[id] then return Λ[id] 15: end function 16: function Crossover(Λ1, Λ2): Generate a random integer x in [0, 1] 18: if x < pc then 19: y ← random integer in [0, nc) child1 ← concatenate(Λ1[0 : y], Λ2[y + 1 : nc − 1]) child2 ← concatenate(Λ2[0 : y], Λ1[y + 1 : nc − 1]) return child1,child2 24: end function 25: function Mutation(childi): for j = 0 → nc − 1 do Generate random x in [0,1] if x < pm then 29: childi[j] ← 1−childi[j] return childi 33: end function C. Les Misérables graph7: 8: 9: 10: 11: id← aux 12: end if 13: end for 14: 17: 20: 21: 22: end if 23: 26: 28: 30: end if 31: end for 32: S Wasserman, K Faust, 10.1017/CBO9780511815478Social Network Analysis: Methods and Applications. Cambridge University PressS. Wasserman and K. 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Candeloro, C. Benedetti, M. G. Genoni, and M. G. A. Paris, Feedback-assisted quantum search by continuous- time quantum walks, Adv. Quantum Technol. 6, 2200093 (2023). Quantum walk in position space with single optically trapped atoms. M Karski, L Förster, J M Choi, A Steffen, W Alt, D Meschede, A Widera, 10.1126/science.1174436Science. 3255937M. Karski, L. Förster, J. M. Choi, A. Steffen, W. Alt, D. Meschede, and A. Widera, Quantum walk in posi- tion space with single optically trapped atoms, Science 325(5937), 174-177 (2009). Probing quantum walks through coherent control of high-dimensionally entangled photons. P Imany, N B Lingaraju, M S Alshaykh, D E Leaird, A M Weiner, 10.1126/sciadv.aba8066Science Advances. 68066P. Imany, N. B. Lingaraju, M. S. Alshaykh, D. E. Leaird, and A. M. Weiner, Probing quantum walks through co- herent control of high-dimensionally entangled photons, Science Advances 6, eaba8066 (2020). A W Young, W J Eckner, N Schine, A M Childs, A M Kaufman, 10.1126/science.abo0608Tweezer-programmable 2D quantum walks in a Hubbard-regime lattice. 377885A. W. Young, W. J. Eckner, N. Schine, A. M. Childs, and A. M. Kaufman, Tweezer-programmable 2D quantum walks in a Hubbard-regime lattice, Science 377(6608), 885 (2022). . I Appendix, I. APPENDIX A. genetic algorithm FIG. 5. Random Graph. Composition of the random graph for. A. genetic algorithm FIG. 5. Random Graph. Composition of the random graph for n=5-10 Results for additional topologies -couplings. Retrieved couplings over 100 runs for a line (a-f) and circle (h-m) network. g) Success rate for line (g) and circle (n). FIG. 6. Results for additional topologies -couplings. Retrieved couplings over 100 runs for a line (a-f) and circle (h-m) network. g) Success rate for line (g) and circle (n). Results for star network, (c-d) results for complete graph with Nr = 1000 (a-c) and Nr = 2500 (b-d). Light red: false negatives, Dark red: true negatives. Light green: true positives, Dark green: false positivesFIG. 9. (a-b) Results for star network, (c-d) results for com- plete graph with Nr = 1000 (a-c) and Nr = 2500 (b-d). Light red: false negatives, Dark red: true negatives, Light green: true positives, Dark green: false positives

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{'abstract': "Control and characterization of networks is a paramount step for the development of many quantum technologies. Even for moderate-sized networks, this amounts to explore an extremely vast parameters space in search for the couplings defining the network topology. Here we explore the use of a genetic algorithm to retrieve the topology of a network from the measured probability distribution obtained from the evolution of a continuous-time quantum walk on the network. Our result shows that the algorithm is capable of efficiently retrieving the required information even in the presence of noise.Networks are a fundamental model to understand the underlying properties of complex systems. They are invaluable tools to describe phenomena happening at different scales ranging from social interactions[1,2], to biological processes[3][4][5][6][7], from the configurations of molecules[8,9], to the structure of internet [10-13] and physical systems alike[14][15][16][17][18]. In the context of quantum technologies, networks constitute the prime structure of communication and computation protocols[19][20][21][22][23]. Understanding how quantum information can be reliably transmitted between distant nodes of a network, or routed among different computational units, is a key step and requires a full characterization of the network's structure. While a direct control may not be attainable with the required accuracy and precision, a straightforward strategy to provide such characterization is that of probing the network with a walker that gathers information on its topology by undergoing an evolution which depends on the network's structure. This is the case of continuous-time quantum walks (CTQWs)[24][25][26][27][28][29][30][31][32][33], which thus emerge as a natural paradigm for tackling this task.Two different scenarios may present: the topology of the network may be known, but an accurate estimation of the coupling strengths between each node may be required. This is tantamount to estimating multiple parameters, and can be address in quantum metrological terms[34][35][36][37]. It might otherwise be the case that the topology of the network is not known in advance. Whether one is interested in characterizing a physical network or a simulated one, this will be relying on an experimental platform controlled with set of experimental parameters Λ exp . These need to be mapped to the associated set of parameters describing the CTQW happening on the network, Λ QW , i.e. the Hamiltonian parameters of the quantum walk which, assuming all coupling strengths are fixed to unity and on-site energies to zero, coincide with the adjacency matrix identifying the topology of the network. In order to asses the evolution of the probe, one has to address an observable, such as * ilaria.gianani@uniroma3.it the spatial probability distribution on the network, which will strongly depend on the network's topology. However, since an analytical description of this distribution for CTQWs is often unattainable, and furthermore the relation between the QW Hamiltonian and its probability distribution is highly non-linear, performing a direct inversion can be involved. At the same time, the parameter space in this instance becomes exceedingly large for this to be treated as an estimation problem. An alternative solution is to cast the issue in terms of a search problem. Having access solely to the initial state of the probe and to the measured experimental distributions at fixed times, the task becomes that of finding an adjacency matrix that matches the evolution. Here we tackle this matter by using a genetic algorithm. We use the algorithm to successfully retrieve different topologies in the ideal case as well as when the measured probabilities are affected by noise.We consider a CTQW with zero on-site energies, defined by the couplings Λ QW = {J xy } between two nodes of the network x and y, such that its Hamiltoian is:(1)We assume that the couplings J xy can take only two values: J xy = 0 if the link between two nodes is off, or J xy = 1 if the link is on, so that each edge is bound to have the same strength. The Hamiltonian thus coincides with the adjacency matrix of the network, hence, determining its parameters amounts to determining the network's topology. The evolution of a walker in the initial state |ψ 0 is described by the unitary operator e −iHt . The probability of occupying a site x at a time t is then:Given an undirected graph of n sites, our objective is that of retrieving the couplings Λ QW = {J 12 , . . . , J (n−1)n }, i.e. a binary string of length n c =n(n−1)/2, having access only to the initial state of the network and to the probabilities p x (t k , Λ QW ) measured at times t k . We tackle this challenge by means of a genetic algorithm (GA). GAs are versatile iterative search algorithms inspired by natural selection and have been extensively arXiv:2301.13842v1 [quant-ph] 31 Jan 2023", 'arxivid': '2301.13842', 'author': ['Claudia Benedetti \nDipartimento di Fisica "Aldo Pontremoli"\nUniversità degli Studi di Milano\n20133MilanItaly\n', 'Ilaria Gianani \nDipartimento di Scienze\nUniversità degli Studi Roma Tre\nVia della Vasca Navale 8400146RomeItaly\n'], 'authoraffiliation': ['Dipartimento di Fisica "Aldo Pontremoli"\nUniversità degli Studi di Milano\n20133MilanItaly', 'Dipartimento di Scienze\nUniversità degli Studi Roma Tre\nVia della Vasca Navale 8400146RomeItaly'], 'corpusid': 256416097, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 14302, 'n_tokens_neox': 12117, 'n_words': 7293, 'pdfsha': 'd30643bb7662994a201cc3978af839fb7af91732', 'pdfurls': ['https://export.arxiv.org/pdf/2301.13842v1.pdf'], 'title': ['Identifying network topologies via quantum walk distributions', 'Identifying network topologies via quantum walk distributions'], 'venue': []}

arxiv

The Highest Redshift Relativistic Jets 2007 C C Cheung L Stawarz A Siemiginowska D E Harris D A Schwartz J F C Wardle D Gobeille N P Lee Kavli Institute for Particle Astrophysics and Cosmology Stanford University 94305StanfordCA Smithsonian Center for Astrophysics Physics Department Harvard 60 Garden St02138CambridgeMA Institute for Astrophysical Research Brandeis University 02454WalthamMA Boston University 725 Commonwealth Ave02215BostonMA The Highest Redshift Relativistic Jets Extragalactic Jets: Theory and Observation from Radio to Gamma Ray ASP Conference Series 2007 We describe our efforts to understand large-scale (10's-100's kpc) relativistic jet systems through observations of the highest-redshift quasars. Results from a VLA survey search for radio jets in ∼30 z>3.4 quasars are described along with new Chandra observations of 4 selected targets. Why High-redshift Jets? It is now well established that X-ray emission is a common feature of kiloparsecscale radio jets (see Harris & Krawczynski 2006, for a recent review and the associated website, http://hea-www.harvard.edu/XJET/). The spectral energy distributions (SEDs) of the powerful quasar jets are predominantly characterized as "optically faint", with the spectra rising between the optical and X-ray bands. Current models for this 'excess' X-ray emission posit either inverse Compton (IC) scattering off CMB photons in a (still) relativistic kpc-scale jet or an additional high-energy synchrotron emitting component. In the simplest scenario, such models have diverging predictions at high redshift. Specifically, we expect a strong redshift dependence in the monochromatic flux ratio, f X /f r ∝ U CMB ∝ (1 + z) 4 for IC/CMB, whereas in synchrotron models, we expect no such dependence, f X /f r ∝ (1 + z) 0 . As a first order test of this simple idea, our approach is to study the highest-redshift relativistic jets. Such jets probe the physics of the earliest (first ∼1 Gyr of the Universe in the quasars studied) actively accreting supermassive black hole systems and 2 Cheung et al. are interesting for other reasons. For instance, the ambient medium in these high-redshift galaxies is probably different (e.g., De Young 2006) and this may manifest in jets with different morphologies, increased dissipation, and slower than their lower-redshift counterparts. Most Chandra studies of quasar jets have so far targeted known arcsecondscale radio jets (e.g., Sambruna et al. 2004;Marshall et al. 2005), as most known examples are at z < ∼ 2 (Liu & Zhang 2002). There are currently only two high-z quasars with well-established kpc-scale X-ray jet detections: GB 1508+5714 at z=4.3 (Siemiginowska et al. 2003;Yuan et al. 2003;Cheung 2004) and 1745+624 at z=3.9 (Cheung et al. 2006). They are observed to have large f X /f r values as expected in the IC/CMB model (Schwartz 2002;Cheung 2004), although the small number of high-z detections preclude any definitive statements (Kataoka & Stawarz 2005;Cheung et al. 2006). We have therefore carried out a VLA survey in search of new radio jets in a sample of high-z quasars ( § 2.1.) and new Chandra observations of a small subset ( § 2.2.). This contribution presents some results from these observations. For the redshifts considered, z=3.4 to 4.7, 1 ′′ corresponds to 7.4 to 6.5 kpc (H 0 = 70 km s −1 Mpc −1 , Ω M = 0.3 and Ω Λ = 0.7). Observations of a High-Redshift Quasar Sample VLA Imaging Survey Using NED, we assembled a sample of z>3.4 flat-spectrum radio quasars for imaging with the VLA. We did not aim for our sample to be a complete one as current samples of lower-z X-ray jets are inhomogenous also. With archival and new VLA observations, we find that radio jets in this redshift range are common with a ∼50% detection rate (Cheung et al. 2005, and in preparation). Examples of new radio jets detected from our observations are shown in Figure 1. Chandra Observations A small percentage of the radio jets from our radio study ( § 2.1.) are extended enough (>2.5" long) to study with Chandra. We observed four of them with short snapshot Chandra observations (Figure 2). We detected bright X-ray counterparts to the jets in the quasars J1421-0643 (z=3.689; Ellison et al. 2001) and GB 1428+4217 (z=4.72; Hook & McMahon 1998); the latter detection is currently the highest-redshift kpc-scale radio and X-ray jet known. We did not detect the X-ray counterparts to the radio jets in 1239+376 (z=3.819; Vermeulen et al. 1996) andJ1754+6737 (z=3.6;Villani & di Serego Alighieri 1999). The 2/4 X-ray jet detection rate of our high-z sample is comparable to that of lower-z samples (Sambruna et al. 2004;Marshall et al. 2005). Discussion and Summary Previous Chandra imaging studies of a number of z>4 radio loud quasars do not reveal significant extended X-ray emission (Bassett et al. 2004;Lopez et al. 2006). However, in these studies, there were no pre-existing information on possible radio structures in the target objects and any definitive statements regarding the nature of the X-ray emission mechanism in jets at high-redshifts may be premature. In fact, in one case where there was evidence of an extended X-ray structure (J2219-2719; Lopez et al. 2006), our VLA observation revealed a radio counterpart (Figure 1). In our approach, we began with a VLA survey of a sample of z>3.4 quasars and found radio jets to be relatively common (∼50% detection rate). These jets are quite luminous; with a confident detection of a 1 mJy knot at 1.4 GHz, this corresponds to luminosities of 1.5 ×10 42 erg s −1 (z=3.4) to 3.1 ×10 42 erg s −1 (z=4.7). Figure 2. Chandra X-ray images (colorscale) with VLA contours overlaid of the four high-z radio jets observed. There are X-ray detections of the top two objects but not of the bottom two ( § 2.2.). With the radio survey results, we found only a few radio jets to have sufficient angular extent to be imaged with Chandra. The detection rate of X-ray counterparts of the high-z radio jets (2/4) is similar to that of lower-z radio jet samples (Sambruna et al. 2004;Marshall et al. 2005). The implications of these observations for models of X-ray emission from large-scale jets will be described in forthcoming publications. Figure 1 . 157.50 57.45 57.40 57.35 57.30 57.25 57.20 57.15 57.10 57.35.55 35.50 35.45 35.40 35.35 35.30 35.25 35.20 35.15 35.27.20 27.15 27.10 27.05 27.00 26.95 26.90 26.85 26.80 26.Examples of newly discovered arcsecond-scale radio jets from our VLA observations ( § 2.1. Hook et al. 2002). The J2219-2719 image is at 1.4 GHz while the rest are at 5 GHz. The beam-sizes are 0.41 ′′ × 0.41 ′′ , 0.73 ′′ × 0.38 ′′ at PA=−8.2 • , 1.13 ′′ × 0.39 ′′ at PA=10.2 • , and 0.75 ′′ × 0.75 ′′ (super-resolved), respectively. The lowest contour levels begin at 0.125 mJy/bm for all images except for J2219-2719 where it is 0.2 mJy/bm, and increase by factors of √ 2.). Clockwise from upper left, the sources are J0624+3856 (z=3.469; Xu et al. 1995), J2042-2223 (z=3.630; Hook et al. 2002), J2220-3336 (z=3.691; Hook et al. 2002), and J2219-2719 (z=3.634; Acknowledgments. 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{'abstract': "We describe our efforts to understand large-scale (10's-100's kpc) relativistic jet systems through observations of the highest-redshift quasars. Results from a VLA survey search for radio jets in ∼30 z>3.4 quasars are described along with new Chandra observations of 4 selected targets.", 'arxivid': '0712.1192', 'author': ['C C Cheung ', 'L Stawarz ', 'A Siemiginowska ', 'D E Harris ', 'D A Schwartz ', 'J F C Wardle ', 'D Gobeille ', 'N P Lee ', '\nKavli Institute for Particle Astrophysics and Cosmology\nStanford University\n94305StanfordCA\n', '\nSmithsonian Center for Astrophysics\nPhysics Department\nHarvard\n60 Garden St02138CambridgeMA\n', '\nInstitute for Astrophysical Research\nBrandeis University\n02454WalthamMA\n', '\nBoston University\n725 Commonwealth Ave02215BostonMA\n'], 'authoraffiliation': ['Kavli Institute for Particle Astrophysics and Cosmology\nStanford University\n94305StanfordCA', 'Smithsonian Center for Astrophysics\nPhysics Department\nHarvard\n60 Garden St02138CambridgeMA', 'Institute for Astrophysical Research\nBrandeis University\n02454WalthamMA', 'Boston University\n725 Commonwealth Ave02215BostonMA'], 'corpusid': 14801932, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 4265, 'n_tokens_neox': 3366, 'n_words': 1717, 'pdfsha': 'd35a04870bd09099d98c3c988da63be3bf64c32d', 'pdfurls': ['https://arxiv.org/pdf/0712.1192v1.pdf'], 'title': ['The Highest Redshift Relativistic Jets', 'The Highest Redshift Relativistic Jets'], 'venue': ['Extragalactic Jets: Theory and Observation from Radio to Gamma Ray ASP Conference Series']}

arxiv

A SEPARATION BETWEEN TROPICAL MATRIX RANKS 8 Dec 2017 Yaroslav Shitov A SEPARATION BETWEEN TROPICAL MATRIX RANKS 8 Dec 2017and phrases Tropical linear algebrasupertropical mathematicsmatrix rank We continue to study the rank functions of tropical matrices. In this paper, we explain how to reduce the computation of ranks for matrices over the 'supertropical semifield' to the standard tropical case. Using a counting approach, we prove the existence of a 01-matrix with many ones and without large all-one submatrices, and we put our results together and construct an n × n matrix with tropical rank o(n 0.5+ε ) and Kapranov rank n − o(n).2010 Mathematics Subject Classification. 15A03, 15A80, 14T05. The tropical arithmetic operations on R are (a, b) → min{a, b} and (a, b) → a+b. One can complete this algebraic structure with an element neutral with respect to addition (it plays the role of an infinite positive element and is denoted by ∞) and get the structure (R, min, +) known as the tropical semiring. This semiring and related structures are being studied since the 1960's because of their applications in optimization theory [35]; the tropical methods also arise naturally in algebraic geometry and lead to important developments in the field (see e.g. [2,6,24]). Other applications of tropical mathematics include operations research [7], discrete event systems [3], automata theory [28], and optimal control [20,23]. This paper is a continuation of the study of tropical rank functions initiated in [8] and developed in [4,30,31,32]. Namely, we are going to focus on the tropical rank and Kapranov rank-the functions arisen from the context of tropical algebraic geometry. (It should be mentioned that there are many more rank functions of tropical matrices that are being extensively studied in the literature, see [1,11] and references therein.) We refer the reader to [4,8,31] for definitions and a detailed discussion of motivation behind these concepts; the rank functions also admit combinatorial descriptions, and we are going to recall them in Section 2. For the purpose of this introduction, we briefly recall that the tropical rank of a matrix is the topological dimension of the tropical convex hull of its columns, and the Kapranov rank is the smallest dimension of tropical linear spaces containing these columns. As we will see, one needs to specify a field F to give the definition of Kapranov rank, and the corresponding function is referred to as the Kapranov rank over F. It is very well known that these functions can be different, but the tropical rank cannot exceed any of the Kapranov ranks (see [8]). The papers [4,8,31] contain a resolution of the following question: For which d, n, r does every d × n matrix of tropical rank less than r have Kapranov rank (over C) less than r as well? This condition is equivalent to the r × r minors of a d × n matrix of variables being what is called a tropical basis of the ideal that they generate. The paper [8] contains an example showing that the 4 × 4 minors of a 7 × 7 matrix are not a tropical basis, and it was asked if it is the case for the minors of the 5 × 5 matrix. The authors of [4] proved that the 4 × 4 minors of a 5 × 5 matrix form a tropical basis, and a complete description of the tuples (d, n, r) for which the answer to the above question is positive was given in [31]. The result below is valid for the Kapranov rank function computed with respect to any infinite field. Theorem 1. Let d, n, r be positive integers, r ≤ min{d, n}. The d-by-n matrices with tropical rank less than r always have Kapranov rank less than r if and only if one of the following conditions holds: (1) r ≤ 3; (2) r = min{d, n}; (3) r = 4 and min{d, n} ≤ 6. The answer to the same question but for finite fields remains unknown, but the corresponding characterization should be different from the one given in Theorem 1. In fact, Example 2.7 in [30] shows that matrices of tropical rank two and larger Kapranov rank exist for any finite ground field. As we see, there are a lot of results describing the cases when the tropical and Kapranov ranks are equal; on the opposite end, there is a result (see [19]) stating that a matrix of tropical rank three can have arbitrarily large Kapranov rank. No non-trivial analogue of this result is known if we compare the behavior of the Kapranov rank functions taken over different fields. Question 2. (See also Question 5 in Section 8 of [8] and Problem 4.1 in [22].) For which d does there exist a matrix with rational Kapranov rank three and real Kapranov rank d? Clearly, this may be possible only if d 3, and for d = 3 the answer is trivially positive. The answer is also positive for d = 4, and to see this, one can construct the cocircuit matrix as in [8] for the matroid corresponding to the Perles configuration (see page 94 of [10]). Nothing is known for d 5. We note that the difference between the tropical rank and Kapranov rank in the above mentioned example is of the order of 0.5 √ n, for an n × n tropical matrix (see Theorem 2.4 in [19]). One can also construct a sequence of n × n matrices whose tropical rank and Kapranov rank differ by εn as a diagonal matrix whose diagonal blocks are equal to any fixed matrix with different ranks. In our paper, we improve these bounds to the asymptotically best possible separation of n − o(n); our result is valid for the Kapranov ranks over all fields. Taking n → ∞ and choosing α n to be (ln n) −1 or any other sequence sufficiently slowly decreasing to 0, we get an n − o(n) separation between the tropical rank and Kapranov rank of an n × n tropical matrix. Since the Kapranov rank is a lower bound for the tropical factorization rank (see [8]), Theorem 3 gives an n − o(n) separation between the tropical rank and factorization rank as well. A similar question is wide open for separations between the conventional rank and factorization rank of nonnegative matrices, and related problems have important applications in optimization and computational complexity theory (see [9]). The following version of this problem is open in both the tropical and nonnegative settings. Question 4. Let k be a fixed constant. Is it correct that, for all n, m satisfying m n k, there exists an n × m matrix with tropical rank k and tropical factorization rank n? Does there exist a nonnegative n × m matrix with conventional rank k and nonnegative rank n? The 'nonnegative part' of this question has negative answer for k 3 (this is easy for k 2 as shown in [5], and the case of k = 3 has been done in [27,33]). If k 4, this problem remains open, see Question 1 in [13]. The 'tropical part' is open already for k = 3, and the problem is non-trivial even in the case k 2 for which Theorem 4.6 in [34] gives a negative answer. Preliminaries. The rank of a supertropical matrix This paper was inspired by the idea of symmetrized semirings (see [1,25]), which are intended to give an analogue of subtraction for those semirings that are not rings. The symmetrized tropical semiring is essentially the set R × R, and we may think of a pair (r 1 , r 2 ) as a formal subtraction r 1 ⊖ r 2 . The tropical operations can be naturally extended to the symmetrized setting as (r 1 , r 2 ) ⊕ (s 1 , s 2 ) = (min{r 1 , r 2 }, min{s 1 , s 2 }) and (r 1 , r 2 ) ⊙ (s 1 , s 2 ) = (min{r 1 + s 1 , r 2 + s 2 }, min{r 1 + s 2 , r 2 + s 1 }) because min is the tropical addition and + is the tropical multiplication. A related structure was introduced by Izhakian and Rowen in [14,15] and became known as the 'supertropical semifield '. Their structure 'is somehow reminiscent of the symmetrized max-plus semiring, and has two kind of elements, the real ones (which can be identified to elements of the max-plus semiring and some ghost elements which are similar to the balanced ones,' as Akian, Gaubert, and Guterman wrote in [1]. We decided to write this paper in terms of the 'supertropical ' structure because it seems to have become more popular nowadays due to a considerable amount of papers on the topic written by Izhakian, Rowen, and their colleagues (see also [16,17] and references therein). As said above, the structure introduced by Izhakian and Rowen belongs to the class most commonly known as 'supertropical semifields' (see [16]), but since it contains non-zero elements without multiplicative inverses, it is not an actual semifield according to the standard definition of the latter. We denote this structure by S = (R τ ∪ R γ ∪ {∞}, ⊕, ⊙), where ∞ is an infinite positive element, and R τ , R γ are two copies of R whose elements are called in the literature 'tangible' and 'ghost ', respectively. Assuming i, j ∈ {τ, γ}, s ∈ S, a, b ∈ R and a > b, we define the operations by ∞ ⊕ s = s ⊕ ∞ = s, ∞ ⊙ s = s ⊙ ∞ = ∞; b j ⊕ a i = a i ⊕ b j = b j , b i ⊕ b j = b γ ; a i ⊙ b j = (a + b) α , where α = τ if i = j = τ , and α = γ otherwise. One can check that ⊕ and ⊙ are commutative and associative operations, and distributivity also holds. Moreover, there is a homeomorphism ν from S to the tropical semiring R defined by ∞ → ∞ and a i → a. One writes c |= d if either c = d or c = d ⊕ g, for some ghost element g; this relation is known in the literature as 'ghost surpassing' relation. Let A = (a ij ) be an n × n supertropical matrix; its permanent is per A = σ∈Sn A(1|σ 1 ) ⊙ . . . ⊙ A(n|σ n ), where S n denotes the symmetric group on {1, . . . , n} and A(p|q) denotes the entry in the pth row and qth column of A. This matrix is said to be tropically non-singular if per A is tangible and tropically singular otherwise. Definition 5. The tropical rank of a supertropical matrix is the largest size of its non-singular square submatrix. In order to recall the definition of Kapranov rank, we need to introduce the field F = F{{t}} of generalized Puiseux series. The elements of F are formal sums a(t) = e∈R a e t e which have coefficients a e in F and whose support Supp(a) = {e ∈ R : a e = 0} is well-ordered (which means that every non-empty subset of Supp(a) has a minimal element). The tropicalization mapping deg : K → R sends a series a to the exponent of its leading term; in other words, we define deg a = min Supp(a) and deg 0 = ∞. Definition 6. The Kapranov rank of a supertropical matrix A is the smallest possible rank of a matrix L whose entries are in F and which satisfies A |= deg L. (Such a matrix L is to be called a lifting of A.) Remark 7. If A is a supertropical matrix without ghost elements, then these rank functions match those of conventional tropical matrices as introduced and studied in [4,8,30,31]. Namely, the tropical rank and Kapranov rank of a matrix T with entries in R coincide with those in Definitions 5 and 6 if the elements in R are replaced by their tangible copies. We finalize the section by proving two results using well known techniques. Proof. Let us expand per A with the first column (as in Lemma 3.2 in [26]). Since ∞ and 0 τ are neutral with respect to ⊕ and ⊙, respectively, we get that per A is the sum of the permanents of the (1, 1) and (2, 1) cofactors of A. Since the permanent of a matrix is linear in its rows (see Lemma 3.13 in [26]), the result follows. Lemma 9. Let A be a non-singular supertropical n × n matrix and a, b ∈ R. Then one of the columns of A can be replaced by v = (a τ b τ ∞ . . . ∞) ⊤ so that the resulting matrix remains non-singular. Moreover, we can choose a column in which one of the first two entries is tangible. Proof. Since permutations of rows and columns and their scaling by elements in R τ cannot affect the non-singularity, we can apply the Hungarian algorithm for the assignment problem corresponding to the matrix ν(A), see [21] for details. Therefore, we can assume without loss of generality that the diagonal entries of A are equal to 0 τ , and the off-diagonal entries are positive. If a < b, then we choose the first column to be replaced by v, and otherwise we replace the second column. Reducing the supertropical case to tropical matrices Let S ∈ S I×J be a supertropical matrix, where I and J denote the row and column indexing sets which we assume to be disjoint. Let us denote by I 1 , I 2 two copies of the set I, and i 1 , i 2 will stand for the elements that correspond to i ∈ I in these copies. Definition 10. We call a tropical matrix T symmetrized if it has I 1 ∪ I 2 as row indexing set and I ∪ J as column indexing set (where I, I 1 , I 2 , J are as above), and satisfies T (i 1 |i) = T (i 2 |i) = 0, T (i 1 |ι) = T (i 2 |ι) = ∞ for all i ∈ I,ι ∈ I \ {i}. Definition 11. Let T be a matrix as in the above definition. We define Σ ∈ S I×J as the matrix obtained from T by putting on the ith place the (supertropical) sum of the i 1 th and i 2 th rows of T and removing the columns with indexes in I. The relation between T and Σ(T ) can be understood in terms of the symmetrized tropical semiring discussed above. As we see, T is obtained from Σ(T ) by replacing every row with a pair of rows which can be thought of as a vector over the symmetrized semiring corresponding to the row of T . Let us illustrate this construction with the example essentially appeared in the paper [29] published in 2010. (Namely, the matrix T (A) below is the one from Example 2.1 in [29] up to permutations of rows and columns and replacing the 4's by the ∞'s, which is not crucial for the argument given in [29].) Example 12. Consider the symmetrized tropical matrix A =         0 ∞ ∞ 0 2 1 ∞ 0 ∞ 2 0 2 ∞ ∞ 0 2 1 0 0 ∞ ∞ 0 ∞ ∞ ∞ 0 ∞ ∞ 0 ∞ ∞ ∞ 0 ∞ ∞ 0         and its supertropical counterpart Σ(A) =   0 γ 2 τ 1 τ 2 τ 0 γ 2 τ 2 τ 1 τ 0 γ   constructed as in Definition 11. If the rows of Σ(A) had indexes 1, 2, 3 (from top to bottom) and its columns had indexes 4, 5, 6 (from left to right), then the rows of A are indexed with 1 1 , 2 1 , 3 1 , 1 2 , 2 2 , 3 2 , and the columns of A with 1, 2, 3, 4, 5, 6. One can check it directly that the tropical rank and Kapranov rank of the matrix Σ(A) above are 1 and 2, respectively; it is proven in [29] that the respective ranks of A are 4 and 5. We can begin proving the main results of this section, which give a general relation between the ranks of A and Σ(A). Theorem 13. Let T be a matrix as in Definition 10. If |F| 3, then Kapranov rank T = Kapranov rank Σ(T ) + |I|. Proof. Let L be a lifting of T with smallest possible rank r; row scalings allow us to assume that L(i 1 |i) = L(i 2 |i) = 1 for all i ∈ I. Let L ′ be the matrix obtained from L by subtracting, for any i, the i 1 th row from i 2 th row. The ranks of L and L ′ are equal, and we have (2.1) L ′ = I * O L , where I and O are, respectively the unit and zero |I| × |I| matrices, L is a lifting of Σ(T ), and * stands for a matrix we need not specify. Therefore, the Kapranov rank of Σ(T ) is at most r − |I|. Conversely, consider a lifting L of Σ(T ) with smallest possible rank ρ. We define the matrix L ∈ F (I 1 ∪I 2 )×(I∪J) as follows. For all i ∈ I, j ∈ J,ι ∈ I \ {i}, we set (i) L(i 1 |i) = L(i 2 |i) = 1 and L(i 1 |ι) = L(i 2 |ι) = 0, (ii) L(i 1 |j) = ζt s , L(i 2 |j) = L(i|j) + ζt s if T (i 1 |j) = T (i 2 |j) = s, (iii) L(i 1 |j) = t s , L(i 2 |j) = L(i|j) + t s if s = T (i 1 |j) > T (i 2 |j), (iv) L(i 1 |j) = t s − L(i|j), L(i 2 |j) = t s if T (i 1 |j) < T (i 2 |j) = s. Since the ground field F contains more than two elements, we can avoid cancellation of leading (degree-s) terms in L(i|j) + ζt s by choosing an appropriate non-zero value of ζ in F. As we see, the constructed matrix L is a lifting of T , and in order to compute its rank we subtract, as above, the i 1 th row from i 2 th row, for any i. We get the matrix L ′ as in (2.1), so the rank of L equals ρ + |I|. Proof. Denote by I 0 , J 0 the sets of row and column indexes of a largest non-singular submatrix C of Σ(T ). Denoting |I| = n and |I 0 | = |J 0 | = r, we observe that the submatrix of T formed by the rows with indexes in I 2 0 ∪I 1 and columns with indexes in I ∪ J 0 looks like (with upper left block having row indexes in I 2 0 ∪ I 1 0 and column indexes in I \ I 0 ) (2.2) Z 2r×(n−r) T ′ U n−r * , where U is the tropical unit matrix (the one with 0's on the diagonal and ∞'s everywhere else), Z is the all-∞ matrix, and T ′ is a symmetrized matrix such that Σ(T ′ ) = C. The permanent of (2.2) equals per(T ′ ), which in turn, according to Proposition 8, equals per(C). Since C is non-singular, it has a tangible permanent, and so does the matrix (2.2), which is therefore non-singular as well. In particular, we get a ' ' inequality for the values in the formulation of the lemma. In order to prove the ' ' inequality, we use Lemma 9 and observe that any of the largest tropically non-singular submatrices of T can be reduced to the form (2.2), and the proof can be finalized as in the previous paragraph. Constructing a tropical matrix In this section, we explain how to construct a tropical matrix with small tropical rank and large Kapranov rank if we are given a 0 − 1 matrix as below. Proof. Any lifting of Φ has ku entries with degrees in (a ijα ), and since these degrees are linearly independent over Q, the corresponding entries should be algebraically independent over F. It remains to note that any matrix of rank ρ has transcendence degree at most 2nρ − ρ 2 and resolve the inequality 2nρ − ρ 2 ku for ρ. Lemma 17. Tropical rank Φ d + kr. Proof. Let H be a square submatrix of Φ of size greater than d + kr. We need to check that H is singular, that is, that the permanent of H seen as a supertropical matrix is either ∞ or a ghost. By Dirichlet's principle, there is a subset I ⊂ I of cardinality ρ r such that, for any i ∈ I, there are two distinct pairs (α i , i) and (β i , i) which appear to be row indexes of H. We denote by H 0 the submatrix of Φ formed by the rows with indexes in i∈I {(α i , i), (β i , i)}; we will be done if we manage to show that every 2ρ × 2ρ submatrix of H 0 is singular. By Theorem 14, we need to show that every ρ×ρ submatrix of Σ(H 0 ) is singular. Removing the columns of Σ(H 0 ) consisting of ∞-entries, we get a matrix H ∈ S I×J such that (1) H(i|j) = 0 γ if M(i|j) = 0, and (2) H(i|j) = a τ ij with a ij ∈ [1, 1 + 1/n] if M(i|j) = 1. Since every ρ × ρ submatrix of M contains a zero, the permanent of every ρ × ρ submatrix of H should have a summand g γ with g 0 + (r − 1)(1 + 1/n) < r. Therefore, the products of tangible entries do not contribute to the permanent of any r × r submatrix of H. Remark 18. Many authors (see [4,8]) consider tropical matrices with finite entries only. We note that the bounds as in Lemmas 16 and 17 will still hold if we replace every ∞ in the entries of Φ by 2. In fact, the resulting matrix can be obtained as D ⊙ Φ, where D is the n × n matrix with 0's on the diagonal and 2's everywhere else. Of course, the tropical rank of D ⊙ Φ is at most that of Φ (see Theorem 9.4 in [1]), and the proof of Lemma 16 reads equally well if we replace Φ by D ⊙ Φ. Constructing the matrix M In this section, we give a probabilistic construction of the matrix as in Definition 15 and finalize the proof of Theorem 3. Proof. Let X be a random d×(d 2 −d) matrix with independent entries each of which is either 0 or 1, and the probability of 0 is q. According to Hoeffding's inequality (see Theorem 1 in [12]), the probability that the number of 1-entries of X does not exceed u is at most exp(−2d −3 (d 3 − d 2 )) < 0.5. Therefore, the condition (1) as in Definition 15 fails with probability less than 0.5. We proceed with condition (2), whose negation means that X has an ⌈r⌉ × ⌈r⌉ submatrix of all ones. The probability that this happens with any particular such submatrix is at most (1−q) r 2 , and the number of these submatrices does not exceed (d 2 − d) r+1 d r+1 . Therefore, the condition (2) which is also less than 0.5 because ln(1 − q)/q < −1. Now we can complete the proof of Theorem 3. Proof of Theorem 3. We define d = ⌊ √ n⌋ and write q = α − 2n −0.25 2 . We can assume without loss of generality that the bound for the tropical rank is less than n because otherwise the result is trivial. In particular, we have that α > 2n −0.25 √ ln n. The numbers d, k, r, u as in Lemma 19 allow us to construct a d 2 × d 2 matrix Φ satisfying the assumptions of Definition 15; we complete Φ to an n × n matrix Φ 0 by adding the copies of existing rows and columns. According to Lemma 16, the Kapranov rank of Φ 0 is at least d 2 − d 4 − d 4 (1 − d −1 )(1 − q − d −1.5 ) d 2 1 − √ q − d −1 + d −1.5 , which is greater than or equal to n(1 − √ q − 2n −0.25 ) = n(1 − α). By Lemma 17, the tropical rank of Φ 0 does not exceed 4d ln d q + d 2.4 √ n ln n q 4 √ n ln n α 2 . Theorem 3 . 3For all n > 1000, α ∈ (0, 0.1), there is an n × n matrix A such that tropical rank (A) 4 √ n ln n α 2 and Kapranov rank (A) n(1 − α). Proposition 8 . 8Let A be an n × n supertropical matrix satisfying A 11 = A 21 = 0 τ and A 31 = . . . = A n1 = ∞. Let B be the (n− 1)× (n− 1) matrix obtained from A by removing the first column and replacing the first two rows by their (supertropical) sum. Then per A = per B. Theorem 14 . 14Let T be a matrix as in Definition 10. Then tropical rank T = tropical rank Σ(T ) + |I|. Definition 15 . 15Numbers (d, k, r, u) are said to be a good tuple if there exists an d × (kd − d) matrix M of zeros and ones such that (1) at least u entries of M are ones;(2) any ρ × ρ submatrix of M contains a zero unless ρ < r.We enumerate the rows and columns of M by disjoint sets I and J and construct the tropical matrix Φ = Φ(M) as follows. Its rows are indexed with {1, . . . , k} × I, its columns with I ∪ J, and its entries are (1) Φ(α, i|i) = 0 and Φ(α, i|î) = ∞ ifî ∈ I \ {i};(2) Φ(α, i|j) = 0 if j ∈ J and M(i|j) = 0;(3) Φ(α, i|j) = a ijα if j ∈ J and M(i|j) = 1, where (a ijα ) are a family of numbers in [1, 1 + 1/(kd)] that are linearly independent over Q. Notice that Φ is an n × n matrix with n = kd. Lemma 16. Kapranov rank Φ n − √ n 2 − ku. Lemma 19 . 19Let q ∈ (0, 0.1) and d 2. Then the numbersk = d, r = 4 ln d/q, u = (1 − q − d −1.5 )(d 3 − d 2 )are a good tuple in the sense ofDefinition 15. Linear independence over tropical semirings and beyond. M Akian, S Gaubert, A Guterman, Contemporary Mathematics. 495M. Akian, S. Gaubert, A. Guterman, Linear independence over tropical semirings and be- yond, Contemporary Mathematics 495 (2009) 1-38. A tropical approach to a generalized Hodge conjecture for positive currents. F Babaee, J Huh, Duke Math. J. 166F. Babaee, J. Huh, A tropical approach to a generalized Hodge conjecture for positive cur- rents, Duke Math. J. 166 (2017) 2749-2813. . F Baccelli, G Cohen, G J Olsder, J.P. Quadrat, Synchronization and Linearity. WileyF. Baccelli, G. Cohen, G.J. Olsder, J.P. Quadrat, Synchronization and Linearity, Wiley, 1992. The 4x4 minors of a 5xn matrix are a tropical basis. M Chan, A N Jensen, E Rubei, Linear Algebra Appl. 435M. Chan, A. N. Jensen, E. Rubei, The 4x4 minors of a 5xn matrix are a tropical basis, Linear Algebra Appl. 435 (2011) 1598-1611. Nonnegative ranks, decompositions, and factorizations of nonnegative matrices. J E Cohen, U G Rothblum, Linear Algebra Appl. 190J. E. Cohen, U. G. Rothblum, Nonnegative ranks, decompositions, and factorizations of nonnegative matrices, Linear Algebra Appl. 190 (1993) 149-168. A tropical proof of the Brill-Noether theorem. F Cools, J Draisma, S Payne, E Robeva, Adv. Math. 230F. Cools, J. Draisma, S. Payne, E. Robeva, A tropical proof of the Brill-Noether theorem, Adv. Math. 230 (2012) 759-776. R A Cuninghame-Green, Minimax algebra. BerlinSpringer-Verlag166R. A. Cuninghame-Green, Minimax algebra, volume 166 of Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, Berlin, 1979. On the rank of a tropical matrix. M Develin, F Santos, B Sturmfels, Discrete and Computational Geometry (E. Goodman, J. Pach and E. WelzlCambridge Univ. PressM. Develin, F. Santos, B. Sturmfels,On the rank of a tropical matrix, in Discrete and Compu- tational Geometry (E. Goodman, J. Pach and E. Welzl, eds.), MSRI Publications, Cambridge Univ. Press, 2005. 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On the nonnegative rank of distance matrices. P Hrubeš, Inform. Process. Lett. 112P. Hrubeš, On the nonnegative rank of distance matrices, Inform. Process. Lett. 112 (2012) 457-461. Tropical arithmetic and matrix algebra. Z Izhakian, Commun. Algebra. 37Z. Izhakian, Tropical arithmetic and matrix algebra, Commun. Algebra 37 (2009) 1445-1468. Supertropical algebra. Z Izhakian, L Rowen, Adv. Math. 225Z. Izhakian, L. Rowen, Supertropical algebra, Adv. Math. 225 (2010) 2222-2286. Supertropical linear algebra. Z Izhakian, M Knebusch, L Rowen, Pacific J. Math. 266Z. Izhakian, M. Knebusch, L. Rowen, Supertropical linear algebra, Pacific J. Math. 266 (2013) 43-75. Layered tropical mathematics. Z Izhakian, M Knebusch, L Rowen, J. Algebra. 416Z. Izhakian, M. Knebusch, L. Rowen, Layered tropical mathematics, J. Algebra 416 (2014) 200-273. . Z Izhakian, A Niv, L Rowen, Supertropical Sln, Linear Multilinear A. Z. Izhakian, A. Niv, L. Rowen, Supertropical SLn, Linear Multilinear A. (2017). Kapranov rank vs. tropical rank. K H Kim, N F Roush, Proc. Amer. Math. Soc. 134K. H. Kim, N. F. Roush, Kapranov rank vs. tropical rank, Proc. Amer. Math. Soc. 134 (2006) 2487-2494. Idempotent analysis and applications. V N Kolokoltsov, V P Maslov, Kluwer Academic PublishersV. N. Kolokoltsov, V. P. Maslov, Idempotent analysis and applications, Kluwer Academic Publishers, 1997. The Hungarian Method for the assignment problem. H W Kuhn, Nav. Res. Logist. Q. 2H. W. Kuhn, The Hungarian Method for the assignment problem, Nav. Res. Logist. Q. 2 (1955) 83-97. Sign patterns with minimum rank 2 and upper bounds on minimum ranks. Z Li, Y Gao, M Arav, F Gong, W Gao, F J Hall, H Van Der Holst, Linear Multilinear A. 61Z. Li, Y. Gao, M. Arav, F. Gong, W. Gao, F. J. Hall, H. van der Holst, Sign patterns with minimum rank 2 and upper bounds on minimum ranks, Linear Multilinear A. 61 (2013) 895-908. Max-plus methods for nonlinear control and estimation. W M Mceneaney, Systems & Control: Foundations & Applications. Boston, MABirkhäuser Boston IncW. M. McEneaney, Max-plus methods for nonlinear control and estimation, Systems & Con- trol: Foundations & Applications, Birkhäuser Boston Inc., Boston, MA, 2006. Enumerative tropical algebraic geometry in R 2. G Mikhalkin, J. Amer. Math. Soc. 18G. Mikhalkin, Enumerative tropical algebraic geometry in R 2 , J. Amer. Math. Soc. 18 (2005) 313-377. Linear systems in (max, +)-algebra. M Plus, Proceedings of the 29th Conference on Decision and Control. the 29th Conference on Decision and ControlHonoluluM. Plus. Linear systems in (max, +)-algebra, in Proceedings of the 29th Conference on De- cision and Control, Honolulu, 1990. Determinantal identities over commutative semirings. P L Poplin, R E Hartwig, Linear Algebra Appl. 387P. L. Poplin, R. E. Hartwig, Determinantal identities over commutative semirings, Linear Algebra Appl. 387 (2004) 99-132. Polygons as Sections of Higher-Dimensional Polytopes. A Padrol, J Pfeifle, Electron. J. Comb. 2224A. Padrol, J. Pfeifle, Polygons as Sections of Higher-Dimensional Polytopes, Electron. J. Comb. 22 (2015) 1.24. Limited Subsets of a Free Monoid. I Simon, Proc. 19th Annual Symposium on Foundations of Computer Science. 19th Annual Symposium on Foundations of Computer SciencePiscataway, N.J., Institute of Electrical and Electronics EngineersI. Simon, Limited Subsets of a Free Monoid, in Proc. 19th Annual Symposium on Foundations of Computer Science, Piscataway, N.J., Institute of Electrical and Electronics Engineers, 1978. Example of a 6-by-6 Matrix with Different Tropical and Kapranov Ranks. Ya, Shitov, arXiv:1012.5507preprintYa. Shitov, Example of a 6-by-6 Matrix with Different Tropical and Kapranov Ranks, preprint (2010) arXiv:1012.5507. On the Kapranov ranks of tropical matrices. Ya, Shitov, Linear Algebra Appl. 436Ya. Shitov, On the Kapranov ranks of tropical matrices, Linear Algebra Appl. 436 (2012) 3247-3253. When do the r-by-r minors of a matrix form a tropical basis?. Ya, Shitov, J. Combin. Theory A. 120Ya. Shitov, When do the r-by-r minors of a matrix form a tropical basis? J. Combin. Theory A 120 (2013) 1166-1201. Mixed subdivisions and ranks of tropical matrices. Ya, Shitov, Proc. Amer. Math. Soc. 142Ya. Shitov, Mixed subdivisions and ranks of tropical matrices, Proc. Amer. Math. Soc. 142 (2014) 15-19. An upper bound for nonnegative rank. Ya, Shitov, J. Comb. Theory A. 122Ya. Shitov, An upper bound for nonnegative rank, J. Comb. Theory A 122 (2014) 126-132. Tropical semimodules of dimension two. Ya, Shitov, St. Petersb. Math. J. 26Ya. Shitov, Tropical semimodules of dimension two, St. Petersb. Math. J. 26 (2015) 341-350. Extremal algebra of positive matrices. N N Vorobyev, Elektron. Informationsverarbeitung und Kybernetik. 3N. N. Vorobyev, Extremal algebra of positive matrices, Elektron. Informationsverarbeitung und Kybernetik 3 (1967) 39-71. . E-Mail Address, yaroslav-shitov@yandex.ruE-mail address: yaroslav-shitov@yandex.ru

