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[ "Sub-millimeter Observations of Giant Molecular Clouds in the Large Magellanic Cloud: Temperature and Density as Determined from J = 3 − 2 and J = 1 − 0 transitions of CO", "Sub-millimeter Observations of Giant Molecular Clouds in the Large Magellanic Cloud: Temperature and Density as Determined from J = 3 − 2 and J = 1 − 0 transitions of CO" ]	[ "Tetsuhiro Minamidani \nDepartment of Astrophysics\nNagoya University\nFuro-cho, Chikusa-ku464-8602NagoyaJapan\n\nDepartment of Physics\nFaculty of Science\nHokkaido University\nKita-kuN10W8, 060-0810SapporoJapan\n", "Norikazu Mizuno \nDepartment of Astrophysics\nNagoya University\nFuro-cho, Chikusa-ku464-8602NagoyaJapan\n", "Yoji Mizuno \nDepartment of Astrophysics\nNagoya University\nFuro-cho, Chikusa-ku464-8602NagoyaJapan\n", "Akiko Kawamura \nDepartment of Astrophysics\nNagoya University\nFuro-cho, Chikusa-ku464-8602NagoyaJapan\n", "Toshikazu Onishi \nDepartment of Astrophysics\nNagoya University\nFuro-cho, Chikusa-ku464-8602NagoyaJapan\n", "Tetsuo Hasegawa \nNational Astronomical Observatory of Japan\n181-8588MitakaTokyoJapan\n", "Ken'ichi Tatematsu \nNational Astronomical Observatory of Japan\n181-8588MitakaTokyoJapan\n", "Masafumi Ikeda \nResearch Center for the Early Univers\nDepartment of Physics\nUniversity of Tokyo\n113-0033TokyoJapan\n", "Yoshiaki Moriguchi \nDepartment of Astrophysics\nNagoya University\nFuro-cho, Chikusa-ku464-8602NagoyaJapan\n", "Nobuyuki Yamaguchi \nNational Astronomical Observatory of Japan\n181-8588MitakaTokyoJapan\n", "Jürgen Ott \nNational Radio Astronomy Observatory\n520 Edgemont Road22903-2475CharlottesvilleVA\n", "Tony Wong \nTelescope National Facility\nPO Box 761710EppingNSWAustralia, Australia\n\nDepartment of Astronomy\nUniversity of Illinois\n221, 61801UrbanaMC, IL\n", "Erik Muller \nTelescope National Facility\nPO Box 761710EppingNSWAustralia, Australia\n", "Jorge L Pineda \nRadioastronomisches Institut\nUniversität Bonn\nAuf dem Hügel 7153121BonnGermany\n", "Annie Hughes \nTelescope National Facility\nPO Box 761710EppingNSWAustralia, Australia\n\nCenter for Supercomputing and Astrophysics\nSwinburne University of Technology\n3122HawthornVICAustralia\n", "Lister Staveley-Smith \nSchool of Physics\nUniversity of Western Australia\n35 Stirling HighwayM013, 6009CrawleyWAAustralia\n", "Ulrich Klein \nRadioastronomisches Institut\nUniversität Bonn\nAuf dem Hügel 7153121BonnGermany\n", "Akira Mizuno \nDepartment of Astrophysics\nNagoya University\nFuro-cho, Chikusa-ku464-8602NagoyaJapan\n\nPresent address: Solar-Terrestrial Environment Laboratory\nNagoya University\nFuro-cho, Chikusa-ku464-8601NagoyaJapan\n", "Silvana Nikolić \nOnsala Space Observatory\n439-92OnsalaSweden\n\nDepartament de Astronomia\nUniversidad de Chile\nCasilla 36-DP.O.Box 4431740Santiago, KrugersdorpChile, South Africa\n", "Roy S Booth \nOnsala Space Observatory\n439-92OnsalaSweden\n", "Arto Heikkilä \nOnsala Space Observatory\n439-92OnsalaSweden\n", "Lars-Åke Nyman \nEuropean Southern Observatory\nSantiago 1919001CasillaChile\n", "Mikael Lerner \nEuropean Southern Observatory\nSantiago 1919001CasillaChile\n", "Guido Garay \nDepartament de Astronomia\nUniversidad de Chile\nCasilla 36-DSantiagoChile\n", "Sungeun Kim \nAstronomy & Space Science Department\nSejong University\n98 Kwangjin-gu143-747Kunja-dong, SeoulKorea\n", "Motosuji Fujishita \nDepartment of Astrophysics\nNagoya University\nFuro-cho, Chikusa-ku464-8602NagoyaJapan\n", "Tokuichi Kawase \nDepartment of Astrophysics\nNagoya University\nFuro-cho, Chikusa-ku464-8602NagoyaJapan\n", "Monicá Rubio \nDepartament de Astronomia\nUniversidad de Chile\nCasilla 36-DSantiagoChile\n", "Yasuo Fukui \nDepartment of Astrophysics\nNagoya University\nFuro-cho, Chikusa-ku464-8602NagoyaJapan\n" ]	[ "Department of Astrophysics\nNagoya University\nFuro-cho, Chikusa-ku464-8602NagoyaJapan", "Department of Physics\nFaculty of Science\nHokkaido University\nKita-kuN10W8, 060-0810SapporoJapan", "Department of Astrophysics\nNagoya University\nFuro-cho, Chikusa-ku464-8602NagoyaJapan", "Department of Astrophysics\nNagoya University\nFuro-cho, Chikusa-ku464-8602NagoyaJapan", "Department of Astrophysics\nNagoya University\nFuro-cho, Chikusa-ku464-8602NagoyaJapan", "Department of Astrophysics\nNagoya University\nFuro-cho, Chikusa-ku464-8602NagoyaJapan", "National Astronomical Observatory of Japan\n181-8588MitakaTokyoJapan", "National Astronomical Observatory of Japan\n181-8588MitakaTokyoJapan", "Research Center for the Early Univers\nDepartment of Physics\nUniversity of Tokyo\n113-0033TokyoJapan", "Department of Astrophysics\nNagoya University\nFuro-cho, Chikusa-ku464-8602NagoyaJapan", "National Astronomical Observatory of Japan\n181-8588MitakaTokyoJapan", "National Radio Astronomy Observatory\n520 Edgemont Road22903-2475CharlottesvilleVA", "Telescope National Facility\nPO Box 761710EppingNSWAustralia, Australia", "Department of Astronomy\nUniversity of Illinois\n221, 61801UrbanaMC, IL", "Telescope National Facility\nPO Box 761710EppingNSWAustralia, Australia", "Radioastronomisches Institut\nUniversität Bonn\nAuf dem Hügel 7153121BonnGermany", "Telescope National Facility\nPO Box 761710EppingNSWAustralia, Australia", "Center for Supercomputing and Astrophysics\nSwinburne University of Technology\n3122HawthornVICAustralia", "School of Physics\nUniversity of Western Australia\n35 Stirling HighwayM013, 6009CrawleyWAAustralia", "Radioastronomisches Institut\nUniversität Bonn\nAuf dem Hügel 7153121BonnGermany", "Department of Astrophysics\nNagoya University\nFuro-cho, Chikusa-ku464-8602NagoyaJapan", "Present address: Solar-Terrestrial Environment Laboratory\nNagoya University\nFuro-cho, Chikusa-ku464-8601NagoyaJapan", "Onsala Space Observatory\n439-92OnsalaSweden", "Departament de Astronomia\nUniversidad de Chile\nCasilla 36-DP.O.Box 4431740Santiago, KrugersdorpChile, South Africa", "Onsala Space Observatory\n439-92OnsalaSweden", "Onsala Space Observatory\n439-92OnsalaSweden", "European Southern Observatory\nSantiago 1919001CasillaChile", "European Southern Observatory\nSantiago 1919001CasillaChile", "Departament de Astronomia\nUniversidad de Chile\nCasilla 36-DSantiagoChile", "Astronomy & Space Science Department\nSejong University\n98 Kwangjin-gu143-747Kunja-dong, SeoulKorea", "Department of Astrophysics\nNagoya University\nFuro-cho, Chikusa-ku464-8602NagoyaJapan", "Department of Astrophysics\nNagoya University\nFuro-cho, Chikusa-ku464-8602NagoyaJapan", "Departament de Astronomia\nUniversidad de Chile\nCasilla 36-DSantiagoChile", "Department of Astrophysics\nNagoya University\nFuro-cho, Chikusa-ku464-8602NagoyaJapan" ]	[]	We have carried out sub-mm 12 CO(J = 3−2) observations of 6 giant molecular clouds (GMCs) in the Large Magellanic Cloud (LMC) with the ASTE 10m submm telescope at a spatial resolution of 5 pc and very high sensitivity. We have identified 32 molecular clumps in the GMCs and revealed significant details of the warm and dense molecular gas with n(H 2 ) ∼ 10 3−5 cm −3 and T kin ∼ 60 K. These data are combined with 12 CO(J = 1 − 0) and 13 CO(J = 1 − 0) results and compared with LVG calculations. The results indicate that clumps we detected	10.1086/524038	[ "https://arxiv.org/pdf/0710.4202v1.pdf" ]	118,475,564	0710.4202	c7e8468f87656d015b6e0d179e142fd79adae5b3	Sub-millimeter Observations of Giant Molecular Clouds in the Large Magellanic Cloud: Temperature and Density as Determined from J = 3 − 2 and J = 1 − 0 transitions of CO 23 Oct 2007 Tetsuhiro Minamidani Department of Astrophysics Nagoya University Furo-cho, Chikusa-ku464-8602NagoyaJapan Department of Physics Faculty of Science Hokkaido University Kita-kuN10W8, 060-0810SapporoJapan Norikazu Mizuno Department of Astrophysics Nagoya University Furo-cho, Chikusa-ku464-8602NagoyaJapan Yoji Mizuno Department of Astrophysics Nagoya University Furo-cho, Chikusa-ku464-8602NagoyaJapan Akiko Kawamura Department of Astrophysics Nagoya University Furo-cho, Chikusa-ku464-8602NagoyaJapan Toshikazu Onishi Department of Astrophysics Nagoya University Furo-cho, Chikusa-ku464-8602NagoyaJapan Tetsuo Hasegawa National Astronomical Observatory of Japan 181-8588MitakaTokyoJapan Ken'ichi Tatematsu National Astronomical Observatory of Japan 181-8588MitakaTokyoJapan Masafumi Ikeda Research Center for the Early Univers Department of Physics University of Tokyo 113-0033TokyoJapan Yoshiaki Moriguchi Department of Astrophysics Nagoya University Furo-cho, Chikusa-ku464-8602NagoyaJapan Nobuyuki Yamaguchi National Astronomical Observatory of Japan 181-8588MitakaTokyoJapan Jürgen Ott National Radio Astronomy Observatory 520 Edgemont Road22903-2475CharlottesvilleVA Tony Wong Telescope National Facility PO Box 761710EppingNSWAustralia, Australia Department of Astronomy University of Illinois 221, 61801UrbanaMC, IL Erik Muller Telescope National Facility PO Box 761710EppingNSWAustralia, Australia Jorge L Pineda Radioastronomisches Institut Universität Bonn Auf dem Hügel 7153121BonnGermany Annie Hughes Telescope National Facility PO Box 761710EppingNSWAustralia, Australia Center for Supercomputing and Astrophysics Swinburne University of Technology 3122HawthornVICAustralia Lister Staveley-Smith School of Physics University of Western Australia 35 Stirling HighwayM013, 6009CrawleyWAAustralia Ulrich Klein Radioastronomisches Institut Universität Bonn Auf dem Hügel 7153121BonnGermany Akira Mizuno Department of Astrophysics Nagoya University Furo-cho, Chikusa-ku464-8602NagoyaJapan Present address: Solar-Terrestrial Environment Laboratory Nagoya University Furo-cho, Chikusa-ku464-8601NagoyaJapan Silvana Nikolić Onsala Space Observatory 439-92OnsalaSweden Departament de Astronomia Universidad de Chile Casilla 36-DP.O.Box 4431740Santiago, KrugersdorpChile, South Africa Roy S Booth Onsala Space Observatory 439-92OnsalaSweden Arto Heikkilä Onsala Space Observatory 439-92OnsalaSweden Lars-Åke Nyman European Southern Observatory Santiago 1919001CasillaChile Mikael Lerner European Southern Observatory Santiago 1919001CasillaChile Guido Garay Departament de Astronomia Universidad de Chile Casilla 36-DSantiagoChile Sungeun Kim Astronomy & Space Science Department Sejong University 98 Kwangjin-gu143-747Kunja-dong, SeoulKorea Motosuji Fujishita Department of Astrophysics Nagoya University Furo-cho, Chikusa-ku464-8602NagoyaJapan Tokuichi Kawase Department of Astrophysics Nagoya University Furo-cho, Chikusa-ku464-8602NagoyaJapan Monicá Rubio Departament de Astronomia Universidad de Chile Casilla 36-DSantiagoChile Yasuo Fukui Department of Astrophysics Nagoya University Furo-cho, Chikusa-ku464-8602NagoyaJapan Sub-millimeter Observations of Giant Molecular Clouds in the Large Magellanic Cloud: Temperature and Density as Determined from J = 3 − 2 and J = 1 − 0 transitions of CO 23 Oct 200717 Present address: Hartebeesthoek Radio Astronomy Observatory,Subject headings: Magellanic Clouds -galaxies: individual (LMC) -ISM: clouds -ISM: molecules -radio lines: ISM -submillimeter We have carried out sub-mm 12 CO(J = 3−2) observations of 6 giant molecular clouds (GMCs) in the Large Magellanic Cloud (LMC) with the ASTE 10m submm telescope at a spatial resolution of 5 pc and very high sensitivity. We have identified 32 molecular clumps in the GMCs and revealed significant details of the warm and dense molecular gas with n(H 2 ) ∼ 10 3−5 cm −3 and T kin ∼ 60 K. These data are combined with 12 CO(J = 1 − 0) and 13 CO(J = 1 − 0) results and compared with LVG calculations. The results indicate that clumps we detected are distributed continuously from cool (∼ 10 -30 K) to warm (∼ higher than 30 -200 K), and warm clumps are distributed from less dense (∼ 10 3 cm −3 ) to dense (∼ 10 3.5−5 cm −3 ).We found that the ratio of 12 CO(J = 3 − 2) to 12 CO(J = 1 − 0) emission is sensitive to and is well correlated with the local Hα flux. We interpret that differences of clump propeties represent an evolutionary sequence of GMCs in terms of density increase leading to star formation.Type I and II GMCs (starless GMCs and GMCs with HII regions only, respectively) are at the young phase of star formation where density does not yet become high enough to show active star formation and Type III GMCs (GMCs with HII regions and young star clusters) represents the later phase where the average density is increased and the GMCs are forming massive stars. The high kinetic temperature correlated with Hα flux suggests that FUV heating is dominant in the molecular gas of the LMC. Subject headings: Magellanic Clouds -galaxies: individual (LMC) -ISM: clouds -ISM: molecules -radio lines: ISM -submillimeter Introduction It is of a fundamental importance in astronomy to understand the evolution of galaxies. Since a major constituent of galaxies is stars, the formation of stars is a fundamental process in galactic evolution. The properties of stars characterize the basic contents of galaxies and their time evolution. We understand from studies of the Milky Way galaxy that giant molecular clouds (GMCs), whose mass ranges from 10 5 to 10 7 M ⊙ , are the principal sites of star formation and that it perhaps holds the true in other galaxies as well. We also recognize that the GMC properties (e.g., L CO -line width relation, index of mass spectrum) are similar among five galaxies in the Local Group according to the spatially resolved studies (Blitz et al. 2006). This supports the idea that studies of GMCs will be useful in understanding the fundamentals of galactic evolution through the formation and evolution of GMCs and star formation therein. Observational studies of GMCs have been most effectively made by the mm interstellar carbon monoxide emission line at 2.6 mm which allows us to probe molecular gas whose density is greater than ∼ 100 cm −3 . We note that the most abundant species, molecular hydrogen, does not have appropriate line emissions in the mm and sub-mm region due to its zero permanent electric dipole moment and large separation between the lowest energy levels, which are not excited significantly in the typical physical conditions of molecular clouds. eters of molecular clouds over much larger ranges than in the mm region by comparing line intensities between different transitions. These sub-mm studies were initiated by the SEST 15m telescope in Chile followed by instruments in Hawaii, Mauna Kea and in the Swiss Alps at an altitude range from 3700 m to 4200 m, including the CSO 10m, JCMT 15m, and KOSMA 3m telescopes, and the AST/RO 1.6m telescope in Antarctica. Subsequently, in the 2000's, the developements of new instruments at an altitude of ∼ 5000 m in Atacama in northern Chile resulted in a superior capability because of the high altitude and dry characteristics of the site. The instruments installed in Atacama include the ASTE 10m, APEX 12m and NANTEN2 4m telescopes. All these instruments are beginning to take new molecular data with significantly better quality than before in terms of noise level as well as angular resolution. It is also noteworthy that the current frequency coverage extends as high as the 800GHz band and even the THz region. Among nearby galaxies we can observe at reasonably high spatial resolutions, the Large and Small Magellanic Clouds offer us a unique opportunity to achieve the highest resolutions due to their unrivaled closeness, 50 -60 kpc. In particular, the Large Magellanic Cloud (LMC) is actively forming stars in clusters and is an ideal laboratory for us to study star formation, particularly massive star formation in star cluster. In the LMC, the metallicity is a factor of ∼ 3 lower than in the Solar neighborhood (Dufour et al. 1982;Dufour 1984;Rolleston et al. 2002). Also, the visual extinctions are lower and the FUV field is stronger in the LMC than in the Milky Way (Israel et al. 1986), characterizing the initial conditions of star formation. The first spatially resolved complete sample of GMCs in a single galaxy has been obtained towards the whole LMC with the NANTEN 4m telescope in 2.6 mm CO emission at 40 pc resolution (Fukui et al. 1999(Fukui et al. , 2007Mizuno et al. 2001). These studies revealed the three types of GMCs in terms of star formation activities; Type I is starless, Type II is with HII regions only, and Type III is associated with active star formation indicated by huge HII regions and young star clusters, where the stars identified are only O stars due to the sensitivity limitation. It also revealed that the lifetime of a GMC is as short as ∼ 30 Myrs (Fukui 2006a;Kawamura et al. 2006Kawamura et al. , 2007. These previous studies naturally place the LMC as one of the prime targets for sub-mm studies to derive the physical parameters of GMCs. Another aspect which deserves our attention is that very young, rich stellar clusters are forming in the LMC. These are so called populous clusters which are very rare in the Milky Way and resemble globulars formed in the primeval Milky Way. The open clusters forming in the Milky Way are small in the number of stars and loose in spatial distribution. Along with the low metallicity of the LMC, it is an interesting possibility to use molecular data to investigate the formation mechanism of super star clusters at the molecular cloud stage. In the past, there have been some studies that used the higher transitions (J = 2 − 1, J = 3 − 2, J = 4 − 3, J = 7 − 6) of CO spectra of the molecular clouds in the LMC (e.g., Sorai et al. 2001;Johansson et al. 1998;Heikkilä et al. 1999;Bolatto et al. 2005;Israel et al. 2003;Kim et al. 2004;Kim 2006). These studies suggest the molecular gas may be warmer and/or denser than in the Milky Way. Johansson et al. (1998) used the SEST 15m telescope to observe the central part of the 30 Doradus nebula (rms ∼ 0.2 K in 0.5 km s −1 velocity resolution for J = 1 − 0 and rms ∼ 1.0 K in 0.5 km s −1 velocity resolution for J = 3 − 2), and the southern HII regions N 158C, N 159, and N 160 with a few prominent CO clouds in the J = 2 − 1 and J = 3 − 2 transitions of CO. They find that the kinetic temperatures are 10 -80 K and the highest temperature is towards 30 Dor. The smallest beam size and grid spacing are 15 ′′ and 11 ′′ respectively in the J = 3 − 2 emission. Heikkilä et al. (1999) used SEST to observe the J = 3 − 2 transition of CO in N 159 and 30 Doradus, as well as other rarer molecular species. This study aimed at obtaining chemical abundance, while it also provides more information on cloud temperature etc, from CO(J = 3 − 2) data. The kinetic temperatures they derived are 50 K in 30 Dor-10, 15 K in 30 Dor-27, and 20 -25 K in N 159W and N 160. Bolatto et al. (2005) employed the AST/RO to observe the 12 CO(J = 4 − 3) transition at 461 GHz with a 109 ′′ beam. They observed 9 regions in the LMC at 6 ′ ×6 ′ field all with HII regions and derived kinetic temperatures from a comparison between the CO(J = 4 − 3) and (J = 1 − 0) transitions. N 48, N 55A, N 79, N 83A, N 113, N 159W, N 167, N 214C, and LIRL 648 are included. They derive temperatures of 100 -300 K and note a trend that higher temperatures occur in moderate density regions, 100 -1000 cm −3 , and the lower temperatures in much denser regions 10 4−5 cm −3 . These studies were preceded by a suggestion that significant amounts of warm molecular gas may exist in the LMC (Israel et al. 2003). Kim et al. (2004) also made similar observations towards an HII region, N 44, and suggest very dense gas of ∼ 10 5 cm −3 . Most recently, Kim (2006) derived T kin = 100 K, n ∼ 10 4.3 cm −3 for 30 Dor from the intensity ratios of 12 CO(J = 7 − 6) to 12 CO(J = 4 − 3) and 12 CO(J = 1 − 0) to 13 CO(J = 1 − 0). In the present study, we aim to obtain sub-mm molecular data at better S/N ratios than in the previous studies to make estimates of temperatures and densities over a large sample in the LMC. We will combine the 12 CO(J = 3 − 2) data obtained with the ASTE telescope and CO(J = 1 − 0) data obtained with the SEST and Mopra telescopes. In order to make reasonable comparisons between the two transitions, J = 3 − 2 and J = 1 − 0, we shall smooth the ASTE results (22 ′′ beam) to the same resolution as the SEST data (45 ′′ ) and use LVG calculations to estimate density and temperature. We shall also employ the 13 CO(J = 1 − 0) data where available to place constraints on the physical parameters. This paper is organized as follows: Section 2 describes the observations. Section 3 and 4 show the results and data analysis, respectively. In section 5, we discuss the physical properties of clumps and evolutional sequence of GMCs. Finally, we present a summary in section 6. Observations Selection of GMCs The present targets were chosen from the NANTEN catalog of 12 CO(J = 1 − 0) GMCs compiled by Fukui et al. (2007). This catalog is based on the 2nd survey, with a factor of ∼ 2 higher sensitivity than the 1st survey . The catalog includes 272 CO clouds 230 of which are detected at three or more observed positions and they are classified into the three Types; 56 (24.3 %) Type I (starless) GMCs, 120 (52.2 %) Type II GMCs (those with HII regions only), and 54 (23.5 %) Type III GMCs (those with HII regions and young star clusters), where "stars" refer only to O stars due to the limited sensitivity of existing observations (Kawamura et al. 2007;Fukui 2006a;Kawamura et al. 2006;Blitz et al. 2006). Among these GMCs, we mainly focus on Type III GMCs whose 12 CO(J = 3 − 2) intensities are expected to be strong due to the highly excited conditions. In addition, relatively high resolution 12 CO and 13 CO J = 1 − 0 data, observed with the SEST 15m or Mopra 22m telescopes, are collected to derive physical properties of molecular clouds through the LVG analysis at a 10 pc scale. In the present study, we observe GMCs in the south-east region of the LMC, which contains 30 Doradus -the largest and most massive HII region in the Local Group. We observe the molecular ridge extending southward from 30 Doradus, and "CO Arc" along the southeastern optical edge (Fukui et al. 1999). 3 Type III GMCs, LMC N J0538-6904 (the 30 Dor region), LMC N J0540-7008 (the N 159 region and the N 171 region), and LMC N J0530-7106 (the N 206 region), are selected as the principal targets, and two Type II GMCs, LMC N J0544-6923 (the N 166 region) and LMC N J0532-7114 (the N 206D region), and a Type I GMC, LMC N J0547-7041 (the GMC 225 region), are included for reference. The locations of the observed GMCs and regions are shown in Figure 1, and their coordinates and the data used in this paper are summarized in Table 1. Hereafter, the region names, which are in the parenthesis above or column 4 of Table 1, are used to identify the regions. 12 CO(J = 3 − 2) Observations of the 12 CO(J = 3 − 2) transition at 345GHz were made with the ASTE (Atacama Submillimeter Telescope Experiment) telescope at Pampa la Bola in Chile (Ezawa et al. 2004) in October 2004. The half power beam width was measured to be 22 ′′ at 345 GHz by observing the planets. This corresponds to 5.3 pc at the distance of the LMC, 50 kpc. The telescope was equipped with a single "cartridge-type" SIS receiver, sensitive from 324 to 384 GHz, which is of a similar design to that for ALMA (Kohno 2005). The spectrometer was an XF-type digital auto-correlator (Sorai et al. 2000), and was used in the wideband mode, which has a bandwidth of 512 MHz with 1024 channels. The spectrometer provided a velocity coverage and resolution of 450 km s −1 and 0.44 km s −1 , respectively, at 345 GHz. We observed 6 GMCs (7 regions) in the Large Magellanic Cloud as shown in Figure 1 and listed in Table 1. These observations were carried out by position switching at a grid spacing of 20 ′′ or 30 ′′ for the entire clouds, and of 10 ′′ or 15 ′′ for the regions around the local peaks of the integrated intensity. The pointing error was measured to be within 7 ′′ in peak to peak by observing CO point sources R Dor or o Cet every 2 hours during this observing term. The spectral intensities were calibrated by employing the standard room-temperature chopperwheel technique. We observed Ori-KL once a day, and N 159W every 2 hours to check the stability of intensity calibration, and the intensity variation during these observations was less than 13 %. We use 0.7 for the main-beam efficiency at 345 GHz, which was measured by observing Jupiter. The system noise temperature was typically 300 K in double-sideband (DSB) including the atmosphere towards the zenith. The typical r.m.s. noise fluctuations were ∼ 0.25 K at a velocity resolution of 0.44 km s −1 for a 1 minute integration for an on-position. In total, about 1400 points were observed in Equatorial coordinates (B1950). Velocities were relative to the Local Standard of Rest (LSR). These observations were made remotely from an ASTE operation room of San Pedro de Atacama, Chile, using the network observation system N-COSMOS3 developed by NAOJ (Kamazaki et al. 2005). 2.3. 12 CO(J = 1 − 0) and 13 CO(J = 1 − 0) Mopra observations A 20 ′ × 120 ′ region, the prominent molecular ridge extending from 30 Doradus southward, was mapped in the J = 1 − 0 transition of 12 CO at a frequency of 115 GHz with the 22m ATNF Mopra telescope, in 5 runs from May to September 2005. This region contains the 30 Dor, N 159 and N 171 regions. The newly implemented on-the-fly (OTF) mode was used, in which the telescope takes data continuously while moving across the sky. Spectra were taken at a 6 ′′ spacing so that the 33 ′′ Mopra beam would be well oversampled in the scanning direction; the row spacing was 8 ′′ , also assuring oversampling. The typical system noise temperature, T sys , was 500 K in the single side band (SSB) towards the zenith. The pointing was checked on the SiO maser R Dor every 2 hours; typical pointing error was less than 5 ′′ rms. The digital correlator was configured to output 1024 channels across 64 MHz in each of two orthogonal polarizations. The velocity resolution and coverage were 0.16 and 160 km s −1 , respectively at 115 GHz. The observing time was about 100 minutes per field (5 ′ ×5 ′ ), providing rms noise fluctuations of ∼ 0.34 K at a velocity resolution of 0.65 km s −1 . Initial spectral processing (baseline fitting and calibration onto a T * A scale) was performed using the livedata task in AIPS++, and the spectra were gridded into data cubes using the AIPS++ gridzilla task. During the gridding, a Gaussian smoothing kernel with a FWHM taken at a half of the beam size was convolved with the data, so the effective resolution of the output cubes was 36 ′′ . The cubes were then rescaled onto a T mb scale using an "extended beam" efficiency of 0.55 (Ladd et al. 2005). This takes into account that sources larger than about 80 ′′ in diameter will couple to both the main beam and the inner error beam of the telescope. The cubes were 2 dimensional Gaussian smoothed to a 45 ′′ beam, which is the beam size of SEST at 115 GHz. SEST observations Observations towards N 206, N 206D and GMC 225 regions in the 12 CO(J = 1 − 0) line (115 GHz) were made in August, 2001 andFebruary, 2002, using the SEST 15m telescope at La Silla, Chile. The HPBW was 45 ′′ at 115 GHz, the front end was the IRAM 115 SIS receiver and the spectrometer was a high-resolution AOS with 2048 channels. The typical system noise temperature was 550 K (SSB). The velocity resolution and coverage were 0.2 and 216 km s −1 respectively at 115 GHz. We mapped these 3 regions in position switching with a grid spacing of 40 ′′ or 20 ′′ . The typical integration time was 1 minute for an onposition, providing rms noise fluctuations of ∼ 0.18 K at a velocity resolution of 0.2 km s −1 . Observations towards N 206, N 206D and GMC 225 regions in the 13 CO(J = 1 − 0) line (110 GHz) were made in February and December, 2002, also using the SEST 15m telescope. We mapped peak positions of 12 CO(J = 1 − 0) in position switching with a grid spacing of 20 ′′ . The system noise temperature was typically 230 K (SSB). The typical integration time was 4 minutes for an on-position, providing rms noise fluctuations of ∼ 0.04 K at a velocity resolution of 0.2 km s −1 . The pointing accuracy was 5 ′′ rms. We checked this by observing SiO maser toward R Dor every 2 hours during this observing term. N 159W was observed periodically for pointing checks and intensity calibration. We use 0.8 for the main-beam efficiency at 115 GHz by assuming main-beam temperature T mb of N 159W to be ∼ 6.5 -6.9 K to keep consistency with former publications (Johansson et al. 1994. Observations toward N 166 region are described separately by Garay et al. (2002). Results We first present the 12 CO(J = 3 − 2) images of the clouds at 5 pc resolution and make an empirical comparison of them with the 12 CO(J = 1 − 0) distribution (section 3.1.). Next, we define molecular clumps and estimate the physical parameters of each clump (section 3.2.). ). The lower panels, Figures 2(d) -(f), show 12 CO(J = 1 − 0) profiles towards the same positions, where the red line indicates 12 CO(J = 3 − 2) profiles convolved to a 45 ′′ Gaussian beam following the method described in Section 4. The 12 CO(J = 3 − 2) intensities from the 30 Dor and N 159 regions are a little stronger than the 12 CO(J = 1 − 0) intensity when convolved to the same resolution. Only towards GMC 225 is the 12 CO(J = 3 − 2) intensity about 50% weaker than the 12 CO(J = 1 − 0) intensity. The peak velocity and line width are nearly the same between the two transitions in these regions. The distributions of the integrated intensities of 12 CO(J = 3 − 2) and 12 CO(J = 1 − 0) are shown in Figures 3 -9. Detailed descriptions of each region are presented in the following. 30 Dor (Figure 3) Figures 3(a) and (b) show the distributions of the 12 CO(J = 3 − 2) and 12 CO(J = 1 − 0) integrated intensities in the 30 Dor region. We see a general trend that the J = 3 − 2 distribution shows more details which are not obvious in the J = 1 − 0 distribution owing to the higher angular resolution and possibly due to the more compact distribution of warmer and denser gas in J = 3 − 2 than in J = 1 − 0. We see three peaks corresponding to the 12 CO(J = 1 − 0) peaks, 30 Dor-10, 30 Dor-6, and 30 Dor-12, reported in Johansson et al. (1998). One of them in the north, 30 Dor-06, which is singly peaked in J = 1 − 0 appears to be resolved into two peaks with the present beam. N 159 (Figure 4) Figures 4(a) and (b) show the 12 CO(J = 3 − 2) and 12 CO(J = 1 − 0) distributions in N 159. We note that N 159W shows the strongest intensity among the present clouds as well as a very compact peak which is not sufficiently resolved with the present beam. Its radius is estimated to be a few pc after de-convolution. N 159E also shows a compact distribution with a hint of a sub peak, while N 159S shows similar distributions both in J = 1 − 0 and J = 3 − 2. The east-west elongation of N 159S may be due to the scanning effect of the OTF mapping, and needs to be confirmed. N 171 (Figure 5) Figures 5(a) and (b) show the 12 CO(J = 3 − 2) and 12 CO(J = 1 − 0) distributions in N 171. We note that the J = 3 − 2 emission is weaker than the J = 1 − 0 emission. There are multiple velocity components, at V LSR = 225 km s −1 , 230 km s −1 , and 240 km s −1 in both J = 3 − 2 and J = 1 − 0 as indicated in Table 2 and Table 1 of Kutner et al. (1997). N 166 (Figure 6) Figures 6(a) and (b) show the 12 CO(J = 3 − 2) and 12 CO(J = 1 − 0) distributions in N 166. The J = 1 − 0 data themselves have already been published by Garay et al. (2002). We see four peaks in both J = 3 − 2 and J = 1 − 0. Of these peaks, three peaks are named Cloud-B, Cloud-C, and Cloud-D, respectively, as reported by Garay et al. (2002), although Cloud-C is resolved into two peaks with the present beam and observing grid. N 206 (Figure 7) Figures 7(a) and (b) show the 12 CO(J = 3 − 2) and 12 CO(J = 1 − 0) distributions in N 206. A J = 1 − 0 peak appears to be resolved into two sub peaks and a north-south filamentary structure with the higher angular resolution of the J = 3 − 2 line. Properties of the clumps We identified clumps according to the following way in the J = 3−2 distributions shown in Figures 3 -9; 1) Pick up local peaks using the integrated intensity. 2) Draw a contour at half of the peak integrated intensity level and identify it as a clump unless it contains other local peaks. 3) When there are other local peaks inside the contour, draw new contours at the 70 % level of each integrated intensity peak. Then, identify clumps separately if their contours do not contain another local peak (the boundary is taken at the "valley" between clumps), or else identify a clump by using the highest contour as a clump boundary. 4) If a spectrum has multiple velocity components with a separation of more than 6 km s −1 , identify these components to be associated with different clumps. As a result, 32 clumps have been identified; 5 clumps in 30 Dor region, 10 clumps in N 159 region, 6 clumps in N 171 region, 5 clumps in N 166 region, 2 clumps in N 206 region, 1 clump in N 206D region, and 3 clumps in GMC 225 region. Their line parameters at the peak positions are shown in Table 2, and their physical properties are listed in Table 3. The clump size, line width, and virial mass range 1.1 -12.4 pc, 4.0 -12.8 km s −1 , and 4.6×10 3 -2.2×10 5 M ⊙ , respectively. The smallest clump identified by the procedure above is detected with 2 observing points and its size is 1.1 pc. There are several other weak emissions (below 6 σ), and they are detected with 1 observing point. These emissions are not identified as clumps. When we assume such small clumps (R ∼ 1 pc, dV ∼ 7 km s −1 ) are exist, their virial mass is 9.3 × 10 3 M ⊙ , and this corresponds to detection limit. There are 18 local peaks above the 6 σ noise level which are not identified as clumps, because the stronger peaks are located near these local peaks. Their virial mass is 6.5 × 10 4 M ⊙ , if their size and line width are similar to identified clumps (R ∼ 7 pc, dV ∼ 7 km s −1 ). This seems to correspond to completeness limit. If the clumps are identified with the 70 % level of the peak integrated intensity level instead of the 50 %, the clump size changes to a half of the original one, althogh the line width does not change. Their virial mass also changes to about a half of the original one. Histograms of their physical properties are presented in Figure 10. Typical values are 7 pc, 7 km s −1 , and 6×10 4 M ⊙ , in size, line width, and virial mass, respectively. Their line widths are larger than those of the GMCs in our galaxy (e.g., Williams & Blitz 1998;Ikeda et al. 1999;Sun et al. 2006). We note a trend that the masses of these clumps are relatively large compared with those of high mass cloud cores in the Milky Way which are of the order of 10 3 M ⊙ (e.g., Burton et al. 2005;Yonekura et al. 2005). Precise comparisons will be necessary using same tracers and resolution in the future, because those galactic studies above are based on optically thin mm dust continuum and C 18 O emission. The clumps in the LMC are also fairly compact, with sizes of several pc or less. Hereafter, the region names and the numbers of clumps are used to identify clumps (e.g., "30 Dor No.1"). Data Analysis Derivation of line intensity ratios The spatial resolution of the present CO data varies depending on the telescope and frequency. We convolved and regridded these data into the same resolution and position using 2 dimensional Gaussian smoothing in order to compare them to derive reliable peak intensity ratios of 12 CO(J = 3 − 2) to 12 CO(J = 1 − 0) (hereafter, R 3−2/1−0 ). We made Gaussian fits to each of the 12 CO(J = 3 − 2) and 12 CO(J = 1 − 0) spectra having a single peak in most cases. We derived peak intensities and FWHM line widths through these fittings. The ratios of the two transitions were then derived as the ratio between peak T mb s. The distributions of these ratios are shown along with the 12 CO(J = 1 − 0) distributions in Figures 11 -17 (a). The youngest stellar clusters SWB0, whose ages are estimated to be less than 10 Myrs, are also shown by red circles (Bica et al. 1996). Averaged R 3−2/1−0 over each clump (hereafter, R 3−2/1−0,clump ) was also derived from the averaged 12 CO(J = 3 − 2) and 12 CO(J = 1 − 0) spectra over the each clump. A summary of the R 3−2/1−0,clump for each clump is shown in Table 4. The histogram in Figure 18 shows that the R 3−2/1−0,clump ranges from 0.2 to 1.6. These ratios will be compared with numerical calculations of radiative transfer in the LVG approximation to derive constraints on density and temperature. LVG analysis Calculations of LVG model To estimate the physical properties of the molecular gas in the LMC, we performed an LVG analysis (Goldreich & Kwan 1974) of the CO rotational transitions. The LVG radiative transfer code simulates a spherically symmetric cloud of constant density and temperature with a spherically symmetric velocity distribution proportional to the radius, and employs a Castor's escape probability formalism (Castor 1970). It solves the equations of statistical equilibrium for the fractional population of CO rotational levels at each density and temperature. It includes the lowest 40 rotational levels of the ground vibrational level and uses the Einstein's A and H 2 impact rate coefficients of Schöier et al. (2005). The present calculations incorporate the lowest 40 rotational levels of CO in the ground vibrational state over a kinetic temperature range of T kin = 5 -200 K and a density range of n(H 2 ) = 10 -10 6 cm −3 . We did not include higher energy levels in the present study, which requires including populations in the lower vibrationally excited states. Therefore, the present work does not cover kinetic temperatures above 200 K, which should be dealt with in the future for analyses of higher sub-mm transitions above J = 4 − 3. This imposes a limit of T kin ∼ 200 K in the present study and even higher temperature is not excluded in general below. We performed calculations for 3 different CO fractional abundances; X(CO) = [CO]/[H 2 ] = 1×10 −6 , 3×10 −6 , and 1×10 −5 , and 3 different 12 CO/ 13 CO abundance ratios of 20, 25, and 30 (Heikkila et al. 1999). 4.2.2. Results from 12 CO(J = 3 − 2) and 12 CO(J = 1 − 0) data for 32 clumps First, we assume that X(CO) is uniform among the clumps and derive density and temperature using the LVG results. Figure 19(a) illustrates that we obtain the following lower limits for kinetic temperature and density. For the 8 clumps with R 3−2/1−0,clump ≥ 1 (30 Dor No.1,2,3,4,N 159 No.1,2,6,8), we estimate T kin >30 K and n(H 2 ) >10 3 cm −3 . For the 24 clumps with R 3−2/1−0,clump <1 (30 Dor No.5,N 159 No.3,4,5,7,9,10,N 171 No.1,2,3,4,5,6,N 166 No.1,2,3,4,5,N 206 No.1,2,N 206D No.1,GMC 225 No.1, 2, 3), we estimate T kin >several K and n(H 2 ) >several×10 2 cm −3 . In either case, the lower limits tend to increase with the R 3−2/1−0,clump . Results from 12 CO(J = 3 − 2), 12 CO(J = 1 − 0), and 13 CO(J = 1 − 0) data for 13 clumps We can better constrain these physical parameters when 13 CO(J = 1 − 0) data are available. Figure 19 shows the general behavior of the loci of constant R 3−2/1−0 and constant R 12/13 (peak intensity ratio of 12 CO(J = 1−0) to 13 CO(J = 1−0)) in the density-temperature plane. It is recognized that the combination of the two will allow us to determine the parameters relatively well since the two lines are nearly "orthogonal" in the plane, except for densities higher than ∼ 10 4 cm −3 (Figure 19). Of the 32 clumps, 13 CO(J = 1 − 0) data are available for 13 clumps, including 4 clumps of R 3−2/1−0,clump ≥ 1 (30 Dor No.1,4,N 159 No.1, 2) and 9 clumps with R 3−2/1−0,clump <1 (N 159 No.4,N 166 No.1,3,4,N 206 No.1,2,N 206D No.1,GMC 225 No.1,3). For these 13 clumps, we made a detailed analysis using the R 12/13 at peaks of 12 CO(J = 1 − 0) and have obtained the best constraints. We summarize the input parameters for the 13 clumps in columns 3-5 of Table 5. All the data refer to the 12 CO(J = 1 − 0) beam size, 45 ′′ ; the higher-transition data has been Gaussian smoothed as described in section 4.1. Clump averaged dv/dr were used for the calculations. R 3−2/1−0,clump and R 12/13 at the peak of 12 CO(J = 1 − 0) were used, and the errors of these ratios are both estimated as ±20 %, which are derived from errors of absolute intensity calibration. These correspond 27 σ and 7 σ noise levels of 12 CO(J = 3 − 2) and 12 CO(J = 1 − 0), respectively. This indicates the errors of intensity ratios are dominated by the error of absolute intensity calibration. The way of the clump definition does not change R 3−2/1−0,clump , but changes dv/dr, and dv/dr is about 2 times as large as the original one, when the clumps are identified with the 70 % level of the peak integrated intensity level. This, however, does not affect the results of the LVG calculations. Figure 22. A fractional CO abundance of X(CO) of 3×10 −6 was used throughout. The horizontal axis is molecular hydrogen density (n(H 2 )), and the vertical axis is the gas kinetic temperature (T kin ). Solid lines are R 3−2/1−0,clump and dashed lines are R 12/13 at 12 CO(J = 1 − 0) peaks. Hatched areas indicate the overlap regions of these two ratios within the errors which is allowed from the observed ratios. It also includes the uncertainty due to a possible variation of the 12 CO/ 13 CO abundance ratio from 20 to 30. Hereafter, we shall call clumps above 30 K "warm" and those below "cold". Clumps with densities greater than 10 3.5 cm −3 are referred to as "dense" and those at lower densities as "less dense". Figures 20(b) and (c) indicate that they are warm (T kin = 30 -200 K) and less dense clumps (n = 10 3 cm −3 ) and cold (T kin = 15 -40 K) and less dense (n = 10 3 cm −3 ) clumps, respectively. We find that R 12/13 is useful to discriminate the temperature difference except in the case of 30 Dor No.1. All the figures for the 13 clumps for all different fractional CO abundances (1×10 −6 , 3×10 −6 , and 1×10 −5 ) are given in Appendix A. In Table 5 we present the results of for the 13 clumps for a fixed fractional CO abundance of X(CO) = 3×10 −6 . The results of previous studies are also summarized in Table 5 for comparison. We find that the present results are the most extensive among these studies in terms of the number of samples, while they are basically consistent with previous study for the individual clumps or clouds. There are relatively large differences in both density and temperature in N 159 No.4 (N 159S) only, and this is because of the high density tracers are used in Heikkilä et al. (1999). Figure 21 summarizes the calculated densities and temperatures for the 13 clumps. We see first that the temperature ranges from 10 K up to more than 200 K and density from 10 3 cm −3 to 10 5 cm −3 . Clumps we detected are distributed continuously from cool (∼ 10 -30 K) to warm (∼ higher than 30 -200 K), and warm clumps are distributed from less dense (∼ 10 3 cm −3 ) to dense (∼ 10 3.5−5 cm −3 ), although cool clumps are all less dense. The three cases shown in Figure 20 represent typical cases, "warm and dense" (T kin ∼ higher than 30 -200 K, n(H 2 ) ∼ 10 3.5−5 cm −3 ), "warm and less dense" (T kin ∼ higher than 30 -200 K, n(H 2 ) ∼ 10 3 cm −3 ), and "cool and less dense" (T kin ∼ 10 -30 K, n(H 2 ) ∼ 10 3 cm −3 ). Effects of X(CO) We will now discuss the possible effect of changing X(CO), as summarized in Table 6. If we adopt X(CO) = 1×10 −6 , the contours of R 3−2/1−0 shift to lower temperature and the contours of R 12/13 shift to higher density. Accordingly, the solution shifts to lower temperature, higher density. If we adopt X(CO) = 1×10 −5 , the R 3−2/1−0 contours shift to higher temperature and the R 12/13 contours shift to lower density. Accordingly, the solution shifts to higher temperature, lower density (Table 6). Next, we vary X(CO) from clump to clump. Heikkilä et al. (1999) estimate X(CO) in 30 Dor N 159 No.1 (N 159W),and N 159 No.4 (N 159S) to be 1×10 −6 , 1×10 −5 , and 3×10 −6 , respectively. Table 6 indicates that 30 Dor No.1 shows similar values to Heikkilä et al. (1999), whereas N 159 No.1 and No.4 become warmer and lower in density than in Heikkilä et al. (1999). This discrepancy may be due to the fact that the high density tracers used by Heikkilä et al. (1999) are not used in the present study. To summarize, the assumption of uniform fractional abundance, X(CO), is fairly good and the present results do not show significant difference even if we adopt the different fractional abundances used by previous authors. Comparisons to Hα flux Relation between R 3−2/1−0 and Hα flux We converted the Hα data (Kim et al. 1999) towards the present clouds using the method given in the Appendix B. The typical background level of the Hα flux is ∼ 10 −12 ergs s −1 cm −2 at the 40 ′′ scale and ranges up to 10 −10 ergs s −1 cm −2 towards strong HII regions ( Figure 22). These data were re-gridded into the 12 CO(J = 3−2) data grids. Hα flux images with the youngest stellar clusters SWB0 (Bica et al. 1996, younger than 10 Myrs) are shown in Figures 11 -17 (b) and the 12 CO(J = 3 − 2) contours from Figures 3 -9 are overlayed for comparison in these figures. Figures 11 -17 indicate a clear trend that the R 3−2/1−0 is enhanced to 1.0 -1.5 towards HII regions or clouds with young clusters, as in 30 Dor and N 159 regions. On the other hand, the ratio is low, less than 1.0, towards clumps with neither HII regions nor clusters, as in GMC 225. We also note that the ratio is enhanced towards the regions where Hα is intense or towards the interfaces between clouds and HII regions. A summary of the averaged Hα flux over each clumps is shown in Table 4. It is not fully guaranteed that the Hα emission is actually in contact with the molecular gas and some of the apparent coincidence could be fortuitous. Nonetheless, previous studies comparing CO and Hα flux indicate a strong correlation between them and lend a support to the assumption that almost all the coincidences indicate actual physical association (Fukui et al. 1999;Yamaguchi et al. 2001;Kawamura et al. 2007). Figure 23 shows the correlation between R 3−2/1−0,clump and the averaged Hα flux over each clump. It is clear that R 3−2/1−0,clump is well correlated with the averaged Hα flux, with a correlation coefficient of ∼ 0.79. This is a fairly good empirical relationship and should be tested to see if it holds true in other galaxies. The relation suggests that higher R 3−2/1−0 reflect higher temperatures or higher densities. It is notable that the clumps with the averaged Hα flux greater than 10 −11 ergs s −1 cm −2 correspond to high R 3−2/1−0,clump of 1.0 -1.5. In the barred spiral galaxy M83, the CO(J = 3 − 2)/(J = 1 − 0) integrated intensity ratio exceeds unity at the nucleus, whereas the ratio gradually decreases to 0.6 -0.7 with distance from center. The ratio is constant through the disk region (Muraoka et al. 2007). Figure 24 shows plots of density and temperature as functions of averaged Hα flux, and gives us another insight into these properties. Where averaged Hα is strong (10 −11 ergs s −1 cm −2 ) the clumps are always warm at around T kin = 100 K or more. On the other hand, when averaged Hα is weak, density is always low but temperature can be either high or low. Comparison with physical properties and Hα emisson The high density clumps show high R 3−2/1−0,clump of 1.0 -1.5 and are associated with strong Hα flux, while the low density clumps show low R 3−2/1−0,clump of 0.5 -1.0 with weak Hα flux. Since we are averaging the Hα intensity in each clump, we may be diluting the localized low Hα flux towards some of the clumps in Type III and II GMCs. The effects of Hα emission or nearby clusters on the molecular gas are perhaps local phenomena as indicated by the comparison in Figures 11 -17 for the individual clumps. Discussions Dense and compact clumps as candidates for proto-cluster condensations We have carried out sub-mm 12 CO(J = 3 − 2) observations of GMCs in the LMC which are most extensive and highly sensitive compared to the previous studies. Six GMCs were selected based on the NANTEN CO survey of the LMC, including 3 Type III GMCs actively forming O stars in addition to 3 Type I/II GMCs which are quiet in O-star formation or cluster formation, although the formation of low to intermediate mass star is not excluded. The spatial resolution of ∼ 5 pc and the high sensitivity allowed us to identify 32 molecular clumps in these GMCs and to reveal significant details of the warm and dense molecular gas with n(H 2 ) ∼ 10 3−5 cm −3 and T kin ∼ 10 -200 K. The typical mass of the molecular clumps is large, in the range of 5×10 3 -2×10 5 M ⊙ with radii of 1 -12 pc. Of all of our objects, N 159 No.1 or -W shows the strongest concentration of mass of ∼ 7×10 4 M ⊙ within a radius of ∼ 5 pc. The masses seem to be larger than those of typical Milky Way GMCs such as those in the eta Car region (e.g., Yonekura et al. 2005), although the propeties of these galactic GMCs are based on optically thin C 18 O data. We suggest that these are good candidates for the precursors of rich super clusters like R136 in 30 Dor which includes more than 10 4 stars in a small volume with a radius of ∼ 1 pc. It is of particular interest to look for even denser gas towards them in higher excitation transitions of the sub-mm region. Density and temperature of the clumps and implications The results of our LVG analysis indicate that clumps are distributed from cool to warm in temperature and from less dense to dense in density. These differences of clump properties in density and temperature show good correspondence with the GMC Types based on the star formation activity, as well as with the Hα emission of ionized gas associated with each clump. Clumps in Type III GMCs are warm (T kin ∼ 30 -200 K) and are either dense (n(H 2 ) ∼ 10 3.5−5 cm −3 ) or less dense (n(H 2 ) ∼ 10 3 cm −3 ). Clumps in Type II GMCs are either warm (T kin ∼ 30 -200 K) or cool (T kin ∼ 10 -30 K) and less dense (n(H 2 ) ∼ 10 3 cm −3 ). Clumps in Type I GMC are cool (T kin ∼ 10 -30 K) and less dense (n(H 2 ) ∼ 10 3 cm −3 ). The physical parameters of clumps are generally correlated with the star formation activity of GMCs and can perhaps be interpreted in terms of evolutionary effects. Our interpretation is that defferences of clump density and temperature represent an evolutionary sequence of GMCs in terms of density increase leading to star formation; Type I/II GMCs are at a young phase of star formation where density has not yet reached high enough values to cause active massive star formation, and Type III GMCs represent the later phase where the average density is higher, including both high and low density sub-types. The high density clumps in Type III GMCs show high R 3−2/1−0,clump of 1.0 -1.5 and are associated with strong Hα flux while the low density clumps in Type III GMCs show low R 3−2/1−0,clump of 0.5 -1.0 with weak Hα flux. We suggest two alternative ideas to explain the density difference of the clumps in Type III GMCs; one is that density is being enhanced by shock compression driven by HII regions and the other is that gravitational condensation of each clump plays a role in the density increase. The former may be difficult because the shock front may occupy a small volume which is likely missed with the present 5 pc beam. It seems thus favorable that the latter scenario is working mainly to enhance density. The present study, which resolved the smaller clumps in GMCs at 5 -10 pc scales, indicates that the clumps may have physical properties affected by local properties such as the Hα distribution. It should be interesting to investigate the variations among these internal clumps and their relation to star formation. FUV Heating of the molecular gas in the LMC The present findings that the R 3−2/1−0,clump is well correlated with Hα flux suggests that the heating of molecular gas by far-ultraviolet (FUV) photons may be effective in the LMC where the dust opacity is lower and the FUV intensity is higher than in the Milky Way. The molecular gas in the Milky Way is mainly heated by cosmic ray protons of ∼ 100 MeV as discussed by a number of authors, although the surface layers of molecular clouds with small visual extinctions at Av ∼ a few mag or less may be dominated by the FUV heating (e.g., Kaufman et al. 1999). Some authors have made detailed calculations of gas heating and cooling under the effects of FUV radiation fileds (Kaufman et al. 1999). We shall try to present a picture that can be applied to the present results below. First, the gas temperature is determined through the balance between the cooling and heating. According to Table 4 in Goldsmith & Langer (1978), the total cooling rate is 6.8×10 −27 T 2.2 ergs cm −3 s −1 for X(CO)/(dv/dr) = 4×10 −5 and n(H 2 ) = 10 3 cm −3 . In 30 Dor region, since X(CO)/(dv/dr) = 3×10 −6 / 0.8 = 3.75×10 −6 and this value is 10 times lower than the value used in Goldsmith & Langer (1978), n(H 2 ) = 10 4 cm −3 can be read 10 3 cm −3 , then the cooling rate is estimated as 1.7×10 −22 ergs cm −3 s −1 for T = 100 K. This value is a factor 2 -3 smaller than that of Galactic clouds with n(H 2 ) = 10 4 cm −3 and T = 50 K. We shall assume that the heating by cosmic ray electrons is not important in the LMC. This assumption is not directly confirmed, but it is consistent with the low non-thermal fraction of the LMC's radio continuun emission (e.g., Hughes et al. 2006) and studies that suggest a significant fraction of cosmic ray electrons are able to escape from low luminosity galaxies (e.g., Bell 2003;Skillman & Klein 1988) The FUV flux (G 0 ) is estimated as 3500 for 30 Doradus Poglitsch et al. 1995;Werner et al. 1978;Israel & Koornneef 1979), and 300 for N 159 Israel et al. 1996;Israel & Koornneef 1979). In Orion, it is estimated as 25 Stacey et al. 1993). The FUV flux in the LMC is larger than that in the Milky Way. PDR models are calculated by Kaufman et al. (1999) which incorporate the chemical and physical processes that form and destruct atoms or molecules, as well as ionization effects. Figuer 1 of Kaufman et al. (1999) shows the kinetic temperature for a molecular gas layer with density of n (cm −3 ) under FUV flux of G 0 at the surface. PDR surface temperatures are estimated as listed in Table 7. These indicate that temperature becomes as high as 100 -300 K on the PDR surface under the conditions of the clumps in Type III GMCs in the LMC. These temperatures are basically consistent with the temperatures of the warm clumps in the present sample. Generally speaking, at a scale of ∼ 10 pc, T kin ∼ 100 K seems to be higher than the kinetic temperatures typical in Milky Way GMCs, where the Milky Way values are usually derived from the 12 CO(J = 1−0) emission only (e.g., eta Car T kin ∼ 50 K by Yonekura et al. 2005). This suggests that the heating of molecular clouds may be stronger in the LMC than in the Milky Way and the molecular temperature may be higher. If this is correct, the lower metallicity, resulting in lower extinction, is the basic cause for the higher temperature in addition to the stronger FUV field in the LMC. We shall note in the end that this higher temperature in the molecular gas possibly leads to an increase of the Jeans mass of molecular clumps, which may favor the formation of rich super clusters in the LMC. This is consistent with the higher mass of the molecular clumps which may represent precursors of the clusters. The present work has undertaken to sample 6 GMCs (7 regions) to have a uniform determination of the density and temperature in the LMC. The number of GMCs is still limited to 6 among ∼ 300 detected with NANTEN. We should make more efforts to collect appropriate data sets in the sub-mm wavelengths to improve our understanding of the cloud properties. NANTEN2, ASTE and others will certainly be powerful in achieving this goal. Summary We summarize the results as follows. 1) We have used the ASTE 10m telescope to obtain the distribution of 12 CO(J = 3 − 2) emission at 345 GHz towards 6 GMCs (7 regions) in the LMC. We have identified 32 clumps in these GMCs at ∼ 5pc resolution. The radius, line width and virial mass are estimated as 1.1 -12.4 pc (7 pc), 4.0 -12.8 km s −1 (7 km s −1 ), and 4.6×10 3 -2.2×10 5 M ⊙ (6×10 4 M ⊙ ), respectively, with the average values in the parenthesis. 2) We have compared the present results with LVG radiative line transfer calculations in order to obtain the density and temperature estimated for the 13 clumps using R 3−2/1−0,clump and R 12/13 at 45 ′′ resolution. The clumps are distributed from cool (∼ 10 -30 K) to warm (more than ∼ 30 -200 K) and from less dense (n(H 2 ) ∼ 10 3 cm −3 ) to dense (n(H 2 ) ∼ 10 3.5−5 cm −3 ) 3) The Hα flux towards these clumps is well correlated with the 12 CO(J = 3−2)/ 12 CO(J = 1 − 0) ratio, R 3−2/1−0,clump , and clumps with Hα fluxes greater than 10 −11 ergs s −1 cm −2 have large R 3−2/1−0,clump of ∼ 1.5. The 12 CO(J = 1 − 0) data were taken with the SEST 15m and Mopra 22m telescopes. 4) The typical mass of the molecular clumps ranges 5×10 3 -2×10 5 M ⊙ with radii of 1 -12 pc. Of all of our objects, N 159 No.1 or -W shows the strongest concentration of mass of ∼ 7×10 4 M ⊙ within a radius of ∼ 5 pc. We suggest that these are good candidates for the precursors of rich super clusters like R136 in 30 Dor. 5) We suggest that differences of clump properties represent an evolutionary seqeunce of GMCs in terms of density increase leading to star formations. Type I/II GMCs are at a young phase of star formation where density has not yet reached high enough values to cause active massive star formation, and Type III GMCs represent the later phase where the average density is higher, including both high and low density sub-types. 6) The high kinetic temperature correlated with Hα flux suggests that FUV heating is dominant in the molecular gas of the LMC. The low fraction of non-thermal radio continuum emission and calculations of PDR models support this suggestion. Furthermore, the high temperature in the molecular gas possibly leads to an increase of the Jeans mass of molecular clumps, which may favor the formation of rich super clusters. ASTE is a joint project between Japan and Chile. The telescope is operated by the ASTE team, including NAOJ, University of Tokyo, Nagoya University, Osaka Prefecture University, and Universidad de Chile. We are grateful to all the members of ASTE team. A. Complete LVG results of all clumps Figures 25 -37 show the complete LVG results described in section 4.2. We present all cases: X(CO) of (a)1×10 −6 , (b)3×10 −6 , and (c)1×10 −5 for 13 clumps. B. Hα flux We shall here describe the method of scaling data values in FITS cube (Hα image in Kim et al. (1999); hereafter, "Kim's FITS"). The data values in Kim's FITS is not flux scale, and the calibration is needed for quantitative comparison with CO clouds and R 3−2/1−0 . The procedure is as follows. 1) Sum up the data values of Kim's FITS inside apertures which are listed in Kennicutt & Hodge (1986) The density plots (Figure 39(a) and (b)) show that higher R 3−2/1−0,clump (>1.0) correspond to higher densities of 10 3 to 10 5 cm −3 , while lower R 3−2/1−0,clump (<1.0) correspond to lower densities of around 10 3 cm −3 . The temperature plots (Figure 39(c) and (d)) show that higher R 3−2/1−0,clump (>0.5) correspond to higher temperatures of >30K, while lower R 3−2/1−0,clump (<0.5) correspond to lower temperatures of <30K. Then, roughly speaking, we can say that clumps with R 3−2/1−0,clump lower than 0.5 have lower densities of around 10 3 cm −3 and lower temperatures of <30K, clumps with R 3−2/1−0,clump of 0.5 to 1.0 have lower density around 10 3 cm −3 and higher temperatures of >30K, and clumps with R 3−2/1−0,clump higher than 1.0 have higher densities of 10 3 to 10 5 cm −3 and higher temperatures of >30K, although ratios, densities, and temperatures are with large error bars. Of course, there are great benefits to using R 12/13 in LVG analyses. We could not obtain the above results with only R 3−2/1−0 , as mentioned in section 4. C.2. Physical properties -R 12/13 (Figure 40) The density (Figure 40(a) and (b)) does not show a significant correlation with R 12/13 . The temperature (Figure 40(c) and (d)) shows a good correlation with R 12/13 , that is, higher ratios indicate higher temperatures. Usually, R 12/13 correspond to density, but in this case, due to the LVG analysis using both R 3−2/1−0 and R 12/13 , larger R 12/13 indicate lower density and lower density tends to higher temperature. C.3. R 12/13 -Hα flux (Figure 41) There is not significant relation between R 12/13 and Hα flux, contrary to R 3−2/1−0 ratio. Yonekura, Y., Asayama, S., Kimura, K., Ogawa, H., Kanai, Y., Yamaguchi, N., Barnes, P. J., & Fukui, Y. 2005, ApJ, 634, 476 This preprint was prepared with the AAS L A T E X macros v5.2. , 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, and 52 K km s −1 . , 6, 8, 10, 12, 14, 16, 18, 20, 22, 24 , 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, and 26 K km s −1 . Observed points are indicated by dots. , 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, and 19 K km s −1 . Observed points are indicated by dots. The vertical axis is kinetic temperature T kin , and the horizontal axis is molecular hydrogen density n(H 2 ). Solid lines are R 3−2/1−0,clump , and dashed lines are R 12/13 . Hatched areas are the regions in which these two ratios overlap within intensity calibration errors of 20% and uncertainty due to a possible variation of 12 CO/ 13 CO abandance ratio from 20 to 30. The vertical axis is kinetic temperature, T kin , and the horizontal axis is molecular hydrogen density, n(H 2 ). Solid lines are R 3−2/1−0,clump , and dashed lines are R 12/13 . Hatched areas are the regions in which these two ratios oberlap within intensity calibration errors and uncertainty due to a possible variation of 12 CO/ 13 CO abandance ratio. X(CO) = (a)1×10 −6 , (b)3×10 −6 , and (c)1×10 −5 . The vertical axis is kinetic temperature, T kin , and the horizontal axis is molecular hydrogen density, n(H 2 ). Solid lines are R 3−2/1−0,clump , and dashed lines are R 12/13 . Hatched areas are the regions in which these two ratios oberlap within intensity calibration errors and uncertainty due to a possible variation of 12 CO/ 13 CO abandance ratio. X(CO) = (a)1×10 −6 , (b)3×10 −6 , and (c)1×10 −5 . The vertical axis is kinetic temperature, T kin , and the horizontal axis is molecular hydrogen density, n(H 2 ). Solid lines are R 3−2/1−0,clump , and dashed lines are R 12/13 . Hatched areas are the regions in which these two ratios oberlap within intensity calibration errors and uncertainty due to a possible variation of 12 CO/ 13 CO abandance ratio. X(CO) = (a)1×10 −6 , (b)3×10 −6 , and (c)1×10 −5 . The vertical axis is kinetic temperature, T kin , and the horizontal axis is molecular hydrogen density, n(H 2 ). Solid lines are R 3−2/1−0,clump , and dashed lines are R 12/13 . Hatched areas are the regions in which these two ratios oberlap within intensity calibration errors and uncertainty due to a possible variation of 12 CO/ 13 CO abandance ratio. X(CO) = (a)1×10 −6 , (b)3×10 −6 , and (c)1×10 −5 . 3. The vertical axis is kinetic temperature, T kin , and the horizontal axis is molecular hydrogen density, n(H 2 ). Solid lines are R 3−2/1−0,clump , and dashed lines are R 12/13 . Hatched areas are the regions in which these two ratios oberlap within intensity calibration errors and uncertainty due to a possible variation of 12 CO/ 13 CO abandance ratio. X(CO) = (a)1×10 −6 , (b)3×10 −6 , and (c)1×10 −5 . The vertical axis is kinetic temperature, T kin , and the horizontal axis is molecular hydrogen density, n(H 2 ). Solid lines are R 3−2/1−0,clump , and dashed lines are R 12/13 . Hatched areas are the regions in which these two ratios oberlap within intensity calibration errors and uncertainty due to a possible variation of 12 CO/ 13 CO abandance ratio. X(CO) = (a)1×10 −6 , (b)3×10 −6 , and (c)1×10 −5 . The vertical axis is kinetic temperature, T kin , and the horizontal axis is molecular hydrogen density, n(H 2 ). Solid lines are R 3−2/1−0,clump , and dashed lines are R 12/13 . Hatched areas are the regions in which these two ratios oberlap within intensity calibration errors and uncertainty due to a possible variation of 12 CO/ 13 CO abandance ratio. X(CO) = (a)1×10 −6 , (b)3×10 −6 , and (c)1×10 −5 . The vertical axis is kinetic temperature, T kin , and the horizontal axis is molecular hydrogen density, n(H 2 ). Solid lines are R 3−2/1−0,clump , and dashed lines are R 12/13 . Hatched areas are the regions in which these two ratios oberlap within intensity calibration errors and uncertainty due to a possible variation of 12 CO/ 13 CO abandance ratio. X(CO) = (a)1×10 −6 , (b)3×10 −6 , and (c)1×10 −5 . Table 1 in Fukui et al.(2006b). Column (2): Name of GMC. Column (3): Type of GMC. Column (4): Region name used in this paper. Column (5)- (7): Coordinates used as reference position in each region for these 12 CO(J = 3 − 2) observations. Column (8) (3): The peak main-beam temperature, T mb , of the 12 CO(J = 3 − 2) spectra derived by using a single Gaussian fitting for a spectrum obtained by averaging all the spectra to the beam within a single clump. The intensities refer to the 12 CO(J = 1 − 0) beam size (45 ′′ ). Column (4): The peak main-beam temperature, T mb , of the 12 CO(J = 1−0) spectra derived by using a single Gaussian fitting for a spectrum obtained by averaging all the spectra within a single clump. Column (5): The ratios of T mb (3-2) to T mb (1-0). Column (6): The Hα flux obtained by averaging within a single clump. This work (Uniform X(CO)) X(CO) 3×10 −6 3×10 −6 3×10 −6 n(H 2 ) (cm −3 ) 3×10 3 -3×10 5 3×10 3 -8×10 5 1×10 3 -6×10 3 T kin (K) >50 >30 20 -60 (Different X(CO)) X(CO) 1×10 −6 1×10 −5 3×10 −6 n(H 2 ) (cm −3 ) 6×10 3 -1×10 6 2×10 3 -2×10 5 1×10 3 -6×10 3 T kin (K) >35 >45 20 -60 Heikkilä et al. 1999 X(CO) 1.4×10 −6 1.2×10 −5 4.6×10 −6 n(H 2 ) (cm −3 ) 1×10 5 3×10 5 1×10 5 T kin (K) 50 25 10 3. 1 . 1Distributions of the 12 CO(J = 3 − 2) emission In Figure 2, typical 12 CO(J = 3 − 2) and 12 CO(J = 1 − 0) profiles of the 30 Dor, N 159, and GMC 225 regions are presented. The upper panels, Figures 2(a) -(c), show 12 CO(J = 3 − 2) profiles. These illustrate the low noise levels of the present data, typically, ∼ 0.25 K rms at 0.44 km s −1 velocity resolution. Among the present observed positions, the 12 CO(J = 3 − 2) intensity is strongest at T mb ∼ 12.3 K towards N 159W (Figure 2(b) Figure 8 ) 8Figures 8(a) and(b)show the 12 CO(J = 3 − 2) and 12 CO(J = 1 − 0) distributions in N 206D. We can see a head-tail structure in J = 3 − 2, although it appears more rounded in J = 1 − 0.3.1.7. GMC 225 (Figure 9)Figures 9(a) and (b) show the 12 CO(J = 3 − 2) and 12 CO(J = 1 − 0) distributions in GMC 225. We note that J = 3 − 2 emissions are weaker than J = 1 − 0. We describe three typical cases; (a) 30 Dor No.1, (b) N 206 No.1, and (c) GMC 225 No.1 are shown in clump of ∼ 1.4, suggesting that it is a warm (T kin = 50 -200 K) and dense (n = 10 3−5 cm −3 ) clump (Figure 20(a)). N 206 No.1(Figure 20(b)) and GMC 225 No.1(Figure 20(c)) show of R 3−2/1−0,clump ∼ 0.7 and ∼0.4, respectively. A part of this study was financially supported by MEXT Grant-in-Aid for Scientific Research on Priority Area (No. 15071202 and No. 15071203) and by JSPS (No. 14102003, core-to-core program 17004, and No. 18684003). T.M. is supported by JSPS Research Fellowships for Young Scientists. M.R. is supported by the Chilean Center for Astrophysics FONDAP No. 15010003. SK was supported in part by Korea Science and Engineering Foundation (KOSEF) under a cooperative agreement with the Astrophysical Research Center of the Structure and Evolution of the Cosmos (ARCSEC). for each listed HII region. 2) Plot cataloged values of Hα flux in Kennicutt & Hodge (1986) as a function of summed values derived in 1). They are well fitted by a power function of y = 5.16 × 10 −15 x 0.9 (c.c. = 0.94) (Figure 38). 3) Convert data values of Kim's FITS to Hα flux scale (ergs s −1 cm −2 ) using the function derived above. C. LVG results in the other planes C.1. Physical properties -R 3−2/1−0,clump (Figure 39) Fig. 2 . 2-12 CO(J = 3 − 2) and 12 CO(J = 1 − 0) profiles at selected points. Note that the vertical scale is main beam temperature, T mb . The velocities (abscissae) are relative to the LSR.(a) 12 CO(J = 3 − 2) profile of 30 Dor. (b) 12 CO(J = 3 − 2) profile of N 159. (c) 12 CO(J = 3 − 2) profile of GMC 225. (d) 12 CO(J = 1 − 0) and smoothed 12 CO(J = 3 − 2) profiles of 30 Dor. (e) 12 CO(J = 1 − 0) and smoothed 12 CO(J = 3 − 2) profiles of N 159. (f) 12 CO(J = 1 − 0) and smoothed 12 CO(J = 3 − 2) profiles of GMC 225.Fig. 3.-(a) Contour map of 12 CO(J = 3 − 2) integrated intensity in the 30 Doradus region. The contour levels are 5, 10, 15, 20, 25, 30, 35, 40, and 45 K km s −1 . Observed points are indicated by dots. (b) Contour map of 12 CO(J = 1 − 0) integrated intensity in the 30 Doradus region. The contour levels are 10, 15, 20, and 25 K km s −1 .Fig. 4.-(a) Contour map of 12 CO(J = 3 − 2) integrated intensity in the N 159 region. The contour levels are 5, 10, 20, 30, 40, 50, 60, 70, 80, 90 and 100 K km s −1 . Observed points are indicated by dots. (b) Contour map of 12 CO(J = 1 − 0) integrated intensity in the N 159 region. The contour levels are 10, 20, 30, and 40 K km s −1 . Fig. 5.-(a) Contour map of 12 CO(J = 3 − 2) integrated intensity in the N 171 region. The contour levels are 8, 12, 16, and 20 K km s −1 . Observed points are indicated by dots. (b) Contour map of 12 CO(J = 1 − 0) integrated intensity in the N 171 region. The contour levels are 8 Fig. 6 . 6-(a) Contour map of 12 CO(J = 3 − 2) integrated intensity in the N 166 region. The contour levels are 6, 8, 10, 12, 14, 16, 18, 20, and 22 K km s −1 . Observed points are indicated by dots. (b) Contour map of 12 CO(J = 1 − 0) integrated intensity in the N 166 region. The contour levels are 8, 10, 12, 14, 16, 18, 20, and 22 K km s −1 . Observed points are indicated by dots.Fig. 7.-(a) Contour map of 12 CO(J = 3 − 2) integrated intensity in the N 206 region. The contour levels are 4 , and 26 K km s −1 . Observed points are indicated by dots. (b) Contour map of 12 CO(J = 1 − 0) integrated intensity in the N 206 region. The contour levels are 4, 6, 8, 12, 14, 16, and 18 K km s −1 . Observed points are indicated by dots.Fig. 8.-(a) Contour map of 12 CO(J = 3 − 2) integrated intensity in the N 206D region. The contour levels are 4, 6, 8, 10, 12, 14, 16, and 18 K km s −1 . Observed points are indicated by dots. (b) Contour map of 12 CO(J = 1 − 0) integrated intensity in the N 206D region. The contour levels are 4 Fig. 9 . 9-(a) Contour map of 12 CO(J = 3 − 2) integrated intensity in the GMC 225 region. The contour levels are 3, 4, 5, 6, 7, and 8 K km s −1 . Observed points are indicated by dots. (b) Contour map of 12 CO(J = 1 − 0) integrated intensity in the GMC 225 region. The contour levels are 3 - 36 - 36Fig. 10.-Histograms of physical properties of 12 CO(J = 3 − 2) clumps: (a)size, (b)line width, and (c)virial mass. Fig. 11 . 11-(a) Color map of R 3−2/1−0 , and (b) Hα flux image of the 30 Doradus region. Contours are 12 CO(J = 3 − 2) integrated intensity. Contour levels are the same as Figure 3a. Thick lines indicate the observed area in 12 CO(J = 3 − 2), and the red circle indicates the position of young cluster (<10 Myrs; SWB0). Fig. 12 . 12-(a) Color map of R 3−2/1−0 , and (b) Hα flux image of the N 159 region. Contours are 12 CO(J = 3 − 2) integrated intensity. Contour levels are the same as Figure 4a. Thick lines indicate observed area of 12 CO(J = 3 − 2), and the red circles indicate the position of young clusters (<10 Myrs; SWB0). Fig. 13 . 13-(a) Color map of R 3−2/1−0 , and (b) Hα flux image of the N 171 region. Contours are 12 CO(J = 3 − 2) integrated intensity. Contour levels are the same as Figure 5a. Thick lines indicate observed area of 12 CO(J = 3 − 2).Fig. 14.-(a) Color map of R 3−2/1−0 , and (b) Hα flux image of the N 166 region. Contours are 12 CO(J = 3 − 2) integrated intensity. Contour levels are the same as Figure 6a. Thick lines indicate observed area of 12 CO(J = 3 − 2).Fig. 15.-(a) Color map of R 3−2/1−0 , and (b) Hα flux image of the N 206 region. Contours are 12 CO(J = 3 − 2) integrated intensity. Contour levels are the same as Figure 7a. Thick lines indicate observed area of 12 CO(J = 3 − 2).Fig. 16.-(a) Color map of R 3−2/1−0 , and (b) Hα flux image of the N 206D region. Contours are 12 CO(J = 3 − 2) integrated intensity. Contour levels are the same as Figure 8a. Thick lines indicate observed area of 12 CO(J = 3 − 2). Fig. 17 . 17-(a) Color map of R 3−2/1−0 , and (b) Hα flux image of the GMC 225 region. Contours are 12 CO(J = 3 − 2) integrated intensity. Contour levels are the same as Figure 9a. Thick lines indicate observed area of 12 CO(J = 3 − 2). Fig. 18 . 18-Histgram of clump averaged peak intensity ratio of 12 CO(J = 3 − 2) to 12 CO(J = 1 − 0) (R 3−2/1−0,clump ). Fig. 19 . 19-Contour plots of LVG analysis for reference. Contours are (a) R 3−2/1−0 , (b) R 12/13 , and (c) a+b. X(CO)=3×10 −6 , dv/dr=1.0km s −1 pc −1 , and abundance ratio of 12 CO/ 13 CO is 25.Fig. 20.-Contour plots of LVG analysis of 3 clumps: (a)30 Dor No.1, (b)N 206 No.1, and (c)GMC 225 No.1. Fig. 21 . 21-Plot of LVG results. The vertical axis is kinetic temperature, T kin , and the horizontal axis is molecular hydrogen density n(H 2 ). Fig. 22 . 22-Histgram of clump averaged Hα flux. Background level is nearly 10 −12 ergs s −1 cm −2 . -49 -Fig. 23.-Plots of R 3−2/1−0,clump as a function of clump averaged Hα flux by (a)region and (b)GMC type, respectively. Fig. 24 . 24-Plots of physical properties as a function of clump averaged Hα flux. (a) n(H 2 ) by region. (b) n(H 2 ) by GMC type. (c) T kin by region. (d) T kin by GMC type. Fig. 25 . 25-Contour plots of LVG analysis of 30 Dor No.1. The vertical axis is kinetic temperature, T kin , and the horizontal axis is molecular hydrogen density, n(H 2 ). Solid lines are R 3−2/1−0,clump , and dashed lines are R 12/13 . Hatched areas are the regions in which these two ratios oberlap within intensity calibration errors and uncertainty due to a possible variation of 12 CO/ 13 CO abandance ratio. X(CO) = (a)1×10 −6 , (b)3×10 −6 , and (c)1×10 −5 .Fig. 26.-Contour plots of LVG analysis of 30 Dor No.4.The vertical axis is kinetic temperature, T kin , and the horizontal axis is molecular hydrogen density, n(H 2 ). Solid lines are R 3−2/1−0,clump , and dashed lines are R 12/13 . Hatched areas are the regions in which these two ratios oberlap within intensity calibration errors and uncertainty due to a possible variation of 12 CO/ 13 CO abandance ratio. X(CO) = (a)1×10 −6 , (b)3×10 −6 , and (c)1×10 −5 .Fig. 27.-Contour plots of LVG analysis of N 159 No.1. Fig. 28 . 28-Contour plots of LVG analysis of N 159 No.2. Fig. 29 . 29-Contour plots of LVG analysis of N 159 No.4. Fig. 30 . 30-Contour plots of LVG analysis of N 166 No.1. Fig. 31 . 31-Contour plots of LVG analysis of N 166 No. Fig. 32 . 32-Contour plots of LVG analysis of N 166 No.4. Fig. 33 . 33-Contour plots of LVG analysis of N 206 No.1. Fig. 34 . 34-Contour plots of LVG analysis of N 206 No.2. Fig. 35 . 35-Contour plots of LVG analysis of N 206D No.1. The vertical axis is kinetic temperature, T kin , and the horizontal axis is molecular hydrogen density, n(H 2 ). Solid lines are R 3−2/1−0,clump , and dashed lines are R 12/13 . Hatched areas are the regions in which these two ratios oberlap within intensity calibration errors and uncertainty due to a possible variation of 12 CO/ 13 CO abandance ratio. X(CO) = (a)1×10 −6 , (b)3×10 −6 , and (c)1×10 −5 .Fig. 36.-Contour plots of LVG analysis of GMC 225 No.1.The vertical axis is kinetic temperature, T kin , and the horizontal axis is molecular hydrogen density, n(H 2 ). Solid lines are R 3−2/1−0,clump , and dashed lines are R 12/13 . Hatched areas are the regions in which these two ratios oberlap within intensity calibration errors and uncertainty due to a possible variation of 12 CO/ 13 CO abandance ratio. X(CO) = (a)1×10 −6 , (b)3×10 −6 , and (c)1×10 −5 .Fig. 37.-Contour plots of LVG analysis of GMC 225 No.3. The vertical axis is kinetic temperature, T kin , and the horizontal axis is molecular hydrogen density, n(H 2 ). Solid lines are R 3−2/1−0,clump , and dashed lines are R 12/13 . Hatched areas are the regions in which these two ratios oberlap within intensity calibration errors and uncertainty due to a possible variation of 12 CO/ 13 CO abandance ratio. X(CO) = (a)1×10 −6 , (b)3×10 −6 , and (c)1×10 −5 .Fig. 38.-Plot of relation between integrated Hα flux and FITS value. The solid line represents the least-squares fit. Fig. 39 . 39-Plots of physical properties as a function of R 3−2/1−0,clump . (a) n(H 2 ) by region. (b) n(H 2 ) by GMC type. (c) T kin by region. (d) T kin by GMC type. Fig. 40 . 40-Plots of physical properties as a function of R 12/13 . (a) n(H 2 ) by region. (b) n(H 2 ) by GMC type. (c) T kin by region. (d) T kin by GMC type. -67 -Fig. 41.-Plots of R 12/13 as a function of clump averaged Hα flux (a) by region and (b) by GMC type. -(10): Telescope used for each observation. a These coordinates are used as reference positions in each regions for this 12 CO(J = 3 − 2) observations b References for the positions. Note. -Column(1): Regions. Column(2): Gas density. Column(3): FUV flux in units of local interstellar value; 1.6×10 −3 ergs cm −2 s −1 . Column(4): Derived PDR surface temperature of the atomic gas. Column(5): References of FUV flux. Table 1 . 1List of observed GMCs and transitionsNote. -Column (1): Running number of GMC used inGMC Position a Telescope Num. 1 Name 1 Type 2 Region Name α(1950) δ(1950) Ref. b 12 CO(J=3-2) 12 CO(J=1-0) 13 CO(J=1-0) (h m s) ( • ′ ′′ ) 186 LMC N J0538-6904 III 30 Dor 5 39 0.2 -69 8 0.0 3 ASTE MOPRA SEST 3 197 LMC N J0540-7008 III N 159 5 40 18.2 -69 47 0.0 3 ASTE MOPRA SEST 3 N 171 5 40 24.1 -70 8 0.0 4 ASTE MOPRA · · · 216 LMC N J0544-6923 II N 166 5 44 52.5 -69 26 39.1 5 ASTE SEST 5 SEST 5 153 LMC N J0530-7106 III N 206 5 31 33.9 -71 10 0.0 6 ASTE SEST SEST 156 LMC N J0532-7114 II N 206D 5 32 52.2 -71 16 0.0 6 ASTE SEST SEST 225 LMC N J0547-7041 I GMC 225 5 48 35.7 -70 40 0.0 6 ASTE SEST SEST Table 2 . 2Observed properties of 12 CO(J = 3 − 2) clumpsPosition Peak properties Region No. α(1950) δ(1950) T mb V LSR ∆V I.I. Another Ident. (h m s) ( • ′ ′′ ) (K) (km s −1 ) (km s −1 ) (K km s −1 ) 30 Dor 1 5 39 8.6 -69 6 15 5.2 250.4 8.4 49.5 30Dor-10 1 2 5 38 54.6 -69 8 0 4.3 246.9 6.9 31.7 30Dor-12 1 3 5 38 49.0 -69 4 30 3.1 251.9 5.4 22.5 4 Table 3 . 3Physical properties of 12 CO(J = 3 − 2) clumpsClump properties Region No. ∆V clump R deconv M vir (km s −1 ) (pc) (×10 4 M ⊙ ) Table 4 . 412 CO(J = 3 − 2)/ 12 CO(J = 1 − 0) ratio and Hα flux Note. -Column (1): Region. Column (2): Running number in each region. ColumnRegion No. T mb (3-2) T mb (1-0) R 3−2/1−0,clump Averaged Hα flux (K) (K) (×10 −12 ergs s −1 cm −2 ) 30 Dor 1 1.63 1.20 1.4 56.49 2 1.76 1.12 1.6 73.37 3 1.56 1.13 1.2 24.09 4 1.98 1.58 1.3 12.83 5 0.73 0.87 0.8 39.71 N 159 1 5.50 5.04 1.1 9.28 2 3.69 3.34 1.1 9.37 3 2.92 3.25 0.9 6.36 4 2.33 3.34 0.7 1.14 5 1.84 2.19 0.8 1.93 6 1.61 1.57 1.0 1.42 7 1.67 1.98 0.9 3.14 8 0.78 0.57 1.4 1.52 9 0.80 0.99 0.8 1.26 10 0.57 0.84 0.7 1.42 N 171 1 1.04 2.20 0.5 1.00 2 0.69 1.55 0.4 0.93 3 1.02 3.09 0.3 0.97 4 1.34 3.89 0.3 0.98 5 0.88 1.92 0.4 0.95 6 0.87 2.72 0.3 0.97 N 166 1 1.97 3.09 0.6 1.05 2 1.59 2.60 0.6 1.01 3 1.17 1.92 0.6 1.06 4 0.79 1.26 0.6 1.31 5 0.62 0.92 0.7 1.06 N 206 1 1.83 2.39 0.8 3.26 2 1.34 2.44 0.5 2.68 N 206D 1 2.26 4.70 0.5 1.11 GMC 225 1 1.04 2.81 0.4 0.93 2 0.55 1.62 0.3 0.91 3 0.77 2.09 0.4 0.92 Table 5 . 5Summary of LVG and MPE analysesNote. -Column (7): Results of LVG analysis Column (8)-(13): Results of previous studies, including both LVG and MEP analyses.n(H 2 ) (cm −3 ) Table 6 . 6Effect of X(CO)30 Dor No.1 N 159 No.1 N 159 No.4 Table 7 . 7Estimated PDR surface temperaturesRegion n G 0 Ts a References (cm −3 ) (K) 30 Dor 10 4 3500 300 1, 2, 3, 4 N 159 10 4 300 100 1, 4, 5 a Estimation is done by usingFigure 1ofKaufman et al. 1999 . 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[ "Intense Electromagnetic Outbursts from Collapsing Hypermassive Neutron Stars", "Intense Electromagnetic Outbursts from Collapsing Hypermassive Neutron Stars" ]	[ "Luis Lehner \nDepartment of Physics\nUniversity of Guelph\nN1G 2W1GuelphOntarioCanada\n\nPerimeter Institute for Theoretical Physics\nN2L 2Y5WaterlooOntarioCanada\n\nCIFAR, Cosmology & Gravity Program\nCanada\n", "Carlos Palenzuela \nCanadian Institute for Theoretical Astrophysics\nM5S 3H8TorontoOntarioCanada\n", "Steven L Liebling \nDepartment of Physics\nLong Island University\n11548New YorkUSA\n", "Christopher Thompson \nCanadian Institute for Theoretical Astrophysics\nM5S 3H8TorontoOntarioCanada\n", "Chad Hanna \nPerimeter Institute for Theoretical Physics\nN2L 2Y5WaterlooOntarioCanada\n" ]	[ "Department of Physics\nUniversity of Guelph\nN1G 2W1GuelphOntarioCanada", "Perimeter Institute for Theoretical Physics\nN2L 2Y5WaterlooOntarioCanada", "CIFAR, Cosmology & Gravity Program\nCanada", "Canadian Institute for Theoretical Astrophysics\nM5S 3H8TorontoOntarioCanada", "Department of Physics\nLong Island University\n11548New YorkUSA", "Canadian Institute for Theoretical Astrophysics\nM5S 3H8TorontoOntarioCanada", "Perimeter Institute for Theoretical Physics\nN2L 2Y5WaterlooOntarioCanada" ]	[]	We study the gravitational collapse of a magnetized neutron star using a novel numerical approach able to capture both the dynamics of the star and the behavior of the surrounding plasma. In this approach, a fully general relativistic magnetohydrodynamics implementation models the collapse of the star and provides appropriate boundary conditions to a force-free model which describes the stellar exterior. We validate this strategy by comparing with known results for the rotating monopole and aligned rotator solutions and then apply it to study both rotating and non-rotating stellar collapse scenarios, and contrast the behavior with what is obtained when employing the electrovacuum approximation outside the star. The non-rotating electrovacuum collapse is shown to agree qualitatively with a Newtonian model of the electromagnetic field outside a collapsing star. We illustrate and discuss a fundamental difference between the force-free and electrovacuum solutions, involving the appearance of large zones of electric-dominated field in the vacuum case. This provides a clear demonstration of how dissipative singularities appear generically in the non-linear timeevolution of force-free fluids. In both the rotating and non-rotating cases, our simulations indicate that the collapse induces a strong electromagnetic transient, which leaves behind an uncharged, unmagnetized Kerr black hole. In the case of sub-millisecond rotation, the magnetic field experiences strong winding and the transient carries much more energy. This result has important implications for models of gamma-ray bursts. Even when the neutron star is surrounded by an accretion torus (as in binary merger and collapsar scenarios), a magnetosphere may emerge through a dynamo process operating in a surface shear layer. When this rapidly rotating magnetar collapses to a black hole, the electromagnetic energy released can compete with the later output in a Blandford-Znajek jet. Much less electromagnetic energy is released by a massive magnetar that is (initially) gravitationally stable: its rotational energy is dissipated mainly by internal torques. A distinct plasmoid structure is seen in our non-rotating simulations, which will generate a radio transient with subluminal expansion, and greater synchrotron efficiency than is expected in shock models. Closely related phenomena appear to be at work in the giant flares of Galactic magnetars.Recent success in studying the behavior of plasmas around magnetized, spinning, stable neutron stars within flat spacetime has been presented in Refs.[7][8][9]. To study arXiv:1112.2622v4 [astro-ph.HE] 5 Sep 2012where σ ab is the conductivity of the fluid. The wellknown scalar Ohm's law is recovered for σ ab = g ab σ.	10.1103/physrevd.86.104035	[ "https://arxiv.org/pdf/1112.2622v4.pdf" ]	119,231,976	1112.2622	928c536224abc5054164b6ab3583ba8a6462f1e7	Intense Electromagnetic Outbursts from Collapsing Hypermassive Neutron Stars Luis Lehner Department of Physics University of Guelph N1G 2W1GuelphOntarioCanada Perimeter Institute for Theoretical Physics N2L 2Y5WaterlooOntarioCanada CIFAR, Cosmology & Gravity Program Canada Carlos Palenzuela Canadian Institute for Theoretical Astrophysics M5S 3H8TorontoOntarioCanada Steven L Liebling Department of Physics Long Island University 11548New YorkUSA Christopher Thompson Canadian Institute for Theoretical Astrophysics M5S 3H8TorontoOntarioCanada Chad Hanna Perimeter Institute for Theoretical Physics N2L 2Y5WaterlooOntarioCanada Intense Electromagnetic Outbursts from Collapsing Hypermassive Neutron Stars We study the gravitational collapse of a magnetized neutron star using a novel numerical approach able to capture both the dynamics of the star and the behavior of the surrounding plasma. In this approach, a fully general relativistic magnetohydrodynamics implementation models the collapse of the star and provides appropriate boundary conditions to a force-free model which describes the stellar exterior. We validate this strategy by comparing with known results for the rotating monopole and aligned rotator solutions and then apply it to study both rotating and non-rotating stellar collapse scenarios, and contrast the behavior with what is obtained when employing the electrovacuum approximation outside the star. The non-rotating electrovacuum collapse is shown to agree qualitatively with a Newtonian model of the electromagnetic field outside a collapsing star. We illustrate and discuss a fundamental difference between the force-free and electrovacuum solutions, involving the appearance of large zones of electric-dominated field in the vacuum case. This provides a clear demonstration of how dissipative singularities appear generically in the non-linear timeevolution of force-free fluids. In both the rotating and non-rotating cases, our simulations indicate that the collapse induces a strong electromagnetic transient, which leaves behind an uncharged, unmagnetized Kerr black hole. In the case of sub-millisecond rotation, the magnetic field experiences strong winding and the transient carries much more energy. This result has important implications for models of gamma-ray bursts. Even when the neutron star is surrounded by an accretion torus (as in binary merger and collapsar scenarios), a magnetosphere may emerge through a dynamo process operating in a surface shear layer. When this rapidly rotating magnetar collapses to a black hole, the electromagnetic energy released can compete with the later output in a Blandford-Znajek jet. Much less electromagnetic energy is released by a massive magnetar that is (initially) gravitationally stable: its rotational energy is dissipated mainly by internal torques. A distinct plasmoid structure is seen in our non-rotating simulations, which will generate a radio transient with subluminal expansion, and greater synchrotron efficiency than is expected in shock models. Closely related phenomena appear to be at work in the giant flares of Galactic magnetars.Recent success in studying the behavior of plasmas around magnetized, spinning, stable neutron stars within flat spacetime has been presented in Refs.[7][8][9]. To study arXiv:1112.2622v4 [astro-ph.HE] 5 Sep 2012where σ ab is the conductivity of the fluid. The wellknown scalar Ohm's law is recovered for σ ab = g ab σ. I. INTRODUCTION Understanding the gravitational collapse of a massive neutron star is of central importance for its connection to some of the most energetic astrophysical phenomena. Such an event may take place within a core-collapse supernova [1], or in the late stage of a binary neutron star merger [2,3], and is widely believed to power some types of gamma-ray bursts. In recent years it has been realized that newly formed stellar mass black holes may be prodigious sources of electromagnetic energy, in addition to driving strong kinetic outflows [1,[4][5][6]. We are faced with the exciting possibility of probing the most extreme forms of gravitational collapse using coordinated measurements of electromagnetic transients and gravitational waves. Refining models that make observational predictions for joint gravitational and electromagnetic radiation is critical in order to establish efficient observation campaigns for both traditional astronomers and gravitational-wave astronomers. It is also a key step toward predicting the delay between peak electromagnetic and gravitational wave emission, the electromagnetic emission pattern (i.e. the beaming angle) and the electromagnetic and gravitational wave spectrum. Our focus here is on the evolution of the neutron star's magnetic field during its collapse to a black hole. We employ fully self-consistent relativistic calculations that follow the dense stellar material as well as the strong electromagnetic and gravitational fields. A particularly intense electromagnetic transient is generated if the initial magnetic field is very strong (∼ 10 15 − 10 16 G), that is, if the star is a magnetar. Rapid rotation will enhance the energy of the transient, to a degree that can only be derived by a full time-evolution of the electromagnetic and gravitational fields. Rapid rotation also provides a context for generating strong magnetic fields: when the neutron star is accreting, the shear layer at its surface is a promising site for dynamo action. A distinct magnetosphere will emerge and hold off the accretion flow if even ∼ 10 −4 −10 −3 of the energy that is dissipated in the shear layer is converted to a poloidal magnetic field. Understanding the evolution of the magnetic field around an isolated star is therefore potentially of key relevance for the collapsar and binary merger scenarios of gamma-ray bursts. the collapse problem, one needs general relativity -to account for the role of spacetime curvature; and general relativistic magnetohydrodynamics (GRMHD) -to determine the internal evolution of the star. The electromagnetic (EM) phenomena outside the star can be approached in a variety of ways: through a full GRMHD calculation, which generally is very expensive given the low matter density; or more approximately using vacuum EM and force-free equations. In this paper, we compare all three approaches (restricting to ideal MHD in the case of GRMHD). Within general relativity, studies paying attention to neutron star collapse have been presented in the context of isolated stellar collapse (e.g. [10][11][12][13][14]) and binary neutron star mergers (e.g. [15][16][17][18][19][20]) with different degrees of realism. The role of electromagnetic fields are typically examined within general relativistic ideal MHD and studies have been presented for single [21][22][23][24][25][26] and binary star systems [17,18,27]. Such simulations have illustrated important details of the extreme dynamical behavior induced by the system which could trigger tremendously energetic phenomena. Newtonian and general relativistic simulations have shed light on how a binary neutron star merger can amplify pulsar-strength magnetic fields by several orders of magnitude [18,[27][28][29]. The resulting hypermassive star is generically unstable to black hole formation, which opens up the tantalizing possibility that the increasing rotation rate and magnetic field strength would drive an intense electromagnetic outflow. To study such a scenario, our goal here is to combine the GRMHD approach with a suitable description of the magneticallydominated region outside a collapsing star, by applying the force-free approximation. Our approach is related to that of Ref. [30], which matched a simplified, analytic, relativistic solution for the interior of a collapsing star (dust-ball) with a numerical solution of the coupled Einstein-Maxwell equations in its exterior. Boundary conditions at the star's surface are provided by the analytic, ideal MHD solution. We go beyond this simplified scenario in three significant ways. First, our stars are evolved through collapse consistently by evolving the ideal GRMHD equations, and we can therefore consider in principle any kind of compact star. In particular, we have studied both rigidly rotating and non-rotating stars, whereas only the nonrotating case was considered in [30]. Second, the magnetosphere is described within the force-free approximation, which, as argued in Ref. [31], is a much more realistic model that, for instance, has been instrumental in understanding pulsar spin down [8]. Third and last, our matching of the exterior solution with the interior one is dynamical. Thus the force-free solution can adapt to time-dependent fields sourced by the star. Our approach thus allows us to examine many interesting scenarios and we apply it here to study the behavior of collapsing, magnetized, compact stars (either rotating or not), the behavior of surrounding plasma (as described within the force-free approximation) and possible electromagnetic radiation induced by the system. We compare our results with recent estimates in Ref. [32] for the nonrotating case which predict that the collapse process is smooth and that the magnetic field remains anchored to the star as a black hole forms, leaving a final black hole with a split-monopole field configuration. Our work, which follows the dynamics of the system, indicates that in both cases the stars radiate significant electromagnetic energy in which reconnection plays a crucial role, and that this radiation ceases shortly after the formation of a black hole. With force-free, such a black hole loses its electromagnetic hair within a dynamical time scale. We describe our hybrid approximation and the evolution equations in Sec. II, followed by a summary of the numerical techniques in Sec. III. Our choice of initial data is described in Sec. IV, and several tests of the hybrid approach are presented in Sec. V. The new results for collapsing, magnetized stars are explained in detail in Sec. VI, while the astrophysical consequences are described in Secs. VII and VIII. We conclude in Sec. IX with some final comments. II. APPROACH In the presence of matter and electromagnetic fields, the Einstein equations must be suitably coupled to both the Maxwell and hydrodynamics equations. This coupling is achieved by considering the stress energy tensor T ab = T fluid ab + T em ab ,(1) with contributions from matter and electromagnetic energy given respectively by T fluid ab = [ρ o (1 + ) + P ] u a u b + P g ab ,(2)T em ab = F a c F bc − 1 4 g ab F cd F cd .(3) A perfect fluid with pressure P , energy density ρ o , internal energy , and four-velocity u a describes the matter state, and the Faraday tensor F ab describes the electromagnetic field. The fluid and electromagnetic components are directly coupled through Ohm's law, which closes the system of equations by defining the electric current 4-vector J a as a function of the other fields. A general relativistic expression can be obtained by considering a multifluid system of charged species [33], leading to a fully non-linear propagation equation for the spatial component of the current. However, it usually suffices to consider a simplified version accounting for an algebraic relation between the current and the fields [34] The equation of motion for the fluid and electromagnetic field are obtained from the conservation laws ∇ a T ab = 0 ; ∇ a (ρ o u a ) = 0 ; (5) ∇ a F ab = J b ; ∇ a * F ab = 0 ; (6) which, together with the Einstein equations G ab = 8πT ab , complete the system of equations governing the dynamics. Once the appropriate form of Ohm's law and the conductivity have been specified, using for instance the algebraic relation of Eq. (4), the resulting equations typically involve vastly different scales, rendering the implementation of these equations quite costly from a computational point of view [114]. Fortunately, for specific regimes certain useful approximations capture the most relevant physics while bypassing the most strenuous difficulties. For our current purposes, involving magnetized neutron stars, the following relevant approximations can be defined and employed in different regimes: • The ideal-MHD equations are obtained by requiring that the current remains finite in the limit of infinite conductivity, σ → ∞. This condition also implies the vanishing of the electric field measured by an observer co-moving with the fluid (F ab u b = 0). This approximation is appropriate for the highly conducting matter expected in neutron stars. However, the numerical evolution of the ideal MHD equations typically fails in low density regions where the inertia of the electromagnetic field is a few orders of magnitude larger than that of the fluid unless sufficient resolution is available. Such resolution requirements increase as the ratio of the electromagnetic to the fluid's inertia (or magnetic to fluid's pressure) grows. Consequently, the approach becomes costlier in regions with decreasing physical relevance with respect to bulk matter motion. Such a situation arises in particular in "vacuum" regions due to the standard practice of maintaining a density floor (a so called atmosphere) in regions of low density to exploit advanced numerical techniques for relativistic hydrodynamics. The atmosphere's density is much smaller than that inside the star, so this approach does not affect the relevant matter physics. However outside the star, the fluid inertia (or pressure) is typically much smaller than that of the electromagnetic field in magnetized cases and one generally encounters a large number of numerical difficulties. Different numerical strategies are often introduced to avoid them but such measures can limit one's ability to extract appropriate physics in these regions. • The force-free Maxwell equations are obtained by assuming the fluid's inertia is much smaller than that of the electromagnetic fields. As a result F ab J b = 0 which in turn implies F ab * F ab = 0 (→ E.B = 0) [31,35]. This assumption therefore allows one to ignore the explicit time-evolution of the fluid as its dynamics are implicitly prescribed by the charge and current distribution. This approximation is well suited to the dynamics of the low density, magnetically dominated plasma surrounding a compact object, but it cannot account for the physics in dense regions. • The vacuum Maxwell equations are trivially recovered assuming no coupling with matter (σ = 0). This is a natural approximation in vacuum regions, far away from the compact objects of interest. As mentioned, computational costs are presently major obstacles to employ the general equations. It is thus highly desirable to define a scheme able to model all the relevant regimes in highly dynamical systems simultaneously since none of these three approaches captures all the expected behavior. In particular we have in mind interacting binary neutron stars, black hole-neutron star binaries as well as the collapse of a magnetized star considered here. All three systems are believed to play an important role in understanding gamma ray bursts and other energetic events driven by compact stellar-mass objects. In what follows, we describe and apply a hybrid approach which, while approximate, can account for the dynamical interaction of both gravitating and electromagnetically driven fluids. Such an approach allows one to study the magnetosphere's behavior, in particular energetics and field topology providing important clues for understanding relevant systems. To do so, we take into account that the electromagnetic inertia of a region with high conductivity (i.e. inside the star) would be orders of magnitude larger than that of the plasma region. Thus, we can ignore the back-reaction of the electromagnetic field in the plasma region onto that inside the star (i.e., a "passive" magnetosphere). We exploit this observation to define our approach, which employs both the ideal and force-free approximations suitably matched around the stellar surface. The matching procedure is such that the star's electromagnetic field provides the boundary conditions for the surrounding region treated with the force-free approximation, but the behavior of the magnetospheric plasma on the high density stellar interior is ignored. Notice that this approach, in a sense, can be regarded as a natural extension of that adopted for studying pulsar-spin down by magnetosphere interactions [7,8] to general relativistic dynamical contexts. In these works, only the behavior of the magnetospheric plasma is studied and the star's influence is accounted for by boundary conditions. We here account for the possible stellar dynamics and consequent influence on the boundary conditions defined for the force-free approximation. In practical terms, we regard our system as described by two sets of electromagnetic fields: (i) the ideal MHD fields {E i , B i } and (ii) the force-free fields {Ẽ i ,B i }, governed by their respective equations. Both sets of fields are defined with respect to observers orthogonal to the spacelike hypersurfaces employed to foliate the spacetime. Thus identifying the appropriate fields for matching is direct. The ideal MHD equations are evolved over the entire computational domain, as is customary. The matching region is determined by fluid density being some value above the vacuum region (though alternatives based on other physical quantities are obviously possible). In particular, the force-free fields are evolved only in regions where ρ o < ρ match . Dynamic boundary conditions are applied to the force-free fields on the surface at which ρ o = ρ match using the ideal MHD fields E i bc = − i jk v j B k ;(7)B i bc = B i .(8) These conditions are applied in the spirit of penalty techniques [36] in which the equations are modified with driving terms at boundary points in order to enforce the desired boundary condition. In particular, we extend the penalty technique across a number of grid points to effect both the boundary conditions and to "turn-off" the evolution of the force-free fields within the higher density regions. Notice that this extension is not mathematically rigorous, but its usage will be justified by examining the solution in different test applications and comparing with the expected behavior. In practice we define a smooth kernel F (x i , x i match ) defined as F (ρ o , ρ match ) = 2 1 + e 2 K (ρo−ρ match )(9) where typically we adopt K ≈ 0.001/ρ atmos and ρ match ≈ 200 − 2000 ρ atmos , being ρ atmos the value for the density of the atmosphere. In the collapsing cases studied, this value is rescaled in time by the ratio of maximum density to the initial maximum density (see appendix). The values found in Eqs. (7)(8) along with the kernel Eq. (9) make their appearance in the equations determining {Ẽ i ,B i } (together with "constraint cleaning" fields {Ψ,φ}, introduced to ensure constraints are well behaved through the evolution [37]) which are ∂ tẼ i = F L βẼ i + ijk ∇ j ( αB k ) − α γ ij ∇ jΨ + α trKẼ i − 4παJ i + λ(1 − F )(E i bc −Ẽ i ) ,(10)∂ tB i = F L βB i − ijk ∇ j ( αẼ k ) − α γ ij ∇ jφ + α trKB i + λ(1 − F )(B i bc −B i ) ,(11)∂ tΨ = F L βΨ − α ∇ iẼ i 4πα q − ασ 2Ψ − λ(1 − F )Ψ ,(12)∂ tφ = F L βφ − α ∇ iB i − ασ 2 φ − λ(1 − F )φ .(13) The final term on the right-hand side of these equations is a penalty factor, which is introduced to impose interface conditions and ensure that a discrete energy norm is bounded. Details of this "penalty technique" are presented in [36] and examples of applications in general relativity can be found in [38]. The above equations, together with the Einstein and GRMHD equations are implemented as described in [25,39,40] to which we refer the reader for further details. III. NUMERICAL TECHNIQUES We adopt finite difference techniques on a regular Cartesian grid to solve the system. To ensure sufficient resolution in an efficient manner we employ adaptive mesh refinement (AMR) via the HAD computational infrastructure that provides distributed, Berger-Oliger style AMR [41,42] with full sub-cycling in time, together with an improved treatment of artificial boundaries [43]. The refinement regions are determined using truncation error estimation provided by a shadow hierarchy [44] which adapts dynamically to ensure the estimated error is bounded within a pre-specified tolerance. The spatial discretization of the geometry and force-free fields is performed using a fourth order accurate scheme satisfying the summation by parts rule, and High Resolution Shock Capturing methods based on the HLLE flux formulae with PPM reconstruction are used to discretize the fluid variables [15,45]. The time-evolution is performed through the method of lines using a third order accurate Runge-Kutta integration scheme, which helps to ensure stability of the numerical implementation [18]. We adopt a Courant parameter of λ = 0.2 so that ∆t l = 0.2∆x l on each refinement level l. On each level, one therefore ensures that the Courant-Friedrichs-Levy (CFL) condition dictated by the principal part of the equations is satisfied. To extract physical information, we monitor several quantities: (i) the matter variables and spacetime behavior, (ii) the electromagnetic field configuration and fluxes, and (iii) the electromagnetic Newman-Penrose (complex) radiative scalar (Φ 2 ). This scalar is computed by contracting the Maxwell tensor with a suitably defined null tetrad Φ 2 = F ab n amb ,(14) and it accounts for the energy carried off by outgoing waves to infinity. The luminosity of the electromagnetic waves is L em = dE em dt = F em dΩ = lim r→∞ r 2 \|Φ 2 \| 2 dΩ .(15) Additionally we monitor the ratio of particular components of the Maxwell tensor Ω F = F tr F rφ = F tθ F θφ ,(16) which, in the stationary, axisymmetric case, can be interpreted as the rotation frequency of the electromagnetic field [35]. IV. INITIAL DATA Initial data for the hybrid equations involve the intrinsic metric (g ij ) and extrinsic curvature (K ij ) on a given hypersurface, as well as the magnetized fluid configuration in terms of its primitive variables (ρ, , v i , B i ). The initial data for the geometry and the fluid of rigidly rotating neutron stars are provided by the LORENE package Magstar [46], which adopts a polytropic equation of state P = Kρ Γ with Γ = 2 rescaled to K = 100. Because the fluid inertia of a neutron star is many orders of magnitude larger than its electromagnetic one, the magnetic field will have a negligible effect on both the geometry and the fluid structure, and so it can be specified freely. Unless noted otherwise, in our simulations we have chosen a dipolar structure for the initial magnetic field. The electric fields are set by assuming the ideal MHD condition Eq. (7), with zero fluid velocity in the exterior region. Additionally, we require initial data for the forcefree fields {Ẽ i ,B i }. Inside the star, they are defined to be exactly the same as their ideal MHD counterparts. Outside the star, the magnetic field is well defined by the dipolar solution, while the electric field is computed by assuming that the magnetosphere rotates rigidly with the star up to R e = 2R s (R s is the stellar radius), and imposing again the ideal MHD condition for the electric field. This configuration provides consistent data for the problem; however such data will not necessarily conform to the physical situation considered and so an unphysical early transient will be generated. V. TESTING THE APPROACH We first establish that the adopted procedure is indeed able to capture correctly the dynamics of relevant systems. Since the ideal MHD approximation is selfconsistently evolved throughout the computational domain, our tests must address the behavior of the forcefree fields. To do so, we first examine the convergence of our implementation and then illustrate that the approach provides the expected behavior by comparing with certain recently studied cases. A. Non-rotating, magnetized star with a dipole magnetic field Adopting a stable and non-rotating stellar solution with mass M = 1.63M and equatorial radius R eq = 8.62 km, from which we remove all initial pressure, we examine convergence of the force-free fields as the star collapses. This scheme contains "standard" sources of Results from FMR solutions at three different resolutions for the same nonrotating collapsing star. Shown are the two differences between the resolutions. The difference inBz along y = 0 in the equatorial plane at t = 0.06ms is first interpolated onto a uniform mesh for each solution. The difference between the medium and high resolutions is smaller than that between the low and medium resolutions, and this decrease as one increases the resolution indicates that the scheme is convergent. (Notice convergence in the central region is affected by initial data errors induced by depleting the pressure to induce the collapse.) error, such as: (i) truncation error of our finite difference approximation of Maxwell's equations, (ii) errors associated with our application of the force-free conditions, (iii) the various numerical errors (truncation and constraint violations) associated with our GRMHD implementation as well as (iv) errors associated with our dynamic boundary condition that matches the force-free fields to the corresponding MHD fields inside the star. A detailed analysis of these errors is delicate and involved; however we illustrate that the fields obtained by this approach converge to a unique solution with increasing grid resolution. We therefore evolve the collapse of a non-rotating star at three different resolutions with fixed mesh refinement (FMR). We subtract a high resolution run (with coarse level grid of 257 3 points) and a medium resolution run (of dimension 193 3 ) and do similarly for the medium and low resolutions (of dimension 129 3 ). These two differences are plotted at t = 0.06ms (i.e., when the radius of the star starts to shrink) in Fig. 1 which shows that the difference decreases with increasing resolution, consistent with convergence. B. Magnetic monopole Next we consider a stationary force-free solution representing a rotating neutron star with a monopole magnetic 20 40 r(km) field [47], and compare to our results. To minimize effects due to oscillations of the star (induced by perturbations to the star induced by the discretization), the geometry and MHD fields are kept to their initial values and are not evolved. The initial data correspond to a rigidly rotating neutron star near the mass shedding limit, with a mass M = 1.84M and an equatorial radius R eq = 12 km, rotating with a period of T = 0.886 ms. The magnetic field is given by B r = αB 0 (R s /r) 2 , regularized conveniently near the origin. As described earlier, the initial electric field satisfies the ideal MHD condition in a rigidly co-rotating magnetosphere which extends initially up to R e ∼ 2 R s . The evolution is performed in a cubic domain of length L = 136 km ≈ 14 R s with only two FMR grids and the star is placed at the origin. The maximum resolution is ∆x = 0.72 km, so that roughly 30 grid points cover the star. This resolution is sufficient for this test as both the geometry and matter variables are kept fixed and only the force-free equations are evolved until reaching a quasi-stationary configuration. The behaviour of different fields across the surface of the star after the solution has settled are displayed in Fig. 2, which plots radial profiles of: the kernel function F , the density ρ o , and the non-trivial components of the magnetic field. As illustrated in the figure, the radial magnetic field only changes outside the star, where there also appears a toroidal component indicating the rotation of the magnetic fields in the magnetosphere. Overall the electromagnetic fields are seen to relax rather quickly to a state approaching the expected one. Another measure of the obtained solution is given by the (normalized) rotational frequency of the magnetic field Ω F in Fig. 3. As time progresses, this value approaches 1 as expected in a radially smooth way. C. Aligned rotator A particularly challenging test of our approach is the aligned rotator solution (see [8]). The aligned rotator is a numerical solution of the force-free equations outside a rotating surface representing a star that demonstrates closed field lines within the light cylinder (LC). This problem has been studied by a number of authors [7,8,[48][49][50] working in flat space with a computational domain consisting only of the stellar exterior. In these works the star's influence is accounted for through suitable boundary conditions derived from the expected electromagnetic field at its surface. These efforts were specifically motivated to obtain the solution in the magnetosphere of isolated pulsars and, in particular, to understand a possible spin-down mechanism that works even when the star's dipolar field is aligned with its angular momentum. As already mentioned, our approach differs in that we solve for the stellar interior, we work within curved space and our computational grid does not conform to the star's geometry [115]. Because of these differences and also because our initial data for the force-free fields is only nonzero in the immediate neighborhood of the star, achieving the expected late-stage stationary solution dynamically constitutes a demanding test. We note that the time and length scales required for such a test with respect to a realistic star are too computationally demanding if we are to capture the main physical aspects of the solution (i.e. field topology, location of the light-cylinder, development of a current sheet, etc). Instead of a realistic star, we adopt a rapidly rotating star so that the light cylinder is brought closer to the star - where there is enough resolution to resolve it-and render the dynamical time scales shorter and easier to follow numerically. To achieve such rapid rotation, we resort to an unstable star but prevent the instability from disrupting the star by artificially freezing both fluid and geometry. Consequently we only evolve the force-free equations and compare the obtained configuration with the expected solution. We adopt a rotating star with a mass M = 1.7M , equatorial/polar radius of R eq = 8.5/6.0 km and rotational period T = 0.64 ms. The light cylinder for this star is located at R LC = c/Ω = 31 km ≈ 3.6 R s . The electromagnetic field is initially set similar to that in the previous monopole test, but with a poloidal magnetic field given by a dipole outside the star. The evolution is performed in a cubic domain of L = 184 km ≈ 22 R s with four FMR levels with resolutions ∆x = (0.25, 0.5, 1.0, 2.0) km. The FMR hierarchy consists of centered cubes with side lengths L = (2.8, 5.6, 11.2, 22)R s , so that there are roughly 70 points across the star in the equatorial plane. We evolve this star until the solution settles to a quasistationary solution. Fig. 4 illustrates the magnetic field topology at three representative times. In the last frame, we show the configuration to which the field settles, illustrating features predicted by known solutions for the aligned rotator. In particular, one can observe a region of closed field lines bounded at the expected radius, denoting the light cylinder. VI. COLLAPSE OF A MAGNETIZED STAR Having tested our hybrid approximation to EM field evolution, we now focus on the collapse of a magnetized star. The star may or may not be rotating, the rotating case being of greater astrophysical relevance. In particular, we are interested in studying how the magnetosphere responds, and the distribution of escaping electromagnetic radiation. Here it is essential to capture both the dynamics of the collapsing star and the surrounding magnetoplasma, since the two are tightly coupled. It is worth recalling that the introduction of rotation introduces a fundamental difference in the radiative output of stationary (noncollapsing) stars as calculated in the electrovacuum and force-free limits: the spindown torque of an aligned rotator vanishes in electrovacuum but does not in the forcefree case. Furthermore, this torque is strong and its magnitude varies only modestly with inclination angle [8]. Previous studies of the magnetosphere of a collapsing star are fairly limited. Some pioneering work in [51] focused on the electrovacuum behavior of a nonrotating star. While this study included the effects of strong gravity, by neglecting plasma or rotation it was not able to capture the winding of the magnetic field during the collapse. The analytic study by [32] estimated the EM output during the collapse by applying the formula for the spindown luminosity of an equilibrium rotator, L sd ∝ B 2 s R 6 s Ω 4 , where R s is the stellar radius, B s the surface magnetic field, and Ω the spin frequency. However, when the collapse time is shorter than the rotation period (as must be the case if the star is to avoid a rotational hang-up), the outer magnetosphere near the light cylinder is not able to follow the change in the surface magnetic field, and this equilibrium formula does not apply. Instead, twisting of the closed magnetic field lines is expected at a radius r max ∼ ct col /3 [116], where t col ≡ R s /\|Ṙ s \| is the collapse time. The correspond- ing toroidal magnetic field is B φ ∼ (Ωr max /c)B(r max ), where B(r max ) ∼ (3R s /ct col ) 3 B s in a dipole geometry. The power injected into the magnetosphere is L sd ∼ 1 6 B 2 s R 2 s c ΩR s c 2 3R s ct col 2(17) which is larger than the equilibrium spindown power by a factor ∼ (Ωt col /3) −2 . To determine the dependence of L sd on R s , equation (17) must be combined with the conservation of magnetic flux, B s R 2 s = constant; and the appropriate scaling between Ω and R s (Ω ∝ R −2 s for self-similar collapse). The collapse does not, in general, follow a simple power-law relation between t col and R s , but in the special case of pressureless collapse from a large radius ( t col ∝ R 3/2 s ) one obtains L sd ∝ R −4 s . It should be re-emphasized that equation (17) represents energy stored in the magnetosphere, and so the maximum amount of energy that can only escape to infinity after the collapse is completed. During the last, relativistic stages of the collapse, additional physical effects arise. Spacetime curvature has a mixed effect on magnetic field winding in the magnetosphere. On the one hand, the rotation frequency is reduced with respect to the Newtonian value; on the other hand, torsional Alfvén waves in the force-free magnetosphere slow down significantly as the horizon approaches the surface of the star. The net result, as we show, is that the magnetic field becomes strongly wound up if the star is rotating close to breakup before the collapse, and develops a/M 0.5 after the collapse. To clarify the influence of plasma and rotation, and to make useful comparisons, we have made a parallel set of runs in the electrovacuum approximation. These are obtained straightforwardly by setting J i = 0 and not enforcing the conditions {E.B = 0, \|E\| < \|B\|} in Eqs. (10)(11)(12)(13). A. Non-rotating stellar collapse The collapse of a magnetized, but non-rotating, neutron star was studied in [30], where a magnetic field frozen into the stellar surface was matched to an exterior, vacuum solution of Maxwell's equations. This calculation followed the transition to a black hole, and the ringdown of the EM field threading the horizon. In reality, the medium outside the star is an excellent electrical conductor. As is argued in [31,35], a combination of large voltages and strong gravitational fields will trigger runaway pair creation near compact objects [31,35]. In a strongly dynamic situation, the number of particles generated can be enhanced even further, e.g. by a Kolmogorov-like transfer of energy from large-scale waves to internal plasma heat [52]. The remnant of a binary merger or a rapidly rotating stellar core collapse is Here t = 0 denotes the time that the horizon appears. The magnetic field lines are dragged by the star during the collapse, producing Alfvén waves in the magnetosphere that carry a small fraction of the magnetospheric energy and stretch the magnetic field lines near the equatorial plane. Most of the EM energy falls into the black hole (the solid white, central rectangle denotes the excision region for the singularity). also a strong neutrino source and, therefore, its magnetosphere will be filled with a much denser baryonic plasma compared with pulsar magnetospheres [53,54]. But in most cases, the plasma around compact objects is magnetically dominated and its dynamics is nearly force free. Therefore, we expect significant differences in the time evolution between the electrovacuum exterior assumed by [30] and a force-free magnetosphere. To anchor the external force-free (or electrovacuum) solution, we adopt a marginally unstable, non-rotating star with a mass M s = 1.63M and a radius R s = 8.62 km. The numerical domain extends up to L = 16 R s and contains three centered FMR grids of sequentially half sizes (and hence twice better resolved). The highest resolution grid has ∆x = 0.18 km. The star collapses to a black hole in ∼ 1 ms [25]. We set t = 0 at the onset of an apparent horizon and display the magnetic field and the density at various times in Fig. 5. This force-free solution has several salient features. There is an early transient in which the dipole magnetic field relaxes to a solution consistent with the physical configuration. Subsequently, as the star collapses, the magnetic field lines are gradually stretched along the equatorial plane. After 1.0 ms an apparent horizon appears in the interior of the star, which grows as it swallows all the remaining fluid in ∼ 0.15 ms. As the outer layers of the star are accreted by the black hole, the stretched magnetic field lines near the equatorial plane reconnect and form closed loops that carry away electromagnetic energy and magnetic flux (note the field loops in the last frame of Fig. 5). The EM field evolution shows qualitative differences in the force-free and electrovacuum runs, especially after the black hole forms (Fig. 6). Starting from a common dipole structure at the beginning of the collapse, the magnetic field in the force-free case becomes radially stretched near the magnetic equator, and maintains a consistent sign. In the electrovacuum evolution, the stellar magnetic field disconnects more readily from the exterior. Changes in the connectivity of the magnetic field follow the appearance of zones where E 2 > B 2 , as is demonstrated in Figs. 10 and 12. For a more quantitative discussion, we consider in Fig. 7 the electric and magnetic fluxes, computed over a surface located at r = 1.5R s , as a function of time. As expected, both the total (signed) fluxes remain small throughout the simulation, indicating that essentially no spurious magnetic/electric charges are created during the collapse. The unsigned magnetic flux decays exponentially in both the electrovacuum and force-free cases. Interestingly, the decay rate in the former case matches the l = 1, m = 0 quasi-normal mode for electromagnetic perturbations of a Schwarzschild black hole with mass M = 1.63M [55]. The decay rate of the magnetic flux in the force-free run is roughly twice the electrovacuum result. One could translate the measured e-folding time t E into an "effective reconnection speed," V rec ≈ GM c −2 t E −1 by observing that reconnection takes place within r (1 → 2)r H (with r H the horizon radius). Our results indicate V rec 0.14c. This result may appear paradoxical at first sight, and deserves some comment. Electrovacuum magnetic fields effectively reconnect (by converting to electric-dominated fields) at the speed of light, as we discuss in more detail in Sec. VI A 4. Why then is the decay faster in the force-free case? The answer appears to reside in the selfinductance of the black hole. As magnetic field lines reconnect through the equator, they generate a strong toroidal EMF. In a vacuum, this EMF sources a magnetic flux of the opposing sign. When conducting matter The magnetic field lines are dragged by the star during the collapse, stretching the magnetic field lines near the equatorial plane until that the fluid is swallowed by the black hole. Afterward, the EM dynamics is mostly described by the Quasi Normal Modes of the system. is present, there is no oscillation in the sign of the flux threading the hole. Rather, reconnection is a monotonic process and, after the formation of an x-point, the magnetic field lines interior to the x-point fall through the horizon. Electromagnetic Output In order to estimate the efficiency with which EM energy is radiated to infinity, we have scaled the timeintegrated luminosity to the peak electromagnetic energy contained in the magnetosphere. This peak energy is reached approximately at the formation of the apparent horizon. In the absence of rotation, it is C peak ∼ 2 times the initial dipole energy, as determined by the conservation of magnetic flux. The measured radiated energy is, in turn, a fraction rad of C peak E dipole,0 . Numerically, E dipole,0 ≈ (2π/3)(B 2 pole /8π)R 3 s = (1/12)B 2 pole R 3 s , and one finds E dipole,0 = 1.4 × 10 47 B 2 pole,15 erg for a (initial) polar field B pole and radius R s ≈ 12 km. Hence E rad ≈ 1.4 × 10 47 C peak rad B 2 pole,15 erg.(18) The radiative efficiency is very small in the force-free case, rad = 0.008; the rest of the peak energy is swallowed by the black hole. This is illustrated in Fig. 8, which shows how rad grows with time in both the forcefree and electrovacuum runs. The radiated energy is E rad ≈ 10 45 B 2 pole,15 ergs, from Eq. (18), expressed in terms of magnetar-strength magnetic fields. Most of this is radiated in a short interval ≈ 1 ms surrounding the collapse, with an average luminosity L ≈ 10 48 B 2 pole,15 erg s −1 . The radiative efficiency measured at r = 1.5R s is an order of magnitude higher in the electrovacuum case. This indicates that a larger proportion of the electromagnetic energy falls into the black hole in the force-free run, instead of escaping to infinity. An important check of this result is to measure the dissipation at current sheets in both the force-free and electrovacuum simulations. In both cases, we find that the integral of E.J is a small fraction of the total EM energy in the magnetosphere. (The constraint E 2 < B 2 is maintained in the force-free case by applying a small enhanced resistivity; hence the reduction in electric field energy appears as E.J dissipation.) An important feature of this radiation in both cases is its predominantly dipolar structure in energy flux (i.e., L ∝ sin 2 (θ)). In the force-free case, energy is radiated in a rather continuous manner and it propagates outwards with a velocity v = 0.89 c, equivalent to a mild Lorentz factor of W = 2.2. For the electrovacuum case the energy is radiated mainly in two long bursts instead of the several, and with shorter periods, resulting in the force-free case. Late Force-Free Evolution and Magnetic Reconnection At later times, when the fluid has completely fallen into the black hole, the field lines that were dragged toward the horizon reconnect near the equatorial plane in a few sequential bursts, expelling most of the remaining magnetic flux. This behavior appears to be associated with the formation of x-type singularities in the magnetic field. Since the field is stretched radially and then reconnects near the horizon, the resulting electromagnetic pulse bunches up in the radial direction, which explains the structure seen in the last panel of Fig. 5. X-point reconnection appears to happen easily in forcefree plasmas; in ohmic plasmas it is associated with inhomogeneities in the electrical resistivity (e.g see [56]). In the absence of a detailed microphysical model for the resistivity, fine details such as these should be treated Time integral of the electromagnetic luminosity, normalized with respect to the peak EM energy of the magnetosphere (i.e., around the formation of the black hole), in both the force-free and electrovacuum simulations. In the electrovacuum case, the net EM output is ∼ 8%, ten times larger than that radiated in the force-free case. with caution, and the calculation should be viewed as illustrative. Vacuum versus Force-Free EM Field Evolution in Axisymmetric, Non-Rotating Collapse Some important features of the vacuum evolution of the electromagnetic field around a collapsing star can be understood by neglecting the effects of spacetime curvature, and by considering a simplified trajectory for the surface of the star. If the collapse starts at a finite time, then an initially potential magnetic field evolves into a hybrid structure that consists of an inner potential magnetic field that matches the surface boundary condition as determined by the conservation of magnetic flux; and a transient electromagnetic wave that propagates into the original field structure. Of especial interest is the appearance of zones within this wave structure that are dominated by the electric field. If the collapse continues for a long time (the final stellar radius R s is small compared with the initial radius), then this zone where E 2 > B 2 extends over a wide range of radius. Since a realistic magnetosphere may contain enough free charges to limit the growth of E · B, this provides a nice example of how the nearly force-free evolution of an electromagnetic field can lead to strong dissipation. In the absence of rotation, a spherically symmetric collapse implies a radial fluid velocity inside the star. As long as the stellar surface contracts with a speed −Ṙ s c, the magnetic field near it is approximately potential. For a pure multipole of order , B(r) = B s (t)R s (t) ∇ P (cos θ) (r/R s ) +1(19) (with B s the magnetic field at the star's surface). A toroidal electric field E φ = −(Ṙ s /c) B θ is present at the surface of the star, assuming its interior to be perfectly conducting. The junction condition at the surface ensures the continuity of E φ . The surface magnetic field therefore increases in accordance with simple flux conservation, ∂B s ∂t +Ṙ s ∂B s ∂r Rs = −2Ṙ s R s B s .(20) At a fixed radius r, the magnetic field grows weaker: in the case of a simple dipole, the stellar magnetic moment scales as µ(R s ) = B s (R s )R 3 s = µ 0 (R s /R s0 ). If the star were to reach infinite density at a finite time t col , then a strong toroidal electric field would develop at r > c(t col − t). The inner potential zone would shrink along with the star as t approaches t col . The external electromagnetic field can then be obtained by rescaling the radius, r → ξ ≡ r/R s (t), and transforming derivatives according to ∂ t X(r, t) → [∂ t + (Ṙ s /R s )∂ ξ ]X(ξ, t). It is simplest to solve for the vector potential A φ , from which the poloidal magnetic field and toroidal electric field are derived. The boundary condition at the surface of the star is R s (t)A φ [R s (t), θ] = B s (t)R 2 s (t) dP dθ = R s0 A φ0 (θ) dP dθ .(21)From Eq. (20), R s0 A φ0 (θ) is constant. Substituting A φ (r, θ) = R s0 A φ0 (θ)g(ξ, t) into the wave equation ∂ 2 t (rA φ ) = ∂ 2 r (rA φ ) − ( + 1) r 2 rA φ ,(22) and adopting a collapse law R s (t) ∝ (t col − t) α (here α = 2/3 for pressureless collapse from a large radius), we find ∂ 2 τ g + 1 α ∂ τ g + 2ξ∂ τ ∂ ξ g = c 2 R 2 s − ξ 2 ∂ 2 ξ g − 1 + 1 α ξ∂ ξ g − c 2 R 2 s ( + 1) ξ 2 g.(23) Here τ = αdt/(t col − t) is a dimensionless time coordinate; hence R s (τ ) = R s0 e −τ . The electromagnetic field is constructed from the solution to Eq. (23) using B r (r, t) = R s0 r 2 g(r/R s , t) sin θ ∂ θ (sin θA φ0 ) ; B θ (r, t) = − R s0 A φ0 r 2 (ξ∂ ξ g) ξ=r/Rs ; E φ (r, t) = −Ṙ s c R s0 A φ0 rR s (∂ τ g + ξ∂ ξ g) ξ=r/Rs .(24) One recovers the usual potential solution A φ (r, θ) = A φ [R s (t), θ] (r/R s ) −( +1) where ξ c/\|Ṙ s \|. A selfsimilar solution is also available in the case of collapse at a uniform speed, α = 1, if the collapse starts at a very large initial radius. Then one can take ∂ τ g = 0 and the electromagnetic field is a function only of r/R s . In this case, the magnetic field can retain a dipolar form out to large distances r c(t col − t) from the star. Restricting to = 1 gives B r (r, t) = R s R s0 2µ 0 cos θ r 3 ; B θ (r, t) = R s R s0 µ 0 sin θ r 3 ; E φ (r, t) = r c(t col − t) B θ (r, t).(25) Although the magnetic field is identical to that sourced by a stationary dipole of magnitude µ(R s ) = µ 0 (R s /R s0 ), the electric field energy dominates outside a distance ∼ (c/\|Ṙ s \|)R s (t) from the star. The similarity solution is accurate out to the larger distance ∼ (c/\|Ṙ s \|) R s0 , where R s0 is the stellar radius at the beginning of the collapse. To understand how this inner solution for the electromagnetic field matches onto the initial potential magnetic field that was present prior to the collapse, or to consider cases other than constantṘ s , one must calculate the full time-dependent solution to (23). We have done this by evolving g(ξ, τ ), ∂ τ g(ξ, τ ) and ∂ ξ g(ξ, τ ) using a centered discretization of Eq. (23) and employing a small Kreiss-Oliger dissipation value O(10 −3 ) to damp high-frequency noise in each of these variables. The resulting electromagnetic field profile is plotted in Fig. 9, with distance normalized to the initial radius of the star. One observes in Fig. 10 the emergence of an extended zone with E 2 > B 2 , as is expected from the similarity solution of Eq. (25). The magnetic field is relatively stronger in the polar regions, since A φ ∝ sin θ → 0 at small θ. Eventually the amplitude of the outgoing wave disturbance becomes large enough that B θ changes sign. After this happens, the magnetic field remains dipolar inside the radius ∼ (c/\|Ṙ s \|)R s0 , but disconnects from the external zone of undisturbed potential field. This transition is illustrated in Fig.11 using two snapshots corresponding to the two most extended field profiles in Fig. 9. The first part of the collapse leads to a re-arrangement of the magnetic field outside the star, while electromagnetic energy flows inward: the Poynting flux, S r = −E φ B θ c −Ṙ s c A 2 φ0 (θ)R 2 s0 R s (t) 4 (∂ ξ g) 2 ξ(26) is negative inside r ∼ (c/\|Ṙ s \|)R s0 . After the horizon forms and reaches the surface of the star, the compression of the magnetic field stops, and some of the trapped magnetic field can be radiated to infinity. The angular distribution of this radiated energy reflects the symmetry of the initial field. In the case of a dipole, the energy is carried away by closed magnetic loops and is mainly channeled through the magnetic equator. We note that the appearance of zones with E 2 > B 2 in the electrovacuum solution implies a fundamental difference with the alternative force-free solution. In this case, E ·B = 0 throughout the collapse, so consistent force-free evolution is obtained only by removing energy from the electric field. From an MHD perspective, such a transition signals the appearance of nearly luminal plasma motions, where the inertia of even a small residue of entrained matter can become important. This simple example shows that there can be profound differences in the macroscopic structure of the electromagnetic fields when even a small amount of conducting matter is present. In the complete absence of free charges, macroscopic zones where E 2 > B 2 are the natural consequence of the time-evolution of an EM field that, initially, is purely magnetic. The quasi-normal mode (QNM) behavior observed shows two remarkable features: an oscillation in the sign of the magnetic flux threading each hemisphere, and the appearance of an equatorial zone with E 2 > B 2 . These two features can be related to each other using a simple planar analogy. Consider an initial magnetic field configuration B = B 0x (−B 0x ) for y > 0 (< 0). This is familiar from studies of conducting fluids, where in the case of uniform FIG. 11: Connectivity of the magnetic field around a collapsing star (uniform dRs/dt). The dipolar magnetic field that is anchored in the star disconnects from the surrounding, undisturbed, potential field when the amplitude of the output wave becomes large enough that B θ reverses sign. Note that E 2 > B 2 (the red sections of the field lines) in the outer part of the inner zone with a dipolar magnetic field line profile. These two frames display the solution at the two latest times shown in Fig. 9 in which B θ becomes negative. resistivity one finds a slow flow of matter to a thin current sheet at y = 0. The sheet thickness is limited by the flow of matter along the sheet to large \|x\| (Sweet-Parker reconnection). But now we are interested in the case where the conducting matter is absent, and the electromagnetic field evolves according to the vacuum wave equation. One finds, instead, a growing zone of pure electric field E = −B 0ẑ for \|y\| < ct, which maintains a uniform sign across the initial magnetic null surface. A pure magnetic field is converted to a pure electric field. In the case of EM fields localized around a black hole, this conversion of magnetic to electric fields occurs at the magnetic equator, where the poloidal field lines merge together. The toroidal electric field that is created sources a poloidal magnetic flux through the horizon of the opposing sign to the pre-existing flux. In this way, a continuing interconversion of magnetic and electric fields can be maintained. FIG. 12: Non-rotating, unstable star (electrovacuum). Electrically-dominated regions (E 2 > B 2 ) are marked in color, at times t = (0.04, 0.13, 0.22, 0.31) ms. Here currents arise in the force-free case, and E 2 < B 2 is maintained through gradual dissipation, resulting in a fundamental different with the electrovacuum evolution. As described by the simple Newtonian model, these regions form near the equatorial plane and close to the collapsing star. As time progresses, they propagate outward in bursts. The grey zone in the center represents the star. B. Rotating stellar collapse The collapse of a rotating, magnetized star produces an interesting generalization of the relativistic wind problem for a stationary star (e.g [8,31]). As in the case of spherical collapse, qualitatively new effects are introduced after the formation of a horizon. We have chosen an unstable, rotating model star with a mass M = 1.84M and equatorial radius R s = 10.6 km. The star rotates with a period T = 0.78ms, so that the light cylinder is initially located at R LC = 37 km ≈ 3.5 R s . The numerical domain and resolution are identical to those employed in the non-rotating case. To remove unphysical transients we evolve the forcefree equations for a couple of periods with both geometry and matter fixed, and afterwards all equations are evolved. This approach ensures the force-free fields relax to a configuration consistent with the physical scenario considered. The expected field configuration emerges during this startup phase, with a closed, corotating magnetosphere extending out to the light cylinder (LC). The Goldreich-Julian (G-J) current structure is present, with an outflow along the polar field lines balanced by a return flow through an equatorial current sheet. The different stages of the collapse are represented in 13, while that the angular velocity of the star is displayed in Fig. 14. As the star contracts, and its rotational frequency increases, the instantaneous LC approaches the star [117]. Differential rotation develops in the magnetosphere, due to the lack of causal contact between the star and the LC, and the magnetic field is wound in the toroidal direction. Furthermore, the deepening of the gravitational potential forces significant changes in the magnetic field profile, by pulling the field lines more tightly toward the star. The poloidal magnetic field strengthens due to flux freezing in the star, just as in the non-rotating case, but now most of the EM energy is in the toroidal component. Near the poles, the field lines twist around, generating a cone-like structure. In general, the magnetic field preserves a stretched dipolar topology for a longer time than in the non-rotating case, up to the point that all the fluid is swallowed by the black hole. As the black hole forms inside the star, and the fluid falls inward, many of the features in the EM field evolution are preserved from the non-rotating runs. Snapshots of the field profile, charge density and Poynting flux, taken at various times close to horizon formation, are displayed in Figs. 15 and 16. Just as in the nonrotating case, the topology of the magnetic field lines changes dramatically: the y-point structure in the magnetic field disappears from the equatorial regions, and a current sheet extends inward to the horizon. This current sheet is subject to spasmodic episodes of reconnection, the details of which may depend on the prescription for the electric resistivity and its variation with radius. In less than a millisecond, the magnetic flux threading the horizon has almost completely vanished. This is qualitatively similar to the rapid evolution of a black hole interacting with a dipolar force-free configuration, as seen by [57]. In contrast with what is argued in [58], magnetic reconnection prevents the emergence of a relatively long-lived split-monopole configuration. In Sec. VIII A we discuss whether force-free or ideal MHD calculations more accurately describe the global structure of the current sheet, and the implications of the slower reconnection that is seen in the MHD calculation of [57]. The evolution of the magnetic flux during the collapse is displayed in Fig. 17, again computed on a surface located at r = 1.5R s . The total (signed) flux is small through most of the simulation, and only at late times does it becomes comparable to the absolute value of the magnetic flux. In the electrovacuum case, the final decay is governed by the main quasi-normal modes of the rotating black hole. One observes faster decay in the force-free run, just as as in the non-rotating case, possibly due to the greater self-inductance of the vacuum black hole. The radiated electromagnetic energy is displayed in Fig. 18, rescaled again with respect to the energy peak in the magnetosphere. The simulations indicate that around 20% of the energy stored in the magnetosphere is radiated during the collapse in the force-free case, similarly to the electrovacuum case. This is in clear contrast to the non-rotating case, where a much larger energy is radiated in the electrovacuum solution than in the force-free one. We note that although the energy of the magnetosphere is similar in the non-rotating and rotating cases (we find C peak = 1.5), the inclusion of rotation leads to a 20−fold enhancement in the EM energy radiated by a collapsing force-free magnetosphere: rad = 0.18. Hence Eq. (18) gives E rad ≈ 1. quadrupolar, since the Newman-Penrose scalar Φ 2 has angular dependence mainly determined by an (l = 2, m = 0) mode. This radiation propagates outwards with a velocity v = 0.88c, which is very similar to what was obtained in the non-rotating case. As displayed in Fig. 19, most of the energy is radiated more efficiently near the angle θ = ±50 o with a peak intensity shortly after the formation of the black hole (For comparison purposes, Fig. 20 illustrates the behavior in the electrovacuum case). Subsequently this emitted radiation fades away rapidly as the magnetic flux is radiated away. VII. IMPLICATIONS FOR ASTROPHYSICAL TRANSIENTS: I. GAMMA-RAY BURSTS We now consider the implications of our simulation results for high-energy transient phenomena. The very luminous gamma-ray bursts (GRBs) are generally believed to result from the formation of a stellar-mass black hole by sudden gravitational collapse, either in the core of a massive star [1,59], or after the merger of a neutron star with another compact object [2,3]. A rapidly rotating magnetar is an interesting alternative [60][61][62]. Our focus here is on ultraluminous EM outflows: the same general mechanism is strongly favored in other contexts such as pulsar synchrotron nebulae [63] and AGN jets [64]. Our calculations focus on the transition between a magnetar and a black hole, as a key part of the engine that drives a GRB. The implication is that a build-up of magnetic flux in the surface layers of a rapidly rotating neutron star (formed, e.g., in a binary merger) can driven an electromagnetic outflow during the collapse of the star, independently of any Blandford-Znajek process operating afterward. As we explain here, this has some advantages over an EM wind operating before the collapse, in that this brief EM transient is likely to be significantly cleaner. We also compare the net EM output with that expected from a post-collapse jet, in the situation where the surface layers of the neutron star, and not the surrounding disk, are the dominant zone for magnetic flux generated by a dynamo process. Our calculations have revealed strongly dissipative processes at work in the EM field surrounding the collapsing star. These include the reconnection of field lines near the magnetic equator, large-amplitude oscillations in the field, and (in vacuum calculations) the formation of extended zones where E 2 B 2 , which in the presence of conducting matter imply relativistic motions of the entrained particles transverse to the magnetic field. The ejection of loops of magnetic field is observed in our non-rotating simulations. The dynamical evolution of these loops has interesting implications for the gamma-ray flares from gravitationally stable magnetars (see Sec. VIII). Although our calculations take into account the presence of plasma implicitly by enforcing E · B = 0, adding the associated force-free current and enforcing \|E\| < \|B\|, we expect that strong heating will occur in practice. Where the magnetic field reaches ∼ 10 15 − 10 16 G, the implied temperature is above ∼ 1 MeV. Such a high density of electrons and positrons is created that macroscopic zones of non-vanishing E · B -as are required by most pulsar discharge models (e.g. [65]) -cannot be maintained. Effective heating can occur by other channels: for example, long-wavelength gradients in the magnetic field are converted efficiently to internal energy if the magnetic field becomes turbulent, so that a wide spectrum of wave motions is formed that extends down to very small scales [66]. The relativistic outflow that is emitted by a rapidly collapsing star should therefore be quite hot. Largescale magnetic fields and a relativistic photon-electronpositron plasma will both contribute substantially to the energy flux. Magnetar outbursts provide a fairly direct example of this phenomenon. A. EM Output Before, During, and After Collapse The star that collapses to form a black hole passes through three distinct phases: a pre-collapse phase during which it emits a magnetized wind if it rotates; the dynamic collapse phase; and, if an orbiting disk is present -as it is following a neutron star merger -an accretion phase that is accompanied by a relativistic jet. Since our current simulations focus on the intermediate step, it is worth examining its relative contribution to the total EM output of the star. A large enough EM output ∆E collapse is measured in our rotating force-free simulation to power some short GRBs, especially if some account is made for beaming. Here the requirement is that the surface magnetic field is strong enough to hold off any accretion flow. At an accretion rateṀ , this implies a polar magnetic field stronger than[118] B pole ∼ 2 Ṁ V c (R NS ) R 2 NS 1/2 ∼ 7 × 10 15 Ṁ M s −1 1/2 G.(27) Here V c is the circular speed, approximated as Keplerian. In this section we take a stellar mass M NS = 2.6 M and a radius R NS = 15 km, as appropriate to a hot and rapidly rotating neutron star. The dipole field energy before the collapse is E dipole,0 = 1 12 B 2 pole R 3 NS = 1.5 × 10 49 Ṁ M s −1 erg. (28) We estimate ∆E collapse ∼ 0.3 E dipole,0 , given that the energy released is about 0.2 times the peak magnetic energy, which in turn is C peak ∼ 1.5 times E dipole,0 . From Eq. (28) we obtain ∆E collapse ∼ 5 × 10 48 Ṁ M s −1 erg.(29) The precise numerical value depends non-linearly on the initial specific angular momentum J/M NS through the factor rad C peak : faster initial spins imply stronger winding of the magnetic field during the collapse. The magnetic field of an isolated neutron star acts as a couple between its reservoir of rotational energy and a dissipative outflow. Even in the case of a (gravitationally stable) magnetar, the magnetic energy begins to dominate the rotational energy only at an advanced age, as the star spins down. Therefore the output of the precollapse phase could be very large compared with the release of EM energy during the collapse. Comparing the spindown energy radiated over a time ∆t with the external magnetic energy (29) gives L sd ∆t E dipole,0 = 0.6 P NS ms −3 R NS 15 km 3 ∆t P NS .(30) Here we have substituted the spindown power of an aligned, force-free rotator [67], L sd = 1 4 B 2 pole R 2 NS c Ω NS R NS c 4 ,(31) where Ω NS = 2π/P NS . An isolated, rotating star would radiate energy equal to E dipole,0 in a few milliseconds.. In the context of binary neutron star mergers, one requires a magnetosphere to emerge from the very strong shear layer near the surface of the merger remnant (see Section VII B for further discussion of how this could happen). If the neutron star is formed hot, a lengthy pre-collapse spindown phase would cause significant difficulties with the application to short GRBs, because the wind generated during the pre-collapse phase is heavily loaded with nucleons and α particles that are driven outward by charged-current absorption of electron-type neutrinos near the neutrinosphere [54,68,69]. The connection between short GRBs and neutron star mergers would also be disfavored if this pre-collapse outflow lasted longer than ∼ 300 ms, given a characteristic short GRB lifetime of 0.03-0.3 seconds [70]. On the other hand, the survival of the merger remnant for ∼ 100 − 300 ms would have the advantage of allowing stronger amplification of the magnetic field before the remnant collapses. The possibility of longer-lived merger remnants for some configurations has been raised by recent simulations that employ a realistic, finite-temperature EOS [71]. Although two merging neutron stars are expected initially to have magnetospheres, the torus formed by the tidal disruption of the lighter star has a pressure vastly exceeding that of a typical pulsar dipole. The torus would, therefore, suppress a magnetosphere around the newly formed merger remnant. The neutrino-driven wind that emerges from the polar regions of the remnant will comb out the magnetic field, but this field need not initially be coherent across the star or dynamically important. By the same token, if a torus were entirely absent, then the magnetic field threading the star would dissipate rapidly after the black hole forms, and the post-collapse phase would contribute negligibly to the output of the star. It is useful to express the EM power in terms of the "open" magnetic flux that connects the surface of the star to the outflowing wind. In the case of an isolated star, this is the flux extending beyond the light cylinder, Φ open (Ω NS ) πB r (R LC )R 2 LC = Φ NS R NS R LC ,(32) where Φ NS = B pole · πR 2 NS is the dipolar magnetic flux threading the star. One can then re-write Eq. (31) as L sd = 1 4π 2 c (Φ open Ω NS ) 2 .(33) After the star forms a distinct magnetosphere, the accretion torus can continue to influence the wind power by modifying the open magnetic flux. Let us suppose that the magnetic pressure dominates the torus ram pressure out to an equatorial distance R A > R NS . Approximating the magnetosphere by a dipole, a fraction (35) During our simulations of the collapse of an isolated star, we observe that the magnetic field lines are strongly twisted, so that most of the closed magnetic flux opens out. Then Φ open Φ NS ∼ R NS R A(34)Φ open → Φ open (Ω H ),(36) where Ω H is the angular velocity of the horizon. A torus also plays an important role after the black hole forms by trapping a certain fraction of the magnetic flux that threads the star. If the magnetic field is strong enough to hold off the torus from the star before the collapse, then it will have a similar effect after the collapse. We therefore calculate the Blandford-Znajek power emerging from the horizon by assuming a uniform flux density out to some radius R A . From Eq. (8.65) of [72], one gets, L BZ ∼ 2 15 Ω F (Ω H − Ω F ) Ω 2 H Ω H R H c 2 R 2 H B 2 c 1 30π 2 c (Φ H Ω H ) 2 ,(37) where B is the flux density threading the region interior to the torus, and Φ H = π B R 2 H . The difference in the normalizations of Eqs. (33) and (37) largely reflects the fact that the torque on the black hole is maximized when the magnetic field has an angular velocity Ω F = Ω H /2. We can now relate Eq. (37) to the EM power generated before the collapse. The spin angular momentum is approximately conserved during the collapse, J I NS Ω NS , and we also set M BH = M NS . Then J GM 2 BH /c = ε I P NS ms −1 R NS 15 km 2 M BH 3 M −1 ,(38)where ε I = I NS /M NS R 2 NS ∼ 0.3. The angular frequency of the black hole is Ω H R H /c = J/M BH R S c = Jc/2GM 2 BH , and is related to Ω NS pre-collapse by Ω H Ω NS 0.3 R NS R H 2 ε I 0.3 .(39) where R Sch = 2GM BH /c 2 is the Schwarzschild radius. One expects B ∼ B pole after the collapse if most of the magnetic flux threading the star is open before the collapse, and the pre-collapse magnetosphere is limited in size. The proportion of the trapped flux threading the hole is Φ H Φ open = R H R A 2 ∼ R H R NS 2 .(40) We can now show that the EM power is suppressed immediately following the collapse. Substituting equations (39) and (40) into (37) gives L BZ L sd 0.01,(41) with a coefficient ( R A /R NS ) 2 (R NS /R A ) 4 (ε I /0.3) 2 . Additional power will flow along the magnetic field lines that thread the ergosphere, but this portion of the black hole magnetosphere will mix with the accretion flow and may be less strongly magnetized. The interesting conclusion here is that the rapid twisting up of the magnetic field during the collapse can generate a larger EM output than a fairly extended jet emission after the collapse. It is worth summarizing the three main sources of this result: i) before the horizon forms, the EM power is proportional to Ω 2 rather than ∼ Ω 2 /4; ii) the magnetic flux remains pinned in the star for a few rotation periods during the collapse (before the onset of the black hole), and then springs out to fill a larger volume; and iii) the relation between rotation frequency and angular momentum is enhanced by a factor ∼ ε −1 I ∼ 3 prior to the collapse; in other words, relativistic gravity has the effect of softening the growth of the rotation frequency as the star collapses. A reduction in the trapped flux (due to outward diffusion of the magnetic field into the torus) would, in this situation, initially increase the Blandford-Znajek power flowing from the horizon. The pressure of the trapped field approximately balances the ram pressure of the accretion flow at some point outside the horizon. Therefore a reduction in the trapped flux allows the flow to reach closer to the black hole, and attain higher pressures, before being interrupted. Although we are considering the flux originating in a dynamo process before the collapse (Sec. VII B), continued flux generation in the torus by the magnetorotational instability [73] could play a role in modulating the jet power. Numerical Comparison Let us consider the EM energy that would be radiated following a binary NS merger, if the remnant survives long enough to form a magnetosphere (Sec. VII B). Taking ∆E collapse ∼ 0.3E dipole,0 , and normalizing to an accretion rate 0.1M s −1 through Eq. (28), gives ∆E collapse ∼ 5 × 10 47 erg. We expect that the gain factor ∆E collapse /E dipole,0 depends non-linearly on the precollapse rotation rate, since the winding of the magnetospheric field results from a competition between differential rotation and torsional wave motion. In addition, ∆E collapse depends indirectly on the torus mass and pressure through the strength of the magnetic field that is required to hold off the torus material from the neutron star surface. The larger accretion rate in a collapsar environment implies a larger transient energy. It is possible to make a direct comparison with the Blandford-Znajek jet that follows the collapse, if the magnetic flux that threads the black hole is left behind by the collapsing magnetar. Combining equations (30) and (41), and taking pre-collapse rotation period and radius 0.8 msec and 15 km, one finds that an energy ∼ 0.3E dipole,0 would be radiated by a BZ jet over ∼ 20 ms. Equivalently, a Blandford-Znajek jet from a ∼ 3 M black hole with Jc/GM 2 BH ∼ 0.7 would generate a power (37) L BZ ∼ 1 × 10 50 B 2 15 erg/s. A recent binary merger simulation with GRMHD and dynamical gravity [74] presented a polar magnetic field ∼ 7 × 10 14 G developing from a much weaker seed field in the torus. The equation of state used gave a fast collapse to a BH (within ∼ 10 ms), and therefore precluded the surface shear dynamo that we have conjectured. (Note that this polar field could be affected by numerical resistivity, e.g. [75]). Baryon Poisoning Comparing the output ∆E collapse with Eq. (29), one sees that the pre-collapse star would release a comparable energy within ∼ 100 ms after forming a magnetosphere. However, it is well known (e.g. [54]) that such an outflow would bear a much higher density of nucleons than a Blandford-Znajek jet from a black hole, due to the absorption of electron-type neutrinos and anti-neutrinos. We expect that this nucleon loading would be strongly suppressed in the dynamical collapse phase, due to i) the relatively short duration of the emission; ii) the strongly wound field geometry (B φ /B P 5); and iii) redshifting effects. Combining these effects suggests a significant suppression 0.01 × 0.1 ∼ 10 −3 in the nucleon loading, so that the mass ejected would essentially be that present in the magnetosphere before the collapse. B. Emergence of a Magnetosphere via Dynamo Action in a Surface Shear Layer The immediate aftermath of a binary neutron star merger is distinguished from disk accretion onto a black hole, in that the velocity shear is strongest where the orbiting material makes a transition from centrifugal to hydrostatic support. Another feature which distinguishes the merger remnant from ordinary accreting neutron stars (X-ray pulsars) is that it does not initially have an ordered magnetosphere. Long after the first stage of the merger is complete, the velocity shear provides a tremendous source of free energy for amplifying a magnetic field. The power dissipated in the material settling onto the neutron star surface is L shear ∼ 1 2Ṁ V 2 c (R NS ) − Ω 2 NS R 2 NS ∼ 3 × 10 52 Ṁ 0.1M s −1 f shear erg s −1 ,(42) where f shear ≡ 1 − [Ω NS R NS /V c (R NS )] 2 . Note that the surface shear becomes more radially concentrated with time: as differential rotation is erased in the interior of the merger remnant, the surface shear is maintained by continuing accretion. The inner part of the shear layer develops positive dΩ/dr and the magnetorotational instability is extinguished. A strong feedback mechanism is present which allows rapid magnetic field growth, but causes this growth to saturate once the star is able to form a magnetosphere that holds off the accretion flow. When the magnetosphere is present, the accreting material follows the magnetic field and reaches the star at the same angular velocity. Shearing of the magnetic field in the outer layers of the star is therefore turned off. This effect is clearly demonstrated in the 3D accretion simulations of [82]. Only a tiny fraction of the accretion energy L shear ∆t must be converted to a poloidal magnetic field to hold off the accretion flow: substituting (27) into the dipole energy (28) gives E dipole L shear ∆t = 2 3f shear ∆t R 3 NS GM NS 1/2 = 6 × 10 −4 ∆t 100 ms −1 f −1 shear . (43) over a duration ∆t. Low-mass X-ray binaries provide a nice example of systems where this feedback appears to operate. The millisecond radio pulsars that are descended from them have magnetic fields that are just strong enough (B ∼ 10 8 −10 9 G) to hold off an Eddington-level accretion flow onto a neutron star -but not much stronger. The absence of persistent pulsations in the majority of LMXBs then requires that the magnetic field be aligned with the angular momentum of the accretion flow. Now let us consider how the magnetic field evolves in the surface shear layer. The field present initially in the merging stars is rapidly amplified by a Kelvin-Helmholtz instability [28,75], or the magnetorotational instability [73]. Rapid growth of the magnetic field on large scales is sensitive to the speed of magnetic reconnection in the fluid, and requires three-dimensional motions. The two-dimensional wrapping of a magnetic field by Kelvin-Helmholtz vortices does not generate net flux; and the initial growth length of the MRI is very small compared with the scale height of the torus. MRI growth is fastest on a scale k −1 ∼ B/(4πρ) 1/2 Ω ∼ 10 −4 (B/10 12 G) r in a torus of mass ∼ 0.01 M [73]. The hydrostatic structure of the inner shear layer makes it easier for the magnetic field to be pinned and retained than it would be in the surrounding torus. The magnetic field threading the shear layer is wound up, and since dΩ/dr > 0, the mean toroidal flux density grows at least in a linear manner. If a hot, rotating, and massive neutron star (M NS ∼ 2.6−3 M ) can survive collapse for more than ∼ 100 ms, as is suggested by recent simulations of [71], then this toroidal field becomes quite strong. Even if the seed poloidal field is as weak as B P,0 ∼ 10 13 G -within the range of pulsar fields -then the toroidal field reaches B φ ∼ [Ω c (R NS ) − Ω NS ]∆t B P,0 = 1 × 10 16 ∆t 100 ms Ω c (R NS ) − Ω NS 10 4 s −1 × B P 0 10 13 G G.(44) Here Ω c (R NS ) is the angular frequency at the surface of the torus. Exponential magnetic field growth becomes possible when the wound-up field can rise buoyantly through the shear layer. The large dissipated power (42) will generate a strongly positive entropy gradient in the inner part of the shear layer. Where the material is convectively stable, only magnetic fields stronger than ∼ (GM NS M shear /R 4 NS ) 1/2 ∼ 1 × 10 17 (M shear /0.1 M ) 1/2 G can directly overcome the pressure of the overlying material. But even in this case, the intense flux of electron-type neutrinos allows a magnetic field stronger than ∼ 10 15 G to rise buoyantly on the Alfvén timescale, by erasing gradients in entropy and electron fraction that impede buoyancy [83]. The buoyancy time is t A ∼ P (4πρ) 1/2 /B φ ∼ 1 (B φ /10 16 G) −1 (ρ/10 14 g cm −3 ) 1/2 ms across a pressure scale height P ∼ R NS /4 ∼ 3 − 4 km. Thus an exponential feedback loop appears quite likely in this situation. One observes that the large-scale dipole magnetic field that emerges may be sensitive to the seed magnetic field, in the sense that a minimal seed field is required for the linearly wound field to reach the buoyancy threshold. Nonetheless, it is also clear that magnetar-strength magnetic fields do not require magnetar-strength seed fields in the presence of persistent surface shear. The strength of the magnetosphere that eventually emerges results from a competition between the finite rate of amplification and the diminishing shear stress at the surface of the star. At an age of ∼ 10(100) ms, a poloidal magnetic field stronger than ∼ 10 16 G (10 15 G) will begin to apply a strong, negative feedback on the surface shear. Once the torus mass drops to the point that accretion is mediated mainly by magnetic stresses, the accretion rate isṀ 0 ∼ M T,0 /t diff,0 ∼ 10 (α/0.1)(M T,0 /0.1 M ) M s −1 . Here M T,0 and t diff,0 ∼ 10(α/0.1) −1 ms are the initial torus mass and diffusion time in this viscous phase, and α is the viscosity coefficient. The accretion rate drops as the torus material spreads outward, asṀ ∼Ṁ 0 (t/t diff,0 ) −4/3 , as long as the torus remains geometrically thick and conserves its angular momentum (e.g. [84]). C. Galactic Magnetars and Delayed Collapse We now consider the observational imprint of a magnetar if it forms by the accretion of a thin layer of strongly sheared material, but is (initially) gravitationally stable. We focus on the rate of energy loss and the spin history of the star. There is a substantial reduction in the energy release by spindown, compared with a star that initially rotated as a solid body, due to the internal rearrangement of angular momentum. Approximate solid-body rotation is attained on the poloidal Alfvén timescale R NS (4πρ) 1/2 /B pole , which is generally much shorter than the magnetic-dipole spindown time t sd = I NS P 2 NS c 3 2π 2 B 2 pole R 6 NS .(45) Most of the shear energy is then dissipated internally. A delayed collapse is possible if the magnetar only slightly exceeds the maximum mass for a non-rotating, zero-temperature neutron star. The loss of rotational support by a magnetic wind would trigger a collapse on the rotational braking time. Quite generally, the accretion of a thin layer of strongly sheared material provides an attractive mechanism for creating magnetars, and so the results of this section should have a broader application to the Galactic magnetar population, even if these stars are well below the maximum mass. Several possible channels are available: the rotation of the accreting material could be generated by an instability of the supernova shock [85][86][87][88]; or the neutron star could be exposed to material with very strong vorticity in a merger event, e.g. when it merges with a companion CO white dwarf, or with the core of an evolving Be star. The spindown power that is released by the magnetar is then reduced significantly compared with a star which rotates uniformly with the same angular velocity as the surface material. The final spin period resulting from the accretion of a layer of mass M shear and rotation period P shear onto a neutron star of total mass M NS is P NS = (2/3)M shear R 2 NS I NS −1 P shear = 10.5 j −1 shear M NS 2 M ms,(46) where j shear ≡ (M shear /0.1 M )(P shear /ms) −1 . As the star spins down, it deposits a rotational energy day. (49) In this situation, only a very strong internal magnetic field (B toroidal 10 17 G) would induce sufficient triaxiality in the star that the gravity wave torque competed with the external electromagnetic torque [89]. VIII. PLASMOID DYNAMICS AND RADIATION: IMPLICATIONS FOR MAGNETAR FLARES Escaping loops of magnetic field are observed in our non-rotating simulations. These loops are formed by magnetic reconnection, which means their size and rate of formation are sensitive to the treatment of resistivity, and to the disregard of stresses associated with bulk plasma flow by the adoption of the force-free equations. An outgoing plasmoid (containing plasma and a closed magnetic field) has a well-defined center of mass: we find that its bulk motion is measurably smaller than the speed of light, V bulk ∼ 0.9c and Γ bulk ∼ 2, at a distance r ∼ 10 times the pre-collapse neutron star radius R NS . This is significantly less relativistic than the bulk motion expected for a gas of freely expanding particles released by the collapsed star (Γ bulk r/R NS ∼ 10). After addressing each of these issues, we make contact with the giant gamma-ray flares of the Galactic magnetars. These appear to involve the ejection of an energetic plasmoid (∼ 10 44 − 10 46 erg), but due to the shearing of the magnetic field lines rather than the collapse of the star [90]. A. Dependence of Reconnection Rate on Resistivity Model When magnetic field lines of an opposing sense are stretched out and forced into contact, the rate at which they reconnect is sensitive to the microscopic model of resistivity. In an ohmic plasma with a uniform resistivity, a long current sheet forms and reconnection is very slow; fast reconnection with an x-point geometry depends on a local maximum in the resistivity [91]. In some contexts, such as the Solar corona, the microscopic explanation for this behavior may be provided by the Hall terms in the conductivity [92]. These are relatively less important if the plasma is loaded with e + /e − pairs, as would be expected in the magnetosphere of a collapsing magnetar. In a fluid, small-scale hydromagnetic turbulence appears to greatly accelerate the reconnection rate [93]; but whether such a process can operate in a low-β plasma of astrophysical dimensions is not yet determined. In the present work we find evidence for relatively fast reconnection of magnetic field lines, V rec ∼ 0.1 c, as calculated in the force-free approximation (Sec. VI A). This results from a change in topology of the field lines (the formation of an x-point). It is not due to strongly enhanced dissipation in an extended equatorial current sheet: E.J dissipation is measured to be small in both the rotating and non-rotating cases. Such a concern arises in force-free calculations of pulsar magnetospheres, where the magnetic field is less dynamic and is anchored in the star. There the absence of fluid pressure support in the current sheet leads to a rapid collapse of the magnetic field toward the sheet, unless explicitly compensated [8,9]. In the present case, after the black hole forms the inflow of magnetic flux toward the equator can continue in a more dynamic manner into the star, or out to the computational boundary. A recent treatment of reconnection in the magnetosphere of a stationary black hole [57] illustrates a slower field decay when employing an ideal MHD treatment, as compared to a force-free approach. (Resistivity in the MHD case arises through the numerical approximation.) Such a slower decay is also observed at late stages after the formation of the black hole in our simulations when comparing force-free and ideal MHD fields [119]; though by this time the field strengths are orders of magnitude below their peak values. A comparison of force-free and MHD reconnection calculations which explores more general resistivity models, and their influence on the reconnection geometry, remains to be developed. B. Magnetic Reconnection Delayed by Plasma Outflow Reconnection is usually studied in a context where the plasma flow speed along the magnetic field is a small fraction of the Alfvén speed: for example, in a steady MHD wind, the conservation of angular momentum implies a slow drift of particles along the spiral magnetic field outside the Alfvén critical point. However, in some contexts, such as magnetar flares, the flow speed can approach the speed of light. Similarly, in our collapse simulations we see large-amplitude motions on the magnetic field loops threading the neutron star, which suggest strong plasma heating. The backreaction of the outflowing plasma on the reconnection of field lines is not taken into account. Even when the magnetic field has a tendency to reconnect through an x-point, reconnection will be delayed until the kinetic pressure ∼ U β 2 of the outflowing pairphoton plasma (with thermal energy density ∼ U ) drops below the Poynting flux that would flow toward the cur-rent sheet in the absence of plasma flow. One requires U β 2 0.1 V A c B 2 8π ,(50) where the coefficient on the right-hand-side is appropriate for fast x-point reconnection. The magnetic field lines are stretched outward by the plasma flow beyond an Alfvén radius [66] R A R NS = B 2 pole R 2 NS c 4L γ 1/4 = 16 B pole 10 15 G 1/2 L γ 10 47 erg s −1 −1/4(51) given R NS ∼ 10 km. Note that, at the peak of the outflow, the pressure of the pair-photon fluid is comparable to the pressure of the stretched magnetic field lines at r = R A , and outside this radius grows as ∼ (r/R A ) 2 with respect to the split-monopole field pressure. For example, in a magnetar giant flare, the ∼ 0.1 s width of the main gamma-ray pulse is comparable to the time for magnetic and elastic stresses to rearrange the stellar interior; but it is ∼ 300 times longer than the flow time out to the Alfvén radius (51), and therefore much longer than the timescale for x-point reconnection at a speed ∼ 0.1c. C. Radio Afterglow from Strongly Magnetized Outflows A magnetically-dominated plasma that is ejected from a collapsing magnetar (or a nearby Soft Gamma Repeater) can be a much stronger source of synchrotron emission than a shocked plasma of comparable energy density. Existing calculations of radio afterglows of short GRBs ( [94] and references therein), as well as calculations of the radio afterglow of SGR giant flares [95], focus on synchrotron emission by a population of non-thermal electrons that are accelerated at a shock that leads the outflow. For the bulk of GRBs, one infers efficiencies of conversion ε e , ε B ∼ 0.1 of bulk kinetic energy to nonthermal electrons and to magnetic fields downstream of the forward shock. These moderate values of ε e and ε B are consistent with the broadband tails of radio-X-ray emission that follow the brief, bright gamma-ray phase. The synchrotron emission from a magneticallydominated plasmoid will be proportionately much brighter, by up to a factor ∼ 100, for two reasons. First, and most obviously, ε B now approaches unity. Second, if the magnetic energy density dominates the thermal energy density, then a cascade process (involving the creation of high-wavenumber Alfvén modes) preferentially heats the electrons [96]. If the plasma is very relativistic, damping is mainly due to charge-starvation of the waves, at wavenumbers where the amplitude of the fluctuating current density begins to exceed en e c [66,77]. (A different damping mechanism operates at higher ion densities: the waves are Landau-damped on the parallel motion of the electrons.) In practice, the relative amplitudes of the bulk synchrotron emission and the shock emission will depend on the degree of disorder in the plasmoid magnetic field. But for magnetar flares, indirect evidence that bulk synchrotron emission dominates comes from a rapid initial drop in radio flux that is consistent with the sudden compression of the plasmoid, followed by rapid adiabatic cooling [95]. D. Applications to Magnetar Outbursts Magnetar outbursts involve more limited releases of energy that leave the original star intact. They are triggered when the footpoints of a ∼ 10 15 G magnetic field are strongly sheared by an elastic instability of a neutron star crust, thereby generating a hot plasma and an intense burst of gamma-rays [90,97]. The first, extremely bright, stage of a giant magnetar flare lasts only ∼ 100 ms and bears a considerable resemblance to a 'classical' gamma-ray burst, demonstrating high temperatures (kT 200 keV) and a significant non-thermal component to the spectrum [98]. (This phase saturates almost all X-ray detectors, but was well-resolved by the Geotail experiment [99] in the 27 August 1998 and 27 December 2004 flares.) The duration and luminosity are consistent with the internal rearrangement of the magnetic field in a neutron star, with a strength ∼ 4−5×10 15 G based on considerations of magnetic field transport and global flare energetics [100], several times stronger than the standard dipole expression for the spindown-derived magnetic field. The giant flares appear to involve the ejection of a plasmoid. The combination of fast variability and extreme luminosity (up to ∼ 10 48 ergs s −1 ) implies, through the usual arguments of gamma-ray opacity [101], that the emitting plasma has expanded to a much larger volume than that of the neutron star. This expanding plasma is an excellent electrical conductor and must carry some of the stellar magnetic field with it. This expected property of magnetar giant flares helps to explain two apparently contradictory phenomena. A straightforward argument based on the theory of thermal fireballs (e.g. [102]) shows that the expanding plasma is moving with a high Lorentz factor at the radius ( 10 8 cm) where it becomes transparent to the gamma-rays [97]. The observation of variability on a timescale δt var ∼ 4−20 ms in the gamma-ray flux implies strong constraints on the baryon rest energy flux. The advected electrons and ions must become transparent to the gamma-rays close enough to the magnetar that the differential lighttravel time r/2Γ 2 c across the outflowing plasma is shorter than δt var . Given a total outflow luminosity L, this implies that the ion rest energy contributes no more than a On the other hand, radio monitoring detected transient emission in the weeks following both flares [103,104]. The emission following the 2004 flare was especially bright, as befitting the much greater energy of the burst, and could be tracked on the sky for more than a year. The expansion of the radio source implies a transverse velocity v ∼ 0.7(D/15 kpc)c [104]. After this, clear evidence is seen for a break in the radio light curve consistent with a transition from uniform expansion to a Sedovlike phase. We infer that the measured proper motion is probably the free expansion velocity, as corrected for relativistic aberration. In principle it is possible for the measured transverse speed to be less than the speed of light, if the intrinsic motion V is nearly luminal but the motion is directed away from the observer (at some angle θ > π/2 with respect to the line of sight to the magnetar): β ⊥,obs = β sin θ/(1 − β cos θ). The observed relative brightness of both the radio and gamma-ray emission argues against this: one measures E γ ∼ 4 × 10 44 (D/15 kpc) 2 in 2004 vs. E γ ∼ 5 × 10 46 (D/15 kpc) 2 in 1998, and a radio flux F ν ∼ 50 mJy vs. 0.3 µJy at 8.5 GHz 1 week after the flare [99,103,105]. (A factor ∼ 2 underestimate of the ∼ 15 kpc distance to SGR 1806-20 appears unlikely given that the source position is in the Galactic plane.) The energy reservoir that powers the early, subluminal stage of the radio afterglow must be composed of something other than electrons and positrons emitted by the star, which would mainly have annihilated during the very brief fireball phase. A reconnected magnetic field is the most plausible delayed carrier of energy, especially given the strong limitations on the baryon flux during the 100 ms gamma-ray pulse. A closed loop of magnetic field carries a finite inertia, and so its center-of-mass frame will not accelerate with distance from the source as does that of a collimated particle beam. The outflowing pair-photon fluid must overcome the tension of the magnetic field lines that are anchored in the star, and so comparable energy can be put into the stretched field, which is pulled into a split-monopole configuration. The energy of the stretched field is concentrated close to the star, B 2 r 3 ∼ r −1 , although not as strongly concentrated as in a static dipole (B 2 r 3 ∼ r −3 ). In a static plasma, the time for reconnection at the Alfvén radius R A is t rec ∼ 3 (R A /100 km)(V A /0.1 c) −1 ms. Comparing this expression with the measured e-folding time of ∼ 30 ms for the tails of the giant flare pulses [99] suggests that reconnection is gated by the decrease in pair-photon pressure. We conclude that, in a magnetar giant flare, most of the released magnetic energy tends to follow the pairphoton pulse, with the delay between the two components being due to continued shearing of the external magnetic as the interior of the magnetar adjusts on the internal Alfvén time of ∼ 0.1 s. IX. CONCLUDING REMARKS Understanding the global behavior of strongly gravitating, dynamical systems, containing dense matter coupled to ultra-strong magnetic fields, is of key importance for a thorough understanding of possible signals from them. In this work, we have presented a new approach to this end by combining the ideal MHD and force-free approximations in a suitable manner within general relativity. We expect that this hybrid scheme represents real progress towards greater realism. The stellar interior utilizes ideal MHD to faithfully model the neutron star without the disadvantages incumbent in the low density exterior. Likewise, the exterior solution uses the forcefree approach and so captures the dynamics of the tenuous plasma. The entire domain is described by a fully nonlinear and fully dynamic general relativity solution necessary for strong-field gravity. Indeed, a key aspect of this approach is its generality. It does not require a prescribed stationary stellar boundary, and can be applied, for example, to dynamical systems such as collapsing stars and non-vacuum compact binaries. We have exploited this approach to study stellar collapse in both rotating and non-rotating cases. Our studies reveal a rich phenomenology in the magnetosphere as the collapse proceeds. In particular, magnetic reconnection plays an important role by inducing strong electromagnetic emission as well as the infall of electromagnetic energy into the black hole, which in a short time loses all its 'hair'. When the star starts off rapidly rotating, the energy of the magnetospheric plasma grows significantly during the collapse. As the star rotates faster, its magnetic field lines do not have time to re-adjust to the increased rotation, and are strongly wound up out to the initial light cylinder. It is worth emphasizing that this conversion of dynamical energy into electromagnetic energy does not depend on resistive effects. Two issues of principle have arisen in performing these calculations. First, we have shown conclusively that the force-free approximation to the evolution of a dilute, relativistic plasma inevitably leads to singularities. These singularities are avoided in our calculations by continuously pruning the electric field. In the simulations that we have run, this procedure appears to cause limited energy dissipation, but its necessity should be kept in mind. The second issue of principle regards the dependence of the reconnection geometry on the resistivity model. Fluid pressure is responsible for slowing down the rate of reconnection unless an x-point geometry is able to form. By neglecting fluid pressure, the force-free approxima-tion clearly facilitates the formation of x-points. In spite of this, we observe very limited numerical dissipation in current sheets. Although the plasma that is represented by our model magnetospheres is, in reality, strongly collisional, it should be kept in mind that fast x-point reconnection still occurs in collisional plasmas in the presence of hydromagnetic turbulence -as has been demonstrated so far in weakly magnetized plasmas [93]. Some further exploration of the reconnection geometry is possible in resistive MHD calculations of stellar collapse by varying the spatial dependence of the resistivity. We have also discussed how the electromagnetic outbursts from collapsing magnetars may have interesting observational effects. In particular, our results are relevant to binary neutron star merger scenarios, in which a hypermassive neutron star forms and collapses to a black hole. Collapse to a black hole can happen either promptly or after many dynamical times, depending on the masses involved and the equation of state describing the stars (see, for instance, [19]). Although Kelvin-Helmholtz and magnetorotational instabilities will create ultrastrong magnetic fields ( 10 14−16 G), we have suggested that the global field (in particular, the amount of magnetic flux threading the merger remnant and eventual black hole) could depend strongly on the lifetime of the remnant. The formation of a distinct magnetosphere would require tapping only ∼ 10 −3 − 10 −4 of the energy dissipated in shear layer at the remnant surface. As such a star collapsed, the magnetosphere would qualitatively follow the behavior outlined here. The electromagnetic output of the collapse could compete with the later emission from a Blandford-Znajek jet emanating from the black hole horizon, and source a powerful electromagnetic counterpart to the gravity wave signal (e.g. [106]). In addition, the magnetic field dynamics that is revealed in our simulations has a number of interesting implications for gamma-ray bursts and magnetar flares, as discussed in Secs. VII and VIII. Beyond the work analyzed here, our approach is readily applicable to other relevant systems and will be applied, in particular, to study binary neutron star systems [107], and black hole-neutron star systems (see e.g. [108]). It is important to stress, however, that our approach is not free of ambiguities, in particular with respect to how and where the matching between the force-free and ideal MHD regions is implemented. To the extent possible, we have tested the robustness of our results versus known solutions, which give us confidence in this approach. FIG. 21: Rotating, unstable star (force-free). Isosurfaces of the kernel function of Eq. (9) at two different times t = −0.30ms, left, and t = −0.07ms, right, while the rotating star collapses. Mapped in color is the density of the star in cgs units in a eridional plane. Black lines mark two isosurfaces of the kernel function, corresponding to F = (0.01, 0.99). MHD atmosphere to co-rotate with the star for R cyl < 2R star , and not to rotate otherwise. Both the location of the stellar surface and the density in the transition layer are dynamical. It is important that the choice of transition density ρ match in Eq. (9) respects the increasing density of the collapsing star. In order to maintain a consistent depth of the transition layer, its position is adjusted as the star collapses by scaling the matching density ρ match in proportion to the peak density within the star: ρ match (t) = ρ match (t = 0)[ρ max (t)/ρ max (t = 0)]. The transition layer is displayed at the beginning and near the end of the collapse in Fig. 21. Notice that with this conservative approach we are underestimating possible rotational effects. The angular frequency Ω F of magnetic field lines anchored near the rotation axis of the star is illustrated in Fig. 22. From the initial condition (in which constancy of Ω F is enforced), a negative gradient in Ω F develops in the transition layer during the earliest stages of the collapse. This negative gradient has then disappeared by the time that the stellar angular velocity has increased by ∼ 20%. above the initial value Ω 0 (see Fig. 14). From then on, the qualitative behavior obtained is consistent with the expected one: during the collapse, the star transfers angular momentum to the magnetic field lines so that Ω F = Ω star near its surface, propagating along the magnetic field lines with a speed ∼ c. The sharp negative gradient appearing in the last stages of the collapse reflects the strong radial gradient in Ω star that appears near the rotation axis, as well as the onset of strong general relativistic effects. As the event horizon arises (prior to the formation of the apparent horizon at t = 0), it disconnects the interior from the exterior solution and causes Ω F to decrease, tending to FIG. 1 : 1Non-rotating, collapsing solution. FIG. 2 : 2Monopole solution. Specific components of the magnetic field displayed along the x = z = 0 line at the initial and final times, together with the normalized density ρ/ρc and the F function. Notice that the radial component has a smooth transition across the surface of the star, while there appears a toroidal component in the magnetosphere. FIG. 3 : 3Monopole solution. The rotation of the magnetic field lines (normalized with respect to its initial value) at different times inside and outside the domain, separated by the kernel function F (continuous line). As time progresses, the rotation frequency approaches the constant value expected for the monopole solution. FIG. 4 : 4Aligned rotator. The fluid density and the magnetic field lines on the x = 0 plane at the four times t = (0, 1/3, 2/3)T . Even before a complete rotational period, the solution exhibits the known properties of the aligned rotator solution. The light cylinder is located roughly at the expected position RLC ≈ 3.6 Rs (large tick marks indicate one stellar radius Rs). The intermediate plot illustrates the transient structure resulting as the LC forms and the initial data, which only extends to 2Rs, relaxes to fill the computational domain. These plots do not show the entire computational domain. FIG. 5 : 5Non-rotating, unstable star (force-free). The fluid density (colors in the central region), the magnetic field lines (blue) and the EM radiation flux density (red) at times t = (−1, −0.2, 0.02, 0.05, 0.1, 0.2) ms (from left to right, top to bottom). FIG. 6 : 6Non-rotating, unstable star (electrovacuum). The fluid density (colors in the central region) and the magnetic field lines (blue) at t = (0.03, 0.06, 0.12, 0.125, 0.13, 0.15) ms. FIG. 7 :FIG. 8 : 78Non-rotating, unstable star. The absolute value of the magnetic flux in the electrovacuum case as a function of time, and the electric and magnetic fluxes in the force-free case. These quantities are integrated over central spheres with radius r = 1.5Rs and normalized with respect to the initial integral of \|ΦB(t = 0)\|. The total signed fluxes remain very small throughout the simulation, and the unsigned magnetic flux decreases as the black hole swallows all the matter which anchors the magnetic field. Non-rotating, unstable star. FIG. 9 : 9Evolving vacuum EM field around a star collapsing to half its initial size (uniform dRs/dt). Top panel: the inner magnetic field tracks the instantaneous dipole of the collapsing star; at a fixed radius, B ∝ µ(Rs) ∝ Rs/Rs0. Time progresses top to bottom. Bottom panel: The zone of rising E φ closely follows the similarity solution of Eq. (25). Time progresses left to right. (Two colours are employed for clarity.) FIG. 10 : 10Relative strength of vacuum electric and magnetic fields around collapsing star (uniform dRs/dt). Different lines illustrate the obtained behavior for different values of θ (from 10 o , top, to 90 o , bottom). The zone where E 2 > B 2 is more extended near the magnetic equator. 4 . 4E 2 > B 2 in the Vector QNM of a Black Hole. FIG. 13 : 13Rotating, unstable star (force-free). Magnetic field configuration (blue lines) and fluid density (marked in red) at times t = (−0.47, −0.17, −0.01, 0.12) ms. As the collapse proceeds the increasing spin rate of the star pulls the external magnetic field in the toroidal direction. FIG. 14 : 14Rotating, unstable star (force-free). Star's angular rotational velocity, measured at the equator, during the collapse. Fig. Fig. 13, while that the angular velocity of the star is displayed in Fig. 14. As the star contracts, and its rotational frequency increases, the instantaneous LC approaches the star[117]. Differential rotation develops in the magnetosphere, due to the lack of causal contact between the star and the LC, and the magnetic field is wound in the toroidal direction. Furthermore, the deepening of the gravitational potential forces significant changes in the magnetic field profile, by pulling the field lines more tightly toward the star. The poloidal magnetic field strengthens due to flux freezing in the star, just as in the non-rotating case, but now most of the EM energy is in the toroidal component. Near the poles, the field lines twist around, generating a cone-like structure. In general, the magnetic field preserves a stretched dipolar topology for a longer time than in the non-rotating case, up to the point that all the fluid is swallowed by the black hole. As the black hole forms inside the star, and the fluid FIG. 15 : 15Rotating, unstable star (force-free). Charge density (blue: positive and red: negative) is plotted at t = (−0.17, −0.01, 0.12, 0.35) ms. The central fluid region marked in black disappears as the horizon emerges. FIG. 16 :FIG. 17 : 1617Rotating, unstable star (force-free). Radial Poynting flux in red at t = (−0.3, −0.17, −0.01, 0.12) ms. The central colored zones mark the stellar fluid. The evolution of the poloidal magnetic field (blue lines) in the equatorial regions is qualitatively similar to that observed in the non-rotating case. Rotating unstable star (force-free). The magnetic flux as a function of time, computed at r = 1.5Rs, normalized with respect to the initial value \|ΦB(t = 0)\|. The unsigned magnetic flux decreases after the black hole formation, similar to the non-rotating case. FIG. 18 :FIG. 19 : 1819Rotating, unstable star. Time integral of the electromagnetic luminosity, normalized to the peak EM energy of the magnetosphere, in both the force-free and electrovacuum cases. The EM output depends weakly on rotation in the electrovacuum calculations, whereas in the force-free case the output is much larger Rotating, unstable star (force-free). The electromagnetic flux within annuli defined symmetrically in the northern and southern hemispheres by concentric cones having apertures in [i15 o , (i + 1)15 o ] (i = 0..5). The highest flux is within θ ∈ [−50 o , 50 o ]. Here, as in the non-rotating case, the radiated energy decays faster than what would be expected from a quasi-normal mode behavior, due to the effects of reconnection. FIG. 20 : 203 × 10 46 B 2 pole,15 erg, resulting in a strong average luminosity of L ≈ 1.3 × 10 49 B 2 pole,15 erg s −1 during the collapse.The distribution of the radiated energy is essentially Rotating, unstable star (electrovacuum). The electromagnetic flux within annuli defined symmetrically in the northern and southern hemispheres by concentric cones having apertures in [i15 o , (i + 1)15 o ] (i = 0..5). The radiated energy decays exponentially with a rate consistent with that expected from a quasi-normal mode behavior. the surrounding shock wave. For a magnetar of polar field B pole , the corresponding spindown luminosity is fairly modest, L sd = 1.4 × 10 45 B pole 10 15 G FIG. 22 : 22Rotating, unstable star (force-free). (Top panel) Angular velocity ΩF of the magnetic field line emanating from θ = 122 • , φ = 0 • as a function of cylindrical radius (normalized with respect to its initial value). (Bottom panel) Shape of such magnetic field line as time progresses. the value Ω F = Ω H /2 expected by the Blandford-Znajek solution of a spinning black hole. t = -0.35ms t = -0.17ms t = -0.10ms t = -0.07ms t = -0.01ms Acknowledgments: It is a pleasure to thank P. Goldreich for insights and discussions during the course of this work. We also thank E. BertiAppendix A: The Transition from Ideal toForce-Free MHD Details of the transition from the ideal MHD regime to the force-free regime merit particular attention, especially in our rotating collapse solutions. Since we match two different formulations of Maxwell's equations coupled to conducting matter, it is important to understand how the choice of matching layer affects the exterior force-free solution. Of course, an unambiguous test can only be provided by a complete resistive MHD solution that can handle the strongly magnetized regions outside the star and, in particular, can follow the large changes in density and rotation that are encountered during the collapse. While work on this direction is in progress, the simpler approach presented here allows us to obtain a first solution to the relativistic magnetosphere in this strongly dynamic situation.The positioning of the matching zone is constrained by competing considerations. On the one hand, if it sits too close to the surface, then the interior MHD solution that sources the exterior force-free solution will be unrealistic, due to the density floor that is applied in the MHD atmosphere. One thus might want to place the transition layer well within the stellar surface. However, such a deep layer might underestimate the magnetic field strength at the base of the magnetosphere, and imply force-free behavior of the magnetic field where that approximation is not justifiedIn particular, we have found that placing the transition zone at too low a density implies an unrealistically large toroidal magnetic field at the base of the force-free zone. Recall that the region exterior to the star, with its tenuous plasma, will have a relatively large magnetization, and should be forced to co-rotate with the interior of the star. Unless prohibitively high resolution is employed, the atmosphere of the MHD solution has a large enough inertia that this condition can be violated. Instead, the magnetic field experiences a non-negligible (and non-physical) differential rotation. 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See however the possibility of doing so with recently developed techniques. 109-111See however the possibility of doing so with recently developed techniques [109-111] This could be achieved by using multipatch methods. 38, 112] howeverThis could be achieved by using multipatch methods [38, 112] however. This is obtained by balancing the spin-up time t col /2 with the time for an Alfvén wave to propagate from the star out to rmax along a dipole field line. This is obtained by balancing the spin-up time t col /2 with the time for an Alfvén wave to propagate from the star out to rmax along a dipole field line. We employ however a different Γ and our implementation does not cope with as large magnetizations as that in. 57We employ however a different Γ and our implementa- tion does not cope with as large magnetizations as that in [57].	[]
[ "Karlsruhe Astrophysical Database of Nucleosynthesis in Stars", "Karlsruhe Astrophysical Database of Nucleosynthesis in Stars", "Karlsruhe Astrophysical Database of Nucleosynthesis in Stars", "Karlsruhe Astrophysical Database of Nucleosynthesis in Stars" ]	[ "I Dillmann \nDepartement Physik und Astronomie\nUniversität Basel\nKlingelbergstrasse 82CH-4056BaselSwitzerland\n", "M Heil ", "F Käppeler ", "R Plag ", "T Rauscher \nDepartement Physik und Astronomie\nUniversität Basel\nKlingelbergstrasse 82CH-4056BaselSwitzerland\n", "F.-K Thielemann \nDepartement Physik und Astronomie\nUniversität Basel\nKlingelbergstrasse 82CH-4056BaselSwitzerland\n", "\nInstitut für Kernphysik\nForschungszentrum Karlsruhe\n3640, D-76021Postfach, KarlsruheGermany\n", "I Dillmann \nDepartement Physik und Astronomie\nUniversität Basel\nKlingelbergstrasse 82CH-4056BaselSwitzerland\n", "M Heil ", "F Käppeler ", "R Plag ", "T Rauscher \nDepartement Physik und Astronomie\nUniversität Basel\nKlingelbergstrasse 82CH-4056BaselSwitzerland\n", "F.-K Thielemann \nDepartement Physik und Astronomie\nUniversität Basel\nKlingelbergstrasse 82CH-4056BaselSwitzerland\n", "\nInstitut für Kernphysik\nForschungszentrum Karlsruhe\n3640, D-76021Postfach, KarlsruheGermany\n" ]	[ "Departement Physik und Astronomie\nUniversität Basel\nKlingelbergstrasse 82CH-4056BaselSwitzerland", "Departement Physik und Astronomie\nUniversität Basel\nKlingelbergstrasse 82CH-4056BaselSwitzerland", "Departement Physik und Astronomie\nUniversität Basel\nKlingelbergstrasse 82CH-4056BaselSwitzerland", "Institut für Kernphysik\nForschungszentrum Karlsruhe\n3640, D-76021Postfach, KarlsruheGermany", "Departement Physik und Astronomie\nUniversität Basel\nKlingelbergstrasse 82CH-4056BaselSwitzerland", "Departement Physik und Astronomie\nUniversität Basel\nKlingelbergstrasse 82CH-4056BaselSwitzerland", "Departement Physik und Astronomie\nUniversität Basel\nKlingelbergstrasse 82CH-4056BaselSwitzerland", "Institut für Kernphysik\nForschungszentrum Karlsruhe\n3640, D-76021Postfach, KarlsruheGermany" ]	[]	The "Karlsruhe Astrophysical Database of Nucleosynthesis in Stars" (KADoNiS) project is an online database for experimental cross sections relevant to the s process and p process. It is available under http://nuclear-astrophysics.fzk.de/kadonis and consists of two parts. Part 1 is an updated sequel to the previous Bao et al. compilations from 1987 and 2000 for (n,γ) cross sections relevant to the big bang and s-process nucleosynthesis. The second part will be an experimental p-process database, which is expected to be launched in winter 2005/06. The KADoNiS project started in April 2005, and a first partial update is online since August 2005. In this paper we present a short overview of the first update of the s-process database, as well as an overview of the status of stellar (n,γ) cross sections of all 32 p isotopes.	10.1063/1.2187846	[ "https://arxiv.org/pdf/0805.4749v1.pdf" ]	14,475,829	0805.4749	a7593f574139a75f673b9a6cf5d8b25a24b717dc	Karlsruhe Astrophysical Database of Nucleosynthesis in Stars May 30, 2008 1 I Dillmann Departement Physik und Astronomie Universität Basel Klingelbergstrasse 82CH-4056BaselSwitzerland M Heil F Käppeler R Plag T Rauscher Departement Physik und Astronomie Universität Basel Klingelbergstrasse 82CH-4056BaselSwitzerland F.-K Thielemann Departement Physik und Astronomie Universität Basel Klingelbergstrasse 82CH-4056BaselSwitzerland Institut für Kernphysik Forschungszentrum Karlsruhe 3640, D-76021Postfach, KarlsruheGermany Karlsruhe Astrophysical Database of Nucleosynthesis in Stars May 30, 2008 1stellar neutron cross sectionsdatabasecompilations processp process PACS: 25 The "Karlsruhe Astrophysical Database of Nucleosynthesis in Stars" (KADoNiS) project is an online database for experimental cross sections relevant to the s process and p process. It is available under http://nuclear-astrophysics.fzk.de/kadonis and consists of two parts. Part 1 is an updated sequel to the previous Bao et al. compilations from 1987 and 2000 for (n,γ) cross sections relevant to the big bang and s-process nucleosynthesis. The second part will be an experimental p-process database, which is expected to be launched in winter 2005/06. The KADoNiS project started in April 2005, and a first partial update is online since August 2005. In this paper we present a short overview of the first update of the s-process database, as well as an overview of the status of stellar (n,γ) cross sections of all 32 p isotopes. STELLAR NEUTRON CAPTURE COMPILATIONS The first collection of stellar neutron capture cross sections was published in 1971 by Allen and co-workers [1]. This paper reviewed the role of neutron capture reactions in the nucleosynthesis of heavy elements and presented also of a list of recommended (experimental or semi-empirical) Maxwellian averaged cross sections at kT= 30 keV (MACS30) for nuclei between carbon and plutonium. The idea of an experimental and theoretical stellar neutron cross section database was picked up again by Bao and Käppeler [2] for s-process studies. This compilation published in 1987 included cross sections for (n,γ) reactions (between 12 C and 209 Bi), some (n,p) and (n,α) reactions (for 33 Se to 59 Ni), and also (n,γ) and (n,f ) reactions for long-lived actinides. A follow-up compilation was published by Beer, Voss and Winters in 1992 [3]. In the update of 2000 this compilation [4] was extended to big bang nucleosynthesis. It now included a collection of recommended MACS30 for isotopes between 1 H and 209 Bi, and -like the original Allen paper -also semi-empirical recommended values for nuclides without experimental cross section information. These estimated values are normalized cross sections derived with the Hauser-Feshbach code NON-SMOKER [5], which account for known systematic deficiencies in the nuclear input of the calculation. Additionally, the database provided stellar enhancement factors and energy-dependent MACS for energies between kT= 5 keV and 100 keV. The most recent KADoNiS version of this compilation has the aim to provide a clearly arranged and user-friendly online database, which is regularly updated and will be in later stages also extended to p-process studies. PART 1: UPDATED BIG BANG AND S-PROCESS DATABASE Included in the present update (status August 2005) were only cross sections, which are already published. Six semi-empirical estimates (see Table 1) were replaced by experimental data, and 20 recommended cross sections were updated by inclusion of new measurements ( Table 2). A full list of measurements with references, which were (will be) included in the update(s) can be found on the KADoNiS homepage in the menu section "Logbook". Future efforts will be focussed on the re-evaluation of semi-empirical cross sections, as well as the inclusion of theoretical results derived with the Hauser-Feshbach code MOST [6]. Another topic will be the re-calculation of cross sections for isotopes, where a recent change in physical properties (e.g. t 1/2 , I γ ...) leads to changes in already measured cross sections. The KADoNiS homepage provides a datasheet with all necessary informations for each isotope similar to the layout in Ref. [4]. On the top of this page the recommended MACS30 for the total and all partial cross sections are shown. In the "Comment" line one can find the previous recommended values, special comments, and the date of the last review. The field "List of all available values" includes the original values as given in the respective publications, renormalized values, year of publication, type of value (theoretical, semi-empirical or experimental), a short comment about the method (accelerator, neutron and reference source), and the (linked) reference(s). This section is followed by the tabulated MACS, reaction rates and stellar enhancement factors for energies between kT= 5 and 100 keV. A "click" on the field "Show/hide mass chain" gives a graphical plot of all available recommended total MACS30 for the isotopic mass chain of the respective element. The bottom part of each datasheet shows a chart of nuclides, which can be zoomed by selecting different sizes (S, M, L, or XL). By clicking on an isotope in this chart, one can easily jump to the respective datasheet. PART 2: EXPERIMENTAL P-PROCESS DATABASE The second part of KADoNiS will be an experimental p-process database and is expected to be launched in winter 2005/06. It will be a collection of the available experimental reaction rates relevant for p-process studies, e.g. (γ,n), (γ,α), (γ,p), (n,p), (n,α), (p,α), and their inverse rates. The role of (n,γ) reactions in the p process was early recognized by Rayet et al. [7]. The (n,γ)↔(γ,n) competition hinders the photodisintegration flux towards lighter nuclei. Additionally the decrease in temperature at later stages of the p process leads to a freezeout (T 9 ≃ 0.3, corresponding to kT= 25 keV) via neutron captures and mainly β + decays, resulting in the typical p-process abundance pattern with maxima at 92 Mo (N=50) and 144 Sm (N=82). The influence of a variation of reaction rates on the final p abundances has been demonstrated repeatedly [8,9]. Thus, it is necessary for p-process studies to know the neutron capture rates for both, at freeze-out energies (kT= 25 keV) and at the p-process energies (kT= 170-260 keV). The (preliminary) results of our extended measuring program of stellar neutron capture cross sections for p nuclei are shown in Table 3. All of our measurements were carried out on natural samples at the Karlsruhe 3.7 MV Van de Graaff accelerator using the activation technique [10,11]. Neutrons were produced via the 7 Li(p,n) 7 Be reaction by bombarding 30 µm thick layers of metallic lithium on a water-cooled copper backing with protons of E p = 1912 keV. The resulting quasi-stellar neutron spectrum approximates a Maxwellian distribution for kT= 25.0 ± 0.5 keV [12]. In all eight cases ( 74 Se, 84 Sr, 96 Ru, 102 Pd, 120 Te, 130,132 Ba, and 174 Hf) we are able to reproduce the previous recommended total cross sections from [4] However, for an inclusion into the planned p-process database, those MACS30 have to be theoretically extrapolated to p-process temperatures. Another step is then the calculation of inverse reaction rates by detailed balance. TABLE 1 . 1List of recommended semi-empirical stellar cross sections, which were now replaced by experimental values.Isotope Old recomm. value [mb] New exp. value [mb] 128 Xe 248 ± 66 262.5 ± 3.7 129 Xe 472 ± 71 617 ± 12 130 Xe 141 ± 51 132.0 ± 2.1 147 Pm 1290 ± 470 709 ± 100 151 Sm 2710 ± 420 3031 ± 68 180 Ta m 1640 ± 260 1465 ± 100 TABLE 2 . 2List of previous and new recommended stellar cross sections, which were updated by inclusion of new experimental values.Isotope Old recomm. value [mb] New recomm. value [mb] 22 Ne 0.059 ± 0.006 0.058 ± 0.004 40 Ar 2.6 ± 0.2 2.6 ± 0.2 96 Ru 238 ± 60 207 ± 8 102 Ru 186 ± 11 151 ± 7 104 Ru 161 ± 10 156 ± 5 110 Cd 246 ± 10 237 ± 2 111 Cd 1063 ± 125 754 ± 12 112 Cd 235 ± 30 187.9 ± 1.7 113 Cd 728 ± 80 667 ± 11 114 Cd 127 ± 5 129.2 ± 1.3 116 Cd 59 ± 2 74.8 ± 0.9 135 Cs 198 ± 17 160 ± 10 139 La 38.4 ± 2.7 31.6 ± 0.8 175 Lu 1146 ± 44 1219 ± 10 176 Lu 1532 ± 69 1639 ± 14 176 Hf 455 ± 20 626 ± 11 177 Hf 1500 ± 100 1544 ± 12 178 Hf 314 ± 10 319 ± 3 179 Hf 956 ± 50 922 ± 8 180 Hf 179 ± 5 157 ± 2 Table 3 3gives an overview of the status of neutron capture cross sections of all 32 p nuclei at kT= 30 keV. The Bao et al. compilation from 2000 [4] provided measured cross sections for 20 isotopes, but 9 of them ( 92,94 Mo, 96 Ru, 124,126 Xe, 130 Ba, 156 Dy, 180 W, and 190 Pt) with uncertainties ≥9%. For the remaining 12 p isotopes ( 74 Se, 84 Sr, 98 Ru, 102 Pd, 120 Te, 132 Ba, 138 La, 158 Dy, 168 Yb, 174 Hf, 184 Os, and 196 Hg) only theoretical predictions were available. within 20%. Thus, only 6 p isotopes ( 98 Ru, 138 La, 158 Dy, 168 Yb, 184 Os, and 196 Hg) remain without any experimental stellar neutron cross section. With exception of 98 Ru and 138 La, all isotopes can be measured with the activation technique. KADoNiS-The Karlsruhe Astrophysical Database of Nucleosynthesis in StarsMay 30, 2008 ACKNOWLEDGMENTSWe thank E. P. Knaetsch, D. Roller and W. Seith for their help during the irradiations at the Van de Graaff accelerator. This work was supported by the Swiss National Science Foundation Grants 2024-067428.01 and 2000-105328. . B Allen, J Gibbons, R Macklin, Adv. Nucl. Phys. 4205B. Allen, J. Gibbons, and R. Macklin, Adv. Nucl. Phys., 4, 205 (1971). . Z Bao, Käppeler, Adndt, 36411Z. Bao, and Käppeler, ADNDT, 36, 411 (1987). . H Beer, F Voss, R Winters, Ap. J. Suppl. 80403H. Beer, F. Voss, and R. Winters, Ap. J. Suppl., 80, 403 (1992). . Z Bao, H Beer, F Käppeler, F Voss, K Wisshak, T Rauscher, ADNDT. 7670Z. Bao, H. Beer, F. Käppeler, F. Voss, K. Wisshak, and T. Rauscher, ADNDT, 76, 70 (2000). . T Rauscher, F.-K Thielemann, H Oberhummer, Ap. J. 45137T. Rauscher, F.-K. Thielemann, and H. Oberhummer, Ap. J., 451, L37 (1995). Hauser-Feshbach rates for neutron capture reactions. S Goriely, 08/26/05S. Goriely, Hauser-Feshbach rates for neutron capture reactions (version 08/26/05), http://www- astro.ulb.ac.be/Html/hfr.html (2005). . M Rayet, N Prantzos, M Arnould, Astron. Astrophys. 227271M. Rayet, N. Prantzos, and M. Arnould, Astron. Astrophys., 227, 271 (1990). . T Rauscher, Nucl. Phys. A. 758549T. Rauscher, Nucl. Phys. A, 758, 549c (2005). . W Rapp, Report FZKA. 6956W. Rapp, Report FZKA 6956, Forschungszentrum Karlsruhe (2004). . W Rapp, M Heil, D Hentschel, F Käppeler, R Reifarth, H Brede, H Klein, T Rauscher, Phys. Rev. C. 6615803W. Rapp, M. Heil, D. Hentschel, F. Käppeler, R. Reifarth, H. Brede, H. Klein, and T. Rauscher, Phys. Rev. C, 66, 015803 (2002). . I Dillmann, M Heil, F Käppeler, T Rauscher, F.-K Thielemann, subm. to Phys. Rev. C. I. Dillmann, M. Heil, F. Käppeler, T. Rauscher, and F.-K. Thielemann, subm. to Phys. Rev. C (2005). . W Ratynski, F Käppeler, Phys. Rev. C. 37595W. Ratynski, and F. Käppeler, Phys. Rev. C, 37, 595 (1988). . T Rauscher, F.-K Thielemann, ADNDT. 7947T. Rauscher, and F.-K. Thielemann, ADNDT, 79, 47 (2001). I Dillmann, M Heil, F Käppeler, R Plag, T Rauscher, F.-K Thielemann, Proceedings Nuclear Physics in Astrophysics II. Nuclear Physics in Astrophysics IIDebrecen/HungaryI. Dillmann, M. Heil, F. Käppeler, R. Plag, T. Rauscher, and F.-K. Thielemann, Proceedings Nuclear Physics in Astrophysics II, Debrecen/Hungary, May 16-20, 2005, subm. to Eur. Phys. J. A (2005).	[]
[ "A Low-Cost Attack against the hCaptcha System", "A Low-Cost Attack against the hCaptcha System" ]	[ "MdImran Hossen [email protected] \nCenter for Advanced Computer Studies\nCenter for Advanced Computer Studies\nUniversity of Louisiana at Lafayette Lafayette\nLAUSA\n", "Xiali Hei [email protected] \nUniversity of Louisiana at Lafayette Lafayette\nLAUSA\n" ]	[ "Center for Advanced Computer Studies\nCenter for Advanced Computer Studies\nUniversity of Louisiana at Lafayette Lafayette\nLAUSA", "University of Louisiana at Lafayette Lafayette\nLAUSA" ]	[]	CAPTCHAs are a defense mechanism to prevent malicious bot programs from abusing websites on the Internet. hCaptcha is a relatively new but emerging image CAPTCHA service. This paper presents an automated system that can break hCaptcha challenges with a high success rate. We evaluate our system against 270 hCaptcha challenges from live websites and demonstrate that it can solve them with 95.93% accuracy while taking only 18.76 seconds on average to crack a challenge. We run our attack from a docker instance with only 2GB memory (RAM), 3 CPUs, and no GPU devices, demonstrating that it requires minimal resources to launch a successful large-scale attack against the hCaptcha system.	10.1109/spw53761.2021.00061	[ "https://arxiv.org/pdf/2104.04683v1.pdf" ]	233,210,092	2104.04683	65b1c0b0a716b0cfc0cff38ae03019515a68aa06	A Low-Cost Attack against the hCaptcha System MdImran Hossen [email protected] Center for Advanced Computer Studies Center for Advanced Computer Studies University of Louisiana at Lafayette Lafayette LAUSA Xiali Hei [email protected] University of Louisiana at Lafayette Lafayette LAUSA A Low-Cost Attack against the hCaptcha System CAPTCHAs are a defense mechanism to prevent malicious bot programs from abusing websites on the Internet. hCaptcha is a relatively new but emerging image CAPTCHA service. This paper presents an automated system that can break hCaptcha challenges with a high success rate. We evaluate our system against 270 hCaptcha challenges from live websites and demonstrate that it can solve them with 95.93% accuracy while taking only 18.76 seconds on average to crack a challenge. We run our attack from a docker instance with only 2GB memory (RAM), 3 CPUs, and no GPU devices, demonstrating that it requires minimal resources to launch a successful large-scale attack against the hCaptcha system. I. INTRODUCTION CAPTCHAs (Completely Automated Public Turing Tests to Tell Computers and Humans Apart) are computergenerated and graded tests that most humans can easily pass, but current computer programs such as Artificial Intelligence (AI) algorithms, cannot pass [42]. CAPTCHAs protect websites from malicious bots and other forms of automated abuse. As a result, the security of CAPTCHAs is critical to defending the Internet against automated attacks. For years, text CAPTCHAs that ask users to recognize distorted texts from the background of an image have been subjected to automated attacks [9], [11], [14], [16], [17], [27], [28], [44], [44], [45], [47]. Successful attacks against text CAPTCHAs underscore that they are no longer secure against current Machine learning (ML) technologies. As a result, text CAPTCHAs have been primarily replaced by image CAPTCHAs. To some extent, image CAPTCHA schemes are considered more robust to automated attacks than their text counterparts. The rationale behind this is that there are still many hard and open problems in the image recognition domain. However, deep learning (DL) algorithms have recently surpassed humans' cognitive ability in a complex visual recognition task [20], putting the security of image CAPTCHAs in question. Researchers have successfully broken several popular real-world image CAPTCHA schemes exploiting DL technologies [22], [39], [43]. hCaptcha is a relatively new but emerging image CAPTCHA service developed by Intuition Machines, Inc. It asks users to select images matching a category/label provided in the challenge instruction to verify that they are humans and not bots. It is becoming increasingly popular on the Internet as an anti-bot solution. On April 8, 2020, Cloudflare announced that they were ditching Google's reCAPTCHA and adopting hCaptcha on their platforms due to privacy concerns and costs of using reCAPTCHA [32]. Unfortunately, to the best of our knowledge, the security of hCaptcha and its ability to resist automated abuses have not yet been formally evaluated. In this paper, we design and develop an end-to-end system to attack the image hCaptcha system. Our attack is highly effective and efficient: it can break hCaptcha challenges with more than 95% accuracy while taking less than 19 seconds on average to crack a challenge. Most importantly, we show that even a resource-constrained adversary can mount a powerful attack using our system. Our attack reinforces the vulnerability of CAPTCHA designs relying on simple image classification tasks as the underlying AI problem to distinguish between humans and bots. In summary, we make the following contributions: • We design and develop a low-cost, end-to-end system to break hCaptcha service. • We evaluate our system against 270 live hCaptcha challenges and achieve the success rate of attack over 95% with the system taking less than 19 seconds to crack a challenge on average. • We provide a preliminary security analysis of the hCaptcha system. Our analysis shows that the hCaptcha service employs minimal to no mechanism to resist automated abuses other than asking users to solve a simple image recognition task. II. HCAPTCHA BACKGROUND hCaptcha is an image CAPTCHA scheme developed by Intuition Machines, Inc. It is intended to be a dropin replacement for Google's popular CAPTCHA service reCAPTCHA [37]. hCaptcha is mainly designed for using human labor to label machine learning datasets for different companies. Unlike reCAPTCHA, hCaptcha pays the publishers (the website owners hosting hCaptcha service) for every visual challenge successfully solved by website visitors. The hCaptcha marketplace runs on the HUMAN Protocol [3], which aims to enable a new generation of machine intelligence to apply human labor to AI model advancement to achieve human parity in task performance. Websites using hCaptcha earn Human Tokens (HMT) whenever users use the hCaptcha widget on the sites. In recent years, hCaptcha has become increasingly popular among publishers, and according to a 2019 report, 10 million people interact with hCaptcha every month on thousands of websites [36]. Websites usually use CAPTCHAs to prevent automated account creation and abuses from malicious bot programs. As such, CAPTCHAs are generally embedded in registration/login forms. When visitors land into a hCaptcha protected webpage, they will need to click on the hCaptcha checkbox "I am Human" to initiate the challenge. After that, they will be prompted with the challenge widget where the actual CAPTCHA test is located. The users are required to select all images matching a description (see Figure 1) to pass a hCaptcha challenge. It is worth noting that the system also provides an "invisible" mode where users will be automatically prompted with a challenge only when they lack enough trust with the hCaptcha system to prove their humanness. III. THREAT MODEL A CAPTCHA scheme is considered broken if a bot can break the CAPTCHA challenges with a success rate higher than 0.01% [13]. Designing a CAPTCHA service with such constraints is very challenging in practice. Elson et al. [15] relaxed the tolerable success rate of attack up to 0.6%. The effectiveness of the attack also depends on the cost of the attack. A powerful adversary who possesses many resources can afford such low success rates and can scale the abuse's impact by attacking the given CAPTCHA system hundreds of thousands of times. Diverting from the above threat model and closely following the threat model in [8], our threat model involves an attacker with limited resources. We will assume the attacker is limited to one computer with a small-size RAM and one IP address. Since such an attacker cannot afford to have a lower success rate, we aim for an accuracy benchmark above 50%. IV. SYSTEM OVERVIEW Our automated CAPTCHA breaking system solves a hCaptcha image challenge in three main steps: 1) Obtaining the challenge, 2) Solving the challenge, and 3) Submitting and verifying the solution. Step 1 and step 3 are browserspecific tasks and automated by controlling a web browser using browser automation software. For step 2, we use an image classifier to classify candidate images in the challenge to find potential target images for the solution. We now discuss in detail the implementation of each step. V. IMPLEMENTATION DETAILS A. Obtaining the hCaptcha Challenge Websites usually embed the hCaptcha widget on the webpages that need protection from bots, spam, and other forms of automated abuse, such as the login/registration forms. One can generally locate the hCaptcha container by its class name h-captcha with a data-sitekey attribute set to the public key, a unique key provided by hCaptcha for each registered page. Our system locates .h-captcha container inside the HTML form. hCaptcha checkbox widget is rendered as an iframe on the webpage. The system switches to the frame and clicks on the "I am Human" checkbox identified by #checkbox. After that, a new iframe element containing the hCaptcha challenge widget pops up. Our system then switches to the challenge widget. The challenge widget contains the actual image challenge that the users must solve to pass the hCaptcha's antibot test. Our system first locates the challenge instruction .prompt-text. The instruction includes the name of the image category/label that users need to select from a set of candidate images/payloads. We can locate the payloads by their identifier .task-image. The source URLs of the images are generated dynamically and can be accessed only for a few seconds. After that period, the URLs expire. Our bot fetches the payloads and stores them on a predefined location of our computer. B. Solving the Challenge We used a deep neural network-based image classifier to classify candidate images in a hCaptcha challenge. Our system provides each image an ID. The images obtained in the previous step and their unique identifiers are sent to the classifier. The classifier returns a label for each image. We process the classifier outputs to filter out the images that do not match the target image category provided in the challenge instruction. For the remaining images, our system keeps the image IDs as a potential solution. We now briefly provide details of our image classifier network. Network Architecture. Our image classifier network follows the Residual Network (ResNet) [21] architecture. ResNet allows building deeper neural networks by utilizing skip connections or shortcuts to jump over some layers. It solves the vanishing gradient issue with traditional deep neural nets, trains faster, and has been proven to produce stateof-the-art performances in several complex vision tasks. While it is possible to build very deep residual neural networks involving as many as 152 layers, we opted to use only 18 layers residual network (ResNet-18) for our task. We made this choice because we want to run the model on a machine that does not necessarily include a GPU for faster computation. Further, training a deeper network such as ResNet-101 takes more time, and running the inference on CPU takes longer. We found that the ResNet-18 model provides decent performance sufficient for our task. In our work, we used a ResNet-18 model pretrained on the ImageNet [34] dataset and finetune it for our task because training the entire network from scratch requires a vast amount of training samples, often a time-consuming and computationally expensive task. We found that hCaptcha challenges show images from only nine classes. We reset the final fully connected (FC) layer of the ResNet-18 such that the size of each output sample is set to nine. Data Collection. With more training samples, deep neural networks learn to extract a better representation of underlying data distribution. At the same time, manually collecting and labeling data is a labor-intensive process. We collected 5000 challenges from 3 hCaptcha protected websites from the period of May 2020 to July 2020. Figure 2 shows the frequencies of different image categories. As shown in Figure 2, we observed only nine image categories (bus and motorbus are considered the same category) frequently appear on hCaptcha challenges. Interesting, data from all of these categories are already available on the OpenImages [25] dataset, a publicly accessible dataset for training machine learning models on various image recognition tasks. Therefore, instead of manually labeling original hCapctha images, we extracted 45000 images from the nine categories to train our ResNet-18 network. We tried to keep the dataset balanced, but some categories have more training samples than others, making the dataset slightly skewed. Training ResNet-18. We split the dataset into three sets: training, validation, and testing sets. The training set is the data that the network will learn from. The validation set is used for fine-tuning the model's hyperparameters. The testing set is used for assessing the model's generality on unseen data. We used the following hyper-parameters for training our network: the batch size of 32, categorical crossentropy as the loss function, Adam optimizer with a learning rate of 0.0001. We trained the model for 40 epochs. The model achieved an accuracy of over 88% on the testing set. The training was performed on a machine running Arch Linux OS with an NVIDIA GeForce RTX 2070 GPU and took about 143 minutes to complete. Note that training the network is mainly a one-time task. However, we may need to retrain the model once in a while if new image categories appear in the challenges. C. Submitting and Verifying the Solution Once the image classifier determines the potential images matching the target label, our system locates corresponding payloads task-image in the challenge widget and performs a mouse click on each of them. Next, it submits the solution by clicking on the submit button, which is identified by .button-submit. Once our bot submits the solution, we need to verify whether the challenge is passed or not. The status of challenge can be monitored via aria-label attribute of #checkbox in hCaptcha widget. The aria-label attribute contains the text string -"You are verified" -when a challenge is successfully solved; otherwise, hCaptcha triggers an error message to indicate a failure. The system identifies error messages via their class name display-error. Further, hCaptcha provides a mechanism for verifying a challenge from the server-side of web applications using the hCaptcha widget. When the user successfully solves a CAPTCHA, the hCaptcha script inserts a unique token, h-captcha-response, into the HTML form data. The server-side needs to check whether the token is valid at the API endpoint URL provided by hCaptcha. The request to the endpoint expects two parameters: hCaptcha secret API key associated with the website and the h-captcha-response token POSTed from the HTML page. Upon receiving the request, the endpoint returns a JSON response. If the token is valid, the "Success" field in the response is set to "True"; otherwise "False". VI. IMPLEMENTATION AND EVALUATION PLATFORM For performing browser-specific tasks, such as visiting and interacting with hCaptcha protected websites, initiating and submitting the hCaptcha challenge, our bot utilizes the puppeteer [5] web automation software. Google develops the puppeteer web automation framework to control the Chrome web browser programmatically. However, we used puppeteer-firefox [6] with the Firefox web browser because it is easier to customize Firefox. Specifically, we used puppeteer-firefox 0.5.1, with Firefox 65.0. Our image classifier network, ResNet-18, was built on top of PyTorch 1.7.0. We ran all of our experiments inside a docker container running the Ubuntu 20.04 image. To simulate a low-resource attack, we configured the container such that the maximum amount of memory (RAM) it could access from the host machine is 2GB. We also set the number of CPU cores the container could use to 3. The host machine on which we ran the container has 8 Intel core i7-8550U (1.8GHz) processors, 16 GB of RAM running Arch Linux OS. Experimental Setting. We used three websites for our experiments: www.hcaptcha.com, 2captcha.com, and one of our own websites, respectively. We did not affect the security/availability of the tested websites by limiting our bot's interactions only to the hCaptcha related components on the hCaptcha protected webpages. Further, we did not send excessive requests within a short time window to prevent DoSing the sites. We accessed the websites from a regular and nonacademic IP address unless otherwise specified. We launched our system with a fresh browser profile during each visit, i.e., no caches or cookies were retained from prior requests. We also did not attempt to change our browser environment's configuration, i.e., using custom User-Agent header, changing screen resolution, etc. VII. ATTACK EVALUATION Accuracy and Speed of Attack. We submitted 270 challenges using our automated system, and it successfully solved 259 of them, resulting in an accuracy or success rate of attack of 95.93%. It takes 18.76 seconds on average to crack a challenge. Figure 3 depicts the breakdown of our system's attack speed by individual modules. Automating browsing-related activities (e.g., initiating the challenge, interacting with checkbox and challenge widget, and submitting and verifying challenge) takes more time. Our deep learning classifier (the Solver) takes 3.79 seconds to classify the images (usually 9) in a challenge, on average. Note that one can further speed up this process by running the inference on a GPU-enabled machine; however, our attack focuses on a low-cost attack, and we show that one can mount a highly accurate attack under minimal resource constraints using our system. We came across only 9 image categories in 270 submitted challenges. Figure 4 shows the frequency and success rate of attack for these image categories. Figure 5 shows the probability distribution of the number of image selections per challenge in all submitted challenges. The majority of the challenges have 2 to 5 images as correct solutions. We also noticed some challenges having as many as 14 images as part of the correct image selections. Solution flexibility. We observed hCaptcha often accepts one or two incorrect image selection(s) while solving a hCaptcha challenge. We manually solved some hCaptcha challenges by choosing different combinations of correct and wrong image selections to test whether hCaptcha provides any solution flexibility. Table I network IP. One might expect an IP address belonging to an academic network might seem less malicious than an IP address that belongs to a Tor exit node. While accessing the hCaptcha-protected webpage, we sent all 200 requests from a single IP address in a row with a 30-second gap between two subsequent requests. Interestingly, our system achieved a success rate of over 90% for all three IP addresses. This suggests that hCaptcha generally does not discriminate against users' IP address types. Adaptability. We also tested whether hCaptcha adapts the challenge difficulty according to the suspiciousness level of the users. For instance, a client who accesses a hCaptcha enabled webpage using web automation software is more likely to a bot than someone accessing the page from a regular browser. In such a scenario, an adaptive CAPTCHA service would escalate the threat level for malicious clients by asking them to solve more complex challenges or solve multiple challenges before accessing the online service. Hossen et al. [22] showed that Google's image reCAPTCHA system adopts such policies to limit the malicious bot program's abuse. We tested if hCaptcha has such measures in place by running several experiments. Our system tried mimicking a regular user browser by overriding several Navigator JavaScript properties in our main experiment. For instance, we set the navigator.webdriver to "False", which is usually set to "True" while using a web automation software, navigator.plugins to a random number, and screen resolution to the size of a regular desktop computer, etc. To test hCaptcha's adaptability, we set up an experiment to control the web browser from the automation software with all default settings. We also ran the browser in headless mode while solving the challenges. The rationale for doing this is that setting the browser to headless mode sends a clear signal that the client is using web automation software. This also signals that the request is likely to be generated from an automated bot program. We solved 100 challenges with this setting. Furthermore, we submitted the same number of challenges using Selenium [7] WebDriver for Firefox as well. Selenium is the most popular web automation software. We analyzed the results for each experimental setting to identify any discrepancies among these different settings. However, we did not notice any distinct pattern that can distinguish the settings. For example, we came across the same nine image categories, achieved similar accuracy (over 90%) in all experimental settings. Further, none of the requests were blocked in any of the experimental settings. Our analysis indicates that hCaptcha solely relies on correct image selections to verify a solution without adapting challenges based on users' threat levels. Blocking. One of the design goals of the hCaptcha system is not to leak detections in real-time. To ensure we did not miss any server-side blocking during our main experiment, we deployed hCaptcha on our website. We created a demo web application that allows a client to register for an account with a user name and password. The registration form is protected by hCaptcha. The web application backend processes the form data only when the client solves a valid CAPTCHA test. Once the user submits the form, the server-side code sends the user response token h-captcha-response POSTed with the form to the hCaptcha backend for verification. If the authentication succeeds, the form is processed. Otherwise, we show a warning to the clients that they failed the robot verification test and their inputs could not be processed. The hCaptcha deployment console allows website owners to adjust the difficulty level 1 of served CAPTCHA tests for the clients accessing the site. It has four difficulty levels: easy, moderate, difficult, and always on. By default, the CAPTCHA difficulty level is set to moderate. We ran our system against our web application and attempted to register for 400 fake accounts by automatically filling out the form and solving the challenges. We were able to create 369 such accounts. That means our bot could crack 369 hCaptcha challenges automatically, resulting in an attack success rate of 92.25%. Next, we increase the difficulty level to difficult and tried to create the same number of fake accounts as before. This time, we were able to register for 354 accounts. Note that all the requests to our web application were sent in a row with only a 1-second delay between two subsequent requests. Further, we followed the same experimental setting mentioned in Section VI during this experiment. During our experiment, only 17 of our attempts (out of the total 800 combined) were blocked by hCaptcha with the message -"Rate limited or network error. Please retry." Besides ratelimiting the bot to a particular session, we did not observe any strict blocking policy by hCaptcha to prevent a malicious client from accessing the service for a specific period. Next, we attempted to trigger blocking deliberately by sending too many requests simultaneously. We launched 50 instances of our bot program concurrently ten times with a 2-second delay between two subsequent iterations against our hCaptcha-enabled webpage. We noticed the hCaptcha system blocked many of our requests with the warning message -"Your computer or network has sent too many requests." Specifically, the number of blockages for the ten iterations are 24,40,48,29,28,26,26,29,30, and 28, respectively. Image Repetition. We found that hCaptcha often repeats images across different challenges. We computed the MD5 hashes of 48330 images collected from the hCaptcha challenges during our analysis and identified 9854 redundant images belonging to 1985 sets of identical images. Cryptographic hash functions such as MD5 may not provide an accurate number of repeated images since the slightest modification in the input will produce a drastic change in the output. As such, we used the perceptual image hash (pHash) [23] algorithm to find similar or completely identical images in the submitted challenges. Interestingly, we found the same 1985 set of images in our pHash analysis as well. That means while repeating the same image across multiple challenges, hCaptcha makes no attempt to modify the image and gives exact copies of it. User Data Collection. Besides asking users to prove their humanness by passing a CAPTCHA challenge, hCaptcha collects information about users' browsers and their devices to assess their susceptibility to being bots. We analyzed the hCaptcha client-side JavaScript library responsible for rendering the challenge on the users' browsers. We found that it checks the following data: browser family (e.g., Chrome, Firefox, Internet Explorer) along with version numbers, Operating System family (e.g., Windows, Linux, Mac OS), OS type (e.g., desktop, mobile). From the web browser, hCaptcha probes the device's screen resolution, the number of plugins installed, mime types, whether it supports canvas and Web Assembly, and whether the device supports touch. In addition to these, hCaptcha also uses dynamic information such as touch events, keypress events, scroll positions, etc. A. Online Attacks We also performed an online attack using three state-ofthe-art online vision API services for image recognition. These services are Google Cloud Vision API [2], Amazon Rekognition [1], and Microsoft Azure Cognitive Vision API [4]. First, we submitted several hundred hCaptcha challenge images from different categories to these services separately and analyzed the classification results. Note that the vision APIs can recognize multiple objects in an image, thus returning multiple labels along with the confidence scores (see Table II). We found that the labels' names are mostly compatible with hCaptcha image classes by manually analyzing the label sets. As a result, we could simply map a label set for an image returned by a vision API directly to the original hCaptcha challenge image class. We developed a proof-of-concept system by replacing our solver module with a particular vision API service. The system works as follows. First, it visits a predefined webpage using hCaptcha. Second, the system initiates the challenge and downloads the images. Third, we send the images to the API service for recognition. Fourth, the system analyzes the label set for each image. If one of the tags in a set matches the hCaptcha target image category, it marks it as a potential solution and saves the image ID. Finally, our system clicks on potential target images in the hCaptcha challenge widget and clicks on the "Submit" button. We then verify whether the challenge has been passed or not using the method described in Section V. We submitted 100 live challenges using each API service. Table III depicts the accuracy and speed of our online attack. All these services achieved an accuracy or success rate of attack over 90%, with Microsoft Azure Cognitive Vision API having the highest success rate (98%). In summary, our original system provides comparable performance to the online vision API-based attack. It is also more cost-effective since those vision services usually incur charges for each API request. VIII. COUNTERMEASURES We discuss potential measures to counter our attack, their limitations, and potential impacts in this section. Use Broader Image Categories. hCaptcha uses only a small number of image categories, making it trivial for an attacker to collect a sufficient amount of data to train a highly accurate image classifier. Expending the image categories will make this process relatively challenging. Note that doing so does not necessarily prevent the attack, hence provides a temporary solution only. Adversarial Examples. Machine learning models, including deep neural networks, have been shown to be vulnerable to adversarial examples [19], [30], [40], specifically crafted inputs that can trick the models into making wrong predictions. Recent work [29], [38] has already demonstrated the efficacy of using adversarial examples in image CAPTCHA designs. Designers can take advantage of this vulnerability by injecting adversarial perturbations in the CAPTCHA challenge images to dupe deep learning-based classifiers, thus lowering the attack accuracy. Resist Web Automation Software. Since most bots rely on the web automation software to launch automated attacks, fingerprinting and resisting requests originating from widely used web automation frameworks will likely lower attackers' success rates. Adaptability. Adapting the challenge based on users' suspiciousness levels and presenting complex challenges to highly suspicious clients and easy ones to users most likely to be humans will discourage malicious bots while providing easy passes to legitimate humans. However, determining the suspiciousness and scoring the requests based on that might require extensive experiments. Commonsense Knowledge. When facing a task that involves higher-order reasoning, machines do not usually perform well. Designers can exploit this weakness by forming the instruction that requires some common sense knowledge to decode what image category needs to be selected to pass a challenge, making the underlying AI problem harder for computers. This, however, may negatively impact the overall usability of the CAPTCHA scheme for humans; therefore, it requires further research to determine whether such a design is practical in the real world. IX. RELATED WORK Image CAPTCHAs. The Asirra CAPTCHA [15], proposed in 2007, relied on the presumed difficulty of automatically distinguishing images of dogs and cats. However, in 2008, Golle et al. [18] developed a machine learning classifier trained on color and texture features, automatically solving Asirra CAPTCHA challenges with a probability of 10.3%. The ARTiFACIAL CAPTCHA scheme proposed by Rui et al. [33] requires users to identify faces and facial features within a heavily distorted image. Zhu et al. [48] demonstrated successful attacks against a series of earlier image CAPTCHA schemes, including ARTiFACIAL. The authors also recommended several guidelines for designing robust image CAPTCHA schemes based on the insights gathered from their attacks. Yardi et al. [46] proposed photo-based authentication for social networks where a user is required to identify subjects who are uniquely known to him/her to pass the CAPTCHA test. However, Polakis et al. [31] showed that the photo-based authentication system could be automatically solved by leveraging publicly available data and face recognition algorithms. Sivakorn et al. [39] demonstrated an attack against the earlier implementation of image reCAPTCHA v2 service by leveraging online image annotation services. While that version of reCAPTCHA v2 is no longer in use and the current version is likely to be immune to such attacks, their attack revealed some interesting insights into reCAPTCHA's advanced risk analysis engine, which determines users' likelihood of being bots using several signals, including users' browser environment. In 2019, Weng et al. [43] demonstrated a series of deep learning-based attacks against different real-world image CAPTCHA services and found them highly vulnerable automated attacks. More recently, Hossen et al. [22] proposed an object detection-based solver that was able to break the latest version of reCAPTCHA v2 challenges with a success rate of over 83%. Their attack also demonstrated that antirecognition techniques such as noise and distortion to render the images unrecognizable to deep learning technologies could be bypassed to a great extent by an advanced attacker. Text CAPTCHAs. The security of text CAPTCHAs has been extensively studied in the literature. Most text CAPTCHAs deployed on the Internet are highly vulnerable to machine learning-based attacks. Mori et al. [27] developed object recognition techniques for breaking Gimpy and EZ-Gimpy CAPTCHAs that are based on recognizing the word in the presence of clutter, obtaining a success rate of 33% and 92%, respectively. Yan et al. [45] presented novel character segmentation techniques to attack the Microsoft CAPTCHA, which was designed to be segmentationresistant at that time. Li et al. [26] conducted a comprehensive study on e-banking CAPTCHA schemes and developed a set of image processing and pattern recognition techniques to break the schemes. Their attacks achieved an almost 100% success rate in most cases. Bursztein et al. [12] evaluated the strengths and weaknesses of text CAPTCHAs and showed that automated attacks could break most of them. In 2014, Bursztein et al. [9] presented a novel approach to solving text CAPTCHAs in a single step using machine learning to attack the segmentation and the recognition problems concurrently. Their approach was generically applicable to all evaluated schemes, achieving a success rate significant enough to consider them broken. Gao et al. [17] proposed a simple, low-cost attack to break a wide range of realword text CAPTCHAs with a success rate ranging from 5% to 77%. Recently, Ye et al. [47] presented a GAN-based approach requiring only a small amount of training samples to break the most widely used text CAPTCHAs. Audio CAPTCHAs. In 2002, Kochanski et al. [24] proposed using the speech recognition problem for the reverse Turing test. They developed a synthetic benchmark for evaluating the efficacy of automated solvers against audio CAPTCHAs. The paper concluded that humans significantly outperform automatic speech recognition (ASR) systems when noise/distortion is injected into spoken digits. Tam et al. [41] tested the security of audio CAPTCHAs from popular websites against several machine learning algorithms and achieved correct solutions for test samples with an accuracy of up to 71%. Bursztein et al. [10] developed an automated solver called Decaptcha that was able to break 75% of eBay audio CAPTCHAs. In 2015, Sano et al. [35] developed an audio reCAPTCHA solver based on speech recognition techniques using hidden Markov models (HMMs). Their attack successfully broke the earlier version of audio reCAPTCHA challenges with 52% accuracy. In 2017, Bock et al. [8] developed the unCaptcha, a low-resource and powerful audio CAPTCHA solver that leverages off-the-shelf speech-to-text services with a novel phonetic mapping technique to break audio reCAPTCHA challenges with over 85% accuracy. X. CONCLUSION We present a low-resource, high success rate attack on the hCaptcha service. Our automated CAPTCHA breaker solves hCaptcha challenges with 95.93% accuracy, making its reverse Turing tests broken. Our security analysis demonstrates that hCaptcha lacks stringent security measures to prevent automated abuses, which will have a severe consequence on the security of online services that rely on hCaptcha to defend against malicious bots. In the future, we plan to investigate the effectiveness of our attack methodology on other image CAPTCHA services relying on image recognition as their underlying AI problem. RESPONSIBLE DISCLOSURE We reported our attack and countermeasures to the hCaptcha security team to help them make the system more robust to automated attacks. They responded that their system would have been pretty confident that our traffic was automated based on the techniques we used, and we would never have observed additional countermeasures. However, we did not notice any measures preventing our bot from passing the image CAPTCHA tests during our experiment. The hCaptcha security team stated that they could not disclose system internals and behavior details since it is proprietary software but mentioned that the website owners would not earn any Human Tokens (HMT) for the traffics flagged as automated by the system even when the bot bypasses the CAPTCHA tests. But the hCaptcha deployment dashboard shows we earned 0.0717 HMT for the challenges that our automated program solved. Figure 1 : 1A hCaptcha challenge widget. Figure 2 : 2The frequency of each image category appears in collected challenges. Figure 3 : 3Cumulative distribution of time required by each module. Figure 4 : 4The accuracy and frequency of each image category in the solved challenges. Figure 5 : 5The probability distribution of no. of images selected per challenge. lists the results of our experiment. For single-prompt CAPTCHAs, users must solve only a single image challenge to pass a test. Double-prompt CAPTCHAs require that two image challenges must be solved subsequently to pass a test. FromTable I, one can see that, to some extent, hCaptcha is flexible while determining a correct solution to a challenge. Influence of IP address. To see whether clients' IP address type affects the attacker's success rate, we submitted 200 challenges separately from three IP addresses. The three IP addresses are an academic IP, a VPN IP, and a Tor Table I : IResults of solution flexibility: Combinations for passing the hCaptcha. Here, n = number of correct image selections, and k = number of wrong image selections.Image selection Constraint Pass (%) n Correct + k Wrong (Single-prompt) k ≤ 1 73.50 n Correct + k Wrong (Double-prompt) k ≤ 1 24.50 (n-1) Correct (Single-prompt) n ≥ 3 71.50 (n-1) Correct (Double-prompt) n ≥ 3 61.50 (n-1) Correct + k Wrong (Single-prompt) k > 0 20.00 (n-1) Correct + k Wrong (Double-prompt) k > 0 30.50 Table II : IIList of labels returned by three image recognition APIs for a sample image from hCaptcha challenge.Image Google Cloud Vision Microsoft Computer Vision Amazon Rekognition Land vehicle, Vehicle, Transport Truck, Car, Mode of transport, Motor vehicle, Trailer truck, Trailer, Asphalt outdoor, truck, road, transport, street, parked, trailer, car, large, lot, parking, front, sitting, driving, side, bed, city, bus, fire, man Truck, Transportation, Vehicle, Tow Truck, Person, Human, Trailer Truck Table III : IIIAttack performance of off-the-shelf vision APIs.Vision API Accuracy (%) Speed (s) Amazon Rekognition 92 16.85 Microsoft Computer Vision 98 14.93 Google Cloud Vision 96 15.28 https://medium.com/@hCaptcha/how-hcaptcha-difficulty-settings-work-13d84279d378 ACKNOWLEDGMENTSThe authors thank the anonymous reviewers for their valuable comments that helped improve this paper. 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[ "Incentive-Compatible Classification ", "Incentive-Compatible Classification " ]	[ "Yakov Babichenko \nTechnion -Israel Institute of Technology Haifa\nIsrael\n", "Oren Dean \nTechnion -Israel Institute of Technology Haifa\nIsrael\n", "Moshe Tennenholtz \nTechnion -Israel Institute of Technology Haifa\nIsrael\n" ]	[ "Technion -Israel Institute of Technology Haifa\nIsrael", "Technion -Israel Institute of Technology Haifa\nIsrael", "Technion -Israel Institute of Technology Haifa\nIsrael" ]	[]	We investigate the possibility of an incentive-compatible (IC, a.k.a. strategy-proof) mechanism for the classification of agents in a network according to their reviews of each other. In the α-classification problem we are interested in selecting the top α fraction of users. We give upper bounds (impossibilities) and lower bounds (mechanisms) on the worst-case coincidence between the classification of an IC mechanism and the ideal α-classification. We prove bounds which depend on α and on the maximal number of reviews given by a single agent, Δ. Our results show that it is harder to find a good mechanism when α is smaller and Δ is larger. In particular, if Δ is unbounded, then the best mechanism is trivial (that is, it does not take into account the reviews). On the other hand, when Δ is sublinear in the number of agents, we give a simple, natural mechanism, with a coincidence ratio of α.	10.1609/aaai.v34i05.6191	[ "https://ojs.aaai.org/index.php/AAAI/article/download/6191/6047" ]	208,176,476	1911.08849	2feddbc12521cb7ec62442e4e6e6d8daf3f7db47	Incentive-Compatible Classification * Yakov Babichenko Technion -Israel Institute of Technology Haifa Israel Oren Dean Technion -Israel Institute of Technology Haifa Israel Moshe Tennenholtz Technion -Israel Institute of Technology Haifa Israel Incentive-Compatible Classification * We investigate the possibility of an incentive-compatible (IC, a.k.a. strategy-proof) mechanism for the classification of agents in a network according to their reviews of each other. In the α-classification problem we are interested in selecting the top α fraction of users. We give upper bounds (impossibilities) and lower bounds (mechanisms) on the worst-case coincidence between the classification of an IC mechanism and the ideal α-classification. We prove bounds which depend on α and on the maximal number of reviews given by a single agent, Δ. Our results show that it is harder to find a good mechanism when α is smaller and Δ is larger. In particular, if Δ is unbounded, then the best mechanism is trivial (that is, it does not take into account the reviews). On the other hand, when Δ is sublinear in the number of agents, we give a simple, natural mechanism, with a coincidence ratio of α. Introduction There are many situations in which peer agents have binary, directed interactions with each other, and in which one side can rate his experience from the interaction, or rather rate the other agent. The following are just a few examples: 1. E-commerce sites in which buyers might also be sellers (e.g., ebay.com, amazon.com). 2. Academic paper reviewers for a conference might themselves be authors of papers submitted to the same conference. 3. Employees in an organisation are sometime asked to fill out a sociometric overview of their fellow friends. In all of the above examples, a coordinator/manager is classifying the agents in the system according to the reviews they received. In the e-commerce example, the top-rated sellers will appear higher and more frequently in search results; the academic conference will only accept a certain toprated portion of the papers. Employees with higher sociometric results have better chances at a promotion. A natural problem arises-in order to maximize one's relative rating, it is a dominant strategy in these situations to give a harsh critique to all interactions. In this paper, we model the agents and their interactions as a directed network and ask whether it is possible to offer an incentive-compatible (IC) mechanism to select a subset which represents the top-rated ("worthy") agents. We measure the quality of a mechanism as the resemblance between the selected set of the mechanism and the set of top-rated agents, in the worst case. We investigate the relation between the quality of the best possible mechanism to two parameters: (i) the maximal number of reviews a single agent can issue (the maximal out-degree in the network, denoted Δ), and (ii) the fastidiousness of the system (the relative size of the selected set, denoted α). Our contribution In this paper we investigate the existence of an αclassification IC mechanism in weighted networks. Weighted networks without any limitation were not considered before as a framework for selection mechanisms ((Kurokawa et al. 2015) considered only Δ-regular weighted networks; the assumption of Δ-regularity somewhat simplifies the optimisation criterion since in this case the optimisation for average in-weights is equivalent to the optimisation for the sum of in-weights.). The most significant novelty of our model is the consideration of mechanisms which classify the agents to "worthy" and "unworthy", in contrast to previous works which only considered k-subset selection (k-selection) mechanisms. Our optimisation criterion is the coincidence between the mechanism's classification and the ideal classification. The difference from k-selection is two-fold. First, we do not know the exact size of the set which we need to select. Second, we try to select as many of the right (truly worthy) agents and not the wrong agents regardless of how high their in-degree is; this is very different from k-selection, which just looks for a subset with high in-degree (even if it is completely disjoint to the optimal subset). We prove upper and lower bounds on the quality of the best possible IC classification mechanism. We chart the behaviour of these bounds as a function of Δ and α. We show that as Δ grows (agents are allowed to review many others), the possibility of an IC classification mechanism narrows down until for large values of Δ the best mechanism is one of the two trivial mechanisms: select all the agents as wor-thy, or select every agent independently with probability 1/2. We show the reverse behaviour with α: as we lower α (the system is more picky about its worthy-classified agents), the quality of the best possible IC classification mechanism decays to zero. On the other hand, for fixed α and for Δ which is negligible with respect to the number of agents, we provide a mechanism with a positive quality. The idea behind this mechanism is based on a well-known practice to partition the agents into three subsets: absolutely worthy, borderline, and absolutely unworthy. Unlike the well-known practice, our mechanism suggests to classify an agent into these categories after ignoring his reviews on others. This makes the mechanism IC, but complicates the performance analysis of the mechanism. Previous works (e.g. (Alon et al. 2011), (Kurokawa et al. 2015)) showed the existence of an optimal k-selection mechanisms when k is large (say k = ω(1) or k = ω(Δ) with regard to the number of agents). As explained above, these mechanisms only select a k-subset of agents with high indegree, while an α-classification mechanism needs to select as many of worthy agents and not unworthy agents. This extra predicament shows in the results as we bound the quality of any IC mechanism away from 1 (i.e., for any α < 1 there is no ideal IC classification mechanism). This also shows the significance of our mechanism which, under reasonable assumptions, selects not only good agents but the right agents. The graphs in Figure 1 summarize our main findings. Graph (a) shows our bounds on the quality of any IC mechanism as a function of α when Δ is negligible with respect to the number of agents. The grey dashed line is the ideal mechanism. The upper bound (red line) shows that for any α < 1, there is a gap between the best possible mechanism and the ideal mechanism, and this disparity is larger for lower values of α. The blue line denotes the quality of our proposed mechanism, and so the green area is what we know about the possible value of the quality of the best IC mechanism. Graph (b) shows the quality of any IC mechanism as a function of α when Δ is not bounded. Here the upper and lower bounds coincide to a single line, that is, we know what is the best possible quality for any value of α. 1: Our bounds for the quality of an IC classification mechanism as a function of α. In (a) Δ = o(n). The red/blue graphs denote our upper/lower bounds and the green area denotes the known feasible quality range. In (b) Δ = n − 1. The single green graph is the quality of any IC mechanism. Related work There have been in the past decade quite a few works on IC (sometimes called strategyproof or impartial) selection mechanisms in networks. The most similar model to ours is (Kurokawa et al. 2015). In that paper the authors considered the k-selection in a weighted Δ-regular network (that is, every agent has Δ out-edges and Δ in-edges), where k and Δ are constants. They considered IC mechanisms which try to optimize the average in-weights of the selected set. Their main result is a probabilistic mechanism which is optimal when Δ is negligible with respect to k. Other papers on this subject only consider unweighted networks, which can be seen as a special case of the weighted networks with weights in {0, 1}. These works can be divided into two flavours: (a) Optimisation works which try to optimize the total indegree of the selected agent or set of agents. Examples of this group are Alon et al.;Fischer and Klimm;Bjelde, Fischer, and Klimm;Bousquet, Norin, and Vetta (2011;2017;. Another example is Babichenko, Dean, and Tennenholtz (2018) in which the authors try to optimize the progeny of the selected agent. These works differ in several parameters of the problem, such as deterministic/probabilistic mechanisms; exact selection (selected set must be of size k) or inexact selection (selected set is of size at most k); and the subfamily of networks considered (all networks/m-regular networks/acyclic networks/etc.). (b) Axiomatic works which define a set of axioms and investigate the possibility/impossibility of mechanisms that fulfil maximal subsets of these axioms. Examples of this group are Holzman and Moulin; Mackenzie; Aziz et al. (2013;2015;2016). In (Altman and Tennenholtz 2008) the authors considered the possibility of complete ranking mechanisms under certain axioms. Paper organization. In Section 2, we formally present our model and main results. In Section 3, we prove our impossibility (upper bound) propositions (Propositions 10 and 11). These proofs rely on an extension of our model to a symmetric, probabilistic mechanism; this extension is formally defined in the beginning of Section 3. In Section 4, we present and prove a mechanism with a positive quality when the number of reviews of a single agent is negligible with respect to the number of agents (Proposition 12). In Section 5, we conclude and discuss our results. Model and main results Let N = [n] be a set of n agents. 1 We represent the interactions between the agents as a directed graph, G(N, E); thus an edge (x, y) means that agent x interacted with agent y and is allowed to review this agent. Let E in (x), E out (x) be the sets of in-edges and out-edges of x, respectively. We assume that each agent in the network is in control of the weights of his outgoing edges. These weights, which are real numbers in the interval [-1,1], 2 represent the reviews of the source agent for the target agents. Thus, the reviews of agent x ∈ N for his interactions are {we\|e ∈ E out (x)}. After all agents submitted reviews on their interactions, 3 we get a weighted, directed graph; from now on we assume all the edges of G are weighted. Based on the reviews that agent x received, {we\|e ∈ E in (x)}, we define his score and his relative ranking in the system. The score of agent x is the average of weights on E in (x): s(x, G) = ⎧ ⎪ ⎨ ⎪ ⎩ e∈E in (x) we \|E in (x)\| , E in (x) = ∅, 0, E in (x) = ∅. (1) The ranking of agent x is the number of agents who strictly have a better score than him: r(x, G) = \|{y ∈ N \|s(y, G) > s(x, G)}\|. 4 For a given real parameter α ∈ (0, 1) we consider as worthy the subset of agents who are in the top α-ranking: 5 Iα(G) = {x ∈ N \|r(x) < αn }. Notice that the size of Iα(G) is at least αn, but might be higher in case of ties. For instance, in the empty graph, Iα(G(N, ∅)) = N for all α. We denote by Iα(G) = N \Iα(G), the subset of unworthy agents. Our goal is to offer an IC mechanism which selects a set which is as similar as possible to the subset of worthy agents. Formally, let G(n) be the family of all [-1,1]-weighted, directed networks on n nodes and let P (N ) be the power set of N . Definition 1. A classification mechanism is a function M : G(n) → P (N ). The set M (G) is the subset of agents which the mechanism M classifies as worthy in the network G. We denote by M (G) = N \M (G) the subset classified as unworthy by the mechanism. Notice that the definition of a mechanism depends on n through its dependence on G(n) and N . We abuse notation and regard a single mechanism M as if it represents a series of mechanisms-one for every natural n. The IC requirement means that an agent's classification is not influenced by his own reviews; that is, changing the weights on the out-edges of an agent does not alter his classification. Definition 2. A classification mechanism M is incentive- compatible if for every n ∈ N, G, G ∈ G(n), x ∈ N , such that: E(G) = E(G ) and ∀e ∈ E\E out (x), we(G) = we(G ), To define a measure for the quality of the mechanism, we first define a measure of coincidence between M (G) and Iα(G): C(M (G), Iα(G)) = 1 n x∈N 1, x∈ (M (G) ∩ Iα(G)) ∪ (M (G) ∩ Iα(G)), −1, otherwise.(2) In other words, C(M (G), Iα(G) gives one point for every agent that M (G) classified correctly and takes one point for every agent which was classified erroneously; the result is normalised by the number of agents. 6 This measure can be somewhat simplified to get, 7 C(M (G), Iα(G)) = 1 n (\|((M (G) ∩ Iα(G)) ∪ (M (G) ∩ Iα(G)))\| − \|M (G) ⊕ Iα(G)\|) = 1 n (\|N \(M (G) ⊕ Iα(G))\| − \|M (G) ⊕ Iα(G)\|) = 1 − 2 n \|M (G) ⊕ Iα(G)\|. Our main theorems imply that the possibility of an IC mechanism which guarantees a fixed level of coincidence depends on two parameters. The first is α. The second is the maximal out-degree in the network (i.e., the maximal reviews an agent can issue), denoted by Δ. Intuitively, if Δ is high, then an unworthy agent might use his influence on the score of Δ worthy agents to improve his ranking and be considered worthy. If α is relatively low, then these manipulations might influence a large portion (or all) of the worthyclassified agents, which makes it harder to find an IC mechanism with high coincidence measure. Our results will show that this intuition is indeed correct. Let G(n, Δ) be the family of all networks on n nodes with maximal out-degree Δ. The quality of a mechanism for given α, Δ, is the limit when n goes to infinity of the worst case of the coincidence measure: (3) We will prove upper and lower bounds on Q α,Δ for any IC classification mechanism. Main results We start by defining two trivial IC mechanisms. 'Trivial' means that they classify the nodes without any regard to the edges' weights. Let M N be the complete mechanism, i.e., M N (G) = N for all G ∈ G(n). Proposition 3. For any α, Δ, Q α,Δ (M N ) = 2α − 1. 6 Other measures might be appropriate for different applications. We discuss two of these alternatives in Section 5. The main conclusions from our results stay the same in these variations. 7 The operator ⊕ is the "exclusive or". Proof. Since \|N ⊕ Iα(G)\| = \|Iα(G)\| = n − \|Iα(G)\| ≤ n(1 − α), we get that for any graph, C(M N (G), Iα(G)) = 1 − 2 n \|N ⊕ Iα(G)\| ≥ 1 − 2n(1 − α) n = 2α − 1. Our second trivial mechanism, M 1/2 , selects every node to be worthy with probability 1/2, independently. We use here the concept of a probabilistic mechanism intuitively; in the beginning of Section 3, we formally extend our model to include this kind of mechanism. Mechanism M 1/2 correctly classifies nodes x with probability 1/2, hence Q α,Δ (M 1/2 ) = 0 for all α, Δ. Our first result is a strong impossibility, saying that when Δ is large, one of the two trivial mechanisms is the best possible. Theorem 4. Proof. This is a direct consequence of Proposition 3, our observation for M 1/2 and Proposition 11. Our second result is a non-trivial, yet quite natural, mechanism with a better quality than the complete mechanism, provided Δ = o(n). The idea behind this mechanism is to recognize three subsets of agents: absolutely worthy, absolutely unworthy, and borderline. The first two subsets contain those agents who will not be able to change their classification, no matter what their reviews will be. The fact that Δ is negligible guarantees that the absolutely worthy and absolutely unworthy sets will include a large portion of the true worthy and unworthy subsets, respectively. The mechanism then classifies the absolutely worthy agents as worthy and the absolutely unworthy as unworthy. If we allow the mechanism to be probabilistic, we can select each of the borderline agents to be worthy with probability 1/2; this strategy assures that these agents will not hurt the quality (but will not help it either). We also provide a deterministic version of this mechanism which always classifies correctly almost half of the borderline agents. The following theorem summarises our knowledge when Δ = o(n). Theorem 5. For any α and Δ = o(n), α ≤ Q α,Δ ≤ 1 2 (1 + α). Proof. The lower bound comes from an analysis of the above-described probabilistic mechanism; see Proposition 13. We also prove the existence of a deterministic version of this mechanism in Proposition 12. The upper bound is proved in Proposition 10. Impossibilities As promised, we start by extending our model to include probabilistic mechanisms. A probabilistic mechanism assigns each node a probability of being worthy. Definition 6. A probabilistic classification mechanism is a function Mp : N × G(n) → [0, 1]. To get a concrete selected set from a probabilistic mechanism, we select each node independently with his assigned probability. In other words, the probability of subset X ⊆ N to be selected under mechanism Mp in the graph G is Mp(x, G)). Pr[X\|Mp(G)] = x∈X Mp(x, G) x / ∈X (1 − (4) The IC requirement translates to the requirement that an agent cannot influence his own selection probability. Definition 7. A probabilistic classification mechanism Mp is incentive-compatible if for every n ∈ N, G, G ∈ G(n), x ∈ N , such that: E(G) = E(G ) and ∀e ∈ E\E out (x), we(G) = we(G ), Mp(x, G) = Mp(x, G ). The coincidence of Mp(G) with Iα(G) is naturally extended using expectation over the selected set: C(Mp(G), Iα(G)) = X∈P (N ) Pr[X\|Mp(G)]C(X, Iα(G)) = 1 n X∈P (N ) Pr[X\|Mp(G)] · x∈N 1, x∈ (X ∩ Iα(G)) ∪ (X ∩ Iα(G)) −1, otherwise. Changing the summation order and inserting (4) we get, C(Mp(G), Iα(G)) = 1 n x∈N X∈P (N \{x}) y∈N \{x} Mp(y, G), y∈ X 1 − Mp(y, G), y / ∈ X · Mp(x, G) − (1 − Mp(x, G)), x ∈ Iα(G) (1 − Mp(x, G)) − Mp(x, G), x ∈ Iα(G). Since X∈P (N \{x}) y∈N \{x} Mp(y, G), y∈ X 1 − Mp(y, G), y / ∈ X = 1, C(Mp(G), Iα(G)) = 1 n x∈N (2Mp(x, G) − 1) · 1, x∈ Iα(G) −1, x ∈ Iα(G) (5) = Iα(G) − Iα(G) n + 2 n x∈N Mp(x, G) · 1, x∈ Iα(G) −1, x ∈ Iα(G).(6) The quality of a probabilistic mechanism can now be defined exactly as in (3): Q α,Δ (Mp) = lim n→∞ min G∈G(n,Δ) C(Mp(G), Iα(G)). From Definitions 6 and 7 and the coincidence definition above, it is clear that the IC (deterministic) mechanisms for which we initially defined our problem are a special case of IC probabilistic mechanisms. Hence we may prove our upper bounds (i.e., impossibilities) for probabilistic mechanisms. We furthermore show that it is enough to consider a subfamily of symmetric, IC, probabilistic mechanisms. Definition 8. A probabilistic mechanism Mp is symmetric if for any network G ∈ G(n) and two isomorphic nodes x, y ∈ N , Claim 9. Let Mp be any IC, probabilistic, classification mechanism. Then there is an IC, symmetric, probabilistic classification mechanism M p with Q α,Δ (M p ) ≥ Q α,Δ (Mp). Proof. Let S(N ) be the set of permutations over N . For π ∈ S(N ), let Gπ be the graph which is isomorphic to G under the automorphism defined by π. We define the mechanism M p : M p (x, G) = 1 n! π∈S(N ) Mp(π(x), Gπ).(7) Mechanism M p is clearly IC, since Mp is IC and otherwise the calculation above is irrelevant of any of the weights in G. It is also symmetric, since for two isomorphic nodes, x, y, the following two sets of the couples are exactly the same: {(π(x), Gπ)\|π ∈ S(N )} = {(π(y), Gπ)\|π ∈ S(N )}. It remains to show that the quality of M p is at least that of Mp. Planting (6) into (7) and changing the summation order we get that for any G, C(M p (G), Iα(G)) = Iα(G) − Iα(G) n + 2 n · n! x∈N π∈S(N ) Mp(π(x), Gπ) · 1, x∈ Iα(Gπ) −1, x ∈ Iα(Gπ) = Iα(G) − Iα(G) n + 1 n! π∈S(N ) ⎡ ⎣ 2 n x∈N Mp(π(x), Gπ) · 1, x∈ Iα(Gπ) −1, x ∈ Iα(Gπ) Let Q = Q α,Δ (Mp). For any > 0, there is n 0 such that for all n > n 0 and for all G ∈ G(n, Δ), C(Mp(G, Iα(G))) ≥ Q − , which means that 2 n x∈N Mp(x, G) · 1, x∈ Iα(G) −1, x ∈ Iα(G) ≥ Q − − Iα(G) − Iα(G) n . Hence, C(M p (G), Iα(G)) ≥ Q − . Since this is true for any , we get that Q α,Δ (M p ) ≥ Q. We are now ready to prove our two impossibility propositions which imply the upper bounds of Theorems 4 and 5. Proposition 10. For all α, Δ, Q α,Δ ≤ 1 2 (1 + α). Proof. Let v be a distinct node. Partition N \{v} into three sets, A, B, C, of sizes (1 − α)n 2 , (1 − α)n 2 , αn − 1, respectively. Let G be the network in which every node in A ∪ B has an out-edge to v, and C is a cycle. Set the weights on the edges from A to v to be 1, the weights on the edges from B to v to be −1 + 1 (1 − α)n , and the weights on C to be 1. For any Hence Iα(G) = C ∪ {v}. Let M be an IC, probabilistic and symmetric classification mechanism. By symmetry, we may denote M (a, G) = μ for any a ∈ A. Using (5) we get that, C(M (G), Iα(G)) ≤ 1 n ((1 − 2μ)\|A\| + \|B\| + \|C\| + 1) = 1 − μ(1 − α).(8) Now choose a distinct node a 0 ∈ A and change the weight on its out-edge to v to be −1 + 1 (1 − α)n ; we refer to this network as G . The score of v has dropped in 2 − 1/(1 − α)n \|A\| + \|B\| = 2 − 1/(1 − α)n (1 − α)n > 1 2(1 − α)n , for n large enough. Hence s(v, G ) < 0. Since the rest of the scores are the same in G and G , we get that Iα(G ) = N \{v}. By IC, M (a 0 , G ) = M (a 0 , G) = μ, and by symmetry M (b, G ) = μ for any b ∈ B. We now get , C(M (G ), Iα(G )) ≤ 1 n (\|A\| + (2μ − 1)\|B\| + C + 1) = 1 − (1 − μ)(1 − α) = α + μ(1 − α).(9) From (8) and (9) , Q α,Δ ≤ min{1 − μ(1 − α), α + μ(1 − α)}. Comparing the two terms to find the optimal value for μ we find that 1 − μ(1 − α) = α + μ(1 − α) ⇐⇒ μ = 1 2 =⇒ Q α,Δ ≤ 1 − 1 − α 2 = 1 + α 2 . Proposition 11. Let m = min 2(1 − α), Δ n . Then Q α,Δ ≤ 1 − m. Proof. Consider two cases: Case I: m ≥ 2α. Let A, B ⊆ N be two disjoint subsets of size nm/2. Let G be the graph in which A ∪ B is a clique and there are no other edges. We set the weights on all the out-edges of nodes in A to be 0, and the weights on all the out-edges of nodes in B to be 1. Hence every a ∈ A has a score of s(a) = \|B\| \|A ∪ B\| , every b ∈ B has a score of s(b) = \|B\| − 1 \|A ∪ B\| and the score of every c / ∈ A ∪ B is 0. Since \|A\| = mn 2 ≥ αn, Iα(G) = A. Let M be an IC, probabilistic, symmetric mechanism. By the symmetry of M , all the vertices of B get the same probability. Denote it μ. Let b be a distinct vertex in B. Let G be the graph we get when we nullify all the weights on the outgoing edges of b. Since b is now isomorphic to the vertices in A, we get by IC and the symmetry of M that ∀a ∈ A, M (a, G ) = M (b, G ) = M (b, G) = μ. We see that there is a trade-off in the value of μ; on the one hand, we need it to be high if we want a good quality on G , but on the other hand, it needs to be low for a good quality on G. The idea of the proof is that for n large enough, we can repeat this process and show that in essence all the vertices in A ∪ B in the graph G (or at least in one of the graphs we get from G after a negligible number of steps) should have approximately the same probability. This implies that Q α,Δ ≤ 1 n (\|A\|(2μ − 1) + \|B\|(1 − 2μ) + \|N \(A ∪ B)\|) = 2μ − 1 n (\|A\| − \|B\|) + 1 − \|A ∪ B\| n = 1 − m. To make this claim precise, suppose for contradiction that Q α,Δ (M ) ≥ 1 − m + for some > 0. For k ≤ X, X to be found, we let A k , B k ⊆ N be two disjoint subsets with sizes \|A k \| = mn/2 + k, \|B k \| = mn/2 − k. Define the graph G k in which A k ∪ B k is a clique, and the weights on the outgoing edges of vertices in A k are all 0, and the weights on the outgoing edges of vertices in B k are all 1. Notice the following: a) The vertices in A k are symmetric, and so are the vertices in B k . b) Iα(G k ) = A k . c) If we nullify the outgoing edges of one of the nodes b ∈ B k then we get the graph G k+1 in which b ∈ A k+1 . By the symmetry of M , we may denote for any k, μ k a = M (a, G k ) for all a ∈ A k , and μ k b = M (b, G k ) for all b ∈ B k . By (c) and IC of M , μ k b = μ k+1 a . By the assumption on the quality of M : 1 n (\|A k \|(2μ k a − 1) + \|B k \|(1 − 2μ k b ) + \|N \(A k ∪ B k )\|) ≥ 1 − m + , 1 n ((mn/2 + k)(2μ k a − 1) + (mn/2 − k)(1 − 2μ k b ) + (1 − m)n) ≥ 1 − m + , (mn + 2k)μ k a − (mn − 2k)μ k b − 2k ≥ n. Summing up this inequality for 0 ≤ k ≤ X and substituting μ k b for μ k+1 a we get, mnμ 0 a − (mn − 2X)μ X b + 2 X k=1 (2k − 1)μ k a − X(X + 1) ≥ X n. Let μmax = max 1≤k≤X μ k a . If X = o(n), then for n large enough mn ≥ 2X. We strengthen the inequality by removing the negative terms on the left-hand side, and replacing all the μ k a with μmax: μmax ⎛ ⎝ mn + 2 X k=1 (2k − 1) ⎞ ⎠ ≥ X n μmax(mn + 2X 2 ) ≥ X n =⇒ μmax ≥ X m + 2X 2 /n . Now taking X = 2m/ we get that for n large enough μmax > 1, which is a contradiction. Case II: m < 2α The proof is very similar. We define three subsets, A, B, C of size nm/2, nm/2 and (α − m/2)n, respectively. 8 Let G be the graph in which A∪B is a clique and C is a cycle. Again we set 8 Since we assume m < 2α, \|C\| > 0. all the weights on the out-edges of the nodes in A to be 0, and weights on all the out-edges of nodes in B to be 1. The edges in the cycle in C also get a weight of 1. Thus now s(c) = 1 > s(a) > s(b) for any c ∈ C, a ∈ A, b ∈ B. Since \|A ∪ C\| = αn, we have Iα(G) = A ∪ C. Let M be an IC, probabilistic, symmetric mechanism. Using the same technique as before, we get that all the vertices in A ∪ B should get the same probability, which we call μ. Since Iα(G) = A ∪ C we can bound as before: Q α,Δ (M ) ≤ 1 n (\|A\|(2μ − 1) + \|B\|(1 − 2μ) + \|N \(A ∪ B)\|) = 1 − m. The formal argument is precisely the same. Non-trivial mechanism In this section we show the existence of a mechanism with quality α, provided Δ = o(n). Specifically, we will show the following: Proposition 12. For any α and any Δ ≤ min{ 1 3 αn, 1 3 (1−α)n}, there is an IC classification mechanism with quality α − 3Δ n . For better exposition, we start by presenting a probabilistic mechanism which is slightly better. Proposition 13. For any α and any Δ ≤ αn, there is an IC probabilistic classification mechanism with quality α − Δ n . For a graph G and x ∈ G, denote by Gx the graph we get when we set all the weights on the out-edges of x to be -1. Let β = α − Δ n . The idea is to partition N into three subsets: W (G) = {x ∈ N \|x ∈ I β (Gx)}, U (G) = {x ∈ N \|x ∈ Iα(Gx)}, B(G) = N \(W ∪ U ) = {x ∈ N \|x ∈ Iα(Gx)\I β (Gx)}. Notice that the definitions of W, U, B are IC, in the sense that the sorting of x to one of these sets does not depend on his own reviews (since we fixed all his reviews to -1 before choosing his set). If x ∈ W , then his ranking in Gx is less than βn: r(x, Gx) < βn. The real reviews of x may increase the score of at most Δ agents, which means that r(x, G) ≤ r(x, Gx) + Δ < βn + Δ = αn; hence x ∈ Iα(G). We have proved that W ⊆ Iα(G). We think of W as the absolutely worthy agents. Similarly, the set U is the set of absolutely unworthy agents: for x ∈ U , r(x, G) ≥ r(x, Gx) ≥ αn (increasing his reviews can only hurt his relative ranking), and U ⊆ Iα(G). The set B contains all the borderline agents: these are the agents which we are not sure how to classify. We define the following probabilistic mechanism: Mp(x, G) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1, x∈ W (G), 0, x∈ U (G), 1/2, x ∈ B(G). Since W, U, B are IC, this mechanism is IC. The mechanism is always correct in the classification of the agents in W ∪ U . We set the probability of agents in B to be 1/2 so that the expected contribution of the agents in B to the coincidence weight on an edge (x, y) with x ∈ B * to -1, we add at most one agent to B which is ranked above x (this might happen when y ∈ W ), or we remove at most one agent from B which is ranked below x (this might happen when y ∈ B and is ranked below him). Thus x must be in the top-half ranked agents in Bx. Since all the agents in U are correctly classified, we get that at least \|B * \| + \|U \| = 1 2 (\|B\| + 2\|U \| − Δ) are classified correctly. Case II: \|Iα(G)\| < αn + 2Δ. Let B * be the bottom 1 2 (\|B\| − \|U \|) − Δ ranked agents in B according to L(B). 11 Since \|B * ∪ U \| = 1 2 (\|B\| + \|U \|) − Δ ≤ 1 2 (1 − β)n − Δ = 1 2 (1 − α)n − 1 2 Δ, and \|Iα(G)\| = n − Iα(G) > (1 − α)n − 2Δ ≥ 1 2 (1 − α)n − 1 2 Δ, we get that B * ⊆ Iα(G). It is therefore enough to show that for any x ∈ B * , x / ∈ B + x . Indeed, lowering the weights on the out-edges of x can advance x in the ranking of B by at most Δ places (if the edges are to nodes in B that are ranked higher than x), and it might also add all the nodes of U to \|B\|; in any case x will still be in the bottom half of Bx. Conclusions We have introduced a generic model for the classification of agents to worthy and unworthy according to their own reviews of each other. We draw two general conclusions regarding the existence of an IC classification mechanism: 1. If Δ is large, there is no good mechanism in the sense that the best mechanisms do not take into consideration the reviews. 2. If Δ is negligible with respect to the number of agents, there is a mechanism with a positive quality, but not with quality 1. Our measure for the coincidence between the selected set and the true worthy agents (2) assumes that every classification/misclassification has the same value/price. In other applications it makes more sense to consider only the classification/misclassification of the worthy agents, which leads to the following measure: Yet another measure might consider all the agents but normalise the classification/misclassification of an agent according to his true set: 12 C(M (G), Iα(G)) = \|M (G) ∩ Iα(G)\| 2\|Iα(G)\| + \|M (G) ∩ Iα(G)\| 2\|Iα(G)\| . We mention here that using these measures does not change our two conclusions above; in other words, our conclusions are intrinsic in the problem and not the result of a specific measure. The exact claims and proofs can be found in the appendix of the full version; see (Babichenko, Dean, and Tennenholtz 2019). Figure Figure 1: Our bounds for the quality of an IC classification mechanism as a function of α. In (a) Δ = o(n). The red/blue graphs denote our upper/lower bounds and the green area denotes the known feasible quality range. In (b) Δ = n − 1. The single green graph is the quality of any IC mechanism. Q α,Δ (M ) = lim n→∞ min G∈G(n,Δ) C(M (G), Iα(G)). Δ ≥ 2(1 − α)n, Q α,Δ = 2α − 1. (b) For α < 1 2 and Δ = (1 − o(1))n, Q α,Δ = 0. a ∈ A, b ∈ B, c ∈ C their scores are: s(c) = 1, s(a) = s(b) = 0, and s(v) = \|A\| + \|B\|(−1 + 1/(1 − α)n) \|A\| + \|B\| = 1 2(1 − α)n > 0. C (M (G), Iα(G)) = 1 \|Iα(G)\| x∈M (G) 1, x∈ Iα(G), −1, x / ∈ Iα(G). We assume n is large as we are generally interested in results which are asymptotic in n. For the same reason we habitually drop floors and ceilings for easier reading. x ∈ M (G) ⇐⇒ x ∈ M (G ).2 A review of '0' is the neutral review. For example, an agent with all-zero in-edges is rated in the same way as an agent with no in-edges (see the definition of s(·) in (1)).3 We can set a zero-weight on all interactions which were not reviewed (see Footnote 2), hence we may assume w.l.o.g. that all interactions have been reviewed.4 From now on, when G is clear from the context, we just write s(x), r(x).5 We think of α as the selectiveness of the system: a lower value for α means that the tag of being worthy is more prestigious. Mp(x, G) = Mp(y, G). That is, for x, y ∈ B, x L y ⇐⇒ (r(x) > r(y)) ∨ ((r(x) = r(y)) ∧ (x < y)).10 To be clear, for x, y ∈ N , let Gx,y be the graph in which we set all the weights on the outgoing edges of x and y to -1. Then Bx(G) = B(Gx) = {y ∈ N \|y ∈ Iα(Gx,y)\I β (Gx,y)}. Notice that \|B * \| > 0 due to our assumption on \|U \|. 12 Though here we need to assume that Iα(G) is not empty, or define the measure for this case separately. measure is zero. We get that for anyG ∈ G(n, Δ)with Δ ≤ αn,This completes the proof of Proposition 13. The idea for the deterministic mechanism is similar. This mechanism selects as worthy all the agents in W and as unworthy all the agents in U . For the agents in B we need to find an IC, deterministic way to classify them such that in every network about half of them are rightly classified. Notice first that if \|U \| ≥ \|B\| − 2Δ thenLet L(B) be a linear ordering of B according to r(·, G) and breaking ties lexicographically. 9 Let B + be the top half. We now complete our definition of mechanism M by setting for every x ∈ B,x . That is, x ∈ B is accepted if and only if it is ranked in the top half of Bx when breaking ties lexicographically. The mechanism does not use the weights on the outgoing edges of x to determine its classification, hence it is IC. We complete the proof of Proposition 12 with the following lemma. Using this lemma we get that for any such graph,as required.Proof of Lemma 14. Consider two cases. Case I: \|Iα(G)\| ≥ αn + 2Δ. In this case B ⊆ Iα\I β , since if x / ∈ Iα(G), then in Gx there are at least \|Iα\| − Δ ≥ αn + Δ agents with a higher score than s(x); hence x / ∈ Iα(Gx). Moreover, for any x ∈ B, Bx ⊆ Iα(G), since for any y / ∈ Iα, there are in Gx,y at least \|Iα\|−2Δ ≥ αn agents with a higher score than s(y). Let B * be the top 1 2 (\|B\| − Δ) ranked agents in B according to L(B). We claim that the agents in B * are all classified by our mechanism to be worthy, which is the correct classification (since B ⊂ Iα(G), as explained). The reason is that when we set the Sum of us: Strategyproof selection from the selectors. N Alon, F Fischer, A Procaccia, M Tennenholtz, Proceedings of the 13th Conference on Theoretical Aspects of Rationality and Knowledge, TARK XIII. the 13th Conference on Theoretical Aspects of Rationality and Knowledge, TARK XIIIAlon, N.; Fischer, F.; Procaccia, A.; and Tennenholtz, M. 2011. 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[ "Online Learning for Combinatorial Network Optimization with Restless Markovian Rewards", "Online Learning for Combinatorial Network Optimization with Restless Markovian Rewards" ]	[ "Yi Gai [email protected] ", "Bhaskar Krishnamachari ", "Mingyan Liu [email protected] \nDepartment of Electrical Engineering and Computer Science\nUniversity of Michigan\n48109Ann ArborMIUSA\n", "Ming Hsieh ", "\nDepartment of Electrical Engineering\nUniversity of Southern California\n90089Los AngelesCAUSA\n" ]	[ "Department of Electrical Engineering and Computer Science\nUniversity of Michigan\n48109Ann ArborMIUSA", "Department of Electrical Engineering\nUniversity of Southern California\n90089Los AngelesCAUSA" ]	[]	Combinatorial network optimization algorithms that compute optimal structures taking into account edge weights form the foundation for many network protocols. Examples include shortest path routing, minimal spanning tree computation, maximum weighted matching on bipartite graphs, etc. We present CLRMR, the first online learning algorithm that efficiently solves the stochastic version of these problems where the underlying edge weights vary as independent Markov chains with unknown dynamics.The performance of an online learning algorithm is characterized in terms of regret, defined as the cumulative difference in rewards between a suitably-defined genie, and that obtained by the given algorithm. We prove that, compared to a genie that knows the Markov transition matrices and uses the single-best structure at all times, CLRMR yields regret that is polynomial in the number of edges and nearly-logarithmic in time.arXiv:1109.1606v1 [cs.LG] 8 Sep 2011 2 problems in section III. In section IV, we present our CLRMR policy, and show that it requires only polynomial storage. We present our novel analysis of the regret of CLRMR policy in section V. In section VI, we discuss examples and show the numerical simulation results, to show that our proposed policy is widely useful for various interesting combinatorial network optimization problems. We finally conclude our paper in section VII.II. RELATED WORKWe summarize below the related work, which has treated a) temporally i.i.d. rewards, b) rested Markovian rewards, and c) restless Markovian rewards.A. Temporally i.i.d. rewardsLai and Robbins [4] wrote one of the earliest papers on the classic non-Bayesian infinite horizon multi-armed bandit problem. They assume K independent arms, each generating rewards that are i.i.d. over time obtained from a distribution that can be characterized by a single-parameter. For this problem, they present a policy that provides an expected regret that is O(K log n), i.e. linear in the number of arms and asymptotically logarithmic in n. Anantharam et al. extend this work to the case when M simultaneous plays are allowed[5]. The work by Agrawal [6] presents easier to compute policies based on the sample mean that also has asymptotically logarithmic regret. The paper by Auer et al. [7] that considers arms with nonnegative rewards that are i.i.d. over time with an arbitrary non-parameterized distribution that has the only restriction that it have a finite support. Further, they provide a simple policy (referred to as UCB1), which achieves logarithmic regret uniformly over time, rather than only asymptotically. Our work utilizes a general Chernoff-Hoeffding-bound-based approach to regret analysis pioneered by Auer et al..Some recent work has shown the design of distributed multiuser policies for independent arms. Motivated by the problem of opportunistic access in cognitive radio networks, Liu and Zhao [8], Anandkumar et al.[9], [10], and Gai and Krishnamachari[11], have developed policies for the problem of M distributed players operating N independent arms.Our work in this paper is closest to and builds on the recent work by Gai et al. which introduced combinatorial multiarmed bandits[2]. The formulation in [2] has the restriction that the reward process must be i.i.d. over time. A polynomial storage learning algorithm is presented in [2] that yields regret that is polynomial in users and resources and uniformly logarithmic in time for the case of i.i.d. rewards.	10.1109/secon.2012.6275789	[ "https://arxiv.org/pdf/1109.1606v1.pdf" ]	1,281,331	1109.1606	41b1fd855597f51785b955578f9c1028611148f0	Online Learning for Combinatorial Network Optimization with Restless Markovian Rewards Yi Gai [email protected] Bhaskar Krishnamachari Mingyan Liu [email protected] Department of Electrical Engineering and Computer Science University of Michigan 48109Ann ArborMIUSA Ming Hsieh Department of Electrical Engineering University of Southern California 90089Los AngelesCAUSA Online Learning for Combinatorial Network Optimization with Restless Markovian Rewards Combinatorial network optimization algorithms that compute optimal structures taking into account edge weights form the foundation for many network protocols. Examples include shortest path routing, minimal spanning tree computation, maximum weighted matching on bipartite graphs, etc. We present CLRMR, the first online learning algorithm that efficiently solves the stochastic version of these problems where the underlying edge weights vary as independent Markov chains with unknown dynamics.The performance of an online learning algorithm is characterized in terms of regret, defined as the cumulative difference in rewards between a suitably-defined genie, and that obtained by the given algorithm. We prove that, compared to a genie that knows the Markov transition matrices and uses the single-best structure at all times, CLRMR yields regret that is polynomial in the number of edges and nearly-logarithmic in time.arXiv:1109.1606v1 [cs.LG] 8 Sep 2011 2 problems in section III. In section IV, we present our CLRMR policy, and show that it requires only polynomial storage. We present our novel analysis of the regret of CLRMR policy in section V. In section VI, we discuss examples and show the numerical simulation results, to show that our proposed policy is widely useful for various interesting combinatorial network optimization problems. We finally conclude our paper in section VII.II. RELATED WORKWe summarize below the related work, which has treated a) temporally i.i.d. rewards, b) rested Markovian rewards, and c) restless Markovian rewards.A. Temporally i.i.d. rewardsLai and Robbins [4] wrote one of the earliest papers on the classic non-Bayesian infinite horizon multi-armed bandit problem. They assume K independent arms, each generating rewards that are i.i.d. over time obtained from a distribution that can be characterized by a single-parameter. For this problem, they present a policy that provides an expected regret that is O(K log n), i.e. linear in the number of arms and asymptotically logarithmic in n. Anantharam et al. extend this work to the case when M simultaneous plays are allowed[5]. The work by Agrawal [6] presents easier to compute policies based on the sample mean that also has asymptotically logarithmic regret. The paper by Auer et al. [7] that considers arms with nonnegative rewards that are i.i.d. over time with an arbitrary non-parameterized distribution that has the only restriction that it have a finite support. Further, they provide a simple policy (referred to as UCB1), which achieves logarithmic regret uniformly over time, rather than only asymptotically. Our work utilizes a general Chernoff-Hoeffding-bound-based approach to regret analysis pioneered by Auer et al..Some recent work has shown the design of distributed multiuser policies for independent arms. Motivated by the problem of opportunistic access in cognitive radio networks, Liu and Zhao [8], Anandkumar et al.[9], [10], and Gai and Krishnamachari[11], have developed policies for the problem of M distributed players operating N independent arms.Our work in this paper is closest to and builds on the recent work by Gai et al. which introduced combinatorial multiarmed bandits[2]. The formulation in [2] has the restriction that the reward process must be i.i.d. over time. A polynomial storage learning algorithm is presented in [2] that yields regret that is polynomial in users and resources and uniformly logarithmic in time for the case of i.i.d. rewards. I. INTRODUCTION The following abstract description of combinatorial network optimization covers many graph theoretic algorithms that form the basis of network protocol design in wired and wireless networks. Given a graph G = (V, E), where each edge e ∈ E is associated with a weight w e , find a structure consisting of a collection of edges satisfying some given property (e.g., a path, a tree, a matching, or an independent set), that maximizes or minimizes the sum of the weights on the selected edges. This kind of linear network combinatorial optimization covers, for instance, shortest path and minimum spanning tree computation used in routing protocols, and maximum-weight matching used for channel scheduling and switching. In practice, the edge weights may correspond to some link quality metric of interest such as packet reception ratio, delay, or throughput. In such a case, the edge weights are often stochastically varying with time. Moreover, the dynamics may not be known a priori. The solution approach to this problem that we advocate here is to combine the estimation and optimization phases jointly via an efficient online learning algorithm. We present in this paper an online learning algorithm that is designed for the setting where the edge weights are modeled by finite-state Markov chains, with unknown transition matrices. We show that this problem can be modeled as a combinatorial multi-armed bandit problem with restless Markovian rewards. To characterize the performance of this algorithm, following the convention in the multi-armed bandit literature, we define a notion of regret, defined as the difference in reward between a suitably defined model-aware genie and that accumulated by the given algorithm over time. Specifically, in this work, we consider a single-action regret formulation, whereby the genie is assumed to know the transition matrices for all edges, but is constrained to stick with one action (corresponding to a particular network structure) at all times 1 . We prove that our algorithm, which we refer to as CLRMR (Combinatorial Learning with Restless Markov Rewards) achieves a regret that is polynomial in the number of Markov chains (i.e., number of edges), and logarithmic with time. This implies that our learning algorithm, which does not know the transition matrices, asymptotically achieves the maximum time averaged reward possible with any single-action policy, even if that policy is given advanced knowledge of the transition matrices. By contrast, the conventional approach of estimating the mean of each edge weight and then finding the desired network structure via deterministic optimization would incur greater overhead and provide only linearly increasing regret over time, which is not asympotically optimal. While recent work has shown how to address multi-armed bandits with restless Markovian rewards in the classic noncombinatorial setting [1], and combinatorial multi-armed bandits in the simpler settings of i.i.d. rewards [2] or rested Markovian rewards [3], this paper is the first to show how to efficiently implement online learning for stochastic combinatorial network optimization when edge weights are dynamically evolving as restless Markovian processes. We perform simulations to evaluate our new algorithm over two combinatorial network optimization problems: stochastic shortest path routing and bipartite matching for channel allocation, and show that its regret performance is substantially better than that of the algorithm presented in [1], which can handle restless Markovian rewards but does not exploit the dependence between the arms, resulting in a regret that grows exponentially in the number of unknown variables. The rest of the paper is organized as follows. We first provide a survey of prior work in section II. We then present a formal model of the combinatorial restless multi-armed bandit B. Rested Markovian rewards There has been relatively less work on multi-armed bandits with Markovian rewards. Anantharam et al. [12] wrote one of the earliest papers with such a setting. They proposed a policy to pick m out of the N arms each time slot and prove the lower bound and the upper bound on regret. However, the rewards in their work are assumed to be generated by rested (i.e. rewards that only evolve when the arms are selected) Markov chains with transition probability matrices defined by a single parameter θ with identical state spaces. Also, for the upper bound the result is achieved only asymptotically. For the case of single users and independent arms, a recent work by Tekin and Liu [13] has extended the results in [12] relaxing the requirement of a single parameter and identical state spaces across arms. They propose to use UCB1 from [7] for the multi-armed bandit problem with rested Markovian rewards and prove a logarithmic upper bound on the regret under some conditions on the Markov chain. In a recent work by Gai et al. [3], learning policies for combinatorial multi-armed bandits with rested Markovian rewards have been studied. Unlike [3], we adopt a model with restless Markovian rewards, which has much broader applications in many network optimization problems. C. Restless Markovian rewards Restless arm bandits are so named because the arms evolve at each time, changing state even when they are not selected. Work on restless Markovian rewards with single users and independent arms can be found in [1], [14]- [16]. In these papers there is no consideration of possible dependencies among arms, as in our work here. Tekin and Liu [1] have proposed a RCA policy that achieves logarithmic single-action regret when certain knowledge about the system is known. We use elements of the policy and proof from [1] in this work, which is however quite different in its combinatorial matching formulation (which allows for dependent arms). Liu et al. [14], [15] adopted the same problem formulation as in [1], and proposed a policy named RUCB, achieving a logarithmic single-action regret over time when certain system knowledge is known. They also extend the RUCB policy to achieve a near-logarithmic regret asymptotically when no knowledge about the system is available. Dai et al. in [16] adopt a stronger definition of regret: the difference in expected reward compared to a modelaware genie. They develop a policy that yields regret of order arbitrarily close to logarithmic for certain classes of restless bandits with a finite-option structure, such as restless MAB with two states and identical probability transition matrices. III. PROBLEM FORMULATION We consider a system with N edges predefined by some application, where time is slotted and indexed by n. For each edge i (1 ≤ i ≤ N ), there is an associated state that evolves as a discrete-time, finite-state, aperiodic, irreducible Markov chain 2 {X i (n), n ≥ 0} with unknown parameters 3 . We denote the state space for the i-th Markov chain by S i . We assume these N Markov chains are mutually independent. The reward obtained from state x (x ∈ S i ) of Markov chain i is denoted as r i x . Denote by π i x the steady state distribution for state x. The mean reward obtained on Markov chain i is denoted by µ i . Then we have µ i = z∈Si,j r i x π i x . The set of all mean rewards is denoted by µ = {µ i }. At each decision period n (also referred to interchangeably as time slot), an N -dimensional action vector a(n), representing an arm, is selected under a policy φ(n) from a finite set F. We assume a i (n) ≥ 0 for all 1 ≤ i ≤ N . When a particular a(n) is selected, the value of r i xi(n) is observed, only for those i with a i (n) = 0. We denote by A a(n) = {i : a i (n) = 0, 1 ≤ i ≤ N } the index set of all a i (n) = 0 for an arm a. We treat each a(n) ∈ F as an arm. The reward is defined as: R a(n) (n) = i∈A a(n) a i (n)r i xi(n)(1) where x i (n) denotes the state of a Markov chain i at time n. When a particular arm a(n) is selected, the rewards corresponding to non-zero components of a(n) are revealed, i.e., the value of r i xi(n) is observed for all i such that a i (n) = 0. The state of the Markov chain evolves restlessly, i.e., the state will continue to evolve independently of the actions. We denote by P i = (p i x,y ) x,y∈S i the transition probability matrix for the Markov chain i. We denote by (P i ) = {(p i ) x,y } x,y∈S i the adjoint of P i on l 2 (π), so (p i ) x,y = p i y,x π i y /π i x . Denotê P i = (P i ) P as the multiplicative symmetrization of P i . For aperiodic irreducible Markov chains,P i s are irreducible [17]. A key metric of interest in evaluating a given policy φ for this problem is regret, which is defined as the difference between the expected reward that could be obtained by the best-possible static action, and that obtained by the given policy. It can be expressed as: R φ (n) = nγ * − E φ [ n t=1 R φ(t) (t)] = nγ * − E φ [ n t=1 i∈A a(t) a i (t)r i xi(t) ](2) where γ * = max a∈F i∈A a(n) a i µ i is the expected reward of the optimal arm. For the rest of the paper, we use * as the index indicating that a parameter is for an optimal arm. If there is more than one optimal arm, * refers to any one of them. We denote by γ a the expected reward of arm a, so γ a = \|Aa\| j=1 a pj µ pj . For this combinatorial multi-armed bandit problem with restless Markovian rewards, our goal is to design policies that perform well with respect to regret. Intuitively, we would like the regret R φ (n) to be as small as possible. If it is sublinear with respect to time n, the time-averaged regret will tend to zero. (1) A straightforward idea is to apply RCA in [1], or RUCB in [14] directly and naively, and ignore the dependencies across the different arms. However, we note that RCA and RUCB both require the storage and computation time that are linear in the number of arms. Since there could be exponentially many arms in this formulation, it is highly unsatisfactory. (2) Unlike our prior work on combinatorial MAB with rested rewards, for which the transitions only occur each time the Markov chains are observed, the policy design for the restless case is much more difficult, since the current state while starting to play a Markov chain depends not only on the transition probabilities, but also on the policy. To deal with the first challenge, we want to design a policy which more efficiently stores observations from the correlated arms, and exploits the correlations to make better decisions. Instead of storing the information for each arm, our idea is to use two 1 by N vectors to store the information for each Markov chain. Then an index for each each arm is calculated, based on the information stored for underlying components. This index is used for choosing the arm to be played each time when a decion needs to be made. To deal with the second challenge, for each arm a we note that the multidimensional Markov chain {X a (n), n ≥ 0} defined by underlying components as X a (n) = (X i (n)) i∈Aa is aperiodic and irreducible. Instead of utilizing the actual sample path of all observations, we only take the observations from a regenerative cycle for Markov chains and discard the rest in its estimation of the index. Our proposed policy, which we refer to as Combinatorial Learning with Restless Markov Reward (CLRMR), is shown in Algorithm 1. Table I summerizes the notation we use for CLRMR algorithm. For Algorithm 1, (x i ) i∈Aa = (ζ i ) i∈Aa means x i = ζ i , ∀i. CLRMR operates in blocks. Figure 1 illustrates one possible realization of this Algorithm 1. At the beginning of each block, an arm a is picked and within one block, this algorithm always play the same arm. For each Markov chain {X i (n)}, we specifiy a state ζ i at the beginning of the algorithm as a state to mark the regenerative cycle. Then, for the multidimentional Markov chain {X a (n)} associated with this arm, the state (ζ i ) i∈Aa is used to define a regenerative cycle for {X a (n)}. Each block is broken into three sub-blocks denoted by SB1, SB2 and SB3. In SB1, the selected arm a is played until the Algorithm 1 Combinatorial Learning with Restless Markov Reward (CLRMR) 1: // INITIALIZATION 2: t = 1, t 2 = 1; 3: ∀i = 1, · · · , N , m i 2 = 0,z i 2 = 0; 4: for b = 1 to N do 5: t := t + 1, t 2 := t 2 + 1; 6: Play any arm a such that b ∈ A a ; denote (x i ) i∈Aa as the observed state vector for arm a; 7: ∀i ∈ A a(n) , let ζ i be the first state observed for Markov chain i if ζ i has never been set;z i 2 :=z i 2 m i 2 +r i x i m i 2 +1 , m i 2 := m i 2 + 1; 8: while (x i ) i∈Aa = (ζ i ) i∈Aa do 9: t := t + 1, t 2 := t 2 + 1; 10: Play arm a; denote (x i ) i∈Aa as the observed state vector; 11: Play an arm a which maximizes max a∈F i∈Aa ∀i ∈ A a(n) ,z i 2 :=z i 2 m i 2 +r i x i m i 2 +1 , m i 2 := m i 2 + 1;a i z i 2 + L ln t 2 m i 2 ; (3) 19: Denote (x i ) i∈Aa as the observed state vector; 20: while (x i ) i∈Aa = (ζ i ) i∈Aa do 21: t := t + 1; 22: Play an arm a and denote (x i ) i∈Aa as the observed state vector; t 2 := t 2 + 1; 26: ∀i ∈ A a(n) ,z i 2 :=z i 2 m i 2 +r i x i m i 2 +1 , m i 2 := m i 2 + 1; 27: while (x i ) i∈Aa = (ζ i ) i∈Aa do 28: t := t + 1, t 2 := t 2 + 1; 29: Play an arm a and denote (x i ) i∈Aa as the observed state vector; 30: b := b + 1, t := t + 1; 34: end while state (ζ i ) i∈Aa is observed. Upon this observation we enter a regenerative cycle, and continue playing the same arm untill (ζ i ) i∈Aa is observed again. SB2 includes all time slots from the first visit of (ζ i ) i∈Aa up to but excluding the second visit to (ζ i ) i∈Aa . SB3 consists a single time slot with the N : number of resources a: vectors of coefficients, defined on set F; we map each a as an arm second visit to (ζ i ) i∈Aa . SB1 is empty if the first observed state is (ζ i ) i∈Aa . So SB2 includes the observed rewards for a regenerative cycle of the multidimentional Markov chain {X a (n)} associated with arm a, which implies that SB2 also includes the observed rewards for one or more regenerative cycles for each underlying Markov chain ∀i ∈ A a(n) ,z i 2 :=z i 2 m i 2 +r i x i m i 2 +1 , m i 2 := m i 2 + 1;A a : {i : a i = 0, 1 ≤ i ≤ N } t: current{X i (n)}, i ∈ A a . The key to the algorithm 1 is to store the observations for each Markov chain instead of the whole arm, and utilize the observations only in SB2 for them, and virtually assemble them (highlighted with thick lines in Figure 1). Due to the regenerative nature of the Markov chain, by putting the observations in SB2, the sample path has exactly the same statics as given by the transition probability matrix. So the problem is tractable. LLR policy requires storage linear in N . We use two 1 by N vectors to store the information for each Markov chain after we play the selected arm at each time slot in SB2. One is Line 1 to line 13 are the initialization, for which each Markov chain is observed at least once, and ζ i is specified as the first state observed for {X i (n)}. (z i 2 ) 1×N in whichz i 2 is After the initialization, at the beginning of each block, CLRMR selects the arm which solves the maximization problem as in (3). It is a deterministic linear optimal problem with a feasible set F and the computation time for an arbitrary F may not be polynomial in N . But, as we show in Section VI, there exist many practically useful examples with polynomial computation time. V. ANALYSIS OF REGRET We summarize some notation we use in the description and analysis of our CLRMR policy in Table II. We first show in Theorem 1 an upper bound on the total expected number of plays of suboptimal arms. Theorem 1: When using any constant L ≥ 56(H + 1)S 2 max r 2 maxπ 2 max / min , we have a:γ a <γ * (γ * − γ a )E[T a (n)] ≤ Z 1 ln n + Z 2 where Z 1 = ∆ max 1 Π min + M max + 1 4N LH 2 a 2 max ∆ 2 min Z 2 = ∆ max 1 Π min + M max + 1 N + πN HS max 3π min To proof Theorem 1, we use the inequalities as stated in Theorem 3.3 from [18] and a theorem from [19]. Lemma 1 (Theorem 3.3 from [18]): Consider a finitestate, irreducible Markov chain {X t } t≥1 with state space S, matrix of transition probabilities P , an initial distribution q and stationary distribution π. Let N q = ( qx πx , x ∈ S) 2 . Let P = P P be the multiplicative symmetrization of P where P is the adjoint of P on l 2 (π). Let = 1 − λ 2 , where λ 2 is the second largest eigenvalue of the matrixP . will be referred to as the eigenvalue gap ofP . Let f : S → R be such that y∈S π y f (y) = 0, f ∞ ≤ 1 and 0 < f 2 2 ≤ 1. IfP is irreducible, then for any positive integer n and all 0 < δ ≤ 1 P n t=1 f (X t ) n ≥ δ ≤ N q exp − nδ 2 28 . Lemma 2: If {X n } n≥0 is a positive recurrent homogeneous Markov chain with state space S, stationary distribution π and τ is a stopping time that is finite almost surely for which X τ = x then for all y ∈ S E τ −1 t=0 I(X t = y)\|X 0 = x = E[τ \|X 0 = x]π y . Proof of Theorem 1: We introduce B i (b) as a counter for the regret analysis to deal with the combinatorial arms. After the initialization period, B i (b) is updated in the following way: at the beginning of any block when a non-optimal arm is chosen to be played, find i such that i = arg min vector of observed states from the j-th block for playing Markov chain î π i x : max{π i x , 1 − π i x } π max : max i,x∈S iπ i x π min : min i,x∈S i π i x π max : max i,x∈S i π i x i : eigenvalue gap, defined as 1 − λ 2 , where λ 2 is the second largest eigenvalue of the multiplicative symmetrization of P i min : min ( B i (b)) 1×N equals the total number of blocks in which we have played non-optimal arms, a:γ a <γ * i i S max : max i \|S i \| r max : max i,x∈S i r i x a max : max i∈Aa,a∈F a i ∆ a : γ * − γ a ∆ min : min γ a ≤γ * ∆ a ∆ max : max γ a ≤γ * ∆ a {X a (n)}: multidimentional Markov chain defined by X a (n) = (X i (n)) i∈Aa ζ a : (ζ i ) i∈Aa ,E[B a (b)] = N i=1 E[ B i (b)].(4) We also have the following inequality for B i (b): (6), B i (b) ≤ m i 2 (t(b − 1)), ∀1 ≤ i ≤ N, ∀b.(5)E[ B i (b)] = b β=N +1 P{ I i (β) = 1} ≤ l + b β=N +1 P{ I i (β) = 1, B i (β − 1) ≥ l} ≤ l + b β=N +1 P{ k∈A a * a * k g k t2(β−1),m k 2 (t(β−1)) ≤ j∈A a(h) a j (b)g j t2(β−1),m j 2 (t(β−1)) , B i (β − 1) ≥ l}.(6) where g i t,s =z i 2 (s)+c t,s and a(β) is defined as a non-optimal arm picked at block β when I i (β) = 1. Note that m i 2 = min j {m j 2 : ∀j ∈ A a(β) }. We denote this arm by a(β) since at each block that I i (β) = 1, we could get different arms. Note that l ≤ B i (β − 1) implies, l ≤ B i (β − 1) ≤ m i 2 (t(β − 1)), ∀j ∈ A a(β) .(7) So we can further derive the upper bound of E[ B i (b)] shown in (15), where h j (1 ≤ j ≤ \|A a * \|) represents the j-th element in A a * ; p j (1 ≤ j ≤ \|A a(β) \|) represents the j-th element in A a(β) or A a(t) . A a(τ ) represents the arm played in the τ -th time slots counting only in SB2. Note that P{ \|Aa * \| j=1 a * hj g hj τ,s h j ≤ \|A a(τ ) \| j=1 a pj (t)g pj τ,sp j } (16) = P{ \|Aa * \| j=1 a * hj (z hj 2 (s hj ) + c τ,s h j ) (17) ≤ \|A a(τ ) \| j=1 a pj (τ )(z pj 2 (s pj ) + c τ,sp j )}(18) = P{At least one of the following must hold: \|Aa * \| j=1 a * hjz hj 2 (s hj ) ≤ γ * − \|Aa * \| j=1 a * hj c τ,s h j ,(19)\|A a(τ ) \| j=1 a pj (τ )z pj 2 (s pj ) ≥ γ a(τ ) + \|A a(τ ) \| j=1 a pj (τ )c τ,sp j , (20) γ * < γ a(τ ) + 2 \|A a(τ ) \| j=1 a pj (τ )c τ,sp j }(21) Now we show the upper bound on the probabilities of inequalities (19), (20) and (21) separately. We first find an upper bound on the probability of (19): P{ \|Aa * \| j=1 a * hjz hj 2 (s hj ) ≤ γ * − \|Aa * \| j=1 a * hj c τ,s h j } = P{ \|Aa * \| j=1 a * hjz hj 2 (s hj ) ≤ \|Aa * \| j=1 a * hj µ hj − \|Aa * \| j=1 a * hj c τ,s h j } ≤ \|Aa * \| j=1 P{a * hjz hj 2 (s hj ) ≤ a * hj (µ hj − c τ,s h j )} = \|Aa * \| j=1 P{z hj 2 (s hj ) ≤ µ hj − c τ,s h j }. ∀1 ≤ j ≤ \|A a * \|, P{z hj 2 (s hj ) ≤ µ hj − c τ,s h j } = P{ x∈S h j ( r hj x m hj x (s hj ) s hj − r hj x π hj x ) ≤ x∈S h j − c τ,s h j \|S hj \| } ≤ x∈S h j P{ r hj x m hj x (s hj ) s hj − r hj x π hj x ≤ − c τ,s h j \|S hj \| } = x∈S h j P{r hj x m hj x (s hj ) − s hj r hj x π hj x ≤ − s hj c τ,s h j \|S hj \| } E[ B i (b)] ≤ l + b β=N +1 P{ min 0<s h 1 ,...,s h \|Aa * \| <t2(β−1) \|Aa * \| j=1 a * hj g hj t2(β−1),s h j ≤ max t2(l)≤sp 1 ,...,sp \|A a(β) \| <t2(β−1) \|A a(β) \| j=1 a pj (β)g pj t2(β−1),sp j } ≤ l + b β=N +1 t2(β−1) s h 1 =1 · · · t2(β−1) s h \|A * \| =1 t2(β−1) sp 1 =t2(l) · · · t2(β−1) sp \|A a(β) \| =t2(l) P{ \|Aa * \| j=1 a * hj g hj t2(β−1),s h j ≤ \|A a(β) \| j=1 a pj (β)g pj t2(β−1),sp j } ≤ l + t2(b) τ =1 τ −1 s h 1 =1 · · · τ −1 s h \|A * \| =1 τ −1 sp 1 =l · · · τ −1 sp \|A a(β) \| =l P{ \|Aa * \| j=1 a * hj g hj τ,s h j ≤ \|A a(τ ) \| j=1 a pj (τ )g pj τ,sp j } (15) = x∈S h j P{r hj x (s hj − y =x m hj y (s hj )) − r hj x s hj (1 − y =x π hj y ) ≤ − s hj c τ,s h j \|S hj \| } = x∈S h j P{ y =x m hj y (s hj ) − y =x π hj y ≥ s hj c τ,s h j r hj x \|S hj \| = x∈S h j P{ s h j t=1 1(Y hj t = x) − s hj (1 − π hj x ) π hj x s hj ≥ s hj c τ,s h j r hj x \|S hj \| } ≤ x∈S h j N q h j τ − L h j 28(\|S h j \|r h j xπ h j x ) 2 (22) ≤ \|S hj \| π min τ − L min 28S 2 max r 2 maxπ 2 max (23) where (22) follows from Lemma 1 by letting δ = s hj c τ,s h j r hj x \|S hj \| , f (Y i t ) = 1(Y i t = x) − (1 − π i x ) π i x . 1(a) is the indicator function defined to be 1 when the predicate a is true, and 0 when it is false.π i x is defined aŝ π i x = max{π i x , 1 − π i x } to guarantee f ∞ ≤ 1. We note that when δ > 1 the deviation probability is zero, so the bound still holds. (23) follows from the fact that for any q i , N q i = q i x π i x , x ∈ S i 2 ≤ \|S i \| x=1 q i x π i x 2 ≤ \|S i \| x=1 q i x 2 π min = 1 π min . Note that all the quantities in computing the indices and the probabilities above come from SB2. Got for every SB2 in a block, the quantities begin with state ζ a and end with a return to ζ a . So for each underlying Markov chain {X i (n)}, i ∈ A a , the quantities got begin with state ζ i and end with a return to ζ i . Note that for all i, Markov chain {X i (n)} could be played in different arms, but the quantities got always begin with state ζ i and end with a return to ζ i . Then by the strong Markov property, the process at these stopping times has the same distribution as the original process. Connecting these intervals together we form a continuous sample path which can be viewed as a sample path generated by a Markov chain with transition matrix identical to the original arm. This is the reason why we can apply Lemma 1 to this Markov chain. Therefore, P{ \|Aa * \| j=1 a * hjz hj 2 (s hj ) ≤ γ * − \|Aa * \| j=1 a * hj c τ,s h j } ≤ HS max π min τ − L min 28S 2 max r 2 maxπ 2 max (24) With a similar derivation, we have P{ \|A a(τ ) \| j=1 a pj (τ )z pj 2 (s pj ) ≥ γ a(τ ) + \|A a(τ ) \| j=1 a pj (τ )c τ,sp j } ≤ \|A a(τ ) \| j=1 P{a pj (τ )z pj 2 (s pj ) ≥ a pj (τ )µ pj + a pj (τ )c τ,sp j } ≤ \|A a(τ ) \| j=1 x∈S p j P{r pj x m pj x (s pj ) − s pj r pj x π pj x ≥ s pj c τ,sp j \|S pj \| } = \|A a(τ ) \| j=1 x∈S p j P{ sp j t=1 1(Y pj t = x) − s pj π pj x π pj x s pj ≥ s pj c τ,sp j r pj x \|S pj \| } ≤ \|A a(τ ) \| j=1 x∈S p j N q p j τ − L p j 28(\|S p j \|r p j xπ p j x ) 2 (25) ≤ HS max π min τ − L min 28S 2 max r 2 maxπ 2 max(26) where (25) follows from Lemma 1 by letting δ = s pj c τ,sp j r pj x \|S pj \| , f (Y i t ) = 1(Y i t = x) − π i x π i x . Note that when l ≥     4L ln t2(b) ∆ a(τ ) Hamax 2     ,(21) is false for τ , which gives, γ * − γ a(τ ) − 2 \|A a(τ ) \| j=1 a pj (τ )c τ,sp j = γ * − γ a(τ ) − 2 \|A a(τ ) \| j=1 a pj L ln t 2 (b) s pj ≥ γ * − γ a(τ ) − Ha max 4L ln t 2 (b) l ≥ γ * − γ a(τ ) − Ha max 4L ln t 2 (b) 4L ln t 2 (b) ∆ a(t) Ha max 2 (27) ≥ γ * − γ a(τ ) − ∆ a(τ ) = 0.(28) Hence, when we let l ≥ (21) is false for all a(τ ). Therefore, we have (29). 4LH 2 a 2 max ln t2(b) ∆ 2 min , Following (29), E[ B i (b)] ≤ 4LH 2 a 2 max ln n ∆ 2 min + 1 + HS max π min ∞ τ =1 2τ − L min −56HS 2 max r 2 maxπ 2 max 28S 2 max r 2 maxπ 2 max (32) = 4LH 2 a 2 max ln n ∆ 2 min + 1 + HS max π min ∞ τ =1 2τ −2 (33) = 4LH 2 a 2 max ln n ∆ 2 min + 1 + πHS max 3π min (33) follows since L ≥ 56(H + 1)S 2 max r 2 maxπ 2 max / min . According to (4), a:γ a <γ * E[B a (b)] = N i=1 E[ B i (b)] ≤ 4N LH 2 a 2 max ln n ∆ 2 min + N + πN HS max 3π min(34) Note that the total number of plays of arm a at the end of block b(n) is equal to the total number of plays of arm a during SB2s (the regenerative cycles of visiting state ζ a ) plus the total number of plays before entering the regenerative cycles plus one more play resulting from the last play of the block which is state ζ a . So we have E[T a (n)] ≤ 1 Π a min + M a max + 1 E[B a (b(n))]. Therefore, a:γ a <γ * (γ * − γ a )E[T a (n)] ≤ ∆ max a:γ a <γ * 1 Π a min + M a max + 1 E[B a (b(n))] (35) ≤ ∆ max 1 Π min + M max + 1 a:γ a <γ * E[B a (b(n))] (36) ≤ Z 1 ln n + Z 2 where Z 1 = ∆ max 1 Π min + M max + 1 4N LH 2 a 2 max ∆ 2 min , Z 2 = ∆ max 1 Π min + M max + 1 N + πN HS max 3π min Now we show our main results on the regret of CLRMR policy as in Theorem 2. Theorem 2: When using any constant L ≥ 56(H + 1)S 2 max r 2 maxπ 2 max / min , the regret of CLRMR can be upper bounded uniformly over time by the following, R CLRM R (n) ≤ Z 3 ln n + Z 4(37) where Z 3 = Z 1 + Z 5 4N LH 2 a 2 max ∆ 2 min Z 4 = Z 2 + γ * ( 1 π min + M max + 1) + Z 5 (N + πN HS max 3π min ) and Z 5 = γ max ( 1 Π min + M max + 1 − 1 π max ) + γ * M * max Proof: Denote the expectations with respect to policy CLRMR given ζ by E ζ . Then the regret is bounded as, R CLRM R ζ (n) = γ * E ζ [T (n)] − E ζ [ T (n) t=1 i∈A a(t) a i (t)r i xi(t) ] + γ * E ζ [n − T (n)] − E ζ [ n t=T (n)+1 i∈A a(t) a i (t)r i xi(t) ] ≤ γ * E ζ [T (n)] − a γ a E ζ [T a (n)] + γ * E ζ [n − T (n)] + a γ a E ζ [T a (n)] − E ζ [ T (n) t=1 i∈A a(t) a i (t)r i xi(t) ] ≤ Z 1 ln n + Z 2 + γ * ( 1 Π min + M max + 1) (38) +   a γ a E ζ [T a (n)] − E ζ [ T (n) t=1 i∈A a(t) a i (t)r i xi(t) ]   .(39) where (38) follows from Theorem 1 and E ζ [n − T (n)] ≤ 1 Πmin + M max + 1. E[ B i (b)] ≤ 4LH 2 a 2 max ln t 2 (b) ∆ 2 min + t2(b) τ =1 τ −1 s h 1 =1 · · · τ −1 s h \|A * \| =1 τ −1 sp 1 =l · · · τ −1 sp \|A a(β) \| =l 2HS max π min τ − L min 28S 2 max r 2 maxπ 2 max (29) Note that a γ a E ζ [T a (n)] − E ζ [ T (n) t=1 i∈A a(t) a i (t)r i xi(t) ] ≤ γ * E ζ [T * (n)] + a:γ a <γ * γ a E ζ [T a (n)] − i∈A a * y∈S i a * i r i y E ζ [ B * (b(n)) j Y i t ∈Y i (j) 1(Y i t = y)] − a:γ a <γ * i∈Aa y∈S i a i r i y E ζ [ B a (b(n)) j Y i t ∈Y i 2 (j) 1(Y i t = y)](40) where the inequality above comes from counting only in Y i 2 (j) instead of Y i (j) in (40). Then applying Lemma 2 to (40), we have E ζ [ B a (b(n)) j Y i t ∈Y i 2 (j) 1(Y i t = y)] = π i y π i ζ i E ζ [B a (b(n))]. So − a:γ a <γ * i∈Aa y∈S i a i r i y E ζ [ B a (b(n)) j Y i t ∈Y i 2 (j) 1(Y i t = y)] ≤ − a:γ a <γ * γ a π max E ζ [B a (b(n))].(41) Also note that a:γ a <γ * γ a E ζ [T a (n)] ≤ a:γ a <γ * γ a ( 1 π a min + M a max + 1)E ζ [B a (b(n))](42) Inserting (41) and (42) into (40), we get a γ a E ζ [T a (n)] − E ζ [ T (n) t=1 i∈A a(t) a i (t)r i xi(t) ] ≤ γ * E ζ [T * (n)] + a:γ a <γ * γ a ( 1 Π a min + M a max + 1 − 1 π max )E ζ [B a (b(n))] − i∈A a * y∈S i a * i r i y E ζ [ B * (b(n)) j Y i t ∈Y i (j) 1(Y i t = y)] = Q * (n) + a:γ a <γ * γ a ( 1 Π a min + M a max + 1 − 1 π max )E ζ [B a (b(n))], where Q * (n) = γ * E ζ [T * (n)] − i∈A a * y∈S i a * i r i y E ζ [ B * (b(n)) j Y i t ∈Y i (j) 1(Y i t = y)] We now consider the upper bound for Q * (n). We note that the total number of time slots for playing all suboptimal arms is at most logarithmic, so the number of time slots in which the optimal arm is not played is at most logarithmic. We could then combine the successive blocks in which the best arm is played, and denote byȲ * (j) the j-th combined block. Denotē b * as the total number of combined blocks up to block b. Each combined blockȲ * starts after dis-continuity in playing the optimal arm, sob * (n) is less than or equal to total number of completed blocks in which the best arm is not played up to time n. Thus, E ζ [b * (n)] ≤ a:γ a <γ * E ζ [B a (b(n))].(43) Each combined blockȲ * consists of two sub-blocks:Ȳ * 1 which contains the state vectors for the optimal arm visited from beginning ofȲ * (empty if the first state is ζ * ) to the state right before hitting ζ * and sub-blockȲ * 2 which contains the rest ofȲ * (a random number of regenerative cycles). Denote the length ofȲ * 1 by \|Ȳ * 1 \| and the length ofȲ * 2 by \|Ȳ * 2 \|. We denoteȲ i 2 (j) by the states for Markov chain i for all i ∈ A a * inȲ * 2 . Therefore we get the upper bound for Q * (n) as Q * (n) = γ * E ζ [T * (n)] − i∈A a * y∈S i a * i r i y E ζ [ B * (b(n)) j Y i t ∈Y i (j) 1(Y i t = y)] (44) ≤ i∈A a * y∈S i a * i r i y π i y E ζ [b * (n) j=1 \|Ȳ * 2 \|] (45) − i∈A a * y∈S i a * i r i y E ζ [b * (n) j=1 Y i t ∈Ȳ i 2 (j) 1(Y i t = y)] (46) + i∈A a * y∈S i γ * E ζ [b * (n) j=1 \|Ȳ * 1 \|] (47) ≤ γ * M * max a:γ a <γ * E ζ [B a (b(n))](48) where the inequality in (45) comes from counting only the rewards obtained in sub-blockȲ i 2 in (44). Also, note that based on Lemma 2, (45) equals (46), and therefore we have the inequality (48). Hence, ∀ζ, R CLRM R ζ (n) ≤ Z 1 ln n + Z 2 + γ * ( 1 π min + M max + 1) (49) + a:γ a <γ * γ a (M a max + 1)E ζ [B a (b(n))] + γ * M * max a:γ a <γ * E ζ [B a (b(n))] ≤ Z 1 ln n + Z 2 + γ * ( 1 π min + M max + 1) + (γ max ( 1 Π min + M max + 1 − 1 π max ) + γ * M * max )E ζ [B a (b(n))] ≤ Z 3 ln n + Z 4 ,(50) where (50) follows from Theorem 1 and (34), and Z 3 = Z 1 + Z 5 4N LH 2 a 2 max ∆ 2 min Z 4 = Z 2 + γ * ( 1 Π min + M max + 1) + Z 5 (N + πN HS max 3π min ). Z 5 is defined as Z 5 = γ max ( 1 Π min + M max + 1 − 1 π max ) + γ * M * max . Theorem 2 shows when we use a constant L ≥ 56(H + 1)S 2 max r 2 maxπ 2 max / min , the regret of Algorithm 1 is upperbounded uniformly over time n by a function that grows as O(N 3 ln n). However, when S max , r max ,π max or min (or the bound of them) are unknown, the upper bound of regret can not be guaranteed to grow logarithmically in n. When no knowledge about the system is available, we extend the CLRMR policy to achieve a regret bounded uniformly over time n by a function that grows as O(N 3 L(n) ln n), using any arbitrarily slowly diverging non-decreasing sequence L(n) in Algorithm 1. Since L(n) could grow arbitrarily slowly, this modified version of CLRMR, named CLRMR-LN, could achieve a regret arbitrarily close to the logarithmic order. We present our analysis in Theorem 3. Theorem 3: When using any arbitrarily slowly diverging non-decreasing sequence L(n) (i.e., L(n) → ∞ as n → ∞), and replacing (3) where n(t 2 ) is the time when total number of time slots spent in SB2 is t 2 , the expected regret under this modified version of CLRMR, named CLRMR-LN policy, is at most R CLRM R−LN (n) ≤ Z 6 L(n) ln n + Z 7(52) where Z 6 and Z 7 are constants. Proof: Replacing c t,s with L(n(t)) ln t s , and replacing L with L(n(t 2 (b))) or L(n(τ )) accordingly in the proof of Theorem 1, (4) to (32) still stand. L(n(τ )) is a diverging non-decreasing sequence, so there exists a constant τ such that for all τ ≥ τ , L(n(τ )) ≥ 56(H + 1)S 2 max r 2 maxπ 2 max / min , which implies τ − L(n(τ )) min −56HS 2 max r 2 maxπ 2 max 28S 2 max r 2 maxπ 2 max ≤ τ −2 . Thus, we have E[ B i (b)] ≤ 4L(n(t 2 (b)))H 2 a 2 max ln n ∆ 2 min + 1 (53) + HS max π min ∞ τ =1 2τ −2 + Z 8 ≤ 4L(n)H 2 a 2 max ln n ∆ 2 min + 1 + πHS max 3π min + Z 8(54) where Z 8 = HS max π min τ −1 τ =1 2τ − L min −56HS 2 max r 2 maxπ 2 max 28S 2 max r 2 maxπ 2 max(55) Then we can according have a:γ a <γ * (γ * − γ a )E[T a (n)] ≤ Z 9 L(n) ln n + Z 2 + ∆ max 1 Π min + M max + 1 N Z 8 . where Z 9 = ∆ max 1 Π min + M max + 1 4N H 2 a 2 max ∆ 2 min .(56) So R CLRM R−LN (n) ≤ Z 6 L(n) ln n + Z 7 ,(57) where Z 6 = Z 9 + Z 5 4N H 2 a 2 max ∆ 2 min Z 7 = Z 2 + γ * ( 1 Π min + M max + 1) + ∆ max 1 Π min + M max + 1 N Z 7(58)+ Z 5 (N + πN HS max 3π min + N Z 7 ). VI. APPLICATIONS AND SIMULATION RESULTS We now present an evaluation of our policy over stochastic versions of two combinatorial network optimization problems of practical interest: stochastic shortest path (for routing), and stochastic bipartite matching (for channel allocation). A. Stochastic Shortest Path In the stochastic shortest path problem, given a graph G = (V, E), with edge weights (D ij ) stochastically varying with time as restless Markov chains with unknown dynamics, we seek to find a path between a given source s and destination t with minimum expected delay. We can apply the CLRMR policy to this problem, with some very minor modifications to the policy and the corresponding regret definition to be applicable to a minimization problem instead of maximization. For the stochastic shortest path problems, each path between s and t is mapped to an arm. Although the number of paths could grow exponentially with the number of Markov chains, \|E\|. CLRMR efficiently solves this problem with polynomial storage \|E\| and regret scaling as O(\|E\| 3 log n). We show the numerical simulation results for the graph in Figure 2. We assume each link has two states with the delay 0.1 on good links, and 1 on bad links. Table III Figure 3 shows the simulation results. We see that our proposed CLRMR performs better than RCA, the algorithm presented in [1] for all L values considered. If we let L = 1512 in this problem, we have that L ≥ 56(H + 1)S 2 max r 2 maxπ 2 max / min . For lower values of L it is not guaranteed by the analysis that the algorithms should yield logarithmic regret. However, numerically, we find that both policies seem to achieve logarithmic regret, and yield much better regret performance, even for much smaller L values. It is unclear whether this can be proved rigorously or whether it is due low probability events not captured in the simulations. B. Stochastic Bipartite Matching for Channel Allocation As a second application, we consider an application in a cognitive radio networks where M secondary users interfering with each other need to be allocated to Q non-conflicting orthogonal channels. We assume that, due to geographic dispersion, each user may see different primary user occupancy behavior in each channel. The availability of spectrum opportunities on each user-channel combination (i,j) over a decision period is modeled as a restless two-state Markov chain. It is easy to show that applying CLRMR to this problem yields storage linear in M Q, and a regret bound that scales as O(min{M, Q} 2 M Q log n), following Theorem 2. We show simulation results comparing CLRMR again with RCA for a system consisting of 9 orthogonal channels, and 5 secondary users. The transition probability matrix used for these scenarios is presented in table IV. The simulation results are shown in Figure 4. As in the stochastic shortest path problem, we find that CLRMR consistently outperforms RCA, for all values of L. Here L = 1135 corresponds to ensuring that L ≥ 56(H + 1)S 2 max r 2 maxπ 2 max / min , which is when the logarithmic regret is guaranteed in theory. However, again, we see that the performance seems to improve in practice with smaller L values, even if it is not be theoretically guaranteed. VII. CONCLUSION We have presented CLRMR, a provably efficient online learning policy for stochastic combinatorial network optimization with restless Markovian rewards. This algorithm is widely applicable to many networking problems of interest, as illustrated by our simulation based evaluation of the policy over two different problems: stochastic shortest path and stochastic maximum weight bipartite matching. One shortcoming of this work is that our focus has been on designing and evaluating the policy with respect to the best single-action policy. However, in general, with restless Markovian rewards, it is possible to further improve performance by developing an algorithm that dynamically switches between different actions over time as the underlying Markov chains evolve. Although this problem is much harder and remains unsolved except in a special case [16], we hope to investigate it further in our future work. Fig. 1 . 1An illustration of CLRMR IV. POLICY DESIGN For the above combinatorial MAB problem with restless rewards, we have two challenges here for the policy design: SB3 IS THE LAST PLAY IN THE WHILE LOOP. THEN A BLOCK COMPLETES. 33: time slot t 2 : number of time slots in SB2 up to the current time slot b: number of blocks up to the current time slot m i 2 : number of times that Markov chain i has been observed during SB2 up to the current time slot z i 2 : average (sample mean) of all the observed values of Markov chain i during SB2 up to the current time slot ζ i : state that determine the regenerative cycles for Markov chain i x i : the observed state when Markov Chain i is played; (x i ) i∈Aa is the observed state vector if arm a is played the average (sample mean) of observed values in SB2 up to the current time slot (obtained through potentially different sets of arms over time). The other one is (m i 2 ) 1×N in which m i 2 is the number of times that {X i (n)} has been observed in SB2 up to the current time slot. i the index of the elements which are among the ones that have been observed least in SB2 in the non-optimal arm). If there is only one such arm, B i (b) is increased by 1. If there are multiple such arms, we arbitrarily pick one, say i , and increment B i by 1. Based on the above definition of B i (b), each time a non-optimal arm is chosen to be played at the beginning of a block, exactly one element in ( B i (b)) 1×N is incremented by 1.So the summation of all number of time slots spent in SB2 up to block b B a (b): total number of blocks within the first b blocks in which arm a is played m i 2 (t 2 (b)): total number of time slots Markov chain i is observed during SB2 up to block b z i 2 (s): the mean reward from Markov chain i when it is observed for the s-th time of only those times played during SB2 T (n): time at the end of the last completed block T a (n): total number of time slots arm a is palyed up to time T (n) m i x (s): number of times that state x occured when Markov chain i has been observed s times Y i 1 (j): vector of observed states from SB1 of the j-th block for playing Markov chain i Y i 2 (j): vector of observed states from SB2 of the j-th block for playing Markov chain i Y i (j): Denote by I i (b) the indicator function which is equal to 1 if B i (b) is added by one at block b. Let l be an arbitrary positive integer. Then we can get the upper bound of E[ B i (b)] shown in Fig. 2 . 2A graph with 19 links and 260 acyclic paths between s and t for stochastic shortest path routing. Fig. 3 . 3Comparison of normalized regret R(n) ln n vs. n time slots for the stochastic shortest path problem. Fig. 4 . 4Comparison of normalized regret R(n) ln n vs. n time slots for Stochastic Bipartite Matching / Channel Allocation Problem. TABLE I NOTATION IFOR ALGORITHM 1 TABLE II NOTATION IIFOR REGRET ANALYSIS counters in summarizes the transition probabilities on each link.Link p 01 , p 10 Link p 01 , p 10 Link p 01 , p 10 e.1 0.2, 0.8 e.8 0.3, 0.8 e.15 0.1, 0.8 e.2 0.3, 0.9 e.9 0.1, 0.9 e.16 0.8, 0.1 e.3 0.2, 0.7 e.10 0.9, 0.1 e.17 0.2, 0.7 e.4 0.7, 0.1 e.11 0.3, 0.8 e.18 0.9, 0.1 e.5 0.3, 0.9 e.12 0.2, 0.7 e.19 0.3, 0.8 e.6 0.2, 0.7 e.13 0.8, 0.1 e.7 0.2, 0.8 e.14 0.4, 0.8 TABLE III TRANSITION PROBABILITIES TABLE IV TRANSITION IVPROBABILITIES p 01 , p 10 FOR EACH USER-CHANNEL PAIR Although a stronger notion of regret can be defined, allowing the genie to vary the action at each time, the problem of minimizing such a stronger regret is much harder and remains open even for simpler settings than the one we consider here. 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[ "QUINTIC THREEFOLDS AND FANO ELEVENFOLDS", "QUINTIC THREEFOLDS AND FANO ELEVENFOLDS" ]	[ "E D Segal ", "Richard P Thomas " ]	[]	[]	The derived category of coherent sheaves on a general quintic threefold is a central object in mirror symmetry. We show that it can be embedded into the derived category of a certain Fano elevenfold. Our proof also generates related examples in different dimensions.	10.1515/crelle-2015-0108	[ "https://arxiv.org/pdf/1410.6829v3.pdf" ]	37,100,477	1410.6829	ff9be1d8aa5a9d351be67ffccd39157ab26f147a	QUINTIC THREEFOLDS AND FANO ELEVENFOLDS 24 Oct 2014 E D Segal Richard P Thomas QUINTIC THREEFOLDS AND FANO ELEVENFOLDS 24 Oct 2014 The derived category of coherent sheaves on a general quintic threefold is a central object in mirror symmetry. We show that it can be embedded into the derived category of a certain Fano elevenfold. Our proof also generates related examples in different dimensions. Introduction Fix a 10-dimensional vector space V ∼ = C 10 . Consider the Grassmannian Gr := Gr(2, V ) ⊂ P ∧ 2 V (1.1) and the Pfaffian variety Pf := Pf 10 = [ω] ∈ P(∧ 2 V ∨ ) : ω ∧5 = 0 ∈ ∧ 10 V ∨ ⊂ P ∧ 2 V ∨ . (1.2) Notice that (1.2) is a quintic hypersurface in P 44 , singular in codimension 5. It is the classical projective dual of (1.1). Now pick a 5-dimensional subspace C 5 ∼ = U ⊂ ∧ 2 V ∨ (1.3) or equivalently a 40-dimensional subspace C 40 ∼ = U ⊥ ⊂ ∧ 2 V. We intersect (1.1) with P(U ⊥ ) ∼ = P 39 and (1.2) with P(U ) ∼ = P 4 . This defines an 11-dimensional linear section of the Grassmannian Y 1 := P(U ⊥ ) ∩ Gr (1.4) and a quintic 3-fold Y 2 := P(U ) ∩ Pf (1.5) respectively. For a generic choice of U , both Y 1 and Y 2 are smooth. Conversely, Beauville [Be,Proposition 8.9] shows that the general smooth quintic threefold Y 2 ⊂ P 4 arises in this way. 1 1 Though not uniquely. Different presentations of a given Y 2 as linear sections of Pf give rise to different dual Fano elevenfolds Y 1 . The moduli space of the Fanos Y 1 is a generically-finite cover of the moduli space of the quintics Y 2 ; in particular h 1 (T Y1 ) = 101 = h 1 (T Y2 ). Moreover their cohomologies are as closely related as possible. By the Lefschetz hyperplane theorem, in degrees less than the middle, the cohomologies of Y 1 and Y 2 are the same as those of their ambient spaces Gr and P 5 respectively. The same is true in degrees higher than the middle after a shift by twice the codimension. Finally in the middle degree, the nonzero pieces of the cohomologies have the same dimensions: h 3,0 h 2,1 h 1,2 h 0,3 = 1 101 101 1 for H 3 (Y 2 ) and h 7,4 h 6,5 h 5,6 h 4,7 = 1 101 101 1 for H 11 (Y 1 ). Our main result categorifies this relation. Theorem A. There is a full and faithful embedding D b (Y 2 ) ֒→ D b (Y 1 ). In fact this is a special case of a more general result, Theorem 2.8 below, which also covers some other interesting examples. Theorem A should have various consequences when combined with mirror symmetry. In particular, the Fukaya categories of Y 1 and Y 2 should also be related after a rescaling of the Novikov parameter q, with the latter a summand of the former. Taking Hochschild cohomologies, we should find that the quantum cohomology ring QH * (Y 2 ) of Y 2 should be a summand of QH * (Y 1 ) after applying a rescaling of the quantum parameter q. Setting q = 0 would recover the embedding of the Hodge diamond of Y 2 into that of Y 1 alluded to above. Theorem A would follow directly from Kuznetsov's beautiful work on homological projective duality [K1, K3] if one could prove [K3,Conjecture 5] for Gr(2,10). In short, Kuznetsov conjectures that (1.1) and (1.2) should be homologically projectively dual varieties [K1] once one replaces Pf with an appropriate categorical crepant resolution of its singularities (which has so far only been found in lower dimensions). This would imply a relation between the derived categories of the linear sections of Gr and of the orthogonal linear sections of Pf. In particular, for P(U ) chosen to avoid the singularities of Pf, we would find that D b (Y 1 ) has a semi-orthogonal decomposition D b (Y 1 ) = A, A(1), . . . , A(4), D b (Y 2 ) , (1.6) where A is the category generated by the exceptional collection Sym 3 S, Sym 2 S, S, O on Y 1 and S is the (restriction to Y 1 of the) universal subbundle on Gr. That is, D b (Y 2 ) ∼ = ⊥ Sym i S(j), 0 ≤ i ≤ 3, 0 ≤ j ≤ 4 = E ∈ D b (Y ) : RHom X (E, Sym i S(j)) = 0 for all 0 ≤ i ≤ 3, 0 ≤ j ≤ 4 . Furthermore the inclusion D b (Y 1 ) ֒→ D b (Y 2 ) should be given 2 by a Fourier-Mukai kernel I Γ , the ideal sheaf of Γ := (φ, P ) ∈ Pf × Gr : ker φ ∩ P = 0 ⊂ Y 1 × Y 2 . Here ker φ ⊂ V denotes the kernel of φ ∈ ∧ 2 V ∨ when thought of as a (skew) linear map V → V ∨ . The correspondence Γ associates φ to the locus of 2-planes P ⊂ V which intersect ker φ nontrivially. Since we deliberately avoid the singularities of Pf the methods of [K1, K2, K3] are surely strong enough to prove Theorem A without finding the right categorical resolution of singularities of Pf. Here however we take a different approach, inspired by string theory. In their paper [HT], Hori and Tong wrote down a non-abelian gauged linear sigma model (GLSM) that gave a physical explanation of the so-called 'Pfaffian-Grassmannian' derived equivalence between two particular Calabi-Yau threefolds. The paper [ADS] gave a mathematical treatment of Hori and Tong's construction at the level of B-brane categories. In this paper we take the techniques and results of [ADS] and apply them to a slightly more general GLSM. This gives us a more general result, Theorem 2.8, which says that we have a derived embedding between certain smooth linear sections of the Pfaffian variety and the dual smooth linear sections of a Grassmannian. Special cases then give the quintic threefold case of Theorem A, the Pfaffian-Grassmannian equivalence, and examples with K3 surfaces and Calabi-Yau 5-folds. Although our terminology is different, our approach is intimately connected with homological projective duality; see [B + ] for another situation in which HPD is realised via GLSMs. This paper is based heavily on [ADS]; the only new technical ingredient is the work in Section 4 to show the vanishing of a Brauer class. Consequently we have made little attempt to make this paper self-contained, and we refer the reader to [ADS] for background, motivation, references and more detailed explanations. Geometric setup and statement of theorem Fix vector spaces V, U and S of dimensions n, k ≤ n 2 and 2 respectively, and consider X = Hom(S, V ) ⊕ (U ⊗ ∧ 2 S) GL(S) . The square brackets indicate that we consider this as an Artin stack (rather than a scheme-theoretic or GIT quotient). We let x and p denote elements of Hom(S, V ) and U ⊗ ∧ 2 S respectively. We have open substacks ι 1 : X 1 = {rank x = 2} ֒−→ X , ι 2 : X 2 = {p = 0} ֒−→ X . The locus X 1 is a variety: the total space of the vector bundle O(−1) ⊗ U −→ Gr(2, V ). The locus X 2 is still an Artin stack; it is a bundle over P(U ) with fibres Hom(S, V ) SL(S) . (2.1) We can rephrase this: we let P be the stack P = (U ⊗ ∧ 2 S) \ {0} GL(S) , (2.2) which is a Zariski-locally trivial bundle of stacks over P(U ), with fibre BSL 2 . Then X 2 is a vector bundle over P. The loci X 1 and X 2 are the semi-stable loci for a positive or negative GIT stability condition, so one of the GIT quotients is X 1 , and the other is the underlying scheme of X 2 . Now fix a surjective linear map 3 A : ∧ 2 V −→ U ∨ . (2.3) This defines an (invariant) function W : X → C by W = p • A • ∧ 2 x. (2.4) We also use W to denote the restriction of this function to X 1 and X 2 . Finally we fix a C * action (an "R-charge") on X by giving x weight zero and p weight 2, so that both X 1 and X 2 are invariant and W has weight 2. Given this data, the three pairs (X , W ), (X 1 , W ) and (X 2 , W ) are all Landau-Ginzburg B-models, as defined in [Se], and we have restriction functors D b (X 1 , W ) ι * 1 ←− D b (X , W ) ι * 2 −→ D b (X 2 , W ) (2.5) between their categories of (global) matrix factorizations. Let Y 1 ⊂ Gr(2, V ) be the zero locus of the section A • ∧ 2 x ∈ Γ Gr(2, V ), O(1) ⊗ U ∨ . (2.6) The critical locus of the function W on X 1 always contains Y 1 , and is equal to Y 1 if and only if Y 1 is a smooth codimension-k complete intersection, i.e. if and only if the section (2.6) is transverse to the zero section. From now on we restrict to generic A for which this is true. By global Knörrer periodicity [Sh,Theorem 3.4], there is a canonical equivalence D b (Y 1 ) ∼ −→ D b (X 1 , W ). (2.7) This describes the left hand side of (2.5). The right hand side is more complicated. Over P(U ) we have a family of 2-forms on V up to scale given to us by A ∨ (2.3). The locus where these have rank < n − 1 is a variety Y 2 ⊂ P(U ), the intersection of the Pfaffian variety Pf ⊂ P(∧ 2 V ∨ ) of degenerate two-forms on V with the linear subspace A ∨ : P(U ) ֒−→ P(∧ 2 V ∨ ). It follows that Y 2 is also the locus where the (degenerate) quadratic form W on the fibres (2.1) of X 2 → P(U ) drops rank. In this situation there is a more complicated version of Knörrer periodicity; see Sections 4, 5 and [ADS]. There is also a corresponding Brauer class, but we show this vanishes in Section 4. The singular locus of Pf is a subvariety of codimension 6 inside P(∧ 2 V ∨ ) when n is even, and codimension 10 when n is odd. Therefore if n is even and k ≤ 6, or n is odd and k ≤ 10, then for a generic choice of A the variety Y 2 is smooth. (If k is larger than these bounds then Y 2 will never be smooth.) Under these assumptions, we prove in Section 5 that D b (Y 2 ) embeds into a certain subcategory of D b (X 2 , W ). Note that the variety Y 1 is Fano if k < n, Calabi-Yau if k = n, and general type if k > n. The canonical bundle of Y 2 is easy to calculate if n is even: Y 2 is Fano for k > n/2, Calabi-Yau for k = n/2 and general type for k < n/2. When n is odd the calculation is a little harder, but the three cases occur when k > n, k = n and k < n respectively. Theorem 2.8. Suppose that (i) k ≤ min(n, 10) if n is odd, or (ii) k ≤ min(n/2, 6) if n is even. Assume also that A is generic, so that both Y 1 and Y 2 are smooth. Then we have an admissible embedding D b (Y 2 ) ֒−→ D b (Y 1 ). Here admissible means that the embedding admits a right adjoint; 4 it follows that D b (Y 1 ) has a semi-orthogonal decomposition whose final term is D b (Y 2 ); c.f. (1.6). We have not attempted to compute the orthogonal piece, however. Setting n = 10, k = 5 gives Theorem A of the Introduction. The case n = k = 7 is the 'Pfaffian-Grassmannian' equivalence, which is the subject of [ADS]. Setting n = 8, k = 4 gives an embedding of the derived category of a Pfaffian quartic K3 into the derived category of a codimension-4 linear section of Gr(2,8). Note that the general quartic in P 3 is Pfaffian [Be,Prop. 7.6]. Setting n = k = 9 we get a novel derived equivalence between Calabi-Yau 5-folds. Grade-restriction windows Recall that n = dim V , and let us set L = n−1 2 for n odd and L = n 2 for n even. Let S be the following set of representations of GL(S): S = Sym l S ∨ ⊗ (det S ∨ ) m : l ∈ 0, L , m ∈ [0, n) if n is odd, or S = Sym l S ∨ ⊗ (det S ∨ ) m : l ∈ 0, L − 2 , m ∈ [0, n) or l = L − 1, m ∈ 0, n 2 if n is even. Each representation induces a vector bundle on X which we denote by the same letters. On restriction to Gr(2, V ) ⊂ X 1 we get the full strong Lefschetz exceptional collection found by Kuznetsov [K2]. This set S is adapted to the 'Grassmannian side' of our set-up; for the 'Pfaffian side' we consider the set T = Sym l S ∨ ⊗ (det S ∨ ) m : l ∈ [0, L), m ∈ [0, k) , (3.1) where k = dim U as before. Notice that T ⊂ S if and only if k ≤ n for n odd, or k ≤ n 2 for n even. (3.2) (There is also a 'reverse' numerical condition that implies that S ⊂ T , but this is less useful to us.) We let G 1 = S and G 2 = T ⊂ D b (X ) be the subcategories of D b (X ) generated by S and T , i.e. the closures of S and T under mapping cones and shifts (but not direct summands). We also let G W 1 and G W 2 ⊂ D b (X , W ) be the subcategories consisting of objects that are (homotopy-equivalent to) matrix factorizations whose underlying vector bundles are direct sums of shifts of the bundles appearing in S and T respectively. Proposition 3.3. The restriction functors ι * 1 : G 1 −→ D b (X 1 ) and ι * 1 : G W 1 −→ D b (X 1 , W ) are both equivalences, and the restriction functors ι * 2 : G 2 −→ D b (X 2 ) and ι * 2 : G W 2 −→ D b (X 2 , W ) are both embeddings. Proof. The statements without W are proved by the same argument as for [ADS,Proposition 4.1]; restriction to X 1 or X 2 does not create any higher Ext groups between the respective sets of vector bundles, and the restriction of S generates D b (X 1 ). The only additional ingredient we need is the appropriate generalisation of [ADS,Lemma 4.4] that certain Ext groups vanish on Gr(2, V ), and this is a routine calculation. The statements with W follow from the statements without W , just as in [ADS,Proposition 4.8]. Under the numerical condition in (3.2) we have that G 2 ⊂ G 1 and G W 2 ⊂ G W 1 . (3.4) Proposition 3.5. The restriction functors ι * 2 : G 1 −→ D b (X 2 ) and ι * 2 : G W 1 −→ D b (X 2 , W ) land in the subcategories ι * 2 (G 2 ) and ι * 2 (G W 2 ) respectively. Given the numerical condition in (3.2), these functors are the right adjoints to the inclusions (3.4). Proof. Consider first the statements without W . On X replace O {p=0} by its Koszul resolution. The resulting sheaves ∧ * (U ⊗ ∧ 2 S) ∨ are all sums of sheaves O(k). Restricting to X 2 the complex becomes acyclic, giving the corresponding relation in D(X 2 ). This relation, tensored with any O(i), shows that any line bundle O(k) lies in ι * 2 (G 2 ). Similarly, tensoring the acyclic complex with Sym l S ∨ shows that Sym l S ∨ (m) lies in ι * 2 (G 2 ) for l, m in the window defining S. Therefore ι * 2 (G 1 ) ⊂ ι * 2 (G 2 ). The statement that ι * 2 is the right adjoint to the inclusion G 2 ⊂ G 1 can be checked on the generators, so we need to know that if E ∈ T and F ∈ S then RHom X (E, F ) = RHom X2 (ι * 2 E, ι * 2 F ). This is proved by checking that the necessary higher Ext groups on X 2 vanish, just as in the proof of Proposition 3.3. As before, the statements with W follow from the statements without W by the techniques of [ADS,Proposition 4.8]. We define BBr(X 2 , W ) ⊂ D b (X 2 , W ) to be the image of G W 2 ; it is hopefully the category of B-branes in some associated SQFT. If we assume the numerical condition from (3.2), then putting together Knörrer periodicity (2.7) with Propositions 3.3 and 3.5 shows that (3.4) gives an embedding BBr(X 2 , W ) ֒−→ D b (Y 1 ) (3.6) as a right-admissible subcategory. To prove Theorem 2.8, it remains to show that D b (Y 2 ) embeds as a right-admissible subcategory of BBr(X 2 , W ). Quadratic bundles arising from symplectic bundles Given a vector bundle equipped with an everywhere non-degenerate quadratic form, Knörrer periodicity implies that the category of matrix factorizations on the total space of the bundle is equivalent to the derived category of the base space, once we twist the latter by a Brauer class. If the bundle admits a Lagrangian subbundle L then the Brauer class vanishes, and the sky-scraper sheaf O L can be used to construct an equivalence between the matrix factorization category and the ordinary derived category of the base. However, the existence of L is a rather stronger condition than the vanishing of the Brauer class. In this section we describe, for quadratic vector bundles of a particular type, an alternative construction which proves the vanishing of the Brauer class and provides the equivalence between the two categories. 4.1. Cleanly intersecting submanifolds of {W = 0}. Before discussing any quadratic vector bundles we make a rather general observation. Let (X, W ) be any Landau-Ginzburg B-model, and let A, B ⊂ W = 0 ⊂ X be submanifolds of the zero locus of W . Assume that A and B intersect cleanly, so A ∩ B is also a submanifold, and we have an excess normal bundle E = T X T A + T B on A ∩ B. Let r denote the rank of E, and let a be the codimension of A ⊂ X. Then in the ordinary derived category D b (X), a standard computation with the Koszul resolution gives the Ext sheaves between O A and O B as Ext i (O A , O B ) = ∧ a−i E ∨ ⊗ det N A/X , a − r ≤ i ≤ a, (4.1) and zero otherwise. Since A and B lie in {W = 0}, the sheaves O A and O B define objects in D b (X, W ). By a minor extension of the argument in [ASS,§A.4], there is a spectral sequence computing the local sheaf of morphisms between them in D b (X, W ), whose 2nd page consists of the sheaves (4.1) and whose differential is given by wedging with the section 5 dW : O A∩B −→ E ∨ . Suppose that this section of E ∨ is transverse to 0 with zero locus Z (which is therefore a component of the critical locus of W ). Then by the 3rd page only one term remains: Ext a−r D b (X,W ) (O A , O B ) = O Z ⊗ det E ∨ ⊗ det N A/X . Thus the spectral sequence collapses to give RHom D b (X,W ) (O A , O B ) = O Z ⊗ K A∩B ⊗ K −1 B [dim A ∩ B − dim B]. (4.2) Here K A∩B and K B denote the canonical bundles, and we have used a − r = dim B − dim A ∩ B. 4.2. Another version of Knörrer periodicity. Let S and V be two symplectic vector spaces. Let θ S ∈ ∧ 2 S be the Poisson bivector on S and Ω V be the symplectic form on V . Then the vector space Hom(S, V ) carries a non-degenerate quadratic form LGr(S) × Hom(S, V ), π * 2 W . Proposition 4.6. The object W : x −→ Ω V , ∧ 2 x (θ S ) .E := Rπ 2 * O N ⊗ (det Λ) − 1 2 dim V ∈ D b Hom(S, V ), W (4.7) is exceptional and generates the category. Proof. Choose a Lagrangian L ⊂ V , giving a maximally-isotropic subspace M ⊂ Hom(S, V ) as in (4.4). Let M = LGr(S) × M be the corresponding maximallyisotropic subbundle of LGr(S) × Hom(S, V ). The functor Rπ 1 * RHom(O M , · ) : D b LGr(S) × Hom(S, V ), π * 2 W −→ D b (LGr(S)) is an equivalence by the simplest version of Knörrer periodicity in families. Moreover the square D b LGr(S) × Hom(S, V ), π * 2 W Rπ1 * RHom(O M , · ) / / Rπ2 * D b (LGr(S)) RΓ D b (Hom(S, V ), W ) R Hom(OM , · ) / / D b (pt) (4.8) commutes by the projection formula. Now we take our tautological maximal isotropic subbundle N (4.5) and compute RHom O M , O N in D b LGr(S) × Hom(S, V ), π * 2 W , using the analysis from Section 4.1. The submanifolds M and N intersect cleanly along the subbundle M ∩ N = Hom(S/Λ, L) with excess normal bundle E = Hom(Λ, V /L) over M ∩ N . We compute that dW x = θ S • x ∨ • Ω V ∈ Hom(V, S) = Hom(S, V ) ∨ , where we consider θ S and Ω V as skew elements of Hom(S ∨ , S) and Hom(V, V ∨ ) respectively. Therefore at a point (Λ, x) of M ∩ N the derivative of π * 2 W is the map θ S • x ∨ • Ω V ∈ Hom(L, S/Λ) = Hom(V /L ∨ , Λ ∨ ), where the last isomorphism follows from the Lagrangian property of L and Λ. This lies in the fibre of E ∨ over (Λ, x). The resulting section of E ∨ has zero locus {x = 0} = LGr(S), so it is transverse to the zero section and we may apply (4.2). By an elementary calculation K M∩N ⊗ K −1 N = (det Λ) 1 2 dim V ⊗ (det L) − 1 2 dim S so (4.2) gives RHom O M , O N = O LGr(S) ⊗ (det Λ) 1 2 dim V ⊗ (det L) − 1 2 dim S − 1 4 dim S · dim V . Consequently, the upper arrow in the square (4.8) takes O N ⊗ (det Λ) − 1 2 dim V to a shift of O LGr(S) . Therefore going the other way round the square shows that E is taken by the lower arrow to the same shift of O pt . In particular, E is isomorphic to a shift of O M in the category D b Hom(S, V ), W . Remark 4.9. The same proof obviously gives another generator and both E and E ′ are supported on this locus. Furthermore, when dim V = 4, we get the skyscraper sheaf on this locus: E ′ = Rπ 2 * O N ⊗ (det Λ) − 1 2 dim V ⊗ π * 1 K LGr(S) .E ′ = Rπ 2 * O N 1 2 dim V − 2 = Rπ 2 * O N = O {rank x≤1} . That this sheaf is an exceptional generator of the category was observed in [ADS]. Since E is canonical, Proposition 4.6 works in families. Let S and V be symplectic vector bundles over a base B, or even vector bundles carrying symplectic forms only up to scale. 6 Then Hom(S, V ) p → B carries a fibrewise non-degenerate quadratic form W up to scale, given by the formula (4.3). We form the bundle E B := Rπ 2 * O N ⊗ (det Λ) − 1 2 rank V . (4.11) This is the global analogue of the object (4.7). Zariski-locally, there is also a version of the object O M of (4.4). The symplectic group is special, so V is Zariski-locally trivial. Therefore, replacing B by an open subset we may assume that V is trivial. Hence it admits a trivial Lagrangian subbundle L, defining a maximally-isotropic subbundle M ⊂ Hom(S, V ) by the formula (4.4). The proof of Proposition 4.6 now applies verbatim: Rp * RHom(O M , E B ) is a shift of a line bundle on our shrunken B. Since D b Hom(S, V ), W is generated over D b (B) by O M , this shows that E B and O M are isomorphic up to a shift and a twist by a line bundle. That is, once we shrink B to ensure that V is trivial, we get the following isomorphism in D b Hom(S, V ), W : E B ∼ = O M ⊗ (det L) − 1 2 rank S − 1 4 rank S · rank V . (4.12) It follows that for any B the two Fourier-Mukai functors D b (B) EB ⊗p * ( · ) / D b Hom(S, V ), W Rp * RHom(EB , · ) o (4.13) are mutual inverses; i.e. the natural adjunction map from their composition to the structure sheaf of the diagonal is a quasi-isomorphism. Again this can be checked locally, where it follows from (4.12) and the corresponding result for O M . Thus (4.13) gives an equivalence with zero Brauer class. Pfaffian side In this section we construct an embedding of D b (Y 2 ) into BBr(X 2 , W ). The method is the one employed in [ADS,§5] supplemented with the construction from Section 4 to produce a global Fourier-Mukai kernel. We let π denote the composition of the projections X 2 −→ P −→ P(U ) of (2.2). The first map is a vector bundle over P with fibre Hom(S, V ); the second is a bundle of stacks BSL 2 . The map A ∨ : U → ∧ 2 V ∨ of (2.3) defines a section of ∧ 2 V ∨ (1) over P(U )i.e. a family of 2-forms on the n-dimensional vector space V , defined up to scale. The variety Y 2 ⊂ P(U ) is the locus where this family drops in rank, either from n to n − 2 (if n is even) or from n − 1 to n − 3 (if n is odd). Let K → Y 2 be the kernel of the family of 2-forms: 0 −→ K −→ V A −→ V ∨ (1). It is a subbundle of the trivial bundle V × Y 2 of rank 2 or 3. Dividing out by K gives a quotient bundle q : V −→ V := V /K. The family of forms A descends to give a family A ∈ Γ ∧ 2 V ∨ (1) of symplectic forms (up to scale) on the vector bundle V → Y 2 . Now consider the vector bundle Hom S, V on the stack P\| Y2 . Since S is 2-dimensional, this carries an associated family of nondegenerate quadratic forms (up to scale) given by the formula (4.3). Via q this pulls back to a family of degenerate quadratic forms on the bundle Hom(S, V ) = X 2 \| Y2 ; this is precisely the restriction of the function W (2.4). We now apply the method of Section 4 to the symplectic bundles S, V over the base B = P\| Y2 to give a object E ∈ D b Hom S, V , W by the formula (4.11). Pulling up to Hom(S, V ) and pushing forward into X 2 gives an object j * q * E ∈ D b (X 2 , W ), where j : X 2 \| Y2 ֒→ X 2 denotes the inclusion map. We claim that O Y2 id −→ Rπ * RHom D b (X2,W ) (j * q * E, j * q * E) (5.1) is a quasi-isomorphism. Again, we can check this locally on Y 2 . We proceed as at the end of Section 4. Even though our base B = P\| Y2 is a stack rather than a scheme, the bundle V is pulled back from the scheme Y 2 . Therefore we can use the same Zariski-locally-trivial argument. We replace P(U ) by an open subset, thus shrinking X 2 and Y 2 by basechange. We may then assume V is trivial and pick a trivial Lagrangian subbundle L ⊂ V . This defines a maximal isotropic subbundle M ⊂ Hom(S, V ) by the formula (4.4), and we get the isomorphism (4.12). That is E is isomorphic to O M up to a shift and a twist by a line bundle. In particular (now that we have shrunk X 2 and Y 2 to produce an M ) we get an isomorphism between j * q * E and j * O q −1 (M) in D b (X 2 , W ) up to a shift and a twist. A key result of [ADS,Proposition 5.3 and Remark 5.13] was that when such an M exists we have O Y2 id −→ Rπ * RHom D(X2,W ) j * O q −1 M , j * O q −1 M is a quasi-isomorphism. Therefore (5.1) is also a quasi-isomorphism over our open set, and hence also globally. Using j * q * E as a Fourier-Mukai kernel, we consider the functor F : D b (Y 2 ) −→ D b (X 2 , W ), F −→ j * π * F ⊗ q * E . We will see in Proposition 5.2 below that j * (π * F ⊗ q * E) really lies in D b (X 2 , W )i.e. that it is quasi-isomorphic to a curved complex of vector bundles rather than just sheaves. This functor has a right adjoint F R : G −→ Rπ * RHom(j * q * E, G) and (5.1) says that F R • F is the identity. Therefore F embeds D b (Y 2 ) as a right-admissible subcategory of D b (X 2 , W ). To conclude the proof of Theorem 2.8 we need only show the following. Proposition 5.2. The image of the functor F is contained in the subcategory BBr(X 2 , W ) ⊂ D b (X 2 , W ). Proof. Recall that E is supported on the locus {rank x ≤ 1} ⊂ Hom(S, V ) (4.10). By pushing-down a twist of the Kozsul resolution of O N we can get a free resolution of E by bundles of the form ∧ a V ∨ ⊗ Sym b S ∨ ⊗ (det S) c ; see for example [Ei,§A2.6]. Furthermore, the symmetric powers of S ∨ that occur lie in the range b ≤ 1 2 rank V . This is precisely the range of symmetric powers included in our set T (3.1), since rank V is either n − 3 (if n is odd) or n − 2 (if n is even). Now the argument proceeds exactly as in [ADS,Proposition 5.8]. It is plausible that F is actually an equivalence between D b (Y 2 ) and BBr(X 2 , W ). periodicity, the category D b (Hom(S, V ), W ) is equivalent to the derived category of a point D b (pt), non-canonically. An equivalence is specified by any exceptional object that generates the category. One option is to choose a Lagrangian L ⊂ V and take the skyscraper sheaf of the corresponding maximally-isotropic subspace:M := Hom(S, L) ⊂ Hom(S, V ). (4.4) This gives an equivalence R Hom(O M , · ) : D b Hom(S, V ), W ∼ −→ D b (pt)sending O M to O pt . Our next result says that in this situation there is a more canonical generator, independent of any choices, and hence equivariant with respect to both Sp(S) and Sp(V ).LetLGr(S) denote the Lagrangian Grassmannian of S, and letLGr(S) π1 ←− LGr(S) × Hom(S, V ) π2 −→ Hom(S, V )denote the projections onto the two factors. The vector bundle π 1 carries a family of non-degenerate quadratic forms π * 2 W and a natural maximally-isotropic subbundle N := Hom(S/Λ, V ), (4.5)where Λ → LGr(S) is the tautological Lagrangian subbundle of S. The skyscraper sheaf O N is an object of D b Consider the case dim S = 2. Then Λ → LGr(S) is just O(−1) → P(S). The image of the map π 2 \| N is exactly {rank x ≤ 1} ⊂ Hom(S, V ) (4.10) π 2 : 2LGr(S) × Y Hom(S, V ) −→ Hom(S, V ) carrying its tautological subbundle Λ ⊂ S. With this we can define the maximallyisotropic subbundle N := Hom(S/Λ, V ) of LGr(S) × Y Hom(S, V ) → LGr(S), and This inclusion differs from the one in (1.6) by some mutations and twists by line bundles. The dual A ∨ : U → ∧ 2 V ∨ will later specialise to the injection (1.3) of the Introduction. By Serre duality the existence of a left adjoint is equivalent to the existence of a right adjoint. This section is well-defined, since W vanishes along A and B. By this we mean a section of ∧ 2 V ∨ ⊗ L for some line bundle L, such that the induced map V → V ∨ ⊗ L is an isomorphism. Acknowledgements. We would like to thank Nick Addington and Will Donovan for allowing us to re-use the arguments of the paper [ADS]. We also thank Nick Addington for useful conversations and for computational help, and Ivan Smith for generous help with the Fukaya category and quantum cohomology. R.T. was partially supported by EPSRC programme grant EP/G06170X/1. N Addington, W Donovan, E Segal, arXiv:1401.3661The Pfaffian-Grassmannian equivalence revisited. N. Addington, W. Donovan and E. Segal, The Pfaffian-Grassmannian equivalence revis- ited, arXiv:1401.3661. N Addington, E Segal, E Sharpe, arXiv:1211.2446D-brane probes, branched double covers, and noncommutative resolutions. N. Addington, E. Segal and E. Sharpe, D-brane probes, branched double covers, and non- commutative resolutions, arXiv:1211.2446. M Ballard, D Deliu, D Favero, M U Isik, L Katzarkov, arXiv:1306.3957Homological Projective Duality via Variation of Geometric Invariant Theory Quotients. M. Ballard, D. Deliu, D. Favero, M. U. Isik and L. Katzarkov, Homological Projective Duality via Variation of Geometric Invariant Theory Quotients, arXiv:1306.3957. Determinantal hypersurfaces. A Beauville, Michigan Math. J. 48A. Beauville, Determinantal hypersurfaces, Michigan Math. J. 48 (2000), 39-64. Commutative algebra with a view toward algebraic geometry. D Eisenbud, Graduate Texts in Mathematics 150. SpringerD. Eisenbud, Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics 150. Springer, 1994. Aspects of non-abelian gauge dynamics in two-dimensional N = (2, 2) theories. K Hori, D Tong, arXiv:hep-th/0609032J. High Energy Phys. 0579K.Hori and D. Tong, Aspects of non-abelian gauge dynamics in two-dimensional N = (2, 2) theories, J. High Energy Phys. 05 (2007) 079. arXiv:hep-th/0609032. Homological projective duality. A Kuznetsov, 157-220. math.AG/0507292Pub. Math. I.H.E.S. 105A. Kuznetsov, Homological projective duality, Pub. Math. I.H.E.S. 105 (2007), 157-220. math.AG/0507292. Exceptional collections for Grassmannians of isotropic lines. A Kuznetsov, math.AG/0512013Proc. London Math. Soc. 97A. Kuznetsov, Exceptional collections for Grassmannians of isotropic lines. Proc. London Math. Soc., 97 (2008), 155-182. math.AG/0512013. Homological projective duality for Grassmannians of lines. A Kuznetsov, math.AG/0610957A. Kuznetsov, Homological projective duality for Grassmannians of lines, math.AG/0610957. Equivalences between GIT quotients of Landau-Ginzburg B-models. E Segal, arXiv:0910.5534Comm. Math. Phys. 304E. Segal, Equivalences between GIT quotients of Landau-Ginzburg B-models, Comm. Math. Phys. 304 (2011), 411-432. arXiv:0910.5534. A geometric approach to Orlov's theorem. I Shipman, arXiv:1012.5282Compos. Math. 148I. Shipman, A geometric approach to Orlov's theorem, Compos. Math. 148 (2012), 1365- 1389. arXiv:1012.5282.	[]
[ "BIVARIANT K-THEORY WITH R Z-COEFFICIENTS AND RHO CLASSES OF UNITARY REPRESENTATIONS", "BIVARIANT K-THEORY WITH R Z-COEFFICIENTS AND RHO CLASSES OF UNITARY REPRESENTATIONS" ]	[ "Paolo Antonini ", "Sara Azzali ", "Georges Skandalis " ]	[]	[]	We construct equivariant KK-theory with coefficients in R and R Z as suitable inductive limits over II1-factors. We show that the Kasparov product, together with its usual functorial properties, extends to KK-theory with real coefficients.Let Γ be a group. We define a Γ-algebra A to be K-theoretically free and proper (KFP) if the group trace tr of Γ acts as the unit element in KK Γ R(A, A). We show that free and proper Γ-algebras (in the sense of Kasparov) have the (KFP) property. Moreover, if Γ is torsion free and satisfies the KK Γ -form of the Baum-Connes conjecture, then every Γ-algebra satisfies (KFP).If α ∶ Γ → Un is a unitary representation and A satisfies property (KFP), we construct in a canonical way a rho class ρ A α ∈ KK 1,Γ R Z (A, A). This construction generalizes the Atiyah-Patodi-Singer K-theory class with R Z coefficients associated to α.	10.1016/j.jfa.2015.06.017	[ "https://arxiv.org/pdf/1504.04495v1.pdf" ]	117,925,870	1504.04495	d861efc59493bcb5af4355e3b2355924824d2b22	BIVARIANT K-THEORY WITH R Z-COEFFICIENTS AND RHO CLASSES OF UNITARY REPRESENTATIONS 17 Apr 2015 Paolo Antonini Sara Azzali Georges Skandalis BIVARIANT K-THEORY WITH R Z-COEFFICIENTS AND RHO CLASSES OF UNITARY REPRESENTATIONS 17 Apr 2015 We construct equivariant KK-theory with coefficients in R and R Z as suitable inductive limits over II1-factors. We show that the Kasparov product, together with its usual functorial properties, extends to KK-theory with real coefficients.Let Γ be a group. We define a Γ-algebra A to be K-theoretically free and proper (KFP) if the group trace tr of Γ acts as the unit element in KK Γ R(A, A). We show that free and proper Γ-algebras (in the sense of Kasparov) have the (KFP) property. Moreover, if Γ is torsion free and satisfies the KK Γ -form of the Baum-Connes conjecture, then every Γ-algebra satisfies (KFP).If α ∶ Γ → Un is a unitary representation and A satisfies property (KFP), we construct in a canonical way a rho class ρ A α ∈ KK 1,Γ R Z (A, A). This construction generalizes the Atiyah-Patodi-Singer K-theory class with R Z coefficients associated to α. Introduction Let V be a closed manifold with fundamental group Γ. The symmetric index of an elliptic pseudodifferential operator D on V is an element of the K-theory of the group C * -algebra ind Γ (D) ∈ K 0 (C * Γ). Atiyah's L 2 -index theorem for covering spaces ( [2]) expresses the triviality of the group trace on this element: it states that the image of the index class ind Γ (D) by the trivial representation of Γ, which is the ordinary index of D, coincides with the image of ind Γ (D) by the group trace of Γ, i.e. the von Neumann index of the associated Γ-invariant operator acting on the universal coverṼ . This property of the group trace in Atiyah's theorem is equivalent to the fact that the Mishchenko bundle with fibre a II 1 -factor can be trivialized. This plays a crucial role in the construction of secondary invariants. Let α ∶ Γ → U n be a unitary representation. Atiyah, Patodi and Singer constructed a class [α] APS in the K-theory of V with R Z-coefficients in a way that the pairing of [α] APS with the K-homology class [D] is equal in R Z to the reduced rho invariant [3,4]. We showed in [1], to which we refer the reader for a more comprehensive literature, that the class [α] APS can be given by a purely K-theoretical construction, using von Neumann algebras. In this description, [α] APS is a relative class, represented by two bundles which become isomorphic when twisted by a bundle with fiber a II 1 -factor. In a sense we showed that In the present paper, we examine general cases for which the same triviality of the trace occurs and associate to them secondary invariants in an appropriate KK-theory with R Z coefficients. To do so, we first have to give a general definition of KK-theory with R and R Z coefficients. It is quite clear that the KK-theory with real coefficients should naturally be defined using II 1 -factors; in other words, one is tempted to put KK R (A, B) = KK(A, B ⊗ M ) where M is a II 1 -factor. We immediately fall into the question: which II 1 -factor should be chosen? If A and B are in the bootstrap category, the group KK(A, B ⊗ M ) is equal to Hom(K * (A) → K * (B) ⊗ R) whence does not depend on the II 1 -factor M . But in general, this group depends (a priori) on M . There is of course a "minimal choice" for M , namely the hyperfinite II 1 -factor R, but we need here factors big enough to contain the group von Neumann algebra of our Γ: then it is not reasonable in this discussion to assume Γ to be amenable, since the typical Γ is a dense subgroup of U n . On the opposite, Ozawa ([22]) proved that there is no "maximal choice" for M . Our solution is to define KK R (A, B) as the inductive limit of KK(A, B ⊗ M ) over all II 1 -factors M . It is not too difficult to give a sense to this inductive limit. One extends also very easily the Kasparov product with all its functorial properties and associativity to the KK R -theory (using von Neumann tensor products). We then define KK R Z as the inductive limit of KK(A, B ⊗Cone(C → M )). The mapping cone exact sequence yields the Bockstein change of coefficients sequence. In the same way, one defines all the equivariant KK-theories with R and R Z coefficients, letting the group (or more generally a group-like object) act trivially on the II 1 -factors. A tracial state τ on a C * -algebra D may be though of as a generalized morphism to a II 1 -factor: in fact, there is a II 1 -factor M and a tracial morphism D → M , so that we obtain a class [τ ] in KK R (D, C). Using Kasparov product, we then obtain morphisms τ * ∶ KK(A, B ⊗ D) → KK R (A, B) for every C * -algebras A and B. Let now Γ be a group. A tracial state on C * Γ defines an element in KK R (C * Γ, C). This group bares a ring structure using the coproduct of Γ, and is actually equal to the equivariant group KK Γ R (C, C). Denote by tr the group trace of Γ and [tr] Γ its class, which is an idempotent of the ring KK Γ R (C, C). As all the Kasparov groups KK Γ R (A, B) are modules over this ring, a natural question is: what is the image of this idempotent on these KK Γ R (C, C) modules? We will be interested on those C * -algebras A endowed with an action of Γ (in short, such an A is called a Γ-algebra), for which the element [tr] Γ acts as a unit element in KK Γ R (A, A). This means that 1 Γ A ⊗ C [tr] Γ = 1 Γ A,R ∈ KK Γ R (A, A) . When this holds, we say that A satisfies property (KFP). This definition is inspired by the commutative case where the deck group Γ of a Galois coveringṼ of a closed manifold V acts on C 0 (Ṽ ). Atiyah's theorem in [2] can be interpreted as stating that the Γ-algebra C 0 (Ṽ ) satisfies property (KFP). In this commutative case, property (KFP) comes from the fact that the action of Γ onṼ is free and proper. This is why algebras satisfying this condition are thought of as being in a sense K-theoretically free and proper. Let ) that we call the rho class associated to A and α. It generalizes the APS class and it is additive with respect to direct sum of representations; its behavior with respect to tensor products is also easy to describe. Furthermore ρ A α is functorial with respect to the algebra A and, more generally, with respect to the action of the KK Γ -groups by Kasparov product. To complete the construction, it is then natural to exhibit classes of algebras satisfying property (KFP). • The first example comes from the Mishchenko bundle, i.e. the already mentioned cocompact regular covering spaceṼ : C 0 (Ṽ ) satisfies property (KFP). In this case, we compare the rho class ρ C 0 (Ṽ ) α with the element [α] AP S . • Based on the example of the Mishchenko bundle, we prove that free and proper Γ-algebras in the sense of Kasparov satisfy property (KFP). • Moreover, property (KFP) is obviously invariant under KK Γ -(sub)equivalence. It follows that if Γ is torsion free and satisfies the KK Γ -form of the Baum-Connes conjecture, then every Γ-algebra satisfies property (KFP). • If Γ is torsion free and has a γ element in the sense of Kasparov, then we can construct in this way the γ part of ρ C α , and therefore of ρ A α for any Γ-algebra A. • Using a result of Guentner-Higson-Weinberger ( [15]), we moreover deduce a construction of ρ C α whenever α(Γ) has no torsion. Finally, we state a weakening of property (KFP), called the the weak (KFP) property, saying that the image of [tr] by Kasparov's descent morphism j Γ in KK R (A⋊Γ, A⋊Γ) is the unit element. Under this condition, we construct a weaker rho classρ A α ∈ KK 1 R Z (A⋊Γ, A⋊Γ). We show that these two constructions are related: if A satisfies property (KFP), thenρ A α = j Γ (ρ A α ). Here is a summary of our paper: • In the first section, we construct the KK-theory with coefficients R and R Z. We also discuss a few other K-theoretic constructions as the torus algebra of a pair of maps ϕ i ∶ A → B and the corresponding six term exact sequence in (equivariant) KK-theory. • In the second section we introduce property (KFP) and construct the rho element associated to a finite dimensional unitary representation. • In the third section we discuss examples of Γ-algebras with property (KFP). • Finally, in section 4, we introduce a weakening of property (KFP) and construct the corresponding weak rho class. 1. K-theoretic constructions 1.1. Some conventions. We will freely use Kasparov's KK-groups and notation from [18,19]. Let us fix a few conventions that will be used throughout the text: • In what follows, by trace on a C * -algebra we will mean a finite (positive) trace. • All traces on von Neumann algebras that we consider, and all morphisms are assumed to be normal. • The suspension of a C * -algebra A is SA = C 0 ((0, 1), A). • If A is a C * -algebra, we denote by 1 A the unit element of the ring KK(A, A). If A is a Γ-algebra, the unit element of the ring KK Γ (A, A) will be denoted by 1 Γ A . If u ∶ A → B is an equivariant morphism, we will denote by [u] Γ ∈ KK Γ (A, B) its class: [u] Γ = u * (1 Γ B ) = u * (1 Γ A ) . • More generally, if E is an equivariant Hilbert B-module endowed with a left action of A through an equivariant morphism A → K(E), we will denote by [E] Γ the corresponding class in KK Γ (A, B). • In what follows, ⊗ means minimal tensor product. Note however that we could do the construction below using maximal tensor products (and normal tensor products wherever von Neumann algebras are involved) as well. However, if Γ is a locally compact group, C * Γ will denote the full group C * -algebra of Γ. • If u ∈ KK Γ (A 1 , B 1 ) and v ∈ KK Γ (A 2 , B 2 ), we will denote by u ⊗ v ∈ KK Γ (A 1 ⊗ A 2 , B 1 ⊗ B 2 ) their external KK-product. • In particular, if x ∈ KK Γ (A, B) and D is a Γ-algebra 1 Γ D ⊗ x ∈ KK Γ (D ⊗ A, D ⊗ B) is the element noted τ D (x) in [18]. Remark 1.1. In this paper, we deal with KK-theory in connection with von Neumann algebras. A von Neumann algebra is (almost) never a separable C * -algebra. On the other hand, recall from [29, Remark 3.2] that if A and B are C * -algebras with A separable then KK(A, B) is the direct limit over the separable subalgebras of B. One then should define KK(A, B) with A not separable as the projective limit over all separable subalgebras of A. Kasparov's KK-group maps to this new KK(A, B) and, with this definition, the Kasparov product ⊗ D ∶ KK(A, D) × KK(D, B) → KK(A, B) is defined for any triple of C * -algebras with no assumptions of separability of any kind, as well as the more general ⊗ D ∶ KK(A 1 , B 1 ⊗ D) × KK(D ⊗ A 2 , B 2 ) → KK(A 1 ⊗ A 2 , B 1 ⊗ B 2 ). 1.2. Torus algebra. Let A, B be C * -algebras and ϕ 0 , ϕ 1 ∶ A → B two * -homomorphisms. Define the corresponding torus algebra to be T (ϕ 0 , ϕ 1 ) = {(a, f ) ∈ A × B[0, 1]; f (0) = ϕ 0 (a) and f (1) = ϕ 1 (a)}. We will use the following straightforward functorial properties: 1] are morphisms joining ϕ 0 to ψ 0 and ϕ 1 to ψ 1 (i.e such that (Φ i (a))(0) = ϕ i (a) and (Φ i (a))(1) = ψ i (a) for all a ∈ A), we construct a homotopy equivalence (1) if ϕ 0 , ϕ 1 ∶ A → B then T (ϕ 0 , ϕ 1 ) ⊗ D = T (ϕ 0 ⊗ id D , ϕ 1 ⊗ id D ). (2) Every f ∶ D → A induces f * ∶ T (ϕ 0 ○ f, ϕ 1 ○ f ) → T (ϕ 0 , ϕ 1 ). (3) Every f ∶ B → D induces f * ∶ T (ϕ 0 , ϕ 1 ) → T (f ○ ϕ 0 , f ○ ϕ 1 ). (4) Let ϕ 0 , ϕ 1 , ψ 0 , ψ 1 ∶ A → B; if ϕ i is homotopic to ψ i , the corresponding torus algebras T (ϕ 0 , ϕ 1 ) and T (ψ 0 , ψ 1 ) are homotopy equivalent. More precisely, if Φ 0 , Φ 1 ∶ A → B[0,T (ϕ 0 , ϕ 1 ) → T (ψ 0 , ψ 1 ) mapping (a, f ) to (a, g), where g(t) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ Φ 0 (1 − 3t) if 0 ≤ t ≤ 1 3 f (3t − 1) if 1 3 ≤ t ≤ 2 3 Φ 1 (3t − 2) if 1 3 ≤ t ≤ 2 3. We have an exact sequence 0 / / SB h / / T (ϕ 0 , ϕ 1 ) p / / A / / 0 (with completely positive lifting), where p ∶ T (ϕ 0 , ϕ 1 ) → A is the morphism (a, f ) ↦ a and h ∶ SB → T (ϕ 0 , ϕ 1 ) is the map f ↦ (0, f ). The corresponding connecting map is [ϕ 0 ] − [ϕ 1 ]. We then have: Proposition 1.2. Let Γ be a discrete group, and let A, B, C, D be Γ-algebras. Let ϕ 0 , ϕ 1 ∶ A → B be Γ-equivariant morphisms. We have six term exact sequences KK Γ (C, D ⊗ SB) h * / / KK Γ (C, D ⊗ T (ϕ 0 , ϕ 1 )) p * / / KK Γ (C, D ⊗ A) (ϕ 0 ) * −(ϕ 1 ) * KK Γ (C, D ⊗ SA) (ϕ 0 ) * −(ϕ 1 ) * O O KK 1,Γ (C, D ⊗ T (ϕ 0 , ϕ 1 )) p * o o KK Γ (C, D ⊗ B). h * o o There is also an analogous exact sequence with reversed arrows for A, B and T (ϕ 0 , ϕ 1 ) on the left hand side. Proof. Although this follows from [18,28] (at least in the non equivariant setting), we wish to outline here that this is a Puppe type exact sequence and holds automatically for every homotopy functor (see [11]). In particular, it holds for the equivariant KK Γ -theory as we will use it. Exactness at A: This is in a sense just tautological; for x ∈ KK Γ (C, D ⊗ A), we have (ϕ 0 ) * (x) = (ϕ 1 ) * (x) if and only if these elements are homotopic i.e. if x is in p * (KK Γ (C, D ⊗ T (ϕ 0 , ϕ 1 ))). Exactness at T : The mapping cone of p is T (ϕ 0 ○e 0 , ϕ 1 ○e 0 ) where e 0 ∶ C 0 ([0, 1), A) → A is the evaluation at 0. As e 0 is homotopic to the evaluation at 1, which is the 0 map, this torus algebra is homotopy equivalent to T (ϕ 0 ○ e 1 , ϕ 1 ○ e 1 ) = SB ⊕ C 0 ([0, 1), A) whence to SB. More precisely, the inclusion of the kernel SB of p in the mapping cone of p is a homotopy equivalence. Use then the Puppe sequence which holds in full generality. Exactness at B: The mapping cone of p is T (φ 0 ,φ 1 ) whereφ i ∶ SA → C 0 ([0, 1), B) are f ↦ ϕ i ○ f . These maps are again homotopic to 0, whence T (φ 0 ,φ 1 ) is homotopy equivalent to SA. Remarks 1.3. (1) Let ϕ 0 , ϕ 1 ∶ A → B be morphisms and u 0 , u 1 ∈ B unitaries. Then we have a natural Morita equivalence between the torus algebras T (ϕ 0 , ϕ 1 ) and T (Ad u 0 ○ ϕ 0 , Ad u 1 ○ ϕ 1 ). Indeed the space E = {(a, f ) ∈ A × B[0, 1]; f (0) = u 0 ϕ 0 (a); f (1) = u 1 ϕ 1 (a)} is the desired Morita equivalence bimodule. If [E] denotes the associated element in KK(T (Ad u 0 ○ ϕ 0 , Ad u 1 ○ ϕ 1 ), T (ϕ 0 , ϕ 1 )), we have p * [E] = [p ′ * ] , (h ′ ) * [E] = [h] where p ∶ T (ϕ 0 , ϕ 1 ) → A and p ′ ∶ T (Ad u 0 ○ ϕ 0 , Ad u 1 ○ ϕ 1 ) → A denote the evaluation morphisms (a, f ) ↦ a, and h ∶ SB → T (ϕ 0 , ϕ 1 ) and h ′ ∶ SB → T (Ad u 0 ○ ϕ 0 , Ad u 1 ○ ϕ 1 ) → A denote the morphisms f ↦ (0, f ). (2) Let A 0 and A 1 be C * -algebras with ϕ 0 ∶ A 0 → B and ϕ 1 ∶ A 1 → B be two morphisms. Recall that the associated double cylinder is defined by Z(ϕ 0 , ϕ 1 ) = {(a 0 , a 1 , f ) ∈ A 0 × A 1 × B[0, 1]; f (0) = ϕ 0 (a 0 ) and f (1) = ϕ 1 (a 1 )}. Given unitaries u 0 , u 1 ∈ B, there is a canonical Morita equivalence Z(ϕ 0 , ϕ 1 ) and Z(Ad u 0 ○ ϕ 0 , Ad u 1 ○ ϕ 1 ). In fact, we may put A = A 0 × A 1 and ψ ∶ A → B, ψ i (a 0 , a 1 ) = ϕ i (a i ), i = 0, 1}; then Z(ϕ 0 , ϕ 1 ) identifies with T (ψ 0 , ψ 1 ). (1) Every finite von Neumann algebra with trace can be embedded in a trace preserving way into a II 1 factor -just take for instance a free product with a II 1 -factor ( [32], [23,Theorem 4.4], [12]). (2) For the same reason, a finite number of II 1 factors can be simultaneously embedded in a trace preserving way into a II 1 factor -just take their (tracial) free product. Indeed, the hyperfinite II 1 -factor is of course the smallest element of (F II 1 (H), ≺). By [22], there is no biggest element in F Note also that if M is a big II 1 -factor acting on a non separable Hilbert space, then for every separable C * -algebra A, and x ∈ KK(A, B ⊗ M ), there exists (by [29, Remark 3.2]) a separable subalgebra N 1 ⊂ B ⊗ M and x 1 ∈ KK(A, N 1 ) whose image is x. We may then construct separable subalgebras B 1 ⊂ B and D 1 ⊂ M and hence a (weakly) separable II 1 -subfactor M 1 ⊂ M such that N 1 ⊂ B 1 ⊗ D 1 ⊂ B 1 ⊗ M 1 ⊂ B ⊗ M . Let thenx 1 denote the image of x 1 ∈ KK R (A, B). If N 2 , x 2 is another such choice, we may find a separable N 3 ⊂ B ⊗ M where x 1 and x 2 coincide and therefore a separable subfactor M 3 ⊂ M such that N 3 ⊂ B ⊗ M 3 . It follows thatx 1 =x 2 . We thus obtain a natural morphism KK(A, B ⊗ M ) → KK R (A, B). (3) If we wish to avoid the use of non separable algebras and the method of [29] to treat the corresponding KK-products, we may also replace this category of II 1 -factors by the category of separable C * -algebras with tracial states, the morphisms in this category being the trace preserving * -homomorphisms. Define then KK R (A, B) as the limit of KK(A, B ⊗D) when D runs over all tracial separable C * -algebras (acting on a given Hilbert space). We may form an inductive system KK(A, B)[D, τ ] as the quotient of KK(A, B ⊗ D) by the subgroup of x such that there exists a C algebra D 1 with a tracial state τ 1 a trace preserving morphism j ∶ D → D 1 such that j (x) = 0. It follows from [29] that the two definitions coincide. (4) The method used to show the existence of the limit in construction 1.5 can be summarized as follows: assume C is a (small) category and F ∶ C → Grp is a functor such that the following properties are satisfied: • for every A, B ∈ Ob C there exists some C ∈ Ob C and arrows A → C and B → C. • For every couple of arrows f, g ∶ A → B there exists some C ∈ Ob C and an arrow h ∶ B → C with F(h ○ f ) = F(h ○ g) . Then we have shown that there is a natural transformation µ ∶ F → F such that F is a directed system and F has a unique limit which is the unique limit of F . 1.4. Traces and KK-theory with real coefficients. A trace on a C * -algebra D can be thought of as a generalized morphism to an abstract II 1 -factor. It therefore gives rise to a natural element of KK R (D, C). Definition 1.7. If D is a separable unital C * -algebra and τ is a trace on D, then there is a II 1 -factor M and a finitely generated projective module E on M with a trace preserving morphism f τ ∶ D → L(E) (with von Neumann dimension dim M (E) = τ (1)); in this way, we define an element [τ ] ∈ KK R (D, C); using the morphism f τ ∶ D → L(E) we also obtain for every pair (A, B) of C * -algebras a morphism τ * ∶ KK(A, B ⊗ D) → KK R (A, B). In particular, we have [τ ] = τ * (1 D ) ∈ KK R (D, C). 1.5. The Kasparov product. Let A, B, D be C * -algebras. If x ∈ KK R (A, D) and y ∈ KK R (D, B), there exist II 1 -factors M and N such that x is the image of x 0 ∈ KK(A, D ⊗ M ) and y is the image of y 0 ∈ KK(D, B ⊗ N ). We then may form the KK-product and obtain an element x 0 ⊗ D y 0 ∈ KK(A, B ⊗ M ⊗ N ). We may then map the minimal tensor product M ⊗N into the corresponding von Neumann tensor product and pass to the limit KK R (A, B). We obtain an element x ⊗ D y ∈ KK R (A, B) which only depends on x and y and not on (M, x 0 ) and (N, y 0 ). This Kasparov product immediately extends to a product ⊗ D ∶ KK R (A 1 , B 1 ⊗ D) × KK R (D ⊗ A 2 , B 2 ) → KK R (A 1 ⊗ A 2 , B 1 ⊗ B 2 ) which has all the usual properties of the Kasparov product (bilinearity, functoriality, associativity...). Remark 1.9. For every II 1 -factor M , the group KK(C, M ) is canonically isomorphic with R. It follows that the ring KK R (C, C) is naturally isomorphic with R. We deduce that KK R (A, B) is naturally a real vector space -and that the KK R -product is R-bilinear. If τ is a trace on a C * -algebra D and s ∈ R + , then [sτ ] = s[τ ] ∈ KK R (D, C). 1.6. KK-theory with R Z coefficients. For every II 1 -factor M denote by i M ∶ C → M the unital inclusion. For the R Z-coefficients we can similarly define: Definition 1.10. Let A, B be C * -algebras we put: KK •,R Z (A, B) ∶= lim M ∈F II 1 (H) KK •+1 (A, B ⊗ C i M ). (1.2) As above, this is a limit along the partially ordered directed set (F II 1 (H), ≺). The mapping cone exact sequence gives rise to a six term Bockstein change of coefficients exact sequence ... → KK(A, B) → KK R (A, B) → KK R Z (A, B) → ... Remark 1.11. Let A, B be C * -algebras and M be a II 1 -factor. Denote by i B M ∶ B → B ⊗ M the map x ↦ x ⊗ 1 M . We have of course C i B M = B ⊗ C i M and Z(i B M , i B M ) = B ⊗ Z(i M , i M ). Using the split exact sequence 0 → C i B M → Z(i B M , i B M ) → B → 0, we find KK(A, B ⊗ Z(i M , i M )) = KK(A, B) ⊕ KK(A, B ⊗ C i M ). We thus get a morphism ϑ ∶ KK(A, B ⊗ Z(i M , i M )) → KK 1 R Z (A, B). Composing with the connecting map ∂ of the Bockstein change of coefficients exact sequence we obtain a map ∂ ○ ϑ = (p 0 ) * − (p 1 ) * ∶ KK(A, B ⊗ Z(i M , i M )) → KK(A, B), where p i ∶ B ⊗ Z(i M , i M ) = Z(i B M , i B M ) → B are the maps (b 0 , b 1 , f ) ↦ b i . 1.7. KK Γ with coefficients. The above realizations of KK R and KK R Z can be extended to the equivariant setting. Definition 1.12. Let Γ be a discrete group. We define KK Γ R (A, B) and KK Γ R Z (A, B) to be respectively the limit of KK Γ (A, B ⊗ M ) and KK Γ (A, B ⊗ C i M ) where M runs over the set F II 1 (H) of II 1 -factors on the separable Hilbert space H -with trivial Γ action, i M ∶ C → M denotes the unital inclusion and C i M its mapping cone. This is again a direct limit along the partially preorderd directed set (F II 1 (H), ≺). Recall from [19] that Kasparov descent morphism is a natural morphism j Γ ∶ KK Γ (A, B) → KK(A ⋊ Γ, B ⋊ Γ). If A, B are Γ-algebras and M a II 1 -factor (with trivial action of Γ), composing the descent morphism together with the natural morphism (B ⊗ M ) ⋊ Γ → (B ⋊ Γ) ⊗ M we obtain a morphism KK Γ (A, B ⊗ M ) → KK(A ⋊ Γ, (B ⊗ M ) ⋊ Γ) → KK(A ⋊ Γ, (B ⋊ Γ) ⊗ M ). This map passes to the inductive limit and defines a descent morphism with real coefficients j Γ R ∶ KK Γ R (A, B) → KK R (A ⋊ Γ, B ⋊ Γ). In the same way, taking coefficients in mapping cone algebras of inclusions i M ∶ C → M and then inductive limit over M ∈ F II 1 (H), we obtain a descent morphism with R Z coefficients j Γ R Z ∶ KK Γ R Z (A, B) → KK R Z (A ⋊ Γ, B ⋊ Γ). The Kasparov product ⊗ D ∶ KK Γ R (A 1 , B 1 ⊗ D) × KK Γ R (D ⊗ A 2 , B 2 ) → KK Γ R (A 1 ⊗ A 2 , B 1 ⊗ B 2 ) is defined exactly as in Section 1.5 with all the functorial properties with respect to algebras. Furthermore a morphism Γ 1 → Γ 2 induces an obvious morphism KK Γ 2 R (A, B) → KK Γ 1 R (A, B) . Also, the mapping cone exact sequence in equivariant KK-theory gives rise to a six term Bockstein change of coefficients exact sequence ... → KK Γ (A, B) → KK Γ R (A, B) → KK Γ R Z (A, B) → ... Remark 1.13. In this paper, we use the equivariant KK R -and KK R Z -theory with respect to a discrete group Γ. Of course, the same constructions can be performed equivariantly with respect to a locally compact group or a Hopf algebra (cf. [5]) or a groupoid (cf. [20]). 1.8. Some remarks on Γ-algebras. Let Γ be a discrete group. We end these comments with some easy observations on Γ-algebras. Trivial action and equivariant Kasparov groups. Lemma 1.14. Let Γ be a discrete group, A a Γ-algebra and let B be a C * -algebra endowed with a the trivial action of Γ. Then there is a canonical isomorphism KK Γ (A, B) ≅ KK(A ⋊ Γ, B). Proof. Note first that by [28, Lemma 1.11] one can take the action of A (resp. A ⋊ Γ) to be non-degenerate in the cycles defining KK Γ (A, B) (resp. KK(A ⋊ Γ; B)). With this in mind the rest of the proof is straightforward because Γ acts by unitary multipliers on A ⋊ Γ. Remark 1.15. Note that the identification in Lemma 1.14 is the composition KK Γ (A, B) j Γ → KK(A ⋊ Γ, B ⊗ C * Γ) (id ⊗ε) * → KK(A ⋊ Γ, B) where ε ∶ C * Γ → C is the trivial representation. The isomorphism KK Γ (A, B) → KK(A ⋊ Γ, B) can be expressed as the composition of Kasparov's descent morphism j Γ ∶ KK Γ (A, B) → KK(A ⋊ Γ, B ⋊ Γ) with the morphism B ⋊ Γ = B ⊗ max C * Γ → B associated with the trivial representation. Inner action and equivariant Morita equivalence. Lemma 1.16. Let Γ be a discrete group, A a C * -algebra and g ↦ u g a morphism of Γ to the group of unitary multipliers of A. Let β be the corresponding inner action of Γ on A (β(a) = u g au * g ). The Γ-algebras A endowed with the trivial action and A endowed with the action α are Morita equivalent in an equivariant way. Proof. We denote by A 1 and A β the algebra A when endowed with the trivial action and the action β, respectively. Let E be the Hilbert A 1 -module A endowed with the action of Γ given by g.x = u g x. As u g (xy) = β g (x)u g y, the action of A by left multiplication on E is equivariant: it follows that E is a Γ-equivariant Morita equivalence between A β and A 1 . 1.9. Kasparov's descent and coproducts. Let A be a Γ-algebra. Denote by δ A ∶ A⋊Γ → (A ⋊ Γ) ⊗ C * Γ the coaction map. Recall that δ A (au g ) = au g ⊗ u g for a ∈ A ⊂ A ⋊ Γ and g ∈ Γ where u g denotes both the element of C * Γ and of (the multiplier algebra of) A ⋊ Γ corresponding to g. Remark 1.17. Let B be a C * -algebra endowed with a trivial Γ-action. Through the iden- tification KK Γ (A, B) ≅ KK(A ⋊ Γ, B), Kasparov's descent morphism is given by x ↦ (δ A max ) * (x ⊗ max 1 C * Γ ) ∶ KK(A ⋊ Γ, B) → KK(A ⋊ Γ, B ⋊ Γ) = KK(A ⋊ Γ, B ⊗ max C * Γ). Here, we denoted by δ A max ∶ A ⋊ Γ → (A ⋊ Γ) ⊗ max C * Γ the morphism defined by the same formula as δ A . 2. K-theoretically "free and proper" algebras and ρ class 2.1. KK-theory elements with real coefficients associated with a trace on C * Γ. As explained in lemma 1.14, for every Γ-algebra A and every C * -algebra B, we may identify ) where B is endowed with the trivial action of B. Replacing in this formula B by B ⊗ M where M is a II 1 -factor and taking the inductive limit, we find for a Γ-algebra A and a C * -algebra B endowed with a trivial Γ action an identification KK(A ⋊ Γ, B) with KK Γ (A, BKK Γ R (A, B) = KK R (A ⋊ Γ, B) -and in the same way, KK Γ R Z (A, B) = KK R Z (A ⋊ Γ, B). A trace τ on C * Γ gives rise to an element [τ ] ∈ KK R (C * Γ, C) (Definition 1.7) and therefore an element [τ ] Γ ∈ KK Γ R (C, C). 2.2. K-theoretically "free and proper" algebras. We now consider a class of Γ-algebras on the K-theory of which "the trace acts as the unit element". Those are the K-theoretically "free and proper" algebras: Definition 2.1. Let Γ be a discrete group. Denote by tr its group tracial state. We say that A satisfies property (KFP) if 1 Γ A ⊗ [tr] Γ is equal to the unit element 1 Γ A,R of the ring KK Γ R (A, A). We will see in the following section important examples of algebras satisfying this property. In particular, we will see that free and proper algebras satisfy this condition (Theorem 3.10). Note however that, if Γ has a γ element in the sense of [19,30], K-theoretically proper algebras should be those for which γ = 1. It is not at all clear to us whether condition (KFP) implies γ = 1. On the other hand, it is worth noting that property (KFP) implies a kind of freeness condition: property (KFP) cannot hold for the algebra C whenever the group Γ has torsion. Indeed, let ε ∶ C * Γ → C be the trivial representation. The algebra C has property (KFP) if and only if the classes [tr] and [ε] in KK Γ R (C, C) = KK R (C * Γ, C) coincide. Assume that g ∈ Γ has finite order m ≠ 1. Then the element p = 1 m(1 + γ + ⋅ ⋅ ⋅ + γ m−1 ) ∈ CΓ is a projection with [p] ⊗ [tr] = tr(p) = 1 m and [p] ⊗ [ε] = ε(p) = 1. Assume A satisfies property (KFP). This means that there is a II 1 -factor N with a morphism λ ∶ C * Γ → N such that tr N ○ λ = tr (where tr N is the normalized trace of N ) and such that the (A, A ⊗ N )-bimodule A ⊗ N where A acts on the left by a ↦ a ⊗ 1 endowed with the action g ↦ g ⊗ λ(g) and g ↦ g ⊗ 1 define the same element in KK Γ (A, A ⊗ N ). Let us make a few comments on this definition: Properties 2.2. (1) If A satisfies property (KFP) and B is any Γ-algebra, then A ⊗ B satisfies property (KFP). Indeed, if 1 Γ A ⊗ [tr] Γ = 1 Γ A,R , then 1 Γ A ⊗ 1 Γ B ⊗ [tr] Γ = (1 Γ A ⊗ [tr] Γ ) ⊗ 1 Γ B = 1 Γ A⊗B,R . (2) Let A, B be Γ-algebras. If A is KK Γ -subequivalent to B and B satisfies property (KFP)then so does A. The assumption means that there exist x ∈ KK Γ (A, B) and y ∈ KK Γ (B, A) satisfying x ⊗ B y = 1 Γ A . Note that y ⊗ C [tr] Γ = [tr] Γ ⊗ C y = (1 Γ B ⊗ [tr] Γ ) ⊗ B y. If B satisfies property (KFP), then y ⊗ C [tr] Γ is the image in KK Γ R (B, A) of y, whence 1 Γ A ⊗[tr] Γ = x ⊗ B (y ⊗ C [tr] Γ ) = 1 Γ A,R . Remark 2.3. Let τ, τ ′ be tracial states on C * Γ. Let [τ ] Γ and [τ ′ ] Γ denote their classes in KK Γ R (C, C). Their Kasparov product [τ ] Γ ⊗ [τ ′ ] Γ is then the class of the trace τ.τ ′ = (τ ⊗ τ ′ ) ○ δ. Note that τ.τ ′ (g) = τ (g)τ ′ (g) for any group element g ∈ Γ, and in particular τ.tr = tr. If A satisfies property (KFP), then 1 Γ A ⊗ [τ ] Γ = 1 Γ A,R for any tracial state τ . Remark 2.4. If A satisfies property (KFP), then the morphism q Γ ∶ A ⋊ Γ → A ⋊ red Γ is KK R invertible. Indeed, given a von Neumann algebra N and a morphism λ ∶ C * Γ → N such that tr N ○λ = tr (where tr N is the normalized trace of N ), the morphism λ ∶ C * Γ → N factors through C * red (Γ); it therefore defines a morphism δ A λ ∶ A ⋊ red Γ → A ⋊ Γ ⊗ N . We obviously have q * Γ [ δ A λ ] = j Γ (1 Γ A ⊗ [tr] Γ ) and (q Γ ) * [ δ A λ ] = j Γ,red (1 Γ A ⊗ [tr] Γ ) . We will use the quite obvious following lemma: Lemma 2.5. Let α ∶ Γ → U n be a unitary representation. The group Γ acts on C(U n ) by left translation via α. Let u ∶ U n → M n (C) be the identity of U n seen as an element of C(U n ) ⊗ M n (C). For g ∈ Γ, we denote by λ g ∈ C(U n ) ⋊ Γ the corresponding element. For g ∈ Γ we have (λ g ⊗ α g )u = u(λ g ⊗ 1) in C(U n ) ⋊ Γ ⊗ M n (C). Proof. Note that, for f ∈ C(U n ) and x ∈ U n we have (λ g f λ −1 g )(x) = f (α −1 g x). Write u = ∑ a j ⊗ b j where a j ∈ C(U n ) and b j ∈ M n (C). By definition of u, we have ∑ j a j (x)b j = x for all x ∈ U n . We have ((λ g ⊗ 1)u(λ −1 g ⊗ 1))(x) = ∑ j a j (α −1 g x)b j = α −1 g x, and the result follows. 2.3. Gluing Kasparov bimodules. To construct ρ A α , we glue two Kasparov bimodules on a mapping cylinder in such a way that they form a Kasparov bimodule on a double cylinder. We discuss here this general construction. Let A, B 0 , B be Γ-algebras and j ∶ B 0 → B an equivariant morphism. Denote by Z j = {(b, f ) ∈ B 0 × B[0, 1]; f (0) = j(b)} the mapping cylinder of j and Z j,j = {(b 0 , b 1 , f ) ∈ B 0 × B 0 × B[0, 1]; f (0) = j(b 0 ), f (1) = j(b 1 )} its mapping double cylinder. Let ev 0 ∶ Z j → B 0 and ev 1 ∶ Z j → B be the maps ev 0 ∶ (b, f ) ↦ b and ev 1 ∶ (b, f ) ↦ f (1). We have a split exact sequence 0 → C j → Z j,j ev 1 → B 0 → 0 where C j = {(b, f ) ∈ B 0 × B[0, 1); f (0) = j(b)} is the mapping cone of j and ev 1 (a, b, f ) = b -the section ∶ B 0 → Z j,j is given by(b) = (b, b, f ) where f (t) = j(b) for all t ∈ [0, 1]. We thus have a group homomorphism Θ ∶ KK Γ (A, Z j,j ) → KK Γ (A, C j ) which is a left inverse of the inclusion C j → Z j,j and vanishes on the image of * . We will identify the double cylinder Z j,j with the algebra Z j = {(x, y) ∈ Z j × Z j ; ev 1 (x) = ev 1 (y)} through the isomorphism Φ ∶ Z j,j → Z j given by: Φ(a, b, f ) = ((a, g), (b, h)) with g(t) = f (t 2), h(t) = f (1 − t 2) (for t ∈ [0, 1]). Given a Hilbert Z j -module E, the Hilbert B 0 -module E ⊗ ev 0 B 0 and the Hilbert-B-module E ⊗ ev 1 B are quotients of E. We denote by ev 0 ∶ E → E ⊗ ev 0 B 0 and ev 1 ∶ E → E ⊗ ev 1 B the quotient maps. Proposition 2.6. Let (E, F ), (E ′ , F ′ ) be Γ-equivariant Kasparov (A, Z j )-bimodules. (1) Let w ∈ L(E ′ ⊗ ev 1 B, E ⊗ ev 1 B) be a unitary equivalence between the Γ-equivariant Kaparov (A, B)-bimodules (ev 1 ) * (E ′ , F ′ ) and (ev 1 ) * (E, F ). Define E ◇ w E ′ = {(ξ, ζ) ∈ E × E ′ ; ev 1 (ξ) = w ev 1 (ζ)}. It is naturally an equivariant Hilbert- (A, Z j )-bimodule. The map (F ◇ F ′ ) ∶ (ξ, ζ) ↦ (F ξ, F ′ ζ) is an (odd) element in L(E ◇ w E ′ ). The pair (E ◇ w E ′ , F ◇ F ′ ) is an equivariant Kasparov (A, Z j )-bimodule (2) If v, w ∈ L(E ′ ⊗ ev 1 B, E ⊗ ev 1 B)(E ◇ v E ′ , F ◇ F ′ ) and (E ◇ w E ′ , F ◇ F ′ ) in KK Γ (A, Z j ) coincide. (3) The class Θ[(E ◇ id E, F ◇ F )] in KK Γ (A, C j ) is 0. (4) Let (E ′′ , F ′′ ) be another Γ-equivariant Kasparov (A, Z j )-bimodule. Let w ∈ L(E ′ ⊗ ev 1 B, E ⊗ ev 1 B) and w ′ ∈ L(E ′′ ⊗ ev 1 B, E ′ ⊗ ev 1 B) be unitary equivalences between the Γ- equivariant Kaparov (A, B)-bimodules (ev 1 ) * (E ′ , F ′ ) and (ev 1 ) * (E, F ) and between (ev 1 ) * (E ′′ , F ′ ) and (ev 1 ) * (E ′ , F ) respectively. Then Θ[(E ◇ ww ′ E ′′ , F ◇ F ′′ )] = Θ[(E ◇ w E ′ , F ◇ F ′ )] + Θ[(E ′ ◇ w ′ E ′′ , F ◇ F ′′ )]. Proof. (1) comes from the fact that w is supposed to commute with A, Γ and wF ′ = F w. Also, it is easily seen that K(E ◇ w E ′ ) = {(T, T ′ ) ∈ K(E) × K(E ′ ); w(T ′ ⊗ ev 1 1) = (T ⊗ ev 1 1)w}. (2) is immediate. (3) Let (E t , F t ) t∈[0,1] be a homotopy joining ( ○ ev 0 ) * (E, F ) to (E, F ): write (E t , F t ) = (h t ) * (E, F ) where h t ∶ Z j → Z j is the map (b, f ) ↦ (b, f t ) where f t (s) = f (st). Then (E t ◇ id E t , F t ◇ F t ) is a homotopy joining ( ○ ev 0 ) * (E, F ) to (E ◇ id E, F ◇ F ). (4) The unitary equivalences u = w 0 0 w ′ , v = 0 id ww ′ 0 ∈ L((E ′ ⊕ E ′′ ) ⊗ ev 1 B, (E ⊕ E ′ ) ⊗ ev 1 B) are homotopic among unitary equivalences. By (2) the classes [((E ⊕ E ′ ) ◇ u (E ′ ⊕ E ′′ ), (F ⊕ F ′ ) ◇ (F ′ ⊕ F ′′ ))] and [((E ⊕ E ′ ) ◇ v (E ′ ⊕ E ′′ ), (F ⊕ F ′ ) ◇ (F ′ ⊕ F ′′ ))] in KK Γ (A, Z j,j ) coincide. But clearly [((E ⊕ E ′ ) ◇ u (E ′ ⊕ E ′′ ), (F ⊕ F ′ ) ◇ (F ′ ⊕ F ′′ ))] = [(E ◇ w E ′ , F ◇ F ′ )] + [(E ′ ◇ w ′ E ′′ , F ◇ F ′′ )] and [((E ⊕ E ′ ) ◇ v (E ′ ⊕ E ′′ ), (F ⊕ F ′ ) ◇ (F ′ ⊕ F ′′ ))] = [(E ◇ ww ′ E ′′ , F ◇ F ′′ )] + [(E ′ ◇ id E ′ , F ′ ◇ F ′ )]. The result follows from (3). Construction of ρ z ∈ KK 1,Γ R Z (A, A) whose image in KK Γ (A, A) is [α] − n1 Γ A . We now construct a class ρ A α ∈ KK 1,Γ R Z (A, A) whose image in KK 1,Γ R Z (A, A) is 1 Γ A,R ⊗ [α] Γ − n1 Γ A,R . The important fact in our construction is that the element ρ A α is independent of all choices. The assumption (KFP) says that there exists a II 1 -factor N (with trivial action of Γ), with a tracial morphism λ = λ N ∶ C * Γ → N , i.e. a morphism such that tr N ○ λ = tr (where tr N is the normalized trace of N ) satisfying 1 Γ A ⊗ [i] = 1 Γ A ⊗ [P λ ] where i = i N ∶ C → N is the unital morphism and P λ is the Γ-equivariant Hilbert C, Nbimodule N with the natural Hilbert N -module structure, and where g ∈ Γ acts by left multiplication by λ(g). Under the isomorphism KK Γ (C, N ) = KK(C * Γ, N ) the class [P λ ] corresponds to the class of the morphism λ. There exists a II 1 -factor M containing N with a morphism C(U n ) ⋊ Γ → M extending the morphism λ ∶ C * Γ → N ⊂ M . Indeed we can take as M (a II 1 -factor containing) the free product of N with L ∞ (U n ) ⋊ Γ with amalgamation over the group von Neumann algebra of Γ. Up to replacing N by M , it follows, thanks to Lemma 2.5, that we may assume that there exists a unitary u ∈ N ⊗ M n (C) satisfying (λ g ⊗ α g )u = u(λ g ⊗ 1) ∈ N ⊗ M n (C) for all g ∈ Γ. Denote by V α the equivariant (C, C) bimodule C n where Γ acts through the action α. Denote also by V n the equivariant (C, C) bimodule C n where Γ acts trivially. We have a unitary equivalence 1 0) and (A ⊗ P λ , 0). More precisely the equivariant Kasparov bimodule (E, F ) is such that: A ⊗ u ∈ L(A ⊗ P λ ⊗ V n , A ⊗ P λ ⊗ V α ) between the Γ-equivariant Kasparov (A, A ⊗ N ) bimodules (A ⊗ P λ ⊗ V n , 0) and (A ⊗ P λ ⊗ V α , 0). Let i ∶ C → N be the unital inclusion. Denote byp 0 ∶ A ⊗ Z i → A and p 1 ∶ A ⊗ Z i → A ⊗ N the maps id A ⊗ev 0 and id A ⊗ev 1 , where ev 0 ∶ Z i → C is the map (a, f ) ↦ a and ev 1 ∶ Z i → N is the map (a, f ) ↦ f (1). As 1 Γ A ⊗ [i] = 1 Γ A ⊗ [P λ ], there exists an equivariant Kasparov-bimodule (E, F ) ∈ E Γ (A, A ⊗ Z i ) joining (A, • the induced equivariant bimodule E ⊗p 0 A is the (A, A)-bimodule A (endowed with the natural Γ-action -and F 0 = 0); • the induced equivariant bimodule E ⊗ p 1 (A ⊗ N ) is the equivariant (A, A ⊗ N )bimodule A ⊗ P λ . We will say that a "proof of the property (KFP)" for A consists of N, λ, (E, F ) where: • N is a II 1 -factor with normalized trace tr N ; • λ ∶ C * Γ → N , is a morphism such that tr N ○ λ = tr; • (E, F ) ∈ E Γ (A, A ⊗ Z i ) is a Kasparov bimodule joining (A, 0) and (A ⊗ P λ , 0). Theorem 2.7. (1) Let (E, F ) ∈ E Γ (A, A ⊗ Z i ) be a Kasparov bimodule joining (A, 0) and (A ⊗ P λ , 0). The class in KK Γ (A, A ⊗ Z i,i ) of (E ⊗ V α , F ⊗ 1) ◇ id A ⊗u (E ⊗ V n , F ⊗ 1) does not depend on the unitary u ∈ N ⊗ M n (C) satisfying (λ g ⊗ α g )u = u(λ g ⊗ 1) for all g ∈ Γ. (2) The image of this class in KK 1,Γ R Z (A, A) through the composition ϑ ∶ KK Γ (A, A ⊗ Z i,i ) Θ → KK Γ (A, A ⊗ C i ) → KK 1,Γ R Z (A, A) does not depend: (a) on (E, F ) ∈ E Γ (A, A ⊗ Z i ) such that E ⊗ A⊗Z i A = A and E ⊗ A⊗Z i A = A ⊗ P λ ; (b) on the II 1 -factor N containing a tracial morphism of λ N ∶ C * Γ → N and a unitary u ∈ N ⊗ M n (C) satisfying (i) 1 Γ A ⊗ [i] = 1 Γ A ⊗ [λ N ] (ii) (λ g ⊗ α g )u = u(λ g ⊗ 1) ∈ N ⊗ M n (C) for all g ∈ Γ. Proof. (1) Choose a u ∈ N ⊗ M n (C) such that (λ g ⊗ α g )u = u(λ g ⊗ 1) ∈ N ⊗ M n (C) for all g ∈ Γ. Then an element v ∈ N ⊗ M n (C) satisfies the same property if and only if it is of the form uw where w is a unitary of the commutant of λ(Γ)⊗1 in N ⊗M n (C). But the set of unitaries in the von Neumann algebra (N ∩ λ(G) ′ ) ⊗ M n (C) is connected. (2) (a) Let (E ′ , F ′ ) ∈ E Γ (A, A ⊗ Z i ) be another Kasparov bimodule joining (A, 0) and (A⊗P λ , 0). Choose a u ∈ N ⊗M n (C) such that (λ g ⊗α g )u = u(λ g ⊗1) ∈ N ⊗M n (C) for all g ∈ Γ and putǔ = id A ⊗u. By Proposition 2.6(3) and 2.6(4), the image by Θ of (E ⊗ V α , F ⊗ 1) ◇ǔ (E ⊗ V n , F ⊗ 1) is opposite to the one of (E ⊗ V n , F ⊗ 1) ◇ǔ * (E ⊗ V α , F ⊗ 1); moreover, by Proposition 2.6(4), the image by Θ of [(E ′ ⊗ V α , F ′ ⊗ 1) ◇ǔ (E ′ ⊗ V n , F ′ ⊗ 1)] + [(E ⊗ V n , F ⊗ 1) ◇ǔ * (E ⊗ V α , F ⊗ 1)] coincides with that of [(E ′ ⊗ V α , F ′ ⊗ 1) ◇ id (E ⊗ V α , F ⊗ 1)] + [(E ⊗ V n , F ⊗ 1) ◇ id (E ′ ⊗ V n , F ′ ⊗ 1)]. Using again 2.6(3) and 2.6(4), we may replace [(E⊗V n , F ⊗1)◇ id (E ′ ⊗V n , F ′ ⊗1)] by −[(E ′ ⊗ V n , F ′ ⊗ 1) ◇ id (E ⊗ V n , F ⊗ 1)]. We end up with Θ([(E, F ) ◇ id (E ′ , F ′ )] ⊗ C ([V α ] − [V n ])). Now, since (p 0 ) * [(E, F ) ◇ id (E ′ , F ′ )] = (p 0 ) * [(E, F )] = 1 A = (p 0 ) * [(E ′ , F ′ )] = (p 1 ) * [(E, F ) ◇ id (E ′ , F ′ )], it follows that Θ([(E, F ) ◇ id (E ′ , F ′ )]) is in the image of an element x ∈ KK Γ (A, A ⊗ SN ) via the inclusion h ∶ SN → C i . As ([V α ] − [V n ])) is the 0 element in KK Γ R (A, A) it follows that the image of x ⊗ C ([V α ] − [V n ])) in KK 1,Γ R (A, A) is the zero element. (b) Let i M,N ∶ N → M be a morphism of II 1 -factors. Denote by i M,N ∶ Z i N → Z i M and i M,N ∶ Z i N ,i N → Z i M ,i M the corresponding morphism, v = (i M,N ⊗id)(u). Let (E, F ) ∈ E Γ (A, A ⊗ Z i N ) such that E ⊗ A⊗Z i N A = A and E ⊗ A⊗Z i N A = A ⊗ P λ N . Put (E M , F M ) = (E, F ) ⊗ i M,N Z i M . One checks immediately that the image by i M,N ∶ Z i N ,i N → Z i M ,i M of [(E ⊗ V α , F ⊗ 1) ◇ǔ (E ⊗ V n , F ⊗ 1)] is [(E M ⊗ V α , F M ⊗ 1) ◇v (E M ⊗ V n , F M ⊗ 1)]. Definition 2.8. Let A a Γ-algebra satisfying property (KFP). We denote by ρ A α ∈ KK 1,Γ R Z (A, A) the image by the map ϑ ∶ KK Γ (A, A ⊗ Z i,i ) → KK 1,Γ R Z (A, A) of an element [(E ⊗ V α , F ⊗ 1) ◇ǔ (E ⊗ V n , F ⊗ 1)] ∈ KK Γ (A, A ⊗ Z i,i )(A, A) → KK Γ (A, A) is [α] Γ − n1 Γ A . Proof. Let N, u and (E, F ) be a "proof of property (KFP) for A", as in Theorem 2.7. Then, by Remark 1.11, the image of ρ A α = ϑ([(E ⊗ V α , F ⊗ 1) ◇ǔ (E ⊗ V n , F ⊗ 1)]) by this connecting map is (p 0 ) * [(E ⊗ V α , F ⊗ 1) ◇ǔ (E ⊗ V n , F ⊗ 1)]) − (p 1 ) * [(E ⊗ V α , F ⊗ 1) ◇ǔ (E ⊗ V n , F ⊗ 1)]). But (p 0 ) * [(E ⊗ V α , F ⊗ 1) ◇ǔ (E ⊗ V n , F ⊗ 1)]) = (p 0 ) * [(E ⊗ V α , F ⊗ 1)] = 1 Γ A ⊗ [α] Γ and (p 1 ) * [(E ⊗ V α , F ⊗ 1) ◇ǔ (E ⊗ V n , F ⊗ 1)]) = (p 0 ) * [(E ⊗ V n , F ⊗ 1)] = n.1 Γ A . 2.5. Properties of the construction ρ A α . 2.5.1. Dependence on α. Proposition 2.10. Let A be a Γ-algebra satisfying property (KFP). Let α 1 , α 2 be finite dimensional unitary representations of Γ. We have (1) ρ A α 1 ⊕α 2 = ρ A α 1 + ρ A α 1 (2) ρ A α 1 ⊗α 2 = dim α 1 ⋅ ρ A α 2 + ρ A α 1 ⊗ C [α 2 ] = dim α 2 ⋅ ρ A α 1 + [α 1 ] ⊗ C ρ A α 2 . Proof. The first statement is obvious. For the second one, put n i = dim α i . Let (E, F ) be as in Theorem 2.7. Thanks to Proposition 2.6(4), we may write (E ⊗ V α 1 ⊗α 2 , F ⊗ 1) ◇ (E ⊗ V n 1 n 2 , F ⊗ 1) = (E ⊗ V α 1 , F ⊗ 1) ◇ (E ⊗ V n 1 , F ⊗ 1) ⊗ V α 2 + (E ⊗ V α 2 , F ⊗ 1) ◇ (E ⊗ V n 2 , F ⊗ 1) ⊗ V n 1 and the result follows. α of Γ, we have f * (ρ A α ) = f * (ρ B α ) ∈ KK 1,Γ R Z (A, B) . This is a particular case of the following more general result. ρ A α ⊗ A x = x ⊗ B ρ B α ∈ KK 1,Γ R Z (A, B). Proof. Let (E, F ) ∈ E Γ (A, B) representing x. Let N, λ, (E A , F A ) be a "proof of the property (KFP) for A". As (E A , F A ) ∈ E Γ (A, A⊗Z i ) is a Kasparov bimodule joining A to A⊗P λ , the Kasparov product (E A , F A )⊗ A (E, F ) can be represented by a Kasparov bimodule (Ẽ,F ) ∈ E Γ (A, B ⊗Z i ) such thatp 0 (Ẽ,F ) = (E, F ) and p 1 (Ẽ,F ) = (E ⊗ P λ , F ⊗ 1). It is then immediately checked (using for instance the connexion and positivity approach of [9]) that [(Ẽ ⊗ V α ,F ⊗ 1) ◇ 1 E ⊗u (Ẽ ⊗ V n ,F ⊗ 1)] is a Kasparov product [(E A ⊗ V α , F A ⊗ 1) ◇ 1 E ⊗u (E ⊗ V n , F ⊗ 1)] ⊗ A [(E, F )] ∈ E Γ (A, A ⊗ Z i,i ). Therefore ρ α A ⊗ A x is the image of the class of [(Ẽ ⊗ V α ,F ⊗ 1) ◇ 1 E ⊗u (Ẽ ⊗ V n ,F ⊗ 1)] in KK 1,Γ R Z (A, B). In the same way, let M, λ M , (E B , F B ) be a "proof of the property (KFP) for B". We may of course assume (replacing M and N if necessary by a factor containing the free product amalgamated over the group von Neumann algebra of Γ) that M = N and λ M = λ. In the same way as above, the Kasparov product (E, F ) ⊗ B (E A , F A ) can be represented by a Kasparov bimodule (Ê,F ) ∈ E Γ (A, B ⊗ Z i ) such thatp 0 (Ê,F ) = (E, F ) and p 1 (Ê,F ) = (E ⊗ P λ , F ⊗ 1). It follows that x ⊗ B ρ α B is the image of the class (A, B). To conclude, note that it follows from the proof of Theorem 2.7 (2a) that the image of [(Ê ⊗ V α ,F ⊗ 1) ◇ 1 E ⊗u (Ê ⊗ V n ,F ⊗ 1)] in KK 1,Γ R Z[(Ě ⊗ V α ,F ⊗ 1) ◇ 1 E ⊗u (Ě ⊗ V n ,F ⊗ 1)] in KK 1,Γ R Z (A, B) does not depend on (Ě,F ) ∈ E Γ (A, B⊗Z i ) joining (E, F ) to (E, F )⊗P λ . Examples 3.1. The Mishchenko bundle. As explained in the introduction, our starting example comes from the Mishchenko bundle associated with a Galois cover of a compact manifold. Proposition 3.1. Assume that the group Γ acts freely, properly and cocompactly on a smooth manifoldṼ . Then C 0 (Ṽ ) satisfies property (KFP). Proof. In [1, Proposition 5.1], we noted that there is a II 1 -factor N with a morphism λ ∶ Γ → N such that tr N ○ λ = tr together with a continuous map ψ fromṼ to the unitary group U (N ) of N such that ψ(gx) = λ(g)ψ(x) for all g ∈ G and x ∈Ṽ . It gives rise to a unitary w ∈ L(C 0 (Ṽ ) ⊗ N ) intertwining the Γ-actions β ⊗ id N and β ⊗ λ on C 0 (Ṽ ) ⊗ N where β is the action of Γ on C 0 (Ṽ ) by translation. Now the Hilbert module C 0 (Ṽ ) ⊗ N endowed with the left action of C 0 (Ṽ ) equal to the right one and the Γ-action β ⊗ id N (and β ⊗ λ) represent respectively the images 1 Γ C 0 (Ṽ ) and 1 Γ C 0 (Ṽ ) ⊗ tr in KK Γ R (C 0 (Ṽ ), C 0 (Ṽ )). These equivariant bimodules being unitarily equivalent they represent the same class in KK Γ R (C 0 (Ṽ ), C 0 (Ṽ )). Thus, for every finite dimensional unitary representation α ∶ Γ → U n , we get an element ρ C 0 (Ṽ ) α ∈ KK 1,Γ R Z (C 0 (Ṽ ), C 0 (Ṽ )) . We now explain the relation of this element with the element [α] APS ∈ K 1 R Z (C(V )) constructed by Atiyah Patodi and Singer [3,4]. Let Y be a locally compact space on which Γ acts and A and B be C 0 (Y ) − Γ-algebras. Denote by RKK Γ (Y ; A, B) the Kasparov equivariant KK-theory group defined in [19, §2]. In the language of KK-theory equivariant by groupoids introduced by Le Gall ([20]), RKK Γ (Y ; A, B) = KKṼ ⋊Γ (A, B). If Γ acts freely and properly on Y there is a descent isomorphism (see [19]) λ Γ ∶ RKK Γ (Y ; A, B) → R(Y Γ; A Γ , B Γ ) where A Γ and B Γ are the algebras of invariant elements. In the language of [20], this is the isomorphism corresponding to the Morita equivalent groupoids Y ⋊ Γ and Y Γ. Let X be a compact space, A a C * -algebra and B be a C(X)-algebra. Then we have a canonical isomorphism RKK(X; A ⊗ C(X), B) = KK (A, B). Let Y be a free and proper and cocompact Γ-space. Put X = Y Γ. Let A be a C * -algebra and let B be a C 0 (Y ) − Γ-algebra. We have a sequence of isomorphisms KK(A, B Γ ) ≃ RKK(X; A ⊗ C(X), B Γ ) ≃ RKK Γ (Y ; A ⊗ C 0 (Y ), B). Using the forgetful map RKK Γ (Y ; A ⊗ C 0 (Y ), B) → KK Γ (A ⊗ C 0 (Y ), B) we obtain a mor- phism U Y,Γ A,B ∶ KK(A, B Γ ) → KK Γ (A ⊗ C 0 (Y ), B) . Using the Morita equivalence of C 0 (Y ) ⋊ Γ with C(X) and of B ⋊ Γ with B Γ we also have a sequence of maps KK Γ (A ⊗ C 0 (Y ), B) j Γ → KK(A ⊗ C 0 (Y ) ⋊ Γ, B ⋊ Γ) ≃ KK(A ⊗ C(X), B Γ ) i * X → KK(A, B Γ ) where i X ∶ C → C(X) is the unital inclusion. We thus obtain a morphism: V Y,Γ A,B ∶ KK Γ (A ⊗ C 0 (Y ), B) → KK(A, B Γ ). Lemma 3.2. The composition V Y,Γ A,B ○ U Y,Γ A,B is the identity of KK(A, B Γ ). Proof. Since X = Y Γ is compact, B Γ is the set of multipliers b of B which are Γ-invariant and satisfy f b ∈ B for all f ∈ C 0 (Y ). If E is a Γ-equivariant Hilbert B module, we may in the same way define the Hilbert B Γ module E Γ , for instance by considering E as a subspace of the Γ-C 0 (Y )-algebra K(E ⊕ B) and then E Γ ⊂ K(E ⊕ B) Γ . We start with a pair (E, F ) representing an element in RKK Γ (Y ; A ⊗ C 0 (Y ), B); by properness of the action (or, equivalently using the isomorphism RKK(X; A ⊗ C(X), B Γ ) ≃ RKK Γ (Y ; A ⊗ C 0 (Y ), B)), we may assume F is Γ-invariant. It then defines an element F 0 ∈ L(E Γ ). The elements (E Γ , F 0 ) and (E, F ) correspond to each other via the isomorphism RKK(X; A ⊗ C(X), B Γ ) ≃ RKK Γ (Y ; A ⊗ C 0 (Y ), B). The element corresponding to (E Γ , F 0 ) via the isomorphism RKK(X; A ⊗ C(X), B Γ ) ≃ KK(A, B Γ ) is of course (E Γ , F 0 ) where we just do not consider the left C(X) action ! The element corresponding to (E, F ) under the forgetful morphism RKK Γ (Y ; A⊗C 0 (Y ), B) → KK Γ (A ⊗ C 0 (Y ), B) is of course (E, F ); its image by j Γ is (E ⋊ Γ, F ⊗ 1) where E ⋊ Γ = E ⊗ B (B ⋊ Γ). The Hilbert (C(X) ⊗ A, B Γ ) bimodule corresponding to it under the Morita equivalences, is then seen to be E Γ -and the Lemma follows. According to Remark 1.13, we define RKK Γ R (V ; A, B) and RKK Γ R Z (V ; A, B) using inductive limits over II 1 -factors. The APS element in K 1 R Z (V ) was described in [1] as an element [α] APS ∈ K 0 (Z i,i ⊗C(V )). Proposition 3.3. The element ρ C 0 (Ṽ ) α is the image of [α] APS under the composition of the isomorphisms K 1,R Z (C(V )) ≃ RKK 1 R Z (V ; C(V ), C(V )) ≃ RKK 1,Γ R Z (Ṽ ; C 0 (Ṽ ), C 0 (Ṽ ) ⊗ Z i,i ) with the forgetful map RKK 1,Γ R Z (Ṽ ; C 0 (Ṽ ), C 0 (Ṽ )) → KK 1,Γ R Z (C 0 (Ṽ ), C 0 (Ṽ )). Proof. In [1, Prop. 5.1 and Prop. 5.2], we represented the class [α] APS as the image under the morphism K 0 (C(V )⊗Z i,i ) → K 1,R Z (C(V )) of the C(V )⊗Z i,i module E = {f ∈ C([0, 1], E + ⊗ N ); f (0) ∈ E + ⊗ 1 , w v f (1) ∈ E − ⊗ 1} where N is a II 1 -factor with tracial inclusion λ ∶ Γ → N , i ∶ C → N is the unital inclusion, E + is the flat vector bundle associated with α and E − is the trivial vector bundle with rank equal to the dimension of α. The unitary w v is given by w v ∶= (ϕ ⊗ 1 E − ) −1 ○ v −1 ○ (ϕ ⊗ 1 E + ) where ϕ ∶ C(V ) ⊗ N → E is an isomorphism, i.e. a trivialization of the flat bundle E associated with λ and v is an isomorphism of flat bundles E ⊗ E − → E ⊗ E + . At the level ofṼ , the trivialization ϕ corresponds to a Γ-equivariant isomorphismφ ∶ V ⊗ N →Ṽ ⊗ P λ and v to an unitary u ∈ N ⊗ M n (C) intertwining λ ⊗ 1 and λ ⊗ α (P λ is as in section 2.4). Let thenÊ denote the C 0 (Ṽ ) ⊗ Z i module {f ∈ C 0 (Ṽ × [0, 1]) ⊗ P λ f (0) ∈φ(C 0 (Ṽ ) ⊗ 1)}. The pair (Ê, 0) defines a Kasparov bimodule in E Γ (C 0 (Ṽ ), C 0 (Ṽ ) ⊗ Z i ) joining (C 0 (Ṽ ), 0) and (C 0 (Ṽ ) ⊗ P λ , 0). We may then perform the "◇" construction as in Theorem 2.7 using the isomorphism u to obtain an element in z ∈ KK Γ (C 0 (Ṽ ), C 0 (Ṽ ) ⊗ Z i,i ) whose image in KK 1,Γ R Z (C 0 (Ṽ ), C 0 (Ṽ )) is ρ C 0 (Ṽ ) α . The element z is of course given by the Hilbert C 0 (Ṽ ) ⊗ Z i,i -modulẽ E ∶= {f ∶ [0, 1] ×Ṽ → C n ⊗ N ∶ f (0, x) ∈ C n × 1 N , w v (x)f (1, x) ∈ C n × 1 N , for every x ∈Ṽ , where w v = (φ ⊗ 1) −1 ○ u −1 ○ (φ ⊗ 1). As the left and right C 0 (Ṽ ) actions onẼ coincide, the element z is in the image of the forgetful map, RKK Γ (Ṽ ; C 0 (Ṽ ), C 0 (Ṽ ) ⊗ Z i,i ) → KK Γ (C 0 (Ṽ ), C 0 (Ṽ ) ⊗ Z i,i ). NowẼ Γ = E, If A is proper and free. Out of the example of the Mishchenko bundle of a compact manifold, we then obtain the following statements. Let Z be a locally finite CW complex realization for the classifying space BΓ and q ∶Z → Z the corresponding covering with group Γ. Write Z = ⋃ Z n , where Z n is an increasing sequence of finite subcomplexes. Denote by N a II 1 -factor containig Γ in a trace preserving way, and E N =Z × Γ N the corresponding bundle over Z with fiber N . (1) The restriction to Z n of the bundle E N is equal in K 0 R (Z n ) to the unit element (corresponding to the trivial bundle). (2) IfX is a free, proper and cocompact Γ-space, then there is a II 1 -factor N with an embedding λ ∶ C * Γ → N such that tr N ○ λ = tr and a unitary w ∈ L(C 0 (X) ⊗ N ) intertwining the Γ-actions β ⊗ id N and β ⊗ λ on C 0 (X) ⊗ N where β is the action of Γ on C 0 (X) by translation. Proof. (1) Since Z n is a finite CW-complex, its K-theory is a finitely generated group. Therefore, using the Rosenberg-Schochet universal coefficient formula ( [27,26]), we find KK R (C 0 (Z n ), C) = Hom(KK(C, C 0 (Z n )); R). Using the Baum-Douglas theory (cf. [6]), we know that the K-homology of Z n is generated by cycles given by elements g * (x) where g is a continuous map g ∶ V → Z n , V is a compact manifold and x ∈ K * (V ) is an element in the K-homology of V . But by Atiyah's theorem for covering spaces ( [2]), the pairing of E N with such a K-homology element coincides with the index. It follows that the class of E N is 1. (2) We have a continuous classifying map f ∶ X → BΓ where X =X. Since X is compact its image sits in some Z n and it follows that the bundle f * (E N ) defines the trivial element in K 0 R (X). Up to replacing N by M k (N ), we may then assume that the bundle f * E N is trivial (as explained in [1,Prop. 5.1]). By definition of E N , this means exactly that there exists a unitary w ∈ L(C 0 (X) ⊗ N ) intertwining the Γactions β ⊗ id N and β ⊗ λ on C 0 (X) ⊗ N . We now recall the definition of free and proper Γ-algebras. Recall ( [19]) that, if X is a locally compact space, a C 0 (X) algebra is a C * -algebra A endowed with a morphism from C 0 (X) into the center of the multiplier algebra of A and such that A = C 0 (X)A. If X is a Γ space, a C 0 (X) − Γ-algebra A is an algebra endowed with compatible structures of C 0 (X)-algebra and Γ-algebra (i.e. the morphism C 0 (X) → ZM(A) is equivariant). Let us recall facts about C 0 (X)-algebras (see [19,20]): Properties 3.5. (1) If A is a C 0 (X)-algebra, we may define for every open subset U ⊂ X the C 0 (U )-algebra A U = C 0 (U )A. For every closed subset F ⊂ X, we put A F = A A F c , which is a C 0 (F ) algebra. In particular, we have a fiber A x for every point x ∈ X. Moreover, there is a natural evaluation map a ↦ a x from A to A x and for a ∈ A, we have a = sup{ a x , x ∈ X}. (2) If A is a C 0 (X)-algebra and f ∶ Y → X is a continuous map, we define a pull back f * (A) which is a C 0 (Y )-algebra. This pull back satisfies f * (A) y = A f (y) for all y ∈ Y . We will use the following facts: (a) If T is a locally compact space and p ∶ X × T → X is the projection then p * (A) = A ⊗ C 0 (T ). (b) If h ∶ Y → Y is a homeomorphism such that f ○ h = f , there is an automorphism θ h ∶ f * (A) → f * (A) such that (θ h (b)) y = b h(y) for every y ∈ Y . Note that by property (1) this equality characterizes θ h (b). Definition 3.6. A Γ-algebra A is said to be free (resp. proper ) if there exists a free (resp. proper) Γ-spaceX such that A is a C 0 (X) − Γ-algebra. Note that if A is free and proper, then there exist a free Γ-spaceX 1 and a proper Γ-spacẽ X 2 such that A is a C 0 (X 1 )−Γ-algebra and a C 0 (X 2 )-Γ-algebra. Then A is a C 0 (X 1 ×X 2 )−Γalgebra, andX 1 ×X 2 is free and proper. We will next prove that every free and proper Γ-algebra is K-theoretically free and proper (Theorem 3.10). Let us start with the cocompact case: Proposition 3.7. IfX is a free, proper and cocompact Γ-space, then every C 0 (X)−Γ-algebra satisfies property (KFP). Proof. Let A be a C 0 (X)−Γ-algebra. Extending the action by left multiplication of C 0 (X)⊗ N on A⊗N to the multiplier algebra, we find a unitaryŵ ∈ L(A⊗N ) intertwining the actionŝ β ⊗ id N andβ ⊗ λ of Γ. It follows that A satisfies property (KFP). As above, let Z be a locally finite CW complex realization for the classifying space BΓ. Write Z = ⋃ Z n , where Z n is an increasing sequence of finite subcomplexes. Using local finiteness, we will also assume that Z n is contained in the interior of Z n+1 . LetX be a a free and proper Γ-space and let A be a (separable) C 0 (X) − Γ-algebra. Put X =X Γ and let q ∶X → X be the quotient. Let also f ∶ X → Z be a classifying map. DefineX n = (f ○ q) −1 (Z n ) ⊂X and Ω = ⋃ n∈NXn ×]n, +∞[. shift (a 0 , a 1 , a 2 , . . .) ↦ (0, a 0 , a 1 , a 2 , . . .). Let A = C 0 (Ω)(A ⊗ C 0 (R)) and A n = C 0 (X n )A. Let A denote the c 0 -sum ⊕ n∈N A n and j ∶ A → A the Lemma 3.8. (1) The Γ-algebras A ⊗ C 0 (R) and A are homotopy equivalent. Proof. (1) Of course A ⊗ C 0 (R) and A ⊗ C 0 (R * + ) are homotopy equivalent. Moreover let h ∶ Z → R + be a (proper) map such that h(z) > n if z ∈ Z n , and putĥ = h ○ f ○ q ∶ X → R * + . The map (x, t) ↦ (x, t +ĥ(x)) is a homeomorphismX × R →X × R. It induces an equivariant automorphism of A ⊗ C 0 (R) mapping A ⊗ C 0 (R * + ) into A which is a homotopy inverse of the inclusion A → A ⊗ C 0 (R * + ). (2) An element in A ⊗ C 0 (R * + ) is a map ξ ∶ R * + → A. Such a ξ is in A if and only if, for all n ∈ V and t ∈]n, n + 1], we have ξ(t) ∈ A n for n < t ≤ n + 1. Associated to ξ is a map ζ ∶ [0, 1] → A defined by ζ(t) n = ξ(n + t) (∈ A n ). It is immediately seen that ζ ∈ T (j, id A ) and that ξ ↦ ζ is an isomorphism. Lemma 3.9. (1) For every n, the algebra A n satisfies property (KFP). Proof. (1) Through the equivariant mapX n →Z n , the algebra A n is aZ n algebra. As Z n is a free proper and cocompact Γ-space, the algebra A n satisfies property (KFP) by prop.3.7. (2) For every Γ-algebra B the map KK Γ R (A, B) ∏ n KK Γ R (A n , B) is an isomorphismthe inverse map associates to a sequence (E n , F n ) of bimodules (with F n bounded) defining elements in KK Γ R (A n , B) the class of (⊕ E n , F ) where F is defined by F ((x n ) n ) = (F n (x n )) n (for (x n ) ∈ ⊕ n E n ) (see [26,27] (A, A). (3) Write A = T (j, id A ) and let p ∶ A → A and h ∶ SA → A be the associated map maps (p ∶ (a, f ) ↦ a and h ∶ f ↦ (0, f )). We have p * (1 Γ A ⊗ C (1 Γ R − [tr] Γ )) = p * (1 Γ A ⊗ C (1 Γ R − [tr] Γ )) = 0 since A satisfies property (KFP). By the torus exact sequence (prop. 1.2), there exists y ∈ KK Γ R (A, SA) such that 1 Γ A ⊗ C (1 Γ R − [tr] Γ ) = h * y. As A satisfies property (KFP), we have y ⊗ SA (1 Γ SA ⊗ C (1 Γ R − [ tr] Γ )) = 0. As the Kasparov product over C is commutative, we find y ⊗ SA (1 Γ SA ⊗ C (1 Γ R − [tr] Γ )) = (1 Γ A ⊗ C (1 Γ R − [tr] Γ )) ⊗ A y whence (1 Γ A ⊗ C (1 Γ R − [tr] Γ )) ⊗ A h * y, which yields 1 Γ A ⊗ C (1 Γ R − [tr] Γ ) ⊗ C (1 Γ R − [tr] Γ ) = (1 Γ A ⊗ C (1 Γ R − [tr] Γ )) ⊗ A (1 Γ A ⊗ C (1 Γ R − [tr] Γ )) = 0. But [tr] Γ is idempotent in KK Γ R (C, C), thus (1 Γ R − [tr] Γ ) 2 = (1 Γ R − [tr] Γ ). (4) The Γ-algebra SA is homotopy equivalent to A. It follows that SA satisfies property (KFP); thus 1 Γ SA ⊗ C (1 Γ R −[tr] Γ ) = 0. The Bott isomorphism x ↦ Sx from KK Γ R (C, D) to KK Γ R (SC, SD) maps 1 Γ A ⊗ C (1 Γ R − [tr] Γ ) to 1 Γ SA ⊗ C (1 Γ R − [tr] Γ ) = 0. It follows that A satisfies property (KFP). We have proved: Theorem 3.10. Every free and proper Γ-algebra satisfies property (KFP). ◻ As a corollary, every Γ-algebra which is KK-subequivalent to a proper and free algebra satisfies property (KFP). 3.3. If Γ satisfies (the KK Γ -form of ) the Baum-Connes conjecture. Let us say that Γ satisfies the KK Γ -form of the Baum-Connes conjecture if there is a proper Γ-algebra Q such that C is KK Γ -subequivalent to Q. This of course implies that Γ is K-amenable ( [10]). On the other hand, Higson and Kasparov proved that every A-T -menable group satisfies this property ( [16,17]). If Γ satisfies the KK Γ -form of the Baum-Connes conjecture every Γ-algebra is KK Γsubequivalent to a proper Γ-algebra (namely A is KK Γ -subequivalent to Q ⊗ A). • If Γ is torsion free. Then Q is automatically free. Then every Γ-algebra satisfies property (KFP). • In the general case. Assume A is free ( 1 ), then Q ⊗ A is free and proper and therefore satisfies property (KFP). It follows that every free Γ-algebra satisfies property (KFP). 3.4. If Γ has a γ element. Recall (cf. [19,30]) that a γ-element for Γ is an element γ ∈ KK Γ (C, C) such that: (1) There exists a proper Γ-algebra Q and elements D ∈ KK Γ (Q, C) and η ∈ KK Γ (C, Q) such that γ = η ⊗ Q D. D and η are respectively the so called Dirac and dual Dirac element; (2) for every proper Γ-algebra A, 1 Γ A ⊗ C γ = 1 Γ A . Recall (cf. [30]), that if γ exists, it is unique. • If Γ is torsion free and has a γ element, then the algebra Q is free and proper, therefore it satisfies property (KFP). It follows that, for every finite dimensional unitary representation α ∶ Γ → U n , we can construct a canonical element ρ Q α ∈ KK 1,Γ R Z (Q, Q) and use the element η and D in order to define a canonical element ρ C α = η ⊗ Q ρ Q α ⊗ Q D ∈ KK 1,Γ R Z (C, C) in "the image of γ". Then for every Γ algebra, we construct ρ A α = 1 Γ A ⊗ C ρ C α . • In the same way as above, if Γ is no longer assumed to be torsion free, we may construct the element ρ Q⊗A α if A is free -and more generally if Q ⊗ A is free. We then use the Dirac and dual Dirac elements to construct a canonical element ρ A α = η ⊗ Q ρ Q⊗A α ⊗ Q D ∈ KK 1,Γ R Z (A, A) in "the image of γ". 3.5. If α(Γ) is torsion free. PutΓ = α(Γ) (with the discrete topology). SinceΓ is linear, it follows from [15] that it has a γ element. We may thus define an element ρ Č α ∈ KK 1,Γ R Z (C, C) whereα is the inclusionΓ ⊂ U n . Then we can use the morphism q ∶ Γ →Γ in order to define ρ C α = q * ρ Č α (and then put, for every Γ-algebra ρ A α = 1 Γ A ⊗ C ρ C α ). Weak (KFP) property 4.1. KK-theory elements with real coefficients associated with a trace on C * Γ. As explained in Lemma 1.14, for every Γ-algebra A and every C * -algebra B, we may identify KK(A ⋊ Γ, B) with KK Γ (A, B) where B is endowed with the trivial action of B. Replacing in this formula B by B ⊗ M where M is a II 1 -factor and taking the inductive limit, we find for a Γ-algebra A and a C * -algebra B endowed with a trivial Γ-action an identification [δ A τ ] ∈ KK R (A ⋊ Γ, A ⋊ Γ): (1) We may put [δ A τ ] = τ * [δ A ] where δ A ∈ KK(A ⋊ Γ, A ⋊ Γ ⊗ C * Γ) is the class of the morphism δ A and τ * ∶ KK(B, C ⊗ C * Γ) → KK R (B, C) is the map associated with the trace τ constructed in Definition 1.7. (2) Starting with [τ ] ∈ KK R (C * Γ, C), we obtain (by tensoring with A ⋊ Γ) an element (1 A⋊Γ ⊗ [τ ]) ∈ KK R (A ⋊ Γ ⊗ C * Γ, A ⋊ Γ) and, with this notation, we have [δ A τ ] = (δ A ) * (1 A⋊Γ ⊗ [τ ]) ∈ KK R (A ⋊ Γ, A ⋊ Γ). (3) Let as usual [τ ] Γ ∈ KK Γ R (C, C) be the element corresponding to [τ ] via the identifica- tion of KK R (C * Γ, C) with KK Γ R (C, C); the element 1 Γ A ⊗ [τ ] Γ ∈ KK Γ R (A, A) satisfies [δ A τ ] = j Γ (1 Γ A ⊗ [τ ] Γ ) where j Γ ∶ KK Γ R (A, A) → KK R (A ⋊ Γ, A ⋊ Γ) is Kasparov's descent. 4.2. Weakly K-theoretically "free and proper" algebras. We now introduce a weakening of (KFP) property Definition 4.2. Let Γ be a discrete group. Denote by tr its group tracial state. We say that the Γ-algebra A satisfies the weak (KFP) property if [δ A tr ] is equal to the unit element 1 A⋊Γ,R of the ring KK R (A ⋊ Γ, A ⋊ Γ). This conditions means that there is a II 1 -factor N with a morphism λ ∶ C * Γ → N such that tr N ○ λ = tr (where tr N is the normalized trace of N ) and such that the maps δ A λ = (id A⋊Γ ⊗λ) ○ δ A ∶ A ⋊ Γ → (A ⋊ Γ) ⊗ N and ι A = id ⊗1 N ∶ A ⋊ Γ → (A ⋊ Γ) ⊗ N define the same element in KK(A ⋊ Γ, A ⋊ Γ ⊗ N ). Let us make a few comments on this definition: Properties 4.3. (1) If a Γ-algebra A satisfies property (KFP), it satisfies the weak (KFP) property. The assumption means that there exist x ∈ KK Γ (A, B) and y ∈ KK Γ (B, A) satisfying Indeed, if 1 Γ A ⊗ [tr] Γ = 1 Γ A,R , then [δ A tr ] = j Γ (1 Γ A ⊗ [tr] Γ ) = j Γ (1 Γ A,R ) = 1 A⋊Γ,R . (2) If C satisfies the weak (KFP) property, it satisfies property (KFP). Indeed, ε * [δ A tr ] = [tr] ∈ KK R (C * Γ, C) = KK Γ R (C, C).x ⊗ B y = 1 Γ A . Note that y ⊗ C [tr] Γ = [tr] Γ ⊗ C y = (1 Γ B ⊗ [tr] Γ ) ⊗ B y. Assume B satisfies the weak (KFP) property. Then j Γ (y ⊗ C [tr] Γ ) is the image in KK R (B ⋊ Γ, A ⋊ Γ) of j Γ (y), whence [δ A tr ] = j Γ (x) ⊗ B⋊Γ j Γ (y ⊗ C [tr] Γ ) = 1 A⋊Γ,R . Remarks 4.4. (1) (see Remark 2.3) If A is satisfies the weak (KFP) property, then [δ A τ ] = 1 A⋊Γ,R for any tracial state τ . (2) (see Remark 2.4) Given a von Neumann algebra N and a morphism λ ∶ C * Γ → N such that tr N ○ λ = tr (where tr N is the normalized trace of N ), then λ factors through C * red (Γ); it therefore defines a morphism δ A λ ∶ A ⋊ red Γ → A ⋊ Γ ⊗ N . Let q Γ ∶ A⋊Γ → A⋊ red Γ be the natural morphism. If A satisfies the weak (KFP) property, then q * Γ [ δ A λ ] = 1 A⋊Γ,R i.e the morphism q Γ ∶ A ⋊ Γ → A ⋊ red Γ has a KK R one sided inverse. 4.3. A construction involving coactions and torus algebras. Throughout this section, the following objects will be fixed: Γ is a discrete group, A is a Γ-algebra, α is a finite dimensional unitary representation of Γ. Moreover, we fix a II 1 -factor N and a morphism λ ∶ C * Γ → N such that tr N ○ λ = tr, where as above tr N denotes the normalized trace of N and tr is the group trace of Γ. We fix two morphisms ι A , δ A λ ∶ A ⋊ Γ → A ⋊ Γ ⊗ N where ι A ∶ x ↦ x ⊗ 1 and δ A λ uses the "coaction" of Γ and is defined in Notation 4.5 below. and g 2 (t) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 0 for t ≤ 1 2 (j A n ⊗ i M,N )f (2 − 2t) for t ≥ 1 2. Both are morphisms from A ⋊ Γ ⊗ SN → A ⋊ Γ ⊗ M n (C) ⊗ SN composed with the inclusion i M,N . Since the ranges of g 1 and g 2 are orthogonal, the sum as morphisms correspond to the sum of the KK-classes. Now the class of g 2 is the class of the morphism Sδ A α ∶ A⋊Γ⊗SN → A ⊗ Γ ⊗ SN induced on the suspension and [Sδ A α ] = 1 SN ⊗ [δ A α ]. The class of g 1 is easily seen to be the opposite of the class of the morphism Sj A n , for which again [Sj A n ] = 1 SN ⊗ [j A n ]. In Remark 1.3, we constructed (using u) a bimodule yielding a Morita equivalence between the torus algebras Z(i n , i n ) and Z(i n , i n ). The KK-class of this bimodule is an element in KK(Z(i n , i n ), Z(i n , i n )). We denote by [E u ] its image as a class in KK(Z(i n , i n ), Z(i, i)), where we use the fact that Z(i n , i n ) = M n ⊗ Z i,i , where i is the inclusion i ∶ C → N . Define (1) We have ψ M α ∶= [∆ u α ] ⊗ [E u ] ∈ KK(T , A ⋊ Γ ⊗ Z i,i ) .(p 0 ) * (ψ M α ) = [δ A α ○ p], (4.2) (p 1 ) * (ψ M α ) = [j A n ○ p] h * (ψ M α ) =h * ([δ A α ] − [j A n ]) ⊗ [S i M,N ] .A α = ϑ(ψ M α ) ∈ KK 1 R Z (T (ι A , δ A λ ), A ⋊ Γ) where ϑ ∶ KK(T (ι A , δ A λ ), A ⋊ Γ ⊗ Z i,i ) → KK 1 R Z (T (ι A , δ A λ ), A ⋊ Γ) is the morphism described in Remark 1.11. Proposition 4.9. (1) The class σ A α doesn't depend on the choices involved in this construction. (2) The image ∂(σ A α ) ∈ KK(T (ι A , δ A λ ), A ⋊ Γ) through the connecting map of the Bockstein exact sequence is ∂(σ A α ) = p * ([δ A α ] − n1 A⋊Γ ). (3) The image h * (σ A α ) ∈ KK 1 R Z (A⋊ Γ ⊗ SN, A⋊ Γ) is the image of ([δ A α ]− n1 A⋊Γ )⊗ 1 SN ∈ KK(A ⋊ Γ ⊗ SN, A ⋊ Γ ⊗ SN ) = KK 1 (A ⋊ Γ ⊗ SN, A ⋊ Γ ⊗ N ) through the composition KK 1 (A ⋊ Γ ⊗ SN, A ⋊ Γ ⊗ N ) → KK 1 R (A ⋊ Γ ⊗ SN, A ⋊ Γ) → KK 1 R Z (A ⋊ Γ ⊗ SN, A ⋊ Γ). Proof. (1) We know that ψ M α does not depend on the choice of u. Furthermore, it behaves well under trace preserving inclusions of II 1 -factors. Since KK 1 R Z (⋅, ⋅) is defined by the limit (1.2) with respect to all these inclusions, every choice of M defines the same class. (2) It follows from Remark 1.11 and Proposition 4.6, in fact: ∂(σ M α ) = (∂ ○ ϑ)(ψ M α ) = [δ A α ○ p] − [j A n ○ p] = p * ([δ A α ] − 1 A⋊Γ ), since [j A n ] = n1 A⋊Γ . (3) It is a consequence of (4.3), because the inclusion SN ↪ C i induces the change of coefficients R → R Z. Remark 4.10. Let N 1 ⊂ N be a sub-factor of N containing λ(Γ). The torus algebra T (ι A , δ A λ ) associated with N 1 is a subalgebra to the one associated with N . Obviously, the element σ A α associated with N 1 is just the restriction of the one of N . Construction ofρ A α . Let A be a Γ-algebra satisfying the weak (KFP) property, and α a finite dimensional unitary representation of Γ. By Remark 4.4, [δ A α ] R = n1 R A⋊Γ where 1 R A⋊Γ denotes the unit element of the ring KK R (A ⋊ Γ, A ⋊ Γ). Using the Bockstein change of coefficients exact sequence, it follows that there exists z ∈ KK 1 R Z (A ⋊ Γ, A ⋊ Γ) whose image in KK(A ⋊ Γ, A ⋊ Γ) is [δ A α ] − n1 A⋊Γ . We now construct a classρ A α ∈ KK 1 R Z (A ⋊ Γ, A ⋊ Γ) whose image in KK(A ⋊ Γ, A ⋊ Γ) is [δ A α ] − n1 A⋊Γ . The important fact in our construction, as for the one given in Definition 2.8 for the stronger context, is that the elementρ A α is independent of all choices. Using the torus T (ι A , δ A λ ) and the corresponding exact sequence (see Section 1.2), this means exactly that there exists a class y ∈ KK(A ⋊ Γ, T (ι A , δ A λ )) with p * (y) = 1 A⋊Γ where p is the projection p ∶ T (ι A , δ A λ ) → A ⋊ Γ. Definition 4.11. We denote byρ A α ∈ KK 1 R Z (A ⋊ Γ, A ⋊ Γ) the product y ⊗ T (ι A ,δ A λ ) σ A α . (1) We have ∂ρ A α = y ⊗ T (ι A ,δ A λ ) ∂(σ A α ) = y ⊗ T (ι A ,δ A λ ) p * ([δ A α ]−n1 A⋊Γ ) by Proposition 4.9. Whence ∂ρ A α = p * y ⊗ A⋊Γ ([δ A α ] − n1 A⋊Γ ) = [δ A α ] − n1 A⋊Γ since p * y = 1 A⋊Γ . (2) Fix first N . We show that if z ∈ KK(A ⋊ Γ, T (ι A , δ A λ )) satisfies p * (z) = 0 then z ⊗ T (δ A λ ,ι A ) σ A α = 0. Using the torus exact sequence, it is enough to show that for every z ∈ KK(A ⋊ Γ, A ⋊ Γ ⊗ SN ) we have h * (z) ⊗ T (ι A ,δ A λ ) σ A α = z ⊗ A⋊Γ⊗SN h * (σ A α ) = 0. But, as A satisfies the weak (KFP) property, [δ A α ] − n1 A⋊Γ is the 0 element in KK R (A ⋊ Γ, A ⋊ Γ) therefore, by Proposition 4.9, h * (σ A α ) = 0. This shows that, given N ,ρ A α is independent on y. Let N 1 be a II 1 -factor containing N . Denote by j ∶ N → N 1 the inclusion. Put λ 1 = j ○ λ ∶ C * Γ → N 1 , ι 1 A = (id A⋊Γ ⊗j) ∶ A ⋊ Γ → A ⋊ Γ ⊗ N 1 and p 1 ∶ T (ι 1 A , δ A λ 1 ) → A ⋊ Γ the morphism (a, f ) ↦ a. Put y 1 = j * (y) ∈ T (ι 1 A , δ A λ 1 ). It follows from remark 4.10 that (N 1 , y 1 ) and (N, y) define the same elementρ A α . Using amalgamated free products, we then see thatρ A α is also independent on N . Remark 4.13. If A satifies property (KFP), it is in fact quite easy to compare the two constructions of ρ A α of Definition 2.8 and the one we just constructed, by showing that ρ A α = j Γ (ρ A α ), where j Γ ∶ KK Γ,1 R Z (A, A) → KK 1 R Z (A⋊Γ, A⋊Γ) is Kasparov's descent morphism. Indeed, the Kasparov descent of a Kasparov bimodule (E, F ) ∈ E Γ (A, A⊗Z i ) joining (A, 0) and (A ⊗ P λ ) is an element y ∈ KK(A ⋊ Γ, T (ι A , δ A λ )) as above; the gluing construction "◇" corresponds to the bimodule ψ M α . It follows that the image by j Γ of (the class of) the bimodule (E ⊗ V α , F ⊗ 1) ◇ id A ⊗u (E ⊗ V n , F ⊗ 1) is y ⊗ T (ι A ,δ A λ ) ψ M α . Taking the images in KK 1 R Z , the result follows. [α] APS can be constructed as a secondary invariant built as a consequence of the triviality of the trace in Atiyah's L 2 -index theorem. Paolo Antonini has received funding from the European Research Council (E.R.C.) under European Union's Seventh Framework Program (FP7/2007-2013), ERC grant agreement No. 291060; Georges Skandalis is funded by ANR-14-CE25-0012-01. We thank these institutions for their support. [α] Γ ∈ KK Γ (C, C) be the class of a unitary representation α ∶ Γ → U n (C) in theKasparov representation ring. Since the regular representation of Γ absorbs, we have [α] Γ ⊗ [tr] Γ = n.[tr] Γ . If A is a Γ-algebra with property (KFP), then the difference 1 Γ A ⊗[α] Γ −n1 Γ A ∈ KK Γ (A, A) vanishes in KK Γ R (A, A), and thus has a lift under the boundary map ∂ ∶ KK 1,Γ R Z (A, A) → KK Γ (A, A) of the Bockstein sequence. Our main construction, Theorem 2.7, is in fact a transgression of [α] Γ , i.e. a canonical lift ρ A α ∈ KK 1,Γ R Z (A, A ( 3 ) 3Given two unital morphisms ϕ, ψ∶ M 1 → M 2 , there exists a II 1 -factor M 3 , namely (a II 1 -factor containing) the corresponding von Neumann HNN extension (see[14, §3]) and a unital embedding χ∶ M 2 → M 3 such that χ ○ ϕ and χ ○ ψ differ by an inner automorphism and thus define the same element of KK(M 1 , M 3 ).Definition 1.4. For A and B separable C * -algebras the diagram F II 1 (H) ∋ M → KK(A, B⊗ M )with values groups has a well defined limit that we take as a definition of:KK R (A, B) ∶= lim M ∈F II 1 (H) KK(A, B ⊗ M ). (1.1)Construction 1.5. To define this limit it is convenient to build a direct system:• for every M from F II 1 (H) definethe group KK(A, B)[M ] to be the quotient of KK(A, B ⊗ M ) with respect to the subgroup of all elements z ∈ KK(A, B ⊗ M ) which become zero under some unital embedding ϕ ∶ M → N . It is a well defined equivalence relation by the property (3) above. • Every unital embedding ϕ∶ M 1 ↪ M 2 of II 1 -factors induces a group homomorphism ϕ * ∶ KK(A, B)[M 1 ] → KK(A, B)[M 2 ] sending the equivalence class of an element [x] ∈ KK(A, B ⊗ M 1 ) to the equivalence class of ϕ * [x] ∈ KK(A, B ⊗ M 2 ). This map is well defined by property (2) and does not depend on ϕ by property (3). We define a relation of partial pre-order in F II 1 (H) saying that M 1 ≺ M 2 if there is a tracial embedding M 1 ↪ M 2 . The partially pre-ordered set (F II 1 (H), ≺) is directed -by property (1) and the map M ↦ KK(A, B)[M ] is a direct system. It follows immediately that the unique limit L of M ↦ KK(A, B)[M ] is a limit for the diagram M ↦ KK(A, B ⊗ M ). ) When A and B are in the bootstrap class, then KK(A, B ⊗ M ) does not depend on the II 1 -factor M and is isomorphic to Hom(K(A), K(B) ⊗ R). In other words, for any II 1 -factor M the map KK(A, B ⊗ M ) → KK R (A, B) is an isomorphism. (2) If A and B are not in the bootstrap category, then there is no best choice for M . II 1 (H), and therefore there is no natural way to choose M and define KK R (A, B) = KK(A, B ⊗ M ). Remark 1. 8 . 8Let τ be a trace on D; by naturality of the Kasparov product, the map τ * ∶ KK(A, B ⊗ D) → KK R (A, B) is given by the product with [τ ] ∈ KK R (D, C). are two unitary equivalences, homotopic among unitary equivalences, then the classes of 2. 5 . 2 . 52Change of algebra. Kasparov product. Proposition 2.11. Let A, B be Γ-algebras satisfying property (KFP). Let f ∶ A → B be an equivariant homomorphism. For every finite dimensional unitary representation Proposition 2 . 12 . 212Let A, B be Γ-algebras satisfying property (KFP) and x ∈ KK Γ (A, B). For every finite dimensional unitary representation α of Γ, we have and the result follows. It follows then from Proposition 3.3 and Lemma 3.2 that we may deduce the class [α] AP S from our class ρ C 0 (Ṽ ) α . ( 2 ) 2The algebra A identifies with the torus algebra T (j, id A ) (see definition 1.2). ( 2 ) 2The algebra A satisfies property (KFP).(3) The algebra A satisfies property (KFP). (4) The algebra A satisfies property (KFP). KK Γ R (A, B) = KK R (A ⋊ Γ, B) -and in the same way, KK Γ R Z (A, B) = KK R Z (A ⋊ Γ, B). Definition 4.1. Given a Γ-algebra A and a trace τ on C * Γ we have three equivalent ways to define the same element ( 3 ) 3Let A, B be Γ-algebras. If A is KK Γ -subequivalent to B and B satisfies the weak (KFP) property then so does A. . Denote byh ∶ A ⋊ Γ ⊗ M n (C) ⊗ SM → A ⋊ Γ ⊗ M n ⊗ Z(i, i) the inclusion f ↦ (0, 0, f ), and let p 0 , p 1 ∶ A ⋊ Γ ⊗ Z i,i → A ⋊ Γ be the natural evaluation maps. not depend on u; (3) ψ M α behaves in a natural way with respect to inclusions i M ,M ∶ M ↪ M of II 1 -factors, namely j M ,M (ψ M α ) = ψ M α where j M ,M is the map induced by i M ,M on the cylinders. Proof. (1) The equality follows directly by Proposition 4.6 and the first Remark 1.3. (2) This follows from the connectedness of the group of unitaries of a von Neumann algebra. In fact, another unitary u ′ in M n (C) ⊗ M satisfying (α g ⊗ λ g )u ′ = u ′ (1 ⊗ λ g ) is of the form uv, where v is a unitary of the von Neumann algebra (M n (C) ⊗ M ) ∩ {λ g , g ∈ Γ} ′ . (3) By part (2), we can construct ψ M α using the unitaryū = (id Mn(C) ⊗i M ,M )(u). Letῑ ∶ C → M be the inclusion in M , and j M ,M ∶ Z(i, i) → Z(ῑ,ῑ) the induced map on the cylinders. In this way, the Morita equivalence [Eū] used in the construction is compatible with the inclusion of cylinders. Definition 4 . 8 . 48We denote by σ A α the element σ ( 1 ) 1The image ofρ A α under the connecting map of the Bockstein change of coefficients exact sequenceKK 1 R Z (A⋊Γ, A⋊Γ) → KK(A⋊Γ, A⋊Γ) is [δ A α ]−n1 A⋊Γ . (2)The classρ A α does not depend on the choices of N and y involved in its construction. Proof. 1.3. KK-theory with real coefficients. It is natural to try to define the KK-theory with real coefficients as KK(A, B ⊗ M ), where M is a II 1 -factor. On the other hand, it is not clear which M we should choose.We give here a natural construction of KK R taking into account all possible such M .Take a separable Hilbert space H and consider the set F II 1 (H) of all II 1 -factors acting on H. Every factor M ∈ F II 1 (H) is endowed with its unique tracial state. Recall the following properties: A α . Let A be a Γ-algebra satisfying property (KFP), and α a finite dimensional unitary representation of Γ. Denote by [α] Γ ∈ KK Γ (C, C) its KK Γ -class.By Remark 2.3, 1 Γ A,R ⊗ [α] = n1 Γ A,R where 1 Γ A,R denotes the unit element of the ring KK Γ R (A, A). Using the Bockstein change of coefficients exact sequence, it follows that there exists as in Theorem 2.7. Proposition 2.9. The image of ρ A α under the connecting map of the Bockstein change of coefficients exact sequence KK 1,Γ R Z Note that we only need to assume that Q ⊗ A is free, which is much weaker: it is just a statement on the torsion elements of Γ. We now construct an element σ A α ∈ KK 1 R Z (T , A ⋊ Γ) where T = T (ι A , δ A λ ) is the corresponding torus algebra. This will be an important ingredient in our construction of the weaker ρ invariant (Definition 4.11).4.3.1. A morphism between torus algebras. There exists a II 1 -factor M containing N with a morphism C(U n ) ⋊ Γ → M extending λ ∶ C * Γ → N ⊂ M . 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The Gamma Element for Groups which Admit a Uniform Embedding into Hilbert Space. Operator theory: Advances and applications. J L Tu, 153J. L. Tu, The Gamma Element for Groups which Admit a Uniform Embedding into Hilbert Space. Operator theory: Advances and applications, 153 (2004), 271-286. Amalgamated free product over Cartan subalgebra. Y Ueda, Pacific. J. of Math. 1912Y. Ueda, Amalgamated free product over Cartan subalgebra. Pacific. J. of Math no. 2 (1991) vol. 191 Symmetries of some reduced free product C-algebras. D Voiculescu, Operator Algebras and their connections with topology and ergodic theory. 1132D. Voiculescu, Symmetries of some reduced free product C-algebras, in Operator Algebras and their connections with topology and ergodic theory, Lecture Notes in Math. 1132 (1985), 566-588. Paolo Antonini Département de Mathématiques Bâtiment 425 Faculté des Sciences d'Orsay, Université Paris-Sud F-91405 Orsay Cedex. Paolo Antonini Département de Mathématiques Bâtiment 425 Faculté des Sciences d'Orsay, Université Paris-Sud F-91405 Orsay Cedex e-mail: [email protected] Germany e-mail: azzali@uni-potsdam. Potsdam, dePotsdam, Germany e-mail: [email protected] Imj-Prg Cnrs, Ufr De Mathématiques, CP 7012 -Bâtiment Sophie Germain 5 rue Thomas Mann, 75205 Paris CEDEX 13. Georges Skandalis Université Paris Diderot, Sorbonne Paris Cité Sorbonne UniversitésUPMC Paris 06,Georges Skandalis Université Paris Diderot, Sorbonne Paris Cité Sorbonne Universités, UPMC Paris 06, CNRS, IMJ-PRG UFR de Mathématiques, CP 7012 -Bâtiment Sophie Germain 5 rue Thomas Mann, 75205 Paris CEDEX 13, France e-mail: [email protected]	[]
[ "Denoising Arterial Spin Labeling Cerebral Blood Flow Images Using Deep Learning", "Denoising Arterial Spin Labeling Cerebral Blood Flow Images Using Deep Learning" ]	[ "Danfeng Xie [email protected] \nTemple University Philadelphia\nTemple University Philadelphia\nTemple University Philadelphia\n19122, 19122, 19122PA, PA, PA\n", "Bai Li \nTemple University Philadelphia\nTemple University Philadelphia\nTemple University Philadelphia\n19122, 19122, 19122PA, PA, PA\n", "Ze Wang [email protected] \nTemple University Philadelphia\nTemple University Philadelphia\nTemple University Philadelphia\n19122, 19122, 19122PA, PA, PA\n" ]	[ "Temple University Philadelphia\nTemple University Philadelphia\nTemple University Philadelphia\n19122, 19122, 19122PA, PA, PA", "Temple University Philadelphia\nTemple University Philadelphia\nTemple University Philadelphia\n19122, 19122, 19122PA, PA, PA", "Temple University Philadelphia\nTemple University Philadelphia\nTemple University Philadelphia\n19122, 19122, 19122PA, PA, PA" ]	[]	Arterial spin labeling perfusion MRI is a noninvasive technique for measuring quantitative cerebral blood flow (CBF), but the measurement is subject to a low signal-to-noise-ratio(SNR). Various post-processing methods have been proposed to denoise ASL MRI but only provide moderate improvement. Deep learning (DL) is an emerging technique that can learn the most representative signal from data without prior modeling which can be highly complex and analytically indescribable. The purpose of this study was to assess whether the record breaking performance of DL can be translated into ASL MRI denoising. We used convolutional neural network (CNN) to build the DL ASL denosing model (DL-ASL) to inherently consider the inter-voxel correlations. To better guide DL-ASL training, we incorporated prior knowledge about ASL MRI: the structural similarity between ASL CBF map and grey matter probability map. A relatively large sample data were used to train the model which was subsequently applied to a new set of data for testing. Experimental results showed that DL-ASL achieved state-of-the-art denoising performance for ASL MRI as compared to current routine methods in terms of higher SNR, keeping CBF quantification quality while shorten the acquisition time by 75%, and automatic partial volume correction.	null	[ "https://arxiv.org/pdf/1801.09672v1.pdf" ]	21,573,791	1801.09672	2e423f039141004e45f2df0a678c2bf6f1d75250	Denoising Arterial Spin Labeling Cerebral Blood Flow Images Using Deep Learning Danfeng Xie [email protected] Temple University Philadelphia Temple University Philadelphia Temple University Philadelphia 19122, 19122, 19122PA, PA, PA Bai Li Temple University Philadelphia Temple University Philadelphia Temple University Philadelphia 19122, 19122, 19122PA, PA, PA Ze Wang [email protected] Temple University Philadelphia Temple University Philadelphia Temple University Philadelphia 19122, 19122, 19122PA, PA, PA Denoising Arterial Spin Labeling Cerebral Blood Flow Images Using Deep Learning Arterial spin labeling perfusion MRI is a noninvasive technique for measuring quantitative cerebral blood flow (CBF), but the measurement is subject to a low signal-to-noise-ratio(SNR). Various post-processing methods have been proposed to denoise ASL MRI but only provide moderate improvement. Deep learning (DL) is an emerging technique that can learn the most representative signal from data without prior modeling which can be highly complex and analytically indescribable. The purpose of this study was to assess whether the record breaking performance of DL can be translated into ASL MRI denoising. We used convolutional neural network (CNN) to build the DL ASL denosing model (DL-ASL) to inherently consider the inter-voxel correlations. To better guide DL-ASL training, we incorporated prior knowledge about ASL MRI: the structural similarity between ASL CBF map and grey matter probability map. A relatively large sample data were used to train the model which was subsequently applied to a new set of data for testing. Experimental results showed that DL-ASL achieved state-of-the-art denoising performance for ASL MRI as compared to current routine methods in terms of higher SNR, keeping CBF quantification quality while shorten the acquisition time by 75%, and automatic partial volume correction. Introduction Arterial spin labeling (ASL) perfusion MRI is a technique for measuring cerebral blood flow (CBF) [5,25]. In ASL, arterial blood water is labeled with radio-frequency (RF) pulses in locations proximal to the tissue of interest, and perfusion is determined by pair-wise comparison with separate images acquired with control labeling using various subtraction approaches [1,17,16]. Limited by the longitudinal relaxation rate (T1) of blood water and the post-labeling transmit process, only a small fraction of tissue water can be labeled, resulting in a very low SNR [26]. To improve SNR, a series of ASL images are usually acquired to take the mean perfusion map for final CBF quantification. Also, various preprocessing and analysis methods such as motion correction and outlier cleaning, have been proposed to further denoise ASL MRI [23,6]. However, those methods typically suffer from two major disadvantages. First, due to very poor original image quality, those methods achieve relative SNR improvement. Second, those methods usually involve a optimization process in the testing stage, which is very time-consuming. Recenlty, Deep learning-based denoising methods emerged and achieves state-of-the-art performance [19,27,15].Instead of modeling explicit image prior, deep learning-based image denoising method learns image prior implicitly. The reasons of using CNN for denoising are as follows: First, CNN with deep or wide architecture [27] has the capacity and flexibility to effectively learning the image prior. Second, various well-developed training strategies and techniques are available to fasten the training process and improve the denoising performance, such as Rectifier Linear Unit (ReLU) [20], dropout [22], residual learning [9] and batch normalization [10]. Third, CNN can be trained on modern powerful GPU using parallel computation, which further improve the run time performance. To the best of our knowledge, this work is the first deep learning-based method for denoising ASL perfusion MRI images. The purpose of this study was to assess the feasibility and efficacy of deep learning-based method for denoising ASL perfusion MRI images. For ease of description, we dubbed the new DL-based ASL denoising method as ASLDLD thereafter. Our model, ASLDLD, was based on Wide Inference Network (WIN) [15], a 5 layer end-to-end Convolutional Neural Networks (CNNs) denoising model with wide structure. The ASLDLD was trained on 3D ASL MRI images acuqired from 280 subjects, where each subject has 20 slices and each slice has 40 control/labeled image pairs. The ASLDLD takes mean of first 10 CBF images without smoothing (meanCBF-10) as the input noisy image and the mean of all 40 CBF images (meanCBF-40) with smoothing and adaptive outlier cleaning [24] as the reference image. In order to fasten and stabilize model learning, we adopt residual learning and batch normalization learning strategies. Furthermore, Grey matter (GM) probability map was incorporated as a regularizer because CBF map shows a similar image contrast to that of a grey matter map. ASLDLD has several advantages in denoising ASL MRI images: 1) the model effectively utilized prior information which significantly improve the denoising performance. 2) Because of the intrinsic of feed-forward CNN architecture, the computaion time is very fast in the test stage, which significantly reduce the computation time. (Contrast to traditional denoising method, which requires very long time to computing.) 3) Comparing traditional ASL denoising methods which requires a large series of label controling image pairs (in our case, 40 pairs), ASLDLD only need 10 pairs of label controlling images. This significantly reduces the acquisition time of ASL MRI, which reduce the chance of head motions and hence reduce the chance of introducing extra noise. The rest of this paper is as follows. In section 2, we discuss about the related works of deep learningbased denoising methods. In section 3, we present the proposed ASLDLD architecture. Section 4 demonstrates the experiment of our methods on data. Last but not least, in section 5, we discuss the main contributions and results of this work. Related Works Imaging denoising is a classic low-level vision problem which have been widely studied in past decades. The image prior modeling often play a central role in image denoising. Traditional methods that used image prior knowledge as regularization techniques, scuh as nonlocal self-similarity models [2,4,3,18], Markov Random Field (MRF) [13,14,21] and spares models [8,7], have shown very promising performance. However, in traditional denoising methods, image prior knowledge are explicitly pre-defined, which are often limited in capturing the full characteristics of image structure and limited in blind image denoising. Neural network based denoising method is another active category of image denoising. The main difference between the neural network based methods and other methods is that neural network typical learn image prior implicitly rather than pre-defined image priors by training directly on pairs of noisy images and corrupted images. The most popular neural network based denoising methods up-to-date are based on Convolutional neural networks(CNNs). CNNs learn a hierarchy of features by a series of convolution, pooling and non-linear activation operations [41,42]. CNNs were originally designed for image classification and object detection, and now are also adopted in image denoising. Jain and Seung [11] demonstrated that convolutional neural networks (CNNs) can be used for image denoising and claimed that CNNs have achieved comparable or even superior performance than the MRF methods. Zhang et al. [27] proposed to incorporate residual learning and batch normalization learning strategies into very deep CNN for denoising. Mao et al. [19] proposed to use skip-layer connection to symmetrically link convolutional and deconvolutional layers, which is able to train even deeper CNN architecture for denoising. Peng and Fang [15] proposed a wider CNN network which has relatively fewer layers but has larger size and number of filters in each layer. They claimed that for low-level vision tasks, the depth of the network is not the key, while the width of the architecture is more important. They state that for denoising tasks, deep learning denoising models learn prior pixel distribution information from original image and then use the learned filter banks to restore degrade images. Thus, the more concentrated convolutions to capture the prior image distribution from noisy images, the better the denoising performances. Residual learning Residual learning is a technique to solve the gradient vanish problem [9]. As the the number of layers increases, the training accuracy of CNN begins to decrease due to gradient vanishing in lower layers. By constructing residual units (i.e., identity shortcuts or skip connections) between a few layers, residual network learns a residual mapping which is much easier to train and prevent gradient vanish. With residual learning strategy, training extremely deep CNN become possible. He et al [9] shows improved performance when using residual learning for image classification and object detection. There are several studies that incorporate residual learning for denoising tasks [19,27,15]. In [19], they used Skip shortcuts to connect from convolutional feature maps to their corresponding deconvolutional feature maps every a few layers, which help ease back-propagation and reuse details. In [27], Zhang et al. proposed DnCNN to using a mapping directly from an input observation to the corresponding reference observation. Batch normalization Batch Normalization is another strategy to fasten the training process and improve the training accuracy. BN was designed to prevent internal covariance shift due to mini-batch stochastic gradient descent (SGD) which changes the distributions of internal non-linearity inputs during training. BN is motivated by the fact that data whitening process improves performance. First, BN normalizes the output of the bottom layer (Conv or ReLU), dimension-wise with zero mean and unit variance within a batch of training images; Second, BN optimally shifts and scales these normalized activations. Methods Model In this section, we present the proposed ASLDLD for denoising ASL MRI images. There are generally two steps to train a CNN model for a specific task: 1) design network architecture and 2) training the network. First, we modify the WIN network to make it suitable to denoise ASL MRI images. We set the depth and width of the network and incorporated grey matter (GM) probability map as a regularizer because CBF map shows a similar image contrast to that of a grey matter map.Second, we adopt residual learning and batch normalization strategy to fasten and stabilize model training. Data ASL data were acquired from 280 subjects using a pseudo-continuous ASL sequence (40 control/labeled image pairs with labeling time = 1.48 sec, post-labeling delay = 1.5 sec, FOV=22 cm, matrix=64x64, TR/TE=4000/11 ms, 20 slices with a thickness of 5 mm plus 1 mm gap). Each pair of control/labeled image is subtracted using[xx] method to generate one CBF image. CBF images were calculated and spatially normalized into the MNI space using ASLtbx [23,24] and SPM12. To maximally show the benefit of ASLDLD, we took the mean of first 10 CBF images without smoothing (meanCBF-10) as the input image while used the mean of all 40 CBF images (meanCBF-40) with smoothing and adaptive outlier cleaning [24] as the reference. The ASLDLD was trained with data from 240 subjects' 3D CBF maps (input and reference). The remained 40 subjects were used as test samples. For each subject, we extract every 3 slice from slice 36 to slice 60 of a 3D CBF image. Thus, the number of total 2d CBF images for training are 240 × 9 = 2160. The normalized CBF image size is 91 × 109 and we set the patch size as 16 × 16 with the stride of 4. The training data set are cropped into 984960 patches in total. Problem formulation The input of our ASLDLD is a noisy observation y = x + n, where x is the latent clean image, y is the noisy image and n is the noise added to x. Rather than directly learn the original unreferenced mapping T (y) → x, the proposed ASLDLD has a skip connection from input-end to output-end to learn the residual mapping T (y) → −n, then it has x = y + T (y). In order to learn the weights Θ of the CNN, we minimize the Mean Squared Error (MSE) between the reference images x i and noisy input images y i . Thus, the loss function of ASLDLD is l(Θ) = 1 2N N i=1 \|\|y i + T (y i ; Θ) − x i \|\| 2 . Furthermore, grey matter (GM) probability map was incorporated as a regularizer because CBF map shows a similar image contrast to that of a grey matter map. To further encode GM prior information to the network, we add GM probability map γ i of each subject during the training. This step can be formulated as \|\|ŷ i − γ i \|\|. Thus the loss function is as follows: l(Θ) = 1 2N N i=1 \|\|y i + T (y i ; Θ) − x i \|\| 2 + α\|\|y i − γ i \|\| . where α is a hyperparameter and we set α to 0.1 in our case. The network architecture of ASLDLD, as shown in Figure 1, was based on WIN, a 5 layer Convolutional Neural Networks (CNNs) with wide structure. The "wide" structure here refers to the relatively larger size of filter (7x7) and the relatively larger number of filters (128) in each layer, while the number of layers (5) is fewer compared to deep CNNs used in high-level tasks. The wider structure was used for two reasons: 1) larger filters can better utilize spatial correlation among neighboring voxels and 2) more filters are able to capture the pixel-level distribution information more effectively [15]. The ASLDLD contains no signal pooling and no fully connected output layer as often used in regular CNNs. Each convolutional layer is followed by a ReLU, except for the last layer. The residual learning and batch normalization technique is further introduced to fasten and stabilize the training performance of ASLDLD. Network architecture Implementation Details The network is trained end-to-end using ADAM [12] with basic learning rate 0.1. Meantime, momentum 0.9, weight decay 1e − 4 and clip gradient 0.1 are also adopted to optimise training process. We train the model using mini batches of size 64. Caffe and MatConvNet packages are used to train the proposed ASLDLD models. All the experiments are running on a PC with Intel(R) Core(TM) i7-5820k CPU @3.30GHz and an Nvidia GeForce 980 Ti GPU. It takes about one day to train the ASLDLD on GPU. ASLDLD was compared with current state-of-the art ASL denoising methods regarding the SNR of the resultant CBF images. Figure 3 shows the resultant CBF maps from one representative subject. ASLDLD yielded superior performances to non-DL-based methods. Especially, ASLDLD recovered CBF signal in the air-brain boundaries as marked by the green boxes and signal loss due to partial volume effects as labeled by the yellow boxes. Slightly better texture was obtained by ASLDLD too. Figure 2 shows the SNR performance of different methods. SNR was calculated as the ratio between the mean signal of a GM region-of-interest (ROI) and the standard deviation of a white matter ROI. Compared to the non-DL-based methods, ASLDLD showed a 38.6% SNR increase (p=1.14e-4). Results Comparison with the state-of-the art In the experiment, we show ASLDLD achieve higher SNR than previous methods. Especially, it should be noted that ASLDLD only need 10 pairs of control labeling images rather than 40 pairs that needed for traditional methods. This significantly reduces the acquisition time of ASL MRI, which reduce the chance of head motions. This also help to reduce the noise introduced by head motions. Furthermore, because traditional methods generally involves a optimization process in the test stage while ASLDLD only has a feed-forward process, ASLDLD significantly reduce denoising time in the final test stages. Conclusion In this study, we showed that DL-based denoising can substantially improve ASL CBF SNR as well as the partial volume effects even only used the mean CBF map of 10 pairs of ASL control/label image acquisitions. 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[ "Order-Revealing Encryption and the Hardness of Private Learning", "Order-Revealing Encryption and the Hardness of Private Learning" ]	[ "Mark Bun [email protected]. \nSchool of Engineering & Applied Sciences\nHarvard University\n\n", "Mark Zhandry \nSchool of Engineering & Applied Sciences\nHarvard University\n\n" ]	[ "School of Engineering & Applied Sciences\nHarvard University\n", "School of Engineering & Applied Sciences\nHarvard University\n" ]	[]	An order-revealing encryption scheme gives a public procedure by which two ciphertexts can be compared to reveal the ordering of their underlying plaintexts. We show how to use order-revealing encryption to separate computationally efficient PAC learning from efficient (ε, δ)-differentially private PAC learning. That is, we construct a concept class that is efficiently PAC learnable, but for which every efficient learner fails to be differentially private. This answers a question of Kasiviswanathan et al. (FOCS '08, SIAM J. Comput. '11).To prove our result, we give a generic transformation from an order-revealing encryption scheme into one with strongly correct comparison, which enables the consistent comparison of ciphertexts that are not obtained as the valid encryption of any message. We believe this construction may be of independent interest.	10.1007/978-3-662-49096-9_8	[ "https://arxiv.org/pdf/1505.00388v1.pdf" ]	2,344,534	1505.00388	00649d157dc22e8d438d4c11bb22da464b6bd4bb	Order-Revealing Encryption and the Hardness of Private Learning May 2, 2015 3 May 2015 Mark Bun [email protected]. School of Engineering & Applied Sciences Harvard University Mark Zhandry School of Engineering & Applied Sciences Harvard University Order-Revealing Encryption and the Hardness of Private Learning May 2, 2015 3 May 2015differential privacylearning theoryorder-revealing encryption * An order-revealing encryption scheme gives a public procedure by which two ciphertexts can be compared to reveal the ordering of their underlying plaintexts. We show how to use order-revealing encryption to separate computationally efficient PAC learning from efficient (ε, δ)-differentially private PAC learning. That is, we construct a concept class that is efficiently PAC learnable, but for which every efficient learner fails to be differentially private. This answers a question of Kasiviswanathan et al. (FOCS '08, SIAM J. Comput. '11).To prove our result, we give a generic transformation from an order-revealing encryption scheme into one with strongly correct comparison, which enables the consistent comparison of ciphertexts that are not obtained as the valid encryption of any message. We believe this construction may be of independent interest. Introduction Many agencies hold sensitive information about individuals, where statistical analysis of this data could yield great societal benefit. The line of work on differential privacy [DMNS06] aims to enable such analysis while giving a strong formal guarantee on the privacy afforded to individuals. Noting that the framework of computational learning theory captures many of these statistical tasks, Kasiviswanathan et al. [KLN + 11] initiated the study of differentially private learning. Roughly speaking, a differentially private learner is required to output a classification of labeled examples that is accurate, but does not change significantly based on the presence or absence of any individual example. The early positive results in private learning established that, ignoring computational complexity, any concept class is privately learnable with a number of samples logarithmic in the size of the concept class [KLN + 11]. Since then, a number of works have improved our understanding of the sample complexity -the minimum number of examples -required by such learners to simultaneously achieve accuracy and privacy. Some of these works showed that privacy incurs an inherent additional cost in sample complexity; that is, some concept classes require more samples to learn privately than they require to learn without privacy [BKN10,CH11,BNS13,FX14,CHS14,BNSV15]. In this work, we address the complementary question of whether there is also a computational price of differential privacy for learning tasks, for which much less is known. The initial work of Kasiviswanathan et al. [KLN + 11] identified the important question of whether any efficiently PAC learnable concept class is also efficiently privately learnable, but only limited progress has been made on this question since then [BKN10,Nis14]. Our main result gives a strong negative answer to this question. We exhibit a concept class that is efficiently PAC learnable, but under plausible cryptographic assumptions cannot be learned efficiently and privately. To prove this result, we establish a connection between private learning and order-revealing encryption. We construct a new order-revealing encryption scheme with strong correctness properties that may be of independent learning-theoretic and cryptographic interest. Differential Privacy and Private Learning We first recall Valiant's (distribution-free) PAC model for learning [Val84]. Let C be a concept class consisting of concepts c : X → {0, 1} for a data universe X. A learner L is given n samples of the form (x i , c(x i )) where the x i 's are drawn i.i.d. from an unknown distribution, and are labeled according to an unknown concept c. The goal of the learner is to output a hypothesis h : X → {0, 1} from a hypothesis class H that approximates c well on the unknown distribution. That is, the probability that h disagrees with c on a fresh example from the unknown distribution should be small -say, less than 0.05. The hypothesis class H may be different from C, but in the case where H ⊆ C we call L a proper learner. Moreover, we say a learner is efficient if it runs in time polynomial in the description size of c and the size of its examples. Kasiviswanathan et al. [KLN + 11] defined a private learner to be a PAC learner that is also differentially private. Two samples S = {(x 1 , b 1 ), . . . , (x n , b n )} and S = {(x 1 , b 1 ), . . . , (x n , b n )} are said to be neighboring if they differ on exactly one example, which we think of as corresponding to one individual's information. A randomized learner L : (X × {0, 1}) n → H is (ε, δ)-differentially private if for all neighboring datasets S and S and all sets T ⊆ H, The original definition of differential privacy [DMNS06] took δ = 0, a case which is called pure differential privacy. The definition with positive δ, called approximate differential privacy, first appeared in [DKM + 06] and has since been shown to enable substantial accuracy gains. Throughout this introduction, we will think of ε as a small constant, e.g. ε = 0.1, and δ = o(1/n). Kasiviswanathan et al. [KLN + 11] gave a generic "Private Occam's Razor" algorithm, showing that any concept class C can be privately (properly) learned using O(log \|C\|) samples. Unfortunately, this algorithm runs in time Ω(\|C\|), which is exponential in the description size of each concept. With an eye toward designing efficient private learners, Blum et al. [BDMN05] made the powerful observation that any efficient learning algorithm in the statistical queries (SQ) framework of Kearns [Kea98] can be efficiently simulated with differential privacy. Moreover, Kasiviswanathan et al. [KLN + 11] showed that the efficient learner for the concept class of parity functions based on Gaussian elimination can also be implemented efficiently with differential privacy. These two techniques -SQ learning and Gaussian elimination -are essentially the only methods known for computationally efficient PAC learning. The fact that these can both be implemented privately led Kasiviswanathan et al. [KLN + 11] to ask whether all efficiently learnable concept classes could also be efficiently learned with differential privacy. Beimel et al. [BKN10] made partial progress toward this question in the special case of pure differential privacy with proper learning, showing that the sample complexity of efficient learners can be much higher than that of inefficient ones. Specifically, they showed that assuming the existence of pseudorandom generators with exponential stretch, there exists for any (d) = ω(log d) a concept class over {0, 1} d for which every efficient proper private learner requires Ω(d) samples, but an inefficient proper private learner only requires O( (d)) examples. Nissim [Nis14] strengthened this result substantially for "representation learning," where a proper learner is further restricted to output a canonical representation of its hypothesis. He showed that, assuming the existence of one-way functions, there exists a concept class that is efficiently representation learnable, but not efficiently privately representation learnable (even with approximate differential privacy). With Nissim's kind permission, we give the details of this construction in Section 5. Despite these negative results for proper learning, one might still have hoped that any efficiently learnable concept class could be efficiently improperly learned with privacy. Indeed, a number of works have shown that, especially with differential privacy, improper learning can be much more powerful than proper learning. For instance, Beimel et al. [BKN10] showed that under pure differential privacy, the simple class of Point functions (indicators of a single domain element) requires Ω(d) samples to privately learn properly, but only O(log d) samples to privately learn improperly. Moreover, computational separations are known between proper and improper learning even without privacy considerations. Pitt and Valiant [PV88] showed that unless NP = RP, k-term DNF are not efficiently properly learnable, but they are efficiently improperly learnable [Val84]. Under plausible cryptographic assumptions, we resolve the question of Kasiviswanathan et al. [KLN + 11] in the negative, even for improper learners. The assumption we need is the existence of "strongly correct" order-revealing encryption (ORE) schemes, described in Section 1.3. Theorem 1.1 (Informal). Assuming the existence of strongly correct ORE, there exists an efficiently computable concept class EncThresh that is efficiently PAC learnable, but not efficiently learnable by any (ε, δ)-differentially private algorithm. We stress that this result holds even for improper learners and for the relaxed notion of approximate differential privacy. We remark that cryptography has played a major role in shaping our understanding of the computational complexity of learning in a number of models (e.g. [Val84,KV94,Kha95,Ser00]). It has also been used before to show separations between what is efficiently learnable in different models (e.g. [Blu94,SG04]). Our Techniques We give an informal overview of the construction and analysis of the concept class EncThresh. We first describe the concept class of thresholds Thresh and its simple PAC learning algorithm. Consider the domain [N ] = {1, . . . , N }. Given a number t ∈ [N ], a threshold concept c t is defined by c t (x) = 1 if and only if x ≤ t. The concept class of thresholds admits a simple and efficient proper PAC learning algorithm L Thresh . Given a sample {(x 1 , c t (x 1 )), . . . , (x n , c t (x n ))} labeled by an unknown concept c t , the learner L Thresh identifies the largest positive example x i * and outputs the hypothesis h = c x i * . That is, L Thresh chooses the threshold concept that minimizes the empirical error on its sample. To achieve a small constant error on any underlying distribution on examples, it suffices to take n = O(1) samples. A For EncThresh to be efficiently PAC learnable, it must be learnable even under distributions that place arbitrary weight on examples corresponding to invalid ciphertexts. To this end, we require a "strong correctness" condition on our ORE scheme. The strong correctness condition ensures that all ciphertexts, even those that are not obtained as encryptions of messages, can be compared in a consistent fashion. This condition is not met by current constructions of ORE, and one of the technical contributions of this work is a generic transformation from weakly correct ORE schemes to strongly correct ones. While a learner similar to L Thresh is able to efficiently PAC learn the concept class EncThresh, we argue that it cannot do so while preserving differential privacy with respect to its examples. Intuitively, the security of the ORE scheme ensures that essentially the only thing a learner for EncThresh can do is output a hypothesis that compares an example to one it already has. We make this intuition precise by giving an algorithm that traces the hypothesis output by any efficient learner back to one of the examples used to produce it. This formalization builds conceptually on the connection between differential privacy and traitor-tracing schemes (see Section 1.4), but requires new ideas to adapt to the PAC learning model. Order-Revealing Encryption Motivated by the task of answering range queries on encrypted databases, an order-revealing encryption (ORE) scheme [BCO11, BLR + 15] is a special type of symmetric key encryption scheme where it is possible to publicly sort ciphertexts according to the order of the plaintexts. More precisely, the plaintext space of the scheme is the set of integers [N ] = {1, ..., N }, 1 and in addition to the private encryption and decryption procedures Enc, Dec, there is a public comparison procedure Comp that takes as input two ciphertexts, and reveals the order of the corresponding plaintexts. The notion of best-possible semantic security, defined in Boneh et al. [BLR + 15], intuitively captures the requirement that, given a collection of ciphertexts, no information about the plaintexts is learned, except for the ordering. Known constructions of order-revealing encryption. Order-revealing encryption can be seen as a special case of 2-input functional encryption. In such a scheme, there are several functions f 1 , . .., f k , and given two ciphertexts c 0 , c 1 encrypting m 0 , m 1 , it is possible to learn f i (m 0 , m 1 ) for all i ∈ [k]. General multi-input functional encryption schemes can be obtained from indistinguishability obfuscation [GGG + 14] or multilinear maps [BLR + 15]. It is also possible to build ORE from singleinput functional encryption with function privacy, which means that f is kept secret. Such schemes can be build from regular single-input schemes without function privacy by work of Brakerski and Segev [BS15], and such single-input schemes can also be built from obfuscation [GGH + 13b] or multilinear maps [GGHZ14]. Unfortunately, the above constructions are insufficient for our purposes. The issue arises from the fact that our learner needs to work for any distribution on ciphertexts, even distributions whose support includes malformed ciphertexts. Unfortunately, previous constructions only achieve a weak form of correctness, which guarantees that encrypting two messages and then comparing the ciphertexts using Comp produces the same result (with overwhelming probability) as comparing the plaintexts directly. This requirement only specifies how Comp works on valid ciphertexts, namely actual encryptions of messages. Moreover, correctness is only guaranteed for these messages with overwhelming probability, meaning even some valid ciphertexts may cause Comp to misbehave. For our learner, this weak form of correctness means, for some distributions that place significant weight on bad ciphertexts, the comparison procedure is completely useless, and thus the learner will fail for these distributions. We therefore need a stronger correctness guarantee. We need that, for any two ciphertexts, the comparison procedure is consistent with decrypting the two ciphertexts and comparing the resulting plaintexts. This correctness guarantee is meaningful even for improperly generated ciphertexts. We note that none of the existing constructions of order-revealing encryption outlined above satisfy this stronger notion. For the obfuscation-based schemes, ciphertexts consist of obfuscated programs. In these schemes, it is easy to describe invalid ciphertexts where the obfuscated program performs incorrectly, causing the comparison procedure to output the wrong result. In the multilinear map-based schemes, the underlying instantiation use current "noisy" multilinear maps, such as [GGH13a]. An invalid ciphertext could, for example, have too much noise, which will cause the comparison procedure to behave unpredictably. Obtaining strong correctness. We first argue that, for all existing ORE schemes, the scheme can be modified so that Comp is correct for all valid ciphertexts. We then give a generic conversion from any ORE scheme with weakly correct comparison, including the tweaked existing schemes, into a strongly correct scheme. We simply modify the ciphertext by adding a non-interactive zero-knowledge (NIZK) proof that the ciphertext is well-formed, with the common reference string added to the public comparison key. Then the decryption and comparison procedures check the proof(s), and only output the result (either decryption or comparison) if the proof(s) are valid. The (computational) zero-knowledge property of the NIZK implies that the addition of the proof to the ciphertext does not affect security. Meanwhile, NIZK soundness implies that any ciphertext accepted by the decryption and comparison procedures must be valid, and the weak correctness property of the underlying ORE implies that for valid ciphertexts, decryption and comparison are consistent. The result is that comparisons are consistent with decryption for all ciphertexts, giving strong correctness. As we need strong correctness for every ciphertext, even hard-to-generate ones, we need the NIZK proofs to have perfect soundness, as opposed to computational soundness. Such NIZK proofs were built in [GOS12]. We note also that the conversion outlined above is not specific to ORE, and applies more generally to functional encryption schemes. Related Work Hardness of Private Query Release. One of the most basic and well-studied statistical tasks in differential privacy is the problem of releasing answers to counting queries. A counting query asks,"what fraction of the records in a dataset D satisfy the predicate q?". Given a collection of k counting queries q 1 , . . . , q k from a family Q, the goal of a query release algorithm is to release approximate answers to these queries while preserving differential privacy. A remarkable result of Blum et al. [BLR08], with subsequent improvements by [DNR + 09, DRV10, RR10, HR10, GRU12,HLM12], showed that an arbitrary sequence of counting queries can be answered accurately with differential privacy even when k is exponential in the dataset size n. Unfortunately, all of these algorithms that are capable of answering more than n 2 queries are inefficient, running in time exponential in the dimensionality of the data. Moreover, several works [DNR + 09, Ull13,BZ14] have gone on to show that this inefficiency is likely inherent. These computational lower bounds for private query release rely on a connection between the hardness of private query release and traitor-tracing schemes, which was first observed by Dwork et al. [DNR + 09]. Traitor-tracing schemes were introduced by Chor, Fiat, and Naor [CFN94] to help digital content producers identify pirates as they illegally redistribute content. Traitor-tracing schemes are conceptually analogous to the example reidentification scheme we use to obtain our hardness result for private learning. Instantiating this connection with the traitor-tracing scheme of Boneh, Sahai, and Waters [BSW06], which relies on certain assumptions in bilinear groups, Dwork et al. [DNR + 09] exhibited a family of 2Õ ( √ n) queries for which no efficient algorithm can produce a data structure which could be used to answer all queries in this family. Very recently, Boneh and Zhandry [BZ14] constructed a new traitor-tracing scheme based on indistinguishability obfuscation that yields the same infeasibility result for a family of n · 2 O(d) queries on records of size d. Extending this connection, Ullman [Ull13] constructed a specialized traitor-tracing scheme to show that no efficient private algorithm can answer more thanÕ(n 2 ) arbitrary queries that are given as input to the algorithm. Dwork et al. [DNR + 09] also showed strong lower bounds against private algorithms for producing synthetic data. Synthetic data generation algorithms produce a new "fake" dataset, whose rows are of the same type as those in the original dataset, with the promise that the answers to some restricted set of queries on the synthetic dataset well-approximate the answers on the original dataset. Assuming the existence of one-way functions, Dwork et al. [DNR + 09] exhibited an efficiently computable collection of queries for which no efficient private algorithm can produce useful synthetic data. Ullman and Vadhan [UV11] refined this result to hold even for extremely simple classes of queries. Nevertheless, the restriction to synthetic data is significant to these results, and they do not rule out the possibility that other privacy-preserving data structures can be used to answer large families of restricted queries. In fact, when the synthetic data restriction is lifted, there are algorithms (e.g. [HRS12,TUV12,CTUW14,DNT14]) that answer queries from certain exponentially large families in subexponential time. One can view the problem of synthetic data generation as analogous to proper learning. In both cases, placing natural syntactic restrictions on the output of an algorithm may in fact come at the expense of utility or computational efficiency. Efficiency of SQ Learning. Feldman and Kanade [FK12] addressed the question of whether information-theoretically efficient SQ learners -i.e., those making polynomially many queriescould be made computationally efficient. One of their main negative results showed that unless NP = RP, there exists a concept class with polynomial query complexity that is not efficiently SQ learnable. Moreover, this concept class is efficiently PAC learnable, which suggests that the restriction to SQ learning can introduce an inherent computational cost. We show that the concept class EncThresh can be learned (inefficiently) with polynomially many statistical queries. The result of Blum et al. [BDMN05] discussed above, showing that SQ learning algorithms can be efficiently simulated by differentially private algorithms, thus shows that EncThresh also separates SQ learners making polynomially many queries from computationally efficient SQ learners. Corollary 1.2 (Informal). Assuming the existence of strongly correct ORE, the concept class EncThresh is efficiently PAC learnable and has polynomial SQ query complexity, but is not efficiently SQ learnable. While our proof relies on much stronger hardness assumptions, it reveals ORE as a new barrier to efficient SQ learning. As discussed in more detail in Section 3.3, even though their result is about computational hardness, Feldman and Kanade's choice of a concept class relies crucially on the fact that parities are hard to learn in the SQ model even information-theoretically. By contrast, our concept class EncThresh is computationally hard to SQ learn for a reason that appears fundamentally different than the information-theoretic hardness of SQ learning parities. Learning from Encrypted Data. Several works have developed schemes for training, testing, and classifying machine learning models over encrypted data (e.g. [GLN13,BPTG14]). In a model use case, a client holds a sensitive dataset, and uploads an encrypted version of the dataset to a cloud computing service. The cloud service then trains a model over the encrypted data and produces an encrypted classifier it can send back to the client, ideally without learning anything about the examples it received. The notion of privacy afforded to the individuals in the dataset here is complementary to differential privacy. While the cloud service does not learn anything about the individuals in the dataset, its output might still depend heavily on the data of certain individuals. In fact, our non-differentially private PAC learner for the class EncThresh exactly performs the task of learning over encrypted data, producing a classifier without learning anything about its examples beyond their order (this addresses the difficulty of implementing comparisons from prior work [GLN13]). Thus one can interpret our results as showing that not only are these two notions of privacy for machine learning training complementary, but that they may actually be in conflict. Moreover, the strong correctness guarantee we provide for ORE (which applies more generally to multi-input functional encryption) may help enable the theoretical study of learning from encrypted data in other PAC-style settings. Preliminaries and Definitions PAC Learning and Private PAC Learning For each k ∈ N, let X k be an instance space (such as {0, 1} k ), where the parameter k represents the size of the elements in X k . Let C k be a set of boolean functions {c : . . represents an infinite sequence of learning problems defined over instance spaces of increasing dimension. We will generally suppress the parameter k, and refer to the problem of learning C as the problem of learning C k for every k. X k → {0, 1}}. The sequence (X 1 , C 1 ), (X 2 , C 2 ), . A learner L is given examples sampled from an unknown probability distribution D over X, where the examples are labeled according to an unknown target concept c ∈ C. The learner must select a hypothesis h from a hypothesis class H that approximates the target concept with respect to the distribution D. More precisely, Definition 2.1. The generalization error of a hypothesis h : X → {0, 1} (with respect to a target concept c and distribution D) is defined by error D (c, h) = Pr x∼D [h(x) = c(x)]. If error D (c, h) ≤ α we say that h is an α-good hypothesis for c on D.(x i , c(x i )), . . . , (x n , c(x n ))), where each x i is drawn i.i.d. from D, algorithm L outputs a hypothesis h ∈ H satisfying Pr[error D (c, h) ≤ α] ≥ 1 − β. The probability here is taken over the random choice of the examples in S and the coin tosses of the learner L. The learner L is efficient if it runs in time polynomial in the size parameter k, the representation size of the target concept c, and the accuracy parameters 1/α and 1/β. Note that a necessary (but not sufficient) condition for L to be efficient is that its sample complexity n is polynomial in the learning parameters. If H ⊆ C then L is called a proper learner. Otherwise, it is called an improper learner. Kasiviswanathan et al. [KLN + 11] defined a private learner as a PAC learner that is also differentially private. Recall the definition of differential privacy: Definition 2.3. A learner L : (X ×{0, 1}) n → H is (ε, δ)-differentially private if for all sets T ⊆ H, and neighboring sets of examples S ∼ S , Pr[L(S) ∈ T ] ≤ e ε Pr[L(S ) ∈ T ] + δ. The technical object that we will use to show our hardness results for differential privacy is what we call an example reidentification scheme. It is analogous to the hard-to-sanitize database distributions [DNR + 09, UV11] and re-identifiable database distributions [BUV14] used in prior works to prove hardness results for private query release, but is adapted to the setting of computational learning. In the first step, an algorithm Gen ex chooses a concept and a sample S labeled according to that concept. In the second step, a learner L receives either the sample S or the sample S −i where an appropriately chosen example i is replaced by a junk example, and learns a hypothesis h. Finally, an algorithm Trace ex attempts to use h to identify one of the rows given to L. If Trace ex succeeds at identifying such a row with high probability, then it must be able to distinguish L(S) from L(S −i ), showing that L cannot be differentially private. We formalize these ideas below. Definition 2.4. An (α, ξ)-example reidentification scheme for a concept class C consists of a pair of algorithms, (Gen ex , Trace ex ) with the following properties. Gen ex (k, n) Samples a concept c ∈ C k and an associated distribution D. Draws i.i. d. examples x 1 , . . . , x n ← R D, and a fixed value x 0 . Let S denote the labeled sample ((x 1 , c(x 1 )), . . . , (x n , c(x n )), and for any index i ∈ [n], let S −i denote the sample with the pair (x i , c(x i )) replaced with (x 0 , c(x 0 )). Trace ex (h) Takes state shared with Gen ex as well as a hypothesis h and identifies an index in [n] (or ⊥ if none is found). The scheme obeys the following "completeness" and "soundness" criteria on the ability of Trace ex to identify an example given to a learner L. Completeness. A good hypothesis can be traced to some example. That is, for every efficient learner L, Pr[error D (c, h) ≤ α ∧ Trace ex (h) = ⊥] ≤ ξ. Here, the probability is taken over h ← R L(S) and the coins of Gen ex and Trace ex . Soundness. For every efficient learner L, Trace ex cannot trace i from the sample S −i . That is, for all i ∈ [n], Pr[Trace ex (h) = i] ≤ ξ for h ← R L(S −i ). We may sometimes relax the completeness condition to hold only under certain restrictions on L's output (e.g. L is a proper learner or L is a representation learner). In this case, we say the (Gen ex , Trace ex ) is an example reidentification scheme for (properly, representation) learning a class C. Theorem 2.5. Let (Gen ex , Trace ex ) be an (α, ξ)-example reidentification scheme for a concept class C. Then for every β > 0 and polynomial n(k), there is no efficient (ε, δ)-differentially private (α, β)-PAC learner for C using n samples when δ < 1 − β − ξ n − e ε ξ. In a typical setting of parameters, we will take α, β, ε = O(1) and δ, ξ = o(1/n), in which case the inequality in Theorem 2.5 will be satisfied for sufficiently large n. Proof. Suppose instead that there were a computationally efficient (ε, δ)-differentially private (α, β)-PAC learner L for C using n samples. Then there exists an i ∈ [n] such that Pr[Trace ex (L(S)) = i] ≥ (1 − β − ξ)/n. However, since L is differentially private, Pr[Trace ex (L(S −i )) = i] ≥ e −ε 1 − β − ξ n − δ > ξ(n), which contradicts the soundness of (Gen ex , Trace ex ). Order-Revealing Encryption Definition 2.6. An Order-Revealing Encryption (ORE) scheme is a tuple (Gen, Enc, Dec, Comp) of algorithms where: • Gen(1 λ , 1 ) is a randomized procedure that takes as inputs a security parameter λ and plaintext length , and outputs a secret encryption/decryption key sk and public parameters params. • Enc(sk, m) is a potentially randomized procedure that takes as input a secret key sk and a message m ∈ {0, 1} , and outputs a ciphertext c. • Dec(sk, c) is a deterministic procedure that takes as input a secret key sk and a ciphertext c, and outputs a plaintext message m ∈ {0, 1} or a special symbol ⊥. • Comp(params, c 0 , c 1 ) is a deterministic procedure that "compares" two ciphertexts, outputting either ">", "<", "=", or ⊥. Correctness. An ORE scheme must satisfy two separate correctness requirements: • Correct Decryption: This is the standard notion of correctness for an encryption scheme, which says that decryption succeeds. We will only consider strongly correct decryption, which requires that decryption always succeeds. For all security parameters λ and message lengths , Pr[Dec(sk, Enc(sk, m) ) = m : (sk, params) ← Gen(1 λ , 1 )] = 1. • Correct Comparison: We require that the comparison function succeeds. We will consider two notions, namely strong and weak. In order to define these notions, we first define two auxiliary functions: In particular, for correctly generated ciphertexts, Comp never outputs ⊥. -Strongly Correct Comparison: This informally requires that comparison is consistent with decryption. For all security parameters λ, message lengths , and ciphertexts c 0 , c 1 , Pr Comp(params, c 0 , c 1 ) = Comp ciph (sk, c 0 , c 1 ) : (sk, params) ← Gen(1 λ , 1 ) = 1. Security. For security, we will consider a relaxation of the "best possible" security notion of Boneh et al. [BLR + 15]. Namely, we only consider static adversaries that submit all queries at once. "Best possible" security is a modification of the standard notion of CPA security for symmetric key encryption to block trivial attacks. That is, since the comparison function always leaks the order of the plaintexts, the left and right sets of challenge messages must have the same order. In our relaxation where all challenge messages are queried at once, we can therefore assume without loss of generality that the left and right sequences of messages are sorted in ascending order. For simplicity, we will also disallow the adversary from querying on the same message more than once. This gives the following definition: • A produces two message sequences m (L) 1 < m (L) 2 < · · · < m (L) q and m (R) 1 < m (R) 2 < · · · < m (R) q • The challenger runs (sk, params) ← Gen(1 λ , 1 ). It then responds to A with params, as well as c 1 , . . . , c q where c i = Enc(sk, m (L) i ) if b = 0 Enc(sk, m (R) i ) if b = 1 • A outputs a guess b for b. We also consider a weaker definition, which only allows the sequences m • A produces a sequence of messages m 1 < m 2 < · · · < m q , and challenge messages m L , m R such that m i < m L < m R < m i+1 for some i ∈ [q − 1]. • The challenger runs (sk, params) ← Gen(1 λ , 1 ). It then responds to A with params, as well as c 1 , . . . , c q where c i = Enc(sk, m i ) and c * = Enc(sk, m L ) if b = 0 Enc(sk, m R ) if b = 1 • A outputs a guess b for b. We now argue that these two definitions are equivalent up to some polynomial loss in security. Proof. We prove that single-challenge security implies many-challenge security through a sequence of hybrids. Each hybrid will only differ in the messages m i that are encrypted, and each adjacent hybrid will only differ in a single message. The first hybrid will encrypt m (L) i , and the last hybrid will encrypt m (R) i . Thus, by applying the single-challenge security for each hybrid, we conclude that the first and last hybrids are indistinguishable, thus showing many-challenge security. Hybrid j for j ≤ q. Hybrid j for j > q. However, we need to make a few technical modifications to ensure that EncThresh is efficiently PAC learnable. 1. In order for the learner to be able to use the comparison function Comp, it must be given the public parameters params generated by the ORE scheme. We address this in the natural way by attaching a set of public parameters to each example. Moreover, we define EncThresh so that each concept is supported on the single set of public parameters that corresponds to the secret key used for encryption and decryption. 2. Only a subset of binary strings form valid (sk, params) pairs that are actually produced by Gen in the ORE scheme. To represent concepts, we need a reasonable encoding scheme for these valid pairs. The encoding scheme we choose is the polynomial-length sequence of random coin tosses used by the algorithm Gen to produce (sk, params). m i = min(m (L) i , m (R) i ) if i ≤ j m (L) i if i > jm i = min(m (L) i , m (R) i ) if i ≤ 2q − j m (R) i if i > 2q − j We now formally describe the concept class EncThresh. Each concept is parameterized by a string r, representing the coin tosses of the algorithm Gen, and a threshold t ∈ [N + 1] for N = 2 . In what follows, let (sk r , params r ) be the keys output by Gen(1 λ , 1 ) when run on the sequence of coin tosses r. Let Notice that given t and r, the concept f t,r can be efficiently evaluated. The description length k of the instance space X k = {0, 1} k is polynomial in the security parameter λ and plaintext length . An Efficient PAC Learner for EncThresh We argue that EncThresh is efficiently PAC learnable by formalizing the argument from the introduction. Because we need to include the ORE public parameters in each example, the PAC learner L (Algorithm 3) for EncThresh actually works in two stages. In the first stage, L determines whether there is significant probability mass on examples corresponding to some public parameters params. Recall that each concept in EncThresh is supported on exactly one such set of parameters. If there is no significant mass on any params, then the all-zeroes hypothesis is a good hypothesis. On the other hand, if there is a heavy set of parameters, the learner L applies Comp using those parameters to learn a good comparator. Theorem 3. 1. Let α, β > 0. There exists a PAC learning algorithm L for the concept class EncThresh achieving error α and confidence 1 − β. Moreover, L is efficient (running in time polynomial in the parameters k, 1/α, log(1/β)). Proof. Fix a target concept f t,r ∈ EncThresh k and a distribution D on examples. First observe that the learner L always outputs a hypothesis with one-sided error, i.e. we always have h ≤ f t,r pointwise. Also observe that f t ,r ≤ f t,r pointwise for any t < t. These both follow from the strong correctness of the ORE scheme. Let (sk r , params r ) denote the keys output by Gen(1 λ , 1 ) when run on the sequence of coin tosses r. Let POS denote the set of examples (params, c) on which f t,r (params, c) = 1. We divide the analysis of the learner in to two cases based on the weight D places on POS. Case 1: D places weight at least α on POS. Definet ∈ [N + 1] as the largestt ≤ t such that error D (ft ,r , f t,r ) ≥ α. Such at is guaranteed to exist since f 0,r is the all-zeros function, and therefore error D (f 0,r , f t,r ) is equal to the weight D places on POS, which is at least α. Suppose ft +1,r ≤ h pointwise. Since h has one-sided error (that is, h ≤ f t,r pointwise), we have error D (ft +1,r , f t,r ) = error D (ft +1,r , h) + error D (h, f t,r ), or error D (h, f t,r ) = error D (ft +1,r , f t,r ) − error D (ft +1,r , h) ≤ error D (ft +1,r , f t,r ) < α. Therefore, it suffices to show that ft +1,r ≤ h with probability at least 1 − β. This is guaranteed as long as L receives a sample (params r , c i , 1) witht ≤ Dec(sk r , c i ) < t. In other words, f t,r (params r , c i ) = 1 and ft ,r (params r , c i ) = 0. Since ft ,r ≤ f t,r pointwise, such samples exactly account for the error between ft ,r and f t,r . Thus since error D (ft ,r , f t,r ) ≥ α, for each i it must be thatt ≤ Dec(sk r , c i ) < t with probability at least α. The learner L therefore receives some sample c i witht ≤ Dec(sk r , c i ) < t with probability at least 1 − (1 − α) n ≥ 1 − β (since we took n ≥ log(1/β)/α). Case 2: D places less than α weight on POS. Then the identically zero hypothesis has error at most α, so the claim holds because 0 ≤ h ≤ f t,r . Hardness of Privately Learning EncThresh We now prove the hardness of privately learning EncThresh by constructing an example reidentification scheme for this concept class. Recall that an example reidentification scheme consists of two algorithms, Gen ex , which selects a distribution, a concept, and examples to give to a learner, and Trace ex which attempts to identify one of the examples the learner received. Our example reidentification scheme yields a hard distribution even for weak-learning, where the error parameter α is taken to be inverse-polynomially close to 1/2. Theorem 3. 2. Let γ(n) and ξ(n) be noticeable functions. Let (Gen, Enc, Dec, Comp) be a statically single-challenge secure ORE scheme. Then there exists an (efficient) (α = 1 2 − γ, ξ)-example reidentification scheme (Gen ex , Trace ex ) for the concept class EncThresh. We start with an informal description of the scheme (Gen ex , Trace ex ). The algorithm Gen ex sets up the parameters of the ORE scheme, chooses the "middle" threshold concept corresponding to t = N/2, and sets the distribution on examples to be encryptions of uniformly random messages (together with the correct public parameters needed for comparison). Let m 1 < m 2 < · · · < m n denote the sorted sequence of messages whose encryptions make up the sample produced by Gen ex (with overwhelming probability, they are indeed distinct). We can thus break the plaintext space up into buckets of the form B i = [m i , m i+1 ). Suppose L is a (weak) learner that produces a hypothesis h with advantage γ over random guessing. Such a hypothesis h must be able to distinguish encryptions of messages m ≤ t from encryptions of messages m > t with advantage γ. Thus, there must be a pair of adjacent buckets B i−1 , B i for which h can distinguish encryptions of messages from B i−1 from encryptions from B i with advantage γ n . This observation leads to a natural definition for Trace ex : locate a pair of adjacent buckets B i−1 , B i that h distinguishes, and output the identity i of the example separating those buckets. Completeness of the resulting scheme, i.e. the fact that some example is reidentified when L succeeds, follows immediately from the preceding discussion. We argue soundness, i.e. that an example absent from L's sample is not identified, by reducing to the static security of the ORE scheme. The intuition is that if L is not given example i, then it should not be able to distinguish encryptions from bucket B i−1 from encryptions from bucket B i . To make the security reduction somewhat more precise, suppose for the sake of contradiction that there is an efficient algorithm L that violates the soundness of (Gen ex , Trace ex ) with noticeable probability ξ. That is, there is some i such that even without example i, the algorithm L manages to produce (with probability ξ) a hypothesis h that distinguishes B i−1 from B i . A natural first attempt to violate the security of the ORE is to construct an adversary that challenges on the message sequences m 1 < · · · < m i−1 < m To overcome this issue, we instead rely on the security of the ORE for sequences that differ on two messages. For the "left" challenge, our adversary samples two messages from the same randomly chosen bucket, B i−1 or B i (in addition to requesting encryptions of m 1 , . . . , m i−1 , m i , . . . , m n ). For the "right" challenge, it samples one message from each bucket B i−1 and B i . Let c 0 and c 1 be the ciphertexts corresponding to thee challenge messages. If h agrees on c 0 and c 1 , then this suggests the messages are from the same bucket, and the adversary should guess "left". On the other hand, if h disagrees on c 0 and c 1 , then the adversary should guess "right". If h distinguishes the buckets B i−1 and B i , this adversary does strictly better than random guessing. On the other hand, even if h fails to distinguish the buckets, the adversary does at least as well as random guessing. So overall, it still has a noticeable advantage at the ORE security game. (L) i < m i+1 , <, m n and m 1 < · · · < m i−1 < m (R) i < m i+1 < · · · < m n , We now give the formal proof of Theorem 3.2. Proof. We construct an example reidentification scheme for EncThresh as follows. The algorithm Gen ex fixes the threshold t = N/2 and samples (sk r , params r ) ← R Gen(1 λ , 1 ), yielding a concept f t,r . Let D be the distribution of (params r , Enc(sk r , m)) for uniformly random m ∈ N , these random messages will be well-spaced. In particular, with overwhelming probability, \|m i+1 − m i \| > 1 for every i, so we assume this is the case in what follows. Gen ex then sets the samples to be (x 1 = (params r , Enc(sk r , m 1 )), . . . , x n = (params r , Enc(sk r , m n ))). Let x 0 = (params r , Enc(sk r , m 0 )) be a "junk" example. The algorithm Trace ex creates buckets B i = [m i , m i+1 ). For each i, let p i = Pr m∈B i ,coins of Enc [h(params r , Enc(sk, m)) = 1]. By sampling random choices of m in each bucket, Trace ex can efficiently compute a good estimatê p i ≈ p i for each i (Lemma 3.3). It then accuses the least i for whichp i−1 −p i ≥ γ n , and ⊥ if none is found. p i = 1 K K j=1 h(x j ) where x j = (params r , Enc(sk r , m j )) for i.i.d. m 1 , . . . , m K ← R B i . Then \|p i − p i \| ≤ γ 4n for every i with probability at least 1 − ξ/4. Proof. By a Chernoff bound, the probability that any givenp i deviates from p i by more than γ 4n is at most 2 exp(−Kγ 2 /8n 2 ) ≤ ξ 4(n+1) . The lemma follows by a union bound. We first verify completeness for this scheme. Let L be a learner for EncThresh using n examples. If the hypothesis h produced by L is ( 1 2 − γ)-good, then there exists i 0 < i 1 such that p i 0 − p i 1 ≥ 2γ. If this is the case, then there must be an i for which p i−1 − p i ≥ 2γ n . Then with probability all but ξ(n)/2 over the estimatesp i , we havep i−1 −p i ≥ γ n , so some index is accused. Now we verify soundness. Fix a PPT L, and let j * ∈ [n]. Suppose L violates the soundness of the scheme with respect to j * , i.e. Pr h← R L(S −j * ),coins of Genex [Trace ex (h) = j * ] > ξ. We will use L to construct an adversary A for the ORE scheme that succeeds with noticeable advantage. It suffices to build an adversary for the static (many-challenge) security of ORE, with Theorem 2.9 showing how to convert it to a single-challenge adversary. This many-challenge adversary is presented as Algorithm 2. (While not explicitly stated, the adversary should halt and output a random guess whenever the messages it samples are not well-spaced.) Let i * be such that m i * = m j * . With probability at least ξ over the parameters (sk r , params r ), the choice of messages, the choice of the hypothesis h, and the coins of Trace ex , there is a gap p i * −1 −p i * ≥ γ n . Hence, by Lemma 3.3, there is a gap p i * −1 − p i * ≥ γ 2n with probability at least ξ 2 . We now calculate the advantage of the adversary A. Fix a hypothesis h. For notational simplicity, let p = p i * −1 and let q = p i * . Let y 0 = h(params r , c 0 i * ) and y 1 = h(params r , c 1 i * ). Then the adversary's success probability is: . . , m n ← R [N ], and let m 1 ≤ · · · ≤ m n be the result of sorting the m j . Let π be the permutation on {1, . . . , n} such that m π(j) = m j . Let m 0 = 0. Let i * = π(j * ) so that m i * = m j * . Algorithm 2 ORE adversary A 1. Sample m 1 , . Construct pairs (m S −j * = (params r , c 1 , χ(m 1 ≤ t)), . . . , (params r , c j * −1 , χ(m j * −1 ≤ t)), (params r , c 0 , 1), (params r , c j * +1 , χ(m j * +1 ≤ t)), . . . , (params r , c n , χ(m n ≤ t)) = (params r , c π(1) , χ(m π(1) ≤ t)), . . . , (params r , c π(j * −1) , χ(m π(j * −1) ≤ t)), (params r , c 0 , 1), (params r , c π(j * +1) , χ(m π(j * +1) ≤ t)), . . . , (params r , c π(n) , χ(m π(n) ≤ t)) Obtain h ← R L(S −j * ). Guess b = 0 if h(params r , c 0 i * ) = h(params r , c 1 i * ). Otherwise guess b = 1. Pr[b = b] = 1 2 (Pr[y 0 = y 1 \|b = 0] + Pr[y 0 = y 1 \|b = 1]) = 1 2 ( 1 2 (p 2 + (1 − p) 2 + q 2 + (1 − q) 2 ) + (1 − pq − (1 − p)(1 − q))) = 1 2 + 1 2 (p − q) 2 . Thus if p − q ≥ γ 2n , then the adversary's advantage is at least γ 2 4n 2 . On the other hand, even for arbitrary values of p, q, the advantage is still nonnegative. Therefore, the advantage of the strategy is at least ξγ 2 8n 2 − negl(k) (the negl(k) term coming from the assumption that the m i sampled where distinct), which is a noticeable function of the parameter k. This contradicts the static security of the ORE scheme. The SQ Learnability of EncThresh The statistical query (SQ) model is a natural restriction of the PAC model by which a learner is able to measure statistical properties of its examples, but cannot see the individual examples themselves. We recall the definition of an SQ learner. Definition 3.4 (SQ learning [Kea98]). Let c : X → {0, 1} be a target concept and let D be a distribution over X. In the SQ model, a learner is given access to a statistical query oracle STAT(c, D). It may make queries to this oracle of the form (ψ, τ ), where ψ : X × {0, 1} → {0, 1} is a query function and τ ∈ (0, 1) is an error tolerance. The oracle STAT(c, D) responds with a value v such that \|v − Pr x∈D [ψ(x, c(x)) = 1]\| ≤ τ . The goal of a learner is to produce, with probability at least 1 − β, a hypothesis h : X → {0, 1} such that error D (c, h) ≤ α. The query functions must be efficiently evaluable, and the tolerance τ must be lower bounded by an inverse polynomial in k and 1/α. The query complexity of a learner is the worst-case number of queries it issues to the statistical query oracle. An SQ learner is efficient if it also runs in time polynomial in k, 1/α, 1/β. [FK12] investigated the relationship between query complexity and computational complexity for SQ learners. They exhibited a concept class C which is efficiently PAC learnable and SQ learnable with polynomially many queries, but assuming NP = RP, is not efficiently SQ learnable. Concepts in this concept class take the form Feldman and Kanade g φ,y (x, x ) = PAR y (x ) if x = φ 0 otherwise. Here, PAR y (x ) is the inner product of y and x modulo 2. The concept class C consists of g φ,y where φ is a satisfiable 3-CNF formula and y is the lexicographically first satisfying assignment to φ. The efficient PAC learner for parities based on Gaussian elimination shows that C is also efficiently PAC learnable. It is also (inefficiently) SQ learnable with polynomially many queries: either the all-zeroes hypothesis is good, or an SQ learner can recover the formula φ bit-by-bit and determine the satisfying assignment y by brute force. On the other hand, because parities are information-theoretically hard to SQ learn, the satisfying assignment y remains hidden to an SQ learner unless it is able to solve 3-SAT. In this section, we show that the concept class EncThresh shares these properties with C. Namely, we know that EncThresh is efficiently PAC learnable and because it is not efficiently privately learnable, it is not efficiently SQ learnable [BDMN05]. We can also show that EncThresh has an SQ learner with polynomial query complexity. Making this observation about EncThresh is of interest because the hardness of SQ learning EncThresh does not seem to be related to the (informationtheoretic) hardness of SQ learning parities. Proposition 3.5. The concept class EncThresh is (inefficiently) SQ learnable with polynomially many queries. As with C there are two cases. In the first case, the target distribution places nearly zero weight on examples with params = params r , and so the all-zeroes hypothesis is good. In the second case, the target distribution places noticeable weight on these examples, and our learner can use statistical queries to recover the comparison parameters params r bit-by-bit. Once the public parameters are recovered, our learner can determine a corresponding secret key by brute force. Lemma 3.6 below shows that any corresponding secret key -even one that is not actually sk r -suffices. The learner can then use binary search to determine the threshold value t. Proof. Let f t,r be the target concept, D be the target distribution, and α be the target error rate. With the statistical query (x × b → b, α/4), we can determine whether the all-zeroes hypothesis is accurate. That is, if we receive a value that is less than α/2, then Pr x∈D [f t,r (x) = 1] ≤ α. If not, then we know that Pr x∈D [f t,r (x) = 1] ≥ α/4, so D places significant weight on examples prefixed with params r . Suppose now that we are in the latter case. Let m = \|params\|. For i = 1, . . . , m, define ψ i (params, c, b) = 1 if params i = 1 and b = 1, and ψ i (params, c, b) = 0 otherwise. Then by asking the queries (ψ i , α/16), we can determine each bit params r i of params r . Now by brute force search, we determine a secret key sk for which (sk, params r ) ∈ Range(Gen). The recovered secret key sk may not necessarily be the same as sk r . However, the following lemma shows that sk and sk r are functionally equivalent: Lemma 3. 6. Suppose (Gen, Enc, Dec, Comp) is a strongly correct ORE scheme. Then for any pair (sk 1 , params), (sk 2 , params) ∈ Range(Gen), we have that Dec sk 1 (c) = Dec sk 2 (c) for all ciphertexts c. With the secret key sk in hand, we now conduct a binary search for the threshold t. Otherwise, if v 1 < v − α/2, we set the next threshold to t 2 = 3N/4, and if v 1 > v + α/2, we set the next threshold to t 2 = N/4. We recurse up to log N = = poly(k) times, yielding a good hypothesis for f t,r . Proof of Lemma 3. 6. Suppose the lemma is not true. First suppose that there exists a ciphertext c such that Dec(sk 1 , c) = p 1 < p 2 = Dec(sk 2 , c). Let c ∈ Enc(sk 1 , p 2 ). Then by strong correctness applied to the parameters (sk 1 , params), we must have Comp(params, c, c ) = "<". Now by strong correctness applied to (sk 2 , params), we must have Dec(sk 2 , c ) > p 2 . Thus, p 1 < Dec(sk 1 , c ) = p 2 < Dec(sk 2 , c ). Repeating this argument, we obtain a contradiction because the message space is finite. Now suppose instead that there is a ciphertext c for which Dec(sk 1 , c) = p ∈ [N ], but Dec(sk 2 , c) = ⊥. Let c ∈ Enc(sk 1 , p ) for some p > p. Then Comp(params, c, c ) = "<" by strong correctness applied to (params, sk 1 ). But Comp(params, c, c ) = "⊥" by strong correctness applied to (params, sk 2 ), again yielding a contradiction. ORE with Strong Correctness We now explain how to obtain ORE with strongly correct comparison, as all prior ORE schemes only satisfy the weaker notion of correctness. The lack of strong correctness is easiest to see with the scheme of Boneh et al. [BLR + 15]. The protocol is built from current multilinear map constructions, which are noisy. If the noise terms grow too large, the correctness of the multilinear map is not guaranteed. The comparison function in [BLR + 15] is computed by performing multilinear operations, and for correctly generated ciphertexts, the operations will give the right answer. However, there exist ciphertexts, namely those with very large noise, for which the comparison function gives an incorrect output. The result is that the comparison operation is not guaranteed to be consistent with decrypting the ciphertexts and comparing the plaintexts. As described in the introduction, we give a generic conversion from any ORE scheme with weakly correct comparison into a strongly correct scheme. We simply modify the encryption algorithm by adding a non-interactive zero-knowledge (NIZK) proof that the resulting ciphertext is well-formed. Then the decryption and comparison procedures check the proof(s), and only output a non-⊥ result (either decryption or comparison) if the proof(s) are valid. Instantiating our scheme. In our construction, we need the (weak) correctness of the underlying ORE scheme to hold with probability one. However, the existing protocols only have correctness with overwhelming probability, so some minor adjustments need to be made to the protocols. This is easiest to see in the ORE scheme of Boneh et al. [ [GGH13a] which may introduce errors. Therefore, the protocol described in [BLR + 15] only achieves the (weak) correctness property with overwhelming probability, whereas we will require (weak) correctness with probability 1 for the conversion. However, it is straightforward to generate the parameters for the protocol in such a way as to completely eliminate errors. Essentially, the parameters in the protocol have an error term that is generated by a (discrete) Gaussian distribution, which has unbounded support. Instead, we truncate the Gaussian, resulting in a noise distribution with bounded support. By truncating sufficiently far from the center, the resulting distribution is also statistically close to the full Gaussian, so security of the protocol with truncated noise follows from the security of the protocol with un-truncated noise. By truncating the noise distribution, it is straightforward to set parameters so that no errors can occur. It is similarly straightforward to modify current obfuscation candidates, which are also built from multilinear maps, to obtain perfect (weak) correctness by truncating the noise distributions. Thus, our scheme has instantiations using multilinear maps or iO. Conversion from Weakly Correct ORE We describe our generic conversion from an order-revaling encryption scheme with weak correctness using NIZKs. We will need the following additional tools: Perfectly binding commitments. A perfectly binding commitment Com is a randomized algorithm with two properties. The first is perfect binding, which states that if Com(m; r) = Com(m ; r ), then m = m . The second requirement is computational hiding, which states that the distributions Com(m) and Com(m ) are computationally indistinguishable for any messages m, m . Such commitments can be built, say, from any injective one-way function. Perfectly sound NIZK. A NIZK protocol consists of three algorithms: • Setup(1 λ ) is a randomized algorithm that outputs a common reference string crs. • Prove(crs, x, w) takes as input a common reference string crs, an NP statement x, and a witness w, and produces a proof π. • Ver(crs, x, π) takes as input a common reference string crs, statement x, and a proof π, and outputs either accept or reject. We make three requirements for a NIZK: • Perfect Completeness. For all security parameters λ and any true statement x with witness w, Pr[Ver(crs, x, π) = accept : crs ← Setup(1 λ ); π ← Prove(crs, x, w)] = 1. • Perfect Soundness. For all security parameters λ, any false statement x and any (invalid) proof π, Pr[Ver(crs, x, π) = accept : crs ← Setup(1 λ )] = 0. • Computational Zero Knowledge. There exists a simulator S 1 , S 2 such that for any computationally bounded adversary A, the quantity is negligible, where Sim(crs, τ, x, w) outputs S 2 (crs, τ, x) if w is a valid witness for x, and Sim(crs, τ, x, w) = ⊥ if w is invalid. NIZKs satisfying these requirements can be built from bilinear maps [GOS12]. The Construction We now give our conversion. Let (Setup, Prove, Ver) be a perfectly sound NIZK and (Gen , Enc , Dec , Comp ) and ORE with weakly correct comparison. We will assume that Enc is deterministic; if not, we can derandomize Enc using a pseudorandom function. Let Com be a perfectly binding commitment. We construct a new ORE scheme (Gen, Enc, Dec, Comp) with strongly correct comparison: • Gen(1 λ , 1 ): run (sk , params ) ← Gen (1 λ , 1 ). Let σ = Com(sk; r) for randomness r, and run crs ← Setup(1 λ ). Then the secret key is sk = (sk , r, crs) and the public parameters are params = (params , σ, crs). • Enc(sk, m): Compute c = Enc (sk , m). Let x c be the statement ∃m,ŝk ,r : σ = Com(ŝk ,r) ∧ c = Enc (ŝk ,m). Run π c = Prove(crs, x c , (m, sk , r) ). Output the ciphertext c = (c , π c ). • Dec(sk, c): Write c = (c , π c ). If Ver(crs, x c , π c ) = reject, output ⊥. Otherwise, output m = Dec (sk , c ). • Comp(params, c 0 , c 1 ); white c b = (c b , π c b ) and params = (params , σ, crs). If Ver(crs, x c b , π c b ) = reject for either b = 0, 1, then output ⊥. Otherwise, output Comp (params , c 0 , c 1 ). Correctness. Notice that, for each plaintext m, the ciphertext component c = Enc (sk , m) is the unique value such that Dec(sk, (c , π)) = m for some proof π. Moreover, the completeness of the zero knowledge proof implies that Enc(sk, m) outputs a valid proof. Decryption correctness follows. For strong comparison correctness, consider two ciphertexts c 0 , c 1 where c b = (c b , π c b ). Suppose both proofs π c b are valid, which means that verification passes when running Comp and so A Separation for Representation Learning In this section, we show how to construct a concept class ValidSig that separates efficient representation learning from efficient private representation learning, assuming only the existence of one-way functions. Here by "representation learning" we mean a restricted form of proper learning where a learner must output a particular representation (i.e. encoding) of a hypothesis h in the concept class C. As with proper learning, this is a natural syntactic restriction to place on a learner: for instance, if one wants to learn linear threshold functions (LTF), it makes sense to require a learner to produce the actual coefficients of an LTF, rather than an arbitrary circuit that happens to compute an LTF. The construction is based on the following elegant idea due to Kobbi Nissim [Nis14]. Suppose H : D → R is a cryptographic hash function with the property that given x 1 , . . . , x n with y = H(x 1 ) = · · · = H(x n ), it is infeasible for an efficient adversary to find another x for which H(x) = y. Consider the concept class HashPoint consisting of the concepts f x (x ) = 1 if H(x) = H(x ) 0 otherwise. for every x ∈ R. The representation of a concept f x is the point x. The concept class HashPoint is very easy to learn (by representation) without privacy: a learner can identify any positive example x i and output the representation x i . Since H(x i ) = H(x), the concept f x i is actually equal to the target concept f x . On the other hand, a learner that identifies an index x * for which f x * = f x cannot be differentially private, since the security of the hash function means it is infeasible to produce such an x * that is not present in the sample. Note that this argument breaks down if one tries to show that HashPoint is not privately properly learnable. While it is infeasible to privately produce a representation x * for which f x * is a good hypothesis, the hypothesis h(x) = χ(H(x) = h(x i )) is equal as a function to every good f x * . Moreover, this hypothesis can be constructed privately as long as the sample contains sufficiently many positive examples. We make this discussion formal by constructing a concept class ValidSig based on super-secure digital signature schemes, which can be constructed from one-way functions. Our use of signatures to derive hardness results for private proper learning is very analogous to prior hardness results for synthetic data generation [DNR + 09, UV11]. Definition 5. 1. A digital signature scheme is a triple of algorithms (Gen, Sign, Ver) where • Gen(1 λ ) produces a key pair (sk, vk). • Sign(sk, m) takes the private signing key sk and a message m ∈ {0, 1} * and produces a signature σ for the message m. • Ver(vk, m, σ) takes the public verification key vk, a message m, and a signature σ, and (deterministically) outputs a bit indicating whether σ is a valid signature for m. The correctness property of a digital signature scheme is that for every (sk, vk) ← R Gen(1 λ ), every message m ∈ {0, 1} * , and every signature σ ← R Sign(sk, m), we have Ver(vk, m, σ) = 1. Definition 5. 2. A digital signature scheme is super-secure under adaptive chosen-plaintext attacks if all efficient adversaries A win the following weak forgery game with negligible probability: • The challenger samples (sk, vk) ← R Gen(1 λ ). • The adversary A is given vk and oracle access to Sign(sk, ·). It adaptively queries the signing oracle, obtaining a sequence of message-signature pairs A. It then outputs a forgery (m * , σ * ). • The value of the game is 1 iff Ver(vk, m * , σ * ) = 1 and (m * , σ * ) / ∈ A. It is known that super-secure digital signature schemes can be constructed from one-way functions [NY89,Rom90,KK05,Gol04]. We now describe our concept class ValidSig. Let (Gen, Sign, Ver) be a super-secure digital signature scheme. We define a concept class ValidSig as follows. Fix the message length . For convenience, we also include the all-zeroes hypothesis in ValidSig, with representation ⊥. Theorem 5. 3. Let α, β > 0. There exists a proper PAC learning algorithm L for the concept class ValidSig achieving error α and confidence 1 − β. Moreover, L is efficient (running in time polynomial in the parameters k, 1/α, log(1/β)). Algorithm 3 Learner L for ValidSig Definition 2.2 (PAC Learning[Val84]). Algorithm L : (X × {0, 1}) n → H is an (α, β)-accurate PAC learner for the concept class C using hypothesis class H with sample complexity n if for all target concepts c ∈ C and all distributions D on X, given an input of n samples S = ( - Comp plain (m 0 , m 1 ) is just the plaintext comparison function. That is, for m 0 < m 1 , Comp plain (m 0 , m 1 ) = " < ", Comp plain (m 1 , m 0 ) = " > ", and Comp plain (m 0 , m 0 ) = " = ". -Comp ciph (sk, c 0 , c 1 ) is a ciphertext comparison function which uses the secret key. If first computes m b = Dec(sk, c b ) for b = 0, 1. If either m 0 = ⊥ or m 1 = ⊥ (in other words, if either decryption failed), then Comp ciph outputs ⊥. If both m 0 , m 1 = ⊥, then the output is Comp plain (m 0 , m 1 ). Now we define our comparison correctness notions: -Weakly Correct Comparison: This informally requires that comparison is consistent with encryption. For all security parameters λ, message lengths , and messages m 0 , m 1 ∈ {0, 1} , Pr Comp(params, c 0 , c 1 ) = Comp plain (m 0 , m 1 ) : (sk, params) ← Gen(1 λ , 1 ) c b ← Enc(sk, m b ) = 1. An ORE scheme (Gen, Enc, Dec, Comp) is statically single-challenge secure if, for all efficient adversaries A, \| Pr[W 0 ] − Pr[W 1 ]\| is negligible, where W b is the event that A outputs 1 in the following experiment: Theorem 2. 9 . 9(Gen, Enc, Dec, Comp) is statically secure if and only if it is statically single-challenge secure. First , notice that all the m i are in order since both sequences m order. Second, the only difference between Hybrid (j − 1) and Hybrid j is that m j = m in Hybrid j. Thus, single-challenge security implies that each adjacent hybrid is indistinguishable. Moreover, for j where m Again, notice that all the m i are in order. Moreover, the only different between Hybrid (2q−j) and Hybrid (2q − j + 1) is that m j = min(m in Hybrid (2q − j) and m j = m (R) j in Hybrid (2q − j + 1). Thus, single-challenge security implies that each adjacent hybrid is indistinguishable. Moreover, for j where m the two hybrids are actually identical.3 The Concept Class EncThresh and its LearnabilityLet (Gen, Enc, Dec, Comp) be a statically secure ORE scheme with strongly correct comparison. We define a concept class EncThresh, which intuitively captures the class of threshold functions where examples are encrypted under the ORE scheme. Throughout this discussion, we will take N = 2 and regard the plaintext space of the ORE scheme to be [N ] = {1, . . . , N }. Ideally we would like, for each threshold t ∈ [N + 1] and each (sk, params) ← Gen(1 λ ), to define a concept f t,sk,params (c) = 1 if Dec sk (c) < t 0 otherwise. f t,r (params, c) = 1 if (params = params r ) ∧ (Dec(sk r , c) = ⊥) ∧ (Dec(sk r , c) < t) 0 otherwise. Algorithm 1 1Learner L for EncThresh1. Request examples {(params 1 , c 1 , b 1 ), . . . , (params n , c n , b n )} for n = log(1/β)/α .2. Identify an i for which b i = 1 and set params * = params i ; if no such i exists, return h ≡ 0.3. Let G = {j : params j = params * , b j = 1}.Let j * ∈ G be an index with Comp(params * , c j , c j * ) ∈ {<, =, ⊥} for all j ∈ G.4. Return h defined byh(params, c) = 1 if (params = params * ) ∧ (Comp(params * , c, c j * ) ∈ {<, =}) 0 otherwise. chosen from B i . Then if h can distinguish B i−1 from B i , the adversary can distinguish the two sequences. Unfortunately, this approach fails for a somewhat subtle reason. The hypothesis h is only guaranteed to distinguish B i−1 from B i with probability ξ. If h fails to distinguish the buckets -or distinguishes them in the opposite direction -then the adversary's advantage is lost. [N ]. Let m 1 , . . . , m n ← R [N ], and let m 1 ≤ · · · ≤ m n be the result of sorting the m i . Let m 0 = 0 and m n+1 = N . Since n = poly(k) Lemma 3 . 3 . 33Let K = 8n 2 γ 2 log(9n/ξ). For each i = 0, . . . , n, let 0 L , m 1 L 01) and (m 0 R , m 1 R ) as follows.Let B 0 = (m i * −1 , m i * ) and B 1 = (m i * , m i * +1 ). Sample m 0 L ≤ m 1 L at random from the same B j , for a random choice of j ∈ {0, 1}. Sample m 0 R ← R B 0 and m 1 R ← R B 1 .3. Challenge on the pair of sequences m 0 , m 1 , . . . , m i * −1 , m 1 L , m 2 L , m i * , . . . , m n and m 0 , m 1 , . . . , m i * −1 , m 1 R , m 2 R , m i * , . . . , m n , receiving ciphertexts c 1 , . . . , c 0 i * , c 1 i * , . . . , c n . For j = j * , let c j = c π(j) so that c j is an encryption of m j . 4. Set t = N/2 and let Recall that we have an estimate v for the weight that f t,r places on positive examples, i.e. \|v − Pr x∈D [f t,r (x) = 1]\| ≤ α/4. Starting at t 1 = N/2, we issue the query (ϕ 1 , α/4) where ϕ 1 (params, c, b) = 1 iff params = params r and Dec(sk, c) < t. Let h t 1 denote the hypothesis h t 1 (params, c) = 1 if (params = params r ) ∧ (Dec(sk, c) = ⊥) ∧ (Dec(sk, c) < t 1 ) 0 otherwise. Thus, the query (ϕ 1 , α/4) approximates the weight h t 1 places on positive examples. Let the answer to this query be v 1 . If \|v 1 − v\| ≤ α/2, then we can halt and output the good hypothesis h t 1 . Pr [ A [Prove(crs,·,·) (crs) = 1 : crs ← Setup(1 λ )] − Pr[A Sim(crs,τ,·,·) (crs) = 1 : (crs, τ ) ← S 1 (1 λ )] For every (vk, m, σ) with m ∈ {0, 1} and Ver(vk, m, σ) = 1, define the concept f vk,m,σ (vk , m , σ ) = 1 if (vk = vk ) ∧ (Ver(vk, m , σ ) = 1) 0 otherwise. simple but important observation about L Thresh is that it is completely oblivious to the actual numeric values of its examples, or even to the fact that the domain is [N ]. In fact, L Thresh works equally well on any totally-ordered domain on which it can efficiently compare examples. In an extreme case, the learner L Thresh still works when its examples are encrypted under an orderrevealing encryption (ORE) scheme, which guarantees that L Thresh is able to learn the order of its examples, but nothing else about them. Up to small technical modifications, our concept class EncThresh is exactly the class Thresh where examples are encrypted under an ORE scheme. BLR + 15]. The Boneh et al. scheme uses noisy multilinear maps More generally, any totally-ordered plaintext space can be considered We now give the proof of Theorem 5. 4. Proof. We construct an example reidentification scheme for ValidSig as follows. The algorithm Gen ex samples (sk, vk) ← R Gen(1 λ ), a message m ∈ {0, 1} , and a signature σ ← R Sign(sk, m), yielding a concept f vk,m,σ . Let D be the distribution of (vk, m, Sign(sk, m)) for random m ← R {0, 1} . Gen ex then samples x 0 , x 1 , . . . , x n i.i.d. from D. Given a representation (vk * , m * , σ * ), the algorithm Trace ex simply identifies an index i for which x i = (vk * , m * , σ * ), and outputs ⊥ if none is found.We first verify completeness for this scheme. Let L be a learner for ValidSig using n examples. If the representation (vk * , m * , σ * ) produced by L represents an (1 − γ)-good hypothesis, then it must be the case that vk * = vk and Ver(vk, m * , σ * ) = 1. Thus, if L violates the completeness condition, it can be used to construct the weak forgery adversary A(Figure 4) that succeeds with noticeable probability ξ.Algorithm 4 Weak forgery adversary A Acknowledgements. We gratefully acknowledge Kobbi Nissim and Salil Vadhan for helpful discussions about this work, and also thank Salil Vadhan for suggestions on its presentation.Proof. We will prove security through a sequence of hybrids. Let A be an adversary with advantage in breaking the static security of (Gen, Enc, Dec, Comp).Hybrid 0. This is the real experiment, where σ ← Com(sk), crs ← Setup(1 λ ), and the proofs π c are answered using Prove and valid witnesses. A has advantage in distinguishing the left and right ciphertexts.Hybrid 1. This is the same as Hybrid 0, except that crs is generated as (crs, τ ) ← S 1 (1 λ ), and all proofs are generated using S 2 (crs, τ, ·). The zero knowledge property of (Setup, Prove, Ver) shows that this is indistinguishable from Hybrid 0.Hybrid 2. This is the same as Hybrid 1, except that σ ← Com(0). Since the randomness for computing σ is not needed for simulation, this change is undetectable using the hiding of Com.Thus the advantage of A in Hybrid 2 is at least − negl for some negligible function negl. Now consider the following adversary cB that attempts to break the security of (Gen , Enc , Dec , Comp ). B simulates A, and forwards the message sequences mproduced by A to its own challenger. In response, it receives params , and ciphertexts c i , where c i encrypts either m, for a random bit b chosen by the challenger. B now generates σ ← Com(0) and (crs, τ ) ← S 1 (1 λ ), and lets params = (params , σ, crs). It also computes π c i ← S 2 (crs, τ, x c i ), and defines c i = (c i , π c i ), and gives params and the c i to A. Finally when A outputs a guess b for b, B outputs the same guess b .We see that the view of A as a subroutine of B is exactly the same view as in Hybrid 2. Thus, b = b with probability at least − negl. The security of (Gen , Enc , Dec , Comp ) implies that this quantity, and hence , must be negligible. Thus A must have negligible advantage in breaking the security of (Gen, Enc, Dec, Comp). Comp (params , c 0 , c 1 ). Verification also passes when decrypting c b , and so Dec(sk, c b ) = Dec (sk , c b ). Comp, Comp(params, c 0 , c 1 ) = Comp (params , c 0 , c 1 ). Verification also passes when decrypting c b , and so Dec(sk, c b ) = Dec (sk , c b ). Since the proofs are valid, c b = Enc (sk , m b ) for some m b for both b = 0, 1. The weak correctness of comparison for. Comp ) implies that Comp (params , c 0 , c 1 ) = Comp plain. Gen , Enc , DecSince the proofs are valid, c b = Enc (sk , m b ) for some m b for both b = 0, 1. The weak correctness of comparison for (Gen , Enc , Dec , Comp ) implies that Comp (params , c 0 , c 1 ) = Comp plain (m 0 , m 1 ). Putting it all together, Comp(params, c 0 , c 1 ) = Comp ciph (sk, c 0 , c 1 ), as desired. Now suppose one of the proofs π c b are invalid. The decryption correctness of (Gen , Enc , Dec , Comp ) then implies that Dec(sk , c b ) = m b , and therefore Dec(sk, c b ) = m b . Thus Comp ciph (sk, c 0 , c 1 ) = Comp plain (m 0 , m 1 ). Then Comp(params, c 0 , c 1 ) = ⊥ and Dec(skThe decryption correctness of (Gen , Enc , Dec , Comp ) then implies that Dec(sk , c b ) = m b , and therefore Dec(sk, c b ) = m b . Thus Comp ciph (sk, c 0 , c 1 ) = Comp plain (m 0 , m 1 ). Putting it all together, Comp(params, c 0 , c 1 ) = Comp ciph (sk, c 0 , c 1 ), as desired. Now suppose one of the proofs π c b are invalid. Then Comp(params, c 0 , c 1 ) = ⊥ and Dec(sk, c b ) = This means Comp ciph (sk, c 0 , c 1 ) = ⊥ = Comp(params, c 0 , c 1 ), as desired. ⊥ , ⊥. This means Comp ciph (sk, c 0 , c 1 ) = ⊥ = Comp(params, c 0 , c 1 ), as desired. Then we use the hiding property of the commitment to replace σ with a commitment to 0. At this point, the entire game can be simulated using an Enc oracle, and so the security reduces to the security of Enc. Security, To prove security, we first use the zero-knowledge simulator to simulate the proofs π c without using a witness (namely, the secret decryption key)Security. To prove security, we first use the zero-knowledge simulator to simulate the proofs π c without using a witness (namely, the secret decryption key). Then we use the hiding property of the commitment to replace σ with a commitment to 0. At this point, the entire game can be simulated using an Enc oracle, and so the security reduces to the security of Enc . Enc , Dec , Comp ) is a (statically) secure ORE, (Setup, Prove, Ver) is computationally zero knowledge, and Com is computationally hiding, then (Gen, Enc, Dec, Comp) is a statically secure ORE. Theorem 4.1. If (GenTheorem 4.1. If (Gen , Enc , Dec , Comp ) is a (statically) secure ORE, (Setup, Prove, Ver) is com- putationally zero knowledge, and Com is computationally hiding, then (Gen, Enc, Dec, Comp) is a statically secure ORE. ((vk n , m n , σ n ), b n )} for n = log(1/β)/α . 2. Identify an i for which b i = 1 and return the representation (vk i , m i , σ i ). If no such i exists. . , Request examples {((vk 1 , m 1 , σ 1 ), b 1 ). return ⊥ representing the all-zeroes hypothesisRequest examples {((vk 1 , m 1 , σ 1 ), b 1 ), . . . , ((vk n , m n , σ n ), b n )} for n = log(1/β)/α . 2. Identify an i for which b i = 1 and return the representation (vk i , m i , σ i ). If no such i exists, return ⊥ representing the all-zeroes hypothesis. Fix a target concept f vk,m,σ ∈ ValidSig k and a distribution D on examples. Let POS denote the set of examples (vk , m , σ ) on which f vk,m,σ (vk , m , σ. Proof. Fix a target concept f vk,m,σ ∈ ValidSig k and a distribution D on examples. Let POS denote the set of examples (vk , m , σ ) on which f vk,m,σ (vk , m , σ Case 2: D places less than α weight on POS. If L gets a positive example. then the analysis of Case 1 applies. Otherwise, the all-zeroes hypothesis is α-goodCase 2: D places less than α weight on POS. If L gets a positive example, then the analysis of Case 1 applies. Otherwise, the all-zeroes hypothesis is α-good. We now prove the hardness of properly privately learning ValidSig by constructing an example. We now prove the hardness of properly privately learning ValidSig by constructing an example Sign, Ver) be a super-secure digital signature scheme. Then there exists an (efficient) (α = 1 − γ, ξ)-example reidentification scheme (Gen ex , Trace ex ) for representation learning the concept class ValidSig. 1. Query the signing oracle on random messages m 1. ) and ξ(n) be noticeable functions. Let (Gen. Theorem 5.4. Let γ(n. m n ← R {0, 1} , obtaining signatures σ 1 , . . . , σ nTheorem 5.4. Let γ(n) and ξ(n) be noticeable functions. Let (Gen, Sign, Ver) be a super-secure digital signature scheme. Then there exists an (efficient) (α = 1 − γ, ξ)-example reidentification scheme (Gen ex , Trace ex ) for representation learning the concept class ValidSig. 1. 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[ "Universality of the fully connected vertex in Laplacian continuous-time quantum walk problems", "Universality of the fully connected vertex in Laplacian continuous-time quantum walk problems" ]	[ "Luca Razzoli [email protected] \nDipartimento di Scienze Fisiche, Informatiche e Matematiche\nUniversità di Modena e Reggio Emilia\nI-41125ModenaItaly\n", "Paolo Bordone [email protected] \nDipartimento di Scienze Fisiche, Informatiche e Matematiche\nUniversità di Modena e Reggio Emilia\nI-41125ModenaItaly\n\nCentro S3\nCNR-Istituto di Nanoscienze\nI-41125ModenaItaly\n", "Matteo G A Paris [email protected] \nDipartimento di Fisica Aldo Pontremoli\nQuantum Technology Lab\nUniversità degli Studi di Milano\nI-20133MilanoItaly\n\nINFN\nSezione di Milano\nI-20133MilanoItaly\n" ]	[ "Dipartimento di Scienze Fisiche, Informatiche e Matematiche\nUniversità di Modena e Reggio Emilia\nI-41125ModenaItaly", "Dipartimento di Scienze Fisiche, Informatiche e Matematiche\nUniversità di Modena e Reggio Emilia\nI-41125ModenaItaly", "Centro S3\nCNR-Istituto di Nanoscienze\nI-41125ModenaItaly", "Dipartimento di Fisica Aldo Pontremoli\nQuantum Technology Lab\nUniversità degli Studi di Milano\nI-20133MilanoItaly", "INFN\nSezione di Milano\nI-20133MilanoItaly" ]	[]	A fully connected vertex w in a simple graph G of order N is a vertex connected to all the other N − 1 vertices. Upon denoting by L the Laplacian matrix of the graph, we prove that the continuous-time quantum walk (CTQW)with Hamiltonian H = γL-of a walker initially localized at \|w does not depend on the graph G. We also prove that for any Grover-like CTQW-with Hamiltonian H = γL + w λ w \|w w\|-the probability amplitude at the fully connected marked vertices w does not depend on G. The result does not hold for CTQW with Hamiltonian H = γA (adjacency matrix). We apply our results to spatial search and quantum transport for single and multiple fully connected marked vertices, proving that CTQWs on any graph G inherit the properties already known for the complete graph of the same order, including the optimality of the spatial search. Our results provide a unified framework for several partial results already reported in literature for fully connected vertices, such as the equivalence of CTQW and of spatial search for the central vertex of the star and wheel graph, and any vertex of the complete graph.	10.1088/1751-8121/ac72d5	[ "https://arxiv.org/pdf/2202.13824v2.pdf" ]	247,158,630	2202.13824	0b37482f6514b5baf34dfffa267b89a75d5fb693	Universality of the fully connected vertex in Laplacian continuous-time quantum walk problems 7 May 2022 Luca Razzoli [email protected] Dipartimento di Scienze Fisiche, Informatiche e Matematiche Università di Modena e Reggio Emilia I-41125ModenaItaly Paolo Bordone [email protected] Dipartimento di Scienze Fisiche, Informatiche e Matematiche Università di Modena e Reggio Emilia I-41125ModenaItaly Centro S3 CNR-Istituto di Nanoscienze I-41125ModenaItaly Matteo G A Paris [email protected] Dipartimento di Fisica Aldo Pontremoli Quantum Technology Lab Università degli Studi di Milano I-20133MilanoItaly INFN Sezione di Milano I-20133MilanoItaly Universality of the fully connected vertex in Laplacian continuous-time quantum walk problems 7 May 2022quantum walksquantum searchGrover searchquantum transportLaplacian matrixgraphs A fully connected vertex w in a simple graph G of order N is a vertex connected to all the other N − 1 vertices. Upon denoting by L the Laplacian matrix of the graph, we prove that the continuous-time quantum walk (CTQW)with Hamiltonian H = γL-of a walker initially localized at \|w does not depend on the graph G. We also prove that for any Grover-like CTQW-with Hamiltonian H = γL + w λ w \|w w\|-the probability amplitude at the fully connected marked vertices w does not depend on G. The result does not hold for CTQW with Hamiltonian H = γA (adjacency matrix). We apply our results to spatial search and quantum transport for single and multiple fully connected marked vertices, proving that CTQWs on any graph G inherit the properties already known for the complete graph of the same order, including the optimality of the spatial search. Our results provide a unified framework for several partial results already reported in literature for fully connected vertices, such as the equivalence of CTQW and of spatial search for the central vertex of the star and wheel graph, and any vertex of the complete graph. Introduction A quantum particle propagating on a discrete space, e.g., on a graph, performs a quantum walk, the quantum analog of classical random walk. Quantum walks are a wellestablished model [1], with already existing physical implementations [2]. Continuoustime quantum walks (CTQWs) were introduced in [3] as a quantum algorithm to traverse decision trees. In a CTQW the state of the walker evolves continuously in time according to the Schrödinger equation under a Hamiltonian which respects the topology of the graph considered. The graph is mathematically represented by the Laplacian matrix L = D − A, which encodes the degree D and the adjacency A of the vertices. Hence, the matrices L and A are usually taken as generators of a CTQW. For regular graphs, A and L are equivalent, since all the vertices have the same degree and thus D is proportional to the identity. For irregular graphs, instead, A and L are not equivalent in general, but it is possible to recover the same probability distributions for certain graphs and depending on the initial states [4]. CTQWs walks inherit the versatility of application from their classical ancestors, but the peculiar features arising from their quantum nature-e.g., the superposition of the quantum walker in their path-make them suitable candidates not only for modeling physical processes, such as coherent transport in complex networks [5] even in biological system [6], but also for applications in quantum technologies. Indeed, they are of use in studying perfect state transfer in quantum spin networks [7,8], which are of utmost importance for quantum communication, they can be used to develop quantum algorithms, such as spatial search [9,10,11] and to solve K-SAT problems [12], and they are universal for quantum computation [13,14]. A number of works have reported equivalent results for Laplacian CTQWs when the fully connected vertex is involved. By fully connected vertex we mean a vertex which is adjacent (connected) to all the other vertices of the graph, as shown in Figure 1. The dynamics of the central vertex of the star graph and that of any vertex of the complete graph are equivalent, showing periodic perfect revivals and strong localization on the initial vertex [15], even in the presence of a perturbation λL 2 [16]. The spatial search of a marked vertex on the complete graph or on the star graph, when the target is the central vertex, are equivalent [17], and the same qualitative results are observed even in the presence of weak random telegraph noise [18]. The quantum-classical dynamical distance is a fidelity-based measure introduced to quantify the differences in the dynamics of classical versus quantum walks on a graph. Such distance turns out to be the same for the complete, star, and wheel graphs when the central vertex is assumed as the initial state for the walker [19]. In this paper we prove the universality of the fully connected vertex in Laplacian CTQWs. This means that when the fully connected vertex of a graph is the initial state of the walk, or when it is the marked vertex (target) of a Grover-like CTQWs (those involved in spatial search or quantum transport), results do not depend on the considered graph G. In other words, those problems formulated on G of order N and on the complete graph of the same order, K N , are equivalent. The present work thus explains the equivalent results between star, wheel, and complete graphs already observed and reported in literature, generalizing the equivalence to the fully connected vertices of any simple graph. The paper is organized as follows. In Section 2 we recall the CTQW model. In Section 3 we briefly review the dimensionality reduction method for quantum walks [20], according to which in Section 4 we prove the equivalence of the Laplacian CTQW of a walker initially localized at a fully connected vertex in any simple graph. Instead, the corresponding CTQWs generated by the adjacency matrix do depend on the graph chosen. Then, in Section 5 we prove that the equivalence applies also to Grover-like CTQWs for a single fully connected marked vertex, focusing on spatial search and quantum transport. In Section 6 we generalize the result to the case of multiple marked vertices. Finally, we present our concluding remarks in Section 7. Continuous-time quantum walks A graph is a pair G = (V, E), where V denotes the non-empty set of vertices and E the set of edges. The order of the graph is the number of vertices, \|V \| = N . A simple graph is an undirected graph containing no self loops or multiple edges. It is mathematically represented by the Laplacian matrix L = D−A, where the adjacency matrix A (A vv = 1 if the vertices v and v are connected, 0 otherwise) is symmetric and describes the connectivity of G and D is the diagonal degree matrix with D vv = deg(v) =: d v the degree of vertex v. According to this, L is real, symmetric, positive semidefinite, and singular (L always admits the null eigenvalue because every row sum and column sum of L is zero, thus det(L) = 0). ‡ The CTQW is the propagation of a free quantum particle when confined to a discrete space, e.g., a graph. The CTQW on a graph G takes place on a N -dimensional Hilbert space H = span({\|v \| v ∈ V }), and the kinetic energy term −∇ 2 /2m is replaced ‡ There are a number of different, all related, definitions of Laplacian of a graph. Sometimes it is useful to normalize the Laplacian matrix L to mitigate the weight of highly connected vertices. Indeed, a large degree results in large diagonal entry, L vv = d v , which dominates the matrix properties because much larger than the off-diagonal entries, L vv = 0, 1. The two matrices commonly known as normalized graph Laplacians are defined as L rw := D −1 L (closely related to a random walk) and L sym : = D −1/2 LD −1/2 (symmetric matrix), with the convention that D −1 vv = 0 for d v = 0 (i.e. , v is an isolated vertex) [21,22]. by γL, where = 1 and γ ∈ R + is the hopping amplitude of the walk. The state of the walker obeys the Schrödinger equation i d dt \|ψ(t) = H\|ψ(t)(1) with Hamiltonian H = γL. Hence, a walker starting in the state \|ψ 0 ∈ H continuously evolves in time according to \|ψ(t) = U (t)\|ψ 0 ,(2) with U (t) = exp[−iHt] the unitary time-evolution operator. The probability to find the walker in a target vertex w is therefore \| w\| exp [−iHt] \|ψ 0 \| 2 . Dimensionality reduction method Method In most CTQW problems encoded on a graph G and a Hamiltonian H, the quantity of interest is the probability amplitude at a certain vertex of G. The graph often contains symmetries that allow us to simplify the problem, reducing the effective dimensionality of the latter. Indeed, the evolution of the system relevant to the problem actually occurs in a subspace, also known as Krylov subspace [23], of the complete N -dimensional Hilbert space H spanned by the vertices of G. This subspace contains the vertex of interest and it is invariant under the unitary time evolution. As a result, the original graph encoding the problem can be mapped onto an equivalent weighted graph of lower order, whose vertices are the basis states of the invariant subspace. The reduced Hamiltonian, i.e., H written in the basis of the invariant subspace, still fully describes the dynamics relevant to the given problem. We can determine the invariant subspace and its basis by means of the dimensionality reduction method for CTQW [20], which we briefly review. The unitary evolution (2) can be expressed as \|ψ(t) = ∞ n=0 (−it) n n! H n \|ψ 0 ,(3) so \|ψ(t) is contained in the subspace I(H, \|ψ 0 ) = span({H n \|ψ 0 \| n ∈ N 0 }). This subspace of H is invariant under the action of the Hamiltonian and, thus, also of the unitary evolution. Naturally, dim I(H, \|ψ 0 ) ≤ dim H = N . If the Hamiltonian is highly symmetrical, then only a small number of powers of H n \|ψ 0 are linearly independent, hence the dimension of I(H, \|ψ 0 ) can be much smaller than N . Let P be the projector onto I(H, \|ψ 0 ). Then U (t)\|ψ 0 = e −iH red t \|ψ 0 ,(4) where H red = P HP is the reduced Hamiltonian. We obtain this using the power series of U (t) and the fact that P 2 = P (projector), P \|ψ 0 = \|ψ 0 , and P U (t)\|ψ 0 = U (t)\|ψ 0 . For any state \|φ ∈ H, solution of the CTQW problem, we have φ\|U (t)\|ψ 0 = φ red \|e −iH red t \|ψ 0 ,(5) where \|φ red = P \|φ is the reduced state. Analogously, using the projector P onto the subspace I(H, \|φ ), we obtain φ\|U (t)\|ψ 0 = φ\|e −iH red t \|ψ 0 red ,(6) with H red = P HP and \|ψ 0 red = P \|ψ 0 . An orthonormal basis of I(H, \|φ ), say {\|e 1 , . . . , \|e m }, can be iteratively obtained, as follows: \|e 1 := \|φ , then \|e n+1 follows from orthonormalizing H\|e n with respect to the previously obtained basis states, {\|e k } k=1,...,n , i.e., \|u n+1 := H\|e n − n k=1 e k \|H\|e n \|e k ⇒ \|e n+1 := \|u n+1 \|u n+1 .(7) The procedure stops when we find the minimum m such that H\|e m ∈ span({\|e 1 , . . . , \|e m }). The projector onto I(H, \|φ ) is therefore P = m n=1 \|e n e n \|. Complete Graph As an example, we review the well-known reduced problem of the CTQW on the complete graph on N vertices, K N , when generated by the Laplacian matrix or by the adjacency matrix. Each pair of vertices is connected by an edge, so any vertex is fully connected and has degree N − 1. The adjacency matrix is (A K ) vv = 1 ∀v = v , the diagonal degree matrix is D K = (N − 1)I, where I is the identity operator, and the Laplacian matrix is L K = D K − A K . Suppose we want to study the CTQW of a walker initially localized at a certain vertex w or, alternatively, for walker starting from any other initial state, to compute the probability amplitude at w. The invariant subspace relevant to problem is I(L K , \|w ) = I(A K , \|w ) = span \|e 1 = \|w , \|e 2 = 1 √ N − 1 v =w \|v .(8) Writing L K and A K in this subspace, we find, respectively, the reduced Laplacian matrix L K,red = N − 1 − √ N − 1 − √ N − 1 1 ,(9) and the reduced adjacency matrix [20] A K,red = 0 √ N − 1 √ N − 1 N − 2 .(10) It is worth noticing that, consistently with L K = D K − A K , we have L K,red = D K,red − A K,red , since D K written in the basis (8) is D K,red = (N − 1)I 2×2 . The steps required to obtain the orthonormal basis (8), the reduced Laplacian matrix (9), and the reduced adjacency matrix (10) for the complete graph are the same as those presented, in a more general case, in the proofs of Theorem 1 and Proposition 1, to which we refer the reader for details. Universality of a CTQW starting from a fully connected vertex In this section we discuss the CTQW generated either by the Laplacian matrix, H = γL, or by the adjacency matrix, H = γA. The hopping amplitude γ plays the role of a time scaling factor in the time-evolution operator exp[−iLγt] or exp[−iAγt]. Therefore, in the following we set γ = 1 so that, together with = 1, time and energy are dimensionless. Laplacian CTQW We will refer to the CTQW generated by the Laplacian matrix L as a Laplacian CTQW. e −iL G t \|w = e −iL G,red t \|w ,(11) is entirely contained in the invariant subspace I(L G , \|w ) = span \|e 1 = \|w , \|e 2 = 1 √ N − 1 v =w \|v ,(12) and is generated by the reduced Laplacian matrix L G,red = N − 1 − √ N − 1 − √ N − 1 1 .(13) Remark 1. We emphasize that dim I(L G , \|w ) = 2 ≤ dim H = N independently of N and of the graph considered. Theorem 1 generalizes what already known for the complete graph in Section 3.2, proving that the CTQW of the fully connected vertex \|w is independent of the graph. Proof. Let H be the N -dimensional Hilbert space of a quantum walker on G. The time evolution of the state \|w generated by L G , exp [−iL G t] \|w , belongs to a subspace of H, I(L G , \|w ) := span({L n G \|w \| n ∈ N 0 }) .(14) The proof makes use of the dimensionality reduction method (Section 3) and consists of two parts. (i) First, we prove Equation (12). (ii) Second, we prove Equation (13). Therefore, if the CTQW of a fully connected vertex w on any graph G satisfy these two conditions, then the statement (11) follows from Equation (4). (i) The first basis state is \|e 1 = \|w . Then we consider L G \|e 1 = (N − 1)\|w − v =w \|v =: (N − 1)\|e 1 − √ N − 1\|e 2 ,(15) where we have used the fact that w is adjacent to all the other vertices, d w = N − 1. The basis state \|e 2 follows from orthonormalizing L G \|e 1 with respect to the previous basis state, \|e 1 . To find the next basis state, we compute L G \|e 2 and then we orthonormalize it with respect to the previous basis states. To compute the projections e n \|L G \|e 2 , with n = 1, 2, it is convenient to use the definition of Laplacian matrix. From Equation (15) we have that e 1 \|L G \|e 2 = − √ N − 1 ,(16) and e 2 \|L G \|e 2 = 1 N − 1 v,v =w (D vv − A vv ) = 1 N − 1 v =w d v − (2M − 2d w ) = 1 N − 1 [2M − (2M − d w )] = 1 ,(17)because D is diagonal, D vv = 0 for v = v , and v,v =w A vv = v∈V v =w A vv − v =w A wv = v,v ∈V A vv − v∈V A vw − d w = 2M − 2d w ,(18)since v =v A vv = v ∈V A vv = d v in a graph with no self loops (a vertex is not adjacent to itself), as in the present case. Summing all the elements of the adjacency matrix, as well as summing the degrees, means counting the edges twice, v,v ∈V A vv = v∈V d v = 2M with M the number of edges. In graph theory the latter is known as the degree sum formula and it implies the handshaking lemma. We can now prove that L G \|e 2 = − √ N − 1\|e 1 + \|e 2 ,(19) therefore that L n G \|w ∈ span({\|e 1 , \|e 2 })∀n ∈ N 0 , by showing that \|λ := (L G − I)\|e 2 + √ N − 1\|e 1 = 0 ,(20) where I is the identity. First, we project it onto \|w w\|λ = 1 √ N − 1 v =w (d v − 1)δ wv − v =w A wv + N − 1 = 1 √ N − 1 (0 − d w + N − 1) = 0 ,(21) and then we project it onto any other vertex state, \|v = w , v \|λ = 1 √ N − 1 v =w (d v − 1)δ v v − v =w A v v + 0 = 1 √ N − 1 d v − 1 − v∈V A v v − A v w = 1 √ N − 1 [d v − 1 − (d v − 1)] = 0 ,(22) where A v w = 1 because w is adjacent to all the other vertices. This proves Equation (20), because the w-th component and any other component, v = w, are null. The statement (12) follows. (ii) We can easily prove Equation (13) by taking the matrix elements (L G,red ) jk := e j \|L G \|e k = (L G,red ) kj ,(23) with j, k = 1, 2, from (i). To summarize, the time evolution of the fully connected vertex state \|w always belongs to the subspace (12) and is fully described by the reduced generator (13) indipendently of the graph G considered. This proves the statement (11), concluding the proof. Corollary 1. Let us consider the Laplacian CTQWs on a graph G 1 and on a graph G 2 both of order N with a fully connected vertex w. Let us assume that the initial states are \|ψ 0,G 1 and \|ψ 0,G 2 , respectively. Then, the probability amplitude of finding the walker at w is the same, w\| exp [−iL G 1 t] \|ψ 0,G 1 = w\| exp [−iL G 2 t] \|ψ 0,G 2 , provided that the two initial states have the same projection onto the subspace I(L G 1 , \|w ) (12). Proof. This directly follows from Equation (6), with \|φ = \|w , and Theorem 1. Adjacency CTQW We will refer to the CTQW generated by the adjacency matrix A as an adjacency CTQW. Proof. The proof makes use of the dimensionality reduction method (Section 3) and consists of three parts. (i.a) First, we prove that dim I(A G , \|w ) ≥ 2 = dim I(A K , \|w ) ,(24) where the subscript K refers to the complete graph and, as known, I(A K , \|w ) is (8). This is a first indication that the CTQW of \|w generated by A G and A K are not equivalent, in general, revealing a first dependence on the graph considered. (i.b) In particular, if the graph G has more than one fully connected vertex and G = K N , then dim I(A G , \|w ) > 2. (ii) Second, we prove that even if I(A G , \|w ) = I(A K , \|w ), the two reduced generators are different, A G,red = A K,red , and thus lead to different time evolutions. (i.a) The first basis state is \|e 1 = \|w . Then we consider A G \|e 1 = v =w \|v =: √ N − 1\|e 2 ,(25) and \|e 2 follows from normalizing A G \|e 1 , as the latter is already orthogonal to \|e 1 . To find the next basis state, we compute A G \|e 2 and then we orthonormalize it with respect to the previous basis states. To compute the projections e n \|A G \|e 2 , with n = 1, 2, it is convenient to use the definition of adjacency matrix. From Equation (25) we have that e 1 \|A G \|e 2 = √ N − 1 ,(26) and, using Equation (18), e 2 \|A G \|e 2 = 1 N − 1 v,v =w A vv = 1 N − 1 (2M − 2d w ) = 2M N − 1 − 2 ,(27) where, we recall, M is the number of edges. We can now study whether or not the state \|α := A G − 2M N − 1 − 2 \|e 2 − √ N − 1\|e 1(28) is null. If null, then the invariant subspace has dimension 2, as A G \|e 2 is a linear combination of \|e 1 and \|e 2 , otherwise it has dimension > 2. First, we project the state (28) onto \|w , observing that w\|α = 0 from Equation (26), and then we project it onto any other vertex state, \|v = w , v \|α = 1 √ N − 1 v =w A v v − 2M N − 1 − 2 v =w δ v v = 1 √ N − 1 (d v − 1) − 2M N − 1 + 2 = 1 √ N − 1 d v + 1 − 2M N − 1 ,(29) where v =w A v v = v∈V A v v − A v w = d v − A v w and A v w = 1 because w is adjacent to all the other vertices. We have proved that the w-th component is null, but the other components v = w depend on v , so they are not null, in general. According to this, A G \|e 2 is not just a linear combination of \|e 1 and \|e 2 , further basis states are required, and so the statement (24) follows. (i.b) Let us now assume that there is another fully connected vertex w = w, d w = N − 1. Then, w\|α = 0 still holds and Equation (29) for v = w reads as w \|α = 1 (N − 1) 3/2 N 2 − N − 2M ,(30) which is null for N = (1 ± √ 1 + 8M )/2. However, N ∈ N requires the solution with the plus sign and √ 1 + 8M = 2m + 1, with m ∈ N 0 . Solving the latter condition with respect to m leads to m = [−1 ± (2m + 1)]/2. The only acceptable solution is m = m, i.e., any positive odd number 2m + 1 can be written as √ 1 + 8M . The degree sum formula, v∈V d v = 2M , allows us to write N = 1 2 1 + 1 + 4 v∈V d v .(31) Now we study whether the Equation (31) admits a solution. The presence of fully connected vertex make the graph connected, and d v ≥ 2 ∀v ∈ V since, by assumption, there are at least two fully connected vertices. The graph satisfying the minimal conditions is the graph with two fully connected vertices, w, w with d w = d w = N − 1, and with all the other N − 2 vertices connected only to w and w , d v = 2 ∀v = w, w . Hence, v∈V d v = 2(N − 1) + (N − 2)2, from which the right-hand side of Equation (31) is f (N ) = 1 2 1 + √ 16N − 23 .(32) If we assume that all the vertices are fully connected, then we get the complete graph. Hence, v∈V d v = N (N − 1), from which Equation (31) holds for any N . However, we are interested in graphs other than the complete one. There is no graph with only N − 1 fully connected vertices, as, otherwise, the remaining vertex is necessarily connected to all the others and so the graph is complete. There is, however, the graph with N − 2 fully connected vertices, obtained by removing one edge from the complete graph. The two non-fully connected vertices thus obtained have degree N − 2. Hence, v∈V d v = 2(N − 2) + (N − 2)(N − 1) , from which the right-hand side of Equation (31) is g(N ) = 1 2 1 + √ 4N 2 − 4N − 7 .(33) All the possible graphs on N vertices having a number 2 ≤ µ ≤ N − 2 of fully connected vertices fall within these two cases. In Figure 2 we study Equation (31), and we observe that there are no solutions, as none of the right-hand sides, f (N ) and g(N ), have intersection with the left-hand side, the line h(N ) = N . We have just proved that, under the assumption of having at least two fully connected vertices and G = K N , more than two basis states are required, therefore dim I(A G , \|w ) > 2. Indeed, while the w-th component is null, the components corresponding to the other fully connected vertex (or vertices) w = w (30) are not, thus A G \|e 2 is not just a linear combination of \|e 1 and \|e 2 . (ii) Let us now assume that there is only one fully connected vertex, w. Then, w\|α = 0 still holds and Equation (29) reads as v \|α = 1 (N − 1) 3/2 (N − 1)d v − v =w d v ,(34)since v∈V d v = v =w d v + (N − 1) = 2M . The above expression is null if (N − 1)d v = v =w d v and the latter condition must apply ∀v = w to make the state (28) null. Therefore, this condition implies that all the vertices, except w, must have the same degree d v . This is the case, e.g., of the star graph [ Figure 1(a)] or the wheel graph [ Figure 1(b)]. We have just proved that if a simple graph has one fully connected vertex, w, and deg(v) = d ∀v ∈ V \ {w}, then all the components of the state (28) are null. Hence, A n G \|w ∈ span({\|e 1 , \|e 2 })∀n ∈ N 0 , because A G \|e 2 = √ N − 1\|e 1 + 2M N − 1 − 2 \|e 2 ,(35) with j, k = 1, 2. Writing A G in the basis {\|e 1 , \|e 2 }, we find that A G,red = 0 √ N − 1 √ N − 1 2M N −1 − 2 .(37) The reduced generator A G,red (37) differs from A K,red (10) in the element (A red ) 22 . We observe that 2M N − 1 − 2 = N − 2 ⇔ M = N (N − 1) 2 ,(38) but only the complete graph has M = N (N − 1)/2 edges. Moreover, also the star graph and the wheel graph differ in that element, as M = N − 1 and M = 2(N − 1), respectively. So, the adjacency CTQW on the graphs which are regular except for the fully connected vertex w are neither equivalent among them, in general, nor to the adjacency CTQW on the complete graph. The reason is that the reduced generators, A G,red and A K,red , are different, as they depend on the number of edges M , and thus they lead to different time evolutions, which, however, belong to the same invariant subspace I(A G , \|w ) = I(A K , \|w ). To summarize, adjacency CTQWs do depend on the given graph G. Considering the adjacency CTQWs of the fully connected vertex state \|w either the time evolutions of it belong to different subspaces (see Equation (24)) or, otherwise, the reduced generators are different, as they depend on the number of edges M . This proves the Proposition 1, concluding the proof. Grover-like CTQWs with single marked vertex Corollary 2. Let w be a fully connected marked vertex of a simple graph G of order N with Laplacian matrix L. Let us consider the Grover-like CTQW where the quantity of interest is the probability amplitude at w. Let H = γL + λ\|w w\| (39) be the Hamiltonian encoding the problem, where γ ∈ R + , λ ∈ C, and H w := λ\|w w\| is the oracle Hamiltonian. Then, given the initial state \|ψ 0 , the probability amplitude at the marked vertex is w\| exp [−iH red t] \|ψ 0 red , where \|ψ 0 red = P \|ψ 0 with P the projector onto the invariant subspace I(H, \|w ) (12) relevant to the problem and the reduced Hamiltonian is H red = γ N − 1 + λ/γ − √ N − 1 − √ N − 1 1 .(40) Grover-like CTQWs on a graph G 1 and on a graph G 2 both of order N result in the same probability amplitude w\| exp [−iH G 1 t] \|ψ 0,G 1 = w\| exp [−iH G 2 t] \|ψ 0,G 2 provided that \|ψ 0,G 1 red = \|ψ 0,G 2 red . Proof. First, we prove that the invariant subspace I(H, \|w ) relevant to the problem is (12) and then that the reduced Hamiltonian is (40). The only effective parameter in the Hamiltonian (39) is the ratio λ/γ. Writing H = γH we understand that γ only determines the timescale of the evolution. Clearly I(H, \|w ) = I(H , \|w ) and \|e 1 = \|w . The oracle H w = (λ/γ)\|w w\| acts nontrivially only onto \|e 1 . Therefore, after orthonormalizing H \|e 1 with respect to \|e 1 , we find the second basis state, \|e 2 defined in Equation (12). We observe that H \|e 2 = L\|e 2 , as H w \|e 2 = 0, thus, according to the proof of Theorem 1, there are no further basis states. Hence, the dynamics relevant to the Grover-like CTQWs for the fully connected vertex belong to the subspace (12). The oracle Hamiltonian H w has a natural representation in such subspace H w,red = λ\|e 1 e 1 \| = λ 0 0 0 .(41) The reduced Hamiltonian (40) follows from summing the reduced Laplacian matrix (13) and the reduced oracle Hamiltonian (41). The remark on the equal probability amplitudes at w depending on the initial state follows from Equation (6), with \|φ = \|w . Grover-like CTQWs of great interest formulated as in the Corollary 2 are spatial search [9], λ = −1, and quantum transport, γ = 1 and λ = −iκ, with κ ∈ R + and i = √ −1 the imaginary unit [24]. In the former, solving the problem amounts to making the walker reach the state \|w with the maximum probability starting from the equal superposition of all vertices. In the latter, the quantity of interest is often the transport efficiency, η = 2κ +∞ 0 w\|ρ(t)\|w dt, the integrated probability of trapping at the vertex w, where ρ(t) is the density matrix of the walker. The transport efficiency can also be read as the complement to 1 of the probability of surviving within the graph, i.e., η = 1 − Tr [lim t→+∞ ρ(t)] [25]. We point out that whenever Im(λ) = 0 the Hamiltonian (39) is a non-Hermitian effective Hamiltonian that leads to non-unitary dynamics. This is useful to phenomenologically model certain processes like, if Im(λ) < 0, the dissipative dynamics in quantum optics [26] or the absorption of an excitation in light harvesting systems [27,28]. Spatial search The Hamiltonian encoding the problem is H = γL − \|w w\| ,(42) where the marked vertex, target of the search, is the fully connected vertex w. Since we have no information about the marked vertex, the initial state is commonly chosen as the equal superposition of all vertices, \|ψ 0 = v∈V \|v / √ N . The goal is to tune the hopping amplitude γ to maximize the probability amplitude at the marked vertex after a period of time of evolution. The time evolution of \|ψ 0 is entirely contained in I(H, \|w ), as \|ψ 0 = (\|e 1 + √ N − 1\|e 2 )/ √ N and so \|ψ 0 red = \|ψ 0 . Hence, not only the success probability of finding w, but also the entire dynamics of the system exp [−iHt] \|ψ 0 is the same on any simple graph G. According to Corollary 2, the results we have for the spatial search on the complete graph, a well-known problem [29,9,17], also apply to the search of w on other graphs. Therefore, if γ = 1/N (optimal value), then the walker reaches w with probability P w (t) = \| w\|e −iHt \|ψ 0 \| 2 = 1 N cos 2 t √ N + sin 2 t √ N (43) equal to one (certainty) at time t * = π √ N /2. Quantum transport The non-Hermitian effective Hamiltonian encoding the problem is H = L − iκ\|w w\| ,(44) where the trapping vertex is the fully connected vertex w and the trapping rate κ ∈ R + (λ = −iκ in (39)). We assume that the initial state is localized at a vertex different from w, \|ψ 0 = \|v = w . Under such assumptions, the transport efficiency of the complete graph is η K = 1/(N − 1) [28]. Hence, according to Corollary 2, all the graphs whose trap is the fully connected vertex w have η = η K . This follows from the fact that η is the overlap of the initial state with the basis states of the invariant subspace I(H, \|w ) [20,25], η = n=1,2 \| e n \|ψ 0 \| 2 = \| e 2 \|ψ 0 \| 2 = 1 N − 1 ,(45) and such invariant subspace is (12) for the problems and graphs under investigation, including the complete graph. Alternatively, we can prove this as follows. We define the integrated probability of trapping within the time interval [0, t], η(t) = 2κ t 0 w\|ρ(τ )\|w dτ ⇒ lim t→+∞η (t) = η ,(46) where w\|ρ(t)\|w = \| w\| exp [−iHt] \|v \| 2 . From Equation (6), the probability amplitude at w, w\|e −iHt \|v = w\|e −iH red t n=1,2 \|e n e n \|v = 1 √ N − 1 e 1 \|e −iH red t \|e 2 ,(47) is independent (i) of the graph under investigation and (ii) of the initial vertex state \|v , provided that v = w. (i) Follows from the fact that the graphs considered have the same basis states and the same reduced Hamiltonian (Corollary 2). (ii) Follows from the fact that all the vertices other than the trap only overlap with \|e 2 , which is the equal superposition of them, and have the same overlap with it. Therefore,η(t) does not depend on the graph under investigation or on the initial vertex state. As a result, in the limit of infinite time we also recover the same transport efficiency η = η K . In this problem the initial state is a vertex state \|v = w and cannot be written as linear combination of the two basis states. Therefore, it evolves differently depending on the given graph. Nevertheless, as just shown, it provides the same dynamics relevant to the problem, i.e., the same (trapped) population at w. be the Hamiltonian encoding the problem, where γ ∈ R + is constant and λ w ∈ C depends on the fully connected vertex. Then, given the initial state \|ψ 0 , the probability amplitude at a marked vertex is w\| exp [−iH red t] \|ψ 0 red , where \|ψ 0 red = P \|ψ 0 with P the projector onto the (µ + 1)-dimensional invariant subspace relevant to the problem, I = span {\|e k = \|w k } k , \|e µ+1 = 1 √ N − µ v / ∈W \|v ,(49) with k = 1, . . . , µ, and the reduced Hamiltonian is H red = γ         ∆ 1 + λ w 1 −1 · · · −1 − ∆ µ −1 . . . . . . . . . . . . . . . . . . . . . −1 . . . −1 · · · −1 ∆ 1 + λ wµ − ∆ µ − ∆ µ · · · · · · − ∆ µ µ         ,(50) where ∆ n = N − n and λ w = λ w /γ. Grover-like CTQWs on a graph G 1 and on a graph G 2 both of order N result in the same probability amplitude w\| exp [−iH G 1 t] \|ψ 0,G 1 = w\| exp [−iH G 2 t] \|ψ 0,G 2 provided that \|ψ 0,G 1 red = \|ψ 0,G 2 red . Remark 2. The dimensionality of the problem can be further reduced if subsets of vertices in W have the same λ, W α = {w ∈ W \| λ w = λ α } such that α W α = W and W α ∩ W β = ∅ ∀α = β. Instead of having one basis state per marked vertex, the equal superposition of all vertex states from the same set W α defines one basis state, \|e Wα = w∈Wα \|w / \|W α \|. This follows from the symmetries of the problem, as they allow to group together identically evolving vertices [30]. The reduced Hamiltonian (50) will change according to the new basis. Proof. We have more than one marked vertex and we cannot apply straightforwardly the dimensionality reduction method, because neither the initial state is unique (except in the spatial search) nor the target state is unique (multiple marked vertices). The Hamiltonian (48) inherits the symmetries of the graph (Laplacian matrix), but each oracle Hamiltonian H w breaks the symmetries involving the corresponding fully connected vertex w. Here we consider the Hamiltonian in the general framework, with no assumptions on λ's. During the time evolution of the system the population at the marked vertices is determined only by the Hamiltonian eigenstates having nonzero overlap with the marked vertices. Our aim is to prove that the subspace E spanned by those eigenstates is the subspace I (49). Let us define the subspace E := span ({\|ε \| H\|ε = ε\|ε ∧ w ∈ W \|ε = 0}) ,(51) where the \|ε are the minimum number of Hamiltonian eigenstates overlapping with the fully connected marked vertices w ∈ W . By mininum we mean that in the case of degenerate eigenspaces more than one eigenstate can have a nonzero overlap with the marked vertices. We can solve this ambiguity by choosing the eigenstate from this degenerate eigenspace which has the maximum possible overlap with the marked vertices and then by orthogonalizing all the other vectors within this eigenspace with respect to it. Therefore, after orthogonalization, the remaining eigenstates in the degenerate space would have zero overlap with the marked vertices. This approach to the problem is explained in [28], where it provides a simple way to compute the efficiency of transport to a trapping vertex on a graph (in the absence of dephasing and dissipation). ∈ E = v / ∈W v\|ε / ∈ E = 0 .(52) Proof. We study the eigenproblem H\|ε = ε\|ε by components in the basis of vertex states, projecting the eigenvalue equation onto a generic \|v v\|H\|ε − ε v\|ε = v [γ(D vv − A vv )] v \|ε + w∈W λ w v\|w w\|ε − ε v\|ε = (γd v − ε + λ w δ vw ) v\|ε − γ v A vv v \|ε = 0 .(53) Let us focus on \|ε / ∈ E and v ∈ W . Then, from Equation (53) , we have v =v v \|ε = v v \|ε = v / ∈W v \|ε = 0 ,(54) as v ∈ W is fully connected, thus A vv = 1 ∀v = v (A vv = 0). The index of summation can be extended to all the vertices v ∈ V or limited to v / ∈ W as v ∈ W \|ε / ∈ E = 0 by definition. v / ∈ W \|ε ∈ E = γ v A vv v \|ε γd v − ε = γ γd v − ε ξ + v / ∈W A vv v \|ε ,(56) where we have defined ξ := v ∈W v \|ε , which does not depend on the v / ∈ W chosen, and we have used A vv = 1 ∀v ∈ W . Indeed, v / ∈ W , thus v = v , and the vertices v are the fully connected ones. Let us start with a particular case. If the vertices v / ∈ W are only connected to the vertices w ∈ W , then d v = µ = \|W \| ∀v / ∈ W and A vv = 0 ∀v, v / ∈ W . Hence, all the components are constant and equal to v / ∈ W \|ε ∈ E = γξ γµ − ε ∀v / ∈ W .(57) In general, instead, we have a system ofμ := N − µ linear equations like (56) inμ unknowns x j := v j / ∈ W \|ε ∈ E , with 1 ≤ j ≤μ,                x 1 − γ γd 1 − ε k =1 A 1k x k = γξ γd 1 − ε . . . xμ − γ γdμ − ε k =μ Aμ k x k = γξ γdμ − ε .(58) We make the following ansatz on the solution x 1 = . . . = xμ = γξ γµ − ε ,(59) based on the analytical solution (57) for a particular case and on numerical evidence for general graphs, including the complete graph. Hence, focusing on the left-hand side of the j-th equation (58), we recover the identity with the right-hand side of the same equation γξ γµ − ε 1 − γ γd j − ε k =j A jk = γξ γµ − ε 1 − γ γd j − ε (d j − µ) = γξ γµ − ε γµ − ε γd j − ε = γξ γd j − ε ,(60) where k =j A jk = d j − µ because the index of summation does not run over all the vertices but runs over the non-marked vertices, hence we get the degree d j lowered by the number of fully connected marked vertices, µ. This identity applies to all j = 1, . . . , N − µ, i.e., to all v / ∈ W . This verifies the correctness of the ansatz (59) and therefore proves the Lemma. According to the previous Lemmas, we now prove that E = I. First, we prove that E ⊆ I. Let c := v / ∈ W \|ε ∈ E (Lemma 2). Then, we can write any \|ε ∈ E as \|ε = v \|v v\|ε = w∈W \|w w\|ε + c v / ∈W \|v = µ n=1 \|e n e n \|ε + c N − µ\|e µ+1 ∈ I , as it is a linear combination of the basis states (49). Second, we prove that I ⊆ E. We start with the basis states \|e j = \|w j for j = 1, . . . , µ \|e j = ε \|ε ε\|e j = ε∈E \|ε ε\|e j ∈ E ,(62) as it is a linear combination of the Hamiltonian eigenstates \|ε ∈ E. The summation over ε denotes the summation over all the Hamiltonian eigenstates. The second equality follows from w j \|ε / ∈ E = 0, by definition. The last basis state is \|e µ+1 = 1 √ N − µ ε v / ∈W \|ε ε\|v = 1 √ N − µ ε∈E v / ∈W \|ε ε\|v + ε / ∈E v / ∈W \|ε ε\|v = 1 √ N − µ ε∈E v / ∈W \|ε ε\|v + 0 ∈ E ,(63) where the last equality follows from Lemma 1. To summarize, I = E (49), since E ⊆ I and I ⊆ E, and this also implies that dim E = dim I = µ + 1. Now that we have the basis of the invariant subspace, we can write the reduced Hamiltonian. Given the Hamiltonian (48), the matrix elements of the reduced Hamiltonian for j, k = 1, . . . , µ are e j \|H\|e k = γ ( e j \|D\|e k − e j \|A\|e k ) + λ w j δ jk = γ(N − 1) + λ w j δ jk − γ ,(64) since d w = N − 1 and the vertices w j and w k are necessarily adjacent, e j \|H\|e µ+1 = −γ e j \|A\|e µ+1 = −γ √ N − µ v / ∈W A w j v = −γ N − µ ,(65) since the basis is orthonormal and A w j v = 1 ∀v / ∈ W ∧ ∀w ∈ W (w is fully connected). The last element is e µ+1 \|H\|e µ+1 = γ N − µ v,v / ∈W L vv = γµ .(66)Indeed, v / ∈W d v = v∈V d v − µ(N − 1) = 2M − µ(N − 1) ,(67) and v,v / ∈W A vv = v,v ∈V A vv − v∈W v ∈V A vv − v / ∈W v ∈W A vv = v,v ∈V A vv − v∈W d v − v / ∈W µ = 2M − µ(N − 1) − (N − µ)µ ,(68) since A vv = 1 ∀v = v ∧ v ∈ W and, we recall, µ = \|W \| and N = \|V \|. Hence, the reduced Hamiltonian (50) follows. Spatial search The Hamiltonian encoding the problem is H = γL − w∈W \|w w\| ,(69) where the marked vertices, the µ possible solutions of the spatial search, are the fully connected vertices w ∈ W . The oracles are unbiased, λ w = −1 ∀w ∈ W in Equation (48), as the solutions are usually assumed to be equivalent [31,32]. The goal is to tune the hopping amplitude γ to maximize success probability P W (t) = w∈W P w (t) after a period of time of evolution. The overall success probability P W is the sum of the probabilities at each w ∈ W because these are equivalent solutions. Solving the problem amounts to finding one of them. The initial value is P W (0) = µ/N , since the initial state is the equal superposition of all vertices. The time evolution of \|ψ 0 is entirely contained in I, as \|ψ 0 = ( µ j=1 \|e j + √ N − µ\|e µ+1 )/ √ N and so \|ψ 0 red = \|ψ 0 . Hence, not only the success probability P W (t), but also the entire dynamics of the system exp [−iHt] \|ψ 0 is the same on any simple graph G. According to Theorem 2, the results we have for the spatial search on the complete graph also apply to the search of w ∈ W on other graphs. The spatial search of µ marked vertices in the complete graph is known to be optimal (P W = 1) for γ = 1/N at time t * = (π/2) N/µ [32]. We point out that in [32] the CTQW is generated by the adjacency matrix, but this is equivalent to using the Laplacian matrix since the complete graph is regular. Hereafter we prove these results on the optimal search without assuming that the graph is complete. Spatial search is a suitable case study to apply Remark 2, as all the fully connected marked vertices have the same λ = −1. Therefore, the Hamiltonian (69) is invariant under permutations of the vertices in W . This symmetry allows us to further reduce the dimensionality of the problem by grouping together such identically evolving vertices in the state \|ẽ 1 = w∈W \|w / √ µ [30]. This state is the solution of the search and is the first basis state of the reduced invariant subspace. Then, it can be shown that H\|ẽ 1 = [γ(N − µ) − 1]\|ẽ 1 − γ µ(N − µ)\|ẽ 2 ,(70)H\|ẽ 2 = −γ µ(N − µ)\|ẽ 1 + γµ\|ẽ 2 ,(71) where \|ẽ 2 := v / ∈W \|v / √ N − µ is the second basis state. Therefore, the orthonormal states \|ẽ 1 and \|ẽ 2 span the invariant subspace relevant to the spatial search. The reduced Hamiltonian is H red = γ N − µ − 1/γ − µ(N − µ) − µ(N − µ) µ .(72) For γ = 1/N , the eigenvalues are ε ± = ± µ/N and the corresponding eigenstates are \|ε ± = √ N ± √ µ 2 √ N ∓ √ N − µ √ N ± √ µ \|ẽ 1 + \|ẽ 2 .(73) The success probability P W (t) = \| ẽ 1 \|e −iHt \|ψ 0 \| 2 = µ N cos 2 µ N t + sin 2 µ N t ,(74) is equal to one (certainty) at time t * = (π/2) N/µ. For µ = 1 we recover the resultsreduced Hamiltonian, success probability, and optimal time-for the spatial search of a single marked vertex discussed in Section 5.1. Quantum transport The non-Hermitian effective Hamiltonian encoding the problem is H = L − i w∈W κ w \|w w\| ,(75) where the µ trapping vertices are the fully connected vertices w ∈ W and have, in general, different trapping rates κ w ∈ R + (λ w = −iκ w in (48)). Accordingly, η := 2 w∈W κ w +∞ 0 w\|ρ(t)\|w dt [24]. We assume \|ψ 0 = \|v / ∈ W , therefore, according to the basis states (49), η = µ+1 n=1 \| e n \|ψ 0 \| 2 = \| e µ+1 \|ψ 0 \| 2 = 1 N − µ .(76) The transport efficiency improves as the number of fully connected traps µ increases and does not depend on the trapping rates. Changing the κ w affects the timescale on which the trapping occurs, not η as it is defined in the limit of infinite time. Moreover, η(t) = 2 w∈W κ w t 0 w\|ρ(τ )\|w dτ does not depend on the initial vertex state \|v , provided that v / ∈ W . Indeed, from Equation (6), the probability amplitude at w ∈ W , w\|e −iHt \|v = w\|e −iH red t µ+1 n=1 \|e n e n \|v = 1 √ N − µ e w \|e −iH red t \|e µ+1 , (77) is independent of v / ∈ W . For µ = 1 we recover the transport efficiency for the single trapping vertex discussed in Section 5.2. Conclusions In this paper we have investigated the role of the fully connected vertex w in continuoustime quantum walks (CTQWs) on simple graphs G of order N . In particular, we have analytically proved that when the dynamics of the walker is governed by the Laplacian matrix, the CTQW starting from the state \|w does not depend on the graph G considered and it is therefore equivalent, e.g., to the CTQW on the complete graph of the same order, K N . Instead, the corresponding adjacency CTQWs do depend on the graph considered. After that, we have investigated Grover-like CTQWs, i.e., systems with Hamiltonian of the form H = γL + w∈W λ w \|w w\|, where W is the subset of vertices made of µ fully connected marked vertices. Here the quantity of interest is the probability amplitude at the vertices w ∈ W . For these systems, we have analytically proved that the probability amplitudes of interest do not depend on the graph considered. In this case, the equivalence concerns the dynamics relevant to the computation of the probability amplitude at w, whereas the full dynamics of the walkers are not necessarily equivalent. As applications of the above results, we have considered spatial search of w ∈ W and quantum transport to w ∈ W . These problems on a simple graph G of order N inherit the results already known for the corresponding problems on the complete graph K N , independently of the considered graph. In particular, the spatial search of equivalent solutions (unbiased oracles) is optimal for γ = 1/N at time t * = (π/2) N/µ, and the full dynamics of the equal superposition of all vertices under the search Hamiltonian on G and on K N are equivalent. Regarding quantum transport of an initially localized excitation, the transport efficiency η increases with the number of fully connected traps as η = 1/ (N − µ), and does not depend on the initial vertex state \|v / ∈ W . Our proofs are based on the notion of Krylov subspaces. We have determined the invariant subspace relevant to the considered Laplacian problems, and the corresponding reduced Hamiltonian, thus reducing the dimensionality of the original problem. Whenever a fully connected vertex is the initial state of the CTQW or a marked vertex of a Grover-like CTQW, results do not depend on the graph considered. Hence, the universality of the fully connected vertex. One of most relevant consequences of our work is that the spatial search of fully connected vertices is always optimal and does not depend on the full topology of the involved graph. We can always find the solution with certainty and we know the parameters, γ and time, to achieve this result. This can be exploited, e.g., in finding the fully connected hubs of a network. Indeed, most often the hub is not connected to all the nodes, but serves as the center of star-shaped subnetwork [33] and our results hold when applied to the subnetwork. More generally, our results provide a coherent and unified framework to understand and extend several partial results already reported in literature for fully connected vertices, and pave the way for further development in the area, e.g., understanding whether universality survives in the presence of chirality [34,35]. Figure 1 . 1Examples of graphs of order N = 8 with at least one fully connected vertex w (orange colored), deg(w) = N − 1. (a) Star graph S N , (b) Wheel graph W N , and (c) Complete graph K N . (d)-(f) Random graphs. Theorem 1 . 1Let G = (V, E) be a simple graph on N = \|V \| vertices and M = \|E\| edges, with Laplacian matrix L G = D − A. Let w ∈ V be a fully connected vertex of G, with degree d w = N − 1. Then, the time-evolution of \|w under the Laplacian matrix is Proposition 1 . 1Let G = (V, E) be a simple graph on N = \|V \| vertices and M = \|E\| edges, with adjacency matrix A G . Let w ∈ V be a fully connected vertex of G, with degree d w = N − 1. Then, the adjacency CTQW of the state \|w does depend on the graph G considered. Figure 2 . 2Graphical solution of Equation(31). The left-hand side (LHS) is N (blue solid line, square). The right-hand side (RHS) is f (N ) (32) (orange dashed line, circles) or g(N ) (33) (yellow dotted line, diamonds). All the possible graphs on N vertices having a number 2 ≤ µ ≤ N − 2 of fully connected vertices result in a RHS which falls within these two cases. Results are shown for N ≥ 4, because the graphs for N = 2, 3 and µ = 2 would be the complete graph K 2 , K 3 , respectively. We observe that there are no intersections between the RHS and the LHS, as highlighted in the log-log plot of N − g(N ) in the inset. Note that, g(N ) ∼ N for large N , but g(N ) never reaches N . Therefore, Equation(31) has no solution. and therefore I(A G , \|w ) = I(A K , \|w ). From (i) we have the matrix elements (A G,red ) jk := e j \|A G \|e k = (A G,red ) kj , 6 . 6Grover-like CTQWs with multiple marked verticesTheorem 2. Let G = (V, E) be a simple graph of order N = \|V \| with M = \|E\| edges. Let W := {v ∈ V \| deg(v) = N − 1 ∧ v is marked} = ∅ bethe set of fully connected marked vertices and let µ := \|W \|, with 1 ≤ µ < N . Let us consider a Grover-like CTQW where the quantities of interest are the probability amplitudes at w ∈ W . Let H = γL + w∈W λ w \|w w\| (48) Lemma 1 . 1The Hamiltonian eigenstates that do not overlap with the marked vertices have projections onto the vertex states that sum to zero, v v\|ε / Lemma 2 . 2The Hamiltonian eigenstates that overlap with the marked vertices have constant projection onto the non-marked vertex states,v / ∈ W \|ε ∈ E = γ γµ − ε v ∈W v \|ε = const ∀v / ∈ W .(55)Proof. From Equation (53), the components under investigation are AcknowledgmentsWork done under the auspices of GNFM-INdAM. The authors thank Claudia Benedetti and Massimo Frigerio for helpful discussions. Portugal, Quantum Walks and Search Algorithms. New YorkSpringerPortugal R 2018 Quantum Walks and Search Algorithms (New York: Springer) . J Wang, K Manouchehri, Physical Implementation of Quantum Walks. SpringerWang J and Manouchehri K 2013 Physical Implementation of Quantum Walks (New York: Springer) . 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[ "An Automated Window Selection Procedure For DFT Based Detection Schemes To Reduce Windowing SNR Loss", "An Automated Window Selection Procedure For DFT Based Detection Schemes To Reduce Windowing SNR Loss" ]	[ "Ç Agatay Candan \nDepartment of Electrical and Electronics Engineering\nMiddle East Technical University (METU)\n06800AnkaraTurkey\n" ]	[ "Department of Electrical and Electronics Engineering\nMiddle East Technical University (METU)\n06800AnkaraTurkey" ]	[]	The classical spectrum analysis methods utilize window functions to reduce the masking effect of a strong spectral component over weaker components. The main cost of side-lobe reduction is the reduction of signal-to-noise ratio (SNR) level of the output spectrum. We present a single snapshot method which optimizes the selection of most suitable window function among a finite set of candidate windows, say rectangle, Hamming, Blackman windows, for each spectral bin. The main goal is to utilize different window functions at each spectral output depending on the interference level encountered at that spectral bin so as to reduce the SNR loss associated with the windowing operation. Stated differently, the windows with strong interference suppression capabilities are utilized only when a sufficiently powerful interferer is corrupting the spectral bin of interest is present, i.e. only when this window is needed. The achieved reduction in the windowing SNR loss can be important for the detection of low SNR targets.	null	[ "https://arxiv.org/pdf/1710.10200v1.pdf" ]	64,623,082	1710.10200	1812a167df63825949a2e4e53763d1438231e231	An Automated Window Selection Procedure For DFT Based Detection Schemes To Reduce Windowing SNR Loss 27 Oct 2017 Ç Agatay Candan Department of Electrical and Electronics Engineering Middle East Technical University (METU) 06800AnkaraTurkey An Automated Window Selection Procedure For DFT Based Detection Schemes To Reduce Windowing SNR Loss 27 Oct 2017Spectral AnalysisWindow FunctionPulse-Doppler RadarTarget Detection The classical spectrum analysis methods utilize window functions to reduce the masking effect of a strong spectral component over weaker components. The main cost of side-lobe reduction is the reduction of signal-to-noise ratio (SNR) level of the output spectrum. We present a single snapshot method which optimizes the selection of most suitable window function among a finite set of candidate windows, say rectangle, Hamming, Blackman windows, for each spectral bin. The main goal is to utilize different window functions at each spectral output depending on the interference level encountered at that spectral bin so as to reduce the SNR loss associated with the windowing operation. Stated differently, the windows with strong interference suppression capabilities are utilized only when a sufficiently powerful interferer is corrupting the spectral bin of interest is present, i.e. only when this window is needed. The achieved reduction in the windowing SNR loss can be important for the detection of low SNR targets. Introduction A common approach, if not the most common, in the spectral analysis of signals is the windowed Fourier transformation, which is the well known periodogram approach. In many applications, including signal-to-noise ratio (SNR) sensitive detection applications, a nominal window is pre-selected and applied irrespective of the operational SNR throughout the deployment of the detection system. Yet, it is known that the desired feature of window function (side-lobe suppression) comes at the cost of output SNR loss. As an example, the frequent Email address: [email protected] (Ç agatay Candan) choice of Hamming window results in an SNR loss of 1.35 dB, which can be important for the detection of targets at low SNR or equivalently for the extension of the instrumented range of a radar system. The main goal of this study is to pose the window selection problem as a hypothesis test and present an automated procedure that keeps the SNR loss due to windowing operation at a minimum. To do that, we assume that the system has a finite number of window functions at its disposal (say rectangle, Hamming, Blackman windows) and aims to select the most suitable window function for each spectral output, i.e. for each discrete Fourier transformation (DFT) bin, through the Bayesian hypothesis testing. The proposed method is based on a single snapshot of data and is applicable in all conventional detection schemes utilizing windowed DFT operation. The spectral analysis is a well established topic of statistical signal processing closely linked with several applications in speech processing, radar signal processing, remote sensing. The main goal of spectral analysis is to detect and accurately estimate the power of each spectral component of the input. The methods to this aim can be categorized as single and multiple snapshot methods as illustrated in Figure 1. The single snapshots methods are, in general, data-independent methods based on the windowed Fourier transformation. Multiple snapshot methods, such as the Capon's method, are data-dependent methods and based on the estimation and minimization (ensemble) average value interference at the output. The multiple snapshot methods use more information, such as the interference/signal autocorrelation function and yield superior results, in general. In certain applications, a-priori information on the signal and interference statistics may not be available or the sensing scheme may not suitable for an ensemble characterization. For the conventional pulse-Doppler radar systems, a single snapshot vector, composed of slow-time samples from a range cell, is available to detect the presence of a moving target in a range cell which can be contaminated with clutter, jammers and potentially other targets. The conventional processing chain typically includes a stage of windowed DFT. A suitable window, say Hamming window having a good side-lobe suppression (43 dB for the Hamming window) is selected to reduce the shadowing effect of strong undesired component over the target component. Unfortunately, this choice brings an SNR loss, which is 1.35 dB for the Hamming window that can be compensated by increasing the transmit power. This requires a factor of 10 (1.35/10) ≈ 1.36 fold increase in the number of transmit elements of a phase array system operating at peak power limitation. SNR should be avoided as much as possible; since their compound effect on the system design, say on the power budget, can be a significant factor affecting the monetary cost of the system. The selection of a suitable window function is an application or scenario specific choice based on some trade-offs. As noted before, the main benefit of windowing is the reduction of the signal sidelobes in the output spectrum at the expense of widened mainlobe (resolution loss) and SNR loss, as documented in [1,Ch.6]. The main goal of this study is to optimize the window selection such that the resulting average SNR loss due to windowing is negligibly small. To this aim, we propose to use a set of conventional windows, say rectangle, Hamming and a Chebyshev windows providing 13, 43, 120 dB side-lobe suppression ratios, pose the window selection as a hypothesis testing problem and apply the principles of the Bayesian hypothesis testing. Upon an analysis of the resulting hypothesis testing based method, we suggest a reduced complexity window selection method and examine its performance. In the literature, the method known as the dual-apodization method (for two windows) and its extension to the multiple windows (multi-apodization) resembles the proposed line of study, [2]. In dual-apodization, the DFT of the input is calculated twice with two different window functions. For a specific DFT bin, the dual apodization output is the output spectrum sample of two windows having the minimum magnitude. The motivation of dual-apodization method can be most easily seen for the noise-free operation. In the absence of noise, the window with the better side-lobe suppression is selected for the side-lobe bins by the mentioned calculation of minimum DFT magnitude. For the bins which are located in the main-lobe, the window with the smaller beamwidth, the higher resolution window, is selected. Hence, the inclusion of a non-linearity in the processing chain, jointly enables both high resolution and small side-lobes, which is not possible with linear processing methods. This method has found applications mainly in the imaging applications [3,4] where SNR is not the most major issue of concern as in the detection applications. To the best of our knowledge, the performance of dual-apodization is not examined from the detection theoretic point of view except the study of [5]. In this work, we present the comparison of the suggested hypothesis testing based window selection method with the multi-apodization method. Background We consider the following signal model, r[n] = √ γ s e j(ωsn+φs) s[n] + √ γ j e j(ω j n+φ j ) j[n] +v[n], n = {0, 1, . . . , N − 1}.(1) Here In the presence of white noise and absence of jamming, the optimal detector, maximizing the output SNR, is the detector matched to the signal vector w = s and the optimal decision statistics for the detection application is \|w H r\| 2 , [6]. This corresponds to the processing of the input data with the rectangle window. When jamming is present, SNR maximizing filter is the whitened matched filter detector, that is w = (I + γ j jj H ) −1 s and the decision statistics is \|s H (I + γ j jj H ) −1 r\| 2 . In this work, we assume that the user does not have the capacity to implement the optimal filter due to the lack of statistical information on the jammer. Instead, user selects the most suitable window from a set of candidate windows to reduce the effect of jamming signal. The decision statistics becomes \|s H D win r\| 2 where D win is a diagonal matrix whose diagonal entries are the samples of the window function selected. To understand the factors effecting the window choice, assume that the target is located at the DC bin. For this target, a very strong jammer in the Region 2 of Figure 2 requires the application of Chebyshev window. Yet, the Hamming window, which has a better SNR loss than Chebyshev window, can be sufficient for a weaker jammer in the same region. On the other hand, even for very strong jammers in Region 1 of Figure 2, the Chebyshev window is not suitable, since there is no side-lobe suppression due to large main-lobe widening of this window. Hence for Region 1 jammers, the possible windows of choice are limited to rectangle or Hamming window. In this study, our goal is to automate such reasoning procedures and present a window selection method with a negligible SNR loss. In Figure 2, the multi-apodization output of three windows is also illustrated. For this case, the multi-apodization output magnitude is simply X M A (e jω ) = min{\|X R (e jω )\|, \|X H (e jω )\|, \|X Ch (e jω )\|}(2) where X R (·), X R (·), X Ch (·) is the spectrum of rectangle, Hamming and Chebyshev windows and X M A (e jω ) is the multi-apodization output, [3,5]. It can be seen from (2) that the multi-apodization output is simply the selection of the magnitude spectrum with the smallest magnitude for each spectral component, [2]. In the absence of noise, the multi-apodization yields jointly high resolution (main-lobe width of rectangle window) and large side-lobe suppression, as shown in Figure 2. For the noisy scenarios, the performance of multi-apodization, in terms of SNR loss, is not immediately clear and a topic of investigation in this study. The multi-apodization idea has found some applications in the synthetic aperture imaging (SAR) applications, [4,7] where SNR loss is not the main concern. The situation is quite different in the pulse-doppler radar systems where much fewer observations (slow time samples) are available for the target detection. An insightful explanation for the multi-apodization method given in [7] states that multi-apodization is, in principle, equivalent to the single snapshot version of Capon's method. In this work, our goal can be more ambitiously stated as to develop a better alternative for the open title of single snapshot Capon's method. Proposed Approach The proposed approach is described with the window functions illustrated in Figure 2. Without any loss of generality, we may assume that the signal of interest is at DC bin and the jamming power is localized either in Region 1 or Region 2 of Figure 2. Our goal is to determine the jammer activity through M-ary hypothesis testing and apply a suitable window based on the jamming activity level. The k'th hypothesis can be written as follows: H k : r = √ γ s e jφs s + √ γ j k e jφ j j + v, k = {1, . . . , M }.(3) The vectors s, j, v refer to signal, jammer and noise vectors as defined in Section 2. The γ s is nuisance parameter of the test appearing in all hypotheses. The critical parameter of the k'th hypothesis is the JNR,γ j k . We assume thatγ j 1 <γ j 2 < . . . <γ j M , that is the weakest jammer case is represented with H 1 . For any observation vector r, the index of the hypothesis with the highest a-posteriori probability can be expressed as k = arg max 1≤k≤M P (H k \|r).(4) We treat the selected hypothesis as an indicator of the jamming activity level and associate a window function for each hypothesis. Note that (4) can also be written in terms of the likelihood ratios as follows, k = arg max 1≤k≤M P (H k \|r) P (H 1 \|r) = arg max 1≤k≤M P (H k ) P (H 1 ) f R (r\|H k ) f R (r\|H 1 ) ,(5) where P (H k ) is the a-priori probability of the k'th hypothesis and the rightmost term is the likelihood ratio of the kth and first hypothesis. In many scenarios, the a-priori probabilities of hypothesis are unknown and are taken as P ( H k ) = 1/M for k = {1, 2, . . . , M }. Then, k reduces to arg max 1≤k≤M f R (r\|H k ) f R (r\|H 1 ) . Window Selection Test Based on the Likelihood Ratio With the definitions given Section 2, f R (r\|H k ) is Gaussian density with zero mean and covariance matrix R k =γ s ss H +γ j k R n j + I, where R n j is the normalized jammer covariance matrix with unit elements on its diagonal. The product of normalized jammer covariance matrix and average jammer power for the k'th hypothesis is denoted as R j k =γ j k R n j . We express the eigendecomposition of R j k as R j k = E j Λ j k E H j . Here, the columns of E j matrix are the eigenvectors spanning the jamming space. The matrix Λ j k is a diagonal matrix having the eigenvalues R j k on its diagonal. The eigenvalues of R j k isγ j k × λ n i , that is the product ofγ j k and the eigenvalues of normalized jammer covariance matrix R n j . The log-likelihood ratio of the k'th and first hypothesis can be written as: log f R (r\|H k ) f R (r\|H 1 ) = log \|R 1 \| − log \|R k \| + r H (R −1 1 − R −1 k )r.(6) Temporarily, we call jammer plus noise covariance matrix for the kth hypothesis as R j k +n = γ j k R j + I = R j k + I, then R k appearing in the likelihood ratio becomes R k =γ s ss H + R j k +n . With this definition, the quadratic product in (6) reduces to r H R −1 k r = r H R −1 j k +n r − 1 1 + s H R −1 j k +n s \|r H R −1 j k +n s\| 2 .(7) The jammer plus noise covariance matrix, R j k +n = R j k + I, can be eigendecomposed as R j k +n = E j Λ j k +n E H j + E n E H n . It should be noted that jammer plus noise covariance matrix is a full rank matrix; therefore, the columns of E n matrix, whose span is the noise space, is orthogonal to the jammer space, the column space of E j matrix. Using the eigendecomposition of R j k +n , we can express s H R −1 j k +n s term appearing in (7) as follows: s H R −1 j k +n s = s H E j Λ −1 j k +n E H j s ≈0 +s H E n E H n s ≈ \|E H n s\| 2 .(8) Here, it is assumed that the signal vector has negligible power in the jammer space. We may consider that the jammer is in Region 2 of Figure 2; therefore the projection of signal power to jammer space is negligible. (The leakage of signal power can be considered as negligible unless the SNR is extremely is high which is a case of little concern for the detection problems.) Similarly, we can express the term r H R −1 j k +n s in (7) as r H R −1 j k +n s = r H E j Λ −1 j k +n E H j s ≈0 +r H E n E H n s ≈ r H E n E H n s.(9) Finally, the term (7) becomes r H R −1 j k +n r = r H (R j k + I) −1 r H inr H R −1 j k +n r = r H E j Λ −1 j k +n E H j r + r H E n E H n r = N j i=1 1 γ j,k λ n i + 1 \|e H i r\| 2 + \|E H n r\| 2 ,(10) where N j is rank of R j k =γ j k R n j and e i is the eigenvector of R j k with the eigenvalueγ j,k λ n i . Using equations (8), (9) and (10), we can simplify the quadratic term appearing on the left side of (7), r H R −1 k r, as r H R −1 k r ≈ (11) r H E j Λ −1 j k +n E H j r + r H E n E H n r − 1 1 + \|E H n s\| 2 \|r H E n E H n s\| 2 . This concludes the simplification of the quadratic term in the log-likelihood ratio expression in (6). The remaining term to be simplified in this equation is the determinant of γ s ss H + R j k +n matrix: R k = ¡ ¢ £ ¤ ¥ ¦ § ¨ © ª « ¬ ® ¯ ° ± ² ³ ´ µ ¶ · ¸ ¹ º » ¼ ½ ¾ ¿ À Á Â Ã Ä Ådet(R k ) = (1 +γ s s H R −1 j k +n s) det(R j k +n ) (a) = (1 +γ s \|E H n s\| 2 ) det(R j k +n ) (b) = (1 +γ s \|E H n s\| 2 ) N i=1 (γ j,k λ n i + 1)(12) In line-(a) of equation (12), the relation s H R −1 j k +n s ≈ \|E H n s\| 2 is utilized one more time (see equation (8)). In line-(b), the determinant is computed via the product of the eigenvalues. With the substitution of relations given by equations (10) and (12) into the log-likelihood ratio given in (6), we get the following expression, log f R (r\|H k ) f R (r\|H 1 ) ≈ (13) N j i=1 log γ j,1 λ n i + 1 γ j,k λ n i + 1 + N j i=1 1 γ j,1 λ n i + 1 − 1 γ j,k λ n i + 1 \|e H i r\| 2 . In the last expression e i and λ n i are the eigenvector of normalized jammer covariance matrix R n j . Next, we need to construct a hypothesis for each candidate window function with a proper selection of hypothesis parameters. On the selection of R n j matrix: We assume that the jammer frequency is assumed to be uniformly distributed in the side-lobe region of the window. As a concrete example, we may consider the side-lobe region of N=16 point Chebyshev window, shown as the Region 2 of Figure 2. This region ranges from 3.175 to 12.84 DFT bin as can be seen from Figure 2. (The same region with unit of radians per sample corresponds to Θ 1 = 2π/N × 3.175 and Θ 2 = 2π/N × 12.84, where N = 16.) The jammer with the frequency θ corresponds to j θ = [1 e jθ e j2θ . . . e j(N −1)θ ] T . The jammer frequency is assumed to be uniformly distributed in the interval [θ 1 , θ 2 ], which is [3.175, 12.84] DFT bins for the Region 2 shown in Figure 2. The auto-correlation matrix for the jammer is E{j θ j H θ }, where the expectation operator E{·} is over the random variable corresponding to the frequency Θ. The mth row, nth column entry of the matrix R n j can be expressed as: R n j mn = θ 2 − θ 1 2π E Θ {j θ j H θ } mn = 1 2π θ 2 θ 1 e j(m−n)θ dθ =    (θ 2 − θ 1 )/2π, m = n exp(j(m−n)θ 2 )−exp(j(m−n)θ 1 ) j2π(m−n) , m = n .(14) On the selection ofγ j k parameter: The candidate windows should have significantly different side-lobe suppression ratios. The rectangle, Hamming and Chebyshev windows of Figure 2 have the peak side-lobe suppression ratios of 13, 40 and 120 dB. For each window, we suggest to use half the peak side-lobe ratio in dB asγ j k parameter. For the mentioned case, this results inγ j 1 = 6.5 dB,γ j 2 = 20 dB andγ j 3 = 60 dB. This choice stems from the consideration that for a jammer of 15 dB JNR, we may almost equally prefer to use either Hamming or rectangle window, but not the Chebyshev window. A Simple Window Selection Test Based On Likelihood Ratio Approximation We present a simpler test based on an approximation to the log-likelihood ratio given in (14). From the definition of normalized jammer covariance matrix given by (14), it can be verified that the eigenvalues of this matrix, denoted as λ n i , are clustered around 0 and 1. (This fact can be checked by noticing that R n j matrix given in (14) is the similarity transform of the discrete prolate spheroidal sequence (DPSS) generating matrix having the mth row, nth column entries sin(θ 2 (m − n))/(π(m − n) whose eigenvalues are known to have a sharp transition between 1 and 0, [8, p.213].) By substituting λ n i ≈ 1 in (14), we get the following expression log f R (r\|H k ) f R (r\|H 1 ) ≈ N j log γ j,1 + 1 γ j,k + 1 + 1 γ j,1 + 1 − 1 γ j,k + 1 N j i=1 \|e H i r\| 2 , = α k + β k r H R n j r.(15) where N j i=1 \|e H i r\| 2 = r H R n j r is an estimate of the jammer power. Hence, the log-likelihood ratio reduces to an affine function of r H R n j r, α k +β k r H R n j r. Hence, the association of window functions to the hypotheses can be simply done by partitioning the real line to M disjoint sets as shown in Figure 3. The decision boundaries, the points A and B, in Figure 3 are the JNR levels for the window switching indicating the boundaries between low, medium, high JNR levels. On the determination of decision boundaries: We assume that the window functions are ordered in the increasing order of interference suppressing capabilities. (For the case shown in Figure 2, the order is rectangle, Hamming and Chebyshev windows, respectively.) The goal is to sequentially set the decision boundaries, that is to determine the boundary for the rectangle and Hamming windows first (point A in Figure 3) and then the boundary for the Hamming and Chebyshev window (point B in Figure 3). It can be said that for a candidate window set with M windows, we need to determine M − 1 boundaries by pairwise comparing consecutive windows ordered in the increasing order of PSL. Figure 4a shows the receiver operating curve (ROC) for the rectangle and Hamming windowed processed received vector r whose signal model is given in (1). Since the signal, interference and noise are assumed to be jointly Gaussian distributed, the detection problem reduces to the problem known as the detection of Gaussian signals in Gaussian noise, [9,Ch.9]. The relation between probability of detection and probability of false alarm, as depicted in Figure 4 is P d = P 1/(1+SJNR) F A , where SJNR refers to the signal-to-jammer-plus-noise-ratio. SJNR at the windowed DFT detector output is written as follows: SJNR k =γ s \|w H k s\| 2 γ j w H k R n j w k + w k 2 .(16) Here w k is the kth window and without any loss of generality, the vector s can be taken as the vector of all ones, that is the signal can be assumed at the DC bin. (If the signal is not at the DC bin, w k should be window function modulated to the frequency of the signal.) From the ROC curve given in (4) In (17), the matrix R n j is the normalized jammer covariance matrix defined in (14). The parameters θ 1 and θ 2 of this matrix is determined by considering the side-lobe suppression In the development of the test based on the likelihood ratio (given in Section 3.1) and its approximate version (given in this section), the interference is assumed to be in the suppression band of all windows, which is the part of spectrum shown as Region 2 of Figure 2. If the interference lies in the transition band of a particular window (shown as Region 1 of Figure 2), then that window should be discarded. As an example, for the interference lying in Region 1 of Figure 2, the Chebyshev window should not be utilized at all; since the Chebyshev window does not provide any improvement in interference suppression in comparison to the Hamming window for this jammer. On the contrary, the Chebyshev window amplifies the interference power in comparison to the Hamming window. Hence, in spite of its superior side-lobe suppression capacity for the Region 2 interferers, the Chebyshev window should be avoided for the Region 1 jammers. We integrate the idea of disabling some of the windows based on the detected jammer region to the existing scheme and present the suggested method in Table 1. For further clarification, we also present a ready-to-use MATLAB code with application examples in [10]. The suggested algorithm can be briefly explained as follows: Step 0 is the initialization step where the stop bands of the windows, the decision boundaries are calculated. In Step 1, the normalized covariance matrix for an interferer lying in the stop-band of each window is calculated. In Step 2, the jamming power residing in the stop band of each window is calculated. Step 3 implements the idea of window disabling based on the jammer spectral location. Step 4 is the quantization of estimated jamming power for the selection of the window function, as illustrated in Figure 3. Numerical Comparisons We present a set of numerical comparisons of the suggested method with the conventional windowed DFT detectors and multi-apodization method. is selected by the system designer and this window is utilized irrespective of the JNR value. JNR case. This study aims to present an alternative method based on a data-adaptive window selection procedure in which SNR loss due to windowing is effectively eliminated. The dashed line in Figure 5a-5b shows the performance of the multi-apodization method. The window selection procedure of this method is established through the relation given in (2). This method can be summarized as follows. The windowed DFT of the input is calculated for all three windows. For a fixed DFT bin of interest, the windowed DFT output with the least magnitude is selected. (It is important to note that the windowed DFT outputs should have the same response when the incoming signal frequency perfectly matches the DFT bin frequency in the absence of noise. This is easily established by equating the DC response of the windows to 1 (equivalently 0 dB) by scaling each window, as shown in Figure 2.) When the dashed line in Figure 5a-5b (multi-apodization method) is compared with the solid lines (conventional windowed DFT detectors), we see that at low JNR values, the multi-apodization method yields a better performance than both Hamming and Chebyshev windows. Figure 5c shows where ω 1 = 2π 16 0.1, ω 2 = 2π 16 6.25 and v[n] is unit variance, zero-mean white noise. We continue to use the SNR definition in Section 2. Under this setting, Table 2 shows the window selection probabilities for the proposed method when signal components forming r[n] has different SNR values. It can be immediately observed that the window selection probabilities vary with the DFT bin of interest. Stated differently, if we are interested in the frequency content of the 11th DFT bin, both components forming r[n], i.e. signals at the bins 0.1 and 6.25, act like jammers corrupting the frequency content of this bin. Case A of Table 2 illustrates the case for weak signal components. It can be noticed that the rectangle window (whose results are shown in the rows indicated by 'R' letter) is selected with a significant majority for the frequency bins close to the signal components (0'th, 6'th and 7'th bins). The Hamming window is selected 14.8% of the time for the DC bin. This is essentially due to interference generated by the signal component at the 6.25 DFT bin. The selection of the Hamming window for 6th or 7th DFT bins is much lower, close to 0.3%, since the signal at DC bin imposes has a 5 dB lower SNR and therefore causes less interference. For the weak jamming signal case, it is assuring to observe that the window with the highest SNR loss, i.e. the Chebyshev window, is not utilized at all. Table 2 i.e. integer valued bins. Due to the mentioned frequency mismatch, the signal energy in the side-lobes of this signal acts as an interference, inhibiting its detection. This problem is known as the parameter mismatch problem (the mismatch of assumed frequencies with the actual signal frequency) and can be reduced by evaluating zero-padded DFT instead of N-point DFT. Case B of We do not further elaborate on this topic in order not to distract the readers from the main message of this study. Cases C and D of Table 2 present the results as the second signal component SNR is further increased. Only in the highest SNR 2 case (Case D), the window preference switches to the Chebyshev window. It should be noted that this change of preference occurs for the bins that the second signal component lies in the stop-band of the Chebyshev window. Stated differently, the bins 4 to 10 do not use Chebyshev at all, since these bins constitute the main lobe and transition region of this window. Hence, for these bins the Chebyshev window can not provide any interference suppression capability. A brief of time of reflection on Table 2 can convince the readers that the suggested method works as if like an experienced operator with the knowledge of jammer specific parameters. Conclusions We present a method without any application specific parameter tuning that automates the window selection procedure for the discrete Fourier transformation (DFT) based detectors. The windowing is, in general, an essential operation to reduce the masking effect of a strong spectral component over the weaker one. Yet, it may come at a significant SNR loss, typically in between 1.5 to 3 dB. The suggested method aims to select the windows with strong sidelobe suppression capabilities, or equivalently the windows with a high SNR loss, only when the situation arises, that is only when it is indeed needed. The numerical results show that the method is capable of switching window function depending on operational JNR level such that the best probability of detection among all detectors is achieved at all JNR values, in spite of the lack of JNR knowledge at the receiving end (see Figures 5 to 7). We believe that the suggested method is a promising contender for the open title of single snapshot version of the Capon's method. An implementation of the method is given in [10]. where k select is the index of the window function to be used for the spectral analysis of α'th DFT bin. (A Matlab implementation is available at [10]. Step 0: Step 1: For the k'th window with the stop-band θ 1 to θ 2 , calculate M k = R n j , 2 ≤ k ≤ L from (14). Step 2: Calculate d k = r H M k r/N for 2 ≤ k ≤ L, where r is the modulated input vector of dimensions N × 1. Step 3: For k = {2, 3, . . . , L − 1}, check whether d k > 2d k+1 condition is satisfied or not. If the condition is not satisfied for any k, set k max = L; else set k max as the lowest k max value for which the condition is satisfied. (window disabling) Step 4: Return k select such thatγ j [k select − 1] ≤ d 2 ≤γ j [k select ]. , where 1 ≤ k select ≤ k max . Figure 1 : 1Conventional and Modern Spectral Analysis Methods losses due to processing, called signal processing loss in the radar signal processing literature, N × 1 column vector with the entries e jωsn , n = {0, 1, . . . , N − 1}. (The vector j is defined similarly.) Throughout this work, we prefer to express the frequency variables, ω s or ω j , with the units of DFT bins, ω s = 2πf s /N , where N is the number of observations. With this definition, f s becomes a real valued parameter in the interval [0, N ). This definition simplifies the description and perception of the numerical values associated with main-lobe width, the frequency difference between signals etc. Figure 2 2shows the magnitude spectrum of rectangle, Hamming and Chebyshev windows of length 16. Figure 2 : 2Magnitude spectrum of Rectangle, Hamming, Chebyshev windows and the multi-apodization spectrum. Figure 3 : 3An illustration for the partitioning of the real line into three parts. Figure 4 : 4The receiver operating curves (ROC) for windowed DFT detectors, PF A = 10 −2 . regions of both windows. As shown in Figure2, the side-lobe suppression region for both rectangle and Hamming windows range from 1.392 to 14.62 in terms of DFT bins. It should be noted that the decision boundaries given by (17) is invariant to SNR and probability of false alarm. Hence, the boundary point can be calculated once and can be utilized for all ROC curves. As a summary, the window selection on the approximate log-likelihood relation is based on the quantization of the metric d = r H R n j r/N . For the case shown in Figure 2, if the d value is smaller than 8.4 dB, then rectangle window is selected; for d values in between 8.4 and 30.5 Hamming window is selected and for d > 30.5 results in the selection of Chebyshev window. Case 1 : 1Interference in the suppression band of all windows: To ease the presentation, we continue with the detectors utilizing the rectangle, Hamming and Chebyshev windows whose magnitude spectrum is illustrated inFigure 2. For this case, it is assumed that the interference lies in Region 2 ofFigure2. More specifically, the interference lies in the band where all three windows have the ability of interference suppression at different capacities.Figures 5a-5b show the variation of the probability of detection versus jammer-to-noise ratio (JNR) for the scenario parameters of SNR = 0 dB, the probability of false alarm of 10 −2 , the number of observations of 16, (N = 16). The frequency of the interfering signal is 4 to 6DFT bins away from the frequency of the signal to be detected and the interfering frequency is uniformly distributed in this interval guaranteeing that the interference is in Region 2 of Figure 2 .Figure 5 :Figure 6 : 256JNR Case 1. Subfigures (a)-(b): ROC curves for the windowed DFT detectors, multi-apodization method and proposed detectors, PF A = 10 −2 . Subfigures (c)-(d): The window selection probabilities of multi-apodization and proposed methods vs. JNR. From Figures 5a-5b, it can be immediately read that the rectangle window presents the best performance (in the sense of detection probability) for sufficiently low JNR values. On the other hand, the performance of rectangle windowed Fourier transformation detector degrades rapidly, once JNR is above 20 dB. It should be clear from these figures that the windows providing higher interference suppression capabilities should be utilized at high JNR levels. In many practical systems, a nominal window function, such as the Hamming window, Case 2. Subfigures (a)-(b): ROC curves for the windowed DFT detectors, multi-apodization method and proposed detectors, PF A = 10 −2 . Subfigures (c)-(d): The window selection probabilities of multi-apodization and proposed methods vs. JNR. Figure 5a -Figure 7 : 5a75b illustrate that the choice of Hamming window brings a sub-optimal performance at low JNR and furthermore does not present a significant gain at excessively high JNR values. Typically, the system designer rules out the possibility of excessive JNR values through another jammer detection mechanism, say a side-lobe blanker, and justifies the SNR loss due to the application of Hamming window as a necessary trade-off between low JNR and mediumCase 3. Subfigures (a)-(b): ROC curves for the windowed DFT detectors, multi-apodization method and proposed detectors, PF A = 10 −2 . Subfigures (c)-(d): The window selection probabilities of multi-apodization and proposed methods vs. JNR. , the probability of window selection for the multi-apodization method. This figure shows that at low JNR values, about 50% of the time rectangle window is selected. This explains the superiority of multi-apodization method over Hamming and Chebyshev windows at low JNR. Similarly, when JNR is extremely high, say 70 dB, the performance of the multi-apodization method approaches Chebyshev window, which is the best choice among the windows examined for excessively high JNR values. Figure 5c illustrates that as JNR increases, the probability of Chebyshev window selection also increases, explaining high JNR behavior of multi-apodization method in Figure 5a-5b. For medium-to-high JNR values, the multi-apodization method presents an acceptable performance, as seen from Figure 5b. The main of reason of less-than-impressive performance is the sluggish change of window selection probabilities as shown in Figure 5c. Stated differently, the multi-apodization method does not react fast enough to JNR increase. The performance of the proposed method with exact and approximate likelihood metric is shown with the dashed lines and triangle markers in Figure 5a/b. It can be immediately noted that the performance with the exact and approximate likelihood metric are almost identical. The performance of suggested method is superior to the windowed based approach at all JNR values and is almost identical to the pointwise maximum of three detectors shown in Figure 5a-5b at all JNR levels. The superior performance in the detection probability can also be explained by the rapid variation of the window selection probabilities given in Figure 5d.Case 2: Interference in the transition band of a window: Here we assume that interference is in the transition band of one of the windows. For this case, the frequency of the interfering signal is assumed to be uniformly distributed 1.5 to 3 DFT bins away from the frequency of the signal to be detected, i.e. in Region 1 of Figure 2. The other simulation parameters are identical to ones in Case 1. Figure 6 Figure 2 , 62presents the results for this scenario. It can be noted the multi-apodization detector and proposed detector approach the performance of Hamming window as JNR increases. From Figure 6c, it can be noticed that the multi-apodization output is dominated by the selection of the Hamming window at high JNR. Similarly, as the dashed red line with the cross marker indicates the suggested method also utilizes Hamming window with increasing JNR, but the proposed method adopts the Hamming window at much smaller JNR values in comparison to the multi-apodization method. The Chebyshev window is not selected at all with the proposed method. It should be intuitively clear that the choice of Chebyshev window should be avoided for this scenario. The next case further elaborates this point. Case 3: Interference in the transition band of a window: Figure 7 presents the results when the interfering frequency is 2.35 to 2.45 bins away from the frequency of the signal to be detected. The interval for the interfering frequency is specially selected to illustrate the disadvantages arising when the interference is in the transition band of a window. From it can be noted that Hamming window presents around 80 dB in the interval of [2.35,2.45] bins; while the Chebyshev window presents a suppression ration of 20 dB. Ideally, the Hamming window should be the most suitable choice for the suppression of a strong interferer in this frequency interval. It should be noted this situation, i.e. the preference of a window having a poorer peak-sidelobe suppression ratio, occurs only when the interfering frequency is in the transition band of one of the windows. Figure 7 7presents the performance of suggested method. The dashed lines in Fig 7d indicate that the Chebyshev window is not selected at all JNR values with the suggested method. It can be said that the suggested method, in effect, eliminates the Chebyshev window choice when the interference is in the transition band of this window as desired. It can also be seen from Figures 7a-7b that high JNR performance of the suggested method is identical to the performance of the Hamming window, as desired. We also note that the method utilizing the exact likelihood (which is not equipped by window disabling feature) only works well when the interferer is in the stop-band of all window functions, as in Case 1.Study of Window Selection Probability: As a final numerical study, we examine the probability of window selection in more depth. The input for this study is assumed to be r[n] = √ γ 1 e j(ω 1 n+φs) + √ γ 2 e j(ω 2 n+φ j ) + v[n], n = {0, 1, . . . , 15} gives the window selection probabilities when the second component with the spectral location of 6.25 DFT bin has an SNR of 15 dB. It can be observed that the weak signal having the frequency of 0.1 DFT bins is processed with the Hamming window 82.8% of the time. The percentage increase from 14.8% (Case A) to 82.8% (Case B) is essentially due to power increase of the second component. One can also notice that the Hamming window utilization for the 6th bin also increases from 0.3% to 13.4% when Cases A and B are compared. The power of the first signal component is identical in both cases; hence this increase can not be explained with the increase of jamming activity due to the first component. The increase in the Hamming window utilization probability is due to the frequency mismatch of the second component (6.25 DFT bin) to frequencies of DFT bins, Order L window functions in the increasing order sidelobe suppression ratio. (Example: Rectangle, Hamming, Chebyshev windows should be ordered as the first, second and third windows, respectively.) b) Calculate the L − 1 decision boundaries for L windows, denoted asγ j [k], by setting w 1 as the kth window function and w 2 as the (k + 1)th window function, k = {1, 2, . . . , L − 1}, in (17). Also setγ j [0] = 0. c) Determine the stop-band of each window with the index k ≥ 2. (Example: In figure 2, the stop-band for the Hamming and Chebyshev windows are 1.392 to 14.62 DFT bins and 3.175 to 12.84 DFT bins, respectively.) d) To select the window for the output DFT bin, α'th DFT bin, 0 ≤ α ≤ N − 1, modulate the input vector r to the zero'th DFT bin, i.e. r ← D α r where D α is a diagonal matrix with the diagonal entries exp(−j 2παn N ), 0 ≤ n ≤ N − 1. s[n] denotes the signal of interest, a complex exponential signal with frequency ω s ; j[n] denotes the intentional or unintentional jamming signal corrupting the observations r[n]; v[n] is zero mean, unit variance complex valued white noise with circularly symmetric Gaussian distribution, v[n] ∼ CN(0, 1). The phase values of signal and jammer components, φ s and φ j , are assumed to be independent random variables with a uniform distribution in [0, 2π). The parameters γ s and γ j are independent, exponentially distributed random variables with mean valuesγ s andγ j , respectively. The input SNR and jammer-to-noise ratio (JNR) is defined as SNR = E{\|s[n]\| 2 } E{\|v[n]\| 2 } =γ s , JNR = E{\|j[n]\| 2 } E{\|v[n]\| 2 } =γ j . (The model given in (1) corresponds to the Rayleigh faded target signal observed under Rayleigh faded rank-1 interference (jammer) and white Gaussian noise.)Equation(1)can be written in vector form as r = √ γ s e jφs s + √ γ j e jφ j j + v, where s is an Table 1 : 1The suggested window selection method. Inputs are r: N × 1 input vector, L window functions, α: DFT bin index of interest. Output is k select Table 2 : 2Rectangle (R), Hamming (H) and Chebyshev (C) window selection probabilities for each DFT bin when input is the superposition of two complex exponential signals of length N = 16 with the frequencies 2π N 0.1 and 2π N 6.25 radians per sample, i.e. frequencies of 0.1 and 6.25 with the units of DFT bins.Case B R 17.8 17.7 15.4 15.2 15.5 33.6 86.6 80 17.6 15.5 15.4 15.4 15.3 15.4 15.3 17.5DFT Bin Index B Porat, A Course in Digital Signal Processing. John Wiley & SonsB. Porat, A Course in Digital Signal Processing, John Wiley & Sons, 1996. Nonlinear apodization for sidelobe control in SAR imagery. H C Stankwitz, R J Dallaire, J R Fienup, IEEE Trans. Aerosp. Electron. Syst. 31H. C. Stankwitz, R. J. Dallaire, J. R. Fienup, Nonlinear apodization for sidelobe control in SAR imagery, IEEE Trans. Aerosp. Electron. Syst. 31 (1995) 267-279. Spatially variant apodization for image reconstruction from partial Fourier data. J A C Lee, D C Munson, IEEE Trans. Image Process. 9J. A. C. Lee, D. C. Munson, Spatially variant apodization for image reconstruction from partial Fourier data, IEEE Trans. Image Process. 9 (2000) 1914-1925. Effect of apodization on SAR image understanding. D Pastina, F Colone, P Lombardo, IEEE Trans. Geoscience and Remote Sensing. 45D. Pastina, F. Colone, P. Lombardo, Effect of apodization on SAR image understanding, IEEE Trans. Geoscience and Remote Sensing 45 (2007) 3533-3551. On the detection of sinusoidal signals under the effect of sinusoidal interference, Master's thesis. B Balcı, Department of Electrical and Electronics Engineering, Middle East Technical University, Advisor: Cagatay CandanB. Balcı, On the detection of sinusoidal signals under the effect of sinusoidal interfer- ence, Master's thesis, Department of Electrical and Electronics Engineering, Middle East Technical University, Advisor: Cagatay Candan, 2010. M A Richards, Fundamentals of Radar Signal Processing. New YorkMcGraw-HillM. A. Richards, Fundamentals of Radar Signal Processing, McGraw-Hill, New York, 2005. Effectiveness of spatially-variant apodization. J A C Lee, J D C Munson, Image Processing, International Conference on. 1J. A. C. Lee, J. D.C. Munson, Effectiveness of spatially-variant apodization, Image Processing, International Conference on 1 (1995) 147-150. Signal Analysis. A Papoulis, McGraw-HillA. Papoulis, Signal Analysis, McGraw-Hill, 1977. . H L V , Trees, Detection, Estimation and Modulation Theory. 3John Wiley -SonsH. L. V. Trees, Detection, Estimation and Modulation Theory, part 3, John Wiley -Sons, 1971. An Automated Window Selection Procedure For Spectral Analysis To Reduce Windowing SNR Loss. C Candan, MATLAB CodeC. Candan, An Automated Window Selection Procedure For Spectral Anal- ysis To Reduce Windowing SNR Loss, (MATLAB Code), 2017. URL: http://users.metu.edu.tr/ccandan/pub.htm.	[]
[ "A SUPPLEMENT TO SCHOLZ'S RECIPROCITY LAW", "A SUPPLEMENT TO SCHOLZ'S RECIPROCITY LAW" ]	[ "Franz Lemmermeyer " ]	[]	[]	In this note we will present a supplement to Scholz's reciprocity law and discuss applications to the structure of 2-class groups of quadratic number fields.	10.4064/aa129-4-2	[ "https://arxiv.org/pdf/1310.6599v1.pdf" ]	119,308,315	1310.6599	632c059cfdacc00955a61978f9bd57ad0c32f7ce	A SUPPLEMENT TO SCHOLZ'S RECIPROCITY LAW 24 Oct 2013 Franz Lemmermeyer A SUPPLEMENT TO SCHOLZ'S RECIPROCITY LAW 24 Oct 2013 In this note we will present a supplement to Scholz's reciprocity law and discuss applications to the structure of 2-class groups of quadratic number fields. Introduction Let us start by fixing some notation: • p and q denote primes ≡ 1 mod 4; • h(d) denotes the class number (in the usual sense) of the quadratic number field with discriminant d; • O p and O q denote the rings of integers in Q( √ p ) and Q( √ q ); • ε p and ε q denote the fundamental units of Q( √ p ) and Q( √ q ), respectively; • [α/p] denotes the quadratic residue symbol in a quadratic number field; recall that it takes values ±1 and is defined for ideals p ∤ 2α by [α/p] ≡ α (N p−1)/2 mod p. Given primes p ≡ q ≡ 1 mod 4 with (p/q) = +1, we have pO q = pp ′ and qO p = qq ′ ; the symbol [ε p /q] does not depend on the choice of q, so we can simply denote it by (ε p /q). Scholz's reciprocity law then says that we always have (ε p /q) = (ε q /p) (for details, see [5,6,7]). Scholz's reciprocity law was first proved by Schönemann [11], and then rediscovered by Scholz [12] (Scholz mentioned his reciprocity law and the connection to the parity of the class number of Q( √ p, √ q ) in a letter to Hasse from Aug. 25, 1928; see [10]). In [13], Scholz found that in fact (ε p /q) = (ε q /p) = (p/q) 4 (q/p) 4 , and showed that these residue symbols are connected to the structure of the 2-class group of Q( √ pq ). Hilbert's Supplementary Laws For extending these results we have to recall the notions of primary and hyperprimary integers (see Hecke [3]). (1) α ≫ 0 is totally positive and α ≡ ξ 2 mod 4 for some ξ ∈ O K ; (2) the extension K( √ α )/K is unramified at all primes above 2∞. If the conditions of Lemma 1 are satisfied, we say that α is primary. Lemma 2. Assume that 2O K = l e1 1 · · · l er r ; then the following assertions are equivalent: (1) α is primary, and α ≡ ξ 2 mod l 2ej +1 j for all j. (2) every prime above 2 splits in the extension K( √ α )/K. If the conditions of Lemma 2 are satisfied, we say that α is hyper-primary. Observe that the conditions in (1) are equivalent to α ≡ ξ 2 mod 4l 1 · · · l r . Also note α is allowed to be a square in Lemma 1 and Lemma 2. (1) (ε/a) = +1 for all units ε ∈ O × k ; (2) a h = (α) for some primary α ∈ O k . Hilbert calls an ideal a with odd norm primary if condition (1) (1) (λ/a) = +1 for all λ ∈ O k whose prime divisors consist only of primes above 2; (2) a h = (α) for some primary α ∈ O k . Hilbert calls ideals satisfying property (1) above hyper-primary. A proof of a generalization of Thm. 2 to arbitrary number fields can be found in [3,Thm 175]. Now we can state Theorem 3. Let p ≡ q ≡ 1 mod 4 be primes with (p/q) = +1. Then pO q = pp ′ and qO p = qq ′ split. The class numbers h(p) and h(q) are odd, and there exist elements π ∈ O q and ρ ∈ O p such that p h(p) = (π) and q h(q) = (ρ). Then the following assertions are equivalent: (1) (ε p /q) = +1; (2) ρ can be chosen primary; (3) h(pq) ≡ 0 mod 4. Proof. Genus theory (see e.g. [7, Chap. 2]) implies that h(p) ≡ h(q) ≡ 1 mod 2. The equivalence (1) ⇐⇒ (2) It is not hard to prove these statements directly using class field theory; below we will do this in an analogous situation. Observe that part (3) of Thm. 3 is symmetric in p and q, which immediately implies Scholz's reciprocity law (ε p /q) = (ε q /p). Note that we can state this reciprocity law in the following form: In the next section we will prove an analogous result connected to Hilbert's Second Supplementary Law of Quadratic Reciprocity. A Supplement to Scholz's Reciprocity Law . Assume that p ≡ q ≡ 1 mod 8 are primes. Then 2 splis in Q( √ p ) and Q( √ q ), and we can write 2O p = ll ′ and 2O q = mm ′ . Now pick elements λ p , λ q such that l h(p) = (λ p ) and m h(q) = (λ q ). Since both fields have units with independent signatures, we may assume that λ p , λ q ≫ 0. The quadratic residue symbol [λ p /q], where qO p = qq ′ , does not depend on the choice of λ p or q, so we may denote it by (λ p /q). Theorem 4. Let p ≡ q ≡ 1 mod 8 be primes with (p/q) = +1, and assume that (ε p /q) = (ε q /p) = +1. Then the following assertions are equivalent: (1) (λ p /q) = +1; (2) ρ can be chosen hyper-primary; (3) the ideal classes generated by the ideals above 2 in F = Q( √ pq ) are fourth powers in Cl(F ); Here is a direct argument using class field theory. Proof. Let F = Q( √ pq ); then F 1 = F ( √ p ) Consider the quadratic extension K = F ( √ ρ ) of F . Then ρ is hyper-primary if and only if the prime l (and, therefore, also its conjugate l ′ ) above 2 splits in K/F . Since K is the unique quadratic subextension of the ray class field modulo q over F , which has degree 2h(p), the prime l will split in K/F if and only if l h(p) = (λ p ) for some λ p ≡ ξ 2 mod q. This shows that (1) ⇐⇒ (2). The symmetry of p and q in the third statement of Thm. 4 then implies Corollary 2. We have (λ p /q) = (λ q /p). While the proof of Thm. 4 required class field theory, the actual reciprocity law in Cor. 2 can be proved with elementary means. We will now give a proof a la Brandler [1]. To this end, write λ p = a+b √ p 2 ; then a 2 − pb 2 = 2 u , where u = h(p) + 2 = 2m + 1 is odd. From a 2 − 2 u = pb 2 we find that a + 2 m √ 2 = π 2 β 2 and a − 2 m √ 2 = π ′ 2 β ′ 2 , where π 2 π ′ 2 = p for some totally positive π 2 ≡ 1 mod 2. Moreover ββ ′ = b and 2a = πβ 2 + π ′ β ′ 2 . Now (π 2 β + β ′ √ p ) 2 = π 2 (πβ 2 + π ′ 2 β ′ 2 + 2y √ p ) = 2π 2 λ. Standard arguments then show that [ π2 ρ2 ] = ( λ q ), where ρ 2 ρ ′ 2 = q. The quadratic reciprocity law in Z[ √ 2 ] shows that [ π2 ρ2 ] = [ ρ2 π2 ], and his implies the following elementary form of the supplement to Scholz's reciprocity law: λ p q = π 2 ρ 2 = ρ 2 π 2 = λ q p . Additional Remarks We close this article with a few remarks and questions. Remark 1. Since p ≡ q ≡ 1 mod 8, we can also write p = N π * 2 and q = N ρ * 2 for elements π * 2 , ρ * 2 ∈ Z[ √ −2 ] with π * 2 ≡ ρ * 2 ≡ 1 mod 2. Then [8, Prop. 2] states that π 2 ρ 2 π * 2 ρ * 2 = p q 4 q p 4 . Under the assumptions of Thm. 4, this means that λ p q = λ q p = π 2 ρ 2 = π * 2 ρ * 2 . Remark 2. Hilbert's Supplementary law as we have stated it applies to all (quadratic) fields with odd class number, not just the fields with prime discriminant. Here we give an example that shows what to expect in this more general situation. Consider primes p ≡ q ≡ 3 mod 4 and primes r ≡ 1 mod 4 with (pq/r) = +1. Let ε pq denote the fundamental unit in k = Q( √ pq ). Then the prime ideals r and r ′ above r in k satisfy r h(pq) = (ρ) for some primary ρ if and only if (ε pq /r) = +1. Since pε pq is a square in k, we have (ε pq /r) = (p/r). Assume now that ρ can be chosen primary, and consider the dihedral extension L/Q with L = Q( √ p, √ q, √ ρ ). Clearly ρ is primary if and only if L/Q( √ pqr ) is cyclic and unramified. It is then easy to show that the quadratic extensions of Q( √ r ) different from Q( √ pq, √ r ) can be generated by a primary element α with prime ideal factorization (pq) h(r) for a suitable choice of prime ideals p and q above p and q, respectively. Note that if pq is primary, then pq ′ is not, since qq ′ = (q) is not primary (we have either q < 0 or q ≡ 3 mod 4). The upshot of this discussion is: if ρ is primary, then exactly one of the ideals pq and pq ′ is primary, say the first one, and then Hilbert's Supplementary Law shows that (ε r /pq) := [ε r /pq] = +1. Conversely, if pq is primary, then (ε r /pq) = (ε pq /r) = +1. We have shown: Proposition 1. Let p ≡ q ≡ 3 mod 4 and r ≡ 1 mod 4 be primes with (pq/r) = +1. Then the following assertions are equivalent: (1) (ε pq /r) = +1; (2) (p/r) = +1; (3) the ideal r in Q( √ pq ) above r is primary; (4) h(pqr) ≡ 0 mod 4; Note that (ε r /pq) is not well defined if (p/r) = −1 since in this case we do not have a canonical way to single out the prime ideals above p and q in Q( √ r ). As an example, consider the case p = 3, q = 7, r = 37; then the elements of norm [2] have generalized Scholz's reciprocity law to higher powers; can the reciprocity law (λ p /q) = (λ q /p) proved above also be generalized in this direction? Our next result is related to the First Supplementary Law of quadratic reciprocity for fields with odd class number; it was stated and proved in a special case by Hilbert ([4]), and proved in full generality by Furtwängler. Nowadays, this result is almost forgotten; for a proof of Hilbert's Supplementary Laws (for arbitrary number fields) based on class field theory, see [9]; Hecke [3, Thm 171] gives a proof based on his theory of Gauss sums and theta functions over algebraic number fields. Theorem 1 (Hilbert's First Supplementary Law). Let a be an ideal of odd norm in some number field k with odd class number h, and let (·/·) denote the quadratic residue symbol in O k . Then the following assertions are equivalent: above is satisfied, i.e., if (ε/a) = +1 for all units ε in k. Hilbert's Second Supplementary Law can be given the following form: Theorem 2 (Hilbert's Second Supplementary Law). Let a be a primary ideal of odd norm in some number field k with odd class number h. Then the following assertions are equivalent: is a special case of Hilbert's First Supplementary Law for fields with odd class number ([4]); observe, however, that Hilbert stated and proved this law only for a very narrow class of fields -the general statement was proved only by Furtwängler. The equivalence (1) ⇐⇒(3)is due to Scholz[13]. Corollary 1 . 1Let p and q satisfy the assumptions of Thm. 3. If the ideals above q in Q( √ p ) are primary, then so are the ideals above p in Q( √ q ). is an unramified quadratic extensions; since ρ is primary, the extension F ( √ ρ )/F is unramified, and it is easily checked that it is the unique cyclic quartic unramified extension of F . Since 2 splits com-pletely in Q( √ p, √ q )/Q, it will split completely in F ( √ ρ )/Q if and only if 2 splits completely in Q( √ p, √ ρ ),which happens if and only if ρ is hyperprimary. On the other hand, the decomposition law in unramified abelian extensions shows that the prime ideals above 2 split completely in F ( √ ρ )/F if and only if their ideal classes are fourth powers in Cl(F ). This proves that (2) ⇐⇒ (3). The equivalence (1) ⇐⇒ (2) is a special case of the Second Supplementary Law of Hilbert's Quadratic Reciprocity Law in number fields with odd class number. ( 5 ) 5there is a unique primary ideal a (up to conjugation) of norm pq in Q( √ r ), and (ε r /pq) := [ε r /a] = +1. Lemma 1. Let K be a number field with ring of integers O K , and let α ∈ O K be an element with odd norm. Then the following assertions are equivalent:1991 Mathematics Subject Classification. Primary 11 R 21; Secondary 11 R 29, 11 R 18. the element −13 + 2 √ 37 ≡ 1 mod 4 is not totally positive) and ±11±. It is easyto check that β = 11+ is primary; now ε r = 6 + √ 37, and [ε r /β] = ( −5 21 ) = +1 as claimed, whereas [ε r /(13 ± 2 √ 37 ] = −1.Remark 3. Above we have seen that, under suitable assumptions, the ideal class generated by a prime above 2 in Q( √ pq ) is a fourth power in the class group if and only if [ π2 ρ2 ] = +1, where π 2 , ρ 2 ∈ Z[ √ 2 ] are elements ≡ 1 mod 2 with norms p and q, respectively. Does an analogous statement hold with 2 replaced by an odd prime ℓ = p, q? Remark 4. Budden, Eisenmenger & Kish21 in the ring of integers in Q( √ 37 ) are ±13±2 √ 37 (these elements are not primary: √ 37 2 √ 37 2 Residuacity properties of real quadratic units. J Brandler, J. Number Theory. 5J. Brandler, Residuacity properties of real quadratic units, J. Number Theory 5 (1973), 271-287 4 A Generalization of Scholz's Reciprocity Law. M Budden, J Eisenmenger, J Kish, J. Théorie Nombr. to appear 6M. Budden, J. Eisenmenger, J. Kish, A Generalization of Scholz's Reciprocity Law, J. Théorie Nombr. Bordeaux, to appear 6 E Hecke, Lectures on the theory of algebraic numbers. Springer-Verlag23E. Hecke, Lectures on the theory of algebraic numbers, Springer-Verlag 1981 1, 2, 3 Über die Theorie des relativquadratischen Zahlkörpers. D Hilbert, Math. Ann. 513D. Hilbert,Über die Theorie des relativquadratischen Zahlkörpers, Math. Ann. 51 (1899), 1-127. 2, 3 Rational quartic reciprocity. F Lemmermeyer, Acta Arith. 674F. Lemmermeyer, Rational quartic reciprocity, Acta Arith. 67 (1994), no. 4, 387-390 1 Rational quartic reciprocity. F Lemmermeyer, Acta Arith. II3F. Lemmermeyer, Rational quartic reciprocity. II, Acta Arith. 80 (1997), no. 3, 273-276 1 From Euler to Eisenstein. F Lemmermeyer, Springer-Verlag13Reciprocity lawsF. Lemmermeyer, Reciprocity laws. From Euler to Eisenstein, Springer-Verlag, 2000 1, 3 Some families of non-congruent numbers. F Lemmermeyer, Acta Arith. 1105F. Lemmermeyer, Some families of non-congruent numbers, Acta Arith. 110 (2003), 15-36 5 Selmer Groups and Quadratic Reciprocity. F Lemmermeyer, Abh. Math. Sem. Hamburg. 76F. Lemmermeyer, Selmer Groups and Quadratic Reciprocity, Abh. Math. Sem. Hamburg 76 (2006), 279-293 2 Die Korrespondenz Hasse-Scholz. F Lemmermeyer, F. Lemmermeyer, Die Korrespondenz Hasse-Scholz, in preparation 1 Ueber die Congruenz x 2 + y 2 ≡ 1 (mod p). Th, Schönemann, J. Reine Angew. Math. 19Th. Schönemann, Ueber die Congruenz x 2 + y 2 ≡ 1 (mod p), J. Reine Angew. Math. 19 (1839), 93-112 1 Zwei Bemerkungen zum Klassenkörperturm. A Scholz, J. Reine Angew. Math. 161A. Scholz, Zwei Bemerkungen zum Klassenkörperturm, J. Reine Angew. Math. 161 (1929), 201-207 1 Über die Lösbarkeit der Gleichung t 2 − Du 2 = −4. A Scholz, Math. Z. 393A. Scholz,Über die Lösbarkeit der Gleichung t 2 − Du 2 = −4, Math. Z. 39 (1934), 95-111 1, 3 Mörikeweg 1, 73489 Jagstzell, Germany E-mail address: [email protected]. deMörikeweg 1, 73489 Jagstzell, Germany E-mail address: [email protected]	[]
[ "Hartree-Fock Theory of Hole Stripe States", "Hartree-Fock Theory of Hole Stripe States" ]	[ "Tae-Suk Kim \nAPCTP\n207-43 Cheongryangri-dong, Dongdaemun-gu130-012SeoulKorea\n", "S R Eric Yang \nDepartment of Physics\nKorea University\nSeoulKorea\n", "A H Macdonald \nDepartment of Physics\nIndiana University\n47405 (BloomingtonIN\n" ]	[ "APCTP\n207-43 Cheongryangri-dong, Dongdaemun-gu130-012SeoulKorea", "Department of Physics\nKorea University\nSeoulKorea", "Department of Physics\nIndiana University\n47405 (BloomingtonIN" ]	[]	We report on Hartree-Fock theory results for stripe states of two-dimensional hole systems in quantum wells grown on GaAs (311)A substrates. We find that the stripe orientation energy has a rich dependence on hole density, and on in-plane field magnitude and orientation. Unlike the electron case, the orientation energy is non-zero for zero in-plane field, and the ground state orientation can be either parallel or perpendicular to a finite in-plane field. We predict an orientation reversal transition in in-plane fields applied along the [233] direction. 73.40.Hm, 71.45.Lr, 73.20.Dx	10.1103/physrevb.62.r7747	[ "https://arxiv.org/pdf/cond-mat/0005374v2.pdf" ]	119,479,869	cond-mat/0005374	b870c2aad7779c7b760bd97397a25afca34a9f9b	Hartree-Fock Theory of Hole Stripe States 30 Jan 2001November 23, 2018) Tae-Suk Kim APCTP 207-43 Cheongryangri-dong, Dongdaemun-gu130-012SeoulKorea S R Eric Yang Department of Physics Korea University SeoulKorea A H Macdonald Department of Physics Indiana University 47405 (BloomingtonIN Hartree-Fock Theory of Hole Stripe States 30 Jan 2001November 23, 2018) We report on Hartree-Fock theory results for stripe states of two-dimensional hole systems in quantum wells grown on GaAs (311)A substrates. We find that the stripe orientation energy has a rich dependence on hole density, and on in-plane field magnitude and orientation. Unlike the electron case, the orientation energy is non-zero for zero in-plane field, and the ground state orientation can be either parallel or perpendicular to a finite in-plane field. We predict an orientation reversal transition in in-plane fields applied along the [233] direction. 73.40.Hm, 71.45.Lr, 73.20.Dx Because of the macroscopic dengeneracy of Landau levels, the physics of two-dimensional (2D) electron systems in strong external fields has been a fertile area for manyparticle physics. Recently [1][2][3][4] the emergence of strongly anisotropic transport properties at low temperatures has been interpreted as evidence for the occurrence of the unidirectional charge-density-wave stripe states predicted by Hartree-Fock theory [5]. For conduction band Landau levels with orbital kinetic energy index n > 2 (filling factor ν > 4), the putative stripe states occur instead of the strongly correlated fluid states responsible for the quantum Hall effect [6]. Although Hartree-Fock theory provides a clear motivation for stripe states [5], it cannot reliably predict the nature of the ground state because the energetic competition with fluid states is delicate [7]. Moreover, the transport properties of stripe states cannot be explained by Hartree-Fock theory, although they are consistent [8] with theories [9][10][11] of quantum-fluctuating stripes. For these reasons, the ability of Hartree-Fock theory to predict [12,13] the low resistance (parallel to stripe) direction in an in-plane magnetic field has played an essential role in establishing the stripe-state explanation of N > 2 anisotropic transport. A recent study [14] in which a reorientation transition in a wide quantum well sample is explained by Hartree-Fock theory is especially convincing in this respect. The present work is motivated by the discovery [15] of anisotropic transport in 2D hole systems grown on GaAs (311)A substrates. In this case, anisotropic transport already occurs for ν ∼ 5/2, demonstrating that there are important differences between the electron and hole cases. The change is not unexpected, given the anisotropy of 2D hole band structure. We have generalized the Hartree-Fock theory of stripe states to the case of valence bands described by a the single-particle Luttinger Hamiltonian [16]. We find that strong orbital-quantumnumber mixing leads to anisotropic effective electronelectron interactions and to a dependence of stripe energy on orientation even in the absence of an in-plane magnetic field. This property favors the formation of a stripe state, consistent with experiment. The ground state orientation is not in general either parallel or perpendicular to the direction of a finite in-plane field when one is present. To The cubic coordinates in terms of which the Luttinger Hamiltonian is usually expressed are related to these coordinates by k a = i u ai k i , a = x, y, z; i = 1, 2, 3 in both direct and reciprocal space. Here the u ai are direction cosines. Each element of the Luttinger Hamiltonian is a quadratic form in the k i 's. The two-dimensional hole gas is created by a GaAs (narrow gap) quantum well flanked by the AlGaAs (wide gap) barriers. Since the barrier lies in the range 100 − 400 meV and the energy scale of interest is ∼ 10 meV, we take the barrier to be infinite. Single particle eigenspinors of the quantum well Luttinger Hamiltonian can be expanded in the form ψ k ( x) = iα c iα ( k) e i k· ρ ζ i (x 3 )χ α ( x). Here the 2D wavevector k = (k 1 , k 2 ) is a good quantum number, and ζ i (x 3 ) ∝ sin(iπx 3 /b) where b is the well width. Bloch functions χ α ( x) are chosen such that when k 1 and k 2 are set to zero the Luttinger Hamiltonian takes a disgonalized form. For the GaAs valence bands we have used the Luttinger model parameters γ 1 = 6.85, γ 2 = 2.1, and γ 3 = 2.9 and retained the first 20 subbands. A typical 2D band structure is illustrated in Fig. 1 by plotting constant energy contours for the lowest energy subband. Since our confinement potential has inversion symmetry, each energy band is doubly degenerate. For wider quantum wells, subband spacing is reduced and subband mixing is strengthened. Since cubic systems are not invariant under 90 • rotations about the [311] direction, the 2D bands have lower than square symmetry. The constant energy contours are elongated in directions with larger effective mass. For the lowest subband, the Fermi sur-face anisotropy is relatively weak near the zone center (Γ point), but gets stronger at larger \| k\|. The effective mass is heavier (lighter) along k 2 than along k 1 direction close to (far away from) the zone center, and is smallest along the direction rotated from k 1 by 45 • at larger \| k\|. As we shall discuss later, the Fermi surface topology is manifested in anisotropic effective electron-electron interactions and ultimately in orientation-dependent stripe state energies. When a magnetic field is applied, the subband spectrum consists of macroscopically degenerate Landau levels whose energies may be evaluated following familiar lines. In the Luttinger Hamiltonian, k 1 and k 2 are replaced by raising and lowering ladder operators, k 1 = i(−a + a † )/ √ 2ℓ and k 2 = (a + a † )/ √ 2ℓ, with ℓ = (hc/eB) 1/2 and a Zeeman term is added to the Luttinger Hamiltonian [16], H Z = −κµ B J · B. Here µ B is the Bohr magneton and κ = 1.2 is an additional Luttinger model parameter. The envelope function eigenspinors can be expanded in the form ψ N X ( x) = niα c niα φ nX ( ρ)ζ i (x 3 )χ α ( x) where φ nX ( ρ) is one of the parabolic band Landau gauge wavefunction generated by the ladder operator algebra, and the Hamiltonian matrix is independent of the guiding center label X. The eigenstates of the Hamiltonian are strong mixture of Landau level n and subband i indices. In diagonalizing the finite-field Luttinger Hamiltonian, three subbands and 30 Landau levels were retained. This procedure is readily generalized to allow for a magnetic field component perpendicular to the growth direction, B = B(x 3 + tan θ[x 1 cos ϕ +x 2 sin ϕ]). In this case, k 1 → k 1 +tan θ sin ϕx 3 /ℓ 2 and k 2 → k 2 −tan θ cos ϕx 3 /ℓ 2 . Fig. 2 illustrates our results for the magnetic field dependence of the Landau level energies for a typical quantum well width and no in-plane field. The Landau levels show very strong nonlinear dispersion with magnetic field. The energy levels are unevenly spaced and the apparently crossed levels are split due to the lack of parity under the inversion symmetry operation. Both Landaulevel and subband mixing are stronger at larger quantum well widths. To model the stripe state seen in Ref. [15] at ν = 5/2, we consider interacting electrons in the third level of Fig. 2 at B ≈ 2.5 Tesla (n h = 1.5×10 11 cm −2 ). The distorted semiclassical cyclotron orbit of a Fermi energy electron with this density is illustrated in Fig.1 for the case of b = 250Å. Semiclassical orbit distortions translate quantum mechanically into mixing of parabolic band Landau levels. For interactions within a Landau level this mixing is described exactly [12] simply by replacing the Coulomb interaction by V eff ( q) = 2πe 2 qǫ(q) {ni} M n1n4,n2n3 (q) F n1n4 ( q)F n2n3 (− q).(1) where ǫ(q) is the dielectric constant arising from polar-ization of the lower filled Landau levels and M n1n4,n2n3 (q) = αβ {ii} c * n1i1α c n4i4α c * n2i2β c n3i3β W (i 1 i 4 , i 2 i 3 ; q) with W (i 1 i 4 , i 2 i 3 ; q) ≡ dz dz ′ e −q\|z−z ′ \| ζ * i1 (z)ζ i4 (z)ζ * i2 (z ′ )ζ i3 (z ′ ). Note that V eff ( q) is always repulsive. The parabolic-band plane-wave matrix elements, F nn ′ ( q), play a crucial role since they are dependent on the 2D angular coordinate of q, φ q : F n1n4 ( q)F n2n3 (− q) ∼ e iπ(n1+n2−n3−n4)φq . It is precisely this effect which gives rise to orientation dependence of the stripe state energy. In an electron gas, n 1 = n 2 = n 3 = n 4 = N and the stripe state has no orientation dependence. In the Hartree-Fock approximation, the energy per electron for the stripe state of a half-filled Landau level is [12] E = 1 2ν * ∞ n=−∞ ∆ 2 n U 2πn aê ,(2) where ∆ n = ν * sin(nν * π) nν * π , e is the direction perpendicular to the stripes, ν * the filling fraction at upper Landau level, and U ( q) = V eff ( q) 2πℓ 2 − d 2 p (2π) 2 V eff ( p) e i(pxqy−pyqx)ℓ 2 . The evaluation of U ( q) is simplified by Fourier expanding the angle dependence of V eff ( q) and taking advantage of inversion symmetry in the 2D plane. The isotropic Fourier component is most dominant near q = 0. Higher Fourier components vanish at q = 0, and oscillate between positive and negative values as a function of q. When summed, the net result is a strong suppression of V eff ( q) at qℓ > 1.5, which helps to make U ( q) more negative (see Figure 3) and favors stripe states. The results we obtain for the orientation dependence of the cohesive energy in a 250Å quantum well are presented in Fig. 4. The ground state stripe orientation is tilted from the [233] axis by φ st ≈ 23 • . We studied the dependence of the ground state φ on well width, b, finding that for b = 100Å and less φ st = 0 • . On the other hand, for b = 150Å and b = 200Å [15], we find φ st ≈ 15 • and φ st ≈ 22 • , respectively. The difference in cohesive energy per electron between φ = 0 and φ st , \|E coh (0 • ) − E coh (φ st )\|, is small: for b = 150, 200, 250Å it is, respectively, 0.7, 2.8, 4.5% of the maximum difference \|E coh (90 • ) − E coh (φ st )\|. For larger values of b = 300Å, 350Å and 400Å, we find φ st ≈ 45 • . The ground state φ st changes gradually from 0 • to 45 • with increasing quantum well width. Note that the dependence of the ground state stripe direction on well width tracks the well-width dependence of the Fermi surface topology. We believe it is band anisotropy, and the orientation dependence of stripe-state energy it leads to, that is primarily responsible for the occurrence of hole stripe states at ν = 5/2. In 2D electron gases, stripe states appear [4,7] at ν = 5/2 only when anisotropy is imposed by adding an in-plane magnetic field. Comparing experiment [4] and in-plane field calculations [12], we find that conduction band stripe states appear at ν = 5/2 when the intrinsic orientation energy per electron is ∼ 0.0002e 2 /ǫ 0 ℓ, comparable to the valence band ν = 5/2 orientation energy per hole at perpendicular fields illustrated in Fig. 4. The valence band orientation energy is also sensitive to in-plane field as illustrated in Fig. 4 for B ≈ 2.5 Tesla. For quantum wells of widths b = 100 − 250Å, increasing in-plane field along the [011] axis ( Fig. 4(a): ϕ = 0 • ) tends to orient the easy axis of striped CDW closer to [233]. On the other hand, when in-plane field is applied along [233] axis ( Fig. 4(b): ϕ = 90 • ) we find that for a sufficiently large well widths and field tilt angles, the ground state orientation can change from near [233] to [011]. In comparing with experiment, it is necessary to use the appropriate sample width and 2D hole density and to account for the corrugations which occur [17] at GaAs/AlGaAs(311)A interfaces. For example, we have investigated the hole density dependence of the orientation energy and found that there is more anisotropy at higher hole densities as expected from Fig. 1. The consequences of corrugation for the stripe states are difficult to quantify, partly because the stripe state period does not match the period of corrugation potential. Nevertheless it is likely that this morphological feature is partly responsible [15] for the factor of ∼ 2 mobility anisotropy at zero field and that it will tend to align the stripes. Taking account of the corrugations, our finding for the 200Å quantum well case [15], is consistent with experimental finding that stripes are aligned along [233]. It is reasonable to conclude that the corrugation contribution to the orientation energy overcomes the small intrinsic value of \|E coh (φ = 0 • ) − E coh (φ st )\|. Intrinsic orientation energies can, however, be altered by tilted fields. For fields along the [011] direction, E coh (φ = 90 • )−E coh (φ = 0 • ) is increased and we would expect little experimental consequence. For fields along the [233] direction, on the other hand, we predict a dramatic reversal of stripe orientations at tilt angle θ ∼ 60 • . At this angle, the intrinsic band-structure effects addressed here favor [011] oriented stripes by ∼ 0.0008e 2 /ǫ 0 ℓ per electron. This number should be compared with the ∼ 0.0001e 2 /ǫ 0 ℓ orientation energy produced [12] by a 25 • field tilt in n = 2 conduction band Landau levels. In the [100] growth conduction band case, stripes orient along [110] directions for perpendicular fields, likely because of MBE growth stabilities analogous to the corrugations discussed above. The anisotropy energy produced by a ∼ 25 • field tilt, in the n = 2 conduction band case is sufficient to overcome this extrinsic anisotropy and reorient the stripes. Based on these comparisons, we conclude that tilted field effects can also overcome extrinsic anisotropy sources in the valence band case. In summary, valence band stripe states have a finite orientation energy even without an in-plane field, favoring their occurrence at ν = 5/2. For quantum well widths b = 100−250Å, intrinsic band effects yield a ground state orientation close to [233] and corrugation effects are likely sufficient to produce the [233] orientation seen in experiment. We predict that for typical quantum well widths, an in-plane field applied along the [233] direction will induce a stripe reorientation transition. We are indebted to J.P. Eisenstein, T. Jungwirth, M. Shayegan, and R. Winkler for valuable discussions. This work was supported in part by the National Science Foundation under grant DMR-9714055, and in part by grant No.1999-2-112-001-5 from the interdisciplinary Research program of the KOSEF. We thank the Center for Theoretical Physics at Seoul National University for providing computing time. FIG. 1 . 1Fermi surface topology for the well width 250Å. The constant energy contour lines (solid lines) are drawn for the lowest subband. k1 and k2 correspond to [011] and [233] directions, respectively. (for illustration, rotated images of solid lines by 90 • are displayed as dashed lines.) The energies for contours are 0.61, 1.1, 1.71, 2.31, 2.92, and 3.53meV from inside to outside. Sample with n h = 1.5 × 10 11 cm −2 corresponds to the third contour (EF = 1.71meV). FIG. 2 .FIG. 4 . 24Landau level dispersion with magnetic field. With the inclusion of the Zeeman term (κ = 1.2), the spin degeneracy is lifted. of width 250Å. Solid line corresponds to no-Landau-level-mixing case with screening. The upper (lower) dashed line represents unscreened (screened) Hartree potential for 2D holes at ν = 5/2. (b) Hartree-Fock (HF) potentials. (c) Magnified view of HF potentials near the minimum where the angle dependence is significant. In (b) and (c), solid line: without Landau-level-mixing; long dashed line: along [011]; short dashed line: along 45 • from x1 axis; dotted line: along [233]. The cohesive energy at ν = 5/2 is drawn as a function of φ (the angle between [233] and the easy axis of the striped CDW) with varying in-plane magnetic field for the confinement potential width 250Å. Solid lines correspond to the case without in-plane fields. θ defines the direction of tilted magnetic fields from the [311] axis. The in-plane fields are pointing along [011] and [233] in (a) and (b), respectively. describe 2D hole gases grown along the [311] direction, it is convenient to choose a Cartesian coordinate system with [011], [233] and [311] direction axes. . M P Lilly, K B Cooper, J P Eisenstein, L N Pfeiffer, K W West, Phys. Rev. Lett. 82394M. P. Lilly, K.B. Cooper, J.P. Eisenstein, L.N. Pfeiffer, and K.W. West, Phys. Rev. Lett. 82, 394 (1999). . R R Du, D C Tsui, H L Stormer, L N Pfeiffer, K W Baldwin, K W West, Solid State Commun. 109389R.R. Du, D.C. Tsui, H.L. Stormer, L.N. Pfeiffer, K.W. Baldwin, and K.W. West, Solid State Commun. 109, 389 (1999). . W Pan, R R Du, H L Stormer, D C Tsui, L N Pfeiffer, K W Baldwin, K W West, Phys. Rev. Lett. 83820W. Pan, R.R. Du, H.L. Stormer, D.C. Tsui, L.N. Pfeiffer, K.W. Baldwin, and K.W. West, Phys. Rev. Lett. 83, 820 (1999). . M P Lilly, K B Cooper, J P Eisenstein, L N Pfeiffer, K W West, Phys. Rev. Lett. 83824M.P. Lilly, K.B. Cooper, J.P. Eisenstein, L.N. 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The sample studied in this experimental sample had a quantum well width b =. M Shayegan, H C Manoharan, S J Papadakis, E P De Poortere, Physica E. 640M. Shayegan, H.C. Manoharan, S.J. Papadakis, and E.P. De Poortere, Physica E 6, 40 (2000). The sample studied in this experimental sample had a quantum well width b = 200Å. . J M Luttinger, Phys. Rev. 1021030J.M. Luttinger, Phys. Rev. 102, 1030 (1956). . R Nötzel, N N Ledentsov, L Däweritz, K Ploog, M Hohenstein, Phys. Rev. B. 453507R. Nötzel, N.N. Ledentsov, L. Däweritz, K. Ploog, and M. Hohenstein, Phys. Rev. B 45, 3507 (1992).	[]