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{'abstract': "We continue to study the rank functions of tropical matrices. In this paper, we explain how to reduce the computation of ranks for matrices over the 'supertropical semifield' to the standard tropical case. Using a counting approach, we prove the existence of a 01-matrix with many ones and without large all-one submatrices, and we put our results together and construct an n × n matrix with tropical rank o(n 0.5+ε ) and Kapranov rank n − o(n).2010 Mathematics Subject Classification. 15A03, 15A80, 14T05.", 'arxivid': '1712.03071', 'author': ['Yaroslav Shitov '], 'authoraffiliation': [], 'corpusid': 119141239, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 10168, 'n_tokens_neox': 8857, 'n_words': 5553, 'pdfsha': '0febc8d2d2e2ec859f1142f58dc6deea6bfbe913', 'pdfurls': None, 'title': ['A SEPARATION BETWEEN TROPICAL MATRIX RANKS', 'A SEPARATION BETWEEN TROPICAL MATRIX RANKS'], 'venue': []}

arxiv

A Classical Background for the Wave Function Prediction in the Infinite System Density Matrix Renormalization Group Method 30 Nov 2009 Hiroshi Ueda Department of Material Engineering Science Graduate School of Engineering Science Osaka University 560-8531OsakaJapan Andrej Gendiar Institute of Electrical Engineering Slovak Academy of Sciences Dúbravská cesta 9SK-841 04BratislavaSlovakia Tomotoshi Nishino Department of Physics Graduate School of Science Kobe University 657-8501KobeJapan A Classical Background for the Wave Function Prediction in the Infinite System Density Matrix Renormalization Group Method 30 Nov 2009Full PaperDMRGPWFRGCTMRGRenormalization We report a physical background of the wave function prediction in the infinite system density matrix renormalization group (DMRG) method, from the view point of two-dimensional vertex model, a typical lattice model in statistical mechanics. Singular value decomposition applied to rectangular corner transfer matrices naturally draws matrix product representation for the maximal eigenvector of the row-to-row transfer matrix. The wave function prediction can be expressed as the insertion of an approximate half-column transfer matrix. This insertion process is in accordance with the scheme proposed by McCulloch recently. Introduction The density matrix renormalization group (DMRG) method is one of the efficient numerical method, which has been applied extensively to one-dimensional (1D) quantum systems and two-dimensional (2D) classical systems. [1][2][3][4] The method is variational in the sense that it assumes a trial state, the matrix product state (MPS), which is written as a product of local tensors. [5][6][7][8][9][10][11][12][13][14][15] Orthogonality of each matrix ensures the numerical stability. One of the bottleneck in the computation of the DMRG method is the diagonalization of super block Hamiltonian. The construction of a good initial vector for this diagonalization is very important. For the finite-system DMRG method, the so-called wave function renormalization scheme provides the answer. 16,17) For the infinite-system DMRG method, Baxter's method of corner transfer matrix (CTM), [18][19][20] which can be reinterpreted from the view point of the DMRG method, 21,22) essentially solves the problem of initial vector. Based on Baxter's CTM method, the product wave function renormalization group (PWFRG) method was proposed, [23][24][25] and has been applied to the study of 1D spin chains. [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40] Recently McCulloch proposed a way of precise wave function prediction, which works better than the PWFRG method especially when the system size is small compared with the correlation length. 41) In this paper we present a physical background for McCulloch's scheme from the view point of 2D vertex model, one of the typical lattice model in statistical mechanics. 20) Although we employ classical lattice model, most of the obtained results can be applicable for 1D quantum systems through the quantum-classical correspondence. Structure of the paper is as follows. In the next section we explain the symmetric vertex model, and express the maximal eigenstate of the row-to-row transfer matrix by use of CTMs. In §3 we consider the area extension of CTMs, introducing an approximate half-column trans- fer matrix. We show the connection between MPS and CTM formulation in §4, where the system size extension scheme by McCulloch is obtained naturally. We summarize the obtained result in the last section, and discuss the remaining problem on the MPS obtained by the finitesystem DMRG method. Eigenstate of row-to-row Transfer Matrix Approximated by Corner Transfer Matrices Throughout this article we consider a square-lattice symmetric vertex model, 20) as an example of 2D classical lattice models. There is a d-state spin variable on each bond, which connects neighboring lattice points. Four spins around a lattice point determine the local Boltzmann weight W , which is called as the vertex weight. We assume that the vertex weight is position independent, and therefore the system is uniform. We also assume that each vertex weight is invariant under exchange of left and right spin variables, and those of up and down spin variables. In other words, we consider the symmetric vertex model in order to simplify the following formulation. As shown on the left side of Fig. 1, we treat a finite size system that has a rectangular shape. This system corresponds to the stack of row-to-row transfer matrices T N , whose width is N , multiplied by an initial vector V N . We choose V N so that it corresponds to the boundary condition at the bottom of the system, where there is a row of boundary spins shown by the cross marks. Those cross marks aligned vertically also represent boundary spins, that are located at the both ends of T N . The row of open circles represents spins on top of the rectangular system. We consider a d N -dimensional vector Ψ N = T N T N T N . . . T N V N , (2.1) where the number of the row-to-row transfer matrix T N is sufficiently large. Under this assumption we can expect that Ψ N is a good approximation of the maximal eigenvector of T N if V N is not orthogonal to that. For a while let us consider the case N = 8; generalization to arbitrary N is straightforward. We label the top spins as q 1 , q 2 , q 3 , q 4 , p 4 , p 3 , p 2 , and p 1 from left to right. The vector elements of Ψ 8 are then written as Ψ 8 (q 1 , q 2 , q 3 , q 4 , p 4 , p 3 , p 2 , p 1 ). Since we have assumed the left-right symmetry for the vertex weight, it is convenient to divide the row-spin into the left half q 1 , q 2 , q 3 , q 4 , where we have counted them from left to right, and the right half p 1 , p 2 , p 3 , p 4 , where we have counted them from right to left. (See right bottom of Fig. 1.) According to this division, we can interpret Ψ 8 as a d 4 -dimensional real symmetric matrix, whose elements can be expressed as Ψ 8 (q 1 q 2 q 3 q 4 |p 1 p 2 p 3 p 4 ). We have used the vertical bar "|" to separate the left and the right indices, and dropped the commas between the spin variables for the book keeping. If necessary, we further abbreviate the matrix notation as Ψ 8 (q|p). We express the left half of the rectangular system by use of the CTM, whose elements are written as C 4 (q 1 q 2 q 3 q 4 |σ 1 σ 2 σ 3 . . .), where σ 1 σ 2 σ 3 . . . represent the half-column spins at the center of the system. In the same manner we can express the right half by the transpose of C 4 , i.e., C T 4 . Joining process of these halves by stitching C 4 and C T 4 via the contraction of the halfcolumn spins can be expressed simply by the product of matrices Ψ 8 = C 4 C T 4 . More precisely, there is a relation Ψ 8 (q|p) = σ C 4 (q|σ) C 4 (p|σ) , (2.2) where we have used the abbreviations q = q 1 q 2 q 3 q 4 , p = p 1 p 2 p 3 p 4 , and σ = σ 1 σ 2 σ 3 . . . . Since we have assumed that the number of T 8 in Eq. (2.1) is sufficiently large, the same for the number of column-spin σ. Although we treat σ, we do not think of them as spins directly treated in numerical calculations, unlike q and p. One of the fundamental mathematical tool in the DMRG method is the singular value decomposition (SVD). 1,2) Let us apply it to the CTM C 4 (q|p) = ξ A 4 (q|ξ) Ω 4 (ξ) U 4 (σ|ξ) , (2.3) where ξ is a d 4 -state block-spin (or an auxiliary) variable, and Ω 4 (ξ) represents the singular values. The matrix A 4 is d 4 -dimensional, and it satisfies the orthogonal relations ξ A 4 (q ′ |ξ) A 4 (q|ξ) = δ(q ′ |q) , q A 4 (q|ξ ′ ) A 4 (q|ξ) = δ(ξ ′ |ξ) , (2.4) where δ(ξ ′ |ξ) is Kronecker's delta, and where δ(q ′ |q) is defined as δ(q ′ |q) = 4 i=1 δ(q ′ i |q i ) . (2.5) The above orthogonal relation can be written shortly as A 4 A T 4 = A T 4 A 4 = I 4 . Column vectors of the rectangular matrix U 4 are also orthogonal with each other, σ U 4 (σ|ξ ′ ) U 4 (σ|ξ) = δ(ξ ′ |ξ) , (2.6) but the row vectors are not ξ U 4 (σ ′ |ξ) U 4 (σ|ξ) = δ(σ ′ |σ) . (2.7) This is because the degree of freedom of σ is far larger than that of q or ξ. Figure 2 is the pictorial representation of SVD applied to C 4 . We often regard the singular values Ω 4 as the diagonal matrix Ω 4 (ξ ′ |ξ) = Ω 4 (ξ) δ(ξ ′ |ξ), and write Eq. (2.3) shortly as C 4 = A 4 Ω 4 U T 4 . For the latter convenience, let us introduce the generalized inverse of the CTM C −1 4 = U 4 Ω −1 4 A T 4 ,(2.8) which satisfies the relation C 4 C −1 4 = A 4 Ω 4 U T 4 U 4 Ω −1 4 A T 4 = A 4 A T 4 = I 4 .(2.9) It should be noted that C −1 4 C 4 is a projection operator U 4 Ω −1 4 A T 4 A 4 Ω 4 U T 4 = U 4 U T 4 (2.10) in the left hand side of Eq. (2.7), where (C −1 4 C 4 ) 2 = C −1 4 C 4 holds. In the context of the DMRG method, small singular values are neglected when it is impossible to store matrix elements during the numerical calculation. This truncation is a kind of decimation in the renormalization group (RG) theory. Under the truncation, the matrices A 4 work as the RG transformation that controls numerical precision. In the next section we do not truncate singular values, in order to avoid complications in notations, and the introduction of truncation is straightforward. Half Column Transfer Matrix and Matrix Product State We introduce a new notation between matrices, the dot product, which contract variables according to Einstein rule. As an example, let us consider P 4 = C −1 3 · C 4 ,(3.1) where q 1 , q 2 , and q 3 are contracted but q 4 is not, since the first three spins are shared by C −1 3 and C 4 . Figure 3 shows this rule graphically. Substituting Eq. (2.3) and (2.8) to C −1 3 · C 4 , we obtain P 4 = (U 3 Ω −1 3 A T 3 ) · (A 4 Ω 4 U T 4 ) = U 3 Ω −1 3 · (A T 3 · A 4 ) Ω 4 U T 4 = U 3 Ω −1 3 ·Ã 4 Ω 4 U T 4 . (3.2) To avoid any confusion, let us write down element of P 4 P 4 (σ ′ |q 4 |σ) (3.3) = ξζ U 3 (σ ′ |ξ) Ω −1 3 (ξ)à 4 (ξ q 4 |ζ) Ω 4 (ζ) U 4 (σ|ζ) , where the new matrixà 4 = A T 3 · A 4 is the renormalized orthogonal matrix A 4 (ξ q 4 |ζ) = q 1 q 2 q 3 A 3 (q 1 q 2 q 3 |ξ) A 4 (q 1 q 2 q 3 q 4 |ζ) , (3.4) which satisfies the relation ξq4à 4 (ξ q 4 |ζ ′ )à 4 (ξ q 4 |ζ) = δ(ζ ′ |ζ) . (3.5) In Eq. (3.4) the group of spins q 1 , q 2 , and q 3 are mapped onto the block spin ξ by the RG transformation A 3 . The obtainedà 4 corresponds to the matrix that constructs MPS, which is constructed by the infinite system DMRG method, as shown later. The P 4 thus obtained has a function of half-column transfer matrix (HCTM), since it extends the width of C 3 by one by way of the dot product C 3 · P 4 = C 3 · (C −1 3 · C 4 ) = (C 3 C −1 3 ) · C 4 = C 4 (3.6) as shown in the left side of Fig. 4. Applying SVD to C 3 and substituting Eq. (3.2), C 3 · P 4 is calculated as (A 3 Ω 3 U T 3 ) · (U 3 Ω −1 3 ·Ã 4 Ω 4 U T 4 ) = A 3 ·Ã 4 Ω 4 U T 4 . (3.7) Since C 3 is again constructed from C 2 and P 3 , as shown in the right side of Fig. 4, we can further decompose C 4 as C 4 = C 2 ·P 3 ·P 4 = A 2 ·Ã 3 ·Ã 4 Ω 4 U T 4 . (3.8) The contraction process by the dot products are shown in the right side of Fig. 5. It should be noted that C 2 · P 4 is not C 3 , since U T 2 contained in C 2 and U 3 contained in P 4 do not matches to give an identity. In this sense, P 4 is an approximation for the half column transfer matrix, optimized for the area extension of C 3 only. Using the decomposition of C 4 in Eq. (3.8), we obtain the matrix product representation of Ψ 8 = C 4 C T 4 . We have Ψ 8 = A 2 ·Ã 3 ·Ã 4 (Ω 4 ) 2à T 4 ·Ã T 3 ·A T 2 = A 2 ·Ã 3 ·Ã 4 Λ 4à T 4 ·Ã T 3 ·A T 2 , (3.9) where Λ 4 = (Ω 4 ) 2 is the singular value of Ψ 8 . (See Fig. 6.) Such a construction of Ψ 8 is equivalent to the MPS ? Fig. 7. Approximate area extension process C App. 5 = C 4 ·P 4 . considered in the context of the infinite system DMRG method. Approximate Area Extension Let us consider a problem of obtaining an approximation of C 5 = C 4 · P 5 without using P 5 . This attempt is equivalent to construct an approximation for C 5 using C 2 , C 3 , or C 4 . One might think that P 4 = C −1 3 · C 4 = U 3 Ω −1 3 ·Ã 4 Ω 4 U T 4 can be of use as an approximation for P 5 . But this idea should be rejected since U T 4 U 3 , which appears in the calculation of C 4 · P 4 , is not an identity. A way to avoid this mismatching is to introduce a spatial reflection of P 4 , which is defined as P 4 = U 4 Ω 4à T 4 · Ω −1 3 U T 3 ,(4.1) and use it as an approximation for P 5 . Leaving the validity of the approximation scheme by the latter discussion, let us calculate the approximate extension C App. 5 = C 4 ·P 4 and write it into the matrix product representation. (See Fig. 7.) We obtain C App. 5 = A 2 ·Ã 3 ·Ã 4 Ω 4 U T 4 U 4 Ω 4à T 4 · Ω −1 3 U T 3 = A 2 ·Ã 3 ·Ã 4 (Ω 4 ) 2à T 4 · Ω −1 3 U T 3 = A 2 ·Ã 3 ·Ã 4 Λ 4à T 4 · Ω −1 3 U T 3 ,(4.2) and from this approximation we can construct Ψ App. 10 = C App. 5 (C App. 5 ) T (4.3)= A 2 ·Ã 3 ·Ã 4 Λ 4à T 4 · Λ −1 3 ·Ã 4 Λ 4à T 4 ·Ã T 3 ·A T 2 ,Ψ App. 10 = A 2 ·Ã 3 ·Ã 4 ·B Λ Λ −1 3 Λ TB T ·Ã T 4 ·Ã T 3 ·A T 2 . (4.5) The extension from Ψ 8 to Ψ App. 10 is the same as the wave function extension scheme proposed by McCulloch, 41) where the approximation for the renormalized wave function is given byΨ App. 10 =B Λ Λ −1 3 Λ TB T . (4.6) We have thus obtained a natural explanations for Mc-Culloch's extension scheme from the view point of 2D vertex model. Up to now we have not considered the effect of basis truncation, which is used in numerical calculation of the infinite system DMRG method. First of all, the extension in Eqs. (4.4) and (4.5) is still efficient under the truncation, as it was shown numerically. 41) We then consider the extension from Ψ N toΨ N +2 in the large system size limit N → ∞. For simplicity, let us assume that the MPS in this limit is uniform, and the system is away from criticality. In this limit we can drop the site index from Eq. (4.1), and can express the approximate transfer matrix as P = U Ω −1 ·Ã Ω U T = U ·S U T ,(4.7) whereS = Ω −1 ·Ã Ω. From the assumed symmetry of the vertex model, both P andS are symmetric P (σ ′ | q |σ) = P (σ| q |σ ′ ) S(σ ′ | q |σ) =S(σ| q |σ ′ ) . (4.8) This symmetry is also expressed in short form asP = P andS =S. Thus at least when the system size N = 2i is large enough, typically several times larger than the correlation length, one can justify the usage of C i ·P i as the approximation for C i · P i+1 . Before closing this section, we consider the MPS expression for Ψ N that is optimized by way of the sweeping process in the finite system DMRG method. The matrix product structure Ψ N = A 2 ·Ã 3 · · ·Ã N 2 Λ N 2à T N 2 · · ·Ã T 3 ·A T 2 (4.9) is similar to that obtained by the infinite system DMRG method, but in this case the matrices satisfies the additional relationà i Λ i = Λ i−1à T i ,(4.10) where bothà i and Λ i differ from those obtained by the infinite system DMRG method. Taking the square root of Λ i , we formally obtain a diagonal matrix Ω i = Λ i . It should be noted that this Ω i is different from that obtained from the SVD applied to C i . Defining S i = Ω −1 i−1 ·Ã i Ω i = Ω i−1à T i · Ω −1 i (4.11) and substituting it to Eq. (4.9), we obtain a new standard form for MPS Ψ N = Ω 1 S 2 ·S 3 · · ·S N 2S T N 2 · · ·S T 3 ·S T 2 Ω 1 ,(4.12) where Ω 1 is just a constant and is not essential. It is then straightforward to obtain the approximation Ψ App. N +2 just by puttingS N 2S T N 2 at the center of the above MPS, where this insertion is a variant of Eq. (4.5). In the thermodynamic limit N → ∞ the matrixS i in Eq. (4.11) is independent on the site index i, and therefore it coincides withS in Eq. (4.8). This symmetric representation of uniform MPS is often of use. Conclusions and Discussions We have considered the wave function prediction in the infinite system DMRG method, when it is applied to the 2D vertex model. Through the singular value decomposition of CTM C i , we obtained the approximate half-column transfer matrix P i . The insertion ofP i naturally explains the wave function prediction proposed by McCulloch, 41) which works better than the product wave function renormalization group (PWFRG) method, [23][24][25] especially when the system size is small. The difference between these two prediction methods can be explained by the shape of finite size system. The PWFRG method treats growing triangular cluster, 24) whereas Mc-Culloch's scheme always treat half-infinite stripe. The relation between CTM and MPS in the finitesystem DMRG method is not so clear. For example, Ψ 8 can be expressed as C 3 C T 5 , but the MPS representation of the optimized Ψ 8 by the finite system DMRG cannot be obtained from the SVD applied to C 3 and C 5 independently. This puzzle is something to do with the targeting scheme for asymmetric vertex model, and also with the determination of optimal RG transformation in the realtime DMRG method, where the density matrix is time dependent. Fig. 1 . 1A finite size vertex model of width N = 8. Cross marks show boundary spins and open circles show row spins, which are the variables of Ψ 8 in Eq. (2.1). The system consists of its lefthalf and the right half, where the vertical stitch corresponds to the half-column spin σ 1 , σ 2 , σ 3 , . . . between them. Fig. 2 . 2The singular value decomposition applied to C 4 . The black square and circle corresponds to the block spin ξ and the singular values Ω 4 (ξ), respectively. The left and the right triangles represent A 4 and U 4 , respectively. Fig. 3 . 3The pictorial representation of P 4 = C −1 3 · C 4 in Eq. (3.1). The circle with the cross mark shows Ω −1 3 . Since P 4 has a function of extending the area of CTM, it can be regarded as an approximation for the half-column transfer matrix. Fig. 4 . 4The area of CTM can be extended by applying the approximate half-column transfer matrices. Fig. 5 . 5Pictorial representation of Eq. (3.8). Fig. 6 . 6Matrix product expression of Ψ 8 in Eq. (3.9). Double circle represent Λ 4 = (Ω 4 ) 2 . Fig. 8 . 8Reorthogonalization process in Eq. (4.3). The rectangular in the lower diagram corresponds to Λ in Eq. (4.4). the new orthogonal matrixB and the right triangular matrix Λ. (See Fig. 8.) Substituting Eq. (4.4) into Eq. (4.3), we get the matrix product expression Fig. 9 . 9Graphical representation of Eq. 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{'abstract': 'We report a physical background of the wave function prediction in the infinite system density matrix renormalization group (DMRG) method, from the view point of two-dimensional vertex model, a typical lattice model in statistical mechanics. Singular value decomposition applied to rectangular corner transfer matrices naturally draws matrix product representation for the maximal eigenvector of the row-to-row transfer matrix. The wave function prediction can be expressed as the insertion of an approximate half-column transfer matrix. This insertion process is in accordance with the scheme proposed by McCulloch recently.', 'arxivid': '0911.5601', 'author': ['Hiroshi Ueda \nDepartment of Material Engineering Science\nGraduate School of Engineering Science\nOsaka University\n560-8531OsakaJapan\n', 'Andrej Gendiar \nInstitute of Electrical Engineering\nSlovak Academy of Sciences\nDúbravská cesta 9SK-841 04BratislavaSlovakia\n', 'Tomotoshi Nishino \nDepartment of Physics\nGraduate School of Science\nKobe University\n657-8501KobeJapan\n'], 'authoraffiliation': ['Department of Material Engineering Science\nGraduate School of Engineering Science\nOsaka University\n560-8531OsakaJapan', 'Institute of Electrical Engineering\nSlovak Academy of Sciences\nDúbravská cesta 9SK-841 04BratislavaSlovakia', 'Department of Physics\nGraduate School of Science\nKobe University\n657-8501KobeJapan'], 'corpusid': 119204451, 'doi': '10.1143/jpsj.79.044001', 'github_urls': [], 'n_tokens_mistral': 9080, 'n_tokens_neox': 7688, 'n_words': 4558, 'pdfsha': '4fa525f5e0d1ba3b9471cf4336b290d1f7f7a6fd', 'pdfurls': ['https://arxiv.org/pdf/0911.5601v1.pdf'], 'title': ['A Classical Background for the Wave Function Prediction in the Infinite System Density Matrix Renormalization Group Method', 'A Classical Background for the Wave Function Prediction in the Infinite System Density Matrix Renormalization Group Method'], 'venue': []}