[ "ON F ω 2 -AFFINE-EXCHANGEABLE PROBABILITY MEASURES", "ON F ω 2 -AFFINE-EXCHANGEABLE PROBABILITY MEASURES" ]	[ "Pablo Candela ", "Diego González-Sánchez ", "Balázs Szegedy " ]	[]	[]	For any standard Borel space B, let P(B) denote the space of Borel probability measures on B. In relation to a difficult problem of Aldous in exchangeability theory, and in connection with arithmetic combinatorics, Austin raised the question of describing the structure of affine-exchangeable probability measures on product spaces indexed by the vector space F ω 2 , i.e., the measures in P(B F ω 2 ) that are invariant under the coordinate permutations on B F ω 2 induced by all affine automorphisms of F ω 2 . We answer this question by describing the extreme points of the space of such affine-exchangeable measures. We prove that there is a single structure underlying every such measure, namely, a random infinite-dimensional cube (sampled using Haar measure adapted to a specific filtration) on a group that is a countable power of the 2-adic integers. Indeed, every extreme affineexchangeable measure in P(B F ω 2 ) is obtained from a P(B)-valued function on this group, by a vertex-wise composition with this random cube. The consequences of this result include a description of the convex set of affine-exchangeable measures in P(B F ω 2 ) equipped with the vague topology (when B is a compact metric space), showing that this convex set is a Bauer simplex. We also obtain a correspondence between affine-exchangeability and limits of convergent sequences of (compact-metric-space valued) functions on vector spaces F n 2 as n → ∞. Via this correspondence, we establish the above-mentioned group as a general limit domain valid for any such sequence.	null	[ "https://arxiv.org/pdf/2203.08915v2.pdf" ]	247,519,053	2203.08915	f5aa32fe0821898e163a41a76337a5e07bf29f84	ON F ω 2 -AFFINE-EXCHANGEABLE PROBABILITY MEASURES 2 Apr 2022 Pablo Candela Diego González-Sánchez Balázs Szegedy ON F ω 2 -AFFINE-EXCHANGEABLE PROBABILITY MEASURES 2 Apr 2022arXiv:2203.08915v2 [math.PR] For any standard Borel space B, let P(B) denote the space of Borel probability measures on B. In relation to a difficult problem of Aldous in exchangeability theory, and in connection with arithmetic combinatorics, Austin raised the question of describing the structure of affine-exchangeable probability measures on product spaces indexed by the vector space F ω 2 , i.e., the measures in P(B F ω 2 ) that are invariant under the coordinate permutations on B F ω 2 induced by all affine automorphisms of F ω 2 . We answer this question by describing the extreme points of the space of such affine-exchangeable measures. We prove that there is a single structure underlying every such measure, namely, a random infinite-dimensional cube (sampled using Haar measure adapted to a specific filtration) on a group that is a countable power of the 2-adic integers. Indeed, every extreme affineexchangeable measure in P(B F ω 2 ) is obtained from a P(B)-valued function on this group, by a vertex-wise composition with this random cube. The consequences of this result include a description of the convex set of affine-exchangeable measures in P(B F ω 2 ) equipped with the vague topology (when B is a compact metric space), showing that this convex set is a Bauer simplex. We also obtain a correspondence between affine-exchangeability and limits of convergent sequences of (compact-metric-space valued) functions on vector spaces F n 2 as n → ∞. Via this correspondence, we establish the above-mentioned group as a general limit domain valid for any such sequence. Introduction Let B be a standard Borel space, with σ-algebra B. Let us denote the standard Borel space of probability measures on B by P(B, B), or by P(B) when the σ-algebra is clear [33, p. 113]. For a countably infinite set T , we denote by (B T , B ⊗T ) the corresponding (standard Borel) product measurable space. In the study of exchangeability in probability theory, a central problem consists in describing the structure of measures in P(B T , B ⊗T ) that are invariant under prescribed symmetries, these symmetries being usually the coordinate permutations on B T induced by a group Γ of permutations of T . Denoting the set of such measures by Pr Γ (B T ) (following the notation in [5]), the aim of such descriptions is to represent any such measure as an image (typically a mixture; see [3, (2.3)]) of simpler measures in Pr Γ (B T ). The most classical result in this topic is de Finetti's theorem [18], which describes Pr Γ (B T ) for Γ the group of all finitely-supported permutations of T . More precisely, in this classical context µ is said to be an exchangeable measure on B T if µ • θ −1 γ = µ for every transformation θ γ : B T → B T of the form (v t ) t∈T → (v γ(t) ) t∈T , where γ : T → T is a permutation that fixes all but finitely many elements of T . De Finetti's theorem states that any such measure µ is a mixture of product measures t∈T λ, λ ∈ P(B). Different notions of exchangeability, involving other groups Γ, lead to various analogues or extensions of de Finetti's theorem. Principal examples of such extensions include the results of Aldous [1,2], Hoover [27,28], Kallenberg [32] and Kingman [34]. Exchangeability theory has various connections beyond probability theory, with combinatorics and ergodic theory among other areas; see [4] for a comprehensive survey, bringing to light in particular the relations with the topic of graph limits; see also [17,19]. This paper concerns notions of exchangeability where T is the infinite discrete cube, which we denote by N , i.e. the subset of {0, 1} N consisting of those sequences with only finitely many non-zero coordinates. We often identify N as a set with the vector space F ω 2 := i∈N F 2 . This direction originates in a problem posed in the 1980s by Aldous [3, §16], which asked for a structural description of Pr Γ (B N ) for Γ = Aut( N ), the group of isometries 1 (or discrete-cube automorphisms) of N . As explained in [3, §16], this problem of Aldous did not seem to yield to previous mainstream methods in this area (see for instance [3,Example (16.20)]). More than two decades later, Austin shed light on the difficulty of the problem by showing in [5] that the space of Aut( N )exchangeable measures on B N (or cube-exchangeable measures, as they are called in [5]) has a markedly less tractable structure than those corresponding to other, more classical, exchangeability notions. More precisely, Austin showed that for any (non-trivial) compact metric space B, the convex set Pr Aut( N ) (B N ) equipped with the vague (a.k.a. weak ) topology is a so-called Poulsen simplex (meaning that its extreme points form a dense subset), whereas for other more classical exchangeability notions, the convex set Pr Γ (B T ) with the vague topology has a more stable structure, in that it is a Bauer simplex (meaning that its extreme points form a closed subset). This motivated the study of other natural notions of exchangeability involving the cube N , especially notions that are stronger than Aut( N )-exchangeability and are thus more likely to have simpler descriptions. In particular, motivated by connections with arithmetic combinatorics, in [5, §5.3] Austin highlighted as an interesting object of study the notion of what we will call affine-exchangeable measures on B N , whose corresponding group of symmetries Γ is the group of (invertible) affine transformations on the cube N (identified with F ω 2 ). Let us denote this group by Aff(F ω 2 ) (following [5]), and note that Aff(F ω 2 ) ∼ = GL(F ω 2 ) ⋉ Z ω 2 . The main results of this paper are, firstly, a solution of the description problem for affine-exchangeability, in the form of a representation theorem for affine-exchangeable measures (Theorem 1.5 below); secondly, an application concerning limit objects for certain convergent sequences of functions, in the spirit of the recent limit theories for graphs and hypergraphs [8,36,37,38]. To introduce this application, let us begin by mentioning that in [41] the third named author proved an analogous result for limits of functions defined on abelian groups, with a notion of convergence involving linear forms of (true) complexity 1 (in the sense of [21]). In this paper we consider limits of sequences of functions (f n : F n 2 → B) n∈N where B is a compact metric space and where the notion of convergence involves systems of linear forms of arbitrary finite complexity. Variants of this notion have been considered previously (e.g. for Boolean functions in [25]); we shall define it in detail in the sequel (in Section 7) but let us briefly outline the notion here. Given a sequence (f n : F n 2 → B) n∈N for B a compact metric space, we can identify any given system of finitely many linear forms over F 2 as a set L ⊂ ). Then, denoting by µ C k (F n 2 ) the Haar probability measure on the group of standard k-dimensional cubes in F n 2 , and noting that any such cube can be viewed as an affine-linear map A : F k 2 → F n 2 , we can define the measure µ L,fn on B L as the pushforward of µ C k (F n 2 ) under the map A → (f n • A(L)) L∈L (see (13)). We then say that the sequence (f n ) n∈N converges if for every finite L ⊂ F ω 2 the measures µ L,fn converge vaguely as n → ∞. Our second main result (Theorem 1.8 below) establishes that every such sequence (f n ) converges to a limit object closely related to our representation theorem for affine-exchangeable measures. Before discussing these results in more detail, let us give some basic examples of affine-exchangeable measures. (v) = x 0 + ∞ i=1 v(i)h i where x 0 ∈ Z 2 , h i ∈ Z 2 for all i ∈ N and v ∈ N . Note that, since every v ∈ N has at most finitely many non-zero coordinates, the last sum above is always well-defined. Letting µ G denote the Haar probability measure on G, we can then define a measure µ ∈ P(B N ) by the formula µ = G v∈ N m(c(v)) dµ G (c),(1) where the product here denotes a countable product of measures. Thus, for any cylinder set S ⊂ B N , i.e. a set of the form S = v∈ N A v ⊂ B N with A v = B only for v in some finite set V ⊂ N (recall that these cylinder sets generate the product σ-algebra B ⊗T ), we have µ(S) = G v∈V m(c(v))(A v ) dµ G (c). It is readily seen that µ is an affineexchangeable measure, using the fact that for every γ ∈ Aff(F ω 2 ) the transformation G → G, c → c • γ preserves the measure µ G . Note also that, since the map sending (x 0 , h 1 , h 2 , . . .) to c(v) = x 0 + ∞ i=1 v(i)h i is an isomorphism Z N 2 → G, we can rewrite formula (1) as follows (where µ Z N 2 is the Haar measure on Z N 2 ): µ = Z N 2 v∈ N m(x 0 + v(1)h 1 + v(2)h 2 + · · · ) dµ Z N 2 (x 0 , h 1 , h 2 , . . .).(2) As mentioned above, the elements c ∈ G can be viewed as infinite-dimensional combinatorial cubes on Z 2 (the integral in (1) or (2) could thereby be viewed as an infinitedimensional, measure-valued, Gowers inner-product of the function m with itself). By generalizing this underlying notion of infinite-dimensional cubes, we soon obtain a large family of examples of affine-exchangeable measures. In particular, we have the following construction where the group Z 2 is replaced by more general filtered abelian groups. Example 1.2. Let Z be a compact abelian group, and let Z • = (Z (j) ) j≥0 be a filtration on Z, i.e., a sequence of closed subgroups of Z with Z (j) ⊇ Z (j+1) for all j ≥ 0 and Z (0) = Z (1) = Z. We can then generalize the group G from Example 1.1 to obtain the group of infinite-dimensional cubes adapted to the filtration Z • . This group, which we shall denote by C ω (Z • ), is the closed subgroup of Z N generated by elements f ∈ Z N of the form f (v 1 , v 2 , . . .) = av j 1 · · · v js where s ∈ N, a ∈ Z (s) and j 1 , . . . , j s ∈ N. In standard terminology related to cubic structures, this construction of the group C ω (Z • ) essentially follows that of the Host-Kra cubes associated with a filtered group, except that here the construction is carried out in a setting of countably infinite dimension (we refer to Part 2 of the book [30] for more background on cubic structures, and to [10, §2.2.1] for a treatment of Host-Kra cubes in a nilspace theoretic context as used in this paper). Let µ C ω (Z•) be the Haar measure on C ω (Z • ). Then for any standard Borel space B and Borel map m : Z → P(B), we can define the following measure in P(B N ) similarly to (1): µ = C ω (Z•) v∈ N m(c(v)) dµ C ω (Z•) (c).(3) It can be checked that µ is affine-exchangeable provided that the filtration Z • satisfies a simple property that we call 2-homogeneity, which states that for every i ∈ N we have 2g ∈ Z (i+1) for every g ∈ Z (i) (see Corollary B.2). More examples can be obtained by further generalizing the underlying cubic structures; we shall see such constructions in detail in the sequel (e.g. in Section 2). In fact, affineexchangeability is a special case of a more general notion of exchangeability involving N , called cubic exchangeability, which was introduced in [14] (and is itself strictly stronger than the notion of Aldous from [3, §16]). We shall now briefly discuss this connection, and the main results concerning cubic exchangeability from [14], as they provide useful ingredients and motivations for the main results in this paper. ON F ω 2 -AFFINE-EXCHANGEABLE PROBABILITY MEASURES 5 A measure on B N is cubic-exchangeable if for each integer k ≥ 0, for any pair of injective discrete-cube morphisms 2 φ 1 , φ 2 : {0, 1} k → N , the two images of the measure under the coordinate projections to φ 1 ({0, 1} k ) and φ 2 ({0, 1} k ) are equal; see [14,Definition 6.3]. 3 The main result in [14] concerning cubic exchangeability is a representation theorem for cubic-exchangeable measures on B N . This theorem describes the extreme points in the space of such measures in terms of a specific construction which involves compact nilspaces; see [14,Theorem 6.7]. Nilspaces are structures in which combinatorial cubes can be defined in great generality, so their connection with exchangeability notions involving N is natural. In particular, nilmanifolds (central examples of compact nilspaces) were already related to certain notions of exchangeability in the work of Frantzikinakis in [19]. Originating in work of Host and Kra (notably [29,31]), nilspaces were introduced in [9], and by now there is a fair amount of literature on these objects [10,11,22,23,24]. We shall recall some basic nilspace theory in this paper, but we defer this to Section 2. On the other hand, the construction of cubic-exchangeable measures involving compact nilspaces is more recent and plays a key role in this paper's main results, so let us recall the construction here. Definition 1.3. [Nilspace construction of extreme cubic-exchangeable measures] Let (B, B) be a standard Borel space, let X be a compact nilspace and let m : X → P(B) be a Borel map. Let C ω (X) denote the space of infinite-dimensional cubes N → X, equipped 4 with its Haar probability measure µ C ω (X) . Then we define the associated cubic-exchangeable measure ζ X,m by the following formula (where the product denotes a countable product of measures): ζ X,m = C ω (X) v∈ N m(c(v)) dµ C ω (X) (c).(4) In [14], this construction was explained from the more statistical point of view on exchangeability, in terms of joint distributions of sequences of random variables. The present paper adopts a more ergodic-theoretic viewpoint on exchangeability, working directly with Γ-invariant measures on product spaces (see e.g. [5]); this viewpoint motivates formula (4), expressing the construction directly as a measure on B N (both viewpoints are useful). Note that the basic example in (1) is a special case of (4), indeed the measure µ in (1) is ζ X,m where the underlying nilspace X consists of the group Z 2 (with the standard 2 A map φ : {0, 1} k → N is a morphism if it extends to an affine homomorphism Z k → i∈N Z; see [14, Definition 6.1]. If φ is injective then its image can be viewed as a k-dimensional affine subcube of N . 3 As such, cubic exchangeability does not involve invariance under a group Γ like other exchangeability notions, although it can be formulated in terms of invariance under a semigroup. 4 The set C ω (X) is a natural generalization for compact nilspaces of the group C ω (Z • ) from Example 1.2. The related notions, in particular the Haar measure on C ω (X), will be explained in Subsection 2.3. abelian cube structure determined by the lower-central series on Z 2 ), and where C ω (X) is the group G in that example. Similarly, the measure in (3) is ζ X,m where X is the so-called group nilspace associated with the filtered group (Z, Z • ), i.e. the nilspace consisting of Z equipped with the Host-Kra cubes determined by the filtration Z • (see e.g. [10, §2.2.1]). To summarize, from previous works we have three exchangeability notions involving the cube N , which are related as follows: 5 An observation leading to the main results of this paper is that the exactness of the above description of cubic exchangeability fails for the stronger notion of affine exchangeability. More precisely, for a measure of the form ζ X,m to be affine-exchangeable, the underlying nilspace X must have a more specific structure than in the general cubicexchangeable setting. These more specific structures are the so-called 2-homogeneous nilspaces, introduced in [12] (and recalled in more detail below). As a consequence, when we replace cubic exchangeability with affine exchangeability, the representation theorem from [14] can be significantly refined, leading to one of the main results of this paper, Theorem 1.5 below. In particular, the potentially varying compact nilspaces X in the representation in [14,Theorem 6.7] can all be replaced here by a single compact nilspace, defined using the group of 2-adic integers. To formalize this, we use the following notation. Pr Aff(F ω 2 ) (B N ) ⊂ {cubic-exchangeable measures in P(B N )} ⊂ Pr Aut( N ) (B N Throughout most of the paper (except in Section 4), we shall denote the group of 2-adic integers by Z (instead of Z 2 ) to avoid an overload of sub-indices in the subsequent notations. For any integer ℓ ≥ 1 we denote by Z •,ℓ the filtration on Z with i-th term Z (i) = Z if i = 0, . . . , ℓ and Z (i) = 2 i−ℓ Z = {2 i−ℓ x : x ∈ Z} for i > ℓ. Definition 1.4. We denote by H the group nilspace consisting of the compact abelian group 6 ∞ ℓ=1 Z N equipped with the cube structure determined by the product filtration H • = ∞ ℓ=1 Z N •,ℓ , that is, the filtration with i-th term H (i) = H for i = 0, 1 and H (i) = (2 i−1 Z N ) × (2 i−2 Z N ) × · · · for i ≥ 2. Recall that a measure µ on a σ-algebra A is said to be concentrated on a set S ∈ A if for every B ∈ A we have µ(B) = µ(B ∩ S). We can now state our main result. 5 A more detailed explanation of these inclusions is given in Lemma 2.24 below. 6 By a slight abuse of notation we shall often denote this group also by H. ON F ω 2 -AFFINE-EXCHANGEABLE PROBABILITY MEASURES 7 Theorem 1.5. Let B be a standard Borel space and let µ ∈ Pr Aff(F ω 2 ) (B N ). Then there is a Borel probability measure κ on P(B N ), which is concentrated on the set 7 of measures {ν ∈ P(B N ) \| ν = ζ H,m for some Borel map m : H → P(B)}, such that µ = P(B N ) ν dκ. In other words, every affine-exchangeable measure is a mixture of measures ζ H,m . There are several equivalent ways to describe these measures ζ H,m , apart from formula (4). Let us give here a particularly simple description that does not involve any nilspace theory: we have ζ H,m = G v∈ N m(c(v)) dµ G (c) where G is a closed subgroup of H F ω 2 determined by a countable set of linear equations. More precisely, if for any map f : F k 2 → H (for any k ∈ N) we denote by σ k (f ) the alternating sum v∈F k 2 (−1) v(1)+···+v(k) f (v), then G is the compact abelian group consisting of all maps c : F ω 2 → H such that for every k and every injective affine homomorphism φ : F k 2 → F ω 2 we have σ k (c • φ) = 0 mod H (k) . This description of G using linear equations (modulo subgroups) is equivalent to the construction in (3) (in particular G = C ω (H • )), by basic results on Host-Kra cubes (see e.g. [10, §2.2.3]). Theorem 1.5 yields a description of the geometry of the set Pr Aff(F ω 2 ) (B N ) for any standard Borel space B. Indeed, as detailed in Section 6, the extreme points of this convex set are exactly the measures of the form ζ H,m . Moreover, the following holds. Theorem 1.6. For any compact metric space B, the convex set Pr Aff(F ω 2 ) (B N ) equipped with the vague topology is a Bauer simplex. This Bauer property was identified in [5, §3.1] to be a common feature of other mainstream exchangeability notions. Theorem 1.6 shows that affine exchangeability also follows this pattern. As mentioned above, Theorem 1.5 has applications concerning limit objects for a notion of convergence for sequences of functions (f n : F n 2 → B) n∈N for a fixed compact metric space B. This notion is analogous to well-known and much studied notions of convergence for sequences of graphs and hypergraphs. These applications rely on a general connection between representation theorems for exchangeable measures in probability theory, and various limit theories in combinatorics; for more background we refer to [4, §2.3], which discusses this connection in the setting of exchangeable random hypergraph colorings. In the present paper, we take the opportunity to develop this connection in the setting of affine exchangeability. To explain this, let us say that a 2-homogeneous compact nilspace X represents affine exchangeability if it can replace H in Theorem 1.5. Given a compact metric space B, let us also say that a compact 2-homogeneous nilspace X is a limit domain for convergent sequences (f n : F n 2 → B) n∈N , if for any such sequence there exists a measurable function m : X → P(B) such that for any finite L ⊂ F ω 2 the measures µ L,fn converge vaguely to 8 ζ X,m • p −1 L . In Section 7, we prove the following result, which connects these concepts. Theorem 1.7. Let X be a compact profinite-step 9 2-homogeneous nilspace. The following statements are equivalent. (i ) X represents affine-exchangeability. (ii ) For every compact 2-homogeneous profinite-step nilspace Y there is a (continuous) fibration ϕ : X → Y. (iii ) X is a limit domain for convergent sequences (f n : F n 2 → B) n∈N , for every compact metric space B. Combining this with Theorem 1.5, we deduce immediately the following result. The group nilspace H, involved in Theorems 1.5 and 1.8, may at first seem quite complicated, and one may wonder whether a simpler object could still yield such representation results. However, statement (ii) in Theorem 1.7 constitutes a strong requirement for a nilspace to be able to represent affine exchangeability, putting constraints on how simple such a nilspace can be. In Section 8 we use this fact to rule out certain other natural candidates. In particular, we were interested in clarifying the relation between our results concerning limit domains and the results in [25]. The latter paper studies a convergence notion for sequences of Boolean functions (f n : F n p → {0, 1}) n∈N , for any prime p, and for p = 2 that notion is equivalent to the one studied in this paper (we detail this equivalence in Appendix E). However, in Subsection 8.1 we prove that the group G ∞ used to define the limit objects in [25] cannot be used as a limit domain in the sense of statement (iii) in Theorem 1.7. The paper has the following outline. In Section 2 we give some background on nilspaces and set up some measure-theoretic tools needed in the sequel, especially the machinery involving the infinite-dimensional cube set C ω (X) on a compact nilspace X, and the associated construction and basic properties of the measures ζ X,m . We take the opportunity, in Subsection 2.4, to give further basic examples illustrating affine-exchangeability, which also help to motivate the subsequent material. In Section 3, we begin proving Theorem 1.5, by first refining the representation theorem [14,Theorem 1.3] in light of more recent results on higher-order Fourier analysis in characteristic p from [13]. This relies especially on the results concerning p-homogeneous nilspaces. More precisely, we prove in Section 3 that the additional strength of affine exchangeability (compared to cubic 8 p L : B N → B L is the projection to the coordinates indexed by L ⊂ F ω 2 (identifying F ω 2 with N ). 9 Meaning that X is the inverse limit of compact nilspaces of finite step; see Subsection 2.2. exchangeability) implies that the nilspaces obtained by applying [14,Theorem 6.7] to an affine-exchangeable measure must be 2-homogeneous nilspaces. This enables us to apply the structure theorem for 2-homogeneous nilspaces obtained in [13], which tells us that any finite 2-homogeneous nilspace is the image, under some nilspace fibration, of a certain finite filtered abelian 2-group. The next part of the argument, carried out in Section 4, involves setting up adequate inverse systems of such filtered 2-groups, in order to prove that for any compact 2-homogeneous nilspace X there is a fibration H → X. These ingredients are then combined in Section 5 to complete the proof of Theorem 1.5. Section 6 is devoted to proving Theorem 1.6. In Section 7 we prove a version of Theorem 1.7 focusing on the relation between limit domains and affine-exchangeability; see Theorem 7.7. This is then used in Section 8 to complete the proof of Theorem 1.7, and thus obtain also Theorem 1.8. In Subsection 8.1 we study the aforementioned relation with [25] concerning alternative limit domains. Finally, let us mention the possibility of extending the methods in this paper to any prime p > 2. Combined with corresponding extensions of prior work (notably the results from [14, §6]) this would yield representation theorems for analogues of affine exchangeability for p > 2. We do not pursue this extension in this paper; see Remark 7.9. Another natural direction would be to restrict the true complexity of the linear forms involved in the definition of convergence of sequences (f n : F n 2 → B) n∈N , i.e., by requiring vague convergence of µ L,fn only for systems of linear forms L of true complexity at most some prescribed finite bound. It can be seen that H is a valid limit domain also to describe the limits of such sequences. Moreover, with more work it is possible to use a simpler version of H in that case. We outline how this can be done in Remark 7.8. Acknowledgements. We thank Hamed Hatami, Pooya Hatami, and Bryna Kra for valuable comments on this paper. All authors received funding from Spain's MICINN project PID2020-113350GB-I00. The second-named author received funding from project Momentum (Lendület) 30003 of the Hungarian Government. The research was also supported partially by the NKFIH "Élvonal" KKP 133921 grant and partially by the Hungarian Ministry of Innovation and Technology NRDI Office within the framework of the Artificial Intelligence National Laboratory Program. Nilspace and measure-theoretic preliminaries In this section we provide a brief introduction to nilspaces and related measure-theoretic aspects needed for the sequel. In particular we introduce the space of infinite-dimensional cubes C ω (X) on a compact nilspace X, and define what we shall call the Haar measure on this space; this underpins the construction of the measures ζ X,m . Finally, we give more background and examples on cubic and affine exchangeability. Brief introduction to nilspaces. For any integer n ≥ 1 we denote by [n] the set {1, 2, . . . , n}. We define n := {0, 1} n , and define 0 := {0}. We denote by 0 n (resp. 1 n ) the element of n with all entries equal to 0 (resp. 1). Following [11, Definition 1.0.1], by compact space we shall mean by default a compact, second-countable, Hausdorff topological space. A map φ : n → m is a discrete-cube morphism [9],[10, Definition 1. 1.1] if for every j ∈ [m] the j-th coordinate of φ(v) is either a constant function of v (equal to 0 or 1) or is equal to v(i j ) or 1 − v(i j ) for some i j ∈ [n] . For n ≤ m we say that a discrete-cube morphism φ : n → m is a face map if φ is injective and exactly m − n coordinates of φ(v) remain constant as v varies through n . Definition 2.1 (Nilspaces [9]). A nilspace is a set X together with a collection of sets C n (X) ⊂ X n , n ∈ N, such that the following axioms are satisfied. (i) (Composition) For any discrete-cube morphism φ : n → m and any c ∈ C m (X), we have c • φ ∈ C n (X). (ii) (Ergodicity) C 1 (X) = X 1 . (iii) (Corner completion) For any n let c ′ : n \ {1 n } → X be such that for all face maps φ : n − 1 → n with 1 n / ∈ φ( n − 1 ) we have c ′ • φ ∈ C n−1 (X). Then there exists c ∈ C n (X) such that c(v) = c ′ (v) for all v ∈ n \ {1 n }. The elements of C n (X) are called the n-cubes on X. The maps c ′ : n \ {1 n } → X satisfying the assumption in the completion axiom are called the n-corners on X, and a cube c satisfying the conclusion of this axiom is a completion of c ′ . We denote the set of all n-corners on X by Cor n (X). We say that X is a k-step nilspace if each c ∈ Cor k+1 (X) has a unique completion. If X is endowed with a topology making it a compact space, and for every n ∈ N the cube set C n (X) is closed relative to the product topology on X n , then we say that X is a compact nilspace. Next we recall the definition of morphisms in the nilspace category, and the special type of morphisms called fibrations (or fiber-surjective morphisms), which are nilspace analogues of surjective homomorphisms between abelian groups. Definition 2.2 (Morphisms and fibrations). Let X, Y be nilspaces. A map ϕ : X → Y is a morphism if ϕ • c ∈ C n (Y) for every c ∈ C n (X). We denote the set of such morphisms by hom(X, Y). We say that ϕ ∈ hom(X, Y) is a fibration if for every c ′ ∈ Cor n (X), for every completion c ∈ C n (Y) of the corner ϕ • c ′ , there exists a completionc ∈ C n (X) of c ′ such that ϕ •c = c. If X and Y are compact nilspaces then morphisms in hom(X, Y) are required to be also continuous maps. The next concept is crucial for the analysis of the structure of nilspaces. Definition 2.3 (Characteristic nilspace factors) . Let X be a nilspace. For every k ≥ 0 and x, y ∈ X we write x ∼ k y if there exist c 1 , c 2 ∈ C k+1 (X) such that c 1 (v) = c 2 (v) for all v = 1 k+1 , c 1 (1 k+1 ) = x, and c 2 (1 k+1 ) = y. We define the k-th characteristic factor X k := X / ∼ k and denote by π k the canonical projection X → X / ∼ k . Then X k equipped with the cube sets C n (X k ) := {π k • c : c ∈ C n (X)} is a k-step nilspace. If X is a compact nilspace then X k is also a compact nilspace and the map π k is continuous and open. Definition 2.4 (Abelian bundle). Let Z be an abelian group and S and B be sets. We say that B is a Z-bundle over S if there exists an action α : B × Z → B, (b, z) → b + z and a (projection) map π : B → S such that the following holds. (i) The action of Z is free, i.e., for any b ∈ B we have {z ∈ Z : b + z = b} = {0 Z }. (ii) The map s → π −1 (s) is a bijection between S and the set of orbits of Z in B. For any integer k ≥ 0, we say that B is a k-fold abelian bundle if there exists a sequence of sets B 0 , . . . , B k with B k = B, and abelian groups Z 1 , . . . , Z k , such that B 0 is a singleton and B i is a Z i -bundle over B i−1 for all i ∈ [k]. Denoting by π i−1,i the projection B i → B i−1 , for each i ∈ [k], we then denote by π i,j the projection π i,i+1 • · · · • π j−1,j : B j → B i for any i ≤ j in [k], and we define π i := π i,k . We call the bundles B i the factors of the bundle B. If Z is a compact abelian group and B, S and Z are compact spaces, we say that B Lemma 2.5 (Compact nilspaces as abelian bundles). Let X be a k-step compact nilspace. Then X along with the maps π i,j : X j → X i for i ≤ j is a k-fold compact abelian bundle. The compact abelian groups Z 1 , . . . , Z k such that X i is a Z i -bundle over X i−1 for each i ∈ [k] are called the structure groups of X. Note that each cube-set C n (X) can also be seen as a k-fold compact abelian bundle. To detail this we recall from [10, Definition 2.2.30] that for any abelian group Z and k ∈ N, the k-step nilspace D k (Z) is Z together with the cube sets C n (D k (Z)) = {c : n → Z \| for every face map φ : k + 1 → n , σ k+1 (c • φ) = 0}, where recall that for any f : k → C we define σ k (f ) = v∈ k (−1) v(1)+···+v(k) f (v). Lemma 2.6. Let X be a k-step compact nilspace with structure groups Z 1 , . . . , Z k and let n ∈ N. Then C n (X) is a k-fold compact abelian bundle, with factors C n (X i ), factor maps π n i,j : C n (X j ) → C n (X i ), and structure groups C n (D 1 (Z 1 )), . . . , C n (D k (Z k )). Every iterated compact abelian bundle B can be endowed with a Borel probability measure, called the Haar measure on B, that generalizes the Haar measure on compact abelian groups. In particular, given a k-step compact nilspace X, the Haar measure on C n (X i ) can be defined as follows (for more details we refer to [11, §2.2.2]). Proposition 2.7. Let X be a compact k-step nilspace. Then for any n ≥ 0 and i ∈ [k] there exists a unique Borel probability measure µ C n (X i ) on C n (X i ) with the following properties: we have µ C n (X i ) = µ C n (X) •(π n i ) −1 and µ C n (X i ) is invariant under the action of C n (D i (Z i (X))). For the proof we refer to [11,Proposition 2.2.5] and [14,Proposition 3.6]. Remark 2.8. We can define a measure µ on X n concentrated on C n (X) by setting µ(A) := µ C n (X) (A ∩ C n (X)) for any Borel set A ⊂ X n . We shall sometimes abuse the notation by denoting µ also by µ C n (X) (i.e. viewing µ C n (X) as a measure on X n ). Finally, let us recall the definition of a specific class of nilspaces which will be used extensively in this paper. Here and throughout the sequel, the group Z n p will be identified with [0, p − 1] n equipped with addition mod p, the usual way. Definition 2.9 (p-homogeneous nilspaces [13]). Let X be a nilspace and let p ∈ N be a prime. We say that X is a p-homogeneous nilspace if for every positive integer n, for every f ∈ hom(D 1 (Z n ), X) the restriction f \| [0,p−1] n is in hom(D 1 (Z n p ), X). We recall more background on these nilspaces in Section 4. For now let us mention that the most basic examples of p-homogeneous nilspaces are given by the elementary abelian p-groups Z n p , and that more examples can be constructed easily (see [13]). These include the nilspace H introduced in Definition 1.4, which is 2-homogeneous. This nilspace is also an example of a compact nilspace that is not of finite step, but rather of what we shall call profinite-step. This is a useful type of infinite-step nilspaces, to which we now turn. Profinite-step nilspaces. In the sequel we often have to deal with compact nilspaces that are not necessarily of finite step. However, we shall always be able to assume that these nilspaces have the following useful property. Definition 2.10. We say that a compact nilspace X is a profinite-step nilspace if it is the inverse limit of compact nilspaces of finite step. Every finite-step compact nilspace is trivially profinite-step, but not all compact nilspaces are profinite-step. For instance, given a filtered group (G, G • ), the associated group nilspace is profinite-step if and only if the filtration G • has the following property. Definition 2.11. We say that a filtration G • = (G (i) ) i≥0 on a group G is non-degenerate if for any pair of distinct elements g, g ′ ∈ G there exists i ∈ N such that g = g ′ mod G (i) . Equivalently, the filtration G • is non-degenerate if ∞ i=0 G (i) = {id G }. Every profinite-step nilspace can be endowed with a unique Borel probability measure, which we shall call its Haar measure, such that for each k ∈ N the pushforward of this measure to the k-th characteristic factor equals the Haar measure on this factor. In order to establish this rigorously, we need to extend the definition of k-fold compact abelian bundles, to define ∞-fold compact abelian bundles. Let us state the main result here and defer the technical (but relatively routine) proofs to Appendix A. Proposition 2.12 (n-cubic Haar measures on compact profinite-step nilspaces). Let X be a compact profinite-step nilspace. Then for every n ≥ 0 there exists a unique measure µ C n (X) on C n (X) such that for every i ∈ N we have µ C n (X) •(π n i ) −1 = µ C n (X i ) . We say that µ C n (X) is the n-cubic Haar measure on C n (X) for every n ≥ 0. Proof. We combine Lemma A.4 with Lemma A.10, applying the latter with P = n and S = ∅. Finally, we need to establish that a fibration between profinite-step compact nilspaces always preserves the Haar measures. Lemma 2.13. Let X, Y be compact profinite-step nilspaces, and let ϕ : X → Y be a fibration. Then, for every n ≥ 0, the map ϕ n : C n (X) → C n (Y), c → ϕ • c preserves the n-cubic Haar measures, i.e. µ C n (X) •(ϕ n ) −1 = µ C n (Y) . Proof. A fibration between nilspaces induces a totally-surjective bundle morphism between the corresponding sets of cubes [22], [10,Lemma 3.3.12], and it therefore preserves the Haar measures as claimed, by Lemma A.6. 2.3. The infinite-dimensional cube set C ω (X) and its Haar measure. As mentioned in the introduction, an important object in this paper is the set of infinitedimensional cubes on a nilspace. Let us define this structure formally. Definition 2.14. Let X be a nilspace. For each n ∈ N let φ n : n → N , v → (v, 0 N\[n] ) = (v(1), . . . , v(n), 0, . . .), and let p n : X N → X n be the projection induced by φ n , i.e. the map (x(v)) v∈ N → (x(φ n (v))) v∈ n . We then define C ω (X) := ∞ n=1 p −1 n (C n (X)).(6) If X is a compact nilspace, then C ω (X) is a compact subset of X N in the product topology. (Note that C ω (X) is non-empty, containing in particular all constant maps N → X.) Recall that a map φ : n → N is a (discrete cube) morphism if it extends to an affine homomorphism Z n → i∈N Z (see [14,Definition 6.1]). Another natural way to think of C ω (X) is that its elements are precisely the functions c : N → X such that for every morphism φ : n → N (for every n) we have c • φ ∈ C n (X). Example 2.15. Let Z be an abelian group and let k ∈ N. Then C ω (D k (Z)) = c : v → k i=0 S={s 1 ,...,s i }∈( N i ) z i,S v(s 1 ) · · · v(s i ) z i,S ∈ Z , where note that the sum over S here is always well-defined since v ∈ N has only finitely many non-zero coordinates. The proof of this equality follows from applying [10, Lemma 2.2.5] to c • φ n for each n ∈ N, and an induction using that the sets D n := n × {0 N\[n] } ⊂ N form an increasing sequence with ∞ n=1 D n = N . We omit the details. Definition 2. 16. Let X be a nilspace. An ω-corner on X (rooted at some vertex v ∈ N ) is a map c ′ : N \ {v} → X such that for every face map 10 φ : n → N with φ( n ) ∋ v, we have c ′ • φ ∈ C n (X). We denote the set of such ω-corners rooted at v by Cor ω v (X). A useful fact about C ω (X) is that if X is profinite-step then this infinite-dimensional cube set satisfies the following form of unique corner-completion. Lemma 2.17. Let X be a compact profinite-step nilspace. Then every ω-corner on X has a unique completion in C ω (X). Proof. Let c ′ : N \ {v} → X be an ω-corner. We need to prove the existence and uniqueness of some c ∈ C ω (X) such that c(w) = c ′ (w) for every w = v. Recall that X = lim ← − X k where X k are the characteristic nilspace factors of X. For every k and any (k + 1)-corner q on X k rooted at 0 k+1 , let q ∈ C k+1 (X k ) be its unique MEASURES 15 completion. Note that π k • c ′ \| k+1 is a (k +1)-corner on X k rooted at 0 k+1 . For every k let ON F ω 2 -AFFINE-EXCHANGEABLE PROBABILITYC k := {q ′ ∈ C k+1 (X) : π k • q ′ = π k • c ′ \| k+1 }. We claim that this is a non-empty compact set. The fact that C k = ∅ follows from the fact that π k : X → X k is a fibration. To see that C k is compact, note that C k = C k+1 (X) ∩ [(π k+1 k ) −1 ({π k • c ′ \| k+1 })] . It follows that the set D k := {c ′′ (0 k+1 ) ∈ X : c ′′ ∈ C k } is a non-empty compact set for every k. The sets D k form a decreasing sequence relative to inclusion, so by compactness of X (the finite intersection property) we have ∞ k=1 D k = ∅. Letting x be any point in ∞ k=1 D k , we see by construction that x completes the corner c ′ . To prove uniqueness, let x, y ∈ X be two points completing c ′ , and denote the corresponding completions in C ω (X) by c x and c y . For every k ≥ 1 we have that π k • c x \| k+1 and π k • c y \| k+1 are cubes in C k+1 (X k ) with equal values at every vertex w ∈ k + 1 \ {0 k+1 }. By uniqueness of corner-completion in X k , we have π k (x) = π k (y). As this holds for every k ≥ 1 and X is profinite-step, it follows that x = y. Lemma 2.18. Let X be a compact profinite-step nilspace and for each k ∈ N let Z k be the k-th structure group of X. Then C ω (X) is an ∞-fold compact abelian bundle with structure groups C ω (D k (Z k )) and projections π N k,i : C ω (X i ) → C ω (X k ) for 0 ≤ k ≤ i. Proof. The first thing that we need to check is that C ω (X k ) is a C ω (D k (Z k ))-compact abelian bundle over C ω (X k−1 ) for every k ∈ N. Thus we need to prove Properties 1 and 2 of [10, Definition 3.2.17]. To prove Property 1, take any c ∈ C ω (X k ) and f ∈ C ω (D k (Z k )). If c +f = c then coordinate-wise we have c (v)+f (v) = c (v) for any v ∈ N . But the action of Z k on X k is free by [10, Theorem 3.2.19] so f = 0. For Property 2, let c ∈ C ω (X k ). We need to show that if c ′ ∈ C ω (X k ) is any other element such that π N k−1,k • c = π N k−1,k • c ′ then there exists f ∈ C ω (D k (Z k )) such that c = c ′ +f . Recall from Definition 2.14 the map φ n : n → N . Note that for any fixed n ∈ N we have π k−1,k • c • φ n = π k−1,k • c ′ • φ n and thus, by [ := c • φ n − c ′ • φ n ∈ C n (D k (Z k )). By definition the maps (f n ) n≥1 are consistent (identifying 1 ⊂ 2 ⊂ · · · ⊂ N according to φ 1 , φ 2 , . . .) and thus if we define f : N → Z k as v → f n (v) where v ∈ n × {0 N\[n] } (for n large enough) we have f ∈ C ω (D k (Z k )) and c = c ′ +f . Next we need to check the conditions of [11, Definition 2. 1.6]. Recall that on C ω (X k ) ⊂ X N k we are using the product topology, and by [11, Proposition 2.1.9], the analogous conditions hold for C n (X k ) instead of C ω (X k ). Then conditions (i) and (ii) of [11, Definition 2.1.6] follow directly from the definitions of C ω (X k ) and C ω (D k (Z k )). The action of C ω (D k (Z k )) on C ω (X k ) is continuous as it is coordinate-wise continuous and we are using the product topology on C ω (X k ) ⊂ X N k . Hence condition (iii) follows. In order to prove (iv) it suffices to check that π N k−1,k is open and continuous. The continuity follows from the continuity of π k−1,k and the definition of the product topology on C ω (X k ) ⊂ X N k . To prove that it is an open map, note that it suffices to prove that images under π N k−1,k of subsets of the form ( v∈ n U v ) × X N \ n k are open (where U v ⊂ X k are open) , as these product sets form a base of the product topology. But for these product sets the openness is already known from the theory for finite-dimensional cubes (see [11,Lemma 2.1.10]). Finally, we need to show that C ω (X) = lim ← − C ω (X k ) (as per Definitions A.1 and A.2). Since X is profinite-step we have X = lim ← − X k with limit maps (fibrations) π k : X → X k . For any c ∈ C ω (X), for each integer k ≥ 0 let c k := π k • c : N → X k . We need to show that c k ∈ C ω (X k ) and that π k−1,k • c k = c k−1 , for each k ∈ N. The latter equality follows from the fact that X is the inverse limit of the X k (which gives the equality coordinate-wise). To see the former claim, note that for each n ∈ N we have (π N k (c)) • φ n = π n k (c • φ n ), and this last map is in C n (X k ) since c • φ n ∈ C n (X) and π k is a morphism, so (π N k (c)) • φ n is an n-cube on X k for every n, which means that π N k (c) ∈ C ω (X k ) as required. This proves the inclusion of C ω (X) in lim ← − C ω (X k ), and the opposite inclusion is similar. Corollary 2.19. Let X be a profinite-step compact nilspace. Then there exists a unique Borel probability measure on C ω (X), which we call the Haar measure and denote by µ C ω (X) , determined by the property that for each integer k ≥ 0 we have µ C ω (X) •(π N k ) −1 = µ C ω (X k ) . Proof. This follows by combining Lemma 2.18 with Lemma A.4. As in Remark 2.8, let us mention similarly here that we shall sometimes abuse the notation and view µ C ω (X) as a measure on X N (concentrated on C ω (X)). Remark 2.20. Note that if X is a compact profinite-step nilspace, the set Cor ω v (X) of ωcorners rooted at v (for any fixed v ∈ N ) is also an ∞-fold compact abelian bundle and thus it has a Haar measure. Furthermore, by Lemma 2.17 the map ψ : C ω (X) → Cor ω v (X), c → c \| N \{v} is a bijective bicontinuous bundle-morphism, so ψ and ψ −1 both preserve the Haar measures. Proposition 2.21. Let X and Y be compact profinite-step nilspaces. Let ϕ : Y → X be a fibration. Then ϕ N : C ω (Y) → C ω (X) preserves the Haar measures. Proof. It suffices to check that ϕ N is a totally-surjective bundle morphism between the corresponding ∞-fold compact abelian bundles. As in the proof of Lemma 2.18, the arguments are routine analogues of existing ones in the literature and we shall not detail them beyond the following outline. First one needs to check the conditions (i) and (ii) of Definition A.5. Condition (i) follows from the fact that given c, c ′ ∈ C ω (X) such that π N k (c) = π N k (c ′ ) we have that for each fixed v ∈ N , π k (c (v)) = (π N k (c))(v) = (π N k (c ′ ))(v) = π k (c ′ (v) ). Then as ϕ is a bundle morphism by [10, Proposition 3.3.2] we have π k (ϕ(c (v))) = π k (ϕ(c ′ (v))). Hence π N k (ϕ N (c)) = π N k (ϕ N (c ′ )). Condition (ii) follows similarly. Continuity follows from that of ϕ. To prove the surjectivity between the i-factors, let f ∈ C ω (D i (Z i )) where Z i is the i-th structure group of Y and c ∈ C ω (Y). It can be checked that ϕ N (c +f ) = ϕ N (c) + α N i (f ) where α i : Z i → Z ′ i is the i-structure homomorphism of ϕ and Z ′ i the i-th structure group of X. It is readily seen using Example 2.15 that the map α N i : C ω (D i (Z i )) → C ω (D i (Z ′ i )) is surjective. Let us record one last feature of the cube set C ω (X) in this subsection. If X is the group nilspace associated with a compact filtered abelian group (Z, Z • = (Z (k) ) k≥0 ), then C ω (X) is a compact abelian subgroup of Z N . Indeed this follows from the definition (6) and the standard fact that the Host-Kra cube-set C n (X) is a closed subgroup of Z n . We then have the following fact. Lemma 2.22. Let X be a compact profinite-step nilspace, and suppose that X is the group nilspace associated with a filtered compact abelian group (Z, Z • ) (thus the filtration Z • is non-degenerate). Then the Haar measure on C ω (X) defined in Corollary 2.19 equals the Haar measure of C ω (X) as a compact abelian group. Proof. By definition and uniqueness of the (group) Haar measure on C ω (X), it suffices to prove that the measure µ C ω (X) given by Corollary 2.19 is invariant under the addition of any c ′ ∈ C ω (X). By Lemma A.6 it is enough to prove that for any c ′ ∈ C ω (X), the map ϕ c ′ : C ω (X) → C ω (X), c → c + c ′ is a totally surjective continuous bundle morphism. The continuity follows from continuity of addition in Z. Recall that in the case of group nilspaces, each projection π k : X → X k equals the quotient map π k : Z → Z / Z (k + 1) , and X k is the group nilspace associated with (Z / Z (k + 1) , (Z (i) / Z (k + 1) ) ∞ i=0 ) (this follows from the definition of the characteristic factors and the definition of the Host-Kra cubes). Clearly the map ϕ c ′ is a bundle morphism, since π k is also a homomorphism of abelian groups. Hence we have to check that the structure homomorphism of (ϕ c ′ ) k : C ω (X k ) → C ω (X k ), c mod Z (k + 1) → c + c ′ mod Z (k + 1) is surjective for every k ≥ 1. The k-th structure group of X can be proved to be Z (k) / Z (k + 1) ([10, Corollary 3.2.16]). Given f ∈ C ω (D k (Z (k) / Z (k + 1) )) we have that (ϕ c ′ ) k (c +f mod Z (k + 1) ) = c + c ′ +f mod Z (k + 1) = (ϕ c ′ ) k (c mod Z (k + 1) ) + f mod Z (k + 1) . Thus the corresponding structure homomorphism is just the identity, which is surjective as required. Remark 2.23. In the sequel we will frequently work with group nilspaces X associated with filtered compact abelian groups (Z, Z • ). In these cases, when there is no risk of confusion, we will say that Z is a nilspace (when the filtration is understood from the context), and its set of n-cubes for n ∈ N ∪ {ω} will be denoted by C n (Z) (or by C n (Z • ) if the filtration needs to be emphasized). By Lemma 2.22, we can use µ C ω (Z) as the (unique) Haar measure, from either of the nilspace or group viewpoints, without ambiguity. Note also that using an argument very similar to the previous one, the (group) Haar measure of C n (X) equals the (nilspace) Haar measure of C n (X) for any n ∈ N. More background on cubic and affine-exchangeability. In this subsection we gather results providing more detailed background on affine and cubic exchangeability, elaborating on remarks and examples from the introduction. We begin by discussing in more detail the inclusions in (5). Lemma 2.24. For every standard Borel space B, we have Pr Aff(F ω 2 ) (B N ) ⊂ {cubic-exchangeable measures on B N } ⊂ Pr Aut( N ) (B N ). (7) Proof. To see the first inclusion, note that given any pair of injective morphisms φ 1 , φ 2 : k → N , their images define two subspaces of dimension k (identifying N with F ω 2 ). Thus, there exists an affine map α ∈ Aff(F ω 2 ) ∼ = GL(F ω 2 ) ⋉ Z ω 2 such that φ 2 = α • φ 1 . For i = 1, 2 let p φ i : B N → B k be the projection (b v ) v∈ N → (b φ i (v) ) v∈ k , and note that p φ 2 = p φ 1 • α ′ −1 where α ′ −1 : B N → B N , (b v ) v∈ N → (b α −1 (v) ) v∈ N . Now, given any µ ∈ Pr Aff(F ω 2 ) (B N ), we have µ • p −1 φ 2 = µ • α ′ • p −1 φ 1 = µ • p −1 φ 1 , where the last equality here uses that µ is affine-exchangeable. Hence µ is cubic-exchangeable, which proves the desired inclusion. To see that this first inclusion can be strict, it suffices to produce a cubic-exchangeable measure that is not affine-exchangeable. For example, if X is the group nilspace associated with a compact filtered abelian group (Z, Z • ), and we set B = X, then the measure µ C ω (X) (seen as a measure on X N ) is cubic-exchangeable, but by Corollary B.2, in order for this measure to be affine-exchangeable, the nilspace X must be 2-homogeneous. A simple example of a compact nilspace that is not 2-homogeneous is the nilspace D 1 (Z 3 ). (7) is proved in [14,Remark 6.6]. This inclusion is strict for instance whenever B is a compact metric space containing at least two elements. To see this, one can use the fact that, on one hand, the convex set of cubic-exchangeable measures on B N is a Bauer simplex relative to the vague topology, i.e. its extreme points form a vaguely closed set (this follows from results in [14] and is recalled in detail in Section 6 below), and on the other hand, it is proved in [5,Theorem 4 The second inclusion in .2] that Pr Aut( N ) (B N ) is a Poulsen simplex, so its set of extreme points need not be closed. Given a nilspace X, the next result gives a criterion involving C ω (X) to decide whether X is 2-homogeneous. Let φ : N → F ω 2 be the bijective map v → v mod 2 (i.e. φ is just the natural identification of N as a set with F ω 2 ). With this, we can view any function c : N → X in a simple way as a function on F ω 2 (just by considering c • φ −1 ), and vice versa. We shall sometimes switch between these two views of such a function c, in a slight abuse of notation. We then have the following result, which ensures in particular that the aforementioned abuse of notation will not be problematic when X is 2-homogeneous. ON F ω 2 -AFFINE-EXCHANGEABLE PROBABILITY MEASURES 19 Lemma 2.25. Let X be a nilspace. Then C ω (X) = hom(D 1 (F ω 2 ), X) if and only if X is 2-homogeneous. Proof. Suppose that X is 2-homogeneous and take any c ∈ C ω (X). We have to prove (via identifying N with F ω 2 ) that c is an element of hom(D 1 (F ω 2 ), X). Let f ∈ C k (D 1 (F ω 2 )). Note that for n = n(f ) ∈ N large enough we have f ( k ) ⊂ F n 2 × {0 N\[n] }. In par- ticular c • f = c \| n ×0 N\[n] • f , where (abusing the notation) we can then consider that f ∈ C k (D 1 (F n 2 )), and that c \| n ×0 N\[n] ∈ C n (X) (by definition of C ω (X)). Then, since X is 2-homogeneous, we have c \| n ×0 N\[n] ∈ hom(D 1 (F n 2 ), X), whence c • f ∈ C k (X) . This proves that c ∈ hom(D 1 (F ω 2 ), X) as required. For the opposite inclusion, let c ′ ∈ hom(D 1 (F ω 2 ), X), and note that any morphism g : k → N can be viewed (via the above identification) as an element of C k (D 1 (F ω 2 )) (for instance, by an explicit description that follows from the definition of g; see [14, Remark 6.2]). Thus, by the morphism property of c ′ , we deduce that c ′ • g ∈ C k (X), so c ′ ∈ C ω (X) as required. The converse is clear, since for any n the cube set C n (X) is a projection of C ω (X) and then the assumption C ω (X) = hom(D 1 (F ω 2 ), X) implies (using the above identification) that C n (X) = hom(D 1 (F n 2 ), X), so X is 2-homogeneous. Finally, let us elaborate on what is perhaps the simplest construction of cubic-exchangeable measures: we take any compact nilspace X, let µ C ω (X) be the Haar measure on C ω (X) constructed in Corollary 2.19, and let µ be the Borel measure on X N defined by µ(A) = µ C ω (X) A ∩ C ω (X) , for any Borel set A ⊂ X N .(8) Note that this is a particularly simple instance of the construction in (4), indeed the measure µ in (8) is the special case of (4) where B = X and m is the Borel function X → P(X) sending each x ∈ X to the Dirac measure δ x . The fact that this measure µ is cubic-exchangeable follows readily from the fact that, from construction of µ C ω (X) , we have that the pushforward of µ on any n-dimensional sub-cube of N (i.e. any image of an injective morphisms n → N ) is always µ C n (X) , for any n. It is then natural to wonder under what conditions this measure µ is not just cubic-exchangeable, but also affine-exchangeable. The following last result of this section gives an answer which further motivates the use of 2-homogeneous nilspaces. Lemma 2.26. Let X be a compact profinite-step nilspace. Then µ C ω (X) (viewed as a measure on X N ) is affine-exchangeable if and only if X is 2-homogeneous. We defer the proof to Appendix B. There is a family of cubic-exchangeable measures which is more general than the construction in (8) but still more specific than that in (4), and which will be very useful in the sequel. This family includes all the measures of the form ζ X,m with the property that the Borel map m : X → P(B) takes values only among Dirac measures. We postpone the discussion on this family to Section 8. 2-homogeneous cubic couplings We recall from [14, Definition 2.18] that given a probability space Ω = (Ω, A, λ) and a set S, a coupling (or self-coupling) of Ω indexed by S is a measure µ on the product measurable space (Ω S , A ⊗S ) such that for each v ∈ S we have µ • p −1 v = λ (where p v is the projection to the v-th coordinate) . Given an injection τ : R → S, we then denote by µ τ the self-coupling of Ω indexed by R obtained as follows: we take the pushforward µ • p −1 τ (R) , and we view it as a measure on Ω R by identifying R and τ (R) (see [14,Definition 2.26]). In this section we shall apply results from [14] concerning cubic couplings. These are measure-theoretic structures defined as follows (see [14,Defintion 3.1]). Definition 3.1. A cubic coupling on a probability space Ω = (Ω, A, λ) is a sequence µ n n≥0 , where the n-th term µ n is a self-coupling of Ω indexed by n , and such that the following axioms hold for all m, n ≥ 0: 1. (Consistency) If φ : m → n is an injective cube morphism, then µ n φ = µ m . 2. (Ergodicity) The measure µ 1 is the independent coupling λ × λ. 3. (Conditional independence) We have 11 ({0} × n − 1 ) ⊥ ( n − 1 × {0}) in µ n . For the purposes of this paper we define the following class of cubic couplings. Definition 3.2 (2-homogeneous cubic coupling). Let (µ n ) n≥0 be a cubic coupling on a probability space (Ω, A, λ). We say that (µ n ) n≥0 is a 2-homogeneous cubic coupling if for every injective affine map T : F m 2 → F n 2 , we have µ n T = µ m . Recalling the concept of p-homogeneous nilspace (Definition 2.9), we can now state the main result of this section, which tells us, roughly speaking, that every 2-homogeneous cubic coupling is essentially the sequence of cubic Haar measures on some 2-homogeneous compact nilspace. Theorem 3.3. Let (µ n ) n≥0 be a 2-homogeneous cubic coupling on a probability space (Ω, A, λ). Then there exists a 2-homogeneous compact nilspace X and a measure-preserving map γ : Ω → X such that for each n ∈ N, the map γ n is measure-preserving (Ω n , µ n ) → (X n , µ C n (X) ). Furthermore, for each n the coupling µ n is relatively independent over the factor generated by γ n . Proof. Recall from [14, Definition 3.31 and Definition 3.40] the definitions of γ k : Ω → X k and γ : Ω → X respectively. Theorem 3.3 follows mainly from [14,Theorem 4.2]. Indeed the latter result tells us that there is a compact profinite-step nilspace X and map γ satisfying the conclusions of Theorem 3.3, and then it only remains to prove that X is 2-homogeneous. In order to do this, by Lemma 2.25 we just need to prove that C ω (X) = hom(D 1 (F ω 2 ), X). To prove this it suffices to show that for every integer n ≥ 0 we have C n (X) = hom(D 1 (F n 2 ), X). Moreover, since X is of profinite-step, the latter is equivalent to proving that for any fixed k ≥ 0 and n ≥ 1 we have C n (X k ) = hom(D 1 (F n 2 ), X k ). Thus (by definition of hom(D 1 (F n 2 ), X k )) we need to check that given c ∈ C n (X k ) and T ∈ C m (D 1 (F n 2 )) we have c • T ∈ C m (X k ) (identifying n with F n 2 the usual way). But as D 1 (F n 2 ) is 2-homogeneous, any T ∈ C m (D 1 (F n 2 ) ) is just an affine-linear map T : F m 2 → F n 2 , so it suffices to check the above composition property for such T . Furthermore, we claim that it suffices to verify the above composition property only for injective affine maps. Indeed, suppose this has been verified and let T : F m 2 → F n 2 be any affine map. Composing with a translation if needed (which is an injective affine transformation) we can assume that T is a linear map. Let f be an invertible linear map on F m 2 such that ker(T • f ) = {0 ℓ } × F m−ℓ 2 for some ℓ ≥ 0. Thus ker(T ) = f ({0 ℓ } × F m−ℓ 2 ) and we define the subspace V := f (F ℓ 2 × {0 m−ℓ }). Note that then F m 2 = ker(T ) ⊕ V . Now let us define p : F m 2 → F ℓ 2 , p(v 1 , . . . , v m ) := (v 1 , . . . , v ℓ ) and i : F ℓ 2 → F m 2 , i(v 1 , . . . , v ℓ ) := (v 1 , . . . , v ℓ , 0 m−ℓ ). It can be checked then that T = T • f • i • p • f −1 . Note that φ := T • f • i is• T = c • φ • p • f −1 where φ and f −1 are affine injective maps and p is a discrete-cube morphism, so the property will follow in general as required. Thus it suffices to prove the result for any such affine injective map φ. Arguing as in the proof of [14,Lemma 4.7], let V := φ(F m 2 ) and ψ : Ω F m 2 → Ω V be the bijection that relabels each coordinate ω v as ω φ(v) . Let ξ : X F m 2 k → X V k be the corresponding bijection between the spaces X F m 2 k and X V k . By Definition 3.2, we have µ m = µ ℓ φ = µ ℓ V • ψ and ψ •(γ m k ) −1 = (γ V k ) −1 • ξ. Thus, the measure support Supp(µ m •(γ m k ) −1 ) is equal to ξ −1 (Supp(µ ℓ V •(γ V k ) −1 )). Now we have to check that for every c ∈ Supp(µ ℓ •(γ ℓ k ) −1 ) we have c • φ ∈ Supp(µ m •(γ m k ) −1 ). Given any open neighborhood U of p V (c) (where p V : X ℓ k → X V k is the usual projection), since µ ℓ V •(γ V k ) −1 is the image of µ ℓ •(γ ℓ k ) −1 under p V , and p V is continuous, we have µ ℓ V •(γ V k ) −1 (U) > 0. Hence p V (c) is in the support Supp(µ ℓ •(γ V k ) −1 ) . The result now follows by the definition of X k in terms of these measure supports, as in [14,Lemma 4.7]. p-homogeneous compact nilspaces as fibration images of H p Throughout this section we fix a prime p ∈ N. The main results in this section concern p-homogeneous nilspaces, which were introduced in [13] and recalled in Definition 2.9 above. Let us refer to [13] for more background and motivation on these nilspaces. Our goal in this section is to prove that there exists a compact p-homogeneous nilspace, that we shall denote by H p , which is "universal" among p-homogeneous nilspaces, in the sense that every other compact profinite-step p-homogeneous nilspace is an image of H p under a fibration. To define H p , we shall use the basic p-homogeneous nilspaces from [13, Definition 1.6], which are denoted by U k,ℓ (or U (p) k,ℓ when we need to specify the prime p). Let us recall their definition here for convenience. For positive integers k ≥ ℓ, let G be the cyclic group of order p ⌊ k−ℓ p−1 ⌋+1 , and let us equip this with the filtration G • = (G (i) ) ∞ i=0 with i-th term G (i) = G for i ∈ [0, ℓ], and G (i) := p ⌊ i−ℓ−1 p−1 ⌋+1 G = {p ⌊ i−ℓ−1 p−1 ⌋+1 x : x ∈ G} for i > ℓ. The nilspace U k,ℓ is then the group nilspace associated with the filtered group (G, G • ). We denote by Q p,k the set of all finite products of nilspaces of the form U (p) k,ℓ for ℓ ∈ [k]. Note that the (k + 1)-step nilspace U k+1,ℓ is an extension (in the nilspace sense; see [10, §3.3.3]) of the k-step nilspace U k,ℓ . Indeed, the corresponding nilspace factor map U k+1,ℓ → U k,ℓ , which we shall denote by π (ℓ) k , is just quotienting by the last nilspace structure group of U k+1,ℓ ; this group is either the trivial group {0} or is Z p , depending on whether ⌊ k+1−ℓ p−1 ⌋ − ⌊ k−ℓ p−1 ⌋ is 0 or 1. This yields a natural way of extending any nilspace in Q p,k to a nilspace in Q p,k+1 , as follows. Definition 4.1. Let X ∈ Q p,k , thus for some integers a ℓ ≥ 0 we have that X is the product nilspace k ℓ=1 (U (p) k,ℓ ) a ℓ . Then we define X + := k ℓ=1 (U (p) k+1,ℓ ) a ℓ ∈ Q p,k+1 . The nilspace factor map π k : X + → X is easily described as a product of the above maps π (1) k , . . . , π (k) k , by applying the adequate such map π (ℓ) k to each component U (p) k+1,ℓ of the product X + . Now, for any fixed ℓ ≥ 1, we define the compact profinite-step nilspace U (p) ∞,ℓ as the inverse limit of the nilspaces U (p) k,ℓ : U (p) ∞,ℓ := lim ← − U (p) k,ℓ .(9) More precisely, the inverse system used here is {π [11, §2.7]). We can rephrase this construction, and thus define H p , in terms of the p-adic integers as follows. (ℓ) i • π (ℓ) i+1 • · · · • π (ℓ) j−1 \| i, j ∈ N, i ≤ j} (see Definition 4.2. For any prime p ∈ N, let Z p denote the group of p-adic integers. For ℓ ∈ N we denote by U (p) ∞,ℓ the compact group-nilspace associated with the filtered group ON F ω 2 -AFFINE-EXCHANGEABLE PROBABILITY MEASURES 23 G, G (ℓ) • where G = Z p and the filtration G (ℓ) • = (G (ℓ) (i) ) ∞ i=0 consists of G (ℓ) (i) = Z p for i ∈ [0, ℓ] and G (ℓ) (i) = p ⌊ i−ℓ−1 p−1 ⌋+1 Z p for i > ℓ. We define H p as the product nilspace ∞ ℓ=1 (U (p) ∞,ℓ ) N . Thus H p is the group nilspace associated with the compact abelian group ∞ ℓ=1 Z N p ∼ = (Z N p ) N , which we equip with the product filtration ∞ ℓ=1 (G (ℓ) • ) N . Note that this filtration is p-homogeneous, whence (using [13, Theorem 1.4]) we have that H p is a p-homogeneous compact nilspace. It is readily seen that H p is also profinite-step. Note that H p is a natural generalization for p > 2 of the nilspace H from Definition 1.4 (indeed the latter nilspace is just H 2 ). We can now state the main result of this section, establishing the above-mentioned universality of H p . We shall obtain this theorem by combining infinitely many applications of a previous structural result obtained in [13,Theorem 1.7]. This will require "gluing" countably many fibrations in a unified way. To do this, we need the following preparation. α ∈ Θ i (X k−1 ) there exists β ∈ Θ i (X) such that π k−1 • β = α • π k−1 . In other words, every translation α ∈ Θ i (X k−1 ) can be lifted through π k−1 to a translation β ∈ Θ i (X). Proof. First note that for i = k the result is trivial, since then Θ k (X k−1 ) = {id}. For i < k, let us use the nilspace T * constructed in [10, §3.3.4], whose structure encodes whether or not the translation α can be lifted as desired. By [10, Lemma 3.3.38] we know that T * is a degree-(k − i) extension of X k−1 ∈ Q p,k−1 . From the definition of T * and the fact that X is p-homogeneous, it also follows that T * is p-homogeneous. As i < k, by [ β ∈ Θ i (X) satisfying π k−1 • β = α • π k−1 as required. Next, we extend this last result to be able to lift translations through fibrations that are more general than π k−1 . Proposition 4.5. Let X ∈ Q p,k , let Y be a finite k-step p-homogeneous nilspace, and let ϕ : X → Y be a fibration. Then for any i ∈ [k] and any translation α ∈ Θ i (Y), there exists β ∈ Θ i (X) such that ϕ • β = α • ϕ. Proof. We argue by induction on k. Note that the case k = 0 is trivial. If i = k then, since Θ k (X) ∼ = Z k (X), Θ k (Y) ∼ = Z k (Y) (by [10, Lemma 3.2 .37]) and since ϕ is a fibration, we can easily lift any such translation using the surjective homomorphism Z k (X) → Z k (Y) induced as a structure morphism by ϕ (see [10, §3.3.2]). If i < k, then given α ∈ Θ i (Y), let α k−1 ∈ Θ i (Y k−1 ) be the translation such that π k−1,Y • α = α k−1 • π k−1,Y where π k−1,Y : Y → Y k−1 is the projection to the (k − 1)-step factor of Y. By induction on k there is β ′ k−1 ∈ Θ i (X k−1 ) such that ϕ k−1 • β ′ k−1 = α k−1 • ϕ k−1 , where ϕ k−1 : X k−1 → Y k−1 is the fibration induced by ϕ between the (k − 1)-step factors (see [10, §3.3.2]). By Proposition 4.4 there is β ′ ∈ Θ i (X) with π k−1,X • β ′ = β ′ k−1 • π k−1,X . Note that π k−1,Y • ϕ • β ′ = π k−1,Y • α • ϕ, so β ′ is a translation of the desired kind but only modulo π k−1,Y . However, this implies that the function f : X → Z k (Y), x → α(ϕ(x)) − ϕ(β ′ (x)) is a well-defined Z k (Y)-valued map, and is in fact a morphism into D k (Z k (Y)). Let φ k : Z k (X) → Z k (Y) be the above-mentioned structure morphism, i.e. the surjective homomorphism such that ϕ(x + z) = ϕ(x) + φ k (z) for all x ∈ X and z ∈ Z k (X). Since X and Y are both finite p-homogeneous nilspaces, this map φ k is a linear map between two finite-dimensional vector spaces over F p . Thus there exists a homomorphism s : Z k (Y) → Z k (X) such that φ k • s is the identity map on Z k (Y). We then define β : X → X by β(x) := β ′ (x) + s(f (x)). It is easily checked that β ∈ Θ i (X) and that ϕ • β = α • ϕ, as required. We now use Proposition 4.5 to show that any fibration between nilspaces in the special class Q p,k can be re-expressed essentially as a coordinate projection, which will be useful in order to carry out the above-mentioned gluing in an orderly way. Proposition 4.6. Let X, Y ∈ Q p,k and let ϕ : X → Y be a fibration. Then X is isomorphic to the product nilspace Y ×Q for some k-step, p-homogeneous finite nilspace Q, and there is a nilspace isomorphism Φ : Y × Q → X such that ϕ(Φ(y, q)) = x for any y ∈ Y, q ∈ Q. Proof. We have Y = k ℓ=1 (U (p) k,ℓ ) a ℓ , where a ℓ ≥ 0 for ℓ ∈ [k]. For each ℓ ∈ [k] and t ∈ [a ℓ ], let α ℓ,t ∈ Θ ℓ (Y) be the translation that acts by adding 1 inside the cyclic group which is the t-th component of the group (U (p) k,ℓ ) a ℓ . By Proposition 4.5 we know that there exists β ℓ,t ∈ Θ ℓ (X) such that ϕ • β ℓ,t = α ℓ,t • ϕ. Let Q := ϕ −1 (0) where 0 ∈ Y is the element with all coordinates equal to 0. It is straightforward to check that Q is a p-homogeneous, k-step, finite nilspace. Let us define MEASURES 25 where y = (y ℓ,t ) ℓ∈[k], t∈[a ℓ ] . Note that this map Φ is thus well-defined, since the order of β ℓ,t as an element of the translation group Θ(X) is the same as the order of y ℓ,t in its corresponding cyclic group, by [13,Proposition 3.11]. More precisely, since y ℓ,t ∈ Z p r for r = ⌊ k−ℓ p−1 ⌋ + 1, we have β p r ℓ,t = id (using that β ℓ,t ∈ Θ ℓ (X) and (Θ ℓ (X)) ℓ≥0 is a phomogeneous filtration by [13,Proposition 3.11]). Thus, it now suffices to prove that Φ is a nilspace isomorphism. The fact that it is a morphism follows from the definitions. To see that it is bijective, note that the following function is an inverse: Φ : Y × Q → X (y, q) → k ℓ=1 a ℓ t=1 β y ℓ,t ℓ,t (q) ON F ω 2 -AFFINE-EXCHANGEABLE PROBABILITYΦ −1 (x) := y, k ℓ=1 a ℓ t=1 β y ℓ,t ℓ,t −1 (y) , where y = ϕ(x). Note also that Φ −1 is a morphism, similarly as above (i.e. from the definitions). We can now explain how we glue various fibrations to obtain a fibration from H p . Proposition 4.7. Let X be a compact nilspace such that X = lim ← − X i where X i is an istep, finite p-homogeneous nilspace for each i. Then there is a strict inverse system of nilspaces (φ i,j : Y j → Y i ) i≤j and a fibration ϕ : lim ← − Y i → X, such that for every i we have Y i ∈ Q p,i , Y i+1 = Y + i × Q i+1 for some Q i+1 ∈ Q p,i+1 and φ i,i+1 = π i • p 1 , where p 1 is the projection to the first component Y + i and π i is the quotient map Y + i → Y i . Proof of Theorem 4.3 using Proposition 4.7. We combine [20,Theorem 5.71] and Proposition 4.7. The nilspace obtained is given as an inverse system lim ← − Y i where the factor maps φ i,i+1 : Y i+1 → Y i have the form of a projection to some coordinates and then taking the quotient map π i . This means that the inverse system has the form 1 ℓ=1 (U (p) 1,ℓ ) a 1,ℓ ← 1 ℓ=1 (U (p) 2,ℓ ) a 1,ℓ × 2 ℓ=1 (U (p) 2,ℓ ) a 2,ℓ ← 1 ℓ=1 (U (p) 3,ℓ ) a 1,ℓ × 2 ℓ=1 (U (p) 3,ℓ ) a 2,ℓ × 3 ℓ=1 (U (p) 3,ℓ ) a 3,ℓ ← · · · where each map φ i,i+1 deletes the coordinate in the last component i+1 ℓ=1 (U (p) i+1,ℓ ) a i+1,ℓ and applies the quotient map π i in the remaining coordinates. Thus, the nilspace obtained has the form ∞ ℓ=1 (U (p) ∞,ℓ ) a ℓ where a ℓ = ∞ k=ℓ a k,ℓ ∈ N ∪ {∞}. By taking a projection that consists in deleting some coordinates, we see that this nilspace is a fiber-surjective image of H p = ∞ ℓ=1 (U (p) ∞,ℓ ) N . Proof of Proposition 4.7. The first part of the proof is very similar to the proof of [13,Proposition 4.9]. We will construct the sequence Y i by induction on i, with the case i = 1 being trivial (as X 1 = D 1 (Z n p )). Suppose that we have already constructed all the factors from Y 1 to Y n−1 , thus we have Y j ∈ Q p,j for j ∈ [n − 1], fibrations ψ j : Y j → X j for j ∈ [n − 1] and maps φ j,ℓ : Y ℓ → Y j (projections to first components followed by quotient maps) for 1 ≤ j ≤ ℓ ≤ n − 1 such that φ j,j = id for all j ∈ [n − 1] and φ j,ℓ • φ ℓ,d = φ j,d for 1 ≤ j ≤ ℓ ≤ d ≤ n − 1. Also, letting γ j,ℓ : X ℓ → X j for 1 ≤ j ≤ ℓ be the fibrations defining the inverse limit of X, we have ψ j • φ j,ℓ = γ j,ℓ • ψ ℓ for all 1 ≤ j ≤ ℓ ≤ n − 1. We can represent the situation with the following diagram: X 1 · · · X n−1 X n Y 1 · · · Y n−1 . γ 1,2 γ n−2,n−1 γ n−1,n ψ 1 ψ n−1 φ 1,2 φ n−2,n−1 Now, the first part of the induction consists in finding a nilspace Y ′ n ∈ Q p,n such that the following diagram commutes: X n−1 X n Y n−1 Y ′ n , γ n−1,n ψ n−1 ρ ν(10) where ν and ρ are fibrations. To do so, we let D be the subdirect product of Y n−1 and X n , i.e., D := {(y, x) ∈ Y n−1 × X n : ψ n−1 (y) = γ n−1,n (x)}. Then, we apply [13, Theorem 1.7] to this nilspace (which is clearly n-step) to obtain Y ′ n . Note that if Y ′ n and ν had the desired properties then we could stop here and proceed to the next step of the induction. However, we cannot guarantee this in general, so let us describe how one can define an appropriate nilspace Y n with the desired properties. Let us start with the following diagram indicating how the definition of Y n will work: X n−1 X n Y n−1 Y ′ n T Y n π n−1 (Y ′ n ) Y n−1 ×Q Y + n−1 ×V + γ n−1,n ψ n−1 ρ ν π n−1 ν n−1 φ ξ p 1 p 2 β Now let us detail the construction. First, note that ν : Y ′ n → Y n−1 is a fibration from an n-step nilspace to an (n − 1)-step nilspace. Thus ν factors through the (n − 1)-step factor π n−1 (Y ′ n ), i.e. we have ν = ν n−1 • π n−1 . Now, as π n−1 (Y ′ n ) ∈ Q p,n−1 , we apply Proposition 4.6 to the fibration ν n−1 and we denote by φ : π n−1 (Y ′ n ) → Y n−1 ×Q the nilspace isomorphism such that ν n−1 • φ −1 (y, q) = y for every y ∈ Y n−1 and q ∈ Q (so ν n−1 • φ −1 is simply the projection to the first component Y n−1 ). Now we apply [13, Theorem 1.7] to Q (which is a p-homogeneous (n − 1)-step finite nilspace) to obtain V ∈ Q p,n−1 and a fibration η : V → Q. We then take the extensions V + and Y + n−1 , and let ξ : Y + n−1 ×V + → Y n−1 ×Q be the corresponding fibration (thus ξ applies the projection π n−1 : Y + n−1 → Y n−1 in the first component, and the fibration η • π n−1 : V + → Q in the second component). Note that Y + n−1 ×V + ∈ Q p,n . Next, we define the subdirect product T := {(y ′ , (y, v)) ∈ Y ′ n ×(Y + n−1 ×V + ) : φ −1 • ξ(y, v) = π n−1 (y ′ )}. By [13,Proposition A.17] we have that T is a degree-n extension of Y + n−1 ×V + by the last structure group Z n (Y ′ n ). By [13,Proposition 4.3] this extension splits. We define Y n = Y + n−1 ×V + × D n (Z n (Y ′ n )) and let β be the isomorphism Y n → T . Since T is a split extension, we can view p 2 • β as projection to the component (y, v) ∈ Y + n−1 ×V + in Y n . We now claim that we can take φ n−1,n : Y n → Y n−1 to be ν • p 1 • β and ψ n := ρ • p 1 • β. To see this, note first that the commutativity of the diagram in (10) implies a similar commutative diagram with φ n−1,n instead of ν and ψ n instead of ρ. Hence, it only remains to check that φ n−1,n (y, v, z) = π n−1 (y). To do so, note that the previous diagram commutes, and therefore φ n−1, n (y, v, z) = ν • p 1 • β(y, v, z) = ν n−1 • π n−1 • p 1 • β(y, v, z) = ν n−1 • φ −1 • ξ • p 2 • β(y, v, z) = ν n−1 • φ −1 • ξ(y, v) = π n−1 (y). Proof of the main result In this section we prove Theorem 1.5. To this end, a useful observation is that given a measure µ ∈ Pr Aff(F ω 2 ) (B N ), we can determine whether µ is of the form ζ X,m by checking whether µ satisfies an equivalent and often more convenient independence property. This observation was already made in the more general context of cubic exchangeability in [14], where this independence property was defined. Let us recall the definition here. First we recall the definition of a face of N from [14, §6]. This involves the notation S (for a countable set S), which denotes the set of sequences in {0, 1} S of finite support. Definition 5.1. A set F ⊂ N is a face of N if there is a set S ⊂ N and an element z ∈ N \ S such that for every element v ∈ F there is a unique v ′ ∈ S such that v(i) = v ′ (i) for i ∈ S and v(i) = z(i) for i ∈ N \ S. We call S the set of free coordinates of F , and we say that F is a finite face if S is finite. Note that finite faces of N can be defined equivalently as the images of face maps n → N , n ∈ N. Two faces of N are independent if they are disjoint and their sets of free coordinates are disjoint. Given a σ-algebra B on a set B, and countable sets S ⊂ T , we denote by B ⊗T S the sub-σ-algebra of B ⊗T consisting of those sets whose indicator functions are independent of coordinates indexed in T \ S; equivalently, letting p S be the coordinate projection B T → B S , we have B ⊗T S = p −1 S (B ⊗S ). Definition 5.2. Let (B, B) be a standard Borel space. We say that a measure µ ∈ P(B N ) has the independence property if for all finite independent faces F 1 , F 2 ⊂ N , the σ- algebras B ⊗ N F 1 , B ⊗ N F 2 are independent according 12 to µ. The next lemma follows from results in [14], but was not stated explicitly in that paper and will be used in the sequel, so we record it here. Following the notation from [33,Ch. 11], given a topology T on a set B, we denote by B(T ) the σ-algebra generated by T . Then, for any Polish topology T on B such that B = B(T ), the set I is a closed subset of P(B N ) in the vague topology induced by T . In particular, the set I is Borel measurable relative to the standard Borel structure on P(B N ). Proof. Let (µ n ) n∈N be a sequence in I converging to µ ∈ P(B N ) in the vague topology induced by T . Let F 1 , F 2 be any finite independent faces in N . We need to show that B ⊗ N F 1 , B ⊗ N F 2 are independent according to µ, i.e., that for every function f 1 ∈ L ∞ (B ⊗ N F 1 ) and f 2 ∈ L ∞ (B ⊗ N F 2 ) we have B N f 1 f 2 dµ = ( B N f 1 dµ)( B N f 2 dµf ′ i such that f i = f ′ i • p F i .B F i , B ⊗F i , µ • p −1 F i ), so there is a continuous functionf i on B F i (relative to the product topology T F i ) such that B N \|f i −f i • p F i \| dµ = B F i \|f ′ i −f i \| d(µ • p −1 F i ) ≤ ε 2 max( f 1 L ∞ , f 2 L ∞ , 1) . We thus have in particular \| B N f 1 f 2 dµ − B N (f 1 • p F 1 )(f 2 • p F 2 ) dµ\| ≤ ε. By definition of vague convergence and the supposed independence property of each µ n , we have B N (f 1 • p F 1 )(f 2 • p F 2 ) dµ = lim n→∞ B N (f 1 • p F 1 )(f 2 • p F 2 ) dµ n = lim n→∞ B N f 1 • p F 1 dµ n B N f 2 • p F 2 dµ n = B N f 1 • p F 1 dµ B N f 2 • p F 2 dµ . Since this last product differs from ( B N f 1 dµ)( B N f 2 dµ) by at most ε, we conclude that \| B N f 1 f 2 dµ − ( B N f 1 dµ)( B N f 2 dµ)\| ≤ 2ε. Letting ε → 0, the result follows. We can now obtain the main result of this section. 12 As usual, given a probability space (Ω, A, µ), two sub-σ-algebras B 1 , B 2 of A are independent according to µ if for every A 1 ∈ B 1 , A 2 ∈ B 2 we have µ(A 1 ∩ A 2 ) = µ(A 1 )µ(A 2 ). ON F ω 2 -AFFINE-EXCHANGEABLE PROBABILITY MEASURES 29 Theorem 5.4. Let B be a standard Borel space and let µ ∈ P(B N ). Then the following statements hold: (i ) µ is affine-exchangeable with the independence property if and only if µ = ζ X,m for some 2-homogeneous profinite-step compact nilspace X and some Borel map m : X → P(B). (ii ) If µ is affine-exchangeable then it is a mixture of affine-exchangeable measures with the independence property. In other words, statement (ii) here tells us that there is a Borel probability measure κ on P(B N ), concentrated on the measurable set I from (11), such that µ = P(B N ) ν dκ(ν). Proof. The argument consists in combining results from [14], in particular [14,Theorem 6.7], with the additional information given here by Theorem 3.3, afforded by the assumption that µ is not just cubic-exchangeable but is affine-exchangeable. The adaptation of the relevant results from [14] goes as follows. First, by a straightforward use of [14, Proposition 6.10] we obtain that µ ∈ P(B N ) is affine-exchangeable if and only if it is the factor of a weak cubic couplingμ (see [14, Definition 6.8]) with the additional property of being 2-homogeneous, i.e. such that for The backward implication is clear because µ directly inherits fromμ the additional 2-homogeneity property, and this property is easily seen to imply affine-exchangeability. every injective affine map T : F m 2 → F n 2 we haveμ • p −1 T (F m 2 ) =μ • p −1 (Indeed, given a 2-homogeneous cubic coupling µ ∈ P(B N ) and an element T ∈ Aff(F ω 2 ), the map T can be seen as an injective map T : F n 2 → F n 2 for some n ≥ 0 large enough. Thus, by Definition 3.2 we have µ n = µ n T where µ n := µ • p −1 n and p n : B N → B n is the projection to the first n coordinates; this is precisely the definition of µ being affine-exchangeable.) Next, we use a straightforward adaptation of [14,Proposition 6.13]. This adaptation states that a cubic-exchangeable measure η on B N that is in addition 2-homogeneous (such as the measureμ obtained above) is a mixture of cubic couplings (viewed as measures on B N ), almost everyone of which is also 2-homogeneous. This adaptation is obtained by going through the proof of [14, Proposition 6.13], replacing the injective cube morphisms φ 1 , φ 2 : k → N by injective affine maps F k 2 → F ω 2 . Thus far we have obtained (similarly as for [14, Theorem 6.14]) that every affineexchangeable probability measure is a mixture of factors of 2-homogeneous cubic couplings (in the sense that a measure µ ∈ P(B N ) can be regarded as a cubic coupling when By Theorem 4.3, for each nilspace X = X ν appearing in this mixture, there exists a fibration ϕ ν : H → X ν , and it is then readily seen that the corresponding measure ζ X ν ,mν is equal to ζ H,mν • ϕν . Indeed, by standard results it suffices to check this equality for cylinder sets S = v∈ N A v ⊂ B N , and for any such set, using the fact that ϕ N ν preserves the Haar measures, we have ζ Xν ,mν (S) = C ω (H) v∈ N m ν • ϕ ν (c(v))(A v ) dµ C ω (H 2 ) (c) = ζ H,mν • ϕν (S) as required. As each map m ν • ϕ ν is Borel H → P(B), we thus deduce that the measure κ in (12) is concentrated on measures of the form ν = ζ H,m for Borel maps m : H → P(B), and this completes the proof. On the geometry of the class of affine-exchangeable measures In this section we prove Theorem 1.6, yielding the Bauer property for affine exchangeability, i.e., that for every compact metric space B the set Pr Aff(F ω 2 ) (B N ) is a Bauer simplex. We begin by recording the following fact concerning cubic-exchangeable measures more generally. (ii ) µ is cubic-exchangeable and satisfies the independence property. (iii ) µ = ζ X,m for some compact profinite-step nilspace X and Borel map m : X → P(B). The ingredients for a proof of this result are all essentially contained in [14], but the lemma is not explicitly established in that paper. We take the opportunity to do so here. Proof. We can now prove the main result of this section. Proof of Theorem 1.6. Let (µ n ) n∈N be a sequence of extreme points in Pr Aff(F ω 2 ) (B N ) converging vaguely to µ ∈ Pr Aff(F ω 2 ) (B N ). By (i) ⇔ (ii) in Proposition 6.2 and Lemma 5.3, it follows that µ is also an extreme point in Pr Aff(F ω 2 ) (B N ). Remark 6.4. An alternative route to establish Theorem 1.6 could be to verify that our main representation result (Theorem 1.5) implies that affine exchangeability satisfies a property called representability, introduced by Austin in [5], and then apply [5, Proposition 3.2], which states that representability implies the Bauer property. The route followed in this section, using our previous results, was chosen in particular to take the opportunity to record Propositions 6.1 and 6.2. Correspondence between representations of affine-exchangeable measures and limits for convergent sequences of functions In this section we show that finding an appropriate integral representation for affineexchangeable measures is equivalent to finding limit objects for convergent sequences of functions, for a concept of convergence that we recall in Definition 7.3 below. This concept is an analogue in arithmetic combinatorics of a well-known convergence notion for sequences of graphs studied in many works (see in particular [8,37]). ON F ω 2 -AFFINE-EXCHANGEABLE PROBABILITY MEASURES 33 Throughout this section let Γ denote the group Aff(F ω 2 ) of invertible affine maps F ω 2 → F ω 2 , and recall that Γ ∼ = GL(F ω 2 ) ⋉ Z ω 2 . Given any function f on F ω 2 and any T ∈ Γ, we use the usual ergodic-theory notation T f to denote the composition f • T . The main result of this section, Theorem 7.7 below, establishes an equivalence between two properties, the first of which is the following. Definition 7.1. Let B be a standard Borel space and let X be a compact profinite-step 2-homogeneous nilspace. We say that X represents (or is a representing nilspace for) Pr Aff(F ω 2 ) (B N ) if for every µ ∈ Pr Aff(F ω 2 ) (B N ) with the independence property, there exists a Borel map m : X → P(B) such that µ = ζ X,m . The second property involves the above-mentioned notion of convergence of functions. This notion has been studied in previous works, notably in [25] in the special case of Boolean functions (taking values in {0, 1}). We shall define this notion for functions taking values in any fixed compact metric space. To this end we shall use the following terminology, much of which is taken from [25] (rephrasing some of it using nilspace theory). Given a finite set L ⊂ F ω 2 , clearly there exists k ∈ N such that L ⊂ F k 2 × {0 N\[k] }. We shall sometimes abuse the notation by identifying F k 2 with the set F k 2 × {0 N\[k] } ⊂ F ω 2 . Note that the set of affine linear maps A : F k 2 → F n 2 with pointwise addition is isomorphic to the abelian group of k-cubes C k (F n 2 ), and we can thereby define a random affinelinear map A using the Haar measure µ C k (F n 2 ) (which here is just the normalized counting measure); we can thereby also apply an element A ∈ C k (F n 2 ) as an affine-linear map to elements L = (λ 1 , . . . , λ k ) ∈ F k 2 . Indeed, for some coefficients a 0 , a 1 , . . . , a k ∈ F n 2 uniquely associated with A we have A(L) = a 0 +λ 1 a 1 +· · ·+λ k a k for every such element L. We can then define the following measures, which underpin the upcoming notion of convergence. Definition 7.2. Let L be a finite subset of F ω 2 and let k ∈ N be such that L ⊂ F k 2 ×{0 N\[k] }. Let B be a compact metric space, and let f : F n 2 → B. We define the probability measure µ L,f ∈ P(B L ) as the pushforward of µ C k (F n 2 ) under the map A → f • A(L) L∈L , that is ∀ Borel set S ⊂ B L , µ L,f (S) := {A ∈ C k (F n 2 ) : (f • A(L)) L∈L ∈ S} \| C k (F n 2 )\| .(13) It follows from the consistency axiom for the cubic coupling (µ C k (F n 2 ) ) k≥0 that, given any such finite set L ⊂ F ω 2 , the definition of µ L,f is independent of k provided that k is large enough so that L ⊂ F k 2 ×{0 N\[k] }. When S is a singleton {g} we shall simplify the notation by writing µ L,f (g) instead of µ L,f ({g}). Remark 7.4. Each point (λ 1 , . . . , λ k ) ∈ F k 2 can be viewed as a linear form F k 2 → F 2 , x → λ 1 x 1 +· · ·+λ k x k . Following [25] we say that this is an affine linear form if λ 1 = 1. For every linear form L = (λ 1 , . . . , λ k ) we can define the affine linear form L = (1, λ 1 , . . . , λ k ) ∈ F k+1 2 . From this viewpoint, in the case of Boolean functions the above notion of convergence can be seen to be equivalent to the following one, familiar in arithmetic combinatorics, and phrased in terms of averages over systems of affine linear forms: a sequence (f n : F n 2 → {0, 1}) n∈N is convergent if for every k ∈ N, for every L = {L 1 , . . . , L r } ⊂ F k 2 , the following averages converge as n → ∞: Λ L (f n ) := E a 0 ,a 1 ,...,a k ∈F n 2 f n (a 0 + L 1 (a 1 , . . . , a k )) · · · f n (a 0 + L r (a 1 , . . . , a k )) = E a 0 ,a 1 ,...,a k ∈F n 2 f n ( L 1 (a 0 , . . . , a k )) · · · f n ( L r (a 0 , . . . , a k )). One implication in the equivalence follows from the fact that Λ L (f n ) = g∈{0,1} L µ L,fn (g), and the opposite implication follows from the fact that each µ L,fn (g) is itself a linear combination of averages Λ L ′ (f n ), as can be shown using Fourier analysis (see e.g. [25, Observation 4.1]). Given any finite set L ⊂ F ω 2 , we denote by p L the coordinate projection B F ω 2 → B L . The second property involved in the main result of this section is the following. Definition 7.5. Let X be a compact profinite-step 2-homogeneous nilspace, and let B be a compact metric space. We say that X is a limit domain for convergent sequences Remark 7.6. It can be seen that X being a limit domain for convergent sequences of Boolean functions (f n : F n 2 → {0, 1}) n∈N is equivalent to any such sequence converging to an ∞-limit object Γ : X → [0, 1] in the sense of [25, Definition 3.2] for X = G ∞ (indeed this limit object can be seen to correspond to the above Borel map m : X → P ({0, 1})). For more details on this correspondence see Appendix E. We are now ready to state and prove the main result of this section. Theorem 7.7. Let X be a compact profinite-step 2-homogeneous nilspace, and let B be a compact metric space. The following statements are equivalent: (i ) X is a representing nilspace for Pr Aff(F ω 2 ) (B N ). (ii ) X is a limit domain for convergent sequences of functions (f n : F n 2 → B) n∈N . Proof. Throughout the proof we use the fact that the group Γ : = Aff(F ω 2 ) ∼ = GL(F ω 2 ) ⋉ Z ω 2 can be regarded as ∞ n=1 Γ n for Γ n := Aff(F n 2 ) ∼ = GL(F n 2 ) ⋉ Z n 2 , where Γ n acts on the first n coordinates of any element of F ω 2 . We split the proof into two parts. Proof of (ii) ⇒ (i). Let µ ∈ Pr Aff(F ω 2 ) (B N ) satisfy the independence property. We shall define a convergent sequence (f n : F n 2 → B) n∈N such that for every finite set L ⊂ F ω 2 the measures µ L,fn converge vaguely to µ • p −1 L . On the other hand, our assumption of (ii) will give us a Borel map m : X → P(B) (dependent on this sequence (f n )), such that µ L,fn converges vaguely to ζ X,m • p −1 L . This will imply that µ • p −1 L = ζ X,m • p −1 L for every finite such set L, whence we will deduce that µ = ζ X,m , confirming (i) as required. To define the desired sequence (f n ) we use the pointwise ergodic theorem for amenable groups [35,Theorem 1.2]. For any function t : B F ω 2 → C let us define the following ergodic average for each n ∈ N: E n (t) : f → E A∈Γn t(Af ) = 1 \|Γ n \| A∈Γn t(f • A).(15) The space of continuous functions B F ω 2 → C equipped with the supremum norm is separable, so we can pick a dense sequence (t i ) i∈N in this space. By the ergodic theorem [35,Theorem 1.2], and the ergodicity of µ (which follows from the independence property, by Proposition 6.2 and Remark 6.3), for every fixed i ∈ N we have E n (t i )(f ) → t i dµ for µ-almost every f ∈ B F ω 2 . Using this for each i, we deduce that there is a set D ⊂ B F ω 2 with µ(D) = 1 such that for every f ∈ D we have lim n→∞ E n (t i )(f ) = t i dµ for all i ∈ N. From the assumed density of (t i ) i∈N , it follows that ∀ f ∈ D, ∀ continuous function t : B F ω 2 → C, lim n→∞ E n (t)(f ) = t dµ.(16) Fix any f ′ ∈ D. For each n ∈ N we define f n : F n 2 → B by f n (v) = f ′ (v, 0 N\[n] ). We claim that for any finite L ⊂ F ω 2 we have µ L,fn → µ • p −1 L in the vague topology (in particular (f n ) n∈N is convergent). To see this, fix any such L, and any k such that L ⊂ F k 2 . Fix any continuous function t : B L → C, and observe that for n ≥ k we have E n (t • p L )(f ′ ) = 1 \|Γn\| A∈Γn t • p L (f ′ • A) = 1 \|Γn\| A∈Γn t(p L (f n • A)) where in the last equation we are abusing the notation, treating p L as a map B n → B L and A as an element of Aff(F n 2 ). Let p k : Γ n → {A ′ : F k 2 → F n 2 \| A ′ is affine linear and injective} be the projection sending A ∈ Γ n to the map A ′ (v) := A(v, 0 n−k ). Note that p k is surjective and in fact for each injective affine map A ′ ∈ C k (F n 2 ) the preimage p −1 k (A ′ ) has the same cardinality . . . , v k ) = a 0 + a 1 v 1 + · · · + a k v k is injective, the linear span of {a 1 , . . . , a k } has dimension k, whence, to extend A ′ to a map A ∈ Γ n of the form A(v 1 , . . . , v n ) = a 0 +a 1 v 1 +· · ·+a k v k +a k+1 v k+1 +· · ·+a n v n , we have 2 n −2 k choices for a k+1 , then 2 n −2 k+1 choices for a k+2 , etc.). Also, if p k (A) = p k (B) then p L (f n • A) = p L (f n • B). n−1 j=k (2 n − 2 j ). (Indeed if A ′ (v 1 , Hence E n (t • p L )(f ′ ) = 1 \|p k (Γ n )\| A ′ ∈p k (Γn) t(p L (f n • A ′ )). Now, while p k (Γ n ) is not exactly C k (F n 2 ) (not every affine map in the latter set is injective), we do have \|p k (Γ n )\| \| C k (F n 2 )\| = 2 n k i=1 (2 n − 2 i−1 ) 2 n(k+1) = k i=1 1 − 1 2 n−i+1 = 1 + o k (1) n→∞ .(17) Indeed, to choose A ′ ∈ p k (Γ n ) we have 2 n choices for a 0 , 2 n − 1 choices for a 1 (all but 0 n ), 2 n − 2 choices for a 2 (all but the subspace generated by a 1 ), and so on. Hence E n (t • p L )(f ′ ) = 1 \|p k (Γ n )\| A ′ ∈C k (F n p ) t(p L (f n • A ′ )) − 1 \|p k (Γ n )\| A ′ ∈C k (F n p ) A ′ not injective t(p L (f n • A ′ )) = (1 + o k (1) n→∞ ) t dµ L,fn + t ∞ o k (1) n→∞ . (18) By (16) the left hand side here converges to t • p L dµ as n → ∞, whence so does the right hand side. This proves that µ L,fn converges vaguely µ • p −1 L , as we claimed. By our assumption that X is a limit domain, there exists a Borel map m : X → P(B) such that µ L,fn converges vaguely to ζ X,m • p −1 L , so µ • p −1 L = ζ X,m • p −1 L . Since the projections p L (for finite sets L) generate the Borel σ-algebra on B F ω 2 , we deduce that µ = ζ X,m . Proof of (i) ⇒ (ii). Suppose that X represents Pr Aff(F ω 2 ) (B N ), and let (f n : F n 2 → B) n∈N be a convergent sequence. We will show that the limits of the measures µ L,fn for finite sets L ⊂ F ω 2 enable the definition of a measure µ ∈ Pr Aff(F ω 2 ) (B N ) with the independence property and such that µ • p −1 L is the vague limit of µ L,fn for every such L, whence (ii) will follow from the representation µ = ζ X,m given by (i). By assumption, for every finite set L ⊂ F ω 2 , the sequence (µ L,fn ) n∈N converges vaguely to a measure in P(B L ), which we shall denote by µ L . We claim that there exists a measure µ ∈ P(B F ω 2 ) such that for any finite L we have µ • p −1 L = µ L . To prove this we shall define a continuous linear functional µ on C(B F ω 2 ), and then apply the Riesz representation theorem [39, p. 464] to identify µ with a measure µ ∈ P(B F ω 2 ) as required ( µ will be the integral with respect to µ). For any finite L ⊂ F ω 2 and continuous t : B L → C we define µ(t • p L ) := t dµ L . Note that if t • p L = t ′ • p L ′ then t dµ L = t ′ dµ L ′ , so µ is well-defined. For any continuous t : B F ω 2 → C, by the Stone-Weierstrass theorem there is a sequence of pairs (L n , f n ) where L n are finite subsets of F ω 2 and t n : B Ln → C, such that t − t n • p Ln ∞ < 1/n for every n. It follows that the sequence ( t n dµ Ln ) n∈N is Cauchy and thus we can define µ(t) := lim n→∞ t n dµ Ln (this limit is easily seen to be the same for every such sequence (t n • p Ln ) n∈N converging to t). The operator µ is easily seen to be linear and bounded (in fact it has norm 1), so we conclude that µ is an element of the dual of C(B F ω 2 ), and we obtain a measure µ as announced, with the property that µ(f ) = f dµ for every f ∈ C(B F ω 2 ) (in particular µ is a probability measure). To prove that µ is affine-exchangeable, since the functions of the form t • p L (for finite L ⊂ F ω 2 and continuous t : B L → C) are dense in the space of continuous functions B F ω 2 → C by the Stone-Weierstrass theorem, it suffices to prove that for every such function and every A ∈ Aff(F ω 2 ) we have t • p L dµ = t • p L • θ A dµ, where θ A is the coordinate permutation on B F ω 2 induced by A. To prove this, we shall use the following notation: given f : Now fix any continuous function t : B L → C and any A ∈ Aff(F ω 2 ). First note that µ L,f = µ C k (F n 2 ) • T −1 f,L,n . If k is large enough so that L ⊂ F k 2 × 0 N\[k] and A can be viewed as an element of Aff(F k 2 ), then t • p L dµ = lim n→∞ t dµ L,fn = lim n→∞ t • T fn,L,n dµ C k (F n 2 ) . The map C k (F n 2 ) → C k (F n 2 ), c → c • A is a surjective homomorphism, so it preserves the Haar measure. Hence t • T fn,L,n (c) dµ C k (F n 2 ) (c) = t • T fn,L,n (c • A) dµ C k (F n 2 ) (c). By definition T fn,L,n (c • A) = (f n • c • A)\| L = φ −1 L,A ((f n • c)\| AL ). Hence we have t • p L dµ = lim n→∞ t • φ −1 L,A ((f n • c)\| AL ) dµ C k (F n 2 ) (c), and this equals lim n→∞ t • φ −1 L,A • p AL dµ AL,fn = (t • φ −1 L,A ) • p AL dµ = t • p L • θ A dµ(t 1 • p L 1 )(t 2 • p L 2 ) dµ = (t 1 • p L 1 ) dµ (t 2 • p L 2 ) dµ. Fix any k such that L 1 ⊔ L 2 ⊂ F k 2 . Then (t 1 • p L 1 )(t 2 • p L 2 ) dµ = lim n→∞ E A∈C k (F n 2 ) (t 1 • T fn,L 1 ,n )(t 2 • T fn,L 2 ,n ). Fix any n ∈ N. Then this last average equals E A∈C k (F n 2 ) t 1 (f n • A\| L 1 ) t 2 (f n • A\| L 2 ).(19) For i = 1, 2 we have that (f n • A)\| L i is a vector indexed by L ∈ L i with values f n (A(L)). Note that there exists two finite faces F 1 and F 2 such that L i ⊂ F i for i = 1, 2, F 1 ∩ F 2 = {K} for some K ∈ F ω 2 and such that F 1 ∪ F 2 = F k 2 , K ∈ L 1 and K ∈ L 2 . Furthermore, by applying if necessary an element of Γ k (which preserves finiteness and independence of faces), we can assume that K = 0 ω , F 1 = {0, 1} d 1 × {0 N\[d 1 ] }, F 2 = {0 d 1 −1 } × {r} × {0, 1} d 2 × {0 N\[d 1 +d 2 ] } (for some r ∈ F 2 ). Clearly k = d 1 + d 2 . Using the bijection between C k (F n 2 ) and {(x, a 1 , . . . , a d 1 +d 2 ) ∈ F n 2 } yielded by the expression A(v) = x + a 1 v 1 + · · · a d 1 +d 2 v d 1 +d 2 , the average in (19) can then be written as follows: E x,a 1 ,...,a d 1 ∈F n 2 a d 1 +1 ,...,a d 1 +d 2 ∈F n 2 t 1 (f n • A\| L 1 ) t 2 (f n • A\| L 2 ). Since the elements of L 1 have coordinates v d 1 +1 , . . . , v d 2 all 0, the first term is independent of the variables a d 1 +1 , . . . , a d 1 +d 2 , and thus the average equals E x,a 1 ,...,a d 1 ∈F n 2 t 1 (f n (x + a 1 v i 1 + · · · + a d 1 v i d 1 )) i∈[I] · E a d 1 +1 ,...,a d 1 +d 2 ∈F n 2 t 2 f n (x + a d 1 +1 v j d 1 +1 + · · · + a d 1 +d 2 v j d 1 +d 2 )) j∈[J] ,(20) where we assume that L 1 = {(v i 1 , . . . , v i d 1 , 0 d 2 ) : i ∈ [I]} for some I ≥ 0 and similarly L 2 = {(0 d 1 , v j d 1 +1 , . . . , v j d 1 +d 2 ) : j ∈ [J]} for some J ≥ 0. Now note that since L 2 ∋ 0 k ,E a d 1 +1 ,...,a d 1 +d 2 ∈F n 2 (t 2 (f n (a d 1 +1 v j d 1 +1 + · · · + a d 1 +d 2 v j d 1 +d 2 )) j∈[J] ). We can now change again the variable a s to a s + y for y ∈ F n 2 , and add an averaging over y ∈ F n 2 , thus proving that this last average equals t 2 dµ L 2 ,fn . Taking this constant out of the average in (20) we conclude that (20) equals t 1 dµ L 1 ,fn t 2 dµ L 2 ,fn . Since this holds for each n, the result follows by letting n → ∞. Remark 7.8. Theorem 7.7 concerns a notion of convergence involving systems of linear forms which can be of arbitrary finite complexity (in the sense of true complexity from [21]). In many situations in higher-order Fourier analysis, we are only interested in working with systems of complexity at most some prescribed finite bound. In this case, it turns out that the correct notion of complexity is the true complexity of L := { L = (1, L) : L ∈ L}. Using the compactness of the set P(B L ), a diagonalization argument, Theorem 7.7, and Theorem 1.5 the following result can be proved. Fix any k ∈ N. Let (f n : F n 2 → B) n∈N be a sequence of functions for some compact metric space B such that µ L,fn converges vaguely for all finite L ⊂ F ω 2 such that the true complexity of L is at most k. Then there exists a measurable function m : H → P(B) such that the following holds. For any finite L ⊂ F ω 2 such that the true complexity of L at most k, the measures µ L,fn converge vaguely to ζ H,m • p −1 L . This result can be further refined in the following sense. Note that H has infinite step, whereas we only required convergence for linear forms up to some finite complexity k. Using an equidistribution theorem (that we omit in this paper), any average over ζ H,m • p −1 L can be written as the limit when n → ∞ of some average over C s (F n 2 ) (where L ⊂ F s 2 × {0} N\[s] for some s ∈ N). Then, using the definition of true complexity it MEASURES 39 can be proved that if the true complexity of L is at most k (for some L ⊂ F ω 2 ) then ζ H,m • p −1 L equals ζ H/H (k+1) ,m k • p −1 L where m k : H/H (k+1) → P(B) is given by the formula m k (π k (x)) := π −1 k (π k (x)) m(y) dµ π −1 k (π k (x)) and π k : H → H/H (k+1) is the quotient map 13 . ON F ω 2 -AFFINE-EXCHANGEABLE PROBABILITY Remark 7.9. Affine exchangeability (or, more precisely, F ω 2 -affine exchangeability) has the following natural analogue for any prime p > 2: a measure µ ∈ P(B F ω p ) is F ω p -affineexchangeable if µ is invariant under the coordinate permutations induced by the group Aff(F ω p ) ∼ = GL(F ω p ) ⋉ Z ω p . As mentioned at the end of the introduction, the main results of this paper have counterparts for F ω p -affine exchangeability, but proving these requires extending previous work. In particular, the main results concerning cubic couplings from [14] would need to be extended to product spaces indexed by F ω p , and these would have to be combined with corresponding extensions of the results from Section 3 in this paper, to obtain structural results involving p-homogeneous nilspaces. This is beyond the scope of the present paper. On representing affine-exchangeability with nilspaces Recall from Definition 7.1 the notion of a nilspace X representing Pr Aff(F ω 2 ) (B N ) for some standard Borel space B. In this section we shall say that X represents affineexchangeability if X represents Pr Aff(F ω 2 ) (B N ) for every standard Borel space B. The main result of this subsection is Theorem 1.7, which we recall here for convenience, establishing counterparts of this notion of representation that concern nilspace theory and limit domains. Theorem 1.7 Let X be a compact profinite-step 2-homogeneous nilspace. The following statements are equivalent. (i ) X represents affine-exchangeability. (ii ) For every compact 2-homogeneous profinite-step nilspace Y there is a (continuous) fibration ϕ : X → Y. (iii ) X is a limit domain for convergent sequences (f n : F n 2 → B) n∈N , for every compact metric space B. From previous main results in this paper (specifically Theorems 5.4 and 4.3) we know that the nilspace H from Definition 1.4 is an example of a nilspace X satisfying properties (i) and (ii) above. To prove Theorem 1.7, we start by studying a class of examples announced at the end of Section 2, which we call deterministic cubic-exchangeable measures. In particular, we show in Proposition 8.4 that the mere assumption that X represents this class of examples already implies a non-trivial property of X, which can be viewed as a measure-theoretic version of property (ii) above. To define deterministic cubic-exchangeable measures, we shall use the following basic result on disintegrations of measures [33, (17.35)]. Lemma 8.1. Let R, S be standard Borel spaces and let f : R → S be a Borel map. Let µ ∈ P(R) and ν = f * µ := µ • f −1 ∈ P(S). Then there is a Borel map S → P(R), s → µ s such that for ν-almost every s ∈ S we have µ s (f −1 {s}) = 1 and µ = S µ s dν(s), i.e., for any Borel set A ⊂ R we have µ(A) = S µ s (A) dν(s). Moreover, if s → κ s is another map with these properties, then for ν-almost every s we have µ s = κ s . We call (µ s ) s∈S the disintegration of µ relative to f . Definition 8.2. Let µ be a cubic-exchangeable measure in P(B N ). We say that µ is deterministic if for every w ∈ N , letting R = B N , S = B N \{w} , letting f : R → S be the coordinate projection and letting ν be the pushforward µ • f −1 , we have that ν-almost every measure µ s in the disintegration of µ relative to f is a Dirac measure. Note that in this definition we can replace "for every w ∈ N " equivalently by "for some w ∈ N ", using that the group Aut( N ) (whose induced coordinate-action leaves µ invariant, by cubic exchangeability) acts transitively on N . Note also that, by Lemma We can now take a first step towards Theorem 1.7, with the following result. Lemma 8.3. Let X be a compact profinite-step nilspace, let B be a standard Borel space, and let m : X → P(B) be a Borel map. Then ζ X,m is deterministic if and only if for µ X -almost-every x ∈ X, the measure m(x) is a Dirac measure on B. Proof. Fix any w ∈ N , let f : B N → B N \{w} be the coordinate projection (i.e. the map that just deletes the w-th coordinate). Forward implication: consider the disintegration ζ X,m = B N \{w} µ s dν(s) relative to f given by Lemma 8.1 (in particular ν = ζ X,m • f −1 ). We are assuming that ζ X,m is deterministic, so µ s is a Dirac measure δ g(s) for ν-almost every s. From the construction (4) of ζ X,m , we have ζ X,m = C ω (X) v∈ N \{w} m(c(v)) × m(c(w)) dµ C ω (X) (c).(21) Note that this last expression can be viewed as a disintegration relative to f . Indeed, first note that the underlying assumption that X is profinite-step implies uniqueness ON F ω 2 -AFFINE-EXCHANGEABLE PROBABILITY MEASURES 41 of ω-corner completion (by Lemma 2.17). By this uniqueness of completion, this last integral can be written as an integral over the space Cor ω (X) of ω-corners "rooted" at w (i.e. the map f restricts to a bijection C ω (X) = hom( N , X) → Cor ω w (X) = hom( N \ {w}, X) that preserves the Haar measures on these two morphism sets; see Remark 2.20). But then, denoting these ω-corners by s, we can see that ν agrees with the measure Cor ω (X) v∈ N \{w} m(s(v)) dµ Cor ω (X) (s), viewed as a measure on B N \{w} as usual. Moreover, comparing with the last integral in (21), we see that ζ X,m = B N \{w} µ s dν(s), where µ s = m(c s (w)), for the unique c s ∈ C ω (X) completing s. By the almost-sure uniqueness of the fiber-measures in the disintegration, and the assumption that each fiber measure µ s is a Dirac measure, it follows that for ν-almost every corner s the measure m(c s (w)) is a Dirac measure. Finally, note that, as s varies ν-uniformly in Cor ω (X), the point c s (w) varies µ X -uniformly in X (just recall that Cor ω (X) = C ω (X) and that the projection p w : C ω (X) → X, c → c(w) is a continuous totally-surjective bundle morphism and by Lemma A.6 it preserves the Haar measure). Thus we are covering almost every point x ∈ X as a value c s (w), and we can therefore deduce that m(x) is a Dirac measure for µ X -almost every x as required. Backward implication: if almost every m(x) is a Dirac measure, then using the view of (21) as a disintegration of ζ X,m relative to f , we see that this is a disintegration with almost every fiber-measure being a Dirac measure, so ζ X,m is deterministic. We deduce the following result, which yields a strong measure-theoretic relation between nilspaces X, Y assuming that ζ X,m , ζ Y,m ′ are the same measure on X N . Proposition 8.4. Let X, Y be compact profinite-step nilspaces, let m be the Borel map Y → P(Y) sending y to δ y , and suppose that ζ X,m ′ = ζ Y,m for some Borel map m ′ : X → P(Y). Then there is a Borel map τ : X → Y such that m ′ (x) = δ τ (x) for µ X -almost-every x ∈ X, and for every integer n ≥ 0 we have µ C n (X) •(τ n ) −1 = µ C n (Y) . Proof. Since by assumption m is Dirac-measure valued, by the backward implication in Lemma 8.3 we have that ζ Y,m is deterministic. Hence, the assumption ζ X,m ′ = ζ Y,m implies that ζ X,m ′ is deterministic, so by the forward implication in Lemma 8.3, we deduce that m ′ (x) is a Dirac measure for µ X -almost-every x ∈ X. We can therefore define a measurable map τ : X → Y such that m ′ (x) = δ τ (x) for µ X -almost-every x ∈ X. To show that τ preserves the Haar measure on every cube set, we use the cubic exchangeability of the measures ζ X,m ′ , ζ Y,m . Take any injective morphism φ : n → N ; for simplicity we can take the morphism embedding n as n × {0 N\[n] }. Then from the formula (4) defining ζ Y,m it is seen directly that the push-forward of ζ Y,m induced by φ on Y n is µ C n (Y) . More precisely, identifying φ( n ) ⊂ N with n , we have an induced identification of Y n with Y φ( n ) , whereby the pushforward of ζ Y,m under the coordinate projection Y N → Y φ( n ) can be identified with µ C n (Y) (viewed as a measure on Y n concentrated on C n (Y) the usual way). On the other hand, the same formula (4) applied to ζ X,m ′ shows that this same push-forward is µ C n (X) •(τ n ) −1 . Since by assumption ζ X,m ′ = ζ Y,m , the result follows. In the proof of Theorem 1.7 we shall use a strengthened version of the above result, in which the map τ is modified on a null set to obtain a continuous cube-surjective map, which is then shown to be a fibration. This technical upgrade using nilspace-theoretic tools is deferred to Appendix D. We are now ready to prove the theorem of this subsection. (iii) ⇒ (ii). The particular case B = H of the assumption tells us that X is a limit domain for convergent sequences of H-valued functions. By Theorem 7.7, the nilspace X represents Pr Aff(F ω 2 ) (H N ). Then, arguing similarly as in the proof of (i) ⇒ (ii) above, using the particular measure µ C ω (H) , we deduce that there is a fibration X → H, whence (ii) follows by Theorem 4.3. (i) ⇒ (iii). This implication follows immediately from Theorem 7.7. 8.1. Some restrictions on the structure of limit domains. Let us record the following consequence of Theorem 1.7 (ii). Corollary 8.5. Let (Z, Z • ) be a filtered compact abelian group such that the associated group nilspace X is a limit domain for convergent sequences (f n : F n 2 → B) n∈N for every compact metric space B. Then there is a fibration ϕ : X → H. This result yields non-trivial restrictions on the objects that could be used as such general limit domains. In this subsection we illustrate these restrictions with a specific example that has appeared in previous work. Throughout this section let G be the compact abelian group ∞ k=1 k ℓ=1 (Z/2 k−ℓ+1 Z) N . This group appeared in [25] (denoted therein by G ∞ ) in connection with convergent sequences of Boolean functions (f n : F n 2 → {0, 1}) n∈N . The results in [25] (specifically [25,Theorem 3.5] with d = ∞) would suggest that, if we equip G with the filtration G • = ∞ k=1 k ℓ=1 U N k,ℓ , then the resulting group nilspace can be used as a representing nilspace for affine exchangeability, or equivalently as a limit domain for convergent sequences of B-valued functions, for every compact metric space B (see Appendix E for more details). However, we will show below that this is not true. More precisely, if (G, G • ) were usable as such a general limit domain then by Corollary 8.5 there would exist a continuous fibration from the associated nilspace onto H, and we shall prove below that such a fibration cannot exist. To prove this, we first note that if there existed such a fibration, then composing it with a projection to a single component U ∞,1 , we would obtain a fibration from (G, G • ) to U ∞,1 . Before proving that such a fibration does not exist, let us prove an analogous fact in the category of abelian groups, namely, that there is no homomorphism from G to the group Z of 2-adic integers. To begin with, we recall the following basic fact. Lemma 8.6. The group of 2-adic integers is torsion-free. From this we can immediately deduce the following. Lemma 8.7. There is no non-constant continuous affine homomorphism G → Z. Proof. Using translations it clearly suffices to prove that there is no non-zero continuous homomorphism G → Z. Suppose that we have a homomorphism ϕ : G → Z. For any x ∈ G with only finitely many non-zero coordinates, we must have ϕ(x) = 0 (otherwise, since x has finite order, then so does ϕ(x), contradicting the previous lemma). As every element of G is the limit of elements of finite order in G, we then deduce by continuity that ϕ is the zero homomorphism, a contradiction. We can generalize this argument to the case of fibrations between nilspaces. Let us start by proving the following lemma. Lemma 8.8. Let H • be a 2-homogeneous filtration of degree k on an abelian group H, and let X be the group nilspace associated with (H, H • ). Then for every morphism ϕ : X → U ∞,1 , there exists a morphism ϕ 1 : π 1 (X) → U ∞,1 such that ϕ = ϕ 1 • π 1 . Proof. We argue by induction on k. The case k = 1 is trivial. Now assume that the result holds for step less than k and let H • be a degree-k filtration. We want to prove that for any g ∈ H and any g k ∈ H (k) we have ϕ(g + g k ) = ϕ(g). Fix any g ∈ H and g k ∈ H (k) . Let a 0 := ϕ(g) and a 1 := ϕ(g + g k ). Recall that for any pair of abelian groups Z, Z ′ and any function f : Z → Z ′ , the forward difference of f with step h ∈ Z is the function ∂ h f (x) := f (x + h) − f (x), x ∈ H. We now claim that ∂ n g k ϕ(g) = 2 n−1 (−1) n (a 0 − a 1 ) for every n ∈ N. Indeed this is clear for n = 1, and for n > 1 we have by induction ∂ n g k ϕ(g) = ∂ g k (2 n−2 (−1) n−1 (ϕ(g) − ϕ(g + g k ))) = 2 n−2 (−1) n−1 (∂ g k ϕ(g) − ∂ g k ϕ(g + g k )) = 2 n−2 (−1) n−1 (a 1 − a 0 − (a 0 − a 1 )) = 2 n−1 (−1) n (a 0 − a 1 ) where we have used that g k has order 2 (since H • is 2-homogeneous). Now, by standard properties of polynomial maps [10, Theorem 2.2.14], since g k ∈ H (k) we have ∂ n g k ϕ(g) ∈ (U ∞,1 ) kn where (U ∞,1 ) j is the j-th term defining the filtration on Z associated with U ∞,1 by Definition 4.2. From this definition it follows that this group is precisely 2 nk−1 Z. Now since a 0 = a 1 , we have for some r ∈ N that a 0 − a 1 = ∞ i=r α i 2 i where α i ∈ {0, 1} for all i ≥ r and α r = 1. In particular the 2-adic order of a 0 − a 1 is at most r ∈ N, and then by the previous paragraph the 2-adic order of ∂ n g k ϕ(g) is at most r + n − 1. But in the paragraph we have also proved that this order is at least nk − 1. This yields a contradiction for n large enough. Thus we have proved that ϕ : X → U ∞,1 factors through π k−1 . By induction on k, we deduce that ϕ factors though π 1 . Let us now adapt the above arguments to the category of nilspaces, to apply it to the filtered group (G, G • ) defined above. Lemma 8.9. Let X be the group nilspace associated with (G, G • ), and let ϕ : X → U ∞,1 be a morphism. Then there is a morphism ϕ 1 : π 1 (X) → U ∞,1 such that ϕ = ϕ 1 • π 1 . Proof. Take g, g ′ ∈ X such that π 1 (g) = π 1 (g ′ ). For any n ∈ N let p n : G → G be the homomorphism that projects onto the subgroup n k=1 k ℓ=1 Z n 2 k−ℓ+1 × {0} N\[n] × ( ∞ k=n+1 k ℓ=1 {0} N ) (by switching the relevant coordinates to 0). Clearly we have p n (g) → g and p n (g ′ ) → g ′ as n → ∞. Furthermore π 1 (p n (g)) = π 1 (p n (g ′ )) for all n (since p n commutes with the projections to the 1-factors). By Lemma 8.8 we have ϕ(g n ) = ϕ(g ′ n ) for all n. Hence, taking the limit as n → ∞ and using the continuity of ϕ we conclude that ϕ(g) = ϕ(g ′ ). Theorem 8. 10. Let X be the group nilspace associated with (G, G • ). There exists no cube-surjective morphism X → H. Proof. If there existed such a morphism ϕ : X → H, then there would exist a cubesurjective morphism ϕ ′ : X → U ∞,1 . By Theorem C.1 we know that ϕ ′ would then be a fibration. But by Lemma 8.9 we also know that ϕ ′ would factor through π 1 , i.e., there would exist a morphism ϕ 1 : π 1 (X) → H such that ϕ ′ = ϕ 1 • π 1 . Then, since ϕ ′ and π 1 would be fibrations, so would ϕ 1 . Thus we would have a fibration ϕ 1 : π 1 (X) → U ∞,1 , and this yields a contradiction because π 1 (X) is a 1-step nilspace whereas U ∞,1 has infinite step (but the step of a fibration's image cannot be larger than the step of its domain). Appendix A. ∞-fold compact abelian bundles The concept of Haar measure on a compact nilspace of finite step is crucial in nilspace theory. A key fact enabling the definition of this measure is that any such nilspace can be expressed as an abelian bundle construction iterated finitely many times (see [11, §2.2.2]). In this paper we need an analogous Haar measure for profinite-step compact nilspaces, and to this end, similarly, we shall use the following notion of abelian bundle. . Then for all 0 ≤ j ≤ i we have π j,i = π j,j+1 • · · · • π i−1,i . Finally, we have that B 0 is a singleton and B = lim ← − B i = {(b k ) ∞ k=0 ∈ ∞ k=0 B k : ∀ 0 ≤ j ≤ i, π j,i (b i ) = b j }. For each i ≥ 0 we denote by π i the limit map B → B i , (b k ) ∞ k=0 → b i . Thus ∞-fold abelian bundles are inverse limits of k-fold abelian bundles. When we carry out this construction in the category of compact abelian bundles, we obtain the following. Proof. Let S := {π −1 i (A) : i ≥ 0, A ⊂ A(B i )} where A(B i ) is the Borel σ-algebra on B i . Note that S is a semi-ring of sets. We define µ B ((π i ) −1 (A)) := µ B i (A). As the functions π j,i : B i → B j for i ≥ j form an inverse system, note that if π −1 i (A) = π −1 j (C) for some i ≥ j, then π −1 i (A) = π −1 i (π −1 j,i (C)). Hence A = π −1 j,i (C) and by [11,Lemma 2.2.6] we know that π j,i preserves the Haar measure. Thus µ B i (A) = µ B j (C). We now check that with this definition the measure µ B is σ-additive on S. Suppose that A ∈ S is of the form A = ⊔ ∞ j=1 A j for some A j ∈ S. In particular ∪ k j=1 A j ⊂ A for any k ≥ 1. We also have A = π −1 r (C) and A j = π −1 r j (C j ) for all j ≥ 0 (since A, A j ∈ S). Letting r ′ ≥ max(r, r 1 , . . . , r k ), we have that these A and these A j can all be seen as subsets of B r ′ . Hence, by additivity of the measure µ B r ′ we have µ B (A) ≥ k j=1 µ B (A j ). Letting k → ∞ we obtain the inequality µ B (A) ≥ ∞ j=1 µ B (A j ) . To prove the reversed inequality, let us fix any ǫ > 0. By [16,Proposition 8.1.12] we know that for all n ≥ 0 and k ≥ 1, the measures µ B k are regular. Thus, there exists a compact set D ⊂ C such that µ Br (C \ D) < ǫ (recall that A = π −1 r (C)). Similarly, for every j ≥ 1 there exists an open set E j ⊃ C j such that µ Br j (E j \ C j ) < ǫ/2 j . As π −1 r (D) ⊂ ∪ ∞ j=1 π −1 r j (E j ), by compactness there is a finite set I ⊂ N such that π −1 r (D) ⊂ ∪ j∈I π −1 r j (E j ) . Now, taking r ′′ ≥ max(r, (r j ) j∈I ), arguing as in the previous paragraph we obtain µ B (A) − ǫ ≤ j∈I (µ B (A j ) + ǫ/2 j ) ≤ ∞ j=1 (µ B (A j ) + ǫ/2 j ). Hence µ B (A) ≤ 2ǫ + ∞ j=1 µ B (A j ). Letting ǫ → 0, the desired inequality follows. We complete the proof by applying Carathéodory's extension theorem. Next, we need to generalize the notion of bundle morphism from [10, Definition 3.3.1]. Definition A.5. Let B, B ′ be ∞-fold abelian bundles. Let B i , B ′ i be the corresponding terms of the inverse limit and Z i , Z ′ i the structure groups. A bundle morphism from B to B ′ is a map ϕ : B → B ′ satisfying the following properties. (i) For every i ≥ 0, if π i (x) = π i (y) then π i (ϕ(x)) = π i (ϕ(y)). Thus there is an induced well-defined map ϕ i : B i → B ′ i . (ii) For every i ≥ 1 there is a homomorphism α i : Z i → Z ′ i such that for every x ∈ B i and z ∈ Z i we have ϕ i (x + z) = ϕ i (x) + α i (z). The bundle morphism is totally surjective if the maps α i are surjective for all i ≥ 1. Totally surjective bundle morphisms for ∞-fold compact abelian bundles generalize the concept of surjective homomorphisms for compact abelian groups. In particular, these maps preserve the Haar measure. f ′ • π i dµ B ′ = f ′ • π i • ϕ dµ B for any such function. This is indeed the case since f ′ • π i • ϕ dµ B = f ′ • ϕ i • π i dµ B ′ = f ′ • ϕ i dµ B ′ i ,dµ B i = f ′ • π i dµ B . Given two ∞-fold compact abelian bundles B, B ′ and ϕ : B → B ′ a continuous totally surjective bundle morphism it can be checked that for every t ∈ B ′ the set ϕ −1 (t) is an ∞-fold compact abelian sub-bundle of B (the proof is a generalization of [10,Lemma 3.3.6]). This means in particular that the i-th limit map on ϕ −1 (t) is the restriction of B's i-th limit map π i to ϕ −1 (t). It is also straightforward to check that the i-th structure group of ϕ −1 (t) is ker(α i ),1 E dµ B = B ′ ϕ −1 (t) 1 E (x) dµ ϕ −1 (t) (x) dµ B ′ (t) where 1 E is the indicator function of E. Proof. Similarly as before, by the Riesz representation and Stone-Weierstrass theorems it suffices to check that B f • π i dµ B = B ′ ϕ −1 (t) f • π i (x) dµ ϕ −1 (t) (x) dµ B ′ (t) for any contin- uous function f : B i → C and i ≥ 1. Note that for any t ∈ B ′ , ϕ −1 (t) f • π i (x) dµ ϕ −1 (t) (x) = ϕ −1 i (π i (t)) f (x ′ ) dµ ϕ −1 i (π i (t)) (x ′ ) = b ∈ B i → ϕ −1 i (b) f (x ′ ) dµ ϕ −1 i (b) (x ′ ) • π i . Hence B ′ ϕ −1 (t) f • π i (x) dµ ϕ −1 (t) (x) dµ B ′ (t) = B ′ i ϕ −1 i (t) f • π i (x ′ ) dµ ϕ −1 i (t ′ ) (x ′ ) dµ B ′ i (t ′ ), and the latter equals B i f dµ B i = B f • π i dµ B by [11, Lemma 2.2.10]. Remark A.8. Note that by standard arguments using approximations by simple functions, we can deduce from Lemma A.7 that for every f ∈ L 1 (B) we have B f dµ B = B ′ ϕ −1 (t) f (x) dµ ϕ −1 (t) (x) dµ B ′ (t). A.1. The cubic coupling property for profinite-step compact nilspaces. The goal of this section is to prove that the n-cubic Haar measures on a compact profinitestep nilspace form a cubic coupling (see Definition 3.1). Let us start proving the ergodicity and consistency axioms. Proposition A.9. Let X be a compact profinite-step nilspace. Then its n-cubic Haar measures satisfy the ergodicity and consistency axioms. Proof. To prove that the ergodicity axiom holds we have to prove that µ C 1 (X) = µ X × µ X . Suppose that X = lim ← − X i where X i is the i-th characteristic factor of X. Let S(X 1 ) := {(π 1 i ) −1 (A) : i ≥ 0, A ∈ A(X i ) 1 } be the semi-ring that generates the σ-algebra on X 1 , where A(X i ) is the Borel σ-algebra on X i . It suffices to check that the equality µ C 1 (X) = µ X × µ X holds on S(X 1 ). But note that µ C 1 (X) ((π 1 i ) −1 (A)) = µ C 1 (X i ) (A) = (µ X i × µ X i )(A) = (µ X × µ X )((π 1 i ) −1 (A)) where in the second equality we have used the ergodicity axiom for X i given by [14,Proposition 3.6]. To establish the consistency axiom, let φ : n → m be an injective discrete-cube morphism (n ≤ m) and p φ : X m → X n the projection (c(v)) v∈ m → (c(φ(v))) v∈ n . We have to prove that µ C m (X) • p −1 φ = µ C n (X) . Similarly as before it is enough to check this for the semi-ring S(X n ). We clearly have π n i • p φ = p φ • π m i . Thus µ C m (X) • p −1 φ •(π n i ) −1 = µ C n (X) •(π m i ) −1 • p −1 φ = µ C m (X i ) • p −1 φ . By the consistency axiom for X i ([14, Proposition 3.6]), we have µ C m (X i ) • p −1 φ = µ C n (X i ) ,f ′ \| P 1 ∩P 2 = 0, there exists a morphism f : n → D k (Z) extending f ′ such that f \| P 1 = 0. The next result extends [11, Lemma 2.2.14]. Proposition A.11. Let X be a compact profinite-step nilspace, let P 1 , P 2 ⊂ n be a good pair in n , and let f : P 1 → X be a morphism. Then the restriction map Ψ : hom f ( n , X) → hom f \| P 1 ∩P 2 (P 2 , X) is a totally surjective continuous bundle morphism. Proof. The proof is very similar to that of [11, Lemma 2.2.14]. Now let us state the main result that will enable us to prove the conditional independence axiom for compact profinite-step nilspaces. Lemma A.12. Let P 0 , P 1 ⊂ n be a good pair. Then for every compact profinite-step nilspace X, the sets P 0 , P 1 are conditionally independent with respect to µ C n (X) . Proof. The proof is very similar to the proof of [20, Lemma 5.108]. We replace Lemma A.28, Proposition 5.57 and Proposition 5.60 from [20] respectively by Proposition A.11, Lemma A.6 and Lemma A.7 from this paper. Corollary A. 13. Let X be a compact profinite-step nilspace. Then its cubic Haar measures satisfy the conditional independence axiom. Proof. It is not hard to check that if P 0 , P 1 ⊂ n are (n − 1)-dimensional faces with P 0 ∩ P 1 = ∅, then they form a good pair. Then the result follows from Lemma A.12. Appendix B. Affine-exchangeable measures on nilspaces and 2-homogeneity In this appendix we prove Lemma 2.26, which we restate here for convenience. Lemma B.1. Let X be a compact profinite-step nilspace. Then µ C ω (X) (viewed as a measure on X N ) is affine-exchangeable if and only if X is 2-homogeneous. Proof. Suppose that X is 2-homogeneous. By Lemma 2.25 this is equivalent to having C ω (X) = hom(D 1 (F ω 2 ), X). Given any T ∈ GL(F ω 2 ) ⋉ Z ω 2 , let us define the map T ′ : C ω (X) → C ω (X) by T ′ (c) = c • T . To see that T ′ (c) is indeed in C ω (X) for every c ∈ C ω (X) = hom(F ω 2 , X), we have to check that for every q ∈ C n (F ω 2 ) we have c • T • q ∈ C n (X). But T • q is clearly in C n (F ω 2 ), so c • T ∈ C ω (X) as required. Moreover, it is seen by a straightforward computation that T ′ defines a totally surjective continuous bundle morphism from the ∞-fold compact abelian bundle C ω (X) to itself. Thus, by Lemma A.6, T ′ preserves the Haar measure, i.e. µ C ω (X) • T ′ −1 = µ C ω (X) . Hence µ C ω (X) is affine-exchangeable. For the converse, suppose that µ C ω (X) (seen as a measure on X N ) is affine-exchangeable, and for any n ∈ N let T ∈ GL(F n 2 ) ⋉ Z n 2 . Considering GL(F n 2 ) ⋉ Z n 2 ⊂ GL(F ω 2 ) ⋉ Z ω 2 in the usual way, we can view T as a transformation in GL(F ω 2 )⋉Z ω 2 (abusing the notation). The map T ′ : C ω (X) → C ω (X), c → c • T preserves the measure µ C ω (X) by assumption. Let ι : n → N be the inclusion map defined by ι(v) := (v, 0 N\[n] ), and let p n : X N → X n be the projection map (b v ) v∈ N → (b ι(v) ) v∈ n . We then have that µ C n (X) = µ C ω (X) • p −1 n is invariant under the action of T ′ , viewing the latter as a map X n → X n . For each k ≥ 0 we know that π n k : X n → X n k preserves the n-cubic Haar measures. We abuse the notation to see T ′ as a function on X n and X n k , and thus T ′ • π n k = π n k • T ′ . We obtain that for every k ≥ 0 the measure µ C n (X k ) is invariant under the action of T ∈ GL(F n 2 ) ⋉ Z n 2 . Note that µ C n (X k ) (T ′ (C n (X k )) = 1. Since T ′ is continuous, T ′ (C n (X k ) is a compact subset of X n k of measure 1. In particular T ′ (C n (X k )) ∩ C n (X k ) is a closed set of measure 1 (because µ C n (X k ) is supported on C n (X k )). Since µ C n (X k ) is a strictly positive Borel measure [11, Proposition 2.2.11], a simple argument deduces that C n (X k ) ⊂ T ′ (C n (X k )) (indeed, otherwise there would be a non-empty open subset of C n (X k ) of zero µ C n (X k )measure). Repeating this argument with T −1 we obtain the opposite inclusion, and thus conclude that T ′ (C n (X k )) = C n (X k ). We have thus obtained that for any n, k ≥ 0, c ∈ C n (X k ), and any T ∈ GL(F n 2 ) ⋉ Z n 2 we have c • T ∈ C n (X k ). Now to complete the proof we need to establish the claim that given any f ∈ C ℓ (F m 2 ) and any c ∈ C m (X k ) we have c • f ∈ C ℓ (X k ) for any ℓ, m, k ≥ 0. This will show that every element of C m (X k ) can be regarded as an element of hom(D 1 (F m 2 ), X k ) (identifying m with F m 2 ). Note that earlier we proved the above claim only for f = T an invertible affine map. We now need the result for any f ∈ C ℓ (F m 2 ), which can be viewed as an affine map (not necessarily injective or surjective) F ℓ 2 → F m 2 (recall that D 1 (F m 2 ) is 2-homogeneous). To prove the claim, first we reduce the proof to the case where f is surjective. If f : F ℓ 2 → F m 2 is not surjective, there exists an invertible transformation T 1 ∈ Aff(F m 2 ) such that T 1 • f (F ℓ 2 ) = F ℓ ′ 2 × 0 m−ℓ ′ . Thus, if p : F m 2 → F ℓ ′ 2 , p(v 1 , . . . , v m ) := (v 1 , . . . , v ℓ ′ ) and i : F ℓ ′ 2 → F m 2 , i(v) := (v, 0 m−ℓ ′ ) we have that f = T −1 1 • i • p • T 1 • f . Now suppose that c ∈ C m (X k ) and we have already proved the result for the case of surjective affine maps. Then c • f = c • T −1 1 • i • p • T 1 • f . But note that c • T −1 1 ∈ C m (X k ) by hypothesis. Then, since i is a discrete-cube morphism, we have (c • T −1 1 ) • i ∈ C ℓ ′ (X k ). Finally, note that p • T 1 • f is an affine surjective morphism and thus by hypothesis (c • T −1 1 • i) •(p • T 1 • f ) ∈ C ℓ (X k ). An analogous argument shows that we can further reduce the proof to the case where f is also injective and then we are done (since we have then reduced to the case of a bijective affine map, which was addressed above). This proves the claim. We can now conclude that C n (X k ) = hom(F n 2 , X k ) and thus X k is 2-homogeneous. As X is the inverse limit of X k , we have that X is also 2-homogeneous. Corollary B.2. Let (Z, Z • ) be a compact filtered abelian group and let X be the associated compact group nilspace. Then µ C ω (X) (seen as a measure on X N ) is affine-exchangeable if and only if the filtration Z • is 2-homogeneous. Proof. By Lemma 2.26 the measure µ C ω (X) is affine-exchangeable if and only if X is 2homogeneous. But for group nilspaces we know that being 2-homogeneous is equivalent to the associated filtration being 2-homogeneous, by [13,Theorem 3.8]. Appendix C. On cube-surjective continuous morphisms Recall that a map ϕ between two nilspaces X, Y is cube-surjective if for every n ≥ 0 the map ϕ n is surjective from C n (X) onto C n (Y). The goal of this section is to prove the following result. Theorem C.1. Let X, Y be compact profinite-step p-homogeneous nilspaces and suppose that ϕ : X → Y is a continuous cube-surjective morphism. Then ϕ is a fibration. To prove this we shall need the following result about finite nilspaces. Theorem C.2. Let X be a k-step finite nilspace, let Y be a d-step nilspace, and let ϕ : X → Y be a cube-surjective morphism. Then ϕ is a fibration. In particular d ≤ k. Indeed, we first need to prove the following particular case: Proposition C.3. Let X be a k-step finite nilspace, let Y be a d-step nilspace with d > k, and suppose that Y is not (d − 1)-step. Then no morphism ϕ : X → Y can be cubesurjective. Proof. First note that Y must be finite as otherwise the result is trivial (since cubesurjective maps are in particular surjective). For i ∈ [k] and j ∈ [d] let Z i (X), Z j (Y) be the structure groups of X and Y respectively. We want to compute the size of C n (X). Assume that n > k. A possible way of doing this consists in first assigning values to the vertices v ∈ n in an order given by \|v\| := v(1) + · · · + v(n). That is, note that the possible values for c(0 n ) are all possible elements of X. Thus we have \| X \| = \| Z 1 (X)\| · · · \| Z k (X)\| possibilities for c(0 n ). Next, for any v ∈ n with \|v\| = 1 note that again we have X possibilities for each such c(v). By the ergodicity axiom all these maps are morphisms from {v ∈ n : \|v\| ≤ 1} to X (see [10,Definition 3.1.4]). Hence, in order to assign a value to all of these vertices we have \| X \| n = (\| Z 1 (X)\| · · · \| Z k (X)\|) n possibilities. Take now any element w ∈ n with \|w\| = 2. Let us say that v ≤ w for v ∈ n if for all i ∈ [n], v(i) ≤ w(i). The map f : {v ∈ n : v ≤ w} \ {w} → X defined by v → c(v) is then a corner in Cor 2 (X) (identifying {v ∈ n : v ≤ w} with \|w\| ). We are interested in how many x ∈ X there are such that the map {v ∈ n : v ≤ w} → X defined as v → c(v) if v = w and w → x is a cube. Clearly if we have two such values x 1 , x 2 ∈ X then by unique completion in X 1 we have π 1 (x 1 ) = π 1 (x 2 ). But the converse is also true by [10, Lemma 3.2.7] and thus the number of possible x ∈ X that complete f equals the size of the fiber π −1 1 (π 1 (x 1 )) for some possible x 1 ∈ X that completes f . Hence, for each w ∈ n with \|w\| = 2 we have \|π −1 1 (π 1 (x 1 ))\| = \| Z 2 (X)\| · · · \| Z k (X)\| possibilities (this follows from [10, Lemma 3.3.6] and [10, Theorem 3.2.19]). As we have n 2 such possible w ∈ n we have a total of (\| Z 2 (X)\| · · · \| Z k (X)\|) ( n 2 ) possibilities once we have fixed the values for \|v\| ≤ 1. Repeating this process, as n > k we have \| C n (X)\| = \| X \| #{pos. for 0 n } \| X \| n #{pos. for \|v\|=1} (\| Z 2 (X)\| · · · \| Z k (X)\|) ( n 2 ) #{pos. for \|v\|=2} · · · \| Z k (X)\| ( n k ) #{pos. for \|v\|=k} . Similarly, for n > d we have \| C n (Y)\| = \| Y \| n+1 (\| Z 2 (Y)\| · · · \| Z d (Y)\|) ( n 2 ) · · · \| Z d (Y)\| ( n d ) . But it is clear that for n large enough, as Z d (Y) = {id} we have \| C n (X)\| < \| C n (Y)\|, whence ϕ n cannot be surjective. Let us recall the following construction of a nilspace modulo a subgroup of the last structure group. Proposition C.4. [13, Proposition A.20] Let Y be a k-step nilspace and let H < Z k (Y) be any subgroup. Let us define the following relation on Y: for y 1 , y 2 ∈ Y, we write y 1 ∼ y 2 if and only if y 1 = y 2 + h for some h ∈ H. Then the following statements hold: (i ) The relation ∼ is an equivalence relation. (ii ) The setỸ := Y / ∼ together with the sets C n (Ỹ) : = {π ∼ • c : c ∈ C n (Y)} is a nilspace. (iii )Ỹ is k-step, with last structure group Z k (Ỹ) = Z k (Y)/H, andỸ k−1 ≃ Y k−1 . Proof of Theorem C.2. We argue by induction on k. The case k = 0 is trivial. To prove the inductive step, by Proposition C.3 we know that d ≤ k. If d ≤ k − 1 then ϕ factors through π k−1 , i.e. ϕ = ϕ k−1 • π k−1 . As π k−1 is a fibration it follows that ϕ k−1 : X k−1 → Y is cube-surjective and thus by induction we have that ϕ k−1 is a fibration. Hence ϕ is a fibration (as a composition of two fibrations). If d = k, then the morphism ϕ k−1 satisfying π k−1 • ϕ = ϕ k−1 • π k−1 is cube-surjective (similarly as in the previous paragraph), so by induction it is a fibration. Therefore it suffices to check that the last structure homomorphism φ k : Z k (X) → Z k (Y) is surjective. Let H := φ k (Z k (X)) < Z k (Y) and define the nilspaceỸ as in Proposition C.4. Now let ψ : X k−1 →Ỹ be defined as π k−1 (x) → π ∼ (ϕ(x)). We claim that this is a cube-surjective morphism. To see this, we first check that ψ is well-defined: for any z ∈ Z k (X) and x ∈ X we have ψ(π k−1 (x + z)) = π ∼ (ϕ(x + z)) = π ∼ (ϕ(x) + φ k (z)) = π ∼ (ϕ(x)) = ψ(π k−1 (x)). To see that ψ is a morphism note that for every c ∈ C n (X) we have ψ • π k−1 • c = π ∼ • ϕ • c ∈ C n (Ỹ). Now, to see that ψ is cube-surjective, let π ∼ • c * be any element of C n (Ỹ) (where c * ∈ C n (Y)). As ϕ is cube-surjective, there is c ′ ∈ C n (X) such that ϕ • c ′ = c * . Thus ψ • π k−1 • c ′ = π ∼ • ϕ • c ′ = π ∼ • c * . We thus have a cube-surjective morphism ψ from a (k − 1)-step nilspace X k−1 to a kstep nilspaceỸ. By Proposition C.3, the step ofỸ is at most k−1, so Z k (Ỹ) = {0}. But by part (iii) of Proposition C.4, we know that {0} = Z k (Ỹ) = Z k (Y)/H = Z k (Y)/φ k (Z k (X)). This implies that φ k (Z k (X)) = Z k (Y), whence φ k is surjective as required. Proof of Theorem C.1. Note that by [13,Proposition 1.5] we know that all p-homogeneous, cfr (i.e. compact and finite-rank) nilspaces are finite. Hence we can apply Theorem C.2 to any cube-surjecive morphism between cfr p-homogeneous nilspaces of finite step. For the general case of the theorem, suppose ϕ : X → Y is continuous and cubesurjective. Let Y = lim ← − Y i be an inverse limit decomposition of Y with each Y i of finite rank and i-step, and limit maps ψ i,Y [20,Theorem 5.71]. Fix any k ∈ N and note that ψ k,Y • ϕ is a cube-surjective continuous morphism X → Y k . In particular, this map factors through the k-th characteristic factor of X, i.e. ψ k, Y • ϕ = (ψ k,Y • ϕ) k • π k , and (ψ k,Y • ϕ) k is cube-surjective as well. Now let π k (X) = lim ← − (π k (X)) i 14 be an inverse limit decomposition of π k (X) with each (π k (X)) i of finite rank and k-step and factor maps ψ k,X k ,i : π k (X) → (π k (X)) i . By [12,Theorem 1.7], there exists j ∈ N such that (ψ k,Y • ϕ) k = ϕ ′ • ψ k,X,j for some morphism ϕ ′ : (π k (X)) j → Y k . It follows from the cube surjectivity of (ψ k,Y • ϕ) k that ϕ ′ is also cube-surjective. But now, since (π k (X)) j and Y k are both finite-rank, k-step, and p-homogeneous it follows from [12, Proposition 1.5] that these nilspaces are finite. Hence ϕ ′ is a fibration by Theorem C.2, and so (since ψ j,X is also a fibration), we conclude that (ψ k,Y • ϕ) k = ϕ ′ • ψ j,X is a fibration. Finally, note that in particular ψ k,Y • ϕ = (ψ k,Y • ϕ) k • π k is a fibration as well. Now from the previous paragraph we have to deduce that ϕ itself is a fibration. We know that ψ k,Y • ϕ is a fibration for every k. Let c ′ be an n-corner on X and supposẽ c ∈ C n (Y) completes the corner ϕ • c ′ ∈ Cor n (Y). For every k, let Q k denote the set of n-cubes on X completing c ′ and whose image under ψ k,Y • ϕ equals ψ k,Y •c. Note that Q k is non-empty for every k because ψ k,Y • ϕ is a fibration. Note also that Q k is closed (hence compact) by standard facts (it is the intersection of the preimage of the cube ψ k,Y •c under ψ k,Y • ϕ n with the set of cubes completing c ′ , so it is the intersection of two closed sets). Finally, note that Q k ⊃ Q k+1 for all k ≥ 0. It then follows from the finite intersection property that ∩ ∞ k=1 Q k is non-empty. Then any cube c in this intersection completes c ′ (by the closeness of C n (X)), and ϕ • c =c because by construction we have this equality modulo ψ k,Y for every k. This proves that ϕ is a fibration. In this section we prove the following result used in the proof of Theorem 1.7. Theorem D.1. Let X, Y be compact profinite-step nilspaces, and suppose that X is the group nilspace associated with a compact abelian filtered group (Z, Z • ). Let τ : Y → X be 14 In this proof we denote the k-characteristic factor of X by π k (X) to avoid confusion with the sub-indices used for the inverse limit. a Borel map such that τ n preserves the n-cubic Haar measures for all n ≥ 0. Then there is a continuous cube-surjective morphism ϕ : Y → X such that τ = ϕ µ Y -a.e. The proof uses the following stability result, whose main ideas are adapted from [15, §4]. Lemma D.2. Let Y be a compact profinite-step nilspace and let X be a k-step compact group nilspace associated with some compact abelian filtered group (Z, Z • ). Let d be any compatible 15 metric on Z. Then there exists δ = δ(X, d) > 0 such that if ψ : Y → X is a continuous morphism with d 1 (ψ, 0) := d(ψ(y), 0) dµ Y (y) < δ then ψ is constant. Proof. First note that since X is k-step we have d 1 (ψ, 0) = d(ψ k (y), 0) dµ Y k (y) where ψ k : Y k → X is such that ψ = ψ k • π k . Hence it suffices to prove the result assuming that Y is k-step as well. Let δ 0 > 0 be a constant to be fixed later. Note that by Markov's inequality, the set of points y ∈ Y such that d(ψ(y), 0) > δ 1/2 0 has Haar measure at most δ 1/2 0 . Fix some y 0 ∈ Y and consider the set C k+1 y 0 (Y) := {c ∈ C k+1 (Y) : c(0 k+1 ) = y 0 }. For every v ∈ k + 1 \ {0 k+1 }, the coordinate projection p v : C k+1 y 0 (Y) → Y, c → c(v) preserves the Haar measure (see [11,Lemma 2.2.17] for the definition of the Haar measure on C k+1 y 0 (Y), and for the measure-preserving property of the projection apply [11, Lemma 2.2.14] with the good pair P 1 = {0 k+1 }, P 2 = {v} inside P = k + 1 ). Thus δ In particular if δ 0 is small enough (depending only on k) there exists c y 0 ∈ C k+1 y 0 (Y) such that d(ψ(c y 0 (v)), 0) ≤ δ 1/2 0 for all v ∈ k + 1 \ {0 k+1 }. Now we recall that by [11, Lemma 2.1.12] the map K : Cor k+1 (X) → X that sends a corner to its unique completion is continuous. As both Cor k+1 (X) and X are compact spaces, this function is uniformly continuous and thus for every ǫ > 0 there exists τ > 0 such that for every c ′ ∈ Cor k+1 (X) if d(c ′ (v), 0) < τ for all v ∈ k + 1 \ {0 k+1 } we have d(K(c ′ ), 0) < ǫ. Then, for some ǫ 0 > 0 to be fixed later, let δ 0 > 0 be small enough (depending only on X and k) so that δ 1/2 0 < τ 0 (ǫ 0 ). Then, by the above facts applied to ψ • c y 0 \| k+1 \{0 k+1 } ∈ Cor k+1 (X), we have d(K(ψ • \| k+1 \{0 k+1 } ), 0) < ǫ 0 . But X is k-step and clearly ψ • c y 0 is a cube completing ψ • c y 0 \| k+1 \{0 k+1 } . Hence K(ψ • c y 0 \| k+1 \{0 k+1 } ) = 15 Meaning that the metric generates the compact second-countable topology on Z. ON F ω 2 -AFFINE-EXCHANGEABLE PROBABILITY MEASURES 55 ψ(y 0 ) and therefore d(ψ(y 0 ), 0) < ǫ 0 . Let us emphasize that for any ǫ 0 we can choose an appropriate δ 0 = δ 0 (X, k) such that the previous bound holds for every y 0 ∈ Y. We now choose ǫ 0 small enough (depending only on X and k) in such a way that [12, Lemma 3.4] is satisfied. Hence ψ is constant and the result follows with δ = δ 0 . Proof of Theorem D.1. We start with the assumption that there is a Borel map τ : Y → X such that for every integer n ≥ 0 we have µ C n (Y) •(τ n ) −1 = µ C n (X) , and our aim is to prove that there is a continuous cube-surjective morphism Y → X that is equal to τ almost surely relative to µ Y . Recall that the nilspace X is the abelian group nilspace generated by some filtered abelian group (Z, Z • = (Z (k) ) ∞ k=0 ). By standard results Z is the inverse limit (in the category of abelian groups) of compact abelian Lie groups, Z = lim ← − Z i . Let p i : Z → Z i be the projection maps. It can be shown that then the nilspace X is the inverse limit of compact nilspaces that we shall denote by X i (i ≥ 1), where X i is the abelian group nilspace associated with the filtered group (Z i /p i (Z (i) ), (p i (Z (j) )/p i (Z (i) )) ∞ j=0 ). Note that each X i is an i-step cfr nilspace, and let us denote by ψ i the limit map X → X i . Hence X = lim ← − X i (as nilspaces). Note that ψ j • τ : Y → X j is Borel measurable and µ C n (X j ) = µ C n (X) •(ψ n j ) −1 = µ C n (Y) •(τ n ) −1 •(ψ n j ) −1 = µ C n (Y) •((ψ j • τ ) n ) −1 for all n ≥ 0. We shall now prove that ψ j • τ agrees µ Y -almost-surely with some cube-surjective continuous morphism. Following [15, §4], for compact profinite-step nilspaces X, Y with a fixed metric d X compatible with the topology on X, let us say that a map φ ′ : Y → X is a (δ, 1)quasimorphism of degree k − 1 if it is a Borel map with the following property: µ C k (Y) c ∈ C k (Y) : ∃ c ′ ∈ C k (X), ∀ v ∈ k , d X φ ′ • c(v), c ′ (v) ≤ δ ≥ 1 − δ.(22) For Borel maps φ, ψ : Y → X we define d 1 (φ, ψ) := Y d X φ(x), ψ(x) dµ Y (x). Fix any j ≥ 0. For any δ > 0 note that ψ j • τ : Y → X j is a (δ, 1)-quasimorphism of degree j. As X j are abelian group nilspaces, we fix a metric d X j on X j that is invariant under addition. By [15,Theorem 4.2] there is a continuous morphism φ δ,j : Y → X j such that d 1 (ψ j • τ, φ δ,j ) ≤ ε, where ε(δ) → 0 as δ → 0. Applying this for each δ n := 1/n, n ∈ N, we obtain a sequence of continuous morphisms φ n,j : Y → X j such that d 1 (ψ j • τ, φ n,j ) → 0 as n → ∞. By the triangle inequality this implies that d 1 (φ m,j , φ n,j ) = d 1 (φ m,j −φ n,j , 0) → 0 as m, n → ∞. By Lemma D.2 we have that φ m,j − φ n,j is constant for n, m ≥ N 0 for some large N 0 = N 0 (X j ). In particular, letting z j,n,m := φ m,j − φ n,j for n, m ≥ N 0 , by compactness of X j we can assume (passing to a subsequence if necessary and relabeling it as (z j,n,m )), that z j,N 0 ,m → z * j,N 0 as m → ∞. Then d 1 (ψ j • τ, φ m,j ) = d 1 (ψ j • τ, φ N 0 ,j + (φ m,j − φ N 0 ,j )), where the left hand side converges to 0 as m → ∞ by the previous paragraph, and the right hand side converges to d 1 (ψ j • τ, φ N 0 ,j + z * j,N 0 ) by construction (and the dominated convergence theorem). Thus ψ j • τ = φ N 0 ,j + z * j,N 0 µ Y -a.e.. Let γ j := φ N 0 ,j + z * j,N 0 and note that this is a continuous morphism Y → X j by construction. We now glue all these maps γ j into a single continuous cube-surjective morphism γ : Y → X. The reduces to proving that the maps γ j are consistent with the given inverse limit expression X = lim ← − X j , that is, given the corresponding factor maps ψ i,j : X j → X i in this inverse limit, we want to prove that for every j ≥ i ≥ 0 we have γ i = ψ i,j • γ j . Since ψ j : X → X j are the limit maps, we have ψ i = ψ i,j • ψ j . Thus ψ i • τ = ψ i,j • ψ j • τ but ψ i • τ = γ i and ψ j • τ = γ j µ Y -a.e.. Hence γ i = ψ i,j • γ j µ Y -a.e.. As X j is j-step, note that {y ∈ Y : γ i (y) = ψ i,j • γ j (y)} = π −1 j ({ỹ ∈ π j (Y) : (γ i ) j (ỹ) = ψ i,j •(γ j ) j (ỹ)}) where π j (Y) is the j-th characteristic factor 16 of Y, (γ i ) j : π j (Y) → X i and (γ j ) j : π j (Y) → X j . 17 As π j preserves the Haar measure we have that (γ i ) j = ψ i,j •(γ j ) j µ π j (Y) -a.e.. Therefore (γ i ) j and ψ i,j •(γ j ) j are continuous functions that agree µ π j (Y) -almost-surely. If there was a pointỹ 0 ∈ π j (Y) such that (γ i ) j (ỹ 0 ) = ψ i,j •(γ j ) j (ỹ 0 ) by continuity we would have an open set U ∋ỹ 0 such that (γ i ) j (ỹ) = ψ i,j •(γ j ) j (ỹ) for allỹ ∈ U. This would contradict the almost-sure equality just established, because µ π j (Y) (U) > 0 by [11,Proposition 2.2.11]. Hence γ i = ψ i,j • γ j , and we can now define γ : Y → X = lim ← − X i as y → (γ j (y)) ∞ j=1 . Clearly this is a continuous morphism. It remains to check that γ is cube-surjective. First we claim that γ = τ µ Y -a.e.. In order to prove this note that {y ∈ Y : τ (y) = γ(y)} = ∪ ∞ j=1 {y ∈ Y : ψ j • τ (y) = γ j (y)}. As the latter set has measure 0 for every j, the claim follows. Next we claim that for all n ≥ 0, µ C n (Y) •(γ n ) −1 = µ C n (X) . Indeed note that by the consistency axiom for cubic couplings it follows that γ n = τ n µ C n (Y) -a.e., and thus we have the desired measure preserving property. Finally we want to prove that for every n ≥ 0 we have γ n (C n (Y)) = C n (X). Suppose for a contradiction that there exists c ∈ C n (X) which is not in the image of γ n . This readily implies that for some j ≥ 1 we have π j • c / ∈ (γ) n j (C n (π j (Y))) where (γ) j : π j (Y) → π j (X) is the map such that π j • γ = (γ) j • π j (this map (γ) j is not to be confused with the map γ j : Y → X j from the previous paragraph). As (γ) n j is continuous and C n (π j (Y)) is compact, we have that (γ) n j (C n (π j (Y))) is a compact and hence closed set. If π j • c / ∈ (γ) n j (C n (π j (Y))) then there must exists an open set V ∋ π j • c in C n (π j (X)) such that V ∩ (γ) n j (C n (π j (Y))) = ∅. But then µ C n (π j (X)) (V ) > 0 by (an argument similar to) [11, Lemma 2.2.11] and µ C n (π j (Y)) •((γ) n j ) −1 (V ) = µ C n (π j (Y)) (∅) = 0. Since for every j we have that µ C n (π j (X)) = µ C n (π j (Y)) •((γ) n j ) −1 (which follows by composing µ C n (Y) •(γ n ) −1 = µ C n (X) with (π n j ) −1 on both sides), we have a contradiction. 16 Here we denote by π j (Y) the j-th characteristic factor of Y to avoid the confusion with X j , which in this argument is not necessarily the j-th characteristic nilspace factor of X. 17 As γ i : Y → X i is a morphism and X i is i-step, this map factors through the j-th characteristic factor of Y for j ≥ i. We denote this map by (γ i ) j : π j (Y) → X i . Similarly for (γ j ) j . Definition E.7 (Consistency, general groups). Let L = {L 1 , . . . , L m } be a system of linear forms, let d ∈ Z ∞ >0 , k ∈ Z ∞ ≥0 and G = ∞ j=1 ( 1 2 n j · Z)/Z for some (n j ∈ Z ≥0 ) j≥1 . We say that a sequence of elements b 1 , . . . , b m ∈ G is consistent with L if for every j ∈ N, b 1 (j), . . . , b m (j) ∈ ( 1 2 n j · Z)/Z ⊂ T is (d(j), k(j))-consistent with L. Let G = ∞ j=1 ( 1 2 n j · Z)/Z for some (n j ∈ Z ≥0 ) j≥1 , and let d ∈ Z ∞ >0 , k ∈ Z ∞ ≥0 . Then the set of points H ⊂ G L that are consistent with L forms a closed subgroup. Furthermore, there exists a 2-homogeneous filtration G • on G such that the group nilspace generated by (G, G • ) equals ∞ j=1 U k(j),ℓ(j) for some ℓ = ℓ k,d ∈ Z ∞ Proof of Lemma E.8. Let b 1 , . . . , b m ∈ G be a sequence of elements consistent with L. Fix any j ∈ N. By Corollary E.6 this means that there exists P j ∈ hom(D 1 (F n 2 ), U k(j),ℓ(j) ) and a point x ∈ (F n 2 ) s such that P j (L i (x)) = b i (j) for all i ∈ [m]. Note that since the L i are affine forms, the values L i (x) are the same thing as images A(L ′ i ) where L ′ i ⊂ F s−1 2 and A ∈ C s−1 (F n 2 ) (recall that L i = (1, L ′ i )). Hence in this case P j (L i (x)) = P j (A(L ′ i )). But as P j • A ∈ hom(D 1 (F s−1 2 ), G), this condition reduces to saying that there exists P ′ j := P j • A ∈ hom(D 1 (F s−1 2 ), G) such that P ′ j (L ′ i ) = b i (j) for all i ∈ [m]. As this holds for every j ∈ N we have proved that the map T : hom(D 1 (F s−1 2 ), G) → G L , f → (f (L ′ 1 ), . . . , f (L ′ m )) is surjective onto H. Moreover, as T is continuous and hom(D 1 (F s−1 2 ), G) is compact, we have that H is a closed group as well. And the fact that T is a surjective homomorphism onto H implies that the Haar measure on H is the pushforward of the Haar measure of the group hom(D 1 (F s−1 2 ), G). F k 2 2× {0 N\[k] } ⊂ F ω 2 for some k ∈ N (where 0 N\[k] denotes the constant 0 sequence with coordinates indexed by N \ [k] Example 1 . 1 . 11Let B be a standard Borel space and fix any function m : Z 2 → P(B). Let G denote the closed subgroup of the compact abelian group Z N 2 consisting of what we can view as infinite-dimensional cubes. More precisely, the elements of G are the functions c : N → Z 2 of the form c Theorem 1 . 8 . 18For any compact metric space B, the nilspace H is a limit domain for convergent sequences of functions (f n : F n 2 → B) n∈N . For the proofs of the claims in this definition, see [10, §3.2] and [11, §2.1]. These characteristic nilspace factors enable us to view nilspaces as iterated abelian bundles, in the sense of the following construction (see [10, Definition 3.2.17]). is a compact Z-bundle over S if in addition to the previous assumptions we have that α is continuous and U ⊂ S is open if and only if π −1 (U) ⊂ B is open. We say that B is a compact k-fold abelian bundle if for every i ∈ [k] the factor B i is a compact Z i -abelian bundle over B i−1 . Composing with a suitable morphism φ : N → N if necessary, we can assume without loss of generality that v = 0 N (if the non-zero coordinates of v are indexed by some finite V ⊂ N, we can set φ(w) := w(i) if i / ∈ V and φ(w) := 1 − w(i) otherwise). an injective linear map. Thus, if we have the composition property for such maps, then c Theorem 4 . 3 . 43Let X be a p-homogeneous compact profinite-step nilspace. Then there exists a fibration ϕ : H p → X. First we record the following fact about how the nilspace factor maps interact with translations on a nilspace. Translations are a basic and important kind of transformations on a nilspace, which can be thought of as generalizations of the translations on a nilpotent group that consist in multiplying by fixed elements. For the basic background concerning the groups of translations Θ i (X) of a nilspace X, we refer to [10, §3.2.4]. Proposition 4 . 4 . 44Let X ∈ Q p,k and let π k−1 : X → X k−1 be the projection to the (k − 1)step nilspace factor. Then for any translation Lemma 5. 3 . 3Let (B, B) be a standard Borel space, and let I = µ ∈ P(B N ) : µ has the independence property . Fix any ε > 0 . 0For i = 1, 2, by [40, Appendix E8] the continuous functions are dense in the L 1 -norm on the probability space ( F m 2 2(recall Definition 3.2). Indeed, for the forward implication, since µ is cubic-exchangeable, by [14, Proposition 6.10] there is some measurable map q : V → B from some standard Borel space V such that µ =μ •(q N ) −1 for some weak cubic couplingμ on V N . Furthermore, from the proof of [14, Proposition 6.10] if N = E ⊔ O where E and O are the sets of even and odd numbers respectively, we have that V := B O . Then we define the coupling ν ′ on V E as the measure µ, using that V E = B N . Given this, we then define q : V → B as the projection on the coordinate 0 O . If the measure µ is affine-exchangeable, then it follows that ν ′ is also affine-exchangeable. The measureμ is then defined as ν ′ composed with doubling the coordinate-indices in N (recall that ν ′ is a measure defined on B E and we need a measure on B N ). The result then follows similarly as in the proof of [14, Proposition 6.10]. restricted to sets of the form n × {0 N\[n] } ⊂ N ). As 2-homogeneous cubic couplings are affine exchangeable (as explained above), and cubic couplings have the independence property (by[14, Lemma 6.15]), this proves statement (ii).To prove statement (i) and thus complete the proof, we now obtain an adaptation of[14, Theorem 6.17] where the assumption is strengthened by replacing cubicexchangeability with affine-exchangeability, and the conclusion is strengthened by adding 2-homogeneity to the resulting cubic coupling. Finally, we apply an adaptation of[14, Lemma 6.18] where the assumption is strengthened by adding the 2-homogeneity property, and as a result we may use Theorem 3.3 from this paper instead of[14, Theorem 4.1], thus obtaining that the resulting nilspace X is 2-homogeneous, as required.We complete the proof of Theorem 1.5 by explaining how it follows from Theorem 5.4.Proof of Theorem 1.5. Theorem 5.4 implies that for every µ ∈ Pr Aff(F ω 2 ) (B N ) there is a Borel probability measure κ on P(B N ), concentrated on the set of measures of the form ν = ζ X,m for X a 2-homogeneous profinite-step nilspace X and Borel map m : X → P(B), such that µ is the mixture µ = P(B N ) ν dκ(ν). Proposition 6 . 1 . 61Let B be a standard Borel space, and let µ ∈ P(B N ). The following statements are equivalent.(i ) µ is an extreme point in the convex set of cubic-exchangeable measures in P(B N ). (ii) ⇒ (iii): this is given by[14, Theorem 6.7 (i)], more specifically by combining[14, Theorem 6.17 and Lemma 6.18]. Note that in the forward implication, the fact that the obtained nilspace X can be assumed to be profinite-step follows from the main structure theorem[14, Theorem 4.1], where the compact nilspace X is obtained as an inverse limit of its characteristic factors (see[14, Remark 4.3 and Definition 3.41]). ( ii) ⇒ (i): suppose that µ has the independence property, and suppose for a contradiction that µ is not an extreme point, i.e., that there exist distinct cubic-exchangeable measures ν 1 , ν 2 ∈ P(B N ) and t ∈ (0, 1) such that µ = tν 1 + (1 − t)ν 2 . By [14, Theorem 6.7 (ii)], each ν i is a mixture of cubic-exchangeable measures with the independence property, i.e. there exist Borel probability measures κ 1 , κ 2 concentrated on I such that ν i = P(B N ) λ dκ i (λ) for i = 1, 2. Note that κ 1 = κ 2 (otherwise ν 1 = ν 2 ). Then κ := tκ 1 + (1 − t)κ 2 is a Borel probability measure on P(B N ) concentrated on I such that µ = P(B N ) λ dκ(λ). Note that κ is not an extreme point of the space of Borel probability measures on P(B N ), so in particular it is not a Dirac measure. On the other hand, since µ and κ-almost-every λ have the independence property, arguing as in the proof of[14, Theorem 6.17] we deduce that κ-almost surely we have λ = µ, so κ is concentrated on a single point in P(B N ), which yields a contradiction since κ is not a Dirac measure. (i) ⇒ (ii): suppose that µ is an extreme point among cubic-exchangeable measures, and note that by [14, Theorem 6.7 (ii)] we have µ = P(B N ) ν dκ(ν) for some Borel probability measure κ on P(B N ) concentrated on cubic-exchangeable measures with the independence property. Then κ is a Dirac measure (otherwise there is a Borel set A such that κ(A) =: t ∈ (0, 1), and then µ = tν 1 + (1 − t)ν 2 , where the probability measures ν 1 = t −1 A ν dκ(ν) and ν 2 = (1−t) −1 P(B N )\A ν dκ(ν) are cubic-exchangeable, contradicting that µ is an extreme point). Hence µ has the independence property.Proposition 6.1 entails the following counterpart for affine-exchangeable measures, involving the compact group nilspace H from Definition 1.4. Proposition 6.2. Let B be a standard Borel space, and let µ ∈ P(B N ). The following statements are equivalent. (i ) µ is an extreme point in the convex set Pr Aff(F ω 2 ) (B N ).(ii ) µ is in Pr Aff(F ω 2 ) (B N ) and satisfies the independence property. (iii ) µ = ζ H,m for some Borel map m : H → P(B).Proof. (i) ⇔ (iii). If µ = ζ H,m then by the implication (iii) ⇒ (i) in Proposition 6.1 the measure µ is an extreme point among cubic-exchangeable measures, and is therefore an extreme point in Pr Aff(F ω 2 ) (B N ) (since every measure in the latter set is cubic-exchangeable). Conversely, suppose that µ in an extreme point in Pr Aff(F ω 2 ) (B N ). By Theorem 1.5 µ is a mixture of measures of the form ζ H,m . Since µ is an extreme point, arguing as in the proof of Proposition 6.1 we see that the measure κ underlying this mixture is a Dirac measure, so µ itself is equal to ζ H,m for some Borel map m : H → P(B). (i) ⇔ (ii). The backward implication here follows from the implication (ii) ⇒ (i) in Proposition 6.1. Conversely, if (i) holds then by the previous paragraph (iii) holds, and then by the implication (iii) ⇒ (ii) from Proposition 6.1 we have that µ satisfies the independence property. Remark 6 . 3 . 63Via the well-known characterization of ergodic measures as the extreme points of the space of Γ-invariant measures, Proposition 6.2 also gives characterizations of the ergodic Γ-invariant measures in P(B N ) for Γ = Aff(F ω 2 ). Definition 7 . 3 . 73Let B be a compact metric space. We say that a sequence of functions (f n : F n 2 → B) n∈N is convergent if for every finite set L ⊂ F ω 2 the measures µ L,fn converge in the vague topology as n → ∞. Note that the special case B = {0, 1} agrees with the case d = ∞ of [25, Definition 3.1]. (f n : F n 2 → B) n∈N if for every such sequence there exists a Borel map m : X → P(B) such that for every finite set L ⊂ F ω 2 the measures µ L,fn converge vaguely to the pushforward of ζ X,m under p L , i.e. to the measure defined by ζ X,m • p −1 L (S) = C ω (X) L∈L m(c(L)) (S) dµ C ω (X) (c), for any Borel set S ⊂ B L . (14) F ω 2 → 2B, and any fixed k such thatL ⊂ F k 2 ×0 N\[k] , we define T f,L,n : C k (F n 2 ) → B L to be the map A → (f • A)\| L ,and for A ∈ Aff(F k 2 ) we define the map φ L,A : B L → B AL by φ L,A (r)(A(L)) := r(L) for any L ∈ L (where AL := {A(L) : L ∈ L}). as required. Finally, we prove that µ satisfies the independence property. Similarly as in the proof of Lemma 5.3, it suffices to prove that for any pair of finite independent faces L 1 , L 2 ⊂ F ω 2 and continuous functions t i : B L i → C for i = 1, 2 we have there must be some s ∈ {d 1 + 1, . . . , d 1 + d 2 } such that for all j ∈ [J] we have that v j s is a non-zero constant c independent of j ∈ [J]. Hence, in (20) we can eliminate the variable x in the second function f n , by the simple change of variables a s → a s − c −1 x, thus showing that this average equals 8. 1 1, each measure µ s in Definition 8.2 is concentrated on f −1 ({s}) (even though it is presented as a measure on B N ), and f −1 ({s}) is measure-theoretically equivalent to B (indeed f −1 ({s}) = B ×{s}). Thus, in particular, the assumption that µ s is a Dirac measure here implies that µ s can be viewed as δ z ∈ P(B) for some z ∈ B. Proof of Theorem 1.7. (i) ⇒ (ii). It suffices to prove (ii) for Y = H. Indeed, if we have this, then for any other 2-homogeneous compact profinite-step nilspace Y, by Theorem 4.3 there is a fibration ϕ ′ : H → Y, so with the fibration ϕ : X → H that we already have we obtain a fibration ϕ ′ • ϕ : X → Y which proves (ii). Let m : H → P(H) be the map x → δ x . Then ζ H,m = µ C ω (H) is an affine-exchangeable measure on H N with the independence property, so by (i) there is a Borel map m ′ : X → P(H) such that ζ H,m = ζ X,m ′ . By Proposition 8.4, there exists a Borel map τ : X → H that preserves the n-cubic Haar measures for all n ≥ 0. By Theorem D.1 the map τ agrees µ X -almosteverywhere with a continuous cube-surjective morphism ϕ : X → H, and by Theorem C.1 the map ϕ is a continuous fibration. (ii) ⇒ (i). If X satisfies (ii) then in particular there is a fibration ϕ : X → H. By Theorem 1.5 the nilspace H represents Pr Aff(F ω 2 ) (B N ) for every standard Borel space B. By Proposition 2.21 the map ϕ N preserves the Haar measures on C ω (Y), C ω (H), whence for any Borel map m : H → P(B) we have ζ H,m = ζ X,m • ϕ , so X also represents affine-exchangeability. Definition A.1 (∞-fold abelian bundle). A set B is an ∞-fold abelian bundle if there exists a sequence of sets (B i ) i≥0 and a sequence of maps π j,i : B i → B j for 0 ≤ j ≤ i such that the following properties hold. For every i ≥ 1 there exists an abelian group Z i such that B i is a Z i -bundle over B i−1 with projection π i−1,i (in the sense of [10, Definition 3.2.17]) Definition A. 2 2(∞-fold compact abelian bundles). An ∞-fold compact abelian bundle is an ∞-fold abelian bundle B = lim ← − B i such that B i is a compact abelian Z i -bundle over B i−1 (in the sense of [11, Definition 2.2.6]) and B is equipped with the compact topology generated by the limit maps π i . Remark A.3. Note that in particular this implies that B is a compact space and the maps π i : B → B i are continuous and open for all i ≥ 0. We can now define the desired notion of Haar measure. Lemma A.4 (Haar measure on ∞-fold compact abelian bundles). Let B = lim ← − B i be an ∞-fold compact abelian bundle. Then there exists a unique measure µ B on B such that for every i ≥ 0 we have µ B • π −1 i = µ B i where µ B i is the Haar measure on the i-fold compact abelian bundle B i (constructed in [11, Proposition 2.2.5]). Lemma A. 6 . 6Let B, B ′ be two ∞-fold compact abelian bundles and let ϕ : B → B ′ be a continuous totally surjective bundle morphism. Then µ B ′ = µ B • ϕ −1 . Proof. By the Riesz representation theorem it suffices to check that for any continuous function f : B ′ → C we have f dµ B ′ = f • ϕ dµ B . By the Stone-Weierstrass theorem, the continuous functions of the form f ′ • π i , for i ≥ 0 and f ′ : B ′ i → C continuous, are dense in the space of continuous functions B → C. Therefore it suffices to check that and by construction this is µ C n (X) •(π n i ) −1 . To verify the conditional independence axiom, we shall need some preparation. The following result generalizes [11, Lemma 2.1.10]. Recall from [10, Definition 3.1.3] the notion of a subset of n having the extension property.Lemma A.10. Let X be a compact profinite-step nilspace, and let n ≥ 0. Let P ⊂ n be a set with the extension property in n and S ⊂ P be a set with the extension property in P . Let f : S → X be a morphism. Then the set hom f (P, X) := {c ∈ hom(P, X) : c \| S = f } is an ∞-fold compact abelian sub-bundle of X P with factors hom π i • f (P, X i ) and structure groups hom S→0 (P, D i (Z i )) for all i ≥ 1, where Z i is the i-th structure group of X.Proof. This follows from [11, Lemma 2.1.10].Recall from [11, Definition 2.2.13] that two subsets P 1 , P 2 of a discrete cube n are said to form a good pair if the following conditions are satisfied: both P 1 and P 1 ∩ P 2 have the extension property in n (see[10, Definition 3.1.3]), and for any k ≥ 1, any abelian group Z, and every morphism f ′ : P 2 → D k (Z) (see [10, Definition 2.2.30]) satisfying Appendix D. Maps preserving the cubic Haar measures are essentially cube-surjective morphisms ) = µ C k+1 y 0 (Y) • p −1 v ({y ∈ Y : d(ψ(y), 0) 0 (Y) ({c ∈ C k+1 y 0 (Y) : ∀ v ∈ k + 1 \ {0 k+1 }, d(ψ(c(v)), 0) µ C k+1 y 0 (Y) ({c ∈ C k+1 y 0 (Y) : ∃ v ∈ k + 1 \ {0 k+1 }, d(ψ(c(v)), 0) Lemma E. 8 . 8Let L = {L 1 , . . . , L m } ⊂ F s 2 × {0} N\[s] be a finite set of affine linear forms where L i = (1, L ′ i ) for all i ∈ [m]and s ≥ 1. >0 and the map T :hom(D 1 (F s−1 2 ), G) → G L , f → (f (L ′ 1 ), . . . , f (L ′ m )) isa surjective homomorphism onto H and thus the Haar measure on H equals the pushforward of the Haar measure on hom(D 1 (F s−1 2 ), G) under the map T . Remark E.9. Note that this result applies in particular to the group G ∞ used in [25, Definition 3.2]. 10, Theorem 3.2.19], we have f n 13 , 13Proposition 4.3 and Lemma 4.4] we have that T * is a split extension of X k−1 . Then [10, Proposition 3.3.39] tells us that there exists ). By the Doob property for Polish spaces [14, Lemma 2.17] , for i = 1, 2 there is B ⊗F i -measurable function for every i ≥ 1. We can now generalize the quotient integral formula [11, Lemma 2.2.10]. Lemma A.7. Let B, B ′ be ∞-fold compact abelian bundles and let ϕ : B → B ′ be a continuous totally surjective bundle morphism. Then for any Borel set E ⊂ B we haveB The set of such measures is shown to be Borel in Lemma 5.3. A face map φ : n → N is an injective morphism such that for some S ∈ N n , the coordinate φ(v)(i) is a constant function of v for every i ∈ S. For R 1 , R 2 ⊂ S, the notation R 1 ⊥ R 2 here denotes orthogonality of the subspaces L 2 (p −1 Ri (A ⊗Ri )), i = 1, 2 inside L 2 (A ⊗S ); see[14, Definition 2.29]. Recall that H is a filtered group with filtration H (j) for j ∈ N by Definition 1.4. -AFFINE-EXCHANGEABLE PROBABILITY MEASURES -AFFINE-EXCHANGEABLE PROBABILITY MEASURES -AFFINE-EXCHANGEABLE PROBABILITY MEASURES 57 Originally in[25] f n would take values in F mn p for some sequence m n → ∞ as n → ∞. For simplicity we have restricted ourselves to the case m n = n. Appendix E. Translation between polynomial and nilspace viewpointsIn this appendix our final goal is to explain in more detail how the notion of limit object for convergent sequences from[25]relates to the definition of limit domain in Section 7.The former definition is phrased using polynomials (see e.g. Definition 3.2 in[25]), and thereby pertains to an approach using polynomials and related tools used previously in[6,7,42,43], whereas the definitions in Section 7 involve the nilspace approach, which in the characteristic-p setting was deployed in[13]. Thus, towards our final goal in this appendix we also take the opportunity to explain how certain key concepts from these two approaches can be related to each other.First it is worth recalling an important relation between the notion of polynomial map and that of a nilspace morphism, which is that the two concepts are equivalent when the nilspaces involved are group nilspaces (see[10, §2.2.2]). In the more specific characteristic-p setting that occupies us here, let us begin by recalling from [43, Definition 1.1] the concept of non-classical polynomials on vector spaces over F p .Definition E.1 (Non-classical polynomials over F p ). Let p be a prime, let n, d ≥ 0 be integers, and let G be an abelian group. A function P :This notion is of particular interest when G = T. The image of any such polynomial P : F n p → T is included in a coset of ( 1 p r · Z)/Z ⊂ T. The least r ≥ 0 with such property is called the depth of the polynomial P (see[25,Lemma 2.5]).Our first observation relating the polynomial and nilspace languages here is the following lemma.Lemma E.2. A function P : F n p → T is a polynomial of degree ≤ k and depth r with P (0) = 0 if and only if P is a morphism from D 1 (F n p ) to the group nilspace U k,ℓ (embedded in T) where ℓ = k − (r + 1)(p − 1).To prove this we use the following description of morphisms from Z n p into the circle group.Proof. Suppose that φ(x 0 ) = 0. Since translation by x 0 is a nilspace automorphism on D 1 (Z n p ), we can suppose without loss of generality that x 0 = 0.We argue by induction on k. For k = 1, the assumptions imply that φ is a group homomorphism Z n p → T, so its image is a subset of H (0) := ( 1 p · Z)/Z and φ is a morphism from D 1 (Z n p ) to the nilspace determined by the standard degree-1 filtration H (1)We have thus shown that pφ ∈ hom(D 1 (Z n p ), UWe now define the following p-homogeneous filtration H • of degree at most k in T:where note that since (U. Also, note that H • is ℓ-fold ergodic for the same ℓ for which U +(p−2) k−1,ℓ ′ is ℓ-fold ergodic. But the latter filtration's structure implies that its ergodicity index ℓ satisfies k − (p − 1) − ℓ = r(p − 1) for some integer r, which is equivalent to k − ℓ = (r + 1)(p − 1), which means that ℓ = ℓ k,p as required. It follows that H • = U k,ℓ .Thus, from (23) we have that the derivative ∂ h i · · · ∂ h 1 φ takes values in H (k) (i) for each i ∈ [1, k] and any elements h 1 , . . . , h i ∈ Z n p , and for i > k this derivative vanishes since φ is a morphism into D k (T). Hence φ ∈ hom(D 1 (Z n p ), U k,ℓ ) as required. In order to prove the last part of the proposition, note that by[13,Theorem 3.8]or[43,Lemma 1.6]we can write φ(v 1 , . . . , v n ) = t∈{0,...,p−1} n w \|t\| \|v 1 \|p t 1 · · · \|vn\|p tn where t = (t 1 , . . . , t n ), \|t\| := t 1 + · · · + t n , \|x\| p ∈ {0, . . . , p − 1} is the residue class of x ∈ Z modulo p, and some coefficients w \|t\| ∈ (U k,ℓ ) \|t\| . If φ(Z n p ) ⊂ p s U k,ℓ (where here we regard U k,ℓ as the group (Z/p ⌊ k−ℓ p−1 ⌋+1 Z)), let w * \|t\| ∈ U k,ℓ+s(p−1) be such that p s w * \|t\| = w \|t\| for each t ∈ {0, . . . , p − 1} n (where note that multiplying by p s is a group homomorphism from U k,ℓ+s(p−1) to U k,ℓ , seen as abelian groups). Then the map φ * (v 1 , . . . , v n ) := t∈{0,...,p−1} n w * \|t\| \|v 1 \|p t 1 · · · \|vn\|p tn can be proved to be in hom(D 1 (Z n p ), U k,ℓ+s(p−1) ) and satisfies p s φ * = φ. The result follows.Proof of Lemma E.2. For the forward implication, from the general correspondence between polynomials and morphisms [10, §2.2.2] we deduce that P ∈ hom(D 1 (F n p ), D k (T)), and then Proposition E.3 gives us that P ∈ hom(D 1 (F n p ), U k,ℓ ) (and the claim about the depth r is clear from its definition). The backward implication is clear by the aforementioned correspondence between morphisms and polynomials.Non-classical polynomials were used in[25]to define a notion of consistency that underpins the notion of convergence for sequences of functions (f n : F n p → {0, 1}) n∈N used in that paper.18For p = 2 this notion will be shown below to coincide with the one given by Definition 7.3 (for B = {0, 1}). In order to explain this we first need to set up some notation. From now on, we focus on the case p = 2. Furthermore, we assume that all non-classical polynomials P : F n 2 → T satisfy 0 ∈ P (F n 2 ) and thus P : F n 2 → ( 1 2 r · Z)/Z for some r ≥ 0. This can be done without affecting the main results of this section.We denote by F ω 2 the set of linear forms and by 1 ×F ω 2 := {v ∈ F ω 2 : v(1) = 1} ⊂ F ω 2 the subset of affine linear forms (those whose first coordinate equals 1). As done previously in this paper, we abuse the notation and view F k 2 as the subsetNote that any L ∈ F k 2 can be viewed as a linear map L :Let us now recall the above-mentioned notion of consistency from[25,Definition 3.3]. The corresponding version of this definition for nilspaces is then the following.Definition E.5 (Consistency, nilspace version). Let L = {L 1 , . . . , L m } be a finite subset of F ω 2 . A sequence of elements b 1 , . . . , b m ∈ T is (k, r)-consistent with L if there exists a morphism P ∈ hom(D 1 (F n 2 ), U k,ℓ ) for some ℓ = ℓ k,r taking values in ( 1 2 r · Z)/Z, and a point x ∈ (F n 2 ) s such that P • L i (x) = b i for every i ∈ [m], where s is such that L ⊂ F s 2 .Corollary E.6. Definitions E.4 and E.5 are equivalent.Proof. 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Rudin, Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No. 12 Interscience Publishers, New York-London 1962. Limits of functions on groups. B Szegedy, Trans. Amer. Math. Soc. 370B. Szegedy, Limits of functions on groups, Trans. Amer. Math. Soc. 370 (2018), 8135-8153. The inverse conjecture for the Gowers norm over finite fields via the correspondence principle. T Tao, T Ziegler, Anal. PDE. 3T. Tao and T. Ziegler, The inverse conjecture for the Gowers norm over finite fields via the corre- spondence principle, Anal. PDE 3 (2010), 1-20. The inverse conjecture for the Gowers norm over finite fields in low characteristic. T Tao, T Ziegler, Ann. Comb. 16T. Tao and T. Ziegler, The inverse conjecture for the Gowers norm over finite fields in low charac- teristic, Ann. Comb. 16 (2012), 121-188. Spain Email address: [email protected] MTA Alfréd Rényi Institute of Mathematics, Reáltanoda utca 13-15. Madrid; Budapest, Hungary, H-1053Universidad Autónoma de Madrid and ICMAT, Ciudad Universitaria de CantoblancoUniversidad Autónoma de Madrid and ICMAT, Ciudad Universitaria de Cantoblanco, Madrid 28049, Spain Email address: [email protected] MTA Alfréd Rényi Institute of Mathematics, Reáltanoda utca 13-15, Budapest, Hun- gary, H-1053 Email address: [email protected] MTA Alfréd Rényi Institute of Mathematics, Reáltanoda utca 13-15. Budapest, Hungary, H-1053Email address: [email protected] MTA Alfréd Rényi Institute of Mathematics, Reáltanoda utca 13-15, Budapest, Hun- gary, H-1053 Email address: szegedyb@gmail. Email address: [email protected]	[]
[ "ENSEMBLE CLUSTERING FOR GRAPHS: COMPARISONS AND APPLICATIONS A PREPRINT", "ENSEMBLE CLUSTERING FOR GRAPHS: COMPARISONS AND APPLICATIONS A PREPRINT" ]	[ "Valérie Poulin [email protected] \nTutte Institute for Mathematics and Computing Ottawa\nTutte Institute for Mathematics and Computing Ottawa\nCanada, Canada\n", "François Théberge [email protected] \nTutte Institute for Mathematics and Computing Ottawa\nTutte Institute for Mathematics and Computing Ottawa\nCanada, Canada\n" ]	[ "Tutte Institute for Mathematics and Computing Ottawa\nTutte Institute for Mathematics and Computing Ottawa\nCanada, Canada", "Tutte Institute for Mathematics and Computing Ottawa\nTutte Institute for Mathematics and Computing Ottawa\nCanada, Canada" ]	[]	We recently proposed a new ensemble clustering algorithm for graphs (ECG) based on the concept of consensus clustering. We validated our approach by replicating a study comparing graph clustering algorithms over benchmark graphs, showing that ECG outperforms the leading algorithms. In this paper, we extend our comparison by considering a wider range of parameters for the benchmark, generating graphs with different properties. We provide new experimental results showing that the ECG algorithm alleviates the well-known resolution limit issue, and that it leads to better stability of the partitions. We also illustrate how the ensemble obtained with ECG can be used to quantify the presence of community structure in the graph, and to zoom in on the sub-graph most closely associated with seed vertices. Finally, we illustrate further applications of ECG by comparing it to previous results for community detection on weighted graphs, and community-aware anomaly detection.Most networks that arise in nature exhibit complex structure [1, 2] with subsets of vertices densely interconnected relative to the rest of the network, which we call communities or clusters. Binary relational data-sets are typically represented as graphs G = (V, E), where vertices v ∈ V represent the entities, and edges e ∈ E represent the relations between pairs of entities. Graph clustering aims at finding a partition of the vertices V = C 1 ∪ . . . ∪ C l into good clusters. This is an ill-posed problem [3], as there is no universal definition of good clusters, leading to a wide variety of graph clustering algorithms [1, 4-10], with different objective functions. In a recent study[11], several state-of-the art algorithms implemented in the igraph [12] package were compared over a wide range of artificial networks generated via the LFR benchmark[13]. We recently introduced a new ensemble clustering algorithm for graphs (ECG), which compared favorably with leading algorithms from that study[14].The ECG algorithm is based on the concept of co-association consensus clustering. It is similar to other consensus clustering algorithms, in particular[15], but differs in two major points: (1) the choice of an algorithm that alleviates the resolution limit issue for the generation step, and (2) the restriction to endpoints of edges for co-occurrences of vertex pairs, which keeps low computational complexity.The rest of the paper is organized as follows. We briefly describe the ECG algorithm in Section 2, where we also recall some results from the previous comparison study. New results are included in the following three sections.In Section 3, we extend our study to a wider variety of graphs by varying the power law exponents of the LFR benchmark. Some of the advantages of ECG are its stability, and its ability to alleviate the well known resolution limit issue. We illustrate those properties in Section 4. We also take a closer look at the edge weights generated by the ECG algorithm, showing that they can be good indicators of the presence (or absence) of community structure in a graph. Applications	10.1007/s41109-019-0162-z	[ "https://arxiv.org/pdf/1903.08012v1.pdf" ]	83,458,572	1903.08012	a678e64371bb4ec9115aa51a5dfd47a1dfe2cec9	ENSEMBLE CLUSTERING FOR GRAPHS: COMPARISONS AND APPLICATIONS A PREPRINT March 20, 2019 Valérie Poulin [email protected] Tutte Institute for Mathematics and Computing Ottawa Tutte Institute for Mathematics and Computing Ottawa Canada, Canada François Théberge [email protected] Tutte Institute for Mathematics and Computing Ottawa Tutte Institute for Mathematics and Computing Ottawa Canada, Canada ENSEMBLE CLUSTERING FOR GRAPHS: COMPARISONS AND APPLICATIONS A PREPRINT March 20, 2019graph clustering · ensemble · consensus We recently proposed a new ensemble clustering algorithm for graphs (ECG) based on the concept of consensus clustering. We validated our approach by replicating a study comparing graph clustering algorithms over benchmark graphs, showing that ECG outperforms the leading algorithms. In this paper, we extend our comparison by considering a wider range of parameters for the benchmark, generating graphs with different properties. We provide new experimental results showing that the ECG algorithm alleviates the well-known resolution limit issue, and that it leads to better stability of the partitions. We also illustrate how the ensemble obtained with ECG can be used to quantify the presence of community structure in the graph, and to zoom in on the sub-graph most closely associated with seed vertices. Finally, we illustrate further applications of ECG by comparing it to previous results for community detection on weighted graphs, and community-aware anomaly detection.Most networks that arise in nature exhibit complex structure [1, 2] with subsets of vertices densely interconnected relative to the rest of the network, which we call communities or clusters. Binary relational data-sets are typically represented as graphs G = (V, E), where vertices v ∈ V represent the entities, and edges e ∈ E represent the relations between pairs of entities. Graph clustering aims at finding a partition of the vertices V = C 1 ∪ . . . ∪ C l into good clusters. This is an ill-posed problem [3], as there is no universal definition of good clusters, leading to a wide variety of graph clustering algorithms [1, 4-10], with different objective functions. In a recent study[11], several state-of-the art algorithms implemented in the igraph [12] package were compared over a wide range of artificial networks generated via the LFR benchmark[13]. We recently introduced a new ensemble clustering algorithm for graphs (ECG), which compared favorably with leading algorithms from that study[14].The ECG algorithm is based on the concept of co-association consensus clustering. It is similar to other consensus clustering algorithms, in particular[15], but differs in two major points: (1) the choice of an algorithm that alleviates the resolution limit issue for the generation step, and (2) the restriction to endpoints of edges for co-occurrences of vertex pairs, which keeps low computational complexity.The rest of the paper is organized as follows. We briefly describe the ECG algorithm in Section 2, where we also recall some results from the previous comparison study. New results are included in the following three sections.In Section 3, we extend our study to a wider variety of graphs by varying the power law exponents of the LFR benchmark. Some of the advantages of ECG are its stability, and its ability to alleviate the well known resolution limit issue. We illustrate those properties in Section 4. We also take a closer look at the edge weights generated by the ECG algorithm, showing that they can be good indicators of the presence (or absence) of community structure in a graph. Applications are presented in Section 5. First, we consider a real graph, and show how ECG weights can be used to zoom-in on significant sub-graphs given some seed vertices. We then use ECG for two recently published applications, respectively for clustering weighted graphs [16], and using graph clustering to find anomalous nodes [17]. We wrap-up in the last section. Previous Results Let G = (V, E) be a graph where V = {1, 2, . . . , n} is the set of vertices, and E ⊆ {(u, v) \| u, v ∈ V, u < v} is the set of edges. We consider undirected graphs. Edges can have weights w(e) > 0 for each e ∈ E. For un-weighted graphs, we let w(e) = 1 for all e ∈ E. The 2-core of a graph G is its maximal subgraph whose vertices have degree at least 2. Let P i = {C 1 i , . . . , C li i } be a partition of V of size l i . We refer to the C j i as clusters of vertices. We use 1 C j i (v) to denote the indicator function for v ∈ C j i . The ECG Algorithm The ECG algorithm is a consensus clustering algorithm for graphs. Its generation step consists of independently obtaining k randomized level-1 partitions from the multilevel-Louvain (ML) algorithm [10]: P = {P 1 , . . . , P k }. Its integration step is performed by running ML on a re-weighted version of the initial graph G = (V, E). The ECG weights are obtained through co-association. The weight of an edge e = (u, v) ∈ E is defined as: W P (u, v) = w * + (1 − w * ) · k i=1 α P i (u,v) k , (u, v) ∈ 2-core of G w * , otherwise(1) where 0 < w * < 1 is some minimum weight and α Pi ( u, v) = li j=1 1 C j i (u) · 1 C j i (v) indicates if the vertices u and v co-occur in a cluster of P i or not. When running the ECG algorithm, the size k of the ensemble and the minimum edge weight w * are the only parameters that need to be supplied. Guidelines for the parameters are given in [14], where we also show that the results are not too sensitive with respect to those parameters. Comparison Study In [14], we re-visited a recently published study of graph clustering algorithms, comparing the best performing algorithms from that study with the ECG algorithm. In general, we found ECG to yield better clusters with respect to all of the measures considered. Moreover, ECG generally found a number of communities much closer to the true value. The algorithms are compared on graphs generated with the LFR benchmarks for undirected and unweighted graphs and with non-overlapping communities. A key parameter when generating an LFR graph G is the mixing parameter µ, which sets the expected proportion of edges in G for which the two endpoints are in different communities. We considered .03 ≤ µ ≤ .75. It was recently shown [18] that graph-agnostic measures such as the adjusted RAND index (ARI) yield high scores for refinements of the true partition, while a graph-aware version (AGRI) gives high scores for coarsenings of the true partition when measuring graph partition similarities. It is thus recommended to use both measures to compare algorithms, as we do throughout this paper. We compared the true communities with those found by the ECG algorithm as well as three other state-of-the-art algorithms: InfoMap (IM) [9], WalkTrap (WT) [5] and multilevel-Louvain (ML) [10]. The quality of the results from ECG are clear from the first two plots of Figure 1, and the number of communities found with ECG remains much closer to the true number as the proportion of noise increases, as shown in the third plot. Those conclusions are illustrative of the results we reported in [14]. Expanding the Comparison In the LFR benchmark [19], three important parameters are: the mixing parameter (µ), the (negative) degree distribution power law exponent (γ 1 ), and the (negative) community size distribution power law exponent (γ 2 ). It is generally recommended to use 2 ≤ γ 1 ≤ 3 and 1 ≤ γ 2 ≤ 2 to model realistic networks [19], [20]. In the previous section, we compared ECG with other state-of-the-art algorithms with the same parameter choices as in [11]. While we considered a wide range for parameter µ, the power law exponents were fixed at γ 1 = 2 and γ 2 = 1. In this section, we summarize the impact of those parameters on the types of networks that are generated, and we re-visit the comparison results, exploring a wider set of graphs. Figure 1: In the first two plots, we compare the accuracy of ECG with state-of-the-art algorithms: InfoMap (IM), WalkTrap (WT) and Louvain (ML). Results from each algorithm are compared with the true communities for LFR graphs with n = 22, 186 vertices, and for various values of µ, the proportion of noisy edges. For each value of µ, we average over 10 LFR graphs; the shaded area shows the standard deviation. We see that ECG outperforms all other algorithms. In the third plot, we look at the ratio of the number of computed vs true communities. We see that ECG remains very close to the desired valueĈ/C = 1, as opposed to the other algorithms. In Figure 2, we show some topological graph differences over 5 choices of parameters (γ 1 , γ 2 ) in the recommended range. We see that for larger values of those parameters, the communities generated are small and of similar size while smaller values for (γ 1 , γ 2 ) yield graphs with more heterogeneous community sizes, which are more realistic. Figure 2: We selected 5 choices for the power law parameters (γ 1 , γ 2 ) which are representative of various types of networks obtained with the LFR benchmark, and we look at the distribution of the sizes of communities. We see that with the largest recommended values (γ 1 , γ 2 ) = (3, 2), we get small communities of homogeneous size. As the exponents decrease, the sizes of the communities get more heterogeneous. All results were obtained by averaging over 10 graphs with 22,186 nodes for every choice of parameters (µ, γ 1 , γ 2 ). In Figure 3, we again compare ECG with IM, WT and ML. For the larger values of (γ 1 , γ 2 ), we see that the ML algorithm does not do very well, with ECG doing much better and IM yielding the best results. As before, we use both the ARI measure and its graph-aware counterpart AGRI. As the exponents decrease, indicative of more heterogeneous community size distribution, we see that the ML algorithm does better, and ECG gives the best results overall. Therefore, by expanding the comparison over a wider range of LFR parameters, we see that ECG generally gives better results, with the exception of graphs with small communities of homogeneous size, where IM is slightly better. Resolution Limit and Stability At the heart of ECG is the fact that we use multiple runs of the single-level Louvain algorithm to build an ensemble of weak (or local) partitionings of the vertices. In this section, we illustrate the two main reasons for this choice. Recall that those graphs have many small communities of similar sizes. As we move toward the right, the Louvain algorithm does better and ECG yields the best results. Those graphs have more heterogeneous community sizes. Resolution Issue: Ring of Cliques Illustration The resolution limit issue is well illustrated by the infamous ring of cliques example, where the n vertices form l cliques (full sub-graphs) of size m, wired together as a ring. For some choices of l and m, grouping pairs of adjacent cliques yields a higher modularity value than the natural choice of each clique forming its own cluster [21]. The latter yields higher modularity if and only if m(m − 1) > l − 2. In [14], we show that choosing a small value for w * in (1) can alleviate this issue. In particular, choosing w * < 1/n avoids the issue altogether. In Figure 4, we look at rings of l cliques of size m = 5, with 1 to 5 edges between contiguous cliques. For the ML algorithm, we see the resolution limit issue when l > 20 (with 1 edge between contiguous cliques), which agrees with the known results. The IM algorithm is stable when only a few edges link the cliques, but quickly becomes unstable as more edges are added, while the ECG algorithm remains very stable keeping the default choice of w * = .05. Figure 4: In each plot, we consider l cliques of size m = 5 where contiguous cliques are linked by 1 to 5 edges, respectively. We compare the number of communities found by the InfoMap (IM), Louvain (ML) and ECG algorithms. The resolution limit phenomenon is clearly seen with the ML algorithm. The IM algorithm fails to find the right number of communities when we increase the number of edges between the cliques, while ECG remains more stable. We further illustrate this stability in Figure 5, where we add up to 15 edges between the cliques of size 5 in a ring with 4 cliques. We see that even when the number of edges linking the cliques is comparable to the number of edges within each clique, the signal obtained with the ECG weights still favours the cliques. This behaviour allow to better identify communities in noisy graphs. In the right plot of Figure 5, we show the case where 15 edges are added between contiguous cliques. Thicker edges are the ones where the ECG weights are above 0.8. We see that most of the clique structure is still captured when looking only at those high weight edges. Figure 5: We add 1 to 15 edges between contiguous cliques in a ring of 4 cliques of size 5, and we look at the effect on the ECG edge weights for edges internal to the cliques, or external edges linking the contiguous cliques. In the right plot, we look at the case with 15 edges between cliques; thick edges are the ones where the ECG weight is 0.8 or above. Weight distribution and community structure We compare the ECG weight distribution over LFR graphs where we vary the mixing parameter. We also compare with a random graph having the same degree distribution as one of the LFR graphs. Bi-modal distribution of the ECG weights near the boundaries (0 and 1) is indicative of strong community structure. We propose a simple community strength indicator (CSI) based on the point-mass Wasserstein distance. For all edges (u, v) ∈ E, with W P (u, v) from (1), we define: CSI = 1 − 2 · 1 \|E\| (u,v)∈E min (W P (u, v), 1 − W P (u, v))(2) such that 0 ≤ CSI ≤ 1, where a value close to 1 is indicative of strong community structure, random weights W P (u, v) yield a value close to 0.5, and CSI = 0 when all W P (u, v) = 0.5. In Figure 6, we see the bi-modal distribution of the weights for low and mid-range choices of µ, along with high CSI values. For larger values of µ, the distribution is not as clear, and there are less and less edges with weight close to 1, which indicate a weak community structure, as confirmed by the CSI values. The random graphs have low weights only, which is indicative of the absence of community structure. This example illustrates how the distribution of edge weights obtained with ECG, along with the proposed CSI, can be used to assess the strength of community structure in a graph. Figure 6: Violin plots of the ECG weight distribution for a family of LFR graphs with n = 22, 186 nodes, parameters γ 1 = 2, γ 2 = 1 and .21 ≤ µ ≤ .75. We also compare with a random graph of the same size and degree distribution as the graph with µ = 0.21. We see the bi-modal distribution over LFR graphs up to a very high noise level. For large µ, the signal gets weaker. It is even weaker for the random graph. The Community Strength Indicator (CSI) is also reported. Stability of ECG So far, we saw that the ECG weights are useful to alleviate the resolution issues of modularity, and can also be used to assess the presence of community structure in a graph. We illustrate another advantage of ECG which is to significantly reduce the instability in the ML algorithm. To test for stability, we run the same algorithm twice on each graph considered, and we compare the two partitions obtained with the ARI (or AGRI) measure. In Figure 7, we did this for the ML and ECG algorithms over LFR graphs with the same parameters as in the previous section. We see that in all cases, ECG greatly improves the stability of the Louvain algorithm. Other Applications of ECG In this Section, we look at a few applications with ECG. ECG weights and a dimmer process Assume that we are interested in some seed vertices in a graph. In large graphs, it is not clear how to properly "zoom in" on the sub-graph showing the main interactions around the seed vertices. Taking the seed's ego-nets (immediate neighbours) may not show all the strong interactions, and taking the entire clusters from a partition which contain the seed vertices may be too large. The weights provided by ECG can be used to define a dimmer-like process around the seed vertices, thus highlighting the sub-graphs that are the most tightly connected to the seeds. Consider a graph G, a seed vertex v and G v ⊂ G the sub-graph of G formed by keeping only the ECG cluster containing vertex v. Given some threshold θ, we delete all edges in G v with ECG weights below θ, and we keep the connected component sub-graph containing vertex v. Increasing θ from 0 to 1 provides a hierarchy of sub-graphs of decreasing size which all contain vertex v. As an illustration of this process, we consider the Amazon co-purchasing graph available from the SNAP repository [22]. This graph has 334,863 nodes and 925,872 edges. There are over 75,000 communities, 5000 of which are identified as the top ones. We picked a vertex v that belongs to one of those top communities 1 . We ran ECG, and isolated the sub-graph G v induced by the vertices in the ECG cluster that contains v. In Figure 8, we gradually increase the threshold θ, keeping only edges in G v with ECG weight above that threshold, and showing the connected component containing v. In the first plot, we set θ = 0, thus showing G v (v is shown with larger size). Vertices in red belong to the same ground truth community as v. While we see a lot of spurious vertices in the first plot, discarding edges with low ECG weights (setting θ = 0.1) yields the second sub-graph, where all ground truth vertices are retained. The last plot shows a more aggressive filtering, where we retain only edges with high ECG weights (setting θ = 0.72). This reveals a tightly connected subset of vertices around the seed vertex v. Application to weighted graphs So far, all of our comparisons for ECG were done over un-weighted graphs. In [16], among other things, the authors study various edge re-weighting schemes and graph clustering algorithms over weighted LFR graphs. They found that Figure 8: We consider a seed vertex (display with larger size) from the Amazon co-purchasing graph. Vertices from the same ground truth communities are displayed in red, and other vertices are displayed in black. From left to right, we display respectively (i) the entire sub-graph obtained from the ECG part that contains the seed vertex, (ii) a connected sub-graph with ECG edge weights above 0.1 containing the seed vertex, and (iii) a connected sub-graph with ECG edge weights above 0.72 containing the seed vertex. While the first plot has many spurious vertices, as we zoom in, most nodes we retain are in the same true community as the seed node. using the GloVe re-weighting function along with the Label Propagation (LP) algorithm [7] gave the best results for identifying communities. We re-created this experiment, using the same graphs available at [23], and the same re-weighting function with the best choice of parameters as reported in Table 1 of [16]. We compared LP, ML and ECG algorithms. While we generally obtained good results with LP, we had to discard some runs as this algorithm sometimes failed to converge. We show our results in Figure 9, where we summarize the ARI and AGRI scores we obtained over all graphs for which the LP algorithm converged. We see that the results are better in general with ECG, and with much improved stability. Figure 9: Results over weighted LFR graphs for 3 graph clustering algorithms: Multilevel-Louvain (ML), Label Propagation (LP) and ECG. Results are obtained over all graphs in [23] for which the LP algorithm converged. GloVe re-weighting as reported in [16] is applied to all graphs. Community-aware anomaly detection In [17], the authors propose CADA, a community-aware method for detecting anomalous vertices. In a nutshell, for each vertex v ∈ V , let N (v) represent the number of neighbors of v, and N c (v) the number of neighbors of v that belong to the most represented community obtained with the IM or ML algorithm. They define: CADA x (v) = N (v) Nc(v) where x ∈ {IM, M L} indicates the clustering algorithm used. They compare their algorithm to other methods by generating LFR graphs with degree exponent γ 1 = 3 and community size exponent γ 2 = 2. As we saw earlier, this choice corresponds to small communities of homogeneous size, where the IM algorithm performs best. We re-visited this approach with ECG, considering different values for the power law exponents, as in section "Expanding the Comparison". We generated LFR graphs with n = 22, 186 nodes and various values for the mixing parameters. For each graph, we introduced 200 random anomalous nodes with the same degree distribution, as in Figure 1 of [17]. In Figure 10, we compare CADA ECG with CADA IM and CADA M L using the areas under the ROC curves (AUC). We see that for large choices of the power law exponents, the IM version does best. This is the only choice of parameters used in [17]. As we decrease the values of the exponents, we see that using ECG becomes a better choice, in particular for large values of µ. This is due to the increased stability and the ability to distinguish the signal from the noise provided by the ECG weights, which we illustrated earlier in section "Resolution Limit and Stability". Conclusion In [14], we proposed ECG, a new graph clustering algorithm based on the concept of consensus clustering, and we compared it to other algorithms by re-creating the study in [11]. In this paper, we compared ECG with state-of-the-art algorihms over a wider range of graphs, showing ECG to be the best performing algorithm in most cases. We provided empirical evidence for the two main advantages of ECG: its ability to greatly reduce the resolution limit issue of modularity, and its high stability. We also illustrated how the edge weights generated in ECG can be used to assess the presence of community structure in graphs. Finally, we favourably applied ECG to three tasks: we showed how to extract relevant sub-graphs around seed vertices, we used ECG to find communities in weighted graphs, and we applied ECG for the task of detecting anomalous vertices in graphs. Figure 3 : 3We measure the quality of the communities found by the InfoMap (IM), WalkTrap (WT), Louvain (ML) and ECG algorithms over LFR graphs with 5 different choice of power law exponents. For the plots on the left, we see that the Louvain algorithms does not do very well, ECG does much better and the InfoMap yields the best results. Figure 7 : 7We compare the stability of the communities found by the Louvain (ML) and ECG algorithms over LFR graphs with 5 different choice of power law exponents. Partitions obtained in distinct runs for each algorithm are compared via the ARI measure. We see the much improved stability with ECG. Conclusions are the same with the AGRI measure (not shown). Figure 10 : 10We compare three flavours of the CADA algorithm, using the InfoMap (IM), Louvain (ML) and ECG. For each value of .3 ≤ µ ≤ .75, we generated 10 LFR graphs of size 22,186, along with 200 random anomalous nodes with the same degree distribution. We considered 5 different choices for the LFR power law exponents. Results are compared via the area under the ROC curve (AUC). vertex 112067 in the minimized data from[22]. Community structure in social and biological networks. M Girvan, M E Newman, Proc. Nat. Acad. of Sci. 9912M. Girvan and M. E. Newman. Community structure in social and biological networks. Proc. Nat. Acad. of Sci., 99(12):7821-7826, 2002. The structure and function of complex networks. M E Newman, SIAM Rev. 45M. E. Newman. The structure and function of complex networks. SIAM Rev., 45:167-256, 2003. Community detection in networks: A user guide. S Fortunato, D Hric, Phys. Rep. 659S. Fortunato and D. Hric. Community detection in networks: A user guide. Phys. Rep., 659:1-44, 2016. Finding community structure in very large networks. A Clauset, M E Newman, C Moore, Phys. Rev. E. 70666111A. Clauset, M. E. Newman, and C. Moore. Finding community structure in very large networks. Phys. Rev. E, 70(6):066111, 2004. Computing communities in large networks using random walks. P Pons, M Latapy, Comp. and Inf. Sci. ISCIS. P. Pons and M. Latapy. Computing communities in large networks using random walks. Comp. and Inf. Sci. ISCIS, pages 284-293, 2005. Finding community structure in networks using the eigenvectors of matrices. M E Newman, Phys. Rev. E. 74336104M. E. Newman. Finding community structure in networks using the eigenvectors of matrices. Phys. Rev. E, 74(3):036104, 2006. Near linear time algorithm to detect community structures in large-scale networks. U N Raghavan, R Albert, S Kumara, Phys. Rev. E. 76336106U. N. Raghavan, R. Albert, and S. Kumara. Near linear time algorithm to detect community structures in large-scale networks. Phys. Rev. E, 76(3):036106, 2007. Statistical mechanics of community detection. J Reichardt, S Bornholdt, Phys. Rev. E. 74116110J. Reichardt and S. Bornholdt. Statistical mechanics of community detection. Phys. Rev. E, 74(1):016110, 2006. Maps of random walks on complex networks reveal community structure. M Rosvall, C T Bergstrom, PNAS105M. Rosvall and C. T. Bergstrom. Maps of random walks on complex networks reveal community structure. PNAS, 105(4):1118-1123, 2007. Fast unfolding of communities in large networks. V D Blondel, J L Guillaume, R Lambiotte, E Lefebvre, J. Stat. Mech. 10008V. D. Blondel, J. L. Guillaume, R. Lambiotte, and E. Lefebvre. Fast unfolding of communities in large networks. J. Stat. Mech., 08(P10008), 2008. A comparative analysis of community detection algorithms on artificial networks. Z Yang, R Algesheimer, C J Tessone, Nat. Sci. Rep. 630750Z. Yang, R. Algesheimer, and C. J. Tessone. A comparative analysis of community detection algorithms on artificial networks. Nat. Sci. Rep., 6:30750, 2016. The igraph software package for complex network research. G Csardi, T Nepusz, Intl. J. of Complex Sys. G. Csardi and T. Nepusz. The igraph software package for complex network research. Intl. J. of Complex Sys., 2006. Benchmark graphs for testing community detection algorithms. A Lancichinetti, S Fortunato, F Radicchi, Phys. Rev. E. 78046110A. Lancichinetti, S. Fortunato, and F. Radicchi. Benchmark graphs for testing community detection algorithms. Phys. Rev. E, 78(046110), 2008. Ensemble clustering for graphs. V Poulin, F Théberge, Complex Networks and Their Applications VII. 1V. Poulin and F. Théberge. Ensemble clustering for graphs. Complex Networks and Their Applications VII, 1:231-243, 2019. Consensus clustering in complex networks. A Lancichinetti, S Fortunato, Nat. Sci. Rep. 2336A. Lancichinetti and S. Fortunato. Consensus clustering in complex networks. Nat. Sci. Rep., 2:336, 2012. Is community detection fully unsupervised? the case of weighted graphs. V Connes, N Dugué, A Guille, Complex Networks and Their Applications VII. 1V. Connes, N. Dugué, and A. Guille. Is community detection fully unsupervised? the case of weighted graphs. Complex Networks and Their Applications VII, 1:256-266, 2019. A community-aware approach for identifying node anomalies in complex networks. T J Helling, J C Scholtes, F Takes, Complex Networks and Their Applications VII. 1T.J. Helling, J.C. Scholtes, and F. Takes. A community-aware approach for identifying node anomalies in complex networks. Complex Networks and Their Applications VII, 1:244-255, 2019. Comparing graph clusterings: Set partition measures vs. graph-aware measures. V Poulin, F Théberge, arXiv:1806.11494V. Poulin and F. Théberge. Comparing graph clusterings: Set partition measures vs. graph-aware measures. arXiv:1806.11494, 2018. Benchmarks for testing community detection algorithms on directed and weighted graphs with overlapping communities. A Lancichinetti, S Fortunato, Phys. Rev. E. 80116118A. Lancichinetti and S. Fortunato. Benchmarks for testing community detection algorithms on directed and weighted graphs with overlapping communities. Phys. Rev. E, 80(1):016118, 2009. A L Barabasi, Network Science. Cambridge University PressA. L. Barabasi. Network Science. Cambridge University Press, 2016. Resolution limit in community detection. S Fortunato, M Barthélemy, Proc. Nat. Acad. Sci. 1041S. Fortunato and M. Barthélemy. Resolution limit in community detection. Proc. Nat. Acad. Sci., 104(1):36-41, 2007. SNAP Datasets: Stanford large network dataset collection. Jure Leskovec, Andrej Krevl, Jure Leskovec and Andrej Krevl. SNAP Datasets: Stanford large network dataset collection. http://snap. stanford.edu/data, Accessed 11 Jan 2019. Weighted community detection, Accessed 18. N Dugué, N. Dugué. 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[ "Zipf's Law in Importance of Genes for Cancer Classification Using Microarray Data", "Zipf's Law in Importance of Genes for Cancer Classification Using Microarray Data" ]	[ "Wentian Li \nLab of Statistical Genetics\nRockefeller University\nBox 19210021New YorkNYUSA\n" ]	[ "Lab of Statistical Genetics\nRockefeller University\nBox 19210021New YorkNYUSA" ]	[]	Microarray data consists of mRNA expression levels of thousands of genes under certain conditions. A difference in the expression level of a gene at two different conditions/phenotypes, such as cancerous versus non-cancerous, one subtype of cancer versus another, before versus after a drug treatment, is indicative of the relevance of that gene to the difference of the high-level phenotype. Each gene can be ranked by its ability to distinguish the two conditions. We study how the single-gene classification ability decreases with its rank (a Zipf's plot). Power-law function in the Zipf's plot is observed for the four microarray datasets obtained from various cancer studies. This power-law behavior in the Zipf's plot is reminiscent of similar power-law curves in other natural and social phenomena (Zipf's law).However, due to our choice of the measure of importance in classification ability, i.e., the maximized likelihood in a logistic regression, the exponent of the power-law function is a function of the sample size, instead of a fixed value close to 1 for a typical example of Zipf's law. The presence of this powerlaw behavior is important for deciding the number of genes to be used for a discriminant microarray data analysis.	10.1006/jtbi.2002.3145	[ "https://arxiv.org/pdf/physics/0104028v1.pdf" ]	14,671,176	physics/0104028	0d8a8791c68af8af1a62875a3eeb8c5a9212e594	Zipf's Law in Importance of Genes for Cancer Classification Using Microarray Data 6 Apr 2001 February 2, 2008 Wentian Li Lab of Statistical Genetics Rockefeller University Box 19210021New YorkNYUSA Zipf's Law in Importance of Genes for Cancer Classification Using Microarray Data 6 Apr 2001 February 2, 2008arXiv:physics/0104028v1 [physics.bio-ph] Microarray data consists of mRNA expression levels of thousands of genes under certain conditions. A difference in the expression level of a gene at two different conditions/phenotypes, such as cancerous versus non-cancerous, one subtype of cancer versus another, before versus after a drug treatment, is indicative of the relevance of that gene to the difference of the high-level phenotype. Each gene can be ranked by its ability to distinguish the two conditions. We study how the single-gene classification ability decreases with its rank (a Zipf's plot). Power-law function in the Zipf's plot is observed for the four microarray datasets obtained from various cancer studies. This power-law behavior in the Zipf's plot is reminiscent of similar power-law curves in other natural and social phenomena (Zipf's law).However, due to our choice of the measure of importance in classification ability, i.e., the maximized likelihood in a logistic regression, the exponent of the power-law function is a function of the sample size, instead of a fixed value close to 1 for a typical example of Zipf's law. The presence of this powerlaw behavior is important for deciding the number of genes to be used for a discriminant microarray data analysis. 1 Introduction 2 Measure of importance of genes in cancer classification: likelihood The measurement we use for the importance of genes in cancer classification is the maximized likelihood, which is proportional to the probability of observing the data when a model is given, and when the parameters in the model are adjusted to give the maximum value. Mathematically, likelihood L is [Edwards, 1972]: L = cP (D\|M, θ),(1) where D is all data points in a data set, M is a model, θ represents all parameters in the model, and c is a proportional constant (often set to be 1). The maximized likelihood iŝ L = max θ L = cP (D\|M,θ),(2) where theˆrepresents a maximization/estimation procedure. The microarray data sets have the form of {x 1i , x 2i , · · · , x pi , y i } (i = 1, 2, · · · N ). Each sample point i (one microarray experiment, one tissue sample) contains measurements of (logarithm of ) mRNA expression level of p genes (p is typical of the order of thousands), and y is a categorical label indicating a condition (e.g. cancerous or normal). The raw data in a microarray experiment could be more complicated: one has to consider background noise, normalization, and controls. These considerations depend on the type of chips: Stanford/cDNA array with two fluorescence dyes [Schena, et al., 1995;Shalon et al., 1996] or Affymetrix/oligonucleotide arrays with only one image intensity to measure but multiple oligonucleotide probes [Fodor, et al., 1991]. In this paper, only the filtered/processed data are used and the subtle issue of scaling/normalization of the raw data is not discussed. The model M is a classification model (classifier, predictor, discriminator, supervised learning machine), that classifies the label y by the gene expression level {x j } (logarithm of an image intensity from the DNA chip). We use a particularly simple classifier, the single-gene logistic regression: P (y i = 1\|x ji ) = 1 1 + e −aj −bj xji . j = 1, 2, · · · p, and i = 1, 2, · · · N. (3) In other words, the probability of a sample being in one class is a "sigmoid" or "logistic" function of the (log) expression level. If the coefficient b is positive, larger expression levels lead to higher probabilities of being the y = 1 label; if b < 0, larger expression level corresponds to the y = 0 label. The likelihood of the whole data set is the product of these model-based probabilities (for a given gene): L j = N i=1 [P (y = 1\|x ji )I yi=1 + (1 − P (y = 1\|x ji ))I yi=0 ] , j = 1, 2, · · · p(4) where I is the indicator function (1 or 0 depending on whether the condition is true or not). Eq.(4) is maximized by adjusting the parameters in the model. The maximized likelihoodL can then be used to rank all genes: the larger theL, the higher the ranking. Li 3 Zipf's plot of classification likelihood The binomial (two-class) logistic regression Eq. (3) has been applied to two microarray data sets. The first is colon cancer data from Princeton University [Alon, et al., 1999]. The expression levels of 2000 genes were available for 62 tissue samples: 40 cancerous and 22 normal. The second data set is leukemia data from Whitehead Institute/MIT [Golub, et al., 1999]. Expression levels of 7129 genes were measured on 72 leukemia samples: 47 obtained from tissues of one subtype of leukemia, acute lymphoblastic leukemia (ALL), and 25 from tissues of another subtype of leukemia, acute myeloid leukemia (AML). Although the number of labels is 2 for both datasets, we distinguish cancerous from normal tissues in the first data set, whereas distinguish one cancer subtype from another in the second data set. For the colon cancer data, the single-gene maximized likelihood Eq.(4) for each gene was calculated and ranked. The likelihood-rank plot (Zipf's plot) is shown in Fig.1 in log-log scale. The power-law behavior of the curve is clearly visible. Fitting the first 600 genes using a generalized form of Zipf's law: L r ∼ 1/r α ,(5) leads to exponent α ≈ 2.18, whereas the fitting of the top 1000 genes leads to α ≈ 2.10. The genes from roughly rank 1000 to 2000 do not seem to follow the same power-law decay of the likelihood. As will be discussed more later in this paper, the exponent α is not an intrinsic quantity for our likelihood-rank plots. The reason is that the likelihood is a product of probabilities of N sample points L r ∝ (p 1 p 2 · · · p N ) r ∼p N r ; if the per-sample averaged likelihoodp r of rank-r gene does not change with the sample size N , the exponent α is then a function of N : α ∼ −N log(p r )/log(r). tissues, and were separated as a training set in [Golub, et al., 1999]. These 38 samples were considered to be more homogeneous, while the rest of the samples were from various sources or other tissue types such as peripheral blood [Golub, et al., 1999], and may not be homogeneous. From Fig.2, the Zipf's plot obtained from the training set seems to follow a generalized form of Zipf's law with a fitting exponent of α ≈ 2.56 from the top 900 genes. The Zipf's plot of all sample points seems to deviate from a power-law trend around rank 100-200. As mentioned earlier, the exponent α from a bigger data set is indeed larger than the exponent from a small data set. When the top genes are examined, it was found that the top-ranking genes for the training set [Li & Yang, 2001] and those for the overall set [Li, et al., 2001] may not be identical. For example, the top performing gene for the 38-sample training set is No.4847, zyxin, a gene encoding the LIM domain protein used in cell adhesion in fibroblasts [Golub, et al., 1999]. On the other hand, the best performing gene for the 72-sample combined set is No. 1834, CD33 antigene which encodes cell surface proteins commonly found in AML leukemia cells (see, e.g., [Lauria, et al., 1994]). The zyxin becomes the rank-4 gene for the combined dataset. This difference reflects a certain degree of inhomogeneity between samples in the training set and those in the remaining set (testing or validation set). Note from Fig.1 that for the training set, genes from rank 3 to 9 exhibits similar likelihood, and form a flat step on the Zipf's plot. Such steps Li 5 are a dominant feature in the Zipf's plot of the frequency of word occurrence in randomly generated texts [Li, 1992]. Classification of multiple cancer classes The logistic regression Eq.(3) for a two-class data set can be generalized to multiple classes: multinomial logistic regression [Agresti, 1996]: P (y i = I\|x ji ) = e −aI −bI xji C K=1 e −aK −bK xji j = 1, 2, · · · p, I = 1, 2, · · · C, i = 1, 2, · · · N ,(6) where the label y can be in one of the C classes, and there are two parameters for each class (though only C − 1 class probabilities are independent). A gene is important (higher maximized likelihood) if it is more able to distinguish all C classes simultaneously. For this multiple-class analysis, we use the microarray data for lymphoma (Stanford University and National Cancer Institute [Alizadeh, et al., 2000] follicular lymphoma (FL), 11 chronic lymphocyte leukemia (CLL). These 3 cancer subtypes plus the normal are the 4 classes to be distinguished. In [Alizadeh, et al., 2000], it is also recommended that two be handled by the logistic regression of paired case-controls (case is a sample with label 1, and control is a sample with label 0) [Breslow & Day, 1980]: P (x(case) ji − x(control) ji ) = 1 1 + e bj (x(case)ji−x(control)ji) , j = 1, 2, · · · p, i = 1, 2, · · · N .(7) Note that the probability is no longer that for observing y -when a case sample and a control sample are paired, their y value is fixed at 1 and 0 -but that of observing the difference of x's. Also note that the first parameter a is now zero. Fig.4 shows the Zipf's plot for genes in the breast cancer microarray data. The top three genes were able to perfectly identify the chemotherapy effect -the expression level is always higher (or lower) before the treatment than after in all 20 samples. The likelihood of a perfect fit is equal to 1. Unlike the previous three plots, the likelihood of genes in Fig.4 does not follow a power-law function, or even a smooth function. There seems to be a gap from the ranking of 19 to the ranking of 20-25. Fitting the two segments (from 4 to 19, and from 25 to 500) by power-law functions leads to different exponents (2.51 versus 1.77). Plots like Fig.4 can be good news for a microarray data analysis, because there seems to be a separation between "relevant" and "irrelevant" genes. Irrelevant genes are not important for distinguishing samples before and after chemotherapy, and may be discarded for further analysis. Of course, this is only a rough description of the gene set. A more systematic approach can be based on the framework of model selection (see, e.g., [Burnham & Anderson, 1998]) or model averaging (see, e.g., [Geisser, 1993]). With model selection, the expression level of different genes can be combined, and adding one gene increases the number of parameters in the model by 1. The appropriate number of genes to be included in a classification is the model with the best "model selection criterion" such as Akaike information criterion [Burnham & Anderson, 1998] or Bayesian information criterion [Raftery, 1995], with a best balance between a larger likelihood value and fewer numbers of parameters [Li & Yang, 2001, Li, et al., 2001. With model averaging, there is in principle no limitation on the number of genes to be included, but irrelevant genes have smaller weights in an averaged classifier [Golub, et al., 1999, Li & Yang, 2001. This makes the effective number of genes used much smaller than the apparent number. Scaling exponent As mentioned earlier, the exponent of the inverse power-law fitting function for Zipf's plot (Figs. 1-4) depends on the sample size. The reason for this is that the measurement for the importance of a gene, the sample classification likelihood, is a function of the sample size. Here we ask the question whether one can define a normalized exponent. SinceL r ∼ 1/r α , we can obviously draw Zipf's plot for the per-sample likelihood:p r =L 1/N r ∼ 1/r (α/N ) . There is another consideration for classification of more than two labels: it is unfair to compare per-sample classification likelihoods when the number of classes differs. Just by random guess, the probability of classifying the label correctly for a binary-class case is 0.5, whereas that for classifying C labels is 1/C. We can normalize the per-sample likelihoodp r by the random-guess probability:l r =p r /(1/C) ∼ C/r (α/N ) with the same scaling exponent. Fig.5 shows the Zipf's plot of normalized per-sample likelihoodl r for all data sets analyzed so far. These curves can be compared in two ways. First, the performance of the top-ranking genes can be compared by theirl r 's. It is clear from Fig.5 that for the leukemia data, the performance obtained from the training set is better than that obtained from the whole set, indicating a possible heterogeneity in the data set. The performance of the top genes for the breast cancer data set is better (in terms of classification) than that for the leukemia data set, which in turn is better than that for the colon cancer data set. It is usually difficult to compare performance between a two-label classification and a multiple-label classification, because it depends on the base-line expectation. Two base-line expectations (called null models) were defined in [Li & Yang, 2001, Li, et al., 2001: one is to randomly guess all classes with equal probability, and another is to guess the class by the proportion of samples with this class in the data set. In Fig.5, the first base-line expectation is used as the normalization factor. The performance of the 4-label classification relative to this base-line expectation for the lymphoma data is considered to be better than that of the 3-label classification, partly due to its low expectation. Different data sets in Fig.5 can also be compared by the rate of falling of likelihood. The fall-off rate, as measured by the scaling exponent α, ranges from 3 to 9 per 100 samples (i.e., α/N ≈ 0.03 − 0.09). It is interesting that scaling exponents obtained from the leukemia data (both the whole data set and the training set) and from the lymphoma data (two DLBCL subtypes plus normal) are almost identical (around 65 -67 per 100 samples). The breast cancer data is different from other data sets for falling faster in the likelihood-rank plot. Discussion Results from Figs. 1-4 established that when microarray data is used to classify cancer tissues, the classification likelihood of individual genes follows a generalized form of Zipf's law. The reason that these power-law functions in the Zipf's plot do not have a scaling exponent equal to 1 is partly because exponent α in Eq.(5) depends on the sample size N . Besides the issue of the exponent value, the overall power-law trend is excellent: a perfect power-law function for all points in Fig.3 (3-label classification, in 3.5 decades), a partial power-law fitting in a range of 2.5 decades (Fig.1), 3 decades (Fig.2), and 2 decades (Fig.3, 4-label classification), respectively. The only poor fitting by a power-law function is Fig.4. The Zipf's plot is flattern out for low-ranking genes in Figs. 1 and 2, while it drops off in Fig.3 (4-label classification). Which functional form is more generic for low-ranking genes is not clear, perhaps our results are sample-size sensitive. Our Zipf's plot can be compared to those obtained from biomolecular sequences. In [Gamow & Ycas, 1955], a Zipf's plot for 20 amino acids usage in protein sequences was presented (in linear-linear scale). Redrawing their data in log-log scale does not show any power-law behavior (result not shown). Recently, it was claimed that oligonucleotide frequencies in DNA sequences follow Zipf's law [Mantegna, et al., 1994]. This paper drew criticism on its strong claim concerning the connection between Zipf's law and human language [Martindale & Konopka, 1995;Israeloff, et al., 1996, Bonhoeffer, et al., 1996a,1996b, Voss, 1996: one of the best counter-examples is Zipf's law in money-typing texts [Li, 1992]. Also, the paper did not show convincingly that the Zipf's plot for oligonucleotide usage was better fitted by a power-law function [Martindale & Konopka, 1996]: the deviation from the power-law fitting function can be gradual and systematic, an indication that the power-law function is not the best choice of fitting function. Finally, the scaling exponent in the power-law fitting function in [Mantegna, et al., 1994] is much smaller than 1 [Li, 1996]. In comparison, our Zipf's plots in Figs 1-5 are a much better example of a generalized form of Zipf's law than those in [Gamow & Ycas, 1955] and [Mantegna, et al., 1994]. One may wonder whether the power-law behavior in Figs 1-5 can be derived by a simple random model. In [Gamow & Ycas, 1955], the Zipf's plot of the frequency of amino acids usage was compared to a "random partition of a unit length" model. Suppose a unit interval is randomly partitioned into p segments (e.g. p = 20 for 20 amino acids). These segments are ranked by their size L (1) = max i (L i ), L (2) = max i (L i ∈ L − L (1) ) · L (p) = min i (L i ), etc. If this random partition is repeated, the mean value of the ranked size can be shown to be [Gamow & Ycas, 1955]: < L (r) >= 1 p p+1−r i=1 1 p + 1 − i r = 1, 2, · · · p(8) Drawing < L (r) > from Eq.(8) versus rank r shows that it is a straight line in linear(y)-log(x) plot, and not a straight line in either log-log or log(y)-linear(x) plots (results not shown). Actually this analytic result may not be applicable to our case, because we keep track of the performance of a given gene on all samples, whereas in Eq.(8) the longest interval (or any given ranking interval) is averaged over all random simulation. In any case, the power-law behavior in Fig.1-5 does not seem to be explainable by a simple random model. We may ask whether Zipf's plot has any practical application for microarray data analysis. In information retrieval [van Rijsbergen, 1975, Salton, 1988 and library/documentation science [Egghe & Rousseau, 1990], Zipf's law is an important foundation that many applications are based upon. It is one of the "bibliometric laws" [White & McCain, 1989] concerning regularities in bibliographies, lists of authors, citation lists, etc. For the purpose of finding relevant, content-bearing words ("keywords"), common (highest-ranking) and rare (lowest-ranking) words should be avoided [Luhn, 1957[Luhn, ,1958]. Do we have a similar situation where the highest-ranking genes may not be interesting for cancer classification? (Lowest-ranking genes are obviously not interesting.) Our ranking system is not really the same as for word usage, since a discrimination or classification ability has already been included in our definition, whereas it is not included in the word usage example. In this sense, our top ranking genes are in fact most efficient for the purpose of cancer classification. On the other hand, if one is more interested in subtle gene effects, not in the known main/dominant effect, it is perhaps useful to remove the well-known genes from the list and examine other genes in a future study. This idea was not tried in our previous analysis [Li & Yang, 2001, Li, et al., 2001. In conclusion, a generalized form of Zipf's law was observed in microarray data for the likelihood on cancer classification using single-gene logistic regression. We suspect that this power-law behavior is generic rather than an exception. A rank-likelihood plot (Zipf's plot) can be a useful quantitative tool for discriminant microarray data analysis. • • • • • • • • • • • •• • • •• •• •• •• • • • •• •••• • •••• •• •••• •• •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Fig. 2 2shows the Zipf's plot of the second data set. Among the 72 samples, 38 are from bone marrow new subclasses of DLBCL can be defined based on the microarray data. These are the germinal centre B-like DLBCL (GC-DLBCL) and the activated B-like DLBCL (A-DLBCL). The GC-DLBCL, A-DLBCL subclasses plus the normal can be the 3 classes to be distinguished. Fig. 3 3shows the Zipf's plot of multinomial logistic regression likelihood for both the 4-class and the 3-class classification. The Zipf's plot for the 3-class multinomial logistic regression follows a perfect powerlaw function for all genes. No deviation from power-law behavior was observed even for the low-ranking genes. The Zipf's plot for the 4-class multinomial logistic regression, on the other hand, does not seem to follow a power-law function. The fall off near the rank-200 gene is perhaps due to the computer roundoff error since the value of likelihood at rank-200 is already as low as 10 −39 (the likelihood is multipled by 10 10 inFig.3). The fitting of the top 200 genes by a power-law function is, however, very good.5 Cancer treatment effectAnother interesting situation is provided by the data set from Stanford University and the Norwegian Radium Hospital[Perou, et al., 2000]. Part of the data is the expression level of 8102 genes measured before and after a 16-week course of doxorubicin chemotherapy[Perou, et al., 2000] on 20 patients. Naively, one may consider these 40 microarray experiments as an example of binary logistic regression. However, logistic regression in Eq.(3) does not apply because it requires samples to be independent of each other, whereas microarray experiments done on the same patient are clearly not independent. This situation can Figure 1 : 1Zipf's plot for colon cancer and normal tissue classification: maximized likelihood of singlevariable logistic regression (Eq.(3)) for the top performing genes vs their ranks (in log-log scale). Figure 2 :Figure 4 :Figure 5 : 245Zipf's plot for leukemia subtype classification: maximized likelihood of single-variable logistic regression (Eq.(3)) for the top genes vs their ranks, in log-log scale. The upper line is obtained from the "training set" which contains 38 samples, and the lower line is from the "training plus testing set" which contains 72 samples. Zipf's plot for the breast cancer treatment effect: maximized likelihood of single-variable paired case-control logistic regression (Eq.(7)) for top genes vs. their rank (in log-log scale). Normalized, per-sample maximum likelihood (l r =L 1/N r /(1/C)) where C is the number of classes (e.g., C = 2 for binomial logistic regression) for top genes vs. gene ranks, for all data sets. The correspondingL r vs rank plots are inFigs. 1-4. ). There are a total 96 tissue samples, with 66 cancerous and 30 normal. Within the cancer samples, there are 46 diffuse large B-cell lymphoma (DLBCL), 9 Acknowledgements: The author would like to thank Yaning Yang for discussion on the random partition of unit interval, Ronald Rousseau and Heting Chu for suggesting references on library sciences, VictoriaHaghighi and Joanne Edington for help with the microarray data. The work was supported by NIH grants K01HG00024 and HG00008. An Introduction to Categorical Data Analysis. A Agresti, Wiley & SonsAgresti A (1996), An Introduction to Categorical Data Analysis (Wiley & Sons). Distinct types of diffuse large B-cell lymphoma identified by gene expression profiling. A A Alizadeh, M B Eisen, Nature. 403Alizadeh AA, Eisen MB, et al. (2000), "Distinct types of diffuse large B-cell lymphoma identified by gene expression profiling", Nature, 403:503-511. . U Alon, N Barkai, D A Notterman, K Gish, S Ybarra, D Mack, A J Levine, Alon U, Barkai N, Notterman DA, Gish K, Ybarra S, Mack D, Levine AJ (1999 Explaining 'linguistic features' of noncoding DNA. S Bonhoeffer, Avm Herz, M C Boerlijst, S Nee, M A Nowak, R M May, Science. 2715245Bonhoeffer S, Herz AVM, Boerlijst MC, Nee S, Nowak MA, May RM (1996a) "Explaining 'linguistic features' of noncoding DNA", Science, 271(5245):14-15. No signs of hidden language in noncoding DNA" (letters). S Bonhoeffer, Avm Herz, M C Boerlijst, S Nee, M A Nowak, Physical Review Letters. 76111977Bonhoeffer S, Herz AVM, Boerlijst MC, Nee S, Nowak MA, May RM (1996b), "No signs of hidden language in noncoding DNA" (letters), Physical Review Letters, 76(11):1977. Statistical Methods in Cancer Research. I -The Analysis of Case-Control Studies (International Agency for Research on Cancer. N E Breslow, N E Day, LyonBreslow NE, Day NE (1980), Statistical Methods in Cancer Research. I -The Analysis of Case-Control Studies (International Agency for Research on Cancer, Lyon). K P Burnham, D R Anderson, Model Selection and Inference. SpringerBurnham KP, Anderson DR (1998), Model Selection and Inference (Springer). Lack of biological significance in the 'linguistic features' of noncoding DNA-a quantitative analysis. C A Chatzidimitriou-Dreismann, Rmf Streffer, D Larhammar, Nucleic Acids Research. 249Chatzidimitriou-Dreismann CA, Streffer RMF, Larhammar D (1996), "Lack of biological significance in the 'linguistic features' of noncoding DNA-a quantitative analysis", Nucleic Acids Research, 24(9):1676-1681. Analysis of Zipf's law: an index approach. Y S Chen, F F Leimkuhler, Information Processing and Management. 23Chen YS, Leimkuhler FF (1987), "Analysis of Zipf's law: an index approach", Information Processing and Management, 23:71-182. Self-similarity in world wide web traffic: evidence and possible causes. M E Crovella, A Bestavros, IEEE/ACM Transactions on Networking. 56Crovella ME, Bestavros A (1997), "Self-similarity in world wide web traffic: evidence and possible causes", IEEE/ACM Transactions on Networking, 5(6):835-846. Likelihood. Awf Edwards, Cambridge Univ PressEdwards AWF (1972), Likelihood (Cambridge Univ Press). L Egghe, R Rousseau, Introduction to Informetrics: Quantitative Methods in Library, Documentation and Information Science. ElsevierEgghe L, Rousseau R (1990), Introduction to Informetrics: Quantitative Methods in Library, Documen- tation and Information Science (Elsevier). Microarrays: their origin and applications. R Ekins, F W Chu, Trends in Biotechmology. 17217Ekins R, Chu FW (1999), "Microarrays: their origin and applications", Trends in Biotechmology, 17:217- Light-directed, spatially addressable parallel chemical synthesisLight-directed, spatially addressable parallel chemical synthesis. S P Fodor, J L Read, M C Pirrung, L Stryer, A T Lu, D Solas, Science. 251Fodor SP, Read JL, Pirrung MC, Stryer L, Lu AT, Solas D (1991), "Light-directed, spatially address- able parallel chemical synthesisLight-directed, spatially addressable parallel chemical synthesis", Science, 251:767-773. Statistical correlation of protein and ribonucleic acid composition. G Gamow, M Ycas, Proceedings of National Academy of Sciences. National Academy of Sciences41Gamow G, Ycas M (1955), "Statistical correlation of protein and ribonucleic acid composition", Proceed- ings of National Academy of Sciences, 41(12):1011-1019. S Geisser, Predictive Inference: An Introduction. Chapman & HallGeisser S (1993), Predictive Inference: An Introduction (Chapman & Hall). Molecular classification of cancer: class discovery and class prediction by gene expression monitoring. T R Golub, D K Sonim, P Tamayo, C Huard, M Gassenbeek, J P Mesirov, H Coller, M L Loh, J R Downing, M A Caligiuri, C D Bloomfield, E S Lander, Science. 286Golub TR, Sonim DK, Tamayo P, Huard C, Gassenbeek M, Mesirov JP, Coller H, Loh ML, Downing JR, Caligiuri MA, Bloomfield CD, Lander ES (1999), "Molecular classification of cancer: class discovery and class prediction by gene expression monitoring", Science, 286:531-536. Initial sequencing and analysis of the human genome. IHGSC (International Human Genome Sequencing Consortium. 409IHGSC (International Human Genome Sequencing Consortium) (2001), "Initial sequencing and analysis of the human genome", Nature, 409:860-921. Can Zipf distinguish language from noise in noncoding DNA?" (letters). N E Israeloff, M Kagalenko, K Chan, Physical Review Letters. 7611Israeloff NE, Kagalenko M, Chan K (1996), "Can Zipf distinguish language from noise in noncoding DNA?" (letters) Physical Review Letters, 76(11):1976. J L Haines, M A Pericak-Vance, Approaches to Gene Mapping in Complex Human Diseases. Wiley-LissHaines JL, Pericak-Vance MA (1998), eds. Approaches to Gene Mapping in Complex Human Diseases (Wiley-Liss). Increased expression of myeloid antigen markers in adult acute lymphoblastic leukemia patients: diagnostic and prognostic implications. F Lauria, D Raspadori, British Journal of Haematology. 87Lauria F, Raspadori D, et al. (1994), "Increased expression of myeloid antigen markers in adult acute lymphoblastic leukemia patients: diagnostic and prognostic implications", British Journal of Haematology, 87:286-292. Random texts exhibit Zipf's-law-like word frequency distribution. W Li, IEEE Transactions on Information Theory. 38Li W (1992), "Random texts exhibit Zipf's-law-like word frequency distribution", IEEE Transactions on Information Theory, 38:1842-1845. . • • • •• • •• • • ••• • •• •• • •• • • • •• •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •, • • • •• • •• • • ••• • •• •• • •• • • • •• •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • . • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 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• • • • • • • • •, • slope= -2.56 ( 1 -> 900 ) training set only (N=38)• • • • • • • • •• ••• •••• ••• ••• •••• ••• •• •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • slope= -2.56 ( 1 -> 900 ) training set only (N=38) . • • • • • • • • •••• •• ••••• ••• •• ••••• •••••••• •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •, • • • • • • • • •••• •• ••••• ••• •• ••••• •••••••• •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • . • • • • • • • • •• ••• •• • •••• ••••• •••• •• •• •••• •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •, classification (GC-DLBCL, A-DLBCL, normal), and the lower line is for the 4-class situation (DLBCL, FL, CLL, normal• • • • • • • • •• ••• •• • •••• ••••• •••• •• •• •••• •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • classification (GC-DLBCL, A-DLBCL, normal), and the lower line is for the 4-class situation (DLBCL, FL, CLL, normal). . •• • ••• • •••• • •• •• •• • •• • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 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• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •	[]
[ "Conditional Symmetry and Reductions for the Two-Dimensional Nonlinear Wave Equation. I. General Case", "Conditional Symmetry and Reductions for the Two-Dimensional Nonlinear Wave Equation. I. General Case" ]	[ "Irina Yehorchenko [email protected] \nInstitute of Mathematics of NAS Ukraine\n\n\nTereshchenkivs'ka Str\n01601Kyiv-4Ukraine\n" ]	[ "Institute of Mathematics of NAS Ukraine\n", "Tereshchenkivs'ka Str\n01601Kyiv-4Ukraine" ]	[]	We present classification of Q-conditional symmetries for the two-dimensional nonlinear wave equations u tt − u xx = F (t, x, u) and the reductions corresponding to these nonlinear symmetries. Classification of inequivalent reductions is discussed.AbstractWe present classification of Q-conditional symmetries for the two-dimensional nonlinear wave equations u tt − u xx = F (t, x, u) and the reductions corresponding to these nonlinear symmetries. Classification of inequivalent reductions is discussed.	null	[ "https://arxiv.org/pdf/1010.4913v2.pdf" ]	118,420,498	1010.4913	a48bc7c9c9f58da610447992ac14257f502166b3	Conditional Symmetry and Reductions for the Two-Dimensional Nonlinear Wave Equation. I. General Case 1 Nov 2010 Irina Yehorchenko [email protected] Institute of Mathematics of NAS Ukraine Tereshchenkivs'ka Str 01601Kyiv-4Ukraine Conditional Symmetry and Reductions for the Two-Dimensional Nonlinear Wave Equation. I. General Case 1 Nov 2010 We present classification of Q-conditional symmetries for the two-dimensional nonlinear wave equations u tt − u xx = F (t, x, u) and the reductions corresponding to these nonlinear symmetries. Classification of inequivalent reductions is discussed.AbstractWe present classification of Q-conditional symmetries for the two-dimensional nonlinear wave equations u tt − u xx = F (t, x, u) and the reductions corresponding to these nonlinear symmetries. Classification of inequivalent reductions is discussed. Introduction Following [1], we discuss conditional symmetries and reductions of the twodimensional nonlinear wave equation u tt − u xx = F (t, x, u)(1) for the real-valued function u = u(t, x); t is the time variable, x is the space variable. In the equation above and further we will use the following designations for the partial derivatives: u t = − ∂u ∂t ; u x = − ∂u ∂x ; u tt = ∂ 2 u ∂t 2 ; u xt = u tx = ∂ 2 u ∂t∂x ; u xx = ∂ 2 u ∂x 2 . Note that the general equation in the class (1) has no invariance operators; however, many well-known particular cases have wide symmetry algebras, see e.g. [2]. The maximal invariance algebra of the equation (1) with general F = F (u) is the Poincaré algebra AP (1, 1) with the basis operators p t = ∂ ∂t , p x = ∂ ∂x , J = tp x + xp t . The invariance algebras of the equation (1) will also include dilation operators e.g. for F = λu k or F = λ exp u. Equations (1) with e.g. F = 0 and F = λ exp u have infinite-dimensional symmetry algebras. Similarity solutions for the equation (1) can be found by symmetry reduction with respect to non-equivalent subalgebras of its invariance algebras. For studies of symmetry and non-classical solutions of the nonlinear wave equation for various space dimensions see [2]- [9]. Here we present results on classification of Q-conditional symmetries for the equation (1) and the relevant reductions in the meaningful cases. It seems that investigation of conditional symmetry now has fallen out of the mainstream of the symmetry analysis of PDE. We would guess that the reason is that practically all interesting equations for which the problem is manageable (mostly for the evolution equations) have been studied already. However, we believe that this problem remains relevant -first, with respect to investigation of "difficult, but interesting" equations (e.g. equations with highest derivatives for all variables of the same order, such as the nonlinear wave equation under study), and with respect to investigation of various related aspects (e.g. geometrical aspects and equivalence). What we mean by conditional symmetry Conditional symmetry in general (additional invariance under arbitrary additional condition) and a narrower concept of the Q-conditional invariance (the additional condition has the form Qu = 0) were initially discussed in the papers [10]- [14]. Later numerous authors developed these ideas into theory and a number of algorithms for studying symmetry properties of equations of mathematical physics. The importance of investigation of the Q-conditional symmetry stems from equivalence of the Q-conditional invariance and reducibility of the equations by means of ansatzes determined by such operators Q (see [15]). Here we will use the following definition of the Q-conditional symmetry: Definition 1. The equation Φ(x, u, u 1 , . . . , u l ) = 0, where u k is the set of all kth-order partial derivatives of the function u = (u 1 , u 2 , . . . , u m ), is called Q-conditionally invariant [5] under the operator Q = ξ i (x, u)∂ x i + η r (x, u)∂ u r if there is an additional condition Qu = 0,(2) such that the system of two equations Φ = 0, Qu = O is invariant under the operator Q. All differential consequences of the condition Qu = 0 shall be taken into account up to the order l − 1. This definition of the conditional invariance of some equation implies in reality a Lie symmetry (see e.g. the classical texts [16,17,18]) of the same equation with a certain additional condition. Conditional symmetries of the multidimensional nonlinear wave equations are specifically discussed in [20,24,25]. Previous work on the problem The particular problem we discuss here was first mentioned by P.Clarkson and E. Mansfield in [21] (the case f = f (u)), where the relevant determining equations were written out but not solved, and studies were continued by M. Euler and N. Euler in [22]. In the latter paper the determining conditions for the Q-conditional invariance were taken without consideration of the differential consequences of the condition Qu = 0, so the resulting operators did not actually present the solution of the problem. Following [21], we will consider the equation (in conic variables) equivalent to (1) of the form u yz = f (y, z, u).(3) We search for the operators of Q-conditional invariance in the form Q = a(y, z, u)∂ y + b(y, z, u)∂ z + c(y, z, u)∂ u . Background of Classification of Conditional Symmetries We will base our classification on the procedure of solution of the determining equations for conditional symmetry and then study equivalence within the resulting classes. Anyway, the equivalence group of the system of the equation (3) and the additional condition a(y, z, u)u y + b(y, z, u)u z + c(y, z, u) = 0 (5) determined by the operator (4) is quite narrow, and the standard classification procedure may not be appropriate. Let us note that the concept of equivalence of Q-conditional symmetries was introduced by R. Popovych in [23]. We study the conditional symmetry for the general case f = f (y, z, u). We will not consider the case f = 0, as equation (3) in this case has a general solution, so its conditional symmetries may be not quite relevant. There are some interesting special partial cases of the equation (3), first of all when f = f (u) and f = r(y, z)u that will be considered in a future paper. Considering the system (3), (5), we can see three inequivalent cases to be studied separately: 1. a = 0, b = 0. Then we can take Q = ∂ z + L(y, z, u)∂ u(6) The case a = 0, b = 0 is equivalent to a = 0, b = 0. In such cases the additional condition reduces equation (3) to a pair of the first-order equations. 2. a = 0, b = 0. In this case we can take Q = ∂ y + K(y, z, u)∂ z + L(y, z, u)∂ u .(7) where K(y, z, u) = 0. This case is obviously the most interesting for consideration. It might seem appropriate to consider separately the case c = 0, but it is easy to check that such systems may be equivalent to the general systems within case 2, so it should not be considered separately. 3. a = 0, b = 0 This case is trivial and in the case if the original equation and the additional condition are compatible, the additional condition just gives a solution for the equation. For cases 1-2 the additional condition Qu = 0 will be represented respectively by the equations u z = L(y, z, u),(8) and u y + K(y, z, u)u z = L(y, z, u).(9) We can start with considering of determining equations for the case 2 with K = 0, having in mind both cases. The determining equations for the conditional symmetry have the form −K 2 u + K uu K = 0,(10)−KL uu + K u K y K + K 2 u L K + K u (L u − K z ) − K uy − LK uu + KK zu = 0, (11) L uy − L uz K + L uu L − L u K y K + K y K z K − K yz − 3K u f − K u L K (L u − K z ) + K u L z − K zu L = 0,(12)−f y −Kf z −Lf u +L yz +L uz L+L u f − K y K (L z −f )−K z f − K u L K (L z −f ) = 0(13) Let us note that these determining equations were first found in [21], though, not studied further. Conditional Symmetry: Main Results Case 1 -K = 0 . Here we have equations u y = L, u yz = f.(14) The determining equations have the form L uy + L uu L = 0, −f y − Lf u + L yz + L uz L + L u f = 0 This case is actually equivalent to a pair of first-order equations u y = L, u z = f − L z L u .(15) If we check directly the compatibility conditions they will coincide with the determining equations of conditional invariance. Case 2.1. K u = 0, K = 0. The determining equations have the form −KL uu = 0, L uy − L uz K + L uu L − L u K y K + K y K z K − K yz = 0, −f y − Kf z − Lf u + L yz + L uz L + L u f − K y K (L z − f ) − K z f = 0 We have K = k(y, z), L = s(y, z)u + d(y, z). Using equivalence transformations, we can put d(y, z) = 0. From the determining equations we get k(y, z) = T y T z ,(16)s(y, z) = T yz T z ,(17) where T = T (y, z) is an arbitrary function. The operator of Q-conditional symmetry then will be Q = ∂ y + T y T z ∂ z + T yz T z u∂ u . In this case the ansatz reducing equation (3) will have the form u = σ(y, z)φ(ω),(18) where ω = ω(y, z) is a new variable, T y ω z + T z ω y = 0, T y σ z + T z σ y = σT yz . The reduced equation will have the form: σ yz φ + φ ′ (ω y σ z + ω z σ y + σω yz ) + φ ′′ σω y ω z = f,(19) where f satisfies the relevant conditions (13). From these conditions we can find the form of the function f up to equivalence: f = T y T z σ 3 Φ(ω, u σ ) + σ yz σ u,(20) where T (y, z) is the same arbitrary function entering expressions (16), (17). At the first glance equation (3) may seem equivalent to some equation of the form f = T y T z Φ(ω, u) reducible with the ansatz u = φ(ω). However, generally that is not the case. The criterion for such reduction has the following form: σ y = kσ z , where k is determined by (16). Let us have a further look at the reduced equation (19). From conditions on ω and σ it is easy to check that ω y σ z + ω z σ y + σω yz = 0, so the reduced equation will have the form φ ′′ σω y ω z = T y T z σ 3 Φ(ω, φ). Note that the reduced equations for this case will not include first-order derivatives. As from conditions on ω and σ σ 2 = T z ω z = − T y ω y , we come to the final form of the reduced equation φ ′′ = −Φ(ω, φ).(21) Equations of the form (21) include many remarkable ODE, equations for many special functions among them. Case 2.2. K u = 0, then K uu K = K 2 u , K = k(y, z)exp(l(y, z)u). We can put k = 1 and prove from the resulting determining conditions l y = l z = 0, so we can put l = 1. Then we can found that L = s(y, z)expu+d(y, z). It is possible to reduce this case to k = 1, and we get the following determining equation for f with arbitrary s and d: f = 1 3 (s y + d z ),(22) so f in this case depends only on y and z, and the equation u yz = f (y, z) is equivalent to the equation u yz = 0. The conditions for s and d have the form 2s yz − sd z + 2s y s − d zz = 0, − s yy + 2d yz + s y d − 2d z d = 0.(23) Conclusions We have considered the equations u yz = f (y, z, u) with f depending on y, z, u. For such general class the only nontrivial case is Case 2.1, K u = 0, K = 0. We have found that in this case the reduced equation has the form φ ′′ = −Φ(ω, φ), including many remarkable equations. We have found a general form of the equation (3) that can be reduced to an ODE by means of an ansatz (18) determined by the conditional symmetry operator (7) f has to be of the form (20). However, for a general equation it may be not straightforward to determine whether f can be reduced to such form. The cases f = f (u), f = r(y, z)u require special consideration, and have more inequivalent cases. Further research may also include study of the general conditional symmetry of the nonlinear wave equation in higher dimensions, as well as description of equivalence classes of conditional symmetries. Introduction Following [1], we discuss conditional symmetries and reductions of the twodimensional nonlinear wave equation u tt − u xx = F (t, x, u)(1) for the real-valued function u = u(t, x); t is the time variable, x is the space variable. In the equation above and further we will use the following designations for the partial derivatives: u t = − ∂u ∂t ; u x = − ∂u ∂x ; u tt = ∂ 2 u ∂t 2 ; u xt = u tx = ∂ 2 u ∂t∂x ; u xx = ∂ 2 u ∂x 2 . Note that the general equation in the class (1) has no invariance operators; however, many well-known particular cases have wide symmetry algebras, see e.g. [2]. The maximal invariance algebra of the equation (1) with general F = F (u) is the Poincaré algebra AP (1, 1) with the basis operators p t = ∂ ∂t , p x = ∂ ∂x , J = tp x + xp t . The invariance algebras of the equation (1) will also include dilation operators e.g. for F = λu k or F = λ exp u. Equations (1) with e.g. F = 0 and F = λ exp u have infinite-dimensional symmetry algebras. Similarity solutions for the equation (1) can be found by symmetry reduction with respect to non-equivalent subalgebras of its invariance algebras. For studies of symmetry and non-classical solutions of the nonlinear wave equation for various space dimensions see [2]- [9]. Here we present results on classification of Q-conditional symmetries for the equation (1) and the relevant reductions in the meaningful cases. It seems that investigation of conditional symmetry now has fallen out of the mainstream of the symmetry analysis of PDE. We would guess that the reason is that practically all interesting equations for which the problem is manageable (mostly for the evolution equations) have been studied already. However, we believe that this problem remains relevant -first, with respect to investigation of "difficult, but interesting" equations (e.g. equations with highest derivatives for all variables of the same order, such as the nonlinear wave equation under study), and with respect to investigation of various related aspects (e.g. geometrical aspects and equivalence). What we mean by conditional symmetry Conditional symmetry in general (additional invariance under arbitrary additional condition) and a narrower concept of the Q-conditional invariance (the additional condition has the form Qu = 0) were initially discussed in the papers [10]- [14]. Later numerous authors developed these ideas into theory and a number of algorithms for studying symmetry properties of equations of mathematical physics. The importance of investigation of the Q-conditional symmetry stems from equivalence of the Q-conditional invariance and reducibility of the equations by means of ansatzes determined by such operators Q (see [15]). Here we will use the following definition of the Q-conditional symmetry: Q = ξ i (x, u)∂ x i + η r (x, u)∂ u r if there is an additional condition Qu = 0,(2) such that the system of two equations Φ = 0, Qu = O is invariant under the operator Q. All differential consequences of the condition Qu = 0 shall be taken into account up to the order l − 1. This definition of the conditional invariance of some equation implies in reality a Lie symmetry (see e.g. the classical texts [16,17,18]) of the same equation with a certain additional condition. Conditional symmetries of the multidimensional nonlinear wave equations are specifically discussed in [20,24,25]. Previous work on the problem The particular problem we discuss here was first mentioned by P.Clarkson and E. Mansfield in [21] (the case f = f (u)), where the relevant determining equations were written out but not solved, and studies were continued by M. Euler and N. Euler in [22]. In the latter paper the determining conditions for the Q-conditional invariance were taken without consideration of the differential consequences of the condition Qu = 0, so the resulting operators did not actually present the solution of the problem. Following [21], we will consider the equation (in conic variables) equivalent to (1) of the form u yz = f (y, z, u).(3) We search for the operators of Q-conditional invariance in the form Q = a(y, z, u)∂ y + b(y, z, u)∂ z + c(y, z, u)∂ u . Background of Classification of Conditional Symmetries We will base our classification on the procedure of solution of the determining equations for conditional symmetry and then study equivalence within the resulting classes. Anyway, the equivalence group of the system of the equation (3) and the additional condition a(y, z, u)u y + b(y, z, u)u z + c(y, z, u) = 0 (5) determined by the operator (4) is quite narrow, and the standard classification procedure may not be appropriate. Let us note that the concept of equivalence of Q-conditional symmetries was introduced by R. Popovych in [23]. We study the conditional symmetry for the general case f = f (y, z, u). We will not consider the case f = 0, as equation (3) in this case has a general solution, so its conditional symmetries may be not quite relevant. There are some interesting special partial cases of the equation (3), first of all when f = f (u) and f = r(y, z)u that will be considered in a future paper. Considering the system (3), (5), we can see three inequivalent cases to be studied separately: 1. a = 0, b = 0. Then we can take Q = ∂ z + L(y, z, u)∂ u(6) The case a = 0, b = 0 is equivalent to a = 0, b = 0. In such cases the additional condition reduces equation (3) to a pair of the first-order equations. 2. a = 0, b = 0. In this case we can take Q = ∂ y + K(y, z, u)∂ z + L(y, z, u)∂ u . where K(y, z, u) = 0. This case is obviously the most interesting for consideration. It might seem appropriate to consider separately the case c = 0, but it is easy to check that such systems may be equivalent to the general systems within case 2, so it should not be considered separately. 3. a = 0, b = 0 This case is trivial and in the case if the original equation and the additional condition are compatible, the additional condition just gives a solution for the equation. For cases 1-2 the additional condition Qu = 0 will be represented respectively by the equations u z = L(y, z, u),(8) and u y + K(y, z, u)u z = L(y, z, u). We can start with considering of determining equations for the case 2 with K = 0, having in mind both cases. The determining equations for the conditional symmetry have the form −K 2 u + K uu K = 0,(10)−KL uu + K u K y K + K 2 u L K + K u (L u − K z ) − K uy − LK uu + KK zu = 0, (11) L uy − L uz K + L uu L − L u K y K + K y K z K − K yz − 3K u f − K u L K (L u − K z ) + K u L z − K zu L = 0,(12)−f y −Kf z −Lf u +L yz +L uz L+L u f − K y K (L z −f )−K z f − K u L K (L z −f ) = 0(13) Let us note that these determining equations were first found in [21], though, not studied further. u y = L, u yz = f.(14) The determining equation has the form −f y − Lf u + L yz + L uz L + L u f + (L uy + L uu L) f − L z L u = 0 This case is actually equivalent to a pair of first-order equations u y = L, u z = f − L z L u .(15) If we check directly the compatibility conditions they will coincide with the determining equation of conditional invariance. L uy − L uz K + L uu L − L u K y K + K y K z K − K yz = 0, −f y − Kf z − Lf u + L yz + L uz L + L u f − K y K (L z − f ) − K z f = 0 We have K = k(y, z), L = s(y, z)u + d(y, z). Using equivalence transformations, we can put d(y, z) = 0. From the determining equations we get k(y, z) = T y T z ,(16)s(y, z) = T yz T z ,(17) where T = T (y, z) is an arbitrary function. The operator of Q-conditional symmetry then will be Q = ∂ y + T y T z ∂ z + T yz T z u∂ u . In this case the ansatz reducing equation (3) will have the form u = σ(y, z)φ(ω),(18) where ω = ω(y, z) is a new variable, T y ω z + T z ω y = 0, T y σ z + T z σ y = σT yz . The reduced equation will have the form: σ yz φ + φ ′ (ω y σ z + ω z σ y + σω yz ) + φ ′′ σω y ω z = f,(19) where f satisfies the relevant conditions (13). From these conditions we can find the form of the function f up to equivalence: f = T y T z σ 3 Φ(ω, u σ ) + σ yz σ u,(20) where T (y, z) is the same arbitrary function entering expressions (16), (17). At the first glance equation (3) may seem equivalent to some equation of the form f = T y T z Φ(ω, u) reducible with the ansatz u = φ(ω). However, generally that is not the case. The criterion for such reduction has the following form: σ y = kσ z , where k is determined by (16). Let us have a further look at the reduced equation (19). From conditions on ω and σ it is easy to check that ω y σ z + ω z σ y + σω yz = 0, so the reduced equation will have the form φ ′′ σω y ω z = T y T z σ 3 Φ(ω, φ). Note that the reduced equations for this case will not include first-order derivatives. As from conditions on ω and σ σ 2 = T z ω z = − T y ω y , we come to the final form of the reduced equation φ ′′ = −Φ(ω, φ).(21) Equations of the form (21) include many remarkable ODE, equations for many special functions among them. Case 2.2. K u = 0, then K uu K = K 2 u , K = k(y, z)exp(l(y, z)u). We can put k = 1 and prove from the resulting determining conditions l y = l z = 0, so we can put l = 1. Then we can found that L = s(y, z)expu+d(y, z). It is possible to reduce this case to k = 1, and we get the following determining equation for f with arbitrary s and d: f = 1 3 (s y + d z ),(22) so f in this case depends only on y and z, and the equation u yz = f (y, z) is equivalent to the equation u yz = 0. The conditions for s and d have the form 2s yz − sd z + 2s y s − d zz = 0, − s yy + 2d yz + s y d − 2d z d = 0. Conclusions We have considered the equations u yz = f (y, z, u) with f depending on y, z, u. For such general class the only nontrivial case is Case 2.1, K u = 0, K = 0. We have found that in this case the reduced equation has the form φ ′′ = −Φ(ω, φ), including many remarkable equations. We have found a general form of the equation (3) that can be reduced to an ODE by means of an ansatz (18) determined by the conditional symmetry operator (7) f has to be of the form (20). However, for a general equation it may be not straightforward to determine whether f can be reduced to such form. The cases f = f (u), f = r(y, z)u require special consideration, and have more inequivalent cases. Further research may also include study of the general conditional symmetry of the nonlinear wave equation in higher dimensions, as well as description of equivalence classes of conditional symmetries. Definition 1 . 1The equation Φ(x, u, u 1 , . . . , u l ) = 0, where u k is the set of all kth-order partial derivatives of the function u = (u 1 , u 2 , . . . , u m ), is called Q-conditionally invariant [5] under the operator Case 2 . 1 . 21K u = 0, K = 0. The determining equations have the form −KL uu = 0, 5 Conditional Symmetry: Main ResultsCase 1 -K = 0 . Here we have equations General results on conditional symmetry for the two-dimensional nonlinear wave equation. I A Yehorchenko, arXiv:0910.1946Yehorchenko I.A. General results on conditional symmetry for the two-dimensional nonlinear wave equation, arXiv:0910.1946. The symmetry and some exact solutions of the nonlinear many-dimensional Liouville, d'Alembert and eikonal equations. W I Fushchych, N I Serov, J. Phys. A. 16Fushchych W.I. and Serov N.I., The symmetry and some exact solutions of the nonlinear many-dimensional Liouville, d'Alembert and eikonal equations,J. Phys. A, 1983, V.16, 3645-3658. Some remarks on similarity and soliton solutions of nonlinear Klein-Gordon equations. M Tajiri, J.Phys.Soc.Japan V. 53Tajiri M., Some remarks on similarity and soliton solutions of nonlinear Klein-Gordon equations, J.Phys.Soc.Japan V.53, 3759-3764. Solution of nonlinear relativistically invariant equations obtained by symmetry reduction. A M Grundland, J Harnad, P Winternitz, J. Math. Phys. 25Grundland A. M., Harnad J., Winternitz P., Solution of nonlinear relativistically invariant equations ob- tained by symmetry reduction. J. Math. Phys., 1984, V. 25, 791-806. Symmetry analysis and exact solutions of nonlinear equations of mathematical physics. W I Fushchych, W M Shtelen, N I Serov, Kyiv, Naukova Dumkain RussianFushchych W.I., Shtelen W.M. and Serov N.I., Symmetry analysis and exact solutions of nonlinear equations of mathematical physics, Kyiv, Naukova Dumka, 1989 (in Russian); On exact solutions of the nonlinear d'Alembert equation in Minkowski space R(1, n). W I Fushchych, A F Barannyk, Dokl. AN Ukr. SSR, Ser.A. 6Fushchych W. I., Barannyk A. F., On exact solutions of the nonlinear d'Alembert equation in Minkowski space R(1, n), Dokl. AN Ukr. SSR, Ser.A, 1990, No 6, 31-34. Partially invariant solutions of nonlinear Klein-Gordon and Laplace equations. L Martina, P Winternitz, J. Math. Phys., V. 33Martina L. and Winternitz P., Partially invariant solutions of nonlinear Klein-Gordon and Laplace equa- tions, J. Math. Phys., V. 33, 1992, 2718-2727. A family of nonlinear Klein-Gordon equations and their solutions. A M Grundland, E Infeld, J. Math. Phys., V. 33Grundland A. M. and Infeld E., A family of nonlinear Klein-Gordon equations and their solutions, J. Math. Phys., V. 33, 1992, 2498-2503. P Bizon, P Breitenlohner, D Maison, A Wasserman, arXiv:0905.3834v1Self-similar solutions of the cubic wave equation. Bizon P., Breitenlohner P., Maison D., Wasserman A., Self-similar solutions of the cubic wave equation, arXiv:0905.3834v1. The construction of special solutions to partial differential equations. P J Olver, P Rosenau, Phys. Lett. A. 114Olver P.J. and Rosenau P., The construction of special solutions to partial differential equations, Phys. Lett. A, 1986, V.114, 107-112. On a reduction and solutions of the nonlinear wave equations with broken symmetry. W I Fushchych, I M Tsyfra, J. Phys. A. 20Fushchych W.I. and Tsyfra I.M., On a reduction and solutions of the nonlinear wave equations with broken symmetry, J. Phys. A, 1987, V.20, L45-L48. Symmetry and exact solutions of nonlinear spinor equations. W I Fushchych, R Z Zhdanov, Phys. Reports. 172Fushchych W.I. and Zhdanov R.Z., Symmetry and exact solutions of nonlinear spinor equations, Phys. Reports, 1989, V.172, 123-174. New similarity solutions of the Boussinesq equation. P Clarkson, M D Kruskal, J. Math. Phys. 30Clarkson P. and Kruskal M.D., New similarity solutions of the Boussinesq equation, J. Math. Phys., 1989, V.30, 2201-2213. Non-classical symmetry reduction: example of the Boussinesq equation. D Levi, P Winternitz, J. Phys. A. 22Levi D. and Winternitz P., Non-classical symmetry reduction: example of the Boussinesq equation, J. Phys. A, 1989, V.22, 2915-2924. A precise definition of reduction of partial differential equations. R Z Zhdanov, I M Tsyfra, R O Popovych, J. Math. Anal. Appl. 1Zhdanov R.Z., Tsyfra I.M. and Popovych R.O., A precise definition of reduction of partial differential equations, J. Math. Anal. Appl., 1999, V.238, N 1, 101-123. Group analysis of differential equations. L V Ovsyannikov, Academic PressNew YorkOvsyannikov L.V., Group analysis of differential equations, New York, Academic Press, 1982. Application of Lie groups to differential equations. P Olver, Springer VerlagNew YorkOlver P., Application of Lie groups to differential equations, New York, Springer Verlag, 1987. Symmetries and differential equations. G W Bluman, S Kumei, Springer VerlagNew YorkBluman G.W. and Kumei S., Symmetries and differential equations, New York, Springer Verlag, 1989. Conditional invariance and exact solutions of the Klein-Gordon-Fock equation. I A Yehorchenko, A I Vorobyova, Dokl. Akad. Nauk Ukrainy. 3Yehorchenko I. A.and Vorobyova A. I., Conditional invariance and exact solutions of the Klein-Gordon-Fock equation. Dokl. Akad. Nauk Ukrainy, 1992, No.3, 19-22. Conditional invariance of the nonlinear equations of d'Alembert, Liouville, Born-Infeld, and Monge-Ampere with respect to the conformal algebra. Symmetry analysis and solutions of equations of mathematical physics. W I Fushchych, M I Serov, Akad. Nauk Ukrain. SSR, Inst. Mat. Fushchych W. I. and Serov M. I., Conditional invariance of the nonlinear equations of d'Alembert, Liouville, Born-Infeld, and Monge-Ampere with respect to the conformal algebra. Symmetry analysis and solutions of equations of mathematical physics, 1988, Akad. Nauk Ukrain. SSR, Inst. Mat., Kyiv, 98-102 . Peter A Clarkson, Clarkson, Peter A; Algorithms for the nonclassical method of symmetry reductions. Elizabeth L Mansfield, solv-int/9401002SIAM Journal on Applied Mathematics. 546Mansfield, Elizabeth L., Algorithms for the nonclassical method of symmetry reductions SIAM Journal on Applied Mathematics, V. 54, no. 6, 1693-1719. 1994, solv-int/9401002 Symmetries for a class of explicitly space-and time-dependent (1+1)-dimensional wave equations. Euler Marianna, Euler Norbert, J. Nonlinear Math. Phys. 1Euler Marianna and Euler Norbert, Symmetries for a class of explicitly space-and time-dependent (1+1)- dimensional wave equations. J. Nonlinear Math. Phys. 1997, V.1, 70-78. Equivalence of Q-conditional symmetries under group of local transformation. R O Popovych, Proceedings of the Third International Conference "Symmetry in Nonlinear Mathematical Physics. the Third International Conference "Symmetry in Nonlinear Mathematical PhysicsKyiv1Proceedings of Institute of MathematicsPopovych R.O., Equivalence of Q-conditional symmetries under group of local transformation, Proceedings of the Third International Conference "Symmetry in Nonlinear Mathematical Physics", Eds. A. Nikitin and V. Boyko, Proceedings of Institute of Mathematics, Kyiv, 2000, V.30, Part 1, 184-189 Conditional symmetry and exact solutions of the multidimensional nonlinear d'Alembert equation. A F Barannyk, D Yu, J. Nonlinear Math. Phys. Barannyk A.F. and Moskalenko Yu.D., Conditional symmetry and exact solutions of the multidimensional nonlinear d'Alembert equation, J. Nonlinear Math. Phys., 1996, V.3, 336-340. New conditional symmetries and exact solutions of the nonlinear wave equation. R Zhdanov, Panchak Olena, J. Phys. A. 31Zhdanov R. and Panchak Olena, New conditional symmetries and exact solutions of the nonlinear wave equation. J. Phys. A, 1998, V. 31, 8727-8734 General results on conditional symmetry for the two-dimensional nonlinear wave equation. I A Yehorchenko, arXiv:0910.1946Yehorchenko I.A. General results on conditional symmetry for the two-dimensional nonlinear wave equation, arXiv:0910.1946. The symmetry and some exact solutions of the nonlinear many-dimensional Liouville, d'Alembert and eikonal equations. W I Fushchych, N I Serov, J. Phys. A. 16Fushchych W.I. and Serov N.I., The symmetry and some exact solutions of the nonlinear many-dimensional Liouville, d'Alembert and eikonal equations,J. Phys. A, 1983, V.16, 3645-3658. Some remarks on similarity and soliton solutions of nonlinear Klein-Gordon equations. M Tajiri, J.Phys.Soc.Japan V. 53Tajiri M., Some remarks on similarity and soliton solutions of nonlinear Klein-Gordon equations, J.Phys.Soc.Japan V.53, 3759-3764. Solution of nonlinear relativistically invariant equations obtained by symmetry reduction. A M Grundland, J Harnad, P Winternitz, J. Math. Phys. 25Grundland A. M., Harnad J., Winternitz P., Solution of nonlinear relativistically invariant equations ob- tained by symmetry reduction. J. Math. Phys., 1984, V. 25, 791-806. Symmetry analysis and exact solutions of nonlinear equations of mathematical physics. W I Fushchych, W M Shtelen, N I Serov, Kyiv, Naukova Dumkain RussianFushchych W.I., Shtelen W.M. and Serov N.I., Symmetry analysis and exact solutions of nonlinear equations of mathematical physics, Kyiv, Naukova Dumka, 1989 (in Russian); On exact solutions of the nonlinear d'Alembert equation in Minkowski space R(1, n). W I Fushchych, A F Barannyk, Dokl. AN Ukr. SSR, Ser.A. 6Fushchych W. I., Barannyk A. F., On exact solutions of the nonlinear d'Alembert equation in Minkowski space R(1, n), Dokl. AN Ukr. SSR, Ser.A, 1990, No 6, 31-34. Partially invariant solutions of nonlinear Klein-Gordon and Laplace equations. L Martina, P Winternitz, J. Math. Phys., V. 33Martina L. and Winternitz P., Partially invariant solutions of nonlinear Klein-Gordon and Laplace equa- tions, J. Math. Phys., V. 33, 1992, 2718-2727. A family of nonlinear Klein-Gordon equations and their solutions. A M Grundland, E Infeld, J. Math. Phys., V. 33Grundland A. M. and Infeld E., A family of nonlinear Klein-Gordon equations and their solutions, J. Math. Phys., V. 33, 1992, 2498-2503. P Bizon, P Breitenlohner, D Maison, A Wasserman, arXiv:0905.3834v1Self-similar solutions of the cubic wave equation. Bizon P., Breitenlohner P., Maison D., Wasserman A., Self-similar solutions of the cubic wave equation, arXiv:0905.3834v1. The construction of special solutions to partial differential equations. P J Olver, P Rosenau, Phys. Lett. A. 114Olver P.J. and Rosenau P., The construction of special solutions to partial differential equations, Phys. Lett. A, 1986, V.114, 107-112. On a reduction and solutions of the nonlinear wave equations with broken symmetry. W I Fushchych, I M Tsyfra, J. Phys. A. 20Fushchych W.I. and Tsyfra I.M., On a reduction and solutions of the nonlinear wave equations with broken symmetry, J. Phys. A, 1987, V.20, L45-L48. Symmetry and exact solutions of nonlinear spinor equations. W I Fushchych, R Z Zhdanov, Phys. Reports. 172Fushchych W.I. and Zhdanov R.Z., Symmetry and exact solutions of nonlinear spinor equations, Phys. Reports, 1989, V.172, 123-174. New similarity solutions of the Boussinesq equation. P Clarkson, M D Kruskal, J. Math. Phys. 30Clarkson P. and Kruskal M.D., New similarity solutions of the Boussinesq equation, J. Math. Phys., 1989, V.30, 2201-2213. Non-classical symmetry reduction: example of the Boussinesq equation. D Levi, P Winternitz, J. Phys. A. 22Levi D. and Winternitz P., Non-classical symmetry reduction: example of the Boussinesq equation, J. Phys. A, 1989, V.22, 2915-2924. A precise definition of reduction of partial differential equations. R Z Zhdanov, I M Tsyfra, R O Popovych, J. Math. Anal. Appl. 1Zhdanov R.Z., Tsyfra I.M. and Popovych R.O., A precise definition of reduction of partial differential equations, J. Math. Anal. Appl., 1999, V.238, N 1, 101-123. Group analysis of differential equations. L V Ovsyannikov, Academic PressNew YorkOvsyannikov L.V., Group analysis of differential equations, New York, Academic Press, 1982. Application of Lie groups to differential equations. P Olver, Springer VerlagNew YorkOlver P., Application of Lie groups to differential equations, New York, Springer Verlag, 1987. Symmetries and differential equations. G W Bluman, S Kumei, Springer VerlagNew YorkBluman G.W. and Kumei S., Symmetries and differential equations, New York, Springer Verlag, 1989. Conditional invariance and exact solutions of the Klein-Gordon-Fock equation. I A Yehorchenko, A I Vorobyova, Dokl. Akad. Nauk Ukrainy. 3Yehorchenko I. A.and Vorobyova A. I., Conditional invariance and exact solutions of the Klein-Gordon-Fock equation. Dokl. Akad. Nauk Ukrainy, 1992, No.3, 19-22. Conditional invariance of the nonlinear equations of d'Alembert, Liouville, Born-Infeld, and Monge-Ampere with respect to the conformal algebra. Symmetry analysis and solutions of equations of mathematical physics. W I Fushchych, M I Serov, Akad. Nauk Ukrain. SSR, Inst. Mat. Fushchych W. I. and Serov M. I., Conditional invariance of the nonlinear equations of d'Alembert, Liouville, Born-Infeld, and Monge-Ampere with respect to the conformal algebra. Symmetry analysis and solutions of equations of mathematical physics, 1988, Akad. Nauk Ukrain. SSR, Inst. Mat., Kyiv, 98-102 . Peter A Clarkson, Clarkson, Peter A; Algorithms for the nonclassical method of symmetry reductions. Elizabeth L Mansfield, solv-int/9401002SIAM Journal on Applied Mathematics. 546Mansfield, Elizabeth L., Algorithms for the nonclassical method of symmetry reductions SIAM Journal on Applied Mathematics, V. 54, no. 6, 1693-1719. 1994, solv-int/9401002 Symmetries for a class of explicitly space-and time-dependent (1+1)-dimensional wave equations. Euler Marianna, Euler Norbert, J. Nonlinear Math. Phys. 1Euler Marianna and Euler Norbert, Symmetries for a class of explicitly space-and time-dependent (1+1)- dimensional wave equations. J. Nonlinear Math. Phys. 1997, V.1, 70-78. Equivalence of Q-conditional symmetries under group of local transformation. R O Popovych, Proceedings of the Third International Conference "Symmetry in Nonlinear Mathematical Physics. the Third International Conference "Symmetry in Nonlinear Mathematical PhysicsKyiv1Proceedings of Institute of MathematicsPopovych R.O., Equivalence of Q-conditional symmetries under group of local transformation, Proceedings of the Third International Conference "Symmetry in Nonlinear Mathematical Physics", Eds. A. Nikitin and V. Boyko, Proceedings of Institute of Mathematics, Kyiv, 2000, V.30, Part 1, 184-189 Conditional symmetry and exact solutions of the multidimensional nonlinear d'Alembert equation. A F Barannyk, D Yu, J. Nonlinear Math. Phys. Barannyk A.F. and Moskalenko Yu.D., Conditional symmetry and exact solutions of the multidimensional nonlinear d'Alembert equation, J. Nonlinear Math. Phys., 1996, V.3, 336-340. New conditional symmetries and exact solutions of the nonlinear wave equation. R Zhdanov, Panchak Olena, J. Phys. A. 31Zhdanov R. and Panchak Olena, New conditional symmetries and exact solutions of the nonlinear wave equation. J. Phys. A, 1998, V. 31, 8727-8734	[]
[ "Chaos in a Jahn-Teller Molecule", "Chaos in a Jahn-Teller Molecule" ]	[ "R S Markiewicz \nPhysics Department and Barnett Institute\nNortheastern U02115BostonMA\n" ]	[ "Physics Department and Barnett Institute\nNortheastern U02115BostonMA" ]	[]	The Jahn-Teller system E ⊗ b1 ⊕ b2 has a particular degeneracy, where the vibronic potential has an elliptical minimum. In the general case where the ellipse does not reduce to a circle, the classical motion in the potential is chaotic, tending to trapping near one of the extrema of the ellipse. In the quantum problem, the motion consists of correlated tunneling from one extremum to the opposite, leading to an average angular momentum reminiscent of that of the better known E ⊗ e dynamic Jahn-Teller system.	10.1103/physreve.64.026216	[ "https://arxiv.org/pdf/cond-mat/0102474v1.pdf" ]	45,883,574	cond-mat/0102474	247588d9bac47fa6a422197c0a5f0badef827399	Chaos in a Jahn-Teller Molecule 26 Feb 2001 R S Markiewicz Physics Department and Barnett Institute Northeastern U02115BostonMA Chaos in a Jahn-Teller Molecule 26 Feb 2001 The Jahn-Teller system E ⊗ b1 ⊕ b2 has a particular degeneracy, where the vibronic potential has an elliptical minimum. In the general case where the ellipse does not reduce to a circle, the classical motion in the potential is chaotic, tending to trapping near one of the extrema of the ellipse. In the quantum problem, the motion consists of correlated tunneling from one extremum to the opposite, leading to an average angular momentum reminiscent of that of the better known E ⊗ e dynamic Jahn-Teller system. In the well-known E ⊗ e Jahn-Teller (JT) effect, a molecule has a two-fold electronic degeneracy coupled to a doubly degenerate vibrational mode. This leads to a 'conical intersection' in the vibronic potential which has a degenerate, circular minimum ('Mexican hat potential'), although higher-order vibronic coupling can break the ring up into three degenerate minima along the trough ('tricorn potential') [1]. Quantum mechanically, the coupled electron-molecular vibrational (vibronic) wave function can tunnel between the three minima, leading to a ground state with a net angular momentum [2]. Remarkably, this 'orbital' angular momentum is quantized in half-integer multiples ofh, indicating the strong coupling of electronic and molecular motions. This quantization is a signature of the Berry phase [3,4] of π associated with the dynamic Jahn-Teller effect; the π Berry phase has been experimentally verified in triangular Na 3 molecules [5]. Points of conical intersection lead to chaotic behavior in the vibrational spectra, manifested quantum mechanically by anomalous level statistics [6]. However, the high symmetry of the E ⊗ e problem precludes chaos [7], so multimode interactions must be included, and the chaos generally appears at high energies (above the conical intersection) where many vibrational modes are excited. Here, it is shown that a simple modification of the symmetry preserves the anomalous Berry phase, yet leads to chaotic behavior at much lower energies, without the need of additional mode coupling. This case is the square X 4 molecule with square planar symmetry, D 4h , corresponding to an E ⊗b 1 ⊕b 2 Jahn-Teller problem [1], Fig. 1. The high symmetry allows two JT modes, with independent frequencies ω i , i = 1, 2, and electron-vibration couplings V i . In the special case ω 1 = ω 2 , V 1 = V 2 , the problem reduces exactly to that of the E ⊗ e molecule. However, there is an intermediate case, which seems not to have been explored till now. When V 1 /ω 1 = V 2 /ω 2 , the two modes have the same JT stabilization energy, E (i) JT = V 2 i /2M ω 2 i , and hence the vibronic potential has an elliptic minimum, which is not circular unless ω 1 = ω 2 . Given the elliptic minimum, the possibility of a periodic orbit arises. However, angular momentum is not conserved. In the present paper I analyze the resulting motion. The phonon modes B i of amplitude Q i are defined as follows. The four atomic positions, Fig. 1a, can be written as r 1 = r 10 − Q 1ŷ + Q 2x , r 2 = r 20 − Q 1x − Q 2ŷ , r 3 = r 30 + Q 1ŷ − Q 2x , r 4 = r 40 + Q 1x + Q 2ŷ ,(1) where the r i0 's are the positions of the X atoms in the undistorted square. The vibronic interaction Hamiltonian is H vib = V 1 Q 1 T x + V 2 Q 2 T y = V 1 Q 1 1 0 0 −1 + V 2 Q 2 0 1 1 0 .(2) Here the electronic operators are represented by the pseudospin T i 's and other factors are included in the electronphonon coupling V i . To the vibronic Hamiltonian must be added an electronic term H el and a phononic part H ph , with H ph = 1 2M P 2 1 + P 2 2 + M 2 ω 2 1 Q 2 1 + M 2 ω 2 2 Q 2 2 ,(3) with ω i the bare phonon frequencies. A spin-orbit coupling can be included [8] H so = λ L · S. For a static JT effect, the momenta P i can be neglected, and the Q i are chosen to minimize the energy, Eqs. 2,3. The solution can be written in terms of the JT energy E (i) JT = V 2 i /(2M ω 2 i ). For E (1) JT = E (2) JT , the lowest energy state consists of a distortion of the mode with larger JT energy only. For instance, if E (2) JT > E (1 JT ), the solution is Q 1 = 0, Q 2 = V 2 /(M ω 2 2 ), E = −E (2) JT . Special cases arise when E (1) JT = E (2) JT ≡ E JT .(5) Eliminating the electrons produces the vibronic potential surfaces E ± = M 2 ω 2 1 Q 2 1 + ω 2 2 Q 2 2 ± V 2 1 Q 2 1 + V 2 2 Q 2 2 + λ 2 4 .(6) When Eq. 5 is satisfied, the lower vibronic surface has a minimum which is degenerate along a trough, similar to the Mexican hat: Q 0 1 = Q 0 0 cos θ ω 1 , Q 0 2 = Q 0 0 sin θ ω 2 ,(7) with Q 0 0 = 2E JT − λ 2 /8E JT , and θ arbitrary. Near the trough, the lower potential surface can be expanded: E − = M 2 α ( g · q) 2 ,(8)with q = (q 1 , q 2 ), q i = Q i − Q 0 i , α = 1 − λ 2 /16E 2 JT and g = (ω 1 cos θ, ω 2 sin θ) -that is, there is a restoring force only 'perpendicular' to the trough. Defining β ω = ω 2 /ω 1 , the electronic eigenvectors are ψ + = cos γψ 1 + sin γψ 2 ψ − = − sin γψ 1 + cos γψ 2 ,(9) where tan γ = ( 1 + δ sin 2 θ − cos θ)/(β ω sin θ) and δ = β 2 ω − 1, Fig.2. By convention, ω 2 is assumed to be the higher frequency (β ω ≥ 1). If additionally ω 1 = ω 2 , the problem reduces to the E ⊗ e problem, and in Eq. 9 γ = θ/2: the electronic wave function is double valued: when θ changes by 2π, γ has only changed by π (the wave functions have changed sign). This sign change is the signature of a Berry phase [3,4], and causes the vibronic orbital angular momentum to take on half-integer values [2]. This can be seen as follows [1]. The z-component of orbital angular momentum is L z = (Q 1 P 2 − Q 2 P 1 )/h, and the operator which commutes with the vibronic hamiltonian, Eq. 2 is j z = L z + 1 2 T z .(10) Since L z is quantized in integers, j 2 has half-integral quanta. Note from Fig. 2 that even when ω 1 = ω 2 , θ must change by 4π to produce a 2π change in γ, suggesting a similar Berry phase. This can be directly demonstrated. The Berry phase is [9] γ B = −s 2π 0 ∂γ ∂θ dθ = −βs 2π 0 dθ 1 + sin 2 θ = −2πs,(11) where s is half an odd integer, introduced to make the total wave function single valued. Thus the Berry phase is π, modulo 2π, for any anisotropy. While this is a standard JT problem, I have not found any detailed analysis of the limit Eq. 5. As a first step, I perform a canonical transformation H ′ = e iS He −iS = H + i[S, H] − ...(12) with S = − V 1 ω 2 1 P 1 T x + V 2 ω 2 2 P 2 T y .(13) The canonical transformation can be performed exactly [10], but for present purposes only the first order result is needed. S, Eq. 13, was chosen to exactly cancel the term linear in Q. It yields a correction H ′ 2 = i[S, H vib ] = − T 0 2 V 2 1 ω 2 1 + V 2 2 ω 2 2 + V 1 V 2 ω 2 1 ω 2 2 AT z ,(14) with T 0 the identity matrix, A = ω 2 2 P 1 Q 2 − ω 2 1 P 2 Q 1 = −ω 2 + L z + ω 2 − (P 1 Q 2 + P 2 Q 1 ),(15) and ω 2 ± = (ω 2 1 ± ω 2 2 )/2. Thus, when ω − = 0, H ′ 2 is proportional to L z T z , and the angular momentum j z = L z + T z /2 is conserved (Eq. 10). For the present case ω − = 0 and j z is not constant. Given the presence of a circular trough in the potential, circulating orbits should be possible: could it be that there is a nonvanishing average < j z > = 0 even though j z is not constant? This possibility can be explored in the related classical Hamiltonian (particle in a non-linear potential well) by numerically integrating the equations of motion Q i = − dE − dQ i = −ω 2 i Q i 1 − 2E JT V 2 1 Q 2 1 + V 2 2 Q 2 2 + λ 2 /4 ≃ −αω 2 i Q i 1 − q 0 ω 2 1 Q 2 1 + ω 2 2 Q 2 2(16) where the last form utilizes the quadratic approximation, Eq. 8, dots indicate time derivatives, and q 2 0 = 2E JT α. The integral is evaluated using a Runge-Kutta routine with initial conditions Q(0) = (q 0 /ω 1 , 0),˙ Q(0) = (0, βq 0 /ω 2 ). In the remaining analysis, I take λ = 0. Given Q 1 (t), Q 2 (t), a winding angle φ is defined such thatφ = Q 1Q2 − Q 2Q1 Q 2 1 + Q 2 2 .(17) If one applies this procedure to the E ⊗ e problem (ω 2 = ω 1 ), the results are quite simple (long dashed line in Fig. 3): φ increases linearly with time, although the frequency is not constant, but varies approximately logarithmically with the velocity parameter β. By contrast, when ω 2 = ω 1 Figure 3 shows that φ is generically a random function of time, with no linearly increasing part indicative of a non-zero < j z >. The various data sets are characterized by the two parameters (ω 2 /ω 1 , β). (The figure utilizes the exact form of Eq. 16; the approximate form yields equivalent results.) The figure also clearly suggests that the motion is chaotic. This is further indicated by the direct time series, inset of Fig. 3. On the other hand, there are certain special values of the initial conditions for which the motion is approximately periodic, and φ does increase linearly with time. These values may most easily be found by plotting φ(T ) vs β for some long time T . Typical examples are illustrated in Fig. 3, while the time series are shown in Fig. 4. Poincare maps (plots of Q 1 vsQ 1 when Q 2 = 0), Figure 5, confirm the chaotic nature. (Note that the curve (7, 0.05394) is almost periodic -see particularly Q 1 (t), Fig. 4d -but the Poincare map is clearly chaotic, Fig. 5d.) While the E ⊗ e limit, β ω = 1, is quasiperiodic (the Poincare map is a smooth closed curve), for β ω = 1 even the special values are weakly chaotic, with the Poincare maps, Fig. 5a, having a finite spread away from smooth curves. The similarity of these special trajectories to scars in e.g., Sinai stadia [11] should be noted. How is this chaotic behavior manifested in the quantum limit? To explore this, it is convenient to first rescale the variables, so that the potential has circular symmetry, and the anisotropy appears in the ionic mass, m i = M (ω 0 /ω i ) 2 , with ω 2 0 = (ω 2 1 + ω 2 2 )/2, and then reduce the problem to one dimension by assuming that the motion is confined to the bottom of the trough and only φ varies. The Hamiltonian becomes H = −h 2 h/(2m + ρ 2 0 ), where ρ 0 is the equillibrium trough radius, m −1 ± = (m −1 2 ± m −1 1 )/2 and h = ∂ 2 φ +α[cos 2φ( 3 2 − ∂ 2 φ ) + 3 sin 2φ∂ φ ] − A 4 cos 4φ,(18) withα = m + /m − = (β 2 ω − 1)/(β 2 ω + 1) and higher order vibronic effects are incorporated in the term proportional to A 4 . Schroedinger's equation can be integrated numerically, letting ψ(φ, t) = ψ(jǫ, nδ) ≡ ψ n j , with ∂ φ ψ = (ψ n j+1 − ψ n j )/ǫ, and [12] ψ n+1 j = e −iHδ/h ψ n j ≃ 1 − iHδ/2h 1 + iHδ/2h ψ n j ,(19) or finally (1 − iγh)ψ n+1 j = (1 + iγh)ψ n j , with γ = hδ/4m + ρ 2 0 . Equation 19 was integrated numerically, assuming an initial gaussian distribution. Figure 6b shows \|ψ(φ, t)\| 2 for a variety of times t. The data can be better understood from Fig. 6a, which plots φ max vs t, where φ max is that value of φ for which \|ψ\| 2 has its maximum value. The wave function remains trapped in one of the effective potential wells, then quickly hops to the next one in a relatively short time. This hopping takes place by the probability spreading over two adjacent wells, as shown in Fig. 6b at times 4, 7, and 11. The tunneling is coherent, so there is a net circulation. Additional information can be found by analyzing the Husimi density [13] ρ H (p, q) = \| < p, q\|ψ > \| 2 , with < p, q\|ψ >= 4 s hπ exp[− s(φ − q) 2 2h − i p h (φ − q 2 )]ψ(φ)dφ,(20) which describes the approximate smearing of ψ in q and p as a function of time. Typical results are shown in Fig. 7, for squeezing parameter s = 1. Thus, the quantum system shows a 'memory' of the classical chaos, in that the wave function shows similar trapping near the points Q 2 = 0. However, whereas the wavefunctions appear to vary stochastically from cycle to cycle, Fig. 6b, the average of the wave function progresses smoothly, Fig. 6a. The main difference is that classically, the wave function can be reflected from a trapping region, reversing its direction of motion, while the quantum wave function always moves in the same direction, similar to the classical problem with special initial conditions. It seems plausible to interpret the special choice of initial conditions as analogous to a Bohr-Sommerfeld quantization condition in the quantum problem. As shown in Fig. 6a, the position of the wave function peak has a step-like component superposed on an average shift with time. This average shift is independent of the mass anisotropy [15], hence corresponding to the same quantized angular momentum as in the isotropic case. This is consistent with the Berry phase remaining π, Eq. 11, but somewhat surprising in light of the classical chaos. Blumel [16] has suggested that this might be a manifestation of quantum localization in angular momentum space [17], while the classical problem leads to angular momentum space diffusion. This possibility will be explored in future work. In addition to its molecular interest, the present results may have condensed matter applications. Berryonic matter [14] has been postulated to explain anomalous properties of Buckyballs and other dynamic JT systems, but based on unit cells of triatomic molecules. Potential applications are greatly expanded for bases of square molecules [15]. I thank J. Jose, F.S. Ham, R. Englman, R. Blumel, and B. Barbiellini for stimulating conversations. FIG. 1 . 1B1 (a) and B2 (b) distortions of a square X4 molecule. FIG. 2 . 2Electronic phase γ vs phononic phase θ, for δ = 0.1 (solid line), 1 (dotdashed line), 10 (dashed line), and 100 (dotted line). FIG. 3 . 3Winding angle φ calculated from Eq. 17, for several values of (βω, β). Inset = time series, Q1, Q2(t) for βω = ω2/ω1 = 2, β = 0.5. FIG. 4 . 4Time series, Q2(t) vs Q1(t) (or vs t, in (d)) for several choices of βω, β): (a) = (2,1), (b) = (3,0.75), (c,d) = (7,0.05394). In frames (a-c), the ellipses are equipotential contours, with the beaded contour representing the FIG. 6 . 6Quantum time evolution, showing (a) position of wave function peak as a function of time, and (b) actual distribution of \|ψ\| 2 at several equally-spaced time intervals (α = 0.3, A4 = 0). FIG. 7 . 7Contour plot of Husimi distribution ρH of data similar to that ofFig. 6at several time intervals. An interwell hopping event occurs between times 90 and 120. I B Bersuker, V Z Polinger, Vibronic Interactions in Molecules and Crystals. BerlinSpringerI.B. Bersuker and V.Z. 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Phys. 955701A. Delon, R. Jost, and M. Lombardi, J. Chem. Phys. 95, 5701 (1991). . H Köppel, W Domcke, L S Cederbaum, Advanc. Chem. Phys. 5759H. Köppel, W. Domcke, and L.S. Cederbaum, Advanc. Chem. Phys. 57, 59 (1984). . C J Ballhausen, Theoret. Chim. Acta (Berl.). 3368C.J. Ballhausen, Theoret. Chim. Acta (Berl.) 3, 368 (1965). . J W Zwanziger, E R Grant, J. Chem. Phys. 872954J.W. Zwanziger and E.R. Grant, J. Chem. Phys. 87, 2954 (1987). The Dynamical Jahn-Teller Effect in Localized Systems. M Wagner, Yu.E. Perlin and M. WagnerNorth Holland155AmsterdamM. Wagner, in "The Dynamical Jahn-Teller Effect in Lo- calized Systems", ed. by Yu.E. Perlin and M. Wagner (North Holland, Amsterdam, 1984), p. 155. . S Sridhar, Phys. Rev. Lett. 67785S. Sridhar, Phys. Rev. Lett. 67, 785 (1991); . A Kudrolli, S Sridhar, A Pandey, R Ramaswamy, Phys. Rev. 4911A. Kudrolli, S. Sridhar, A. Pandey, and R. Ramaswamy, Phys. Rev. E49, 11 (1994). . H J Korsch, H Wiescher, Computational Physics. K.H. Hoffman and M. Schreiber225SpringerH.J. Korsch and H. Wiescher, in "Computational Physics", edited by K.H. Hoffman and M. Schreiber (Springer, Berlin, 1996), p. 225. . K Husimi, Proc. Phys. Math. Soc. Jpn. 22264K. Husimi, Proc. Phys. Math. Soc. Jpn. 22, 264 (1940). . N Manini, E Tosatti, S Doniach, Phys. Rev. 513731N. Manini, E. Tosatti, and S. Doniach, Phys. Rev. B51, 3731 (1995). . R S Markiewicz, C Kusko, unpublishedR.S. Markiewicz and C. Kusko, unpublished. . R Blumel, personal communicationR. Blumel, personal communication. . F Borgonovi, G Casati, B Li, Phys. Rev. Lett. 774744F. Borgonovi, G. Casati, and B. Li, Phys. Rev. Lett. 77, 4744 (1996); . K M Frahm, D L Shepelyansky, Phys. Rev. Lett. 781440K.M. Frahm and D.L. Shepelyansky, Phys. Rev. Lett. 78, 1440 (1997).	[]
[ "Noun phrase reference in Japanese-to-English machine translation", "Noun phrase reference in Japanese-to-English machine translation" ]	[ "Francis Bond \nNTT Communication Science Laboratories\n\n", "Kentaro Ogura \nNTT Communication Science Laboratories\n\n", "Tsukasa Kawaoka \nNTT Communication Science Laboratories\n\n" ]	[ "NTT Communication Science Laboratories\n", "NTT Communication Science Laboratories\n", "NTT Communication Science Laboratories\n" ]	[]	This paper shows the necessity of distinguishing different referential uses of noun phrases in machine translation. We argue that differentiating between the generic, referential and ascriptive uses of noun phrases is the minimum necessary to generate articles and number correctly when translating from Japanese to English. Heuristics for determining these differences are proposed for a Japanese-to-English machine translation system. Finally the results of using the proposed heuristics are shown to have raised the percentage of noun phrases generated with correct use of articles and number in the Japanese-to-English machine translation system ALT-J/E from 65% to 77%.	null	[ "https://www.aclanthology.org/1995.tmi-1.1.pdf" ]	1,330,445	cmp-lg/9601008	ab593377a63843bc6ac3a4a71aae05cfa9d34460	Noun phrase reference in Japanese-to-English machine translation Francis Bond NTT Communication Science Laboratories Kentaro Ogura NTT Communication Science Laboratories Tsukasa Kawaoka NTT Communication Science Laboratories Noun phrase reference in Japanese-to-English machine translation This paper shows the necessity of distinguishing different referential uses of noun phrases in machine translation. We argue that differentiating between the generic, referential and ascriptive uses of noun phrases is the minimum necessary to generate articles and number correctly when translating from Japanese to English. Heuristics for determining these differences are proposed for a Japanese-to-English machine translation system. Finally the results of using the proposed heuristics are shown to have raised the percentage of noun phrases generated with correct use of articles and number in the Japanese-to-English machine translation system ALT-J/E from 65% to 77%. Introduction Determining the referential property of noun phrases is essential not only to understanding a text, but also to decide how to generate it in English. This paper proposes a heuristic algorithm to determine the referential properties of noun phrases in a Japanese text. The original motivation of the research was to improve the quality of English output by NTT Communication Science Laboratories' Japanese to English machine translation system ALT-J/E (Ikehara et al., 1991;. We expect, however, that the results will also be useful for text extraction and general text understanding. In this paper we use the term noun phrase reference to describe the relation between a noun phrase and what it stands for when it is used. We distinguish between three uses of noun phrases, two referential and one non-referential. A noun phrase can be used to refer in two different ways: GENERIC where a noun phrase is used to refer to a whole class, and REFERENTIAL where a noun phrase refers to a particular entity or entities. A third use is ASCRIPTIVE where a noun Now at Dōshisha University, Kyoto, JAPAN: <[email protected]>. phrase is used not to refer to anything but rather, normally with a copula verb, to ascribe a property to some referent. Although ASCRIPTIVE noun phrases are non-referring, we will refer to all three uses under the general term of noun phrase reference. This three-way distinction of noun phrase reference was introduced in Bond et al. (1994) and used as a, base to determine the countability and number of noun phrases in Japanese-to-English machine translation. In this paper we define exactly what is meant by the three kinds of reference and show how the distinction is essential in the generation of articles. This paper is structured as follows. First, we define the three kinds of referentiality which we distinguish and justify the definitions on theoretical and practical grounds, comparing them with those suggested by other researchers. We then describe in detail a heuristic method for determining noun phrase reference in Japanese sentences. Next, we show how the distinction is used in a Japanese to English machine translation system to generate articles and number. Finally, we look at experimental results gained by implementing the proposed methods and compare them to those achieved by an earlier version of the same system, and by other systems. Definition of noun phrase reference Noun phrase reference is of fundamental importance in any discussion of meaning (Lyons 1977). In English, it is also important in determining how articles should be used. In this section we give a more detailed definition of the three kinds of noun phrase reference under discussion and compare them with the definitions used in other machine translation systems. Generic: Noun phrases with generic reference denote an entire class: e.g. mammoth in Mammoths are extinct. In English generic noun phrases can normally be expressed in three ways, as discussed in Section 4 . 1 . Referential: Referential noun phrases are those that refer to some entity or entities in the discourse world: e.g. mammoth in There is a mammoth in my garden! Referential noun phrases are plural if there is more than one discrete referent, and are marked for definiteness. Ascriptive: Ascriptive noun phrases are used with a copula verb, or in an appositive expression, to ascribe a property to their subject: e.g. a mammoth in That animal is a mammoth. Because ascriptive noun phrases are nonreferring they cannot be the antecedent of other noun phrases. Zelinsky-Wibbelt (1992) distinguishes between GENERIC and IDENTIFYING, which appear to be equivalent to our GENERIC and REFERENTIAL. Zelinsky-Wibbelt's examples do contain ascriptive noun phrases, for example a human being in 'A spectator is a human being', instead they appear to be treated as adjective phrases in the rules (for example in their rule 14 (p. 797 op cit) where the complement of the copulative predicate with a generic subject is an evaluative adjective phrase). If the definition of adjective phrase has been expanded to include ASCRIPTIVE noun phrases 1 then our analysis is compatible. Unfortunately there is no discussion in Zelinsky-Wibbelt as to how effective their rules are when actually used in a machine translation system so we cannot make a quantitative comparison. Murata and Nagao (1993) distinguish between GENERIC and NON-GENERIC (which is further divided into DEFINITE and INDEFINITE), using heuristics similar to rewriting rules in expert systems. They make no distinction between REFER-ENTIAL and ASCRIPTIVE for non-generic noun phrases. This leaves open the possibility for conflict with their rule that a noun phrase will be definite if it has been presented previously. Consider the following sentence 2 : zō-wa honyūrui da-si, manmosu-mo honyūrui da. 'Elephant-TOP mammal be-and mammoth-ALSO mammal be.' Elephants are mammals and mammoths are also mammals. This will become Elephants are mammals and mammoths are also the mammals using the rules given. Distinguishing between REFERENTIAL and ASCRIPTIVE prevents this kind of problem from occurring. We compare their results to ours quantitatively in Section 5. Determination of noun phrase reference All proper nouns are, by definition, REFERENTIAL. The algorithm used to determine the referential property of noun phrases headed by common nouns is shown in Figure 1. The algorithm presented is based on single sentences, it does not address the considerable problems of using information from outside the sentence being considered 3 . It is possible for the algorithm to be applied to the Japanese parse tree as part of the semantic analysis 4 . In ALT-J/E, however, the algorithm is applied after the semantic analysis has finished, during the transfer stage, because much of the semantic information is stored in the transfer dictionaries where the combination of Japanese and English makes it easy to disambiguate word senses. The overall process of translation in ALT-J/E is divided into seven parts. First, the system splits the Japanese text into morphemes and assigns parts of speech. Second, it parses the segmented text, often giving multiple possible interpretations. 1 We feel this expanded definition is plausible, since the copula and ascriptive noun phrase combination fulfills the same semantic role as the copula and adjective phrase, that is, to ascribe a property. 2 Examples are given with the (romanized) Japanese original, a gloss and the human translation. The examples have been simplified to exemplify points more clearly; a new translation has been made for each simplified sentence. Japanese particles are glossed as follows: TOP for wa which marks the topic, OBJ for o which marks the object and GEN for no which shows a genitive relation. 3 Algorithms to use contextual information from outside the sentence are currently being implemented. 4 For information retrieval it is obviously essential to determine the referentiality of noun phrases as part of the source language analysis. Third, it rewrites complicated Japanese expressions into simpler ones. Fourth, ALT-J/E semantically evaluates the various interpretations. Fifth, syntactic and semantic criteria are used to select the best interpretation. Sixth, the selected interpretation is transferred into English. Finally, the English sentence is adjusted to give the correct inflectional forms. The algorithm described in this section has been implemented as part of the sixth stage. However, it could be implemented as part of the fifth stage. Rules are applied in the order shown in Figure 1, with later rules over-ruling earlier ones. The default assumption is that a noun phrase will be used to refer to some specific entity or entities in the discourse world, i.e. that it is REFERENTIAL. There are five rules that are applied at the sentence level, which use the meanings of verbs combined with the semantic categories of nouns 5 . These can all be overridden by subsequent rules. The subjects of verbs that predicate over an entire class, and the objects of verbs which predicate EMOTIVE ACTION or EMOTIVE STATE, are GENERIC. Verbs that trigger these rules, e.g. evolve, die out are marked in the lexicon (Bond et al., 1993). For copulas, the subject is GENERIC if its semantic category is a descendant of the semantic category of the object, while it's complement is taken to be ASCRIPTIVE by default 6 . Finally, appositive noun phrases will be judged to be ASCRIPTIVE, as though they were the complement of a copula. Note that these rules are only applied if the noun phrase in question is a common noun. In sentence 1, the semantic category of meeting place is ACTUAL PLACE, which is a child of the semantic category of Aoi hall PUBLIC PLACE. Aoi hall, however, is a proper noun so the rule is not applied. ( 1 ) Jap: kaijō-wa Aoi-kaikan φ. Gloss: meeting place-TOP Aoi hall is Eng: The meeting place is the Aoi Hall The next level of rules (level 3) applies to noun phrases modified by embedded sentences. Japanese makes no phonological, morphological, or syntactic distinctions between restrictive and non-restrictive relative clauses (Kuno 1973, p. 235). This algorithm uses a simple heuristic: a noun phrase modified by a tensed embedded sentence is REFERENTIAL. The next level of rules (level 4) is based on post-modification in the Japanese sentence. The use of some setsubiji 'suffixes' 7 implies that their modificant is 5 The meanings of nouns are given in terms of a semantic hierarchy of 2,800 nodes. Each node is called a semantic category. Edges in the hierarchy represent IS-A relationships, so that the child of a semantic category IS-A instance of it. For example, ORGAN IS-A BODY-PART . 6 If the complement is later judged to be REFERENTIAL by a subsequent rule it is equivalent, to judging that the copula has been used equatively. 7 setsubiji are a Japanese part of speech made up of suffixes that cannot stand alone, but change the meaning of the word they modify. (a) A noun phrase whose head is modified by a demonstratives or numeral is REFERENTIAL: kono otoko 'this man', futari-no otoko 'two men' (b) A noun phrase whose head is modified by the genitive construction is REF- ERENTIAL: hana-no saki 'the tip of my nose' 6. A noun phrase with a 'unique' referent is REFERENTIAL: chikyū 'the earth' Figure 1: Determination of noun phrase referentiality GENERIC. For example muke 'aimed at' in josei-muke-no-zasshi 'woman aimedat GEN magazine' a magazine, aimed at women. Similarly the construction Ato-iu-no-wa 'things called A' implies that its modificant is GENERIC. It can in fact be thought of as a pseudo-particle, the whole construction acting as a single marker which has the effect of marking its modificant as being a generic noun phrase used as the topic 8 . The next level of rules (level 5) makes a noun phrase whose head is modified by a demonstrative, numeral or the genitive construction NP-no 'NP's' REFER-ENTIAL. Note that only noun phrases modified by no judged to be genitive are REFERENTIAL. Partitive constructions such as ōkami-no-mure 'pack of wolf' a pack of wolves are not included in this judgment. The genitive construction may be translated into English in a variety of ways including a prepositional phrase headed by 'of', a possessive phrase with a clitic in the determiner position, or a possessive pronoun. Finally (level 6), noun phrases headed by nouns that are marked in the lexicon as likely to have a unique referent, such as chikyū 'the earth' are assumed to be REFERENTIAL. The algorithm presented in this section is only heuristic. Further work remains to be done to refine it. In particular: using the wa/ga distinction in conjunction with noun anaphora relations to distinguish between GENERIC and REFERENTIAL. and improving the rules at level 3 for relative clauses. 4 Using noun phrase referentiality to select articles and determine number Knowledge of a noun phrase's referential use is essential when translating from Japanese to English, as it plays a large part in determining how a noun phrase is expressed in English. In this section we show how articles and number are generated differently for the three different referentialities in the machine translation system ALT-J/E. Correct generation of articles and number is important not only to express meaning accurately, but because it is one of the major factors in determining the readability of Japanese-to-English translations. Translation of generic noun phrases A GENERIC noun phrase (with a countable head noun) can generally be expressed in three ways (Huddleston, 1984). We call these GEN 'a', where the noun phrase is indefinite: A mammoth is a mammal: GEN 'the', where the noun phrase is definite: The mammoth is a mammal; and GEN φ, where there is no article: 8 In ALT-J/E the entire construction (and the similar construction A-to-iu-mono-wa 'things called A') is rewritten during the Japanese rewriting stage into a pseudo-particle , which marks its modificant as being a generic noun phrase in the ha-case (TOPIC). It is not however necessary to do this, as shown in Murata (1993), where this construction is found by matching against the Japanese dependency structure. Mammoths are mammals. Uncountable nouns and pluralia tantum can only be expressed by GEN φ (eg: Furniture is expensive). They cannot take GEN 'a' and they do not take GEN 'the', because then the noun phrase would normally be interpreted as having definite reference. Nouns that can be either countable or uncountable take only GEN φor 'a': Cake is delicious/ Cakes are delicious, A cake is a kind of food. These combinations are shown in Table 1. Noun phrases that cannot be used to show GENERIC reference are marked with an asterisk (). The use of all three kinds of GENERIC noun phrases is not acceptable in some contexts, for example a mammoth evolved. Sometimes a noun phrase can be ambiguous, for example / like the elephant, where the speaker could like a particular elephant, or all elephants. Because the use of GEN φ is acceptable in all contexts, ALT-J/E generates all GENERIC noun phrases as such, that is as bare noun phrases. The number of the noun phrase depends on the Countability preference of the noun phrase heading it and there will be no article. Translation of referential noun phrases The Countability and number of REFERENTIAL noun phrases can be determined with heuristics that use information from the Japanese sentence along with knowledge of English Countability stored in the lexicon. This is described in Bond et al. (1994). According to Quirk et al. (1985:p 265), for REFERENTIAL noun phrases: The definite article the is used to mark the phrase it introduces as referring to something which can be identified uniquely in the contextual or general knowledge shared by speaker and hearer. Whether or not a REFERENTIAL noun phrase is definite or not is determined using heuristic criteria based on whether there is enough information to uniquely identify the noun phrase's referent, such as the following: • if the head noun is marked in the lexicon as being unique: the earth • if the noun phrase is made logically unique by a modifier: the best price • if the noun phrase's referent is restrictively described: the man who came to dinner, the aim of this research • direct and indirect anaphoric reference: I saw a cat and a dog. The dog chased the cat. As the above criteria are only meaningful for REFERENTIAL noun phrases, it is essential to determine whether the noun phrase is referential as a first step. When it has been determined whether a noun phrase is definite or indefinite, then articles can be generated 9 . In the final stage of processing, if there is no determiner, definite noun phrases take the definite article the. Indefinite countable singular noun phrases will take the indefinite article a/an, while indefinite countable plural and uncountable noun phrases will take the zero article φ. This is summarized in Table 2. Translation of ascriptive noun phrases The countability and number of predicativeASCRIPTIVE noun phrases matches that of their subject, and the countability and number of two appositive noun phrases match each other as described in Bond et al. (1994), with the following proviso. If one element is plural and the other is a collective noun such as group, then they need not match. For example, many insects, a whole swarm, . . . as opposed to many insects, bees I think, . . . . ALT-J/E makes the simplifying assumption that all ASCRIPTIVE noun phrases are indefinite. Therefore, articles will be generated in the same way as for indefinite REFERENTIAL noun phrases. Countable singular noun phrases will therefore take the indefinite article a/an, and countable plural and uncountable noun phrases will take the zero article φ. Results The processing described above has been implemented in ALT-J/E. The rules were designed using data from a specially constructed set of test sentences collected by the authors. The algorithm was evaluated on a collection of newspaper articles from the Nikkei-Sangyou newspaper by an English native speaker not connected with the development of the algorithm. The results are summarized in Table 3. 70% 46% 65% 5% New shows the results using the proposed method. Old shows the results using the unmodified system. We tested the system on newspaper articles, in the articles tested, there were an average of 7 noun phrases in each sentence. The articles were translated by ALT-J/E and the raw output examined by an English native speaker. Each noun phrase was given one of the following scores: For the purpose of evaluating the generation of articles and number, noun phrases that were either the BEST possible translation, or that had a problem only with STRUCTURE/CHOICE OF TRANSLATION, were judged to be successful. A thirdparty evaluator gave the success rates as 77% for the system with the proposed method and 65% for the original system. The method of evaluation described above does not give a reproducible, absolute level of success. It does, however, successfully show the overall level of improvement/degradation, and help to identify the remaining problems. Our initial evaluation was done by the the authors, who found the success rates at the noun phrase level to be 92% for the proposed method and 76% for the system as it used to be. Nakazawa points out that this shows that the evaluation method is not reproducible (personal communication May 1995). Because the goal is to produce a translation, which is new text, there is no objective target 10 This includes any major problems not connected with articles or number, such as outputing Japanese characters or spelling errors. to compare the results with. This is a perennial problem for machine translation output. Knight and Chander (1994) in a small pilot study showed that humans could replace articles (a/an and the) in an English text in which the articles had been replaced by blanks with an accuracy of around 95%. Raw machine translation output is less coherent, than normal English text and so deciding which article is appropriate is an even harder task. Discussion In this section we discuss the remaining errors and compare the results to two other systems. 168 of the 717 noun phrases in the machine translation of the newspaper articles had some problem. An brief analysis of the errors is given in Table 4. Testing on the newspaper articles revealed one major heuristic that had been overlooked in the algorithm presented in section 3: some nouns when heading a construction such as 'N-of-NP' carry an implication that the complement NP has GENERIC reference: for example, the applications of databases. This rule will be added to the algorithm at level 5, reducing the number of errors by around 8%. Apart from this there were no major changes that needed to be made to the algorithm. Overall, the largest sources of errors are problems with the source language analysis and dictionaries (22% each). These are not problems with the proposed algorithm but with the machine translation system as a whole. Another major source of errors is the translation of numerical expressions (12%). The processing for handling numerical expressions is currently being overhauled. The errors caused by lack of information in the dictionaries are solvable immediately, which will reduce the number of errors by around 20%. In the generation of articles and numbers for REFERENTIAL noun phrases some of the errors can simply be solved by the addition of new rules: for example, adding rules which use the meaning of adverbs to determine number or rules using pre-head modifiers to determine definiteness. The problems of common sense deduction and indirect anaphora, however, require a large scale knowledge base and inference rules. While both are being researched at the moment, they are unlikely to be implemented soon. We estimate that the number of errors caused by insufficiencies in the generation of articles and numbers for REFERENTIAL noun phrases can be reduced at least a quarter, thus reducing the total number of errors by around 8%,. Combining the above figures, we predict it is possible to reduce the errors by around 30%, bringing the total success rate to 84% for a window test. To go beyond this needs new processing to improve the source language analysis, the translation of numerical expressions and more use of contextual inferences. In addition examining even this small sample of text we came up with one major addition to the algorithm for determining noun phrase reference. Therefore the algorithm needs to be tested on a wider range of texts before the rules can be considered comprehensive. We have started testing the algorithm on a larger corpus of newspaper articles and are investigating methods for automatically learning rules. In Murata and Nagao (1993) success rates of 68.9% for referential property and 85.6% for number were given for unknown texts of the same genre as that used in development of the rules. Their approach seems effective, although we predict the lack of a ASCRIPTIVE class will cause problems. It is impossible to directly compare our results as Murata and Nagao's testing was all carried out in Japanese by the developers, so the problems of actually generating the English and getting an impartial evaluation were not addressed. Setting these considerations aside, when we separate our results for noun phrase reference (counting as failures noun phrases with errors in article use, noun phrase reference or the use of possessive determiners), and countability and number (counting as failures noun phrases with errors in number or countability), our proposed algorithm gave success rates of 74% and 85% respectively. Another approach is that of Knight and Chander (1994), who proposed using an automated post-editor to correct articles. Their prototype has a success rate for learning to replace articles when they have been removed from English texts of 78%. At present however the prototype cannot be used to post-edit output from a typical machine translation system as it assumes the knowledge that an article should be used in a given position, which is not normally available, and that the generation rules can function using machine translation output, which has not been shown. Conclusion This paper proposes a method that uses the information available in a Japanese sentence to identify a noun phrase as being used either GENERICALLY, REFEREN-TIALLY or ASCRIPTIVELY. This distinction is shown to be both theoretically justified and practically useful. The three way distinction in noun phrase reference is used as a base to determine a noun phrase's number and to generate appropriate articles and possessive pronouns when translating from Japanese to English. Incorporating this method into the machine translation system ALT-J/E helped to improve the percentage of noun phrases with correctly generated articles and number from 65% to 77%. It is shown that the proposed method can be straightforwardly extended to increase the success rate to 84%. Several problems remain to be explored. We consider the following to of primary importance: 1. Extension of the algorithm to translate texts as coherent passages, not just as single sentences. 2. Improvement of the reproducibility of the evaluation method. 3. Investigation of the coverage of the algorithm on a wider collection of texts. 1 . 1The default is REFERENTIAL 2. Sentence level rules (a) the subject of a verb marked in the lexicon as predicating over an entire class is GENERIC: manmosu-wa zetsumetsu-shita 'Mammoths died out' (b) if the semantic category of the subject of a copula is a descendant of the semantic category of the object then the subject is GENERIC: manmosu-wa dōbutsu-da 'Mammoths are animals'(a) A noun phrase whose head is modified by a tensed relative clause is REF-ERENTIAL:kinou kita otoko 'the man who came yesterday'4. Post-modification by setsubiji 'suffixes' and joshi-sōtōgo 'pseudo-particles'(a) the modificant of muke 'aimed at', yō 'for' .. .is GENERIC: josei-muke-no zasshi 'A magazine for women' (b) the modificant of -to-iu-no-wa 'things called' is GENERIC: kikai hon'yaku-to-iu-no-wa muzukashii 'Machine translation is difficult' 5. Modification by demonstratives, numerals and the genitive construction no 'of'(c) the object of a verb which predicates EMOTIVE ACTION or EMOTIVE STATE is GENERIC: watashi-wa manmosu-wo suki-da 'I like mammoths' (d) the complement of a copula is ASCRIPTIVE: manmosu-wa dōbutsu-da 'Mammoths are animals' (e) appositive noun phrases are ASCRIPTIVE: denwagaisha-no NTT 'NTT, a telephone company' 3. Modification by embedded sentences Table 1 : 1Genericness and CountabilityGEN Noun Countability Preference type Countable Both Uncountable 'a' a mammoth a cake a furniture 'the' the mammoth the cake the furniture φ mammoths cake/cakes furniture Table 2 : 2Generation of articles for referential noun phrases.Noun Phrase Number Definite Indefinite Countable singular the a/an Countable plural the φ Uncountable the φ Table 3 : 3Correct Generation of Articles and NumberTest Sentences Newspaper Articles NPs (240) Sentences (120) NPs (717) Sentences (102) New: 94% 90% 77% 15% Old: Table 4 : 4Errorsin the generation of articles and number (168 noun phrases from newspaper articles)Problem AreaFreq. Description of error Analysis error 22% The Japanese noun phrase was parsed incorrectly so the rules did not trigger. Dictionary errors 22% The dictionary entry was incomplete. Kbits of networks per second should be a 384 Kbit/s network Reference 8% There needs to be a rule to make database GENERIC in expressions like: the strategic applications of databases which is currently translated as the strategic applications of a database Reference 5% Miscellaneous errors in determining noun phrase reference. Number 9% In some cases rules using common sense and inference are needed to determine the number correctly: for example sales counter should be plural in the sales counter of telephone companies through out the country Number 2% There are no rules to deduce number from information given by adverbs: for example prices should be plural in The price is 5 yen and 15 yen respectively Articles 7% The rules for deciding whether a noun has been restrictively described by an embedded sentence are too coarse. Articles 6% There needs to be a rule for indirect anaphora. two models should be definite in NTT introduced video-tel 111 and video-tel 222 in June. Two models are the first to have video receivers. Articles 3% There needs to be a rule to make a noun phrase definite if its pre-head modifier restricts it sufficiently: for example NTT will enter a video rental business Articles 4% Miscellaneous errors in determining whether a noun phrase is definite or not.Numerical 12% Complicated numerical expressions are Expressions translated badly: f o r e x a m - ple 384 As well as generating definite and indefinite articles, ALT-J/E also generates possessive pronouns(Bond et al. 1995) and some/any for REFERENTIAL noun phrases when appropriate. AcknowledgmentsThe paper has benefited greatly from the comments of the anonymous reviewers for TMI, Graham, Monique and Mitsuyo Bond, Satoru Ikehara, Roly Sussex and especially Tsuneko Nakazawa. We would like to thank Toshiaki Nebashi, Kazuya Fukamachi and Yoshitake Ichii for their invaluable help in implementing the processing described here. Countability and number in Japanese to English machine translation. [ References, Bond, Proceedings of the 15th International Conference on Computational Linguistics (COLING-94). the 15th International Conference on Computational Linguistics (COLING-94)References [BOND et al. (1994)] BOND, FRANCIS, KENTARO OGURA, and SATORU IKE- HARA. 1994. Countability and number in Japanese to English machine trans- lation. In Proceedings of the 15th International Conference on Computational Linguistics (COLING-94), 32-38. Possessive pronouns as determiners in Japanese-to-English machine translation. [ Bond, Proceedings of the 2nd Pacific Association for Computational Linguistics Conference. the 2nd Pacific Association for Computational Linguistics Conference95[BOND et al. (1995)] -----, KENTARO OGURA, and SATORU IKEHARA. 1995. Possessive pronouns as determiners in Japanese-to-English machine transla- tion. 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(1991)] IKEHARA, SATORU, SATOSHI SHIRAI, AKIO YOKOO, and HIROMI NAKAIWA. 1991. Toward an MT system without pre-editing -effects of new methods in ALT-J/E. In Proceedings of MT Summit III, 101-106. Automated postediting of documents. [ Knight, Kevin Chander ; Knight, Ishwar Chander, AAAI-94. AAAI. [KNIGHT AND CHANDER (1994)] KNIGHT, KEVIN, and ISHWAR CHANDER. 1994. Automated postediting of documents. In AAAI-94. AAAI. Susumu Kuno, The Structure of the Japanese Language. Cambridge, Massachusetts, and London, EnglandMIT PressKUNO (1973)[KUNO (1973)] KUNO, SUSUMU. 1973. The Structure of the Japanese Language. Cambridge, Massachusetts, and London, England: MIT Press. . John Lyons, Cambridge University Press2CambridgeLYONS, JOHN. 1977. Semantics, volume 2. Cambridge: Cam- bridge University Press. Research into the determination of referential property and number of nouns using Japanese structure as a guide. Bachelor's thesis. Masaki Murata, Kyoto, JapanKyoto Universityin JapaneseMURATA, MASAKI, 1993. Research into the determination of referential property and number of nouns using Japanese structure as a guide. Bachelor's thesis, Kyoto University, Kyoto, Japan. (in Japanese) Determination of referential property and number of nouns in Japanese sentences for machine translation into English. Murata, Nagao, Proceedings of the Fifth International Conference on Theoretical and Methodological Issues in Machine Translation (TMI-93). the Fifth International Conference on Theoretical and Methodological Issues in Machine Translation (TMI-93)and MAKOTO NAGAO[MURATA AND NAGAO (1993)] -----------, and MAKOTO NAGAO. 1993. Determi- nation of referential property and number of nouns in Japanese sentences for machine translation into English. In Proceedings of the Fifth International Conference on Theoretical and Methodological Issues in Machine Translation (TMI-93), 218-25. 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Fifth International Conference on Theoretical and Methodological Issues in Machine Translation (TMI-93)[SHIRAI et al. (1993)] SHIRAI. SATOSHI, SATORU IKEHARA, and TSUKASA KAWAOKA. 1993. Effects of automatic rewriting of source language within a Japanese to English MT system. In Proceedings of the. Fifth International Conference on Theoretical and Methodological Issues in Machine Translation (TMI-93). Exploiting linguistic iconism for article selection in machine translation. Cornelia Zelinsky-Wibbelt ; Zelinsky-Wibbelt, Proceedings of the. 14th International Conference on Computational Linguistics (COLING-92. the. 14th International Conference on Computational Linguistics (COLING-92[ZELINSKY-WIBBELT (1992)] ZELINSKY-WIBBELT, CORNELIA. 1992. Exploit- ing linguistic iconism for article selection in machine translation. In Proceedings of the. 14th International Conference on Computational Linguistics (COLING- 92). 792-798.	[]
[ "β Functions of Orbifold Theories and the Hierarchy Problem", "β Functions of Orbifold Theories and the Hierarchy Problem" ]	[ "Csaba Csáki [email protected] \nDepartment of Physics\nUniversity of California\n94720BerkeleyCA\n\nTheoretical Physics Group\nLawrence Berkeley National Laboratory University of California\n94720BerkeleyCA\n", "Witold Skiba [email protected] \nDepartment of Physics\nUniversity of California at San Diego\nLa Jolla92093CA\n", "John Terning [email protected] \nDepartment of Physics\nUniversity of California\n94720BerkeleyCA\n\nTheoretical Physics Group\nLawrence Berkeley National Laboratory University of California\n94720BerkeleyCA\n" ]	[ "Department of Physics\nUniversity of California\n94720BerkeleyCA", "Theoretical Physics Group\nLawrence Berkeley National Laboratory University of California\n94720BerkeleyCA", "Department of Physics\nUniversity of California at San Diego\nLa Jolla92093CA", "Department of Physics\nUniversity of California\n94720BerkeleyCA", "Theoretical Physics Group\nLawrence Berkeley National Laboratory University of California\n94720BerkeleyCA" ]	[]	We examine a class of gauge theories obtained by projecting out certain fields from an N = 4 supersymmetric SU(N) gauge theory. These theories are non-supersymmetric and in the large N limit are known to be conformal.Recently it was proposed that the hierarchy problem could be solved by embedding the standard model in a theory of this kind with finite N. In order to check this claim one must find the conformal points of the theory. To do this we calculate the one-loop β functions for the Yukawa and quartic scalar couplings. We find that with the β functions set to zero the one-loop quadratic divergences are not canceled at sub-leading order in N; thus the hierarchy between the weak scale and the Planck scale is not stabilized unless N is of the order 10 28 or larger. We also find that at sub-leading orders in N renormalization induces new interactions, which were not present in the original Lagrangian.	10.1103/physrevd.61.025019	[ "https://arxiv.org/pdf/hep-th/9906057v2.pdf" ]	16,766,010	hep-th/9906057	64b30e0422a0700913327f4aa9ebf1e26d3a9813	β Functions of Orbifold Theories and the Hierarchy Problem Jun 1999 Csaba Csáki [email protected] Department of Physics University of California 94720BerkeleyCA Theoretical Physics Group Lawrence Berkeley National Laboratory University of California 94720BerkeleyCA Witold Skiba [email protected] Department of Physics University of California at San Diego La Jolla92093CA John Terning [email protected] Department of Physics University of California 94720BerkeleyCA Theoretical Physics Group Lawrence Berkeley National Laboratory University of California 94720BerkeleyCA β Functions of Orbifold Theories and the Hierarchy Problem Jun 1999arXiv:hep-th/9906057v2 15 * Research fellow, Miller Institute for Basic Research in Science. We examine a class of gauge theories obtained by projecting out certain fields from an N = 4 supersymmetric SU(N) gauge theory. These theories are non-supersymmetric and in the large N limit are known to be conformal.Recently it was proposed that the hierarchy problem could be solved by embedding the standard model in a theory of this kind with finite N. In order to check this claim one must find the conformal points of the theory. To do this we calculate the one-loop β functions for the Yukawa and quartic scalar couplings. We find that with the β functions set to zero the one-loop quadratic divergences are not canceled at sub-leading order in N; thus the hierarchy between the weak scale and the Planck scale is not stabilized unless N is of the order 10 28 or larger. We also find that at sub-leading orders in N renormalization induces new interactions, which were not present in the original Lagrangian. Introduction The study of conformal symmetry has a long history in particle physics. Recently it has attracted renewed interest due to the work of Maldacena [1] on the correspondence between string theory on anti-de Sitter backgrounds and four dimensional conformal field theories, and further work on the orbifold projections of these theories [2][3][4][5][6][7][8][9][10][11][12]. An interesting result of this work [2][3][4][5][6] is that non-supersymmetric gauge theories obtained by orbifolding an N = 4 SUSY SU(N) gauge theory are conformal in the large N limit. Additional non-supersymmetric conformal theories can be obtained from a similar construction in type 0 string theories [13,14]. Although conformal theories are seemingly quite esoteric, the idea of using static or slowly running couplings to generate a large hierarchy of scales has cropped up many times in particle phenomenology. Attempts to use approximate conformal symmetry in phenomenology have included such diverse topics as: electroweak symmetry breaking (walking technicolor) [15,16,17], the hunt for light composite scalars [16,18,19] (including the search for a Goldstone boson of spontaneously broken scale invariance * [18]), dynamical supersymmetry breaking [21], and the cosmological constant problem [22]. Most recently Frampton and Vafa [11,12] have conjectured that orbifold theories are conformal at finite N, and further proposed that embedding the standard model in an orbifold theory can solve the naturalness problem of the electroweak scale (stabilizing the large hierarchy of scales without fine-tuning). This sudden appearance of such a simple solution to a long standing problem is quite surprising, so it seems worthwhile to discuss the underlying ideas of this scenario in some detail. It has been previously noted [23] that conformal symmetry can remove the quadratic divergences that are responsible for destabilizing the hierarchy between the weak scale and a more fundamental scale like the Planck scale. In a conformal theory we must insist on regulators (like dimensional regularization) that respect conformal invariance or include counterterms that maintain the symmetry. With such a regularization quadratic divergences are impossible (since there is no cutoff scale on which they could depend). Such a resolution of the naturalness problem is of course only valid if the theory is exactly conformal (i.e. physics is the same at any length scale). In the real world we know that physics is not conformal below the weak scale, and we expect that the fundamental theory of gravity will not be scale invariant since gravity has an intrinsic scale associated with it. Thus the best we can hope for phenomenologically is a theory that is approximately scale invariant in some energy range. That is we can only have an effective conformal theory that is valid above some infrared cutoff (which must be above the weak scale) and below some ultraviolet (UV) cutoff M (which must be at or below the Planck scale). From the perspective of the fundamental theory there is some non-conformal physics above (or near) the scale M (e.g. heavy particles or massive string modes) which we can integrate out of the theory. Studying the sensitivity of the effective theory to the cutoff M is equivalent to studying sensitivity of the low-energy physics to the details of the very high-energy physics. If we believe that there is a new fundamental scale of physics beyond the weak scale then in a "natural theory" we would like to see that the weak scale is not quadratically sensitive to changes in the high scale. The two known solutions to the naturalness problem are to either lower the UV cutoff of the effective theory to the weak scale (e.g. technicolor, large extra dimensions) or to arrange cancelations of the quadratic divergences order-byorder in perturbation theory (e.g. supersymmetry). One might expect that an effective conformal theory would fall into the latter category, however the vanishing of β functions does not imply the cancellation of quadratic divergences, they are independent [23]. To see that they are independent one need only consider supersymmetric theories where quadratic divergences cancel independently of the values of β functions. In this paper we consider a class of N = 4 orbifold theories [11,12] at one loop. We explicitly calculate the β functions, solve for the couplings by imposing that the β functions vanish, and calculate the quadratic divergences. We find that the quadratic divergences do not cancel for finite N. We also discuss new interactions that are induced by renormalization group (RG) running, and remark on some open questions. The Orbifold Theories In this section we review the construction of N = 4 orbifold theories, and present the matter content and Lagrangian for the particular models that we will be considering in this paper. One starts with an N = 4 supersymmetric SU(N) gauge theory. The field content of this theory is (all fields are in the adjoint representation): gauge bosons A µ , which are singlets of the SU(4) R global symmetry, four copies of (two-component) Weyl fermions Ψ i , i = 1, 2, 3, 4, which transform as the fundamental 4 under the SU(4) R , and six copies of (real) scalars Φ ij which transform as the antisymmetric tensor 6 of SU(4) R . In the procedure of orbifolding (discussed in detail in Refs. [2][3][4][5][6][7][8][9][10]) one chooses a discrete subgroup Γ of the SU(4) R symmetry of order \|Γ\|, and also embeds this subgroup into the gauge group (chosen here to be SU(N\|Γ\|)) as N copies of its regular representation (for a very clear explanation of this embedding see Ref. [9]). Orbifolding then means projecting out all fields from the theory which are not invariant under the action of the discrete group Γ. If Γ is a generic subgroup of SU(4) R , then one obtains a non-supersymmetric theory. If Γ is embedded in an SU(3) subgroup of SU(4) R then one obtains an N = 1 supersymmetric theory, while if Γ is embedded in an SU(2) subgroup of SU(4) R one obtains an N = 2 supersymmetric theory. For a compilation of results on discrete subgroups of SU(3) and SU(4) see Refs. [24] and [25]. We are interested only in the non-supersymmetric theories, in which case Γ must be a subgroup of SU(4). In order to simplify the analysis of the β functions, we restrict our attention in this paper to the case when Γ is Abelian, Γ = Z k . In this case we start with an SU(Nk) gauge group, and after orbifolding we obtain an SU(N) k theory. Let us denote the k-th root of unity e 2πi k by ω. An embedding of Z k into SU(4) R is specified by the transformation properties of the fundamental representation: 4 → diag (ω k 1 , ω k 2 , ω k 3 , ω k 4 ) 4. This embedding is an SU(4) subgroup if k 1 + k 2 + k 3 + k 4 = 0 mod k (in order to insure that the determinant is one), moreover k 1 , k 2 , k 3 , k 4 = 0 mod k so that we obtain a non-supersymmetric theory. In order to simplify our calculations, we will assume in this paper that no two k i 's are equal, and also that k i +k j = 0 mod k. With the assumption that k i + k j = 0 mod k one can avoid the presence of adjoint scalars, and thus all fermions and scalars will be in bifundamental representations. The assumption that no two k i 's are equal implies that there is only a single field with given gauge quantum numbers. This is probably the simplest and most symmetric orbifold theory that one can consider. However, we believe that the conclusions we draw from these particular orbifolds could be generalized to more complicated embeddings. With this choice of embedding of the discrete group we get the following field content for our orbifold theory: -gauge bosons A µ for every gauge group SU(N), -(two-component) fermions Ψ(m, m + k i ) ≡ Ψ αm β m+i which transform as fundamentals under the m-th SU(N) factor in the SU(N) k product and as antifundamental under m + k i (m is arbitrary, i = 1, 2, 3, 4, and m + i is a short hand for m + k i ), -complex scalars Φ(m, m + l i ) ≡ Φ αm β m+i which transform as fundamentals under the m-th group and as antifundamentals under m + l i (m is arbitrary, l i = k i + k 4 and i = 1, 2, 3). Note that for the scalars a different shorthand is employed, m + i represents m + k i + k 4 . The Lagrangian of orbifold theories is obtained from the original Lagrangian by retaining only terms containing fields invariant under the discrete symmetry. We give the N = 4 Lagrangian in the Appendix. The Yukawa couplings in the orbifold theory are given by L Y ukawa = −Y m,i<j Ψ αm β m+i Ψ β m+i γp Φ † γp αm + h.c. , (2.1) where in the above sum m + i is again a shorthand for m + k i , and p = m + k i + k j . Note that, unlike in the supersymmetric theory, there is no factor of √ 2 appearing in this coupling. The quartic scalar couplings are given by L quart = − 1 2 m,j<i λ 1 φ αm β m+i φ γ m+i αm φ δm γ m+i φ β m+i δm − λ 3 φ αm β m−i φ γ m+i αm φ δm γ m+i φ β m−i δm +λ 4 φ αm β m−i φ γ m+j αm φ δm γ m+j φ β m−i δm + φ αm β m+i φ γ m−j αm φ δm γ m−j φ β m+i δm +λ 5 φ αm β m+i φ γ m+j αm φ δm γ m+j φ β m+i δm + φ αm β m−i φ γ m−j αm φ δm γ m−j φ β m−i δm −2λ 2 φ αm β m−j φ γ m+i αm φ δ m+i−j γ m+i φ β m−j δ m+i−j + φ αm β m−i φ γ m+j αm φ δ m−i+j γ m+j φ β m−i δ m−i+j , (2.2) where we have used the shorthand notation m + i = m + l i = m + k i + k 4 in the above sums. In N = 1 language, the λ 1 , λ 3 , and λ 4 couplings are descendants of the D-terms, while the λ 2 coupling is a descendant of the superpotential term, and λ 5 receives contributions from both terms. In our normalization λ 5 is twice the superpotential coupling minus the D-term coupling. The Lagrangian obtained by orbifolding the N = 4 theory corresponds to "degenerate" values of couplings: Y 2 = λ 1 = λ 2 = λ 3 = λ 4 = λ 5 = g 2 , where g is the gauge coupling. However, as we will see below, for these values of the couplings the β functions do not vanish. Therefore, if the theory is indeed conformal for finite N, one has to assume that there will be a different set of couplings for which all the β functions vanish. However, for generic values of the quartic scalar couplings the potential is unbounded from below, while when all couplings are identical the potential is positive definite (as guaranteed by the supersymmetry of the theory it was projected from). We will assume that the ratios of the couplings are sufficiently close to one at the zeros of the β functions so that the potential is bounded. We will see later that this is true in the large N limit. The Renormalization Group Equations To calculate the one-loop β functions we rely heavily on the work of Machacek and Vaughn [26] who summarized one-loop results and derived two-loop β functions for a general field theory. We first calculated the N = 4 SUSY β functions for the gauge, Yukawa and quartic couplings despite the fact that they are related by supersymmetry. In order for this calculation to be useful for the non-supersymmetric orbifold theories one has to refrain from using the superfield formalism and instead deal separately with component scalar, fermion, and gauge boson fields. There is a term by term correspondence between the N = 4 theory and the orbifolded theory in the large N limit [6]. The fact that all the β functions vanish when SUSY relations are imposed between the various couplings provides strong cross checks on the calculation. At one-loop the gauge β function vanishes identically [11], so at one-loop the gauge coupling is a free parameter. The general one-loop β function for the Yukawa couplings is [26]: (4π) 2 β a Y = 1 2 Y † 2 (F )Y a + Y a Y 2 (F ) + 2Y b Y †a Y b +Y b Tr Y †b Y a − 3g 2 m {C m 2 (F ), Y a } (3.1) where Y a ij is the Yukawa coupling of scalar a to fermions i and j, Y 2 (F ) = Y †a Y a ,(3.2) and C m 2 (F ) is the quadratic Casimir of the fermion fields transforming under the m-th gauge group. Thus the first term in Eq. (3.1) represents scalar loop corrections to the fermion legs, the second term 1PI corrections from the Yukawa interactions themselves, the third term fermion loop corrections to the scalar leg, and the last term represents gauge loop corrections to the fermion legs. The Yukawa β function can be derived by projecting the N = 4 result graph by graph (see the Appendix). The only changes are that \|Γ\|N is replaced by N and the fermions are in bifundamental representations rather than the adjoint. Thus we find: (4π) 2 β Y = 6NY 3 − 6 N 2 − 1 N g 2 Y , (3.3) so β Y vanishes when Y = Y * ≡ g 1 − 1 N 2 . (3.4) Note that this result is independent of the values of the quartic scalar couplings. In the notation of Machacek and Vaughn [26] the β function for a quartic scalar coupling at one-loop is given by (4π) 2 β λ = Λ 2 − 4H + 3A + Λ Y − 3Λ S ,(3.5) where Λ 2 corresponds to the 1PI contribution from the quartic interactions themselves and should not be confused with a mass scale, H corresponds to the fermion box graphs, A to the two gauge boson exchange graphs, Λ Y to the Yukawa leg corrections, and finally Λ S corresponds to the gauge leg corrections. The contributions to Λ 2 , H, and Λ Y can be found by simply projecting the N = 4 results (see the Appendix). The contributions to Λ S can be found by noting that the scalars are bifundamentals rather than adjoints. The gauge boson exchange term, A, can be calculated by a simple manipulation of the gauge generators, which is explained in the Appendix. We find: (4π) 2 β λ 1 = N(4λ 2 1 + λ 2 3 + 2λ 2 4 + 2λ 2 5 − 16Y 4 + 16λ 1 Y 2 ) +3 N 2 − 4 N g 4 − 12 N 2 − 1 N g 2 λ 1 , (3.6) (4π) 2 β λ 2 = N(−2λ 2 λ 4 − 2λ 2 λ 5 + 8Y 4 − 16λ 2 Y 2 ) + 12 N 2 − 1 N g 2 λ 2 , (3.7) (4π) 2 β λ 3 = N( 1 2 λ 2 3 − 2λ 1 λ 3 + 2λ 4 λ 5 − 8λ 3 Y 2 ) +3 N 2 − 4 2N g 4 + 6 N 2 − 1 N g 2 λ 3 , (3.8) (4π) 2 β λ 4 = N( 1 2 λ 2 5 + 2λ 2 2 + λ 2 4 + 2λ 1 λ 4 − λ 3 λ 5 − 8Y 4 + 8λ 4 Y 2 ) +3 N 2 − 4 2N g 4 − 6 N 2 − 1 N g 2 λ 4 , (3.9) (4π) 2 β λ 5 = N( 1 2 λ 2 5 + 2λ 2 2 − λ 3 λ 4 + λ 4 λ 5 + 2λ 1 λ 5 − 8Y 4 + 8λ 5 Y 2 ) +3 N 2 − 4 2N g 4 − 6 N 2 − 1 N g 2 λ 5 . (3.10) Finding the general solution for the simultaneous zeroes of the β λ functions is obviously a complicated problem, here we choose to focus on the solutions for the couplings that reduce in the large N limit to the N = 4 SUSY fixed point, i.e. λ i * → g 2 . At order 1/N 4 there are two such solutions which are given by: λ 1 * ≈ g 2 1 − 5 8N 2 + 459 1024N 4 + . . . λ 2 * ≈ g 2 1 − 19 16N 2 − 387 2048N 4 + . . . λ 3 * ≈ g 2 1 − 7 4N 2 − 423 512N 4 + . . . (3.11) λ 4 * ≈ g 2 1 − 5 8N 2 + 459 1024N 4 + . . . λ 5 * ≈ g 2 1 − 5 8N 2 + 459 1024N 4 + . . . and λ 1 * ≈ g 2 1 − 19 16N 2 + 225 8192N 4 + . . . λ 2 * ≈ g 2 1 − 47 32N 2 − 1467 16384N 4 + . . . λ 3 * ≈ g 2 1 − 5 8N 2 − 153 4096N 4 + . . . (3.12) λ 4 * ≈ g 2 1 − 1 16N 2 + 5067 8192N 4 + . . . λ 5 * ≈ g 2 1 − 1 16N 2 + 5067 8192N 4 + . . . We should note that the zeroes of the β functions are not true fixed points. This is because we have not included all possible quartic couplings allowed by gauge invariance, we have only included the quartic couplings that arise from the projection from the N = 4 theory. Examples of operators that do not appear in the tree-level Lagrangian of these orbifold theories include φ αm β m+i φ β m+i αm φ γm δ m+i φ δ m+i γm and φ αm β m+i φ β m+i αm φ γm δ m−i φ δ m−i γm . (3.13) Such gauge invariant operators are induced, for example, by two gauge boson exchange diagrams. In the non-supersymmetric theory there is no symmetry or non-renormalization theorem that prevents these operators from appearing via RG evolution. A full calculation would require considering all possible quartic interactions, and finding the simultaneous zeroes of all β functions. However, if the fixed point values of some of these new couplings are nonzero then, as we will see, we loose the special large N behavior of the pure projected theory. We will proceed as follows: we assume that the effective "conformal" theory is embedded in a more fundamental theory at a scale M (e.g. some set of particles of mass M are integrated out of the theory at this scale), we assume that the theory has been arranged such that the β functions for Y and λ i vanish, and that at this particular renormalization scale, M, all other quartic couplings vanish. We can then compute the proper 1PI contribution to the mass of any particular scalar. We will only keep the quadratically divergent piece. The quadratic divergence is given by m 2 φ = N(2λ 1 − λ 3 + 2λ 4 + 2λ 5 ) + 3 N 2 − 1 N g 2 − 8Y 2 N M d 4 p (2π) 4 1 p 2 . (3.14) Plugging in our solutions for the zeroes of the β functions we have (to lowest non-vanishing order in N) for both cases: m 2 φ = 3g 2 N M 2 16π 2 . (3.15) Note that, as expected, the terms linear in N canceled. Thus we see that there is a technically unnatural hierarchy in this set of theories. In order to keep the scalars light a mass counterterm must be tuned, order by order, to cancel quadratic divergences. Alternatively, N has to be taken extremely large. For m = m weak ≈ 1 TeV, M = M P l ≈ 10 18 GeV, g 2 4π ≈ 1 30 we find that one would need N ≈ 10 28 . We now briefly comment on the possible effects of including other quartic operators like those displayed in Eq. (3.13). There is a contribution to Λ 2 of (3.5) of order N 2 (λ new ) 2 , the contribution to A is of order g 4 (see Appendix). Thus the form of the β function is: (4π) 2 β new k = N 2 a ij k λ new i λ new j + Nb ij k λ new i λ j + c ij k λ i λ j + 16Nλ new k Y 2 +d k 3(1 + 2 N 2 )g 4 − 12 N 2 − 1 N g 2 λ new k ,(3.16) where we have taken the coupling λ new k to have the same sign and normalization as λ 1 . In the above formula, d k is an integer, depending on how many gauge groups the scalar fields share (see Appendix). Thus we expect λ new k to be of order g 2 /N at a fixed point. The contribution of the graphs arising from these operators to the quadratic divergence is of order N 2 , so the contribution to m 2 φ is of order g 2 N. Thus the inclusion of these additional operators seems to make the naturalness problem much worse. It may be possible to cancel the quadratic divergence order by order, but a priori there seems to be no reason for such a cancellation to occur at a fixed point of the theory. Using the methods presented above one can also calculate the two-loop gauge β function. The two-loop piece of the gauge β function in a general gauge theory is given by [26]: β (2) g = − g 3 (4π) 4 34 3 (C 2 (G)) 2 − 1 2 4C 2 (F ) + 20 3 C 2 (G) S 2 (F ) − 4C 2 (S) + 2 3 C 2 (G) S 2 (S) g 2 + Y 4 (F ) , (3.17) where C 2 (G) is the Casimir of the adjoint, C 2 (F )S 2 (F ) is the sum over (twocomponent) fermions of the Casimir times the Dynkin index in the given representation, C 2 (S)S 2 (S) is the same for complex scalars, while Y 4 (F ) is the contribution of the Yukawa couplings defined by TrY a Y †a t A t B = Y 4 (F )δ AB , (3.18) where Y a are the Yukawa coupling matrices for the scalar field a, and t A are the gauge generators in the representation of the fermion fields. For the orbifold theory considered above these expressions are given by C 2 (G) = N, C 2 (F )S 2 (F ) = 4(N 2 − 1), C 2 (G)S 2 (F ) = 4N 2 , C 2 (S)S 2 (S) = 3(N 2 − 1), C 2 (G)S 2 (S) = 3N 2 , Y 4 (F ) = 24N 2 Y 2 . (3.19) Note that Eq. (3.17) is independent of the quartic scalar couplings. At the one-loop fixed point of the Yukawa coupling, which is also independent of the values of the quartic scalar couplings, Y 2 = N 2 −1 N 2 g 2 . Using this value we find that the leading order terms in N cancel, and the sub-leading pieces give (3.20) thus the theory is not asymptotically free. If the theory is indeed conformal, then the fixed point would necessarily be a UV fixed point. In order to check whether the theory is conformal or not, one would need to study the three-loop gauge β function. If the three-loop term turns out to be negative and of O(N 2 ), then there will be a perturbative UV fixed point, since the fixed point will be If this theory turns out to be conformal with a perturbative fixed point, then this could provide an interesting example of a theory with a non-trivial UV fixed point. Such a theory could then serve as a counter example to the conjecture presented in Ref. [28]. β (2) g = 4g 5 (4π) 4 > 0, Conclusions In this paper we have considered a particular class of non-supersymmetric orbifold theories obtained from finite N = 4 theories. Our calculations are summarized by equations (3.3), (3.6)-(3.10) and (3.15). We calculated the one-loop β functions and found the simultaneous zeroes that approach the SUSY fixed point in the large N limit. At one-loop the theory possesses quadratic divergences in sub-leading orders in N and therefore cannot stabilize the weak scale without N being unreasonably large. RG running also generates new operators (quartic scalar couplings) which are not present in the tree-level orbifold Lagrangian. These new couplings will shift the fixed point values of the original operators, and also contribute to the quadratic divergences themselves. It is possible, but unlikely, that with these new couplings all quadratic divergences vanish. The difficulty in canceling the quadratic divergences stems from the fact that the contributions of the new operators to the quadratic divergence is more important in the 1/N expansion than the divergences we have discussed here. We think that a cancellation is unlikely to occur, but the importance of the problem merits further investigation which would involve the renormalization of the full set of operators allowed by symmetries. Appendix A N = 4 β Functions N = 4 supersymmetric theories are finite, therefore the β function vanishes to all orders in perturbation theory. In terms of component fields the N = 4 Lagrangian has three different kinds of couplings: gauge, Yukawa, and quartic scalar. Even though these couplings are related by N = 4 supersymmetry it is useful to calculate their β functions separately. In the orbifold theories different couplings are not related by supersymmetry, yet N = 4 results are helpful in the calculation of the non-supersymmetric β functions. The N = 4 theory can be thought of as an N = 1 theory with three adjoint chiral superfields and a superpotential for these fields. When the N = 4 theory is expressed in terms of N = 1 component fields the SU(4) R global symmetry is not explicit in the Lagrangian, only its SU(3) × U(1) subgroup is manifest. In terms of components the Lagrangian is given by L N =4 = − 1 4 F µν F µν − iλ a σ µ D µ λ a − iΨ a i σ µ D µ Ψ a i + D µ φ †a i D µ φ a i + − √ 2gf abc (φ †c i λ a Ψ b i −Ψ c iλ a φ b i ) − Y √ 2 ǫ ijk f abc (φ c i Ψ a j Ψ b k +Ψ c iΨ a j φ † b k ) + g 2 2 (f abc φ b i φ † c i )(f ade φ d j φ † e j ) − Y 2 2 ǫ ijk ǫ ilm (f abc φ b j φ c k )(f ade φ †d l φ †e m ), (A.1) where a, . . . , e = 1, . . . , N 2 −1 are the adjoint gauge indices, while i, . . . , m = 1, 2, 3 are SU(3) flavor indices. The SU(N) structure constant is denoted by f abc , λ is the (two-component) gaugino, Ψ i are the (two-component) adjoint fermions, and φ i are the three complex adjoint scalars. Meanwhile g is the gauge coupling and Y is the coupling of the superpotential term for the chiral superfields. The above Lagrangian is N = 4 supersymmetric for Y = g. In order to easily identify the origin of different terms in the calculation it is instructive to keep Y explicit in the Lagrangian. The one-loop (as well as two-loop) β functions are known for a general field theory [26]. In order to use the formulae given in Ref. [26] one needs to calculate certain group theoretic factors. This calculation can be conveniently carried out using the method of Cvitanovic [27], in which one draws a separate "group theory diagram" for every Feynman diagram. Evaluating these group theory diagrams will then amount to calculating the group theory coefficients needed for the general formulas of the β functions of [26]. Since all fields are in the adjoint representation every Yukawa coupling carries a factor f abc while every quartic scalar coupling carries a factor f abc f ade . In order to obtain the group theory diagrams one replaces every factor of if abc with a cubic vertex (see Fig. 1). The diagram obtained this way does not have to coincide with the actual form of the Feynman diagram that one is evaluating. Using the Lagrangian (A.1) and the above rules of calculating the group theory factors one can obtain the various β functions for the N = 4 theory. The one-loop β function for the gauge coupling is given by (4π) 2 β g = −g 3 ( 11 3 C 2 (G) − 2 3 S 2 (F ) − 1 3 S 2 (S)), (A.2) where C 2 (G) is the Casimir of the adjoint, S 2 (F ) is the Dynkin index of the (two-component) fermions, and S 2 (S) is the Dynkin index for the complex scalars. For the N = 4 theory C 2 (G) = N, S 2 (F ) = 4N, S 2 (S) = 3N, and thus β g = 0. The one-loop β function for the Yukawa coupling Y a in a general gauge theory is given by the formula (4π) 2 β a Y = 1 2 (Y † 2 (F )Y a + Y a Y 2 (F )) + Y b Y †a Y b + Y b TrY †b Y a −3g 2 {C 2 (F ), Y a }. (A.3) In the case of the N = 4 theory we evaluate the β function of the vertex − √ 2gf abc φ †c i λ a Ψ b i . In the projected orbifold theory all Yukawa couplings are equal due to the Z k symmetry of the theory, thus we can use any of the N = 4 vertices to obtain the projected result. For this coupling the different Figure 2: Contributions to the β function of the quartic scalar couplings of the fields φ a 1 φ †b 1 φ c 2 φ †d 2 in the N = 4 theory. The ordering of the fields in the above diagrams is clockwise, with φ a 1 in the upper left corner. The meaning of the above group theory diagrams is explained in Fig. 1. terms in the above β function are: 1 2 (Y † 2 (F )Y a + Y a Y 2 (F )) = (4Ng 2 + 2NY 2 ) √ 2g, Y b Y †a Y b = (−4NY 2 ) √ 2g, Y b TrY †b Y a = (2Ng 2 + 2NY 2 ) √ 2g, −3g 2 {C 2 (F ), Y a } = (−6Ng 2 ) √ 2g. (A.4) The sum of these terms adds up to zero independently of the value of Y , which can be understood in the following way: for Y = g we have an N = 1 supersymmetric theory with three adjoint fermions and a non-vanishing superpotential. Since we have chosen the β function of the Yukawa coupling involving the gaugino, therefore the Yukawa β function has to be proportional to the gauge β function for any value of Y . The one-loop β function of the gauge coupling is independent of Y therefore the cancellation has to happen for a generic value of Y . This provides an independent check of our result. Finally we calculate the one loop β functions for the quartic scalar couplings. The general formula for an arbitrary gauge theory is given by Figure 3: Contributions to the β function of the quartic scalar couplings of the fields φ a 1 φ †b 1 φ c 1 φ †d 1 in the N = 4 theory. The ordering of the fields in the above diagrams is clockwise, with φ a 1 in the upper left corner. The meaning of the above group theory diagrams is explained in Fig. 1. (4π) 2 β quartic = Λ 2 + 3A − 4H + Λ Y − 3Λ S . (A.5) ( + ) ( + ) 2 Λ = Y Λ = ( We calculate two different combinations of quartic β functions in the N = 4 theory: one for the coupling of the operator φ a 1 φ †b 1 φ c 2 φ †d 2 , for which the contributions are given in Fig. 2, and another for the operator φ a 1 φ †b 1 φ c 1 φ †d 1 the contributions to which are given in Fig. 3. Combining these results according to Eq. (A.5) one finds that these β functions indeed vanish for the N = 4 theory. Cancellation of various terms occurs after decomposing the "gluon box" diagrams [27] in a complete basis of group theory tensors using the results given in Fig. 4. In order to project the N = 4 theory down to the orbifolded theory it is convenient to make use of large N double-line notation, since all our fields are bifundamentals. To do this we need two SU(N) identities: These results are taken from [27]. if abc = 2Tr (T a T b T c − T c T b T a ) , To keep the fields canonically normalized after changing from the single index basis to the double index basis we need to rescale φ a = √ 2φ j i (T a ) i j . (A.8) Using these identities and representing δ i j by a line with an arrow we can obtain the large N results given in Fig. 5. At tree-level the effect of orbifolding is similar to taking the above large N limit, the only difference is that different oriented lines can correspond to different gauge groups. The appropriate combination of gauge groups for each vertex can be read off from the projected Lagrangian (2.2). Once we have the tree-level vertices we can simply calculate all the diagrams relevant to the β functions. Additionally we can apply the projection rules to the N = 4 diagrams involving quartic or Yukawa couplings, however sub-leading terms in N can be generated in loops, and these terms must be kept. This procedure provides a check on the calculation. The double line notation is also convenient for gauge diagrams, however 1/N terms are already present in the gauge boson propagator so a little more care must be taken. We illustrate the use of the double line notation in the calculation of the proper correction to the quartic coupling from two gauge boson exchange. For simplicity, we consider the case of two different scalar fields that share one gauge group. The calculation proceeds by using the identity (A.7) and is depicted in Fig. (6). g 2 = 2O(1/N 2 ) and higher loop corrections to the gauge β function can be neglected. For any other case there cannot be a perturbative fixed point. For example if the three-loop term is O(N), then any putative fixed point can only be seen by summing all planar diagrams. Such a fixed point could exist independent of the sign of the three-loop term. Figure 1 : 1The group theory Feynman diagrams for the Yukawa couplings and the quartic scalar couplings of the N = 4 theory. Figure 4 : 4The diagrammatic representation of the SU(N) group theory identities needed to show that the N = 4 β functions of the quartic couplings do indeed vanish. The first line gives the decomposition of the "gluon box diagram" in terms of a complete set of tensors, the second line is the Jacobi identity, while the third line is an identity relating different combinations of the d and f tensors. A single unconnected line corresponds to δ a b . Figure 5 : 5The large N rules for adjoints. Figure 6 : 6The proper correction to quartic couplings from gauge boson exchange. 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[ "Region Growing Curriculum Generation for Reinforcement Learning", "Region Growing Curriculum Generation for Reinforcement Learning" ]	[ "Artem Molchanov \nUniversity of Southern\nCalifornia 2 Nvidia\n", "Karol Hausman \nUniversity of Southern\nCalifornia 2 Nvidia\n", "Stan Birchfield \nUniversity of Southern\nCalifornia 2 Nvidia\n", "Gaurav Sukhatme \nUniversity of Southern\nCalifornia 2 Nvidia\n" ]	[ "University of Southern\nCalifornia 2 Nvidia", "University of Southern\nCalifornia 2 Nvidia", "University of Southern\nCalifornia 2 Nvidia", "University of Southern\nCalifornia 2 Nvidia" ]	[]	Learning a policy capable of moving an agent between any two states in the environment is important for many robotics problems involving navigation and manipulation. Due to the sparsity of rewards in such tasks, applying reinforcement learning in these scenarios can be challenging. Common approaches for tackling this problem include reward engineering with auxiliary rewards, requiring domain-specific knowledge or changing the objective. In this work, we introduce a method based on region-growing that allows learning in an environment with any pair of initial and goal states. Our algorithm first learns how to move between nearby states and then increases the difficulty of the startgoal transitions as the agent's performance improves. This approach creates an efficient curriculum for learning the objective behavior of reaching any goal from any initial state. In addition, we describe a method to adaptively adjust expansion of the growing region that allows automatic adjustment of the key exploration hyperparameter to environments with different requirements. We evaluate our approach on a set of simulated navigation and manipulation tasks, where we demonstrate that our algorithm can efficiently learn a policy in the presence of sparse rewards.	null	[ "https://arxiv.org/pdf/1807.01425v1.pdf" ]	49,570,858	1807.01425	73abd0733e0125c123e6a0b23472772a9c343e5f	Region Growing Curriculum Generation for Reinforcement Learning Artem Molchanov University of Southern California 2 Nvidia Karol Hausman University of Southern California 2 Nvidia Stan Birchfield University of Southern California 2 Nvidia Gaurav Sukhatme University of Southern California 2 Nvidia Region Growing Curriculum Generation for Reinforcement Learning Learning a policy capable of moving an agent between any two states in the environment is important for many robotics problems involving navigation and manipulation. Due to the sparsity of rewards in such tasks, applying reinforcement learning in these scenarios can be challenging. Common approaches for tackling this problem include reward engineering with auxiliary rewards, requiring domain-specific knowledge or changing the objective. In this work, we introduce a method based on region-growing that allows learning in an environment with any pair of initial and goal states. Our algorithm first learns how to move between nearby states and then increases the difficulty of the startgoal transitions as the agent's performance improves. This approach creates an efficient curriculum for learning the objective behavior of reaching any goal from any initial state. In addition, we describe a method to adaptively adjust expansion of the growing region that allows automatic adjustment of the key exploration hyperparameter to environments with different requirements. We evaluate our approach on a set of simulated navigation and manipulation tasks, where we demonstrate that our algorithm can efficiently learn a policy in the presence of sparse rewards. INTRODUCTION In recent years, deep reinforcement learning (Deep RL) has enjoyed success in many different applications, including playing Atari games [Mnih et al., 2013], controlling a humanoid robot to perform various manipulation tasks [Chebotar et al., 2017b;Chebotar et al., 2017a] and beating the world champion in Go [Silver et al., 2016]. The success and wide range of use cases of RL algorithms is partly due to the very general description of the problem that RL aims to solve, i.e., to learn autonomous behaviors given a high-level specification of a task by interacting with the environment. Such highlevel specification is provided by a reward function, which must be sufficiently descriptive as well as easy to optimize for an RL algorithm to learn efficiently. These requirements make the design of the reward function challenging in practice, creating a bottleneck for even a wider set of applications for RL algorithms. The problem of designing a reward function has been tackled in various ways. These include: i) learning the reward function from human demonstrations in the field of inverse reinforcement learning (IRL) [Levine et al., 2011;Abbeel and Ng, 2004], ii) initializing the reinforcement learning process with demonstrations in imitation learning [Chebotar et al., 2017b;Kalakrishnan et al., 2012], and iii) creating reward shaping functions that aim to guide the RL process to high-reward regions [Chebotar et al., 2017a;Popov et al., 2017]. Even though all of these methods have shown promising solutions to the problem of reward function design, they present other significant challenges such as the requirement of domain expertise or access to demonstration data. Ideally, one would like to learn from a simple sparse binary reward that indicates completion of the task. Such a reward signal is natural for many goal-oriented tasks. It allows significant reduction of engineering effort, and in some cases can be used to learn very complicated skills from human feedback, where design of the reward function is very hard [Christiano et al., 2017]. Despite being attractive, such a reward function creates significant difficulties for learning. This is due to the fact that it is very unlikely for an agent to generate the exact sequence of actions leading to solving the task from random exploration [Duan et al., 2016]. Recent efforts focus on learning from such sparse reward signals by constructing a curriculum from a continuous set of tasks Florensa et al., 2017]. These methods exploit the simple intuition that tasks initialized closer to the goal should be easier to solve. Proximity to the goal is defined either explicitly or through the number of random actions needed to reach the state from the goal [Florensa et al., 2017]. Nevertheless, all of these methods have a common disadvantage: they are designed for either singlestart or single-goal scenarios. In this paper, we address the situation in which the task contains both a continuous set of goals and a continuous set of initial conditions, thus broadening the applicability of our algorithm to a wide range of problems. In addition, we introduce a method to adaptively adjust expansion of the growing region, eliminating manual tuning of a key exploration hyperparameter whose optimal value varies across different environments. RELATED WORK Intrinsic motivation. Learning from sparse rewards is a long-standing goal in RL. The most established way of coping with such scenarios has been reward shaping [Chebotar et al., 2017a], which requires extensive engineering and domain specific knowledge. To address this problem, various researchers proposed curiosity and intrinsic motivation [Schmidhuber, 2010;Oudeyer et al., 2007] as a more general way of guiding learning in the absence of the main reward. Intrinsic motivation is typically introduced in the form of auxiliary rewards or loss components incentivizing exploration, that are not connected to the main objective. Such incentives could be based on counting visited states and/or maintaining a state-visitation density model Ostrovski et al., 2017;Martin et al., 2017], prediction error [Stadie et al., 2015, prediction error-improvement of the learned model [Lopes et al., 2012], predictive model uncertainty [Houthooft et al., 2016, neuro-correlation [Schossau et al., 2016] or learning auxiliary tasks [Jaderberg et al., 2016]. Despite a vast variety of approaches, many curiosityinspired methods are prone to creating additional local minima in the learned objective function. Curriculum learning. Another approach to learning in the presence of sparse rewards is to construct a curriculum of the task instances to ease the learning process. In this case, the agent initially learns from easy scenarios, where the chance of acquiring positive reward is relatively high, and the difficulty of the presented tasks is gradually increased until the final task is learned. The main advantage of such an approach is that the agent learns on the final objective directly, and thus avoids the problems of curiosity-driven methods. Traditionally, curriculum design has been explored from the perspective of manually engineered schedules in both supervised tasks [Zaremba and Sutskever, 2014;Bengio et al., 2015] and reinforcement learning scenarios [Wu and Tian, 2017;Heess et al., 2017]. More recently, there have also been multiple approaches for automated curriculum generation for RL. [Svetlik et al., 2017] create curriculum in the form of an acyclic graph based on a transfer potential metric, [Sharma et al., 2017] explored task sampling based on their current performance, and [Matiisen et al., 2017] utilized task performance improvement as a basis for task sampling. All of these approaches, as opposed to our method, are designed to perform well only in a discrete set of tasks with dense rewards. Another related approach is presented in the recent work by [Sukhbaatar et al., 2017] based on the idea of self play between two agents. The first agent plays the role of a teacher that sets the tasks for the second agent, who plays the role of a student who tries to repeat the teacher's actions or reverse the environment to its original state. As mentioned by the authors and confirmed in [Florensa et al., 2017], the asymmetric structure of this method often leads to a biased exploration resulting in the teacher and the student becoming stuck in a small subspace of the task. Our method avoids such situations by using random exploration to expand the set of goals and the initial conditions to the appropriate level of difficulty. Another related piece of work is that of , who consider the problem of generating multiple goals of the appropriate level of difficulty using generative adversarial networks (GANs) [Goodfellow et al., 2014]. Their approach is designed to learn a goal distribution and, thus, in its straightforward form, cannot learn to generalize to multiple initial conditions. In addition, since their approach contains a learned generative model, it tends to struggle when the dimensionality of the task representation is large and the number of examples is very limited, which is usually the case for robotics. We address this problem by generating tasks through the interaction with the environment. The approach most related to ours is the concurrent work of [Florensa et al., 2017]. We exploit similar core principles and assumptions, i.e., we utilize Brownian motion for growing the current task region and generate curriculum through reverse exploration of new tasks. We extend this approach to multi-goal and multi-start scenarios with infinitely many start-goal pairs, and present results in environments with sparse rewards. In addition, we address the question of controlling expansion of the growing region. Our algorithm adaptively changes the key exploration hyperparameter for environments with significantly different optimal settings. These contributions lead to improved resampling efficiency and eliminate the need of expensive hyperparameter tuning. Background We consider a reinforcement learning problem where an agent is represented by a global policy that aims to reach any goal in an environment. This section introduces a formal definition of the problem and our framework. Markov decision process We consider a discrete-time, finite-horizon Markov decision process (MDP) defined by a tuple M = (S, G, A, P, r, ρ 0 , T ), in which S is the agent state set, A is the action set, P : S × A → R n is the transition probability distribution, r : S × G × A → R is a bounded reward function dependent on the goal state, where G represents the goal set; ρ 0 : S → R n is the initial state distribution, and T ∈ N is the time horizon. Our aim is to learn a stochastic policy π θ : S × A × G → R m parameterized by θ. We would like to point out that in order to communicate the goal to the agent, our formulation requires the policy to be conditioned on the goal g specified by the environment, i.e. π θ = π(a t \|s t , g). The objective is to maximize the expected return, η ρ0 (π θ ) = E s0∼ρ0,g∼ρg R(π θ , s 0 , g) with the expected reward starting at s 0 being R(π θ , s 0 , g) := E τ \|s0 [ T t=0 r(s t , a t , g)], where τ = (s 0 , a 0 , . . .) denotes the trajectory generated by executing actions a t ∼ π θ (a t \|s t , g) sampled from the policy under environment dynamics s t+1 ∼ P(s t+1 \|s t , a t ). Goal-dependent sparse reward function In this work, we consider the problem of reaching any goal state g ∼ U(G) in the environment from any initial state ρ 0 ∼ U(S 0 ), where U denotes a uniform distribution. For this purpose, we define a sparse binary reward function dependent on the goal: r(s t , a t , g) = 1{s t ∈ S g } ,(1) where S g ⊂ S is a set of states corresponding to the goal g. We note that although the binary reward function in Eq. (1) is typically defined through some distance metric , our learning algorithm does not explicitly utilize this metric. Assumptions In this work we exploit several assumptions: Assumption 1 The agent can be initialized at an arbitrary state s ∈ S. This assumption is a common requirement for many algorithms [Florensa et al., 2017;Kearns et al., 2002] in the RL setting, especially those that exploit uniform initialization to generalize to multiple initial states. Assumption 2 At least one initial state is provided to the algorithm, which we call a seed state. Assumption 3 For every state s ∈ S, there exists a function g = f g (s) that maps any state in the environment to the corresponding goal representation. This assumption is required since, in our algorithm, states encountered by the agent should be converted to the corresponding goal representations. Assumption 4 For any pair of states s 1 , s 2 ∈ S there exists a trajectory that moves the agent from s 1 to s 2 . In other words, the agent can reach any state from any other state. We note that although we explicitly introduced Assumption 4, it does not prevent our algorithm from being applied to a wider set of tasks where some states might not be mutually reachable. For example, if isolated or irreversible pairs of states exist, the algorithm nevertheless applies to all the reachable states, which depends on the initial state provided. Approach The main difficulty of training an RL agent in a sparse reward setting arises from the fact that it is unlikely for the agent to accomplish the task using random exploration if the initial state is far from the goal state. In this work, we take advantage of the intuition also exploited by [Florensa et al., 2017] that the agent has a higher chance of success if the goal is located in close proximity to the initial state. In particular, if one could initialize learning by generating goal states that are close to the initial states, then the initial learning stages should progress much faster. Since it can be highly nontrivial to engineer a correct distance metric directly in the observation space, we define the proximity of points by the number of actions it takes to reach one point from another. Taking this into consideration, we propose the idea of gradually-growing reachability regions for generating a curriculum in a multi-goal setting. Our algorithm consists of two agents: a sampler and a learner. The sampler uses short chains of random actions to arrive at a state that is then added to the currently-explored set of points, which we refer to as the reachability region. This region is defined as the area where the learner has already mastered transitions between all pairs of points. As learning progresses, the sampler removes already-learned states from the reachability region and adds new points that have not been explored yet. This generates a natural curriculum for learning a global reaching policy, i.e., a policy capable of moving the agent between any two states in the environment. In the following, we first discuss the sampler and then describe the learner. Filtering states In the first part of the sampling algorithm, we focus on a criterion that indicates whether a particular set of states has been mastered. In order to select which states have already been mastered, we retain statistics of rewards received by the agent on every state within the current reachability region. We choose to follow a simple approach in which we only retain statistics of the points in the role of starting states, as opposed to retaining statistics on start-goal pairs. We define thresholds R min and R max that prevent states from being too hard or too easy, respectively. We refer to the set of all states in the current reachability region as s. We keep a history of rewards in a vector r and associate them with start states. If the average reward for a state in r does not exceed the R min and R max thresholds we use the state for further resampling. This behavior is implemented in a helper function FilterStates that takes s, r, R min , and R max as input and returns the retained set of states as s. Adaptive state resampling As previously mentioned, we define the proximity of the points through the action space, i.e., points are close to each other if they are reachable via short random trajectories. We use Brownian motion to sample new states to grow the region of learned state-goal pairs. A major challenge of this approach is the selection of the variance for exploration. Poorly selected variance can result in either a very spread set of points that are hard to learn from, or a set of points that are too easily mastered, which in both cases results in a slow learning progress of the RL algorithm. We adjust the sampling variance σ dynamically using a method that is inspired by the integral part of a PID controller. Our approach adjusts the variance such that average reward in the current iteration (r avg ) is close to a user-provided target reward (R pref ). In particular, every time before resampling, we update the sampling variance (σ) according to the following procedure: δ σ ← Clip(k σ * (r avg − R pref ), − δ σ max , δ σ max ) σ ← Clip(σ + δ σ , σ min , σ max )(2) where Clip(x, α, β) min(max(x, α), β), k σ is the control coefficient, δ σ max is the maximum change of variance, and σ min/max are the variance limits. Thus, if the success ratio systematically exceeds the preferred value, our method increases the variance, promoting faster exploration and vice versa. We encode Eq. (2) in the helper function UpdateVariance that takes σ and r avg as inputs and returns the new sampling variance. Resampling a set of new states is implemented in the helper function ResampleStates that takes the current set of states s, the set of old mastered states s old , and the variance σ as inputs and returns the the new set of states. Resampling is carried out in two stages. First, we create an oversampled set of states by performing Brownian-motion rollouts, which we refer to as sampling rollouts. We use random actions generated by the sampler agent using N (0, σ 2 ) and collect states visited by the agent. Each of these rollouts is initialized at one of the states from the growing oversampled set. This set is initialized with the states retained in s after filtering. At the second stage, we sample N new states uniformly from the oversampled set and add them to N old states sampled uniformly from s old to form the new current set of states. Policy training Algorithm 1: Policy Training Input : s seed : seed state, N : iterations, K: sampling period, π 1 : initial policy, σ: initial sampling variance Output: π N +1 : policy 1 s old , s, r ← {s seed }, {s seed }, s train , g train , r, r avg ← Rollouts(π i , s, s old ) 15 π i+1 ← UpdatePolicy(π i , s train , g train , r) 16 end Algorithm 1 describes the policy training procedure including both the sampler and the learner agents. The sampler agent updates the reachability region (lines 5 -13), while the learner follows a its own learning strategy (lines 14 -15). Our method starts by initializing the current state set s, the corresponding vector of history of rewards r and the pool of the previously learned states s old (line 1). The sampler uses a fixed update period K (line 5) to adjust the variance according to Eq. (2) (line 7) and proceeds to the filtering stage to find good states to propagate from (line 9). Once the filtering is finished, the sampler resamples a new set of states using Brownian motion (line 12). The learner performs policy rollouts in every iteration (line 14) using the helper function Rollouts. This function follows a special start-goal pair sampling strategy. Start states for the rollouts are sampled uniformly from the current state set s, whereas the goals are sampled from either s (with probability P new ) or s old (with probability 1 − P new ). Once the batch of samples used for the user-chosen RL algorithm is accumulated, we update the policy (line 15). Our approach is agnostic to the choice of agent optimization method; we only require that this method provides the UpdatePolicy function. In our experiments we use Left: Maze environment. The square represents the cube that the agent has to push to a goal state. Black lines represent the walls of the maze. Right: SparseReacher environment. The two-link manipulator has to touch the goal marker. TRPO [Schulman et al., 2015] as one of the most robust RL algorithms with an implementation available online. Experiments We implemented this approach in Python and applied it to two representative environments. We show empirically that this technique successfully trains agents in our multi-goal scenarios. Furthermore, we demonstrate that our dynamic variance selection is less sensitive to hyperparameters than other alternatives. In all of our experiments, we use the following parameters across all environments: R max = 0.9, R min = 0.3, K = 5, N new = 135, N old = 65, P new = 0.6, R pref = 0.7, k σ = 2.0, δ σ max = 0.5, σ min = 0.1, σ max = 1.0. Environments The SparseReacher is an environment with a two-link manipulator based on the Reacher-v0 environment from Ope-nAI Gym [Brockman et al., 2016]. We use it in a sparse reward setting: the agent receives a positive binary reward only when it touches the goal marker. This corresponds to the situation where the robot's end effector is not further than 2 cm from the center of the goal marker. In addition, the Cartesian velocities of the robot must be lower than 0.2 m/s. The episode ends as soon as the positive reward is acquired. As we observed in our experiments, such sparse reward makes this environment significantly more challenging, especially when the goal is to learn a policy that can reach any point in the robot's workspace. The goal in the Maze environment is to bring a cube of size h cube to a goal location. The agent receives a reward only if the center of the cube lies still within an -radius of the goal location. The episode ends as soon as the positive reward is acquired. We define a variable time step in the environment that is dependent on the time it takes for the cube to settle after a force is applied. The table is constrained by the size h table and surrounded by walls, such that the cube cannot fall off the table. This environment has continuous action space that consists of two components of the force F x , F y applied to the center of the cube, parallel to the table plane. Observations contain a 7-dimensional cube pose where the rotation is encoded as quaternion. We define a goal representation as a simple 2d position on the table. This environment is challenging due to several aspects. First, the search space increases with h 2 table , thus, the probability to encounter the target by chance is very small. Second, the cube has relatively complex dynamics compared to a simple point mass: it can be pushed or rolled depending on the direction and amount of force applied, and it exhibits a complex behavior when it comes into a contact with the wall. Third, the cube cannot simply roll over the goal to acquire a positive reward-it must stop at the precise location of the goal. (a) i = 5 (b) i = 125 (c) i = 135 (d) i = 450 Both environments are shown in Fig. 1. For each environment, we select a single seed state to expand the growing region. For the Maze environment, we explicitly pick the most challenging scenario of the seed state located at the end of the central corridor since the policy has to learn how to precisely navigate inside of the narrow corridor entrance. Both environments can be naturally used in both single-and multigoal settings. We also note that in every training scenario, in addition to the sparse reward, we add a very small negative reward for every time step to promote shorter episodes. Fig. 2 demonstrates how the region expansion proceeds during learning in the maze environment. In particular, it shows an interesting phenomenon associated with variance adaptation that we refer to as region clustering. During expansion, if the new set of points was selected too aggressively, our algorithm responds by decreasing the variance of the region expansion. Since this event by definition happens due to a poor performance (see Eq. (2)), there will be very few available states to sample from. Thus, the algorithm forms a cluster of newly resampled states located around a few states that passed through the filtering stage (see Fig. 2b). Later, as the learner agent improves, those clusters grow and connect, forming a single region which is illustrated in Fig. 2c. Such behavior helps the learner to create new growing regions in isolated areas. Reachability regions SparseReacher Our results for the multi-goal version of the SparseReacher environment are shown in Fig. 3a. We execute the learning process several times and provide the average reward for each iteration over six executions. We test our algorithm with two constant resampling variances (σ = 0.1 and σ = 0.5) and with our adaptive variance. We also provide results for the case that does not use a reachability region, but instead samples start and goal states uniformly over the environments. The environment is conservative and requires small exploration variances; we found that a constant variance of σ = 0.1 performs much better than a variance of σ > 0.5. Our adaptive variance selection achieves a slightly higher average reward than the best hand-tuned constant variance. The simple uniform state sampling performs as well as our reachability approach when a bad constant variance is applied. Maze The experimental results for the multi-goal version of the Maze environments are shown in Fig. 3b. As before, we provide the average reward for each iteration over six executions. This environment requires more exploration than the SparseReacher environment; we found that when using a constant variance a value of σ = 1.0 the agent performs best, while σ < 0.25 results in a very poor learning performance. The reward of our adaptive variance selection is comparable to the best hand-tuned constant variance. Uniform state-goal sampling performed surprisingly well, but as we can see our approach clearly indicates the benefits of generating a curriculum for learning. We would also like to note that uniform sampling exhibits the best of its capabilities to learn transitions in this environment, whereas we selected the worst seed state for our algorithm specifically to emphasize benefits of our dynamic reachability region. Hyperparameter adaptation The environments that we selected are representative in the spectrum of requirements for growing region expansion. The multi-goal version of the SparseReacher environment is more conservative and requires small exploration variances, whereas the Maze environment benefits from aggressive exploration, and hence high variances perform well. For example, Fig. 3a shows that, under constant variance, the learner completely fails to improve when the variance is set to a high value. On the other hand, Fig. 3b shows the opposite for the : Reward for different algorithm variants for the multi-goal case. The data is averaged over 6 executions. "Uniform" refers to uniform random sampling of the start and goal states with no reachability region. "var x" uses the reachability regions for the sampler agents, but uses a constant σ = x for action selection. "var adapt" is our full algorithm using the reachability regions and adaptive σ. Maze, where the optimal variance value is close to the maximum value. Our adaptive variance approach performs similar to the optimal constant variance. Given that we have the same set of exploration hyperparameters for both environments, our approach eliminates the need to tune the key hyperparameter of the region growing curriculum learning method. Fig. 4 shows the sampling variance evolution over training. Initially, our algorithm picks the largest and the smallest variance values for the Maze and the SparseReacher environments, respectively. In the case of SparseReacher, it keeps a low variance at the beginning, since random initialization of the policy weights results in actions of large magnitude. As the agent keeps learning, the exploration is gradually relaxed. Our algorithm regulates the variance in such a way that allows the learner to maintain the proper exploration pace, resulting in steep learning curves. We also find this idea connected to the approach proposed by [Berthelot et al., 2017] in the context of adversarial learning. In our scenario, since there is no loss for the sampler, we apply the equilibrium principle through balancing the success ratio for the learner. We also evaluated the single-goal variations of our environments, where the seed state represents the only goal in each environment. In this scenario, variance adaptation showed similar benefits. On average the single-goal SparseReacher was learned 20-50% faster with variance adaptation than with manual tuning of a constant variance. For the Maze environment our algorithm is able to match the performance of the version with the constant sampler variance. Conclusions and Future work In this work we proposed a novel algorithm for learning a global policy capable of moving an agent in environments with any pair of start-goal transitions. Our algorithm is based on the idea of region growing, and it is capable of automatic adjustment of the region expansion to achieve an appropriate pace of learning without extensive hyperparameter tuning. To apply our approach to real robotic systems in the future, we plan to address the following shortcomings. First, the algorithm could substantially benefit from parallel learning of a reversing policy, allowing it to return to some safe states within the current growing region. Second, the current version of our algorithm is sensitive to the choice of the seed state. For example, in experiments with the Maze environment, we observed that the success rate can be twice as good depending on the location of the seed state. This phenomenon occurs because for some seeds the policy can avoid learning how to reach states in the hardest subspace of the environment, namely, the corridor. In the future, we plan to address this problem by using a principle of skill chaining to learn a set of policies for different state regions. FilterStates(s, r, R min , R max ) 10 s old ← s old ∪ s Figure 1 : 1Environments with seed states used in our experiments. Figure 2 : 2Illustration of state propagation for the maze multi-goal environment. Circles represent the current states in the reachability region. Images are ordered from left to right in the order of learning progress. The leftmost image depicts the beginning of training and the rightmost image shows the state set at the end of training. The middle two plots show the phenomenon of state clustering. Colors encode average reward associated with the states, where red refers to high reward and blue to low reward. Figure 3 3Figure 3: Reward for different algorithm variants for the multi-goal case. The data is averaged over 6 executions. "Uniform" refers to uniform random sampling of the start and goal states with no reachability region. "var x" uses the reachability regions for the sampler agents, but uses a constant σ = x for action selection. "var adapt" is our full algorithm using the reachability regions and adaptive σ. Figure 4 : 4Variance adaptation for different environments in the multi-goal scenario. The lines show the average and the shaded region the standard deviation over 6 executions. Combining model-based and model-free updates for trajectory-centric reinforcement learning. 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[ "Face Recognition Methods & Applications", "Face Recognition Methods & Applications" ]	[ "Divyarajsinh N Parmar \nP.G. Student of Computer Engineering\n\n\nAsst.Prof. Dept.of Computer Engineering C.U. Shah College of Engg. & Tech Wadhwan city\nIndia\n", "Brijesh B Mehta [email protected] \nAsst.Prof. Dept.of Computer Engineering C.U. Shah College of Engg. & Tech Wadhwan city\nIndia\n" ]	[ "P.G. Student of Computer Engineering\n", "Asst.Prof. Dept.of Computer Engineering C.U. Shah College of Engg. & Tech Wadhwan city\nIndia", "Asst.Prof. Dept.of Computer Engineering C.U. Shah College of Engg. & Tech Wadhwan city\nIndia" ]	[]	Face recognition presents a challenging problem in the field of image analysis and computer vision. The security of information is becoming very significant and difficult. Security cameras are presently common in airports, Offices, University, ATM, Bank and in any locations with a security system. Face recognition is a biometric system used to identify or verify a person from a digital image. Face Recognition system is used in security. Face recognition system should be able to automatically detect a face in an image. This involves extracts its features and then recognize it, regardless of lighting, expression, illumination, ageing, transformations (translate, rotate and scale image) and pose, which is a difficult task. This paper contains three sections. The first section describes the common methods like holistic matching method, feature extraction method and hybrid methods. The second section describes applications with examples and finally third section describes the future research directions of face recognition.	null	[ "https://arxiv.org/pdf/1403.0485v1.pdf" ]	2,867,674	1403.0485	623ed70463df79231e87a8a2fa3d4913494463b5	Face Recognition Methods & Applications Divyarajsinh N Parmar P.G. Student of Computer Engineering Asst.Prof. Dept.of Computer Engineering C.U. Shah College of Engg. & Tech Wadhwan city India Brijesh B Mehta [email protected] Asst.Prof. Dept.of Computer Engineering C.U. Shah College of Engg. & Tech Wadhwan city India Face Recognition Methods & Applications Face RecognitionHolistic Matching MethodsFeature-based (structural) MethodsHybrid Methods Face recognition presents a challenging problem in the field of image analysis and computer vision. The security of information is becoming very significant and difficult. Security cameras are presently common in airports, Offices, University, ATM, Bank and in any locations with a security system. Face recognition is a biometric system used to identify or verify a person from a digital image. Face Recognition system is used in security. Face recognition system should be able to automatically detect a face in an image. This involves extracts its features and then recognize it, regardless of lighting, expression, illumination, ageing, transformations (translate, rotate and scale image) and pose, which is a difficult task. This paper contains three sections. The first section describes the common methods like holistic matching method, feature extraction method and hybrid methods. The second section describes applications with examples and finally third section describes the future research directions of face recognition. I. INTRODUCTION Face recognition systems have been conducted now for almost 50 years. Face recognition is one of the researches in area pattern recognition & computer vision due to its numerous practical applications in the area of biometrics, Information security, access control, law enforcement, smart cards and surveillance system. The first large scale application of face recognition was carried out in Florida. Biometric-based techniques have emerged as the most promising option for recognizing individuals in recent years since, instead of certifying people and allowing them access to physical and virtual domains based on passwords, PINs, smart cards, plastic cards, tokens, keys and so, these methods examine an individual's physiological and/or behavioral characteristics in order to determine and/or ascertain his/her identity. Passwords and PINs are difficult to remember and can be stolen or guessed; cards, tokens, keys and the like can be misplaced, forgotten, or duplicated; magnetic cards can become corrupted and unclear. However, an individual's biological traits cannot be misplaced, forgotten, stolen or forged [1]. In order to develop a useful and applicable face recognition system several factors need to be take in hand. 1. The overall speed of the system from detection to recognition should be acceptable. 2. The accuracy should be high 3. The system should be easily updated and enlarged, that is easy to increase the number of subjects that can be recognized. II. Face Recognition Methods In the beginning of the 1970's, face recognition was treated as a 2D pattern recognition problem [2]. The distances between important points where used to recognize known faces, e.g. measuring the distance between the eyes or other important points or measuring different angles of facial components. But it is necessary that the face recognition systems to be fully automatic. Face recognition is such a challenging yet interesting problem that it has attracted researchers who have different backgrounds: psychology, pattern recognition, neural networks, computer vision, and computer graphics. The following methods are used to face recognition. 1. Holistic Matching Methods 2. Feature-based (structural) Methods 3. Hybrid Methods Holistic Matching Methods: In holistic approach, the complete face region is taken into account as input data into face catching system. One of the best example of holistic methods are Eigenfaces [8] (most widely used method for face recognition), Principal Component Analysis, Linear Discriminant Analysis [7] and independent component analysis etc. Holistic example The first successful demonstration of machine recognition of faces was made by Turk and Pentland [2] in 1991 using eigenfaces. Their approach covers face recognition as a twodimensional recognition problem. The flowchart in Figure 1 illustrates the different stages in an eigenface based recognition system. (1) The first stage is to insert a set of images into a database, these images are names as the training set and this is because they will be used when we compare images and when we create the eigenfaces. to the weights of those already in the system. If the input image's weight is over a given threshold it is considered to be unidentified. The identification of the input image is done by finding the image in the database whose weights are the closest to the weights of the input image. The image in the database with the closest weight will be returned as a hit to the user of the system [2]. Feature-based (structural) Methods: In this methods local features such as eyes, nose and mouth are first of all extracted and their locations and local statistics (geometric and/or appearance) are fed into a structural classifier. A big challenge for feature extraction methods is feature "restoration", this is when the system tries to retrieve features that are invisible due to large variations, e.g. head Pose when we are matching' a frontal image with a profile image. [5] Distinguishes between three different extraction methods: I. Generic methods based on edges, lines, and curves II. Feature-template-based methods III. Structural matching methods that take into consideration geometrical Constraints on the features. Hybrid Methods: Hybrid face recognition systems use a combination of both holistic and feature extraction methods. Generally 3D Images are used in hybrid methods. The image of a person's face is caught in 3D, allowing the system to note the curves of the eye sockets, for example, or the shapes of the chin or forehead. Even a face in profile would serve because the system uses depth, and an axis of measurement, which gives it enough information to construct a full face. The 3D system usually proceeds thus: Detection, Position, Measurement, Representation and Matching. Detection -Capturing a face either a scanning a photograph or photographing a person's face in real time. Position -Determining the location, size and angle of the head. Measurement -Assigning measurements to each curve of the face to make a template with specific focus on the outside of the eye, the inside of the eye and the angle of the nose. Representation -Converting the template into a code -a numerical representation of the face and Matching -Comparing the received data with faces in the existing database. In Case the 3D image is to be compared with an existing 3D image, it needs to have no alterations. Typically, however, photos that are put in 2D, and in that case, the 3D image need a few changes. This is tricky, and is one of the biggest challenges in the field today. III. Face Recognition Applications Face recognition is also useful in human computer interaction, virtual reality, database recovery, multimedia, computer entertainment, information security e.g. operating system, medical records, online banking., Biometric e.g. Personal Identification -Passports, driver licenses , Automated identity verification -border controls , Law enforcement e.g. video surveillances , investigation , Personal Security -driver monitoring system, home video surveillance system. Applications & examples: Face Identification: Face recognition systems identify people by their face images. Face recognition systems establish the presence of an authorized person rather than just checking whether a valid identification (ID) or key is being used or whether the user knows the secret personal identification numbers (Pins) or passwords. The following are example. To eliminate duplicates in a nationwide voter registration system because there are cases where the same person was assigned more than one identification number. The face recognition system directly compares the face images of the voters and does not use ID numbers to differentiate one from the others. When the top two matched faces are highly similar to the query face image, manual review is required to make sure they are indeed different persons so as to eliminate duplicates. Access Control: In many of the access control applications, such as office access or computer logon, the size of the group of people that need to be recognized is relatively small. The face pictures are also caught under natural conditions, such as frontal faces and indoor illumination. The face recognition system of this application can achieve high accuracy without much co-operation from user. The following are the example. Face recognition technology is used to monitor continuously who is in front of a computer terminal. It allows the user to leave the terminal without closing files and logging out. When the user leaves for a predetermined time, a screen saver covers up the work and disables the mouse & keyboard. When the user comes back and is recognized, the screen saver clears and the previous session appears as it was left. Any other user who tries to logon without authorization is denied. Security: Today more than ever, security is a primary concern at airports and for airline staff office and passengers. Airport protection systems that use face recognition technology have been implemented at many airports around the world. The following are the two examples. In October, 2001, Fresno Yosemite International (FYI) airport in California deployed Viisage's face recognition technology for airport security purposes. The system is designed to alert FYl's airport public safety officers whenever an individual matching the appearance of a known terrorist suspect enters the airport's security checkpoint. Anyone recognized by the system would have further investigative processes by public safety officers. Computer security has also seen the application of face recognition technology. To prevent someone else from changing files or transacting with others when the authorized individual leaves the computer terminal for a short time, users are continuously authenticated, checking that the individual in front of the computer screen or at a user is the same authorized person who logged in. IV. Conclusion & Scope for Future Research It is our opinion that research in face recognition is an exciting area for many years to come and will keep many scientists and engineers busy. In this paper we have given concepts of face recognition methods & its applications. The present paper can provide the readers a better understanding about face recognition methods & applications. In the future, 2D & 3D Face Recognition and large scale applications such as e-commerce, student ID, digital driver licenses, or even national ID is the challenging task in face recognition & the topic is open to further research. ( 2 ) 2The second stage is to create the eigenfaces. Eigenfaces are made by extracting characteristic features from the faces. The input images are normalized to line up the eyes and mouths. They are then resized so that they have the same size. Eigenfaces can now be extracted from the image data by using a mathematical tool called Principal Component Analysis(PCA). (3) When the eigenfaces have been created, each image will be represented as a vector of weights. (4) The system is now ready to accept entering queries. (5) The weight of the incoming unknown image is found and then compared Divyarajsinh N Parmar et al ,Int.J.Computer Technology & Applications,Vol 4 (1),84-86 IJCTA \| Jan-Feb 2013 Available [email protected] Figure 1 : 1Flow chart of the eigenface-based algorithm. Divyarajsinh N Parmar et al ,Int.J.Computer Technology & Applications,Vol 4 (1),84-86 IJCTA \| Jan-Feb 2013 Available [email protected] Image database investigations:Searching image databases of licensed drivers, benefit recipients, missing children, immigrants and police bookings. identity verification: Electoral registration, banking, electronic commerce, identifying newborns, national IDs, passports, employee IDs.Surveillance: Like security applications in public places, surveillance by face recognition systems has a low user satisfaction level, if not lower. Free lighting conditions, face orientations and other divisors all make the deployment of face recognition systems for large scale surveillance a challenging task. The following are some example of facebased surveillance. To enhance town center surveillance in Newham Borough of London, this has 300 cameras linked to the closed circuit TV (CCTV) controller room. The city council claims that the technology has helped to achieve a 34% drop in crime since its facility. Similar systems are in place in Birmingham, England. In 1999 Visionics was awarded a contract from National Institute of Justice to develop smart CCTV technology.General A Survey of Face Recognition Techniques. R Jafri, H R Arabnia, Journal of Information Processing Systems. 52R. Jafri, H. R. Arabnia, "A Survey of Face Recognition Techniques", Journal of Information Processing Systems, Vol.5, No.2, June 2009. Face Recognition. C A Hansen, NorwayInstitute for Computer Science University of TromsoC. A. Hansen, "Face Recognition", Institute for Computer Science University of Tromso, Norway. Visual identification of people by computer. M D Kelly, Stanford, CA, USAStanford UniversityPhD thesisM. D. Kelly. Visual identification of people by computer. PhD thesis, Stanford University, Stanford, CA, USA, 1971. Computer Recognition of Human Faces. T Kanade, 47T. Kanade. Computer Recognition of Human Faces, 47, 1977. Face recognitions literature survey. W Zhao, R Chellappa, P J Phillips, & A Rosenfeld, ACM Computing Surveys. 354W. Zhao, R. Chellappa, P. J. Phillips & A. Rosenfeld, "Face recognitions literature survey", ACM Computing Surveys, Vol. 35, No. 4, December 2003, pp. 399-458. Digital Image processing Using MATLAB. C Gonzalez, R E Woods, S Liddins, C. Gonzalez, R. E. Woods, S. liddins, "Digital Image processing Using MATLAB". Face Recognition Using Principal Component Analysis and Linear Discriminant Analysis on Holistic Approach in Facial Images Database. S Suhas, A Kurhe, . P Dr, Khanale, IOSR Journal of Engineering. 2S. Suhas, A. Kurhe, Dr.P. Khanale, "Face Recognition Using Principal Component Analysis and Linear Discriminant Analysis on Holistic Approach in Facial Images Database", IOSR Journal of Engineering e-ISSN: 2250-3021, p-ISSN: 2278-8719, Vol. 2, Issue 12 (Dec. 2012), \|\|V4\|\| PP 15-23 Face Recognition Using Eigenfaces. M A Turk, A P Pentland, M. A. Turk and A. P. Pentland, "Face Recognition Using Eigenfaces", 1991. A Comparative study of Face Recognition with PCA and Cross-Correlation Technique. S Asadi, Dr D V Subba, R V Saikrishna, IJCA10S. Asadi, Dr. D. V. Subba R. V. Saikrishna, "A Comparative study of Face Recognition with PCA and Cross-Correlation Technique", IJCA(0975-8887), Volume 10-No.8, November 2010. Face Recognition Based on Nonlinear Feature Approach. E A Abusham, A T B Jin, W E Kiong, American Journal of Applied Sciences. E. A. Abusham, A. T. B. Jin, W. E. Kiong, "Face Recognition Based on Nonlinear Feature Approach", American Journal of Applied Sciences, 2008. A New Distance Measure for Face Recognition System. A Nigam, P Gupta, Fifth International Conference on Image and Graphics. A. Nigam, P. Gupta, "A New Distance Measure for Face Recognition System", 2009 Fifth International Conference on Image and Graphics . N Divyarajsinh, Parmar, Int.J.Computer Technology & Applications. 41Divyarajsinh N Parmar et al ,Int.J.Computer Technology & Applications,Vol 4 (1),84-86 . Ijcta \|, IJCTA \| Jan-Feb 2013 . Available [email protected]. Available [email protected]	[]