arxiv

Deep ChArUco: Dark ChArUco Marker Pose Estimation Danying Hu Magic Leap, Inc Daniel Detone ddetone@magicleap.com Magic Leap, Inc Vikram Chauhan vchauhan@magicleap.com Magic Leap, Inc Igor Spivak ispivak@magicleap.com Magic Leap, Inc Tomasz Malisiewicz tmalisiewicz@magicleap.com Magic Leap, Inc Deep ChArUco: Dark ChArUco Marker Pose Estimation ChArUco boards are used for camera calibration, monocular pose estimation, and pose verification in both robotics and augmented reality. Such fiducials are detectable via traditional computer vision methods (as found in OpenCV) in well-lit environments, but classical methods fail when the lighting is poor or when the image undergoes extreme motion blur. We present Deep ChArUco, a real-time pose estimation system which combines two custom deep networks, ChArUcoNet and RefineNet, with the Perspective-n-Point (PnP) algorithm to estimate the marker's 6DoF pose. ChArUcoNet is a two-headed markerspecific convolutional neural network (CNN) which jointly outputs ID-specific classifiers and 2D point locations. The 2D point locations are further refined into subpixel coordinates using RefineNet. Our networks are trained using a combination of auto-labeled videos of the target marker, synthetic subpixel corner data, and extreme data augmentation. We evaluate Deep ChArUco in challenging low-light, high-motion, high-blur scenarios and demonstrate that our approach is superior to a traditional OpenCV-based method for ChArUco marker detection and pose estimation. Introduction In this paper, we refer to computer-vision-friendly 2D patterns that are unique and have enough points for 6DoF pose estimation as fiducials or markers. ArUco markers [1,2] and their derivatives, namely ChArUco markers, are frequently used in augmented reality and robotics. For example, Fiducial-based SLAM [3,4] reconstructs the world by first placing a small number of fixed and unique patterns in the world. The pose of a calibrated camera can be estimated once at least one such marker is detected. But as we will see, traditional ChArUco marker detection systems are surprisingly frail. In the following pages, we motivate and explain our recipe for creating a state-of-the-art Deep ChArUco marker detector based on deep neural networks. We focus on one of the most popular class of fiducials in augmented reality, namely ChArUco markers. In this paper, we highlight the scenarios under which traditional computer vision techniques fail to detect such fiducials, and present Deep ChArUco, a deep convolutional neural network system trained to be accurate and robust for ChArUco marker detection and pose estimation (see Figure 1). The main contributions of this work are: 1. A state-of-the-art and real-time marker detector that improves the robustness and accuracy of ChArUco pattern detection under extreme lighting and motion 2. Two novel neural network architectures for point ID classification and subpixel refinement 3. A novel training dataset collection recipe involving auto-labeling images and synthetic data generation Overview: We discuss both traditional and deep learning-based related work in Section 2. We present ChArUcoNet, our two-headed custom point detection network, and RefineNet, our corner refinement network in Section 3. Finally, we describe both training and testing ChArUco datasets in Section 4, evaluation results in Section 5, and conclude with a discussion in Section 6. Related Work Traditional ChArUco Marker Detection A ChArUco board is a chessboard with ArUco markers embedded inside the white squares (see Figure 2). ArUco markers are modern variants of earlier tags like ARTag [5] and AprilTag [6]. A traditional ChArUco detector will first detect the individual ArUco markers. The detected ArUco markers are used to interpolate and refine the position of the chessboard corners based on the predefined board layout. Because a ChArUco board will generally have 10 or more points, ChArUco detectors allow occlusions or partial views when used for pose estimation. In the classical OpenCV method [7], the detection of a given ChArUco board is equivalent to detecting each chessboard inner corner associated with a unique identifier. In our experiments, we use the 5 × 5 ChArUco board which contains the first 12 elements of the DICT_5x5_50 ArUco dictionary as shown in Figure 2. . ChArUco = Chessboard + ArUco. Pictured is a 5x5 ChArUco board which contains 12 unique ArUco patterns. For this exact configuration, each 4x4 chessboard inner corner is assigned a unique ID, ranging from 0 to 15. To goal of the computer vision algorithm is to detect these unique 16 corners and IDs. Deep Nets for Object Detection Deep Convolutional Neural Networks have become the standard tool of choice for object detection since 2015 (see systems like YOLO [8], SSD [9], and Faster R-CNN [10]). While these systems obtain impressive multi-category object detection results, the resulting bounding boxes are typically not suitable for pose inference, especially the kind of high-quality 6DoF pose estimation that is necessary for augmented reality. More recently, object detection frameworks like Mask-RCNN [11] and PoseCNN [12] are building pose estimation capabilities directly into their detectors. Deep Nets for Keypoint Estimation Keypoint-based neural networks are usually fullyconvolutional and return a set of point-specific heatmaps. Deep Nets for keypoint estimation are popular in the human pose estimation literature. Since for a rigid object, as long as we can repeatably detect a smaller yet sufficient number of 3D points in the 2D image, we can perform PnP to recover the camera pose. Albeit indirectly, keypoint-based methods do allow us to recover pose using a hybrid deep (for point detection) and classical (for pose estimation) system. One major limitation of most keypoint estimation deep networks is that they are too-slow because of the expensive upsampling operations in hourglass networks [13]. Another relevant class of techniques is those designed for human keypoint detection such as faces, body skeletons [14], and hands [15]. Deep Nets for Interest Point Detection The last class of deep learning-based techniques relevant to our discussion are deep interest point detection systemsmethods that are deep replacements for classical systems like SIFT [17] and ORB [18]. Deep Convolutional Neural Networks like DeTone et al's SuperPoint system [16] are used for joint interest point and descriptor computation. SuperPoint is a single real-time unified CNN which performs the roles of multiple deep modules inside earlier deep learning for interest-point systems like the Learned Invariant Feature Transform (LIFT) [19]. Since SuperPoint networks are designed for real-time applications, they are a starting point for our own Deep ChArUco detector. A major challenge addressed in this paper is how to convert general purpose descriptor-based architectures into point detectors with a predetermined number of pattern-specific ids. Deep ChArUco: A System for ChArUco Detection and Pose Estimation In this section we describe the fully convolutional neural network we used for ChArUco marker detection. Our network is an extension of SuperPoint [16] which includes a custom head specific to ChArUco marker point identification. We develop a multi-headed SuperPoint variant, suit- Figure 4. Two-Headed ChArUcoNet and RefineNet. ChArUcoNet is a SuperPoint-like [16] network for detecting a specific ChArUco board. Instead of a descriptor head, we use a point ID classifier head. One of the network heads detects 2D locations of ChArUco boards in X and the second head classifies them in C. Both heads output per-cell distributions, where each cell is an 8x8 region of pixels. We use 16 unique points IDs for our 5x5 ChArUco board. ChArUcoNet's output is further refined via a RefineNet to obtain subpixel locations. able for ChArUco marker detection (see architecture in Figure 4). Instead of using a descriptor head, as was done in the SuperPoint paper, we use an id-head, which directly regresses to corner-specific point IDs. We use the same point localization head as SuperPoint -this head will output a distribution over pixel location for each 8x8 pixel region in the original image. This allows us to detect point locations at full image resolution without using an explicit decoder. Defining IDs. In order to adapt SuperPoint to CharUco marker detection, we must ask ourselves: which points do we want to detect? In general, there are multiple strategies for defining point IDs (see Figure 3). For simplicity, we decided to use the 4x4 grid of interior chessboard corners for point localization, giving a total of 16 different point IDs to be detected. The ID classification head will output a distribution over 17 possibilities: a cell can belong to one of the 16 corner IDs or an additional "dustbin" none-of-the-above class. This allows a direct comparison with the OpenCV method, since both classical and deep techniques attempt to localize the same 16 ChArUco board-specific points. ChArUcoNet Network Architecture The ChArUcoNet architecture is identical to that of the SuperPoint [16] architecture, with one exception -the descriptor head in the SuperPoint network is replaced with a ChArUco ID classification head C as shown in Figure 4. The network uses a VGG-style encoder to reduce the dimensionality of the image. The encoder consists of 3x3 convolutional layers, spatial downsampling via pooling and non-linear activation functions. There are three maxpooling layers which each reduce the spatial dimensionality of the input by a factor of two, resulting in a total spatial reduction by a factor of eight. The shared encoder outputs features with spatial dimension H c × W c . We define H c = H/8 and W c = W/8 for an image sized H ×W . The keypoint detector head outputs a tensor X ∈ R Hc×Wc×65 . Let N id be the number of ChArUco points to be detected (e.g. for a 4x4 ChArUco grid N c = 16). The ChArUco ID classification head outputs a classification tensor C ∈ R Hc×Wc×(Nc+1) over the N c classes and a dustbin class, resulting in N c + 1 total classes. The ChArUcoNet network was designed for speed-the network weights take 4.8 Megabytes and the network is able to process 320 × 240 sized images at approximately 70fps using an 8GB NVIDIA GPU. RefineNet Network Architecture To improve pose estimation quality, we additionally perform subpixel localization -we refine the detected integer corner locations into subpixel corner locations using Re-fineNet, a deep network trained to produce subpixel coordinates. RefineNet, our deep counterpart to OpenCV's cornerSubPix, takes as input a 24 × 24 image patch and outputs a single subpixel corner location at 8× the resolution of the central 8 × 8 region. RefineNet performs softmax classification over an 8× enlarged central region -Re-fineNet finds the peak inside the 64 × 64 subpixel region (a 4096-way classification problem). RefineNet weights take up only 4.1 Megabytes due to a bottleneck layer which converts the 128D activations into 8D before the final 4096D mapping. Both ChArUcoNet and RefineNet use the same VGG-based backbone as SuperPoint [16]. For a single imaged ChArUco pattern, there will be at most 16 corners to be detected, so using RefineNet is as expensive as 16 additional forward passes on a network with 24 × 24 inputs. Pose Estimation via PnP Given a set of 2D point locations and a known physical marker size we use the Perspective-n-Point (PnP) algorithm [20] to compute the camera pose. PnP requires knowledge of K, the camera intrinsics, so we calibrate the camera before collecting data. We calibrated the camera until the reprojection error fell below 0.15 pixels. We use OpenCV's solvePnPRansac to estimate the final camera pose in our method as well as the OpenCV baseline. ChArUco Datasets To train and evaluate our Deep ChArUco Detection system, we created two ChArUco datasets. The first dataset focuses on diversity and is used for training the ChArUco detector (see Figure 5). The second dataset contains short video sequences which are designed to evaluate system performance as a function of illumination (see Figure 7). Training Data for ChArUcoNet We collected 22 short video sequences from a camera with the ChArUco pattern in a random but static pose in each video. Some of the videos include a ChArUco board taped to a monitor with the background changing, and other sequences involve lighting changes (starting with good lighting). Videos frames are extracted into the positive dataset with the resolution of 320 × 240, resulting in a total of 7, 955 gray-scale frames. Each video sequence starts with at least 30 frames of good lighting. The ground truth of each video is auto-labeled from the average of the first 30 frames using the classical OpenCV method, as the OpenCV detector works well with no motion and good lighting. The negative dataset contains 91, 406 images in total, including 82, 783 generic images from the MS-COCO dataset 1 and 8, 623 video frames collected in the office. Our in-office data contains images of vanilla chessboards, and adding them to our negatives was important for improving overall model robustness. We collect frames from videos depicting "other" ChArUco markers (i.e., different than the target marker depicted in Figure 2). For these videos we treated the classifier IDs as negatives but treated the corner locations as "ignore." Data Augmentation for ChArUcoNet With data augmentation, each frame will undergo a random homographic transform and a set of random combination of synthetic distortions under certain probability (see Table 1) during the training stage, which dramatically increases the diversity of the input dataset. The order and the extent of the applied distortion effects are also randomly selected for each frame. For example, Figure 5 Synthetic Subpixel Corners for RefineNet We train RefineNet using a large database of synthetically generated corner images. Each synthetic training image is 24×24 pixels and contains exactly one a ground-truth corner within the central 8 × Evaluation Data Evaluation and Results We compare our Deep ChArUco detector against a traditional OpenCV-based ChArUco marker detector in a frameby-frame manner. We first evaluate both systems' ability to detect the 16 ChArUco markers for a fixed set of images, under increasing blur and lighting changes (synthetic effects). Then, on real sequences, we estimate the pose of the ChArUco board based on the Perspective-n-Point algorithm and determine if the pose's reprojection error is below a threshold (typically 3 pixels). Below, we outline the metrics used in our evaluation. • Corner Detection Accuracy (accuracy of ChArU-coNet) • ChArUco Pose Estimation Accuracy (combined accuracy of ChArUcoNet and RefineNet) A corner is correctly detected when the location is within a 3 pixel radius of the ground truth, and the point ID is identified correctly based on ChArUcoNet ID classifier. The corner detection accuracy is the ratio between the number of accurately detected corners and 16, the total number of marker corners. The average accuracy is calculated as the mean of detection accuracy across 20 images with different static poses. To quantitatively measure the pose estimation accuracy in each image frame, we use the mean reprojection error re as defined below: re = n i=1 |PC i − c i | n ,(1) where P is the camera projection matrix containing intrinsic parameters. C i represents the 3D location of a detected corner computed from the ChArUco pose, c i denotes the 2d pixel location of the corresponding corner in the image. n (≤ 16) is the total number of the detected ChArUco corners. Evaluation using synthetic effects In this section, we compare the overall accuracy of the Deep ChArUco detector and the OpenCV detector under synthetic effects, in which case, we vary the magnitude of the effect linearly. The first two experiments are aimed to Figure 9. Synthetic Motion Blur Test. We compare Deep ChArUco with the OpenCV approach on 20 random images from our test-set while increasing the amount of motion blur. evaluate the accuracy of ChArUcoNet output, without relying on RefineNet. In each of our 20 synthetic test scenarios, we start with an image taken in an ideal environment -good lighting and random static pose (i.e., minimum motion blur), and gradually add synthetic motion blur and darkening. Synthetic Motion Blur Test In the motion blur test, a motion blur filter along the horizontal direction was applied to the original image with the varying kernel size to simulate the different degrees of motion blur. In Figure 9, we plot average detection accuracy versus the degree of motion blur (i.e., the kernel size). It shows that Deep ChArUco is much more resilient to the motion blur effect compared to the OpenCV approach. 1. 1. 1. 1. 1. 1. 1. 1. 1. 0. ] Synthetic Lighting Test In the lighting test, we compare both detectors under different lighting conditions created synthetically. We multiply the original image with a rescaling factor of 0.6 k to simulate increasing darkness. In Figure 11, we plot average detection accuracy versus the darkness degree, k. Figure 10 shows an example of increasing darkness and the output of both detectors. We note that Deep ChArUco is able to detect markers in many cases where the image is "perceptually black" (see last few columns of Figure 10). Deep ChArUco detects more than 50% of the corners even when the brightness is rescaled by a factor of 0.6 9 ∼ .01, while the OpenCV detector fails at the rescaling factor of 0.6 4 ∼ .13. Figure 11. Synthetic Lighting Test. We compare Deep ChArUco with the OpenCV approach on 20 random images from our test-set while increasing the amount of darkness. Evaluation on real sequences First, we qualitatively show the accuracy of both detectors in real video clips captured in different scenarios as described in section 4.4, "Evaluation Data." Figure 13 shows the results of both detectors under extreme lighting and motion. Notice that the Deep ChArUco detector significantly outperforms the OpenCV detector under these extreme scenarios. Overall, our method detects more correct keypoints where a minimum number of 4 correspondences is necessary for pose estimation. In our large experiment, we evaluate across all 26, 000 frames in the 26-video dataset, without adding synthetic effects. We plot the fraction of correct poses vs. pose correctness threshold (as measured by reprojection error) in Figure 12. Overall, we see that the Deep ChArUco system exhibits higher detection rate (97.4% vs. 68.8% under a 3 pixel reprojection error threshold) and lower pose error compared to the traditional OpenCV detector. For each sequence in this experiment, Table 3 lists the ChArUco detection rate (where re < 3.0) and the mean re . For sequences at 1 and 0.3 lux, OpenCV is unable to return a pose-they are too dark. For sequences with shadows, Deep ChArUco detects a good pose 100% of the time, compared to 36% for OpenCV. For videos with motion blur, Deep ChArUco works 78% of the time, compared to 27% for OpenCV. For a broad range of "bright enough" scenarios ranging from 3 lux to 700 lux, both Deep ChArUco and OpenCV successfully detect a pose 100% of the time, but Deep ChArUco has slightly lower reprojection error, re on most sequences. 2 Deep ChArUco Timing Experiments At this point, it is clear that Deep ChArUco works well under extreme lighting conditions, but is it fast enough for real-time applications? We offer three options in network configuration based on the application scenarios with different requirements: • ChArUcoNet + RefineNet: This is the recommended configuration for the best accuracy under difficult conditions like motion blur, low light, and strong imaging noise, but with longest post processing time. • ChArUcoNet + cornerSubPix: For comparable accuracy in well lit environment with less imaging noise, this configuration is recommended with moderate post processing time. • ChArUcoNet + NoRefine: This configuration is preferred when only the rough pose of the ChArUco pattern is required, especially in a very noisy environment where cornerSubPix will fail. The processing time is therefore the shortest as the image only passes through one CNN. Configurations Approx. fps (Hz) ChArUcoNet + RefineNet 17.5 ChArUcoNet + cornerSubPix 67.5 ChArUcoNet + NoRefine 71.2 Table 2. Deep ChArUco Timing Experiments. We present timing results for ChArUcoNet running on 320 × 240 images in three configurations: with RefineNet, with an OpenCV subpixel refinement step, and without using refinement. We compare the average processing speed of 320 × 240 sized images using each of the above three configurations in Table 2. The reported framerate is an average across the evaluation videos described in Section 4.4. Experiments are performed using one 8GB NVIDIA GPU. Since ChArU-coNet is fully convolutional, it is possible to apply the network to different image resolutions, depending on computational or memory requirements. To achieve the best performance with larger resolution images, we can pass a lowresolution image through ChAuRcoNet to roughly localize the pattern and then perform subpixel localization via Re-fineNet in the original high-resolution image. Conclusion Our paper demonstrates that deep convolutional neural networks can dramatically improve the detection rate for ChArUco markers in low-light, high-motion scenarios where the traditional ChArUco marker detection tools inside OpenCV often fail. We have shown that our Deep ChArUco system, a combination of ChArUcoNet and Re-fineNet, can match or surpass the pose estimation accuracy of the OpenCV detector. Our synthetic and real- Table 3. Deep ChArUco vs OpenCV Individual Video Summary. We report the pose detection accuracy (percentage of frames with reprojection error less than 3 pixels) as well as the mean reprojection error, re, for each of our 26 testing sequences. Notice that OpenCV is unable to return a marker pose for images at 1 lux or darker (indicated by nan). data experiments show a performance gap favoring our approach and demonstrate the effectiveness of our neural network architecture design and the dataset creation methodology. The key ingredients to our method are the following: ChArUcoNet, a CNN for pattern-specific keypoint detection, RefineNet, a subpixel localization network, a custom ChArUco pattern-specific dataset, comprising extreme data augmentation and proper selection of visually similar patterns as negatives. The final Deep ChArUco system is ready for real-time applications requiring marker-based pose estimation. Furthermore, we used a specific ChArUco marker as an example in this work. By replacing the ChArUco marker with another pattern and collecting a new dataset (with manual labeling if the automatic labeling is too hard to achieve), the same training procedure could be repeated to produce numerous pattern-specific networks. Future work will focus on multi-pattern detection, integrating ChArU-coNet and RefineNet into one model, and pose estimation of non-planar markers. Figure 14. Shadow Sequences. We report the pose accuracy vs. reprojection error threshold on the following sequences: shadow 1, shadow 2, and shadow 3. The results on shadow sequences indicate that Deep ChArUco is very robust to nuisance factors such as cast shadows. See top of Figure 13 for examples of difficult shadows-Deep ChArUco is relatively unaffected by such shadows while OpenCV rarely detects point IDs behind a shadow. Figure 15. Motion blur Sequences. We report the pose accuracy vs. reprojection error threshold on the following sequences: motion 1, motion 2, and motion3 3 (see bottom of Figure 13). For motion blur sequences, we see that the traditional method is slightly better when a pose threshold of 1 pixel or less is chosen. This suggests that Deep ChArUco could benefit from training with examples of real (non-synthetic) blur. Figure 16. Extreme Low Light Sequences. We report the pose accuracy vs. reprojection error threshold on the sequences at 1 lux and below. OpenCV completely fails. Figure 17. Normal Light Sequences. We report the pose accuracy vs. reprojection error threshold on the sequences between 3 and 700 lux. From 3 to 5 lux, Deep ChArUco shows a visible improvement, while for a higher lux, both methods perform similarly. Figure 1 . 1Deep ChArUco is an end-to-end system for ChArUco marker pose estimation from a single image. Deep ChArUco is composed of ChArUcoNet for point detection (Section 3.1), Re-fineNet for subpixel refinement (Section 3.2), and the Perspectiven-Point (PnP) algorithm for pose estimation (Section 3.3). For this difficult image, OpenCV does not detect enough points to determine a marker pose. Figure 2 2Figure 2. ChArUco = Chessboard + ArUco. Pictured is a 5x5 ChArUco board which contains 12 unique ArUco patterns. For this exact configuration, each 4x4 chessboard inner corner is assigned a unique ID, ranging from 0 to 15. To goal of the computer vision algorithm is to detect these unique 16 corners and IDs. Figure 3 . 3Defining ChArUco Point IDs. These three examples show different potential structures in the pattern than could be used to define a single ChArUco board. a) Every possible corner has an ID. b) Interiors of ArUco patterns chosen as IDs. c) Interior chessboard of 16 ids, from id 0 of the bottom left corner to id 15 of the top right corner (our solution). Figure 5 . 5ChArUco Training Set. Examples of ChArUco dataset training examples, before and after data augmentation. shows frames 1 MS-COCO 2014 train: http://images.cocodataset.org/zips/train2014.zip Figure 6 . 6RefineNet Training Images. 40 examples of synthetically generated image patches for training RefineNet. from the training sequences (top row) and augmented with a set of distortions (bottom row). 8 pixel region. For examples of such training image patches, see Figure 6. For•Figure 7 . 7evaluation, we captured 26 videos of 1000 frames at 30Hz from a Logitech R webcam (see examples in Figure 7). Each video in this set focuses on one of the following effects: • Lighting brightness (20 videos with 10 different lighting configurations) Shadow ChArUco Evaluation Set. Examples of frames from the ChArUco evaluation set. From left to right, each frame focuses on lighting (10lux), shadow, motion blur. Figure 8 . 8Synthetic Motion Blur Test Example. Top row: input image applied with varying motion blur effect from kernel size 0 to 10; middle row: corners and ids detected by OpenCV detector, with detection accuracy [1. 1. 1. 1. 1. 0.125 0. 0. 0. 0. 0. 0. ]; bottom row: corners and ids detected from the Deep ChArUco, with detection accuracy [1. 1.1. 1. 1. 1. 1. 1. 1. 1. 1. 1.] Figure 8 shows an example of increasing motion blur and the output of both detectors. Both the visual examples and resulting plot show that OpenCV methods start to completely fail (0% detection accuracy) for kernel sizes of 6 and larger, while Deep ChArUco only degrades a little bit in performance (94% detection accuracy), even under extreme blur. Figure 10 . 10Synthetic Lighting Test Example. Top row: input image applied with a brightness rescaling factor 0.6 k with k from 0 to 10; middle row: corners and ids detected by OpenCV detector with detection accuracy [1. 1. 1. 1. 0. 0. 0. 0. 0. 0. 0.]; bottom row: corners and ids detected from the Deep ChArUco with detection accuracy [1. Figure 12 . 12Deep ChArUco vs OpenCV across entire evaluation dataset. Pose accuracy vs. reprojection error re threshold is computed across all 26, 000 frames in the 26 videos of our evaluation set. Deep ChArUco exhibits higher pose estimation accuracy (97.4% vs. 68.8% for OpenCV) under a 3 pixel reprojection error threshold. Figure 13 . 13Deep ChArUco vs OpenCV Qualitative Examples. Detector performance comparison under extreme lighting: shadows (top) and motion (bottom). Unlike OpenCV, Deep ChArUco appears unaffected by cast shadows. Table 1 . 1Synthetic Effects Applied For Data Augmentation.During training we transform the images to capture more illumination and pose variations. Videodeep acc cv acc deep re cv re0.3lux 100 0 0.427 nan 0.3lux 100 0 0.388 nan 1lux 100 0 0.191 nan 1lux 100 0 0.195 nan 3lux 100 100 0.098 0.168 3lux 100 100 0.097 0.164 5lux 100 100 0.087 0.137 5lux 100 100 0.091 0.132 10lux 100 100 0.098 0.106 10lux 100 100 0.097 0.105 30lux 100 100 0.100 0.092 30lux 100 100 0.100 0.088 50lux 100 100 0.103 0.101 50lux 100 100 0.102 0.099 100lux 100 100 0.121 0.107 100lux 100 100 0.100 0.118 400lux 100 100 0.086 0.093 400lux 100 100 0.085 0.093 700lux 100 100 0.102 0.116 700lux 100 100 0.107 0.120 shadow 1 100 42.0 0.254 0.122 shadow 2 100 30.1 0.284 0.130 shadow 3 100 36.9 0.285 0.141 motion 1 74.1 16.3 1.591 0.154 motion 2 78.8 32.1 1.347 0.160 motion 3 80.3 31.1 1.347 0.147 For per-video analysis on the 26 videos in our evaluation dataset, please see the Appendix. Appendix Aruco: a minimal library for augmented reality applications based on opencv. R Munoz-Salinas, Universidad de CórdobaR. Munoz-Salinas, "Aruco: a minimal library for aug- mented reality applications based on opencv," Univer- sidad de Córdoba, 2012. Automatic generation and detection of highly reliable fiducial markers under occlusion. S Garrido-Jurado, R Muñoz-Salinas, F J Madrid-Cuevas, M J Marín-Jiménez, Pattern Recognition. 476S. Garrido-Jurado, R. Muñoz-Salinas, F. J. Madrid- Cuevas, and M. J. Marín-Jiménez, "Automatic gener- ation and detection of highly reliable fiducial markers under occlusion," Pattern Recognition, vol. 47, no. 6, pp. 2280-2292, 2014. Improved structure from motion using fiducial marker matching. J Degol, T Bretl, D Hoiem, Proceedings of the European Conference on Computer Vision (ECCV). the European Conference on Computer Vision (ECCV)J. DeGol, T. Bretl, and D. Hoiem, "Improved structure from motion using fiducial marker matching," in Pro- ceedings of the European Conference on Computer Vi- sion (ECCV), 2018, pp. 273-288. Real-time single camera slam using fiducial markers. 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You only look once: Unified, real-time object detection. J Redmon, S Divvala, R Girshick, A Farhadi, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionJ. Redmon, S. Divvala, R. Girshick, and A. Farhadi, "You only look once: Unified, real-time object detec- tion," in Proceedings of the IEEE conference on com- puter vision and pattern recognition, 2016, pp. 779- 788. Ssd: Single shot multibox detector. W Liu, D Anguelov, D Erhan, C Szegedy, S Reed, C.-Y Fu, A C Berg, SpringerW. Liu, D. Anguelov, D. Erhan, C. Szegedy, S. Reed, C.-Y. Fu, and A. C. Berg, "Ssd: Single shot multibox detector," in European conference on computer vision. Springer, 2016, pp. 21-37. Faster r-cnn: Towards real-time object detection with region proposal networks. S Ren, K He, R Girshick, J Sun, Advances in neural information processing systems. S. Ren, K. He, R. Girshick, and J. Sun, "Faster r-cnn: Towards real-time object detection with region pro- posal networks," in Advances in neural information processing systems, 2015, pp. 91-99. Mask r-cnn. K He, G Gkioxari, P Dollár, R Girshick, Computer Vision (ICCV. K. He, G. Gkioxari, P. Dollár, and R. Girshick, "Mask r-cnn," in Computer Vision (ICCV), 2017 IEEE Inter- national Conference on. IEEE, 2017, pp. 2980-2988. Posecnn: A convolutional neural network for 6d object pose estimation in cluttered scenes. Y Xiang, T Schmidt, V Narayanan, D Fox, arXiv:1711.00199arXiv preprintY. Xiang, T. Schmidt, V. Narayanan, and D. Fox, "Posecnn: A convolutional neural network for 6d object pose estimation in cluttered scenes," arXiv preprint arXiv:1711.00199, 2017. Stacked hourglass networks for human pose estimation. A Newell, K Yang, J Deng, European Conference on Computer Vision. SpringerA. Newell, K. Yang, and J. Deng, "Stacked hourglass networks for human pose estimation," in European Conference on Computer Vision. 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Cambridge university press, 2003.

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{'abstract': "ChArUco boards are used for camera calibration, monocular pose estimation, and pose verification in both robotics and augmented reality. Such fiducials are detectable via traditional computer vision methods (as found in OpenCV) in well-lit environments, but classical methods fail when the lighting is poor or when the image undergoes extreme motion blur. We present Deep ChArUco, a real-time pose estimation system which combines two custom deep networks, ChArUcoNet and RefineNet, with the Perspective-n-Point (PnP) algorithm to estimate the marker's 6DoF pose. ChArUcoNet is a two-headed markerspecific convolutional neural network (CNN) which jointly outputs ID-specific classifiers and 2D point locations. The 2D point locations are further refined into subpixel coordinates using RefineNet. Our networks are trained using a combination of auto-labeled videos of the target marker, synthetic subpixel corner data, and extreme data augmentation. We evaluate Deep ChArUco in challenging low-light, high-motion, high-blur scenarios and demonstrate that our approach is superior to a traditional OpenCV-based method for ChArUco marker detection and pose estimation.", 'arxivid': '1812.03247', 'author': ['Danying Hu \nMagic Leap, Inc\n\n', 'Daniel Detone ddetone@magicleap.com \nMagic Leap, Inc\n\n', 'Vikram Chauhan vchauhan@magicleap.com \nMagic Leap, Inc\n\n', 'Igor Spivak ispivak@magicleap.com \nMagic Leap, Inc\n\n', 'Tomasz Malisiewicz tmalisiewicz@magicleap.com \nMagic Leap, Inc\n\n'], 'authoraffiliation': ['Magic Leap, Inc\n', 'Magic Leap, Inc\n', 'Magic Leap, Inc\n', 'Magic Leap, Inc\n', 'Magic Leap, Inc\n'], 'corpusid': 54462763, 'doi': '10.1109/cvpr.2019.00863', 'github_urls': [], 'n_tokens_mistral': 10197, 'n_tokens_neox': 8845, 'n_words': 5475, 'pdfsha': 'dac84caf366843c812f532f0050475f5841db662', 'pdfurls': ['https://arxiv.org/pdf/1812.03247v1.pdf'], 'title': ['Deep ChArUco: Dark ChArUco Marker Pose Estimation', 'Deep ChArUco: Dark ChArUco Marker Pose Estimation'], 'venue': []}

arxiv

Spatially Covariant Theories of a Transverse, Traceless Graviton, Part I: Formalism 5 Aug 2011 Justin Khoury Center for Particle Cosmology Department of Physics & Astronomy University of Pennsylvania 209 South 33rd Street19104PhiladelphiaPA Godfrey E J Miller Center for Particle Cosmology Department of Physics & Astronomy University of Pennsylvania 209 South 33rd Street19104PhiladelphiaPA Andrew J Tolley Department of Physics Case Western Reserve University 10900 Euclid Ave44106ClevelandOH Spatially Covariant Theories of a Transverse, Traceless Graviton, Part I: Formalism 5 Aug 2011 General relativity is a covariant theory of two transverse, traceless graviton degrees of freedom. According to a theorem of Hojman, Kuchař, and Teitelboim, modifications of general relativity must either introduce new degrees of freedom or violate the principle of general covariance. In this paper, we explore modifications of general relativity that retain the same number of gravitational degrees of freedom, and therefore explicitly break general covariance. Motivated by cosmology, the modifications of interest maintain spatial covariance. Demanding consistency of the theory forces the physical Hamiltonian density to obey an analogue of the renormalization group equation. In this context, the equation encodes the invariance of the theory under flow through the space of conformally equivalent spatial metrics. This paper is dedicated to setting up the formalism of our approach and applying it to a realistic class of theories.Forthcoming work will apply the formalism more generally. Introduction For nearly a century, general relativity has been the most successful paradigm for interpreting and understanding classical gravitational phenomena, and to this day there have been no unequivocal refutations of general relativity. Nonetheless, there are compelling reasons to study alternative gravitational theories. Perhaps the most obvious reason is to explain empirical anomalies, most notably the observed cosmic acceleration. While it is true that this phenomenon can be understood in terms of a cosmological constant in pure general relativity, this approach has the draw- Unfortunately, there is no unambiguous evidence for such scalar degrees of freedom. One motivation for this paper is the possibility that cosmic acceleration might be directly associated with the graviton degrees of freedom. Apart from any attempt to understand empirical anomalies, there remains a compelling theoretical reason to study alternatives: to determine which features of general relativity are essential to its experimental success, and which features are merely incidental. To analyze the theory in this manner, we must know what freedom we have to modify the theory while retaining its explanatory power. The two transverse, traceless graviton degrees of freedom are a key feature of general relativity. Though graviton exchange has never been measured and gravitational waves have never been directly detected, there is substantial indirect evidence for the existence of these degrees of freedom. It is logically possible that additional gravitational degrees of freedom exist, but at the same time there is no unambiguous evidence for them. It is therefore natural to ask whether and how we can modify general relativity while preserving the same number of degrees of freedom. In this paper, we construct manifestly consistent modified theories of gravity that retain the same local degrees of freedom as general relativity. To evade the consequences of the theorem that general relativity is the unique theory of a massless spin-2 particle [1,2], our theories break local Lorentz symmetry explicitly. Theories in which Lorentz symmetry is only broken spontaneously necessarily rely on additional local degrees of freedom. These appear in the broken phase as massless Goldstone modes; an example of such a theory is ghost condensation [3]. Perhaps the biggest obstacle to modifying general relativity is the opaqueness of the theory. General relativity as formulated by Einstein and Hilbert is a covariant theory, which means that the equations of motion for the spacetime metric g µν take the same form in any coordinate system. Unfortunately, invariance under coordinate transformations implies that the theory contains a great deal of gauge arbitrariness, and the underlying dynamical degrees of freedom of the theory have proven difficult to isolate. In fact, the notorious elusiveness of the physical degrees of freedom is partially responsible for the difficulty of quantizing general relativity. This gauge arbitrariness can be understood most clearly by treating general relativity as a constrained field theory. By writing the metric g µν in ADM form 1 and discarding a boundary term, the Einstein-Hilbert action can be rewritten in canonical form as a theory of a spatial metric h ij and a conjugate momentum tensor π ij subject to four first class constraints H µ [4]. While this form of the theory is not written in terms of manifestly diffeomorphism covariant objects, the covariance of the theory can still be inferred from the covariance "algebra" satisfied by the H µ , which act as the generators of spacetime diffeomorphisms under the action of the Poisson bracket [5]. By representing the gauge symmetries of general relativity as constraints on the phase space of the theory, it becomes straightforward to count degrees of freedom. According to the standard counting prescription, it follows from the presence of four first class constraints H µ in a theory of six canonical coordinates h ij that general relativity contains two local degrees of freedom. 2 In the passage to quantum theory, these transverse, traceless degrees of freedom become the two polarizations of the graviton. To isolate the physical graviton degrees of freedom, one would have to solve the four constraints H µ . By taking the configuration space for the metric to be Wheeler's superspace, it is possible to solve the three momentum constraints H i by fiat, but the Hamiltonian constraint H 0 has thus far defied solution in general. Unless the Hamiltonian constraint can be solved, the gauge-arbitrariness of general relativity cannot be eliminated. Fortunately, though no general solution to the Hamiltonian constraint has been found, it is possible to 1 i.e., in terms of a spatial metric h ij , a lapse N ≡ N 0 , and a shift N i . 2 See section 2.1 for more detail. solve it in certain circumstances by imposing an appropriate gauge-fixing condition. In 1974, Hojman, Kuchař, and Teitelboim (HKT) proved that general relativity with a cosmological constant is the unique covariant theory of a spatial metric h ij and its conjugate momentum π ij ; in fact, general relativity is the minimal representation of the covariance algebra [6,7]. It follows that alternative covariant theories of gravitation must introduce additional degrees of freedom beyond the two graviton degrees of freedom in general relativity [8]. Conversely, alternative theories of a spatial metric h ij and its conjugate momentum π ij cannot be covariant. To modify general relativity, one must either introduce new degrees of freedom or violate the principle of general covariance. We wish our theories to retain the same local degrees of freedom as general relativity, so in accordance with the theorem of HKT, our theories cannot be diffeomorphism covariant. This aspect of our approach is not necessarily a defect. The covariance property of general relativity may be intellectually appealing, but the form invariance of the equations of motion is purchased at the cost of substantial gauge redundancy. Moreover, since we do not observe exact spacetime symmetry in our universe, this property of general relativity is not necessarily key to the success of the theory. Simply put, on cosmological scales there is a strong asymmetry between the past and the future, and the observable universe has a preferred rest frame; these observations are conventionally understood as a result of spontaneous symmetry breaking, but explicit symmetry breaking is another logical possibility. That being said, on cosmological scales in the cosmological rest frame there is substantial evidence for spatial homogeneity and isotropy. To maximize the verisimilitude of our treatment, the theories we consider will retain explicit covariance under spatial diffeomorphisms. To summarize, we will attempt to modify general relativity while preserving 1) the number of graviton degrees of freedom, and 2) covariance under spatial diffeomorphisms. In this paper, we develop a general framework within which to explore the freedom we have to modify general relativity while retaining these two desirable properties. Concretely, we will begin by recasting general relativity in spatially covariant form, by solving the Hamiltonian constraint while preserving the momentum constraints. We will solve the Hamiltonian constraint by imposing a cosmologically motivated gauge constraint: we will take the determinant of the spatial metric to be the measure of time. This operation destroys the manifest diffeomorphism covariance and local Lorentz covariance of the theory. We emphasize that this gauge breaks down in the general case when the determinant of the spatial metric is allowed to evolve non-monotonically, but it is a natural choice when considering perturbative corrections to FRW spacetime. By solving the Hamiltonian constraint, the determinant of the spatial metric and the trace of the momentum tensor drop out of the phase space of the theory. We thereby obtain general relativity as a theory of a unit-determinant metrich ij and a traceless conjugate momentum tensorπ ij subject to three first class momentum constraintsH i , which act as the generators of spatial diffeomorphisms. By the standard counting prescription, the presence of three first class constraintsH i in a theory of five canonical coordinatesh ij guarantees that spatially covariant general relativity contains two degrees of freedom, as it should. 3 Our strategy for modifying general relativity relies on the fact that any theory of five canonical coordinates subject to three first class constraints contains two degrees of freedom. To modify general relativity, we will modify the functional form of the physical Hamiltonian density on the reduced phase space (h ij ,π ij ), subject to the condition that the momentum constraintsH i remain first class; to ensure the consistency of the modification, we will also demand that the constraintsH i remain preserved by the equations of motion. Any theory that satisfies these two restrictions will retain manifest spatial covariance, and by the counting prescription will necessarily contain two graviton degrees of freedom. In this paper, we introduce the formalism necessary to pursue this program of modification and apply the formalism to a class of realistic theories. Forthcoming work will apply the formalism developed here to a broader class of theories [9]. The literature abounds with many and varied approaches to the pursuit of modified gravity theories, but covariant modifications of general relativity that introduce additional degrees of freedom have been the most widely explored. The well-known method for finding covariant theories is to construct a diffeomorphism-invariant Lagrangian density out of manifestly covariant objects by contracting all free spacetime indices. Using this technique, all manner of theories have been explored: scalar-tensor theories [10], theories with higherorder curvature terms [11,12,13], theories of massive gravity [14,15,16,17,18,19], higherdimensional gravity theories [20,21,22,23], galileons [24,25,26], chameleons [27,28,29,30], symmetrons [31,32], etc. For a comprehensive review of Lorentz-invariant massive gravity theories with detailed references, see [2]. For a comprehensive review of observational tests of modified gravity, see [33]. 3 See section 3.4 for more detail. Non-covariant approaches have been tried as well, but there is no single unifying procedure for the construction of such theories. In general, the natural procedure for understanding non-covariant theories depends on which symmetries survive in the theory. For example, in [34] Lorentz-violating massive graviton theories were classified by assuming the graviton mass to be invariant under the three-dimensional Euclidean group. A prominent recent example of a non-covariant metric theory is Hořava-Lifshitz gravity [35,36,37], 4 in which the phase space constraints are chosen to satisfy a non-relativistic version of the covariance algebra. Also of note is the work of Barbour and collaborators on theories of conformally equivalent spatial metrics [38]. This paper is organized as follows. In section 2, we cover the basic concepts of constrained field theory in the context of analyzing the phase space and constraint structure of general relativity. In section 3, we show how to impose our cosmological gauge condition and solve the Hamiltonian constraint to obtain a consistent spatially covariant formulation of general relativity. In section 4, we introduce the formalism of our approach to modifying gravity in the context of theories with an ultralocal physical Hamiltonian density. In section 5, we apply our method to derive consistency relations for a class of realistic theories which includes general relativity. General Relativity as a Constrained Field Theory In this section, we will analyze general relativity by treating it as a constrained field theory. In particular, we will examine its phase space and constraint structure, and count its local degrees of freedom. Our starting point is the Einstein-Hilbert action with a cosmological constant, S = dt d 3 x √ −g R (4) − 2Λ .(1) From this action, the general covariance of the theory is manifest, but the counting of degrees of freedom is not. The metric tensor g µν has ten components, but the theory has only two independent local degrees of freedom. To facilitate the counting of degrees of freedom, it is conceptually simplest to rewrite the action in a manner which makes the counting manifest, 4 The original incarnation [35] of Hořava-Lifshitz gravity struggled with consistency issues [36] which were resolved in [37] by imposing a consistent constraint algebra. i.e., canonical form. To this end, the metric g µν must first be expressed in ADM form, in terms of a lapse N ≡ N 0 , a shift N i , and a spatial metric h ij : ds 2 = g µν dx µ dx ν = −N 2 dt 2 + h ij (dx i + N i dt)(dx j + N j dt) .(2) Up to a boundary term, the action of general relativity is S = dt d 3 x √ hN K ij K ij − K 2 + R − 2Λ .(3) In this expression, indices are lowered with h ij and raised with its inverse h ij , R ≡ R (3) is the Ricci scalar of the metric h ij , the extrinsic curvature tensor K ij is defined by K ij ≡ 1 2 N −1 ḣ ij − ∇ i N j − ∇ j N i ,(4)K ≡ h ij K ij , and ∇ i ≡ ∇ (3) i is the covariant spatial derivative with respect to the metric h ij . To obtain the canonical action, one must first define the momentum conjugate to the spatial metric π ij ≡ δL δḣ ij = √ h K ij − Kh ij ;(5) the momentum π ij is three-tensor density of unit weight. 5 By inverting the relation between π ij and K ij , one can rewrite the action of general relativity in canonical form as S = dt d 3 x π ijḣ ij − N µ H µ ,(6) where H 0 ≡ − √ h(R − 2Λ) + 1 √ h π ij π ij − 1 2 (π i i ) 2 , H i ≡ −2h ij ∇ k π jk .(7) Variation of the action with respect to h ij and π ij yields Hamilton's equations, h ij (x) = δH δπ ij (x) ,π ij (x) = − δH δh ij (x) ,(8) where the Hamiltonian H is H = d 3 x N µ H µ .(9) 5 According to the standard convention, the weight of a tensor density is the number of times √ h multiplies the underlying tensor. To evaluate the above variational derivatives, one must use the relations δh ij (x) δh kl (y) = δπ kl (x) δπ ij (y) = δ kl ij δ 3 (x − y) ,(10) where δ kl ij ≡ 1 2 δ k i δ l j + δ l i δ k j .(11) Defining the Poisson bracket {A, B} ≡ d 3 z δA δh mn (z) δB δπ mn (z) − δA δπ mn (z) δB δh mn (z) ,(12) the equation of motion for any quantity A(h ij , π ij , t) can be written aṡ A = ∂A ∂t + {A, H} = ∂A ∂t + d 3 y N ν (y) {A, H ν (y)} .(13) If A has no explicit dependence on time, its evolution is generated by its Poisson bracket with the H µ . Variation of the action with respect to N µ yields the four constraints H µ ∼ 0 .(14) The symbol ∼ denotes weak equality, or equality after the constraints H µ ∼ 0 have been enforced. For example, if X = Y + λ µ H µ , then X ∼ Y . Since the constraints define a surface in phase space, weak equality is also termed equality on the constraint surface. As an aside, it follows from (9) and (14) that H ∼ 0; the vanishing of the Hamiltonian on the constraint surface is a feature common to covariant theories. There is no π µṄ µ term that would allow us to compute a variational expression foṙ N µ , so the time evolution of N µ is unconstrained by the action. The four functions N µ are thus arbitrary until and unless we gauge-fix them. Constraint Properties & Degrees of Freedom Before examining the constraints more closely, we pause to review some terminology first introduced by Dirac for describing constrained theories. A quantity whose Poisson bracket with each of the constraints vanishes (identically or weakly) is termed first class; a quantity whose Poisson bracket fails to vanish weakly with at least one constraint is termed second class. A first class constraint has vanishing Poisson bracket with all constraints, while a second class constraint has non-vanishing Poisson bracket with at least one other constraint. In most cases of interest, first class constraints generate gauge symmetries under the action of the Poisson bracket. Second class constraints can usually be solved, either implicitly (by using the "Dirac bracket") or explicitly (by expressing some phase space variables in terms of others). In general relativity, the constraints H µ generate spacetime diffeomorphisms. By direct calculation -see appendix A for details -it is possible to prove that the constraints H µ are first class, {H µ (x), H ν (y)} ∼ 0. This means that the symmetry generators are closed under the action of the Poisson bracket, as they should be in order to consistently represent any kind of a symmetry. In particular, {H 0 (x), H 0 (y)} = H i (x)∂ x i δ 3 (x − y) − H i (y)∂ y i δ 3 (x − y) , {H 0 (x), H i (y)} = H 0 (y)∂ x i δ 3 (x − y) , {H i (x), H j (y)} = H j (x)∂ x i δ 3 (x − y) − H i (y)∂ y j δ 3 (x − y) .(15) Obeying this covariance "algebra" is the necessary and sufficient condition for the H µ to consistently generate spacetime diffeomorphisms [5]. In fact, four first class constraints obeying this algebra are guaranteed to arise in any covariant field theory; in canonical form, general covariance of the action is encoded in this constraint algebra. For the constraints to be consistent with the equations of motion, the constraints must be preserved by the equations of motion, i.e.,Ḣ µ ∼ 0. Since ∂H µ /∂t = 0, applying the equations of motion to H µ yieldṡ H µ (x) = d 3 y N ν (y){H µ (x), H ν (y)} .(16) From the first class character of the constraints, it follows thatḢ µ ∼ 0, as desired. The Hamiltonian formulation of GR is a theory of a spatial metric h ij and its conjugate momentum π ij , so the theory contains twelve canonical (or six real) variables. However, these variables are not independent. First, they are related by the four constraints H µ ∼ 0. Second, the equations of motion for h ij and π ij depend on the four arbitrary functions N µ ; to gauge-fix the N µ would require imposing four gauge-fixing constraints. 6 · h ij ′ s + 6 · π ij ′ s − 4 · H µ ′ s − 4 · N µ ′ s = 4 canonical DoF .(17) The theory therefore has four canonical (or two real) degrees of freedom. Spatially Covariant General Relativity We would like to depart from general relativity by modifying the equations of motion for the two graviton degrees of freedom. Ideally, we would like to solve all four gauge constraints, go down to the physical phase space, and modify the theory at that level. In this way, we would circumvent all the difficulties of consistently modifying a constrained field theory. Unfortunately, we do not know how to do this. One possible approach is to modify the equations of motion for the phase space variables h ij and π ij . However, the counting of degrees of freedom in general relativity relies on the fact that the four constraints H µ satisfy a consistent first class algebra, namely the covariance algebra of equation (15), and we know from the HKT theorem that any modification of the action for h ij and π ij will destroy this algebra. If we modify the action for the phase space variables h ij and π ij , we must impose an alternative constraint structure that consistently constrains the phase space to the same degree as the covariance algebra; this is the approach taken in [37]. Since we take the point of view that full spacetime covariance is a spurious symmetry, we do not wish our theory to contain a constraint structure that implies the same degree of redundancy as the covariance algebra. Though spacetime symmetry is manifestly broken on cosmological scales (whether spontaneously or explicitly), there is strong evidence for spatial homogeneity and isotropy, so we will attempt to modify general relativity while preserving the manifest spatial covariance of the theory. To obtain a spatially covariant formulation of general relativity to modify, we will solve the Hamiltonian constraint H 0 while leaving the three momentum constraints H i intact. The Hamiltonian constraint is famously hard to solve in general, but we are interested in using our theories in a cosmological context, so we will solve it using a gauge-fixing constraint which is well-defined on an expanding FRW background. Metric Decomposition Before gauge-fixing, we decompose the metric h ij into a conformal factor Ω ≡ h 1/3 and a unit-determinant metrich ij , i.e., h ij = Ωh ij .(18) Note that Ω = ( √ h) 2/3 is a three-scalar density of weight 2/3, whileh ij is a three-tensor density of weight −2/3. The scalar density we will work with is not the conformal factor Ω, but the volume factor ω ≡ √ h = Ω 3/2 , which is a scalar density of unit weight. We choose ω because its conjugate momentum, π ω ≡ δL δω = 2π i i 3ω = − 4 3 K ,(19) is a three-scalar and hence invariant under a rescaling of Ω or ω; this fact will simplify matters in sections 4 and 5. The momentum conjugate toh ij is π ij ≡ δL δḣ ij = Ω π ij − 1 3 h ij π k k = ωΩ K ij − 1 3 Kh ij ,(20) which is a traceless three-tensor density of weight 5/3; the quantitỹ π ij T ≡π ij ωΩ(21) is the corresponding traceless three-tensor. By defining the traceless projection tensorδ kl ij δ kl ij ≡ δ kl ij − 1 3h ijh kl , = δ kl ij − 1 3 h ij h kl ,(22) we can writeπ ij more compactly as π ij = Ωδ ij kl π kl = Ωωδ ij kl K kl .(23) The phase space variables h ij and π ij can thus be written as h ij = ω 2/3h ij , π ij = ω −2/3πij + 1 2h ij ω 1/3 π ω .(24) The decomposition of the spatial metric into a volume factor and a unit-determinant metric is completely general. Though the corresponding conjugate momenta were derived by taking variational derivatives of the Einstein-Hilbert Lagrangian, the decomposition of the momentum tensor into its trace part and its traceless part is likewise completely general. Those familiar with the techniques of numerical relativity may be reminded of the York-Lichnerowicz conformal decomposition or the BSSNOK (Baumgarte, Shapiro, Shibata, Nakamura, Oohara, and Kojima) formalism [39]. Cosmological gauge To solve the constraint H 0 , we must first gauge-fix the lapse N with a gauge-fixing constraint χ for which {H 0 , χ} ≁ 0; this renders H 0 second class, and hence solvable. Since we wish to retain explicit spatial covariance, our constraints H i must remain first class. In a cosmological context, it is natural to use the volume factor of the spatial metric as a clock, so that t = t(ω); we call this cosmological gauge. As mentioned in the introduction, cosmological gauge is only valid when the determinant of the spatial metric evolves monotonically, so this procedure is only valid when considering perturbative corrections to FRW spacetime. When the evolution of ω is monotonic, t(ω) is an invertible function, so this gauge is equivalent to taking the volume factor ω to be a function of time, i.e., ω = ω(t). To impose cosmological gauge, we add to the canonical action of general relativity a gauge-fixing constraint χ ≡ ω − ω(t) ,(25) along with a corresponding Lagrange multiplier λ. The new gauge-fixed action is S ′ = dt d 3 x π ijḣ ij − N µ H µ − λχ .(26) Varying the action with respect to λ then reproduces the constraint χ ∼ 0 .(27) By direct calculation -see appendix B for details -one can verify that {H 0 (x), χ(y)} = 1 2 π i i (x)δ 3 (x − y) ;(28) the constraints H 0 and χ are thus second class, so we expect to be able to solve them. The only wrinkle is that {H i (x), χ(y)} = h(x)∂ x i δ 3 (x − y) ,(29) so the constraints H i are also second class! By shuffling our constraints slightly, we can obtain a set of two second-class constraints and three first class constraints, and thereby preserve explicit spatial covariance. Indeed, since 2 h(x)∂ x i H 0 (x) π k k (x) , χ(y) ∼ h(x)∂ x i δ 3 (x − y) ,(30) it follows that the combinationH i ≡ H i − 2 √ h ∂ i H 0 π k k = H i − 2 √ h ∇ i H 0 π k k (31) obeys H i (x),H j (y) ∼ H i (x), H 0 (y) ∼ H i (x), χ(y) ∼ 0 .(32) The interpretation of this result is simple. The H i 's generate spatial diffeomorphisms, while H 0 generates time translation. A generic spatial diffeomorphism will alter the conformal factor of the spatial metric. If the conformal factor is taken to be the measure of time, then the H i 's, by altering the conformal factor, will generate time translation, while H 0 , by generating time translation, will alter the conformal factor. TheH i 's generate spatial diffeomorphisms that preserve the conformal factor, so they must differ from the H i 's by the gradient of a compensating time translation term. From the definition ofH i , it is apparent that demanding χ ∼ 0 and H µ ∼ 0 is equivalent to demanding χ ∼ 0, H 0 ∼ 0, andH i ∼ 0. The latter set of constraints has the virtue that theH i are first class, and can thus consistently represent symmetries. We therefore take our five constraints to be the two second class constraints χ and H 0 and the three first class constraintsH i . Using H i =H i + 2 √ h∇ i H 0 /π k k , the gauge-fixed action can be rewritten in terms ofH i as S ′ = dt d 3 x π ijḣ ij − N 0 H 0 − N iH i − 2 √ hN i ∇ i H 0 π k k − λχ .(33) Upon integration by parts, the action becomes S ′ = dt d 3 x π ijḣ ij −Ñ H 0 − N iH i − λχ ,(34) whereÑ ≡ N − 2 √ h π k k ∇ i N i .(35) Variation of the action S ′ with respect to h ij and π ij yields Hamilton's equations, h ij (x) = δH ′ δπ ij (x) ,π ij (x) = − δH ′ δh ij (x) ,(36) where the new Hamiltonian H ′ is H ′ = d 3 x Ñ H 0 + N iH i + λχ .(37) The equation of motion for any quantity A(h ij , π ij , t) is thereforė A = ∂A ∂t + {A, H ′ } = ∂A ∂t + d 3 y Ñ (y){A, H 0 (y)} + N i (y){A,H i (y)} + λ(y){A, χ(y)} ,(38) where the Poisson bracket is defined as in (12). Variation of the action S ′ with respect tõ N , λ, and N i yields the five constraints H 0 ∼ 0 , χ ∼ 0 ,H i ∼ 0 .(39) The action does not contain time derivatives of the Lagrange multipliers, so at first their evolution appears unconstrained. Since theH i are first class, the three functions N i are indeed arbitrary until and unless we gauge-fix them. The evolution ofÑ and λ, however, will be determined by demanding the consistency of H 0 and χ with the equations of motion. For the constraints to be consistent with the equations of motion, they must be preserved by the equations of motion; we therefore demand thatḢ i ∼ 0,Ḣ 0 ∼ 0, andχ ∼ 0. Since theH i are first class and ∂ tHi = 0, it follows at once thatḢ i ∼ 0. Since ∂ t H 0 = 0, {H 0 (x), H 0 (y)} ∼ 0, and {H 0 (x),H i (y)} ∼ 0, it follows thaṫ H 0 (x) ∼ d 3 y λ(y){H 0 (x), χ(y)} ∼ 1 2 λ(x)π i i (x) .(40) On a flat FRW background, 6 K = 3ȧ/a and hence π i i = −2ωK = −6ȧa 2 . Since we are only considering gravity on an expanding background, we assume that π i i (x) ≁ 0 more generally. The demandḢ 0 ∼ 0 thus implies λ ∼ 0 . (41) Since {χ(x), χ(y)} = 0, {χ(x),H i (y)} ∼ 0, and ∂χ/∂t = −ω(t), it follows thaṫ χ(x) ∼ −ω(t) + d 3 yÑ (y){χ(x), H 0 (y)}, ∼ −ω(t) − 1 2Ñ (x)π i i (x) .(42) Since π i i ≁ 0, demandingχ ∼ 0 allows us to solve forÑ, N ∼ −2ω(t) π i i .(43) The functionsÑ and λ are thus not arbitrary. Since N =Ñ + 2 √ h(∇ i N i )/π k k , the lapse N has not been completely gauge-fixed, but its arbitrariness stems solely from its dependence on the three arbitrary functions N i . As a check, let us revisit the counting of degrees of freedom in cosmological gauge. For these purposes, the only effect of gauge-fixing is to replace the first class constraint H 0 ∼ 0 and the arbitrary function N with the second class constraints H 0 ∼ 0 and χ ∼ 0. This modifies the left hand side of equation (17), but does not change the final tally. 6 · h ij ′ s + 6 · π ij ′ s − 1 · H 0 − 1 · χ − 3 ·H i ′ s − 3 · N i ′ s = 4 canonical DoF .(44) After gauge-fixing, the theory still has four canonical (or two real) degrees of freedom. Solving H 0 and χ In this section, we will solve the constraints H 0 and χ to obtain a spatially covariant formulation of general relativity as a theory of a unit-determinant metrich ij and its conjugate momentumπ ij . This will set the stage for modifying general relativity in section 4. Since χ and H 0 are second class, they can be solved explicitly to yield expressions for ω and π ω in terms of t,h ij ,π ij , and spatial derivatives. "Solving" for ω is trivial: ω = ω(t). Solving for π ω requires us to take a square root and pick a sign, which amounts to picking either an expanding or a contracting background. We pause to emphasize once again that our procedure is only valid in a cosmological context, when the conformal factor of the spatial metric can be assumed to be evolving monotonically. To pick the sign corresponding to an expanding background, first recall that π ω = − 4 3 K .(45) On a flat FRW background, K = 3ȧ/a and hence π ω = −4ȧ/a. An expanding FRW background therefore corresponds to π ω < 0. Returning to the general case, we choose π ω < 0 to obtain π ω = π GR ≡ − 8 3 π ijπ ij ω 2 −R ω 2/3 + 2Λ ,(46) where indices are raised and lowered withh ij , andR is the Ricci scalar forh ij . Substituting these results for ω and π ω back into the action S ′ yields the action of general relativity on the reduced phase space (h ij ,π ij ), S ′′ = dt d 3 x π ijḣ ij + π ωω − N iH i ,(47)whereH i = −2h ij∇kπ jk − ω∇ i π ω ,(48) and∇ i is the covariant derivative with respect toh ij . This action yields the new Hamiltonian H ′′ = d 3 x −ωπ ω + N iH i .(49) The term π ijḣ ij has split into the termπ ijḣ ij and a contribution −ωπ ω to the physical Hamiltonian density. Variation of the action with respect toh ij andπ ij yieldṡ h ij (x) = δH ′′ δπ ij (x) ,π ij (x) = − δH ′′ δh ij (x) .(50) To evaluate these variational derivatives, one must use the relations δh ij (x) δh kl (y) =δ kl ij δ 3 (x − y) , δπ ij (x) δh kl (y) = − 1 3h ijπkl δ 3 (x − y) ,(51) and δh ij (x) δπ kl (y) = 0 , δπ ij (x) δπ kl (y) =δ ij kl δ 3 (x − y) .(52) Defining the Poisson bracket appropriate to the reduced phase space, {A, B} ≡ d 3 x δA δh ij (x) δB δπ ij (x) − δA δπ ij (x) δB δh ij (x) ,(53) any quantity A(h ij ,π ij , t) obeys the equation of motioṅ A = ∂A ∂t + {A, H ′′ } .(54) Variation of the action with respect to N i yields the three constraints H i ∼ 0 .(55) As before, the time evolution of N i is unconstrained by the action; in the absence of a gauge-fixing procedure, the three functions N i are arbitrary. Constraint Properties & Degrees of Freedom By lengthy direct calculation, it is possible to prove that the constraintsH i are first class, i.e., {H i (x),H j (y)} ∼ 0. Furthermore, by applying the equations of motion toH i , it is possible to show thatḢ i ∼ 0, so the constraints are preserved by the equations of motion. We defer demonstrations of these two facts to section 5, where we will examine general relativity in the context of a class of realistic theories. This is an important consistency check, because a priori it is not clear that our procedure for solving the Hamiltonian constraint will yield a consistent action on the reduced phase space. As a final check, we revisit the counting of degrees of freedom in spatially covariant general relativity. After imposing cosmological gauge and solving the Hamiltonian constraint H 0 ∼ 0, general relativity is a theory of a unit-determinant spatial metrich ij and its traceless conjugate momentumπ ij , so the theory contains ten canonical (or 5 real) variables. This reduction in the size of the phase space is compensated by a corresponding reduction in the number of constraints and arbitrary functions: the theory contains three first class constraintsH i ∼ 0, and its equations of motion involve three arbitrary functions N i . 5 ·h ij ′ s + 5 ·π ij ′ s − 3 ·H i ′ s − 3 · N i ′ s = 4 canonical DoF .(56) Spatially covariant general relativity thus contains four canonical (or two real) degrees of freedom, the same number as fully covariant general relativity. Ultralocal Modified Gravity We have two criteria in mind for our modified theories of gravity: two graviton degrees of freedom, and manifest spatial covariance. Our starting point is the action of spatially covariant general relativity, which has both of these properties. To modify general relativity, we will change the functional form of the scalar quantity π ω , which in general relativity obeys π ω = π GR . This yields the action S = dt d 3 x π ijḣ ij +ωπ ω − N iH i , H i = −2h ij∇kπ jk − ω∇ i π ω ,(57) where π ω is an unspecified scalar function of t, the phase space variablesh ij andπ ij , and spatial derivatives. This action leads to the equation of motioṅ A = ∂A ∂t + {A, H} ,(58) where the Hamiltonian H is H = d 3 x −ωπ ω + N iH i ,(59) and the Poisson bracket is {A, B} ≡ d 3 x δA δh ij (x) δB δπ ij (x) − δA δπ ij (x) δB δh ij (x) .(60) Retaining the manifest spatial covariance of the theory amounts to demanding 1) that the modifiedH i remain first class, i.e., {H i (x),H j (y)} ∼ 0 ,(61) and 2) that the modified constraints be preserved by the modified equations of motion, i.e., H i ∼ 0 .(62) Any theory satisfying these two points will be manifestly covariant under spatial diffeomorphisms, with the constraintsH i acting as the generators of the gauge symmetry. Moreover, the presence of three first class constraintsH i on the phase space (h ij ,π ij ) guarantees that such a theory contains two local degrees of freedom, exactly as desired. In the remainder of the paper, we examine two classes of theories. First, for pedagogical purposes, we treat the case when π ω is an ultralocal 7 function of time t and the phase space variablesh ij andπ ij ; in other words, π ω will not contain spatial derivatives. Second, to make contact with general relativity, we treat the more realistic case when π ω also depends onR, the Ricci scalar ofh ij . Forthcoming work will examine more general classes of scalar momenta [9]. In this section, we use the ultralocal case to introduce the formalism needed to determine when the constraintsH i remain first class and when the constraints are preserved by the equations of motion. In section 5, we apply the formalism to the realistic case. In both the ultralocal and the realistic case, the consistency of the constraints with the equations of motion requires π ω to satisfy an analogue of the renormalization group equation; scalar momenta satisfying this equation are manifestly invariant under rescaling of the volume factor ω. In the ultralocal case, this is the only consistency condition that arises. In the realistic case, demanding that the constraintsH i satisfy a first class algebra is equivalent to demanding that π ω obey a rather complicated differential equation. Constraint Algebra In this section, we will compute the Poisson bracket {H i (x),H a (y)} assuming that π ω is an ultralocal function, and use the result to determine when the constraintsH i remain first class. To simplify the calculation of {H i (x),H a (y)}, we splitH i into a tensor part J i and a scalar part K i . Concretely, we define the vector densities J i ≡ −2h ij∇kπ jk , K i ≡ −ω∇ i π ω ,(63) in terms of whichH i becomes simplyH i = J i + K i .(64) 7 Local functions depend on the value of fields in a neighborhood around a point, so they can be written in terms of fields and derivatives of fields. Ultralocal functions only depend on the value of fields at a point, so they do not contain derivatives. The ultralocal limit is commonly used to analyze long distance cosmological perturbations. The idea of using the cosmological gauge in the ultralocal limit, sometimes referred to as the separate universes approach, is treated in [40]. The Poisson bracket {H i (x),H a (y)} can then be written as the sum of more manageable brackets, {H i (x),H a (y)} = {J i (x), J a (y)} + {K i (x), K a (y)} + {J i (x), K a (y)} + {K i (x), J a (y)} .(65) To simplify the evaluation of these component Poisson brackets, we introduce the smoothing functionals F J ≡ d 3 x f i J i , F K ≡ d 3 x f i K i , G J ≡ d 3 y g a J a , G K ≡ d 3 y g a K a ,(66) where the functions f i and g i are time-independent smoothing functions. We then compute the brackets {F J , G J } = d 3 x d 3 y f i (x)g a (y){J i (x), J a (y)} , {F J , G K } + {F K , G J } = d 3 x d 3 y f i (x)g a (y) {J i (x), K a (y)} + {K i (x), J a (y)} , {F K , G K } = d 3 x d 3 y f i (x)g a (y){K i (x), K a (y)} .(67) We make the key assumption that the smoothing functions decay so rapidly at infinity that when we integrate by parts inside the smoothing functionals, the boundary term vanishes identically; the smoothing functions are otherwise arbitrary. With the freedom to integrate by parts at will, it is straightforward to compute variational derivatives of the smoothing functionals, and thereby to obtain explicit expressions for their Poisson brackets. By comparing these explicit expressions to the formal expressions in equation (67), we will derive explicit expressions for the Poisson brackets involving J i and K i . To compute variational derivatives of the smoothing functional F J , first integrate by parts to obtain F J = 2 d 3 xh ijπ jk∇ k f i ,(68) from which it follows that δF J = d 3 x 2π jk (∇ k f i )δh ij + 2h ij (∇ k f i )δπ jk + 2h ijπ jk δ∇ k f i .(69) The first two terms in this integral are in a convenient form for taking variational derivatives with respect toh ij andπ jk , but the third term requires finessing. To evaluate δ∇ k f i , expand the covariant derivative as∇ k f i = ∂ k f i +Γ i kr f r , whereΓ i jk is the connection of the metric h ij . It follows immediately that δ∇ k f i = f r δΓ i kr . The identity δΓ i kr = 1 2h im ∇ r δh km + ∇ k δh rm − ∇ m δh rk(70) thus implies that 2π jkh ij δ∇ k f i = f iπjk∇ i δh jk , so equation (69) becomes δF J = d 3 x 2π jk (∇ k f i )δh ij + 2h ij (∇ k f i )δπ jk + f iπjk∇ i δh jk .(71) Integrating by parts, this reduces to δF J = d 3 x 2π jk (∇ k f i )δh ij −∇ i (f iπjk )δh jk + 2h ij (∇ k f i )δπ jk .(72) From this expression, it is straightforward to compute variational derivatives of F J , δF J δh mn = 2δ mn ijπ jk∇ k f i −∇ i f iπmn − 2 3π mn∇ i f i , δF J δπ mn = 2δ jk mnh ij∇k f i .(73) The corresponding results for G J are δG J δh mn = 2δ mn abπ bc∇ c g a −∇ a (g aπmn ) − 3π mn∇ a g a , δG J δπ mn = 2δ bc mnh ab∇c g a . The variational calculation for the smoothing functional F K is less straightforward. After integrating by parts, F K becomes F K = ω d 3 x ∂ i f i π ω ,(75) from which it follows that δF K = ω d 3 x ∂ i f i δπ ω .(76) To evaluate δπ ω in full generality would be very difficult, so we will make some simplifying assumptions about the form of π ω . In this section, we will assume that π ω is an ultralocal function of t,h ij , andπ ij . To facilitate calculations, we will enumerate all the scalars that can be built by contracting factors ofh ij against factors ofπ ij . We begin by recursively defining Π ij (n), the linked chain of n factors ofπ ij . The chain of zero factors ofπ ij is simply Π ij (0) ≡h ij .(77) The process of adding a link to the chain is defined by Π ij (n + 1) ≡π i k Π kj (n) .(78) By closing the chain, one obtains scalars, φ(n) ≡ Π i i (n) .(79) The φ(n) are the only scalars that can be built out of connected contractions ofh ij andπ ij . For an arbitrary ultralocal function π ω , it follows that δπ ω = ∞ n=2 ∂π ω ∂φ(n) δφ(n) .(80) The variational derivatives of F K are thus δF K δh mn = ω ∂ i f i ∞ n=2 n ∂π ω ∂φ(n) δ mn jk Π(n) jk − 1 3π mn φ(n − 1) , δF K δπ mn = ω ∂ i f i ∞ n=2 n ∂π ω ∂φ(n)δ jk mn Π(n − 1) jk .(82) Similarly, the variational derivatives of G K are δG K δh mn = ω (∂ a g a ) ∞ m=2 m ∂π ω ∂φ(m) δ mn bc Π(m) bc − 1 3π mn φ(m − 1) , δG K δπ mn = ω (∂ a g a ) ∞ m=2 m ∂π ω ∂φ(m)δ bc mn Π(m − 1) bc .(83) We emphasize that these results for F K and G K rely on the ultralocality assumption, and will be modified in section 5. We are now in a position to compute the Poisson brackets of the smoothing functionals, from which we will extract the Poisson brackets of the vector densities J i and K i . • {J i (x), J a (y)} To obtain the bracket {J i (x), J a (y)}, we first compute {F J , G J }. Combining the F J and G J variations into the bracket {F J , G J } yields {F J , G J } = 2 d 3 z ∇ c f i ∇ i g a h abπ bc − ∇ k g a ∇ a f i h ijπ jk + ∇ k f i ∇ a g ah ijπ jk − ∇ c g a ∇ i f ih abπ bc .(84) After integrating by parts, using the definition J i = −2h ij∇kπ jk , and using the identity ∇ i∇j −∇ j∇i V a =R a bij V b , this reduces to {F J , G J } = d 3 z f i J a∇i g a − g a J i∇a f i + 2f i g aπjk R jika +R jaik .(85) From the symmetries of the Riemann tensor 8 and the traceless momentum tensor 9 , it follows that 0 =π jk R jika +R jaik , so the last term in the integrand vanishes. The connection terms inside the remaining covariant derivatives cancel to yield {F J , G J } = d 3 z f i J a ∂ i g a − g a J i ∂ a f i .(86) To extract the bracket {J i (x), J a (y)} from this result, first relabel dummy indices {F J , G J } = d 3 x f i J a ∂ i g a − d 3 y g a J i ∂ a f i .(87) Under the spatial derivatives in this equation, insert the identities g a (x) = d 3 y δ 3 (x − y)g a (y) , f i (y) = d 3 x δ 3 (x − y)f i (x) ,(88) to obtain {F J , G J } = d 3 x d 3 y f i (x)g a (y) J a (x)∂ x i δ 3 (x − y) − J i (y)∂ y a δ 3 (x − y) .(89) 8R abcd =R cdab ,R abcd = −R bacd = −R abdc . 9πij =π ji . Comparing this expression to equation (67) yields the identity {J i (x), J a (y)} = J a (x)∂ x i δ 3 (x − y) − J i (y)∂ y a δ 3 (x − y) .(90) This is the same algebra obeyed by the H i in equation (15). This result is completely independent of our choice of π ω , and will carry over unchanged into section 5. • {J i (x), K a (y)} + {K i (x), J a (y)} To obtain {J i (x), K a (y)} + {K i (x), J a (y)}, we first compute {F J , G K } + {F K , G J }. Assembling the F J and G K variations into the Poisson bracket {F J , G K } yields {F J , G K } = −ω d 3 z (∂ a g a ) ∞ m=2 ∂π ω ∂φ(m) mΠ(m − 1) bc∇i f iπbc .(91) By expanding the covariant derivative, simplifying the ensuing total derivative of π ω , and recalling that K i = −ω∇ i π ω , this expression reduces to {F J , G K } = d 3 z f i K i ∂ a g a − ω d 3 z ∂ i f i (∂ a g a ) ∞ m=2 ∂π ω ∂φ(m) mφ(m) .(92) Similarly, {F K , G J } = − d 3 z g a K a ∂ i f i + ω d 3 z ∂ i f i (∂ a g a ) ∞ m=2 ∂π ω ∂φ(m) mφ(m) ,(93) so the sum of the two brackets simplifies considerably, {F J , G K } + {F K , G J } = d 3 z f i K i ∂ a g a − g a K a ∂ i f i .