[ "Study of band bending effect in Dye Sensitized Solar Cell through Constant-Current-Discharging Voltage Decay", "Study of band bending effect in Dye Sensitized Solar Cell through Constant-Current-Discharging Voltage Decay" ]	[ "Xiaoqi Wang \nPhysics Department\nShanghai University\n200444ShanghaiChina\n", "Chuanbing Cai \nPhysics Department\nShanghai University\n200444ShanghaiChina\n" ]	[ "Physics Department\nShanghai University\n200444ShanghaiChina", "Physics Department\nShanghai University\n200444ShanghaiChina" ]	[]	A measurement method of constant-current-discharging voltage decay is established to characterize the band bending effect in the heterojunction of conducting glass/TiO 2 for typical dye-sensitized solar cells. Furthermore, a dark-state electron transport regarding the TiO 2 conduction band bending is proposed based upon the viewpoints of thermionic emission mechanism, which suggests an origin of the band bending effect in a theoretical model. This model quantitatively agrees well with our experimental results and indicates that both the Fermi level decay in TiO 2 and the potential difference across the heterojunction will lead to the TiO 2 conduction band bending downwards.	null	[ "https://arxiv.org/pdf/1210.8217v3.pdf" ]	93,917,856	1210.8217	bd11d77bcf050ef22cbd79d5a4ad200da2b4b054	Study of band bending effect in Dye Sensitized Solar Cell through Constant-Current-Discharging Voltage Decay Xiaoqi Wang Physics Department Shanghai University 200444ShanghaiChina Chuanbing Cai Physics Department Shanghai University 200444ShanghaiChina Study of band bending effect in Dye Sensitized Solar Cell through Constant-Current-Discharging Voltage Decay (Chuanbing Cai)voltage decaydischargephotoanodedye sensitized solar cellband bending Authors' A measurement method of constant-current-discharging voltage decay is established to characterize the band bending effect in the heterojunction of conducting glass/TiO 2 for typical dye-sensitized solar cells. Furthermore, a dark-state electron transport regarding the TiO 2 conduction band bending is proposed based upon the viewpoints of thermionic emission mechanism, which suggests an origin of the band bending effect in a theoretical model. This model quantitatively agrees well with our experimental results and indicates that both the Fermi level decay in TiO 2 and the potential difference across the heterojunction will lead to the TiO 2 conduction band bending downwards. Introduction Dye-sensitized solar cell (DSSC) and quantum dot sensitized solar cell (QDSSC) have drawn increasing attention during the past decades [1][2][3][4]. A lot of efforts have been made on the aspect of photoanode, such as the blocking layer [5], the electrode materials [6] and photoanode decoration [7], and so forth. Recently, it is observed that at the interface of photoanode, i.e. the conducting glass(FTO)/TiO 2 , the conduction band of TiO 2 will bend downwards with the quasi-Fermi level of TiO 2 varies even when the cell is under illumination [8][9][10]. This phenomenon implies the FTO/TiO 2 as a kind of rectifying contact, which may have an influence on the current subjected to the electric migration process. It is now believed, however, that the trap-limited diffusive process plays a crucial role in the photovoltaic current and covers the influence of the electric migration process which is considered negligible [10][11][12][13][14]. To address this issue on the band bending effect, the transport measurement is suggested to be achieved in the dark state, in which the term of electron diffusive process vanishes. Nevertheless, in a composite DSSC there are still some problems in distinguishing the band bending effect from the others, the recombination process at the interface of TiO 2 /electrolyte and the redox reaction at electrolyte/counter electrode, both of which are dominant in the most conventional measurements, e.g. Cyclic Voltammtry (CV) [15] and Electric Impedance Spectrum (EIS) [16]. To overcome such a difficulty, it is required to establish a method which can effectively clarify the band bending effect. Consider that the open-circuit voltage decay (OCVD), a useful method processed in the dark state, is frequently utilized in characterization of the recombination process at the interface of TiO 2 /electrolyte [17][18]. The TiO 2 thin film appears to be an electron reservoir due to the injection of photo-exited electrons under illumination. As the light is cut off in the case of OCVD, these electrons will release and recombine with electrolyte, see the arrow (R) in the Fig. 1(a). Consequently, the potential level E FTO of FTO goes down as the Fermi level E F of TiO 2 decays [19]. It can be proposed to apply a small discharge current on the FTO as the voltage decay, giving rise to less electron leakage from TiO 2 , see the arrow (-I) in the Fig. 1(a). Moreover, the applied current will cause a potential difference between FTO and TiO 2 , implying the information on the band bending effect. In this article, we firstly propose an effective method named constant current discharging voltage decay (CCDVD) and then model the dark-state transport concerning the band bending effect, so as to characterize the interface of the heterojunction FTO/TiO 2 , with respect to the understanding of band bending effect observed in DSSC. The efficient achieved is around 4 %. More details for the sample preparation and structure can be found in our previous publications [20]. Experimental details Preparation of the DSSC sample Principle for the method CCDVD OCVD is the powerful method that characterizes the dynamical process of recombination in DSSC. There are a lot of work, both in experimental [17] and theoretical studies [18], in this issue. Under illumination, electrons are injected from the photo-excited dyes to TiO 2 thin film, giving rise to the shift upwards of the Fermi level (E F ). As the light is cut off, the injected electrons will recombine with electrolyte, resulting in the E F decay [18]. Moreover, the decay of potential level (E FTO ) in FTO is coincidence with that of E F in TiO 2 , because there is no net charge transfer through the FTO/TiO 2 interface, seen in the Fig. 1 (b). As for the measurement of CCDVD, a discharge current is applied on the photoanode as the voltage decays, resulting in a potential difference between FTO and TiO 2 , as shown in the Fig. 1(c). The applied current depends not only on the characterizations of junction interface of FTO/TiO 2 but also on the potential difference of E FTO and E F , so that it appears to be a function of E FTO , E F and P, where P denotes some characteristic parameters for junction interface. Alternatively, the E FTO can also be written as a function f by inversely calculation, that is E FTO =f(I, E F , P). As the applied current can be set constant I C and E FTO is measurable, it suggests that once E F is derived the parameters P regarding the characteristic information of FTO/TiO 2 interface can be clarified. A validity condition addressed here to ensure the method effective is that the applied discharge current is too lower than recombination current to affect the variation of E F , so that the decay of E F in the CCDVD can be identical to that shown in the OCVD. Therefore, in the limit of I C →0, there is ( , , ) (0, , ) (0, , ) F C F FTO E I t E t E t   P P P( 1 ) It is noted that OCVD in principle is the special case of CCDVD at condition I C =0. By eliminating the time parameter t, one can derive the time independent relationship of E FTO (I C , P) v.s. E F (I C , P) by the experimental results of E FTO (I C , P) and E FTO (0, P), allowing us to further characterize the junction interface. In the following measurement of CCDVD, the constant current is supplied by Keithley 2420 digital meter and the voltage decay is recorded by an oscilloscope. are recorded by the oscilloscope, where has been suggested the redox level is grounded, shown in Fig. 2 Measurement of CCDVD OCVD, i.e. E FTO (I C )E F (I C )=E FTO (0) . ii) As E F moves down, the electrons in TiO 2 reduces much, and consequently a difference of E FTO and E F appears on account of preserving the constant discharge current, that indicates a drop of E FTO in the decay measurement. iii) As E F approximates to 0 V, however, the recombination process gradually goes weaker than the discharging process, so that E F decay is gradually dominated by the constant current. This stage has to be excluded in our consideration because it is out of the validity condition of CCDVD, which requires the recombination process dominates the E F decay. Therefore, only the data of E FTO above 0 V will be taken into account in the results discussion. and then gradually drops off, that are the stages i) and ii) identical to the observation in the Fig. 2(b). These behaviors appear to be related to the interface properties, and are supposed to give us more important information with respect to the band bending effect. Modeling for heterojunction transport of FTO/TiO 2 To further understand the band bending effect from the experimental results of CCDVD, one is required to have basic knowledge of the relationship of E FTO vs. E F for such heterojunction FTO/TiO 2 , that is the explicit expression for E FTO =f(I, E F , P). Analysis for the conduction band bending In general, bands will bend locally when FTO and TiO 2 come in contact, because the two Fermi levels of the materials will equilibrate to the same level through a local exchange of charge carriers [21], the same level that is named quasi-equilibrium state. The photoanode FTO/TiO 2 is a n + -n type semiconductor heterojunction, and the conduction band edge of TiO 2 bends downwards by several tenths electron volts (eV) of , leading to an accumulation layer [10,22], shown in the Fig. 4(a). In the case of OCVD, as the Fermi level E F decreases due to the recombination process, the junction FTO/TiO 2 has experienced a series of quasi-equilibrium state, and in which E FTO is always equal to E F [23]. For each state of quasi-equilibrium, the band edge (E B ) will shift downwards and reach a new equilibrium position, indicating a derivation of E B from its initial value E B0 , as shown in the Fig. 4(b). In the method of CCDVD, E B is considered to further bend owing to the potential difference of E FTO and E F appears, as illustrated in the Fig. 4(c). Therefore, the variation of E B is suggested to be a function both of the TiO 2 Fermi level E F and of the difference (E FTO -E F ), with a first-order form,     1 1 0 B B F O C F T O F E E E E E E         (2) where E B0 and E OC denote the initial values of E B and E F respectively, that is E B0 =E C + and E OC =-qV OC . While  -1 denotes E B /E F and  -1 the E B /E FTO , indicating what the extent of band bending is as E F and E FTO varies, respectively, and they are the characteristic parameters of the junction FTO/TiO 2 mentioned above. Transport modeling regarding the band bending It is generally believed that under illumination electrons are injected in the TiO 2 thin film by the excited dye molecules and then diffuse through the photoanode, suggesting that a diffusive term plays a significant role in the current [10][11][12][13][14]. This term, however, vanishes in the measurement of CCDVD because the light is cut off as the voltage decays. Due to the accumulation layer at junction interface, we address the transport issue based upon the viewpoints of thermionic emission model. A number of electrons that accumulate in the accumulation layer will participate in transport from TiO 2 to FTO. Meanwhile, a few electrons in the FTO will also by absorbing thermal energies transfer in the opposite direction, as depicted in the Fig. 4(c). The observed net current can then be described by the transmission function and the electron state density, and is expressed by [23,24], the probability that electron appears in energy level E 1 as the chemical potential is E 2 , I=q[ TF N C F(E B ,E F )- FT N F F(E C ,E FTO )],way. F(E) is the Boltzmann distribution function F(E 1 ,E 2 )= exp[(E 1 -E 2 )/k B T] where k B and T are the Boltzmann constant and the absolute temperature, respectively. In equilibrium, i.e. E FTO =E F and I=0, one have [23], so that the current can be rewritten  FT N F = TF N C exp[(E B -E C )/k B T]1 FTO F F B B B E E E E k T k T TF C I q N e e                           ( 3 ) Substituting the band edge E B by Eq. (2), one derives the explicit expression describing the current flowing through the heterojunction of FTO/TiO 2 . 1 exp 1 exp FTO F C C B FTO F B E E I q n k T E E k T                                         ( 4 ) In the Eq. (4), we have defined  C = eff exp(-E OC /k B T) and n c =N C exp[(E C -E F )/k B T], Where,  eff = FT exp(/k B T), is an effective transmission rate independent on the variation of E FTO and E F . As the current is set constant I C , the relationship of E FTO and E F can then be deduced, seen in Eq. (5). By means of the method CCDVD, E F and E FTO are measurable at conditions of I C =0 and I C 0, respectively, that allows one to derive the parameters  and , and to further understand the band bending effect observed in DSSC.   ln 1 1 exp 1 C FTO B C C FTO F C B F C I q E k T N E E E k T E E                                                        (5) Results and discussion For the purpose of characterization of junction FTO/TiO 2 , with respect to the understanding of band bending effect, the Eq. (5) [9,10]. Although this effect has an influence on the dark-state current as we modeled, it is still covered in the photovoltaic transport measurements, in which the photocurrent depends mainly on the electron diffusive process due to a large concentration gradient of photo-excited electrons. Conclusion In a summary, the method of voltage decay with a constant discharging current is on the variation Fermi level in TiO 2 but also to a small extent (10%) on the potential difference across the junction. It is believed that the present method can be developed as an effective technology to characterize various photoanode heterostructures, giving rise to crucial information hard to be realized by conventional measurements. The inset re-plots the result of OCVD in the time region of 10 seconds. A 5×5 mm 2 monolayer of TiO 2 nanoparticles (size ~20 nm) with thickness of about 4 m was prepared by screen printing on the F-doped transparent conducting glass (FTO) substrate. And then it was sealed by a 25 m-thick plastic spacer together with the platinized CE after sensitizing in the dye of N719 for 20 h. The resultant cell was filled with the electrolyte including 0.1 M LiI, 0.1M I 2 , 0.5 M 4-tert-butyl pyridine, and 0.6 M 2-Dimethyl-3-propylimidazolium iodide in acetonitrile. The photovoltaic performance for the studied DSSC sample is measured by Keithley 2420. An AM1.5 light was provided by a commercial solar simulator (SAN-EI XES-151S) equipped with a 150W Xenon lamp. The short-circuit current density (J SC ) and the open-circuit voltage (V OC ) are approximately 9.5 mA/cm 2 and 0.67 V, respectively. Figure 2 ( 2a) illustrates the CCDVD measurement for a prepared dye-sensitized solar cell. Keithley 2420 meter is parallel-connected with the circuit and services as a constant current source. As a series of discharge current applied, the decays of E FTO Figure 3 3summarizes the relationship of E FTO vs. E F plots in a serious of constant current applied from 1A to 5 A, where E F is identical to the E FTO (I C =0) and the experimental data are represented by various symbols. As the Fermi level E F decreases the measured voltage E FTO first moves down along the line with a slope of 1 where q is the elementary charge, N F and N C are the states of electron density in region of FTO and TiO 2 , respectively.  FT is the transmission rate of electrons transfer in the direction from FTO to TiO 2 , and  TF denotes the transmission rate of transfer in the opposite established to study the effect of the conduction band bending observed in the photoanode FTO/TiO 2 of DSSC. The method CCDVD, a general version of the open circuit photovoltage decay (OCVD), allows us to derive the relationship of E FTO vs. E F in the dark state. To further understand the band bending effect, the explicit expression for the heterojunction transport is deduced based upon the viewpoint of thermionic emission model and agrees well with the experimental results, clarifying the conduction band bending of TiO 2 depends not only to a significant extent (70%) Fig. 1 ( 1a) Schematic diagram for heterojunction of FTO/TiO 2 . The leftwards and rightwards arrows denote the direction of electrons flowing. (b) Energy level diagram of FTO/TiO 2 in the OCVD measurement. The downwards arrows denote the time evolution. The dashed lines denote the Fermi levels in respective materials. E ele is the redox level of electrolyte. (c) Energy level diagram in the CCDVD measurement. Fig. 2 ( 2a) Schematic diagram for the method of constant-current-discharging voltage decay. (b) The voltage decays of CCDVD with various constant current from 0 to 10 A. Fig. 3 3The relationship of E FTO vs. E F concluded fromFig. 2. The hollow symbols denote experimental data and the lines are the fitting curves. The inset table lists the temperature used in fitting and the square average root (SAR) of error rate of fitting. Fig. 4 4Energy level diagram for the n + -n type junction of FTO-TiO 2 . (a) in equilibrium state under illumination; (b) quasi-equilibrium state in the case of OCVD; (c) non-equilibrium state in the case of CCDVD. The red arrows denote direction of electrons transfer. Figure 1 Figure 2 Figure 3 Figure 4 1234Figure 1 (b). The curves of E FTO decay can be characterized by three stages. i) In first a few microseconds, the high E F suggests a large enough electron density in TiO 2 to maintain such a small discharge current that a decay in E FTO coincides with the variation of E F identical to the observation in the measurement of temperature, it does affect the fitting results a little, that is possibly because the transport model proposed here is based upon the viewpoint of thermionic emission sensitive to the temperature perturbation. For our DSSC samples, it is derived that of the E F decrease, and it will also bend by no more than 10% (1/) times of the potential difference between E FTO and E F . It can thus be suggested, even under illumination, that the TiO 2 conduction band will bend downwards as the Fermi level of TiO 2 decreases, just as what observed in the previous experiments of light interference reflectionis utilized to fitting our experimental data, as plotted by the solid lines in the Fig. 3. The temperatures and Error rate are listed in the table inserted in Fig. 3. Although there is less change in the  eff N C =4.79810 18 cm -3 s -1 , =1.408, =10.204, and E C approximates to -0.86 V. In general, the electron state density in the conduction band of TiO 2 approximates 610 20 cm -3 [18], so that the effective transmission rate for our FTO/TiO 2 interface is around 810 -3 s -1 . More importantly, these results reveal the information of the band bending effect; the conduction band of TiO 2 will shift downwards by approximately 70% (1/) times Acknowledgement:This work is partly sponsored by the Innovation Funds for Ph.D. . 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X Q Wang, C B Cai, Y F Wang, W Q Zhou, Y M Lu, Z Y Liu, L Hu, S Y Dai, Appl. Phys. Lett. 9511112Wang X Q, Cai C B, Wang Y F, Zhou W Q, Lu Y M, Liu Z Y, Hu L H and Dai S Y 2009 Appl. Phys. Lett. 95 011112 S Sze, Semiconductor Devices Physics and Technology. New YorkJohn Wiley & SonsSze S M Semiconductor Devices Physics and Technology 1985 John Wiley & Sons New York For a heterojunction, such as FTO/TiO 2 , the accumulation layer appears as the conduction band of TiO 2 bends downwards. it is contrast to the depletion layer where the band bends upwards, referred to the reference 21For a heterojunction, such as FTO/TiO 2 , the accumulation layer appears as the conduction band of TiO 2 bends downwards; it is contrast to the depletion layer where the band bends upwards, referred to the reference 21 . J Bisquert, G G Belmonte, Pitarch A Bolink, H J , Chem. Phys. Lett. 422184Bisquert J, Belmonte G G, Pitarch A and Bolink H J 2006 Chem. Phys. Lett. 422 184 The Boltzmann distribution function is a proper approximation as the Fermi level far below the conduction band. If the Fermi level either near or above the conduction band the Fermi-Dirac function will be required. The Boltzmann distribution function is a proper approximation as the Fermi level far below the conduction band. If the Fermi level either near or above the conduction band the Fermi-Dirac function will be required	[]
[ "Desingularization Explains Order-Degree Curves for Ore Operators", "Desingularization Explains Order-Degree Curves for Ore Operators" ]	[ "Shaoshi Chen ", "Maximilian Jaroschek ", "Manuel Kauers ", "Michael F Singer ", "\nDept. of Mathematics / NCSU Raleigh\nRISC / Joh\nRISC / Joh\nKepler University\n27695, 4040LinzNCUSA, Austria\n", "\nDept. of Mathematics / NCSU Raleigh\nKepler University\n4040, 27695LinzNCAustria, USA\n" ]	[ "Dept. of Mathematics / NCSU Raleigh\nRISC / Joh\nRISC / Joh\nKepler University\n27695, 4040LinzNCUSA, Austria", "Dept. of Mathematics / NCSU Raleigh\nKepler University\n4040, 27695LinzNCAustria, USA" ]	[]	Desingularization is the problem of finding a left multiple of a given Ore operator in which some factor of the leading coefficient of the original operator is removed. An order-degree curve for a given Ore operator is a curve in the (r, d)-plane such that for all points (r, d) above this curve, there exists a left multiple of order r and degree d of the given operator. We give a new proof of a desingularization result by Abramov and van Hoeij for the shift case, and show how desingularization implies order-degree curves which are extremely accurate in examples.	10.1145/2465506.2465510	[ "https://arxiv.org/pdf/1301.0917v1.pdf" ]	17,013	1301.0917	c150970619dad32001848b1f1fac638fb358e4f1	Desingularization Explains Order-Degree Curves for Ore Operators 5 Jan 2013 Shaoshi Chen Maximilian Jaroschek Manuel Kauers Michael F Singer Dept. of Mathematics / NCSU Raleigh RISC / Joh RISC / Joh Kepler University 27695, 4040LinzNCUSA, Austria Dept. of Mathematics / NCSU Raleigh Kepler University 4040, 27695LinzNCAustria, USA Desingularization Explains Order-Degree Curves for Ore Operators 5 Jan 2013Categories and Subject Descriptors I12 [Computing Methodologies]: Symbolic and Alge- braic Manipulation-Algorithms General Terms Algorithms Keywords Ore Operators, Singular Points Desingularization is the problem of finding a left multiple of a given Ore operator in which some factor of the leading coefficient of the original operator is removed. An order-degree curve for a given Ore operator is a curve in the (r, d)-plane such that for all points (r, d) above this curve, there exists a left multiple of order r and degree d of the given operator. We give a new proof of a desingularization result by Abramov and van Hoeij for the shift case, and show how desingularization implies order-degree curves which are extremely accurate in examples. INTRODUCTION We consider linear operators of the form L = ℓ0 + ℓ1∂ + · · · + ℓr∂ r , where ℓ0, . . . , ℓr are polynomials or rational functions in x, and ∂ denotes, for instance, the derivation d dx or the shift * Supported by the National Science Foundation (NSF) grant CCF-1017217. † Supported by the Austrian Science Fund (FWF) grant Y464-N18. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Copyright 20XX ACM X-XXXXX-XX-X/XX/XX ...$10.00. operator x → x + 1. (Formal definitions are given later.) Operators act in a natural way on functions. They are used in computer algebra to represent the functions f which they annihilate, i.e., L · f = 0. Multiplication of operators is defined in such a way that the product of two operators acts on a function like the two operators one after the other: (P L)·f = P ·(L·f ). Therefore, if L is an annihilating operator for some function f , and if P is any other operator, then P L is also an annihilating operator for f . We are interested in turning a given operator L into a "nicer" one by multiplying it from the left by a suitable P , for two different flavors of "nice". First, we consider the problem of removing factors from the leading coefficient ℓr of L. This is known as desingularization and it is needed for computing the values of f at the roots of ℓr (provided it is defined there). Desingularization of differential operators is classical [9], and for difference operators, Abramov and van Hoeij [2,1] give an algorithm for doing it. We give below a new proof of (a slightly generalized version of) their results. Secondly, we consider the problem of producing left multiples with polynomial coefficients of low degree. Unlike the situation for commutative polynomials, a left multiple P L of L may have polynomial coefficients even if P has rational function coefficients with nontrivial denominators and the polynomial coefficients of L have no common factors. In such situations, it may happen that the degrees of the polynomial coefficients in P L are strictly less than those in L. This phenomenon can be exploited in the design of fast algorithms because a small increase of the order can allow for a large decrease in degree and therefore yield a smaller total size of the operator ("trading order for degree"). Degree estimates supporting this technique have been recently given for a number of different computational problems [4,7,6,3]. Although limited to special situations, these estimates can overshoot by quite a lot. Below we derive a general estimate for the relation between orders and degrees of left multiples of a given operator L from the results about desingularization. This estimate is independent of the context from which the operator L arose, and it is fairly accurate in examples. OVERVIEW Before discussing the general case, let us illustrate the concepts of desingularization and trading order for degree on a concrete example. Consider the differential operator L = −(45 + 25x − 35x 2 − x 3 + 2x 4 ) + 2(33 − 9x − 3x 2 − x 3 )∂ (1 + x)(23 − 20x − x 2 + 2x 3 )∂ 2 ∈ É[x][∂], where ∂ = d dx . That L is desingularizable at (a root of) p := 23 − 20x − x 2 + 2x 3 means that there is some other operator P ∈ É(x)[∂] such that P L has coefficients in É[x] and its leading coefficient no longer contains p as factor. Such a P is called a desingularizing operator for L at p and P L the corresponding desingularized operator. In our example, P = 299 p ∂ + 1035 − 104x − 136x 2 p ∈ É(x)[∂] is a desingularizing operator for L at p, the desingularized operator is P L = (−2350 − 2055x + 104x 2 + 136x 3 ) + (2151 + 281x + 136x 2 )∂ + (1932 + 931x − 240x 2 − 136x 3 )∂ 2 + 299(1 + x)∂ 3 . A desingularizing operator need not exist. For example, it is impossible to remove the factor x + 1 from the leading coefficient of L by means of desingularization. In Section 3 we explain how to check for a given operator L and a factor p of its leading coefficient whether a desingularizing operator exists, and if so, how to compute it. Desingularization causes a degree drop in the leading coefficient but may affect the other coefficients of the operator in an arbitrary fashion. However, a desingularizing operator can be turned into an operator which lowers the degrees of all the coefficients. To this end, multiply P from the left by some polynomial q ∈ É[x] for which the coefficients of pqP have low degree modulo p, i.e., for which qP = 1 p P1 + P2 where P1, P2 ∈ É[x] [∂] and P1 has low degree coefficients. In our example, a good choice is q = (−43 + 34x)/299, i.e. P1 = (−22x + 29) + (−43 + 34x)∂, P2 = − 2312 299 . Since P L has polynomial coefficients, so does 1 p P1L = qP L − P2L = (−10 − 165x + 22x 2 ) + (201 + 65x − 34x 2 )∂ + (−100 + 109x − 22x 2 )∂ 2 − (1 + x)(43 − 34x)∂ 3 . This operator has degree deg we have (q0 + q1∂ + q2∂ 2 )P = 1 x (L) + deg x (P1) − deg x (p) = 2, compared to deg x (L) + deg x (P ) =p 3 Q1 + Q2, where Q1 = (841 + 580x − 436x 2 − 148x 3 + 59x 4 + 12x 5 − 4x 6 ) + (1697 − 528x − 752x 2 + 120x 3 + 127x 4 − 12x 5 − 4x 6 )∂ + (x − 7)(−9 + x + 2x 2 )p ∂ 2 + p 2 ∂ 3 , Q2 = 16(779+374x) 89401 − 272(69+34x) 89401 ∂. Set Q := 1 p 3 Q1. Then, since P L has polynomial coefficients, so does QL = (q0 + q1∂ + q2∂ 2 )P L − Q2L = (2 + x) + (−3 + x)∂ − (8 + 2x)∂ 2 + (2 − 2x)∂ 3 + (6 + x)∂ 4 + (1 + x)∂ 5 . Its degree is deg x (L) + deg x (Q1) − 3 deg x (p) = 1. As the factor x + 1 cannot be removed from L, we cannot hope to reduce the degree even further. We have thus found that the region of all points (r, d) ∈ AE 2 such that there is a left É(x)[∂]-multiple of L of order r and with polynomial coefficients of degree at most d is given by ((2, 4) + AE 2 ) ∪ ((3, 2) + AE 2 ) ∪ ((5, 1) + AE 2 ). In Section 4 we explain the construction of the operators Q that turn a desingularizing operator into one that lowers all the degrees as far as possible, and we give a formula that describes the points (r, d) for which such a Q exists. PARTIAL DESINGULARIZATION In this section we discuss under which circumstances an operator L admits a left multiple P L in which a factor of the leading coefficient of L is removed. This is of interest in its own right, and will also serve as the starting point for the construction described in the following section. In view of this latter application, we cover here a slightly generalized variant of desingularization, which not only applies to the case where a factor can be completely removed, but also cases where only the multiplicity of the factor can be lowered. Example 1. In the shift case (i.e., ∂x = (x+1)∂), consider the operator L = (3 + x)(9 + 7x + x 2 ) − (33 + 70x + 47x 2 + 12x 3 + x 4 )∂ + (2 + x) 2 (3 + 5x + x 2 )∂ 2 . The factor (x + 2) 2 in the leading coefficient cannot be removed completely. Yet we can find a multiple in which x + 2 appears in the leading coefficient (in shifted form) with multiplicity one only. One such left multiple of L is (402 + 208x + 25x 2 ) − (514 + 743x + 258x 2 + 25x 3 )∂ + (233 + 378x + 183x 2 + 25x 3 )∂ 2 − 9(3 + x)∂ 3 . We speak in this case of a partial desingularization. The general definition is as follows. We formulate it for operators in an arbitrary Ore algebra Ç := A[∂] := A[∂; σ, δ] where A is a Ã-algebra (in our case typically A = Ã[x] or A = Ã(x)), Ã is a field, σ : A → A is an automorphism and δ : A → A a σ-derivation, i.e., a Ã-linear map satisfying the skew Leibniz rule δ(pq) = δ(p)q + σ(p)δ(q) for p, q ∈ A. For any f ∈ A, the multiplication rule in A[∂; σ, δ] is ∂f = σ(f )∂ + δ(f ). We write deg ∂ (L) for the order of L ∈ A[∂], and if A = Ã[x], we write deg x (L) for the maximum degree among the polynomial coefficients of L. For general information about Ore algebras, see [5]. Definition 2. Let L ∈ Ã[x][∂; σ, δ] and let p ∈ Ã[x] be such that p \| lc ∂ (L) ∈ Ã[x] . We say that p is removable from L at order n if there exists some P ∈ Ã(x)[∂] with deg ∂ (P ) = n and some w, v ∈ Ã[x] with gcd(p, w) = 1 such that P L ∈ Ã[x][∂] and σ −n (lc ∂ (P L)) = w vp lc ∂ (L). We then call P a p-removing operator for L, and P L the corresponding premoved operator. p is simply called removable from L if it is removable at order n for some n ∈ AE. If gcd(p, lc ∂ (L)/p) = 1, we say desingulariz[able\|ing\|ed] instead of remov[able\|ing\|ed ], respectively. The backwards shift σ −n in the definition above is introduced in order to compensate the effect of the term ∂ n in P on the leading coefficient on L (i.e., lc ∂ (∂ n L) = σ n (lc ∂ (L)).) Moreover, observe that in this definition, removing a polynomial p does not necessarily mean that the p-removed operator has no roots of (some shift of) p in its leading coefficient. If L contains some factors of higher multiplicity, as in the example above, then removal of a polynomial is defined so as to respect multiplicities. Also observe that in the definition we allow that some new factors w are introduced when p is removed. This is only a matter of convenience. We will see below that we may always assume v = w = 1, i.e., if something can be removed at the cost of introducing new factors into the leading coefficient, then it can also be removed without introducing new factors. The justification rests on the following lemma. Lemma 3. Let L ∈ Ã[x][∂; σ, δ], let p ∈ Ã[x] with p \| lc ∂ (L) be removable from L, and let P ∈ Ã(x)[∂;σ,δ] be a p-removing operator for L with deg ∂ (P ) = n. 1. If U ∈ Ã[x][∂] with gcd(lc ∂ (U ), σ n+deg ∂ (U ) (p)) = 1, then U P is also a p-removing operator for L. If P = P1 +P2 for some P1 ∈ Ã(x)[∂] with deg ∂ (P1) = n and P2 ∈ Ã[x][∂] , then P1 is also a p-removing operator for L. 3. There exists a p-removing operator P ′ with deg ∂ (P ′ ) = n and with pσ −n (lc ∂ (P ′ L)) = lc ∂ (L). Proof. Let v, w ∈ Ã[x] be as in Definition 2, i.e., gcd(p, w) = 1 and vpσ −n (lc ∂ (P L)) = w lc ∂ (L). 1. Since P L is an operator with polynomial coefficients, so is U P L. Furthermore, with u = lc ∂ (U ) and m = deg ∂ (U ) we have vpσ −n−m (lc ∂ (U P L)) = σ −n−m (u)w lc ∂ (L). Since gcd(u, σ n+m (p)) = 1, we have gcd(σ −n−m (u)w, p) = 1, as required. 2. Clearly, P2 ∈ Ã[x][∂] implies P2L ∈ Ã[x][∂]. Since also P L ∈ Ã[x][∂], it follows that P1L = (P − P2)L = P L − P2L ∈ Ã[x][∂]. If deg ∂ (P2) < n, then we have lc ∂ (P L) = lc ∂ (P1L), so there is nothing else to show. If deg ∂ (P2) = n, then lc ∂ (P1L) = lc ∂ (P L) − lc ∂ (P2L) and therefore vpσ −n (lc ∂ (P1L)) = vpσ −n (lc ∂ (P L) − lc ∂ (P2L)) = (w − vpσ −n (lc ∂ (P2))) lc ∂ (L). Since gcd(p, w − vpσ −n (lc ∂ (P2))) = gcd(p, w) = 1, the claim follows. 3. By the extended Euclidean algorithm we can find s, t ∈ Ã[x] with 1 = sw + tpv. Then σ n (s)P is p-removing of order n by part 1 (σ n (s) is obviously coprime to σ n (p)), and its leading coefficient is σ n sw pv = 1 σ n (pv) − σ n (t). By part 2 we may discard the polynomial part σ n (t), obtaining a p-removing operator P ′ with the desired property. The lemma implies that if there is a p-removing operator at all, then there is also one in which all the denominators are powers of σ n (p) (because any factors coprime with p can be cleared according to part 1), and where all numerators have smaller degree than the corresponding denominators (because polynomial parts can be removed according to part 2). Similarly as in the proof of part 3, we can also reduce the problem of removing a composite polynomial to the problem of removing powers of irreducible polynomials. For exam- ple, if p = p1p2 is removable from L, where p1, p2 ∈ Ã[x] are coprime, then obviously both p1 and p2 are removable. Conversely, if p1 and p2 are removable, and if P1, P2 are removing operators of orders n1, n2 with lc ∂ (P1) = 1/σ n 1 (p1) and lc ∂ (P2) = 1/σ n 1 (p2), then for n = max{n1, n2} and u1, u2 ∈ Ã[x] with u1σ n (p2) + u2σ n (p1) = gcd(σ n (p1), σ n (p2)) = 1 the operator P := u1∂ n−n 1 P1 + u2∂ n−n 2 P2 ∈ Ã(x)[∂] is such that P L ∈ Ã[x][∂] and lc ∂ (P L) = lc ∂ (L)/σ n (p). In summary, in order to determine whether a polynomial p = p k 1 1 p k 2 2 · · · p km m is removable from an operator L, it suffices to be able to check for an irreducible polynomial pi and a given ki ≥ 1 whether p k i i is removable. Let now p be an irreducible polynomial and k ≥ 1. If there exists a p k -removing operator, then it can be assumed to be of the form P = p0 σ n (p) e 0 + p1 σ n (p) e 1 ∂ +· · ·+ pn−1 σ n (p) e n−1 ∂ n−1 + 1 σ n (p) k ∂ n , for some e0, . . . , en−1 ∈ AE, and p0, . . . , pn−1 ∈ Ã[x] with deg x (pi) < ei deg x (p). In order to decide whether such an operator exists, it is now enough to know a bound on n as well as a bound e on the exponents ei, for if n and e are known, we can make an ansatz pi = e−1 j=0 pi,jx j with undetermined coefficients pi,j, then calculate P L and rewrite all its coefficients in the form a/σ n (p) e + b for some polynomials a, b depending linearly on the undetermined pi,j, then compare the coefficients of the various a's with respect to x to zero and solve the resulting linearly system for the pi,j. How the bounds on n and e are derived depends on the particular Ore algebra at hand. In this paper, we give a complete treatment of the shift case ((σp)(x) = p(x + 1), δ = 0) and make some remarks about the differential case (σ = id, δ = d dx ). For other cases, see the preprint [8]. Shift Case In this section, let Ã[x][∂] denote the Ore algebra of recurrence operators, i.e., σ is the automorphism mapping x to x + 1 and δ is the zero map. This case was studied by Abramov and van Hoeij [2,1]. We give below a new proof of their result, and extend it to the case of partial desingularization. For consistency with the differential case, we formulate the result for the leading coefficients, while Abramov and van Hoeij consider the analogous for the trailing coefficients. Of course, this difference is immaterial. We proceed in two steps. First we give a bound on the order of a removing operator (Lemma 4), and then, in a second step, we provide a bound on the exponents in the denominators (Theorem 5). As explained above, it is sufficient to consider the case of removing powers of irreducible polynomials, and we restrict to this case. Lemma 4. Let L = ℓ0 + ℓ1∂ + · · · + ℓr∂ r ∈ Ã[x][∂] with ℓ0, ℓr = 0, and let p be an irreducible factor of lc ∂ (L) such that p k is removable from L for some k ≥ 1. Let n ∈ AE be s.t. gcd(σ n (p), ℓ0) = 1 and gcd(σ m (p), ℓ0) = 1 for all m > n. Then p k is removable at order n from L. Proof. By assumption on L, there exists a p k -removing operator P , say of order m, and by the observations following Lemma 3 we may assume that P = p0 σ m (p) e 0 + p1 σ m (p) e 1 ∂ + · · · + pm σ m (p) em ∂ m , for ei ∈ AE and pi ∈ Ã[x] with deg x (pi) < ei deg x (p) (i = 0, . . . , m). We may further assume gcd(σ m (p), pi) = 1 for i = 0, . . . , m (viz. that the ei are chosen minimally). Suppose that m > n. We show by induction that then e0 = e1 = · · · = em−n−1 = 0, so that pi = 0 for i = 0, . . . , m − n − 1, i.e., the operator P has in fact the form P = pm−n σ m (p) e m−n ∂ m−n + · · · + pm σ m (p) em ∂ m . Thus ∂ n−m P ∈ Ã(x)[∂] is a p k -removing operator of order n. Consider the operator T := r+m by the choice of p0 and the assumption in the lemma, respectively, and this leaves no possibility for cancellation. Assume now, as induction hypothesis, that e0 = e1 = · · · = ei−1 = 0 for some i < m − n. Then from i=0 ti∂ i := P L ∈ Ã[x][∂].ti = pi σ m (p) e i σ i (ℓ0) + pi−1 σ m (p) e i−1 σ i−1 (ℓ1) + · · · + p0 σ m (p) e 0 ℓi = pi σ m (p) e i σ i (ℓ0) it follows that σ m (p) e i \| piσ i (ℓ0). By the choice of pi we have gcd(σ m (p), pi) = 1 and by the assumption in the lemma we have gcd(σ m−i (p), ℓ0) = 1 (because m − i > n), so it follows that ei = 0. Inductively, we obtain e0 = e1 = · · · = em−n−1 = 0, which completes the proof. It can be shown that p cannot be removed from L if σ n (p) is coprime with the trailing coefficient of L for all n ∈ AE by a variant of [2, Lemma 3.], so the above lemma covers all situations where removing of a factor is possible. In order to formulate the result about the possible exponents in the denominator, it is convenient to first introduce some notation. Let us call two irreducible polynomials p, q ∈ Ã[x] \ {0} equivalent if there exists n ∈ such that σ n (p)/q ∈ Ã. We write [q] for the equivalence class of q ∈ Ã[x] \ {0}. If p, q are equivalent in this sense, we write p ≤ q if σ n (p)/q ∈ Ã for some n ≥ 0, and p > q otherwise. The irreducible factors of a polynomial u ∈ Ã[x] can be grouped into equivalence classes, for example u = (x − 4)(x − 1) 3 x(x + 1) 2 (2x − 5)(2x + 3) 2 (2x + 9) × (x 2 + 5x + 1)(x 2 + 11x + 25) 3 . For any monic irreducible factor p of u ∈ Ã[x], let vp(u) denote the multiplicity of p in u, and define v<p(u) := max{ vq(u) \| q ∈ [p] : p > q }. For example, for the particular u above we have vx−4(u) = 1, v<x−4(u) = 0, v<x+1(u) = 3, and so on. Besides being applicable not only to desingularization but also removal of any factors, the following theorem also refines the corresponding result of Abramov and van Hoeij in so far as their version only covers the case of desingularizing L at some p with v>p(lc ∂ (L)) = 0 whereas we do not need this assumption. Theorem 5. Let L = ℓ0 + ℓ1∂ + · · · + ℓr∂ r ∈ Ã[x][∂] with ℓ0, ℓr = 0, and let p be an irreducible factor of ℓr such that p k is removable from L for some k ≥ 1. Let n ∈ AE be such that gcd(σ n (p), ℓ0) = 1 and gcd(σ m (p), ℓ0) = 1 for all m > n. Then there exists a p k -removing operator P for L and p of the form Let e = max{e1, . . . , en} andP := n i=0p i∂ i := σ n (p) e P . Thenpi = σ n (p) e−e i pi (i = 0, . . . , n) and σ n (p) e T =P L and gcd(p0, . . . ,pn, σ n (p)) = 1. P = p0 σ n (p) e 0 + p1 σ n (p) e 1 ∂ + · · · + pn σ n (p) en ∂ n , Abbreviating v := v<p(lc ∂ (L)), assume that e > k + n v. We will show by induction that thenpi contains σ n (p) with multiplicity more than i v for i = n, n − 1, . . . , 0, which is inconsistent with gcd(p0, . . . ,pn, σ n (p)) = 1. First it is clear thatpn = σ n (p) e pnσ n (ℓr) contains σ n (p) with multiplicity ≥ e − k > nv, because P is p k -removing. Suppose now as induction hypothesis that there is an i ≥ 0 such that σ n (p) j v+1 \|pj for j = n, n − 1, . . . , i + 1. Consider the equality σ n (p) e ti+r =piσ i (ℓr) +pi+1σ i+1 (ℓr−1) + · · · +pnℓr−n, where we use the convention ℓj := 0 for j < 0. The induction hypothesis implies that σ n (p) (i+1)v+1 \|pj for j = n, n − 1, . . . , i + 1. Furthermore, since (i + 1)v ≤ nv < e, we have σ n (p) (i+1)v+1 \| σ n (p) e ti+r. Both facts together imply σ n (p) (i+1)v+1 \|piσ i (ℓr). The definition of v ensures that σ n (p) is contained in σ i (ℓr) with multiplicity at most v, so it must be contained inpi with multiplicity more than (i + 1)v − v = i v, as claimed. Differential Case In this section Ã[x][∂] refers to the Ore algebra of differential operators, i.e., σ = id and δ = d dx . Let L ∈ Ã[x][∂] and suppose for simplicity that p = x is a factor of lc ∂ (L). In [1], the authors show that L can be desingularized at x if and only if x = 0 is an apparent singularity, that is, if and only if L(y) = 0 admits deg ∂ (L) linearly independent formal power series solutions. The authors furthermore give an algorithm to find an operator P such that if ξ is either an ordinary point of L or an apparent singularity of L, then ξ is an ordinary point of P L. Therefore this algorithm desingularizes all the points that can be desingularized. The authors also give a sharp bound for deg ∂ (P ). The authors furhtermore give some indications concerning partial desingularizations. It would be interesting to give a complete algorithm for partial desingularizations. ORDER-DEGREE CURVES We now turn to the construction of left multiples of L with polynomial coefficients of small degree, and to the question of how small these degrees can be made. As already indicated in Section 2, we start from an operator P which removes some factor from the leading coefficient of L, say it removes a polynomial p of degree k. According to Lemma 3, we may assume that lc ∂ (P ) = 1/σ deg ∂ (P ) (p) and that all other coefficients of P are rational functions whose numerators have lower degree than the corresponding denominators. Thus we already have deg x (P L) ≤ deg x (L) − 1. Furthermore, if q is any polynomial with deg x (q) < deg x (p) = k, then multiplying P by q (from left) and removing polynomial parts by Lemma 3.2 gives another operator Q with deg x (QL) ≤ deg x (L) − 1. All the operators Q obtained in this way form a Ã-vector space of dimension k. Within this vector space we search for elements where deg x (QL) is as small as possible. Forcing the coefficients of the highest degrees to zero gives a certain number of linear constraints which can be balanced with the number of degrees of freedom offered by the coefficients of q, as illustrated in the figure below. As long as we force fewer than k terms to zero, we will find a nontrivial solution. If we want to eliminate k terms or more in order to get a result of even lower degree, we need more variables. We can create k more variables if instead of an ansatz qP we make an ansatz (q0 + q1∂)P for some q0, q1 ∈ Ã[x] with deg x (q0), deg x (q1) < k. Again removing all polynomial parts from the rational function coefficients we obtain a vector space of operators Q with deg x (QL) ≤ deg x (L) − 1 whose dimension is 2k. The additional degrees of freedom can be used to eliminate more high degree terms, the result being an operator of lower degree but higher order. If we let the order increase further and for each fixed order use all the available degrees of freedom to reduce the degrees to minimize the degrees of the polynomial coefficients, a hyperbolic relationship between the order and the degree of QL emerges. In Theorem 9 below, we make this relationship precise, taking into account that for a given operator L the leading coefficient may contain several factors p that are removable at different orders n. The resulting region of all points (r, d) ∈ AE 2 for which there exists a left multiple of L of order r with polynomial coefficients of degree at most d is then given by an overlay of a finite number of hyperbolas. Before turning to the proof of this theorem, let us illustrate its basic idea with the example operators from Section 2. Example 6. Let L ∈ É[x][∂], p ∈ É[x] , and P ∈ É(x)[∂] be as in Section 2. Recall that p is an irreducible cubic factor of lc ∂ (L) and that P is a p-removing operator for L. We have P = p 1 p ∂ + p 0 p for some p1, p0 ∈ É[x] with deg x (p1) = 0 and deg x (p0) = 2. We have seen in Section 2 that there is an operator Q ∈ É(x)[∂] of order 3 such that QL ∈ É[x][∂] and deg x (QL) = 1. Our goal here is to explain why this operator exists. Make an ansatz Q1 = (q0 + q1∂ + q2∂ 2 )P with undetermined polynomials q0, q1, q2. After expanding the product and applying commutation rules, Q1 has the form p1q2 p ∂ 3 + (. . .)q2 + (. . .)q1 p 2 ∂ 2 + (. . .)q2 + (. . .)q1 + (. . .)q0 p 3 ∂ + (. . .)q2 + (. . .)q1 + (. . .)q0 p 3 , where the (. . .) are certain polynomials whose precise form is irrelevant for our purpose. The coefficients of Q2 depend linearly on the undetermined polynomials q0, q1, q2. If we choose their degree to be deg x (p) − 1 = 2, then we have 3(2 + 1) = 9 variables for the coefficients of q0, q1, q2. Choosing a higher degree would give more variables but also introduce undesired solutions such as q0 = q1 = q2 = p, for which the reduction modulo p 3 leads to the useless result Q2 = 0. This cannot happen if we enforce deg x (qi) < deg x (p). The operator p −3 Q2L has degree deg x (Q2) + deg x (L) − 3 deg x (p) = deg x (Q2) − 5, which is equal to 1 if deg x (Q2) = 6. A priori, the degree of Q2 in x may be up to deg x (p 3 ) − 1 = 8. In order to bring it down to 6, we equate the coefficients of x i ∂ j for i = 7, 8 and j = 0, . . . , 3 to zero. This gives 8 equations. As there are more variables than equations, there must be a nontrivial solution. For formulating the proof of the general statement, it is convenient to work with an alternative formulation of removability, which is provided in the following lemma. Through- Proof. "⇐": P0 = 1 σ n (p) lc ∂ (P ) P is a p-removing operator. "⇒": Start from a p-removing operator of the form P0 = n−1 i=0 pi σ n (p) e i ∂ i + 1 σ n (p) ∂ n , and set P = σ n (p) e P0 where e = max{e0, . . . , en−1, 1} ≥ 1. Because of P0L ∈ Ã[x][∂] it follows that P L ∈ σ n (p) e Ã[x][∂] = σ n (p) lc ∂ (P )Ã[x][∂]. The next lemma is a generalization of Bezout's relation to more than two coprime polynomials, which we will also need in the proof. Lemma 8. Let u1, . . . , um ∈ Ã[x] be pairwise coprime and u = u1u2 · · · um, and let v1, . . . , vm ∈ Ã[x] be such that deg x (vi) < deg x (ui) (i = 1, . . . , m). If m i=1 vi u ui = 0 then v1 = v2 = · · · = vm = 0. Proof. Since the ui are pairwise coprime, ui ∤ u/ui for all i. However, ui \| u/uj for all j = i. Both facts together with m i=1 viu/ui = 0 imply that ui \| vi for all i. Since deg x (vi) < deg x (ui), the claim follows. Proof. Let r ≥ deg ∂ (L), and set s := r − deg ∂ (L) so that s = deg ∂ (Q). We may assume without loss of generality that s is such that 1 − n i r−deg ∂ (L)+1 = 1 − n i s+1 > 0 for all i by simply removing all the pi for which 1 − n i s+1 ≤ 0 from consideration. We thus have s ≥ ni for all i. Lemma 7 yields operators Pi ∈ Ã[x][∂] of order ni with PiL ∈ σ n i (pi) lc(Pi)Ã[x][∂]. Set q = m i=1 s−n i j=0 σ j+n i (pi)σ j (li), where li = lc ∂ (Pi). Consider the ansatz Q1 = m i=1 s−n i j=0 qi,j q σ j+n i (pi)σ j (li) ∂ j Pi for undetermined polynomial coefficients qi,j (i = 1, . . . , m; j = 0, . . . , ni) of degree less than deg x (pi This means that we can replace the coefficients in Q1 by their remainders upon division by q without violating any of the mentioned properties of Q1. Also observe that any operator Q2 obtained in this way is nonzero unless all the qi,j are zero, because if k is maximal such that at least one of the q i,k is nonzero, then lc ∂ (Q1) = m i=1 q i,k q σ k+n i (pi)σ k (li) σ k (li) = m i=1 q i,k q σ k+n i (pi) is nonzero by Lemma 8. Furthermore, lc ∂ (Q1) ≡ 0 mod q because deg x (q i,k ) < deg x (pi) implies deg x (lc ∂ (Q1)) < deg x (q). The ansatz for the qi,j gives m i=1 (s−ni +1) deg x (pi) variables. Plug this ansatz into Q1 and reduce all the polynomial coefficients modulo q, obtaining an operator Q2 of degree less than deg x (q) = m i=1 (s − ni + 1)(deg x (pi) + deg x (li)) . Then for each of the s + 1 polynomial coefficients in Q2 equate the coefficients of the terms x j for j > m i=1 (s − ni) deg x (pi) + deg x (li) + m i=1 ni deg x (pi) s + 1 to zero. This gives altogether (s + 1) m i=1 (s−ni+1) deg x (pi)+ deg x (li) − 1 − deg x (li) − m i=1 (s−ni) deg x (pi)+ deg x (li) − m i=1 ni deg x (pi) s + 1 = (s + 1) m i=1 deg x (pi) − 1 − m i=1 ni deg x (pi) s + 1 equations. The resulting linear system has a nontrivial solution because #vars − #eqns = m i=1 (s − ni + 1) deg x (pi) − (s + 1) m i=1 deg x (pi) − 1 − m i=1 ni deg x (pi) s + 1 = − m i=1 ni deg x (pi) − (s + 1) −1 − m i=1 ni deg x (pi) s + 1 > − m i=1 ni deg x (pi) + s + 1 s + 1 m i=1 ni deg x (pi) = 0. deg x (L) + deg x (Q2) − deg x (q) ≤ deg x (L) + m i=1 (s − ni)(deg x (pi) + deg x (li)) + m i=1 ni deg x (pi) s + 1 − m i=1 (s − ni + 1)(deg x (pi) + deg x (li)) ≤ deg x (L) − m i=1 deg x (pi) + m i=1 ni deg x (pi) s + 1 = deg x (L) − m i=1 1− ni s + 1 deg x (pi) , as required. (The final step uses the facts ⌊−x⌋ = −⌈x⌉ and ⌈x + n⌉ = ⌈x⌉ + n for x ∈ Ê and n ∈ .) Example 10. d := deg x (QL) ≤ 4 − 1 − 1 r − 2 + 1 + 3 = r + 2 r − 1 . This hyperbola precisely predicts the order-degree pairs we found in Section 2: − 10n(8 + 5n)(9 + 5n)(11 + 5n)(12 + 5n)p(n)∂ 3 , where ∂ represents the shift operator and p is a certain irreducible polynomial of degree 10. This polynomial is removable of order 1. Therefore, by the theorem, we expect left multiples of L of order r and degree bounded by 16 − 1 − 1 r − 3 + 1 + 10 = 6r − 2 r − 2 . In the figure below, the curve d = 6r−2 r−2 (solid) is contrasted with the estimate d = 8r−1 r−2 (dashed) derived last year for this example [6] as well as the region of all points (r, d) for which a left multiple of L of order r and degree d exists (gray). The new curve matches precisely the boundary of the gray region, even including the very last degree drop (which is not clearly visible on the figure): for r = 12 we have 6r−2 r−2 = 7 and for r = 13 we have 6r−2 r−2 ≈ 6.9 < 7. for all r ∈ AE. Again, this estimate is accurate, while the estimate 24r−9 r−2 derived in [7] overshoots. Operators coming from applications tend to have leading coefficients that contain a single irreducible polynomial of large degree which can be removed at order 1, besides factors that are not removable. But Theorem 9 also covers the more general situation of factors that are only removable of higher order, and even the case of several polynomials that are removable at several orders. As an example for this general situation, consider the operator L = 8(1+x)(1+2x) 3 (37+3z) 7 (14+32x+26x 2 +7x 3 ) 7 − 9(1+3x) 9 (2+3x) 2 (1+x+5x 2 +7x 3 ) 7 ∂, where ∂ represents the shift operator. From its leading coefficient, the polynomial (1 + x + 5x 2 + 7x 3 ) 7 is removable at order 1, and in addition, (1 + 3x) 7 is removable at order 12. The remaining factors are not removable. According to Theorem 9 we expect that L admits left multiples of order r and degree 32 − 21 1 − 1 r + − 7 1 − 3 1 + , for all r ∈ AE. It turns out that this prediction is again accurate for every r. Observe that in this example the curve is a superposition of two hyperbolas. In conclusion, we believe that removable factors provide a universal explanation for all the order-degree curves that have been observed in recent years for various different contexts. We have derived a formula for the boundary of the gray region associated to a fixed operator L, which, although formally only a bound, happens to be exact in all the examples we considered. This does not immediately imply better complexity estimates or faster variants of algorithms exploiting the phenomenon of order-degree curves, because usually L is not known in advance but rather the desired output of a calculation, and therefore we usually have no information about the removable factors of lc ∂ (L). However, we now know what we have to look at: in order to improve algorithms based on trading order for degree, we need to develop a theory which provides a priori information about the removable factors of lc ∂ (L). In other words, our result reduces the task of better understanding order-degree curves to the task of better understanding what causes the appearance of removable factors in operators coming from applications. From p 0 σ 0m (p) e 0 ℓ0 = t0 ∈ Ã[x] it follows that e0 = 0, because gcd(σ m (p), p0) = gcd(σ m (p), ℓ0) = 1 for some ei ∈ AE and pi ∈ Ã[x] with 1. deg x (pi) < ei deg x (p) and gcd(σ n (p), pi) = 1, and 2. ei ≤ k + n v<p(lc ∂ (L)) for i = 0, . . . , n − 1, and pn = 1, en = k.Proof. Lemmas 3 and 4 imply the existence of an operator P with all the required properties except possibly the exponent estimate in item 2. Let P be such an operator, and consider the operator T := r+n i=0 ti∂ i := P L ∈ Ã[x][∂]. out the section, Ã[x][∂] = Ã[x][∂; σ, δ] is an arbitrary Ore algebra. Lemma 7. p ∈ Ã[x] is removable from L ∈ Ã[x][∂] at order n if and only if there exists P ∈ Ã[x][∂] with deg ∂ (P ) = n and P L ∈ σ n (p) lc ∂ (P )Ã[x][∂]. Theorem 9 . 9Let L ∈ Ã[x][∂], and let p1, . . . , pm ∈ Ã[x] be factors of lc ∂ (L) which are removable at orders n1, . . . , nm, respectively, so that the σ n i (pi) are pairwise coprime. Let r ≥ deg ∂ (L) and d ≥ deg x (L) x (pi) , where we use the notation (x) + := max{x, 0}. Then there exists an operator Q ∈ Ã(x)[∂]\{0} such that QL ∈ Ã[x][∂] and deg ∂ (QL) = r and deg x (QL) = d. Consider the sequence (an) ∞ n=0 defined byan = k Γ(2n + k)Γ(n − k + 2) Γ(2n − k)Γ(n + 2k) (n ∈ AE).Zeilberger's algorithm finds an annihilating operator L of the form 9(1 + n)(1 + 3n)(2 + 3n) 2 (3n + 4)p(n + 1) + (. . . degree 16. . . )∂ + (. . . degree 15. . . )∂ 2 the minimal order telescoper L for the hyperexponential term in Example 15.2 in[7]. It has order 3 and degree 40. The leading coefficient contains an irreducible polynomial p of degree 23 at order 1 and otherwise only non-removable factors. Theorem 9 therefore predicts left multiples of L of degree r and degree By construction, the solution gives rise to an operator Q2 ∈ Ã[x][∂] of order at most n with polynomial coefficients of for which Q2L ∈ qÃ[x][∂]. Thus if we set Q = 1 q Q2 ∈ Ã(x)[∂], we have deg ∂ (QL) = deg ∂ (L)+s = r and deg x (QL)degree at most m i=1 (s − ni)(deg x (pi) + deg x (li)) + m i=1 ni deg x (pi) s + 1 , is at most 1. Consider again the example from Section 2. There we started from an operator L ∈ Ã[x][∂] of order 2 and degree 4 for which there exists a desingularizing operator P of order 1 which removes a polynomial p of degree 3. According to the theorem, for every r ≥ 2 exists an operator Q ∈ Ã[x][∂] with QL ∈ Ã[x][∂], deg ∂ (QL) ≤ r and Apparent singularities of linear difference equations with polynomial coefficients. Sergei A Abramov, Moulay A Barkatou, Mark Van Hoeij, AAECC. 17Sergei A. Abramov, Moulay A. Barkatou, and Mark van Hoeij. Apparent singularities of linear difference equations with polynomial coefficients. AAECC, 17:117-133, 2006. Desingularization of linear difference operators with polynomial coefficients. Sergei A Abramov, Mark Van Hoeij, Proceedings of ISSAC'99. ISSAC'99Sergei A. Abramov and Mark van Hoeij. Desingularization of linear difference operators with polynomial coefficients. In Proceedings of ISSAC'99, pages 269-275, 1999. Fast computation of common left multiples of linear ordinary differential operators. Alin Bostan, Frederic Chyzak, Ziming Li, Bruno Salvy, Proceedings of ISSAC'12. ISSAC'12Alin Bostan, Frederic Chyzak, Ziming Li, and Bruno Salvy. Fast computation of common left multiples of linear ordinary differential operators. In Proceedings of ISSAC'12, pages 99-106, 2012. Differential equations for algebraic functions. Alin Bostan, Frédéric Chyzak, Bruno Salvy, Grégoire Lecerf, Schost, Proceedings of ISSAC'07. ISSAC'07Alin Bostan, Frédéric Chyzak, Bruno Salvy, Grégoire Lecerf, andÉric Schost. Differential equations for algebraic functions. In Proceedings of ISSAC'07, pages 25-32, 2007. An introduction to pseudo-linear algebra. Manuel Bronstein, Marko Petkovšek, Theoretical Computer Science. 1571Manuel Bronstein and Marko Petkovšek. An introduction to pseudo-linear algebra. Theoretical Computer Science, 157(1):3-33, 1996. Order-degree curves for hypergeometric creative telescoping. Shaoshi Chen, Manuel Kauers, Proceedings of ISSAC'12. ISSAC'12Shaoshi Chen and Manuel Kauers. Order-degree curves for hypergeometric creative telescoping. In Proceedings of ISSAC'12, pages 122-129, 2012. Trading order for degree in creative telescoping. Shaoshi Chen, Manuel Kauers, Journal of Symbolic Computation. 478Shaoshi Chen and Manuel Kauers. Trading order for degree in creative telescoping. Journal of Symbolic Computation, 47(8):968-995, 2012. Taming apparent singularities via Ore closure. Frederic Chyzak, Philippe Dumas, Ha Le, Jose Martin, Marni Mishna, Bruno Salvy, in preparationFrederic Chyzak, Philippe Dumas, Ha Le, Jose Martin, Marni Mishna, and Bruno Salvy. Taming apparent singularities via Ore closure. in preparation. E L Ince, Ordinary Differential Equations. DoverE. L. Ince. Ordinary Differential Equations. Dover, 1926.	[]
[ "PRELIMINARY GROUP CLASSIFICATION AND SOME EXACT SOLUTIONS OF 2−HESSIAN EQUATION", "PRELIMINARY GROUP CLASSIFICATION AND SOME EXACT SOLUTIONS OF 2−HESSIAN EQUATION" ]	[ "Mahdieh Yourdkhany ", "Mehdi Nadjafikhah ", "Megerdich " ]	[]	[]	We study the class of 3-dimensional nonlinear 2−hessian equations u xx u yy + u xx u yy + u yy u zz − u 2 xy − u 2 yz − u 2 xz − f (x, y, z) = 0, where f is an arbitrary smooth function of the variables (x, y, z). We perform preliminary group classification on 2−hessian equation. In fact, we find additional equivalence transformation on the space (x, y, z, u, f ), with the aid of N. Bila's method, then we take their projections on the space (x, y, z, f ), so we prove an optimal system of one-dimensional Lie subalgebras of this equation is generated by A 1 , · · · , A 12 , which introduced in theorem(2), ultimately, A number of new interesting nonlinear invariant models are obtained which have non-trivial invariance algebras. The result of these works is a wide class of equations which summarized in table. So at the end of this work, some exact solutions of 2−hessian equation are presented. The paper is one of the few applications of an algebraic approach to the group classification using Lie method.	null	[ "https://arxiv.org/pdf/1902.02702v1.pdf" ]	119,691,236	1902.02702	e8608127c27decc7654abc6e8d46b3fc4a970736	PRELIMINARY GROUP CLASSIFICATION AND SOME EXACT SOLUTIONS OF 2−HESSIAN EQUATION 7 Feb 2019 February 8, 2019 Mahdieh Yourdkhany Mehdi Nadjafikhah Megerdich PRELIMINARY GROUP CLASSIFICATION AND SOME EXACT SOLUTIONS OF 2−HESSIAN EQUATION 7 Feb 2019 February 8, 2019Hessian equationOptimal systemPreliminary group classification AMS Classification 2010: 53C1053C1253A5535A3076M6058J70 We study the class of 3-dimensional nonlinear 2−hessian equations u xx u yy + u xx u yy + u yy u zz − u 2 xy − u 2 yz − u 2 xz − f (x, y, z) = 0, where f is an arbitrary smooth function of the variables (x, y, z). We perform preliminary group classification on 2−hessian equation. In fact, we find additional equivalence transformation on the space (x, y, z, u, f ), with the aid of N. Bila's method, then we take their projections on the space (x, y, z, f ), so we prove an optimal system of one-dimensional Lie subalgebras of this equation is generated by A 1 , · · · , A 12 , which introduced in theorem(2), ultimately, A number of new interesting nonlinear invariant models are obtained which have non-trivial invariance algebras. The result of these works is a wide class of equations which summarized in table. So at the end of this work, some exact solutions of 2−hessian equation are presented. The paper is one of the few applications of an algebraic approach to the group classification using Lie method. Introduction Nowadays it is generally accepted that a huge number of real processes arising in physics, biology, chemistry, etc. can be described by nonlinear PDEs. And the most powerful methods for costruction of exact solutions for a wide ranges of nonlinear PDEs are symmetry-based methods, and these methods originated from the Lie method, so the basic part of the theory, is infinitesimal method of Sophus Lie [14], that is connection between continuous transformation groups and algebras of their infinitesimal generators. This method leads to techniques in the group-invariant solutions and conservation laws of differential equations [18,11,20]. In fact the method that we proposed, the method of preliminary group classification, is a conclusion of Lie infinitesimal method, and is defined and related to the theory of group classification of differential equations. this method is proposed in [1] and is developed for deferential equation in [10,4]. The main idea of preliminary group classification is based on extension of the kernel of admitted Lie groups that are obtained by the transformations from the corresponding equivalence Lie group. The problem of finding inequivalent cases of such extension of symmetry can reduce to the classification of inequivalent subgroups of the equivalence Lie group(In particular, if a Lie group is finite-parameter, then one can use an optimal systems of its subgroups). we use equivalence transformations and the theory of classification of finite-dimensional Lie algebras. In this paper we study point symmetry and equivalence classification of HESI equation, leading to a number of new interesting nonlinear invariant models associated to non-trivial invariance algebras. A complete list of these models are given for a finite-dimensional equivalence algebra derived for HESI equation. To obtain these goals we perform algorithms that is explained in references [20,3,12,5], and we use similar works in [13,10,15,16,17,9,22]. For the local solution: The existence of C ∞ local solutions of HESI equation In R 3 is studied in [23], and the solution is in the following form: u(x, y, z) = 1 2 (τ 1 x 2 + τ 2 y 2 + τ 3 z 2 ) + ε 5 ω(ε −2 (x, y, z)), where ε and τ i are arbitrary constants, and ω is a given smooth function. So at the end of this work, with the symmetry group of the equation, this solutions transforms to another solution of HESI equation. About HESI equation: Based on refrences [6,23], k-Hessian equations are a family of PDEs in n-dimensional space equations that can be written as S k [u] = f , where 1 k n, S k [u] = σ k (λ(D 2 u)), and λ(D 2 u) = (λ 1 , · · · , λ n ), are the eigenvalues of the Hessian matrix D 2 u (∂ i ∂ j u) 1 i,j n and σ k (λ) = i 1 <···<i k λ i 1 · · · λ i k , is a kth elementry symmetric polynomial. The k-Hessian equations include the Laplace equation, when k = 1, And the Monge-Ampere equation, when k = n. Here we study 2−hessian equation in three dimensions, and f is an arbitrary function of x, y, z (HESI equation) : S 2 [u] := u xx u yy + u xx u yy + u yy u zz − u 2 xy − u 2 yz − u 2 xz ,(1) this equation is a fully nonlinear elliptic partial diferential equation, that is related to intrinsic curvature for three-dimensional manifolds. In fact, the 2−hessian equation is unfamiliar outside Riemannian geometry and elliptic regularity theory, that is closely related to the scalar curvature operator, which provides an intrinstic curvature for a three-dimensional manifold. Geometric PDEs have been used widely in image analysis [21]. In particular, the Monge-Ampere equation in the context of optimal transportation has been used in three dimensional volume-based image registration [8]. The 2−hessian operator also appears in conformal mapping problems. Conformal surface mapping has been used for two-dimmensional image registration [2,7], but does not generalize directly to three dimensions. Quasi-conformal maps have been used in three dimensions [24,25]. However, these methods are still being developed. Principal Lie Algebra The symmetry approach to the classification of admissible partial differential equations depends heavily on a useful way of describing transformation groups that keep invariant the form of a given partial differential equation. This is done via the well-known infinitesimal method developed by Sophus Lie [18,19,20].Given a partial differential equation, the problem of finding its maximal (in some sense) Lie invariance group reduces to solving determining equations that is an over-determined system of linear partial differential equation. We consider the 2−hessian equation as the form: HESI : S 2 [u] = f (x, y, z),(2) where u = u(x, y, z) is dependent variable and x, y, z are independent variables and f is arbitrary function. Considering the total space E = X × U with local coordinate (x, y, z, u) which x, y, z ∈ X and u ∈ U . The solution space of equation (2) is a subvariety S ∆ ⊂ J 2 (R 3 , R) of the second order of jet bundle of 3−dimensional submanifolds of E. The 1−parameter Lie group of infinitesimal transformations on E is as follows:x = x + tξ(x, y, z, u) + O(t 2 ),ỹ = y + tζ(x, y, z, u) + O(t 2 ), z = z + tη(x, y, z, u) + O(t 2 ),ũ = u + tφ(x, y, z, u) + O(t 2 ),(3) where t is the group parameters and ξ, ζ, φ and η are the infinitesimals of the transformations for the independent and dependent variables, resp. So the corresponding infinitesimal generators have the following form generally V = ξ(x, y, z, u)∂ x + ζ∂ y + η∂ z + φ∂ u . So based on Theorem 2.31 of [19], V is a invariant point transformation if pr (2) V [HESI] = 0. Where pr (2) V is the second order prolongation of the vector field V , that means: pr (2) V = V + φ x (x, y, z, u (2) ) ∂ ux + · · · + φ xz (x, y, z, u (2) ) ∂ uxz ,(5) in which u (2) = (u, u x , u y , u z , u xx , u xy , u xz , u yy , u yz , u zz ) and φ J (x, y, z, u (2) ) = D J φ − 3 i=1 ξ i ∂u ∂x i + 3 i=1 ξ i ∂u J ∂x i(6) where (ξ 1 , ξ 2 , ξ 3 ) = (ξ, ζ, η) and (x 1 , x 2 , x 3 ) = (x, y, z), further J = (j 1 , · · · , j k ) is a k-th order multiindex, and j i s adopt x or y or z, for each 1 i k, then D J denotes the total derivatives for the multi-index J, and the J-th total derivative is as D J = D j 1 · · · D j k , where D i = ∂ x i + J ∂u J ∂x i ∂ u J , (x 1 , x 2 , x 3 ) = (x, y, z). So pr (2) V acts on Eq. (2) and with replacing u yy with equivalent expression of HESI equation we have the following system as determining equation: φ xx = φ xy = φ xz = φ xu = φ yy = φ yz = φ yu = φ zz = φ zu = 0, φ uu = ξ u = ξ yy = ξ yz = ξ zz = ζ u = ζ zz = η u = η zz = 0, ξ x = η z , ζ x = −ξ y , ζ y = η z , η x = −ξ z , η y = −ζ z , f x ξ + f y ζ + f z η + 2f (2η z − φ u ) = 0.(7) where f is arbitrary function.with solving above relations we have: ξ = c 6 x + c 7 y + c 8 z + c 9 , ζ = c 10 z + c 6 y − c 7 x + c 11 , η = −c 10 y + c 6 z − c 8 x + c 12 , φ = c 1 x + c 2 u + c 3 y + c 4 z + c 5 , f x ξ + f y ζ + f z η + 2f (2η z − f φ u ) = 0.(8) which c i , i = 1, · · · , 12 are arbitrary costants. So if f (x, y, z) = 0 the last equation of relations (8) will be removed and based on the first four equations in relations (8), we have 12-dimensional symmetry group, but if f (x, y, z) = 0 we substitute the first four equations of (8) in the last one, and obtain the following condition: (c 6 x + c 7 y + c 8 z + c 9 )f x + (c 10 z + c 6 y − c 7 x + c 11 )f y + (−c 10 y + c 6 z − c 8 x + c 12 )f z + (−2c 2 + 4c 6 )f = 0. (9) There isn't c 1 , c 3 , c 4 , c 5 in condition (9), so these coeficients are free,that means equation (2) have 4−dimentional symmetry group at minimum. So we conclude the following theorem from above relations: Theorem 1 : The HESI equation(Eq.(2)) admits symmetry group of dimension 4 to 12, for different choises of given function f (x, y, z). These equations have the common following vectors as infinitesimal generators: V 1 = ∂ u , V 2 = x∂ u , V 3 = y∂ u , V 4 = z∂ u .(10) Then the Lie algebra g generated with the vectors (10) is called the principal Lie algebra for Eq.(2). Now we want to specify the coefficient f such that Eq.(2) admits an extension of the principal algebra g. therefore, we do not solve the determining equation, instead we obtain a partial group classification of Eq.(2) via so-called method of preliminary group classification. This method was suggested in [1] and applied when an equivalence group is generated by a finitedimensional Lie algebra g E . The essential part of the method is the classification of all nonsimilar subalgebras of g E . Actually the classification is based on finite-dimentional equivalence algebra g E . Equivalence Transformations with a nondegenerate change of the variables x, y, z an equation of the form HESI equation convert to an equation of the same form, but with different f (x, y, z). The set of all equivalence transformatioms forms an equivalence group E. We shal find a subgroup E c of it with infinitesimal method. We suppose an operator of the group E c is in the form: Y = ξ(x, y, z, u)∂ x + ζ∂ y + η∂ z + φ∂ u + ψ(x, y, z, u, f )∂ f .(11) So from the invariance conditions of Eq.(2) written as the following system: S 2 [u] = f (x, y, z), f u = 0.(12) Note that f and u are considered as differential variables; u on the space (x, y, z) and f on the space (x, y, z, u). The coordinates ξ, ζ, η, φ of operator (11) are funtions of x, y, z, u, while the coordinate ψ is function of x, y, z, u, f . as usual way we should solve the following system that obtained of the invariance conditions: pr (2) Y (S 2 [u] = f (x, y, z)), pr (2) Y (f u ) = 0.(13) Where pr (2) Y is the second order prolongation of the vector field Y . But, to obtain the operator Y of the group E c we use of N. Bila's method in ref. [3]. The base of our procedure is theorem (1) of [3], then this theorem and it's results can be summarized as the following three-steps procedure: step 1: Find the determining equations of the extended classical symmetries related to the Eq.(2). For the meaning of the extended classical symmetries, a vector V = ξ(x, y, z, u, f )∂ x + ζ∂ y + η∂ z + φ∂ u + ψ∂ f .(14) is said the extended classical symmetry operator assosiated with HESI Equation and the determining equations of the extended classical symmetries related to the HESI Equation is the following equation: pr (2) V [HESI] = 0,(15) where ξ, ζ, η, φ and ψ are functions of x, y, z, u and f , and pr (2) V is V + J φ J (x, y, z, u (2) , f (2) ) ∂ u J + J ψ J (x, y, z, u (2) , f (2) ) ∂ f J(16) where u (2) = (u, u x , · · · , u zz ) and f (2) = (f, f x , · · · , f zz ): and the coefficients obtain from: φ J (x, y, z, u (2) , f (2) ) = D J φ − 3 i=1 ξ i ∂u ∂x i + 3 i=1 ξ i ∂u J ∂x i ψ J (x, y, z, u (2) , f (2) ) = D J ψ − 3 i=1 ξ i ∂f ∂x i + 3 i=1 ξ i ∂f J ∂x i(17) where (ξ 1 , ξ 2 , ξ 3 ) = (ξ, ζ, η) and (x 1 , x 2 , x 3 ) = (x, y, z), further J = (j 1 , · · · , j k ) is a k-th order multiindex, and j i s adopt x, y or z, for 1 i k, then D J denotes the total derivatives for the multi-index J, and the J-th total derivative is as D J = D j 1 · · · D j k , that total derivative operator with respect to i is as following D i = ∂ x i + J ∂u J ∂x i ∂ u J + J ∂u f ∂x i ∂ f J , Note: pr (2) V is determined by taking into account that u and f are both dependent variables, exactly as one would proceed in finding the classical Lie symmetries for a system without arbitrary functions. So with solving equation (15) we gain: ξ = c 6 x + c 9 y + c 7 z + c 8 , ζ = c 10 z + c 6 y − c 9 x + c 11 , ψ = 2f (−2c 6 + c 3 ), η = −c 10 y + c 6 z − c 7 x + c 12 , φ = c 1 x + c 3 u + c 2 y + c 5 z + c 4 ,(18) which c i , i = 1, · · · , 12 are arbitrary costants. step 2: Augment the system of step 1 with the following conditions: ∂ξ ∂u = 0, ∂ζ ∂u = 0, ∂η ∂u = 0, ∂ψ ∂u = 0.(19) As we seen in relations (18) above conditions are satiesfied. step 3: Augment the system of steps 1 and 2 with the following conditions: ∂ξ ∂f = 0, ∂ζ ∂f = 0, ∂η ∂f = 0, ∂ψ ∂f = 0.(20) As we seen in relations (18) above conditions are satisfied too. Ultimately, The class of equations (2) has a finite continuous group of equivalence transformations generated by the following infinitesimal operators: Y 1 = ∂ x , Y 2 = ∂ y , Y 3 = ∂ z , Y 4 = ∂ u , Y 5 = x∂ u , Y 6 = y∂ u , Y 7 = z∂ u , Y 8 = z∂ x − x∂ z , Y 9 = y∂ x − x∂ y , Y 10 = z∂ y − y∂ z , Y 11 = u∂ u + 2f ∂ f , Y 12 = x∂ x + y∂ y + z∂ z − 4f ∂ f .(21) Moreover, in the group of equivalence transformations are included also discrete transformations, i.e., reflections (x, y, z, u, f ) → −(x, y, z, u, f ). Sketch of the method of preliminary group classification In many applications of group analysis, most of extensions of the principal Lie algebra admitted by an equation are obtained from the equivalence algebra g E . We call these extension E -extension of the principal Lie algebra. The classification of all nonequivalent equations admitting E -extension of the principal Lie algebra is called a preliminary qroup classification. What we obtain also is not necessarily the largest equivalence group but, it can be any subgroup of the qroup of all equivalence transformations. The application of this method is effective and simple when it is based on a finite-dimensional equivalence algebra g E . So we take finite dimensional algebra g 12 spanned on the basis (21) and use it for preliminary group classification. The function f of Eq.(2) depends on the variables x, y, z, so we don't construct any prolongations of operators (11). But we take projections on the space (x, y, z, f ). The nonzero projections of (21) are: Z 1 = pr(Y 1 ) = ∂ x , Z 2 = pr(Y 2 ) = ∂ y , Z 3 = pr(Y 3 ) = ∂ z , Z 4 = pr(Y 8 ) = z∂ x − x∂ z , Z 5 = pr(Y 9 ) = y∂ x − x∂ y , Z 6 = pr(Y 10 )z∂ y − y∂ z , Z 7 = pr(Y 11 ) = 2f ∂ f , Z 8 = pr(Y 12 ) = x∂ x + y∂ y + z∂ z − 4f ∂ f .(22) It's clear that there aren't the minimal infinitesimal generators (10), among above vectors. The Lie algebra generated with the vectors in (22) is denoted by g 8 . The essence of the preliminary method is based on the following two proposition: Proposition 1 : Let g m be a m−dimensional subalgebra of g 8 . Suppose Z (i) , i = 1, · · · , m be a basis of g m and Y (i) is the elements of the algebra g 12 , such that Z (i) = pr(Y (i) ), that means, if Z (i) = 8 α=1 e α i Z α ,(23) then with respect to (21) and (22): Y (i) = e 1 i Y 1 + e 2 i Y 2 + e 3 i Y 3 + e 4 i Y 8 + e 5 i Y 9 + e 6 i Y 10 + e 7 i Y 11 + e 8 i Y 12 .(24) If function f = f (x, y, z) be invariant with respect to the algebra g m , then the HESI equation admits the operators X (i) = projection of Y (i) on (x, y, z, u).(25) Proposition 2 : Let equations S 2 [u] = f (x, y, z),(26)S 2 [u] = f ′ (x, y, z),(27) be constructed according to proposition (1) with subalgebras g m and g m ′ , respectively. If g m and g m ′ are similar subalgebras in g 12 then equations (26) and (27) are equivalent with respect to the equivalence group G 12 generated by g 12 . According to above propositions, continuation of the preliminary group classification of Eq.(2) with respect to the finite-dimensional algebra g 12 , is reduced to the algebraic problem of constructing of nonsimilar subalgebras of g 8 , or optimal systems of subalgebras. note: In this paper we just solve the problem of preliminary group classification with respect to one-dimensional subalgebras. Adjoint group for algebra g 8 We determine a list or optimal system, of conjuacy inequivalent subalgebras with the property that any other subalgebra is equivalent to a unique member of the list under some element of the adjoint representation, i.e.h = Ad(g)h for some g of a considered Lie group, see [18,19,20]. The adjoint action is given by the Lie series Ad(exp(εY i ))Y j = Y j − ε[Y i , Y j ] + ε 2 2 [Y i , [Y i , Y j ]] − · · · ,(28) The commutator and adjoint representations of g 8 are listed in tables 1 and 2. Z 1 Z 2 Z 3 Z 4 Z 5 Z 6 Z 7 Z 8 Z 1 0 0 0 −Z 3 −Z 2 0 0 Z 1 Z 2 0 0 0 0 Z 1 −Z 3 0 Z 2 Z 3 0 0 0 Z 1 0 Z 2 0 Z 3 Z 4 Z 3 0 −Z 1 0 −Z 6 Z 5 0 0 Z 5 Z 2 −Z 1 0 Z 6 0 −Z 4 0 0 Z 6 0 Z 3 −Z 2 −Z 5 Z 4 0 0 0 Z 7 0 0 0 0 0 0 0 0 Z 8 −Z 1 −Z 2 −Z 3 0 0 0 0 0: Ad(exp(ε i Y i ))Y j Z 1 Z 2 Z 3 Z 4 Z 5 Z 6 Z 7 Z 8 Z 1 Z 1 Z 2 Z 3 ε 1 Z 3 + Z 4 ε 1 Z 2 + Z 5 Z 6 Z 7 −ε 1 Z 1 + Z 8 Z 2 Z 1 Z 2 Z 3 Z 4 −ε 2 Z 1 + Z 5 ε 2 Z 3 + Z 6 Z 7 −ε 2 Z 2 + Z 8 Z 3 Z 1 Z 2 Z 3 −ε 3 Z 1 + Z 4 Z 5 −ε 3 Z 2 + Z 6 Z 7 −ε 3 Z 3 + Z 8 Z 4 cos(ε 4 )Z 1 − sin(ε 4 )Z 3 Z 2 sin(ε 4 )Z 1 + cos(ε 4 )Z 3 Z 4 cos(ε 4 )Z 5 + sin(ε 4 )Z 6 − sin(ε 4 )Z 5 + cos(ε 4 )Z 6 Z 7 Z 8 Z 5 cos(ε 5 )Z 1 − sin(ε 5 )Z 2 sin(ε 5 )Z 1 + cos(ε 5 )Z 2 Z 3 cos(ε 5 )Z 4 − sin(ε 5 )Z 6 Z 5 sin(ε 5 )Z 4 + cos(ε 5 )Z 6 Z 7 Z 8 Z 6 Z 1 cos(ε 6 )Z 2 − sin(ε 6 )Z 3 sin(ε 6 )Z 2 + cos(ε 6 )Z 3 cos(ε 6 )Z 4 + sin(ε 6 )Z 5 − sin(ε 6 )Z 4 + cos(ε 6 )Z 5 Z 6 Z 7 Z 8 Z 7 Z 1 Z 2 Z 3 Z 4 Z 5 Z 6 Z 7 Z 8 Z 8 e ε8 Z 1 e ε8 Z 2 e ε8 Z 3 Z 4 Z 5 Z 6 Z 7 Z 8 6 Construction of the optimal system of one-dimensional subalgebras of g 8 Theorem 2 : An optimal system of one-dimensional Lie algebras of g 8 in HESI equation are as follows: A 1 = Z 7 , A 2 = ±Z 1 + Z 7 , A 3 = γ 1 Z 6 + Z 7 , A 4 = ±Z 1 + γ 2 Z 6 + Z 7 , A 5 = α 1 Z 4 + Z 7 , A 6 = ±Z 2 + α 2 Z 4 + Z 7 , A 7 = α 3 Z 4 + γ 3 Z 6 + Z 7 , A 8 = ±Z 1 + α 4 Z 4 + γ 4 Z 6 + Z 7 , A 9 = α 5 Z 4 + β 1 Z 5 + Z 7 , A 10 = ±Z 3 + α 6 Z 4 + β 2 Z 5 + Z 7 , A 11 = α 7 Z 4 + β 3 Z 5 + γ 5 Z 7 + Z 8 , A 12 = ±Z 2 + α 8 Z 4 + β 4 Z 5 + γ 6 Z 7 + Z 8 ,(29) Where α i , i = 1, ...8 and β i , i = 1, · · · , 4 and γ i , i = 1, · · · , 6 are arbitrary constants. Proof: We will start with Z = 8 i=1 a i Z i , suppose Z is a nonzero vector field of g 8 , we want simplify as many of the coefficients a i , i = 1, · · · , 8 as possible through proper Adjoint applications on Z. We proceed this simplifications through following cases: note: The coefficients a 7 and a 8 don't change at all. Case 1: At first, we assume that a 8 = 0, so with scalling on Z, we can suppose that a 7 = 1, then we have Z = 6 i=1 a i Z i + Z 7 . therefore for different values of a 5 = 0, when it is either zero or nonzero, we have cases 1.1 and 1.2. Case 1.1: If a 8 = a 5 = 0, so Z = a 1 Z 1 + a 2 Z 2 + a 3 Z 3 + a 4 Z 4 + a 6 Z 6 + Z 7 . Then for different values of a 4 , when it is either zero or nonzero, we have cases 1.1.a and 1.1.b. Case 1.1.a: If a 8 = a 5 = a 4 = 0, so Z = a 1 Z 1 + a 2 Z 2 + a 3 Z 3 + a 6 Z 6 + Z 7 . Then for different values of a 6 , when it is either zero or nonzero, we have cases 1.1.a1 and 1.1.a2. Case 1.1.a1: If a 8 = a 5 = a 4 = a 6 = 0, so Z = a 1 Z 1 + a 2 Z 2 + a 3 Z 3 + Z 7 . Then for different values of a 3 , when it is either zero or nonzero, the coefficient can be vanished; when a 3 = 0, with effecting Ad(exp(cot −1 (a 1 /a 3 )Z 4 )) on Z. Then we have Z = a 1 Z 1 + a 2 Z 2 + Z 7 . Now if a 2 = 0 or a 2 = 0; by effecting Ad(exp(cot −1 (a 1 /a 2 )Z 5 )) on Z, we can make the coefficient of Z 2 vanished.Then we have Z = a 1 Z 1 + Z 7 . So if a 1 = 0, then Z = Z 7 , so we have A 1 . And if a 1 = 0, with Ad(exp(ln(±1/a 1 )Z 8 )) change the coefficient of Z 1 equal ±1, so Z = ±Z 1 + Z 7 , therefore we have A 2 . Case 1.1.a2: If a 8 = a 5 = a 4 = 0, but a 6 = 0, so Z = a 1 Z 1 + a 2 Z 2 + a 3 Z 3 + a 6 Z 6 + Z 7 . Then for different values of a 3 , when it is either zero or nonzero, the coefficient can be vanished; when a 3 = 0, with applying Ad(exp(−a 3 /a 6 )Z 2 ) on Z. So we have Z = a 1 Z 1 + a 2 Z 2 + a 6 Z 6 + Z 7 . Similarly, the coefficient a 2 is either zero or we make it vanished with effecting Ad(exp(a 2 /a 6 )Z 3 ) on Z. Then we have Z = a 1 Z 1 + a 6 Z 6 + Z 7 . So a 1 = 0 or a 2 = 0, if a 1 = 0, so Z = a 6 Z 6 + Z 7 , and we have A 3 . And if a 1 = 0, with Ad(exp(ln(±1/a 1 )Z 8 )) change the coefficient of Z 1 equal ±1, so Z = ±Z 1 + +a 6 Z 6 + Z 7 , therefore we have A 4 . Case 1.1.b: If a 8 = a 5 = 0, but a 4 = 0, so Z = a 1 Z 1 + a 2 Z 2 + a 3 Z 3 + a 4 Z 4 + a 6 Z 6 + Z 7 . Then for different values of a 3 , when it is either zero or nonzero, the coefficient can be vanished; when a 3 = 0, with effecting Ad(exp(−a 1 /a 4 )Z 1 )) on Z. Then we have Z = a 1 Z 1 + a 2 Z 2 + a 4 Z 4 + a 6 Z 6 + Z 7 . Therefore, for different values of a 6 , when it is either zero or nonzero, we have cases 1.1.b1 and 1.1.b2. Case 1.1.b1: If a 8 = a 5 = a 3 = a 6 = 0, but a 4 = 0, so Z = a 1 Z 1 + a 2 Z 2 + a 4 Z 4 + Z 7 . Then either a 1 is zero or nonzero, but the coefficient can be vanished; when a 1 = 0, with effecting Ad(exp(a 1 /a 4 )Z 3 )) on Z. Then we have Z = a 2 Z 2 + a 4 Z 4 + Z 7 . Ultimately, a 2 = 0 or a 2 = 0; If a 2 = 0, so we have Z = a 4 Z 4 + Z 7 .Then we have A 5 . And if a 2 = 0, with Ad(exp(ln(±1/a 2 )Z 8 )) change the coefficient of Z 2 equal ±1, so Z = ±Z 2 + a 4 Z 4 + Z 7 , therefore we have A 6 . Case 1.1.b2: If a 8 = a 5 = a 3 = 0, but a 6 = 0 and a 4 = 0, so Z = a 1 Z 1 + a 2 Z 2 + a 4 Z 4 + a 6 Z 6 + Z 7 . Then for different values of a 2 , when it is either zero or nonzero, the coefficient can be vanished; when a 2 = 0, with applying Ad(exp(a 2 /a 6 )Z 3 ) on Z. So we have Z = a 1 Z 1 + a 4 Z 4 + a 6 Z 6 + Z 7 . Similarly, the coefficient a 1 is either zero or nonzero, if a 1 = 0, so Z = a 4 Z 4 + a 6 Z 6 + Z 7 , and we have A 7 . And if a 1 = 0, with Ad(exp(ln(±1/a 1 )Z 8 )) change the coefficient of Z 1 equal ±1, so Z = ±Z 1 + a 4 Z 4 + +a 6 Z 6 + Z 7 , therefore we have A 8 . Case 1.2: If a 8 = 0 but a 5 = 0, so Z = a 1 Z 1 + a 2 Z 2 + a 3 Z 3 + a 4 Z 4 + a 5 Z 5 + a 6 Z 6 + Z 7 . So we have different values of a 2 , when it is either zero or nonzero, the coefficient can be vanished; when a 2 = 0, with applying Ad(exp(−a 2 /a 5 )Z 1 ) on Z. Then again we have two cases a 1 = 0 or a 1 = 0; If a 1 = 0, with applying Ad(exp(a 1 /a 5 )Z 2 ) on Z, make it vanished. so we have Z = a 3 Z 3 + a 4 Z 4 + a 5 Z 5 + a 6 Z 6 + Z 7 . Again we have two cases a 6 = 0 or a 6 = 0; If a 6 = 0, with applying Ad(exp(cot −1 (a 4 /a 6 ))Z 5 ) on Z, make it vanished. so we have Z = a 3 Z 3 + a 4 Z 4 + a 5 Z 5 + Z 7 . Ultimately, If a 3 = 0, then Z = a 4 Z 4 + a 5 Z 5 + Z 7 .Then we have A 9 . And if a 3 = 0, then with Ad(exp(ln(±1/a 3 )Z 8 )) change the coefficient of Z 3 equal ±1, so Z = ±Z 3 + a 4 Z 4 + a 5 Z 5 + Z 7 .Then we have A 10 . Case 2: If a 8 = 0, so with scalling on Z, we can suppose that a 8 = 1, then we have Z = 7 i=1 a i Z i +Z 8 . So we have different values of a 1 , when it is either zero or nonzero, the coefficient can be vanished; when a 1 = 0, with applying Ad(exp(a 1 Z 1 )) on Z. So we reduce Z to Z = a 2 Z 2 + a 3 Z 3 + a 4 Z 4 + a 5 Z 5 + a 6 Z 6 + a 7 Z 7 + Z 8 . Now if a 3 = 0 or a 3 = 0; by effecting Ad(exp(cot −1 (a 2 /a 3 )Z 6 ) on Z, make the coefficient of Z 3 vanished. so we have Z = a 2 Z 2 + a 4 Z 4 + a 5 Z 5 + a 6 Z 6 + a 7 Z 7 + Z 8 . Again we have two cases a 6 = 0 or a 6 = 0; If a 6 = 0, with applying Ad(exp(− cot −1 (a 5 /a 6 ))Z 4 ) on Z, make it vanished. So we have Z = a 2 Z 2 + a 4 Z 4 + a 5 Z 5 + a 7 Z 7 + Z 8 . Ultimately, If a 2 = 0, then Z = a 4 Z 4 + a 5 Z 5 + a 7 Z 7 + Z 8 . Then we have A 11 . And if a 2 = 0, then with Ad(exp(ln(±1/a 2 )Z 8 )) change the coefficient of Z 2 equal ±1, so Z = ±Z 2 + a 4 Z 4 + a 5 Z 5 + a 7 Z 7 + Z 8 . Then we have A 12 . There is not any more possible cases, and the proof is complete. ✷ 7 Equations admitting an extension by one of the principal Lie algebra Now based on propositions (1) and (2), and with the optimal system (29), we obtain all nonequivalent equations of the form equation (2), that admitting extension of principal Lie algebra g by one operator V 5 , that means every equation of the form equation (2) admits symmetry group of dimension 4 with infinitesimal generators (10), also together with a fifth operator V 5 . for every case, when this extension occurs, we indicate the corresponding coefficients f and additional operator V 5 . The algorithm will be clarified with these examples: First example: Consider the operator A 3 = γ 1 Z 6 + Z 7 , which γ 1 = 0, from (29), so A 3 = γ 1 z∂ y − γ 1 y∂ z + 2f ∂ f .(30) Invariants are found from the following equation (see [19]): dy γ 1 z = − dz γ 1 y = df 2f ,(31) and are the following functions: I 1 = x, I 2 = y 2 + z 2 , I 3 = f exp(− 2 γ 1 tan −1 ( y z )).(32) It follows f = exp( 2 γ 1 tan −1 ( y z ))H(x, y 2 + z 2 ),(33) where H is arbitrary function. By applying the formulas (23), (24) and (25) on the operator A 3 we obtain the additional operator V 5 = γ 1 z∂ y − γ 1 y∂ z + u∂ u . Thus, the equation HESI : S 2 [u] = exp( 2 γ 1 tan −1 ( y z ))H(x, y 2 + z 2 ),(34) admits the five-dimensional algebra g 5 , that is generated with the following vectors V 1 = ∂ u , V 2 = x∂ u , V 3 = y∂ u , V 4 = z∂ u , V 5 = γ 1 z∂ y − γ 1 y∂ z + u∂ u .(35) Second example: Consider the operator A 1 = Z 7 from (29), so A 1 = 2f ∂ f . Invariants are the following functions: I 1 = x, I 2 = y, I 3 = z.(36) So, there are no invariant functions f = f (x, y, z) because the necessary condition for existence of invariant solutions based on ref. [20] (section 19.3) is not satisfied; that means invariants (36) can't be solved with respect to f . We continue calculations on some operators of (29), and show results in table 3, that is the preliminary group classification of equation (2), which admit an extension g 5 of the principal Lie algebra g. The results of classification: f V 5 A 2 exp(±2x)H(y, z) ±∂ x + u∂ u A 3 (γ 1 =0) exp( 2 γ 1 tan −1 ( y z ))H(x, y 2 + z 2 ) γ 1 z∂ y − γ 1 y∂ z + u∂ u A 4 (γ 2 =0) exp( 2 γ 2 tan −1 ( y z )) H(y 2 + z 2 , x ∓ 1 γ 2 tan −1 ( y z )) ±∂ x + γ 2 z∂ y − γ 2 y∂ z + u∂ u A 5 (α 1 =0) exp( 2 α 1 tan −1 ( x z ))H(y, x 2 + z 2 ) α 1 z∂ x − α 1 x∂ z + u∂ u A 6 (α 2 =0) exp(±2y)H(x, z) ±∂ y + u∂ u A 6 (α 2 =0) exp( 2 α 2 tan −1 ( x z )) H(x 2 + z 2 , y ∓ 1 α 2 tan −1 ( x z )) ±∂ y + α 2 z∂ x − α 2 x∂ z + u∂ u A 7 (α 3 =0,γ 3 =0) exp( 2 α 3 2 + γ 3 2 tan −1 ( α 3 x + γ 3 y z α 2 3 + γ 2 3 )) H(y − γ 3 α 3 x, ((1 − γ 3 2 α 3 2 )x 2 + 2γ 3 α 3 xy + z 2 )) α 3 z∂ x + γ 3 z∂ y −(α 3 x + γ 3 y)∂ z + u∂ u A 9 (α 5 =0,β 1 =0) exp( 2 β 1 tan −1 ( x y ))H(z, x 2 + y 2 ) β 1 y∂ x − β 1 x∂ y + u∂ u A 9 (α 5 =0,β 1 =0) exp( −2 β 2 1 + α 2 5 tan −1 ( β 1 y + α 5 z x α 2 5 + β 2 1 )) H(z − α 5 β 1 y, (x 2 + (1 − α 2 5 β 2 1 )y 2 + 2α 5 β 1 yz)) (α 5 z + β 1 y)∂ x −β 1 x∂ y − α 5 x∂ z + u∂ u A 10 (α 6 =β 2 =0) exp(±2z)H(x, y) ±∂ z + u∂ u A 10 (α 6 =0,β 2 =0) exp( 2 β 2 tan −1 ( x y )) H(x 2 + y 2 , z ∓ 1 β 2 tan −1 ( x y )) ±∂ z + β 2 y∂ x − β 2 x∂ y + u∂ u A 11 (α 7 =β 3 =γ 5 =0) x −4 H( y x , z x ) x∂ x + y∂ y + z∂ z A 11 (α 7 =β 3 =0,γ 5 =0) x 2c−4 H( y x , z x ) x∂ x + y∂ y + z∂ z A 12 (α 8 =β 4 =γ 6 =0) x −4 H( y±1 x , z x ) x∂ x + (y ± 1)∂ y + z∂ z A 12 (α 8 =β 4 =0,γ 6 =0) x 2c−4 H( y±1 x , z x ) x∂ x + (y ± 1)∂ y + z∂ z Some Local Solutions Based on the following theorem of [23], The 2−hessian equation in R 3 , S 2 [u] = f (x, y, z, u, Du) on Ω ⊂ R 3 ,(37) which f ∈ C ∞ (Ω × R × R 3 ), Du = (∂ 1 u, · · · , ∂ n u), has C ∞ local solutions in R 3 , and the solution is in the following form: u(x, y, z) = 1 2 (τ 1 x 2 + τ 2 y 2 + τ 3 z 2 ) + ε 5 ω(ε −2 (x, y, z)),(38) where ε and τ i are arbitrary constants, and ω is a given smooth function. Theorem 3 Assume that f ∈ C ∞ (Ω × R × R 3 ), then for any Z 0 = (x 0 , u 0 , p 0 ) ∈ Ω × R × R 3 , we have that (1) If f (Z 0 ) = 0, then (37) admits a 1-convex C ∞ local solution which is not convex. (2) If f ≥ 0 near Z 0 , then (37) admits a 2-convex C ∞ local solution which is not convex. If f (Z 0 ) > 0, (37) admits a convex C ∞ local solution. (3) If f (Z 0 ) < 0, then (37) admits a 1-convex C ∞ local solution which is not 2-convex. Moreover, the equation (37) is uniformly elliptic with respect to the above local solutions. In this part we obtain the one-parameter groups generated by some operators, and since these groups are symmetry groups of HESI equation, then the above solution transforms to another solution of HESI equation. • Case 1: The operator V 1 = ∂ u that produces one-parameter group G 1 = (x, y, z, t + u), t ∈ R, for equation S 2 [u] = f (x, y, z), transforms solution(38) to the following solution u(x) = 1 2 (τ 1 x 2 + τ 2 y 2 + τ 3 z 2 ) − t + ε 5 ω(ε −2 (x, y, z)), where ε, τ i ∈ R. • Case 2: The operator V 2 = x∂ u that produces one-parameter group G 2 = (x, y, z, tx + u), t ∈ R, for equation S 2 [u] = f (x, y, z), transforms solution(38) to the following solution u(x) = 1 2 (τ 1 x 2 + τ 2 y 2 + τ 3 z 2 ) − tx + ε 5 ω(ε −2 (x, y, z)), where ε, τ i ∈ R. In such a manner are the following cases: • Case 3: The operator V 3 = y∂ u , G 3 = (x, y, z, ty + u), t ∈ R, for equation S 2 [u] = f (x, y, z); So u(x) = 1 2 (τ 1 x 2 + τ 2 y 2 + τ 3 z 2 ) − ty + ε 5 ω(ε −2 (x, y, z)), where ε, τ i ∈ R. • Case 4: The operator V 4 = z∂ u , G 4 = (x, y, z, tz + u), t ∈ R, for equation S 2 [u] = f (x, y, z); So u(x) = 1 2 (τ 1 x 2 + τ 2 y 2 + τ 3 z 2 ) − tz + ε 5 ω(ε −2 (x, y, z)), where ε, τ i ∈ R. • Case 5: The operator V 5 = ±∂ x + u∂ u , G 5 = (x ± t, y, z, tu + u), t ∈ R, for equation S 2 [u] = exp(±2x)H(y, z); So u(x) = 1 2(t + 1) τ 1 (x ± t) 2 + τ 2 y 2 + τ 3 z 2 + 2ε 5 ω(ε −2 (x ± t, y, z)) , where ε, τ i ∈ R. • Case 6: The operator V 5 = γ 1 z∂ y − γ 1 y∂ z + u∂ u , G 5 = (x, z sin(γ 1 t) + y cos(γ 1 t), z cos(γ 1 t) − y sin(γ 1 t), tu + u), t ∈ R, for equation S 2 [u] = exp( 2 γ 1 tan −1 ( y z ))H(x, y 2 + z 2 ); So u(x) = 1 2(t + 1) τ 1 x 2 + τ 2 (z sin(γ 1 t) + y cos(γ 1 t)) 2 + τ 3 (z cos(γ 1 t) − y sin(γ 1 t)) 2 + 2ε 5 ω(ε −2 (x, z sin(γ 1 t) + y cos(γ 1 t), z cos(γ 1 t) − y sin(γ 1 t))) , where ε, τ i ∈ R. • Case 7: The operator V 5 = ±∂ x + γ 2 z∂ y − γ 2 y∂ z + u∂ u , G 5 = (x ± t, z sin(γ 2 t) + y cos(γ 2 t), z cos(γ 2 t) − y sin(γ 2 t), tu + u), t ∈ R, for equation S 2 [u] = exp( 2 γ 2 tan −1 ( y z ))H(y 2 + z 2 , x ∓ 1 γ 2 tan −1 ( y z )); So u(x) = 1 2(t + 1) τ 1 (x ± t) 2 + τ 2 (z sin(γ 2 t) + y cos(γ 2 t)) 2 + τ 3 (z cos(γ 2 t) − y sin(γ 2 t)) 2 + 2ε 5 ω(ε −2 (x ± t, z sin(γ 2 t) + y cos(γ 2 t), z cos(γ 2 t) − y sin(γ 2 t))) , where ε, τ i ∈ R. • Case 8: The operator V 5 = α 1 z∂ x − α 1 x∂ z + u∂ u , G 5 = (z sin(α 1 t) + x cos(α 1 t), y, z cos(α 1 t) − x sin(α 1 t), tu + u), t ∈ R, for equation S 2 [u] = exp( 2 α 1 tan −1 ( x z ))H(y, x 2 + z 2 ); So u(x) = 1 2(t + 1) τ 1 (z sin(α 1 t) + x cos(α 1 t)) 2 + τ 2 y 2 + τ 3 (z cos(α 1 t) − x sin(α 1 t)) 2 + 2ε 5 ω(ε −2 (z sin(α 1 t) + x cos(α 1 t), y, z cos(α 1 t) − x sin(α 1 t))) , where ε, τ i ∈ R. • Case 9: The operator V 5 = ±∂ y + u∂ u , G 5 = (x, y ± t, z, tu + u), t ∈ R, for equation S 2 [u] = exp(±2y)H(x, z); So u(x) = 1 2(t + 1) τ 1 x 2 + τ 2 (y ± t) 2 + τ 3 z 2 + 2ε 5 ω(ε −2 (x, y ± t, z)) , where ε, τ i ∈ R. • Case 10: The operator V 5 = ±∂ y + α 2 z∂ x − α 2 x∂ z + u∂ u , G 5 = (z sin(α 2 t) + x cos(α 2 t), y ± t, z cos(α 2 t) − x sin(α 2 t), tu + u), t ∈ R, for equation S 2 [u] = exp( 2 α 2 tan −1 ( x z ))H(x 2 + z 2 , y ∓ 1 α 2 tan −1 ( x z )); So u(x) = 1 2(t + 1) τ 1 (z sin(α 2 t) + x cos(α 2 t)) 2 + τ 2 (y ± t) 2 + τ 3 (z cos(α 2 t) − x sin(α 2 t)) 2 + 2ε 5 ω(ε −2 (z sin(α 2 t) + x cos(α 2 t), y ± t, z cos(α 2 t) − x sin(α 2 t))) , where ε, τ i ∈ R. • Case 11: The operator V 5 = β 1 y∂ x − β 1 x∂ y + u∂ u , G 5 = (y sin(β 1 t) + x cos(β 1 t), y cos(β 1 t) − x sin(β 1 t), z, tu + u), t ∈ R, for equation S 2 [u] = exp( 2 β 1 tan −1 ( x y ))H(z, x 2 + y 2 ); So u(x) = 1 2(t + 1) τ 1 (y sin(β 1 t) + x cos(β 1 t)) 2 + τ 2 (y cos(β 1 t) − x sin(β 1 t)) 2 + τ 3 z 2 + 2ε 5 ω(ε −2 (y sin(β 1 t) + x cos(β 1 t), y cos(β 1 t) − x sin(β 1 t), z)) , where ε, τ i ∈ R. • Case 12: The operator V 5 = ±∂ z + u∂ u , G 5 = (x, y, z ± t, tu + u), t ∈ R, for equation S 2 [u] = exp(±2z)H(x, y); So u(x) = 1 2(t + 1) τ 1 x 2 + τ 2 y 2 + τ 3 (z ± t) 2 + 2ε 5 ω(ε −2 (x, y, z ± t)) , where ε, τ i ∈ R. • Case 13: The operator V 5 = ±∂ z + β 2 y∂ x − β 2 x∂ y + u∂ u , G 5 = (y sin(β 2 t) + x cos(β 2 t), y cos(β 2 t) − x sin(β 2 t), z ± t, tu + u), t ∈ R, for equation S 2 [u] = exp( 2 β 2 tan −1 ( x y ))H(x 2 + y 2 , z ∓ 1 β 2 tan −1 ( x y )); So u(x) = 1 2(t + 1) τ 1 (y sin(β 2 t) + x cos(β 2 t)) 2 + τ 2 (y cos(β 2 t) − x sin(β 2 t)) 2 + τ 3 (z ± t) 2 + 2ε 5 ω(ε −2 (y sin(β 2 t) + x cos(β 2 t), y cos(β 2 t) − x sin(β 2 t), z ± t)) , where ε, τ i ∈ R. • Case 14: The operator V 5 = x∂ x + y∂ y + z∂ z , G 5 = (e t x, e t y, e t z, u), t ∈ R, for equation S 2 [u] = x 2c−4 H( y x , z x ); So u(x) = 1 2 (τ 1 e 2t x 2 + τ 2 e 2t y 2 + τ 3 e 2t z 2 ) + ε 5 ω(ε −2 (e t x, e t y, e t z)), where ε, τ i ∈ R. • Case 15: The operator V 5 = x∂ x + (y ± 1)∂ y + z∂ z , G 5 = (e t x, e t y ± e t ∓ 1, e t z, u), t ∈ R, for equation S 2 [u] = x 2c−4 H( y±1 x , z x ); So u(x) = 1 2 (τ 1 e 2t x 2 + τ 2 (e t y ± e t ∓ 1) 2 + τ 3 e 2t z 2 ) + ε 5 ω(ε −2 (e t x, e t y ± e t ∓ 1, e t z)), where ε, τ i ∈ R. Conclusion In this paper, we performed preliminary group classification on equation, by studying the class of 3dimensional nonlinear 2−hessian equations S 2 [u] = f (x, y, z), and investigating the algebraic structure of the symmetry groups for the equation. Then, we obtained an optimal system of one-dimensional Lie subalgebras of this equation, with the aid of propositions (1) and (2). The result of these work is a wide class of equations which summarized in table 3. And at the end of this work, some exact solutions of 2−hessian equation are presented. Of course, the results in table 3 can be continued for remainder vectors, and it is possible to obtain the corresponding reduced equations for all the cases in the classification in up comming works. Table 1 : 1Commutators table for g 8 : [Z i , Z j ] Table 2 : 2Adjoint table for g 8 Table 3 : 3The equation E : S 2 [u] = f has V 5 as its additional operator v.r.t A s Nonlocal symmetries. Heuristic approach. 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[ "ANALYSIS OF THE BINARY ASYMMETRIC JOINT SPARSE FORM", "ANALYSIS OF THE BINARY ASYMMETRIC JOINT SPARSE FORM" ]	[ "Clemens Heuberger ", "Sara Kropf " ]	[]	[]	We consider redundant binary joint digital expansions of integer vectors. The redundancy is used to minimize the Hamming weight, i.e., the number of nonzero digit vectors. This leads to efficient linear combination algorithms in abelian groups, which are for instance used in elliptic curve cryptography.If the digit set is a set of contiguous integers containing zero, a special syntactical condition is known to minimize the weight. We analyze the optimal weight of all non-negative integer vectors with maximum entry less than N . The expectation and the variance are given with a main term and a periodic fluctuation in the second order term. Finally, we prove asymptotic normality.   2 0 . Definition 1.2. The Hamming weight h(ε L . . . ε 0 ) of a digit expansion (ε L . . . ε 0 ) is the number of nonzero columns ε i = 0. Example 1.2. Continuing with Example 1.1, we have the Hamming weight h   1011 0020 2001   = 3. 2010 Mathematics Subject Classification. 11A63; 94A60 68W40 60F05.	10.1017/s0963548314000352	[ "https://arxiv.org/pdf/1303.2819v2.pdf" ]	35,268,714	1303.2819	0014d5c95911e279706862e2714ebbfcf16a2cf3	ANALYSIS OF THE BINARY ASYMMETRIC JOINT SPARSE FORM Clemens Heuberger Sara Kropf ANALYSIS OF THE BINARY ASYMMETRIC JOINT SPARSE FORM We consider redundant binary joint digital expansions of integer vectors. The redundancy is used to minimize the Hamming weight, i.e., the number of nonzero digit vectors. This leads to efficient linear combination algorithms in abelian groups, which are for instance used in elliptic curve cryptography.If the digit set is a set of contiguous integers containing zero, a special syntactical condition is known to minimize the weight. We analyze the optimal weight of all non-negative integer vectors with maximum entry less than N . The expectation and the variance are given with a main term and a periodic fluctuation in the second order term. Finally, we prove asymptotic normality.   2 0 . Definition 1.2. The Hamming weight h(ε L . . . ε 0 ) of a digit expansion (ε L . . . ε 0 ) is the number of nonzero columns ε i = 0. Example 1.2. Continuing with Example 1.1, we have the Hamming weight h   1011 0020 2001   = 3. 2010 Mathematics Subject Classification. 11A63; 94A60 68W40 60F05. Introduction We deal with integer representations of vectors of integers called joint representations. Definition 1.1. For base 2, dimension d and a digit set D ⊆ Z, the dimension-d joint representation of a vector n ∈ Z d is a word (ε L . . . ε 0 ) with ε i ∈ D d and n = value(ε L . . . ε 0 ) with value(ε L . . . ε 0 ) = L i=0 ε i 2 i . Such representations can be used for computing a linear combination m 1 P 1 + · · · + m d P d of points P i of an elliptic curve, or more generally an abelian group (cf. [9]). For every nonzero digit ε i , an elliptic curve addition is performed. Since these are expensive, we want to minimize the number of nonzero digits. On the other side, every nonzero column vector ε ∈ D d corresponds to a precomputed point. The number of doublings corresponds to the length of the expansion. Each ε i in the expansion (ε L . . . ε 0 ) is called a column vector of the expansion. The Hamming weight of an integer depends on the representation we use. For example, we have two representations of 4 = value(12) = value(100) with Hamming weight h(12) = 2 and h(100) = 1. But since we always use a specific digit expansion in this paper, we just write h(n) for the Hamming weight of this digit expansion. This specific digit expansion is the asymmetric joint sparse form (short AJSF) as presented by Heuberger and Muir in [7]. The AJSF is the unique dimension-d joint integer representation in base 2 with digit set D l,u = {a ∈ Z \| l ≤ a ≤ u} described in Theorem 2 (see [7, Theorem 6.1]). There, Heuberger and Muir proved that the AJSF is colexicographically minimal and has minimal Hamming weight among all representations with this digit set D l,u . The width-w nonadjacent form [8,1] and the simple joint sparse form [5] are special cases of the asymmetric joint sparse form. For the width-w nonadjacent form, we use l = −2 w−1 + 1, u = 2 w−1 − 1 and dimension 1. The simple joint sparse form has digit set D −1,1 and dimension 2. The special case of Theorem 1 for the simple joint sparse form has been proved in [5]. For further results on syntactically defined optimal digit expansions, we refer to [7] and the references therein. We compute the expected value, the variance and the asymptotic distribution of the Hamming weight of the AJSF. We obtain a main term plus a periodic fluctuation and an error term, similar to the asymptotic estimates of digital sums in [3]. The definitions and algorithms of the AJSF are recalled in Section 2. In Section 3, we construct a transducer from this algorithm. In Theorem 5, we explicitly describe this transducer to compute the Hamming weight. In Section 4, we prove the following Theorem 1 about the asymptotic normal distribution of the Hamming weight. We use the discrete probability space {n ∈ Z \| 0 ≤ n < N } d with uniform distribution as a probabilistic model, in contrast to [7]. There, only residue classes modulo powers of 2 have been considered in the "full-block-length" analysis. and w is the unique integer such that 2 w−1 ≤ u − l + 1 < 2 w . Furthermore, for d = 1, the function Ψ 1 (x) is nowhere differentiable. General formulas for e l,u,d for d = 2 are given in [7, Table 3]. For d ∈ {1, 2, 3, 4}, general formulas for e l,u,d and v l,u,d are given in [6]. For higher dimension or the variance, the question of non-differentiability of the periodic fluctuations remains open. In the last Section 5, we further investigate the error term of the expected value and the variance of the Hamming weight in the case of the width-w nonadjacent form. In this case, we have δ = log 2 1 + 3π 2 w 3 for sufficiently large w, see Theorem 6. Preliminaries First, we define some properties of the digit set. Definition 2.1. Let D l,u = {a ∈ Z \| l ≤ a ≤ u} for l ≤ 0 and u ≥ 1 be the digit set. It contains u − l + 1 digits. We define w to be the unique integer s.t. 2 w−1 ≤ u − l + 1 < 2 w . Because 0, 1 ∈ D l,u , we have w ≥ 2. The digit set contains at least a complete set of residues modulo 2 w−1 . However, some residues modulo 2 w are not contained. Thus we define the following sets: Definition 2.2. Let unique(D l,u ) = {a ∈ D l,u \| u − 2 w−1 < a < l + 2 w−1 }, nonunique(D l,u ) = D l,u \ unique(D l,u ), upper(D l,u ) = {a ∈ D l,u \| u − 2 w−1 < a ≤ u}. The sets unique(D l,u ) and nonunique(D l,u ) contain the unique respectively non-unique residues modulo 2 w−1 . The set upper(D l,u ) is a complete set of representatives modulo 2 w−1 . Without loss of generality, we can restrict l to be greater than −2 w−1 . Otherwise, we would take the digit set D −u,−l where we have −2 w−1 < −u ≤ −1. Then every representation of a vector n of integers with digit set D l,u would correspond to a representation of −n with digit set D −u,−l by changing the sign of each digit. By this transformation, the weight of the representation does not change. Theorem 2 ([7]). Let D l,u be a digit set and n ∈ Z d (with n ≥ 0 if l = 0). Then there exists exactly one representation (ε L . . . ε 0 ) (up to leading 0's) of n, such that the following conditions are satisfied: (1) Each column ε j is 0 or contains an odd digit. (2) If ε j = 0 for some j, then ε j+w−2 = · · · = ε j+1 = 0. (3) If ε j = 0 and ε j+w−1 = 0 for some j, then (a) there is an i ∈ {1, . . . , d} such that ε j+w−1,i is odd and ε j,i ∈ unique(D l,u ), (b) if ε j,i ∈ nonunique(D l,u ), then ε j+w−1,i ≡ u + 1 mod 2 w−1 , (c) if ε j,i ∈ upper(D l,u ) ∩ nonunique(D l,u ), then ε j+w−1,i ≡ u mod 2 w−1 . Definition 2.3. The digit expansion described in Theorem 2 is called asymmetric joint sparse form (short AJSF) of n with digit set D l,u . Example 2.1. The AJSF of (7, 11) T with digit set D −2,3 is 1001 1003 , where1 is the digit −1. Thus its Hamming weight is h(7, 11) = 2. We also consider the width-w nonadjacent form (cf. [8,1]). Definition 2.4. The width-w nonadjacent form (short w-NAF) of an integer n is a radix-2 representation (ε L . . . ε 0 ) of n with the digit set D w := {0, ±1, ±3, . . . , ±(2 w−1 − 3), ±(2 w−1 − 1)} and the following property: If ε i = 0, then ε i+1 = . . . = ε i+w−1 = 0. The AJSF is a generalization of the w-NAF. In the 1-dimensional case, only odd digits and 0 are used in the AJSF due to Theorem 2. After a nonzero digit, there are w − 1 zeros. Thus, for l = −2 w−1 + 1 and u = 2 w−1 − 1, we obtain the w-NAF. It is known that the w-NAF representation exists and is unique for every integer (cf. [8]). In [7], Heuberger and Muir introduce the AJSF, provide an algorithm to compute it, and prove its minimality with respect to the Hamming weight. Theorem 3 ([7] ). The AJSF has minimal Hamming weight among all digit expansions of an integer vector n with digit set D l,u . Algorithm 3 in [7] computes the AJSF in dimension d for an integer vector n. We present a slightly modified version of Algorithm 3 in [7] as Algorithm 1. The modification takes into account that we are only interested in the weight. Furthermore, those iterations of the while loop where the output is already predetermined are skipped. For simplicity, we write n + a, for a vector n and an integer a, to denote that we add a to every coordinate of the vector n. I unique = {j ∈ {1, 2, . . . , d} \| a j ∈ unique(D l,u )} 11: I nonunique = {j ∈ {1, 2, . . . , d} \| a j ∈ nonunique(D l,u )} 12: if m j ≡ 0 mod 2 for all j ∈ I unique then 13: for j ∈ I nonunique such that m j is odd do In the 1-dimensional case, we can further simplify Algorithm 1. If I unique = ∅, then I nonunique = ∅. Thus the else branch in line 17 will not be processed. Algorithm 2 is the simplified version for the 1-dimensional case. Construction of the transducers In this section, we describe the construction of the transducers for the computation of the Hamming weight. We start with the easiest case, the w-NAF. We will then modify the ideas to deal with the asymmetric case of D l,u -expansions in dimension 1. We finally generalize the approach to the d-dimensional D l,u -expansions. All transducers and automata take a (joint) binary expansion as input and read from right to left. The output of the transducers is a sequence of 0's and 1's. Then the computed Hamming weight is the number of 1's in this output. Lemma 1. Let w ≥ 2. The transducer in Figure 1 calculates the weight h(n) of the w-NAF of an integer n. Proof. Let n = value(n L . . . n 0 ) with n j ∈ {0, 1} be the standard binary expansion of n and (ε K . . . ε 0 ) be the w-NAF representation of n. If n ≡ 0 mod 2, then ε 0 = 0 and we stay in the initial state. Otherwise, we have ε 0 = 0 and the weight is h(ε 0 ) = 1. Since we have a w-NAF representation, the next w − 1 digits fulfill ε 1 = ε 2 = . . . = ε w−1 = 0, no matter what the corresponding n j , j = 1, . . . , w − 1 are. The sign of the digit ε 0 depends on n w−1 mod 2. If n w−1 ≡ 0 mod 2, then ε 0 > 0 and we go to state w with the next input n−ε0 2 w with carry 0. If n w−1 ≡ 1 mod 2, then ε 0 < 0 and we therefore have a carry of 1 and go to state w + 1. There, reading an input of 1 and having a carry of 1 results in the same outcome as reading 0, but the carry remains 1. Reading an input 0 with carry 1 is equivalent to reading an input 1 with a carry 0, so we are in state 1 again. In the next step, we construct a transducer for the Hamming weight of the 1-dimensional AJSF. Therefore, we need the following automaton to compare integers. Proof. The states are (s, t) with s, t ∈ {0, 1}. The label s signifies the carry of the addition a + b which still has to be processed. The label t corresponds to the truth value of the expression (a + b) mod 2 i > c mod 2 i where i is the number of read digits up to now. So the automaton accepts the input if it stops in state (0, 0) where there is no carry anymore and a + b > c is false. The initial state is (0, 0). Therefore, there is a path from (0, 0) to (s, t) in Automaton 2 with input label   α i−1 . . . α 0 β i−1 . . . β 0 γ i−1 . . . γ 0   if and only if s = value(α i−1 . . . α 0 ) + value(β i−1 . . . β 0 ) 2 i and t = (value(α i−1 . . . α 0 ) + value(β i−1 . . . β 0 )) mod 2 i > value(γ i−1 . . . γ 0 ) . Here, we use Iverson's notation, that is [expression] is 1 if expression is true and 0 otherwise. From this, the rules for the transitions follow. There is a transition (s, t) (α,β,γ) T −−−−−→ (s , t ) if and only if s = α+β+s 2 and t = [(α + β + s) mod 2 > γ − t]. Theorem 4. There exists a transducer with input and output alphabet {0, 1}, having less than 4w − 2 states, where one state is initial and final, that computes the Hamming weight of the AJSF from the binary expansion of an integer. Proof. We construct a transducer performing the same calculation as Algorithm 2. It will look similar to the transducer in Figure 1. We start at state 0. Then there is a vertical block of states with w − 1 rows having states (0, 0) i , (1, 0) i , (0, 1) i and (1, 1) i in each row i = 1, . . . , w − 1. After this block, we either go back to state 0, or to a similar state 1, or again to the block of states (see Figure 3). We call the states 0 and 1 the looping states. Their labels signify the carry which is to be processed. The state 0 is also the final state. The block of states corresponds to the if statement in line 11 in Algorithm 2. In this line, we have to check the inequality (n − l) mod 2 w−1 ≤ u − l − 2 w−1 . A first step to this aim is to compare n +l ≤ũ withl := −l andũ := u − l − 2 w−1 . Therefore, we use Automaton 2. Next, we examine the binary expansions ofũ andl. Since we have assumed that l > −2 w−1 , we know that the length of the binary expansion ofl is at most w − 1. Furthermore, −1 ≤ũ < 2 w−1 . In the caseũ = −1, the set nonunique(D l,u ) is empty and we have no choices for the digits. We will return to this case later. Then the length of the binary expansion ofũ is at most w − 1. Let (l w−2 . . . l 0 ) and (u w−2 . . . u 0 ) be the binary expansions ofl respectivelyũ. Now we can verify n +l mod 2 w−1 ≤ũ by checking the label t of the state (s, t) after reading w − 1 digits from the binary expansion of (n,l,ũ) T in Automaton 2. If t = 0, then the inequality is true, otherwise it is false. Since the length ofũ is less than or equal to w − 1, there are no digits ofũ left. Only a possible carry of the addition n +l is left. This carry is the label s of the current state (s, t). Therefore, we have checked n +l mod 2 w−1 ≤ũ. To ensure that we read exactly w − 1 digits, the transducer in Figure 3 has w − 1 copies of the four states of Automaton 2. 0, 0 0, 1 1, 0 1, 1 (0, 1, 0) T (1, 0, 0) T (0, 0, 1) T (1, 0, 1) T (0, 1, 1) T (1, 1, 0) T (1, 1, 1) T (1, 1, 0) T (0, 0, 1) T (0, 0, 0) T (0, 0, 1) T (1, 1, 0) T (1, 1, 1) T (0, 0, 0) T (0, 0, 0) T , (0, 0, 1) T (0, 1, 1) T , (1, 0, 1) T (1, 0, 1) T , (0, 1, 1) T (1, 0, 0) T , (0, 1, 0) T (1, 1, 1) T (0, 1, 0) T , (1, 0, 0) T (0, 1, 1) T , (1, 0, 1) T (0, 0, 0) T (1, 1, 1) T , (1, 1, 0) T (1, 0, 0) T , (0, 1, 0) T The transitions start in a state of the i-th copy and go to an appropriate state of the (i + 1)-th copy while reading the i-th digit of the expansion. In the if statement in line 11 in Algorithm 2, we must also check the other condition m ≡ 1 mod 2. Let (s, t) w−1 be the current state at the end of the block of states. We know that m = (n+l)−(n+l) mod 2 w−1 2 w−1 . Therefore, the least significant digit of m is simply the next digit of the addition n +l. Since there are no digits of the expansion ofl left, we only have to look at the next digit ε of n and consider the carry s. Thus we have m ≡ s + ε mod 2. If the inequality of the if statement is satisfied, that is if t = 0, then whatever digit ε we read next, the transducer starts from a looping state again. If m is even, then the next written digit is 0 anyway. If m is odd, we can change the digit in the representation (because it is non-unique) and m becomes even too. We only have to remember the carry. If s = 0 or s = 1 and we read ε = 0, then there will be no carry propagation and we continue with state 0. If s = 1 and we read ε = 1, then there is a carry propagation and we start at state 1. If the inequality is not satisfied, that is if t = 1, and m ≡ s + ε mod 2 is odd, then we have to start with the w − 1 transitions of Automaton 2 immediately. If m is even however, then the transducer starts from a looping state again. In both cases, we have to consider the carry propagation as well. At state s ∈ {0, 1}, we stay in state s as long as we read s. If we read 1 − s we start with the w − 1 transitions of Automaton 2. In the caseũ = −1, the set nonunique(D l,u ) is empty. Therefore, we have t = 1 in each state, and the initial state of Automaton 2 has to be (0, 1). Let (u w−2 . . . u 0 ) = (0 w−1 ). Then we have a transition from s to (s , t ) 1 with input label 1 − s if and only if there is a transition from (s, 1) to (s , t ) with input label (1 − s, l 0 , u 0 ) T in Automaton 2. To summarize, we have the following transitions in the transducer in Figure 3 for s, s , t, t , ε ∈ {0, 1} and i ∈ {1, . . . , w − 2}: 0 1 0, 0 1 1, 0 1 0, 1 1 1, 1 1 0, 0 w−1 1, 0 w−1 0, 1 w−1 1, 1 w−1 0 \| 1 0, 1 \| 0 0 \| 0 0 \| 0 1 \| 0 1 \| 0 1 \| 1 1 \| 1 0 \| 1 0 \| 0 1 \| 0• s ε\|0 − − → s if s = ε • s ε\|1 − − → (s , t ) 1 if s = ε and (s, [ũ = −1]) (ε,l0,u0) T − −−−−− → (s , t ) is a transition in Automaton 2 • (s, t) i ε\|0 − − → (s , t ) i+1 if (s, t) (ε,li,ui) T − −−−−− → (s , t ) is a transition in Automaton 2 • (s, t) w−1 ε\|0 − − → s if t = 0 or ε + s ≡ 0 mod 2, and s = ε+s 2 • (s, t) w−1 ε\|1 − − → (s , t ) 1 if t = 1, ε + s ≡ 1 mod 2 and (s, [ũ = −1]) (ε,l0,u0) T − −−−−− → (s , t ) is a transition in Automaton 2. We note that there is only one accessible state in the first row because the transitions 0 1 have both the same target state. This target state depends on l and u. Finally, we restrict the transducer to the states which are actually accessible from the initial state. Now we can describe the last state of the path with input label (ε L . . . ε 0 ), a binary expansion. The following lemma can easily be proved by induction. 1\|1 − − → (s, t) 1 and 1 0\|1 − − → (s, t)Lemma 3. Let k i , s i , t i for i ≥ 0 and a i , f i for i ≥ 1 be sequences with s 0 = 0, t 0 = 1. The states (s i , t i ) w−1 are the states in the last row of the path. The integers k i count how often we circle in a looping state after the state (s i , t i ) w−1 . The integers f i are the positions of the nonzeros in the AJSF and a i is the digit at position f i . Forũ ≥ 0, these sequences satisfy the following recursions for i ≥ 1: k i =          max{k ∈ N \| (ε fi+k+w−2 . . . ε fi+w−1 ) = (0 k−1 1) or (0 k )} if s i = t i = 0, max{k ∈ N \| (ε fi+k+w−2 . . . ε fi+w−1 ) = (1 k ) or (0 k )} if s i = 1, t i = 0, max{k ∈ N \| (ε fi+k+w−2 . . . ε fi+w−1 ) = (0 k )} if s i = 0, t i = 1, max{k ∈ N \| (ε fi+k+w−2 . . . ε fi+w−1 ) = (1 k )} if s i = t i = 1, f i = k 0 + · · · + k i−1 + (i − 1)(w − 1), s i = value(ε fi+w−2 . . . ε fi+1 1) +l 2 w−1 , t i = value(ε fi+w−2 . . . ε fi+1 1) +l mod 2 w−1 >ũ , a i = −l + (value(ε fi+1−1 . . . ε fi+1 1) +l mod 2 ki+w−1 ). Then we have value(ε fi+w−2 . . . ε fi+1 1) = a i + 2 w−1 s i (1 − [s i = 1 ∧ t i = 0 ∧ ε fi+w−1 = 0]). 0 1 1, 0 1 0, 0 3 1, 0 3 1, 0 2 1 \| 1 0 \| 1 0, 1 \| 0 0 \| 0 1 \| 0 0, 1 \| 0 1 \| 0 0 \| 0 0 \| 0 1 \| 0L − f i + 1, f i ≤ L ≤ f i + w − 2 and s = value(ε L . . . ε fi+1 1) + value(l j−1 . . . l 0 ) 2 j , t = (value(ε L . . . ε fi+1 1) + value(l j−1 . . . l 0 )) mod 2 j > value(u j−1 . . . u 0 ) . There is a path from 0 to s with input label (ε L . . The transducer can be seen in Figure 4, where all non-accessible states are gray. We recall that a reset sequence of a transducer is a sequence (n L . . . n 0 ) such that there exists a state s with the following property: For all states t, if the transducer is in state t and the next input is (n L . . . n 0 ), then the transducer is in state s. Now we generalize this transducer to arbitrary dimension d. Theorem 5. There exists a transducer to compute the Hamming weight of the AJSF for the joint binary expansion of a d-dimensional vector of integers as input. It has one state which is initial and final, input and output alphabet {0, 1} and less than 8 d w states. The word 0 4w is a reset sequence of this transducer. It leads to the initial and final state of the transducer. Proof. We construct a transducer calculating the weight of AJSF. In order to explain the structure of this transducer, we first consider a provisional transducer implementing a simpler version of the Algorithm 1 which omits the else branch in line 17, see also the algorithm on page 306 of [7]. The resulting provisional transducer is similar to the transducer in Figure 3. For every vector s ∈ {0, 1} d , there is a state. These states are called looping states. The vector s signifies the carry at each coordinate. The state (0, . . . , 0) T is the initial state. Furthermore, there is a block of states. The states inside the block have the labels (s, t) i where s, t ∈ {0, 1} d , and i is the row in the block. The coordinates of s and t have the same meaning as in the proof of Theorem 4, that is: s is the carry of the addition n +l and t signifies whether the digit is in nonunique(D l,u ) or not. If s ∈ {0, 1} d is a looping state, then there is a loop with label s \| 0 at this state. Because if we read ε = s, then we have ε + s ≡ 0 mod 2, the output is 0 and the carry propagates to the next step. If we read ε = s, then we start with the w − 1 transitions of Automaton 2 in Figure 2 in each coordinate. These w − 1 transitions are processed independently for every coordinate. Therefore, we need 4 d states in each row and w − 1 rows to process exactly w − 1 transitions of Automaton 2. After the last row of the block of states, we either go back to a looping state or again start with the block of states immediately. Let (s, t) w−1 be the current state in the last row and ε the next input digit. As in the 1-dimensional case we have m ≡ ε + s mod 2. If for every coordinate j, t j = 1 implies that m j is even, then we have to process the if branch in line 12. We write this condition as the scalar product t · (s + ε mod 2) = 0. In this case, the next output digit will be 0 and we go on to a looping state s where the new carry is s = s+ε 2 . If t · (s + ε mod 2) = 0 does not hold, then we would have to process the else branch in line 17. But since we skip this part for now, we simply have to restart the transducer with the input m in the case t · (s + ε mod 2) > 0. We know that m is the original next input ε plus the carry s. In this case, s = ε, otherwise t · (s + ε mod 2) > 0 would be false. Therefore, there is a transition • s ε\|0 − − → s if ε = s • s ε\|1 − − → (s , t ) 1 if ε = s and ∀j : (s j , [ũ = −1]) (εj ,l0,u0) T − −−−−−− → (s j , t j ) is a transition in Automaton 2 • (s, t) i ε\|0 − − → (s , t ) i+1 if ∀j : (s j , t j ) (εj ,li,ui) T − −−−−−− → (s j , t j ) is a transition in Automaton 2 • (s, t) w−1 ε\|0 − − → s if t · (s + ε mod 2) = 0 and s = s+ε 2 • (s, t) w−1 ε\|1 − − → (s , t ) 1 if t · (s + ε mod 2) > 0 and s ε\|1 − − → (s , t ) 1 is a transition in this transducer. This transducer does the same as Algorithm 1 without the else branch in line 17. In the casẽ u = −1 we are finished because in the else branch nothing is done. Otherwise we must consider the else statement. Let (s, t) w−1 be the current state in the last row and ε be the next input digit. To process the else branch, t · (s + ε mod 2) > 0 must hold in the state (s, t) w−1 . Otherwise, we would process the if branch. First let us examine one coordinate j. If t j = 1, nothing is done in the else branch because the digit at this coordinate is unique. If t j = 0, we have to decide whether m j ≡ u + 1 mod 2 w−1 . Here, m j mod 2 w−1 corresponds to the next w − 1 input digits plus the carry s j from the current state (s, t) w−1 . So we just have to compare the input letters plus the carry with the binary expansion of u + 1 mod 2 w−1 or, equivalently, we compare m j − l mod 2 w−1 withṽ = u − l + 1 mod 2 w−1 . If they are not the same at some point, then we just go on like we did in the provisional transducer. If they are the same, we have to process the else branch. There we would have taken m j − 1 as the next input of the algorithm instead of m j . Therefore we have to decide where we would be in the provisional transducer, when starting in (s, t) w−1 and the input is the original input minus 1. This case only happens if originally the next nonzero digit is unique, but changing the current digit ensures that the next nonzero digit is non-unique. Nevertheless, the next digit will not be 0, since this is the case when the if branch is processed. Therefore we would start in (s, t) w−1 with original input minus 1 and immediately go to the block of states again. Otherwise, the next digit would be 0. Thus after w − 1 transitions, we are again in a state (s , t ) w−1 in the last row. Since the next digit is non-unique, we have t j = 0. To determine the value of s j , we have to decide whether there is a carry at position w − 1 in the addition of m j − 1 andl. We have m j − 1 mod 2 w−1 = u + k2 w−1 for k ∈ Z. Since 0 ≤ u ≤ 2 w − 1, we have k ∈ {0, −1}. Then the carry is s j = (m j − 1) mod 2 w−1 +l mod 2 w−1 2 w−1 = u +l + k2 w−1 2 w−1 = 1 + k, because 2 w−1 ≤ u +l < 2 w . Therefore, we have s j = 0 if u ≥ 2 w−1 , 1 if u < 2 w−1 . As a result, the state (s , t ) w−1 where we would be in the provisional transducer has ( u < 2 w−1 , 0) in the j-th coordinate. To remember that we can change the j-th coordinate at the end of the block, we have to use a second identical block {j}. Let ∅ be the first block, which already exists in the provisional transducer. Let (s, t) C i be a state in block C. At the end of block ∅, we go to block {j} if t · (s + ε mod 2) > 0 and t j = 0. Otherwise, we go to a looping state or to the block ∅. If we find out that m j ≡ u + 1 mod 2 w−1 in block {j}, then we go back to the appropriate state in block ∅. At the end of block {j} in the state (s, t) {j} w−1 , we go to the same states as we would go from the state with ( u < 2 w−1 , 0) ∅ w−1 in the j-th coordinate. Up to now, we only considered one coordinate. Now we combine this approach for all coordinates. Since for each coordinate, we have to remember whether we are allowed to change it or not, we need one block for every subset of coordinates. Let block C ⊆ {1, . . . , d} be the block where we can change the coordinates in C. The states in block C are denoted by (s, t) C i . The block ∅ is the block which already exists in the provisional transducer. The block {1, . . . , d} is not accessible since we need at least one unique coordinate and only non-unique coordinates can be changed. If we are in block C = ∅ and we find out that not every coordinate j ∈ C satisfies m j ≡ u + 1 mod 2 w−1 , we go to the appropriate state in block C = C \ {j ∈ {1, . . . , d} \| m j ≡ u + 1 mod 2 w−1 }. At the end of block C in state (s, t) C w−1 , we can change the coordinates in C and all other coordinates remain the same. Therefore, we go to the same states as we would go from (s,t) ∅ w−1 wheres j = u < 2 w−1 ,t j = 0 for j ∈ C and all other coordinates stay the same, that is s j = s j andt j = t j for j ∈ C. Let • s ε\|0 − − → s if s = ε • s ε\|1 − − → (s , t ) ∅ 1 if s ε\|1 − − → (s , t ) 1 is a transition in the provisional transducer • (s, t) C i ε\|0 − − → (s , t ) C i+1 if (s, t) i ε\|0 − − → (s , t ) i+1 is a transition in the provisional transducer and C = C \ {j : s j + ε j + l i mod 2 = v i } • (s, t) ∅ w−1 ε\|0 − − → s if t · (s + ε mod 2) = 0 and s = s+ε 2 • (s, t) ∅ w−1 ε\|1 − − → (s , t ) C 1 if t · (s + ε mod 2) > 0, (s, t) w−1 ε\|1 − − → (s , t ) 1 is a transition in the provisional transducer and C = {j : s j + ε j + l 0 mod 2 = v 0 and t j = 0} • (s, t) C w−1 ε\|1 − − → (s , t ) C 1 if C = ∅ and (s,t) ∅ w−1 ε\|1 − − → (s , t ) C 1 is a transition in this transducer withs j = u < 2 w−1 ,t j = 0 for j ∈ C ands j = s j ,t j = t j for j ∈ C • (s, t) C w−1 ε\|0 − − → s if C = ∅ and (s,t) ∅ w−1 ε\|0 − − → s is a transition in this transducer with s j = u < 2 w−1 ,t j = 0 for j ∈ C ands j = s j ,t j = t j for j ∈ C. Finally, we restrict the transducer to the states which are actually accessible from the initial state. Due to the construction of the transducer, the sequence 0 4w leads to the initial and final state from any state. It is possible to define similar sequences to those in Lemma 3, but since this requires more than one page, we omit this here. Proof of Theorem 1 This section contains the proof of Theorem 1 which is a generalization of Theorem 6 in [5]. With the transducer in Theorem 5, we can compute the asymptotic Hamming weight. Therefore, we use the following lemma which can be proved by induction on L. We define f (m 1 , . . . , m d ) := e ith(m1,...,m d ) . The matrices M ε1,...,ε d for ε i ∈ {0, 1} are defined as follows: The (j, k)-th entry of the matrix M ε1,...,ε d is e ith if there is a transition from state j to k with input label (ε 1 , . . . , ε d ) T and output label h. The entry is 0 if there is no transition from state j to k with this input label. The ordering of states is considered to be fixed in such a way that the initial state is the last state. Then we have f (m 1 , . . . , m d ) = v T L p=0 M m1,p,...,m d,p M 4w 0,...,0 v(1) for v T = (0, . . . , 0, 1) and m i = L p=0 m i,p 2 p . The product describes all possible paths from any state to any other state, using edges with input labels corresponding to the input (m 1 , . . . , m d ). The exponent of the entries of the matrix product is the sum of output labels on these paths. Since we are interested in paths starting and ending in state (0, . . . , 0) T , we multiply by v T from the left and v from the right. The factor M 4w 0,...,0 is due to the reset sequence from Theorem 5 and ensures that we stop at the final state. We further define the following summatory functions . . , d}. The first index C of B C,D is the set of coordinates where the digit is 0. The second index D is the set of coordinates where the digit is 1. All other coordinates in (C ∪ D) c can be any digit. By construction, we have M ε1,...,ε d 1 = 1, where · · · 1 denotes the row sum norm of a matrix. We conclude that B C,D 1 ≤ 2 d−\|C\|−\|D\| . As a special matrix we define A = B ∅,∅ . Furthermore we define the functions E(N ) =G C (N ) := 0≤mi<N i ∈C mi=N i∈C M (m 1 , . . . , m d ) for every set C ⊆ {1, . . . , d}. Then we have F (N ) = G ∅ (N ), and the functions satisfy the following recursion formulas due to 2-multiplicativity G C (2N ) = εi=0,1 i ∈C εi=0 i∈C 2mi+εi<2N i ∈C 2mi+εi=2N i∈C M (2m 1 + ε 1 , . . . , 2m d + ε d ) = B C,∅ G C (N ), G C (2N + 1) = εi=0,1 i ∈C εi=1 i∈C 2mi+εi<2N +1 i ∈C 2mi+εi=2N +1 i∈C = D⊆C c εi=0,1 i ∈C∪D εi=0 i∈D εi=1 i∈C M ε1,...,ε d mi<N i ∈C∪D mi=N i∈C∪D M (m 1 , . . . , m d ) = D⊆C c B D,C G C∪D (N ). From this recursion, we can determine G C (N ) inductively because all functions G C required for computing G C have C C. Therefore, we have the following recursion formula for F (N ) = G ∅ (N ) F (2N + ε) = AF (N ) + εH(N ) for N ≥ 1, ε ∈ {0, 1} and H(N ) = ∅ =D⊆{1,...,d} B D,∅ G D (N ). If we define H(0) = G ∅ (1), we can use Lemma 4 and get F L p=0 ε p 2 p = L p=0 ε p A p H   L j=p+1 ε j 2 j−p−1   .(2) Here, H(N ) is considered to be a known function because it is a sum of functions G C (N ), which are recursively known by Lemma 4. From the definition of G C (N ), we can derive the growth rates of the functions G C (N ) and H(N ). We have G C (N ) 1 = O(N d−\|C\| ) and H(N ) 1 = O(N d−1 ). Next, we investigate the eigenvalues of the matrix A. We first consider the case t = 0. In this case, A is the adjacency matrix of the underlying graph of the transducer in Theorem 5. Therefore, it has an eigenvalue 2 d with eigenvector (1, . . . , 1) T . By the theorem of Perron-Frobenius, there is a unique dominant eigenvalue µ(0) of A which is easily seen to be primitive as every state is reachable from any other state in exactly 4w steps. As A 1 ≤ 2 d and the largest eigenvalue is always at most A 1 , µ(0) = 2 d is the largest eigenvalue. We denote the modulus of the second largest eigenvalue by β(0). Since eigenvalues are continuous, for t in a suitable neighborhood of 0, A has a unique dominant eigenvalue µ(t) and the modulus β(t) of the second largest eigenvalue fulfills β(t) < \|µ(t)\|. Now we want to split up (2) into two parts, one for the dominating eigenvalue and one for the remaining eigenvalues. Therefore, let J = T −1 AT be a Jordan decomposition of A where J has been sorted such that it has µ(t) in the upper left corner. We define Λ := T diag(µ(t) −1 , 0, . . . , 0)T −1 and R := T (J − diag(µ(t), 0, . . . , 0))T −1 . Then A p = µ L Λ L−p + R p holds for p ≤ L. Further, we define Λ(x 0 , x 1 , . . .) = ∞ p=0 x p Λ p H   p−1 j=0 x j 2 p−1−j   ,(3)R L p=0 ε p 2 p = L p=0 ε p R p H   L j=p+1 ε j 2 j−p−1   . The function Λ is well defined on the infinite product space {0, 1} N because it is dominated by a geometric series. We extend Λ to a function on [1, 2) by setting Λ So we have E(N ) = N d+a1t+a2t 2 +O(t 3 ) Ψ(log 2 N, t) + O(N d−δ )(4) with a 1 and a 2 depending on the Taylor expansion of log 2 µ(t) at t = 0. If we insert t = 0 in (4), we obtain ψ 0 = Ψ(log 2 N, 0) = 1 + O N −δ . The function Ψ(x, t) is periodic in x with period 1 and is well defined for all x ∈ R + . To prove continuity in x, we first note that continuity for x ∈ [0, 1) with x = log 2 y where y is not a dyadic rational follows from (3). To prove it for x = log 2 y with y = L p=0 ε p 2 −p a dyadic rational with ε L = 1, we observe that the two one-sided limits exist due to (3). Next, we prove that they are the same. Therefore, we look at the two sequences N k = y2 L+k andÑ k = y2 L+k − 1. Then lim k→∞ 2 {log 2 N k } = (ε 0 ε 1 . . . ε L−1 10 ω ) and lim k→∞ 2 {log 2Ñk } = (ε 0 ε 1 . . . ε L−1 01 ω ). If we insert these two sequences in (4), we get O(N d−1 k ) = E(N k ) − E(Ñ k ) = N d k Ψ(log 2 N k , t) −Ñ d k Ψ(log 2Ñk , t) + O(N d−δ k ), and hence lim k→∞ Ψ({log 2 N k }, t) = lim k→∞ Ψ({log 2Ñk }, t). Therefore, Ψ(x, t) is continuous in x. In t, Ψ(x, t) is also continuous because the eigenvalues of a matrix are continuous. Furthermore, the function Ψ(x, t) is arbitrarily often differentiable in t because it is dominated by a geometric series. By the same argument as above, these derivatives are continuous in x. The first and second derivative of E(N ) with respect to t at t = 0 imply that the expected value of the Hamming weight is + Ψ 2 (log 2 N ) + O(N −δ log 2 N ) with v l,u,d = −2a 2 log 2 and Ψ 2 (log 2 N ) = − ∂ 2 ∂t 2 Ψ(log 2 N, t)\| t=0 . From that, we calculate the variance which is 1 N d mi<N h 2 (m 1 , . . . , m d ) − 1 N d mi<N h(m 1 , . . . , m d ) 2 = v l,u,d log 2 N − Ψ 2 1 (log 2 N ) + Ψ 2 (log 2 N ) + O(N −δ log 2 N ). We first compute the characteristic functionĝ N (t) of the random variable 1 + O t 3 log 3/2 N ψ log 2 N, t v l,u,d log 2 N + 1 N dR N, t v l,u,d log 2 N e −it e l,u,d √ v l,u,d √ log 2 N . Sinceĝ N (t) is a characteristic function, we have 1 = ψ 0 + r 0 for r 0 = 1 N dR (N, 0) and ψ 0 = Ψ(log 2 N, 0). We know that 1 N dR (N, t) = O(N −δ ). Next, we can estimate the difference from g N (t) to the characteristic functionf (t) = e − t 2 2 of the normal distribution with mean 0 and variance 1, which is N ). Therefore, the Berry-Esseen inequality (cf. [11]) implies \|ĝ N (t) −f (t)\| = e − t 2 2 1 + O t 3 log 3/2 N ψ 0 + O t √ log N + 1 N dR N, t v l,u,d log 2 N exp −it e l,u,d √ v l,u,d log 2 N − (ψ 0 + r 0 )e − t 2 2 = O t √ log N , for t = o( √ logP h(m 1 , . . . , m d ) − e l,u,d log 2 N v l,u,d log 2 N < x = 1 √ 2π x −∞ e − y 2 2 dy + O 1 4 √ log N . For a specific digit set and dimension, we can compute the constants e l,u,d and v l,u,d explicitly. Example 4.1. We consider the digit set D −2,3 in dimension 2. See Example 3.2 and Figure 5 for the transducer. The adjacency matrix A of the underlying graph of this transducer is given in Table 1, where z = e it . The characteristic polynomial of A is −(x − 1)x 7 x 2 − 2z x 3 − x 2 − xz − 2z 2 x 5 − x 4 − 7x 3 z − 20x 2 z + 6xz 2 − 24z 2 . At t = 0, the dominating eigenvalue µ(0) = 4 is a root of the fourth factor. Therefore the Taylor expansion of µ(t) around t = 0 is µ(t) = 4 + 128i 89 t − 673216 2114907 t 2 + O(t 3 ). Hence the expected value of the Hamming weight is 32 89 log 2 N + O(1) and the variance is 63200 2114907 log 2 N + O(1). In order to determine the constants e l,u,d and v l,u,d giving mean and variance in general, we rephrase the results of the "full-block-length" analysis in [7] in a probability model which is easily compared with our main results. 0 ≤ m i < 2 k . A =                                                                           Then E W k = e l,u,d k + O(1) and V W k = v l,u,d k + O( √ k) for the constants given in Theorem 1. Proof. For j < k, we denote the j-th digit of the AJSF of a random vector m = (m 1 , . . . , m d ) T by X j , where we assume equidistribution of all vectors m = (m 1 , . . . , m d ) T with 0 ≤ m i < 2 k . In [7, § 6.2], the random variables X j denoting the j-th digit of a random AJSF has been considered, where the probability measure was defined to be the image of the Haar measure on the space of d-tuples of 2-adic integers under the AJSF, i.e., equidistribution on all residue classes modulo 2 l for all l has been assumed. Furthermore, W j was defined to be the weight of the first j digits. From Algorithm 1, it is clear that X j only depends on m modulo 2 j+w . This implies that X j and X j are identically distributed for j < k − w. Therefore W k−w and W k−w are identically distributed, too. Furthermore, we always have \| W k − W k−w \| ≤ w. By [7, Theorem 6.7], we have E W k−w = EW k−w = e l,u,d (k − w) + O(1) and V W k−w = VW k−w = v l,u,d (k − w) + O(1). We conclude that E W k = E W k−w + O(1) = EW k−w + O(1) = e l,u,d k + O(1), V W k = V W k−w + V( W k − W k−w ) + 2 Cov( W k−w , W k − W k−w ) = VW k−w + O( √ k), where the Cauchy-Schwarz inequality has been used in the form Cov( W k−w , W k − W k−w ) ≤ V W k−w V( W k − W k−w ). In the next lemma, we prove that the function Ψ 1 (x) is non-differentiable at any real number in the 1-dimensional case. The proof uses the method presented by Tenenbaum [10], see also Grabner and Thuswaldner [4]. Lemma 6. Let d = 1. Then the function Ψ 1 (x) in Theorem 1 is nowhere differentiable. Proof. Let g(N ) = 2 −1−4w N c c+1 be a positive integer valued function with c ∈ Z and c > 1 δ − 1. We have g(N ) = o(N ) and N 1−δ log N = o(g(N )). Assume Ψ 1 is differentiable at x ∈ [0, 1). Let 2 x = ∞ p=0 ε p 2 −p be the standard binary digit expansion choosing the representation ending on 0 ω in the case of ambiguity. Further, let x k , y k and N k be such that 2 x k = k p=0 ε p 2 −p , N k = 2 k(c+1)+x k ∈ Z and 2 k(c+1)+y k = N k + g(N k ). Then we have x − x k = O(2 −k ), y k − x k = log 2 1 + g(N k ) N k = 1 log 2 g(N k ) N k + O g(N k ) 2 N 2 k , lim k→∞ y k = x. Because of the choice of g(N ), we have g(N k ) < 2 ck−4w , g(N k ) N k = Θ(2 −k ). We have h(2 p+4w n + m) = h(n) + h(m) for p ≥ 0 and m < 2 p because 0 4w is a reset sequence leading to the initial state (see Theorem 5). Due to (5) and the periodicity and continuity of Ψ 1 , we have N k ≤n<N k +g(N k ) h(n) = g(N k )h(N k ) + n<g(N k ) h(n)(6) = g(N k )h(N k ) + e l,u,1 g(N k ) log 2 g(N k ) + g(N k )Ψ 1 c c + 1 x + o(g(N k )). On the other hand, we have N k ≤n<N +g(N k ) h(n) = e l,u,1 (N k + g(N k )) log 2 (N k + g(N k )) + (N k + g(N k ))Ψ 1 (log 2 (N k + g(N k ))) − e l,u,1 N k log 2 N k − N k Ψ 1 (log 2 N k ) + O(N 1−δ k log N k ) = e l,u,1 N k (y k − x k ) + N k (Ψ 1 (y k ) − Ψ 1 (x k )) + e l,u,1 g(N k )(k(c + 1) + y k ) + g(N k )Ψ 1 (y k ) + o(g(N k )). If we divide by g(N k ) in (6) and (7), then we obtain h(N k ) + e l,u,1 log 2 g(N k ) + Ψ 1 c c + 1 x = e l,u,1 (y k − x k ) N k g(N k ) + N k g(N k ) (Ψ 1 (y k ) − Ψ 1 (x k )) + e l,u,1 (k(c + 1) + y k ) + Ψ 1 (y k ) + o(1). Now we can write the difference of the Ψ 1 on the right-hand side in terms of the derivative Ψ 1 (y k ) − Ψ 1 (x k ) = Ψ 1 (x)(y k − x k ) + o(x − x k ) + o(\|y k − x\|), and we get h(N k ) = −Ψ 1 c c + 1 x + e l,u,1 log 2 + Ψ 1 (x) log 2 + e l,u,1 (k + x c + 1 ) + Ψ 1 (x) + o(1). Next, we take the difference of two subsequent terms h(N k+1 ) − h(N k ) = e l,u,1 + o(1), where the left-hand side is an integer. We have e l,u,1 ∈ Z because 0 < e l,u,1 = 1 w−1+λ < 1. Therefore the right-hand side of (8) is not an integer for k large enough. This contradicts our assumption that Ψ 1 is differentiable in x. Asymptotic distribution of the w-NAF In this section, we specialize the result of Theorem 1 to the w-NAF. Theorem 6. The weight h(n) of the w-NAF of the integer n with equidistribution on {n ∈ Z \| 0 ≤ n < N } is asymptotically normally distributed. There exists a δ > 0 such that the mean is 1 w + 1 log 2 N + Ψ 1 (log 2 N ) + O(N −δ log N ) and the variance is 2 (w + 1) 3 log 2 N − Ψ 2 1 (log 2 N ) + Ψ 2 (log 2 N ) + O(N −δ log 2 N ), where Ψ 1 and Ψ 2 are continuous, 1-periodic functions on R. If w is large enough, then δ = log 2 1 + 3π 2 w 3 . In particular, we have P   h(n) − log 2 N w+1 2 (w+1) 3 log 2 N < x   = 1 √ 2π x −∞ e − y 2 2 dy + O 1 4 √ log N for all x ∈ R. This follows from the following Lemma 7 and Theorem 1. Lemma 7. The characteristic polynomial of the matrix A of the transducer in Figure 1 is (x − 1)(x w − x w−1 − 2 w−1 e it ). The largest eigenvalue µ(t) is unique around t = 0 and µ(0) = 2. Furthermore, for large enough w and t = 0, there is exactly one simple eigenvalue in T k = {x ∈ C : \|x\| ≥ 2 polynomial, we found w distinct roots of the polynomial z w +z −2. Thus, we found all roots of this polynomial. We only have to investigate 0 ≤ k ≤ w 2 because the coefficients of the polynomial are real. Let z ∈ T k be the fixed point of f k . For ρ = exp( 2πi w ), we have \|z − ρ k \| = O( 1 w ). Therefore, z = ρ k + O( 1 w ). For k ≤ w α with a fixed α ∈ 1 2 , 1 , we have z = f (z) = 1 + 2πik w + O k 2 w 2 . Iterating, we successively get z = 1 + 2πik w − 2π 2 k 2 w 2 − 2πik w 2 + O k 3 w 3 and \|z\| = 1 + 4π 2 k 2 w 3 + O k 3 w 4 . Therefore, for large w, only the fixed points for k = w−1, 0, 1 are in the disk {z ∈ C : \|z\| ≤ 1+ 10π 2 w 3 }. Thus for k ≥ w α , the fixed point of f k is not in the disk {z ∈ C : \|z\| ≤ 1 + 10π 2 w 3 } for large w. Theorem 1 . 1The Hamming weight h(m 1 , . . . , m d ) of the AJSF of an integer vector (m 1 , . . . , m d ) T over the digit set D l,u in dimension d with equidistribution of all vectors (m 1 , . . . , m d ) T with 0 ≤ m i < N for an integer N is asymptotically normally distributed. There exist constants e l,u,d , v l,u,d ∈ R and δ > 0, depending on u, l and d, such that the expected value is e l,u,d log 2 N + Ψ 1 (log 2 N ) + O(N −δ log N ) and the variance is v l,u,d log 2 N − Ψ 2 1 (log 2 N ) + Ψ 2 (log 2 N ) + O(N −δ log 2 N ), where Ψ 1 and Ψ 2 are continuous, 1-periodic functions on R. In particular, we have P h(m 1 , . . . , m d ) − e l,u,d log 2 N v l,u,d log 2 x ∈ R. For d = 1, we have e l,u,1 = 1 w − 1 + λ and v l,u,1 = (3 − λ)λ (w − 1 + λ) 3 , where λ = 2(u − l + 1) − (−1) l − (−1) u 2 w Algorithm 1 1Algorithm to compute the weight of the AJSF with digit set D l,u Input: A vector of integers n ∈ Z d , integers l ≤ 0, u > 0, n ≥ 0 if l = 0 Output: Weight h(n) n j ∈ I nonunique such that m j ≡ u + 1 mod 2 = m 25: end while 26: return h The if branch in line 3 of Algorithm 1 makes the digit at the current position a zero column if possible. If this is not possible, the else branch in line 6 chooses the smallest digit in each component which is congruent to the input. In the inner if and else branches, the algorithm checks if we should change any non-unique digits. In the if statement in line 12, we check whether we can make the (w − 1)-st digit after the current digit 0. Otherwise, in the else statement in line 17, we check whether we can increase the redundancy at the (w − 1)-st digit after the current digit by changing any non-unique digits at the current position. Algorithm 2 Figure 1 . 21Algorithm to compute the weight of the 1-dimensional AJSF with digit set D l,u Input: Integers n, l ≤ 0, u > 0, n ≥ 0 if l m ≡ 1 mod 2 and (n − l) mod 2 w−1 ≤ u − l − 2 Transducer to compute the Hamming weight of a w-NAF representation. Lemma 2 . 2Automaton 2 in Figure 2 accepts the input of three integers a, b, c if and only if a + b ≤ c. The binary expansions of a, b and c must have the same length where leading 0's are allowed. Remark 3.1. Automaton 2 could be simplified by combining the input a + b and merging the states (1, 0) and (0, 1). But since we use this automaton in Theorems 4 and 5 with the input of the transducer as a and some fixed parameter as b, this would complicate the constructions in Theorems 4 and 5. Figure 2 . 2Automaton 2 to compare three integers a, b and c, accepts if a + b ≤ c. Figure 3 . 3Transducer to compute the weight of the AJSF with digits in D l,u . The transitions into and inside the block of states depend on l and u. Figure 4 . 4Transducer to compute the weight of the AJSF with digits in D −3,11 . The non-accessible states are gray.There is a path from 0 to (s, t) j with input label (ε L . . . ε 0 ) if and only if j = . ε 0 0) if and only if f i + w − 1 ≤ L ≤ f i+1 − 1 and s = s i or s = 0, s i = 1 and (ε L . . . ε fi+w−1 ) = (0 L−fi−w+2 ).Forũ = −1 the only difference is t i = 1 and t = 1. Example 3. 1 . 1For l = −3 and u = 11, we have w = 4,l = (011) 2 andũ = u − l − 2 w−1 = (110) 2 . → (s , t ) 1 in this transducer. This ensures that, when restarting the transducer with input m, we immediately go on to the state (s , t ) 1 . Hence, we have a transition (s, t) w−1 ε\|1 − − → (s , t ) 1 in the provisional transducer. Altogether, for s, s , t, t ∈ {0, 1} d , i ∈ {1, . . . w − 2}, j ∈ {1 . . . d} and ε ∈ {0, 1} d , we have the following transitions in this provisional transducer: (v w−2 . . . v 0 ) be the binary expansion ofṽ. Further let s, s , t, t ∈ {0, 1} d , C, C {1, . . . , d}, j ∈ {1 . . . d}, i ∈ {1, . . . , w − 2} and ε ∈ {0, 1} d . Then altogether there are the following transitions in the final transducer: Example 3 . 2 . 32In Figure 5, there is a sketch of the transducer computing the weight of the AJSF over D −2,3 in dimension 2. The labels of transitions are omitted in the figure and the transitions Figure 5 . 5Transducer to compute the Hamming weight of the AJSF in dimension 2 over the digit set D −2,3 .going back at the end of a block or inside a block are gray. We have w = 3,ũ = (01),l = (10) andṽ = (10). Lemma 4 . 4Let A 0 , A 1 be matrices in C n×n , H : N → C n×n be any function and G : N → C n×n be a function which satisfies the recurrence relationG(2N + ε) = A ε G(N ) + εH(N )for N ≥ 1 and ε ∈ {0, 1}. M 0≤m1,...,m d <N f (m 1 , . . . , m d ), F (N ) = 0≤m1,...,m d <N M (m 1 , . . . , m d ), with M (m 1 , . . . , m d ) = L p=0 M m1,p,...,m d,p . In other words, this last equation says that the function M (m 1 , . . . , m d ) is 2-multiplicative (cf. [2]). By (1), we have E(N ) = v T F (N )M 4w 0,...,0 v. To write down a recursion formula for F (N ), we need the following matrices ε1,...,ε d for disjoint C, D ⊆ {1, . ∞ p=0 x p 2 −p := Λ(x 0 , x 1 , . . .) with the standard binary expansion and choosing the representation ending on 0 ω in the case of ambiguity.Then we haveF L p=0 ε p 2 p = µ L Λ(ε L , ε L−1 , . . . , ε 0 , 0 ω ) + R(ε L , . . . , ε 0 )andE(N ) = µ(t) log 2 N Ψ(log 2 N, t) +R(N, t) with Ψ(x, t) = µ(t) −{x} v T Λ(2 {x} )M 4w 0,...,0 v andR(N, t) = v T R(N )M 4w 0,...,0 v. Furthermore, there is a δ ∈ (0, 1] such that log 2 β(t) < d − δ in a suitable neighborhood of 0. Then we have \|R(N, t)\| = O(N d−δ ). 1 , . . . , m d ) = e l,u,d log 2 N + Ψ 1 (log 2 N ) + O(N −δ log N ) (5) with e l,u,d = −ia 1 log 2 and Ψ 1 (log 2 N ) = −i ∂ ∂t Ψ(log 2 N, t)\| t=0 , and 1 N d mi<N h 2 (m 1 , . . . , m d ) = v l,u,d log 2 N + e 2 l,u,d log 2 2 N + 2e l,u,d log 2 N Ψ 1 (log 2 N ) Z = h(m 1 , . . . , m d ) − e l,u,d log 2 N v l,u,d log 2 N , Lemma 5 . 5Let k ≥ w be a positive integer. Let W k be the Hamming weight of the AJSF of a random vector m = (m 1 , . . . , m d ) T with equidistribution of all vectors m = (m 1 , . . . , m d ) T with 2 w 2α−2 + O(w 4α−4 ), \|2 − z\| ≥ \|2 − ρ k \| − \|z − ρ k \| = 1 + 4π 2 w 2α−2 + O(w 4α−4 + w −1 ), \|f k (z)\| = exp 1 w log \|2 − z\| ≥ 1 + 4π 2 w 2α−3 + O(w 4α−5 + w −2 ). Table 1 . 1Adjacency matrix of the transducer in Example 4.1 M (2m 1 + ε 1 , . . . , 2m d + ε d ) Acknowledgement. We thank the anonymous referees for their constructive comments and for encouraging us to prove the non-differentiability of Ψ 1 .w \| ≤ π 2w } for each k = 0, . . . , w − 1. Additionally there is the obvious eigenvalue x = 1. The eigenvalues with the second largest absolute value are in T 1 and T w−1 . For each eigenvalue at t = 0, an expansion in 1 w can be computed with arbitrarily small error term. Proof. The characteristic polynomial of A is obtained by Laplace expansion. 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[ "ON THE POWER OF STANDARD INFORMATION FOR TRACTABILITY FOR L 2 -APPROXIMATION IN THE AVERAGE CASE SETTING", "ON THE POWER OF STANDARD INFORMATION FOR TRACTABILITY FOR L 2 -APPROXIMATION IN THE AVERAGE CASE SETTING" ]	[ "Heping Wang " ]	[]	[]	We study multivariate approximation in the average case setting with the error measured in the weighted L 2 norm. We consider algorithms that use standard information Λ std consisting of function values or general linear information Λ all consisting of arbitrary continuous linear functionals. We investigate the equivalences of various notions of algebraic and exponential tractability for Λ std and Λ all for the absolute error criterion, and show that the power of Λ std is the same as that of Λ all for all notions of algebraic and exponential tractability without any condition. Specifically, we solve Open Problems 116-118 and almost solve Open Problem 115 as posed by E.Novak and H.	10.1016/j.jco.2021.101618	[ "https://arxiv.org/pdf/2101.05200v1.pdf" ]	231,592,623	2101.05200	2a1f3157d654debd7128727ae1de203474870025	ON THE POWER OF STANDARD INFORMATION FOR TRACTABILITY FOR L 2 -APPROXIMATION IN THE AVERAGE CASE SETTING 12 Jan 2021 Heping Wang ON THE POWER OF STANDARD INFORMATION FOR TRACTABILITY FOR L 2 -APPROXIMATION IN THE AVERAGE CASE SETTING 12 Jan 2021Problems 116-118 and almost solve Open Problem 115 as posed by E.Novak and H.Woźniakowski in the book: Tractability of Multivariate Problems, Vol-ume III: Standard Information for Operators, EMS Tracts in Mathematics, Zürich, 2012. We study multivariate approximation in the average case setting with the error measured in the weighted L 2 norm. We consider algorithms that use standard information Λ std consisting of function values or general linear information Λ all consisting of arbitrary continuous linear functionals. We investigate the equivalences of various notions of algebraic and exponential tractability for Λ std and Λ all for the absolute error criterion, and show that the power of Λ std is the same as that of Λ all for all notions of algebraic and exponential tractability without any condition. Specifically, we solve Open Problems 116-118 and almost solve Open Problem 115 as posed by E.Novak and H. Introduction In this paper, we study multivariate approximation APP = {APP d } d∈N in the average case setting, where APP d : F d → G d with APP d f = f is the continuous embedding operator, F d is a separable Banach function space on D d equipped with a zero-mean Gaussian measure µ d , G d is a weighted L 2 space on D d , D d is a Lebesgue measurable subset of R d , and the dimension d is large or even huge. We consider algorithms that use finitely many information evaluations. Here information evaluation means continuous linear functional on F d (general linear information) or function value at some point (standard information). We use Λ all and Λ std to denote the class of all continuous linear functionals and the class of all function values, respectively. For a given error threshold ε ∈ (0, 1), the information complexity n(ε, d) is defined to be the minimal number of information evaluations for which the average case error of some algorithm is at most ε. Tractability is aimed at studying how the information complexity n(ε, d) depends on ε and d. There are two kinds of tractability based on polynomial convergence and exponential convergence. The algebraic tractability (ALG-tractability) describes how the information complexity n(ε, d) behaves as a function of d and ε −1 , while the exponential tractability (EXP-tractability) does as one of d and (1 + ln ε −1 ). The existing notions of tractability mainly include strong polynomial tractability (SPT), polynomial tractability (PT), quasi-polynomial tractability (QPT), weak tractability (WT), (s, t)-weak tractability ((s, t)-WT), and uniform weak tractability (UWT). In recent years the study of algebraic and exponential tractability has attracted much interest, and a great number of interesting results have been obtained (see [22,23,24,34,6,35,29,30,5,4,15,27,36,9,1,19,28] and the references therein). This paper is devoted to investigating the equivalences of various notions of algebraic and exponential tractability for Λ std and Λ all in the average case setting (see [24,Chapter 24]). The class Λ std is much smaller and much more practical, and is much more difficult to analyze than the class Λ all . Hence, it is very important to study the power of Λ std compared to Λ all . There are many paper devoted to this field. For example, for the randomized setting, see [24,33,16,11,2,16,3,20]; for the average case setting, see [24,7,18,38]; for the worst case setting, see [24,32,8,17,25,26,13,14,10,21,12]. In [7,24,38] the authors obtained the equivalences of various notions of algebraic and exponential tractability in the average case setting for Λ std and Λ all for the normalized error criterion without any condition. Meanwhile, for the absolute error criterion under some conditions on the initial error, the equivalences of ALG-SPT, ALG-PT, ALG-QPT, ALG-WT were also obtained in [24]. Xu obtained in [38] the equivalences of ALG-PT, ALG-QPT, ALG-WT for Λ all and Λ std under much weaker conditions. This gives a partial solution to Open problems 116-118 in [24]. Xu also obtained in [38] the equivalences of ALG-(s, t)-WT, ALG-UWT, and various notions of EXP-tractability under some conditions on the initial error. In this paper we obtain the equivalences of various notions of algebraic and exponential tractability for Λ all and Λ std in the average case setting for the absolute error criterion without any condition, which means the above conditions are unnecessary. This completely solves Open problems 116-118 in [24]. We also give an almost complete solution to Open Problem 115 in [24]. This paper is organized as follows. In Subsections 2.1 we introduce the approximation problem in the average case setting. The various notions of algebraic and exponential tractability are given in Subsection 2.2. Our main results Theorems 2.1-2.5 are stated in Subsection 2.3. In Section 3, we give the proofs of Theorems 2.1 and 2.2. After that, in Section 4 we show the equivalences of the notions of algebraic tractability for the absolute error criterion without any condition. The equivalence results for the notions of exponential tractability for the absolute error criterion are proved in Section 5. Preliminaries and Main Results Average case setting. For d ∈ N, let F d be a separable Banach space of d-variate real-valued functions on D d equipped with a zero-mean Gaussian measure µ d , G d = L 2 (D d , ρ d (x)dx) be a weighted L 2 space, where D d is a Borel measurable subset of R d with positive Lebesgue measure, ρ d is a probability density function on D d . We consider the multivariate approximation problem APP = {APP d } d∈N in the average case setting which is defined via the continuous linear operator (2.1) APP d : F d → G d with APP d f = f. We suppose that function value at some point x ∈ D d is well defined continuous linear functional on F d . That is, we suppose that Λ std ⊂ Λ all = (F d ) * , where (F d ) * is the dual space of F d . It is well known that, in the average case setting with the average being with respect to a zero-mean Gaussian measure, adaptive choice of the above information evaluations do not essentially help, see [31]. Hence, we can restrict our attention to nonadaptive algorithms, i.e., algorithms A n,d f of the form (2.2) A n,d f = φ n,d (L 1 (f ), L 2 (f ), . . . , L n (f )), where L i ∈ Λ, i = 1, . . . , n, Λ ∈ {Λ all , Λ std }, and φ n,d : R n → G d is an arbitrary measurable mapping from R n to G d . The average case error for the algorithm A n,d of the form (2.2) is defined as e avg (A n,d ) := F d APP d f − A n,d f 2 G d µ d (df ) 1/2 . The nth minimal average case error for Λ ∈ {Λ all , Λ std } is defined by e avg (n, d; Λ) := inf A n,d with Li∈Λ e avg (A n,d ), where the infimum is taken over all algorithms of the form (2.2). For n = 0, we use A 0,d = 0. We obtain the so-called initial error e avg (0, d) defined by e avg (0, d) := e avg (0, d; Λ all ) = e avg (0, d; Λ std ) = F d APP d f 2 G d µ d (df ) 1/2 . We set Γ d := (e avg (0, d; Λ all )) 2 = (e avg (0, d; Λ std )) 2 . It follows from [31,Chapter 6] and [24] that e avg (n, d; Λ all ) are described through the eigenvalues and the eigenvectors of the covariance operator C ν d : G d → G d of the induced measure ν d = µ d S −1 d of µ d . Here, µ d is a zero-mean Gaussian measure of F d , so that ν d is a zero-mean Gaussian measure on the Borel sets of G d . The operator C ν d is self-adjoint, non-negative definite, and the trace of C ν d is finite. Let (λ k,d , η k,d ) ∞ k=1 denote the eigenpairs of C ν d satisfying λ 1,d ≥ λ 2,d ≥ . . . λ n,d · · · ≥ 0. That is, {η k,d } ∞ k=1 is an orthonormal basis in G d , and C ν d η k,d = λ k,d η k,d , k ∈ N. From [31,24] we get that the nth minimal average case error is e avg (n, d; Λ all ) = ∞ k=n+1 λ k,d 1/2 , and it is achieved by the optimal algorithm A * n,d f = n k=1 f, η k,d G d η k,d . That is, (2.3) e avg (n, d; Λ all ) = F d f − A * n,d f 2 G d µ d (df ) 1/2 = ∞ k=n+1 λ k,d 1/2 . The trace of C ν d is just the square of the initial error e avg (0, d) given by trace(C ν d ) = Γ d = (e avg (0, d)) 2 = G d g 2 G d ν d (dg) = ∞ k=1 λ k,d < ∞. The information complexity can be studied using either the absolute error criterion (ABS) or the normalized error criterion (NOR). In the average case setting for ⋆ ∈ {ABS, NOR} and Λ ∈ {Λ all , Λ std }, we define the information complexity n ⋆ (ε, d; Λ) as (Γ d ) 1/2 , for ⋆=NOR. Since Λ std ⊂ Λ all , we get e avg (n, d; Λ all ) ≤ e avg (n, d; Λ std ). It follows that for ⋆ ∈ {ABS, NOR}, (2.5) n ⋆ (ε, d; Λ all ) ≤ n ⋆ (ε, d; Λ std ). Notions of tractability. In this subsection we briefly recall the various tractability notions in the average case setting. First we introduce all notions of algebraic tractability. Let APP = {APP d } d∈N , ⋆ ∈ {ABS, NOR}, and Λ ∈ {Λ all , Λ std }. In the average case setting for the class Λ, and for error criterion ⋆, we say that APP is • Algebraic strongly polynomially tractable (ALG-SPT) if there exist C > 0 and non-negative number p such that (2.6) n ⋆ (ε, d; Λ) ≤ Cε −p , for all ε ∈ (0, 1). The exponent ALG-p ⋆ (Λ) of ALG-SPT is defined as the infimum of p for which (2.6) holds; • Algebraic polynomially tractable (ALG-PT) if there exist C > 0 and nonnegative numbers p, q such that n ⋆ (ε, d; Λ) ≤ Cd q ε −p , for all d ∈ N, ε ∈ (0, 1); • Algebraic quasi-polynomially tractable (ALG-QPT) if there exist C > 0 and non-negative number t such that (2.7) n ⋆ (ε, d; Λ) ≤ C exp(t(1 + ln d)(1 + ln ε −1 )), for all d ∈ N, ε ∈ (0, 1). The exponent ALG-t ⋆ (Λ) of ALG-QPT is defined as the infimum of t for which (2.7) holds; • Algebraic uniformly weakly tractable (ALG-UWT) if lim ε −1 +d→∞ ln n ⋆ (ε, d; Λ) ε −α + d β = 0, for all α, β > 0; • Algebraic weakly tractable (ALG-WT) if lim ε −1 +d→∞ ln n ⋆ (ε, d; Λ) ε −1 + d = 0; • Algebraic (s, t)-weakly tractable (ALG-(s, t)-WT) for fixed s, t > 0 if lim ε −1 +d→∞ ln n ⋆ (ε, d; Λ) ε −s + d t = 0. Clearly, ALG-(1, 1)-WT is the same as ALG-WT. If APP is not ALG-WT, then APP is called intractable. If the nth minimal error decays faster than any polynomial and is exponentially convergent, then we should study tractability with ε −1 being replaced by (1 + ln 1 ε ), which is called exponential tractability. Recently, there have been many papers studying exponential tractability (see [5,4,37,28,15,9,1,19]). In the definitions of ALG-SPT, ALG-PT, ALG-QPT, ALG-UWT, ALG-WT, and ALG-(s, t)-WT, if we replace 1 ε by (1 + ln 1 ε ), we get the definitions of exponential strong polynomial tractability (EXP-SPT), exponential polynomial tractability (EXP-PT), exponential quasi-polynomial tractability (EXP-QPT), exponential uniform weak tractability (EXP-UWT), exponential weak tractability (EXP-WT), and exponential (s, t)-weak tractability (EXP-(s, t)-WT), respectively. We now give the above notions of exponential tractability in detail. Let APP = {APP d } d∈N , ⋆ ∈ {ABS, NOR}, and Λ ∈ {Λ all , Λ std }. In the average case setting for the class Λ, and for error criterion ⋆, we say that APP is • Exponential strongly polynomially tractable (EXP-SPT) if there exist C > 0 and non-negative number p such that (2.8) n ⋆ (ε, d; Λ) ≤ C(ln ε −1 + 1) p , for all ε ∈ (0, 1). The exponent EXP-p ⋆ (Λ) of EXP-SPT is defined as the infimum of p for which (2.8) holds; • Exponential polynomially tractable (EXP-PT) if there exist C > 0 and nonnegative numbers p, q such that n ⋆ (ε, d; Λ) ≤ Cd q (ln ε −1 + 1) p , for all d ∈ N, ε ∈ (0, 1); • Exponential quasi-polynomially tractable (EXP-QPT) if there exist C > 0 and non-negative number t such that (2.9) n ⋆ (ε, d; Λ) ≤ C exp(t(1 + ln d)(1 + ln(ln ε −1 + 1))), for all d ∈ N, ε ∈ (0, 1). The exponent EXP-t ⋆ (Λ) of EXP-QPT is defined as the infimum of t for which (2.9) holds; • Exponential uniformly weakly tractable (EXP-UWT) if lim ε −1 +d→∞ ln n ⋆ (ε, d; Λ) (1 + ln ε −1 ) α + d β = 0, for all α, β > 0; • Exponential weakly tractable (EXP-WT) if lim ε −1 +d→∞ ln n ⋆ (ε, d; Λ) 1 + ln ε −1 + d = 0; • Exponential (s, t)-weakly tractable (EXP-(s, t)-WT) for fixed s, t > 0 if lim ε −1 +d→∞ ln n ⋆ (ε, d; Λ) (1 + ln ε −1 ) s + d t = 0. Main results. We shall give main results of this paper in this subsection. We remark that for multivariate approximation problem results and proofs in the average case setting are in full analogy with ones in the randomized setting (see [20]). For the convenience of the reader, we provide details of all proofs. The authors in [7,24,38] used the mean value theorem and iterated Monte Carlo methods to obtain the relation between e avg (n, d; Λ std ) and e avg (n, d; Λ all ). We use the mean value theorem and the method used in [10,20] to get an inequality between e avg (n, d; Λ std ) and e avg (n, d; Λ all ). See the following theorem. Theorem 2.1. Let δ ∈ (0, 1), m, n ∈ N be such that m = n 48( √ 2 ln(2n) − ln δ) . Then we have (2.10) e avg (n, d; Λ std ) ≤ 1 + 4m n 1 2 1 √ 1 − δ e avg (m, d; Λ all ), where ⌊x⌋ denotes the largest integer not exceeding x. Based on Theorem 2.1, we obtain two relations between the information complexities n ⋆ (ε, d; Λ std ) and n ⋆ (ε, d; Λ all ) for ⋆ ∈ {ABS, NOR}. Theorem 2.2. For ⋆ ∈ {ABS, NOR} and ω > 0, we have (2.11) n ⋆ (ε, d; Λ std ) ≤ C ω n ⋆ ( ε 4 , d; Λ all ) + 1 1+ω , where C ω is a positive constant depending only on ω. Similarly, for sufficiently small ω, δ > 0 and ⋆ ∈ {ABS, NOR}, we have In the average case setting, for the normalized error criterion, [24,Theorems 24.10,24.12,and 24.6] gives the equivalences of ALG-PT (ALG-SPT), ALG-QPT, ALG-WT for Λ all and Λ std , and shows that the exponents of ALG-SPT and ALG-QPT for Λ all and Λ std are same; [38, Theorems 3.4 and 3.5] gives the equivalences of ALG-(s, t)-WT, ALG-UWT for Λ all and Λ std . (2.12) n ⋆ (ε, d; Λ std ) ≤ C ω,δ n ⋆ ( ε A δ , d; Λ all ) + 1 1+ω , where A δ := 1 + 1 12 ln 1 δ 1 2 1 √ 1−δ , C ω, For the absolute error criterion, [24, Theorems 24.11, 24.13, and 24.6] gives the equivalences of ALG-PT (ALG-SPT), ALG-QPT, ALG-WT for Λ all and Λ std under some conditions on the initial error. Novak and Woźniakowski posed Open problems 116-118 in [24] which ask whether the above conditions are necessary. Xu obtained in [38, Theorems 3.1-3.5] the equivalences of ALG-PT, ALG-QPT, ALG-WT, ALG-(s, t)-WT, ALG-UWT for Λ all and Λ std under much weaker conditions. This gives a partial solution to Open problems 116-118 in [24]. In this paper we obtain the equivalences of ALG-SPT, ALG-PT, ALG-QPT, ALG-WT, ALG-(s, t)-WT, ALG-UWT for Λ all and Λ std in the average case setting for the absolute error criterion without any condition, which means the above conditions are unnecessary. This solves Open problems 116-118 in [24]. See the following theorem. • The exponents ALG-p ABS (Λ) of ALG-SPT for Λ all and Λ std are same, and the exponents ALG-t ABS (Λ) of ALG-QPT for Λ all and Λ std are also same. For exponential convergence in the average case setting, we first give an almost complete solution to Open Problem 115 in [24]. In this paper we obtain the equivalences of EXP-SPT, EXP-PT, EXP-QPT, EXP-WT, EXP-(s, t)-WT, EXP-UWT for Λ all and Λ std in the average case setting for the absolute error criterion without any condition, which means the above conditions are unnecessary. We also show that the exponents of EXP-SPT and EXP-QPT for Λ all and Λ std are same for the normalized or absolute error criterion. See the following theorem. • for ⋆ ∈ {ABS, NOR}, the exponents EXP-p ⋆ (Λ) of EXP-SPT for Λ all and Λ std are same, and the exponents EXP-t ⋆ (Λ) of EXP-QPT for Λ all and Λ std are also same. Combining the obtained results in [24,38] with Theorems 2.3 and 2.4 we obtain the following corollary. • the exponents of SPT and QPT are the same for Λ all and Λ std , i.e., for ⋆ ∈ {ABS, NOR}, ALG−p ⋆ (Λ all ) = ALG−p ⋆ (Λ std ), ALG−t ⋆ (Λ all ) = ALG−t ⋆ (Λ std ), EXP−p ⋆ (Λ all ) = EXP−p ⋆ (Λ std ), EXP−t ⋆ (Λ all ) = EXP−t ⋆ (Λ std ). 3. Proofs of Theorems 2.1 and 2.2 Let us keep the notation of Subsection 2.1. For any m ∈ N, we define the functions h m,d (x) and ω m,d on D d by h m,d (x) := 1 m m j=1 \|η j,d (x)\| 2 , ω m,d (x) := h m,d (x) ρ d (x), where {η j,d } ∞ j=1 is an orthonormal basis in G d = L 2 (D d , ρ d (x)dx). Then ω m,d is a probability density function on D d , i.e., D d ω m,d (x) dx = 1. We define the corresponding probability measure µ m,d by µ m,d (A) = A ω m,d (x) dx, where A is a Borel subset of D d . We use the convention that 0 0 := 0. Then {η j,d } ∞ j=1 is an orthonormal system in L 2 (D d , µ m,d ), wherẽ η j,d := η j,d h m,d . For X = (x 1 , . . . , x n ) ∈ D n d , we use the following matrices (3.1) L m = L m (X) =      η 1,d (x 1 ) η 2,d (x 1 ) · · · η m,d (x 1 ) η 1,d (x 2 ) η 2,d (x 2 ) · · · η m,d (x 2 ) . . . . . . . . . η 1,d (x n ) η 2,d (x n ) · · · η m,d (x n )      and H m = 1 n L * m L m , where A * is the conjugate transpose of a matrix A. Note that N (m) := sup x∈D d m k=1 \| η k,d (x)\| 2 = m. According to [10, Propositions 5.1 and 3.1] we have the following results. P( H m − I m > 1 2 ) ≤ (2n) √ 2 exp − n 48m , where L m , H m are given by (3.1), I m is the identity matrix of order m, and L denotes the spectral norm (i.e. the largest singular value) of a matrix L. Furthermore, if H m − I m ≤ 1/2, then (3.2) ( L * m L m ) −1 ≤ 2 n . Remark 3.2. From Lemma 3.1 we immediately obtain (3.3) P H m − I m ≤ 1/2 ≥ 1 − δ if (3.4) m = n 48( √ 2 ln(2n) − ln δ) ≥ 1, holds, where ⌊x⌋ denotes the largest integer not exceeding x. Now let m, n ∈ N satisfy (3.4), x 1 , . . . , x n be independent and identically distributed sample points from D d that are distributed according to the probability measure µ m,d , and L m , H m be given by (3.1). We consider the conditional distribution given the event H m − I m ≤ 1/2 and the conditional expectation E(X H m − I m ≤ 1/2) = Hm−Im ≤1/2 X(x 1 , . . . , x n ) dµ m,d (x 1 ) . . . dµ m,d (x n ) P H m − I m ≤ 1/2 of a random variable X. If H m − I m ≤ 1/2 for some X = (x 1 , . . . , x n ) ∈ D n d , then L m = L m (X) has the full rank. The algorithm is a weighted least squares estimator (3.5) S m X f = arg min g∈Vm \|f (x i ) − g(x i )\| 2 h m,d (x i ) , which has a unique solution, where V m := span{η 1,d , . . . , η m,d }. It follows that S m X f = f whenever f ∈ V m . Algorithm Weighted least squares regression. Input: X = (x 1 , . . . , x n ) ∈ D n d set of distinct sampling nodes, f = f (x 1 ) √ h m,d (x 1 ) , . . . , f (x n ) √ h m,d (x n ) T weighted samples of f evaluted at the nodes from X, m ∈ N m < n such that the matrix L m := L m (X) from (3.1) has full (column) rank. Solve the over-determined linear system L m ( c 1 , · · · , c m ) T =f via least square, i.e., compute ( c 1 , · · · , c m ) T = ( L * m L m ) −1 L * mf . Output: c = ( c 1 , · · · , c m ) T ∈ C m coefficients of the approximant S m X (f ) := m k=1 c k η k,d which is the unique solution of (3.5). Proof of Theorem 2.1. We use the above notation. Let m, n ∈ N satisfy (3.4), x 1 , . . . , x n be independent and identically distributed sample points from D d that are distributed according to the probability measure µ m,d , H m − I m ≤ 1/2, and S m X (f ) be defined as above. We estimate f − S m X (f ) 2 G d for f ∈ F d . We set H d = L 2 (D d , µ m,d ). We recall that {η j,d } ∞ j=1 is an orthonormal basis in G d = L 2 (D d , ρ d (x)dx), and hence {η j,d } ∞ j=1 is an orthonormal system in H d = L 2 (D d , µ m,d ), wherẽ η j,d := η j,d h m,d , η j,d ,η k,d H d = η j,d , η k,d , G d = δ i,j . For f ∈ F d ⊂ G d , we have f = ∞ k=1 f, η k,d G d η k,d . We note that f − A * m,d (f ) is orthogonal to the space V m with respect to the inner product ·, · G d , and A * m,d (f ) − S m X (f ) = S m X (f − A * m,d (f )) ∈ V m := span{η 1,d , . . . , η m,d }, where A * m,d (f ) = m k=1 f, η k,d G d η k,d . It follows that f − S m X (f ) 2 G d = f − A * m,d (f ) 2 G d + S m X (f − A * m,d (f )) 2 G d = g 2 G d + S m X (g) 2 G d , where g := f − A * m,d (f ). We recall that S m X (g) = m k=1 c k η k,d , c = ( c 1 , . . . , c m ) T = ( L * m L m ) −1 ( L m ) * g, where g := ( g(x 1 ), · · · , g(x n )) T , g := g h m,d . Since {η k,d } ∞ k=1 is an orthonormal system in G d , we get S m X (g) 2 G d = c 2 2 = (( L m ) * L m ) −1 ( L m ) * g 2 2 ≤ (( L m ) * L m ) −1 · ( L m ) * g 2 2 ≤ 4 n 2 ( L m ) * g 2 2 , where · 2 is the Euclidean norm of a vector. We have ( L m ) * g 2 2 = m k=1 n j=1 η k,d (x j ) · g(x j ) 2 = m k=1 n j=1 n i=1 η k,d (x j ) g(x j ) η k,d (x i ) g(x i ). It follows that J = Hm−Im ≤ 1 2 ( L m ) * g 2 2 dµ m,d (x 1 ) . . . dµ m,d (x n ) ≤ D n d ( L m ) * g 2 2 dµ m,d (x 1 ) . . . dµ m,d (x n ) ≤ m k=1 n i,j=1 D n d η k,d (x j ) g(x j ) η k,d (x i ) g(x i ) dµ m,d (x 1 ) . . . dµ m,d (x n ), Noting that for i = j and 1 ≤ k ≤ m, D n d η k,d (x j ) g(x j ) η k,d (x i ) g(x i ) dµ m,d (x 1 ) . . . dµ m,d (x n ) = \| g, η k,d H d \| 2 = \| g, η k,d G d \| 2 = 0, and h m,d (x) = 1 m m k=1 \|η k,d (x)\| 2 , we continue to get J ≤ n m k=1 η k,d · g 2 H d = n m k=1 D n d \| g(x) η k,d (x)\| 2 ρ d (x)h m,d (x) dx = n m k=1 D n d \|g(x)η k,d (x)\| 2 h m,d (x) ρ d (x) dx = n D n d m\|g(x)\| 2 ρ d (x) dx = nm · g 2 G d . Hence, we have Hm−Im ≤ 1 2 f − S m X (f ) 2 G d dµ m,d (x 1 ) . . . dµ m,d (x n ) ≤ g 2 G d + 4 n 2 J ≤ (1 + 4m n ) g 2 G d = (1 + 4m n ) f − A * m,d (f ) 2 G d .E F d f − S m X (f ) 2 G d µ d (df ) H m − I m ≤ 1/2 = F d E f − S m X (f ) 2 G d H m − I m ≤ 1/2 µ d (df ) = F d Hm−Im ≤ 1 2 f − S m X f 2 G d dµ m,d (x 1 ) . . . dµ m,d (x n ) µ d (df ) P( H m − I m ≤ 1 2 ) ≤ 1 + 4m n 1 1 − δ F d f − A * m,d (f ) 2 G d µ d (df ) = 1 + 4m n 1 1 − δ (e avg (m, d; Λ all )) 2 . By the mean value theorem, we conclude that there are sample points X * = {x 1 * , . . . , x n * } such that H * m − I m ≤ 1 2 and F d f − S m X * (f ) 2 G d µ(df ) = E F d f − S m X (f ) 2 G d µ d (df ) H m − I m ≤ 1/2 . We obtain that (e avg (n, d; Λ std )) 2 ≤ F d f − S m X * (f ) 2 G d µ(df ) ≤ 1 + 4m n 1 1 − δ (e avg (m, d; Λ all )) 2 . This completes the proof of Theorem 2.1. We stress that Theorem 2.1 is not constructive since we do not know how to choose the sample points X * = {x 1 * , . . . , x n * }. We only know that there exist X * = {x 1 * , . . . , x n * } for which the average case error of the weighted least squares algorithm S m X * enjoys the average case error bound of Theorem 2.1. Proof of Theorem 2.2. Applying Theorem 2.1 with δ = 1 2 √ 2 , we obtain (3.7) e avg (n, d; Λ std ) ≤ 1 + 4m n 1 2 2 √ 2 2 √ 2 − 1 1 2 e avg (m, d; Λ all ), where m, n ∈ N, and m = n 48 √ 2 ln(4n) . Since 1 + 4m n ≤ 1 + 1 √ 1 − δ e avg (m, d; Λ all ) ≤ 1 + 1 12 √ 2 ln(2n) + ln 1 δ 1 2 1 √ 1 − δ e avg (m, d; Λ all ) ≤ 1 + 1 12 ln 1 δ 1 2 1 √ 1 − δ e avg (m, d; Λ all ) = A δ e avg (m, d; Λ all ), where A δ = 1 + 1 12 ln 1 δ 1 2 1 √ 1−δ . Using the same method used in the proof of (3.9), we have n ⋆ (ε, d; Λ std ) ≤ min n \| e avg (m, d; Λ all ) ≤ ε A δ CRI d . We note that n ≤ 48 √ 2 ln(2n) + ln 1 δ (m + 1). Taking logarithm on both sides, and using the inequalities ln x ≤ x 4 for x ≥ 9 and a + b ≤ ab for a, b ≥ 2, we get ln n ≤ ln 48 + ln √ 2 ln(2n) + ln 1 δ + ln(m + 1) + ln 1 δ n ⋆ ( ε A δ , d; Λ all ) + 1 . Since for sufficiently small ω, δ > 0, there holds sup x≥1 48(4(ln 48 + ln ln 1 δ + ln x) + ln 1 δ ) x ω = C ω,δ < +∞, we get (2.12). Theorem 2.2 is proved. Equivalence results of algebraic tractability First we consider the equivalences of ALG-PT and ALG-SPT for Λ std and Λ all in the average case setting. The equivalent results for the normalized error criterion can be found in [7] and [24,Theorem 24.10]. For the absolute error criterion, [24,Theorem 24.11] shows the equivalence of ALG-PT under the condition (4.2) Γ d ≤ exp(Cd v ) for all d ∈ N, some C > 0, and some v ≥ 0. We obtain the following equivalent results of ALG-PT and ALG-SPT without any condition. Hence, the condition (4.1) or (4.2) is unnecessary. This solves Open Problem 117 as posed by Novak and Woźniakowski in [24]. • ALG-PT for Λ all is equivalent to ALG-PT for Λ std . • ALG-SPT for Λ all is equivalent to ALG-SPT for Λ std . In this case, the exponents of ALG-SPT for Λ all and Λ std are the same. Proof. It follows from (2.5) that ALG-PT (ALG-SPT) for Λ std means ALG-PT (ALG-SPT) for Λ all . It suffices to show that ALG-PT (ALG-SPT) for Λ all means that ALG-PT (ALG-SPT) for Λ std . Suppose that ALG-PT holds for Λ all . Then there exist C ≥ 1 and non-negative p, q such that (4.3) n ABS (ε, d; Λ all ) ≤ Cd q ε −p , for all d ∈ N, ε ∈ (0, 1). It follows from (2.11) and (4.3) that n ABS (ε, d; Λ std ) ≤ C ω Cd q ( ε 4 ) −p + 1 1+ω ≤ C ω (2C4 p ) 1+ω d q(1+ω) ε −p(1+ω) , which means that ALG-PT holds for Λ std . If ALG-SPT holds for Λ all , then (4.3) holds with q = 0. We obtain n ABS (ε, d; Λ std ) ≤ C ω (2C4 p ) 1+ω ε −p(1+ω) , which means that ALG-SPT holds for Λ std . Furthermore, since ω can be arbitrary small, we get ALG−p ABS (Λ std ) ≤ ALG−p ABS (Λ all ) ≤ ALG−p ABS (Λ std ), which means that the exponents of ALG-SPT for Λ all and Λ std are the same. This completes the proof of Theorem 4.1. Next we consider the equivalence of ALG-QPT for Λ std and Λ all in the average case setting. The result for the normalized error criterion can be found in [24,Theorem 24.12]. For the absolute error criterion, [24,Theorem 24.13] shows the equivalence of ALG-QPT under the condition lim sup d→∞ Γ d < ∞. Xu obtained in [38, Theorem 3.2] the equivalence of ALG-QPT under the weaker condition (4.2). We obtain the following equivalent results of ALG-QPT without any condition. Hence, the condition (4.2) is unnecessary. This solves Open Problem 118 as posed by Novak and Woźniakowski in [24]. Theorem 4.2. We consider the problem APP = {APP d } d∈N in the average case setting for the absolute error criterion. Then, ALG-QPT for Λ all is equivalent to ALG-QPT for Λ std . In this case, the exponents of ALG-QPT for Λ all and Λ std are the same. Proof. Similar to the proof of Theorem 4.1, it is enough to prove that ALG-QPT for Λ all implies ALG-QPT for Λ std . Suppose that ALG-QPT holds for Λ all . Then there exist C ≥ 1 and non-negative t such that (4.4) n ABS (ε, d; Λ all ) ≤ C exp(t(1 + ln d)(1 + ln ε −1 )), for all d ∈ N, ε ∈ (0, 1). For sufficiently small δ > 0 and ω > 0, it follows from (2.12) and (4.4) that n ran,⋆ (ε, d; Λ std ) ≤ C ω,δ n wor,⋆ ( ε A δ , d; Λ all ) + 1 1+ω ≤ C ω,δ C exp t(1 + ln d) 1 + ln ε A δ −1 ) + 1 1+ω ≤ C ω,δ (2C) 1+ω exp (1 + ω)t(1 + ln d)(1 + ln A δ + ln ε −1 ) ≤ C ω,δ (2C) 1+ω exp (1 + ω)t(1 + ln A δ )(1 + ln d)(1 + ln ε −1 ) , where t * = (1 + ω)(1 + ln A δ )t, A δ = 1 + 1 12 ln 1 δ 1 2 1 √ 1−δ . This implies that ALG-QPT holds for Λ std . Furthermore, taking the infimum over t for which (4.4) holds, and noting that lim (1 + ω)(1 + ln A δ ) = 1, we get that ALG−t ABS (Λ std ) ≤ ALG−t ABS (Λ all ). It follows from (2.5) that ALG−t ABS (Λ std ) ≤ ALG−t ABS (Λ all ) ≤ ALG−t ABS (Λ std ), which means that the exponents ALG-t ABS (Λ all ) and ALG-t ABS (Λ std ) are equal if ALG-QPT holds for Λ all . This completes the proof of Theorem 4.2. Now we consider the equivalence of ALG-WT for Λ std and Λ all in the average case setting . The result for the normalized error criterion can be found in [24,Theorem 24.6]. For the absolute error criterion, [24,Theorem 24.6] shows the equivalence of ALG-WT under the condition We obtain the following equivalent results of ALG-WT without any condition. Hence, the condition (4.5) is unnecessary. This solves Open Problem 116 as posed by Novak and Woźniakowski in [24]. We obtain the following equivalent results of ALG-(s, t)-WT for fixed s, t > 0 and ALG-UWT for the absolute error criterion without any condition. Hence, the condition (4.6) or (4.7) is unnecessary. Proof. Again it is enough to prove that ALG-(s, t)-WT for Λ all implies ALG-(s, t)-WT for Λ std . Suppose that ALG-(s, t)-WT holds for Λ all . Then we have (4.8) lim ε −1 +d→∞ ln n ABS (ε, d; Λ all ) ε −s + d t = 0. It follows from (2.11) that for ω > 0, ln n ABS (ε, d; Λ std ) ε −s + d t ≤ ln C ω n ABS (ε/4, d; Λ all ) + 1 1+ω ε −s + d t ≤ ln(C ω 2 1+ω ) ε −s + d t + 4 s (1 + ω) ln n ABS (ε/4, d; Λ all ) (ε/4) −s + d t . Since ε −1 + d → ∞ is equivalent to ε −s + d t → ∞, by (4.8) we get that lim ε −1 +d→∞ ln(C ω 2 1+ω ) ε −s + d t = 0 and lim ε −1 +d→∞ ln n ABS (ε/4, d; Λ all ) (ε/4) −s + d t = 0. We obtain lim ε −1 +d→∞ ln n ABS (ε, d; Λ std ) ε −s + d t = 0, which implies ALG-(s, t)-WT for Λ std . The proof of Theorem 4.4 is finished. Theorem 4.5. We consider the problem APP = {APP d } d∈N in the average case setting for the absolute error criterion. Then ALG-UWT for Λ all is equivalent to ALG-UWT for Λ std . Proof. By definition we know that APP is ALG-UWT if and only if APP is ALG-(s, t)-WT for all s, t > 0. Since by Theorem 4.4 ALG-(s, t)-WT for Λ std is equivalent to ALG-(s, t)-WT for Λ all for all s, t > 0, we get the equivalence of ALG-UWT for Λ std and Λ all . Theorem 4.5 is proved. Proof of Theorem 2.3. Theorem 2.3 follows from Theorems 4.1-4.5 immediately. Equivalence results of exponential tractability First we consider exponential convergence. Assume that there exist two constants A ≥ 1 and q ∈ (0, 1) such that (5.1) λ n,d ≤ Aq n e avg (0, d; Λ all ) = Aq n Γ d . It follows that e avg (n, d; Λ all ) ≤ A 1 − q q n+1 Γ d . Novak and Woźniakowski proved in [24,Corollary 24.5] that there exist two constants C 1 ≥ 1 and q 1 ∈ (q, 1) independent of d and n such that (5.2) e avg (n, d; Λ std ) ≤ C 1 A 1 − q q √ n 1 Γ d . If A, q in (5.1) are independent of d, then n NOR (ε, d; Λ all ) ≤ C 2 (ln ε −1 + 1), and n NOR (ε, d; Λ std ) ≤ C 3 (ln ε −1 + 1) 2 . Novak and Woźniakowski posed the following Open Problem 115: (1) Verify if the upper bound in (5.2) can be improved. (2) Find the smallest p for which there holds n NOR (ε, d; Λ std ) ≤ C 4 (ln ε −1 + 1) p . We know that p ≤ 2, and if (5.1) is sharp then p ≥ 1. The following theorem gives a confirmative solution to Open Problem 115 (1). We improve enormously the upper bound q √ n 1 in (5.2) to q n ln(4n) 2 in (5.5), where q 1 , q 2 ∈ (q, 1). e avg (n, d; Λ std ) ≤ 4A 1 − q q n 48 √ 2 ln(4n) +1 Γ d ≤ 4A 1 − q q n 48 √ 2 ln(4n) Γ d = 4A 1 − q q n ln(4n) 2 e avg (0, d; Λ all ). This completes the proof of Theorem 5.1. Now we consider the equivalences of various notions of exponential tractability for Λ std and Λ all for the absolute error criterion in the average case setting. First we consider the equivalences of EXP-PT and EXP-SPT for Λ std and Λ all . The results for the normalized error criterion can be found in [38,Theorem 4.1]. For the absolute error criterion, [38,Theorem 4.1] shows the equivalences of EXP-PT and EXP-SPT under the condition (4.2). We obtain the following equivalent results of EXP-PT and EXP-SPT without any condition. Theorem 5.2. We consider the problem APP = {APP d } d∈N in the average case setting. Then • for the absolute error criterion, EXP-PT (EXP-SPT) for Λ all is equivalent to EXP-PT (EXP-SPT) for Λ std ; • if EXP-SPT holds for Λ all for the absolute or normalized error criterion, then the exponents of EXP-SPT for Λ all and Λ std are the same. Proof. Again, it is enough to prove that EXP-PT for Λ all implies EXP-PT for Λ std for the absolute error criterion. Suppose that EXP-PT holds for Λ all . Then there exist C ≥ 1 and non-negative p, q such that (5.6) n ABS (ε, d; Λ all ) ≤ Cd q (ln ε −1 + 1) p , for all d ∈ N, ε ∈ (0, 1). It follows from (2.11) and (5.6) that n ABS (ε, d; Λ std ) ≤ C ω Cd q (ln( ε 4 ) −1 + 1) p + 1 1+ω ≤ C ω (2C) 1+ω (1 + ln 4) p(1+ω) d q(1+ω) (ln ε −1 + 1) p(1+ω) , which means that EXP-PT holds for Λ std . If EXP-SPT holds for Λ all , then (5.6) holds with q = 0. We obtain n ABS (ε, d; Λ std ) ≤ C ω (2C) 1+ω (1 + ln 4) p(1+ω) (ln ε −1 + 1) p(1+ω) , which means that EXP-SPT holds for Λ std . Furthermore, if EXP-SPT holds for Λ all for the absolute or normalized error criterion and p * = EXP−p ⋆ (Λ all ) for ⋆ ∈ {ABS, NOR}, then for any ε > 0, there is a constant C ε ≥ 1 for which n ⋆ (ε, d; Λ all ) ≤ C ε (ln ε −1 + 1) p * +ε holds. Using the same method, we get n ⋆ (ε, d; Λ std ) ≤ C ω (2C ε ) 1+ω (1 + ln 4) (p * +ε)(1+ω) (ln ε −1 + 1) (p * +ε)(1+ω) , Noting that ε, ω can be arbitrary small, we have for ⋆ ∈ {ABS, NOR}, We obtain the following equivalent results of EXP-QPT without any condition. EXP−p ⋆ (Λ std ) ≤ EXP−p ⋆ (Λ all ) ≤ EXP−p ⋆ (Λ std ), Theorem 5.4. We consider the problem APP = {APP d } d∈N in the average case setting. Then, for the absolute error criterion EXP-QPT for Λ all is equivalent to EXP-QPT for Λ std . If EXP-QPT holds for Λ all for the absolute or normalized error criterion, then the exponents of EXP-QPT for Λ all and Λ std are the same. Proof. Again, it is enough to prove that EXP-QPT for Λ all implies EXP-QPT for Λ std for the absolute error criterion. Suppose that EXP-QPT holds for Λ all for the absolute or normalized error criterion. Then there exist C ≥ 1 and non-negative t such that for ⋆ ∈ {ABS, NOR}, (5.7) n ⋆ (ε, d; Λ all ) ≤ C exp(t(1 + ln d)(1 + ln(ln ε −1 + 1))), for all d ∈ N, ε ∈ (0, 1). For sufficiently small ω > 0 and δ > 0, it follows from (2.12) and (5.7) that n ⋆ (ε, d; Λ std ) ≤ C ω,δ n ⋆ ( ε A δ , d; Λ all ) + 1 1+ω ≤ C ω,δ C exp t(1 + ln d) 1 + ln(ln ε −1 + ln A δ + 1)) + 1 1+ω ≤ C ω,δ (2C) 1+ω exp (1 + ω)t(1 + ln d)(1 + ln(ln A δ + 1) + ln(ln ε −1 + 1)) ≤ C ω,δ (2C) 1+ω exp t * (1 + ln d)(1 + ln(ln ε −1 + 1)) , (5.8) where t * = (1 + ω)(1 + ln(ln A δ + 1))t and A δ = 1 + 1 12 ln 1 The inequality (5.8) with ⋆ = ABS implies that EXP-QPT holds for Λ std for the absolute error criterion. Next, we suppose that EXP-QPT holds for Λ all for the absolute or normalized error criterion. Taking the infimum over t for which (5.7) holds, and noting that lim (δ,ω)→(0,0) (1 + ω)(1 + ln(ln A δ + 1)) = 1, by (2.5) we obtain that EXP−t ⋆ (Λ std ) ≤ EXP−t ⋆ (Λ all ) ≤ EXP−t ⋆ (Λ std ). which means that the exponents EXP-t ⋆ (Λ all ) and EXP-t ⋆ (Λ std ) are equal. This completes the proof of Theorem 5.4. Next, we consider the equivalences of EXP-(s, t)-WT and EXP-WT for Λ std and Λ all in the average case setting. The results for the normalized error criterion can be found in [38,Theorems 4.3 and 4.4]. For the absolute error criterion, [38,Theorem 4.3] shows the equivalence of EXP-WT under the condition(4.5). Meanwhiles, [38,Theorem 4.4] shows the equivalence of EXP-(s, t)-WT under the condition (4.6). We obtain the following equivalent results of EXP-(s, t)-WT and EXP-WT for the absolute error criterion without any condition. Theorem 5.5. We consider the problem APP = {APP d } d∈N in the average case setting for the absolute error criterion. Then for fixed s, t > 0, EXP-(s, t)-WT for Λ all is equivalent to EXP-(s, t)-WT for Λ std . Specifically, EXP-WT for Λ all is equivalent to EXP-WT for Λ std . Proof. Again, it is enough to prove that EXP-(s, t)-WT for Λ all implies EXP-(s, t)-WT for Λ std . Suppose that EXP-(s, t)-WT holds for Λ all . Then we have (5.9) lim ε −1 +d→∞ ln n ABS (ε, d; Λ all ) (1 + ln ε −1 ) s + d t = 0. It follows from (2.11) that for ω > 0, ln n ABS (ε, d; Λ std ) (1 + ln ε −1 ) s + d t ≤ ln C ω n ABS (ε/4, d; Λ all ) + 1 1+ω (1 + ln ε −1 ) s + d t ≤ ln(C ω 2 1+ω ) (1 + ln ε −1 ) s + d t + (1 + ln 4) s (1 + ω) ln n ABS (ε/4, d; Λ all ) (1 + ln(ε/4) −1 ) s + d t . Since ε −1 + d → ∞ is equivalent to (1 + ln ε −1 ) s + d t → ∞, by (5.9) we get that lim ε −1 +d→∞ ln(C ω 2 1+ω ) (1 + ln ε −1 ) s + d t = 0 and lim ε −1 +d→∞ ln n ABS (ε/4, d; Λ all ) (1 + ln(ε/4) −1 ) s + d t = 0. We obtain lim ε −1 +d→∞ ln n ABS (ε, d; Λ std ) (ln ε −1 ) s + d t = 0, which implies that EXP-(s, t)-WT holds for Λ std . Specifically, EXP-WT is just EXP-(s, t)-WT with s = t = 1. This completes the proof of Theorem 5.5. Finally, we consider the equivalences of EXP-UWT for Λ std and Λ all in the average case setting. The results for the normalized error criterion can be found in [38,Theorems 4.5]. For the absolute error criterion, [38,Theorem 4.5] shows the equivalence of EXP-UWT under the condition (4.7). We obtain the following equivalent result of EXP-UWT for the absolute error criterion without any condition. Theorem 5.6. We consider the problem APP = {APP d } d∈N in the average case setting for the absolute error criterion. Then, EXP-UWT for Λ all is equivalent to EXP-UWT for Λ std . Proof. By definition we know that APP is EXP-UWT if and only if APP is EXP-(s, t)-WT for all s, t > 0. Then Theorem 5.6 follows from Theorem 5.5 immediately. Proof of Theorem 2.4. Theorem 2.4 follows from Theorems 5.2 and 5.4-5.6 immediately. ⋆ (ε, d; Λ) := n avg,⋆ (ε, d; Λ) := inf{n e avg (n, d; Λ) ≤ ε CRI d }, δ is a positive constant depending only on ω and δ. Theorem 2 . 3 . 23Consider the problem APP = {APP d } d∈N in the average case setting for the absolute error criterion. Then • ALG-SPT, ALG-PT, ALG-QPT, ALG-WT, ALG-(s, t)-WT, ALG-UWT for Λ all is equivalent to ALG-SPT, ALG-PT, ALG-QPT, ALG-WT, ALG-(s, t)-WT, ALG-UWT for Λ std ; In the average case setting for the normalized error criterion, Xu obtained in [38, Theorems 4.1-4.5] the equivalences of EXP-SPT, EXP-PT, EXP-QPT, EXP-WT, EXP-(s, t)-WT, EXP-UWT for Λ all and Λ std , however, he did not show that the exponents of EXP-SPT and EXP-QPT for Λ all and Λ std are same. For the absolute error criterion, Xu also obtained in [38, Theorems 4.1-4.5] the equivalences of EXP-SPT, EXP-PT, EXP-QPT, EXP-WT, EXP-(s, t)-WT, EXP-UWT for Λ all and Λ std under weak conditions on the initial error. Theorem 2 . 4 . 24Consider the problem APP = {APP d } d∈N in the average case setting. Then • for the absolute error criterion, EXP-SPT, EXP-PT, EXP-QPT, EXP-WT, EXP-(s, t)-WT, EXP-UWT for Λ all is equivalent to EXP-SPT, EXP-PT, EXP-QPT, EXP-WT, EXP-(s, t)-WT, EXP-UWT for Λ std ; Corollary 2 . 5 . 25Consider the approximation problem APP = {APP d } d∈N for the absolute or normalized error criterion in the average case setting. Then • ALG-SPT, ALG-PT, ALG-QPT, ALG-WT, ALG-(s, t)-WT, ALG-UWT for Λ all is equivalent to ALG-SPT, ALG-PT, ALG-QPT, ALG-WT, ALG-(s, t)-WT, ALG-UWT for Λ std ; • EXP-SPT, EXP-PT, EXP-QPT, EXP-WT, EXP-(s, t)-WT, EXP-UWT for Λ all is equivalent to EXP-SPT, EXP-PT, EXP-QPT, EXP-WT, EXP-(s, t)-WT, EXP-UWT for Λ std ; Lemma 3. 1 . 1Let n, m ∈ N. Let x 1 , . . . , x n ∈ D d be drawn independently and identically distributed at random with respect to the probability measure µ m,d . Then it holds that ≤ Cd v for all d ∈ N, some C > 0, and some v ≥ 0, and the equivalence of ALG-SPT under the condition (4.1) with v = 0. Xu obtained in [38, Theorem 3.1] the equivalence of ALG-PT under the weaker condition Theorem 4 . 1 . 41We consider the problem APP = {APP d } d∈N in the average case setting for the absolute error criterion. Then, in[38, Theorem 3.3] the equivalence of ALG-QPT under the much weaker condition. Theorem 4 . 3 . 43We consider the problem APP = {APP d } d∈N in the average case setting for the absolute error criterion. Then, ALG-WT for Λ all is equivalent to ALG-WT for Λ std .Proof. The proof is identical to the proof of Theorem 4.4 with s = t = 1 for the absolute error criterion. We omit the details.Finally, we consider the equivalences of ALG-(s, t)-WT and ALG-UWT for Λ std and Λ all in the average case setting. The results for the normalized error criterion can be found in[38, Theorems 3.4 and 3.5]. For the absolute error criterion,[38, Theorem 3.4] shows the equivalence of ALG-(s, t)-WT under the Theorem 4. 4 . 4We consider the problem APP = {APP d } d∈N in the average case setting for the absolute error criterion. Then for fixed s, t > 0, ALG-(s, t)-WT for Λ all is equivalent to ALG-(s, t)-WT for Λ std . a + b) ≤ ln(1 + a) + ln(1 + b), a, b ≥ 0. It follows that n ⋆ (ε, d; Λ std ) = min n e avg (n, d; Λ std ) ≤ εCRI d ≤ min n 4e avg (m, d; Λ all ) ≤ εCRI d1 12 √ 2 ln(4n) ≤ 2, by (3.7) we get (3.8) e avg (n, d; Λ std ) ≤ 4e avg (m, d; Λ all ). = min n \| e avg (m, d; Λ all ) ≤ ε 4 CRI d . (3.9) We note that m = n 48 √ 2 ln(4n) ≥ n 48 √ 2 ln(4n) − 1. This inequality is equivalent to (3.10) 4n ≤ 192 √ 2(m + 1) ln(4n). Taking logarithm on both sides of (3.10), and using the inequality ln x ≤ 1 2 x for x ≥ 1, we get ln(4n) ≤ ln(m + 1) + ln(192 √ 2) + ln ln(4n), and 1 2 ln(4n) ≤ ln(4n) − ln ln(4n) ≤ ln(m + 1) + ln(192 √ 2). It follows from (3.10) that (3.11) n ≤ 96 √ 2(m + 1)(ln(m + 1) + ln(192 √ 2)). By (3.9) and (3.11) we obtain n ⋆ (ε, d; Λ std ) ≤ 96 √ 2 n ⋆ ( ε 4 , d; Λ all ) + 1 ln n ⋆ ( ε 4 , d; Λ all ) + 1 + ln(192 √ 2) . (3.12) Since for any ω > 0, sup x≥1 96 √ 2(ln x + ln(192 √ 2)) x ω = C ω < +∞, we obtain (2.11). For sufficiently small δ > 0 and m, n ∈ N satisfying m = n 48( √ 2 ln(2n) − ln δ) , by Theorem 2.1 we have e avg (n, d; Λ std ) ≤ 1 + 4m n 1 2 Theorem 5.1. Let m, n ∈ N and(5.3) m = n 48 √ 2 ln(4n) . Then we have (5.4) e avg (n, d; Λ std ) ≤ 4e avg (m, d; Λ all ). Specifically, if (5.1) holds, then we have (5.5) e avg (n, d; Λ std ) ≤ 4A 1 − q q n ln(4n) 2 e avg (0, d; Λ all ) , where q 2 = q 1 48 √ 2 ∈ (q, 1). Proof. Inequality (5.4) is just (3.8), which has been proved. If (5.1) holds, then by (5.3) and (5.4) we get which means that the exponents of EXP-SPT for Λ all and Λ std are the same. This completes the proof of Theorem 5.2.Remark 5.3. We remark that if (5.1) holds with A, q independent of d, then the problem APP is EXP-SPT for Λ all in the average case setting for the normalized error criterion, and the exponent EXP−p NOR (Λ all ) ≤ 1.If (5.1) is sharp, then EXP−p NOR (Λ all ) = 1.Open Problem 115 (2) is equivalent to finding the exponent EXP−p NOR (Λ std ) of EXP-SPT. By Theorem 5.2 we obtain that if (5.1) holds, then EXP−p NOR (Λ std ) ≤ 1, and if (5.1) is sharp, then EXP−p NOR (Λ std ) = 1.This solves Open Problem 115 (2) as posed by Novak and Woźniakowski in[24].Next we consider the equivalence of EXP-QPT for Λ std and Λ all in the average case setting. The result for the normalized error criterion can be found in[38,. For the absolute error criterion,[38, Theorem 4.2] shows the equivalence of EXP-QPT under the condition (4.2). Acknowledgment This work was supported by the National Natural Science Foundation of China (Project no. 11671271). 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Email address: [email protected] School of Mathematical Sciences, Capital Normal University, Beijing 100048, China. Email address: [email protected]	[]
[ "A BALANCED EXCITED RANDOM WALK", "A BALANCED EXCITED RANDOM WALK" ]	[ "Itai Benjamini ", "Gady Kozma ", "Bruno Schapira " ]	[]	[]	The following random process on Z 4 is studied. At first visit to a site, the two first coordinates perform a (2-dimensional) simple random walk step. At further visits, it is the last two coordinates which perform a simple random walk step. We prove that this process is almost surely transient. The lower dimensional versions are discussed and various generalizations and related questions are proposed.	10.1016/j.crma.2011.02.018	[ "https://arxiv.org/pdf/1009.0741v1.pdf" ]	119,167,689	1009.0741	829f5283e6cdee350711eb498f0175aa07338fe2	A BALANCED EXCITED RANDOM WALK 3 Sep 2010 Itai Benjamini Gady Kozma Bruno Schapira A BALANCED EXCITED RANDOM WALK 3 Sep 2010 The following random process on Z 4 is studied. At first visit to a site, the two first coordinates perform a (2-dimensional) simple random walk step. At further visits, it is the last two coordinates which perform a simple random walk step. We prove that this process is almost surely transient. The lower dimensional versions are discussed and various generalizations and related questions are proposed. Introduction Excited random walk as defined by Benjamini and Wilson [BW] has a bias in some fixed direction, a feature which is highly useful in its analysis. See e.g. [MPRV] and references within. Attempts to relax the dependence of the proof structure on monotonicity resulted in a number of works where the walker has competing drifts. See [ABK, KZ, H]. One motivation was to get closer to standard models of reinforced random walks on Z d , which are symmetric in nature. We think for instance on the question of recurrence vs. transience of 1-reinforced random walks, which is still widely open (see the surveys [MR] and [Pem]). With the same goal in mind, we started exploring excited-like models where the walker is in addition also a martingale or a bounded perturbation of one, and posed some questions in 2007 [K], which, it seems, are all still open. Progress on this kind of models was achieved in [KRS], but there the laws were not nearest-neighbors. Here we describe and solve one such model of a nearest-neighbor walk in 4 dimensions. We describe a general form of the model in any dimension d ≥ 2, but we will actually only deal with 4 dimensions here. So one has first to choose arbitrarily two integers d 1 ≥ 1 and d 2 ≥ 1 such that d = d 1 + d 2 . Then we define the process (S n , n ≥ 0) on Z d as a mixture of two simple random walks in the following sense. Set S n = (X n , Y n ), where X n ∈ Z d1 is the set of the first d 1 coordinates of S n and Y n ∈ Z d2 is the set of the last d 2 coordinates. Now the rule is the following. First S 0 = 0. Next if S visits a site for the first time then only the X component performs a simple random walk step, that is: P[S n+1 − S n = (0, . . . , 0, ±1, 0, . . . , 0) \| F n ] = 1/(2d 1 ), where the ±1 can be at any of the first d 1 coordinates. Otherwise, only Y performs a simple random walk step: P[S n+1 − S n = (0, . . . , 0, ±1, 0, . . . , 0) \| F n ] = 1/(2d 2 ), if S already visited the site S n in the past, where this time the ±1 can be at any of the last d 2 coordinates. We call this process S the M (d 1 , d 2 )-random walk. Here we say that a process is transient if almost surely any site is visited only finitely many times. It is said to be recurrent if almost surely it visits all sites infinitely often. We will prove the following, Theorem 1. The M (2, 2)-random walk is transient. The proof of Theorem 1 is elementary, and uses only basic estimates on the standard 2-dimensional simple random walk. It relies on finding good upper bounds for the probability of return to the origin and then use Borel-Cantelli Lemma (what makes however the proof nontrivial is that the two components X and Y are not independent). Note that the canonical projections of S on Z d1 and Z d2 are usual (time changed) simple random walks. So if d = 4, and if d 1 or d 2 equals 3, then S is automatically transient, since the simple random walk on Z 3 is transient. Likewise if d is larger than 5, then for any choice of d 1 and d 2 (larger than 1), the resulting process will be transient. Thus the question of recurrence vs transience is only interesting in dimension less than 4. In dimension 3 there are two versions: d 1 = 1 and d 2 = 2 or d 1 = 2 and d 2 = 1. We conjecture that in both cases S will be transient, also because it is a 3-dimensional process. Proving this seems nontrivial, but notice that a possible intermediate step between the dimension 3 and 4 could be to consider the analogue problem on the discrete 3-dimensional Heisenberg group, which is generated by 2 elements (and their inverses), yet balls of radius r has size order r 4 . Let us make some comments now on the 2-dimensional case. As a 2-dimensional process, we believe that M (1, 1) is recurrent. Observe however that this is not true when starting from any configuration of visited sites. Indeed if we start with a vertical line of visited sites, then the process will be trapped in this line, and if the line does not include the origin, the process will not return there. It is also not difficult to construct starting environments such that the first coordinate of the process will tend almost surely toward +∞. For example if the initial configuration is the "trumpet" {(x, y) : \|y\| < e x } then the walker will drift to infinity in the x direction 1 . Of course it is not possible for the random walk to create these environments in finite time, so it is not an obstacle for recurrence, but it may be interesting to keep this in mind. Another problem concerns the limiting shape of the range (i.e. the set of visited sites) of the process. Based on heuristics and some simulations, we believe that it is a vertical interval. This problem is closely related to the question of evaluating the size of the range R n at time n. Indeed the horizontal displacement of the process at time n is of order √ #R n , whereas its vertical displacement is always of order √ n. So another formulation of the problem would be to show that R n is sublinear. By the way we mention a related question. Assume that at each step, one can decide, conditionally on the past, to move the first coordinate or the second coordinate (and then perform a 1-dimensional simple random walk step). Then what is the best strategy to maximize the range? In particular is it possible for the range to be of size roughly n, or at least significantly larger than n/ ln n, which is the size of the range of the simple random walk? A possible generalization of our model would be to consider multi-excited versions, in the spirit of Zerner [Z]. In this case one should first decompose d as d = d 1 + · · · + d m , for some m ≥ 2 and d i ≥ 1, with i ≤ m. Then at i th visit to a site only the i th component of S performs a simple random walk step, if i < m, and at further visits only the m th component moves. In dimension 4 for instance the case d 1 = 2 and d 2 = d 3 = 1 seems interesting and nontrivial. Another interesting case is d ≥ 3 and d i = 1 for each i ≤ d (even the case d very large seems nontrivial). Another related problem is the following. Take two symmetric laws µ 1 and µ 2 on Z 4 . Decide that at first visit to a site the jump of the process has law µ 1 , and at further visits it has law µ 2 . Then is it true that if the support of µ 1 and µ 2 both generate Z 4 , then the process is transient? Acknowledgments: This work was done while BS was a visitor at the Weizmann Institute and he thanks this institution for its kind hospitality. Proof of the theorem The theorem is a direct consequence of the following proposition. Proposition 1. There exists a constant C > 0 such that for any n > 1, P [0 ∈ {S n , . . . , S 2n }] ≤ C ln ln n ln n 2 . Indeed assuming this proposition we get k≥0 P [0 ∈ {S 2 k , . . . , S 2 k+1 }] < +∞, and we can conclude by using the Borel-Cantelli lemma. So all we have to do is to prove this proposition. Proof of Proposition 1. For any n ≥ 1, denote by r n the cardinality of the range of S at time n. The next lemma will be needed: Lemma 1. For any M > 0, there exists a constant C > 0, such that P n/(C ln n) 2 ≤ r n ≤ 99n/100 = 1 − o(n −M ). Proof. Note first that for any k, if S k and S k+1 were not already visited in the past, then S k+2 = S k with probability at least 1/4. In particular for any k, there is probability at least 1/4 that S is not at a fresh site at one of the time k, k + 1 or k + 2. Then a standard use of the Azuma-Hoeffding inequality gives the desired upper bound on r n . We now prove the lower bound. Let c > 0 be fixed. Let (U n , n ≥ 0) be a simple random walk on Z 2 . For any n ≥ 1 and x ∈ Z 2 , denote by N n (x) the number of visits of U to x before time n. A simple and standard calculation (see e.g. [LL,Proposition 4.2.4]) shows that there exists a constant C > 0 such that the probability to not visit x in the next n steps after a given visit is ≥ C/ log n. Using the strong Markov property one gets that the probability to make k + 1 visits by time n is ≤ exp(−Ck/ log n) and hence there exists some C ′ > 0 depending on M such that P[N n (x) ≥ C ′ (ln n) 2 ] = o(n −M−2 ). Moreover since U makes nearest neighbor jumps, before time n it stays in a ball of radius n. Thus if N * n = sup x N n (x), then P[N * n ≥ C ′ (ln n) 2 ] = n 2 × o(n −M−2 ) = o(n −M ). Thus if r n,U is the size of the range of U at time n, we get P r n,U ≤ n/(C ′ (ln n) 2 ) = o(n −M ). Let's come back to the original process S = (X, Y ) now. We just observe that at time n one of the X or Y component performed n/2 steps. Since each of these components is a simple random walk, we deduce from the previous estimate, that before time n, X or Y will visit at least n/(2C ′ (ln n) 2 ) sites, with probability at least 1 − o(n −M ). This gives the desired lower bound for r n and concludes the proof of the lemma. We can finish now the proof of Proposition 1. As noticed in the introduction, observe that the X and Y components are time changed simple random walks. Specifically we have the equality in law: ((X k , Y k ), k ≥ 0) = ((U (r k ), V (k − r k )), k ≥ 0), where U and V are two independent simple random walks on Z 2 (and where by abuse of notation we also denote by r k , the size of the range of the (U, V ) process at k-th' step). By using Lemma 1 and the independence of U and V , we get Thus Proposition 1 follows from the following lemma: Lemma 2. Let U be the simple random walk on Z 2 and let t ∈ [n/(ln n) 3 , 2n]. Then P [0 ∈ {U (t), . . . , U (2n)}] = O ln ln n ln n .(1) Proof. This lemma is standard, but we give a proof for reader's convenience. First let \| · \| denotes some norm on R 2 . Since t ≥ n/(ln n) 3 , it is well known (see e.g. [LL,Theorem 2 .1.1]) that P \|U (t)\| ≤ √ n (ln n) 3 = O((ln n) −1 ). Moreover for any \|x\| ≥ 4 (see e.g. [L,Proposition 1.6.7]), P x [τ 0 < τ \|x\|(ln \|x\|) 4 ] = O ln ln \|x\| ln \|x\| , where P x denotes the law of U starting from x and for any r ≥ 0, τ r = inf{k > 0 : r < \|U (k)\| ≤ r + 1}. But if \|x\| ≥ √ n/(ln n) 3 , then n ln n = O \|x\|(ln \|x\|) 4 and in particular (see e.g. [LL, Proposition 2.1.2]) P τ \|x\|(ln \|x\|) 4 ≤ 2n = O(n −1 ). Notice finally that if \|x\| ≥ √ n/(ln n) 3 , then ln ln \|x\| ln \|x\| = O ln ln n ln n . The lemma follows by using the strong Markov property. The proof of Theorem 1 is now finished. Remark. The proof shows actually that for any finite initial configuration of visited sites, M (2, 2) is transient. This is of course not always the case if this configuration is infinite. For instance if we decide that all sites of the form (0, 0, * , * ) are already visited at time 0, then the X component will never move and M (2, 2) will not be transient. P [ 0 ∈ 0{S n , . . . , S 2n }] ≤ P 0 ∈ {U (n/(C ln n) 2 ), . . . , U (2n)} ×P [0 ∈ {V (n/100), . . . , V (2n)}] + o(n −M ). We will not prove any of these claims, as they are somewhat off-topic Excited random walk against a wall. G Amir, I Benjamini, G Kozma, Probab. Theory and Related Fields. 140Amir G., Benjamini I, Kozma G.: Excited random walk against a wall, Probab. Theory and Related Fields 140, (2008), 83-102. Excited random walk. I Benjamini, D B Wilson, electronic). 8Benjamini I., Wilson D. B.: Excited random walk, Electron. Comm. Probab. 8 (elec- tronic), (2003), 86-92. Excited against the tide: A random walk with competing drifts. M Holmes, arXiv:0901.4393preprintHolmes M.: Excited against the tide: A random walk with competing drifts, preprint, arXiv:0901.4393. H Kesten, O Raimond, Schapira Br, arXiv:0911.3886Random walks with occasionally modified transition probabilities, preprint. Kesten H., Raimond O., Schapira Br.: Random walks with occasionally modified transition probabilities, preprint, arXiv:0911.3886. Positively and negatively excited random walks on integers, with branching processes. E Kosygina, M P W Zerner, Electron. J. Probab. 13Kosygina E., Zerner M. P. W.: Positively and negatively excited random walks on integers, with branching processes, Electron. J. Probab. 13, (2008), 1952-1979. Problem session. G Kozma, Oberwolfach report 27/2007, Non-classical interacting random walks. Kozma, G.: Problem session, in: Oberwolfach report 27/2007, Non-classical interacting random walks. www.mfo.de Intersections of random walks, Probability and its Applications. G F Lawler, Birkhäuser Boston, Inc219ppBoston, MALawler G. F.: Intersections of random walks, Probability and its Applications. Birkhäuser Boston, Inc., Boston, MA, (1991), 219 pp. Random walk: a modern introduction. G F Lawler, V Limic, Cambridge Studies in Advanced Mathematics. 123Cambridge University PressLawler G. F., Limic V.: Random walk: a modern introduction, Cambridge Studies in Advanced Mathematics 123, Cambridge University Press, Cambridge (2010) M Menshikov, S Popov, A Ramirez, M Vachkovskaia, arXiv:1001.1741On a general manydimensional excited random walk, preprint. Menshikov M., Popov S., Ramirez A., Vachkovskaia M.: On a general many- dimensional excited random walk, preprint, arXiv:1001.1741 F Merkl, S W W Rolles, Linearly edge-reinforced random walks, Dynamics & stochastics. Beachwood, OH48Inst. Math. Statist.Merkl F., Rolles S.W.W.: Linearly edge-reinforced random walks, Dynamics & stochas- tics, 66-77, IMS Lecture Notes Monogr. Ser. 48, Inst. Math. Statist., Beachwood, OH, (2006). A survey of random processes with reinforcement, Probab. Surv. 4 (electronic). R Pemantle, Pemantle R.: A survey of random processes with reinforcement, Probab. Surv. 4 (elec- tronic), (2007), 1-79. M P W Zerner, Multi-excited random walks on integers, Probab. Theory and Related Fields. 133Zerner M. P. W.: Multi-excited random walks on integers, Probab. Theory and Related Fields 133, (2005), 98-122.	[]
[ "GLOBAL EXISTENCE, SCATTERING AND BLOW-UP FOR THE FOCUSING NLS ON THE HYPERBOLIC SPACE", "GLOBAL EXISTENCE, SCATTERING AND BLOW-UP FOR THE FOCUSING NLS ON THE HYPERBOLIC SPACE", "GLOBAL EXISTENCE, SCATTERING AND BLOW-UP FOR THE FOCUSING NLS ON THE HYPERBOLIC SPACE", "GLOBAL EXISTENCE, SCATTERING AND BLOW-UP FOR THE FOCUSING NLS ON THE HYPERBOLIC SPACE" ]	[ "Valeria Banica ", "Thomas Duyckaerts ", "Valeria Banica ", "Thomas Duyckaerts " ]	[]	[]	We prove global well-posedness, scattering and blow-up results for energysubcritical focusing nonlinear Schrödinger equations on the hyperbolic space. We show in particular the existence of a critical element for scattering for all energy-subcritical power nonlinearities. For mass-supercritical nonlinearity, we show a scattering vs blowup dichotomy for radial solutions of the equation in low dimension, below natural mass and energy thresholds given by the ground states of the equation. The proofs are based on trapping by mass and energy, compactness and rigidity, and are similar to the ones on the Euclidean space, with a new argument, based on generalized Pohozaev identities, to obtain appropriate monotonicity formulas.	10.4310/dpde.2015.v12.n1.a4	[ "https://arxiv.org/pdf/1411.0846v2.pdf" ]	119,612,242	1411.0846	4b1f4beacf64093f2a400903dd69453ae66cc108	GLOBAL EXISTENCE, SCATTERING AND BLOW-UP FOR THE FOCUSING NLS ON THE HYPERBOLIC SPACE 14 Nov 2014 Valeria Banica Thomas Duyckaerts GLOBAL EXISTENCE, SCATTERING AND BLOW-UP FOR THE FOCUSING NLS ON THE HYPERBOLIC SPACE 14 Nov 2014 We prove global well-posedness, scattering and blow-up results for energysubcritical focusing nonlinear Schrödinger equations on the hyperbolic space. We show in particular the existence of a critical element for scattering for all energy-subcritical power nonlinearities. For mass-supercritical nonlinearity, we show a scattering vs blowup dichotomy for radial solutions of the equation in low dimension, below natural mass and energy thresholds given by the ground states of the equation. The proofs are based on trapping by mass and energy, compactness and rigidity, and are similar to the ones on the Euclidean space, with a new argument, based on generalized Pohozaev identities, to obtain appropriate monotonicity formulas. Introduction The nonlinear Schrödinger equations on manifolds have been intensively studied in the last decades. Most works concern local existence, blow-up in finite time, small data scattering and existence of wave operators. Recently, results on scattering for all solutions in the defocusing case were obtained on hyperbolic space [BaCaSt08,IoSt09,IoPaSt12], more general rotationally symmetric manifolds [BaCaDu09], flat manifolds such as exterior domains [PlVe09,IvPl10,PlVe12,KiViZh12] and product spaces R × T 2 , R n × T [HaPa14,TzVi14]. Let us also note the recent work on long range effects on R × T n [HaPaTzVi13P]. The purpose of this work is to initiate the study of "large" data -that is out of the perturbative framework of the small data theory -for focusing NLS on manifolds. More precisely, we are interested with the focusing NLS on the hyperbolic space H n : (1) i∂ t u + ∆ H n u + \|u\| p−1 u = 0, u ↾t=0 = u 0 ∈ H 1 (H n ), where n ≥ 2, ∆ H n is the (negative) Laplace-Beltrami operator on H n and the power p is energy-subcritical: 1 < p < 1 + 4 n−2 (1 < p < ∞ if n = 2). It follows from Strichartz estimates that equation (1) is locally well-posed in the Sobolev space H 1 = H 1 (H n ) [Ba07]: for u 0 ∈ H 1 , there exists a unique maximal solution u ∈ C 0 (−T − (u 0 ), T + (u 0 )), H 1 satisfying the following blow-up criterion: (2) T + (u 0 ) < ∞ =⇒ lim t→T + (u 0 ) u(t) H 1 = +∞. The mass of a solution (where µ is the standard measure on H n ) and its energy (4) E(u(t)) = 1 2 H n \|∇ H n u(t, x)\| 2 dµ(x) − 1 p + 1 H n \|u(t, x)\| p+1 dµ(x), are conserved. In the defocusing case (equation (1) with a minus sign in front of the nonlinearity), it was proved in [BaCaSt08,BaCaDu09,IoSt09] that all solutions with initial data in H 1 scatter to a solution of the linear Schrödinger equation in both time directions (see also [IoPaSt12] for the energy-critical case in space dimension 3). Note that this holds for all p such that 1 < p ≤ 1 + 4 n−2 , in contrast with the Euclidean setting scattering results where a lower bound larger than 1 is imposed on p. This is a consequence of the stronger long-time dispersion for the linear Schrödinger equation in H n compared to R n , which translates into a wider range of exponents for the Strichartz estimates. More precisely, global Strichartz estimates on H n are available for all exponents of Strichartz estimates on R d , d ≥ n [BaCaSt08, IoSt09,AnPi09]. Using this fact the scattering results are proved for all the range of exponents p allowed on R d , d ≥ n, so for 1 < p ≤ 1 + 4 n−2 . In the focusing case, for the same reasons as above, scattering remains valid for small data in H 1 , and wave operators also exist for all energy-subcritical p. However, solutions with larger initial data do not always scatter. If p ≥ 1 + 4 n , blow-up in finite time may occur [Ba07,MaZh07]. Furthermore, for any p > 1, there exist nonzero time-periodic solutions of (1). The aim of this article is to obtain sharp global existence, scattering and blow-up results in terms of geometric objects that are specific to H n . Before stating our main results, we recall known ones on the focusing Schrödinger equation on R n , n ≥ 1: i∂ t u + ∆ R n u + \|u\| p−1 u = 0, u ↾t=0 = u 0 ∈ H 1 (R n ), where 4 n + 1 < p, and, if n ≥ 3, p < 4 n−2 + 1. Fixing µ < 0, the equation −∆f − µf = \|f \| p−1 f, x ∈ R n has a unique radial, positive solution in H 1 (R n ) that we will denote by R µ . Let s c = n 2 − 2 p−1 ∈ (0, 1) be the critical Sobolev exponent, M (u) and E(u) be the invariant masses and energy, defined as in (3), (4) with the integrals on R n . Then (see [Stu91,HoRo07,HoRo08,DuHoRo08,FaXiCa11,Gu13,AkNa13]): Theorem A. Assume 1 + 4 n < p, and p < 1 + 4 n−2 if n > 3. Let u 0 ∈ H 1 (R n ) be such that (5) E(u 0 ) sc M (u 0 ) 1−sc < E(R µ ) sc M (R µ ) 1−sc . Let, for t in the maximal interval of existence of u, δ(u(t)) = ∇u(t) sc L 2 u(t) 1−sc L 2 − ∇R µ sc L 2 R µ 1−sc L 2 . Then δ(u(t)) = 0, and the sign of δ(u(t)) is independent of t. Furthermore, (a) If δ(u 0 ) < 0 then u scatters in both time directions. (b) If δ(u 0 ) > 0 and either u 0 is radial and p ≤ 5, or \|x\| 2 \|u 0 \| 2 < ∞ then u blows up in finite time. Our aim is to obtain a scattering/blow-up dichotomy for equation (1), in the spirit of the above theorem, under an optimal mass/energy threshold. Our main motivation is to clarify the influence of the geometry on the dynamics of focusing Schrödinger equations. Note that the choice of the ground state, i.e. of the parameter µ < 0 in the above theorem is not relevant. Indeed, As a consequence, for all λ < (n−1) 2 4 and f ∈ H n , the following inequality holds (7) H n \|f \| p+1 2 p+1 ≤ D λ H n \|∇ H n f \| 2 − λ H n \|f \| 2 . The best constant in (7) is attained for a positive, radial function Q λ ∈ H 1 (H n ), solution of the equation −∆Q λ − λQ λ = \|Q λ \| p−1 Q λ which we will call ground state (see [MaSa08] and §2.3 for details). This ground state is not always known to be unique if n = 2: when it is not we will denote by Q λ one of the ground states corresponding to p and λ. In all cases, we let Q λ the set of all ground states, that is the set of all solutions of the above equation that are also positive, radial minimizers for (7). Before stating our main results, we introduce some notations. If f ∈ H 1 (H n ), we denote by f 2 H λ = H n \|∇ H n f \| 2 dµ − λ H n \|f \| 2 dµ, which, for any λ < (n−1) 2 4 , is a norm on H 1 (H n ) equivalent to the usual H 1 norm. This is due to the spectrum of ∆ H n , implying ∇f 2 2 ≥ (n−1) 2 4 f 2 2 . We define E λ (f ) = 1 2 f 2 H λ − 1 p + 1 f p+1 L p+1 . In particular E 0 = E. Note that E λ (u(t)) is independent of t for any solution u of (1). We denote also δ λ (f ) = f 2 H λ − Q λ 2 H λ . When the ground state Q λ is not unique, δ λ does not depend on the choice of Q λ ∈ Q λ (see (19), (20)). Theorem 1 (Trapping and global existence). Let n ≥ 2, λ < (n−1) 2 4 and u 0 ∈ H 1 (H n ). If 1 < p < 1 + 4 n−2 and E λ (u 0 ) ≤ E λ (Q λ ) then δ λ (u(t)) does not change sign. Moreover, under these hypothesis, (a) If δ λ (u 0 ) = 0, then there exists θ ∈ R and an hyperbolic isometry h such that We refer to §2.1 below for the definition of the group of hyperbolic isometries. u 0 = e iθ Q(h·), Q ∈ Q λ . (b) If δ λ (u 0 ) < 0 then the solution u is global in time. (c) If δ λ (u 0 ) > 0 Remark 1.1. The statement about global existence in Theorem 1 is relevant only if p ≥ 1 + 4 n . If 1 < p < 1 + 4 n , it follows from the Gagliardo-Nirenberg inequality on H n (which is the same than on R n ) that all solutions of (1) are global in time. Remark 1.2. In the mass-critical case p = 1 + 4 n a sharp global existence result based on the mass is known. On the one hand, global existence occurs for initial data of mass less than 1/C G−N where C G−N is the best constant of Gagliardo-Nirenberg inequality on H n . On the other hand blow-up solutions can be constructed as in [BaCaDu11,RaSz11] from the Euclidean ground state, hence of mass the inverse of the best constant of Gagliardo-Nirenberg inequality on R n . Very recently the two constants have been proved to coincide, yielding a mass threshold for blow-up [Mu14]. Remark 1.3. We note that in the case p > 1 + 4 n , the analog of Theorem 1 is valid on R n with almost the same proof. One can prove, for example, a global existence condition similar to Theorem 1 (b) for the NLS equation on R n depending on a parameter µ < 0 and using the ground states R µ defined above. However, fixing an initial data u 0 and using the scaling to obtain the optimal value for µ, one would exactly obtain the scale-invariant criteria of Theorem A. In this sense, the conditions of Theorem 1 are the natural analogs, for the hyperbolic space, of the conditions of Theorem A. We conjecture that in the context of Theorem 1, with the stronger assumptions E λ (u 0 ) < E λ (Q λ ), and p ≥ 1 + 4 n , solutions such that δ λ (u 0 ) < 0 are global and scatter, whereas solutions such that δ λ (u 0 ) > 0 blow up in finite time. This is false if 1 < p < 1 + 4 n : all solutions are global, and there exist values of λ and initial data u 0 such that E λ (u 0 ) < E λ (Q λ ), δ λ (u 0 ) < 0, and the corresponding solution u does not scatter (see Proposition 1.5 below). In this work we prove the conjecture in space dimensions n = 2 and n = 3, for radial data (i.e. depending only on the distance to the origin of H n ): Let us note that Theorem 2 implies that for p, n as in the theorem, the ground states are (orbitally) unstable: indeed, one can check as a consequence of this theorem that the solution with initial data αQ λ blows up in finite time if α > 1, and scatters if α ∈ (0, 1). Theorem 2. Assume n ∈ {2, 3}, p ≥ 3 if n = 2, 7 3 ≤ p < 5 if n = 3. Let λ < (n−1) 2 4 and u 0 ∈ H 1 rad (H n ). Assume E λ (u 0 ) < E λ (Q λ ). Then Let us say a few words about the proof of Theorem 2. To prove the scattering result, we use the compactness-rigidity method initiated in [KeMe06]. The compactness step consists in proving the existence of a nonscattering solution of (1) with minimal energy. More precisely: Theorem 3 (Existence of the critical element). Assume 1 < p < 1 + 4 n−2 (p > 1 if n = 2) and λ < (n−1) 2 4 . There exists a global radial solution v c of equation (1) such that {v c (t, ·), t ∈ R} has compact closure in H 1 (H n ), E λ (v c (0)) ≤ E λ (Q λ ), v c (0) H λ ≤ Q λ H λ , and, for any u 0 ∈ H 1 (H n ) radial, if E λ (u 0 ) < E λ (v c (0)), u 0 H λ ≤ Q λ H λ , then the solution u of equation (1) scatters in both time directions. We stated Theorem 3 in a radial setting. A nonradial version is available (see Proposition 3.12 p. 28). Remark 1.4. Note that v c exists in all dimensions and for all energy-subcritical exponent p. Again, this contrasts with the Euclidean case where for p close to one there is no small data scattering, and thus no critical solution in the above sense. The proof of Theorem 3 follows the line of the corresponding proof on R n (see [Ke06,KeMe06,TaViZh08,HoRo07]). The main ingredient of the proof is a profile decomposition adapted to the energy-subcritical equation (1). We construct this profile decomposition in Section 3.2, using Fourier analysis on the hyperbolic spaces, in the spirit of the analogous construction in the energy-critical setting, given in [IoPaSt12]. The rigidity step in the proof of Theorem 2 (a) consists in proving that the critical element v c given by Theorem 3 (under the assumptions on n and p in Theorem 2), satisfies E λ (v c ) = E λ (Q λ ) (see Proposition 4.1 p. 33). Similarly to the Euclidean case, we use a localized version of the following virial-type identity: (8) ∂ 2 t \|u(t, x)\| 2 r 2 dµ(x) = G(u(t)), where, r is the distance to the origin of H n , and, if f is radial, G(f ) = 8 f 2 H + 2(n − 1)(n − 3) H n \|f \| 2 r cosh r − sinh r sinh 3 r dµ(x) − 4(p − 1) p + 1 H n \|f \| p+1 1 + (n − 1) r cosh r sinh r dµ(x). A crucial property of G is that it is positive for solutions satisfying the assumptions of Theorem 2 (a). However, unlike in the Euclidean setting where the analogous property follows quite easily from the characterization of the ground states R µ as maximizers for the Gagliardo-Nirenberg inequality and the trapping of solutions below the ground state mass and energy, the proof of this property is quite intricate (see Section 4). The key new ingredient of this proof is a generalized Pohozaev identity satisfied by the minimizers of G(f ) under the constraint E λ (f ) = E λ (Q λ ). It is in this part of the proof that the assumption n = 2, 3 is needed. We think that the radiality assumption and the assumption n = 2, 3 are technical, and that Theorem 2 remains valid for any n ≥ 2, without symmetry, provided p ≥ 1 + 4 n . On the other hand, the hypothesis p ≥ 1 + 4 n is crucial, as emphasized by the following proposition: Proposition 1.5. If n ≥ 3 and 1 < p < 1+ 4 n , there exists λ < (n−1) 2 4 such that E λ (v c (0)) < E λ (Q λ ). We do not know what is v c in this case (a ground state Q ν with ν = λ, or some other type of solution). Proposition 1.5 follows from the fact that if p < 1 + 4 n , for some λ < (n−1) 2 4 the solution e itλ Q λ is stable (in the set of orbits of the minimizers of the energy at this mass). It is an open question if all ground states are stable if p < 1 + 4 n (this is stated in [ChMa10], but with a gap in the proof, see Remark 2.15 below). The proof of the blow-up part of Theorem 2 follows the classical proof of the so-called Glassey criterion on R n [VlPeTa71, Gl77], using the virial identity (8). For this, we prove that G is negative for solutions of (1) satisfying the assumptions of Theorem 2 (b), using arguments that are similar to the ones of the proof of the positivity of G in the other regime of Theorem 2. We refer to [Ba07] and [MaZh07] for other blow-up criteria, recalled in Proposition 5.1 p. 39. Note that in these criteria, the threshold given by Q λ does not appear. In [MaZh07] a variant of virial identity (8) is used, based on another weight than r 2 . Unfortunately this different weight does not seem useful in the setting of Theorem 2. It is also possible to construct blow-up solutions with an explicit behavior, starting from the Euclidean ground state: see [BaCaDu09], [RaSz11], [Bo12], [Go13] for the construction of conformal and log-log type blow-up solutions on H n or on related manifolds. The outline of the article is as follows. In Section 2, we prove Theorem 1, after some preliminaries and reminders on the hyperbolic space, Cauchy theory for equation (1) and ground states. Section 3 is dedicated to the existence of the critical solution (Theorem 3 and its nonradial analog). Following a standard scheme, we construct an adapted profile decomposition (see §3.2), which follows from an improved Sobolev inequality, proved in §3.1. The critical solution is constructed in §3.3. Proposition 1.5 is proved in §3.4. In Section 4, we conclude the proof of Theorem 2 (a) (scattering) by proving the rigidity part of the argument. Section 5 concerns the proof Theorem 2 (b) (blow-up). Acknowledgment. The authors would like to thank Benoît Pausader for very useful clarifications on spectral projectors on the hyperbolic space. Both authors were partially supported by the French ANR project SchEq ANR-12-JS-0005-01. H n = x ∈ R n+1 : [x, x] = 1 and x 0 > 0 , where [·, ·] is the bilinear form [x, y] = x 0 y 0 − x 1 y 1 − . . . − x n y n on R n+1 . The hyperbolic space H n is endowed with the metric g induced by the Minkowski metric −(dx 0 ) 2 + (dx 1 ) 2 + . . . + (dx n ) 2 . We will denote by 0 the origin (1, 0, . . . , 0) of H n , and dµ the induced measure. We shall use often radial coordinates on the hyperbolic space, x = (cosh r, sinh r ω) where r = d(x, 0 H n ), ω ∈ S n−1 . In such coordinates, the Laplacian writes ∆ H n = ∂ 2 r + (n − 1) cosh r sinh r ∂ r + 1 sinh 2 r ∆ S n−1 . To lighten notations, we will often write ∆ instead of ∆ H n . We denote by G = SO(n, 1) the group of hyperbolic isometries, that is the group of (n + 1) × (n + 1) matrices that leave the form [·, ·] invariant. For any h ∈ G, the mapping x → h · x restricts to an isometry of H n . The group G acts transitively on H n . We introduce the following notation, which is the quadratic form associated to the socalled shifted Laplacian on hyperbolic space, whose bottom of the spectrum is zero, (9) f 2 H = H n \|∇ H n f \| 2 − (n − 1) 2 4 H n \|f \| 2 . By (6), · H is a norm on C ∞ c (H n ). We will denote by H the closure of C ∞ c (H n ) in L p+1 (H n ) for the norm · H . It is a Hilbert space which is included in L p+1 . 2.2. Cauchy theory. We give here some results related to well-posedness and scattering for equation (1). We omit most of the proofs, that are classical. 2.2.1. Strichartz estimates on the hyperbolic space. We will denote by q ′ the conjugate exponent of q ∈ [1, ∞]. We recall from [BaCaSt08, AnPi09, IoSt09] the wider range of Strichartz estimates on the hyperbolic space: Theorem 2.1. Let, for j = 1, 2, (q j , r j ) ∈ (q, r) ∈ [2, ∞) × (2, ∞) : 2 q ≥ n 2 − n r ∪ (∞, 2) . If u 0 ∈ L 2 (H n ), F ∈ L q ′ 2 (R, L r ′ 2 (H n )), then, denoting by u the solution of i∂ t u + ∆u = F, u ↾t=0 = u 0 , we have u L q 1 (Rt,L r 1 (H n )) ≤ C u 0 L 2 + C F L q ′ 2 (R,L r ′ 2 (H n ) . Let I be an interval. If 1 < p ≤ 1 + 4 n , we define S 0 (I) = L p+1 (I, L p+1 (H n )) S 1 (I) = u ∈ S 0 (I) : ∇u ∈ S 0 (I) N 0 (I) = L p+1 p (I, L p+1 p (H n )) N 1 (I) = u ∈ N 0 (I) : ∇u ∈ N 0 (I) . Note that p+1 p = (p + 1) ′ . If 1 + 4 n ≤ p < 1 + 4 n−2 , we let (following [FaXiCa11]) a = 2(p − 1)(p + 1) 4 − (n − 2)(p − 1) , b = 2(p − 1)(p + 1) n(p − 1) 2 + (n − 2)(p − 1) − 4 , q = 4(p + 1) n(p − 1) , (so that pb ′ = a), and S 0 (I) = L a (I, L p+1 (H n )) S 1 (I) = u ∈ L q (I, L p+1 (H n )) : ∇u ∈ L q (I, L p+1 (H n )) N 0 (I) = L b ′ (I, L p+1 p (H n )) N 1 (I) = u ∈ L q ′ (I, L p+1 p (H n )) : ∇u ∈ L q ′ (I, L p+1 p (H n )) . One easily checks that the definitions coincide when p = 4 n + 1. If I = (a, b), we will write S 0 (a, b) instead of S 0 ((a, b)), and similarly for S 1 , N 0 , N 1 . Proposition 2.2. If t 0 ∈ R ∪ {±∞}, e it∆ u 0 S j (R)∩L ∞ (R,H 1 ) ≤ C u 0 H 1 , j = 0, 1, (10) t t 0 e i(t−s)∆ f (s)ds S 0 (R) ≤ C f N 0 (R) (11) t t 0 e i(t−s)∆ f (s)ds S 1 (R)∩L ∞ (R,H 1 ) ≤ C f N 1 (R) (12) and t → t t 0 e i(t−s)∆ f (s)ds ∈ C 0 (R, H 1 ), if f ∈ N 1 (R), +∞ 0 e −is∆ f (s)ds H 1 ≤ C f N 1 (0,∞) . (13) Proposition 2.3. \|u\| p−1 u − \|v\| p−1 v N 0 (I) ≤ C u − v S 0 (I) u p−1 S 0 (I) + v p−1 S 0 (I) (14) \|u\| p−1 u N 1 (I) ≤ C u S 1 (I) u p−1 S 0 (I) .(15) Remark 2.4. If p > 2, we can of course obtain a Lipschitz bound similar to (14) for the N 1 -norm. Sketch of proof of Proposition 2.2. The inequalities are obtained from Theorem 2.1 as follows. If p < 1 + 4 n , the pair (p + 1, p + 1) is not an Euclidean admissible but it enters the wider range of Strichartz exponents on the hyperbolic space from Theorem 2.1. This yields inequalities (10)-(13) if p < 1 + 4 n . In the case 1 + 4 n ≤ p < 1 + 4 n−2 , the Proposition follows from the Euclidean-type Strichartz estimates and Sobolev inequalities, as for instance in [FaXiCa11]. The only delicate point is the Strichartz estimate for non-admissible couples (11); these can be obtained from dispersion in the spirit of Lemma 2.1 in [CaWe92] (see also [Fo05]). We sketch it (with t 0 = 0) for completeness. By Lemma 3.3 of [IoSt09], ∀t = 0, e it∆ u 0 L p+1 ≤ C \|t\| 2 q u 0 L p+1 p . Thus ∀t = 0, t 0 e i(t−s)∆ f (s) ds L p+1 p ≤ C t 0 1 \|t − s\| 2 q f (s) L p+1 p ds, and the result follows from the classical Riesz potential inequality (see e.g. Ch. 5 of [Ste70Bo]). Proposition 2.3 follows immediately from Hölder inequality and we omit it. In view of Propositions 2.2 and 2.3, the well-posedness of equation (1) in H 1 is classical (see [Ka87]). Recall that a solution u of (1), defined on a maximal interval of existence (T − (u), T + (u)) satisfies the following blow-up criterion T + (u) < ∞ =⇒ lim t→T + (u) u(t) H 1 = 0. 2.2.2. Scattering results. Proposition 2.5 (Existence of wave operators). Let v 0 ∈ H 1 (H n ). Then there exists a solution u of (1) such that T + (u) = +∞ and lim t→∞ e it∆ v 0 − u(t) H 1 = +∞. This follows by a fixed point in the closed subset of S 0 (T, +∞): B T,ε = u ∈ S 1 (T, +∞) ∩ S 0 (T, +∞) : u S 1 (T,+∞) + u S 0 (T,+∞) ≤ ε , for T large, ε > 0 small, using again Propositions 2.2 and 2.3. We omit the details of the classical proof. Proposition 2.6 (Sufficient condition for scattering). Let u be a solution of (1) with maximal time of existence T + and such that u S 0 (0,T + ) < ∞. Then T + = +∞ and u scatters forward in time to a linear solution: there exists v 0 ∈ H 1 such that lim t→+∞ e it∆ v 0 − u(t) H 1 = 0. We skip the standard proof. Proposition 2.7 (Long time perturbation theory). Let M > 0. There exists constants ε 0 > 0, C > 0 depending on M with the following properties. Let 0 < T ≤ ∞ and u 0 ∈ H 1 ,ũ ∈ C 0 ((0, T ), H 1 ) ∩ S 0 (0, T ), e ∈ N 0 (0, T ) such that i∂ tũ + ∆ũ + \|ũ\| p−1ũ = e. Assume ũ S 0 (0,T ) ≤ M, e N 0 (0,T ) + e it∆ (u 0 −ũ(0)) S 0 (0,T ) = ε ≤ ε 0 . Then the solution u of (1) with initial data u 0 is defined on (0, T ) and u −ũ S 0 (0,T ) ≤ Cε. This type of result that goes back to [CoKeStTaTa08, Lemma 3.10], is by now standard. In the case p > 1 + 4 n , in view of the Strichartz estimates of Proposition 2.2, the proof is exactly the same as in [FaXiCa11,Proposition 4.7] (simply replacing R n by H n ). In the case 1 < p ≤ 1 + 4 n , it can be easily adapted, using Propositions 2.2 and 2.3. We apply the preceding proposition withũ = e it∆ u 0 to get: Corollary 2.8 (Small data theory). There exists ε 1 such that if u 0 ∈ H 1 satisfies e it∆ u 0 S 0 (R) = ε ≤ ε 1 then the corresponding solution u of (1) is global and satisfies u − e it∆ u 0 S 0 (R) ≤ Cε p . 2.3. Ground states on the hyperbolic space. We review here results on ground states for NLS on the hyperbolic space and additional variational properties. Most of these results come from [MaSa08], see also [Wa14]. See [MuTa98] for a previous work in space dimension 2 and [ChMa10] for similar existence results. Consider the equation on H n (16) ∆ H n f + λf + \|f \| p−1 = 0, Positive solutions to (16) can be constructed as solution to the following minimizing problems: (17) 1 D λ = min f ∈H 1 \{0} f 2 H λ f 2 L p+1 Then (Theorems 5.1 and 5.2 of [MaSa08]): Theorem 2.9. The minimizing problem (17) has a solution if (18) (n = 2 or 1 < p < 1 + 4 n − 2 ) and λ < (n − 1) 2 4 . In this case, any minimizer is radial up to hyperbolic symmetries, positive up to multiplication by a unit complex number, and satisfies equation (16). We will denote by Q λ the set of positive, radial minimizers for (17) that are solutions to equation (16), and Q λ an arbitrary element of Q λ . We note that Q λ is not always known to be unique (see below), however our statements will never depend on the choice of Q λ . Let Q ∈ Q λ . Multiplying (16) by ϕ(εr)Q (where ϕ is a radial smooth compactly supported function equal to 1 around 0), integrating by parts and letting ε → 0, we get (19) ∀Q ∈ Q λ , Q 2 H λ = Q p+1 L p+1 As a consequence, (20) ∀Q ∈ Q λ , D λ = Q 2 L p+1 Q 2 H λ = Q 1−p L p+1 . In particular, the values of E λ (Q), Q H λ and Q L p+1 do not depend on the choice of Q in Q λ . The following theorem follows from Theorems 1.2 and 1.3 of [MaSa08]: Theorem 2.10 (Uniqueness). Assume (18). If n ≥ 3, or n = 2 and λ ≤ 2(p+1) (p+3) 2 equation (16) has only one positive solution up to hyperbolic isometries. Remark 2.11. Uniqueness in the case n = 2, 2(p+1) (p+3) 2 < λ ≤ 1 4 is an open question. Theorem 2.12 (Nonexistence). If p > 1, λ ≥ (n−1) 2 4 , then equation (16) has no positve solution in H 1 . Let us mention that for the critical value λ = (n−1) 2 4 , equation (16) has a solution which is in H (see (9)), but not in H 1 , and solution to the minimization problem (17). For λ > (n−1) 2 4 the nonexistence theorem 2.12 remains valid if H 1 is replaced by H. We next give a result that is specific to the mass-critical case, and will be needed in the proof of Proposition 1.5. Proposition 2.13. Assume 1 < p < 1 + 4 n . Then there exists α 0 > 0 such that for all α > α 0 , the infimum (21) inf u∈H 1 (H n ) u 2 L 2 =α 2 1 2 u 2 H − 1 p + 1 u p+1 L p+1 = e(α) is attained by a radial, positive function. If n ≥ 3, this function is equal to Q λ for some λ < (n−1) 2 4 . Finally, any radial minimizing sequence converges (up to a subsequence) to a radial minimizer. Proof. First we note that Gagliardo-Nirenberg inequality implies the infimum in (21) to be finite if 1 < p < 1 + 4 n . By a rearrangement procedure ( [Dr05] and [ChMa10, Section 3]) or moving planes technics [MaSa08, Section 2] , the infimum in (21) can be restricted to H 1 radial functions such that u L 2 = α. Using as in [ChMa10] the change of functions v = sinh r r n−1 2 u, we are reduced to minimize R n 1 2 \|∇v\| 2 + (n − 1)(n − 3) 8 R n \|v\| 2 r 2 − sinh 2 r r 2 sinh 2 r − 1 p + 1 R n \|v\| p+1 r sinh r (p−1)(n−1) 2 , on all radial function in H 1 (R n ) such that v L 2 = α. Since ∇\|u\| L 2 ≤ ∇u L 2 , with strict inequality if u is not positive up to a constant factor, minimizers are positive (up to a constant factor). Using, as in [ChMa10], the concentration-compactness method, or simply the compactness of the radial embedding of H 1 in L p , it is easy to prove that if e(α) < 0, the infimum is attained. Fixing u ∈ H 1 (H n ) with u L 2 = 1, we obtain lim α→∞ α 2 2 u 2 H − α p+1 p + 1 u p+1 L p+1 = −∞. Thus e(α) is negative for large α, concluding the proof of the existence of a minimizer S α for (21). The compactness of minimizing sequences also follows. Since S α is solution to the minimization problem (21), there exists a Lagrange multiplier λ such that −∆S α − λS α = \|S α \| p−1 S α . By the considerations above, we can assume that S α is radial and positive. If n ≥ 3, using Theorems 2.12 and 2.10, we obtain λ < (n−1) 2 4 and S α = Q λ , concluding the proof. Remark 2.14. In [ChMa10, Section 5], it is claimed that the infimum (21) is attained for all α > 0. This cannot be true, since it would contradict the small data scattering for equation (1) for 1 < p < 1 + 4 n , proved in [BaCaSt08, Section 4]. Note that for small α > 0, one can prove (using Poincaré-Sobolev (6) and Gagliardo-Nirenberg inequalities) that e(α) = 0, whereas it is claimed and used in the proof of [ChMa10] than e(α) < 0 for any α > 0. Remark 2.15. One can deduce from Proposition 2.13, following [CaLi82], the orbital stability of the set of all solutions e itλ Q λ of (1), with Q λ minimizer for (21), that is of mass α = Q λ L 2 (see [ChMa10, Section 6]). Note that the proof of Proposition 2.13 does not imply that any Q λ is a minimizer for the problem (21). In particular, Proposition 2.13 and the method of [CaLi82] do not yield stability for all ground states solutions e itλ Q λ as seems to say [ChMa10, Proposition 6.3]. We refer to [LaSuSo14Pa] for the study of ground states stability for wave maps on the hyperbolic plane: in this case also the situation is quite different from the Euclidean setting. Let us also mention that uniqueness of minimizers for (21) and uniqueness of a minimal mass ground state are open questions. A similar issue appears in the context of combined power-type nonlinear Schrödinger equation [KiOhPoVi14]. 2.4. Trapping and global well-posedness. In this section we prove Theorem 1. We use a classical trapping argument, that goes back to [PaSa75] in the context of the Klein-Gordon equation (see e.g. [Stu91] for NLS). We start by proving the following stationary lemma: Lemma 2.16. Assume p > 1, λ < (n−1) 2 4 , and 1 < p < 1 + 4 n−2 if n ≥ 3. Then if E λ (f ) ≤ E λ (Q λ ) and f 2 H λ ≤ Q λ 2 H λ we have (22) f 2 H λ ≤ E λ (f ) E λ (Q λ ) Q λ 2 H λ . In particular there is no function f such that E λ (f ) < E λ (Q λ ) and f 2 H λ = Q λ 2 H λ . Proof. Recall from subsection 2.3 the variational definition of D λ . By (19), (20) Q λ 2 H λ = Q λ p+1 L p+1 D λ = Q λ 2(1−p) p+1 H λ (23) E λ (Q λ ) = Q λ 2 H λ p − 1 2(p + 1) . (24) Therefore (25) E λ (f ) = 1 2 f 2 H λ − 1 p + 1 f p+1 L p+1 ≥ 1 2 f 2 H λ − D p+1 2 λ p + 1 f p+1 H λ = a f 2 H λ , where (in view of (23)) a(x) = 1 2 x − Q λ 1−p H λ p + 1 x p+1 2 . In particular, b(x) = a(x) − p − 1 2(p + 1) x vanishes at x = 0 and at x = Q λ 2 H λ , increases on 0, 2 p+1 2 p−1 Q λ 2 H λ and decreases on 2 p+1 2 p−1 Q λ 2 H λ , Q λ 2 H λ so it is a positive function on the whole interval [0, Q λ 2 H λ ]. Since f 2 H λ ≤ Q λ 2 H λ , combining with (25) we have obtained that E λ (f ) ≥ a f 2 H λ ≥ p − 1 2(p + 1) f 2 H λ . Dividing this estimate by the value (24) of E λ (Q λ ) we obtain (22). Proof of Theorem 1. Let u 0 be as in Theorem 1. If δ λ (u 0 ) = 0, then by (22), E λ (u 0 ) = E λ (Q λ ). Thus u 0 is a minimizer for Poincaré-Sobolev inequality and by Theorem 2.9, u 0 (x) = e iθ Q(h(x)), for some Q ∈ Q λ , θ ∈ R, and h ∈ G, which gives Case (a). As a consequence of Case (a), if δ λ (u 0 ) = 0, then δ λ (u(t)) = 0 for all t in the domain of existence of u, which proves by continuity that δ λ (u(t)) does not change sign. We next assume that δ λ (u 0 ) is negative, and thus that δ λ (u(t)) is negative for all t. This ensures that the H norm of u(t) is bounded in time. By mass conservation we deduce that the H 1 norm of u is bounded and global well-posedness follows from the blow-up criterion (2) mentioned in the introduction. This proves case (b). In case (c) δ λ (u(t)) > 0 for all t in the domain of existence of u and thus E λ (u 0 ) ≤ E λ (Q λ ) = 1 2 Q λ 2 H λ − 1 p + 1 Q λ p+1 L p+1 < 1 2 u(t) 2 H λ − 1 p + 1 Q λ p+1 L p+1 . If the solution u scatters for positive times, then lim t→∞ u(t) L p+1 = 0, and for any ǫ > 0 there exists t large such that 1 2 u(t) 2 H λ < E λ (u(t)) + ǫ = E λ (u 0 ) + ǫ, so we get a contradiction by taking ǫ = 1 2(p+1) Q λ p+1 L p+1 . 3. Construction of the critical solution 3.1. Some spaces of functions and inequalities. 3.1.1. Preliminaries on Fourier analysis on hyperbolic space. We will mostly use the notations of [IoPaSt12]. We refer to this article for more details. We define the Fourier transform on H n , following the general definition of the Fourier transform on symmetric spaces given in [He65]. For ω ∈ S n−1 , λ ∈ R, we define the Fourier transform of f ∈ L 1 (H n ) bŷ f (λ, ω) = H n f (x)[x, (1, ω)] iλ−ρ dµ(x), ρ = n − 1 2 . We have ∆ H n f (λ, ω) = − λ 2 + ρ 2 f (λ, ω). The Fourier inversion formula reads (26) f (x) = ∞ −∞ S n−1f (λ, ω)[x, (1, ω)] −iλ−ρ \|c(λ)\| −2 dλdω, where the Harish-Chandra function c(λ) is defined by \|c(λ)\| −2 = 1 2 \|Γ(ρ)\| 2 \|Γ(2ρ)\| 2 \|Γ(ρ + iλ)\| 2 \|Γ(iλ)\| 2 . We note that \|c(λ)\| −2 is of the order λ n−1 as λ → ∞, and λ 2 as λ → 0. A version of Plancherel theorem is also available on H n : the Fourier transform f → f extends to an isometry of L 2 (H n ) onto L 2 (−∞, ∞) × S n−1 ), \|c(λ)\| −2 dλdω , and, for f, g ∈ L 2 (H n ), H n f (x)g(x)dµ = ∞ −∞ S n−1f (λ, ω)ĝ(λ, ω)\|c(λ)\| −2 dλdω. We will use the spectral projectors P m , m > 0 defined as follows (27) P m = − 1 m 2 ∆e 1 m 2 ∆ , that is P m f (λ, ω) = 1 m 2 λ 2 + ρ 2 e − λ 2 +ρ 2 m 2f (λ, ω). For s ∈ R, we define the Sobolev space H s (H n ) as the closure of C ∞ 0 (H n ) for the norm f H s = (−∆) s/2 f L 2 . Note that f 2 H s ≈ ∞ −∞ H n ρ 2 + λ 2 s f (λ, ω) 2 \|c(λ)\| −2 dλdω. 3.1.2. A refined subcritical Sobolev inequality. Recall from (27) the definition of the spectral projector P m . For s ∈ (0, n/2], we define the Banach space B s as the closure of C ∞ 0 (H n ) for the B −( n 2 −s),∞ ∞ Besov-type norm: u B s = sup m≥1 m s−n/2 P m f L ∞ (H n ) . Lemma 3.1. For 0 < s ≤ n/2, there exists C > 0 such that for all f ∈ H s , one has f ∈ B s and f B s ≤ C f H s (28) ∀m > 0, \|P m f (x)\| ≤ C 1 m 2 + m n 2 −s e − ρ 2 m 2 f H s . (29) Proof. Let f ∈ H s . By the definition of P m and Fourier inversion formula (26), (30) \|P m f (x)\| ≤ ∞ −∞ S n−1 1 m 2 λ 2 + ρ 2 e − λ 2 +ρ 2 m 2 f (λ, ω) \|c(λ)\| −2 [x, (1, ω)] −ρ dλ dω ≤ ∞ −∞ S n−1 (λ 2 + ρ 2 ) s f (λ, ω) 2 \|c(λ)\| −2 dλ dω × 2 ∞ 0 1 m 4 (λ 2 + ρ 2 ) 2−s e − 2(λ 2 +ρ 2 ) m 2 \|c(λ)\| −2 dλ S n−1 [x, (1, ω)] −2ρ dω. Using that \|c(λ)\| −2 ∼ λ 2 as λ → 0, we obtain ∞ 1 λ 2 + ρ 2 2−s e − 2(λ 2 +ρ 2 ) m 2 m 4 \|c(λ)\| 2 dλ ≤ Ce − 2ρ 2 m 2 ∞ 1 e − 2λ 2 m 2 λ 4−2s λ n−1 m 4 dλ ≤ Ce − 2ρ 2 m 2 m n−2s ∞ 1/m e −2σ 2 σ 3−2s+n dσ ≤ Cm n−2s e − 2ρ 2 m 2 .(32) Finally, we claim that the spherical function-like integral S n−1 [x, (1, ω)] −2ρ dω = C π 0 (cosh \|x\| − sinh \|x\| cos α) −2ρ sin n−2 α dα, is uniformly bounded in \|x\|. Indeed, we have (for some constant c n > 0) F (r) = c n π 0 (cosh r − cos θ sinh r) 1−n sin n−2 θ dθ = c n π 0 (cosh r − sinh r + (1 − cos θ) sinh r) 1−n sin n−2 θ dθ = c n π 0 e (n−1)r (1 + e r sinh r(1 − cos θ)) 1−n sin n−2 θ dθ. We have max r∈[0,1] F (r) < ∞. Assuming r ≥ 1, we obtain: Combining this with (30), (31) and (32), we obtain (29). Inequality (28) follows. We next prove a refined Sobolev inequality which generalizes [IoPaSt12, Lemma 2.2,ii)] which treats the case s = 1, n = 3. It is in the spirit of the refined Sobolev embedding on Euclidean space in [GeMeOr96]. More precisely, the inequality we prove is the analog on the hyperbolic space of the inequality f L α (R n ) ≤ C f 2 α H n 2 − n α (R n ) f 1− 2 α B − n α ,∞ ∞ (R n ) , 2 < α < ∞. Proposition 3.2. Let 0 < s < min{2, n/2} and α such that 1 α = 1 2 − s n . There is a constant C > 0 such that for all f ∈ H s , (33) f L α ≤ C f 2 α H s f 1− 2 α B s . Proof. We use a method based on spectral calculus that goes back to [ChXu97]. By the definition (27) of P m (34) A 0 1 m P m (f ) dm = A 0 1 m 3 (λ 2 + ρ 2 )e − λ 2 +ρ 2 m 2f dm = 1 2 e − λ 2 +ρ 2 A 2f , for any A > 0. Step 1. Low-frequency bound. Here we prove the desired estimate for the low frequencies part e ∆ f = 1 0 1 m P m (f ) dm. Since e t∆ f L ∞ ≤ f L ∞ (see for instance [GrNo98]) and e t∆ f L 2 ≤ e −ρ 2 t f L 2 (this follows from (−∆f, f ) L 2 ≥ ρ 2 f 2 L 2 ), we get e t∆ f L α ≤ e −ct f L α , where c = 2ρ 2 α . Then e ∆ f L α ≤ C 1 0 1 m P m (f ) L α dm ≤ C 1 0 1 m 3 e ( 1 m 2 − 1 2 )∆ P √ 2 (f ) L α dm ≤ C 1 0 e c( 1 2 − 1 m 2 ) m 3 P √ 2 (f ) L α dm ≤ C P √ 2 (f ) L α ≤ C P √ 2 (f ) 2 α L 2 P √ 2 (f ) 1− 2 α L ∞ , and we conclude (33) by noting that since we are at low frequencies, P √ 2 (f ) L 2 ≤ C f H s . Now we shall treat the high frequencies. We let (35) g = 2 ∞ 1 1 m P m (f ) dm = f − e ∆ f in view of (34). To complete the proof of the proposition, in view of Step 1, we need to prove g L α ≤ C f 2 α H s f 1− 2 α B s , which we will do in two steps. We first introduce some notations. Let, for R > 0, A R = R n 2 − s 4 f B s 2 n−2s . We write g = g ≤A R + g >A R , where, if A R ≥ 1, g ≤A R = 2 A R 1 1 m P m (f ) dm, g >A R = 2 ∞ A R 1 m P m (f ) dm. and if A R < 1, g ≤A R = 0, g >A R = g. Step 2. In this step, we prove: (36) g α L α ≤ C ∞ 0 R α−3 g >A R 2 L 2 dR. Indeed, if A R ≥ 1, by the definition of B s and g ≤A R \|g ≤A R \| ≤ 2 A R 0 1 m m n 2 −s f B s dm = 4A n 2 −s R n − 2s f B s = R 2 . This inequality remains valid if A R < 1 since the left-hand side is zero. Thus µ ({\|g\| > R}) ≤ µ ({\|g >A R \| > R/2}) ≤ 4 R 2 g >A R 2 L 2 . g α L α = α ∞ 0 R α−1 µ ({\|g\| > R}) dR ≤ C ∞ 0 R α−3 g >A R 2 L 2 dR. Hence (36). Step 3. By (34) and the definition of g >A R , g >A R (λ, ω) = 1 − e − λ 2 +ρ 2 B 2 R f (λ, ω), where B R = max(A R , 1). By (36), (37) g α L α ≤ C ∞ 0 R α−3 ∞ 0 S n−1 1 − e − λ 2 +ρ 2 B 2 R 2 f (λ, ω) 2 \|c(λ)\| −2 dω dλ dR = C ∞ 0 S n−1 f (λ, ω) 2 \|c(λ)\| −2 ∞ 0 R α−3 1 − e − λ 2 +ρ 2 A 2 R 2 dR dω dλ. By the definition of A R and the change of variable in R r = R( n 2 − s) 4 f B s (λ 2 + ρ 2 ) n−2s 4 = A 2 R λ 2 + ρ 2 n−2s 4 , we deduce (38) ∞ 0 R α−3 1 − e − λ 2 +ρ 2 A 2 R 2 dR ≤ C f α−2 B s (λ 2 + ρ 2 ) n−2s 4 (α−2) ∞ 0 r α−3 1 − e −r − 4 n−2s 2 dr. Note that α − 3 > −1 and, as r goes to infinity, r α−3 1 − e −r − 4 n−2s 2 ≈ r 4s−8 n−2s −1 , which proves (using that s < n 2 and s < 2) that the integral at the right-hand side of (38) is finite. Going back to (37), we obtain g α L α ≤ C f α−2 B s ∞ 0 S n−1 (λ 2 + ρ 2 ) s \|f (λ, ω)\| 2 \|c(λ)\| −2 dω dλ = C f α−2 B s f 2 H s which concludes the proof. An interpolation inequality. Proposition 3.3. There exists θ ∈ (0, 1) and a constant C > 0, both depending on p, such that (39) e it∆ f S 0 (R) ≤ C e it∆ f θ L ∞ t B s f 1−θ H s , where s = n 2 − n p+1 ∈ (0, 1). Proof. First case: 1 < p ≤ 4 n + 1. In this case, S 0 (R) = L p+1 (R × H n ). By the refined Sobolev inequality (33), e it∆ f L ∞ t L p+1 x ≤ C e it∆ f 2 p+1 L ∞ t H s e it∆ f 1− 2 p+1 L ∞ t B s , where by definition, s = n 2 − n p+1 ∈ (0, (1 + 2/n) −1 ). Hence (40) e it∆ f L ∞ t L p+1 x ≤ C f 2 p+1 H s e it∆ f 1− 2 p+1 L ∞ t B s . Moreover, by the Strichartz inequalities on H n (see Theorem 2.1), for all γ with 2 < γ < 2 + 4 n , (41) e it∆ f L γ t L γ x ∩L γ t L β x ≤ C f L 2 , where β = 2nγ nγ−4 . Note that lim γ→2 β = +∞ if n = 2 2n n−2 if n ≥ 3 . Choosing γ > 2 close enough to 2, we obtain 2 < γ < p + 1 < β. For these value of γ, (41) implies (42) e it∆ f L γ t L p+1 x ≤ C f L 2 . Combining (40), (42), and Hölder's inequality we obtain (39) with θ = 1 − 2 p+1 1 − γ p+1 . Second case: 4 n + 1 < p and p < 1 + 4 n−2 if n ≥ 3. In this case S 0 (R) = L a (R, L p+1 ) where a = 2(p − 1)(p + 1) 4 − (n − 2)(p − 1) . By Strichartz estimates (Theorem 2.1), (43) e it∆ f L γ t L p+1 x ≤ C f L 2 , where γ = 4(p+1) n(p−1) < a if p > 1 + 4 n . By the generalized Sobolev inequality (33), with α = p + 1, s = n 2 − n p+1 ∈ (0, 1), ∀t, e it∆ f L p+1 ≤ C e it∆ f 2 p+1 H s e it∆ f 1− 2 p+1 B s . Hence (44) e it∆ f L ∞ t L p+1 x ≤ C f 2 p+1 H s e it∆ f 1− 2 p+1 L ∞ t B s . Combining (43) and (44) and using γ < a, we obtain (39) in this case also. Profile decomposition. 3.2.1. Linear profile decomposition. Recall from §3.1.1 the definition of the isometry group G. We denote by d the geodesic distance on H n . Define, for f ∈ H 1 , f Σ = sup m≥1,t∈R m 1−n/2 log(m + 2) P m e it∆ f L ∞ (H n ) . In particular we have for all s ∈ (0, 1), for all t ∈ R, (45) e it∆ f B s ≤ C s f Σ . By (29), f Σ ≤ C f H 1 . Proposition 3.4 (Subcritical profile decomposition). Let (f k ) k be a bounded sequence in H 1 (H n ). Then there exists a subsequence of (f k ) k (that we still denote by (f k ) k ), a family (ϕ j ) j≥1 of functions in H 1 (H n ) and, for each j ≥ 1, a sequence (t j,k , h j,k ) k in R × G such that j≥1 ϕ j 2 H 1 < ∞ (46) j = j ′ =⇒ lim k→∞ d(h j,k · 0, h j ′ ,k · 0) + \|t j,k − t j ′ ,k \| = +∞ (47) ∀j ≥ 1, e −it j,k ∆ f k (h −1 j,k ·) −−−⇀ k→∞ ϕ j weakly in H 1 (48) and, denoting by r J,k = f k − J j=1 e it j,k ∆ ϕ j (h j,k ·) we have (49) lim J→∞ lim k→∞ r J,k Σ + e it∆ r J,k S 0 (R) = 0. We refer for example to [MeVe98], [Ke01] for profile decompositions for the Schrödinger equation on R n . The H 1 -critical profile decomposition on the space H 3 was constructed in [IoPaSt12] (see also [LaSuSo14Pb] for the analogous result for the wave equation on H n ). In this setting, profiles might concentrate at one point of H n , and become solutions of the Schrödinger equation on the Euclidean space. In our case, this is prevented by the subcriticality of the problem. Notation 3.5. In what follows, we will often extract subsequences from a given sequence. To lighten notations, we will always, as in the preceding proposition, use the same notation for the extracted subsequence and the original sequence. Remark 3.6. It follows from (48) that if the f k are all radial, then we can assume that h j,k is the identity of H n for all j, k, and that all profiles ϕ j are radial. Definition 3.7. If (ϕ j ; (t j,k , h j,k ) k ) j≥1 satisfies the conclusions of Proposition 3.4, we say that it is a profile decomposition for the sequence (f k ) k . We postpone the proof of the profile decomposition, and state the following Pythagorean expansions, to be proved at the end of this section: Proposition 3.8. Let λ < (n−1) 2 4 . Let (f k ) k be a bounded sequence in H 1 that admits a profile decomposition (ϕ j ; (t j,k , h j,k ) k ) j≥1 . Then ∀J ≥ 1, lim k→∞ f k 2 H λ − J j=1 ϕ j 2 H λ − r J,k 2 H λ = 0 (50) lim k→∞ f k p+1 L p+1 − +∞ j=1 e −it j,k ∆ ϕ j p+1 L p+1 = 0.(51) To prove Proposition 3.4 and Proposition 3.8, we need the following lemma: Lemma 3.9. Let f, g ∈ H 1 (H n ), (t k , h k ) k and (t ′ k , h ′ k ) two sequences in R × G such that lim k→∞ d(h k · 0, h ′ k · 0) + \|t k − t ′ k \| = +∞. Then ∀λ < (n − 1) 2 4 , lim k→∞ e it k ∆ f (h k ·), e it ′ k ∆ g(h ′ k ·) H λ = 0 (52) lim k→∞ e it k ∆ f (h k x) e it ′ k ∆ g(h ′ k x) p dx = 0. (53) Proof. Proof of (52). By density we can assume, without loss of generality, f, g ∈ C ∞ 0 (H n ). We have e it k ∆ f (h k ·), e it ′ k ∆ g(h ′ k ·) H λ = − (∆ + λ)e it k ∆ f (h k ·), e it ′ k ∆ g(h ′ k ·) L 2 = −(∆ + λ)f, e i(t ′ k −t k )∆ g(h ′ k • h −1 k ·) L 2 . If \|t k − t ′ k \| → ∞ as k → ∞, then e i(t ′ k −t k )∆ g L ∞ → 0 and the result follows. If not, we can assume without loss of generality: lim k→∞ t ′ k − t k = θ ∈ R, and (52) is equivalent to (54) lim k→∞ (∆ + λ)f, (e iθ∆ g)(h ′ k • h −1 k ·) L 2 = 0. Furthermore (55) lim d(0, h ′ k • h −1 k · 0) = +∞. If θ = 0, the support of (∆ + λ)f and (e iθ∆ g)(h ′ k • h −1 k ·) are disjoint for large k and (54) follows. If not, one can approximate e iθ∆ g, in L 2 , by compactly supported functions which yields (54), arguing again on the supports. Proof of (53). Note that by Sobolev embeddings and conservation of the H 1 -norm for the linear equation, the sequences e it k ∆ f L p+1 k and e it ′ k ∆ g L p+1 k are bounded. Furthermore, lim k→∞ t k = ±∞ ⇒ lim k→∞ e it k ∆ f L p+1 = 0 and lim k→∞ t ′ k = ±∞ ⇒ lim k→∞ e it ′ k ∆ g L p+1 = 0. In both cases, (53) holds. Arguing by contradiction and extracting subsequences, we are reduced to prove (53) when t k and t ′ k have finite limits as k goes to infinity. Time translating, we can also assume that these limits are both 0, and we see that it is sufficient to prove: ∀f, g ∈ H 1 (H n ), lim k→∞ \|f (h k · x)\|\|g(h ′ k · x)\| p dx = 0, provided (55) holds. This follows by approximating f and g, in L p+1 , by compactly supported functions and arguing on the supports. We next prove Proposition 3.4. Proof. We shall use the following general abstract concentration-compactness result (see Proposition 2.1, Definition 2.2 and Theorem 2.3 of [ScTi02]): ScTi02]). Let H be a separable Hilbert space and D a group of unitary operators in H such that if (g k ) k ∈ D N does not converge weakly to zero, then there exists a strongly convergent subsequence of g k such that s-lim k g k = 0. If (f k ) k ∈ H N is a bounded sequence, then (extracting subsequences in k), there exist ϕ j ∈ H, (g j,k ) k ∈ D N , j ≥ 1 such that Theorem B ([(56) j≥1 ϕ j 2 ≤ lim sup k→∞ f k 2 , (57) (g j,k ) −1 g j ′ ,k −−−⇀ k→∞ 0 for j = j ′ ,(58)(g j,k ) −1 f k −−−⇀ k→∞ ϕ j , and for all φ ∈ H, (59) lim J→∞ lim k→∞ sup g∈D g f k − J j=1 g j,k ϕ j , φ = 0. Note that in [ScTi02, Theorem 2.3], (59) is stated without parameter J and with an infinite sum. However (59) follows easily from the proof in [ScTi02]. We apply this result for H = H 1 (H n ) and D = {g : H 1 (H n ) → H 1 (H n ) , g(f )(x) = e it∆ f (h · x), (t, h) ∈ R × G}. The hypothesis on D is satisfied in view of Lemma 3.9. Indeed, if g k = (t k , h k ) k ∈ D does not converge weakly to zero, then by taking g ′ k = (0, Id) ∈ D, the conclusion (52) ensures that d(h k · 0, 0) + \|t k \| does not tend to +∞. Therefore, in view of the definition of G, there exists strongly convergent subsequences of (t k ) k and (h k ) k in R and G. This implies that g k has a strong limit (θ, h) ∈ D. Now we transcribe the results of this theorem in our context. The statements (56) and (58) imply directly (46) and (48). If the conclusion of (47) does not hold, than by the same argument used above to check the assumption on D, we obtain a contradiction with (57). Therefore (47) is satisfied. We are left with proving (49). By the interpolation inequality (39), it is sufficient to consider only the · Σ norm in (49). We will deduce (49) from the following lemma, proved below. Lemma 3.10. Let (r k ) k be a bounded sequence in H 1 (H n ). Assume that for all sequence (t k , h k ) k in R × G, (60) e it k ∆ r k (h k ·) −−−⇀ k→∞ 0 weakly in H 1 (H n ).e it∆ r J ℓ(ν) ,k(ν) (h·) , φ α H 1 ≤ 1 2 ν r J ℓ(ν) ,k(ν) Σ ≥ ε 2 . As a consequence of the first inequality and the density of (φ α ) α∈N in H 1 , we obtain that for all sequence (t ν , h ν ) ν in R × G, e itν ∆ r J ℓ(ν) ,k(ν) (h ν ·) −−−⇀ ν→∞ 0 weakly in H 1 (H n ). This proves that (r J ℓ(ν) ,k(ν) ) ν contradicts Lemma 3.10, concluding the proof. Proof of Lemma 3.10. To prove Lemma 3.10, we argue by contradiction. Assume that (61) does not hold. Then there exist a subsequence of (r k ) k (still denoted by ( r k ) k ), ε > 0, a sequence (t k , x k , m k ) k in R × H n × [1, ∞) such that (63) ∀k, P m k e it k ∆ r k (x k ) m 1−n/2 k log(2 + m k ) ≥ ε. Combining with (29) for s = 1, we deduce ∀k, ε log(2 + m k ) ≤ C r k H 1 , which proves that the sequence (m k ) k is bounded. Extracting subsequences, we can assume (64) lim k→∞ m k = m ∈ [1, +∞). Since G acts transitively on H n , we can choose for all k an isometry h k ∈ G such that h k (0) = x k . Let g k (x) = e it k ∆ r k (h k · x). By (63) and (64), and since h k commutes with P m , there exists ε ′ > 0 such that for large k (65) \|P m g k (0)\| ≥ ε ′ . By assumption (60), (66) g k −−−⇀ k→∞ 0 weakly in H 1 . It follows from the inequality (29) that f → P m (f )(0) is a continuous linear form on H 1 (H n ), which combined with (65) and (66) yields a contradiction. The proof of Lemma 3.10 (and thus of Proposition 3.4) is complete. Proof of Lemma 3.8. We first prove (50). We have f k = J j=1 e it j,k ∆ ϕ j (h j,k ·) + r J,k , and thus f k 2 H 1 = J j=1 ϕ j 2 H 1 + r J,k 2 H 1 + A k + B k , where A k = 2 1≤j<j ′ ≤J e it j,k ∆ ϕ j (h j,k ·), e it j ′ ,k ∆ ϕ j ′ (h j ′ ,k ·) , B k = 2 J j=1 e it j,k ∆ ϕ j (h j,k ·), r J,k . The sum A k goes to 0 as k goes to infinity by the orthogonality of the profiles ensured by Lemma 3.9 and (47). Moreover, the term B k equals 2 J j=1 ϕ j , e −it j,k ∆ f k (h −1 j,k ·) − ϕ j H 1 − 2A k , which goes to 0 as k goes to infinity by (48). We next prove (51). By the refined Sobolev embedding (33) applied to s = n(p−1) 2(p+1) ∈ (0, 1) and (45), r J,k L p+1 ≤ e it∆ r J,k L ∞ L p+1 ≤ r J,k 2 p+1 H 1 e it∆ r J,k p−1 p+1 L ∞ B s ≤ r J,k 2 p+1 H 1 r J,k p−1 p+1 Σ , so using (49), lim J→∞ lim k→∞ r J,k L p+1 = 0. By the Poincaré-Sobolev inequality (6) and (46), j≥1 ϕ j p+1 L p+1 < ∞. We are thus reduced to prove that for a fixed J, lim k→∞ J j=1 (e −it j,k ∆ ϕ j )(h −1 j,k x) p+1 dx − J j=1 (e −it j,k ∆ ϕ j )(h −1 j,k x) p+1 dx = 0. This last property follows from the inequality (67) ∀(a 1 , . . . , a J ) ∈ [0, +∞) J , J j=1 a p+1 j − J j=1 a j p+1 ≤ C J 1≤j,j ′ ≤J j =j ′ a p j a j ′ , and the limit (53) of Lemma 3.9, which concludes the proof. Nonlinear profiles and scattering. Let (ϕ j , (t j,k , h j,k ) k ) j≥1 be a profile decomposition for a bounded sequence in H 1 . Extracting subsequences, we can assume: (68) ∀j ≥ 1, lim k→∞ t j,k = τ j ∈ [−∞, +∞]. For any j, we denote by U j the nonlinear profile associated to ϕ j and the sequence (t j,k ) k . This is by definition the unique solution of (1) such that t j,k ∈ I max (U j ) for large k and (69) lim k→∞ e it j,k ∆ ϕ j − U j (t j,k ) H 1 = 0. Assuming (68), there always exists a nonlinear profile U j : this follows from the local Cauchy theory if τ j ∈ R, and from the existence of wave operators (see Proposition 2.5) if τ j = ±∞. Note that if T + (U j ) is finite, then τ j < T + (U j ), and similarly, if T − (U j ) is finite, T − (U j ) < τ j . We next prove: Proposition 3.11. Let (f k ) k≥1 be a bounded sequence in H 1 that admits a profile decomposition (ϕ j , (t j,k , h j,k ) k ) j≥1 . Assume that for all j ≥ 1, the corresponding nonlinear profile U j scatters forward in time. Then for large k, the solution u k of (1) with initial data f k at t = 0 scatters forward in time. Furthermore, lim k→∞ u k S 0 (0,+∞) < ∞. Proof. This is a standard consequence of the long-time perturbation theory (Proposition 2.7) applied to u k and u J,k = J j=1 U j,k , where U j,k (t, x) = U j (t + t j,k , h j,k · x). We sketch the proof for 1 < p ≤ 4 n +1; recall that in this case S 0 (I) = L p+1 (I, L p+1 ), N 0 (I) = L p p+1 (I, L p p+1 ). We refer to [FaXiCa11] for a very close proof, in the Euclidean setting, in the case p > 4 n + 1. Step 1. Uniform bound on the S 0 norm. We first prove that there is a constant M > 0, depending on the sequence (f k ) k , but not on J, such that (70) ∀J, lim k→∞ u J,k S 0 (0,+∞) ≤ M. To this purpose we first use inequality (67), (71) +∞ 0 \|u J,k \| p+1 dµ(x) dt − J j=1 +∞ 0 H n \|U j,k \| p+1 dµ(x) dt ≤ C 1≤j,j ′ ≤J j =j ′ +∞ 0 H n \|U j,k (x)\| p \|U j ′ ,k (x)\| dµ(x) dt. We next prove (72) j = j ′ =⇒ lim k→∞ +∞ 0 H n \|U j,k \| p \|U j ′ ,k \| dµ(x) dt = 0. The term in the limit (72) equals +∞ 0 H n \|U j (t j,k + t, h j,k · x)\| p \|U j ′ (t j ′ ,k + t, h j,k · x)\| dµ(x) dt We first note that (73) ∀f, g ∈ L p+1 (R × H n ), lim k→∞ R H n \|f (t j,k + t, h j,k · x)\| p \|g(t j ′ ,k + t, h j ′ ,k · x)\| dµ(x) dt = 0. Indeed, this is obvious, arguing on the supports, and using the pseudo-orthogonality (48) of the parameters, if f and g are compactly supported. The general case follows by density. If U j and U j ′ are globally defined, (72) follows immediately from (73) with f = U j and g = U j ′ . If U j and U j ′ are not globally defined backward in time, (72) follows from (73) with f = χ t≥τ j U j and g = χ t≥τ j ′ U j ′ , where χ t≥A is the characteristic function of [A, +∞), and τ j , τ j ′ are defined in (68). The other cases are similar. Combining (71) and (72), we get lim k→∞ u J,k p+1 S 0 (0,+∞) = lim k→∞ J j=1 U j (t j,k + ·, h j,k ·) p+1 S 0 (0,+∞) = J j=1 U j p+1 S(τ j ,+∞) , which yields (70) since +∞ j=1 U j p+1 S(τ j ,+∞) is finite by (46) and the small data theory for (1). Step 2. End of the proof. Fix J ≥ 1 such that lim k→∞ e it∆ r J,k S 0 (R) ≤ ε(M ) 4 , where ε(M ) is given by Proposition 2.7. Recall the notation u J,k = J j=1 U j,k . Then i∂ t u J,k + ∆u J,k + \|u J,k \| p−1 u J,k = J j=1 U j,k p−1 J j=1 U j,k − J j=1 \|U j,k \| p−1 U j,k e J,k . We have e J,k 1+ 1 p N 0 (0,+∞) = e J,k 1+ 1 p L 1+ 1 p ≤ C +∞ 0 H n 1≤j,j ′ ≤J j =j ′ \|U j,k \| p−1 U j ′ ,k 1+ 1 p , which goes to 0 as k goes to infinity, using the pseudo-orthogonality (48) of the sequence of parameters and a proof similar to the one in Step 1. Choosing k 0 large, so that for k ≥ k 0 e J,k N 0 (0,+∞) + e it∆ r J,k S 0 (0,+∞) ≤ ε(M ) 2 , we obtain by using (10) and (69) that for k ≥ k 0 , e it∆ f k − e it∆ u J,k (0) S 0 (0,+∞) = e it∆ r J,k S 0 (0,+∞) ≤ ε(M ) 2 , and the results follows from long-time perturbation theory (Proposition 2.7) applied to u k and u J,k . 3.3. Existence of the critical solution. In this section we shall prove Theorem 3. We will also prove the nonradial version of this result: Proposition 3.12 (Nonradial critical element). Let λ < (n−1) 2 4 , 1 < p < 1 + 4 n−2 . There exists a global solution u c of equation (1) and a family (h(t)) t∈R of elements of G such that {u c (t, h(t)·), t ∈ R} has compact closure in H 1 (H n ), E λ (u c (0)) ≤ E λ (Q λ ), u c (0) H λ ≤ Q λ H λ , and, if E λ (u(0)) < E λ (u c (0)), u(0) H λ ≤ Q λ H λ , then the solution u of equation (1) scatters in both time directions. We will use the compactness/rigidity method initiated in [KeMe06]. We fix λ < (n−1) 2 4 . Let 0 < ω ≤ 1. We introduce the following set: K ω = f ∈ H 1 (H N ) : E λ (f ) ≤ ωE λ (Q λ ) and f 2 H λ ≤ Q λ 2 H λ . (note that K ω also depends on λ, which will be fixed in all this subsection). Theorem 1 and Lemma 2.16 yield the following facts. First, the set K ω is invariant with respect to the nonlinear evolution (1). Second, if u 0 ∈ K ω , then its evolution through equation (1) is global in time. Third, if u 0 ∈ K ω , then u 0 H λ ≤ ω 1 2 Q λ H λ . Therefore, for ω small enough, starting with u 0 ∈ K ω we obtain a scattering solution of (1), in view of the small data theory (Corollary 2.8). We can then define ω 0 = sup 0 < ω ≤ 1, u 0 ∈ K ω =⇒ the solution u of (1) scatters in both time directions . Since u λ = e itλ Q λ is a non scattering solution of (1) it follows that ω 0 ≤ 1. Note that if ω 0 = 1, then this solution is a critical element in the sense of Proposition 3, and Proposition 3 follows. We shall now focus on the remaining cases ω 0 < 1 and prove: Proposition 3.13. Let λ, ω 0 and K ω 0 be as above. Assume ω 0 < 1. Then there exists a solution u c of (1) such that u c (0) ∈ K ω 0 u c S 0 (−∞,0) = u c S 0 (0,+∞) = +∞, and there exists a h : R → G such that K = u c (t, h(t)·), t ∈ R has compact closure in H 1 (H n ). Similarly, define K ω as the subset of the elements of K ω that are radially symmetric, and defineω 0 as ω 0 , replacing K ω by K ω . Proposition 3.14. Assumeω 0 < 1. Then there exists a radially symmetric solution v c of (1) such that v c (0) ∈ Kω 0 v c S 0 (−∞,0) = v c S 0 (0,+∞) = +∞, and K = v c (t), t ∈ R has compact closure in H 1 (H n ). Note that ω 0 ≤ω 0 . We conjecture that if p ≥ 1 + 4 n , ω 0 = 1. We will show later (see Section 4) thatω 0 = 1 if n = 3 and p ≥ 7 3 , or if n = 2 and p ≥ 3. The proofs of Propositions 3.13 and 3.14 are by now standard. We give the proof of Proposition 3.13 for the sake of completeness. In view of Remark 3.6, the proof of Proposition 3.14 is the same, assuming that all the functions are radial and taking off the isometries h(t). We first prove a preliminary result. We denote by U (t) the nonlinear evolution (1): if u 0 ∈ H 1 , u(t) = U (t)u 0 is the unique solution of (1), with maximal time of existence I max (u 0 ) = (T − (u 0 ), T + (u 0 )). Lemma 3.15. Assume λ < (n−1) 2 4 . Let (f k ) k be a sequence in H 1 (H n ) such that lim k→∞ E λ (f k ) = ω 0 E λ (Q λ ), f k H λ ≤ Q λ H λ and U (t)f k S 0 (T − (f k ),0) k→∞ −→ ∞, U (t)f k S(0,T + (f k )) k→∞ −→ ∞. Then there exists a subsequence of (f k ) k (that we still denote by (f k ) k ), a sequence (h k ) k ∈ G N , and V ∈ H 1 (H n ) with E λ (V ) = ω 0 E λ (Q λ ), V H λ ≤ Q λ H λ , such that f k (h k ·) − V H 1 k→∞ −→ 0. Proof. Extracting subsequences, we can assume by Proposition 3.4 that the sequence (f k ) k has a profile decomposition (ϕ j ; (t j,k , h j,k ) k ) j≥1 . By the Pythagorean expansion (50) for large k, and since f k H λ < Q λ H λ , we obtain (74) ∀j ≥ 1, ϕ j 2 H λ ≤ Q λ 2 H λ . By the Poincaré-Sobolev inequality and the value of its best constant (20) we have E λ (ϕ j ) ≥ ϕ j H λ 2 2 1 − 2 p + 1 ϕ j p−1 H λ Q λ p−1 H λ , so we get (75) ∀j, ϕ j = 0 =⇒ E λ (ϕ j ) > 0. Combining the Pythagorean expansions (50) and (51) with the assumption f k ∈ K ω 0 , we obtain that for all J ≥ 1, (76) 1 2 J j=1 ϕ j 2 H λ + lim k→∞   1 2 r J,k 2 H λ − j≥1 1 p + 1 e −it j,k ∆ ϕ j p+1 L p+1   ≤ ω 0 E λ (Q λ ). Extracting again subsequences, we can assume that for all j, there exists a nonlinear profile U j associated to (ϕ j , (t j,k ) k ) (see §3.2.2). Using the conservation of the mass and energy for each of this nonlinear profiles, we can write (76): (77) J j=1 E λ (U j ) + lim k→∞   1 2 r J,k 2 H λ − j≥J+1 1 p + 1 e −it j,k ∆ ϕ j p+1 L p+1   ≤ ω 0 E λ (Q λ ). Note that for large J, j≥J+1 e −it j,k ∆ ϕ j p+1 L p+1 ≤ C j≥J+1 ϕ j p+1 H 1 which is finite, independent of k and goes to 0 as J → ∞. Furthermore, by (75), E λ (U j ) is nonnegative (and positive if ϕ j = 0). Thus (77) implies: +∞ j=1 E λ (U j ) ≤ ω 0 E λ (Q λ ). By (75), if there are more than two indexes j such that ϕ j = 0, we obtain E λ (U j ) < ω 0 E λ (Q λ ) for all j ≥ 1. In view of (74) and Theorem 1, we deduce that for all j, U j is globally defined and satifies U j (0) ∈ K ω for some ω < ω 0 . By the definition of ω 0 , all the nonlinear profiles U j scatter in both time directions. From Proposition 3.11, we deduce that f k scatters in both time directions and (78) lim k→∞ U (t)f k S 0 (R) < ∞, which contradicts our assumptions. Thus there is at most one nonzero profile, say ϕ 1 , and r J,k = r 1,k for all J ≥ 1. Going back to (77), we see that if lim k→∞ r 1,k H 1 > 0, then E λ (U 1 ) < ω 0 E λ (Q λ ). Arguing as before, we would obtain again that (78) holds, a contradiction. Thus lim k→∞ r 1,k H 1 = 0, E λ (U 1 ) = ω 0 E λ (Q λ ). Hence (letting V = ϕ 1 , t k = −t 1,k , h k = h −1 1,k ), f k (h k ·) − e −it k ∆ V H 1 k→∞ −→ 0. It remains to prove that t k is bounded. If t k k→∞ −→ −∞ then by using Strichartz and Sobolev estimates we get e it∆ f k S 0 (0,∞) ≤ e it∆ (f k (h k ·) − e −it k ∆ V ) S 0 (0,∞) + e it∆ V S 0 (−t k ,∞) k→∞ −→ 0. Corollary 2.8 insures that for k large, U (t)f k scatters forward in time in H 1 , and its S 0 (0, ∞) norm is bounded from above by a constant independent of k. This contradicts the hypothesis, so the limit of t k cannot be −∞. In the same manner the limit cannot be ∞. In conclusion the limit of the sequence t k is finite and we conclude by using the H 1 continuity of the free Schrödinger evolution. Proof of Proposition 3.13. Step 1. Existence of u c . Since ω 0 < 1, by its definition we obtain a sequence of numbers ω k approaching ω 0 with ω 0 ≤ ω k < 1 and a sequence of functions f k such that E λ (f k ) ≤ ω k E λ (Q λ ), f k H λ ≤ Q p H λ whose global evolution U (t)f k through equation (1) satisfies U (t)f k S(R) = ∞. There exists a sequence t k such that U (t − t k )f k S(−∞,0) k→∞ −→ ∞, U (t − t k )f k S(0,∞) k→∞ −→ ∞. To simplify notations, we denote by f k the translations in time U (−t k )f k . In view of the global in time result of Theorem 1 we have U (t)f k H λ < Q λ H λ for all t ∈ R. We have lim k→∞ E λ (f k ) ≤ ω 0 E λ (Q λ ), and we claim that equality holds. Otherwise there exists k such that E λ (f k ) < ω 0 E λ (Q λ ), and by definition of ω 0 we obtain that U (t)f k scatters in both time directions, which is not true. Therefore we can apply Lemma 3.15 to conclude that there exists u 0c ∈ H 1 with E λ (u 0c ) = ω 0 E λ (Q λ ), u 0c H λ ≤ Q λ H λ and a sequence (h k ) ∈ G N such that f k (h k ·) − u 0c H 1 k→∞ −→ 0. By Proposition 2.7 applied to U (t)f k (h k ·) and U (t)u 0c , and since lim k→∞ U (t)f k S(0,∞) = lim k→∞ U (t)f k S(−∞,0) = ∞, we obtain U (t)u 0c S(0,∞) = U (t)u 0c S(−∞,0) = ∞. Thus u c = U (t)u 0c does not scatter in H 1 neither forward nor backward in time. Step 2. We show that there exists h : R → G such that the set {u c (t, h(t)·), t ∈ R} has compact closure in H 1 . By a standard lifting argument, it is sufficient to prove that for all sequence of times (t k ) k , there exists a subsequence of (t k ) k (still denoted by (t k ) k ) and a sequence (h k ) k ∈ G N such that (u(t k , h k ·)) k converges in H 1 . In view of Lemma 2.16, u 0c satisfy the assumptions of the global existence result Theorem 1, so it follows that {u c (t k ), k ∈ N} is a bounded set of H 1 . Also, by the mass and energy conservations, E λ (u c (t k )) = E λ (u 0c ) = ω 0 E λ (Q λ ). From Step 1 we know that U (t)u(t k ) does not scatter in H 1 neither forward nor backward in time. Then in view of Proposition 2.6 we obtain that U (t)u(t k ) S(0,∞) = U (t)u(t k ) S(−∞,0) = ∞. Therefore we can apply Lemma 3.15 to obtain the existence of V ∈ H 1 and a sequence (h k ) k ∈ G N such that u c (t k , h k ·) − V H 1 k→∞ −→ 0. This concludes the proof. 3.4. Mass-subcritical case. We conclude this section by proving Proposition 1.5. We assume n ≥ 3, 1 < p < 1 + 4 n . By Proposition 2.13, there exists λ < (n−1) 2 4 , α > 0 such that Q λ is a minimizer for (21). We will prove that E λ (v c (0)) < E λ (Q λ ) by contradiction, in the spirit of the proof of the stability of the orbital stability of the ground states by Cazenave and Lions [CaLi82]. Assume E λ (v c (0)) = E λ (Q λ ). For β > 0, we will consider u β , the solution of (1) with initial data u β (0) = βQ λ . Then E λ (u β (0)) = β 2 2 Q λ 2 H λ − β p+1 p + 1 Q λ p+1 L p+1 , and thus (using the equality Q λ 2 H λ = Q λ p+1 L p+1 ), β < 1 =⇒ E λ (u β (0)) < E λ (Q λ ) = E λ (v c ) and u β (0) H λ < Q λ H λ . By the definition of v c , we deduce that the solution u β scatters in both time directions if β < 1. In particular, u β is global and (79) lim t→∞ u β (t) L p+1 = 0. Furthermore, by Theorem 1, again if β < 1, (80) ∀t ∈ R, u β (t) H λ ≤ Q λ H λ , u β (t) L p+1 ≤ Q λ L p+1 Let k be an integer, and β k = 1 − 2 −k . By (79), there exists t k such that (81) u β k (t k ) L p+1 ≤ 2 −k . Let f k = 1 β k u β k (t k ). Then, by mass conservation, f k L 2 = 1 β k u β k (0) L 2 = Q λ L 2 = α. By energy conservation, E(β k f k ) = E(u β k (0)) k→∞ −→ E(Q λ ) = e(α), Thus, using also (80) E(f k ) = 1 2 − β 2 k 2 ∇f k 2 L 2 − 1 p + 1 − β p+1 k p + 1 f k p+1 L p+1 + E(β k f k ) k→∞ −→ E(Q λ ). Finally, we have obtained that (f k ) k is a minimizing sequence for the minimization problem (21). By Proposition 2.13, f k converges (extracting subsequences if necessary) to a minimizer, a contradiction with (81). The rigidity argument In this subsection we shall prove the following proposition, which, together with Proposition 3.14 will imply the Theorem 2, (a). Proposition 4.1. Let n ∈ {2, 3}, 1 + 4 n ≤ p < 1 + 4 n−2 and λ < (n−1) 2 4 . Let u be a radial solution of (1) such that {u(t)} is a compact subset of H 1 rad . If u 0 is radial, E λ (u 0 ) < E λ (Q λ ) and u 0 H λ ≤ Q λ H λ , then u ≡ 0. In order to prove this proposition we shall need some additional information. We first recall the classical virial formula: ∂ 2 t H n \|u(t)\| 2 r 2 = G(u(t)), v(r) = sinh r r n−1 2 u(r), wee see that it is enough to show that H 1 rad (R n ) is compactly embedded in L p+1 (w(r)dr) with w(r) = r sinh r (p−1)(n−1) 2 for the first embedding result and in L p+1 (w(r)dr) with w(r) = (1 + (n − 1) r cosh r sinh r )w(r) for the second one. This follows immediately from the compact embedding of H 1 rad (R n ) into L p+1 (R n ) (see [We82, Compactness Lemma p. 570]). We shall use the following crucial lemma. Lemma 4.3. Let n ∈ {2, 3}, 1 + 4 n ≤ p < 1 + 4 n−2 and λ < (n−1) 2 4 . Then inf f ∈H 1 rad , E λ (f )≤E λ (Q λ ), f H λ ≤ Q λ H λ G(f ) = 0, and the minimizing sequences converge (after extraction) in H to the constant zero function or to e iθ Q for some θ ∈ R, Q ∈ Q λ . Multiplying the equation by ϕ ∂ r f + ∂rϕ 2 f , integrating from 0 to infinity and taking the real part, we obtain by integration by parts that 0 = ∞ 0 \|∂ r f \| 2 ∂ r ϕ − (n − 1) cosh r sinh r ϕ + \|f \| 2 − ∂ 3 r ϕ 4 + n − 1 4 ∂ r cosh r sinh r ∂ r ϕ + 1 2 ∂ r gϕ + ∞ 0 \|f \| p+1 − p − 1 2(p + 1) h∂ r ϕ + 1 p + 1 ∂ r h ϕ . We choose ϕ(r) = r sinh r n−1 , so that ∂ 3 r ϕ = 2(n − 1) 2 sinh n−1 r + (n − 1)(2n − 5) sinh n−3 r − (n − 1)(n − 3)r cosh r sinh n−4 r + (n − 1) 2 ∂ r cosh 2 r sinh 2 r ϕ , ∂ r cosh r sinh r ∂ r ϕ = (n − 1) sinh n−1 r + (n − 2) sinh n−3 r + (n − 1)∂ r cosh 2 r sinh 2 r ϕ , and get, using that f 2 H = H n \|∂ r f \| 2 dµ − (n−1) 2 4 H n \|f \| 2 , 0 = f 2 H + H n \|f \| 2 (n − 1)(n − 3) 4 r cosh r − sinh r sinh 3 r + r 2 ∂ r g + H n \|f \| p+1 − p − 1 2(p + 1) h(1 + (n − 1) r cosh r sinh r ) + 1 p + 1 r∂ r h . It follows then that G(f ) = \|f \| p+1 4(p − 1) p + 1 1 + (n − 1) r cosh r sinh r (h − 1) − 8 p + 1 r∂ r h − 4 \|f \| 2 r∂ r g. In order to obtain that G(f ) > 0 we want to have the coefficients of \|f \| p+1 and of \|f \| 2 positive. The coefficient of \|f \| p+1 is 16(p − 1) (16 − µ)(p + 1) (n − 1)r 2 sinh 2 r (p − 1)(n − 1) cosh 2 r + 2 + 2(n − 1)(p − 4) r cosh r sinh r + p − 5 . From the behavior near r = 0 we see that p ≥ 1 + 4 n is a necessary condition for positivity. Moreover, since the coefficient of p is positive, in order to show that the function is positive for p ≥ 1 + 4 n , it is enough to show it for p = 1 + 4 n , which is equivalent to (2n − 2)r 2 cosh 2 r + nr 2 − (3n − 4)r cosh r sinh r − 2 sinh 2 r ≥ 0. This function vanishes at r = 0 and its first four derivatives are (4n−4)r cosh 2 r+(4n−4)r 2 cosh r sinh r+2nr−3n cosh r sinh r−(3n−4)r(cosh 2 r+sinh 2 r), (4n−4) cosh 2 r+4nr cosh r sinh r+(4n−4)r 2 (cosh 2 r+sinh 2 r)+2n+(−6n+4)(cosh 2 r+sinh 2 r), (−12n + 8) cosh r sinh r + (12n − 8)r(cosh 2 r + sinh 2 r) + (16n − 16)r 2 cosh r sinh r, (80n − 64)r cosh r sinh r + (16n − 16)r 2 (cosh 2 r + sinh 2 r). All these derivatives vanish at r = 0 and the fourth derivative is positive. Therefore we have the initial inequality for all r ≥ 0 and all n, so m = G(f ) > −4 \|f \| 2 r∂ r g. Since ∂ r g = − 4(n − 1)(n − 3) 16 − µ r sinh 2 r − 3r cosh 2 r + 3 cosh r sinh r sinh 4 r , its sign is given by n − 3, so in particular, in dimensions n ≤ 3 we obtain m = G(f ) > 0 = G(0), which contradicts the fact that f is a minimizer. Therefore this case is excluded. Case 4: f = 0, E λ (f ) < E λ (Q λ ) and f H λ = Q λ H λ . This case is excluded by (22). Summarizing we have obtained that m = 0, that the only minimizers are the constant zero function and e iθ Q, for some Q ∈ Q λ and θ ∈ R, and that minimizing sequences tend in H to a minimizer. We are now able to prove Proposition 4.1. Proof. We suppose that u 0 is not the constant null function. Recall here that Theorem 1 insures us that if the initial data satisfies to E λ (u 0 ) < E λ (Q λ ) and u 0 H λ ≤ Q λ H λ , then these properties will be preserved in time. In view of Lemma 4.3, inf t G(u(t)) ≥ 0. and equality holds if there is a sequence of times (t n ) such that G(u(t n )) → 0 and u(t n ) tends in H to the constant zero function or to e iθ Q for some Q ∈ Q λ , θ ∈ R. It follows that there exists δ 0 > 0 such that (88) G(u(t)) > δ 0 , ∀t ∈ R. Indeed, otherwise there is a sequence of times (t n ) such that G(u(t n )) → 0 and u(t n ) tends to the constant zero function or to e iθ Q for some Q ∈ Q λ , θ ∈ R, strongly in H, and thus, by compactness of {u(t)} in H 1 rad , strongly in H 1 . In particular, E λ (u 0 ) = E λ (u(t n )) which tends to 0 or to E λ (Q λ ). The second case contradicts the hypothesis E λ (u 0 ) < E λ (Q λ ). In the first case, E λ (u 0 ) = 0 and by using the variational inequality (22) this contradicts the fact that we have supposed that u 0 is not the null function. Now we recall that the classical virial computation yields for radial functions: (89) ∂ 2 t H n \|u(t)\| 2 h = H n \|∂ r u\| 2 − (n − 1) 2 4 \|u\| 2 4∂ 2 r h − 2 p − 1 p + 1 \|u\| p+1 ∆h + \|u\| 2 ((n − 1) 2 ∂ 2 r h − ∆ 2 h). Let ϕ be a smooth positive decreasing radial function supported in B(0, 2), valued 1 in B(0, 1). We shall use the above formula with the weight h R (r) = r 2 ϕ r R and R ≥ 1. Note that when all derivatives fall on r 2 , then we recover G(u(t)) with the weight ϕ r R . Otherwise at least one derivative in space falls on ϕ r R , so the integral is restricted to the region R ≤ r ≤ 2R. More precisely, ∂ 2 r h R = (∂ 2 r r 2 )ϕ r R + 4r R ϕ ′ r R + r 2 R 2 ϕ ′′ r R , ∆h R = (∆r 2 )ϕ r R + (n − 1) cosh r sinh r r 2 R ϕ ′ r R + 4r R ϕ ′ r R + r 2 R 2 ϕ ′′ r R , and similar computations show that ((n − 1) 2 ∂ 2 r h R − ∆ 2 h R ) − (n − 1) 2 (∂ 2 r r 2 )ϕ r R − (∆ 2 r 2 )ϕ r R ≤ C r 1 R≤r≤2R . Therefore we obtain, using also the fact that ϕ ′ ≤ 0, ∂ 2 t H n \|u(t)\| 2 h R = ∂ t 4ℑ H n u(t)∇u(t)∇h R ≥ G(u(t)) − C H n ∩{\|x\|≥R} \|∇u(t, x)\| 2 + \|u(t, x)\| 2 + \|u\| p+1 . Therefore for R large enough, we get, using the compactness of {u(t)} and (88) that ∂ t 4ℑ H n u(t)∇u(t)∇h R ≥ δ 0 2 . Integrating in time, and using Cauchy-Schwarz inequality and then Hardy's inequality as above, we get t δ 0 2 ≤ C(1 + R 2 ) sup τ ∈(0,t) u(τ ) 2 L 2 (r≤2R) + ∇u(τ ) 2 L 2 (r≤2R) ≤ C(R, λ) sup τ ∈(0,t) u(τ ) H λ . Therefore, we obtain a contradiction by letting t go to infinity, and the Proposition follows. 5. Blow-up 5.1. Previous blow-up results on hyperbolic space. In this section we recall the known results on blow-up for equation (1). These results are based on the method of Glassey [Gl77,VlPeTa71]. If u is a general (not necessarily radial) solution of (1), and h a radial weight, we have the following virial identity which generalizes (89): ∂ 2 t H n \|u(t)\| 2 h = H n 4\|∂ r u\| 2 ∂ 2 r h + 4 \|∇ S n−1 u\| 2 sinh 2 r ∂ r h cosh r sinh r − \|u\| 2 ∆ 2 h − 2 p − 1 p + 1 \|u\| p+1 ∆h. (b) [MaZh07] If u 0 is of finite variance, not necessarily radial, p ≥ 1+ 4 n−1 and E(u 0 ) < 0. We refer to [Ba07,MaZh07] for the proofs. Let us mention that both proofs are based on the preceding virial identity, with h(r) = r 2 for (a), and with h(r) = r 0 s 0 sinh n−1 τ dτ ds sinh n−1 s , that satisfies ∆h = 1, for (b). In [Ba07], the blow-up sufficient condition is stated as: E(u 0 ) < inf r>0 ∆ 2 H n r 2 16 u 0 2 L 2 . Condition (a) follows, since ∆ 2 H n r 2 = 2(n − 1) 2 − 2(n − 1)(n − 3) r cosh r − sinh r sinh 3 r and sup r>0 r cosh r − sinh r (sinh r) 3 = 1 3 , inf r>0 r cosh r − sinh r (sinh r) 3 = 0. Remark 5.2. By Proposition 5.1 (a), radial solutions with positive energy, small with respect to the L 2 norm always blow up, which seems better than the analoguous blow-up sufficient condition in the Euclidean setting. However, it is more natural to write this in term of the following conserved modified energy: E m (u(t)) = 1 2 u(t) 2 H − 1 p + 1 u(t) p+1 L p+1 = E(u(t)) − (n − 1) 2 8 u(t) 2 L 2 , which takes into account the fact that the bottom of the spectrum of −∆ H n is (n−1) 2 4 . The blow-up sufficient condition of Proposition 5.1 (a) can be rewritten as In dimension n = 2 and n = 3, we find a negative energy criterion similar to Glassey's criterion in the Euclidean setting. In higher dimension, we can only show that a stronger condition implies blow-up. Technically this is due to the term (n − 1)(n − 3) H n r cosh r − sinh r sinh 3 r \|u\| 2 which has a bad sign in the virial identity, in dimension n ≥ 4. Note that the assumption n ∈ {2, 3} in Theorem 2 comes from the same technical problem (see e.g. the proof of Lemma 4.3, Case 3). 5.2. Blow-up criterion in the finite variance case. In the particular case h = r 2 , we can write the virial formula as (90) ∂ 2 t H n \|u(t)\| 2 r 2 = G(u(t)), with G(f ) defined in (82). In this section we obtain Theorem 2, (b) in the finite variance case as a consequence of the following proposition. particular, f is a local maximizer for G(f ) under the only constraint E λ (f ) = E λ (Q λ ). We Proof. By Proposition 5.3, it is sufficient to prove (95) when ∇u(t) L 2 is large, i.e. (96) ∃M, δ > 0, ∀t ∈ (T − , T + ), ∇u(t) L 2 ≥ M =⇒ G(u(t)) ≤ −δ ∇u(t) 2 L 2 . Using the definition (83) of G(u(t)) and the assumption n ≤ 3, we obtain G(u(t)) ≤ 8 u(t) 2 H − 4n(p − 1) p + 1 u(t) p+1 L p+1 = 4n(p − 1)E(u(t)) + (8 − 2n(p − 1)) ∇u(t) 2 L 2 − (n − 1) 2 4 u(t) 2 L 2 , and (96) follows, since by our assumptions 8 − 2n(p − 1) < 0. We next prove Theorem 2 (b). We will only sketch the proof, which is close to the corresponding proof on R n once (95) is known. Let ϕ : (0, ∞) → (0, +∞) be a smooth function such that ϕ(r) = r 2 if r ≤ 1, ϕ(r) is constant for r ≥ 2 and ϕ ′′ (r) ≤ 2 for all r > 0. Let R ≥ 1 and h R (r) = R 2 ϕ(r/R). By the choice of ϕ, the integrand in the definitions of a, b and c is zero for r ≤ R. We claim a ≤ C R u 2 L 2 , \|b\| ≤ Ce −R/C u p−1 2 H 1 u p+3 2 L 2 , \|c\| ≤ C R u 2 L 2 .(98) We first assume (98) and prove that u blows up in finite time. Combining Proposition 5.5, (97) and (98), we obtain that for all t in the domain of existence of u, ∂ 2 ∂t 2 H n \|u(t)\| 2 h R ≤ −δ u(t) 2 H 1 + Ce −R/C u(t) p−1 2 H 1 u(t) p+3 2 L 2 + C R u(t) 2 L 2 . Using the conservation of the mass and Young's inequality together with the assumption p ≤ 5 we deduce (for a constant C > 0 that depends on the mass and energy of u), ∂ 2 ∂t 2 H n \|u(t)\| 2 h R ≤ −δ u(t) 2 H 1 + Ce −R/C u(t) 2 H 1 + C R . then the solution u does not scatter in any time direction. (a) If δ λ (u 0 ) < 0 then the solution u is global and scatters in both time directions. (b) If δ λ (u 0 ) > 0, and H n r 2 \|u 0 \| 2 dµ < ∞ or 1 + 4 n < p ≤ 5 then the solution u blows up in finite positive and negative times. Furthermore, using \|c(λ)\| −2 ∼ λ n−1 as λ → ∞ F (change of variable σ = e r θ. This concludes the proof since 2(1 − n) + n − 2 = −n ≤ −2. 3.10, we prove (49) by contradiction. If (49) does not hold, there exists ε > 0 and a sequence of positive integers (J ℓ ) ℓ≥0 such that (62) lim ℓ→∞ J ℓ = +∞ and ∀ℓ ≥ 0, lim k→∞ r J ℓ ,k Σ ≥ ε. Let (φ α ) α∈N be a countable, dense family in H 1 (H n ). Let ν ∈ N. By (59) and (62), there exists indexes ℓ(ν) and k(ν) with the following properties ∀α ∈ {0, . . . , ν}, sup t∈R h∈G Proposition 5. 1 . 1Blow-up occurs in the following cases: (a)[Ba07] If u 0 is radial, of finite variance, p ≥ Combining the virial identity (89) with h(r) = h R (r), and the definition (83) of G, we obtain(97) ∂ 2 ∂t 2 \|u(t)\| 2 h R − G(u(t)) = a + b + c, with a = 4 H n \|∂ r u\| 2 − (n − 1) 2 4 \|u\| 2 ϕ ′′ (r/R) − 2 , b = −2 p − 1 p + 1 H n \|u\| p+1 ϕ ′′ (r/R) − 2 + (n − 1)cosh r sinh r Rϕ ′ (r/R) 2 (r) ϕ ′′ (r/R) − 2 + cosh r sinh 3 r (n − 1)(n − 3) Rϕ ′ (r/R) − 2r . where (82) G(f ) = 16E(f ) + 8 H n \|∇ S n−1 u(t)\| 2 r cosh r − sinh r sinh r 3 − H n \|u(t)\| 2 ∆ 2 H n r 2 − H n \|u(t)\| p+1 2(p − 1) p + 1 ∆ H n r 2 − 16 p + 1 ,where ∆ H n r 2 = 2 + 2(n − 1) r cosh r sinh r , ∆ 2 H n r 2 = 2(n − 1) 2 − 2(n − 1)(n − 3)r cosh r − sinh r sinh 3 r for radial functions, we have:H n \|f \| 2 r cosh r − sinh r sinh 3 r dµ(x) − 4(p − 1) p + 1 H n \|f \| p+1 1 + (n − 1) r cosh r sinh r dµ(x).Note that the third term is well defined for u ∈ H 1 , in view of the following lemma, that will be of use also later.Lemma 4.2. Let n ≥ 2 and 1 < p < 1 + 4 n−2 . The space H 1 rad (H n ) is compactly embedded in L p+1 (H n ) and in L p+1 (1 + (n − 1) r cosh r sinh r )dr, H n . Proof. By the change of function Proof. We denote m the infimum and we consider a minimizing sequence f k ,Since (f k ) k is bounded in H 1 , we can suppose that (up to a subsequence) there exists a radial weak limit f of f k in H 1 ,Thus f k ⇀ f in H also. As a consequence,By Lemma 4.2 we obtain thatFinally, by Hölder's inequality,In view of the expression of G it follows that −∞ < m andThus G(f ) = m, so f is a minimizer. We shall distinguish five cases.Case 0: f = 0. In this case, m = 0. By(85),Case 1:and since f is not identically zero it follows thatso this case is excluded.By Theorem 2.9 it follows that f = e iθ Q, for some θ ∈ R, Q ∈ Q λ . The function e itλ Q is a solution of (1), so the virial formula yieldsthus (f k ) k converges strongly to e iθ Q in H 1 (and thus in H).Case 3:. Indeed, to avoid that f * is an isolated point in the set E λ (f ) = E λ (Q λ ) we might argue as follows. There exists locally a curve through f * that is in the set E λ (f ) = E λ (Q λ ): otherwise f * is a local extremum for E λ (f ), so −∆f * − λf * − \|f * \| p−1 f * = 0, which contradicts, by using Poincaré-Sobolev inequalities,Therefore we obtain the existence of a Lagrange multiplier µ such that f solvesIf µ ≥ 0 we multiply with f , integrate and obtain, which contradicts the fact that f is a minimizer.If µ < 0 we note that the equation on f is of typewith explicit variable radial coefficients g(r) and h(r):Proposition 5.3. Let n ∈ {2, 3}, p ≥ 1 + 4 n and λ < (n−1) 24. Let u be a radial solution ofTheorem 2 (b) in the finite variance case follows immediately from Proposition 5.3. Indeed, in this case and under the assumptions of Theorem 2 (b), the function t → H n r 2 \|u(t)\| 2 is positive and strictly concave on (T − (u), T + (u)), which proves that T + (u) and T − (u) must be finite. We will treat the case of infinite variance in the next subsection. It remains to prove Proposition 5.3.Proof of Proposition 5.3. Theorem 1 insures that if the initial data satisfies E λ (u 0 ) ≤ E λ (Q λ ) and u 0 H λ > Q λ H λ , then these properties will be preserved in time. We thus defineWe will show that m = 0, then improve this estimate to get the conclusion of the proposition. We divide the proof into 4 steps.Step 1. We shall first prove that the supremum in (91) can be restricted to the set E λ (f ) = E λ (Q λ ) . More precisely, we shall prove the following result.Proof. We consider the family of functions {σf } σ∈[0,1] . If E λ (σf ) < E λ (Q λ ) for all σ ∈ [0, 1], then by Lemma 2.16 and a simple continuity argument, it follows that σf H λ > Q λ H λ for all σ ∈]0, 1], which is in contradiction with Q λ H λ > 0. This yields the existence of σ * ∈]0, 1[ such that E λ (σ * f ) = E λ (Q λ ) and E λ (σf ) < E λ (Q λ ) for σ ∈]σ * , 1]. So for σ ∈]σ * , 1], again by Lemma 2.16 and a simple continuity argument we have σf H λ > Q λ H λ . If σ ∈]σ * , 1], we haveThe inequality is strict since f = 0. Hence, using that p < 1 + 4 n ,Integrating between σ * and 1, we get:Since σf H λ > Q λ H λ for σ ∈]σ * , 1] we obtain that σ * f H λ ≥ Q λ H λ so we can set f * = σ * f .Step 2. Maximizer for an equivalent maximization problem.Let. In virtue of Lemma 5.4 we obtain thatIn this step we prove that there exists a maximizer f for the maximization problemNote thatby our assumptions on p and n. We consider now a maximizing sequence f k ,From the above upper-bound onit follows that (f k ) k is bounded in H 1 , and we can suppose that (up to a subsequence) there exists a radial weak limit f of f k in H 1 ,By the compactness Lemma 4.2,and, using also Hölder's inequalityIn view of the expression of H λ it follows that m < ∞ and that ( (n−1) 2By the weak convergence we obtainso combining this with the L p+1 convergence,Moreover, since f k satisfy the constraints in (92), it follows that f k L p+1 ≥ Q λ L p+1 and by the L p+1 convergence we get f L p+1 ≥ Q λ L p+1 . Now using Poincaré-Sobolev inequality (6),H λ . Therefore we have obtained that f is a solution of the maximization problem (92).To conclude this step, we must prove E λ (f ) = E λ (Q λ ). Indeed, if f does not satisfy this constraint, then E λ (f ) < E λ (Q λ ) and, letting f * be as in Lemma 5.4, we haveStep 3. Proof that the maximum is zero. In the following we shall prove that f = e iθ Q for some θ ∈ R, Q ∈ Q λ which implies m = 0. We suppose that f = e iθ Q for any θ ∈ R and any Q ∈ Q λ . By the definition of Q λ and Theorem 2.9 we get f 2 H λ > Q λ 2 H λ . In derive the equation G ′ (f ) = µE ′ λ (f ), with the Lagrange multiplier µ ∈ R. This is precisely equation (86). In particular, (87) writesSince p ≥ 1 + 4 n , the right-hand side of (94) is positive. So, in view of the hypothesis n ∈ {2, 3}, λ < (n−1) 2 4 , we must have µ > 16. Then, recalling the computation in Case 3 of the proof of Lemma 4.3, but with the opposite sign for 16 − µ, we getwhich contradicts the definition of m.Step 4. Conclusion of the proof. By Lemma 5.4, for all t in the domain of existence of u, there exists u * (t) with E λ (u * ) = E λ (Q λ ), u * H λ > Q λ H λ and such thatsince byStep 3, G(u * (t)) ≤ 0. This concludes the proof of Proposition 5.3.5.3.Blow-up criterion in the infinite variance case. We next assume (in addition to the preceding assumptions on p, n and λ), 1 + 4 n < p ≤ 5, and prove Theorem 2 (b) without the finite variance assumption. The proof relies on a localized version of the virial identity (90) in the spirit of[OgTs91b]. To use this localized version, we need the following refinement of Proposition 5.3:Proposition 5.5. Let n ∈ {2, 3}, p > 1 + 4 n and λ < (n−1) 24. Let u be a radial solution of (1) with u 0 in H λ . Then, if E λ (u 0 ) < E λ (Q λ ) and u 0 H λ > Q λ H λ , there exists δ > 0, depending only on the conserved mass and energy of u, such that for all t in the maximal interval of existence (T − , T + ) of u,Note that u(t) 2 H 1 is bounded from below (by the conserved mass of u). Chosing R large, we obtainThus \|u(t)\| 2 h R is a positive, strictly concave function on the domain of existence of u, which proves that u blows up in finite time. It remains to prove (98). 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Banica) Laboratoire de Mathématiques et de Modélisation d'Évry (UMR 8071), Département de Mathématiques, Université d'Évry, 23 Bd. de FranceFrance E-mail address: [email protected]) Laboratoire de Mathématiques et de Modélisation d'Évry (UMR 8071), Département de Mathématiques, Université d'Évry, 23 Bd. de France, 91037 Evry, France E-mail address: [email protected] (T. Duyckaerts) Laboratoire Analyse, Géométrie et Applications (UMR 7539), Département de Mathématiques, Institut Galilée, Université Paris 13, 99 Av. J.-B. Clément 93430 Villeta- neuse, France E-mail address: [email protected]	[]
[ "Separating the Effect of Independent Interference Sources with Rayleigh Faded Signal Link: Outage Analysis and Applications", "Separating the Effect of Independent Interference Sources with Rayleigh Faded Signal Link: Outage Analysis and Applications" ]	[ "Arshdeep S Kahlon ", "Member, IEEESebastian S Szyszkowicz ", "Senior Member, IEEEShalini Periyalwar ", "Member, IEEEHalim Yanikomeroglu " ]	[]	[]	We show that, for independent interfering sources and a signal link with exponentially distributed received power, the total probability of outage can be decomposed as a simple expression of the outages from the individual interfering sources. We give a mathematical proof of this result, and discuss some immediate implications, showing how it results in important simplifications to statistical outage analysis.We also discuss its application to two active topics of study: spectrum sharing, and sum of interference powers (e.g., lognormal) analysis.	10.1109/wcl.2012.071612.120392	[ "https://arxiv.org/pdf/1201.5434v1.pdf" ]	8,553,497	1201.5434	e493a820872f8b8ffee437397aaed7eb8b5f4816	Separating the Effect of Independent Interference Sources with Rayleigh Faded Signal Link: Outage Analysis and Applications 26 Jan 2012 Arshdeep S Kahlon Member, IEEESebastian S Szyszkowicz Senior Member, IEEEShalini Periyalwar Member, IEEEHalim Yanikomeroglu Separating the Effect of Independent Interference Sources with Rayleigh Faded Signal Link: Outage Analysis and Applications 26 Jan 2012Index Terms wireless interferenceoutage probabilityRayleigh fadingspectrum sharingheterogeneous networks We show that, for independent interfering sources and a signal link with exponentially distributed received power, the total probability of outage can be decomposed as a simple expression of the outages from the individual interfering sources. We give a mathematical proof of this result, and discuss some immediate implications, showing how it results in important simplifications to statistical outage analysis.We also discuss its application to two active topics of study: spectrum sharing, and sum of interference powers (e.g., lognormal) analysis. There may be cases when we wish to study the outage at a receiver due to the sum of independent interfering signal powers, yet the distribution of the constituent interfering powers is unknown. Such a case can be considered in a spectrum sharing scenario where two or more heterogeneous networks share the same spectrum [7]- [10]. Throughout this paper, we only consider spectrum sharing without any spectrum sensing or cognition, which implies that the secondary network necessarily increases the outage probability of primary receivers. The operator of the primary network may be interested in obtaining insights into the additional outage that a receiver would suffer from the deployment of a heterogeneous secondary network, in order to determine the feasibility of spectrum sharing. In this work, we show that in the case of independent interfering powers following any distribution, and an independent signal power with exponential power distribution, it is possible to separate the outage effect of each interferer. We show this result analytically and exactly, and discuss some of its more immediate consequences for the simplification of outage analysis. In Section II, we give the general outage problem as it is often formulated. In Section III, we introduce our main expression for the total outage probability and the mathematical result it is based on, and make some general observations on its consequences to outage analysis. In Section IV, we show how our result can concretely simplify calculations in two important research topics: 1) primary/secondary network sharing scenarios and 2) sum of lognormals modeling, before concluding in section V. II. PROBLEM FORMULATION Consider a wireless device receiving a useful signal with power S, and suffering from a total received interference of power I. We assume that S is exponentially distributed (due, notably, to Rayleigh fading), while I can be written as I = N i=1 I i ,(1) where {I i } is the set of the N independent received interference powers (which may originate from individual transmitters, entire networks or parts thereof, or thermal noise), and the interference powers are assumed to add incoherently (in power) [1]- [3], [6], [11], and are treated as additive white Gaussian noise as far as outage is concerned [2], [11]. We also assume the signal power to be independent of the interference powers. The outage probability ε on the signal link is obtained from the cumulative distribution function (CDF) of the power ratio S/I: ε = P S I < β = P S N i=1 I i < β ,(2) where β is the outage threshold in terms of the signal to interference (and noise) power ratio. III. ANALYSIS A. Mathematical Result We introduce a result on random variables (RVs) that will allow us to separate the summation of interference powers under our assumptions. Consider {X i } a set of N independent RVs, and Y an independent exponentially distributed RV. We may then write P N i=1 X i < Y = N i=1 P (X i < Y ) .(3) The proof is in the Appendix. B. Separability of the Interference Powers Applying (3) to the outage expression (2), identifying X i = I i and Y = S/β, and inverting the inequalities gives ε = 1 − N i=1 (1 − ε i ) , ε i = P S I i < β .(4) We have thus expressed the total outage probability ε as a simple algebraic expression of the partial outage probabilities ε i that would have been caused by each individual interfering source separately (given the same outage threshold β). This generalises (and simplifies) similar results in [12] and in [13] (and references therein), where all the signals are Rayleigh faded. C. Some Useful Consequences Some interesting observations immediately result from (4): 1) The difficulty of finding the CDF of the ratio of the sum of N independent RVs and the exponentially distributed RV disappears, and reduces to that of finding M CDFs of the ratio of that exponentially distributed RV and each independent RV, where M ≤ N is the number of statistically different RVs (if all the sources have the same statistics, then M = 1). 2) The total outage probability can be obtained directly from the partial outage probabilities, without the need to know the models of the underlying interferences, which is useful when the partial outage probabilities are obtained from simulation, field measurements, or even usage statistics of a working network. 3) If the interference powers are statistically dependent (due, e.g., to correlated shadowing [2], [4], [6]), our result can still be useful if the interferers can be grouped in such a way that the interferences are independent across the groups. Then, in order to calculate the outage probability, the CDFs of the sum interferences (or the partial outage probabilities) need to be found only within those groups, but not globally. Our result has important implications in simplifying the analysis of outage caused by multiple interference sources. In the next section, we show how our result can concretely be applied to two research directions that have already received much attention. IV. APPLICATIONS TO CURRENT RESEARCH Our result in (4) has immediate applications in simplifying various outage calculations, e.g., when a secondary network shares the spectrum without sensing the primary network's activity, thereby increasing the outage probability. The result also simplifies outage probability calculations when the interference is modeled as the sum of independent lognormal RVs. A. Spectrum Sharing between Primary and Secondary Network A direct application of our result with N = 2 can be seen in the spectrum sharing scenario, where we want to find the additional outage at a primary network receiver due to the deployment of a secondary network, while avoiding the potentially complex task of characterising the interference from the primary network. Consider a typical primary receiver, experiencing an outage probability ε 1 due to co-channel interfering primary transmitters (in the absence of the secondary network). We call ε T the maximum outage probability allowed at a primary receiver. It then follows from (4) that the secondary network must be designed in such a way that the outage probability ε 2 caused by its interference alone (in the absence of the primary network's co-channel interferers) satisfies ε 2 ≤ ε T − ε 1 1 − ε 1 . (5) B. Outage Analysis Using the Sum of Lognormal Random Variables The study of the distribution of the sum of several lognormal RVs has received much research attention for several decades [2], [3], and still attracts significant interest [4]- [6]. The research is primarily (but not exclusively) in the field of wireless communications, where it is motivated by the model where each interference source suffers (possibly correlated) lognormal shadowing. In this case, each I i is modeled as a lognormal RV, and the challenge is to find the sum distribution of I. However, no closed-form solution exists [4], [5] even for the simplest cases, and in fact there exist many approximating methods that trade-off accuracy against simplicity. It is important to see the context of this research: the goal of finding the sum distribution of the interference is not necessarily an end in itself. Its main use is as an intermediate step in finding the distribution of the signal-to-interference-(and possibly noise)-power ratio, and hence the outage probability [2], [3], [11]. Our result (4) shows that, given independent interference powers and an exponentially distributed received signal power, the unsolved problem of a sum of lognormal RVs disappears, and essentially reduces to the problem of the outage from a single lognormal interferer: ε i = P S I i < β = P √ S · I i −1/2 < β .(6) Now, √ S follows a Rayleigh distribution, while I i −1/2 is an independent lognormal RV, hence the problem reduces the computation of M ≤ N probabilities from the Suzuki distribution 1 , given M statistically distinct lognormal RVs. The result can also be extended to the case where the interference powers are not lognormal: notably they may include small-scale fading (lognormal-times-fading power, as in [11]), and path loss based on random positions [6]. It remains the case that the most difficult probability calculation, i.e., the summation of random powers, need not be performed. V. CONCLUSION We showed that, under the assumption of independent received interference powers, and an exponentially distributed received signal power (e.g., due to Rayleigh fading), the outage probability due to all the interfering sources can easily be decomposed into the partial outage probabilities as would be caused by the interferers individually. It is therefore not necessary to know the distribution of the total interference power to find the outage probability of the system, nor in fact even that of the individual interference powers, as long as the corresponding partial outage probabilities are known. Our result makes important simplifications in the calculation of outage probability, which is applicable in a variety of scenarios, and notably in the case of spectrum sharing, as well as in the case of sum of lognormal RVs interference modeling. It naturally extends to include noise powers as well. It has the advantage of being simple and exact, and can be used in practical scenarios with possibly complex and intractable interfering sources, in order to get insights into the effects of those various interference sources. APPENDIX Proof of (4): We can write the left hand side of (3) as E X 1 ,X 2 ,...,X N P N i=1 X i < Y X 1 , X 2 , . . . , X N . Let µ be the mean of the exponentially distributed RV Y . Then we can write the above as E X 1 ,X 2 ,...,X N exp −µ N i=1 X i = E X 1 ,X 2 ,...,X N N i=1 exp (−µX i ) .(8) Now, since X 1 , X 2 , . . . , X N are independent RVs, we can write the above as N i=1 E X i (exp (−µX i )) ,(9) which is equivalent to N i=1 P (X i < Y ) .(10) January 27, 2012DRAFT This is a well established numerical calculation: e.g., the SuzukiDistribution[µ, ν] function in Wolfram Mathematica. A mathematical theory of network interference and its applications. M Win, P Pinto, L Shepp, Proc. IEEE. IEEE97M. Win, P. Pinto, and L. Shepp, "A mathematical theory of network interference and its applications," Proc. IEEE, vol. 97, pp. 205-230, Feb. 2009. Outage probabilities in the presence of correlated lognormal interferers. A A Abu-Dayya, N C Beaulieu, IEEE Trans. Veh. Technol. 43A. A. Abu-Dayya and N. C. Beaulieu, "Outage probabilities in the presence of correlated lognormal interferers," IEEE Trans. Veh. Technol., vol. 43, pp. 164-173, Feb. 1994. An optimal lognormal approximation to lognormal sum distributions. N Beaulieu, Q Xie, IEEE Trans. Veh. Technol. 53N. Beaulieu and Q. Xie, "An optimal lognormal approximation to lognormal sum distributions," IEEE Trans. Veh. Technol., vol. 53, pp. 479-489, Mar. 2004. Smolyak's algorithm: A simple and accurate framework for the analysis of correlated log-normal power-sums. M Di Renzo, L Imbriglio, F Graziosi, F Santucci, IEEE Commun. Lett. 139M. Di Renzo, L. Imbriglio, F. Graziosi, and F. Santucci, "Smolyak's algorithm: A simple and accurate framework for the analysis of correlated log-normal power-sums," IEEE Commun. Lett., vol. 13, no. 9, pp. 673-675, Sept. 2009. Accurate computation of the MGF of the lognormal distribution and its application to sum of lognormals. C Tellambura, D Senaratne, IEEE Trans. Commun. 585C. Tellambura and D. Senaratne, "Accurate computation of the MGF of the lognormal distribution and its application to sum of lognormals," IEEE Trans. Commun., vol. 58, no. 5, pp. 1568-1577, May 2010. Aggregate interference distribution from large wireless networks with correlated shadowing: An analytical-numerical-simulation approach. S Szyszkowicz, F Alaca, H Yanikomeroglu, J Thompson, IEEE Trans. Veh. Technol. 606S. Szyszkowicz, F. Alaca, H. Yanikomeroglu, and J. Thompson, "Aggregate interference distribution from large wireless networks with correlated shadowing: An analytical-numerical-simulation approach," IEEE Trans. Veh. Technol., vol. 60, no. 6, pp. 2752-2764, July 2011. A survey of dynamic spectrum access. Q Zhao, B Sadler, IEEE Signal Process. Mag. 243Q. Zhao and B. Sadler, "A survey of dynamic spectrum access," IEEE Signal Process. Mag., vol. 24, no. 3, pp. 79-89, May 2007. Spectrum sharing between cellular and mobile ad hoc networks: transmission-capacity trade-off. K Huang, V Lau, Y Chen, IEEE J. Sel. Areas Commun. 277K. Huang, V. Lau, and Y. Chen, "Spectrum sharing between cellular and mobile ad hoc networks: transmission-capacity trade-off," IEEE J. Sel. Areas Commun., vol. 27, no. 7, pp. 1256-1267, Sept. 2009. Aggregate interference modeling in cognitive radio networks with power and contention control. Z Chen, C Wang, X Hong, J Thompson, S A Vorobyov, X Ge, H Xiao, F Zhao, IEEE Trans. Commun. submitted toZ. Chen, C. Wang, X. Hong, J. Thompson, S. A. Vorobyov, X. Ge, H. Xiao, and F. Zhao, "Aggregate interference modeling in cognitive radio networks with power and contention control," submitted to IEEE Trans. Commun., Aug. 2010, [Online] Available: http://arxiv.org/abs/1008.1043. Identification of spectrum sharing opportunities for a finite field secondary network through an exact outage expression under Rayleigh fading. A Kahlon, S Szyszkowicz, S Periyalwar, H Yanikomeroglu, IEEE International Symposium on Personal Indoor and Mobile Radio Communications. A. Kahlon, S. Szyszkowicz, S. Periyalwar, and H. Yanikomeroglu, "Identification of spectrum sharing opportunities for a finite field secondary network through an exact outage expression under Rayleigh fading," in IEEE International Symposium on Personal Indoor and Mobile Radio Communications, Sept. 2011, pp. 1-5. An approximation of the outage probability in Rayleigh-lognormal fading scenarios. C Fischione, M , D' Angelo, UC Berkeley, Tech. Rep. C. Fischione and M. D'Angelo, "An approximation of the outage probability in Rayleigh-lognormal fading scenarios," UC Berkeley, Tech. Rep., Mar. 2008. Investigations into cochannel interference in microcellular mobile radio systems. Y.-D Yao, A Sheikh, IEEE Trans. Veh. Technol. 417Y.-D. Yao and A. Sheikh, "Investigations into cochannel interference in microcellular mobile radio systems," IEEE Trans. Veh. Technol., vol. 41, no. 7, pp. 114-123, May 1992. Capacity outage probability for multi-cell processing under Rayleigh fading. V Garcia, N Lebedev, J.-M Gorce, IEEE Commun. Lett. 158V. Garcia, N. Lebedev, and J.-M. Gorce, "Capacity outage probability for multi-cell processing under Rayleigh fading," IEEE Commun. Lett., vol. 15, no. 8, pp. 801-803, Aug. 2011.	[]
[ "Resistors in dual networks", "Resistors in dual networks" ]	[ "Martina Furrer [email protected] \nBerufsbildungszentrum des Kantons Schaffhausen\n8200SchaffhausenSwitzerland\n", "Norbert Hungerbühler [email protected] \nDepartment of Mathematics\nETH Zürich\n8092ZürichSwitzerland\n", "Simon Jantschgi [email protected] \nUniversity of Zurich\n8050ZürichSwitzerland\n" ]	[ "Berufsbildungszentrum des Kantons Schaffhausen\n8200SchaffhausenSwitzerland", "Department of Mathematics\nETH Zürich\n8092ZürichSwitzerland", "University of Zurich\n8050ZürichSwitzerland" ]	[ "Electronic Journal of Graph Theory and Applications" ]	Let G be a finite plane multigraph and G its dual. Each edge e of G is interpreted as a resistor of resistance R e , and the dual edge e is assigned the dual resistance R e := 1/R e . Then the equivalent resistance r e over e and the equivalent resistance r e over e satisfy r e /R e + r e /R e = 1. We provide a graph theoretic proof of this relation by expressing the resistances in terms of sums of weights of spanning trees in G and G respectively.	10.5614/ejgta.2020.8.2.6	[ "https://ijc.or.id/index.php/ejgta/article/download/657/pdf_141" ]	21,698,201	1805.01380	f7984fde2f6f24e12bf74ed93fcd727090364256	Resistors in dual networks 2020 Martina Furrer [email protected] Berufsbildungszentrum des Kantons Schaffhausen 8200SchaffhausenSwitzerland Norbert Hungerbühler [email protected] Department of Mathematics ETH Zürich 8092ZürichSwitzerland Simon Jantschgi [email protected] University of Zurich 8050ZürichSwitzerland Resistors in dual networks Electronic Journal of Graph Theory and Applications 82202010.5614/ejgta.2020.8.2.6dual graphselectrical networksequivalent resistance Mathematics Subject Classification: 05C0505C1215A1515A18 Let G be a finite plane multigraph and G its dual. Each edge e of G is interpreted as a resistor of resistance R e , and the dual edge e is assigned the dual resistance R e := 1/R e . Then the equivalent resistance r e over e and the equivalent resistance r e over e satisfy r e /R e + r e /R e = 1. We provide a graph theoretic proof of this relation by expressing the resistances in terms of sums of weights of spanning trees in G and G respectively. Introduction The systematic study of electrical resistor networks goes back to the German physicist Gustav Robert Kirchhoff in the middle of the 19th century. In particular, Kirchhoff's node and loop laws, and Ohm's law allow to fully describe the electric current and potential in a given static network of resistors and voltage sources. In the course of his investigations, Kirchhoff discovered the Matrix Tree Theorem, which states that the number of spanning trees in a graph G is equal to any cofactor of the Laplacian matrix of G. Surprisingly, this purely graph theoretic fact has a deep connection to the physical question of the equivalent resistance between two vertices of an electric network. In the simplest situation, a finite simple graph G can be interpreted as an electrical network by considering each edge as a resistor of 1 Ohm. One is then interested in the resulting equivalent resistance between any two vertices. For example, a method developed by Van Steenwijk [8] has recently been generalized to calculate the resistance between any two vertices of a symmetrical polytope (see [6]). For general graphs the key observation for solving the problem goes back to Kirchhoff: The equivalent resistance over an edge e in the graph G is given by the quotient of the number of spanning trees containing the edge e divided by the total number of spanning trees in G. More generally, we may consider a finite multigraph G and assign to each edge e of G a weight R e > 0 interpreted as resistance of e. Also in this case, the equivalent resistance between two vertices can be expressed in terms of sums of weights of spanning trees (see Section 2). Consider a cube as a graph with unit resistance on each edge and the dual polyhedron, the octahedron, in the same way. The equivalent resistance over an edge of the cube turns out to be 7/12 Ohm, the equivalent resistance over an edge of the octahedron is 5/12 Ohm. Observe, that these values add up to 1! The same phenomenon occurs for the dodecahedron with equivalent resistance of 19/30 Ohm over each edge and the dual graph, the icosahedron, with 11/30 Ohm, or the rhombic dodecahedron with equivalent resistance of 13/24 Ohm over each edge and the dual graph, the cuboctahedron, with 11/24 Ohm. This is not just a coincidence: Suppose that a planar graph and its dual are both interpreted as electrical networks with unit resistance for all edges. Now, if r e is the equivalent resistance over an edge e and r e is the equivalent resistance over the dual edge e , then r e + r e = 1 (see [2, Exercise 7, Section 10.5], [9, Theorem 2.3]). The aim of this article is to generalize this formula to plane networks with arbitrary resistors (see Theorem 5) and to give a graph theoretic proof. Preliminaries Let G be a finite connected graph with n ≥ 3 vertices and without loops. Multiple edges are allowed. Each edge e is considered as a resistor of resistance R e > 0. Then, we consider the weighted Laplace matrix L = ( ij ) of G defined as ij :=      − 1 Re , where the sum runs over all edges e between the vertices i and j 1 , a ii , if i = j, where the diagonal values a ii are chosen such that the sum of all rows of L vanishes. The weight of a subgraph H of G is defined as Π(H) := e an edge of H 1 R e . The set of all spanning trees of a graph G will be denoted by S(G). We recall the following: 1 By convention, an empty sum is 0. Proposition 1. (i) The value of each cofactor L ij of L is the sum of the weights of all spanning trees of G. (ii) Let L ij,ij denote the determinant of the matrix L with rows i, j and columns i, j deleted, and let e be an edge between the vertices i and j. Then the quotient L ij,ij /R e equals the sum of the weights of all spanning trees of G which contain the edge e. Proof. The first part of the proposition follows directly from a general version of the Matrix Tree Theorem (see, e.g., [7,Theorem VI.27]). For the second part we proceed as follows: Observe, that by using the edge e between the vertices i and j we can split up the sum of the weights of all spanning trees of G as follows: T ∈S(G) Π(T ) = T ∈S(G) e∈T Π(T ) + T ∈S(G) e / ∈T Π(T ).(1) Furthermore, the second sum on the right-hand side of (1) corresponds to the sum of the weights of all spanning trees of G − e, i.e., G with edge e removed. Using the first part of the proposition, we get from (1) that T ∈S(G) e∈T Π(T ) = L ii − L e ii , where L e = ( e hk ) is the weighted Laplace matrix of G − e. We have e ij = e ji = ij + 1 R e , e ii = ii − 1 R e , e jj = jj − 1 R e and e hk = hk for all other h, k. The term L ii − L e ii can be computed using Laplace's cofactor expansion. Expanding both L ii and L e ii along the j-th row yields T ∈S(G) e∈T Π(T ) = k =i jk (L ii ) jk − k =i e jk (L e ii ) jk . Since the cofactors (L ii ) jk and (L e ii ) jk are equal for all k we are left with T ∈S(G) e∈T Π(T ) = ( jj − e jj )(L ii ) jj = L ij,ij R e . 2 Remark. The second part of Proposition 1 follows also quite easily from the All Minors Matrix Tree Theorem (see [4]). The connection to the equivalent resistance is given by the following Proposition 2. Let e be an edge between the vertices i and j. Then the equivalent resistance r e over the edge e is given by r e = L ij,ij L 11 .(2) Remark. Recall, that by Proposition 1(i) the denominator in (2) can be replaced by any other cofactor L hk . Proof. Observe that the Laplace matrix of the weighted multigraph G corresponds to the Laplace matrix of a weighted simple graph H where the multiple edges e 1 , . . . e k between each two vertices i and j of G are collapsed to a single edge e with weight R e = 1 1 Re 1 + . . . + 1 Re k . However this value corresponds exactly to the equivalent resistance of the parallel resistors R e 1 , . . ., R e k . Thus, the equivalent resistance over the vertices i and j in G equals the equivalent resistance over the vertices i and j in H and the claim follows from [1, Remark 2.1, Equation (16)] for simple graphs, and Proposition 1. From now on we assume that G is a finite planar multigraph with dual graph G . Recall that in general G depends on the embedding of G in the plane. Definition 3. Let G be the dual of a planar embedded multigraph G, and let G be interpreted as an electrical network by associating to each edge e a resistance R e > 0. For each edge e of G, we define the electrical resistance R e of the dual edge e to be the conductance of e, i.e., R e := 1/R e . Then, G equipped with these resistances is called the dual electrical network of G. Observe that the Laplace matrix L = ( ij ) of the dual electrical network G is given by ij :=      − 1 R e = − R e ,L ij = L ij Π(G ),(3) where Π(G ) = R e is the total weight of G . (ii) The product R e L ij,ij equals the sum of the weights of the spanning trees in G which contain the dual edge e of edge e. Proof. We only have to show equation (3), since all other statements follow from Proposition 1. Let Ψ denote the canonical bijection from S(G ) to S(G) given by Ψ(T ) = {e ∈ G\|e ∈ G − T }. Observe that the weight of a spanning tree T of G can be expressed as Π(T ) = Π(G ) Π(G − T ) .(4) Using the bijection Ψ and Definition 3 in (4), namely the fact, that the electrical resistance of an edge in G is equal to the conductance of the dual edge in G, we get Π(T ) = Π(G )Π(Ψ(T )).(5) The characterization of L ij and L ij as the sum of the weights of all spanning trees of G and G respectively leads, together with (5), to L ij = T ∈S(G ) Π(T ) = T ∈S(G ) Π(G )Π(Ψ(T )) = = Π(G ) T ∈S(G) Π(T ) = Π(G )L ij , where we have used the bijectivity of Ψ in the penultimate equality. This completes the proof. The sum formula in dual networks The main result is now the following: Theorem 5. Let R e be the resistance of an edge e and R e = 1/R e the resistance of the dual edge e in the dual electrical network. Let r e denote the equivalent resistance over edge e and r e denote the equivalent resistance over edge e . Then r e R e + r e R e = 1. For a proof of this formula based upon physical arguments see [5]. Here, we provide a purely graph theoretic proof. Proof. Let e be an edge between the vertices i and j in G, and we may assume that the vertices are numbered such that e also runs between the vertices i and j in G . In a first step, we are going to derive a new expression for L ij,ij . Let Ψ be the canonical bijection from S(G ) to S(G) as defined in the proof of Proposition 4. By part (ii) of Proposition 4 we have L ij,ij = 1 R e T ∈S(G ) e ∈T Π(T ).(6) The identity (5) and the fact that Ψ(T ) does not contain the edge e if T contains the dual edge e allow to rewrite the right-hand side of (6) as follows 1 R e T ∈S(G ) e ∈T Π(T ) = Π(G ) R e T ∈S(G) e / ∈T Π(T ).(7) Furthermore, the sum on the right-hand side of (7) can be expressed as the difference between the sum of the weights of all spanning trees of G and the sum of the weights of the spanning trees that contain the edge e. Therefore, it holds that L ij,ij = Π(G ) R e T ∈S(G) Π(T ) − T ∈S(G) e∈T Π(T ) .(8) Using Proposition 1 in (8) yields the following identity: L ij,ij = Π(G ) R e L ii − L ij,ij R e .(9) Now, in a second step, it follows from Proposition 2 and Definition 3, that r e R e + r e R e = L ij,ij L ii R e + R e L ij,ij L ii .(10) Using the first part of Proposition 4 and (9), we can rewrite the right-hand side of (10) and simplify the resulting expression to arrive at r e R e + r e R e = L ij,ij L ii R e + R e Π(G ) Re (L ii − L ij,ij Re ) L ii Π(G ) = 1, as claimed. Example 6. Let us consider the following electrical network: 1 2 3 4 R 1 R 2 R 3 R 5 R 4 The corresponding Laplace matrix L is L =     1 R 1 + 1 R 2 − 1 R 2 0 − 1 R 1 − 1 R 2 1 R 2 + 1 R 3 − 1 R 3 0 0 − 1 R 3 1 R 3 + 1 R 4 + 1 R 5 − 1 R 4 − 1 R 5 − 1 R 1 0 − 1 R 4 − 1 R 5 1 R 1 + 1 R 4 + 1 R 5     The cofactor L 11 = 1 R 1 R 2 R 3 R 4 R 5 R 4 (R 1 + R 2 + R 3 ) + R 5 (R 1 + R 2 + R 3 + R 4 ) corresponds indeed to the total weight of all spanning trees of G as one easily checks directly. For the edge e between the vertices 3 and 4 with resistance R 4 , we get L 34,34 = 1 R 1 R 2 + 1 R 1 R 3 + 1 R 2 R 3 which, divided by R 4 , gives the sum of the weights of the spanning trees which contain e, as stated in Proposition 1. The dual network looks as follows: 1 2 3 4 R 1 R 2 R 3 R 5 R 4 1/R 4 1/R 5 1/R 1 1/R 3 1/R 2 1 2 3 and the corresponding Laplace matrix is L =   R 1 + R 2 + R 3 + R 4 −R 4 −(R 1 + R 2 + R 3 ) −R 4 R 4 + R 5 −R 5 −(R 1 + R 2 + R 3 ) −R 5 R 1 + R 2 + R 3 + R 5   . The cofactor L 11 = R 4 (R 1 + R 2 + R 3 ) + R 5 (R 1 + R 2 + R 3 + R 4 ) is the total weight of the spanning trees of G . And indeed, we have L 11 = L 11 R 1 R 2 R 3 R 4 R 5 as predicted by Proposition 4. Furthermore, we get L 12,12 = R 1 + R 2 + R 3 + R 5 which gives according to Proposition 1, after multiplication by R 4 , the total weight of the trees in G which contain the dual edge e . Now, the equivalent resistances over edge e and e respectively are, according to Proposition 2, r 4 = L 34,34 L 11 = R 4 R 5 (R 1 + R 2 + R 3 ) R 4 (R 1 + R 2 + R 3 ) + R 5 (R 1 + R 2 + R 3 + R 4 ) r 4 = L 12,12 L 11 = R 1 + R 2 + R 3 + R 5 R 4 (R 1 + R 2 + R 3 ) + R 5 (R 1 + R 2 + R 3 + R 4 ) and finally indeed, with R 4 = 1/R 4 , r 4 R 4 + r 4 R 4 = 1. Open Problems It would be interesting to investigate graphs which are embedded in a compact surface of positive genus and their respective duals. In particular the results in [3] might help to generalize Theorem 5 for such graphs. However, the direct analogue is false, so one would have to expect some sort of a correction term in the formula. Another interesting question would be to ask if Theorem 5 holds for infinite periodic planar graphs and their duals. . where the sum runs over all edges e between the vertices i and j of G ,a ii , if i = j,where the diagonal values a ii are chosen such that the sum of all rows of L vanishes. Similarly the weight of a subgraph H of G is (i) The value of an arbitrary cofactor L ij of L is equal to the sum of the weights of all spanning trees in G and therefore AcknowledgementWe would like to thank the referees for their valuable remarks and suggestions which greatly helped to improve this article. Identities for minors of the Laplacian, resistance and distance matrices of graphs with arbitrary weights. P Ali, F Atik, R B Bapat, Linear Multilinear Algebra. 682P. Ali, F. Atik, and R.B. Bapat, Identities for minors of the Laplacian, resistance and distance matrices of graphs with arbitrary weights, Linear Multilinear Algebra 68 (2) (2020), 323- 336. Graphs and Matrices. R B Bapat, Hindustan Book AgencyLondon; New DelhiUniversitextsecond editionR.B. Bapat, Graphs and Matrices, Universitext. Springer, London; Hindustan Book Agency, New Delhi, second edition, 2014. Spanning trees of dual graphs. N Biggs, J. Combin. Theory Ser. B. 11N. Biggs, Spanning trees of dual graphs, J. Combin. Theory Ser. B 11 (1971), 127-131. A combinatorial proof of the all minors matrix tree theorem. S Chaiken, SIAM J. Algebraic Discrete Methods. 33S. Chaiken, A combinatorial proof of the all minors matrix tree theorem, SIAM J. Algebraic Discrete Methods 3 (3) (1982), 319-329. M Furrer, Widerstandssumme in dualen Netzwerken. Mentorierte Arbeit, 2017, ETH ZürichM. Furrer, Widerstandssumme in dualen Netzwerken, Mentorierte Arbeit, 2017, ETH Zürich. Resistor networks based on symmetrical polytopes. J Moody, P K Aravind, Electron. J. Graph Theory Appl. 31J. Moody and P.K. Aravind, Resistor networks based on symmetrical polytopes, Electron. J. Graph Theory Appl. 3 (1) (2015), 56-69. W T Tutte, Graph theory, volume 21 of Encyclopedia of Mathematics and its Applications. Crispin St. J. A. Nash-WilliamsCambridgeCambridge University PressReprint of the 1984 originalW.T. Tutte, Graph theory, volume 21 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2001, With a foreword by Crispin St. J. A. Nash- Williams, Reprint of the 1984 original. Equivalent resistors of polyhedral resistive structures. F J Van Steenwijk, American Journal of Physics. 661F.J. van Steenwijk, Equivalent resistors of polyhedral resistive structures, American Journal of Physics 66 (1) (1998), 90-91. An identity on resistance distances. Y J Yang, Material and Manufacturing Technology IV. Trans Tech Publications748Y.J. Yang, An identity on resistance distances, In Material and Manufacturing Technology IV, volume 748 of Advanced Materials Research, pages 1024-1027. Trans Tech Publications, 10, 2013.	[]
[ "Ensemble of Neural Classifiers for Scoring Knowledge Base Triples The Lettuce Triple Scorer at WSDM Cup 2017", "Ensemble of Neural Classifiers for Scoring Knowledge Base Triples The Lettuce Triple Scorer at WSDM Cup 2017", "Ensemble of Neural Classifiers for Scoring Knowledge Base Triples The Lettuce Triple Scorer at WSDM Cup 2017", "Ensemble of Neural Classifiers for Scoring Knowledge Base Triples The Lettuce Triple Scorer at WSDM Cup 2017" ]	[ "Ikuya Yamada \nStudio Ousia\nNara Institute of Science and Technology\nNara Institute of Science and Technology\n\n", "Motoki Sato [email protected] \nStudio Ousia\nNara Institute of Science and Technology\nNara Institute of Science and Technology\n\n", "Hiroyuki Shindo [email protected] \nStudio Ousia\nNara Institute of Science and Technology\nNara Institute of Science and Technology\n\n", "Ikuya Yamada \nStudio Ousia\nNara Institute of Science and Technology\nNara Institute of Science and Technology\n\n", "Motoki Sato [email protected] \nStudio Ousia\nNara Institute of Science and Technology\nNara Institute of Science and Technology\n\n", "Hiroyuki Shindo [email protected] \nStudio Ousia\nNara Institute of Science and Technology\nNara Institute of Science and Technology\n\n" ]	[ "Studio Ousia\nNara Institute of Science and Technology\nNara Institute of Science and Technology\n", "Studio Ousia\nNara Institute of Science and Technology\nNara Institute of Science and Technology\n", "Studio Ousia\nNara Institute of Science and Technology\nNara Institute of Science and Technology\n", "Studio Ousia\nNara Institute of Science and Technology\nNara Institute of Science and Technology\n", "Studio Ousia\nNara Institute of Science and Technology\nNara Institute of Science and Technology\n", "Studio Ousia\nNara Institute of Science and Technology\nNara Institute of Science and Technology\n" ]	[]	This paper describes our approach for the triple scoring task at the WSDM Cup 2017. The task required participants to assign a relevance score for each pair of entities and their types in a knowledge base in order to enhance the ranking results in entity retrieval tasks. We propose an approach wherein the outputs of multiple neural network classifiers are combined using a supervised machine learning model. The experimental results showed that our proposed method achieved the best performance in one out of three measures (i.e., Kendall's τ ), and performed competitively in the other two measures (i.e., accuracy and average score difference).	null	[ "https://arxiv.org/pdf/1703.04914v2.pdf" ]	95,523	1703.04914	56fa4e733ba87296061dab3eedac730aa3235b2e	Ensemble of Neural Classifiers for Scoring Knowledge Base Triples The Lettuce Triple Scorer at WSDM Cup 2017 5 Apr 2017 Ikuya Yamada Studio Ousia Nara Institute of Science and Technology Nara Institute of Science and Technology Motoki Sato [email protected] Studio Ousia Nara Institute of Science and Technology Nara Institute of Science and Technology Hiroyuki Shindo [email protected] Studio Ousia Nara Institute of Science and Technology Nara Institute of Science and Technology Ensemble of Neural Classifiers for Scoring Knowledge Base Triples The Lettuce Triple Scorer at WSDM Cup 2017 5 Apr 2017 This paper describes our approach for the triple scoring task at the WSDM Cup 2017. The task required participants to assign a relevance score for each pair of entities and their types in a knowledge base in order to enhance the ranking results in entity retrieval tasks. We propose an approach wherein the outputs of multiple neural network classifiers are combined using a supervised machine learning model. The experimental results showed that our proposed method achieved the best performance in one out of three measures (i.e., Kendall's τ ), and performed competitively in the other two measures (i.e., accuracy and average score difference). INTRODUCTION In the last decade, huge online structured knowledge bases (KBs) such as Wikidata [11], Freebase [4], and DBpedia [1] have emerged. These KBs contain an enormous number of entities (e.g., people) and their types (e.g., professions and nationalities). 1 These data enable users to easily formulate a complex query to a KB such as querying a list of all scientists who are nationals of Japan. However, the KB also contains many entity types that are rarely useful for humans when querying a KB. For example, Barack Obama has four professions listed in Freebase, namely Politician, Lawyer, Law professor, and Author, but it is considered that people primarily want to retrieve Barack Obama as a Politician. Recently, Bast et al. [2] addressed this problem by assigning a relevance score to each pair consisting of an entity and its type in KB. These scores enable us to enhance the ranking results of entity retrieval tasks by sorting the results based on these relevance scores. In this paper, we describe our approach for this task. We use multiple neural network classifiers with the objective of predicting the probability of an entity type when a KB entity is given. Notably, we introduce an attention mechanism to our neural network model in order to enable the model to prioritize a small number of relevant features. In addition, we use another supervised machine learning model (i.e., gradient boosted regression trees (GBRT) [6]) to convert the outputs of these classifiers into the final relevance scores. The proposed method was applied to the triple scoring task at the WSDM Cup 2017 [3,7] The results demonstrated that our method achieved the best results in one out of three measures (i.e., Kendall's τ ), and exhibited competitive performance in the other two measures (i.e., accuracy and average score difference). OUR APPROACH Given a KB entity e and its target type t, our method predicts a score that represents the relevance of e belonging to t. Here, we adopt a two-step approach: the first step is a classification step that aims to estimate the probability of e belonging to t (P (t\|e)) using multiple neural network-based classifiers. The second step is a scoring step that uses a supervised machine learning model to convert the outputs of these classifiers to the target relevance score. In accordance with the task specifications for WSDM Cup 2017, our model assigns relevance scores to pairs of people and their professions, and people and their nationalities. Classification Step We train the classifier by using all the KB entities that only have a single type, as in the previous work by Bast et al. [2]. This configuration enables us to address this problem as a multi-class classification of entities over all possible types. It is important to note that, because our objective is assigning relevance scores to entities with multiple types, entities with only a single type can be safely used as training data. Model We use sets of words and entities that are relevant to e as inputs to the classifier. We adopt the neural bag-of-items model with a simple item-level attention mechanism [9] to derive the representation of the set of items (i.e., words or entities). Specifically, given a set of items, x1, x2, ..., xN , we first compute the weighted sum of their corresponding embedding as follows: c = N i=1 a(xi)vx i ,(1) Here, vx ∈ R dw is an embedding of x, and a(x) is a function that computes the item-level attention weight for x, which is defined as the following softmax function: a(x) = exp(wa ⊤ ux + ba) N j=1 exp(wa ⊤ ux j + ba) ,(2) where wa ∈ R da is a weight vector, ba ∈ R is a bias, and ux ∈ R da is an attention embedding of x. The function a(x) aims to capture the importance of the item x, thereby allowing the model to focus on a small number of relevant items. Finally, we adopt a multi-layer perceptron (MLP) classifier with a single hidden layer with l units, ReLU non-linearity, and dropout with a probability p. Using Eq. (1), we compute two feature vectors cw and ce using the sets of words and entities, respectively. We then build a feature vector by concatenating L2-normalized versions of vectors cw \|\|cw \|\| and ce \|\|ce\|\| 2 , and feed the vector to MLP. Corpus As explained in the previous section, we train the classifier by using sets of words and entities relevant to e. To extract words and entities relevant to e, we use the following two sources: (1) the corresponding Wikipedia articles of e (denoted by article), and (2) Wikipedia sentences that contain a link anchor that corresponds to e (denoted by sentence). In both cases, words are extracted simply by tokenizing the text, and entities are the referent entities of link anchors in the text. Further, in the latter case, we restrict the words to the contextual words of the link anchor in a window of length m. 3 We extracted Wikipedia articles directly from the July 2016 Wikipedia dump obtained from Wikimedia Downloads 4 . We also used the public wiki-sentences dataset 5 to obtain the Wikipedia sentences. In addition, we used words and entities that appear five times or more in the corpus, and simply ignored the other words and entities. Training All parameters used in this model were initialized randomly and updated using back-propagation. We trained the model using stochastic gradient descent (SGD) and the learning rate was controlled by Adam [8]. The batch size was fixed as 100, the training consisted of one epoch, and the categorical cross-entropy was used for the loss function. We used a NVIDIA Tesla K80 GPU to train the model. Regarding hyper-parameters, the number of embedding dimensions dw and da were 300 and 10, respectively; the number of units in the hidden layer l was 2,000, and the dropout probability p was 0.5. We also selected the context window size of the link anchors m from 5 and 10. In addition, we optionally introduced class weights to the loss function because the distribution of the target type was highly imbalanced. We adopted a weighted loss function based on the class weight heuristic implemented in Scikit-learn 6 . We trained classifiers with various configurations. Table 1 shows the list of configurations used to train the classifiers. For each of the two corpora (i.e., article and sentence), we created eight classifiers with different training configurations, such as class weights and an attention mechanism in the enabled or disabled states 7 , using either both words and entities or only entities as input, and changing the context window size. In addition, we trained these classifiers for both the profession and nationality domains. Therefore, the total number of classifier instances was 32. 2 We also tested the vector averaging ( c N ) rather than L2 normalization; however, L2 normalization, in general, performed marginally more accurate in terms of the classification accuracy. 3 We do not include the words within the anchor text. 4 https://dumps.wikimedia.org/ 5 We downloaded the dataset from the Web site of the WSDM Cup 2017: http://www.wsdm-cup-2017.org/triple-scoring.html 6 https://github.com/scikit-learn/ 7 We disabled the attention mechanism by simply replacing a(x) in Eq.(1) by 1. Scoring Step We converted the outputs of the above-mentioned classifiers into relevance scores by adopting gradient boosted regression trees (GBRT) [6]. Given an entity e and a type t, our scoring model predicts the relevance score ranging from 0 to 7. We experimented with two models of GBRT: the regression model and the binary classification model. The regression model directly learns the target scores ranging from 0 to 7, whereas the binary classification model is trained using a modified dataset where the training instances with scores less than or equal to 2 are relabeled as false, while those with scores greater than or equal to 5 are relabeled as true, and the other instances are excluded from the training. During the inference stage, the regression model outputs an integer value that is the closest to the estimated score. The binary classification model predicts 5 if the predicted result is true, and predicts 2 otherwise. Moreover, we use exactly the same features for these two models. Features We compute the features based on two types of outputs of each classifier, the probability P (t\|e) and the unnormalized version of P (t\|e), which is the corresponding input value to the softmax layer of the MLP. For each of the two values, we compute three features, the value itself, and the difference between the value and the minimum and the maximum value among all valid types. It should be noted that the maximum value corresponds to the output value of the predicted type of the classifier. Further, we observe that some pairs of types co-occur very frequently in the KB (e.g., Singer and Singer-songwriter). In order to incorporate this into the model, we also use the point-wise mutual information (PMI) on the type co-occurrence data in the KB. In particular, we add the feature representing the PMI score between the target type t and the type predicted by each classifier when these two types are not equal. Moreover, apart from the classifier outputs, we also include the number of valid types associated with e in the feature set. Dataset We train our model by using the dataset obtained from the WSDM Cup web site. This dataset comprises two domains, professions and nationalities, of person entities retrieved from Freebase. The profession dataset and the nationality dataset contain relevance scores for 515 and 162 entity-type pairs with 134 and 77 distinct entities, respectively. We then use this dataset for feature selection and parameter tuning as described below. Training We train the regression and classification models for both the profession and the nationality domains. Feature selection is used to select a subset of the most relevant features. We first perform a greedy forward feature selection based on the performance of 10-fold cross validation, and simply select the set of features that perform the best. We also tune the hyper-parameters of GBRT using the selected features and the 10-fold cross validation, and use the hyper-parameters that provide the best performance. In addition, the performance is evaluated using the mean absolute error for the regression model and the accuracy for the binary classification model. Implementation We implemented the classifier described in Section 2.1 using Python, Keras 8 , and Theano [10]. Further, our scoring model de- 8 scribed in Section 2.2 was implemented using Python and Scikitlearn. We also used Hyperopt 9 for performing the hyper-parameter search of GBRT. EXPERIMENTS In this section, we first describe the performance evaluation of the classifiers presented in Section 2.1. Then, we present the official results of the triple scoring task at the WSDM Cup 2017. Evaluating Classifiers In order to independently evaluate the performances of the proposed classifiers, we randomly selected 10% of the KB entities with a single type, and measured the classification accuracy using these selected entities. Table 1 lists the accuracies of the classifiers corresponding to various training configurations presented in Section 2.1.3. As can be seen in the table, the attention mechanism effectively improved the performance, whereas the use of class weights degraded the accuracy in general. Further, the classifiers trained with the article corpus generally performed more accurately than those trained with the sentence corpus. We also found in our experiments that incorporating the outputs of classifiers that achieve lower accuracies often improved the performance of the scorer. Therefore, the strategy we adopted used the outputs of various classifiers rather than focusing on the outputs of a single accurate classifier. Further, in order to investigate how the attention model works in practice, we inspected the words and entities having large attention weights wa ⊤ ux in Eq.(2). Table 2 and Table 3 presents the top 10 words and entities with large weights, respectively. These weights were extracted from classifier 1, which was trained for the profession domain. It appeared that our classifier effectively focused on words and entities that strongly indicate a profession. For example, the top words included various professions, such as physicists and economists, and all the top entities were lists or categories that were strongly associated with a profession. 9 Competition Results We submitted our proposed method to the triple scoring task at the WSDM Cup 2017. In this competition, the submitted methods were evaluated based on the following three measures: • Accuracy, which is the percentage for which the estimated score differs from the score from the ground truth by at most 2. • Average score difference, which is the average score difference between the estimated scores and the ground truth scores. • Kendall's τ , which is the average Kendall's τ score 10 between the estimated scores and the ground truth scores. The τ score is computed for each entity, and the final score is averaged over all entities. Experiments were conducted using the 710 entity-type pairs containing the instances of 513 profession pairs and 197 nationality pairs. We used different scoring models trained with the corresponding dataset for each domain. Note that the accuracy described here is different from the accuracy used to evaluate the classifiers in the previous section. List of drummers 10 Category:Tennessee Titans players compared with the other top five methods proposed by competitors in terms of accuracy. The table lists the accuracies (acc), the average score differences (asd), and the Kendall's τ scores (tau). Our regression model achieved the best performance in terms of Kendall's τ scores among all the methods, and performed competitively in the accuracy and the average score difference. Further, the performance of our binary classification model was superior, particularly in terms of accuracy. CONCLUSIONS In this study, we proposed an approach for assigning a relevance score to each entity-type pair in a given KB. We trained neural network-based multiple classifiers by directly using the KB data, and converted the results of these classifiers into target relevance scores using a supervised machine learning model (i.e., GBRT). It is worth noting that the item-based attention model we introduced to the neural network model had not been applied to this kind of task previously. The experimental results confirmed the superiority of our approach; we achieved the best performances in terms of Kendall's τ scores, and performed competitively in terms of the accuracy and average score difference. We publicized the source code of our proposed method at https://github.com/wsdm-cup-2017/lettuce to enable it to be used for further academic research. https://github.com/fchollet/kerasCorpus type ID Word Entity Attention Class weight Window Accuracy (profession) Accuracy (nationality) Article 1 - - 85.4% 94.7% 2 - - - 84.5% 94.3% 3 - 73.3% 91.4% 4 - - 70.8% 90.9% 5 - - - 83.6% 94.3% 6 - - - - 82.5% 93.5% 7 - - 73.1% 90.4% 8 - - - 70.5% 89.4% Sentence 9 - 5 80.6% 90.4% 10 - - 5 79.5% 89.2% 11 5 56.4% 82.6% 12 - 5 55.6% 80.7% 13 - 10 79.0% 91.4% 14 - - 10 78.4% 90.3% 15 10 55.6% 83.4% 16 - 10 51.4% 81.8% Table 1: Various configurations used to train the classifiers. http://hyperopt.github.io/hyperopt/Rank Top words 1 physicists 2 economists 3 mathematicians 4 psychologists 5 draftexpress 6 novelists 7 bàsquet 8 botanists 9 aoni 10 barristers Table 2 : 2Top 10 words with large attention weights. Table 4 4contains the official results of our methods based on the regression model (reg) and the binary classification model (clf) Members of the United States House of Representatives from New York 2 List of Major League Baseball career stolen bases leaders 3 Category:Liberal Party of Australia members of the Parliament of AustraliaRank Top entities 1 Category:4 Category:Shooters at the 2012 Summer Olympics 5 Category:American science writers 6 Category:National Hockey League first round draft picks 7 Category:Cleveland Browns players 8 Category:American anthropologists 9 Table 3 : 3Top 10 entities with large attention weights.Name Acc Asd Tau Our method (reg) 0.77 1.59 0.29 Our method (clf) 0.82 1.76 0.36 bokchoy1 0.87 1.63 0.33 bokchoy2 0.82 1.50 0.32 radicchio 0.80 1.69 0.40 catsear 0.80 1.86 0.41 cress 0.78 1.61 0.32 Table 4 : 4Experimental results of our methods compared with the other top five methods submitted to WSDM Cup 2017. Entities and their types can be easily extracted from KB triples where their subjects refer to entities and their objects are the corresponding types. Here, the target triple is a triple describing a relation of which the object can be one among a limited set of values such as the nationalities of people. Following Bast et al.[2], we used the modified version of Kendall's τ score proposed in Fagin et al.[5] DBpedia: A Nucleus for a Web of Open Data. The Semantic Web. S Auer, C Bizer, G Kobilarov, J Lehmann, R Cyganiak, Z Ives, S. Auer, C. Bizer, G. Kobilarov, J. Lehmann, R. Cyganiak, and Z. Ives. DBpedia: A Nucleus for a Web of Open Data. The Semantic Web, pages 722-735, 2007. Relevance scores for triples from type-like relations. H Bast, B Buchhold, E Haussmann, SIGIR. ACMH. Bast, B. Buchhold, and E. Haussmann. Relevance scores for triples from type-like relations. In SIGIR, pages 243-252. ACM, 2015. Overview of the Triple Scoring Task at the WSDM Cup. H Bast, B Buchhold, E Haussmann, WSDM Cup. H. Bast, B. Buchhold, and E. Haussmann. Overview of the Triple Scoring Task at the WSDM Cup 2017. In WSDM Cup, 2017. Freebase: A Collaboratively Created Graph Database for Structuring Human Knowledge. K Bollacker, C Evans, P Paritosh, T Sturge, J Taylor, SIGMOD. K. Bollacker, C. Evans, P. Paritosh, T. Sturge, and J. Taylor. Freebase: A Collaboratively Created Graph Database for Structuring Human Knowledge. In SIGMOD, pages 1247-1250, 2008. Comparing and Aggregating Rankings with Ties. R Fagin, R Kumar, M Mahdian, D Sivakumar, E Vee, PODS. 47R. Fagin, R. Kumar, M. Mahdian, D. Sivakumar, and E. Vee. Comparing and Aggregating Rankings with Ties. In PODS, page 47, 2004. Greedy Function Approximation: A Gradient Boosting Machine. J H Friedman, The Annals of Statistics. 295J. H. Friedman. Greedy Function Approximation: A Gradient Boosting Machine. The Annals of Statistics, 29(5): 1189-1232, 2001. WSDM Cup 2017: Vandalism Detection and Triple Scoring. S Heindorf, M Potthast, H Bast, B Buchhold, E Haussmann, WSDM. ACM. S. Heindorf, M. Potthast, H. Bast, B. Buchhold, and E. Haussmann. WSDM Cup 2017: Vandalism Detection and Triple Scoring. In WSDM. ACM, 2017. Adam: A Method for Stochastic Optimization. D Kingma, J Ba, arXiv:1412.6980arXiv preprintD. Kingma and J. Ba. Adam: A Method for Stochastic Optimization. arXiv preprint arXiv:1412.6980, 2014. Not All Contexts Are Created Equal: Better Word Representations with Variable Attention. W Ling, Y Tsvetkov, S Amir, R Fermandez, C Dyer, A W Black, I Trancoso, C.-C Lin, EMNLP. W. Ling, Y. Tsvetkov, S. Amir, R. Fermandez, C. Dyer, A. W. Black, I. Trancoso, and C.-C. Lin. Not All Contexts Are Created Equal: Better Word Representations with Variable Attention. In EMNLP, pages 1367-1372, 2015. arXiv:1605.02688Theano Development Team. Theano: A Python Framework for Fast Computation of Mathematical Expressions. arXiv preprintTheano Development Team. Theano: A Python Framework for Fast Computation of Mathematical Expressions. arXiv preprint arXiv:1605.02688, 2016. Wikidata: A Free Collaborative Knowledgebase. D Vrandečić, M Krötzsch, Communications of the ACM. 5710D. Vrandečić and M. Krötzsch. Wikidata: A Free Collaborative Knowledgebase. Communications of the ACM, 57(10):78-85, 2014.	[ "https://github.com/scikit-learn/", "https://github.com/wsdm-cup-2017/lettuce", "https://github.com/fchollet/kerasCorpus" ]