(94) Integrating by parts and invoking the identity ∂ i K a = ∂ a K i yields {F J , G K } + {F K , G J } = d 3 z f i K a ∂ i g a − g a K i ∂ a f i .(95) To extract the quantity {J i (x), K a (y)} + {K i (x), J a (y)}, relabel dummy indices and insert the identities in equation (88) to obtain {F J , G K } + {F K , G J } = d 3 x d 3 y f i (x)g a (y) × K a (x)∂ x i δ 3 (x − y) − K i (y)∂ y a δ 3 (x − y) . (96) Combined with equation (67), this result implies that {J i (x), K a (y)} + {K i (x), J a (y)} = K a (x)∂ x i δ 3 (x − y) − K i (y)∂ y a δ 3 (x − y) .(97) This expression depends strongly on the assumed form for π ω . This result is modified heavily in section 5.1, when π ω is allowed to depend onR. • {K i (x), K a (y)} To obtain {K i (x), K a (y)}, we first compute the bracket {F K , G K }. Substituting the F K and G K variations into the Poisson bracket {F K , G K } yields {F K , G K } = ω 2 ∂ i f i (∂ a g a ) ∞ m=2 ∞ n=2 mn ∂π ω ∂φ(m) ∂π ω ∂φ(n) × Π(n) bc Π(m − 1) bc − Π(m) jk Π(n − 1) jk .(98) From the definition of the momentum chain Π(n) ij , it follows that Π(n) bc Π(m − 1) bc = Π(m) jk Π(n − 1) jk = φ(n + m − 1). The terms of the sum thus vanish order by order, so the bracket reduces to {F K , G K } = 0 .(99) By comparing this result to equation (67), it is apparent that {K i (x), K a (y)} = 0 .(100) When π ω is an ultralocal function of the phase space variables, the Poisson bracket {K i (x) , K a (y)} vanishes identically. This will not be the case when π ω depends nontrivially onR, as in section 5.1. By substituting equations (90), (97), and (100) into equation (65), and recalling thatH i = J i + K i , we obtain {H i (x),H j (y)} =H j (x)∂ x i δ 3 (x − y) −H i (y)∂ y j δ 3 (x − y) .(101) This is the same algebra obeyed by the H i in equation (15), and by the J i in equation (90). SinceH i ∼ 0, this result implies that {H i (x),H j (y)} ∼ 0, so the constraintsH i are first class. To establish this result, we assumed only that π ω was an arbitrary ultralocal function of t,h ij , andπ ij ; we showed that this was equivalent to making π ω a function of t and the scalars φ(n) defined in equation (79). Evidently, π ω can be made any ultralocal function of the phase space variables and the momentum constraints will remain first class. Consistency of Constraints with Equations of Motion In this section, we will compute the time derivativeḢ i assuming that π ω is an ultralocal function, and use the result to determine when the constraintsH i are preserved by the equations of motion. The time evolution ofH i is determined by the equation of motioṅ H i = ∂H i ∂t + {H i , H} ,(102) where H = d 3 x −ωπ ω + N iH i .(103) SinceH i = J i + K i and ∂J i /∂t = 0, it follows that ∂H i /∂t = ∂K i /∂t. Recalling that K i = −ω∂ i π ω , the first term in equation (102) becomes ∂H i ∂t = −∂ i ωπ ω + ω ∂π ω ∂t .(104) To simplify the bracket {H i , H}, we define Π ω ≡ d 3 x π ω ,(105) so that H can be written as H = −ωΠ ω + d 3 x N iH i .(106) From the first class character of the constraintsH i , it follows that {H i , H} ∼ −ω{H i , Π ω }. SinceH i = J i + K i , the second term in equation (102) becomes {H i , H} ∼ −ω{J i , Π ω } −ω{K i , Π ω } .(107) To compute the brackets {J i , Π ω } and {K i , Π ω }, we first compute the smoothing functional brackets {F J , Π ω } = d 3 x f i (x){J i (x), Π ω } {F K , Π ω } = d 3 x f i (x){K i (x), Π ω } .(108) We have already done all the work needed to evaluate these two brackets: since Π ω can be obtained from G K by the substitution ∂ a g a → ω −1 , brackets involving Π ω can be obtained by applying this substitution to brackets involving G K . • {J i , Π ω } To compute the bracket {J i , Π ω }, we first compute the bracket {F J , Π ω }. Applying ∂ a g a → ω −1 to equation (92) and integrating by parts yields {F J , Π ω } = d 3 x f i ∂ i −π ω + ∞ m=2 mφ(m) ∂π ω ∂φ(m) .(109) It follows by comparing this result with equation (108) that {J i , Π ω } = ∂ i −π ω + ∞ m=2 mφ(m) ∂π ω ∂φ(m) . (110) • {K i , Π ω } To compute the bracket {K i , Π ω }, we first compute the bracket {F K , Π ω }. By applying the transformation ∂ a g a → ω −1 , equation (99) becomes {F K , Π ω } = 0 .(111) Along with equation (108), this implies that {K i , Π ω } = 0 .(112) By substituting equations (110) and (112) into equation (107), we obtain {H i , H} ∼ω∂ i π ω − ∞ m=2 mφ(m) ∂π ω ∂φ(m) .(113) Upon inserting equations (113) and (104) into the equation of motion (102), theω∂ i π ω terms cancel to yieldḢ i ∼ −∂ i ω ∂π ω ∂t +ω ∞ m=2 mφ(m) ∂π ω ∂φ(m) .(114) DemandingḢ i ∼ 0 implies the consistency condition ω ∂π ω ∂t +ω ∞ m=2 mφ(m) ∂π ω ∂φ(m) ∼ f (t) ,(115) where f (t) is an arbitrary function of time. We observe that the equation of motion (58) is invariant under π ω → π ω + g(t), where g(t) is an arbitrary function of time, so we are free to apply this transformation to simplify our consistency condition. If we choose g(t) so that ωg ′ (t) = f (t), the consistency condition becomes ω ∂π ω ∂t +ω ∞ m=2 mφ(m) ∂π ω ∂φ(m) ∼ 0 .(116) By assumption, ω(t) is an invertible function of time, so ∂/∂t =ω ∂/∂ω. Our consistency condition can thus be written as ∆π ω ∼ 0 ,(117) where we have defined the operator ∆ ≡ ω ∂ ∂ω + ∞ m=2 mφ(m) ∂ ∂φ(m) .(118) To rule out the possibility of a π ω which satisfies ∆π ω ∼ 0 while ∆π ω = 0, we note that the constraintsH i contain one power of spatial derivatives, while by assumption the scalar momentum π ω is ultralocal. To satisfy ∆π ω ∼ 0, the quantity ∆π ω would need to depend on the constraintsH i , and would thus need to contain at least one power of spatial derivatives. However, applying ∆ to π ω does not increase the number of spatial derivatives. It follows that ∆π ω cannot contain any spatial derivatives, and thus cannot depend onH i . The consistency condition can therefore be promoted to ∆π ω = 0 .(119) To obtain the most general solution to this equation, we first note that ∆ (ω −n φ(n)) = 0, which motivates us to defineφ (n) ≡ φ(n) ω n (t) .(120) The most general solution to the condition ∆π ω = 0 is an arbitrary function of theφ(n). The explicit time dependence of π ω is thus determined by its dependence on the phase space variables. To understand this result, we return briefly to the phase space (h ij , π ij ). To construct three-scalars out of the tensor h ij and the traceless tensorπ ij T , we begin by recursively defining Π ij T (n), a chain of n factors ofπ ij T linked together by factors of h ij . In analogy with our construction of the φ(n) of equation (79), we define Π ij T (0) ≡ h ij = Ω −1 Π ij (0) ,(121) and Π ij T (n + 1) ≡π ia T h ab Π bj (n) = ω −1πia f ab Π bj (n) ,(122) from which it follows that Π ij T (n) = Ω −1 ω −n Π ij (n). The contraction h ij Π ij T (n) yields the desired scalars, φ T (n) ≡ h ij Π ij T (n) = φ(n) ω n .(123) The φ T (n) are the only scalars that can be built out of fully connected contractions of h ij andπ ij T . In the presence of the constraint ω ∼ ω(t), it follows that φ T (n) ∼φ(n) .(124) In other words, theφ(n) are the scalars on the phase space (h ij ,π ij ) which have the correct conformal weight to have been derived from three-scalars on the phase space (h ij , π ij ). It follows that theφ(n) are invariant under a rescaling ω → µω of the volume factor ω, and the condition ∆π ω = 0 is thus analogous to a renormalization group equation. Summary In this section, we developed a formalism for testing when our modified theories of gravity lead to a consistent first class constraint algebra, and hence contain two degrees of freedom. To develop the formalism, we made the simplifying assumption that the scalar momentum π ω is an ultralocal function of time t and the phase space variablesh ij andπ ij . This assumption is sufficient to guarantee that the constraintsH i remain first class. However, for the constraints to be consistent with the equations of motion, π ω must be invariant under renormalization of the volume factor ω. Concretely, π ω must obey the renormalization group equation ∆π ω = 0 ,(125) where ∆ ≡ ω ∂ ∂ω + ∞ m=2 mφ(m) ∂ ∂φ(m) .(126) Satisfying this equation completely fixes the dependence of π ω on ω(t). In the next section, we will apply the methods of this section to generalize this result to a more realistic class of scalar momenta. Realistic Modified Gravity The ultralocal ansatz has the virtue of simplifying calculations, but it has the defect of being manifestly unphysical: the laws of nature are local, not ultralocal. In this section, we will apply the formalism developed in the last section to theories in which π ω depends on spatial derivatives of the metrich ij through a dependence on the Ricci scalarR. Since the π GR of spatially covariant general relativity belongs to this class, we call it the "realistic" class. As we will demonstrate, realistic π ω must obey stringent consistency conditions in order for thẽ H i to generate a consistent first class constraint algebra. Constraint Algebra In this section, we will compute {H i (x), H a (y)} assuming that π ω is a function of t, the phase space variablesh ij andπ ij , and the Ricci scalarR. We will then use the result to determine when the constraintsH i remain first class. As before, we decomposeH i into a tensor part J i ≡ −2h ij∇kπ jk and a scalar part In this section, we assume that π ω is a function of t,h ij ,π ij , andR. To simplify calculations, note that this is equivalent to making π ω a function of t,R, and the φ(n) defined in equation (79). It follows from this assumption that K i ≡ −ω∇ i π ω . Computing {H i (x), H a (y)}δπ ω = ∞ n=2 ∂π ω ∂φ(n) δφ(n) + ∂π ω ∂R δR .(127) Substituting this result into equation (76), using the identity δR = −R jk δh jk +∇ k∇j δh jk , and integrating by parts yields δF K = ω d 3 x ∂ i f i ∞ n=2 ∂π ω ∂φ(n) δφ(n) − ∂π ω ∂RR jk δh jk +ω d 3 x∇ j∇k ∂ i f i ∂π ω ∂R δh jk .(128) Using equation (81), it is now straightforward to compute the variational derivatives of F K , δF K δh mn = ω ∂ i f i ∞ n=2 n ∂π ω ∂φ(n) δ mn jk Π(n) jk − 1 3π mn φ(n − 1) −ω ∂ i f i ∂π ω ∂Rδ mn jkR jk + ωδ mn jk∇ j∇k ∂ i f i ∂π ω ∂R , δF K δπ mn = ω ∂ i f i ∞ n=2 n ∂π ω ∂φ(n)δ jk mn Π(n − 1) jk .(129) The corresponding results for G K are δG K δh mn = ω (∂ a g a ) ∞ m=2 m ∂π ω ∂φ(m) δ mn bc Π(m) bc − 1 3π mn φ(m − 1) −ω (∂ a g a ) ∂π ω ∂Rδ mn bcR bc + ωδ mn bc∇ b∇c (∂ a g a ) ∂π ω ∂R , δG K δπ mn = ω (∂ a g a ) ∞ m=2 m ∂π ω ∂φ(m)δ bc mn Π(m − 1) bc .(130) We are now in a position to compute the brackets involving K i . • {J i (x), K a (y)} + {K i (x), J a (y)} To compute {J i (x), K a (y)} + {K i (x), J a (y)}, we first compute {F J , G K } + {F K , G J }. We begin by substituting equations (73) and (130) into the bracket {F J , G K }. After expanding and simplifying a total derivatives of φ(n), {F J , G K } turns into {F J , G K } = −ω d 3 z f i (∂ a g a ) ∞ m=2 ∂π ω ∂φ(m)∇ i φ(m) + 2ω d 3 z (∂ a g a ) ∂π ω ∂RR k i∇ k f i −2ω d 3 z ∇ k f i ∇ i∇ k (∂ a g a ) ∂π ω ∂R + 2 3 ω d 3 z ∂ i f i ∇ c∇ c (∂ a g a ) ∂π ω ∂R −ω d 3 z ∂ i f i (∂ a g a ) 2 3R ∂π ω ∂R + ∞ m=2 mφ(m) ∂π ω ∂φ(m) .(131) To finesse this expression, integrate by parts, use the identities∇ i∇j V i =∇ j∇i V i + R ij V i and 2∇ jR j i =∇ iR , simplify a total derivative of π ω , use the identity K i = −ω∇ i π ω , and expand to obtain {F J , G K } = d 3 z f i K i ∂ a g a + 4 3 ω d 3 z∇ k ∂ i f i ∇ k (∂ a g a ) ∂π ω ∂R −ω d 3 z ∂ i f i (∂ a g a ) 2 3R ∂π ω ∂R + ∞ m=2 mφ(m) ∂π ω ∂φ(m) .(132) Similarly, {F K , G J } = − d 3 z g a K a ∂ i f i − 4 3 ω d 3 z∇ k (∂ a g a )∇ k ∂ i f i ∂π ω ∂R +ω d 3 z ∂ i f i (∂ a g a ) 2 3R ∂π ω ∂R + ∞ m=2 mφ(m) ∂π ω ∂φ(m) ,(133) so the sum of the two brackets reduces to {F J , G K } + {F K , G J } = d 3 z f i K i ∂ a g a − d 3 z g a K a ∂ i f i + 4 3 ω d 3 z∇ k ∂ i f i ∇ k (∂ a g a ) ∂π ω ∂R − 4 3 ω d 3 z∇ k (∂ a g a )∇ k ∂ i f i ∂π ω ∂R .(134) After integrating by parts, expanding, and using the identity ∂ i K a = ∂ a K i , this becomes {F J , G K } + {F K , G J } = d 3 z f i K a ∂ i g a − d 3 z g a K i ∂ a f i + d 3 z ∂ i f i (∂ k ∂ a g a ) M k − d 3 z (∂ a g a ) ∂ k ∂ i f i M k ,(135) where M k ≡ − 4 3 ω∇ k ∂π ω ∂R .(136) To extract the bracket {J i (x), K a (y)} + {K i (x), J a (y)}, integrate by parts, relabel dummy indices, and insert the identities in equation (88) to yield {F J , G K } + {F K , G J } = d 3 x d 3 y f i (x)g a (y) K a (x)∂ x i δ 3 (x − y) − K i (y)∂ y a δ 3 (x − y) + d 3 x d 3 y f i (x)g a (y)∂ x i −M k (x)∂ x k ∂ x a δ 3 (x − y) − d 3 x d 3 y f i (x)g a (y)∂ y a −M k (y)∂ y k ∂ y i δ 3 (x − y) .(137) By comparing this expression to equation (67), it is clear that {J i (x), K a (y)} + {K i (x), J a (y)} =K a (x)∂ x i δ 3 (x − y) − K i (y)∂ y a δ 3 (x − y) +∂ x i −M k (x)∂ x k ∂ x a δ 3 (x − y) −∂ y a −M k (y)∂ y k ∂ y i δ 3 (x − y) .(138)• {K i (x), K a (y)} To compute {K i (x) , K a (y)}, we first compute the bracket {F K , G K }. Substituting equations (129) and (130) into the bracket {F K , G K } yields {F K , G K } = ω 2 d 3 z (∂ a g a ) ∂π ω ∂π jk∇ j∇k ∂ i f i ∂π ω ∂R −ω 2 d 3 z ∂ i f i ∂π ω ∂π jk∇ j∇k (∂ a g a ) ∂π ω ∂R ,(139) where ∂π ω ∂π jk =δ bc jk ∞ n=2 n ∂π ω ∂φ(n) Π(n − 1) bc .(140) After integrating by parts and expanding, the bracket becomes {F K , G K } = d 3 z ∂ i f i (∂ k ∂ a g a ) N k − d 3 z (∂ a g a ) ∂ k ∂ i f i N k ,(141) where N k ≡ ω 2 ∂π ω ∂R∇ j ∂π ω ∂π jk − ω 2 ∂π ω ∂π jk∇ j ∂π ω ∂R .(142) To extract the bracket {K i (x), K a (y)}, integrate by parts, relabel dummy indices, and insert the identities in equation (88) to obtain {F K , G K } = d 3 x d 3 y f i (x)g a (y)∂ x i −N k (x)∂ x k ∂ x a δ 3 (x − y) − d 3 x d 3 y f i (x)g a (y)∂ y a −N k (y)∂ y k ∂ y i δ 3 (x − y) .(143) Comparing this expression to equation (67), it follows that {K i (x), K a (y)} =∂ x i −N k (x)∂ x k ∂ x a δ 3 (x − y) −∂ y a −N k (y)∂ y k ∂ y i δ 3 (x − y) .(144) By substituting equations (90), (138), and (144) into equation (65), and recalling thatH i = J i + K i , we obtain the identity {H i (x),H j (y)} =H j (x)∂ x i δ 3 (x − y) −H i (y)∂ y j δ 3 (x − y) +∂ x i −I k (x)∂ x k ∂ x j δ 3 (x − y) − ∂ y j −I k (y)∂ y k ∂ y i δ 3 (x − y) ,(145) where I k ≡ M k + N k , or I k = ω 2 ∂π ω ∂R∇ j ∂π ω ∂π jk − ω 2 ∂π ω ∂π jk∇ j ∂π ω ∂R − 4 3 ω∇ k ∂π ω ∂R .(146) Expanding the derivatives in this expression and using the fact thatH i ∼ 0 yields {H i (x),H j (y)} ∼ − I k (x) + I k (y) ∂ x i ∂ x j ∂ x k δ 3 (x − y) − ∂ x i I k (x) ∂ x j ∂ x k δ 3 (x − y) + ∂ y j I k (y) ∂ y i ∂ y k δ 3 (x − y) .(147) The three terms of this equation are algebraically independent, so the necessary and sufficient condition for the Poisson bracket {H i (x),H j (y)} to vanish is I k ∼ 0 .(148) In the ultralocal case the constraints were automatically first class, but to generate a first class constraint algebra in the realistic case, the scalar momentum π ω must obey the fearsome looking differential equation I k ∼ 0. As a check, we will now compute the I k arising from the π GR of spatially covariant general relativity. Recall from equation (46) that π GR = − 8 3 ω −2 φ(2) − ω −2/3R + 2Λ .(149) Since π GR is a function only of φ(2) andR, its partial derivative with respect toπ ij simplifies, ∂π GR ∂π jk = 2 ∂π GR ∂φ(2)π jk .(150) After substituting this relation into the definition of I k and recalling that J i = −2h ij∇kπ jk , I k becomes I k (π GR ) = − 4 3 ω∇ k ∂π GR ∂R − ω 2 ∂π GR ∂R ∂π GR ∂φ(2) J k + 2ω 2π jk ∂π GR ∂R∇ j ∂π GR ∂φ(2) − ∂π GR ∂φ(2)∇ j ∂π GR ∂R .(151) Upon substituting the derivatives ∂π GR ∂φ(2) = 4 3ω 2 1 π GR , ∂π GR ∂R = − 4 3ω 2/3 1 π GR ,(152) into I k (π GR ), the term in parentheses vanishes. By using the relations K i = −ω∇ i π ω and H i = J i + K i , we obtain I k (π GR ) = 16 9ω 2/3 π 2 GR H k .(153) SinceH i ∼ 0, the scalar momentum π GR satisfies I k ∼ 0. The constraintsH i of spatially covariant general relativity thus generate a first class constraint algebra. Consistency of Constraints with Equations of Motion In this section, we will compute the time derivativeḢ i for realistic π ω assuming that the constraintsH i are first class, and use the result to determine when the constraintsH i are also preserved by the equations of motion. The analysis ofḢ i proceeds exactly as in the ultralocal case until we arrive at the expressioṅ H i = −∂ i ωπ ω + ω ∂π ω ∂t −ω{J i , Π ω } −ω{K i , Π ω } ,(154) where as before Π ω ≡ d 3 x π ω .(155) The point of departure from the ultralocal case is the evaluation of the two Poisson brack- ets {J i , Π ω } and {K i , Π ω }. To compute them, we first compute the smoothing functional brackets{F J , Π ω } and {F K , Π ω }. As in the ultralocal case, we will obtain brackets involving Π ω by applying the substitution ∂ a g a → ω −1 to brackets involving G K . • {J i , Π ω } To obtain the bracket {J i , Π ω }, we first compute the bracket {F J , Π ω }. Applying ∂ a g a → ω −1 to equation (132) and integrating by parts yields {F J , Π ω } = d 3 x f i ∂ i −π ω + 2 3R ∂π ω ∂R + ∞ m=2 mφ(m) ∂π ω ∂φ(m) − ω −1∇ k M k ,(156) where as before M k = − 4 3 ω∇ k ∂π ω ∂R .(157) It follows from an application of equation (108) that {J i , Π ω } = ∂ i −π ω + 2 3R ∂π ω ∂R + ∞ m=2 mφ(m) ∂π ω ∂φ(m) − ω −1∇ k M k .(158)• {K i , Π ω } To compute the bracket {K i , Π ω }, we first compute the bracket {F K , Π ω }. After substituting ∂ a g a → ω −1 and integrating by parts, equation (141) becomes {F K , Π ω } = d 3 x f i ∂ i −ω −1 ∇ k N k ,(159) where as before N k = ω 2 ∂π ω ∂R∇ j ∂π ω ∂π jk − ω 2 ∂π ω ∂π jk∇ j ∂π ω ∂R .(160) Comparing with equation (108) yields {K i , Π ω } = ∂ i −ω −1 ∇ k N k .(161) After substituting equations (158) and (161) into the equation of motion (154) and recalling that I k = M k + N k , we obtaiṅ H i = −∂ i ω ∂π ω ∂t + 2 3ωR ∂π ω ∂R +ω ∞ m=2 mφ(m) ∂π ω ∂φ(m) −ωω −1∇ k I k .(162) Since theH i are assumed to be first class, it follows necessarily that I k ∼ 0. Demandinġ H i ∼ 0 thus implies the consistency condition ω ∂π ω ∂t + 2 3ωR ∂π ω ∂R +ω ∞ m=2 mφ(m) ∂π ω ∂φ(m) ∼ f (t) ,(163) where f (t) is an arbitrary function of time. Recall once again that the equation of motion (58) is invariant under π ω → π ω + g(t), where g(t) is an arbitrary function of time. By choosing a function g(t) such that ωg ′ (t) = f (t), the consistency condition becomes ω ∂π ω ∂t + 2 3ωR ∂π ω ∂R +ω ∞ m=2 mφ(m) ∂π ω ∂φ(m) ∼ 0 .(164) Since ω(t) is assumed to be an invertible function of time, ∂/∂t =ω ∂/∂ω. In analogy with our approach in the ultralocal case, we rewrite the consistency condition as ∆π ω ∼ 0 ,(165) where we have redefined the operator ∆ as ∆ ≡ ω ∂ ∂ω + 2 3R ∂ ∂R + ∞ m=2 mφ(m) ∂ ∂φ(m) .(166) To rule out the possibility of a π ω which satisfies ∆π ω ∼ 0 while ∆π ω = 0, we note that the constraintsH i contain a term∇ i π ω , making the constraints higher order in spatial derivatives than π ω itself. However, by examining a series expansion of π ω in the parameterR, one can verify that applying ∆ to π ω does not alter its order in spatial derivatives. 10 It follows that ∆π ω cannot depend onH i . The condition ∆π ω ∼ 0 is therefore equivalent to the apparently stronger condition ∆π ω = 0 .(167) Since ∆(ω −n φ(n)) = 0 and ∆(ω −2/3R ) = 0, we are led to define the quantities φ(n) ≡ φ(n) ω n (t) ,R ≡R ω 2/3 .(168) The most general solution to the condition ∆π ω = 0 is an arbitrary function ofR and thē φ(n). In this manner, the dependence of π ω on ω(t) is determined by its dependence on the phase space variables. As before, to understand this result, we return briefly to the phase space (h ij , π ij ). As shown in section 4.2, the only scalars that can be built out of the tensor h ij and the traceless tensorπ ij T are the φ T (n) = ω −n φ(n). If we impose the gauge-fixing constraint ω ∼ ω(t), then φ T (n) ∼φ(n); likewise, the Ricci scalar R of the metric h ij obeys R ∼R. 11 This means that R and theφ(n) have the correct conformal weight to have been derived from three-scalars on the phase space (h ij , π ij ). The scalarsR and theφ(n) are thus invariant under a rescaling 10 Spatial derivatives enter π ω solely throughR, so the derivative expansion of π ω can be written π ω = ∞ k=0 c kR k , where the coefficients c k depend on ω and the φ(n). Applying the ∆ operator to π ω changes the functional form of the c k , but does not generate higher order powers ofR. 11 See equation (259) in appendix C. ω → µω of the volume factor ω, so once again ∆π ω = 0 is revealed to be analogous to a renormalization group equation. As a check, we will now apply the renormalization group equation to the scalar momentum π GR of spatially covariant general relativity. Since π GR = − 8 3 φ (2) −R + 2Λ ,(169) the scalar momentum π GR satisfies the condition ∆π GR = 0; this implies that the constraints of the theory are preserved by the equations of motion. Combined with the result that I k (π GR ) ∼ 0, which implies that the constraints are also first class, it is now clear within the context of our formalism that the constraintsH i of spatially covariant general relativity generate a consistent first class algebra. This result justifies the assertions we made in the first paragraph of section 3.4. Summary In this section, we applied the formalism developed in section 4 to determine when scalar momenta π ω built out ofh ij ,π ij , andR yield a consistent first class constraint algebra. To ensure the first class character of the constraintsH i , it is necessary and sufficient for π ω to obey the condition I k ∼ 0 ,(170) where I k = ω 2 ∂π ω ∂R∇ j ∂π ω ∂π jk − ω 2 ∂π ω ∂π jk∇ j ∂π ω ∂R − 4 3 ω∇ k ∂π ω ∂R .(171) If ∂π ω /∂R = 0, then I k = 0, so ultralocal scalar momenta satisfy this condition trivially. The scalar momentum π GR of spatially covariant general relativity depends essentially onR, and thus satisfies this condition non-trivially. To guarantee the preservation of the constraintsH i by the equations of motion, the scalar momentum π ω must also be invariant under renormalization of the volume factor ω. This requires π ω to obey the renormalization group equation ∆π ω = 0 ,(172) where ∆ ≡ ω ∂ ∂ω + 2 3R ∂ ∂R + ∞ m=2 mφ(m) ∂ ∂φ(m) .(173) This is a generalization of the renormalization group equation (125) to include a possible dependence of π ω onR. The scalar momentum π GR satisfies this condition in addition to the first, so the constraints of spatially covariant general relativity generate a consistent first class constraint algebra. Conclusions In this paper, we developed a general formalism for verifying the consistency of spatially covariant modified theories of the transverse, traceless graviton degrees of freedom. It was a long road, so it is worth retracing our steps to see the logic of our path. In section 2, we showed how to express general relativity as a theory of a spatial metric h ij and its conjugate momentum π ij . In this language, the general covariance of the theory is represented on the phase space (h ij , π ij ) by the algebra of the four constraints H µ . In section 3, we showed how to obtain a spatially covariant version of general relativity. We began in section 3.1 by splitting the phase space (h ij , π ij ) into the phase space (ω, π ω ) of the spatial volume factor and the phase space (h ij ,π ij ) of the unit-determinant metric. In the context of cosmology on an FRW background, it is natural to represent time diffeomorphism symmetry on the phase space (ω, π ω ) and to represent spatial diffeomorphisms on the phase space (h ij ,π ij ); in section 3.2, we showed how to achieve this splitting using a cosmological gauge condition. On an expanding background, ω drops out of the dynamical phase space of the theory, and its conjugate momentum π ω becomes the scalar Hamiltonian density on the phase space (h ij ,π ij ); in section 3.3, we showed how to reduce the phase space by solving the Hamiltonian constraint in cosmological gauge. By successfully projecting the degrees of freedom of general relativity onto the reduced phase space (h ij ,π ij ), we have shown how to represent the graviton dynamics of general relativity on the class of conformally equivalent spatial metrics. To modify general relativity, we simply modified the functional form of the scalar momentum π ω while retaining the explicit spatial diffeomorphism symmetry generated by the three constraintsH i . In section 4, we considered the case in which π ω is an ultralocal function of the phase space quantitiesh ij andπ ij . In this case, the consistency of the constraintsH i imposes a single non-trivial condition on the form of π ω , namely that it must satisfy a renormalization group equation with flow parameter ω. The renormalization group equation encodes the fact that π ω must be invariant under flow through the space of conformally equivalent spatial metrics. In section 5, we applied our formalism to the more realistic case in which π ω is also allowed to depend onR, the Ricci scalar of the metric h ij . In this case, π ω must satisfy a corresponding renormalization group equation, but its form is further constrained by a differential equation that relates its dependence onR to its dependence on the phase space variablesh ij andπ ij . As a proof of principle, this paper demonstrates the possibility of consistently modifying the graviton equations of motion, but more remains to be done. In forthcoming work [9], we will apply our formalism to search for realistic alternatives to general relativity. In particular, we will examine scalar momenta π ω with a more general dependence on derivative quantities, such as the Ricci tensorR ij of the unit-determinant metrich ij , which fully determines the spatial curvature in cosmological gauge. After deriving consistency conditions in the more general case, we will attempt to solve them perturbatively to obtain valid scalar momenta π ω related to the π GR of general relativity by the deformation of a continuous parameter. If H 0 ≡ − √ h(R − 2Λ) + 1 √ h π ij π ij − 1 2 (π i i ) 2 H i ≡ −2h ij ∇ k π jk .(175) Our object in this section is to derive the constraint algebra {H 0 (x), H 0 (y)} = H i (x)∂ x i δ 3 (x − y) − H i (y)∂ y i δ 3 (x − y) {H 0 (x), H i (y)} = H 0 (y)∂ x i δ 3 (x − y) {H i (x), H j (y)} = H j (x)∂ x i δ 3 (x − y) − H i (y)∂ y j δ 3 (x − y) .(176) To evaluate these Poisson brackets, we first define the smoothing functionals F H ≡ d 3 x f 0 (x)H 0 (x) , F ≡ d 3 x f i (x)H i (x) , G H ≡ d 3 y g 0 (y)H 0 (y) , G ≡ d 3 y g a (y)H a (y) ,(177) where the functions f 0 , f i , g 0 , and g i are time-independent smoothing functions. We then compute the brackets {F H , G H } = d 3 x d 3 y f 0 (x)g 0 (y){H 0 (x), H 0 (y)} {F H , G} = d 3 x d 3 y f 0 (x)g a (y){H 0 (x), H a (y)} {F, G} = d 3 x d 3 y f i (x)g a (y){H i (x), H a (y)} .(178) As in section 4.1, we assume that the smoothing functions decay so rapidly that they eliminate all boundary terms generated by integration by parts, but that they are otherwise arbitrary. This greatly simplifies the explicit evaluation of the brackets of the smoothing functionals. By comparing the explicit forms of the brackets to the implicit forms in equation (178), we will derive explicit formulae for the brackets of the H µ 's. To simplify the calculation of the variational derivatives of F H , we will split the Hamiltonian constraint H 0 into a kinetic piece H T and a potential piece H V . Explicitly, we have H 0 = H T + H V , where H T ≡ 1 √ h h ik h jl − 1 2 h ij h kl π ij π kl , H V ≡ − √ h(R − 2Λ) .(179) Similarly, F H = F T + F V , where F T ≡ d 3 x f 0 (x)H T (x) , F V ≡ d 3 x f 0 (x)H V (x) .(180) Before taking variational derivatives, we exploit our freedom to integrate by parts to pull the covariant derivatives off the metric variation δh ij : δF V = d 3 x f 0 1 2 H V h ij + √ hR ij δh ij + d 3 x √ h h ij ∇ k ∇ k f 0 − ∇ i ∇ j f 0 δh ij .(185) It follows that δF V δh mn = f 0 1 2 H V h mn + √ hR mn + √ h h mn ∇ k ∇ k f 0 − ∇ m ∇ n f 0 δF V δπ mn = 0 .(186) Similarly, δG V δh mn = g 0 1 2 H V h mn + √ hR mn + √ h h mn ∇ k ∇ k g 0 − ∇ m ∇ n g 0 δG V δπ mn = 0 .(187) Before computing δF , we integrate by parts inside F : F = 2 d 3 x h ij π jk ∇ k f i .(188) This simplifies the variational calculation: δF = 2 d 3 x (∇ k f i )π jk δh ij + 2 d 3 x (∇ k f i )h ij δπ jk + 2 d 3 x π jk h ij δ∇ k f i .(189) To evaluate δ∇ k f i , first expand the covariant derivative as ∇ k f i = ∂ k f i + Γ i ka f a . It follows that δ∇ k f i = f a δΓ i ka . The identity δΓ l ki = 1 2 h lm (∇ i δh km + ∇ k δh im − ∇ m δh ik )(190) implies that 2π jk h ij δ∇ k f i = f i π jk ∇ i δh jk . Substituting this result into the expression for δF and integrating by parts yields δF = 2 d 3 x (∇ k f i )π jk δh ij − d 3 x ∇ i (f i π jk )δh jk + 2 d 3 x (∇ k f i )h ij δπ jk .(191) It follows that δF δh mn = 2(∇ k f i )π jk δ mn ij − ∇ i (f i π mn ) δF δπ mn = 2(∇ k f i )h ij δ jk mn .(192) Likewise, δG δh mn = 2(∇ c g a )π bc δ mn ab − ∇ a (g a π mn ) δG δπ mn = 2(∇ c g a )h ab δ bc mn . We are now in a position to compute the Poisson brackets of interest. • {H i (x), H j (y)} To calculate {H i (x) , H j (y)}, we will first calculate {F, G}. Substituting equations (192) and (193) into the Poisson bracket yields {F, G} = 2 d 3 z h ab π bc ∇ c f i (∇ i g a ) − 2 d 3 z h ij π jk (∇ k g a ) ∇ a f i − 2 d 3 z (∇ c g a ) ∇ i f i h ab π bc + 2 d 3 z ∇ k f i ∇ a g a h ij π jk .(194) After integrating by parts, applying the identity (∇ i ∇ j − ∇ j ∇ i ) u a = R a bij u b , and recalling that H i = −2h ij ∇ k π jk , this bracket becomes {F, G} = d 3 z f i H a ∇ i g a − g a H i ∇ a f i + 2 d 3 z f i g a π jk (R jika + R jaik ) .(195) It follows from the symmetry (R abcd = R cdab ) and antisymmetry (R abcd = −R bacd = −R abdc ) properties of the Riemann tensor that R jaik = −R kija . The symmetry property (π ij = π ji ) of the momentum tensor then implies that π jk (R jika + R jaik ) = 0, so {F, G} = d 3 z f i H a ∇ i g a − g a H i ∇ a f i .(196) Upon expanding the covariant derivatives, the connection terms cancel, yielding {F, G} = d 3 z f i H a ∂ i g a − g a H i ∂ a f i .(197) To extract the Poisson brackets {H i (x), H j (y)}, first relabel integration variables, {F, G} = d 3 x f i (x)H a (x)∂ x i g a (x) − d 3 y g a (y)H i (y)∂ y a f i (y) ,(198) then use the identities g a (x) = d 3 y δ 3 (x − y)g a (y) , f i (y) = d 3 x δ 3 (x − y)f i (x) ,(199) to write {F, G} = d 3 x d 3 y f i (x)g a (y) H a (x)∂ x i δ 3 (x − y) − H i (y)∂ y a δ 3 (x − y) .(200) By comparing this expression to (178), we obtain the identity {H i (x), H j (y)} = H j (x)∂ x i δ 3 (x − y) − H i (y)∂ y j δ 3 (x − y) .(201)• {H 0 (x), H i (y)} To calculate {H 0 (x), H i (y)}, we will first calculate {F H , G} = {F T , G} + {F V , G}. Substituting equations (182) and (193) into the bracket {F T , G} yields {F T , G} = d 3 z f 0 ∇ a (g a H T ) .(202) Assembling equations (186) and (193) into the bracket {F V , G} yields {F V , G} = d 3 z f 0 (∇ a g a ) H V + 2 d 3 z √ hf 0 (∇ c g a )R c a + 2 d 3 z √ h (∇ a g a ) ∇ c ∇ c f 0 − 2 d 3 z √ h (∇ c g a ) ∇ a ∇ c f 0 .(203) After integrating the last two terms by parts, the identity (∇ a ∇ c − ∇ c ∇ a )g a = R ac g a implies that {F V , G} = d 3 z f 0 (∇ a g a ) H V + 2 d 3 z √ hR c a ∇ c f 0 g a .(204) By integrating the last term by parts, using the identity 2∇ c R c a = ∇ a R = ∇ a (R−2Λ), and recalling that H V = √ h(2Λ − R), the bracket becomes {F V , G} = d 3 z f 0 ∇ a (g a H V ) .(205) Combining {F V , G} with {F T , G} and recalling that H 0 = H T + H V yields {F H , G} = d 3 z f 0 ∇ a (g a H 0 ) .(206) Since g a is a three-vector and H 0 / √ h is a three-scalar, ∇ a (g a H 0 ) = ∂ a (g a H 0 ) ,(207) from which it follows that {F H , G} = d 3 z f 0 ∂ a (g a H 0 ) .(208) To extract the bracket {H 0 (x), H i (y)}, first relabel the variable of integration, {F H , G} = d 3 x f 0 (x)∂ x a (g a (x)H 0 (x)) ,(209) then use the identity g a (x)H 0 (x) = d 3 y δ 3 (x − y)g a (y)H 0 (y) to write {F T , G} = d 3 x d 3 y f 0 (x)g a (y)H 0 (y)∂ x a δ 3 (x − y) .(211) By comparing this expression to (178), we obtain the identity {H 0 (x), H i (y)} = H 0 (y)∂ x i δ 3 (x − y) .(212) • {H 0 (x), H 0 (y)} To calculate {H 0 (x), H 0 (y)}, we will first calculate {F H , G H } = {F T , G T } + {F T , G V } + {F V , G T } + {F V , G V } .(213) It is straightforward to verify that the brackets {F T , G T } and {F V , G V } vanish identically. To compute {F T , G V }, substitute equations (182) and (187) into the Poisson bracket to obtain {F T , G V } = 2 d 3 z f 0 π mn ∇ m ∇ n g 0 − d 3 z f 0 g 0 1 √ h 1 2 H V π k k + 2 √ hR mn π mn .(214) Likewise, {F V , G T } = −2 d 3 z g 0 π mn ∇ m ∇ n f 0 + d 3 z f 0 g 0 1 √ h 1 2 H V π k k + 2 √ hR mn π mn .(215) The sum of the four brackets reduces to {F H , G H } = 2 d 3 z f 0 π mn ∇ m ∇ n g 0 − g 0 π mn ∇ m ∇ n f 0 . After integrating by parts and recalling that H i = −2h ij ∇ k π jk , the bracket becomes {F H , G H } = d 3 z f 0 H i ∇ i g 0 − g 0 H i ∇ i f 0 .(217) Upon expanding the covariant derivatives in terms of partial derivatives and connection terms, the connection terms cancel to yield {F H , G H } = d 3 z f 0 H i ∂ i g 0 − g 0 H i ∂ i f 0 .(218) To extract the bracket {H 0 (x), H 0 (y)}, relabel integration variables and use the identities g 0 (x) = d 3 y δ 3 (x − y)g 0 (y) , f 0 (y) = d 3 x δ 3 (x − y)f 0 (x)(219) Appendix B: Constraint brackets after imposing χ We begin with the four constraints H µ . After introducing the gauge-fixing constraint χ ≡ √ h − ω(t) ,(222) we need to compute the brackets of each of the five constraints (including χ) with χ. We introduce the smoothing functionals F χ ≡ d 3 x f χ (x)χ(x) , G χ ≡ d 3 y g χ (y)χ(y) ,(223) where f χ and g χ are arbitrary rapidly-decaying smoothing functions. We then compute the brackets {F χ , G χ } = d 3 x d 3 y f χ (x)g χ (y){χ(x), χ(y)} {F H , G χ } = d 3 x d 3 y f 0 (x)g χ (y){H 0 (x), χ(y)} {F, G χ } = d 3 x d 3 y f i (x)g χ (y){H i (x), χ(y)} .(224) The variation δF χ is δF χ = d 3 x f χ 1 2 √ hh ij δh ij ,(225) so δF χ δh mn = f χ 1 2 √ hh mn , δF χ δπ mn = 0 . Likewise, δG χ δh mn = g χ 1 2 √ hh mn , δG χ δπ mn = 0 .(227) It follows at once that {F χ , G χ } = 0 .(228) Comparing with (224), we obtain the identity {χ(x), χ(y)} = 0 . We now turn to the brackets of χ with the H µ . • {H 0 (x), χ(y)} We split {F H , G χ } into {F H , G χ } = {F T , G χ } + {F V , G χ }. Assembling equations (182) and (226) into the Poisson bracket {F T , G χ } yields {F T , G χ } = d 3 z f 0 g χ 1 2 π k k .(230) The bracket {F V , G χ } vanishes identically, so {F H , G χ } = d 3 x f 0 g χ 1 2 π k k .(231) To extract the bracket {H 0 (x), χ(y)}, use the identity g χ (x) = d 3 y g χ (y)δ 3 (x − y) , which yields {F H , G χ } = d 3 x d 3 y f 0 (x)g χ (y) 1 2 π k k (x)δ 3 (x − y).(233) Comparing to (224), we obtain the identity {H 0 (x), χ(y)} = 1 2 π k k (x)δ 3 (x − y).(234) • {H i (x), χ(y)} From equation (192), it follows that {F, G χ } = − d 3 z g χ √ h∇ i f i .(235) Integrating by parts and using the fact that g χ is a scalar, this bracket becomes {F, G χ } = d 3 x f i √ h∂ i g χ .(236) To extract the bracket {H i (x), χ(y)}, use the identity g χ (x) = d 3 y g χ (y)δ 3 (x − y) to write {F, G χ } = d 3 x d 3 y f i (x)g χ (y) h(x)∂ x i δ 3 (x − y) .(238) Comparing to (224), we obtain the identity {H i (x), χ(y)} = h(x)∂ x i δ 3 (x − y) .(239) and the metricg µν ≡ |g| −1/d g µν (241) so that g µν = Ωg µν . By construction, the signature ofg µν is the same as that of g µν . Denote the determinant of g µν byg. From the definition ofg µν , it follows thatg = g/|g|, sog = ±1, depending on the signature of g µν . We therefore callg µν a unit-determinant metric. The inverse metrics are related by g µν =g µν Ω −1 . We denote the covariant derivative with respect to g µν by ∇ µ , and the covariant derivative with respect tog µν by∇ µ . The connection Γ λ µν defined by g µν is Γ λ µν = 1 2 g λσ (∂ µ g νσ + ∂ ν g µσ − ∂ σ g µν ) ,(243) while the connectionΓ λ µν defined byg µν is Γ λ µν = 1 2g λσ (∂ µgνσ + ∂ νgµσ − ∂ σgµν ) .(244) The connectionΓ λ µν obeysΓ λ µν = Γ λ µν − C λ µν , where C λ µν = δ λσ µν − 2g λσg µν ∂ σ log Ω . For convenience, we can write Ω in terms of a scalar field ϕ and a constant Ω 0 as Ω ≡ Ω 0 e 2ϕ ,(246) in which case C λ µν = δ λ µ∇ ν ϕ + δ λ ν∇ µ ϕ −g µν∇ λ ϕ . The Riemann tensor of g µν is R λ κµν = ∂ µ Γ λ κν − ∂ ν Γ λ κµ + Γ λ µσ Γ σ κν − Γ λ νσ Γ σ κµ ,(248) while the Riemann tensor ofg µν is Using Γ λ µν =Γ λ µν + C λ µν , the Riemann tensor R λ κµν can be rewritten as R λ κµν =R λ κµν + C λ µσ C σ κν − C λ νσ C σ κµ + ∂ µ C λ κν +Γ λ µσ C σ κν −Γ σ µκ C λ σν − ∂ ν C λ κµ −Γ λ νσ C σ κµ +Γ σ νκ C λ µσ . Using∇ µ C λ κν −∇ ν C λ κµ = ∂ µ C λ κν +Γ λ µσ C σ κν −Γ σ µκ C λ σν − ∂ ν C λ κµ −Γ λ νσ C σ κµ +Γ σ νκ C λ σµ ,(251)R λ κµν becomes R λ κµν =R λ κµν + C λ µσ C σ κν − C λ νσ C σ κµ +∇ µ C λ κν −∇ ν C λ κµ .(252) The Ricci tensor of g µν is R µν = R λ µλν ; the Ricci tensor ofg µν isR µν =R λ µλν . Tracing equation (252) appropriately yields R µν =R µν + C λ λσ C σ µν − C λ νσ C σ µλ +∇ λ C λ µν −∇ ν C λ µλ .(253) We now express R µν in terms ofR µν and derivatives of ϕ. Recalling that δ µ µ = d, we find C λ λσ = d∇ σ ϕ C λ µσ C σ νλ = (d + 2)(∇ µ ϕ)(∇ ν ϕ) − 2g µν (∇ α ϕ)(∇ α ϕ) , so R µν =R µν + (d − 2)(∇ µ ϕ)(∇ ν ϕ) − (d − 2)g µν (∇ σ ϕ)(∇ σ ϕ) − (d − 2)∇ µ∇ν ϕ −g µν∇σ∇ σ ϕ .(255) The Ricci scalar for g µν is R = g µν R µν ; the Ricci scalar forg µν isR =g µνR µν . In terms of covariant derivatives of ϕ, we have ΩR =R − (d − 1)(d − 2)(∇ α ϕ)(∇ α ϕ) − 2(d − 1)∇ σ∇ σ ϕ .(256) In three dimensions, the Weyl tensor vanishes, so the Riemann tensor is completely determined by the Ricci tensor and the metric via R lkmn = 1 d − 2 (g lm R kn − g ln R km − g km R ln + g kn R lm ) − 1 (d − 1)(d − 2) (g lm g kn − g ln g km )R . In this case, it suffices to compute the Ricci tensor. When d = 3, our previous formulas reduce to R ij =R ij + (∇ i ϕ)(∇ j ϕ) −g ij (∇ k ϕ)(∇ k ϕ) −∇ i∇j ϕ −h ij∇k∇ k ϕ , ΩR =R − 2(∇ k ϕ)(∇ k ϕ) − 4∇ k∇ k ϕ .(258) The condition ω ∼ ω(t) amounts to ϕ ∼ ϕ(t), so in cosmological gauge we have R ij ∼R ij , ΩR ∼R .(259) back that we have no handle on what physics might set the magnitude of the cosmological constant. It is currently an outstanding theoretical challenge to determine what physical degrees of freedom are associated with late-time acceleration. Dynamical theories of dark energy postulate scalar degrees of freedom similar to those invoked to account for inflation. Since φ( 0 ) 0= 3 and φ(1) = 0, δφ(0) = δφ(1) = 0. For n ≥ 2, the variational derivatives of the φ(n) are δφ(n)(x) δh mn (y) = δ mn ab nΠ(n) ab − 1 3π mn nφ(n − 1) δ 3 (x − y) , δφ(n)(x) δπ mn (y) =δ ab mn nΠ(n − 1) ab δ 3 (x − y) . is then a matter of computing the four brackets in equation (65). The result for {J i (x), J a (y)} carries over unchanged from equation (90), but we will have to revisit the brackets involving K i . To do so, we will first evaluate the smoothing functional brackets {F J , G K } + {F K , G J } and {F K , G K }. By comparing the ensuing explicit expressions to the formal expressions in equation (67), we will derive explicit expressions for the Poisson brackets involving K i . Our analysis of the variational derivatives of the smoothing functional F K defined in equation (66) proceeds exactly as in the ultralocal case up to equation (76), where the quantity δπ ω arises. we discover non-trivial modifications of general relativity that contain only two degrees of freedom, it could open up new lines of theoretical and experimental research. A null result, on the other hand, would serve as further evidence of the uniqueness of general relativity. It will be interesting to see just how far we can push this program. to write{F H , G H } = d 3 x d 3 y f 0 (x)g 0 (y) H i (x)∂ x i δ 3 (x − y) − H i (y)∂ y i δ 3 (x − y) . (220)By comparing this expression to (178), we obtain the identity{H 0 (x), H 0 (y)} = H i (x)∂ x i δ 3 (x − y) − H i (y)∂ y i δ 3 (x − y) . Appendix A: Covariant Constraint Algebra of GRRecall the Poisson bracket of GR{A, B} ≡ d 3 z δA δh mn (z)δB δπ mn (z) − δA δπ mn (z) δB δh mn (z) (174) and the constraints A spatially-flat FRW spacetime corresponds to N = 1, N i = 0, and h ij = a 2 (t)δ ij . Computing the variation δF T is straightforward:It follows thatLikewise,Keeping in mind that δR = −δh ij R ij + ∇ j ∇ i δh ij − ∇ k ∇ k h ij δh ij , computing δF V is just as straightforward:Appendix C: Conformal DecompositionConsider a metric g µν in a number of dimensions d. Denote the determinant of g µν by g.Define the positive conformal factor Ω ≡ |g| 1/d > 0 (240) . 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{'abstract': 'General relativity is a covariant theory of two transverse, traceless graviton degrees of freedom. According to a theorem of Hojman, Kuchař, and Teitelboim, modifications of general relativity must either introduce new degrees of freedom or violate the principle of general covariance. In this paper, we explore modifications of general relativity that retain the same number of gravitational degrees of freedom, and therefore explicitly break general covariance. Motivated by cosmology, the modifications of interest maintain spatial covariance. Demanding consistency of the theory forces the physical Hamiltonian density to obey an analogue of the renormalization group equation. In this context, the equation encodes the invariance of the theory under flow through the space of conformally equivalent spatial metrics. This paper is dedicated to setting up the formalism of our approach and applying it to a realistic class of theories.Forthcoming work will apply the formalism more generally.', 'arxivid': '1108.1397', 'author': ['Justin Khoury \nCenter for Particle Cosmology\nDepartment of Physics & Astronomy\nUniversity of Pennsylvania\n209 South 33rd Street19104PhiladelphiaPA\n', 'Godfrey E J Miller \nCenter for Particle Cosmology\nDepartment of Physics & Astronomy\nUniversity of Pennsylvania\n209 South 33rd Street19104PhiladelphiaPA\n', 'Andrew J Tolley \nDepartment of Physics\nCase Western Reserve University\n10900 Euclid Ave44106ClevelandOH\n', 'Justin Khoury \nCenter for Particle Cosmology\nDepartment of Physics & Astronomy\nUniversity of Pennsylvania\n209 South 33rd Street19104PhiladelphiaPA\n', 'Godfrey E J Miller \nCenter for Particle Cosmology\nDepartment of Physics & Astronomy\nUniversity of Pennsylvania\n209 South 33rd Street19104PhiladelphiaPA\n', 'Andrew J Tolley \nDepartment of Physics\nCase Western Reserve University\n10900 Euclid Ave44106ClevelandOH\n'], 'authoraffiliation': ['Center for Particle Cosmology\nDepartment of Physics & Astronomy\nUniversity of Pennsylvania\n209 South 33rd Street19104PhiladelphiaPA', 'Center for Particle Cosmology\nDepartment of Physics & Astronomy\nUniversity of Pennsylvania\n209 South 33rd Street19104PhiladelphiaPA', 'Department of Physics\nCase Western Reserve University\n10900 Euclid Ave44106ClevelandOH', 'Center for Particle Cosmology\nDepartment of Physics & Astronomy\nUniversity of Pennsylvania\n209 South 33rd Street19104PhiladelphiaPA', 'Center for Particle Cosmology\nDepartment of Physics & Astronomy\nUniversity of Pennsylvania\n209 South 33rd Street19104PhiladelphiaPA', 'Department of Physics\nCase Western Reserve University\n10900 Euclid Ave44106ClevelandOH'], 'corpusid': 12259964, 'doi': '10.1103/physrevd.85.084002', 'github_urls': [], 'n_tokens_mistral': 36741, 'n_tokens_neox': 31291, 'n_words': 19281, 'pdfsha': '8bc0a6bce39b429ef4bde6b217e9db803a77f7ea', 'pdfurls': ['https://arxiv.org/pdf/1108.1397v1.pdf'], 'title': ['Spatially Covariant Theories of a Transverse, Traceless Graviton, Part I: Formalism', 'Spatially Covariant Theories of a Transverse, Traceless Graviton, Part I: Formalism', 'Spatially Covariant Theories of a Transverse, Traceless Graviton, Part I: Formalism', 'Spatially Covariant Theories of a Transverse, Traceless Graviton, Part I: Formalism'], 'venue': []}

arxiv

Learning Algorithm Generalization Error Bounds via Auxiliary Distributions 2 Oct 2022 Member, IEEE, SaeedAminian ⋆ Gholamali Member, IEEEMasiha ⋆ Senior Member, IEEELaura Toni Senior Member, IEEEMiguel R D Rodrigues Learning Algorithm Generalization Error Bounds via Auxiliary Distributions 2 Oct 20221Index Terms Generalization Error BoundsMutual InformationGeneralized α-Jensen-Shannon Informationα-Rényi Information Generalization error boundaries are essential for comprehending how well machine learning models work. In this work, we suggest a creative method, i.e., the Auxiliary Distribution Method, that derives new upper bounds on generalization errors that are appropriate for supervised learning scenarios. We show that our general upper bounds can be specialized under some conditions to new bounds involving the generalized α-Jensen-Shannon, α-Rényi (0 < α < 1) information between random variable modeling the set of training samples and another random variable modeling the set of hypotheses. Our upper bounds based on generalized α-Jensen-Shannon information are also finite. Additionally, we demonstrate how our auxiliary distribution method can be used to derive the upper bounds on generalization error under the distribution mismatch scenario in supervised learning algorithms, where the distributional mismatch is modeled as α-Jensen-Shannon or α-Rényi (0 < α < 1) between the distribution of test and training data samples. We also outline the circumstances in which our proposed upper bounds might be tighter than other earlier upper bounds. I. INTRODUCTION N UMEROUS methods have been proposed in order to describe the generalization error of learning algorithms. These include VC-based bounds [2], algorithmic stability-based bounds [3], algorithmic robustness-based bounds [4], PAC-Bayesian bounds [5]. Nevertheless, for a number of reasons, many of these generalisation error bounds are unable to describe how different machine-learning techniques can generalise: some of the bounds depend only on the hypothesis class and not on the learning algorithm; existing bounds do not easily exploit dependencies between different hypotheses; or do not exploit dependencies between the learning algorithm input and output. More recently, methods that make use of information-theoretic tools have also been developed to describe the generalisation of learning techniques. Such methods frequently incorporate the many components related to the learning problem by expressing the generalisation error in terms of certain information measurements between the learning algorithm input (the training dataset) and output (the hypothesis). In particular, building upon pioneering work by Russo and Zou [6], Xu and Raginsky [7] have derived generalization error bounds involving the mutual information between the training set and the hypothesis. Bu et al. [8] have derived tighter generalization error bounds involving the mutual information between each individual sample in the training set and the hypothesis. Meanwhile, bounds using chaining mutual information have been proposed in [9], [10]. Other authors have also constructed information-theoretic based generalization error bounds based on other information measures such as α-Rényi divergence for α > 1, f -divergence, and maximal leakage [11]. In [12], an upper bound based on α-Rényi divergence for 0 < α < 1 is derived by using the variational representation of α-Rényi divergence. Bounds based on the Wasserstein distance between the training sample data and the output of a randomized learning algorithm [13], [14] and Wasserstein distance between distributions of an individual sample data and the output of the learning algorithm are proposed in [15], and tighter upper bounds via convexity of Wasserstein distance are proposed in [16]. Upper bounds based on conditional mutual information and individual sample conditional mutual information are proposed in [17] and [18], respectively. The combination of conditioning and processing techniques can provide tighter generalization error upper bounds [19], [20]. An exact characterization of the generalization error for the Gibbs algorithm in terms of symmetrized KL information is provided in [21], and is extended for transfer learning scenario in [22]. Generalization error bounds have also been developed to address scenarios where the training data distribution differs from the test data distribution, known as Distribution Mismatch. This scenario -which also links to out-of-distribution generalizationhas attract various contributions in recent years such as [23]- [25]. In particular, Masiha et al. [26] provides information-theoretic generalization error upper bounds in the presence of training/test data distribution mismatch, using rate-distortion theory. The Covariate-shift as a distribution mismatch scenario in semi-supervised learning is studied in [27]. In this work, we propose an auxiliary distribution method (ADM) to characterize the generalization error upper bound of supervised learning algorithms in terms of novel information measures. Our new bounds offer two advantages over existing ones: (1) Some of our bounds -such as the generalized α-Jensen-Shannon information ones -are always finite, whereas conventional mutual information ones (e.g., [7]) may not be; (2) In contrast to mutual information-based bounds, our bounds-such as the α-Rényi information for 0 < α < 1-are finite for some deterministic supervised learning algorithms; (3) We also apply ADM to provide an upper bound on generalization error of supervised learning algorithms under distribution mismatch between test and training data distributions. The distribution mismatch between test and training dataset is measured by some novel divergences between the distributions of test and training data samples. In summary, our main contributions are as follows: 1) We suggest a novel method, i.e., ADM, that uses auxiliary distributions over the spaces of the hypothesis and data sample to obtain upper bounds on the generalization error. 2) Using this approach that one can derive new generalization error bounds expressed via generalized α-Jensen-Shannon divergence which is known to be finite. 3) Using ADM, we offer an upper bound based on α-Rényi divergence for 0 < α < 1 with the same decay rate as the mutual information-based upper bound. Furthermore, in contrast to the mutual information-based bounds, the α-Rényi divergence bounds for 0 < α < 1 are finite when the hypothesis (output of the learning algorithm) is a deterministic function of at least one data sample. 4) Using ADM, we also provide generalization error upper bound under training and test data distribution mismatch. It turns out that training and test distribtion mismatch is captured in our upper bounds via generalized α-Jensen-Shannon or α-Rényi divergences. It is noteworthy to add that, although the generalized α-Jensen-Shannon measure does not appear to have been used to characterize the generalization ability of learning algorithms, these information-theoretic quantities as well as α-Rényi measure for 0 < α < 1, have been employed to study some machine learning problems, including the use of • generalized α-Jensen-Shannon as a loss function under label noise scenario [28], and Jensen-Shannon divergence (generalized α-Jensen-Shannon divergence for α = 1/2) in adversarial learning [29] and active learning [30]. • α-Rényi divergence in feature extraction [31] and image segmentation based on clustering [32]. II. PROBLEM FORMULATION A. Framework of Statistical Learning We analyse a standard supervised learning setting where we wish to learn a hypothesis given a set of input-output examples that can then be used to predict a new output given a new input. In particular, in order to formalize this setting, we model the input data (also known as features) using a random variable X ∈ X where X is the input space, and we model the output data (also known as predictors or labels) using a random variable Y ∈ Y where Y is the output space. We also model input-output data pairs using a random variable Z = (X, Y ) ∈ Z = X × Y where Z is drawn from Z per some unknown distribution µ. We also let S = {Z i = (X i , Y i ) n i=1 } be a training set consisting of n input-output data points drawn i.i.d. from Z according to µ. We denote hypotheses by a random variable W ∈ W (W : X → Y) where W is a hypothesis class. Finally, we represent a learning algorithm via a Markov kernel that maps a given training set S onto hypothesis W defined on the hypothesis class W according to the probability law P W |S . We introduce a (non-negative) loss function ℓ : W × Z → R + that measures how well a hypothesis predicts an output given an input. We can now define the population risk and the empirical risk associated with a given hypothesis as follows: L P (w, µ) Z ℓ(w, z)dµ(z),(1)L E (w, s) 1 n n i=1 ℓ(w, z i ),(2) respectively. We can also define the (expected) generalization error as follows: gen(P W |S , µ) E PW,S [gen(W, S, µ)],(3) where gen(w, s, µ) L P (w, µ) − L E (w, s). This (expected) generalization error quantifies by how much the population risk deviates from the empirical risk. This quantity cannot be computed directly because µ is unknown, but it can often be (upper) bounded, thereby providing a means to gauge various learning algorithms' performance. We also analyse a supervised learning scenario under distribution mismatch, where training and test data are drawn from different distributions (µ and µ ′ , respectively). In particular, we define the population risk based on test distribution µ ′ as follows: L P (w, µ ′ ) Z ℓ(w, z)dµ ′ (z).(4) We define the mismatched(expected) generalization error as follows: gen(P W |S , µ, µ ′ ) E PW,S [gen(W, S, µ, µ ′ )],(5) where gen(w, s, µ, µ ′ ) L P (w, µ ′ ) − L E (w, s). Our goal in the sequel will be to derive (upper) bounds on the generalization errors (3) and the mismatched(expected) generalization error (5) expressed via various information-theoretic measures. B. Auxiliary Distribution Method We now describe our main method to derive the generalization error upper bounds, i.e., the ADM. Consider P and Q as two distributions defined on a measurable space X and let f : X → R be a measurable function. Assume that we can use an asymmetric information measure T (P Q) between P and Q to construct the following upper bound: |E P [f (X)] − E Q [f (X)]| ≤ F (T (P Q)),(6) where F (·) is a given non-decreasing concave function. Consider R as an auxiliary distribution on the same space X . We can use following upper bound instead of (6): |E P [f (X)] − E Q [f (X)]| ≤ |E P [f (X)] − E R [f (X)]| + |E Q [f (X)] − E R [f (X)]| ≤ F (T (P R)) + F (T (Q R))(7) From concavity of F , we have F (T (P R)) + F (T (Q R)) ≤ 2F T (P R)/2 + T (Q R)/2(8) We assume that T satisfies a reverse triangle inequality as follows: min R T (P R) + T (Q R) ≤ T (P Q).(9) Considering R * ∈ arg min R T (P R) + T (Q R), we have |E P [f (X)] − E Q [f (X)]| ≤ 2F T (P R * )/2 + T (Q R * )/2 .(10) We can also provide another upper bound based on T (R P ) and T (R Q) instead of T (P R) and T (Q R): |E P [f (X)] − E Q [f (X)]| ≤ |E R [f (X)] − E P [f (X)]| + |E R [f (X)] − E Q [f (X)]| ≤ F (T (R P )) + F (T (R Q)).(11) ConsideringR ∈ arg min R T (R P ) + T (R Q), we have |E P [f (X)] − E Q [f (X)]| ≤ 2F T (R P )/2 + T (R Q)/2 .(12) Via this ADM approach -taking T (· ·) to be a KL divergence -we can derive generalization error bounds involving KL divergences as follows: αD KL (P W,Zi P W,Zi ) + (1 − α)D KL (P W ⊗ µ P W,Zi ),(13)αD KL ( P W,Zi P W,Zi ) + (1 − α)D KL ( P W,Zi P W ⊗ µ),(14) where P W,Zi , P W,Zi and P W ⊗ µ are an auxiliary joint distribution over the space Z × W, the true joint distribution of the random variables W and Z i and the product of marginal distributions of random variables W and Z i , respectively. Inspired by the ADM, we use the fact that KL divergence is asymmetric and satisfies the reverse triangle inequality [33]. Hence, we can choose the auxiliary joint distribution, P W,Zi , to derive new upper bounds which are finite or tighter under some conditions. C. Information Measures In our characterization of generalization error upper bounds, we will use the information measures between two distributions P X and P X ′ on a common measurable space X , summarized in Table I. The last two divergences are Generalized α-Jensen-Shannon divergence 1 , α-Rényi divergence, which can be characterized by Equations (13), (14) respectively (See their characterizations as a convex combination of KL-divergences in Lemmas 2 and 3). They are the main divergences discussed in this paper and defined in Table I. KL divergence, Symmetrized KL divergence, Bhattacharyya distance, and Jensen-Shannon divergence can be obtained as special cases of the first three divergences in Table I. D KL (P X P X ′ ) X P X (x) log P X (x) P X ′ (x) dx Symmetrized KL divergence [36] D SKL (P X P X ′ ) D KL (P X P X ′ ) + D KL (P X ′ P X ) Jensen-Shannon divergence [37] D JS (P X P X ′ ) 1 2 D KL P X P X +P X ′ 2 + 1 2 D KL P X ′ P X +P X ′ 2 Bhattacharyya distance [38] D B (P X P X ′ ) − log X P X (x)P X ′ (x)dx Generalized α-Jensen-Shannon divergence [37], [39] D α JS (P X P X ′ ) αD KL (P X αP X + (1 − α)P X ′ ) + (1 − α)D KL (P X ′ αP X + (1 − α)P X ′ ) α-Rényi divergence for α ∈ [0, ∞) [40] Dα(P X P X ′ ) 1 α−1 log X P α X (x)P 1−α X ′ (x)dx In addition, in our generalization error characterizations, we will also use various information measures between two random variables X and X ′ with joint distribution P XX ′ and marginals P X and P X ′ . These information measures are summarized in Table II. Note that all these information measures are zero if and only if the random variables X and X ′ are independent. Mutual information I(X; X ′ ) D KL (P X,X ′ P X ⊗ P X ′ ) Lautum information [41] L(X; X ′ ) D KL (P X ⊗ P X ′ P X,X ′ ) Symmetrized KL Information [42] I SKL (X; X ′ ) I(X; X ′ ) + L(X; X ′ ) Generalized α-Jensen-Shannon information (0 < α < 1) I α JS (X; X ′ ) D α JS (P X,X ′ P X ⊗ P X ′ ) Jensen-Shannon information [43] I JS (X; X ′ ) D JS (P X,X ′ P X ⊗ P X ′ ) α-Rényi information I α R (X; X ′ ) Dα(P X,X ′ P X ⊗ P X ′ ) Sibson's α-Mutual information [44] I α S (X; X ′ ) min Q X ′ Dα(P X,X ′ P X ⊗ Q X ′ ) D. Notations In this work, we adopt the following notation in the sequel. Calligraphic letters denote spaces (e.g. Z), Upper-case letters denote random variables (e.g., Z), and lower-case letters denote a realization of random variable (e.g. z). We denote the distribution of the random variable Z by P Z , the joint distribution of two random variables (Z 1 , Z 2 ) by P Z1,Z2 , and the αconvex combination of the joint distribution P Z1,Z2 and the product of two marginals P Z1 ⊗P Z2 , i.e. αP Z1 ⊗P Z2 +(1−α)P Z1,Z2 for α ∈ (0, 1), by P (α) Z1,Z2 . We denote the derivative of a real-valued function f (x) with respect to its argument x by f ′ (·). We also adopt the notion log(·) for the natural logarithm. E. Definitions We offer some standard definitions that will guide our analysis in the sequel. Definition 1: The cumulant generating function (CGF) of a random variable X is defined as Λ X (λ) log E[e λ(X−EX) ].(15) Assuming Λ X (λ) exists, it can be verified that Λ X (0) = Λ ′ X (0) = 0, and that it is convex. Definition 2: For a convex function ψ defined on the interval [0, b), where 0 < b ≤ ∞, its Legendre dual ψ ⋆ is defined as ψ ⋆ (x) sup λ∈[0,b) λx − ψ(λ) .(16) The following lemma characterizes a useful property of the Legendre dual and its inverse function. 1 a.k.a. capacitory discrimination [34] for α = 1/2 Lemma 1: [45,Lemma 2.4] Assume that ψ(0) = ψ ′ (0) = 0. Then, the Legendre dual ψ ⋆ (x) of ψ(x) defined above is a non-negative convex and non-decreasing function on [0, ∞) with ψ ⋆ (0) = 0. Moreover, its inverse function ψ ⋆−1 (y) = inf{x ≥ 0 : ψ ⋆ (x) ≥ y} is concave, and can be written as ψ ⋆−1 (y) = inf λ∈[0,b) y + ψ(λ) λ , b > 0.(17) Importantly, using these results, we can characterize the tail behaviour of Sub-Gaussian random variables. A random variable X is σ-sub-Gaussian, if ψ(λ) = σ 2 λ 2 2 is an upper bound on Λ X (λ), for λ ∈ R. Then by Lemma 1, ψ ⋆−1 (y) = 2σ 2 y.(18) The tail behaviour of sub-Exponential and sub-Gamma random variables are introduced in Appendix A. III. AUXILIARY DISTRIBUTION BASED GENERALIZATION ERROR UPPER BOUNDS We now offer a series of bounds on the expected generalization error of supervised learning algorithms based on different information measures using the ADM coupled with KL divergence. We also suggest upper bounds based on Triangular Discrimination and α-KL-Pythagorean information measures in Appendix F. A. Generalized α-Jensen-Shannon based Upper Bound In the following Theorem, we provide a new generalization error upper bound based on KL divergence by applying ADM, and using KL divergences terms, D KL (P W ⊗ µ P W,Zi ) and D KL (P W,Zi P W,Zi ). Theorem 3: (Proved in Appendix B-A) Assume that under an auxiliary joint distribution P W,Zi -Λ ℓ(W,Zi) (λ) exists, it is upper bounded by ψ + (λ) for λ ∈ [0, b + ), 0 < b + < +∞, and it is also upper bounded by ψ − (−λ) for λ ∈ (b − , 0], ∀i = 1, · · · , n. Also assume that ψ + (λ) and ψ − (λ) are convex functions and ψ − (0) = ψ + (0) = ψ ′ + (0) = ψ ′ − (0) = 0. Then, it holds that: gen(P W |S , µ) ≤ 1 n n i=1 ψ ⋆−1 + (A i ) + ψ ⋆−1 − (B i ) ,(19)−gen(P W |S , µ) ≤ 1 n n i=1 ψ ⋆−1 − (A i ) + ψ ⋆−1 + (B i ) ,(20) where A i = D KL (P W ⊗µ P W,Zi ), B i = D KL (P W,Zi P W,Zi ), ψ ⋆−1 − (x) = inf λ∈[0,−b−) x+ψ−(λ) λ and ψ ⋆−1 + (x) = inf λ∈[0,b+) x+ψ+(λ) λ . Remark 4: Note that Theorem 3 applies to sub-Gaussian (18), sub-Exponential and sub-Gamma assumptions on loss function CGF, introduced in Appendix A. Theorem 3 can be used not only to recover existing generalization error bounds, but also to offer new ones. For example, we can immediately recover the mutual information bound [7] from the following results. Example 5: Choose P W,Zi = P W ⊗ µ for i = 1, · · · , n. It follows immediately from Theorem 3 that: gen(P W |S , µ) ≤ 1 n n i=1 ψ ⋆−1 − (I(W ; Z i )),(21)−gen(P W |S , µ) ≤ 1 n n i=1 ψ ⋆−1 + (I(W ; Z i )).(22) Example 6: Choose P W,Zi = P W,Zi for i = 1, · · · , n. It also follows immediately from Theorem 3 that: gen(P W |S , µ) ≤ 1 n n i=1 ψ ⋆−1 + (L(W ; Z i )),(23)−gen(P W |S , µ) ≤ 1 n n i=1 ψ ⋆−1 − (L(W ; Z i )).(24) The result in Example 5 is the same as a result appearing in [8] whereas the result in Example 6 extends the result appearing in [46]. By repeatedly using ADM, the conclusion in Theorem 3 can be extended to many auxiliary distributions. In this study, we take into account just one auxiliary distribution and use ADM just once. Building upon Theorem 3, we are also able to provide a generalisation error upper bound based on a convex combination of KL terms, i.e., αD KL (P W ⊗ µ P W,Zi ) + (1 − α)D KL (P W,Zi P W,Zi ), that relies on a certain σ-sub-Gaussian tail assumption. Theorem 7: (Proved in Appendix B-B) Assume that the loss function is σ (α) -sub-Gaussian-under the distribution P W,Zi ∀i = 1, · · · , n-Then, it holds ∀α ∈ (0, 1) that: |gen(P W |S , µ)| ≤ 1 n n i=1 2σ 2 (α) (αA i + (1 − α)B i ) α(1 − α) ,(25) where A i = D KL (P W,Zi P W,Zi ) and B i = D KL (P W ⊗ µ P W,Zi ) . We now first offer a Lemma connecting certain KL divergences to the generalized α-Jensen-Shannon information. Lemma 2: (Proved in Appendix B-C) Consider an auxiliary distribution P W,Zi . Then, the following equality holds: αD KL (P W ⊗ µ P W,Zi ) + (1 − α)D KL (P W,Zi P W,Zi ) =(26)I α JS (W ; Z i ) + D KL (P (α) W,Zi P W,Zi ). Note that the proof is inspired by [47]. The result in Theorem 7 and Lemma 2 paves the way to offer a tighter version of the generalization error bound appearing in Theorem 7 based on choosing an appropriate auxiliary distribution, as well as recover existing ones. Proposition 8: (Proved in Appendix B-D) Assume that the loss function is σ (α) -sub-Gaussian-under the distribution P (α) W,Zi ∀i = 1, · · · , n-Then, it holds ∀α ∈ (0, 1) that: |gen(P W |S , µ)| ≤ 1 n n i=1 2σ 2 (α) I α JS (W ; Z i ) α(1 − α) , ∀α ∈ (0, 1).(27) This bound in Proposition 8 results from minimizing the bound in Theorem 7 over the joint auxiliary distribution P W,Zi . Such an optimal joint auxiliary distribution is P (α) W,Zi := αP W P Zi + (1 − α)P W,Zi . It turns out that we can immediately recover existing bounds from Proposition 8 depending on how we choose α. Remark 9: The generalization error upper bound based on Jensen-Shannon information in [1] can be immediately recovered by considering α = 1 2 in Proposition 8. Remark 10: The generalization error upper bound based on mutual information in [7] can be immediately recovered by considering α → 1 in Proposition 8. Remark 11: The generalization error upper bound based on lautum information in [46] can be immediately recovered by considering α → 0 in Proposition 8. Note that we can also establish how the bound in Proposition 8 behaves as a function of the number of training samples. This can be done by using P W,Zi = P W ⊗ µ in Lemma 2, leading up to (1 − α)I(W ; Z i ) = I α JS (W ; Z i ) + D KL (P (α) W,Zi P W ⊗ µ).(28) and in turn to the following inequality I α JS (W ; Z i ) ≤ (1 − α)I(W ; Z i ), ∀α ∈ (0, 1).(29) Now, we prove the convergence rate of the upper bound in Proposition 8 using (29). Proposition 12: (Proved in Appendix B-E) Assume the hypothesis space is countable and the data samples, {Z i } n i=1 , are i.i.d.|gen(P W |S , µ)| ≤ σ (α) 2 h(α) α(1 − α) , ∀α ∈ (0, 1),(30) where h(α) = −α log(α) − (1 − α) log(1 − α). This proposition shows that, unlike mutual information-based and lautum information-based generalisation bounds that currently exist (e.g. [7], [8], [9], and [11]) the proposed generalised α-Jensen-Shannon information generalisation bound is always finite. Corollary 14: (Proved in Appendix B-G) Consider the assumptions in Proposition 8. Then, it follows that: |gen(P W |S , µ)| ≤ 2σ (1/2) 2 log(2).(31) This result derives by optimizing the bound in (30) with respect to alpha, where the minimum is achieved at α = 1/2. Also, this result applies independently of whether the loss function is bounded or not. Naturally, it is possible to show that the absolute value of the expected generalization error is always upper bounded as follows |gen(P W |S , µ)| ≤ (b − a) for any bounded loss function within the interval [a, b]. If we consider the bounded loss functions in the interval [a, b], then our upper bound (31) would be 2 log(2)(b − a) which is less than total variation constant upper bound, 2(b − a) presented in [15], [48]. Note also that this result cannot be immediately recovered from existing approaches such as [11,Theorem. 2]. For example, if we consider the upper bound based on Jensen-Shannon information, then there exist f -divergence based representations of the Jensen-Shannon information as follows: D JS (P X , P X ′ ) = dP X f dP X ′ dP X ,(32)with f (t) = t log(t) − (1 + t) log( 1+t 2 ) . However, [11,Theorem. 2] requires that the function f (t) associated with the f -divergence is non-decreasing within the interval [0, +∞), but such a requirement is naturally violated by the function f (t) = t log(t) − (1 + t) log( 1+t 2 ) associated with the Jensen-Shannon divergence. B. α-Rényi Based Upper Bound Next, we provide a new generalization error upper bound based on KL divergence by applying ADM, and using the following KL divergences terms, D KL ( P W,Zi P W ⊗ µ) and D KL ( P W,Zi P W,Zi ). Proposition 15: Suppose that Λ ℓ(W,Z) (λ) ≤ γ + (λ) and Λ ℓ(W,Zi) (λ) ≤ φ + (λ), i = 1, · · · , n for λ ∈ [0, a + ), 0 < a + < +∞ and λ ∈ [0, c + ), 0 < c + < +∞, under P W ⊗ µ and P W,Zi , resp. We also have Λ ℓ(W,Z) (λ) ≤ γ − (−λ) and Λ ℓ( W , Zi) (λ) ≤ φ − (−λ), i = 1, · · · , n for λ ∈ (a − , 0], −∞ < a − < 0 and λ ∈ (c − , 0], −∞ < c − < 0 under P W ⊗ µ and P W,Zi , resp. Assume that γ + (λ), φ + (λ), γ − (λ) and φ − (λ) are convex functions, γ − (0) = γ + (0) = γ ′ + (0) = γ ′ − (0) = 0 and φ − (0) = φ + (0) = φ ′ + (0) = φ ′ − (0) = 0. Then, the following upper bounds hold, gen(P W |S , µ) ≤ 1 n n i=1 γ ⋆−1 − (D i ) + φ ⋆−1 + (C i ) ,(33)− gen(P W |S , µ) ≤ 1 n n i=1 φ ⋆−1 − (C i ) + γ ⋆−1 + (D i ) ,(34) where D i = D KL ( P W,Zi P W ⊗µ), C i = D KL ( P W,Zi P W,Zi ), γ ⋆−1 − (x) = inf λ∈[0,−a−) x+γ−(λ) λ , γ ⋆−1 + (x) = inf λ∈[0,a+) x+γ+(λ) λ , φ ⋆−1 − (x) = inf λ∈[0,−c−) x+φ−(λ) λ and φ ⋆−1 + (x) = inf λ∈[0,c+) x+φ+(λ) λ . Proof: The proof approach is similar to Theorem 3 by considering different cumulant generating functions and their upper bounds. Inspired by the upper bound in Proposition 15, we are able to provide an upper bound on generalization error instantly that is dependent on the convex combination of KL divergence terms, i.e., αD KL ( P W,Zi P W,Zi ) + (1 − α)D KL ( P W,Zi P W ⊗ µ), and assuming σ-sub-Gaussian tail distribution. Theorem 16: (Proved in Appendix C-A) Assume that the loss function is σ-sub-Gaussian under distribution P W ⊗ µ and γ-sub-Gaussian under P W,Zi ∀i = 1, · · · , n-Then, it holds for ∀α ∈ (0, 1) that: |gen(P W |S , µ)| ≤ (35) 1 n n i=1 2(ασ 2 + (1 − α)γ 2 ) (αC i + (1 − α)D i ) α(1 − α) , where C i = D KL ( P W,Zi P W ⊗ µ) and D i = D KL ( P W,Zi P W,Zi ) . Akin to Theorem 7, the result in Theorem 16 paves the way to offer new tighter generalization error upper bound as well as recover existing ones. We next offer a Lemma connecting certain KL divergences to the α-Rényi information. Lemma 3: ( [40, Theorem 30], Proved in Appendix C-C) Consider an arbitrary distribution P W,Zi . Then, the following equality holds for 0 ≤ α ≤ 1: αD KL ( P W,Zi P W ⊗ µ) + (1 − α)D KL ( P W,Zi P W,Zi ) = (36) (1 − α)I α R (W ; Z i ) + D KL P W,Zi (P Zi ⊗ P W ) α (P W,Zi ) (1−α) W×Z (dP Zi ⊗ dP W ) α (dP W,Zi ) (1−α) i . We now offer a tighter version of the generalization error bound appearing in Theorem 16 based on choosing an appropriate auxiliary distribution. Proposition 17: (Proved in Appendix C-B) Consider the same assumptions in Theorem 16. The following upper bound for ∀α ∈ (0, 1) holds: |gen(P W |S , µ)| ≤ 1 n n i=1 2(ασ 2 + (1 − α)γ 2 ) I α R (W ; Z i ) α ,(37) This bound in Proposition 17 results from minimizing the bound in Theorem 16 over the joint auxiliary distributionP W,Zi . Such an optimal joint auxiliary distribution is P W,Zi = (P Zi ⊗ P W ) α (P W,Zi ) (1−α) W×Z (dP Zi ⊗ dP W ) α (dP W,Zi ) (1−α) . Remark 18: If the hypothesis, W , is a deterministic function of data sample Z i , then I(W ; Z i ) = ∞. However, by considering the α-Rényi information for α ∈ [0, 1), we have I α R (W ; Z i ) < ∞. It turns out that we can also immediately recover existing bounds and derive new bound in terms of Bhattacharyya distance from Proposition 17 depending on how we choose α. Remark 19: We can derive the generalization error upper bound based on Bhattacharyya distance by considering α = 1 2 in Proposition 17, |gen(P W |S , µ)| ≤ 2 n n i=1 (σ 2 + γ 2 )D B (P W,Zi P W ⊗ µ), Remark 20: We can recover the generalization error upper bound based on mutual information in [7] by considering α → 1 in Proposition 17. Remark 21: We can recover the generalization error upper bound based on lautum information in [46] by considering α → 0 in Proposition 17. Now, by considering P W,Zi = P W,Zi , we have that: αI(W ; Z i ) = (1 − α)I α R (W ; Z i )(38)+ D KL P W,Zi (P Zi ⊗ P W ) α (P W,Zi ) (1−α) W×Z (dP Zi ⊗ dP W ) α (dP W,Zi ) (1−α) . Since that KL divergence is non-negative, based on Lemma 3 and the monotonicity of D α with respect to α, we have: I α R (W ; Z i ) ≤ min 1, α 1 − α I(W ; Z i ).(39) The result in (39) implies that our generalization error bound based on α-Rényi information in Proposition 17 exhibits the same decay rate as upper bound based on mutual information [7]. Proposition 22: (Proved in Appendix C-D) Assume the hypothesis space is countable and the data samples are i.i.d. Then, the upper bounds based on α-Rényi information in Proposition 17 exhibit a decay rate of O( 1 √ n ). Now, we provide an upper bound based on Sibson's α-mutual information. Theorem 23: (Proved in Appendix C-E) Assume that the loss function is σ-sub-Gaussian under distribution µ for all w ∈ W and γ-sub-Gaussian under P W,Zi , ∀i = 1, · · · , n. Then, it holds that: |gen(P W |S , µ)| ≤ 1 n n i=1 2(ασ 2 + (1 − α)γ 2 ) I α S (W ; Z i ) α .(40) The upper bound based on α-Rényi divergence could also be derived using variational representation of α-Rényi divergence in [49]. This approach is applied in [12] by considering the sub-Gaussianity under P Zi and P Zi|W . Our approach is more general paving the way to offer an upper bound based on α-Sibson's mutual information in Theorem 23. Since that, I α S (W ; Z i ) = min QW D α (P W,Zi Q W ⊗ µ) (41) ≤ D α (P W,Zi P W ⊗ µ) = I α R (W ; Z i ),(42) the upper bound in Theorem 23 is tighter than the upper bound in Proposition 17. It is worthwhile mentioning that we assume the loss function is σ-sub-Gaussian under P W ⊗ µ distribution in Proposition 17. However, in Theorem 23, we consider the loss function is σ-sub-Gaussian under µ distribution for all w ∈ W. We can also apply generalized Pinsker's inequality [40] to bounded loss functions for bounding the generalization error using the α-Rényi information between data samples, S, and hypothesis, W . Proposition 24: (Proved in Appendix C-F) Consider ℓ(w, z) be a bounded loss function i.e. |ℓ(w, z)| ≤ b. Then |gen(P W |S , µ)| ≤ 2b 2 nα I α R (W ; S), ∀α ∈ (0, 1].(43) Considering the bounded loss function can help to provide an upper bound based on α-Sibson's mutual information between S and W . Proposition 25: Consider ℓ(w, z) be a bounded loss function, i.e., |ℓ(w, z)| ≤ b. Then |gen(P W |S , µ)| ≤ 2b 2 nα I α S (W ; S), ∀α ∈ (0, 1].(44) As Proposition 25 shows that the generalization error could be upper bounded in terms of α-Sibson's mutual information between dataset, S, and hypothesis, W , it is interesting to consider the regularized empirical risk minimization (ERM) with α-Sibson's mutual information for 0 < α < 1: min P W |S E[L S (W )] + 1 β I α S (W ; S),(45) where β > 0 is a parameter that balances fitting and generalization. Since the optimization problem in (45) is dependent on the data generating distribution, µ, we propose to relax the problem in (45) by replacing α-Sibson's mutual information, i.e. I α S (W ; S), with the upper bound D α (P W |S Q W |P S ) as follows: min P W |S E[L S (W )] + 1 β D α (P W |S Q W |P S ),(46) where Q W is an arbitrary distribution on hypothesis space. Lemma 4: The optimization problem considered in (46) is a convex optimization problem. Proof: The first term in objective E[L S (W )] is linear in term of P W |S and the second term 1 [40,Theorem 11]. Problem (46) does not yield a closed form solution for P W |S . Therefore, we use the following upper bound on D α (P Q) β D α (P W |S Q W |P S ) is convex in P W |S for 0 < α < 1 due toD α (P Q) ≤ min{1, α 1 − α }D KL (P Q), 0 ≤ α ≤ 1, to derive the following regularized ERM problem: min P W |S E[L S (W )] + min{1, α 1−α } β D KL (P W |S Q W |P S ).(47) which yields the following Gibbs algorithm [21] P [α] W |S=s (dw) = e −β min{1, α 1−α } [Ls(w)] Q W (dw) E QW e −β min{1, α 1−α } [Ls(w)] ,(48) Inspired by [21, Theorem 1], we provide the following exact characterization of expected generalization error under Gibbs algorithm. Proposition 26: (Proved in Appendix C-G) Under the following Gibbs algorithm, P [α] W |S=s (dw) = e −β min{1, α 1−α } [Ls(w)] Q W (dw) E QW e −β min{1, α 1−α } [Ls(w)] ,(49) then the exact characterization of expected generalization error using symmetrized KL information (See Table II) is as follows: gen(P [α] W |S , µ) = I SKL (W ; S) β min(1, α 1 − α ).(50) C. Comparison of Proposed Bounds A summary of upper bounds on generalization error under various σ-sub-Gaussian assumptions is provided in Table III. Note that any bounded loss function l : [7]. In fact, for bounded functions, we have: W × Z → [a, b] is ( b−a 2 )-sub-Gaussian under all distributionsσ = γ = σ (α) = (b − a) 2 .(51) We next compare the upper bounds based on generalized α-Jensen-Shannon information, Proposition 8, with the upper bounds based on α-Rényi information, Proposition 17. The next proposition showcases that the generalized α-Jensen-Shannon information bound can be tighter than the α-Rényi based upper bound under certain conditions. 2σ 2 I(W ; Z i ) No Lautum information ( [46]) P W,Z i , ∀i = 1, . . . , n 1 n n i=1 2γ 2 L(W ; Z i ) No Generalized α-Jensen-Shannon information (Proposition 13) P (α) W,Z i , ∀i = 1, . . . , n 1 n n i=1 2σ 2 (α) I α JS (W ;Z i ) α(1−α) Yes (σ (α) 2 h(α) α(1−α) ) α-Rényi information (0 ≤ α < 1) (Proposition 17) P W ⊗ µ and P W,Z i , ∀i = 1, . . . , n 1 n n i=1 2(ασ 2 + (1 − α)γ 2 ) I α R (W ;Z i ) α No Proposition 27: (Proved in Appendix D) Consider the same assumptions in Proposition 8. Then, it follows that generalized α ′ -Jensen-Shannon bound given by: |gen(P W |S , µ)| ≤ 1 n n i=1 2σ 2 (α ′ ) I α ′ JS (W ; Z i ) α ′ (1 − α ′ ) , 0 ≤ α ′ ≤ 1(52) is tighter than the α-Rényi based upper bound for 0 ≤ α ≤ 1, given by: |gen(P W |S , µ)| ≤ 1 n n i=1 2(ασ 2 + (1 − α)γ 2 ) I α R (W ; Z i ) α ,(53) provided that αh(α ′ ) (1−α ′ )α ′ ≤ I α R (W ; Z i ) holds for i = 1, · · · , n and σ (α ′ ) = σ = γ. Remark 28: The condition in Proposition 27, i.e. αh(α ′ ) (1−α ′ )α ′ ≤ I α R (W ; Z i ), could be tightened by considering α ′ = 1 2 and considering the upper bound based on Jensen-Shannon information. Remark 29: If we consider α → 1 and α ′ = 1 2 in Proposition 27, then the upper bound based on Jensen-Shannon information is tighter than ones based on mutual information [8] provided that 4 log(2) ≤ I(W ; Z i ) for all i = 1, · · · , n and σ = σ JS . IV. AUXILIARY DISTRIBUTION BASED GENERALIZATION ERROR UPPER BOUNDS UNDER DISTRIBUTION MISMATCH In this section, we extend our results in Section III under distribution mismatch, where the training data distribution differs from the test data. In the following Propositions, we provide an upper bound on generalization error under distribution mismatch in terms of generalized α-Jensen-Shannon divergence and α-Rényi divergence, respectively. Proposition 30: (Proved in Appendix E-A) Assume that the loss function is σ (α) -sub-Gaussian-under the distributions P (α) W,Zi ∀i = 1, · · · , n and αµ + (1 − α)µ ′ for all w ∈ W-Then under distribution mismatch, it holds ∀α ∈ (0, 1) that: |gen(P W |S , µ, µ ′ )| ≤ 2σ 2 (α) D α JS (µ ′ µ) α(1 − α)(54)+ 1 n n i=1 2σ 2 (α) I α JS (W ; Z i ) α(1 − α) , ∀α ∈ (0, 1). Proposition 31: (Proved in Appendix E-B) Assume that the loss function is σ-sub-Gaussian under distributions µ and µ ′ for all w ∈ W and also γ-sub-Gaussian under P W,Zi ∀i = 1, · · · , n-. The following upper bound for ∀α ∈ (0, 1) holds under distribution: |gen(P W |S , µ, µ ′ )| ≤ 2(ασ 2 + (1 − α)γ 2 ) D α (µ ′ µ) α(55)+ 1 n n i=1 2(ασ 2 + (1 − α)γ 2 ) I α R (W ; Z i ) α . The distributional mismatch between the test and training samples is characterised in [26,Theorem 5] as KL divergence between test and training samples distributions, i.e., D KL (µ ′ µ). However, using ADM based on the assumption that the loss function is σ (α) -sub-Gaussian under αµ + (1 − α)µ ′ for all w ∈ W, we can explain the distributional mismatch in terms of generalized α-Jensen-Shannon divergence, which is finite (See Proposition 30). In Proposition 31, the distributional mismatch is modeled as α-Rényi divergence, i.e., D α (µ ′ µ). If µ ′ is not absolutely continuous with respect to µ 2 , then we have D KL (µ ′ µ) = ∞. However, for α-Rényi divergence (0 < α < 1), we just require that the mutual singularity [40], i.e., µ ′ ⊥ ⊥ µ, does not hold which is more tolerant with respect to absolutely continuity condition. V. NUMERICAL EXAMPLE We now illustrate that some of our proposed bounds can be indeed tighter than existing ones in a simple toy example. As the upper bound based on triangular discrimination is looser than the upper bound based on Jensen-Shannon, we consider the generalized α-Jensen-Shannon and α-Rényi information only. Our example setting involves the estimation of the mean of a Gaussian random variable Z ∼ N (β, σ 2 ) based on two i.i.d. samples Z 1 and Z 2 . We consider the hypothesis (estimate) given by W = tZ 1 + (1 − t)Z 2 for 0 < t < 1. We also consider the loss function given by ℓ(w, z) = min((w − z) 2 , c 2 ). In view of the fact that the loss function is bounded within the interval [0, c 2 ], it is also c 2 2 -sub-Gaussian so that we can apply the generalization error upper bounds based on mutual information and generalized α-Jensen-Shannon information and α-Rényi information for ∀α ∈ (0, 1) as follows: gen(P W |Z1,Z2 , P Z ) ≤ c 2 4 2I(W ; Z 1 ) + 2I(W ; Z 2 ) ,(56)gen(P W |Z1,Z2 , P Z ) ≤ c 2 4 2 I α JS (W ; Z 1 ) α(1 − α) + 2 I α JS (W ; Z 2 ) α(1 − α) ,(57)gen(P W |Z1,Z2 , P Z ) ≤ c 2 4 2 I α R (W ; Z 1 ) α + 2 I α R (W ; Z 2 ) α .(58) It can be immediately shown that W ∼ N (β, σ 2 (t 2 +(1−t) 2 )) and (W, Z 1 ) and (W, Z 2 ) are jointly Gaussian with correlation coefficients ρ 1 = t √ t 2 +(1−t) 2 and ρ 2 = (1−t) √ t 2 +(1−t) 2 . Therefore, it also be shown that the mutual informations appearing above are given by I(W ; Z 1 ) = − 1 2 log(1 − ρ 2 1 ) and I(W ; Z 2 ) = − 1 2 log(1 − ρ 2 2 ) . In contrast, the generalized α-Jensen-Shannon informations appearing above can be computed via an extension of entropic based formulation of Jensen-Shannon measure as follows [37]: I JS (W ; Z i ) = (59) h P (α) W,Zi − (αh(P W ) + αh(P Zi ) + (1 − α)h(P Zi,W )), -with h(·) denoting the differential entropy -where h(P Zi ) = 1 2 log(2πσ 2 ), h(P W ) = 1 2 log(2πσ 2 (t 2 + (1 − t) 2 )), h(P W,Zi ) = log(2πσ 2 (t 2 + (1 − t) 2 )(1 − ρ 2 i )), whereas h P (α) W,Zi can be computed numerically. Fig.1 depicts the true generalization error, the mutual information based bound in (56), and the generalized α-Jensen-Shannon information based bound for α = 0.25, 0.5, 0.75 in (57) for values of t ∈ (0, 0.5], considering σ 2 = 1, 10, µ = 1, c = σ 4 . It can be seen that for α = 0.75 the generalized α-Jensen-Shannon information bound is tighter than mutual information bound. Now, for α = 0.5, which is equal to traditional Jensen-Shannon information, if we consider t < 0.25 then Jensen-Shannon information bound is tighter than the mutual information bound; in contrast, for t > 0.25, the mutual information bound is slightly better than the Jensen-Shannon information bound. This showcases indeed that our proposed bounds can be tighter than existing ones in some regimes. Fig.2 also depicts the true generalization error, the mutual information based bound in (56), and the α-Rényi information based bound for α = 0.25, 0.5, 0.75 in (58). It can be seen that the α-Rényi based bound is looser than mutual information based bound. If the learning algorithm is a function of some samples, the mutual information based bound is infinite and the α-Rényi information based bound is bounded. VI. CONCLUSION AND FUTURE WORKS We have introduced a new approach to obtain information-theoretic upper bounds on the generalization error associated with supervised learning problems. Our approach can be used to recover the existing bounds and derive new ones based on generalized α-Jensen-Shannon, α-Rényi information measures. Our upper bounds based on generalized α-Jensen-Shannon information measure are bounded by a finite value. Unlike mutual information-based bounds, our upper bound based on α-Rényi information for α ∈ (0, 1) under a deterministic learning process is finite-value. Notably, it is shown that the new generalized α-Jensen-Shannon information can be tighter in some regimes in comparison to existing bounds. For future works, we propose the PAC-Bayesian extension of our bounds based on Generalized α-Jensen-Shannon and α-Rényi divergence for 0 < α < 1. The conditional technique based on individual sample measures [18] could also be applied to our upper bounds. APPENDIX A SUB-EXPONENTIAL AND SUB-GAMMA We introduce the tail behaviour of two random variables, including sub-Exponential and sub-Gamma. • Sub-Exponential: A random variable X is (σ 2 e , b)-sub-Exponential, if ψ(λ) = σ 2 e λ 2 2 is an upper bound on Λ X (λ), for 0 ≤ |λ| ≤ 1 b and b > 0. Using Lemma 1, we have ψ ⋆−1 (y) = 2σ 2 e y, if y ≤ σ 2 e 2b ; by + σ 2 e 2b , otherwise. (60) • Sub-Gamma: A random variable X is Γ(σ 2 s , c s )-sub-Gamma [50], if ψ(λ) = λ 2 σ 2 s 2(1−cs|λ|) is an upper bound on Λ X (λ), for 0 < |λ| < 1 cs and c s > 0. Using Lemma 1, we have ψ ⋆−1 (y) = 2σ 2 s y + c s y.