[ "Lags of the KiloHertz Quasi-Periodic Oscillations in the transient source XTE J1701−462", "Lags of the KiloHertz Quasi-Periodic Oscillations in the transient source XTE J1701−462" ]	[ "Valentina Peirano \nKapteyn Astronomical Institute\nUniversity of Groningen\nP.O. BOX 8009700 AVGroningenThe Netherlands\n", "Mariano Méndez \nKapteyn Astronomical Institute\nUniversity of Groningen\nP.O. BOX 8009700 AVGroningenThe Netherlands\n" ]	[ "Kapteyn Astronomical Institute\nUniversity of Groningen\nP.O. BOX 8009700 AVGroningenThe Netherlands", "Kapteyn Astronomical Institute\nUniversity of Groningen\nP.O. BOX 8009700 AVGroningenThe Netherlands" ]	[ "MNRAS" ]	We analysed 14 observations with kilohertz quasi-periodic oscillations (kHz QPOs) of the neutron star X-ray binary XTE J1701−462, the first source to show a clear transition between atoll and Z-like behaviour during a single outburst. We calculated the average cross-spectrum of both atoll and Z-phase observations of XTE J1701−462 between a reference/hard band (6.1 -25.7 keV) and a subject/soft band (2.1 -5.7 keV) to obtain, using a novel technique, the average time lags of the lower and upper kHz QPOs. During the atoll phase, we found that at the frequency of the lower kHz QPO the soft photons lag behind the hard ones by 18 ± 8 s, whereas during the Z phase the lags are 33 ± 35 s, consistent with zero. This difference in the lags of both phases suggests that in XTE J1701−462, as observed in other sources, the lags decrease with increasing luminosity. We found that for both the atoll and Z phase observations the fractional rms amplitude increases with energy up to ∼10 keV and remains more or less constant at higher energies. Since these changes in the variability of XTE J1701−462 occur within the same outburst, properties like the mass of the neutron star or the inclination of the system cannot be responsible for the differences in the timing properties of the kHz QPOs in the atoll and Z phase. Here we suggest that these differences are driven by a Comptonizing component or corona, possibly oscillating in a coupled mode with the innermost regions of the accretion disc.	10.1093/mnras/stac1071	[ "https://arxiv.org/pdf/2204.06623v1.pdf" ]	248,177,801	2204.06623	895def2b766a42ca40c5335880eca4989a67b8a8	Lags of the KiloHertz Quasi-Periodic Oscillations in the transient source XTE J1701−462 2022 Valentina Peirano Kapteyn Astronomical Institute University of Groningen P.O. BOX 8009700 AVGroningenThe Netherlands Mariano Méndez Kapteyn Astronomical Institute University of Groningen P.O. BOX 8009700 AVGroningenThe Netherlands Lags of the KiloHertz Quasi-Periodic Oscillations in the transient source XTE J1701−462 MNRAS 0002022Accepted XXX. Received YYY; in original form ZZZPreprint 15 April 2022 Compiled using MNRAS L A T E X style file v3.0accretion, accretion discs -stars:neutron -X-ray:binaries -X-ray:individual:XTE J1701−462 We analysed 14 observations with kilohertz quasi-periodic oscillations (kHz QPOs) of the neutron star X-ray binary XTE J1701−462, the first source to show a clear transition between atoll and Z-like behaviour during a single outburst. We calculated the average cross-spectrum of both atoll and Z-phase observations of XTE J1701−462 between a reference/hard band (6.1 -25.7 keV) and a subject/soft band (2.1 -5.7 keV) to obtain, using a novel technique, the average time lags of the lower and upper kHz QPOs. During the atoll phase, we found that at the frequency of the lower kHz QPO the soft photons lag behind the hard ones by 18 ± 8 s, whereas during the Z phase the lags are 33 ± 35 s, consistent with zero. This difference in the lags of both phases suggests that in XTE J1701−462, as observed in other sources, the lags decrease with increasing luminosity. We found that for both the atoll and Z phase observations the fractional rms amplitude increases with energy up to ∼10 keV and remains more or less constant at higher energies. Since these changes in the variability of XTE J1701−462 occur within the same outburst, properties like the mass of the neutron star or the inclination of the system cannot be responsible for the differences in the timing properties of the kHz QPOs in the atoll and Z phase. Here we suggest that these differences are driven by a Comptonizing component or corona, possibly oscillating in a coupled mode with the innermost regions of the accretion disc. INTRODUCTION Kilohertz quasi-periodic oscillations (kHz QPOs), fast and highly coherent variability in the emission of a source, have been detected in neutron star low-mass X-ray binaries (LMXBs) since the first kHz QPO was observed in 1996 (Strohmayer et al. 1996;van der Klis et al. 1996). Among the different types of variability observed in the emission of neutron star LMXBs (see van der Klis 2004, for a review), kHz QPOs have the highest frequency in the power density spectra (PDS), with central frequencies spanning from 250 to 1200 Hz . Quasi-periodic oscillations in the emission of neutron star LMXBs are characterised by three basic parameters: their central frequency, central , their quality factor, = central /FWHM, where FWHM is the full width at half the maximum of the power of the QPO (see Belloni et al. 2002), and their fractional rms amplitude, rms. Usually kHz QPOs appear in pairs in the PDS and are called, respectively, upper and lower kHz QPOs, according to their relative Fourier frequency. Studying the phenomena behind kHz QPOs has remained of great interest due to the tight relation between the short timescales of the variability and the dynamical timescales of the inner accretion flow close to neutron stars (Stella et al. 1999;Psaltis & Norman 2000;Psaltis 2001). Understanding the nature of kHz QPOs, and the mechanism that produces them, can potentially shed light onto the physics of neutron stars and the study of environments under the ★ E-mail: [email protected] influence of their strong gravity (see e.g. Miller et al. 1998b;van der Klis 2005;Psaltis 2008). Multiple models have been proposed to explain the characteristics of kHz QPOs, however, none of them can represent all of the QPO properties simultaneously in a consistent way. Dynamical models that specifically try to explain the Fourier frequencies of kHz QPOs and relations among them are, for example: kHz QPOs as Keplerian oscillators under the influence of a rotating frame of reference (Osherovich & Titarchuk 1999;Titarchuk 2003), the sonic-point beatfrequency model (Miller et al. 1998a;Lamb & Miller 2003), kHz QPOs as resonances between the orbital and radial epicyclic relativistic frequencies in the accretion disc (Kluzniak & Abramowicz 2001) and the relativistic precession model (Stella & Vietri 1997). Deeper studies of spectral-timing properties of kHz QPOs can provide a more complete picture of the phenomena that produce the variability and its nature. For instance, by studying the correlation between the emission in different energy ranges using higher order Fourier techniques like the cross-spectrum, it is possible to constrain the emission processes involved in the variability (see Uttley et al. 2014, for a review of Fourier spectral timing techniques). Energy-dependent time lags are among these higher order timing products used to study the X-ray variability in LMXBs. The time lags of kHz QPOs were first studied by Vaughan et al. (1997) and Kaaret et al. (1999). Both groups found that, at the frequency of the lower kHz QPOs in 4U 1608−52 and 4U 1636−53, respectively, the soft Xray photons consistently lag behind the hard X-ray photons by around 25 s 1 . Since then, spectral-timing studies have been performed on other low mass X-ray binaries to measure lags more precisely, finding soft lags at the frequency of the lower kHz QPO (see e.g. de Avellar et al. 2013;Barret 2013;Troyer & Cackett 2017) and lags that are consistent with zero or slightly hard at the frequency of the upper kHz QPO (see e.g. de Avellar et al. 2013de Avellar et al. , 2016Peille et al. 2015;Troyer et al. 2018, for studies of the lags of both kHz QPOs). The nature of the lags between the soft and hard X-ray bands at the QPO frequencies is still a subject of debate, with no clear interpretation of the phenomena causing them and their relation to the physics of the source. For a long time Comptonization has been considered to be the source of the hard X-ray component in the spectra of X-ray binaries (Thorne & Price 1975;Shapiro et al. 1976;Sunyaev & Titarchuk 1980), and the interaction of soft photons with the Comptonizing region, or corona, to be a potential mechanism responsible for the observed time lags (Lee & Miller 1998). In principle, inverse Compton scattering leads to hard lags, since the most energetic photons are the ones that suffered the most number of scatterings and are, therefore, the ones that emerge last from the system. Soft lags are explained in Comptonization models by considering an oscillation in the temperature of the source of the soft photons produced through feedback by an oscillation in the temperature of the corona, with a fraction of the Comptonized photons returning to the soft photon source. This effect will ultimately delay the soft photons with respect to the hard photons, producing the lag that we observe (Lee et al. 2001). Self-consistent models that consider the changes in temperature of the corona and the aforementioned feedback onto the soft photons source have been subsequently proposed, and are capable of explaining the energy-dependent fractional rms amplitude and time lags (Kumar & Misra 2014Karpouzas et al. 2020). Neutron star LMXBs, where kHz QPOs appear, are generally divided into two classes: Atoll and Z sources (Hasinger & van der Klis 1989), depending on the path that the LMXB traces in the colourcolour diagram (CCD). The differences in the evolution of a source in the CCD are believed to be related to the mass accretion rate onto the neutron star and to the geometry of the accretion flow Done et al. 2007). Until 2006, with the discovery of XTE J1701−462 (Remillard et al. 2006), it was considered that atoll and Z sources were two types of intrinsically different neutron star systems, with different spectral and timing behaviour (van Straaten et al. 2003;Reig et al. 2004). However, XTE J1701−462 showed both atoll-like and Z-like behaviour during one single outburst (Homan et al. 2007;Lin et al. 2009a;Homan et al. 2010) becoming the first observed system to display a clear transition between the two classes. Sanna et al. (2010) studied the properties of the kHz QPOs in both phases of XTE J1701−462, and found that the quality factor and the fractional rms amplitude of both the lower and the upper kHz QPOs are consistently higher, at a given QPO frequency, in the atoll than in the Z phase (see also Barret et al. 2011). Sanna et al. (2010 proposed that the spectral properties of XTE J1701−462 suggest that many of the intrinsic differences believed to exist between atoll and Z sources (e.g. magnetic field or inclination) cannot explain the differences observed in the timing properties of kHz QPOs in both phases. The behaviour of the lag of the lower kHz QPO in atoll sources has indeed been observed to depend on energy in LMXBs, as suggested by the Comptonization with feedback model, becoming softer 1 When the soft photons lag behind the hard ones, the lags are called soft. When the hard photons are the ones lagging behind the soft ones, the lags are called hard. with increasing energy (see e.g. Troyer et al. 2018). Peirano & Méndez (2021) studied 8 atoll LMXBs and found that the slopes of the best-fitting linear model to the time-lag spectrum and the total rms amplitude of the lower kHz QPO exponentially decrease with increasing luminosity of the source, suggesting that the mechanism responsible for the lower kHz QPO depends on the properties of the corona. Peirano & Méndez (2021) also found that for the upper kHz QPO the slope of the time-lag spectrum is consistent with zero for all sources, concluding that the upper kHz QPOs have a different origin to the lower kHz QPOs (see e.g. de Avellar et al. 2013;Peille et al. 2015). The transient source XTE J1701−462 offers an unprecedented perspective if studied in a similar way, considering that during the transition from Z-like to atoll-like behaviour, XTE J1701−462 also experimented a significant change in luminosity (Sanna et al. 2010). To date, no study of the energy-dependent lags and their dependence on luminosity has been performed for XTE J1701−462. In this paper we combine previous studies of this source and other atoll LMXBs, with an analysis of the spectral-timing properties of the kHz QPOs observed during the 2006−2007 outburst of XTE J1701−462. We study and compare the energy dependence of the fractional rms amplitude, intrinsic coherence and lags at the frequency of the kHz QPOs in the atoll and Z phases of the outburst of XTE J1701−462. We also study the dependence of the average lags on the luminosity of the source, putting this result into context with the behaviour observed in other LMXBs. In § 2 we describe the observations and the methods used in the analysis of the data, in § 3 we show the results of such analysis, and in § 4 we discuss the scientific implications of these results. OBSERVATIONS AND DATA ANALYSIS Observations There are 866 observations in the public archive of the Rossi Xray Timing Explorer (RXTE; Bradt et al. 1993) of the source XTE J1701−462 collected using the Proportional Counter Array (PCA; Jahoda et al. 2006). From these observations here we studied the 14 observations that have kHz QPOs in their power spectra. We selected these observations using the criteria described in Sanna et al. (2010). Following Homan et al. (2010), we considered that XTE J1701−462 was in the Z phase of the outburst from the first time it was observed (Remillard et al. 2006) in January 2006, until April 2007. After this date, and until the source went into the quiescent phase, we considered XTE J1701−462 to be in the atoll phase of the outburst. Following this criterion there are 6 observations in the Z phase of the source (ObsIDs: 93703-01-02-04, 93703-01-02-05, 93703-01-02-08, 93703-01-02-11, 93703-01-03-00 and 93703-01-03-02) and 8 observations in the atoll phase (ObsIDs: 91442-01-07-09, 92405-01-01-02, 92405-01-01-04, 92405-01-02-03, 92405-01-02-05, 92405-01-03-05, 92405-01-40-04 and 92405-01-40-05) that show kHz QPOs. Hereafter, we will refer to the observations during the Z phase of XTE J1701−462 as Z observations, and to the observations during the atoll phase as atoll observations. Fourier timing analysis To study the kHz QPOs we computed Leahy-normalised PDS of each observation, using event-mode data with at least 250 s time resolution, covering the full energy range of the instrument (nominally 2 − 60 keV). For some Z observations event-mode data were not available for the entire energy range; in these cases we used a A multi-lorentzian model plus a constant accounting for the Poisson level has been consistently used to describe the shape of the PDS of low mass X-ray binaries (e.g. Nowak 2000;Belloni et al. 2002;Pottschmidt et al. 2003;Ribeiro et al. 2017). This model describes well the shape of the variability present in different types of sources, but is independent of the underlying physics that cause them, making it specially fit to compare the variability behaviour without making assumptions about the nature of the mechanism that produces the oscillations. The different components of the variability in the PDS can be labelled considering the strength, width and central frequency of the Lorentzian functions used to fit them (e.g. van Straaten et al. 2002). We used the convention ℓ and to label the lower and upper kHz QPOs, respectively. For a more detailed review of different kinds of variability observed in X-ray binaries see . We used a Lorentzian model with one or two Lorentzian functions plus a constant to fit and characterise the kHz QPOs in the PDS of observations in both the atoll and Z phase of XTE J1701−462. For the atoll observations we fitted the PDS between 600 and 900 Hz, rebinning the data by a factor of 64; and for the Z observations we fitted the PDS between 400 and 1200 Hz, rebinning the data by a factor of 128. Since we observed only one kHz QPO during the atoll phase, which Sanna et al. (2010) identified as the lower kHz QPO, we used only one Lorentzian function to characterise it during the fit. In the Z phase we observed two simultaneous kHz QPOs, from which we identified the one at the lowest central frequency as the lower kHz QPO and the one at the highest central frequency as the upper kHz QPO. For the Z observations we used two Lorentzian functions to describe each kHz QPO during the fit. Inspection of the PDS of observations in both the atoll and Z phases showed that the lower kHz QPOs has a higher quality factor in the atoll phase than in the Z phase, confirming the results of Sanna et al. (2010). We found that, in the PDS of the atoll observations, the central frequency of the lower kHz QPOs varies considerably throughout the length of one entire observation. To trace these changes in central frequency, we constructed dynamical power spectra of the atoll observations using 16-s segments PDS and assigning a unique ℓ central frequency to each segment. When necessary, we combined multiple contiguous 16-s PDS until it was possible to identify a unique central frequency value for those segments combined. Using these central frequencies, we split every PDS into 8 different frequency selections, with limits in frequency listed in Table 4. We shift-and-added (Méndez et al. 1998) together every 16-s PDS of each frequency selection into one unique PDS, shifting the kHz QPOs to a frequency in the centre of the corresponding frequency selection. In contrast, when analysing the PDS of the Z observations, because the QPOs are weaker and broader than in the atoll phase, it is not Table 2. Energy and channel equivalent ranges of the bands used in the calculation of the fractional rms amplitude. These ranges were selected to be as close as possible to the ones defined by Ribeiro et al. (2019). The energies correspond to the Epoch 5 of the instrument. possible to significantly detect changes in the central frequency of the kHz QPOs within the length of a single observation. For this reason we analysed the PDS of every full observation during the Z phase separately (see Table 3). In Fig. 1 and Fig. 2 we show examples of the PDS from two Z observations (91442-01-07-09, with a more significant lower kHz QPO, and 92405-01-02-03, with a more significant upper kHz QPO) and the PDS of the 830−840 Hz frequency selection of the atoll observations. To calculate the average cross-spectra, G = Re[G] + Im[G], we computed complex Fourier transforms for both the atoll frequency selections and the Z observations in two different bands: a reference (hard) band and a subject (soft) band (see Table 1 for the limits in RXTE channels and equivalent energy of the bands that correspond to the Epoch 5 of the instrument, as all our observations were performed towards the end of the RXTE mission). Equivalently to the atoll PDS procedure described in the previous paragraph, we calculated the atoll complex Fourier transforms of each frequency selection in Table 4 shift-and-adding (using the already defined central frequencies) 16s segments with a time resolution of 250 s, and a minimum and maximum Fourier frequency of 0.0625 Hz and 2048 Hz, respectively. Examples of real and imaginary parts of these averaged cross-spectra are shown in the second and third panels of Fig. 1 and Fig. 2, for the atoll and Z phase, respectively. The errors reported for both the real and imaginary parts of the cross-spectra are given by Eq. (13) in Ingram (2019). Using these averaged cross-spectra and following the procedure in Nowak et al. (1999), we calculated the phase lag and intrinsic coherence as a function of the Fourier frequency for each phase (bottom panels in Fig. 1 and Fig. 2). Considering that we use the hard band as the reference band, positive values of the lags represent the soft photons lagging the hard ones, while negative values represent hard photons arriving after the soft ones. Average phase lag of the kHz QPOs To calculate the time and phase lags of kHz QPOs it is common to select a frequency range (using, for example, the FWHM as criterion) and average the frequency-dependent real and imaginary parts of the cross-spectrum of an observation over this frequency range (see e.g. de Avellar et al. 2013;Troyer et al. 2018). This approach gives an accurate enough value when the lag is significant enough in the cross-spectrum, which is the case for the lower kHz QPO in the atoll observations of XTE J1701−462, but it can be a limitation 1 ± 0.3 9.0 ± 0.5 5.5 ± 0.9 0.12 ± 0.12 830 − 840 835.0 ± 0.2 8.7 ± 0.2 6.3 ± 0.5 0.18 ± 0.06 840 − 850 845.1 ± 0.3 7.6 ± 0.5 5.7 ± 1.0 −0.06 ± 0.12 850 − 950 900.2 ± 0.5 6.7 ± 0.5 6.8 ± 1.4 −0.19 ± 0.14 Joint fit 0.10 ± 0.04 Table 4. Properties of the kHz QPOs detected in the atoll phase of XTE J1701−462. The fractional rms amplitude was calculated over the full energy range of the instrument, between 2 and 60 keV. The phase lag, between the reference-hard band and the subject-soft band in Table 1, was calculated as described in § 2.3, first for each frequency selection individually and then in a joint fit of all frequency selections to obtain a unique value for the atoll phase of the source. Subscript letter ℓ denotes lower kHz QPOs. when studying the Z observations, where the kHz QPOs are less significant and frequency-dependent lags are consistent with zero. In this paper we applied a novel technique to calculate the average lags of the the lower and upper kHz QPOs of XTE J1701−462, considering the entire region in the PDS where the QPO is present. We computed the average lag of the kHz QPOs directly from a joint multi-Lorentzian fit to the real and imaginary parts of the crossspectra of each phase, calculated as described in § 2.2. For the Z phase, we jointly fitted the cross-spectra from all Z observations (both linked and separately) fixing the values of the central frequency and the FWHM to the corresponding best-fitting parameters that describe the kHz QPOs in the Z-phase PDS (see Table 3). To directly obtain the average lags from the fit, we allowed the normalisation of the real part and the phase lag to vary and computing the normalisation of the imaginary part as Im[G] = Re[G] tan(Δ ). This procedure is justified because the power in the full band is equal to the sum of the square of the real and imaginary parts of the Fourier transform of the full band light curve, and the signals in the two bands used to calculate the cross-spectrum are highly correlated in the frequency range of the QPOs. Because this method assumes that the lags in the frequency range over which the QPO is significant are constant, our procedure takes the full QPO profile to measure the lags. For the Z observations 91442-01-07-09 and 92405-01-01-02 we only included the lower kHz QPO in the fit, as the upper kHz QPO is not significant in the PDS. Similarly, for the observation 92405-01-01-04 we considered only the upper kHz QPO in the fit, as the lower kHz QPO is not significant in the PDS (see Sanna et al. 2010). For the atoll phase we followed a similar procedure to the one of the Z phase, but we used the cross-spectra of each frequency selection, instead of the individual atoll observations, and the corresponding best-fitting parameters to the PDS (see Table 4) to perform the joint fit of the real and imaginary parts of the cross-spectra. In all the fits we considered the so-called channel cross-talk (see section 2.4.2 in Lewin et al. 1988), a consequence of deadtime producing correlation between energy channels, by adding a constant function to the multi-Lorentzian model, that varies independently while fitting both the real and imaginary parts of the cross-spectrum. We find that the constant component of the imaginary part is consistent with zero for all fits, which is as expected since the channel cross-talk only contributes to the real part of the cross-spectrum. The technique we used in this paper makes the lag comparison between the atoll and Z phases of XTE J1701−462 consistent, avoiding bias by selecting an arbitrary frequency range in the cross-spectrum related to the FWHM of the QPO, given that the FWHM of the QPO in Z and atoll phases is significantly different (Sanna et al. 2010). Energy-dependent rms Additionally, we calculated the PDS of both atoll and Z observations for a set of energy ranges in order to study the fractional rms spectrum Real and imaginary parts of the cross-spectrum of the two Z observations. The red solid line shows the joint best-fitting Lorentzian functions. Fourth and fifth panels: Phase lag and intrinsic coherence as a function of Fourier frequency. In the fourth panel, the grey solid line indicates zero phase lag. In the fifth panel, the the grey solid lines indicate coherence equal to one (perfect coherence between signals) and equal to zero (completely incoherent signals). of the QPO. We selected the energy ranges to be as close as possible to the ones used in Ribeiro et al. (2019), where they defined the channel limits considering the drift in energy-to-channel relation of the ranges used in de Avellar et al. (2016). While for the atoll observations we were able to use exactly the same energy ranges as in Ribeiro et al. (2019), in the Z observations the ranges had to be adapted due to the data structure of the observations. The limits of the energy ranges used in this paper are given in Table 2, where the equivalent channel ranges correspond to the Epoch 5 of the instrument like in Table 1. Again these PDS were calculated using the same shift-and-add procedure described previously to obtain one single PDS per phase and energy range, with a Fourier frequency range from 0.0625 Hz to 2048 Hz. To calculate the fractional rms amplitude, we fitted the highfrequency region (frequencies higher than 400 Hz) of the PDS calculated in each energy band in Table 2 with either one -for the atoll phase -or two -for the Z phase -Lorentzian functions plus a constant Phase lag and intrinsic coherence as a function of Fourier frequency. In the fourth panel, the grey solid line indicates zero phase lag. In the fifth panel, the the grey solid lines indicate coherence equal to one (perfect coherence between signals) and equal to zero (completely incoherent signals). to account for the Poisson noise. We considered the integral power obtained from the fit of the Lorentzian function as equivalent to the total power at the frequency of the kHz QPO, QPO . The fractional rms amplitude in percent units is then given by, rms = 100 √︄ QPO + + %,(1) where is the source count rate -equivalent to the total count rate minus the background count rate, . The errors reported for the rms calculations correspond to 1 , every time the kHz QPOs were detected with at least 3 significance. In those cases in which the power of the QPO was consistent with zero or not significant enough when calculating the rms, we report the 95% confidence upper limit, which we calculated by fixing both the central frequency and the width of the Lorentzian function to the values we obtained from the fit on the entire energy range (in Table 3 for the Z phase and Table 4 for the atoll phase). RESULTS In this section we show the results of the Fourier analysis of the XTE J1701−462 observations during the atoll and Z phases of the outburst. In § 3.1 we show the PDS of the source during both phases and the results of the fitting procedure to identify the kHz QPOs. In § 3.2 we show the analysis of the phase lags and intrinsic coherence of the signal as a function of Fourier frequency in the atoll and Z phases. In § 3.3 we examine the behaviour of the phase lags vs QPO frequency for each observation during the Z phase and for each frequency selection of the atoll phase. In § 3.3 we study the relation between the average time lags of the lower kHz QPOs during the atoll and Z phase of XTE J1701−462 and the luminosity of the source, comparing our results with other atoll sources. Finally, in § 3.5 we explore the fractional rms amplitude dependence upon energy in each phase of the outburst. QPO identification and properties To distinguish between the lower and the upper kHz QPO, we adopted the kHz QPO identification used in Sanna et al. (2010). In the Z phase of XTE J1701−462 the identification is straightforward: two peaks appear in the PDS at high frequencies, one corresponding to the lower and one to the upper kHz QPO. In contrast, during the atoll phase only one peak is significant enough in each observation, corresponding to the lower kHz QPO (see figure 2 in Sanna et al. 2010, where a second higher frequency peak appears after applying the shift-and-add method to all atoll phase observations combined). In the top panels of Fig. 1 and Fig. 2 we show the resulting bestfitting Lorentzian model to the PDS of two Z observations (91442-01-07-09 and 92405-01-02-03) and one frequency selection (between 830 and 840 Hz) of the atoll observations, respectively, as examples of the analysis performed over all the observations of XTE J1701−462. The resulting best-fitting parameters to each Z observation PDS are listed in Table 3, while the best-fitting parameters to each frequency selection PDS of the atoll observations are listed in Table 4. Phase lag and coherence In Table 3 and Table 4 the best-fitting values of the phase lag, calculated following the procedure described in § 2.3, of both lower and upper kHz QPOs in the Z and atoll phases, are shown. In the atoll phase the average phase lag over all the selections of the lower kHz QPO frequency is 0.08 ± 0.04 radians, which means that the soft photons lag the hard ones by approximately 16 s at this frequency. In the Z phase the average phase lag at the frequency of the lower kHz QPO is 0.05 ± 0.15 radians and at the frequency of the upper kHz QPO is 0.25 ± 0.16 radians, both consistent with zero. The bestfitting Lorentzian functions to the real and imaginary parts of the cross-spectra of two Z observations and one frequency selection of the atoll observations are shown in the middle panels of Fig. 1 and Fig. 2. Since the data from each observation in the Z phase and frequency selection in the atoll phase look very similar, we show these three examples to illustrate the analysis process. In the bottom panel of Fig. 2 is apparent that the coherence is well constrained (small error bars) around the central frequency of the QPO in the atoll phase, within a close to symmetric frequency range around ∼835 Hz. Immediately outside of this frequency range the errors of the coherence increase noticeably in the plot. In the bottom panels in Fig. 1, while we also observe smaller errors in coherence at the central frequency of the kHz QPOs, the frequency range at which the coherence appears more constrained in the Z phase is narrower (2021), the blue diamond corresponds to XTE J1701−462 during its atoll phase and the red diamond to XTE J1701−462 during its Z phase. The solid black line indicates the best-fitting exponential model to the data. The solid grey line indicates the best-fitting exponential model to the atoll sample in Peirano & Méndez (2021). The value of the luminosity of XTE J1701−462 during its atoll and Z phases was extracted from Fig. 5 in Sanna et al. (2010). Residuals, (data-model)/error, are also shown. and less symmetric than it is in the atoll phase. This difference in the behaviour of the coherence of the signal, and considering that during the Z phase the kHz QPOs appear weaker and broader than during the atoll phase, makes the technique we used here to calculate the lags (see § 2.3) especially suitable to appropriately compare the two phases of XTE J1701−462. Phase lags and QPO frequency In Fig. 3 we show the relation between the phase-lag of the lower and upper kHz QPOs of XTE J1701−462 and the QPO frequency, using the data in Table 3 and Average time lags and luminosity In Fig. 4 we show the dependence of the average time-lag of the lower kHz QPO in the atoll and Z phases of XTE J1701−462 upon luminosity, and compare it to the data of the 8 atoll sources in Peirano & Méndez (2021). The solid lines in the figure represent the bestfitting exponential model, Δ ℓ = −( / Edd )/ , to only the 8 atoll sources in Peirano & Méndez (2021) (in grey), with = 36.5 ± 7.1 s and = 0.09 ± 0.03; and to the same 8 atoll sources plus XTE J1701−462 in both phases (in black), with = 36.5 ± 6.2 s and = 0.09±0.03. From Fig. 4 is clear that, as the luminosity increases, the average time-lags decrease exponentially for the 8 atoll sources in Peirano & Méndez (2021) and XTE J1701−462 in the atoll phase. In the Z phase of XTE J1701−462 the time-lags are not as well constrained as in the atoll phase, however, the residuals show that the lags are consitent with the same trend of the atoll sources in Peirano & Méndez (2021) and the atoll observations of XTE J1701−462. We extracted the luminosity of XTE J1701−462 during its Z and atoll phases from Fig. 5 in Sanna et al. (2010). To calculate the luminosity, Sanna et al. (2010) used a distance of 8.8 kpc, estimated by Lin et al. (2009b) using the Type-I X-ray bursts that occured during the 2006-2007 outburst of XTE J1701−462, and the 2 − 50 keV flux from the source normalised by Edd = 2.5 × 10 38 erg s −1 , which corresponds to the Eddington luminosity of a 1.9 M neutron star accreting gas with cosmic abundance. Sanna et al. (2010) studied the dependence of the fractional rms amplitude of the kHz QPOs upon the QPO frequency of XTE J1701−462, both in the atoll or Z phases, and showed that the fractional rms amplitude of the lower kHz QPO is consistently higher in the atoll than in the Z phase (see Fig. 4 Sanna et al. 2010). Similarly, Sanna et al. (2010) found that the quality factor of the lower kHz QPO is significantly higher in the atoll than in the Z phase. Fractional rms amplitude vs energy In Fig. 5 we show the fractional rms amplitude of both kHz QPOs as a function of energy for the atoll and Z phases, for the different energy bands defined in Table 2. In the figure, the error-bars represent the 1 uncertainty and the arrows represent the 95% confidence upper limits. In Fig. 5 it is apparent that the fractional rms amplitude of the lower kHz QPO is consistently higher in the atoll than in the Z phase for all energies. In the Z phase observations, the dependence of the fractional rms amplitude upon energy for the lower and the upper kHz QPOs has no apparent significant differences. In all cases the fractional rms amplitude increases with energy up to a certain value and then remains constant for higher energies. To study the shape of the relation between the fractional rms amplitude and the photon energy, we fitted a broken line model to the data, given by the following equation, rms( ) = 1 , if < break 2 + ( 1 − 2 ) break , if ≥ break ,(2) where break is the break energy (up until which the rms value increases), and 1 and 2 are the slopes of the lines before and after the break energy, respectively. We first performed a joint fit of the fractional rms amplitude data of all kHz QPOs (the lower kHz QPO in the atoll phase, and the lower and upper kHz QPOs in the Z phase) using the model given by Eq. (2) and linking both break and 2 for all the data-sets. This fit yielded a reduced chi-squared 2 = 1.037 for 9 degrees of freedom. Since the resulting best-fitting value of 2 = 0.27 ± 0.13 was not significantly different from zero, we performed a new fit linking only break and fixing 2 to zero for all data-sets. This new fit yielded a reduced chi-squared 2 = 1.400 for 10 degrees of freedom. The F-test between the broken-line model with break and 2 linked and the broken-line model with only break linked and 2 = 0 gives a probability of 0.06. In other words, there is no statistically significant advantage in considering the slope after the break 2 different from zero in the fit. The best-fitting parameters of the fit with break linked and 2 fixed to zero are given in Table 5 and the resulting fit is shown in Fig. 5 for both the Z and atoll phases. From this figure, and the values in Table 5, we found that the difference between the slope before the break 1 of the lower kHz QPO in the atoll and Z phases has a significance of ∼ 14 , which means that the dependence upon energy of the fractional rms amplitude of the lower kHz QPO is significantly different for both phases. Similarly, we contrasted the slope before the break for the lower and upper kHz QPOs in the Z phase only, finding that there is no significant difference between them. In the fit of the fractional rms amplitude vs energy we used the 1 error as the uncertainty of the variable to fit. When the fitted Lorentzian in the corresponding energy band PDS yielded a negative integral power of the QPO in comparison with the Poisson level, we fixed the fractional rms amplitude to zero and we considered the uncertainty to be the 95% confidence upper limit to perform the fit. In the cases where the integral power of the QPO was positive, but not significantly different from zero, we considered the original fitted value of the integral QPO power and its 1 error as valid during the fit. The binning in time we performed when calculating the PDS to have a Nyquist frequency of 2048 Hz reduces the variability dominantly at high frequencies (see section 4.3 van der Klis 1989), which can be relevant for the frequency ranges at which kHz QPOs are observed. Equation 4.7 in van der gives the correction factor for this effect as follows, = / sin / = /2 Nyq sin /2 Nyq ,(3) where is the frequency of the QPO, is the length of the observation, is the number of power spectra binned together when calculating the PDS and Nyq = 1/2Δ is the Nyquist frequency of the PDS, with Δ = / , the length of each data segment. The correction factor, for Nyq = 2048 Hz, of the lower and upper kHz QPO in the Z observations is ∼1.04 and ∼1.09, respectively, and of the lower kHz QPO in the atoll observations is ∼1.07. These values are all within the errors reported in Fig. 5, therefore not affecting the results we present here. DISCUSSION We studied the timing properties of the kHz QPOs of the transient neutron-star LMXB XTE J1701−462 and characterised, for the first time, the frequency dependent QPO lags simultaneously for the atoll and Z phases of the source. We calculated, using a novel technique, the average lags at the frequency of the QPO both in the Z and atoll phases and discovered that during the atoll phase the time lags of the lower kHz QPO are soft, with the soft photons lagging the hard ones by around 16 s. During the Z phase, the lags of both the lower and upper kHz QPOs are consistent with zero. We also found that the intrinsic coherence of the signal is more well constrain at the frequency of the lower kHz QPO in the atoll than in the Z phase. Additionally, we explored the behaviour of the phase lags at different QPO frequencies and observed that while the lags of the lower kHz QPO in the atoll observations of XTE J1701−462 have a slight dependence upon QPO frequency, the phase lags of both the lower and upper kHz QPOs in the Z observations do not. We also studied the dependence of the average time-lags of the lower kHz QPO upon luminosity during both the Z and atoll phases. We found that the average lags follow the same trend with luminosity of other atoll neutron-star systems, with the lags decreasing exponentially with increasing luminosity. Finally, we studied the fractional rms amplitude dependence upon energy of the lower and upper kHz QPOs in each phase of XTE J1701−462 and discovered that, as observed in other LMXBs, the fractional rms amplitude of the lower kHz QPO increases with energy up to approximately 10 keV and then remains constant at higher energies. Soft lags and the corona During the atoll phase of XTE J1701−462 we observe more clearly that, in the presence of the lower kHz QPO, the photons in the soft band, between 2.1 and 5.7 keV, lag behind the photons in the hard band, between 6.1 and 25.7 keV. Inverse Compton scattering with feedback onto the soft photon source (see Lee et al. 2001;Kumar & Misra 2016;Karpouzas et al. 2020) can explain these soft lags in a scenario where it is not the disc, but the corona that is responsible for the variability we observe. Indeed, the corona dominates the X-ray emission at high energies, where we observe the variability reaching its maximum amplitude (see Fig. 5, where the fractional rms amplitude for the lower kHz QPO in the atoll phase of XTE J1701−462 reaches its maximum at around 10 keV). Inverse Compton scattering occurs in the corona when electrons transfer energy to photons coming from the soft energy source in an LMXB (either the disc or the surface of the neutron star), producing a delay in the emission of hard photons, with higher energies. A fraction of these Comptonized photons return to the disc and are emitted again at lower energies and later times, producing the soft lags we observe (Lee et al. 2001;Karpouzas et al. 2020). This model not only explains the soft lags we see in the lower kHz QPOs in our data, but also can describe the difference in sign of the lags of the upper kHz QPOs in other sources (see e.g. de Avellar et al. 2013;Peille et al. 2015) and the behaviour of the lags of other variability components (see e.g. Miyamoto et al. 1988;Ford et al. 1999). Variability in the Z and atoll phases We observe multiple differences between the variability properties of the kHz QPOs in the Z and atoll phases of XTE J1701−462. More evident in Fig. 5 is the consistently higher fractional rms amplitude of the lower kHz QPO in the atoll phase of the source when compared with the lower and upper kHz QPOs present in the Z phase. This discrepancy in the timing properties of the kHz QPOs in both phases of XTE J1701−462 are comparable with observed differences in the variability of atoll and Z LMXBs. For example, Méndez (2006) studied 12 LMXBs and found that in the Z sources of his sample the maximum rms amplitude of both kHz QPOs and the maximum quality factor of the lower kHz QPO are consistently lower than in the atoll sources. In Fig. 1 and Fig. 2, we also observe a difference in the intrinsic coherence of the signal at the frequency of the QPOs, between both phases. In the figures it is clear that the frequency range within which the uncertainty of the intrinsic coherence is smaller, is not as well constrained in the Z than in the atoll observations. During the atoll phase of the source, the intrinsic coherence remains visibly more stable when the QPO is present in the PDS. The difference in the intrinsic coherence of the signal, although consistent with other LMXBs (see e.g. Barret et al. 2011), makes using it as a criterion to measure the average lag over a frequency range, and compare both phases consistently, difficult. Here we used a novel technique to calculate the average lags at the frequency of the QPO (see § 2.3), that allows for a more consistent comparison between the atoll and Z phases. In this method we use the Lorentzian function that describes the kHz QPOs in the PDS to find the amplitudes of the real and imaginary parts of the cross-spectra, without averaging them over a certain arbitrary frequency range defined by the coherence of the signal (which will, particularly in the case of the Z observations, lead to inconsistencies in the obtained lags). We also studied the behaviour of the phase lags of the lower and upper kHz QPOs with respect to the QPO frequency, for both the atoll and Z phases. In Fig. 3 we observe a weak dependence upon QPO frequency of the lags of the lower kHz QPO in the atoll phase. Despite the fact that the trend of these lags is also consistent with a constant model, the slight increase and then decrease with frequency we see in Fig. 3 is consistent within errors with the relations shown by Barret (2013) and de Avellar et al. (2013) for the lags of the lower kHz QPOs in 4U 1608−522 and 4U 1636−53, respectively. The phase lags of the lower and upper kHz QPOs in the Z phase do not show a trend with QPO frequency in Fig. 3, which is also in agreement with previous studies where it has been shown that the lags of the upper kHz QPO are constant with QPO frequency (see e.g. de Avellar et al. 2013;Peille et al. 2015). The large error of the average lags of the Z observations that we obtained using the technique described in § 2.3, together with the differences in the behaviour of the intrinsic coherence of the signal in the Z and atoll phases and the lower fractional rms amplitude of the variability in the Z phase (already observed by Sanna et al. (2010)), suggest that the mechanism responsible for the kHz QPOs is affected during the transition from a Z-like source to an atoll-like source of XTE J1701−462. Differences in the properties of kHz QPOs of atoll and Z sources, like the ones we observe before and after the transition of XTE J1701−462, are believed to be related to the geometry of the accretion flow and the mass accretion rate of the source (see Méndez 2006, for a more detailed discussion). Sanna et al. (2010) discarded changes in the magnetic field, neutron-star mass and inclination of the system as responsible for the variations we observe in the high-frequency variability in our data, as these changes cannot occur within the timescale of the transition of XTE J1701−462. Furthermore, Sanna et al. (2010) studied how the maximum quality factor and rms amplitude of the lower kHz QPOs in both the Z and atoll phases of XTE J1701−462 depended upon the luminosity of the source and found that this relation follows the trend found by Méndez (2006) for a sample of 12 atoll and Z LMXBs. These results characterise XTE J1701−462 as a unique case to study, as the behaviour of its variability follows the relations observed in other sources, while many of the fundamental properties of the source remain constant. Peirano & Méndez (2021) studied the relation between the slope of the time-lag spectrum, , and the luminosity of the source for eight atoll LMXBs and found that decreases exponentially with increasing luminosity. These authors found a similar relation between the average time-lags and the luminosity of the source. Considering these results, it is interesting to check whether the relation of the average time-lags and luminosity holds for the Z and atoll phases of XTE J1701−462. Lags and the atoll-and Z-phase luminosity In Fig. 4 we combine the data from Peirano & Méndez (2021) with the data of XTE J1701−462, using the luminosities for the Z and atoll phases of XTE J1701−462 given by Sanna et al. (2010). In the top panel of the figure it is apparent that XTE J1701−462, during its atoll phase, follows the trend of the other atoll sources very closely. Indeed, the black solid line, that represents the best-fitting exponential model to all data points, lies almost exactly over the grey solid line that represents the best-fitting exponential model to only the atoll sources in Peirano & Méndez (2021). During the Z phase the average lags have a large uncertainty, however, in the bottom panel of Fig. 4 it is apparent that during the Z phase of XTE J1701−462 the relation between the average lags of the lower kHz QPO and luminosity also holds. Peirano & Méndez (2021) suggested that the similar relations be-tween the slope of the time-lag spectrum and the total rms amplitude with the luminosity of the source (both decrease exponentially with increasing luminosity) imply that there is one single property of the system in these LMXBs that drives the behaviour of the variability. Since we observe a similar dependence upon luminosity of the average lags in both Z and atoll phases of XTE J1701−462, we can conclude that a similar effect is taking place in this source while it transitioned from one phase to the other. This relation suggests that it is the corona that is responsible for the changes we observe in the variability, as its contribution in the energy spectrum changes with luminosity 2 . This scenario fits with the inverse Compton scattering model with feedback onto the soft photon source (see Lee et al. 2001;Kumar & Misra 2016;Karpouzas et al. 2020) we described in § 4.1, as in this model is also the corona the one that drives the properties of the variability. Our results confirm what was suggested in Peirano & Méndez (2021), where they described the mechanism that modulates the kHz QPOs as a coupled mode of oscillation between the corona and the disc. In this "coupled oscillation mode", changes in the luminosity -if used as a proxy of the properties of the corona -when XTE J1701−462 transitions from the Z to the atoll-like behaviour, the fractional rms amplitude and the lags of the kHz QPOs should decrease, exactly as we observe in our data. Studies of the behaviour of time and phase lags with luminosity for Z neutron star LMXBs, that are located in the high-luminosity end of Fig. 4, could help elucidate the real nature of the mechanism responsible for the high-frequency variability we observe. To date no systematic study of the timing properties of Z sources has been performed, but the results shown in the present paper suggest that the average lags of the lower kHz QPO should decrease exponentially as the luminosity increases. A more in depth analysis of the results presented in this paper, through the glass of the model described in Karpouzas et al. (2020) and García et al. (2021) could also help understand the changes in the geometry of the accretion flow that can be driving the changes in the properties of the variability when XTE J1701−462 is transitioning from the Z to the atoll phase. Figure 1 . 1Top panel: PDS of the observations 91442-01-07-09 and 92405-01-02-03 of XTE J1701−462 during its Z phase. The fitted Lorenztian function to the kHz QPO is shown in red on top of the histogram. Second and third panels: Figure 2 . 2Top panel: Shifted-and-added PDS of XTE J1701-462 during its atoll phase, for the observations where the frequency of the lower kHz QPO was between 830 and 840 Hz. Second and third panels: Real and imaginary parts of the cross-spectrum of the atoll frequency selection. The red solid line shows the joint best-fitting Lorentzian function. Fourth and fifth panels: Figure 3 . 3Phase-lag of the lower (red squares) and upper (purple triangles) kHz QPO of XTE J1701−462 Z observations and the lower (blue circles) kHz QPOs of XTE J1701−462 atoll observations as a function of QPO frequency. The error-bars of the atoll-phase lower kHz QPO lag represent the frequency bins ranges fromTable 4. The diamonds show the average lag value of the lower and upper kHz QPOs in the Z phase anf the lower kHz QPO in the atoll phase, with frequency equal to the mean frequency of each data group. The solid grey line indicates the zero lag. Figure 4 . 4Average time-lag of the lower kHz QPO as a function of luminosity. The grey circles correspond the atoll LMXBs from Peirano & Méndez Figure 5 . 5Fractional rms amplitude as a function of energy for the atoll phase lower kHz QPO (blue circles) and the Z phase lower (red squares) and upper (purple triangles) kHz QPOs of XTE J1701−462. The solid lines indicate the best-fitting broken-line model, where the break energy, break , is marked by the vertical grey line. The arrows indicate upper limits. Table 3 . 3Properties of the kHz QPOs detected in the Z phase of XTE J1701−462. The fractional rms amplitude was calculated over the full energy range of the instrument, between 2 and 60 keV. The phase lag, between the reference-hard band and the subject-soft band inTable 1, was calculated as described in § 2.3, first for each Z observation individually and then in a joint fit of all Z observations to obtain a unique value for the Z phase of the source. Subscript letters ℓ and denote lower and upper kHz QPOs, respectively.Z phase ℓ ObsID central (Hz) rms (%) FWHM (Hz) Phase lag central (Hz) rms (%) FWHM (Hz) Phase lag 91442-01-07-09 641.5 ± 1.8 1.3 ± 0.2 12.2 ± 1.2 0.39 ± 0.34 − − − − 92405-01-01-02 617.8 ± 16.8 2.9 ± 0.7 102.4 ± 57.5 0.11 ± 0.54 − − − − 92405-01-01-04 − − − − 755.3 ± 8.6 3.6 ± 0.4 117.2 ± 31.7 0.37 ± 0.20 92405-01-02-03 620.2 ± 17.6 3.2 ± 0.6 172.1 ± 69.4 −0.20 ± 0.65 925.5 ± 5.0 2.1 ± 0.3 43.1 ± 16.2 0.29 ± 0.41 92405-01-02-05 598.6 ± 7.2 2.0 ± 0.3 67.5 ± 23.4 0.36 ± 0.22 850.2 ± 12.6 2.2 ± 0.4 110.1 ± 45.7 0.01 ± 0.30 92405-01-03-05 611.2 ± 14.9 3.1 ± 0.7 116.2 ± 54.6 −0.67 ± 0.75 916.7 ± 8.8 2.9 ± 0.5 75.8 ± 30.9 −0.12 ± 0.29 92405-01-40-04 651.8 ± 9.2 3.0 ± 0.5 91.5 ± 31.8 −0.44 ± 0.29 914.0 ± 8.8 3.2 ± 0.5 103.8 ± 35.5 −0.36 ± 0.52 92405-01-40-05 637.6 ± 8.1 3.5 ± 0.5 94.2 ± 28.2 0.24 ± 0.29 914.0 ± 6.6 3.3 ± 0.4 78.2 ± 22.8 0.52 ± 0.24 Joint fit 0.13 ± 0.14 0.24 ± 0.14 Atoll phase ℓ Frequency selection range (Hz) central (Hz) rms (%) FWHM (Hz) Phase lag 600 − 660 621.5 ± 0.6 8.3 ± 0.9 5.5 ± 1.7 −0.06 ± 0.25 660 − 700 671.6 ± 1.3 9.8 ± 1.1 13.3 ± 3.9 0.02 ± 0.15 700 − 750 725.3 ± 0.4 10.2 ± 0.5 9.2 ± 1.2 0.09 ± 0.10 750 − 800 775.6 ± 0.4 9.8 ± 0.8 5.5 ± 1.2 0.07 ± 0.20 800 − 830 815. Table 4 . 4The lags of the lower kHz QPO dur- ing the atoll phase of XTE J1701−462 show a marginal dependence on QPO frequency (consistent with the results of Barret (2013) and de Avellar et al. (2013)), however, considering the errors of the mea- sured lags, their overall trend remains constant with increasing QPO frequency. The lags of the lower and upper kHz QPOs during the Z phase observations show no clear dependence upon QPO frequency. The diamonds in the plot show the average lags we obtained from the joint fit of the cross-spectra of each phase (as described in § 2.3). The value of the frequency for these average lags corresponds to the mean frequency of all the data points of each kHz QPO, and is meant to be used only as a reference. V.Peirano et al. MNRAS 000, 1-11(2022) ACKNOWLEDGEMENTSThe authors wish to thank Federico García and Kevin Alabarta for useful discussions that helped develop the data analysis methods used in this paper. They also thank the referee for constructive comments that helped improve the manuscript. This research has made use of data and software provided by the High Energy Astrophysics Science Archive Research Center (HEASARC), which is a service of the Astrophysics Science Division at NASA/GSFC.DATA AVAILABILITYThe data underlying this article are publicly available at the website of the High Energy Astrophysics Science Archive Research Center (HEASARC, https://heasarc.gsfc.nasa.gov/). . M G B De Avellar, M Méndez, A Sanna, J E Horvath, 10.1093/mnras/stt1001Monthly Notices of the Royal Astronomical Society. 4333453de Avellar M. G. B., Méndez M., Sanna A., Horvath J. E., 2013, Monthly Notices of the Royal Astronomical Society, 433, 3453 2 We use here the luminosity as a proxy for the properties of the corona as done in Peirano & Méndez. 2 We use here the luminosity as a proxy for the properties of the corona as done in Peirano & Méndez (2021). . 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[ "Superconductivity under pressure: application of the functional derivative", "Superconductivity under pressure: application of the functional derivative" ]	[ "G I González-Pedreros \nDepartamento de Física\nGAM\nCinvestav. Av. IPN 250807360Ciudad de MéxicoMéxico\n", "R Baquero \nDepartamento de Física\nGAM\nCinvestav. Av. IPN 250807360Ciudad de MéxicoMéxico\n" ]	[ "Departamento de Física\nGAM\nCinvestav. Av. IPN 250807360Ciudad de MéxicoMéxico", "Departamento de Física\nGAM\nCinvestav. Av. IPN 250807360Ciudad de MéxicoMéxico" ]	[]	In this paper, we calculate the superconducting critical temperature as a function of pressure, T c (P ), using a method based on the functional derivative of the critical temperature with the Eliashberg function, δT c /δα 2 F (ω). The coulomb electron-electron repulsion parameter, µ * (P ), at each pressure is obtained in a consistent way by solving the linearized Migdal-Eliashberg equation.This method requires as the starting input only the knowledge of T c (P ) at the starting pressure. It applies to superconductors for which the Migdal-Eliashberg equations hold. We study Al, a typical BCS weak coupling superconductor with a low T c . Our results of T c (P ) as a function of pressure for Al show an excellent agreement with the calculations of Profeta et al. (Phys. Rev. Lett.96, 047003 (2006)) which agree well with experiment. . PACS numbers: 63.20.kd,74.62.Fj	null	[ "https://arxiv.org/pdf/1708.03301v1.pdf" ]	103,994,628	1708.03301	be15c6e8f5440dea3c4be4ea16a45c0f0d251f60	Superconductivity under pressure: application of the functional derivative (Dated: May 8, 2018) 10 Aug 2017 G I González-Pedreros Departamento de Física GAM Cinvestav. Av. IPN 250807360Ciudad de MéxicoMéxico R Baquero Departamento de Física GAM Cinvestav. Av. IPN 250807360Ciudad de MéxicoMéxico Superconductivity under pressure: application of the functional derivative (Dated: May 8, 2018) 10 Aug 2017BAQ/ 02 2017/ DFM .superconductivitycritical temperaturepressure 1 In this paper, we calculate the superconducting critical temperature as a function of pressure, T c (P ), using a method based on the functional derivative of the critical temperature with the Eliashberg function, δT c /δα 2 F (ω). The coulomb electron-electron repulsion parameter, µ * (P ), at each pressure is obtained in a consistent way by solving the linearized Migdal-Eliashberg equation.This method requires as the starting input only the knowledge of T c (P ) at the starting pressure. It applies to superconductors for which the Migdal-Eliashberg equations hold. We study Al, a typical BCS weak coupling superconductor with a low T c . Our results of T c (P ) as a function of pressure for Al show an excellent agreement with the calculations of Profeta et al. (Phys. Rev. Lett.96, 047003 (2006)) which agree well with experiment. . PACS numbers: 63.20.kd,74.62.Fj I. INTRODUCTION To determine the superconducting critical temperature, T c , as a function of pressure we use the density functional theory (DFT) and the density functional perturbation theory [1][2][3] (DFPT) to get the electron and the phonon band structures and the Eliashberg function α 2 F (ω) from first principles. We use the Quantum Espresso suite codes 4 for that purpose. This method applies to superconductors for which the Migdal-Eliashberg (ME) equations 5,6 are valid to describe their superconducting properties as the electron-phonon ones. There is a set of parameters that influence each other when the ME equations are used, namely, the critical temperature, T c , the electron-phonon interaction parameter, λ, the Coulomb electron-electron repulsion parameter, µ * , and the frequency at which the sum over the Matsubara frequencies is stopped, the so-called, cut-off frequency, ω c , which can actually be fixed numerically. We can take λ from specific heat and T c from resistivity experiments, for example. Then, µ * can be fitted to T c by solving the Linear Migdal-Eliashberg (LME) equation. In cases, where two of these parameters are unknown (usually T c and µ * ) a problem arises. To a certain extend, this is an unsolved problem. Oliveira et al. 7 presented a formulation of this problem that does not use the parameter µ * . There are several suggestions in the literature on how to estimate this parameter. From the solution of the LME equation, we can get the coulomb electron-electron repulsion parameter, µ * , as long as we know T c , assuming that the Eliashberg function is known and ω c is fixed. There are other ways to estimate µ * . Morel and Anderson 8 suggest the following analytic formula µ * = µ 1 + µln E el ω ph(1) where the dimensionless parameter µ = V N (E F ) is the product of the averaged screened Coulomb interaction, V, and the density of states at the Fermi energy, N (E F ); E el and ω ph are the electron and phonon energy scales, respectively. Further, Bennemann and Garland 9 , 19 and 0.249 20 which differ considerably from each other. Smith In this paper, we consider fcc Al. We start our calculation from the data at ambient pressure, say P i , where T c (P i ), the crystal structure of the system and the lattice parameters at the first pressure are known. We first optimize the lattice parameters using the Quantum Espresso code 4 . So we start with lattice parameters that minimize the energy as a function of the volume. We then obtain α 2 F (ω, P i ). µ * (P i ) is fitted to T c (P i ) solving the LME equation. We fix ω c =10 ω max , the maximum phonon frequency. We solve, at P i , the LME equation using the Mc Master programs 14,16,[21][22][23][24] . We obtain, at P i , µ * (P i ) and then the functional derivative δT c /δα 2 F (ω, P i ) 25 . We define a next pressure, say P i+1 and obtain the Eliashberg function at this new pressure. The T c (P i+1 ) is obtained from the value of the functional derivative at P i and the difference in the Eliashberg functions at the two pressures considered (see below for details). From the knowledge of T c (P i+1 ) we fit the value of µ * (P i+1 ) by solving the LME equation which we then use to obtain δT c /δα 2 F (ω, P i+1 ). This procedure can be repeated to get T c at other pressures. One has to be careful with the magnitude of the interval at which we calculate the next pressure since the information carried through the functional derivative could become meaningless for too large pressure intervals. The rest of the paper is organized as follows. In Section II, we present the theory that supports our method. The method is described in detail in Section III. In Section IV, we report some technical details used in the calculation. In the next section V, we present our results and compare them with other work, namely, with the known successful calculations of Profeta et al. 26 , and with experiment [27][28][29] . We present our conclusions in a final Section VI. II. THE THEORY As we mentioned above, we solve the LME equation to fit µ * (P ) to the calculated value of T c (P ). On the imaginary axis, the LME equation is ρ∆ n = πT m (λ mn − µ * ) − δ nm \|ω n \| πT ∆ m ,(2)ω n = ω n + πT m λ mn sgn(ω n ),(3)ω n = (2n − 1)πT,(4)∆ n =∆ n ρ + \|ω n \| ,(5)λ mn = 2 ∞ 0 dωωα 2 F (ω) ω 2 + (ω n − ω m ) 2 .(6) where T is the temperature,∆ n is the gap function, ω n is the Matsubara frequency, ρ is the pair breaking parameter and n = 0, ±1, ±2, .... In particular, λ nn ≡ λ is the electronphonon coupling constant. The numerical solution of the LME, Eq. (2) requires the summation over the Matsubara frequencies to be stopped at ω c as we mentioned before. The error caused by this restriction can be compensated 25 by replacing the true Coulomb repulsion parameter µ by the pseudorepulsion parameter µ * which we mentioned above and used in our calculations . Bergmann and Rainer 25 suggest a cut-off frequency ten times the maximum phonon frequency, ω max . Other authors consider that 3-7 could be enough 16,30 . The proper cut-off can be fixed numerically by studying the contribution of the last term in the summation. The Eliashberg function is defined as follows α 2 F (ω) = 1 N ( F ) mn qν δ(ω − ω qν ) k \|g qν,mn k+q,k \| 2 ×δ( k+q,m − F )δ( k,n − F ),(7) where g qν,mn k+q,k is the electron-phonon coupling matrix element, k+q,m and k,n are the energy of the quasi-particles in bands m and n with wave vectors k + q and k, respectively. ω qν is the phonon energy with momentum q and branch ν. N ( F ) is the electronic density of states at the Fermi energy, F . From the first order derivative of the self-consistent Kohn-Sham 31,32 (KS) potential, V KS , with respect to the atomic displacements u sR for the s th atom in the position R, the electronphonon matrix element can be obtained as g qν,mn k+q,k = h 2ω qν 1/2 ψ k+q,m \|∆V qν KS \|ψ k,n ,(8) where ∆V qν KS is the self-consistent first variation of the KS potential and ψ k,n is the n th valence KS orbital of wave vector k. The functional derivative of T c with respect to α 2 F (ω), δT c /δα 2 F (ω), is central to this work. With the algorithm of Bergmann and Rainer 25 and Leavens 21 the functional derivative can be calculated. Several authors have worked this calculation from the solution of the LME equation 14,16,22,23 , as well as Baquero et all. 24 and Yamsun et all. 30 as we mentioned before. δT c δα 2 F (ω) = − δρ/δα 2 F (ω) (∂ρ/∂T ) Tc(9) Ounce the functional derivative, δT c /δα 2 F (ω), is known the change in T c , ∆T c , caused by a change in α 2 F (ω) can be obtained directly as we show next. The transition temperature of a superconductor depends on the effective interaction with the existing phonons in the system. To have a high frequency phonon is not enough for a system to have a high-Tc as it can be seen in Al where a 41 meV peak in the phonon spectrum is notorious. The functional derivative δT c /δα 2 F (ω) shows how the different phonon frequencies participate in defining the T c . As a function of the dimensionless variablehω/K B T c it presents a maximum at about 7-8 which turns out to be universal for the conventional superconductors 16,22 as Al where the electron-phonon interaction is known to be the mechanism. This defines the so called optimum frequency, ω opt . This is actually the most important phonon frequency as far as the magnitude of T c is concerned. At any frequency, it shows how sensitive T c is to a change in α 2 F (ω) at this particular frequency. By applying pressure, we induce changes in the Eliashberg function. When α 2 F (ω) is changed by a certain amount the difference in the Eliashberg function, ∆α 2 F (ω), together with the functional derivative allow to calculate the change in T c , ∆T c , which is given by the formula 25,33 To get T c,P i+1 at the next pressure we start from Eq.10 and use the next Eq.11 ∆T c,P i+1 ,P i = ∞ 0 δT c δα 2 F (ω) (α 2 F (ω, P i+1 ) − α 2 F (ω, P i ) dω.(10)T c,P i+1 = T c,P i + ∆T c,P i+1 ,P i(11) which can in turn be used to fit µ * (P i+1 ) using the LME equation. This procedure can be repeated at will. Our results are presented in several Tables below. III. TECHNICAL DETAILS The electron and phonon (PHDOS) densities of states for Al have been calculated using the DFT and the DFPT with plane waves (PW) pseudo-potentials 1-3,37 . To calculate the density of states (DOS) a kinetic energy cut-off of 50 Ry was used. Our calculations were performed using the generalized gradient approximation (GGA) and the norm conserving pseudo-potential together with the plane wave self-consistent field (PWSCF) 38 . For the electronic and vibrational calculations we used a 32x32x32 and 16x16x16 Monkhorst-Park 39 (MP) k mesh, respectively. The PHDOS was obtained from individual phonons calculated on a 8x8x8 MP q mesh using the tetrahedron method 40 . We used the Quantum Espresso code 4 for all these calculations. IV. RESULTS AND DISCUSSION We now apply the method to the weak coupling superconductor Al. We have taken T c =1.8K 28 at ambient pressure which is our starting pressure. The Eliashberg function was obtained using the Espresso code. λ was calculated directly from it, µ * was fitted to T c using the LME equation. The functional derivative at this starting pressure was calculated using the Mc Master programs 14,16,21-24 . From these starting data, we can obtain the variation of T c with pressure by applying the method just described. Other authors have worked in this problem. Namely, Dacorogna et all. 41 have calculated the T c as a function of pressure. They calculated self-consistently the phonon frequencies and the electron-phonon coupling. µ * was fitted to obtain T c at ambient pressure. Then for the variation with pressure they use the empirical relation of Bennemann and Garlandand 9 . We got µ * fitting it to T c through the LME equation at each pressure 26 and in good agreement with experiment. We present our results in Fig. 1 and in the next Table I and Profeta et all. 26 . d) We took our input data from 28 . In Table I, we consider a variation of pressure, P , from 0-6 GPa. We first compare our results for T c as a function of pressure, P , with the ones of Profeta et al. 26 . The agreement is excellent. In the next column we present the result from experiment 27,28 . The trent is reproduced quite well. Next, we show the variation of the electron-phonon interaction parameter, λ. It always diminishes with pressure. The decrement in the value of it is not exactly equal for all intervals of pressure since it varies from 0.0091 between P = 0 and P = 0.5 GPa to 0.0051 between P = 5 to P = 5.5 GPa. The electron-electron repulsion parameter, µ * , behaves somehow differently according to our calculations, since it presents a minimum. At 0 GPa, its value is 0.14154 and decreases steadily to a minimum value of 0.13856 at 4 GPa. Increasing the pressure, µ * increases and reaches a value of 0.15205 at 6 GPa. The minimum of the decrement in λ arises between P = 5 and P = 5.5 GPa and so it does not correlate with the minimum in µ * . The lattice parameter, a, diminishes steadily with pressure. Upon a 0.5 GPa enhancement in pressure it changes with a difference around 0.0148 Bohr. This decrement in the lattice constant is higher at low pressure and smaller at high pressure. The minimum occurs at P = 6 GPa. So, this behavior does not seem to correlate either with the behavior of the electron-electron repulsion parameter µ * . Further, if we look at the contribution of each phonon mode (two transverse and one longitudinal) to the behavior of T c under pressure by taking only the corresponding energy interval of α 2 F (ω) into account and apply to this part only our method, we get the result that they all contribute lowering the T c . This behavior is not universal. Some preliminary results for Nb give evidence of a different behaviour. V. CONCLUSIONS We presented in this paper an application of the functional derivative of the critical temperature with the Eliashberg function, δT c /δα 2 F (ω), to calculate T c as a function of pressure. We applied the method to superconducting Al. We get an excelent agreement with the successful calculations of Profeta et al. 26 which are in agreement with experiment. This work can be extended to calculate the thermodynamics under pressure (the thermodynamic critical field, H(0), the jump in the specific heat and the gap, for example). This is the subject of our next work. VI. ACKNOWLEGMENTS This work was performed using the facilities of the super-computing center (Xiuhcoatl) at CINVESTAV-México. González-Pedreros acknowledges the support of Conacyt-México through a PhD scholarship. r There are several papers in the literature that deal with the functional derivative for different purposes. For example, Bergmann and Rainer 25 discuss how T c is influenced by different parts of α 2 F (ω) and apply their findings to several crystalline and amorphous superconductors. Mitrovic 34 considers it as a diagnostic tool to analyze the behavior of T c as a function of an external variable. Allen and Dynes 35 study in detail the case of Pb, Baquero et all. 36 took several Eliashberg functions from experiment to study the changes in T c when N b 3 Ge is taken off stoichiometry. Yansun et all. 30 investigated the superconducting properties of Li as a function of pressure at the interval of pressure where it undergoes three phase transitions. Mitrovic 34 developed a general formalism to calculate the functional derivative of T c with respect to α 2 F (ω) for a superconductor with several bands with isotropic intra-band and inter-band interactions. FIG. 1 . 1( color on line) T c [K] under pressure for Al: FDM is the present work, SCDFT is the linear interpolation of the data reported by Profeta et al. 26 . Also the curves identified as Gusber and Webb 28 and Levy and Olsen 27 are interpolations to their experimental data. instead. So our values are consistent with the Mc Millan-Eliashberg linear equation and no further approximation is needed. In a recent work, Profeta et all. 26 studied the behaviour of T c for Al as a function of pressure and obtained a good agreement with experiment. The experimental results we compare with are the ones of Gubser and Webb 28 and Sundqvist and Rapp 29 . Our results are in excellent agreement with the ones of Profeta et all. 10 and Neve et all.11 give semi-empirical formulas to estimate the behavior of the Coulomb pseudo-potential as a function of pressure, Liu et all.12 and Freericks et all.13 calculate µ * scaled to the maximum phonon frequency, meaning to replace ω ph in Eq.(1)by ω max , the maximum phonon frequency. Daams and Carbotte 14 fit µ * solving the LME equation using the experimental value of T c . In a more recent work Bauer et all.15 calculated corrections to µ * based on the Hubbard-Holstein model. There is no consensus concerning the proper way to estimate or to calculate µ * under pressure or even at ambient pressure. Forexample, for Nb at ambient pressure, a set of different values for µ * are reported : 0.117 16 , 0.13 11 , 0.14 17 , 0.183 18 , 0.21 Table I - IProperties of superconducting Al under pressureP[GPa] T F DM c [K] a T SCDF T c [K] b T Exp c [K] c λ µ * a[Bohr] 0.0 1.18 d r 1.18 1.18 0.4259 0.14154 7.6460 0.5 1.078 1.06 0.90 0.4168 0.14087 7.6297 1.0 0.969 0.96 0.79 0.4084 0.14024 7.6141 1.5 0.872 0.87 0.70 0.4009 0.13967 7.5989 2.0 0.791 0.78 0.62 0.3942 0.13950 7.5841 2.5 0.706 0.71 0.56 0.3870 0.13914 7.5695 3.0 0.634 0.63 0.46 0.3805 0.13888 7.5553 3.5 0.567 0.57 0.37 0.3743 0.13872 7.5415 4.0 0.505 0.50 0.29 0.3680 0.13856 7.5280 4.5 0.448 0.42 0.22 0.3623 0.13876 7.5148 5.0 0.392 0.40 0.16 0.3569 0.13935 7.5019 5.5 0.324 0.38 0.10 0.3518 0.14190 7.4892 6.0 0.204 0.35 0.07 0.3460 0.15205 7.4766 TABLE I. a) our results b) Profeta et all. 26 . c) Lineal interpolation of the experimental data read from Gusber and Webb Ref. 28 , and Levy and Olsen 27 . 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[ "With Great Dispersion Comes Greater Resilience: Efficient Poisoning Attacks and Defenses for Linear Regression Models", "With Great Dispersion Comes Greater Resilience: Efficient Poisoning Attacks and Defenses for Linear Regression Models" ]	[ "Jialin Wen \nEast China Normal University\nChina\n", "Benjamin Zi ", "Hao Zhao \nThe University of New South Wales and CSIRO-Data61\nAustralia\n", "Minhui Xue \nThe University of Adelaide\nAustralia\n", "Alina Oprea \nNortheastern University\nUSA\n", "Haifeng Qian \nEast China Normal University\nChina\n" ]	[ "East China Normal University\nChina", "The University of New South Wales and CSIRO-Data61\nAustralia", "The University of Adelaide\nAustralia", "Northeastern University\nUSA", "East China Normal University\nChina" ]	[]	With the rise of third parties in the machine learning pipeline, the service provider in "Machine Learning as a Service" (MLaaS), or external data contributors in online learning, or the retraining of existing models, the need to ensure the security of the resulting machine learning models has become an increasingly important topic. The security community has demonstrated that without transparency of the data and the resulting model, there exist many potential security risks, with new risks constantly being discovered.In this paper, we focus on one of these security riskspoisoning attacks. Specifically, we analyze how attackers may interfere with the results of regression learning by poisoning the training datasets. To this end, we analyze and develop a new poisoning attack algorithm. Our attack, termed Nopt, in contrast with previous poisoning attack algorithms, can produce larger errors with the same proportion of poisoning data-points. Furthermore, we also significantly improve the state-of-the-art defense algorithm, termed TRIM, proposed by Jagielsk et al. (IEEE S&P 2018), by incorporating the concept of probability estimation of clean datapoints into the algorithm. Our new defense algorithm, termed Proda, demonstrates an increased effectiveness in reducing errors arising from the poisoning dataset through optimizing ensemble models. We highlight that the time complexity of TRIM had not been estimated; however, we deduce from their work that TRIM can take exponential time complexity in the worst-case scenario, in excess of Proda's logarithmic time. The performance of both our proposed attack and defense algorithms is extensively evaluated on four real-world datasets of housing prices, loans, health care, and bike sharing services. We hope that our work will inspire future research to develop more robust learning algorithms immune to poisoning attacks.	10.1109/tifs.2021.3087332	[ "https://arxiv.org/pdf/2006.11928v5.pdf" ]	219,969,323	2006.11928	a8dc0bc206dcf71c0dec7e3afac8ab5c1e1c53f5	With Great Dispersion Comes Greater Resilience: Efficient Poisoning Attacks and Defenses for Linear Regression Models Jialin Wen East China Normal University China Benjamin Zi Hao Zhao The University of New South Wales and CSIRO-Data61 Australia Minhui Xue The University of Adelaide Australia Alina Oprea Northeastern University USA Haifeng Qian East China Normal University China With Great Dispersion Comes Greater Resilience: Efficient Poisoning Attacks and Defenses for Linear Regression Models 1Index Terms-Data Poisoning Attacks and DefensesLinear Regression ModelsComplexity With the rise of third parties in the machine learning pipeline, the service provider in "Machine Learning as a Service" (MLaaS), or external data contributors in online learning, or the retraining of existing models, the need to ensure the security of the resulting machine learning models has become an increasingly important topic. The security community has demonstrated that without transparency of the data and the resulting model, there exist many potential security risks, with new risks constantly being discovered.In this paper, we focus on one of these security riskspoisoning attacks. Specifically, we analyze how attackers may interfere with the results of regression learning by poisoning the training datasets. To this end, we analyze and develop a new poisoning attack algorithm. Our attack, termed Nopt, in contrast with previous poisoning attack algorithms, can produce larger errors with the same proportion of poisoning data-points. Furthermore, we also significantly improve the state-of-the-art defense algorithm, termed TRIM, proposed by Jagielsk et al. (IEEE S&P 2018), by incorporating the concept of probability estimation of clean datapoints into the algorithm. Our new defense algorithm, termed Proda, demonstrates an increased effectiveness in reducing errors arising from the poisoning dataset through optimizing ensemble models. We highlight that the time complexity of TRIM had not been estimated; however, we deduce from their work that TRIM can take exponential time complexity in the worst-case scenario, in excess of Proda's logarithmic time. The performance of both our proposed attack and defense algorithms is extensively evaluated on four real-world datasets of housing prices, loans, health care, and bike sharing services. We hope that our work will inspire future research to develop more robust learning algorithms immune to poisoning attacks. I. INTRODUCTION With the widespread adoption of Machine Learning (ML) algorithms, it has been elevated out of the exclusive use of high-tech companies [1]. Services such as "Machine Learning as a Service" (MLaaS) [2] can assist companies without domain expertise in ML to solve business problems with ML. However, in the MLaaS setting, there exist poisoning attacks, in which malicious MLaaS providers can either manipulate the integrity of the training data supplied by the company or compromise the integrity of the training process. Alternatively, in a collaborative setting, whereby a model holder solicits data contributions from multiple parties for online training or retraining of an existing model, a malicious participant may provide poisoned training samples in their submission, thereby infecting the resulting model for all parties. In such poisoning attacks, the attacker's objective may be to indiscriminately alter prediction results, create a denial of service, or cause specific targeted mis-predictions during test time. The attacker seeks to create these negative effects while preserving correct predictions on the remaining test samples to bypass detection. An inconspicuous attack may produce dire consequences, thus necessitating to study poisoning attacks on ML. A conceptual example of poisoning attacks is illustrated in Figure 1. Many poisoning attacks have been proposed and demonstrated against different ML architectures. Specially, grey-box attacks, in which the attacker has no knowledge of the training set but has an alternative dataset with the same distribution as the training set, have been proposed against Support Vector Machines (SVMs) [3], Deep Neural Networks (DNNs) [4], Logistic Regression (LR) [5], Graph-based classification [6], and Recommender systems [7]. These works have shown that poisoning attacks are effective in interfering with the accuracy of producing classifications, or recommendations by either indiscriminately altering prediction results, or causing specific mis-predictions at test time. However, the aforementioned attacks target models producing a label prediction; in this work, we shall focus on models that perform regression, the prediction of a numerical value. Thus, with a different functional objective of regression, an attacker's objective for poisoning a regression model may also differ. For example, an attacker may want to increase or decrease the predicted value, or may want to maximize the dispersion of the training set. Ma et al. [8] first propose a white-box poisoning attack against linear regression, aimed at manipulating the trained model by adversarially modifying the training set. Additionally, Jagielski et al. [1] propose several white-box and grey-box 1 poisoning attacks against linear regression, which aims to increase the loss function on the original training set. Our attack contributions. We scrutinize poisoning attacks on linear regression by improving and redefining the attacker's objective in existing attack models and establish a new attack optimization problem for linear regression. Our new attack, termed Nopt, is observed to be more efficient than the state of the art [1] (IEEE S&P 2018), termed Opt, in maximizing the dispersion of the training set, for the same proportion of poisoned data-points. Intuitively, the key difference between Opt [1] and Nopt is that the optimization evaluated for each subsequent poisoning point is performed on a dataset that includes all previous poisoning points, thereby creating a new poisoning point that maximizes the loss of the collective training dataset of original clean and poisoning points. To defend against poisoning attacks on regression learning, defense mechanisms have been proposed [9,10,11,12,13]. One such approach treats poisoning attack data-points as outliers, which can be counteracted with data sanitization techniques [14,15,16] (i.e., input validation and removal). Another approach is through robust learning [1,10,17,18], as learning algorithms based on robust statistics are intrinsically less sensitive to outlying training samples; the robustness can be realized through bounded losses or specific kernel functions. Ma et al. [8] leverage differential privacy as a defensive measure against poisoning attacks on linear regression. Jagielski et al. [1] propose a defense algorithm against regression learning poisoning attacks, named TRIM. TRIM offers high robustness and resilience against a large number of poisoning attacks. We highlight that the time complexity of TRIM had not been estimated; however, we deduce from their work that TRIM can take exponential time complexity in the worst-case scenario, in excess of our defense (termed Proda)'s logarithmic time. Our defense contributions. We define a new defense algorithm against poisoning attacks, termed Proda. We are the first to introduce the concept of probability estimation of unpolluted data-points into the defense algorithm. Proda demonstrates an increased effectiveness in reducing errors arising from the poisoning dataset. Additionally, the time complexity of Proda is also lower than the state-of-the-art defense of TRIM [1] (IEEE S&P 2018). The key insight of achieving better efficacy is that we have prior knowledge that from the sets of points we randomly sample, there must be a group of points that all belong to the unpolluted dataset. Therefore, when comparing the minimum mean squared error (MSE) values of some groups of data-points, even if there are poisoning points in the group of the smallest MSE, they must conform to the original distribution and rules of the training dataset, and will have little impact on altering the regression model. In this paper, we systematically study the poisoning attack and its defense for linear regression models. We define a new poisoning attack on linear regression to maximize the dispersion of the training set. Additionally, we develop a new probabilistic defense algorithm against poisoning attacks, named Proda. We extensively evaluate poisoning attacks and defenses across different four regression models (Ordinary Least Squares, Ridge Regression, LASSO, Elastic-Net) trained on multiple datasets originating from different fields, including house pricing, loans, pharmaceuticals, and bike sharing services. In summary, the overall contributions of this paper are as follows. • We develop a new grey-box poisoning attack against regression models, termed Nopt. Nopt outperforms the state-of-the-art attack, termed Opt, proposed by Jagielsk et al. [1] (IEEE S&P 2018). • We prove that the state-of-the-art defense, termed TRIM [1], is estimated to have exponential time complexity in the worst-case scenario, in excess of our Proda's logarithmic time. • We further overhaul TRIM and propose to date the most effective defense against poisoning attacks through optimizing ensemble models, termed Proda. The performance of both our proposed attack and defense algorithms is extensively evaluated on four real-world datasets of housing prices, loans, health care, and bike sharing services. To the best of our knowledge, we are among the first to systematically design, develop, and evaluate the poisoning attack and defense for linear regression models. We hope that our work will inspire future research to develop more robust learning algorithms immune to poisoning attacks. II. PRELIMINARIES In this section, we first introduce linear regression and then take a deep dive into defining the threat model of this paper. A. Linear Regression Linear Regression [1] is a supervised machine learning algorithm, frequently used to analyze complex data relationships. Linear regression uses inherent statistical features of the training dataset to quantitatively determine mutually dependent relationships between two or more variables. By learning these relationships, the resulting linear regression model can produce a numerical output on an unseen input. Specifically, in linear regression, the model after training is a linear function f (x, θ) = w T + b, which seeks to regress the value of y for a given input x. The real parameter vector θ = (w, b) of dimension d+1 consists of the feature weights w and the bias b, of dimensionality d and 1, respectively. However, the true value is noted as y = f (x, θ) + e, containing e, the error between the true value and the predicted value. Assuming that e is Independent and Identically Distributed (IID), the mean is 0, the variance is fixed, and that the noise term e satisfies the Gaussian distribution: g(y i \|x i ; θ) = 1 √ 2πσ exp(− (y i − θ T x i ) 2 2σ 2 ).(1) Then, the maximum likelihood function of the model parameters can be obtained as the product of all training sets: L(θ) = m i=1 g(y i \|x i ); θ)) = m i=1 1 √ 2πσ exp(− (y i − θ T x i ) 2 2σ 2 ).(2) The maximum value of L(θ) is maintained when the log function is applied, log L(θ). log L(θ) = log m i=1 1 √ 2πσ exp(− (y i − θ T x i ) 2 2σ 2 ) = m log 1 √ 2πσ − 1 σ 2 · 1 2 m i=1 (y i − θ T x i ) 2 .(3) In order to obtain the maximum likelihood, the latter term of Equation (3) is to be minimized. We note that the maximum likelihood of linear regression can also be converted into the minimum value of the least squares, the most common mathematical form of the loss function: L(D tr , θ) = 1 2 m i=1 (f (x i , θ) − y i ) 2 + λΩ(w),(4) where D tr is the training data, Ω(w) is a regularization term penalizing large weight values, and λ is the regularization parameter used to prevent overfitting. 2 The primary difference between popular linear regression methods is in the choice of the regularization term. In this paper, we study the following four regression models: Ordinary Least Squares (OLS), with no regularization, Ridge regression, which uses l 2 −norm regularization, LASSO, which uses l 1 −norm regularization, and Elastic-net regression, which uses a combination of l 1 −norm and l 2 −norm regularization. We elaborate on the regularization term in context of the more common minimum least squares form of the loss function. MSE = 1 m m i=1 (f (x i , θ) − y i ) 2 .(5) Our proposed attack and defense hinge on its relative effectiveness in comparison with existing methods. We shall inspect effectiveness in two aspects. Firstly, the degree of poisoning by comparing the loss function of the poisoned model with the non-poisoned model when trained on the same dataset, as a successful poisoning attack, will have increased the dispersion of the points, and thus the resulting learned regression line. The specific metric used to quantify the effect of the poisoning attack will be the Mean Squared Error (MSE) (see Equation (5)) of the true value from the predicted value. Secondly, specific to our defense, the time complexity of the deploying defense shall be experimentally measured in seconds. B. Threat Model The core objective of a poisoning attack is to corrupt the learning model generated from the training phase, such that predictions on unseen data will greatly differ in the testing phase. However, depending on whether the goal is to produce predictions that greatly differ on specific subsets of input data, while preserving predictions on the remaining subsets, or if predictions are to be altered indiscriminately, the poisoning attack is categorized as either an Integrity attack, or an Availability attack. A similar deconstruction of attacks is found in backdoor poisoning attacks [19,20,21,22,23]. In this work, we consider a poisoning availability attack. 1) Attack Assumptions: There is a sliding scale of knowledge that is assumed to be available to the attack, from whitebox to grey-box and black-box attacks. Under the assumptions of a white-box attack, the attacker has access to the training data D tr , the learning algorithm L, and the trained parameters θ. Black-box attacks have no knowledge about the internal construction of the model, with only input and output access to the model. However, situated between white-box and blackbox, in our grey-box setting, 3 the attacker has no knowledge of the training set D tr but has an alternative dataset D tr that has the same distribution as the pristine training set D tr . Internal to the model, the learning algorithm L is known; however, the trained parameters θ are not. We do note that an attacker, can approximate θ by optimizing L on D tr [13,24]. It is known that black-box attacks are more practical in real-world adversaries with less knowledge required about the model. However, in this work we adopt the grey-box setting. Under the grey-box setting, we assume the adversary has no information about the structure L or parameters θ of linear regression, and does not have access to any large training dataset. 2) Poisoning Rates: As visualized in Figure 1, a poisoning attack is performed by injecting poisoned data into the training set before the regression model is (re)trained. The influence of an attacker on the resulting model is limited by an upper bound on the proportion α = n p /N of poisoned data (D p of size n p ) to the original clean data (D o of size n o ) in the training dataset (D N of size N = n p + n o ) [1]. An attacker has complete control of the poisoning samples, as such input feature values and responses can be arbitrarily set within known bounds (These feature bounds may either be derived from D tr or D tr , or assumed if the data is normalized.). Consistent with restrictions imposed by our settings of MLaaS, online learning, and retraining, prior works rarely consider poisoning rates larger than 20%, as the attacker is limited to being able to control only a small fraction of the training data [1]. Thus, in this paper, we shall investigate poisoning rates up to a maximum of α = 0.2. This maximum is motivated by prior works [1], as poisoning rates higher than 20% have rarely been considered, since the attacker is assumed to be capable of controlling only a small fraction of the training data. This is motivated by application scenarios, such as crowdsourcing and network traffic analysis, in which attackers can only reasonably control a small fraction of participants and network packets, respectively. Moreover, learning a sufficiently-accurate regression function in the presence of higher poisoning rates would be an ill-posed task, as the poisoning attack would be trivial [1]. 3) Defense Assumptions: We shall also be investigating defenses. To the defender, the model is a white box (as they are the model holder), the only additional item of information a defender may not know is the poisoning rate of an attacker. It is possible for the defender to derive the poisoning rate from the size of the update data provided to it. In the event of an inability to derive the poisoning rate, it has been argued that a poisoning rate of α = 0.2 is representative of an upper limit of poisoning attacks, and can be assumed as a worst-case scenario. III. POISONING ATTACKS BASED ON OPTIMIZATION Previous works have discussed a poisoning attack strategy, which is applicable not only to linear regression, but also to classification algorithms. Those poisoning attacks aim to maximize the test error. However, as we have discussed, the attack objective on linear regression is different from the attack objectives for classification algorithms, with the latter seeking to only produce a specific wrong answer. Therefore, we define a new poisoning attack, Nopt poisoning attack. By establishing a new poisoning optimization algorithm for linear regression, this attack will force the model to receive a more dispersed training dataset. With a more dispersed training set this will result in larger losses and/or poor convergence on the regression task, which may erode confidence in the model holder, or simply result in worse prediction confidence in practice. A. Definition of Nopt Poisoning Attack In this section, we define a new form of linear regression attack, the Nopt attack. Previously in Section II-A, we observed that the loss function of linear regression is the sum of squares from each point to a regression model. When more points are added, the loss should also increase. However, when the added points are distributed in the similar manner as the pristine data, we obtain (without considering regularization): L(D N ) L(D o ) = N n o .(6) That is, the ratio of the loss function L(D N ) of the new training set D N to the loss function L(D o ) of the original training set D o should be equal to the ratio of the size of the new training set N to the size of the original training set n o when the poisoning data is distributed in a similar way as the original training set. Additionally, as the new training set D N is made up of the original training set D o and poisoning data set D p , we formulate our attack as follows: E = L(D N ) L(D o ) − N n o = L(D o ∪ D p ) L(D o ) − (n o ∪ n p ) n o .(7) As we can see from Equations (6) and (7), when the added poisoning points are distributed in the same way as the original data, E = 0. The further a data point is added away from the original regression line, the higher the value of E. Therefore, the value of E is directly related to the rate of poisoning. B. Application of Nopt Poisoning Attack Previously, Jagielski et al. [1] established a bilevel optimization problem to find the set of poisoning points that maximize the loss function of the original data-points. In this section, we compose our new poisoning attack by optimizing the objective function: arg max Dp E(D tr ∪ D p , θ (p) ), s.t. θ (p) ∈ arg min θ L(D tr ∪ D p , θ).(8) The Nopt poisoning attack searches for poisoning datapoints D p by maximizing E, optimizing the loss function with respect to the poisoning training dataset. Intuitively, the key difference between the work [1] and Nopt is that the optimization evaluated for each subsequent poisoning point is performed on a dataset D tr ∪ D p that includes all previous poisoning points, thereby creating a new poisoning point that maximizes the loss of the collective training dataset of original clean and poisoning points. Algorithm 1 outlines the Nopt poisoning attack. As our loss function takes the same form as that of [1], the process to find the optimal points, we adopt the same gradient descent approach. In summary, vector x c is updated through a line search along the direction of the gradient from the outer objective E (evaluated at the current iteration). The algorithm finishes when the outer objective E yields no further changes. Figure 3, illustrates the iterative process of Nopt to find out the poisoning points. Figure 2 provides a visual representation of a contrived example in which our attack is applied with different poisoning rates. If the abscissa of the poisoning point must be located in the feasible domain, then we can still obtain an optimal solution within the feasible domain, and E will converge. Algorithm 1: Nopt poisoning attack algorithm. Input: D = D tr (white-box) or D = D tr (grey-box), L, E, the initial poisoning attack samples D This application scenario is realistic as the attacker's goal was to create poisoning data-points to interfere with the original data-points without making the poisoning points appear abnormal, and thus compromises the secrecy of the attack. Therefore the attacker should determine the feasible domain before determining the location of the poisoning point through optimization. Through Equation (7), we obtain the set of poisoning points D p with a specified poisoning degree E . (0) p = (x c , y c ) p c=1 , a small positive constant ε. 1: i ← 0 (iteration counter); 2: θ (i) ← arg min θ L(D ∪ D (i) p ); 3: repeat 4: e (i) ← E(θ (i) ); 5: θ (i+1) ← θ (i) ; 6: for c = 1, . . . , p do 7: x (i+1) c ←linesearch (x (i) c , ∇ xc E(D ∪ D (i+1) p , θ (i+1) )); 8: θ (i+1) ← arg min θ L(D ∪ D (i+1) p ); 9: e (i+1) ← E(θ (i+1) ); 10: end for 11: i ← i + 1; 12: until e (i) − e (i+1) < ε Output: The final poisoning attack sample D p ← D (i) p . Training set Dtr Dp(1) Training set Dtr∪Dp (1) Dp(2) · · ·Dp(p-1) Training set Dtr∪Dp(p-1) Dp (p) Training set Dtr∪Dp Nopt Nopt Nopt C. Gradient Computation The loss function we have defined in Equation (8) takes the same form as the loss function found in the work [1]. Consequently, the steps of the derivation and computation of the gradients are summarized below. Algorithm 1 reduces to a gradient-ascent task with a line search. To compute the gradient ( xc E(θ (p) )), whilst capturing the relationship between θ and the poisoning point x c , the chain rule can be used: xc E = xc θ(x c ) T · θ E,(9) where the first term captures the dependency of learned θ and the point x c , and the second term is the derivative of the outer objective with respect to the regression parameters θ. To solve xc θ(x c ) in the bilevel optimization problem, the inner learning problem is replaced with its Karush-Kuhn-Tucker (KKT) equilibrium condition, namely θ L(D tr ∪ D p , θ) = 0, while searching for x c . This replacement is necessary as approximations will be required to solve the inner problem, particularly when the inner problem is not convex (when the inner problem is convex, it may be solved via its closed form expression). Imposing the derivative with respect to x c satisfies this condition, xc θ L(D tr ∪ D p , θ) = 0. It is seen that L depends explicitly on x c and implicitly through θ. One final application of the chain rule produces the linear system: xc θ L + xc θ T · 2 θ L = 0.(10) For our specific form of L given in Equation (4), the derivative follows: ∂ω T ∂xc ∂b ∂xc Σ + λg µ µ T 1 = − 1 n M ω ,(11) where Σ = 1 n Σ i x i x T i , µ = 1 n Σ i x i , and M = ωx T c + (f (x c ) − y c )Id. To jointly optimize the feature values x c associated with their responses y c , we need to consider the optimization of z c = (x c , y c ). To do this, we replace zc by xc through expanding xc θ by incorporating derivatives with respect to y c : zc θ = ∂ω ∂xc ∂ω ∂yc ∂b ∂xc ∂b ∂yc ,(12) and, accordingly, we update Equation (11) as: zc θ T = − 1 n M ω −x T c −1 Σ + λg µ µ T 1 −1 .(13) Therefore, when Algorithm 1 is used to implement this attack, both x c and y c are to be updated along the gradient xc E (cf. Algorithm 1, line 7). With this, we have to use tools to perform the optimization for our Nopt poisoning attack. Following the proposal of the defense algorithm shown in Section IV, in Section V we will evaluate our attack in comparison to previous poisoning attacks. We shall demonstrate that our subtle change in Equation (8) produces larger errors compared to the previous poisoning attack on linear regression models, for the same given poisoning rates. IV. DEFENSE AGAINST POISONING ATTACKS In this section, we propose a new probabilistic defense algorithm, named Proda, which is designed to deal with regression learning poisoning attacks. We shall analyze its time complexity and efficiency. The objective of the Proda algorithm is similar to TRIM [1], which is to find the original training set through a subset solving algorithm, instead of trying to identify the poisoning set of the algorithm. If we randomly select γ points, the probability of all γ points belonging to the unpolluted training set is: P 1 = (1 − α) γ ,(14) where α is the poisoning rate of the training set. If we randomly select β groups of γ points, the probability that no group of points belongs to the unpolluted training set is: P = (1 − P 1 ) β .(15) When P ≤ ε, a small positive constant, there must be a group of points all belonging to the unpolluted training set. We note that ε = 10 −5 shall be used later in our experiment after (16), there must be a group of γ points that all belong to the unpolluted training set. (1 − (1 − α) γ ) β ≤ ε,(16) where α represents the poisoning rate. So the value of β is: β = log 1−(1−α) γ ε.(17) The steps of Proda algorithm are illustrated in Figure 4, and formally detailed in Algorithm 2. However, intuitively, the Proda algorithm proceeds as follows: 1) Given α and γ, calculate the value of β. 2) Choose β groups of γ points at random. (This step can ensure that there must be a group of γ points that all belong to the unpolluted training set). 3) Each group of γ points is subjected to linear regression, resulting in β lines. 4) For each line, take the n points closest to this line. 5) Each group of n points is subjected to linear regression, resulting in β lines, and we obtain β groups of MSEs. 6) Find n points corresponding to the smallest MSE. As Proda requires the pre-selection of γ, in Figure 5, we graphically display the effect of different selected γ on the resulting regression line. A. Efficiency According to Equation (7), we know that from the sets of points we randomly sample, there must be a group of points Algorithm 2: Proda algorithm. Input: Training data D = D tr ∪ D p of \|D\| = N ; number of attack points p = α · n; γ; ε 1: β = ε/(1 − a γ ) 2: i ←1 (iteration counter) 3: for i ≤ β do 4: J (i) ← a random subset of size γ ∈ {1, . . . , a}; 5: L (i) ← arg min θ L(J (i) , θ); 6: list (i) ← distance of N points to L (i) ; 7: Q (i) ← sorted(list (i) )[: n]; 8: S (i) ← arg min θ L(Q (i) , θ); 9: M (i) ← the MSE between S (i) and Q (i) ; 10: i ← i + 1; 11: end for 12: M (j) ←min(M (i) ); Output: The final optimizing sets Q (j) . that all belong to the unpolluted dataset. Therefore, when comparing the MSE minimum values of β groups, even if there are poisoning points in the group of the smallest MSE, they must conform to the original distribution and rules of the training dataset, and will have little impact on altering the regression model. Therefore, we argue that the Proda algorithm has good efficacy in mitigating the poisoning attack. An empirical evaluation on the efficiency of different defense algorithms will be demonstrated in Section V. The goal is to select γ points that all belong to the original training set and that these points sufficiently reproduce the linear relationship of the original training set. Our algorithm can guarantee that the γ selected points are likely to all belong to the original training set, but whether these points can represent the linear regression trend of the original training set depends on the value of γ. It is known that for one-dimensional inputs, two points are required to define a linear relationship, between the input (R) and the one dimensional output (R), and for two dimensional inputs, three points, to define the linear relationship between the input (R 2 ) and output (R). Therefore, assuming the size of uncontaminated training sets is m = n o , and the feature dimensionality of the training dataset is d, the minimum value of γ also needs to be at least one feature dimension greater than the input training set (R d ), i.e., γ d + 1. When γ = d + 1, we can only guarantee that the line obtained by the defense algorithm is d-dimensional. Therefore, it is very difficult to make randomly selected γ points in training set of m points that conform to the original trends of the dataset. Assuming that the points in the unpoisoned training set are evenly distributed, we can obtain the relationship between the MSE of the original training set D tr on its corresponding line θ tr and the MSE of the γ points D γ on its corresponding line θ γ , as shown in Equation (18): MSE(D tr , θ tr ) ≤ m γ · MSE(D γ , θ γ ).(18) As γ increases, the difference between MSE(D tr , θ tr ) and m γ · MSE(D γ , θ γ ) becomes smaller, defense algorithms would also be more efficient, but the corresponding time complexity will become worse (We will analyze the time complexity of the defense algorithm in detail in Section IV-B.). While knowledge of α appears to be an essential parameter for Proda, the defense algorithm will still operate for an assumed value of α (0.2, a safe assumption due to practical bounds [1]). Recall that Proda creates groupings of points and selects the set of points least likely to contain poisoning points as a representative set for the dataset. Thus, if there are less poisoning points than assumed, the representative set should still only contain clean points representative of the data distribution. The largest consequence of assuming a worst case scenario for α is the increase in the time complexity, as we shall discuss in the next section; however, this still remains smaller than that of the competing defense of TRIM. B. Time Complexity We now compute the time complexity of our proposed defense. According to Equation (7), we know that the probability of at least one group of γ points all belonging to the unpolluted data set is: P u = 1 − (1 − (1 − α) γ ) β .(19) When P ≤ ε, P u approaches 1. The value of β satisfies: β log 1−(1−α) γ (1 − P u ).(20) That is, at least β times of random point selection can be carried out to obtain a high probability of obtaining a group of γ points which are all uncontaminated data. Thus, the time complexity of Proda algorithm is: T (n) = O(n) × log 1−(1−α) γ (1 − P u ),(21) where O(n) is the time complexity of the linear regression algorithm. We note that if the feature dimensionality of the training data set is d, and γ d + 1, the minimum time complexity can be found as T (n) = O(n)×log 1−(1−α) d+1 (1− P u ). In contrast, for the time complexity of the TRIM algorithm, which we obtain by inspecting the TRIM algorithm, the worst case of TRIM involves the traversal of all n training subsets; thus, TRIM may iterate M m times. 4 Therefore, when the time complexity of TRIM is compared to our Proda algorithm, a huge improvement is expected. V. EXPERIMENTAL ANALYSIS Note that our poisoning attack algorithm in Section III is stated without any assumptions on the training data distribution. In practice, such information on training data is typically unavailable to attackers. Moreover, an adaptive attacker can also inject poisoning samples to modify the mean and covariance of training data. Thus, we argue that our attack algorithm results are stronger than prior works as we rely on fewer assumptions except for the work [1] (IEEE S&P 2018) of which the assumptions the same as ours. To evaluate the effectiveness of our attacks and defenses, we have selected two key metrics: MSE to measure the effects of poisoning as a result of the attacks and defenses (or the "success" of the attack), and the time complexity of each configuration; this time is computed by multiplying the number of iterations required for each configuration (we assume average of 1000 iterations per 1µs on our computer hardware). Our attack and defense algorithms were implemented in Python 3.7, leveraging the NumPy and scikit-learn libraries. We use a conventional cross-validation method to split the datasets into three equally-sized sets for training, validation, and testing. To ensure data splitting biases are not introduced, we repeat each experiment and average results over 5 independent runs. The remainder of this section is laid out as follows. We first describe the datasets used in our experiments in Section V-A. Followed by Section V-B, we compare results obtained from our poisoning optimization algorithm with previous poisoning attacks on the datasets we have obtained, across four different types of regression models. Finally, we present the results of our Proda algorithm and compare it with previous defenses in Section V-C. A. Datasets We first introduce the four publicly available datasets used in our evaluation. • Housing Prices [25], a dataset used for predicting the price of the house at the time of sale given attributes of the house structure, and location information. • Loans [26], a lending dataset that seeks to estimate the appropriate interest rate of a loan given information about the total loan size, interest rate, amount of principal paid off, and the borrower's personal information such as credit status, and state of residence. , a pharmaceuticals dataset that estimates the dosage of Warfarin for a patient depending on physical attributes of said patient, such as age, height, and weight. • Bike Sharing [28], the Capital Bikeshare system dataset estimates the number of vehicles within a certain time period, given hourly information of rental bikes and environmental variables like weather. The aforementioned datasets, with the exception of the bike sharing dataset, have been used in previous evaluations of poisoning attacks on regression models [1]. 5 All datasets are pre-processed in the same manner, with categorical variables one-hot encoded, and numerical features normalized between 0 and 1. This produces 275, 89, 204, and 15 features for Housing, Loans, Pharm, and Bike Sharing, respectively. In total, each dataset contains 1460, 887383, 4683, and 17389 records, respectively. However, due to computational limitations, the only first 5, 4, 3, and 8 features of the Housing Prices, Loans, Pharm, Bike Sharing datasets, respectively, were used in the defense evaluation. B. Nopt Poisoning Attack We now perform experiments on the four selected regression datasets to evaluate our newly proposed attack. In addition, we compare our results to MSEs of the clean dataset and the optimization attack as proposed by Jagielski et al. [1]. We use MSE as the metric for assessing the effectiveness of an attack, and also compute the attacks' time complexity. We vary the poisoning rate between 4% and 20% at intervals of 4% with the goal of inferring the trend in attack success. Figures 6, 7, 8 and 9 show the MSE of each attack on OLS, Ridge, LASSO, and Elastic-net regression, respectively. We note that Jagielski et al. [1] had only evaluated Ridge and LASSO regression. We plot results for the clean dataset (called Unpoison), optimization attack, proposed by Jagielski et al. [1] (called Opt), in addition to our Nopt attack. The horizontal coordinate is the poisoning rate α, that is, the proportion of pollution data in the new training dataset, and the vertical coordinate is the MSE of the model trained on the poisoned dataset. Overall it can be observed from the diagrams that Nopt is able to achieve larger MSE (in comparison to Opt) for the same poisoning rate. While every configuration of the regression type, dataset and poisoning rate observes Nopt exceeding Opt, we note that Figures 7(c) and 8(c) show a relatively similar increase of MSE between the two attacks. We also observe that OPT is more suitable for Ridge Regression and Lasso Regression on PARM Dataset; However, NOPT is seen to produce more stable attack performance. In Table I, we detail the specific MSEs of our new attack (Nopt) and the optimization attack proposed by Jagielski et al. [1] (Opt). For this table, we reproduce the numerical value of the MSE at a fixed poisoning rate of α = 0.2. From Table I we can observe that when the poisoning rate is α = 0.2, our attack (Nopt) is again consistently higher than that of the previous attack algorithm (Opt). Our results confirm that the optimization framework we design demonstrates increased effectiveness when poisoning both different linear regression models and across datasets. The Nopt attack can achieve MSEs with a factor of 1.3 higher than the Opt attack in the House dataset, a factor of 1.5 higher in the Loan dataset, a factor of 1.2 higher than Opt in the Pharm dataset, and a factor of 4.0 higher than Opt in the bike sharing dataset. We note that the primary difference between NOPT and OPT is the difference in the objective function. The objective function of the OPT algorithm is determined to make the model post-poisoning deviate a maximal amount from the original model. The objective function of NOPT, however, is to perturb the original training set to become more disordered, thereby increasing the degree of dispersion of the training data set, and thus affecting the MSE of the final model on the training data set. Therefore, it is expected that the NOPT algorithm will produce more effective poisoning than the OPT algorithm, a property we have demonstrated through experimentation. C. Proda Defense Algorithm In this section, we evaluate if our Proda algorithm can effectively defend against the optimized attack (Nopt), the attack that produced the highest MSE in Section V-B. We shall use two measures in the experiment: the difference of MSE and time complexity. We evaluated the MSE among the resulting dataset of the Proda algorithm, the resulting dataset of TRIM algorithm, and the clean dataset. We also evaluated the time complexity of our defense by measuring the running time of the algorithm. For the probability algorithm, 1) Defense Efficiency: When the experiment was performed, γ and α are assumed known to be the defender (Recall from Section II-B2, α is known by the defender, as they can assume that every sample submitted to the learning process is likely to be malicious; in the event that the α is unknown and cannot be found by the defender, the defender can assume an upper bound of 0.2 poisoning rate.). Therefore in specific cases, β, the intermediate variable, would not change. So, if γ is given, MSE varies only with the poisoning rate ( Figure 11, for a fixed γ = 50). If a poisoning rate α is given, MSE varies only with γ ( Figure 10, for a fixed α = 0.2). We remark that the value of γ has been set to a specific value instead of a percentage of the training set. Empirically, we observed the selection of γ had no clear correlation with the size of the training sets, instead the value of γ is more strongly associated with the number of training set features. Specifically, we observed that the minimum value of γ should be greater than the number of training set features (d + 1). Consequently, we have analyzed in more detail the relationship between γ and the number of features in the training set. Figure 10 shows the resulting MSE of the poisoning attack when setting different values of γ for the defense algorithm. We can see that when γ is small, the defense efficacy of Proda is less than that of TRIM (with the exception of the bike sharing dataset, where the MSE is the same), but with an increase of γ, the effectiveness of Proda in removing the influence of poisoning samples increases, however at the cost of additional computational time (see Table III). As we have previously described in Section IV, Proda seeks to find a subset of data that contains only clean samples, as the presence of poisoning points will increase the MSE of the resulting regression (in the overall model and in each subset evaluated by Proda), poisoning points that greatly increase the . Both TRIM and Proda algorithms have smaller MSE results than clean data sets. When the γ value is large, the defensive effect of the two algorithms converge. In the bike sharing dataset, two lines of TRIM and Proda overlap. The Proda algorithm achieves a satisfying defense effect when γ is set to be the minimum value d + 1. There is no general value of γ, as can be seen from Fig. 10. For each dataset, as γ increases, the MSE of the training set when a defense is applied will plateau. Therefore, when increasing γ no longer significantly improving the MSE, we fix this value as the appropriate γ value. dispersion of the subset (and thus MSE) will be discarded by Proda. In Figure 10(b), when γ is equal to twice the number of features, the MSE of the probability algorithm result is lower than the MSE of the clean dataset, while the remaining datasets observe lower MSE compared to the clean dataset for all values of γ. This indicates that a proper γ can induce not only an effective defense result observed, but it also has assisted in the generalization of the model. We assert that the result is obtained as Proda also removes poor training points that may exist in the clean dataset. Interestingly, Proda on the Loan's dataset at very small values of γ demonstrates an MSE larger than that of the clean dataset, given that the Loans dataset contains the largest number of records. Selecting too small a set of points to act as the representative set will negatively impact the resulting regression. It can be seen from Figures 10 and 11 that the resulting MSE of TRIM algorithm is lower than the resulting MSE of the probability algorithm and lower than the MSE of the clean dataset until γ is much larger. We note that we have not evaluated other defenses since Jagielski et al. [1] have shown that TRIM, their state of the art, is capable of outperforming prior defense mechanisms. In the bike sharing dataset, we can see from Figure 10(d) that three lines of Unpoison, TRIM, and Proda are almost compressed to one line, indicating their performances resemble each other. Proda has achieved a satisfying defense effect when the parameter γ is set as a minimum threshold, i.e., γ = d + 1. Although the three lines overlap almost completely all the way to γ = 41, if we further zoom in, the performance of Proda's defense will boost as the γ increases. An unknown poisoning rate α. Oftentimes the defender has zero knowledge of the poisoning rate α used by an attacker; however, due to the construction of the Proda algorithm, it can still protect against poisoning attacks of different poisoning rates, when the defender assumes a worst case scenario of α = 0.2 (the argued largest realistic poisoning rate [1]). In Table II, we demonstrate results for the Proda algorithm run with a known α (as previously seen in Figure 11), and the Proda algorithm when executed with an assumed α = 0.2, 0.1. From the table, we can observe that there is a small decrease in the MSE, when the assumed poisoning rate is larger than the true rate of α. Conversely, in the event that a defender underestimates the poisoning rate α used by an attacker (e.g., a conservative estimate was used), we can observe from Table II and Figure 12 that once the real poisoning rate (α) exceeds the defender's assumed poisoning rate, the MSE of the model exceeds that of the clean MSE. However, this increase still remains below that of the undefended MSE. This trend can be consistently observed across all datasets. Thus, it is recommended that the poisoning rate is set higher than any realistic poisoning rate for the strongest defense at the expense of time complexity (as we shall analyze in the following section); however, even a conservative estimate (for less time complexity) will still yield MSE reductions compared to no defense. Instead the defender assumes a worst case scenario of α = 0.2. Like Figure 11, both Proda and TRIM algorithms display good defensive performances at various poisoning rates. In the house price dataset and pharm dataset, the two trends overlap. In the bike sharing dataset, Unpoison, TRIM, and Proda trends overlap. 2) Time Complexity: In Section IV-B, we analyzed the time complexity of Proda and TRIM, and found that the worst case of Proda is superior to TRIM. In both Tables III and IV, we detail the respective computed time complexities of Proda and TRIM, for different values of γ (see Table III), and different values of α (see Table IV). We reiterate that the time complexity indicated in these tables is computed by the number of iterations established in Section IV-B with an assumed average processing speed of 1000 iterations per 1µs on our computing hardware. In Table III, for a fixed α = 0.2, the time complexity of Proda algorithm is directly related to γ and the number of features (recall that 5, 4, 3, and 8 features were used in House, Loan, Pharm, and Bike Sharing datasets, respectively), while the time complexity of the TRIM algorithm is related to the training data size (The size of our clean training set is 300.). From the average and worst-case time complexities shown in Table IV, it can be observed that TRIM is faster than the Proda on the average case; however, our Proda algorithm provides an upper bound on the time complexity. In Table IV, we observe time complexities for a fixed γ = 50, but a varying poisoning rate α, we can see that the worst-case time complexity of the two algorithms increases with the increase of the poisoning rate α; however, as we have shown earlier in Section IV-B, Proda is bounded by the worst-case scenario. Note that TRIM algorithm uses an iterative search to find the smallest MSE subset as its defense. Suppose the size of the poisoned training set is k. The worst-case scenario for TRIM is to exhaust k (1−α)k subsets to converge. In the case of a poisoning rate of α = 0.2, TRIM will not terminate until it has compared MSEs of k 0.8k subsets ( 1.6 k subsets, given Stirling's approximation). In contrast, Proda algorithm squarely selects the smallest MSEs of 1−αk clean data-points from our randomly chosen subsets; hence, the time complexity of Proda algorithm is dependent only on the value of the parameter γ we define, consequently offering a substantial reduction in the worst-case time complexity of such a defense. 3) Effectiveness of Proda against Opt and Nopt: Both Opt and Nopt add previously poisoned data points during the optimization phase; thus it is desirable to compare whether the two different attack algorithms will perform differently against the same defense of Proda. For all four datasets and across three values of α (4%, 12%, 20%), it is observed in Table V that there is little to no difference between attack algorithms. VI. DISCUSSION AND RELATED WORK In this section, we discuss the limitations of this paper and survey the related work. A. Limitations Supervised machine learning algorithms can solve common regression and classification problems, but the training data of supervised machine learning may potentially be manipulated by attackers seeking to interfere with the training process for their own nefarious purposes. For example, an attacker may add poisoning data to interfere with a supervised machine learning algorithm and likewise, a defender can use data optimization to prevent data poisoning attacks. In data poisoning attacks, attackers may have a variety of goals to interfere the regression result. In this work we focus on maximizing the dispersion of the training set. As we have discussed earlier, an investigation into the possibility of a targeted attack is left for future work. When defending against poisoning attacks, the Proda algorithm seeks to find a subset of points that minimize the linear regression loss function. In prior work the TRIM [1], a defense algorithm, has been proposed against linear regression poisoning attacks. Both TRIM [1] and our work are based on the premise that the loss function of linear regression greatly increases with the addition of even a single poisoned point, a method that differs from other approaches which seek to classify or isolate poisoned points. What our approaches differ is that the worst-case time complexity of our defense algorithm Proda is better than TRIM [1]. While we have experimentally demonstrated the effectiveness in removing the effect of poisoning attacks, it is possible that Proda can be retooled to detect the subset of poisoning points. Our Proda algorithm has a parameter γ, which controls the precision and time complexity of the defense algorithm. The setting of the parameter γ is data-dependent; however, we have established that γ should not be less than one more than the dimension of the dataset d + 1. Experimentally, we observed that a larger γ will produce lower MSE at the expense of a higher time complexity. It can be seen from Figure 10 that when γ approaches d 2 , Proda algorithm can obtain basically stable defense efficiency. The specific value of γ should be chosen based on a time complexity acceptable to the defender. B. Related Work Poisoning attacks. Data poisoning attacks are a general class of attacks that manipulate the training data of a machinelearning system such that the learned model behaves in a way dictated by the attacker. Such poisoning techniques have been studied in various applications, such as anomaly detection [10] and email spam filtering [9]. It has also been shown that data poisoning attacks are indifferent to the underlying machine learning algorithms (SVMs [3], regression [1,29,30], graphbased approaches [6,31,32], neural networks [19,33,34,35,36], and federated learning [37]). While the aforementioned works primarily compromise classification tasks, there has also been work on poisoning recommender systems [38,39,40]. Most relevant to ours are attacks against regression tasks [1,5], whereas our attack outperforms the prior works. Defenses against poisoning attacks. Many defense mechanisms have been proposed to defend against poisoning attacks [9,10,11,13]. One core approach to mitigating poisoning attacks is to recognize that the poisoned data is intentionally dissimilar to the clean training data (as to inflict the greatest change to the learning process). Due to the limitation on the proportion of poisoned data to clean data an attacker has control over, the poisoned data may be treated as outliers, and mitigated with data sanitization techniques [14,15,16]. The other means to defend against potentially poisoned data is to use robust learning algorithms [1,10,17,18]. These algorithms are designed to limit the sensitivity to any single sample within the training data. Differential privacy has also been investigated as a means to mitigate data poisoning attacks [8]. Specifically for regression models, the closest to this work is TRIM [1], a defense capable of managing a large number of poisoning data points. Like Proda, TRIM finds a subset of training data that minimizes the model loss, and uses this subset as a representative set to train the non-poisoned model. Presently, the defense methods against deep learning poisoning attacks can be divided into three stages: data and feature modification, model modification, and output defense. Data and feature modification [9,10,11,13] primarily refers to the processing of data or features before it is accepted as input into the model to achieve the defense objective. A core tenet to mitigating poisoning attacks is to recognize that the poisoned data is intentionally dissimilar to the clean training data (as to inflict the greatest change to the learning process). Due to limitations to the proportion of poisoned data to clean data an attacker can control, the poisoned data is the minority class and may be treated as outliers, and mitigated with data sanitization techniques [14,15,16]. Another method to defend against potentially poisoned data is to use robust learning algorithms [1,10,17,18]. These algorithms are designed to limit the model's sensitivity to any single sample within the training data. Differential privacy has also been investigated as a means to mitigate data poisoning attacks [8]. Specifically for regression models, TRIM [1] is a defense capable of managing a large number of poisoning data points. Like Proda, TRIM finds a subset of training data that minimizes the model loss, and uses this subset as a representative set to train the non-poisoned model. From the perspective of preprocessing features, Shen et al. [41] propose a method to automatically produce shielding features to identify and neutralize abnormal distributions. Directly protecting the model involves the modification of the deep learning model. Liu et al. [42] propose a pruning algorithm to reduce the size of the backdoor networks by eliminating dormant neurons on pure inputs. Iandola et al. [43] propose a fine-tuning defense in which a potentially poisoned neural network is updated to disable backdoor triggers. DeepInspect [44], a detection framework, is proposed to leverage a conditional generation model to learn the probability distribution of potential triggers from the queried model to retrieve fingerprints left behind during the insertion of the backdoor. An output defense mitigates an attack by analyzing the output behavior of the potentially poisoned deep learning model. Yang et al. [45] propose a loss-based method, which would trigger an accuracy check if the loss of the target model exceeded a threshold multiple times. Hitaj et al. [46] presents an integrated defense, by combining the prediction results of different models to judge the prediction categories of samples. Chandola et al's work detects poisoned inputs through Support Vector Machines (SVM) and decision trees [47]. Zhao et al. [48] propose the multi-task model defense and analyze the output results of the model through data cleaning to improve the robustness of multi-task joint learning. VII. CONCLUSION We systematically study the poisoning attack and its defense for regression models in MLaaS setting. We have proposed a modification to an attack optimization framework that requires no additional knowledge of the training process, yet produces better offensive results. Our attack allows attackers to carry out both white-box and grey-box attacks and is capable of increasing the dispersion of the poisoned training set. However, as attackers may have many goals in the interference of the regression outcome, the possibility of targeted attacks remains in question. To respond to a more powerful poisoning attack developed in this paper, we designed a probabilistic defense algorithm, Proda, which can be tuned to effectively mitigate poisoning attacks on the regression model while significantly reducing the worst-case time complexity. We highlight that the time complexity of the state-of-the-art defense, TRIM, had not been estimated; however, we deduce from their work that TRIM can take exponential time complexity in the worst-case scenario, in excess of Proda's logarithmic time. Finally, we hope that our work will inspire future research to develop more robust learning algorithms that are not susceptible to poisoning attacks. policies, either expressed or implied, of the Combat Capabilities Development Command Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation. Benjamin Zi Hao Zhao is pursuing his Ph.D. at the School of Electrical Engineering and Telecommunications at the University of New South Wales and CSIRO-Data61. His current research interests are authentication systems, and security and privacy with machine learning. His work has received the ACM AsiaCCS best paper award. Minhui Xue is a Lecturer (a.k.a. Assistant Professor) of School of Computer Science at the University of Adelaide. He is also an Honorary Lecturer with Macquarie University. He is the recipient of the ACM SIGSOFT distinguished paper award and IEEE best paper award, and his work has been featured in the mainstream press, including The New York Times and Science Daily. Fig. 1 : 1The poisoning attack on linear regression Fig. 2 : 2Impact on learned regression line as a result of poisoning attacks at different poisoning rates. The original unpoisoned data is shown in blue, while poisoning data is shown in red. Fig. 3 : 3The workflow of the poisoning attack algorithm smallest MSE of n points for β models Fig. 4 : 4The workflow of the defense algorithm considering trade-offs between accuracy and time complexity. This means if the value of β follows the rules of Equation Fig. 5 : 5Different parameters γ used in the Proda algorithm. The original unpoisoned data-points are shown in blue, poisoning datapoints in red, and defensive data-points generated by the Proda algorithm in green, where green data-points are subsets of blue ones. The Proda algorithm screens out the subsets of uncontaminated datasets. Fig. 6 :Fig. 8 :Fig. 9 : 689Mean Squared Error (MSE) of poisoning attacks on OLS regression on the four datasets MSE of attacks on LASSO regression on the four datasets MSE of attacks on Elastic-net regression on the four datasets • Pharm [27] Fig. 10 : 10The effects of increasing γ on the efficiency of defense algorithms on our four datasets (The poisoning algorithm is Nopt on LASSO regression, poisoning rate α = 0.2.) Fig. 11 : 11The effects of different poisoning rates α on the efficiency of defense algorithms on our four datasets (The poisoning algorithm is Nopt of LASSO regression, γ = 50.) Both Proda and TRIM algorithms display good defensive performances at various poisoning rates. In the house price dataset, pharm and bike dataset, the two trends overlap. Fig. 12 : 12Comparison of defense performance when α is varied for Proda on four datasets (The poisoning attack used for this evaluation is Nopt on Ridge regression with γ = 50. The size of our clean training set is 300.). However, the defender does not possess knowledge about the attacker's poisoning rate, instead assuming a poisoning rate of 0.1 (10%) and 0.2 (20%).TABLEII: Comparison of defense algorithm when α is known and unknown (The poisoning algorithm is Nopt of LASSO regression, γ = 70.). The effects of different poisoning rates α on the efficiency of defense algorithms in which the attacker does not posses knowledge of the poisoning rate α (The poisoning algorithm is Nopt of LASSO regression, γ = 70.). He currently serves on the Program Committee of IEEE Symposium on Security and Privacy (Oakland) 2021, ACM CCS 2021, USENIX Security 2021, NDSS 2021, ICSE 2021, ESORICS 2021, and PETS 2021 and 2020. Alina Oprea is an Associate Professor at Northeastern University in the Khoury College of Computer Sciences. She was the recipient of the Technology Review TR35 award for research in cloud security in 2011 and the recipient of the Google Security and Privacy Award 2019. She currently serves as Program Committee co-chair for the IEEE Symposium on Security and Privacy 2021. She served as Program Committee co-chair for the IEEE Symposium on Security and Privacy 2021 and 2020, as well as NDSS 2019 and 2018. Haifeng Qian is a professor at the Software Engineering Institute of East China Normal University, Shanghai, China. He received a BS degree and a master degree in algebraic geometry from the Department of Mathematics at East China Normal University, in 2000 and 2003, respectively, and the PhD degree from the Department of Computer Science and Engineering at Shanghai Jiao Tong University in 2006. His main research interests include network security, cryptography, and algebraic geometry. [email protected]) are the corresponding authors of this paper.Haifeng Qian ([email protected]) and Minhui Xue (Training Data Linear Regression poiso n data Target Model Poison Data add For Prediction Ideal Training Phase TABLE I : IComparison of MSE between Opt and our newly proposed Nopt poisoning algorithms. It can be observed that in all configurations, our attack achieves a larger MSE. the default value of ε in our experiments is 10 −5 .Dataset Regression MSE after Poisoning (α = 0.2) Opt Nopt House Prices OLS 0.055 0.07 Ridge 0.07 0.10 LASSO 0.085 0.10 Elastic-net 0.04 0.08 Loans OLS 0.061 0.081 Ridge 0.043 0.095 LASSO 0.062 0.094 Elastic-net 0.032 0.091 Pharm OLS 0.077 0.10 Ridge 0.134 0.145 LASSO 0.11 0.14 Elastic-net 0.031 0.11 Bike Sharing OLS 0.04 0.16 Ridge 0.036 0.15 LASSO 0.049 0.15 Elastic-net 0.0413 0.157 different γ values will produce different operation results and time complexities; thus our experiments may also evaluate results for different values of γ, however unless otherwise stated, TABLE III : IIIComparison of defense algorithm time performance when γ is varied for Proda on four datasets (The poisoning attack used for this evaluation is Nopt on Ridge regression with the poisoning rate α = 0.2. The size of our clean training set is 300.).Dataset Algorithm γ Time Complexity Time Complexity (average, s) (worst case, s) House Prices TRIM - 0.021 ≥ 5 80 Proda 6 0.037 0.037 28 5.946 5.946 50 806.646 806.646 Loans TRIM - 0.021 ≥ 5 80 Proda 5 0.028 0.028 25 3.041 9.294 45 264.318 264.318 Pharm TRIM - 0.021 ≥ 5 80 Proda 4 0.021 0.021 30 9.294 9.294 60 7512.528 7512.528 Bike Sharing TRIM - 0.021 ≥ 5 80 Proda 9 0.079 0.079 25 3.041 3.041 41 108.261 108.261 TABLE IV : IVComparison of defense algorithm time performance when the poisoning rate α is varied on four datasets (The poisoning attack is Nopt on Ridge regression with a fixed γ = 50. The size of our clean training set is 300.).Dataset Algorithm α Time Complexity Time Complexity (average, s) (worst case, s) House Prices TRIM 4% 0.021 ≥ 5 18 12% 0.021 ≥ 5 43 20% 0.021 ≥ 5 80 Proda 4% 0.143 0.143 12% 12.162 12.162 20% 806.646 806.646 Loans TRIM 4% 0.021 ≥ 5 18 12% 0.021 ≥ 5 43 20% 0.021 ≥ 5 80 Proda 4% 0.143 0.143 12% 12.162 12.162 20% 806.646 806.646 Pharm TRIM 4% 0.021 ≥ 5 18 12% 0.021 ≥ 5 43 20% 0.021 ≥ 5 80 Proda 4% 0.143 0.143 12% 12.162 12.162 20% 806.646 806.646 Bike Sharing TRIM 4% 0.021 ≥ 5 18 12% 0.021 ≥ 5 43 20% 0.021 ≥ 5 80 Proda 4% 0.143 0.143 12% 12.162 12.162 20% 806.646 806.646 TABLE V : VComparison of MSE after defense between Opt and our newly proposed Nopt poisoning algorithms when the defense algorithm is the same (The poisoning algorithm is Nopt of LASSO regression. The size of our clean training set is 300. The defense used for this evaluation is Proda, γ = 70.).Dataset α MSE MSE MSE MSE MSE (Clean) Opt Nopt (Opt after Proda) (Nopt after Proda) House Prices 4% 0.03 0.055 0.07 0.012 0.012 12% 0.03 0.07 0.10 0.003 0.003 20% 0.03 0.085 0.10 0.005 0.005 Loans 4% 0.038 0.061 0.081 0.029 0.028 12% 0.038 0.043 0.095 0.029 0.029 20% 0.038 0.062 0.094 0.028 0.028 Pharm 4% 0.036 0.077 0.10 0.004 0.003 12% 0.036 0.134 0.145 0.004 0.004 20% 0.036 0.11 0.14 0.004 0.003 Bike Sharing 4% 0.015 0.04 0.16 1.89 × 10 −19 1.96 × 10 −19 12% 0.015 0.036 0.15 1.72 × 10 −19 1.69 × 10 −19 20% 0.015 0.049 0.15 1.87 × 10 −19 1.78 × 10 −19 The assumptions of[1] is the same as ours, albeit previously mislabelled as a black-box attack. Equation(4)is an instantiation of Equation(1)in[1] for our alternative loss E, as we shall explain later in Section III-A. There are many types of grey-box attacks; this is our own grey-box setting. No time complexity analysis was provided in TRIM's proposal[1]. 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Data poisoning attacks on multi-task relationship learning. M Zhao, B An, Y Yu, S Liu, S J Pan, Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence, (AAAI). the Thirty-Second AAAI Conference on Artificial Intelligence, (AAAI)M. Zhao, B. An, Y. Yu, S. Liu, and S. J. Pan, "Data poisoning attacks on multi-task relationship learning," in Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence, (AAAI), 2018, pp. 2628-2635. Jialin Wen is pursuing her Master's degree at School of Computer Science of Technology of East China Normal University. She focuses primarily on the areas of machine learning and security, specifically exploring the robustness of machine learning models against various adversarial attacks. Jialin Wen is pursuing her Master's degree at School of Computer Science of Technology of East China Normal University. She focuses primarily on the areas of machine learning and security, specif- ically exploring the robustness of machine learning models against various adversarial attacks.	[ "https://github.com/jagielski/" ]
[ "Center-Periphery Structure in Communities: Extracellular Vesicles", "Center-Periphery Structure in Communities: Extracellular Vesicles", "Center-Periphery Structure in Communities: Extracellular Vesicles", "Center-Periphery Structure in Communities: Extracellular Vesicles" ]	[ "Eleanor Wedell \nDepartment of Computer Science\nUniversity of Illinois Urbana-Champaign\n61801UrbanaIL\n", "Minhyuk Park \nDepartment of Computer Science\nUniversity of Illinois Urbana-Champaign\n61801UrbanaIL\n", "Dmitriy Korobskiy \nNTT DATA\n22102McLeanVA\n", "Tandy Warnow \nDepartment of Computer Science\nUniversity of Illinois Urbana-Champaign\n61801UrbanaIL\n", "George Chacko \nDepartment of Computer Science\nUniversity of Illinois Urbana-Champaign\n61801UrbanaIL\n\nOffice of Research\nGrainger College of Engineering\nUniversity of Illinois Urbana-Champaign\n61801UrbanaIL\n", "Eleanor Wedell \nDepartment of Computer Science\nUniversity of Illinois Urbana-Champaign\n61801UrbanaIL\n", "Minhyuk Park \nDepartment of Computer Science\nUniversity of Illinois Urbana-Champaign\n61801UrbanaIL\n", "Dmitriy Korobskiy \nNTT DATA\n22102McLeanVA\n", "Tandy Warnow \nDepartment of Computer Science\nUniversity of Illinois Urbana-Champaign\n61801UrbanaIL\n", "George Chacko \nDepartment of Computer Science\nUniversity of Illinois Urbana-Champaign\n61801UrbanaIL\n\nOffice of Research\nGrainger College of Engineering\nUniversity of Illinois Urbana-Champaign\n61801UrbanaIL\n" ]	[ "Department of Computer Science\nUniversity of Illinois Urbana-Champaign\n61801UrbanaIL", "Department of Computer Science\nUniversity of Illinois Urbana-Champaign\n61801UrbanaIL", "NTT DATA\n22102McLeanVA", "Department of Computer Science\nUniversity of Illinois Urbana-Champaign\n61801UrbanaIL", "Department of Computer Science\nUniversity of Illinois Urbana-Champaign\n61801UrbanaIL", "Office of Research\nGrainger College of Engineering\nUniversity of Illinois Urbana-Champaign\n61801UrbanaIL", "Department of Computer Science\nUniversity of Illinois Urbana-Champaign\n61801UrbanaIL", "Department of Computer Science\nUniversity of Illinois Urbana-Champaign\n61801UrbanaIL", "NTT DATA\n22102McLeanVA", "Department of Computer Science\nUniversity of Illinois Urbana-Champaign\n61801UrbanaIL", "Department of Computer Science\nUniversity of Illinois Urbana-Champaign\n61801UrbanaIL", "Office of Research\nGrainger College of Engineering\nUniversity of Illinois Urbana-Champaign\n61801UrbanaIL" ]	[]	Clustering and community detection in networks are of broad interest and have been the subject of extensive research that spans several fields. We are interested in the relatively narrow question of detecting communities of scientific publications that are linked by citations. These publication communities can be used to identify scientists with shared interests who form communities of researchers. Building on the wellknown k-core algorithm, we have developed a modular pipeline to find publication communities. We compare our approach to communities discovered by the widely used Leiden algorithm for community finding. Using a quantitative and qualitative approach, we evaluate community finding results on a citation network consisting of over 14 million publications relevant to the field of extracellular vesicles. * EW and MP contributed equally to this manuscript	10.1162/qss_a_00184	[ "https://arxiv.org/pdf/2111.07410v1.pdf" ]	244,117,106	2111.07410	39e90ca8ce2552dd36f14942529138324ffbeaeb	Center-Periphery Structure in Communities: Extracellular Vesicles November 16, 2021 Eleanor Wedell Department of Computer Science University of Illinois Urbana-Champaign 61801UrbanaIL Minhyuk Park Department of Computer Science University of Illinois Urbana-Champaign 61801UrbanaIL Dmitriy Korobskiy NTT DATA 22102McLeanVA Tandy Warnow Department of Computer Science University of Illinois Urbana-Champaign 61801UrbanaIL George Chacko Department of Computer Science University of Illinois Urbana-Champaign 61801UrbanaIL Office of Research Grainger College of Engineering University of Illinois Urbana-Champaign 61801UrbanaIL Center-Periphery Structure in Communities: Extracellular Vesicles November 16, 2021 Clustering and community detection in networks are of broad interest and have been the subject of extensive research that spans several fields. We are interested in the relatively narrow question of detecting communities of scientific publications that are linked by citations. These publication communities can be used to identify scientists with shared interests who form communities of researchers. Building on the wellknown k-core algorithm, we have developed a modular pipeline to find publication communities. We compare our approach to communities discovered by the widely used Leiden algorithm for community finding. Using a quantitative and qualitative approach, we evaluate community finding results on a citation network consisting of over 14 million publications relevant to the field of extracellular vesicles. * EW and MP contributed equally to this manuscript Introduction Community detection in networks is of broad interest and has been discussed in comprehensive reviews (Fortunato and Castellano, 2009;Fortunato, 2010;Javed et al., 2018). At a high level, the community detection problem amounts to identifying groups within a complex network that share some common properties. As observed by Coscia et al. (2011), this definition suffers from some degree of imprecision given the diverse ways in which communities can be defined and a richness of perspectives. For example, a community detection approach may focus on vertex similarity or edge-density; disjoint, overlapping or hierarchical community structure; and static versus dynamic networks. Newman (2006) had also noted distinctions between community discovery and graph partitioning. Thus, the context of the question being asked and the techniques being employed tend to determine the flavor of community detection in a study. From the perspective of scientometrics, detecting a research community can be framed as a community finding problem in which communities of publications defining areas of research are discovered in the scientific literature. First, the scientific literature is modeled as a network with publications as nodes and citations as directed edges (Boyack and Klavans, 2019). From this network, an area of research is defined by a community of publications-a sufficiently citation-dense area in the network. Citation networks of scientific literature can be constructed using different approaches. Direct citations were used to build clusters of articles from a dataset of over 10 million publications (Waltman and van Eck, 2012); this methodology was also used in building citation maps from 19 and 43 million publications (Boyack and Klavans, 2014). Direct citation, bibliographic coupling, and co-citation have been compared for their relative value in identifying research fronts (Boyack and Klavans, 2010), with a hybrid approach involving bibliographic coupling and textual similarity performing the best. A subsequent study conducted at a larger scale and with improved evaluation criteria suggested that direct citation was the most promising (Klavans and Boyack, 2017). We use direct citations in this study. Once constructed, citation networks can be analyzed using different community finding or clustering approaches (Boyack and Klavans, 2019;Ahlgren et al., 2020;Traag et al., 2019;Sciabolazza et al., 2017;Waltman and van Eck, 2012;Šubelj et al., 2016). Of these approaches, a recent development is the availability of the Leiden algorithm, which offers better partitioning and performance (Traag et al., 2019) compared to a preceding approach, the Louvain algorithm (Blondel et al., 2008), which seeks to optimize the modularity quality function (Newman, 2006). We are interested in how research communities form and collaborate around research questions (Kuhn, 1970;Crane, 1972). Collaboration within the scientific community goes beyond co-authorship, and includes building upon prior discovery by other researchers; citations between publications thus indicate this more general form of collaboration, and the publication communities detected in this process are evidence of such collaborations. Further, the authors of the publications in the publication communities represent communities of researchers working on related questions. Historical studies have estimated the size of research communities to be in the order of a few hundreds (Price and Beaver, 1966;Crane, 1972;Kuhn, 1970;Mullins, 1985). However, this question merits re-examination in the modern scientific enterprise; expanded, globalized, and electronically connected. Since researchers can work on more than one problem and be considered members of more than one research community, to find author communities we begin by finding communities of publications. Then, for each publication community, the authors of the publications in the community constitute a researcher community organized around the questions defined by the publication community. We are interested in articles linked by citation as the products of a research community rather than articles clustered by textual similarity, therefore, we use direct citations to discover communities. In an earlier study (Chandrasekharan et al., 2021), we explored this approach to detect publication communities and subsequently their author communities in networks of biology literature. We used an ensemble technique to find publication communities coupled with limited qualitative analysis, where the publication communities were significantly overlapping in clusters identified by the Leiden algorithm and also the Markov Clustering (MCL) algorithm (Van Dongen, 2008). We were specifically interested in whether the author communities we found exhibited substructure indicating the influence of a few individuals associated with the majority of publications while exhibiting different degrees of collaboration within and across subgroups (Price and Beaver, 1966;Crane, 1972). While we did detect such center-periphery substructure (Breiger, 2014, p. 60) in both publication and author communities, our findings were potentially limited by the clustering methods we used, Leiden and MCL, since neither is designed to detect or require substructure when identifying communities. Here, we aim to investigate more carefully whether publication clusters exhibiting center-periphery structure exist in citation networks of the scientific literature. The central idea is that each community contains core nodes representing the "center" of centerperiphery organization and additional nodes representing the "periphery"; our approach first finds the core nodes and then augments the cluster to include peripheral nodes. Since prior community detection methods are not explicitly designed to detect such communities, we propose a new modular pipeline that specifically aims to find such communities. To find the core node sub-community, we combine two approaches represented in the clustering and community detection literature: first, that valid communities should have positive modularity score (Fortunato and Barthelemy, 2007), and second that each community should be dense, which is expressed by the ratio between the average node degree and the number of nodes in the network. Thus, our approach combines three ideas from the literature: the center-periphery model of authorship communities extended here to a model for publication communities, positive modularity for individual communities, and sufficient citation density for individual communities. By distinguishing between the core and non-core nodes, we can require that the positive modularity and sufficient citation density requirements hold for the subclusters of core nodes but not necessarily for the clusters that combine both core and non-core nodes. This distinction potentially enables us to better model real world community structure. Our modular pipeline begins by finding clusters of core nodes, building on the ideas of Giatsidis et al. (2011) who quantified the cohesiveness of a community using the "kcore" concept from graph theory (Matula and Beck, 1983). Although optimizing the total modularity in the clustering has drawbacks (as demonstrated in Fortunato and Barthelemy (2007)), we also required that the clusters individually have modularity scores above 0; this is a relatively mild criterion that seeks internal cohesion, and has been considered in the prior literature (e.g., Newman and Girvan (2004); Fortunato and Barthelemy (2007) to be evidence of a valid community. We tested our pipeline on a network of over 14 million publications that we constructed by harvesting citing and cited articles from a seed set defined by the keyword "exosome". This keyword captures articles from the field of extracellular vesicles, which may be important for intercellular communication and development of some diseases, as well as having potential for therapy (Edgar, 2016;Kalluri and LeBleu, 2020;Raposo et al., 2021). We chose extracellular vesicles (Raposo et al., 2021) as the focus of this study for two reasons: first, it is a large research area, and second, because it is rapidly expanding-it has seen spectacular numbers of publications each year since 2010, and therefore represents an excellent test case for community finding methods in the modern scientific enterrprise. Expecting that not all communities discovered in such a large network would be directly relevant to exosomes, we use 1,218 cited references from 12 relevant review articles as expert-identified markers for specificity in the communities we discover. We report our findings in the following sections. Materials and Methods Data A citation network consisting of 14,695,475 nodes and 99,663,372 edges was generated using the Dimensions bibliography (Hook et al., 2018). Briefly, a "seed" set, S, was obtained by performing a text search for the term "exosome" with years of publication restricted to 2010 or earlier. This constraint was applied to allow every element in the seed set to have accumulated at least 10 years of citations. The search retrieved 11,156 publications of type article from Dimensions. To capture publications proximal by citation to the seed set, a network was constructed using a protocol we labelled SABPQ. First we start with the seed set S. Set A is the set of publications that cite at least one publication from set S. Similarly, set B is the set of publications that are cited by at least one publication from set S. Once the sets S, A, and B are identified, then we define set P as the set of publications that cite at least one publication from the set S ∪ A ∪ B. Similarly, set Q is the set of publications that are cited by at least one publication from the set S ∪ A ∪ B. The network contains directed edges defined by citations; if publication x cited y then we created an edge from x to y. The SABPQ protocol was implemented using Dimensions on BigQuery in Google Cloud Services. The data was then exported to Google Cloud Storage Bucket and subsequently exported to a PostgreSQL database for further analysis. Marker Nodes and Specificity As marker nodes for our analysis, we used 1,218 articles cited in 12 recent reviews on extracellular vesicles and exosome biology (van Niel et al., 2018;Kalluri and LeBleu, 2020;Verdi et al., 2021;Ghoroghi et al., 2021;Lananna and Imai, 2021;Busatto et al., 2021;He et al., 2021;Schnatz et al., 2021;Le Lay et al., 2021;Leidal and Debnath, 2021;Clancy et al., 2021;Raposo et al., 2021). All 1,218 markers are present in our constructed network with 14,695,475 nodes. Any community containing at least one marker node was considered relevant. Marker nodes were matched to clusters using identifiers in our network. For example, the marker node with title "Tumour exosome integrins determine organotropic metastasis" and DOI 10.1038/nature15756 is identified as node 4431204 in our network. The complete list of marker nodes is available on our Github site (Park et al., 2021). Clustering methods Leiden We used version 1.1.0 of the Java implementation for the Leiden algorithm (Traag et al., 2019) provided by the Centre for Science and Technology Studies and available in Github (Traag, 2021). We ran Leiden in default mode, which means that the quality function being optimized was the Constant Potts Model rather than modularity. Leiden includes a parameter for the resolution value, which we vary in our experiments from 0.05 to 0.95. New clustering methods Here we describe at a high level the clustering and communityfinding methods we developed in our study. The methods we developed and use in this study are freely available in Github, and the locations of these software and exact commands we used are provided in the supplementary materials. We also provide software version numbers and commands for the existing codes that we use in the supplementary materials. We note here that the codes we developed rely on NetworKit (Staudt et al., 2016), which is an open-source Python module designed for scalable network analysis. In our approach, the objective was to produce a set of clusters each of which has core nodes and peripheral nodes, consistent with the "center-periphery" structure described earlier. These clusters are considered to be "publication communities", with two types of members: core members that are densely connected to each other and peripheral (non-core) members connected to the core members but with fewer edges within a cluster. Our approach requires values for k and p, where k specifies a minimum connectivity between the core nodes, and p indicates a minimum connectivity between each non-core ("periphery") node and the core nodes. These parameter values for k and p are provided by the user, and different choices for these parameters will produce different clusterings. The required minimum connectivity k between core nodes is related to the k-core concept in graph theory, which we now describe. A k-core of a network N is a largest connected subnetwork A of N such that every node in A is adjacent to at least k other nodes in A (Seidman, 1983;Matula and Beck, 1983;Pittel et al., 1996). The k-cores can be calculated in polynomial time (Matula and Beck, 1983), and our new clustering methods build on these algorithms. We are also interested in the modularity scores of the clusters that are produced by each method, as given in Definition 1: Definition 1. The modularity of a single cluster s within a network N , denoted by mod(s), is given by mod(s) = l s L − d s 2L 2 where l s is the number of internal edges in cluster s, d s is the sum of the degrees of the nodes inside s, and L is the number of total edges in the network N (Fortunato and Barthelemy, 2007). The total modularity of a clustering is the sum of the modularity scores of its clusters. Rather than aiming to maximize the total modularity of the clustering, we will only require that each cluster have positive modularity; as noted in Fortunato and Barthelemy (2007), this approach aims to detect valid communities. We now define some additional terms that we will use in designing new clustering methods: Definition 2. Given a network N , a clustering C, and a cluster C drawn from C where C is partitioned into core nodes and non-core nodes, we will say: • C is k-valid if and only if each core node in C is adjacent to at least k other core nodes in C • C is m-valid if and only if the sub-cluster induced by the core nodes is connected and has a positive modularity score • C is p-valid if and only if each non-core node is adjacent to at least p core nodes in C • C is kmp-valid if and only if it is k-valid, m-valid, and p-valid • The clustering C is kmp-valid if and only if every cluster C ∈ C is kmp-valid Note that if a cluster does not contain non-core nodes then it is vacuously p-valid. The clustering methods that we develop seek to produce kmp-valid clusters, so that we can interpret these clusters as communities with center-periphery structure. Moreover, we are interested in clusterings that produce a large number of kmp-valid clusters, as well as those that include as many nodes as possible in the kmp-valid clusters (which by definition must be non-singleton when k > 1). Hence we explore different techniques that seek to optimize these two opposing criteria. We also require positive modularity in the core node sub-clusters for each cluster. This is a relatively mild requirement that avoids cases where the core node sub-cluster may be k-valid and connected but might not reflect a preference for itself over the outside. Consider the case where a 10-clique, a complete graph on 10 nodes, is contained in a clique with 20 nodes. This 10-clique would satisfy k-validity for k ≤ 9 and would be connected, but would not have positive modularity. By enforcing positive modularity, we would avoid returning such clusters. This example illustrates the advantage of enforcing positive modularity even though the probability of it occurring in a real world network is likely to be small. We also note that enforcing positive modularity in the core node sub-cluster (or even in the final cluster that contains both core and non-core nodes) is not the same as trying to maximize the sum of the modularity scores of the individual clusters (total modularity score). In other words, enforcing positive modularity does not have the same vulnerability to the resolution limit that was established for the modularity criterion, which seeks to maximize the total modularity score (Fortunato and Barthelemy, 2007). Four-Stage kmp-Clustering We designed a four-stage pipeline that is designed to enable the user to explore different clustering options, while guaranteeing that the output is a kmp-valid clustering. The input is a network N and values for the parameters k and p. At a high level, the first stages aim to construct the core member components. The second stage extracts valid sub-clusters from those generated by the first stage. The third stage augments these clusters with additional members, most likely non-core members, though some might qualify as core members, and the fourth stage assigns core or non-core status to the nodes, and retains only those clusters that are kmp-valid. We begin with a description of the overall multi-stage structure of our new clustering methods; note that stages 2 and 3 are optional. • Stage 1: Cluster the network N into disjoint clusters (core members), so that each non-singleton cluster is k-valid and m-valid. • Stage 2: Attempt to break each non-singleton cluster produced in Stage 1 into a set of pairwise disjoint clusters, each of which is k-valid at minimum. • Stage 3: For each non-singleton cluster, add unclustered nodes, nodes that are not in any non-singleton cluster, as non-core (peripheral) members, provided that they are adjacent to at least p core nodes in the selected cluster. This is the augmentation stage, which adds non-core nodes to the clusters produced in the earlier stage. • Stage 4: Process the clustering that is received so that each final cluster is partitioned into core and non-core members, and so that the clustering is kmp-valid. Thus, Stages 1 and 2 together are directed at finding core members of clusters, with Stage 1 directed at clusters with large numbers of core nodes and Stage 2 aimed at extracting smaller clusters within these larger clusters. At the end of Stage 1, all clusters will be k-valid and m-valid. If the optional Stage 2 is applied, the clusters it produces will be k-valid and connected, but may no longer have positive modularity. Neither of these stages introduces any non-core nodes, and so the output of each stage is vacuously p-valid. Stage 3 augments the clusters to include non-core nodes; by design the clusters will be connected, k-valid, and p-valid, but depending on the outcome of Stage 2, they may not have positive modularity and so may not be m-valid. Stage 4 is designed to ensure that all final clusters are kmp-valid, and so may modify or discard clusters found in the earlier stages. However, after Stage 4 is run, the output clustering is guaranteed to be kmp-valid. Furthermore, the clusters produced are parsed into core and non-core nodes. We now describe the techniques we have developed for each stage. For Stage 1, we present iterative k-core (IKC) clustering, a method that is inspired by the k-core concept in graph theory. For Stage 2, we also present two different techniques: recursive Graclus (RG) and iterative Graclus (IG), both of which are based on the Graclus (Dhillon et al., 2007) clustering method used in its default setting and applied to split a graph into two parts. Stage 3 is implemented using a straightforward algorithm that we describe below. In contrast to the earlier stages, Stage 4 involves multiple steps, and is described below. This multi-stage design provides a flexible framework by allowing different techniques to be used at each stage. Stage 1: Iterative k-core clustering (IKC) To motivate the IKC algorithm, we first describe the technique that computes k-cores and then describe the iterative method. k-core For a network N and specified positive integer value for k, a k-core of a network N is the largest connected subnetwork A of N such that every node in A is adjacent to at least k other nodes in A. Note that for every network that does not have any isolated vertices (i.e., nodes of degee 0), each connected component is a 1-core of the network. The distribution of k-core sizes also indicate how quickly a network shrinks as k increases (Leskovec and Horvitz, 2008). The identification of the k-core for a maximum achieved value of k in a network has been proposed as a quality measure for community finding that measures cohesiveness and suggests collaboration (Giatsidis et al., 2011). This idea is reiterated in Kong et al. (2019); Malliaros et al. (2019), who discuss applications of the k-core in biology and real world networks. The k-cores can be calculated in polynomial time (Matula and Beck, 1983), as follows. First, we calculate the degree of every node in the network. Then, every node of degree less than k is deleted from the graph, and this process repeats until every node has degree at least k (i.e., every node is adjacent to at least k other nodes). Every connected component that remains is called a k-core of the network. As an example, given a network that contains two connected components: one is a clique of size 100 and the other has a node x that is adjacent to 2000 other nodes, each of which is only adjacent to x (and so have degree 1). Note that for all k with 2 ≤ k ≤ 99, the k-core of this network is the clique of size 100. k-core clustering The simple k-core clustering method takes as input a network N and a value for k, and computes the k-cores of the network. The set of the k-cores is returned as the clustering. By construction, the simple k-core clustering method produces clusters that are k-valid and connected. However, it does not constrain the clusters to have positive modularity. Iterative k-core clustering To improve on the simple k-core clustering technique for our purposes, we developed an iterative k-core (IKC) algorithm. The input to the IKC algorithm is a network N and a positive integer k. IKC then operates as follows: • We will construct a bin B of clusters that will be returned by the IKC clustering. In this step, we initialize B to be the empty set. • We run the k-core labelling algorithm, which labels every node in the network with a non-negative integer. We let L be the largest label found in this labelling. If L < k, then IKC exits, and returns the clusters in the bin B.Otherwise, the L-cores (i.e., the connected components that are labelled by L) of the network are evaluated as potential clusters. • An L-core A is added to the bin B if and only if A has positive modularity. • The L-core is then deleted from the network, and the residual network is recursively analyzed by IKC. The stopping condition is when all the nodes have been deleted from the network. This procedure produces a collection of clusters, each of which has the following properties: (i) each cluster is connected and has positive modularity, and hence each cluster is m-valid and (ii) each cluster is k-valid, which means every node in the cluster is adjacent to at least k other nodes in the cluster. Stage 2: Finding clusters within clusters using Graclus In Stage 2 we seek to discover k-valid clusters that exist within larger k-valid clusters. For this, we use the Graclus clustering method (Dhillon et al., 2007). Graclus has been previously used to cluster bibliometric data (Dhillon et al., 2007;Devarakonda et al., 2020;Subelj et al., 2016), but here we use it to split a given cluster into two sub-clusters. We use Graclus to optimize its default criterion, which is the normalized cut criterion. In this setting, we seek a partition of a given cluster C into two parts C 1 and C 2 so as to minimize links(C 1 , C 2 ) links(C 1 , C) + links(C 1 , C 2 ) links(C 2 , C) , where links(A, B) denotes the number of edges with one endpoint in A and the other endpoint in B. As is the case in other optimization methods, local search in Graclus helps the optimizer escape poor local minima; its default setting does no iterations but this can be modified by specifying the number of iterations. In this study we explore both the default mode, l = 0 (no iterations) and l = 2000 iterations. We implement our use of Graclus in two different ways: recursively and iteratively. Thus, we use Graclus in four different ways. Recursive Graclus The input to this method is a clustering of the network N and the value for the parameter k. We create a bin B of clusters (setting it initially to the empty set). We take a cluster C from the clustering and apply Graclus recursively using either the default mode or the local search mode. The result of this is a division of the cluster C into two non-empty sets A 1 and A 2 . If A 1 has positive modularity and is k-valid, then we add A 1 to B (the bin we have created), and similarly for subset A 2 . If neither A 1 nor A 2 is added to B, add cluster C to B and delete C from the network, effectively removing it from further consideration by Recursive Graclus. The stopping condition for Recursive Graclus is that all nodes have been deleted from the network. When the stopping condition is reached, the final output of Recursive Graclus is the set of clusters in bin B. Though all clusters that are produced by Recursive Graclus are guaranteed to be k-valid and have positive modularity, they may not be m-valid since this requires that the core node subsets be connected. Iterative Graclus To use Graclus iteratively, we follow a similar procedure as in Recursive Graclus but with two key differences. The first is that the user provides a parameter for the number of iterations, so that the procedure must stop after that number of iterations, if it hasn't already stopped. The second difference is that, in contrast to Recursive Graclus, the procedure is guaranteed to produce clusters that are k-valid and m-valid in each iteration, as we now describe. When we apply Graclus, in either default or local search mode, to split a cluster into two sub-clusters, each of the created sub-clusters is parsed into core and non-core nodes (using a variant of the algorithm described for Stage 4, see Section 2.2.6). If the core node set is empty, the sub-cluster will be discarded. However, if the core is non-empty then the core node set is by definition k-valid, and is then evaluated further. Each core node set is divided into its connected components, and each connected component that has positive modularity is passed to the next iteration. Any cluster that does not end up producing a sub-cluster that is passed to the next iteration is added to the bin B as in Recursive Graclus and is then deleted from the network. Iterative Graclus stops when one of two conditions occurs: the number of allowed iterations has been completed, or all the nodes have been deleted from the network. By design, the output of Iterative Graclus is a set of clusters each of which is k-valid and m-valid (thus, every cluster is connected, has positive modularity, and every node is adjacent to at least k nodes in its cluster). Stage 3: Augmentation The purpose of the augmentation step is to assemble the periphery of center-periphery structures. The input to Stage 3 is a set of clusters, so that each non-singleton cluster is k-valid and m-valid. Here we allow all nodes that are not in any non-singleton cluster to be added to some cluster as long as it is adjacent to at least p core nodes in the (single) cluster to which it is added. In this study, we set p = 2 to ensure that we captured publications that are linked by co-citation or bibliographic coupling to core nodes in a community. If no such cluster exists such that a node can be added to it in a p-valid manner, the node remains unclustered. We add x to the cluster C that maximizes N C (x) \|C\| where N C (x) is the number of core node neighbors of node x in cluster C and where \|C\| denotes the number of nodes in cluster C. In other words, we add node x to the cluster where x has proportionally the most core node neighbors. As an example, suppose C 1 and C 2 are clusters of core nodes and that x is not yet added as a non-core node to any cluster. Suppose x has 5 neighbors in cluster C 1 and 10 nodes in cluster C 2 , where \|C 1 \| = 1000 and \|C 2 \| = 20. This procedure would add x as a non-core member to C 2 since 50% of the nodes in C 2 are neighbors of x while only 5/1000 = 0.5% of the nodes in C 1 are neighbors of x. Stage 4: Parsing clusterings to produce kmp-valid clusters Although Stage 2 is guaranteed to produce k-valid clusters, it does not always produce mvalid clusters. Furthermore, the impact of Stage 3 (the augmentation step) is to add nodes to clusters that can participate as non-core nodes, and it is possible for a node added to a cluster in Stage 3 to have sufficient neighbors in its cluster to qualify for core membership. Hence, the result of these three stages is a set of clusters that needs to be "parsed" in order to know which nodes are core members, which nodes are non-core members, and whether the clusters are kmp-valid (as defined in Definition 2). Parsing and modifying a single cluster Here we describe how we perform this parsing on a given cluster C (taken from a clustering C) and values for k and p. • Step 1: We label every node in the cluster C using the k-core labelling algorithm, applied only to the subnetwork defined by C. We let C denote the subset of nodes in C, each of whose labels is at least k, and we put the remaining nodes into a bin B(C). • Step 2: We compute the connected components of C , and delete the components that do not have positive modularity; the retained components thus have positive modularity, and are referred to as C-derived clusters (to indicate their derivation from the original cluster C). The clusters produced in this step will be the core node members in the final clusters we produce at the end of Step 3. • Step 3: We augment the C-derived clusters using the bin B(C) (i.e., we find noncore members to add) as follows. We examine each node x in bin B(C) to see if it has at least p neighbors in at least one C-derived cluster; if so, we select the best C-derived cluster A (using the algorithm from Stage 3) and add the node x to A nc (where "nc" refers to "non-core"). After all nodes are examined and processed, we let A = A ∪ A nc (for each C-derived cluster A) and output the set of all such clusters A as the "final clusters" derived from cluster C. Note that this process indicates the parsing of each final cluster A into core (A) and non-core (A nc ) nodes. • Step 4: We return all the final clusters derived from C. Theorem 1. For any clustering C of a network N , and any positive integers for k and p (with p < k), the output of kmp-processing is a clustering that is kmp-valid. Therefore, the output of the four-stage clustering method is kmp-valid for all networks N and values for k and p. Proof. Let C be an arbitrary clustering. Hence, some of its clusters may not be k-valid, may not be connected, and may not have positive modularity. We will prove that after the kmp-processing, all the clusters are kmp-valid. Specifically, we will prove that the parsing produced in Step 3 into core and non-core satisfies kmp-validity. First, note that by construction the clusters (referred to as C ) that are produced in Step 1 have the property that every node in these clusters is adjacent to at least k other nodes in their cluster. Hence, treating each cluster as only containing core nodes, these clusters are k-valid. In Step 2, these clusters are divided into components and the components are retained only if they have positive modularity; hence the C-derived clusters that are produced are k-valid and m-valid, under the interpretation that they contain only core nodes. In Step 3, the C-derived clusters are augmented. This augmentation step maintains connectivity, and so the final clusters are connected. It remains to establish that after parsing any final cluster into core and non-core nodes, the final cluster would be k-valid (i.e., every core node would be adjacent to at least k other core nodes), p-valid (i.e., every non-core node would be adjacent to at least p core nodes), and the core node subcluster would have positive modularity, and hence also be m-valid. Let A be some final cluster. By construction, it is formed by taking a C-derived cluster A produced in Step 2, and then augmenting it. Thus, A = A ∪ A nc , where A nc is the set of nodes that are added during the augmentation step. We will establish that applying Stage 4 kmp-parsing to this cluster A will not change its decomposition into core and non-core (i.e., A will still be the core nodes and A nc will be the non-core nodes). Note that by construction, all the nodes in A are adjacent to at least k other nodes in A. Hence, when A is kmp-parsed, the nodes that are identified as core nodes will contain all the nodes in A and then possibly some nodes from A nc . Independent of whether there are new core nodes, A will be p-valid. If there are no new core nodes, therefore, then A will be kmp-valid. Here we show that in fact no node in A nc will be labelled as core, and so there are no new core nodes. Let x ∈ A nc with A a C-derived cluster. Hence, x is drawn from bin B(C). By Step 1, the label assigned to x during Step 1 (when the nodes in cluster C were labelled) was a value L 1 that is strictly less than k. Since A is a C-derived cluster, A ⊆ C. Consider the label L 2 assigned to x by the k-core labelling of A . Since A ⊆ C, it follows that L 2 ≤ L 1 . Since L 1 < k, it follows that L 2 < k. Hence, x will not be placed in the core for A , and the final clusters output in Step 4 are therefore kmp-valid. Additional uses of kmp-parsing The Stage 4 kmp-parsing routine is also used in modified forms in other aspects of this study: • Strict kmp-parsing: This strict parsing routine evaluates each cluster in a clustering for being kmp-valid: those clusters that are kmp-valid are retained and all others are discarded. We use this strict parsing routine to evaluate Leiden. • Using kmp-parsing to extract core node clusters: We also have a variant where use the parsing to extract only the core nodes within each cluster, and then return the components of the core node sub-clusters that have positive modularity score; this variant is used within Iterative Graclus. Multidimensional Scaling Analysis We used Multidimensional Scaling analysis (MDS) to visualize clusters of marker nodes. For the distance between marker nodes x and y, we calculated the number of clusterings in which x and y were not in the same cluster. Using the matrix of pairwise distances, we produced a 2-dimensional visualization using metric MDS. See supplementary materials for additional details. Results & Discussion Properties of the citation network Exosomes are an area of investigation within the larger field of extracellular vesicles in biology (Raposo et al., 2021;Kalluri and LeBleu, 2020). This field has been exponentially expanding, as evidenced by a keyword search for "exosome" in the Dimensions bibliography yielding a count of less than 100 publications in 1990 and earlier, 11,100 publications from 1991 through 2010, and 115,300 publications from 2011-2021 (rounded). To find communities in this research area, we constructed and analyzed a large citation network that captured articles concerning exosomes and extracellular vesicles. The citation network we built consisted of 14,695,475 publications in 13 components, of which the largest component accounts for 99.998% of the network. The network was generated through amplifying a seed set of 11,156 articles (Materials and Methods). The degree distribution of the nodes in this network is typical of citation networks with a few nodes of high degree and many nodes of low degree (Figure 1). Roughly 68% of the nodes have degree at most five and the 90th, 95th, and 99th percentiles of degree counts are 6113, 9186, and 24510. The highest degree is 256,836 and corresponds to an article describing an assay for protein measurement. The nodes in this dataset were roughly distributed by year of publication as follows-1990 or earlier (1.17 million), 1991-2010 (6.05 million), and 2011-2021 (6.99 million), suggesting not only a rapid growth of exosome publications in the post-1990 period but also a substantial increase in publications linked by citation to the seed articles. . The k-core clustering algorithm was applied to the exosome citation network for multiple values of k (x-axis). The size of the k-core is shown on the ordinate (y-axis). A single component is returned in each case, with the exception of k=1 and k=2. Node coverage decreases as k increases, and is approximately 36% and 21% when k = 5 and k = 10 respectively. The single cluster at k=56 has 3,630 nodes, amounting to node coverage of 0.02%. At k=57 or greater, the entire network dissolves, thus the degeneracy of the network is 56. Clusters shown have positive modularity (mod +ve) when k = 1 or k >= 40 and are colored teal. By definition, the connected components of the network are the 1-cores so 13 clusters are returned for k = 1. For k = 2, only two clusters are returned. Clusters of size 100 or less (0.001% of the network) are not included in this plot and pertain only to k=1 or k=2. Inset: k-core sizes sampled at intervals of 5 are displayed using a linear scale. Results of clustering methods We now present results using different clustering methods, including the k-core clustering method, Iterative k-core clustering, our four-stage pipeline, and the Leiden algorithm. As noted earlier, we were explicitly interested in discovering citation-dense regions with center-periphery structure that reflected cohesiveness and collaboration (Giatsidis et al., 2011;Breiger, 2014). In this case, collaboration refers to the recognition of prior work by others in the community through citations. We were not interested in communities that consisted of a single heavily cited article or a single article that cited many references (Chandrasekharan et al., 2021, p. 193). Results for the Leiden algorithm We first used the Leiden algorithm, which guarantees connected communities (Traag et al., 2019), as a benchmark for community finding. We consider Leiden as a reference method in the scientometrics community, and have previously used it (Chandrasekharan et al., 2021) in combination with Markov Clustering (Van Dongen, 2008) to detect communities in the immunology and ecology literature. In the present study, we used the Leiden software (Traag, 2021) with the Constant Potts Model as quality function, and with varying resolution factors to cluster the exosome citation network. The resolution factor is designed to modify the clustering, as it determines the required minimum density within communities. For this citation network, at most of the resolution values that were tested, the Leiden algorithm generated a relatively large number of small clusters (Table 1). At the smallest resolution factor value we examined (0.05), Leiden produced 488,285 non-singleton clusters and 6,323,695 singletons and the largest cluster comprised 960 nodes. Node coverage, defined as the ratio of nodes in non-singleton clusters to the total number of nodes in the network was 57% with resolution factor set to 0.05. As we increased the resolution value, the number of singleton clusters increased, the sizes of non-singleton clusters decreased, and we observed a progressive drop in node coverage (down to 16% at resolution factor 0.25 and to 9% for resolution factor 0.50). We also screened the clusters generated by Leiden for kmp-validity (Materials and Methods, Definition 2) at k=5 or 10 and p = 2, the values that we use in when applying our pipeline. For the resolution value of 0.05, the node coverage is 3.16% and 2.54% when k is 5 or 10 respectively (Supplementary Materials, Tables 3 and 4), a large drop from 57% for the Leiden clustering before restriction to kmp-valid clusters. For larger resolution values, restriction to kmp-valid clusters results in even smaller node coverage. In addition, again using resolution value 0.05, after restriction to kmp-valid clusters, the number of clusters is much smaller: only 4,076 for k = 5 and 2,320 for k = 10, a large drop from 488,285. Thus, restriction to kmp-valid clusters greatly reduces the node coverage and number of clusters compared to Leiden before restriction, and this holds for all resolution values. Moreover, this analysis shows that only a small fraction of the clusters produced by Leiden at any resolution value are kmp-valid (e.g., less than 1% for resolution value 0.05). In summary, while Leiden was able to efficiently cluster our network into a large number of small communities that represented between 8% and 57% of the network depending on the resolution factor employed, only a small fraction of the communities exhibited kmpvalidity. This is not surprising since the Leiden algorithm and optimization criterion was not designed to produce kmp-valid communities, and these trends indicated that Leiden had limited utility for our purposes. Results for k-core clustering To identify citation-dense regions of the network, we ran the k-core clustering method on our network for different values of k ≥ 1 (Figure 2). The largest value for k for which a cluster is returned is 56, indicating that the degeneracy of the network is 56. The network has no isolated vertices and so k = 0 and k = 1 produce the same output, which is 13 clusters, each corresponding to a connected component in the network. When k = 2, two clusters were returned. For each 3 < k ≤ 56, only a single cluster was returned, hence only k = 0, 1, 2 produced more than one cluster, and in each of these cases, a single cluster dominated in size. An interesting trend in this analysis is how k impacts the modularity scores of the clusters. By definition, the components in the graph all have positive modularity, and so when k = 1 the clusters all have positive modularity. For larger values of k, the modularity scores do not become positive until k = 40, and then all subsequent values of k produce clusters with positive modularity scores. Cluster size decreased monotonically as k increased to 56, however we observe a pattern of relatively stable core sizes at lower values of k followed by a more rapid decrease as k increases above 40. Leskovec and Horvitz (2008) reported similar findings about changes in core sizes on a much larger network of instant messaging data that consisted of 180,000,000 nodes. These authors suggest that the rapid decrease in core size occurs once nodes on the fringe of the network are removed. On our network, this more rapid decrease of cores sizes also coincides with the appearance of positive modularity of clusters; modularity was not reported in Leskovec and Horvitz (2008). Thus, we are mainly interested in the k-cores for large values of k: they have positive modularity, small changes in k result in large changes to their sizes, and they have been identified in the prior literature as what is left after the fringe of the network is removed. While the simple k-core clustering method identifies citation-dense areas in the network suggesting cohesiveness and collaboration, it has two limitations: it does not ensure positive modularity for every cluster it produces, and, on our data, for every k ≥ 3 it produced only a single large cluster. We note that for k = 0, 1, 2, it produced two or more clusters, one of which was very large. These limitations impact the ability to find multiple communities of interest in the exosome literature, especially considering the possibility that some of the communities of interest could be contained within larger clusters with non-positive modularity. Results for Iterative k-core (IKC) We designed Iterative k-core (IKC) to improve on the k-core clustering algorithm. The input to IKC is the network N and the parameter k. The first cluster that is found in the network is the L-core where L is the largest label assigned to any node. If L ≥ k, then the L-core is produced as a cluster and removed from the network, and otherwise the algorithm stops. If the algorithm has not stopped, it is run recursively on the reduced network. Therefore, IKC(k) will contain all the clusters in IKC(k ) for k ≤ k . We explored IKC varying k between 5 and 50, and examined the distribution of cluster sizes generated. We also recorded the minimum degree in each cluster. Since IKC clusters satisfy k-validity, the nodes in the clusters are all "core" members, and so this minimum degree is also the Minimum Core Degree (MCD) of the cluster. As expected, increasing k results in decreases in cluster size and increases in the MCD value ( Figure 3). In order to include as much of the network as is reasonable and still have sufficient density to define community structure, we selected k = 5 and k = 10 for IKC. While IKC discovered km-valid communities that were trivially p-valid, it too has limitations. It identifies only core nodes with modest node coverage of 7.38% and 4.22% at k=5 and k = 10, respectively. It also generates some large clusters with lower MCD values that leave open the question of whether denser kmp-valid communities exist within them. Results for the Four-Stage Clustering Pipeline Citation Network Augmentation + kmp parsing Figure 4: Our four-stage clustering pipeline takes as input a citation network and produces clusters based on values selected for parameters k and p. Boxes adjacent to edges indicate stages in the pipeline; boxes with blue borders indicate tests that are performed to determine which clusters are passed to the next stage. In Stage 1, it runs the Iterative k-core (IKC) algorithm for the selected value of k; clusters that are km-valid are then passed to the next stage. Stage 2 (optional) breaks the clusters from the first stage into smaller clusters, using either recursive Graclus or Iterative Graclus (and in each case, using either the default version or a heuristic search version). Clusters that pass the required validity check (k-validity for Recursive Graclus and km-validity for Iterative Graclus) are then passed to the next stage. These clusters are then enlarged with additional nodes in Stage 3, the "Augmentation" step. All clusters produced are p-valid at this point, and are passed to Stage 4, kmp-parsing, which produces a set of kmp-valid clusters, each of which is parsed into their core subcluster and non-core subcluster. To address the limitations of IKC, we designed a four-stage pipeline (Figure 4). This four-stage pipeline is guaranteed to produce a kmp-valid clustering of the network, for user-provided values of k and p. In Stage 1 we use IKC with k = 5 and k = 10. Stage 2 breaks the clusters found in Stage 1 into smaller clusters; for this stage we apply either Recursive Graclus or Iterative Graclus, and for each of these we use Graclus either in default mode or in local search mode. Thus, for each setting of k we have four versions of the four-stage pipeline: Iterative and Recursive Graclus run in either default or local search mode. In Stage 3, the clusters produced by Stage 2 are augmented by the addition of nodes that satisfy p-validity for p = 2. In Stage 4, we parse the output of Stage 3 and retain only those clusters that are kmp-valid. Stage 2 Stage 3 Table 2: Cluster statistics for 12 variants of Four-Stage clustering. All results include Stages 1 and 4, but some pipelines do not use Stages 2 or 3; all clusterings are kmp-valid. Top: results for k = 5, Bottom: results for k = 10. Stage 2 is performed either using Iterative Graclus (IG) or Recursive Graclus (RG), which are each run with either 0 or 2000 local search iterations. Node coverage refers to the percentage of network nodes contained in non-singleton clusters and singletons refers to the number of nodes in singleton clusters. All other statistics refer to non-singleton clusters, with the last three columns refering to the sizes of non-singleton clusters. Specific noteworthy trends include: (a) all clusterings that use Stage 3 have node coverage above 18%, (b) all clusterings have at least one very large cluster, (c) Stage 2 choice impacts maximum cluster size, and (d) setting k = 5 produces more clusters with a smaller median cluster size than setting k = 10. We show results for the different versions of this four-stage pipeline in Table 2, where the rows correspond to different ways of setting k and running Stages 2 and 3. Using IKC alone without Stages 2 and 3 resulted in 276 and 119 non-singleton clusters and had total node coverage of 7.38% and 4.22%, with maximum cluster sizes of 345,139 and 213,670, for k = 5 and k = 10, respectively. Skipping Stage 2 (breaking down clusters) but adding Stage 3 (augmentation) greatly increases the node coverage to at least 33% for both settings of k but also increases the maximum cluster size to 856,623 and 964,503 for k = 5 and k = 10, respectively. This approach also has large median cluster sizes, especially for k = 10 where the median is 1638. Thus, using Stage 3 (augmentation) but not also Stage 2 produces high node coverage and large cluster sizes. Using all four stages, and hence using Stage 2, reduces the node coverage to values that range from 18.51% to 27.61% and also reduces the median and maximum cluster sizes, but the choice of how Stage 2 is run has a significant impact. Node coverage is higher when using Recursive Graclus rather than Iterative Graclus. Using Recursive Graclus in default mode rather than local search mode tends to produce a smaller number of non-singleton clusters that are also somewhat larger; for example, when k = 10, default usage of Graclus that doesn't employ the local search strategy produces a median cluster size of 1,014 as opposed to 761 when using 2000 local search iterations. The choice between k = 5 and k = 10 also impacts results, with k = 10 producing a much smaller number of non-singleton clusters that are substantially larger than the results for k = 5. For example, using default Recursive Graclus at k = 10 produces 359 non-singleton clusters with median size 1014 while the same setting for k = 5 produces 2,261 non-singleton clusters with median size 145. Comparison between different clustering methods We now compare the different clustering outputs with respect to node coverage, number of non-singleton clusters, and the distribution of non-singleton cluster sizes. As our discussion of the different variants of the four-stage pipeline revealed, how Stages 2 and 3 are run and the value for k impact these statistics. If node coverage is the most important criterion, then skipping Stage 2 but using Stage 3 is recommended. However, since these approaches produce very large clusters, including Stage 2 is more likely to be desirable. Among the techniques that use Stage 2, Iterative Graclus tends to produce smaller clusters, but Recursive Graclus produces higher node coverage; the choice between these should be made based on the specific question that is being addressed. Similarly, how k is set should depend on the features of the citation network, and picking larger values of k may be suitable under some conditions. A comparison to Leiden is also helpful: before restricting to the kmp-valid clusters, Leiden has very high node coverage (57%) for resolution value 0.05, but after restricting to kmp-valid clusters, node coverage drops to 3.16% and 2.54% when k is 5 or 10, respectively. In contrast, the four-stage pipeline has node coverage that varies from 18.51% to 27.61%, depending on k and how Stage 2 is performed (Table 2). Thus, a potential user of this approach is presented with options that can be used to address contextual needs. For example, after considering features of the source data and the purpose of community finding, a user may choose Leiden, IKC, IKC with augmentation, or the complete pipeline with its kmp-parsing requirements. Marker Node Analysis The network we constructed for this study has more than 14 million nodes. We assumed that some of the communities discovered would represent areas of investigation peripheral to exosomes and extracellular vesicles. Consequently, we used a set of 1,218 independently selected articles from the extracellular vesicle literature, all of which are present in the network, as marker nodes (Materials and Methods) and used them to identify clusters of interest. Any community containing at least one marker node was considered relevant; however, we were particularly interested in communities with high numbers of marker nodes. For IKC clustering at either k=5 or k=10 (Figure 2), two clusters contained 256 and 227 marker nodes respectively, and together accounted for approximately 40% of the 1,218 markers. The first of these clusters, with 256 marker nodes, was the 56-core of the network, and comprised 3,630 nodes. The second, with 227 marker nodes, exhibited an MCD (minimum core degree) value of 12 and consisted of 213,670 nodes. To visualize the distribution of the entire set of marker nodes, we used a multidimensional scaling approach using the frequency of co-occurrence in clusters across 12 different clustering methods as the measure of similarity between publications. Each of these two sets of marker nodes are found in dense and clearly defined clusters after multidimensional scaling ( Figure 5). These two clusters were similar in having a large number of marker nodes but were otherwise different from each other with respect to MCD and size; therefore, we used the two sets of 256 and 227 markers as examples for further study. A second criterion considered was robustness to clustering method, which we measure by the frequency with which marker nodes were found co-located in the same cluster across the 12 clusterings we studied. We began this evaluation by first determining which of these nodes are always placed in non-singleton clusters for all 12 clustering methods. We found that 27 of 256 and 35 of 227 marker nodes respectively were always placed in non-singleton clusters, and studied these two sets further. The first set (A) consisting of 27 marker nodes contained 17 articles and 8 reviews that were published between 2006 and 2019 with 23 of these published in 2010 or later. Based upon inspection of titles and journal, the contents of articles in set A spanned basic cell biology, the role of exosomes in cancer, and exosome isolation methods with some variation in terms of being descriptive or mechanistic. The second set (B) of 35 markers contained 26 articles and 9 reviews published between 2013 and 2021, largely focused on a basic and translational studies of exosomes in nervous tissue but also including a few articles on exosomes in pregnancy and exosome isolation methods. We then examined the publications in sets A and B for co-occurrence in the same cluster across all 12 clusterings. We found that 8 articles from set A were always found in the same cluster and 4 articles from set B were always found in the same cluster. While the numbers of markers in these sets are small, they serve to identify a larger community, which can then be characterized further by other techniques, such as detailed scholarly examination or textual content analysis. We identified the smallest cluster across the 12 clusterings that contained the 8 articles from A; the selected cluster (Cluster 1) consisted of 73 articles and was focused on extracellular vesicles in cancer. We performed a similar analysis for the 4 articles from B, and found a cluster (Cluster 2) of 145 articles that was focused on extracellular vesicles in the nervous system. Thus, both these clusters were relatively homogenous with respect to article content, one focused on extracellular vesicles in cancer and the other on extracellular vesicles in the nervous system. Our use of a single label for each cluster should be considered a subjective approximation; an alternate view is that Cluster 2 also includes articles on cancer and has some focus on astrocytes. Both clusters were derived from the IKC(5)-Iterative Graclus branch of the pipeline. We then extracted the authors of articles in these two clusters (Table 3). Cluster 1 involved 356 authors of which 9 were authors of at least 5 articles in the cluster and one person was an author of 17 articles in the cluster. However, 301 authors (84.6%) had contributed to only one article in the cluster. Cluster 2 involved 742 authors of which 4 were authors of at least 5 articles in the cluster. One author had contributed 7 articles and 650 authors (87.6%) had contributed only one article each. Interestingly, the two publication clusters share 14 authors. Thus, the two author communities defined by two disjoint publication clusters overlap. These trends are strikingly similar to the observations of Price and Beaver (1966) in that the authors segregate into a small number with large numbers of papers in the cluster and many with only one paper in the cluster. Although we only examined two clusters, both exhibited the center-periphery structure described in Price and Beaver (1966); our sample is too small to draw conclusions beyond suggesting that the trends observed may be true for other clusters, and this should be evaluated. We also found discrete co-author groups within these clusters ( Figure 6). Cluster 1 featured 4 non-overlapping co-author groups where authors were linked to each other if they had co-authored at least two articles in the cluster. Cluster 2 featured 17 such discrete coauthor groups suggesting, despite its larger size, that influence within the group was more distributed considering the larger number of co-author groups and the smaller number of articles written by individual authors. These examples are provided to illustrate the potential utility of the pipeline and the use of marker nodes. Figure 6: Co-author clusters in two communities identified using marker nodes. Two clusters were selected for analysis because they contained the greatest number of marker nodes that were co-located across all 12 clustering methods, and were the smallest of such clusters (see text). Discrete co-author groups are found in both clusters when inclusion in a co-author groups requires at least two instances of intra-cluster co-authorship between two authors. (a) Cluster 1 consists of 73 articles contributed by 356 authors. This cluster contains 4 non-overlapping co-author groups, with 5, 8, 11, and 28 authors. (b) Cluster 2 consists of 145 articles authored by 356 authors. This cluster contains 17 nonoverlapping groups, with four groups of 2 authors, three groups of 3 authors, three groups of 4 authors, two groups of 5 authors, four groups of 7 authors, and one group of 18 authors. Cluster 1 73 47 16 356 9 17 Cluster 2 145 129 31 742 4 7 Table 3: Statistics on the two selected small clusters. Cluster 1 and Cluster 2 were selected based on marker nodes: the greatest number of marker nodes that were found co-located in the same cluster across all 12 clustering methods were used to define their common clusters, and the smallest of the common clusters for each set was returned. Both clusters are kmp-valid for k = 5 and p = 2. Articles refers to the total number of nodes in the cluster, core nodes refers to the number of these nodes that are core, and markers refers to the number of nodes that are markers. Authors denotes the total number of authors of articles in the cluster, Auth 5 refers to the number of authors that have at least 5 publications in the cluster, and Max Pubs refers to the largest number of publications in the cluster by any single author. Articles Core Nodes Markers Authors Auth 5 Max Pubs Conclusions Based on historical studies of research communities, we posed corresponding properties for the graphical structure of communities in networks. We developed an analytic pipeline to ask whether communities of publications with center-periphery substructure exist in citation networks. In designing a four-stage pipeline to find communities of this form, we were implicitly asking whether the information encoded in the graphical structure of communities can be used to make inferences on the social structure of these communities, for example, discrete co-author groups. We examined these questions using a citation network representative of exosomes and extracellular vesicles, a field that has rapidly expanded in recent years. In this citation network, our pipeline found many publication communities that exhibit center-periphery structure. This finding supports our hypothesis that communities of this type exist within the extracellular vesicles research community, and shows that the pipeline we used can find such communities. Whether such communities exist in other citation networks and whether our pipeline is successful at finding such communities are important questions that future work should address. Our pipeline is designed to enable investigators to interrogate their data with different options for each stage and different settings for k and p. As we observed in our study, changes to the settings for the parameters k and p as well as how each stage was performed produced clusterings that differ from each other in terms of the the node coverage as well as the number and sizes of non-singleton clusters, effectively providing different views of the network. Thus, the specific question of interest and the properties of the citation network are important in choosing how to set these parameters. Alternatively, the pipeline can be used to generate many different clusterings, and the investigator may assess community structure through an integrative analysis that does not depend on a single clustering method. We also saw significant differences between clusterings produced by our pipeline and those produced by the Leiden software. While there is some overlap in the range of cluster sizes generated, our pipeline tends to generate much larger clusters than the Leiden software, and all of our pipeline clusterings produced at least one very large cluster with more than 19,000 nodes and, in the case of Recursive Graclus, the largest cluster contained more than 500,000 nodes. Given our focus on the small clusters produced in our various pipelines, we did not explore large publication clusters. It is not clear to us what insights can be gained from clusters that have tens of thousands of nodes. We speculate that they may reflect the many connections in a rapidly expanding field, and we also consider them as tempting targets for future versions of Stage 2, which seeks to break up the large clusters. Further work is clearly needed. This study suggests several directions for future work. For method development, our four-stage pipeline is designed to enable substitutions to how each stage is performed; as noted above, breaking up large clusters might be more successfully executed using new approaches rather than either recursive or iterative Graclus. We also note that all the clustering methods we developed produce disjoint clusters, yet publications may be expected to be members of more than one research community. Some clustering methods have been developed that can produce overlapping clusters (Rossetti, 2020), and exploring this approach in the context of large citation networks is likely to provide additional insights. Methods that can combine information from multiple clustering methods could lead to better insight into community structure, and so principled development of ensemble methods (a standard approach in machine learning) is another direction for future research. One of the most intriguing directions for future research is the life cycle of these research communities, both in terms of how ideas and questions being focused on by the research community as a whole change over time and how authors move between different communities over time. This is challenging to study because the network evolves through growth each year and community finding in dynamic graphs is a promising direction (Rossetti, 2020). Additional clusters could be examined for in-depth examination based, for example, on specific authors, articles of particular importance, funding sources, or clusters identified by marker node co-located in many but not all 12 clusterings. Other insights could be obtained by examining the relationship between publication communities (e.g., citations between communities) in a single clustering or comparisons of communities obtained using different clustering methods; such investigations would help elucidate how the research ideas and communities relate to each other, and the extent to which these communities are hierarchically organized. For extracellular vesicles, insight into community life cycles can be obtained by studying, over time, communities containing key studies such as the ones on transferrin recycling (Harding et al., 1983;Pan and Johnstone, 1983), the observation that B-lymphocytes secrete antigen-presenting vesicles (Raposo et al., 1996), a report of exosome mediated trans-fer of mRNA and microRNA (Ratajczak et al., 2006;Valadi et al., 2007), and the biological effects of transferring exosomes between lean and obese mice (Ying et al., 2017). Our analysis was based on a single set of marker nodes; extending the set of marker nodes and annotating each marker node with respect to content would allow finer-grained evaluation. Exploration of author communities associated to these publication communities would shed additional insights into the social structure of the research community, potentially identifying authors with high influence within a particular emerging research area, and others that are highly influential across several areas. We close with comments about the high-level approach we took to understanding community structure. Our approach relied entirely on the graphical structure of the citation network. This restriction was used in order to provide a scalable approach that did not rely on any other information or expert knowledge; however, by design this limits which communities can be detected (McCain, 1986). Textual analysis, relationships between authors based on institutions, other social interactions such as conference presentations and sources of funding could be used to used to supplement citation data and would likely lead to a different set of publication or author communities with potentially different properties. Understanding author role is also important, and again our reliance on citations to identify influential researchers is biased towards well-cited and well-funded authors. Perhaps one of the benefits, therefore, of our approach to community detection is that we can use it to find small and thematically-focused publication communities and hence identify those authors who are influential within these small communities. Nevertheless, we propose that while scalable methods for community detection may generally tend to rely on purely graph-theoretic properties, mixed method approaches support more a more nuanced understanding of social structures and dynamics within the scientific enterprise. Data Availability Access to the bibliographic data analyzed in this study requires access from Digital Science. Code generated for this study is freely available from our Github site (Park et al., 2021). Figure 1 :Figure 2 : 12Intra-network degree distribution of the 14,695,475 nodes in the exosome citation network. x-axis: log degree, y-axis: log node count. Cluster sizes using the simple k-core clustering algorithm. The input exosome citation network (Materials and Methods) consists of 13 components summing to 14,695,475 nodes. Of these, a single component accounts for 14,695,226 articles (99.998% of the network) Figure 3 : 3Empirical statistics of the Iterative k-core (IKC) clustering methods (varying k) on the Exosome network. The y-axis shows the cluster sizes (logarithmic scale) and the x-axis shows the Minimum Core Degree (MCD) values, where the MCD of a cluster is the minimum degree of any node in the cluster. By design, IKC(k) contains all the clusters of IKC(k ) if k ≤ k ; thus, the panels showing IKC at lower values of k contain greater numbers of clusters. Figure 5 : 5Multidimensional scaling (MDS) of 1,218 marker nodes based on the frequency with which they are placed in the same cluster in 12 clustering outputs. The circled beige cluster (bottom right) corresponds to the 256 markers found using either IKC(5) or IKC(10) in a single cluster of minimum cluster degree (MCD) of 56, while the circled blue cluster (top) corresponds to the 227 markers found in a different cluster with an MCD value of 12. AcknowledgmentsWe thank Valerie King from the University of Victoria for directing us to the k-core literature. We thank Phil Stahl from Washington University in St. Louis for helpful discussions and for drawing our attention to recent reviews of the extracellular vesicle literature. We thank Digital Science, Google, the Grainger Foundation, and the Thomas and Stacey Siebel Foundation.ORCID IDs• Eleanor Wedell 0000-0002-7911-9156• Minhyuk Park 0000-0002-8676-7565• Dmitriy Korobskiy 0000-0002-7909-0218• Tandy Warnow 0000-0001-7717-3514• George Chacko 0000-0002-2127-1892Competing InterestsThe authors have no competing interests. Dimensions data were made available by Digital Science through the free data access for scientometrics research projects program. Digital Science personnel did not participate in conceptualization, experimental design, review of results, or conclusions presented. 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[ "Glottal Source Estimation using an Automatic Chirp Decomposition", "Glottal Source Estimation using an Automatic Chirp Decomposition" ]	[ "Thomas Drugman \nTCTS Lab\nFaculté Polytechnique de Mons\nBelgium\n", "Baris Bozkurt \nDepartment of Electrical & Electronics Engineering\nIzmir Institute of Technology\nTurkey\n", "Thierry Dutoit \nTCTS Lab\nFaculté Polytechnique de Mons\nBelgium\n" ]	[ "TCTS Lab\nFaculté Polytechnique de Mons\nBelgium", "Department of Electrical & Electronics Engineering\nIzmir Institute of Technology\nTurkey", "TCTS Lab\nFaculté Polytechnique de Mons\nBelgium" ]	[]	In a previous work, we showed that the glottal source can be estimated from speech signals by computing the Zeros of the Z-Transform (ZZT). Decomposition was achieved by separating the roots inside (causal contribution) and outside (anticausal contribution) the unit circle. In order to guarantee a correct deconvolution, time alignment on the Glottal Closure Instants (GCIs) was shown to be essential. This paper extends the formalism of ZZT by evaluating the Z-transform on a contour possibly different from the unit circle. A method is proposed for determining automatically this contour by inspecting the root distribution. The derived Zeros of the Chirp Z-Transform (ZCZT)-based technique turns out to be much more robust to GCI location errors.	10.1007/978-3-642-11509-7_5	null	933,093	2005.07897	31fddb2cce1e0e4453f1aa7a5673175ee0dddeef	Glottal Source Estimation using an Automatic Chirp Decomposition Thomas Drugman TCTS Lab Faculté Polytechnique de Mons Belgium Baris Bozkurt Department of Electrical & Electronics Engineering Izmir Institute of Technology Turkey Thierry Dutoit TCTS Lab Faculté Polytechnique de Mons Belgium Glottal Source Estimation using an Automatic Chirp Decomposition In a previous work, we showed that the glottal source can be estimated from speech signals by computing the Zeros of the Z-Transform (ZZT). Decomposition was achieved by separating the roots inside (causal contribution) and outside (anticausal contribution) the unit circle. In order to guarantee a correct deconvolution, time alignment on the Glottal Closure Instants (GCIs) was shown to be essential. This paper extends the formalism of ZZT by evaluating the Z-transform on a contour possibly different from the unit circle. A method is proposed for determining automatically this contour by inspecting the root distribution. The derived Zeros of the Chirp Z-Transform (ZCZT)-based technique turns out to be much more robust to GCI location errors. Introduction The deconvolution of speech into its vocal tract and glottis contributions is an important topic in speech processing. Explicitly isolating both components allows to model them independently. While techniques for modeling the vocal tract are rather well-established, it is not the case for the glottal source representation. However the characterization of this latter has been shown to be advantageous in speaker recognition [1], speech disorder analysis [2], speech recognition [3] or speech synthesis [4]. These reasons justify the need of developing algorithms able to robustly and reliably estimate and parametrize the glottal signal. Some works addressed the estimation of the glottal contribution directly from speech waveforms. Most approaches rely on a first parametric modeling of the vocal tract and then remove it by inverse filtering so as to obtain the glottal signal estimation. In [5], the use of the Discrete All-Pole (DAP) model is proposed. The Iterative Adaptive Inverse Filtering technique (IAIF) described in [6] isolates the source signal by iteratively estimating both vocal tract and source parts. In [7], the vocal tract is estimated by Linear Prediction (LP) analysis on the closed phase. As an extension, the Multicycle closed-phase LPC (MCLPC) method [8] refines its estimation on several larynx cycles. In a fundamentally different point of view, we proposed in [9] a non-parametric technique based on the Zeros of the Z-Transform (ZZT). ZZT basis relies on the observation that speech is a mixed-phase signal [10] where the anticausal component corresponds to the vocal folds open phase, and where the causal component comprises both the glottis closure and the vocal tract contributions. Basically ZZT isolates the glottal open phase contribution from the speech signal, by separating its causal and anticausal components. In [11], a comparative evaluation between LPC and ZZT-based decompositions is led, giving a significant advantage for the second technique. This paper proposes an extension to the traditional ZZT-based decomposition technique. The new method aims at separating both causal and anticausal contributions by computing the Zeros of a Chirp Z-Transform (ZCZT). More precisely, the Z-transform is here evaluated on a contour possibly different from the unit circle. As a result, we will see that the estimation is much less sensitive to the Glottal Closure Instant (GCI) detection errors. In addition, a way to automatically determine an optimal contour is also proposed. The paper is structured as follows. Section 2 reminds the principle of the ZZT-based decomposition of speech. Its extension making use of a chirp analysis is proposed and discussed in Section 3. In Section 4, a comparative evaluation of both approaches is led on both synthetic and real speech signals. Finally we conclude in Section 5. ZZT-based Decomposition of Speech For a series of N samples (x(0), x(1), ...x(N − 1)) taken from a discrete signal x(n), the ZZT representation is defined as the set of roots (zeros) (Z 1 , Z 2 , ...Z N −1 ) of the corresponding Z-Transform X(z): X(z) = N −1 n=0 x(n)z −n = x(0)z −N +1 N −1 m=1 (z − Z m )(1) The spectrum of the glottal source open phase is then computed from zeros outside the unit circle (anticausal component) while zeros inside it give the vocal tract transmittance modulated by the source return phase spectrum (causal component). To obtain such a separation, the effects of the windowing are known to play a crucial role [12]. In particular, we have shown that a Blackman window centered on the Glottal Closure Instant (GCI) and whose length is twice the pitch period is appropriate in order to achieve a good decomposition. Chirp Decomposition of Speech The Chirp Z-Transform (CZT), as introduced by Rabiner et al [13] in 1969, allows the evaluation of the Z-transform on a spiral contour in the Z-plane. Its first application aimed at separating too close formants by reducing their bandwidth. Nowadays CZT reaches several fields of Signal Processing such as time interpolation, homomorphic filtering, pole enhancement, narrow-band analysis,... As previously mentioned, the ZZT-based decomposition is strongly dependent on the applied windowing. This sensitivity may be explained by the fact that ZZT implicitly conveys phase information, for which time alignment is known to be crucial [14]. In that article, it is observed that the window shape and onset may lead to zeros whose topology can be detrimental for accurate pulse estimation. The subject of this work is precisely to handle with these zeros close to the unit circle, such that the ZZT-based technique correctly separates the causal (i.e minimum-phase) and anticausal (i.e maximum-phase) components. For this, we evaluate the CZT on a circle whose radius R is chosen so as to split the root distribution into two well-separated groups. More precisely, it is observed that the significant impulse present in the excitation at the GCI results in a gap in the root distribution. When analysis is exactly GCI-synchronous, the unit circle perfectly separates causal and anticausal roots. On the opposite, when the window moves off from the GCI, the root distribution is transformed. Such a decomposition is then not guaranteed for the unit circle and another boundary is generally required. Figure 1 gives an example of root distribution for a natural voiced speech frame for which an error of 0.6 ms is made on the real GCI position. It is clearly seen that using the traditional ZZT-based decomposition (R = 1) for this frame will lead to erroneous results. In contrast, it is possible to find an optimal radius leading to a correct separation. In order to automatically determine such a radius, let us have the following thought process. We know that ideally the analysis should be GCI-synchronous. When this is not the case, the chirp analysis tends to modify the window such that its center coincides with the nearest GCI (to ensure a reliable phase information). Indeed, evaluating the chirp Z-transform of a signal x(t) on a circle of radius R is equivalent to evaluating the Z-transform of x(t) · exp(log(1/R) · t) on the unit circle. It can be demonstrated that for a Blackman window w(t) of length L starting in t = 0: w(t) = 0.42 − 0.5 · cos( 2πt L ) + 0.08 · cos( 4πt L ),(2) the radius R necessary to modify its shape so that its new maximum lies in position t * (< L) is expressed as: R = exp[ 2π L · 41 tan 2 ( πt * L ) + 9 25 tan 3 ( πt * L ) + 9 tan( πt * L )) ].(3) In particular, we verify that R = 1 is optimal when the window is GCIcentered (t * = L 2 ) and, since we are working with two-period long windows, the optimal radius does not exceed exp(± 50π 17L ) in the worst cases (the nearest GCI is then positioned in t * = L 4 or t * = 3L 4 ). As a means for automatically determining the radius allowing an efficient separation, the sorted root moduli are inspected and the greatest discontinuity in the interval [exp(− 50π 17L ), exp( 50π 17L )] is detected. Radius R is then chosen as the middle of this discontinuity, and is assumed to optimally split the roots into minimum and maximum-phase contributions. Experimental Results This Section gives a comparative evaluation of the following methods: the traditional ZZT-based technique: R = 1, the proposed ZCZT-based technique: R is computed as explained at the end of Section 3 (see Fig. 2), the ideal ZCZT-based technique: R is computed from Equation 3 where the real GCI location t * is known. This can be seen as the ultimate performance one can expect from the ZCZT-based technique. Among others, it is emphasized how the proposed technique is advantageous in case of GCI location errors. Results on Synthetic Speech Objectively and quantitatively assessing a method of glottal signal estimation requires working with synthetic signals, since the real source is not available for real speech signals. In this work, synthetic speech signals are generated for different test conditions, by passing a train of Liljencrants-Fant waves [15] through an all-pole filter. This latter is obtained by LPC analysis on real sustained vowel uttered by a male speaker. In order to cover as much as possible the diversity one can find in real speech, parameters are varied over their whole range. Table 1 summarizes the experimental setup. Note that since the mean pitch during the utterances for which the LP coefficients were extracted was about 100 Hz, it reasonable to consider that the fundamental frequency should not exceed 60 and 180 Hz in continuous speech. To evaluate the performance of our methods, two objective measures are used: the determination rate on the glottal formant frequency F g : As one of the main feature of the glottal source, the glottal formant [10] should be preserved after estimation. The determination rate consists of the percentage of frames for which the relative error made on F g is lower than 20%. the spectral distortion: This measure quantifies in the frequency-domain the distance between the reference and estimated glottal waves (here noted x and y by simplification), expressed as: Figure 3 compares the results obtained for the three methods according to their sensitivity to the GCI location. The proposed ZCZT-based technique is clearly seen as an enhancement of the traditional ZZT approach when an error on the exact GCI position is made. Electroglottograph (EGG) informative about the GCI positions. Both next panels compare respectively the detected glottal formant frequency F g and the radius for the three techniques. In the middle panel, deviations from the constant F g can be considered as errors since F g is expected to be almost constant during three pitch periods. It may be noticed that the traditional ZZT-based method degrades if analysis is not achieved in the GCI close vicinity. Contrarily, the proposed ZCZT-based technique gives a reliable estimation of the glottal source on a large segment around the GCI. Besides the obtained performance is comparable to what is carried out by the ideal ZCZT. In Figure 5 the glottal source estimated by the traditional ZZT and the proposed ZCZT-based method are displayed for four different positions of the window (for the vowel /a/ from the same file). It can be observed that the proposed technique (solid line) gives a reliable estimation of the glottal flow wherever the window is located. On the contrary the sensivity of the traditional approach can be clearly noticed since its glottal source estimation turns out to be irrelevant when the analysis is not performed in a GCI-synchronous way. SD(x, y) = π −π (20 log 10 \| X(ω) Y (ω) \|) 2 dω 2π(4) Results on Real Speech Conclusion This paper proposed an extension of the ZZT-based technique we proposed in [9]. The enhancement consists in evaluating the Z-transform on a contour possibly different from the unit circle. For this we considered, in the Z-plane, circles whose radius is automatically determined by detecting a discontinuity in the root distribution. It is expected that such circles lead to a better separation of both causal and anticausal contributions. Results obtained on synthetic and real speech signals report an advantage for the proposed ZCZT-based technique, mainly when GCIs are not accurately localized. As future work, we plan to characterize the glottal source based on the proposed framework. Fig. 1 . 1Example of root distribution for a natural speech frame. Left panel : representation in the Z-plane, Right panel : representation in polar coordinates. The chirp circle (solid line) allows a correct decomposition, contrarily to unit circle (dotted line). Fig. 2 . 2Determination of radius R (dashed line) for ZCZT computation by detecting, within the bounds exp(± 50π 17L ) (dotted lines), a discontinuity (indicated by rectangles) in the sorted root moduli (solid line). Figure 4 4displays an example of decomposition on a real voiced speech segment (vowel /e/ from BrianLou4.wav of the Voqual03 database, F s = 16kHz). The top panel exhibits the speech waveform together with the synchronized (compensation of the delay between the laryngograph and the microphone) differenced Fig. 3 . 3Comparison of the traditional ZZT (dashdotted line), proposed ZCZT (solid line) and ideal ZCZT (dotted line) based methods on synthetic signals according to their sensitivity to an error on the GCI location. Left panel: Influence on the determination rate on the glottal formant frequency. Right panel: Influence on the spectral distortion. 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Examples of glottal source estimation using either the traditional ZZT or the proposed ZCZT-based method. Top panel: a voiced speech segment (solid line) with the synchronized differenced EGG (dashed line) and four different positions of the window (dotted line). Panels (a) to (d): for the corresponding window location, two cycles of the glottal source estimation achieved by the traditional ZZT. dotted line) and by the proposed ZCZT-based technique (solid lineFig. 5. Examples of glottal source estimation using either the traditional ZZT or the proposed ZCZT-based method. Top panel: a voiced speech segment (solid line) with the synchronized differenced EGG (dashed line) and four different positions of the window (dotted line). Panels (a) to (d): for the corresponding window location, two cycles of the glottal source estimation achieved by the traditional ZZT (dotted line) and by the proposed ZCZT-based technique (solid line). P Alku, J Svec, E Vilkman, F Sram, Glottal wave analysis with pitch synchronous iterative adaptive inverse filtering. 11P. Alku, J. Svec, E. Vilkman, F. Sram: Glottal wave analysis with pitch synchronous iterative adaptive inverse filtering, Speech Communication, vol. 11, issue 2-3, pp. 109-118, 1992. Automatic glottal inverse filtering from speech and electroglottographic signals. D Veeneman, S Bement, IEEE Trans. on Signal Processing. 33369377D. Veeneman, S. BeMent: Automatic glottal inverse filtering from speech and elec- troglottographic signals, IEEE Trans. on Signal Processing, vol. 33, pp. 369377, 1985. Speaker characteristics from a glottal airflow model using glottal inverse filtering. D Brookes, D Chan, Proc. Institue of Acoust. 15501508D. Brookes, D. Chan: Speaker characteristics from a glottal airflow model using glottal inverse filtering, Proc. Institue of Acoust., vol. 15, pp. 501508, 1994. Dutoit: Zeros of Z-Transform Representation With Application to Source-Filter Separation in Speech. B Bozkurt, B Doval, C Dalessandro, T , IEEE Signal Processing Letters. 124B. Bozkurt, B. Doval, C. DAlessandro, T. Dutoit: Zeros of Z-Transform Representa- tion With Application to Source-Filter Separation in Speech IEEE Signal Processing Letters, vol. 12, no. 4, 2005. The voice source as a causal/anticausal linear filter. B Doval, C Dalessandro, N Henrich, Proceedings ISCA ITRW VOQUAL03. ISCA ITRW VOQUAL03B. Doval, C. dAlessandro, N. Henrich: The voice source as a causal/anticausal linear filter, Proceedings ISCA ITRW VOQUAL03, pp. 15-19, 2003. Doval: A comparative evaluation of the Zeros of Z-transform representation for voice source estimation, The Interspeech07. N Sturmel, C D'alessandro, B , N. Sturmel, C. D'Alessandro, B. Doval: A comparative evaluation of the Zeros of Z-transform representation for voice source estimation, The Interspeech07, pp. 558-561, 2007. Appropriate windowing for group delay analysis and roots of Z-transform of speech signals. B Bozkurt, B Doval, C D'alessandro, T Dutoit, Proc. of the 12th European Signal Processing Conference. of the 12th European Signal essing ConferenceB. Bozkurt, B. Doval, C. D'Alessandro, T. Dutoit: Appropriate windowing for group delay analysis and roots of Z-transform of speech signals, Proc. of the 12th European Signal Processing Conference, 2004. L Rabiner, R Schafer, C Rader, The Chirp-Z transform Algorithm and Its Application. 48L. Rabiner, R. Schafer, C. Rader: The Chirp-Z transform Algorithm and Its Ap- plication, Bell System Technical Journal, vol. 48, no.5, pp. 1249-1292, 1969. Short-time homomorphic analysis. J Tribolet, T Quatieri, A Oppenheim, IEEE International Conference on Speech and Signal Processing. 2J. Tribolet, T. Quatieri, A. Oppenheim: Short-time homomorphic analysis, IEEE International Conference on Speech and Signal Processing, vol. 2, pp.716-722, 1977. G Fant, J Liljencrants, Q Lin, A four parameter model of glottal flow, STL-QPSR4. G. Fant, J. Liljencrants, Q. Lin: A four parameter model of glottal flow, STL- QPSR4, pp. 1-13, 1985.	[]
[ "New Gauss-Bonnet black holes with curvature induced scalarization in the extended scalar-tensor theories", "New Gauss-Bonnet black holes with curvature induced scalarization in the extended scalar-tensor theories" ]	[ "Daniela D Doneva \nTheoretical Astrophysics\nEberhard Karls University of Tübingen\n72076TübingenGermany\n\nINRNE -Bulgarian Academy of Sciences\n1784SofiaBulgaria\n", "Stoytcho S Yazadjiev \nTheoretical Astrophysics\nEberhard Karls University of Tübingen\n72076TübingenGermany\n\nDepartment of Theoretical Physics\nFaculty of Physics\nSofia University\n1164SofiaBulgaria\n" ]	[ "Theoretical Astrophysics\nEberhard Karls University of Tübingen\n72076TübingenGermany", "INRNE -Bulgarian Academy of Sciences\n1784SofiaBulgaria", "Theoretical Astrophysics\nEberhard Karls University of Tübingen\n72076TübingenGermany", "Department of Theoretical Physics\nFaculty of Physics\nSofia University\n1164SofiaBulgaria" ]	[]	In the present paper we consider a class of extended scalar-tensor-Gauss-Bonnet (ESTGB) theories for which the scalar degree of freedom is excited only in the extreme curvature regime. We show that in the mentioned class of ESTGB theories there exist new black hole solutions which are formed by spontaneous scalarization of the Schwarzaschild balck holes in the extreme curvature regime. In this regime, below certain mass, the Schwarzschild solution becomes unstable and new branch of solutions with nontrivial scalar field bifurcate from the Schwarzschild one. As a matter of fact, more than one branches with nontrivial scalar field can bifurcate at different masses but only the first one is supposed to be stable. This effect is quite similar to the spontaneous scalarization of neutron stars. In contrast with the standard spontaneous scalarization of neutron stars which is induced by the presence of matter, in our case the scalarization is induced by the curvature of the spacetime.	10.1103/physrevlett.120.131103	[ "https://arxiv.org/pdf/1711.01187v1.pdf" ]	19,089,229	1711.01187	e6276d2d8b3672bbc5ddf59abd69654ee805f119	New Gauss-Bonnet black holes with curvature induced scalarization in the extended scalar-tensor theories Daniela D Doneva Theoretical Astrophysics Eberhard Karls University of Tübingen 72076TübingenGermany INRNE -Bulgarian Academy of Sciences 1784SofiaBulgaria Stoytcho S Yazadjiev Theoretical Astrophysics Eberhard Karls University of Tübingen 72076TübingenGermany Department of Theoretical Physics Faculty of Physics Sofia University 1164SofiaBulgaria New Gauss-Bonnet black holes with curvature induced scalarization in the extended scalar-tensor theories numbers: 0440Dg0450Kd0480Cc In the present paper we consider a class of extended scalar-tensor-Gauss-Bonnet (ESTGB) theories for which the scalar degree of freedom is excited only in the extreme curvature regime. We show that in the mentioned class of ESTGB theories there exist new black hole solutions which are formed by spontaneous scalarization of the Schwarzaschild balck holes in the extreme curvature regime. In this regime, below certain mass, the Schwarzschild solution becomes unstable and new branch of solutions with nontrivial scalar field bifurcate from the Schwarzschild one. As a matter of fact, more than one branches with nontrivial scalar field can bifurcate at different masses but only the first one is supposed to be stable. This effect is quite similar to the spontaneous scalarization of neutron stars. In contrast with the standard spontaneous scalarization of neutron stars which is induced by the presence of matter, in our case the scalarization is induced by the curvature of the spacetime. I. INTRODUCTION The historic direct detection of gravitational waves has opened a new era in physics, giving a powerful tool for exploring the strong-gravity regime, where spacetime curvature is extreme. General Relativity is well-tested in the weak-field regime, whereas the strong-field regime still remains essentially unexplored and unconstrained. There are both phenomenological and theoretical reasons for the modification of the original Einstein quations. For example, predictions based on General Relativity and the Standard Model of particle physics fail to explain the accelerated expansion of the Universe. It is also well known that the pure general relativity is not a renormalizable theory which poses severe obstacles to the efforts of quantizing gravity. The renormalization at one loop demands that the Einstein-Hilbert action be supplemented with all the possible algebraic curvature invariants of second order [1]. On the other hand, the attempts to construct a unified theory of all the interactions, naturally lead to scalar-tensor type generalizations of general relativity with an additional dynamical scalar field and with Lagrangians containing various kinds of curvature corrections to the usual Einstein-Hilbert Lagrangian coupled to the scalar field [2]- [5]. The most natural modifications of this class are the extended scalar-tensor theories (ESTT) where the usual Einstein-Hilbert action is supplemented with all possible algebraic curvature invariants of second order with a dynamical scalar field nonminimally coupled to these invariants The equations of the ESTT in their most general form are of order higher than two. This in general can lead to the Ostrogradski instability and to the appearance of ghosts. However, there is a particular sector of the ESTT, namely the sector where the scalar field is coupled exactly to the Gauss-Bonnet invariant, for which the field equations are of second order as in general relativity and the theory is free from ghosts. Due to these reasons, in the present paper we shall focus on the extended scalar-tensor-Gauss-Bonnet (ESTGB) gravity as a natural modification of general relativity and a natural extension of the standard scalar-tensor theories. A particular model of ESTGB gravity, the so-called Einstein-dilaton-Gauss-Bonnet(EdGB) gravity, was extensively studied in the literature. The nonrotating black holes in EdGB gravity with a coupling function αe γϕ and vanishing potential for the dilaton field, with α and γ being constants, were studied perturbatively or numerically in [6]- [9]. It was shown that the EdGB black holes exist only for α > 0 and when the black hole mass is greater than certain lower bound proportional to the paremeter α. The slowly rotating EdGB black holes were studied in [9], [10] and [11]. The rapidly rotating EDGB black holes were constructed numerically in [12]- [14]. The rotating EdGB black holes can exist only when the mass and the angular momentum fall in certain domain depending on the coupling constant. Another interesting fact about the EdGB black holes is that they can exceed the Kerr bound for the angular momentum. In the present paper we shall consider a class of ESTGB theories with a scalar coupling functions for which the scalar degree of freedom is excited only in the extreme curvature regime. In particular we shall show that in the mentioned class of ESTGB theories there exist new black hole solutions which are formed by spontaneous scalarization of the Schwarzaschild balck holes in the extreme curvature regime. In contrast with the standard spontaneous scalarization [15]- [17] which is induced by the presence of matter, in our case the scalarization is induced by the curvature of the spacetime. II. BASIC EQUATIONS AND SETTING THE PROBLEM The general action of ESTGB theories in vacuum is given by S = 1 16π d 4 x −g R − 2∇ µ ϕ∇ µ ϕ − V(ϕ) + λ 2 f (ϕ)R 2 GB ,(1) where R is the Ricci scalar with respect to the spacetime metric g µν , ϕ is the scalar field with a potential V(ϕ) and a coupling function f (ϕ) depending only on ϕ, λ is the Gauss-Bonnet coupling constant having dimension of length and R 2 GB is the Gauss-Bonnet invariant 1 . The action yields the following field equations R µν − 1 2 Rg µν + Γ µν = 2∇ µ ϕ∇ ν ϕ − g µν ∇ α ϕ∇ α ϕ − 1 2 g µν V(ϕ),(2)∇ α ∇ α ϕ = 1 4 dV(ϕ) dϕ − λ 2 4 d f (ϕ) dϕ R 2 GB ,(3) where ∇ µ is the covariant derivative with respect to the spacetime metric g µν and Γ µν is defined by Γ µν = −R(∇ µ Ψ ν + ∇ ν Ψ µ ) − 4∇ α Ψ α R µν − 1 2 Rg µν + 4R µα ∇ α Ψ ν + 4R να ∇ α Ψ µ −4g µν R αβ ∇ α Ψ β + 4R β µαν ∇ α Ψ β (4) with Ψ µ = λ 2 d f (ϕ) dϕ ∇ µ ϕ.(5) In what follows we shall focus on the case V(ϕ) = 0. We consider further static and spherically symmetric spacetimes as well as static and spherically symmetric scalar field configurations. The spacetime metric then can be written in the standard form ds 2 = −e 2Φ(r) dt 2 + e 2Λ(r) dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ).(6) The dimensionally reduced field equations (2) are the following 2 r 1 + 2 r (1 − 3e −2Λ )Ψ r dΛ dr + (e 2Λ − 1) r 2 − 4 r 2 (1 − e −2Λ ) dΨ r dr − dϕ dr 2 = 0,(7)2 r 1 + 2 r (1 − 3e −2Λ )Ψ r dΦ dr − (e 2Λ − 1) r 2 − dϕ dr 2 = 0,(8)d 2 Φ dr 2 + dΦ dr + 1 r dΦ dr − dΛ dr + 4e −2Λ r 3 dΦ dr dΛ dr − d 2 Φ dr 2 − dΦ dr 2 Ψ r − 4e −2Λ r dΦ dr dΨ r dr + dϕ dr 2 = 0,(9)d 2 ϕ dr 2 + dΦ dr − dΛ dr + 2 r dϕ dr − 2λ 2 r 2 d f (ϕ) dφ (1 − e −2Λ ) d 2 Φ dr 2 + dΦ dr dΦ dr − dΛ dr + 2e −2Λ dΦ dr dΛ dr = 0,(10) with Ψ r = λ 2 d f (ϕ) dϕ dϕ dr .(12) In the present paper we are interested in ESTGBT with coupling function f (ϕ) satisfying the conditions 2 d f dϕ (0) = 0 and b 2 = d 2 f dϕ 2 (0) > 0. Without loss of generality we can put b = 1 and this can be achieved by rescaling the coupling parameter λ → bλ and by redefining the coupling function f → b −2 f . In addition, since the theory depends only on d f (ϕ) dϕ , we can also impose f (0) = 0. The natural and the important question is whether the class of ESTGBT defined above admits (static and spherically symmetric) black hole solutions. From the dimensionally reduced field equations (7)- (10) it is clear that the usual Schwarzschlild black hole solution is also a black hole solution to the ESTGBT under consideration with a trivial scalar field ϕ = 0. We shall however show that the Schwarzschild solution within the certain range of the mass is unstable in the framework of the ESTGBT under consideration. For this purpose we consider the perturbations of the Schwarzschild solution with mass M within the framework of the described class of EST-GBT. It is not difficult to see that in the considered class of ESTGBT the equations governing the perturbations of the metric δg µν are decoupled from the equation governing the perturbation δϕ of the scalar field. The equations for metric perturbations are in fact the same as those in the pure Einstein gravity and therefore we shall focus only on the scalar field perturbations. The equation governing the scalar perturbations is 2 (0) δϕ + 1 4 λ 2 R 2 GB(0) δϕ = 0,(13) where 2 (0) and R 2 GB(0) are the D'alambert operator and the Gauss-Bonnet invariant for the Schwarzschild geometry. Taking into account that the background geometry is static and spherically symmetric, the variables can be separated in the following way δϕ = u(r) r e −iωt Y lm (θ, φ),(14) with Y lm (θ, φ) being the spherical harmonics. After substituting in (13) we find g(r) r d dr r 2 g(r) d dr u(r) r + ω 2 + g(r) − l(l + 1) r 2 + 1 4 λ 2 R 2 GB(0) u(r) = 0,(15) where g(r) = 1 − 2M r and R 2 GB(0) = 48M 2 r 6 for the Schwarzschild solution. By introducing the tortoise coordinate dr * = dr g(r) which maps the domain r ∈ (2M, +∞) to r * ∈ (−∞, +∞), the equation can be cast in the Schrödinger form d 2 u dr 2 * + [ω 2 − U(r)]u = 0(16) with a potential U(r) = 1 − 2M r 2M r 3 + l(l + 1) r 2 − λ 2 12M 2 r 6 .(17) A sufficient condition for the existence of an unstable mode is [18] +∞ −∞ U(r * )dr * = ∞ 2M U(r) 1 − 2M r dr < 0.(18) For the spherically symmetric perturbations the above condition gives M 2 < 3 10 λ 2 . Therefore we can conclude that the Schwarzshild black holes with mass satisfying M 2 < 3 10 λ 2 are unstable within the framework of the ESTGBT under consideration. Stated differently, the Schwarzshild black holes become unstable when the curvature of the horizon exceeds a certain critical valuein terms of the Kretschmann scalar of the horizon K H , the instability occurs when K H > 25 3λ 4 . This result naturally leads us to the conjecture that, in our class of ESTGBT and in the interval where the Schwarzschild solution is unstable, there exist black solutions with nontrivial scalar field. In the next sections we numerically prove that such black hole solutions really exist and present some of their basic properties. III. NUMERICAL SETUP In order to obtain the black hole solutions with a nontrivial scalar field we solve numerically the system of reduced field equations (7)-(10). The system of differential equations is transformed in the following way in order to simplify the numerical calculations. Using equations (7) and (8) one can straightforward obtain expressions for the metric function Λ and its derivative that depend only on the metric function Φ, the scalar field ϕ and their derivatives. In this way the system of equations (9)-(10) decouples from the rest of the equations and we are left with two second order partial differential equations for Φ and ϕ. Let us point out that after a solution for Φ and ϕ is found the metric function Λ can be constructed directly without the need for integration due to the particular form of equation (8). One can notice that in all of the reduced field equations the metric function Φ do not enter directly but only through its derivatives. That is why instead of solving two second order equations one can solve one second order equation for the scalar field ϕ and one first order equation for the first derivative of the metric function dΦ/dr. Later Φ can be found by simply integrating the resulting dΦ/dr with the appropriate boundary conditions. Even though this might look like a very small simplification it is very important since it reduces the number of the shooting parameters from two to one and thus simplifies a lot the search for bifurcations of the Schwarzschild solution. Let us discuss now the boundary and the regularity conditions. They come from the requirements for asymptotic flatness at infinity and the regularity at the black hole horizon r = r H . As usual the asymptotic flatness imposes the following asymptotic conditions Φ\| r→∞ → 0, Λ\| r→∞ → 0, ϕ\| r→∞ → 0 .(19) The very existence of black hole horizon requires e 2Φ \| r→r H → 0, e −2Λ \| r→r H → 0.(20) The regularity of the scalar field and its first and second derivatives on the black hole horizon gives one more condition, namely dϕ dr H + 2λ 2 r H d f dϕ (ϕ H ) dϕ dr 2 H + 2λ 2 r 3 H d f dϕ (ϕ H ) = 0.(21) From (21) we can express the value of the first derivative d f dϕ (ϕ H ) as a function of the value of scalar field on the horizon and the horizon radius, namely dϕ dr H = r H 4λ 2 d f dϕ (ϕ H )   −1 ± 1 − 24λ 4 r 4 H d f dϕ (ϕ H ) 2   .(22) In this expression we have to chose the plus sign since only in this case we can recover the trivial solution in the limit ϕ H → 0. The requirement for positiveness of the expression inside the square root in the above expression imposes restriction on the possible solutions with nontrivial scalar field, i.e. black hole solutions exist only when r 4 H > 24λ 4 d f dϕ (ϕ H ) 2 .(23) For Schwarzschild ϕ = 0 and since the coupling function by definition satisfies d f dϕ (0) = 0, this condition is automatically satisfied. For black holes with nontrivial scalar field, though, eq. (23) can be violated as we will see below. We use a shooting method to find the black hole solutions. First, the first order differential equation for dΦ/dr and the second order equation for ϕ are solved using the above boundary conditions. The input parameter which determines the black hole solutions is r H and we have one shooting parameter -the value of the scalar field at the horizon. This shooting parameter is determined by the boundary condition of ϕ at infinity (19). The metric function Φ is calculated afterwards also with a shooting procedure using the obtained dΦ/dr and the boundary conditions (19). As we commented, after we have found the solutions for Φ and ϕ, Λ can be determined directly without the need for integration using eq. (8). The mass of the black hole M and the dilaton charge D are obtained through the asymptotics of the functions Λ, Φ and ϕ, namely Λ ≈ M r + O(1/r 2 ), Φ ≈ − M r + O(1/r 2 ), ϕ ≈ D r + O(1/r 2 ).(24) IV. RESULTS The only thing that remains to be fixed is the explicit form of the coupling function f (ϕ). In the present paper we consider the following function f (ϕ) = 1 12 1 − exp(−6ϕ 2 ) ,(25) that is chosen in such a way that we have both non-negligible deviations from the Schwarzschild solution and the condition for the existence of solutions with nontrivial scalar field (23) is fulfilled for large enough range of parameters. We have explicitly checked that other choices of f (ϕ) that lead to similar results are of course possible but exploring a large variety of f (ϕ) functions is out of the scope of the present paper. Instead, we will focus on examining in detail the black hole solutions with non-trivial scalar field and the non-uniqueness of the solutions. As discussed above, the Schwarzschild solution with zero scalar field is always a solution of the field equations but in a certain region of the parameter space it becomes unstable in the framework of the ESTGBT under consideration and new solutions with nontrivial scalar field appear. Moreover, there can be regions where more than one solution with nontrivial scalar field exists and this corresponds roughly speaking to the appearance of more than one bound state of the potential in the perturbation equation 3 (16). The different branches of solutions will have scalar field with different number of zeros similar to the eigenfunctions of the perturbation equation (16). Finding the solutions with nontrivial scalar field might be sometimes numerically difficult and it is of great help to know the exact points of bifurcation. In the previous sections we discussed that for M 2 < 3 10 λ 2 the Schwarzschild black holes are unstable but this is only a sufficient condition for instability and the true point of the first bifurcations is actually at a little bit larger masses. In order to find the points of bifurcation we can use the fact that they are the same as the points where new unstable modes of eq. (16) appear (for a detailed discussion see [17]). That is why instead of solving the reduced field equations, we can determine the bifurcations points using the perturbation equation (16), that is numerically easier. Since we are interested in unstable modes, ω 2 should be negative which leads to the fact that the boundary conditions are zero both at the black hole horizon and infinity (for more details and derivation see [17]). Therefore, we have a self-adjoint Sturm-Liouville problem. We employed a shooting procedure to find the eigenvalues and the eigenfunction of eq. (16) and determined the regions of the parameter space where the Schwarzschild solution is stable, where it is unstable and only one unstable mode is present, where two unstable modes are present and so on. This means that we have determined the points of bifurcation of the Schwarschild solution which significantly simplifies the search for black holes with nontrivial scalar field. The obtained black hole solutions are plotted in Fig. 1 where only the first three bifurcations of the Schwarzschild solution are shown in order to have better visibility. We will call the Schwarzschild solution the trivial branch of solutions (with trivial scalar field) while the rest of the branches of black holes with nontrivial scalar field will be called nontrivial branches (with nontrivial scalar field). As one can see, all the nontrivial branches start from a bifurcation point at the trivial branch and they span either to M = 0 (the first nontrivial branch) or they are terminated at some nonzero M (all the other nontrivial branches). The reason for termination of the branches at nonzero M is that beyond this point the condition (23) is violated. We should point out as well, that calculating black holes with nontrivial scalar field for very small M is very difficult from a numerical points view since the scalar field increases significantly and goes to infinity as M approaches zero. As discussed above, the different nontrivial branches of solutions are characterized by different number of zeros of the scalar field. For the first branch (the red dashed line in Fig. 1) there are no zeros of ϕ as one can see in the left panel of Fig. 2, the next one (green line) has one zero while the third one (blue line) has two zeros as one can see in Fig. 3. For smaller values of M there are more bifurcation points but our investigations show the corresponding nontivial branches would be even shorter and that is why we have not plotted them. Moreover, it is expected that only the first nontrivial branch characterized by a scalar field without zeros will be stable while the rest of the branches correspond to unstable solutions. One can notice as well that Fig. 1 is symmetric with respect to the x-axis that can be shown easily analytically also using the field equations with the particular coupling function (25). Thus for a fixed M the solutions with positive and negative values of ϕ H would naturally have opposite signs of the dilaton charge, but they have the same metric functions and thus mass. The components of the metric g tt and g rr , as well as the scalar field as function of the normalized radial coordinate r H /λ are shown for some representative solutions with different r H /λ in Figs. 2 and 3 for the first tree nontrivial branches. These figures demonstrate what we have commented above -the different branches of nontrivial solutions are characterized by different numbers of zeros of the scalar field. As one can see, the metric of the first nontrivial branch can deviate significantly not only qualitatively but also quantitatively from the Schwarzschild one. We have not plotted g tt and g rr for the next nontrivial branches since they are almost indistinguishable from the pure general relativistic case. The dilaton charge as a function of the mass is shown in Fig. 4 and as a function of ϕ H -in Fig. 5. As one can see, while the dependence ϕ H (M/λ) is monotonic for the first nontrivial branch and ϕ H increases significantly for small masses, D(M/λ) has an extremum (either minimum or maximum depending on the sign of ϕ H ) and tends to zero for small masses. Fig. 6. The first three nontrivial branches of black holes are plotted in addition to the corresponding dependence in the Schwarzschild case. This graph can be used to better judge how strong the deviations from pure general relativity are. As one can see, only the first branch of nontrivial solutions (with scalar field which has no zeros) deviates significantly from the Schwarzschild case and the deviations are the largest for intermediate masses. (1) is not just one forth of the horizon area and its definition is a little bit more complicated. We adopt the entropy formula proposed by Wald in [19], [20] namely S H = 2π H ∂L ∂R µναβ µν αβ ,(26) where L is the Lagrangian density and αβ is the volume form on the 2-dimensional cross section H of the horizon. In our case we find S H = 1 4 A H + 4πλ 2 f (ϕ H ).(27) The entropy as a function of the black holes mass is plotted in Fig. 7. As one can see the first nontrivial branch has entropy larger than the Schwarschild one and it is therefore thermodynamically more stable. This is an expected results since for masses smaller that the point of the first bifurcation the Schwarschild solution get unstable and there should be another one. The second and the third nontrivial branches on the other hand have lower entropy compared to the pure general relativistic case, which means that they are most probably unstable. The same is expected to apply for the rest of nontrivial branches that exist for smaller masses. The dynamical stability of our black hole solutions will be investigated in a future work. V. CONCLUSION In the present paper we have studied black hole solutions in a particular class of ESTGB theories described by a coupling function f (ϕ) that satisfies the conditions d f dϕ (0) = 0 and d 2 f dϕ 2 (0) > 0. We have shown that for such theories an effect similar to the spontaneous scalarization of neutron stars exists -the Schwarzschild solution becomes unstable below certain mass and new branches of black hole solutions with nontrivial scalar field appear that bifurcate from the Schwarzschild one at certain masses. The first branch of nontrivial solutions is characterized by a scalar field that has no zeroes while the scalar field has one zero for the second branch, two zeros for the third branch and so on. The general expectation, though, is that only the first branch of solutions would be stable and the rest would be unstable. The main difference with the spontaneous scalarization of neutron stars is that the scalar field is not sourced the by matter, but instead by the extreme curvature of the spacetime around black holes. This places the considered solutions amongst the very few examples of scalarized black holes. We have explicitly constructed such solutions with nonzero scalar field and it was shown that the first branch of nontrivial solutions is thermodynamically more stable compared to the Schwarzshild one. The results presented in the current paper are for a particular coupling function that can produce non-negligible deviations from pure general relativity. We have tested, though, several other functions satisfying the above given conditions for f (ϕ) and the results are qualitatively very similar. It was demonstrated that the behavior of the scalar field at the horizon ϕ H and the dilaton charge D (i.e. the coefficient in front of the 1/r term in the asymptotic expansion of the scalar field at infinity) are qualitatively different. While ϕ H monotonically increase with the decrease of the mass (for positive ϕ H ), the dilaton charge first increases and after reaching a maximum it decreases to zero. As expected, the A H (M) dependence exhibits similar behavior to D, i.e. the black hole solutions tend to the Schwarzschild one for very small masses and for larger masses close to the bifurcation point and the maximum deviation is observed for intermediate masses. We should note that the above given observations are true only for the first nontrivial branch characterized by scalar field without zeros. The rest of the branches are terminated at some nonzero mass because beyond that mass they violate condition (23). We have studied the behavior of the black hole entropy and the results show that the first nontrivial branch has higher entropy than the Schwarzschild black holes and it is thermodynamically more stable while the rest of the branches have lower entropy. Thus, the general expectation is that the first branch of solutions is stable and it is the one that would realize in practice because of the instability of the Schwarzschild solutions. The other nontrivial branches are supposed to be unstable. Of course, this can be rigorously proven only if one considers the linear perturbations of the solutions with nontrivial scalar field that would be done in a future publication. Finally, let us briefly comment on the following. The black holes we are considering posses a nontrivial scalar field, and thus they have scalar "hair". When the branch of the solution is fixed then this "hair" is secondary which means that the dilaton charge is not an independent parameter but instead it depends on black hole mass. However, the number of the branches is an independent parameter introducing a new "hair" of discrete type. One may adopt the view that only the stable branch has to be considered getting rid in this way of the discrete "hair". In our opinion the classification of black hole solutions presented in the present work is rather subtle and needs a much deeper analytical investigation. We shall discuss this problem in a future publication. FIG. 1 : 1The scalar field at the horizon as a function of the black hole mass. The right figure is a magnification of the left one.The area of the black hole horizon, A H = 4πr 2 H and the normalized A H functions of the mass in FIG. 2 : 2The scalar field and the g tt and g rr components of the metric as functions of the normalized radial coordinate r/r H for several black hole solutions that belong to the first nontrivial branch with different values of r H /λ. FIG. 3 : 3The scalar field as a function of the normalized radial coordinate r/r H for two representative solution from the second and the third branch of nontrivial solutions. The components of the metric g tt and g rr are not shown since they are almost indistinguishable from the Schwarzschild case.This observation is similar to the behavior of the dilaton change which tends to zero for very large and very small masses, reaching maximum for intermediate values of M/λ.Up to now we have shown and discussed the first three branches of nontrivial solutions and as we commented there are more branches that bifurcate at smaller values of M/λ. In order to have an indicator for the stability of the black hole branches one can study the entropy of the black holes. The black hole entropy in the presence of a Gauss-Bonnet term in the action FIG. 4 : 4The dilaton charge of the black hole as a function of its mass. The right figure is a magnification of the left one. FIG. 5 : 5The dilaton charge of the black hole as a function of the scalar field at the horizon. The right figure is a magnification of the left one. FIG. 6 : 6The area of the black hole horizon A H as a function of the mass. In the right figure the black hole area is normalized to the corresponding value in the Schwarzschild limit, i.e. FIG. 7 : 7The entropy of the black hole as a function of its mass. . The Gauss-Bonnet invariant is defined by R 2 GB = R 2 − 4R µν R µν + R µναβ R µναβ where R is the Ricci scalar, R µν is the Ricci tensor and R µναβ is the Riemann tensor We consider here the case when the cosmological value of the scalar field is zero, ϕ ∞ = 0. In the present paper we consider only spherically symmetric solutions and therefore l = 0. AcknowledgementsDD would like to thank the European Social Fund, the Ministry of Science, Research and the Arts Baden-Württemberg for the support. DD is indebted to the Baden-Württemberg Stiftung for the financial support of this research project by the Eliteprogramme for Postdocs. 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[]	[ "Joachim Dzubiella \nDepartment of Physics\nHelmholtz Zentrum Berlin für Materialien und Energie\nHahn-Meitner-Platz 114109Berlin, Germany\n\nHumboldt University Berlin\nNewtonstr. 1512489BerlinGermany\n" ]	[ "Department of Physics\nHelmholtz Zentrum Berlin für Materialien und Energie\nHahn-Meitner-Platz 114109Berlin, Germany", "Humboldt University Berlin\nNewtonstr. 1512489BerlinGermany" ]	[]	How interface geometry dictates water's thermodynamic signature in hydrophobic association the date of receipt and acceptance should be inserted later Abstract As a common view the hydrophobic association between molecular-scale binding partners is supposed to be dominantly driven by entropy. Recent calorimetric experiments and computer simulations heavily challenge this established paradigm by reporting that water's thermodynamic signature in the binding of small hydrophobic ligands to similar-sized apolar pockets is enthalpy-driven. Here we show with purely geometric considerations that this controversy can be resolved if the antagonistic effects of concave and convex bending on water interface thermodynamics are properly taken into account. A key prediction of this continuum view is that for fully complementary binding of the convex ligand to the concave counterpart, water shows a thermodynamic signature very similar to planar (large-scale) hydrophobic association, that is, enthalpy-dominated, and hardly depends on the particular pocket/ligand geometry. A detailed comparison to recent simulation data qualitatively supports the validity of our perspective down to subnanometer scales. Our findings have important implications for the interpretation of thermodynamic signatures found in molecular recognition and association processes. Furthermore, traditional implicit solvent models may benefit from our view with respect to their ability to predict binding free energies and entropies.That the reasoning above may not be transferable to concave solute geometries has been anticipated by Carey, Chen, and Rossky [12] who concluded that "the enthalpy and entropy, associated with displacement of the solvent on substrate binding [to a concave protein pocket] should not follow 'conventional' entropy driven hydrophobic interactions." Indeed, experimental studies of synthetic host-guest systems where hydrophobic confinement is involved exhibit enthalpy-driven association[45].More recently, systematic molecular dynamics computer simulations by Setny, Baron, and McCammon have unequivocally demonstrated that the hydrophobic binding between generic uncharged ligands and pockets is indeed strongly driven by enthalpy and results from the expulsion of disordered water from the weakly hydrated cavity[42]. The experimental relevance of this finding has been highlighted shortly after by Englert et al. who presented evidence that indeed the displacement of a few, unstructured water molecules from a hydrophobic pocket creates an enthalpic signature in the binding of phosphonamidate to the thermolysin protein[20]. In fact, it becomes more and more evident that numerous apolar protein cavities may be weakly hydrated or even dehydrated [54] which has nontrivial effects on ligand binding affinity[50]. A better understanding of these unconventional processes is fundamentally important for the interpretation of virtually all ligand-binding processes to hydrophobic pockets in molecular recognition[23,35,28].	10.1007/s10955-011-0217-8	[ "https://arxiv.org/pdf/1104.5569v1.pdf" ]	52,885,946	1104.5569	9fa4d7418ce1c789352574ebb27d3ab9dce0d39d	29 Apr 2011 Joachim Dzubiella Department of Physics Helmholtz Zentrum Berlin für Materialien und Energie Hahn-Meitner-Platz 114109Berlin, Germany Humboldt University Berlin Newtonstr. 1512489BerlinGermany 29 Apr 2011arXiv:1104.5569v1 [cond-mat.soft] Noname manuscript No. (will be inserted by the editor) How interface geometry dictates water's thermodynamic signature in hydrophobic association the date of receipt and acceptance should be inserted later Abstract As a common view the hydrophobic association between molecular-scale binding partners is supposed to be dominantly driven by entropy. Recent calorimetric experiments and computer simulations heavily challenge this established paradigm by reporting that water's thermodynamic signature in the binding of small hydrophobic ligands to similar-sized apolar pockets is enthalpy-driven. Here we show with purely geometric considerations that this controversy can be resolved if the antagonistic effects of concave and convex bending on water interface thermodynamics are properly taken into account. A key prediction of this continuum view is that for fully complementary binding of the convex ligand to the concave counterpart, water shows a thermodynamic signature very similar to planar (large-scale) hydrophobic association, that is, enthalpy-dominated, and hardly depends on the particular pocket/ligand geometry. A detailed comparison to recent simulation data qualitatively supports the validity of our perspective down to subnanometer scales. Our findings have important implications for the interpretation of thermodynamic signatures found in molecular recognition and association processes. Furthermore, traditional implicit solvent models may benefit from our view with respect to their ability to predict binding free energies and entropies.That the reasoning above may not be transferable to concave solute geometries has been anticipated by Carey, Chen, and Rossky [12] who concluded that "the enthalpy and entropy, associated with displacement of the solvent on substrate binding [to a concave protein pocket] should not follow 'conventional' entropy driven hydrophobic interactions." Indeed, experimental studies of synthetic host-guest systems where hydrophobic confinement is involved exhibit enthalpy-driven association[45].More recently, systematic molecular dynamics computer simulations by Setny, Baron, and McCammon have unequivocally demonstrated that the hydrophobic binding between generic uncharged ligands and pockets is indeed strongly driven by enthalpy and results from the expulsion of disordered water from the weakly hydrated cavity[42]. The experimental relevance of this finding has been highlighted shortly after by Englert et al. who presented evidence that indeed the displacement of a few, unstructured water molecules from a hydrophobic pocket creates an enthalpic signature in the binding of phosphonamidate to the thermolysin protein[20]. In fact, it becomes more and more evident that numerous apolar protein cavities may be weakly hydrated or even dehydrated [54] which has nontrivial effects on ligand binding affinity[50]. A better understanding of these unconventional processes is fundamentally important for the interpretation of virtually all ligand-binding processes to hydrophobic pockets in molecular recognition[23,35,28]. Introduction One of the major driving forces in biomolecular association and self-assembly is the hydrophobic effect, mediated by structural rearrangements of the surrounding water molecules and thus ubiquitous in every biological reaction. Much progress has been made in the last decade in understanding the hydrophobic effect, in particular, by the distinction between the qualitatively different hydration behavior at small and large length scales [33,13]. Water's bulk-like hydrogen network is only moderately distorted around small, convex solutes upon hydration while strongly penalized by configurational freedom. Consequently, below a cross-over scale of ∼1 nm water's thermodynamic signature to hydrophobic association is entropy dominated, in contrast to large-scale enthalpy-driven association. This is often exemplified by the hydration or association of small, apolar solutes, such as methane. It has thus become a common view that small-scale hydrophobic association is entropy-driven. In the theoretical description of apolar association and recognition, interface models based only on surface tension and solvent-accessible surface area (SASA) have been traditionally surprisingly insightful, e.g., SASA-like models [39,23,35] or scaled particle theory (SPT) [4,27,2]. However, they are expected to fail in cases where only a few, disordered water molecules are involved [28], as in a highly concave, apolar pocket, and "the corresponding free energy seems unlikely to be suitably described by an interfacial tension" [12]. To tackle this specific problem, Young et al. [53] introduced a method based on the displacement of quasilocalized waters upon ligand binding and recently introduced a term attributable to the occupation of the dehydrated regions by ligand atoms [50]. However, in this paper we argue that surface area based models may remain valid even for high concave curvature if the antagonistic effects of concave vs. convex bending on water interface thermodynamics are properly taken into account. Our reasoning is based on a generalization of capillarity theory, resembling SPT, but extended to inhomogeneous and high local curvature and coupled to weak dispersion interactions [10,18]. Since this capillarity theory involves the minimization of solutesolvent interfacial area, the existence of dehydrated ('dry') states of apolar pockets can be in principle captured [44]. The basic model is briefly motivated in the next section while the thermodynamic consequences and a detailed comparison to the computer simulations by Setny et al. [42] are discussed in the Results section. The qualitative agreement between the interface model predictions and the simulations support the general validity of our predictions down to subnanometer length scales. In particular, an important conclusion is that for fully complementary apolar pocket-ligand binding, water carries a similar thermodynamic signature as planar (large-scale) hydrophobic association independent of the particular pocket/ligand geometry. Methods Basic framework Let V be the volume occupied by water in three-dimensional space W. In the volume void of water W \ V, the solvent is displaced by the solute atoms and possibly by capillary effects. The free energy of the solvent interface surrounding the apolar solutes was proposed to be a geometric functional of V of the form [18] G[V] = P W\V dV + γ ∂V [1 − 2δH(r)]dA + ρ 0 V U sw (r)dV(1) where V and ∂V denote volume and surface integrals over V or its boundary ∂V, respectively. P is the liquid bulk pressure, γ(T ) the temperature-dependent liquid-vapor surface tension, ρ 0 (T ) the water bulk density, and δ(T ) the coefficient of the curvature correction to the surface tension linear in mean curvature H(r). The latter is defined by the mean of the two local principal surface curvatures 1/R 1 and 1/R 2 , that is H = (1/R 1 + 1/R 2 )/2. In the last term in (1), the water density couples to the inhomogeneous, nonelectrostatic solute-water energy potential U sw (r). Typically in molecular modeling, a Lennard-Jones (LJ) potential is used to express the Pauli repulsion and dispersion attraction between a solute atom and a water molecule. Thus, U sw (r) may represent the sum of the LJ interactions of N s solute atoms fixed at positions r i acting on bulk water possibly present at r, i.e, U sw (r) = Ns i U LJ (r − r i ). Since the LJ coupling is proportional to density and volume, it can be interpreted as a local mechanical pressure exerted by the solute atoms on the solvent interface. Without curvature correction and dispersion interactions, eq. (1) reduces to the macroscopic interfacial free energy well known in thermodynamics since Gibbs [40]. The curvature correction The curvature correction term linear in integrated mean curvature H in (1) has been justified for various reasons. While first derived by Tolman 1949 for spherical symmetry on thermodynamic grounds [47], it is an integral part of the phenomenological SPT [2] which has proven extremely successful in describing the solvation thermodynamics of spherical solutes in simple liquids. SPT even serves as a good fit for the hydration free energy of hard spheres in water [33] and can be used for more sophisticated cavity hydration analysis using revised SPT [2,3]. However, more than 30 years ago, Boruvka and Neumann extended Gibbs' classical theory of capillarity to arbitrary surface geometries having high, inhomogeneous curvature distributions [10]. Within this generalization, the curvature correction in (1) was derived on the basis of fundamental equations of Gibbs' dividing surfaces making use of extensive geometric properties. In fact, a mathematical theorem well-known in differential geometry, the Hadwiger theorem, states that a functional defined on a set of three-dimensional bodies satisfying certain constraints, such as motion invariance, continuity, and additivity, must be a linear combination of the four mathematical measures volume, surface area, and the surface integral of mean and Gaussian curvatures [25,34]. The Hadwiger perspective is actually equivalent to Boruvka and Neumann's generalized capillarity theory, and the three constraints above can be understood as a more precise definition for the conventional term extensive [31]. Indeed, the application of geometric functionals akin to (1) to solvation processes resulted into the first successful morphological description of the thermodynamics of confined fluids, such as hard-sphere solvents [31,26] or water in protein side chain packing [51]. In another line of developments, the free energy (1), minimized by δG[V]/δV = 0 for a given solute geometry in water, yielded a proper description of the hydration free energy and interactions of small apolar solutes [17]. In particular, capillary evaporation between extended hydrophobic plates found in computer simulations was quantitatively captured for the first time in an implicit description. Remarkably, the free energy and polymodal hydration behavior of water in the binding between an apolar pocket and a ligand found in MD simulations could be quantified and interpreted in terms of topologically distinct interfaces [44]. In other words the (stable or metastable) 'dry' state observed in many protein cavities [54] is a local minimum of the interfacial free energy functional (1). These works underline the validity of the geometric functional and its potential not only to describe 'classical' macroscopic capillary effects but even for the quantitative or at least qualitative description of aqueous hydration in small-scale apolar confinements of arbitrary geometry. It is important to remark that the use of only the simple morphological measures linear in volume, surface area, and integrated curvature in (1) should be strictly valid only if all intrinsic (correlation) length scales are small compared to the (confining) system size [31]. Close to the critical point, for instance, correlations become long-ranged and the geometric expansion in (1) may break down [10,46]. Given the promising results cited above and presented in this work, however, the quantitative validity range of (1) for water at ambient conditions seemingly reaches to surprisingly small length scales. Obtaining thermodynamic coefficients by fitting to the hydration of LJ spheres In order to obtain an estimate for values of the correction coefficient δ(T ) we minimize eq. (1) and compare the solution to MD simulation results of the hydration free energies of the relatively large (in terms of atomic sizes) apolar and spherical atomic solutes xenon and methane [37]. For spherical geometry and negligible pressure contributions on small scales, eq. (1) reduces to a function of the interface radius R given by G = 4πR 2 γ(1 − 2δ/R) + 4πρ 0 ∞ R U LJ (r)r 2 dr.(2) The minimization ∂G/∂R = 0 yields 2γ R 1 − δ R − ρ 0 U LJ (R) = 0 (3) which is solved by the optimal interface radius R 0 , and U LJ = 4ǫ[(σ/r) 12 −(σ/r) 6 ] is the Lennard-Jones (LJ) interaction. Plugging back R 0 into (2) gives the hydration free energy of the solute. The parameter δ(T ) is then determined by comparing the resulting hydration free energy to those of methane and xenon obtained from explicit-water computer simulations for a variety of temperatures and water models [37]. The input parameters in (2) are the liquid-vapor surface tension of a planar interface γ(T ) and the liquid density ρ 0 (T ). The values for γ(T ) and ρ 0 for SPC/E and TIP4P are taken from the works of Chen and Smith [16] and Paschek [37] and are summarized in Tab. I. This fitting procedure at T = 300 K yields δ = 0.086 ± 0.009 nm and δ = 0.076 ± 0.008 nm for SPC/E and TIP4P water, respectively, comparable to the values found in previous studies on the hydration of hard spheres [27,4,22]. From repeating this fitting procedure for temperatures T = 275 K and T = 325 K, we estimate the slope of δ(T ) by the finite difference ∂δ/∂T (T = 300 K) = [δ(T = 325 K) − δ(T = 275 K)]/(50 K) and find values T ∂δ/∂T = −0.090 ± 0.003 nm and −0.144 ± 0.005 nm for SPC/E and TIP4P water at T = 300 K, respectively. All values found for T = 300 K are summarized in Tab. I. As a consistency check of the fitting procedure we have calculated the hydration entropy of OPLS methane [30] using eq. (4) below. Taking R 0 = 0.31 nm for OPLS methane from minimization of (2) and the TIP4P thermodynamic parameters at T = 300 K, we obtain T S = −14.2 kJ/mol what indeed compares favorably with the result from Paschek [37]. Note that also the value of R 0 is consistent with the effective radius estimated from water density distribution averaged in the MD simulations, see the discussion in III.C. Results and discussion Interface thermodynamics Let us now have a closer look on the thermodynamic consequences of description (1). Identifying (1) with a free energy in the Gibbs ensemble (N P T ) with enthalpy and entropy defined via G = H − T S, the entropy S is given by [10,11] S = − ∂G ∂T N,P = − ∂γ ∂T ∂W [1 − 2δH]dA + 2γ ∂δ ∂T ∂W HdA,(4) where the contribution of the van der Waals term in eq. (1) has been neglected. At ambient conditions the relative water density changes with T are typically very small, and it is well-known that van der Waals interactions contribute primarily to the enthalpy [13]. Note that for the creation of a planar liquid-vapor interface the curvature terms can be neglected and simply G = γA and T S = −T A∂γ/∂T . Given a surface with unit area A = 1 nm 2 and the surface tension of real water at T = 300 K, we obtain G = 43 kJ/mol, T S = 29 kJ/mol, and H = G + T S = 72 kJ/mol. Thus, we recall that the creation of a planar hydrophobic interface is strongly penalized by enthalpy which can be attributed to the breaking of hydrogen bonds. The water interfacial entropy increases due to the increased number of configurational degrees of freedom of the water molecules. This is well-known as the thermodynamic signature of large-scale hydrophobicity [13]. What is the consequence of curvature? By definition the mean curvature H changes sign when going from a convex surface to a concave one. Here, we follow the convention that a locally convex solute curvature (e.g., a spherical solute such as an argon atom) has a positive mean curvature, H > 0. Consequently, H < 0 for concave curvature (as for a droplet or a hemispherical protein pocket). With that fixed, the change of the free energy with curvature is governed by the correction coefficient δ(T ). For real water no experimental efforts have been undertaken so far to determine δ(T ). However, good water models exists with respect to the description of material and thermodynamics properties, such as the liquid density and the liquid-vapor surface tension. The SPC/E [7] and TIP4P [29] models are examples for such well-performing water models, and qualitative agreement to real water can be expected for curvature effects to the surface tension. By minimizing eq. (1) (1) to hydration free energies of hard spheres of a wide range of radii up to 1 nm [27,4,22]. As a first consequence, the solvation free energy (1) per unit area increases with increasing solute concavity and decreases with increasing solute convexity. Since δ is of Angstrom size, the correction is notably for curvature radii below ≃ 1 − 2 nm, a realm where water interfaces are intrinsically rough due to surface reconstructions of the hydrogen bonding network [41]. What about the balance between entropy and enthalpy? For this we need an estimate for the temperature-dependence of both the surface tension γ(T ) and the coefficient δ(T ), as emphasized by Ashbaugh and Pratt within SPT [2,3]. From the work of Chen and Smith [16] we estimate T (∂γ/∂T ) = Ashbaugh and Pratt also found a negative slope of δ(T ) of about −7 · 10 −4 nm/K for hydration of a hard spherical cavity in SPC/E water. However, they employed a fitting formula different than (1) for large curvatures based on revised SPT [2]. Recently, Graziano used eq. (1) on Ashbaugh and Pratt's data to find ∂δ/∂T = −3.7 · 10 −4 nm/K at T = 300 K' [24]. We first exemplify curvature effects for spherical symmetry for which eq. (4) reduces to S/A = −∂γ/∂T + (∂γ/∂T )2δ/R + 2γ(∂δ/∂T )/R,(5) where the radius R > 0 for convexity and R < 0 for concavity. Let us discuss numbers at hand of the [13], i.e., negative hydration entropies. The existence of a crossover-scale larger than atomic radii may be special for water-like fluids as has been recently emphasized by Ashbaugh [3] analyzing eq. (5) taken from SPT. The special nature of the water hydrogen-bond network may lead to the relatively large (positive) value of δ when compared to LJ fluids [8,49] and contrasting the negative values of organic solvents [3]. However, for local solute concavity, per definition R < 0, and the entropy per area (3) increases with 1/\|R\|. Because the total interfacial free energy G/A rises with concave bending, increasing entropy also means a more strongly increasing enthalpic penalty. In this perspective, the physical picture emerges that for concave bending of a planar liquid-solute interface more and more hydrogen-bonds adjacent to the bent interface are broken [6], thus the enthalpic solvation penalty grows, and more configurational freedom (disorder) adjacent to the interface is created. In other words, friendly hydrogen-bonding neighbors are squeezed away in a concave environment and an interfacial water molecule has more freedom to fluctuate. The general trend makes sense as such as a extremely small apolar cavity would bear only one or two water molecules with a high entropy gain when compared to the more ordered bulk network. It indeed conforms with the experimental and simulation picture in which a few water molecules in strictly apolar cavities are typically characterized as being disordered. They possess enhanced configurational freedom (entropy) due to highly unsaturated hydrogen bonding as firstly shown by Chau for generic concave geometries [14] followed by studies for more complex cavities [12,48,43,53,21,55,42]. Hydrophobic association The free energy of hydrophobic association, ∆G, along a reaction distance r between two approaching solutes is formally given by the difference in hydration free energies between infinite separation and the final distance r 0 , i.e., ∆G = G(r = ∞) − G(r 0 ).(6) Analogously it holds for the entropy ∆S = S(r 0 ) − S(r = ∞). As frequently rationalized in literature large scale hydrophobic association is enthalpy driven, see e.g., the association of planar hydrophobic plates [56], for which the curvature corrections in (4) hardly plays a role. In this case, ∆G ≃ γ∆A, plus an enthalpic van der Waals contribution, and ∆A is the desolvated plate area. As a consequence, ∆S ≃ −∂γ/∂T ∆A and we obtain the signature of large-scale association, that is enthalpy-driven. Below a crossover scale, see the discussion above, the association of two small convex solutes is entropy-driven, as (convex) high-curvature parts of the solute-solvent interface are cut away upon association. It is by now a well-established view in the physical and biological chemistry communities that small-scale hydrophobic association is always entropy driven. That this view is generally not acceptable has been demonstrated by computer simulations and experiments of small-scale apolar pocket-ligand binding [45,12,42,20]. To rationalize this within our perspective, let us first analyze the binding of a small solute (say methane) with radius R = 0.3 nm and a planar hydrophobic interface. Let us assume that upon association half of the methane is desolvated and additionally a planar interfacial part of area πR 2 . The water-contributed binding free energy would be G ww = −2πR 2 γ(1 − 2δ/R) − πR 2 γ. Taking the values for SPC/E water at T = 300 K we obtain G ww = −19.3 kJ/mol. The entropy is given by (1) and (4) cancel each other due to the opposite sign of the local mean curvature. Fig. 1 (a) and (b) show two illustrating sketches of spherical and a more complex pocket-ligand geometry, respectively. Consequently, within perspective (1), for fully complementary apolar pocket-ligand (pl) binding, water's thermodynamic signature is the same as for large-scale hydrophobicity, i.e., ∆S pl /∆A = −∂γ/∂T,(7) that is, dominated by enthalpy. ∆A is the total solute-solvent interface area which vanishes upon association, i.e., twice the desolvated area of the ligand-water interface upon fully complementary binding. Intriguingly, eq. (4) predicts that this result is generally valid independent of the particular geometry of the pocket and ligand, as long they are fully complementary. However, this strong statement has to be weakened by following two notes. First, as sketched in Fig. 1 (a) and (b) edge effects may occur resulting in nonvanishing corrections. However, these corrections can be included by a full numerical minimization of eq. (1) for a given geometry [44]. Secondly, we emphasize again that (1) should be strictly correct only for radii of curvature larger than typical correlation lengths in the fluid. Thus the quantitative validity of prediction (7) for small-scale pocket-ligand binding in water deserves further investigation. Comparison to computer simulation results of apolar pocket-ligand binding In the following we attempt a more detailed comparison between the geometric description (1) and the results from the simulation setup of Setny et al. [42] . Local interface curvatures in the latter are large and no quantitative agreement may be anticipated. However, the geometrical analysis will be illustrative and will illuminate the qualitative validity of description (1) at hand of a well-defined model setup for apolar pocket-ligand binding. To directly refer to the geometry found in the MD simulation we estimate effective interface radii of the reference (r = ∞) and bound (r = r 0 ) states from average water density distributions calculated in the MD simulation. The density distributions by Setny et al. [42] are reprinted in Fig. 2. Spherical interface radii are then fitted to the loci r 1/2 where the water density is half the bulk density ρ(r 1/2 ) ≃ ρ 0 /2. The solute pocket features (hemi)spherical geometry and is larger than the ligand, see Fig. 2. From the water density distribution by Setny et al. [42], we estimate an average concave radius of curvature of about R 1 = −0.7 ± 0.1 nm for the pocket-water interface with the ligand being far away. In the simulations the ligand is represented by a (convex) sphere modeled by the OPLS united-atom force field for methane [30] in TIP4P water. From the average water density distribution one finds (1) without the van der Waals term and for spherical geometry together with (6) leads to the following expression for the binding free energy contributed by water-water (ww) interactions only: G ww = U ww − T S ww = −2π\|R 1 \|hγ(1 − 2δ/R 1 ) − 4πR 2 2 γ(1 − 2δ/R 2 ) + πl 2 γ,(8) with R 1 < 0 and R 2 > 0. Analogously, the water-contributed binding entropy reads S ww = 2π\|R 1 \|h [∂γ/∂T (1 − 2δ/R 1 ) − 2γ(∂δ/∂T )/R 1 ] + 4πR 2 2 [∂γ/∂T (1 − 2δ/R 2 ) − 2γ(∂δ/∂T )/R 2 ] − πl 2 ∂γ/∂T.(9) Using the thermodynamic parameters summarized in Tab. I for TIP4P at T = 300 K, we find that the water-water contribution to the total free energy is G ww = −32.6 ± 7.9 kJ/mol, to the entropy T S ww = −16.6 ± 12.8 kJ/mol, and thus the water-water energy U ww = G ww + T S ww = −49.8 ± 20.7 kJ/mol. The error mainly reflects the uncertainty we assigned to the lengths R 1 , R 2 , and l. a bit ambitious at that stage of modeling. As can be seen in Fig. 2, for instance, the newly created interface is not totally flat but may exhibit some curvature which, however, is difficult to quantify given the statistical fluctuations in the density distributions from the MD simulations. For a better quantitative performance check of description (1), a full minimization for the current pocket-ligand geometry must be performed. This is feasible but requires more detailed numerical attention due to the appearance of two free energy branches of hydration for the unbound states [43,44]. The two branches correspond to solvated (wet) and desolvated (dry) states of the pocket separated by an energy barrier. Additionally a thorough analysis of the van der Waals contribution to the enthalpy profile H(r) would be highly desirable. While this is out of scope of the current communication, a few interesting conclusions on the performance of (1) may be drawn by employing a few preliminary results [44]: the pocket-water interface profiles resulting from the minimization of (1) for the unbound reference state (r = ∞) and the bound state (r = r 0 ) of Setny's pocket-ligand system are also plotted in Fig. 2. We find that for the wet-pocket branch in the reference state, the radius of curvature of the pocket-solute interface is R 1 = −0.55 ± 0.1 nm. For the dry-pocket branch in the reference state the interface is almost flat with R 1 = −2.50 ± 0.2 nm. The MD simulation result consistently fits right between them as a result from sampling both branches in equilibrium [43]. For the bound state, minimization of (1) leads to a final, slightly concave interface with radius R = −1.5 ± 0.1 nm, see the bottom-right panel in Fig. 2. Compared to the MD results, the interface seems a bit shifted outwards from the pocket but with similar overall curvature. If we apply this curvature correction to the πl 2 term in (9) we obtain an additional ≃ +4 kJ/mol to G ww and ≃ +10 kJ/mol to T S ww , still yielding a quantitative agreement within the MD error bar and not at all altering the qualitative picture. Thus, the minimzation of (1) is fully consistent with the water density profiles and the thermodynamics calculated in the MD simulations. The enthalpy-dominated thermodynamics can be rationalized by the antagonistic effects of the concave and convex association geometry. We have demonstrated that the remarkable enthalpy-driven hydrophobic association in apolar (convex) pocket -(concave) ligand binding geometries can be rationalized by a surface-area model which properly accounts for the right sign and magnitude of the curvature correction contributions. Only the fitting of δ(T ) and its slope to hydration free energies of simple spherical solutes such as methane or xenon have been input to the model. The key prediction of this perspective is that water in fully complementary apolar pocket-ligand binding exhibits a thermodynamic signature very similar to large-scale (planar) hydrophobic association, that is, enthalpy-dominated, and hardly depends on the particular pocket/ligand geometry. The trends discussed in this paper qualitatively agree with recent computer simulations [1,42] and thereby support the validity of our conclusions down to subnanometer length scales. One interesting remark concerns the validity of prediction (7) for solvents in general. As (1) is a totally generic, geometrical description, the binding entropy in solvophobic pocket-ligand systems may be hardly geometry-dependent in any solvent, as long as the radii of the confining curvatures are larger than typical solvent correlation lengths. This can be easily tested with simple solvents, such as hard-sphere [32,36] or LJ solvents. For hard spheres the validity of the geometrical description (1) has been confirmed already for confining containers including concavity [31]. For LJ solvents, where also capillary evaporation may occur [9], the sign asymmetry in concave vs. convex solvation has unequivocally be proven by computer simulations [8] in agreement with microscopic statistical mechanics approaches [49]. Note that the special nature of the water hydrogen-bond network may lead to the relatively large (positive) value of δ when compared to LJ fluids [8,49] and contrasting the negative values of organic solvents [3]. We have investigated purely apolar and mostly spherical geometries. The reality in biomolecular systems is considerably more complex. Protein pockets, for instance, ubiquitously feature local polar groups such as backbone carbonyls or weakly polar side chain groups. Their effects on water binding makes the interpretation of the thermodynamic signature much more subtle due enhanced enthalpic contributions which in turn may compensate entropy or not [52,53,38,55]. Indeed, a single electrostatic charge placed in the simulated pocket or centered in the ligand of the system of Setny et al. may significantly change water's entropic signature [5], even depending on the sign of charge. Additionally, biological pockets exhibit strongly inhomogeneous high-curvature distributions. These subtleties obviously pose major challenges to surface-tension based hydration models as (1), in particular with respect to the inclusion of polarity [19,15], higher order curvature corrections [19], or their numerical evaluation [17]. Our work also suggests that traditional SASA models [39], which are often applied to predict binding free energies in molecular recognition processes [23,35], may require revision of surface definitions with respect to local curvature for improving their quantitative performance and predicting binding entropies. Table 1 Water parameters used in this work for a temperature T = 300 K for the SPC/E and TIP4P models. The water liquid bulk density ρ0 is taken from the work of Paschek [37]. The liquid-vapor surface tension γ(T ) and its slope have been taken from Chen and Smith [16]. The curvature parameter δ(T ) and its slope have been calculated by minimizing and fitting eq. (2) to the hydration free energy of xenon and methane for temperatures 275, 300, and 325 K obtained from explicit-water computer simulations [37]. and fitting it to hydration free energies of hydrophobic (hard-sphere-like or LJ) solutes obtained from explicit-water computer simulations (see Methods), we find positive values of δ = 0.086 nm for the SPC/E water model and δ = 0.076 nm for TIP4P at T = 300 K. This is in good agreement with literature values where 0.09±0.02 nm are typically obtained by fitting −23. 4 4kJ/mol/nm 2 for SPC/E water and T (∂γ/∂T ) = −36.7 kJ/mol/nm 2 for TIP4P water for T = 300 K. (For real water, T (∂γ/∂T ) = −27.9 kJ/mol/nm 2 ). From our fitting procedure (seeMethods)for temperatures between 275 and 325 K, we find negative values for the slope of the coefficient δ(T ) of about ∂δ/∂T = −4.8 · 10 −4 and −3.1 · 10 −4 nm/K for TIP4P and SPC/E at T = 300 K, respectively. S ww = 2πR 2 [(∂γ/∂T )(1−2δ/R)−2γ(∂δ/∂T )/R]+πR 2 (∂γ/∂T ). Summing up, the result is a negligible T S ww = +0.6 kJ/mol contributing to the total energy, i.e, the favorable entropic contribution from desolvation of the sphere and the unfavorable contribution from desolvation of the plane roughly compensate each other. Thus, the main driving force is enthalpy-driven. This in qualitative agreement with recent computer simulations on hydrophobic model enclosures, where apolar association turns from entropy-driven to enthalpy-driven when one of two small solutes is replaced by one or two planar hydrophobes[1].As argued above, concavity further increases the disorder of interfacal water with respect to a planar interface. Thus, in the geometric perspective formulated by (1), the puzzling thermodynamic controversy apparently arising in small-scale apolar pocket-ligand binding can be resolved if the antagonistic effects of concave and convex bending on water interface thermodynamics are taken into account. If the (concave) pocket and the (convex) solute exactly complement each other geometry-wise, in other words, the surrounding solute-water interface contours are identical, the curvature corrections in R 2 = 2+0.30 ± 0.01 nm. For the bound state, r 0 ≃ −0.35 nm (in Setny's definition ξ ≡ r 0 ). Upon ligand binding the concave pocket-solvent interface with radius R 1 is cut away by the ligand with a chord of length 2l = 1.08 ± 0.04 nm, as estimated from Fig. 2. The replaced interface area is then given by 2π\|R 1 \|h with h = \|R 1 \|− R 2 1 − l 2 = 0.068 nm. Further inspection of Fig. 2 reveals that the ligand with interface area 4πR 2 2 dehydrates while a new, nearly flat interface with area πl 2 is created. Application of As a result, water's signature in the hydrophobic association of the pocket and the ligand is driven by enthalpy in qualitative agreement with the MD simulations. In our perspective this is rationalized by the displacement of highly concave interface parts upon ligand binding. The actual numbers we found are not too far from those found by Setny et al. in their computer simulations, where U ww = −37.0 ± 17.3 kJ/mol and T S = −12.6 ± 17.3 kJ/mol. Note, however, that the simple calculation above still draws an idealized geometry and additionally edge effects have been neglected. Thus, a serious comparison of numbers is Fig. 1 1Illustrative sketches of fully complementary apolar pocket-ligand association geometries. Convex interface curvature are shown by lines colored in green while concave interface parts are colored in red. In (a) the ligand and pocket are both spherical with the same absolute radius \|R1\| and fully complement each other upon binding, apart from small edge effects. In (b) pocket and ligand geometries are more complex but cancel each other upon binding, so that the final state is indistinguishable from (a). In both cases the intrinsic entropic signature of the water is predicted to be given mostly by ∆S pl /∆A = −∂γ/∂T , independent of the particular ligand geometry and scaling with the desolvated interface area ∆A. Fig. 2 2Color-coded water density (ρ * ) distributions from MD computer simulations of the generic pocket-ligand system by Setny et al.[42]. The pocket-ligand distance ξ along the z-axis is equivalent to reaction-coordinate rin the main text. The spherical ligand with interface radius R2 > 0 (thick green line) desolvates upon binding (top left to bottom right). At the same time it displaces the concave hemispherical pocket-water interface with radius R1 < 0 (thick black line), with \|R2\| < \|R1\|, leaving an almost flat interface (dashed black line in the bottom right panel) with length 2l. The interface profiles drawn by red lines are the solute-solvent interface lines obtained from a full minimization of eq. 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[ "SUBMITTED TO THE IEEE INTERNATIONAL CONFERENCE ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING (ICASSP) PARTITIONING RELATIONAL MATRICES OF SIMILARITIES OR DISSIMILARITIES USING THE VALUE OF INFORMATION", "SUBMITTED TO THE IEEE INTERNATIONAL CONFERENCE ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING (ICASSP) PARTITIONING RELATIONAL MATRICES OF SIMILARITIES OR DISSIMILARITIES USING THE VALUE OF INFORMATION" ]	[ "Isaac J Sledge \nDepartment of Electrical and Computer Engineering\nUniversity of Florida\n\n", "Student Member, IEEEJosé C Príncipe \nDepartment of Electrical and Computer Engineering\nUniversity of Florida\n\n\nDepartment of Biomedical Engineering\nUniversity of Florida\n\n", "IEEELife Fellow " ]	[ "Department of Electrical and Computer Engineering\nUniversity of Florida\n", "Department of Electrical and Computer Engineering\nUniversity of Florida\n", "Department of Biomedical Engineering\nUniversity of Florida\n" ]	[]	In this paper, we provide an approach to clustering relational matrices whose entries correspond to either similarities or dissimilarities between objects. Our approach is based on the value of information, a parameterized, information-theoretic criterion that measures the change in costs associated with changes in information. Optimizing the value of information yields a deterministic annealing style of clustering with many benefits. For instance, investigators avoid needing to a priori specify the number of clusters, as the partitions naturally undergo phase changes, during the annealing process, whereby the number of clusters changes in a data-driven fashion. The global-best partition can also often be identified.	10.1109/icassp.2018.8462453	[ "https://arxiv.org/pdf/1710.10381v1.pdf" ]	3,760,616	1710.10381	4b9e3f04338d889b3023cc9d9d2b8fa953c860b1	SUBMITTED TO THE IEEE INTERNATIONAL CONFERENCE ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING (ICASSP) PARTITIONING RELATIONAL MATRICES OF SIMILARITIES OR DISSIMILARITIES USING THE VALUE OF INFORMATION Isaac J Sledge Department of Electrical and Computer Engineering University of Florida Student Member, IEEEJosé C Príncipe Department of Electrical and Computer Engineering University of Florida Department of Biomedical Engineering University of Florida IEEELife Fellow SUBMITTED TO THE IEEE INTERNATIONAL CONFERENCE ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING (ICASSP) PARTITIONING RELATIONAL MATRICES OF SIMILARITIES OR DISSIMILARITIES USING THE VALUE OF INFORMATION 1 In this paper, we provide an approach to clustering relational matrices whose entries correspond to either similarities or dissimilarities between objects. Our approach is based on the value of information, a parameterized, information-theoretic criterion that measures the change in costs associated with changes in information. Optimizing the value of information yields a deterministic annealing style of clustering with many benefits. For instance, investigators avoid needing to a priori specify the number of clusters, as the partitions naturally undergo phase changes, during the annealing process, whereby the number of clusters changes in a data-driven fashion. The global-best partition can also often be identified. INTRODUCTION The clustering of vector-based data is a critical problem, as it is encountered in many applications that involve analysis with little to no prior knowledge about the data [1]. The clustering of similarityand dissimilarity-based relational data is also important [2]. A given representation of objects may not be readily defined in terms of features yet it can be characterized by the relationships between the objects [3][4][5]. This type of representation is common in many fields, including bioinformatics, computer vision, and psychology. Regardless of the data representation, clustering is often formulated by defining a cost function to be minimized. Traditional approaches for optimizing these cost functions rely on coordinate descent to produce partitions of the data. Such approaches tend to converge only to sub-optimal solutions, as many of the cost functions are non-convex and hence contain many local minima. One way to circumvent becoming trapped in local minima during clustering is to employ simulated annealing [6]. Simulated annealing works by generating a sequence of random partitions. The decision to accept a given partitions depends on the probability of the resulting configuration. The cost is therefore not always always minimized in a monotonic fashion. The process may iteratively jump from one local-best solution to a region with a worse one. In the limit, simulated annealing will eventually reach a global minimizer. It requires a sufficiently slow cooling of parameters that influence the sequence generation for this to occur, though. This slow rate can be non-conducive for some applications. Another option for avoiding local minima during clustering is to rely on deterministic annealing [7]. Deterministic annealing bears some resemblance to simulated annealing. It inherits some positive attributes of simulated annealing. It is guaranteed to reach the global minimum, for instance. This occurs despite replacing the random walk approach to generating partitions with an expectation. It also comes with other benefits. For example, while an adjustment of parameter values is necessary during the optimization process, the cooling rate can be much quicker than in simulated annealing. This makes it attractive for many real-world applications. Due to its beneficial properties, deterministic annealing has been applied to the problem of clustering vector-based data. Several information-theoretic clustering algorithms are given by Rose This work was funded by ONR grant N00014-15-1-2013. The first author was additionally funded by a University of Florida Research Fellowship, a Robert C. Pittman Fellowship, and an ASEE Naval Research Enterprise Fellowship. and his colleagues [8,9]. In each of these frameworks, they obtained a parameterized, Shannon-entropy-based [10] free-energy expression that describes the quality of a particular partition. They have shown that the free energy is minimized by the most probable set of cluster representatives at a given parameter value. At high parameter values, there is only one local minimum, which is global by default. This minimum corresponds to a crisp partition where each vector-based data point belongs to a single cluster. As the parameter value is lowered, more clusters emerge and the cluster memberships become increasingly fuzzy. A hierarchy of partitions, with decreasing average costs, is obtained as the process undergoes a series of phase transitions. The annealing process tracks the global minimum across each phase change. While deterministic annealing has proved useful for clustering vectorial data, there has yet to be a formulation of it for clustering relational representations. In this paper, we provide an informationtheoretic formulation [11] for such data types, which is the value of information due to Stratonovich [12]. The value of information [13,14] is a type of free-energy criterion that describes the largest reduction in costs associated with a given amount of information. In the context of clustering, the number of groups is implicitly dictated by the information amount. High amounts of information lead to a small number of clusters with many elements. A potentially good qualitative partitioning of the relationships is often observed in such cases. Lower amounts of information yield larger number of clusters with fewer elements per cluster. The partitions can be qualitatively poor, as clusters are unnecessarily split. Determining the 'right' amount of information is therefore crucial. When optimized, the value of information yields a deterministic annealing process for updating the cluster memberships. A hierarchy of partitions, corresponding to differing amounts of information, is produced through the annealing process. Partitions from this hierarchy that quantize the data well can be automatically identified through analysis of a rate-distortion-like curve. This allows investigators to sidestep needing to a priori specify the number of clusters. The cluster count, and hence the 'right' amount of information, is determined in a data-driven fashion. Our approach does not require manually setting any parameters, which is a novelty of our method compared to existing, vector-data-based deterministic annealing clustering schemes. Unlike the approaches given by Rose et al. [8,9], the value of information relies on Shannon mutual information [10], not Shannon entropy. The resulting partition update equations therefore contain extra terms that account for the cluster population statistics. These additional terms lead to more qualitatively appealing partitions of the data. They also avoid producing coincident clusters as the annealing process undergoes phase changes. The approaches of Rose et al. sometimes yield coincident clusters; this would also occur if we extended their work to the relational-data case, which we show. METHODOLOGY Our approach for relational clustering can be described as follows. Given a relational-matrix-based representation of a weighted graph, we seek to partition it to produce a reduced-size graph, which we refer to as an accumulation matrix. The vertices in the accumulation matrix are are analogous to cluster prototypes in the vector-data case. There is a one-to-many mapping of a vertex from the accumulation matrix to the vertices of the relational matrix. The edges of the accumulation matrix arXiv:1710.10381v1 [cs.AI] 28 Oct 2017 codify the relative dissimilarity between pairs of prototypes. There are many possible accumulation matrices that can be formed for a given relational matrix. We would like to find one that minimizes some distortion measure. But, due to the different sizes of the relational and accumulation matrices, defining a distortion is difficult. We therefore specify how to construct a so-called composite matrix, that sidesteps this difficulty, and define an appropriate distortion between it and the original relational matrix. We provide an objective function for finding the optimal accumulation matrix from both the composite matrix and the relational matrix. We also encounter another issue: uncovering the optimal accumulation matrix is not trivial due to the binary-valuedness of the one-to-many mappings. We relax the binary assumption and offer an alternate objective function, which is based on the value of information. Optimization of the value of information yields a deterministic annealing process for finding part of the accumulation matrix. In the limit, the solution of deterministic annealing approaches the global solution of the original objective function. A hierarchy of possible partitions, each with a different number of clusters, are produced as intermediate solutions. Preliminaries In what follows, we rely on the concept of a relational matrix. A relational matrix is an adjacency-based representation of a directed graph. Definition 1.1. A relational matrix R is a matrix R n×n + given by (Vπ, Eπ, Π), where Vπ is a set of n vertices, Eπ is a set of edge connections between all pairs of vertices, and [Π]i,j = πi,j, are positive, symmetric, reflexive weights assigned to the graph edges. The subscripts on the vertices and edges represent the dependence on the particular weight matrix. For any relational matrix, we define the notion of an outgoing vector πi,1:n = [πi,1, . . . , πi,n] for the ith vertex by its weights on the outgoing edges. We assume that πi,j = 0 in the outgoing vector if and only if there is no directed edge from the ith vertex to the jth vertex. The outgoing vectors provide basis of comparison between vertex pairs. If two relational matrices, Rπ and Rϕ are of the same size, then this comparison can be performed on πi,1:n and πj,1:n according to a measure g : R n + × R n + → R+. If, however, they are of different sizes, then a composite matrix must be formed so that the distortion between the two matrices can be assessed. Definition 1.2. A partition function ψ is a mapping between two index sets such that ψ −1 (Z1:m) is a partition of Z1:n. That is, ψ −1 (j) ⊂ Z1:n, ψ −1 (j) ∩ ψ −1 (k) = ∅, for j = k, and ψ −1 (1) ∪ . . . ∪ ψ −1 (m) = Z1:n. It can be seen that a partition induces a binary accumulation matrix [Ψ]i,j = ψi,j, where ψi,j = 1 if i ∈ ψ −1 (j) and ψi,j = 0 if i / ∈ ψ −1 (j). Therefore, [Ψ] 1:n,k = i∈ψ −1 (k) ei, where ei is the ith unit vector. Definition 1.3. Given relational matrices (Vπ, Eπ, Π), with n vertices, and (Vϕ, Eϕ, Φ), with m vertices, R ϑ is the composite relational matrix (V ϑ , E ϑ , Θ), which satisfies the conditions (i) The vertex set V ϑ = Vπ ∪ Vϕ is the union of all vertices in Rπ and Rϕ. For simplicity, the composite vertex set is indexed such that the first m nodes are from Rϕ and the remaining n nodes are from Rπ. (ii) The edges in R ϑ are one-to-many mappings from the vertices in Rϕ to the vertices in Rπ. Each vertex in Rϕ represent groups of vertices from Rπ. Although R ϑ has m+ n vertices, we can represent its weighting matrix by Θ = [ϑ 1,1:m , ϑ 2,1:m , . . . , ϑ m,1:m ] . The outgoing vectors ϑi,1:n are of the same direction as πi,1:n. (iii) The partition function ψ provides an accumulation relation between the edge weights of Rϕ and R ϑ , which is given by ϕ j,k = i∈ψ −1 (k) ϑ j,k , ∀j, k. Note that, for our application, the two relational matrices will almost always be of different sizes. The first matrix, Rπ, will be the matrix specified by an investigator. The accumulation matrix, Rϕ, will essentially be a partitioning of Rπ. The objects of Rϕ correspond to relational cluster prototypes; the weights of Rϕ correspond to prototype-prototype distances. Each object in Rπ will map to some prototype in Rϕ. We can now assess the distortion of any Rπ and Rϕ. First, we define the weighted distance between corresponding outgoing vectors of Rπ and the composite matrix R ϑ assigned by the partition ψ, n i=1 p(i)g(πi,1:n, ϑ ψ(i),1:n ). Here, p(i) are a set of weights, which can be viewed as probabilities. We then use this weighted distance to define the quantization distortion between Rπ and Rϕ, which is q(Rπ, Rϕ) = min R ϑ ∈R m×n + n i=1 p(i)g(πi,1:n, ϑ ψ(i),1:n ) R ϑ ∈ Rπϕ where Rπϕ is the set of all composite matrices for Rπ and Rϕ. This objective function is over all possible sets of binary partitions. Suppose that we have a relational matrix Rπ ∈ R n×n + . We would like to find another relational matrix Rϕ ∈ R m×m + that provides a coarse representation of Rπ. Rϕ is referred to as an accumulated relational matrix. Definition 1.4. Suppose that we have a relational matrix Rπ = (Vπ, Eπ, Π) with n vertices, where the weight matrix is given by Π = [π 1,1:n , π 2,1:n , . . . , π n,1:n ] . An accumulation relational matrix is given by another relational matrix Rϕ = (Vϕ, Eϕ, Φ) that has m vertices, with a weight matrix Φ = [ϕ 1,1:m , ϕ 2,1:m , . . . , ϕ m,1:m ] , m ≤ n. An accumulation relational matrix satisfies arg min Ψ∈R n×m + , Rϕ∈R m×m + q(Rπ, Rϕ) [Ψ] 1:n,k = i∈ψ −1 (k) ei , for the positive distortion measure q : R n×n + × R m×m + → R+ given above. It is immediate that at least one minimizer for this function exists, since the number of possible binary partitions is finite. Due to the binary nature of the partitions, finding an accumulation relational matrix has an NP-hard computational complexity. Partitioning Relational Matrices We hence seek approximately optimal accumulation relational matrices that are more computationally tractable to produce. To do this, we decompose the optimization problem in definition 1.4, which is outlined by the following definition. Definition 1.5. Suppose that we have a relational matrix Rπ = (Vπ, Eπ, Π) with n vertices, where the weight matrix is given by Π = [π 1,1:n , π 2,1:n , . . . , π n,1:n ] . Suppose that we also have a composite relational matrix R ϑ = (V ϑ , E ϑ , Θ) with n + m vertices, where the weight matrix is given by Θ = [ϑ 1,1:m , ϑ 2,1:m , . . . , ϑ m,1:m ] . An accumulation relational matrix Rϕ = (Vϕ, Eϕ, Φ) that has m vertices, can be constructed in a two-step fashion: (i) Vertex grouping: Solve the optimization problem given in (1) for the positive distortion measure q given above. This has the effect of partitioning the n vertices of the relational matrix Rπ into m groups. To each group, a representative super-vertex is ascribed such that the average pairwise distance between a vertex and a super-vertex is minimized. (ii) Edge aggregation: Obtain Rϕ from the following expression: ϕ j,k = i∈ψ −1 (k) ϑ j,k using the optimal weights ϑi,j and partition matrix Ψ from step (i). To address the vertex grouping problem in the first step, we utilize the value of information. The value of information is an informationtheoretic criterion originally proposed by Stratonovich. We have previously shown how it can be used for addressing the explorationexploitation problem in reinforcement learning [15][16][17]. We have also arg min Ψ∈R n×m + , R ϑ ∈R m×n + n i=1 p(i)g(πi,1:n, ϑ ψ(i),1:n ) R ϑ ∈ Rπϕ, [Ψ] 1:n,k = i∈ψ −1 (k) ei (1) min P ∈R m×n + minj n i=1 p(i)g( demonstrated that using the value of information leads to a clustering of the state-action space according to the value function. For the problem of clustering relational data, the value of information can be used to quantify the expected amount of decrease in the matrix-matrix distortion associated with changes in information. Information, in a clustering context, corresponds to how finely we are partitioning the data. Low amounts of information correspond to many clusters and fuzzy cluster memberships. High amounts of information correspond a single cluster and crisp cluster memberships. The choice of the 'right' amount of information for a given dataset, and hence the number of clusters, can be made automatically by processing a rate-distortion-like curve that relates the distortion of Rπ and Rϕ to information. We utilize the value of information, given in (2), to induce a soft partitioning of the relational matrix. This leads to the notion of a soft accumulation matrix. The problem of finding soft accumulation matrices is specified by the above constrained optimization problem. To effectively solve this problem, we form the Lagrangian and differentiate it. This provides a grouped coordinate descent procedure for specifying the non-negative association weights. Proposition 1.1. For a given relational matrix Rπ, the accumulation relational matrix Rϕ can be found from the non-negative association weights determined by the following alternating updates p (k) (j) ← n i=1 p(i)p(j\|i), p(j\|i) ← p(j)e −βg(π i,1:n ,ϑ j,1:m ) m j=1 p(j)e −βg(π i,1:n ,ϑ j,1:m ) , which are iterated until convergence. Here, β ≥ 0 is a Lagrange multiplier that emerges from the Shannon mutual information constraint. The variable p(j) corresponds to the cluster population statistics. Substituting the association weights from proposition 1.1 into the Lagrangian yields F (Rπ, R ϑ ) = − 1 β n i=1 p(i)log m j=1 e −βg(π i,1:n ,ϕ j,1:m ) . At each grouped-coordinate descent iteration, the Lagrange multiplier β is fixed and a local minimum of the Lagrangian is found. That is, the representative outgoing vectors ϑj,1:m are computed using the following implicit equation ∇ ϑ j,1:m F (Rπ, R ϑ ) = n i=1 p(i)p(j\|i)∇ ϑ j,1:m g(πi,1:n, ϑj,1:m) = 0. This equation can be solved using gradient descent methods where the solutions from the previous iterations are used as the starting values for the current iteration. These computations are repeated as the multiplier β is increased, leading to an annealing-like process. For small values of β, this procedure finds the global minimum of the Lagrangian. This minimum is tracked as β is iteratively increased. The effects of β are as follows. As β tends to zero, minimizing the Lagrangian is approximately same as minimizing the negative Shannon information. Shannon information is known to be convex and hence has a global minimizer. In this case, the weights are approximately uniform, p(j\|i) ≈ m −1 ∀i, j, so all outgoing vectors ϑj,1:m are coincident. There are hence many clusters, and every object in Rπ has the same fuzzy membership to each cluster. As β is increased, the soft accumulation matrix becomes more crisp. Smaller number of clusters are formed. Moreover, the annealing process exhibits a series of phase transitions where the outgoing vectors ϑj,1:m are insensitive to changes in β except at critical values. The number of distinct outgoing vectors in the composite relational matrix increases at these critical values. When β approaches infinity, the information constraint is essentially ignored. Thus, minimizing the Lagrangian is the same as minimizing the relational-matrix distortion q between Rπ and Rϕ. We therefore obtain an almost-crisp partition, p(j\|i) ≈ 1 ∀i, j. We also begin to recover the relationalmatrix distortion function over binary partitions. This crisp partition contains only a single cluster. Determining Number of Clusters. Our approach to clustering relational matrices entails iterating over a range of β values from small to large. Each value of β between two critical values leads to accumulation matrices that define partitions with different number of clusters. This entire process yields a hierarchy of partitions. Investigators are often interested in obtaining only a single partition, containing the 'right' number of clusters, that 'best' fits the observations. We hence consider an automated heuristic of choosing a parsimonious partition from the hierarchy that is produced. Our heuristic is based on comparing the amount of information r against the relational-matrix dissimilarity between Rπ and Rϕ. Such a comparison leads to a rate-distortion like curve, which often contains a knee-like region for some moderate information amount. Our studies have shown that partitions around the knee region often qualitatively partition the data well with few to no unnecessary clusters. A good partition in this region can be easily detected via: (i) Iterating over each point along this curve. For each point, we fit two linear functions that bisect it: one of which is a leastsquares fit to the part of the curve that is to the left of the bisector and one that is fit to the part of the curve to the right. (ii) Finding the point that leads to the lowest sum of leastsquared errors for the two linear functions, which almost always corresponds to the knee. The partition corresponding to this amount of information r (as quantified by β) is returned. EXPERIMENTS To assess our approach, we consider three relational datasets from real-world applications. We have previously analyzed these datasets in [18,19], in the context of relational cluster validity. (i) RGD-30: This data was formed from a combination of cDNA microarray gene expressions and gene ontology similarities of 30 genes related to cell apoptosis in human lymphomas. -198 . For each row, the first plot is the relational matrix of dissimilarities. Dark colors correspond to low dissimilarity, while lighter colors correspond to higher dissimilarity. The second and third plots are the crisp partitions returned for out method when using Shannon mutual information and Shannon entropy, respectively, to specify the constraint for the value of information (VOI) criterion. We converted the fuzzy partitions to crisp partitions for display purposes. The fourth plot provides a rate-distortion curve which compares the free-energy magnitude to the amount of information. We converted the information amount to the number of clusters to ease interpretations of these plots. Circles in these plots are provided to highlight the number of clusters chosen by our automated knee-detection heuristic. (ii) RHGP-194: This data was constructed by applying gene ontology similarity measures to 194 human gene products. The data are composed of three groups or 21, 86, and 87 gene products from the myotubularin, receptor precursor, and collagen alpha chain protein families, respectively. (iii) RATA-198: This data was created from a combination of gene ontology similarity and microarray gene expressions on 198 genes from the Arabidopsis Thaliana plant. The plant was subject to a variety of stresses from insects and stress controls. Note that the relationships for each of these datasets cannot be considered as pairwise distances between latent set of vectors. They were generated completely from similarity measures. We hence cannot expect to perform multi-dimensional scaling [20,21], to be able to apply vector-data deterministic annealing clustering algorithms, without disrupting the cluster structure [22]. Results and Discussions We applied our clustering approach to the above three relational datasets. Note that our approach has no parameters that must be manually set. Corresponding results are presented in figure 1. For each dataset, we thresholded the fuzzy association weights and overlaid the resulting crisp partition on top of the relationships. We expect to see these partitions segment the dark, blocky structures along the main diagonal. These blocky structures correspond to compact, low-dissimilarity object groups [?]. Light values on the offdiagonal indicate that these groups are well separated. We re-ordered the dissimilarities according to the visual assessment of cluster tendency algorithm [23][24][25] to better highlight the latent data structure. In figure 1, we also provide results for the case where Shannon entropy is used as a constraint for the free-energy criterion instead of Shannon mutual information. This is a direct extension of Rose's deterministic annealing method for the relational-data case. Clustering Results. For RGD-30 and RHGP-194, the value of information with the Shannon mutual information constraint returned a fuzzy partitions with c = 4 and c = 3 clusters, respectively. These partitions are consistent with the dark blocks present along the main diagonal; compact, well-separated clusters are therefore being properly identified. As we explain in the online appendix 1 , this partitioning of genes best aligns with their biological functionality. RATA-198 was, comparatively, more challenging due to the sparse nature of the gene relationships. There were genes that naturally grouped into many small clusters that were compact and wellseparated. A total of c = 12 clusters were identified by our method. This partitioning aligns with the visual interpretation of cluster structure according to the re-ordered dissimilarity plot: compact, wellseparated groups are properly segmented. Some of these clusters also have biological significance, as we explain in the online appendix. The value of information with the Shannon entropy constraint returned fuzzy partitions with c = 9, c = 12, and c = 24 clusters, respectively, for RGD-30, . This approach had the tendency to over-segment the data. Objects that were slight outliers were frequently assigned to a singleton cluster, which led to a large number of 'unnecessary' clusters. Coincident partitions were also returned, which redundantly described the natural object groupings. Manually aggregating these coincident clusters led to similar object groupings as the mutual-information-constrained approach. Cluster Validity Results. To quantitatively assess the goodness of our results, we applied twenty relational cluster validity indices [18] to the hierarchy of crisp and fuzzy partitions produced. These included the generalized Dunn's indices [26], modified Hubert's statistics [27], and the Xie-Beni index [28]. For RGD-30, almost every index selected c = 4 as the best estimate for the number of clusters. For RHGP-194, a majority of these indices chose c = 3 as the best cluster count estimate. These results agree with both the visual partitioning of the data and the corresponding knees of the rate-distortion curves. We can hence conclude that our approach is identifying the cluster structure well for these two datasets. Some of the validity indices for RGD-30 favored partitions with c = 6 or 7 clusters. Such partitions separated each of the genes in the bottom, right corner of the relational matrix into singleton clusters. Likewise, for RHGP-194, some indices selected partitions with c = 4 or c = 5. These partitions identified the cluster sub-structure for the bottom-right block. The conflicting findings for both datasets are not necessarily incorrect, as there is biological evidence to support such partitions of the data. However, such partitions do not lead to a parsimonious set of well-separated clusters. For RATA-198 there was no clear validity index consensus. Some indices favored c = 6 or c = 8 clusters, which is not completely consistent with a visual inspection of the groups highlighted by the re-ordered relationships. Other indices suggested that c = 12 or c = 14 clusters best describe the data, which better aligns with our results. Our previous studies with this data indicate that there are viable explanations for each of these cluster counts. CONCLUSIONS In this paper, we have proposed a deterministic-annealing-based approach to relational clustering. Our approach is based upon producing a type of partition matrix, known as an accumulation matrix, that quantizes the original relational matrix. We rely on an informationtheoretic criterion, the value of information, to specify a computationally feasible procedure for finding globally optimal accumulation matrices. The value of information trades off against the amount of information against the quantization fit of the accumulation matrix to the original relational data. Ranges of information amounts lead to different number of clusters. The information amount also dictates the fuzziness of the partitions. Both of these properties are data-dependent: the best values for one dataset may not work for another. We hence provided a heuristic for choosing the 'right' amount of information, and hence a parsimonious partition, in a data-driven fashion; no parameters need to be set by investigators when using our clustering approach. This heuristic performed well for the complicated datasets that we considered. Definition 1 . 6 . 16Suppose that we have a relational matrix Rπ = (Vπ, Eπ, Π) with n vertices, where the weight matrix is given by Π = [π 1,1:n , π 2,1:n , . . . , π n,1:n ] . A soft accumulation relational matrix is given by another relational matrix Rϕ = (Vϕ, Eϕ, Φ) that has m vertices, with a weight matrix Φ = [ϕ 1,1:m , ϕ 2,1:m , . . . , ϕ m,1:m ] , m ≤ n. 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[ "Shift of percolation thresholds for epidemic spread between static and dynamic small-world networks", "Shift of percolation thresholds for epidemic spread between static and dynamic small-world networks" ]	[ "J K Ochab \nSmoluchowski Institute of Physics\nJagellonian University\nReymonta 430-059KrakwPoland\n", "P F Góra \nSmoluchowski Institute of Physics\nJagellonian University\nReymonta 430-059KrakwPoland\n" ]	[ "Smoluchowski Institute of Physics\nJagellonian University\nReymonta 430-059KrakwPoland", "Smoluchowski Institute of Physics\nJagellonian University\nReymonta 430-059KrakwPoland" ]	[]	The aim of the study was to compare the epidemic spread on static and dynamic small-world networks. The network was constructed as a 2-dimensional Watts-Strogatz model (500 × 500 square lattice with additional shortcuts), and the dynamics involved rewiring shortcuts in every time step of the epidemic spread. The model of the epidemic is SIR with latency time of 3 time steps. The behaviour of the epidemic was checked over the range of shortcut probability per underlying bond φ = 0 − 0.5. The quantity of interest was percolation threshold for the epidemic spread, for which numerical results were checked against an approximate analytical model. We find a significant lowering of percolation thresholds for the dynamic network in the parameter range given. The result shows that the behaviour of the epidemic on dynamic network is that of a static small world with the number of shortcuts increased by 20.7 ± 1.4%, while the overall qualitative behaviour stays the same. We derive corrections to the analytical model which account for the effect. For both dynamic and static small-world we observe suppression of the average epidemic size dependence on network size in comparison with finite-size scaling known for regular lattice. We also study the effect of dynamics for several rewiring rates relative to latency time of the disease. * [email protected] † Also at Mark Kac Complex Systems Research Centre. 1 nature, and secondly, the contact network of individuals affected by the disease may change in time as the epidemic spreads. These features make epidemiological models a part of larger studies of dynamics on complex networks, but also dynamics of complex networks.Research findings of the epidemic spread on dynamic networks include its behaviour on adaptive networks, where the susceptible are able to avoid contact with the infected [5]), however a coupling between the epidemic and the network dynamics does not necessarily exist. For instance, in [1], spread of the aforementioned plant diseases is modelled by vectors performing random walk on the network, thus infecting individuals on their paths; Saramäki and Kaski [12] utilise SIR (Susceptible-Infectious-Removed) mechanism on a dynamically changing small-world contact network, although mainly time development of the epidemic is of their interest. Likewise, in[15](where focus is on the average epidemic size in time) nodes of the contact network can swap their edges at a given rate, preserving the degree distribution. It is also worth to note[11], where disease spread was simulated on a weighted contact network produced from real day-to-day encounters (as weights represent the frequency of encounters, the dynamics has been in a sense projected onto static weighted network).While dynamic network models have been applied in the recent research, it seems that we lack comparative study on how the dynamics of the network influences the process that takes place on it. The aim of this paper is to find and quantify this effect for SIR epidemic spread on static and dynamic smallworld networks. Based on known analytical calculations for static small-world network [7] we derive corrections accounting for the dynamics of the network, and check the results against numerical agent-based simulations.	10.1140/epjb/e2011-10975-6	[ "https://arxiv.org/pdf/1011.2985v1.pdf" ]	15,806,182	1011.2985	ad934342d6a73153e1946af3125ceab1e82768b0	Shift of percolation thresholds for epidemic spread between static and dynamic small-world networks 12 Nov 2010 November 15, 2010 J K Ochab Smoluchowski Institute of Physics Jagellonian University Reymonta 430-059KrakwPoland P F Góra Smoluchowski Institute of Physics Jagellonian University Reymonta 430-059KrakwPoland Shift of percolation thresholds for epidemic spread between static and dynamic small-world networks 12 Nov 2010 November 15, 2010 The aim of the study was to compare the epidemic spread on static and dynamic small-world networks. The network was constructed as a 2-dimensional Watts-Strogatz model (500 × 500 square lattice with additional shortcuts), and the dynamics involved rewiring shortcuts in every time step of the epidemic spread. The model of the epidemic is SIR with latency time of 3 time steps. The behaviour of the epidemic was checked over the range of shortcut probability per underlying bond φ = 0 − 0.5. The quantity of interest was percolation threshold for the epidemic spread, for which numerical results were checked against an approximate analytical model. We find a significant lowering of percolation thresholds for the dynamic network in the parameter range given. The result shows that the behaviour of the epidemic on dynamic network is that of a static small world with the number of shortcuts increased by 20.7 ± 1.4%, while the overall qualitative behaviour stays the same. We derive corrections to the analytical model which account for the effect. For both dynamic and static small-world we observe suppression of the average epidemic size dependence on network size in comparison with finite-size scaling known for regular lattice. We also study the effect of dynamics for several rewiring rates relative to latency time of the disease. * [email protected] † Also at Mark Kac Complex Systems Research Centre. 1 nature, and secondly, the contact network of individuals affected by the disease may change in time as the epidemic spreads. These features make epidemiological models a part of larger studies of dynamics on complex networks, but also dynamics of complex networks.Research findings of the epidemic spread on dynamic networks include its behaviour on adaptive networks, where the susceptible are able to avoid contact with the infected [5]), however a coupling between the epidemic and the network dynamics does not necessarily exist. For instance, in [1], spread of the aforementioned plant diseases is modelled by vectors performing random walk on the network, thus infecting individuals on their paths; Saramäki and Kaski [12] utilise SIR (Susceptible-Infectious-Removed) mechanism on a dynamically changing small-world contact network, although mainly time development of the epidemic is of their interest. Likewise, in[15](where focus is on the average epidemic size in time) nodes of the contact network can swap their edges at a given rate, preserving the degree distribution. It is also worth to note[11], where disease spread was simulated on a weighted contact network produced from real day-to-day encounters (as weights represent the frequency of encounters, the dynamics has been in a sense projected onto static weighted network).While dynamic network models have been applied in the recent research, it seems that we lack comparative study on how the dynamics of the network influences the process that takes place on it. The aim of this paper is to find and quantify this effect for SIR epidemic spread on static and dynamic smallworld networks. Based on known analytical calculations for static small-world network [7] we derive corrections accounting for the dynamics of the network, and check the results against numerical agent-based simulations. Introduction The epidemic modelling has become a significant and needed branch of complex systems research, as we have witnessed the recent epidemic threats and outbreaks of human diseases (H5N1 and H1N1 influenzas [10,8] or severe acute respiratory syndrome [9,2]) or animal (foot-and-mouth disease [6]) and plant diseases alike (e.g. Dutch elm disease [14] or rhizomania [13]). There are two crucial characteristics of the epidemic spread that make it complicated to be modelled on the one hand, and costly to be prevented in reality on the other: firstly, a number of infectious diseases exhibit long-range transmissions of varied 2 Model Network We adopt Watts-Strogatz model of a small-world network [16]: first we take a 2-dimensional square lattice with N = L 2 nodes and 2N undirected edges. To avoid some finite-size effects we impose periodic boundary conditions for the lattice (i.e. we get a torus). Then, we add a number of undirected edges between random pairs of nodes. The number of additional edges ('shortcuts') is set as 2φN , hence φ is shortcut probability per underlying bond. Network with φ = 0 is just a regular lattice. For nonzero φ we call the network a static small-world. The third type of network is a dynamic small-world. One can construct it by randomly distributing shortcuts in every time step of simulation. Here, we choose 2φN nodes randomly, and keep them fixed for the whole run of the epidemic. In every time step we randomly launch shortcuts anchored in these nodes, which means the dynamics consists in rewiring one end of these shortcuts. For the sake of simplicity we allow for multiple shortcuts being incident with the same node, for shortcuts leading to nearest neighbours, and for loops being formed. The construction of the source nodes launching shortcuts allows for an easier interpretation of the network: the fixed nodes could correspond to centres of activity that can be identified as in the real world networks. Epidemic The SIR (Susceptible-Infectious-Removed) model is adopted, where the disease is transmitted along the edges of the network in discrete time steps. The probability p of infecting a susceptible node by an infectious neighbour during one time step is set equal for short-and long-range links, both static and dynamic. The latency time l of the disease is measured in discrete time units (we take l = 3, 4). Thus, an infectious node can transmit disease to susceptible nodes with probability p every turn for the period of l turns, and after that time it is removed, i.e. it cannot infect nor be reinfected. Every simulation starts with only one initially infecting node, all others being susceptible, and it ends when no node in the infectious state is left. Sample snapshots of the epidemic time development are presented in Fig.2. Grassberger [4] related the probability of infection to the probability T in bond percolation through T = l t=1 p(1 − p) t−1 = 1 − (1 − p) l , where T is the so called transmissibility (it is the total probability of a node infecting one of its neighbours during the whole latency time). In the case of 2-dimensional square lattice the bond percolation threshold is T c = 0.5. 3 Numerical data Parameters of simulations The linear lattice size used for most calculations is L = √ N = 500. In Section 5.2 we take sizes L = 50, 63, 79, 100, 126, 158, 199, 251, 315, 397, 500. The disease latency is set to l = 3 (for faster simulations reported in Sec. 5) or l = 4 (in Sec. 5.3 in order to get larger set of dynamic rates). The range of probability p scanned is p = 0.05 − 0.22 (depending on φ) with resolution of 1/1024, which translates to around T = 0.15 − 0.5. For every p and φ the epidemic is run 1024 times with random distributions of shortcuts each time. The fraction of shortcuts is φ = 0 − 0.5, with steps of 0.025. The simulations are performed for both static and dynamic small-world network. Calculating percolation threshold In the study of the epidemic spread on networks, we stick to the percolation theory as a reference point. In the theory, a percolation threshold would be the value of p that generates an epidemic cluster spanning between the boundaries of the whole system. Otherwise, it is possible to define percolation as the point at which a cluster of macroscopic size forms (i.e. it occupies a finite fraction of the system for N → ∞). We employ the latter to define percolation threshold (numerically) as the point at which the average epidemic's size divided by N rises above a certain value (here, set to 0.00115). The average is taken over a number of reruns for different shortcut drawings. As we can perform simulations only for finite sizes, we take the results for a relatively large network of √ N = 500. The choice of the threshold value is taken so as to calibrate the results for the static network to the previously confirmed analytical result. We take as the theoretical model [7], where the generating function and series expansion methods were used to find the approximate position of bond percolation transition in 2D small-world network, which corresponds to the epidemic spread on what we refer to as static small-world. Theoretical analysis We can account for the change between static and dynamic networks analytically using the model known for static small-world network [7]. As the original theory has no time variable, it would be a hard task to introduce dynamics explicitly. The solution, however, is astonishingly simple. One can estimate the average number of nodes infected through shortcuts during latency time l: N stat = φ stat N · T = φ stat N · l t=1 p(1 − p) t−1 ,(1) i.e. the number of shortcuts in the static network multiplied by the total probability of infecting a neighbouring node (this probability is the same for both regular links and shortcuts). The analogous expression for the dynamic network is found easily N dyn = φ dyn N · l i=1 i l i p i (1 − p) l−i = φ dyn N · lp ,(2) where the sum is an average number of infections transmitted by a single source of dynamic shortcuts for a given latency time. It comes from the fact that a dynamic shortcut can pass infection several times (the factor p i ), while in the static case a node could infect only once (since nodes cannot be reinfected in the SIR model). This expression predicts lowering percolation thresholds, although numerical values of the shift are considerably smaller than the ones obtained from simulations. Figures 3(a)-3(c) explain why the above expression is not yet correct: it is derived only for the source nodes passing the disease on, while it disregards the fact that the node may itself become infected via long-range link. Since on the static network there is no difference between shortcuts' source and target nodes, we can attach the factor φN/2 to both infection graphs presented in Fig.3(a). For dynamic network, the graphs in Fig.3(b)-3(c) for infecting a source node through a regular link and through a dynamic link give different counts of how many shortcuts were used. The former was given in Eq.2 as lp, and the latter actually utilises the same formula, but with the substitution l → l + 1. In total, we get N dyn = φ dyn N/2 · lp + φ dyn N/2 · (l + 1) p .(3) We assume that N dyn = N stat if the epidemic on both networks has the same percolation threshold. Thus, we can obtain the ratio of the two shortcut densities r(p, l) = φ stat /φ dyn = p (l + 1/2) T = p (l + 1/2) 1 − (1 − p) l ,(4) where p is the probability of infection in one time step and l latency time of a disease. Now, we can calculate T c (rφ)) numerically, just as we do it with the fitted Fig.4. The ratio in Eq.4 was used to plot the lower solid line in Fig.4. T c [(1 + v)φ)] in Results Shift of percolation thresholds In Figure 4 we plot numerical and theoretical values of percolation thresholds T c for both static and dynamic small-worlds. The resulting T c (φ) data points for static small-world network agree with the analytical approximation [7], which confirms the validity of calibration procedure. As the lower dataset marks the effect of network dynamics, the difference between the two networks proves to be systematic and significant. The dashed line is a fit T c [(1 + v)φ] of the analytical model for the static network, where the fitted parameter v may be interpreted as a virtual percentage of additional shortcuts needed to obtain the dynamic network percolation thresholds. It follows from the fit that percolation thresholds for dynamic network are lower as if the shortcut density were (1+v)φ (where v = 0.207 ± 0.014 is the fitted parameter). Nonetheless, qualitatively the epidemic on dynamic small world behaves in the same way as on the static one for the given range of parameters (φ = 0.5 corresponds to every node in the network having on average two additional links). The analytical correction slightly exceeds the values of simulation data points, but the overall agreement is satisfactory. The difference between the analytical solution and the observed behaviour does not exceed the shift between static and dynamic networks obtained from simulations. The discrepancy might be due to the method of calculating percolation thresholds from numerical data or due to the approximate nature of the correction. Suppression of finite-size scaling The primary motivation of checking finite-size scaling for the system was to utilise it to determine the percolation thresholds very accurately (as the shift of thresholds observed in Fig.4 is relatively small), and to arrive at threshold value for infinite system size. Yet, it is worth noting at this point that the knowledge of thresholds for infinite system sizes would not usually be appropriate for evaluation of risks in the real epidemic, given the sizes of some real networks. To study the size of finite-size effects is thus vital on its own right. In Figure 5(b) the convergence of the average epidemic size to the threshold behaviour can be observed, and the significant dependence on system size ranges up to the epidemic size of around 0.5N and interval of transmissibility of length around 0.08 (the numbers are very rough estimates). As presented in Fig.5(a) T N = T ∞ − N −1/ν = T ∞ − L −2/ν ,(5) where T N are the values of transmissibility for a given system size N and a set section position, and T ∞ is the percolation threshold for infinite system size. For regular lattice T ∞ is fitted correctly for various section positions as 0.500±0.005 (the error may vary for different sections, but does not exceed the given value). It appears that the dependence on system size for small-world networks (both static and dynamic) is dissimilar to the one of regular lattices, as can be seen in Fig.5.2 (φ = 0.05). It is suppressed to smaller values of the average epidemic size. For the shortcuts density φ = 0.5 the dependence on system size is already visible only below the epidemic size of 0.03. Because the dependence of the epidemic size on size of the system becomes of the order of magnitude of statistical fluctuations (the quality of the data can already be seen in the Fig.5.2), any attempts to utilise finite-size scaling for determining percolation threshold are not viable. Indeed, the errors do not allow us to check if the same form of finite-size dependence as in Eq.5 holds. 5.3 Dependence on the rate of dynamics One can generalise the theoretical analysis for various rates of dynamics, given the formula in Eq.2. To explain this, let us notice that there are two time scales in the model: the latency time l of the infection and the duration 1/d between consecutive rewirings of dynamic links (both measured in discrete time steps of the epidemic spread). As the choice of latency l only rescales the total probability of infection T = T (p, l), we can dispose of it, and the crucial parameter ld that accounts for the shift of percolation thresholds is defined as the number of shortcut movements during latency time. Obviously, for a static network we get d = 0, while for all the above analysis of dynamic network we have ld = 3 (l = 3 and the rewiring was performed every turn, so d = 1). Depending on the interpretation of the model, we could also consider d > 1. However, if p is to be the probability of infection during one time step it is reasonable that shortcuts rewiring faster than one time step would infect with appropriately smaller probability, and there would be no further shift of percolation thresholds. Since the epidemic spreads with discrete time, which results in sums as in Eq.2, we are interested in rational numbers d ∈ [0, 1]∩Q, particularly of the form 1/i, i ∈ Z. What we need is N dyn calculated in a similar way to that in Eq.2. Here, we take l = 4, d = 1, 1/2, . . . , 1/7, and we plot both the numerical and theoretical results for φ = 0.25 in Fig.7. Theoretical derivation is to be found in the Appendix. The theoretical approach gives slightly exceeding values (the scale should be noted), which is the same effect as discussed at the end of Section 5.1. Discussion We have shown that introducing dynamics of the long-range links in a smallworld network significantly lowers an epidemic threshold in terms of probability of disease transmission, although the overall dependence on number of shortcuts stays the same. Consequently, the risk of an epidemic outbreak is higher than in any calculations involving static models. The effect remains secondary to the influence that the introduction of additional of shortcuts has on the spread of the disease. It should be noted that the shift of percolation thresholds depends on the relative measure of dynamics of the network with respect to the process on the network (rewiring rate and latency time, respectively). Any accurate analytical calculation or simulation should take this quantity as a significant parameter, to be estimated for a particular disease and type of the network. As in reality we consider only finite-size networks, and real epidemic sizes do not usually reach values of the order of even 10% of the system size, the information on finite-size effects seemed very much needed. That the epidemic outbreak magnitude does not depend on the system size for small-world networks as much as it does for regular lattices means that we should not expect the epidemic outbreaks below transmissibility threshold value. Thus, finite-size effects seem to become secondary, as well. The usefulness of such a model for risk prediction still depends on our knowledge of the probability of transmission (p or T ) of a given disease, which is not easy to obtain for diseases spreading outside of well controlled environments like hospitals. Relatively good estimates, thanks to the nature of transmission, exist for syphilis. Transmissibility of the disease is reviewed in [3], where authors give values ranging from 9.2% to 63% per partner, and decide on 60% as the lower boundary for untreated disease. This seems to be well above the epidemic threshold, irrespective of very different network topology for such diseases. However, this also shows that errors on estimates of transmission probabilities exceed the effect of threshold shifting studied here. Though the 2-dimensional network structure used here may be said to correspond mainly to that of plantations, it is worth noting its generality: nodes may be interpreted as plants, animals or humans, but also on a larger scale as farms, households, or cities and airports; in turn, long-range links could mean wind (on farms), disease vectors, occasional human contacts, or airline connections. Still, it has some other fairly realistic characteristics: according to [11], who analysed the structure of human social interactions, 'the majority of encounters (76.70%; 75.26-78.07) occur with individuals never again encountered by the participant during the 14 days of the survey.' This may mean that about 24% of the repeated contacts corresponds roughly to our regular underlying lattice with z = 4 neighbours for each node, while the 76% correspond to around 3z dynamic contacts distributed over over 14 days. This gives on average φ ≈ 0.20 for simulation with daily time steps, which lies within the parameter range studied in this paper. A 0 (p, 5) = · · · · · A 1 (p, 5) = ·\| · · · · + · · · · \|· = 2 · \| · · · · A 2 (p, 5) = · · \| · · · + · · · \| · · = 2 · ·\| · · · (7) A 11 (p, 5) = ·\| · · · \| · A 12 (p, 5) = ·\| · ·\| · · + · · \| · ·\|· = 2 · \| · ·\| · · where the symbol '\|' marks rewiring, and '·' one epidemic time step during latency period. For instance ·\| · · would correspond to three turns with one rewiring, during which either 0, 1 or 2 infections are possible. The derivation involves only very easy combinatorics, but for longer latency periods one would need to repeat these calculations to obtain more terms and different prefactors. Now, one can easily obtain expressions for N dyn for any 1/d ∈ Z. Below we give only the general expression for 1/d ≥ l: where l = 4. The first term in the brackets corresponds to Fig.3(b) and the second to Fig.3(c). For greater numbers of rewiring per turn d, we need to consider the terms A 11 , A 12 . The result is plotted against simulated data in Fig.7. N dyn = φ dyn N 2 d{[ Figure 1 : 1(a) Regular 2D grid with periodic boundary conditions (torus). (b) Watts-Strogatz 2D small-world network: 2D grid with shortcuts added to it. (c) Dynamic small-world: all the long-range links connected to a set of source nodes randomly rewire in time. Figure 2 : 2(Colour online) Snapshots of the epidemic spread slightly above percolation threshold. L = 512, the number of shortcuts is 10 (which gives φ = 2 · 10 −5 ). t gives the epidemic's time steps. The snapshots for t = 364, 694 show a dynamic infection (the two joined blue lines appear). Figure 3 : 3(a) Infections through static shortcuts are symmetric. (b) Infection of the dynamic shortcuts' source through regular lattice. (c) Infection of the dynamic shortcuts' source through a shortcut. , one may check that sections of the plot for a given average epidemic size obey Figure 4 : 4(Colour online) Circles dataset -the static small-world. Squares dataset -the dynamic network. The solid blue (upper) line is the analytic approximation [7] for T c (φ) and the dashed line gives T c [(1 + v)φ)], with the fit parameter v = 0.207±0.014. The solid purple (lower) line represents theoretical approximation from Sec. 5.3. Error bars are of the size of the plot markers, unless visible. scaling of the form Figure 5 : 5The behaviour of the epidemic outbreak magnitude for various system sizes (linear size vary between L = 50 − 500, left-and rightmost data points in (b), respectively). (a) Finite-size scaling T N = T ∞ − L −2/ν on regular lattice. The points correspond to values of T at the level of the epidemic size 0.1. (b) The extent of size dependence for regular lattice. Figure 6 : 6For dynamic small world size dependence of the epidemic outbreak magnitude is suppressed. Inset shows enlarged region around percolation threshold. Figure 7 : 7Dependence on dynamics for φ = 0.25, latency l = 4. AcknowledgmentsThis work is supported by the International PhD Projects Programme of the Foundation for Polish Science within the European Regional Development Fund of the European Union, agreement no. MPD/2009/6.Appendix: Dependence on the rate of dynamicsBelow we present the way to calculate N dyn for latency periods l = 4, 5 (in the simulation we set l = 4, but we need to take into account also the process fromFig.3(c), which in a sense increases latency by 1). Let us definewhere we substituted T (1) for p on the right-hand sides, and we leave out the argument p in T (p, l) to simplify the notation. Those quantities correspond to the average number of infections during one latency period depending on when the rewiring takes place. 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[ "Fermionic T-duality and momenta noncommutativity ", "Fermionic T-duality and momenta noncommutativity " ]	[ "B Nikolić \nInstitute of Physics\nUniversity of Belgrade\nP.O.Box 5711001BelgradeSerbia\n", "B Sazdović \nInstitute of Physics\nUniversity of Belgrade\nP.O.Box 5711001BelgradeSerbia\n" ]	[ "Institute of Physics\nUniversity of Belgrade\nP.O.Box 5711001BelgradeSerbia", "Institute of Physics\nUniversity of Belgrade\nP.O.Box 5711001BelgradeSerbia" ]	[]	In this article we establish the relationship between fermionic T-duality and momenta noncommuativity. This is extension of known relation between bosonic Tduality and coordinate noncommutativity. The case of open string propagating in background of the type IIB superstring theory has been considered. We perform T-duality with respect to the fermionic variables instead to the bosonic ones. We also choose Dirichlet boundary conditions at the string endpoints, which lead to the momenta noncommutativity, instead Neumann ones which lead to the coordinates noncommutativity. Finally, we establish the main result of the article that momenta noncommutativity parameters are just fermionic T-dual fields.	10.1103/physrevd.84.065012	[ "https://arxiv.org/pdf/1103.4520v2.pdf" ]	118,494,813	1103.4520	b3d1676a71d466206ff52b41461fb7f73d1d017f	Fermionic T-duality and momenta noncommutativity * 15 Jun 2011 January 16, 2013 B Nikolić Institute of Physics University of Belgrade P.O.Box 5711001BelgradeSerbia B Sazdović Institute of Physics University of Belgrade P.O.Box 5711001BelgradeSerbia Fermionic T-duality and momenta noncommutativity * 15 Jun 2011 January 16, 2013number(s): 1110Nx0420Fy1110Ef1125-w In this article we establish the relationship between fermionic T-duality and momenta noncommuativity. This is extension of known relation between bosonic Tduality and coordinate noncommutativity. The case of open string propagating in background of the type IIB superstring theory has been considered. We perform T-duality with respect to the fermionic variables instead to the bosonic ones. We also choose Dirichlet boundary conditions at the string endpoints, which lead to the momenta noncommutativity, instead Neumann ones which lead to the coordinates noncommutativity. Finally, we establish the main result of the article that momenta noncommutativity parameters are just fermionic T-dual fields. Introduction Two theories that are dual to one another can be viewed as being physically identical [1]. An important kind of duality is so called T-duality, where T stands for target space-time. This means that we can switch the target space with its dual without loosing the physical content of the theory. When the open string endpoints are attached to D-brane, its world-volume becomes noncommutative manifold [2,3,4,5,6]. The noncommutativity parameter is proportional to the Neveu-Schwarz antisymmetric field B µν , while in the supersymmetric case the noncommutative parameters are proportional to the Ω odd parts of NS-R field, Ψ α −µ , and R-R field strength, F αβ s . The noncommutative (super)space is startting point in studying properties of the noncommutative (super) Yang-Mills theories [7]. Recently a new kind of T-duality was discovered, the fermionic T-duality [8]. It consists in certain non-local redefinitions of the fermionic variables of the superstring mapping a supersymmetric background to another supersymmetric background. Technically fermionic T-duality is similar to the bosonic one, except that dualization is performed along fermionic directions, θ α andθ α . Ref. [8] also shows that T-duality maps gluon scattering amplitudes in the original theory to Wilson loops in the dual theory. They also investigated connection between "dual conformal symmetry" and integrability. The articles [9], focussing more on integrability, deal with fermionic T-duality also, using Green-Schwarz string on AdS 5 × S 5 . From slightly different point of view most of the results of the Ref. [8] have been obtained. The present article is motivated by the fact that for the specific solution of the boundary conditions some of the bosonic T-dual background fields coincide with noncommutativity parameters [10,11]. In these articles type IIB superstring theory in pure spinor formulation has been considered. Performing Buscher T-duality [12] along all bosonic directions x µ , the background fields of the T-dual theory have been found. On the other hand, consequences of the particular boundary conditions at the open string endpoints have been investigated: the Neumann boundary conditions for bosonic coordinates and preserving half of the initial N = 2 supersymmetry for fermionic ones. It turned out that coordinates noncommutativity parameters are the bosonic T-dual fields. So, the particular choice of duality (along bosonic directions) corresponds to the particular choice of boundary conditions. In the present article we are looking for such boundary conditions which produce noncommutativity parameters equal to the fermionic T-dual background fields. The article is organized in the following way. First, we introduce the action of the pure spinor formulation for type IIB superstring theory keeping quadratic terms. Then, we perform canonical analysis in the light-cone coordinates. Because of reparameterization invariance we can take any timelike or lightlike coordinate as evolution parameter. For lightlike evolution parameter the Lagrangian is linear in velocities, and there are primary constraints which we will use as suitable introduced currents. There are two cases for consideration: 1) τ → σ − and σ → σ + and 2) τ → σ + and σ → −σ − . Canonical Hamiltonian with timelike evolution parameter can be written in the Sugawara form of the currents. In the case of open string action principle, besides equations of motion, produces boundary conditions. Choosing Dirichlet boundary conditions and treating them as canonical constraints [5,6], we obtain the initial coordinates and momenta in terms of the effective ones, which are odd under world-sheet parity transformation Ω : σ → −σ. It turns out that momenta are noncommutative, while the coordinates are commutative. The source of noncommutativity is the presence of the effective coordinates in the solution for initial momenta. The noncommutativity parameters are fermionic T-dual background fields. At the end we give some concluding remarks. Type IIB superstring and fermionic T-duality In this section we will introduce the action of type IIB superstring theory in pure spinor formulation and perform fermionic T-duality [8]. The action of type IIB superstring theory in pure spinor formulation (up to the quadratic terms [13,14,15,10,11] and neglecting ghost terms as in Ref. [14]) is of the form S = κ Σ d 2 ξ∂ + x µ Π +µν ∂ − x ν (2.1) + Σ d 2 ξ −π α ∂ − (θ α + Ψ α µ x µ ) + ∂ + (θ α +Ψ α µ x µ )π α + 1 2κ π α F αβπ β , where the world sheet Σ is parameterized by ξ m = (ξ 0 = τ , ξ 1 = σ) and ∂ ± = ∂ τ ± ∂ σ . Superspace is spanned by bosonic coordinates x µ (µ = 0, 1, 2, . . . , 9) and fermionic ones, θ α andθ α (α = 1, 2, . . . , 16). The variables π α andπ α are canonically conjugated momenta to θ α andθ α , respectively. All spinors are Majorana-Weyl ones and Π ±µν = B µν ± 1 2 G µν . On the equations of motion for fermionic momenta π α andπ α we obtain π α = − 1 2 ∂ +ηα ,π α = 1 2 ∂ − η α ,(2.2) where we introduce useful notation η α ≡ 4κ(F −1 ) αβ (θ β + Ψ β µ x µ ) ,η α ≡ 4κ(θ β +Ψ β µ x µ )(F −1 ) βα . (2.3) Using these relations the action gets the form S(∂ ± x, ∂ − θ, ∂ +θ ) = κ Σ d 2 ξ∂ + x µ Π +µν ∂ − x ν (2.4) + 2κ Σ d 2 ξ∂ + θ α +Ψ α µ x µ (F −1 ) αβ ∂ − θ β + Ψ β ν x ν . Now we will perform fermionic T-duality presented in Ref. [8]. We suppose that the action has a global shift symmetry in θ α andθ α directions. So, we introduce gauge fields (v α + , v α − ) and (v α + ,v α − ) and make a change in the action ∂ − θ α → D − θ α ≡ ∂ − θ α + v α − , ∂ +θ α → D +θ α ≡ ∂ +θ α +v α + .(2.5) In addition we introduce the Lagrange multipliers ϑ α andθ α which will impose that field strengths of gauge fields v α ± andv α ± vanish (2.6) and the full action is of the form S gauge (ϑ, v ± ,θ,v ± ) = 1 2 κ Σ d 2 ξθ α (∂ + v α − − ∂ − v α + ) + 1 2 κ Σ d 2 ξ(∂ +v α − − ∂ −v α + )ϑ α ,S ⋆ (x, θ,θ, ϑ,θ, v ± ,v ± ) = S(∂ ± x, D − θ, D +θ ) + S gauge (ϑ,θ, v ± ,v ± ) . (2.7) If we vary with respect to the Lagrange multipliers ϑ α andθ α we obtain ∂ + v α − − ∂ − v α + = 0 and ∂ +v α − − ∂ −v α + = 0 which gives v α ± = ∂ ±θ α , v α ± = ∂ ± θ α . (2.8) Substituting these expression in (2.7) we obtain the initial action (2.4). Now we can fix θ α andθ α to zero and obtain the action quadratic in the fields v ± and v ± S ⋆ = κ Σ d 2 ξ∂ + x µ Π +µν + 2Ψ α µ (F −1 ) αβ Ψ β ν ∂ − x ν (2.9) +2κ Σ v α + (F −1 ) αβ v β − +v α + (F −1 ) αβ Ψ β ν ∂ − x ν + ∂ + x µΨα µ (F −1 ) αβ v β − + 1 2 κ Σ d 2 ξθ α (∂ + v α − − ∂ − v α + ) + 1 2 κ Σ d 2 ξ(∂ +v α − − ∂ −v α + )ϑ α , which can be integrated out classically. On the equations of motion for v ± andv ± we obtain, respectively ∂ −θα = 0 ,v α + = 1 4 ∂ +θβ F βα − ∂ + x µΨα µ , (2.10) ∂ + ϑ α = 0 , v α − = − 1 4 F αβ ∂ − ϑ β − Ψ α µ ∂ − x µ . (2.11) Substituting these expression in the action S ⋆ we obtain the dual action ⋆ S(∂ ± x, ∂ − ϑ, ∂ +θ ) = κ Σ d 2 ξ∂ + x µ Π +µν ∂ − x ν ,(2.12)+ κ 8 Σ d 2 ξ ∂ +θα F αβ ∂ − ϑ β − 4∂ + x µΨα µ ∂ − ϑ α + 4∂ +θα Ψ α µ ∂ − x µ , from which we read the dual background fields (denoted by stars) ⋆ B µν = B µν + (ΨF −1 Ψ) µν − (ΨF −1 Ψ) νµ , ⋆ G µν = G µν +2 (ΨF −1 Ψ) µν + (ΨF −1 Ψ) νµ , (2.13) ⋆ Ψ αµ = 4(F −1 Ψ) αµ , ⋆Ψ µα = −4(ΨF −1 ) µα ,(2. 14) ⋆ F αβ = 16(F −1 ) αβ . (2.15) Let us note that two successive dualizations give the initial background fields. Canonical structure of the theory The main technical problem is to perform complet consistency procedure for the constraints because it has infinite many steps. So, it is useful to find such basic variables (currents), which Poisson brackets with Hamiltonian are as simple as possible. Following the idea of Ref. [16], we can obtain these currents as canonical constraints when lightlike direction is evolution parameter. It turns that they are good basis for all canonical supervariables, and that they have simple Poisson brackets as well with Hamiltonian as among them. Canonical analysis with light-like evolution parameter Because of world sheet reparametrization invariance, any timelike or lightlike coordinate could be chosen as evolution parameter. The action (2.4) is linear in derivatives with respect to the light-cone coordinates ∂ ± . So, in order to get some canonical constraints, we have two possibilities: 1) σ − → τ and σ + → σ, and 2) σ + → τ and σ − → −σ, where σ ± = 1 2 (τ ± σ). In the first case, σ − → τ and σ + → σ, the world-sheet action gets the form S = 2κ Σ d 2 ξ x ′µ Π +µνẋ ν + 2(θ ′α +Ψ α µ x ′µ )(F −1 ) αβ (θ β + Ψ β νẋ ν ) . (3.1) The canonical momenta conjugated to the variables x µ , θ α andθ α π µ = ∂L ∂ẋ µ = 2κ −Π −µν x ′ν + 1 2κη ′ α Ψ α µ , (3.2) π α = ∂ L L ∂θ α = −η ′ α ,π α = ∂ L L ∂˙θ α = 0 ,(3.3) do not depend on the τ -derivatives, and consequently, there are primary constraints J −µ = j −µ −η ′ α Ψ α µ , J −α = π α +η ′ α ,J −α =π α ,(3.4) where we introduce j ±µ = π µ + 2κΠ ±µν x ′ν . (3.5) If we use the notation J −A = (J −µ , J −α ,J −α ) and the basic Poisson algebra {x µ (σ), π ν (σ)} = δ µ ν δ(σ−σ) , {θ α (σ), π β (σ)} = θ α (σ),π β (σ) = −δ α β δ(σ−σ), (3.6) algebra of the constraints gets the form {J −A , J −B } = −2κ ⋆ G AB δ ′ ,(3.7) where ⋆ G AB =    ⋆ G µν 1 2 ⋆Ψ µγ 1 2 ( ⋆ Ψ T ) µδ 1 2 ( ⋆ΨT ) αν 0 1 8 ( ⋆ F T ) αδ 1 2 ⋆ Ψ βν − 1 8 ⋆ F βγ 0    . (3.8) Let us note that ⋆ G AB obeys graded symmetrization rule ⋆ G AB = (−) AB ⋆ G BA . In the second case, σ + → τ and σ − → −σ, the action is of the form S = −2κ Σ d 2 ξ ẋ µ Π +µν x ′ν + 2(θ α +Ψ α µẋ µ )(F −1 ) αβ (θ ′β + Ψ β ν x ′ν ) . (3.9) Similarly, we obtain primary constraints J +µ = j +µ +Ψ α µ η ′ α , J +α = π α ,J +α =π α + η ′ α ,(3.10) where the corresponding algebra of the constraints J +A = (J +µ , J +α ,J +α ) is 11) with the same coefficient as in the first case. {J +A , J +B } = 2κ ⋆ G AB δ ′ ,(3. It is easy to check that {J +A , J −B } = 0 ,(3.12) so that we obtain two independent Abelian Kac-Moody algebras {J ±A , J ±B } = ±2κ ⋆ G AB δ ′ , {J ±A , J ∓B } = 0 . (3.13) Note that the algebra of the constraints closes on the fermionic T-dual background fields (2.13)-(2.15) (except ⋆ B µν ). Because the action is linear in time derivative in both cases, the canonical Hamiltonian density is zero, H c = 0, and the total Hamiltonian takes the form H T ± = dσH T ± = dσλ A ± J ±A ,(3.14) where λ A are Lagrange multipliers. With the help of (3.13) it is easy to check thaṫ J ±A = {J ±A , H T ± } = ∓2κ ⋆ G AB λ ′B ± .(3.15) Consequently, there are no more constraints and for s det ⋆ G AB ∼ det ⋆ G det ⋆ F 2 = 0, all constraints, except the zero modes, are of the second class. Canonical structure with time like evolution parameter -From Kac-Moody to Virasoro algebra Following reasons of Ref. [16] we are going to formulate canonical structure with time-like evolution parameter τ = ξ 0 using the structure with light-like ones τ = σ + and τ = σ − . We construct energy-momentum tensor components in Sugawara form as bilinear combination of the currents J ±A T ± = ∓ 1 4κ J ±A ( ⋆ G −1 ) AB J ±B ,(3.16) where ( ⋆ G −1 ) AB =    G µν −Ψ µγ −Ψ µδ Ψ αν −Ψ α ρ Ψ ργ − 1 2 (F αδ + 2Ψ α ρΨ ρδ ) Ψ βν 1 2 (F T ) βγ − 2Ψ β ρ Ψ ργ −Ψ β ρΨ ρδ    ,(3.17) is inverse of supermatrix ⋆ G AB . Note that the currents J ±A , which was canonical constraints for lightlike evolution parameter, are not canonical constraints for timelike evolution parameter. Here, the canonical constraints are only energy-momentum tensor components. They satisfy two independent Virasoro algebras {T ± (σ), T ± (σ)} = − [T ± (σ) + T ± (σ)] δ ′ , {T ± (σ), T ∓ (σ)} = 0 ,(3.18) which are equivalent to the algebra of world-sheet diffeomorphisms. The Hamiltonian for τ = ξ 0 is given by H c = dσH c , H c = T − − T + . (3.19) By straightforward calculation we can prove {H c , J ±A } = ∓J ′ ±A . (3.20) Using (3.17) we can obtain the expressions of energy-momentum tensors in terms of the components T ± = ∓ 1 4κ G µν J ±µ J ±ν ∓ 1 2κ J ±α Ψ αµ J ±µ ∓ 1 2κJ ±αΨ αµ J ±µ (3.21) ± 1 4κ J ±α Ψ α Ψ β J ±β ± 1 4κ J ±α F αβ + 2Ψ αΨβ J ±β ± 1 4κJ ±αΨ αΨβJ ±β . We can check that our construction is equivalent to that of Refs. [15,11] obtained by prime calculation. Here we used the relation between currents J ±A and the current I ±µ introduced in Refs. [15,11] I ±µ = J ±µ + J ±α Ψ α µ −Ψ α µJ±α . The currents ⋆ J A ± , where index is raised by ( ⋆ G −1 ) AB , are of the form ⋆ J A ± ≡ ( ⋆ G −1 ) AB J ±B =    ⋆ J µ ± ⋆ J α ± ⋆J β ±    =    J µ ± − Ψ µα J ±α −Ψ µαJ ±α Ψ αν J ±ν − Ψ α µ Ψ βµ J ±β − 1 2 (F αβ + 2Ψ α µΨ βµ )J ±β Ψ βν J ±ν + 1 2 (F γβ − 2Ψ βµ Ψ γ µ )J ±γ −Ψ βµΨ γ µJ±γ    . (3.23) Let us note that ⋆ J µ ± = G µν I ±ν . Boundary conditions as a canonical constraints In this section we will look for such solution of the boundary conditions that corresponding noncommutativity parameters are just the background fields of the fermionic T-dual theory (2.13)-(2.15). Choice of the boundary conditions and canonical consistency procedure Varying the Hamiltonian (3.19) we obtain δH c = δH (R) c − γ (0) µ δx µ + 1 4κ J +α F αβ δη β + 1 4κ δη α F αβJ −α \| π 0 ,(4.1) where δH (R) c is regular term, without derivatives of coordinates and momenta variations, andγ (0) µ = Π +µν ⋆ J ν − + Π −µν ⋆ J ν + . (4.2) Because the Hamiltonian is time translation generator it must have well defined functional derivatives with respect to the coordinates and momenta. Consequently, we get the boundary condition γ (0) µ δx µ + 1 4κ J +α F αβ δη β + 1 4κ δη α F αβJ −α \| π 0 = 0 .(4.3) We will choose Dirichlet boundary conditions (fixed string endpoints) x µ \| π 0 = const. , η α \| π 0 = const. ,η α \| π 0 = const. , (4.4) which solve boundary condition (4.3). They can be expressed in more suitable form in terms of the currents γ (0) µ \| π 0 = 0 , γ (0) µ ≡ J +µ + J −µ , γ (0) α \| π 0 = 0 , γ (0) α ≡ J +α + J −α ,(4.5)γ (0) α \| π 0 = 0 ,γ (0) α ≡J +α +J −α . In fact, on the equations of motion for momenta π µ , π α andπ α we have (4.6) which means that string endpoints velocities are zero J ±µ = κG µν ∂ ± x ν + 1 2Ψ α µ ∂ ± η α + 1 2 ∂ ±ηα Ψ α µ , J ±α = − 1 2 ∂ ±ηα ,J ±α = 1 2 ∂ ± η α ,2κG µνẋ ν = γ (0) µ + γ (0) α Ψ α µ −Ψ α µγ (0) α , −η α = γ (0) α ,η α =γ (0) α . (4.7) Following method developed in Refs. [5,6] we will consider the expressions γ (0) A = (γ (0) µ , γ (0) α ,γ (0) α ) as the canonical constraints. Applying Dirac consistency procedure we obtain infinite set of the constraints γ (n) A \| π 0 = 0 , γ (n) A ≡ H c , γ (n−1) A , (n = 1, 2, 3, . . .) (4.8) where γ (n) µ = (−1) n ∂ n σ J +µ + ∂ n σ J −µ , γ (n) α = (−1) n ∂ n σ J +α + ∂ n σ J −α , (n = 1, 2, 3, . . .) (4.9) γ (n) α = (−1) n ∂ n σJ +α + ∂ n σJ −α . With the help of the relation (3.20), using Taylor expansion Γ A (σ) = ∞ n=0 σ n n! γ (n) A \| 0 , Γ A = (Γ µ , Γ α ,Γ α ) (4.10) we rewrite these infinite sets of consistency conditions at σ = 0 in compact, σ dependent form Γ µ (σ) = J +µ (−σ) + J −µ (σ) , (4.11) (4.13) In the similar way we can write the consistency conditions at σ = π. If we impose 2π periodicity of the canonical variables, the solution of the constraints at σ = 0 also solve the constraints at σ = π. Γ α (σ) = J +α (−σ) + J −α (σ) , (4.12) Γ α (σ) =J +α (−σ) +J α (σ) . Because of the relation (4.14) there are no other constraints in the theory and the consistency procedure is completed. Using the algebra of the currents (3.13) we obtain the algebra of the constraints (4.15) and conclude that they are of the second class because the metric ⋆ G AB defined in (3.8) is nonsingular for det ⋆ G µν = 0 and det ⋆ F αβ = 0. In bosonic case the algebra of the constraints closes on bosonic T-dual fields. In the particular case the algebra of the constraints (4.15) closes on fermionic T-dual background fields (3.8). {H c , Γ A } = Γ ′ A ≈ 0 ,{Γ A (σ), Γ B (σ)} = −4κ ⋆ G AB δ ′ , Solution of the boundary conditions Γ A Let us first introduce new variables symmetric and antisymmetric under world-sheet parity transformation Ω : σ → −σ. For bosonic variables and fermionic momenta we use standard notation [6] q µ (σ) = P s x µ (σ) ,q µ (σ) = P a x µ (σ) , (4.16) p µ (σ) = P s π µ (σ) ,p µ (σ) = P a π µ (σ) (4.17) p α (σ) = P a π α (σ) ,p α (σ) = P aπα (σ) , (4.18) while for fermionic coordinates we use subscript a θ α a (σ) = P a θ α (σ) ,θ α a (σ) = P aθ α (σ) , (4.19) where the projectors on Ω even and odd parts are P s = 1 2 (1 + Ω) , P a = 1 2 (1 − Ω) . (4.20) Now we are ready to solve the constraint equations Γ µ (σ) = 0 , Γ α (σ) = 0 ,Γ α (σ) = 0 . (4.21) We obtain initial variables in terms of the effective ones x µ (σ) =q µ (σ) , π µ =p µ − 2κ ⋆ B µνq ′ν + κ 2 ⋆Ψ αµ θ ′α a +θ ′α a ⋆ Ψ αµ , (4.22) θ α (σ) = θ α a (σ) , π α =p α − κ 8θ ′β a ⋆ F βα + κ 2 ⋆Ψ αµq ′µ , (4.23) θ α (σ) =θ α a (σ) ,π α =p α − κ 8 ⋆ F αβ θ ′β a − κ 2 ⋆ Ψ αµq ′µ ,(4.24) where the fermionic dual background fields (with stars) are defined in (2.13)-(2.15). We can reexpress these solutions in terms of the initial background fields too x µ (σ) =q µ (σ) , π µ =p µ − 2κB µνq ′ν − 1 2Ψ α µ (η ′ a ) α + 1 2 (η ′ a ) α Ψ α µ , θ α (σ) = θ α a (σ) , π α =p α − 1 2 (η ′ a ) α , (4.25) θ α (σ) =θ α a (σ) ,π α =p α − 1 2 (η ′ a ) α , where (η a ) α ≡ 4κ(F −1 ) αβ (θ β a + Ψ β µq µ ) , (η a ) α ≡ 4κ(θ β a +Ψ β µq µ )(F −1 ) βα ,(4.26) are Ω odd projections of the variables (2.3). Note that, as a difference of all previous cases, our basic effective variablesq µ ,p µ , θ α a ,p α ,θ α a andp α are Ω odd and the solution for momenta is nontrivial. From basic Poisson bracket {x µ (σ), π ν (σ)} = δ µ ν δ(σ −σ) , (4.27) we obtain the corresponding one in Ω odd subspace {q µ (σ) ,p ν (σ)} = 2δ µ ν δ a (σ ,σ) , (4.28) where δ a (σ,σ) = 1 2 [δ(σ −σ) − δ(σ +σ)] ,(4.29) is antisymmetric delta function. The factor 2 in front of antisymmetric delta function comes from the fact that Ω-odd functions on the interval [−π, π],q µ andp ν , are restricted on the interval [0, π] (see [17]). Similarly, using basic Poisson algebra of fermionic variables {θ α (σ), π β (σ)} = {θ α (σ),π β (σ)} = −δ α β δ(σ −σ) ,(4.30) we have {θ α a (σ) ,p β (σ)} = −2δ α β δ a (σ ,σ) , θ α a (σ) ,p β (σ) = −2δ α β δ a (σ ,σ) . (4.31) The momentap µ ,p α andp α are canonically conjugated to the coordinatesq µ , θ α a andθ α a , respectively, in Ω odd subspace. Momenta noncommutativity relations When the Neumann boundary conditions have been used [10,11], the solution for the super momenta was trivial while the solution for the super coordinates depended not only on the effective coordinates but also on the effective momenta. This was a source of the coordinate noncommutativity which corresponded to the bosonic T-duality. In the present case, with Dirichlet boundary conditions, the solution for the super coordinates is trivial while the solution for the super momenta depends not only on the effective momenta but also on the effective coordinates. This is a source of momenta noncommutativity which will correspond to the fermionic T-duality. Instead to calculate Dirac brackets in the initial phase space associated with constraints Γ A , we will calculate the equivalent brackets in the reduced phase space. We will put the subscript D to distinguish them from Poisson ones of initial phase space. With the help of the solution (4.22)-(4.24) we find that all supercoordinates are commutative, while the D brackets of momenta have a form {π µ (σ), π ν (σ)} D = 4κ ⋆ B µν ∂ σ δ(σ +σ) ,(5.1) {π µ (σ), π α (σ)} D = κ ⋆Ψ µα ∂ σ δ(σ +σ) , (5.2) {π µ (σ),π α (σ)} D = −κ ⋆ Ψ αµ ∂ σ δ(σ +σ) , (5.3) {π α (σ),π β (σ)} D = − κ 4 ⋆ F βα ∂ σ δ(σ +σ) , (5.4) {π α (σ), π β (σ)} D = {π α (σ),π β (σ)} D = 0 . (5.5) If we define the variables Π µ (σ) = σ 0 dσ 1 π µ (σ 1 ) , Π α = σ 0 dσ 1 π α (σ 1 ) ,Π α = σ 0 dσ 1πα (σ 1 ) ,(5.6) the noncommutativity relations turn to the standard form {Π µ (σ), Π ν (σ)} D = 4 κ ⋆ B µν θ(σ +σ) , (5.7) {Π µ (σ), Π α (σ)} D = κ ⋆Ψ µα θ(σ +σ) , (5.8) Π µ (σ),Π α (σ) D = −κ ⋆ Ψ αµ θ(σ +σ) , (5.9) Π α (σ),Π β (σ) D = − κ 4 ⋆ F βα θ(σ +σ) , (5.10) {Π α (σ), Π β (σ)} D = Π α (σ),Π β (σ) D = 0 , (5.11) where θ(x) =      0 if x = 0 1/2 if 0 < x < 2π . 