(61) APPENDIX B PROOF OF SECTION III-A A. Proof of Theorem 3 The proofs of the bounds to gen(P W |S , µ) and −gen(P W |S , µ) are similar. We therefore focus on the later. Let us consider the Donsker-Varadhan variational representation of KL divergence between two probability distributions α and β on a common space Ψ given by [51]: D KL (α β) = sup f Ψ f dα − log Ψ e f dβ,(62) where f ∈ F = {f : Ψ → R s.t. E β [e f ] < ∞}. We can now use the Donsker-Varadhan representation to bound D KL (P W,Zi P W,Zi ) for λ ∈ (b − , 0] as follows: D KL (P W,Zi P W,Zi ) ≥ (63) E PW,Z i [λℓ(W, Z i )] − log E PW,Z i [e λℓ(W,Zi) ] ≥ λ(E PW,Z i [ℓ(W, Z i )] − E PW,Z i [ℓ(W, Z i )]) − ψ − (−λ),(64) where the last inequality is due to: Λ ℓ(W,Zi) (λ) = (65) log E PW,Z i [e λ(ℓ(W,Zi)−E P W,Z i [ℓ(W,Zi)]) ] ≤ ψ − (−λ). It can then be shown from (64) that the following holds for λ ∈ (b − , 0]: E PW,Z i [ℓ(W, Z i )] − E PW,Z i [ℓ(W, Z i )] ≤ (66) inf λ∈[0,−b−) D KL (P W,Zi P W,Zi ) + ψ − (λ) λ = ψ ⋆−1 − (D KL (P W,Zi P W,Zi )).(67) It can likewise also be shown by adopting similar steps that the following holds for λ ∈ [0, b + ): E PW,Z i [ℓ(W, Z i )] − E PW,Z i [ℓ(W, Z i )] ≤ (68) inf λ∈[0,b+) D KL (P W,Zi P W,Zi ) + ψ(λ) λ = ψ ⋆−1 + (D KL (P W,Zi P W,Zi )).(69) We can similarly show using an identical procedure that: E PW ⊗µ [ℓ(W, Z i )] − E PW,Z i [ℓ(W, Z i )] ≤ ψ ⋆−1 + (D KL (P W ⊗ µ P W,Zi ))(70)E PW,Z i [ℓ(W, Z i )] − E PW ⊗µ [ℓ(W, Z i )] ≤ ψ ⋆−1 − (D KL (P W ⊗ µ P W,Zi )).(71) Finally, we can immediately bound the generalization error by leveraging (70) and (66) as follows: gen(P W |S , µ) = 1 n n i=1 E PW ⊗µ [ℓ(W, Z i )] − E PW,Z i [ℓ(W, Z i )] = 1 n n i=1 E PW ⊗µ [ℓ(W, Z i )] − E PW,Z i [ℓ(W, Z i )]+ E PW,Z i [ℓ(W, Z i )] − E PW,Z i [ℓ(W, Z i )] ≤ 1 n n i=1 ψ ⋆−1 + (A i ) + ψ ⋆−1 − (B i ) , where A i = D KL (P W ⊗ µ P W,Zi ) and B i = D KL (P W,Zi P W,Zi ). B. Proof of Theorem 7 The assumption that the loss function is σ-sub-Gaussian under the distribution P W,Zi implies that ψ ⋆−1 − (y) = ψ ⋆−1 + (y) = 2σ 2 y, [8]. Consider arbitrary auxiliary distributions { P W,Zi } n i=1 defined on W × Z. gen(µ, P W |S ) = E PW PS [L S (W )] − E PW,S [L S (W )] = 1 n n i=1 E PW PZ i [ℓ(W, Z i )] − E PW Z i [ℓ(W, Z i )] (72) ≤ 1 n n i=1 E PW PZ i [ℓ(W, Z i )] − E PW Z i [ℓ(W, Z i )](73) Using the assumption that the loss function ℓ(w, z i ) is σ 2 α -sub-Gaussian under distribution P W,Zi and Donsker-Varadhan representation for D KL (P W Zi P W,Zi ), we have: λ E PW,Z i [ℓ(W, Z i )] − E PW,Z i [ℓ(W, Z i )] ≤ (74) D KL (P W Zi P W,Zi ) + λ 2 σ 2 α 2 . ∀λ ∈ R Using the assumption loss that the function ℓ(w, z i ) is σ 2 α -sub-Gaussian under distribution P W,Zi and Donsker-Varadhan representation for D KL ( P W,Zi P W P Zi ), we have: λ ′ E PW PZ i [ℓ(W, Z i )] − E PW,Z i [ℓ(W, Z i )] ≤ (75) D KL (P W P Zi P W,Zi ) + λ ′ 2 σ 2 (α) 2 . ∀λ ′ ∈ R Now if we consider λ < 0, then we can choose λ ′ = α α−1 λ. Hence we have: E PW,Z i [ℓ(W, Z i )] − E PW,Z i [ℓ(W, Z i )] ≤(76)D KL (P W,Zi P W,Zi ) |λ| + |λ|σ 2 α 2 . ∀λ ∈ R − and, E PW PZ i [ℓ(W, Z i )] − E PW,Z i [ℓ(W, Z i )] ≤ (77) D KL (P W P Zi P W,Zi ) λ ′ + λ ′ σ 2 α 2 . ∀λ ′ ∈ R + Now sum up two Inequalities (76) and (77). E PW PZ i [ℓ(W, Z i )] − E PW Z i [ℓ(W, Z i )] ≤(78) αD KL (P W,Zi P W,Zi ) + (1 − α)D KL (P W P Zi P W,Zi ) α|λ| + |λ|σ 2 α 2 + |λ| α 1−α σ 2 α 2 , ∀λ ∈ R − . Similarly, using an identical approach, we also obtain: − E PW PZ i [ℓ(W, Z i )] − E PW Z i [ℓ(W, Z i )] ≤ (79) αD KL (P W,Zi P W,Zi ) + (1 − α)D KL (P W P Zi P W,Zi ) αλ + λσ 2 α 2 + λ α 1−α σ 2 α 2 , ∀λ ∈ R + . Considering (78) and (79), we have a nonnegative parabola in λ, whose discriminant must be nonpositive, and we have ∀α ∈ (0, 1): E PW PZ i [ℓ(W, Z i )] − E PW Z i [ℓ(W, Z i )] 2 ≤ (80) 2σ 2 (α) αD KL (P W,Zi P W,Zi ) + (1 − α)D KL (P W P Zi P W,Zi ) α(1 − α) . Now using (72), we prove the claim. C. Proof of Lemma 2 αD KL (P W ⊗ P Zi P W,Zi ) + (1 − α)D KL (P W,Zi P W,Zi ) (81) = W×Z α(dP W ⊗ dP Zi ) log(dP W ⊗ dP Zi ) (82) + W×Z (1 − α)dP W,Zi log(dP W,Zi ) − W×Z ((α(dP W ⊗ dP Zi ) + (1 − α)dP W,Zi ) log(d P W,Zi )) = W×Z α(dP W ⊗ dP Zi ) log(dP W ⊗ dP Zi )(83)+ W×Z (1 − α)dP W,Zi log(dP W,Zi ) − dP D. Proof of Proposition 8 As shown in [52], and by considering the Lemma 2 we have min PW,Z i αD KL (P W ⊗ µ P W,Zi ) + (1 − α)D KL (P W,Zi P W,Zi ) = (85) min PW,Z i I α JS (W ; Z i ) + D KL (P (α) W,Zi P W,Zi ). As we have 0 ≤ D KL (P (α) W,Zi P W,Zi ), therefore, the minimum of (25) is achieved with P W,Zi = P E. Proof of Proposition 12 Using (28), I α JS (W ; Z i ) ≤ (1 − α)I(W ; Z i ),(86) we have: |gen(P W |S , µ)| ≤ 1 n n i=1 2σ 2 (α) I α JS (W ; Z i ) α(1 − α) (87) ≤ 1 n n i=1 2σ 2 (α) I(W ; Z i ) α (88) ≤ 2σ 2 (α) n i=1 I(W ; Z i ) αn (89) ≤ 2σ 2 (α) I(W ; S) αn (90) ≤ 2σ 2 (α) H(W ) αn ,(91) where the final result, would follow from the finite hypothesis space. F. Proof of Proposition 13 This proposition follows from the fact that I α JS (W, Z i ) ≤ h(α) for i = 1, · · · , n. Now, we prove that I α JS (W, Z i ) ≤ h(α). I α JS (W, Z i ) = (92) αD KL (P W ⊗ P Zi P (α) W,Zi ) + (1 − α)D KL (P W,Zi P (α) W,Zi ) = α W×Z dP W ⊗ dP Zi log dP W ⊗ dP Zi dP (α) W,Zi(93)+ (1 − α) W×Z dP W,Zi log dP W,Zi dP (α) W,Zi ≤ α W×Z dP W ⊗ dP Zi log dP W ⊗ dP Zi α(dP W ⊗ dP Zi ) (94) + (1 − α) W×Z dP W,Zi log dP W,Zi (1 − α)dP W,Zi = −α log(α) − (1 − α) log(1 − α) (95) = h(α). (96) G. Proof of Corollary 14 We first compute the derivative of h(α) α(1−α) with respect to α ∈ (0, 1) d h(α) α(1−α) dα = log(1 − α) α 2 − log(α) (1 − α) 2 .(97) Now for α = 1 2 , we have d h(α) α(1−α) dα = 0. APPENDIX C PROOFS OF SECTION III-B A. Proof of Proposition 16 Consider arbitrary auxiliary distributions { P W,Zi } n i=1 defined on W × Z. Then, gen(P W |S , µ) = E PW PS [L S (W )] − E PW,S [L S (W )] = 1 n n i=1 E PW PZ i [ℓ(W, Z i )] − E PW Z i [ℓ(W, Z i )] (98) ≤ 1 n n i=1 E PW PZ i [ℓ(W, Z i )] − E PW Z i [ℓ(W, Z i )] (99) ≤ 1 n n i=1 E PW,Z i [ℓ(W, Z i )] − E PW Z i [ℓ(W, Z i )](100)+ E PW,Z i [ℓ(W, Z i )] − E PW PZ i [ℓ(W, Z i )] . Using the assumption that loss function ℓ(w, z i ) is γ 2 -sub-Gaussian under distribution P W,Zi and Donsker-Varadhan representation we have: λ E PW,Z i [ℓ(W, Z i )] − E PW Z i [ℓ(W, Z i )] ≤ (101) D KL ( P W,Zi P W Zi ) + λ 2 γ 2 2 , ∀λ ∈ R. Using the assumption that ℓ(w, Z) is σ 2 -sub-Gaussian under P W ⊗ P Zi , and again Donsker-Varadhan representation we have: λ ′ E PW,Z i [ℓ(W, Z i )] − E PW PZ i [ℓ(W, Z i )] ≤ (102) D KL ( P W,Zi P W P Zi ) + λ ′ 2 σ 2 2 . ∀λ ′ ∈ R Note that E PW PZ i [ℓ(W, Z i ) − E PZ i [ℓ(W, Z i )]] = 0. Now if we consider λ > 0, then we choose λ ′ = α α−1 λ. Hence we have E PW,Z i [ℓ(W, Z i ) − E PW,Z i [ℓ(W, Z i )]] ≤ (103) D KL ( P W,Zi P W Zi ) λ + λγ 2 2 , ∀λ ∈ R + . Using the assumption that ℓ(w, Z) is σ 2 -sub-Gaussian and again Donsker-Varadhan representation, (103) and (104), to obtain − E PW,Z i [ℓ(W, Z i ) − E PW PZ i [ℓ(W, Z i )]] ≤ (104) D KL ( P W,Zi P W P Zi ) |λ ′ | + |λ ′ |σ 2 2 , ∀λ ′ ∈ R − . Now sum up two Inequalities E PW PZ i [ℓ(W, Z i )] − E PW Z i [ℓ(W, Z i )] ≤ (105) αD KL ( P W,Zi P W,Zi ) + (1 − α)D KL ( P W,Zi Q W P Zi ) αλ + λγ 2 2 + λ α 1−α σ 2 2 , ∀λ ∈ R + . Considering (105), we have a nonnegative parabola in λ, whose discriminant must be nonpositive, and we have: E PW PZ i [ℓ(W, Z i )] − E PW Z i [ℓ(W, Z i )] ≤ (106) 2(ασ 2 + (1 − α)γ 2 ) (αC i + (1 − α)D i ) α(1 − α) , where C i = D KL ( P W,Zi P W ⊗ µ) and D i = D KL ( P W,Zi P W,Zi ) . Now using (98), we prove the claim. B. Proof of Proposition 17 Using Lemma 3, we have: min PW,Z i αD KL ( P W,Zi P W ⊗ µ) + (1 − α)D KL ( P W,Zi P W,Zi ) = (107) (1 − α)I α R (W ; Z i ) + min PW,Z i D KL P W,Zi (P Zi ⊗ P W ) α (P W,Zi ) (1−α) W×Z (dP Zi ⊗ dP W ) α (dP W,Zi ) (1−α) . Now by considering the d P W,Zi = (dPZ i ⊗dPW ) α (dPW,Z i ) (1−α) W ×Z (dPZ i ⊗dPW ) α (dPW,Z i ) (1−α) , the KL term would be equal to zero. Now, using Proposition 16, we prove the final result. C. Proof of Lemma 3 Our proof is based on [40,Theorem 30]. For 0 ≤ α ≤ 1, we have: αD KL ( P W,Zi P W ⊗ µ) + (1 − α)D KL ( P W,Zi P W,Zi ) (108) = W×Z d P W,Zi log(d P W,Zi ) (109) − W×Z P W,Zi log((dP W ⊗ dP Zi ) α (dP W,Zi ) (1−α) ) = W×Z d P W,Zi log(d P W,Zi ) (110) − W×Z d P W,Zi log (dP W ⊗ dP Zi ) α (dP W,Zi ) (1−α) + log W×Z (dP W ⊗ dP Zi ) α (dP W,Zi ) (1−α) − log W×Z (dP W ⊗ dP Zi ) α (dP W,Zi ) (1−α) = − log W×Z (P W ⊗ dP Zi ) α (dP W,Zi ) (1−α) (111) + W×Z d P W,Zi log(d P W,Zi ) − W×Z d P W,Zi log (dP W ⊗ dP Zi ) α (dP W,Zi ) (1−α) W×Z (dP W ⊗ dP Zi ) α (dP W,Zi ) (1−α) = (1 − α)I α R (W ; Z i ) (112) + D KL P W,Zi (P Zi ⊗ P W ) α (P W,Zi ) (1−α) W×Z (dP Zi ⊗ dP W ) α (dP W,Zi ) (1−α) . D. Proof of Proposition 22 Using (39), I α R (W ; Z i ) ≤ α 1 − α I(W ; Z i ),(113) and considering the hypothesis space is countable and the upper bound in Proposition 17 we have: |gen(P W |S , µ)| ≤ 1 n n i=1 2(ασ 2 + (1 − α)γ 2 ) I α R (W ; Z i ) α (114) ≤ 1 n n i=1 2(ασ 2 + (1 − α)γ 2 ) min 1 α , 1 1 − α I(W ; Z i ) (115) ≤ 2(ασ 2 + (1 − α)γ 2 ) min 1 α , 1 1 − α n i=1 I(W ; Z i ) n (116) ≤ 2(ασ 2 + (1 − α)γ 2 ) min 1 α , 1 1 − α I(W ; S) n (117) ≤ 2(ασ 2 + (1 − α)γ 2 ) min 1 α , 1 1 − α H(W ) n (118) ≤ 2(ασ 2 + (1 − α)γ 2 ) min 1 α , 1 1 − α log(k) n , where (116) follows from Jensen inequality and (117) follows from i.i.d assumption for Z i 's. E. Proof of Theorem 23 Consider arbitrary auxiliary distributions {P W Zi } n i=1 defined on W × Z. gen(P W |S , µ) = E PW PS [L S (W )] − E PW,S [L S (W )] = 1 n n i=1 E PW PZ i [ℓ(W, Z i )] − E PW Z i [ℓ(W, Z i )](119)≤ 1 n n i=1 E PW PZ i [ℓ(W, Z i )] − E PW Z i [ℓ(W, Z i )] .(120) Using the assumption centered loss function ℓ(w, z i ) − E PZ i [ℓ(w, Z i )] is γ 2 -sub-Gaussian under distribution P W Zi and Donsker-Varadhan representation by considering function ℓ(w, z i ) − E PZ i [ℓ(w, Z i )] we have: λ EP W Z i [ℓ(W, Z i ) − E PZ i [ℓ(W, Z i )]] (121) − E PW Z i [ℓ(W, Z i ) − E PZ i [ℓ(W, Z i )]] ≤ D KL (P W Zi P W Zi ) + λ 2 γ 2 2 . ∀λ ∈ R Note that E PW Z i [ℓ(W, Z i ) − E PZ i [ℓ(W, Z i )]] = E PW Z i [ℓ(W, Z i )] − E PW PZ i [ℓ(W, Z i )]. Using the assumption that ℓ(w, Z) is σ 2 -sub-Gaussian under P Zi for all w ∈ W, and again Donsker-Varadhan representation by considering function ℓ(w, z i ) − E PZ i [ℓ(w, Z i )] we have: λ ′ EP W Z i [ℓ(W, Z i ) − E PZ i [ℓ(W, Z i )]] (122) − E QW PZ i [ℓ(W, Z i ) − E PZ i [ℓ(W, Z i )]] ≤ D KL (P W Zi Q W P Zi ) + λ ′ 2 σ 2 2 . ∀λ ′ ∈ R Note that E QW PZ i [ℓ(W, Z i ) − E PZ i [ℓ(W, Z i )]] = 0. Now if we consider λ > 0, then we choose λ ′ = α α−1 λ. Hence we have EP W Z i [ℓ(W, Z i ) − E PZ i [ℓ(W, Z i )]] (123) − E PW Z i [ℓ(W, Z i ) − E PZ i [ℓ(W, Z i )]] ≤ D KL (P W Zi P W Zi ) λ + λγ 2 2 , ∀λ ∈ R + . Using the assumption ℓ(w, Z) is σ 2 -sub-Gaussian and again Donsker-Varadhan representation, − EP W Z i [ℓ(W, Z i ) − E PZ i [ℓ(W, Z i )]] ≤(124)D KL (P W Zi Q W P Zi ) |λ ′ | + |λ ′ |σ 2 2 , ∀λ ′ ∈ R − . Now sum up the two Inequalities (123) and (124) to obtain, E PW PZ i [ℓ(W, Z i )] − E PW Z i [ℓ(W, Z i )] ≤ (125) αD KL (P W Zi P W Zi ) + (1 − α)D KL (P W Zi Q W P Zi ) αλ + λγ 2 2 + λ α 1−α σ 2 2 , ∀λ ∈ R + . Taking infimum onP W Zi and using [40,Theorem 30] that states (1 − α)D α (P 1 P 2 ) = inf R {αD KL (R P 1 ) + (1 − α)D KL (R P 2 )} Now, we have: (1 − α)D α (P W Zi Q W P Zi ) = (126) inf PW Z i {αD KL (P W Zi P W Zi ) + (1 − α)D KL (P W Zi Q W P Zi )} and taking infimum on Q W , we have: inf QW D α (P W Zi Q W P Zi ) = I α S (Z i ; W ).(127) Using (127) in (125), we get: E PW PZ i [ℓ(W, Z i )] − E PW Z i [ℓ(W, Z i )] ≤(128)(1 − α)I s α (Z i ; W ) λα + λγ 2 2 + λ α 1−α σ 2 2 ∀λ ∈ R + . Using the same approach for λ ∈ R − , we have: E PW Z i [ℓ(W, Z i )] − E PW PZ i [ℓ(W, Z i )] ≤ (129) (1 − α)I s α (Z i ; W ) |λ|α + |λ|γ 2 2 + |λ| α 1−α σ 2 2 , ∀λ ∈ R − . Considering (128) and (129), we have a nonnegative parabola in λ, whose discriminant must be nonpositive, and we have: E PW PZ i [ℓ(W, Z i )] − E PW Z i [ℓ(W, Z i )] ≤ (130) 2(ασ 2 + (1 − α)γ 2 ) I α S (Z i ; W ) α . Now using (98), we prove the claim. F. Proof of Proposition 24 There is a generalization of Pinsker's inequality in [40] as follows: Theorem 32 (generalization of Pinsker's inequality): P − Q 2 T V ≤ 2 α D α (P Q), α ∈ (0, 1](131) Denote f : X → R a bounded function |f | ≤ L, then E P [f (X)] − E Q [f (X)] = (132) f (x)(P (dx) − Q(dx)) ≤ sup x f (x) · |P (dx) − Q(dx)| ≤ L 2 α D α (P Q). Let P = P W,S , Q = P W P S and f (w, s) = L µ (w) − L s (w). Then, we have the final result, gen(P W |S , µ) = E[L µ (W ) − L S ] ≤ 2b 2 nα D α (P W,S P W P S )(133) G. Proof of Proposition 26 It can be shown that the symmetrized KL information can be written as I SKL (W ; S) = E PW,S [log(P α W |S )] − E PW ⊗PS [log(P α W |S )].(134) Note that P W,S and P W ⊗ P S share the same marginal distribution, hence we have E PW,S [log π(W )] = E PW [log π(W )], and E PW,S [log V (S, α)] = E PS [log V (S, α)]. Then, combining the following Gibbs algorithm, P [α] W |S=s (dw) = e −β min{1, α 1−α } [Ls(w)] Q W (dw) E QW e −β min{1, α 1−α } [Ls(w)] ,(135) with (134) completes the proof. APPENDIX D PROOF OF PROPOSITION 27 It follows from I α JS (W ; Z i ) ≤ h(α ′ ) α ′ (1 − α ′ ) , that if we have αh(α ′ ) α ′ (1−α ′ ) ≤ I α R (W ; Z i ) for all i = 1, · · · , n, then the results holds for σ = γ = σ JS . PROOF OF SECTION IV We first propose the following Lemma to provide an upper bound on the expected generalization error under distribution mismatch. Lemma 5: The following upper bound holds on expected generalization error under distribution mismatch between the test and training distributions: |gen(P W |S , µ, µ ′ )| ≤ (136) |gen(P W |S , µ)| + |E PW ⊗µ ′ [ℓ(W, Z)] − E PW ⊗µ [ℓ(W, Z)]|. Proof: We have: |gen(P W |S , µ, µ ′ )| (137) = |E PW,S [L P (W, µ ′ ) − L P (W, µ) + L P (W, µ) − L E (E, S)]| ≤ |E PW,S [L P (W, µ ′ ) − L P (W, µ)]| + |gen(P W |S , µ)| = |E PW ⊗µ ′ [ℓ(W, Z)] − E PW ⊗µ [ℓ(W, Z)]| + |gen(P W |S , µ)| A. Proof of Proposition 30 In Lemma 5, the generalization error under distribution mismatch can be upper bounded by two terms. Considering Proposition 8, we can provide the upper bound based on generalized α-Jensen-Shannon information over |gen(P W |S , µ)|. We can also provide an upper bound on the term |E PW ⊗µ ′ [ℓ(W, Z)] − E PW ⊗µ [ℓ(W, Z)]| in Lemma 5 by applying ADM using a similar approach as in Proposition 8 and using the generalized α-Jensen-Shannon divergence as follows: |E PW ⊗µ ′ [ℓ(W, Z)] − E PW ⊗µ [ℓ(W, Z)]| ≤ (138) 2σ 2 (α) D α JS (P W ⊗ µ ′ P W ⊗ µ) α(1 − α) = (139) 2σ 2 (α) D α JS (µ ′ µ) α(1 − α) .(140) B. Proof of Proposition 31 Based on Lemma 5, the generalization error is upper bounded by two terms (See Equation (136)). We can provide the upper bound based on α-Rényi information over |gen(P W |S , µ)| using Proposition 17. We can also provide an upper bound on the term |E PW ⊗µ ′ [ℓ(W, Z)] − E PW ⊗µ [ℓ(W, Z)]| by applying ADM using a similar approach as in Proposition 17 and using α-Rényi divergence as follows: |E PW ⊗µ ′ [ℓ(W, Z)] − E PW ⊗µ [ℓ(W, Z)]| ≤ (141) 2(ασ 2 + (1 − α)γ 2 ) D α (P W ⊗ µ ′ P W ⊗ µ) α = (142) 2(ασ 2 + (1 − α)γ 2 ) D α (µ ′ µ) α .(143) APPENDIX F GENERALIZATION ERROR UPPER BOUNDS BASED ON α-KL-PYTHAGOREAN AND TRIANGULAR DISCRIMINATION MEASURES In this section, we provide an upper bound based on a new divergence measure, α-KL-Pythagorean divergence, which is inspired by the Pythagorean inequality for KL divergence. We also provide an upper bound based on triangular Discrimination by applying auxiliary distribution to Chi-square divergence. The upper bound based on triangular Discrimination is also finite. The triangular Discrimination measure has been employed in many machine learning problems including PAC-Bayesian learning, [53] and surrogate loss functions [54]. A. Preliminaries In our characterization of generalization error upper bounds in this section, we will use the information measures between two distributions P X and P X ′ on a common measurable space X in Table IV. The new measure α-KL-Pythagorean divergence can be characterized by a convex combination of KL-divergences as defined in Definition (149). Chi-square divergence and Triangular Discrimination 3 are other divergences that are related to our discussion on the ADM (See Sections F-C). The information measures based on these divergences are summarized in Table V. α-KL-Pythagorean divergence (new measure) Kα(P X P X ′ ) inf Q X ∈P(X ) αD KL (P X Q X ) + (1 − α)D KL (Q X P X ′ ) Chi-square divergence [56] χ 2 (P X P X ′ ) X (P X (x)−P X ′ (x)) 2 P X ′ (x) dx Triangular Discrimination [34] ∆(P X P X ′ ) X (P X (x)−P X ′ (x)) 2 P X (x)+P X ′ (x) dxI ∆ (X; X ′ ) ∆(P X,X ′ P X ⊗ P X ′ ) α-KL-Pythagorean information I Kα (X; X ′ ) Kα(P X,X ′ P X ⊗ P X ′ ) B. α-KL-Pythagorean Based Upper bound We now provide a different generalization bound, relying on KL divergence terms D KL ( P W,Zi P W ⊗µ) and D KL (P W,Zi P W,Zi ), whereP W,Zi is an auxiliary distribution, that can be ultimately expressed in terms of a new divergence measure that we will also discuss in the sequel. Corollary 33: Considering the same assumptions in Theorem 3 and Proposition 15, the following expected generalization error upper bounds hold: gen(P W |S , µ) ≤ 1 n n i=1 ψ ⋆−1 + (B i ) + φ ⋆−1 + (D i ) ,(144)− gen(P W |S , µ) ≤ 1 n n i=1 ψ ⋆−1 − (B i ) + φ ⋆−1 − (D i ) ,(145) where B i = D KL (P W,Zi P W,Zi ), and D i = D KL ( P W,Zi P W ⊗ µ). Proof: The proof is straightforward and similar to Theorem 3. In Section III-A, we provided generalized α-Jensen-Shannon information based upper bound using the following convex combination of two KL divergences terms, D α JS (P Q) = min R αD KL (P R) + (1 − α)D KL (Q R),(146) In turn, in Section III-B, the α-Rényi based upper bound is derived by using the following convex combination of two KL divergences terms, (1 − α)D α (P Q) = (147) min R αD KL (R P ) + (1 − α)D KL (R Q). Now, we define a new divergence measure inspired by the following convex combination of KL divergence terms: αD KL (P R) + (1 − α)D KL (R Q).(148) Definition 34: Assume P ≪ Q. Then, the α-KL-Pythagorean divergence is defined as follows: K α (P Q) inf R∈P(X ) αD KL (P R) + (1 − α)D KL (R Q).(149) We will be assuming that K α (P Q) is well-defined i.e. there is a unique minimizer R ∈ P(X ), where P(X ) is the set of all probability distributions over space X . Note that for many sample spaces X (E.g. compact metric space), P(X ) is a closed set under some topology. Therefore, the optimization problem in (149)) is convex, exhibiting a unique solution. We provide the main properties of α-KL-Pythagorean divergence in the following Theorem. Theorem 35: The α-KL-Pythagorean divergence has the following properties: (i). Non-negativity: K α (P Q) ≥ 0 and equality holds if and only if P = Q. (ii). Joint convexity: K α (P Q) is a jointly convex function of (P, Q). (iii). Monotonicity: K α (P XY Q XY ) ≥ K α (P X Q X ). (iv). Data processing: Consider a channel that produces Y given X based on the conditional law P Y |X . Let P Y (resp. Q Y ) denote the distribution of Y when X is distributed as P X (resp. Q X ), then K α (P Y Q Y ) ≤ K α (P X Q X ). (v). Supermodularity: Let P X P Y |X −−−→ P Y and Q XY = Q X Q Y . Then K α (P XY Q XY ) ≥ K α (P X Q X ) + K α (P Y Q Y ). Proof: Proof of (i): If α = 0 or 1, the claim is obvious. Otherwise, non-negativity of K α follows from non-negativity of KL-divergence. K α (P Q) = 0 iff there exist distribution R such that D KL (P R) = 0 and D KL (R Q) = 0. The only way is P = Q. Proof of (ii): Let define K α (P, Q, R) := αD KL (P R) + (1 − α)D KL (R Q). Consider (P 1 , Q 1 , R 1 ) and (P 2 , Q 2 , R 2 ) are two points in P(X ) × P(X ) × P(X ). K α (αP 1 + (1 − α)P 2 , αQ 1 + (1 − α)Q 2 , αR 1 + (1 − α)R 2 ) = αD KL (βP 1 + (1 − β)P 2 αR 1 + (1 − α)R 2 ) (150) + (1 − α)D KL (αR 1 + (1 − α)R 2 βQ 1 + (1 − β)Q 2 ) (a) ≤ β αD KL (P 1 R 1 ) + (1 − α)D KL (R 1 Q 1 ) (151) + (1 − β) [αD KL (P 2 R 2 ) + (1 − α)D KL (R 2 P 2 )] = βK α (P 1 , Q 1 , R 1 ) + (1 − β)K α (P 2 , Q 2 , R 2 ),(152) where (a) comes from jointly convexity of KL-divergence. So K α (P, Q, R) is a jointly convex function of (P, Q, R). We know from the elementary convex analysis that min a∈A φ(a, b) is convex when φ(a, b) is jointly convex function of (a, b) and A is a convex set. So K α (P Q) = inf R∈P(X ) K α (P, Q, R) is a convex function of (P, Q). Proof of (iii): K α (P XY Q XY ) = (153) inf RXY ∈P(X ×Y) αD KL (P XY R XY ) + (1 − α)D KL (R XY Q XY ) (a) ≥ inf RXY ∈P(X ×Y) αD KL (P X R X ) + (1 − α)D KL (R X Q X ) (154) = K α (P X Q X ), where (a) comes from monotonicity of KL divergence D KL (P XY Q XY ) ≥ D KL (P X Q X ). Proof of (iv): K α (P Y Q Y ) (155) (a) ≤ K α (P XY Q XY ) = K α (P X P Y |X Q X P Y |X ) = inf RXY ∈P(X ×Y) αD KL (P X P Y |X R XY )(156)+ (1 − α)D KL (R XY Q X P Y |X ) (b) ≤ inf RX ∈P(X ) αD KL (P X P Y |X R X P Y |X )(157)+ (1 − α)D KL (R X P Y |X Q X P Y |X ) = K α (P X Q X ),(158) where (a) comes from Part (iii) and (b) is derived by considering a smaller subset of the joint distribution like R XY = R X P Y |X in the general minimization. The last equality is derived from chain rule for KL divergence. Proof of (v): K α (P XY Q XY ) = (159) inf RXY ∈P(X ×Y) αD KL (P XY R XY ) + (1 − α)D KL (R XY Q XY ) (a) ≥ inf RXY ∈P(X ×Y) α[D KL (P X R X ) + D KL (P Y R Y )](160)+ (1 − α)[D KL (R X Q X ) + D KL (R Y P Y )] (b) ≥ inf RX ∈P(X ) αD KL (P X R X ) + (1 − α)D KL (R X Q X ) (161) + inf RY ∈P(Y) αD KL (P Y R Y ) + (1 − α)D KL (R Y Q Y ) = K α (P X Q X ) + K α (P Y Q Y )(162) where (a) comes from Supermodularity of KL-divergence as follows: D KL (P XY Q XY ) ≥ D KL (P X Q X ) + D KL (P Y Q Y ). (b) comes from the fact that min a f (a) + g(a) ≥ min a f (a) + min a g(a). It is worthwhile to mention that by considering α = 1 2 , the K α divergence reduces to Bregman symmetrized centroid which is studied in [57], [58]. Recalling Lemmas 2 and 3, we can offer a similar result as follows: αD KL (P Q) + ln dQ(x) dR(x) [α · dP (x) − (1 − α) · dR(x)] (163) = αD KL (P R) + (1 − α)D KL (R Q). Actually, there is no closed-form solution R * for K α but if we write the Lagrange function for this minimization problem by considering the constraint X dR * (x) = 1, then R * can be characterized by Lambert function: Lemma 6: Assume that distributions P , Q have non-degenerate densities. Then dR * (x) = α 1−α dP (x) W α 1−α dP (x) dQ(x) e λ 1−α +1 ,(164) where W (·) is Lambert function and λ should be determined in such a way that X dR * (x) = 1. Proof: Introduce one slack variable λ which is Lagrangian's multiplier for E Q [dR(X)/dQ(X)] = 1 and write Lagrange function as follows: L(R, λ) = αD KL (P R) + (1 − α)D KL (R Q) + λ E Q [ dR dQ ] − 1 .(165) So if there exists a solution (R * , λ * ) then it should satisfy KKT condition: ∂ ∂(dR(x)) L(R * , λ * ) = 0, ∂ ∂(dR(x)) L(R * , λ * ) = − α dP (x) dR * (x) + (1 − α) log dR * (x) dQ(x) + 1 − α + λ * = 0(166) Taking expectation on R * from (166), we get λ * = (α − 1)D KL (R * Q) + 2α − 1. From Equation (166), we get α 1 − α e 1+ λ 1−α +1 dP (x) dQ(x) = α 1 − α dP (x) dR * (x) e α 1−α dP (x) dR * (x) .(167) Using Lambert function W (z) which is the inverse of Z(w) = we w , we have dR * (x) = α 1−α dP (x) W α 1−α dP (x) dQ(x) e λ * 1−α +1 .(168) It can be shown that by choosing R = P or R = Q, we have K α (P Q) ≤ min{α, 1 − α}D KL (P Q). We now offer new upper bound in the expected generalization error using the new information measure. + (1 − α)D KL ( P W,Zi P W P Zi ). Then |gen(P W |S , µ)| ≤ (170) 1 n n i=1 2(ασ 2 + (1 − α)γ 2 K ) I Kα (W ; Z i ) α(1 − α) . Proof: gen(µ, P W |S ) = E PW PS [L S (W )] − E PW,S [L S (W )] = 1 n n i=1 E PW PZ i [ℓ(W, Z i )] − E PW Z i [ℓ(W, Z i )] = 1 n n i=1 E PW PZ i [ℓ(W, Z i )] − EP α W Z i [ℓ(W, Z i )] + EP α W Z i [ℓ(W, Z i )] − E PW Z i [ℓ(W, Z i )].(171) Using the assumption that the loss function ℓ(w, z i ) is γ 2 -sub-Gaussian under distributionP α W,Zi and Donsker-Varadhan representation by considering function ℓ(w, z i ) we have: λ E PW Z i [ℓ(w, z i )] − EP α W,Z i [ℓ(w, z i )] ≤ (172) D KL (P W Zi P α W,Zi ) + λ 2 γ 2 2 , ∀λ ∈ R. Using the assumption ℓ(w, Z) is σ 2 -sub-Gaussian under P Zi for all w ∈ W, and again Donsker-Varadhan representation by considering function ℓ(w, z i ) we have: λ ′ EP α W,Z i [ℓ(w, z i )] − E PW PZ i [ℓ(w, z i )] ≤ (173) D KL ( P W,Zi P W P Zi ) + λ ′ 2 σ 2 2 . ∀λ ′ ∈ R Now if we consider λ < 0, then we choose λ ′ = α 1−α λ. Sum up two Inequalities (172) and (173). E PW PZ i [ℓ(w, Z i )] − E PW Z i [ℓ(w, z i )] ≤ (174) αD KL (P W,Zi P α W,Zi ) + (1 − α)D KL (P α W,Zi P W P Zi ) α|λ| + |λ|γ 2 2 + |λ| α 1−α σ 2 2 ∀λ ∈ R − Similar to (174), we can write the equation for λ > 0 and in conclusion we get: E PW PZ i [ℓ(w, Z i )] − E PW Z i [ℓ(w, z i )] ≤ 2(ασ 2 + (1 − α)γ 2 ) (αC i + (1 − α)D i ) α(1 − α) 1 n n i=1 2 α(1 − α) (ασ 2 + (1 − α)γ 2 )I Kα (W ; Z i )(175) where C i = D KL (P W,Zi P α W,Zi ) and D i = D KL (P α W,Zi P W × P Zi ) . In [59], for the class of algorithms with bounded input/output mutual f -information, they give a novel connection between the f -distortion function and the generalization error. Moreover, this leads to a new upper bound on the generalization error using the f -distortion function that strictly improves over the previous bounds in [7], [8]. The properties of the f -distortion function defined with super-modular f -divergences help its evaluation when the number of data samples is large. Similar to the upper bounds based on mutual f -information, the sharpest possible upper bound on the generalization error given an upper bound R on I Kα (S; A(S)) is U IK α 1 (R) sup P W |S : IK α (S;W )≤R E[L µ (W ) − L S (W )].(176) Similarly, assuming I Kα (Z i ; W ) ≤ R i for 1 ≤ i ≤ n, the sharpest bound is gen (µ, A) ≤ 1 n n i=1 U IK α 2 (R i ),(177) where U IK α 2 (R) sup P W |Z : IK α (Z;W )≤R E[L µ (W ) − ℓ(W, Z)].(178) The bound U IK α 2 (d) ≤ 2 α(1 − α) (ασ 2 + (1 − α)γ 2 K ) min{α, 1 − α} log(k) n ,(185) where (a) follows from Jensen inequality and (b) follows from the Supermodularity of K α in Part (v) of Theorem 35. (c) derives from the fact that K α (P Q) ≤ min{α, 1 − α}D KL (P Q). (d) comes from the fact that I(W ; S) ≤ H(W ) ≤ log(k). We acknowledge that it may be difficult to verify our sub-Gaussian assumption underlying Proposition 36 in view of the fact that our distribution is characterized indirectly via (164). To bypass this issue, we use another distribution instead of the minimizer in the definition of the α-KL-Pythagorean in Theorem 40. Let us definê R = arg min R∈Γ D KL (R Q),(186) where Γ is a closed convex set in P(X ). Now, considering the α-KL-Pythagorean divergence definition (149), we have: K α (P Q) ≤ αD KL (P R ) + (1 − α)D KL (R Q).(187) In our scenario of upper bounding generalization error, we can define P η W,Zi = arg min µW,Z i ∈Γ ǫ i D KL (µ W,Zi P W P Zi ),(188) where Γ ǫ i := {µ W,Zi : |E µW,Z i [ℓ(W, Z i ) − E PZ i [ℓ(W, Z i )]]| ≥ ǫ}. Claim 39: If P W,Zi ∈ Γ ǫ i i.e. ǫ is sufficiently small such that it is a lower bound for |gen(P W |Zi , µ)|, then αD KL (P W,Zi P η W,Zi ) + (1 − α)D KL (P η W,Zi P W P Zi ) ≤ max{α, 1 − α}I(W ; Z i ).(189) Proof of Claim 39: We use the following lemma to prove Claim 39. Lemma 7 (Pythagorean inequality for KL [36]): Assume R * ∈ P(X ) exists such that D KL (R * Q) = min R∈Γ D KL (R Q). Then for every P ∈ Γ we have D KL (P Q) ≥ D KL (P R * ) + D KL (R * Q).(190) Choose Γ = Γ ǫ i and Q = P W P Zi and P = P W,Zi . The claim is established. To prove Lemma 7, choose R θ := (1 − θ)R * + θP for θ ∈ (0, 1). Since R * is a minimizer of D KL (R Q), we have 0 ≤ d dθ D KL (R θ Q)| θ=0 (191) = D KL (P Q) − D KL (P R * ) − D KL (R * Q). We provide an upper bound using P η W,Zi as auxiliary distribution in (148). Theorem 40: Assume that the loss function is σ-sub-Gaussian under P W ⊗ µ and γ η -sub-Gaussian under some P η W,Zi for each i = 1, . . . , n dP η W,Zi (w, z) = e η(ℓ(w,zi)−Lµ(w)) dP W (w)dP Zi (z) E PW PZ i e η(ℓ(W,Zi)−Lµ(W )) , where η ≥ 0 is chosen in such way that E P η W,Z i [ℓ(W, Z i ) − E PW PZ i [ℓ(W, Z i )]] = ǫ. Then |gen(P W |S , µ)| ≤ 1 n n i=1 2(ασ 2 + (1 − α)γ 2 η ) α(1 − α) K ⋆ α ,(192)where K ⋆ α = αD KL (P W,Zi P η W,Zi ) + (1 − α)Λ * ℓ(W ,= 1 n n i=1 E PW PZ i [ℓ(W, Z i )] − E PW,Z i [ℓ(W, Z i )].(196) Using the assumption that the loss function ℓ(w, z i ) is γ 2 η -sub-Gaussian under distribution P η W,Zi and Donsker-Varadhan representation by considering function ℓ(w, z i ) we have: λ E PW Z i [ℓ(w, z i )] − E P η W,Z i [ℓ(w, z i )] ≤(197) D KL (P W Zi P η W,Zi ) + λ 2 γ 2 η 2 , ∀λ ∈ R. Using the assumption that ℓ(w, Z) is σ 2 -sub-Gaussian under P Zi for all w ∈ W, and again Donsker-Varadhan representation by considering function ℓ(w, z i ) we have: λ ′ E P η W,Z i [ℓ(w, z i )] − E PW PZ i [ℓ(w, z i )] ≤ (198) D KL (P η W,Zi P W P Zi ) + λ ′ 2 σ 2 2 , ∀λ ′ ∈ R. Now if we consider λ < 0, then we choose λ ′ = α 1−α λ. Sum up two Inequalities (197) and (198). E PW PZ i [ℓ(w, Z i )] − E PW Z i [ℓ(w, z i )] ≤ (199) αD KL (P W,Zi P η W,Zi ) + (1 − α)D KL (P η W,Zi P W P Zi ) αλ + λγ 2 η 2 + λ α 1−α σ 2 2 ∀λ ∈ R − . For λ < 0, we have the similar equation to (199). Minimizing on λ and using Equation (193), we get |gen(P W |S , µ)| ≤ 1 n n i=1 2(ασ 2 + (1 − α)γ 2 η ) α(1 − α) K ⋆ α ,(200) where K ⋆ α = αD KL (P W,Zi P η W,Zi ) + (1 − α)Λ * ℓ(W ,Zi) (ǫ). C. Auxiliary Distribution Based Generalization Error Upper Bounds Using Chi-square divergence + 1) Yes (4 √ 2σ ∆ ) In this section we apply the ADM to the upper bounds based on the Chi-square divergence. Theorem 41: Let the loss function be σ-sub-Gaussian under the distribution P W,Zi ∀i = 1, · · · , n. Then it holds that, |gen(P W |S , µ)| ≤ 2σ n n i=1 ( √ χ A + √ χ B ) ,(201) where χ A = χ 2 (P W ⊗ µ P W,Zi ) + 1 and χ B = χ 2 (P W,Zi P W,Zi ) + 1. Proof: In a similar approach to [60], we have: |E PW,Z i [ℓ(W, Z i )] − E PW,Z i [ℓ(W, Z i )]| (202) = |E PW,Z i [ℓ(W, Z i ) − E PW,Z i [ℓ(W, Z i )]]| ≤ E PW,Z i [|ℓ(W, Z i ) − E PW,Z i [ℓ(W, Z i )]| 2 ](203) × χ 2 (P W,Zi P W,Zi ) + 1 ≤ 2σ χ 2 (P W,Zi P W,Zi ) + 1, and |E PW ⊗µ [ℓ(W, Z i )] − E PW,Z i [ℓ(W, Z i )]| (205) = |E PW ⊗µ [ℓ(W, Z i ) − E PW,Z i [ℓ(W, Z i )]]| ≤ E PW,Z i [|ℓ(W, Z i ) − E PW,Z i [ℓ(W, Z i )]| 2 ](206) × χ 2 (P W ⊗ µ P W,Zi ) + 1 ≤ 2σ χ 2 (P W ⊗ µ P W,Zi ) + 1, where the second inequality is based on Hölder inequality, [11] and [60], and last inequality is valid in view of the fact that the loss function is σ-sub-Gaussian under P W,Zi and [61, Lemma 1.4]. Finally, we can immediately bound the generalization error by leveraging (204) χ 2 (P W ⊗ µ P W,Zi ) + 1 + χ 2 (P W,Zi P W,Zi ) + 1 . Remark 42: The ADM can be applied to generalization error upper bound based on the power information measure [60], in a similar way to Theorem 41. It is shown, [43], that the following holds between two distributions P X and P X ′ , 1 2 ∆(P X P X ′ ) = χ 2 (P X P X + P X ′ 2 ) = χ 2 (P X ′ P X + P X ′ 2 ). Considering the upper bound in Theorem 41, the following upper bound is derived by assuming the average distribution, PW,Z i +Pw⊗µ 2 , as auxiliary distribution and using (208). Corollary 43: Consider the loss function is σ ∆ -sub-Gaussian under the distribution PW ⊗µ+PW,Z i 2 , ∀i = 1, . . . , n . Then it holds that: |gen(P W |S , µ)| ≤ 4σ ∆ n n i=1 I ∆ (W ; Z i ) 2 + 1.(209) Proof: This corollary follows immediately by setting P W,Zi = PW,Z i +Pw⊗µ 2 in Theorem 41 and using Jensen inequality. Using that the triangular discrimination measure is finite, i.e., I ∆ (W ; Z i ) ≤ 2, we can show that the upper bound in Corollary 43 is finite. Proposition 44: Consider the assumptions in Corollary 43. Then, it holds that: |gen(P W |S , µ)| ≤ 4 √ 2σ ∆ .(210) Proof: Let's prove the constant upper bound for triangular discrimination based on chi-square divergence, 1 2 ∆(P x P X ′ ) = χ 2 (P x PX +P X ′ 2 ) = χ 2 (P X ′ PX +P X ′ 2 ) ≤ 1. Using chi-square definition we have: χ 2 (P x P X + P X ′ 2 ) = X 2 P X (x) P X (x) + P X ′ (x) P X (x)dx − 1 (211) ≤ X 2P X (x)dx − 1 = 1.(212) Now, we have: I ∆ (W ; Z i ) ≤ 2.(213) Using (213) in Corollary 43, completes the proof. The comparison of all proposed upper bound is shown in Table VI. We compare the upper bounds based on generalized α-Jensen-Shannon information for α = 1/2 with the upper bounds based on Triangular Discrimination information, Corollary 43 in the following Proposition. Proposition 45: Consider the assumptions in Proposition 13 and Corollary 43. Then it follows that the generalized α-Jensen-Shannon upper bound for α = 1/2 given by: |gen(P W |S , µ)| ≤ 2 n n i=1 σ 2 (α) I JS (W ; Z i ),(214) is tighter than the triangular discrimination upper bound given by: |gen(P W |S , µ)| ≤ 2 n n i=1 4σ 2 ∆ I ∆ (W ; Z i ) 2 + 1 .(215) Proof: As we have σ ∆ = σ (α) for α = 1/2, then we need to prove that I JS (W ; Z i ) ≤ I ∆ (W ; Z i ) + 2. It is shown in [34], that we have I JS (W ; Z i ) ≤ log (2) 2 I ∆ (W ; Z i ). It completes the proof. Then, the bound in Proposition 8 shows a decay rate of O( 1 √ n ). The value of this new proposed bound presented in Proposition 8 in relation to existing bounds can also be further appreciated by offering two additional results. Proposition 13: (Proved in Appendix B-F) Consider the assumptions in Proposition 8. Then, it follows that: Fig. 1 .Fig. 2 . 12True generalization error, Generalized α-Jensen-Shannon based bound for α = 0.25, 0.5, 0.75, and Mutual Information based bound. True generalization error, α-Rényi based bound for α = 0.25, 0.5, 0.75, and Mutual Information based bound. = I α JS (W ; Z i ) + D KL (P (α)W,Zi P W,Zi ). in Theorem 7, completes the proof. [ℓ(W, Z i ) − L µ (W )] and then η is the maximizer in the last equation.gen(µ, P W |S ) = E PW PS [L S (W )] − E PW,S [L S (W )] I 1−α) ) α-Rényi information (0 ≤ α < 1) (Proposition 17) P W ⊗ µ and P W,Z i , ∀i = 1, . . . ασ 2 + (1 − α)γ 2 ) I α R (W ;Z i ) α No α-KL-Pythagorean information (Theorem 36) P W ⊗ µ and P Kα W,Z i , ∀i = 1, . . . 1−α) (ασ 2 + (1 − α)γ 2 K )I Kα (W ; Z i ) No Triangular Discrimination information (Corollary 43) P W ⊗µ+P W,Z i 2 , ∀i = 1, . . . ∆ (W ;Z i ) 2 |E|E PW ⊗µ [ℓ(W, Z i )] − E PW,Z i [ℓ(W, Z i )PW ⊗µ [ℓ(W, Z i )] − E PW,Z i [ℓ(W, Z i )]|+ |E PW,Z i [ℓ(W, Z i )] − E PW,Z i [ℓ(W, Z i ) TABLE I DIVERGENCE IMEASURES DEFINITIONSDivergence Measure Definition KL divergence [35] TABLE II INFORMATION IIMEASURES DEFINITIONSInformation Measure Definition TABLE III GENERALIZATION IIIERROR UPPER BOUNDSUpper Bound Measure Assumptions Bound Is finite? Mutual information ( [8]) P W ⊗ µ 1 n n i=1 TABLE IV DIVERGENCE IVMEASURES DEFINITIONS Divergence Measure Definition TABLE V VINFORMATION MEASURES DEFINITIONS Information Measure Definition Triangular Discrimination information Proposition 36: Assume that the loss function is σ-sub-Gaussian under P W ⊗ µ and γ K -sub-Gaussian under P Kα W,Zi for each i = 1, . . . , n such thatP Kα W,Zi = arg min PW,Z i αD KL (P W,Zi P W,Zi ) (169) Zi) (ǫ) and Λ * ℓ(W ,Zi) (See Definition 2) is the convex conjugate of Λ ℓ(W ,Zi) (CGF is defined in 1).Proof: Consider the following auxiliary distributionsdP η W,Zi (w, z i ) = e η(ℓ(w,zi)−Lµ(w)) dP W (w)dP Zi (z) E PW PZ i e η(ℓ(W,Zi)−Lµ(W )) for i ∈ [1 : n] defined on W × Z. PW PZ i e η(ℓ(W,Zi)−Lµ(W )) − log E PW PZ i e η(ℓ(W,Zi)−Lµ(W )) η ′ ǫ − log E PW PZ i e η ′ (ℓ(W,Zi)−Lµ(W ))(195)= Λ * ℓ(W ,Zi) (ǫ).D KL (P η W,Zi P W P Zi ) (193) = E P η W,Z i log e η(ℓ(W,Zi)−Lµ(W )) E = η d dη log E PW PZ i e η(ℓ(W,Zi)−Lµ(W )) (194) = sup η ′ where ǫ E P η W,Z i TABLE VI GENERALIZATION VIERROR UPPER BOUNDS 2σ 2 I(W ; Z i ) NoLautum information ([46]) P W,Z i , ∀i = 1, . . . , nUpper Bound Measure Assumptions Bound Is finite? Mutual information ( [8]) P W ⊗ µ 1 n n i=1 1 n n i=1 µ ′ is absolutely continuous with respect to µ and this is written as µ ′ ≪ µ, when µ ′ is a distribution on Z with the support which is a subset of the µ support. a.k.a. Vincze-Le Cam distance [55, p.47] ACKNOWLEDGMENTThe authors would like to thank Patrick Thiran for several helpful discussions. Gholamali Aminian is supported in part by the Royal Society Newton International Fellowship, grant no. NIF\R1 \192656, the UKRI Prosperity Partnership Scheme (FAIR) under the EPSRC Grant EP/V056883/1, and the Alan Turing Institute. Part of this work was completed while Saeed Masiha was studying at Sharif university of technology.(R) is easier to compute than U IK α 1 (R) because the optimization problem in (178) is for a single symbol Z whereas the optimization problem in (176) is for a sequence S of n symbols.Theorem 37:Proof: Using supermodularity of K α , we haveWe also havewhere (181) follows from the definition of UK α (P W,S P W P S ) n (c)(ασ 2 + (1 − α)γ 2 K ) min{α, 1 − α}I(W ; S) n Jensen-shannon information based characterization of the generalization error of learning algorithms. G Aminian, L Toni, M R D Rodrigues, 2020 IEEE Information Theory Workshop (ITW). G. Aminian, L. Toni, and M. R. D. Rodrigues, "Jensen-shannon information based characterization of the generalization error of learning algorithms," in 2020 IEEE Information Theory Workshop (ITW), pp. 1-5, 2021. 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{'abstract': 'Generalization error boundaries are essential for comprehending how well machine learning models work. In this work, we suggest a creative method, i.e., the Auxiliary Distribution Method, that derives new upper bounds on generalization errors that are appropriate for supervised learning scenarios. We show that our general upper bounds can be specialized under some conditions to new bounds involving the generalized α-Jensen-Shannon, α-Rényi (0 < α < 1) information between random variable modeling the set of training samples and another random variable modeling the set of hypotheses. Our upper bounds based on generalized α-Jensen-Shannon information are also finite. Additionally, we demonstrate how our auxiliary distribution method can be used to derive the upper bounds on generalization error under the distribution mismatch scenario in supervised learning algorithms, where the distributional mismatch is modeled as α-Jensen-Shannon or α-Rényi (0 < α < 1) between the distribution of test and training data samples. We also outline the circumstances in which our proposed upper bounds might be tighter than other earlier upper bounds.', 'arxivid': '2210.00483', 'author': ['Member, IEEE, SaeedAminian ⋆ Gholamali ', 'Member, IEEEMasiha ⋆ ', 'Senior Member, IEEELaura Toni ', 'Senior Member, IEEEMiguel R D Rodrigues '], 'authoraffiliation': [], 'corpusid': 252683733, 'doi': '10.48550/arxiv.2210.00483', 'github_urls': [], 'n_tokens_mistral': 38834, 'n_tokens_neox': 33783, 'n_words': 18444, 'pdfsha': '6fc90a854988c34eaa7ae5c14a561f5281e5c17a', 'pdfurls': ['https://export.arxiv.org/pdf/2210.00483v1.pdf'], 'title': ['Learning Algorithm Generalization Error Bounds via Auxiliary Distributions', 'Learning Algorithm Generalization Error Bounds via Auxiliary Distributions'], 'venue': []}