1 if x = 2π (5.12) Separating the mean value of momenta Π A (σ) = Π mv A + P A (σ) , Π mv A = 1 π π 0 dσΠ A (σ) , we obtain that only integrals of the momenta at the string endpoints are noncommutative {P µ (σ), P ν (σ)} D = Θ µν ∆(σ +σ) ,(5.13) {P µ (σ), P α (σ)} D =Θ µα ∆(σ +σ) , (5.14) P µ (σ),P α (σ) D = Θ αµ ∆(σ +σ) , (5.15) P α (σ),P β (σ) D = Θ αβ ∆(σ +σ) ,(5.16){P α (σ), P β (σ)} D = P α (σ),P β (σ) D = 0 ,(5.17) where the noncommutativity parameters are defined as Θ µν = 2κ ⋆ B µν ,Θ µα = κ 2 ⋆Ψ µα , Θ αµ = − κ 2 ⋆ Ψ αµ , Θ αβ = − κ 8 ⋆ F βα ,(5.18) and ∆(x) = 2θ(x) − 1 =      −1 if x = 0 0 if 0 < x < 2π . 1 if x = 2π (5.19) Therefore, all background fields of the fermionic T-dual theory (2.13)-(2.15), except ⋆ G µν , appear as noncommutativity parameters for the solution of boundary conditions (4.4). Concluding remarks In the present article we considered the relationship between fermionic T-duality and noncommutativity in type IIB superstring theory. We used the pure spinor formulation of the theory keeping all terms up to the quadratic ones and neglecting ghost terms. Our goal was to find such solution of the boundary conditions which will produce fermionic T-dual fields as noncommutativity parameters. First, we performed fermionic T-duality in the way described in Refs. [8]. Comparing initial and dualized theory, we found the expressions for fermionic T-dual background fields. Varying the canonical Hamiltonian and demanding that it has well defined functional derivatives with respect to the coordinates and momenta, we obtain the boundary condition (4.3). In order to satisfy them, for all supercoordinates we chose Dirichlet boundary conditions. Treating these conditions as canonical constraints and reexpressing them in terms of useful introduced currents we were able to examine consistency of the constraints. For nonsingular dual metric ⋆ G µν and nonsingular dual R-R field strength ⋆ F αβ all constraints are of the second class. Instead to use Dirac brackets we solved the second class constraints. We took Ω odd parts of canonical variables as independent effective variables, and expressed the Ω even ones in terms of them. We found that the solution of supercoordinates was trivial, because they depended only on its Ω odd projections. The solutions for supermomenta depend both on effective supercoordinates and effective supermomenta. So, as a difference of the previous investigations [5,10,11,6,14,15], here supercoordinates are commutative while integrals of supermomenta are noncommutative. Similar as in previous investigations, noncommutativity appears only at the string endpoints, and not in the string interior. Noncommutativity parameters at σ = 0 and σ = π have opposite signs. Let us comment relation between fermionic T-dual background fields defined in (2.13)-(2.15) with noncommutative parameters corresponding to the Dirichlet boundary conditions (4.4). All noncommutativity parameters, up to the some constant multipliers, are equal to the fermionic T-dual fields. Because noncommutativity relations close on ∆(σ+σ) which is symmetric under σ ↔σ, the noncommutativity parameter symmetric in spacetime indices is absent. Therefore, only T-dual metric tensor ⋆ G µν does not appear as noncommutativity parameter. As well as in the previous cases, dual fields appear in the algebra of constraints (4.15). Because here the algebra closes on ∂ σ δ(σ −σ) which is antisymmetric under σ ↔σ, the background field antisymmetric in space-time indices is absent. So, the D-brackets of σ-dependent constraints Γ A close on all T-dual background fields except ⋆ B µν . 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[ "Information loss problem and a 'black hole' model with a closed apparent horizon", "Information loss problem and a 'black hole' model with a closed apparent horizon" ]	[ "Valeri P Frolov [email protected] \nTheoretical Physics Institute\nUniversity of Alberta\nT6G 2G7EdmontonABCanada\n" ]	[ "Theoretical Physics Institute\nUniversity of Alberta\nT6G 2G7EdmontonABCanada" ]	[]	In a classical description the spacetime curvature inside a black hole infinitely grows. In the domain where it reaches the Planckian value and exceeds it the Einstein equations should be modified. In the absence of reliable theory of quantum gravity it is instructive to consider simplified models. We assume that a spacetime curvature is limited by some value (of the order of the Planckian one). We use modified Vaidya metric, proposed by Hayward, to describe the black hole evaporation process. In such a spacetime the curvature near r = 0 remains finite, it does not have an event horizon and its apparent horizon is closed. If the initial mass of such a 'black hole' is much larger than the Planckian one its properties (as seen by an external observer) are practically the same as properties of the 'standard' black hole with the event horizon. We study outgoing null rays in the vicinity of the outer apparent and introduce a notion of quasi-horizon. We demonstrate that particles, trapped inside a 'black hole' during the evaporation process, finally may return to external space after the evaporation is completed. We also demonstrate that such quanta would have very large blue-shift. The absence of the event horizon makes it possible restoration of the unitarity in evaporating black holes.	10.1007/jhep05(2014)049	[ "https://arxiv.org/pdf/1402.5446v3.pdf" ]	119,266,024	1402.5446	39700f718466e9411e4e74f33e0b807abcad1266	Information loss problem and a 'black hole' model with a closed apparent horizon 1 Mar 2014 Valeri P Frolov [email protected] Theoretical Physics Institute University of Alberta T6G 2G7EdmontonABCanada Information loss problem and a 'black hole' model with a closed apparent horizon 1 Mar 2014Prepared for submission to JHEP In a classical description the spacetime curvature inside a black hole infinitely grows. In the domain where it reaches the Planckian value and exceeds it the Einstein equations should be modified. In the absence of reliable theory of quantum gravity it is instructive to consider simplified models. We assume that a spacetime curvature is limited by some value (of the order of the Planckian one). We use modified Vaidya metric, proposed by Hayward, to describe the black hole evaporation process. In such a spacetime the curvature near r = 0 remains finite, it does not have an event horizon and its apparent horizon is closed. If the initial mass of such a 'black hole' is much larger than the Planckian one its properties (as seen by an external observer) are practically the same as properties of the 'standard' black hole with the event horizon. We study outgoing null rays in the vicinity of the outer apparent and introduce a notion of quasi-horizon. We demonstrate that particles, trapped inside a 'black hole' during the evaporation process, finally may return to external space after the evaporation is completed. We also demonstrate that such quanta would have very large blue-shift. The absence of the event horizon makes it possible restoration of the unitarity in evaporating black holes. Introduction Information loss puzzle is one of the longstanding problems of the black hole physics. A black hole can be formed by matter in a pure quantum state. In the subsequent process of its quantum evaporation its radiation is described by a (thermal) density matrix. If the black hole completely disappears what is left is its emitted radiation. Hence, in such a process the pure initial quantum mechanical state is transformed in a state described by the density matrix, so that unitarity would be violated. In order to describe such processes in quantum mechanics the latter should be modified. For this purpose Hawking in 1976 [1] introduced a so called superscattering operator. The information loss puzzle (the loss of the unitarity in processes involving black holes) was (and still is) the subject which has attracted a lot of attention (see e.g. [2][3][4][5][6][7][8][9] and references therein). The string theory strongly advocates the point of view that the unitarity should be restored after the evaporation of the black hole. These arguments are based on the AdS/CFF duality according to which the evolution of the black hole in the bulk asymptotically AdS spacetime is in correspondence with the unitary evolution of the corresponding quantum field on the boundary (see e.g. [10] and references therein). However attempts to find the mechanism of the unitarity restoration results in new problems. In order to solve the latter it was recently proposed a 'firewall' model [11]. According to this model any attempt by a falling into a black hole observer to determine exact quantum state of the matter results in the creation of the firewall which change the structure of the black hole itself. However this drastic approach seams to be not very appealing. Critical discussion of the firewall model can be found in [12,13]. Hawking in his recent paper [14] proposed to solve the information loss puzzle by assuming that in the gravitational collapse an apparent is formed but the event horizon does not exists. Such models have been studied earlier. A black hole model with a closed apparent horizon was proposed more than 30 years ago in [15,16]. Later similar models have been widely discussed (see e.g. papers [17][18][19][20][21][22][23] and references therein). Such models also naturally arise in study of the gravitational collapse in quantum gravity [24][25][26][27][28]. The purpose of the present paper is to review main features of the models with a closed apparent horizon. For this purpose we use a simple model, proposed by Hayward [19], which allows one to study these properties in details. The paper is organized as follows. In section 2 we discuss our main assumptions. Sections 3-4 describe properties of the model based on the modified Vaidya metric. In section 5 we introduce a notion of a quasi-horizon, which is a natural generalization of the event horizon for evaporating black holes. Namely for slow decrease of the 'black hole' mass it is 'almost null'. It also serves as a separatrice between outgoing null rays that reach infinity and those that propagates to the center of the 'black hole'. Behavior of null rays in the vicinity of the inner horizon is considered in section 6. Global properties of the 'black hole' model with the closed apparent horizon are discussed in section 7. The last section 8 contains general discussion of the models with a closed apparent horizon in relation to the information loss puzzle. We also discuss some of the further problems of such models and their possible solutions. In the Appendix we demonstrate how a simple massive thin shell model can be used for an effective description of the quantum particle creation by a black hole. Assumptions We consider a formation of a spherical black hole in a gravitational collapse of a massive object and its subsequent quantum evaporation. We assume that the initial mass M of the black hole is much larger than the Planckian mass m P l , so that the black hole is a classical object. We also assume that its gravitational field during all the evolution (including the final state of the evaporation) as well in the black hole interior is described by a metric tensor g µν . In the domain where quantum corrections are small it obeys the Einstein equations. At the initial stage of the evaporation one can use quasi-classical description, so that the Hawking process results in the positive energy flux of created particles to infinity and (in accordance with the conservation law) by the negative energy flux through the horizon. The latter decreases the mass of the black hole. The main contribution to this process is due to massless fields, so that we describe the energy flux to future null infinity by properly chosen null fluid stress-energy tensor. We assume that the negative energy flux through black hole horizon into its interior can also be approximated the null fluid with negative energy. Both incoming and outgoing fluxes are the result of the quantum process of particle creation in the vicinity of the horizon. To match these two fluxes and make the model consistent one should introduce a transition region between them. We assume that this region is narrow and use a model of a massive thin shell. We demonstrate in the Appendix that such a shell can be adjusted so that for slowly evolving black holes its effective stress-energy tensor is small. In the spacetime domain, where the curvature becomes comparable with the Planckian one, the classical Einstein equations should be modified as a result of the effects of the vacuum polarization and intensive quantum particle creation. For the black hole with the gravitational radius r S = 2M the curvature reaches the Planckian value at r = r 0 which can be found from the equation r S r 3 0 ∼ l −2 P l . (2.1) In the spacetime domain where r < r 0 = r s (l P l /r s ) 2/3 the curvature calculated for the classical solution exceeds the Planckian one. During the initial stage of the evaporation this region is deep inside the black hole. The modification of the classical Einstein equations are also certainly required in order to describe the final stage of the evaporation. Namely in these domains the metric is expected to be quite different from its 'classical form'. Since we do not have reliable theory of gravity valid at the Planckian scales we use special ansatz for the metric in these domains. We specify a model so that it satisfies the limiting curvature principle [29,30]. This prescription certainly contains an ambiguity. However even consideration of such simple models is quite instructive and allows one to study their robust predictions. A model The negative energy flux through the horizon, that accompanies the particle creation, reduces the mass of the black hole. We use Vaidya solution to describe it dS 2 = −F dV 2 + 2dV dr + r 2 dω 2 , F = 1 − 2M − (V ) r . (3.1) The Ricci tensor for this metric is R µν = 2Ṁ − r 2 V ,µ V ,ν . (3.2) Since V ,µ is a null vector, one has R 2 ≡ R µν R µν = 0 and R = 0, while the square of the Riemann tensor is R 2 ≡ R µναβ R µναβ = 48M − (V ) 2 r 6 . (3.3) Since we assumed that the curvature is limited by the Planckian value, the metric Eq.(3.1) should be modified at r ∼ r 0 = l P l (M − /l P l ) 1/3 . To describe an evaporating black hole with a regular interior we use a modified Vaidya metric, proposed by Hayward [19] 1 This metric has the same form as Eq.(3.1) with a modified function F F = 1 − 2M − (V )r 2 r 3 + 2M − (V )b 2 . (3.4) Here b is the cut-off parameter of the order of the Planck length. For M − b and r r 0 the modified metric remains practically the same as earlier. However for r r 0 F ∼ 1 − (r/b) 2 . (3.5) This implies that the geometry near r = 0 is regular and r = 0 is a regular timelike line. Denote q = 2M − b 2 r 3 ,(3.6) then the curvature invariants for the modified Vaidya metric are of the form R 2 = 12q 2 (1 − 4q + 18q 2 − 2q 3 + 2q 4 ) b 4 (1 + q) 6 , (3.7) R 2 = 18q 4 (5 − 2q + 2q 2 ) b 4 (1 + q) 6 , (3.8) R = 6q 2 (2q − 1) b 2 (1 + q) 3 . (3.9) For small q one has R 2 ∼ 12q 2 b 4 = 48M 2 − r 6 , R 2 ∼ 18q 4 b 4 = 288M 4 − b 4 r 12 , R ∼ − 6q 2 b 2 = − 24M 2 − b 2 r 6 . (3.10) In the limit b → 0 one restores the expressions for the original Vaidya metric. In the opposite case (when q → ∞) one has R 2 ∼ 24b −4 , R 2 ∼ 36b −4 , R ∼ 6b −2 . (3.11) These relations show that the curvature of the modified Vaidya metric is limited and its maximal value is of the order of b −2 . Before studying properties of the modified Vaidya metric it is convenient to present it in the dimensionless form. For this purpose we use the cut-off parameter b as a natural length scale and introduce the following dimensionless quantities v = V b , ρ = r b , µ(v) = M − (V ) b , ds 2 = b −2 dS 2 . (3.12) The dimensionless form of the modified Vaidya metric is ds 2 = −f dv 2 + 2dvdρ + ρ 2 dω 2 , f = 1 − 2µ(v)ρ 2 ρ 3 + 2µ(v) . (3.13) -4 - Apparent horizon The apparent horizon for the metric Eq.(3.13) is determined by the condition (∇ρ) 2 ≡ f = 0 . (4.1) This gives the a following equation for the position of the apparent horizon ρ(v) at the moment of advanced time v when the mass is µ(v). We write this relation in the form 2µ(v) = Q(ρ) , Q(ρ) = ρ 3 ρ 2 − 1 . (4.2) The function Q is positive for ρ > 1, it has minimum at ρ = ρ * = √ 3 and it grows infinitely when either ρ → 1 or ρ → ∞. This means that the apparent horizon does not exist for µ < µ * 2 , where µ * = 1 2 Q(ρ * ) = 3 √ 3 4 . (4.3) (See also [19] and discussion therein) For µ(v) > µ * the equation Eq.(4.2) has two solutions 1 < ρ − (v) < ρ + (v). We call the solution ρ − (v) an inner brunch of the apparent horizon (or simply inner horizon, and the solution ρ + (v) its outer brunch (or outer horizon). The inner horizon is located in the region 1 < ρ − (v) < ρ * = √ 3, that is in the spacetime domain where the curvature is of the order of the Planckian one. The outer horizon for large µ(v) is located close to 2µ(v). The apparent horizon is closed (see Figure 2). The region inside the closed horizon is called T − -region. To specify the model one needs to choose the function µ(v). We shall do it later. At the moment we only assume that the 'black hole' is formed as a result of the spherical collapse of the null fluid, and after its formation its mass monotonically decreases as a result of the Hawking process and finally vanishes. For such a scenario the function µ(v) behaves as follows: it vanishes before v 0 (the moment when the collapse started), monotonically grows during the collapse and reaches its maximum value µ 0 at v = 0. After this it monotonically decreases until it vanishes at v 1 (the end of the evaporation). After this the mass µ(v) is identically zero. For such a scenario the apparent horizon is represented by a closed line in (v, ρ) plane. It appears and disappears at the moments v * − and v * + , respectively ( v 0 < v * − < v * + < v 1 ) . At the moments v * ± the inner and outer brunches of the apparent horizon coincide ρ ± (v * ± ) = ρ * . (4.4) Using Eq.(4.2) in the vicinity of this points one finds ρ ∼ ρ * + λ\|v − v * ± \| 1/2 , λ = (4/3) 3/4 \|µ * \| 1/2 . (4.5) 2 It is interesting to notice that in the theory of gravity with quadratic in curvature corrections there exists a critical mass, so that an apparent horizon is not formed if the mass a collapsing object is smaller than the critical one. This was demonstrated in [15,16] for a model of massive null shells. It is plausible that the existence of such a mass gap is a generic property of models with a closed apparent horizon. It would be interesting to study this problem for recently proposed class of singularity and ghost free theories of gravity [31]. 0 v 1 v 0 ' 0   ' 0   ' 0   ' 0   *  *  ' 0   ' 0   Figure 1. Apparent horizon structure. For large µ one can solve Eq.(4.2) perturbatively and get ρ + = 2µ − 1 2µ − 1 4µ 3 + . . . , (4.6) ρ − = 1 + 1 4µ + 5 32µ 2 + 1 8µ 3 + . . . . (4.7) To describe Hawking evaporation of the black hole we use the following approximation µ 3 (v) = µ 3 0 − v , (4.8) so that the rate of the mass loss is dµ dv = − 1 3µ 2 . (4.9) Restoring dimensionality one obtains dM dV = −C m P l t P l m 2 P l M 2 ,(4.10) with C = 1/3. For a realistic black hole the coefficient C depends on the number and statistics of the particles that are emitted. In our model Eq.(4.8) we neglect these details. For this choice of µ(v) the model contains two parameters: maximal mass of the 'black hole' µ 0 and time v 0 of its formation. For large mass µ(v) the rate of mass loss Eq.(4.9) is very small and the evaporation process is adiabatic with very high accuracy. Figure 1 demonstrates the structure of the apparent horizon for the model. Figure 2 shows the geometry of a flow generated by outgoing null rays in the modified Vaidya metric. An external observer is not able to get information from the interior of such an object during all the time of its evaporation. However after the evaporation ends this becomes possible. This means that the event horizon does not exist in our model. According to the standard definition in the absence of the event horizon there is no black hole. This standard definition is well adapted for the proof of many important results about black holes, but it is based on the assumption that one can have information about future evolution of the matter in the Universe until future time infinity. On the other hand the apparent horizon is defined for any given moment of time. Its existence indicates that the gravitational field is so strong that even light inside the apparent horizon moves to the 'center'. Namely this property (very strong gravitational field) is important for explanation of the observable properties of black holes, that can be registered by a distant observer. However in the case of evaporating black holes the apparent horizon is a timelike surface. This means that it can be penetrated by light and particles in both directions, so that it looses the property of the boundary of the no-escape region. In the next section we define a quasi-horizon which, for slowly evolving black holes, has the property of the boundary of 'no-return' domain and which is practically a null surface. These features make quasi-horizon similar to the event horizon, but its definition does not require knowledge about infinite future. In order to make it clear that the objects with a closed apparent horizon that we discuss in this paper do not have the event horizon, we use quotation marks and write a 'black hole'. Quasi-horizon Consider an outgoing null ray passing through the apparent horizon at time v • when its size is ρ • = ρ(v • ). If the mass of the black hole decreases with time, then for slightly later time v > v • the gravitational field at ρ • becomes weaker. Hence such a ray propagates with increase of the radius r and finally it escapes to infinity. In order to characterize the boundary of the no-escape region we introduce a notion of a quasi-horizon. For black holes with changing mass this is a natural generalization of the notion of the event horizon. The first order differential equation for outgoing null rays is of the form dρ dv = 1 2 f (ρ, v) . (5.1) We define the quasi-horizon by the condition that d 2 ρ/dv 2 for such a ray vanishes. This gives the following equation 2∂ v f + f ∂ ρ f = 0 . (5.2) Quasi-horizon for an evaporating black hole is shown at Figure 2. For a static metric ∂ v f = 0 and the quasi-horizon coincides with the apparent and event horizons, f = 0. For large µ(v) one has f = 1 − 2µ(v) ρ ,(5.3) and equation (5.2) for ρ > 0 has a solution ρ = ρ qh = 4µ(v) 1 + 1 − 16µ (v) . (5.4) Let us assume that \|µ \| \|µ \|, 3 then dρ qh dv ≈ 4µ 1 + 1 − 16µ (v) , (5.5) f (ρ qh , v) = 1 2 (1 − 1 − 16µ (v)) . (5.6) It is easy to check that the relation Eq.(5.1) is valid for ρ qh (v) up to the terms of the order of \|µ \|. Hence for large µ(v) the surface ρ = ρ qh is practically null. Let us discuss the properties of null rays propagating close to the quasi-horizon. We consider the 'black hole' at time v when its mass is much larger than the Planckian one, µ(v) 1. The radius of outer horizon is close to 2µ(v) (see Eq.(4.6)), so that one can neglect the quantity 2µ(v)/ρ 3 and write f = 1 − 2µ(v) ρ . (5.7) Let us fix the moment of the advanced time v • and denote µ(v • ) = µ • , (dµ(v)/dv)\| v• = µ • , (5.8) v − v • = 2µ • x , ρ = 2µ • y , z = −[x + (2µ • ) −1 ] . (5.9) In these notations and for the chosen approximation the equation Eq.(5.1) for the outgoing null rays takes the form dy dz = − 1 2 − µ • z y . (5.10) The equation (5.10) for null rays near slowly evolving apparent horizon was studied in [32]. It allows a following solution − ln(z/z • ) = B(q) − B(q • ) , q = y/z , (5.11) B(q) = q + q + − q − ln \|q − q + \| − q − q + − q − ln \|q − q − \| .(5.12) Here q ± = − 1 4 1 ± 1 − 16µ • . (5.13) For µ • < 0 one has q + < 0 and q − > 0. Besides this general solution, there exist a special solution corresponding to the degenerate case q = q − =const. This solutions is y = −q − x + (2µ • ) −1 . (5. 14) It is easy to check that it coincides with the quasi-horizon. For q → q − the function B(q) becomes infinite negative. Thus for outgoing null rays near the quasi-horizon one can approximate B(q) as follows B(q) − B(q • ) ≈ − q − q + − q − ln q − q − q • − q − . (5.15) In this approximation a solution of Eq.(5.11) is y = (1 + 2µ • x) ŷ + (y • −ŷ)(1 + 2µ • x) β . (5.16) Hereŷ = −q − /(2µ • ) = −(2q + ) −1 and β = q + −q − q − is a negative number. If the initial value of y coincides with the critical value, y • =ŷ, the solution takes the form y = q − z, that is it coincides with the quasi-horizon. If y • >ŷ it deviates from the critical solution to the right and the outgoing null ray propagates to infinity, while in the opposite case, y • <ŷ, the ray moves to the center. The parameter β controls the rate of expansion and contraction of the rays near the quasi-horizon. If one traces backward in time outgoing null rays one finds that for such rays near a slowly evolving black holes the quasi-horizon plays the role of the attractor. For small µ • one has q + = − 1 2 + . . . , q − = −2µ • + . . . ,(5.y − 1 ≈ (y • − 1)(1 + 2µ • x) β . (5.19) In the limit µ • → 0 it can be rewritten as ln y − 1 y • − 1 = lim µ • →0 (2µ • β)x = x/2 . (5.20) Thus for a static black hole of mass M one has r − 2M = (r • − 2M ) exp[κ(V − V • )] ,(5.21) where κ = (4M ) −1 is the surface gravity. This relation correctly reproduces to a well known result for a static black hole. Apparent horizon Quasihorizon 2 1 v  Figure 3. Quasi-horizon and apparent horizon for a slowly evolving 'black hole'. An outgoing photon 1 emitted outside the quasi-horizon propagates to infinity. A similar photon 2 emitted inside the quasi-horizon remains in the T -region during the time of the 'black hole' evaporation. Inner horizon Let us discuss properties of the metric Eq.(3.13) near the inner horizon. First let us notice that in the limit µ 1 this metric only slightly differs from the de Sitter metric ds 2 = −(1 − ρ 2 )dv 2 + 2dvdρ + ρ 2 dω 2 , (6.1) However, in the exact de Sitter metric outgoing null rays never (in time v) cross the horizon ρ = 1. Important difference of the metric Eq. (3.13) in the vicinity of ρ = 1 is that it is time dependent, and after the evaporation of the black hole the inner horizon disappears. As earlier we denote by ρ − (v) the solution of Eq.(4.2) obeying conditions 1 < ρ − < ρ * = √ 3. Let us fix a moment of advanced time v • and denote µ(v • ) = µ • , µ (v • ) = µ • , v − v • = 2µ • x , µ ≈ µ • (1 + 2µ • x) , ρ = ρ − • + y(x) . (6.2) Here ρ − • is the value of the radius of the inner horizon at time v • . Substituting these relations into the equation for outgoing rays Eq.(5.1) and keeping the linear in x and y leading terms one obtains the following equation dy dx = Ay + Bx , A = − 4µ • − 3ρ • 2ρ • , B = − µ • ρ • µ • . (6.3) A solution of this equation is y = − B A 2 − B A x + Ce Ax . (6.4) Since ρ • ≈ 1 µ • one can slightly simplify expression Eq.(6.4) and write it in the form The temperature of the thermal radiation registered by a distant observer at ime U is (8πM ) −1 [32]. A ray 2 represents a 'partner' of the emitted Hawking quantum. It moves in T − region in the direction of increasing of r. After it reaches the inner horizon, it propages in its close vicinity until a complete evaporation of the black hole. y ≈ ρ − qh + Ce −(v−v•) , (6.5) ρ − qh = ρ − • + µ • 4µ 3 • − µ • 2µ • x . One can check that Eq.(6.6) is the equation for the inner quasi-horizon. Relation Eq.(6.5) implies that ρ − qh is an attractor for outgoing null rays propagating in its vicinity. 7 Global properties of the model Null rays Let us discuss general properties of the proposed black hole model with closed apparent horizon. First of all, any null ray and particle that fall into such a 'black hole' finally return to external space after the 'black hole' evaporates. This means that in such a model there is no event horizon. However, if the initial mass is much larger than the Planckian one the properties of such an object are practically identical to properties of an evaporating black hole during long period of time of its evaporation.The quasi-horizon effectively plays the role of the event horizon. Any particle or photon that crosses the quasi-horizon inevitably moves to the center until it reaches a region with Planckian curvature. In this region it propagates exponentially close to the inner quasi-horizon, and remains there until the time when the mass of the 'black hole' reaches the Planckian value. After this it propagates back to an external observer. Frequency shift The inner horizon is an attractor for outgoing null rays in its vicinity. The exponential convergence of such rays to the inner horizon results in the exponential increase of 'photon' frequency. This effect can be easily analyzed. Denote by α an affine parameter along such a ray. One of the geodesic equations in this affine parametrization gives d 2 v dα 2 = −κ dv dα 2 , κ = 1 2 ∂ ρ f . (7.1) Eq.(7.1) can be written in the form d dv dv dα = −κ dv dα . (7.2) For our model the surface gravity κ of the inner horizon with high accuracy is −1, so that a solution of Eq. (7.2) is dv dα = dv dα v• exp(v − v • ) . (7.3) To restore dimensionality in this relation we use a transformation V = bv , λ = bα . (7.4) We use also a scale ambiguity in the choice of the 'new' affine parameter λ and choose it so that dV dλ V• = ω • ,(7.5) where ω • is a 'physical' frequency of the 'photon' at time V • . Then Eq.(7.3) determines the frequency ω of a 'photon' propagating in the vicinity of the inner horizon at later time V ω = ω • exp[(V − V • )/b] .(7.6) This confirms the above conclusion that the frequency of photons propagating close to the inner horizon becomes exponentially blue-shifted [33]. Discussion We presented a simple model of a 'black hole' with a closed apparent horizon. We assume that the 'black hole' is isolated and no matter falls into it after its formation. However the Hawking radiation reduces the mass. If the initial mass M of the 'black hole' is much larger than the Planckian mass, M/m P l 1, this process is very slow. This adiabatic phase ends after the time of the order ∼ t P l (M/m P l ) 3 . The model is constructed so that the curvature invariants in the interior of the 'black hole' are limited. As a result the apparent horizon in the adopted model is closed and the metric in the 'core' of the 'black hole' is close to the de Sitter metric. This makes this model similar to the one proposed in [34,35]. However their global properties are different. Instead of a new universe formation inside the black hole, the black hole interior for the present model remains causally connected with the original external space. The outer apparent horizon for an evaporating 'black hole' is a timelike surface. Hence it looses the main property of the event horizon as a no-escape surface. We introduced a new notion -the quasi-horizon, and demonstrated that it is much better generalization of the event horizon for time-dependent 'black holes' than the apparent horizon. The quasihorizon separates two families of outgoing null rays: (1) The rays that enter T − domain and remain there until complete evaporation of the 'black hole', and (2) the rays that escape from the vicinity of the 'black hole' and reach infinity. For adiabatic evolution of the 'black hole' the quasi-horizon is practically a null surface. One can define a 'black hole' as an object that has a quasi-horizon. It should be emphasized that the quasi-horizon is defined locally, and one should not wait infinite time and know all the details of the future evolution of the Universe, in order to make a decision whether one has a 'black hole' or does not. This makes this new definition of the 'black hole' more attractive than the 'standard' one and more useful for the discussion of evaporating black holes. Main feature of the presented model is that after the complete evaporation of the 'black hole' all the information concerning the matter, that originally have produced it as well as particles created inside it in the Hawking evaporation process, becomes available to an external observer. This opens a possibility of the restoration of the unitarity in evaporating 'black holes'. One of the potential difficulty of the proposed model is strong blue-shift effect for rays propagating in the vicinity of the inner horizon [33]. These rays are focused and produce exponentially thin beams of ultra-high frequency. Let us notice that this phenomenon has common features with a well known transplanckian problem in black holes [36] and in the inflating universe [37]. Namely if one traces backward in time outgoing rays close to the event horizon of the black hole, one discovers a similar effect of focusing and exponentially large blue-shift. One can say that in models with a closed apparent horizon there exists some kind of duality between properties of the inner and outer horizons. This duality implies that one can expect that there exist an inverse Hawking process of annihilation of two quanta of positive and negative energy propagating on both sides of the inner horizon. One also needs to remember that the proposed model is certainly over-simplified. We used a classical metric to describe the spacetime geometry. In quantum gravity such a metric is an effective one, that arises as a result of averaging of the quantum fluctuating metric and it obeys some (unknown at the moment) effective gravitational equations. Our assumption was that solutions of these modified Einstein equations obey the principle of the limiting curvature. But this description can be valid only when quantum fluctuations are small. The inner 'core' of the 'black hole' is of the Planckian size and as a result of focusing light rays propagate at much smaller distance from the inner horizon. One can expect that quantum fluctuations of the metric are large at these scales and the description in terms of the effective metric may not be suitable. Another important feature of the models with closed apparent horizon is that the matter, that contains information about the state of the collapsing matter and phases of the Hawking quanta, is compressed in a small domain of the Planckian size and has very high (Planckian) density. This means that the non-linear quantum interaction between particles as well as their interaction with gravity becomes important. Certainly the presented model is too simplified to address these problems. Let us also mention that if the low energy gravity is an emergent phenomenon, then for proper description of the matter in the inner 'core' and near the inner horizon one needs to use the language of the fundamental background theory of some heavy constituents (e.g strings). All these problems are connected with unknown properties of the quantum gravity, the theory which does not exist at the moment. However a proposed model of a 'black hole' with a closed apparent horizon allows one to address them and test some of possible new ideas. A Near horizon geometry As a result of the Hawking process a black hole emits radiation to infinity. Massless particles production dominates in this process. The outgoing energy flux can be effectively described by properly chosen null fluid. The conservation law requires that this radiation is accompanied by the negative energy flux through the horizon, which we also approximate by a null fluid. In order to make a model consistent one needs to assume that between the two regions with pure outgoing and pure incoming fluxes there exists a transition region, corresponding to the domain where the particle are created. We assume that this region is narrow and approximate it by a massive thin shell (see also discussion in [19]. In this appendix we demonstrate consistency of such a model and show that for slowly evolving black holes the parameters of the shall can be very small. To describe the metric near a horizon we use the Vaidya solution which we write in the form dS 2 ε = −F ε dZ 2 ε − 2εdZ ε dr + r 2 dω 2 , F ε = 1 − 2M ε (Z ε ) r . (A.1) Here dω 2 is the metric on a unit sphere and ε = ±1. For ε = +1 Z + is a retarded null time U , while for ε = −1 Z − is an advanced null time V . The Ricci tensor calculated for this metric is R ε µν = − 2ε r 2 dM ε dZ ε Z ε;µ Z ε;ν . (A.2) If dM ε /dZ ε < 0 then according to the Einstein equations the corresponding stress-energy tensor for ε = +1 T + µν = − 1 4πr 2 dM + dU U ,µ U ,ν (A.3) describes outgoing null fluid with positive energy density. The corresponding stress-energy tensor for ε = −1 and assume that outside this surface the metric is dS 2 + while inside it is dS 2 − . We also assume that the null coordinate V in the inner region is chosen so that on Γ it coincides with the corresponding value of U , so that one has T − µν = 1 4πr 2 dM − dV V ,µ V ,V \| Γ = U \| Γ = Z . (A.6) The intrinsic 3-geometries induced by the metrics dS 2 ± on Γ are The tetrad components of the extrinsic curvature are K ε iĵ = e μ i e ν j n ε µ;ν . (A. 15) We choose a special form of Γ and put R = 2M (Z)(1 + w), where w is a dimensionless positive small parameter. Then calculations give the following expressions for the nonvanishing components of K ε iĵ K ε tt = A + εB , K ε θθ = −2wA + εB , K ε φφ = −2wA + εB , (A.16) where A = 1 4M w(1 + w) 3 , B = M √ 1 + w √ wM . (A.17) The jumps of the extrinsic curvature at Γ determine parameters (mass and pressure) of the shell. These jumps are One can interpret the corresponding distribution of matter as being connected with a region of particle creation. In any case for a slow change of the black hole parameters, that is when \|M \| 1, the influence of this matter on the black hole geometry is proportional \|M \| and hence is extremely small and can be neglected. Figure 2 . 2'Outgoing' radial null rays in the modified Vaidya metric. The horizontal axis is v and the vertical one is ρ. The vector field l µ = (1, f /2, 0, 0) tangent to the outgoing null rays is shown by small arrows. The plot is constructed for µ 0 = 4 and v 0 = −3. A closed solid line shows a position of the apparent horizon. A dashed line is a quasi-horizon. Figure 4 . 4Outgoing null rays. A outgoing null ray 1 moves outside of the quasi-horizon. At later time it leaves the vicinity of the black hole and propagates to infinity. A far distant observer registers that it happens at some moment of the retarded time U . In the WKB approximation such a ray describes a trajectory of a 'quantum' of the emitted in the process of the Hawking radiation. Figures 4 and 5 illustrate global properties of different types of null rays in the metric (3.13). Figure 5 . 5Incoming null ray 3 (at v = 30) passes through ρ = 0 and after moves as an outgoing ray. It moves along the inner horizon until a complete evaporation of the black hole. null fluid with negative energy density. Consider a spherical surface Γ described by the equation Q(r, Z) = r − R(Z) = 0 , (A.5) dZ 2 + R 2 dω 2 . (A.7)Here R = dR/dZ. Both metrics are identical when the following condition is satisfiedM + (Z) − M − (Z) = 2RR . (A.8)In what follows we assume that this condition is satisfied and write M ± in the formM ε (Z) = M (Z) + εRR , (A.9)so that the induced metric on Γ (which is the same for ε = ±1) isdσ 2 = −(1 − 2M (Z) R )dZ 2 + R 2 dω 2 . (A.10)Denote by n ε µ a unit normal vector to the surface Γ. Simple calculations given ε µ = α −1 (−R , 1, 0, 0) , (A.11) α ε = 1 − 2M ε (Z ε )/r + 2εR (Z ε ) . (A.12)We denote by e μ t a unit future directed vector tangent to Γ e μ t = β −1 (1, R , 0, 0) , β = α ε \| Γ = 1 − [ Ktt] = [Kθθ] = [Kφφ] = 2B . (A.18) The author is grateful to James Bardeen. After the first version of this paper appeared in archive, he attracted my attention to the paper by Sean Hayward, who has used the same form of the metric for the discussion of properties of 'black holes' with a closed apparent horizon. This condition certainly is valid for µ 1. Really, µ ∼ µ −2 and µ ∼ µ −5 and \|µ \|/µ 2 ∼ µ −1 1. 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[ "Self-sustained current oscillations in the kinetic theory of semiconductor superlattices A. Carpio", "Self-sustained current oscillations in the kinetic theory of semiconductor superlattices A. Carpio" ]	[ "E Cebrián ", "L L Bonilla ", "\nDepartamento de Matemáticas y Computación\nUniversidad de Burgos\n09001BurgosSpain\n", "\nG. Millán Institute of Fluid Dynamics, Nanoscience and Industrial Mathematics\nUniversidad Carlos III de Madrid\nAvenida de la Universidad 3028911LeganésSpain\n", "\nDepartamento de Matemática Aplicada\nFac. Matemáticas\nUniversidad Complutense de Madrid\n28040MadridSpain\n" ]	[ "Departamento de Matemáticas y Computación\nUniversidad de Burgos\n09001BurgosSpain", "G. Millán Institute of Fluid Dynamics, Nanoscience and Industrial Mathematics\nUniversidad Carlos III de Madrid\nAvenida de la Universidad 3028911LeganésSpain", "Departamento de Matemática Aplicada\nFac. Matemáticas\nUniversidad Complutense de Madrid\n28040MadridSpain" ]	[]	We present the first numerical solutions of a kinetic theory description of selfsustained current oscillations in n-doped semiconductor superlattices. The governing equation is a single-miniband Boltzmann-Poisson transport equation with a BGK (Bhatnagar-Gross-Krook) collision term. Appropriate boundary conditions for the distribution function describe electron injection in the contact regions. These conditions seamlessly become Ohm's law at the injecting contact and the zero charge boundary condition at the receiving contact when integrated over the wave vector. The time-dependent model is numerically solved for the distribution function by using the deterministic Weighted Particle Method. Numerical simulations are used to ascertain the convergence of the method. The numerical results confirm the validity of the Chapman-Enskog perturbation method used previously to derive generalized drift-diffusion equations for high electric fields because they agree very well with numerical solutions thereof.	10.1016/j.jcp.2009.07.008	[ "https://arxiv.org/pdf/0907.3807v1.pdf" ]	7,555,333	0907.3807	45698e02d69f41d7346f2194c11bdd5eab3ed40b	Self-sustained current oscillations in the kinetic theory of semiconductor superlattices A. Carpio 22 Jul 2009 E Cebrián L L Bonilla Departamento de Matemáticas y Computación Universidad de Burgos 09001BurgosSpain G. Millán Institute of Fluid Dynamics, Nanoscience and Industrial Mathematics Universidad Carlos III de Madrid Avenida de la Universidad 3028911LeganésSpain Departamento de Matemática Aplicada Fac. Matemáticas Universidad Complutense de Madrid 28040MadridSpain Self-sustained current oscillations in the kinetic theory of semiconductor superlattices A. Carpio 22 Jul 2009Preprint submitted to Elsevier 22 July 2009Semiconductor superlatticeBoltzmann-BGK-Poisson kinetic equationcontact boundary conditionsself-sustained current oscillationsparticle methods PACS: 7363Hs0560Gg8535Be0260Lj We present the first numerical solutions of a kinetic theory description of selfsustained current oscillations in n-doped semiconductor superlattices. The governing equation is a single-miniband Boltzmann-Poisson transport equation with a BGK (Bhatnagar-Gross-Krook) collision term. Appropriate boundary conditions for the distribution function describe electron injection in the contact regions. These conditions seamlessly become Ohm's law at the injecting contact and the zero charge boundary condition at the receiving contact when integrated over the wave vector. The time-dependent model is numerically solved for the distribution function by using the deterministic Weighted Particle Method. Numerical simulations are used to ascertain the convergence of the method. The numerical results confirm the validity of the Chapman-Enskog perturbation method used previously to derive generalized drift-diffusion equations for high electric fields because they agree very well with numerical solutions thereof. Introduction When non-interacting electrons in the conduction band of a material are subject to a constant electric field E, their positions should oscillate with a frequency proportional to the electric field, ω B = eEl/ , where −e < 0, and l are the charge of the electron, the Planck constant and the crystal period. These coherent Bloch oscillations (BO) and the associated current were predicted by Zener in 1934 [21]. Scattering limits the observability of BO: to observe them, their period should be smaller than the scattering time τ , so that E > /(elτ ). In standard materials, the fields required to observe BO are too large and therefore damped Bloch oscillations were not found until 1992 in experiments with undoped semiconductor superlattices [11], which have much larger periods than natural crystals. Semiconductor superlattices are artificial one-dimensional crystals formed by epitaxial growth of layers belonging to two different semiconductors that have similar lattice constants [4]. They were synthesized following Esaki and Tsu's idea that these artificial crystals would be useful to realize BO or related high frequency oscillations [8]. The difference in the energy gaps of the component semiconductors makes the conduction band of the superlattice to be a periodic succession of barriers and wells with typical periods of several nanometers. The electronic spectrum of a superlattice (SL) consists of a succession of minibands and minigaps generated by its periodicity. Tayloring the size of barriers and wells and the negative doping density of the latter, it is possible to achieve SLs with wide minibands and to populate only the lowest one. Electrons moving in this miniband have energies that are periodic functions of their wave numbers and are scattered by phonons, impurities and other electrons. When an appropriate dc voltage is held between the ends of one such SL with finitely many periods, it is possible to obtain high-frequency self-sustained oscillations of the current through the structure [4]. These oscillations are caused by repeated formation of electric field pulses at the injecting contact of the SL that move forward and disappear at the receiving contact. Thus they are transit-time oscillations whose frequency is inversely proportional to the SL length: they are similar to the Gunn effect in bulk semiconductors [18] and are different from BO. These Gunn-type oscillations have been observed in experiments with GaAs/AlAs SL (and with other SL based on III-V semiconductors) since 1996 and are the basis of fast oscillator devices [13]. The connection between the existence of Gunn-type oscillations and the suppression of Bloch oscillations is not yet well understood despite theoretical and experimental efforts [4]. Although mathematical models at the level of semiclassical kinetic theory go back to the 1970s [19], their analysis has been based on simplified reduced ordinary differential equations [14,15] which typically ignore space charge effects. Electron transport in a single miniband SL can be described by a kinetic equation coupled to a Poisson equation approximately describing the electric potential due to the other electrons [3]. A simple kinetic equation [14] contains an energy-dissipating collision term of Bhatnagar-Gross-Krook (BGK) type [1] and a simple energy-conserving (but momentum-dissipating) collision term. This model does not include coupling to the Poisson equation. An important point is that the dispersion relation between miniband energy and momentum is periodic because this periodicity gives rise to a relation between electron drift velocity and electric field which has a maximum value [4]. Then the drift velocity decreases as the field increases for large field values (negative differential mobility) and this in turn causes the Gunn-type self-sustained current oscillations (SSCO) for appropriate bias and contact boundary conditions [4]. These features are absent in the more usual Boltzmann-Poisson systems with parabolic band dispersion relations. Recently, Bonilla et al [3] have derived a nonlinear drift-diffusion equation from the KSS-BGK kinetic model coupled to the Poisson equation, which we will call the BGK-Poisson system. They use a Chapman-Enskog perturbation method in a particular limit in which the collision terms are of the same order as the term containing the electric field and dominate all other terms in the kinetic equation. Then stable SSCO are obtained by numerically solving the drift-diffusion equation with appropriate boundary and initial conditions [3]. However, no one has solved numerically the kinetic equation directly and shown that self-oscillations are among its solutions or studied the relation between these solutions and those of the limiting drift-diffusion equation. These are the problems tackled in the present paper and solving them could be a step in more precise studies of stable current oscillations in superlattices and other low dimensional solid state systems. We solve the BGK-Poisson kinetic equation model by means of a deterministic weighted particle method that has been used in the past to solve Boltzmann equations with non-periodic energy band dispersion relations [20]. Particle methods (see a recent one in [10]) are appropriate to study our system of equations because their solutions may present large gradients: the electric field pulses obtained by simulating the approximate drift-diffusion equations have a smooth leading front but a steep trailing back front [4]. The present work paves the way to numerically solving interesting problems in nanoelectronics and spintronics that are described by related quantum kinetic equations with more than one miniband [2]. The Model Our model for electron transport in a single miniband SL is a Boltzmann-Poisson system with BGK collision term [1] plus appropriate boundary and initial conditions. The governing equations are: ∂ t f + v(k)∂ x f + eF ∂ k f = −ν en f − f F D − ν imp 2 [f − f (x, −k, t)], (1) ǫ∂ 2 x V = e l (n − N D ) , F = ∂ x V,(2)n = l 2π π l − π l f (x, k, t) dk = l 2π π l − π l f F D (k; µ(n)) dk,(3)f F D (k; µ) = m * k B T π 2 ln 1 + exp µ − ε(k) k B T ,(4) with x ∈ [0, L] and f periodic in k with period 2π/l. Here l, L = Nl, N, ǫ, f , n, N D , k B , T , V , −F , m * and −e < 0 are the SL period, the SL length, the number of SL periods, the dielectric constant, the one-particle distribution function, the 2D electron density, the 2D doping density, the Boltzmann constant, the lattice temperature, the electric potential, the electric field, the effective mass of the electron, and the electron charge, respectively. We shall describe boundary and initial conditions later. The first term in the right hand side of Eq. (1) represents energy relaxation towards a 1D effective Fermi-Dirac distribution f F D (k; µ(n)) [3] (local equilibrium) due to e.g. phonon scattering. ν en is the collision frequency, taken as constant for simplicity. Here, µ(n) is the chemical potential that is a function of n resulting from solving equation (3) when (4) is substituted in it. A similar BGK model with a Boltzmann local distribution function was proposed by Ignatov and Shashkin [14,15]. The second term in the right hand side of Eq. (1) accounts for impurity elastic collisions with the constant collision frequency ν imp , which conserve energy but dissipate momentum [19,3,4]. Transfer of lateral momentum due to impurity scattering [12] is ignored in this model. We assume the simple tight-binding miniband dispersion relation, ε(k) = ∆ 2 (1 − cos kl) =⇒ v(k) = 1 dε dk = l∆ 2 sin kl,(5) where ∆ is the miniband width. The exact and Fermi-Dirac distribution functions, f and f F D , have the same electron density n, according to (3). The latter equation is solved for the chemical potential µ in terms of n, which yields the function µ(n). When (1) is integrated over k, we obtain the charge continuity equation, ∂ t n + l e ∂ x J n = 0, with(6)J n = e 2π π/l −π/l v(k) f (x, k, t) dk,(7) where J n is the electron current density. Voltage bias condition Using the Poisson equation (2) to eliminate n, we obtain the following form of Ampère's law: ǫ ∂ t F + J n = J(t),(8) where J(t) is the total current density. The total current density can be obtained from the voltage bias condition: Φ(t) ≡ 1 L L 0 F (x, t) dx = φ,(9) where Φ(t)L is the voltage between the two contacts at the end of the SL and Φ(t) is an average field. For dc voltage bias, Φ(t) is a fixed constant φ. If we integrate (8) over x and use (9), we obtain J(t) = 1 L L 0 J n (x, t) dx.(10) Boundary conditions The boundary conditions give the distribution function f on the contacts at x = 0 and x = L through the distribution function inside the semiconductor. For fixed \|k\|, there are two possible characteristic curves at a point (x, t): one for k > 0 and another one for k < 0. With k < 0 the characteristic curve for x → 0+ and t > 0 is given by the initial condition whereas it is given by the distribution function at the contact (x = 0) if k > 0. Then, for x = 0 we need to specify the distribution function at the contact for k > 0, f + , whereas for x = L we need to specify the distribution function at the contact for k < 0, f − . Instead of inventing a theory for injecting and collecting contacts, we use a top-down approach proposed in Ref. [4]: we know that the following boundary conditions appropriately describe current self-oscillations in the drift-diffusion equation for the electric field, J n (0, t) = σ F (0, t), (11) n(L, t) = N D ,(12) where σ > 0 is the constant contact conductivity and the left hand side of Eq. (11) is the electron current density. We will use boundary conditions for f such that they become (11) and (12) when we integrate them according to the definitions (3) and (7) of n and J n respectively: f + = 2π σF e∆ − f (0) π l 0 v(k)f (0) dk 0 − π l v(k)f − dk,(13) for x = 0, and f − = f (0) (l/2π) 0 − π l f (0) dk   N D − l 2π π l 0 f + dk    ,(14) for x = L. Note that the integral over k of (13) times e v(k)/(2π) yields (11) and the integral over k of (14) times l/(2π) yields (12). In these equations, f (0) is the leading order approximation for the distribution function in the Chapman-Enskog method [3,4]: f (0) (k; n, F ) = ∞ j=−∞ f (0) j exp (ıjkl) ,(15) where f (0) j = 1 − ıjϕ/τ e 1 + j 2 ϕ 2 f F D j f F D j = l π π l 0 f F D cos(jkl) dk ϕ = F F M F M = ν en (ν en + ν imp ) el τ e = 1 + ν imp ν en . Eq. (15) is the solution of (1) when we drop the x and t derivatives of f (see [3]). If we use the electric potential V instead of the field F = ∂V /∂x (recall that the true electric field is −F ), the following boundary conditions for V are compatible with (9): V (0, t) = 0, V (L, t) = φL = L 0 F (x, t) dx.(16) Initial condition We select (15) as our initial condition for the distribution function. The initial electric field is assumed to be constant, F (x, 0) = φ, where φ is the average field. If we start from other initial conditions, the evolution of the current and other magnitudes are similar to those presented here after about 0.3 ps. Recapitulating, the equations governing our model are (1) -(4) for the unknowns f and V with initial condition (15) and boundary conditions (13), (14) and (16). If we use the field F instead of the electric potential V , the voltage bias condition (9) for F replaces (16). Nondimensional equations We use the scales defined in Table 1 to nondimensionalize the Boltzmann-BGK-Poisson kinetic equations. These scales are based on the hyperbolic scaling explained in Ref. Table 1 : Hyperbolic scaling. [3] F V k x t µ n, f F M F M x 0 1/l x 0 t 0 k B T N D F a = F F M V a = V F M x 0 k a = kl x a = x x 0 t a = t t 0 µ a = µ k B T n a = n N D , f a = f N Dx 0 = ǫF M l eN D , t 0 = x 0 v M , v M = ∆lℑ 1 M 4 τ e ℑ 0 M , ℑ j M = 1 2π π −π cos(jk a ) ln 1 + exp M − δ + δ cos(k a ) dk a , where M verifies 1 = α 2π π −π ln 1 + exp M − δ + δ cos(k a ) dk a , with α = m * k B T π 2 N D . Numerical values for these parameters will be given in Section 5. Equations (1) -(4) have the following nondimensional form ∂ t a f a + ∆l 2 v M sin(k a )∂ x a f a + τ e η F a ∂ k a f a = 1 η f F Da (k a ; µ a (n a )) − 1 + ν imp 2ν en f a + ν imp 2ν en f a (x a , −k a , t a ) ,(17)∂ 2 x a V a = ∂ x a F a = n a − 1 (18) n a = 1 2π π −π f a (x a , k a , t a ) dk a = 1 2π π −π f F Da (k a ; µ a (n a )) dk a (19) f F Da (k a ; µ a ) = α ln [1 + exp (µ a − δ + δ cos(k a ))] (20) η = v M ν en x 0 δ = ∆ 2k B T . The dimensionless boundary conditions are, for x a = 0: f a+ = βF a − f a(0) π 0 sin (k a ) f a(0) dk a 0 −π sin (k a ) f a− dk a (21) with β = 2π σF M e∆N D and for x a = L/x 0 : f a− = f a(0) (1/(2π)) 0 −π f a(0) dk a   1 − 1 2π π 0 f a+ dk a   .(22) The boundary conditions for the electric potential V a are V a (0, t a ) = 0, V a (L a , t a ) = φ a L a ≡ φ F M L x 0 .(23) 8 The dimensionless initial condition is f a(0) (k a ; n a ) = ∞ j=−∞ exp (ıjk a ) 1 − ıjF a /τ e 1 + j 2 (F a ) 2 f F Da j (n a ) (24) f F Da j (n a ) = 1 π π 0 f F Da (k a ; µ a (n a )) cos(jk a ) dk a with x a ∈ [0, L a = L/x 0 ] and f a periodic in k a with period 2π. Besides the electron current density, J n , it is convenient to calculate the average energy E (and its nondimensional version, E a ), defined as E a = E/(k B T ): E a = π/l −π/l ε(k)f (x, k, t)dk k B T π/l −π/l f (x, k, t)dk = δ π −π (1 − cos k a )f a (x a , k a , t a )dk a π −π f a (x a , k a , t a ) dk a .(25) From now on we drop the superscript a. The Deterministic Weighted Particle Method The most widely used numerical method used for solving Boltzmann equations is the Monte-Carlo Method [17]. This stochastic method yields data with a lot of numerical noise. The deterministic Weighted Particle Method (WPM) is an interesting alternative because it yields the distribution function (and therefore its moments: electron density, average energy and current density) at each time during the transient regimes with much less noise than the Monte Carlo simulation; cf. [20,7,5] (a numerical analysis of WPM can be found in [6] and in [16] for the special case of the BGK equation of gas dynamics). The WPM relies on a particle description of the distribution function, which means that f (x, k, t) is written as a sum of delta functions f (x, k, t) ≈ N i=1 ω i f i (t)δ(x − x i (t)) ⊗ δ(k − k i (t)) where ω i , f i (t), x i (t) and k i (t) are, respectively, the (constant) control volume, the weight, the position and the wave vector of the ith particle. N is the number of numerical particles. In the WPM, the motion of particles is governed by collisionless dynamics, whereas the collisions are accounted for by the variation of weights. Large gradients in the solution profile arise from appropriate particles acquiring large weights, not by accumulating many particles in the large gradient regions. The evolution of the particles is determined by their positions and wave vectors which are the characteristic curves of the convective part of the equation. Their equations are: d dt k i (t) = τ e η F i (t), d dt x i (t) = ∆l 2 v M sin (k i (t)) (26) where F i (t) = F (x i (t) , t) denotes the electric field at the instantaneous position of the i-th particle. The evolution of the weight f i (t) is given by the ordinary differential equation: d dt f i (t) = 1 η − 1 + ν imp 2ν en f i (t) + ν imp 2ν en f (x i (t), −k i (t), t) + f F D i (t) (27) with f F D i (t) the Fermi-Dirac distribution evaluated for the i-th particle. The system of ordinary differential equations (26) -(27) is now solved by using a modified (semi-implicit) Euler method: f n i = f n−1 i + dt 1 η − 1 + ν imp 2ν en f n−1 i + ν imp 2ν enf n−1 i + f F D,n−1 i (28) withf n−1 i = f (x n−1 i , −k n−1 i , t n−1 ), k n i = k n−1 i + dt τ e η F n−1 i ,(29)x n i = x n−1 i + dt ∆l 2 v M sin (k n i ) .(30) For stability reasons, we use k n i to update x n i . The standard Euler method would use k n−1 i to update x n i but this would require using unpractically small time steps to have a stable scheme. The same problem appears when we employ explicit Runge-Kutta or multi-step methods. To select the initial positions and wave vectors in the modified Euler method, we build a grid in the domain The boundary conditions are taken into account as follows: • If k n i > π, we set k n i = k n i − 2π. If k n i < −π, we set k n i = k n i + 2π. • If x n i > L, we set x n i = x n i − L and f n−1 i = f + i . If x n i < 0, we set x n i = x n i + L and f n−1 i = f − i . Here f + i and f − i are calculated by discretization of the integrals in (21) and (22) using the composite Simpson's rule on an equally spaced mesh K m ′ with step ∆k. To calculate x i , k i and f i at the next time step t n+1 , we need to update the electric field and the Fermi-Dirac distribution in the equations for the particles. According to Eqs. (2) and (3), this updating requires an interpolation procedure to generate an approximation of the distribution function on a regular mesh X m , K m ′ which is then used to approximate the electric field and the chemical potential. To approximate the values of the distribution function over the mesh, f n m,m ′ , we use the following weighted mean of its values for the particles, f n i : f n m,m ′ = N i=1 f n i W i m,m ′ N i=1 W i m,m ′ (31) where W i m,m ′ = max 0, 1 − \|X m − x n i \| ∆x · max 0, 1 − \|K m ′ − k n i \| ∆k and ∆x and ∆k are the spatial and wave vector steps. An approximation for the density (19) and average energy (25) at the mesh points, n (X m , t n ) ≈ n n m and (k B T ) −1 E (X m , t n ) ≈ (k B T ) −1 E n m , are obtained using the composite Simpson's rule and the interpolated values of the distribution function on the mesh. We calculate the nondimensional chemical potential µ by using a Newton-Raphson iterative scheme to solve equations (19) and (20): µ p = µ p−1 − g (µ p−1 ) g ′ (µ p−1 ) (32) with g (µ) = n − α 2π π −π ln [1 + exp (µ − δ + δ cos (k))] dk g ′ (µ) = − α 2π π −π exp (µ − δ + δ cos (k)) 1 + exp (µ − δ + δ cos (k)) dk. The initial guess for µ is obtained by plotting g(µ) and selecting a value near its zero. g(µ) and g ′ (µ) are evaluated using the composite Simpson's rule. Once we know the chemical potential µ, Eq. (20) provides the Fermi-Dirac distribution function at mesh points, f F D (K m ′ ; µ(n n m )), which is then interpolated to get the Fermi-Dirac weight function for the particles, f F D,n i : f F D,n i = X m+1 − x n i ∆x K m ′ +1 − k n i ∆k f F D (K m ′ ; µ(n n m )) + x n i − X m ∆x K m ′ +1 − k n i ∆k f F D K m ′ ; µ(n n m+1 ) + X m+1 − x n i ∆x k n i − K m ′ ∆k f F D (K m ′ +1 ; µ(n n m )) + x n i − X m ∆x k n i − K m ′ ∆k f F D K m ′ +1 ; µ(n n m+1 ) ,(33)provided the particle i is in [X m , X m+1 ] × [K m ′ , K m ′ +1 ]. To compute the electric field at time t n , we use finite differences to discretize the Poisson equation on the grid X m : V n m+1 − 2V n m + V n m−1 (∆x) 2 = n n m − 1,(34)F n m = V n m+1 − V n m−1 2 ∆x .(35) Here V (0, t n ) = 0 and V (L, t n ) = φL as indicated by (23). V n m and F n m denote our approximations of V (X m , t n ) and F (X m , t n ) on the equally spaced mesh X m . Finally, the electric field is interpolated at the location of the particle i F n i = X m+1 − x n i ∆x F n m + x n i − X m ∆x F n m+1 ,(36) provided the particle i is in [X m , X m+1 ]. The total current density J is given by Eq. (10), whose nondimensional version is J(t) = ς L L 0   π −π sin(k)f (x, k, t) dk   dx,(37) in which ς = l∆ 4π v M . We use the composite Simpson rule to approximate J(t n ). Summarizing, at each time step t n : (1) Calculate the boundary conditions (21) and (22) with data at time t n−1 . (2) Compute f n i , k n i and x n i according to (28), (29) and (30), respectively, by using their values at t n−1 . (3) Evaluate the distribution function f n m,m ′ at the mesh points (X m , K m ′ ) by the weighted mean (31). (4) Compute the electron density (19) and nondimensional average energy (25) at the mesh points. We have observed that the costlier processes are 2 (computation of f n i using (28)) and 6 (computation of the Fermi-Dirac weight function f F D,n i ): these two processes take about 50% of the overall computation time and they are equally costly. After these processes, 3, 5 and 7 have the largest computational cost (each takes between 10% and 19% of the overall computation time). Numerical results We have used the parameter values of [9]. Numerical solutions of the nonlinear drift-diffusion equation derived from the Boltzmann-BGK model show that there is a stable stationary state for voltage bias below a certain threshold. Above this critical voltage, stable self-sustained oscillations of the current appear. These oscillations are due to the periodic generation of electric field pulses at the injecting contact and their motion towards the receiving contact. We have observed the same phenomena in our numerical solutions of the Boltzmann-BGK kinetic equations. Firstly, we present a typical case of selfsustained current oscillations accompanied by the motion and recycling of an electric field dipole wave, corresponding to a 157-period 3.64 nm GaAs/0.93 nm AlAs SL at 14 K, with ∆ = 72 meV, N D = 4.57×10 10 cm −2 , ν imp = 2ν en = 18×10 12 Hz and dimensionless dc average field φ = 1 [9]. The constant conductivity is 2.5 Ω cm −1 and the effective mass is m * = (0.067d W + 0.15d B )m 0 /l, where m 0 = 9.109534 × 10 −31 kg is the electron rest mass. Using these numer- For these parameter values, we consider 140800 particles and a mesh of 440 grid points for x and 80 points for k. The time step (dt) is 0.002 ps. Figure 1 shows the self-oscillations of the current, and Figure 2 the corresponding electric field pulse at different times. We observe how the electric field pulses are periodically created at the injecting contact x = 0, move to the end of the SL and disappear at the receiving contact. In Fig. 2, we have depicted the field profiles at the times marked (a) -(e) in Fig.1. We observe that the total current density reaches its maximum value when the electric field pulse is about to disappear at the collector. The electric field as a function of time and position is shown in Figure 3, both during one oscillation period in Fig. 3(a) and during several periods in Fig. 3(b). The ratio from the maximum to the minimum current in Fig.1 is 2.6 whereas the same ratio calculated by solving the drift-diffusion equation derived in [9] is 2.1 (cf. dashed line in Fig. 1(a) [9]). Measured in units of t 0 (which has a different numerical value in [9]), the oscillation period is 104.3 in Fig. 1 whereas the drift-diffusion equation yields 113.8. Comparing Fig. 1(b) of [9] with our Fig. 2, we find that at the time (c) the solution of the BGK-Poisson equation produces a pulse far from the contacts which is 11 x 0 wide and 7 F M tall whereas the drift-diffusion equation yields a similar pulse which is 10.7 x 0 wide and 6.8 F M tall (cf. dashed line in Fig. 1(b) of [9]). Thus the agreement between the simulation of the BGK-Poisson system and that of the drift-diffusion equation is very good considering the approximations made in the derivation of the latter from the former. Figure 4 shows the dimensionless electron density. We see the profile during several times belonging to one oscillation period as a function of position in Fig. 4(a) and as a function of the time and position in Fig. 4(b). The electron density profile corresponding to an electric field pulse is that of a traveling dipole wave such that n > 1 behind the peak of the electric field and 0 < n < 1 ahead of the peak. Comparison with Fig. 3(a) shows that the local maximum of the electron density is reached somewhat later than the peak of the electric field pulse. of distance at different instants of one oscillation period. The average energy profile is pulse-like. Its local maximum is always quite close to the peak of the electric field during each oscillation period. Fig. 5(b) shows the average energy profile as a function of position and time during one oscillation period. In Figure 6, we have depicted snapshots of the distribution function f (x, k, t) for different times as marked in Fig. 1 (30 ps, 36 ps, 42 ps, 48 ps, 54 ps, 60 ps) during one period of the self-oscillations. The structure of the distribution function is shown more clearly in the density plots depicted in Fig. 7 for the same times. The electron density profiles at the these times are shown in Figure 8. We observe that the distribution function has a local maximum at location of the peak of electron density. Similarly, f and n have local minima at the same positions. The distribution function has a local maximum at a positive k (cf. Fig. 7), and this situation persists from the initial time onwards; cf. Figure 9. Fig. 3(a), we observe that the particle positions oscillate with very small amplitudes when the electric field has a local maximum at their locations and these amplitudes become larger once the pulse has surpassed the particles. In contrast with these great changes in oscillation amplitude of the particle positions, the wave vectors of the particles oscillate with almost constant amplitudes, as shown in Fig. 10(b), 11 and 12. Since the evolution of the particle wave vector is more regular than the evolution of the particle position, we can save mesh points on the wave vectors. Recalling that the wave vector is a periodic variable, its boundary condition is as follows: when one particle goes out of the domain at k = π, it is reintroduced at k = −π. This condition can be readily observed in Figures 11 and 12. Fig. 13(a), we need more particles for the method to converge whereas in the second case, Fig. 13 Lastly, Figure 16 shows the evolution of the total current density for different time steps in simulations with 90000 particles and 260 mesh grid points for the position and 80 for the wave vector. We observe that our results are similar for time steps dt = 8 × 10 −4 ps and smaller. Figures 14 to 16 show that the shape of J(t) is similar for different mesh points and time steps: the device behavior is qualitatively correct even if we take fewer mesh points or larger time steps than needed to attain a numerically precise current vs time graph. Smaller M k , M x and larger dt result in slightly smaller oscillation periods and slightly larger oscillation amplitudes. Our numerical simulations have been carried out using a Matlab code in a computer with a Genuine Intel(R) CPU T2050 @ 1.60GHz processor with a 1595 MHz speed. Several computation times for time steps dt of 0.008 and 0.002 ps and 10000 time steps are shown in Table 2. Clearly, the time the computer takes to calculate one time step dt decreases as the number of par-ticles, M x or M k decrease. Except for the last row in Table 2, all rows satisfy N/(M x M k ) ≥ 2.25, and the corresponding particle numbers and x and k mesh points produce accurate enough results. Conclusion We have proposed a deterministic weighted particle method to numerically solve for the first time the semiclassical Boltzmann-BGK-Poisson system of equations with periodic miniband energy dispersion relation. This system describes vertical electron transport in a GaAs/AlAs superlattice under dc voltage bias conditions. When using appropriate values for the injecting contact conductivity and voltage, we find a stable self-sustained oscillation of the current through the structure which corresponds to periodic nucleation of electric field pulses at the injecting contact that then move to the receiving contact. The pulses have a large electron density on their trailing edges which implies large gradients of the electric field there. These gradients are well resolved by particles having large weights there, which is one of the advantages of using the weighted particle numerical method. Our results agree with experimental observations [13,4] and confirm the validity of the Chapman-Enskog perturbation method used to derive a drift-diffusion equation for high electric fields [3]. In fact, the electric field profile and the total current density obtained by numerically solving the the drift-diffusion equation [9] agree very well with the numerical solution of the kinetic equations obtained in the present work. Having solved the kinetic equations directly, we can obtain the evolution of the distribution function and its relevant moments such as electron density, current density and average energy. The present work paves the way to numerically solving interesting problems in nanoelectronics and spintronics that are described by related quantum kinetic equations with more than one miniband [2]. [ 0 , 0L] × [−π, π] and choose the values (x 0 i , k 0 i ) as the cell centers. The weights f 0 i are then chosen according to (24). ( 5 ) 5Calculate the chemical potential (32) and compute the Fermi-Dirac distribution (20) at the mesh points. (6) Interpolate the Fermi-Dirac distribution (20) at the mesh points to obtain the Fermi-Dirac weight function f F D,n i according to (33). (7) Compute the electric field at the mesh points by solving the finite difference discretization of the Poisson equation, (34) and (35) and interpolate at the particles according to (36). (8) Calculate the current by evaluation of (37) using the composite Simpson's rule. Fig. 1 . 1Total current density versus time plot exhibiting self-sustained oscillations. Units are written in parentheses. The oscillation period is 24 ps and the ratio between the maximum and the minimum current is 2.6. At the times marked (a) -(f) within one oscillation period (30, 36, 42, 48, 54 and 60 ps, respectively), we shall depict the electric field profile, the electron density profile, the distribution function and the density plots thereof inFigures 2, 8, 6 and 7, respectively.ical values, the scales of space, time, velocity, electric field and dimensionless chemical potential defined in Section 3 take on the following values:x 0 = 15.92 nm, t 0 = 0.23 ps, v M = 68.33 km/s, F M = 22.45 kV/cm, M = 7.11. Fig. 2 . 2Electric field versus position at different times within one period of the oscillation. Far from the contacts, at time (c), the electric field pulse is 320 nm wide and 139 kV/cm tall. Thus it occupies about 45% of the SL extension. At time (e), the electric field has a maximum value of 312 kV/cm. Figure 5 (Fig. 3 . 53a) depicts the nondimensional average energy, E/(k B T ) as a function Evolution of the electric field F (x, t) during (a) one period and (b) several periods of the self-oscillation. Fig. 4 . 4Electron density profiles during one oscillation period. Fig. 5 . 5Nondimensional average energy profiles, E/(k B T ), at different times of one oscillation period. Fig. 6 . 6(a) -(f) Distribution function versus position and wave vector at the different times of one oscillation period as marked in Fig. 1. Figure 10 ( 10a) shows the time evolution of the position for six particles, whereas Figure 10(b) shows the wave vector vs position for the same particles. The motion of the particles is a superposition of an uniform motion and an oscillation about it. Comparing Figure 10(a) with Fig. 7 . 7(a) -(f) Density plots of the distribution function versus position and wave vector at the different times of one oscillation period as marked in Fig. 1. Fig. 8 . 8Electron density profile for the different times as marked inFig. 1. Fig. 9 .Fig. 10 . 910Distribution function versus wave vector at t = 0 ps. Time evolution of the position for six particles whose initial dimensionless wave vector is −3.126: (a) position vs time, (b) wave vector vs position for the same particles.6 Convergence of the methodWe have checked the convergence of the method in terms of the number of particles N, mesh points M x and M k , and time step dt. Since the Fermi-Dirac weights in (28) and the electric field in (29) have to be calculated by interpolation over mesh points, all these parameters are important for the convergence of the calculations over particles and of the calculations over the mesh. The physical parameters are the same as in the previous section. Fig. 11 . 11Time evolution of the particle with initial dimensionless position and wave vector (13.986, −3.126) during several oscillation periods. Fig. 12 . 12Time evolution of the particle with initial dimensionless position and wave vector (13.986, −3.126) during one oscillation period. Figure 13 shows 13the evolution of the current density for different number of particles N. We have kept the time step and the wave number mesh points fixed at the values M k = 80 and dt = 0.008 ps. We observe that we need different N for convergence of the calculation depending on the value of M x . InFig. 13(a), for M x = 440 position mesh points, we have chosen N so that N/(M x M k ) takes on the values 1.5, 1.84, 2.25 and 3, whereas in Fig. 13(b), M x = 360 and N is chosen so that N/(M x M k ) takes on the values 1.5, 2.25 and 3. We observe that for dt = 0.008 ps and N/(M x M k ) ≥ 2.25 the results do not change if we increase the number of particles. In particular, for N = 64800, N/(M x M k ) = 1.84 if M x = 440 and N/(M x M k ) = 2.25 if M x = 360. In the first case shown in Fig. 13 . 13(b) shows that we do not improve our results by increasing N. The convergence range of N/(M x M k ) depends slightly on the time step dt: if dt = 0.002 ps, M x = 440, M k = 80, our calculations yield indistinguishable curves J(t) for N/(M x M k ) = 3, 4, but not for N/(M x M k ) = 2.25. Thus we have found that it is advisable to select N so that 3 ≤ N/(M x M k ) ≤ 4.5: numerical results are indistinguishable when N/(M x M k ) ≥ 3 and the computational cost is not very large if we keep N/(M x M k ) ≤ 4.5. Figures 14 and 15 show the evolution of the total current density in simulations Current versus time for different number of particles N when M k = 80 and dt = 0.008 ps. (a) M x = 440 and (b) M x = 360. Fig. 14 . 14Current versus time for different number of wave vector mesh points when M x = 520, N = 200000 and dt = 0.008 ps. Fig. 15 . 15Current versus time for different number of position mesh points when M k = 80, N = 480000 and dt = 0.008 ps. 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[ "THE MILNOR INVARIANTS OF CLOVER LINKS", "THE MILNOR INVARIANTS OF CLOVER LINKS" ]	[ "Kodai Wada ", "Akira Yasuhara " ]	[]	[]	J.P. Levine introduced a clover link to investigate the indeterminacy of the Milnor invariants of a link. It is shown that for a clover link, the Milnor numbers of length at most 2k + 1 are well-defined if those of length at most k vanish, and that the Milnor numbers of length at least 2k + 2 are not well-defined if those of length k + 1 survive. For a clover link c with the Milnor numbers of length at most k vanishing, we show that the Milnor number µc(I) for a sequence I is well-defined up to the greatest common devisor of µc(J) ′ s, where J is a subsequence of I obtained by removing at least k + 1 indices. Moreover, if I is a non-repeated sequence with length 2k + 2, the possible range of µc(I) is given explicitly. As an application, we give an edge-homotopy classification of 4-clover links.	10.1142/s0129167x16501081	[ "https://arxiv.org/pdf/1507.01385v1.pdf" ]	119,690,186	1507.01385	0004dad2e757e34faded8b756222383aad84d56f	THE MILNOR INVARIANTS OF CLOVER LINKS 6 Jul 2015 Kodai Wada Akira Yasuhara THE MILNOR INVARIANTS OF CLOVER LINKS 6 Jul 2015arXiv:1507.01385v1 [math.GT] J.P. Levine introduced a clover link to investigate the indeterminacy of the Milnor invariants of a link. It is shown that for a clover link, the Milnor numbers of length at most 2k + 1 are well-defined if those of length at most k vanish, and that the Milnor numbers of length at least 2k + 2 are not well-defined if those of length k + 1 survive. For a clover link c with the Milnor numbers of length at most k vanishing, we show that the Milnor number µc(I) for a sequence I is well-defined up to the greatest common devisor of µc(J) ′ s, where J is a subsequence of I obtained by removing at least k + 1 indices. Moreover, if I is a non-repeated sequence with length 2k + 2, the possible range of µc(I) is given explicitly. As an application, we give an edge-homotopy classification of 4-clover links. Introduction The Milnor invariant introduced by J. Milnor [7], [8]. For an oriented ordered n-component link L in the 3-sphere S 3 with peripheral information, the Milnor number µ L (I), which is an integer, is specified by a finite sequence I in {1, 2, . . . , n}. In [6], J. P. Levine examined the Milnor invariants from the point of view of based links, in order to understand the indeterminacy. A based link is a link for which some peripheral information is specified, i.e., meridians (the weakest) or both meridians and longitudes (the strongest). It is known that these invariants are completely well-defined for the strongest form of basing (disk links [5] or string links [2]). As basing only slightly stronger than the specification of meridians, he introduced an n-clover link which is an embedded graph consisting of n loops, each loop connected to a vertex by an edge in S 3 . The Milnor number of a clover link is defined up to flatly isotopy and it is shown that those of length ≤ k vanish implies those of length ≤ 2k + 1 are completely non-indeterminate [6]. The first author [10] redefined the Milnor number of a clover link up to ambient isotopy as follows. Given an n-clover link c, we construct an n-component bottom tangle γ(F c ) by using a disk/band surface F c of c. In [6], Levine defined the Milnor number of a bottom tangle. Therefore we define the Milnor number µ c of an nclover link c to be the Milnor number µ γ(Fc) . (In [6], a bottom tangle is called a string link. The name 'bottom tangle' follows K. Habiro [3].) In [10], it is shown that the same result as Levine [6] holds while there are infinitely many choices of γ(F c ) for c. Unfortunately, the Milnor numbers of length at least 2k+2 are not well-defined if those of length k + 1 survive. In this paper, we show that the Milnor number µ c (I) of a sequence I modulo δ k c (I) is well-defined if the Milnor numbers of length ≤ k vanish, where δ k c (I) is the greatest common devisor of µ c (J) ′ s, where J is range to over all subsequences of I obtained by removing at least k + 1 indices. In fact, we have the following theorem. Theorem 1.1. Let c be an n-clover link and l c a link which is the disjoint union of loops of c. If the Milnor numbers of l c for sequences with length ≤ k vanish, then the residue class of µ c (I) modulo δ k c (I) is an invariant for any sequence I. Remark 1.2. In contrast to the µ-invariant for a link in S 3 , we do not need to take cyclic permutation for getting δ k c (I). It is an important property that for non-repeated sequences, the Milnor invariants of links are link-homotopy invariants [7]. The first author showed that the Milnor numbers for any non-repeated sequence with length ≤ 3 give an edge-homotopy classification of 3-clover links [10], where edge-homotopy [9] is an equivalence relation, which is a generalization of link-homotopy, generated by crossing changes on the same spatial edge. We also discuss giving an edge-homotopy classification of 4-clover links. The Milnor numbers for non-repeated sequences with length 4 could be useful to have an edge-homotopy classification of 4-clover links. But they are not well-defined in general. Hence we consider the set of all Milnor numbers of length 4 for all disk/band surfaces of c. More generally, we define the following set; H c (2k + 2, j) =      S∈S 2k+1 j µ γ(Fc) (Sj)X S F c : a disk/band surface of c      for each integer j (1 ≤ j ≤ n), where S 2k+1 j is the set of length-(2k + 1) nonrepeated sequences without containing j and for a sequence S = i 1 i 2 . . . i 2k+1 , X S = X i1 X i2 · · · X i 2k+1 is a monomial in non-commutative variables X 1 , ..., X n . Since H c (2k + 2, j) consists of the Milnor numbers µ γ(Fc) (Sj) (S ∈ S 2k+1 j ) for all disk/band surfaces of c, it is an invariant of c. While it seems too big to handle H c (2k + 2, j), we have the following theorem. Theorem 1.3. Let c be an n-clover link and F c a disk/band surface of c. If the Milnor numbers of l c for non-repeated sequences with length ≤ k vanish, then we have the following: H c (2k + 2, j) =          \|J\|=\|I\|=k JIl∈S 2k+1 j µ γ(Fc) (Jj)µ γ(Fc) (Il)   is∈{J} m lis X J<s (X isIl − X islI − X Ilis + X lIis )X Js< + m lj (X IlJ − X lIJ − X JIl + X JlI ) + S∈S 2k+1 j µ γ(Fc) (Sj)X S m pq = m qp ∈ Z      , where for a sequence J = i 1 . . . i m , {J} = {i 1 , . . . , i m } and J <s (resp. J s< ) is a subsequence i 1 . . . i s−1 (resp. i s+1 . . . i m ) of J for 1 ≤ s ≤ m, and X J<1 and X Jm< are defined to be 1. This theorem implies that the set H c (2k+2, j) is obtained from the Milnor numbers of γ(F c ) for any single disk/band surface F c of c, that is, H c (2k + 2, j) is specified explicitly. By the following corollary we have that H c (2k + 2, j) is not only an invariant of c but also an edge-homotopy invariant. It is the definition that the Milnor numbers of length 1 are zero. If k = 1, then the theorem above holds without the condition. The following example follows directly from Theorem 1.3. + µ γ(Fc) (14)µ γ(Fc) (23)(m 34 − m 13 − m 24 + m 12 ) +µ γ(Fc) (13)µ γ(Fc) (24)(m 34 − m 23 − m 14 + m 12 ) X 231 + µ γ(Fc) (13)µ γ(Fc) (24)(m 23 − m 34 − m 12 + m 14 ) +µ γ(Fc) (12)µ γ(Fc) (34)(m 23 − m 24 − m 13 + m 14 ) X 312 + µ γ(Fc) (14)µ γ(Fc) (23)(m 13 − m 34 − m 12 + m 24 ) +µ γ(Fc) (12)µ γ(Fc) (34)(m 24 − m 23 − m 14 + m 13 ) X 321 + S∈S 3 4 µ γ(Fc) (S4)X S m pq ∈ Z . This together with the theorem below gives us an edge-homotopy classification of 4-clover links, see Remark 1.7. Remark 1.7. By Example 1.5, we are able to determine whether H c (4, 4)∩H c ′ (4, 4) is empty or not. Hence by combining Example 1.5 and Theorem 1.6, we obtain an edge-homotopy classification of 4-clover links. The Milnor numbers of clover links In this section we define the Milnor numbers for clover links. γ = γ 1 ∪ γ 2 ∪ · · · ∪ γ n defined by Levine [6] is a tangle with ∂γ i = {( 2i−1 2n+1 , 1 2 , 0), ( 2i 2n+1 , 1 2 , 0)} ⊂ ∂[0, 1] 3 for each i (= 1, 2, . . . , n). A spatial graph is an embedded graph in S 3 . Let C n be a graph consisting of n oriented loops e 1 , e 2 , . . . , e n , each loop e i connected to a vertex v by an edge f i (i = 1, 2, . . . , n) , see Figure 2.1. An n-clover link in S 3 is a spatial graph of C n [6]. The each part of a clover link corresponding to e i , f i and v of C n are called the leaf, stem and root, denoted by the same notations respectively. [4] defined disk/band surfaces of spatial graphs. For a spatial graph Γ, a disk/band surface F Γ of Γ is a compact, oriented surface in S 3 such that Γ is a deformation retract of F Γ contained in the interior of F Γ . Note that any disk/band surface of a spatial graph is ambient isotopic to a surface constructed by putting a disk at each vertex of the spatial graph, connecting the disks with bands along the spatial edges. We remark that for a spatial graph, there are infinitely many disk/band surfaces up to ambient isotopy. Given an n-clover link, we construct an n-component bottom tangle using a disk/band surface of the clover link as follows: (1) For an n-clover link c, let F c be a disk/band surface of c and let D be a disk which contains the root. From now on, we may assume that the intersection D ∩ n i=1 f i and orientations of the disks are as illustrated in Figure 2.2. (3) Since the 3-ball is homeomorphic to [0, 1] 3 , we obtain an oriented ordered (3) and (4) While there are infinitely many disk/band surfaces of a clover link which satisfy the condition (1) above, they are related by certain local moves as follows. 0, 0) and G q the qth lower central subgroup of G. Let α i and λ i be the ith meridian and ith longitude of γ respectively as illustrated in Figures 2.5. We assume that λ i is trivial in G/G 2 . Since the quotient group G/G q is generated by α 1 , α 2 , . . . , α n [1], γ is represented by α 1 , α 2 , . . . , α n modulo G q . We consider the Magnus expansion of λ j . The Magnus expansion is a homomorphism (denoted E) from a free group α 1 , α 2 , . . . , α n to the formal power series ring in non-commutative variables X 1 , X 2 , . . . , X n with integer coefficients defined as follows. f 1 f 2 f n D e in-component bottom tangle γ(F c ) from (∂F c \N (D)) \ n i=1 S 1 i as illustrated inγ = γ 1 ∪ γ 2 ∪ · · · ∪ γ n be an oriented ordered n- component bottom tangle in [0, 1] 3 with ∂γ i = {( 2i−1 2n+1 , 1 2 , 0), ( 2i 2n+1 , 1 2 , 0)} ⊂ ∂[0, 1] 3 for each i (= 1, 2, . . . , n). Let G be the fundamental group of [0, 1] 3 \ γ with a base point p = ( 1 2 ,α 1 α i α n p λ 1 λ i λ n pE(α i ) = 1 + X i , E(α −1 i ) = 1 − X i + X 2 i − X 3 i + · · · (i = 1, 2, . . . , n) . For a sequence I = i 1 i 2 . . . i k−1 j (i m ∈ {1, 2, . . . , n}, k ≤ q), we define the Milnor number µ γ (I) to be the coefficient of X i1i2···i k−1 in E(λ j ) (we define µ γ (j) = 0), which is an invariant [6]. (In [6], the set of λ i 's, without taking the Magnus expansion, is called the Milnor's µ-invariant.) For a bottom tangle γ = γ 1 ∪ γ 2 ∪ · · · ∪ γ n , an oriented link L(γ) = L 1 ∪ L 2 ∪ · · · ∪ L n in S 3 can be defined by In this section we will give a proof of Theorem 1.1. An n-component tangle u = u 1 ∪ u 2 ∪ · · · ∪ u n is an n-component string link if for each i (= 1, 2, . . . , n), the boundary L i = γ i ∪ a i ,∂u i = {( 2i−1 2n+1 , 1 2 , 0), ( 2i−1 2n+1 , 1 2 , 1)} ⊂ ∂[0, 1] 3 . In particular, u is trivial if for each i (= 1, 2, . . . , n), u i = {( 2i−1 2n+1 , 1 2 )} × [0, 1] in [0, 1] 3 . Here we introduce a SL-move [10] given by a string link u which is a transformation of an n-component bottom tangle γ Figure 3.7. = γ 1 ∪ γ 2 ∪ · · · ∪ γ n with ∂γ i = {( 2i−1 2n+1 , 1 2 , 0), ( 2i 2n+1 , 1 2 , 0)} ⊂ ∂[0, 1] 3 . (1) Let u = u 1 ∪ u 2 ∪ · · · ∪ u′ i = {( 2i 2n+1 , 1 2 , 0), ( 2i 2n+1 , 1 2 , 1)} ⊂ ∂[0, 1] 3 , seeu 1 u ′ 1 u i u ′ i u n u ′ n Figure 3.7. (2) Let γ ′ = γ ′ 1 ∪ γ ′ 2 ∪ · · · ∪ γ ′ n be an n-component bottom tangle in [0, 1] 3 defined by γ ′ i = h 0 (u i ∪ u ′ i ) ∪ h 1 (γ i ) for i = 1, 2, . . . , n, where h 0 , h 1 : ([0, 1]×[0, 1])×[0, 1] → ([0, 1]×[0, 1])×[0, 1] are embeddings defined by h 0 (x, t) = (x, 1 2 t) and h 1 (x, t) = (x, 1 2 + 1 2 t) for x ∈ ([0, 1] × [0, 1]) and t ∈ [0, 1]. We say that γ ′ is obtained from γ by a SL-move. For example, see Figure 3.8. We note that if u is trivial, a SL-move is just adding full-twists or nothing. A SL-move is determined by a string link and a number of full-twists, that is, 'SL' stands for String Link. Proof. The proof is by induction on the length \|I\| = q. If q ≤ k, the proposition clearly holds. We note that, by the induction hypothesis, δ k γ (I) = δ k γ ′ (I). Denote respectively by α i , λ i (resp. α ′ i , λ ′ i ) the ith meridian and ith longitude of γ (resp. γ ′ ) for 1 ≤ i ≤ n. Let E X (resp. E Y ) be the Magnus expansion in non-commutative variables X 1 , . . . , X n (resp. Y 1 , . . . , Y n ) obtained by replacing α i by 1 + X i (resp. α ′ i by 1 + Y i ) for 1 ≤ i ≤ n. Fix j, by the assumption, the Milnor numbers for γ and γ ′ of length ≤ k vanish, so E X (λ j ) and E Y (λ ′ j ) can be written respectively in the form ..., Y n )) are terms of degree ≥ k. Here we define a set of polynomials; E X (λ j ) = 1 + F j (X) and E Y (λ ′ j ) = 1 + F ′ j (Y ), where F j (X)(= F j (X 1 , ..., X n )) and F ′ j (Y )(= F ′ j (Y 1 ,D k j = ν(i 1 . . . i m )Y i1 · · · Y im ν(i 1 . . . i m ) ≡ 0 mod δ k γ (i 1 . . . i m j), m < q ν(i 1 . . . i m ) ∈ Z, m ≥ q . Then it is enough to show F ′ j (Y ) − F j (Y ) ∈ D k j . The following claims are shown by similar to the assertions (16) and (18) in [8]. degree ≥ 2k + 1 is obtained from F j (Y ) by inserting at least k + 1 variables. By E Y (u ±1 i ) − 1 is at least k + 1. Set E Y (u i ) = 1 + G i (Y ) and E Y (u −1 i ) = 1 + G i (Y ), where G i (Y ) and G i (Y ) mean the terms of degree ≥ k + 1. Since α i = u −1 i α ′ i u i (where α i is assumed to be an element of π 1 ([0, 1] 3 \ γ ′ )), we have E Y (α i ) = E Y (u −1 i α ′ i u i ) = (1 + G i (Y ))(1 + Y i )(1 + G i (Y )) = (1 + G i (Y ))(1 + G i (Y )) + (1 + G i (Y ))Y i (1 + G i (Y )) = 1 + Y i + Y i G i (Y ) + G i (Y )Y i + G i (Y )Y i G i (Y ). Hence E Y (λ j ) is obtained from E X (λ j ) by substituting X i for Y i + Y i G i (Y ) + G i (Y )Y i + G i (Y )Y i G i (Y ). Set E Y (λ j ) = 1 + H j (Y ), where H j (Y ) is the terms of degree ≥ k. Note that terms of degree ≤ 2k of H j (Y ) − F j (Y ) vanish, and that any term of H j (Y ) − F j (Y ) of γ γ ′ SL-move λ 1 λ i λ n λ ′ 1 λ ′ i λ ′ n α 1 α i α n α ′ 1 α ′ i α ′ n u 1 u i u n β 1 β i β nClaim 2, H j (Y ) − F j (Y ) ∈ D k j . Since λ ′ j = u j λ j u −1 j (where λ j is assumed to be an element of π 1 ([0, 1] 3 \ γ ′ )), E Y (λ ′ j ) = E Y (u j λ j u −1 j ) = (1 + G j (Y ))(1 + H j (Y ))(1 + G j (Y )) = 1 + H j (Y ) + H j (Y )G j (Y ) + G j (Y )H j (Y ) + G j (Y )H j (Y )G j (Y ). It follows from Claims 1 and 2 that we have F ′ j (Y ) − H j (Y ) = H j (Y )G j (Y ) + G j (Y )H j (Y ) + G j (Y )H j (Y )G j (Y ) ∈ D k j . Since H j (Y ) − F j (Y ) ∈ D k j , by Claim 1, we have F ′ j (Y ) − F j (Y ) ∈ D k j . This completes the proof. In this section we will give proofs of Theorems 1.3, 1.6 and Corollary 1.4. Proposition 4.1. Let γ be an n-component bottom tangle and γ ′ a bottom tangle obtained from γ by a SL-move which is given by a string link u. If the Milnor numbers of γ and γ ′ for non-repeated sequences with length ≤ k vanish, then we have the following: S∈S 2k+1 j (µ γ ′ (Sj) − µ γ (Sj))Y S = \|J\|=\|I\|=k JIl∈S 2k+1 j µ γ (Jj)µ γ (Il) is∈{J} µ u (li s )Y J<s (Y isIl − Y islI − Y Ilis + Y lIis )Y Js< + \|J\|=\|I\|=k JIl∈S 2k+1 j µ γ (Jj)µ γ (Il)µ u (lj)(Y IlJ − Y lIJ − Y JIl + Y JlI ). Proof. We compare with the Magnus expansions of the jth longitudes of γ and γ ′ . Denote respectively by α i , λ i (resp. α ′ i , λ ′ i ) the ith meridian and ith longitude of γ (resp. γ ′ ) for 1 ≤ i ≤ n. Let E X (resp. E Y ) be the Magnus expansion in non-commutative variables X 1 , . . . , X n (resp. Y 1 , . . . , Y n ) obtained by replacing α i by 1 + X i (resp. α ′ i by 1 + Y i ) for 1 ≤ i ≤ n. By the assumption, the Minor numbers for γ and γ ′ of degree k coincide, hence denote by E X (λ j ) = 1 + I∈S k j µ γ (Ij)X I + r j (X) + O X (2) and E Y (λ ′ j ) = 1 + I∈S k j µ γ (Ij)Y I + r ′ j (Y ) + O Y (2), where r j (X) and r ′ j (Y ) mean the terms of degree ≥ k + 1 and O X (2) (resp. O Y (2)) denotes the terms which contain X i (resp. Y i ) at least 2 times for some i(= 1, 2, . . . , n). Let f j (X) = I∈S k j µ γ (Ij)X I and f j (Y ) = I∈S k j µ γ (Ij)Y I . Let u i be the ith longitude of u and let First, we observe E Y (β l ) and E Y (α i ), where α i is assumed to be an element of β i = [λ ′ i , α ′ i ] = λ ′ i −1 α ′ i −1 λ ′ i α ′ i ,π 1 ([0, 1] 3 \γ ′ ). Since E Y (λ ′ l )E Y (λ ′ l −1 ) = 1, set E Y (λ ′ l −1 ) = 1−f l (Y )+r ′ l (Y )+O Y (2), where r ′ l (Y ) is the terms of degree ≥ k + 1. Observe that E Y (β l ) = E Y (λ ′ l −1 α ′ l −1 λ ′ l α ′ l ) = (1 − f l (Y ) + r ′ l (Y ) + O Y (2))(1 − Y l + O Y (2))(1 + f l (Y ) + r ′ l (Y ) + O Y (2))(1 + Y l ) = 1 + f l (Y )Y l − Y l f l (Y ) + O(k + 2) + O Y (2) = 1 + I∈S k l µ γ (Il)Y I Y l − Y l I∈S k l µ γ (Il)Y I + O(k + 2) + O Y (2) = 1 + I∈S k l µ γ (Il)(Y Il − Y lI ) + O(k + 2) + O Y (2). This implies that E Y (u i ) is obtained from E Z (u i ) by substituting Z l for I∈S k l µ γ (Il)(Y Il − Y lI ) + O(k + 2) + O Y (2). So we have E Y (u i ) = 1 + l =i µ u (li) I∈S k l µ γ (Il)(Y Il − Y lI ) + O(k + 2) + O Y (2). Let g i (Y ) = l =i µ u (li) I∈S k l µ γ (Il)(Y Il − Y lI ). Then we have E Y (u −1 i ) = 1 − g i (Y ) + O(k + 2) + O Y (2). Since α i = u −1 i α ′ i u i , we have E Y (α i ) = E Y (u −1 i α ′ i u i ) = (1 − g i (Y ) + O(k + 2) + O Y (2))(1 + Y i )(1 + g i (Y ) + O(k + 2) + O Y (2)) = 1 + Y i + Y i g i (Y ) − g i (Y )Y i + O(k + 3) + O Y (2) = 1 + Y i + l =i µ u (li) I∈S k l µ γ (Il)(Y iIl − Y ilI − Y Ili + Y lIi ) + O(k + 3) + O Y (2). Now we consider the difference d j (Y ) = E Y (λ j ) − (1 + f j (Y ) + r j (Y ) + O Y (2)). Since E Y (λ j ) is obtained from E X (λ j )(= 1+f j (X)+r j (X)+O X (2)) by substituting X i for Y i + l =i µ u (li) I∈S k l µ γ (Il)(Y iIl − Y ilI − Y Ili + Y lIi ) + O(k + 3) + O Y (2), all terms of degree ≤ 2k of d j (Y ) − O Y (2) vanish. The terms of degree 2k + 1 in d j (Y ) − O Y (2) is obtained from f j (Y ) by substituting Y i for l =i µ u (li) I∈S k l µ γ (Il)(Y iIl − Y ilI − Y Ili + Y lIi ) for some i ∈ {1, 2, . . . , n}. It follows that d j (Y ) − (O Y (2) + O(2k + 2) + O Yj ) = J∈S k j µ γ (Jj) is∈{J} Y J<s l =is µ u (li s ) JIl∈S 2k+1 j µ γ (Il)(Y isIl − Y islI − Y Ilis + Y lIis ) Y Js< = \|J\|=\|I\|=k JIl∈S 2k+1 j µ γ (Jj)µ γ (Il) is∈{J} µ u (li s )Y J<s (Y isIl − Y islI − Y Ilis + Y lIis )Y Js< , where O Yj means the terms which contain Y j at least one time. Finally, we observe the difference E Y (λ ′ j ) − (1 + f j (Y ) + r j (Y ) + O Y (2)). Since λ ′ j = u j λ j u −1 j (where λ j is assumed to be an element of π 1 ([0, 1] 3 \ γ ′ )), we have E Y (λ ′ j ) = E Y (u j λ j u −1 j ) = 1 + (1 + g j (Y ))(f j (Y ) + r j (Y ) + d j (Y ))(1 − g j (Y )) + O(2k + 2) + O Y (2) = 1 + f j (Y ) + r j (Y ) + d j (Y ) + g j (Y )f j (Y ) − f j (Y )g j (Y ) + O(2k + 2) + O Y (2). So we have E Y (λ ′ j ) − (1 + f j (Y ) + r j (Y ) + O Y (2)) = d j (Y ) + \|J\|=\|I\|=k JIl∈S 2k+1 j µ γ (Jj)µ γ (Il)µ u (lj)(Y IlJ − Y lIJ − Y JIl + Y JlI ) +O(2k + 2) + O Y (2) = \|J\|=\|I\|=k JIl∈S 2k+1 j µ γ (Jj)µ γ (Il) is∈{J} µ u (li s )Y J<s (Y isIl − Y islI − Y Ilis + Y lIis )Y Js< + \|J\|=\|I\|=k JIl∈S 2k+1 j µ γ (Jj)µ γ (Il)µ u (lj)(Y IlJ − Y lIJ − Y JIl + Y JlI ) +O(2k + 2) + O Y (2) + O Yj . This completes the proof. In order to prove Corollary 1.4 and Theorem 1.6, we need the following lemma given in [10]. Proof of Corollary 1.4. Let c and c ′ be n-clover links. We assume that they are edge-homotopic. By Lemma 4.2, there exist disk/band surfaces F c and F c ′ of c and c ′ respectively such that γ(F c ) and γ(F c ′ ) are link-homotopic. This implies that µ γ(Fc) (I) = µ γ(F c ′ ) (I) for any non-repeated sequence I [7], [2]. On the other hand, by Theorem 1.3, the set H c (2k + 2, j) (resp. H c ′ (2k + 2, j)) is obtained from the Milnor numbers of γ(F c ) (resp. γ(F c ′ )) for F c (resp. F c ′ ). Hence we have H c (2k + 2, j) = H c ′ (2k + 2, j). Proof of Theorem 1.6. Suppose that two 4-clover links c and c ′ are edge-homotopic. By Corollary 1.4, H c (4, 4) = H c ′ (4, 4)( = ∅). By Lemma 4.2 there exist disk/band surfaces F c and F c ′ such that γ(F c ) and γ(F c ′ ) are link-homotopic. This implies that the Milnor numbers of γ(F c ) and γ(F c ′ ) are equal for any non-repeated sequence [7], [2]. Since the Milnor numbers of length ≤ 3 are always well-defined for clover links (see Remark 2.3), we have µ c (I) = µ c ′ (I) for any non-repeated sequence I with \|I\| ≤ 3. Conversely if H c (4, 4) ∩ H c ′ (4, 4) = ∅, then there exist disk/band surfaces F c and F c ′ of c and c ′ respectively such that In particular, µ γ(Fc) (1234) = µ γ(F c ′ ) (1234) and µ γ(Fc) (2134) = µ γ(F c ′ ) (2134). According to the link-homotopy classification theorem for string links by N. Habegger and X. S. Lin [2], for two 4-component string links (bottom tangles) that have common values of the Milnor numbers for non-repeated sequences with length ≤ 3, they are link-homotopic if and only if their Milnor numbers for sequences 1234 and 2134 coincide, see also [11,Theorem 4.3]. This together with the hypothesis implies that γ(F c ) and γ(F c ′ ) are link-homotopic. Therefore c and c ′ are edge-homotopic by Lemma 4.2. This completes the proof. The Milnor µ-invariant µ L (I) is the residue class of µ L (I) modulo the greatest common devisor of µ L (J)'s, where J is obtained from proper subsequence of I by permuting cyclicly. The length of the sequence I is called the length of µ L (I) and denoted by \|I\|. His original definition of the Milnor invariant eliminates the indeterminacy of the possible variations of the Milnor numbers caused by different choices of peripheral elements. Date: July 7, 2015. The second author is partially supported by a JSPS Grant-in-Aid for Scientific Research (C) (#26400081). Corollary 1. 4 . 4For a clover link c, if the Milnor numbers of l c for non-repeated sequences with length ≤ k vanish, then H c (2k + 2, j) is an edge-homotopy invariant of c. Example 1 . 5 . 15For a 4-clover link c and a disk/band surface F c of c, we have H c (4, 4) = µ γ(Fc) (14)µ γ(Fc) (23)(m 13 − m 34 − m 12 + m 24 ) +µ γ(Fc) (12)µ γ(Fc) (34)(m 24 − m 23 − m 14 + m 13 ) X 123 + µ γ(Fc) (14)µ γ(Fc) (23)(m 34 − m 13 − m 24 + m 12 ) +µ γ(Fc) (13)µ γ(Fc) (24)(m 34 − m 23 − m 14 + m 12 ) X 132 + µ γ(Fc) (13)µ γ(Fc) (24)(m 23 − m 34 − m 12 + m 14 ) +µ γ(Fc) (12)µ γ(Fc) (34)(m 23 − m 24 − m 13 + m 14 ) X 213 Theorem 1 . 6 . 16Let c and c ′ be 4-clover links. They are edge-homotopic if and only if H c (4, 4) ∩ H c ′ (4, 4) = ∅ and µ c (I) = µ c ′ (I) for any non-repeated sequence I with \|I\| ≤ 3. 2. 1 . 1A construction of bottom tangles. An n-component tangle is a properly embedded disjoint union of n arcs in the 3-cube [0, 1] 3 . An n-component bottom tangle Figure 2 . 1 . 21the graph C n L. Kauffman, J. Simon, K. Wolcott and P. Zhao Figure 2 2Figure 2.2. ( 2 ) 2Let N (D) be the regular neighborhood of D andN (D) the interior of N (D). Since S 3 \N (D) is homeomorphic to the 3-ball, F c \N (D) can be seen as a disjoint union of surfaces in the 3-ball. Hence ∂F c \N (D) is a disjoint union of n-arcs and n Figure 2 . 3 . 23of Figure 2.3. We call γ(F c ) an n-component bottom tangle obtained from F c . A method for obtaining a bottom tangle from a diskband surface of a clover link Lemma 2.1. [10, Proposition 2.5] For an n-clover link c, any two disk/band surfaces F c and F ′ c are transformed into each other by adding full-twists to bands(Figure 2.4 (a)) and a single move illustrated inFigure 2.4 (b). Figure 2 . 4 . 24Two local moves of disk/band surfaces 2.2. Milnor invariants. Let us briefly recall from [6] the definition of the Milnor number of a bottom tangle. Let Figure 2 . 25. meridians and longitudes L where a i is a line segment connecting ( 2i−1 2n+1 ,1 2 , 0) and ( 2i 2n+1 , 1 2 , 0), seeFigure 2.6. We call L(γ) the closure of γ. On the other hand, for any oriented link in S 3 , there is a bottom tangle γ L such that the closure of γ L is equal to L.So we define the Milnor number of L to be the Milnor number of γ L . Let ∆ L (I) be the greatest common devisor of µ L (J) ′ s, where J is obtained from proper subsequence of I by permuting cyclicly. The Milnor invariant µ L (I) is the residue class of µ L (I) modulo ∆ L (I). We note that for a sequence I, if we have ∆ L (I) = 0, then the Milnor invariant µ L (I) is equal to the Milnor number µ γL (I). Now we define the Milnor number of a clover link.Definition 2.2. Let c be an n-clover link and F c a disk/band surface of c. Let γ(F c ) be the n-component bottom tangle obtained from F c . For a sequence I, the Milnor number µ c (I) of c is defined to be the Milnor number µ γ(Fc) (I). Remark 2.3. While µ c (I) depends on a choice of F c , the first author [10] proved the following result: Let l c be a link which is the disjoint union of leaves of c. If the Milnor numbers of l c for sequences with length ≤ k vanish, then µ c (I) is well-defined for any sequence I with \|I\| ≤ 2k + 1. 3. Proof of Theorem 1.1 n be an oriented ordered n-component string link in [0, 1] 3 . For each i (= 1, 2, . . . , n), we consider an arc u ′ i which is parallel to the ith component u i of u with opposite orientation and ∂u Proposition 3 . 1 . 31Let γ be an n-component bottom tangle and γ ′ a bottom tangle obtained from γ by a SL-move. If the Milnor numbers of γ and γ ′ for sequences with length ≤ k vanish, then for any sequence I, µ γ ′ (I) ≡ µ γ (I) mod δ k γ (I), where δ k γ (I) is the greatest common devisor of µ γ (J) ′ s for a proper subsequence J of I which is obtained by removing at least k + 1 indices. Figure 3 . 8 . 38An example of a SL-move Claim 1 . 1D k j is a two-sided ideal of the formal power series ring in non-commutative variables Y 1 , . . . , Y n with integer coefficients.Claim 2. If at least k variables are inserted anywhere in a term µ(i 1 i 2 . . . i m j)Y i1i2···im , then the resulting term belongs to D k j . Let u i be the ith longitude of a string link which gives the SL-move, see Figure 3.9. In the proof of [10, Lemma 2.6], it is shown that the degree of each term in Figure 3 . 9 . 39Figure 3.9. Proof of Theorem 1.1. By Lemma 2.1, any two disk/band surfaces F c and F ′ c of an n-component clover link c are transformed into each other by the moves (a) and (b) illustrated in Figure 2.4. So two bottom tangles γ(F c ) and γ(F ′ c ) are transformed into each other by a SL-move. Since the both closures L(γ(F c )) and L(γ(F ′ c )) are ambient isotopic to l c , by the hypothesis of Theorem 1.1, 0 = µ lc (J) = µ γ(Fc) (J) = µ γ(F ′ c ) (J) for any sequence J with \|J\| ≤ k. Hence by Proposition 3.1, µ γ(F ′ c ) (I) ≡ µ γ(Fc) (I) mod δ k γ(Fc) (I) for any sequence I. This completes the proof. 4. Proof of Theorems 1.3 and 1.6 see Figure 3.9. Let E Z be the Magnus expansion in non-commutative variables Z 1 , . . . , Z n obtained by replacing β i by 1 + Z i for 1 ≤ i ≤ n. Then we have E Z (u i ) = 1 + l =i µ u (li)Z l + O(2). Proof of Theorem 1.3. By Lemma 2.1, any two disk/band surfaces F c and F ′ c of an n-component clover link c are transformed into each other by the moves (a) and (b) in Figure 2.4. So two bottom tangles γ(F c ) and γ(F ′ c ) are transformed into each other by a SL-move. Since the both closures L(γ(F c )) and L(γ(F ′ c )) are ambient isotopic to l c and the hypothesis of Theorem 1.3, 0 = µ lc (J) = µ γ(Fc) (J) = µ γ(F ′ c ) (J) for any sequence J with \|J\| ≤ k. Since µ u (pq) is the 'linking number' of the pth component and the qth component of u, µ u (pq) = µ u (qp) and the set {µ u (pq) \| u : a string link} = Z for any p and q. This and Proposition 4.1 give us the Thorem 1.3. ] Two n-clover links c and c ′ are edge-homotopic if and only if there exist disk/band surfaces F c and F c ′ of c and c ′ respectively such that the two bottom tangles γ(F c ) and γ(F c ′ ) are link-homotopic. (F c ′ ) (S4)X S . Commutator calculus and link invariants. K T Chen, Proc. Amer. Math. Soc. 3K. T. Chen, Commutator calculus and link invariants, Proc. Amer. Math. Soc. 3 (1952), 44-55. The classification of links up to link-homotopy. N S Habegger; X, Lin, J. Amer. Math. Soc. 3N. Habegger; X. S. Lin, The classification of links up to link-homotopy, J. Amer. Math. Soc. 3 (1990), 389-419. Bottom tangles and universal invariants. K Habiro, Algebr. Geom. Topol. 6K. Habiro, Bottom tangles and universal invariants, Algebr. Geom. Topol. 6 (2006), 1113- 1214. . L Kauffman, ; J Simon, L. Kauffman; J. Simon; . K Wolcott, K. Wolcott; Invariants of theta-curves and other graphs in 3-space. P Zhao, Topology Appl. 49P. Zhao, Invariants of theta-curves and other graphs in 3-space, Topology Appl. 49 (1993), 193-216. Cobordisme d'enlacements de disques (French). J Ledimet, Mém. Soc. Math. France (N.S.) No. 32J. LeDimet, Cobordisme d'enlacements de disques (French), Mém. Soc. Math. France (N.S.) No. 32 (1988). The µ-invariants of based links. J P Levine, Differential topology. Siegen; BerlinSpringerJ. P. Levine, The µ-invariants of based links, Differential topology (Siegen, 1987), 87-103, Lecture Notes in Math., 1350, Springer, Berlin, (1988). Link groups. J Milnor, Ann. of Math. 2J. Milnor, Link groups, Ann. of Math. (2) 59 (1954), 177-195. Isotopy of links, Algebraic geometry and topology, A symposium in honor of S. Lefschetz. J Milnor, Princeton University PressPrinceton, N. J.J. Milnor, Isotopy of links, Algebraic geometry and topology, A symposium in honor of S. Lefschetz, pp. 280-306. Princeton University Press, Princeton, N. J. (1957). . K Taniyama, Cobordism, Topology. 3K. Taniyama, Cobordism, homotopy and homology of graphs in R 3 , Topology 33 (1994), 509-523. Milnor invariants and edge-homotopy classification of clover links. K Wada, preprintK. Wada, Milnor invariants and edge-homotopy classification of clover links, preprint (2015). Self delta-equivalence for links whose Milnor's isotopy invariants vanish. A Yasuhara, Trans. Amer. Math. Soc. 361A. Yasuhara, Self delta-equivalence for links whose Milnor's isotopy invariants vanish, Trans. Amer. Math. Soc. 361 (2009), 4721-4749. Nishi-Waseda 1-6-1, Shinjuku-ku. Nukuikita-machi. Department of Mathematics, School of Education, Waseda University ; Japan Department of Mathematics, Tokyo Gaugei UniversityJapan E-mail address: [email protected], [email protected] of Mathematics, School of Education, Waseda University, Nishi-Waseda 1-6-1, Shinjuku-ku, Tokyo, 169-8050, Japan Department of Mathematics, Tokyo Gaugei University, 4-1-1 Nukuikita-machi, Koganei- shi, Tokyo, 184-8501, Japan E-mail address: [email protected], [email protected]	[]
[ "Visual Time Series Forecasting: An Image-driven Approach", "Visual Time Series Forecasting: An Image-driven Approach" ]	[ "Srijan Sood [email protected] ", "J P Morgan ", "Zhen Zeng [email protected] ", "J P Morgan ", "Naftali Cohen [email protected] ", "J P Morgan ", "Tucker Balch [email protected] ", "J P Morgan ", "Manuela Veloso [email protected] ", "J P Morgan ", "Srijan Sood ", "Zhen Zeng ", "Naftali Cohen ", "Tucker Balch ", "Manuela Veloso ", "\nAI Research New York\nNYUSA\n", "\nAI Research New York\nNYUSA\n", "\nAI Research New York\nNYUSA\n", "\nAI Research New York\nNYUSA\n", "\nAI Research New York\nNYUSA\n" ]	[ "AI Research New York\nNYUSA", "AI Research New York\nNYUSA", "AI Research New York\nNYUSA", "AI Research New York\nNYUSA", "AI Research New York\nNYUSA" ]	[]	Time series forecasting is essential for agents to make decisions. Traditional approaches rely on statistical methods to forecast given past numeric values. In practice, end-users often rely on visualizations such as charts and plots to reason about their forecasts. Inspired by practitioners, we re-imagine the topic by creating a novel framework to produce visual forecasts, similar to the way humans intuitively do. In this work, we leverage advances in deep learning to extend the field of time series forecasting to a visual setting. We capture input data as an image and train a model to produce the subsequent image. This approach results in predicting distributions as opposed to pointwise values. We examine various synthetic and real datasets with diverse degrees of complexity. Our experiments show that visual forecasting is effective for cyclic data but somewhat less for irregular data such as stock price. Importantly, when using image-based evaluation metrics, we find the proposed visual forecasting method to outperform various numerical baselines, including ARIMA and a numerical variation of our method. We demonstrate the benefits of incorporating vision-based approaches in forecasting tasks -both for the quality of the forecasts produced, as well as the metrics that can be used to evaluate them.	10.1145/3490354.3494387	[ "https://arxiv.org/pdf/2011.09052v3.pdf" ]	235,731,609	2107.01273	e2cda0f49cc491e38396135d93d993f08f2eb059	Visual Time Series Forecasting: An Image-driven Approach Virtual EventCopyright Virtual EventNovember 3-5, 2021 Srijan Sood [email protected] J P Morgan Zhen Zeng [email protected] J P Morgan Naftali Cohen [email protected] J P Morgan Tucker Balch [email protected] J P Morgan Manuela Veloso [email protected] J P Morgan Srijan Sood Zhen Zeng Naftali Cohen Tucker Balch Manuela Veloso AI Research New York NYUSA AI Research New York NYUSA AI Research New York NYUSA AI Research New York NYUSA AI Research New York NYUSA Visual Time Series Forecasting: An Image-driven Approach USAVirtual EventNovember 3-5, 202110.1145/3490354.3494387ACM Reference Format: 2021. Visual Time Series Forecasting: An Image-driven Approach. In 2nd * Both authors contributed equally to this research. ACM ISBN 978-1-4503-9148-1/21/11. . . $15.00 Figure 1: Typical workstation of a professional trader. Credit: Photoagriculture/ Shutterstock.com ACM International Conference on AI in Finance (ICAIF'21), November 3-5, 2021, Virtual Event, USA. ACM, New York, NY, USA, 9 pages. https://doi.org/CCS CONCEPTSComputing methodologies → Image representations;Math- ematics of computing → Time series analysis KEYWORDS time-series forecasting, image representations, neural networks, ARIMA, visualizations Time series forecasting is essential for agents to make decisions. Traditional approaches rely on statistical methods to forecast given past numeric values. In practice, end-users often rely on visualizations such as charts and plots to reason about their forecasts. Inspired by practitioners, we re-imagine the topic by creating a novel framework to produce visual forecasts, similar to the way humans intuitively do. In this work, we leverage advances in deep learning to extend the field of time series forecasting to a visual setting. We capture input data as an image and train a model to produce the subsequent image. This approach results in predicting distributions as opposed to pointwise values. We examine various synthetic and real datasets with diverse degrees of complexity. Our experiments show that visual forecasting is effective for cyclic data but somewhat less for irregular data such as stock price. Importantly, when using image-based evaluation metrics, we find the proposed visual forecasting method to outperform various numerical baselines, including ARIMA and a numerical variation of our method. We demonstrate the benefits of incorporating vision-based approaches in forecasting tasks -both for the quality of the forecasts produced, as well as the metrics that can be used to evaluate them. INTRODUCTION AND RELATED WORK Time series forecasting is a standard statistical task that concerns predicting future values given historical information. Conventional forecasting tasks range from uncovering simple periodic patterns to forecasting intricate nonlinear patterns. The prevailing and most widely used forecasting techniques include linear regression, exponential smoothing, and ARIMA (e.g., [12,20,25]). In recent years, modern approaches emerge as tree-based algorithms, ensemble methods, neural network autoregression, and recurrent neural networks (e.g., [12]). These methods are useful for highly nonlinear and inseparable data but are often considered less stable than the more traditional approaches (e.g., [20,21]). In the last few years, deep learning approaches have been applied in the domain of time series analysis, for forecasting [4,13,31,32], as well as unsupervised approaches for pre-training, clustering, and distance calculation [1,27,33,35]. The common theme across these works is their use of stacked autoencoders (with different variations -vanilla, convolutional, recurrent, etc.) on numeric time series data. Autoencoders have also shown promise in the computer vision domain across tasks as image denoising [3,16], image compression [2], and image completion and in-painting [23,26]. This study is motivated by a financial application. Traders execute trades while observing financial time series images as charts on their desktop screens ( Figure 1). When it comes to financial time series, the data is consumed in its numeric form, but decisions are often augmented by visual representations. arXiv:2011.09052v3 [cs.CV] 19 Nov 2021 This paper presents a new perspective on numerical time series forecasting by transforming the problem completely into the computer-vision domain. We capture input data as images and build a network that outputs corresponding subsequent images. To the best of our knowledge, this is the first study that aims at explicit visual forecasting of time series data as plots. Previous researches leveraged computer vision for time-series data but focused on classifying trade patterns [7,8], numeric forecast [6], learning weights to combine multiple statistical forecasting methods [22], and video prediction for multivariate economic forecasting [37]. We follow up on these approaches but focus on an explicit regression-like image prediction task. This work presents a few advantages. Visual time series forecasting is a data-driven non-parametric method, not constrained to a predetermined set of parameters. Thus, the approach is flexible and adaptable to many data forms, as shown by application across various datasets. This bears a stark contrast with classical time series forecasting approaches that are often tailored to the particularity of the data in hand. The main advantage of this method is that its prediction is independent of other techniques. This is important as it was repeatedly shown that an aggregate of independent techniques outperforms the best-in-class method (e.g., [12,14,17]). Secondly, visual predictions result in inherent uncertainty estimates as opposed to pointwise estimates, as they represent distributions over pixels as opposed to explicit value prediction. In addition, financial time series data are often presented and act upon without having access to the underlying numeric information (e.g., financial trading using the smartphone applications). Thus, it seems viable to examine the value in inferring using visualizations alone. Lastly, as will be discussed later on, we show that transforming the continuous numeric data to a discrete bounded space using visualization results in robust and stable predictions. We evaluate predictions using multiple metrics. When considering object-detection metrics such as Intersection-over-Union (IoU), visual forecasting outperforms the corresponding numeric baseline. However, when utilizing more traditional time-series evaluation metrics as the symmetric mean absolute percentage error (SMAPE), we find the visual view to perform similarly to its numerical baselines. DATASETS This paper uses four datasets, two synthetic and two real, with varying degrees of periodicity and complexity to examine the utility of forecasting using images. Synthetic Data We generate two different series for the synthetic datasets: multiperiodic data sampled from harmonic functions and mean-reverting data generated following the Ornstein-Uhlenbeck process. 2.1.1 Harmonic data. The first dataset is derived synthetically and is designed to be involved but still with a prominent, repeated signal. We synthesized the time series with a linearly additive two-timescale harmonic generating function, = ( 1 + 1 ) sin(2 / 1 + 1 ) + ( 2 + 2 ) sin(2 / 2 + 2 ), where the time varies from = 1 to = , and denotes the total length of the time series. The multiplicative amplitudes 1 and 2 are randomly sampled from a Gaussian distribution N (1, 0.5), while the amplitude of the linear trends 1 and 2 are sampled from a uniform distribution U (−1/ , 1/ ). The driving time scales are short ( 1 ) and long ( 2 ) relative to the total length of . Thus, 1 ∼ N ( /5, /10), while 2 ∼ N ( , /2). Lastly, the phase shifts 1 and 2 are sampled from a uniform distribution U (0, 2 ). We generated and used 42,188 examples as a train set, 4,687 for the validation set, and 15,625 for the test set. Each time series differ concerning the possible combination of tuning parameters. Panel a) in Figure 2 shows three sampled examples of the harmonic data and it is easy to see that the synthetic time series consist of two time-scales: short oscillations that are composed on a much longer wave trains. 2.1.2 OU Data. We synthesized mean-reverting time series based on Ornstein-Uhlenbeck (OU) process as described in [5]. A meanreverting time series tends to drift towards a fundamental mean value. We chose to synthesize the mean-reverting time series to resemble the characteristics of financial interest rates or volatility. OU's stochastic nature makes it noisy on fine scales but predictable on the larger scale, which is the focus of this study. Specifically, we generated the OU dataset following the equation adopted from [5] with, ∼ N ( + ( −1 − ) − , 2 2 (1 − −2 )), where is the mean value that the time series reverts back to, and 0 starts at . We used mean reversion rate ∼ N (8 −8 , 4 −8 ) with units ns −1 , and a volatility value ∼ N (1 −2 , 5 −3 ). Overall, we generated the time series by sampling at every minute. We generated and used 45,000 examples as a train set, 5,000 for the validation set, and 15,000 for the test set. Similar to the Harmonic data, each time series differ concerning the possible combination of tuning parameters. Figure 2(b) shows three samples of the OU data. One can see that the OU data tend to be noisy with uncorrelated ups and downs, but on larger scales, the data is concentrated in the middle of the image as values drift toward the mean due to its reversion constraint. Real Data Along with the synthetically generated data, we use two real-world time series datasets. 2.2.1 ECG data. The ECG data is measured information from 17 different people adopted from MIT-BIH Normal Sinus Rhythm Database [15]. We curated 18 hours of data for each subject after manually examining the data's consistency and validity by analyzing the mean and standard deviation of the time series data for each subject (not shown). For each subject, we consider segments of 2.56 seconds (corresponding to 320 data points) sampled randomly from the data. These are then downsampled to 80 data points to be on-par with the other datasets. 13 out of the 17 subjects are used a training data while the other 4 are used as out-of-sample testing data. Overall, we sampled 42,188 examples for the training set, while from the test data, we sampled 4,687 as a validation set and 15,625 as a test set. Panel c) in Figure 2 shows three sampled examples of the ECG data. One can see that the data has prominent spikes about every second, which makes the data predictable. However, there is noticeable noise between spikes that is much harder to predict. Figure 2 shows three sampled examples of the financial data. Here, one can see that the data is much less predictable than the previous three. Although financial data is persistent with sequentially related information, it is hard to spot repeated signals that will make the data predictable. Indeed, the prevailing theory of financial markets argues that markets are very efficient, and their future movements are notoriously hard to predict, especially given price information alone (e.g., [29]). Complexity of Time-Series Data To provide a reference for how the time series across our datasets vary, we measured the complexity of each dataset using a standard measure called Weighted Permutation Entropy [10] (WPE). The larger WPE, the more complex the data is. Input Output Autoencoder Figure 4: An overview of our problem setup. The blue region shows a 75% overlap between the input and output ; the forecast regionˆis denoted in red. As shown in Figure 3, we can see that the datasets cover a broad range of complexity. As expected, the simplest data is Harmonic with its deterministic periodicity. The ECG data is also periodic but more complex due to its irregularities between the spikes. The Financial data is filled with almost random movement of fine scales, therefore, more complex than both the Harmonic and ECG. The OU data exhibits even more random oscillations and abrupt changes compared to other datasets, thus it is measured as the most complex dataset. However, on the larger scale, the OU data bounces around a hidden mean value with bounded noise, making it possible to predict future value mean and ranges, as we will show later in Section 5. PROBLEM STATEMENT Given a time series signal, our goal is to produce a visual forecast of its future. We approach this problem by first converting the numeric time series into an image (as explained later in Section 4.1) and then producing a corresponding forecast image using deep-learning techniques. Let be the set of images of input time series signals, and be the set of corresponding forecast output images. The overlap constant defines the overlap fraction between the input image ∈ and the forecast ∈ , where = 1 implies = , ∀ ∈ , and = 0 implies that ∩ = ∅, ∀ ∈ , i.e., and are distinct. In our experiments, we use = 0.75 which means the first 75% of the forecast image is simply a reconstruction of the later 75% of the input image , and the rest 25% of corresponds to visual forecasting of the future, as shown in Figure 4. We chose = 0.75 such that the reconstructed overlap region (first 75% in ) serves as a sanity check on the effectiveness of a forecasting method, and the prediction region (later 25% in ) provides forecasting into the near future. METHOD 4.1 Data Preprocessing Given a 1-d numeric time series = [ 0 , · · · , ] with ∈ R, we convert into a 2-d image by plotting it out, with being the horizontal axis and being the vertical axis 1 . We standardize each converted image through following pre-processing steps. First, pixels in are scaled to [0, 1] and negated (i.e., = 1 − /255) so that the pixels corresponding to the plotted time series signal are bright (values close to 1), whereas the rest of the background pixels become dark (values close to 0). Note that there can be multiple bright (non-zero) pixels in each column due to anti-aliasing while plotting the images. Upon normalizing each column in such that the pixel values in each column sum to 1, each column can be perceived as a discrete probability distribution (see Figure 6). Columns represent the independent variable time, while rows capture the dependent variable: pixel intensity. The value of the time series at time is now simply the pixel index (row) at that time (column) with the highest intensity. Predictions are made over normalized data. To preserve the ability to forecast in physical units, we utilize the span of the input raw data values to transform forecasts to the corresponding physical scales. Image-to-Image Regression As mentioned in Section 1, recent work has seen the extensive use of autoencoders in both the time series and computer vision domains. Following these, we extend the use of autoencoders to our image-to-image time series forecasting setting. We use a simplistic convolutional autoencoder to produce a visual forecast image with the continuation of an input time series image, by learning an undercomplete mapping • , = ( ( )), ∀ ∈ , where the encoder network (·) learns meaningful patterns and projects the input image into an embedding vector, and the decoder network (·) reconstructs the forecast image from the embedding vector. We purposely do not use sequential information or LSTM cells as we wish to examine the benefits of framing the regression problem in an image setting. We call this method VisualAE, the architecture for which is shown in Figure 5. We used 2D convolutional layers with a kernel size of 5 × 5, stride 2, and padding 2. All layers are followed by ReLU activation and batch normalization. The encoder network consists of 3 convolutional layers which transform a 80 × 80 × 1 input image to 10 × 10 × 512, after which we obtain an embedding vector of length 512 using a fully connected layer. This process is then mirrored for the decoder network, resulting in a forecast image of dimension 80 × 80. We will explain the loss function for training in detail in the next section. Loss Functions One challenge with the converted time series images is that the majority of the information gets concentrated on fine lines, leaving most of the image blank. This is propagated downstream to the loss function that aims to quantify the dissimilarity of two sparse matrices. We care about the likelihood of pixel intensity in a particular location (row) in each column of the forecast image. This can be achieved by leveraging metrics that compare two probability distributions. We do so in a column-wise manner: the loss to compare target ground-truth (GT) image with prediction imageˆis the sum of column-wise distances between the two, ( ,ˆ) = ∑︁ =1 ( ,ˆ), where is any distance measure between two distributions ( andˆin this case), and is the width of images. This process is depicted in Figure 6. Measures such as the Kullback-Leibler Divergence have been extensively used as loss functions ( [17]), as they provide a way of computing the distance from an approximate distribution to a true distribution . In this study, following [19], we choose to be the Jensen-Shannon Divergence (JSD), which is a symmetric, more stable version of the Kullback-Leibler Divergence having the property that ( ∥ ) = ( ∥ ). Here, JSD is computed as ( ∥ ) = 1 2 ( ∥ ) + 1 2 ( ∥ ) where = 1 2 ( + ). EXPERIMENTS We experimented with four datasets: Harmonic, OU, ECG, and Financial, as they cover a wide range of complexity and predictability in time series data, as discussed in Section 2.3. In this study we use the PyTorch Lightning framework [11,28] for implementation and Nvidia Tesla T4 GPUs in our experiments. We benchmark the proposed method against three baseline methods as we describe below. As described in Section 3, there is a 75% overlap between input and output (overlap ratio = 0.75). Each sample contains 80 datapoints; we aim to forecast the last 20 datapoints (last 25%) of the output image. This is shown in Figures 4 and 7, where for the predicted image, the first 75% region (in blue) denotes the reconstructed input, while the last 25% (in red) region is the visual forecast. All metrics reported are averaged over the unseen red forecast region. Methods 5.1.1 VisualAE. This is the proposed method as discussed in Section 4.2. We train on images with size 80 × 80. We use a batch size of 128 and early stopping after 15 consecutive non-improving validation epochs to avoid overfitting during training. We start with a learning rate of 0.1, which is decayed by a factor of 0.1 (till 1e−4) after every 5 non-improving validation epochs. NumAE (Numeric AE). We also train an autoencoder network to produce numeric forecasts of the original numerical time series signal. The numeric input and output time series are standardized using min-max normalization (with bounds obtained from the input to avoid leakage to future). The autoencoder is trained to predict the output time series by minimizing the Huber loss [18]. The architecture, though similar to Figure 5, is shallower (as the dimension of numeric input is much smaller than the images), and uses 1D convolutional layers of kernel size of 5 × 5, stride 2 and padding 2. All layers are followed by ReLU activation and batch normalization. The encoder part consists of a series of 2 convolutional layers (of /2 and /4 filters, where is the length of the signal) and a fully connected layer, which gives us a latent representation of embedding length /4. The decoder is a mirrored encoder. We use a batch size of 128, along with a learning rate of 0.01 which is decayed by a factor of 0.1 after every 5 consecutive nonimproving validation epochs. We also utilize early stopping, as described earlier. ARIMA. Autoregressive Integrated Moving Average (ARIMA) models are a class of methods that are designed to capture autocorrelations in the data using a combination approach of autoregressive model, moving average model, and differencing (e.g., [36]). The purpose of each of these three features is to make the model fit the data as well as possible. We used auto arima from pdmaria 2 library in our experiments. RandomWalk. We used the random walk without drift model as a naive numeric forecasting baseline for comparison (e.g., [34]). Specifically, this model assumes that the first difference of the time series data is not time-dependent, and follows a Gaussian distribution N (0, ). Given a numeric input time series { 0 , · · · , −1 , }, in order to predict { +1 , · · · , + }, we first estimates as = √︂ E =1 [( − −1 ) 2 ] and the prediction at future time + follows + ∼ N ( , √ ). 2 http://alkaline-ml.com/pmdarima/ This results in a naive numeric forecast that simply extrapolates the last observed value into the future. If we wish to obtain the corresponding image, this forecast is accompanied with a growing uncertainty cone obtained through the equation above. Forecast Accuracy Metrics We use a variety of measures to assess the accuracy of forecast predictions from each method. Some of these metrics are extensively used in the time series forecasting domain, whereas the others we extend from the overarching machine learning field to this task. The baseline methods ARIMA, NumAE and RandomWalk produce continuous numeric forecasts, whereas our method Visu-alAE produces an image. Accordingly, we convert this image back to a numeric forecast which we can use to assess predictions using the metrics described in Section 5.2.1. Similarly, to leverage the image based metrics described in Section 5.2.2, we transform the numeric predictions of the baseline methods into images using the process described in Section 4.1. We discuss the interplay between these metrics across Section 5.3, with further details in Section 5.3.2. Numeric Measures. SMAPE. The Symmetric Mean Absolute Percentage Error (or SMAPE) is a widely used measure of forecast accuracy [24]. It is calculated as: 1 ∑︁ =1 \| −ˆ\| (\| \| + \|ˆ\|)/2 whereˆis the forecast, the corresponding observed groundtruth, and is the length of the time series. It ranges from 0.0 to 2.0, with lower values indicating better forecasts. MASE. The Mean Absolute Scaled Error (or MASE) is another commonly used measure of forecast accuracy [24]. It is the mean absolute error of a forecast divided by the mean absolute first-order difference of actuals, calculated as \| \| 1 −1 =2 \| − −1 \| where the numerator \| \| is the mean absolute error of the forecast, and the denominator is the mean absolute first-order difference over the ground-truth data period . Errors less than 1 imply that the forecast performs better than the naive one-step method, with lower values indicating better predictions. Image based Measures. In addition to utilizing tradition forecasting error metrics, we can measure the similarity between the predicted image and the ground-truth image in our setting to evaluate forecast accuracy. We do this through two metrics: JSD. Jensen-Shannon Divergence can be used to measure the similarity between image pairs, as described in Section 4.3. This ranges from 0.0 to 1.0, with lower values indicating a better forecast. IoU. We extend the Intersection-over-Union (IoU) metric, which is commonly practiced in the object-detection literature [9,30], to the purpose of measuring forecast accuracy. This ranges from 0.0 to 1.0, with higher values indicating better forecasts. We compute IoU pairwise for each corresponding column in the ground-truth and predicted image. This is done by obtaining the Table 1: Summary of various metrics on out-of-sample data with mean ± standard deviation for the forecast region (annotated as red region in Figure 4 and Figure 7). VisualAE is our proposed method, RandomWalk, NumAE, and ARIMA are the baselines. Lower SMAPE/MASE/JSD error (or higher IoU score) implies better prediction accuracy. 1D bounding boxes of non-zero pixels for each column and then calculating IoU of corresponding columns from the ground-truth and predicted image. Results All reported metrics mentioned in Section 5.2 are over the unseen future prediction region (in red in Figure 4). For both VisualAE and NumAE, we averaged these metrics over five independently trained models with different random weight initializations. We demonstrate that the proposed method VisualAE outperforms baseline methods NumAE, RandomWalk, and ARIMA across all four datasets when evaluated using image-based metrics (such as IoU). However, as we will discuss in this section, traditional numeric metrics are inconsistent with this finding. We demonstrate the value of using a visual approach to time-series forecasting, and how image-based evaluation metrics can help address some of the caveats of traditional numeric metrics. We report the mean and standard deviation of various prediction accuracy metrics in Table 1. VisualAE achieves higher IoU scores than all baselines across the four datasets. The same holds true for JSD (with the exception of RandomWalk scoring better in the Financial dataset). The numeric metrics are often inconsistentwithin themselves (SMAPE and MASE) -as well as across the four datasets. According to the numeric metrics, VisualAE is a close second (if not similar) to NumAE, with the exception of the ECG dataset, where VisualAE performs the best, and the OU dataset, where ARIMA and VisualAE perform similarly to NumAE . We will now discuss the characteristics of benchmarked methods and metrics, along with our overarching findings with the aid of Table 1 and Figure 7. Dataset-wise breakdown of approaches. Harmonic Data: The Harmonic dataset are dominated by cyclic patterns (see left column of Figure 2). As explained in section 2, each time series in the Harmonic dataset is a mixture (superposition) of two randomly generated individual sinusoids, and each sinusoid exhibits short cyclic patterns along with long term damping or magnifying trends. ARIMA performs well but not best because on many occasions, individual sequences don't span the full range of variability needed to tune the model's parameters. This is a barrier for ARIMA as it doesn't have the capability to cross-learn between multiple independent time-series. The naive forecasting baseline RandomWalk can only learn time-independent stepwise value changes, thus cannot model cyclic patterns. NumAE and VisualAE capture these patterns well, as observed in Figure 7(a) and Table 1, with NumAE performing slightly better according to numeric metrics, and VisualAE taking the lead in image-based metrics. OU Data: It is hard to predict the exact daily changes in the OU data owing to the random process' nature. However, over a larger scale, the OU data is predictable as a mean-reverting process. Visually, we expect majority of the values to concentrate around the mean value of the time series with some noise. RandomWalk extrapolates the last observed value, whereas ARIMA extrapolates the in-sample mean value as a steady line for each sample; Nu-mAE predicts a slightly jagged version of the same. This becomes evident in the abnormally large MASE error, which is sensitive to division by a small term = 1 − 4 when the naive one-step denominator approaches 0. The SMAPE metric appears to be similarly non-informative, as it cannot disambugate the performance of the four methods. The intricacies of these forecasts are captured in the IoU and JSD metrics, according to which VisualAE performs the best. This is evident in Figure 7(b), where we show that Visu-alAE concentrates on the hidden mean value, and was also able to partially recover the range of the noise -unlike the other baselines. ECG Data: ECG time series are periodical with intermittent spikes, and hence inherently predictable. They have relatively constant frequency and do not posess much time dependent uncertainty. Figure 7(c) shows that VisualAE is able to capture these cyclic patterns well, as evidenced by all metrics -image-based and numeric. VisualAE is able to handle data with sharp and abrupt changes, and better recovers the heart beat spikes as compared to NumAE . Similar to the Harmonic dataset, ARIMA is unable to capture the spiky patterns in ECG dataset, and RandomWalk simply extrapolates the last value. Financial Data: Financial time series are the most challenging to forecast amongst the four datasets. According to the prevailing literature (e.g., [29]), financial data is close to random on short scales and shows no apparent periodicity on large scales. Figure 7(d) shows that similar to the OU predictions, NumAE and ARIMA predicted the future with a weak linear trend, while VisualAE outperformed with a predicted curve that captures some of the finer details along with the overall nonlinear trend. This is captured by the IoU metric, but if judged according to SMAPE and MASE, RandomWalk would be the best-performer, tied with NumAE . This is rather concerning, as solely using numeric metrics would lead us to misleading conclusions, further demonstrating the benefit of using a visual approach in conjunction with traditional numeric methods. 5.3.2 Insights: Numeric vs. Image based Metrics . As discussed in the previous section, numeric metrics are often not consistent with the image-based ones, and sometimes do not agree amongst each other (e.g., see Table 1: SMAPE & MASE values for OU dataset). They are sensitive and often fail to recognize good quality forecasts (e.g., RandomWalk reportedly performing the best for the Financial dataset). Picking a percentage error such as SMAPE also carries the inability to compare forecast method quality across series (e.g., the low errors in the Financial dataset do not capture that it is in fact the most challenging to predict). The IoU metric is able to capture this information across the datasets, along with preserving rank-ordering of forecast quality amongst the four methods. Figure 8(i) shows an example for the Harmonic dataset, where according to MASE and SMAPE, the Nu-mAE forecast (column b) is the best performer. This is disputed by IoU, according to which the VisualAE forecast (column d) is better, and a qualitative visual inspection also suggests the same. Similarly, in Figure 8(ii), we see a hard-to-predict example of the Financial dataset. Just looking at MASE and SMAPE metrics would suggest that both NumAE and VisualAE forecasts are of similar quality, whereas a visual inspection shows that VisualAE captures that long-term trend whereas NumAE absolutely does not. Once again, the IoU measure captures this difference, reinforcing our belief that a two-pronged approach of utilizing both numeric and visual approaches holds immense value for the field of time series forecasting. SUMMARY AND CONCLUSION To the best of our knowledge, this study is the first to explicitly forecast time series using visual representations of numeric data. We show that image-based measures can capture prediction quality more consistently than traditional numeric metrics. The proposed visual forecasting approach, albeit simplistic, performs well across datasets. Our findings show promising results for both periodic time series (including abrupt spikes in ECG) and irregular financial data. We believe that leveraging visual approaches holds immense promise for the field of time series forecasting in the future, especially when used in conjunction with traditional methods. Figure 2 : 2Sampled examples of the four datasets Harmonic, OU, ECG, and Financial. Figure 3 : 3Distribution of dataset's complexity measured using Weighted Permutation Entropy. Figure 5 : 5The architecture of undercomplete convolutional autoencoder network used in this study. Figure 6 : 6A depiction of comparison of two sample column probability distributions = [0.01, 0.1, 0.75, 0.13, 0.01] andˆ= [0.02, 0.63, 0.2, 0.12, 0.03]. Figure 7 : 7Example out-of-sample forecast predictions using the baseline methods RandomWalk, NumAE, ARIMA, and the proposed method VisualAE. The blue region indicates overlap between the input and output, whereas the red area denotes the future forecast. We show the reconstructed (or fitted) and forecast time series in blue and red region respectively. Figure 8 : 8IoU metric better captures visual forecast accuracy compared to traditional numeric metrics SMAPE and MASE. We plotted each time series with bounded intervals. The interval for -axis is [0 − , + ], whereas the interval for -axis is [min( ) − , max( ) + ], where = 10 −6 . Disclaimer: This paper was prepared for information purposes by the Artificial Intelligence Research group of J. P. Morgan Chase & Co. and its affiliates ("J. P. 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[ "Towards a manifestly SL(2,Z)-covariant action for the type IIB (p, q) super-five-branes", "Towards a manifestly SL(2,Z)-covariant action for the type IIB (p, q) super-five-branes" ]	[ "Anders Westerberg \nDAMTP\nUniversity of Cambridge\nSilver StreetCB3 9EWCambridgeUK\n", "Niclas Wyllard \nDAMTP\nUniversity of Cambridge\nSilver StreetCB3 9EWCambridgeUK\n" ]	[ "DAMTP\nUniversity of Cambridge\nSilver StreetCB3 9EWCambridgeUK", "DAMTP\nUniversity of Cambridge\nSilver StreetCB3 9EWCambridgeUK" ]	[]	We determine a manifestly SL(2,Z)-covariant κ-symmetric action for the type IIB (p, q) five-branes as a perturbative expansion in the world-volume field strengths within the framework where the brane tension is generated by a world-volume field. In this formulation the Lagrangian is expected to be polynomial; we construct the κ-invariant action to fourth order in the world-volume field strengths.	10.1088/1126-6708/1999/06/006	[ "https://arxiv.org/pdf/hep-th/9905019v1.pdf" ]	119,337,678	hep-th/9905019	a745c32138804a2583551b49e44905cc373b26df	Towards a manifestly SL(2,Z)-covariant action for the type IIB (p, q) super-five-branes May 1999 Anders Westerberg DAMTP University of Cambridge Silver StreetCB3 9EWCambridgeUK Niclas Wyllard DAMTP University of Cambridge Silver StreetCB3 9EWCambridgeUK Towards a manifestly SL(2,Z)-covariant action for the type IIB (p, q) super-five-branes May 1999arXiv:hep-th/9905019v1 4 We determine a manifestly SL(2,Z)-covariant κ-symmetric action for the type IIB (p, q) five-branes as a perturbative expansion in the world-volume field strengths within the framework where the brane tension is generated by a world-volume field. In this formulation the Lagrangian is expected to be polynomial; we construct the κ-invariant action to fourth order in the world-volume field strengths. Introduction Type IIB superstring theory is known to have a non-perturbative SL(2,Z) symmetry [1] under which the p-branes of the theory fall into representations. The strings transform in a doublet ("(p, q) strings") [2,3], whereas the three-brane is an SL(2,Z) singlet [4]. The five-branes again belong to a doublet as argued in ref. [3]. Supergravity solutions for these (p, q) five-branes were constructed in ref. [5]. In addition, there are seven-branes in the theory which should transform in a triplet under SL(2,Z) [6]. There exist formulations of type IIB supergravity (to which the above branes couple) in which the SL(2) symmetry is manifest [7,8], a fact that can be exploited to construct world-volume actions for the p-branes of the type IIB theory displaying manifest SL (2) symmetry. This programme has been completed for the strings in refs [9,10] and for the three-brane in ref. [11]. So far, however, such formulations are lacking for the higher-dimensional branes. It is the purpose of this note to investigate the case of a manifestly SL(2)-covariant action for the five-brane doublet. The world-volume theory of a (p, q) five-brane should be described on shell by a sixdimensional vector super-multiplet. However, in order to make the SL(2) symmetry manifest, one needs to introduce additional dynamical fields in the action. Our treatment will be within the framework where the brane tension is generated by a world-volume field-in the present case a complex six-form field strength. In this formulation the Lagrangian is 1 [email protected] 2 [email protected] expected to be a polynomial function of the gauge-invariant world-volume field strengths. To regain the correct counting for the degrees of freedom, auxiliary duality relations are then imposed at the level of the equations of motion. For the three-brane, for instance, a complex world-volume two-form field strength (or two real ones) satisfying a non-linear duality relation is required [11]. In ref. [11], some aspects of a manifestly covariant formulation of a (p, q) five-brane action were presented; we continue this study here using the constructive method of refs [11][12][13]. The analysis is complicated by the fact that the tension form is complex and by a "non-canonical" structure of the auxiliary duality relations, forcing us to resort to a perturbative treatment. The main result of our investigations is the determination of the κ-symmetric, manifestly SL(2)-covariant action and the associated projection operator for the type IIB (p, q) five-branes to fourth order in the world-volume field strengths. In the next section we discuss some facts about the background type IIB supergravity theory and some general features of the world-volume theory of the five-brane doublet. In section 3 we then discuss the method used to construct the action and present our results. Finally, we list our conventions in a short appendix. Preliminaries The type IIB supergravity theory in ten dimensions [7,14] is chiral and has a U(1) Rsymmetry. In the complex superspace formulation [7] the two Majorana-Weyl spinorial superspace coordinates are combined into a complex Weyl spinor. The theory, furthermore, has an SL(2,R) symmetry at the classical level, which is broken down to SL(2,Z) by non-perturbative quantum effects. By gauging the U(1) R-symmetry it is possible to formulate the theory in a way which makes the SL(2) symmetry manifest. In this formulation the scalars of the theory belong to the coset space SL(2,R)/U(1). More precisely, the scalars form a 2×2 matrix Í 1Í 1 Í 2Í 2 (2.1) on which SL(2,R) acts from the left and U(1) acts locally from the right, both group actions leaving invariant the constraint i 2 ǫ rs Í rÍ s = 1 (here ǫ 12 = −1). From the components of the above matrix one can construct the one-forms Q = 1 2 ǫ rs dÍ rÍ s , P = 1 2 ǫ rs dÍ r Í s ,(2.2) which have special significance [7]. Both are SL(2,R)-invariant, whereas under the local U(1) transformation Í r → Í r e iϑ they transform as Q → Q + dϑ and P → P e i2ϑ , respectively. Hence, the real one-form Q is a U(1) connection, while P has U(1) charge +2 (the U(1) charge of Í r is normalised to +1). They furthermore satisfy dQ − i P ∧P = 0 , dP − 2i P ∧ Q = 0 ,(2.3) the second equation showing that P is U(1)-covariantly constant (the U(1)-covariant derivative is D = d − ie Q, where e denotes the U(1) charge, and acts from the right). The two physical scalars of the theory are encoded in the projective invariant Í 1 /Í 2 , which in our conventions later will be identified with −τ = −C 0 + i e −φ . In addition to the vielbein and the scalars, there are a four-form potential whose field strength is non-linearly self-dual and two two-form potentials. When dealing with p-branes with p > 3 one needs to use a formulation of the supergravity theory in which the Poincaré duals to all field strengths are included on the same footing as the original forms. In the present case we therefore also have two six-form potentials (for the seven-branes the eight-form potentials become important too). The SL(2) doublet Í r discussed above serves as a bridge between quantities transforming in the fundamental representation of SL(2) and SL(2)-invariant (complex) quantities which are charged under the gauged U(1) R-symmetry. Examples include the two-and six-form potentials above, which can be expressed in terms of the SL(2) invariant forms 2 = Í r C 2;r and 6 = Í r C 6;r , both of U(1) charge +1 (here C 2;1 = B 2 , C 2;2 = C 2 and similarly for C 6;r ). We use calligraphic letters to denote complex quantities with U(1) charge +1. Complex conjugation is indicated with a bar and the corresponding quantities have U(1) charge −1. The background field strengths which we need are À 3 = Í r dC 2;r , H 5 = dC 4 + 1 2 Im( 2À3 ) , À 7 = Í r dC 6;r + x 2 H 5 − (1−x) C 4 À 3 + 1 2 ( 1 3 −x) Im( 2À3 ) 2 , (2.4) where, following ref. [13], we have introduced a free parameter x in the definition of À 7 . These fields satisfy the Bianchi identities DÀ 3 + iÀ 3 P = 0 , dH 5 − i 2 À 3À3 = 0 , DÀ 7 + iÀ 7 P + À 3 H 5 = 0 . (2.5) The constraints which have to be imposed in the superspace approach are at dimension 0 T αβ a = Tᾱ β a = i (γ a ) αβ , À aαβ = 2 (γ a ) αβ , H abcᾱβ = −H abcαβ = (γ abc ) αβ , À abcdeαβ = 2i (γ abcde ) αβ . (2.6) Here the barred indices on the left-hand sides refer to components corresponding to the basis form Eᾱ = E α ; since barred and un-barred indices are of the same type, the bars have been dropped on the right-hand side (see also the appendix). For fermionic backgrounds one also needs the dimension 1/2 constraints P α = 2i Λ α , À abᾱ = 2i (γ ab Λ) α , À abcdefᾱ = 2 (γ abcdef Λ) α , (2.7) where Λ α is the dilatino superfield of U(1) charge + 3 2 . The two sets of constraints given above put the theory on-shell. Note that we have only displayed the non-vanishing components that are relevant for our calculations. An expedient way to obtain the constraints for À 7 is by translating the results of ref. [15] into the complex formulation used here (taking into account the sign misprint corrected in ref. [13]). The particular choice of gauge we have made use of in converting between the real and complex formulations is Í 1 = −e 1 2 φ C 0 + ie − 1 2 φ and Í 2 = e 1 2 φ . Let us next consider the world-volume gauge field content of the (p, q) five-brane theory. In order to be able to formulate the world-volume action in a manifestly SL(2)covariant manner one needs the following gauge-invariant world-volume forms: 2 = Í r dA 1;r − 2 , F 4 = dA 3 − C 4 + 1 2 Im(C 2¯ 2 ) , 6 = Í r dA 5;r − 6 + x 2 F 4 − (1−x) C 4 2 + 1 2 ( 2 3 −x) Im( 2¯ 2 ) 2 + 1 2 ( 1 3 −x) Im( 2¯ 2 ) 2 . (2.8) The Bianchi identities for these fields are D 2 + i¯ 2 P + À 3 = 0 , dF 4 + H 5 + 1 2 Im(¯ 2 À 3 ) = 0 , D 6 + i¯ 6 P + À 7 − x À 3 F 4 + (1−x) H 5 2 + 1 2 ( 2 3 −x) 2 Im( 2À3 ) = 0 . (2.9) A crucial ingredient of supersymmetric brane actions is κ-symmetry, a local world-volume symmetry for which the variation parameter κ is a target-space spinor satisfying κ = P + ζ = 1 2 (1l + Γ)ζ, where P + is a projection operator of half-maximal rank. It is generally accepted that the background theory being on shell is both a necessary and sufficient condition for κ-invariance, although the necessity part has been explicitly proven only in a few cases. We will only investigate κ-symmetry for on-shell backgrounds. The variations of the induced metric and the world-volume form fields under a κ-transformation can be shown to be δ κ g ij = 2 E (i a E j) B κ α T αB b η ab + c.c. , δ κ 2 = −i¯ 2 i κ P − i κ À 3 , δ κ F 4 = −i κ H 5 + 1 2 Im(¯ 2 i κ H 3 ) , δ κ 6 = −i¯ 6 i κ P − i κ À 7 + x F 4 i κ À 3 − (1−x) 2 i κ H 5 + 1 2 ( 2 3 −x) 2 Im(¯ 2 i κ À 3 ) . (2.10) The next step is to compute the variation of the action under a κ-transformation. On general grounds the action is taken to be of the form S = d 6 ξ √ −g λ 1 + Φ( 2 ,¯ 2 , F 4 ) − * 6 * ¯ 6 ,(2.11) where λ is a Lagrange multiplier for the constraint Υ = 1 + Φ( 2 ,¯ 2 , F 4 ) − * 6 * ¯ 6 ≈ 0. For more details on actions of this type, see refs [9-13, 16, 17]. The function Φ is required to have U(1) charge zero but is otherwise unconstrained at this stage. It is often convenient to rewrite the action in "form language" as S = λ * 1 + * Φ( 2 ,¯ 2 , F 4 ) + 6 * ¯ 6 ; (2.12) this form of the action is better suited for the derivation of the duality relations supplementing it. These relations are constrained by compatibility with the equations of motion encoded in (2.11) and the Bianchi identities (2.9) to take the form [11] − 2x Re( * 6 * ¯ 2 ) = K 4 := δΦ δF 4 , (1−x) * 6 * F 4 + i 6 * [Re( * 6¯ 2 ) ∧ 2 ] = Ã 2 := δΦ δ¯ 2 (2.13) (for further details, see refs [11][12][13]). These relations are a crucial ingredient in the κsymmetry analysis to be discussed next. As we shall see, their complicated structure makes this analysis difficult. The method and the result In this section we determine the action and its associated duality relations from the requirement of κ-symmetry. The analysis is significantly more complicated than for the cases considered previously in the literature as a consequence of the non-canonical structure of the duality relations (2.13) and the fact that the tension form is complex. To make the problem tractable we will use a perturbative approach and expand the action in powers of the field strengths. At first sight it appears that adopting such a procedure would not be possible since there are identities which follow from the duality relations that mix terms of different orders. However, once these identities too are treated in a perturbative order-by-order fashion the procedure becomes consistent. In order to establish the κ-invariance of the action (2.11), it is sufficient to show that the variation of the constraint Υ = 1 + Φ( 2 ,¯ 2 , F 4 ) − * 6 * ¯ 6 ≈ 0 vanishes. Using a scaling argument, this variation is found to be δ κ Υ = (Ã 2 ·δ κ 2 + Ã 2 ·δ κ¯ 2 ) + K 4 ·δ κ F 4 + ( 6 ·δ κ¯ 6 +¯ 6 ·δ κ 6 ) − 1 2 (Ã (i l j)l + Ã (i l¯ j)l ) + 2 4! K (i lmn F j)lmn + 2·3 6!¯ (i lmnpq j)lmnpq δ κ g ij . (3.1) By inserting the explicit expressions (2.13) for the K's, as well as the supergravity on-shell constraints given in eqs (2.6) and (2.7), we obtain (δ κ Υ) (1/2) =P i * 6 * γ 6 − * [ * 6 F 4 + i 2 Re( * 6¯ 2 ) ∧ 2 ]·γ 2 κ + i * 6 * [ 6 − (1−x) 2 ∧ F 4 ] + 1 6 * [Re( * 6¯ 2 ) ∧ 2 ∧ 2 ] κ + c.c. , (δ κ Υ) (0) = 2iĒ i * 6 ( * γ 5 ) i − i * [ * 6 F 4 + i 2 Re( * 6¯ 2 ) ∧ 2 ] ij γ j κ + − i 6 Re( * 6 * ¯ ijkl ) γ jkl + [{\| * 6 \| 2 + x * Re( * 6¯ 2 ) ∧ F 4 } g ij − Re( * 6¯ 2 ) (i l * F 4 j)l ]γ j − i 6 Im[ * {Im( * 6¯ 2 ) ∧ 2 } (i l¯ j)l ]γ j κ + c.c. . (3.2) The next step is to insert the projected spinor parameter κ = P + ζ into these variations using an appropriate Ansatz for the projection operator, and then examine the irreducible components of the expression obtained by expanding the products of γ-matrices (for more details on the method and similar calculations see ref. [13]). It turns out that the parameter x is fixed to the value 2 3 in the process, a value which corresponds to the field strengths used in ref. [11] after taking into account some differences in conventions. (Actually, it is difficult to conclusively rule out the possibility that x could remain a free parameter; this would, however, require a very intricate form of P + .) A major complication of the analysis arises from the fact that, in contrast to all previously considered cases, an overall factor of the tension form can not be factored out from the κ-variation of the constraint. The reason for this can be traced to the fact that there are two linearly independent tension forms ( * 6 and * ¯ 6 ). This furthermore turns out to lead to the result that one does not get the duality relations in a simple form from any component; rather, one finds the duality relations entangled with various identities implied by them. Although this makes the problem difficult, it is still amenable to a perturbative approach, by means of which we have determined the action and the associated projection operator to fourth order in the world-volume fields. Higher-order corrections to the action, if present, are expected to appear at sixth order only. The projection operator is found to be 2 * γ 6 P ± ζ = * γ 6 ζ ∓ 2i 3 * F 4 ·γ 2 ζ + i 3 * 2 ·γ 4ζ + * 6ζ + O(F 5 ) ,(3.3) with O(F 5 ) denoting terms of total order five in the world-volume field strengths 2 ,¯ 2 and F 4 . The final expression for the action is S = d 6 ξ √ −g λ 1 + 1 3 2 ·¯ 2 + 2 3 F 4 ·F 4 + 1 6 (β−2) * ( 2 ∧ F 4 ) * (¯ 2 ∧ F 4 ) + 1 6 (1−β) ( 2 ∧¯ 2 )·( * F 4 ∧ * F 4 ) + 1 6 (β− 2 3 ) 2 ·¯ 2 F 4 ·F 4 + 1 6 β ( 2 ∧ * F 4 )·(¯ 2 ∧ * F 4 ) + O(F 6 ) − * 6 * ¯ 6 ,(3.4) which is to be supplemented by the duality relations −Re( * 6 * ¯ 2 ) = F 4 + 1 4 (β−2) Re[ * ( 2 ∧ F 4 ) * ¯ 2 ] + 1 4 (1−β) * ( 2 ∧¯ 2 ) ∧ * F 4 + 1 4 (β− 2 3 ) ( 2 ·¯ 2 ) F 4 + 1 4 β Re[ * ( 2 ∧ * F 4 ) ∧¯ 2 ] + O(F 5 ) , * 6 * F 4 + i 2 * [Re( * 6¯ 2 ) ∧ 2 ] = 2 + 1 2 (β−2) ( 2 · * F 4 ) * F 4 + 1 2 (β− 2 3 ) (F 4 ·F 4 ) 2 − 1 2 (1−β) * [ * ( * F 4 ∧ * F 4 ) ∧ 2 ] − 1 2 β * [ * ( * F 4 ∧ 2 ) ∧ * F 4 ] + O(F 5 ) . (3.5) Here β is a free parameter (see below). Two ubiquitous identities in the κ-symmetry calculations are [ 2 ,¯ 2 ] = 0 and [ 2 , * F 4 ] = 0, where 2 ,¯ 2 and * F 4 are viewed as matrices and the bracket is a matrix commutator. Another important result needed to verify κ-symmetry is the relation Im ( * 6 * ¯ 2 ) = − 1 6 2 ∧¯ 2 + 2 3 * F 4 ∧ * F 4 + O(F 6 ) . (3.6) These identities can be shown to follow from the duality relations (3.5). They are also required in order for P + to have the correct properties. In addition, one needs to use the fact that the following relation holds when the duality relations are satisfied: Φ ≈ 1 9 2 ·¯ 2 + 4 9 F 4 ·F 4 + O(F 6 ) . (3.7) This relation follows from the expression for Φ encoded in (3.4) combined with the identity 2x Re (Ã 2 ·¯ 2 ) + (1−x) K 4 ·F 4 = 0 (for x = 2 3 ), which can readily be derived from the form of the duality relations (2.13). Once one has shown that the duality relations imply the above commutator identities, it is straightforward to check that the terms in the duality relations proportional to β vanish for purely algebraic reasons, showing that β can chosen arbitrarily. (It is likely that β will be fixed in the complete action.) Let us also mention that one can change the appearance of the fourth-order terms. For instance, it follows from the duality relations given above that 2 ∧¯ 2 − * F 4 ∧ * F 4 = O(F 4 ); adding this expression squared to the action does not violate κ-symmetry to fourth order. In order to see that the tension of the (p, q) branes described by the action (3.4) works out correctly one proceeds analogously to the discussion in ref. [10]. To get agreement with the formula for the tension first obtained in ref. [3], one has to take into account the fact that when transforming to the string frame the tension receives an additional overall factor e −φ compared to the string case (recall that g string mn = e 1 2 φ g Einstein mn ). The question arises to what extent the above action differs from the complete action. It appears likely that the modifications, if any, should be rather minor. Furthermore, it is not clear whether P + has to be modified (the above expression, obtained from a fairly general Ansatz, certainly looks deceptively simple). However, if (3.3) is the complete result, it seems difficult to modify the action without ruining the property (3.7). It appears that some new input is needed to make further progress; this is especially true in the case of the search for a manifestly SL(2,Z)-covariant formulation of the seven-branes. These are known [6] to form a triplet and couple to the eight-form potentials dual to the three scalars which belong to the SL(2,R)/U(1) coset. Perhaps one way to make further progress is via T-duality; it may be possible to derive T-duality rules which relate duality covariant actions in the M/type IIA and type IIB theories, and in this way make the problem more tractable. It would also be desirable to have a more uniform description of the manifestly SL(2,Z)-covariant type-IIB-brane actions. AcknowledgementsThe work of A.W. and N.W. was supported by the European Commission under the contracts FMBICT972021 and FMBICT983302, respectively. A.W. would like to thankMartin Cederwall for collaboration on the project which initiated the present study.A ConventionsWe employ a complex superspace notation in which a one-form is expanded in a local inertial-frame basis as Ω 1 = E A Ω A = E a Ω a + E α Ω α + Eᾱ Ωᾱ, with Eᾱ = E α . The relation to the real formulation used in ref.[15]and Ωᾱ = 1 2 (Ω 1α +i Ω 2α ). Given these translation rules our spinor conventions follow those of ref.[15]. Moreover, complex conjugation of a bispinor reverses the order of the spinors. For higher forms we use the additional convention Ω n = 1..An . The exterior derivative d acts from the right, so that d(Ω m ∧Ω n ) = Ω m ∧dΩ n + (−1) n dΩ m ∧Ω n . (We usually suppress the symbol ∧ when no confusion should arise.) The world-volume forms (which are bosonic) follow the same conventions and hence obey the same rules. Furthermore, we do not distinguish notationally between a target-space form Ω n and its pull-back to the world-volume, the components of which are given byThe Hodge dual of a world-volume n-form is defined bywhere g is the determinant of the induced metric g ij =∂ i Z m ∂ j Z n g mn (with mostly-plus signature) and ǫ i 1 ...i 6 is the totally antisymmetric tensor density satisfying ǫ 01...5 = +1. World-volume γ-matrices are defined as the pull-backs γ i = E i a Γ a . Their symmetrised product obeys the Clifford algebra {γ i , γ j } = 2 g ij 1l inherited from the target-space, while their antisymmetrised products can be combined into the formswhere γ i 1 ...in = γ [i 1 . . . γ in] and the antisymmetrisation is of weight one. And finally, we use the notationfor the scalar product of two world-volume n-forms. Enhanced gauge symmetries in superstring theories. 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[ "Solvable model of quantum phase transitions and the symbolic-manipulation-based study of its multiply degenerate exceptional points and of their unfolding", "Solvable model of quantum phase transitions and the symbolic-manipulation-based study of its multiply degenerate exceptional points and of their unfolding" ]	[ "Miloslav Znojil [email protected] \nNuclear Physics Institute ASCR\n250 68ŘežCzech Republic\n" ]	[ "Nuclear Physics Institute ASCR\n250 68ŘežCzech Republic" ]	[]	It is known that the practical use of non-Hermitian (i.e., typically, PT −symmetric) phenomenological quantum Hamiltonians H = H † requires an efficient reconstruction of an ad hoc Hilbert-space metric Θ = Θ(H) which would render the time-evolution unitary. Once one considers just the N−dimensional matrix toy models H = H (N ) , the matrix elements of Θ(H) may be defined via a coupled set of N 2 polynomial equations. Their solution is a typical task for computer-assisted symbolic manipulations. The feasibility of such a model-completion construction is illustrated here via a discrete square well model H = p 2 + V endowed with a k−parametric close-to-the-boundary interaction V . The model is shown to possess (possibly, multiply degenerate) exceptional points marking the phase transitions which are attributable, due to the exact solvability of the model at any N < ∞, to the loss of the regularity of the metric. In the parameter-dependence of the energy spectrum near these singularities one encounters a broad variety of alternative, topologically non-equivalent scenarios.	10.1016/j.aop.2013.05.016	[ "https://arxiv.org/pdf/1305.4822v1.pdf" ]	118,683,059	1305.4822	9a06e89a9342f359106a8a80ed6561a639790025	Solvable model of quantum phase transitions and the symbolic-manipulation-based study of its multiply degenerate exceptional points and of their unfolding 21 May 2013 Miloslav Znojil [email protected] Nuclear Physics Institute ASCR 250 68ŘežCzech Republic Solvable model of quantum phase transitions and the symbolic-manipulation-based study of its multiply degenerate exceptional points and of their unfolding 21 May 2013 It is known that the practical use of non-Hermitian (i.e., typically, PT −symmetric) phenomenological quantum Hamiltonians H = H † requires an efficient reconstruction of an ad hoc Hilbert-space metric Θ = Θ(H) which would render the time-evolution unitary. Once one considers just the N−dimensional matrix toy models H = H (N ) , the matrix elements of Θ(H) may be defined via a coupled set of N 2 polynomial equations. Their solution is a typical task for computer-assisted symbolic manipulations. The feasibility of such a model-completion construction is illustrated here via a discrete square well model H = p 2 + V endowed with a k−parametric close-to-the-boundary interaction V . The model is shown to possess (possibly, multiply degenerate) exceptional points marking the phase transitions which are attributable, due to the exact solvability of the model at any N < ∞, to the loss of the regularity of the metric. In the parameter-dependence of the energy spectrum near these singularities one encounters a broad variety of alternative, topologically non-equivalent scenarios. Introduction In the two most recent collections [1] of papers on the applicability of non-Hermitian operators in quantum physics one can find multiple samples of the advantages which are provided by the use of manifestly non-Hermitian effective quantum Hamiltonians H = H † in several areas of phenomenology. Among these advantages one of the key roles is played by the capability of these sufficiently general phenomenological Hamiltonians of mimicking the quantum phase transitions and/or an onset of quantum chaos in many-body systems, etc. The growth of popularity of this area of research motivated also our present study in which we intend to pay particular attention to the role of computer-assisted symbolic manipulations. From the point of view of mathematics an explanation of the deepest essence of at least some of the above-mentioned phenomena is not too difficult since many of them are simply caused by the so called spontaneous breakdown of certain symmetries. For the most elementary illustration let us recall, e.g., paper [2] where we explained that and how the spontaneous breakdown of the combined parity and time-reversal symmetry (conveniently abbreviated as PT −symmetry in the physics literature [3]) plays the role of a trigger of transition between the observability and non-observability of the energy in an elementary toy model of quantum dynamics. The message delivered by this and similar elementary examples is nontrivial and unexpectedly deep showing, e.g., that the possibility of transitions between different dynamical regimes is closely connected to the presence of branch-point singularities, say, on the Riemann energy surface E(g) in the complex plane of coupling (or of any other tunable parameter) g. In the closely related Kato's monograph [4] on the mathematics of perturbation theory the latter singularities were systematically studied via finite-dimensional matrix models and they were also given the well-chosen name of "exceptional points" (EP). The same finite-dimensional-matrix methodical strategy will be also accepted in what follows. In Introduction let us also mention that in a broader mathematical setting of the geometric singularity theory one can find the same (or at least very similar) concepts in the Thom's classical theory of catastrophes [5] (with multisided applications [6]) as well as in its multiple newer descendants: pars pro toto let us mention our recent proposals [7,8] of the simplest possible quantum analogues of such a classical singularity classification pattern. In the mathematically narrower square-root-branch EP context the studies of the Riemannsurface singularities found particularly numerous explicit applications in quantum physics. In the context of perturbative quantum field theory and in a way enhancing our understanding of quantum anharmonic oscillations in potentials V (x) = x 2 + gx 2+δ the singularities of this class became widely known under a nickname of "Bender-Wu singularities" [9]. In optics, the alternative theoretical identification of EPs with the points of non-Hermitian degeneracies [10] encountered an enormous experimental popularity recently [11]. This success was supported not only by the availability of innovated metamaterials possessing anomalous refraction indices but also by the underlying analogies between the Maxwell and Schrödinger equations in the dynamical regime of phase transitions [12]. After a return to the standard quantum mechanics of stable systems or to the atomic, molecular or nuclear phenomenology [13], the studies of concrete models reveal the existence of EP hypersurfaces ∂D playing the role of certain natural horizons of observability of quantum systems (i.e., of certain separation boundaries between different phases), with numerous important physical as well as mathematical consequences [14,15]. In our present paper we shall be inspired by this particular problem. We shall describe some of its aspects in detail, emphasizing that their clarification finds a very natural methodical support in the symbolic as well as advanced numerical manipulations mediated, typically, by MAPLE [16] and/or by similar, mostly commercially available software. The concept of hidden Hermiticity of Hamiltonians During practically all of the history of quantum theory it has been overlooked that its applicability is restricted by the use of concrete representations of the physical Hilbert space H (P ) made in parallel with a concrete self-adjoint representation of observables (say, generators h (P ) of the unitary evolution with time). A criticism of such a paradigm emerged, e.g., in Refs. [3,17]. The change of the paradigm has been encouraged by the practical needs of applications of quantum theory in nuclear physics. Besides the often cited Dyson-inspired non-Hermitian variational approach to the so called interacting boson models of heavy nuclei [14] one might also recall another manifestly non-Hermitian variational method based on the judicious, Hilbert-space-metricemploying coupling of clusters [18], etc. One of the main obstacles of the necessary conceptual separation of the simultaneous choices of the Hilbert spaces and Hamiltonian operators was the difficulty of its implementation in calculations. Only too often, people prefer the choice of the simplest Hilbert-space representations (with, say, H (P ) ≡ L 2 (R d )) and of the simplest dynamics (cf. also the critical commentary in [19] in this context). On abstract level, the amended quantum theory admitting a broader class of quantum dynamics may be found summarized in [20]. In the spirit of Ref. [14] one finds the simultaneous introduction of space H (P ) and self-adjoint observable h (P ) overrestrictive. The information about dynamics is separated into the choice of Hilbert space H (F ) (which remains friendly, cf. the superscript) and a given Hamiltonian operator H (which need not necessarily remain self-adjoint in the same space, H = H † ). The emerging apparent puzzle ("does one violate the requirements of unitarity and Stone's theorem?") has an elementary explanation [3,14]: The initial Hilbert space (say, H (F ) ≡ L 2 (R)) is reclassified as auxiliary and unphysical. In parallel, the apparently non-Hermitian Hamiltonian (take, for illustration, just the most popular Bessis' imaginary cubic oscillator H (ICO) = p 2 +ix 3 [21] with real spectrum [22]) is also reclassified. As "crypto-Hermitian", i.e., by definition [20], as self-adjoint in another, "standard" Hilbert space H (S) . One makes the Hamiltonian H self-adjoint in H (S) by using just an ad hoc redefinition of the inner product in H (F ) . Once we have the two different operators H and H † = H acting on the ket vectors \|φ in H (F ) , we simply change ψ\|φ (F ) → ψ\|φ (S) := ψ\|Θ\|φ (F ) .(1) The self-adjoint and positive definite operator Θ = Θ † > 0 may be perceived as playing the role of the Hilbert space metric (the mathematical conditions are listed, say, in [14]). Thus, the usual Hermiticity of observables Λ is now required in H (S) , Λ ‡ := Θ −1 Λ † Θ = Λ .(2) For the Hamiltonian Λ 0 = H (with real spectrum), in particular, the hidden Hermiticity property H † Θ = Θ H(3) is often re-read as an implicit, ambiguous [14] definition of a suitable metric Θ = Θ(H). An update of the concept of solvability The concept of solvability is often restricted to the availability of the closed-form eigenstates of ordinary differential Hamiltonians [23]. The phenomenologically oriented search for the measurable aspects of quantum systems forced many physicists to search for various extensions of the concept. The question reemerged, recently, in connection with the new wave of interest in phase transitions described in terms of the spontaneous breakdown of antilinear symmetries. Typically, the same PT −symmetric Hamiltonian H = H(λ) is considered before and after the phase transition. During the phase transition itself such a Hamiltonian becomes "anomalous" (i.e., some of its eigenvalues get complex). In other words, one leaves the physical domain of parameters D. During the application of such an idea to the above-mentioned imaginary cubic oscillator H (ICO) people encountered serious mathematical (i.e., technical [24] as well as much more serious conceptual [25]) difficulties. A partial escape out of such a trap has been found in the exceptional differential-operator and boundary-delta-function model H (Robin) of Refs. [26] which proved sufficiently representative though still completely solvable [27]. A more systematic realization of the project (i.e., in the language of mathematics, of an exhaustive solution of the operator Eq. (3)) has been revealed and described, in Refs. [28,29], as based on the use of suitable finite-dimensional toy models. Their solvability opened new perspectives in a choice of the Hilbert-space metric in Eq. (2). The key point was that one did not need to start from a metric which would be given a priori. Even in a generalized (e.g., PT −symmetric) setting one was suddenly able to reconstruct the metric from the given Hamiltonian H = H † non-numerically. One of the key difficulties now emerged in connection with the ambiguity of the general solution Θ = Θ κ (H) of Eq. (3). Here the subscripted (multi)index κ numbers the alternatives (see [14] for an exhaustive discussion of this point). The new problem only remains reasonably tractable in the finite-dimensional Hilbert spaces with dim H (F,S) = N < ∞. In these cases it appears sufficient [30] to solve the conjugate Schrödinger equation H † \|Ξ n = E † n \|Ξ n(4) (note that by assumption the spectrum is real and discrete here). One then defines the general metric by the formula Θ κ (H) = N n=1 \|Ξ n κ n Ξ n \| .(5) An arbitrary optional N−plet of coefficients κ n > 0 is admitted. In other words, the solvability of the model proves crucial under the N < ∞ assumption. Oscillator-type solvable models Even if one decides to work with dimensions N < ∞, serious technical difficulties with the analysis of spectra of H and/or with the explicit construction of the metrics already emerge at dimensions as low as N = 4 [31]. It is recommended to work with the matrices of specific forms as sampled by the well-motivated non-Hermitian and PT-symmetric version of the popular Bose-Hubbard complex Hamiltonian H (BH) [32] and by some other realistic physical models [33]. Alas, strictly speaking, many of the realistic choices of these computationally tractable (i.e., complex and, typically, tridiagonal or pentadiagonal) Hamiltonians H cease to be solvable, in spite of their frequent merit of being well described by perturbation theory [32]. In this sense the first decisive step toward the exactly solvable family has only been made in Ref. [34] where we picked up and studied the anharmonic-oscillator-related real and tridiagonal anharmonic-like matrices H (N ) (AT M ) =             1 − N g 1 0 0 . . . 0 − g 1 3 − N g 2 0 . . . 0 0 − g 2 5 − N . . . . . . . . . 0 0 − g 3 . . . g 2 0 . . . . . . . . . . . . N − 3 g 1 0 0 . . . 0 − g 1 N − 1            (6) which appeared particularly construction-friendly. With the purpose of their further necessary simplification at the larger N we imposed an additional requirement g N −1 = g 1 , g N −2 = g 2 , . . . , having enhanced the symmetry of the underlying multiparametric coupling pattern perceivably. We were rewarded by the discovery of the exact solvability of the resulting model at all N < ∞ [34]. This discovery proved heavily dependent on the availability of the symbolic manipulations with polynomials in MAPLE. Pars pro toto let us recall that for the very localization of the degenerate EP value of the very first coupling g N −1 = g 1 := √ D (i.e., of the first coordinate of the vertex of the boundary manifold ∂D) at the not too large sample dimension N = 8 we had to determine this particular EP value (equal, incidentally, exactly to √ 7) as a unique (sic!) root of the sixteenth-degree secular-like polynomial equation 314432 D 17 − 5932158016 D 16 + 4574211144896 D 15 + 3133529909492864 D 14 + +917318495163561932 D 13 + 167556261648918275684 D 12 + +14670346929744822064505 D 11 + 720991093724510065469933 D 10 + +62429137451114251409236415 D 9 + 676326278232758784369966787 D 8 + +40525434802944282153115803370 D 7 + 2361976444746440513605248930610 D 6 − −145759836636885012145070948315366 D 5 + +8129925258122948689157916436170874 D 4 + −68875673245487669398850290405642067 D 3 + +235326754101824439936800228806905073 D 2 − −453762279414621179815552897029039797 D+ +153712881941946532798614648361265167 = 0 . This and similar polynomials were generated by means of the Groebner-basis technique as implemented in MAPLE. Needless to add, even the very proof of the uniqueness of this root (which we never published, due to its length) required an even more extensive use of the MAPLE software. Let us also add that the similar computer-assisted constructions of the EP boundaries had to be performed just at a few not too large dimensions N. Due to the immanent friendliness of our highly symmetric toy models H (N ) (T AM ) we were able to extrapolate the resulting closed formulae to all N, we clarified their structure and we pointed out their relevance in Ref. [35]. The readers may find more details therein. Classical-orthogonal-polynomials-related solvable models The above family of solvable physics-motivated quantum models H (N ) (T AM ) has been complemented by the other, mathematically motivated and classical-orthogonal-polynomials-related models of Refs. [36]. Subsequently [37] we added more details of the symbolic-manipulation-assisted constructions of the necessary Hilbert-space metrics Θ for the underlying specific Hamiltonians. For illustration we choose there the Gegenbauer-related family of Hamiltonians                 0 1/2 a −1 0 0 . . . 0 2 a 2 a+2 0 (2 a + 2) −1 0 . . . . . . 0 2 a+1 2 a+4 0 (2 a + 4) −1 . . . 0 0 0 2 a+2 2 a+6 . . . . . . 0 . . . . . . . . . . . . 0 (2 a + 2N − 4) −1 0 . . . 0 0 2 a+N −1 2 a+2N −2 0                 .(7) This enabled us to explain that besides the above-mentioned key contribution of computer facilities to the feasibility of the symbolic-manipulation constructions of the eligible metrics Θ, an equally important role appeared to be played by the MAPLE-supported numerical software which enables one to control the numerical precision needed in the, in general, very ill-conditioned task of the localization of the eigenvalues of Θ. Yielding the guarantee of the necessary strict positivity of all of these eigenvalues and of their inverse values. The resulting explicit knowledge of the boundaries of the domain at which the eigenvalues of the metric Θ = Θ κ (H) were losing their positivity appeared to be of a particular relevance in the quantum analogue of the classical theory of catastrophes as described in Ref. [8]. Quantum-graph solvable models Just for completeness let us also mention our quantum-graph proposals [38] in which the third family of the solvable quantum models emerged after a suitable discretization of coordinates, x ∈ R → x j , j ∈ Z. This trick helped us to make the approximate models tractable by the standard tools of linear algebra. The simplest dynamically nontrivial though still topologically trivial model of the latter discretecoordinate family dates already back to Ref. [28]. In this paper the most elementary special case was characterized by the following next-to-trivial tridiagonal matrix Hamiltonian H (N ) (λ) =             2 −1 − λ 0 . . . 0 0 − 1 + λ 2 −1 0 . . . 0 0 −1 2 . . . . . . . . . . . . 0 . . . . . . −1 0 0 . . . . . . −1 2 −1 + λ 0 0 . . . 0 −1 − λ 2             .(8) In our present paper we intend to generalize such a one-parametric boundary-interaction family, keeping in mind, i.a., the not yet explored possibility of connecting this and similar N < ∞ quantum Hamiltonians and bound-state spectra with their respective analogues as defined and derived in continuous limits [39]. New, k−parametric boundary-interaction toy model The highly restricted flexibility of the exactly solvable one-parametric discrete square well model (8) of Ref. [28] is disappointing. This disappointment is only partially compensated by the immanent merit of the possible connection of the model to its continuous N → ∞ limit and analogue as proposed and described in Refs. [26]. On the other hand, one of the key shortcomings of the one-parametric and discrete N < ∞ models (8) is that they do not allow us to perform any EP-degeneracy fine-tuning, found and accessible in several multiparametric N < ∞ toy models, say, of Refs. [32,34]. In our recent paper [40] we turned attention, therefore, to the two-and three-parametric extensions of the above-mentioned model (8). We revealed that the extended models remain solvable. In our present further extension of the latter paper we shall introduce and study, therefore, the entirely general family of the k−parametric and N by N dimensional matrix quantum Hamiltonians H (N ) (λ, −µ, . . .) =               2 −1 − λ 0 . . . . . . 0 − 1 + λ 2 −1 + µ 0 . . . . . . 0 −1 − µ 2 −1 − ν 0 . . . . . . 0 −1 + ν 2 . . . . . . . . . . . . . . . . . . −1 − µ 0 . . . . . . −1 + µ 2 −1 + λ 0 . . . . . . 0 0 −1 − λ 2               .(9) They contain an antisymmetrized and sign-changing sequence of the couplings λ, µ, . . . entering the elements −1 − λ, −1 + µ, −1 − ν, . . . , −1 + ν, −1 − µ, −1 + λ of the upper diagonal and, mutatis mutandis, also the elements of the lower diagonal. The construction of the simplest Hermitizing metric Our first result may be now formulated as the statement that models (9) remain solvable at any number of parameters k. In order to illustrate the contents of such a result, let us now take the sufficiently representative N = 11 sample of Hamiltonian (9) We may feel inspired by papers [28,41] and use the diagonal ansatz for the metric, Θ (diag) j,j , j = 1, 2, . . . , N = {z 1 , z 2 , z 3 , z 4 , 1, 1, . . . , 1, z 4 , z 3 , z 2 , z 1 } . Its insertion converts the crypto-Hermiticity condition (3) into a set of coupled nonlinear algebraic equations. Their more or less routine solution (using symbolic manipulations) leads to the unambiguous step-by-step elimination and specification of the unknown metric-matrix elements, z 4 = 1 + ρ 1 − ρ ≡ f (−ρ) , z 3 = −1 + ν − ρ + ν ρ −1 + ρ − ν + ν ρ = f (−ρ) f (ν) , z 2 = − −1 − µ + ν + ν µ − ρ − ρ µ + ν ρ + ν ρ µ 1 − ρ + ν − ν ρ − µ + ρ µ − ν µ + ν ρ µ = f (−ρ) f (ν) f (−µ) plus, finally, z 1 = = 1 − λ + µ − µ λ − ν + ν λ − ν µ + ν µ λ + ρ − ρ λ + ρ µ − ρ µ λ − ν ρ + ν ρ λ − ν ρ µ + ν ρ µ λ 1 − ρ + ν − ν ρ − µ + ρ µ − ν µ + ν ρ µ + λ − ρ λ + ν λ − ν ρ λ − µ λ + ρ µ λ − ν µ λ + ν ρ µ λ = = f (−ρ) f (ν) f (−µ) f (λ). The extrapolation pattern is now obvious, yielding the proof of the following, entirely general Proposition 1. Every k−parametric Hamiltonian H (N ) = H (N ) (λ 1 , λ 2 , . . . , λ k ) of Eq. (9) with the real parameters λ 1 = +λ, λ 2 = −µ, λ 3 = +ν etc which are all smaller than one in absolute value is self-adjoint in Hilbert space H (S) ∼ R N using the diagonal non-Dirac Hilbert space metric Θ (diag) = I which differs from the unit matrix just by its 2k outermost diagonal matrix elements Θ (diag) kk = Θ (diag) N +1−k,N +1−k = f (λ k ), Θ (diag) k−1k−1 = Θ (diag) N +2−k,N +2−k = f (λ k )f (λ k−1 ), . . . , Θ (diag) 11 = Θ (diag) N,N = f (λ k )f (λ k−1 ) . . . f (λ 1 ) where f (x) := (1 − x)/(1 + x). Remark 1. It is more than appropriate to add here that once we managed to construct the positive and invertible diagonal metric Θ (diag) , we need not bother about the proof of the reality of the spectrum of the related crypto-Hermitian matrix H (N ) = H (N ) (λ 1 , λ 2 , . . . , λ k ) anymore. Indeed, the latter matrix is, by construction, self-adjoint in the "new auxiliary" Hilbert space H (N A) ∼ R N which is assumed endowed with the (S) −superscripted inner product (1) where Θ = Θ (diag) = I. Following paper [41] one should add that the latter space need not coincide with the ultimate physical Hilbert space H (S) of Ref. [20]. Still, the (hidden) Hermiticity H = H ‡ in H (N A) implies the reality of the energy spectrum of course. Non-equivalent tridiagonal Hermitizing metrics In the spirit of paper [28] let us now recall the highly ambiguous nature of the general, N−parametric, spectral-expansion-resembling formula (5) for the metric and let us make use of this great flexibility in assuming that there might exist some still sufficiently elementary next-to-diagonal metric of the form Θ (tridiag) = Θ (diag) + v P (10) in which the N−dimensional and real diagonal metric Θ (diag) = Θ (diag) (λ 1 , . . . , λ k ) of Proposition 1 is "perturbed" by a suitable bidiagonal pseudometric. For the sake of clarity let us first set N = 11 and insert the ansatz P =                                                              in Eq. (3) again. In a more or less routine manner we may again solve the resulting set of the N 2 = 121 coupled algebraic equations yielding the following unique result, t 4 = 1 + ρ ≡ (1 − ρ) z 4 , t 3 = −1 + ν − ρ + ν ρ −1 + ρ ≡ (1 + ν) z 3 , t 2 = −1 − µ + ν + ν µ − ρ − ρ µ + ν ρ + ν ρ µ −1 + ρ − ν + ν ρ ≡ (1 − µ) z 2 plus, similarly and finally, t 1 = (1 + λ) z 1 . It is rather easy to generalize now this construction and to reformulate it into a detailed proof of the following P kk+1 = P N −k,N +1−k = (1 + λ k ) f (λ k ), P k−1k = P N +1−k,N +2−k = (1 + λ k−1 ) f (λ k )f (λ k−1 ), . . . , P 12 = P N −1,N = (1 + λ 1 ) f (λ k )f (λ k−1 ) . . . f (λ 1 ) where the function f (x) := (1 − x)/(1 + x) is the same as above. Remark 2. Whenever the real parameter v in formula (10) remains sufficiently small, the resulting tridiagonal metric Θ (tridiag) remains "acceptable", i.e., safely positive and invertible. For the larger values of v the analysis is more difficult. One must proceed in full methodical parallel with the analogous problem as studied in Ref. [37] and mentioned also in paragraph 3.2 above. Near the EP boundary ∂D some of the elements of the diagonal metric Θ (diag) of Proposition 1 will, for the consistency reasons, almost vanish or almost diverge. We may expect, therefore, that the most elementary parametric path of couplings λ = µ = . . . = t will certainly cross the EP boundary at t = ±1. This expectation is confirmed by Fig. 1 in which the t−dependence of the real spectrum as well as the phase-transition-marking loss of its reality at t = ±1 are displayed, for illustration, at N = 11 and k = 4. The most interesting features illustrated by the latter picture seem to be the t−independence of the exceptional level E = 0, the presence of the two outer "spectator-like" levels and, last but not least, the nine-tuple EP degeneracy which occurs at t = ±1. In a complementary step of analysis one can easily switch to symbolic manipulations and derive the corresponding exact secular equation E 11 − 10 − 8 t 2 E 9 + 36 − 58 t 2 + 22 t 4 E 7 − 56 − 136 t 2 + 104 t 4 − 24 t 6 E 5 + + 35 − 114 t 2 + 132 t 4 − 62 t 6 + 9 t 8 E 3 − 6 − 24 t 2 + 36 t 4 − 24 t 6 + 6 t 8 E = 0 . On this ground we may confirm the above-mentioned graphical result rigorously. Indeed, this secular equation degenerates to the trivial relation E 11 − 2 E 9 = 0 at the two EP-marking parameters t = ±1, etc. A typology of the unfoldings of the EP degeneracies In an attempt of exploring the small vicinity of the maximal EP degeneracy at λ (M EP ) = µ (M EP ) = ν (M EP ) = ρ (M EP ) = 1 let us now fix one of the individual parameters near the EP boundary ∂D and let us keep the selected parameters, one by one, t−independent. This change will define the four new phase-transition parametrizing paths of the couplings. One may expect that the nine-fold degeneracy of the energies of Fig. 1 at t = ±1 will unfold in different ways forming the alternative phase-transition patterns. In Fig. 2 we see the first t−dependent spectrum in which we fixed the innermost coupling ρ = 9/10 and in which we kept the other three couplings in the same form as above, λ = µ = ν = t. With the two outer, "spectator" real levels left out of the picture we see that the eight innermost levels remain degenerate at t = −1 while the degeneracy of the remaining pair gets shifted rather far to the left. In the subsequent picture provided by Fig. 3 we see the more thoroughly modified t−dependence of the spectrum which is caused by the move to the next scenario in which we choose the constant ν = 9/10 while keeping the remaining couplings variable as above, λ = µ = ρ = t. Ignoring now still the two outermost spectator levels as less relevant, we observe that another outer pair of the real levels has got separated at t = −1 and that it only becomes degenerate and complexified more to the left. Marginally, it is worth noticing that at the values of t ≪ −1 which already lie out of the picture (i.e., very far to the left) the previously shifted second outer pair gets merged with the respective upper or lower "spectators" so that merely the single, exceptional constant energy E = 0 remains real in the \|t\| ≫ 1 asymptotic region. A return to the asymptotic reality of the triplet of the energies (including again two outermost spectators, i.e., not visible in Fig. 4) is the phenomenon which characterizes, a bit unexpectedly, the next choice of µ = 9/10 together with λ = ν = ρ = t. A graphical explanation is provided by the parallel change of behaviour of the levels near the MEP degeneracy. In Fig. 4 we see that the sequential unfolding of this degeneracy further continues in a way which is a bit more subtle. In fact, the "expectable" complexification of the further two nontrivial innermost energy trajectories gets replaced by their mere crossing, followed by the two separate subsequent complexifications which only occur again a bit later, i.e., further to the left. After the last possible choice of λ = 9/10 with µ = ν = ρ = t all the nontrivial levels get again complex at the sufficiently large \|t\| ≫ 1. The whole real spectrum is shown in Fig. 5 where we see that the complexification now involves the sextuplet of outer energy levels. The extreme phase transition pattern of Fig. 1 is now changed most thoroughly. The original degeneracy of the spectrum survives just in the weakest form at t = ±1. This and similar observations may be also deduced from the t = ±1 form of the secular equation, E 11 − 119 50 E 9 + 7961 10000 E 7 − 361 5000 E 5 = 0 . Due to the unexpected exact solvability of this t = ±1 algebraic equation one can confirm the expected presence of the five degenerate inner roots E = 0 and also the less expectable double degeneracy of the two other, non-vanishing roots E = ± √ 19/10 which moved away from zero. The last though, possibly, just marginal surprise is that the last two non-degenerate, outer energy roots E = ± √ 2 did not move after the change of path at all. Summary In an overall applied-mathematics context and, more explicitly, within the framework of the use of the crypto-Hermitian representations of observables in quantum mechanics [20] our present paper and main model-building message may be read as based on the following three methodical assumptions, viz., • {1} the requirement of the feasibility of constructive considerations • {2} a correspondence-principle connection • {3} an offer of insight. We interpreted point {1} as our convenient restriction of the Hilbert spaces in question to the finite-dimensional ones. In section 4 we realized the second assumption {2} via a "derivation" of our main difference-operator N < ∞ toy-model example from its differential-operator N = ∞ predecessor of Ref. [26] (cf. also Ref. [28] for more details). Thirdly, in connection with item {3} it is worth emphasizing that our present results just reconfirmed that the symbolic manipulations and the related advanced software (like MAPLE, etc) became, with time, an inseparable condition sine qua non of all the similar, manifestly constructive (i.e., basically, applied-linear-algebra) projects. of our study was firmly rooted in the underlying physics. Briefly, it was aimed at a constructive analysis of the parametric domain D of the unitarity of the underlying hypothetical physical quantum system. Such a global purpose and aim have been achieved in several directions. Firstly, in contrast to the most common and plainly Hermitian toy models (where, typically, D ≡ R d has no accessible boundary) we (re-)emphasized that the domains D of our present interest were, typically, compact, possessing a nontrivial EP boundary alias horizon ∂D = ∅. Secondly, we demonstrated that the constructive study of many features of these horizons may be rendered feasible via an interactive use of a suitable graphical software, of a sufficiently advanced computer arithmetics and, first of all, of certain extensive symbolic manipulation package. Speaking in technical terms we used MAPLE and profited from its Gröbner basis facilities, etc. Thus, although our original motivation came from the physical background (concerning, typically, the questions of the stability of quantum systems), our concrete main tasks (typically, the manipulations with secular polynomials) and results (typically, the recurrent construction of non-Dirac metrics Θ = I, etc) were of a more mathematical nature. Within this framework, a subsequent return to physics might be inspired, in the nearest future, by the "next-step" transition to the complex-matrix model-building. Such a next-step enrichment of the tunable dynamics would already lead us very close to experimental setups. Typically, they might be explored, using coupled waveguides, in a way sampled, more concretely, in Ref. [42]. In conclusion let us mention that in the future study of the model the emphasis may be expected to get shifted, formally speaking, beyond the horizons ∂D and out of the quantumstability domains D. The emergent complex spectra and unstable physical mechanisms seem to open new challenging questions. Curiously enough, we are witnessing an enormous increase of interest in similar phenomena not only in the theoretical studies of quantum catastrophes [8] and in many innovative and non-standard practical quantum-model analyses [13, 32, 43, 44] but even far beyond the quantum physics itself and, in particular, as we already mentioned, in classical optics [11,45]. containing, say, the four nonvanishing couplings in the matrix H (11) (λ, −µ, ν, −ρ) of the bidiagonal form with vanishing main diagonal and with the upper diagonal j+1 (λ, −µ, ν, −ρ) , j = 1, 2, . . . , N − 1 = {−λ, µ, −ν, ρ, 0, 0, −ρ, ν, −µ, λ} and with the lower diagonal such that 1 + H (11) j+1,j (λ, −µ, ν, −ρ) , j = 1, 2, . . . , N − 1 = {λ, −µ, ν, −ρ, 0, 0, ρ, −ν, µ, −λ} . Proposition 2 . 2Every k−parametric Hamiltonian H (N ) = H (N ) (λ 1 , λ 2 , . . . , λ k ) is also self-adjoint in another Hilbert space H (S) ∼ R N where a tridiagonal non-Dirac Hilbert space metric Θ (tridiag) = I is used in the form of a positive definite linear combination (10) of the diagonal metric of preceding Proposition with the bidiagonal pseudometric P. In the latter matrix the main diagonal vanishes while its non-vanishing upper and lower diagonals have the same form filled with units, with the exception of the 2k outermost matrix elements 5 The descriptive and spectral properties of the model 5.1 The EP degeneracy phenomenon from H (N ) (λ, −µ, . . .) using N = 11 Figure 1 : 1Real eigenvalues of the k = 4 Hamiltonian H (11) (t, −t, t, −t) and the graphical localization of the degenerate phase-transition EP points at t = ±1. Figure 2 : 2The nine central real eigenvalues of the k = 4 Hamiltonian H (11) (t, −t, t, −0.9) and the "weakest" unfolding of the degeneracy near the phase-transition point t = −1. Figure 3 : 3The central real eigenvalues of the k = 4 Hamiltonian H (11) (t, −t, 0.9, −t) and the second form of the unfolding of the degeneracy near the phase-transition point t = −1. Figure 4 : 4Our present concrete results were aimed, basically, at a deeper understanding of the spectral features of Hamiltonians sampled by the new model H (N ) (λ, −µ, . . .) of section 4. The motivation The central real eigenvalues of the k = 4 Hamiltonian H (11) (t, −0.9, t, −t) and the third form of the unfolding of the degeneracy near the phase-transition point t = −1. Figure 5 : 5Real eigenvalues of the k = 4 Hamiltonian H (11) (0.9, −t, t, −t) and the strongest form of the unfolding of the MEP degeneracy at the phase-transition points t = ±1. AcknowledgmentsResearch supported by the GAČR grant Nr. 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[ "Relaxed Magnetohydrodynamics with Ideal Ohm's Law Constraint (arxiv v2)", "Relaxed Magnetohydrodynamics with Ideal Ohm's Law Constraint (arxiv v2)" ]	[ "R L Dewar \nMathematical Sciences Institute\nThe Australian National University\nCan-berra2601ACTAustralia\n", "Z S Qu \nMathematical Sciences Institute\nThe Australian National University\nCan-berra2601ACTAustralia\n" ]	[ "Mathematical Sciences Institute\nThe Australian National University\nCan-berra2601ACTAustralia", "Mathematical Sciences Institute\nThe Australian National University\nCan-berra2601ACTAustralia" ]	[]	The gap between a recently developed dynamical version of relaxed magnetohydrodynamics (RxMHD) and ideal MHD (IMHD) is bridged by approximating the zero-resistivity "Ideal" Ohm's Law (IOL) constraint using an augmented Lagrangian method borrowed from optimization theory. The augmentation combines a pointwise vector Lagrange multiplier method and global penalty function method and can be used either for iterative enforcement of the IOL to arbitrary accuracy, or for constructing a continuous sequence of magnetofluid dynamics models running between RxMHD (no IOL) and weak IMHD (IOL almost everywhere). This is illustrated by deriving dispersion relations for linear waves on an MHD equilibrium.	10.1017/s0022377821001355	[ "https://arxiv.org/pdf/2106.12348v3.pdf" ]	235,606,319	2106.12348	9af3dbcdfb749c465dc4cac26f87016824cda39c	Relaxed Magnetohydrodynamics with Ideal Ohm's Law Constraint (arxiv v2) December 20, 2021 R L Dewar Mathematical Sciences Institute The Australian National University Can-berra2601ACTAustralia Z S Qu Mathematical Sciences Institute The Australian National University Can-berra2601ACTAustralia Relaxed Magnetohydrodynamics with Ideal Ohm's Law Constraint (arxiv v2) December 20, 2021 The gap between a recently developed dynamical version of relaxed magnetohydrodynamics (RxMHD) and ideal MHD (IMHD) is bridged by approximating the zero-resistivity "Ideal" Ohm's Law (IOL) constraint using an augmented Lagrangian method borrowed from optimization theory. The augmentation combines a pointwise vector Lagrange multiplier method and global penalty function method and can be used either for iterative enforcement of the IOL to arbitrary accuracy, or for constructing a continuous sequence of magnetofluid dynamics models running between RxMHD (no IOL) and weak IMHD (IOL almost everywhere). This is illustrated by deriving dispersion relations for linear waves on an MHD equilibrium. Introduction Basics In this paper choosing constraint equations is central to our approach to developing new fluid models. The concept of a constraint equation occurs in both the variational approach to classical mechanics [see e.g. Goldstein (1980)] and optimization theory [see e.g. Nocedal & Wright (2006)]. While both traditionally treat finite-dimensional systems, the language and techniques of these fields can also help in understanding the infinite-dimensional dynamics of non-dissipative continuous media. In the following we shall distinguish between a hard constraint, i.e. one that is enforced exactly, a soft constraint, one that is enforced only approximately, and a weak version of a hard constraint, one that is enforced as the limiting case of a sequence of soft constraints (formulating such a method being the goal of this work, which it is hoped will lead to a physical regularization 1 of MHD that allows reconnection). We also distinguish between microscopic, i.e. acting within each fluid element or infinitesimal parcel of fluid, and macroscopic constraints, i.e. global within a spatial domain Ω of the fluid (or subdomain if the system is partitioned into multiple regions). The mathematical model we seek to regularize is Ideal MHD (IMHD), a special case in the general field of magnetohydrodynamics (MHD). In the general, resistive case Ohm's Law is E = ηj, where E [u] def = E + u × B(1) is the electric field observed in the local frame of each fluid element, E being the electric field in the lab frame. These elements are advected in the fluid velocity field u(x, t) (i.e.ẋ = u at each spatial point x and time t). Also B(x, t) is the magnetic field, η is the resistivity and j(x, t) is the electric current density (N.B. j = ∇ × B/µ 0 in standard non-relativistic MHD, where µ 0 is the vacuum permeability constant used in SI electromagnetic units). We have exhibited u as an explicit argument for use later in the paper, while leaving dependencies on x, t, E, and B implicit. To get IMHD, set η = 0 so that E = 0, giving what is often called the Ideal Ohm's Law (IOL): E + u × B = 0 .(2) While E is not usually explicit in the IMHD equations, this is only because it is eliminated between (2), after taking the curl of both sides, and the Maxwell-Faraday induction equation ∇ × E = −∂ t B ,(3) to give the IMHD magnetic-field propagation equation ∂ t B + ∇ × (u × B) = 0 .(4) With the "pre Maxwell" Ampère's Law j = ∇ × B/µ 0 and ∇ · B = 0, the Maxwell-Faraday equation (3) plays the important role of preserving Galilean invariance [Hosking & Dewar (2015, Sec. 5.4); Webb & Anco (2019)], independent of whether or not the IOL equation is enforced. Thus we shall retain it in the following development of a dynamical relaxation theory. Equation (3) can be viewed as a holonomic constraint on E, and likewise ∇ · B = 0 is a holonomic constraint on B, i.e. we can remove these constraints from consideration by expressing the constrained variables in terms of fewer unconstrained variables. Here these are the vector and scalar potentials A and Φ, respectively, in terms of which B = ∇ × A ,(5)E = −∂ t A − ∇Φ .(6) These imply ∇ · B = 0 and also (3), as is easily seen by calculating ∇ × E. We restrict the choice of gauge to be such that Φ is a spatially single-valued potential and such that ∂ t A = 0 in equilibrium cases in a frame (the LAB frame) where ∂ t · = 0, so A has no effect on E in that static case. Of course the vector potential still does play an explicit role in describing plasma equilibria because the magnetic flux threading a loop is¸A · dl. Dynamically, only ∂ t A contributes to inductive e.m.f.s¸E · dl around closed loops. In our case, we assume e.m.f.s are zero around any loop on the boundary ∂Ω -the trapped-flux boundary condition of RxMHD [see Appendix B of Dewar et al. (2015)]. Aside from this restriction, there is still considerable gauge freedom in A. If we choose Coulomb gauge, ∇ · A = 0, the potential representation is an example of the Helmholtz decomposition of an arbitrary vector field into the sum of curl-free and divergence-free vector fields, but we shall not make this gauge choice except in Sections 5.5 and 6 -we shall treat the magnetic helicity term carefully in our general derivation of the conservation form momentum equation in order to make it gauge invariant. Methodology: Variational principles and Euler-Lagrange equations In mechanics and optimization theory there are objective functions whose extrema -maxima, minima and saddle points -are given by Euler-Lagrange (EL) equations, which are found by setting first derivatives of these functions to zero. In mechanics such functions are Hamiltonians whose extrema give stable or unstable equilibria, or actions, time integrals of Lagrangians, whose extrema give physical time evolution equations (Hamilton's Principle). The main aim of this paper is to use an infinite-dimensional generalization of Hamilton's Principle in which partial derivatives are replaced by functional derivatives [see e.g. Morrison (1998)] of action integrals incorporating the IOL constraint, and also global entropy, magnetic-helicity and cross-helicity constraints. These functional derivatives are with respect to the basic physical fields, e.g. Φ, A, u, etc., describing the state of the system and are set to zero to find a set of Euler-Lagrange equations which together are sufficient to describe the dynamics of the system. For brevity we shall refer e.g. to the equation found by setting the functional derivative with respect to Φ as the "δΦ-EL equation". Relaxation See Appendix A for a brief history of the variational approach to finding relaxed plasma equilibrium states by minimizing the IMHD energy functional using one or more IMHD invariants as global constraints. This construction implies immediately that such relaxed magnetostatic states are a special subset of all possible IMHD equilibria, most of which, being of higher energy, are likely to be more unstable than relaxed states. In this paper we instead seek to find a time-dependent variational formulation for relaxed plasma systems going through a dynamical phase as they transition from one equilibrium state to another (e.g. due to boundary deformations). Thus, instead of minimizing energy, we use Hamilton's variational Principle, widely regarded as the most fundamental principal in all mathematical physics, from general relativity through classical mechanics to quantum field theories (for instance connecting symmetries and conservation laws by Noether's theorem). As we are attempting to establish a new classical field theory related to, but different from, ideal magnetohydrodynamics (IMHD), it is appropriate to seek new magnetofluid models by modifying the IMHD Hamilton's Principle. Following this precept, Dewar et al. (2020) derived a new dynamical magnetofluid model, Relaxed MagnetoHydroDynamics (RxMHD), from Hamilton's Action Principle using a phase-space version of the magnetofluid Lagrangian with a noncanonical momentum field u, physically identified as the lab-frame mass-flow velocity, and a kinematically constrained velocity field v (the fluid velocity relative to a magnetic-field-aligned flow). The resulting Euler-Lagrange equations generalize from statics to dynamics the usual relaxation-by-energyminimization concept developed by Taylor (1986) for flowless plasma equilibria, and its generalization to equilibria with steady flow by various authors : Finn & Antonsen (1983); Hameiri (1998); Vladimirov et al. (1999); Hameiri (2014); Dennis et al. (2014b). These generalized Taylor equilibria were shown by Dewar et al. (2020) to be consistent with RxMHD when time derivatives are set to zero. However, specific cases of equilibria with flows not aligned with the magnetic field have been limited to axisymmetric equilibria, whereas in this paper we aim to treat more general, non-axisymmetric (3-D) equilibria with flow, as well as time-dependent problems such as the calculation of the spectrum of normal modes of oscillation of 3-D relaxed equilibria. The advection equation for B, (4), implies the "frozen-in flux constraint", which, as discussed by Newcomb (1958), preserves the topology of magnetic field lines. This prevents field-line breaking and reconnection from forming new structures, such as magnetic islands, and this frustration of topological changes leads to singularities developing as time tends toward infinity Grad (1967). Though in this paper we proceed in a formal way by simply inserting constancy constraints of selected IMHD invariants as postulates, historically the heuristic assumption motivating relaxation theory is that, if it would be energetically favourable to do so, and on a long enough timescale, "nature will find a way" for reconnection to occur, either due to the magnifying effect of large gradients on small but finite resistivity at singularities, or through "anomalous" phenomena such as turbulence. Thus in the RxMHD of Dewar et al. (2020) the continuum of local frozen-in flux constraints is replaced by only two constraints involving B, the two global IMHD invariants magnetic helicity and cross helicity. However, as will be argued in Subsection 3.2, there is reason to believe that, for general three-dimensional equilibria with non-integrable magnetic field dynamics, imposing (2) as a hard constraint would lead to an ill-posed variational principle with no smooth extremum. In this case we regularize the problem by approaching an IOL-constrained state through a sequence of softly constrained states where the IOL constraint is not exactly satisfied. For a dynamical relaxed MHD theory to be fully satisfactory we require it to be well-posed mathematically and desire it to agree with ideal MHD in two cases: (i) on the boundary ∂Ω(t), because MRxMHD interfaces are regarded as arbitrarily thin sheets of IMHD fluid; and (ii) in an equilibrium state with steady flow, when one imagines any transient non-ideal behaviour to have died away, justifying the Principle of IMHD-Equilibrium Consistency ]. This Consistency Principle was satisfied by the one flowing equilibrium test case Dewar et al. (2020) looked at using their RxMHD formulation, the rigidly rotating axisymmetric steady-flow equilibrium. However RxMHD does not enforce the IOL constraint (2), so there is no reason to believe that ideal consistency would necessarily apply to more general relaxed equilibria. [Indeed, Dewar et al. (2020) showed that small dynamical perturbations about an equilibrium exhibited no tendency to preserve the IOL constraint.] Specifically, we are interested in non-axisymmetric relaxed steady-flow toroidal equilibria such as may occur in stellarators. The elliptic nature of RxMHD (when flows are small) makes it reasonable to assume that smooth solutions of the RxMHD equations exist for such equilibria. We argue in Subsection 3.2 that, generically, magnetic field and fluid flow lines on these smooth RxMHD solutions will be chaotic so their ergodic properties will exhibit complexity on all scales. While RxMHD offers no impediment to the formation of such fractal structure [one of the principal motivations for the develoment of the SPEC code, Hudson et al. (2012)] the same is not true for IMHD where the topological constraints arising from its frozen-in-flux properties (see above) force the formation of singularities. The ability of the SPEC equilibrium code to study difficult physical problems ], and subtle fundamental problems involving chaos ], motivates our current endeavour to extend the RxMHD formalism on which it is based to make it closer to IMHD but to retain sufficient topological relaxation to allow magnetic island formation and chaos, thus allowing further extension of SPEC to hande time-dependent problems in three-dimensional geometries. Background flow We define a fully relaxed RxMHD equilibrium as one where the electrostatic potential Φ has relaxed to a constant value throughout a volume Ω, so E = 0. As Finn & Antonsen (1983) recognized, this would occur in the extreme case where magnetic field lines fill Ω ergodically, because dotting both sides of (2) with B gives the derivative along B as B · ∇Φ = 0. As we shall see, constant Φ implies purely parallel flow, u = u , whose magnitude is constrained by the steady-flow continuity equation ∇ · [(ρ/B)Bu ] = B · ∇(ρu /B) = 0, where ρ is mass density. For consistency again with the (unachievable) fully ergodic limit, we define fully relaxed parallel flow as such that ρu /B = const. We denote this special parallel flow velocity as u Rx def = ν Ω B µ 0 ρ ,(7) where ν Ω is a constant throughout Ω -its significance in the RxMHD formalism is explained below: In the variational u,v dynamical relaxation formalism of Dewar et al. (2020), EL equations for u, v, B and pressure p are derived variationally from Hamilton's Principle, while the mass continuity equation is built in as a holonomic constraint. The fully relaxed flow u Rx occurs in these EL equations, with ν Ω arising as the Lagrange multiplier for the magnetic-helicity constraint in the phase-space Lagrangian. Specifically, the EL equation arising from free variations of u is u = u Rx + v ,(8) so v is the relative flow, the fluid velocity relative to the fully relaxed flow velocity u Rx . Noting from (7) that ∇·(ρu Rx ) = 0, we see that ∇·(ρu) = ∇·(ρv). Thus the continuity equation holds for both u and v, i.e. both flows are microscopically mass-conserving. Also, u × B = v × B, so E [u] = E [v] . In order to preserve (8) in the variational formulation (see later), the version of the IOL constraint we shall be using in the body of this paper is E [v] = 0, which becomes equivalent to E [u] = 0 only after the Euler-Lagrange equations are derived. Domains and boundaries For most purposes in this paper it is sufficient to restrict attention to plasma within a single domain Ω(t) that is closed, of genus at least 1, and whose boundary ∂Ω(t) is smooth, gapless, perfectly conducting and time-dependent. However we note this is part of a larger project, the development of Multiregion Relaxed MHD (MRxMHD) Dewar et al. (2015), in which Ω is but a subregion of a larger plasma region, partitioned into multiple relaxation domains physically separated by moving interfaces. As ∂Ω(t) is the union of the inward-facing sides of the interfaces Ω(t) shares with its neighbours, it transmits external forcing to the restricted subsystem within Ω(t) and imparts equal and opposite reaction forces on the neighbouring subdomains. We take the interfaces to be perfectly flexible and impervious to mass and heat transport. We also take them to be impervious to magnetic flux like a superconductor, implying the tangentiality condition n · B = 0 on ∂Ω ,(9) where B def = ∇ × A is the magnetic field and n is a unit normal at each point on ∂Ω (here and henceforth leaving the argument t implicit in Ω, n etc.). Also, to conserve magnetic fluxes trapped within Ω, loop integrals of the vector potential A within the interfaces must be conserved [see e.g. Dewar et al. (2015)]. Layout of this paper The phase-space Lagrangian variational approach to deriving ideal MHD equations is briefly reviewed in Section 2, then some general implications of the IOL when it is a hard constraint\ are discussed in Section 3 including speculations in Subsection 3.2 on the implications of chaos and ergodic theory on flows in three-dimensional systems, in Subsection 3.1 the E × B drift is derived. In Section 4 the adaptation of the augmented Lagrangian penalty function method from optimization theory to the physical purpose of approximating the IOL constraint is discussed as a softly constrained optimization problem in Subsction 4.1.1, and the specific Lagrangian density constraint term for this method is given in Subsection 4.1. The entropy, magnetic helicity and crosshelicity conservation constraints used in Relaxed MHD theory are discussed in Subsection 4.2, and the complete phase space Lagrangian to be used in this paper is constructed in Subsection 4.3. In Section 5 the Euler-Lagrange equations, including an equation of motion in momentum conservation form, are derived formally in Subsection 5.1, and in specific forms in Subsections 5.2-5.7 where the IOL constraint term provides new contributions that vanish only when the constraint is satisfied. In addition to the momentum equation form, an equation of motion in Bernoulli form is derived. A physical interpretation of the Lagrange multiplier for the IOL constraint in terms of a polarization field is also mentioned. Section 6 illustrates the implementaton of the augmented Lagrangian method for linear waves propagating on an IOL-compliant equilibrium in the WKB approximation. A continuous family of dispersion relations for wave residuals C ranging from zero in the IMHD case to its value in the RxMHD case, where it is the perturbed Lagrange multiplier λ that is set to zero. The Conclusion, Section 7, briefly summarizes what has been achieved in this paper and what more needs to be done. More detail on derivations of equations is available as online Supplementary Material in an unabridged version of this paper. A brief historical overview of MHD relaxation theory is given in Appendix A and some useful vector and dyadic calculus identities are derived in Appendix B, in particular the little-known identity (139), which is crucial for getting the general form of the momentum equation (54) into a general conservation form, (57). Ideal MHD in phase space The mathematical foundation on which our dynamical relaxation formalism is built is a noncanical form (which we call the u, v picture) of the canonical MHD Hamiltonian, and a Phase-Space Lagrangian (PSL). Here we review how Hamilton's action principle leads to IMHD when microscopic constraints on entropy and magnetic flux are applied. Later we show how RxMHD arises when these are replaced by global constraints using the same PSL formalism. Both ideal and relaxed MHD starts from the canonical MHD Hamiltonian H MHD [x, π, t] =ˆΩ H MHD dV , with H MHD (x, π, t) def = π 2 2ρ + p γ − 1 + B 2 2µ 0 ,(10) where π (x, t) is the canonical momentum density, the analogue of p in finitedimensional classical dynamics. The analogue of q is not x the Eulerian independent variable but r, the Lagrangian position with respect to a given reference frame. We do not make this explicit as we shall always work in the Eulerian picture, but the Lagrangian picture in the background does manifest in interpreting variations. [This is discussed in more detail by Dewar et al. (2020).] For instance, the analogue of the variation δq at fixed t is ∆x = ξ(x, t), the Lagrangian fluid displacement in Eulerian representation, and the analogue of the variationq δt is v(x, t) δt, which we shall refer to as the Lagrangian velocity field (not always the same as the Eulerian velocity u). Both ideal and RxMHD also use the constrained kinematic variation Newcomb (1962), δv = ∂ t ξ + v · ∇ξ − ξ · ∇v .(11) They also use the mass density variation δρ = −∇ · (ρ ξ) = −ρ∇ · ξ − ξ · ∇ρ ,(12) which is an expression of the microscopic conservation of mass and can be found by integrating the perturbed continuity equation ∂ t ρ + ∇ · (ρv) = 0 ⇔ dρ dt = −ρ∇ · v(13) along varied Lagrangian trajectories r(t\|x 0 ) [Frieman & Rotenberg (1960)] and expressing this Lagrangian variation in the Eulerian picture [Newcomb (1962)]. Instead of seeking a Poisson bracket to get phase-space dynamics from H [see e.g. Morrison (1998)], we instead work directly with the canonical phase-space Lagrangian (PSL) density L MHD , L MHD ph [x, v, π] =ˆΩ π · v − H MHD (x, π, t) dV , def =ˆΩL MHD (x, π, t) dV =ˆΩ π · v − π 2 2ρ − p γ − 1 − B 2 2µ 0 dV ,(14) and the corresponding canonical phase-space action, S MHD ph def =¨ΩL MHD dV dt(15) as the primary tools, deriving Euler-Lagrange (EL) equations from Hamilton's Principle of stationary action, δS MHD ph = 0 ,(16) varying phase space paths under appropriate constraints. We have used the subscript notation · ph on the Lagrangian L ph and the action S ph to make it clear the PSL defined in (14) is fundamentally different from the more usual configuration space Lagrangian and action. This is because, in (15), π is now regarded as freely variable, so the dimensionality of the space of allowed variations is doubled in the phase-space action principle. For instance, varying π in (14) gives the δπ-EL equation δS ph /δπ = v − π/ρ = 0, i.e., multiplying by ρ, the analogue of p = mq is seen to be π = ρv, as expected. Likewise, using the microscopic holonomic constraints of entropy and flux, δp = −γp∇ · ξ − ξ · ∇p and δB = ∇ × (ξ × B), respectively, one can verify that the Euler-Lagrange equation arising from Lagrange-varying x (i.e. varying ξ) is just the IMHD equation of motion. However, as it is not customary in fluid mechanics to work with canonical momenta, we follow Burby (2017) in exploiting the freedom afforded by the PSL to work with a velocity-like phase-space variable u, obtained by the noncanonical change of variable π = ρu. Then the canonical Hamiltonian density (10) becomes the noncanonical Hamiltonian density H MHD nc (x, u, t) = ρu 2 2 + p γ − 1 + B 2 2µ 0 ,(17) and the PSL density in noncanonical form becomes, from (14), L MHD nc (x, u, v, t) def = ρu · v − H MHD nc (x, u, t) = ρu · v − ρu 2 2 − p γ − 1 − B 2 2µ 0 .(18) 8 As neither p nor B depends on u, the δu-EL equation is δS MHD ph /δu = ρv−ρu = 0, i.e. in IMHD we have v = u. The IMHD equation of motion, which can be written in conservation form as ∂ t (ρu) + ∇ · T MHD = 0 ,(19) where T MHD def = ρuu + p + B 2 2µ 0 I − BB µ 0 ,(20) follows as it does for the ξ-EL equation in the canonical form. It will shown below that an isothermal version of IMHD can be derived by replacing the holonomic variational constraint δp = −γp∇ · ξ − ξ · ∇p with a global entropy conservation constraint, giving thermal relaxation, a more realistic model for hot plasmas than the microscopic entropy constraints implied by δp = −γp∇ · ξ − ξ · ∇p. Implications of the IOL constraint In this section we examine consequences of applying the IOL constraint (2) in the form (see Subsection 1.4) E [v] = 0, which can be written −E = v × B or ∂ t A + ∇Φ = v × B. 3.1 E × B drift As E = E + v × B we have the two identities B × E = −E × B + (B 2 I − BB) · v ,(21) and v × E = −E × v − (v 2 I − vv) · B .(22) Equation (21) leads to a decomposition of the relative fluid flow into a component v tangential to B at x, and a component v ⊥ , its projection onto the plane transverse to B, the "E × B drift," v ⊥ = E ⊥ × B B 2 .(23) It is usually safe to assume B 2 = 0 anywhere in toroidally confined plasmas, so the representation (23) generally applies everywhere, and to both equilibrium and dynamic ES MHD cases. 2 The equilibrium ergodicity problem Resolving the IOL onto the vectors B and v (or u) eliminates the v × B term, so in equilibrium, when ∂ t A = 0 these components of the IOL imply B · ∇Φ = 0 ,(24) u · ∇Φ = 0 . This means Φ = const on stream lines as well as magnetic field lines. As a consequence, level sets of Φ are invariant under magnetic and fluid flow. For instance, if Φ has smoothly nested level surfaces in a region then both u and B lie in the local tangent plane at each point on each isopotential surface -the magnetic and fluid flows are both locally integrable. In the opposite extreme, Finn & Antonsen (1983) [after Eq. (29)] conclude from the constancy of Φ along a field line that "if the turbulent relaxation has ergodic field lines throughout the plasma volume," then ∇Φ = 0, which implies that u × B = 0 -the fluid flows along magnetic field lines. As already mentioned, we call such field-aligned steady flows fully relaxed equilibria (though the converse does not apply -field-aligned flows can be integrable). However, field-aligned flow equilibria exclude many applications of physical interest -in particular tokamaks with strong toroidal flow. For such axisymmetric equilibria Dewar et al. (2020) show RxMHD can give the same axisymmetric relaxed solutions with cross-field flow as found by Finn & Antonsen (1983) and Hameiri (1983), but without needing the angular momentum constraint used by these authors. Unlike Finn & Antonsen (1983) we are not appealing to turbulence to justify relaxation, but, in fully three dimensional (3-D) plasmas, we may be able to appeal to the existence of chaotic magnetic field and stream lines. However "chaotic" is not the same as "ergodic" -while chaotic flows do involve ergodicity, this is in an infinitely complicated way, visualized in Figure 1] in terms of the fractal ergodic partition of [Mezić & Wiggins (1999); Levnajić & Mezić (2010). (This figure is generated for an iterated area-preserving map, but magnetic fieldline flows being flux preserving, the magnetic field-line return map of a Poincaré section onto itself in a magnetic containment device is similar.) A similar problem involving chaos and ergodicity arises in magnetohydrostatics, Hudson et al. (2012), where the equilibrium condition ∇p = j × B implies B · ∇p = 0, analogously to (24) for Φ, so the fractal ergodic partition for field-line flow is as relevant to the pressure p as it is for the potential Φ. In their MRxMHD equilibrium code Hudson et al. (2012) solved the puzzle posed by Grad (1967) (i.e. how to formulate the three-dimensional IMHD equilibrium problem so as to avoid a "pathological" pressure profile) by using a much simpler ergodic partition obtained by aggregating contiguous elements of the fractal ergodic partition into a finite number of constant-pressure "relaxation regions" Ω i , with pressure changing (discontinuously) only across the interfaces between the Ω i s. The code was thus named the Stepped Presssure Equilibrium Code (SPEC). Continuity of Electrostatic Potential One might think that an analogous "stepped potential equilibrium" could provide a solution to the problem of finding a non-trivial but tractable solution of ∇ Φ def = (B/B) · ∇Φ = 0 in a chaotic magnetic-field-line flow. Unfortunately however we must restrict ∇ ⊥ Φ def = (I − BB/B 2 ) · ∇Φ to square-integrable functions in order to keep the E × B drift (23) from acquiring a δ-function component. This rules out having steps in Φ because δ-functions are not square integrable, so stepped potentials would make the kinetic energy integral infinite. However, this does not necessarily imply Φ is constant in weakly chaotic regions with a finite measure of KAM surfaces -perhaps weak KAM theory [see e.g. Fathi (2009)] would allow fractal potential profiles having finite kinetic energy associated with them. As a way to handle non-constant Φ computationally we propose using a penalty or augmented Lagrangian method [see e.g. Nocedal & Wright (2006)]. That is, we treat Hamilton's Principle as a constrained saddle-point optimization problem and add a penalty functional to the Hamiltonian, which regularizes the variational problem by approaching the (perhaps fractal) IMHD "feasible region" of configuration space from outside, in the less-constrained space on which RxMHD is defined [which is smoother, see Figure 1 of Dewar et al. (2020)]. Another approach might be a time-evolution code with added dissipation such that the long-time solution is attracted to one having chaotic regions of constant pressure interspersed with integrable regions with changing pressure. This can be viewed as a steepest-descents solution of the same optimization problem. Constraints and Constrained Optimization In this section we first discuss the new aspect of variational relaxation theory introduced in this paper, namely the imposition of Ideal Ohm's Law (IOL) as a constraint. We then review use the Lagrange multiplier method in Subsection 4.3 for imposing the conservation of entropy, magnetic helicity and cross helicity as hard constraints, causing the EL equations, and hence the conserved quantities, to be parametrized by the triplet of multipliers τ Ω , µ Ω , and ν Ω (the subscripts Ω indicating they are constant throughout Ω, but may jump across ∂Ω if there are adjacent relaxation regions as in MRxMHD). Augmented Lagrangian constraint method In implementing the IOL constraint we propose to adapt the Augmented Lagrangian method from finite-dimensional optimization theory, as described by Nocedal & Wright (2006, §17.3), or for Banach spaces [see e.g. Kanzow et al. (2018) and references therein]. This is a hybrid numerical method that combines two constraint approaches: the Lagrange multiplier method and the penalty function method. We shall use the Lagrange multiplier method in Subsection 4.3 for imposing the conservation of entropy, magnetic helicity and cross helicity as hard constraints, causing the EL equations, and hence the conserved quantities, to be parametrized by the triplet of multipliers τ Ω , µ Ω , and ν Ω (the subscripts Ω indicating they are constant throughout Ω, but may jump across ∂Ω if there are adjacent relaxation regions as in MRxMHD). To impose the IOL as a hard constraint using the Lagrange multiplier method we would "simply" add λ · (E + v × B) to the Lagrangian density, solve the resultant EL equations to give E+v×B as a function of the Lagrange multiplier λ, and then solve for λ such that E + v × B = 0. Apart from the unavoidable complication that λ is not just a 3-vector but also is a function of x and t, so infinite dimensional, there is the more fundamental problem, flagged in Subsection (3.2), that the limit E + v × B → 0 is likely singular in 3-D equilibria because E, v, and B presumably tend toward being fractal functions. Thus there is good reason to believe the hard IOL constraint problem is ill-posed in 3-D systems such as stellarators, which leads us to seek a soft IOL constraint approach in order to regularize the Hamilton's Principle optimization problem. We build in the Maxwell-Faraday induction constraint (3) as a hard constraint by using the potential representations (6) , E = −∇Φ − ∂ t A, and (5), B = ∇ × A. Thus the set of primary variables subject to variation during an optimization is X = {r, u, p, A, Φ} ,(26) where r is the Lagrangian fluid-element position field discussed in Section 2. [Note we have not included ρ and v as a independent variables because they are functionals of r, with variations given by (12) and (11).] The simplest soft IOL constraint approach is to add 1 2 µ P Ω (E + v × B) 2 to the Hamiltonian density (thus subtracting it from our Lagrangian density), where µ P Ω → +∞ is a penalty multiplier. In this limit the penalty term is supposed to dominate all other terms in the Hamiltonian or Lagrangian and enforce IOL feasibility through a sequence of infeasible solutions. However, this method is clearly ill-conditioned numerically, leading us to resort to the "best of both worlds" augmented Lagrangian method described below. The IOL as a softly constrained optimization problem In implementing the parallel IOL constraint we propose to adapt the Augmented Lagrangian method from finite-dimensional optimization theory, as described by Nocedal & Wright (2006, §17.3), or for Banach spaces [see e.g. Kanzow et al. (2018) and references therein]. This is a hybrid numerical method that combines two constraint approaches: the Lagrange multiplier method and the penalty function method sketched above. As well as adapting notation and methods from optimization theory we have borrowed the terms feasible region, meaning the range in which the vector X of variables to be solved for is such that a set of equality constraints c i [X] = 0 are satisfied [also inequality constraints c i [X] > 0, but we do not consider this case]. The infeasible region, is its complement, where one or more constraints are violated. By hard constraint we mean one where X must be in the feasible region, and by soft constraint we mean one where X need only be in some neighbourhood of the feasible region, which is useful both practically and for regularizing when, as in MHD, defining the boundary between feasible and infeasible is complicated by the possibility of singular behaviour like current sheets and reconnection points. We now formulate two related physical tasks, the simpler one being 1. The equilibrium problem: In toroidal plasma confinement theory the most physically desirable states are stable, time-independent equilibria, i.e. minima of a Hamiltonian functional H[X], kinetic plus potential energy within a static boundary ∂Ω. We seek a numerical algorithm that starts from an initial guess for the physical fields X and iterates to extremize (minimize if seeking a stable equilibrium) a Hamiltonian, under the IOL equality constraint, (2). Finding a stable equilibrium can be stated concisely as the optimization problem To treat the implementation of the IOL in Hamilton's Principle, a constrained saddle point optimization problem, we shall use the set of values of the components of E (x ∈ Ω) Note the identities Equilibrium min X H[X] subject to C[X](x)] = 0, ∀ x ∈ Ω and b.c.s ∀ x ∈ ∂Ω ,(27)where C def = E + v × B .(28)∂C ∂E = I , ∂C ∂v = I × B and ∂C ∂B = −I × v .(29) We seek a soft form of the equilibrium constraint, i.e. a formulation such that C → 0, ∀ x ∈ Ω, where → denotes a limiting process whereby X moves from the infeasible class of states where C = 0 toward the feasible class defined pointwise as C = 0, ∀ x ∈ Ω(t), or, in a weak form, as \|\|C\|\| = 0. Such a soft constraint procedure is provided by the augmented Lagrangian (or, rather, Hamiltonian in the Equilibrium problem) as prescribed by Nocedal & Wright (2006, §17.3), H A def =ˆΩ H − λ · C + 1 2 µ P Ω C 2 dV ,(30) where λ is a Lagrange multiplier and the spatial constant µ P Ω ≥ 0 is a penalty multiplier of the non-negative quadratic penalty 1 2´Ω C 2 dV . Nocedal & Wright (2006, §17.3) give an iterative algorithmic framework that combines the advantages of both the Lagrange multiplier and penalty function methods. In their algorithm the user provides an increasing sequence µ P Ω \| n , : n = 0, 1, 2, . . . penalty multipliers and adjusts λ to solve for the minima of Hamiltonians with the Laggrange muliplier and penalty terms. The iteration update rule for the sequence of Lagrange multipliers and corresponding constraint residuals {λ n , C n : n = 0, 1, 2, . . .} is λ n+1 = λ n − µ P Ω \| n C n .(31) In the following sections we shall take the iteration index n as implicit unless needed for clarity, with the updated λ as given by the RHS of (31) denoted by λ * def = ∂ ∂C λ · C − µ P Ω 2 C 2 = λ − µ P Ω C ,(32) wich is a "best estimate" of the optimum Lagrange multiplier given the current estimate and penalty multiplier. The more difficult second physical task is: 2. The time evolution problem: This is similar to Task 1 except we seek an evolution, a dynamical path in space-time Ω t ×[t 1 , t 2 ] given a time-dependent boundary ∂Ω t , the objective function for extremization now being an action integral. The stable minima of the Hamiltonian now becoming saddle points of the coresponding action functional S[X], kinetic minus potential energy. This task can be summarized as the pseudo optimization problem Dynamics extr X S[X] subject to C [X] (x, t) = 0, ∀ x ∈ Ω t , t ∈ [t 1 , t 2 ] ,(33) under the same boundary conditions as for equilibrium at each time t. Although sometimes called the "Principle of Least Action", Hamilton's Principle is often not an optimization problem but rather a saddlepoint problem, where the stationary point of S[X] cannot be found by a descent algorithm. [This is well known in nonlinear Hamiltonian dynamics [Meiss (1992)] where periodic orbits are classified as (action) minimizing orbits, which are hyperbolic (unstable), or as minimax orbits, which are elliptic (stable).] Although "extremum" or "extremization" is not quite correct either, as extremum strictly means "maximum or mininimum", it is convenient to use the abbreviation "extr" as an abbeviation for these words and add the rider "depending on direction of traversal" (implying also the existence of neutral directions between max and min), so as to include saddle points. To find saddle points requires some form of Newton method, needing at least estimates of the second variation (Hessian matrix) rather than a descent method. The augmented Lagrangian method still works if we solve (33) at each iteration, so here again we adopt it to solve for a stationary point of the augmented phase spoce action functional (18) S A ph def =¨Ω ρu · v − H + L C Ω dV dt,(34) where the augmented penalty constraint density L C Ω is defined by L C Ω def = λ · C − µ P Ω C 2 2 ,(35) with λ and µ P Ω are taken as external parameters in the application of Hamilton's Principle at each iteration, giving a sequence of regularized magnetofluid models. When µ P Ω = 0, the pure Lagrange multiplier method, feasible critical points of H A might be saddle points with descending directions in the infeasible sector even if they are physically stable ideal equilibria where the IMHD Hamiltonian is minimized. When λ = 0, the pure penalty function method, feasible stable equilibria could be approximated arbitrarily well in the limit as µ P Ω tends to infinity, but this becomes an increasingly ill-posed optimization problem. (It does however have the attractive feature of providing a continous family of relaxed MHD models running from the RxMHD of Dewar et al. (2020) when µ P Ω = 0 to a subset of weak IMHD when µ P Ω → +∞.) Remarks: (i) Task 1 can be treated as a subclass of Task 2 in which time derivatives are set to zero and t is taken to be an irrelevant constant, but the Hamiltonian is more appropriate than the Lagrangian for treating it as an optimization problem. (ii) The iteration method for implementing constraints is implicit, meaning that the state variables in the n th iteration need to be found by solving Euler-Lagrange equations, taking it for granted the Euler-Lagrange equations can be solved and any sub-iterations required have converged. We shall not discuss detailed implementation issues here, except to remark that time evolution over a large time interval can be implemented numerically in an outer time-stepping loop in which a large time interval is split into multiple short time intervals (timesteps) [t i , t i+1 ], within each of which constraint iterations are repeated until converged to the required accuracy. Thus the evolutions required in implementing the constraint iterations are over short time intervals, with each initial guess being the converged evolution from the previous timestep and the evolution representable to sufficent accuracy on a low-dimensional interpolation basis (e.g. dimension 2 for piecewise-linear representation of the full evolution) -the increase in difficulty in going from Task 1 to Task 2 may not be as great as at first it appears to be. Global constraints for isothermal RxMHD and IMHD We shall always retain the microscopic holonomic constraints (Section 2) on ρ and v, but we relax the infinite number of microscopic dynamical constraints on p and B imposed in IMHD by replacing these constraints with only three macroscopic hard constraints. These three constraints, described below, are chosen to be quantities that are exact invariants under IMHD dynamics in order to ensure that relaxed equilibria are subset of all ideal equilibria. Further, as we seek plasma relaxation formalisms applicable in arbitrary 3-D toroidal geometries, we invoke only the MHD invariants least dependent on integrability of the fluid and magnetic field line flows, the conservation of total mass M Ω def =´Ω ρ dV being the most fundamental (whose conservation is built in microscopically) . While these global invariants are not as well conserved as mass under small resistive, viscous and 3-D chaos efects, in the spirit of Taylor (1986) we assume they are sufficiently robust that postulating their conservation produces a model that is useful in appropriate applications. We can get IMHD by retaining all the microscopic holonomic constraints of Section 2, but it seems more physically relevant to almost collisionless hot plasmas with high thermal conductivity along magnetic field lines to relax the plasma thermally by relaxing the microscopic dynamical constraint on p and replacing it with the first global constraint below (entropy) to give isothermal IMHD. As just indicated, our first global constraint is the adiabatic-ideal-gas thermodynamic invariant, total entropy S Ω [ρ, p] def =ˆΩ ρ γ − 1 ln κ p ρ γ dV ,(36) where κ is, for our purposes, an arbitrary dimensionalizing constant, though it can be identified physically through a statistical mechanical derivation of (36) [see e.g. Dewar et al. (2015)]. Its functional derivatives are δS Ω δρ = 1 γ − 1 ln κ p ρ γ − γ γ − 1 ,(37) δS Ω δp = 1 γ − 1 ρ p .(38) We also impose conservation of the magnetic helicity 2µ 0 K Ω , where, following Bhattacharjee & Dewar (1982), we define the invariant K Ω as K Ω [A] def = 1 2µ 0ˆΩ A · B dV(39) giving, with help of (134), the functional derivative δK Ω δA = B µ 0 .(40) As discussed by Hameiri (2014), in single-fluid IMHD we do not have a separate fluid helicity invariant, but do have the cross helicity µ 0 K X Ω , which can be derived from a relabelling symmetry in the Lagrangian representation of the fields, see e.g. Ch. 7 of Webb (2018). Analogously to our other constraint parameters containing B, we include µ −1 0 in the definition of the cross helicity functional, K X Ω [u, A] def = 1 µ 0ˆΩ u · B dV ,(41) which, like P Ω and S Ω , has two functional derivatives δK X Ω δu = B µ 0 , δK X Ω δB = u µ 0 .(42) IOL-constrained Phase-Space Lagrangians and Actions As foreshadowed, our recipe for constructing a non-dissipative relaxed magnetofluid model is to start with the IMHD noncanonical Hamiltonian, (17), but to relax many, but not all, of the microscopic constraints to which it is subject when deriving the IMHD Euler-Lagrange equations. Specifically, to retain the basic compressible Euler-fluid backbone of our relaxed MHD model Dewar et al. (2015) we keep the microscopic kinematic and mass conservation constraints, 11 and (12). However we delete the microscopic ideal gas and flux-frozen magnetic field variational constraints, δp = −γp∇·ξ −ξ ·∇p and δB = ∇×(ξ ×B), replacing these infinities of constraints with only the three robust IMHD global invariants (36-41). These global constraints are imposed by adding the global-invariantsconstraint (GIC) Lagrange multiplier term L GIC Ω def = τ Ω ρ γ − 1 ln κ p ρ γ + µ Ω A · B 2µ 0 + ν Ω u · B µ 0(43) to L MHD nc to form the RxMHD PSL density ] L Rx Ω def = L MHD nc + L GIC Ω = ρu · v − ρu 2 2 − p γ − 1 − B 2 2µ 0 + τ Ω ρ γ − 1 ln κ p ρ γ + µ Ω A · B 2µ 0 + ν Ω u · B µ 0 .(44) In 43 the Lagrange multipliers τ Ω , µ Ω , and ν Ω are spatially constant throughout Ω, but can change in time to enforce constancy respectively of total entropy, magnetic helicity and cross helicity in Ω. By removing the infinite numbers of microscopic constraints on p and B that are imposed in IMHD, in the RxMHD formalism Dewar et al. (2020) we greatly increased the variationally feasible region of the state space, thus allowing the system to access a lower energy equilibrium. In fact, as the Eulerian fields δp(x, t) and δA(x, t) are now locally free variations at each point x, we have added two infinities of degrees of freedom, which turns out to be too many as the IOL constraint embedded in IMHD is entirely lost in RxMHD. Thus we reduce the degrees of freedom of RxMHD by imposing a soft penalty-function IOL constraint using the augmented Lagrangian constraint density L C Ω , (35). As the IOL constraint applies pointwise throughout Ω, giving an infinite number of constraints, on Φ and B. Adding the constraint term we get the full Lagrangian density with augmented constraint L A Ω def = L Rx Ω + L C Ω .(45) We shall also have need to define the gauge-invariant part of the Lagrangian density by substracting off the magnetic helicity term, L A− Ω def = L A Ω − µ Ω A · B 2µ 0 .(46) (For derivatives of the Lagrangian density with respect to anything other than A, B, x, or t, L A Ω and L A− Ω can be used interchangeably.) The augmented phase-space action integral is S A Ω =ˆdtˆΩdV L A Ω .(47) As in (18), the fluid velocity u is treated as a noncanonical momentum variable that is freely variable in the phase-space version of Hamilton's Principle, δS A Ω = 0, and v is a relative flow whose variation with respect to ξ obeys the kinematical constraint (11). It is also the flow appearing in the mass conservation constraint equations (12) and (13). 5 Euler-Lagrange (EL) equations Formal view of EL equations The utility of Hamilton's action-principle approach is that a complete set of equations for our physical variables is provided by the EL equations following from the general variation of the generic augmented action S A Ω , δS A Ω =ˆdtˆΩdV δu · ∂L A Ω ∂u + δA · ∂L A Ω ∂A + δB · ∂L A Ω ∂B + δE · ∂L A Ω ∂E + δp ∂L A Ω ∂p + δρ ∂L A Ω ∂ρ + δv · ∂L A Ω ∂v def =ˆdtˆΩdV δu · δS A Ω δu + δΦ δS A Ω δΦ + δA · δS A Ω δA + δp δS A Ω δp + ξ · δS A Ω δx(48) where the top equation on the RHS is simply an integral over the first variation of L A Ω and the second RHS equation defines the functional derivatives with respect to the independent variables by matching the corresponding terms in the top RHS equation after the variations of the explicit variables in L A Ω are expanded and integrations by parts where necessary -ignoring boundary terms as we can assume the support of the variations does not include the boundary [note that there are no δλ or δµ P Ω terms as λ and µ P Ω are taken as given -see discussion around (35)].For instance δS A Ω /δx is the sum of the terms linear in ξ obtained from δρ and δv given in (12), and (11)respectively. (Note: For notational convenience δx is used in the denominator of the functional derivative as an alternative to the Lagrangian variation of x, denoted everywhere else as ∆x or ξ. It does not denote the Eulerian variation of x, which is by definition zero.) Inspecting (35) we see that L C Ω contains E and B but does not contain u, p, or ρ, and no term in L Rx Ω contains E, ∇u, ∇p, or ∇ρ, so the corresponding functional derivatives of S A Ω are are simply partial derivatives of L A Ω , e.g. the δu-and δp-EL equations are δS A Ω δu = ∂L A Ω ∂u = 0 ,(49)δS A Ω δp = ∂L A Ω ∂p = 0 .(50) The δA-EL equation is best displayed by splitting L A Ω into the gauge-invariant part L A− Ω , (46), and the magnetic helicity constraint term µ Ω A · B/2µ 0 in order to make manifest the explicitA-dependence. Thus ∂L A Ω ∂A = µ Ω B 2µ 0 , ∂L A Ω ∂B = ∂L A− Ω ∂B + µ Ω A 2µ 0(51) The δA-EL equation is then found by using these results in the lemma (134) to give δS A Ω δA = µ Ω B 2µ 0 + ∇ × ∂L A− Ω ∂B + ∇ × µ Ω A 2µ 0 + ∂ ∂t ∂L A− Ω ∂E = 0 , i.e. ∂ ∂t ∂L A− Ω ∂E + ∇ × ∂L A− Ω ∂B = − µ Ω B µ 0 .(52) The δΦ-EL equation is, using (135), δS A Ω δΦ = ∇ · ∂L A Ω ∂E = 0 .(53) and, using (11) , δv = ∂ t ξ + v · ∇ξ − ξ · ∇v, the ∆x-EL [or ξ-EL -see (48)] equation is δS A Ω δx = −∂ t Π − ∇ · (vΠ) − (∇v) · Π + ∇ ρ ∂L A Ω ∂ρ − (∇ρ) ∂L A Ω ∂ρ = 0 ,(54) where Π def = ∂L A Ω ∂v(55) is a new canonical momentum density (cf. π in Subsection (14)). To get a more transparent version we now derive a canonical-momentum conservation form of the equation of motion, the existence of which is implied by Noether's theorem and translational invariance (within Ω, i.e. not including ∂Ω). To do this we transform (54) into the same form as (22) Formal conservation-form momentum equation ∂ t Π + ∇ · v Π + I L A Ω − ρ ∂L A Ω ∂ρ = ∇L A Ω − (∇v) · Π − (∇ρ) ∂L A Ω ∂ρ ,(56) Local translational invariance implies the only x dependence of L A Ω is through its component fields, so the chain rule gives ∇L A Ω = (∇u) · ∂L A Ω ∂u + (∇p) ∂L A Ω ∂p + (∇B) · ∂L A Ω ∂B + (∇A) · ∂L A Ω ∂A + (∇E) · ∂L A Ω ∂E + (∇ρ) ∂L A Ω ∂ρ + (∇v) · Π + (∇λ) · ∂L A Ω ∂λ , which can be simplified slightly because the two terms on the top line of the RHS vanish by (49) and (50) . Using also (51) we get ∇L A Ω = (∇B) · ∂L A− Ω ∂B + µ Ω 2µ 0 [(∇B) · A + (∇A) · B] + (∇E) · ∂L A Ω ∂E + (∇ρ) ∂L A Ω ∂ρ + (∇p) ∂L A Ω ∂p + (∇v) · Π + (∇λ) · ∂L A Ω ∂λ = − ∂ ∂t ∂L A Ω ∂E × B − ∇ · ∂L A− Ω ∂B × I × B + µ Ω 2µ 0 ∇ (A · B) + (∇E) · ∂L A Ω ∂E + (∇ρ) ∂L A Ω ∂ρ + (∇v) · Π + (∇λ) · ∂L A Ω ∂λ = ∇ · ∂L A− Ω ∂E E − ∂L A− Ω ∂B × I × B + I µ Ω A · B 2µ 0 − ∂ ∂t ∂L A− Ω ∂E × B + (∇ρ) ∂L A− Ω ∂ρ + (∇v) · Π + (∇λ) · ∂L A− Ω ∂λ , where we used the identity (139) (53), to reduce all but the last three terms to divergence form. Eliminating these ∇ρ and ∇v terms between those in (56) and ∇L A Ω above, and also cancelling the A · B terms, gives a general momentum equation in gauge-independent conservation form on the LHS, but with the ∇λ term on the RHS acting as an external forcing term, ∂ t Π + ∂L A− Ω ∂E × B + ∇ · T = (∇λ) · ∂L A− Ω ∂λ ,(57) (where LHS/RHS denote "left/right-hand side"). Here the tensor T is given by T = vΠ + ∂L A− Ω ∂B × I × B − ∂L A− Ω ∂E E + I L A− Ω − ρ ∂L A− Ω ∂ρ .(58) [See (140) for a dyadic identity that is useful for interpreting the second term on the RHS.] This construction illustrates that the momentum conservation form is a general property of any translation-invariant Lagrangian formulation (by Noether's theorem) and thus is preserved even with our augmented penalty function constraint (except for the forcing term from the symmetry-breaking Lagrange multiplier). It is not manifestly symmetric but we expect it to be symmetrizable from local rotational invariance [Dewar (1970), Dewar (1977)]. We now examine the implications of our EL equations in more detail. Explicit Variation of Eulerian velocity From the δu-EL equation (49) , ρ(v − u) + ν Ω B µ 0 = 0 ,(59) which is equivalent to the relative flow formula (8) given in the Introduction, thus both motivating and justifying substituting v for u [see text below (8)]. We shall use this below in the form v = u − u Rx for eliminating v when required. (Recall u Rx def = ν Ω B/µ 0 ρ.) N.B. Taking the divergence of both sides of 8, the EL equation (59), we have ∇ · (ρv) = ∇ · (ρu). Thus, as noted below (8), u obeys the same continuity equation as v, (13). That is, ∂ t ρ + ∇ · (ρu) = 0 .(60) Variation of pressure From the δp-EL equation (50) 1 γ − 1 τ Ω ρ p − 1 = 0 , which leads to the isothermal equation of state p = τ Ω ρ .(61) A related result is sometimes useful: From (45) after a little algebra, ∂L A− Ω ∂ρ = u · v − u 2 2 − τ Ω ln ρ ρ Ω , def = u · v − h Ω(62) where the Bernoulli "head" h Ω is defined by h Ω = u 2 2 + τ Ω ln ρ ρ Ω ,(63) with ρ Ω a spatially constant reference density that need not be given as it does not contribute to the ∆x-EL, (54). It has the property that ρ∇h Ω = ρ∇ 1 2 u 2 + ∇p. Explicit Variation of scalar potential From the δΦ-EL equation (53) and (73), ∇ · ∂L C Ω ∂E = ∇ · ∂L C Ω ∂C = 0 i.e. ∇ · λ * = 0 ,(64) where λ * = λ − µ P Ω C is as defined in (32). Comparing the update rule (31), λ\| n+1 = λ\| n − µ P Ω \| n C\| n , with (32) we identify λ * as the updated λ for initializing the next iteration, i.e. λ\| n+1 = λ * \| n . As ∇ · λ * \| n = 0 we therefore have ∇ · λ n+1 = 0, and likewise for ∇ · λ\| n+2 and all subsequent Lagrange multipliers in the iteration sequence. In fact, assuming integer n is a typical step in the iteration, we must also conclude ∇ · λ\| n = 0 ∀ n, including 0 and ∞ . Thus we can eliminate both λ and λ * from (32) by taking the divergence of both sides to give ∇ · C = 0 , which, being a homogeneous equation, provides no driving term for C. While, from (28), (66) implies an inhomogeneous equation for E, ∇ · E = −∇ · (u × B) ,(67) this is also implied by the IOL, again showing that we cannot determine nonfeasibility by taking divergences only. However, we also have an expression for ∇ × E from the Maxwell-Faraday induction equation (3), which, combined with (28), gives the inhomogeneous equation ∇ × C = −∂ t B + ∇ × (u × B) .(68) Thus the non-feasibility parameter C can be viewed as driven by the departure from the ideal MHD magnetic-field evolution equation. This is seen better by rewriting (68) as an evolution equation for B, ∂ t B = ∇ × (u × B) − ∇ × C .(69) When C = 0 this is the IMHD evolution eqation for B, irrespective of our magnetic and cross-helicity constraints and confirms that allowing C = 0 is sufficient to relax the flux-freezing topological constraints of IMHD. However, to satisfy (3) automatically we use the potential representations (6), E = −∇Φ − ∂ t A,and (5), B = ∇ × A, so (28) becomes C = −∇Φ − ∂ t A + v × (∇ × A) ,(70) showing C as the discrepancy between the potential representations of −E and of v × B (equivalently u × B after Euler-Lagrange equations are derived). As (70) implies (69), the latter is now not an independent equation and is useful only for insight. In potential representation (67) becomes the Poisson equation ∇ 2 Φ = ∇ · (u × B) ,(71) where Coulomb gauge, ∇ · A = 0, has been adopted to eliminate the explicit unknown A, though it is still implicit through B. The solution of this elliptic differential equation, using the Dirichlet boundary conditions discussed after (28), is such that Φ is a smooth function. However, as discussed in Section 3.2 the electric and magnetic field lines it defines are not generically integrable and thus may represent chaotic flows in 3-D geometries. Assuming u and A are determined using other Euler-Lagrange equations, so (71) can be solved for Φ, (70) gives us C and then λ * from (32). This will be illustrated in Subsection 6.2.2. Explicit Variation of vector potential (23) From (46), (44) and (29) we have ∂L A− Ω ∂B = ∂ ∂B − B 2 2µ 0 + ν Ω B · u µ 0 + C · λ − µ P Ω C 2 2 = − B µ 0 + ν Ω u µ 0 + ∂C ∂B · λ * = − B µ 0 + ν Ω u µ 0 − v × λ (72) and ∂L A− Ω ∂E = λ .(73) Inserting these identities in the δA-EL equation (52) gives ∂λ * ∂t − ∇ × (v × λ * ) = 1 µ 0 (∇ × B − µ Ω B − ν Ω ∇ × u) ,(74) displayed as an inhomogeneous hyperbolic equation for the Lagrange-multiplier field λ * . However, it can also be displayed as an inhomogeneous elliptic equation for B by multiplying both sides with −µ 0 and rearranging to give ∇ × B = µ Ω B + ν Ω ω + µ 0 [∂ t λ * − ∇ × (v × λ * )] ,(75) where ω def = ∇ × u is the fluid vorticity. Apart from the terms in λ * this is the RxMHD modified Beltrami equation found by Dewar et al. (2020). The relation between Φ, A, λ * and C appears somewhat difficult to untangle in general so we shall defer detailed analysis of these equations to Section 6, where the WKB aproximation makes the task easier. Suffice it here simply to count equations to give confidence that the problem can be solved in principle -the four independent equations for these four unknowns are, in order of occurrence, (32), (70), (71) and (75). [Unless we set ν Ω = 0, ρ occurs through the u Rx in v, in which case we need to add (7), (8) and (60) to the list.] When solved, all variables should be known in terms of u, whose evolution can then be determined from the ξ-EL equation. A final remark: Taking the divergence of both sides the δA Euler-Lagrange equation (75) verifies that it propagates the δΦ Euler-Lagrange equation (64), ∇ · λ * = 0. That is, if ∇ · λ * = 0 initially, it will remain so even if λ * changes as the plasma evolves in time, and at each step in the iteration to converge C → 0. So the two Euler-Lagrange equations are consistent, though otherwise independent. Electric current We can also identify the electric current, j def = ∇ × B/µ 0 , so (75) can be written j = µ Ω µ 0 B + ν Ω µ 0 ω − ∇ × (v × λ * ) + ∂λ * ∂t = µ Ω µ 0 B + ν Ω µ 0 ∇ × u + B ρ × λ * − ∇ × (u × λ * ) + ∂λ * ∂t .(76) The first term on the RHS of (75) is the usual parallel electric current term of the linear-force-free (Beltrami) magnetic field model, the second term is a vorticitydriven current Yokoi (2013) term, while the last term is a new IOL constraint current which, (taking into account the EL equation ∇ · λ * = 0) maintains the divergence-free nature of j as required to maintain quasi-neutrality). Physical interpretation of estimated Lagrange multiplier In the special case µ Ω = ν Ω = 0, v = u, if we make the identification λ * = P (76) becomes identical with the representation of j in terms of the electrostatic dipole moment per unit volume or polarization vector P [see e.g. §1-10 of Panofsky & Phillips (1962)]. This representation is as given in eq. (12) of Calkin (1963) and eq. (1.2) of Webb & Anco (2017), specialized to the MHD case of a quasineutral moving medium, where ∇ · P = 0 [consistently with (64)]. Calkin (1963) goes on to derive an IMHD action principle in terms of Clebsch potentials, but these are not globally defined in a 3-D plasma with non-integrable magnetic fields. Our derivation shows the Clebsch representation is not needed to apply this polarization representation for j in an action principle if we apply the Lagrangian variational approach of Newcomb (1962). [See also Webb & Anco (2019) for a discussion of the equivalence of Lagrangian and Eulerian variational approaches.] Explicit Lagrangian variation of fluid element position This final Euler-Lagrange equation will in principle provide sufficient equations to solve for the unknowns. Equations of motion From(29) ∂ v C = I × B, thus Π = ρu + I × B · λ − µ P Ω C , = ρu + B × λ * .(77) The Euler-Lagrange equation obtained from setting δS A Ω /δx = 0 in (54) thus becomes ∂ t (ρu + B × λ * ) + ∇ · [v (ρu + B × λ * )] + (∇v) · (ρu + B × λ * ) = ρ∇ ∂L A Ω ∂ρ = ρ∇ (u · v − h Ω ) ,(78) by (62). Cancelling the ρ(∇v) · u occurring on both sides and rearranging, we have ρ∂ t u + ρv · ∇u − ρ (∇u) · v = −ρ∇h Ω − ∂ t (B × λ * ) − ∇ · [v (B × λ * )] − (∇v) · (B × λ * ) ,(79) where we used (13), ∂ t ρ + ∇ · (ρv) = 0, to cancel all derivatives of ρ. Thus, dividing both sides by ρ we have the compact Bernoulli-like form ∂ t u + ω × v = −∇h Ω − a λ ,(80) the residual acceleration term containing λ * being a λ def = ρ −1 [∂ t (B × λ * ) + ∇ · [v (B × λ * )] + (∇v) · (B × λ * )] = ∂ t w + v · ∇w + (∇v) · w ,(81) where w def = B × λ * ρ ,(82) again using ∂ t ρ + ∇ · (ρv) = 0. In the special case µ Ω = ν Ω = 0, v = u we can use the identification in Subsection 5.6.2 of λ * as the polarization field P to write w = B × P /ρ. We can then recognize (80) as the Eulerian equation of motion, eq. (23) of Calkin (1963), thus providing a physical interpretation of our equations of motion in terms of a Lagrange multiplier field. Check: Calkin's (23) can be written as ∂ t (u + w) + [∇ × (u + w)] × u = −∇ (h Ω + u · w) i.e. ∂ t u + ω × u = −∇h Ω − ∇ (u · w) − ∂ t w − (∇ × w) × u = −∇h Ω − a P , where a P def = ∂ t w + (∇ × w) × u + ∇ (u · w) =∂ t w + u · ∇w − (∇w) · u + (∇u) · w + (∇w) · u = ∂ t w + u · ∇w + (∇u) · w ≡ a λ Conservation form Now consider the conservation form (57) where, from (29), (55) and (35) Π + ∂L A− Ω ∂E × B = ρu + ∂L C Ω ∂v + ∂L C Ω ∂E × B = ρu + ∂C ∂v · λ * + ∂C ∂E · λ * × B (83) = ρu + (1−) B × λ * .(84) Starting with the coefficient of I in the tensor T, (58), and referring to (44), (45) and (46) we find L A− Ω − ρ ∂L A− Ω ∂ρ = − p γ − 1 − −γτ Ω ρ γ − 1 − B 2 2µ 0 + ν Ω u · B µ 0 + L C Ω = p − B 2 2µ 0 + ν Ω u · B µ 0 + λ · C − µ P Ω C 2 2 .(85) The penultimate term in T is − ∂L C Ω ∂E E = −λ * E(86) which consists of a symmmetric E E term and a non-symmetric E u×B term. The preceding term of T is, using (72)and (140), ∂L A− Ω ∂B × I × B = (−B + ν Ω u − µ 0 v × λ * ) × I × B µ 0 = B µ 0 (−B + ν Ω u − µ 0 v × λ * ) − I µ 0 −B 2 + ν Ω u · B − µ 0 v × λ * · B , and the remaining, first term is vΠ = u − ν Ω B µ 0 ρ (ρu + B × λ * ) . Thus, combining all terms, (57) becomes ∂ t (ρu) + ∇ · (T MHD + T Res ) = (∇λ) · C ,(87) where T MHD is the momentum transport plus stress tensor for both IMHD and RxMHD, Dewar et al. (2020), T MHD = ρuu + p + B 2 2µ 0 I − BB µ 0 ,(88) the terms in ν Ω that might have contributed to T MHD in the RxMHD case having cancelled. The new term T Res is the "internal" residual stress contribution arising when action-extremizing solutions are infeasible, i.e. when the IOL constraint is not satisfied exactly, T Res def = λ · C − µ P Ω C 2 2 − λ * · v × B I − Bv × λ * + vB × λ * − λ * E = λ * · C − λ * · u × B + µ P Ω C 2 2 I + Bλ * × u + uB × λ * + λ * u × B − λ * C .(89) (Interestingly, the ν Ω cancellation also occurred in deriving T Res when v was replaced by u − ν Ω B/µ 0 ρ.) The "external" residual force on the RHS of (87) obviously vanishes for feasible solutions. However it is not obvious that T Res vanishes when C = 0 as it involves the unknown converged Lagrange multiplier λ\| ∞ (= λ * \| ∞ as C\| ∞ = 0). However it is easy to verify that both the diagonal and off-diagonal terms of T Res not involving C explicitly do vanish if λ * is proportional to B pointwise, implying at least in this case T Res = 0 if and only if C = 0 (assuming µ P Ω = 0). Momentum and angular momentum conservation When a trial solution is IOL-infeasible, i.e. C = 0, T Res is not a symmetric tensor, indicating it imparts both an isotropic pressure force and a torque on the plasma, presumably tending to change u in such a way as to "bend" the flow toward conformity with the Ideal Ohm's Law. There is a cyclic symmetry in T Res among the three terms in λ * , u, B that indicate that the magnetic field is coupled to E in a similar fashion as u, and indeed we see from (76) that that there is a "dynamo" term depending on E in j that modifies B, by Ampère's Law. Linearized dynamics in the WKB approximation As indicated in the Introduction, the present paper is a step toward a multiregion RxMHD dynamics code in which the primary role of the relaxed fluid dynamics within an annular toroidal domain Ω is twofold a) to regularize IMHD by relaxing the topological constraint forbidding magnetic reconnection, so magnetic islands can form at resonances rather than singularities, and b) to transmit pressure disturbances across the thin layer of plasma between the two disjoint interfaces forming the boundary ∂Ω, thereby coupling the interfaces and endowing them with the plasma's inertia. This section derives, in the WKB approximation, dispersion relations for the waves that transmit these disturbances. Linearization Thus, as a simple first step toward understanding the dynamical implications of the RxMHD equations we linearize around a steady, (∂ t → 0), IOL-compliant C (0) = 0, λ (0) * = λ (0) solution of the Euler-Lagrange equations in a domain Ω with either fixed boundaries or with only low-amplitude, short-wavelength perturbations on ∂Ω. Thus, insert in these equations the ansatz u = u (0) + αu (1) + O(α 2 ), where α is the amplitude expansion parameter, and similarly for other perturbations except we use their potential representations for B (1) and E (1) as this is important for enforcing 3. The entropy, helicity and cross-helicity integrals are conserved at O(α), with therefore no perturbation in the Lagrange multipliers. Thus here we take τ Ω , µ Ω , and ν Ω as time-independent constants. Also, from here on we take the superscript (0) to be implicit, e.g. ρ means ρ (0) , u means u (0) , λ means λ (0) etc. While we assume the background equilibrium obeys the IOL, we do not assume the augmented-Lagrangian iteration for our perturbations is fully converged, so C (1) = 0 and our two successive Euler-Lagrange iterates are not equal, λ (1) * = λ (1) . Linearization of Lagrange multiplier determination Focusing first on the novel part of the calculation we list the linearizations of immediate relevance to the Augmented Lagrangian determination of the updated Lagrange multiplier field λ * . From (5) and (6), B (1) = ∇ × A (1) and E (1) = −∇Φ (1) − ∂ t A (1) , so (28) becomes C (1) = −∇Φ (1) − ∂ t A (1) + u × ∇ × A (1) + u (1) × B(90) with Φ (1) to be determined from (71), which used the Euler-Lagrange equation from the Φ variation in its derivation. This becomes ∇ 2 Φ (1) = ∇ · u (1) × B + u × ∇ × A (1) .(91) When C (1) is found, the updated Lagrange multiplier is determined from the linearization of (32), λ (1) * = λ (1) − µ P Ω C (1) . While one occurrence of A (1) in (91) has been eliminated by assuming Coulomb gauge, ∇ · A (1) = 0, it still arises in the ∇ × A (1) term arising from B (1) . Thus we also need the linearization of the δA Euler-Lagrange equation to give us A (1) . We use the modified Beltrami form (75) ∇ × ∇ × A (1) = µ Ω ∇ × A (1) + ν Ω ∇ × u (1) − µ 0 ∇ × v (1) × λ + µ 0 ∂ t λ (1) * − ∇ × v × λ (1) * ,(92) where v (1) = u (1) − ν Ω µ 0 ∇ × A (1) ρ − ρ (1) ρ B ρ ,(93) with ρ (1) to be determined from ∂ t ρ (1) + ∇ · (ρu (1) + ρ (1) u) = 0 ,(94) and u (1) to be treated as the one unknown in terms of which all other physical perturbations are to be expressed. Wave perturbations in WKB approximation Eikonal ansatz and natural basis vectors For short wavelength, high frequency velocity perturbations we use the eikonal ansatz u (1) = u(x, t) exp iϕ(x, t) ε ,(95) with similar notations for linear perturbations of other quantities, ε being the WKB (local plane-wave) expansion parameter. The instantaneous local values of wave vector and frequency as seen in the LAB frame are then defined as k def = ∇ϕ and ω(x, t) def = −∂ t ϕ. In the following development we shall also encounter two "shifted" frequencies: ( 1) ω u k def = ω − k · u(96) the Doppler-shifted frequency of the wave as seen in the local rest frame of a fluid element, velocity u(x, t), and (2) ω v k def = ω − k · v(97) the same as frequency (1) except with u replaced by the relative velocity v = u − u Rx ≡ u − ν Ω B/µ 0 ρ. Taking ϕ and equilibrium quantities to vary on O(1) spatial and temporal scales, ω, k, ∂ t u, ∇u, µ Ω , ν Ω etc. are O(1), but ∂ t u (1) , ∇u (1) etc. are large, O(αε −1 ), relative to u (1) = O(α)ε 0 , and similarly for spatio-temporal derivatives of ρ (1) and B (1) . In order for B (1) to be the same order as u (1) , the potentials Φ (1) andA (1) must be O(αε), so we write Φ (1) = ε i Φ(x, t) exp iϕ(x, t) ε and A (1) = ε i A(x, t) exp iϕ(x, t) ε ,(98) (5) and (6) giving then B = k × A and E = −k Φ + ω A. As in Dewar et al. (2020) our strategy is to express all perturturbations in terms of u, in order to find a 3 × 3 matrix eigenvalue equation whose roots give the dispersion relations for the three propagating wave branches. We shall also include a forcing term of the form similar to (95) in the equation of motion for u (1) so that this 3 × 3 matrix appears also as a response function, with the dispersion relations giving the location of its poles. We shall find it useful to expand vectors and dyadics in the orthonormal MHD-wave basis e 1 def = k ⊥ k ⊥ , e 2 def = B B ≡ e B and e 3 def = ≡ e 1 × e 2 ≡ k × B \|k × B\| ,(99)where k ⊥ def = P ⊥ · k, so k = k ⊥ e 1 + k e 2 (where k ⊥ def = \|k ⊥ \|, k def = k · e B ), u Rx = u Rx e 2 (where u Rx def = u Rx ), k × B A = k ⊥ c A e 3 , P ⊥ = e 1 e 1 + e 3 e 3 and I = e 1 e 1 + e 2 e 2 + e 3 e 3 . (There is of course a problem if k ⊥ = 0, but we are interested in low-frequency MHD waves around k = 0 where \|k ⊥ \|is maximal.) IOL Constraint in WKB approximation We now use (95) and (98) in the linearizations in Subsection 6.1.1, working to leading order in ε(for instance µ Ω B (1) will be dropped as higher order in ε than other terms in (92)). Then (90) becomes C = −k Φ + ω A + u × k × A + u × B = −k Φ − u · A + ω u k A + u × B ,(100) Also (91) becomes k 2 Φ = k · u × B + k · u × k × A , = k 2 u · A + k · u × B ,(101) assuming Coulomb gauge, k · A = 0. Inserting (101) in (100) gives C = I − kk k 2 · u × B + ω u k A(102) Next, (92) becomes k 2 A = −ν Ω k × u + µ 0 k × ( v × λ) + µ 0 ω λ * + k × v × λ * , = −ν Ω k × u − µ 0 k · v λ + µ 0 ω v k λ * ,(103) where we used v = u − u Rx ≡ u − ν Ω B/µ 0 ρ. Finally, (94) and (93) become ρ ρ = k · u ω u k ,(104)v = u + ν Ω µ 0 ρ k · u ω u k B − k × A(105) hence k · v = 1 + k · u Rx ω u k k · u . = ω v k ω u k k · u .(106) Substituting (106) in (103) gives k 2 A = −ν Ω k × u − µ 0 ω v k ω u k k · u λ + µ 0 ω v k λ * ,(107) which in (102) then gives C = I − kk k 2 · u × B − ν Ω ω u k k × u k 2 + µ 0 ω v k k 2 ω u k λ * − k · u λ .(108) Treating C for the moment as a known and solving for λ * we have λ * = k · u ω u k λ + ν Ω µ 0 k × u ω v k + k 2 I − kk × B · u + k 2 C µ 0 ω u k ω v k .(109) Normally we do not need to know the residual IOL error term C exactly, but to convince ourselves it can be made arbitrarily small by iteration, replace λ * with its explicit form from the linearization of (32), λ − µ P Ω C, and collect both C terms on the left: 1 + µ 0 ω u k ω v k µ P Ω k 2 C = − I − kk k 2 × B · u − ν Ω ω u k k × u k 2 + µ 0 ω v k k 2 ω u k λ − k · u λ ,(110) which confirms the implication in Subsection (23) that we have enough equations to determine C, and hence Φ, A and λ * , in terms of u.) Dividing both sides of (110) by the large penalty multiplier µ P Ω we see that C is smaller than the other terms and the previous iterate of C by an O 1/µ P Ω factor. Thus the sequence . . . , C n , C n+1 , C n+2 , . . . will converge exponentially toward 0, or super-exponentially if µ P Ω is increased appropriately at each step. Also λ * will converge to λ\| ∞ . However, this linearized calculation is sufficiently simple that we do not actually need to carry out the iteration as we can find λ\| ∞ analytically from 110 by setting its LHS to zero and solving for λ = λ\| ∞ . Or, if we want to investigate the hypothesis that terminating the iteration at finite n, so that C = 0, will regularize MHD we can prescribe C and use (109) to give λ * . To provide a continuous sequence of dynamical fluid models running between the unconstrained RxMHD perturbation dynamics of Dewar et al. (2020) to the converged, C = 0, present model we choose C = ε Rx C 0 ,(111) with the "relaxedness" parameter ε Rx running from 0 (IMHD, IOL-compliant) to 1 (RxMHD, may be IOL-infeasible). We shall later also have use of the complementary "ideality" parameter ε I def = 1 − ε Rx . Here C 0 is the IOL error for unconstrained, ε I = 0, ε Rx = 1, RxMHD perturbations, which we can find by setting λ = 0, µ P Ω = 0, and thus λ * = 0, in (108) to give C 0 = I − kk k 2 · u × B − ν Ω ω u k k × u k 2 − µ 0 λω v k k · u k 2 ,(112) which is a linear tensor function of the form C 0 = B C 0 (k, B) · u, where the factor B is taken out to make C 0 dimensionless. By inspection of (112), C 0 def = − 1 k 2 B µ 0 ω v k λk + ν Ω ω u k k × I + k 2 I − kk × B .(113) The linear tensor form of C 0 implies C, λ and λ * are of similar form, C = B C(k, B) · u, λ = Λ(k, B) · u, λ * = Λ * (k, B) · u ,(114) where from Λ = Λ * + µ P Ω B C, and from (109) and (114), Λ * = λ k ω u k + ν Ω µ 0 k × I ω v k + k 2 I − kk × B + k 2 BC µ 0 ω u k ω v k = ε I λ k ω u k + ν Ω µ 0 k × I ω v k + k 2 I − kk × B µ 0 ω u k ω v k ,(115) using C = ε Rx C 0 , 111. Short-wavelength dynamical RxMHD equations We now consider the linearized equation of motion with the forcing term mentioned in Subsection 6.2.1, a specific force (i.e. force/mass density) we denote as f (1) . Thus the linearized (80) with forcing term becomes ∂ t u (1) + ω (1) × v + ω × v (1) = −a (1) λ − ∇h (1) Ω + f (1) ,(116) where, from (63), h (1) Ω = u · u (1) + τ Ω ρ (1) ρ ,(117) from (81), a (1) λ =∂ t w (1) + v · ∇w (1) + (∇v) · w (1) + v (1) · ∇w + ∇v (1) · w,(118) and, from (82), w (1) = B (1) × λ ρ + B × λ (1) * ρ − ρ (1) B × λ ρ 2 .(119) Using the WKB representations in (98) and in (95), and analogous representations for ρ (1) , B (1) , w (1) , a (1) λ and f (1) in the linearizations above, and working to leading order in ε as before we have −ω u + (k × u) × v = −k u + τ Ω k ω u k · u − a λ + f ,(120) which we shall show can be written as D (ω, k) · u = − f(121) where, noting canceling of ku · u terms, D · u = ω u − k · u u − u Rx × (k × u) − a λ − τ Ω kk ω u k · u(122) From (118), a λ = − ω v k w + k v · w .(123) and, from (119), (104) and (107), ρ w = k × A × λ − ρ ρ B × λ + B × λ * = k k 2 × −ν Ω k × u − µ 0 ω v k ω u k k · u λ − k · u ω u k B × λ + µ 0 ω v k k 2 (λ · k I − kλ) + B × I · Λ * · u ,(124) where Λ * was given in (115). WKB RxMHD response matrix To simplify the calculation of the response we expand around a relaxed equilibrium with λ = 0, such as the axisymmetric tokamak equilibrium in Dewar et al. (2020), which has a steady flow field u that is the vector sum of an arbitrary rigid toroidal rotation carried by v and an axisymmetric magnetic-field-aligned flow u Rx proportional to ν Ω , which equilibrium was shown to satisfy the IOL without needing a Lagrange multiplier. In such a case w = 0 and (124) becomes w = B × Λ * · u/ρ. Then (123) becomes a λ = − ε I B ρ × ν Ω µ 0 k × I + k 2 I − kk × B µ 0 ω u k · u = − ε I u Rx × (k × u) − ε I B × k 2 I − kk × B µ 0 ρ ω u k · u .(125) In (121) the u Rx terms not involving ε Rx cancel, giving D = ω u k I + ε I B × k 2 I − kk × B µ 0 ρ ω u k − ε Rx ku Rx − k · u Rx I − τ Ω kk ω u k = ω u k + ε Rx k · u Rx I − ε Rx k u Rx + ε I c A × k 2 I − kk × c A ω u k − c 2 s kk ω u k = ω u k + ε Rx k · u Rx I − ε Rx k u Rx + ε I k ⊥ × c A k ⊥ × c A − k 2 c 2 A P ⊥ ω u k − c 2 s kk ω u k ,(126) where u Rx = ν Ω B/µ 0 ρ is defined in (7), c A def = B/ (µ 0 ρ) 1/2 is the Alfvén veloc- ity, c s = τ 1/2 Ω is the isothermal sound speed, and we have used (140) to write c A × I × c A = c A c A − c 2 A I ≡ −c 2 A P ⊥ . To represent D as a matrix we project onto the orthonormal basis 99, which can be written e 1 = k ⊥ /k ⊥ , e 2 = c A /c A and e 3 = k ⊥ × c A / (k ⊥ c A ) . We thus have D = ω u k + ε Rx k u Rx I − ε Rx u Rx k ⊥ e 1 + k e 2 e 2 − ε I c 2 A ω u k k 2 e 1 e 1 + k 2 e 3 e 3 − c 2 s ω u k k ⊥ e 1 + k e 2 k ⊥ e 1 + k e 2 ,(127) which can be represented as the block-diagonal matrix D = D MS 0 0 ω u k + ε Rx k u Rx − ε I k 2 c 2 A /ω u k ,(128) with the 1 × 1 Alfvén block on the lower right and the 2 × 2 magnetosonic block, D MS = ω u k + ε Rx k u Rx − ε I k 2 c 2 A + k 2 ⊥ c 2 s /ω u k −k ⊥ ε Rx u Rx + k c 2 s /ω u k −k k ⊥ c 2 s /ω u k ω u k − k 2 c 2 s /ω u k ,(129) upper left. Limiting cases Consider first the ideal, fully converged case C = 0 (ε I = 1, ε Rx = 0) and use (140) to write k 2 I − kk = −k × I × k so D = ω u k I − B × (k × I × k) × B µ 0 ρ ω u k − τ Ω kk ω u k = ω u k I − (kB − k · BI) · (Bk − k · BI) µ 0 ρ ω u k − τ Ω kk ω u k , which, apart from the definitions of D differing by a factor of ρ ω u k , agrees with the IMHD form, eq. (75), of Dewar et al. (2020). In the pure RxMHD case ε I = 0, ε Rx = 1, Apart from the definitions of D again differing by a factor of ρ ω u k , this agrees with the RxMHD form, eq. (88), of Dewar et al. (2020). D = ω u k I − ku Rx − k · u Rx I − τ Ω kk ω u k . Thus ε Rx parametrizes a continuous interpolation between RxMHD and IMHD. Dispersion relations -Alfvén branches Multiplying the first factor of the determinant det D = ω u k + ε Rx k u Rx − ε I k 2 c 2 A /ω u k det D MS(130) by ω u k gives the dispersion relation for the Alfvén-wave branch(es) as the quadratic equation (ω u k ) 2 + ε Rx k u Rx ω u k − (1 − ε Rx ) k 2 c 2 A = 0 .(131) The general solution of the quadratic equation is Qualitative analysis is more informative: As ε Rx → 0 the dispersion relations for the two branches approach the Doppler-shifted Alfvén-wave dispersion relations ω − k · u = ±k c A . Also, inspection shows that ω − k · u → 0 as k → 0 quite generally, and when \|ε Rx \| 1 the modification of the dispersion departure from the standard Alfvén-wave dispersion relation is essentially determined by the product ε Rx u Rx . Thus, when when \|ε Rx \| 1 and the parallel flow parameter is at most Alfvénic, u Rx /c A ≤ O(1), ε Rx and u Rx will have little effect on the Alfvén-wave branches. ω u k = k 2 −ε Rx u Rx ± 4ε I c 2 A + ε Rx u Rx 2 1/2(132) The plots in figure 2 give a visualization of the dependence of the dispersion relation on ε Rx . The figure is for a case where u Rx = 0, when (132) simplifies to ω u k = ± √ ε I k c A . (We call the + solution the principal branch.) The square root term √ ε I = √ 1 − ε Rx gives rise to a singular dependence on ε Rx at ε Rx = 1 but the vicinity of IMHD is regular. Dispersion relations -Magnetosonic branches The magnetosonic dispersion relations are obtained by setting det D MS = 0, where Figure 4: Showing transition of the fast-magnetosonic-branch dispersion relation: ω vs. k ; from IMHD (ε Rx = 0) to RxMHD (ε Rx = 1). Fixed parameters (in arb. units) are k = 1, u = 0, u Rx = 0, c s = 1 and c A = 5. (Colour online. The vertical ordering of the lines in the k > 0 half plane coincides with that of ε Rx in the legend.) det D MS = c 2 A ε I k 2 k 2 c 2 s − ω 2 − ω c 2 s k 2 − ω 2 (ε Rx k u Rx + ω) ω 2 . The solution of the quartic equation ω 2 det D MS = 0 is extremely complicated but the figures 3 and 4 give an overview of the ε Rx dependence. Again, the limit ε Rx → 1 is clearly singular in the slow magnetosonic case but not ε Rx → 0. This regularity around ideal MHD means our dispersion relation analysis is too crude to reveal the potential regularizing effect of softening the IOL constraint. Conclusion Invoking the augmented Lagrangian version of the penalty function method for constrained optimization, we have sketched out what we hope is a practical computational approach for iteratively solving the Relaxed MHD (RxMHD) Euler-Lagrange equations of Dewar et al. (2020)with added Ideal Ohm's Law (IOL) constraint terms. This method depends crucially on the existence of a Lagrange multiplier field to be found using the augmented Lagrangian iteration algorithm borrowed from finite-dimensional optimization theory. A formal proof of convergence in may in general be difficult, but a practical approach will be to test the algorithm by perturbing away from IOL-feasible relaxed equilibria in simple geometries. A suitable such starting point is the rigidly rotating axisymmetric tokamak equilibrium discussed by Dewar et al. (2020). In this paper have illustrated the construction of the Lagrange multiplier field for linearized wave perturbations in the short-wavelength WKB approximation. To find the constrained momentum equation we have used a little-known dyadic identity to derive a general conservation form. Substituting the constrained-RxMHD Lagrangian into this general form reveals residual terms in the stress tensor and a fictitious external force that should tend to zero uniformly in Ω if the constraint iteration converges so as to satisfy the IOL equality constraint. However in non-axisymmetric, three-dimensional (3-D) plasma confinement systems such as stellarators and real tokamaks with field errors and intentionally resonant magnetic perturbations, there is good physical reason to believe uniform pointwise convergence is impossible. In such cases the best we can hope for is convergence in an L 2 -norm, which will provide a weak-form regularization to cope with the singularities to which IMHD is prone in 3-D. This regularization should break the frozen-in flux condition of IMHD on small scales and allow interesting behaviour to be simulated without raising the order of the PDEs as adding resistivity does. Potential applications include reconnection events and the conjectured formation of equilibrium fractal magnetic and fluid flow patterns in 3-D systems. Other potential physical phenomena to investigate in 3-D systems include the linear normal mode spectrum, nonlinear saturation, bifurcations to oscillatory modes, and the effect of quasisymmetry [Nührenberg & Zille (1988 [Vanneste & Wirosoetisno (2008)]. Also, to improve the physical applicability of relaxed MHD it will be important to extend the handling of thermal relaxation beyond isotropic pressure. Relaxation parallel to the magnetic field is very reasonable physically but perpendicular relaxation has forced the use of discontinuous pressure profiles in the MRxMHD-based SPEC code described by Hudson et al. (2012). Thus it will be important to build on the work of Dennis et al. (2014a) to include an anisotropic pressure tensor in a weakly IOL-feasible model. N.B. An unabridged version of this paper with more detail on derivations of equations is available online as Supplementary Material at <link to be inserted by editors>. Appendices A A very brief history of relaxed MHD The term relaxation in the physical sciences generally connotes a process by which a system tends toward an equilibrium state: thermodynamic, chemical, electrodynamic, mechanical, or a combination of these. For example, in a closed, constant energy system initially out of thermodynamic equilibrium, relaxation occurs as the entropy increases toward a maximum. In an open system at a temperature above that of a surrounding heat bath, relaxation occurs as heat carries energy out of the system, so its thermal energy tends toward a minimum. In an open system with unbalanced mechanical forces, potential energy is converted into kinetic energy, which in turn is dissipated by friction into heat that is lost to the outside world, thus minimizing total energy, thermal and potential. This is the paradigm implicit in our use of the term "relaxation", the assumption that a relaxed state is defined by the minimum of a Hamiltonian. In plasma physics the first use of the term may have been in the paper by Chandrasekhar & Woltjer (1958), which proposes two variational principles other than maximizing entropy or minimizing energy: maximum energy for given mean-square current density and minimum dissipation for a given magnetic energy. The common element in these, and the minimum energy at constant magnetic helicity principle used by Woltjer (1958a) and Taylor Taylor (1974) is the derivation of a "linear-force-free" magnetic field obeying the Beltrami equation ∇ × B = µB, with µ constant, as the outcome. The Chandrasekhar and Woltjer work was in the context of plasma astrophysics, justifying the force-free assumption (where the force density in question is j × B) basically on the assumption the plasma has low β def = p/ B 2 /2µ 0 and no confining forces that are strong compared with gradients of magnetic pressure. In contrast Taylor considered a toroidal terrestrial plasma confined in a metal shell and driven by a strong induced current, creating a turbulent state from which the plasma relaxes. Taylor regards the relaxation mechanism as the breaking of the microscopic IMHD topological invariants leaving only the global magnetic helicity as conserved. Taylor (1974) is uncommital as to the exact details of this breaking of microscopic invariants and is content to use successful comparison with experiment of the predictions flowing from his derivation of the Beltrami equation as sufficient validation of his elegantly simple model, a general philosophy we also adopt. However in his later review, Taylor (1986) gives some more detail on the decay mechanism, citing some turbulence simulations and invokes turbulence scale length arguments to explain why it is energy that is minimized rather than magnetic helicity. Moffatt (2015) has recently critically reviewed the arguments of Taylor (1986) from a more modern perspective. Woltjer (1958b) pointed out there were other global IMHD invariants beyond magnetic helicity, in particular his eq. (2), the cross helicity involving both flow and magnetic field. Bhattacharjee & Dewar (1982) pointed that in an axisymmetric system an infinity of additional global invariants could be generated by taking moments of A · B with powers of a flux function, and used lower moments to generate more physical pressure and current profiles for tokamak equilibria than the very restricted profiles given by Taylor's relaxation principle. Hudson et al. (2012) developed multi-region relaxed MHD (MRxMHD), a generalization of single-region Taylor relaxation by inserting thin IMHD barrier interface tori to frustrate global Taylor relaxation. This generalization is appropriate to non-axisymmetric equilibria in stellarators and in tokamaks with symmetry-breaking perturbations, where magnetic field-line flow can be chaotic even without turbulence. This MRxMHD formulation is implemented in the now well-established Stepped-Pressure Equilibrium Code (SPEC). A relaxation approach for finding equilibria with flow by adding a constraint additional to conservation of magnetic helicity, conservation of cross helicity, was used by Finn and Antonsen Finn & Antonsen (1983) using an entropy-maximization relaxation principle [see also the contemporaneous paper by Hameiri Hameiri (1983)]. However, they show this leads to the same equations as energy minimization. Thus we take, as in IMHD, the entropy in Ω to be conserved and follow Taylor in defining relaxed states as energy minima. Pseudo-dynamical energy-descent relaxation processes that conserve topological invariants have been developed, Vallis et al. (1989); Vladimirov et al. (1999) but we stay within the framework of conservative classical mechanics by developing a dynamical formalism, RxMHD, that includes relaxed equilibria as stationary points of a relaxation Hamiltonian, with Lagrange multipliers to constrain chosen macroscopic invariants, but which also allows non-equilibrium motions, most easily done using Hamilton's action principle. Stability can also be examined by taking the second variation of the Hamiltonian, Vladimirov et al. (1999) but in this paper, as in Dewar et al. (2015) and Dewar et al. (2020)we deal only with first variations. However the SPEC code implements a Newton method for finding energy minima and saddle points by calculating a Hessian matrix, which is the second variation of the MRxMHD energy. Combined with a model kinetic energy obtained by loading all mass onto the interfaces between the relaxation regions. This has recently been used successfully by Kumar et al. (2021, Submitted 2021 for calculating the spectrum of some linear eigenmodes in a tokamak, but comparison between the model kinetic energy and our new dynamical relaxation theory is desirable for determining the domain of applicability of the mass loading model. B Some vector and dyadic identities In the body of this paper we have used the usual coordinate-free vector (and dyadic) calculus notations, but in this appendix we derive some identities that are more easily proved using elementary tensor notation. Assuming an arbitrary fixed orthonormal basis {e i }, i = 1, 2, 3 mod 3, a vector, a say, is represented as a = a i e i , the summation convention for contraction over repeated dummy indices being assumed throughout. Thus dot and cross products are represented as a · b = a i b i and a × b = e i ε i,j,k a j b k , respectively, where the alternating Levi-Civita tensor ε ijk is 1 or −1 according as {i, j, k} is an even or odd permutation of {1, 2, 3}, or 0 if it is neither (e.g. if there are repeated integers). Also the operations of grad and curl acting on scalar and vector functions f and f , respectively, are represented as ∇f = ∂f /∂x def = e i ∂ i f and ∇ × a def = e i ε ijk ∂ j a k , where ∂ i def = ∂/∂x i . We use parentheses to limit the scope of the rightward differentiation of such operators. NB Left-right ordering is more important in vector notation. E.g. the dyadics ab and ba are distinct, but a i b j = b j a i . First we derive three useful identities involving gradients with respect to B ≡ B i e i , and the unit vector parallel to B, e B (x) ≡ B(x)/B(x). (By "parallel to B" we mean locally tangent to the magnetic field line passing though any point x. Henceforth the dependence on x is implicit as these identities concern functions purely of B.): = ∂f ∂A + ∇ × ∂f ∂B + ∂ ∂t ∂f ∂E ,(134) where f is an arbitrary scalar-valued function of A, B = ∇ × A, and E = −∂ t A − ∇Φ from (6), A being an arbitrary vector field. Varying A δF =¨ ∂f ∂A · δA + ∂f ∂B · ∇ × δA − ∂f ∂E · ∂ t δA dV dt =¨ ∂f ∂A i δA i + ∂f ∂B i ε i,j,k ∂ j δA k − ∂f ∂E i ∂ t δA i dV dt =¨ ∂f ∂A i δA i − ε k,j,i ∂ j ∂f ∂B k δA i + ∂ t ∂f ∂E i δA i dV dt , i k & ibp =¨ ∂f ∂A i + ε i,j,k ∂ j ∂f ∂B k + ∂ t ∂f ∂E i δA i dV dt , ε k,j,i = −ε i,j,k =¨ ∂f ∂A + ∇ × ∂f ∂B + ∂ ∂t ∂f ∂E ·δA dV dt def =ˆδ F δA · δA dV dt , where stands for "have swapped dummy indices" and "ibp" stands for "have integrated by parts" (neglecting surface terms on the assumption that the supports of variations do not include the boundary). Lemma 3. Variational derivative of functional F above is δF δΦ = ∇ · ∂f ∂E ,(135)) · f = ∇ · [f ∇Φ] − (∇Φ)∇ · f ,(136) where f is an arbitrary vector field, e.g. ∂L/∂∇Φ. Derivation: Follows directly from fact ∇∇Φ is a symmetric dyadic, proved in first line below, (∇∇Φ) · f = e i (∂ i ∂ j Φ)f j = e i (∂ j ∂ i Φ)f j , ∂ i ∂ j = e i (∂ j ∂ i Φf j ) − e i (∂ i Φ)∂ j f j = ∇ · [f ∇Φ] − (∇Φ)∇ · f , where stands for "have commuted operators". Corollary 1. For E = −∇Φ − ∂ t A, (∇E) · f = ∇ · [f E] − E∇ · f − f × ∂ t B .(137) Derivation: Muliplying each side of (136) by −1, writing −∇Φ = E + ∂ t A and subtracting ∂ t A from both sides, the lemma (136) becomes (∇E) · f = ∇ · [f (E + ∂ t A)] − (E + ∂ t A)∇ · f − (∇∂ t A) · f , = ∇ · [f E] − E∇ · f + ∇ · [f ∂ t A] − (∂ t A) ∇ · f − (∇∂ t A) · f = ∇ · [f E] − E∇ · f + f · (∇∂ t A) − (∇∂ t A) · f = ∇ · [f E] − E∇ · f − f × (∇ × ∂ t A) = ∇ · [f E] − E∇ · f − f × ∂ t B . Lemma 5. For A an arbitrary vector field and B = ∇ × A, (∇B) · f = −∇ · [f × (∇A) T ] + (∇A) · ∇ × f ,(138) where f is an arbitrary vector field, e.g. ∂L/∂B. [N.B. For a more useful form see the corollary (139) below.] Derivation: Corollary 2. (∇∇ × A) · f = e i (∂ i ε j,k,l ∂ k A l )f j = e i (∂ k ε j,k,l ∂ i A l )f j , ∂ i ∂ k = e i ∂ k [(∂ i A l )ε j,k,l f j ] − e i (∂ i A l )ε j,k,l ∂ k f j = −∂ k [ε k, ( ∇B) · f = (∇ × f ) × B − ∇ · [f × I × B](139) Derivation: Because of the identity ∇ · (f × ∇A) − (∇ × f ) · ∇A = 0 (which is easily proven using the properties of the scalar product and the identity ∇ × ∇ = 0) we can add ∇ · (f × ∇A) − (∇A) T · (∇ × f ) to the RHS of (138) to antisymmetrize ∇A and thus to eliminate it in favour of ∇ × A = B: (∇B) · f = ∇ · f × ∇A − (∇A) T + ∇A − (∇A) T · ∇ × f = −∇ · [f × I × (∇ × A)] + (∇ × f ) × (∇ × A) the second term in the second line following from Alternatively, verify without using vector potential but assuming ∇ · B = 0: RHS = (∇ × f ) × B − ∇ · [f × e i e i × B] = (∇ × f ) × B − (∇ × f ) · e i e i × B + f · e j × ∇B × e j = −e m f ε j,i,k ε j,m,l ∂ k B l = −e m f i (δ i,m δ k,l − δ i,l δ k,m ) ∂ k B l = e k f i ∂ k B i − e i f i ∂ k B k = (∇B) · f − f ∇ · B = LHS , where we used the Levi-Civita tensor contraction result ε a,b,c ε a,j,k = δ b,j δ c,k − δ b,k δ c,j , where here δ is the Kronecker symbol. 4 Lemma 6. f × I × g = gf − f ·g I Derivation: (f × I × g) i,j = f k e i · e k × e l e l × e m · e j g m = f k ε i,k,l ε l,m,j g m = f k ε l,i,k ε l,m,j g m = f k (δ i,m δ k,j − δ i,j δ k,m ) g m = f j g i − f k g k δ i,j Verification: a · LHS = a · f × I × g = a × f · I × g = (a × f ) × g = (a · g) f − (f · g) a = a · RHS ∀ a . LHS · b = f × I · (g × b) = f × (g × b) = (f · b) g − (f · g) b = RHS · b ∀ b . Figure 1 : 1Ergodic partition of iterates of the standard map as depicted inFig. 11ofLevnajić & Mezić (2010). (Reprinted with permission from Chaos.) is the generalization ofNocedal & Wright (2006)'s finite set of equality constraint functions {c i } (as a 3-vector it is finite-dimensional but as a function of x it is infinite dimensional).For the purposes of the present paper H is the noncanonical version, H MHD nc , of the Hamiltonian, H MHD defined in (10) plus the global constraint terms described in the next subsection, 4.2. The ideal boundary conditions (b.c.s) are E + v × B = 0, n · v = 0, n · B = 0 on ∂Ω and Φ = const on each disjoint component of ∂Ω (think plates of a capacitor or electrodes of a vacuum tube). A general form of the equation of motion is provided by the ξ-EL equation (54), which agrees with (21) of Dewar et al. (2020) in the special case of their λ = [ρ], V = [0], and Λ = [1]. , (∇B) · f = (∇ × f ) × B − ∇ · [f × I × B] of Appendix B, with f = ∂LA− Ω /∂B and the δA-EL equation (52). Also the identity (137) (∇E)·f = ∇·[f E]−E∇·f −f ×∂ t B with f = ∂L A− Ω /∂E, and the δΦ-EL equation Figure 2 : 2Showing transition of the Alfvén-branch dispersion relation: ω vs. k ; IMHD (ε Rx = 0) to RxMHD (ε Rx = 1: We have used ε Rx = 0.99 for clarity as the ε Rx = 1 line coincides with the k axis). Fixed parameters (in arb. units) are k = 1, u = 0, u Rx = 0, c s = 1 and c A = 5. (Colour online. The vertical ordering of the lines in the k > 0 half plane coincides with that of ε Rx in the legend.) Figure 3 : 3Showing transition of the slow-magnetosonic-branch dispersion relation: ω vs. k ; from IMHD (ε Rx = 0) to RxMHD (ε Rx = 1). Fixed parameters (in arb. units) are k = 1, u = 0, u Rx = 0, c s = 1 and c A = 5. (Colour online. The vertical ordering of the lines in the k > 0 half plane coincides with that of ε Rx in the legend.) ); Burby et al. (2020); Rodriguez et al. (2020); Constantin et al. (2021)] on 3-D equilibria with flow Lemma 1 . 1The gradients of B, B and e B with respect to B are, in terms of the identity dyadic I def = i e i e i , the unit tangent vector e B , and P ⊥ def = I − e B e B , the projector onto the plane perpendicular to B, ∂B ∂B = I, ∂B ∂B = e B , and ∂e B ∂B = P ⊥ B . (133) Derivations: Using the notations ∂ B · ≡ e i ∂ Bi ≡ ∂ · /∂B, we have the obvious identity ∂ B B = I. Applying this first identity to B ≡ (B · B) 1/2 we find the second identity, ∂ B B = (2I · B) /2B = B/B = e B . The third identity follows from the first two: ∂ B (B/B) = I/B − Be B /B 2 = (I − e B e B ) /B. Lemma 2. Variational derivative of functional F [A, Φ] =˜f (A, B, E)dV dt is δF δA j,l f j (∇A l )] + (∇A l )ε l,k,j ∂ k f j , anticyclic perms. of j, k, l = −∇ · [f × (∇A) T ] + (∇A) · ∇ × f ,where (∇A) T is the transpose of the dyadic ∇A. f × ∇A − (∇A) T = f × e i e i · ∇A − (∇A) T = −f × e i e i × (∇ × A) of Dewar et al.(2020) 3 by subtracting ∇L A Ω from both sides, giving, after a little rearrange- ment, Derivation: Varying Φ neglecting surface term as above.Two useful identities, closely related to integration by parts, for deriving conservation forms of Euler-Lagrange equations for freely variable fields [members of the set denoted η inDewar et al. (2020)] are Lemma 4. For scalar fields, e.g. Φ (∇∇ΦδF =¨ − ∂f ∂E · ∇δΦ dV dt =¨ ∇ · ∂f ∂E δΦ dV dt def =ˆδ F δΦ δΦ dV dt , We use regularization in the physics sense -adjusting for incipient singular behaviour in a way that is consistent with physics on scales outside the strict domain of applicability of a mathematical model. This goes somewhat beyond the mathematical sense of adjusting a problem to avoid ill-posedness. The case u 2 = 0 may well occur in plasma containment devices so using (22) to make a decomposition of B in terms of u analogous to the reverse in (23) seems less useful. Unfortunately the seemingly general stress tensor (27) derived byDewar et al. (2020) was limited to scalar fields like Φ. Appendix B derives (138) to handle vector fields like A. This contraction result was obtained using the helpful tool at https://demonstrations. wolfram.com/ProductOfTwoLeviCivitaTensorsWithContractions/ . AcknowledgmentsWe gratefully acknowledge useful discussions with Naoki Sato on constraint options, Robert MacKay for suggesting the relevance of weak KAM theory to 3-D MHD equilibrium theory, Joshua Burby for discussions of an earlier version of this paper, Markus Hegland for a discussion of regularization, Lindon Roberts for references on infinite-dimensional augmented Lagrangian optimization methods and Matthew Hole for reading the manuscript. We also thank Zoran Levnajić and Igor Mezić for consenting to use of their visualizations of chaos in ourFig. 1. Finally we thank an anonymous referee for pointing out the interpretation of the IOL Lagrange multiplier as an electrostatic polarization field.The work of ZQ was supported by the Australian Research Council under grant DP170102606 and the Simons Foundation/SFARI (560651, AB). RLD and ZQ also acknowledge travel support from the Simons Foundation/SFARI (560651, AB). Energy principle with global invariants. 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[ "ON PARABOLIC SUBGROUPS OF ARTIN GROUPS", "ON PARABOLIC SUBGROUPS OF ARTIN GROUPS" ]	[ "Philip Möller ", "Luis Paris And ", "Olga Varghese " ]	[]	[]	Given an Artin group A Γ , a common strategy in the study of A Γ is the reduction to parabolic subgroups whose defining graphs have small diameter, i.e. showing that A Γ has a specific property if and only if all "small" parabolic subgroups of A Γ have this property. Since "small" parabolic subgroups are the puzzle pieces of A Γ one needs to study their behavior, in particular their intersections. The conjecture we address here says that the class of parabolic subgroups of A Γ is closed under intersection. Under the assumption that intersections of parabolic subgroups in complete Artin groups are parabolic, we show that the intersection of a complete parabolic subgroup with an arbitrary parabolic subgroup is parabolic. Further, we connect the intersection behavior of complete parabolic subgroups of A Γ to fixed point properties and to automatic continuity of A Γ using Bass-Serre theory and a generalization of the Deligne complex.	10.1006/jabr.1997.7098	[ "https://arxiv.org/pdf/2201.13044v1.pdf" ]	122,061,209	2201.13044	bfde51705437869f47b05d220cb55d02df99b32c	ON PARABOLIC SUBGROUPS OF ARTIN GROUPS Philip Möller Luis Paris And Olga Varghese ON PARABOLIC SUBGROUPS OF ARTIN GROUPS arXiv:2201.13044v1 [math.GR] 31 Jan 2022Parabolic subgroups of Artin groupsfixed point propertiesautomatic continuitylocally compact Hausdorff groupsCAT(0) cube complexes 2010 Mathematics Subject Classification Primary: 20F36; Secondary: 20F6522D05 Given an Artin group A Γ , a common strategy in the study of A Γ is the reduction to parabolic subgroups whose defining graphs have small diameter, i.e. showing that A Γ has a specific property if and only if all "small" parabolic subgroups of A Γ have this property. Since "small" parabolic subgroups are the puzzle pieces of A Γ one needs to study their behavior, in particular their intersections. The conjecture we address here says that the class of parabolic subgroups of A Γ is closed under intersection. Under the assumption that intersections of parabolic subgroups in complete Artin groups are parabolic, we show that the intersection of a complete parabolic subgroup with an arbitrary parabolic subgroup is parabolic. Further, we connect the intersection behavior of complete parabolic subgroups of A Γ to fixed point properties and to automatic continuity of A Γ using Bass-Serre theory and a generalization of the Deligne complex. Introduction One class of groups that is studied from algebraic, geometric and combinatoric perspective is the class consisting of Artin groups (also known as Artin-Tits groups). Given a finite simplicial graph Γ with the vertex set V (Γ), the edge set E(Γ) and with an edge-labeling m : E(Γ) → {2, 3, 4, . . .}, the associated Artin group A Γ is defined as 1.1. Intersections of parabolic subgroups. Given a subset X ⊂ V (Γ) of the vertex set, we write A X for the subgroup generated by X. It was proven by van der Lek in [Vdl83] that A X is canonically isomorphic to the Artin group A X where we denote by X the induced subgraph of X in Γ. A group of the form A X is called a standard parabolic subgroup of A Γ , and a subgroup conjugate to a standard parabolic subgroup is simply called a parabolic subgroup. We see the parabolic subgroups of an Artin group as puzzle pieces of the whole group and we are in particular interested in their intersection behavior. It was proven by van der Lek in [Vdl83] that the class of standard parabolic subgroups is closed under intersection and it is conjectured that the same result holds for the class consisting of all parabolic subgroups. Here A Γ := V (Γ) \| vwv . . . Date: February 1, 2022. The first author is funded by a stipend of the Studienstiftung des deutschen Volkes and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy EXC 2044-390685587, Mathematics Münster: Dynamics-Geometry-Structure. This work is part of the PhD project of the first author. The second author is supported by the French project "AlMaRe" (ANR-19-CE40-0001-01) of the ANR. The third author is supported by DFG grant VA 1397/2-1. our focus is mainly on parabolic subgroups where the diameter of the defining graph is small. We say that X ⊂ V (Γ) is free of infinity if {v, w} ∈ E(Γ) for all v, w ∈ X, v = w. In this case the subgroup A X is called complete standard parabolic subgroup. A subgroup conjugate to a complete standard parabolic subgroup is called a complete parabolic subgroup. Usually, it is quite hard to prove that an Artin group A Γ has a specific property, but it is sometimes possible to reduce a conjecture about A Γ to complete standard parabolic subgroups of A Γ . The following reduction principle was formulated by Godelle and the second author in [GP12A]. Reduction principle (RP). Let P be a property of a group and let A Γ be an Artin group. If all complete standard parabolic subgroups of A Γ have property P, then A Γ has property P. Here we are interested in intersections of parabolic subgroups of A Γ and our aim is to reduce the intersection conjecture of parabolic subgroups to complete standard parabolic subgroups of A Γ . Hence we are interested in the following properties of A Γ : • Property (Int): For each free of infinity subset Y ⊂ V (Γ) and for all parabolic subgroups P 1 , P 2 of A Y , the intersection P 1 ∩ P 2 is a parabolic subgroup. • Property (Int+): For all complete parabolic subgroups P 1 , P 2 of A Γ , the intersection P 1 ∩ P 2 is a parabolic subgroup. • Property (Int++): For all parabolic subgroups P 1 , P 2 of A Γ , the intersection P 1 ∩ P 2 is a parabolic subgroup. Unfortunately, we cannot prove that property (Int) implies property (Int++). However, we can prove that property (Int) implies property (Int+-), which is a property between properties (Int+) and (Int++), and which is defined as follows. • Property (Int+-): For each complete parabolic subgroup P 1 and for each parabolic subgroup P 2 of A Γ , the intersection P 1 ∩ P 2 is a parabolic subgroup. It is easily checked that property (Int++) implies property (Int+) and that property (Int+) implies property (Int). We think that all these properties are actually equivalent and, even more, that they always hold. We show: Theorem 1.1. Let A Γ be an Artin group. If A Γ has property (Int), then A Γ has property (Int+-). The crucial ingredient of the proof of Theorem 1.1 is Bass-Serre theory. If Γ is not complete, then A Γ is an amalgam of smaller standard parabolic subgroups and we use the action on the corresponding Bass-Serre tree to show the result of the theorem. For each Artin group A Γ there is an associated Coxeter group W Γ . It is obtained by adding the relations v 2 = 1 for all v ∈ V (Γ). Hence, the Coxeter group W Γ associated to A Γ is given by the following presentation W Γ := V (Γ) \| v 2 = 1, (vw) m({v,w}) = 1 for all v ∈ V (Γ), {v, w} ∈ E(Γ) . An Artin group is of spherical type if the associated Coxeter group is finite and an Artin group is of FC-type if all complete standard parabolic subgroups are of spherical type. It was proven by Cumplido et al. in [CGGW19] that intersections of parabolic subgroups in a finite type Artin group are parabolic, therefore as an immediately corollary we obtain the following result. Corollary 1.2. Let A Γ be an Artin group of FC-type and P 1 , P 2 be two parabolic subgroups. If P 1 is complete, then P 1 ∩ P 2 is parabolic. This is a generalisation of Theorem 3.1 in [Mor21], which states that the intersection of two complete parabolic subgroups of an Artin group of FC-type is parabolic. 1.2. Automatic continuity. In a remarkable article [Dud61] Dudley was interested in the relation between locally compact Hausdorff groups and free (abelian) groups. Using a special length function on the target groups he showed that any group homomorphism from a locally compact Hausdorff group into a free (abelian) group is continuous. Inspired by this result Conner and Corson defined the notion of lcH-slenderness [CC19]. A discrete group G is called lcH-slender if any group homomorphism from a locally compact Hausdorff group into G is continuous. Our focus here is on continuity of group homomorphisms from locally compact Hausdorff groups into Artin groups. Many types of Artin groups are known to be lcH-slender such as right-angled Artin groups [KV19], [CK20], [MV20], Artin groups of spherical type [KV19] and more generally Artin groups of FC-type [KMV21]. We conjecture: Conjecture. All Artin groups are lcH-slender. Since automatic continuity of group homomorphisms from locally compact Hausdorff groups into "geometric" groups and fixed point properties of these geometric groups are strongly connected, see [MV20], we are interested in fixed point properties of subgroups of Artin groups, in particular in properties FA ′ and FC ′ . Recall, a group G is called an FA ′ -group if any simplicial action of G on a tree without inversion is locally elliptic, i.e. any element in G acts as an elliptic isometry, that means every isometry has a fixed point. Finite groups are special cases of groups having property FA ′ but there are also many examples of infinite groups having this property, for instance divisible and compact groups [CM11]. We conjecture: Conjecture. All Artin groups do not have non-trivial FA ′ -subgroups. We define a class A of Artin groups as follows: an Artin group A Γ is contained in the class A iff A Γ has property (Int++). Examples of Artin groups in the class A are right-angled Artin groups [DKR07], Artin groups of spherical type [CGGW19], and large-type Artin groups [CMV21]. We reduce the above conjecture for the class A to complete Artin groups. Proposition 1.3. Let A Γ be an Artin group in the class A and let H ⊂ A Γ be a subgroup. If H is an FA ′ -group, then H is contained in a complete parabolic subgroup of A Γ . Since a complete right-angled Artin group is isomorphic to a free abelian group and free abelian groups do not have non-trivial FA ′ -subgroups we obtain: Corollary 1.4. Right-angled Artin groups do not have non-trivial FA ′ -subgroups. The proof of Proposition 1.3 is of geometric nature. It is known that if A Γ is not complete, then A Γ in an amalgam of non-trivial parabolic subgroups A Γ 1 * A Γ 3 A Γ 2 and this group acts on the Bass-Serre tree corresponding to this splitting. We show that the subgroup H has a global fixed vertex and therefore it is contained in a conjugate of one of the factors in the amalgam. We proceed to decompose the factors in the amalgam until the defining graphs of these subgroups are complete. The main tool for showing that H has a global fixed vertex is the following algebraic result concerning parabolic subgroups of Artin groups. Note that, as in many other cases, the corresponding result for Coxeter groups is known to be true (see e.g. [Qi07]). Proposition 1.5. Let A Γ be an Artin group and gA Ω g −1 , hA ∆ h −1 be two parabolic subgroups such that gA Ω g −1 ⊂ hA ∆ h −1 . Then the cardinalities of V (Ω) and V (∆) satisfy \|V (Ω)\| ≤ \|V (∆)\| and, if \|V (Ω)\| = \|V (∆)\|, then gA Ω g −1 = hA ∆ h −1 . As immediate corollary we have: Corollary 1.6. Let A Γ be an Artin group. If A Γ is in the class A, then an arbitrary intersection of parabolic subgroups is a parabolic subgroup. In particular, for a subset B ⊂ A Γ there exists a unique minimal (with respect to inclusion) parabolic subgroup containing B. By definition, a locally compact Hausdorff group L is called almost connected if the quotient L/L • , where L • is the connected component of L, is compact. Using the fact that any almost connected locally compact Hausdorff group has property FA ′ [Alp82] we show: Theorem 1.7. Let A Γ be an Artin group in the class A. (1) Let ψ : L → A Γ be a group homomorphism from a locally compact Hausdorff group L into A Γ . If L is almost connected, then ψ(L) is contained in a complete parabolic subgroup of A Γ . (2) If all complete standard parabolic subgroups of A Γ are lcH-slender, then A Γ is lcHslender. Associated to an Artin group A Γ is a generalization of the Deligne complex, the so called clique-cube complex C Γ , whose vertices are cosets of complete standard parabolic subgroups of A Γ . We describe the construction of this cube complex and some important properties of it in Section 4. The group A Γ acts on C Γ via left-multiplication and preserves the cubical structure of C Γ . We use this action to show under "weaker" assumptions on parabolic subgroups of an Artin group than on Artin groups in the class A that principle RP holds for the property of being lcH-slender. Since the puzzle pieces of the complex C Γ are cosets of complete standard parabolic subgroups of A Γ and hence the stabilizers of these vertices are complete parabolic subgroups, this proof relies heavily on their "good" behavior. A generalization of the fixed point property FA ′ is the property FC ′ . A group G has property FC ′ if every cellular action of G on a finite dimensional CAT(0) cube complex is locally elliptic. Examples of FC ′ -groups are finite, divisible and compact groups, see [CM11]. It is known that torsion free CAT(0) cubical groups do not have non-trivial FC ′ -subgroups. Hence, any FC ′ -subgroup of a right-angled Artin group is trivial. We conjecture: Conjecture. All Artin groups do not have non-trivial FC ′ -subgroups. We define a class B of Artin groups as follows: A Γ is in the class B iff A Γ has property (Int). We show: Proposition 1.8. Let A Γ be an Artin group in the class B. Let H ⊂ A Γ be a subgroup. If H is a FC ′ -group, then H is contained in a complete parabolic subgroup of A Γ . Using the result of the Main Theorem in [MV20] we reduce the assumptions regarding parabolic subgroups in Theorem 1.7 to complete parabolic subgroups. Theorem 1.9. Let A Γ be an Artin group in the class B. (1) Let ψ : L → A Γ be a group homomorphism from a locally compact Hausdorff group L into A Γ . If L is almost connected, then ψ(L) is contained in a parabolic complete subgroup of A Γ . (2) If all complete standard parabolic subgroups of A Γ are lcH-slender, then A Γ is lcHslender. In particular principle RP holds for Artin groups in the class B for the property of being lcH-slender. Groups associated to edge-labeled graphs In this section we review the basics of simplicial graphs, Coxeter groups and Artin groups, which are relevant for our applications. 2.1. Graphs. We will be working with non-empty simplicial graphs. The basic, standard definitions can be found in [Die17]. Since the definition of the star and the link of a vertex sometimes vary we recall their definitions. Given a graph Γ = (V, E) and a vertex v ∈ V we define two subgraphs of Γ, the star of v, denoted by st(v) and the link of v denoted by lk (v) in the following way: The star of v is the subgraph generated by all vertices connected to v and v, i.e. the subgraph induced by {w ∈ V \|w = v or {v, w} ∈ E}. We obtain the link of v from the star of v by removing the vertex v and all edges that have v as an element. Recall, if X ⊆ V is a subset of the vertex set, then the subgraph generated or induced by X, denoted by X , is defined as the graph (X, F ), where {v, w} ∈ F if and only if {v, w} ∈ E. v w st(v) lk(w) We say a graph Γ = (V, E) is complete if every pair of vertices v, w ∈ V , v = w is connected by an edge, that is {v, w} ∈ E. We often denote the vertex set of Γ by V (Γ) and the edge set by E(Γ). Coxeter and Artin groups. Let Γ be a finite simplicial graph with an edge-labeling m : E(Γ) → N ≥2 . (1) The Coxeter group W Γ is defined as W Γ := V (Γ) \| v 2 , (vw) m({v,w}) for all v ∈ V (Γ) and whenever {v, w} ∈ E(Γ) . (2) The Artin group A Γ is defined as A Γ := V (Γ) \| vwv . . . m({v,w})−letters = wvw . . . m({v,w})−letters whenever {v, w} ∈ E(Γ) . In particular, W Γ is the quotient of A Γ by the subgroup normally generated by the set {v 2 \| v ∈ V (Γ)}. The epimorphism θ : A Γ → W Γ induced by v → v is called natural projection and the kernel of θ, denoted by CA Γ , is called colored Artin group. Given a subset X ⊂ V (Γ) of the vertex set, we write A X resp. W X for the subgroup generated by X, and we set CA X = CA Γ ∩ A X . The group A X is called standard parabolic subgroup of A Γ resp. W X is called standard parabolic subgroup of W Γ . Proposition 2.1. Let Γ be a finite simplicial graph with edge-labeling m : E(Γ) → {2, 3, . . .} and A Γ resp. W Γ be the associated Artin resp. Coxeter group. Let X be a subset of V (Γ). (1) The subgroup W X is canonically isomorphic to W X [Bou68]. (2) The subgroup A X is canonically isomorphic to A X [Vdl83]. As a consequence we get: Corollary 2.2. Let Γ be a finite simplicial graph with edge-labeling m : E(Γ) → {2, 3, . . .}. Let X be a subset of V (Γ). Then the subgroup CA X of CA Γ is canonically isomorphic to CA X . For colored Artin groups there exists a very useful tool to project to standard parabolic subgroups: Proposition 2.3 ([GP12B] ). Let A Γ denote an Artin group and X ⊂ V (Γ) a subset. Then CA X is a retract of CA Γ in the sense that there is a homomorphism π X : CA Γ → CA X satisfying π X (a) = a for all a ∈ CA X . We say that A X resp. W X is a complete standard parabolic subgroup if the subgraph X is complete. From our point of view the puzzle pieces of a general Artin group are complete standard parabolic subgroups, in the sense that using amalgamation one can decompose any Artin group in complete standard parabolic subgroups. Lemma 2.4. Let A Γ be an Artin group and Γ 1 , Γ 2 two induced subgraphs of Γ. If Γ 1 ∪ Γ 2 = Γ, then A Γ ∼ = A Γ 1 * A Γ 1 ∩Γ 2 A Γ 2 . The proof of this lemma follows easily by analyzing the presentation of A Γ and the canonical presentation of the amalgam. In particular, if there exist two vertices v, w ∈ V (Γ) such that {v, w} / ∈ E(Γ), then (1) A Γ ∼ = A st(v) * A lk(v) A V −{v} , (2) A Γ ∼ = A V −{v} * A V −{v,w} A V −{w} . Before we proceed to introduce further properties of Coxeter and Artin groups we discuss one example of a decomposition into an amalgam. Let Γ be as follows: v w x y 2 3 4 5 The associated Artin group is given by the presentation A Γ = v, w, x, y \| vw = wv, wxw = xwx, xyxy = yxyx, yvyvy = vyvyv . The vertices v and x in Γ are not connected by an edge, hence A Γ ∼ = A st(v) * A lk(v) A V −{v} , where A st(v) = v, w, y \| vw = wv, vyvyv = yvyvy , A lk(v) = w, y , A V −{v} = w, x, y \| wxw = xwx, xyxy = yxyx . We continue to decompose the amalgamated parts until the special subgroups are complete. We obtain A Γ ∼ = ( v, w \| vw = wv * v v, y \| vyvyv = yvyvy ) * w,y ( w, x \| wxw = xwx * x x, y \| xyxy = yxyx ) . Hence Lemma 2.4 gives us an algebraic tool to reduce some questions about Artin groups to complete standard parabolic subgroups. 2.3. Parabolic subgroups of Coxeter and Artin groups. Coxeter groups are fundamental, well understood objects in group theory, but there are many open questions concerning Artin groups. It is conjectured that most properties of Coxeter groups carry over to Artin groups. Remember that the conjugates of standard parabolic subgroups in A Γ resp. W Γ are called parabolic subgroups. Proposition 2.5. [Qi07, proof of Lemma 3.3] Let W Γ be a Coxeter group and gW Ω g −1 , hW ∆ h −1 be two parabolic subgroups such that gW Ω g −1 ⊂ hW ∆ h −1 . Then the cardinalities of V (Ω) and V (∆) satisfy \|V (Ω)\| ≤ \|V (∆)\| and, if \|V (Ω)\| = \|V (∆)\|, then gW Ω g −1 = hW ∆ h −1 . We now move to Proposition 1.5 from the introduction, namely the corresponding version of the above proposition for Artin groups. Proposition 2.6. Let A Γ be an Artin group and gA Ω g −1 , hA ∆ h −1 be two parabolic subgroups such that gA Ω g −1 ⊂ hA ∆ h −1 . Then \|V (Ω)\| ≤ \|V (∆)\| and, if \|V (Ω)\| = \|V (∆)\|, then gA Ω g −1 = hA ∆ h −1 . The remaining of the subsection is dedicated to the proof of the above proposition. First, we need to understand how to lift an inclusion of the form gW Ω g −1 ⊂ W ∆ to the corresponding Artin group (see Lemma 2.8). For that we will use the following classical result on Coxeter groups. If W Γ is a Coxeter group, then we denote by lg : W Γ → N the word length with respect to V (Γ). For more information on the length function of Coxeter groups see [Hum90]. Proposition 2.7. [Bou68] Let W Γ be a Coxeter group and X, Y subsets of V (Γ). Let g ∈ W Γ . There exists a unique element g 0 in the double coset W X · g · W Y of minimal length, and each element g ′ ∈ W X · g · W Y can be written in the form g ′ = h 1 g 0 h 2 with h 1 ∈ W X , h 2 ∈ W Y and lg(g ′ ) = lg(h 1 ) + lg(g 0 ) + lg(h 2 ). Moreover, lg(g 0 h) = lg(g 0 ) + lg(h) for all h ∈ W Y and lg(hg 0 ) = lg(h) + lg(g 0 ) for all h ∈ W X . Let Γ be a finite simplicial graph with edge-labeling m : E(Γ) → {2, 3, . . .}. We consider a set-section ι : W Γ → A Γ of the natural projection θ : A Γ → W Γ which is defined as follows. Let g ∈ W Γ . Recall that an expression g = v 1 · · · v l over V (Γ) is called reduced if lg(g) = l. Choose a reduced expression g = v 1 · · · v l of g and define ι(g) to be the element of A Γ represented by the same word v 1 · · · v l . By Tits [Tits69] this definition does not depend on the choice of the reduced expression. Note that ι is not a group homomorphism, but, if g, h ∈ W Γ are such that lg(gh) = lg(g) + lg(h), then ι(gh) = ι(g) ι(h). Lemma 2.8. Let Γ be a finite simplicial graph with edge-labeling m : E(Γ) → {2, 3, . . .}. Let X, Y be two subsets of V (Γ) and g ∈ W Γ such that gW X g −1 ⊂ W Y . Then ι(g)A X ι(g) −1 ⊂ A Y , and, if \|X\| = \|Y \|, then ι(g)A X ι(g) −1 = A Y . Proof. By Proposition 2.7, we can write g in the form g = h 1 g 0 h 2 , where h 1 ∈ W Y , h 2 ∈ W X , g 0 is the element of minimal length in the double coset W Y · g · W X , and lg(g) = lg(h 1 ) + lg(g 0 ) + lg(h 2 ). Then ι(g) = ι(h 1 )ι(g 0 )ι(h 2 ), and, since h 1 ∈ W Y and h 2 ∈ W X , we also obtain g 0 W X g −1 0 ⊂ W Y . Let v ∈ X. There exists f v ∈ W Y such that g 0 v = f v g 0 . By Proposition 2.7, lg(g 0 ) + 1 = lg(g 0 v) = lg(f v g 0 ) = lg(f v ) + lg(g 0 ), hence lg(f v ) = 1, that is, f v ∈ Y . Moreover, by the definition itself of ι, ι(g 0 )v = ι(g 0 v) = ι(f v g 0 ) = f v ι(g 0 ), hence ι(g 0 )vι(g 0 ) −1 = f v . So, ι(g 0 )Xι(g 0 ) −1 ⊂ Y . This implies that, on the one hand, ι(g 0 )A X ι(g 0 ) −1 ⊂ A Y , and, on the other hand, if \|X\| = \|Y \|, then ι(g 0 )Xι(g 0 ) −1 = Y and therefore ι(g 0 )A X ι(g 0 ) −1 = A Y . So, since ι(h 1 ) ∈ A Y and ι(h 2 ) ∈ A X , ι(g)A X ι(g) −1 = ι(h 1 )ι(g 0 )ι(h 2 )A X ι(h 2 ) −1 ι(g 0 ) −1 ι(h 1 ) −1 = ι(h 1 )ι(g 0 )A X ι(g 0 ) −1 ι(h 1 ) −1 ⊂ ι(h 1 )A Y ι(h 1 ) −1 = A Y , and we have equality if \|X\| = \|Y \|. With this tool we can now prove the main proposition of this section. Proof of Proposition 2.6. Recall that θ : A Γ → W Γ denotes the natural projection to the corresponding Coxeter group. By applying θ we have θ(g)W Ω θ(g) −1 ⊂ θ(h)W ∆ θ(h) −1 , hence, by Proposition 2.5, \|V (Ω)\| ≤ \|V (∆)\|. Now, we assume that \|V (Ω)\| = \|V (∆)\| and we prove that gA Ω g −1 = hA ∆ h −1 . Without loss of generality we can assume that h = 1, else we conjugate both sides with h −1 and replace g by h −1 g. Setḡ = θ(g). We haveḡW Ωḡ −1 ⊂ W ∆ and \|V (Ω)\| = \|V (∆)\|, hence, by Lemma 2.8, ι(ḡ)A Ω ι(ḡ) −1 = A ∆ . Thus, we obtain gA Ω g −1 ⊂ ι(ḡ)A Ω ι(ḡ) −1 and after conjugating with ι(ḡ) −1 we obtain kA Ω k −1 ⊂ A Ω , where k = ι(ḡ) −1 g. Note that θ(k) = 1, that is, k ∈ CA Γ , hence we can apply the retraction map π Ω : CA Γ → CA Ω to k. We have the following commutative diagram: CA Ω / / A Ω / / W Ω CA Ω / / A Ω / / W Ω where the vertical arrow W Ω → W Ω is the identity and the vertical arrows CA Ω → CA Ω and A Ω → A Ω are conjugations by k. Using the retraction map π Ω : CA Γ → CA Ω we show that the left vertical arrow is an isomorphism. For a ∈ CA Ω we define a ′ := kπ Ω (k −1 )aπ Ω (k)k −1 . Then a ′ ∈ kCA Ω k −1 ⊂ CA Ω , and a ′ = π Ω (a ′ ) = a. Hence, the arrow is surjective. Note that the arrow is also injective, since the map is a conjugation. Since we know that the arrows CA Ω → CA Ω and W Ω → W Ω are isomorphisms, by applying the five lemma we get that the arrow A Ω → A Ω is an isomorphism as well, which means that kA Ω k −1 = A Ω . By conjugating with ι(ḡ) we conclude that gA Ω g −1 = A ∆ . 2.4. Parabolic closure. Let A Γ be an Artin group. Suppose that a subset B ⊂ A Γ is contained in a unique minimal parabolic subgroup of A Γ (minimal with respect to ⊂). Then this parabolic subgroup is called the parabolic closure of B and is denoted by P C Γ (B). In the case that the intersection of two parabolic subgroups is parabolic, we show that the parabolic closure exists. Proposition 2.9. Let A Γ be an Artin group. If A Γ has property (Int++), then an arbitrary intersection of parabolic subgroups is a parabolic subgroup. In particular, for a subset B ⊂ A Γ there exists P C Γ (B). Proof. Let N be a set consisting of parabolic subgroups of A Γ . Our goal is to show that N is parabolic. We write N using indices in an index set I as N = i∈I {P i } where each P i is a parabolic subgroup of A Γ . We construct a chain: P i 1 ⊃ P i 1 ∩ P i 2 ⊃ P i 1 ∩ P i 2 ∩ P i 3 ⊃ ... We show that this chain stabilizes: Each proper inclusion in the chain is of the form gA ∆ 1 g −1 hA ∆ 2 h −1 since finite intersection of parabolic subgroups are assumed to be parabolic. Due to Proposition 2.6 the cardinality of the vertices in the defining graphs has to strictly reduce at each step. Since Γ is finite, this can only happen finitely many times. Thus there exists n ∈ N such that N = P i 1 ∩ . . . ∩ P in and by assumption this finite intersection is parabolic. For the in particular statement, we consider the set M = {P ⊂ A Γ \|PQ j 1 ⊃ Q j 1 ∩ Q j 2 ⊃ Q j 1 ∩ Q j 2 ∩ Q j 3 ⊃ ... This chain stabilizes by the above proof. We define R = j∈J Q j which is a parabolic subgroup since the chain stabilizes. That R is minimal is obvious from its definition. Note that R is also unique, more precisely: assume that there exists another minimal parabolic subgroup R ′ with B ⊂ R ′ . By assumption R ∩ R ′ is a parabolic subgroup containing B. Since R and R ′ are minimal, we have R ∩ R ′ = R and R ∩ R ′ = R ′ , hence R = R ′ . In particular, the parabolic closure of a subset B ⊂ A Γ is the parabolic subgroup aA Γ ′ a −1 containing B for \|V (Γ ′ )\| minimal. The proofs of the following lemmata are of the same flavor as the proof of the above proposition so we omit the details. Lemma 2.10. Let A Γ be an Artin group and B be a subset of a complete parabolic subgroup. If A Γ has (Int+-), then there exists P C Γ (B). Lemma 2.11. Let A Γ be an Artin group and B 1 and B 2 be subsets of A Γ . Assume that P C Γ (B 1 ) and P C Γ (B 2 ) exist. If B 1 ⊂ B 2 , then P C Γ (B 1 ) ⊂ P C Γ (B 2 ). Remember that we defined two classes of Artin groups in the introduction as follows: (1) An Artin group A Γ is contained in the class A iff A Γ has property (Int++). (2) An Artin group A Γ is contained in the class B iff A Γ has property (Int). Due to previous work of multiple authors, we know that many classes of Artin groups fall into A or B, for example: In the next section we show that an Artin group in the class B already has property (Int+-). This implies: Corollary 2.12. (1) Let A Γ be in the class A. Then for any subset B ⊂ A Γ there exists P C Γ (B). (2) Let A Γ be in the class B. For a subset B ⊂ A Γ of a complete parabolic subgroup there exists P C Γ (B). Bass-Serre theory meets Artin groups Let ψ : G → Isom(T ) be a group action via simplicial isometries on a tree T without inversion. One can consider a tree as a metric space by assigning each edge the length one. If we write 'x ∈ T ' we implicitly assume that we metrized the tree in that way. For a subset A ⊂ G we denote by Fix(ψ(A)) := {x ∈ T \| ψ(a)(x) = x for all a ∈ A} the fixed point set of ψ(A). Note that if Fix(ψ(A)) is non-empty, then it is a subtree of T . Further, for an edge e ∈ E(T ) the stabilizer of e, denoted by stab(e) is defined as stab(e) := {g ∈ G \| ψ(g)(e) = e} and resp. for a vertex v ∈ V (T ) the stabilizer stab(v) := {g ∈ G \| ψ(g)(v) = v}. Given an amalgam A * C B, by the Bass-Serre theory, there is a simplicial tree T A * C B on which A * C B acts simplicially without a global fixed point. The Bass-Serre tree T A * C B is constructed as follows: the vertices of T A * C B are cosets of A and B and two vertices gA and hB, g, h ∈ A * C B are connected by an edge iff gA∩hB = gC. For more information about amalgamated products and Bass-Serre theory see [Ser03]. How the Bass-Serre tree changes with amalgamation can be seen in [Ser03,p. 35], where the Bass-Serre tree is drawn for Z/4Z * Z/2Z Z/6Z. The group G = A * C B acts on its Bass-Serre tree via left-multiplication. For this action the stabilizer of a vertex gA is given by gAg −1 and the stabilizer of an edge between gA and hB is given by gCg −1 (since gA ∩ hB = gC). 3.1. Intersection of parabolic subgroups. In this section we prove Theorem 1.1 from the introduction. Our proof is of geometric nature with main tool being Bass-Serre theory. We will be using a slightly different notation for the vertices and edges of the Bass-Serre tree associated Proof of Theorem 1.1. We assume that A Γ has Property (Int). We take a complete parabolic subgroup P 1 and a parabolic subgroup P 2 of A Γ , and we prove that P 1 ∩ P 2 is a parabolic subgroup by induction on the number of pairs {s, t} ⊂ V (Γ) satisfying {s, t} / ∈ E(Γ). If there is no such a pair, then Γ itself is complete and then property (Int) suffices for saying that P 1 ∩ P 2 is a parabolic subgroup. So, we can assume that there exists a pair {s, t} ⊂ V (Γ) such that {s, t} / ∈ E(Γ) and that the inductive hypothesis holds. We set I = V (Γ) \ {s}, J = V (Γ) \ {t}, and K = V (Γ) \ {s, t}. By Lemma 2.4 we have the amalgamated product A Γ ∼ = A I * A K A J , which leads to the construction of the Bass-Serre tree T associated to this splitting. to A Γ = A X * A Z A Y . There exist subsets X, Y ⊂ V (Γ) and elements g, h ∈ A Γ such that X is free of infinity, P 1 = gA X g −1 , and P 2 = hA Y h −1 . Since X is free of infinity, we have either X ⊂ I or X ⊂ J, hence there exists a vertex u 0 = v(a 0 , U 0 ) of T such that P 1 = gA X g −1 ⊂ stab(u 0 ) = a 0 A U 0 a −1 0 . From here the proof is divided into two cases. Case 1 : {s, t} ⊂ Y . Then either Y ⊂ I or Y ⊂ J, hence there exists a vertex v 0 = v(b 0 , V 0 ) of T such that P 2 = hA Y h −1 ⊂ stab(v 0 ) = b 0 A V 0 b −1 0 . We denote by d the distance in T between u 0 and v 0 and we argue by induction on d. Suppose d = 0, that is, u 0 = v 0 . Then a −1 0 P 1 a 0 is a complete parabolic subgroup of A U 0 and a −1 0 P 2 a 0 is a parabolic subgroup of A U 0 . The number of pairs {i, j} satisfying {i, j} ∈ E(Γ) is strictly smaller in U 0 than in V (Γ), hence, by the induction hypothesis, (a −1 0 P 1 a 0 ) ∩ (a −1 0 P 2 a 0 ) = a −1 0 (P 1 ∩ P 2 )a 0 is a parabolic subgroup of A U 0 , and therefore a parabolic subgroup of A Γ . So, P 1 ∩ P 2 is a parabolic subgroup of A Γ . Now we assume that d ≥ 1 and that the inductive hypothesis on d holds. Let (u 0 , u 1 , . . . , u d ) be the unique geodesic in T connecting u 0 with v 0 = u d . For i ∈ {1, . . . , d} we denote by e i the edge connecting u i−1 with u i . Note that, since P 1 ⊂ stab(u 0 ) and P 2 ⊂ stab(v 0 ), we have P 1 ∩ P 2 ⊂ stab(u i ) for all i ∈ {0, 1, . . . , d} and P 1 ∩ P 2 ⊂ stab(e i ) for all i ∈ {1, . . . , d}. We set u 1 = v(a 1 , U 1 ) and e 1 = e(c 1 ). u 0 = v(a 0 , U 0 ) u 1 = v(a 1 , U 1 ) v 0 = u d e ( c 1 ) The group a −1 0 gA X g −1 a 0 is a complete parabolic subgroup of A U 0 and a −1 0 c 1 A K c −1 1 a 0 is a parabolic subgroup of A U 0 , hence, by the induction hypothesis (on the number of pairs {i, j} such that {i, j} is not an edge), (a −1 0 gA X g −1 a 0 ) ∩ (a −1 0 c 1 A K c −1 1 a 0 ) = a −1 0 ((gA X g −1 ) ∩ (c 1 A K c −1 1 ))a 0 is a parabolic subgroup, and therefore (gA X g −1 ) ∩ (c 1 A K c −1 1 ) is a parabolic subgroup. It is complete because it is contained in P 1 = gA X g −1 . So, there exist g 1 ∈ A Γ and X 1 ⊂ V (Γ) such that X 1 is free of infinity and (gA X g −1 ) ∩ (c 1 A K c −1 1 ) = g 1 A X 1 g −1 1 . Moreover, g 1 A X 1 g −1 1 ⊂ c 1 A K c −1 1 ⊂ a 1 A U 1 a −1 1 = stab(u 1 ) . By the induction hypothesis (on d), it follows that P 1 ∩ P 2 = (gA X g −1 ) ∩ (hA Y h −1 ) = (gA X g −1 ) ∩ (c 1 A K c −1 1 ) ∩ (hA Y h −1 ) = (g 1 A X 1 g −1 1 ) ∩ (hA Y h −1 ) is a parabolic subgroup of A Γ . This finishes the proof of Case 1. Case 2: {s, t} ⊂ Y . We set Y I = Y \ {s} = Y ∩ I, Y J = Y \ {t} = Y ∩ J and Y K = Y \ {s, t} = Y ∩ K. As for A Γ we have the amalgamated product A Y = A Y I * A Y K A Y J . We denote by T Y the Bass-Serre tree associated to this amalgamated product. Claim. We have an embedding of T Y into T which, for each a ∈ A Y , sends v(a, Y I ) to v(a, I), v(a, Y J ) to v(a, J), and e(a) to e(a). Proof of the Claim. Let a, b ∈ A Y . We need to show the following equivalencies: aA Y I = bA Y I ⇔ aA I = bA I , aA Y J = bA Y J ⇔ aA J = bA J , aA Y K = bA Y K ⇔ aA K = bA K . We prove the first one. The others can be proved in the same way. Recall that, by Van der Lek [Vdl83], A Y I = A Y ∩I = A Y ∩ A I . So, since a −1 b ∈ A Y , aA Y I = bA Y I ⇔ a −1 b ∈ A Y I = A Y ∩ A I ⇔ a −1 b ∈ A I ⇔ aA I = bA I . This completes the proof of the Claim. Let v 0 be the unique vertex of hT Y at minimal distance from u 0 . Note that v 0 may be equal to u 0 . u 0 hT Y v 0 Since P 1 stabilizes u 0 and P 2 stabilizes hT Y (as a set, not pointwise), P 1 ∩ P 2 stabilizes v 0 , that is, P 1 ∩ P 2 ⊂ stab(v 0 ) due to the minimality of the distance between u 0 and v 0 . Let a ∈ A Y and V ∈ {I, J} such that v 0 = v(ha, V ). We have stab(v 0 ) ∩ P 2 = (haA V a −1 h −1 ) ∩ (hA Y h −1 ) = (haA V a −1 h −1 ) ∩ (haA Y a −1 h −1 ) = ha(A V ∩ A Y )a −1 h −1 = haA V ∩Y a −1 h −1 , hence P 1 ∩ P 2 = P 1 ∩ stab(v 0 ) ∩ P 2 = P 1 ∩ (haA V ∩Y a −1 h −1 ) . Since {s, t} ⊂ V ∩ Y , we conclude by Case 1 that P 1 ∩ P 2 is a parabolic subgroup. 3.2. Serre's property FA and a generalization of this fixed point property. We use Bass-Serre theory in this section to prove Proposition 1.3 and Theorem 1.7. Let us first recall the definitions of properties FA and FA ′ and the basic implications we need later on. A group G is said to have property FA if every simplicial action of G on any tree without inversions has a global fixed vertex. A weaker property is property FA ′ , here we require that for every action of G on any tree without inversions every element has a fixed point. Proof. Due to [Ser03, Thm. 15] a countable group cannot have property FA if it has a quotient isomorphic to Z. Given an Artin group A Γ with standard generating set V = {v 1 , ..., v n } we define a homomorphism ϕ : A Γ → Z by setting ϕ(v i ) = 1 for every i ∈ {1, ..., n}. Due to the universal property of group presentations this defines a homomorphism, since all relations are in the kernel. Furthermore this map is surjective, which means that A Γ indeed has a quotient isomorphic to Z, namely A Γ / ker(ϕ). Moving on to property FA ′ we need the following lemma to show that FA ′ subgroups of Artin groups have to be contained in complete parabolic subgroups. This allows us to prove the following proposition: Proposition 3.4. Let ψ : G → Isom(T ) be a simplicial action on a tree T without inversion. If Fix(ψ(g)) = ∅ for all g ∈ G, then either (1) Fix(ψ(G)) = ∅, or (2) there exists a sequence of edges (e i ) i∈N such that stab(e 1 ) stab(e 2 ) .... Additionally, if G is countable, then G = i∈N stab(e i ). Proof. We differentiate two cases: Case 1: There exists a subgroup H ⊂ G such that Fix(ψ(H)) consists of exactly one vertex v. Then for g ∈ G we know by Lemma 3.3 that Fix(ψ( H, g )) = ∅, hence ψ(g)(v) = v. Since g is arbitrary we have Fix(ψ(G)) = {v}. Case 2: For any subgroup H ⊂ G such that Fix(ψ(H)) = ∅ the fixed point set Fix(ψ(H)) contains always an edge. Let g 1 be in G. By assumption the fixed point set of ψ(g 1 ) contains at least one edge e 1 ∈ Fix(ψ(g 1 )). If e 1 ∈ Fix(ψ(G)), then we are done. Otherwise there exists g 2 ∈ G − stab(e 1 ). By Lemma 3.3 the fixed point set Fix(ψ( stab(e 1 ), g 2 )) is non-empty and by assumption this set contains at least one edge e 2 ∈ Fix(ψ( stab(e 1 ), g 2 )). We obtain stab(e 1 ) stab(e 2 ). If e 2 ∈ Fix(ψ(G)) = ∅, then we are done. Otherwise there exists g 3 ∈ G − stab(e 2 ) and we proceed as before. By this construction we obtain an ascending chain of stab(e i ). Now there are two possibilities: this chain either stabilizes or it does not. If it stabilizes, then there exists an edge e n such that G = stab(e n ) and thus e n ∈ Fix(ψ(G)). If the chain does not stabilize, then we end up in case (2) of Proposition 3.4 and if G is countable, then it is straightforward to verify that it is possible to write G = i∈N stab(e i ). This can be done with a slight modification of the chain construction by always picking a specific element g m ∈ G − stab(e m−1 ). Corollary 3.5. Let G = A * C B denote an amalgamated free product and H ⊂ G be a subgroup. If H is an FA ′ -group, then (1) H is contained in a conjugate of A or B or (2) there exists a sequence of elements (g i ) i∈N , g i ∈ G such that g 1 Cg −1 1 ∩ H g 2 Cg −1 2 ∩ H . . . Additionally, if H is countable, then H = i∈N g i Cg −1 i ∩ H. Proof. The group G acts on its Bass-Serre tree without inversion. Restricting this action to H and applying Proposition 3.4 shows the corollary since the stabilizers of vertices are conjugates of A or B and the stabilizers of edges are conjugates of C intersected with H. Proposition 3.6. Let A Γ be an Artin group in the class A and let H ⊂ A Γ be a subgroup. If H is an FA ′ -group, then H is contained in a conjugate of a special complete subgroup of A Γ . Proof. Let A Γ denote an Artin group in the class A and H ⊂ A Γ an F A ′ subgroup. If Γ is not complete, then decompose A Γ as an amalgamated product according to Lemma 2.4, thus A Γ = A st(v) * A lk(v) A V −v . We now apply Corollary 3.5 to the amalgamated product A Γ = A st(v) * A lk(v) A V −v . If we are in case (1), we are in a conjugate of one of the factors and we repeat the process of decomposing the factor into an amalgam. If we are in case (2), we obtain a proper infinite chain g 1 A lk(v) g −1 1 ∩ H g 2 A lk(v) g −1 2 ∩ H ... and since A Γ is countable we know that H is countable, hence H = i∈N g i A lk(v) g −1 i ∩ H. By Lemma 2.11 we can transform the above chain into the following chain of parabolic subgroups P C Γ (g 1 A lk(v) g −1 1 ∩ H) ⊂ P C Γ (g 2 A lk(v) g −1 2 ∩ H) ⊂ ... and we also have H ⊂ i∈N P C Γ (g i A lk(v) g −1 i ∩ H) . We can now apply Proposition 2.6 to see that the above chain of parabolic subgroups stabilizes, say at P C Γ (g n A lk(v) g −1 n ∩ H). Then we have H ⊂ P C Γ (g n A lk(v) g −1 n ∩ H) ⊂ g n A lk(v) g −1 n ⊂ g n A st(v) g −1 n . If the subgraph st(v) is complete we are done, if not then we decompose A st(v) again and proceed as before. This process stops at some point since the graph Γ is finite and we remove at least one vertex at every step of the decomposition. Before moving to the proof of Theorem 1.7 we recall a basic result about lcH-slender groups. By definition, a discrete group G is called lcH-slender if every group homomorphism from a locally compact Hausdorff group into G is continuous. From here on we will be dealing with locally compact Hausdorff groups and their basic properties. Everything that will be used can be found in [Str06] and theorems will be cited, however not every basic property will be cited in that way. Lemma 3.7. Let G be an lcH-slender group. Then G is torsion free. Proof. Suppose there exists a non-trivial torsion element g ∈ G. Without loss of generality we can assume that g has order p ∈ N and p is a prime number (if it is composite, there is a power of g having a prime order). Then g ∼ = Z/pZ. Now consider the group N Z/pZ. This is a compact topological group and also a vector space. The space N Z/pZ is a linear subspace. We define ψ : N Z/pZ → Z/pZ by setting ψ(x) := m∈N x m . Now we take a basis B of N Z/pZ and extend it to a basis C of N Z/pZ. We then extend the map ψ to a map ϕ : N Z/pZ → Z/pZ via linear extension of: ϕ(c j ) := ψ(c j ) if c j ∈ B 0 if c j ∈ C − B The linearity of this map guarantees this is a group homomorphism. However this map cannot be continuous because of Theorem 3.8. Let A Γ be an Artin group in the class A. (1) Let ψ : L → A Γ be a group homomorphism from a locally compact Hausdorff group L into A Γ . If L is almost connected, then ψ(L) is contained in a conjugate of a special complete subgroup of A Γ . (2) If all special complete subgroups of A Γ are lcH-slender, then A Γ is lcH-slender. Proof. It is known that an almost connected locally compact Hausdorff group has property F A ′ [Alp82, Cor. 1]. Thus, by Proposition 1.3, it follows that ψ(L) is contained in a complete parabolic subgroup since property F A ′ is preserved under images of homomorphisms. Let ϕ : L → A Γ be a group homomorphism from a locally compact Hausdorff group L into an Artin group A Γ that lies in the class A. We give A Γ the discrete topology. By (1) we know that the image of the connected component L • under ϕ is contained in a parabolic complete subgroup gA ∆ g −1 of A Γ . By assumption, any special complete subgroup of A Γ is lcH-slender, therefore the restricted group homomorphism ϕ \|L • : L • → gA ∆ g −1 is continuous. Since the image of a connected group under a continuous group homomorphism is connected, the group ϕ(L • ) is trivial and therefore the map ϕ factors through ψ : L/L • → A Γ . Since the group L/L • is totally disconnected there exists a compact open subgroup K ⊂ L/L • by van Danzig's Theorem [Bou89,III §4,No.6]. By (1) we know that ψ(K) is contained in a parabolic complete subgroup hA Ω h −1 . By assumption we know that ψ \|K is continuous. The image of a compact set under a continuous map is always compact, hence ψ(K) is finite. We also know that a lcH-slender group is always torsion free due to Lemma 3.7, thus ψ(K) is trivial. Hence the map ψ has open kernel, since it contains the compact open group K and is therefore continuous. Since the quotient map π : L → L/L • is continuous, so is ϕ = ψ • π. The clique-cube complex In this section we will be using CAT(0) spaces and their basic properties. For the definition and further properties of these metric spaces we refer to [BH99]. Associated to an Artin group A Γ is a CAT(0) cube complex C Γ where the dimension is bounded above by the cardinality of V (Γ). We describe the construction of this cube complex that is closely related to the Deligne complex introduced in [CD95]. For an Artin group A Γ we consider the poset {aA ∆ \| a ∈ A Γ and ∆ is a complete subgraph of Γ or ∆ = ∅} . This poset is ordered by inclusion. We now construct the cube complex C Γ in the usual way: The vertices are the elements in the poset and two vertices aA ∆ 1 bA ∆ 2 span an n-cube if \|∆ 2 \| − \|∆ 1 \| = n. The group A Γ acts on C Γ by left multiplication and preserves the cubical structure. Moreover, the action is strongly cellular i.e. the stabilizer group of any cube fixes that cube pointwise. Therefore, if a subgroup H ⊂ A Γ has a global fixed point in C Γ , then there exists a vertex in C Γ which is fixed by H. The action is cocompact with the fundamental domain K, which is the subcomplex spanned by all cubes with vertices 1A ∆ for ∆ ⊂ Γ a complete subgraph. However the action will in general not be proper, since the stabilizer of a vertex gA X for X = ∅ is the parabolic complete subgroup gA X g −1 . Theorem 4.1 ([GP12B], Thm. 4.2). The clique-cube complex C Γ is a finite-dimensional CAT(0) cube complex. It was proven in [LV20, Thm. A] that any cellular action of a finitely generated group on a finite dimensional CAT(0) cube complex via elliptic isometries always has a global fixed point. In general, this result is not true for not finitely generated groups, not even for actions on trees. Example 4.2 ([Ser03] Thm. 15). Consider the group Q and pick an infinite sequence of elements (g i ) i∈N such that g 1 g 1 , g 2 .... This is possible since Q is not finitely generated. Set G i := g 1 , g 2 , ..., g i for i ∈ N and define a graph Γ in the following way. The set of vertices of Γ is the disjoint union of Q/G n , i.e. the vertices are the cosets qG n and there is an edge between two vertices if and only if they correspond to consecutive Q/G n and Q/G n+1 and correspond under the canonical projection Q/G n → Q/G n+1 . One can now check that this is a tree with a natural action of Q. The key feature is that if there was a fixed point P , then there exists an n ∈ N such that P ∈ Q/G n and therefore Q = G n , a contradiction. We conjecture that the structure of Artin groups does not allow such an example, that means: Conjecture. Let A Γ be an Artin group and Φ : A Γ → Isom(C Γ ) be the action on the associated clique-complex via left multiplication. Let H ⊂ A Γ be a subgroup. If Φ(h) is elliptic for all h ∈ H, then Fix(Φ(H)) is non-empty, thus Φ(H) is contained in a complete parabolic subgroup. Before we prove that this conjecture holds for Artin groups in the class B we discuss a tool for proving that some fixed point sets are non-empty. We have Fix(Φ(H)) = h∈H Fix(Φ(h)) . By definition, a family of subsets (A i ) i∈I of a metric space is said to have the finite intersection property if the intersection of each finite subfamily is non-empty. Monod proved in [Mon06, Thm. 14] that a family consisting of bounded closed convex subsets of a complete CAT(0) space with the finite intersection property has a non-empty intersection. We consider the family consisting of fixed point sets (Fix(φ(h))) h∈H . Any fixed point set is closed and convex. Since the CAT(0) space C Γ is a finite dimensional cubical complex we know by [LV20, Thm. A] that this family has the finite intersection property, but in general a fixed point set does not need to be bounded as the following example shows. Thus we need a different strategy in order to prove that Fix(φ(H)) is non-empty. Now we consider the corresponding Artin group A Γ and the clique-complex C Γ . For the clique-complex the fundamental domain K is given by A {a} A ∅ A {c} A {a,b} A {b} A {b,c} Let Φ : A Γ → Isom(C Γ ) denote the natural action by left multiplication. We want to show that Fix(Φ(b)) is unbounded. First notice that since Γ is not complete, the clique-complex is unbounded itself. First we calculate K ∩ Fix(Φ(b)): Since b = a n and b = c k for any n, k ∈ Z, b does not fix In general we know that b lies in the center of A Γ , so following the pattern from above it is easy to see that Φ(b) fixes vertices of the following form: wA ∆ , where w ∈ A Γ is an arbitrary element and ∆ is a subgraph of Γ containing the vertex b. That means any copy of the 'top edge' in the fundamental domain is fixed by Φ(b). Since K is a fundamental domain for the action and C Γ is unbounded, it immediately follows that Fix(Φ(b)) is unbounded, too. Let A Γ be in the class B and H be a subgroup that is contained in a parabolic complete subgroup. Since the action of A Γ on C Γ is strongly cellular, we can write the parabolic closure of a subgroup H as P C Γ (H) := v∈V (C Γ ) {stab(v) \| H ⊂ stab(v)}, which coincides with the notion defined in Chapter 2. Now we claim that for a subset B of a parabolic complete subgroup we have Fix(Φ(B)) = Fix(Φ(P C Γ (B))) . The inclusion ⊃ is clear since B ⊂ P C Γ (B). For the other directions, let v ∈ Fix(Φ(B)). It suffices to check that v ∈ Fix(Φ(P C Γ (B))) since the action is strongly cellular. Since B fixes the vertex v, we have B ⊂ stab(v). Due to the definition of the parabolic closure, we therefore obtain P C Γ (B) ⊂ stab(v), and thus P C Γ (B) fixes v as well, or in other words, v ∈ Fix(Φ(P C Γ (B))). So, our chain transforms into Fix(Φ(P C Γ (H 1 ))) Fix(Φ(P C Γ (H 2 ))) ... On the other hand, since H i ⊂ H i+1 for all i, we have the chain P C Γ (H 1 ) ⊂ P C Γ (H 2 ) ⊂ ... Since Γ is finite, this chain stabilizes by Proposition 2.6. So, there exists an index j such that P C Γ (H i ) = P C Γ (H j ) for all i ≥ j, hence Fix(Φ(P C Γ (H i ))) = Fix(Φ(P C Γ (H j ))) for all i ≥ j, which is a contradiction. Therefore Φ(H) has a global fixed vertex. Since the stabilizer of a vertex is a complete parabolic subgroup, Φ(H) is contained in such a group. This implies Proposition 1.8 from the introduction, since an FC ′ group acts locally elliptically on every CAT(0) cube complex. Corollary 4.5. Let K denote a compact group and A Γ an Artin group in the class B, further let φ : K → A Γ be a group homomorphism. Then φ(K) is contained in a complete parabolic subgroup. Proof. Due to [MV20,Cor. 4.4] the compact group K acts locally elliptically on C Γ . Thus by Proposition 4.4 the image of K under φ is contained in a complete parabolic subgroup. 4.1. Proof of Theorem 1.9. First we recall the result of the Main Theorem in [MV20]. Theorem 4.6. Let Φ : L → Isom(X) be a group action of an almost connected locally compact Hausdorff group L on a complete CAT(0) space X of finite flat rank. If (1) the action is semi-simple, (2) the infimum of the translation lengths of hyperbolic isometries is positive, (3) any finitely generated subgroup of L which acts on X via elliptic isometries has a global fixed point, (4) any subfamily of {Fix(Φ(l)) \| l ∈ L} with the finite intersection property has a non-empty intersection, then Φ has a global fixed point. Proof of Theorem 1.9. The proof of the second part is very similar to the proof of Theorem 1.7. Let ψ : L → A Γ be a group homomorphism from an almost connected locally compact Hausdorff group L into an Artin group A Γ that is contained in the class B. Further, let Φ : A Γ → Isom(C Γ ) be the action on the associated clique-cube complex via left-multiplication. Since the dimension of C Γ is finite we know that the flat rank of C Γ is also finite. Further, the first three conditions are satisfied by any cellular action on a finite dimensional CAT(0) cube complex [Bri99, Thm. A and Prop.], [LV20, Thm. A], hence also the action ψ • Φ : L → A Γ → Isom(C Γ ) satisfies these conditions. By Proposition 4.4 it follows that any subfamily of {Fix(ψ • Φ(l)) \| l ∈ L} with the finite intersection property has a non-empty intersection. Hence, the action ψ • Φ has a global fixed point. Since the action is strongly cellular there exists a vertex gA ∆ ∈ C Γ that is fixed by this action and therefore ψ(L) is contained in the stabilizer of this vertex that is equal to gA ∆ g −1 . Let ϕ : L → A Γ be a group homomorphism from a locally compact Hausdorff group L into an Artin group A Γ that lies in the class B. Once again we give A Γ the discrete topology. By the above paragraph we know that the image of the connected component L • under ϕ is contained in a parabolic complete subgroup gA ∆ g −1 of A Γ . Now follow the same argument as given in the proof of Theorem 1.7 (2). = wvw . . . m({v,w})−letters whenever {v, w} ∈ E(Γ) . The most common examples of Artin groups are braid groups and Artin groups A Γ where E(Γ) = ∅ or m(E(Γ)) = {2}, those are called right-angled Artin groups. is a parabolic subgroup and B ⊂ P } and we write M using indices in an index set J as M = j∈J {Q j }, where each Q j is a parabolic subgroup of A Γ . We have ( 1 ) 1Artin groups of spherical type, right-angled Artin groups and large type Artin groups are in the class A [CGGW19, DKR07, CMV21]. (2) Artin groups of FC type are in the class B [Mor21]. For the very small example of G = Z/4Z * Z/6Z (a portion of) the Bass-Serre tree looks like this (suppose G = a, b , o(a) = 4, o(b) Notation 3 . 1 . 31Given an Artin groupA Γ = A X * A Z A Y , we write vertices in the Bass-Serre tree T as v(a, X) := aA X and v(b, Y ) := bA Y .For the edges we simply associate an edge e(a) to each a ∈ A Γ with endpoints v(a, X) and v(a, Y ). We have e(a) = e(b) if and only if aA Z = bA Z and every edge of T has this form. Proposition 3 . 2 . 32Artin groups do not have property FA. Lemma 3.3 ([Ser03], Prop. 26). Let ψ : G → Isom(T ) be a simplicial action on a tree T without inversion. Let A and B be subgroups of G. If Fix(ψ(A)) = ∅, Fix(ψ(B)) = ∅ and Fix(ψ(ab)) = ∅ for all a ∈ A and b ∈ B, then Fix(ψ( A, B )) = ∅. the following reason: If it was continuous, then ϕ −1 (0) would be an open neighborhood V of the identity, since {0} ⊂ Z/pZ is an open subset. So (after reordering the factors) V contains a set A 1 × A 2 × ... × A m × n>m Z/pZ. But then the elements x = (0, 0, ...) and y= (0, ..., 0, 1, 0, 0, ...), where the 1 is in coordinate m + 1, both lie in V and we have 0 = ϕ(x) = ϕ(y) = ϕ(x) + 1 = 1 which is a contradiction. A {a} or A {c} and it also can't fix A ∅ . However b fixes all the vertices in the top row, that is Φ(b)(A {a,b} ) = A {a,b} , Φ(b)(A {b} ) = A {b} and Φ(b)(A {b,c} ) = A {b,c} and also the edges between those. Proposition 4. 4 . 4Let A Γ be in B and Φ : A Γ → Isom(C Γ ) be the action on the associated clique-complex via left multiplication. Let H ⊂ A Γ be a subgroup. If Φ(h) is elliptic for all h ∈ H, then Fix(Φ(H)) is non-empty and therefore Φ(H) is contained in a complete parabolic subgroup. Proof. If Φ(H) had no global fixed point, then we could construct an infinite chain Fix(Φ(H 1 )) Fix(Φ(H 2 )) ..., where H i := h 1 , h 2 , ..., h i , since any finitely generated group acting locally elliptically already has a global fixed point due to [LV20, Thm. A]. Locally compact groups acting on trees. R Alperin, Pacific J. Math. 1001R. Alperin, Locally compact groups acting on trees. Pacific J. Math. 100 no. 1, (1982), 23-32. N Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines. 1337N. Bourbaki,Éléments de mathématique. Fasc. XXXIV. 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Symposia Mathematica (INDAM, Rome, 1967/68), vol. 1, pp. 175-185, Academic Press, London (1969). . Philip Möller, Einsteinstraße. 62Department of Mathematics, University of MünsterPhilip Möller, Department of Mathematics, University of Münster, Einsteinstraße 62, Münster (Germany) Email address: philip.moeller@uni-muenster. deMünster (Germany) Email address: [email protected] . Luis Paris, Imb, Cnrs, address: [email protected] Franche-Comté, 21000 Dijon (France) EmailUniversitéLuis Paris, IMB, UMR 5584 du CNRS, Université Bourgogne Franche-Comté, 21000 Dijon (France) Email address: [email protected]	[]
[ "Kolmogorov complexity as intrinsic entropy of a pure state: Perspective from entanglement in free fermion systems", "Kolmogorov complexity as intrinsic entropy of a pure state: Perspective from entanglement in free fermion systems" ]	[ "Ken K W Ma \nDepartment of Physics\nNational High Magnetic Field Laboratory\nFlorida State University\n32306TallahasseeFloridaUSA\n", "Kun Yang \nDepartment of Physics\nNational High Magnetic Field Laboratory\nFlorida State University\n32306TallahasseeFloridaUSA\n" ]	[ "Department of Physics\nNational High Magnetic Field Laboratory\nFlorida State University\n32306TallahasseeFloridaUSA", "Department of Physics\nNational High Magnetic Field Laboratory\nFlorida State University\n32306TallahasseeFloridaUSA" ]	[]	We consider free fermion systems in arbitrary dimensions and represent the occupation pattern of each eigenstate as a classical binary string. We find that the Kolmogorov complexity of the string correctly captures the scaling behavior of its entanglement entropy (EE). In particular, the logarithmically-enhanced area law for EE in the ground state and the volume law for EE in typical highly excited states are reproduced. Since our approach does not require bipartitioning the system, it allows us to distinguish typical and atypical eigenstates directly by their intrinsic complexity. We reveal that the fraction of atypical eigenstates which do not thermalize in the free fermion system vanishes exponentially in the thermodynamic limit. Our results illustrate explicitly the connection between complexity and EE of individual pure states in quantum systems.	10.1103/physrevb.106.035143	[ "https://export.arxiv.org/pdf/2202.02852v2.pdf" ]	246,634,391	2202.02852	fa1c46859a657eafd1b08794d118a4cfe021a2d1	Kolmogorov complexity as intrinsic entropy of a pure state: Perspective from entanglement in free fermion systems 25 Jul 2022 Ken K W Ma Department of Physics National High Magnetic Field Laboratory Florida State University 32306TallahasseeFloridaUSA Kun Yang Department of Physics National High Magnetic Field Laboratory Florida State University 32306TallahasseeFloridaUSA Kolmogorov complexity as intrinsic entropy of a pure state: Perspective from entanglement in free fermion systems 25 Jul 2022(Dated: July 27, 2022) We consider free fermion systems in arbitrary dimensions and represent the occupation pattern of each eigenstate as a classical binary string. We find that the Kolmogorov complexity of the string correctly captures the scaling behavior of its entanglement entropy (EE). In particular, the logarithmically-enhanced area law for EE in the ground state and the volume law for EE in typical highly excited states are reproduced. Since our approach does not require bipartitioning the system, it allows us to distinguish typical and atypical eigenstates directly by their intrinsic complexity. We reveal that the fraction of atypical eigenstates which do not thermalize in the free fermion system vanishes exponentially in the thermodynamic limit. Our results illustrate explicitly the connection between complexity and EE of individual pure states in quantum systems. INTRODUCTION Statistical mechanics of isolated quantum system is a topic of tremendous current interest [1][2][3][4][5][6][7][8][9][10][11][12][13]. Without an external heat bath, definitions of standard thermodynamic quantities such as temperature and entropy become subtle and often ambiguous in such systems [13][14][15]. For example, every pure state has zero von Neumann entropy [16]. On the other hand, according to the eigenstate thermalization hypothesis (ETH) [1][2][3][4], a highly excited eigenstate of a nonintegrable system should be "thermal" and thus have the same temperature and corresponding entropy density of a (mixed) thermal state with the same energy density (and other conserved charge densities if present). One way to resolve this tension is to partition the system (usually in position space) and focus on the smaller subsystem, which is in a mixed state and has a nonzero von Neumann entropy associated with its reduced density matrix. This is known as the entanglement entropy (EE). Indeed, it has been shown that the overwhelming majority of (or typical) free fermion eigenstates give rise to thermal reduced density matrices, a property termed eigenstate typicality [17,18]. An immediate consequence is that EE equals the corresponding thermal entropy in these cases. Eigenstate typicality also plays an important role in the dynamical generation of entanglement in free fermion systems [19]. This (by now standard) way of revealing the thermal nature of a pure state is unsatisfactory in several aspects. First of all, in principle EE depends on the way the system is partitioned, while entropy should be an intrinsic property of a state. Free fermion states are good examples of this: They are highly entangled in real space, but are product states with zero EE in a momentum space partitioning. Second, entanglement is a unique property of quantum mechanics [20], while the notion of entropy was first introduced in classical statistical thermodynamics, where all individual (or micro; not an ensemble of) states are pure. As a result, the von Neumann definition of entropy would always be zero there, regardless of whether one considers the whole universe or a subset of it. While one may object that the universe is intrinsically quantum, we can always consider semiclassical pure states that are well-described by classical physics, whose EE can be made arbitrarily small. Over the years, various alternative definitions of entropy have been introduced, in an attempt to reveal the intrinsic thermal properties of a state, either mixed or pure [21][22][23][24][25][26][27][28][29]. Meanwhile, the applications of classical and quantum Kolmogorov complexity make it possible to quantify the complexity of quantum states . It has been suggested that physical entropy should be a reflection of the complexity of a state, and quantify the amount of information carried by (or "hidden" in) it [26][27][28][29][30][31]. Furthermore, it has been shown that the von Neumann entropy of a probabilistic source (or density matrix) and the average quantum Kolmogorov complexity of the qubit strings generated by the source should coincide [32][33][34][35][36][37][38][39][40][41]. Nevertheless, a concrete example of the connection between Kolmogorov complexity and nonzero EE of individual pure states remains elusive. In this paper, we show that the classical Kolmogorov complexity of free fermion states has the same scaling behavior as their bipartite EE, thus directly relating EE to the intrinsic complexity of such pure states. Furthermore, Kolmogorov complexity is a quantitative measure of how typical a state is. This not only provides a systematic way to distinguish between typical and atypical eigenstates in the free fermion system from their occupation patterns, but also allows us to demonstrate that the fraction of atypical eigenstates which do not thermalize in the free fermion system vanishes exponentially in the system size in the thermodynamic limit. Our results shed light on the quantification of typical and atypical (the non-interacting version of scar [53]) states, which is important in understanding thermalization and the emergence of statistical mechanics in pure states. A BRIEF REVIEW OF KOLMOGOROV COMPLEXITY Given a binary string x, its Kolmogorov complexity is defined as the length of the shortest possible description of x [54][55][56][57]. Specifically, one can consider the "two-part codes" which consist of a universal Turing machine and a program [57]. Then, the plain Kolmogorov complexity of x is defined as [58] C(x) = min {l(T ) + l(p) : T (p) = x} + O(1).(1) The program p is executed by the universal Turing machine T , which outputs the string x and halts. Here, l(p) denotes the length of p in bits. It is obvious that the shortest possible program that can reconstruct x depends on the choice of T . Nevertheless, using another Turing machine (or computer) can only lead to a difference in C(x) bounded from above by a finite constant that is independent of l(p). In other words, this is a change in O(1). Furthermore, the length of the self-delimiting encoding of T , i.e. l(T ), is independent of l(p). Therefore, it is common to simply focus on l(p) and view it as the Kolmogorov complexity of x. Roughly speaking, all irregularities in the string x are reflected by l(p). With the above definition, we now summarize some important results for C(x). Although the value of C(x) cannot be computed from any program, C(x) is bounded from above. Consider a string which is random and has no simple description. To output the string, the best one can do is to take the entire string as the input and ask the Turing machine to copy the input to the output. Hence, the Kolmogorov complexity of any string satisfies C(x) ≤ l(x) + O(1). It is expected that a typical string is random and has C(x) ≃ l(x). We use the symbol ≃ when the relationship holds up to the leading order. As the O(1) term becomes negligible for sufficiently long strings, it will be dropped for convenience. On the other hand, some strings are easy to describe. For example, consider the string "11 · · · 1" where the bit "1" is repeated n times. We abbreviate the string as 1 n . This abbreviation immediately shows that the string is very simple and can be reconstructed from a very short input. Specifically, one can define a Turing machine which prints "1" for n times. Now, we simply need log n bits to specify the binary representation of n in the program p [59]. Alternatively, we can say that the string 1 n is highly compressible by encoding it as the binary representation of n. Hence, the string 1 n has Kolmogorov complexity, C(1 n ) ≃ log n.(2) A string x is called c-incompressible if its Kolmogorov complexity satisfies C(x) ≥ l(x) − c. Note that the upper bound C(x) ≤ l(x) + O(1) always holds. Denote the set of all binary strings as B = {Λ, 0, 1, 00, 01, 10, 11, · · ·}, where Λ is the empty string. The total number of binary strings with lengths shorter than N − c is N −c−1 i=0 2 i = 2 N −c − 1. (3) When one encodes x, the final result must be an element in B. Notice that different elements in B may correspond to different encodings of the same string. Hence, the largest possible fraction of strings with length N that is c compressible is 2 N −c − 1 2 N = 2 −c , for N → ∞.(4) This result implies that most of the strings cannot be compressed by a significant amount. Therefore, simple strings do exist but they are rare and atypical. Furthermore, x is said to be Kolmogorov random if it cannot be compressed by one bit. From the pigeonhole principle [60], there must be at least one string for every length N that is incompressible. Moreover, the difficulty in describing x depends on the information y that is already specified to the program. This leads to the concept of conditional Kolmogorov complexity, denoted as C(x\|y). The difference between C(x) and C(x\|y) is the most noticeable in simple strings. For example, suppose that the length of the string N is given. Then, 1 N has a conditional Kolmogorov complexity, C(1 N \|N ) = c,(5) where c is a constant. Another related example for our later discussion is the string which has a fixed number of ones in its elements. When both the length of the string N and the number of ones in the string n are given, then C(x\|N, n) log N n . Applying Stirling's approximation, one has C(x\|N, n) N H(n/N ). Here, H(α) = −α log α − (1 − α) log(1 − α)(7) is the Shannon entropy of a Bernoulli distribution [61]. ENTANGLEMENT ENTROPY IN A 1D FREE FERMION SYSTEM In gapped systems described by local Hamiltonians and most of the gapless systems in d > 1 dimensions, ground state EE satisfies an area law and scales with the surface area of the subsystem, S ∼ L d−1 [62][63][64]. This originates from local or short-distance entanglement. Here, d is the dimensionality of the system and L is the typical length of the subsystem in any direction. However, EE of free fermions in the ground state satisfies S ∼ dL d−1 log L [65][66][67][68][69], whereas a volume law S ∼ L d is satisfied in the vast majority of highly excited eigenstates [17]. We first consider the system of n free spinless fermions in one dimension, and show that the Kolmogorov complexity of eigenstates has the same scaling behavior as their bipartite EE. We assume that there are N different single-particle eigenstates in the momentum space, where N is proportional to the volume (in 1D, length) of the system. In the following discussion, we are only interested in the thermodynamic limit, in which both n and N are infinite but the ratio α = n/N is fixed. Now, each many-body eigenstate can be described by an occupation pattern (n 1 , n 2 , · · · , n N ). Here, n i = 1 if the single-particle eigenstate with momentum k i is occupied by a fermion. Otherwise, n i = 0. This description resembles a binary string x with length N that has n ones in its elements. From Eq. (6), the Kolmogorov complexity of a typical occupation pattern is asymptotically equal to the Shannon entropy, i.e. C(x\|α) ≃ N H(n/N ). The scaling behavior of C(x\|α) agrees with the volume law of EE in typical eigenstates [17]. Here, we reemphasize that EE in these states is also the thermal entropy since the typical eigenstates are thermal [17]. Later, we will have further discussion of typical and atypical eigenstates. What happens if we apply the above argument to the ground state of the system? Suppose that the singleparticle eigenstates with the n smallest momenta are occupied. This occupation pattern leads to the binary string 1 n 0 N −n . From Eq. (2), this string has a Kolmogorov complexity C(x GS \|α) ≃ log n ≃ log N ∝ log L. Here, x GS denotes the ground state occupation pattern. For a more generic Hamiltonian, the free-fermion ground state may possess m > 1 Fermi surfaces (pairs of points in 1D). When m ≪ n, the Kolmogorov complexity of the occupation pattern satisfies C(x GS \|α) ≃ m log N [70]. The above results correctly reproduce the scaling behavior of the ground state EE without bipartitioning the system! Since the thermal entropy should be extensive and scale as L in one dimension, the EE of the ground state is not the thermal entropy. In both ground state and typical eigenstates, the Kolmogorov complexity of the occupation pattern agrees with the scaling behaviors of EE. ENTANGLEMENT ENTROPY IN HIGHER DIMENSIONAL FREE FERMION SYSTEMS For free fermion systems in d > 1 dimensions, we can still assign a label to each single-particle eigenstate in the momentum space. Each label takes a value between 1 to N , with the values of all labels being different. Fig. 1 illustrates an example of labeling the single-particle eigenstates in the two-dimensional momentum space. We assume that the labeling scheme is a piece of information that is already specified to the program. Now, it becomes very straightforward to generalize the previous discussion on typical eigenstates to d > 1 dimensions. Again, the occupation pattern for a typical eigenstate satisfies C(x\|α) ≃ N H(α) ∝ L d . Here, L d is the volume of the system. Just as with the one-dimensional system, the result resembles the volume law of EE in a typical eigenstate. One may naively think that the previous argument on the ground state in one-dimensional system can also be directly generalized to higher dimensions. This will lead to a Kolmogorov complexity that scales as log N ∝ d log L, which does not agree with the EE of the ground state, S ∼ dL d−1 log L [65][66][67][68]. However, the naive generalization breaks down because the occupation pattern 1 n 0 N −n contains no information about the shape of the Fermi surface (FS)! Therefore, we need to develop a suitable approach for describing the FS with its shape in the form of a one-dimensional binary string. The tool we employ is graph theory [71]. A graph is an ordered pair G(V, E) comprising a set of vertices V and a set of edges E. A graph G ′ (V ′ , E ′ ) is a subgraph of G if and only if V ′ ⊆ V and E ′ ⊆ E. Usually, one represents the adjacency matrix as a binary string to describe the graph. Since there are at most N (N − 1)/2 edges, the Kolmogorov complexity of a typical graph scales as N 2 . Meanwhile, basic (simple) graphs exist. An example is the complete graph, in which each vertex is connected to all other vertices. This graph has Kolmogorov complexity O(1) [72]. Now, the FS encloses n points (including the points on the FS) in the momentum space. In particular, the points on the FS and the edges connecting them form a cycle subgraph of the aforementioned complete graph. Obviously, this subgraph describes the shape of the FS. To describe this cycle subgraph, we need to specify the labels of the vertices on the FS in the order of their connection [72]. It takes no more than log N ∝ log L d bits to specify each label. An example for the two-dimensional system is given in Fig. 1. We assume that the system is nearly isotropic, such that the number of vertices lying on the FS scales as n (d−1)/d ∼ L d−1 . After describing the FS, a suitably defined Turing machine can fill in 1 for the string elements which label the points inside the FS and 0 otherwise. Therefore, the occupation pattern for the ground state has Kolmogorov complexity, C(x GS \|α) ≃ n (d−1)/d log N ∝ dL d−1 log L.(8) This agrees with the scaling behavior of EE of free fermions in the ground state [65][66][67][68]. What happens if n = N , corresponding to a band insulator? In this case, all single-particle eigenstates are occupied, and there is no FS. From Eq. (5), we know that the occupation pattern has C(x\|n) = O(1). The occupation pattern can be specified by describing the cycle graph that connects all outermost vertices, which has C(G) = O(1) [72], consistent with the simple result above. In this case, EE is actually dominated by local or short-distance entanglement (not directly related to the complexity of the global state) that gives rise to the area law. It is illuminating to compare the above case with disordered free fermions, where there is no FS even for the metallic phase. In this case the ground state EE always satisfies the area law [73,74]. Since momentum is no longer a good quantum number, the previous graph theoretic description of the FS becomes unsuitable. Instead, each single-particle eigenstate is labeled by its eigenenergy. The Kolmogorov complexity of the global manybody ground state scales as log N ∼ d log L just as in the 1D case, but is subdominant compared with the area law contribution. We conjecture that this logarithmic term would show up as a subleading correction in the EE in the metallic phase, while it is absent in the insulating phase. It would be very interesting to test this numerically. Although our focus in this paper is the free fermion system, the methodology can be easily generalized to other systems. An obvious example is free boson states. At zero temperature all bosons condense into momentum k = k 0 [75]. This ground state can be described by specifying the label of the single-particle eigenstate (see Fig. 1) being occupied by the bosons. Thus, C(x GS \|α) ≃ log N . This result agrees with the scaling behavior of bipartite EE obtained in Refs. [76,77], and there is no arealaw contribution in this case. Suppose that the system is perturbed by a weak interaction between bosons. In this (more generic) case there is an area law term in the ground state EE, while the logarithmic term from the condensate becomes a subleading contribution, which comes from the spontaneously broken continuous symmetry it represents and the corresponding quantum fluctuations of the order parameter and Goldstone modes [78,79]. Such behavior is consistent with the scenario described in the paragraph above. It is worthwhile to mention that such subleading contributions are in some sense more important than the leading area law contribution in EE, as they reflect the intrinsic complexity of the global state. A famous example is the topological EE [80,81], which captures the topological nature of the ground state. TYPICAL AND ATYPICAL EIGENSTATES Previously, we observed that it is much easier to describe the occupation pattern for the ground state than the typical eigenstates for free fermions. Now, we make the distinction more explicit. We define a state as typical if and only if the Kolmogorov complexity of its occupation pattern scales as the number of particles or the system size. On the other hand, the occupation pattern of an atypical state has a Kolmogorov complexity that scales as o(N ). Since satisfying the volume law in EE is a necessary condition for thermalization, the above definition and our previous results directly imply that atypical states do not thermalize. What can we say about the population of atypical eigenstates? Following the reasoning in Eq. (3), we know that the largest possible number of atypical eigenstates is 2 o(N )+1 − 1. The number is actually smaller as the entanglement entropy of an eigenstate cannot be lower than that of the ground state. For the entire energy spectrum, there are N n ≃ 2 N H(n/N ) different manybody eigenstates for the free fermion systems. Suppose that α = n/N is sufficiently away from 0 or 1, so that H(n/N ) is not close to zero. In the thermodynamic limit, the largest possible fraction of atypical eigenstates in the entire spectrum, lim N →∞ 2 o(N )+1 − 1 2 N H(n/N ) → 0,(9) vanishes exponentially. This justifies our definitions of typical and atypical eigenstates based on the Kolmogorov complexity of their occupation patterns. CONCLUSION AND DISCUSSION To conclude, using free fermion systems as our primary examples, we have demonstrated explicitly the connection between the intrinsic complexity and entanglement entropy of individual pure states. Specifically, we have shown that the Kolmogorov complexity of the fermion occupation pattern successfully reproduces the logarithmically-enhanced area law and the volume law of EE for the ground state and typical eigenstates, respectively. In the latter case, the Kolmogorov complexity asymptotically agrees with the Shannon entropy in the thermodynamic limit [82]. Interestingly, our result suggests an alternative explanation to the logarithmic enhancement in the ground state EE. By representing the Fermi surface as a graph, the logarithmic term originates from the number of bits required to specify a vertex on the FS. Furthermore, we distinguish between typical and atypical eigenstates by the Kolmogorov complexity of their occupation patterns. Based on this, we deduced that the fraction of atypical eigenstates in the entire spectrum vanishes exponentially in the thermodynamic limit. As pointed out in Ref. [18], these atypical states can be easily eliminated by mixing with the typical states when the fermions interact. It is expected that most of the states in the interacting system would satisfy the volume law of EE and become thermal. On the other hand, quantum states with low EE, which are analogous to scar states [53], may still persist with low probabilities. Our present approach cannot prove or disprove the strong ETH, which postulates that all highly excited states in nonintegrable systems are thermal [83]. We leave this important problem for future studies. Last but not least, we should clarify that the intrinsic complexity of a generic pure state may not be quantified by the classical Kolmogorov complexity. To serve the purpose, the concept of quantum Kolmogorov complexity was introduced [32][33][34][35][36][37][38][39][40][41][42][43][44]57]. Nevertheless, explicit examples of the connection between intrinsic complexity and entanglement entropy in realistic physical systems remain elusive. The simple free fermion system allows us to define its occupation patterns in momentum space which take a disentangled form (i.e., behave as classical-like objects), and quantify the intrinsic complexity of its eigenstates by classical Kolmogorov complexity. This further allows us to demonstrate its connection to the entanglement entropy. In fact, Kolmogorov complexity was employed in studying the physical entropy of classical systems, in particular, the Boltzmann gas [26]. Its relevance to entanglement entropy in quantum systems that may have classical-like descriptions of their wave functions in some basis is revealed in this paper. Therefore, we believe that our work provides an important step in the research direction of connecting intrinsic complexity and entanglement entropy in (quantum) pure states. FIG. 1 : 1Labeling single-particle eigenstates (the black dots) and representing the ground state (GS) occupation pattern in the two-dimensional free fermion system as a binary string. Here, the square Fermi surface is a subgraph of the complete graph formed by the 49 vertices. All the edges in the complete graph are skipped for better illustration. To describe the Fermi surface, one needs to specify the labels 9 − 13, 16 − 20, 23 − 27, 30 − 34, and 37 − 41 in order of their connection. 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M Li, P Vitányi, An Introduction to Kolmogorov Complexity and Its Applications. BerlinSpringer4th ed.M. Li and P. Vitányi, An Introduction to Kolmogorov Complexity and Its Applications, 4th ed. (Springer, Berlin, 2019). Using K(x), the joint Kolmogorov complexity K(x1, x2) satisfies the subadditive condition: K(x1, x2) ≤ K(x1) + K(x2) + O(1). 57In a self-delimiting program, x is encoded as a prefix-free code. This feature provides a more rigorous connection between the Kolmogorov complexity and Shannon entropy [61To be more precise, one should consider the prefix Kol- mogorov complexity, K(x). In a self-delimiting program, x is encoded as a prefix-free code [57]. Using K(x), the joint Kolmogorov complexity K(x1, x2) satisfies the sub- additive condition: K(x1, x2) ≤ K(x1) + K(x2) + O(1). This feature provides a more rigorous connection between the Kolmogorov complexity and Shannon entropy [61]. Since the self-delimiting program p * requires an additional O(log [l(p)]) bits to store the length of p where p is the non self-delimiting program, K(x) and C(x) are different. Compared with the leading order term l(p), \|K(x)−C(x)\| is always a subleading-order term. that does not affect our results in the main textSince the self-delimiting program p * requires an addi- tional O(log [l(p)]) bits to store the length of p where p is the non self-delimiting program, K(x) and C(x) are different. Compared with the leading order term l(p), \|K(x)−C(x)\| is always a subleading-order term that does not affect our results in the main text. To follow the convention in information theory. log is defined as the logarithm with base twoTo follow the convention in information theory, log is defined as the logarithm with base two. P G L Dirichlet, R Dedekind, Lectures on Number Theory. Providence, RIAmerican Mathematical SocietyP.G.L. Dirichlet and R. 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[ "Capturing the Severity of Type II Errors in High-Dimensional Multiple Testing", "Capturing the Severity of Type II Errors in High-Dimensional Multiple Testing" ]	[ "Li He \nMerck Research Laboratories\nWest Point19486PA\n", "Zhigen Zhao \nDepartment of Statistics\nTemple University\n19122PhiladelphiaPA, US\n\nDepartment of Statistics\nTemple University\n19122PhiladelphiaPA, US\n" ]	[ "Merck Research Laboratories\nWest Point19486PA", "Department of Statistics\nTemple University\n19122PhiladelphiaPA, US", "Department of Statistics\nTemple University\n19122PhiladelphiaPA, US" ]	[]	The severity of type II errors is frequently ignored when deriving a multiple testing procedure, even though utilizing it properly can greatly help in making correct decisions. This paper puts forward a theory behind developing a multiple testing procedure that can incorporate the type II error severity and is optimal in the sense of minimizing a measure of false non-discoveries among all procedures controlling a measure of false discoveries. The theory is developed under a general model allowing arbitrary dependence by taking a compound decision theoretic approach to multiple testing with a loss function incorporating the type II error severity. We present this optimal procedure in its oracle form and offer numerical evidence of its superior performance over relevant competitors.	10.1016/j.jmva.2015.08.005	[ "https://arxiv.org/pdf/1403.5609v1.pdf" ]	27,383,369	1403.5609	e9c7c7c07166a7b296ea8a2e1b8a4cdaa009f89d	Capturing the Severity of Type II Errors in High-Dimensional Multiple Testing 22 Mar 2014 Li He Merck Research Laboratories West Point19486PA Zhigen Zhao Department of Statistics Temple University 19122PhiladelphiaPA, US Department of Statistics Temple University 19122PhiladelphiaPA, US Capturing the Severity of Type II Errors in High-Dimensional Multiple Testing 22 Mar 2014arXiv:1403.5609v1 [stat.ME]Bayes ruleCompound decision theoryOracle procedureMultiple testingWeighted marginal false discovery rateWeighted marginal false non-discovery rate The severity of type II errors is frequently ignored when deriving a multiple testing procedure, even though utilizing it properly can greatly help in making correct decisions. This paper puts forward a theory behind developing a multiple testing procedure that can incorporate the type II error severity and is optimal in the sense of minimizing a measure of false non-discoveries among all procedures controlling a measure of false discoveries. The theory is developed under a general model allowing arbitrary dependence by taking a compound decision theoretic approach to multiple testing with a loss function incorporating the type II error severity. We present this optimal procedure in its oracle form and offer numerical evidence of its superior performance over relevant competitors. Introduction Simultaneous testing of multiple hypotheses is an integral part of analyzing high-dimensional data from modern scientific investigations like those in genomics, brain imaging, astronomy, and many others, making multiple testing an area of current importance and intense statistical research. A variety of multiple testing methods have been put forward in the literature from both frequentist and Bayesian perspectives. However, the theories behind the developments of these methods are mostly driven by the overreaching goal of controlling an overall measure of type I errors or false discoveries, with other fundamentally important statistical issues often being ignored. For instance, in many of the aforementioned experiments there is a cost associated with the error of making a false discovery or missing a true discovery, and this cost increases with increasing severity of that error. This is an important issue not often taken into account when developing multiple testing procedures. A Bayesian decision theoretic approach can yield a powerful multiple testing method not only incorporating costs of false and missed discoveries but also simultaneously addressing dependency, optimality, and multiplicity Cai (2007, 2009)). This motivates us to take a similar approach, but in a more general framework that conforms more to the present problem, that is, to address the aforementioned issue related to severity of errors. Before explaining this generalization, let us first briefly outline the approach taken in Cai (2007, 2009). Given a set of observations X = (X 1 , . . . , X m ) ∼ f (x, θ), where θ = (θ 1 , . . . , θ m ) ∈ {0, 1} m , consider the problem of deciding between H i : θ i = 0 andH i : θ i = 1 simultaneously for i = 1, . . . , m, assuming that X i \| θ i ind ∼ (1− θ i )f 0 (x i )+θ i f 1 (x i ), for some given densities f 0 and f 1 , and θ i ∼ Bernoulli(1− π 0 ). Cai (2007, 2009) started with the following uniformly weighted 0-1 loss function: L λ (δ(X), θ) = 1 m m i=1 {λ(1 − θ i )δ i (X) + θ i (1 − δ i (X))} ,(1.1) for a decision rule δ(X) = (δ 1 (X), . . . , δ m (X)) ∈ {0, 1} m , where λ is the relative cost of making a false discovery (type I error) to that of missing a true discovery (type II error) and assumed to be constant over all the hypotheses. They considered the Bayes rule associated with this loss function and showed that it is also optimal from a multiple testing point of view. Specifically, given any α ∈ (0, 1), there exists a λ ≡ λ(α) for which it controls the marginal false discovery rate, mFDR = E [ m i=1 δ i (X)(1 − θ i )] E [ m i=1 δ i (X)] , at α, and minimizes the marginal false non-discovery rate, mFNR = E [ m i=1 {1 − δ i (X)}θ i ] E [ m i=1 {1 − δ i (X)}] , among all decision rules defined in terms of statistics satisfying a monotone likelihood ratio condition (MLR) and controlling the mFDR at α. They expressed this optimal procedure in an alternative form using hypothesis specific test statistics defined in terms of the local FDR measure [lfdr, Efron (2010)], and called it the oracle procedure. They provided numerical evidence showing that their oracle procedure can outperform its competitors, such as those in Benjamini and Hochberg (1995) and Genovese and Wasserman (2002). Clearly, the loss function used in the above formulation is somewhat simplistic. It gives equal importance to all type I errors as well as to all type II errors. While it might be reasonable to treat the type I errors equally in terms of severity and attach a fixed cost to all of them, it is often unrealistic to do so for type II errors. For instance, in a microarray experiment, there might be a fixed cost of doing a targeted experiment to verify that each gene is active and the loss due to making a false discovery might be that cost (which is being wasted in case the gene is found to be inactive). However, it would be unrealistic to assume that the loss in identifying a truly active gene as inactive does not depend on how strong is the expected signal that has been missed. In fact, it might reasonably be proportional to the difference (Duncan, 1965;Scott and Berger, 2006;Waller and Duncan, 1969) or even to the squared difference between the expected values of the missed and no signals. In other words, the above formulation needs to be generalized conforming it more to the reality in modern high-dimensional multiple testing. With that in mind, we consider testing H i : µ i = µ i0 against its one or two-sided alternative, for some specified values µ i0 , simultaneously for i = 1, . . . , m, under the following model: X \| µ, θ ∼ f (x \| µ), with µ = (µ 1 , . . . , µ m ), θ = (θ 1 , . . . , θ m ) µ i \| θ i ∼ (1 − θ i )I(µ i = µ i0 ) + θ i h(µ i − µ i0 ) (1.2) θ i ∼ Bernoulli(1 − π 0 ), given a density h, and under the following more general loss function: (1.3) L λ,s (δ(X), µ, θ) = 1 m m i=1 {λ(1 − θ i )δ i (X) + s(µ i − µ i0 )θ i (1 − δ i (X))} . We do not impose any dependence restriction on X, µ or θ. It is assumed that there is only a baseline cost λ 1 for each type I error (which, as argued above, is reasonable for a point null hypothesis). For each type II error, however, we assume that the cost is λ 2 , the baseline cost, times s(µ i − µ i0 ), a function s of µ i − µ i0 such that s(0) = 0 and is non-decreasing as µ i moves away from µ i0 . We call s(·) the severity function for type II errors. Through this function, a penalty is being imposed on making a type II error for each H i ; the larger the value of \|µ i − µ i0 \| is, the more severe this penalty is. The λ equals λ 1 /λ 2 , the relative baseline cost of a type I error to a type II error. In other words, λ/s(µ i − µ i0 ) is the relative cost of a type I error to a type II error. The specific choice of s(·) will depend on how fast we want the cost of the type II error to increase as µ i moves away from µ i0 . Our proposed loss function (1.3) is a non-uniformly weighted 0-1 loss function giving less and less weight to the type I error relative to the type II error as the type II error gets more and more severe as measured by the severity function. In this paper, we focus on deriving the theoretical form of an optimal multiple testing procedure from the Bayes rule under this general loss function. Given a severity function s, this Bayes rule provides an optimal multiple testing procedure in the sense of minimizing a measure of non-discoveries subject to controlling a measure of false discoveries at a specified level for a suitably chosen λ. These measures of false discoveries and false non-discoveries are of course different from the mFDR and mFNR, respectively, since we now need to account for the weights or penalties attached to the type II errors through the severity function that is not necessarily equal to one. We define these newer error rates as weighted mFDR and weighted mFNR and establish the aforementioned optimality result through these rates. We study the performance of this oracle optimal procedure with its relevant competitors through two numerical studies. The remainder of the paper is organized as follows. The development of the Bayes rule under the loss function (1.3), its characterization as an optimal multiple testing procedure in the framework of weighted false discovery and false non-discovery rates, and our oracle multiple testing procedure are given in the next section. In Section 3, we present the results of two numerical studies providing evidence of this oracle procedure's superior performance over its relevant competitors. We end the paper with some concluding remarks in Section 4. Optimal Rules Assuming that our problem is that of testing H i : µ i = 0 simultaneously for i = 1, . . . , m under the model (1.2) and the loss function L λ,s in (1.3), we do the following in this section: (i) determine the Bayes rule; (ii) show that the Bayes rule with an appropriately chosen λ provides an optimal multiple testing procedure in the sense of minimizing a measure of false non-discoveries among all rules that control a measure of false discoveries at a specified level; and (iii) express this optimal multiple testing procedure in terms of some test statistics to define the oracle procedure in this paper. The Bayes rule Let us first define w i (X) = E [s(µ i ) \| θ i = 1, X] , (2.1) the average severity of type II errors conditional on the data X and θ i = 1. Then, we have the following: Theorem 2.1. Consider testing H i : µ i = 0 simultaneously for i = 1, . . . , m under the model (1.2) and the loss function (1.3). Then, the decision rule δ * (X) = (δ * 1 (X), . . . , δ * m (X)), where δ * i (X) =          1 if P (θ i = 0 \| X) < w i (X) λ P (θ i = 1 \| X) 0 if P (θ i = 0 \| X) > w i (X) λ P (θ i = 1 \| X) , (2.2) is the Bayes rule. Proof. For any rule δ(X) = (δ 1 (X), . . . , δ m (X)), we have E [L λ,s (θ, µ, δ(X)) \| X] = 1 m m i=1 {λδ i (X)P (θ i = 0 \| X) + [1 − δ i (X)]E [s(µ i )I(θ i = 1) \| X]} = 1 m m i=1 {λδ i (X)P (θ i = 0 \| X) + [1 − δ i (X)]E [s(µ i ) \| θ i = 1, X] P (θ i = 1 \| X)} = 1 m m i=1 {w i (X)P (θ i = 1 \| X) + δ i (X) [λP (θ i = 0 \| X) − w i (X)P (θ i = 1 \| X)]} . Since the first term is constant with respect to δ, given X, it is clear that δ * (X) in (2.2) is the rule for which this conditional expectation is the minimum among all δ, and hence is Bayes. Optimal Multiple Testing Procedure Here we show that the aforementioned Bayes rule with an appropriately chosen λ provides an optimal multiple testing procedure in the sense of minimizing a measure of false non-discoveries among all rules that control a measure of false discoveries at a specified level. These measures of false discoveries and false non-discoveries are defined for any multiple testing rule δ as mFDR * (δ) = E [ m i=1 δ i (X)(1 − θ i )w * (θ i , µ i )] E [ m i=1 δ i (X)w * (θ i , µ i )] , (2.3) and mFNR * (δ) = E [ m i=1 {1 − δ i (X)}θ i w * (θ i , µ i )] E [ m i=1 {1 − δ i (X)} w * (θ i , µ i )] , (2.4) respectively, where w * (θ, µ) = 1 if θ = 0 s(µ) if θ = 1. With w * (θ i , µ i ) representing a weight associated with the ith hypothesis, these measures of false discoveries and false non-discoveries can be referred to as weighted mFDR and weighted mFNR, respectively. When w * (θ, µ) ≡ 1, they reduce to the corresponding mFDR or mFNR. Theorem 2.2. Consider the model in (1.2). Suppose there exists a test- ing procedure δ 0 (X) = (δ 10 (X), . . . , δ m0 (X)) such that δ i0 (X) is defined as in (2.2) and mFDR * (δ 0 ) = α. Let δ(X) be any other rule such that mFDR * (δ) ≤ α. Then mFNR * (δ 0 ) ≤ mFNR * (δ). Proof. First note that m i=1 E {δ i0 (X) − δ i (X)} P (θ i = 0 \| X) − w i (X) λ P (θ i = 1 \| X) ≤ 0, (2.5) according to (2.2), and m i=1 E {δ i0 (X) − δ i (X)} P (θ i = 0 \| X) − α 1 − α w i (X)P (θ i = 1 \| X) ≥ 0, (2.6) from the assumption, mFDR * (δ) ≤ α = mFDR * (δ 0 ). From (2.5) and (2.6), we get m i=1 E {δ i0 (X) − δ i (X)} w i (X)P (θ i = 1 \| X) 1 λ − α 1 − α ≥ 0, which implies that m i=1 E [δ i0 (X)w i (X)P (θ i = 1 \| X)] ≥ m i=1 E [δ i (X)w i (X)P (θ i = 1 \| X)] , (2.7) since α 1 − α = m i=1 E [δ i0 (X)P (θ i = 0 \| X)] m i=1 E [δ i0 (X)w i (X)P (θ i = 1 \| X)] ≤ 1 λ . Thus, we have from (2.7) E m i=1 1 − δ i0 (X) m i=1 E [{1 − δ i0 (X)} w i (X)P (θ i = 1 \| X)] − 1 − δ i (X) m i=1 E [{1 − δ i (X)} w i (X)P (θ i = 1 \| X)] {P (θ i = 0 \| X)− w i (X) λ P (θ i = 1 \| X) ≥ 0. This implies that 1 − mFNR * (δ 0 ) mFNR * (δ 0 ) ≥ 1 − mFNR * (δ) mFNR * (δ) , that is, mFNR * (δ 0 ) ≤ mFNR * (δ), as desired. Remark 2.1. Theorem 2.2 improves the work of Sun and Cai (2007) in the following sense: 1) it accommodates situations where penalties or weights associated with type II errors can be assessed through a severity function and incorporated into the development of a multiple testing procedure; 2) it provides a rule that is optimal among all procedures controlling the mFDR* at level α without any distributional restriction on the corresponding test statistics. Next, we will prove the existence of such a procedure δ 0 (X). We can express the optimal procedure δ 0 (X) in Theorem 2.2 in terms of the following test statistics: T i (X) = P (θ i = 0 \| X) P (θ i = 0 \| X) + w i (X)P (θ i = 1 \| X) , i = 1, . . . , m. (2.8) The statistic T i will be referred to as generalized local fdr (Glfdr). It reduces to the usual definition of the local fdr (Lfdr) of Efron (2004) under independence and to the test statistic defined in Sun and Cai (2009) under arbitrary dependence when s(µ) = 1. We consider decision rules of the form δ(T, c) = (δ(T 1 , c), . . . , δ(T m , c)), where δ(T i , c) = 1 if T i ≤ c 0 if T i > c,(2.9) with c being such that mFDR * (δ(T, c)) ≤ α. This will be our proposed oracle procedure. Before we state this oracle procedure more explicitly in terms of the distributions of T i 's, we give the following proposition asserting the existence of such a c. In this paper we assume that X is continuous and hence mFDR * (δ(T, c)) is continuous in c. Proposition 2.1. For the decision rule in (2.9) with T i defined in (2.8), mFDR * (δ(T, c)) is non-decreasing in c. We will prove this proposition by making use of the following two lemmas. c > 0, if dH 1 (t)/dH 0 (t) is non-decreasing (non-increasing) in t. Proof. The ratio can be expressed as the expectation, E H * c ϕ(T ), of the non-decreasing function ϕ( T ) = dH 1 (T )/dH 0 (T ), where H * c is such that dH * c (t) = δ(t, c)dH 0 (t)/E H 0 [δ(T, c)] . Since δ(t, c) is totally positive of order two (TP 2 ) in (t, c), that is, it satisfies the inequality δ(t, c) δ(t ′ , c ′ ) ≥ δ(t, c ′ ) δ(t ′ , c), ∀t < t ′ , c < c ′ , the lemma follows from the following result (Karlin and Rinott, 1980): The expectation of a non-decreasing (non-increasing) function of a random variable Y ∼ g(y, θ), with g(y, θ) being TP 2 in (y, θ), is non-decreasing (nonincreasing) in θ. Remark 2.2. Sun and Cai (2007) derived the above result for the collection of decisions based on the test statistics satisfying the MLR condition. Note that our proof, which is different, does not rely on any such condition. Lemma 2.2. Given two distributions f 0 (x) and f 1 (x) of a random vector X, define T (X) = af 0 (X)/{af 0 (X) + bf 1 (X)}, for any constants a, b > 0. Let H i (t) = P f i (T (X) ≤ t), 0 < t < 1, for i = 0, 1. Then, dH 1 (t)/dH 0 (t) = a(1 − t)/bt. Proof. Since [T (X) − t] [I(T (X) ≤ t) − I(T (X) ≤ t ± ǫ)] ≤ 0, ∀0 < t < 1, ǫ > 0, by taking expectations of both sides in this inequality with respect to X ∼ a a + b f 0 (x) + b a + b f 1 (x), we have a(1 − t) [H 0 (t) − H 0 (t ± ǫ)] ≤ bt [H 1 (t) − H 1 (t ± ǫ)] , ∀0 < t < 1, ǫ > 0. The desired result then follows by letting ǫ → 0. Proof of Proposition 2.1. Let G i,µ denote the conditional distribution of T i (X) given θ i = j and µ, for j = 0, 1. Then, from (2.3), we note that mFDR * (δ(T, c)) = π 0 m i=1 G i,0 (c) π 0 m i=1 G i,0 (c) + (1 − π 0 ) m i=1 G i,1 (c) , where G i,0 (c) = G (0) i,µ (c)h(µ\|θ i = 0)dµ and G i,1 (c) = s(µ i )G (1) i,µ (c)h(µ\|θ i = 1)dµ, with h(µ\|θ i = 0) and h(µ\|θ i = 1) representing the joint distribution of µ conditionally given θ i = 0 and θ i = 1, respectively. 1 − mFDR * (δ(T, c)) mFDR * (δ(T, c)) = E [ m i=1 δ(T i , c)θ i ω * (θ i , µ i )] E [ m i=1 δ(T i , c)(1 − θ i )ω * (θ i , µ i )] = E [ m i=1 δ(T i , c)s(µ i )I(θ i = 1)] E [ m i=1 δ(T i , c)I(θ i = 0)] = 1 − π 0 π 0 1 m m i=1 β i E G 1 [δ(T, c)] E G 0 [δ(T, c)] , (2.10) where G 1 (t) = m i=1 w iGi,1 (t), G 0 (t) = 1 m m i=1 G i,0 (t),G i,1 (t) = G i,1 (t)/β i , and w i = β i / m j=1 β j , with β i = s(µ i )h(µ\|θ i = 1)dµ. The proposition will be proved from Lemma 2.1 if we can show that dG 1 (t)/dG 0 (t) is a nonincreasing function of t, since the left hand side of proposition (2.10) is a decreasing function of mFDR * (δ(T, c)). Since T i (X) = π 0 f i,0 (X)/{π 0 f i,0 (X) + (1 − π 0 )β i f * i,1 (X)}, and G i,0 and G i,1 are the cdf's of T i (X) under the distributions f i,0 (x) = f (x \| θ i = 0) and f * i,1 (x) = 1 β i s(µ i )f (x \| θ i = 1, µ)h(µ\|θ i = 1)dµ, respectively, we see from Lemma 2.2 that dG i,1 (t) = π 0 (1−π 0 )β i 1 t − 1 dG i,0 (t), for any 0 < t < 1. Thus, m i=1 β i dG 1 (t) = m i=1 β i dG i,1 (t) = m i=1 β i π 0 (1 − t) β i (1 − π 0 )t dG i,0 (t) = mπ 0 1 − π 0 1 t − 1 dG 0 (t), implying that dG 1 (t)/dG 0 (t) is non-increasing in t ∈ (0, 1), as desired. Thus, the proposition is proved. Given Proposition 2.1, we are now ready to define our oracle procedure in the following: Definition 2.1 (The Oracle Procedure). Consider the multiple testing procedure δ(T, c * ), where c * = sup {t : mFDR * (δ(T, t)) ≤ α} . (2.11) This is a generalized version of the oracle procedure of Sun and Cai (2007). It is developed not only under any dependence structure among (X, µ) but also it allows the alternatives to vary across tests and each type II error to be weighted by a measure of severity. Moreover, for its optimality, any specific property, like the monotone likelihood ratio property that Sun and Cai (2007) assumed, for the underlying test statistics is not required. Remark 2.3. Let f dr i (X) = P (θ i = 0\|X) and d i (X) = f dr i (X)/T i (X). Then, it is to be noted that the mFDR * (δ(T, t)) can be expressed as follows: m i=1 E [I(T i (X) < t)f dr i (X)] m i=1 E [I(T i (X) < t)f dr i (X) + I(T i (X) < t)(1 − f dr i (X))w i (X)] = m i=1 E [I(T i (X) < t)T i (X)d i (X)] m i=1 E [I(T i (X) < t)d i (X)] . Numerical Studies Related to the Oracle Procedure We carried out two numerical studies to see how our procedure in its oracle form compares with its relevant competitors for the problem of testing µ i = 0 against µ i = 0, i = 1, . . . , m, with s(µ) = µ 2 , under the following model. Let (X i , µ i , θ i ), i = 1, . . . , m, be such that X i \| µ i , θ i ind ∼ N(µ i , 1) µ i \| θ i ind ∼ (1 − θ i )I(µ i = 0) + θ i h(µ i ) θ i iid ∼ Bernoulli(1 − π 0 ). (3.1) Often a multiple testing procedure can be seen as first ranking the hypotheses according to a measure of significance, based on some test statistic, p-value, or local fdr, before choosing a cut-off point for the significance measure to determine which hypotheses are to be declared significant subject to control over a certain error rate, such as FDR or mFDR, at a specified level. Such ranking plays an important role in a procedure's performance, and can itself be used as a basis to compare with another procedure controlling a different error rate. More specifically, between two procedures providing the same number of discoveries, the one with better ranking should provide more true discoveries. The first numerical study was designed to make such ranking comparison between the Sun and Cai (2007) and our oracle procedures that control two different measures of false discoveries, even though one is a generalized version of the other. Towards understanding what significance measure is being used to rank the hypotheses in our procedure, we note that under the independence model (3.1), the mFDR * (δ(T, t)) given in Remark 2.3 reduces to the following: mF DR * (δ(T, t)) = E (I(T (X) ≤ t)T (X)d(X)) E (I(T (X) ≤ t)d(X)) , with T (X) ≡ T 1 (X) and d(X) ≡ d 1 (X). The numerator and denominator expectations in the above ratio can be approximated (for large m) by 1 m m i=1 (I(T i (X) ≤ t)T i (X)d i (X)) and 1 m m i=1 (I(T i (X) ≤ t)d i (X)) , respectively, resulting in a measure of mF DR * (δ(T, t)) at t as follows: mF DR * (δ(T, t)) = m i=1 I(T i (X) ≤ t)T i (X)d i (X) m i=1 I(T i (X) ≤ t)d i (X) . Let T (1) , . . . , T (m) be the ordered versions of T 1 (X), . . . , T m (X), and H (i) and d (i) (X) be respectively the null hypothesis and the d-value corresponding to T (i) (X). Then, our oracle procedure can be described approximately as follows: Find k = max j : j i=1 T (i) (X)d (i) (X) j i=1 d (i) (X) ≤ α ,(3.2) and reject H (i) for all i = 1, . . . , k. In other words, our procedure can be seen as ranking the hypotheses according to the increasing values of T i (X), the Glfdr scores corresponding to the H i 's, before determining the cut-off point t ∈ {T (1) (X), . . . , T (m) (X)} to control the mFDR; whereas, the Sun-Cai oracle procedure does the same in terms of the lfdr scores. The second numerical study was conducted to see how well our oracle procedure with the cut-off point chosen subject to controlling the mFDR compares with Sun-Cai's oracle procedure and the p-value based oracle procedure in Genovese and Wasserman (2002) in terms of the acceptance region, the mFDR, and the mFNR. Numerical Study 1 We considered using a measure of non-discoveries to compare the rankings provided by the Sun-Cai and our oracle procedures. More specifically, we wanted to see how these procedures compare in terms of not discovering the most important signals (i.e., the signals that are truly and highly significant), given the same number of discoveries made by each of them. The measure of non-discoveries is defined with weights assigned to the signals according to their magnitudes using our chosen severity function s(µ) = µ 2 to capture these most important signals with greater certainty. With that in mind, we generated m = 1, 000 observations according to the model (3.1). Here we chose π 0 = 0.95 and h(µ i ) = π 11 N(µ − , τ 2 ) + π 12 N(µ + , τ 2 ), with π 11 = 0.2, µ − = −1.5, µ + = 1, and τ = 0.5. We then calculated the values of Glfdr given in (2.8), which can be written for this model as Glf dr i = π 0 φ(x i ) π 0 φ(x i )+π 1 H(x i ) with H(x i ) = = π 11 1 √ 1 + τ 2 φ x i − µ − √ 1 + τ 2 τ 2 1 + τ 2 + (τ 2 x i + µ − ) 2 (1 + τ 2 ) + π 12 1 √ 1 + τ 2 φ x i − µ + √ 1 + τ 2 τ 2 1 + τ 2 + (τ 2 x i + µ − ) 2 (1 + τ 2 ) . We ordered these values of Glfdr increasingly as Glf dr (1) ≤ · · · ≤ Glf dr (m) . Let H (i) be the null hypothesis corresponding to Glf dr (i) , for i = 1, . . . , m. For each given R = 1, 2, · · · , m, we marked the first R null hypothesis to be rejected and the rest to be accepted. With θ (i) = 0 or 1 indicating whether the null hypothesis H (i) is true or false (with µ (i) being the true signal), respectively, we then calculated the weighted type II errors m j=R+1 θ (j) µ 2 (j) . We replicated these steps 2,000 times and averaged the 2,000 values of the weighted type II errors before obtaining the simulated value of β * (R), the expected weighted type II errors (or non-discoveries) given R rejections (or discoveries). The red curve in Figure 1 shows the plot of β * (R) against R. The similar plot was obtained for the lf dr score and is shown using the green curve in this figure. As seen from this figure, between the Sun-Cai and our oracle procedures, ours can potentially be more powerful in the sense of producing a smaller amount of weighted type II errors associated with missing the most important signals. Numerical Study 2 We chose π 0 = 0.8, h(µ i ) = π 11 I(µ i = µ − )+π 12 I(µ i = µ + ) with µ − = −3, µ + = 4, and let π 11 vary in (0, 1). This model was also considered in Example 1, Section 3.2, of Sun and Cai (2007) and was chosen here to make the comparison with the Sun and Cai (2007) procedure meaningful. The rejection region for our oracle procedure is {X i : X i ≤ c l or X i ≥ c u } for each H i , with the cut-offs c l and c u being determined following the steps for their calculations as below: • For a given 0 < t < 1, solve the following equation for z to obtain c tπ 1 [π 11 µ 2 1 exp(µ 1 z − 1 2 µ 2 1 ) + π 12 µ 2 2 exp(µ 2 z − 1 2 µ 2 2 )] − π 0 (1 − t) = 0 • Calculate mF DR * = π 0 Ψ(c (t) l , c (t) u ) π 0 Ψ(c (t) l , c (t) u ) + π 1 {π 11 µ 2 1 Ψ(c (t) l − µ 1 , c (t) u − µ 1 ) + π 12 µ 2 2 Ψ(c (t) l − µ 2 , c (t) u − µ 2 )} , where Ψ(c (t) l , c (t) u ) = 1 − Φ(c (t) u ) + Φ(c (t) l ), and Φ is the cdf of N(0, 1). • Repeat the above two steps until we find t * such that the mFDR* converges to α. • c l and c u are then determined as c Once c l and c u are determined, the mFNR * of the oracle procedure is calculated as follows: mF NR * = π 1 {π 11 µ 2 1 [1 − Ψ(c l − µ 1 , c u − µ 1 )] + π 12 µ 2 2 [1 − Ψ(c l − µ 2 , c u − µ 2 ]} π 0 [1 − Ψ(c l , c u )] + π 1 {π 11 µ 2 1 [1 − Ψ(c l − µ 1 , c u − µ 1 )] + π 12 µ 2 2 [1 − Ψ(c l − µ 2 , c u − µ 2 )]} . (3.3) For the p-value based procedure, the rejection region for H i is {X i : \|X i \| ≥ c} where c is determined according to Genovese and Wasserman (2002). The oracle method of Sun and Cai (2007) is the special case of ours with s(µ) = 1. The results of this numerical study are shown in Figure 2. As seen from Figure 2(a), the rejection regions corresponding to our oracle procedure are much wider than those corresponding to both of the other two oracle procedures. From Figures 2(b) and 2(c), we see that while the Sun-Cai oracle procedure has smaller mFNR and mFNR * than those of the p-value based oracle procedure for almost all values of π 11 , ours has the smallest mFNR and mFNR * among all three for each value of π 11 . For instance, the ratio of the mFNR* of our procedure to that of the Sun-Cai oracle procedure can be as small as 0.15. It is thus demonstrated that our proposed approach can potentially be more powerful than the other two approaches. Concluding Remarks The decision theoretic approach to a multiple testing problem is not new. Other relevant work includes Sarkar et al. (2008) and Peña et al. (2011). Nevertheless, the idea of incorporating the severity of type II errors has not been fully explored previously in the literature. We have developed the theory behind our idea from a compound decision theoretic point of view considering a loss function that incorporates the type II error severity. The consideration of type II error severity into the loss function allows us to re-formulate the work of Sun and Cai (2007) in a more general framework involving newer, generalized forms of marginal false discovery and marginal false non-discovery rates. Newer theoretical results generalizing and often improving the existing ones are given in this process. We now have the theory for developing a much wider class of multiple testing procedures constructed from a decision theoretic point of view. Some of the newer methods in this class, those corresponding to non-constant type II error severity, are seen The data are generated according to (3.1) with π 0 = 0.8, π 11 varying from 0 to 1, µ 1 = −3, and µ 2 = 4. For all three procedures, the level of control α is set at 0.05. to have better performance in their oracle forms, as shown in our numerical studies, than those with constant type II error severity (i.e., those in Sun and Cai (2007) and some standard p-value based procedures). The idea of weighting hypotheses or p-values while developing multiple testing methods in an FDR but non-decision theoretic framework has been proposed before. Benjamini and Hochberg (1997) considered weighting the hypotheses in the original definition of the FDR to define the weighted FDR and proposed a weighted version of their 1995 FDR controlling method, the so-called BH method, that controls this weighted FDR. Genovese et al. (2006), on the other hand, weighted each p-value and developed a BH type method controlling the usual FDR based on these weighted p-values. Our concern in this paper has been to define weighted versions of not only the marginal FDR but also the marginal FNR from their original definitions before providing a theoretical framework for the development of our procedure. Our approach to defining weighted mFDR and weighted mFNR is similar to Benjamini and Hochberg (1997). We attach weights to the hypotheses, although they are chosen to effectively act only on the false nulls. More specifically, we have . The weight is assigned to a false null hypothesis according to its signal strength. It does not depend on whether acceptance or rejection of the false null contributes to a measure of false non-discoveries or false discoveries in the form of a penalty or boon. It is important to point out that our weights for all the hypotheses don't add up to m, contrary to what one might conclude from Benjamini and Hochberg (1997). In fact, a careful study of Benjamini and Hochberg (1997) would reveal that such a restriction on the weights is not necessary in their paper, even though they have assumed it. Derivation of an optimal multiple testing procedure incorporating type II error severity in its oracle form has been our primary focus in this paper. Now that we have this oracle procedure, a data-driven version of it with similar optimal property can potentially be constructed. However, construction of such an optimal data-driven procedure depends heavily on the underlying model and the chosen severity function, requiring newer efforts and techniques. We therefore leave this for a future communication. Also, a more comprehensive study of the procedure in terms of its sensitivity under varying choice of the severity function is also on our agenda for future research. Acknowledgement Figure 1 : 1Simulated average weighted type II errors. Figure 2 : 2Comparison of the three procedures: (i) Our oracle procedure controlling the mFDR * (red), (ii) the oracle procedure ofSun and Cai (2007) controlling the mFDR (blue), and (iii) the p-value based oracle procedure of Genovese and Wasserman (2002) (green). mFDR * (δ(T,c)) = E [ m i=1 I(T i < c, θ i = 0)] E [ m i=1 I(T i < c, θ i = 0) + m i=1 I(T i < c, θ i = 1)s(µ i )], and mFNR * (δ(T,c)) =E [ m i=1 I(T i > c, θ i = 1)s(µ i )] E [ m i=1 I(T i > c, θ i = 1)s(µ i ) + m i=1 I(T i > c, θ i = 0)] The research of Li He is supported by Merck Research Fellowship. Sanat K. Sarkar's research is supported by NSF Grants DMS-1006344 and DMS-1208735. Zhigen Zhao's research is supported by NSF Grant DMS-1208735. 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[ "Riemann's ξ-function: A GGC representation", "Riemann's ξ-function: A GGC representation", "Riemann's ξ-function: A GGC representation", "Riemann's ξ-function: A GGC representation" ]	[ "Nicholas G Polson \nUniversity of Chicago\n\n", "Nicholas G Polson \nUniversity of Chicago\n\n" ]	[ "University of Chicago\n", "University of Chicago\n" ]	[]	A GGC (Generalized Gamma Convolution) representation for Riemann's ξ-function and its reciprocal are constructed.	null	[ "https://arxiv.org/pdf/1806.07964v7.pdf" ]	119,732,977	1806.07964	364b6eca0db198cc4588fc784a3410e195ba1fa3	Riemann's ξ-function: A GGC representation 10 Jul 2018 July 8, 2018 Nicholas G Polson University of Chicago Riemann's ξ-function: A GGC representation 10 Jul 2018 July 8, 2018arXiv:1806.07964v6 [math.CA]RHGGCZeta functionXi function A GGC (Generalized Gamma Convolution) representation for Riemann's ξ-function and its reciprocal are constructed. Introduction Riemann (1859) defines the ζ-function via the analytic continuation of ∞ n=1 n −s on the region Re(s) > 1 and the ξ-function by ξ(s) = 1 2 s(s − 1)π − 1 2 s Γ ( 1 2 s) ζ(s). (1) Polson (2017) provides a GGC (Generalized Gamma Convolution) representation for Riemann's ξ-function and its reciprocal. Theorem 1 provides a modification of the representation which applies to the reciprocal ξ-functions ξ(1)/ξ(1+ √ s) and ξ( 1 2 )/ξ( 1 2 + √ s). Bondesson (1992) defines the GGC class of probability distributions on [0, ∞) whose Laplace transform (LT) takes the form, for s > 0, E e −sH = exp −as + (0,∞) log z z + s U (dz) with (left-extremity) a ≥ 0 and U (dz) a non-negative measure on (0, ∞) (with finite mass on any compact set of (0, ∞)) such that (0,1) \| log t\|U (dz) < ∞ and (0, ∞) z −1 U (dz) < ∞. The sigma-finite measure U on (0, ∞) is chosen so that the exponent φ(s) = (0,∞) log(1 + s/z)U (dz) = ∞ 0 ∞ 0 (1 − e −sz )t −1 e −tz U (dz) < ∞(2) and U is often referred to as the Thorin measure, which can have infinite mass. The corresponding Lévy measure is t −1 (0,∞) e −tz U (dz). D = τ >0 Y τ where Y τ ∼ Exp(τ ). The rest of the paper is outlined as follows. Section 2 derives the GGC representation of Riemann's ξ-function and its reciprocal based on the results of Polson (2017). The function ξ(1)/ξ(1+ √ s) is expressed as the LT of a GGC distribution. Section 3 concludes with a discussion. Riemann's ξ-function and GGC representations The following Theorem is based on Polson (2017, Theorem 1) and provides our GGC representation of Riemann's ξ-function and its reciprocal. First, by definition, ξ(α + s) = (α − 1 + s)π − 1 2 (α+s) Γ (1 + 1 2 (α + s)) ζ(α + s) (3) ξ (α) = (α − 1)π − 1 2 α Γ (1 + 1 2 α) ζ (α) .(4) Theorem 1. Riemann's ξ-function satisfies, for α > 1 and s > 0, ξ(α + s) ξ(α) = exp ξ ′ (α) ξ(α) s + ∞ 0 (e −sx + sx − 1)e −αx µ ξ (dx) x (5) µ ξ (dx) x = e x x dx + 1 x(e 2x − 1) dx + n≥2 Λ(n) log n δ log n (dx).(6) Riemann's reciprocal ξ-function satisfies ξ(α) ξ(α + s) = exp − ξ ′ (α) ξ(α) s − ∞ 0 (1 − e − 1 2 s 2 t ) ν α (t) t dt (7) := E(exp(−s 2 H ξ α )) (8) where H ξ α has a GGC distribution. The measure ν α (t) = ∞ 0 e −tz U α (z)dz is completely monotone, and U α (z) is given by U α (z) = 1 √ 2π ∞ 0 2 sin 2 (x z/2)e −αx µ ξ (dx) 1 √ πz .(9) Proof. From the decomposition in (3), for α > 1 and s > 0, ξ(α + s) ξ(α) e −s ξ ′ (α) ξ(α) = 1 + s α − 1 e − s α−1 · Γ (1 + 1 2 (α + s)) Γ (1 + 1 2 α) e −s 1 2 ψ(1+ 1 2 α) · ζ(α + s) ζ (α) e −s ζ ′ ζ (α) .(10) Write ζ and Γ as Lévy representations (Polson, 2017), for α > 1 and s > 0, ζ(α + s) ζ(α) e − ζ ′ (α) ζ(α) s = exp ∞ 0 (e −sx + sx − 1)e −αx µ ζ (dx) x (11) µ ζ (dx) x = p µ ζ p (dx) x = n≥2 Λ(n) log n δ log n (dx)(12) where Λ(n) is the von Mangoldt function. For s > 0, the Gamma function is Γ (1 + 1 2 (α + s)) Γ (1 + 1 2 α) e −s 1 2 ψ(1+ 1 2 α) = exp ∞ 0 (e −sx + sx − 1)e −αx µ ζ (dx) x (13) µ Γ (dx) = dx e 2x − 1 .(14) For s > 0 and α > 1, the first term on rhs of (10) is 1 + s α − 1 e − s α−1 = exp ∞ 0 (e −sx + sx − 1)e −αx µ 1 (dx) x (15) µ 1 (dx) = e x dx.(16) This follows from ∞ 0 e −(α−1)x dx = 1/(α − 1) and Frullani's identity log 1 + s α − 1 = ∞ 0 (e −sx − 1)e −αx e x x dx. Now, taking derivatives of log ξ(s) at s = α, with ψ(s) = Γ ′ (s)/Γ(s), gives ξ ′ ξ (α) = 1 α − 1 − 1 2 log π + ζ ′ ζ (α) + 1 2 ψ (1 + 1 2 α) .(17) Combining terms with µ ξ (dx) := µ 1 (dx) + µ Γ (dx) + µ ζ (dx), gives the result ξ(α + s) ξ(α) = exp ξ ′ (α) ξ(α) s + ∞ 0 (e −sx + sx − 1)e −αx µ ξ (dx) x (18) µ ξ (dx) x = e x x dx + 1 x(e 2x − 1) dx + n≥2 Λ(n) log n δ log n (dx).(19) The reciprocal ξ-function follows from the identities (Polson, 2017), for s > 0 and x > 0, e −sx + sx − 1 = ∞ 0 (1 − e − 1 2 s 2 t )(1 − e − 1 2 x 2 /t ) x √ 2πt 3 dt (20) 1 − e − 1 2 x 2 /t √ t = ∞ 0 e −tz 2 sin 2 (x z/2) √ πz dz.(21) Therefore, for s > 0 and α > 1 ξ(α) ξ(α + s) = exp − ξ ′ (α) ξ(α) s − ∞ 0 (1 − e − 1 2 s 2 t ) ν α (t) t dt(22) Let U α (dz) = U α (z)dz from (9), so, by construction, (0,∞) log(1 + s/z)U α (dz) < ∞. Using the identity s 1 2 = 1 √ 2π ∞ 0 (1 − e − 1 2 st ) ν(t) t dt where ν(t) := t − 1 2 = ∞ 0 e −tz dz √ πz(23) leads to the GGC representation, with c α = ξ ′ (α) ξ(α) / √ 2π and α > 1, ξ(α) ξ(α + √ s) = exp − ∞ 0 (1 − e − 1 2 st ) ν ξ α (t) t dt where ν ξ α (t) = c α ν(t) + ν α (t) (24) = E(exp(−sH ξ α )). (25) Now ν ξ α (t) = ∞ 0 e −tz U ξ α (z)dz, where U ξ α (z) = c α / √ πz + U α (z),As H ξ α D = τα>0 Y τ is GGC with Y τα ∼ Exp(τ α ) this provides a Hadamard factorisation of the ξ-function, see Polson (2018) and Roynette and Yor (2005). The measure ν α (t) = ν 1 α (t) + ν Γ α (t) + ν ζ α (t) can be explicitly calculated as ν 1 α (t) = 1 √ 2πt ∞ 0 (1 − e −x 2 /2t )e −(α−1)x dx (26) ν Γ α (t) = 1 √ 2πt ∞ 0 1 − e −x 2 /2t 1 − e −2x e −(α+2)x dx (27) ν ζ α (t) = 1 √ 2πt n≥2 Λ(n) n α 1 − e −(log 2 n)/2t (28) = 1 √ 2πt p prime log p    r≥1 1 p αr 1 − e −(r 2 log 2 p)/2t    .(29) The following Corollary holds as the GGC property is closed under limits as α → 1. Corollary 1. The reciprocal ξ-function satisfies, for s > 0, ξ(1) ξ(1 + √ s) = E(exp(−sH ξ 1 )).(30) Given ξ(1) = 1 2 and ξ ′ (1) > 0, this implies c 1 < ∞ and (0,∞) log(1 + s/z)U 1 (dz) < ∞. ξ( 1 2 ) ξ( 1 2 + √ s) = E(exp(−sH ξ 1 2 )).(31) Proof. Now (18) holds for Re(s) > 1 − α, and hence transforming ξ(α + s) ξ(α) = exp ξ ′ (α) ξ(α) s + ∞ 0 (e −sx + sx − 1)e −αx µ ξ (dx) x (32) with s → s − (α − 1 2 ) for s > 1 2 , yields ξ( 1 2 + s) ξ(α) = exp ξ ′ (α) ξ(α) (s − (α − 1 2 )) + ∞ 0 (e −sx − 1)e ((α− 1 2 )−α)x µ ξ (dx) x(33)+ ∞ 0 (s − (α − 1 2 ))e −αx µ ξ (dx) + ∞ 0 (e (α− 1 2 )x − 1)e −αx µ ξ (dx) x (34) = exp ξ ′ (α) ξ(α) ( 1 2 − α) + ∞ 0 (e −( 1 2 −α)x + ( 1 2 − α)x − 1)e −αx µ ξ (dx) x (35) + ξ ′ (α) ξ(α) − ∞ 0 (e − 1 2 x − e −αx )µ ξ (dx) s + ∞ 0 (e −sx + sx − 1)e − 1 2 x µ ξ (dx) x .(36) The linear term in s is zero as (ξ ′ /ξ)(α) = ∞ 0 (e − 1 2 x − e −αx )µ ξ (dx) for α > 1 with (ξ ′ /ξ)( 1 2 ) = 0. Therefore, using (20), ξ( 1 2 + s) ξ( 1 2 ) = exp ∞ 0 (e −sx + sx − 1)e − 1 2 x µ ξ (dx) x (37) = exp ∞ 0 (1 − e − 1 2 s 2 t ) ν 1 2 (t) t dt(38) where the Thorin measure, U 1 2 , is given by ν 1 2 (t) = ∞ 0 e −tz U 1 2 (z)dz = 1 √ 2πt ∞ 0 (1 − e − 1 2 x 2 /t )e − 1 2 x µ ξ (dx)(39)= ∞ 0 e −tz 1 √ 2π 2 z ∞ 0 2 sin 2 (x z/2)e − 1 2 x µ ξ (dx) dz.(40) Hence, there exists a GGC, H ξ 1 2 , and a HCM condition for the reciprocal ξ-function ξ( 1 2 ) ξ( 1 2 + √ s) = E(exp(−sH ξ 1 2 )). This provides an alternative derivation of Theorem 2 in Polson (2017). This is known as Thorin's condition for the Riemann Hypothesis (Bondesson, 1992, p.128). is completely monotone by Bernstein's theorem. Hence there exists a GGC distribution, H ξ α , as required. Dantzig pairs and Wald couples of random variables. Polsonand to the prime number theorem via the behavior of ζ(1 + scalculations are also related to van Dantzig pairs and Wald couples of random variables (Polson, 2018) and to the prime number theorem via the behavior of ζ(1 + s). Classes of infinitely divisble distributions and densities. L Bondesson, Z. Wahrsch. Verw. Gebiete. 57Bondesson, L. (1981). Classes of infinitely divisble distributions and densities. Z. Wahrsch. Verw. Gebiete, 57, 39-71. Generalised Gamma Convolutions and Related Classes of Distributions and Densities. L Bondesson, Springer-VerlagNew YorkBondesson, L. (1992). Generalised Gamma Convolutions and Related Classes of Distributions and Densities. Springer-Verlag, New York. The Spectral Decomposition of a Diffusion Hitting Time. J T Kent, Ann. Prob. 10Kent, J. T. (1982). The Spectral Decomposition of a Diffusion Hitting Time. Ann. Prob., 10, 207-219. N G Polson, arXiv:1708.02653On Hilbert's 8th problem. Polson, N. G. (2017). On Hilbert's 8th problem. arXiv: 1708.02653. N G Polson, arXiv:1804.10043Wald couples and Hadamard Factorisation. van Dantzig pairsPolson, N. G. (2018). van Dantzig pairs, Wald couples and Hadamard Factorisation. arXiv: 1804.10043. Über die Anzahl der Primzahlen unter einer gegebenen Grösse. Monatsberichte der Berliner Akademie. B Riemann, Riemann, B. (1859).Über die Anzahl der Primzahlen unter einer gegebenen Grösse. Monatsberichte der Berliner Akademie. Infinitely Divisible Wald's Couples: Examples linked with the Euler gamma and the Riemann zeta functions. B Roynette, M Yor, Annales de l'institut Fourier. 554Roynette, B. and M. Yor (2005). Infinitely Divisible Wald's Couples: Examples linked with the Euler gamma and the Riemann zeta functions. Annales de l'institut Fourier, 55 (4), 1219-1283. The Theory of the Riemann Zeta-function. E C Titchmarsh, Oxford University PressTitchmarsh, E.C. (1974). The Theory of the Riemann Zeta-function. Oxford University Press.	[]
[ "TITCHMARSH-WEYL THEORY FOR VECTOR-VALUED DISCRETE SCHRÖDINGER OPERATORS", "TITCHMARSH-WEYL THEORY FOR VECTOR-VALUED DISCRETE SCHRÖDINGER OPERATORS" ]	[ "Raj Keshav \nDepartment of Mathematics\nEmbry-Riddle Aeronautical University Daytona Beach\n32114-3900FLU.S.A\n", "Acharya [email protected] \nDepartment of Mathematics\nEmbry-Riddle Aeronautical University Daytona Beach\n32114-3900FLU.S.A\n" ]	[ "Department of Mathematics\nEmbry-Riddle Aeronautical University Daytona Beach\n32114-3900FLU.S.A", "Department of Mathematics\nEmbry-Riddle Aeronautical University Daytona Beach\n32114-3900FLU.S.A" ]	[]	We develop the Titchmarsh-Weyl theory for vector-valued discrete Schrödinger operators and show that the Weyl m functions associated with these operators map complex upper half plane to the Siegel upper half space. We also discuss about the Weyl disk and Weyl circle corresponding to these operators.	null	[ "https://arxiv.org/pdf/1708.04306v1.pdf" ]	119,323,887	1708.04306	d0f8e5fa4a0825c5f131c4f7ee89d695c95d4034	TITCHMARSH-WEYL THEORY FOR VECTOR-VALUED DISCRETE SCHRÖDINGER OPERATORS 14 Aug 2017 Raj Keshav Department of Mathematics Embry-Riddle Aeronautical University Daytona Beach 32114-3900FLU.S.A Acharya [email protected] Department of Mathematics Embry-Riddle Aeronautical University Daytona Beach 32114-3900FLU.S.A TITCHMARSH-WEYL THEORY FOR VECTOR-VALUED DISCRETE SCHRÖDINGER OPERATORS 14 Aug 2017Discrete Schrödinger operatorTitchmarsh-Weyl m-function AMS (MOS) Subject Classification: 39A7047A0534B20 We develop the Titchmarsh-Weyl theory for vector-valued discrete Schrödinger operators and show that the Weyl m functions associated with these operators map complex upper half plane to the Siegel upper half space. We also discuss about the Weyl disk and Weyl circle corresponding to these operators. Introcuction The goal of this paper is to extend the Titchmarsh-Weyl theory for vector valued discrete Schrödinger operators. We consider a discrete Schrödinger equation in d− dimensional space of the form y(n + 1) + y(n − 1) + B(n)y(n) = zy(n), z ∈ C (1. 1) where y(n) = [y 1 (n) y 2 (n), . . . y d (n)] t ( t stands for a transpose), is a vector valued sequence in l 2 (I, C d ). Usually I = Z or I = N. Here l 2 (I, C d ) is a Hilbert space of square summable vector valued sequences with the inner product u, v = n∈I u(n) * v(n), where " * " stands for conjugate transpose and B(n) is a symmetric d×d matrix. We denote the space of all d × d complex matrices by C d×d . The equation (1.1) can be generalized to a d−dimensional Jacobi equation of the form A(n)y(n + 1) + A(n − 1)y(n − 1) + B(n)y(n) = zy(n), z ∈ C (1. is called a block Jacobi matrix. Some studies about the block Jacobi matrix can be found in the paper [10]. Equation (1.1) is a particular case of Jacobi equation with A(n) ≡ 1. The equation (1.1) induces a discrete Schrödinger operator J on l 2 (I, C d ) as J y(n) = y(n + 1) + y(n − 1) + B(n)y(n). It can be easily observed that if B(n) is a Hermitian matrix, B(n) * = B(n), then J is a self-adjoint operator on l 2 (N, C d ). Then, the spectrum of J is a set of real numbers: σ(J) ⊂ R. To get a solution of the equation (1.1), we may fix any two vectors c, d ∈ C d at two consecutive sites, that is, we fix the values u(k) = c, u(k + 1) = d and evolve according to (1.1). In particular, we fix u(0) and u(1) then any u(n) is obtained by solving the difference equation (1.1) using transfor matrices: T (m; z) = zI − B(m) −I I 0 (1.3) where I is an d × d identity matrix. Let A(n; z) = T (n; z) × · · · × T (1, z) × I. (1.4) Then, u solves (1.1) for every n if and only if u(n + 1) u(n) = A(n; z) u(1) u(0) (1.5) This matrix can also be used to get a solution at cite n from cite m as u(n + 1) u(n) = A(n, m; z) u(m + 1) u(m) . For every pair of vectors c, d ∈ C d , there exists a solution of (1.1), therefore, the space of solutions of (1.1) is a 2d-dimensional vector space. In [1], it is shown that are exactly d linearly independent solutions of (1.1) that are in l 2 (N, C d ). It is now convenient to fix a basis of the solution space of (1.1). An easier way is to prescribe a pair of initial conditions. For z ∈ C, let U (n, z) = (u 1 (n), u 2 (n), . . . , u d (n)), V (n, z) = (v 1 (n), v 2 (n), . . . , v d (n)) (1.6) where u i (n) = [u 1,i (n) u 2,i (n) . . . u d,i (n)] t v i (n) = [v 1,i (n) v 2,i (n) . . . v d,i (n) ] t are solutions of (1.1). Thus, both of the sets U (n, z) and V (n, z) consists of d linearly independent solutions of (τ − z)u(n) = 0, where τ is the expression on the left side of equation (1.1). For our convenience, we call these sets as matrix valued solutions of (1.1). We further suppose that these solutions satisfy the following initial conditions (1.7) U (0, z) = −I, V (0, z) = 0, U (1, z) = 0, V (1, z) = I. By iterating the difference equation, we see that for fixed n ∈ N, U (n, z), V (n, z) are polynomial of degree n − 2 over C d×d . So U (n, z) = U (n,z) and V (n, z) = V (n,z). We generalize the equation (1.5) for the matrix valued solutions U (n, z), V (n, z) as U (n + 1, z) V (n + 1, z) U (n, z) V (n, z) = A(n; z) U (1, z) V (1, z) U (0, z) V (0, z) = A(n; z) 0 I −I 0 = A(n; z)J, where J = 0 I −I 0 Lemma 1.1. Suppose n ∈ N 0 = N ∪ {0}, and W (z) = U (n + 1, z) V (n + 1, z) U (n, z) V (n, z) then W t JW = W JW t = J Proof. Notice that T (n; z) t JT (n; z) = T (n; z)JT (n; z) t = J for any n so that A(n; z) t JA(n; z) = A(n; z)JA(n; z) t = J. Then W t JW = (A(n; z)J) t JA(n; z) = J t A(n; z) t JA(n; z)J = J t JJ = J Exactly the same way we can see: W JW t = J Definition 1.2. The Wronskian of any two sequences f (n, z), g(n, z) ∈ l 2 (N, C d ) is defined by W n (f, g) = [f * (n + 1,z)g(n, z) − f * (n,z)g(n + 1, z)]. This definition incorporate with the definition in one dimensional space and in the continuous case. In [1], it is shown that for fixed z ∈ C, if f (n, z), g(n, z) ∈ l 2 (N, C d ) are any two solutions of (1.1) then W n (f, g) is independent of n. Moreover, the Wronskian W n is linear in both arguments. For f (n, z), g(n, z) ∈ l 2 (N 0 , C d ) the Green's identity corresponding to equation (1.1) is given by n j=0 f * (τ g) − (τ f ) * g (j) = W 0 (f , g) − W n (f , g). We extend the definition of Wronskian and the Green's identity for the matrix valued solutions U (n, z), V (n, z), each contains d linearly independent solutions of (1.1) for fixed z ∈ C. W n (U, V ) = [U * (n + 1,z)V (n, z) − U * (n,z)V (n + 1, z)]. It is shown in [1] that the Wronskian W n (U, V ) is a matrix independent of n ∈ N. We extend the Green's Identity for these matrix valued solutions. N j=0 F (j, z) * (τ G(j, z)) − (τ F (j, z)) * G(j, z) = W 0 (F , G) − W N (F , G). (1.8) Again, the proof of the Green's identity can be found in [1]. Titchmarsh-Weyl m function The theory of Titchmarsh-Weyl m functions is very important tool in the spectral theory of Jacobi and Schrödinger operators. In order to study the asymptotic behavior of solutions of Jacobi and Schrödinger equations, one need to study these m functions. Moreover, the absolutely continuous, singular continuous and essential spectrum of the operators associated with these equations are well explained in terms of m functions. These m functions were first introduced in 1910 by H. Weyl in [16] for Sturn-Liouville differential equations. It was further studied by E. C. Titchmarsh in [15] and established the connection between the analyticity of the solution and the spectrum of the operator of Sturn-Liouville differential equations. For further history of m function, one can see [6]. The theory of m functions in one dimensional space has been widely studied, some of which can be found in the papers [2,3,8,11,13,14]. The Titchmarsh-Weyl m function for the vector-valued discrete Schrödinger operators associated to the equation (1.1) is defined in terms of solutions as follows. Definition 2.1. Let z ∈ C + = {z ∈ C : Im(z) > 0}. The Titchmarsh-Weyl m function is defined as the unique complex matrix M (z) ∈ C d×d such that F (n, z) = U (n, z) + V (n, z)M (z) (2.1) where U (n, z), V (n, z) are matrix valued solutions consisting of d linearly independent solutions with initial values (1.7) and the matrix valued solution F (n, z) is a set of d linearly independent solutions of (1.1) that are in l 2 (N, C d ). This definition, is in fact well defined. As we mentioned above that there are only d linearly independent solutions in l 2 (N 0 , C d ), if there is another M (z) satisfying the above conditions then the solutions from both U (n, z) and V (n, z) will be in l 2 (N 0 , C d ). The solution V (n, z) is such that V (0, z) = 0 which implies that V (n, z) is the set of eigen-functions for the self adjoint operator J. This contradicts that the spectrum of J is a set of real numbers. Theorem 2.2. [1] Let z ∈ C + . If (τ − z)F = 0 and F is a d × d matrix valued solution whose d columns are linearly independent solutions of (1.1) that are in l 2 (N, C d ). Then M (z) = −F (1, z)F (0, z) −1 . (2.2) Moreover, M (z) = (m ij (z)) d×d ∈ C d×d , m ij (z) = δ j , (J − z) −1 δ i . (2.3) Proof. If the matrix valued soulution F is given by ( 2.1) then F (0, z) = −I and F (1, z) = M (z). So (2.2) holds. Suppose G(n, z) is any d×d matrix valued solution then it is a constant (matrix) multiple of the solution set F (n, z) from (2.1) because (2.1) is a set of d linearly independent solutions. That is, G(n, z) = F (n, z)C where C is a d × d scalar invertible matrix. F (n, z) = G(n, z)C −1 so that −G(1, z)G(0, z) −1 = − F (1, z)CC −1 F (0, z) −1 = − F (1, z)F (0, z) −1 =M (z). Let F (n, z) as in (2.2) and let g i = (J − z) −1 δ i where δ i (n) ∈ l 2 (N, C d ) such that the values of δ i (n) = 0 if i = 0 and δ i (i) = [1, 0, . . . 0] t . Then (J − z)g i = δ i . So (τ − z)g i (n) = 0 for n ≥ 2. Moreover g i ∈ l 2 for all i = 1, 2, ......., d. Let G(n, z) = [g 1 , g 2 , ......., g d ]. Then G(n, z) = F (n, z)C, C ∈ C d×d . By comparing values at n = 1, G(1, z) = [g 1 (1), g 2 (1), ........., g d (1)]. Here g 1 (1) = (J − z) −1 δ 1 (1) and g 1 = [g 11 , g 21 , ..., ..., ..., g d1 ] t , g i1 = δ i , g 1 , i = 1, 2, ...., d. Then M (z) = G(1, z)C −1 and M (z) = (m ij (z)) = ( δ j , (J − z) −1 δ i )C −1 . To find the value of C, we compare values at n = 2. m ij (z) = R 1 t − z dµ ij where µ ij is a spectral measure for the vectors δ j and δ i . Therefore, M (z) = R 1 t − z dµ, µ = (µ ij ) d×d and M (z) = R 1 t − z dµ = R 1 t − z dµ ij d×d The matrix valued measure µ is a spectral measure of the d−dimensional discrete Schrödinger operator J. For each i, j the entries m i,j (z) maps complex upper half plane to itself. For if z ∈ C + , Im m ij (z) = 1 2i (m ij (z) − m ij (z)) = R y \|t−z\| 2 dµ ij > 0 Suppose M (z)m ij (z) = δ j , (J − z) −1 δ i = (J −z)(J −z) −1 δ j , (J − z) −1 δ i = (J −z) −1 δ j , (J −z) * (J − z) −1 δ i Since J is self adjoint, (J −z) * = (J − z) m ij (z) = (J −z) −1 δ j , δ i = δ i , (J −z) −1 δ j =m ji (z) =m ji (z) for all i, j. Hence M (z) t = M (z). Thus we proved the following proposition. Let S be a subspace of C d×d , consisting of all symmetric matrices with positive definite imaginary part. That is, S = {M ∈ C d×d : 1 2i (M − M * ) > 0} The space S is called a Seigel upper half space. From above discussion we proved Theorem 2.4. For z ∈ C + , the map z → M (z) maps complex upper half plane C + to Seigel upper half space S. Titchmarsh-Weyl circles and disks In this section, we define the Titchmarsh-Weyl circles and disks. We consider the equation (1.1) on a compact interval [0, N ]. Suppose U (n, z), V (n, z) are the matrix valued solutions of (1.1) with initial values (1.7). For z ∈ C + , define a matrix valued solution F (n, z) = U (n, z) + V (n, z)M β N (z) satisfying a boundary condition β 2 F (N, z) + β 1 F (N + 1, z) = 0 where β = [β 1 , β 2 ] ∈ R d×2d , β 1 , β 2 ∈ R d×d , β t β = I, βJβ t = 0. (3.1) The unique coefficient M β N (z) is called the Weyl m function on the interval [0, N ]. On solving we see that, M β N (z) = − β 2 V (N, z) + β 1 V (N + 1, z) −1 β 2 U (N, z) + β 1 U (N + 1, z) . (3.2) Note that β 2 V (N, z) + β 1 V (N + 1, z) is invertible. Since z, N.β varies, M β N (z) becomes a function of these arguments, and since U, V are matrix polynomials with entries meromorphic functions of z. Lemma 3.1. The weyl m function M β N (z) on [0, N ] is symmetric. Proof. Let U(z) = U (N + 1) U (N ) = A(N ; z) U (1) U (0) = A(N ; z) 0 −I and V(z) = V (N + 1) V (N ) = A(N ; z) V (1) V (0) = A(N ; z) I 0 Using equation (3.2), the Weyl m function can be written as M β N (z) = −(βV(z)) −1 (βU(z)). Suppose E = βU(z) and F = βV(z) so that M β N (z) = −F −1 E. Now, M β N (z) T − M β N (z) = F −1 E − (F −1 E) T = F −1 [F E T − EF T ]F −T = F −1 [βV(βU) T − βU(βV) T ]F −T = F −1 β[VU T − UV T ]β T F −T = F −1 β A(N ; z) I 0 A(N ; z) 0 −I T − A(N ; z) I 0 A(N ; z) 0 −I T β T F −T = −F −1 β A(N ; z)JA(N ; z) T β T F −T = −F −1 βJβ T F −T = 0 Lemma 3.2. For a matrix valued solution F (n, z) = U (n, z) + M β N (z)V (n, z) of (1.1) we have W N (F , F ) = 2i Im M − 2i Im z N j=0 F (j, z) * F (j, z). Proof. We use the Greens identity (1.8) with G = F. N j=0 F (j, z) * (τ F (j, z)) − (τ F (j, z)) * F (j, z) = W 0 (F , F ) − W N (F , F ) (z −z) N j=0 F (j, z) * F (j, z) = W 0 (F , F ) − W N (F , F ) For F (n, z) = U (n, z) + M β N (z)V (n, z), using the linearity of the Wronskian we get N j=0 F (j, z) * (τ F (j, z)) − (τ F (j, z)) * F (j, z) = W 0 (F , F ) − W N (F , F ) (z −z) N j=0 F (j, z) * F (j, z) = W 0 (F , F ) − W N (F , F ) = W 0 (U + V M , U + V M ) − W N (F , F ) = W 0 (U , U ) + W 0 (U , V M ) + W 0 (V M , U ) + W 0 (V M , V M ) − W N (F , F ). Then we have Here W 0 (U , U ) = W 0 (V M , V M ) = 0, W 0 (V M , U ) = −M , W 0 (U , V M ) = M (z −z) N j=0 F (j, z) * F (j, z) = M −M − W N (F , F ) 2i Im z N j=0 F (j, z) * F (j, z) = 2i Im M − W N (F , F ) W N (F , F ) = 2i Im M − 2i Im z N j=0 F (j, z) * F (j, z) The condition on β in the boundary condition (3.1) implies that β 1 and β 2 are invertible. Observe that Proof. By lemma 3.1M β N (z) is symmetric. Since M β N (z) ∈ C(N, z), E(M, N ) = 0. It follows that −iW N (F , F ) = 0. By lemma 3.2 we have Equation (3.2) is written as M β N (z) = − β 2 V (N, z) + β 1 V (N + 1, z) −1 β 2 U (N, z) + β 1 U (N + 1, z) = − β −1 1 β 2 V (N, z) + V (N + 1, z) −1 β −1 1 β 2 U (N, z) + U (N + 1, z) = − γV (N, z) + V (N + 1, z) −1 γU (N, z) + U (N + 1, z) , γ = β −1 1 β 2 ∈ R d×d .E(M, N ) = −i[F (N + 1, z) * , F (N, z) * ]J F (N + 1, z) F (N, z) = −iW N (F , F ) = −2 Im M + 2 Im z N j=0 F (j, z) * F (j, z).2 Im M − 2 Im z N j=0 F (j, z) * F (j, z) = 0. That is Im M Im z = N j=0 F (j, z) * F (j, z) > 0. This implies that Im M is positive definite. F (j, z) * F (j, z) ≤ −2 Im M + 2 Im z N +1 j=0 F (j, z) * F (j, z) = E(M, N + 1) ≤ 0. This shows that M ∈ D(N, z). Hence the result. From above we have, N ) can be written as E(M, N ) = −i[I, M * ] U (N + 1, z) * U (N, z) * V (N + 1, z) * V (N, z) * J U (N + 1, z) V (N + 1, z) U (N, z) V (N, z) I M = −i[I, M * ] W N (Ū , U ) W N (Ū , V ) W N (V , U ) W N (V , V ) I M = −i[W N (Ū , U ) + W N (Ū , V )M + M * W N (V , U ) + M * W N (V , V )M ] Using W N (V , V ) * = −W N (V , V ) and W N (V , U ) * = −W N (Ū , V ), E(M,E(M, N ) = −i [M − W N (V , V ) −1 W N (Ū , V ) * ] * W N (V , V )[M − W N (V , V ) −1 W N (Ū , V ) * ] (3.4) +W N (Ū , U ) + W N (Ū , V )W N (V , V ) −1 W N (Ū , V ) * Lemma 3.6. For z ∈ C + , W N (Ū , V )W N (V , V ) −1 W N (Ū , V ) * + W N (Ū , U ) = −W N (V,V ) −1 Proof. Let W = W * JW. Notice that W * JW = W N (Ū , U ) W N (Ū , V ) W N (V , U ) W N (V , V ) . From lemma 1.1 we see that W * JW = J. Then W t JW = (W * JW ) t J(W * JW ) = W t J t W * t JW * JW = −W t JW = J. On the other hand, W t JW = W N (Ū , U ) t W N (V , U ) t W N (Ū , V ) t W N (V , V ) t J W N (Ū , U ) W N (Ū , V ) W N (V , U ) W N (V , V ) = W N (U,Ū ) * W N (V,Ū ) * W N (U,V ) * W N (V,V ) * J W N (Ū , U ) W N (Ū , V ) W N (V , U ) W N (V , V ) By direct computation we see that −W N (V,V ) * W N (Ū , U ) + W N (U,V ) * W N (V , U ) = −I (3.5) −W N (V,V ) * W N (Ū , V ) + W N (U,V ) * W N (V , V ) = 0. (3.6) From equation (3.6) we have W N (Ū , V ) * = W N (V , V ) * W N (U,V )W N (V,V ) −1 = −W N (V , V )W N (U,V )W N (V,V ) −1 Using this we get W N (Ū , V )W N (V , V ) −1 W N (Ū , V ) * + W N (Ū , U ) = W N (Ū , V )W N (V , V ) −1 [−W N (V , V )W N (U,V )W N (V,V ) −1 ] + W N (Ū , U ) = −W N (Ū , V )W N (U,V )W N (V,V ) −1 + W N (Ū , U ) Also from equation (3.5) we get, W N (U,V ) * W N (V , U ) = −I + W N (V,V ) * W N (Ū , U ) −W N (U,V ) * W N (Ū , V ) * = −I + W N (V,V ) * W N (Ū , U ) (W N (Ū , V )W N (U,V )) * = I − W N (V,V ) * W N (Ū , U ) W N (Ū , V )W N (U,V ) = I − W N (Ū , U ) * W N (V,V ) W N (Ū , V )W N (U,V ) = I + W N (Ū , U )W N (V,V ) Then, W N (Ū , V )W N (V , V ) −1 W N (Ū , V ) * + W N (Ū , U ) = −(I + W N (Ū , U )W N (V,V ))W N (V,V ) −1 + W N (Ū , U ) = −W N (V,V ) −1 . Using Using lemma (3.6) and equation (3.4) we can express E(M, N ) in the form E(M, N ) = −i [M − W N (V , V ) −1 W N (Ū , V ) * ] * W N (V , V ) [M − W N (V , V ) −1 W N (Ū , V ) * ] − W N (V,V ) −1 . Thus it can be expressed as Proof. By Green's identity we have E(M, N ) = −[(M − C N (z)) * R(N, z) −2 (M − C N (z)) − R(N,z) 2 ] where C N (z) = W N (V , V ) −1 W N (Ū ,2 Im z N j=0 V (j, z) * V (j, z) = iW N (V , V ) = R(N, z) −2 > 0. Also R(N, z) −2 is nondecreasing. Thus, R(N, z) is non increasing and so lim N →∞ R(N, z) exists. This shows that C N (z) is a Cauchy sequence, hence converges. Let C 0 (z) = lim N →∞ C N (z) and R 0 (z) = lim N →∞ R(N, z). Define D 0 (z) = {M ∈ C d×d : (M − C 0 (z)) * R 0 (z) −2 (M − C 0 (z)) ≤ R 0 (z) 2 then D 0 (z) = ∩ N ≥1 D(N, z). F (j, z) * F (j, z) ≤ Im M Im z . Taking limit as N → ∞ we get So E(M, N ) ≤ 0 for all N and hence M ∈ D 0 (z). Similar explanation also proves (2). (n), B(n) are sequences of d × d matrices. If I = N The equation (1.2) can be written in the First (J − z)G(1, z) = (δ 1 , δ 2 , ......, δ It follows that G(2, z) + B(1)G(1, z) − zG(1, z) = I G(2, z) = (z − B(1))G(1, z) + I.............(i) Also, F (2, z) = (z − B(1))F (1, z) − F (0, and (ii), we get −F (0, z)C = I and so I.C = I =⇒ C = I. allows us to connect the m function with a matrix valued Borel measure using functional calculus for these resolvent operators δ j , (J − z) −1 δ i , where δ i (n) ∈ l 2 (N, C d ) such that the values of δ i (n) = 0 if i = 0 and δ i (i) = [1, 0, . . . 0] t By functional calculus, denotes the complex conjugate of M (z) obtained by taking the complex conjugate of each entries of M (z). Then by integral representation of m ij (z), we have m ij (z) = m ij (z)) so that M (z) = M (z). Also, M (z) = (m ij (z)) = ( δ j , (J − z) −1 δ i ) so that Proposition 2. 3 . 3M (z) * = M (z), The imaginary part of M (z) is Im M (z) = 1 2i (M (z) − M (z) * ) and it is clear from the above observation that Im M (z) > 0. Again solving for γ we have, γ = −F (N + 1, z)F (N, z) −1 . Observe that ℑγ = 1 2i (γ − γ * ) = 0 Let W(N, z, M ) = U (N + 1, z) V (N + 1, z) U (N, z) V (N, z) I M . Define a matrix function E(M, N ) = −iW(N, z, M ) * JW(N, z, M ) . Let z ∈ C + . The set D(N, z) = {M ∈ C d×d \|E(M, N ) ≤ 0} and C(N, z) = {M ∈ C d×d \|E(M, N ) = 0} are respectively called the Weyl disk and Weyl circle. Clearly, C N (z) = {M β N (z) : β ∈ R d×d , satisfying (3.1)}. Thus for any complex symmetric matrix M ∈ C d×d M ∈ C(N, z) ⇐⇒ Im(−F (N + 1, z)F (N, z) −1 ) = 0 Theorem 3.4. The map z → M β N (z) maps complex upper half plane to Seigel half space. Lemma 3. 5 ( 5Nesting property of Weyl disks). Let z ∈ C + . ThenD(N + 1, z) ⊂ D(N, z), N ∈ N Proof. Let M ∈ D(N + 1, z). From (3.3) we have E(M, N ) = −2 Im M + 2 Im z N j=0 V ) * and R(N, z) = (iW N (V , V )) −1/2 . So the equation of Weyl circle can be written as (M − C N (z)) * R(N, z) −2 (M − C N (z)) = R(N,z) 2 Theorem 3.7. For all z ∈ C + , lim N →∞ R(N, z) exists and lim N →∞ R(N, z) ≥ 0. Theorem 3 . 8 . 38For all z ∈ C + , lim N →∞ C N (z) exists.Proof. For any M ∈ C(N, Z) we have(M − C N (z)) * R(N, z) −2 (M − C N (z)) = R(N,z) 2 which follows that R(N, z) −1 (M − C N (z))R(N,z) −1 * R(N, z) −1 (M − C N (z))R(N,z) Suppose M ∈ C N +1 (z) ⊂ C N (z) then we have M = C N +1 (z) + R(N + 1, z)U N +1 R(N + 1,z) and M = C N (z) + R(N, z)U N R(N,z)Equating and taking operator norm on both sides we getC N +1 (z) − C N (z) = R(N + 1, z)U N +1 R(N + 1,z) − R(N, z)U N R(N,z) ≤ R(N + 1, z)U N +1 R(N + 1,z) − R(N, z)U N +1 R(N + 1,z) + R(N, z)U N +1 R(N + 1,z) − R(N, z)U N R(N + 1,z) + R(N, z)U N R(N + 1,z) − R(N, z)U N R(N,z) ≤ R(N + 1, z) − R(N, z) U N +1 R(N + 1,z) + R(N, z) U N +1 − U N R(N + 1,z) + R(N, z) U N R(N + 1,z) − R(N, z) Theorem 3 . 9 . j=0 F 39j=0Let z ∈ C + and M ∈ C d×d . Then for F (N, z) = U (N, z) + V (N, z)M we have (1) M is inside D 0 (z) if and only if ∞ N =1 F (N, z) * F (N, z) ≤ Im M Im z(2) M is on the boundary of D 0 (z) if and only if ∞ N =1 F (N, z) * F (N, z) = Im M Im z Proof. Let M ∈ D 0 (z). Then M ∈ D(N, z) for all N. So from (3.3) we have E(M, N ) = −2 Im M + 2 Im z N (j, z) * F (j, z) FF j=1 F j=1(N, z) * F (N, z) ≤ Im M Im z .Conversely, for any N we have, (j, z) * F (j, z) ≤ ∞ (j, z) * F (j, z) ≤ Im M Im z . TITCHMARSH-WEYL THEORY FOR VECTOR-VALUED DISCRETE SCHRÖDINGER OPERATORS Acknowledgement:The author would like to thank the Department of Mathematics and the Office of Sponsored Research, Embry-Riddle Aeronautical University for support. A note on multidimensional discrete Schrödinger operators. K R Acharya, The Nepal Math. Sc. Report. 341K. R. Acharya, A note on multidimensional discrete Schrödinger operators, The Nepal Math. Sc. Report Vol 34, No.1, 2016, 1-10. J Behrndt, J Rohleder, arXiv: 12085224v2Titchmarsh-Weyl Theory for Schrödinger operators on unbounded domain. J.Behrndt, J. Rohleder, Titchmarsh-Weyl Theory for Schrödinger operators on unbounded do- main, arXiv: 12085224v2. WeylTitchmarsh M-function asymptotics for matrixvalued Schrodinger operators. S L Clark, F Gesztesy, Proc. London Math. Soc. 3701724S. L. Clark and F. Gesztesy, WeylTitchmarsh M-function asymptotics for matrixvalued Schrodinger operators , Proc. London Math. Soc. (3) 82 (2001), no. 3, 701724. H L Cycon, R G Froese, W Kirsch, B Simon, Schrödinger Operators, With Applications to Quantum Mechanics and Global Geometry. SpringerH. L. Cycon, R.G. Froese, W. Kirsch, B. Simon, Schrödinger Operators: With Applications to Quantum Mechanics and Global Geometry, Springer, 2008. The analytic theory of matrix orthogonal polynomials. D Damanik, A Pushnitski, B Simon, Surv. Approx. Theory. 4D. Damanik, A. Pushnitski, B. Simon, The analytic theory of matrix orthogonal polynomials. Surv. Approx. Theory, 4: 1-85, 2008. A personal history of the m-coefficient. W N Everitt, J. Comput. Appl. Math. 1711-2W. N. Everitt, A personal history of the m-coefficient, J. Comput. Appl. Math. 171(2004), no. 1-2, 185197. Scattering theory and matrix orthogonal polynomials on the real line. J S Geronimo, Circuits Systems Signal Process. 13-4J. S. Geronimo, Scattering theory and matrix orthogonal polynomials on the real line, Circuits Systems Signal Process., 1(3-4): 472-495, 1982. On matrix-valued Herglotz functions. F Gesztesy, E Rsekanovskii, Math. Machr. 218F. Gesztesy, E. Rsekanovskii, On matrix-valued Herglotz functions. Math. Machr., 218: 61-138, 2000. Zeros of the Wronskian and renormalized oscillation theory. F Gesztesy, B Simon, G Teschl, Amer. J. Math. 118F. Gesztesy, B. Simon, G. Teschl, Zeros of the Wronskian and renormalized oscillation theory, Amer. J. Math. 118 (1996), 571 -594. Equivalence classes of block Jacobi matrices. R Kozhan, Proc. Amer. Math. Soc. 139R. Kozhan, Equivalence classes of block Jacobi matrices. Proc. Amer. Math. Soc., (139) 799-805, 2011. The absolutely continuous spectrum of Jacobi Matrices. C Remling, Annals of Math. 174C. Remling, The absolutely continuous spectrum of Jacobi Matrices, Annals of Math., 174, 125-171, 2011. The absolutely continuous spectrum of one-dimensional Schrödinger operators. C Remling, Math. Phys. Anal. Geom. 104C. Remling, The absolutely continuous spectrum of one-dimensional Schrödinger operators, Math. Phys. Anal. Geom., 10(4), 359-373, 2007. m-functions and the absolutely continuous spcetrum of one-dimensional almost periodic Schrödinger operators, Differential equation. B Simon, North-Holland Math. Stud519Birmingham, Ala; North-Holland, AmsterdamB. Simon, m-functions and the absolutely continuous spcetrum of one-dimensional almost peri- odic Schrödinger operators, Differential equation (Birmingham, Ala., 1983), 519, North-Holland Math. Stud. 92, North-Holland, Amsterdam, 1984. Jacobi Operators and Completely Integrable Nonlinear Latices. G , Mathematical Monographs and Surveys. 72American Mathematical SocietyG. Teschl, Jacobi Operators and Completely Integrable Nonlinear Latices, Mathematical Mono- graphs and Surveys, Vol.72, American Mathematical Society, Providence, 2000. E C Titchmarsh, Eigenfunction Expansions Associated with Second-order Differential Equations, Part I, Second Edition. OxfordClarendon PressE.C. Titchmarsh, Eigenfunction Expansions Associated with Second-order Differential Equa- tions, Part I, Second Edition, Clarendon Press, Oxford, 1962. Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen. H , Math. Ann. 682H. Weyl,Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen En- twicklungen willkürlicher Funktionen, Math. Ann., 68 (1910), no. 2, 220-269.	[]
[]	[ "\nLaboratory of Particle Detection and Electronics is with the State Key Laboratory of Particle Detection and Electronics\nUniversity of Science and Technology of China\n230026HefeiChina\n", "\nLi is with the State Key Laboratory of Particle Detection and Electronics\nUniversity of Science and Technology of China\n230026HefeiChina\n", "\nis with the State Key Laboratory of Particle Detection and Electronics\nUniversity of Science and Technology of China\n230026HefeiChina\n", "\nUniversity of Science and Technology of China\n230026HefeiChina\n" ]	[ "Laboratory of Particle Detection and Electronics is with the State Key Laboratory of Particle Detection and Electronics\nUniversity of Science and Technology of China\n230026HefeiChina", "Li is with the State Key Laboratory of Particle Detection and Electronics\nUniversity of Science and Technology of China\n230026HefeiChina", "is with the State Key Laboratory of Particle Detection and Electronics\nUniversity of Science and Technology of China\n230026HefeiChina", "University of Science and Technology of China\n230026HefeiChina" ]	[]	An ultra-high-speed waveform digitizer prototype based on gigabit Ethernet has been developed. The prototype is designed to read out signals of detectors to realize the accurate measurement of various physical quantities for plasma diagnostics. The prototype includes an ultra-high-speed analog-to-digital converter (ADC) used to realize high speed digitization, a Xilinx Kintex-7 field-programmable gate array (FPGA) used for system configuration and digital signal processing, a DDR3 memory bar for data storage, and a gigabit Ethernet transceiver for interfacing with a computer. The sampling rate of the prototype is up to 5Gsps with 10-b resolution. The features of the prototype are described in detail.Index Terms-plasma diagnostics, ultra-high-speed waveform digitizer, gigabit Ethernet.	null	[ "https://arxiv.org/pdf/1806.08025v1.pdf" ]	115,517,930	1806.08025	a3e0c172f2c3df4bc5ebdc45c8b13eaa69edea23	Laboratory of Particle Detection and Electronics is with the State Key Laboratory of Particle Detection and Electronics University of Science and Technology of China 230026HefeiChina Li is with the State Key Laboratory of Particle Detection and Electronics University of Science and Technology of China 230026HefeiChina is with the State Key Laboratory of Particle Detection and Electronics University of Science and Technology of China 230026HefeiChina University of Science and Technology of China 230026HefeiChina 8 First Author et al.: Title  An ultra-high-speed waveform digitizer prototype based on gigabit Ethernet has been developed. The prototype is designed to read out signals of detectors to realize the accurate measurement of various physical quantities for plasma diagnostics. The prototype includes an ultra-high-speed analog-to-digital converter (ADC) used to realize high speed digitization, a Xilinx Kintex-7 field-programmable gate array (FPGA) used for system configuration and digital signal processing, a DDR3 memory bar for data storage, and a gigabit Ethernet transceiver for interfacing with a computer. The sampling rate of the prototype is up to 5Gsps with 10-b resolution. The features of the prototype are described in detail.Index Terms-plasma diagnostics, ultra-high-speed waveform digitizer, gigabit Ethernet. 1.25 Gsps each [4]. Four independent analog signal conditioning circuits enable the ADC to work in four-channel mode (1.25Gsps 4), two-channel (2.5Gsps 2) or one-channel mode (5Gsps *1) in order to meet the needs of different measurements. An external trigger input channel enables the prototype to synchronize with the experimental system. The clock circuit provides sampling clock with ultra-low jitter. A DDR3 memory bar with 4GB capacity is employed to cache date from ADC, which makes the storage depth greatly increased, up to 600ms. Data communication is accomplished using gigabit Ethernet. Control of the whole board is done by a Xilinx Kintex-7 FPGA, including prototype configuration, data processing and transmitting. A. Analog Conditioning Circuit Protection circuit, single-ended to differential conversion circuit, and anti-aliasing filter form the analog conditioning circuit, which is exhibited in Fig. 2. A TVS diode is placed at the front end of the circuit to protect the circuit from being damaged by electro-static discharge (ESD). The π-type resistor network, which is comprised of R1, R2 and R3, is designed for termination and amplitude adjustment for signal from detector. Diode D2 and D3 is adopted to avoid damage to the circuit caused by large signals. Given the background that most ultra-high-speed ADCs use differential inputs to suppress common-mode noise and interference, a fully differential amplifier is equipped to convert the single-ended detector signal to differential level to match the requirement of ADC. Before the analog signal enters the ADC, an anti-aliasing filter is used to filter out the high frequency noise to improve the signal-to-noise ratio (SNR). B. Clock Generation Circuit In the field of ultra-high-speed analog digital conversion, the jitter of sampling clock have a significant impact on effective number of bits (ENOB) [5], As the frequency of input signal increases, the jitter of the sampling clock will significantly limit the ENOB. The total jitter of sampling is contributed by ADC aperture time itself and the jitter of sampling clock. In order to achieve high accuracy, an ultra-low jitter phase-locked loop (PLL) is designed to synthesize the sampling clock from the 100MHz clock, which is generated from a precision crystal oscillator. An ordinary crystal oscillator produces another 100MHz clock used for system configuration. Simulation result shows that the jitter of sampling clock is less than 60fs. C. Signal Flow in FPGA The block diagram of the signal flow in FPGA of the prototype is shown in Fig. 3. LVDS interface transmits the digital signal from ADC to FPGA. Then, a dynamic phase alignment (DPA) module is embedded to stabilize the transmission link [6]. When the trigger signal arrives, the signal processing module will transfer the data to DDR3 memory bar through DDR3 controller, which is generated via the core generator of the Xilinx ISE software. After the signal is cached, data will be upload to PC via the gigabit Ethernet. 1) DPA module Since the speed of the interface between the ADC and FPGA reaches 1.25G, the time window of data reception is only about 400 ps. Considering the differences between the line delay and the gate delay of the signal in the chip, coupled with temperature and voltage fluctuations, it is difficult to receive the data accurately and steadily. Therefore, we developed a DPA module based on IODELAY technology for aligning the data signals with the clock signal automatically. The DPA module consists of 2 parts, bit-alignment and word-alignment. The goal of the bit-alignment procedure is to position the captured clock edge in the center of the data eye to provide maximum margin, while the word-alignment procedure aligns the output pattern from the ISERDES to a specific training pattern. Fig. 4 illustrates the state transition diagram of the DPA algorithm. When the prototype is powered on, it will enter the self-test state. At this stage, the ADC is configured to send a fixed training pattern. Then increase the value of IODELAY in the FPGA to find the left and right edges of the signal. Next, set the IODELAY to the center of the left and right boundaries to get maximum margin. Finally, use BITSLIP to remove word skew and align all channels to a specific word pattern. After DPA is completed, the ADC is switched to normal sampling mode. 2) UDP interface A physical layer chip 88E1116 is used to achieve gigabit Ethernet. The data transfer protocol is based on UDP. Compared with TCP, UDP has higher transmission efficiency and lower resource consumption. The data format of an Ethernet packet is shown in Fig. 7(a), it is composed of a 6 bytes destination MAC address, a 6 bytes source MAC address, a 2 bytes packet type, 46-1500 bytes data and a 4 bytes CRC check. When the type is 0x0800, the data which contains 46-1500 bytes is an IP Datagram. The IP Datagram consists of a 20 byte header and a UDP packet. UDP packets include source port, destination port, packet length, and checksum and user data. In this prototype, the length of user data is 1024 bytes per packet. III. TEST A. Data Transmission Performances Test A network test module is built in the FPGA for performance evaluation of the gigabit Ethernet. It can accept the data request command from the host computer, and reply a specific number packets of data to each request. The number of packets ranging from 1 to 256 is set by the data request command. Each packet contains 1024 bytes of user data. A counter for outputting the binary code is used as a data source. These data are checked by the host computer in real time and assess data rate, packet loss rate and code error rate. The results of the data transmission rate test is shown in Table. I. Each test lasts for one hour. As the N (number of packets sent for each data request command) increases, the transmission speed rises accordingly. When transmitting 256 packets of data each time, the average Ethernet transfer rate reaches up to 813Mb/s, which is less than 1Gb/s. The reasons are mainly attributed to the following. On the one hand, the hardware does not upload Ethernet packet when the host computer sends the data request command. On the other hand, the UDP also occupies part of the bandwidth, which is mainly the tens of bytes of each packet header and the last few bytes of each packet used for CRC check. These reasons decrease the usable bandwidth on a gigabit Ethernet network. Within the 8-hour test, no packet loss and error code are detected. This indicates that the data interface has superior stability. B. ENOB Test Because ENOB takes into account the effects of noise, harmonics and other factors on the sampling accuracy, it can comprehensively indicate the performance of the waveform digitizer prototype well. To evaluate the ENOB of the prototype, a series of sine wave signals with frequency ranging from 10.7MHz to 1034MHz are used. All signals pass through a bandpass filter before being connected to the prototype to reduce the impacts of the noise and harmonics of the signal generator itself. For each test, 98240 sampling points are used to calculate ENOB through Fast Fourier Transform (FFT). Test results are shown in Fig. 6, ENOB reaches 8.4 b at 10.7MHz. When the input frequency increases, the ENOB drops to 7.9 b at 1034MHz. which is much better than most commercial oscilloscopes. IV. CONCLUSION In this paper, an ultra-high-speed waveform digitizer prototype based on gigabit Ethernet has been built for plasma diagnostics. The sampling rate of the prototype is as high as 5Gsps. ENOB is more than 7.9 b within 1G bandwidth. Besides, it has a large DDR3 memory cache of 4GB, all the data can be upload quickly via gigabit Ethernet interface. It can perform comprehensive physical information measurements such as time measurement, particle identification, energy measurements and so on for plasma diagnostics. Compared with commercial oscilloscopes, it has better performance and smaller volume. Fig. 1 . 1Schematic of the prototype. Fig. 2 .Fig. 3 . 23Analog conditioning circuit. The signal flow in FPGA of the prototype. Fig. 4 . 4The state transition diagram of the DPA algorithm. Fig. 5 . 5The data format of an Ethernet packet. TABLE I UNITS IFOR MAGNETIC PROPERTIESN Data rate ( Mbps ) 1 44 2 86 4 165 8 189 16 340 32 433 64 615 128 736 256 813 Fig. 6. Results of ENOB test. Neutron time of flight energy spectrometer for ICF ion temperature diagnostic. Tang Zheng-Yu, Acta Phys. Sin. 82Tang Zheng-Yu, "Neutron time of flight energy spectrometer for ICF ion temperature diagnostic," Acta Phys. Sin., vol. 8, no. 2, pp. 913-918, Dec. 1999. Application of lasers to the production of high-temperature and high-pressure plasma. R E Kidder, Nucl. Fusion. 81R.E. Kidder, "Application of lasers to the production of high-temperature and high-pressure plasma," Nucl. Fusion, vol. 8, no. 1, pp. 3-12, 1968. Nuclear diagnostics for petawatt experiments. M A Stoyer, Rev. Sci. Instrum. 721M. A. Stoyer, "Nuclear diagnostics for petawatt experiments," Rev. Sci. Instrum., vol. 72, no. 1, pp. 767-772, Jan. 2001. . Ev10aq190a, Inc, Uk, EV10AQ190A, E2V Inc., UK, 2013. Aperture Uncertainty and ADC System Performance. B Brannon, A Barlow, Analog Devices Inc. B. Brannon and A. Barlow, "Aperture Uncertainty and ADC System Performance," Analog Devices Inc., 2000, [Online]. Available: http://www.bdtic.com/DownLoad/ADI/AN-501.pdf Dynamic phase alignment for networking applications. T Y Yeoh, Xilinx IncT. Y. Yeoh, "Dynamic phase alignment for networking applications," Xilinx Inc., Jul. 2005.	[]
[ "Chemical separation of primordial Li + during structure formation caused by nanogauss magnetic field", "Chemical separation of primordial Li + during structure formation caused by nanogauss magnetic field" ]	[ "Motohiko Kusakabe \nInstitute for Cosmic Ray Research\nUniversity of Tokyo\n277-8582KashiwaChibaJapan\n", "Masahiro Kawasaki \nInstitute for Cosmic Ray Research\nUniversity of Tokyo\n277-8582KashiwaChibaJapan\n\nKavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU)\nTODIAS\nUniversity of Tokyo\n5-1-5 Kashiwanoha277-8583KashiwaJapan\n" ]	[ "Institute for Cosmic Ray Research\nUniversity of Tokyo\n277-8582KashiwaChibaJapan", "Institute for Cosmic Ray Research\nUniversity of Tokyo\n277-8582KashiwaChibaJapan", "Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU)\nTODIAS\nUniversity of Tokyo\n5-1-5 Kashiwanoha277-8583KashiwaJapan" ]	[ "Mon. Not. R. Astron. Soc" ]	During the structure formation, charged and neutral chemical species may have separated from each other at the gravitational contraction in primordial magnetic field (PMF). A gradient in the PMF in a direction perpendicular to the field direction leads to the Lorentz force on the charged species. Resultantly, an ambipolar diffusion occurs, and charged species can move differently from neutral species, which collapses gravitationally during the structure formation. We assume a gravitational contraction of neutral matter in a spherically symmetric structure, and calculate fluid motions of charged and neutral species. It is shown that the charged fluid, i.e., proton, electron and 7 Li + , can significantly decouple from the neutral fluid depending on the field amplitude. The charged species can, therefore, escape from the gravitational collapse. We take the structure mass, the epoch of the gravitational collapse, and the comoving Lorenz force as parameters. We then identify a parameter region for an effective chemical separation. This type of chemical separation can reduce the abundance ratio of Li/H in early structures because of inefficient contraction of 7 Li + ion. Therefore, it may explain Li abundances of Galactic metal-poor stars which are smaller than the prediction in standard big bang nucleosynthesis model. Amplitudes of the PMFs are controlled by a magneto-hydrodynamic turbulence. The upper limit on the field amplitude derived from the turbulence effect is close to the value required for the chemical separation.	10.1093/mnras/stu2115	[ "https://arxiv.org/pdf/1404.3485v3.pdf" ]	119,203,642	1404.3485	43f50ad529d18d4c92262cac7535562a33835618	Chemical separation of primordial Li + during structure formation caused by nanogauss magnetic field 1-30 (20XX Motohiko Kusakabe Institute for Cosmic Ray Research University of Tokyo 277-8582KashiwaChibaJapan Masahiro Kawasaki Institute for Cosmic Ray Research University of Tokyo 277-8582KashiwaChibaJapan Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU) TODIAS University of Tokyo 5-1-5 Kashiwanoha277-8583KashiwaJapan Chemical separation of primordial Li + during structure formation caused by nanogauss magnetic field Mon. Not. R. Astron. Soc 0001-30 (20XXAccepted xxx. Received xxx; in original form xxxPrinted 9 (MN L A T E X style file v2.2)atomic processes -hydrodynamics -magnetic fields -plasmas -Galaxy: abundances -early Universe During the structure formation, charged and neutral chemical species may have separated from each other at the gravitational contraction in primordial magnetic field (PMF). A gradient in the PMF in a direction perpendicular to the field direction leads to the Lorentz force on the charged species. Resultantly, an ambipolar diffusion occurs, and charged species can move differently from neutral species, which collapses gravitationally during the structure formation. We assume a gravitational contraction of neutral matter in a spherically symmetric structure, and calculate fluid motions of charged and neutral species. It is shown that the charged fluid, i.e., proton, electron and 7 Li + , can significantly decouple from the neutral fluid depending on the field amplitude. The charged species can, therefore, escape from the gravitational collapse. We take the structure mass, the epoch of the gravitational collapse, and the comoving Lorenz force as parameters. We then identify a parameter region for an effective chemical separation. This type of chemical separation can reduce the abundance ratio of Li/H in early structures because of inefficient contraction of 7 Li + ion. Therefore, it may explain Li abundances of Galactic metal-poor stars which are smaller than the prediction in standard big bang nucleosynthesis model. Amplitudes of the PMFs are controlled by a magneto-hydrodynamic turbulence. The upper limit on the field amplitude derived from the turbulence effect is close to the value required for the chemical separation. INTRODUCTION In the standard cosmology, abundances of light elements, i.e., hydrogen, helium, lithium, and very small amounts of other nuclides, evolve during big bang nucleosynthesis (BBN) at the redshift of z ∼ 10 9 (Fields 2011). Lithium abundance predicted in standard BBN (SBBN) model (Coc et al. 2012;Coc, Uzan, & Vangioni 2013), however, disagrees with that determined by spectroscopic observations of metal-poor stars (MPSs) (Meléndez & Ramírez 2004;Asplund et al. 2006). The observational number ratio of lithium and hydrogen is 7 Li/H= (1 − 2) × 10 −10 (Spite & Spite 1982;Ryan et al. 2000;Meléndez & Ramírez 2004;Asplund et al. 2006;Bonifacio et al. 2007;Shi et al. 2007;Aoki et al. 2009;González Hernández et al. 2009;Sbordone et al. 2010;Monaco et al. 2010Monaco et al. , 2012Mucciarelli, Salaris, & Bonifacio 2012). It is 2-4 times lower than the prediction in SBBN model with the baryon-to-photon ratio from the observation of the cosmic microwave background radiation by Wilkinson Microwave Anisotropy Probe (WMAP) (Spergel et al. 2003(Spergel et al. , 2007Larson et al. 2011;Hinshaw et al. 2013). The formations of atom and molecules proceed in the redshift range of z 10 4 (Saslaw & Zipoy 1967;Peebles & Dicke 1968;Lepp & Shull 1984;Dalgarno & Lepp 1987;Galli & Palla 1998;Vonlanthen et al. 2009). Since lithium has a low ionization potential, it remains ionized when the recombination of hydrogen occurs (Dalgarno & Lepp 1987). The relic abundance of Li + is, therefore, high (Galli & Palla 1998). A recent study (Vonlanthen et al. 2009) shows that abundances of Li and Li + are almost equal at z = 10. Magnetic fields exist in various astronomical objects, such as Sun, Galaxy, galactic cluster (see (Grasso & Rubinstein 2001) for a review). Magnetic field have possibly existed in the early universe. The origin of the magnetic fields is, however, not determined yet. Magnetic fields can be generated through electric currents induced by a velocity difference of electrons and ions (Biermann 1950;Browne 1968). Such an electric current is produced in a rotating gas system because of different viscous resistances of electrons and ions (Browne 1968). This current creates poloidal magnetic field. Similarly, the drift current can be produced from gravitation working on electrons and ions, and it can generate a magnetic field (Browne 1968(Browne , 1982(Browne , 1985. It has been noted (Harrison 1969), however, that these batteries (Biermann 1950;Browne 1968) can not generate a large magnetic field since the time-scale of field generation is much larger than the age of the universe (Spitzer 1948;Hoyle & Ireland 1960;Harrison 1969). The primordial magnetic field (PMF) can be generated at a couple of epochs in the early universe, i.e., the inflation, electroweak and quark-hadron transitions, and reionization (see Grasso & Rubinstein 2001;Widrow 2002;Widrow et al. 2012, and references therein). The PMF generation, however, most probably occurs around the cosmological recombination epoch (Harrison 1970;Matarrese et al. 2005;Takahashi et al. 2005;Ichiki et al. 2006Ichiki et al. , 2007Takahashi, Ichiki, & Sugiyama 2008;Fenu, Pitrou, & Maartens 2011;Maeda, Takahashi, & Ichiki 2011). In the evolution of primordial density perturbation, the magnetic field can be perturbatively generated at second order through the vorticity (Matarrese et al. 2005) and the anisotropic stress of photon (Takahashi et al. 2005;Ichiki et al. 2006). These generation processes can be calculated rather precisely with use of the cosmological perturbation theory. Recent calculation (Fenu et al. 2011) shows that the comoving amplitude of generated field on cluster scales, i.e., 1 Mpc, is about 3 × 10 −29 G at redshift z = 0. Effects of PMFs on Galaxy formation have been studied (Rees & Reinhardt 1972;Wasserman 1978;Coles 1992). Effects on Galactic angular momentum and Galactic magnetic fields have been also investigated utilizing magneto-hydrodynamic (MHD) equations (Wasserman 1978;Coles 1992). It was found that a magnetic field can trigger a large density fluctuation with an overdensity of δ = 1. Such a large fluctuation is produced in a structure with a scale LB if the comoving field amplitude measured in the present intergalactic medium (IGM) is as large as B0(LB) ∼ 10 −9 (LB/1 Mpc) G (Wasserman 1978;Coles 1992). It has been suggested that an inhomogeneous magnetic field causes a streaming velocity of baryon relative to dark matter, and resultantly an infall of baryon in potential wells of dark matter may be inhibited. Cosmological structure formation is thus affected by the inhomogeneous field (Coles 1992). In this paper, we study a chemical separation of charged and neutral species triggered by a PMF during the structure formation. Neutral chemical species collapse gravitationally during the structure formation. Motions of charged species can, however, decouple from that of neutral species by PMF, and an ambipolar diffusion occurs. If the PMF has a gradient in a direction perpendicular to the field direction in the early universe, an electric current of charged species necessarily exists in the direction perpendicular to both of the field lines and the gradient direction. The Lorentz force working on the charged species then causes a velocity difference between charged and neutral species in the direction of the field gradient. This velocity difference enables an ambipolar diffusion. Therefore, it is possible that 7 Li + ions did not collapsed, while neutral 7 Li atoms gravitationally collapsed into structures. We suggest that the ambipolar diffusion provides a possible explanation of the small Li abundance in MPSs. The situation of the 7 Li + depletion due to PMFs and structure collapse studied in this paper is analogous to that of the charged grain depletion in the star-forming magnetic molecular clouds (MCs). The chemical separation by an ambipolar diffusion has been studied for the case of the gravitational collapse in dusty interstellar MCs (e.g. Ciolek & Mouschovias 1994, 1996. In the MCs, the abundance of charged dust grains which is a component of their plasma is reduced since the magnetic field retards the infall of the grains while the neutral particles collapse to form a protostellar core (Ciolek & Mouschovias 1994). The depletion of the grain abundance by the magnetic field is a very important phenomenon since information on the star formation mechanism can in principle be obtained from the ratio of observed abundances of grains in the core and the envelope of MC (Ciolek & Mouschovias 1996). The organization of this paper is as follows. In Sec. 2 we describe the model of chemical separation during a gravitational collapse of a structure. In Sec. 3 we introduce physical quantities used in this study, and typical numerical values relevant to the structure formation. In Sec. 4 we show results of calculations of the chemical separation caused by the magnetic field. In Sec. 5 we comment on the magnetic field amplitude. In Sec. 6 we comment on a possible generation of a magnetic field gradient during the gravitational collapse. In Sec. 7 we identify a parameter region required for a successful chemical separation. In Sec. 8 we briefly mention a later epoch of the structure formation and possible reactions neglected in this study. We suggest that the chemical separation of the 7 Li + ion can reduce the abundance ratio 7 Li/H in the early structure. Another theoretical constraint on the magnetic field amplitude is also described. In Sec. 9 we summarize this study. In Appendix A we show drift velocities of protons and electrons in a structure, equations for ions and electrons which should be satisfied in equilibrium states, and typical values of variables required for an efficient chemical separation. In Appendix B we show supplemental Figure 1. Illustration of chemical separation in a collapsing structure. In the left panel, the large solid circle delineates a collapsing structure, thin arrows are magnetic field lines, and open arrows are gravitational accelerations. The structure is axisymmetric with respect to the field direction, and can be seen as a large coil indicated with a dashed lines. The right panel shows an enlarged view of the cross section. The three axes of the cylindrical coordinate are defined. We assume a magnetic field along the z-axis (the thin arrow), and a gradient of the field amplitude in the −r direction (the filled thick arrow). There is a φ component of the ∇ × B term or an azimuthal electric current (mark ⊗). Charged fluid with this current in the magnetic field receives the Lorentz force (F L ) in the r direction. Resultantly, the charged fluid has a radial velocity relative to the neutral fluid. The Lorentz force is then balanced with a friction force by neutral fluid (F fric ). results for the calculations of the chemical separation. In this paper, the Boltzmann's constant (kB) and the light speed (c) are normalized to be unity. MODEL We focus on the leaving of ionic species behind forming structures at redshift z = O(10). First, let us define the initial state of the model structure. The structure has a uniform density and parallel magnetic field. The field amplitude has a gradient in a direction perpendicular to the field lines. This simple condition is assumed as one example case that ionized chemical species can have bulk velocities different from that of neutral hydrogen. It is considered that the relevant magnetic field has been generated by motions of charged species which exist outside the structure originally. In order to precisely follow the evolution of the spatial field distribution in the structure, the evolution of electric circuit including both inside and outside of the structure should be considered (Alfven 1981, Chaps. III and V). In this calculation, however, we do not treat the outer region, and use a boundary condition. Second, the structure is axisymmetric in a cylindrical coordinate system (r, φ, z) with an axis of symmetry taken to be z-axis. Azimuthal components of all physical quantities, therefore, do not depend on the azimuthal angle φ. The outer boundary of the structure exists at r = rstr. Figure 1 is an illustration of the physical concept of chemical separation. In the left panel, the large solid circle is a boundary of collapsing structure, thin arrows are magnetic field lines, and open arrows are gravitational accelerations. Since the axial symmetry is assumed, this structure itself can be roughly regarded as a large coil as indicated with a dashed lines. The right panel shows an enlarged cross-sectional view of the structure. The three axes of the cylindrical coordinate are defined in the panel. A magnetic field exits along the z-axis (the thin arrow), and the field amplitude has a gradient in the −r direction (the filled thick arrow). Then, there is a φ component of the ∇ × B term or an azimuthal electric current (mark ⊗). The combination of this current and the magnetic field generates the Lorentz force (F L) on the charged fluid in the r direction. As a result, the charged fluid has a radial velocity relative to the neutral fluid. The Lorentz force is then balanced with a friction force by neutral fluid which depends on the radial relative velocity (F fric ). Fluid and electromagnetic equations The following equations are adopted. (i) equation of continuity for neutral matter: ∂ρn ∂t + ∇ · (ρnvn) = 0,(1) where ρn and vn are the density and the fluid velocity of the neutral matter, respectively, and t is the cosmic time. The neutral matter is mainly composed of neutral hydrogens. (ii) equation of continuity for ionized species i: ∂ρi ∂t + ∇ · (ρivi) = 0,(2) where ρi and vi are the density and the velocity of charged species i. The finite differential expression in the cylindrical coordinate system is ∆ρi ∆t = − 1 r ∆ (rρivir) ∆r − ∆ (ρiviz) ∆z .(3) (iii) force equations of proton and electron: Equations of motion are given by Dvp Dt = − 1 ρp ∇Pp + e mp (E + vp × B) + 1 τpn (vn − vp) + 1 τpe (ve − vp) ,(4)Dve Dt = − 1 ρe ∇Pe − e me (E + ve × B) + 1 τen (vn − ve) − 1 τep (ve − vp) ,(5) where D/(Dt) is the material time-derivative, vj , Pj, and mj are the velocity, pressure, and particle mass of species j, respectively, e is the electronic charge, E is the electric field, B = (Br, B φ , Bz) is the magnetic field in a cylindrical coordinate, and τ −1 ab is the energy loss rate of a through the scattering with b, or the slowing-down rate of relative velocity of a and b [cf. Eq. (55)]. In the force equations for charged species, a term of cosmological redshift is neglected since the time-scale relevant to the redshift is much larger than those for others. We neglect terms of pressure gradient in this paper. In the steady state, the force equations reduce to the form of E = −vp × B − ρp τpn (vn − vp) enp − ρp τpe (ve − vp) enp ,(6)E = −ve × B + ρe τen (vn − ve) ene − ρe τep (ve − vp) ene .(7) We neglected an effect of ∇B drift since it would be small. The force of ∇B is given by F∇B = − mjv 2 j⊥ ∇B 2B = 1.55 × 10 −47 GeV 2 mj GeV v j⊥ 6.59 × 10 4 cm s −1 2 ∇B/B kpc −1 ,(8) where v j⊥ is the velocity of j perpendicular to the B direction. On the other hand, the friction force from neutral species on charged species is given by F fric = mj (vn − vj) τjn = 2.01 × 10 −41 GeV 2 mj GeV mH mH + mj vnr − vjr 1.61 km s −1 nH 5.69 × 10 −3 cm −3 (σv) jn 10 −9 cm 3 s −1 ,(9) where (σv) ab is the product of the momentum transfer cross section σ and the velocity v at the reaction of a+b [cf. Eq. (53)]. Since the equation F fric ≫ F∇B holds, the effect of the field gradient is much smaller than that of the friction. (iv) Faraday's law of induction: ∂B ∂t = −∇ × E.(10) The following equation is derived with Eqs. (6) and (10) ∂B ∂t = ∇ × (vp × B) − mp e ∇ × (vp − vn) τpn + (vp − ve) τpe .(11) Note that the azimuthal component of magnetic field is always much smaller than Bz in the setup of this study. (v) electric current density: j = enpvp − eneve.(12) The Lorentz force term is balanced with the friction term from neutral matter [cf. Eqs. (6), (7) and (12)]: j × B = ρp τpn (vp − vn) + ρe τen (ve − vn) ,(13) In the steady state, the force balance in the radial direction leads to j φ Bz = enp (αpn + αen) (vpr − vnr),(14) where we used α ab = ma/(eτ ab ) and additionally assumed the conditions, np = ne and jr = jz = 0. The latter condition is derived from the fact that the radial and longitudinal fluid velocities of ions and electrons are essentially the same, vpr = ver and vpz = vez because of the charge neutrality of the system. (vi) Ampere's law: ∇ × B = 4πj(15) (vii) Gauss's law for magnetism: ∇ · B = 0.(16) The divergence of the Maxwell-Faraday equation [Eq. (10)] is given by ∂(∇ · B) ∂t = −∇ · (∇ × E) = 0.(17) When the Gauss's law is satisfied at the initial time, it remains satisfied because of this equation. (viii) force equation of neutral matter: When the pressure gradient term is neglected, the force equation for neutral particles is given (Ciolek & Mouschovias 1993) by ∂ (ρnvn) ∂t + ∇ · (ρnvnvn) = ρng + ρn τnp (vp − vn) + ρn τne (ve − vn) ,(18) where g is the gravitational acceleration. The second and third terms in the right-hand side (RHS) represent frictions from protons and electrons, respectively, working on the neutral particles. Using the balance between the Lorentz force and friction force [Eq. (13)], the Ampere's equation [Eq. (15)], and the Newton's law of action and reaction [Eq. (60)], we can transform the friction terms to the Lorentz-force term (Ciolek & Mouschovias 1993) as ∂ (ρnvn) ∂t + ∇ · (ρnvnvn) = ρng + (∇ × B) × B 4π .(19) When the azimuthal component of magnetic field is negligibly small, i.e., B φ ≃ 0, the second term in RHS is described as (∇ × B) × B = ∂Br ∂z − ∂Bz ∂r   Bz 0 −Br   .(20) Under the spherical symmetry, the gravitational acceleration is given by where r sph is the position vector from the centre in a spherical coordinate, rstr is the radius at the boundary of the structure, G is the gravitational constant, M (r sph ) is the mass contained inside the radius r sph , ρm is the matter density, and Mstr is the mass of the structure. In the equation the energy density is normalized to the value at the turnround (ztur = 16.5; see Sec. 3). On the other hand, the amplitude of the first term in RHS of Eq. (19) for gravitation is estimated as ρng ≃ ρ b g = 4.72 × 10 −89 GeV 5 ρ b 1.27 × 10 −26 g cm −3 ρm 7.64 × 10 −26 g cm −3 2/3 Mstr 10 6 M⊙ 1/3 ,(22) where ρ b is the baryon density. The amplitude of the second term in RHS of Eq. (19) for the Lorentz force is estimated to be (∇ × B) × B 4π ∼ 1 4π B 2 LB = 4.09 × 10 −89 GeV 5 B 10 −7 G 2 LB 597 pc −1 ,(23) where LB is the length scale of coherent magnetic field. The ratio of the gravitational and Lorentz terms, Eqs. (22) and (23), respectively, is related to the mass-to-magnetic flux ratio for the gravitational collapse of a structure with a frozen-in magnetic field (Mouschovias & Spitzer 1976;Ciolek & Mouschovias 1993). The critical mass-to-magnetic flux ratio has been determined from a numerical calculation (Mouschovias & Spitzer 1976) as M b ΦB crit = 0.126 G 1/2 ,(24) where M b is the total baryonic mass of the structure, and ΦB = πBr 2 str is the total magnetic flux through the structure. This ratio is invariant in comoving coordinates if the ambipolar diffusion is negligible. Above the critical ratio a gravitational collapse can occur while below the ratio the collapse cannot. The ratio of the gravitational and Lorentz terms can be rewritten in the form ρ b g \|(∇ × B) × B/ (4π)\| = 3π 2 G M M b M b ΦB 2 ,(25) where we supposed LB ∼ rstr. The factor M/M b takes into account that not only the baryon but also the dark matter contributes to the gravitation of the system. This factor is absent in the case of collapsing MC since effects of the dark matter mass is negligible. The combination of the critical mass-to-magnetic flux ratio [Eq. Galactic infall model We assume that the second term in RHS of Eq. (19) is negligible, and that the initial density is exactly uniform inside a sphere. This setup defines a toy model of collapsing structure. Then, Eqs. (1) and (19) are spherically symmetric, and Eq. (19) describes a free fall of spherical material. A gas heating associated with virialization is neglected, and the gas temperature is assumed to evolve adiabatically after it decoupled from the temperature of the cosmic background radiation (CBR) at z ∼ 200 (Peebles 1993). It is then given by T = 2.3 K[(1 + z)(1 + δ) 1/3 /10] 2 , where δ ≡ (ρm −ρm)/ρm(26) is the ratio of overdensity of matter relative to the cosmological average densityρm (Loeb & Zaldarriaga 2004). We assume that the baryon density is proportional to the matter density. This approximation is good as long as any radiative astrophysical objects such as first stars do not form yet. The free fall of the sphere controlled by a self gravity is described by the Lagrangian equation of motion, i.e., ∂ 2 r sph /∂t 2 = −GM (r sph )/r 2 sph (Hunter 1962;Peacock 1999). The radius and velocity are then related to the time t as described (Peacock 1999) by r sph = AG(1 − cos θG),(27)t = BG(θG − sin θG),(28)vn = ∂r sph ∂t = AG BG sin θG 1 − cos θG ,(29) where the condition A 3 G = GM B 2 G is satisfied. The parameters with the subscript G are used for the gravitational collapse and distinguished from parameters without the subscript. The velocity evolution, vn(t) = \|vn(t)\|, is given by this equation set. The assumption of the initial uniform density corresponds to a constant BG value for any AG. Then, the velocity depends on the radius parameter AG only. In this case, a homologous evolution occurs, and the density is alway independent of the spatial coordinate. Every mass shell satisfies ρn(t) = ρn,i[r sph,i /r sph (t)] 3 ,(30) where ρn(t) and ρn,i are the densities at time t and initial time ti, respectively, and r sph,i is the radius at initial time. In the present assumption, the ratio r sph,i /r sph (t) is position-independent. For a given time t, θG and corresponding r sph and vn are derived. Velocities of charged species For the system composed mainly of protons, electrons, and neutral matter, the total plasma force equation holds (Ciolek & Mouschovias 1993): ρn τnp (vp − vn) + ρn τne (ve − vn) = (∇ × B) × B 4π .(31) Thus, a velocity difference of charged and neutral species is related to the Lorentz force operating on whole charged species that is mostly composed of protons and electrons. In general, matters in astrophysical objects are nearly complete chargeneutral. We, therefore, assume that fluid velocities of protons and electrons are equal as for r-and z-components. Charged species are then considered as one component as long as motions in r-and z-directions are concerned. Since the proton density is larger than the electron density by a factor of mp/me = 1836, the friction on proton is the predominant in the total fluid. The following relation is derived from a balance of the friction and the Lorentz force: v p,(rz) = v n,(rz) + τnp (1 + τnp/τne) ρn [(∇ × B) × B] ,(rz) 4π ,(32) where vector components in the r-z plane are represented by subscript (rz). The factor (1 + τnp/τne) in the denominator is neglected because of τnp/τne ≪ 1. ρn 1 τnp + 1 τne v nφ + [(∇ × B) × B] φ 4π = 0.(33) The proton velocity vp is derived from the rotation of magnetic field, ∇ × B, using the Ampere's equation [Eq. (15)], as in studies of MCs (Ciolek & Mouschovias 1993Basu & Mouschovias 1994;Mouschovias, Ciolek, & Morton 2011). The rotation of the magnetic field is related to the electric current density. Both physical quantities have existed from the start time of calculation (see Sec. 5). Using Eqs. (12) for the current density, and (A7) for the velocities of protons and electrons, the φ-component of the Ampere's equation gives the azimuthal proton velocity as v pφ = ∂zBr − ∂rBz 4πenp(1 + αpn/αen) . The assumption of vpr = ver and vpz = vez correspond to no current density in the r-and z-directions. Then, the Ampere's equation does not give constraints on velocities of charged species in the r-z plane. Table 1 shows adopted masses of atoms and ions (H, H + , Li, and Li + ) which are derived with atomic and electronic mass data (Audi, , and ionization energies of j or binding energies of j + and e − , BE(j + ,e) (Martin et al. 2011): BE(H + ,e)=13.5984 eV (Johnson & Soff 1985) and BE(Li + ,e)=5.3917 eV (Lorenzen & Niemax 1982). Reaction cross sections σin are taken from Glassgold, Krstić, & Schultz (2005); Schultz et al. (2008) for i =H + and Krstić & Schultz (2009) for 7 Li + . Linear interpolations are utilized with velocities taken as parameters. Cross sections for energies lower than the minimum energy of data (Emin) are given by the value at the energy E = Emin, while those for energies larger than the maximum energy (Emax) are given by the value at E = Emax. Atomic mass and cross section data Initial conditions We take a typical comoving magnetic field value Bz0 as an input parameter. The initial magnetic field is then assumed to be Bzz i (r) = Bz0(1 + zi) 2 (1.5 − r/rstr) for 0 r 1.4rstr and Bzz i (r) = 0.1Bz0(1 + zi) 2 for 1.4rstr r, where Bzz i (r) is the z-component of magnetic field in IGM at the initial redshift zi at radius r. We note that in this calculation the ambipolar diffusion is caused by the magnetic pressure gradient [Eq. (32)]. The pressure gradient does not depend on the amplitude of the magnetic field alone. However, we fix the pattern of the gradient distribution, and take the comoving field value Bz0 as the only free parameter. As for initial velocities of protons and Li + , the radial and z-components are assumed to be the same as those of hydrogens. The φ-component of proton velocity is given by Eq. (34) with the initial B(r) distribution. Initial chemical abundances are taken from values at z = 10 calculated in the model of homogeneous universe (Vonlanthen et al. 2009): H + /H= 6.52 × 10 −5 , and Li + /Li= 1.0. We assume the Li nuclear abundance in SBBN model, Li/H= 5.2 × 10 −10 (Kawasaki & Kusakabe 2012). The chemical number fractions relative to hydrogen are then given by H + /H= 6.52 × 10 −5 , Li/H= 2.6 × 10 −10 , and Li + /H= 2.6 × 10 −10 . A precise calculation of ionic motions should include chemical reactions coupled to the hydrodynamical calculation of the structure formation. This is, however, beyond the scope of this paper. Boundary conditions Boundary conditions are important to describe plasma motions since a plasma inside some region is affected by not only physical parameters inside the region but also those outside the region (Alfven 1981, Chaps. III and V). We adopt the following conditions. Radial velocity components of any species j are zero on the symmetrical axis: vjr(r = 0) = 0.(35) The density and the recession velocity of neutral hydrogens, and number fractions of chemical species are initially given by the cosmic average values. Outside the structure, the magnetic field is supposed to exist homogeneously in the z-direction. In addition, the field amplitude evolves by redshift in the homogeneous universe. We, however, just assume that physical variables such as ion velocities connect smoothly at the structure boundary, and do not treat the conjunction. Calculations are performed for a contraction of material with a homogeneous overdensity of infinite size. As for a treatment for edges of computation domain, an origin and outer edge points are defined. Because of the symmetry, constraints on velocities, vj = 0 (for any j), always holds at the origin. In every time step, values of ρi (for ionic species i) and B at r = 0 (on z-axis) and on the plane of z = 0 are reset to be values calculated for the next innermost grid points, e.g. B(0, z, φ) = B(∆r, z, φ) and B(r, 0, φ) = B(r, ∆z, φ), respectively. Values of ρi and B at the outer edge points are always given by the average value of collapsing matter. Because of the axial symmetry, the radial and azimuthal components of the magnetic field are zero on z axis. At outer edge points of maximum r and z values, the densities of protons, electrons, and 7 Li + are fixed to values derived for the homogeneous contraction. In addition, at the edge points the magnetic field components are fixed as Br = 0 , B φ = 0, and Bzz(rstr) = 0.1Bz0(1 + z) 2 (1 + δ) 2/3 . Calculation The time step is determined so that changes in magnetic field and densities of ionized species in each step are much smaller than their amplitudes. In the time integration of variables A(a), the spatial differentiation is estimated with a finite difference method using the central difference. The difference is derived from quantities evaluated at intermediate positions between grid points with intervals of ∆a, i.e., ∂A(a)/∂a = [A(a + ∆a/2) − A(a − ∆a/2)]/∆a. The number of grid points is 260 (r direction) ×102 (z direction), and the spacing is ∆r = ∆z = 5.97 pc. The computational region is, therefore, 0 r 1.55 kpc and 0 z 0.603 kpc. The initial time is 9.29Myr, and the ending time is 474Myr, respectively, after big bang. In our calculation code, time evolutions of physical variables are calculated as follows. PHYSICAL QUANTITIES (i) cosmological parameters The ΛCDM (dark energy Λ and cold dark matter) model is adopted for the cosmic expansion history. Parameter values are taken from analysis of WMAP9 CBR data (ΛCDM model (Hinshaw et al. 2013)) 1 : The Hubble parameter is H0 = 70.0 ± 2.2 km s −1 Mpc −1 , and energy density parameters of matter and baryon are Ωm = 0.279 ± 0.025 and Ω b = 0.0463 ± 0.0024, respectively. The energy density parameter is defined by Ω k ≡ ρ k /ρc, where ρ k is the density of species k = m and b and ρc ≡ 3H 2 0 /(8πG) is the critical density. The present temperature of CBR is Tγ0 = 2.7255 K (Fixsen 2009). The primordial abundances of hydrogen, helium, and lithium are taken from calculation of SBBN model (Kawasaki & Kusakabe 2012) with the mean value of baryon density parameter Ω b described above, and the neutron lifetime 878.5 ± 0.7stat ± 0.3sys s (Serebrov & Fomin 2010): mass fractions of hydrogen and helium are X = 0.753 and Y = 0.247, respectively, and the number ratio of lithium to hydrogen is Li/H=5.2×10 −10 . (ii) redshift (z) versus time (t) relation a(t) = 1 1 + z(t) = Ωm 1 − Ωm 1/3 sinh 3 1 − Ωm 2 H0t 2/3 ,(36)t = 2H −1 0 3 1 − Ωm sinh −1 1 1 + z 3/2 1 − Ωm Ωm 1/2 ,(37) where a(t) is the scale factor of the universe. (iii) baryon density ρ b = ρcΩ b (1 + z) 3 (1 + δ) = 1.27 × 10 −26 g cm −3 h 0.700 2 Ω b 0.0463 1 + z 17.5 3 1 + δ 5.55 ,(38) where h ≡ H0/(100 km s −1 Mpc −1 ) is the reduced Hubble constant, and 1 + δ = ρm/ρm is the density normalized to the universal average valueρm. It has been assumed that the baryon density is proportional to the matter density. (iv) hydrogen number density nH ∼ = n b X = ρ b m b X = 5.69 × 10 −3 cm −3 h 0.700 2 Ω b 0.0463 1 + z 17.5 3 X 0.75 1 + δ 5.55 ,(39) where n b is the total baryon density, and m b = 0.938 GeV is the baryon mass. (v) matter density ρm = ρcΩm(1 + z) 3 (1 + δ) = 7.64 × 10 −26 g cm −3 h 0.700 2 Ωm 0.279 1 + z 17.5 3 1 + δ 5.55 .(40) (vi) spherical collapse model The mass of the structure is Mstr = 10 6 M⊙. The collapse of the structure finishes at the redshift z col = 10, or the cosmic time t col = 0.483 Gyr. The turnround then occurs at ztur = 16.5, ttur = t col /2 = 0.242 Gyr. The model structure is assumed to be a uniform density sphere with the radius at turnround of L = 597 pc Mstr 10 6 M⊙ 1/3 h 0.700 −2/3 Ωm 0.279 −1/3 1 + ztur 17.5 −1 ,(41) which derives from Mstr = (4πL 3 /3)ρm(1 + δ) with density contrast 1 + δ = 9π 2 /16 at turnround. The comoving length scale is L0 = (1 + ztur)L = 10.4 kpc. The parameter AG specifies the distance from the structure centre. When the AG value is chosen as 2AG = L at the structure boundary at turnround, the BG value is fixed to be BG = A 3 G GMstr = 1 6π 3 Gρm = 76.7 Myr h 0.700 −1 Ωm 0.279 −1/2 1 + ztur 17.5 −3/2 .(42) (vii) typical amplitude of magnetic field in the background universe Bz(z) ∼ Bz0(1 + z) 2 = 3.06 × 10 −8 G Bz0 10 −10 G 1 + z 17.5 2 ,(43) where Bz0 is the z-component of the field value measured at present age, i.e., redshift z = 0. (viii) Larmor frequency of ion Ωi = ZieB mi = 28.4Zi yr −1 B 10 −10 G mi 1 GeV −1 ,(44) where Zi is the charge number of ion i. (ix) gyration radius of ion Ri,g = miv i⊥ ZieB = v i⊥ Ωi = 1.08Z −1 i × 10 −8 pc v i⊥ 3.00 × 10 4 cm s −1 B 10 −10 G −1 mi 1 GeV ,(45) where v i⊥ is the velocity of i in the direction perpendicular to the magnetic field. (x) cosmic recession velocity v(r sph , z) = H(z)r sph ∼ H0Ω 1/2 m (1 + z) 3/2 r sph = 1.61 km s −1 h 0.700 Ωm 0.279 1/2 1 + z 17.5 3/2 r sph 596 pc ,(46) where r sph (z) is the radius in a spherical coordinate at redshift z, and the matter dominated universe was assumed for the Hubble expansion rate at z 10. (xi) gas temperature T (z) = 22 K 1 + z 17.5 2 1 + δ 5.55 2/3 ,(47) where the amplitude is taken from the calculation in Loeb & Zaldarriaga (2004). (xii) thermal average velocity of ion v i,th = 8T πmi = 6.59 × 10 4 cm s −1 T 22 K 1/2 mi 1 GeV −1/2 .(48) (xiii) momentum transfer cross section of p+H at the relative velocity v rel = 1.61 km s −1 σpn = 1.4 × 10 −14 cm 2 . (xiv) momentum transfer cross section of 7 Li + +H at v rel = 1.61 km s −1 σ7n = 1.3 × 10 −14 cm 2 .(50) (xv) elastic scattering cross section of e+H at v rel = 1.61 km s −1 We approximately take the elastic scattering cross section (Moiseiwitsch 1962): σen ≈ σ en,el = 41πa 2 0 = 3.6 × 10 −15 cm 2 ,(51) where a0 = 5.29 × 10 −9 cm is the Bohr radius. This relative velocity v rel = 1.61 km s −1 corresponds to the centre of mass energy 7.37 µeV. The recession velocity is smaller than the electron thermal velocity, v e,th = 29.1 km s −1 (T /22 K) 1/2 [Eq. (48)]. (xvi) Thomson scattering cross section σeγ = 8πe 4 3m 2 e = 6.65 × 10 −25 cm 2 . Thomson scattering between electron and CBR is neglected since it does not occur so frequently, and its momentum transfer is negligible. The momentum transfer rate of electrons, i.e., nγ σeγ , multiplied by the fractional change in electron momentum at one scattering ∼ O(Tγ /me), is much smaller than that of the e+H scattering. (xvii) momentum transfer rate of charged particles through the scattering with hydrogen τ −1 in = mH mH + mi nH (σv) in = 0.180 kyr −1 mH mH + mi nH 5.69 × 10 −3 cm −3 (σv) in 10 −9 cm 3 s −1 ,(53) where (σv) ab is the product of the cross section σ and the velocity v in the reaction of a + b. In the equation, we have assumed that the reaction of i with neutral matter is dominated by that of i+H, and neglected reactions with other neutral atoms. The factor mH/(mH + mi) is equal to the ratio of ionic momenta in the center of mass and laboratory systems. The velocity is given by the larger of the hydrodynamic velocity difference, \|va − v b \|, and the thermal mean velocity v ab,th = 8T πµ ab ,(54) where µ ab is the reduced mass of the a + b system. (xviii) friction parameter A parameter representing the friction effect on a species a from a species b is defined as α ab = ma eτ ab .(55) (xix) energy loss rate via the Coulomb scattering When the velocity of the incident electron measured in the rest frame of ion i, i.e., w, is much smaller than the root mean square velocity of the target ion particle, the slowing-down time (the inverse of the energy loss rate) of electrons via the scattering with ions is given (Spitzer 2006 ) by τei = 3 4 2π meµeiT 3/2 e 4 m 3/2 i ni ln Λ ,(56) where µei ∼ me is the reduced mass of the e + i system. The quantity ln Λ is related to the cutoff scale of the scattering length, and is given by ln Λ ≡ ln h/p0 = ln 3 2e 3 T 3 πne 1/2 = 21.5 + 3 2 ln T 22 K − 1 2 ln nH 5.69 × 10 −3 cm −3 − 1 2 ln χ H + 6.52 × 10 −5 ,(57) where h is the Debye shielding distance, p0 is the impact parameter at a scattering through which an electron is deflected by the angle of π/2, and χ H + = n H + /nH is the ionization degree of hydrogen. The energy loss rate is then given by τ −1 ei = 2.21 × 10 −2 s −1 T 22 K −3/2 mi mp 3/2 nH 5.69 × 10 −3 cm −3 χ H + 6.52 × 10 −5 ln Λ 21.5 .(58) The parameter αei is given by αei ≡ me eτei = 4 2π 3 e 3 m 3/2 i Z 2 i ni ln Λ µeiT 3/2 .(59) We use the Newton's law of action and reaction (Ciolek & Mouschovias 1993), i.e., ρa τ ab = ρ b τ ba .(60) The following relation then holds in the case of np = ne: αpe = αep.(61) The parameter αep = αpe is given [Eq. (59)] by αep = 1.26 × 10 −9 G T 22 K −3/2 nH 5.69 × 10 −3 cm −3 χ H + 6.52 × 10 −5 ln Λ 21.5 .(62) We apply Eq. (60) to the e+ 7 Li + system, and derive α7e = αe7 ne n7 = 4 2π 3 e 3 m 3/2 7 np ln Λ µe7T 3/2 ∼ m7 mp 3/2 αep.(63) We also apply Eq. (60) to the p+ 7 Li + system, and derive α7p = αp7 np n7 = 4 2π 3 e 3 m 3/2 7 np ln Λ µp7T 3/2 = α7e µe7 µp7 ∼ 8me 7mp α7e ≪ α7e.(64) The friction from the p+ 7 Li + scattering is then neglected. (xx) escape fraction of ion The fraction of an ionic species escaping through the outer boundary of the structure during the structure formation is estimated as Fi,esc(t) = ∆Mi(t) Mi = t t iṀ i(t ′ )dt ′ Mi ,(65) where Mi is the total mass of ion i initially contained in the structure before the contraction, ∆Mi(t) is the total mass of ion i which escaped from the structure by time t, and ti is the initial time which should be larger than the time of the primordial nucleosynthesis ∼200 s. We have assumed the spherical symmetry in the infall of neutral hydrogens, and the axial symmetry in the ion infall. The mass loss rate is then given bẏ Mi(t) = 2πr 2 str (t) π 0 sin θ dθ ρi(t, rstr(t), θ) vi,esc(t, rstr(t), θ),(66) where rstr(t) is the structure radius in a spherical coordinate at time t, θ = tan −1 (r/z) is the angle between the position vector and the symmetrical z axis, and ρi(t, rstr(t), θ) is the density at position (rstr(t), θ) at time t. The variable vi,esc(t, rstr(t), θ) is the escape velocity defined by vi,esc(t, rstr(t), θ) = (vi − vn) ·r = sin θ [vir(t, rstr(t), θ) − vnr(t, rstr(t), θ)] + cos θ [viz(t, rstr(t), θ) − vnz(t, rstr(t), θ)] ,(67) wherer is the unit vector with the direction of the position vector r. When we roughly assume that the density in the structure is homogeneous, the mass loss rate reduces tȯ Mi(t) = 3Mi 2rstr(t) π 0 sin θ dθ vi,esc(t, rstr(t), θ).(68) The escape fraction of ion i is then given by Fi,esc(t) = 3 2 t t i dt ′ 1 rstr(t ′ ) π 0 sin θ dθ vi,esc(t ′ , rstr(t ′ ), θ) = 2 ln t ln t i vi,esc(t ′ , rstr(t ′ )) µ H(t ′ )rstr(t ′ ) d ln t ′ ,(69) where vi,esc(t ′ , rstr(t ′ )) µ = 1 2 1 −1 dµ vi,esc(t ′ , rstr(t ′ ), cos −1 µ)(70) is the average value of the escape velocity. The recession velocity at the structure boundary, r sph = rstr(t), is H(t)rstr(t). Then, in Eq. (69) the time integration is dominated by the epoch when the escape velocity is a significant fraction of the recession velocity. We note that the escape of ions from cosmological structures is similar to that of charged dust grains from MCs. Equations (66) and (68) for the time evolution of ionic mass in cosmological structures by the ambipolar diffusion during gravitational contraction is conceptually the same as equations (3a) and (3b) of Ciolek & Mouschovias (1996) for that of the mass fraction of charged dust grains in MCs. RESULT We assume two cases of magnetic field amplitudes, Bz0 = 3 × 10 −10 G (Case 1) and 3 × 10 −11 G (Case 2). The former value is so large that charged chemical species escape from a gravitational collapse of neutral atoms, while the latter is not. The mass of the structure is 10 6 M⊙ in the both cases. The electric current density j is determined from rotation of the magnetic field through the Ampere's equation [Eq. (15)]. The friction from inflowing neutral hydrogens determines the radial velocities of charged species through a balance between the friction and Lorentz forces. The structure mass is chosen for the following reason. The chemical separation of charged and neutral species proceeds when the gravitational collapse of structures enhances the matter density. In the ΛCDM cosmological model, smaller structures form earlier. Larger structures such as galaxies form through collisions and mergers of smaller structures. Here we consider only structures such that they collapse at the redshift of z = 30 − 10, and baryonic matter can form astrophysical highdensity objects in the structures after their collapses. Then, masses of such structures should be larger than ∼ 10 6 − 10 8 M⊙ (Tegmark et al. 1997). Significant fractions of baryonic matters in large structures which are observed today, therefore, have experience that they enhanced their densities at gravitational contractions of small structures with nearly the minimum masses. We then assume a small structure with mass 10 6 M⊙ as a first structure. Although the merger is a dominant cause of the formation of large structures, a part of baryonic matter is expected to have flown into the structures along filament structures (T. Ishiyama, 2013; private communications). It is, therefore, not to say that almost all material experienced the density enhancement at gravitational collapses of near-spherical structures. In this section, we show results of time evolutions for average densities of chemical species, spatial distributions of the densities and azimuthal magnetic field. Results of other physical variables are described in Appendix B. where Aj is the mass number of j andχj ≡ (nj/nH) is the initial cosmic average value for the number ratio of j to hydrogen. Solid lines show analytical curves of hydrogen densities in the structure (upper line) and IGM (lower). They are calculated based on the following assumption: Outside the structure, the density is given by cosmic average density: Average densities versus time ρ O H (t) =ρH(t) = ρ O H (ti) 1 + z 1 + zi 3 ,(71) where z and zi are redshifts corresponding to time t and ti, respectively. Inside the structure, on the other hand, the density is given by ρ I H (t) = ρ b (ti)X r sph (ti) r sph (t) 3 = 3 4π Mstr A 3 G (1 − cos θG) 3 Ω b Ωm X = 7.57 × 10 −26 (1 − cos θG) −3 g cm −3 Mstr 10 6 M⊙ AG 298 pc −3 Ω b 0.0463 Ωm 0.279 −1 X 0.75 .(72) When effects of magnetic field are small, curves of H + and 7 Li + should be nearly the same as that of hydrogen. The dashed line shows an analytical curve for charged species, such as proton and 7 Li + , based on the following assumption: The species can collapse gravitationally along the axis of magnetic field (z-axis), and just expands across the field at the same velocity as the cosmic average expansion. In this case, its density evolves as ρ I i (t) = ρ I i (ti) r sph (ti) r sph (t) 1 + z 1 + zi 2 = Aiχi ρ I H (t) 1/3 ρ O H (t) 2/3 ,(73) where it was assumed that hydrogen densities inside and outside the structure, i.e., ρ I H (t) and ρ O H (t), respectively, are almost equal at the initial time ti(≪ t). The ti value has been taken to be enough small. In Case 1, charged species in the structure are diluted at the intermediate phase with low densities. This dilution can be measured as the ratio between the normalized densities of p (and 7 Li + ) and hydrogen. The ratio reduces when the density becomes low around the turnround. In the early and late phases of high densities, dilutions do not proceed effectively since the motions of charged and neutral species are strongly coupled in high density environments. Eventually, the 7 Li + ion is diluted in the structure by a factor of ∼ 4 in the end of the calculation. This dilution history is qualitatively applied to Case 2. The dilution factor is, however, much smaller in Case 2. (5), and 474 Myr (6). Times 1-3 are in an expanding phase, and times 4-6 are in a collapsing phase. Note that structure sizes or densities of neutral hydrogen are the same at times 1 and 6, 2 and 5, and 3 and 4, respectively. Solid and dashed lines correspond to the regions inside and outside of the structure, respectively. The initial gradient of Bz causes an expansion of the charged-species fluid. Since the Bz value is large at a small radius, the expansion is fast in the region of small r. Accordingly the magnetic field amplitude and its gradient rapidly decrease in the inner region of small r. Since the gradient of Bz is not assumed in the outer region of large r, Case 2 (filled triangles), respectively, in the structure as a function of cosmic time t. The densities are normalized by the factor of nuclear mass number A j times the initial cosmic average value for the number ratio of j and hydrogenχ j . Solid lines show analytical curves of hydrogen densities in the structure (upper line) and IGM (lower). It was assumed that inside the structure, the density is determined by the gravitational free fall of matter, and that outside the density is given by the cosmic average density. The dashed line shows an analytical curve for charged species, such as proton and 7 Li + , based on the following assumption: The species can collapse gravitationally along the axis of magnetic field, but expands across the field exactly following the cosmic average expansion. charged particles do not move in the outer region. Then, high density shells forms at the boundaries between the inner and outer regions as seen in this figure as bumps. The curves for the densities of charged species have oscillatory structures as well as the bumps caused by the assumed initial condition. The charged species inside the structure are diluted more efficiently in Case 1 than in Case 2 because of the stronger magnetic field. Figure 4 shows the magnetic field (z-component) as a function of radius for Case 1 (left panel) and Case 2 (right panel). Solid and dashed lines correspond to values inside and outside the structure, respectively. Additionally to the effect of expansion and collapse of neutral hydrogens, weakening of magnetic field is observed in the small r region. This dilution is cased by outward movements of charged species (Fig. 3). It is seen that outgoing charged species decrease the Bz value in the small r region more significantly in Case 1 than in Case 2. Chemical separation Magnetic field In Case 1, the magnetic field at time 6 is almost uniform inside the collapsing structure. The information for the initial condition of the field gradient (Sec. 2.5) is thus wiped out by the diffusion of the charged plasma and magnetic field through the contracting neutral matter. It is expected that when the initial magnetic field gradient is large enough and that an ambipolar diffusion effectively operates as in Case 1, the result is possibly not so sensitive to the type or the details of the initial magnetic field gradient. This is because if the field evolves by the ambipolar diffusion, the initial conditions will be forgotten after a certain amount of time. MAGNETIC FIELD AMPLITUDE In preceding sections, a magnetic field generation is neglected. The generation, however, proceeds through a drift current creation in the structure, although its effect is insignificant as explained below. The magnetic field in a structure evolves (Browne 1982) as dB dt = 1 4πσe ∇ 2 B ≈ B 4πσeL 2 B ,(74) Thus, large inductances of large astrophysical objects result in very long diffusion times. The generation of magnetic field in the cosmological time-scale is, therefore, impossible. The self-inductance of astrophysical objects with length scale LB is given by L ind (LB) ∼ µmLB, where µm is the magnetic permeability. When some electromotive force is created in the structure, an electric current is produced at an approximately constant production rate. The rate is inversely proportional to the inductance. The magnetic energy W stored in a coil, that is the structure itself in the present case, is proportional to the electric current squared, W = L ind (LB)I 2 /2 ∼ µmLBI 2 , where I ∼ jL 2 B is the electric current. The magnetic energy per volume, L 3 B , is then proportional to the length scale squared: W/L 3 B ∼ µmLB(jL 2 B ) 2 /L 3 B ∼ µmj 2 L 2 B . The generation of magnetic field on a large scale of LB, therefore, requires large amount of source energy density. For the reason above, the magnetic field is never generated effectively by an electric current associated with dynamical friction. The Ampere's equation [Eq. (15)] relates an electric current density to a magnetic field as If a magnetic field and an electric current density existed from the beginning of the structure formation, and the Lorentz force is enough large to realize a separation between charged and neutral species, charged species possibly do not collapse gravitationally. The charged species, therefore, do not participate in structure formations. Although charged species move differently from neutral species, scatterings between charged and neutral species efficiently transfer the kinetic energy of neutral species to charged species. The velocity difference between H and proton has been assumed to be the typical cosmological recession velocity in the present case [Eq. (46)]. This velocity corresponds to the proton temperature of T ∼ mpv 2 rel /6 = 52.3 K. The scatterings then gradually increase the temperature of charged species as a function of time. Resultantly, it is expected that the friction time-scale, τei [cf. Eq. (58)], increases. The equilibrium amplitude of the magnetic field is then related with the velocity difference [Eqs. (14) and (77) where ∆vr = vpr − vnr is the velocity difference. In the second line, we assumed typical physical values estimated at the gravitational turnround z = ztur = 16.5, and the critical value of ∆vr given by the cosmological recession velocity at the turnround [Eq. (46)]. The corresponding comoving magnetic field is Bz0 = Bz(ztur)/(1 + ztur) 2 = 0.171 nG. We note that the minimum amplitude of magnetic field which can support the charged species against the dynamical friction is larger when a larger structure is considered (Sec. 5.2). If the time-scale of field generation were shorter than the dynamical time of the system, an azimuthal electric current density is gradually induced by F × B drifts. A poloidal magnetic field is then generated. Magnetic fields in astronomical objects can be related to electric currents existing in their interiors. In general, the fields are generated by electric currents which themselves are formed by motions of charged species, vi, in regions with finite amplitudes of magnetic fields. This process for an amplification of magnetic field is called self-exciting dynamo, and is thought to operate in the Sun, Earth, other planets, interstellar clouds, and Galaxy (Alfven 1981, pp. 86-88). The dynamo effectively operates if a primary field exists initially, and has its origin different from the dynamo. One of requirements for a self-exciting dynamo is an enough energy release inside the object to energize the dynamo (Alfven 1981, pp. 114-115). Two stream instability A relative motion of an electron fluid to an ion fluid can cause a micro-instability (Woods 2004). If the relative velocity exceeds a critical value, a turbulence is triggered. When the temperatures of electron and proton are equal, the critical values of relative velocity is v rel = FCe, where F ≈ 0.604 is a factor fixed for the maximum growth rate of instability, and Ce ≡ (Te/me) 1/2 is a measure for thermal speed of electron. However, the relative velocity is much smaller than the electron thermal velocity even when the azimuthal electric current density is so high that the radial velocity difference of protons and hydrogens is equal to the cosmic recession velocity at the turnround. Then, the instability does not occur. The relative velocity is given by v rel = v pφ − v eφ ≈ αpn Bz HL = 3.81 × 10 −4 cm s −1 χ H + 6.52 × 10 −5 −1/2 H 2.70 × 10 3 km s −1 Mpc −1 1/2 (σv) pH 2.3 × 10 −9 cm 3 s −1 1/2 L LB 1/2 ,(79) where Eqs. (53) Magnetic field generation in molecular cloud We roughly check an amplitude of magnetic field generated through a drift current in MCs. For this purpose, we take physical quantities at the surface of MCs in Model A of Ciolek & Mouschovias (1994): nH ∼ 2.6 × 10 3 cm −3 , χ H + ∼ 10 −10 , L ∼ 4.3 pc, B ∼ 35.3 µG, \|vnr\| ∼ 1.9 × 10 3 cm s −1 , σpn(∆vr ∼ 10 3 cm s −1 ) ∼ 1.6 × 10 −13 cm 2 . The αpn value [Eqs. (53) and (55)] is then estimated to be αpn = 2.93 × 10 −12 G nH 10 3 cm −3 (σv) pH 3.1 × 10 −10 cm 3 s −1 . (81) The amplitude of generated field is then given [Eqs. (14) and (77) This is much smaller than the initial magnetic field assumed in a MC, Beq,c0 = 35.3 µG. The field generation, therefore, does not affect at all the total amplitude of magnetic field during the time evolution of the model MC. GENERATION OF A MAGNETIC FIELD GRADIENT In the present calculation, gradients of the magnetic field in the r-direction are assumed in the initial conditions. Practically, charged species of ions and electrons move outward only in special configurations of magnetic fields as in this setting. A gradient of the magnetic field can, however, be generated through the gravitational collapse of a structure even if the initial magnetic field amplitude is coherent and homogeneous. Figure 5 shows an illustration for the creation of magnetic field gradient in the r-direction. The upper direction on the plane of paper is defined as the z-axis, and open circles correspond to boundaries of a structure at an early epoch before the gravitational contraction (left part) and at a late epoch during the contraction (right part). Thin arrows show magnetic field lines, open thick arrows indicate directions of the gravity, and filled thick arrows indicate directions of the field gradient. The baryon density inside the structure increases relative to that of outside, as a function of time. Since field lines are initially frozen into the charged plasma, the Bz value increases inside the structure. A field gradient is then generated in the r-direction near the boundary (right part). Consequently, the Lorentz force is produced in the r-direction with its strength proportional to [(∇ × B) × B]r ∼ −(∂rBz) Bz [Eq. (20)]. PARAMETER REGION FOR CHEMICAL SEPARATION Scales of structure and magnetic domain In the ΛCDM model, large structures such as galaxies and galactic clusters are formed through collisions and mergers of smaller structures. When we consider gravitational collapses of structures with scales smaller than that of Galaxy, effects of the magnetic field on motions of charged and neutral species are quantitatively different from that of larger structures. For example, a smaller velocity difference is needed for charged species to escape from gravitational collapse of smaller structures at the time of turnround, i.e., H(ztur)r sph . We note that the typical scale of the magnetic domain in which the field direction is coherent should be larger than the system scale. If the scale of the magnetic domain is smaller than the system scale, average radial velocities of charged species are roughly the same as that of neutral species although there are fluctuations in velocities caused by the magnetic field existing over small scales. We, therefore, have a constraint on the comoving LB0 value, i.e., LB0 L(1 + z) [cf. Eq. (41)]. Constraints The condition for the gravitational collapse of neutral matter is that the gravitation [the first term in RHS of Eq. (19)] is larger than the Lorentz force (the second term). It is clear that magnetic fields on the scale larger than 600 pc with amplitude less than ∼ 10 −7 G do not affect the gravitational collapse of neutral atoms [Eqs. (22) and (23)]. When the amplitude and the spatial scale of magnetic field satisfy the condition, neutral species can collapse gravitationally. Using Eqs. (22) This constraint is independent of the turnround redshift ztur. The condition to suppress the gravitational collapse of charged species is that the Lorentz force is larger than the friction from neutral hydrogens for the velocity difference given by the cosmic recession velocity at the turnround. In this case the equation, i.e., vpr − vnr > H(ztur)r sph , holds. When the amplitude and the spatial scale of magnetic field satisfy the condition, charged species can get left in IGM typically. The friction on proton is the predominant friction working on the whole charged fluid in the radial direction. The proton velocity is then related to the velocity of neutral matter [Eq. (32)]. Using Eqs. (23) The condition on the comoving value is also given by In Eqs. (85) and (86), the reaction rate (σv)pH is a function of the turnround redshift and the structure mass. It is given by the value for the cosmic recession velocity [Eq. (46)] at the boundary of the structure [Eq. (41)]. Figure 6 shows constraints on the comoving Lorentz force B 2 z0 /LB0 as a function of the turnround redshift ztur. Solid lines correspond to lower limits from the condition that charged species do not contract along with neutral hydrogen [Eq. (86)]. Dashed lines correspond to upper limits from the condition for the gravitational collapse of neutral hydrogens [Eq. (84)]. For respective constraints, lines are shown for three cases of the structure mass, Mstr = 10 6 (the lowest lines), 10 9 (the middle lines), and 10 12 M⊙ (the highest lines). For Mstr = 10 6 M⊙, we find a parameter region for a successful chemical separation at B 2 z0 /LB0 10 −20 G 2 kpc −1 at redshift 1 + ztur 30. For Mstr = 10 9 M⊙, a similar interesting parameter region exists at B 2 z0 /LB0 10 −19 G 2 kpc −1 and 1 + ztur 15. For the most massive case of Mstr = 10 12 M⊙, no region is found at relatively high redshifts of 1 + ztur ∼ 10. In this way, at gravitational collapses of heavier objects, it is more difficult to separate the motions of neutral and charged particles. Figure 7 shows constraints on the comoving Lorentz force as a function of the structure mass Mstr. Solid lines correspond to lower limits from the condition for the motion of charged species [Eq. (86)] for three cases of the turnround redshift, 1 + ztur = 17.5, 25.4, and 33.3 (corresponding to the collapse redshift z col = 10, 15, and 20, respectively). The dashed line shows the upper limit from the condition for the gravitational collapse of neutral hydrogens, which are independent of the turnround redshift [Eq. (84)]. It can be seen that the chemical separation is more difficult in structures which collapse earlier. We find parameter regions for the chemical separation at Mstr O(10 8 ) M⊙ for the latest collapse case of 1 + ztur = 17.5, Mstr O(10 7 ) M⊙ for 1 + ztur = 25.4, and Mstr O(10 6 ) M⊙ for 1 + ztur = 33.3. DISCUSSION Later epoch of the structure formation We comment on a possibility of chemical separation in a later epoch of structure formation. Depending on the virialization temperature of the collapsing structure, the ionization degree after the virialization can be smaller than that during the gravitational collapse because of the high density. The baryon density in the late epoch is, on the other hand, much larger than that during the collapse. Then, the larger friction force must be balanced by the Lorentz force originating from a larger magnetic field. For a fixed structure mass, the gravitation term [the first term in RHS of Eq. (19)] roughly scales as ∝ ρ (22)]. On the other hand, the Lorentz force term (the second term) scales as ∝ B 2 /LB ∝ (1 + δ) 5/3 if we roughly assume adiabatic contractions of charged species and magnetic domains in the early epoch of structure formation. Therefore, it is expected that if an ambipolar diffusion does not occur in the early structure formation epoch, it does not also in a later epoch as long as a magnetic field generation does not operate during the structure formation. 5/3 b ∝ (1 + δ) 5/3 [Eq. Chemical reactions Lithium atoms can be ionized by a ultraviolet (UV) photon as Li + γ → Li + + e − .(87) They can be ionized also through a collision with an H + ion, which is generated by UV photons or cosmic rays: Li + H + → Li + + H.(88) The ionization potential of Li is I(Li) = 5.39 eV which corresponds to the temperature T = 2I(Li)/3 ∼ 4 × 10 4 K. Some proportion of Li atoms can be also easily ionized by external UV sources or a gas heating at the virialization of structures. The Li + ions produced secondarily in this way can then be trapped by magnetic field, and possibly be left out of forming structures. Such a contribution to a resulting lithium abundance in the collapsed structure, however, operates after the gravitational collapse considered in this paper. They are then neglected here. Li abundance of MPS Astronomical observations indicate primordial abundances of D (Pettini & Cooke 2012), 3 He (Bania, Rood, & Balser 2002), and 4 He (Izotov & Thuan 2010; Aver, Olive, & Skillman 2010) consistent with those predicted in SBBN model. Primordial 7 Li abundance is inferred from spectroscopic observations of metal-poor halo stars. We adopt log( 7 Li/H)= −12 + (2.199 ± 0.086) determined with a 3D nonlocal thermal equilibrium model ). This estimation corresponds to the 2σ range of 1.06 × 10 −10 < ( 7 Li/H) MPS < 2.35 × 10 −10 . This Li abundance level is ∼ 3-4 times smaller than the SBBN prediction (Coc et al. 2012(Coc et al. , 2013, and the dispersion of observed Li abundance is small. Since the observed 7 Li abundance is not so different from the SBBN prediction, it is naturally expected that SBBN model successfully describes the outline of primordial light element synthesis. The Li abundances in MPSs can be affected by several physical processes operating after the BBN epoch. The abundance ratio of Li and H in MPSs is then expressed as (Li/H) MPS = (Li/H) SBBN F dep ,(90) where (Li/H) SBBN is the abundance ratio in SBBN model, and F dep is the depletion factor associated with 1) modified BBN models including exotic long-lived particles or changed expansion rate, 2) the structure formation as considered in this paper, 3) the virialization of the structure, 4) the formation of observed MPSs, and 5) the stellar processes in surfaces of MPSs occurring from the star formation until today. Generally, cosmological processes change elemental abundances universally, while astrophysical processes do locally depending on physical environments of respective stars. It is, therefore, difficult to explain the discrepancy in 7 Li abundance with astrophysical processes which result in large dispersions in the abundance. The depletion factor from the chemical separation during the structure formation can be described by F dep ≡ [(n7 Li + n7 Li + )/(nH + n H + )]str [(n7 Li + n7 Li + )/(nH + n H + )]uni ≈ χ7 Li,uni + χ Li + ,str (χ7 Li + χ7 Li + ) uni ,(91) where quantities with subscripts, 'uni' and 'str', are values of the homogeneous early universe after the cosmological recombination, and those of the collapsed structure in the late universe, respectively. In the second line, it was assumed that the primordial ionization degree of hydrogen is negligibly small, i.e., χ H + ≪ 1, and that values of the number ratio χ7 Li are equal in the homogeneous early universe and the structure. We suppose the initial abundance ratio of 7 Li + / 7 Li ∼ 1 as suggested from a chemical history of homogeneous early universe (Vonlanthen et al. 2009). The chemical separation via the ambipolar diffusion can only dilute the charged 7 Li + . The depletion factor is, therefore, 1/2 at minimum when the primordial 7 Li + is completely expelled from the structure. This factor would be smaller if the initial 7 Li + abundance in the gravitational structure formation is larger for some reason. For example, even a small intensity of ionizing photon of 7 Li would quickly transform 7 Li to 7 Li + without absorption by neutral hydrogen (Sec. 8.2). On the other hand, the depletion factor would be larger if the chemical separation is less efficient. The Li abundance of MPSs may not be explained by the chemical separation only. In that case, we need another depletion mechanism. As an example, a rotationally induced mixing model (Pinsonneault et al. 1999(Pinsonneault et al. , 2002 for MPSs is chosen here since dispersions as well as depletion factors are predicted theoretically only in this model among stellar depletion models. Since the predicted depletion factor is proportional to the dispersion factor, the depletion factor is constrained from observed dispersions. Pinsonneault et al. estimated the depletion factor: '0.13 dex, with a 95 % range extending from 0.0 to 0.5 dex' (Pinsonneault et al. 2002). This model explains a part of the Li abundance discrepancy although the complete solution by this mechanism only seems almost impossible. The Li abundances in MPSs may, therefore, be explained by the combination of the ambipolar diffusion during the structure formation and the rotationally induced mixing in stars. Stellar Li abundances in metal-poor globular clusters (GCs) have also been measured. For example, GC M4 was studied using high-resolution spectra with GIRAFFE at Very Large Telescope. The Li abundance in turn-off stars is then found to be log( 7 Li/H)= −12 + (2.30 ± 0.02 + 0.10) (Mucciarelli et al. 2011). All Li abundances measured so far are summarized in Fig. 3 of Mucciarelli et al. (2011), and they are consistent with abundances in metal-poor halo stars at present. If the ambipolar diffusion studied in this paper caused the small Li abundances of MPSs, however, reduction factors of MPSs can reflect respective histories of parent structure of MPSs. In a modern model calculation for GC formation, the Galaxy formation results from a continuous process of merging and accretion which is realized in a hierarchical structure formation scenario (Kravtsov & Gnedin 2005). In the model, GCs form at densest regions of filaments in a large-scale structure. Other constraint on PMF Theoretical and observational constraints on the cosmic magnetic field have been summarized in Durrer & Neronov (2013). The magnetic field strength in the interesting parameter region found in this study (Sec. 7) looks somewhat higher than the theoretical upper limit from the effect of dissipation of magnetic field through the processing by MHD turbulence. The propagation length of Alfvén wave is given by λB ∼ vAt, where vA is the Alfvén speed. This length scale corresponds to "the size of largest processed eddies" (Durrer & Neronov 2013) by MHD turbulence. The Alfvén speed during the matter dominated epoch of the homogeneous universe is given by vA = B/ 4πρ b with ρ b ∝ (1 + z) 3 the baryon density [Eq. (38)]. Note that the density used in the Alfvén speed is that of fluid with a frozen-in magnetic field. The density is then given by the total density if the fluid is fully ionized or if the neutral fluid is effectively coupled to the charged fluid through the collision so that the magnetic field can be considered frozen also into the neutral fluid. The physical states considered in this paper are ones in which the matter is only weakly ionized and the coupling of the charged and neutral fluids is effective. Although the ambipolar diffusion reduces the magnetic pressure gradient until the Lorentz force becomes comparable to the gravitation [cf. Eq. (25)], the coupling is effective after then. Therefore, the total fluid has a frozen-in magnetic field and its density is used in the Alfvén speed. The distance is then given by λB ∼ B0 (4πρ b0 ) 1/2 (1 + z) 1/2 2 3H0Ω 1/2 m (1 + z) 3/2 = 2 3/2 3 3/2 B0 m Pl H 2 0 Ω 1/2 b Ω 1/2 m (1 + z) ,(92) where m Pl is the Planck mass. Consequently, the comoving propagation length λB0 = λB(1 + z) is constant. Since the magnetic fields on scales shorter than λB0(B0) decay, there is a maximum amplitude of magnetic field which escapes from this decay for a given λB0. From the above equation, an upper limit on the B0 value is derived as B0 1.3 × 10 −10 G h 0.700 2 Ω b 0.0463 1/2 Ωm 0.279 1/2 λB0 10 kpc .(93) This upper limit is lower than the field value required for the chemical separation [Eqs. (84) and (86)] (by a factor of ∼ two for λB0 = 10 kpc). However, Eq (93) is just a rough estimate, and realistic limits should be derived in precise calculations in future. It is interesting that the upper limit caused by the MHD processing in the early universe is near to the interesting field strength. It indicates that relatively large magnetic field in the early universe may have been reduced by the MHD effect to the level which is most appropriate for the chemical separation causing the lithium problem. During the gravitational collapse of structures, the Alfvén speed increases as ∝ (1 + δ) 1/6 if the dissipation of magnetic field is not operative. The dissipation scale in collapsed structures which are decoupled from the cosmic expansion is then given by λ str B (z) ∼ 2 3/2 3 3/2 B0(1 + z) 1/2 (1 + δ) 1/6 m Pl H 2 0 Ω 1/2 b Ω 1/2 m (1 + z) 3/2 .(94) The contraction increases the dissipation scale slightly. The MHD effect then becomes significant in a large density environment. Therefore, after the collapse, the magnetic field strength can be decreased further. SUMMARY We considered a possible effect of PMFs on motions of charged and neutral chemical species during the formation of first structures at redshift z = O(10). We assumed that the PMF has a gradient in a direction perpendicular to the field direction. This gradient is realized by an electric current density in the direction perpendicular to both directions of the field lines and the gradient. The Lorentz force on the charged species then causes a velocity difference between charged and neutral species in the direction of the field gradient. Resultantly, a velocity of charged species can be different from that of neutral species which collapses gravitationally during the structure formation. Therefore, 7 Li + ions may have possibly escaped from gravitational collapse of early structures. Calculations for fluid motions of charged and neutral species were performed through a simple estimation using fundamental fluid and electromagnetic equations. We assumed a gravitational contraction of neutral matter in a spherically symmetric structure. In addition, we utilized a cylindrical coordinate, and assumed a gradient of the altitudinal (z-component) magnetic field in the radial direction. Related physical quantities are listed, and their typical values are given in Sec. 3. Some analytical equations are introduced in Appendix A. When the amplitude of magnetic field is sufficiently large, the charged fluid significantly decouples from the neutral fluid. It is then possible that during the gravitational contraction of structure mainly composed of neutral hydrogens, contractions of protons, electrons, and 7 Li + ions do not occur. Although fluid motions of charged chemical species are solved for only H + , e, and 7 Li + in this study, other charged species are expected to have similar motions. Because of large inductances of large astronomical structures, the generation of magnetic field is never efficient during the structure formation at z ∼ 10. Therefore, only PMFs which existed from the start of the structure formation can trigger the chemical separation. The chemical separation requires the magnetic field gradient in a direction perpendicular to the field direction. Although such a gradient was assumed in the initial condition in this study, it may be produced associated with a density gradient during the gravitational contraction of structures without any initial field gradient. Based on the calculated result of the chemical separation, we derived a parameter region for a successful chemical separation taking the structure mass, the turnround redshift of the gravitational collapse, and the comoving Lorenz force, i.e., B 2 z0 /LB0, as parameters. It was found that the parameter region can be constrained to be very narrow. If such a chemical separation has occurred during the structure formation, the primordial 7 Li + , which was produced via the recombination of 7 Li 2+ but survived against its recombination during the cosmological recombination epoch, possibly does not participate in the gravitational contraction. The abundance ratio of Li/H in early structures, which are progenitors of the Galaxy, can then be smaller than that inferred from SBBN model. Therefore, the chemical separation may have caused the Li problem of the MPSs. The amplitude of the PMFs required for the chemical separation was estimated. It is close to (somewhat smaller than) an upper limit determined from the effect of MHD turbulence on the decay of field amplitude. This fact indicates the following possibility: The PMF was generated via some mechanism operating in the extremely early universe. The field amplitude was modulated by the MHD effect to the value appropriate to the chemical separation. APPENDIX A: SOLUTIONS OF VARIABLES FROM THE FORCE BALANCE A1 Drifts in the expanding universe We consider two different cases of weak and strong Lorentz forces. A1.1 Weak Lorentz force Firstly, we suppose that a magnetic field is so weak that collisional momentum transfers from hydrogens to electrons and protons result in very small velocity differences despite the existence of the weak Lorentz force. This case typically satisfies the condition Bz ≪ (4πenpαpnHLLB) 1/2 [cf. Eqs. (78) or (32)]. Because of effective scatterings between protons, electrons, and neutral hydrogens, velocities of p and e are almost identical to that of neutral hydrogens, i.e., vp = ve = vn. The force balances for p and e [Eqs. (4) and (5) with an assumption D/Dt = 0 and a neglect of ∇Pp and ∇Pe terms] give the value of electric field: E ∼ = −vn × B = −   0 vnzBr − vnrBz 0   ,(A1) where it was assumed that neither magnetic field nor neutral hydrogen velocity has an azimuthal component. A1.2 Strong Lorentz force Secondly, we suppose that a magnetic field is strong. The collisional momentum transfers between charged species and hydrogens with large radial relative velocities are then counterbalanced by the Lorentz force, i.e., Bz ∼ (4πenpαpnHLLB) 1/2 . The protons and electrons receive dynamical frictions from hydrogens with different amplitudes determined by momentum transfer cross sections. In such a case, protons and electrons are promoted to start drifting in directions opposite to each other with velocities, vDj = F j × B/(Zj eB 2 ), where F j is the friction force [cf. Eqs. (4) and (5)]. The both drift directions are perpendicular to the direction of the friction force. Drift velocities of protons and electrons are given by vDp = 1 B 2   [−αpnv pφ + αpe (v eφ − v pφ )] Bz [αpn (vnz − vpz)] Br − [αpn (vnr − vpr)] Bz [αpnv pφ − αpe (v eφ − v pφ )] Br   ,(A2)vDe = − 1 B 2   − [αenv eφ + αep (v eφ − v pφ )] Bz [αen (vnz − vez)] Br − [αen (vnr − ver)] Bz [αenv eφ + αep (v eφ − v pφ )] Br   .(A3) However, these drifts never complete effectively because of large inductances of large astrophysical objects (Sec. 5). A2 Equilibrium state A2.1 Proton and electron It is expected that bulk motions of chemical species and electromagnetic fields are in equilibrium states at all times during the early epoch of structure formation. In the equilibrium state, force balance equations for proton (p or H + : the dominant component of ion) and electron [Eqs. (6) and (7)] include nine unknown parameters: components of vectors E, vp, and ve. Because of nearly complete neutrality for local charge, fluid velocities of ions and electrons should be approximately equal as for r-and z-components. The number of independent parameters is then reduced to be seven: Er, E φ , Ez, vpr = ver, vpz = vez, v pφ , and v eφ . The following conditions have been imposed additionally: v nφ = 0 and B φ = 0. Then, three equations for Er, E φ , and Ez are obtained: Er = −v pφ Bz − αpn(vnr − vpr) = −v eφ Bz + αen(vnr − vpr),(A4)E φ = − (vpzBr − vprBz) + αpnv pφ − αpe (v eφ − v pφ ) = − (vpzBr − vprBz) − αenv eφ − αpe (v eφ − v pφ ) ,(A5)Ez = v pφ Br − αpn(vnz − vpz) = v eφ Br + αen(vnz − vpz).(A6) From the second equality of Eq. (A5), we instantaneously find v eφ = − αpn αen v pφ .(A7) From the second equalities of Eqs. (A4) and (A6), vpr and vpz values are given, respectively: vpr = vnr + Bz αen v pφ , (A8) vpz = vnz − Br αen v pφ .(A9) Insertion of Eqs. (A8) and (A9) in the first equality of Eq. (A5) leads to an expression for the parameter v pφ : v pφ = αen (E φ + Brvnz − Bzvnr) B 2 + αenαpn + αpe (αen + αpn) . The v pφ value is given by the rotation of B field [Eq. (34)]. We thus have seven equations for seven variables. Therefore, the solutions can be obtained. A2.2 General singly-ionized ions From the force equation for singly-ionized ions (i =H + , Li + , ...), the electric field is given by E = −vi × B − ρi τin (vn − vi) eni − ρi τie (ve − vi) eni .(A11) The left-hand side corresponds to the term of electric field, and the first, second and third terms in the RHS correspond to the Lorentz force, the frictions from H and electrons, respectively. The friction from protons is neglected in the force equation for i = p since the friction parameter αip is ∼ me/mp times smaller than αie [cf. Eq. (64)]. For the case of 7 Li + , the force balance equation is somewhat different from that of proton because of differences in the ion masses and the momentum transfer cross sections. We do not assume the condition of charge neutrality since abundance of i can be negligibly small (for example, the primordial number ratio of 7 Li + /H is about 2.6 × 10 −10 (Vonlanthen et al. 2009)). We have assumed that v nφ = 0 and B φ = 0. Then, three equations for Er, E φ , and Ez are obtained: Er = −v iφ Bz − αin (vnr − vir) − αie (ver − vir) ,(A12)E φ = − (vizBr − virBz) + αinv iφ − αie (v eφ − v iφ ) ,(A13)Ez = v iφ Br − αin (vnz − viz) − αie (vez − viz) .(A14) Using these three equations with values of E and ve derived in Appendix A2.1, the velocity of i is solved to be   vir v iφ viz   = 1 αi (α 2 i + B 2 r + B 2 z )   α 2 i + B 2 r αiBz BrBz −αiBz α 2 i αiBr BrBz −αiBr α 2 i + B 2 z       Er E φ Ez   +   αinvnr+ αiever αiev eφ αinvnz+ αievez     ,(A15) where αi ≡ αin + αie was defined. A3 Typical case for an effective chemical separation at the turnround We define, as a guide, a typical case of effective chemical separation in which the radial velocity difference between charged and neutral species is exactly equal to the recession velocity at the turnround. The velocity difference is determined from the balance between the Lorentz force and the friction from neutral hydrogens in the r-direction. We assume a strong magnetic field in z-direction, i.e., Bz ≫ α ab , Br. In this case, ions and electrons possibly do not move to the structure centre, and their radial velocities can be as large as the cosmic recession velocity. Velocities of charged species at the radius r sph are then matched to the cosmic recession velocity, i.e., vpr = Hr sph . The proton velocities and the electric field are then determined. Eqs. (A10) and (A9), respectively, give relations of v pφ = αen Bz (Hr sph − vnr) , (A17) vpz = vnz − Br Bz (Hr sph − vnr) ∼ vnz.(A18) The equilibrium electric field is then derived: Figure B1 shows relaxation time-scales in collisions with hydrogens, τ7n (solid line) for 7 Li + , τpn (dashed line) for H + , and τen (dotted line) for e − , as a function of cosmic time t. In this calculation, a relaxation time τ ab is defined as the time it takes a species a to change its velocity toward that of b until a velocity difference of a and b becomes smaller than the thermal relative velocity of the a+b system. When the velocity difference, \|va − v b \|, is smaller than the thermal velocity, the τ ab value is approximated by the value evaluated at the thermal velocity. In the present calculations for Case 1 and 2, any velocity differences were found to be smaller than thermal velocities at almost all position and time. The relaxation time-scales are then determined only from the thermal velocity in the structure. Therefore, they do not depend on radius. Since the relaxation time is inversely proportional to the target number density [Eq. (53)], the τ ab values increase during the expanding phase (t 0.242 Gyr), while they progressively decrease during the contracting phase (t 0.242 Gyr). Cross sections are evaluated with linear interpolations of the adopted data. This approximation causes a non-smooth behavior in the τ7n curve. In addition to a similar non-smoothness, zigzags are seen in the τpn curve, which result from fluctuations in the cross section. The smooth shape of the τen curve reflects the constant cross section assumed for center of mass collision energy smaller than the lowest energy data point of ∼ 15 meV. Figure B2 shows the relaxation time τpe of H + as a function of radius for Case 1 (left panel) and Case 2 (right panel). Solid and dashed lines correspond to values inside and outside the structure, respectively. In the whole calculation time, the relative velocity of proton and electron is given by the thermal velocity since the relative fluid velocity is smaller than the thermal velocity. The τpe values at large radii are almost constant with time. The reason comes from a constant reaction rate for a given thermal velocity, and a constant density of electrons. The thermal velocity is assumed to be proportional to the square root of gas temperature T ∝ ρ 2/3 n [Eq. (47)], while the proton and electron number densities scale as np,e ∝ ρp,e. The relaxation time is then given by τpe = (mp/me)τep ∝ T 3/2 /np ∝ ρn/ρp [cf. Eqs. (55), (58), and (61)]. Inside the structure, this quantity increases with time since the number abundances of proton and other charged species decrease (Fig. 3). B2 Relative velocities Figure B3 shows the radial velocity difference of H + and H, i.e., vpr − vnr, as a function of radius for Case 1 (left panel) and Case 2 (right panel). Solid and dashed lines correspond to values inside and outside the structure, respectively. The velocity difference of 7 Li + and H, i.e., v7r − vnr, is the same as that of H + and H. The conditions, Bz ≫ αie ≫ αin, Br (for i = p and 7 Li + ) [cf. Eqs. (43), (53), (55), (62), and (63)] and vjr, vjz ≫ v jφ (for any species j), are satisfied in the present case. Under these conditions, the approximate relation v7r ∼ vpr is satisfied [see Eq. (A22)]. Strictly, ions such as the 7 Li + ion have radial velocities very slightly different from that of proton under an electric field in which radial motions of protons and electrons are balanced 2 . The velocity difference is larger in Case 1 than in Case 2. The time evolution is also different since the movement of charged species is larger and the Bz value in small radii evolves more significantly for the larger magnetic field in Case 1. Er = (αpn − αen) (Hr sph − vnr) ,(A19)E φ = αenαpn + αpe (αen + αpn) + B 2 r Bz (Hr sph − vnr) − Brvnz + BzHr sph ,(A20) B1 Relaxation time-scales One can intuitively understand that the radial velocity of the lithium ions is essentially the same as that of the protons as follows. Under the conditions of Bz ≫ αie ≫ αin, Br, the Hall parameter of lithium ions is much larger than 1. The Hall parameter is the dimensionless ratio between the Larmor frequency and the collision rate given by where j is the target particle at the collision, and j =n and e for the case of i= 7 Li. Since βi 1 is satisfied in the present situation, the 7 Li ions gyrate about a magnetic field line more than several times before being knocked off the line by colliding with a neutral particle. The lithium ions, therefore, move with the magnetic field and the proton-electron plasma. Figure B4 shows the azimuthal velocity of proton v pφ as a function of radius for Case 1 (left panel) and Case 2 (right panel). Solid and dashed lines correspond to values inside and outside the structure, respectively. This quantity scales as (∇ × B) φ [Eq. (34)]. Velocities at small radii are larger since a gradient of Bz is assumed at small radii in the initial condition. The velocity is roughly independent of time or hydrogen density if there is no weakening of magnetic field (cf. Fig. 4). This is because v pφ ∝ (Bz/LB)/np ∝ 1/(L 3 B np) is nearly constant. The movement of charge species relative to neutral hydrogen, however, gradually reduces the field gradient ∂rBz. This effect reduces the velocity as a function of time. Shapes of the curves are similar between Cases 1 and 2. Amplitudes and the time evolutions of the velocities are, however, different for the same reason described for Fig. B3. Figure B5 shows the azimuthal velocity of 7 Li + v 7φ as a function of radius for Case 1 (left panel) and Case 2 (right panel). Solid and dashed lines correspond to values inside and outside the structure, respectively. This quantity reflects the force balance as described in Eq. (A15). Because of the conditions, Bz ≫ αie ≫ αin, Br (for i = p and 7 Li + ) and vjr, vjz ≫ v jφ (for any species j), the azimuthal velocity has a limit value of v 7φ = [(αpn + αin)/Bz](vpr − vnr), which is different from both of v pφ and v eφ . B3 Electric field Figure B6 shows the radial component of electric field Er as a function of radius for Case 1 (left panel) and Case 2 (right panel). Solid and dashed lines correspond to values inside and outside the structure, respectively. The Er value is given by where Eqs. (A4) and (A8) were used in the first equality, and Eqs. (20) and (32) were used in the second equality. It thus scales as ∝ (∂rBz)Bz/ρn, and decreases during the expanding phase (curves 1-3), while it increases during the contracting phase (curves 4-6). Figure B7 shows the azimuthal component of electric field E φ as a function of radius for Case 1 (left panel) and Case 2 (right panel). Solid and dashed lines correspond to positive values inside and outside the structure, respectively. Dot-dashed and dotted lines correspond to negative values inside and outside the structure, respectively. In the case of strong magnetic field of Bz ≫ α ab , Br, the relation E φ ∝ vprBz holds in this calculation [Eq. (A5)]. According to the relation, the E φ value decreases with time. The radial velocity is large in the early epoch (curve 1), which is approximately equal to the cosmic expansion velocity. The velocity is negative and its amplitude is large in the late epoch (curve 6), which is given by free fall velocity of the structure. (24)] and the ratio M/M b = 6.03 leads to the value of ρ b g/\|(∇×B)×B/(4π)\| = 2.83. Therefore, roughly speaking, a gravitational collapse occurs if M b /ΦB > (M b /ΦB)crit while a collapse does not occur if M b /ΦB < (M b /ΦB)crit. The φ-component of Eq. (31), on the other hand, does not give an equation with v pφ when the balance relation between v pφ and v eφ [Eq. (A7)] is satisfied. In this case, terms of v pφ and v eφ cancel with each other [cf. Eqs. (55)and(60)]. The equation then reduces to For a time t, the velocity [Eq. (29)] and the density [Eq. (30)] of neutral matter, the overdensity of matter [Eq. (26)] and the temperature [Eq. (47)] are derived. For respective reactions, the code evaluates thermal mean velocities [Eq. (54)] and relative fluid velocities. Then, the friction time-scales [Eqs. (53) and (56)] and the friction parameters [Eq. (55)] are derived using the law of action and reaction [Eq. (60)]. Besides, the electric field [Eqs. (A4-A6)], the velocities of protons [Eqs. (32) and (34)], electrons [Eq. (A7)], and Li + [Eq. (A15)] are calculated. Finally, the magnetic field [Eq. (11)] and the densities of charged species [Eq. (3)] are obtained by time integrations of their change rates. Figure 2 2shows densities of hydrogens (open circles), proton and 7 Li + for Case 1(open diamonds) and Case 2 (filled triangles), respectively, averaged over the structure volume as a function of cosmic time t. The densities are normalized as ρj/(Ajχj ), Figure 3 3shows normalized densities of hydrogen (straight lines), proton and 7 Li + (curves) as a function of radius from the structure centre for Case 1 (left panel) and Case 2 (right panel). Density distributions are drawn for six different times: t=9.29 Myr (denoted by number 1), 102 Myr (2), 195 Myr (3), 288 Myr (4), 381 Myr Figure 2 . 2Calculated average densities of hydrogen (open circles), proton and 7 Li + for Case 1 (open diamonds) and Figure 3 . 3Normalized densities of hydrogen (straight lines), proton and 7 Li + (curves) as a function of radius from the structure centre for Case 1 (left panel) and Case 2 (right panel) at t=9.29 Myr (1), 102 Myr (2), 195 Myr (3), 288 Myr (4), 381 Myr (5), and 474 Myr (6). Solid and dashed lines correspond to the regions inside and outside the structure, respectively. Figure 4 . 4Magnetic field (z-component) as a function of radius for Case 1 (left panel) and Case 2 (right panel) at t=9.29 Myr (1), 102 Myr (2), 195 Myr (3), 288 Myr (4), 381 Myr (5), and 474 Myr (6). Solid and dashed lines correspond to values inside and outside the structure, respectively. electron conductivity(Grasso & Rubinstein 2001). The magnetic field on a length scale LB diffuses during the early structure formation in the typical time-scale of τ diff (LB) .89 × 10 −13 cm −2 s −1 Bz 10 −10 G LB 597 pc −1 . 4πLBj φ Bz ∼ 4πLBenp(αpn + αen)(vpr − vnr) , (55), (78), (A7), and (A17) were used. On the other hand, the Ce value is given by Figure 5 . 5Illustration for the creation of magnetic field gradient in the radial direction. The upper direction is defined as the z-axis, and open circles correspond to boundaries of a structure at an early epoch before the gravitational contraction (left part) and at a late epoch during the contraction (right part). Thin arrows show magnetic field lines, open thick arrows indicate directions of the gravity, and filled thick arrows indicate directions of the field gradient. value is related to the proper value by B 2 z0 /LB0 = (1 + z) −5 B 2 z /LB . The condition on the comoving value is then , (39), (40), (41), (46), and(78), the condition is derived: Figure 6 . 6Constraints on the comoving Lorentz force as a function of the turnround redshift. Solid lines show lower limits from the condition that charged species do not contract along with neutral hydrogen. Dashed lines show upper limits from the condition for the gravitational collapse of neutral hydrogen. Lines are drawn for three cases of the structure mass, Mstr = 10 6 , 10 9 , and 10 12 M ⊙ . Figure 7 . 7Constraints on the comoving Lorentz force as a function of the structure mass. Solid lines show lower limits from the condition that charged species do not contract along with neutral hydrogen. Lines are drawn for three cases of the turnround redshift, 1+ztur = 17.5, 25.4, and 33.3 (corresponding to the collapse redshift z col = 10, 15, and 20, respectively). The dashed line shows the upper limit from the condition for the gravitational collapse of neutral hydrogen independently of the turnround redshift. Figure B1 . B1Relaxation time-scales in collisions with hydrogen, τ 7n (solid line) for 7 Li + , τpn (dashed line) for H + , and τen (dotted line) for e − , as a function of cosmic time t for both Cases 1 and 2. Figure B2 . B2Relaxation time τpe of H + in collisions with electron as a function of radius for Case 1 (left panel) and Case 2 (right panel) at t=9.29 Myr (1), 102 Myr (2), 195 Myr (3), 288 Myr (4), 381 Myr (5), and 474 Myr (6). Solid and dashed lines correspond to values inside and outside the structure, respectively. Figure B3 . B3Radial velocity difference between H + and H, vpr − vnr, as a function of radius for Case 1 (left panel) and Case 2 (right panel) at t=9.29 Myr (1), 102 Myr (2), 195 Myr (3), 288 Myr (4), 381 Myr (5), and 474 Myr (6). Solid and dashed lines correspond to values inside and outside the structure, respectively. Figure B4 . B4Azimuthal velocity of proton v pφ as a function of radius for Case 1 (left panel) and Case 2 (right panel) at t=9.29 Myr (1), 102 Myr (2), 195 Myr (3), 288 Myr (4), 381 Myr (5), and 474 Myr (6). Solid and dashed lines correspond to values inside and outside the structure, respectively. Figure B5 . B5Azimuthal velocity of 7 Li + v 7φ as a function of radius for Case 1 (left panel) and Case 2 (right panel) at t=9.29 Myr (1), 102 Myr (2), 195 Myr (3), 288 Myr (4), 381 Myr (5), and 474 Myr (6). Solid and dashed lines correspond to values inside and outside the structure, respectively. Figure B6 . B6Radial electric field as a function of radius for Case 1 (left panel) and Case 2 (right panel) at t=9.29 Myr (1), 102 Myr (2), 195 Myr (3), 288 Myr (4), 381 Myr (5), and 474 Myr (6). Solid and dashed lines correspond to values inside and outside the structure, respectively. Er = (αpn − αen) (vpr − vnr) Figure B7 . B7Azimuthal electric field as a function of radius for Case 1 (left panel) and Case 2 (right panel) at t=9.29 Myr (1), 102 Myr (2), 195 Myr (3), 288 Myr (4), 381 Myr (5), and 474 Myr (6). Solid and dashed lines correspond to positive values inside and outside the structure, respectively. Dot-dashed and dotted lines correspond to negative values inside and outside the structure, respectively. Table 1 . 1Mass of chemical species.species mass (GeV) H 0.93878 H + 0.93827 Li 6.53536 Li + 6.53485 Under the assumption of Eq. (A16), the velocity of the singly charged ion i [Eq. (A15)] has an approximate form of vir ≈ Hr sph , (A22) APPENDIX B: DETAILED RESULTS OF PHYSICAL VARIABLES In this section, we show supplemental results of calculations performed in Sec. 4.Ez = = − (αpn − αen) Br Bz (Hr sph − vnr) . (A21) v iφ ≈ αin − αpn + αen Bz (Hr sph − vnr) , (A23) viz ≈ vnz − Br Bz (Hr sph − vnr) . (A24) c 20XX RAS, MNRAS 000, 1-30 WWW: http://lambda.gsfc.nasa.gov. c 20XX RAS, MNRAS 000, 1-30 Inhomogeneous chemical abundances in cosmic plasma including solar flares have been considered(Alfven 1981, pp. 82-84). One of their mechanisms is the mass dependent gravitational drift resulting in isotope separation of single element. In this study, we treat species-dependent dynamical frictions operating in low density plasma with a very small ionization degree of hydrogen, as realized in the early universe. The frictions induce a chemical separation of various singly charged ionic species as one kind of separations. Its effect is, however, negligible because of a strong electric coupling of positively charged ions and negatively charged electron. c 20XX RAS, MNRAS 000, 1-30 This paper has been typeset from a T E X/ L A T E X file prepared by the author. c 20XX RAS, MNRAS 000, 1-30 ACKNOWLEDGMENTSWe are indebted to the referee, Glenn E. Ciolek, for instructive comments on astrophysical plasma physics. We are grateful to Tomoaki Ishiyama for instructive information on structure formation. This work has been supported by Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sports, and Culture (MEXT), Japan, No.25400248 and No.21111006, and by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. Cosmic plasma. 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[ "Coevolutionary dynamics of a variant of the cyclic Lotka-Volterra model with three-agent interactions", "Coevolutionary dynamics of a variant of the cyclic Lotka-Volterra model with three-agent interactions" ]	[ "Filippo Palombi \nENEA-Italian National Agency for New Technologies, Energy and Sustainable Economic Development\nVia Enrico Fermi 4500044FrascatiItaly\n", "Stefano Ferriani \nEnergy and Sustainable Economic Development Via Martiri di Monte Sole 4\nENEA-Italian National Agency for New Technologies\n40129BolognaItaly\n", "Simona Toti \nISTAT-Italian National Institute of Statistics\nVia Cesare Balbo 1600184RomeItaly\n" ]	[ "ENEA-Italian National Agency for New Technologies, Energy and Sustainable Economic Development\nVia Enrico Fermi 4500044FrascatiItaly", "Energy and Sustainable Economic Development Via Martiri di Monte Sole 4\nENEA-Italian National Agency for New Technologies\n40129BolognaItaly", "ISTAT-Italian National Institute of Statistics\nVia Cesare Balbo 1600184RomeItaly" ]	[]	We study a variant of the cyclic Lotka-Volterra model with three-agent interactions. Inspired by a multiplayer variation of the Rock-Paper-Scissors game, the model describes an ideal ecosystem in which cyclic competition among three species develops through cooperative predation. Its rate equations in a well-mixed environment display a degenerate Hopf bifurcation, occurring as reactions involving two predators plus one prey have the same rate as reactions involving two preys plus one predator. We estimate the magnitude of the stochastic noise at the bifurcation point, where finite size effects turn neutrally stable orbits into erratically diverging trajectories. In particular, we compare analytic predictions for the extinction probability, derived in the Fokker-Planck approximation, with numerical simulations based on the Gillespie stochastic algorithm. We then extend the analysis of the phase portrait to heterogeneous rates. In a well-mixed environment, we observe a continuum of degenerate Hopf bifurcations, generalizing the above one. Neutral stability ensues from a complex equilibrium between different reactions. Remarkably, on a two-dimensional lattice, all bifurcations disappear as a consequence of the spatial locality of the interactions. In the second part of the paper, we investigate the effects of mobility in a lattice metapopulation model with patches hosting several agents. We find that strategies propagate along the arms of rotating spirals, as they usually do in models of cyclic dominance. We observe propagation instabilities in the regime of large wavelengths. We also examine three-agent interactions inducing nonlinear diffusion."Three at play. That'll be the day!" (a child in Wings of desire [W. Wenders, 1987]) arXiv:1905.05591v1 [q-bio.PE] 14 May 2019 1 Reactions considered in Ref.[31] can be in fact recast in the form of Eq. (I.1) for k = = 5, 9. Yet, their rates are somewhat intertwined in that they are functions of the pairwise rates δ 0 , δ 1 .	10.1140/epjb/e2020-100552-5	[ "https://arxiv.org/pdf/1905.05591v1.pdf" ]	153,312,845	1905.05591	136254751a366b066fa0400306c2adeb28d51ae8	Coevolutionary dynamics of a variant of the cyclic Lotka-Volterra model with three-agent interactions Filippo Palombi ENEA-Italian National Agency for New Technologies, Energy and Sustainable Economic Development Via Enrico Fermi 4500044FrascatiItaly Stefano Ferriani Energy and Sustainable Economic Development Via Martiri di Monte Sole 4 ENEA-Italian National Agency for New Technologies 40129BolognaItaly Simona Toti ISTAT-Italian National Institute of Statistics Via Cesare Balbo 1600184RomeItaly Coevolutionary dynamics of a variant of the cyclic Lotka-Volterra model with three-agent interactions (Dated: May 15, 2019) We study a variant of the cyclic Lotka-Volterra model with three-agent interactions. Inspired by a multiplayer variation of the Rock-Paper-Scissors game, the model describes an ideal ecosystem in which cyclic competition among three species develops through cooperative predation. Its rate equations in a well-mixed environment display a degenerate Hopf bifurcation, occurring as reactions involving two predators plus one prey have the same rate as reactions involving two preys plus one predator. We estimate the magnitude of the stochastic noise at the bifurcation point, where finite size effects turn neutrally stable orbits into erratically diverging trajectories. In particular, we compare analytic predictions for the extinction probability, derived in the Fokker-Planck approximation, with numerical simulations based on the Gillespie stochastic algorithm. We then extend the analysis of the phase portrait to heterogeneous rates. In a well-mixed environment, we observe a continuum of degenerate Hopf bifurcations, generalizing the above one. Neutral stability ensues from a complex equilibrium between different reactions. Remarkably, on a two-dimensional lattice, all bifurcations disappear as a consequence of the spatial locality of the interactions. In the second part of the paper, we investigate the effects of mobility in a lattice metapopulation model with patches hosting several agents. We find that strategies propagate along the arms of rotating spirals, as they usually do in models of cyclic dominance. We observe propagation instabilities in the regime of large wavelengths. We also examine three-agent interactions inducing nonlinear diffusion."Three at play. That'll be the day!" (a child in Wings of desire [W. Wenders, 1987]) arXiv:1905.05591v1 [q-bio.PE] 14 May 2019 1 Reactions considered in Ref.[31] can be in fact recast in the form of Eq. (I.1) for k = = 5, 9. Yet, their rates are somewhat intertwined in that they are functions of the pairwise rates δ 0 , δ 1 . I. INTRODUCTION Cyclic competition is distinctively associated with closed relational chains, describing aspects of the struggle for life such as feeding, hunting, mating. The prototype is a system made of three different species, interacting with one another like children playing Rock-Paper-Scissors (RPS), the famous game where paper (P) wraps rock, scissors (S) cut paper and rock (R) crushes scissors, see Fig. 1. Such schemes are at heart of biological systems spanning a wide range of length scales and complexity. Examples include the repressilator [1], the E. coli colicin E2 system [2], the common side-blotched lizard [3], several plant systems [4][5][6] and so forth. In principle, the absence of apex predators and bottom preys in cyclic chains allows species to dominate in * Corresponding author: [email protected] turn. As soon as one of them outperforms the others, it becomes itself a source of nourishment for the next one along the chain. This feature suggests that cyclic competition may serve as a fundamental mechanism facilitating coexistence and biodiversity. Experiments performed in Ref. [2] on three cyclically interacting strains of E. coli confirmed this thesis and also made it more precise: species can coexist provided ecological processes (interaction and dispersal) develop locally. In practice, to ensure coexistence in the experiments, colonies of different strains had to first grow in separate spatial domains and then be left free to invade neighboring colonies. Theoretical insight into the role of locality and indi-vidual mobility for coexistence was achieved in a groundbreaking paper [7] thanks to an evolutionary model based on a three-state RPS game. Results were then generalized to a four-state variant of the model [8]. These studies revealed that: • individual mobility promotes coexistence by inducing self-organization of the strategies into spiral waves, traveling across the environment; • stochastic noise, arising in finite populations, produces local inhomogeneities; nevertheless, it cannot prevent the creation of spiral waves; • beyond a critical threshold, individual mobility leads to species extinction. Subsequent research dealt with a plethora of either induced or independent issues. They include the effects of competition on pattern formation [9], the observation of multi-armed spirals [10], the emergence of convective instability [11], the role of stochastic noise [12] and mutations [13,14], the analysis of coexistence and extinction basins [15][16][17], the effects of uniform and nonuniform intra-specific competition [18,19], the influence of directional mobility on coexistence [20] and so forth. We omit to mention other relevant developments only for the sake of conciseness, while we refer the reader to Ref. [21] for a systematic review of concepts and results. A common assumption in all the above-mentioned studies is that predation is a pairwise interaction, involving a single predator and a single prey. While this is the case for many living organisms, either animals or microbes, it is not so for others. An alternative strategy, favored by natural selection, is cooperative predation. Mammals such as wolves [22], chimpanzees [23], dolphins [24] and lions [25] cooperate in hunting. Some insects, including ants [26], behave analogously. Even bacteria can practice group hunting. Among them we mention Saprospira [27], Myxococcus Xanthus [28,29] and Lysobacter [30]. They essentially "require a quorum of predators to degrade the prey, using excreted hydrolytic enzymes" [30]. Spatially structured models of cyclic competition featuring group predation were considered in Refs. [31][32][33][34]. Specifically, the authors of Ref. [31] studied two variants of a three-state lattice model in which predation entails simultaneous pairwise interactions. One of them assumes that agents interact with their four von-Neumann neighbors, the other with their eight Moore neighbors [35]. They found that increasing the interaction range can decelerate the propagation of predators and even revert the direction of species invasion contrary to its natural definition. The authors of Ref. [32] observed the emergence of mesoscopic subgroups of coexisting species in a fivestate lattice model based on the Rock-Paper-Scissors-Lizard-Spock game. They developed a mean-field theory to show that group interactions at the mesoscopic scale must be taken into account to justify the observed states of coexistence. The authors of Ref. [33] studied a stochastic lattice version of a model introduced by Lett, Auger, Gaillard [36]. In this model, the abundances of preys and predators are constant, while the fractions of each population using either an individual or a collective strategy coevolve. Their results include a complex phase diagram in which four different strategies (corresponding to preys/predators behaving collectively/individually in all possible combinations) dominate or coexist. Remarkably, in the pure coexistence phase, they found cyclic dominance of the four strategies. Finally, the same authors further explored the spatial version of the model to quantify some geometrical and percolative properties of the clusters formed by the four strategies [34]. The reader will recognize that none of these approaches considers ab initio multiagent microscopic reactions with independent rates, such as X 1 X 2 . . . X k → Y 1 Y 2 . . . Y , for k, ≥ 3 , X i , Y j ∈ {R, P, S, ∅} (I.1) (in the literature ∅ conventionally denotes an empty site) 1 . In this paper, we study a simple variant of the cyclic Lotka-Volterra model [37][38][39][40][41][42][43][44] with three-agent interactions, like Eq. (I.1) for k = = 3. To explore the model, we use mathematical techniques developed in similar contexts, including nonlinear analysis of bifurcations, stochastic partial differential equations and numerical simulations. We find that the underlying rate equations in a wellmixed environment exhibit a reactive fixed point falling in the universality class of the Hopf bifurcations. As such, they induce a macroscopic phenomenology qualitatively similar to models already studied in the literature. Nonetheless, the internal dynamics of the model is original and interesting. Two opposing forces drive the system. Interactions involving two preys plus one predator pull it towards the reactive fixed point, thus playing an equilibrating role. By contrast, interactions involving two predators plus one prey push the system away from the reactive fixed point, thus producing a polariz-ing effect. The relative strength of the two forces controls the evolution of the system in a spatially structured version of the model, with patches hosting several motile agents (metapopulation model). When polarizing interactions dominate over equilibrating ones, coexistence is achieved through the development of spatiotemporal patterns, taking the usual form of rotating spiral waves. We also find that spatial topology is critical for the evolution of species. Indeed, the phase portrait of the model changes drastically on a two-dimensional lattice with single agent per site and nearest-neighbor interactions. Locality makes the reactive fixed point stable for every choice of reaction rates. The disappearance of Hopf bifurcations indicates that local patches hosting several agents are essential for the development of patterns. The paper is organized as follows. First of all, in section II, we define the model and study its equations in a well-mixed environment for homogeneous rates. In section III, we estimate the magnitude of the stochastic noise affecting finite agent populations. As known [37], fluctuations turn neutrally stable orbits into diverging trajectories, eventually resulting in the extinction of two species. We compare the extinction probability, derived in the Fokker-Planck approximation, with numerical simulations. In section IV, we discuss the phase portrait of the model for heterogeneous rates. We also compare results in a well-mixed environment with those on a twodimensional lattice. In section V, we introduce mobility reactions in a lattice metapopulation model with patches hosting several agents. In section VI, we examine threeagent interactions inducing nonlinear diffusion. Finally, in section VII, we draw conclusions. II. RATE EQUATIONS Some time ago we happened to observe three children playing a three-player variant of RPS in the hall of their school. Intrigued by the game, we asked them about it. In order not to leave anyone out-they proudly explained-they were playing all at once. The rules of the game were as follows. On each round, children had to deliver simultaneously one of the usual hand signals, representing R, P and S. Round by round they received payoffs, based on cyclic dominance and depending on the combination of delivered signals, as we report in Table I 2 . 2 While writing the paper we realized that essentially the same threeplayer variant of RPS is described in Ref. [45]. Similar variants can outcome payoffs R R R, P P P, S S S all players receive 0 points R R P, P P S, S S R the dominant player receives 1 point R R S, P P R, S S P each dominant player receives 1/2 point R P S all players receive 0 points To translate the game into the language of evolutionary game theory, we assume a population of N 1 agents, each adopting one of the competing strategies. We let r, p, s denote the relative abundances of R, P, S respectively. As such they fulfill r + p + s = 1. In our model, densities evolve in time as a consequence of microscopic interactions inspired by Table I. More precisely, in place of payoffs we consider stochastic reactions mediated by a dominance-replacement mechanism, namely R R P → R P P with rate d RRP , P P S → P S S " d PPS , S S R → S R R " d SSR , R R S → R R R " d RRS , P P R → P P P " d PPR , S S P → S S S " d SSP , (II.1) where rates (transition probabilities per unit time) are independent of one another. For the time being, we leave out transitions in which agents carry three different strategies before interacting. We also leave out transitions in which they all carry the same strategy, since such configurations yield no payoff in the classic formulation of the game. Rate equations (RE) including all contributions listed in Eqs. (II.1) read aṡ r = F r (r, p, s) ,ṗ = F p (r, p, s) ,ṡ = F s (r, p, s) ; F r = r 2 (d RRS s − d RRP p) + r d SSR s 2 − d PPR p 2 , F p = p 2 (d PPR r − d PPS s) + p d RRP r 2 − d SSP s 2 , F s = s 2 (d SSP p − d SSR r) + s d PPS p 2 − d RRS r 2 . (II.2) Altogether, these cubic equations depend on six parameters. We can absorb one of them into a redefinition of time, thus obtaining five effective parameters. Moreover, we can drop the equation forṡ provided we insert be found on various websites. In a scene of Sonatine (ソナチネ), a 1993 film by T. Kitano, three yakuza gangsters play a three-player variant of RPS on the beach. s = 1 − r − p into those forṙ andṗ. Eqs. (II.2) describe correctly the evolution of strategies, induced by Eqs. (II.1), in a well-mixed environment as N → ∞. They represent a variant of the cyclic Lotka-Volterra equations [7,[38][39][40][41][42][43][44] encompassing three-agent interactions. To study the model, we first consider a simplified version in which rates are homogeneous. More precisely, we let d RRP = d PPS = d SSR ≡ d e , d RRS = d PPR = d SSP ≡ d p . (II.3) As can be seen, homogeneity is not complete. Eq. (II.3) follows from separating reactions into two disjoint sets. The first three reactions in Eq. (II.1) correspond to the first line of Eq. (II.3). They are functionally homogeneous in that they involve two preys and one predator in the initial state. Their occurrence produces a local change of majority. The last three reactions in Eq. (II.1) correspond to the second line of Eq. (II.3). They start with two predators and one prey. Their occurrence results in a further increase of the local majority. The two groups of reactions play antagonistic roles. When the system is in a macroscopic state in which a strategy dominates, the former contributes to equilibrating it, whereas the latter contributes to further polarizing it. Resulting RE have three absorbing fixed points, (r 1 , p 1 ) = (1, 0), (r 2 , p 2 ) = (0, 1), (r 3 , p 3 ) = (0, 0) and a reactive one, (r * , p * ) = (1/3, 1/3). Since we are interested in the behavior of the system near the latter point, we let r = 1/3 + x r and p = 1/3 + x p . Upon expanding the RE in Taylor series to first order around x r = x p = 0, we obtaiṅ x r = − d e 3 x r − d e + d p 3 x p , x p = d p 3 x p + d e + d p 3 x r . (II.4) The eigenvalues of the Jacobian matrix are λ = 1 6 (d p − d e ) + i 2 √ 3 (d p + d e ) , λ = 1 6 (d p − d e ) − i 2 √ 3 (d p + d e ) . (II.5) The real part vanishes for d e = d p , i.e. when equilibrating and polarizing forces compensate exactly. For this choice of rates we have a Hopf bifurcation. Starting near (r * , p * ), the system spirals inwards for d e > d p , whereas it spirals outwards for d e < d p . For d e = d p (= 1) it trav-els on neutrally stable orbits enclosing the fixed point. They have angular frequency ω 0 = 1/ √ 3 and first integral rps = constant. To determine the type of the Hopf bifurcation, we need to bring the RE to normal form according to a standard procedure, detailed, e.g., in Refs. [46,47]. To this aim, we let d e = 1 and d p = 1 + . We perform a linear change of variables, namely we let y = Sx with S = 1/ √ 3 −1/ √ 3 1 1 . (II.6) It yields equationṡ y r = Re(λ)y r − Im(λ)y p + f r (y r , y p , ) , y p = Im(λ)y r + Re(λ)y p + f p (y r , y p , ) , (II.7) where the nonlinear functions f r and f p read as f r (y r , y p , ) = √ 3 4 (2 + )y 2 r + 2 y r y p − √ 3 4 (1 + )y 2 p − 3 4 y 3 r − 3 4 y r y 2 p , (II.8) f p (y r , y p , ) = 4 y 2 r − √ 3 2 (2 + )y r y p − 4 y 2 p − 3 4 y 3 p − 3 4 y p y 2 r . (II.9) Next, we perform a change to complex variables z = y r + iy p andz = y r − iy p . Inverse relations are given by y r (z,z) = (z +z)/2 and y p (z,z) = (z −z)/2i. Inserting them into the above formulas yieldṡ z = λz + f (z,z) , (II.10) with f (z,z) = √ 3 2 1 + 2 + i 4 z 2 − 3 4 ≡ Rz 2 − Szz 2 . (II.11) As can be seen, f has quadratic and cubic terms. We can remove the former by performing an additional change of variable, viz. z → z + Az 2 . We choose A such that (λ − 2λ)A = R. After some algebra 3 we arrive aṫ z = λz − c( )z\|z\| 2 , (II.12) (II.14) Since a( ) ≈ (5/6) as → 0, we conclude that the Hopf bifurcation is degenerate. In polar coordinates z = ρe iθ Eq. (II.12) turns intȯ ρ = 6 ρ − a( )ρ 3 ,θ = 2 + 2 √ 3 − b( )ρ 2 . (II.15) In Fig. 2, we compare by numerical integration Eq. (II.15) with the full RE for = 0.1. In both plots, dashed and dotted lines correspond respectively to the leading order approximation (LO) and the full next-to-leading order (NLO) version of Eq. (II.15). We always choose initial conditions with densities lying near the reactive fixed point. We observe no limit cycle. The asymptotic saturation of ρ in Fig. 2 (a) corresponds to heteroclinic cycles approaching the absorbing fixed points. A Hopf bifurcation of the same type as as Eqs. (II.12)-(II.14) characterizes the dynamics of the May-Leonard model [48]. In that model, the bifurcation point corresponds to the vanishing of dominance-removal reactions. Therefore, the May-Leonard model is well-defined only on one side of the bifurcation. Ours makes sense on both sides. More importantly, on the unstable side, both models feature heteroclinic cycles. Owing to this similarity, their spatially structured versions, discussed respectively in Ref. [11] and section V, are phenomenologically equivalent. III. STOCHASTIC NOISE Intuition suggests that multiagent interactions should produce larger stochastic fluctuations than pairwise ones since they involve more fluctuating degrees of freedom. A non-trivial and surprising consequence is that strategies have to fight longer before one of them prevails on the others. As a result, the probability of coexistence increases. Using the Fokker-Planck equation, we derive an accurate estimate of the magnitude of the stochastic noise in our model. We follow Ref. [37], where calculations are fully detailed. The idea is to consider a specific setting in which the RE predict neutrally stable orbits, namely d e = d p = 1 in our case. For N < ∞ the conservation law rps = const. is broken by O(N −1 ) terms. Hence, the system follows an erratic trajectory, interpolating between different neutrally stable orbits. Eventually, it ends up on one of the absorbing fixed points. We can derive the intrinsic magnitude of the stochastic noise from the Master Equation ∂ t P(φ, t) = δφ {w(φ + δφ → φ)P(φ + δφ, t) − w(φ → φ + δφ)P(φ, t)} , (III.1) where φ = (r, p) and w(φ → φ ) denotes the transition probability per unit time (rate) from φ to φ . The sum over δφ includes all possible microscopic transitions characterizing the model. We choose conventionally the macroscopic time unit as the interval including N reactions. In Table II, we list rates and density variations corresponding to this choice (one has to replace s = 1−r −p in all expressions). Eq. (III.1) yields an exact description for N < ∞. Unfortunately, we cannot solve it analytically. We can obtain an effective approximation by means of the Kramers-Moyal expansion, which, upon truncation to reaction w N δr N δp R R P → R P P deN r 2 p −1 1 P P S → P S S deN p 2 s 0 −1 S S R → S R R deN s 2 r 1 0 R R S → R R R dpN r 2 s 1 0 P P R → P P P dpN p 2 r −1 1 S S P → S S S dpN s 2 p 0 −1∂ t P(φ, t) = − i=r,p ∂ i [α i (φ)P(φ, t)] + 1 2 i,j=r,p ∂ i ∂ j [B ij (φ)P(φ, t)] . (III.2) The functions α i and B ij , respectively known as drift and diffusion functions, are given by α i (φ) = δφ δv i w(φ → φ + δφ) , (III.3) B ij (φ) = δφ δφ i δφ j w(φ → φ + δφ) . (III.4) Fromα r (φ) = r(s − p) , (III.5) α p (φ) = p(r − s) , (III.6) B rr (φ) = 1 N r(p + s − 2ps) , (III.7) B rp (φ) = B pr (φ) = − 1 N rp(r + p) , (III.8) B pp (φ) = 1 N p(r + s − 2rs) . (III.9) Just like in section II, we let φ = (1/3, 1/3) + x, then we expand the whole FPE around x = 0 (Van Kampen's linear noise approximation [49]). Accordingly, we obtain ∂ t P(x, t) = − i,j=r,p ∂ i [A ij x j P(x, t)] + 1 2 B ij ∂ i ∂ j P(x, t) , (III.10) with A = − 1 3 1 2 −2 1 , B = 2 27N 2 −1 −1 2 . (III.11) As a final step, we perform another change of variables, namely we let x → y = Sx, with S = √ 3 2 2ω0 ω0 0 1 . Ac-cordingly, A and B turn into A →Ã = SAS −1 = ω 0 0 −1 −1 0 , (III.12) B →B = SBS T = 1 9N 1 0 0 1 . (III.13) The diffusion matrixB is now diagonal. Hence, the FPE takes the simplified form ∂ t P(y, t) = −ω 0 [y p ∂ r − y r ∂ p ]P(y, t) + 1 18N [∂ 2 r + ∂ 2 p ]P(y, t) . (III.14) The diffusion constant D = 1/(18N ) is smaller than the analogous constant in the original cyclic Lotka-Volterra model by a factor of 2/3, whereas ω 0 is the same, see Eq. (25) of Ref. [37] 4 . The reduction factor is rather suggestive, as it equals the ratio of simultaneous players in the two models. The value of 2/3 can be easily explained. The original model has three microscopic reactions, ours has twice this number. Each reaction contributes positively to the stochastic noise. This yields a multiplicative factor of 2 in the diffusion matrix. Moreover, the contribution from each reaction is quadratic in the original model while cubic in ours. In the linear noise approximation, we calculate the diffusion matrix at the reactive fixed point. This yields an additional multiplicative factor of 1/3 and that is all. In general, the more agents partake in the interactions, the larger the number of possible microscopic reactions. The magnitude of the induced stochastic noise depends eventually on combinatorial factors, including the number and degree of reactions. Of course, it depends as well on the value of the strategy densities at equilibrium. We can calculate the probability P ext that two species go extinct after a certain time in the Fokker-Planck approximation as the authors of Ref. [37] do. There is no need to repeat the derivation here, since it applies identically to our model. The final formula is LT{P ext (u)} = 1 sI 0 (R √ Ds) , (III.15) where LT stands for Laplace Transform, u = t/N is a scaling variable measuring time in units of N and I 0 denotes the Bessel function of first kind and 0th order. R represents the distance traveled by the system on its random walk from the reactive fixed point to one of the absorbing fixed points. We can regard it as the radius of an absorbing sphere. The authors of Ref. [37] adopt three possible definitions of R, namely R 0 = 1/3, R 1 = 1/ √ 3 and R 2 = (R 0 + R 1 )/2. They yield three different probability functions. Fig. 3 shows them together with the results of numerical simulations based on Gillespie's algorithm [50,51]. To compute the extinction probability from Eq. (III.15) we use a numerical implementation of the inverse LT. Moreover, we expand I 0 asymptotically to the 10th order, as also the authors of Ref. [37] do. To simulate extinction times correctly, we need to rescale all rates by appropriate volume factors. Notice that all reactions in our model involve two reactants of the same species and one of another. For such reactions, the right definition of the reaction parameters c XXY in Gillespie's algorithm is c XXY = 2d XXY /N 2 , for X,Y ∈ {R, P, S} [50,51]. Similar to the original Lotka-Volterra model, the analytic prediction that best fits numerical data is the one corresponding to R 2 . As anticipated, the extinction probability is uniformly lower in the cyclic Lotka-Volterra model with three-agent interactions than with two-agent ones, although the former are intrinsically noisier than the latter. This result is only apparently counterintuitive. In fact, it has a simple interpretation. When three agents interact, strategies fluctuate longer around the reactive fixed point before one of them prevails on the others. Group interactions help the system stay in equilibrium. Hence, we conclude, they promote species coexistence. IV. HETEROGENEOUS RATES The reactive fixed point is symmetric for homogeneous rates, independently of whether d e > d p or d e < d p . Things become more interesting as soon as we break homogeneity. Unfortunately, studying the model in full generality is not simple, due to the high dimensionality of the parameter space. A reasonable compromise is to let d RRP = d PPS = d SSR ≡ d e = 1 and to leave all other rates unconstrained. Under this assumption, the RE still exhibit four fixed points, three absorbing plus one reactive. The former coincide with the vertices of the ternary diagram, as can be easily checked from Eqs. (II.2). The latter has a complex algebraic structure. Indeed, it reads r * = (dSSP−dPPR)X 2 +(1+2dPPR)X−dPPR 1−dPPR+(dPPR−dRRS)X , p * = (dRRS−dSSP)X 2 −(2+dRRS)X+1 1−dPPR+(dPPR−dRRS)X , (IV.1) with X fulfilling the cubic equation 0 = a3X 3 + a2X 2 + a1X + a0 , a0 = 1 − dRRS d 2 PPR , a1 = −2 dRRS + dPPR − 3 − dSSP dPPR + 2 dRRS dPPR + dSSP + 3 dRRS d 2 PPR − d 2 PPR , (IV.2) a2 = 3 dRRS − 3 dSSP − 2 dRRS dSSP − 3 dRRS dPPR − 3 dRRS d 2 PPR + d 2 RRS + 3 dRRS dPPR dSSP + 2 dSSP dPPR + 2 d 2 PPR , a3 = d 2 SSP dPPR + dSSP dPPR + dRRS dSSP − 3 dRRS dPPR dSSP + dRRS d 2 PPR + dRRS dPPR − d 2 SSP − d 2 RRS − d 2 PPR + d 2 RRS dSSP . We wish to examine how strategies rank at equilibrium depending on the polarizing rates. To this aim, we proceed like the authors of Ref. [31]. Specifically, we denote by (A,B,C) a domain in parameter space for which a * ≤ b * ≤ c * , for a * , b * , c * a permutation of r * , p * , s * ≡ 1 − r * − p * and A, B, C the corresponding permutation of R, P, S. In accordance with Ref. [31], we adopt the name of phase for all such domains, although this word is somewhat misleading (at least in our case). Indeed, both the equilibrium densities and their derivatives are smooth functions. In particular, they do not show discontinuities at phase transitions. In Fig. 4, we report the phase structure for a sequence of values of d RRS , ranging from 0.2 to 2.1 in steps of 0.1. In each plot, we notice a white region and a variously colored one. The former corresponds to unstable spirals, diverging from the reactive fixed point (they eventually turn into heteroclinic cycles), whereas the latter corresponds to stable spirals, converging to it. A continuum of Hopf bifurcations lies along the lines separating colored and white regions, as we discuss below. Colored regions split into six phases or less, depending on d RRS . Each phase makes contact with all others for d PPR = d SSP = d RRS . The contact point lies in the stable region only for d RRS ≤ 1. The overall surface occupied by stable phases reduces as d RRS increases (the larger d RRS the stronger the polarizing force corresponding to fixed values of d PPR and d SSP ). A glance to plot (i), corresponding to d RRS = 1, suggests a recipe for predicting the phase of the system for given values of d RRS , d PPR , d SSP . It consists of three steps: 1. rank the polarizing rates in ascending order (e.g. d PPR < d RRS < d SSP ); 2. extract the losing strategy from each rate label (it yields (R, S, P) in the above example); 3. turn each strategy into the immediately inferior one (it finally yields (S, P, R)). The recipe has a straightforward interpretation: the more a species is preyed on by its predator, the more its prey has room to develop. Its main drawback is that it is only approximately exact due to nonlinear effects. For instance, the transition line separating (S, R, P) from (R, S, P) should bisect the phase plane for d RRS = 1, while it does not. Moreover, all other transition lines develop a slope for d RRS = 1. We notice that by no reason the phases of Fig. 4 should be symmetric for d SSP ↔ d PPR . The only symmetry of the system is the cyclic one, namely R ← P ← S ← R. If we fix d RRS , we lose it. Altogether, the phase structure looks more complex than observed in Ref. [31], although our model features simpler multiagent interactions. As far as we understand, the rationale behind this is that we consider fully independent rates, while the authors of Ref. [31] build multiagent rates as accumulated payoffs, depending on the pairwise rates. As a consequence, they explore a twodimensional manifold embedded in a larger and more complex phase space. We can provide a better characterization of the transition lines separating white and colored regions. The Jacobian of the linearized RE has two complex conjugate eigenvalues λ r ± iλ i . We can express both the real and the imaginary part as functions of the fixed-point densities, namely λ r = 1 2 (d RRP − d RRS )r 2 * + 1 2 (d PPS − d PPR )p 2 * + 1 2 (d SSR − d SSP )s 2 * + (d PPR − d RRP )r * p * + (d SSP − d PPS )p * s * + (d RRS − d SSR )r * s * , (IV.3) λ i = 1 2 {c r4 r 4 * + c p4 p 4 * + c s4 s 4 * + c r3p1 r 3 * p * + c r3s1 r 3 * s * + c p3r1 p 3 * r * + c p3s1 p 3 * s * + c s3r1 s 3 * r * + c s3p1 s 3 * p * + c r2p2 r 2 * p 2 * + c r2s2 r 2 * s 2 * + c p2s2 p 2 * s 2 * + c r2ps r 2 * p * s * + c p2sr p 2 * s * r * + c s2rp s 2 * r * p * } 1/2 , (IV.4) with coefficients reading cr4 = (dRRP + dRRS) 2 , cp4 = (dPPS + dPPR) 2 , cs4 = (dSSR + dSSP) 2 , cr3p1 = 4(dRRS + dRRP)(dPPR − dRRP) , cr3s1 = 4(dRRP + dRRS)(dSSR − dRRS) , cp3r1 = 4(dPPS + dPPR)(dRRP − dPPR) , cp3s1 = 4(dPPR + dPPS)(dSSP − dPPS) , cs3r1 = 4(dSSR + dSSP)(dRRS − dSSR) , cs3p1 = 4(dSSP + dSSR)(dPPS − dSSP) , cr2p2 = 4d 2 RRP + 4d 2 PPR − 2dRRSdPPR − 2dRRSdPPS − 2dRRPdPPS − 10dRRPdPPR , (IV.5) cr2s2 = 4d 2 RRS + 4d 2 SSR − 2dRRSdSSP − 2dRRPdSSP − 2dRRPdSSR − 10dRRSdSSR , cp2s2 = 4d 2 PPS + 4d 2 SSP − 2dPPSdSSR − 2dPPRdSSR − 2dPPRdSSP − 10dPPSdSSP , cr2ps = −4(dRRSdPPS − 2dPPRdSSR + 2dPPRdRRS + 2dSSRdRRP + dSSPdRRP + dRRSdSSP + 2dRRSdRRP + dPPSdRRP) , cp2sr = −4(2dPPRdSSP + dRRSdPPS − 2dSSPdRRP + 2dPPSdRRP + dPPRdSSR + dSSRdPPS + 2dPPRdPPS + dPPRdRRS) , cs2rp = −4(2dRRSdSSP + 2dSSRdPPS + dSSPdRRP + dPPRdSSR − 2dRRSdPPS + 2dSSRdSSP + dSSRdRRP + dPPRdSSP) . Both (r * , p * ) and λ r depend nonlinearly upon d RRS , d PPR , d SSP . Any point, belonging to a transition line separating white and colored regions in Fig. 4, corresponds by definition to a reactive fixed point yielding λ r = 0. Keeping (r * , p * ) fixed means imposing two constraints. Since we have three degrees of freedom, we remain with one. We conclude that there is a one-dimensional (non-planar) manifold, characterized by (r * , p * ) = const., crossing the transition line at the chosen point. All other points belonging to this manifold correspond to reactive fixed points yielding λ r = 0. Apart from possible exceptions (that we never observed in our numerical experiments), the manifold splits into two parts, one having λ r > 0, the other λ r < 0. Therefore, the system undergoes a Hopf bifurcation along the manifold for λ r = 0. Hence, the RE exhibit a continuum of Hopf bifurcations at all transition lines of Fig. 4. Unfortunately, we have been unable to clarify whether RE at the bifurcation points have first integrals, like rps for homogeneous rates and, if affirmative, how they look analytically. So far, we have assumed that d RRP = d PPS = d SSR = 1. We now relax this constraint to examine another feature of Eqs. (II.2). We known that the cyclic Lotka-Volterra model with pairwise interactions has only neutrally stable orbits [37]. Moreover, the ensemble of fixed points fills the ternary diagram, as can be seen from Eq. (4) of Ref. [37]: for each pair (r * , p * ), there exists a choice of rates for which (r * , p * ) is a reactive fixed point. The reader may ask whether this feature holds similarly in our model. The problem is non-trivial because (r * , p * ) has a complex dependency on three-agent rates, as Eq. (IV. 1) shows. To answer the question, we let H denote the ensemble of neutrally stable fixed points corresponding to a given choice of equilibrating rates, namely H(d RRP , d PPS , d SSR ) = {(r, p) : F r = F p = 0 , λ r = 0 and λ i = 0 d RRP , d PPS , d SSR , (IV.6) where integration over polarizing rates is understood. In Fig. 5, we reconstruct H numerically for several values of d RRP , d PPS , d SSR . Continuous lines correspond to sequences of polarizing rates. Points lying between neighboring lines belong to the ensembles as well. They just correspond to polarizing rates we did not consider numerically. Plot (i) shows that H(1, 1, 1) is a proper subset of the ternary diagram, symmetric under cyclic permutations. Changing one or two rates distorts its shape: the smaller the rates, the closer H shifts towards the boundaries. By extrapolation, Fig. 5 suggests that dRRP,dPPS,dSSR H(d RRP , d PPS , d SSR ) covers the whole diagram. However, one-to-one correspondence between fixed points and reaction rates is lost: several distinct sets of rates yield the same reactive fixed point. Therefore, we conclude, our model has a more complex algebraic structure than the cyclic Lotka-Volterra model with pairwise interactions. Besides, Fig. 5 highlights that the the law of the weakest, first described in Ref. [52], holds here as well. Take for instance plots (g)-(h)-(i): the lower d PPS , the closer H approaches the (p, s) boundary. Recall that d PPS mediates the equilibrating transition P P S → P S S. As such, it yields a relative measure of the strength of S versus P. Any neutrally stable orbit surrounds the corresponding fixed point and flows counterclockwise on the ternary diagram. Therefore, the lower d PPS the higher the probability that the system leaves its orbit by stochastic noise and falls eventually on the (p, s) boundary, where r = 0. When this happens, the dynamics terminates with s = 1. Hence, S, the weakest strategy, survives, while R and P go extinct. We finally investigate how the phase structure of the model changes off a well-mixed environment. To this aim, we make use of the Gillespie algorithm to simulate the dynamics on a two-dimensional lattice. We consider a square grid with N = L × L sites and periodic boundary conditions. a relative majority on the lattice, whereas R has a relative majority in a well-mixed environment. We conclude that changing spatial topology induces distortive effects analogous to those observed in Ref. [31]. As we discussed in section III, stochastic noise perturbates the evolution of strategies for N < ∞. On a two-dimensional lattice, stable spirals do not converge exactly. Global densities end up fluctuating erratically around the fixed point. Fig. 7 (c)-(d) shows this effect for two sets of rates, both corresponding to unstable equilibria in a well-mixed environment. In both plots, continuous lines are representative trajectories of the global densities, while dashed/dotted lines represent averages over 100 sample trajectories. As can be seen, stochastic noise averages to zero. The amplitude of similar erratic oscillations was studied with full detail in a model featuring species mutations [53]. In that context, a resonance amplification, occurring at a specific frequency, influenced fluctuations. The resonant frequency was estimated by the power spectrum method in the Fokker-Planck approximation. Here, we have less analytic infor-mation concerning the reactive fixed point. Hence, we do not attempt such an analysis. We only report that we observed a very light dependence of the fluctuation amplitude upon the polarizing rates in our numerical experiments. However, this result might depend on the excessive coarseness of the grid of values we chose. V. AGENT MOBILITY In the previous sections, we have allowed strategies to propagate as a result of predation. Now, we let them diffuse explicitly via additional pair exchange reactions, namely X Y → Y X , X, Y ∈ {R, P, S} , (V.1) all occurring with rate γ 2 . We set up a lattice metapopulation model along the lines of Refs. [13,15,[54][55][56]. Specifically, we consider a two-dimensional square lattice with N = L × L sites and periodic boundary conditions. We interpret lattice sites as patches having a carrying . capacity of M ≤ ∞ agents. We consider all reactions listed in Eq. (II.1) as local processes, meaning that they always involve agents lying on the same patch. For simplicity, we assume homogeneous rates, as specified by Eqs. (II.3). By contrast, we interpret exchange reactions as bilocal processes, involving agents lying on two neighboring patches. Finally, we choose the lattice spacing to be h = 1/L so that the whole lattice has length one. As N → ∞, the density field φ ≡ (r, p) (x, t) is governed by stochastic partial differential equations (SPDE), reading as ∂ t r = D 2 ∆r + F r (r, p, s) + i=r,p C ri ξ i , ∂ t p = D 2 ∆p + F p (r, p, s) + i=r,p C pi ξ i , (V.2) where D 2 = γ 2 /N is a scaling diffusion constant (in order to keep it finite we have to scale γ 2 ∝ N as N → ∞), ∆ = ∂ 2 x + ∂ 2 y is the two-dimensional Laplace operator and ξ i denotes uncorrelated Gaussian noise. The matrix C fulfills CC T = B, where B is the diffusion matrix of the FPE. In section III, we obtained a simplified formula for B, holding for d e = d p = 1. The most general expression to be used in Eqs. (V.2) reads B r,r = d e M (r 2 p + s 2 r) + d p M (r 2 s + p 2 r) , (V.3) B r,p = B p,r = − d e M r 2 p − d p M p 2 r , (V.4) B p,p = d e M (r 2 p + p 2 s) + d p M (p 2 r + s 2 p) . (V.5) We compute C from B via Cholesky decomposition whenever B is a positive definite matrix. If φ falls on a vertex of the ternary diagram, we let C ij = 0 by continuity. The dynamics of the system is trivial for d e > d p . In this case, φ converges uniformly to the reactive fixed point (1/3, 1/3) on all patches, up to small fluctuations, independently of the initial conditions. A convenient way to studying the dynamics for d p ≥ d e is to set up initial conditions such that each strategy occupies exclusively a finite portion of the lattice, as originally proposed and implemented in Refs. [14,57]. We proceed identically, namely for x = (x, y) we let φ(x, 0) =                    (0, 0) for 0 ≤ x < L/2 and L/2 ≤ y < L , (0, 1) for 0 ≤ x < L/2 and 0 ≤ y < L/2 , (1, 0) for L/2 ≤ x < L . (V.6) The advantage of such initial conditions is related to the special role of the four lattice points x = (0, 0), (L/2, 0), (0, L/2), (L/2, L/2). Here, all strategies meet and give rise to spiral waves for t > 0. Just for the sake of completeness, we briefly review why this happens. To this aim, we introduce the topological current J µ (x, t) = 1 2 µνρ ab ∂ ν φ a (x, t)∂ ρ φ b (x, t) . (V.7) We assume that Greek indices take values {0, 1, 2}, while Latin indices take values {1, 2}. Repeated indices are conventionally understood to be summed over their respective domains. The symbols µνρ and ab denote totally antisymmetric tensors with three and two components respectively. We let ∂ 0 ≡ ∂ t . It takes no effort to show that J µ fulfills the local conservation law 0 = ∂ µ J µ = ∂ 0 J 0 + ∂ k J k . We define the topological charge in the thermodynamic limit as Q(t) = dx J 0 (x, t) = dx (∂ 1 r ∂ 2 p−∂ 1 p ∂ 2 r) . (V.8) From the conservation of J µ and the periodicity of the boundary conditions, it follows that dQ/dt = dx ∂ 0 J 0 = − dx ∂ k J k = 0, hence Q is invariant in time. We then let J ± 0 (x, t) = max{±J 0 (x, t), 0}. We notice that J ± 0 is strictly positive only near points around which the three strategies follow cyclically in counterclockwise/clockwise order. In particular, J + 0 (x, 0) is positive (infinite) for x = (0, L/2), (L/2, 0), while J − 0 (x, 0) is positive (infinite) for x = (0, 0), (L/2, L/2) and J ± 0 (x, 0) = 0 elsewhere. As a result, we have four topological charges, two positive and two negative, localized on the four mentioned points, yielding 0 = Q(0) = Q(t) for all t > 0. While the topological density is sharply peaked for t = 0, it becomes somewhat smooth for t > 0. This corresponds to the appearance of two spirals plus two anti-spirals originating from the four points. Whether these objects last forever or disappear sooner or later is not a matter of topology; it depends on Eqs. (V.2). In Fig. 8 we report a sequence of snapshots of φ in RGB representation. The sequence refers to the following choice of parameters: • N = 2048 × 2048; • γ 2 = 16 (D 2 3.81 × 10 −6 ); • d e = 1 , d p = 1.2 k , for k = 0, 1, . . . , 14; • M = ∞. All pictures represent φ in the central area of the lattice, namely for L/4 < x 1 , x 2 < 3L/4. Plot (a) corresponds to t = 0, all others to t = 400. The latter time is sufficiently large to ensure that transient effects have disappeared and the system has evolved to a steady state. To obtain Fig. 8, we integrated Eqs. (V.2) numerically via the ETD2RK scheme, introduced in Ref. [58]. We summarize below the main features of the plots: • chaotic patterns with blurred and stretched shapes are predominant for < 1; • blurring reduces as increases; • small spirals emerge from chaos for 1; • a central spiral arises in a background of smaller spirals for 2; • the propagation radius of the central spiral increases progressively, until it becomes largest for ≈ 10; • for even larger the central spiral keeps definitively stable; • the wavelength of spatial patterns decreases monotonically as increases. Patterns characterized by very similar behavior emerge in the spatial version of the May-Leonard model, featuring different particle interactions but analogous Hopf bifurcation [11]. This similarity provides a confirmation that macroscopic phenomena induced by cyclic competition are robust. They depend only on the type of the Hopf bifurcation, whereas the details of the interactions have no qualitative (and little quantitative) influence [7,59]. We follow Ref. [11] for the analysis of spiral waves in terms of the complex Ginzburg-Landau equation (CGLE). In our case, the CGLE reads (after rescaling the complex amplitude and shifting its phase) ∂ t z = D 2 ∆z + λ r z − [1 + iα( )] z\|z\| 2 , (V.9) with λ r = /6 being the real part of the eigenvalues of the linearized RE, see Eq. (II.5), and with α( ) a function parameter given by α( ) = b( ) a( ) = 3 √ 3 2 (2 + )( 2 + 3 + 3) (4 2 + 15 + 15) . (V.10) It is important to recall that the CGLE is accurate only in the vicinity of a supercritical Hopf bifurcation. In that case, the role of the Laplacian is to synchronize limit cycles at neighboring lattice sites, thus giving rise to coherent spatiotemporal patterns. In our model, we have heteroclinic cycles, just like in Ref. [11]. Hence, the CGLE is not guaranteed to describe correctly the dynamics of the system. Nonetheless, we can compare predictions of the CGLE with numerical observations. In particular, in the approach of Ref. [60] the wavelength ( ) of spiral waves and their propagation speed v( ) read Fig. 9 shows a comparison of Eq. (V.11) with numerical data corresponding to N = 512 × 512, M = ∞ and γ 2 = 1, 2, 4. With regard to the wavelength, the agreement is perfect at all tested scales up to a multiplicative constant µ = 1.55 ÷ 1.60, depending on γ 2 but not on . This constant includes nonlinear effects which are not properly captured by the CGLE. Surprisingly, µ is very similar in size to the analogous constant in Ref. [11]. As for the propagation speed, the agreement becomes very good only in the asymptotic regime → ∞. Moreover, the multiplicative constant µ is slightly smaller. This yields evidence that nonlinear effects may change from one observable to another 5 . An intuitive explanation of why the RGB representation of φ blurs as → 0 follows from the observation thatρ andθ in Eq. (II.15) are monotonic functions of at LO. This feature has straightforward consequences for the dynamics in the metapopulation model. Indeed, the smaller , the closer φ keeps wandering chaotically around the reactive fixed point before walking over heteroclinic cycles. As a result, RGB colors representing φ shift progressively to gray (the color corresponding to equal densities), while spatial patterns fade out. ( ) = 2πα( ) D 2 /λ r 1 + α( ) 2 − 1 , v( ) = 2 D 2 λ r , (V.11) Closer inspection suggests that two different physical mechanisms interfere with the spatiotemporal coherence 5 A different theoretical approach can be found in Ref. [14]. Here, and v are expressed in the plane wave approximation as functions of \|z\| 2 . The latter quantity is computed from numerical solutions of the CGLE via global averaging over the whole space. of rotating spirals in regimes of intermediate and small , as discussed in Refs. [11,14,57]. For 2 5 small disturbances propagate radially together with the wavefronts. They intensify while traveling away from the core of spirals until they result in a far-field break-up of spatial coherence (convective instability). As → 0, the blurring of φ becomes very strong. In this limit, coherent propagation is fully compromised. Small disturbances grow locally. Their overall effect is to twist and stretch spatial patterns (absolute instability). 6). The increase rate of the spiral radius with M indicates relatively slow convergence to the asymptotic limit, represented by Fig. 8 (m). VI. THREE-AGENT CHASE REACTIONS Spatially structured games, admitting empty sites, usually assume that agents move via site hopping and pair exchange. These two processes correspond to dis-tinct mobility reactions, namely X ∅ → ∅ X , occurring with rate γ h , (VI.1) X Y → Y X , occurring with rate γ e , (VI.2) with X, Y ∈ {R, P, S}. As N → ∞, they induce differential variations of the strategy densities, quantified by δa = γ h N ∆a + (γ h − γ e ) N [a∆(b + c) − (b + c)∆a] , (VI.3) for (a, b, c) a permutation of (r, p, s). Eq. (VI.3) reduces to Gaussian diffusion for γ h = γ e . nonlinear terms, arising for γ h = γ e , can disrupt the stability of spiral waves by producing perturbations that result in the far-field break-up of spatial coherence [13]. Hence, they could be important to clarify in which circumstances growing bacterial colonies develop coherent patterns and in which they do not [13]. Anyway, hopping and pair exchange yield homogeneous and isotropic diffusion, regardless of how γ h and γ e are chosen. In this section, we address the issue of whether we can introduce nonlinear diffusion in our model. Since we admit no empty sites, Eq. (VI.1) is ruled out. Hence, Eq. (VI.3) makes sense no more. The mobility operator induced by pair exchange is the Laplacian, δa ex = (γ 2 /N )∆a for a = r, p, s. We have three possibilities: either we give up species homogeneity in pair exchange, or we break isotropy, or, less trivially, we introduce more complex mobility reactions. Going back to Table I and Eq. (II.1), we realize that we can tentatively use for our purpose transitions where agents carry initially three different strategies. Given a lattice site x and two nearest neighbors y 1 , y 2 , we introduce the chase reactions X(x) Y(y 1 ) Z(y 2 ) → Y(x) Z(y 1 ) X(y 2 ) , (VI.4) for (X, Y, Z) an even permutation of (R, P, S) and X(x) Y(y 1 ) Z(y 2 ) → Z(x) X(y 1 ) Y(y 2 ) , (VI.5) for (X, Y, Z) an odd permutation of (R, P, S). Incidentally, the interpretation of Eqs. (VI.4)-(VI.5) as reactions where species chase one another cyclically is not the only possible one. We could, equivalently and perhaps more imaginatively, regard them as an evolutionary version of the playground singing game Ring a ring of roses. Anyway, we assume that all such reactions occur with rate γ 3 . Fig. 11 shows the initial state corresponding to all possible permutations. We wish to derive the mobility operator arising from Eqs. (VI.4)-(VI.5), as N → ∞. To this aim, it is sufficient to focus on one of the strategies, e.g. R. The average variation of r(x), due to the above reactions, receives two negative and two positive contributions, corresponding respectively to initial configurations 1.-2. and 5.-6., namely δr chase (x \| y 1 , y 2 ) = k=1,2,5,6 δr k (x \| y 1 , y 2 ) , (VI.6) with δr 1 (x \| y 1 , y 2 ) = −γ 3 r(x)p(y 1 )s(y 2 ) , (VI.7) δr 2 (x \| y 1 , y 2 ) = −γ 3 r(x)s(y 1 )p(y 2 ) , (VI.8) δr 5 (x \| y 1 , y 2 ) = γ 3 s(x)r(y 1 )p(y 2 ) , (VI.9) δr 6 (x \| y 1 , y 2 ) = γ 3 s(x)p(y 1 )r(y 2 ) . (VI.10) In principle, there are six possible ways of choosing y 1 , y 2 given x. An elegant expression for the mobility operator follows provided we discard configurations (a)-(b) of Fig. 6 and keep all others. The variation of r(x) corresponding to configuration (c) of Fig. 6 reads as δr c (x) = − γ 3 r(x) [p(x + hx)s(x + hŷ) +s(x + hx)p(x + hŷ)] + γ 3 s(x) [p(x + hx)r(x + hŷ) +r(x + hx)p(x + hŷ)] . (VI.11) Expanding it in Taylor series around h = 0 yields δr c (x) = δr (0) c (x) + h δr (1) c (x) + h 2 δr (2) c (x) + O(h 3 ) , (VI.12) with δr (0) c (x) = 0 , (VI.13) δr (1) c (x) = −γ 3 r(x)[p(x)∂ y s(x) + p(x)∂ x s(x)] + γ 3 s(x)[p(x)∂ y r(x) + p(x)∂ x r(x)] , (VI.14) δr (2) c (x) = − 1 2 γ 3 r(x) p(x)∆s(x) + 2[∂ x p(x)][∂ y s(x)] + 2[∂ y p(x)][∂ x s(x)] + 1 2 γ 3 s(x) p(x)∆r(x) + 2[∂ x p(x)][∂ y r(x)] + 2[∂ y p(x)][∂ x r(x)] . (VI.15) We find analogous expressions δr k (x) for k = d, e, f, corresponding respectively to initial configurations (d), (e), (f) of Fig. 6. For the sake of conciseness, we leave the reader with the exercise of deriving them. Upon adding δr c and δr d , the mixed-derivative terms at O(h 2 ) cancel, hence we obtain δr (0) c + δr (0) d = 0 , (VI.16) δr (1) c (x) + δr (1) d (x) = − 2γ 3 r(x)p(x) ∂ y s(x) + 2γ 3 s(x)p(x) ∂ y r(x) , (VI.17) δr (2) c (x) + δr with scaling diffusion constant D 3 = 2γ 3 /N . Eq. (VI.22) yields a continuous chase operator. It is just one of several possible definitions. We could produce others, for instance, by differently choosing configurations from Fig. 6. Unsurprisingly, δr chase is nonlinear (cubic) in the strategy densities. At first sight, the expression in square brackets looks smaller than ∆r(x) because two densities multiply the Laplace operators. In view of this, we should reasonably expect that Eq. (VI. 22) contributes as a small perturbation to Eq. (V.2) for γ 3 ≈ γ 2 . To make δr chase comparable in strength to δr ex , we should let γ 3 larger that γ 2 by a factor of about 3÷5. Generally, reaction-diffusion equations with nonlinear diffusion operators cannot be mapped onto a CGLE. Hence, the formation of stable spirals is ruled out. Fig. 12 shows six snapshots of φ corresponding to N = 2048 × 2048, γ 2 = 0, γ 3 = 64, 128, 256, = 9.70 and M = ∞. We obtained them by numerically integrating the SPDE with random initial conditions. To this aim, we used a semi-implicit Runge-Kutta scheme of order two, with time step dt = 0.005 and mobility reactions represented in configuration space (exponential time differencing is not possible in this case). Plots (a)-(b)-(c) show φ at an early stage. They correspond respectively to t = 5, 17, 36. As can be seen, strategy densities try to arrange initially into spatial patterns with size and shapes depending on γ 3 . In particu-lar, as γ 3 increases, patterns grow. Soon, chaos replaces all regular patterns. Plots (d)-(e)-(f) represent φ for t = 400. No sign of spatial coherence is left. Patterns are chaotic for all values of γ 3 . They keep evolving rapidly in time. Their size increases with γ 3 as expected, while their boundaries, as we observed in unreported snapshots, can be equally sharp or blurred. Fig. 13 shows four snapshots of φ corresponding to N = 3072 × 3072, γ 2 = 36, γ 3 = 18, 36, 54, 72, = 9.70 and M = ∞. We obtained them by numerically integrating the SPDE with initial conditions given by Eq. (V.6). To this aim, we used the same integration scheme as explained above. We chose = 9.70 since we know from Fig. 8 (o) that the evolution of the strategy densities for γ 3 = 0 yields a fully extended and perfectly stable central spiral. Hence, this choice allows us to assess the effects of the chasing operator in a controlled set up. Similar to stochastic noise, nonlinear diffusion is responsible for the far-field break-up of the central spiral. The break-up mechanism seems to be essentially the same: nonlinear disturbances propagate from the core outwards; they grow in size while propagating; spatial coherence breaks down as soon as disturbances overcome the carrier signal. However, a comparison with Fig. 10 highlights two differences regarding the breaking pattern: δr (1) e (x) + δr (1) f (x) = 2γ 3 r(x)p(x) ∂ y s(x) − 2γ 3 s(x)p(x) ∂ y r(x) , (VI.20) δr (2) c (x) + δr (2) d (x) = γ 3 [s(x)p(x)∆r(x) − r(x)p(x) • stochastic noise deforms the profile of wavefronts near the break-up radius. This effect is absent in Fig. 13, where all wavefronts up to the break-up radius are nearly perfect. If our interpretation is correct, the effects of nonlinear diffusion reveal suddenly, whereas stochastic noise is somewhat progressive; • in Fig. 10, spatial coherence is essentially lost beyond the break-up radius. Small spirals fill the environment without large emerging patterns. This behavior is at odds with Fig. 13, where spatial coherence reappears beyond the break-up radius for γ 3 γ 2 . External patterns consist of almost-linear wavefronts. Nearly oriented towards the main lattice directions, they form cusps along the main diagonals. Moreover, they resemble the boxy-shaped patterns of Fig. 12 (c). The effect of coherence recovery reduces progressively as γ 3 increases. Paradoxically, nonlinear diffusion has the power to rectify spiral waves. In this paper, we do not go beyond the above qualitative observations. Our goal was to show that mobility reactions, different from hopping and pair exchange, can have non-trivial and interesting consequences on the propagation of spiral waves. Reaction-diffusion equations with nonlinear diffusion mechanisms have attracted much attention in the past years, thanks to their potential applications to many natural phenomena. We refer the reader to the Refs. [61][62][63][64][65] for in-depth studies. VII. CONCLUSIONS To summarize, we have investigated several aspects of a variant of the cyclic Lotka-Volterra model, featuring three-agent interactions. We aimed at studying cyclic dominance, mediated by cooperative predation, in a simple theoretical setting. nonlinear analysis of the underlying rate equations in a well-mixed environment has revealed the existence of degenerate Hopf bifurcations. They occur for specific values of the rate constants. More precisely, in our model, reactions involving two preys and one predator equilibrate the system, while reactions involving two predators and one prey polarize it. Bifurcations correspond to non-trivial equilibria between the rates of the former and the latter reactions. If equilibrating and polarizing reactions have homogeneous rates, respectively d e and d p , the rate equations bifurcate for d e = d p . This condition describes predators hunting in a group or alone with equal propensity. In a metapopulation model with patches hosting several agents, rotating spiral waves appear only for d e < d p , i.e., when the propensity for cooperative predation is stronger than for individual hunting. Theoretical methodologies used in this paper have been developed elsewhere in the literature. We have just applied them to our model to compare features of the underlying dynamics with other existing models. In particular, we have derived the magnitude of the stochastic noise at the bifurcation point for homogeneous rates (where the rate equations predict neutrally stable orbits), to make a comparison with the original cyclic Lotka-Volterra model [37]. Three-agent interactions are intrinsically noisier than two-agent ones since they involve more fluctuating degrees of freedom. Nevertheless, the extinction probability is uniformly lower in our model than in Ref. [37]. This apparent paradox has a simple solution. When three agents interact, strategies fluctuate longer around the reactive fixed point before one of them prevails on the others. Stochastic noise has no preferred direction. Hence, it acts as an equilibrating force. Group interactions help the system stay in equilibrium. Doing so, they promote species coexistence. Similarly, we have studied the phase portrait for heterogeneous rates, to make a comparison with a model in which group interactions involve an agent and its four von-Neumann neighbors or its eight Moore neighbors [31]. Although we consider only three interacting agents, the phase portrait of our model shows a richer structure. As far as we understand, the reason for this result is that our reaction rates are fully independent. Besides, we have shown that spatial topology plays a critical role in shaping the phase portrait. Indeed, Hopf bifurcations disappear on a two-dimensional lattice (with one agent per lattice site) as a consequence of the locality of the interactions. Then, we have studied the effects of individual mobility in a lattice metapopulation model, with patches hosting several agents. It turns out that our rotating spirals are qualitatively similar to those arising in the spatially structured version of the May-Leonard model [11]. Although our model differs from Ref. [11] in that we assume no empty sites, no birth and no selection-removal interactions, the observed similarity is not surprising, for three reasons. First of all, models of cyclic dominance without mutations usually undergo degenerate Hopf bifurcations. Secondly, it is usually possible to map a system of reaction-diffusion equations, with supercritical Hopf bifurcation, onto a complex Landau-Ginzburg equation [7,59]. The map is possible provided diffusion is linear. Finally, the latter equation yields a good description, even when the bifurcation is degenerate. As a result, the regimes of species coexistence in our model and the spatially structured May-Leonard model fall in the same universality class. As such, they are in one-toone correspondence. To conclude, we have shown that one can build nonlinear continuous mobility operators starting from threeagent chase reactions on a lattice. We have focused on one of several possible definitions. In particular, we have studied the effects of chase reactions on the propagation of spiral waves. Similar to nonlinear diffusion, arising in models where hopping and pair exchange occur with different rates, chase reactions produce far-field break-up of spiral waves. However, the breaking pattern is peculiar. Nonlinear diffusion could play a role in explaining structural patterns in growing bacterial colonies. Our study shows that group interactions provide viable mechanisms for both predation and dispersal. ACKNOWLEDGMENTS F.P. acknowledges his little daughter Giulia for (re)inventing RPS with three players and for suggesting it as a research topic. The computing resources used for our numerical simulations and the related technical support have been provided by the CRESCO/ENEAGRID High Performance Computing infrastructure and its staff [66]. CRESCO (Computational RESearch centre on COmplex systems) is funded by ENEA and by Italian and European research programmes. Fig. 1 - 1[color online] RPS cyclic chain. Fig. 2 - 2[color online] Comparison between the RE and their Hopf normal form.with c( ) = a( Fig. 3 - 3[color online] Extinction probability. Fig. 4 - 4Phase structure of the model in a well-mixed environment for heterogeneous rates. Fig. 6 - 6[color online] An agent ( ) and two interacting neighbors ( ) on a two-dimensional lattice. Fig. 7 0 Fig. 5 - 705(a)-(b) shows phases corresponding to d RRS = 0.2 and d RRS = 1.0 for N = 256 × 256. The most remarkable difference with respect to the well-mixed environment is the absence of bifurcations. In both plots the phase plane is entirely filled by stable fixed points with global densities (here meaning strategy fractions over the whole lattice) spiralling inwards. A comparison with plots (a) and (i) ofFig. 4indicates that the relative position of phases is essentially the same in the region of stable equilibrium, even if their shapes are deformed. For instance, for d RRS = 1.0, d PPR < 1, d SSP 1, P hasd RRP = 0.1, d PPS = 0.1, d SSR = 1.0 d RRP = 0.1, d PPS = 0.5, d SSR = 1.0 d RRP = 0.1, d PPS = 1.0, d SSR = 1.0 d RRP = 0.5, d PPS = 0.1, d SSR = 1.0 d RRP = 0.5, d PPS = 0.5, d SSR = 1.0 d RRP = 0.5, d PPS = 1.0, d SSR = 1.0 d RRP = 1.0, d PPS = 0.1, d SSR = 1.0 d RRP = 1.0, d PPS = 0.5, d SSR = 1.0 d RRP = 1.0, d PPS = 1.0, d SSR = 1.Ensemble H of neutrally stable fixed points. Fig. 7 - 7[color online] (a)-(b) Phase structure of the model on a two-dimensional lattice. (c)-(d) Representative trajectories (continuous lines) and averages (dashed/dotted lines) of global densities over 100 sample trajectories. Both plots correspond to equilibrating rates (dRRP, dPPS, dRRS) = (1, 1, 1) Fig. 8 - 8Snapshots of the strategy density field in RGB representation for N = 2048 × 2048, γ2 = 16, M = ∞, t = 400. Fig. 9 - 9Comparison of wavelength and propagation speed of spiral waves from the CGLE with data obtained from numerical integration of Eqs. (V.2) for N = 512 × 512 and M = ∞. Fig. 10 - 10Effects of stochastic noise on the propagation of spiral waves. In Fig. 10, three snapshots of φ, corresponding respectively to M = 256 (a), 128 (b) and 64 (c), illustrate how stochastic noise induced by finite M perturbates the stability of the central spiral wave. We obtained all plots from the numerical integration of Eqs. (V.2) for N = 2048 × 2048, γ 2 = 16, = 6.43, t = 400 and initial conditions given by Eq. (V. Fig. 11 - 11[color online] Configurations of three neighboring chasing agents. [s(x)p(x)∆r(x) − r(x)p(x)∆s(x)] . (VI.18) Analogously, adding δr e and δr f yields Fig. 12 - 12Chaotic patterns induced by chasing reactions. Fig. 13 - 13Effects of chasing reactions on the propagation of spiral waves. TABLE I - IA three-player variant of RPS. TABLE II - IITransition rates and density variations.second order, yields the Fokker-Planck equation (FPE) Table II it follows that As can be seen, the O(h) contributions, Eqs. (VI.17), (VI.20), are equal and opposite, while the O(h 2 ) ones, Eqs. (VI.18), (VI.21), are just equal. Consequently, the former cancel whereas the latter add up. At the end, we get δr chase (x) = D 3 [s(x)p(x)∆r(x) − r(x)p(x)∆s(x)]∆s(x)] . (VI.21) + O(N −3/2 ) , (VI.22) Recall that y = x + Ax 2 cannot be inverted exactly. By iteration we have x = y −Ax 2 = y −A(y −Ax 2 ) 2 = y −Ay 2 +2A 2 y 3 +O(y 4 ) . The drift term in Eq. 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[ "Dynamic interferometry measurement of orbital angular momentum of light", "Dynamic interferometry measurement of orbital angular momentum of light" ]	[ "Hailong Zhou \nWuhan National Laboratory for Optoelectronics\nHuazhong University of Science and Technology\n430074WuhanChina\n", "Lei Shi \nWuhan National Laboratory for Optoelectronics\nHuazhong University of Science and Technology\n430074WuhanChina\n", "Xinliang Zhang \nWuhan National Laboratory for Optoelectronics\nHuazhong University of Science and Technology\n430074WuhanChina\n", "Jianji Dong \nWuhan National Laboratory for Optoelectronics\nHuazhong University of Science and Technology\n430074WuhanChina\n" ]	[ "Wuhan National Laboratory for Optoelectronics\nHuazhong University of Science and Technology\n430074WuhanChina", "Wuhan National Laboratory for Optoelectronics\nHuazhong University of Science and Technology\n430074WuhanChina", "Wuhan National Laboratory for Optoelectronics\nHuazhong University of Science and Technology\n430074WuhanChina", "Wuhan National Laboratory for Optoelectronics\nHuazhong University of Science and Technology\n430074WuhanChina" ]	[]	We present a dynamic interferometry to measure the orbital angular momentum (OAM) of beams. An opaque screen with two air slits is employed, which can be regarded as the Young's double-pinhole interference. When the OAM beams with an annular intensity distribution vertically incident, the far-field interference patterns depend on the phase difference of the light in the two pinholes. We scan the angle between the two slits, the output intensity at center changes alternatively between darkness and brightness. Utilizing this characteristic, we can measure the OAM of light. This scheme is very simple and lowcost. In addition, it can measure very large topological charge of OAM beams due to the continuously scanning. PACS numbers: 42.50.Tx, 07.60.Ly. Light beams carrying orbital angular momentum (OAM) are associated with an azimuthal phase structure exp( ) il , where  is the angular coordinate and l is the azimuthal index, defining the topological charge (TC) of the OAM beams [1]. These beams have an OAM of l per photon ( is Planck's constant divided by 2 ). In recent years, OAM	null	[ "https://arxiv.org/pdf/1407.4218v1.pdf" ]	119,270,443	1407.4218	d8fac0736ca9607e0ed7430926b8e1bae3df95fc	Dynamic interferometry measurement of orbital angular momentum of light Hailong Zhou Wuhan National Laboratory for Optoelectronics Huazhong University of Science and Technology 430074WuhanChina Lei Shi Wuhan National Laboratory for Optoelectronics Huazhong University of Science and Technology 430074WuhanChina Xinliang Zhang Wuhan National Laboratory for Optoelectronics Huazhong University of Science and Technology 430074WuhanChina Jianji Dong Wuhan National Laboratory for Optoelectronics Huazhong University of Science and Technology 430074WuhanChina Dynamic interferometry measurement of orbital angular momentum of light We present a dynamic interferometry to measure the orbital angular momentum (OAM) of beams. An opaque screen with two air slits is employed, which can be regarded as the Young's double-pinhole interference. When the OAM beams with an annular intensity distribution vertically incident, the far-field interference patterns depend on the phase difference of the light in the two pinholes. We scan the angle between the two slits, the output intensity at center changes alternatively between darkness and brightness. Utilizing this characteristic, we can measure the OAM of light. This scheme is very simple and lowcost. In addition, it can measure very large topological charge of OAM beams due to the continuously scanning. PACS numbers: 42.50.Tx, 07.60.Ly. Light beams carrying orbital angular momentum (OAM) are associated with an azimuthal phase structure exp( ) il , where  is the angular coordinate and l is the azimuthal index, defining the topological charge (TC) of the OAM beams [1]. These beams have an OAM of l per photon ( is Planck's constant divided by 2 ). In recent years, OAM beams have been widely used in a variety of interesting applications, such as optical microscopy [2], micromanipulation [3,4], quantum information [5,6], free-space and fiber optical communication [7][8][9]. The TC of OAM beams characterizes the corresponding mode [7][8][9] and the magnitude of optical torque [10,11], so the capability of distinguishing different OAM states is very important. There are several existing methods to detect optical vortices, such as interference with a plane wave [12], self-homodyne detection [13], Cartesian to log-polar coordinate transformation [14,15], diffraction patterns of various apertures or slits [16][17][18][19][20][21][22][23][24][25]. Interference with a plane wave is the most common way to measure the TC of OAM beams in the laboratory but needs an additional reference beam. Selfhomodyne detection converts the TC to the voltage and is convenient to process by computer or other digital processing system. While it employed 90 0 or 180 0 hybrids, it is inconvenience to couple beams to waveguide and the operation bandwidth is restricted by hybrids. In addition, it is only applicable for high pure OAM beams with a small TC because only two sampling points are employed. Cartesian to log-polar coordinate transformation can detect multiple modes at the same time. Nonetheless, The method has a limitation due to the overlap of the spots for different OAM states Diffraction patterns of various apertures or slits such as square and triangular slits [21][22][23] can also be used to measure the TC but only for small TC (<20 and <10 for square and triangular slits respectively). Double-slit interference [24,25] does not need additional reference light and is a linear model. In our previous report, we integrated the function of polarization filter into the device. However, this model strongly relies on the fringe resolution, so it is still difficult to distinguish large TC. The phenomenon of the other methods [16][17][18][19][20] is not intuitional or regular, so the patterns of different OAM beams are difficult to identify. In this letter, we put forward a dynamic interferometry to measure OAM of beams. An opaque screen with two air slits is employed, which can be regarded as the Young's double-pinhole interference. When the OAM beams with an annular intensity distribution vertically irradiate the screen, the far-field interference patterns depend on the phase difference of the light in the two pinholes. The intensity is converted to the voltage by a photoelectric detector (PD) and it is convenient to process by digital processing system. We scan the angle between the two slits, the output intensity at center changes alternatively between darkness and brightness. Utilizing this characteristic, we can measure the OAM of light. This scheme is very simple and low-cost. In addition, it can measure very large TC of OAM beams due to the continuous scanning. It also has a quite large tolerance for central deviation and mode impurity. [ ( , )] ( , ) f x y F u v  F ,(1) where []  F denotes the spatial Fourier transform operation only on the transverse coordinates ( , ) xy and ( , ) uv is the coordinate of spatial frequencies. So the two interferential waves in the receiving plane 22 ( , ) xy can be wrote as 22 ( , ) [ ( cos , sin ) exp( )] ( , ) exp( ) exp[ 2 ( cos sin )] i i i i i i i E x y f x R y R jl F u v jl j R u v              F ,(2) where R is the radius of the OAM beams in the screen and ( 1, 2) i i   is the azimuth angle of the two slits (the angle  between the slits equals 21   ). 2 x u f   , 2 y v f   where  is the input wavelength and f is the focus length of the lens. Near the optical axis ( 22 ( , ) xy  (0, 0) ) the interferential waves is simplified by 22 ( , ) ( , ) exp( ) ii E x y F u v jl  .(3) So the interferential intensity distribution is From Eq. (4), one can see that the intensity distribution has a relationship of cosine with the TC multiplying the angle between the slits. And the electrical signal from the PD will be proportional to the received power. So we can determine the TC of the input OAM beams by scanning the angle. I x y E E l  .(4 In the following, we will prove the aforementioned derivation by scalar diffraction theory [26]. The wave at the receiving plane is the Fraunhofer diffraction of the one behind the screen, so the intensity can expresses as 2 2 2 2 2 2 ( , ) ( , ) ( , ) exp[ ( )] in j I x y E x y P x y xx yy dxdy f      ,(5) where in E is the input OAM wave and ( , ) P x y is the transmittance function of the opaque screen with two air slits. We set 1  = 0 and assume the input beam is an ordinary Gaussian vortex beam, expressed as [27] 22 1 ( , ) ( / ) exp( / )exp( ) l in U r r w r w il   ,(6) where ( , ) r  are two-dimensional polar coordinates corresponding to rectangular coordinates ( , ) xy and w is the waist size. The radius of the OAM beams can be deduced that = ( / 2) R w l . We set the wavelength of input beams as 1550 nm. The waist size equals 5 mm, the focal length of lens is 200 mm and the width of slits is d = 1 mm. Figure Thus this peak will split into two. This characteristic is very beneficial to distinguish the period. To clarify more clearly, Normalized power Angle between slits/2(rad) Angle between slits(degree) The main advantage of the dynamic interferometry measurement is its ability to detect OAM mode with a very large TC benefited from the continuously scanning and it is convenient to process by digital processing system. For example, when an OAM beam with a TC equal to  40 vertically incidents, the received signals are still precise as shown in Fig. 3(a-b). The main factor which limits the magnitude of detected OAM is the width of slits. When the width is too large to sample the 2 changes of phase, it will miss some peaks, thus some key information will be lost. This scheme also has a quite large tolerance for central deviation and mode impurity.  Phase All analysis above is not relevant to the sign of TCs. In fact, it fails to distinguish the sign in current approach. This characteristic can be also inferred from Eq. (4) where the cosine function is even symmetric. To distinguish the sign of TC, we must break the even symmetric. It can be realized by tinily tilted light incidence. Assume that the OAM beam tilted irradiates in 2 ( , ) xz plane. In this case, the cosine term in Eq. (4) turns into 2 cos( + x l k x  ) ( x k is a constant). Then Eq. (4) turns to be relevant to the sign of TC. Figure 5 presents the phase in the screen and the received power when the OAM beam tinily tilted irradiates with a TC equal to 6 or -6. The black circle represents the location of the maximum intensity. It can be seen that the phase distributions are quite uneven and are related to the sign, that is, the phase change in upper half plane is slower than the one in lower half plane when TC > 0 (Fig. 5(a)) and is inverse when TC < 0 (Fig. 5(b)). So the received power of the two modes is exactly inverse when scanning the angle. One can see that the power petal is wider when the corresponding phase change is slower. Although the tilted incidence can detect the sign of TC, the measured magnitude of TC is prone to be wrong because the phase distribution in the screen is sensitive to the incidence angle. A little incline of incidence will add a tightly phase grating to the OAM beams and seriously break the helical phase structure. Thus the received power will be chaotic since the helical phase structure is broken, and it is hard to sample the fast phase variation accurately. As shown in Fig. 6(a), the times of phase change in 2 along the maximum intensity (black circle) become a very large value when the OAM beams with TC = 6 incident with a little tilted angle, which will cause an erroneous measurement results shown in Fig. 6(b). So we must precisely tune the incidence angle, or only use the tinily tilted incidence to determine the sign and measure the magnitude accurately by vertically incidence. The main difficulty of the proposed scheme is how to achieve the scanning of the angle between two slits. It can be realized by using spatial amplitude modulation devices such as binary-amplitude spatial light modulator [28,29]. Here we suggest a simple and equivalent method. The equivalent structure is shown in the inset of Fig. 1 From Eq. (5) and Eq. (7), it proves the two structures have the same function and the simulations also indicate the received power of the two structures is almost the same. In addition, the double-arm structure is easy to fabricate and easy for scanning. In conclusion, we put forward a dynamic interferometry to measure OAM beams. An opaque screen with two air slits is employed, which can be regarded as the Young's double-pinhole interference. When the OAM beam with an annular intensity distribution vertically incidents, the far-field interference patterns depend on the phase difference of the light in the two pinholes. The intensity is converted to the voltage by a PD and it is convenient to process by digital processing system. We scan the angle between the two slits, the output intensity at center changes alternatively between darkness and brightness. Utilizing this characteristic, we can measure the OAM of light. This scheme is very simple and low-cost. In addition, it can measure very large TC of OAM beams due to the continuously scanning. This scheme also has a quite large tolerance for central deviation and mode impurity. Besides, our scheme can be extended to measure any engineering structured light which is characterized by azimuthal phase distribution, such as optical vortices and so forth. This work was partially supported by the Program for New Century Excellent Talents FIG. 1 . 1Schematic structure of the dynamic interferometry with double slits. Inset, equivalent double arms. Figure 1 1presents the schematic structure of the proposed setup. The setup is composed by an opaque screen with two air slits, a Fourier transform lens and a PD. When the OAM beams with an annular intensity distribution vertically incident on the screen, only two pinholes have light transmitted shown as the black circles inFig. 1. The lens is used to shift the interference patterns to center by Fourier transform, then the PD converts the intensity to the voltage. The voltage is minimum when the interference is destructive and is maximum when the interference is constructive. Assume the waves in the two pinholes are plane with different phase and the distributions of amplitudes are ( , ) f x y around their own center. We define that 2 presents the dependence of the received power of the PD on the angle between the slits. From Fig. 2(a), we can see the receive signals are periodic and the period equals 2 . The number of peaks or troughs in a period equals the TC of input OAM beams. The results agree well with the theoretical derivation by Eq. (4) except the first split peak. This phenomenon can be easily understood. When the angle is near 2 m  ( m is an arbitrary integer), the two slits overlap so that the transmitted light energy decreases. In the most extreme case when the angle equals 2 m  , there is only one output wave and the energy is approximately equal to the quarter of the other peak power. Fig. 2 ( 2b) shows the normalized power in polar coordinates, the number of petals represent the corresponding OAM beams and the first one splits into two owing to the overlap of the two slits. Other examples are also given when TCs equal 1, 2, 3, 10     , as shown in Fig. 2(c). FIG. 2. Received power dependence on the slit angle in (a) Cartesian and (b-c) polar coordinates when different OAM beams vertically irradiate. FIG. 3 . 3Received power when TC equals  40. (a) Cartesian coordinate, (b) polar coordinate For example, when setting d = 15 mm and TC =  40, Figure 4(a) presents the received signals. One can see that the signals deteriorate and only 38 peaks appear. When keeping increasing the slit width or the TC of OAM, the number of lost peaks will increase because the slits cannot sample the phase precisely. But the slit width of 1 mm is precise enough for most applications. Figure 4 ( 4b) presents the output signals when TC =  6 and the central deviation is half of radius (4.3 mm). Even so, the scanning power signal exhibits a TC of 6 or -6 regardless of the peak distortion and different petal widths. The distortion is caused by the nonuniform phase distributions around the center axis. As shown inFig. 4(b), the hot picture displays the phase of incoming OAM beam in the screen, where the red circle represents the location of the maximum intensity and the black lines indicate the two slits. The OAM beam is obviously off-centered. When the angle of the second slit scans from 0 to 2 , the changing rate of phase in left side is slower than the one in right side. So the number of peaks in left side is less. The key element of this scheme is to count how many times the phase changes by 2 , so it has a tolerance for mode impurity as long as the mode impurity does not influence the times of phase change by 2 . For example,Fig. 4(c)shows the received power when the purity of TC = 6 is 80% and the ones of the other two modes (TC =7, 5) are both 10%. The width and the amplitude of received power distribution are changed by other OAM modes, but we can still infer the main mode is the one whose TC equals 6 or -6.FIG. 4. Received power when (a) d = 15 mm. (b) Shift 0.5 R . (c) Interferometry measurement of impure mode superposition with 80% OAM6, 10% OAM7, and 10% OAM5. FIG. 5 . 5Phase in the screen and received power when tinily tilted incidence. FIG. 6 . 6(a) Phase and (b) received power of TC = 6 in the screen under a little tilted angle incidence. in Ministry of Education of China (Grant No. NCET-11-0168), a Foundation for the Author of National Excellent Doctoral Dissertation of China (Grant No. 201139), the National Natural Science Foundation of China (Grant No. 11174096) and the Fundamental Research Funds for the Central Universities, HUST: 2014YQ015. ) OAM beams PD  Inset Screen with two slits Lens . The double-arm structure is complementary to the double-slit structure, it means the sum of the transmittance functions of the two structures is 1. Since there is no intensity at center for OAM beams, the received intensity in this case can expresses as(b)TC=-6 (a)TC=6 Phase Received power Phase Received power (b) (a) 22 2 2 2 2 2 2 22 ( , ) (0,0) 2 ( , ) ( , )[1 ( , )]exp[ ( )] 2 ( , ) ( , ) exp[ ( )] in in xy j I x y E x y P x y xx yy dxdy f j E x y P x y xx yy dxdy f              . (7) . 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[ "Higgs alignment and novel CP -Violating observables in two-Higgs-doublet models", "Higgs alignment and novel CP -Violating observables in two-Higgs-doublet models" ]	[ "Ian Low \nDepartment of Physics and Astronomy\nNorthwestern University\n60208EvanstonIllinoisUSA\n", "Nausheen R Shah \nDepartment of Physics and Astronomy\nWayne State University\n48201DetroitMichiganUSA\n", "Xiao-Ping Wang \nSchool of Physics\nBeihang University\n100191BeijingChina\n\nBeijing Key Laboratory of Advanced Nuclear Materials and Physics\nBeihang University\n100191BeijingChina\n", "\n1a High Energy Physics Division\nArgonne National Laboratory\n60439ArgonneIllinoisUSA\n" ]	[ "Department of Physics and Astronomy\nNorthwestern University\n60208EvanstonIllinoisUSA", "Department of Physics and Astronomy\nWayne State University\n48201DetroitMichiganUSA", "School of Physics\nBeihang University\n100191BeijingChina", "Beijing Key Laboratory of Advanced Nuclear Materials and Physics\nBeihang University\n100191BeijingChina", "1a High Energy Physics Division\nArgonne National Laboratory\n60439ArgonneIllinoisUSA" ]	[]	Null results from searches for new physics at the Large Hadron Collider (LHC) tend to enforce the belief that new particles must be much heavier than the weak scale. We undertake a systematic study of the interplay between Higgs alignment and CP -violation in complex two-Higgs-doublet models, which enables us to construct a CP -violating scenario where new Higgs bosons are close to the weak scale after including stringent constraints from the electric dipole moment and measurements at the LHC. In addition, we propose a novel potential signal of CP -violation in the Higgs-to-Higgs decays, h3 → h2h1, where h3, h2, and h1 are the heaviest, second heaviest and the Standard Model-like neutral Higgs bosons, respectively. The decay could manifest itself in triple boson final states in h1h1h1 and h1h1Z, which are quite distinct and provide unique venues for new measurements at the LHC.	10.1103/physrevd.105.035009	[ "https://arxiv.org/pdf/2012.00773v3.pdf" ]	246,276,388	2012.00773	40b2b015756441547aef577c2839fa35f0d5778c	Higgs alignment and novel CP -Violating observables in two-Higgs-doublet models Ian Low Department of Physics and Astronomy Northwestern University 60208EvanstonIllinoisUSA Nausheen R Shah Department of Physics and Astronomy Wayne State University 48201DetroitMichiganUSA Xiao-Ping Wang School of Physics Beihang University 100191BeijingChina Beijing Key Laboratory of Advanced Nuclear Materials and Physics Beihang University 100191BeijingChina 1a High Energy Physics Division Argonne National Laboratory 60439ArgonneIllinoisUSA Higgs alignment and novel CP -Violating observables in two-Higgs-doublet models Null results from searches for new physics at the Large Hadron Collider (LHC) tend to enforce the belief that new particles must be much heavier than the weak scale. We undertake a systematic study of the interplay between Higgs alignment and CP -violation in complex two-Higgs-doublet models, which enables us to construct a CP -violating scenario where new Higgs bosons are close to the weak scale after including stringent constraints from the electric dipole moment and measurements at the LHC. In addition, we propose a novel potential signal of CP -violation in the Higgs-to-Higgs decays, h3 → h2h1, where h3, h2, and h1 are the heaviest, second heaviest and the Standard Model-like neutral Higgs bosons, respectively. The decay could manifest itself in triple boson final states in h1h1h1 and h1h1Z, which are quite distinct and provide unique venues for new measurements at the LHC. I. INTRODUCTION CP -violation (CPV) is a critical ingredient for the matter-antimatter asymmetry in the Universe [1] and its presence is of existential significance. However, the amount of CPV in the Standard Model (SM), via the Cabbibo-Kobayashi-Maskawa mechanism, is insufficient to generate the observed baryon asymmetry [2,3]; new sources of CPV must be present outside of the SM. A two-Higgs-doublet model (2HDM) [4] is not only one of the simplest extensions of the SM which may provide new sources for CPV [5][6][7], but also the prototype employed in numerous more elaborate new physics models [8]. There is vast literature on CPV and 2HDMs. However, the majority of these studies focus on detecting a CPeven and CP -odd mixture in a mass eigenstate through angular correlations or asymmetries in kinematic distributions [9][10][11][12][13][14][15][16][17], which requires significant experimental resources and statistics. 1 On the other hand, there are two major results derived from data collected at the LHC: 1) null searches for new particles beyond the SM, and 2) a SM-like 125 GeV Higgs. The first result suggests that new particles, if present, should be much heavier than the weak scale, while the latter implies a dominantly CPeven 125 GeV Higgs. In light of these considerations, it becomes clear that we must reevaluate the possibility of CPV in 2HDM under the assumption of a SM-like 125 GeV Higgs, which is dubbed the alignment limit [19][20][21]. Of particular interest is the "alignment without decoupling" limit, where new Higgs bosons could still be present near the weak scale [22][23][24]. This has been done only under limited purview in the past [16,25,26] but we aim to achieve a comprehensive and analytical understanding. Specifically we emphasize there are two distinct sources of CPV in 2HDM; in the mixing and in the decay of the Higgs bosons. Kinematic distributions are only sensitive to CPV in the mixing. This realization allows us to construct a benchmark scenario where new Higgs bosons are not far above the weak scale, at around 500 GeV or lighter, and propose a novel signature of CPV, without recourse to angular correlations or electric dipole moment (EDM) signals, in the Higgs-to-Higgs decay, (h 3 → h 2 h 1 → 3h 1 ), whose existence is sufficient to establish CPV in complex two-Higgs-doublet models (C2HDMs). 2 The presence of such an observable is nontrivial, as this decay channel vanishes in the exact alignment limit. Our benchmark survives constraints from EDMs [29][30][31][32][33] and collider measurements, and could be discovered at the LHC in the near future. II. THE HIGGS BASIS The most general potential for a 2HDM [34][35][36] in terms of the two hypercharge-1, SU (2) doublet fields Φ a = (Φ + a , Φ 0 a ) T , a = {1, 2}, is given by V = m 2 1 Φ † 1 Φ 1 + m 2 2 Φ † 2 Φ 2 − m 2 12 Φ † 1 Φ 2 + H.c. + λ 1 2 (Φ † 1 Φ 1 ) 2 + λ 2 2 (Φ † 2 Φ 2 ) 2 + λ 3 (Φ † 1 Φ 1 )(Φ † 2 Φ 2 ) + λ 4 (Φ † 1 Φ 2 )(Φ † 2 Φ 1 ) + λ 5 2 (Φ † 1 Φ 2 ) 2 + λ 6 (Φ † 1 Φ 1 )(Φ † 1 Φ 2 ) +λ 7 (Φ † 2 Φ 2 )(Φ † 1 Φ 2 ) + H.c. .(1) 2 In models with additional CP -even scalars beyond the 2HDM, such decays may be present without CPV [27,28]. However, the mass spectrum in this case is different from that of 2HDM. We assume a vacuum preserving the U (1) em gauge symmetry and adopt a convention where both scalar vacuum expectation values (VEVs) are real and non-negative, Φ 1 = 1 √ 2 0 v 1 , Φ 2 = 1 √ 2 0 v 2 ,(2) where v 2 1 + v 2 2 ≡ v = 246 GeV. It is customary to define an angle β through tan β = v 2 /v 1 . We choose to study the alignment limit [22][23][24] in the Higgs basis [37], which is defined by two doublet fields, H i , i = {1, 2}, having the following property H 0 1 = v/ √ 2 , H 0 2 = 0 .(3) We will parametrize the Higgs basis doublets as H 1 = (G + , (v + φ 0 1 + iG 0 )/ √ 2) T and H 2 = (H + , (φ 0 2 + ia 0 )/ √ 2) T , where G + and G 0 are the Goldstone bosons. The neutral fields are φ 0 1 , φ 0 2 and a 0 , and the charged field is H + . Moreover, our phase convention is such that φ 0 2 and a 0 are the CP -even and CP -odd eigenstates, defined with respect to the fermion Yukawa couplings. There is a residual U (1) redundancy in the Higgs basis, labeled by H 2 → e iη H 2 , which leaves Eq. (3) invariant and motivates writing the scalar potential in terms of H 2 ≡ e iη H 2 [38], V = Y 1 H † 1 H 1 + Y 2 H † 2 H 2 + Y 3 e −iη H † 1 H 2 + H.c. + Z 1 2 (H † 1 H 1 ) 2 + Z 2 2 (H † 2 H 2 ) 2 + Z 3 (H † 1 H 1 )(H † 2 H 2 ) + Z 4 (H † 1 H 2 )(H † 2 H 1 ) + Z 5 2 e −2iη (H † 1 H 2 ) 2 + Z 6 e −iη (H † 1 H 1 )(H † 1 H 2 ) +Z 7 e −iη (H † 2 H 2 )(H † 1 H 2 ) + H.c. .(4) In the above, different choices of parameters truly represent physically distinct theories [38]. The potentially complex parameters are {Y 3 , Z 5 , Z 6 , Z 7 }. The minimization of the scalar potential gives Y 1 = −Z 1 /2v 2 and Y 3 = −Z 6 v 2 /2. The first relation can be viewed as the definition of v in the Higgs basis, while the second relation implies there are only three independent complex parameters, usually taken to be {Z 5 , Z 6 , Z 7 }. If one can find a choice of η such that all parameters in Eq. (4) are real after imposing the minimization condition, the vacuum and the bosonic sector of the 2HDM is CP -invariant. This can happen if and only if [39] Im(Z * 5 Z 2 6 ) = Im(Z * 5 Z 2 7 ) = Im(Z * 6 Z 7 ) = 0 . Otherwise, CP invariance is broken. In a 2HDM the most general Higgs-fermion interactions result in tree-level flavor-changing neutral currents, which can be removed by imposing a discrete Z 2 symmetry [40][41][42], Φ 1 → Φ 1 and Φ 2 → −Φ 2 . In addition, the Z 2 symmetry can be broken softly by mass terms, leading to λ 6 = λ 7 = 0 in Eq. (1). In the Higgs basis, the existence of a softly broken Z 2 symmetry is guaranteed through the condition [38,43], (Z 1 − Z 2 ) [(Z 3 + Z 4 )(Z 6 + Z 7 ) * − Z 2 Z * 6 − Z 1 Z * 7 +Z * 5 (Z 6 + Z 7 )] − 2(Z 6 + Z 7 ) * (\|Z 6 \| 2 − \|Z 7 \| 2 ) = 0 . (6) Equation. (6) assumes Z 6 + Z 7 = 0 and Z 1 = Z 2 , and eliminates two real degrees of freedom. In the end there are a total of nine real parameters in a complex 2HDM. III. THE ALIGNMENT LIMIT The alignment limit [19][20][21] is defined by the limit where the scalar carrying the full VEV in the Higgs basis is aligned with the 125 GeV mass eigenstate [22][23][24], in which case the observed Higgs boson couples to the electroweak gauge bosons with SM strength. The mass- squared matrix M 2 in the φ 0 1 −φ 0 2 −ā 0 basis, where H 2 = (H + , (φ 0 2 + iā 0 )/ √ 2) T , can be diagonalized by an orthogonal matrix R relating φ = (φ 0 1 ,φ 0 2 ,ā 0 ) T to the mass eigenstates h = (h 3 , h 2 , h 1 ) T , h = R · φ [38], R = R 12 R 13 R 23 =   c 12 −s 12 0 s 12 c 12 0 0 0 1     c 13 0 −s 13 0 1 0 s 13 0 c 13     1 0 0 0c 23 −s 23 0s 23c23   .(7) Here we have used the notation c ij = cos θ ij , s ij = sin θ ij , c 23 = cosθ 23 ands 23 = sinθ 23 . An important observation is thatθ 23 [44] rotates betweenφ 0 2 andā 0 , which corresponds to the phase rotation H 2 → e iθ23 H 2 . Therefore the effect of theθ 23 rotation is to shift the η parameter labelling the Higgs basis. In the end the combination that appears in the physical couplings is θ 23 ≡ η +θ 23 . This motivates defining [38] M 2 ≡ R 23 M 2 R T 23 = v 2   Z 1 Re[Z 6 ] −Im[Z 6 ] Re[Z 6 ] Re[Z 5 ] + A/v 2 − 1 2 Im[Z 5 ] −Im[Z 6 ] − 1 2 Im[Z 5 ] A/v 2   ,(8)whereZ 5 = Z 5 e −2iθ23 ,Z 6/7 = Z 6/7 e −iθ23 , and A = Y 2 + v 2 (Z 3 + Z 4 − Re[Z 5 ])/2. Alignment is achieved by the conditions Re[Z 6 ] = Im[Z 6 ] = 0. M 2 can be diagonalized by just two angles: R M 2 R T = diag (m 2 h3 , m 2 h2 , m 2 h1 ) and R = R 12 R 13 =   c 12 c 13 −s 12 −c 12 s 13 s 12 c 13 c 12 −s 12 s 13 s 13 0 c 13   ,(9) which relates the mass eigenbasis (h 1 , h 2 , h 3 ) to the CP - eigenbasis (φ 0 1 , φ 0 2 , a 0 )   h 3 h 2 h 1   = R   φ 0 1 φ 0 2 φ 0 3   = R   φ 0 1 c 23 φ 0 2 − s 23 a 0 s 23 φ 0 2 + c 23 a 0   . (10) θ 23 will be important when discussing CP -conservation. Recall φ 0 1 carries the full SM VEV and exact alignment is when φ 0 1 coincides with a mass eigenstate. We choose to align φ 0 1 with h 1 , which can be achieved by setting c 13 = 0 and θ 13 = π/2 in Eq. (9). We also impose the ordering, m h1 ≤ m h2 ≤ m h3 so that m h1 = 125 GeV. Small departures from alignment can be parametrized by writing θ 13 = π/2 + , 1, R =   − c 12 −s 12 −c 12 (1 − 2 /2) − s 12 c 12 −s 12 (1 − 2 /2) 1 − 2 /2 0 −   .(11) Equation (6) Re[Z 5 ] = 1 v 2 c 2θ12 m 2 h2 − m 2 h3 + 2 m 2 h3 c 2 12 + m 2 h2 s 2 12 − m 2 h1 ,(12)Im[Z 5 ] = 1 v 2 s 2θ12 1 − 2 2 m 2 h2 − m 2 h3 ,(13)Re[Z 6 ] = 2v 2 s 2θ12 m 2 h3 − m 2 h2 ,(14) Im [Z 6 ] = v 2 m 2 h1 − m 2 h2 c 2 12 − m 2 h3 s 2 12 ,(15)g h1h2h3 = v Re[Z 7 e −2iθ12 ] .(16) From the above we see that the mass splitting between h 3 and h 2 is determined at leading order in by ∆m 2 23 ≡ (m 2 h3 − m 2 h2 ) = v 2 \|Z 5 \|. Therefore, in general, an O(v 2 ) splitting can be achieved with \|Z 5 \| ∼ O(1) . Further, the CPV coupling g h1h2h3 is nonzero away from exact alignment and for nonzero Re[Z 7 e −2iθ12 ]. Hence the decay (h 3 → h 2 h 1 ) may be achieved for reasonable choices of parameters, which however are constrained from LHC and EDM constraints, as will be discussed later. In the Z 2 basis, where each field in the model has a well-defined Z 2 charge, the Yukawa interactions must also respect the Z 2 invariance, which necessitates assigning Z 2 charges to SM fermions as well [45,46]. Two distinct possibilities exist in the literature, leading to type I [47,48] and type II [48,49] models which differ by interchanging tan β with cot β in the Yukawa couplings. Importantly tan β is a derived parameter [38,50] which strongly depends on the mass spectrum for type II 2HDM. In the left panel of Fig. 2 we show contours of tan β in the m h2 -m h3 plane. For our parameter region of interest, tan β ∼ 1 except when m h2 and m h3 are degenerate. We focus on type II models with tan β ∼ O(1), in which region type I and type II models have similar Yukawa couplings. IV. TWO CP -CONSERVING LIMITS The condition for CP invariance in Eq. (5) can be realized as follows [6,38]: CPC1 : Im[Z 5 ] = Im[Z 6 ] = Im[Z 7 ] = 0 ,(17) CPC2 : Im[Z 5 ] = Re[Z 6 ] = Re[Z 7 ] = 0 .(18) In CPC1, M 2 in Eq. (8) is block-diagonal; M 2 13 = M 2 23 = 0, in which case φ 0 1 andφ 0 2 defined in Eq. (10) are CP -even and can mix in general, whereasφ 0 3 is CPodd. This can be achieved by θ 23 = 0 so thatφ 0 3 = a 0 in Eq. (10). Further, neither of the two CP -even states can mix with the CP -odd state. From Eq. (9) we see θ 13 controls the mixing between φ 0 1 andφ 0 3 , which implies θ 13 = π/2 in the CP -conserving limit. This coincides with the exact alignment limit = 0. The mixing betweenφ 0 2 andφ 0 3 is dictated by θ 12 and can be removed by θ 12 = 0 or π/2, which corresponds to h 3 = a 0 or h 2 = a 0 , respectively. Therefore, CPC1 is reached by θ 13 = π 2 , θ 23 = 0 , θ 12 = {0, π/2} , Im[Z 7 ] = Im[Z 7 ] = 0 .(19) One sees from Eqs. (13) and (15) that Im[Z 5 ] = Im[Z 6 ] = 0 under the choice of parameters in Eq. (19). It can be further checked that fermionic couplings of the mass eigenstates follow from their CP -property and the EDM constraints vanish as expected [51]. In CPC2, M 2 12 = M 2 23 = 0 and M 2 is again blockdiagonal. In this case φ 0 1 can mix withφ 0 3 , since they are both CP -even. The CP -odd state isφ 0 2 . Referring back to Eq. (10) we see that this requires θ 23 = π/2. In contrast to the CPC1 scenario, the mixing angle θ 13 , which controls alignment, can now be arbitrary. Turning off mixing betweenφ 0 2 andφ 0 3 again implies θ 12 = 0 or π/2. Hence CPC2 is represented by: θ 23 = π/2 , θ 12 = {0, π/2} , Re[Z 7 ] = Im[Z 7 ] = 0 . (20) Again one can check that Im[Z 5 ] = Re[Z 6 ] = 0 and couplings of the mass eigenstates to the fermions behave as expected from their CP quantum numbers. There is an important distinction between these two scenarios. In CPC1 the CP -conserving limit coincides with the alignment limit because misalignment introduces a small CP -odd component to the SM-like Higgs boson. Then the stringent EDM limits on CPV also constrain the misalignment, ∼ O(10 −4 ), thereby forcing the 125 GeV Higgs to be almost exactly SM-like [51]. This is consistent with the findings in Refs. [25,26,52]. To the contrary, in CPC2 the SM-like Higgs boson only contains a CP -even non-SM-like component. Therefore EDM limits do not constrain misalignment. 3 Equations. (17) and (18) also make it clear that there are two sources of CPV in 2HDM;Z 5 andZ 6 enter into the scalar mass-squared matrix in Eq. (8), whileZ 7 does not. When Im[Z 5 ] = Im[Z 6 ] = 0 or Im[Z 5 ] = Re[Z 6 ] = 0, there is no CPV in the scalar mixing matrix and each mass eigenstate h i is also a CP -eigenstate: two are CPeven and one is CP -odd. In this case, CPV could still be present through nonzero Re[Z 7 ] or Im[Z 7 ] and will manifest in the bosonic interactions of the Higgs bosons. In light of these considerations, we construct a benchmark which interpolates between the CPC1 and CPC2 limits, In Fig. 1 we show the tan β contours on the m h3 − m h2 plane, for the region of parameter space close to our benchmark; our benchmark has tan β ∼ 2.3. With these parameters, h 1 is mostly CP -even, while h 2 and h 3 are CP -mixed states. In our benchmark the charged Higgs and h 3 are degenerate in mass so as to be consistent with precision electroweak measurements, which include the oblique parameters S, T and U [53][54][55]. Conventional wisdom has it that a charged Higgs lighter than 800 GeV is constrained by b → sγ measurements [56,57]. However, more recent results [58] argued that the theoretical uncertainty leaves more room for the new physics contribution. So in our analysis, we set the charged Higgs mass at 420 GeV, which is considered safe for the b → sγ 3 We emphasize that this statement concerns the EDM constraints on the alignment. It was pointed out in Ref. [25] that the Z 2 condition in Eq. (6) would force CPV to vanish in the exact alignment limit. m H ± =420 GeV m h3 =420 GeV m h2 =280 GeV θ 12 =0 θ 23 =1.23 Z 3 =0.1 κ F(h 1) κg ( h1 ) h2/ 3 → h1 h1 (L H C ) h2/3→ Zh1 (LH C) κV(h1) H ± (LH C) Figure 2. LHC constraints on \| \| from Higgs couplings with gluons (κg), vector bosons (κV ), fermions (κF ) and photons (κγ), as well as searches for H + → tb (cyan) , h 2/3 → Zh1(magenta) and h 2/3 → h1h1 (orange). Stars denote our benchmark point. measurement [33]. V. LHC/EDM CONSTRAINTS In Fig. 2 we show the LHC constraints on \| \| and Re [Z 7 ]. For Higgs coupling measurements we use recent results from both ATLAS [59,60] and CMS [61], which constrain κ i = g measured i /g SM i , i = g, V, F, γ. Blue, gray, red and green shaded regions correspond to regions excluded by constraints coming from κ g , κ V , κ F , and κ γ , respectively. The cyan shaded region is excluded due to searches for H + → tb [62,63], which requires tan β ≥ 2 and is satisfied by our benchmark point, tan β = 2.3. For m h3(h2) = 420(280) GeV the experimental limit from double Higgs production [64] is shown as the orange shaded region in Fig. 2 and the limit from h 3 /h 2 → Zh 1 search [65,66] is given by the magenta shaded region in Fig. 2. We also checked that LHC limits on heavy Higgs decays to tt final states [67] are not relevant for our benchmark. In our analysis, we consider both constraints from the electron EDM (eEDM) [32,68,69] and neutron EDM (nEDM) [70]. The most recent constraints are \|d n \| < 1.8 × 10 −26 e cm (90%C.L.) (22) \|d e \| < 1.1 × 10 −29 e cm (90%C.L.). The dominant contribution for both eEDM and nEDM are the two-loop Barr-Zee(BZ) diagrams [71][72][73][74][75][76][77]. The BZ diagrams to d f (f = e, d, and u) and d C q (q = d and u) contain contributions from fermion-loops, Higgs bosonloops, and gauge boson-loops [16] d f = d f (fermion) + d f (Higgs) + d f (gauge),(23) and each contribution includes d f (X) = d γ f (X) + d Z f (X) + d W f (X).(24) For the nEDM, the relevant formula for d n [73] is d n = 0.79d d − 0.2d u + e g s 0.59d C d + 0.3d C u ,(25) where g s is the QCD gauge coupling constant. Both fermion-loops and gauge boson-loops contributions are related to the CP property of the Yukawa interactions, which are parametrized by θ 23 , θ 12 and . The Higgsloops contributions are both related to the CP property of Yukawa interaction and the coupling of g H ± H ∓ hj , which not only depends on , θ 23 , θ 12 , but also depends on Re[Z 7 ]. After consider both eEDM and nEDM, we found that the eEDM constraints are stronger than those from the neutron EDM, so we only show the relevant plots for eEDM. In Fig. 3 contours for the eEDM and the experimental constraints on the most relevant parameters are shown: θ 23 vs (left) and Re[Z 7 ] (right). The solid red line denotes d e = 0, while the dashed red lines bound the experimentally allowed region \|d e \| < 1.1 × 10 −29 e cm (90%C.L.) [32]. We fix the mass spectrum as for the LHC constraints, and again choose θ 12 = 0. While not shown, EDM constraints are minimized when the masses are degenerate [38]. However, regardless of the mass spectrum, eEDM constraints severely limit the CPV components of the mass eigenstates. This can be seen from the limits on d e tracking the behavior expected from our analysis of CPC1 and CPC2. Small values of θ 23 (CPC1 limit) can only be obtained for small values of \| \|, but for \|θ 23 \| ∼ π/2 (CPC2), is effectively unconstrained. Further, small values of Re[Z 7 ] are obtained for values of θ 23 ∼ π/2 (CPC2 limit), but larger values are allowed as θ 23 decreases. Additionally, we see that in regions far from CPC1 and CPC2, d e can be 0 due to cancellations between various contributions. This is the region where our benchmark resides. VI. COLLIDER PHENOMENOLOGY With the generically small CPV components allowed in the mass eigenstates due to experimental constraints, directly probing the CP nature of the mass eigenstates will be challenging. However, the decay (h 3 → h 2 h 1 ) could provide a smoking gun signature for CPV in 2HDMs. If kinematically accessible, this signal is maximized for maximum possible misalignment and largest possible Re[Z 7 ] [cf. Eq. (16)], as allowed from LHC and where eEDM constraints are minimized. Further, we are interested in the possibility of both additional Higgs bosons being within reach of the LHC, which motivates the benchmark presented in Eq. (21). Fig. 4 shows the branching ratios of h 3 (top panel) and h 2 (bottom panel). Gray hatching denotes mass spectra in tension with eEDM constraints.One particular decay mode we would like to focus on is the CPV Higgs-to-Higgs decay, h 3 → h 2 h 1 , which in our benchmark has a branching fraction BR(h 3 → h 2 h 1 ) ∼ 1.5%. As can be seen from Eq. (16), the coupling controlling this decay is CP -violating. Under the assumption that there are two CP -even and one CP -odd scalars in 2HDM, it is easy to see that such a decay mode is forbidden if CP is conserved. From Fig. 4 we also see that h 2 can decay into h 1 h 1 and h 1 Z, giving rise to rates for h 1 h 1 Z and h 1 ZZ at the HL-LHC with L=3000 fb −1 are 2 × 10 4 and 2 × 10 5 . h 3 → h 1 h 1 h 1 and h 3 → h 1 h 1 Z final states via h 3 → h 2 h 1 . Alternatively, h 2 could be produced directly through the gluon fusion mechanism. In this case, simultaneous observations of h 2 → h 1 h 1 and h 2 → h 1 Z would be an unambiguous signal of CP -violation. The events rates for h 1 h 1 and h 1 Z are 4 × 10 4 and 9 × 10 5 at the HL-LHC. These triboson signatures have not been searched for at the LHC and represent excellent opportunities to pur-sue CPV in 2HDMs at a high-energy collider. Moreover, the relatively light mass of h 2 and its decays into two 125 GeV Higgs bosons also imply a significant discovery potential in the near future. VII. CONCLUSION Motivated by the SM-like nature of the 125 GeV Higgs and null searches for new particles at the LHC, we present a systematic study of Higgs alignment and CPV in C2HDMs and distinguish two distinct sources of CPV in the scalar sector. The outcome is the construction of a new CP violating scenario where additional Higgs bosons could be light, below 500 GeV, and stringent EDM limits and current collider searches may still be evaded. In particular, we propose a smoking gun signal of CPV in C2HDMs in the h 1 h 2 h 3 coupling through the Higgs-to-Higgs decays, (h 3 → h 2 h 1 → 3h 1 ), without resorting to the challenging measurements of kinematic distributions. The existence of this decay in C2HDMs is indicative of CPV and the final state in three 125 GeV Higgs bosons is quite distinct, which has not been searched for at the LHC. A ballpark estimate demonstrates the great potential for discovery at the high-luminosity LHC. Figure 1 . 1tan β contours in the m h 2 -m h 3 plane. The relevant parameters are specified in Eq. (21). Stars denote our benchmark point. {Z 3 , 3Re[Z 7 ], θ 12 , θ 23 , } = {0.1, 3, π/2, 1.23, 0.1}, {m h3 , m h2 , m H ± } = {420, 280, 420} GeV . Figure 3 . 3Contours for eEDM (de) in θ 23 vs. \| \| (top), and Re[Z 7 ] (bottom) plane. Only regions within the dashed red lines are experimentally allowed \|de\| < 1.1 × 10 −29 e cm (90%CL)[32]. Thick red line denotes \|de\| = 0. Note different scales for the left/right axes and legends. Stars denote our benchmark point. Figure 4 . 4Branching ratios for h3 (top) and h2 (bottom) for the listed parameters. Gray dashed lines denote mass spectra in tension with eEDM constraints for chosen set of parameters. remains the same after we changing {Z 5 , Z 6 , Z 7 } into {Z 5 ,Z 6 ,Z 7 }. We can use Eq. (6) to eliminate Z 2 and Im[Z 7 ] and choose nine input parameters {v, m h1 , m h2 , m h3 , m H ± , θ 12 , θ 13 , Z 3 , Re[Z 7 ]}. Some important relations are, in the approximate alignment limit, The main production channel for both h 2 and h 3 is gluon fusion. At the √ s = 13 TeV LHC[74]:σ(gg → h 2 ) 5.8 pb , σ(gg → h 3 ) 2.7 pb . (26)The large production rate for h 3 stems from its sizable CP -odd component. Therefore, for an integrated luminosity of L = 3000 fb −1 , we will have approximately 500 CPV triple Higgs events from h 3 → h 1 h 1 h 1 , which is a smoking gun signature of CP -violation in C2HDMs. The decay modes of h 3 , include h 3 → h 2 Z and h 3 → h 1 h 2 , will gives rise to the final state h 1 ZZ and h 1 h 1 Z,from h 2 → h 1 Z, h 2 → ZZ and h 2 → h 1 h 1 . The event BR m H ±=420 GeV m h2 =280 GeV Z 3 =0.1 10 -1 1 m h 2 [GeV] BR m H ±=420 GeV m h3 =420 GeV Z 3 =0.1200 400 600 800 1000 10 -3 10 -2 10 -1 1 m h 3 [GeV] Re[Z7]=3 θ23=1.23 ϵ=0.1 θ12=0 h 2 h 1 h 2h 2 h 1 h 1 h 2 Z H ± W ∓ W ± W ∓ Z Z tt bb gg 200 250 300 350 10 -3 10 -2 Re[Z7]=3 θ23=1.23 ϵ=0.1 θ12=0 h 1 h 1 h1Z tt bb gg γγ An exception is Ref.[18] which proposed a combination of three different decay channels. is supported by U.S. Department of Energy under Contracts No. DE-SC0021497. I.L. is supported in part by the U.S. Department of Energy under contracts No. DE-AC02-06CH11357 at Argonne and No. DE-SC0010143 at Northwestern. X.P.W. is supported by NSFC under Grant No.12005009. I.L. also acknowledges the hospitality from the National. Marcela We, N.R.S.Kai-Feng Carena, N.R.S.Cheng-Wei Chen, N.R.S.Howie Chiang, N.R.S.George Haber, N.R.S.Shinya Wei-Shu Hou, N.R.S.Kanemura, N.R.S.Center for Theoretical Sciences at National Tsing Hua University and National Taiwan UniversityJia Liu, and Carlos Wagner for useful discussions and comments. in Taiwan where part of this work was performedWe would like to thank Marcela Carena, Kai-Feng Chen, Cheng-Wei Chiang, Howie Haber, George Wei- Shu Hou, Shinya Kanemura, Jia Liu, and Carlos Wag- ner for useful discussions and comments. N.R.S. is sup- ported by U.S. Department of Energy under Contracts No. DE-SC0021497. 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[ "On Static Charged Black Holes in Type IIA on a Nearly-Kähler Coset", "On Static Charged Black Holes in Type IIA on a Nearly-Kähler Coset" ]	[ "Tetsuji Kimura \nKEK Theory Center\nInstitute of Particle and Nuclear Studies\nHigh Energy Accelerator Research Organization Tsukuba\n305-0801IbarakiJapan\n" ]	[ "KEK Theory Center\nInstitute of Particle and Nuclear Studies\nHigh Energy Accelerator Research Organization Tsukuba\n305-0801IbarakiJapan" ]	[]	We study static, spherically symmetric black hole solutions in four-dimensional N = 2 gauged supergravity with one vector multiplet and one hyper-tensor multiplet. This is derived from massive type IIA theory compactified on the nearly-Kähler coset space G 2 /SU (3). It is well-known that the Romans' mass parameter yields the Stückelberg-type deformation of the gauge field strengths in the four-dimensional system. This deformation requires that all the (covariant) derivatives of the scalar fields must vanish and the two-form field is closed. It turns out that charged solutions are forbidden. This implies that only AdS vacua or Schwarzschild-AdS black holes are allowed as the static, spherically symmetric solutions.	10.1143/ptp.128.873	[ "https://arxiv.org/pdf/1203.1544v2.pdf" ]	119,237,440	1203.1544	568eaa0a8ab5826628683b3f4c37e28663b149b0	On Static Charged Black Holes in Type IIA on a Nearly-Kähler Coset 17 Apr 2012 Tetsuji Kimura KEK Theory Center Institute of Particle and Nuclear Studies High Energy Accelerator Research Organization Tsukuba 305-0801IbarakiJapan On Static Charged Black Holes in Type IIA on a Nearly-Kähler Coset 17 Apr 2012 We study static, spherically symmetric black hole solutions in four-dimensional N = 2 gauged supergravity with one vector multiplet and one hyper-tensor multiplet. This is derived from massive type IIA theory compactified on the nearly-Kähler coset space G 2 /SU (3). It is well-known that the Romans' mass parameter yields the Stückelberg-type deformation of the gauge field strengths in the four-dimensional system. This deformation requires that all the (covariant) derivatives of the scalar fields must vanish and the two-form field is closed. It turns out that charged solutions are forbidden. This implies that only AdS vacua or Schwarzschild-AdS black holes are allowed as the static, spherically symmetric solutions. Introduction Searching asymptotically Anti-de Sitter (AdS) black hole solutions in four-dimensional N = 2 gauged supergravity [1] has been developed for a long period. This is quite interesting because the value of the cosmological constant in gauged supergravity is non-trivial; i.e., the cosmological constant is given by the Fayet-Iliopoulos parameters or the expectation value of the scalar potential. In the context of AdS/CFT correspondence, the AdS black hole configuration provides many significant features of condensed matter physics. In pure supergravity without any matter fields, supersymmetric (non)-rotating AdS black holes with (un)usual topology were investigated by [2]. Soon after that, gauged supergravity with vector multiplets was applied to search static AdS black holes with naked singularity [3][4][5]. Recently, supersymmetric static AdS black holes with regular horizon were found [6][7][8]. It is also interesting to find an AdS black hole solution of gauged supergravity in the presence of hypermultiplets [9]. There are two features. One is that the existence of hypermultiplets excludes the Fayet-Iliopoulos parameters, which support non-vanishing cosmological constant in the gauged supergravity only with vector multiplets. The other is that some scalar fields in hypermultiplets appears as two-form fields in the system. Caused by the gauging, they cannot be dualized back to the original scalar fields in the hypermultiplets. The multiplet containing two-form field(s) is referred to as the hyper-tensor multiplet. In addition, these two-form fields deform the gauge field strengths by the Stückelberg-type coupling [10,11]. In order to control the feature of the gauged supergravity with multiplets of the hyper-sector as well as vector multiplets, we study AdS black holes in the framework of the string theory compactification scenarios. Exploiting four-dimensional N = 2 gauged supergravity is of importance because this system appears as the low energy effective theory of type II string theory via flux compactifications [12][13][14][15]. The deformation parameters in gauging of the four-dimensional supergravity are provided by the NSNS-and the RR-flux charge parameters on the internal space. Indeed the Stückelberg-type deformation is realized by the RR flux charges [13][14][15]. In particular, the Romans' mass parameter yields AdS vacua [16]. In this work we focus on the coset spaces G 2 /SU (3) [17,18]. This is one of the nearly-Kähler manifold with torsion and the SU (3)-structure. This is useful because the moduli space of this coset space has been studied very well. In addition, this coset space provides the simplest matter contents in the four-dimensional system; one vector multiplet and one hypertensor multiplet with the Stückelberg-type deformation. We are now ready to search black holes of Reissner-Nordström-AdS type in four-dimensional N = 2 gauged supergravity with one vector multiplet and one hyper-tensor multiplet given by the flux compactification on G 2 /SU (3). Previously the author studied it under the (covariantly) constant condition [9]. This condition prohibits the existence of charged solutions. In this paper, we assume that the configuration is static and spherically symmetric. Surprisingly the static condition eventually derives the (covariantly) constant condition [9]. This implies that the static condition itself forbids Reissner-Nordström-AdS black hole solutions of the gauged supergravity in the presence of the two-form field with the Stückelberg-type deformation [10,11,[13][14][15]. The organization of this paper is as follows. In section 2 we briefly exhibit the feature of the coset space G 2 /SU (3). Next we write down the equations of motion for the gauge fields, the twoform field and the gravitational field which play a central role in the main analysis. Third we restrict the system to the static configuration. Here we introduce two functions in order to describe the static field strengths. In section 3 we show that all the fields are (covariantly) constant in the static configuration: The equations of motion for the gauge fields intertwine the two-form field with the two functions. The equation for the two-form field provides the differential equations among the two functions and the scalar fields. Finally, the Einstein equation reveals that all the scalar fields are (covariantly) constant because each term quadratic in the (covariant) derivatives of the scalar fields is positive semi-definite. We also find that the two-form field is closed. The (covariantly) constant condition only allows neutral solutions such as AdS vacua or Schwarzschild-AdS black holes. In section 4 is devoted to conclusion. Gauged supergravity with two-form field The deformation parameters in gauging of the four-dimensional supergravity are provided by the NSNS-flux charge parameters {e ΛI , e Λ I , m Λ I , m ΛI } and the RR-flux charge parameters {e RΛ , m Λ R } on the internal space [13][14][15]. The ranges of the labels Λ and I are Λ = 0, 1, . . . , n V and I = 0, 1, . . . , n H , where n V denotes the number of the vector multiplets, whilst n H indicates the number of multiplets in the hyper-sector, respectively. The Romans' mass parameter is involved as m 0 R in the above flux charge parameters. In this section we briefly exhibit the feature of N = 2 abelian gauged supergravity with B-field derived from type IIA compactification on the nearly-Kähler coset space G 2 /SU (3). The details of the derivation can be seen in [14,15]. Profile from coset space G 2 /SU(3) First of all let us consider the generic feature of the gauged supergravity associated with the type IIA compactification on G 2 /SU (3) [17,18]. The indices Λ and I run only Λ = 0, 1 and I = 0, respectively. The following flux charge parameters involve the profile of this compactification: e 10 = 2 √ 3I , m Λ R e Λ0 = 0 , m 0 R = 0 , e R0 = 0 ,(2.1) whilst other flux charges such as e Λ 0 , e 00 , m 1 R and e R1 , are zero. m 0 R is interpreted as the Romans' mass parameter. The value I denotes the volume of the coset space. In this compactification, the moduli space of the vector multiplet is given by SU (1, 1)/U (1). This is governed by the cubic prepotential F(X): F ≡ I X 1 X 1 X 1 X 0 . (2.2a) In terms of the local coordinate t ≡ X 1 /X 0 , we describe the Kähler potential K V : K V ≡ − log i(X Λ F Λ − X Λ F Λ ) = − log − iI(t − t) 3 . (2.2b) It is quite useful to introduce the period matrix N ΛΣ on this moduli space: N ΛΣ ≡ F ΛΣ + 2i (ImF) ΛΓ X Γ (ImF) Σ∆ X ∆ X Π (ImF) ΠΞ X Ξ = I 2 t 2 (t + 3t) −3t(t + t) −3t(t + t) 3(3t + t) . (2.2c) This pertains to the generalization of the gauge coupling constant and the theta-angle in N = 2 supersymmetric system. In the hyper-sector, there exists only one multiplet. Notice that the non-vanishing m 0 R makes the axion field a in the hypermultiplet be dualized to the two-form field B µν , referred to as the B-field. We call the multiplet with the constituents {ϕ, ξ 0 , ξ 0 , B µν } the hyper-tensor multiplet. Equations of motion Here we exhibit the equations of motion for the gauge fields A Λ µ , the B-field B µν , and the gravitational field g µν , which we will utilize exhaustively in the next section 1 : 0 = ǫ σµνρ 2 √ −g ∂ µ F Λνρ − ǫ σµνρ 2 √ −g ∂ µ B νρ (e RΛ − e Λ0 ξ 0 ) − e 2ϕ e Λ0 D σ ξ 0 , (2.3a) 0 = 1 √ −g ∂ µ e −4ϕ √ −gH µρσ + ǫ µνρσ √ −g D µ ξ 0 D ν ξ 0 − D µ ξ 0 D ν ξ 0 + (e RΛ − e Λ0 ξ 0 )F Λ µν + 2m Λ R µ ΛΣ F Σρσ − ǫ µνρσ √ −g m Λ R ν ΛΣ F Σ µν , (2.3b) E µν ≡ R µν − 1 2 R g µν = 1 4 g µν µ ΛΣ F Λ ρσ F Σρσ − µ ΛΣ F Λ µρ F Σ νσ g ρσ − g µν g tt ∂ ρ t∂ ρ t + 2g tt ∂ µ t∂ ν t − g µν ∂ ρ ϕ∂ ρ ϕ + 2∂ µ ϕ∂ ν ϕ − e −4ϕ 24 g µν H ρσλ H ρσλ + e −4ϕ 4 H µρσ H ν ρσ − e 2ϕ 2 g µν D ρ ξ 0 D ρ ξ 0 + D ρ ξ 0 D ρ ξ 0 + e 2ϕ D µ ξ 0 D ν ξ 0 + D µ ξ 0 D ν ξ 0 − g µν V . (2.3c) Here we have introduced various functions: µ ΛΣ = ImN ΛΣ and ν ΛΣ = ReN ΛΣ are the generalization of the gauge coupling constant, and the field dependent theta-angle, respectively; g tt is the Kähler metric defined by g tt = ∂ t ∂ t K V ; and E µν is the Einstein tensor. The field strength of the B-field is given as H µνρ = 3∂ [µ B νρ] . Notice that, due to the presence of the RR-flux charges m Λ R , the gauge field strengths are deformed to [14] F Λ µν = 2∂ [µ A Λ ν] + m Λ R B µν . (2.4) The covariant derivatives of the scalar fields ξ 0 and ξ 0 are given by D µ ξ 0 = ∂ µ ξ 0 , D µ ξ 0 = ∂ µ ξ 0 − e Λ0 A Λ µ . (2.5) They are derived from abelian gauge symmetry of the RR potentials in the ten-dimensional type IIA theory, and from the geometrical structure of the six-dimensional internal space [14]. Due to the absence of the flux charges e Λ 0 on the coset space G 2 /SU (3), the covariant derivative Dξ 0 is reduced to the ordinary derivative [18]. The dual gauge field strengths F Λµν are F Λµν ≡ ν ΛΣ F Σ µν + √ −g 2 ǫ µνρσ µ ΛΣ F Σρσ , (2.6) where we used the constant tensor whose normalization is ǫ 0123 = +1, and its contravariant form ǫ 0123 = −1 in a generic curved spacetime. The scalar potential V is given as [15] V = g tt D t P + D t P + + g tt D t P 3 D t P 3 − 2\|P + \| 2 + \|P 3 \| 2 . (2.7a) Due to the absence of the flux charges e Λ 0 , the scalar field ξ 0 does not contribute to the scalar potential V . The triplet of the Killing prepotentials P a [15] is explicitly described as P + = −2e ϕ L 1 e 10 , P − = −2e ϕ L 1 e 10 , P 3 = e 2ϕ L 0 e R0 − L 1 e 10 ξ 0 − M 0 m 0 R , (2.7b) L 0 = e K V /2 , L 1 = t e K V /2 , M 0 = −I t 3 e K V /2 , (2.7c) D t P a = ∂ t + 1 2 ∂ t K V P a (2.7d) Static setup So far we exhibited generic feature of the gauged supergravity. Here we prepare the static, spherically symmetric metric: ds 2 = −e 2A(r) dt 2 + e −2A(r) dr 2 + e 2C(r) r 2 dθ 2 + sin 2 θ dφ 2 . (2.8a) We impose the time-independent condition on an arbitrary field X such as 0 = ∂ t X . (2.8b) The electric and magnetic charges are defined in terms of the field strengths: p Λ ≡ 1 4π dθdφ F Λ θφ , (2.9a) q Λ ≡ 1 4π dθdφ F Λθφ = 1 4π dθdφ ν ΛΣ F Σ θφ + √ −g µ ΛΣ F Σtr . (2.9b) Since we concentrate on the static configuration of the electric field and the magnetic fields, we consider the following set of relations between the gauge fields and the B-field: F Λ θφ = ∂ θ A Λ φ − ∂ φ A Λ θ + m Λ R B θφ ≡ f Λ (θ, φ) sin θ , (2.10a) F Λ tr = −∂ r A Λ t + m Λ R B tr ≡ e −2C r 2 g Λ (θ, φ) , (2.10b) for the non-vanishing components of the field strengths, and 0 = ∂ r F Λ θφ , 0 = ∂ r F Λθφ , (2.10c) 0 = F Λ φr = ∂ φ A Λ r − ∂ r A Λ φ + m Λ R B φr ,(2.10d) 0 = F Λ rθ = ∂ r A Λ θ − ∂ θ A Λ r + m Λ R B rθ , (2.10e) 0 = F Λ tθ = −∂ θ A Λ t + m Λ R B tθ , (2.10f) 0 = F Λ tφ = −∂ φ A Λ t + m Λ R B tφ , (2.10g) for the vanishing components, respectively. Note that the fields are independent of time coordinate because we consider the static configuration (2.8b). Analysis In this section we carefully analyze the equations of motion ( Equation of motion for gauge fields First we evaluate the equation of motion for the gauge fields (2.3a). Each component provides a powerful constraint among the fields and the functions: (2.3a) σ=t : 0 = e −2C r 2 sin θ (e RΛ − e Λ0 ξ 0 )H rθφ + e Λ0 e 2ϕ−2A D t ξ 0 , (3.1a) (2.3a) σ=r : 0 = −(e RΛ − e Λ0 ξ 0 ) e −2C r 2 sin θ H θφt − e Λ0 e 2ϕ+2A D r ξ 0 , (3.1b) (2.3a) σ=θ : 0 = − e −2C r 2 sin θ ∂ φ F Λtr − (e RΛ − e Λ0 ξ 0 )H φtr + e Λ0 e 2ϕ D θ ξ 0 sin θ , (3.1c) (2.3a) σ=φ : 0 = e −2C r 2 sin θ ∂ θ F Λtr − (e RΛ − e Λ0 ξ 0 )H θtr − e Λ0 e 2ϕ sin θ D φ ξ 0 . (3.1d) Multiplying m Λ R to the above equations and using the identity m Λ R e Λ0 = 0 due to the flux compactification [14], we extract the following forms: 0 = m Λ R e RΛ H rθφ , (3.2a) 0 = m Λ R e RΛ H θφt , (3.2b) m Λ R ∂ φ F Λtr = m Λ R e RΛ H φtr , (3.2c) m Λ R ∂ θ F Λtr = m Λ R e RΛ H θtr . (3.2d) Furthermore, the expressions (2.10) enable us to rewrite the components of the three-form H µνρ in terms of the functions f Λ (θ, φ) and g Λ (θ, φ): m Λ R e RΛ H φtr = m Λ R e RΛ ∂ φ B tr + ∂ r B φt = e RΛ ∂ φ ∂ r A Λ t + e −2C r 2 g Λ (θ, φ) + ∂ r − ∂ φ A Λ t = e −2C r 2 ∂ φ e RΛ g Λ (θ, φ) , (3.3a) m Λ R e RΛ H θtr = m Λ R e RΛ ∂ r B θt + ∂ θ B tr = e RΛ ∂ r − ∂ θ A Λ t + ∂ θ ∂ r A Λ t + e −2C r 2 g Λ (θ, φ) = e −2C r 2 ∂ θ e RΛ g Λ (θ, φ) . (3.3b) Substituting them into (3.2), we obtain the explicit forms: H φtr = 1 m Σ R e RΣ e −2C r 2 ∂ φ e RΛ g Λ (θ, φ) , (3.4a) H θtr = 1 m Σ R e RΣ e −2C r 2 ∂ θ e RΛ g Λ (θ, φ) , (3.4b) 0 = ∂ φ (m Λ R µ ΛΣ )f Σ (θ, φ) + m Λ R ν ΛΣ − e RΣ g Σ (θ, φ) , (3.4c) 0 = ∂ θ (m Λ R µ ΛΣ )f Σ (θ, φ) + m Λ R ν ΛΣ − e RΣ g Σ (θ, φ) . (3.4d) Under the above description, the equations of motion (3.1) are further reduced to 0 = D t ξ 0 = −e Λ0 A Λ t , (3.5a) 0 = D r ξ 0 = ∂ r ξ 0 − e Λ0 A Λ r , (3.5b) 0 = e −2C r 2 ∂ φ µ ΛΣ f Σ (θ, φ) + ν ΛΣ − e RΛ − e Λ0 ξ 0 m Γ R e RΓ e RΣ g Σ (θ, φ) + e Λ0 e 2ϕ sin θ D θ ξ 0 , (3.5c) 0 = e −2C r 2 ∂ θ µ ΛΣ f Σ (θ, φ) + ν ΛΣ − e RΛ − e Λ0 ξ 0 m Γ R e RΓ e RΣ g Σ (θ, φ) − e Λ0 e 2ϕ sin θ D φ ξ 0 . (3.5d) The equation (3.5a) with the flux charge condition (2.1) imposes a strong condition on g Λ (θ, φ): e Λ0 F Λ tr = −∂ r (e Λ0 A Λ t ) = 0 , ∴ e Λ0 g Λ (θ, φ) = 0 . (3.6) Notice that g 0 (θ, φ) still remains non-trivial. Equation of motion for B-field Next task is to investigate the equation of motion for the B-field (2.3b). Owing to the expressions (2.10) and (3.4), each component of the equation is described in terms of the functions f Λ (θ, φ) and g Λ (θ, φ) in the following way: (2.3b) [ρσ]=[tr] : 0 = − 1 m Σ R e RΣ e −2C r 2 ∂ θ e −4ϕ sin θ ∂ θ e RΛ g Λ (θ, φ) − 1 m Σ R e RΣ e −2C r 2 sin θ ∂ φ e −4ϕ ∂ φ e RΛ g Λ (θ, φ) + 2 m Λ R ν ΛΣ − (e RΣ − e Σ0 ξ 0 ) f Σ (θ, φ) − (m Λ R µ ΛΣ )g Σ (θ, φ) sin θ − 2 ∂ θ ξ 0 D φ ξ 0 − D θ ξ 0 ∂ φ ξ 0 , (3.7a) (2.3b) [ρσ]=[tθ] : 0 = sin θ m Σ R e RΣ ∂ θ e RΛ g Λ (θ, φ) ∂ r e −4ϕ−2C r 2 + 2∂ r ξ 0 D φ ξ 0 , (3.7b) (2.3b) [ρσ]=[tφ] : 0 = 1 m Σ R e RΣ 1 sin θ ∂ φ e RΛ g Λ (θ, φ) ∂ r e −4ϕ−2C r 2 − 2∂ r ξ 0 D θ ξ 0 , (3.7c) (2.3b) [ρσ]=[θφ] : 0 = 2e −4C r 4 sin θ (m Λ R µ ΛΣ )f Σ (θ, φ) + (m Λ R ν ΛΣ − e RΣ )g Σ (θ, φ) , (3.7d) where we used (3.6). Then the equations (3.4c) and (3.4d) become trivial: 0 = (m Λ R µ ΛΣ )f Σ (θ, φ) + (m Λ R ν ΛΣ − e RΣ )g Σ (θ, φ) . (3.8) We can learn that the covariant derivatives of the RR-axion fields are related to the derivatives of the function g Λ (θ, φ). Indeed the equations (3.7b) and (3.7c) will contribute to the evaluation of the Einstein equation in a crucial way. Einstein equation So far we analyzed the equations of motion in terms of arbitrary functions A(r) and C(r) in the static metric (3.9). From now on, we focus only on the metric of Reissner-Nordström-AdS type: e 2A(r) ≡ κ − 2η r + Z 2 r 2 + r 2 ℓ 2 , e 2C(r) ≡ 1 , (3.9) where the black hole parameters of mass and charges are given by η and Z 2 = Q 2 e +Q 2 m , respectively. The parameter ℓ gives the negative cosmological constant Λ c.c. = −3/ℓ 2 . The diagonal components of the Einstein tensor E µν under the metric (3.9) become simple: g tt E tt = e 2A 1 r 2 (1 − e −2(A+C) ) + 2 r (A ′ + 3C ′ ) + C ′ (2A ′ + 3C ′ ) + 2C ′′ = − Z 2 r 4 + 3 ℓ 2 , (3.10a) g rr E rr = e 2A 1 r 2 (1 − e −2(A+C) ) + 2 r (A ′ + C ′ ) + C ′ (2A ′ + C ′ ) = − Z 2 r 4 + 3 ℓ 2 , (3.10b) g θθ E θθ = e 2A 2 r (A ′ + C ′ ) + 2(A ′ ) 2 + C ′ (2A ′ + C ′ ) + A ′′ + C ′′ = Z 2 r 4 + 3 ℓ 2 , (3.10c) g φφ E φφ = e 2A 2 r (A ′ + C ′ ) + 2(A ′ ) 2 + C ′ (2A ′ + C ′ ) + A ′′ + C ′′ = Z 2 r 4 + 3 ℓ 2 . (3.10d) On the other hand, substituting the vanishing conditions (2.10) and (3.2), we see the reduced components of the right-hand side of the Einstein equation (2.3c): g tt E tt = − 1 2 µ ΛΣ F Λ tr F Σtr + 1 2 µ ΛΣ F Λ θφ F Σθφ − g tt ∂ r t∂ r t + ∂ θ t∂ θ t + ∂ φ t∂ φ t − ∂ r ϕ∂ r ϕ + ∂ θ ϕ∂ θ ϕ + ∂ φ ϕ∂ φ ϕ + e −4ϕ 4 H trθ H trθ + e −4ϕ 4 H trφ H trφ − e 2ϕ 2 ∂ r ξ 0 ∂ r ξ 0 + ∂ θ ξ 0 ∂ θ ξ 0 + ∂ φ ξ 0 ∂ φ ξ 0 − e 2ϕ 2 D θ ξ 0 D θ ξ 0 + D φ ξ 0 D φ ξ 0 − V , (3.11a) g rr E rr = − 1 2 µ ΛΣ F Λ tr F Σtr + 1 2 µ ΛΣ F Λ θφ F Σθφ − g tt − ∂ r t∂ r t + ∂ θ t∂ θ t + ∂ φ t∂ φ t − − ∂ r ϕ∂ r ϕ + ∂ θ ϕ∂ θ ϕ + ∂ φ ϕ∂ φ ϕ + e −4ϕ 4 H trθ H trθ + e −4ϕ 4 H trφ H trφ − e 2ϕ 2 − ∂ r ξ 0 ∂ r ξ 0 + ∂ θ ξ 0 ∂ θ ξ 0 + ∂ φ ξ 0 ∂ φ ξ 0 − e 2ϕ 2 D θ ξ 0 D θ ξ 0 + D φ ξ 0 D φ ξ 0 − V , (3.11b) g θθ E θθ = 1 2 µ ΛΣ F Λ tr F Σtr − 1 2 µ ΛΣ F Λ θφ F Σθφ − g tt ∂ r t∂ r t − ∂ θ t∂ θ t + ∂ φ t∂ φ t − ∂ r ϕ∂ r ϕ − ∂ θ ϕ∂ θ ϕ + ∂ φ ϕ∂ φ ϕ + e −4ϕ 4 H trθ H trθ − e −4ϕ 4 H trφ H trφ − e 2ϕ 2 ∂ r ξ 0 ∂ r ξ 0 − ∂ θ ξ 0 ∂ θ ξ 0 + ∂ φ ξ 0 ∂ φ ξ 0 − e 2ϕ 2 − D θ ξ 0 D θ ξ 0 + D φ ξ 0 D φ ξ 0 − V , (3.11c) g φφ E φφ = 1 2 µ ΛΣ F Λ tr F Σtr − 1 2 µ ΛΣ F Λ θφ F Σθφ − g tt ∂ r t∂ r t + ∂ θ t∂ θ t − ∂ φ t∂ φ t − ∂ r ϕ∂ r ϕ + ∂ θ ϕ∂ θ ϕ − ∂ φ ϕ∂ φ ϕ − e −4ϕ 4 H trθ H trθ + e −4ϕ 4 H trφ H trφ − e 2ϕ 2 ∂ r ξ 0 ∂ r ξ 0 + ∂ θ ξ 0 ∂ θ ξ 0 − ∂ φ ξ 0 ∂ φ ξ 0 − e 2ϕ 2 D θ ξ 0 D θ ξ 0 − D φ ξ 0 D φ ξ 0 − V . (3.11d) Combining (3.10) with (3.11), we evaluate the derivatives of the fields. First, the difference between the time component and the radial component gives g tt E tt − g rr E rr = 0 = −2e 2A(r) g tt \|∂ r t\| 2 + (∂ r ϕ) 2 + e 2ϕ 2 (∂ r ξ 0 ) 2 . (3.12) This is a strong condition. Because each term in the right-hand side is positive semi-definite, we immediately obtain 0 = ∂ r t , 0 = ∂ r ϕ , 0 = ∂ r ξ 0 ,(3.13) This makes the derivative of the function g Λ (θ, φ) vanish through the equations (3.7b) and (3.7c): 0 = ∂ θ e RΛ g Λ (θ, φ) , 0 = ∂ φ e RΛ g Λ (θ, φ) , which implies that g Λ is a constant and the components of the three-form H in (3.4) vanish: g Λ = (constant) , 0 = H trθ = H trφ . (3.14) Combining (3.2), we find that all the components of H µνρ vanish once the static condition is imposed on the system. In a similar way, we also evaluate the following three equations: g rr E rr + g θθ E θθ = 6 ℓ 2 = − 2 r 2 sin 2 θ g tt \|∂ φ t\| 2 + (∂ φ ϕ) 2 + e 2ϕ 2 (∂ φ ξ 0 ) 2 + e 2ϕ 2 (D φ ξ 0 ) 2 − 2V , (3.15a) g rr E rr − g θθ E θθ = − 2Z 2 r 4 = 1 r 4 µ ΛΣ f Λ (θ, φ)f Σ (θ, φ) + g Λ g Σ − 2 r 2 g tt \|∂ θ t\| 2 + (∂ θ ϕ) 2 + e 2ϕ 2 (∂ θ ξ 0 ) 2 + e 2ϕ 2 (D θ ξ 0 ) 2 , (3.15b) g θθ E θθ − g φφ E φφ = 0 = 1 r 2 g tt \|∂ θ t\| 2 + (∂ θ ϕ) 2 + e 2ϕ 2 (∂ θ ξ 0 ) 2 + e 2ϕ 2 (D θ ξ 0 ) 2 − 1 r 2 sin 2 θ g tt \|∂ φ t\| 2 + (∂ φ ϕ) 2 + e 2ϕ 2 (∂ φ ξ 0 ) 2 + e 2ϕ 2 (D φ ξ 0 ) 2 . (3.15c) Note that we have used the vanishing condition of all the components of H µνρ . Since the scalar fields t, ϕ and ξ 0 are independent of the radial coordinate r (3.13), the period matrix N ΛΣ = ν ΛΣ + iµ ΛΣ (2.2c) are also independent of r. The scalar potential V defined in (2.7a) does not depend on r, either. Then the square bracket in the right-hand side of (3.15a) and the second line in the right-hand side of (3.15b) must vanish: 0 = g tt \|∂ θ t\| 2 + (∂ θ ϕ) 2 + e 2ϕ 2 (∂ θ ξ 0 ) 2 + e 2ϕ 2 (D θ ξ 0 ) 2 = g tt \|∂ φ t\| 2 + (∂ φ ϕ) 2 + e 2ϕ 2 (∂ φ ξ 0 ) 2 + e 2ϕ 2 (D φ ξ 0 ) 2 . (3.16) This condition satisfies (3.15c) consistently. Because each term in this equation is positive semidefinite, we obtain 0 = ∂ θ t = ∂ φ t , 0 = ∂ θ ϕ = ∂ φ ϕ , (3.17a) 0 = ∂ θ ξ 0 = ∂ φ ξ 0 , 0 = D θ ξ 0 = D φ ξ 0 . (3.17b) Due to the constant condition of g Λ and t, the equation (3.8) tells us that f Λ becomes constant. Their values are described in terms of the electric and magnetic charges through (2.9): f Λ = p Λ , g Λ = −(µ −1 ) ΛΣ q Σ − ν ΣΓ p Γ . (3.18) Let us summarize the analysis. Assuming only the static configuration (2.8b), (2.8b), and (2.10), we obtain the (covariantly) constant forms of the scalar fields and the constant gauge field strengths as showed in (3.2), (3.5), (3.13), (3.14), (3.17) and (3.18): 0 = H µνρ = 3∂ [µ B νρ] , 0 = D µ ξ 0 , 0 = D µ ξ 0 = ∂ µ ξ 0 , 0 = ∂ µ t , (3.19a) F Λ θφ = p Λ sin θ , F Λ tr = − 1 r 2 (µ −1 ) ΛΣ q Σ − ν ΣΓ p Γ . (3.19b) This leads to the black hole parameters in such a way as Z 2 = − µ ΛΣ 2 f Λ f Σ + g Λ g Σ = − 1 2 p Λ µ ΛΣ p Σ + q Λ − ν ΛΓ p Γ (µ −1 ) ΛΣ q Σ − ν Σ∆ p ∆ , (3.20a) Λ c.c. = − 3 ℓ 2 = V . (3.20b) Constant solution Let us proceed the analysis under the field configuration (3.19) 2 . The covariantly constant condition 0 = D µ ξ 0 in (3.19) gives the vanishing condition of the field strengths: 0 = [∂ µ , ∂ ν ] ξ 0 = e Λ0 F Λ µν ,(3.21) i.e., F 1 µν = 0. Here we again used the flux charge condition m Λ R e Λ0 = 0 in (2.1). The above condition implies 0 = p 1 , 0 = (µ −1 ) 10 q 0 + (µ −1 ) 11 q 1 − (µ −1 ν) 1 0 p 0 . (3.22) Substituting this into (3.7a), we obtain p 0 = m 0 R e R0 q 0 , (3.23a) (µ −1 ) 11 q 1 = −q 0 (µ −1 ) 10 − m 0 R e R0 (µ −1 ν) 1 0 , (3.23b) 0 = q 0 (µ −1 ) 11 (µ −1 ) 00 − [(µ −1 ) 01 ] 2 − 2 m 0 R e R0 (µ −1 ) 11 (µ −1 ν) 0 0 − (µ −1 ) 01 (µ −1 ν) 1 0 + m 0 R e R0 2 (µ −1 ) 11 (µ + νµ −1 ν) 00 − [(µ −1 ν) 1 0 ] 2 . (3.23c) Since the value in the square bracket of (3.23c) is non-zero [9], the electric charge q 0 must vanish. Substituting this into the above equations, we eventually find that all the charges must be zero: q 0 = 0 , q 1 = 0 , p 0 = 0 , p 1 = 0 . The gauge field strengths given in (3.19) also vanish. This leads to the vanishing black hole charges Z 2 = 0. Then we conclude that the static, spherically symmetric configuration in the gauged supergravity derived from type IIA theory compactified on G 2 /SU (3) provides only neutral solutions such as AdS vacua or Schwarzschild-AdS black holes [9]. Conclusion In this work we studied static, spherically symmetric, asymptotically AdS black hole solutions in four-dimensional N = 2 gauged supergravity in the presence of one vector multiplet and one hyper-tensor multiplet. This system is associated with the type IIA theory compactified on the nearly-Kähler coset space G 2 /SU (3). The Romans' mass yields the Stückelberg-type deformation in the gauge field strengths. Then we found an intrinsic relation between the gauge fields A Λ µ and the B-field. Eventually all the scalar fields should satisfy the (covariantly) constant condition, and the B-field must be closed. Furthermore, the (covariantly) constant condition leads to the vanishing black hole charges. It turns out that only the possible solutions are AdS vacua or Schwarzschild-AdS black hole as analyzed in [9]. In the main analysis we fixed the sign of the cosmological constant to be negative in (3.9). But this did not affect the evaluation of the scalar fields in the Einstein equation (3.15a). In addition, the topology of the horizon is not crucial in the equations (3.15b) and (3.15c). The primary reason was that the H µνρ did vanish through the equations (3.7) and (3.12). This eventually removed the dependence of the B-field in the equations (3.15), and the independence of the angular coordinates (3.17) were realized. Thus the result (3.19) could be applied to static, asymptotically flat (or de Sitter) black holes with unusual topology such as two-torus or hyperbolic surface (see, for instance, [2]). 2.3) under the static setup (2.8). Since the B-field is incorporated into the equation of motion and the gauge field strengths (2.4), we can find a series of restrictions on the B-field via the equation of motion for the gauge fields. The equation of motion for the B-field further gives rise to differential equations among the set of functions {f Λ (θ, φ), g Λ (θ, φ), C(r)}, and the scalar fields {ϕ, ξ 0 , ξ 0 } of the hyper-tensor multiplet. Utilizing these restrictions, we evaluate the Einstein equation in the static, spherically symmetric (AdS) black hole spacetime. Recombining the components of the Einstein equation, we finally obtain the (covariantly) constant condition. See[9] for the Lagrangian and the equations of motion for other bosonic fields. The essential points have already been discussed in[9]. This work reveals that, as far as we concern the metric (2.8a) with (3.9), time-independent configurations must be forbidden to build a charged (AdS) black hole in the presence of the hypertensor multiplet with the Stückelberg-type deformation. It seems to be inevitable to search timedependent configurations. A typical one is the stationary, rotating charged black hole referred to as the Kerr-Newman-(AdS) black hole. N = 2 supergravity and N = 2 super Yang-Mills theory on general scalar manifolds: Symplectic covariance, gaugings and the momentum map. L Andrianopoli, M Bertolini, A Ceresole, R Auria, S Ferrara, P Fré, T Magri, hep-th/9605032J. Geom. Phys. 23111L. Andrianopoli, M. Bertolini, A. Ceresole, R. D'Auria, S. Ferrara, P. Fré and T. Magri, "N = 2 supergravity and N = 2 super Yang-Mills theory on general scalar manifolds: Symplectic co- variance, gaugings and the momentum map," J. Geom. Phys. 23 (1997) 111 [hep-th/9605032]. Supersymmetry of Anti-de Sitter black holes. 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Kimura, "Schwarzschild-AdS black holes in N = 2 geometric flux compactification," arXiv:1108.1113 [hep-th]. D = 4, N = 2 gauged supergravity in the presence of tensor multiplets. G Agata, R Auria, L Sommovigo, S Vaulà, hep-th/0312210Nucl. Phys. B. 682243G. Dall'Agata, R. D'Auria, L. Sommovigo and S. Vaulà, "D = 4, N = 2 gauged supergravity in the presence of tensor multiplets," Nucl. Phys. B 682 (2004) 243 [hep-th/0312210]. N = 2 supergravity Lagrangian coupled to tensor multiplets with electric and magnetic fluxes. R Auria, L Sommovigo, S Vaulà, hep-th/0409097JHEP. 041128R. D'Auria, L. Sommovigo and S. Vaulà, "N = 2 supergravity Lagrangian coupled to tensor multiplets with electric and magnetic fluxes," JHEP 0411 (2004) 028 [hep-th/0409097]. Flux compactifications in string theory: A Comprehensive review. M Graña, hep-th/0509003Phys. Rept. 42391M. Graña, "Flux compactifications in string theory: A Comprehensive review," Phys. Rept. 423 (2006) 91 [hep-th/0509003]. 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[]	[ "Carlos Luiz \nInstituto de Física\nUniversidade Federal do Rio de Janeiro\nCaixa Postal 6852821945-970Rio de JaneiroRJBrazil\n", "Ryff [email protected] \nInstituto de Física\nUniversidade Federal do Rio de Janeiro\nCaixa Postal 6852821945-970Rio de JaneiroRJBrazil\n" ]	[ "Instituto de Física\nUniversidade Federal do Rio de Janeiro\nCaixa Postal 6852821945-970Rio de JaneiroRJBrazil", "Instituto de Física\nUniversidade Federal do Rio de Janeiro\nCaixa Postal 6852821945-970Rio de JaneiroRJBrazil" ]	[]	An Einstein-Podolsky-Rosen (EPR)-like argument using events separated by a time-like interval strongly suggestes that measuring the polarization state of a photon of an entangled pair changes the polarization state of the other distant photon. Trough a very simple demonstration, the Wigner-D'Espagnat inequality is used to show that in order to prove Bell's theorem neither the assumption that there is a well-defined space of complete states λ of the particle pair and well-defined probability distribution ρ(λ) over this space nor the use of counterfactuals is necessary. These results reinforce the viewpoint that quantum mechanics implicitly presupposes some sort of nonlocal connection between the particles of an entangled pair. As will become evident from our discussion, relinquishing realism and/or free will cannot solve this apparent puzzle.PACS numbers: 03.65.Ta., 03.65.Ud.PACS numbers:	null	[ "https://arxiv.org/pdf/quant-ph/0510101v2.pdf" ]	15,263,321	quant-ph/0510101	02d5edce3a52b21d937e5efda77ea23ee72e8c95	Carlos Luiz Instituto de Física Universidade Federal do Rio de Janeiro Caixa Postal 6852821945-970Rio de JaneiroRJBrazil Ryff [email protected] Instituto de Física Universidade Federal do Rio de Janeiro Caixa Postal 6852821945-970Rio de JaneiroRJBrazil arXiv:quant-ph/0510101v2 4 Apr 2006 Bell's Theorem Reexamined An Einstein-Podolsky-Rosen (EPR)-like argument using events separated by a time-like interval strongly suggestes that measuring the polarization state of a photon of an entangled pair changes the polarization state of the other distant photon. Trough a very simple demonstration, the Wigner-D'Espagnat inequality is used to show that in order to prove Bell's theorem neither the assumption that there is a well-defined space of complete states λ of the particle pair and well-defined probability distribution ρ(λ) over this space nor the use of counterfactuals is necessary. These results reinforce the viewpoint that quantum mechanics implicitly presupposes some sort of nonlocal connection between the particles of an entangled pair. As will become evident from our discussion, relinquishing realism and/or free will cannot solve this apparent puzzle.PACS numbers: 03.65.Ta., 03.65.Ud.PACS numbers: I. INTRODUCTION As emphasized by Schrödinger, entanglement is the characteristic trait of quantum mechanics. 1 Einstein, Podolsky, and Rosen (EPR) used entangled states to try to prove that this theory is incomplete, 2 and Bell made things clearer by showing that no local hidden variable theory can mimic quantum mechanics. 3 Apparently, measuring the state of a particle can instantaneously change the state of another particle that can be arbitrarily distant from the first. But no superluminal telegraph can be devised using this phenomenon. The correlations become evident only when the results, gathered at two different spatial regions, are compared. In principle, one observer cannot know what kind of experiment the other is performing. That is, there is no detectable contradiction with special relativity. Bell's theorem has been extended to real situations, 4 proofs have been introduced that do not rely on inequalities, 5 longdistance experimental tests of entanglement have been performed, 6 and the use of entangled particles for cryptographic purposes has been proposed. 7 However, the mystery remains, 8 and even conflicting points of view on the conceptual significance of Bell's theorem have been presented. 9 Here I will advocate the viewpoint that quantum mechanics is an intrinsically nonlocal theory, that is, it implicitly presupposes that measuring the state of a distant particle of an entangled pair can instantaneously affect the other's state. II. THE WEIRDNESS OF QUANTUM ENTANGLEMENT Let us consider the following situation: a source S emits pairs of entangled photons, ν 1 and ν 2 , in the state \|ψ = 1 √ 2 (\|a, k \|a, −k + \|a ⊥ , k \|a ⊥ , −k ) = 1 √ 2 (\|b, k \|b, −k + \|b ⊥ , k \|b ⊥ , −k ) = 1 √ 2 (\|c, k \|c, −k + \|c ⊥ , k \|c ⊥ , −k ) = ...,(1) where \|a, k (\|a ⊥ , −k ) represents a photon with polarization parallel (perpendicular) to a following direction k (−k), and so on. From (1) we see that quantum mechanical formalism does not allow us to assign definite polarization states to the photons. This can only be done when a polarization measurement is performed. For example, if photon ν 1 (ν 2 ) is detected in state \|a , then we immediately know that the other photon of the pair, ν 2 (ν 1 ), has been "forced" into the same state \|a . We also see that (1) does not allow us to predict the outcome of the measurement-state \| a or \| a ⊥ , for example. This strongly suggests that quantum mechanics implicitly presupposes some sort of superluminalactually, infinite-speed-interaction: measuring the state of one of the two particles instantaneously changes the state of the other. According to Bell, "in these EPR experiments there is the suggestion that behind the scenes something is going faster than light," and Bohm declared: "I would be quite ready to relinquish locality; I think it is an arbitrary assumption." 10 An important point to be emphasized is that the correlations become evident when one reads the results that have been automatically registered. The observer's consciousness does not seem to play any role in the process. If this were not so, mind states of distant observers would have to communicate to reproduce the quantum correlations. Although the discussion has been centered on the nonlocal aspects of quantum mechanics, the important question is knowing if a measurement performed on one of the photons of an entangled pair can change the state of the other. To examine this problem, it is preferable to consider time-like events. Let us imagine that the path followed by ν 2 is modified so that the first observer, Alice (A), after measuring the state of ν 1 , can inform the second observer, Bob (B), about her result before he detects ν 2 . Naturally, assuming that the detection of ν 1 cannot change the polarization state of ν 2 , it is irrelevant whether we consider space-like or time-like events. On the other hand, if there is some sort of nonlocal connection between the photons, it may be more illuminating to discuss situations in which there is no doubt about which one has been detected first. An aleatory sequence of photons in states \|a and \|a ⊥ is indistinguishable from another aleatory sequence of photons in states \|b and \|b ⊥ , or \|c and \|c ⊥ , and so on. Therefore, the detection of photons ν 2 provides us no information about the orientation of the polarizer on which photons ν 1 are impinging. That is, without using a classical communication channel, A cannot use entangled states to send information to B. But time-like events allow us to try to clarify the following question: What does it really mean when we say-in agreement with (1)-that measuring the state of the first photon forces the second into a well-defined polarization state? When we are dealing with space-like events, it is not possible to assign an objective and well-defined polarization state to ν 2 just before it impinges on the polarizer, since the question "Which photon was really detected first?" is meaningless. On the other hand, in the case of time-like events, A can send a message informing about the state of ν 2 , and B can then perform a measurement to check if the information is correct. It seems that EPR's criterion-"If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity." 2 -is perfectly valid here. Therefore, depending on the orientation of the polarizer on which ν 1 impinges, ν 2 will reach its polarizer in two possible states: \|a and \|a ⊥ , or \|b and \|b ⊥ , or \|c and \|c ⊥ , etc. However, assuming that the detection of ν 1 has no influence on the polarization state of ν 2 , there is no reason for ν 2 to be found only in the states \|a and \|a ⊥ , for instance. On the other hand, if there is an influence, it still must be present when space-like events are considered, since the very same correlations can be observed. From this point of view, there must be a nonlocal connection between ν 1 and ν 2 , unless we assume that a photon can somehow be in different polarization states simultaneously. We can also consider the following reasoning: If the outcome of the first measurement is aleatory, but that of the second becomes well determined (after we know the result of the first), then, considering the symmetry of the situation, either (a) the first measurement was not aleatory or (b) there must be some connection between ν 1 and ν 2 . In other words, if quantum mechanics is complete, it must be nonlocal. III. BELL'S THEOREM The above notions, however, clash with the spirit of relativity, and we may conjecture that some still unknown local theory exists that perfectly mimics quantum theory. Bell's theorem shows that this cannot be so. As a consequence, it seems that our conceptions of space and time must be revised; specifically, the concept of distance does not seem to be valid for a system of entangled particles. As has been emphasized, one way to demonstrate Bell's inequality is to assume "that there is a well-defined space of complete states [λ] of the particle pair and a well-defined probability distribution [ρ(λ)] over this space when an experimental procedure for specifying an ensemble of pairs is given." 11 Although this may sound like a reasonable assumption, it can be considered an unnecessarily restrictive one, and arguments for dispensing with it have been presented, but they have used counterfactuals, thereby raising some serious criticisms. 9,11 Counterfactual reasoning is based on the assumption that we could have performed another measurement on one of the particles of an entangled pair (instead of the measurement actually performed, for example) without changing the outcome of the measurement performed on the other, distant particle (locality assumption). But it can be argued that in this case we would be in another Universe, which invalidates the counterfactual-local reasoning. In particular, assuming that the present is rigidly determined by the past, two physical events, even separated by a space-like interval, can be interconnected in such a way that we cannot change one without changing the other. The introduction of free will, although very reasonable for some, only makes the argument more disputable. Many different versions of Bell's theorem have been presented, but they either introduce ρ(λ) or use counterfactuals. Therefore, there seems to be no compelling theoretical evidence that entanglement implies that quantum mechanics is inherently nonlocal. However, as we will see, to derive Bell's theorem we do not need to assume that there is a well-defined ρ (λ) or, alternatively, that counterfactual reasoning is valid. IV. THE WIGNER-D'ESPAGNAT INEQUALITY Our purpose is to answer the question: can a local theory mimic the predictions of quantum mechanics? In other words, we are interested in situations represented by (1). This expression shows us, in an ideal situation, what correlations must be observed if the polarizations are measured. That is, we are considering "latent" probabilities, so to speak. Actually, there may be no polarizers, that is no experiment to determine the polarization of the photons. What we want to know is whether a local theory may have states with the same latency as the states represented by (1). Initially, let us imagine a situation in which two-channel polarizers I and II, on which ν 1 and ν 2 impinge, respectively, have the same orientation. According to (1), whenever ν 1 is transmitted (reflected), ν 2 must also be transmitted (reflected). Therefore, assuming locality (i.e., what happens to ν 1 cannot affect ν 2 , and vice versa), whether ν 1 and ν 2 will be transmitted or reflected is already determined before they impinge on the polarizers; otherwise, we could have a situation in which one photon is transmitted and the other is reflected. In other words, perfect correlations and locality imply a strong form of determinism. Assuming that the source has no information about the orientations of the polarizers, each photon pair has to be emitted with "instructions," so to speak, for all possible orientations. For instance, transmission in case of orientation a, reflection in case of orientation b, and so on. Or, put another way, the outcome of a potential experiment is determined by the photon's hidden variable state and the polarizer orientation, and nothing else. As we will see, it is impossible to mimic the predictions of quantum mechanics in this case. It is important to mention that in 1982 Itamar Pitowsky published a paper in which the EPR-Bell "paradox" was supposedly solved. 12 His point was that the derivation of Bell's inequality was valid only when a well-defined ρ(λ) could be introduced, which was not the case for his model. But, as shown by Alan Macdonald,13 in the particular case of the Pitowsky model, there is another and very simple way to obtain these inequalities. Actually, a similar derivation had already been presented by Bernard D'Espagnat in his article on quantum theory and realism in Scientific American, 14 and before that by Wigner. 15 It is perhaps the simplest and most satisfactory proof of Bell's inequality, but seldom presented or mentioned; probably because it is only valid for perfect correlations. Although it is already very simple, it still can be simplified even further, as we will see. Let us assume that N 0 pairs of photons are emitted. Now let N (a, b, c) [N (a ⊥ , b, c)] represent the number of photon pairs in which the photons are "prepared" to be transmitted when impinging on a polarizer oriented parallel to b or c, and to be transmitted [reflected] if oriented parallel to a. Then, we must have N (a, b, c) + N (a ⊥ , b, c) = N (b, c) ,(2) where N (b, c) represents the number of photon pairs prepared to be transmitted when impinging on a polarizer oriented parallel to b or c. We also must have N (a, c) ≥ N (a, b, c)(3) and N (a ⊥ , b) ≥ N (a ⊥ , b, c).(4) From (2), (3) and (4) we obtain N (a, c) + N (a ⊥ , b) ≥ N (b, c),(5) which is the Wigner-D'Espagnat inequality. 16 According to quantum mechanics, we must have N (a, c) = (N 0 /2) cos 2 (a, c), N (a ⊥ , b) = (N 0 /2) sin 2 (a, b), and N (b, c) = (N 0 /2) cos 2 (b, c). Thus, choosing (a, b) = (b, c) = 30 0 , and (a, c) = 60 0 , we obtain 0.5 ≥ 0.75, violating inequality (5). V. DISCUSSION As our discussion based on time-like events and entangled particles has made evident, quantum mechanics is intrinsically a nonlocal theory, and this conclusion is equally valid for space-like events. Measuring the polarization state of one of the photons of an entangled state, represented by (1), instantaneously forces the other, distant photon into a well-defined polarization state. Paradoxical as it may sound, this seems to be true independently of the Lorentz frame used to describe the events (it is important to remember, however, that special relativity is not necessarily incompatible with the idea of a preferred frame). 17 In other words, if quantum mechanics is complete, it must be nonlocal. Since this seems to go against the spirit of special relativity, it is important to investigate the possibility of quantum mechanics being only a manifestation of a deeper local theory. As we have seen, perfect correlations, together with locality, leads to determinism, and determinism leads to the violation of a Bell inequality. Many derivations of Bell's inequalities are based on the assumption that there is a well-defined space of complete states λ of the particle pair and a well-defined probability distribution ρ(λ) over this space. Since the violation of these inequalities might be considered only one proof that this assumption is false (we might have an infinite and non denumerable number of hidden variables, for example), it is important to show that it is unnecessary. As has been emphasized, no counterfactual definiteness needs to be introduced for this; we only need determinism. Although counterfactual definiteness presupposes determinism, the converse is not necessarily true. Strictly speaking, some kinds of determinism-as in the present paper, for example, in which the outcome of the experiment depends only on the photon hidden variable state and the orientation of the polarizer-may imply a sort of virtual contrafactualness, valid in the realm of imagined experiments, but as a consequence and not as a basic assumption. Actually, counterfactual definiteness was introduced in an attempt to avoid the use of hidden variable states. 18 A delicate point related to Bell's inequalities involves realism. It was implicit in our assumption of hidden variable states. It is evident that if realism is abandoned, it becomes difficult to explain the predicted correlations assuming locality. Therefore, abandoning realism does not solve the EPR puzzle. Of course, solipsism is a logical alternative but very unsatisfactory as a predictive tool and difficult to maintain consistently in real life. It seems that physics has little to contribute to this longstanding philosophical debate. Another delicate point is related to the use of a free-will assumption to derive Bell's inequalities. According to Bell, "In the analysis [of EPR experiments] it is assumed that free will is genuine, and as a result of that one finds that the intervention of the experimenter at one point has to have consequences at a remote point, in a way that influences restricted by the finite velocity of light would not permit. If the experimenter is not free to make this intervention, if that is also determined in advance, the difficulty disappears." 19 As has become evident from our discussion, abandoning free will-which plays no role in our demonstration-is not a solution to EPR paradox. In conclusion: We have reexamined Bell's theorem using a different approach and trying to answer the question: Can a local theory mimic the predictions of quantum mechanics? We have assumed that no information about the orientation of the polarizers is contained in the state of the emitted pair of photons. In other words, Nature is governed by physical laws; nothing that might sound like a sort of Big Conspiracy exists. Actually, there may be no polarizers, that is, no experiment to determine the polarization of the photons. We only know what correlations must be observed if the polarizations are measured. This is in agreement with expression (1), which only expresses potentialities. Strictly speaking, we are not discussing whether Nature is nonlocal or not-although experimental evidence strongly suggests it is 20 -but whether quantum mechanics is nonlocal or not. In our demonstration, neither the assumption that there is a well-defined space of complete states of the particle pair and a well-defined probability distribution over this space nor the use of counterfactuals is needed. 