arxiv

Space-Time Description of Scalar Particle Creation by a Homogeneous Isotropic Gravitational Field 26 Nov 2008 Yu V Pavlov Institute of Mechanical Engineering A. Friedmann Laboratory for Theoretical Physics St. Petersburg Russia Russian Acad. Sci 61 Bolshoy pr199178St. PetersburgRussia Space-Time Description of Scalar Particle Creation by a Homogeneous Isotropic Gravitational Field 26 Nov 2008PACS number: 0462+v, 0370+k We give the generalization of the method of the space-time description of particle creation by a gravitational field for a scalar field with nonconformal coupling to the curvature. The space-time correlation function is obtained for a created pair of the quasi-particles, corresponding to a diagonal form of the instantaneous Hamiltonian. The case of an adiabatic change of the metric of homogeneous isotropic space is analyzed. We show that the created pairs of quasi-particles in de Sitter space should be interpreted as pairs of virtual particles. PACS number: 04.62.+v, 03.70.+k H(η) = dµ(J) Ω( INTRODUCTION Quantum field theory in curved space-time is nowadays a sufficiently deeply elaborated area of theoretical physics (see the monographs [1,2]) with important applications in cosmology and astrophysics. In particular, creation of particles with GUT-scale masses by the gravitational field of the early Universe may be used for an explanation of the observed visible and dark matter density [3]. In the description of particle creation by the gravitational field, a widely used method is that of instantaneous Hamiltonian diagonalization [1] suggested by A. A. Grib and S. G. Mamayev [4]. A detailed study of the created pair formation process was performed in [5] for the case of a scalar field conformally coupled to the curvature. The method of a spacetime correlation function suggested in [5] made it possible to distinguish real created particles from virtual ones, to reveal the role of horizons in particle creation etc. In quantum field theory in curved space-time, one frequently considers scalar field nonconformally coupled to gravity, in particular, the minimally coupled one. In such cases, the contributions related to nonconformal coupling may be dominant both in the particle creation effect [6] and in the vacuum averages values of the stress-energy tensor (see, e.g., [7]). Such a coupling with the curvature also takes place in the case of massive vector mesons (the longitudinal components [1]). Conformal invariance is lacking * E-mail: yuri.pavlov@mail.ru in the graviton equations as well [8]. Therefore, it appears to be necessary to generalize the method of studying the particle creation process suggested in [5] to the nonconformal case. Moreover, some authors [9] consider the absence of such a generalization as an argument in favor of choosing only the conformal coupling in the wave equation for a scalar field. The present paper suggests a generalization of the space-time description of particle creation [5] to the case of nonconformal coupling. In Section 2, we perform quantization of a general-type nonconformally coupled scalar field in homogeneous isotropic space. In Section 3, we carry out diagonalization of the generalized Hamiltonian built in [10], which, in the nonconformal case, allows one to solve the wellknown problem of an infinite density of quasiparticles created [11]. In Section 4, we build a space-time correlation function for quasiparticles corresponding to the diagonal form of the instantaneous Hamiltonian and study the case of an adiabatically changing metric. In Section 5, we consider particle creation and the space-time correlation function in de Sitter space. The conclusion briefly sums up the results of the paper. We use the system of units in whichh = c = 1. The signs of the curvature tensor and the Ricci tensor are chosen in such a way that R ik = R l ilk , R i jkl = ∂ l Γ i jk − ∂ k Γ i jl + Γ i nl Γ n jk − Γ i nk Γ n jl , where Γ i jk are the Christoffel symbols. SCALAR FIELD IN CURVED SPACE Consider a complex scalar field ϕ(x) with mass m, the Lagrangian (1) and the corresponding equation of motion L(x) = |g| g ik ∂ i ϕ * ∂ k ϕ − (m 2 + V g ) ϕ * ϕ ,(∇ i ∇ i + V g + m 2 ) ϕ(x) = 0 ,(2) where ∇ i are covariant derivatives in N -dimensional space-time with the metric g ik , g = det(g ik ), V g is a function of invariant combinations of the metric tensor g ik and its partial derivatives. Eq. (2) is conformally invariant if m = 0 and V g = ξ c R, where R is the scalar curvature and ξ c = (N − 2)/ [ 4 (N − 1)] (conformal coupling). The case V g = 0 corresponds to minimal coupling. An arbitrary V g leads to the advent of third-and higherorder derivatives of the metric in the metric stressenergy tensor of the scalar field and consequently in the Einstein equations. It is well known that additional terms with higherorder derivatives appearing in equations lead to radical changes in the theory even if the coefficients of these terms are small. If one requires that the metric stress-energy tensor should not contain derivatives of the metric of orders higher than two, then the following function is admissible as V g : V g = ξR + ζR 2 GB ,(3) where R 2 GB def = R lmpq R lmpq − 4R lm R lm + R 2(4) (the Gauss-Bonnet coupling [12]). Let us note that for N = 4, with constant ϕ(x), the contribution to the metric stress-energy tensor from the term with R 2 GB is absent because the corresponding variation derivative vanishes [13]. But for a variable ϕ(x), a contribution from such terms could be taken into account if the constant ζ with the dimension (mass) −2 is nonzero. Accounting for a possible coupling between a scalar field and the Gauss-Bonnet invariant R 2 GB may play an important role in the early Universe; effects from a nonzero value of the parameter ζ in scalar field equations may appear in black-hole radiation, may affect the parameters of the so-called boson stars etc. The question of the values of the parameters ζ and ξ are ultimately related to the area of the experiment. Furthermore, without specifying V g , let us consider an N -dimensional homogeneous isotropic space-time, choosing the metric in the form ds 2 = g ik dx i dx k = a 2 (η) (dη 2 − dl 2 ) ,(5) where dl 2 = γ αβ dx α dx β is the metric of an (N − 1)dimensional space of constant curvature K = 0, ±1. The complete set of solutions to Eq. (2) in the metric (5) may be found in the form ϕ(x) =φ (x) a (N −2)/2 (η) = a −(N −2)/2 (η) g λ (η)Φ J (x) ,(6)where g ′′ λ (η) + Ω 2 (η) g λ (η) = 0 ,(7)Ω 2 (η) = (m 2 + V g − ξ c R)a 2 + λ 2 ,(8)∆ N −1 Φ J (x) = − λ 2 − N − 2 2 2 K Φ J (x) ,(9) the prime denotes a derivative with respect to the conformal time η, and J is the set of indices (quantum numbers) numbering the eigenfunctions of the Laplace-Beltrami operator ∆ N −1 in (N −1)- dimensional space. According to the Hamiltonian diagonalization method [1] (see the case of an arbitrary function V g in [14]), the functions g λ (η) should obey the following initial conditions: g ′ λ (η 0 ) = i Ω(η 0 ) g λ (η 0 ) , |g λ (η 0 )| = Ω −1/2 (η 0 ) .(10) To perform quantization, let us expand the field ϕ(x) in the complete set of solutions (6) ϕ(x) = dµ(J) ϕ (+) J a (+) J + ϕ (−) J a (−) J ,(11) where dµ(J) is a measure on the set of quantum numbers, ϕ (+) J (x) = g λ (η) Φ * J (x) √ 2 a (N −2)/2 (η) , ϕ (−) J (x) = ϕ (+) J (x) * ,(12) and require that the standard commutation relations hold for a (±) J and * a (±) J . Let us build the Hamiltonian as the canonical one for the variablesφ(x) andφ * (x), for which the equation of motion does not contain their first-order derivatives with respect to the time η [14]. Recall that the equations of motion do not change after adding a full divergence ∂J i /∂x i to the Lagrangian density L(x). Let us choose, in the coordinate system (η, x), the vector (J i ) = ( √ γcφ * φ (N − 2)/2, 0, . . . , 0 ), where γ = det(γ αβ ), c = a ′ /a. Then, using the Lagrangian density L ∆ (x) = L(x) + ∂J i /∂x i , we obtain for the momenta canonically conjugate toφ andφ * : π ≡ ∂L ∆ ∂φ ′ = √ γφ * ′ , π * ≡ ∂L ∆ ∂φ * ′ = √ γφ ′ ,(13) respectively. Integrating the Hamiltonian density h(x) =φ ′ π +φ * ′ π * − L ∆ (x) over the hypersurface Σ: η = const, we obtain the following expression for the canonical Hamiltonian: H(η) = Σ d N −1 x √ γ φ * ′φ′ + γ αβ ∂ αφ * ∂ βφ + m 2 + V g a 2 − N − 2 4 2c ′ + (N − 2)c 2 φ * φ (14) (see a justification of such a choice of the Hamiltonian in [10,14] and in Section 3). The Hamiltonian (14) may be written in terms of the operators a (±) J and * a (±) J in the following way: H(η) = dµ(J) E J (η) * a (+) J a (−) J + * a (−) J a (+) J + + F J (η) * a (+) J a (+) J + F * J (η) * a (−) J a (−) J ,(15) where E J = |g ′ λ | 2 + Ω 2 |g λ | 2 2 , F J = ϑ J 2 g ′ λ 2 + Ω 2 g 2 λ ,(16) and we have chosen such eigenfunctions Φ J (x) that, for arbitrary J, there is suchJ that Φ * J (x) = ϑ J ΦJ (x), |ϑ J | = 1 , (J = J, ϑJ = ϑ J ). Such a choice is possible due to completeness and orthonormality of the set Φ J (x). In spherical coordinates of a homogeneous isotropic space, if J = {λ, l, . . . , m}, we haveJ = {λ, l, . . . , −m}, ϑ J = (−1) m (see [1]). HAMILTONIAN DIAGONALIZATION The Hamiltonian (15) will be diagonal at the time instant η 0 with respect to the operators a J (η), connected with * a (±) J , a (±) J by time-dependent Bo- goliubov transformations:    a (−) J = α * J (η) b (−) J (η) − β J (η)ϑ J b (+) J (η) , * a (−) J = α * J (η) * b (−) J (η) − β J (η)ϑ J * b (+) J (η) ,(17) where the functions α J (η) = αJ (η) and β J (η) = βJ (η) satisfy the initial conditions |α J (η 0 )| = 1, β J (η 0 ) = 0 and the identity |α J (η)| 2 − |β J (η)| 2 = 1. Substituting (17) and the conjugate expressions to (15), one can obtain an expression for the Hamiltonian having the same form (15) but with the replacement * a (±) J , a (±) J → * b (±) J , b (±) J and E J → b E J = E J (|α J | 2 + |β J | 2 )− 2Re (F J α J β * J ϑ * J ), (18) F J → b F J = −2α J β J ϑ J E J + α 2 J F J + β 2 J ϑ 2 J F * J . (19)α J = iχ J g * ′ λ − iΩ g * λ 2 √ Ω , β J = iχ J g ′ λ − iΩ g λ 2 √ Ω ,(20) where χ J = χJ is an arbitrary complex function of time with a unit absolute value. Therefore further we will use the operators c (+) J (η) = χ J (η) b (+) J (η), c (−) J (η) = χ * J (η) b (−) J (η),(21)dc (±) J dη = i H(η), c (±) J + Ω ′ 2Ω ϑ (∓1) J c (∓) J .(23) The second term in the right-hand side of (23) is connected with re-definition of the particle notion at each time instant. An expansion of the field operatorφ(x) in the operators c (±) J (η): Consider the question of particle creation in a nonstationary metric. We suppose that the quantized scalar field is in the state |0 , annihilated by the operators a ϕ(x) = dµ(J) √ 2Ω Φ * J (x) c (+) J (η) + ΦJ (x) c (−) J (η)(24)J (η) |0 η = * c (−) J (η) |0 η = 0 .(25) The state |0 contains, at the time instant η, |β J (η)| 2 pairs of particles and antiparticles corresponding to the operators c (±) J (η) (see [1]). The density of the created particle pairs may be calculated (for the quasi-Euclidean metric with K = 0) by the formula n(η) = B N 2a N −1 ∞ 0 S λ (η) λ N −2 dλ,(26) where B N = 2 N−3 π (N−1)/2 Γ((N − 1)/2) −1 , Γ(z) is the gamma function, S λ (η) = |β λ (η)| 2 (in a homogeneous isotropic space, |β J | ≡ |β λ |). For N = 4 and K = 0, −1, for the number density of the particle pairs created, the following formula is valid (see [1]): n(η) = 1 2π 2 a 3 ∞ 0 S λ (η) λ 2 dλ,(27) For K = 1 (spherical space), the set of eigenfunctions of the Laplace-Beltrami operator ∆ N −1 is discrete, and the formula for the number density of the created particle pairs N = 4 has the form (see [1]) n(η) = 1 2π 2 a 3 ∞ λ=1 S λ (η)λ 2 .(28) Using (20) and that the function g λ (η)g * λ ′ (η) − g ′ λ (η)g * λ (η) is a first integral of the equation (7), equal to −2i according to the initial conditions (10), we obtain: S λ (η) = 1 4Ω |g ′ λ | 2 + Ω 2 |g λ | 2 − 1 2 .(29) As it was shown in [10], S λ (η) ∼ λ −6 as λ → ∞. Therefore, in four-dimensional space-time, the number density of particles created, defined by the Hamiltonian (14) diagonalization method, is finite in the nonconformal case as well. Let us note that a divergent expression for the number density of created nonconformal scalar particles, obtained with another choice of the Hamiltonian in Ref. [11], has been one of the reasons for a criticism of the Hamiltonian diagonalization method as a whole in the well-known book [2]. THE SPACE-TIME CORRELATION FUNCTION To study the space-time characteristics of the created quasiparticles, we apply the approach suggested in [5]. We use the notion of a particle's localized state introduced by Newton and Wigner [15]. By analogy with the Newton-Wigner operator for a free field (see, e.g., [16]), we introduce creation operators of a localized state of a particle and an antiparticle: * ϕ (+) 1 (η, x) = a − N−1 2 (η) dµ(J) Φ * J (x) * c (+) J (η), ϕ (+) 1 (η, x) = a − N−1 2 (η) dµ(J) Φ * J (x) c (+) J (η) . (30) By analogy with the case of a conformal scalar field considered in [1], we introduce the operator of particle number in the volume V N V = V d N −1 x (N−1) g * ϕ (+) 1 (η, x) ϕ (−) 1 (η, x),(31) where (N−1) g = det (N−1) g αβ , and (N−1) g αβ = a 2 (η)γ αβ is the induce metric tensor on the hypersurface η = const. Using (17), (21), (30) and the properties of the eigenfunctions Φ J (x) (see, e.g., [1], § 9.1), it can be shown that the expression for the number density of the particles created n = 0| N V |0 / V(32) obtained with the aid of (31), reproduces Eqs. (26)-(28). As in [5], we consider as a characteristic of the spatial distribution of the quasiparticle pairs created, the matrix element R 0 (η, x, x ′ ) = 0 η | ϕ (−) 1 (η, x) * ϕ (−) 1 (η, x ′ ) |0 0 η | 0 ,(33) which has the meaning of the probability amplitude that a quasiparticle created is located at the point x at the time instant η while the antiquasiparticle is at the point x ′ . Using (17), (20), (21) and (30), we obtain: R 0 (η, x, x ′ ) = 1 a N −1 (η) dµ(J)Φ J (x)Φ * J (x ′ )P λ (η),(34) where P λ (η) = (iΩg λ − g ′ λ )/(iΩg λ + g ′ λ ) .(35) The function P λ (η) satisfies the following equation and initial condition: P ′ λ + 2iΩP λ + Ω ′ 2Ω (P 2 λ − 1) = 0 , P λ (η 0 ) = 0 . (36) Consider the case that the metric is changing adiabatically: 1 Ω Ω ′ Ω 2 ′ ≪ Ω ′ Ω 2 ≪ 1 .(37) Furthermore, we denote M = √ Ω 2 − λ 2 /a. In the general case, M = M (η). For a conformally coupled scalar field, M = m. If M = const, the conditions (37) hold ifḣ(t)/M 2 ≪ h/M ≪ 1, where h(t) =ȧ(t)/a is the Hubble parameter. If the conditions (37) and a ′ (η 0 ) = 0 are satisfied, an approximate solution to (36) has the form P J (η) ≈ −iΩ ′ /(4Ω 2 ) .(38) Let us find an expression for the space-time correlation function (33) in the approximation considered in the quasi-Euclidean metric (i.e., ds 2 = a 2 (η)(dη 2 − dx α dx α )). The eigenfunctions of the Laplace operator ∆ N −1 in the coordinates x α are Φ J (x) = (2π) (1−N )/2 exp(−iλ α x α ) ,(39) where −∞ < λ a < +∞, α λ 2 α = λ 2 . Consequently, J (λ=const) Φ * J (x)Φ J (x ′ ) = λ 2π N −1 2 J (N −3)/2 (λρ) ρ (N −3)/2 , (40) where = d → λ δ(| → λ | − λ), ρ = |x − x ′ |, and J ν (z) are Bessel functions. Substituting (38), (40) into (34) and performing integration, we obtain R 0 (η, x, x ′ ) = −i(M a) 2 ′ 16π 2 a 3 M 2πr N −4 2 K N −4 2 (M r),(41) where r = ρa and K ν (z) are MacDonald's functions. If M = const, which is the case for a conformally coupled scalar field and for an arbitrary coupled field in de Sitter space, from (41) we obtain R 0 (t, x, x ′ ) = −iM 2 h(t) 8π 2 M 2πr N −4 2 K N −4 2 (M r). (42) Let us further consider spatial sections with K = ±1. For N = 4, the space-time metric may be written in the form dl 2 = dχ 2 + f 2 (χ) dϑ 2 + sin 2 ϑ dϕ 2 ,(43) where f (χ) = sinh(χ), χ, sin(χ) for K = −1, 0, +1, respectively. Meanwhile, J (λ=const) Φ * J (x)Φ J (x ′ ) = λ 2π 2 sin λρ f (ρ) ,(44) where ρ is the geodesic distance between the points x and x ′ (see [1]). Therefore, from (34) and (38) we obtain in the hyperbolic (K = −1) case: R 0 (η, x, x ′ ) = −i(M a) 2 ′ 16π 2 a 3 ρ sinh ρ K 0 (M r).(45) If |z| ≫ 1, then K ν (z) ∼ π/(2z) e −z . Consequently, if the metric is changing adiabatically, the function |R 0 | in the quasi-Euclidean (see (41)) and hyperbolic space-time decreases exponentially with growing ρ at spacings r ≫ 1/M , i.e., exceeding the Compton wavelength. In spherical case, the spacing ρ changes in the range 0 ≤ ρ ≤ π. A substitution of (38) and (44) into (34) leads to R 0 (η, x, x ′ ) = −i(M a) 2 ′ 16π 2 a 3 sin ρ ∞ λ=1 λ sin λρ (M 2 a 2 + λ 2 ) 3/2 . (46) The sum in the right-hand side of (46), after a chain of transformations, may be presented in the form ∞ n=0 (ρ+2πn) K 0 (M a(ρ+2πn)) − − ∞ n=1 (2πn−ρ) K 0 (M a(2πn−ρ)) .(47) From the asymptotic properties of the function K 0 (z) it follows that, for ρ ≪ 1 and M a ≫ 1 (the distance between particles od a pair and a particle's Compton wavelength are much smaller than the curvature radius of space), in this representation one could retain only the term ρ K 0 (M aρ), and therefore Eq. (46) takes the form R 0 (η, x, x ′ ) = −i(M a) 2 ′ 16π 2 a 3 ρ sin ρ K 0 (M r) .(48) Therefore, in the spherical case |R 0 | also decreases exponentially with growing ρ at distances r ≫ 1/M , exceeding the Compton wavelength. Thus if the metric of a homogeneous isotropic state is changing adiabatically, the space-time correlation function R 0 (η, x, x ′ ) is exponentially small for r ≫ 1/M . This indicates that the corresponding quasiparticles are virtual pairs with the characteristic correlation length equal to 1/M . Real particle creation is exponentially small and does not manifest itself in perturbation theory. In both cases, the correlation function decreases exponentially at distances between the quasiparticles exceeding the Compton wavelength l C = 1/M . Thus real particles creation in de Sitter space does not occur. The quasiparticle pairs being created, whose density has been calculated and shown in Fig. 1, should be interpreted as pairs of virtual particles. As has been noticed in [1], the absence of real particle creation in de Sitter space is confirmed by the local nature of the vacuum stress-energy tensor and by vanishing of the imaginary part of the effective Lagrangian. CONCLUSIONS In this paper, for a scalar field nonconformally coupled to the curvature, we gave a generalization of the method of space-time description of particle creation by the gravitational field. In a homogeneous isotropic space, we have introduced the creation operators (30) of localized one-particle states and the operator (31) of particle number in a specified volume. We have obtained the expressions (34) and (35) for the space-time correlation function (33) of a pair of created quasiparticles corresponding to a diagonal form of the instantaneous Hamiltonian. We have analyzed the case of adiabatic changes in the metric of a homogeneous isotropic space. The expressions (41), (42), (45) and (46) have been obtained for the correlation function of a pair of quasiparticles created. It has been shown that the correlation function exponentially decreases at spaces exceeding the Compton wavelength, and consequently real particle creation is suppressed. Particle creation in de Sitter space has been considered, and, from, the properties of the space-time correlation function for a pair of quasiparticles created, it has been concluded that such a pair should be interpreted as a pair of virtual particles. J , * a (±) J , under the conditions (10). Diagonalization of the Hamiltonian at an arbitrary time instant η is carried out in terms of the operators * J (η) at the time η, i.e., b F J (η) = 0, (for Ω 2 (η) > 0), it follows: which, due to (17) and (20), do not depend on the specific choice of the functions χ J (η). The operators * of (21), taking into account (17) and their conjugate expressions, into (15) leads to the following expression for the Hamiltonian:. (22)Thus the energy of quasiparticles corresponding to the diagonal form of the Hamiltonian (15) is equal to the oscillator frequency Ω(η) (unlike the Hamiltonian built from the metric stress-energy tensor of a nonconformal scalar field[6]).Using(7), (17), (20), and (22), one can verify that the operators c (±) J (η) obey Heisenberg's equations of motion: follows from (11), (12), (17), and (20). The equations of motion hold for each mode in (24) separately. , i.e., in the vacuum state for the instant η 0 . At the time η, the vacuum state is the state |0 η , defined by the equalities c (−) Figure 1 : 1The number density of quasiparticles created in de Sitter space. Figure 2 : 2The correlation function for M/H = 0.01.where r = a(t)|x ′ − x|, i.e., the correlation function, expressed in terms of r (the "physical" distance between the quasiparticles in a pair) is timeindependent.Examples of numerical calculations for N = 4, M/H = 0.01 and M/H = 0.2 are given in Figs. 2 and 3, respectively. Figure 3 : 3The correlation function for M/H = 0.2. Let us consider de Sitter space and take the metric in the form (5) with K = 0 and a = a 1 e Ht = − 1 Hη ,t ∈ (−∞, +∞) ⇔ η ∈ (−∞, 0). Solutions to Eq. (7) with V g = ξR and the conditions (10) for η 0 → −∞ have the formwhere H(2)ν (z) is a Hankel function, α 0 is an arbitrary real constant,andFurthermore, assuming m 2 + (ξ − ξ c )R > 0, from (26), (29) and (50) we obtaini.e., the created particle number density in de Sitter space is time-independent! The result of a numerical computation for the function F N (M/H) in the case N = 4 is represented inFig. 1. At M/H ≫ 1, the metric is changing adiabatically and, as shown in Section 4, real particle creation does not occur.In the general case, substituting the exact solution (50) into (34) and (35) and using (40), we ob-ACKNOWLEDGMENTSThe author thanks Prof. A.A. Grib for helpful discussions. The work has been financially supported by RNP Grant 2.1.1.6826. A A Grib, S G Mamayev, V M Mostepanenko, Vacuum Quantum Effects in Strong Fields. Moscow; St.PetersburgFriedmann Lab. Publin Russian; English translationA. A. Grib, S. G. Mamayev, and V. M. Moste- panenko, Vacuum Quantum Effects in Strong Fields (Energoatomizdat, Moscow, 1988, in Russian; English translation: Friedmann Lab. Publ., St.Petersburg, 1994). N D Birrell, P C W Davies, Quantum Fields in Curved Space. CambridgeCambridge Univ. PressN. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space (Cambridge Univ. Press, Cambridge, 1982). . A A Grib, Yu V Pavlov, Int. J. Mod. Phys. D. 11433A. A. Grib and Yu. V. Pavlov, Int. J. Mod. 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{'abstract': 'We give the generalization of the method of the space-time description of particle creation by a gravitational field for a scalar field with nonconformal coupling to the curvature. The space-time correlation function is obtained for a created pair of the quasi-particles, corresponding to a diagonal form of the instantaneous Hamiltonian. The case of an adiabatic change of the metric of homogeneous isotropic space is analyzed. We show that the created pairs of quasi-particles in de Sitter space should be interpreted as pairs of virtual particles. PACS number: 04.62.+v, 03.70.+k H(η) = dµ(J) Ω(', 'arxivid': '0811.4236', 'author': ['Yu V Pavlov \nInstitute of Mechanical Engineering\nA. Friedmann Laboratory for Theoretical Physics\nSt. Petersburg\nRussia\n\nRussian Acad. Sci\n61 Bolshoy pr199178St. PetersburgRussia\n'], 'authoraffiliation': ['Institute of Mechanical Engineering\nA. Friedmann Laboratory for Theoretical Physics\nSt. Petersburg\nRussia', 'Russian Acad. Sci\n61 Bolshoy pr199178St. PetersburgRussia'], 'corpusid': 16954116, 'doi': '10.1134/s020228930804004x', 'github_urls': [], 'n_tokens_mistral': 9234, 'n_tokens_neox': 7909, 'n_words': 4638, 'pdfsha': '2f7c922558a4405d55477ee1090bcfbadc16411e', 'pdfurls': ['https://arxiv.org/pdf/0811.4236v1.pdf'], 'title': ['Space-Time Description of Scalar Particle Creation by a Homogeneous Isotropic Gravitational Field', 'Space-Time Description of Scalar Particle Creation by a Homogeneous Isotropic Gravitational Field'], 'venue': []}