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[ "Simulating Transmission Scenarios of the Delta Variant of SARS-CoV-2 in Australia", "Simulating Transmission Scenarios of the Delta Variant of SARS-CoV-2 in Australia", "Infectious ; Diseases -Surveillance, Prevention and Treatment, a section of the journal Frontiers in Public Health" ]	[ "Teruya Maki ", "Mikhail Prokopenko [email protected] ", "Sheryl L Chang \nFaculty of Engineering\nCentre for Complex Systems\nThe University of Sydney\nSydneyNSW, Australia\n", "Oliver M Cliff \nFaculty of Engineering\nCentre for Complex Systems\nThe University of Sydney\nSydneyNSW, Australia\n\nSchool of Physics\nThe University of Sydney\nSydneyNSWAustralia\n", "Cameron Zachreson \nFaculty of Engineering\nCentre for Complex Systems\nThe University of Sydney\nSydneyNSW, Australia\n\nSchool of Computing and Information Systems\nThe University of Melbourne\nParkvilleVICAustralia\n", "Mikhail Prokopenko \nFaculty of Engineering\nCentre for Complex Systems\nThe University of Sydney\nSydneyNSW, Australia\n\nSydney Institute for Infectious Diseases\nThe University of Sydney\nWestmeadNSWAustralia\n", "\nKindai University\nJapan\n" ]	[ "Faculty of Engineering\nCentre for Complex Systems\nThe University of Sydney\nSydneyNSW, Australia", "Faculty of Engineering\nCentre for Complex Systems\nThe University of Sydney\nSydneyNSW, Australia", "School of Physics\nThe University of Sydney\nSydneyNSWAustralia", "Faculty of Engineering\nCentre for Complex Systems\nThe University of Sydney\nSydneyNSW, Australia", "School of Computing and Information Systems\nThe University of Melbourne\nParkvilleVICAustralia", "Faculty of Engineering\nCentre for Complex Systems\nThe University of Sydney\nSydneyNSW, Australia", "Sydney Institute for Infectious Diseases\nThe University of Sydney\nWestmeadNSWAustralia", "Kindai University\nJapan" ]	[ "Frontiers in Public Health \| www.frontiersin.org" ]	An outbreak of the Delta (B.1.617.2) variant of SARS-CoV-2 that began around mid-June 2021 in Sydney, Australia, quickly developed into a nation-wide epidemic. The ongoing epidemic is of major concern as the Delta variant is more infectious than previous variants that circulated in Australia in 2020. Using a re-calibrated agent-based model, we explored a feasible range of non-pharmaceutical interventions, including case isolation, home quarantine, school closures, and stay-at-home restrictions (i.e., "social distancing.") Our modelling indicated that the levels of reduced interactions in workplaces and across communities attained in Sydney and other parts of the nation were inadequate for controlling the outbreak. A counter-factual analysis suggested that if 70% of the population followed tight stay-at-home restrictions, then at least 45 days would have been needed for new daily cases to fall from their peak to below ten per day. Our model predicted that, under a progressive vaccination rollout, if 40-50% of the Australian population follow stay-at-home restrictions, the incidence will peak by mid-October 2021: the peak in incidence across the nation was indeed observed in mid-October. We also quantified an expected burden on the healthcare system and potential fatalities across Australia.ACKNOWLEDGMENTSWe are thankful for support provided by High-Performance Computing (HPC) service (Artemis) at the University of Sydney.	10.3389/fpubh.2022.823043	[ "https://arxiv.org/pdf/2107.06617v5.pdf" ]	244,709,411	2107.06617	1a09537fa06fb48b7a0cce84fe2182cc773ac755	Simulating Transmission Scenarios of the Delta Variant of SARS-CoV-2 in Australia 1 February 2022 Teruya Maki Mikhail Prokopenko [email protected] Sheryl L Chang Faculty of Engineering Centre for Complex Systems The University of Sydney SydneyNSW, Australia Oliver M Cliff Faculty of Engineering Centre for Complex Systems The University of Sydney SydneyNSW, Australia School of Physics The University of Sydney SydneyNSWAustralia Cameron Zachreson Faculty of Engineering Centre for Complex Systems The University of Sydney SydneyNSW, Australia School of Computing and Information Systems The University of Melbourne ParkvilleVICAustralia Mikhail Prokopenko Faculty of Engineering Centre for Complex Systems The University of Sydney SydneyNSW, Australia Sydney Institute for Infectious Diseases The University of Sydney WestmeadNSWAustralia Kindai University Japan Simulating Transmission Scenarios of the Delta Variant of SARS-CoV-2 in Australia Infectious ; Diseases -Surveillance, Prevention and Treatment, a section of the journal Frontiers in Public Health Frontiers in Public Health \| www.frontiersin.org 108230431 February 202210.3389/fpubh.2022.823043Specialty section: This article was submitted to Received: 26 November 2021 Accepted: 20 January 2022ORIGINAL RESEARCH Edited by: Reviewed by: Satoshi Mitarai, Japan Anti-Tuberculosis Association, Japan *Correspondence: Citation: Chang SL, Cliff OM, Zachreson C and Prokopenko M (2022) Simulating Transmission Scenarios of the Delta Variant of SARS-CoV-2 in Australia. Front. Public Health 10:823043.COVID-19SARS-CoV-2Delta (B16172) variantcomputational epidemiologyagent-based modelsocial distancingvaccinationhealthcare burden An outbreak of the Delta (B.1.617.2) variant of SARS-CoV-2 that began around mid-June 2021 in Sydney, Australia, quickly developed into a nation-wide epidemic. The ongoing epidemic is of major concern as the Delta variant is more infectious than previous variants that circulated in Australia in 2020. Using a re-calibrated agent-based model, we explored a feasible range of non-pharmaceutical interventions, including case isolation, home quarantine, school closures, and stay-at-home restrictions (i.e., "social distancing.") Our modelling indicated that the levels of reduced interactions in workplaces and across communities attained in Sydney and other parts of the nation were inadequate for controlling the outbreak. A counter-factual analysis suggested that if 70% of the population followed tight stay-at-home restrictions, then at least 45 days would have been needed for new daily cases to fall from their peak to below ten per day. Our model predicted that, under a progressive vaccination rollout, if 40-50% of the Australian population follow stay-at-home restrictions, the incidence will peak by mid-October 2021: the peak in incidence across the nation was indeed observed in mid-October. We also quantified an expected burden on the healthcare system and potential fatalities across Australia.ACKNOWLEDGMENTSWe are thankful for support provided by High-Performance Computing (HPC) service (Artemis) at the University of Sydney. INTRODUCTION Strict mitigation and suppression measures eliminated local transmission of SARS-CoV-2 during the initial pandemic wave in Australia (March-June 2020; peaked around 500 cases per day, i.e., around 20 daily cases per million) (1), as well as a second wave that developed in the southeastern state of Victoria (June-September 2020; peaked around 700 cases per day, i.e., around 30 daily cases per million) (2, 3) 1 . Several subsequent outbreaks were also detected and managed quickly and efficiently by contact tracing and local lockdowns, e.g., a cluster in the Northern Beaches Council of Sydney, New South Wales (NSW) totalled 217 cases and was controlled in 32 days by locking down only the immediately affected suburbs (December 2020-January 2021) (5). Overall, successful pandemic response was facilitated by effective travel restrictions and stringent stay-at-home restrictions (i.e., "social distancing, ") underpinned by a high-intensity disease surveillance (6)(7)(8)(9)(10). Unfortunately, the situation changed in mid-June 2021, when a highly transmissible variant of concern, B.1.617.2 (Delta), was detected. The first infection was recorded on June 16 in Sydney, and quickly spread through the Greater Sydney area. Within ten days, there were more than 100 locally acquired cumulative cases, triggering stay-at-home (social distancing) restrictions imposed in Greater Sydney and nearby areas (11). By July 9 (23 days later), the locally acquired cases reached 439 in total (5), and a tighter lockdown was announced (11). Further restrictions and business shut-downs, including construction and retail industries, were announced on 17 July (12). By then, the risk of a prolonged lockdown had become apparent (13), with the epidemic spreading to the other states and territories, most notably Victoria (VIC) and the Australian Capital Territory (ACT). The incidence peaked, around 2,750 daily cases, i.e., around 100 daily cases per million, only by mid-October 2021, and stabilised in November within the range between 1,200 and 1,600 daily cases, i.e., between 45 and 65 daily cases per million (5), before a new surge of infections in December 2021 due to the Omicron variant (B.1.1.529). The difficulty of controlling the third epidemic wave (June-November 2021) is attributed to a high transmissibility of the B.1.617.2 (Delta) variant, which is known to increase the risk of household transmission by approximately 60% in comparison to the B.1.1.7 (Alpha) variant (14). This transmissibility was compounded by the initially low rate of vaccination in Australia, with around 6% of the adult population double vaccinated before the Sydney outbreak and only 7.92% of adult Australians double vaccinated by the end of June 2021 (15), with this fraction increasing to 67.24% by 15 October 2021 and 83.01% by 13 November 2021 (16). Several additional factors make the Sydney outbreak and the third pandemic wave in Australia (June-November 2021) an important case study, in which the system complexity and the search space formed by possible interventions can be reduced. Because previous pandemic waves were eliminated in Australia, the Delta variant has not been competing with other variants. Secondly, the level of acquired immunity to SARS-CoV-2 in the Australian population was low at the onset of the outbreak, given that (a) the pre-existing natural immunity was limited by cumulative confirmed cases of around 0.12%, and (b) immunity acquired due to vaccination did not extend beyond 6% of the adult population. Furthermore, the school winter break in NSW (28 June-9 July) coincided with the period of social distancing restrictions announced on 26 June, with school premises remaining mostly closed beyond 9 July. Thus, the epidemic suppression policy of school closures is not a free variable, further reducing the search space of available control measures. This study addresses several important questions. Firstly, we investigate a feasible range of key non-pharmaceutical interventions (NPIs): case isolation, home quarantine, school closures, and social distancing, available to control virus transmission within a population with a low immunity. Social distancing (SD) is interpreted and modelled in a broad sense of comprehensive stay-at-home restrictions, comprising several specific behavioural changes that reduce the intensity of interactions among individuals (and hence the virus transmission probability), including physical distancing, mobility reduction, mask wearing, and so on. Our primary focus is a "retrodictive" estimation of the average (unknown) SD level under which the modelled transmission and suppression dynamics can be best matched to the observed incidence data. An identification of the SD level helps to distinguish and evaluate the distinct and time-varying impacts of NPIs and vaccination campaigns. Secondly, in a counter-factual mode, we quantify under what conditions the initial outbreak could have been suppressed, aiming to clarify the extent of required NPIs during an early outbreak phase with low vaccination coverage, in comparison to previous pandemic control measures successfully deployed in Australia. This analysis highlights the challenges associated with imposing very tight restrictions which would be required to suppress the high transmissible Delta variant. Finally, we offer and validate a projection for the peak of case incidence across the nation, formed in response to a progressive vaccination campaign rolling out concurrently with the strict lockdown measures adopted in NSW, VIC, and ACT. In doing so, we predict the expected hospitalisations, intensive care unit (ICU) demand, and potential fatalities across Australia. Importantly, this analysis shows that a 10% increase in the average SD level reduces the clinical burden approximately 3fold, and the potential fatalities approximately 2-fold. METHODS We utilised an agent-based model (ABM) for transmission and control of COVID-19 in Australia that has been developed in our previous work (1,17) and implemented within a large-scale software simulator (AMTraC-19). The model was cross-validated with genomic surveillance data (6), and contributed to policy recommendations on social distancing that were broadly adopted by the World Health Organisation (18). The model separately simulates each individual as an agent within a surrogate population composed of about 23.4 million software agents. These agents are stochastically generated to match attributes of anonymous individuals (in terms of age, residence, gender, workplace, susceptibility, and immunity to diseases), informed by data from the Australian Census and the Australian Curriculum, Assessment and Reporting Authority. In addition, the simulation follows the known commuting patterns between the places of residence and work/study (19)(20)(21). Different contact rates specified within diverse social contexts (e.g., households, neighbourhoods, communities, and work/study environments) explicitly represent heterogeneous demographic and epidemic conditions (see Supplementary Material: Agent-based model). The model has previously been calibrated to produce characteristics of the COVID-19 pandemic corresponding to the ancestral lineage of SARS-CoV-2 (1, 17), using actual case data from the first and second waves in Australia, and re-calibrated for B.1.617.2 (Delta) variant using incidence data of the Sydney outbreak (see Supplementary Material: Model calibration). Each epidemic scenario is simulated by updating agents' states in discrete time. In this work, we start from an initial distribution of infection, seeded by imported cases generated by the incoming international air traffic in Sydney's international airport (using data from the Australian Bureau of Infrastructure, Transport, and Regional Economics) (19,20). At each time step during the seeding phase, this process probabilistically generates new infections within a 50 km radius of the airport (covering the area within Greater Sydney's boundaries), in proportion to the average daily number of incoming passengers (using a binomial distribution and data from the Australian Bureau of Infrastructure, Transport, and Regional Economics) (19). A specific outbreak, originated in proximity to the airport, is traced over time by simulating the agents interactions within their social contexts, computed in 12-h cycles ("day" and "night.") Once the outbreak size (cumulative incidence) exceeds a pre-defined threshold (e.g., 20 detected cases), the travel restrictions (TR) are imposed by the scenario, so that the rest of infections are driven by purely local transmissions, while no more overseas acquired cases are allowed (presumed to be in effective quarantine). Case-targeted nonpharmaceutical interventions (CTNPIs), such as case isolation (CI) and home quarantine (HQ), are applied from the outset. A scenario develops under some partial mass-vaccination coverage, implemented as either a progressive rollout, or a limited prepandemic coverage, as described in Supplementary Material: Vaccination modelling. The outbreak-growth phase can then be interrupted by another, "suppression, " threshold (e.g., 100 or 400 cumulative detected cases) which triggers a set of general NPIs, such as social distancing (SD) and school closures (SC). Every intervention is specified via a macro-distancing level of compliance (i.e., SD = 0.8 means 80% of agents are socially distancing), and a set of micro-distancing parameters (quantifying context-specific interaction strengths, e.g., moderate or tight restrictions) that indicate the level of social distancing within a specific social context (households, communities, workplaces, etc.). For instance, for those agents that are compliant, contacts (and thus likelihood of infection) can be reduced during a lockdown to SD w = 0.1 within workplaces and SD c = 0.25 within communities, whilst maintaining contacts SD h = 1.0 within households. To re-iterate, "social distancing" modelled in this study comprises a range of restrictions that reduce the intensity of interactions among individuals, including mask wearing, physical distancing by several metres, mobility, and so on. We do not estimate a relative importance of these specific NPI approaches, each of which separately contributes to reducing SARS-CoV-2 transmission (22)(23)(24)(25)(26)(27), focusing instead on a differentiation between the effects of NPIs and vaccination campaigns. RESULTS Using the ABM calibrated to the Delta (B.1.617.2) variant, we varied the macro-and micro-parameters (for CI, HQ, SC, and SD), aiming to match the incidence data recorded during the Sydney outbreak in a retrodiction mode. As shown in Figure 1, the modelling horizon was set to July 25 and assumed a progressive vaccination rollout in addition to a tighter lockdown being imposed at 400 cases (corresponding to July 9). Construction works were temporarily paused across Greater Sydney during 19-30 July 2021 (inclusive), with the temporary "construction ban" lifted on 28 July (28,29). Within the considered timeline, the actual incidence growth rate has reduced from β I = 0.098 (17 June -13 July), to β II = 0.076 (17 June -25 July), to β III = 0.037 (16-25 July), as detailed in Supplementary Material: Growth rates. The closest match to the actual incidence data over the entire period was produced by a moderate macro-level of social distancing compliance, SD = 0.5, or even a lower level (SD = 0.4) for the period up to 13 July (see Figure 1 and Supplementary Material: Sensitivity of outcomes for moderate restrictions, Supplementary Figure 2; also see section 4 for a comparison of these SD levels with real-world mobility reductions). The match is not exact-with the actual incidence growth rate changing several times during this period-perhaps as a consequence of restrictions being imposed heterogeneously across different local government areas. Importantly, however, the growth in actual incidence during the period of the comprehensive lockdown restrictions (16-25 July) is best matched by a higher compliance level, SD = 0.6. This match is also reflected by proximity of the corresponding growth rate β 0.6 = 0.029 to the incidence growth rate β III = 0.037. The considered SD levels were based on moderately reduced interaction strengths within community, i.e., SD c = 0.25, see Table 1, which were inadequate for outbreak suppression even with high macro-distancing such as SD = 0.7. Furthermore, we considered moderate-to-high macro-levels of social distancing, 0.5 ≤ SD ≤ 0.9, while maintaining CI = 0.7 and HQ = 0.5, in a counter-factual mode by reducing the micro-parameters (the interaction strengths for CI, HQ, SC, and SD) within their feasible bounds. Again, the control measures were triggered by cumulative incidence exceeding 400 cases (corresponding to a tighter lockdown imposed on July 9). An effective suppression of the outbreak within a reasonable timeframe is demonstrated for macro-distancing at SD ≥ 0.7, coupled with the lowest feasible interaction strengths for most interventions, i.e., NPI c = 0.1 (where NPI is one of CI, HQ, SC, and SD), as shown in Figure 2 and summarised in Table 1. For SD = 0.8, new cases fall below 10 per day approximately a month (33 days) after the peak in incidence, while for SD = 0.7 this period reaches 45 days 2 . Social distancing at SD = 0.9 is probably infeasible (as this assumes that 90% of the population consistently stays at home), but would reduce the new cases to below 10 a day within four weeks (25 days) following the peak in incidence. 25 July for (A) (log-scale) incidence (crosses), and (B) cumulative incidence (circles); with an exponential fit of the incidence's moving average (black solid: β II , and black dotted: β III ). Vertical dashed marks align the simulated days with the outbreak start (17 June, day 9), initial restrictions (27 June,day 19), and tighter lockdown (9 July, day 31). Traces corresponding to each social distancing (SD) compliance level are shown as average over 10 runs (coloured profiles for SD varying in increments of 10%, i.e., between SD = 0.0 and SD = 1.0). 95% confidence intervals for incidence profiles, for SD ∈ {0.4, 0.5, 0.6}, are shown as shaded areas. Each SD intervention, coupled with school closures, begins with the start of tighter lockdown, when cumulative incidence exceeds 400 cases (B: inset). The alignment between simulated days and actual dates may slightly differ across separate runs. Case isolation and home quarantine are in place from the outset. Supplementary Material (Sensitivity of suppression outcomes for tight restrictions) presents results obtained for the scenarios which assume a limited pre-pandemic vaccination coverage (immunising 6% of the population). A positive impact of the partial progressive rollout which covers up to 40% of the population by mid-September is counterbalanced by a delayed start of the tighter lockdown, with the 12-day delay leading to a higher peak-incidence, as can be seen by comparing Figure 2 and Supplementary Figure 4. For example, for SD = 0.8, a scenario following the limited pre-pandemic vaccination, but imposing control measures earlier, demonstrates a reduction of incidence below 10 daily cases in four weeks after the peak in incidence (Supplementary Figure 4), rather than 33 days under progressive rollout (Figure 2). For SD = 0.9 the suppression periods differ by about one week: 17 days (Supplementary Figure 4) against 25 days (Figure 2). However, this balance is nonlinear, as shown in Table 2: for SD = 0.7, the suppression period under the pre-pandemic vaccination scenario approaches 55 days (Supplementary Figure 4), in contrast to the progressive rollout scenario achieving suppression earlier, in 45 days (Figure 2). This is, of course, explained by the longer suppression period under SD = 0.7, during which a progressive rollout makes a stronger impact. We then considered feasible scenarios tracing the epidemic spread at the national level for the period between mid-June and mid- . Traces corresponding to feasible social distancing (SD) compliance levels are shown as average over 10 runs (coloured profiles for SD varying in increments of 10%, i.e., between SD = 0.5 and SD = 0.9). Vertical lines mark the incidence peaks (dotted) and reductions below 10 daily cases (dashed). Each SD intervention, coupled with school closures, begins with the start of tighter lockdown, when cumulative incidence exceeds 400 cases (i.e., simulated day 31). The alignment between simulated days and actual dates may slightly differ across separate runs. Case isolation and home quarantine are in place from the outset. symptomatic children, σ c = 0.268 (see Supplementary Material: Model calibration). The actual incidence curve is traced between the profiles formed by SD = 0.4 and SD = 0.5, with the latter providing the best match. The model projection for incidence peaking across the nation in the range between approximately 1,500 and 5,000 daily cases pointed to early to mid-October. This projection is validated by the actual profiles, as shown in Supplementary Figures 8-10, and summarised in Table 3 and Supplementary Tables 9, 10. The scenario developing under SD = 0.5 offers the best match with the actual dynamics again. As expected, the unvaccinated cases form a vast majority among the hospitalisations, ICU occupancy and fatalities (cf. Tables 9, 10). Importantly, a comparison across the three moderate levels of social distancing, SD ∈ {0.4, 0.5, 0.6} shows that with a 10% increase in the level of social distancing, the hospitalisations and ICU demand reduce approximately 3fold, and the fatalities reduce at least two times. These effects of a 10% increase in the social distancing adherence on the clinical burden and the potential fatalities are robust with respect to changes in the vaccine efficacy against infectiousness, as shown in Supplementary Figure 12, and Tables 9, 10. Supplementary DISCUSSION Despite a relatively high computational cost, and the need to calibrate numerous internal parameters, ABMs capture the natural history of infectious diseases in a good agreement with the established estimates of incubation periods, serial/generation intervals, and other key epidemiological variables. Various ABMs have been successfully used for simulating actual and counter-factual epidemic scenarios based on different initial conditions and intervention policies (30)(31)(32)(33)(34). Our early COVID-19 study (1) modelled transmission of the ancestral lineage of SARS-CoV-2 characterised by the basic reproduction number of R 0 ≈ 3.0 (adjusted R 0 ≈ 2.75). This study compared several NPIs and identified the minimal SD levels required to control the first wave in Australia. Specifically, a compliance at the 90% level, i.e., SD = 0.9 (with SD w = 0 and SD c = 0.5) was shown to control the disease within 13-14 weeks. This relatively high SD compliance was required in addition to other restrictions (TR, CI, HQ), set at moderate levels of both macro-distancing (CI = 0.7 and HQ = 0.5), and interaction strengths: CI w = HQ w = CI c = HQ c = 0.25, CI h = 1.0, and HQ h = 2.0 (1). The follow-up work (17) quantified possible effects of a mass-vaccination campaign in Australia, by varying the extents of pre-pandemic vaccination coverage with different vaccine efficacy combinations. This analysis considered hybrid vaccination scenarios using two vaccines adopted in Australia: BNT162b2 (Pfizer/BioNTech) and ChAdOx1 nCoV-19 (Oxford/AstraZeneca). Herd immunity was shown to be out of reach even when a large proportion (82%) of the Australian population is vaccinated under the hybrid approach, necessitating future partial NPIs for up to 40% of the population. The model was also calibrated to the basic reproduction number of the ancestral lineage (R 0 ≈ 3.0, adjusted R 0 ≈ 2.75), and used the same moderate interaction strengths as the initial study (1) (except SD c = 0.25, reduced to match the second wave in Melbourne in 2020). In this work, we re-calibrated the ABM to incidence data from the ongoing third pandemic wave in Australia driven by the Delta variant. The reproductive number was calibrated to be at least twice as high (R 0 = 5.97) as the one previously estimated for pandemic waves in Australia. We then explored effects of available NPIs on the outbreak suppression, under a progressive vaccination scenario. The retrodictive modelling identified that the current epidemic curves, which continued to grow (until mid-October 2021), can be closely matched by moderate social distancing coupled with moderate interaction strengths within community (SD in [0.4, 0.5], SD c = 0.25), as well as moderate compliance with case isolation (CI = 0.7, CI w = CI c = 0.25) and home quarantine (HQ = 0.5, HQ w = HQ c = 0.25). The estimate of compliance has briefly improved to SD ≈ 0.6 during the period of comprehensive lockdown measures, announced on July 17, but returned to SD ≈ 0.5 in early August. We note that the workers delivering essential services are exempt from lockdown restrictions. The fraction of the exempt population can be inferred conservatively as 4% (strictly essential) (35), more comprehensively as approximately 19% (including health care and social assistance; public administration and safety; accommodation and food services; transport, postal and warehousing; electricity, gas, water and waste services; financial and insurance services), but can reach more significant levels, around 33%, if all construction, manufacturing, and trade (retail/wholesale) are included in addition (36). The latter, broad-range, case limits feasible social distancing levels to approximately SD ≈ 0.7. However, even with these inclusions, there is a discrepancy between the level estimated by ABM (SD in [0.4, 0.5]) and the broad-range feasible level (SD ≈ 0.7). This discrepancy would imply that approximately 20-25% of the population have not been consistently complying with the imposed restrictions, while 30-35% may have been engaged in services deemed broadly essential (other splits comprising 50-60% of the "non-distancing" population are possible as well). The inferred levels of social distancing are supported by realworld mobility data (37). Specifically, when compared to baseline (i.e., the median value for the corresponding day of the week, during the 5-week period 3 January-6 February 2020, as set by data provider to represent the pre-pandemic levels), the reports for July 16 showed 31% reduction of mobility at workplaces, and 37% reduction of mobility in retail and recreation settings, with concurrent 65% reduction of mobility on public transport. On July 21, the mobility reductions were reported as 43% (workplaces), 41% (retail and recreation), and 72% (public transport). The extent of the mobility reduction in workplaces, as well as retail and recreation, closely matched the social distancing levels estimated by the model (approximately 40%). The partial reductions in mobility across workplaces, retail, and recreation have since been maintained around 40-50% on average (37). According to numerous reports (38)(39)(40), the infection spread among essential workers was substantial, and the interactions within workplaces and community contributed to the disease transmission stronger than contacts in public transport. Moderate levels of compliance (SD in [0.4, 0.6]) would be inadequate for suppression of even less transmissible coronavirus variants (1). The Delta variant demands a stronger compliance and a reduction in the scope of essential services (especially, in a setting with low immunity). Specifically, our results indicate that an effective suppression within a reasonable timeframe can be demonstrated only for very high compliance with social distancing (SD ≥ 0.7), supported by dramatically reduced, and practically infeasible, interaction strengths within the community and work/study environments (NPI c = NPI w = 0.1). Importantly, a significant fraction of local transmissions during the Sydney outbreak in NSW, as well as during the following outbreak in Melbourne, VIC [which started on 13 July 2021, was initially suppressed, but then resumed its growth on 4 August 2021 (5)], occurred in the suburbs characterised by socioeconomic disadvantage profiles, as defined by The Australian Bureau of Statistics' Index of Relative Socio-economic Advantage and Disadvantage (IRSAD) (38,39,41). To a large extent, the epidemic spread in these suburbs was driven by structural factors, such as higher concentrations of essential workers, high-density housing, shared and multi-generational households, etc. Thus, even a combination of government actions (e.g., a temporary inclusion of some services previously deemed essential under the lockdown restrictions (28,29), while providing appropriate financial support to the affected businesses and employees), and a moderate community engagement with the suppression effort, proved to be insufficient for the outbreaks' suppression. Obviously, the challenges of suppressing emerging variants of concern can be alleviated by a growing vaccination uptake. However, in Australia, the vaccination rollout was initially limited by various supply and logistics constraints. Furthermore, as our results demonstrate, a progressive vaccination rollout reaching up to 40% of the population (i.e., approximately 50% of adults) was counter-balanced by a delayed introduction of the tighter control measures. This balance indicated that a comprehensive mass-vaccination rollout plays a crucial role over a longer term and should preferably be carried out in a preoutbreak phase (17). Ultimately, the epidemic peak in NSW during the lockdown period was reached only when about a half of the adults were double vaccinated by mid-September (i.e., 49.6% on 15 September 2021) (16). Across the nation, the peak in incidence was observed by mid-October (as predicted by the model), once approximately two thirds of adults were double vaccinated (16), also in concordance with the model (see Supplementary Material: Vaccination modelling). A post-lockdown increase in infections is expected when the stay-at-home orders are lifted in recognition of immunising 70%, and then 80%, of adults (42). However, a detailed analysis of a possible post-lockdown surge in infections, the resultant increased demand on the healthcare system, and potential fatalities, is outside of the scope for this study. While the model was not directly used to inform policy, it forms part of the information set available to health departments, and we hope that its policy relevance can contribute to rapid and comprehensive responses in jurisdictions within Australia and overseas. A failure in reducing the size of the initial outbreak, due to a delayed vaccination rollout, challenging socioeconomic profiles of the primarily affected areas, inadequate population compliance, and a desire to maintain and restart socioeconomic activities, has generated a substantial pandemic wave affecting the entire nation (43-45). Study Limitations In modelling the progressive vaccination rollout, we assumed a constant weekly uptake rate of 3%, while the rollout was accelerating. The rate of progressive vaccination is expected to vary, being influenced by numerous factors, such as access to national stockpiles, dynamics of social behaviour, and changing medical advice. In addition, we did not consider a diminishing vaccine efficacy, given that the temporal scope of the study was limited to a relatively short period of 6 months (June-November 2021) during which a progressive rollout was modelled. Thus, only a relatively small fraction of the population vaccinated during the very first few months would be experiencing a tangibly diminished vaccine efficacy (with respect to the Delta variant) (46). Nevertheless, the study included a sensitivity analysis of the vaccine efficacy across three static levels. Another limitation is that the surrogate ABM population which corresponds to the latest available Australian Census data from 2016 (23.4M individuals, with 4.45M in Sydney) is smaller than the current Australian population (25.8M, with 4.99M in Sydney). We expect low sensitivity of our results to this discrepancy due to the outbreak size being three orders of magnitude smaller than Sydney population. Finally, the model does not directly represent in-hotel quarantine and in-hospital transmissions. Since the frontline professionals (health care and quarantine workers) were vaccinated in a priority phase carried out in Australia in early 2021, i.e., before the Sydney outbreak, this limitation is expected to have a minor effect. Overall, as the epidemiology of the Delta variant continues to be refined with more data becoming available, our results may benefit from a retrospective analysis. DATA AVAILABILITY STATEMENT We used anonymised data from the 2016 Australian Census obtained from the Australian Bureau of Statistics (ABS). These datasets can be obtained publicly, with the exception of the work travel data which can be obtained from the ABS on request. It should be noted that some of the data needs to be processed using the TableBuilder: https://www.abs.gov. au/websitedbs/censushome.nsf/home/tablebuilder. The actual incidence data are available from the health departments across Australia (state, territories, and national), and at: https://www. covid19data.com.au/. Other source and supplementary data, including simulation output files, are available at Zenodo (47). The source code of AMTraC-19 is also available at Zenodo (48). AUTHOR CONTRIBUTIONS MP conceived and co-supervised the study and drafted the original article. SC and MP designed the computational experiments, re-calibrated the model, and estimated hospitalisations, ICU occupancy, and potential fatalities. CZ implemented simulations of progressive vaccination and social distancing policies. SC carried out the computational experiments, verified the underlying data, and prepared all figures. All authors had full access to all the data in the study, contributed to the editing of the article, read, and approved the final article. FIGURE 1 \| 1Moderate restrictions (NSW; progressive vaccination rollout; suppression threshold: 400 cases): a comparison between simulation scenarios and actual epidemic curves, under moderate interaction strengths (CI c = CI w = 0.25, HQ c = HQ w = 0.25, SD c = 0.25, SC = 0.5). A moving average of the actual time series up to November 2021, constrained by moderate levels of social distancing, SD ∈ {0.4, 0.5, 0.6}, under partial CTNPIs (CI = 0.7 and HQ = 0.5), see Supplementary Table 4. A progressive vaccination rollout was simulated concurrently with the continuing restrictions (see Supplementary Material: Vaccination modelling). Our Australia-wide model was calibrated by 31 August 2021, adopting a higher fraction of FIGURE 2 \| 2Tight restrictions (NSW; progressive vaccination rollout; suppression threshold: 400 cases): counter-factual simulation scenarios, under lowest feasible interaction strengths (CI c = CI w = 0.1, HQ c = HQ w = 0.1, SD c = 0.1, SC = 0.1), for (A) (log scale) incidence (crosses), and (B) cumulative incidence (circles) Figure 3 and Supplementary Figure 11 . 11The corresponding levels of simulated and actual vaccination coverage reached across Australia are shown in Supplementary Material: Vaccination modelling. Using the Australia-wide model, we quantified the expected demand in terms of hospitalisations (occupancy) and the intensive care units (ICUs), and the number of potential fatalities across the nation. The estimation methods are described in Supplementary Material: Hospitalisations and fatalities. The projections obtained for the three feasible levels of social distancing, SD ∈ {0.4, 0.5, 0.6}, are shown in FIGURE 3 \| 3Moderate restrictions (Australia; progressive vaccination rollout; suppression threshold: 400 cases): a comparison between simulation scenarios and actual epidemic curves up to November 13, under moderate interaction strengths (CI c = CI w = 0.25, HQ c = HQ w = 0.25, SD c = 0.25, SC = 0.5). A moving average of the actual time series for (A) (log scale) incidence (crosses), and (B) cumulative incidence (circles). Traces corresponding to social distancing levels SD ∈ {0.4, 0.5, 0.6} are shown for the period between 16 June and 13 November, as averages over 10 runs (colored profiles). 95% confidence intervals are shown as shaded areas. For each SD level, minimal and maximal traces, per time point, are shown with dotted lines. Peaks formed during the suppression period for each SD profile are identified with coloured dashed lines. Each SD intervention, coupled with school closures, begins with the start of initial restrictions. The alignment between simulated days and actual dates may slightly differ across separate runs. Case isolation and home quarantine are in place from the outset. TABLE 1 \| 1The macro-distancing parameters and interaction strengths: retrodiction ("moderate") and counter-factual ("tight.").Macro-distancing Interaction strengths Intervention Compliance levels Household Community Workplace/School moderate → high moderate → tight moderate → tight TABLE 2 \| 2Comparison of control measures: projected lockdown duration after the incidence peak, until new cases fall below 10 per day.Vaccination Vaccination Lockdown trigger Post-peak duration (days) scenario uptake (cumulative cases) SD = 0.7 SD = 0.8 SD = 0.9 Pre-pandemic 6% 100 55 28 17 Progressive → 40% 400 45 33 25 TABLE 3 \| 3Estimates (across Australia) of the peak demand in hospitalisations and ICUs; and cumulative fatalities (15 October 2021).Scenario Peak hospitalisations: Peak ICU demand: Cumulative fatalities: mean and 95% CI mean and 95% CI mean and 95% CI SD = 0.4 4805 [4282, 5257] 812 [731, 885] 1201 [1057, 1326] SD = 0.5 1604 [1358, 1844] 272 [230, 312] 539 [479, 624] SD = 0.6 533 [476, 579] 91 [80, 99] 235 [209, 256] Actual 1551 (28 September) 308 (12 October) 596 (15 October) In describing a "wave" we follow the definition based on two key features: (i) an epidemic wave comprises upward and/or downward periods; and (ii) the increase during an upward period, as well as the decrease during a downward period, must be substantial over a period of time(4). Frontiers in Public Health \| www.frontiersin.org February 2022 \| Volume 10 \| Article 823043 A post-peak period duration for each SD level is obtained using the incidence trajectory averaged over ten simulation runs.Frontiers in Public Health \| www.frontiersin.org FUNDINGSUPPLEMENTARY MATERIALThe Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fpubh. 2022.823043/full#supplementary-material Conflict of Interest: The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.Publisher's Note: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.Copyright © 2022 Chang, Cliff, Zachreson and Prokopenko. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms. Modelling transmission and control of the COVID-19 pandemic in Australia. S L Chang, N Harding, C Zachreson, O M Cliff, M Prokopenko, 10.1038/s41467-020-19393-6Nat Commun. 11Chang SL, Harding N, Zachreson C, Cliff OM, Prokopenko M. Modelling transmission and control of the COVID-19 pandemic in Australia. Nat Commun. 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[ "CDMA Technology for Intelligent Transportation Systems", "CDMA Technology for Intelligent Transportation Systems" ]	[ "Rabindranath Bera [email protected] \nSikkim Manipal Institute of Technology\nSikkim Manipal University\n737132MajitarRangpo, East SikkimIndia\n", "Jitendranath Bera \nDepartment of Applied Physics\nUniversity of Calcutta\n92 Acharya Prafulla Chandra Road700, 009KolkataIndia\n", "Sanjib Sil \nInstitute of Radio physics & Electronics\nUniversity of Calcutta\n92 Acharya Prafulla Chandra Road700, 009KolkataIndia\n", "Dipak Mondal \nSikkim Manipal Institute of Technology\nSikkim Manipal University\n737132MajitarRangpo, East SikkimIndia\n", "Sourav Dhar \nSikkim Manipal Institute of Technology\nSikkim Manipal University\n737132MajitarRangpo, East SikkimIndia\n", "Debdatta Kandar \nDepartment of Electronics &\nTelecommunication Engineering Jadavpur University\n700, 032KolkataIndia\n" ]	[ "Sikkim Manipal Institute of Technology\nSikkim Manipal University\n737132MajitarRangpo, East SikkimIndia", "Department of Applied Physics\nUniversity of Calcutta\n92 Acharya Prafulla Chandra Road700, 009KolkataIndia", "Institute of Radio physics & Electronics\nUniversity of Calcutta\n92 Acharya Prafulla Chandra Road700, 009KolkataIndia", "Sikkim Manipal Institute of Technology\nSikkim Manipal University\n737132MajitarRangpo, East SikkimIndia", "Sikkim Manipal Institute of Technology\nSikkim Manipal University\n737132MajitarRangpo, East SikkimIndia", "Department of Electronics &\nTelecommunication Engineering Jadavpur University\n700, 032KolkataIndia" ]	[]	Scientists and Technologists involved in the development of radar and remote sensing systems all over the world are now trying to involve themselves in saving of manpower in the form of developing a new application of their ideas in Intelligent Transport system( ITS). The world statistics shows that by incorporating such wireless radar system in the car would decrease the world road accident by 8-10% yearly. The wireless technology has to be chosen properly which is capable of tackling the severe interferences present in the open road. A combined digital technology like Spread spectrum along with diversity reception will help a lot in this regard. Accordingly, the choice is for FHSS based space diversity system which will utilize carrier frequency around 5.8 GHz ISM band with available bandwidth of 80 MHz and no license. For efficient design , the radio channel is characterized on which the design is based. Out of two available modes e.g. Communication and Radar modes, the radar mode is providing the conditional measurement of the range of the nearest car after authentication of the received code, thus ensuring the reliability and accuracy of measurement. To make the system operational in simultaneous mode, we have started the ' Software Defined Radio' approach for best speed and flexibility.	null	[ "https://arxiv.org/pdf/0705.2084v1.pdf" ]	1,530,997	0705.2084	0038944498c0736ff5973788ce2143e360d91027	CDMA Technology for Intelligent Transportation Systems Rabindranath Bera [email protected] Sikkim Manipal Institute of Technology Sikkim Manipal University 737132MajitarRangpo, East SikkimIndia Jitendranath Bera Department of Applied Physics University of Calcutta 92 Acharya Prafulla Chandra Road700, 009KolkataIndia Sanjib Sil Institute of Radio physics & Electronics University of Calcutta 92 Acharya Prafulla Chandra Road700, 009KolkataIndia Dipak Mondal Sikkim Manipal Institute of Technology Sikkim Manipal University 737132MajitarRangpo, East SikkimIndia Sourav Dhar Sikkim Manipal Institute of Technology Sikkim Manipal University 737132MajitarRangpo, East SikkimIndia Debdatta Kandar Department of Electronics & Telecommunication Engineering Jadavpur University 700, 032KolkataIndia CDMA Technology for Intelligent Transportation Systems ISMDS-CDMACDMA2000MC-CDMAMIMO CDMAIMCNFHSSMSCACICCI Scientists and Technologists involved in the development of radar and remote sensing systems all over the world are now trying to involve themselves in saving of manpower in the form of developing a new application of their ideas in Intelligent Transport system( ITS). The world statistics shows that by incorporating such wireless radar system in the car would decrease the world road accident by 8-10% yearly. The wireless technology has to be chosen properly which is capable of tackling the severe interferences present in the open road. A combined digital technology like Spread spectrum along with diversity reception will help a lot in this regard. Accordingly, the choice is for FHSS based space diversity system which will utilize carrier frequency around 5.8 GHz ISM band with available bandwidth of 80 MHz and no license. For efficient design , the radio channel is characterized on which the design is based. Out of two available modes e.g. Communication and Radar modes, the radar mode is providing the conditional measurement of the range of the nearest car after authentication of the received code, thus ensuring the reliability and accuracy of measurement. To make the system operational in simultaneous mode, we have started the ' Software Defined Radio' approach for best speed and flexibility. Introduction Speed limit in the super highways is generally not imposed on the cars moving at their highest possible speeds. As a result, it often results in severe accidents and deaths. A CDMA radar based collision avoidance system can therefore be thought of which is to be fitted in the cars. This paper will highlight the detailed development of such radar for collision avoidance of cars. CDMA Technology and its several versions are also popular for communication. It can also be exploited for a wide range of applications including range measurement, material penetration and low probability of interception. DS-CDMA, CDMA2000, MC-CDMA and MIMO CDMA [2], [3] are the different versions of same technology. The heart of such CDMA technology is the spread spectrum technology using PN sequence coding. The CDMA based digital radar technology will give rise to several advantages over conventional radars so that it can be used in ITS application successfully. Additionally, the same technology can also be explored to meet the communication need in ITS application [1]. The Intelligent Mobile Campus Network (IMCN) [4][5] The above mentioned two applications of CDMA in ITS can be further expanded in an IMCN which is modeled as shown in figure A. There are 4 cells namely cell1, cell2, cell3 and cell4 where each cell is defined as the geographical area of typically 100 meter over which a wireless communication is to be established between a mobile user and a fixed Base station. cell 1 and cell 2 are the two neighboring cells whereas cell3 and cell4 are another two remote neighboring cells. The four base stations will be placed on the rooftop of each building. All the four Base stations have their wireless connectivity with their respective wireless mobile handsets using the carrier frequency near 5.8GHz. The two neighboring base stations are connected by an MSC (Master Switching Center). So to have a total integrity among the four base stations two MSC namely MSC1 and MSC2 are required. As shown in figure A, MSC1 connects cell1 and cell2 whereas MSC2 connects cell3 and cell4. Each MSC is physically separated by a distance of 200 meter or more and is linked with the wireless network using 12 GHz microwave carrier. Thus the total system will provide full duplex communication with higher data rate of 64 Mbits/s approximately. Thus the total IMCN is too complex requiring a knowledge of several technologies for its design and successful implementation. This paper will highlight only the CDMA technology and other relevant technologies used for the communication and radar applications utilized in the car. The authors are encouraged to exploit the latest digital communication and digital radar technology in their design and implementation. CDMA Technology in Car A radio mounted on the car will normally face a lots of problems like: 1. Active Interferences comprising both adjacent as well as co-channel interference ( ACI & CCI). 2. Passive Interference coming from multi path . Multi path transmission When the handset and base station are within line of sight, the primary propagation will usually be the line of sight and secondary propagations due to reflections will be less significant [6]. Reflected propagations become more significant if the line of sight is obstructed. Figure 1 illustrates a simplified multi path propagation. Whenever there is more than one significant impinging wave (with different phases) on a mobile receive antenna, the receiver will be subject to varying signal levels as it moves around. This is caused by constructive and destructive addition of the impinging waves due to their different phase offsets. This mechanism is called multi path fading. Simulation of Space Diversity The idea of antenna diversity is that if receive antenna A is experiencing a low signal level due to fading, also called a deep fade, Antenna B will probably not suffer from the same deep fade, provided the two antennas are displaced in position or in polarity. A Matlab based simulation is conducted in the Laboratory considering slow fading and the received signal strength variation is illustrated in figure 2. The option to select the best antenna significantly improves performance in outdoor environments, but does not necessarily increase the maximum line-of-sight range of a product. Figure 3 illustrates the effect of selecting the best antenna. Antenna diversity is implemented by equipping the base station or handset (or both) with two antennas. Various selection schemes can be implemented, depending on the actual antenna setup. Preamble antenna diversity, also known as fast antenna diversity, has proven its use in fast frequency hopping systems. Preamble antenna diversity is implemented by comparing the RSSI value of each antenna in the beginning of each receive burst. Experiments on Frequency Diversity A delay spread multi path propagation study experiment was carried over the sea near Sagardwip Island ,The Bay of Bengal, West Bengal. A LOS link was set up over the sea saline water and two carriers at 12 & 13 GHz were transmitted simultaneously and received from a distance of about 5 km. The experiment was successfully conducted and the interesting results related to justification of using FHSS technology will be highlighted The interesting observations are as follows: a) A typical problem of fading with a fade depth of 30 dB is noticed at 12.5 GHz lasts for about half an hour depending on the sea water condition. The fading time varies from day to day. The LOS link data of signal strength at Kakdwip, received at Sagardwip over the river with saline water near Bay of Bengal is shown in fig.4. b) With a deep interest to observe whether the above fading is effective at the same time to other neighboring frequency, we have transmitted two radio frequencies one at 12.5 GHz and another at 11.5 GHz and three kinds of observations are noticed and shown in Fig.4. Region III: reception at 11.5 GHz is faded by approximately 30 dB while 12.5 GHz reception remains steady. The reason for the above interesting observation is that the same path specified by sea water produces different time delay for two frequencies such that for one frequency signal strengths are additive and subtractive for other. This complementary nature of fading at two different frequencies can be exploited to mitigate fading problem in a FREQUENCY DIVERSITY SYSTEM. Justification for Spread Spectrum (SS) Technology In another experimental set up a SS based radio is tested for its Interference rejection capability. Figure5 illustrates the interference suppression capability of SS radio unit and is the right justification for the choice of SS in the physical layer. The region below the curve is the jamming free region . Figure5 . Interference suppression capability of SS radio Vs. Frequency curve. Choice of Technology for ITS application The above three Simulation and experimental facts can be referred to choose the proper technology for ITS application. Diversity reception may be the best choice to ( both frequency as well as space diversity ) restrict the active and passive interference to an acceptable level which are very strong in ITS application. Therefore, we like to incorporate the FHSS ( frequency hopping spread spectrum ) technology as a frequency diversion method and two antenna instead of single as space diversity reception system with a separation of 25 cm between them. Choice of frequency To avoid the end user radio license problem, the best choice of radio carrier both for communication and radar should be the ISM band frequencies. There are 3 band of frequencies allotted for ISM bands, namely: 900 MHz, 2.4 GHz and 5.8 GHz.. The following Table 1 will dictate the choice of frequency. The above table is self explanatory which will justify our choice of frequency for ITS application to be at 5.8 GHz. Before launching any kind of radio application with the above specification (e.g. Communication or radar) the two channel characteristics , bandwidth and Power , constitute the primary resources available to the designer [7]. Accordingly, in an experimental laboratory set up (consists of developed radio and several Instrument like Pulse generator, Distortion analyzer, spectrum Analyzer, Digital Storage Oscilloscope etc.), the following results are established as shown in figure 6. The radio is best usable in the range of 350-600 micro second PRT range. Specifications of the Radio Radar Modes of Operation A 13 bit code is transmitted from the Car using omni directional antenna so that signal from the nearby cars can be echoed back towards the space diversity Figure 6; Channel Characterization of the Radio reception system using two antenna. The space diversity antenna is helping us in determining the nearest car and rejecting reflection from distant car. The 'start' bit preceding the actual code is also helping in authentication of signal received i.e. if start bit followed by the received code then only the delay between transmitted and received code will be measured . It is then translated into the distance of the nearest car. Conclusion The IMCN system is thus operational in its basic form for its two modes of operation supporting several mobile users over a distance of 500 meter or more. Lots of R&D efforts to be imparted for its commercialization. Further Extension The above radio model is successfully implemented at SMIT but is operational in 'either or' mode i.e. the same radio is either used as communication device or as radar installed in the Car. To make the system operational in simultaneous mode, we have started the ' Software Defined Radio' approach which is very much useful today for best speed and flexibility. The block diagram of revised ITS system is as shown in figure7. It utilizes two radio front end namely Radio I and Radio II serving the purposes of communication and radar respectively. At the backend, FPGA based signal processor will be used which will be finally controlled and monitored by PC. The other major components are A/D and D/A converter, Flash and SDRAM and 4 channel receivers. 1: A typical problem of fading with a maximum fade depth of 30 dB is noticed at both the frequencies. It lasts for about half an hour depending on the sea water condition. The fading time varies from day to day. 2: The close observation of the results reveals that the signals of 11.5 & 12.5 GHz radio frequencies are not degraded at the same time, rather at different times. 3: The plots of the above observations are divided into 3 separate regions: Region I: reception at both the frequencies is stable & normal Figure 4 . 4Frequency Diversity experiment Fig 4 : 4Region II: reception at 12.5 GHz is faded by approximately 30 dB while 11.5 GHz reception remains steady. . (MAX.),dBm [FREQ.12.5 GHZ] SIG. (MAX. ), dBM [FREQ.11.5 GHZ] Table 1 : 1Choice of Frequency for ITS .Freq. span (MHz) Available bandwidth ∆f ( MHz) for frequency diversity Value of wavelength λ ( cm) Effectiveness of space diversity Remarks 900- 930 30 33 Not effective ∆f less, λ more. 2400- 2480 80 12.5 Effective ∆f more, λ less 5760- 5840 80 5.172 More effective ∆f more, λ least Ultra Wideband Radar Technology. James D Taylor, CRC PressJames D. Taylor, "Ultra Wideband Radar Technology"., CRC Press, 2001 On the System Design Aspects of Code Division Multiple Access (CDMA) Applied to Digital Cellular and Personal Communications Networks. Proc. 41 st IEEE VTC'91. A. Salmasi and K. Gilhousen41 st IEEE VTC'91"On the System Design Aspects of Code Division Multiple Access (CDMA) Applied to Digital Cellular and Personal Communications Networks," A. Salmasi and K. Gilhousen, Proc. 41 st IEEE VTC'91, 1991. Code Division Multiple Access -A Modern Trends in Multimedia Mobile Communication. D Kandar, R Bera, A R Sardar, S Kandar, S S Singh, S K Sarkar, International Conference on Services Management ICSM2005 held at IIMT. Gurgaon, Delhi"Code Division Multiple Access -A Modern Trends in Multimedia Mobile Communication", D. Kandar, R. Bera ,A. R. Sardar, S. Kandar, S. S. Singh and S. K. Sarkar, International Conference on Services Management ICSM2005 held at IIMT, Gurgaon, Delhi during March 11-12,2005. Wireless communication and networks. W. StallingsPearson Education Asia publication"Wireless communication and networks", W. Stallings, Pearson Education Asia publication, pp 320-333,2002 Computer networking -a top down approach featuring the internet. J A Kurose, K W Ross, Pearson Education Asia publication" Computer networking -a top down approach featuring the internet", J.A. Kurose and K. W. Ross, Pearson Education Asia publication, pp 480-487,2003 Introduction to 3G mobile Communications. J Korhonen, Artech HouseJ. Korhonen, "Introduction to 3G mobile Communications", Artech House ,2001 Digital communication systems. Simon Haykin, John Wiley & Sons"Digital communication systems", Simon Haykin, John Wiley & Sons, 2004. The University of Calcutta, in the year 1982,1985 & 1997 respectively. Currently working as Professor and Head of the Deparment. Intelligent Mobile Campus Networking ( IMCN) at Sikkim Manipal Institute of Technology, Sikkim. Figure 7: The block diagram of revised ITS system. Authors Information Rabindranath Bera: Born in 1958 at Kolaghat , West Bengal, B. Tech, M. Tech & Ph.D (Tech) from the Institute of Radiophysics & Electronics. Sikkim; KolkataElectronics & Communication Engineering, Sikkim Manipal University ; D.( Engg.) from Jadavpur University ; Dept. of Applied Physics, The University of Calcutta. Microwave/ Millimeter wave based Broadband Wireless Mobile Communicationthe year 2005. Currently working as Reader. Remote Sensing and Embedded System are the area of specialisatoionFigure A: Intelligent Mobile Campus Networking ( IMCN) at Sikkim Manipal Institute of Technology, Sikkim. Figure 7: The block diagram of revised ITS system. Authors Information Rabindranath Bera: Born in 1958 at Kolaghat , West Bengal, B. Tech, M. Tech & Ph.D (Tech) from the Institute of Radiophysics & Electronics, The University of Calcutta, in the year 1982,1985 & 1997 respectively. Currently working as Professor and Head of the Deparment, Electronics & Communication Engineering, Sikkim Manipal University, Sikkim, Microwave/ Millimeter wave based Broadband Wireless Mobile Communication and Remote Sensing are the area of specialisatoion. Jtendranath Bera: Born in 1969 at Sajinagachi, West Bengal, B. Tech, M. Tech from the Dept. of Applied Physics, The University of Calcutta, in the year 1993,1995 respectively, and Ph.D.( Engg.) from Jadavpur University, Kolkata. In the year 2005. Currently working as Reader, Dept. of Applied Physics, The University of Calcutta. Microwave/ Millimeter wave based Broadband Wireless Mobile Communication, Remote Sensing and Embedded System are the area of specialisatoion. Currently working as Asst. Professor, International Institute of Information Technology , Kolkata. Microwave/ Millimeter wave based Broadband Wireless Mobile Communication, Remote Sensing are the area of specialisatoion. Dipak Mondal: Born in 1976 at Baruipur, West Bengal, B. Tech, M. Tech from the Institute of Radiophysics & Electronics, the University of Calcutta. Sanjib Sil, Born in 1965 at Kolkata, West Bengal, B. Tech from IETE ( 1989), M. Tech from BIT, Meshra. The University of Calcutta ; Currently working as Lecturer, Dept. of Electronics & Comm. Engg., Sikkim Manipal UniversityPh.D. registration from the Institute of Radiophysics & Electronics. in the year 2002. Microwave/ Millimeter wave based Broadband Wireless Mobile Communication, Remote Sensing are the area of specialisatoionSanjib Sil: Born in 1965 at Kolkata, West Bengal, B. Tech from IETE ( 1989), M. Tech from BIT, Meshra (1991). Ph.D. registration from the Institute of Radiophysics & Electronics, The University of Calcutta ( 2002). Currently working as Asst. Professor, International Institute of Information Technology , Kolkata. Microwave/ Millimeter wave based Broadband Wireless Mobile Communication, Remote Sensing are the area of specialisatoion. Dipak Mondal: Born in 1976 at Baruipur, West Bengal, B. Tech, M. Tech from the Institute of Radiophysics & Electronics, the University of Calcutta, in the year 2002, 2004 respectively. Currently working as Lecturer, Dept. of Electronics & Comm. Engg., Sikkim Manipal University, Microwave/ Millimeter wave based Broadband Wireless Mobile Communication, Remote Sensing are the area of specialisatoion. Sourav Dhar, Visveswaraiah Technological University in the year 2002, M. Tech from Sikkim Manipal Instiutte Of Technology, Sikkim Manipal University in the year 2005. Currently working as Lecturer, Dept. of Electrical & Electronics, SMIT. West Bengal, B. E from Bangalore Institute of TechnologyBorn in 1980 at Raiganj. Broadband Wireless Mobile Communication is the area of specialisatoionSourav Dhar: Born in 1980 at Raiganj, West Bengal, B. E from Bangalore Institute of Technology, Visveswaraiah Technological University in the year 2002, M. Tech from Sikkim Manipal Instiutte Of Technology, Sikkim Manipal University in the year 2005. Currently working as Lecturer, Dept. of Electrical & Electronics, SMIT, Broadband Wireless Mobile Communication is the area of specialisatoion. Born in 1977 at Deulia, West Bengal, B.Sc. ( Honours) from The University of Calcutta in the year 1997, M. Sc from Vidyasagar University in the year 2001. Debdatta Kandar, Currently working as Research Fellow in Jadavpur UniversityDebdatta Kandar: Born in 1977 at Deulia, West Bengal, B.Sc. ( Honours) from The University of Calcutta in the year 1997, M. Sc from Vidyasagar University in the year 2001. Currently working as Research Fellow in Jadavpur University.	[]
[ "Cosmic infrared background from Population III stars and its effect on spectra of high-z gamma-ray bursts", "Cosmic infrared background from Population III stars and its effect on spectra of high-z gamma-ray bursts" ]	[ "A Kashlinsky " ]	[]	[]	We discuss the contribution of Population III stars to the near-IR (NIR) cosmic infrared background (CIB) and its effect on spectra of high-z high-energy gamma-ray bursts (GRBs) and other sources. It is shown that if Population III were massive stars, the claimed NIR CIB excess will be reproduced if only ∼ 4 ± 2% of all baryons went through these stars. Regardless of the precise amount of the NIR CIB from them, they likely left enough photons to provide a large optical depth for high-energy photons from distant GRBs. Observations of such GRBs are expected following the planned launch of NASA's GLAST mission. Detecting such damping in the spectra of high-z GRBs will then provide important information on the emissions from the Population III epoch and location of this cutoff may serve as an indicator of the GRB's redshift. We also point out the difficulties of unambiguously detecting the CIB part originating from Population III in spectra of low z blazars.	10.1086/498243	[ "https://arxiv.org/pdf/astro-ph/0508089v4.pdf" ]	15,272,906	astro-ph/0508089	7ba030a60e10c8e3f8fa285e7589b113f760adfd	Cosmic infrared background from Population III stars and its effect on spectra of high-z gamma-ray bursts arXiv:astro-ph/0508089v4 10 Feb 2006 A Kashlinsky Cosmic infrared background from Population III stars and its effect on spectra of high-z gamma-ray bursts arXiv:astro-ph/0508089v4 10 Feb 2006Subject headings: cosmology: theory -cosmology: observations -diffuse radiation -gamma-rays: bursts -gamma rays: theory We discuss the contribution of Population III stars to the near-IR (NIR) cosmic infrared background (CIB) and its effect on spectra of high-z high-energy gamma-ray bursts (GRBs) and other sources. It is shown that if Population III were massive stars, the claimed NIR CIB excess will be reproduced if only ∼ 4 ± 2% of all baryons went through these stars. Regardless of the precise amount of the NIR CIB from them, they likely left enough photons to provide a large optical depth for high-energy photons from distant GRBs. Observations of such GRBs are expected following the planned launch of NASA's GLAST mission. Detecting such damping in the spectra of high-z GRBs will then provide important information on the emissions from the Population III epoch and location of this cutoff may serve as an indicator of the GRB's redshift. We also point out the difficulties of unambiguously detecting the CIB part originating from Population III in spectra of low z blazars. Introduction Zero-metallicity Population III stars (hereafter P3) are thought to have preceded the normal metal-enriched stellar populations, but because they would be located at high z they are inaccessible to direct observations by current telescopes. If massive, they are expected to have left a significant level of diffuse radiation shifted today into IR, and it was suggested that the cosmic infrared background (CIB) contains a significant contribution from P3 in near-IR, both its mean level and anisotropies (see review by Kashlinsky, 2005 and references therein). This has recently received strong support from measurements of CIB anisotropies in deep Spitzer/IRAC images . If P3 are responsible for even a fraction of the claimed NIR CIB they would provide a high comoving density of photons all the way to the P3 era. In this Letter we analyze effects of such photons on spectra of high-z high-energy gamma-ray bursts (GRBs) and blazars that would be observed with the upcoming GLAST LAT instrument to 300 GeV. We show that the entire claimed NIR CIB excess (NIRBE) can be explained if only ∼4% of the baryons have gone through P3 stars. This would result in ∼ 0.1(1 + z) 3 photons/cm 3 whose present day energy is between 1 and 4µm. Such photons would provide a large optical depth due to photon-photon absorption for GRBs (and other sources) at energies that will be probed with GLAST. Detecting this spectral damping in forthcoming GRB observations will provide an important test of the P3 era parameters. CIB from Population III Ay wavelengths > ∼ 10µm the total flux produced by the observed galaxies matches the levels of the CIB within its uncertainties, but in the near-IR (NIR) the claimed levels of the CIB are substantially higher than the net fluxes produced by galaxies out to flux limits where this contribution saturates (see review by Kashlinsky 2005 for details). Fig. 1 shows the CIB excess levels (filled circles) over the net flux from galaxies observed in deep surveys (open symbols); the caption discusses the details. The excess is significant at 1µm λ 4µm, the range we term NIR, and its bolometric flux is (Kashlinsky 2005): F NIRBE = 29 ± 13 nW m 2 sr ; F CIB excess (λ > ∼ 10µm) < ∼ 10 nW m 2 sr(1) At λ > ∼ 10µm we evaluated the upper limits shown in Fig. 1 described in the caption. The wavelengths > ∼ 10µm contribute little, so we adopt the value of F NIRBE for what follows. It was suggested that the NIRBE is produced by massive P3 stars at high z ( 10) (Santos et al 2002;Magliochetti et al 2003;Cooray et al 2004;Kashlinsky et al 2004). Significant energy release by P3 is suggested from the recent measurement of CIB anisotropies in deep exposure Spitzer data . Because P3 stars, if massive, would radiate at the Eddington limit, where L ∝ M, the total flux produced by them is largely model-independent (Rees 1978;Kashlinsky et al 2004). We reproduce briefly the argument from Kashlinsky et al (2004): Each star would produce flux L 4πd 2 L , where d L is the luminosity distance. Because for massive stars L ∝ M, the total comoving luminosity density from P3 is n(L)LdL ∝ Ω baryon f * 3H 2 0 8πG , where n(L) is their luminosity function and f * is the mass fraction of baryons locked in P3 at any given time. In the flat Universe, the volume per unit solid angle subtended by cosmic time dt is dV = c(1 + z)d 2 L dt. Finally, these stars would radiate at efficiency ǫ (≃ 0.007 for hydrogen burning). This then leads to the closed expression for the total bolometric flux from these objects: F bol = 3 8π c 5 /G 4πR 2 H (1 + z) −1 ǫf 3 Ω baryon ≃ 4 × 10 7 1 z 3 ǫf 3 Ω baryon h 2 nW m 2 sr(2) Here f 3 is the mean mass fraction of baryons locked in P3 stars and z 3 ≡ 1 (1+z) −1 is a suitably averaged redshift over their era. The total flux is a product of the maximal luminosity produced by any gravity-bound object, c 5 /G, distributed over the surface of the Hubble radius, R H =cH −1 0 , and the fairly understood dimensionless parameters. From WMAP observations we adopt Ω baryon h 2 =0.0224 (Bennett et al 2003) and, since the massive stars are fully convective, their efficiency is close to that of hydrogen burning (Schaerer 2002), ǫ=0.007. Requiring that P3 stars are responsible for the flux given by eq. 1 leads to: f 3 = (4.2 ± 1.9) × 10 −3 z 3 0.0224 Ω baryon h 2 0.007 ǫ(3) Assuming z 3 ≃10 this is somewhat less than the > ∼ 5% value suggested by Madau & Silk (2005) and considerably less than the > ∼ 10% value of Dwek et al (2005). Within the NIRBE uncertainty, only > ∼ 2% of the baryons had to go through P3. This is not unreasonable considering that primordial clouds are not subject to many of the effects inhibiting star formation at the present epochs, such as magnetic fields, turbulent heating etc. The only criterion for P3 formation seems to be that primordial clouds turning-around out of the primordial density field have the virial temperature, T vir , that can enable efficient formation of and cooling by molecular hydrogen (Abel 2002;Bromm et al 1999). Assuming spherical collapse of gaussian fluctuations and the ΛCDM model from WMAP observations (Bennett et al 2003) the fraction of collapsed haloes at z=10 with T vir ≥(400,2000)K is (2.6, 5) × 10 −2 in good agreement with eq. 3 as can be derived from Fig. 2 in Kashlinsky et al (2004). Eq. 3 was evaluated from 1 to 4µm, but with significant CIB excess flux outside that range, f 3 would increase. However, at wavelengths < ∼ 0.1z 3 µm the high-z emissions would be below the Lyman break and would be reprocessed to λ > ∼ 1µm (Santos et al 2002). If z 3 < 10, which is unlikely in light of WMAP polarization data (Kogut et al 2003), the rest frame Lyman break may be redshifted to < ∼ 1µm, but the possible extra CIB excess from <1µm will be compensated by f 3 in eq. 3 decreasing with z 3 . At longer wavelengths, the CIB excess given by eq. 1 can at most increase f 3 by < ∼ 30%. The above estimate is subject to two caveats: First, it assumes the NIR CIB at the levels given by eq. 1, which were derived assuming a specific set of zodiacal light models. The latter may carry large systematic uncertainties, which are not included in the formal uncertainties in eq. 3. Our estimate of f 3 is proportional to the CIB excess summarized in Fig. 1 and eq. 1 and would change accordingly if these are superseded by future measurements. Secondly, this estimate assumes that P3 were massive stars leading to the effective efficiency, ǫ, very close to that of the hydrogen burning, ǫ=0.007. If P3 were less massive than ∼ 50 − 100M ⊙ , the effective ǫ of their energy release would decrease by a factor of a few (Siess et al 2002) requiring significantly larger values of f 3 for a given F NIRBE . Independently of the CIB considerations, if P3 were less massive than ∼ 240M ⊙ , their fraction must be very small in order not to overproduce the metallicities of the poorest Population II stars (Heger et al 2003), leading to CIB fluxes from P3 significantly lower than eq. 1. 3. Optical depth to photon-photon absorption at high energies at high z: application to forthcoming GRB measurements If P3 at early epochs produced even a fraction of the claimed NIRBE, they would supply abundant photons at high z. The present-day value of I ν =1 MJy/sr corresponds to the comoving number density of photons per logarithmic energy interval, d ln E, of n γ = 4π c Iν h Planck =0.6 cm −3 and if these photons come from high z then n γ ∝ (1 + z) 3 . These photons also had higher energies in the past, ≃(0.1-0.3)(1 + z)eV, providing an abundance of absorbers for sufficiently energetic photons at high redshifts via γγ CIB → e + e − (Akhiezer & Berestetskii 1965 ;Nikishov 1962). Stecker & de Jager (1993) have pioneered applications of the γ-γ absorption to constraining the present-day CIB from high-energy spectra of low-z blazars. Madau & Phinney (1996) and Salamon & Stecker (1998) have considered effects by evolving normal galaxy populations on potential future intermediate z ( > ∼ 0.5) blazars. However, as shown in Sec. 2, P3 stars are likely to have provided a far more abundant source of photons at high z to interact with high-energy gamma-ray photons. GRBs are the obvious objects whose high-energy emissions would be damped by the absorption from the NIRBE photons. This effect was difficult to detect with EGRET because of its low sensitivity. Observations by EGRET have detected only 6 GRB's with one of them being a record energy 18 GeV photon (Hurley et al 1994). A successor to the Gamma-Ray Observatory, NASA's GLAST is to be launched in 2007. Its Large-Area Telescope (LAT) will provide significantly improved sensitivity needed for detecting high-z GRBs out to E=300 GeV with better than 10% energy resolution above 100 MeV, and its large field-of-view should detect ∼ 100-300 GRBs/yr. It is expected that ∼50 of these would have enough high-energy photons to measure spectral indices at E >0.1 GeV with uncertainty better than 0.1 1 . We show in this section that the photons produced by the P3 stars should leave a detectable signature by damping the high-energy part of the spectra of high-z GRBs. We denote with E and E the present day energies of the CIB and GRB photons respectively and the primes refer to rest-frame energies, e.g. E ′ = E(1 + z). The photonphoton absorption, being electromagnetic in nature, has cross-section ∼ that for the Thompson scattering, σ T ; it is σ(E ′ , E ′ , x) = 3 16 σ T (1 − β 2 )[2β(β 2 − 2) + (3 − β 4 ) ln( 1+β 1−β )], where β = 1 − 2m 2 e c 4 E ′ E ′ (1−x) and x = cos θ. The cross-section has a sharp cutoff as β → 1, peaks at ≃ 1 4 σ T at β ≃ 0.7, and is σ ∝ β for β < ∼ 0.6. The mean free path of GRB photons in the presence of CIB would be (n γ CIB σ) −1 ∼ 0.8(σ T /σ)(1MJy/sr/I ν )(1 + z) −3 Mpc. The right vertical axis in Fig. 1 shows the product of σ T cH −1 0 and the comoving photon number density for given I ν . The figure also marks the regions defined by the photon-photon absorption threshold. Discussion in Sec.2 makes it plausible that, if the NIRBE originates from P3 stars, it must have a sharp truncation corresponding to the redshifted Lyman break, or ≃ 0.1z 3 µm. Thus we assume that, at least at high z, there are few photons at wavelengths that are shifted today to <1µm. At longer wavelengths there is no observational evidence for CIB excess over that from "ordinary" galaxies containing Population I and II, but at the same time only spectra of the sources of very high energy γ-rays at very high z would be affected by that range of the CIB. Thus it appears that there exists a narrow wavelength window of 1µm < ∼ λ < ∼ 4µm in which the CIB photons can interact with high-E, high-z GRBs and probe the emissions from the, so far putative, P3 era. We assume a flat Universe dominated by the cosmological constant and that the NIRBE photons originated from P3 at redshifts higher than that of the GRBs, so that n γ ∝ (1 + z) 3 . In this case, the optical depth due to photon-photon absorbtion is: dτ GRB (E) dz = R H (1 + z) Ω m (1 + z) + (1 − Ω m )(1 + z) −2 1 −1 dx E ′ max 2m 2 e c 4 E ′ (1−x) σ(β)n γ CIB ( E ′ 1 + z ) dE ′ E ′ (4) where n γ CIB is the present-day photon density corresponding to the observed CIB excess. Fig. 2 shows the resultant optical depth and contributions to it from different z. We adopted E max =1.24 eV (1 µm) and the form of n γ CIB corresponding to the solid line in Fig. 1. The onset of τ >1 occurs rapidly at E > ∼ m 2 e c 4 /(1eV)(1 + z) −2 ∼ 261(1 + z) −2 GeV. The GRB spectra at these energies should either be strongly damped or there had to be only negligible energy releases from the P3 era. Given the high values of τ this would still hold even if P3 era produced only a small part of F NIRBE , but assumes that the latter has a Lyman cutoff redshifted to ∼1µm today. If the P3 era extended to lower z, the Lyman cutoff would occur at the observer wavelength ∼ 0.1z 3 µm and GRB spectra would be damped at proportionately lower E. Thus, the location of this cutoff may also serve as an indicator of the GRB's redshift. With the advertised LAT energy resolution of < 10% at E > ∼ 1 GeV, one could determine GRB redshifts from the damping by the P3 photons to better than ∼ 5%. How robust is this result? First, photons due to optical counts of galaxies at much later times do not affect much the GRB spectra because τ ∼ zn γ σ T R H at low z and they contribute n γ (z = 0)σ T R H ∼ a few as Fig. 1 shows. Furthermore, unlike the P3 photons for which n γ ∝ (1 + z) 3 , those contributed by galaxies would produce a still smaller contribution at earlier times compared to P3. Second, even if the NIRBE is not entirely cosmological, Fig. 2 shows that P3 emissions should still lead to a very large optical depth, which scales as τ ∝ F NIRBE . Thus P3 would likely be the dominant contributors to the optical depth to GRBs at high energies. Finally, the magnitude of τ at a given bolometric CIB flux should not be sensitive to the CIB excess spectrum because the (already narrow) range of 1-4 µm available to dampen GRBs at high z is further decreased by (1 + z). Fig. 2 shows that the optical depth from P3 is very high, τ > ∼ 10 2 − 10 3 F NIRBE 30 nW/m 2 /sr which, when combined with the sudden onset of the optically thick regime, would lead to an identifiable damping by the P3 photons. The damping will affect progressively lower energy part of the rest-frame GRB spectra as one moves to higher z. At these energies, the GRB emission is likely produced by the inverse Compton and the synchrotron self Compton components and is expected to be high (or even dominate) for the typical Lorentz factors involved (Piran 2004 and references therein); e.g. Dingus et al (1998) construct an average spectrum from four EGRET burst and find the differential photon index of dn γ /dE ∝ E −(1.95±0.25) out to ∼10GeV. It may be sufficient to use GLAST observations of fairly low z GRBs (say, z ∼2-4 determined spectroscopically in afterglow observations) to establish the existence of the CIB from P3 epochs; if positive then the higher z GRBs can be used to further verify that it comes from the P3 epochs and/or calibrate the z determinations from the damping. Is Population 3 detectable with low z blazars? Although spectra of more and more distant blazars are now measured with new instruments, such as HESS 2 , these blazars are still too close for unambiguous detection of the P3 emissions. The farthest blazars with known spectra are at z ≃0.13-0.18 and the spectra extend to E <3 TeV (Dwek et al 2005, Aharonian et al 2005. The right panel in Fig. 2 shows the optical depth of a blazar at z=0.18 produced by the γ-γ absorption 1) due to observed galaxies, 2) due to NIRBE from DIRBE and IRTS, and 3) omitting the IRTS-based point at 1.65 µm and assuming the lower end of the DIRBE-based results. Even if the entire NIRBE is correct and originates from P3, the additional damping from it is small and is more pronounced only at E >1 TeV, where measurements and interpretation are difficult. This is because P3 contribute to the background over a short range of wavelengths longward of ∼ 1µm. If P3 were to contribute only a fraction of the NIRBE, their emissions will not contribute appreciably to the observed spectra of z ∼ 0.1 − 0.2 blazars, but would be seen in the high-z GRBs. If GLAST collects a large sample of blazars and other AGNs at z > ∼ 1, Fig. 2 shows that they can also be used to probe the emissions from the P3 era. Table 5 and beyond). Briefly, the net CIB flux is adopted from Cambresy et al (2001) at 1.25µm, from Matsumoto et al (2005) at 1.65 µm, from Gorjian et al (2000) and Matsumoto et al (2005) at 2.2 µm, and from Dwek & Arendt (1998) and Wright & Reese (2000) at 3.5 µm and from Matsumoto et al (2005) at 4 µm. The flux from OG is taken from HST counts out to 2.2 µm (open squares from Madau & Pozzetti 2000) and from Spitzer/IRAC counts at 3.6 and 4.5 µm (open diamonds, Fazio et al 2004). At λ > ∼ 10µm no CIB excess was observed and the levels of CIB are consistent with the net contribution from OG. The upper limits on the CIB excess there are shown where net flux from ordinary galaxies is known from SCUBA and ISO measurements. The CIB level at 450 and 850 µm was taken from Fixsen et al (1998). At 12 and 24 µm we adopted the lowest upper limits on the net CIB flux using γ-ray blazar observations (Stanev & Franceschini 1998;Renault et al 2001). They are also largely irrelevant for computations of the GRB photons absorption: vertical bars with left-pointing arrows show the range where photon-photon absorption is possible for the redshifts and energies indicated. The thick light-shaded solid and dashed lines show the CIB excess spectrum used in computing the optical depth shown in Fig. 2. The net τ vs the GRB photon energy for the GRB redshifts shown in the panel. Solid, dotted, short-dashed, dash-dotted, dash-triple-dotted and long-dashed lines correspond to increasing order in z. The range of redshifts was chosen to avoid overlap between GRBs and P3 era; the latter is assumed to have ended by z=10. Only NIRBE from Fig. 1 is assumed in the calculations. This assumption is fairly safe at larger z as this component gets progressively larger the ordinary galaxies emissions, but at z ≃ 1 the latter can still contribute (Madau & Phinney 1996). Right: Optical depth to photon-photon absorption for a source at z=0.18. Line notations correspond to Fig. 1. Dotted line assumes only ordinary galaxies measured in deep counts and that their photons originated at z ≥ 0.18. Thick lightshaded lines correspond to the NIRBE: solid line assumes both the DIRBE-and IRTS-based claims at the central points of the measurements and dashed line assumes the "minimal" NIRBE with only the DIRBE-based points (i.e. the 1.65 µm point is omitted) and the CIB levels corresponding to the lower end of the error bars in Fig. 1. This work was supported by the NSF under Grant No. AST-0406587. I thank David Band and Demos Kazanas for useful discussions and comments on the manuscript. 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[ "QUADRATIC POLYNOMIALS REPRESENTED BY NORM FORMS", "QUADRATIC POLYNOMIALS REPRESENTED BY NORM FORMS" ]	[ "T D Browning ", "D R Heath-Brown " ]	[]	[]	Let P (t) ∈ Q[t] be an irreducible quadratic polynomial and suppose that K is a quartic extension of Q containing the roots of P (t). Let N K/Q (x) be a full norm form for the extension K/Q. We show that the variety P (t) = N K/Q (x) = 0 satisfies the Hasse principle and weak approximation. The proof uses analytic methods.	10.1007/s00039-012-0168-5	[ "https://arxiv.org/pdf/1109.0232v1.pdf" ]	14,755,588	1109.0232	3e20dbce13ce894c2eb02cc29a332427668f3779	QUADRATIC POLYNOMIALS REPRESENTED BY NORM FORMS Sep 2011 T D Browning D R Heath-Brown QUADRATIC POLYNOMIALS REPRESENTED BY NORM FORMS Sep 2011 Let P (t) ∈ Q[t] be an irreducible quadratic polynomial and suppose that K is a quartic extension of Q containing the roots of P (t). Let N K/Q (x) be a full norm form for the extension K/Q. We show that the variety P (t) = N K/Q (x) = 0 satisfies the Hasse principle and weak approximation. The proof uses analytic methods. Introduction Let k be a number field with set of valuations Ω k . Given an algebraic variety X defined over k we have the obvious inclusions X(k) ∆ − → X(A k ) ⊆ ν∈Ω k X(k ν ), where ∆ is the diagonal embedding of the set X(k) of k-rational points into the set X(A k ) of adèles of X. Moreover the set X(A k ) is empty if and only if ν∈Ω k X(k ν ) is empty and clearly provides a local obstruction to the existence of k-rational points on X. Recall that a class X of algebraic varieties X defined over k is said to satisfy the Hasse principle if X(k) = ∅ whenever X(A k ) = ∅. Likewise X is said to satisfy weak approximation if whenever it is non-empty the image of X(k) under ∆ is dense in X(A k ) in the product of ν-adic topologies. This paper is concerned with the Hasse principle and weak approximation for the class of varieties satisfying the Diophantine equation P (t) = N K/k (x 1 , . . . , x n ) = 0, (1.1) where N K/k is a full norm form for an extension K/k of number fields, and P (t) is a polynomial over k. Thus if [K : k] = n and we fix a basis {ω 1 , . . . , ω n } for K as a vector space over k, then N K/k (x 1 , . . . , x n ) := N K/k (x 1 ω 1 + · · · + x n ω n ). Throughout this paper we will use N K/k to denote a norm form, and N K/k to denote the corresponding field norm. Progress on this problem has been limited, and we begin by discussing what is known in the simplest cases. A crude measure of difficulty is given by the number of distinct roots of P (t) over an algebraic closurek. When P (t) is a non-zero constant polynomial the Hasse principle for (1.1) is known as the "Hasse norm principle". The validity of the Hasse norm principle for cyclic extensions K/k was established by Hasse himself, but for non-cyclic extensions there can be counterexamples. There is an extensive literature on the subject and it is known, for example, that the Hasse norm principle holds if the field K has prime degree over k (Bartels [1], for example); or the Galois group of N/k is dihedral, where N is the normal closure of K over k (Bartels [2]); or the extension K/k is Galois and every Sylow subgroup of the Galois group is cyclic (Gurak [16,Corollary 3.2]);. Following the work of Colliot-Thélène and Sansuc [6,Proposition 9.1], we also have simple sufficient conditions to ensure that "weak approximation for norms" holds, by which we mean that weak approximation holds for (1.1) when P (t) is a non-zero constant polynomial. Let N be the normal closure of K over k. Then weak approximation holds if either the degree [K : k] is prime, or if the Galois group of N/k has cyclic Sylow subgroups. In particular the latter result implies that it suffices for K/k to be cyclic. The next case to consider is that in which P (t) = ct d for some c ∈ k × and some positive degree d. In this situation the Hasse principle and weak approximation may fail. However (1.1) is a principal homogeneous space under an algebraic k-torus, and the work of Sansuc [23] and Voskresenskiȋ [24] shows that the Brauer-Manin obstruction is the only obstruction to the Hasse principle and weak approximation on any smooth projective model of this variety. When [K : k] = 2 and P (t) has degree 3 or 4 then (1.1) defines a Châtelet surface. The arithmetic of such surfaces is well-understood. The Hasse principle and weak approximation may fail, but it has been shown by Colliot-Thélène, Sansuc and Swinnerton-Dyer [9] that all such failures are explained by the Brauer-Manin obstruction. The same conclusion is available when [K : k] = 3 and deg P (t) 3, by work of Colliot-Thélène and Salberger [4]. There have also been investigations into (1.1) when P (t) factors completely over k, with at most two roots. In this case one may write P (t) = c(t − a) u (t − b) v , with a, b, c ∈ k and u, v ∈ N. It is known that one has the Hasse principle and weak approximation whenever the Brauer-Manin obstruction is empty, providing that we work over the ground field k = Q. This was first proved under the assumption that gcd(u, v, n) = 1, by Heath-Brown and Skorobogatov [18], a condition that was subsequently removed by Colliot-Thélène, Harari and Skorobogatov [8]. While all the previous work described had been purely algebraic, the approach used by Heath-Brown and Skorobogatov combined analytic machinery, in the form of the Hardy-Littlewood circle method, with the previous descent approach to the Brauer-Manin obstruction. The circle method can be adapted, with some effort, to apply to ground fields other than k = Q. However, for simplicity, this possibility was not explored in [18]. Very little is known about other polynomials P (t). When P (t) is a non-zero separable polynomial with degree at least 2, it is conjectured that the Hasse principle and weak approximation hold whenever the Brauer-Manin obstruction is empty. When K/k is cyclic and Schinzel's Hypothesis is granted, work of Colliot-Thélène, Skorobogatov and Swinnerton-Dyer [11,Theorem 1.1] yields a positive answer to this question. Note that this result is already a special case of earlier work of Colliot-Thélène and Swinnerton-Dyer [10] on pencils of Severi-Brauer varieties, but this connection is only made clear in the discussion [11, page 10]. Note that when K/k is cyclic the Brauer-Manin obstruction is known to be empty if P (t) is irreducible over k. (This follows from Corollary 2.6(c) of Colliot-Thélène, Harari and Skorobogatov [8], which shows that the Brauer group contains only vertical elements when K/k is cyclic. However it is not hard to show that the vertical part of the Brauer group is trivial when P (t) is irreducible over k, using the remark on page 76 of [8]. ) Finally we mention that there is potential for tackling the case in which P (t) is an arbitrary polynomial which splits completely over k into d linear factors, at least in the case k = Q, by using ideas from the work of Green and Tao [14] together with the main theorem from Green, Tao and Ziegler [15]. In this case the methods of Heath-Brown and Skorobogatov [18] reduce the problem (1.1) to one involving a system of equations N K/Q (x i ) + a i N K/Q (x 0 ) = c i N K/Q (y i ) = 0, (1 i d − 1). In the language of [14], this is a system of linear forms of finite complexity. The machinery of Green and Tao, and of Green, Tao and Ziegler, allows one to handle such systems when the norms are replaced by primes, and it seems reasonable to hope that a variant of the method would allow one to handle the problem above. This plan was first mentioned to us by Professor Wooley. It will be apparent from the foregoing survey that the most obvious open case is that in which P (t) is an irreducible quadratic over k, and this is the goal of the present paper. Again we shall be dependent on techniques from analytic number theory which have not been fully developed for ground fields other than k = Q, so we shall confine attention to this latter case. With this restriction our goal will be to establish the Hasse principle and weak approximation for P (t) = N K/Q (x 1 , . . . , x n ) = 0, (1.2) under suitable assumptions on the extension K/Q. Let \| · \| ν denote the ν-adic norm, which we extend to vectors by setting \|z\| ν := max 1 i m \|z i \| ν , if z = (z 1 , . . . , z m ). When ν = ∞ we will simply write \| · \| ∞ = \| · \|. With this in mind the following is our main result. Theorem 1. -Let P (t) ∈ Q[t] be an irreducible quadratic polynomial and let K be a quartic extension of Q containing a root of P (t). Suppose that, for every ν ∈ Ω Q , we are given a solution (t (ν) , x (ν) ) ∈ Q 5 ν of (1.2). Let S ⊂ Ω Q be any finite subset and let ε > 0. Then there is a solution (t, x) ∈ Q 5 of (1.2) such that \|t − t (ν) \| ν < ε, \|x − x (ν) \| ν < ε,(1. 3) for every ν ∈ S. Thus the Hasse principle and weak approximation hold for (1.2). It is interesting to note that our result is both unconditional and concerns field extensions K/Q which may be non-cyclic. This marks a departure from the sort of results achieved in [11]. In fact our theorem answers in the affirmative a question posed by Colliot-Thélène, during the 2005 Bremen workshop "Rational points on curves -explicit methods", about the Hasse principle for (1.2) in the special case that K is a biquadratic extension containing a root of P (t) (cf. the questions at the close of §2 in the work of Colliot-Thélène, Harari and Skorobogatov [8]). The proof of Theorem 1 relies on techniques from analytic number theory and is inspired by work of Fouvry and Iwaniec [12], who proved that there are infinitely many primes p of the form a 2 +q 2 , with q also prime. More generally they showed how to produce primes of the form a 2 + q 2 with q from any sufficiently dense set. Our argument involves many complexities of detail, but also one major simplification, since we have only to produce integers in N K/Q (K × ), rather than primes. We can generalise our result mildly, to include the case in which P (t) = cQ(t) u for an odd positive integer u, where Q(t) ∈ Q[t] is an irreducible quadratic polynomial. This is achieved by establishing a bijection between solutions of (1.2) and solutions of the corresponding equation in which P (t) is quadratic. To do this we begin by choosing e, f ∈ Z for which eu + 4f = 1. The equation (1.2) becomes cQ(t) u = N K/Q (x), and raising to the power e we obtain c e Q(t) 1−4f = N K/Q (x) e , whence c e Q(t) = N K/Q (x) e Q(t) 4f . Since K has degree 4 over Q, we deduce that P 0 (t) := c e Q(t) is a norm from K whenever P (t) = cQ(t) u is a norm from K. The converse deduction is similar. Thus if c e Q(t) = N K/Q (x) then, raising both sides to the power u we find that c 1−4f Q(t) u = N K/Q (x) u , whence cQ(t) u = c 4f N K/Q (x) u . Thus P (t) is a norm from K whenever P 0 (t) is a norm from K. It is now immediate from Theorem 1 that we have the Hasse principle and weak approximation for P (t) = cQ(t) u . By a simple change of variable we may assume that P (t) = c(1 − at 2 ) in Theorem 1, where c is a non-zero rational and a is a square-free integer. Let L := Q( √ a). The fields to which our theorem applies take the shape K = L( √ β), with β ∈ L. In particular we have L ⊆ K in the statement of Theorem 1. It turns out that most of our argument carries over to an arbitrary degree n extension K of Q that contains L as a subfield. Given P (t) = c(1 − at 2 ) as above, we suppose that (t (ν) , x (ν) ) ∈ Q n+1 ν are solutions of (1.2), for each ν ∈ Ω Q . Then we want to determine conditions on K, beyond the hypothesis L ⊆ K, such that for any finite set S ⊂ Ω Q we can find a solution (t, x) ∈ Q n+1 of (1.2) for which the weak approximation condition (1.3) holds for each ν ∈ S. In pursuing this goal we may assume that {ω 1 , . . . , ω n } is an integral basis for the ring of integers o K , with ω 1 = 1. By the transitivity of norms we have N K/Q = N L/Q • N K/L , since L ⊆ K. Hence any norm from K to Q is also a norm from L to Q. We will make frequent use of this fact in our work. Since (1 − at 2 ) = N L/Q (1 + t √ a) it follows from the hypotheses of the theorem that the equation c = N L/Q (u + v √ a) can be solved for u, v ∈ Q ν for any ν ∈ Ω Q . The Hasse norm principle therefore implies that there exists δ ∈ L × such that c = N L/Q (δ) −1 . Thus it will suffice to work with the equation 1 − at 2 = N L/Q (δ)N K/Q (x) = 0,(1.4) rather than (1.2), with square-free a ∈ Z and non-zero δ ∈ L. We are then given a finite set S ⊂ Ω Q and a solution (t (ν) , x (ν) ) ∈ Q n+1 ν of this equation for every ν ∈ Ω Q and we wish to establish the existence of a solution (t, x) ∈ Q n+1 such that (1.3) holds for every ν ∈ S. Our plan is to achieve this by arranging that 5) where N K/L (y 1 , . . . , y n ) := N K/L (y 1 ω 1 + · · · + y n ω n ). N K/Q (w)(1 + t √ a) = δN K/L (y) = 0,(1. Then if β 1 = y 1 ω 1 + · · · + y n ω n and β 2 = w 1 ω 1 + · · · + w n ω n we obtain (1.4) on taking x 1 ω 1 + · · · + x n ω n to correspond to the element β = β 1 β −2 2 . It will be convenient to write x = y.w −2 for the vector x produced by this construction. In fact we will establish the following result, which demonstrates the Hasse principle and weak approximation for (1.5), for any number field K of degree n that contains L. Theorem 2. -Let δ ∈ L × and assume that L ⊆ K. Suppose that, for every ν ∈ Ω Q , we are given a solution (t (ν) , w (ν) , y (ν) ) ∈ Q 2n+1 ν of (1.5). Let S ⊂ Ω Q be any finite subset and let ε > 0. Then there is a solution (t, w, y) ∈ Q 2n+1 of (1.5) such that \|t − t (ν) \| ν < ε, \|w − w (ν) \| ν < ε, \|y − y (ν) \| ν < ε, for every ν ∈ S. In § 2 we will show how Theorem 1 follows from this result when K is a quadratic extension of L. This will be achieved via the following result. Lemma 1. -Let K be a quadratic extension of L. Let S ⊂ Ω Q be a finite set and let ε > 0 be given. Suppose that the equation c(1 − at 2 ) = N K/Q (x) = 0 (1.6) has solutions (t (ν) , x (ν) ) everywhere locally. Then there exists δ = δ ε ∈ L with c = N L/Q (δ) −1 such that 1 + t √ a = δN K/L (x) = 0 (1.7) has solutions (t (ν) 0 , x (ν) 0 ) everywhere locally, with \|t (ν) − t (ν) 0 \| ν < ε, \|x (ν) − x (ν) 0 \| ν < ε, for every ν ∈ S. We should emphasise here that when we speak of local solutions we are thinking of zeros over Q ν of the polynomial, defined over Q ν , which specifies the equation. In particular, elements of the completions L µ , for µ \| ν, do not occur. It has been suggested to us by Professor Colliot-Thélène that the open descent method of Colliot-Thélène and Skorobogatov [7] might be used to establish a variant of Lemma 1 in which K is an arbitrary finite extension of L. The proposed lemma would then give the same conclusion as Lemma 1, but under the assumption that the solutions (t (ν) , x (ν) ) of (1.6) produce an adèlic point orthogonal to the Brauer group of the variety. Once combined with Theorem 2, this should demonstrate that the Brauer-Manin obstruction to the Hasse principle and to weak approximation is the only one for (1.2), when P (t) ∈ Q[t] is an irreducible quadratic polynomial and K is an arbitrary extension of Q containing a root of P (t). However this would still leave open the difficult problem of calculating the Brauer group, which our route avoids. By Lemma 1, given local solutions of (1.6), we may produce an equation (1.5) in which w = (1, 0, 0, . . . , 0) and y = x, and which has corresponding local solutions suitably close to those of (1.6). We may then use Theorem 2 to produce a global solution of (1.5) close to the given local solutions. Finally, taking the norm from L to Q we obtain a suitable global solution of (1.6). It should be pointed out that this argument uses the fact that the map from (w, y) to x = y.w −2 is continuous for \| · \| ν providing that we avoid a neighbourhood of w = 0. We stress that the only point in the paper where we use our assumption that K/L is quadratic occurs in the proof of Lemma 1. We proceed to indicate the initial steps in our treatment of (1.5) in Theorem 2. A suitable value of t ∈ Q will exist, providing that Tr L/Q δN K/L (y) = 2N K/Q (w) = 0. (1.8) Moreover, we will then have t = 1 2 Tr L/Q N K/Q (w) −1 δN K/L (y) √ a . Thus if we have a solution of (1.8) in which y and w are sufficiently close to y (ν) and w (ν) it will be automatic that the corresponding solution t will be close to t (ν) . It follows that we have only to establish a suitable Hasse principle and weak approximation result for (1.8). We must make a further manoeuvre before reaching our fundamental equation. As mentioned above, the work of Fouvry and Iwaniec handles primes. Using standard machinery, prime numbers are dealt with by means of "Type I sums" and "Type II sums". Of these, Type II sums involve bilinear forms in which the prime number is replaced by a product of integers uv, which have to lie in suitable ranges. In our case we can insist that our norm N K/L (y) is a product N K/L (u)N K/L (v), thereby eliminating the need to consider Type I sums. Indeed, since we can specify the sizes of u and v, the treatment of the Type II sums will also be simplified somewhat. Thus instead of attacking (1.8) we shall consider the Diophantine equation Tr L/Q δN K/L (u)N K/L (v) = 2N K/Q (w) = 0, with the aim of finding suitably localised solutions (u, v, w) ∈ Z n × Z n × Z n . Let σ denote the non-trivial automorphism of L and suppose that {1, τ } is a Z-basis for o L , and hence also a Q-basis for L. For technical reasons it will be convenient to replace the trace Tr L/Q by a "skew-trace" Tr(x, y) := Tr L/Q (xy σ D −1 L ) for x, y ∈ L, where D L = τ − τ σ . Thus (D L ) is the different of L/Q. On writing x = δN K/L (u), y = N K/L (v)D L σ our condition becomes Tr(x, y) = 2N K/Q (w) = 0. (1.9) We will count suitably restricted solutions of this equation. If N is the number of such solutions we can write N = x∈o L y∈o L α(x)β(y)λ Tr(x, y) . Here the function α(x), respectively β(y), counts appropriately restricted representations of x by δN K/L (u), respectively of y by N K/L (v)D L σ . Moreover λ(l) counts suitably constrained solutions of l = 2N K/Q (w). Our expression for N can be viewed as a bilinear form. We have good techniques for estimating these, going back to the works of Vinogradov. However the methods are designed to produce upper bounds for bilinear forms in which we expect cancellation, while our problem is to establish an asymptotic formula for an expression in which all the terms are non-negative. We shall therefore split α(x) into two parts α(x) =α(x) + α 0 (x), and write N = M + E , where M is a main term, and contains the contribution fromα(x), while E is an error term, and is the corresponding expression involving α 0 (x). Thus we will needα(x) to be a sufficiently simple function that we can compute M directly. Moreover we will want α 0 (x) to produce sufficient cancellation on average, so that a bilinear form estimation of E can be achieved. The underlying principle here is exactly that which Linnik [22] developed in his "dispersion method". In § 4 we will describe a general procedure for producing an approximation of the typê α(x). In our context x runs over the ring o L , but we will begin by presenting the method as it applies to sequences indexed by Z, since we hope this will prove to be of independent interest. With this choice made, our proof that the bilinear form E makes a satisfactory overall contribution to the asymptotic formula is the subject of § 7. It is this part of our argument which is based on ideas from the work of Fouvry and Iwaniec [12], who provide a general framework for estimating sums of this sort. Finally, the asymptotic evaluation of M will be executed in § 8 and § 9. Acknowledgements. -Some of this work was done while the authors were visiting the Hausdorff Institute in Bonn and the Institute for Advanced Study in Princeton, and while the second author was visiting the Mathematical Sciences Research Institute in Berkeley. The hospitality and financial support of these bodies is gratefully acknowledged. While working on this paper the first author was supported by EPSRC grant number EP/E053262/1. Deduction of Theorem 1 The goal of this section is to prove Lemma 1. As explained in the previous section, this is enough to allow the deduction of Theorem 1 from Theorem 2. Let P (t) = c(1 − at 2 ), where c is a non-zero rational and a is a square-free integer. For the moment let K be an arbitrary number field of degree n containing L = Q( √ a). In particular n is even. Fix any δ 0 ∈ L × such that c = N L/Q (δ 0 ) −1 and let S ⊂ Ω Q be finite. We may assume that S contains the archimedean valuation, together with any valuations that become ramified in K, and any valuations ν ∈ Ω Q for which v µ (δ 0 ) = 0 for some µ ∈ Ω L above ν. We proceed to look for a suitable δ, fulfilling the conditions of Lemma 1, by examining values δ = δ 0 γ σ γ −1 . We will see that when ν ∈ S any γ ∈ L × is acceptable. Thus our first task is to find a value γ which works for the "bad" places ν ∈ S. For such valuations we claim that there exist γ (ν) 1 , γ (ν) 2 ∈ Q ν , with γ (ν) = γ (ν) 1 + γ (ν) 2 √ a = 0, such that 1 + t (ν) √ a = δ 0 γ (ν) σ γ (ν) −1 N K/L (x (ν) ). It then suffices to choose γ ∈ L × so that γ is close to γ (ν) for each ν ∈ S, since then we may take t (ν) 0 = t (ν) and find a suitable x (ν) 0 close to x (ν) . To establish the claim we begin by noting that the form N K/L decomposes as N K/L = N 1 + N 2 √ a, (2.1) over L, where N 1 , N 2 ∈ Q[x 1 , . . . , x n ] are forms of degree n/2. Setting δ 0 = δ 1 + δ 2 √ a and γ (ν) = c 1 + c 2 √ a, and multiplying through by γ (ν) , we see that the equation becomes (c 1 + c 2 √ a)(1 + t (ν) √ a) = (c 1 − c 2 √ a)(δ 1 + δ 2 √ a)(N 1 (x (ν) ) + N 2 (x (ν) ) √ a). (2.2) Thus our problem is to show the existence of (c 1 , c 2 ) ∈ Q 2 ν satisfying this, given the condition (1.4), namely 1 − at (ν) 2 = (δ 2 1 − aδ 2 2 )(N 1 (x (ν) ) 2 − aN 2 (x (ν) ) 2 ) = 0. (2.3) If we set A 1 = δ 1 N 1 (x (ν) ) + aδ 2 N 2 (x (ν) ) and A 2 = δ 1 N 2 (x (ν) ) + δ 2 N 1 (x (ν) ) for convenience, then (2.2) becomes a pair of conditions c 1 (1 − A 1 ) + c 2 (at (ν) + aA 2 ) = c 1 (t (ν) − A 2 ) + c 2 (A 1 + 1) = 0. (2.4) We need to find a solution c 1 , c 2 of these, with c 2 1 − ac 2 2 = 0. In doing so we may assume (2.3), which becomes 1 − at (ν) 2 = A 2 1 − aA 2 2 = 0. However the determinant of the system (2.4) is (1 − A 1 )(A 1 + 1) − (at (ν) + aA 2 )(t (ν) − A 2 ) = (1 − at (ν) 2 ) − (A 2 1 − aA 2 2 ) = 0. Moreover, if c 1 = ± √ ac 2 = 0 one readily deduces from (2.2) that 1 − at (ν) 2 = 0, which is impossible. This suffices for the proof of the claim. In handling the case ν ∈ S the following lemma will be useful. It will be proved at the end of the section. Lemma 2. -Let ν ∈ Ω Q be a finite place, unramified in K. Suppose that β = b 1 + b 2 √ a ∈ L × is a unit in L µ for each place µ of L above ν. Then β = N K/L (x) for some x ∈ Q n ν , by which we mean that if N 1 and N 2 are as in (2.1) then b 1 = N 1 (x) and b 2 = N 2 (x). While Lemma 2 is valid for arbitrary extensions K of L, we now make the assumption that K is a quadratic extension of L. In particular we have n = 4 in the above discussion. When ν ∈ S our task is to show that if f (t) := (1 + t √ a)γ(γ σ ) −1 δ −1 0 , then there exists t (ν) 0 ∈ Q ν such that f (t (ν) 0 ) is of the form N K/L (x (ν) 0 ) for some vector x (ν) 0 ∈ Q 4 ν . Since ν ∈ S there are no weak approximation conditions to be satisfied. We begin by considering the case in which ν ∈ S is inert in L/Q. Then v ν (γ σ γ −1 ) = 0 for any γ, and v ν (δ 0 ) = 0 by the choice of S, whence γ(γ σ ) −1 δ −1 0 will be a unit in L ν . Lemma 2 then shows that there exists x (ν) 0 ∈ Q 4 ν such that f (0) = N K/L (x (ν) 0 ) . It therefore suffices to take t (ν) 0 = 0 in (1.7). Finally we must deal with the case ν ∈ S, with ν split in L. Suppose ν splits as µ 1 and µ 2 = µ σ 1 in L. Write p for the rational prime associated to ν. Let v µ 1 (γ) = e 1 , v µ 2 (γ) = e 2 . Let p 1 and p 2 = p σ 1 be the prime ideals associated to µ 1 and µ 2 respectively, so that (γ) = p e 1 1 p e 2 2 g for some ideal g coprime to both p 1 and p 2 . Choose α 1 ∈ p 1 \ (p 2 1 ∪ p 2 ), and set α 2 = α σ 1 . We write e = 2 + \|e 1 \| + \|e 2 \| and h 1 = 1 + 1 2 (\|e 1 \| + \|e 2 \| + e 2 − e 1 ), h 2 = 1 + 1 2 (\|e 1 \| + \|e 2 \| + e 1 − e 2 ), so that e, h 1 and h 2 are integers, and e is strictly positive. Since α i ∈ L and K/L is quadratic, we have N K/L (α i ) = α 2 i . We may then calculate that v µ 1 (1 + p −e √ a) = −e, v µ 1 (γ) = e 1 , v µ 1 (γ σ ) = e 2 , and v µ 1 N K/L (α h 1 1 α h 2 2 ) = 2h 1 v µ 1 (α 1 ) + 2h 2 v µ 1 (α 2 ) = 2h 1 , whence v µ 1 f (p −e )N K/L (α h 1 1 α h 2 2 ) = −e + e 1 − e 2 + 2h 1 = 0. Similarly we have v µ 2 f (p −e )N K/L (α h 1 1 α h 2 2 ) = 0. Thus Lemma 2 tells us that f (p −e )N K/L (α h 1 1 α h 2 2 ) can be written as N K/L (y) for some y ∈ Q 4 ν . It follows that f (p −e ) takes the form N K/L (x 0 ) for some x 0 ∈ Q 4 ν , whence t (ν) 0 = p −e is acceptable in (1.7). This completes the proof of Lemma 1. It now remains to establish Lemma 2, for which we return to an arbitrary extension K of degree n over Q, which contains L as a subfield. It will be convenient to use the notation i µ for the embedding of L into the completion L µ , and similarly for valuations of Q and K. Thus our hypothesis is that i µ (β) is a unit for every µ \| ν. For each such µ we choose a place λ(µ) ∈ Ω K such that λ(µ) \| µ. Then the extension of local fields K λ(µ) /L µ is unramified, whence i µ (β) must be a norm from K λ(µ) (see Gras [13,Corollary 1.4.3, part (ii), page 75], for example). For each µ above ν we may therefore write i µ (β) = N K λ(µ) /Lµ (y λ(µ) ), for appropriate elements y λ(µ) ∈ K λ(µ) . If λ ∈ Ω K lies above µ but is different from λ(µ) we take y λ = i λ (1), so that i µ (1) = N K λ /Lµ (y λ ). We now use weak approximation to find elements y (j) ∈ K such that i λ (y (j) ) − y λ λ < 1 j for every λ ∈ Ω K above ν. We note in particular that the sequence y (j) converges with respect to each valuation λ above ν. We now have lim j→∞ N K λ /Lµ (i λ (y (j) )) = i µ (β), if λ = λ(µ) for some µ \| ν, i µ (1), otherwise, where the limit is with respect to \| · \| µ . We therefore conclude that λ\|µ N K λ /Lµ (i λ (y (j) )) → i µ (β). However, according to Gras [13, Proposition 2.2, page 93] we have λ\|µ N K λ /Lµ (i λ (y (j) )) = i µ (N K/L (y (j) )), so that i µ (N K/L (y (j) )) → i µ (β). Since this holds for all µ above ν it follows that N 1 (x (j) ) → b 1 and N 2 (x (j) ) → b 2 , the convergence being with respect to \| · \| ν , where y (j) = x (j) 1 ω 1 + · · · + x (j) n ω n . Finally, since the sequence y (j) ∈ K converges for every valuation λ above ν, the sequence x (j) ∈ Q n must converge in Q ν , yielding the required vector x ∈ Q n ν . Preliminaries for Theorem 2 We are now ready to begin the proof of Theorem 2, which concerns the Hasse principle and weak approximation for the variety (1.5). For the remainder of the paper let K be an arbitrary number field of degree n that contains L = Q( √ a). As noted, it will be convenient to work with the equivalent variety (1.8). For ease of reference we repeat the definition here:- Tr L/Q δN K/L (y) = 2N K/Q (w) = 0. (3.1) We are presented with local solutions y (ν) , w (ν) for every valuation ν, and wish to find a global solution which approximates these for every ν ∈ S. We claim that it suffices to consider the case in which y (ν) , w (ν) are integral for every finite ν ∈ S. Indeed, let us suppose that we can solve (3.1) with the local conditions \|y − y (ν) \| ν < ε, \|w − w (ν) \| ν < ε,(3.2) providing that y (ν) , w (ν) are integral for all finite ν ∈ S. Let us suppose further that we have a general set of values for y (ν) , w (ν) satisfying (3.1). Then we may choose an integer N ∈ N such that N 2 y (ν) , N w (ν) are integral for all finite ν ∈ S. We note that these values will still satisfy (3.1). Then, by our assumption, we can find a global solution y, w of (3.1) which satisfies \|y − N 2 y (ν) \| ν < ε\|N 2 \| ν , \|w − N w (ν) \| ν < ε\|N \| ν for all ν ∈ S. It follows that N −2 y, N −1 w is a solution of (3.1) fulfilling the condition (3.2). This establishes our claim. We now use weak approximation for Z n to produce vectors y (M ) , w (M ) ∈ Z n such that \|y (M ) − y (ν) \| ν < ε, \|w (M ) − w (ν) \| ν < ε for all finite ν ∈ S. Thus (3.2) becomes y ≡ y (M ) (mod M ), w ≡ w (M ) (mod M ) (3.3) for an appropriate modulus M ∈ N. Having suitably re-interpreted the weak approximation conditions for the finite places we turn our attention to the infinite place. Here we use a similar re-scaling argument to conclude that if Y, W ∈ N satisfy W ≡ 1 (mod M ) and Y = W 2 , then a solution y, w of (3.1) which satisfies both (3.3) and the ν = ∞ constraints \|y − Y y (R) \| < εY, \|w − W w (R) \| < εW, (3.4) gives rise to a solution Y −1 y, W −1 w of (3.1) which meets the original condition (3.2). Since y (R) and w (R) cannot vanish we may choose ε sufficiently small that neither of y or w can be zero in (3.4). As in § 1 we replace the vector y by u and v to produce the variety (1.9), whose definition it is convenient to repeat here:- J : Tr δN K/L (u), (N K/L (v)D L ) σ = 2N K/Q (w) = 0. (3.5) It is clear that if u and v are sufficiently close to u (ν) := y (ν) and v (ν) := (1, 0, 0, . . . , 0) (3.6) in Q ν then y will be suitably close to y (ν) . We therefore assume that (3.5) has local solutions u (ν) , v (ν) , w (ν) for all places ν, with v (ν) given by (3.6), and we aim to find a global solution such that \|u − u (ν) \| ν < ε, \|v − v (ν) \| ν < ε, \|w − w (ν) \| ν < ε,(3.7) for ν ∈ S. Since u (ν) and v (ν) are integral at all finite ν ∈ S we can re-interpret the corresponding conditions as congruences u ≡ u (M ) (mod M ), v ≡ v (M ) (mod M ) with integer vectors u (M ) and v (M ) for which v (M ) := (1, 0, 0, . . . , 0). (3.8) For technical reasons we will move u (R) in (3.5) very slightly, and make a corresponding adjustment in w (R) to compensate, so as to ensure that ∂N K/Q (u (R) ) ∂u i = 0, (1 i n). (3.9) For the infinite place we replace the parameter Y by two further values U and V satisfying U V = Y and impose the conditions \|u − U u (R) \| < εU, \|v − V v (R) \| < εV instead of \|y − Y y (R) \| < εY . We can now summarise our conclusions in the following result. Lemma 3. -Suppose we are given local solutions of (3.5) for every valuation ν of Q, subject to the condition (3.6). Let ε > 0 also be given. Then there is a modulus M ∈ N having \|M \| ν < 1 for all finite ν ∈ S, and a solution ( u (M ) , v (M ) , w (M ) ) of (3.5) over Z/M Z satisfying (3.8), having the following property. Let V, H 0 be integer parameters with H 0 ≡ V ≡ 1 (mod M ). Let H = H 2 0 and suppose that V H 1. If U = HV, W = H 1/2 V, then any solution (u, v, w) ∈ Z 3n of (3.5) satisfying u ≡ u (M ) (mod M ), v ≡ v (M ) (mod M ), w ≡ w (M ) (mod M ),(3. 10) and \|u − U u (R) \| < εU, \|v − V v (R) \| < εV, \|w − W w (R) \| < εW, (3.11) gives rise to a global solution of (3.5) satisfying (3.7). Moreover, for every finite place ν ∈ S there is a solution (u, v, w) ∈ Z 3n ν of (3.5) satisfying (3.10). It might help the reader at this point to say more about the rôle of the parameters H and V . We shall think of H as being a small fixed power of V . When we estimate error terms in our analysis we cannot afford to lose any power V θ of V , unless θ can be taken arbitrarily small. On the other hand there will be certain points in our argument where we will lose factors of V η with arbitrary small η > 0. This will not matter since we will make a key saving which is a power of H, so that there is a net gain overall. With this in mind, many of our estimates will involve factors of the type V O(η) . These involve the standard convention that there are implicit order constants for each occurrence of the O(·) notation, which need not be the same on each occasion. Since we are taking the degree n of K to be fixed, we will allow these implicit order constants to depend on n. Recalling that H V we may replace terms involving any combinations of H η , U η and W η by V O(η) . The number η will be a sufficiently small positive constant, which will be fixed throughout the proof. We could have chosen to specify its value at the outset of the argument, but we feel it is more instructive merely to impose the condition that η is sufficiently small, at various points in the proof. We are now ready to cast our problem in terms of a bilinear form. If R is any ring it will be convenient to write J (R) for the set of solutions (u, v, w) of the equation Henceforth we will allow the constants implied by the notations ≪, ≫ and O(·) to depend on u (R) , v (R) , w (R) , u (M ) , v (M ) , w (M ) , M, δ, L, K, and ε, which are to be regarded as fixed once and for all. In our work we will restrict the values over which v runs by stipulating that if N K/L (v) = N 1 (v) + N 2 (v)τ then gcd(N 1 (v), N 2 (v)) = 1. (3.12) In particular it follows that gcd(a, b) = O(1) for (N K/L (v)D L ) σ = a + bτ. (3.13) We are now ready to specify the sets over which we will sum. In the case of the variable u there is a technical point to be dealt with in § 5. For the time being we give ourselves independent linear forms L 1 (u), . . . , L n (u) whose rôle will become clear later. Let G be a further parameter, tending to infinity with V , which we assume is in the range 1 G H. We then define the regions U := u ∈ R n : max 1 i n \|L i (u) − U L i (u (R) )\| < G −1 U , V := v ∈ R n : \|v − V v (R) \| < G −1 V , W := {w ∈ R n : \|w − W w (R) \| < G −1 W }. (3.14) In order to interpret our counting function N (H, V ) as a bilinear form, we let α(x) := #{u ∈ U ∩ Z n : u ≡ u (M ) (mod M ), δN K/L (u) = x} (3.15) and β(y) := # v ∈ V ∩ Z n : v ≡ v (M ) (mod M ), (3.12) holds and (N K/L (v)D L ) σ = y , (3.16) for x, y ∈ o L . Lastly we define the function λ(l) := # w ∈ W ∩ Z n : w ≡ w (M ) (mod M ), 2N K/Q (w) = l ,(3.17) on Z. Notice that u (R) , v (R) and w (R) are all non-zero, whence u, v and w will be non-zero throughout U , V and W , if G is large enough. It follows in particular that α, β and λ are supported on non-zero x, y ∈ o L and l ∈ Z. We now define the bilinear form N (G, H, V ) := x∈o L y∈o L α(x)β(y)λ Tr(x, y) . It is easy to check that N (H, V ) N (G, H, V ) whenever G ≫ ε −1 , where the implied constant is allowed to depend on the linear forms L 1 , . . . , L n , a convention that we adhere to for the remainder of the paper. It now suffices to demonstrate that N (G, H, V ) > 0 for large values of G. Note that we will ultimately take G = log V. Although we have framed N (G, H, V ) as a bilinear form, it is not an upper bound for N (G, H, V ) that we seek but an asymptotic formula as G → ∞. As indicated in the introduction we will begin by extracting a main term from our expression for N (G, H, V ). Instrumental in this will be finding a decomposition α(x) =α(x) + α 0 (x), for an appropriate approximationα(x) to α(x). We will then write N (G, H, V ) = M (G, H, V ) + E (G, H, V ), (3.18) where M (G, H, V ) := x∈o L y∈o Lα (x)β(y)λ Tr(x, y) is regarded as the main term and E (G, H, V ) := x∈o L y∈o L α 0 (x)β(y)λ Tr(x, y) is the error term. The handling of E (G, H, V ) will be executed in § 7 and the estimation of M (G, H, V ) will be the subject of § 8 and § 9. Our treatment of E (G, H, V ) requires bounds for x∈R x≡x 0 (mod h) α 0 (x) uniformly for small moduli h and square regions R. Thus our approximationα will have to be such that α 0 averages to zero over all congruence classes to small moduli. This will be achieved via a quite general procedure described in the next section. Our estimate for E (G, H, V ) will also require bounds for x∈o L \|α 0 (x)\| 2 , y∈o L \|β(y)\| 2 and l∈Z \|λ(l)\| 2 , for which we have the following result. Lemma 4. -For any η > 0 we have x∈o L \|α(x)\| 2 ≪ η U n+η , y∈o L \|β(y)\| 2 ≪ η V n+η , and l∈Z \|λ(l)\| 2 ≪ η W n+η . Since \|α 0 (x)\| 2 2\|α(x)\| 2 + 2\|α(x)\| 2 we will also require a bound for x∈o L \|α(x)\| 2 . This will be established in § 5. We conclude this section by proving Lemma 4. We will discuss the upper bound for the case of β, the remaining estimates being dealt with similarly. Let h(v) = (N K/L (v)D L ) σ . We shall show that if v ′ is given, then there are O η (V η ) choices of v for which h(v) = h(v ′ ). We set ̟ = h(v ′ ), so that ̟ ∈ o L \{0}. If ρ = v 1 ω 1 +· · ·+v n ω n ∈ o K , then ρ \| ̟ in o K ,≪ ξ V ξ \|N L/Q (̟)\| ξ ≪ ξ V ξ (V n ) ξ = V (1+n)ξ for any ξ > 0. If we now take ξ = η/(1 + n) it follows that the number of v corresponding to a given v ′ is O η (V η ), as required. The fact that a given value ̟ = h(v) is attained O(V η ) times will be used at various points in the rest of the paper without further comment. Similarly, we shall use estimates O(V η ) for the number of representations of ̟ as N K/L (u) with u ∈ U , or of l as 2N K/Q (w) with w ∈ W . A general approximation principle Our goal now is to split α(x) into two parts α(x) =α(x)+α 0 (x), whereα(x) is a sufficiently simple function that we can compute M (G, H, V ) directly. Moreover we will want α 0 (x) to produce sufficient cancellation on average, so that a bilinear form estimation of E (G, H, V ) can be achieved. We start by describing a general procedure for producing an approximation of the typê α(x). In our context x runs over the ring o L , but we will begin by presenting the method as it applies to sequences indexed by Z, since we hope this will prove to be of independent interest. The underlying ideas are perhaps not new. In particular there are certain similar features in recent work of Brüdern [3]. However we have not found anything in the literature which exactly meets our needs. We begin by supposing that we are given a sequence k(1), . . . , k(N ) of complex numbers. We aim to show how to approximate k(n) locally by a functionk(n). By this we mean that for any congruence class a (mod q) the sum S(a, q) := n N n≡a (mod q) k(n) will be approximated byŜ (a, q) := n N n≡a (mod q)k (n), at least for small values of q. Naturally we can do this by merely settingk(n) = k(n), but we seek a functionk(n) which is defined in terms of the density of the sequence k(n) in congruence classes to small moduli. As an example of what we have in mind, consider the sequence k(n) = Λ(n), where Λ(n) is the von Mangoldt function. If we choose any constant A 3 and set Λ Q (n) := q Q µ(q) ϕ(q) a (mod q) gcd(a,q)=1 e q (an), with Q = (log x) A , then n x n≡b (mod h) Λ(n) = n x n≡b (mod h) Λ Q (n) + O A (x(log x) −A/2 ) (4.1) uniformly for h Q, for all residue classes b (mod h). This result follows from Heath-Brown [17, Lemma 1]. We first introduce our fundamental hypotheses. We assume that we have an arithmetic function ρ(a, q) and a "smooth" function ω(n), together with a bound E 1 such that \|S(a, q) − ρ(a, q)S\| E, (4.2) with S = n N ω(n), for all residue classes a to moduli q Q. We observe that a (mod rs) a≡b (mod r) S(a, rs) = S(b, r) for all b, r, s, and we therefore impose the natural condition that a (mod rs) a≡b (mod r) ρ(a, rs) = ρ(b, r) (4.3) for all b, r, s. Since we can always re-scale the functions ρ and ω in (4.2) there is no loss in generality in assuming that ρ(0, 1) = 1. (4.4) Although it is not necessary in general, it will prove convenient to assume that ρ(a, q) ∈ R, ρ(a, q) 0 for all pairs a, q. We will also require a smoothness condition on the function ω(n), which we formulate as the bound n N n≡a (mod q) ω(n) − q −1 S W, (4.5) for all residue classes a to moduli q Q 2 . Our choice fork(n) will be motivated by the treatment of major arcs in the circle method. If we consider the exponential sum Σ(α) := n N k(n)e(αn), then when α is close to a/q one would use the major-arc approximation    c (mod q) ρ(c, q)e q (ac)       n N ω(n)e((α − a/q)n)    . When α is not close to a/q the above expression tends to be small. Hence it is reasonable to approximate Σ(α) by q Q a (mod q) gcd(a,q)=1    c (mod q) ρ(c, q)e q (ac)       n N ω(n)e((α − a/q)n)    for all real α. Picking out the coefficient of e(αn) in the above expression we are therefore led to suggest the choicê k(n) := ω(n) q Q a (mod q) gcd(a,q)=1 e q (−an) c (mod q) ρ(c, q)e q (ac). (4.6) We proceed to investigate the sumŜ(b, h) with h Q. We havê S(b, h) = n N n≡b (mod h) ω(n) q Q c (mod q) ρ(c, q) a (mod q) gcd(a,q)=1 e q (a(c − n)). The final sum over a (mod q) is a Ramanujan sum, for which the standard evaluation as d\|gcd(q,c−n) dµ(q/d) producesŜ (b, h) = q Q c (mod q) ρ(c, q) d\|q dµ(q/d) n N n≡b (mod h) n≡c (mod d) ω(n). The simultaneous congruences n ≡ b (mod h) and n ≡ c (mod d) are only soluble if gcd(d, h) divides c− b, in which case there is a unique solution e (mod [d, h]) say. Thus if gcd(d, h) \| c− b our hypothesis (4.5) shows thatŜ (b, h) = M + E , with M = S q Q c (mod q) ρ(c, q) d\|q gcd(d,h)\|c−b dµ(q/d) [d, h] and \|E \| W q Q c (mod q) ρ(c, q) d\|q d\|µ(q/d)\|. In view of (4.3) and (4.4) we have c (mod q) ρ(c, q) = ρ(0, 1) = 1, whence the crude bound d\|q d\|µ(q/d)\| q 2 yields \|E \| W Q 3 . Turning to the main term M we observe in general that d\|q gcd(d,h)\|k dµ(q/d) [d, h] = h −1 d\|q gcd(d,h)\|k µ(q/d)(d, h) = h −1 d\|q µ(q/d) e\|d,h,k e f \|d/e,h/e µ(f ) = h −1 e\|q,h,k e f \|q/e,h/e µ(f ) d\|q ef \|d µ(q/d) = h −1 e\|q,h,k e f \|q/e,h/e µ(f ) g\|q/(ef ) µ q/(ef ) g . The final sum vanishes unless ef = q, in which case we must have q \| h. It then follows that d\|q gcd(d,h)\|k dµ(q/d) [d, h] = h −1 e\|q,k eµ(q/e) = h −1 a (mod q) gcd(a,q)=1 e q (ak). Inserting this result into our formula for the main term M , we see that M = Sh −1 q\|h c (mod q) ρ(c, q) a (mod q) gcd(a,q)=1 e q (a(c − b)) . Using (4.3) we have ρ(c, q) = d (mod h) d≡c (mod q) ρ(d, h), whence M = Sh −1 d (mod h) ρ(d, h) q\|h a (mod q) gcd(a,q)=1 e q (a(d − b)). As q runs over divisors of h, and a runs over residue classes coprime to q, the fractions a/q run over the entire set n h : 0 n < h . We therefore deduce that M = Sh −1 d (mod h) ρ(d, h) n (mod h) e h (n(d − b)) = Sρ(b, h), since the summation over n produces the value h when d ≡ b (mod h), and the value 0 otherwise. In view of (4.2) we may now summarise our results as follows. Lemma 5. -With the above assumptions we have \|S(b, h) −Ŝ(b, h)\| W Q 3 + E for all h Q and all residue classes b modulo h. Thusk(n) approximates k(n) well, in congruence classes to small moduli. It may be instructive to consider the effect of this procedure on the sequence k(n) = Λ(n) that we discussed earlier. Taking ρ(a, q) = 1/ϕ(q), if gcd(a, q) = 1, 0, otherwise, we readily deduce from the Siegel-Walfisz theorem that for any A 1 there exists a constant C A > 0 such that n N n≡a(mod q) Λ(n) = ρ(a, q)N + O N exp(−C A (log N ) 1/2 ) , uniformly for q (log N ) A . On taking ω(n) = 1 we see that E ≪ N exp(−C A (log N ) 1/2 ) is admissible in (4.2). Since W ≪ 1 in (4.5), the approximation in (4.1) is a trivial consequence of Lemma 5 with Λ Q (n) = ω(n) q Q a (mod q) gcd(a,q)=1 e q (−an) b (mod q) gcd(b,q)=1 e q (ab) ϕ(q) = q Q µ(q) ϕ(q) a (mod q) gcd(a,q)=1 e q (an), for q (log N ) A . In applications it may be important to know about the size ofk(n), and we therefore investigate the mean square Σ := n N \|k(n)\| 2 . If we write c a,q : = b (mod q) ρ(b, q)e q (ab) thenk (n) = ω(n) q Q a (mod q) gcd(a,q)=1 c a,q e q (−an). Thus, if we assume that \|ω(n)\| ω 0 for all n ∈ N, then the dual large sieve produces Σ ω 2 0 n N q Q a (mod q) gcd(a,q)=1 c a,q e q (−an) 2 ω 2 0 (N + Q 2 ) q Q a (mod q) gcd(a,q)=1 \|c a,q \| 2 . In view of (4.2) we have \|Sc a,q − T (a, q)\| qE, with T (a, q) := b (mod q) S(b, q)e q (ab) = n N k(n)e q (an). It follows that \|Sc a,q \| 2 2\|T (a, q)\| 2 + 2q 2 E 2 , and hence that \|S\| 2 Σ 2ω 2 0 (N + Q 2 )     E 2 Q 4 + q Q a (mod q) gcd(a,q)=1 \|T (a, q)\| 2     . We now apply the standard large sieve to deduce that \|S\| 2 Σ 2ω 2 0 (N + Q 2 )   E 2 Q 4 + (N + Q 2 ) n N \|k(n)\| 2   . We express this result formally in the following lemma. Lemma 6. -We have n N \|k(n)\| 2 2\|S\| −2 ω 2 0 (N + Q 2 )   E 2 Q 4 + (N + Q 2 ) n N \|k(n)\| 2   . Moreover n N \|k(n)\| 2 ≪ ω 0 N \|S\| 2   N + n N \|k(n)\| 2   , providing that EQ 2 N . Thus under suitable circumstances the L 2 -norm ofk(n) will have order of magnitude bounded by the L 2 -norm of k(n). Having described the situation for sequences k(n) indexed by Z we return to our original problem, in which we have a sequence α(x) with x running over o L . We will describe the situation as it applies to a quite general function α, and only later, in § 5, restrict to the function (3.15). We shall consider sums in which x runs over a region R, say, and lies in a congruence class y (mod q), where y ∈ o L and q ∈ N. We therefore set S(y, q) := x∈R x≡y (mod q) α(x) (4.7) andŜ (y, q) := x∈R x≡y (mod q)α (x), and assume that we have functions ρ(y, q) and ω(x) such that ρ(y, q) is non-negative and \|S(y, q) − ρ(y, q)S\| E (4.8) for q Q, with E 1 and S := x∈R ω(x). As before we will assume that y (mod rs) y≡z (mod r) ρ(y, rs) = ρ(z, r) (4.9) and ρ(0, 1) = 1. (4.10) Finally, we shall suppose that x∈R x≡y (mod q) ω(x) − q −2 S W,(4.11) for all residue classes y to moduli q Q 2 . In order to defineα(x) we will need an analogue of the coprimality condition gcd(a, q) = 1 which occurs in (4.6). It turns out that the correct generalisation is to require that there is no non-trivial common divisor of y and q in N. Of course this is not the same as requiring gcd(y, q) = 1 in o L . We write * y (mod q) to denote a sum in which y ∈ o L runs over residue classes modulo q such that y and q have no non-trivial common divisor in N. We also need an analogue for the exponential function e q (a). Recall that {1, τ } is a Z-basis for o L , and hence also a Q-basis for L. If x = a + bτ ∈ L, with a, b ∈ Q, and if q ∈ N, we define e (L) (x) := e(b) and e (L) q (x) := e q (b). These exponential functions have the property that if y ∈ o L then x (mod q) e (L) q (xy) = q 2 , if q \| y, 0, if q ∤ y. Thus, for the analogue of the Ramanujan sum we have We are now ready to specify our approximation to α(x). We set α(x) := ω(x) q Q * y (mod q) e (L) q (−yx) z (mod q) ρ(z, q)e (L) q (yz). (4.13) An argument precisely analogous to that used for Lemma 5 then produces the following result. To produce an analogue of Lemma 6 we shall assume that R = {a + bτ ∈ o L : \|a − a 0 \|, \|b − b 0 \| < N } for certain a 0 , b 0 . Then we will have a large sieve inequality for o L , taking the form q Q * y (mod q) x∈R c x e (L) q (xy) 2 ( √ 2N + Q) 4 x∈R \|c x \| 2 . (4.14) This follows from the two dimensional large sieve of Huxley [20,Theorem 1]. Under the condition that Q 2 N , which we now impose, we will then have ( √ 2N + Q) 4 ≪ N 2 . We proceed to argue as before to deduce from the dual of the above estimate that x∈R \|α(x)\| 2 ≪ ω 2 0 N 2 q Q * y (mod q) \|c y,q \| 2 , with c y,q = z (mod q) ρ(z, q)e (L) q (yz). This time the estimate (4.8) yields \|Sc y,q − T (y, q)\| q 2 E, with T (y, q) = x∈R α(x)e (L) q (xy). Continuing as before we then obtain \|S\| 2 x∈R \|α(x)\| 2 ≪ ω 2 0 N 2 E 2 Q 7 + N 2 x∈R \|α(x)\| 2 . This gives us the following conclusion. Lemma 8. -Suppose that R = {a + bτ ∈ o L : \|a − a 0 \|, \|b − b 0 \| < N } for certain a 0 , b 0 , and assume that there is a constant ω 0 such that \|ω(x)\| ω 0 for all x ∈ R. Then x∈R \|α(x)\| 2 ≪ ω 0 N 2 \|S\| 2 N 2 + x∈R \|α(x)\| 2 , providing that Q 2 N and E 2 Q 7 N 4 . The functionsα(x) and α 0 (x) In this section we will apply Lemmas 7 and 8 to the function (3.15). It will be important for us to produce functionsα and ω which depend only on the set U , and not on the set R, since we require results which are uniform in R. We will write x ∈ o L in the form x = x 1 +x 2 τ with x 1 , x 2 ∈ Z. We shall also write δN K/L (u) as N 1 (u) + N 2 (u)τ . The reader should observe that this notation does not coincide with that used temporarily in (2.1), nor that used in our discussion of (3.12). We let R be a square in the (x 1 , x 2 )-plane, with sides parallel to the x 1 and x 2 axes, and side-length ρ ≪ U n/2 . This corresponds, by abuse of notation, to the square R used in defining the sums S(y, q) in (4.7). Extending this abuse of notation we shall allow δN K/L (u) to denote the ordered pair (N 1 (u), N 2 (u)). We will specify the function ω(x) = ω(x 1 , x 2 ) and verify its properties at the end of this section. For the time being we content ourselves with describing the key features as follows. Lemma 9. -There is a continuously differentiable function ω : R 2 → [0, ∞), depending on U , for which ω(x + h) − ω(x) ≪ U −n/2 \|h\| for all x, h ∈ R 2 , and such that R ω(x 1 , x 2 ) dx 1 dx 2 = M n meas{u ∈ U : δN K/L (u) ∈ R},(5.1) for every square R as above. Furthermore ω is supported on a disc of radius O(U n/2 ) and satisfies ω(x) ≪ 1 throughout this disc. Moreover there is a disc of radius ≫ G −1 U n/2 around the point U n/2 δN K/L (u (R) ) on which we have ω(x) ≫ G 2−n . Although it is a real measure which occurs on the right-hand-side of (5.1) what occurs naturally for us is the corresponding cardinality S 0 := #{u ∈ U ∩ Z n : δN K/L (u) ∈ R}. We proceed to establish a relation between the two. For each u ∈ U ∩ Z n we let S(u) := {y ∈ R n : u i y i < u i + 1, (1 i n)} and U (−) := {S(u) : u ∈ Z n , S(u) ⊆ U }. Thus U (−) ⊆ U , and the number of integer vectors in U but not U (−) is O(U n−1 ). In particular, S 0 = #{u ∈ U (−) ∩ Z n : δN K/L (u) ∈ R} + O(U n−1 ). (5.2) Now if u ∈ U (−) ∩ Z n and y ∈ S(u) then δN K/L (y) = δN K/L (u) + O(U n/2−1 ). Thus there are squares R 1 and R 2 of sides ρ 1 and ρ 2 respectively, such that \|ρ 1 − ρ\| ≪ U n/2−1 and \|ρ 2 − ρ\| ≪ U n/2−1 , and for which δN K/L (y) ∈ R 1 ⇒ δN K/L (u) ∈ R ⇒ δN K/L (y) ∈ R 2 whenever y ∈ S(u). We deduce that #{u ∈ U (−) ∩ Z n : δN K/L (u) ∈ R} meas{y ∈ U (−) : δN K/L (y) ∈ R 1 } and #{u ∈ U (−) ∩ Z n : δN K/L (u) ∈ R} meas{y ∈ U (−) : δN K/L (y) ∈ R 2 }. However U (−) is contained in U and differs from it by a set of measure O(U n−1 ), whence meas{y ∈ U (−) : δN K/L (y) ∈ R 1 } meas{y ∈ U : δN K/L (y) ∈ R 1 } + O(U n−1 ) and meas{y ∈ U (−) : δN K/L (y) ∈ R 2 } meas{y ∈ U : δN K/L (y) ∈ R 2 }. According to (5.1) we may then deduce that #{u ∈ U (−) ∩ Z n : δN K/L (u) ∈ R} M −n R 1 ω(x 1 , x 2 ) dx 1 dx 2 + O(U n−1 ) and #{u ∈ U (−) ∩ Z n : δN K/L (u) ∈ R} M −n R 2 ω(x 1 , x 2 ) dx 1 dx 2 . However the squares R 1 and R 2 each differ from R by a set of measure O(U n−1 ), and furthermore ω ≪ 1. Thus #{u ∈ U (−) ∩ Z n : δN K/L (u) ∈ R} = M −n R ω(x 1 , x 2 ) dx 1 dx 2 + O(U n−1 ). Finally, if we compare this with (5.2), we deduce that S 0 = M −n R ω(x 1 , x 2 ) dx 1 dx 2 + O(U n−1 ). (5.3) We next consider x∈R x≡y (mod q) ω(x). To each point x counted in the above sum we associate the square R( x) = [x 1 , x 1 + q) × [x 2 , x 2 + q). If t ∈ R(x) then ω(t) − ω(x) ≪ qU −n/2 by Lemma 9, whence R(x) ω(t) dt 1 dt 2 = q 2 ω(x) + O(q 3 U −n/2 ). We sum this for points x ∈ R with x ≡ y (mod q). Since there are O(q −2 U n ) such points if q U n/2 we deduce that x∈R x≡y (mod q) ω(x) = q −2 x∈R x≡y (mod q) R(x) ω(t) dt 1 dt 2 + O(qU n/2 ). The union of the squares R(x) will be a square R ′ say, whose sides are within a distance q of the sides of R. Thus R and R ′ differ by a set of measure O(qU n/2 ), if q U n/2 . Since ω(t) ≪ 1 for all t this produces x∈R x≡y (mod q) ω(x) = q −2 R ′ ω(t) dt 1 dt 2 + O(qU n/2 ) = q −2 R ω(t) dt 1 dt 2 + O(qU n/2 ). Comparing this with (5.3) we therefore obtain M −n x∈R x≡y (mod q) ω(x) = q −2 S 0 + O(U n−1 ), providing that n 4 and q U . In particular, when q = 1 we obtain S = M n S 0 + O(U n−1 ), (5.4) so that (4.11) holds with W ≪ U n−1 and any Q U 1/2 . We next consider the condition (4. If u ∈ T (r, w) then u * will belong to T (r, 0) unless either u is within a distance O(r) of the boundary of U , or δN K/L (u) is within a distance O(rU n/2−1 ) of the boundary of R. A pair (N 1 , N 2 ) arises at most O η (U η ) times as a value of δN K/Q (u) with u ∈ U ∩ Z n , and hence it follows that #T (r, 0) = #T (r, w) + O η (rU n−1+η ) (5.5) for any η > 0. If we now sum for all w (mod r) we deduce that r n #T (r, 0) = #{u ∈ U ∩ Z n : δN K/L (u) ∈ R} + O η (r n+1 U n−1+η ) and hence that r n #T (r, 0) = S 0 + O η (r n+1 U n−1+η ). Substituting this back into (5.5) we deduce that #T (r, w) = r −n S 0 + O η (rU n−1+η ). We therefore see that The conditions (4.9) and (4.10) are now readily checked, and we see that (4.8) follows from (5.4) with E ≪ η Q n+1 U n−1+η . S(y, q) = M −n ρ(y, q)S 0 + O η (q n+1 U n−1+η ), We are now ready to apply Lemma 7, which produces the following result. α 0 (x) ≪ η Q n+1 U n−1+η , for all q Q U 1/2 and all y (mod q). In our application the square R will vary and so it is crucial that Lemma 10 is uniform in squares of side length O(U n/2 ). As to Lemma 8, we will only need a result for the L 2 -norm taken over all x. Hence we choose R to be a square centred at the origin and with side ρ = cU n/2 , in which the constant c is taken sufficiently large that x ∈ R wheneverα(x) = 0. We may clearly take ω 0 ≪ 1 by Lemma 9 and x∈R \|α(x)\| 2 ≪ η U n+η by Lemma 4. Our remaining task is thus to estimate S from below. However our choice of R ensures that S 0 = #(U ∩ Z n ) ≫ U n G −n , whence (5.4) yields S ≫ U n G −n , assuming that G U 1/(n+1) , say. Consequently we deduce from Lemma 8 the following bound. Lemma 11. -For any constant η > 0 we have x∈o L \|α(x)\| 2 ≪ η U n+η G O(1) providing that Q n+5 U 1−η and G U 1/(n+1) . Moreover we then have x∈o L \|α 0 (x)\| 2 ≪ η U n+η G O(1) . The final estimate follows immediately from Lemma 4, since x∈o L \|α 0 (x)\| 2 ≪ x∈o L \|α(x)\| 2 + x∈o L \|α(x)\| 2 . We end this section by proving Lemma 9. We first show that the map from C n to C 2 given by u → (N 1 (u), N 2 (u)) is non-singular at any point for which N K/Q (u) = 0, by which we mean that ∇N 1 (u) and ∇N 2 (u) are only proportional if N K/Q (u) = 0. Clearly it suffices to do the same for the map u → (N 1 (u) + τ N 2 (u), N 1 (u) + τ σ N 2 (u)). However N 1 (u) + τ N 2 (u) (respectively, N 1 (u) + τ σ N 2 (u)) is a product of δ (respectively, δ σ ) with certain conjugates of u 1 ω 1 + · · · + u n ω n . Thus we can write our map in the form u → (Λ 1 (u) · · · Λ n/2 (u) , Λ n/2+1 (u) · · · Λ n (u)), with suitable linear forms Λ i , which will be linearly independent. Hence if we set v i = Λ i (u) for 1 i n it will suffice to show that the map v → (v 1 · · · v n/2 , v n/2+1 · · · v n ) is non-singular whenever v 1 · · · v n = 0. This however is trivial. Since N K/Q (u (R) ) = 0, we deduce that there are indices i and j such that ∂N 1 (u (R) ) ∂u i ∂N 2 (u (R) ) ∂u j > ∂N 1 (u (R) ) ∂u j ∂N 2 (u (R) ) ∂u i . We suppose for notational simplicity that i = 1 and j = 2, the other cases being similar. By continuity we then have J(u) := ∂N 1 (u) ∂u 1 ∂N 2 (u) ∂u 2 − ∂N 1 (u) ∂u 2 ∂N 2 (u) ∂u 1 ≫ U n−2 (5.7) throughout U , if G is large enough. We now write v = (u 1 , u 2 ) and u = (v, w), where w = (u 3 , . . . , u n ). We will find it convenient to number the entries in w as (w 3 , . . . , w n ). By the Implicit Function Theorem, if N 1 (v, w) = x 1 , N 2 (v, w) = x 2 with (v, w) ∈ U , we can express v in terms of x and w as v = v(x, w). We may now calculate that meas{u ∈ U : δN K/L (u) ∈ R} = x∈R w∈W (x) J(v(x, w), w) −1 dw 3 · · · dw n dx 1 dx 2 , where W (x) := {w ∈ R n−2 : (v(x, w), w) ∈ U }. We therefore define ω(x) := M n w∈W (x) J(v(x, w), w) −1 dw 3 · · · dw n ,(5.8) so that (5.1) immediately follows. It is clear from the definition of W (x) that ω is supported on the set of values δN K/L (u) as u runs over U . Thus the support is contained in a disc of radius O(U n/2 ) as claimed in the lemma. Moreover meas{W (x)} meas{w ∈ R n−2 : (v, w) ∈ U for some v} ≪ U n−2 , (5.9) whence (5.7) yields ω(x) ≪ 1 as required. The function v(x, w) will be continuously differentiable with respect to x and w. Moreover it will be weighted-homogeneous of degree 1 in w and degree 2/n in x. Since ∂x k /∂v l ≪ U n/2−1 for 1 k, l 2 we deduce from (5.7) that ∂v k ∂x l ≪ J(u) −1 U n/2−1 ≪ U 1−n/2 , (1 k, l 2). (5.10) We observe from (3.9) that ∂N K/Q (u (R) )/∂w 3 does not vanish, whence ∂N j (u (R) )/∂w 3 = 0 for at least one of j = 1 or j = 2. Moreover ∂N j (v(x, w), w) ∂w 3 = ∂x j ∂w 3 = 0 for j = 1, 2, whence ∂N j ∂v 1 ∂v 1 ∂w 3 + ∂N j ∂v 2 ∂v 2 ∂w 3 + ∂N j ∂w 3 = 0, (j = 1, 2). Thus at least one of ∂v 1 /∂w 3 and ∂v 2 /∂w 3 is non-vanishing at u (R) . We suppose that ∂v 1 /∂w 3 = 0, the alternative case being similar. By continuity we then have \|∂v 1 /∂w 3 \| ≫ 1 for \|u − u (R) \| ≪ G −1 , if G is large enough. It follows that ∂v 1 ∂w 3 ≫ 1 (5.11) throughout U , since the partial derivative is homogeneous in w, of weight zero. We can now investigate ω( x + h) − ω(x). On W (x + h) ∩ W (x) we have J(v(x + h, w), w) −1 −J(v(x, w), w) −1 ≪ U 4−2n \|J(v(x + h, w), w) − J(v(x, w), w)\|, by (5.7). Moreover J(v(x + h, w), w) − J(v(x, w), w) ≪ U n−3 \|v(x + h, w) − v(x, w)\|, since J(u) is a form in u of degree n − 2. It then follows from (5.10) that J(v(x + h, w), w) −1 − J(v(x, w), w) −1 ≪ U 2−3n/2 \|h\| on W (x + h) ∩ W (x) . We therefore see from (5.9) that the contribution to ω(x + h) − ω(x) from the set W (x + h) ∩ W (x) will be O(U −n/2 \|h\|). This is satisfactory. On the remaining range we merely use the bound J −1 ≪ U 2−n . It therefore suffices to estimate the measure of the set of w ∈ R n−2 for which either (v(x + h, w), w) ∈ U or (v(x, w), w) ∈ U , but not both. By substituting x ′ = x + h and h ′ = −h if necessary, we may suppose that (v(x, w), w) ∈ U and (v(x + h, w), w) ∈ U . In view of (5.10) this means that (v(x, w), w) lies in U at a distance O(U 1−n/2 \|h\|) from the boundary of U . Let U x (h) denote the set of such points w. As yet we have not completely specified the set U in (3.14), and it is now time to do so. For indices i 3 we merely choose L i (u) = w i . Then if (v(x, w), w) lies in U at a distance O(U 1−n/2 \|h\|) from the edge defined by L i we see that the corresponding w i runs over an interval of length O(U 1−n/2 \|h\|), so that the contribution to meas(U x (h)) is ≪ U 1−n/2 \|h\|U n−3 = U n/2−2 \|h\|. We take the remaining linear forms L i to be v 1 and v 1 + λv 2 . Here λ is a non-zero constant chosen sufficiently small that ∂(v 1 + λv 2 ) ∂w 3 ≫ 1 throughout U . In view of (5.11) such a choice will be possible, since we have ∂v 2 /∂w 3 ≪ 1. Suppose now that w 4 , w 5 , . . . , w n are fixed. Then if (v(x, w), w) lies in U at a distance O(U 1−n/2 \|h\|) from the edge defined by the linear form L i = v 1 we see that w 3 is confined to an interval of length O(U 1−n/2 \|h\|), and similarly for the edge defined by L i = v 1 + λv 2 . It follows that the contribution to meas(U x (h)) is O(U n/2−2 \|h\|) in these cases too. Since J −1 ≪ U 2−n we deduce that ω(x + h) − ω(x) ≪ U −n/2 \|h\| as required. It remains to establish the lower bound for ω(x). It is clear that J(u) is homogeneous of degree n − 2, whence J(u) ≪ U n−2 for all relevant u. In view of (5.8) it therefore suffices to show that meas{W (x)} ≫ G 2−n U n−2 on a suitable set of values x. Now Moreover if \|x\| ≪ U n/2 and \|w\| ≪ U then v(x, w) − v U n/2 δN K/L (u (R) ), U u (R) 3 , . . . , U u (R) n ≪ U \|U −n/2 x − δN K/L (u (R) )\| + max 3 i n \|U −1 w i − u (R) i \| , by the homogeneity properties of v(x, w). We therefore see from the definition (3.14) of U that there is a small constant c > 0 such that (v(x, w), w) ∈ U whenever we have \|w i − U u (R) i \| cG −1 U for 3 i n and \|x − U n/2 δN K/L (u (R) )\| cG −1 U n/2 . We therefore have meas{W (x)} ≫ G 2−n U n−2 in the above region, as required. A large sieve bound for α 0 (x) From Lemma 10 we know that α 0 (x) is evenly distributed in all congruence classes for moduli q Q. The condition that Q U 1/2 causes no problems. However a more serious constraint on the size of Q comes from the fact that we cannot handleα if Q is too large. The goal of the present section is to show that in fact α 0 is evenly distribution for "almost all" congruence classes, for much larger values of q. This will enable us to get equidistribution for large moduli, on average, while keeping Q sufficiently small thatα can be adequately handled. Our equidistribution result will be achieved by a quite general large sieve argument, motivated by (but, we believe, simpler than) that used by Fouvry and Iwaniec [12, § 10]. A related procedure is given by Iwaniec and Kowalski [21,Theorem 17.5]. However we have found it more convenient to use additive characters directly, rather than to switch to multiplicative ones as they do. The reader might care to note that a somewhat different argument, also of a very general kind, appears in Heath-Brown [17, § 2]. We assume that we have a function α 0 (x) defined on o L , satisfying x∈R x≡y (mod q) α 0 (x) W 0 ,(6.α 0 (x), for any square S. We proceed to consider S 1 (Q 0 ) := q Q 0 q 2 y (mod q) \|Σ(R; q, y)\| 2 . If K(t) := x∈R α 0 (x)e (L) (tx) for t ∈ L then x∈R x≡y (mod q) α 0 (x) = q −2 b (mod q) e (L) q (−by)K(b/q), and we deduce that S 1 (Q 0 ) = q Q 0 b (mod q) \|K(b/q)\| 2 . We now write the fraction b/q in lowest terms as c/h, say. Then S 1 (Q 0 ) = h Q 0 * c (mod h) \|K(c/h)\| 2 #{q Q 0 : h \| q} Q 0 h Q 0 h −1 * c (mod h) \|K(c/h)\| 2 . The contribution from terms h > Q is at most Q 0 Q −1 h Q 0 * c (mod h) \|K(c/h)\| 2 Q 0 Q −1 ( √ 2N + Q 0 ) 4 x∈R \|α 0 (x)\| 2 , by the two dimensional large sieve in the form (4.14). For the remaining terms with h Q we observe that * c (mod h) \|K(c/h)\| 2 c (mod h) \|K(c/h)\| 2 = h 2 y (mod h) \|Σ(R; q, y)\| 2 h 4 W 2 0 , by our assumption (6.1). Thus terms with h Q contribute at most Q 0 Q 4 W 2 0 to S 1 (Q 0 ). This enables us to conclude as follows. Lemma 12. -Under the assumption (6.1) we have q Q 0 q 2 y (mod q) x∈R x≡y (mod q) α 0 (x) 2 ≪ Q 0 Q 4 W 2 0 + Q 0 (N 2 + Q 4 0 ) Q x∈R \|α 0 (x)\| 2 . We will require a form of this estimate in which we have a maximum over different squares R. We assume that α 0 (x) is supported on a set S = {a + bτ ∈ o L : (a, b) ∈ (−N, N ] 2 } and proceed to cover this with K 2 smaller squares, each contained in S and of the type R i = {a + bτ ∈ o L : (a − a i , b − b i ) ∈ (−N/K, N/K] 2 }, for appropriate pairs (a i , b i ). Here K N is a positive integer parameter which we will specify in due course. Now any square R ⊆ S, with sides aligned with the coordinate axes, will include a union of certain of the squares R i , outside which there are only O(N 2 K −1 ) points of o L . If we now require that \|α 0 (x)\| A 0 (6.2) for all x, then \|Σ(R; q, y)\| O( N 2 K −1 A 0 ) + i K 2 \|Σ(R i ; q, y)\| . Thus if S 2 (Q 0 ) := q Q 0 q 2 y (mod q) max R \|Σ(R; q, y)\| 2 , then S 2 (Q 0 ) ≪ N 4 K −2 A 2 0 Q 5 0 + K 2 i K 2 q Q 0 q 2 y (mod q) \|Σ(R i ; q, y)\| 2 . Lemma 12 then yields q Q 0 q 2 y (mod q) \|Σ(R i ; q, y)\| 2 ≪ Q 0 Q 4 W 2 0 + Q 0 (N 2 K −2 + Q 4 0 ) Q x∈R i \|α 0 (x)\| 2 . Since i K 2 x∈R i \|α 0 (x)\| 2 x∈R \|α 0 (x)\| 2 , we deduce that S 2 (Q 0 ) ≪ N 4 K −2 A 2 0 Q 5 0 + K 4 Q 0 Q 4 W 2 0 + K 2 Q 0 (N 2 K −2 + Q 4 0 ) Q x∈S \|α 0 (x)\| 2 . We now choose K = A 0 Q 3 0 , which yields the following conclusion. Lemma 13. -Under the assumptions (6.1) and (6.2) we have q Q 0 q 2 y (mod q) max R x∈R x≡y (mod q) α 0 (x) 2 ≪ N 4 Q −1 0 + A 4 0 Q 13 0 Q 4 W 2 0 + Q 0 Q −1 N 2 x∈S \|α 0 (x)\| 2 , providing that A 0 Q 5 0 N . We apply this last result to our particular situation. In view of the conditions in Lemmas 10 and 11 we take N of order U n/2 , Q n+5 U 1−η and G U 1/(n+1) , so that the value W 0 = Q n+1 U n−1+η is admissible. In order to estimate α 0 (x) we note that α(x) ≪ η U η for any η > 0. Moreover (4.13) yields \|α(x)\| Q 3 ω(x), since (4.9) and (4.10) imply z (mod q) ρ(z, q) = 1. (6.3) Since ω(x) ≪ 1 by Lemma 9, we will certainly have α 0 (x) ≪ η Q 3 U η . Taking A 0 ≪ η Q 3 U η , the right hand side in Lemma 13 is therefore ≪ η U 2n Q −1 0 + Q 13 0 Q 2n+18 U 2n−2+O(η) + Q 0 Q −1 U 2n+η G O(1) , by Lemma 11. This enables us to conclude as follows. Lemma 14. -If Q Q 0 U 1/(n+16) and G U 1/(n+1) , then q Q 0 q 2 y (mod q) max R x∈R x≡y (mod q) α 0 (x) 2 ≪ η Q 0 Q −1 U 2n+O(η) G O(1) , for any η > 0. We end by establishing a trivial bound for the above sum, which provides an instructive comparison. If q Q 0 U n/2 then x∈R x≡y (mod q) α 0 (x) 2 ≪ #{x ∈ R : x ≡ y (mod q)} x∈R x≡y (mod q) \|α 0 (x)\| 2 ≪ U n q −2 x∈R x≡y (mod q) \|α 0 (x)\| 2 , whence y (mod q) max R x∈R x≡y (mod q) α 0 (x) 2 ≪ U n q −2 y (mod q) x≡y (mod q) \|α 0 (x)\| 2 = U n q −2 x∈o L \|α 0 (x)\| 2 . (6.4) We therefore obtain the trivial bound q Q 0 q 2 y (mod q) max R x∈R x≡y (mod q) α 0 (x) 2 ≪ η Q 0 U 2n+η G O(1) , via Lemma 11. Thus Lemma 14 provides a saving which is a power of Q, providing that Q is larger than a suitable power of U η G. Bilinear forms in dimension 2 The estimation of bilinear forms is one of the cornerstones of analytic number theory and can be traced back to the work of Vinogradov. Given finite sequences u m , v n ∈ C and a matrix A = (a m,n ) of complex numbers, the essential problem is to estimate the double sum It may happen that the sum itself is small. In other cases one can give an asymptotic evaluation with a main term M (n 1 , n 2 ) say. One may then complete the analysis via an estimation of the sum n 1 n 2 v n 1 v n 2 M (n 1 , n 2 ). In our work we will require an analogue of this procedure for sequences indexed by elements of Z 2 , rather than by Z. In this endeavour we are inspired by the analytic machinery developed for the Gaussian integers Z[i] by Fouvry and Iwaniec [12, § 9]. We will provide a completely self-contained account of the method, which has the advantage of being slightly more general in scope. u T Av = Let M ∈ SL 2 (Z) and let α, β : Z 2 → C be arbitrary functions with finite L 2 -norms α 2 :=   x∈Z 2 \|α(x)\| 2   1/2 , β 2 :=   y∈Z 2 \|β(y)\| 2   1/2 . Furthermore, let λ : Z → C be such that λ 2 := l∈Z \|λ(l)\| 2 1/2 is also finite. Our objective is to estimate the double sum S(α, β; λ) := x∈Z 2 (E) y∈Z 2 α(x)β(y)λ x T My , where (E) indicates that the summation over y is restricted by the condition gcd(y 1 , y 2 ) E. In the analysis of this section the implied constants in our O(·) and ≪ notation will be allowed to depend implicitly on the coefficients of the matrix M. However they will be uniform in all other parameters. It will be convenient to introduce a norm on elements x ∈ R 2 via \|x\| = max{\|x 1 \|, \|x 2 \|}. We should observe that this particular choice of norm is important, since it will be significant for us that the unit ball is a square. Let A, B 1 with A B. We will think of A and B as being large, with A/B also large, but of considerably smaller order than B. We will suppose that α, β are supported on the sets A = {x ∈ Z 2 : \|x\| A}, B = {y ∈ Z 2 : B \|y\| 2B},(7.1) respectively. Let M 0 denote the maximum of the moduli of the entries of M, and let M = 0 1 −1 0 M. We now define B = {z ∈ Z 2 : (2M 0 ) −1 B \|z\| 4M 0 B} and we observe that My ∈ B whenever y ∈ B. An application of Cauchy's inequality now gives \|S(α, β; λ)\| 2 β 2 2 · λ 2 2 (E) y∈B l∈Z x∈Z 2 x T My=l α(x) 2 . Enlarging the range of summation for y we therefore deduce that \|S(α, β; λ)\| β 2 · λ 2 · T (α) 1/2 ,(7.2) where T (α) := (E) y∈Z 2 My∈ B l∈Z x∈Z 2 x T My=l α(x) 2 . Opening up the inner sum we obtain T (α) = (E) y∈Z 2 My∈ B w∈Z 2 w T My=0 (α * α)(w), with (α * α)(w) := x,x ′ ∈Z 2 x−x ′ =w α(x)α(x ′ ). The vectors y take the form eu where u = (u 1 , u 2 ) ∈ Z 2 is primitive, and where e E is a positive integer. Since det M = 1, we see that the general solution of the linear equation w T Mu = 0 is w = c Mu for c ∈ Z. Hence T (α) = e E * u∈Z 2 e Mu∈ B c∈Z (α * α) c Mu = e E * ez∈ B c∈Z (α * α)(cz), where * z denotes summation for primitive vectors z = (z 1 , z 2 ). We note that Cauchy's inequality yields \|(α * α)(w)\| α 2 2 for any w ∈ Z 2 . Thus the contribution to the above sum from terms with c = 0 is O( α 2 2 B 2 ), whence T (α) = T 0 (α) + O( α 2 2 B 2 ), (7.3) with T 0 (α) := e E * ez∈ B c∈Z\{0} (α * α)(cz). We now use the Möbius function to pick out the condition gcd(z 1 , z 2 ) = 1, giving T 0 (α) = e E c∈Z\{0} ∞ b=1 µ(b)T 0 (α; b, c, e), where T 0 (α; b, c, e) := ec −1 w∈ B bc\|w (α * α)(w). For ec −1 w ∈ B we deduce that (2M 0 ) −1 e −1 cB \|w\| 4M 0 e −1 cB. (7.4) Moreover we will have (α * α)(w) = 0 unless \|w\| 2A. Thus we can restrict the summation over c to the range 0 < \|c\| 4M 0 AB −1 E = C, say. We may now handle certain ranges of b and c trivially. To this end we give ourselves a parameter K 1 which we will choose in due course. In view of (7.4) the number of available vectors w ≡ 0 (mod bc), corresponding to a particular triple b, c, e is O(B 2 b −2 e −2 ). Since \|(α * α)(w)\| α 2 2 we deduce that the contribution to T 0 (α) from b > K is ≪ e E c C b>K α 2 2 B 2 b −2 e −2 ≪ α 2 2 ABEK −1 . Similarly the contribution from \|c\| CK −1 is ≪ e E \|c\| C/K b α 2 2 B 2 b −2 e −2 ≪ α 2 2 ABEK −1 . It follows that T 0 (α) = e E b K µ(b) C/K<\|c\| C T 0 (α; b, c, e) + O( α 2 2 ABEK −1 ). Combining this with (7.2) and (7.3) therefore leads to the conclusion that S(α, β; λ) ≪ β 2 · λ 2 α 2 B + α 2 (ABEK −1 ) 1/2 + T 1 (α) 1/2 for any A B 1 and K 1, with T 1 (α) := e E b K C/K<\|c\| C \|T 0 (α; b, c, e)\|. We now open up the convolution α * α to obtain T 0 (α; b, c, e) = x ′ ∈Z 2 α(x ′ ) x∈R x≡x ′ (mod bc) α(x), where R = {x ∈ Z 2 : (2M 0 ) −1 e −1 cB \|x − x ′ \| 4M 0 e −1 cB}. Thus \|T 0 (α; b, c, e)\| x ′ ∈Z 2 \|α(x ′ )\| max R x∈R x≡x ′ (mod bc) α(x) , where R now runs over all squares with sides aligned to the axes. It follows from Cauchy's inequality that \|T 0 (α; b, c, e)\| 2 α 2 2 x ′ ∈A max R x∈R x≡x ′ (mod bc) α(x) 2 ≪ α 2 2 (1 + A 2 (bc) −2 )T 2 (α; \|bc\|), with T 2 (α; q) := u (mod q) max R x∈R x≡u (mod q) α(x) 2 . (7.5) If we insist that E B then 1 + A\|bc\| −1 1 + AKC −1 ≪ BKE −1 , whence T 1 (α) ≪ E b K C/K<c C α 2 BKE −1 T 2 (α; \|bc\|) 1/2 . Hence, if τ (q) denotes the usual divisor function, and we set T 3 := q CK q 2 T 2 (α; q),(7.6) a further application of Cauchy's inequality yields T 1 (α) ≪ BK α 2    C/K<q CK τ (q) 2 q −2    1/2 T 1/2 3 ≪ BK α 2 KC −1 (log C) 3 1/2 T 1/2 3 ≪ (BK) 3/2 (AE) −1/2 (log A) 3/2 α 2 T 1/2 3 . We may now conclude as follows. Lemma 15. -Let A B E 1 and define C = 4M 0 AB −1 E. Let α, β : Z 2 → C be functions supported on the sets (7.1), and let λ : Z → C. Define T 2 (α; q) and T 3 as in (7.5) and (7.6). Then for any K 1 we have S(α, β; λ) ≪ β 2 · λ 2 α 2 B + α 2 (ABEK −1 ) 1/2 + T 1 (α) 1/2 , with T 1 (α) ≪ (BK) 3/2 (AE) −1/2 (log A) 2 α 2 T 1/2 3 . Before preceding further it may be helpful to comment on the above estimate. For the purposes of this illustration we shall suppose that K is chosen so that max{E 2 , log 2 A} K A/B. (7.7) If we merely estimate S(α, β; λ) via Cauchy's inequality, using the fact that x T My = l has O(AB log A) solutions x, y for each given l, we are led to a trivial bound S(α, β; λ) ≪ α 2 · β 2 · λ 2 AB log A. Hence the first term in the above lemma saves at least A/B. Similarly the second term saves at least K/E. Both these are at least K 1/4 . To analyse the third term we use the argument in (6.4) to deduce that T 2 (α; q) ≪ A 2 q −2 α 2 2 for q ≪ A. We therefore obtain the trivial bound T 1 (α) ≪ α 2 2 A 1/2 B 3/2 C 1/2 K 2 E −1/2 log 2 A ≪ α 2 2 ABK 2 log 2 A ≪ α 2 2 ABK 3 . For comparison, in order for the third term in Lemma 15 to produce a comparable saving to that in the first two terms, one would wish to replace the above by T 1 (α) ≪ α 2 2 ABK −1 , say. The type of saving we require is exactly that given by Lemma 14, providing that we work with S(α 0 , β; λ). We now show how Lemma 15 can be applied to estimate Tr(x, y) , E (G, H, V ) = x∈o L y∈o L α 0 (x)β(y)λwhen G = log V . We have yet to specify the parameter Q used in the definition ofα, but we shall do this shortly. Under suitable hypotheses we will see in § 8 and § 9 that the main term M (G, H, V ) has order at least (log V ) 1−3n H n V 2n . Our goal is to show that E (G, H, V ) is smaller than this by at least a power of H. If x = x 1 + x 2 τ and y = y 1 + y 2 τ , then Tr(x, y) = x T My, where M = 0 −1 1 0 . By abuse of notation we will write α 0 (x) = α 0 (x) and β(y) = β(y). From our definitions of α,α and ω in (3.15), (4.13) and (5.8) we see that if α 0 (x) = 0 then x = δN K/L (u) for some u ∈ U . Moreover, from (3.14) we deduce that δN K/L (u) = U n/2 δN K/L (u (R) ) + o(U n/2 ) as G → ∞. Since δN K/L (u (R) ) = 0 it follows that α(x) is supported on a suitable set \|x\| A with A = c 1 U n/2 for a certain constant c 1 > 0. We may analyse the support of β similarly. Since N K/L (v (R) ) = 0 we deduce that β(y) is supported on a set B \|y\| 2B with B = c 2 V n/2 for a suitable positive constant c 2 . Moreover, β(y) is also supported on vectors y with gcd(y 1 , y 2 ) ≪ 1, by (3.13). We may therefore take E of order 1. Estimates for β 2 and λ 2 are given by Lemma 4, while Lemma 11 handles α 0 2 . Inserting these bounds into Lemma 15 and adopting the assumption (7.7), together with the bounds Q n+5 U 1−η and G U 1/(n+1) , we then see that B + (ABEK −1 ) 1/2 ≪ U n/4 V n/4 K −1/2 and (BK) 3/2 (AE) −1/2 (log A) 2 ≪ U −n/4 V 3n/4 K 5/2 , so that E (G, H, V ) ≪ η V (n+η)/2 W (n+η)/2 U (n+η)/2 · U n/4 V n/4 K −1/2 + U −n/8 V 3n/8 K 5/4 · U (n+η)/4 T 1/4 3 G O(1) . We now apply Lemma 14, along with the bounds C ≪ AB −1 E and G = O η (V η ), to deduce that T 3 = q CK q 2 T 2 (α; q) ≪ η CKQ −1 U 2n V O(η) ≪ η KQ −1 U 5n/2 V −n/2+O(η) , providing that Q CK U 1/(n+16) . (7.8) Note that when G = log V the condition G U 1/(n+1) is automatic for large V . Moreover, if Q U 1/(n+16) then we automatically have Q n+5 U 1−η , if η is small enough. Thus, on recalling that U = HV and W = H 1/2 V , we deduce that E (G, H, V ) ≪ η H n/4 V n+O(η) H 3n/4 V n K −1/2 + H 3n/4 V n K 3/2 Q −1/4 = H n V 2n+O(η) (K −1/2 + K 3/2 Q −1/4 ). It is now apparent that we should take K = Q 1/8 , which will satisfy (7.7) and (7.8) if Q ≪ H 4n/7 and (log V ) 16 ≪ Q ≪ H −4n U 8/(n+16) . We conclude as follows. Lemma 16. -Let G = log V and assume that (log V ) 16 ≪ Q ≪ min{H n/2 , H −5n U 8/(n+16) }. Then we have E (G, H, V ) ≪ η Q −1/16 H n V 2n+O(η) . Estimation of the main term The purpose of this section and the next is to produce a satisfactory estimate for the sum M = M (G, H, V ) = x∈o L y∈o Lα (x)β(y)λ Tr(x, y) , as G → ∞, whereα is the approximation to α which we constructed in (4.13). Our goal will be to demonstrate that M has order G −2n H n V 2n , which simple heuristics suggest to be the expected size of N (G, H, V ). In fact we will fall a little short of this, showing that M ≫ G 1−3n H n V 2n when suitable constraints are placed on the parameters Q, G, H and V . Let (3.12) holds}, V 1 := {v ∈ V ∩ Z n : v ≡ v (M ) (mod M ),W 1 := {w ∈ W ∩ Z n : w ≡ w (M ) (mod M )}, (8.1) where V , W are given by (3.14). Opening up the functions β and λ from (3.16) and (3.17), respectively, we find that M = v∈V 1 w∈W 1 x∈o L Tr L/Q (xN K/L (v))=2N K/Q (w) α(x) = v∈V 1 w∈W 1 M (v, w), say. Recall that D L = τ − τ σ where {1, τ } is a Z-basis for o L . It will be convenient to make the observation that 2 Tr L/Q (τ 2 ) − Tr L/Q (τ ) 2 = D 2 L , which is a non-zero integer. By enlarging the weak approximation set S in Theorem 2, if necessary, we may clearly assume that M contains any prime divisors of 2D 2 L . We now set in motion our analysis of M (v, w), for given v ∈ V 1 and w ∈ W 1 . Suppose that N K/L (v) decomposes as N 1 (v) + N 2 (v)τ , for suitable forms N 1 , N 2 of degree n/2. Then (3.12) demands that that N 1 (v) and N 2 (v) be coprime. Moreover, in view of (3.8) and our convention that ω 1 = 1 in the integral basis {ω 1 , . . . , ω n } for K over Q, it is clear that N 1 (v) ≡ 1 (mod M ), N 2 (v) ≡ 0 (mod M ). Let us write x = x 1 + x 2 τ . The constraint in the x summation is equivalent to a 1 x 1 + a 2 x 2 = b, for integers a 1 , a 2 , b such that a 1 = Tr L/Q (N K/L (v)) = 2N 1 (v) + Tr L/Q (τ )N 2 (v),a 2 = Tr L/Q (τ N K/L (v)) = Tr L/Q (τ )N 1 (v) + Tr L/Q (τ 2 )N 2 (v), b = 2N K/Q (w). (8.2) In order to analyse M (v, w) we will need to recall the expression forα(x). Let the function ω(x) = ω(x 1 , x 2 ) be given by (5.8), with key properties as in Lemma 9. Then for any parameter Q 1 and x ∈ o L we havê α(x) = ω(x) q Q * t (mod q) e (L) q (−tx) z (mod q) ρ(z, q)e (L) q (tz), where ρ(z, q) is given by (5.6) and the notation * means that the sum is taken over t = t 1 + t 2 τ , with t 1 , t 2 ∈ Z/qZ such that gcd(q, t 1 , t 2 ) = 1. Define c q (t) := z (mod q) ρ(z, q)e (L) q (tz), (8.3) for t (mod q). It is clear from (6.3) that \|c q (t)\| 1. Inserting our expression forα(x) and breaking the inner sum into congruence classes modulo q, we see that M (v, w) = q Q * t (mod q) c q (t) r (mod q) e (L) q (−tr)L r (v, w), where L r (v, w) := a 1 x 1 +a 2 x 2 =b x≡r (mod q) ω(x). We proceed to investigate the summation conditions on x. We first prove that a 1 and a 2 are not both zero, that b is non-zero, and that gcd(a 1 , a 2 ) = 2 κ , κ := 1, if 2 \| Tr L/Q (τ ), 0, if 2 ∤ Tr L/Q (τ ). (8.4) In particular gcd(a 1 , a 2 ) \| b. To establish the claim we observe that gcd(a 1 , a 2 ) is a common divisor of 2a 2 − Tr L/Q (τ )a 1 = D 2 L N 2 (v) and Tr L/Q (τ 2 )a 1 − Tr L/Q (τ )a 2 = D 2 L N 1 (v). However v and w are in the regions (3.14) with N K/L (v (R) ) and N K/Q (w (R) ) non-zero. Thus if G is large enough we will have N K/L (v) and N K/Q (w) non-zero. Hence b will be non-zero and similarly, since D L = 0, the numbers a 1 and a 2 cannot both be zero. We also see that gcd(a 1 , a 2 ) \| D 2 L gcd(N 1 (v), N 2 (v)) = D 2 L . Since N 1 (v) ≡ 1 (mod p) and N 2 (v) ≡ 0 (mod p) for any prime divisor p of 2D 2 L , the claim readily follows. We now write a ′ i = 2 −κ a i for i = 1, 2, and b ′ = 2 −κ b, so that the first condition on x becomes a ′ 1 x 1 + a ′ 2 x 2 = b ′ . Thus the sum will be empty unless a ′ 1 r 1 + a ′ 2 r 2 ≡ b ′ (mod q), as we now assume. For future use we note that this is equivalent to demanding that 2 −κ Tr L/Q (rN K/L (v)) ≡ 2 1−κ N K/Q (w) (mod q). (8.5) We may therefore write a ′ 1 r 1 + a ′ 2 r 2 = b ′ + qk for some k ∈ Z. Then, if we take x = r + qy, the summation conditions on x translate into the requirement that a ′ 1 y 1 + a ′ 2 y 2 = −k. Since a ′ 1 is coprime to a ′ 2 we can find integers a 1 , a 2 such that a ′ 1 a 1 + a ′ 2 a 2 = 1. We then have a ′ 1 (y 1 + ka 1 ) + a ′ 2 (y 2 + ka 2 ) = 0, so that y 1 + ka 1 = −ma ′ 2 and y 2 + ka 2 = ma ′ 1 for some integer m. It follows that the summation conditions a 1 x 1 + a 2 x 2 = b and x ≡ r (mod q) are satisfied if and only if x 1 and x 2 take the forms x 1 = r 1 − a 1 kq − a ′ 2 qm and x 2 = r 2 − a 2 kq + a ′ 1 qm respectively. Our conclusion is therefore that L r (v, w) = m∈Z f (m), if (8.5) holds, where f (m) := ω r 1 − a 1 kq − a ′ 2 qm, r 2 − a 2 kq + a ′ 1 qm . We would now like to replace the discrete summation over m by a continuous integral. For this a relatively crude approach is available to us through Lemma 9. Thus if v ∈ [0, 1] it follows that f (m + v) − f (m) ≪ U −n/2 q max{\|a ′ 1 \|, \|a ′ 2 \|}. Moreover Lemma 9 tells us that ω is supported on a disc of radius O(U n/2 ), whence f is supported on an interval with length O(U n/2 /(q max{\|a ′ 1 \|, \|a ′ 2 \|})). Recalling that U V in Lemma 3 and max{\|a ′ 1 \|, \|a ′ 2 \|} ≪ V n/2 , this therefore produces the conclusion ∞ −∞ f (m) dm − m∈Z f (m) ≪ 1 + qU −n/2 max{\|a ′ 1 \|, \|a ′ 2 \|} ≪ q. Assuming that a 2 = 0 the change of variables m = q −1 (−2 κ x + r 1 /a ′ 2 − a 1 kq/a ′ 2 ) now yields L r (v, w) = 2 κ q I(v, w) + O(q), for r satisfying (8.5), with I(v, w) := ∞ −∞ ω(a 2 x, −a 1 x + b/a 2 ) dx. (8.6) Here a 1 , a 2 , b depend on v and w and are given by (8.2). If a 2 = 0 we reverse the rôles of a 1 and a 2 to produce an integral involving ω(−a 2 x + b/a 1 , a 1 x). Recall the estimate \|c q (t)\| 1 that we recorded above. Inserting our estimate for L r (v, w) into that for M (v, w), it now follows that M (v, w) = 2 κ I(v, w) q Q 1 q * t (mod q) c q (t) r (mod q) (8.5) holds e (L) q (−tr) + O Q 6 . We now sum both sides over all v ∈ V 1 and w ∈ W 1 . On observing that #V 1 = O(G −n V n ) and #W 1 = O(G −n W n ), we see that the overall contribution from the error term is ≪ Q 6 G −2n V n W n ≪ Q 6 H n/2 V 2n . This will be satisfactory if Q is sufficiently small compared to H. Our work so far has shown that M = 2 κ v∈V 1 w∈W 1 I(v, w) q Q 1 q C + O Q 6 H n/2 V 2n ,(8.7) where V 1 , W 1 are given by ( Opening up (8.3), we find that C = r (mod q) (8.5) holds z (mod q) ρ(z, q) * t (mod q) e (L) q (t(z − r)) . We have seen that the condition (8.5) can be written a ′ 1 r 1 + a ′ 2 r 2 ≡ b ′ (mod q), for non-zero integers a ′ 1 , a ′ 2 , b ′ such that gcd(a ′ 1 , a ′ 2 ) = 1. The inner sum is a Ramanujan sum and thus it follows from (4.12), combined with (4.9), that C = u\|q u 2 µ(q/u) r (mod q) a ′ 1 r 1 +a ′ 2 r 2 ≡b ′ (mod q) z (mod q) z≡r (mod u) ρ(z, q) = u\|q u 2 µ(q/u) r (mod q) a ′ 1 r 1 +a ′ 2 r 2 ≡b ′ (mod q) ρ(r, u). Let q = uv. Writing r = r ′ + ur ′′ for r ′ (mod u) and r ′′ (mod v) we see that the sum over r is equal to r ′ (mod u) a ′ 1 r ′ 1 +a ′ 2 r ′ 2 ≡b ′ (mod u) ρ(r ′ , u)# r ′′ (mod v) : a ′ 1 r ′′ 1 + a ′ 2 r ′′ 2 ≡ g(r ′ ) (mod v) , where g(r ′ ) = u −1 (b ′ − a ′ 1 r ′ 1 − a ′ 2 r ′ 2 ) . Since a ′ 1 and a ′ 2 are coprime it follows that there are precisely v possibilities for r ′′ . The definition (5.6) of ρ(r ′ , u) therefore reveals that This is the polynomial that underpins the variety introduced in (3.5). We proceed to insert this expression for C into (8.7). We will use the Möbius function to remove the coprimality condition (3.12), which is implicit in the definition (8.1) of V 1 . We therefore arrive at the estimate M = 2 κ M n ∞ k=1 µ(k) q Q u\|q uµ(q/u) [M, u] n s (mod [M,u]) s≡u (M ) (mod M ) × (v,w)∈V 2 ×W 1 2 −κ F (v;w;s)≡0 (mod u) I(v, w) + O Q 6 H n/2 V 2n ,(8.8) where I(v, w) is given by (8.6) and V 2 is defined as for V 1 , but with the condition (3.12) replaced by k \| N K/L (v). This latter condition is taken componentwise as k \| N 1 (v) and k \| N 2 (v), in the usual way. The sum over k is empty when k ≫ V n/2 so that we only need consider values k ≪ V n/2 . However we need to reduce this range further. Since N K/L (v (R) ) = 0 it follows from (8.2) and the definition of V that max{\|a 1 \|, \|a 2 \|} ≫ V n/2 . Perhaps the most obvious way to deal with I (∆) is to approximate the sums over w and v by integrals. A simple change of variables would then permit us to extract the dependence of I (∆) on p, q, s and ∆. Instead of this it turns out that we can manage with a relatively crude direct comparison of I (∆) with I (1), as we proceed to show. It follows from our conditions of summation that ∆ ≪ ku KQ. It will be convenient to make the additional hypothesis KQ V, (8.13) which implies in particular that ∆ ≪ V . In view of (8.6) we have I (∆) = ∞ −∞ w∈W ∩Z n w≡q (mod ∆) v∈V ∩Z n v≡p (mod ∆) f (v, w) dx, where if a i = a i (v) and b = b(w) are given by (8.2) then f (v, w) := ω a 2 (v)x, −a 1 (v)x + b(w) a 2 (v) . We will denote by T (p, q; ∆) the integrand that appears in this expression for I (∆). Using an approach based on the proof of Lemma 10 in § 5 we compare T (p, q; ∆) with T (0, 0; ∆). For this we will assume without loss of generality that max i=1,2 \|a i (v)\| = \|a 2 (v)\| in the sum over v. The alternative possibility is accommodated by a simple change of variables in the integral over x. Hence the definition of V ensures that a 2 (v) has order of magnitude V n/2 and by Lemma 9 we have x ≪ H n/2 in I (∆). Under the change of variables w = w ′ + q we see that (8.13). We may therefore conclude from Lemma 9 that b(w) a 2 (v) − b(w ′ ) a 2 (v) ≪ ∆\|w ′ \| n−1 \|a 2 (v)\| ≪ ∆W n−1 V n/2 , since \|w ′ \| ≪ ∆ + W ≪ W byT (p, q; ∆) = v∈V ∩Z n v≡p (mod ∆) w ′ +q∈W ∩Z n w ′ ≡0 (mod ∆) f (v, w ′ ) + O ∆W n−1 U n/2 V n/2 . Note that the number of v appearing in the outer sum is O(∆ −n V n ). For w ′ such that w ′ + q ∈ W it is clear that w ′ ∈ W unless w ′ is within a distance ∆ of the boundary of W . Invoking Lemma 9 to deduce that f is bounded absolutely, and recalling that U = HV and W = H 1/2 V , it easily follows that \|T (p, q; ∆) − T (p, 0; ∆)\| ≪ (∆ −1 V ) n (∆ −1 W ) n−1 + ∆ −2n+1 V n/2 W 2n−1 U n/2 ≪ ∆ −2n+1 H (n−1)/2 V 2n−1 , by (8.13). We now repeat the above process by considering the effect of a change of variables v = v ′ +p in T (p, 0; ∆). Recalling that ∆ ≪ KQ V , by (8.13), we obtain a i (v) − a i (v ′ ) ≪ ∆\|v ′ \| n/2−1 ≪ ∆V n/2−1 , for i = 1, 2. In particular it follows that b(w) a 2 (v) − b(w) a 2 (v ′ ) ≪ ∆V n/2−1 W n \|a 2 (v)a 2 (v ′ )\| ≪ ∆V n/2−1 W n V n = ∆H n/2 V n/2−1 , for any w ∈ W . An application of Lemma 9 reveals that T (p, 0; ∆) = w∈W ∩Z n w≡0 (mod ∆) v ′ +p∈V ∩Z n v ′ ≡0 (mod ∆) f (v ′ , w) + O ∆H n/2 V n/2−1 U n/2 . To control the error term we note that the total number of available v ′ , w in the sums is ≪ W n ∆ n · ∆ + V ∆ + 1 n ≪ ∆ −2n V n W n . We conclude that \|T (p, 0; ∆) − T (0, 0; ∆)\| ≪ (∆ −1 W ) n (∆ −1 V ) n−1 + ∆ −2n+1 H n/2 V 3n/2−1 W n U n/2 ≪ ∆ −2n+1 H n/2 V 2n−1 , whence T (p, q; ∆) = T (0, 0; ∆) + O ∆ −2n+1 H n/2 V 2n−1 . Summing the latter estimate over p, q modulo ∆ gives T (0, 0; 1) = ∆ 2n T (0, 0; ∆) + O ∆H n/2 V 2n−1 , which we substitute back in to get T (p, q; ∆) = ∆ −2n T (0, 0; 1) + O ∆ −2n+1 H n/2 V 2n−1 . Now the outer integral in our expression for I (∆) is over an interval of length O(H n/2 ). We have therefore arrived at the following result. This result will allow us to separate out what is in effect the "singular integral" associated to our counting problem. In the notation of (8.6) it is given by σ ∞ (G, H, V ) := I (1) = w∈W ∩Z n v∈V ∩Z n I(v, w). (8.14) It follows from (8.9) that σ ∞ (G, H, V ) ≪ (G −1 V ) n · (G −1 W ) n · H n/2 = G −2n H n V 2n . (8.15) It is interesting to compare the present situation with the singular integrals arising from typical applications of the Hardy-Littlewood circle method. These are expressed as volumes that reflect the real density of solutions. It transpires that we will be able to provide a lower bound for σ ∞ (G, H, V ) which essentially matches the upper bound (8.15) without first approximating the sum by an integral. Nonetheless crucial use will be made of the fact that where ∆ = [M, u, k]. We claim that N M (k, u) ≪ ξ ∆ 3n−1+ξ k ,(8.17) for any ξ > 0. To see this we write A = Tr L/Q δN K/L (p)N K/L (s) for fixed p, s modulo ∆. Then the number of q (mod ∆) contributing to N M (k, u) is at most N ≪∆ n N ≡A (mod u) #{q ∈ [1, ∆] n : 2N K/Q (q) = N } ≪ ξ ∆ n+ξ/2 u . Moreover there are ∆ n possibilities for s (mod ∆) and the number of available p (mod ∆) is at most N ≪∆ n N ≡0 (mod k 2 ) #{p ∈ [1, ∆] n : N K/Q (p) = N } ≪ ξ ∆ n+ξ/2 k 2 . (8.18) It therefore follows that N M (k, u) ≪ ξ ∆ n+ξ/2 u · ∆ n · ∆ n+ξ/2 k 2 = ∆ 3n+ξ uk 2 . Noting that uk ≫ ∆, we easily arrive at (8.17). The extraction of σ ∞ = σ ∞ (G, H, V ) now follows on combining Lemmas 17 and 18 with (8.10) and (8.17). Putting these together, and assuming that KQ V , we therefore deduce that M = σ ∞ 2 κ M n k K µ(k) q Q u\|q uµ(q/u) ∆ 3n N M (k, u) + O η H n V 2n+η Q 6 H n/2 + Q 2+η K + Q 2+η K η V , where N M (k, u) is given by (8.16). Since KQ V it is clear that the error term is ≪ η H n V 2n+2η Q 6 H n/2 + Q 2 K . We now show that the summation over k can be extended to infinity with acceptable error. It follows from (8.15) and (8.17) that the error in so doing is ≪ η G −2n H n V 2n k>K q Q u\|q u ∆ 3n · ∆ 3n−1+η/2 k ≪ η Q 1+η H n V 2n K 1−η . This error term is subsumed by that above when KQ V . Choosing K = H n/2 Q ,M = 2 κ M n σ ∞ S(Q) + O η Q 6 H n/2 V 2n+2η , where σ ∞ = σ ∞ (G, H, V ) is given by (8.14) and S(Q) := q Q ∞ k=1 µ(k) u\|q uµ(q/u) ∆ 3n N M (k, u), with ∆ = [M, u, k] and N M (k, u) given by (8.16). The singular series and integral It remains to produce satisfactory lower bounds for the quantities S(Q) and σ ∞ . For the former our strategy will be to show that as Q tends to infinity the sum S(Q) converges to a limit S which is a product of local factors. We will then show that each of these factors is positive. In view of (8.17) the sum f (q) := ∞ k=1 µ(k) u\|q uµ(q/u) ∆ 3n N M (k, u) is absolutely convergent. We rearrange it as f (q) = u\|q uµ(q/u) ∞ k=1 µ(k)∆ −k 1 \|uM µ(k 1 ) [M, u] 3n k 2 µ(k 2 ) k 3n 2 N M (k 1 k 2 , u). The function N M (k 1 k 2 , u) factors as N M (k 1 k 2 , u) = N M (k 1 , u)# p, q, s (mod k 2 ) : k 2 \| N K/L (p) , Lemma 3 ensures the existence of p-adic integer solutions with v ≡ (1, 0, . . . , 0) (mod p) and N K/Q (w) = 0. It follows that p ∤ N K/L (v). Finally we prove, by contradiction, that any such point must be non-singular. For otherwise we would have ∇ w F (v; w; u) = −2∇ w N K/Q (w) = 0 where ∇ w = ( ∂ ∂w 1 , . . . , ∂ ∂wn ) . It would then follow from Euler's identity that w.∇ w N K/Q (w) = nN K/Q (w) = 0, a contradiction. Thus we have suitable non-singular points for every prime p, and hence it follows that σ p > 0 for all primes p. We remark that, by combining (9.2), (9.5) and (9.7), we have S = p σ * p , with σ * p := lim β→∞ p −(3n−1)β M (p β , p µ ). Thus S is a standard product of local densities. It remains to establish Lemmas 20 and 21. To handle Lemma 20 we observe that the variety defined by F (x) = 0 takes the simple form X 1 · · · X n + X n+1 · · · X 2n + X 2n+1 · · · X 3n = 0 over F p . This makes it clear that we have a hypersurface of projective dimension 3n−2, whose singular locus has projective dimension 3n − 7. Here we use the fact that the singular locus consists of points where two or more coordinates vanish from each of the sets {X 1 , . . . , X n }, {X n+1 , . . . , X 2n }, and {X 2n+1 , . . . , X 3n }. According to the result of Hooley [19] In the set on the right we have p \| Tr L/Q (δN K/L (p)N K/L (q)) − 2N K/Q (s) and p \| N K/L (p), from which it follows that p \| 2N K/Q (s). It follows, by the argument leading to (8.18) that the number of possible p (mod p) is O ξ (p n−2+ξ ) and that the number of possible s (mod p) is O ξ (p n−1+ξ ). We then see that #{x (mod p) : p \| F (x), p \| N(x)} ≪ ξ p 3n−3+2ξ for any fixed ξ > 0, which gives us the required bound (9.6). Turning to the proof of Lemma 21 we will begin by supposing that α max{2µ − 1, 2}. (9.9) It follows that α > max{µ, 1}, whence (9.7) yields f 0 (p α , p µ ) = p −(3n−1)α 1 − R(p)p −n M (p α , p µ ) − p 3n−1 M (p α−1 , p µ ) . (9.10) It will be appropriate to observe at this point that 1 − R(p)/p n ≫ 1, which follows from (9.1). We proceed to compare M (p e , p µ ) with M (p e+1 , p µ ), using Hensel lifting. For t < e we define S t (p e , p µ ) := {x (mod p e ) : x ≡ x 0 (mod p µ ), p e \| F (x), p ∤ N(x), p t ∇F (x)}. When t < e/2 and t e− max{µ, 1} one sees that if x ∈ S t (p e , p µ ) then x+ p e−t y ∈ S t (p e , p µ ) for all y (mod p t ). Thus S t (p e , p µ ) is composed of cosets modulo p e−t . Moreover x + p e−t y will be in S t (p e+1 , p µ ) for exactly p 3n−1 choices of y (mod p). It follows that each coset modulo p e−t in S t (p e , p µ ) lifts to exactly p 3n−1 cosets modulo p e+1−t in S t (p e+1 , p µ ), and hence that #S t (p e+1 , p µ ) = p 3n−1 #S t (p e , p µ ) (9.11) for t < e/2 and t e − max{µ, 1}. We now write T (p t ) = #{x (mod p t ) : p t \|∇F (x)}, whence #{x (mod p e ) : p t \|∇F (x)} = p 3n(e−t) T (p t ) for t e. Then for any non-negative integer τ e we have M (p e , p µ ) = 0 t<τ #S t (p e , p µ ) + #    x (mod p e ) : x ≡ x 0 (mod p µ ), We therefore wish to use (9.11) for every value t < τ . This will require that τ − 1 < e/2 and τ − 1 e − max{µ, 1}. This condition is equivalent to requiring that τ − 1 (e − 1)/2 and τ − 1 e − max{µ, 1}, or that τ (e + 1)/2 and τ e − max{µ − 1, 0}. Thus if τ = τ (e) = min e − max{µ − 1, 0} , e + 1 2 then (9.11) yields M (p e+1 , p µ ) − p 3n−1 M (p e , p µ ) 2p 3n(e+1−τ ) T (p τ ). p e \| F (x), p ∤ N(x), p τ \|∇F (x)    , We proceed to insert this bound into (9.10). If e = α−1 we find that τ (e) = [α/2] providing that α max{2µ − 1, 1}. In fact we have made the stronger assumption (9.9), and we deduce that f 0 (p α , p µ ) ≪ p α−3n[α/2] T (p [α/2] ). (9.12) We have therefore reduced our problem to one of providing a suitable upper bound for T (p t ). We now recall that F (x) = 2 −κ Tr L/Q (δN K/L (p)N K/L (q)) − 2N K/Q (s) . If we set G(p, q) = Tr L/Q (δN K/L (p)N K/L (q)) it then follows that T (p t ) T 1 (p t )T 2 (p t ), (9.13) where T 1 (p t ) := #{(p, q) (mod p t ) : ∇ p G(p, q) ≡ ∇ q G(p, q) ≡ 0 (mod p t )} and T 2 (p t ) := #{s (mod p t ) : ∇N K/Q (s) ≡ 0 (mod p t )}. We begin by explaining our estimation of T 2 (p t ). The treatment of T 1 (p t ) will then be in the same spirit, but a little more complicated. Over Q we have ∂ ∂s j N K/Q (s) = n i=1 c ij µ i , where µ i := L 1 (s) · · · L i−1 (s)L i+1 (s) · · · L n (s). Let C denote the matrix (c ij ) i,j n and write D = det C, so that D 2 is the discriminant of K. Suppose now that p t divides ∇N K/Q (s). Then if µ is the column vector with elements µ i we will have Cµ ≡ 0 (mod p t ). This divisibility relation may be interpreted in the ring of integers for N . We now pre-multiply by the matrix C adj , whose entries are algebraic integers, and use the fact that C adj C = (det C)I to deduce that Dµ ≡ 0 (mod p t ). We conclude that p t \| Dµ i for each index i, where divisibility is again within the ring of integers of N . Then p tn \| D n i µ i = D n N K/Q (s) n−1 , and hence p ⌈tn/(n−1)⌉ \| D 2 N K/Q (s) whenever p t \| ∇N K/Q (s). Let σ = max ⌈tn/(n − 1)⌉ − v p (D 2 ) , 0 . Since t σ 2t we then have T 2 (p t ) = p n(t−σ) #{s (mod p σ ) : p t \| ∇N K/Q (s)} p n(t−σ) #{s ∈ N n ∩ (0, p σ ] n : p σ \| N K/Q (s)}. Suppose that (p) splits over K as (p) = p e 1 1 · · · p er r with N K/Q (p i ) = p f i , for 1 i r. Let α = n j=1 s j ω j , so that α is a non-zero element of o K . If we now set v i = v p i (α) we deduce that p σ \| r i=1 N K/Q (p i ) v i , whence σ f i v i . It then follows that σ f i g i , where g i = min{v i , σ}. Thus for each element s there are non-negative integers g i σ for which f i g i σ and n j=1 s j ω j ∈ p g 1 1 · · · p gr r . This condition restricts s to a lattice Λ say, with p σ Z n ⊆ Λ ⊆ Z n , and with p σ \| det(Λ). The number of choices for g 1 , . . . , g r is at most (σ + 1) r (σ + 1) n . It therefore follows that T 2 (p t ) p n(t−σ) (σ + 1) n p nσ−σ (2t + 1) n p nt−σ ≪ (2t + 1) n p nt−⌈tn/(n−1)⌉ , (9.14) since D is fixed. We now examine T 1 (p t ) in a similar way. The form G(p, q) takes the shape δ n/2 i=1 L i (p) n/2 i=1 L i (q) + δ σ n i=n/2+1 L i (p) n/2 i=n/2+1 L i (q). If we replace δL 1 (p) by L 1 (p) and δ σ L n/2+i (p) by L n/2+1 (p) we find that ∂G(p, q) ∂p j = n i=1 c ij µ i , where c ij = ∂L i (x) ∂x j and µ i = L 1 (p) · · · L i−1 (p)L i+1 (p) · · · L n/2 (p) n/2 i=1 L i (q) for 1 i n/2 and µ i = L n/2+1 (p) · · · L n/2+i−1 (p)L n/2+i+1 (p) · · · L n (p) n i=n/2+1 L i (q) for n/2 < i n. Thus p t \| Dµ i for 1 i n, where D is the determinant of the matrix C = (c ij ) i,j n . In particular D is a non-zero algebraic integer. Taking the product for i n yields p tn \| D n N K/Q (p) n/2−1 N K/Q (q) n/2 . By symmetry we also obtain p tn \| D n N K/Q (q) n/2−1 N K/Q (p) n/2 , and hence p 2tn \| D 2n N K/Q (p) n−1 N K/Q (q) n−1 . It follows that p σ \| N K/Q (p)N K/Q (q) with σ = max ⌈2tn/(n − 1)⌉ − v p (D 4 ) , 0 . We can now complete the argument as before. This time there will be two sets of nonnegative exponents g i and g ′ i , say, corresponding to p and q respectively, and such that i f i (g i + g ′ i ) σ. Thus the factor (σ + 1) n must be replaced by (σ + 1) 2n . For each such set of exponents we find that (p, q) is restricted to a sublattice of Z 2n of index at least p σ , and we deduce as before that T 1 (p t ) ≪ (4t + 1) 2n p 2nt−⌈2tn/(n−1)⌉ . We now see that, for points (v, w) satisfying (9.16), the line (a 2 x, −a 1 x + b/a 2 ) meets the disc B(δN K/L (U u (R) ); ρ) in a segment of length ≫ ρ, so that J(v; w) ≫ ρ max{\|a 1 \|, \|a 2 \|} ≫ ρ V n/2 ≫ G −1 H n/2 . The number of available points (v, w) is ≫ G −2n (V W ) n and we therefore conclude from (9.15) that σ ∞ ≫ G 1−3n H n V 2n . This should be compared with the upper bound (8.15). Bringing together our lower bounds for S and σ ∞ in Lemma 19, we deduce that M ≫ G 1−3n H n V 2n + O η Q 6 H n/2 V 2n+2η , provided that H n/2 V . This therefore leads to the following conclusion. (3.5) in which each of the vectors has coordinates lying in R. In the light of Lemma 3 we introduce the counting functionN (H, V ) := #{(u, v, w) ∈ J (Z) : (3.10) and (3.11) hold}. and the conjugates of ρ are all O(V ) in absolute magnitude. The number of admissible elements ρ is therefore Lemma 7 . 7-With the above assumptions we have\|S(z, h) −Ŝ(z, h)\| W Q 4 + Efor all h Q and all residue classes z modulo h. 8). In view of the definition (3.15) we see thatS(y, q) = w #{u ∈ U ∩ Z n : u ≡ w (mod [M, q]), δN K/L (u) ∈ R},where w runs over vectors modulo [M, q] for which w ≡ u (M ) (mod M ) and δN K/L (w) ≡ y (mod q). For a general modulus r U we proceed to compare the sizes of the setsT (r, x) := {u ∈ U ∩ Z n : u ≡ x (mod r), δN K/L (u) ∈ R}for the values x = w and 0. For u = (u 1 , . . . , u n ) let u * = (r[u 1 /r], . . . , r[u n /r]). [M, q] n # s (mod [M, q]) : s ≡ u (M ) (mod M ), δN K/L (s) ≡ y (mod q) . (5.6) Lemma 10 . 10-Let R be a square with side ρ 1 satisfying ρ ≪ U n/2 . Then if η is any positive constant we have x∈R x≡y (mod q) . v n a m,n .The standard procedure is to use Cauchy's inequality to remove the dependence on u = (u m ).For the second term on the right one expands the square and reverses the order of summation, so as to use suitable information about the sum m a m,n 1 a m,n 2 . 8.1), and C := * t (mod q) c q (t) r (mod q) (8.5) holds e (L) q (−tr). C = M n q u\|q uµ(q/u) [M, u] n # s (mod [M, u]) : s ≡ u (M ) (mod M ), 2 −κ F (v; w; s) ≡ 0 (mod u) , where F (v; w; s) := Tr L/Q (δN K/L (s)N K/L (v)) − 2N K/Q (w). The next phase of the argument concerns a detailed analysis of K k,u (s). It is natural to break the sum over v and w into congruence classes modulo ∆ = [M, u, k]. LetI (∆) = I (p, q, s; ∆) := w∈W ∩Z n w≡q (mod ∆) v∈V ∩Z n v≡p (mod ∆) sum is over (p, q) ∈ (Z/∆Z) 2n for which (p, q, s) ≡ (v (M ) , w (M ) , u (M ) ) (mod M ) (8.11)and 2 −κ F (p; q; s) ≡ 0 (mod u), k \| N K/L (p).(8.12) Lemma 18 . 18-Assume that KQ V . Then we have I (∆) = ∆ −2n I (1) + O(∆ −2n+1 H n V 2n−1 ). σ ∞ (G, H, V ) features a sum over points close to a non-singular real point on the variety (3.5). For given k and u, let N M (k, u) := # {p, q, s (mod ∆) : (8.11), (8.12) hold} , (8.16) 3n N M (k, u) and write k = k 1 k 2 where k 1 \| uM and gcd(k 2 , uM ) = 1. Then [M, u, k] = [M, the number of projective points modulo p differs from (p 3n−1 − 1)/(p − 1) by an amount O(p 3n−4 ), whence the number of points in A n (F p ) is p 3n−1 + O(p 3n−3 ). It follows that M (p, 1) = p 3n−1 + O(p 3n−3 ) − #{x (mod p) : p \| F (x), p \| N(x)}. M (p e , p µ ) − 0 t<τ #S t (p e , p µ ) #{x (mod p e ) : p τ \|∇F (x)} = p 3n(e−τ ) T (p τ ).Similarly we haveM (p e+1 , p µ ) − 0 t<τ #S t (p e+1 , p µ ) p 3n(e+1−τ ) T (p τ ). in the normal closure N , say, of K. Indeed our original choice of the basis ω 1 , . . . , ω n ensures that the c ij are algebraic integers. We now have Lemma 22 . 22-Let G = log V . Assume that H n/2 V and Q H n/12 V −η/2 .Then we have M (G, H, V ) ≫ (log V ) 1−3n H n V 2n .Recalling (3.18) and our choiceG = log V , it is now time to select parameters Q, H, V such that E (G, H, V ) = o(M (G, H, V )). We will choose Q = H (n−1)/12 , with which choice Lemma 22 implies that M (G, H, V ) ≫ (log V ) 1−3n H n V 2n , if V 6η H V 2/n .In line with Lemma 3 we let V run through large integers congruent to 1 modulo M . Next we choose H 0 = 1 + M [V 1/(10n 2 ) ], which is a positive integer congruent to 1 modulo M . But then H = H 2 0 has order V 1/(5n 2 ) and so Q = H (n−1)/12 satisfies the conditions of Lemma 16. This implies that the required estimate for E (G, H, V ) holds and so completes the proof of Theorem 2. the hypothesis (8.13) becomes H n/2 V . Putting everything together we have therefore established the following result. Lemma 19. -Assume that H n/2 V . Then we have The integration in(8.6) is therefore over an interval of length ≪ V −n/2 U n/2 = H n/2 , by Lemma 9. A second application of Lemma 9 to bound the size of ω now yieldsHence there is an absolute constant c > 0 such that the overall contribution to M from terms with k > K is ≪ H n/2K<k≪V n/2 q Q u\|qwhere M k (X) := #{v ∈ Z n : \|v\| X, k \| N K/L (v)}.ClearlyThe overall contribution to the main term from k > K is therefore seen to beThis allows us to truncate the summation over k in (8.8) with acceptable error. We now replace the sum over s in(8.8)by one in which the variable runs modulo[M, u, k]. This has the effect of multiplying it by [M, u, k] −n [M, u] n . We may therefore summarise our findings in the following result.Lemma 17. -Let K 1 and let κ be given by(8.4). Then we havewhere ∆ := [M, u, k] andHere W 1 is given by (8.1), I is given by(8.6) and V 2 is defined as for V 1 in (8.1), but with (3.12) replaced by k \| N K/L (v).where ∆ =[M, u]in N M (k 1 , u). We write R(k) := # p (mod k) : k \| N K/L (p) and observe that R(k) ≪ ξ k n−2+ξ and R(k) < k n for k > 1, (9.1) by the argument used for(8.18), and the fact that N K/Q (1, 0, 0, . . . , 0) = 1. It follows thatWe therefore see thatis a multiplicative function in the two variables q and M . We write q = p α and M = p µ , where µ = v p (M ) vanishes for all but finitely many primes. ThenProviding that this product converges we will be able to deduce thatand in order to prove that S > 0 it will suffice to show that σ p > 0 for every prime p.Our treatment of the singular series will depend on the following two lemmas, which we will prove later in this section. It will be convenient to write x for the vector (p, q, s) andWe now prove that the product (9.4) converges. Lemma 21 yieldsWe note also that µ is non-zero only for the primes in S, and that f 0 (1, 1) = 1 Thus a bound f 0 (p, 1) ≪ p −3/2 will suffice to establish absolute convergence. In general we havein view of (9.3). The final sum over q vanishes unless u = p β , and if β max{µ, 1} our expression reduces to(9.7)Taking β = 1 and µ = 0 we obtainThe required estimate f 0 (p, 1) ≪ p −3/2 now follows from Lemma 20, since R(p)/p n ≪ p −3/2 by (9.1). This completes the proof of absolute convergence.We turn now to proof that σ p > 0 for every prime p. By (9.7) we haveThus σ p > 0 providing that M (p β , p µ ) ≫ p,µ p (3n−1)β as β → ∞. Using Hensel's lemma it will follow that σ p > 0 if the varietyintroduced in (3.5), has a non-singular point over Z p which satisfies the constraintsThe function M (p, 1) counts points over F p which lie on the above variety and satisfy the constraints (9.8), whether they are non-singular or not. The number of singular points will be O(p 3n−2 ), whence the estimate (9.6) shows that there will be a suitable non-singular point providing that p p 0 , say. We can arrange that the set S includes all primes p < p 0 , so that In view of (9.13) and (9.14) we deduce that T (p t ) ≪ (4t + 1) 3n p 3nt−⌈tn/(n−1)⌉−⌈2tn/(n−1)⌉ ≪ (4t + 1) 3n p 3nt−3t−2 , since if m ∈ Z we have ⌈θ⌉ m + 1 for any real number θ > m. It now follows from (9.12) that f 0 (p α , p µ ) ≪ (2α + 1) 3n p α−3[α/2]−2 , and this suffices for Lemma 21 when (9.9) holds. Finally, it is clear in fact that we still have this estimate if 2 α 2µ − 1, since if µ > 0 then p belongs to the finite set of divisors of M . This therefore completes the proof of Lemma 21.Our final task in this section is to establish a lower bound for σ ∞ to complement the upper bound in(8.15). For any c ∈ R 2 let B(c; ρ) ⊂ R 2 denote the box centered on c with side length 2ρ. The final part of Lemma 9 implies that there exists ρ ≫ G −1 U n/2 such that ω(x 1 , x 2 ) ≫ G 2−n for every (x 1 , x 2 ) ∈ B(δN K/L (U u (R) ); ρ). Here we view δN K/L (U u (R) ) as a vector (c 1 , c 2 ) in R 2 . 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Sansuc, Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, J. reine angew. Math. 327 (1981), 12-80. Algebraic groups and their birational invariants (translated from Russian by B. Kunyavskiȋ). V E Voskresenskiȋ, Translations of Math. Monographs. 179American Math. SocV.E. Voskresenskiȋ, Algebraic groups and their birational invariants (translated from Russian by B. Kunyavskiȋ). Translations of Math. Monographs 179, American Math. Soc. 1998. . T D Browning, Bristol, BS8 1TW, United Kingdom E-mailSchool of Mathematics, University of BristolT.D. Browning, School of Mathematics, University of Bristol, Bristol, BS8 1TW, United Kingdom E-mail : [email protected] . D R Heath-Brown, Oxford, OX1 3LB, United Kingdom E-mailMathematical Institute, University of OxfordD.R. Heath-Brown, Mathematical Institute, University of Oxford, Oxford, OX1 3LB, United Kingdom E-mail : [email protected]	[]
[ "MULTI-SLICE NET: A NOVEL LIGHT WEIGHT FRAMEWORK FOR COVID-19 DIAGNOSIS", "MULTI-SLICE NET: A NOVEL LIGHT WEIGHT FRAMEWORK FOR COVID-19 DIAGNOSIS" ]	[ "Harshala Gammulle \nThe Signal Processing\nArtificial Intelligence and Vision Technologies (SAIVT)\nQueensland University of Technology\nAustralia\n", "Tharindu Fernando \nThe Signal Processing\nArtificial Intelligence and Vision Technologies (SAIVT)\nQueensland University of Technology\nAustralia\n", "Sridha Sridharan \nThe Signal Processing\nArtificial Intelligence and Vision Technologies (SAIVT)\nQueensland University of Technology\nAustralia\n", "Simon Denman \nThe Signal Processing\nArtificial Intelligence and Vision Technologies (SAIVT)\nQueensland University of Technology\nAustralia\n", "Clinton Fookes \nThe Signal Processing\nArtificial Intelligence and Vision Technologies (SAIVT)\nQueensland University of Technology\nAustralia\n" ]	[ "The Signal Processing\nArtificial Intelligence and Vision Technologies (SAIVT)\nQueensland University of Technology\nAustralia", "The Signal Processing\nArtificial Intelligence and Vision Technologies (SAIVT)\nQueensland University of Technology\nAustralia", "The Signal Processing\nArtificial Intelligence and Vision Technologies (SAIVT)\nQueensland University of Technology\nAustralia", "The Signal Processing\nArtificial Intelligence and Vision Technologies (SAIVT)\nQueensland University of Technology\nAustralia", "The Signal Processing\nArtificial Intelligence and Vision Technologies (SAIVT)\nQueensland University of Technology\nAustralia" ]	[]	This paper presents a novel lightweight COVID-19 diagnosis framework using CT scans. Our system utilises a novel twostage approach to generate robust and efficient diagnoses across heterogeneous patient level inputs. We use a powerful backbone network as a feature extractor to capture discriminative slice-level features. These features are aggregated by a lightweight network to obtain a patient level diagnosis. The aggregation network is carefully designed to have a small number of trainable parameters while also possessing sufficient capacity to generalise to diverse variations within different CT volumes and to adapt to noise introduced during the data acquisition. We achieve a significant performance increase over the baselines when benchmarked on the SPGC COVID-19 Radiomics Dataset, despite having only 2.5 million trainable parameters and requiring only 0.623 seconds on average to process a single patient's CT volume using an Nvidia-GeForce RTX 2080 GPU.	10.1109/icas49788.2021.9551157	[ "https://arxiv.org/pdf/2108.03786v1.pdf" ]	236,956,756	2108.03786	96429ffbae51de68469d24bee69b22a239c77a95	MULTI-SLICE NET: A NOVEL LIGHT WEIGHT FRAMEWORK FOR COVID-19 DIAGNOSIS Harshala Gammulle The Signal Processing Artificial Intelligence and Vision Technologies (SAIVT) Queensland University of Technology Australia Tharindu Fernando The Signal Processing Artificial Intelligence and Vision Technologies (SAIVT) Queensland University of Technology Australia Sridha Sridharan The Signal Processing Artificial Intelligence and Vision Technologies (SAIVT) Queensland University of Technology Australia Simon Denman The Signal Processing Artificial Intelligence and Vision Technologies (SAIVT) Queensland University of Technology Australia Clinton Fookes The Signal Processing Artificial Intelligence and Vision Technologies (SAIVT) Queensland University of Technology Australia MULTI-SLICE NET: A NOVEL LIGHT WEIGHT FRAMEWORK FOR COVID-19 DIAGNOSIS Index Terms-COVID19 DiagnosisDeep LearningComputed TomographyMedical Imaging This paper presents a novel lightweight COVID-19 diagnosis framework using CT scans. Our system utilises a novel twostage approach to generate robust and efficient diagnoses across heterogeneous patient level inputs. We use a powerful backbone network as a feature extractor to capture discriminative slice-level features. These features are aggregated by a lightweight network to obtain a patient level diagnosis. The aggregation network is carefully designed to have a small number of trainable parameters while also possessing sufficient capacity to generalise to diverse variations within different CT volumes and to adapt to noise introduced during the data acquisition. We achieve a significant performance increase over the baselines when benchmarked on the SPGC COVID-19 Radiomics Dataset, despite having only 2.5 million trainable parameters and requiring only 0.623 seconds on average to process a single patient's CT volume using an Nvidia-GeForce RTX 2080 GPU. INTRODUCTION Although Reverse Transcription Polymerase Chain Reaction (RT-PCR) is considered the global standard SARS-CoV-2 (COVID-19) diagnosis, this test is very time consuming and has a high false negative rate, which in turn yields significant challenges in preventing the spread of the infection [1, 2]. As such, Computed Tomography (CT) imaging has been identified as a fast, simple and reliable diagnosis tool due to the existence of discriminative patterns associated with the COVID-19 infection within the CT scans. However, recent literature has shown that COVID-19 lung manifestations show substantial similarities with Community Acquired Pneumonia (CAP), complicating the diagnosis process [1]. To this end several deep learning based frameworks have been introduced to automate diagnosis, where models are trained to uncover discriminative patterns embedded within the data and which cannot be identified by the naked-eye. This paper presents the QUT SAIVT team's 1 framework for the 2021 IEEE ICASSP Signal Processing Grand Challenge 1 https://research.qut.edu.au/saivt/ (SPGC) -"COVID-19 Radiomics". This challenge dataset has been constructed to motivate machine learning practitioners to develop robust and reliable systems to classify patients into COVID-19, CAP and NORMAL diagnosis classes using a heterogeneous set of CT scans. In particular, these CT scans are composed of different slice thicknesses, radiation doses, and noise levels, in addition to featuring patients with various comorbidities and different surgical histories. While volumetric CT scans provide a comprehensive illustration of lung abnormalities and their structure, patient level diagnosis from heterogeneous CT volumes faces several challenges as noise and variation between scans can lead to misclassification of individual CT slices. Hence, simplistic score-level/ feature-level [1, 3] aggregation performs poorly as there is a tendency for some slices to be misclassified. Structures such as 3D-CNNs have also been used to regress volumetric CT inputs directly to the final diagnosis decision [2,4,5]. While this allows the model to extract and operate over feature vectors that represent the entire lung of the patient, these models have a very high-dimensional parameter space (tens of millions of trainable parameters) and are prone to over-fitting when trained using datasets with patients (individual samples) in the order of hundreds. To alleviate these challenges we propose a novel twostage framework where features from individual slices are aggregated to a patient level diagnosis via an efficient, lightweight 1D-CNN based model. As novel contributions, (1) our design exploits slice-level features from adjacent slices at different granularities, combining and compressing these discriminative features, prior to classification; (2) has fewer trainable parameters, enabling effective training from a smaller set of volumetric CT scans; (3) our method allows us to seamlessly process examples with a variable number of slices, and even allows the model to learn from incomplete/partial scans; and (4) due to the use of a pretrained backbone (feature extractor) to extract features from the individual CT slices, the backbone can be swapped or modified. Hence, the proposed two-stage framework is not limited to CT lung classification tasks, but can be easily adapted to any diagnosis task which requires aggregation of heterogeneous information across different samples. SPGC COVID-19 RADIOMICS DATASET The SPGC COVID-19 Radiomics Dataset is one of the largest datasets containing COVID-19, Community Acquired Pneumonia (CAP), and normal cases, and is captured in different medical centers with various imaging settings. The dataset comprises volumetric CT scans of 307 patients (171 COVID-19, 60 CAP, and 76 NORMAL patients). All captured slices in the CT scans are in the Digital Imaging and Communications in Medicine (DICOM) format. The data is acquired using a SIEMENS, SOMATOM Scope scanner with the normal radiation dose and the slice thickness of 2mm. Apart from this patient level labelling, a small subset (i.e 55 COVID-19, and 25 CAP) were analyzed and the individual slices were labeled to indicate evidence of infection. In total 4,993 slices were identified as being indicative of infection. From this dataset, 30% of the data was randomly selected and provided as a validation set. The validation set contains 98 patients (55 COVID, 19 CAP, and 24 NORMAL). The test set consists of three subsets where they consist of 35 COVID, 20 CAP, and 35 NORMAL patients. Test dataset labels are withheld, however, we report the challenge evaluation released by the organisers. METHODOLOGY We propose a deep network approach, Multi-slice Net, which performs the lung infection classification from the volumetric chest CT scans. The proposed framework is shown in Fig. 1, and is composed of a backbone for slice level feature extraction and a network to aggregate these features from a patient to a single score (Multi-Slice Network). Feature Extractor/ Backbone One of the key motivations of the proposed approach is to minimise pre-processing. Hence, aside from converting individual DICOM files to JPG format, no pre-processing steps are performed. In contrast to existing state-of-theart approaches [1, 2] which perform lung detection and segmentation during pre-processing, the proposed framework applies the feature extractor directly to the JPG slice images. Extracting features from CT slices that capture discriminative infection-related information is crucial for infection classification. We utilise the squeeze-and-excitation ResNet50 (SE-ResNet50) model [6], pre-trained on the ImageNet dataset [7].The SE-ResNet50 extends the original ResNet50 architecture with the aid of squeeze and excitation operations. In particular, the squeeze operation extracts global information from each of the channels of the input while the excitation act as a bottleneck, adaptively recalibrating the importance of each channel. We fine-tune the SE-ResNet50 model, though the first 6 layers are frozen. For fine-tuning, the subset of patients with slice level annotations are used. This subset contains 55 COVID, and 25 CAP patients. We also randomly selected slices from 15 NORMAL patients for the fine-tuning data. The constructed dataset contains of 2482 COVID, 742 CAP, and 1820 NORMAL slices for training and 1333 COVID, 436 CAP and 840 NORMAL slices for validation. For the compatibility with the pre-trained backbone network, the input CT slices of shape 512 × 512 × 1 are resized to 224 × 224 × 1 and replicated 3 times (224 × 224 × 3), before being fed to the backbone SE-ResNet50. To reduce over-fitting we used data augmentation and added Random Horizontal Flips with 50% probability, and randomly changed the brightness, contrast and saturation of the input by a factor of upto 0.4. The network is trained using the Adam [8] optimiser with a learning rate of 1e−5 using Categorical Cross-Entropy Loss for 100 epochs. We used class weights to balance the impact of the minority classes. After fine-tuning, we use the model with best validation accuracy and extract the features from the penultimate layer of SE-ResNet50, with a feature dimensionality of 2048. Multi-Slice Network The features extracted from the backbone then fed to the proposed Multi-Slice Net to obtain a patient-level infection classification. Fig. 2 illustrates our approach. Multi-Slice Net iterates through the features extracted from all the CT slices that belong to a particular patient, aggregating them, and generates a single feature descriptor representing the patient. This feature is then fed through a series of dense layers to obtain the final patient diagnosis. In the challenge dataset, the CT scans of each patient have a varying number of slices. As such, we designed the network to handle a variable number of slices with the aid of fully convolution network. Let the number of CT slices for a particular subject be l, then, the input (I) of the Multi-Slice Net takes the shape (l, 2048). Our Multi-Scale Net consists of a temporal convolution layer followed by 4 dilated residual blocks, composed of dilated convolutions. Inspired by [9,10], we doubled the dilation factor at each layer, and the number of convolutional filters used at each layer is 64. The output of the fourth dilated residual block is then passed through a max-pooling layer with the kernel size of l, which encodes the input sequence into a single feature with dimensionality of 64. This feature is then passed through the classification network which is composed of two dense layers with sizes 32 and 3 (number of infection classes) respectively. The use of temporal convolution allows our network to interrogate the slice level features at different granularities, comparing and contrasting features of neighbouring slices. By aggregating these features to a single vector, the most salient features from the patient are passed to the classifier. This network is trained using the Adam [8] optimiser with a learning rate of 1e−4 using Categorical Cross-Entropy Loss for 100 epochs. Slice Level Features Backbone/Feature Extractor CT Slices Multi-Slice Net Patient Level Classification EVALUATION RESULTS In this section, we first present evaluation results for the finetuning process of the feature extractor (Sec. 4.1). In Sec. 4.2 we report patient level diagnosis performance using Multi-Slice Net (MS-Net). Slice Level Classification Performance (Backbone Networks) We evaluate several network architectures to determine an appropriate backbone for feature extraction. When fineturning these networks, we initialised them with their respective ImageNet weights and fine-tuned them for 100 epochs using the Adam optimiser, a learning rate of 1e−5 and the categorical cross entropy loss. Note that for the fine-tuning process we utilised a subset of the SPGC COVID-19 Radiomics Dataset provided by the organisers which had slice level annotations (see Sec. 3.1 for details). Method Validation Sensitivity Validation Accuracy COVID CAP NORMAL DenseNet [11] 32.80% 83.66% 71.52% 63.52% ResNet-18 [12] 60.84% 61.39% 90.82% 71.02% SqueezeNet [13] 76.72% 71.06% 89.95% 79.15% ResNet-50 [12] 72.08% 78.47% 88.80% 79.92% SE-ResNet-50 [6] 79.50% 84.58% 96.02% 86.63% Table 1. Slice-level classification accuracy using CT slices from a subset of the SPGC COVID-19 Radiomics dataset. We report class-level sensitivity for COVID, Community Acquired Pneumonia (CAP), and NORMAL classes and overall accuracy (percentage of correct predictions). Tab. 1 provides results for the ResNet-18 [12], ResNet-50 [12], SqueezeNet [13], DenseNet [11] , and SE-ResNet-50 [6] architectures when fine-tuned to obtain a slice level diagnosis. We observe superior performance from the SE-ResNet-50 architecture, despite of the fact that it has been introduced for channel level feature re-calibration on RGB inputs. Despite the need to replicate a single channel CT slice image three times to satisfy the 3-channel requirement of the network, we observe a significant performance increase between ResNet-50 and SE-ResNet-50. We believe this is a result of the removal of redundant/replicated information in channels through the squeeze and excitation blocks of SE-ResNet-50, allowing the classification layers to better focus on informative spatial attributes of the input. Patient-Level Evaluation Evaluation results with respect to the validation set of SPGC COVID-19 Radiomics dataset are provided in Tab. 2. We report the results of the baseline model provided by the challenge organisers as well as results for MS-Net with different backbones. Our framework outperforms the baseline system, especially when considering the COVID detection sensitivity. We observe similar performance between the ResNet and SE-ResNet backbones, despite the significant performance gap between these methods with respect to slice level evaluations. In Tab. 3 we provide results across testing subsets of the SPGC COVID-19 Radiomics dataset. Despite the lightweight architecture we observe that our framework has achieved competitive results for all classes across all subsets. As the ground truth labels of the test data is not available we cannot compare our performance with existing state-of-the-art models. However, we note that this framework achieved 9th place (from 17 competitive systems) in the SPGC COVID-19 Radiomics challenge. Furthermore, one important characteristic of the proposed method is its consistent performance across the different classes. Despite the heterogeneous test sets, including different slice thicknesses, radiation dose, patient level differences, our lightweight system has been able to achieve consistent performance. Another noteworthy aspects of the proposed approach is the ability to seamlessly switch between different backbone networks. Due to our two-stage approach, the architecture of MS-Net does not require any changes when changing the backbone feature extractor. Moreover, the backbone can be trained in a separate dataset, even without any patient-level data (i.e multiple-slices per patient). As the proposed MS-Net has fewer trainable parameters it can be tuned later with a small scale dataset with fewer patient-level annotations. In addition, we highlight that the MS-Net architecture is not limited to slice level feature aggregation from CT scans. It could be utilised for any aggregation task where features from different spatial or temporal locations need to be aggregated. Network Complexity The majority of the trainable parameters in our framework lie within the backbone feature extractor (SE-ResNet-50), which has 2.5 million trainable parameters (the first six layers are frozen during fine-tuning). MS-Net has only 207,683 trainable parameters due to its careful design. Despite the parameter heavy design of the backbone, the plug and play nature of MS-Net allows the backbone to be pre-trained on a completely different data corpus, and fine-tuned for the task at hand using a smaller dataset. It generates 268 patient level predictions (each of which has a variable number of slices, between 100 and 200, per patient) in 166.9619 seconds. This includes inference for both the backbone network for feature extraction and MS-Net to obtain patient-level predictions. Therefore, on average it takes only 0.6229 seconds to process a CT volume. In future works we will be investigating better backbone architectures to further improve our model's performance, while maintaining it's light weight nature. CONCLUSION We present a novel light weight framework for COVID-19 diagnosis. Our approach uses a two-stage architecture, composed of a backbone network for feature extraction from individual CT scan slices, and a network to aggregate these slice level features for patient-level diagnosis. Considering the limited data availability of complete patient-level CT volumes, we design a light-weight network to aggregate the slice-level features for patient-level diagnosis. This system is evaluated using the SPGC COVID-19 dataset and achieves competitive results. One prominent attribute of our design is the plug-and-play nature of the aggregation network, which allows the backbone to be trained on a completely different dataset and then tuned on a smaller dataset for the task at hand with patient-level annotations. 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[ "Teaching DevOps in academia and industry: reflections and vision", "Teaching DevOps in academia and industry: reflections and vision" ]	[ "Evgeny Bobrov \nInnopolis University\nRussian Federation\n", "Antonio Bucchiarone \nFondazione Bruno Kessler\nTrentoItaly\n", "Alfredo Capozucca \nUniversity of Luxembourg\n\n", "Nicolas Guelfi \nUniversity of Luxembourg\n\n", "Manuel Mazzara \nInnopolis University\nRussian Federation\n", "Sergey Masyagin \nInnopolis University\nRussian Federation\n" ]	[ "Innopolis University\nRussian Federation", "Fondazione Bruno Kessler\nTrentoItaly", "University of Luxembourg\n", "University of Luxembourg\n", "Innopolis University\nRussian Federation", "Innopolis University\nRussian Federation" ]	[]	This paper describes our experience of delivery educational programs in academia and in industry on DevOps, compare the two approaches and sum-up the lessons learnt. We also propose a vision to implement a shift in the Software Engineering Higher Education curricula.	10.1007/978-3-030-39306-9_1	[ "https://arxiv.org/pdf/1903.07468v1.pdf" ]	81,976,967	1903.07468	3766ff5fa893336c23b8cbed8287cb38d9e5db56	Teaching DevOps in academia and industry: reflections and vision Evgeny Bobrov Innopolis University Russian Federation Antonio Bucchiarone Fondazione Bruno Kessler TrentoItaly Alfredo Capozucca University of Luxembourg Nicolas Guelfi University of Luxembourg Manuel Mazzara Innopolis University Russian Federation Sergey Masyagin Innopolis University Russian Federation Teaching DevOps in academia and industry: reflections and vision This paper describes our experience of delivery educational programs in academia and in industry on DevOps, compare the two approaches and sum-up the lessons learnt. We also propose a vision to implement a shift in the Software Engineering Higher Education curricula. Introduction DevOps is a natural evolution of the Agile approaches [1,2] from the software itself to the overall infrastructure and operations. This evolution was made possible by the spread of cloud-based technologies and the everything-as-a-service approaches. Adopting DevOps is however more complex than adopting Agile [3] since changes at organisation level are required. Furthermore, a complete new skill set has to be developed in the teams [4]. The educational process is therefore of major importance for students, developers and managers. DevOps way of working has introduced a set of software engineering activities and corresponding supporting tools that has disrupted the way individual developers and teams produce software. This has led both the world of research and industry to review software engineering life-cycle and all the supporting techniques to develop software in continuous operation and evolution. If we want to enclose DevOps in one word, it is continuous. Modelling, integration, testing, and delivery are significant part of DevOps life-cycle that, respect to enterprise or monolithic applications developed some years ago, must be revised continuously to permit the continuous evolution of the software and especially an easy adaptability at context changes and new requirements. Adopting the De-vOps paradigm helps software teams to release applications faster and with more quality. In this paper, we consider two sides of the same coin that are the usage of DevOps in academia and in industry. Research in traditional software engineering settings has mainly focused on providing batch automation, as in the case of translation and re-engineering of legacy code [5], or on helping developers keep track of their changes, as in the case of version control [6]. The radically new development processes, introducing with the DevOps, have required major changes to traditional software practices [7]. New versions of software components are developed, released, and deployed continuously to meet new requirements and fix problems. A study performed by Puppet Labs in 2015 1 testifies that using DevOps practices and automated deployment led organisations to ship code 30 times faster, complete deployments 8,000 times faster, have 50% fewer failed deployments, and restore service 12 times faster than their peers. Due to the dramatically growing of the DevOps supporting tools 2 , has seen a big change in the role played by the software engineers of a team. The latter today have the complication of covering both management and development aspects of a software product. They are part of a team and have the following responsibilities: (1) to be aligned with the new technologies to ensure that the the high-performance software is released using smart tools to specify, develop, deploy and execute scalable software systems, (2) to define procedures to guarantee the high security level of the running code, (3) to monitor the software in operation and guarantee the right level of adaptability. As long as DevOps became a widespread philosophy, the necessity of education in the field become more and more important, both from the technical and organisational point of view [4]. This paper describes parallel experiences of teaching both undergraduate and graduate students at the university, and junior professional developers in industry. There are similarities and differences in these two activities, and each side can learn from the other. We will discuss here some common issues and some common solutions. We also propose a vision to implement a shift in the Software Engineering Higher Education curricula. The paper is organised as follows: after this introduction of the context in Section 1, we first discuss the experience gained in teaching DevOps at the university (Section 2). We then present the key elements of training and consultancies delivered in industry on the same subject (Section 3) and we analyse similarities and differences in Section 4. Section 5 proposes a vision to implement a shift in the Software Engineering Higher Education curricula. Finally, in Section 6 we present our conclusion. Teaching in Academia DevOps experienced significant success in the industrial sector, but still received minor attention in higher education. One of the few and very first courses in Europe focusing on DevOps was delivered at the university of Luxembourg [8]. This course is part of a graduate programme aimed at students pursuing a degree in computer science. Students following this programme either continue their development either in the private sector or doing a PhD at the same university (most of the cases). Therefore, most of the courses in such a programme are designed as a sequence of theoretical lectures and assessed by a mid-term and final exam. Our course is the exception in the programme as it is designed according to the Problem-based learning (PBL) method. Organisation and delivery Following a problem-based approach, the learning of the students is centred on a complex problem which does not have a single correct answer. The complex problem addressed by the course corresponds to the implementation of a Deployment Pipeline, which needs to satisfy certain functional and non-functional requirements. These requirements are: This means that students work in groups all along the course duration to produce a solution to the given problem. By working in groups students are immerse in a context where interactions problems may arise, and so allowing them to learn soft-skills to deal with such as problems. Therefore, the success to achieve a solution to the problem depends on not only the technical abilities, but also the soft-skills capacities each group member either has already had or is able to acquire during the course. Notice that DevOps is not only about tools, but also people and processes. Thus, soft-skills capabilities are a must for future software engineers working expected to work in a DevOps-oriented organisations. - Structure The course is organised as a mix of lectures, project follow-up sessions (aimed at having a close monitoring of the work done for each group member and helping solve any encountered impediments), and checkpoints (sessions where each group presents the advances regarding the projects objectives). Lectures are aimed at presenting the fundamental DevOps-related concepts required to implement a Deployment Pipeline (Configuration Management, Build Management, Test Management, and Deployment Management). Obviously, the course opens with a general introduction to DevOps and a (both procedural and architectural) description of what a Deployment Pipeline is. In the first project follow-up session each group presents the chosen product they will use to demonstrate the functioning of the pipeline. The remaining of the course is an interleaving between lectures and follow-up sessions. The first check-point takes place at the fifth week, and the second one at the tenth week. The final checkpoint, where each group has to make a demo of the Deployment pipeline, takes place at the last session of the course. Execution Most of the work done by the students to develop the Deployment Pipeline was done outside of the course hours due to the limited in-class time assigned to the course. However, examples (e.g. virtual environments creation, initial setup and provisioning) and references to well-documented tools (e.g. Vagrant, Ansible, GitLab, Jenkins, Maven, Katalon) provided during the sessions helped students on moving the project ahead. Moreover, students had to possibility to request support either upon appointment or simply signalling the faced issues with enough time in advance to be handled during a follow-up session. the teaching. Nevertheless, the staff was closely supervising the deployment pipeline development by both monitoring the activity on the groups working repositories and either asking technical questions or requesting live demos during the in-class sessions. Assessment As described in [8], each kind of activity is precisely specified, so it lets students know exactly what they have to do. This also applies to the course assessment: while the project counts for 50% of the final grade, the other half is composed of a report (12.5%) and the average of the checkpoints (12.5%). The aim at requesting to each group submit a report is to let students face with the challenge of doing collaborative writing in the same way most researchers do nowadays. Moreover, this activity makes the course to remain aligned with programmes objectives: prepare the student to continue a research career. It is also in this direction the we have introduced peer-reviewing: each student is requested to review (at least one) no-authored report (this activity also contributes to the individual grading of the student). Despite of these writing and reviewing activities may seem specific to the programme where the course fits, we do believe that they also contribute to the development of the required skills software engineers need to have. Latest experience and feedback Based on our latest experience the relevant points to highlight are: (1) the positive feedback obtained from students, (2) the absence of drops out, and (3) the quality of the achieved project deliverables. Regarding the first point, the evidence was found through a survey filled out by students once the course was over: 100% strongly agreed that the course was well organised and ran smoothly, 75% (25%) agreed (strongly agreed) the technologies used in the course were interesting, and 75% was satisfied with the quality of this course. We are very happy about the second point as it was one of the objectives (i.e. reduced the number of drops out -it used to reach up to 70%) when we decided to redesign the course to its current format. Moreover, the absence of drops out can also be confirmed by the fact that (based on the survey) 75% of the students would advise other students to take the course, if it were optional. Last, but not least, the survey also helped to confirm that PBL is the right pedagogical approach to tackle subjects like DevOps (and any others related to software engineering): 100% of the students agreed that they would like to have more project-oriented courses like this one. The third relevant point was about the quality of the project deliverables: considering the limited time to present and work out the subjects related to a Deployment Pipeline, each group succeed to provide deliverables able to meet the given functional and non-functional requirements. Teaching in Industry Our team is specialised in delivering corporate training for management and developers and has long experience of research in the service-oriented area [9,10,11]. In recent years we have provided courses, training and consultancies to a number of companies with particular focus on east Europe [12]. For example, only in 2018 more than 400 hours of training were conducted involving more than 500 employees in 4 international companies. Although we cannot share the details of the companies involved, they are mid to large size and employ more than 10k people. The trainings are typically focusing on: -Agile methods and their application [3] -DevOps philosophy, approach and tools [13] -Microservices [14,15] Organisation and delivery In order for the companies to absorb the DevOps philosophy and practice, our action has to focus on people and processes as much as on tools. The target group is generally a team (or multiple teams) of developers, testers and often mid-management. We also suggest companies to include representatives from businesses and technical analysts, marketing and security departments. These participants could also benefit from participation and from the DevOps culture. The nature of the delivery depends on the target group: sessions for management focus more on effective team building and establishment of processes. When the audience is a technical team, the focus goes more on tools and effective collaboration within and across the teams. Structure The events are typically organised in several sessions run over a one-day to threeday format made or frontal presentations and practical sessions. The sessions are generally conducted at the office of the customer in a space suitably arranged after the previous discussion with the local management. Whenever possible the agenda and schedule of the activities have to be shared in advance. In this way, the participants know what to expect, and sometime a preparatory work is required. Limitations of the set-up One of the limitations we had to cope with, often but not always, is the fact that bilateral previous communication with teams is not always possible or facilitated, and the information goes through some local contact and line manager. At times this demands for an on-the-fly on-site adaptation of the agenda. In order to collect as much information as possible on the participants and the environment, we typically send a survey to be completed a few days in advance, and we analyse question by question to give specific advice depending on the answers. Lessons learnt and optimisation In retrospective, the most effective training for DevOps and Agile were those in which the audience consisted of both management and developers. Indeed the biggest challenge our customer encountered was not how to automatise existing processes, but in fact how to set up the DevOps approach itself from scratch. Generally, technical people know how to set up automatisation, but they may have partial understanding about the importance and the benefits for the company, for other departments, the customer and ultimately for themselves. It is important therefore to show the bigger picture and help them understanding how their work affects other groups, and how this in turn affects themselves in a feedback loop. The presence of management is very useful in this process. The technical perspective is often left for self-study or for additional sessions. Latest experience and feedback The feedback from participants surpassed our expectation. In synthesis, this are the major achievements of the past sessions: -Marketers now understand how they may use A/B testing and check the hypothesis -Security engineers find positive to approve small pieces of new features, not the major releases -Developers developed ways to communicate with other departments and fulfil their needs step by step based on the collaboration -Testers shifted their focus on product testing (integration-, regression-, soak-, mutation-, penetration-testing) rather than unit testing, and usually set future goals for continuing self-education on the subject Often multiple session can be useful. The primary objective is to educate DevOps ambassadors, but it is also important to create an environment that can support the establishment of DevOps processes and the realisation of a solid DevOps culture, when every department welcome these changes. This does not typically happen in a few days. Discussion The experience of teaching in both an academic and industrial context emphasised some similarities and some differences that we would like to discuss here. Understanding these two realities may help in offering better pedagogical programme from the future since each domain can be cross-fertilised by the ideas taken by the other. What we have seen in terms of similarities: -Pragmatism: Both students and developers appreciate hands-on sessions -Hype: Interest and curiosity in the topic has been seen both in academia and industry, demonstrating the relevance of the topic -Asymmetry: Classic education and developers training put more important on Development than Operations and presenting the two sides as interrelated strengthen the knowledge and increase efficacy What we have seen in terms of differences: -Learners initial state: based on the academic curriculum where the course is included, it is possible to know (or at least to presume) the already acquired knowledge for the participant students. This may not be the case in a corporate environment, where the audience is generally composed by people with different profiles and backgrounds. -Learners attitude: students too often are grade-focused, developers are interested in the approach as long as can improve their working conditions, manager see things in terms of cost saving -Pace of education: short and intense in a corporate environment, can be long and diluted in academia -Assessment/measure of success: classic exam-based at the university, a corporate environment often does not require a direct assessment at the end of the sessions and the success should be observed in the long run -Expectation: corporate audience is more demanding. This may nor be a surprise given the costs and what is a stake. Students are also subject to a cost, but it is more moderate and spread over a number of course attended in one year. Vision After reporting experiences in teaching DevOps-based courses in both academic and industrial environments (reflection), in this section we will look at the future and we will describe our vision for the modernisation of university curricula in Computer Science, in particular for the Software Engineering tracks. While our vision and conclusions can be effectively applied in every Higher Education institution, we are here considering a specific case study: Innopolis University, a new IT educational institution in the Russian Federation. This is the reality we have more direct experience of. In [16] the first five years of Innopolis University and the development of the internationalisation strategy is discussed, while [17] presents some teaching innovations and peculiarities of the university. At Innopolis University students have a 4-year bachelor, the firs two years are fundamental, and a specific track is chosen at the third year (Software Engineering, Data Science, Security and Network Engineering or Artificial Intelligence and Robotics). There are also 2-year Master Programs, following exactly the same four tracks. The last two years of the bachelors are characterised by a fewer number of courses. Moreover, some of these courses are elective, and delivered either by academic or industrial lecturers. These elective courses are aimed at covering specific topics required by industry. While working with industry we realised that the obstacles for the full adoption of DevOps are not only of technical nature, but also of mindset. This issue is difficult to solve since companies need to establish a radically new culture and transfer it to the new employees who join the company with a legacy mindset. The same situation may occur for fresh graduates. Classic curricula are very often based on the idea of system as a monolith and process as a waterfall. Of course, in the last twenty years, innovations have been added to the plan of study worldwide. However, when focusing on the first two years of Bachelor education, it can be seen that the backbone of the curricula is still outdated (due to legacy reasons, and sometimes, ideological ones). It is therefore necessary to explain students the DevOps values from scratch, establishing clear connections of every course with DevOps, and describing how fundamental knowledge works within the frame of this philosophy. Furthermore, Computer Science curricula have a strong emphasis on the "Dev" part, but cover the "Ops" part only marginally, for example as little modules inside courses such as Operating Systems and Databases. To cover the "Ops" part we need to teach how to engineer innovative software systems that can react to changes and new needs properly, without compromising the effectiveness of the system and without imposing cumbersome a priori analyses. To this end, we need to introduce courses on learning and adaptation theories, algorithms and tools, since they are becoming the key enablers for conceiving and operating quality software systems that can automatically evolve to cope with errors, changes in the environment, and new functionalities. At the same time, to continuously assess the evolved system, we need also to think to teach validation and verification techniques pushing more them at runtime. The DevOps philosophy is broad, inclusive, and at the same time, flexible enough to work as a skeleton for Software Engineering education. This is what drives our vision and we described in the next parts of this section. Phases of Software Engineering Education The DevOps philosophy presents recurring and neat phases. It has been shown that companies willing to establish a strong DevOps culture have to pay attention to every single phase [18]. Missing a phase, or even a simple aspect of it, might lead to poor overall results. This attention to every single phase should also be applied also to university education. In this interpretation (or proposal), every phase corresponds to a series of concepts and a skill-set that the student has to acquire along the process. It is therefore possible to organise the educational process and define a curriculum for software engineering using the DevOps phases as a backbone (Figure 1 summarises these phases). This path would allow students to realise the connection between different courses and apply the knowledge in their future career. The plan described here is what we are considering to experiment at Innopolis University, expanding the experience acquired on the delivery of specialised DevOps courses to the entire plan of study. We will use the idea described in [19] as a backbone for curriculum innovation. Fig. 1. DevOps Phases We consider ideal an incremental and iterative approach for bachelors to fully understand and implement the DevOps philosophy. We utilise the following taxonomy: 1. How to code 2. How to create software 3. How to create software in a team 4. How to create software in a team that someone needs 5. How to create software in a team that business needs In details, this is the path we propose for the bachelor 3 programme, based on Agile and DevOps according to the taxonomy: 1. The first three semesters are devoted to fundamental knowledge of hard and soft skills, which are essential to create software, especially following Agile and DevOps. We want to educate the next generations of students providing them not only with knowledge of programming languages and algorithms, but also with software architectures, design patterns and testing. This way students know how to create quality software fulfilling the essential nonfunctional requirements (such as reliability, maintainability, and scalability). 2. The fourth semester has a software project course (to be considered as an introduction to the software engineering track) based on the trial and error approach without any initial constraints and thorough analysis of identified problems in the second part of the semester. 3. The fifth semester has a new iteration of the software project course with a deep understanding of the Agile philosophy and the most popular Agile frameworks. 4. The sixth semester is based on the same project that has been created earlier and adds automaton, optimisation of the Development, and it introduces the Operational part and the feedback concept. 5. During the last two semesters (i.e. seventh and eighth), students start to work with real customers from industry and try to establish all processes and tools learnt in the previous three years. 6. During the third and fourth years, we propose additional core and elective courses in order to explore deeper modern technologies, best practices, patterns and frameworks. Transition towards the new curriculum In this section we will address the transition from the current curriculum to the new one identifying the iterations and steps year by year until the full implementation, and we will emphasise the role of industry in this process. For the last years since foundation (2012), the curriculum for Software Engineers at Innopolis University was mostly waterfall-based with a clear focus on hard skills. Each course was delivering methods and tools specific of a certain phase, but not always the "fil rouge" between courses was emphasised. Courses connecting the dots and providing the basis for an iterative and incremental approach are now under development. The first four semesters of the bachelor provide the prerequisites for Software Engineering (and for Computer Science in general), whereas the last four semesters are track-based (see Fig.2 and Fig.3). The transition is planned to happen in 5-year time: -Year 1. Make minor changes to the curriculum, targeting in particular two courses: Software Project for second-year spring semester, and Project for Software Engineers at the third year, fall semester. The first one has to be adapted to teach students how to establish processes and develop software according to Agile. The second one will be increased by adding the possibility to collaborate with industry and develop actual projects. The students interact with industry representatives and define project objectives with industry under the control of the university. -Year 2. Work more closely with industry and add more elective courses covering skills required by companies. A course on DevOps will be added to the spring semester of the third year of the bachelor to be intended as a continuation of Software Project. The content of some courses will be adjusted to contain DevOps philosophy. Conclusions Ultimately, DevOps [2,13] and the microservices architectural style [14,15] with its domains of interests [21,22,23,24,25] may have the potential of changing how companies run their systems in the same way Agile has changed the way of developing software. The critical importance of such cultural change should not be undervalued. It is in this regard that higher education institutions should put a major effort to fine tune their curricula and cooperative programme in order to meet this challenge. In terms of pedagogical innovation, the authors of this paper have experimented for long with novel approaches under different forms [17]. However, DevOps represents a newer and significant challenge. Despite of the fact current educational approaches in academia and industry show some similarities, they are indeed significantly different in terms of attitude of the learners, their expectation, delivery pace and measure of success. Similarities lay more on the perceived hype of the topic, its typical pragmatic and applicative nature, and the minor relevance that education classically reserves to "Operations". While similarities can help in defining a common content for the courses, the differences clearly suggest a completely different nature of the modalities of delivery. From the current experience we plan to adjust educational programs as follows: -University teaching: trying to move the focus out of final grade, emphasising more the learning aspect and give less importance to the final exam, maybe increasing the relevance of practical assignments. It may be also useful to intensify the theoretical delivery to keep the attention higher and have more time for hand-on sessions. Ultimately, our vision is to build a Software Engineering curricula on the backbone derived from the DevOps philosophy. -Corporate training: it is important not to focus all the training activity as a frontal session university-like. Often the customers themselves require this classical format, maybe due to the influence of their university education. We believe that this makes things less effective and we advocate for a change of paradigm. Finally, we have described our vision for the transition to the new curriculum at Innopolis University. In terms of educational innovation, other realities are moving fast and we should not be shy in proposing curricula drastic changes. Fig. 2 . 2Curriculum of Year 1 and Year 2 Fig. 3 . 3Curriculum for Software Engineering track -Year 3. Update fundamental courses at the first and second year according to the Software Engineering Body of Knowledge (SWEBOK) standard[20] (chapters Mathematical Foundations, Computing Foundations and Engineering Foundations). Furthermore, soft skills courses such as personal software process, critical writing and effective presentations will be added to the first three semesters.-Year 4. Follow the SWEBOK and deliver the most essential knowledge areas. -Year 5. 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[ "Spin chains and classical strings in two parameters q-deformed AdS 3 ×S 3", "Spin chains and classical strings in two parameters q-deformed AdS 3 ×S 3" ]	[ "Wen-Yu Wen [email protected] \nCenter for High Energy Physics\nDepartment of Physics\nChung-Yuan Christian University\nChung-Li 320Taiwan, R.O.C\n\nLeung Center for Cosmology and Particle Astrophysics\nNational Taiwan University\n106TaipeiTaiwan\n", "Shoichi Kawamoto [email protected] \nCenter for High Energy Physics\nDepartment of Physics\nChung-Yuan Christian University\nChung-Li 320Taiwan, R.O.C\n" ]	[ "Center for High Energy Physics\nDepartment of Physics\nChung-Yuan Christian University\nChung-Li 320Taiwan, R.O.C", "Leung Center for Cosmology and Particle Astrophysics\nNational Taiwan University\n106TaipeiTaiwan", "Center for High Energy Physics\nDepartment of Physics\nChung-Yuan Christian University\nChung-Li 320Taiwan, R.O.C" ]	[]	In this paper, we study the spin chain and string excitation in the two-parameters qdeformed AdS 3 ×S 3 proposed by Hoare[4]. We obtain the deformed spin chain model at the fast spin limit for choices of deformed parameters. General ansatz for giant magnons are studied in great detail and complicated dispersion relation is treated perturbatively. We also study several types of hanging string solutions and their charges and spins are analyzed numerically. At last, we explore its pp-wave limit and find its solution only depends on the difference of deformed parameters.	10.1016/j.cjph.2019.12.020	[ "https://arxiv.org/pdf/1911.01567v1.pdf" ]	207,869,846	1911.01567	f43f983e9f86fb38fc5733f1fc41e2478d841aca	Spin chains and classical strings in two parameters q-deformed AdS 3 ×S 3 November 6, 2019 5 Nov 2019 Wen-Yu Wen [email protected] Center for High Energy Physics Department of Physics Chung-Yuan Christian University Chung-Li 320Taiwan, R.O.C Leung Center for Cosmology and Particle Astrophysics National Taiwan University 106TaipeiTaiwan Shoichi Kawamoto [email protected] Center for High Energy Physics Department of Physics Chung-Yuan Christian University Chung-Li 320Taiwan, R.O.C Spin chains and classical strings in two parameters q-deformed AdS 3 ×S 3 November 6, 2019 5 Nov 2019 In this paper, we study the spin chain and string excitation in the two-parameters qdeformed AdS 3 ×S 3 proposed by Hoare[4]. We obtain the deformed spin chain model at the fast spin limit for choices of deformed parameters. General ansatz for giant magnons are studied in great detail and complicated dispersion relation is treated perturbatively. We also study several types of hanging string solutions and their charges and spins are analyzed numerically. At last, we explore its pp-wave limit and find its solution only depends on the difference of deformed parameters. Introduction and summary Since AdS/CFT correspondence was first advocated two decades ago [1], the dictionary of correspondence has been getting more elaborated and precise. One of the sharpest correspondence has been observed in the correspondence between a single trace operator of large dimension in N = 4 super Yang-Mills theory in large-N limit and a sting/brane configuration in AdS 5 × S 5 spacetime [2]. Integrability has played a prominent role in this correspondence; the problem of finding anomalous dimension of the single trace operator leads to a problem of an integrable spin chain, and the sigma model on the string world sheet also exhibits integrability. The sigma model may be deformed with integrable structure intact and this deformation would correspond to a choice of new background that preserves integrability. This integrable deformation has been attracting attention and stimulates many works [2,3]. Among many works that focus on one parameter deformation of the sigma model, Hoare proposed a two-parameter deformation of sigma models on the world-sheet of superstrings on AdS 3 × S 3 × M 4 with M 4 being T 4 or S 1 × S 3 [4], and a deformed metric on AdS 3 × S 3 was also presented based on the sigma-model deformation. In this paper, we consider some string solutions on this deformed background and analyze the dispersion relation. With this new deformed metric, we are interested in investigating several types of string solution in this background. The deformed metric involves two deformation parameter, κ + and κ − , and we can assume κ + ≥ κ − without loss of generality due to the symmetry κ + ↔ κ − . By taking κ − → 0 with κ + fixed, it becomes a one parameter deformation case [5], while κ − → κ + limit provides a known squashed S 3 case [6]. We thus explore the effect of the two parameter deformation near these special point. The dispersion relations turn out to be complicated but some explicit forms are to be given as a perturbative series. The organization of the paper is as follows: In Section 2, the spin chain Hamiltonian obtained from the sigma model on this back ground is discussed. In Section 3, long string solutions in this background and its dispersion relation is analyzed. The PP-wave limit is briefly discussed in Section 4. Section 5 serves summary and discussion. The appendix A summarizes some facts on the two-parameter deformed geometry. 2 Spin chain from q 2 -deformed AdS 3 (S 3 ) The Heisenberg spin chain could be derived from the sigma model in the fast spinning limit, and its Hamiltonian agrees with the one-loop calculation of anomalous dimensions in N = 4 super Yang-Mills theory [7]. Although this quantity may no longer be protected in the deformed theory with less symmetry, one can still study the effect of deformation to the spin chain Hamiltonian and dynamics from gravity side. Let us consider the deformed metric in R × S 3 ∈ AdS 3 × S 3 [4]. In the deformed coordinates (95) and (96), after a twist of angular coordinates ϕ = φ 1 + φ 2 and φ = φ 1 −φ 2 for simplicity, we consider a spinning string at the center of AdS by rotating coordinate φ 1 → t +φ 1 and setting ρ = 0. The metric then reads ds 2 = 1 1 + κ 2 − cos 2 θ + κ 2 + sin 2 θ dθ 2 − (κ + − κ − ) 2 cos 2 θ sin 2 θ dt 2 + 1 + (κ − cos 2 θ + κ + sin 2 θ) 2 2dtdφ 1 + dφ 2 1 + 1 + (κ − cos 2 θ − κ + sin 2 θ) 2 dφ 2 2 + cos 2θ + κ 2 − cos 4 θ − κ 2 + sin 4 θ 2dtdφ 2 + 2dφ 1 dφ 2 .(1) The case of one-parameter deformation, i.e. κ − = 0, has been studied in [8]. Here we will focus on those novel cases where κ − = 0. With the choice of gauge t = κ 0 τ and taking fast-moving limit: κ ± → 0,Ẋ µ → 0, κ 0 → ∞, but κ 0 κ ± and κ 0Ẋ µ being kept fixed, one obtains the pull-back string action: S = T 2 dτ dσ C sin 2 2θ + 2κ 0 (φ 1 + cos 2θφ 2 ) − θ 2 −φ 2 1 − φ 2 2 − 2 cos 2θφ 1 φ 2 ,(2) where C = −κ 2 0 (κ + − κ − ) 2 /4. Using one of the Virasoro constraints, 2κ 0 (φ 1 + cos 2θφ 2 ) = 0,(3) one obtains a q−deformed spin chain with Hamiltonian density H = θ 2 + sin 2 2θφ 2 2 − C sin 2 2θ(4) This is more or less a Heisenberg XXZ spin chain. To see that, one first rescales θ → θ/2, φ 2 → φ 2 /2 and defines a new Hamiltonian H ≡ 4(H + C). Then one obtains a spin chain interacting with magnetic field along z-axis, that is H = n · n + ( n · B) 2 ,(5) where the spin vector n = (sin θ cos φ 2 , sin θ sin φ 2 , cos θ) and an external magnetic field B = (0, 0, 2 \|C\|). Similar result was obtained before in the case of deformed three-sphere [9]. Open string solutions One-spin giant magnon solution We start with a simple case of the basic giant magnon solution that has one spin in S 3 , by following [13]. Because of Z 2 symmetry of κ + ↔ κ − , we assume κ + ≥ κ − without loss of generality; the κ − → 0 limit corresponds to the one-parameter deformation case. We take the ansatz t =κτ , ψ = Ψτ , ρ = ρ(τ ) ,(6)ϕ =ω(τ + h(y)) , θ = θ(y) , φ = 0 ,(7) where y = σ − vτ and 0 < v < 1. The equation of motion for t admits a constant ρ solution. We choose ρ = 0 which is consistent with the equation of motion. The action and the Virasoro constraints are I = −T 2 dτ dy κ 2 + (1 − v 2 )g θθ θ 2 + ω 2 g ϕϕ (1 − v 2 )h 2 + 2vh − 1 ,(8)0 = − κ 2 + (1 + v 2 )g θθ θ 2 + ω 2 g ϕϕ (1 + v 2 )h 2 − 2vh + 1 ,(9)0 = − vg θθ θ 2 + ω 2 g ϕϕ h (1 − vh ) ,(10) where the prime denotes y derivative. The h equation, once integrated, leads to ω 2 g ϕϕ (1 − v 2 )h + v = C ,(11) where C is the constant of integration. By eliminating θ 2 from two Virasoro constraints, one finds C = vκ 2 . Thus, h = v 1 − v 2 κ 2 ω 2 g ϕϕ − 1 .(12) The θ equation is θ 2 = ω 2 g θθ g ϕϕ (1 − v 2 ) 2 κ 2 ω 2 − g ϕϕ g ϕϕ − v 2 κ 2 ω 2 .(13) In terms of r = cos θ coordinate (0 ≤ r ≤ 1), r 2 = ω 2 (1 − v 2 ) 2 [1 + κ 2 − (1 − r 2 ) + κ 2 + r 2 ] 2 1 + κ 2 − (1 − r 2 ) κ 2 ω 2 − g ϕϕ g ϕϕ − v 2 κ 2 ω 2 ,(14) where g ϕϕ = (1 − r 2 ) 1 + κ 2 − (1 − r 2 ) 1 + κ 2 − (1 − r 2 ) + κ 2 + r 2 .(15) Note that g ϕϕ is a monotonically decreasing function of r from g ϕϕ = 1 (r = 0) to g ϕϕ = 0 (r = 1) under the condition κ + ≥ κ − . In order to have a physical solution, r has to be real. Namely r 2 ≥ 0. This leads to the condition, v 2 κ 2 ω 2 ≤ g ϕϕ ≤ κ 2 ω 2 .(16) Thus, v 2 ≤ 1 and v 2 ≤ ω 2 /κ 2 are required. We require that there exists two turning points; this implies ω ≥ κ. Two roots are given by g ϕϕ (r min ) = κ 2 ω 2 , g ϕϕ (r max ) = v 2 κ 2 ω 2 .(17) A general root for the condition g ϕϕ (r 0 ) = C is given by r 2 0 = 1 2κ 2 − 2κ 2 − + 1 + C(κ 2 + − κ 2 − ) − 2κ 2 − + 1 + C(κ 2 + − κ 2 − ) 2 − 4κ 2 − (1 + κ 2 − )(1 − C) ,(18) where we have taken the − branch solution since it has a smooth κ − → 0 limit, r 2 0 = (1 − C)/(1 + κ 2 + C), which agrees with the expression in [13]. Thus, r min = 2κ 2 − + 1 + κ 2 ω 2 (κ 2 + − κ 2 − ) 2κ 2 − 1 − 1 − 4κ 2 − (1 + κ 2 − )(1 − κ 2 ω 2 ) [2κ 2 − + 1 + κ 2 ω 2 (κ 2 + − κ 2 − )] 2 ,(19)r max = 2κ 2 − + 1 + v 2 κ 2 ω 2 (κ 2 + − κ 2 − ) 2κ 2 − 1 − 1 − 4κ 2 − (1 + κ 2 − )(1 − v 2 κ 2 ω 2 ) [2κ 2 − + 1 + v 2 κ 2 ω 2 (κ 2 + − κ 2 − )] 2 .(20) Now we calculate the various conserved charges. The angular momentum J(= J 1 ) is J =2T rmax r min dr \|r \| g ϕϕ ω(1 − vh ) =2T rmax r min dr g rr (r)g ϕϕ (r) g ϕϕ (r) − g ϕϕ (r max ) g ϕϕ (r min ) − g ϕϕ (r) .(21) We also have 2π = π −π dσ = π −π dy = 2 rmax r min dr \|r \| =2 1 − v 2 ω rmax r min dr g rr g ϕϕ (r) − g ϕϕ (r max ) g ϕϕ (r min ) − g ϕϕ (r) .(22) For later convenience, we define g ϕϕ (r) − g ϕϕ (r 0 ) = r 2 0 − r 2 u 1 (r 2 )u 1 (r 2 0 ) u 3 (r 2 0 ; r 2 ) ,(23)u 1 (r 2 ) =1 + κ 2 + r 2 + κ 2 − (1 − r 2 ) , u 2 (r 2 ) = 1 + κ 2 + + κ 2 − (1 − r 2 ) , (24) u 3 (r 2 0 ; r 2 ) =(1 + κ 2 − )u 2 (r 2 0 ) − κ 2 − u 1 (r 2 0 )r 2 .(25) The energy is given by (recall that g tt (ρ = 0) = −1) E = − P t =T π −π dσ (−g tt )∂ τ t = −2πκT .(26) Namely, the constant κ is related to the energy as κ = − E 2πT .(27) Infinite J limit Following [13], we consider the infinite J giant magnon. This corresponds to r min = 0, namely, ω 2 = κ 2 = E 2 (2πT ) 2 .(28) In this case, the spin J and the constant factor 2π are J = 2T u 1 (r 2 max ) rmax 0 dr 1 u 1 (r 2 ) r 2 max − r 2 r 2 u 3 (r 2 max ; r 2 )u 4 (r 2 ) u 2 (r 2 ) .(29)2π = 2 1 − v 2 ω u 1 (r 2 max ) rmax 0 dr 1 r 2 (r 2 max − r 2 ) u 4 (r 2 ) u 3 (r 2 max ; r 2 )u 2 (r 2 ) ,(30) where both of them are divergent quantities. We need to choose a constant K such that 2πK − J becomes finite. This K will be related to the energy E and it will give a dispersion relation E − J = finite. The divergence is due to the lower end of the integral r = 0, and in order to cancel this divergence K is chosen to be K = ωT .(31) This is the same result as in [13] except the overall sign. (In [13], their J has an opposite sign. But we may choose the sign of K for J to have a positive value.) We choose the positive root of ω = E/(2πT ), and E − J =2T rmax 0 dr 1 r u 1 (r 2 max ) 1 + κ 2 − (1 − r 2 ) (r 2 max − r 2 )u 2 (r 2 )u 3 (r 2 max ; r 2 ) r 2 max u 2 (r 2 max ) − (r 2 max − r 2 )u 3 (r 2 max ; r 2 ) u 1 (r 2 ) .(32) We may want to represent the dispersion relation E − J as a function of the momentum (or the deficit angle of the string configuration), but the current expression is too complicated to analyze analytically. We thus consider perturbative corrections with respect to κ − . First, r 2 max =r 2 max,0 1 + v 4 κ 2 + (1 + κ 2 + ) (1 + v 2 κ 2 + ) 2 κ 2 − − v 6 κ 2 + (1 + κ 2 + ) 1 + (2 − v 2 )κ 2 + (1 + v 2 κ 2 + ) 4 κ 4 − + O(κ 6 − ) ,(33) where r 2 max,0 = 1−v 2 1+v 2 κ 2 + . In order to avoid a unnecessary divergence, we first expand the integrands and integrate them to r max , and then expand the results in terms of κ − . This leads to E − J =2T κ 4 + + κ 2 + κ 2 − + κ 4 − κ 5 + sinh −1 κ + r max,0 − κ 2 − r max,0 1 + κ 2 + r 2 max,0 2κ 2 + (1 + κ 2 + ) 2 + 3 κ + 2 − 2κ 2 + r 2 max,0 + κ 4 − r max,0 1 + κ 2 + r 2 max,0 24κ 4 + (1 + κ 2 + ) 2 2κ 6 + r 2 max,0 24r 4 max,0 − 56r 2 max,0 + 35 + κ 4 + −8r 4 max,0 + 20r 2 max,0 − 15 + 16κ 2 + r 2 max,0 − 3 − 24 + O(κ 6 − ) ,(34) where we have used v 2 = (1 − r 2 max,0 )/(1 + κ 2 + r 2 max,0 ). The momentum corresponds to the angle spanned by asymptotic directions of the string, p =∆ϕ = dϕ = 2 rmax r min dr \|r \| ϕ =2v rmax 0 dr u 1 (r 2 max ) r 1 − r 2 u 2 (r 2 ) r 2 max − r 2 u 3 (r 2 max ; r 2 )u 4 (r 2 ) .(35) In the κ − → 0 limit, it is reduced to r max,0 = sin p 2 [13]. Small κ − corrections for p are calculated as p =2 arcsin r max,0 − κ 2 − r max,0 1 − r 2 max,0 2κ 2 + r 2 max,0 + 1 κ 2 + + 1 − κ 4 − r max,0 1 − r 2 max,0 48κ 4 + r 6 max,0 − 8κ 2 + 7κ 2 + − 6 r 4 max,0 + 6 − 60κ 2 + r 2 max,0 − 9 12 (κ 2 + + 1) 2 + O(κ 6 − ) .(36) This relation can be inverted to r max,0 = sin p 2 + · · · . By substituting this into E − J result, we obtain perturbative corrections to the dispersion relation, E − J =2T κ 4 + + κ 2 + κ 2 − + κ 4 − κ 5 + sinh −1 κ + sin p 2 + κ 2 − sin p 2 16κ 2 + (κ 2 + + 1) κ 2 + sin 2 p 2 + 1 κ 4 + sin p 2 15 sin 3 p 2 − 10 sin 2 p 2 − 20 sin p 2 + 7 − 2κ 2 + sin 2 p 2 + 3 sin p 2 + 8 − 16 + κ 4 − sin p 2 1536κ 4 + (κ 2 + + 1) 2 κ 2 + sin 2 p 2 + 1 3/2 ×+ O(κ 6 − ) .(37) The first term is a generalization of the result of [13]; the coefficient has some κ − corrections. The other terms are corrections in terms of sin p 2 . It is interesting to see how this complicated dispersion relation can be obtained via a dual gauge theory, but at this moment it is not clear. Two-spin "spiky" string solution We next try the two spin ansatz of [16], t =τ + h 1 (y) , ρ = ρ(y) , ψ = ω[τ + h 2 (y)] , φ = Ωτ , θ = π 2 ,(38) where y = σ − vτ . In this case, as we will see, we find several types of hanging string solutions, rather than a spiky string solution. The action is I = −T 2 dτ dy g tt (1 − v 2 )h 2 1 + 2vh 1 − 1 + g ρρ (1 − v 2 )ρ 2 + g ψψ ω 2 (1 − v 2 )h 2 2 + 2vh 2 − 1 + 2g tψ ω (1 − v 2 )h 1 h 2 + v(h 1 + h 2 ) − 1 − Ω 2 .(39) The equations of motion for h 1 and h 2 , after once integrated, are g tt (1 − v 2 )h 1 + v + ωg tψ (1 − v 2 )h 2 + v = −c 1 ,(40)ω 2 g ψψ (1 − v 2 )h 2 + v + ωg tψ (1 − v 2 )h 1 + v = ω 2 c 2 ,(41) where c 1 and c 2 are constants of integration. The negative sign for c 1 and an extra ω 2 for c 2 are for later convenience. Thus, h 1 = 1 1 − v 2 1 − κ 2 + sinh 2 ρ cosh 2 ρ c 1 + ωκ + κ − c 2 − v ,(42)h 2 = 1 1 − v 2 − κ + κ − c 1 ω + 1 + κ 2 − cosh 2 ρ sinh 2 ρ c 2 − v .(43) By eliminating ρ 2 from two Virasoro constraints, 0 =g tt (1 + v 2 )h 2 1 − 2vh 1 + 1 + g ρρ (1 + v 2 )ρ 2 + g ψψ ω 2 (1 + v 2 )h 2 2 − 2vh 2 + 1 + 2g tψ ω (1 + v 2 )h 1 h 2 − v(h 1 + h 2 ) + 1 + Ω 2 ,(44)0 =g tt h 1 1 − vh 1 − g ρρ vρ 2 + ω 2 g ψψ h 2 1 − vh 2 + ωg tψ (h 1 + h 2 − 2vh 1 h 2 ) ,(45) we obtain g tt (1 − v 2 )h 1 + v + ω 2 g ψψ (1 − v 2 )h 2 + v + ωg tψ (1 − v 2 )(h 1 + h 2 ) + 2v + vΩ 2 = 0 .(46) Using (40) and (41), we find the relation for the constants of integration, c 1 − ω 2 c 2 = vΩ 2 .(47) From now on, we set Ω = 1 for simplicity. We take the solution c 1 = v and c 2 = 0 [17] which satisfies the condition for forward propagation of the string, dt dτ = v 1 − v 2 1 v − 1 − κ 2 + sinh 2 ρ cosh 2 ρ c 1 − κ + κ − ωc 2 > 0 .(48) The solutions are now h 1 = − (1 + κ 2 + ) v 1 − v 2 tanh 2 ρ , h 2 = − 1 + κ + κ − ω v 1 − v 2 .(49) ρ can be obtained from these two solution and the second Virasoro constraint, (ρ ) 2 = tanh 2 ρ (1 − v 2 ) 2 f (ρ) ,(50)f (ρ) =(ωκ + − κ − ) 2 cosh 4 ρ + (1 + κ 2 + ) 1 − ω 2 + v 2 (κ 2 + − κ 2 − ) cosh 2 ρ − v 2 (1 + κ 2 + ) 2 .(51) Note that ρ → ±∞ for ρ → ∞. We may write ρ = ± tanh ρ 1 − v 2 f (ρ) ,(52) and f (ρ) = 0 has two roots as a function of cosh 2 ρ as cosh 2 ρ ± = 1 + κ 2 + 2(ωκ + − κ − ) 2 − 1 + ω 2 − v 2 (κ 2 + − κ 2 − ) ± (1 − ω) 2 + v 2 (κ + − κ − ) 2 (1 + ω) 2 + v 2 (κ + + κ − ) 2 ,(53) where ωκ + − κ − = 0 is assumed. Let us examine the allowed region of ρ. The original metric has a singularity at (recall that κ + ≥ κ − is assumed) ρ s = sinh −1 1 + κ 2 − κ 2 + − κ 2 − ,(54) It is not difficult to check that ρ ± and ρ s satisfy the following inequalities, cosh 2 ρ − ≤ 0 , ρ s ≥ ρ + .(55) Therefore, ρ − is not real. Since (ρ ) 2 → ∞ for ρ → ∞, ρ is real for ρ + ≤ ρ. Thus, the possible classical solutions exist in the following regions: (I) : ρ + ≤ ρ ≤ ρ s (II) : ρ s ≤ ρ < ∞(56) where the region (I) is valid if ρ + is real. This condition for ρ + to be real is ω ≥ −κ + κ − + (1 − v 2 )(1 + κ 2 + )(1 + κ 2 − ) , ω ≤ −κ + κ − − (1 − v 2 )(1 + κ 2 + )(1 + κ 2 − ) .(57) Otherwise, there exists no turning point, and we consider (I) : 0 ≤ ρ ≤ ρ s .(58) This can also be viewed as a hanging string solution from the singular surface to the center of AdS. The results with real ρ + cases are summarized in Fig. 1 and 2. It contains the both region (I) and (II), and they are separated by the locations of the singular surfaces (the horizontal dashed lines in the figures). Fig. 1 shows the profiles for a general parameter setting, while Fig. 2 displays two special cases. The left one is for κ + = κ − , and then the singular surface is pushed to infinity. The right one is the case with κ − = 0 which is reduced to the one-parameter deformation result. When ω is small ρ + becomes imaginary. So the solutions run from ρ = 0 to ρ s (and further). This case is summarized in Fig. 3. E = −T dσ g ttṫ =T 1 − v 2 ρs ρ + dρ ρ 1 + κ 2 − cosh 2 ρ (cosh 2 ρ − v 2 (1 − κ 2 + sinh 2 ρ)) 1 + κ 2 − cosh 2 ρ − κ 2 + sinh 2 ρ ,(59)S =T dσ g ψψψ = ωT 1 − v 2 ρs ρ + dρ ρ sinh 2 ρ(1 − κ 2 + sinh 2 ρ) 1 + κ 2 − cosh 2 ρ − κ 2 + sinh 2 ρ 1 + v 2 ω κ + κ − ,(60)J 2 =T dσ g φφφ =T ρs ρ + dρ ρ ,(61) and J 1 = 0. The lower end ρ + should be understood as ρ + = 0 if it takes a complex value. The dispersion relation is expressed as E − J 2 Ω = S ω + K(κ + , κ − ) ,(62) where the reminder function K(κ + , κ − ) is K(κ + , κ − ) =T ω(1 − v 2 ) dy sinh 2 ρ ωκ 2 + cosh 2 ρ + ω(1 + v 2 κ 2 + )κ 2 − cosh 2 ρ − v 2 κ + κ − (1 − κ 2 + sinh 2 ρ) 1 + κ 2 − cosh 2 ρ − κ 2 + sinh 2 ρ ,(63) which vanishes in the κ ± → 0 limit as it should, and the dispersion relation of the undeformed case in [17] is recovered. In the case of the region (I), the conserved charges are finite (ρ s works as a natural cutoff). PP-wave limit Following [10,11], we will take the pp-wave limit. First we perform the coordinate transformation in (95) and (96) z = 2 √ 2e ρ 0 −ρ , x ± = e ρ 0 ∓θ 0 (ψ ± t)(65) and then take following limits: send both ρ 0 , θ 0 → ∞ but keep their difference finite such that e θ 0 −ρ 0 ≡ 2µ. We would also like to have a controllable way to incorporate the deformation. A simple but nontrivial limit is to send κ ± → 0, but instead keep κ ± e ρ 0 finite. The metric after scaling becomes ds 2 pp 1 z 2 [2dx + dx − − µ 2 z 2 dx 2 + − µ 2 z 2 ∆ 2 dx 2 + + dz 2 ],(66) where ∆ ≡ 2(κ + − κ − )e 2ρ 0 . We briefly look at two types of solutions in this background. Moving straight string solution We consider the following ansatz [10], x + = τ , x − = V τ , z = z(σ) .(67) By looking at a point with constant z (a point on the world-sheet with a fixed σ), ds 2 = 1 z 2 2V − µ 2 z 2 + ∆ 2 z 2 dτ 2 .(68) we find a condition for a part of the string not to travel faster than light; namely, z has to satisfy the condition, The Nambu-Goto action is z 2 + ∆ 2 z 2 ≥ 2V µ 2 ,(69)I = −T d 2 σ z z µ 2 z 2 + ∆ 2 z 2 − 2V ,(70) where we use z ≥ 0 and assume z ≥ 0; namely z(σ) is monotonic in σ. With respect to the choice of the parameters, different ranges of z are allowed in the condition (69): • If V 2 ≤ µ 4 ∆ 2 , arbitrary z (and then σ) satisfies the condition. The solution is z = σ, 0 ≤ σ < ∞, and the string reaches the boundary. It covers the above light-like solution. Note that for ∆ = 0, the string can reach the boundary for V > 0. • If V 2 > µ 4 ∆ 2 , we have two branches: one is z = σ, σ 1 ≤ σ < ∞ with σ 1 = V µ 2 1 + 1 − µ 4 ∆ 2 V 2 ,(71) which is similar to the straight string solution of [10]; a folded string with the spike not reaching the boundary. The other is z = σ, 0 ≤ σ ≤ σ 2 with σ 2 = V µ 2 1 − 1 − µ 4 ∆ 2 V 2 ,(72) which is a folded string solution that reaches to the boundary and it turns back at a point in the bulk. Now we focus on the solution with σ 1 ≤ σ ≤ ∞, which is a generalization of the straight line solution. Note that this case corresponds to a small ∆. The conserved charges are P + =µT − σ 4 − 2V µ 2 σ 2 + ∆ 2 2σ 2 + 1 2 log − V µ 2 + σ 2 + σ 4 − 2V µ 2 σ 2 + ∆ 2 b a ,(73)P − = T 2µ∆ log 2∆ 2 − 2V µ 2 σ 2 + 2∆ σ 4 − 2V µ 2 σ 2 + ∆ 2 σ 2 b a ,(74) where a and b represent the two ends of the string. For σ 1 ≤ σ < ∞, we introduce a large R for a cutoff. P + =µT 1 2 − 1 + log 2R 2 + O(R −4 ) − 1 2 log σ 2 1 − V µ 2 ,(75)P − = T 2µ∆ log 1 − ∆µ 2 V − log 1 − µ 2 ∆ 2 V σ 2 1 .(76) A standard dictionary of the folded string solution in AdS space reads [18] P + =P t + P θ = E − S , P − = −P t + P θ = −(E + S) .(77) Then S = − 1 2 (P + + P − ) = \|P − \| 2 + µT 4 1 + log ∆ 2R 2 coth 2µ∆\|P − \| T 1 − tanh 2 2µ∆\|P − \| T .(78) This relation can be inverted in the large S limit. We first consider a small ∆ limit and take inversion, \|P − \| =2S + µT 2 − 1 + ln 4µR 2 \|P − \| T + µ 2 ∆ 2 \|P − \| 2 3T + O(∆ 4 ) =2S + µT 2 ln S + ln 8µR 2 T − 1 + µ 2 T 2 8S ln S + ln 8µR 2 T − 1 + O(S −2 ) + ∆ 2 4µ 3 S 2 3T + 2µ 4 S 3 ln S + ln 8µR 2 T − 1 2 + O(ln S) + O(∆ 4 ) .(79) Thus, the dispersion relation is expressed as E − S =P + = µT 2 − 1 + ln \|P − \| + ln 4µR 2 T + µ 3 \|P − \| 2 3T ∆ 2 + O(∆ 4 ) = µT 2 ln S + ln 8µR 2 T − 1 + µ 2 T 2 8S ln S + ln 8µR 2 T − 1 + ∆ 2 4µ 3 S 2 3T + 2µ 4 S 3 ln S + ln 8µR 2 T − 1 2 + · · · .(80) In the large-S limit, ∆ correction terms become dominant and the standard relation E − S ∝ ln S is spoiled. It may suggest that two limits (S → ∞ and ∆ → 0) are not interchangeable. Periodic spike solutions We next consider the ansatz for the periodic spike solution [10], x + = τ , x − = σ , z = z(ξ) .(81) Here ξ = τ − 1 η 2 0 σ = − v − 1 √ 2 x − vt , η 2 0 = v − 1 v + 1(82) and v > 1. A point with fixed z is therefore traveling in x direction with speed v. By using ∂ τ = ∂ ξ and ∂ σ = − 1 background is characterized by a parameter which is essentially the difference of two deformation parameter. When this parameter is small, we can evaluate the dispersion relation for the moving straight string solution. We also briefly looked at the existence of periodic spike solutions. In the appendix, we briefly discuss the PP-wave limit near the singular surface and find a similar structure that appears in the PP-wave limit near the boundary. This would imply another clue for the similarity between the theories near the singular surface and the AdS boundary. The solutions obtained in this paper are generalization of the solutions in the one parameter deformed background. Although the results including two deformation parameter is complicated and not so illuminating, we can consider some future directions. First, we may calculate various physical quantities with these classical solutions; the holographic entanglement entropy or the complexity from the Wheeler-de Wit patch are some examples. At least in the limit of the parameters discusses in this paper, we will be able to evaluate these values and observe how their behavior changes due to the deformations. The dual gauge theory corresponding to this geometry is still unclear. We also hope that some hints on the dual gauge theory side are obtained through further study of the classical solutions presented in this paper. Note that this metric has a Z 2 symmetry [4], r → √ 1 − r 2 , ϕ ↔ φ, and κ + ↔ κ − (equivalent to θ → π 2 − θ φ 1 → φ 1 , φ 2 → −φ 2 , and κ + ↔ κ − ), and the we can assume that κ ≥ κ − in this paper. By changing the coordinates,ρ → sinh ρ and r → sin θ, we obtain ds 2 A 3 = 1 1 + κ 2 − cosh 2 ρ − κ 2 + sinh 2 ρ dρ 2 − cosh 2 ρ 1 + κ 2 − cosh 2 ρ dt 2 + sinh 2 ρ(1 − κ 2 + sinh 2 ρ)dψ 2 + 2κ + κ − sinh 2 ρ cosh 2 ρdtdψ ,(95)ds 2 S 3 = 1 1 + κ 2 − cos 2 θ + κ 2 + sin 2 θ dθ 2 + cos 2 θ 1 + κ 2 − cos 2 θ dϕ 2 + sin 2 θ(1 + κ 2 + sin 2 θ)dφ 2 + 2κ − κ + sin 2 θ cos 2 θdϕdφ .(96) The AdS part of the metric has a singularity (or rather a surface of singularity) at ρ = ρ s with ρ s being a solution of cosh 2 ρ s = 1 + κ 2 + κ 2 + − κ 2 − .(97) It can be checked that this is the curvature singularity as in the case of one-parameter deformed geometry. From a point in the bulk, the singular surface can be reached in a finite coordinate time t but it takes infinite affine time as also in the one-parameter deformation case, The coordinate time from the center to the singular surface reads Interestingly, this result is independent of the second deformation parameter κ − and precisely agrees with that of one-parameter deformation case [19]. Especially, in a limit, κ − → κ + (κ + ≥ κ − ), the singular surface is pushed to infinity, ρ s → ∞, but the coordinate time is still t = arccot κ + . Under the undeformed limit κ + → 0, we have t = π/2. Next we consider the affine parameter t A . In the massless condition, 0 = p 2 = G tt p t p t + G ρρ p ρ p ρ with E = −p t and p ρ = G ρρ dρ dt A leads to − 1 + κ 2 − cosh 2 ρ − κ 2 + sinh 2 ρ cosh 2 ρ 1 + κ 2 − cosh 2 ρ E 2 + 1 1 + κ 2 − cosh 2 ρ − κ 2 + sinh 2 ρ dρ dt A 2 = 0 .(99) Thus, t A = ρs 0 dρ E cosh ρ 1 + κ 2 − cosh 2 ρ 1 + κ 2 − cosh 2 ρ − κ 2 + sinh 2 ρ = − 1 2E(κ 2 + − κ 2 − ) 2κ − ln 1 + κ 2 − cosh 2 ρ + κ − sinh ρ 1 + κ 2 − + κ + ln 1 + κ 2 − cosh 2 ρ − κ 2 + sinh 2 ρ 1 + κ 2 − cosh 2 ρ − κ 2 + sinh 2 ρ + 2κ + sinh ρ 1 + κ 2 − cosh 2 ρ ρ→ρs → ∞ . (100) Thus, it takes infinite affine time to reach the singular surface. Note that in the κ − limit, we recover the one parameter deformation case t A = 1 Eκ + arctanh κ + sinh ρ ρ→ρs = ∞ .(101) New coordinates By following [19], we may define a new coordinate χ as cosh χ = √ −g tt = cosh ρ 1 + κ 2 − cosh 2 ρ 1 + κ 2 − cosh 2 ρ − κ 2 + sinh 2 ρ ,(102) which covers 0 ≤ χ ≤ ∞ for 0 ≤ ρ ≤ ρ s . With this new variable, the AdS part of the metric can be expressed as g tt dt 2 = − cosh 2 χdt 2 ,(103)g ψψ dψ 2 = − 1 2κ 2 + κ 2 − (κ 2 + + κ 2 − + 2κ 2 + κ 2 − ) + (κ 4 + + κ 4 − ) cosh 2 χ − (κ 2 + + κ 2 − ) f 1 (κ + , κ − ; χ) dψ 2 ,(104)g tψ dtdψ = 1 + (κ 2 + + κ 2 − ) cosh 2 χ − f 1 (κ + , κ − ; χ) 2κ + κ − dtdψ ,(105) g ρρ dρ 2 = 1 + (κ 2 + + κ 2 − ) cosh 2 χ + f 1 (κ + , κ − ; χ) 2f 1 (κ + , κ − ; χ) dχ 2 , f 1 (κ + , κ − ; χ) =4κ 2 − (1 + κ 2 + ) cosh 2 χ + 1 + (κ 2 + − κ 2 − ) cosh 2 χ 2 . With these new coordinates, we may revisit the PP-wave limit. The standard pp-wave limit corresponds to zooming up the near boundary region of the AdS space, which is a part seen by a fast rotating string. We now consider a similar limit in this new coordinate, which corresponds to zooming up near the singular surface. The definition of the new coordinates and the limit are the same, where we use χ instead of ρ that covers inside region bounded by the singular surface ρ = ρ s : z = 2 √ 2e χ 0 −χ , x ± = e χ 0 ∓θ 0 (ψ ± t) .(108) Taking the following limit: ρ 0 , θ 0 → ∞ , e θ 0 −χ 0 = 2µ = fixed. Plugging these in to the aforementioned new metric, and taking the limit κ ± → 0 with κ ± e 2ρ 0 = µ ± kept finite (∆ = 2(µ + − µ − )), we find that the reduced metric is ds 2 1 z 2 dz 2 − µ 2 z 2 dx 2 + + 2dx + dx − − µ 2 ∆ 2 z 2 dx 2 + .(110) In this simplified limit, this is equivalent to the PP-wave limit in the original coordinates as discussed in Section 4. It might suggest that the properties near the AdS boundary and the singular surface are quite similar at least in this limit. Figure 1 : 1Hanging string solutions with v = 0.1. (Left) ω = 1.3 and κ + = 0.5, and varied values of κ − = 0, 0.3, 0.5. (from darker color to brighter). The horizontal dashed lines indicate the locations of the corresponding singular surfaces; ρ s = 1.44364 (κ − = 0), 1.68735 (κ − = 0.3), and ∞ (κ − = 0.5). (Right) Varying ω = 1.02, 1.04, 1.06, 1.08 (from the outermost darkest one to inside) with fixed κ − = 0.1 and κ + = 0.2. The value of ρ s = 2.45875. Dispersion relation The conserved charges, in the region (I), are Figure 2 : 2Hanging string solutions in special cases. (Left) κ + = κ − case. Namely ρ s = ∞. The parameters are v = 0.1, ω = 1.3, and κ + = κ − = 0.01, 0.1, 0.3, 0.5 (from the innermost darkest color to outside). (Right) κ − = 0 case; namely the one-parameter deformation case. The parameters are v = 0.1, ω = 1.3, and κ + = 0.1, 0.3, 0.5 (from the innermost darkest color to outside). The corresponding locations of the singular surface are ρ s = 2.99822, 1.9189, 1.44364. Figure 3 : 3For example, for ω = 1, v = 0.1, κ − = 0.1 and κ + = 0.2 (this is in the region (I), E/T = 322.691, S/T = −39.427, J 2 /T = 63.829 , K/T = 298.289 .(64)For small values of κ ± , we plot the energy, the spins and the reminder function inFigure 4. The reality condition (57) for ω = 1.0 and v = 0.1 suggests that the real values exist about κ ± ≤ 0.2. Hanging string solutions with the case (I) . In this case, the solutions reach the center of AdS, ρ = 0. (Left) Varying κ − . v = 0.1, ω = 0.9 and κ + = 0.5, and varied values of κ − = 0, 0.3, 0.5. (from the innermost darkest one to outside). The corresponding values of ρ s are ρ s = 1.44364, 1.68735, ∞. (Right) Varying ω. v = 0.1, κ − = 0.1, κ + = 0.2 and ω = 0.7, 0.8, 0.9 (from the innermost darkest one to outside). The corresponding ρ s = 2.45875. Figure 4 : 4The energy, the spins and the reminder function in the region (I), for 0 ≤ κ − ≤ κ + ≤ 0.05. ω = 1.0 and v = 0.1. The scale of the vertical axes is divided by 1000. and the string may not reach the boundary. AcknowledgmentsWe are grateful to Dr. Shogo Kuwakino for his contribution in the early stage of this project. This work is supported in parts by the Taiwan's Ministry of Science and Technology (grant No. 106-2112-M-033-007-MY3 for WYW and grant No. 107-2112-M-033-008 and 108-2112-M-033-003 for SK) and the National Center for Theoretical Science.withThis can be integrated aswith z 1 = √ 2η 0 /µ and z 0 is a constant of integration, z 2 F = z 2 0 . Here, we look at z equation of motion,Now we apply the following relations that are from x − equation of motion (and its ξ derivative),and thenwhere in the first equality the first relation of (87) is used and in the last the second used. Namely, z equation of motion is also satisfied. The turning point is given by z = z 0 . The position of the spike isfor η 2 0 > µ 2 ∆. If η 2 0 ≤ µ 2 ∆, the denominator inside the square root does not take zero, and the position of the spike is z = 0; namely the spike reaches to the boundary. In this case, we do not investigate the dispersion relation in detail, but we present the expressions of the conserved charges,wherez 0 andz 1 are the turning points of the string solution and the factor two comes from the configuration being folded. It is easy to check that they come back to (33) and (34) of[10]in the ∆ → 0 limit.ConclusionIn this paper, we have considered the classical string solutions in the two parameter deformation of AdS space constructed by Hoare[4]. We first observe that in the fast spinning limit, the string Hamiltonian coincides with a spin chain Hamiltonian in a small deformation parameter limit. Therefore, the integrability may be preserved in this background at least in the case of small deformation. We further construct the giant magnon solutions and the hanging string solutions. They are obtained as a generalization of the one-parameter deformation case. We also derive the expression of the conserved charges. It however turns out that the dispersion relation takes a fairly complicated form in the two parameter deformation; we then consider a perturbative expansion of the dispersion relation. In this geometry, there appears the surface of singularity whose location is determined by the two deformation parameters. In the hanging string case, there appear several types of solutions; hung from the singular surface, stretching between the boundary and the singular surface, or reaching to the center of AdS space. We also numerically evaluated the energy and the spins in this case. Finally, we consider the PP-wave limit of this background. The PP-wave A Two-parameter deformation of AdS 3 × S 3 geometryIn this appendix, we present a briefly summary of the two-parameter q-deformed AdS 3 × S 3 geometry presented by Hoare[4]and the singular surface of this geometry. 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[ "The Wideband Slope of Interference Channels: The Small Bandwidth Case", "The Wideband Slope of Interference Channels: The Small Bandwidth Case" ]	[ "Minqi Shen ", "Anders Høst-Madsen " ]	[]	[]	This paper studies the low-SNR regime performance of a scalar complex K-user interference channel with Gaussian noise. The finite bandwidth case is considered, where the low-SNR regime is approached by letting the input power go to zero while bandwidth is small and fixed. We show that for all δ > 0 there exists a set with non-zero measure (probability) in which the wideband slope per user satisfies S0 < 2 /K + δ. This is quite contrary to the large bandwidth case [1], where a slope of 1 per user is achievable with probability 1. We also develop an interference alignment scheme for the finite bandwidth case that shows some gain.	10.1109/tit.2019.2927483	[ "https://arxiv.org/pdf/1207.4252v1.pdf" ]	5,882,412	1207.4252	312202c698684d3457315857d33e57f11678a5b5	The Wideband Slope of Interference Channels: The Small Bandwidth Case 18 Jul 2012 Minqi Shen Anders Høst-Madsen The Wideband Slope of Interference Channels: The Small Bandwidth Case 18 Jul 2012arXiv:1207.4252v1 [cs.IT] 1 This paper studies the low-SNR regime performance of a scalar complex K-user interference channel with Gaussian noise. The finite bandwidth case is considered, where the low-SNR regime is approached by letting the input power go to zero while bandwidth is small and fixed. We show that for all δ > 0 there exists a set with non-zero measure (probability) in which the wideband slope per user satisfies S0 < 2 /K + δ. This is quite contrary to the large bandwidth case [1], where a slope of 1 per user is achievable with probability 1. We also develop an interference alignment scheme for the finite bandwidth case that shows some gain. I. INTRODUCTION This paper and the companion paper [1] study the bandwidth-power trade-off of a K-user interference channel in the low-SNR (signal-to-noise) regime, where explicitly SNR P BN 0 .(1) Bandwidth and input power, two important design parameters, are related by the function R E b N0 , where E b N0 is the transmitted energy per bit, and R is the spectral efficiency. The concept of the low-SNR regime was introduced by S. Verdú in the 2002 paper [2]. A system working in this regime is characterized by very small spectral efficiency, so that the R E b N0 curve can be closely approximated by its first-order approximation, which is determined by two measures: the minimum energy per bit E b N0 min and the wideband slope S 0 . E b N0 min is the minimum transmitted energy per bit required by reliable communication, which is generally achieved at zero spectral efficiency; and S 0 is the first-order slope of R E b N0 as E b N0 approaches E b N0 min . These two measures are defined by E b N 0 min = lim SNR↓0 SNR R (SNR) (2) S 0 lim E b N 0 ↓ E b N 0 min R E b N0 10 log 10 E b N0 − 10 log 10 E b N0 min 10 log 10 2, Further manipulations in [2] show that E b N0 min and S 0 can be determined by the first and second order derivative of R (SNR) at zero SNR: E b N 0 min = log e 2 R (0) ,(4)S 0 = − 2 Ṙ (0) 2 R (0) ,(5) The authors are with the Department of Electrical Engineering, University of Hawaii Manoa, Honolulu, HI 96822 (e-mail: {minqi,ahm}@hawaii.edu. This work was supported in part by NSF grant CCF 1017823. This paper was presented in part at the 49th annual Allerton Conference on Communication, Control and Computing, September 2011 (Urbana-Champaign, IL). whereṘ (0) andR (0) are the first-order and the second-order Taylor expansion coefficients for SNR → 0.Ṙ (0) = dR(SNR) dSNR SNR=0 andR (0) = d 2 R(SNR) dSNR 2 SNR=0 if R (SNR) is differentiable. What is interesting is that there are two distinct ways to approach the low-SNR regime, which have very different impacts on the performance of the interference channel Although approaching the low-SNR regime by letting B → ∞ is emphasized in previous papers (hence the term "wideband slope"), it is not the only way. As can be noted from the definition of SNR (1), SNR approaches zero if either B → ∞ or P → 0. Consider a point-to-point AWGN channel with spectral efficiency R = log 1 + P BN 0 . The low-SNR results are based on a Taylor series of log(1 + x) , as also seen by (4)(5); therefore as long as SNR = P BN0 → 0 in any manner, low-SNR results such as minimum energy per bit and wideband slope are unchanged. The key is that the spectral efficiency R → 0, not that B → ∞. For the interference channel, on the other hand, different results are obtained depending on how the low-SNR regime is approached. In the first approach, let B → ∞ while P is fixed and finite. We call this the large bandwidth regime. In [1] we proved that in this case a wideband slope of K was achievable with probability one by using channel delays. In the second approach, let P → 0 while B is fixed and finite. In this case, the rate BR in bits/s must necessarily approach 0 as well, and we therefore call this the low-rate regime. This is the case considered in this paper, and as will be seen the results are quite different than the the case in [1]. To put the results of this paper in context, consider the completely symmetric channel: the channel between receiver pairs (i, j) is the same for all 1 ≤ i, j ≤ K, both i = j and i = j. We call this channel the 1-channel. The capacity of this channel is fully known: because of the symmetry all receivers must be able to decode all messages, and the capacity is therefore given by the MAC (multiple access channel) bound into one of the nodes. For this channel, FDMA (frequency division multiple access) or TDMA (time division multiple access) is optimum, and the degrees of freedom [3] is 1 (1/K per user) while the wideband slope is 2 (2/K per user). A key question is if this channel is typical. For degrees of freedom the answer is no: the results in [4] and [5] show that the degrees of freedom is K/2 ( 1 2 per user) almost everywhere for a scalar channel. Thus, the degrees of freedom is discontinuous in 1, and in fact almost everywhere. Similarly, [3] shows that for time-varying channels, the degrees of freedom is K/2 with probability one. In [1] we proved analogously that in the large bandwidth regime the wideband slope is K(1 per user) with probability one for a line-of-sight channel. Thus, also the wideband slope is discontinuous in 1 and again in fact discontinuous with probability one. The main result of this paper is that in the low-rate regime the wideband slope is upper semi-continuous in 1. That is, for any δ > 0 there exists an open setC δ of channels so that 1 ∈ cl(C δ ) (cl means closure) and S 0 ≤ 2 + δ inC δ . While this does not give a complete characterization of the wideband slope as in [1], it does show that interference alignment in the low-rate regime does not give the same dramatic gain in performance as in the large bandwidth and high SNR regimes. We still show that interference alignment can outperform TDMA, but in line with the outer bound, not by much. II. SYSTEM MODEL AND PRELIMINARIES In [1] we derived the following baseband model for the interference channel (in a line-of-sight model): y j [n] = C jj x j [n] + i =j C jixi [n − n ji ] + z j [n] wherex i [n] = ∞ m=−∞ x i [m]sinc(n − m + δ ji ).(6) and n ji = τ ji B + 1 2 (7) δ ji = τ ji B − τ ji B + 1 2(8) are the symbol and fractional delays, respectively. It was these delays that allowed interference alignment in [1] as B → ∞. In the present paper we keep B fixed; we will further assume that B is so small that the delays are insignificant, n ji = 0, δ ji ≈ 0, and we therefore arrive at the usual model for the interference channel, y j [n] = C jj x i [n] + i =j C ji x i [n] + Z j [n],(9) where C ji is a complex scalar and the noise Z j is i.i.d. (independent, identically distributed) circularly symmetric complex random variable with distribution CN (0, BN 0 ); since B does not play any role in the rest of the paper we will put B = 1 and omit it from future formulas. Notice that the model (9) is valid also for a non line-of-sight model, as long as delays along all paths are insignificant. A. Circularly Asymmetric Signaling To characterize the Shannon capacity region of the model (9), most research restricts the inputs to be circularly symmetric, i.e., the the real part of the input Re {x j } and the imaginary part of the input Im {x j } are i.i.d.. However, [6] shows that circularly asymmetric signaling achieves higher degree of freedom in the high-SNR regime. Although the specific interference alignment technique they proposed is not applicable to the low-SNR regime, that work still has inspired our interference alignment for the low-SNR regime. In section IV, we will see that circularly asymmetric signaling indeed benefits system performance. In circularly asymmetric signaling, the transmitters are allowed to allocate power on real and imaginary dimensions, and the real part of the input Re {x j } is allowed to be correlated with the imaginary part of the input Im {x j }, while in circularly symmetric signaling, Re {x j } and Im {x j } are required to be i.i.d.. To characterize such transmission schemes, it is more convenient to consider the scalar complex channel as a two-dimensional vector real channel. Following [6], we extend (9) into an equivalent two-dimensional real channel, Y j = \|C jj \| X j + K i=1,i =j \|C ji \| U ji X i + Z j (10) where U ji cos (φ ji ) − sin (φ ji ) sin (φ ji ) cos (φ ji ) is the rotation matrix with angle φ ji , and the 2 × 1 vector white Gaussian noise is Z j ∼ N 0, N0 2 I 2×2 . Notice that we let receiver j be phase-synchronized with the received x j so that φ jj = 0. Without without loss of generality, we can assume N 0 = 1 whenever convenient. The input signal X j is related to the scalar complex model by: X j = Re {x j } Im {x j } . We assume that an 2 nRj , n code is used at receiver j, for j = 1, · · · , K. At the transmitter j, the input message W j is drawn uniformly randomly from the index set 1, · · · , 2 nRj , and a deterministic function yields the length n transmitted codeword X n j (W j ). The codebook of user j is composed by the set of codewords X n j (1) , · · · , X n j 2 nRj . We require each user to satisfy power constraint Pj /B per second per Hz. Recall that we may assume B = 1. Denote the ith entry of X n j by X (i) j . Therefore the input must satisfy constraint 1 n n i=1 E X (i) j X (i) j T V j ,(11) where Tr (V j ) = P j , j = 1, · · · , K. For any two given matrices A and B, the notation A B means that the matrix B − A is positive semi-definite. Notice that given the assumption B = N 0 = 1, we have SNR j = P j BN 0 = P j(12) Corresponding to the X n j (W j ) codebook , we also define four Gaussian random variables X ′ jG , X jG , Y ′ jG , and Y jG as follows for later use. Let X ′ jG be i.i.d. vector Gaussian random variable, X ′ jG ∼ N 0, V ′ j , where V ′ j = 1 N N n=1 X n j X n j H , V ′ j V j given power constraint (11). Let X jG be i.i.d. vector Gaussian random variable, X jG ∼ N (0, V j ). Y ′ jG and Y jG are defined as Y ′ jG = \|C jj \| X ′ jG + K i=1,i =j \|C ji \| U ji X ′ iG + Z j Y jG = \|C jj \| X jG + K i=1,i =j \|C ji \| U ji X iG + Z j B. Performance Criterion And Performance Measures For more than two users it is complicated to compare complete slope regions, and we are therefore looking at a single quantity-the sum slope S 0 , to characterize performance. The formal definitions are as follows. Definition 1 (Sum slope). S 0 is defined as the first-order slope of the R sum E b N0 sum curve, where R sum K j=1 R j and E b N0 sum K j=1 Pj N0B K j=1 Rj . It characterizes the wideband slope of R sum as E b N0 sum approaches its minimum value E b N0 min : E b N 0 min = lim Psum↓0 K j=1 P j K j=1 R j · N 0 B (13) S 0 lim E b N 0 sum ↓ E b N 0 min R sum E b N0 10 log 10 2 10 log 10 E b N0 sum − 10 log 10 E b N0 min (14) Denote the sum power constraint by P sum = K j=1 P j . Under the assumption that N 0 B = 1, E b N0 min and S 0 can be obtained from the first and second order derivatives of R sum (P sum ): E b N 0 min = log e 2 R sum (0) ; (15) S 0 = − 2 Ṙ sum (0) 2 R sum (0) .(16) Notice that constraints on P j or R j are required for a well-posed problem; otherwise the best low-SNR performance is achieved by allocating all power to the user with largest direct link gain so that E b N0 min is minimized. Such a solution is just a single user solution and gives no insight into the interference channel. To fix this insufficiency while keeping our problem relatively simple to analyze, we require the interference channel to work under the equal-power constraint, which is defined as Definition 2. Equal power constraint is the case where the sum rate R sum is maximized under the constraint P 1 = P 2 = · · · = P K . Given (3), we can see that if two systems achieve equal E b N0 min value, the E b N0 value of the system with higher wideband slope approaches its minimum value faster, and the system is therefore more spectrally efficient. On the other hand, we should notice that the priority in the low-SNR regime is to minimize E b N0 min . Based on this observation, we make the following statement: The results in [7] reveal that the optimal achievable minimum energy per bit E b N0 min of an interference channel is equal to that of its corresponding interference-free channel. The first-order optimality criterion under the equal power constraint is stated in the following lemma. Lemma 4. The optimal minimum energy per bit of the interference channel defined by (9) is E b N 0 min = K log e 2 K j=1 \|C jj \| 2 (17) under the equal power constraint. Given Remark 3, any achievable scheme or capacity outer bound gives valid bound on the sum slope only if it has correct E b N0 min values, stated in Theorem 4. For performance measure we use ∆S 0 = S 0 S 0,no interference . The quantity S 0,no inteference is the wideband slope of the corresponding interference-free channel: R j = log 1 + \|C jj \| 2 P j . We can interpret ∆S 0 as the loss in wideband slope due to interference. Under the equal power constraint, S 0,no interference the sum slope of the interference-free channel, and S 0,T DMA and S 0,T IN the sum slope achieved by TDMA and treating interference as noise (TIN) respectively, are listed as follows for comparison purposes; they can be obtained directly obtained from (4-5) S 0,no interference = 2 j \|C jj \| 2 2 j \|C jj \| 4(18) The R sum (P sum ) achieved by TIN is R sum (P sum ) = K j=1 log 1 + \|C jj \| 2 P sum K + P sum i =j \|C ji \| 2 , which gives S 0,T IN = 2 j \|C jj \| 2 2 K j=1 \|C jj \| 4 + 2 i =j \|C ji \| 2 \|C jj \| 2 ; (19) ∆S 0 = K j=1 \|C jj \| 4 K j=1 \|C jj \| 4 + 2 i =j \|C ji \| 2 \|C jj \| 2(20) The R sum (P sum ) achieved by TDMA is R sum (P sum ) = 1 K K j=1 log 1 + \|C jj \| 2 P sum , which gives S 0 = 2 K j \|C jj \| 2 2 j \|C jj \| 4 (21) ∆S 0 = 1 K (22) III. GENERALIZED Z-CHANNEL OUTER BOUND In this section, we develop a new outer bound on the wideband slope for a set of the 2-dimensional vector channels defined by (10), under the equal power constraint. The outer bound is specific to the low-rate regime. The outer bound is derived from the sum Shannon capacity of a type of generalized Z-channel, which is constructed by elimination of a subset of the interference links. In Section III-A, we show that for a subset of channels C, the optimal sum capacity of their corresponding Z-channels can be achieved by i.i.d. 2-dimensional vector Gaussian inputs. Further, assuming that channel coefficients C ji is drawn from i.i.d. continuous distribution, the set C has non-zero probability. In Section III-B, the Z-channel outer bound is used to derive an outer bound on the wideband slope. A. Generalized Z-Channel And Its Sum Capacity We define the generalized Z-channel corresponding to the interference channel (10) aŝ Y j = \|C jj \| X j + K i=j+1 \|C ji \| U ji X i + Z j .(23)Figure 1. Generalized Z-channel Eliminating a subset of interference links will not reduce channel capacity and therefore, the sum capacity outer bound for the generalized Z-channel is also a sum capacity outer bound for the interference channel. To derive the Z-channel sum capacity, we provide receiver j, j = 2, · · · , K with side information S n j = S n j1 , · · · , S n j(j−1) T , where S n jp = \|C pj \| U pj X n j + K i=j+1 \|C pi \| U pi X n i + W n jp (24) p = 1, · · · , j − 1. . The entries in the length n noise vector W n jp are i.i.d 2 × 1 vector Gaussian noise with the same marginal distribution as Z j . Further, they satisfy the following properties • W n j(j−1) , · · · , W n j1 are independent of all input length n codewords X n i , i = 1, · · · , K; • Z j , W j(j−1) , · · · , W j1 are jointly Gaussian random variables, with zero mean and covariance matrix K Sj =          I A j(j−1) · · · A j1 A j1 A T j(j−1) I A (j−1)(j−2) · · · A (j−1)1 . . . . . . . . . . . . A T j1 I A 21 A T j1 A T (j−1)1 · · · A T 21 I         (25) To guarantee such multivariate Gaussian random variable exists, A jk should be chosen such that for all j = 1, · · · , K K Sj 0(26) We emphasize the following property of K Sj , which will play a key role in the proof of the main result. Lemma 5. The distributions of S n (j−1)p S n (j−1)(p−1) , · · · , S n (j−1)1 , X n (j−1) and S n jp S n j(p−1) , · · · , S n j1 are equal. Proof of Lemma 5 is in Appendix A. Lemma 6. The distributions ofŶ n j−1 S n (j−1)(j−2) , · · · , S n (j−1)1 , X n (j−1) and S n j(j−1) S n j(j−2) , · · · , S n j1 are equal. The proof of Lemma 6 is almost identical to the proof of Lemma 5 and will therefore be omitted. Define the average covariance matrix of the input at transmitter as V j 1 n n i=1 E X (i) j X (i) j T for any length n input sequence X n j . It must satisfy the power constraint defined in (11), i.e.,Ṽ j V j . The next lemma states how to choose A jk . Lemma 7. Let A jp , j = 2, · · · , K and p = 1, · · · , j − 1 be A jp = \|C pj \| 2 \|C jj \| 2 U (−φ pj ) + \|C pj \| 2 \|C jj \| 2 K i=j+1 \|C ji \| 2 U (φ ji ) V i U (−φ pj − φ ji ) − K i=j+1 \|C ji \| 2 \|C pi \| 2 U (φ ji ) V i U (−φ pi ) (27) If A jp defined by (27) satisfy K Sj 0, then X jG →Ŷ jG → S j1G , · · · , S j(j−1)G T (28) forms a Markov chain for all j = 2, · · · , K. Here X jG andŶ jG are defined in section II-A; the proof of Lemma 7 is in Appendix B. For a channel realization, denote its channel coefficients by C {C ji ; i, j = 1, · · · , K}. In the following lemma, we state a sufficient condition on C so that K Sj 0 if A jp is chosen according to (27). Lemma 8. For any 0 < α < 1 there exist some ǫ α , ǫ ′ α > 0 and ǫ ′′ α (C) > 0 so that if C ∈ C α C ij : \|C ij \| 2 \|C jj \| 2 − α < ǫ α , \|φ ji \| < ǫ ′ α } (29) P j < ǫ ′′ α (C)(30) then K Sj 0 for A jp chosen according to (27). Proof of Lemma 8 is in Appendix C Our main result of this section is stated in the following theorem. Theorem 9. For every interference channel realization C ∈ C = α∈(0,1) C α defined by (29) there exists an ǫ ′′ α (C) > 0 so that if P j < ǫ ′′ α (C) the sum capacity of its corresponding Z-channel is given by K j=1 R j ≤ C sum = max Tr (V j ) ≤ P j V j 0, j = 1, · · · , K K j=1 I X jG ;Ŷ jG (31) = max Tr (V j ) ≤ P j V j 0, j = 1, · · · , K K j=1 log   I + K i=j \|C ji \| 2 V i     I + K i=j+1 \|C ji \| 2 V i   −1 (32) Because the sum capacity of the interference channel is outer bounded by the sum capacity of the generalized Z-channel, (31) is an outer bound for the sum capacity of the interference channel. Proof of Theorem 9 is in Appendix D. Note that the bound in Theorem 9 is valid for P j < ǫ ′′ α (C), and it therefore bounds the actual capacity for suitably low SNR. However, we will mainly use it to bound the wideband slope, a weaker result. B. Sum Slope Outer Bound for the Interference Channel Given the capacity in Theorem 9, we have following result on the low-rate performance of the interference channel. Theorem 10. For the interference channel (9), the sum capacity is outer bounded by (31) for low SNR. Under the equal power constraint, the minimum energy per bit of this upper bound satisfy the requirement imposed by Remark 3, which is E b N 0 min = K log 2 K j=1 \|C jj \| 2 (33) For channel realizations C ∈ C = α∈(0,1) C α defined as (29) it therefore gives the following valid upper bound on the sum slope: S 0 ≤   K j=1 \|C jj \| 2   2 (34) × max Tr V j ≤ 1 V j 0   K j=1 \|C jj \| 4 Tr V 2 j (35) +2 K−1 j=1 K i=j+1 \|C jj \| 2 \|C ji \| 2 Tr V j U jiVi U † ji   −1(36) Proof of Theorem 10 is in Appendix E. Theorem 11. For the symmetric channel where C jj = 1, C ji = α ∈ (0, 1) , the sum slope is bounded by S 0 ≤ 2K αK + (1 − α) Proof of Theorem 11 is in Appendix F. As discussed in the introduction, the wideband slope in the point C = 1 is 2 K per user, achievable by TDMA. Theorem 11 shows that the point C = 1 is not exceptional in the low-rate regime: for α close to 1 (from below) the channel with C jj = 1, C ji = α has slope close to 2 K . However, the set of channels C jj = 1, C ji = α still has Lebesgue measure zero, i.e., if the channel coefficients are drawn from a continuous distribution, this set has probability zero. The main result of the paper is the following theorem that shows that the set of channels with slope close to 2 K can be be extended to a set of non-zero measure. Theorem 12. For all σ > 0, there exists an open setC σ ⊂ C K(K−1) with 1 ∈ cl C σ , so that for C ∈C σ S 0 ≤ 2 + σ,(37) If the magnitude and phase of the channel coefficients are drawn from continuous random distribution, P r C σ > 0. And as σ → 0, lim σ→0 ∆S 0 = 1 K Because ∆S 0 achieved by TDMA is 1 K , when σ is small, TDMA transmission scheme is almost optimal for channels inC σ . Proof of Theorem 12 is in Appendix G. IV. SUM SLOPE ACHIEVABLE SCHEME In the previous section, we have shown that there exist a set of channels C σ , P r C σ > 0, for which TDMA is almost optimal. However, we also notice that the probability that a channel realization is not in C σ is likewise greater than zero. Therefore, it is natural to ask the question: for channels not inC σ , can we find achievable schemes better than TDMA or Treating Interference as Noise (TIN)? In section IV-A, we propose a circularly asymmetric transmission scheme and analyze its theoretical performance. Simulation results are shown in section IV-B. We will also discuss possible improvements of this scheme. A. One-Dimensional Gaussian Signaling In this section, we use the complex scalar channel model defined in (9). We define a one-dimensional Gaussian signaling transmission scheme and analyze its performance. The idea is to align interference as much as possible. Definition 13. One-dimensional Gaussian signaling transmission scheme • At transmitter j, let input sequence be x j [n] = w j [n] e jθj , where w j [n] is drawn from i.i.d real Gaussian random variable with distribution N (0, SNR j ), and the phase θ j is a prior chosen design parameter, unchanged for all n during the transmission. • At receiver j, interference is treated as noise. We call this one-dimensional because every transmitter only transmits along e jθj , therefore only one dimension is used out of the two-dimensional signal space. Our objective is to find the set of phases θ = {θ 1 , · · · , θ K } that maximize the achievable wideband slope S 0 . The achievable S 0 for any θ is stated in the next lemma. For computational convenience, we return to the equivalent two-dimensional real channel model. In the equivalent 2-dimensional real channel model, the input X j has covariance matrixw V j = P j cos 2 θ sin 2θ 2 sin 2θ 2 sin 2 θ , rank (V j ) = 1. We denote the normalized covariance matrix byV j = Vj Pj . Lemma 14. For the equivalent 2-dimensional real channel model defined by (10), the sum slope achieved by the one-dimensional Gaussian signaling is S 0 = K j=1 \|C jj \| 2 2 K j=1 \|C jj \| 4 + K j=1 K i =j \|C jj \| 2 \|C ji \| 2 + f (θ) ,(38) where f (θ) K j=1 K i =j \|C jj \| 2 \|C ji \| 2 cos 2 (φ ji − θ j + θ i ) .(39) Proof: Treating interference as noise at the receiver, the achievable sum rate 13 is R sum = K j=1 1 2 log I 2 + 2 K P sum \|C jj \| 2 U (φ jj )V j U 2 (−φ jj ) + K i=1,i =j \|C ji \| 2 U (φ ji )V i U (−φ ji )   (40) − 1 2 log I 2 + 2 K P sum K i=1,i =j \|C ji \| 2 U (φ ji )V i U (−φ ji )  (41) under the equal power constraint where SNR j = SNRs K . Combining (4), (5) and (40), we havė R s (0) = K j=1 \|C jj \| 2 K (42) −R s (0) = 2 K j=1 \|C jj \| 4 K 2 + 2 K j=1 K i =j \|C jj \| 2 \|C ji \| 2 K 2 + 2 K 2 K j=1 K i =j \|C jj \| 2 \|C ji \| 2 · cos 2 (φ ji − θ j + θ i )) .(43)Given S 0 = 2Ṙ 2 s (0) −Rs(0) , (38) follows. Given (38), maximizing S 0 is equivalent to finding the set of θ j that minimizes f (θ) . Denote this optimization problem by P (θ), which is defined as min f (θ) subject to θ j ∈ [−π, π] . Notice that θ j mod 2π will not affect the value of f (θ). Given the definition of the objective function in (39), the constraint θ j ∈ [−π, π] can be discarded. Therefore, P (θ) can be solved using standard numerical methods for unconstrained optimization problems. B. Simulation Results and Discussions In this section, we simulate the performance of the one-dimensional signaling scheme in a 10-user interference channel with unit direct link gains and symmetric weak interference link gains, i.e., \|C jj \| 2 = 1 and \|C ji \| 2 = a < 1 for all i, j = 1, · · · , 10; the phases φ ji is drawn from U [−π, π] in each channel realization. This performance will be compared with existing achievable schemes: treating interference as noise and TDMA. The simulation results are presented below. We can see that when α, the ratio between the direct link gain and the interference link gain, is close to 1, then with non-zero probability the one dimensional Gaussian signaling transmission scheme performs better than TDMA. In Figure 3, we compare the median value of S 0 achieved by one-dimensional interference alignment scheme with the performance of treating interference as noise and TDMA. The main result of this paper can be summarized as follows. In the low rate regime, the wideband slope is (upper semi-) continuous in the point 1, the point where all channels are identical, and where the wideband slope (per user) is 2 K . This does not give a full characterization of the wideband slope. However, it is a stark contrast to the large bandwidth regime [1], where a wideband slope of 1 is achievable almost everywhere, implying discontinuity in the point 1. It is also a contrast to the high SNR regime, where 1 2 DoF per user is achievable almost everywhere [3], [4], [5], and where the DoF is discontinuous almost everywhere. The results in [1] and [3], [4] were obtained by using interference alignment, and the result in this paper implies that interference alignment does not give the dramatic gains in the low rate regime seen elsewhere. Yet, we show that interference alignment can still give some gain. One implication of the result is that in networks, as opposed to point-to-point channels, it is important how the low SNR regime is approached. This may effect how networks are designed and operates for maximum energy efficiency. Freund S n (j−1)p = C (j−1)j U p(j−1) X n (j−1) + K i=j \|C pi \| U pi X n i + W n (j−1)p S n (j−1)(p−1) = C (j−1)j U (p−1)(j−1) X n (j−1) + K i=j C (p−1)i U (p−1)i X n i + W n (j−1)(p−1) . . . S n (j−1)1 = C (j−1)j U 1(j−1) X n (j−1) + K i=j \|C 1i \| U 1i X n i + W n (j−1)1 When X n (j−1) is given, it can be subtracted from S n (j−1)p , S n (j−1)(p−1) , · · · , S n (j−1)1 to givê S n (j−1)p = S n (j−1)p − C (j−1)j U p(j−1) X n (j−1) = K i=j \|C pi \| U pi X n i + W n (j−1)p (44) S n (j−1)(p−1) = S n (j−1)(p−1) − C (j−1)j U (p−1)(j−1) X n (j−1) = K i=j C (p−1)i U (p−1)i X n i + W n (j−1)(p−1)(45) . . . S n (j−1)1 = S n (j−1)1 − C (j−1)j U 1(j−1) X n (j−1) = K i=j \|C 1i \| U 1i X n i + W n (j−1)1 (46) while S n jp = K i=j \|C pi \| U pi X n i + W n jp (47) S n j(p−1) = K i=j C (p−1)i U (p−1)i X n i + W n j(p−1)(48) . . . S n j1 = K i=j \|C 1i \| U 1i X n i + W n j1 .(49) We know that Z j , W j(j−1) , · · · , W j1 are jointly Gaussian random variables, with zero mean and covariance matrix K Sj equal to:          I A j(j−1) · · · A j2 A j1 A T j(j−1) I A (j−1)(j−2) · · · A (j−1)1 . . . . . . . . . . . . A T j1 I A 21 A T j1 A T (j−1)1 · · · A T 21 I          which is defined in (27) . It is clear that the covariance matrices of the jointly Gaussian random variables W (j−1)p , W (j−1)(p−1) , · · · , W (j−1 and W jp , W j(p−1) , · · · , W j1 are the same: cov W (j−1)p , W (j−1)(p−1) , · · · , W (j−1)1 = cov W jp , W j(p−1) , · · · , W j1 =          I A p(p−1) · · · A p2 A p1 A T p(p−1) I A (p−1)(p−2) · · · A (p−1)1 . . . . . . . . . . . . A T p2 I A 21 A T p1 A T (p−1)1 · · · A T 21 I         (50) Comparing (44)~(46) and (47)~(49), we can see that distribution of S n (j−1)p S n (j−1)(p−1) , · · · , S n (j−1)1 , X n (j−1) and S n jp S n j(p−1) , · · · , S n j1 are equal as long as W n (j−1)p W n (j−1)(p−1) , · · · , W n (j−1)1 and W n jp W n j(p−1) , · · · , W n j1 have the same distribution. Recall that W ji is i.i.d. Gaussian random variables which is independent from the input signals X n . Therefore given (50), Lemma 5 is proved. APPENDIX B PROOF OF LEMMA 7 Lemma 7 is proved using the following lemma from [8]. cov X jG , S j = cov X jG ,Ŷ jG cov Ŷ jG −1 cov Ŷ jG , S j(51) Given (23), (24) and the independence of W jp and X i , the left hand side of (51) is LHS =       \|C 1j \| 2 V j U (−φ 1j ) \|C 2j \| 2 V j U (−φ 2j ) . . . C (j−1)j 2 V j U −φ (j−1)j       T and the right hand side is RHS = \|C jj \| 2 V j U (−φ jj )   K i=j \|C ji \| 2 U (φ ji ) V i U (−φ ji ) + I   −1       K i=j \|C ji \| 2 \|C 1i \| 2 U (φ ji ) V i U (−φ 1i ) + A j1 K i=j \|C ji \| 2 \|C 2i \| 2 U (φ ji ) V i U (−φ 2i ) + A j2 . . . K i=j+1 \|C ji \| 2 C (j−1)i 2 U (φ ji ) V i U −φ (j−1)i + A j(j−1)       T In order for LHS = RHS, we must have \|C pj \| 2 V j U (−φ pj ) = \|C jj \| 2 V j U (−φ jj )   K i=j \|C ji \| 2 U (φ ji ) V i U (−φ ji ) + I   −1   K i=j \|C ji \| 2 \|C pi \| 2 U (φ ji ) V i U (−φ pi ) + A jp   Solving the equation above, we have A jp = \|C pj \| 2 \|C jj \| 2 U (φ jj − φ pj ) + \|C pj \| 2 \|C jj \| 2 K i=j+1 \|C ji \| 2 U (φ ji ) V i U (φ jj − φ pj − φ ji ) − K i=j+1 \|C ji \| 2 \|C pi \| 2 U (φ ji ) V i U (−φ pi ) APPENDIX C PROOF OF LEMMA 8 First, consider the simple case where \|Cpj\| 2 \|Cjj \| 2 = α, φ ji = 0 and P j = 0, that is, K xj = 0. For this case, given (27) we have A ji = B = α 0 0 α for all i, j. It is easy to check that the eigenvalues of K Sj =       I B · · · B B T I · · · B . . . . . . . . . B T · · · I       are λ 1 = 1 − α and λ 2 = 1 + (j − 1) α, with multiplicity 2 (j − 1) and 2 respectively. Therefore, K Sj is positive definite if 0 < α < 1. Now let us consider the case where φ ji and P j are small but non-zero, and \|Cpj\| 2 \|Cjj \| 2 are not necessarily equal to α. Denote the (p, q) th element of B by b pq . It is well known that the eigenvalues of symmetric matrix are locally (Lipschitz) continuous [9] with respect to its elements. Therefore, corresponding to every α ∈ (0, 1), for anyǫ > 0, there exist some strictly positive real numbers ǫ α , ǫ ′ α and ǫ ′′ α (C) such that if \|Cpj\| 2 \|Cjj \| 2 − α < ǫ α , \|φ ji \| < ǫ ′ α , and P j < ǫ ′′ α (C) then every eigenvalues λ s of K Sj satisfies \|λ s − λ 1 \| <ǫ or \|λ s − λ 2 \| <ǫ. The bound on P j may depend C to ensure that the two last terms in (27) are of bounded variation. For any 0 < α < 1 we can always find someǫ > 0 that guarantees λ s > 0, and K Sj is positive definite as a result. APPENDIX D PROOF OF THEOREM 9 First we state a useful result from [8]. Lemma 16. ([8, Lemma 2]) Let X n = (X 1 , · · · , X n ) and Y n = (Y 1 , · · · , Y n ) be two sequences of random vectors, and let X ′ G , X G , Y ′ G , and Y G be Gaussian vectors with covariance matrices satisfying cov X ′ G Y ′ G = 1 n n i=1 cov X i Y i cov X G Y G then we have h (X n ) ≤ nh (X ′ G ) ≤ nh (X G ) h ( Y n \| X n ) ≤ nh ( Y ′ G \| X ′ G ) ≤ nh ( Y G \| X G ) By Fano's inequality, the sum capacity of the generalized Z-channel (23) must satisfy n K j=1 R j − nǫ (a) ≤ I X n 1 ;Ŷ n 1 + K j=2 I X n j ;Ŷ n j , S j (b) = h Ŷ n 1 − h Ŷ n 1 X n 1 + K j=2 I X n j ; S j + K j=2 I X n j ;Ŷ n j S j (c) = h Ŷ n 1 − h Ŷ n 1 X n 1 + K j=2 j−1 p=1 I X n j ; S jp S n j(p−1) , · · · , S n j1 + K j=2 I X n j ;Ŷ n j S j (d) = h Ŷ n 1 − h Ŷ n 1 X n 1 + K j=2 j−1 p=1 h S n jp S n j(p−1) , · · · , S n j1 −h S n jp S n j(p−1) , · · · , S n j1 , X n j + K j=2 h Ŷ n j S n j(j−1) , S n j(j−2) , · · · , S n j1 −h Ŷ n j S n j(j−1) , S n j(j−2) , · · · , S n j1 , X n j (e) ≤ nh Ŷ 1G − h Ŷ n 1 X n 1 +h (S n 21 ) − h ( S n 21 \| X n 2 ) + K j=3 j−1 p=j−1 h S n jp S n j(p−1) , · · · , S n j1 + K j=3 j−2 p=1 h S n jp S n j(p−1) , · · · , S n j1 − K−1 j=3 j−1 p=1 h S n jp S n j(p−1) , · · · , S n j1 , X n j − K j=K j−1 p=1 h S n jp S n j(p−1) , · · · , S n j1 , X n j + K j=2 h Ŷ n j S n j(j−1) , S n j(j−2) , · · · , S n j1 − K j=2 h Ŷ n j S n j(j−1) , S n j(j−2) , · · · , S n j1 , X n j (f ) = nh Ŷ 1G − h Ŷ n 1 X n 1 +h (S n 21 ) + K j=3 h S n j(j−1) S n j(j−2) , · · · , S n j1 + K j=3 j−2 p=1 h S n jp S n j(p−1) , · · · , S n j1 −h S n (j−1)p S n (j−1)(p−1) , · · · , S n (j−1)1 , X n (j−1) − K−1 p=1 h S n Kp S n K(p−1) , · · · , S n K1 , X n K + K j=2 h Ŷ n j S n j(j−1) , S n j(j−2) , · · · , S n j1 − K j=2 h Ŷ n j S n j(j−1) , S n j(j−2) , · · · , S n j1 , X n j (g) = nh Ŷ 1G − h Ŷ n 1 X n 1 +h (S n 21 ) + K j=3 h S n j(j−1) S n j(j−2) , · · · , S n j1 − K−1 p=1 h S n Kp S n K(p−1) , · · · , S n K1 , X n K + K j=2 h Ŷ n j S n j(j−1) , S n j(j−2) , · · · , S n j1 − K j=2 h Ŷ n j S n j(j−1) , S n j(j−2) , · · · , S n j1 , X n j (h) = nh Ŷ 1G + K j=3 h S n j(j−1) S n j(j−2) , · · · , S n j1 −nh W K(K−1) , · · · , W K1 + K j=2 h Ŷ n j S n j(j−1) , S n j(j−2) , · · · , S n j1 − K j=2 h Ŷ n j S n j(j−1) , S n j(j−2) , · · · , S n j1 , X n j (i) = h Ŷ n 1 − h Ŷ n 1 X n 1 , and the chain rule which gives I X n j ;Ŷ n j , S j = I X n j ; S j + I X n j ;Ŷ n j S j .. (c) is from the chain rule, which gives I X n j ; S j = j−1 p=1 I X n j ; S jp S n j(p−1) , · · · , S n j1 . (d) is from the expansion of mutual information. = nh Ŷ 1G − nh W K(K−1) , · · · , W K1 + K j=2 h Ŷ n j S n j(j−1) , S n j(j−2) , · · · , S n j1 −nh N K \| W K(K−1) , · · · , W K1 (j) = nh Ŷ 1G − nh N K , W K(K−1) , · · · , W K1 + K j=2 h Ŷ n j S n j(j−1) , S n j(j−2) , · · · , S n j1 (k) ≤ nh Ŷ 1G − nh N K , W K(K−1) , · · · , W K1 +n K j=2 h Ŷ jG S j(j−1)G , S j(j−2)G , · · · , S j1G (l) = nh Ŷ 1G − nh N K , W K(K−1) , · · · , W K1 +n K j=2 h Ŷ jG −n K j=2 h S j(j−1)G , S j(j−2)G , · · · , S j1G +n K j=2 h S j(j−1)G , S j(j−2)G , · · · , S j1G Ŷ jG (m) = nh Ŷ 1G − nh N K , W K(K−1) , · · · , W K1 +n K j=2 h Ŷ jG −n K j=2 h S j(j−1)G , S j(j−2)G , · · · , S j1G +n K−1 j=2 h S j(j−1)G , S j(j−2)G , · · · , S j1G Ŷ jG +nh S K(K−1)G , S K(K−2)G , · · · , S K1G Ŷ KG (n) = nh Ŷ 1G − nh N K , W K(K−1) , · · · , W K1 +n K j=2 h Ŷ jG −n K j=2 h S j(j−1)G , S j(j−2)G , · · · , S j1G +n K−1 j=2 h S (j+1)(j−1)G , S (j+1)(j−2)G , · · · , S (j+1)1G S (j+1)jG +nh W K(K−1) , W K(K−2) , · · · , W K1 N K = nh Ŷ 1G − nh N K , W K(K−1) , · · · , W K1 +n K j=2 h Ŷ jG −n K j=2 h S j(j−1)G , S j(j−2)G , · · · , S j1G +n K j=3 h S j(j−2)G , S j(j−3)G , · · · , S j1G S j(j−1)G +nh W K(K−1) , W K(K−2) , · · · , W K1 N K = nh Ŷ 1G − nh N K , W K(K−1) , · · · , W K1 +n K j=2 h Ŷ jG − h (S 21G ) −n K j=3 h S j(j−1)G , S j(j−2)G , · · · , S j1G − h S j(j−2)G , S j(j−3)G , · · · , S j1G S j(j−1)G +nh W K(K−1) , W K(K−2) , · · · , W K1 N K = nh Ŷ 1G − nh N K , W K(K−1) , · · · , W K1 +n K j=2 h Ŷ jG − h (S 21G ) −n K j=3 h S j(j−1)G +nh W K(K−1) , W K(K−2) , · · · , W K1 N K = nh Ŷ 1G − nh (N K ) +n K j=2 h Ŷ jG − h (S 21G ) −n K j=3 h S j(j−1)G = n K−1 j=1 h Ŷ jG − h S (j+1)jG +nh Ŷ KG − nh (N K ) = n K j=1 I X jG ;Ŷ jG (e) is from the inequality h Ŷ n 1 ≤ nh Ŷ 1G . It holds because Gaussian random variable maximize entropy under given power constraint, and line 2 to line 6 in (e) is equivalent to line 2 and line 3 in (d). (f) is from the following equation: −h ( S n 21 \| X n 2 ) − K−1 j=3 j−1 p=1 h S n jp S n j(p−1) , · · · , S n j1 , X n (k) is from Lemma 16. (l) is from the formula h ( X\| Y ) = h (X) + h ( Y \| X) − h (Y ). (m) is from K j=2 h S j(j−1)G , S j(j−2)G , · · · , S j1G Ŷ jG = + K−1 j=2 h S j(j−1)G , S j(j−2)G , · · · , S j1G Ŷ jG +h S K(K−1)G , S K(K−2)G , · · · , S K1G Ŷ KG (n) Combining Lemma 7 and Lemma 8, we know that for channels in C α , if the power constraint P j satisfies P j ≤ ǫ ′′ α , then X jG →Ŷ jG → S j(61) form a Markov chain, and the following equality holds: h S j(j−1)G , S j(j−2)G , · · · , S j1G Ŷ jG = h S j(j−1)G , S j(j−2)G , · · · , S j1G Ŷ jG , X jG = h S (j+1)(j−1)G , S (j+1)(j−2)G , · · · , S (j+1)1G S (j+1)jG(62) We can conclude that the achievable sum capacity of the generalized Z-channel must satisfy K j=1 R j ≤ max Tr (V j ) ≤ SNR j V j 0, j = 1, · · · , K K j=1 I X jG ;Ŷ jG (63) = max Tr (V j ) ≤ SNR j V j 0, j = 1, · · · , K K j=1 log   I + K i=j \|C ji \| 2 V i     I + K i=j+1 \|C ji \| 2 V i   −1(64) Notice that for Z-channel, this sum capacity outer bound is achievable because the expression above is identical to the sum capacity achieved by treating interference as noise. Since the generalized Z-channel is obtained by eliminating some of the interference links from the interference channel, (63) is an outer bound for the sum capacity of the interference channel. Theorem 9 is proved. APPENDIX E PROOF OF THEOREM 10 In Theorem 9, we have proved that the sum capacity (31) of the generalized Z-channel is achieved by i.i.d. Gaussian input, K j=1 R j ≤ max Tr (V j ) ≤ P j V j 0, j = 1, · · · , K K j=1 I X jG ;Ŷ jG (65) = max Tr (V j ) ≤ P j V j 0, j = 1, · · · , K K j=1 log   I + K i=j \|C ji \| 2 V i     I + K i=j+1 \|C ji \| 2 V i   −1(66) Define the normalized covariance matrixV j = Vj Pj , Tr V j = 1. Consider the equal power constraint where P j = Psum /K for all users. For an expression of the form log \|I + xA\|, let the eigenvalue of matrix A be 0 ≤ λ i (A) < ∞. Then log \|I + xA\| = n i=1 log (1 + xλ i (A)) = n i=1 xλ i (A) − 1 2 x 2 λ 2 i (A) + o x 2 = xTr (A) − 1 2 x 2 Tr A 2 + o x 2(67) The second equation uses Taylor's theorem for several variables atλ i (A) = xλ i (A), since when x → 0, xλ i (A) → 0 as well. Combining (67), (4), (5) and (31), we find (33) and (34). APPENDIX F PROOF OF THEOREM 11 To maximize the right hand side of (34), we need to solve the following optimization problem minV 1 ,···,VK K j=1 \|C jj \| 4 Tr V 2 j +2 K−1 j=1 K i=j+1 \|C jj \| 2 \|C ji \| 2 Tr V j U jiVi U † ji (68) s.t. Tr V j = 1 V j 0.(69) First, consider a simple case where the channel is strictly symmetric: φ ji = 0, \|C jj \| 2 = 1 and \|C ji \| 2 = α < 1 for all i, j. Tr V 2 j + 2α K−1 j=1 K i=j+1 Tr V jVi (70) s.t. Tr V j = 1 V j 0.(71) Let the 2 × 2 real positive definite matrixV j beV j = k j1 k j3 k j3 k j2 .(72) Substituting (72) into (70), we construct a non-linear optimization problem from (70) on standard form: min k11,k12,k13,···,kK1,kK2,kK3 K j=1 k 2 j1 + k 2 j2 + 2k 2 j3 (73) +2α K−1 j=1 K i=j+1 (k j1 k i1 + k j2 k i2 + 2k j3 k i3 ) s.t. −k j1 ≤ 0 (74) −k j2 ≤ 0(75)k 2 j3 − k j1 k j2 ≤ 0 (76) k j1 + k j2 = 1(77) f or all j = 1, · · · , K The optimal solution of the problem defined by (73)~(77) is also the optimal solution of the problem defined by (70). Denote the optimization problem defined by (73)~(77) as P k , where k = (k 11 , k 12 , k 13 , · · · , k K1 , k K2 , k K3 ) represents the set of feasible solutions. Notice that while any positive k j1 , k j2 with k j1 +k j2 ≤ 1 satisfies the power constraint, we require constraint (77) to be an equality. Because only when it is satisfied with equality, the system can achieve correct E b N0 min0 . Denote the objective function in (73) by f (k). Construct the Lagrangian function for problem (73) as F (k, u 1 , u 2 , u 3 , v) = f (k) − K j=1 u j1 k j1 − K j=1 u j2 k j2 + K j=1 u j3 k 2 j3 − k j1 k j2 + K j=1 v j (k j1 + k j2 − 1) .(78) To find a optimal solution for this problem, we use Karush-Kuhn-Tucker (KKT) sufficient condition. It is stated as followed. Theorem 17. (KKT Sufficient Condition[10]) Consider an optimization problem (P ) defined as min x f (x) subject to g k (x) ≤ 0, k = 1, · · · , m h l (x) = 0, l = 1, · · · , n, with Lagrangian function L (x, u, v) = f (x) + g (x) T u + h (x) T v Let x be a feasible solution of (P ), and suppose (x, u, v) satisfy ∇ x L (x, u, v) = 0 u ≥ 0 u k g k (x) = 0 Then if f (x) is a pseudoconvex function, g k (x), k = 1, · · · , m are quasiconvex functions, and h l (x), l = 1, · · · , n are linear functions, then x is a global optimal solution. Given P k , it is clear that the objective function f (k) is a convex function, the equality constraints (77) are linear, and the sets of inequality constraints (74), (75), and (76) are convex. Notice that a convex function is a special case of pseudoconvex and quasiconvex. Comparing the standard problem (P ) in Theorem 17 with our optimization problem (P K ), we can conclude that any feasible k satisfying ∇ k F (k, u 1 , u 2 , u 3 , v) = 0 u 1 , u 2 and u 3 ≥ 0 u j1 k j1 = 0 u j2 k j2 = 0 u j3 k 2 j3 − k j1 k j2 = 0 is a global optimal for P k . Solving ∇ k F (k, u 1 , u 2 , u 3 , v) we have ∇F ∇k j1 = 2k j1 + 2α K i=1,i =j k i1 − u j1 − u j3 k j2 + v j = 0 ∇F ∇k j2 = 2k j2 + 2α K i=1,i =j k i2 − u j2 − u j3 k j1 + v j = 0 ∇F ∇k j3 = 4k j3 + 4α K i=1,i =j k i3 + 2u j3 k j3 = 0. It is easy to check that k j1 = k j2 = 1 2 , k j3 = 0 while the Lagrange multipliers u j1 = u j2 = u j3 = 0, and v j = −1−α (K − 1) satisfy KKT condition. Therefore, k j1 = k j2 = 1 2 , k j3 = 0, i.e.V xj = 1 2 0 0 1 2 is a global optimal solution. Substitute this optimal solution into the formula of sum slope (34), the sum slope has upper bound S 0 ≤ 2K αK + (1 − α) APPENDIX G PROOF OF COROLLARY 12 Before proving this result, we state existing results for general parametric optimization problems. A general parametric optimization problem P (t) depending on parameters t ∈ R r is defined by min f (x, t) subject to x ∈ R n g i (x, t) ≤ 0, i = 1, · · · , s g i (x, t) = 0, i = s + 1, · · · , m where f and g i are real functions. Denote the parametric feasible region by A (t) { x\| x ∈ R n ; g i (x, t) ≤ 0 if i = 1, · · · , s; g i (x, t) = 0 if i = s + 1, · · · , m} . And denote the parametric optimal value function by ν (t) inf x∈A(t) f (x, t). The following theorem gives the sufficient condition under which ν (t) is a continuous function of t. Theorem 18 (Theorem 3, p.70, [11]). Suppose that 1) the function f is continuous on x × t; 2) the correspondence A is continuous on t; 3) the subsets A (t) are non empty and compact Then the optimal value function ν (t) is continuous and the correspondence optimal solution set is upper semi-continuous. Let C correspond to t, and let the k as that defined in Appendix F correspond to x of Theorem 18. It is easy to see that the objective function of (68) is continuous on k × C, while the feasible region A (C) is non empty, compact, and independent of C. Therefore, all three conditions in Theorem 18 are satisfied and the optimal value function f (k, C) is continuous on C. Further, in Theorem 11 we have shown that when C o = C : φ ji = 0, \|C jj \| 2 = 1, \|C ji \| 2 = α , the optimal value of the objective function of the optimization problem P k (C o ) is f (k, C o ) = 2K αK + (1 − α) . Given the continuity of f (k, C o ) provided by Theorem 18 , for any σ, there exist σ 1 , σ 2 , σ 3 such that for the channels C ∈C σ , where the setC σ is defined asC σ = {C : \|φ ji \| < σ 1 \|C jj \| 2 − 1 < σ 2 \|C ji \| 2 − α < σ 3 Remark 3 . 3To make fair comparison of the wideband slopes between different systems, they must have equal E b N0 min in the first place. Fig. 2 2illustrates the empirical cumulative distribution functions of the sum slope achieved by the one-dimensional interference alignment scheme at different a values. For comparison, S 0 achieved by treating interference as noise are also shown, and TDMA always achievesS 0 = 2 for all a value. Figure 2 . 2Empirical cumulative distribution functions of S 0 achieved by treating interference as noise (TIN), interference alignment (INTA) and TDMA under different a values. Figure 3 . 3the median value of S 0 achieved by INTA, and the achievable S 0 of TIN and TDMA as a function of a V. CONCLUSION Lemma 15 ([ 8 , 158Lemma 4, p5037]). Let X, Y and Z be jointly Gaussian vectors.If cov (Y ) is invertible, then X → Y → Zforms a Markov chain if and only if cov (X, Z) = cov (X, Y ) cov (Y ) −1 cov (Y , Z) Given Lemma 15 and the fact that cov Ŷ jG is invertible, X jG →Ŷ jG → S j forms a Markov chain if and only if a) is from Fano's inequality. (b) is from the expansion of mutual information: I X n 1 ;Ŷ n 1 //ocw.mit.edu/courses/sloan-school-of-management/15-084j-nonlinear-programming-spring-2004/index.htm [11] J. Aubin, Mathematical methods of game and economic theory, ser. Dover books on mathematics. Dover Publications, 2007. [Online]., "15.084j nonlinear programming, spring 2004. (massachusetts institute of technology: Mit open- courseware), http://ocw.mit.edu (accessed 16 jul, 2012). license: Creative commons by-nc-sa." [Online]. Available: http:Available: http://books.google.com/books?id=COhFPgAACAAJ APPENDIX A PROOF OF LEMMA 5 Given (24), we have h S n (j−1)p S n (j−1)(p−1) , · · · , S n (j−1)1 , X n (j−1) .(g) is from Lemma 5. Because random variables S n (j−1)p S n (j−1)(p−1) , · · · , S n (j−1)1 , X n (j−1) and S n jp S n j(p−1) , · · · , S n j1 have the same marginal distribution, h S n (j−1)p S n (j−1)(p−1) , · · · , S n (j−1)1 , X n (j−1) and h S n jp S n j(p−1) , · · · , S n j1 are equal, which gives K j=3(h) Given S Kp = \|C pK \| U pK X K + W Kp , the summation in the third line after (g) givesIt is also easy to see that S n 21 andŶ n 1 X n 1 have same marginal distribution, therefore(i) Now combine the second and the last terms after (h):(h-1) is from Lemma 6. Given that random variablesŶ n j−1 S n (j−1)(j−2) , · · · , S n (j−1)1 , X n (j−1) and S n j(j−1) S n j(j−2) , · · · , S n j1 have the same marginal distribution, we have h S n j(j−1) S n j(j−2) , · · · , S n j1 = h Ŷ n (j−1) S n (j−1)(j−2) , · · · , S n (j−1)1 , X n (j−1) , (j) From chain rule of entropy, we know thatthe optimal value of the objective function of the optimization problem P k (C) satisfiesNotice that C α is defined in Theorem 9. Because 1 ∈ cl C σ , as α → 1, for any positive σ, there existsC σ , such that for C ∈C σ its sum slope satisfiesIf the magnitude and phase of the channel coefficients are drawn from continuous random distribution, P r C σ > 0.And as σ → 0, lim σ→0 ∆S 0 = 1 K The wideband slope of interference channels: The large bandwidth case. M Shen, A Høst-Madsen, IEEE Transactions on Information Theory. SubmittedM. Shen and A. Høst-Madsen, "The wideband slope of interference channels: The large bandwidth case," IEEE Transactions on Information Theory, Submitted, available at http://arxiv.org/abs/1010.5661. Spectral efficiency in the wideband regime. S Verdú, IEEE Transactions on Information Theory. 486S. Verdú, "Spectral efficiency in the wideband regime," IEEE Transactions on Information Theory, vol. 48, no. 6, pp. 1319-1343, 2002. Interference alignment and degrees of freedom of the k-user interference channel. V Cadambe, S Jafar, Information Theory, IEEE Transactions on. 54V. Cadambe and S. Jafar, "Interference alignment and degrees of freedom of the k-user interference channel," Information Theory, IEEE Transactions on, vol. 54, no. 8, pp. 3425 -3441, aug. 2008. Real interference alignment with real numbers. A S Motahari, S O Gharan, A K Khandani, IEEE Transactions on Information Theory. SubmittedA. S. Motahari, S. O. Gharan, and A. K. Khandani, "Real interference alignment with real numbers," IEEE Transactions on Information Theory, Submitted, available online at http://arxiv.org/abs/0908.1208. Degrees of freedom of the interference channel: A general formula. Y Wu, S Shamai, S Verdu, 2011 IEEE International Symposium on. Information Theory ProceedingsY. Wu, S. Shamai, and S. Verdu, "Degrees of freedom of the interference channel: A general formula," in Information Theory Proceedings (ISIT), 2011 IEEE International Symposium on, 31 2011-aug. 5 2011, pp. 1362 -1366. Interference alignment with asymmetric complex signaling -settling the Host-Madsen-Nosratinia conjecture. V R Cadambe, S A Jafar, C Wang, abs/0904.0274CoRR. V. R. Cadambe, S. A. Jafar, and C. Wang, "Interference alignment with asymmetric complex signaling -settling the Host-Madsen-Nosratinia conjecture," CoRR, vol. abs/0904.0274, 2009. Suboptimality of TDMA in the low-power regime. G Caire, D Tuninetti, S Verdú, IEEE Transactions on Information Theory. 504G. Caire, D. Tuninetti, and S. Verdú, "Suboptimality of TDMA in the low-power regime," IEEE Transactions on Information Theory, vol. 50, no. 4, pp. 608-620, 2004. Capacity regions and sum-rate capacities of vector gaussian interference channels. X Shang, B Chen, G Kramer, H Poor, IEEE Transactions on Information Theory. 5610X. Shang, B. Chen, G. Kramer, and H. Poor, "Capacity regions and sum-rate capacities of vector gaussian interference channels," IEEE Transactions on Information Theory, vol. 56, no. 10, pp. 5030 -5044, oct. 2010. Matrix analysis, ser. Graduate texts in mathematics. R Bhatia, Springer10] RR. Bhatia, Matrix analysis, ser. Graduate texts in mathematics. Springer, 1997. [Online]. Available: http://books.google.com/books?id=f0ioPwAACAAJ [10] R.	[]
[ "Multisymplectic Lagrangian and Hamiltonian Formalisms of Classical Field Theories", "Multisymplectic Lagrangian and Hamiltonian Formalisms of Classical Field Theories" ]	[ "Integrability Symmetry ", "Geometry " ]	[]	[ "Methods and Applications SIGMA" ]	This review paper is devoted to presenting the standard multisymplectic formulation for describing geometrically classical field theories, both the regular and singular cases. First, the main features of the Lagrangian formalism are revisited and, second, the Hamiltonian formalism is constructed using Hamiltonian sections. In both cases, the variational principles leading to the Euler-Lagrange and the Hamilton-De Donder-Weyl equations, respectively, are stated, and these field equations are given in different but equivalent geometrical ways in each formalism. Finally, both are unified in a new formulation (which has been developed in the last years), following the original ideas of Rusk and Skinner for mechanical systems.	10.3842/sigma.2009.100	[ "http://emis.u-strasbg.fr/journals/SIGMA/2009/100/sigma09-100.pdf", "http://www.emis.de/journals/SIGMA/2009/100/sigma09-100.pdf", "http://www.maths.tcd.ie/EMIS/journals/SIGMA/2009/100/sigma09-100.pdf" ]	11,341,628	math-ph/0506022	f3b26278eef0803f6811ce718c94a213b96c4951	Multisymplectic Lagrangian and Hamiltonian Formalisms of Classical Field Theories 2009 Integrability Symmetry Geometry Multisymplectic Lagrangian and Hamiltonian Formalisms of Classical Field Theories Methods and Applications SIGMA 525200910.3842/SIGMA.2009.100Received July 02, 2009, in final form October 30, 2009;classical field theoriesLagrangian and Hamiltonian formalismsfiber bundlesmultisymplectic manifolds 2000 Mathematics Subject Classification: 70S0555R1053C80 This review paper is devoted to presenting the standard multisymplectic formulation for describing geometrically classical field theories, both the regular and singular cases. First, the main features of the Lagrangian formalism are revisited and, second, the Hamiltonian formalism is constructed using Hamiltonian sections. In both cases, the variational principles leading to the Euler-Lagrange and the Hamilton-De Donder-Weyl equations, respectively, are stated, and these field equations are given in different but equivalent geometrical ways in each formalism. Finally, both are unified in a new formulation (which has been developed in the last years), following the original ideas of Rusk and Skinner for mechanical systems. Introduction In recent years much work has been done with the aim of establishing the suitable geometrical structures for describing classical field theories. There are different kinds of geometrical models for making a covariant description of classical field theories described by first-order Lagrangians. For instance, we have the so-called k-symplectic formalism which uses the k-symplectic forms introduced by Awane [4,5,6], and which coincides with the polysymplectic formalism described by Günther [46] (see also [84]). A natural extension of this is the k-cosymplectic formalism, which is the generalization to field theories of the cosymplectic description of non-autonomous mechanical systems [75,76]. Furthermore, there are the polysymplectic formalisms developed by Sardanashvily et al. [39,91] and Kanatchikov [52], which are based on the use of vector-valued forms on fiber bundles, and which are different descriptions of classical field theories than the polysymplectic one proposed by Günther. In addition, soldering forms on linear frame bundles are also polysymplectic forms, and their study and applications to field theory constitute the k-symplectic geometry developed by Norris [85,86,87]. There also exists the formalism based on using Lepagean forms, used for describing certain kinds of equivalent Lagrangian models with non-equivalent Hamiltonian descriptions [64,65,66,67]. Finally, a new geometrical framework for field theories based on the use of Lie algebroids has been developed in recent works [72,82,83]. In this work, we consider only the multisymplectic models [18,41,43,68,79], first introduced by Tulczyjew and other authors [37,40,60,61]. They arise from the study of multisymplectic manifolds and their properties (see [14,15] for general references, and Appendix A.1 for a brief review); in particular, those concerning the behavior of multisymplectic Lagrangian and Hamiltonian systems. The usual way of working with field theories consists in stating their Lagrangian formalism [3,11,17,26,27,37,39,40,92], and jet bundles are the appropriate domain for doing so. The construction of this formalism for regular and singular theories is reviewed in Section 2. The Hamiltonian description presents different kinds of problems. For instance, the choice of the multimomentum bundle for developing the theory is not unique [29,30], and different kinds of Hamiltonian systems can be defined, depending on this choice and on the way of introducing the physical content (the "Hamiltonian") [23,25,47,48,78,88]. Here we present one of the most standard ways of defining Hamiltonian systems, which is based on using Hamiltonian sections [16]; although this construction can also be done taking Hamiltonian densities [16,39,79,91]. In particular, the construction of Hamiltonian systems which are the Hamiltonian counterpart of Lagrangian systems is carried out by using the Legendre map associated with the Lagrangian system, and this problem has been studied by different authors in the (hyper) regular case [16,92], and in the singular (almost-regular) case [39,69,91]. In Section 3 we review some of these constructions. Another subject of interest in the geometrical description of classical field theories concerns the field equations. In the multisymplectic models, both in the Lagrangian and Hamiltonian formalisms, these equations can be derived from a suitable variational principle: the so-called Hamilton principle in the Lagrangian formalism and Hamilton-Jacobi principle in the Hamiltonian formulation [3,23,26,30,37,40], and the field equations are usually written by using the multisymplectic form in order to characterize the critical sections which are solutions of the problem. In addition, these critical sections can be thought of as being the integral manifolds of certain kinds of integrable multivector fields or Ehresmann connections, defined in the bundles where the formalism is developed, and satisfying a suitable geometric equation which is the intrinsic formulation of the systems of partial differential equations locally describing the field [26,27,28,69,92]. All these aspects are discussed in Sections 2 and 3 (furthermore, a quick review on multivector fields and connections is given in Appendix A.2). Moreover, multivector fields are also used in order to state generalized Poisson brackets in the Hamiltonian formalism of field theories [34,50,51,52,88]. In ordinary mechanics there is also a unified formulation of Lagrangian and Hamiltonian formalisms [94], which is based on the use of the Whitney sum of the tangent and cotangent bundles (the velocity and momentum phase spaces of the system). This formalism has been generalized for non-autonomous mechanics [7,20,45] and recently for classical field theories [24,71]. The main features of this formulation are explained in Section 4. Finally, an example showing the application of these formalisms is analyzed in Section 5. A last section is devoted to make a discussion about the current status on the research on different topics concerning the multisymplectic approach to classical field theories. We ought to point out that there are also geometric frameworks for describing the noncovariant or space-time formalism of field theories, where the use of Cauchy surfaces is the fundamental tool [42,44,74]. Nevertheless we do not consider these topics in this survey. As a review paper, this work recovers results and contributions from several previous papers, such as [16,24,26,27,28,30,47,69,71,88], among others. In this paper, manifolds are real, paracompact, connected and C ∞ , maps are C ∞ , and sum over crossed repeated indices is understood. 2 Lagrangian formalism Lagrangian systems A classical field theory is described by the following elements: First, we have the configuration fibre bundle π : E → M , with dim M = m and dim E = n + m, where M is an oriented manifold with volume form ω ∈ Ω m (M ). π 1 : J 1 π → E is the first-order jet bundle of local sections of π, which is also a bundle over M with projectionπ 1 = π • π 1 : J 1 π −→ M , and dim J 1 π = nm + n + m. We denote by (x ν , y A , v A ν ) (ν = 1, . . . , m; A = 1, . . . , n) natural coordinates in J 1 π adapted to the bundle structure and such that ω = dx 1 ∧ · · · ∧ dx m ≡ d m x. Second, we give the Lagrangian density, which is aπ 1 -semibasic m-form on J 1 π and hence it can be expressed as L = £(π 1 * ω), where £ ∈ C ∞ (J 1 π) is the Lagrangian function associated with L and ω. The bundle J 1 π is endowed with a canonical structure, V ∈ Ω 1 (J 1 π) ⊗ Γ(J 1 π, V(π 1 )) ⊗ Γ(J 1 π,π 1 * TM ), which is called the vertical endomorphism [26,37,40,92] (here V(π 1 ) denotes the vertical subbundle with respect to the projection π 1 , and Γ(J 1 π, V(π 1 )) the set of sections in the corresponding bundle). Then the Poincaré-Cartan m and (m + 1)-forms associated with L are defined as Θ L := i(V)L + L ∈ Ω m (J 1 π), Ω L := −dΘ L ∈ Ω m+1 (J 1 π). We have the following local expressions (where d m−1 x α ≡ i ∂ ∂x α d m x): Θ L = ∂£ ∂v A ν dy A ∧ d m−1 x ν − ∂£ ∂v A ν v A ν − £ d m x, Ω L = − ∂ 2 £ ∂v B ν ∂v A α dv B ν ∧ dy A ∧ d m−1 x α − ∂ 2 £ ∂y B ∂v A α dy B ∧ dy A ∧ d m−1 x α + ∂ 2 £ ∂v B ν ∂v A α v A α dv B ν ∧ d m x + ∂ 2 £ ∂y B ∂v A α v A α − ∂£ ∂y B + ∂ 2 £ ∂x α ∂v B α dy B ∧ d m x. (1) Definition 1. (J 1 π, Ω L ) is said to be a Lagrangian system. The Lagrangian system and the Lagrangian function are said to be regular if Ω L is a multisymplectic (m + 1)-form (i.e., 1-nondegenerate) [16,26]. Elsewhere they are singular (or non-regular). The regularity condition is locally equivalent to det ∂ 2 £ ∂v A α ∂v B ν (ȳ) = 0, ∀ȳ ∈ J 1 π. We must point out that, in field theories, the notion of regularity is not uniquely defined (for other approaches see, for instance, [9,21,22,64,66,67]). Lagrangian f ield equations The Lagrangian field equations can be derived from a variational principle. In fact: Definition 2. Let (J 1 π, Ω L ) be a Lagrangian system. Let Γ(M, E) be the set of sections of π. Consider the map L : Γ(M, E) −→ R, φ → M (j 1 φ) * Θ L , where the convergence of the integral is assumed. The variational problem for this Lagrangian system is the search of the critical (or stationary) sections of the functional L, with respect to the variations of φ given by φ t = σ t • φ, where {σ t } is a local one-parameter group of any compact-supported Z ∈ X V(π) (E) (the module of π-vertical vector fields in E), that is: d dt t=0 M j 1 φ t * Θ L = 0. This is the Hamilton principle of the Lagrangian formalism. j 1 Z ∈ X(J 1 π) the canonical lifting of Z to J 1 π; and, if Z ∈ X V(π) (E), then j 1 Z ∈ X V(π 1 ) (J 1 π) (see [26] for the details). Therefore d dt t=0 M (j 1 φ t ) * Θ L = d dt t=0 V (j 1 φ t ) * Θ L = d dt t=0 V [j 1 (σ t • φ)] * Θ L = d dt t=0 V (j 1 φ) * [(j 1 σ t ) * Θ L ] = V (j 1 φ) * lim t→0 (j 1 σ t ) * Θ L − Θ L t = V (j 1 φ) * L(j 1 Z)Θ L = V (j 1 φ) * [i(j 1 Z)dΘ L + d i(j 1 Z)Θ L ] = − V (j 1 φ) * [i(j 1 Z)Ω L − d i(j 1 Z)Θ L ] = − V (j 1 φ) * i(j 1 Z)Ω L + V d[(j 1 φ) * i(Z)ΘL] = − V (j 1 φ) * i(j 1 Z)Ω L + ∂V (j 1 φ) * i(j 1 Z)Θ L = − V (j 1 φ) * i(j 1 Z)Ω L , as a consequence of Stoke's theorem and the hypothesis made on the supports of the vertical fields. Thus, by the fundamental theorem of the variational calculus we conclude that d dt t=0 V (j 1 φ t ) * Θ L = 0 if, and only if, (j 1 φ) * i(j 1 Z)Ω L = 0, for every compact-supported Z ∈ X V(π) (E) . However, as compact-supported vector fields generate locally the C ∞ (E)-module of vector fields in E, it follows that the last equality holds for every Z ∈ X V(π) (E). Now, suppose φ ∈ Γ(M, E) is a critical section; that is, (j 1 φ) * i(j 1 Z)Ω L = 0, for every Z ∈ X V(π) (E) , and consider X ∈ X(J 1 E), which can be written as X = X φ + X v where X φ is tangent to the image of j 1 φ and X v isπ 1 -vertical, both in the points of the image of j 1 φ. However, X v = (X v −j 1 (π 1 * X v ))+j 1 (π 1 * X v ), where j 1 (π 1 * X v ) is understood as the prolongation of a vector field which coincides with π 1 * X v on the image of φ. Observe that π 1 * (X v −j 1 (π 1 * X v )) = 0 on the points of the image of j 1 φ. Therefore (j 1 φ) * i(X)ΩL = (j 1 φ) * i(Xφ)ΩL + (j 1 φ) * i(Xv − j 1 (π 1 * X v ))Ω L + (j 1 φ) * i(j 1 (π 1 * X v ))Ω L . However, (j 1 φ) * i(Xφ)ΩL = 0, because X φ is tangent to the image of j 1 φ, hence Ω L acts on linearly dependent vector fields. Nevertheless, (j 1 φ) * i(Xv − j 1 (π 1 * X v ))Ω L = 0, because X v − j 1 (π 1 * X v ) is π 1 -vertical and Ω L vanishes on these vector fields, when it is restricted to j 1 φ. Therefore, as φ is stationary and π 1 * X v ∈ X V(π) (E), we have M (j 1 φ) * i(X)ΩL = M (j 1 φ) * i(j 1 (π 1 * X v ))Ω L = 0. The converse is a consequence of the first paragraph, since the condition (j 1 φ) * i(X)ΩL = 0, ∀ X ∈ X(J 1 π), holds, in particular, for j 1 Z, for every Z ∈ X V(π) (E). (2 ⇔ 3) If X = α ν ∂ ∂x ν + β A ∂ ∂y A + γ A ν ∂ ∂v A ν ∈ X(J 1 π) , taking into account the local expression (1) of Ω L , we have i(X)ΩL = (−1) η α ν ∂ 2 £ ∂v B µ ∂v A η dv B µ ∧ dy A ∧ d m−2 x ην + ∂ 2 £ ∂y B ∂v A η dy B ∧ dy A ∧ d m−2 x ην − ∂ 2 £ ∂v B µ ∂v A η v A η dv B µ ∧ d m−1 x ν − ∂ 2 £ ∂y B ∂v A η v A η − ∂£ ∂y B + ∂ 2 £ ∂x η ∂v B η dy B ∧ d m−1 x ν + β A ∂ 2 £ ∂v B µ ∂v A η dv B µ ∧ d m−1 x η + ∂ 2 £ ∂y A ∂v B η − ∂ 2 £ ∂y B ∂v A η dy B ∧ d m−1 x η + ∂ 2 £ ∂y A ∂v B η v B η − ∂£ ∂y A + ∂ 2 £ ∂x η ∂v A η d m x + γ A ν − ∂ 2 £ ∂v A ν ∂v B η dy B ∧ d m−1 x η + ∂ 2 £ ∂v A ν ∂v B η v B η d m x but if φ = (x µ , y A (x η )), then j 1 φ = (x µ , y A (x η ), v A (x η )) = x µ , y A (x η ), ∂y A ∂x µ (x η ) , and hence (j 1 φ) * i(X)ΩL = (−1) η+ν α η ∂ ∂x µ ∂£ ∂v A µ • j 1 φ − ∂£ ∂y A • j 1 φ ∂(y A • φ) ∂x η d m x + β A ∂ ∂x µ ∂£ ∂v A µ • j 1 φ − ∂£ ∂y A • j 1 φ d m x, and, as this holds for every X ∈ X(J 1 π), we conclude that (j 1 φ) * i(X)ΩL = 0 if, and only if, the Euler-Lagrange equations (4) hold for φ. (3 ⇔ 4) Using the local expressions (1) of Ω h and (3) for X L , and taking f = 1 as a representative of the class {X L }, from the equation (5), we obtain that 0 = F B µ − v B µ ∂ 2 £ ∂v A ν ∂v B µ ,(7)0 = ∂£ ∂y A − ∂ 2 £ ∂x µ ∂v A µ − ∂ 2 £ ∂y B ∂v A µ F B µ − ∂ 2 £ ∂v B ν ∂v A µ G B νµ + ∂ 2 £ ∂y A ∂v B µ F B µ − v B µ ,(8) but, if X L is holonomic, it is semiholonomic and then F B µ = v B µ . Therefore the equations (7) are identities, and the equations (8) are 0 = ∂£ ∂y A − ∂ 2 £ ∂x µ ∂v A µ − ∂ 2 £ ∂y B ∂v A µ v B µ − ∂ 2 £ ∂v B ν ∂v A µ G B νµ .(9) Now, for a section φ = (x µ , y A (x η )), if j 1 φ = x µ , y A (x η ), ∂y A ∂x µ (x η ) is an integral section of X L , then G A νµ = ∂ 2 y A ∂x ν ∂x µ , and the equations (9) are equivalent to the Euler-Lagrange equations for φ. (3 ⇔ 5) The proof is like in the above item: using the local expressions (1) of Ω L and (2) for ∇ L , we prove that the equation (6) holds for an integrable connection if, and only if, the Euler-Lagrange equations (4) hold for its integral sections. Semi-holonomic (but not necessarily integrable) locally decomposable multivector fields and connections which are solution to the Lagrangian equations (5) and (6) respectively are called Euler-Lagrange multivector fields and connections for (J 1 π, Ω L ). If (J 1 π, Ω L ) is regular, Euler-Lagrange m-multivector fields and connections exist in J 1 π, although they are not necessarily integrable. If (J 1 π, Ω L ) is singular, in the most favourable cases, Euler-Lagrange multivector fields and connections only exist in some submanifold S → J 1 π, which can be obtained after applying a suitable constraint algorithm (see [70]). Hamiltonian formalism Multimomentum bundles. Legendre maps As we have pointed out in the introduction, the construction of the Hamiltonian formalism of field theories is more involved than the Lagrangian formulation. In fact, there are different bundles where the Hamiltonian formalism can be developed (see, for instance, [29], and references therein). Here we take one of the most standard choices. First, Mπ ≡ Λ m 2 T * E, is the bundle of m-forms on E vanishing by the action of two π-vertical vector fields (so dim Mπ = nm + n + m + 1), and is diffeomorphic to the set Aff(J 1 π, Λ m T * M ), made of the affine maps from J 1 π to Λ m T * M (the multicotangent bundle of M of order m [15]) [16,30]. It is called the extended multimomentum bundle, and its canonical submersions are denoted κ : Mπ → E;κ = π • κ : Mπ → M. As Mπ is a subbundle of Λ m T * E, then Mπ is endowed with a canonical form Θ ∈ Ω m (Mπ) (the "tautological form"), which is defined as follows: let (x, α) ∈ Λ m 2 T * E, with x ∈ E and α ∈ Λ m 2 T * x E; then, for every X 1 , . . . , X m ∈ T (x,α) (Mπ), Θ((x, α); X 1 , . . . , X m ) := α(x; T (x,α) κ(X 1 ), . . . , T (q,α) κ(X m )). Then we define the multisymplectic form Ω := −dΘ ∈ Ω m+1 (Mπ). They are known as the multimomentum Liouville m and (m + 1)-forms If we introduce natural coordinates (x ν , y A , p ν A , p) in Mπ adapted to the bundle π : E → M , and such that ω = d m x, the local expressions of these forms are Θ = p ν A dy A ∧ d m−1 x ν + pd m x, Ω = −dp ν A ∧ dy A ∧ d m−1 x ν − dp ∧ d m x. Now we denote by J 1 π * the quotient Mπ/π * Λ m T * M , with dim J 1 π * = nm + n + m. We have the natural submersions τ : J 1 π * → E;τ = π • τ : J 1 π * → M. Furthermore, the natural submersion µ : Mπ → J 1 π * endows Mπ with the structure of an affine bundle over J 1 π * , with (π • τ ) * Λ m T * M as the associated vector bundle. J 1 π * is usually called the restricted multimomentum bundle associated with the bundle π : E → M . Natural coordinates in J 1 π * (adapted to the bundle π : E → M ) are denoted by (x ν , y A , p ν A ). Definition 3. Let (J 1 π, Ω L ) be a Lagrangian system. The extended Legendre map associated with L, FL : J 1 π → Mπ, is defined by ( FL(ȳ))(Z 1 , . . . , Z m ) := (Θ L )ȳ(Z 1 , . . . ,Z m ), where Z 1 , . . . , Z m ∈ T π 1 (ȳ) E, andZ 1 , . . . ,Z m ∈ TȳJ 1 π are such that Tȳπ 1Z α = Z α . The restricted Legendre map associated with L is FL := µ • FL : J 1 π → J 1 π * . In natural coordinates we have: FL * x α = x α , FL * y A = y A , FL * p α A = ∂£ ∂v A α , FL * p = £ − v A α ∂£ ∂v A α , FL * x α = x α , FL * y A = y A , FL * p α A = ∂£ ∂v A α . Then, observe that FL * Θ = Θ L , and FL * Ω = Ω L . Definition 4. (J 1 π, Ω L ) is regular (hyper-regular) if FL is a local (global) diffeomorphism. Elsewhere it is singular. (This definition is equivalent to that given above.) (J 1 π, Ω L ) is almost-regular if 1. P := FL(J 1 π) is a closed submanifold of J 1 π * (natural embedding  0 : P → J 1 π * ). 2. FL is a submersion onto its image. 3. The fibres FL −1 (FL(ȳ)), ∀ȳ ∈ J 1 π, are connected submanifolds of J 1 π. The (hyper)regular case In the Hamiltonian formalism of field theories, there are different ways of introducing the physical information (the "Hamiltonian"). For instance, we can use connections in the multimomentum bundles in order to obtain a covariant definition of the so-called Hamiltonian densities (see, for instance, [16,39,79,91]). Nevertheless, the simplest way of defining (regular) Hamiltonian systems in field theory consists in considering the bundleτ : J 1 π * → M and then giving sections h : J 1 π * → Mπ of the projection µ, which are called Hamiltonian sections and carry the physical information of the system. Then we can define the differentiable forms Θ h := h * Θ ∈ Ω m (J 1 π * ), Ω h := −dΘ h = h * Ω ∈ Ω m+1 (J 1 π * ) which are the Hamilton-Cartan m and (m + 1) forms of J 1 π * associated with the Hamiltonian section h. The couple (J 1 π * , Ω h ) is said to be a Hamiltonian system. In a local chart of natural coordinates, a Hamiltonian section is specified by a local Hamilto- nian function h ∈ C ∞ (U ), U ⊂ J 1 π * , such that h(x ν , y A , p ν A ) ≡ (x ν , y A , p ν A , p = −h(x γ , y B , p η B ) ). Then, the local expressions of the Hamilton-Cartan forms associated with h are Θ h = p ν A dy A ∧ d m−1 x ν − hd m x, Ω h = −dp ν A ∧ dy A ∧ d m−1 x ν + dh ∧ d m x.(10) Notice that Ω h is 1-nondegenerate; that is, a multisymplectic form (as a simple calculation in coordinates shows). Now we want to associate Hamiltonian systems to the Lagrangian ones. First we consider the hyper-regular case (the regular case is analogous, but working locally). If (J 1 π, Ω L ) is a hyper-regular Lagrangian system, then we have the diagram J 1 π FL FL¨¨¨¨B E J 1 π * Mπ c µ T h It is proved [16] thatP := FL(J 1 π) is a 1-codimensional imbedded submanifold of Mπ ( 0 :P → Mπ denotes is the natural embedding), which is transverse to µ, and is diffeomorphic to J 1 π * . This diffeomorphism is µ −1 , when µ is restricted toP, and also coincides with the map h := FL • FL −1 , when it is restricted onto its image (which is justP). Thus h and (J 1 π * , Ω h ) are the Hamiltonian section and the Hamiltonian system associated with the hyper-regular Lagrangian system (J 1 π, Ω L ), respectively. Locally, the Hamiltonian section h( x ν , y A , p ν A ) = (x ν , y A , p ν A , p = −h(x γ , y B , p γ B )) is specified by the local Hamiltonian function h = p ν A (FL −1 ) * v A ν − (FL −1 ) * £. Then we have the local expressions (10) for the corresponding Hamilton-Cartan forms and, of course, FL * Θ h = Θ L , and FL * Ω h = Ω L . The Hamiltonian field equations can also be derived from a variational principle. In fact: Definition 5. Let (J 1 π * , Ω h ) be a Hamiltonian system. Let Γ(M, J 1 π * ) be the set of sections ofτ . Consider the map H : Γ(M, J 1 π * ) −→ R, ψ → M ψ * Θ h , where the convergence of the integral is assumed. The variational problem for this Hamiltonian system is the search for the critical (or stationary) sections of the functional H, with respect to the variations of ψ given by ψ t = σ t • ψ, where {σ t } is the local one-parameter group of any compact-supported Z ∈ X V(τ ) (J 1 π * ) ( the module ofτ -vertical vector fields in J 1 π * ), that is: d dt t=0 M ψ * t Θ h = 0. This is the so-called Hamilton-Jacobi principle of the Hamiltonian formalism. The Hamilton-Jacobi principle is equivalent to find distributions D of J 1 π * such that: 1. D is m-dimensional. 2. D isτ -transverse. 3. D is integrable (that is, involutive). 4. The integral manifolds of D are the critical sections of the Hamilton-Jacobi principle. As in the Lagrangian formalism, D are associated with classes of integrable andτ -transverse m-multivector fields {X } ⊂ X m (J 1 π * ) or, what is equivalent, with connections in the bundlē π : J 1 π → M , whose expressions are X = m ν=1 f ∂ ∂x ν + F A ν ∂ ∂y A + G ρ Aν ∂ ∂p ρ A , (f ∈ C ∞ (J 1 π * ) non-vanishing),(11)∇ = dx µ ⊗ ∂ ∂x µ + F A µ ∂ ∂y A + G ρ Aµ ∂ ∂p ρ A .(12) Then we have: Theorem 2. The following assertions on a section ψ ∈ Γ(M, J 1 π * ) are equivalent: 1. ψ is a critical section for the variational problem posed by the Hamilton-Jacobi principle. 2. ψ * i(X)Ωh = 0, ∀ X ∈ X(J 1 π * ). 3. If (U ; x ν , y A , p ν A ) is a natural system of coordinates in J 1 π * , then ψ satisfies the Hamilton-De Donder-Weyl equations in U ∂(y A • ψ) ∂x ν = ∂h ∂p ν A • ψ, ∂(p ν A • ψ) ∂x ν = − ∂h ∂y A • ψ.(13) 4. ψ is an integral section of a class of integrable andτ -transverse multivector fields {X h } ⊂ X m (J 1 π * ) satisfying that i(Xh)Ωh = 0, ∀ X h ∈ {X h }.(14) 5. ψ is an integral section of an integrable connection ∇ h in J 1 π * satisfying the equation i(∇h)Ωh = (m − 1)Ω h .(15) Proof . This proof is taken from [23,28], and [30]. (1 ⇔ 2) Let Z ∈ X V(τ ) (J 1 π * ) be a compact-supported vector field, and V ⊂ M an open set such that ∂V is a (m − 1)-dimensional manifold and thatτ (supp (Z)) ⊂ V . Then d dt t=0 M ψ * t Θ h = d dt t=0 V ψ * t Θ h = d dt t=0 V ψ * (σ * t Θ h ) = V ψ * lim t→0 σ * t Θ h − Θ h t = V ψ * L(Z)Θh = V ψ * (i(Z)dΘ h + d i(Z)Θh) = − V ψ * (i(Z)Ω h − d i(Z)Θh) = − V ψ * i(Z)Ωh + V d[ψ * i(Z)Θh] = − V ψ * i(Z)Ωh + ∂V ψ * i(Z)Θh = − V ψ * i(Z)Ωh, as a consequence of Stoke's theorem and the hypothesis made on the supports of the vertical fields. Thus, by the fundamental theorem of the variational calculus we conclude that d dt t=0 V ψ * t Θ h = 0 if, and only if, ψ * i(Z)Ωh = 0, for every compact-supported Z ∈ X V(τ ) (J 1 π * ). However, as compact-supported vector fields generate locally the C ∞ (J 1 π * )-module of vector fields in J 1 π * , it follows that the last equality holds for every Z ∈ X V(τ ) (J 1 π * ). Now, if p ∈ Im ψ, then T p J 1 π * = V p (τ ) ⊕ T p (Im ψ). So if X ∈ X(J 1 π * ), then X p = (X p − T p (ψ •τ )(X p )) + T p (ψ •τ )(X p ) ≡ X V p + X ψ p , and therefore ψ * i(X)Ωh = ψ * i(X V )Ω h + ψ * i(X ψ )Ω h = ψ * i(X ψ )Ω h = 0, since ψ * i(X V )Ω h = 0, by the conclusion in the above paragraph. Furthermore, X ψ p ∈ T p (Im ψ), and dim (Im ψ) = m, being Ω h ∈ Ω m+1 (J 1 π * ). Hence we conclude that ψ * i(X)Ωh = 0, for every X ∈ X(J 1 π * ). The converse is obvious taking into account the reasoning of the first paragraph, since the condition ψ * i(X)Ωh = 0, ∀ X ∈ X(J 1 π * ), holds, in particular, for every Z ∈ X V(τ ) (J 1 π * ). (2 ⇔ 3) If X = α ν ∂ ∂x ν + β A ∂ ∂y A + γ ν A ∂ ∂p ν A ∈ X(J 1 π * ) , taking into account the local expression (10) of Ω h , we have i(X)Ωh = (−1) η α η dp ν A ∧ dy A ∧ d m−2 x ην − ∂h ∂p ν A dp ν A ∧ d m−1 x η + β A dp ν A ∧ d m−1 x ν + ∂h ∂y A d m x + γ ν A −dy A ∧ d m−1 x ν + ∂h ∂p ν A ∧ d m x but if ψ = (x ν , y A (x η ), p ν A (x η )), then ψ * i(X)Ωh = (−1) η+ν α η ∂(y A • ψ) ∂x ν − ∂h ∂p ν A ψ ∂(p ν A • ψ) ∂x η d m x + β A ∂(p ν A • ψ) ∂x ν + ∂h ∂y A ψ d m x + γ ν A − ∂(y A • ψ) ∂x ν + ∂h ∂p ν A ψ d m x, and, as this holds for every X ∈ X(J 1 π * ), we conclude that ψ * i(X)Ωh = 0 if, and only if, the Hamilton-De Donder-Weyl equations (13) hold for ψ. (3 ⇔ 4) Using the local expressions (10) of Ω h and (11) for X h , and taking f = 1 as a representative of the class {X h }, the equation (14), in coordinates, is F A ν = ∂h ∂p ν A , G ν Aν = − ∂h ∂y A . This result allows us to assure the local existence of (classes of) multivector fields satisfying the desired conditions. The corresponding global solutions are then obtained using a partition of unity subordinated to a covering of J 1 π * made of local natural charts. Now, if ψ(x) = (x ν , y A (x γ ), p ν A (x γ )) is an integral section of X h , then ∂(y A • ψ) ∂x ν = F A ν • ψ, ∂(p ρ A • ψ) ∂x ν = G ρ Aν • ψ. Thus, combining both expressions we obtain the Hamilton-De Donder-Weyl equations (13) for ψ. (3 ⇔ 5) The proof is like in the above item: using the local expressions (10) of Ω h and (12) for ∇ h , we prove that the equation (15) holds for an integrable connection if, and only if, the Hamilton-De Donder-Weyl equations (13) hold for its integral sections. Theτ -transverse locally decomposable multivector fields and connections which are solution to the Hamiltonian equations (14) and (15) respectively (but not necessarily integrable) are called Hamilton-De Donder-Weyl multivector fields and connections for (J 1 π * , Ω h ). Hence, the existence of Hamilton-De Donder-Weyl multivector fields and connections for (J 1 π * , Ω h ) is assured, although they are not necessarily integrable. Finally, we can establish the equivalence between the Lagrangian and Hamiltonian formalisms in the hyper-regular case: Proof . This proof is taken from [28] and [30]. Bearing in mind the diagram J 1 π FL E J 1 π * π 1 τ j 1 φ ψ π φ E M s C t t t t t t t t t 0 T c if φ is a solution to the Lagrangian variational problem then (j 1 φ) * i(X)ΩL = 0, for every X ∈ X(J 1 π) (Theorem 1, item 2); therefore, as FL is a local diffeomorphism, 0 = (j 1 φ) * i(X)ΩL = (j 1 φ) * i(X)(FL * Ω h ) = (j 1 φ) * FL * (i(FL −1 * X)Ω h ) = (FL • j 1 φ) * i(X )Ω h ), which holds for every X ∈ X(J 1 π * ) and thus, by the item 2 of Theorem 2, ψ ≡ FL • j 1 φ is a solution to the Hamiltonian variational problem. Conversely, let ψ ∈ Γ(M, J 1 π * ) be a solution to the Hamiltonian variational problem. Reversing the above reasoning we obtain that (FL −1 • ψ) * i(X)ΩL = 0, for every X ∈ X(J 1 π), and hence σ ≡ FL −1 • ψ ∈ Γ(M, J 1 E) is a critical section for the Lagrangian variational problem. Then, as we are in the hyper-regular case, σ must be an holonomic section, σ = j 1 φ [27,69,92], and since the above diagram is commutative, φ = τ 1 • ψ ∈ Γ(M, E). The equivalence between the Lagrangian and the Hamiltonian formalisms can be stated also in terms of multivector fields and connections (see [28]). The almost-regular case Now, consider the almost-regular case. LetP := FL(J 1 π), P := FL(J 1 π) (the natural projections are denoted by τ 1 0 : P → E andτ 1 0 := π • τ 1 0 : P → M ), and assume that P is a fibre bundle over E and M . Denote by 0 :P → Mπ the natural imbedding, and by FL 0 and FL 0 the restrictions of FL and FL to their images, respectively. So, we have the diagram J 1 π FL 0 FL 0¨¨¨¨B E P P T c h Pμ E E  0  0 J 1 π * Mπ c µ d d d d τ 0τ © M Now, it can be proved that the µ-transverse submanifoldP is diffeomorphic to P [69]. This diffeomorphism is denotedμ :P → P, and it is just the restriction of the projection µ toP. Then, taking h P :=μ −1 , we define the Hamilton-Cartan forms Θ 0 h = ( 0 • h P ) * Θ ∈ Ω m (P), Ω 0 h = −dΘ 0 h ( 0 • h P ) * Ω ∈ Ω m+1 (P), which verify that FL * 0 Ω 0 h = Ω L . Then h P is also called a Hamiltonian section, and (P, Ω 0 h ) is the Hamiltonian system associated with the almost-regular Lagrangian system (J 1 π, Ω L ). In general, Ω 0 h is a pre-multisymplectic form and (P, Ω 0 h ) is the Hamiltonian system associated with the almost-regular Lagrangian system (J 1 π, Ω L ). In this framework, the Hamilton-Jacobi principle for (P, Ω 0 h ) is stated like above, and the critical sections ψ 0 ∈ Γ(M, P) can be characterized in an analogous way than in Theorem 2. If Ω 0 h is a pre-multisymplectic form, Hamilton-De Donder-Weyl multivector vector fields and connections only exist, in the most favourable cases, in some submanifold S → J 1 π, and they are not necessarily integrable. As in the Lagrangian case, S can be obtained after applying the suitable constraint algorithm [70]. Then, the equivalence theorem follows in an analogous way than above. It is important to point out that the analysis of the Hamiltonian description of non-regular field theories is far to be completed and, in fact, there is a lot of topics under discussion. For instance, there are some kinds of singular Lagrangian systems for which the construction of the associated Hamiltonian formalism (following the procedure that we have presented here) is ambiguous and, in order to overcome this trouble, a different notion of regularity must be done, which involve the use of Lepagean forms [64,66,67]. Neverthelees, the analysis of this and other problems exceeds the scope of this work. Unif ied Lagrangian-Hamiltonian formalism 4.1 Geometric framework The extended and the restricted jet-multimomentum bundles are W := J 1 π × E Mπ, W r := J 1 π × E J 1 π * , with natural coordinates (x α , y A , v A α , p α A , p) and (x α , y A , v A α , p α A ). We have natural projections (submersions) µ W : W → W r , and ρ 1 : W → J 1 π, ρ 2 : W → Mπ, ρ E : W → E, ρ M : W → M, ρ r 1 : W r → J 1 π, ρ r 2 : W r → J 1 π * , ρ r E : W r → E, ρ r M : W r → M.(16) Definition 6. The coupling m-form in W, denoted by C, is an m-form along ρ M which is defined as follows: for everyȳ ∈ J 1 y E, withπ 1 (ȳ) = π(y) = x ∈ E, and p ∈ M y π, let w ≡ (ȳ, p) ∈ W y , then C(w) := (T x φ) * p, where φ : M → E satisfies that j 1 φ(x) =ȳ. Then, we denote byĈ ∈ Ω m (W) the ρ M -semibasic form associated with C. The canonical m-form Θ W ∈ Ω m (W) is defined as Θ W := ρ * 2 Θ, and is ρ E -semibasic. The canonical (m + 1)-form is the pre-multisymplectic form Ω W : = −dΘ W = ρ * 1 Ω ∈ Ω m+1 (W). There existsĈ ∈ C ∞ (W) such thatĈ =Ĉ(ρ * M ω), andĈ(w) = (p + p α A v A α )d m x. Local expressions of Θ W and Ω W are the same than for Θ and Ω. LetL := ρ * 1 L ∈ Ω m (W), andL =L(ρ * M ω), withL = ρ * 1 L ∈ C ∞ (W) . We define the Hamiltonian submanifold  0 : W 0 → W by W 0 := {w ∈ W \|L(w) =Ĉ(w)}. The constraint function defining W 0 iŝ C −L = p + p α A v A α −L x ν , y B , v B ν = 0. There are projections which are the restrictions to W 0 of the projections (16), as it is shown in the following diagram: J 1 π ρ 0 1 Q ρ 1 T k ρ r 1 W 0  0 E W µ W E W r ρ 0 2 ρ 2 ρ r 2 ρ 0 2ρ r 2 µ Mπ J 1 π * c s C t t t t t t t t t ) c (x α , y A , v A α , p α A ) are local coordinates in W 0 , and ρ 0 1 (x α , y A , v A α , p α A ) = (x α , y A , v A α ),  0 (x α , y A , v A α , p α A ) = (x α , y A , v A α , p α A , L − v A α p α A ), ρ 0 2 (x α , y A , v A α , p α A ) = (x α , y A , p α A ), ρ 0 2 (x α , y A , v A α , p α A ) = (x α , y A , p α A , L − v A α p α A ). It is proved that W 0 is a 1-codimensional µ W -transversal submanifold of W, diffeomorphic to W r . As a consequence, W 0 induces a Hamiltonian section of µ W ,ĥ : W r → W, which is locally specified by giving the local Hamiltonian functionĤ = −L + p α A v A α ; that is,ĥ(x α , y A , v A α , p α A ) = (x α , y A , v A α , p α A , −Ĥ).µ −1μ E Ë¨¨¨B  h J 1 π * Mπ c µ ' ' ρ r 2 ρ 2 W r W T h (For hyper-regular systems we haveP = Mπ and P = J 1 π * .) We define the forms Θ 0 :=  * 0 Θ W = ρ 0 * 2 Θ ∈ Ω m (W 0 ), and Ω 0 :=  * 0 Ω W = ρ 0 * 2 Ω ∈ Ω m+1 (W 0 ), whose local expressions are Θ 0 = (L − p α A v A α )d m x + p α A dy A ∧ d m−1 x α , Ω 0 = d(p α A v A α − L) ∧ d m x − dp α A ∧ dy A ∧ d m−1 x α , (W 0 , Ω 0 ) (equiv. (W r ,ĥ * Ω 0 ) ) is a pre-multisymplectic Hamiltonian system. Field equations A Lagrange-Hamilton problem consists in finding sections ψ 0 ∈ Γ(M, W 0 ) such that ψ * 0 i(Y0)Ω0 = 0, ∀ Y 0 ∈ X(W 0 ).(17) Taking Y 0 ∈ X V(ρ 0 2 ) (W 0 ) we get the first constraint submanifold  1 : W 1 → W 0 , W 1 = {(ȳ, p) ∈ W 0 \| i(V0)(Ω0) (ȳ,p) = 0, for every V 0 ∈ V(ρ 0 2 )}, and sections solution to (17) W 1 = {(ȳ, FL(ȳ)) ∈ W \|ȳ ∈ J 1 π} , and W 1 is diffeomorphic to J 1 π. Theorem 4. (see diagram (18)) If ψ 0 : M → W 0 is a section fulfilling equation (17), then ψ 0 = (ψ L , ψ H ) = (ψ L , FL • ψ L ), where ψ L = ρ 0 1 • ψ 0 , and: 1. ψ L is the canonical lift of the projected section φ = ρ 0 E • ψ 0 : M → E (that is, ψ L is a holonomic section). 2. ψ L = j 1 φ is a solution to the Lagrangian problem, and µ • ψ H = µ • FL • ψ L = FL • j 1 φ is a solution to the Hamiltonian problem. Conversely, for every section φ : M → E such that j 1 φ is a solution to the Lagrangian problem (and hence FL • j 1 φ is a solution to the Hamiltonian problem) we have that ψ 0 = (j 1 φ, FL • j 1 φ), is a solution to (17). W ρ 1 )  0 T d d d d d d ρ 2 W 0 Mπ ρ 0 1 C  1 T E ρ 0 2 J 1 π ρ 1 1 ' W 1 ρ 1 2 E J 1 π * Mπ π 1 ρ 1 E τ 1 ψ L = j 1 φ ψ H = FL • j 1 φ ψ 1 ψ 0 φ E M c s C t t t t t t t t t Q T T T (18) Proof . This proof is taken from [24]. See also [71]. 1. Taking ∂ ∂p α A as a local basis for the ρ 0 1 -vertical vector fields, and a section ψ 0 , we have i ∂ ∂p α A Ω 0 = v A α d m x − dy A ∧ d m−1 x α =⇒ 0 = ψ * 0 i ∂ ∂p α A Ω 0 = v A α (x) − ∂y A ∂x α d m x, and thus the holonomy condition appears naturally within the unified formalism, and it is not necessary to impose it by hand to ψ 0 . Thus we have that (17) take values in W 1 , we can identify them with sections ψ 1 : M → W 1 . These sections ψ 1 verify, in particular, that ψ * 1 i(Y1)Ω1 = 0 holds for every Y 1 ∈ X(W 1 ). Obviously ψ 0 =  1 • ψ 1 . Moreover, as W 1 is the graph of FL, denoting by ρ 1 ψ 0 = x α , y A , ∂y A ∂x α , ∂L ∂v A α , since ψ 0 takes values in W 1 , and hence it is of the form ψ 0 = (j 1 φ, FL • j 1 φ), for φ = (x α , y A ) = ρ 0 E • ψ 0 . 2. Since sections ψ 0 : M → W 0 solution to1 = ρ 0 1 •  1 : W 1 → J 1 π the diffeomorphism which identifies W 1 with J 1 π, if we define Ω 1 =  * 1 Ω 0 , we have that Ω 1 = ρ 1 * 1 Ω L . In fact; as (ρ 1 1 ) −1 (ȳ) = (ȳ, FL(ȳ)), for everyȳ ∈ J 1 π, then (ρ 2 0 •  1 • (ρ 1 1 ) −1 )(ȳ) = FL(ȳ) ∈ Mπ, and hence Ω L = ρ 2 0 •  1 • (ρ 1 1 ) −1 * Ω = ρ 1 1 −1 * •  * 1 • ρ 2 * 0 Ω = ρ 1 1 −1 * •  * 1 Ω 0 = ρ 1 1 −1 * Ω 1 . Now, let X ∈ X(J 1 π). We have (j 1 φ) * i(X)ΩL = (ρ 0 1 • ψ 0 ) * i(X)ΩL = (ρ 0 1 •  1 • ψ 1 ) * i(X)ΩL = (ρ 1 1 • ψ 1 ) * i(X)ΩL = ψ * 1 i((ρ 1 1 ) −1 * X)(ρ 1 * 1 Ω L ) = ψ * 1 i(Y1)Ω1 = ψ * 1 i(Y1)( * 1 Ω 0 ) = (ψ * 1 •  * 1 ) i(Y0)Ω0 = ψ * 0 i(Y0)Ω0,(19) where Y 0 ∈ X(W 0 ) is such that Y 0 =  1 * Y 1 . But as ψ * 0 i(Y0)Ω0 = 0, for every Y 0 ∈ X(W 0 ), then we conclude that (j 1 φ) * i(X)ΩL = 0, for every X ∈ X(J 1 π). Conversely, let j 1 φ : M → J 1 π such that (j 1 φ) * i(X)ΩL = 0, for every X ∈ X(J 1 π), and define ψ 0 : M → W 0 as ψ 0 := (j 1 φ, FL • j 1 φ) (observe that ψ 0 takes its values in W 1 ). Taking into account that, on the points of W 1 , every Y 0 ∈ X(W 0 ) splits into Y 0 = Y 1 0 + Y 2 0 , with Y 1 0 ∈ X(W 0 ) tangent to W 1 , and Y 2 0 ∈ X V(ρ 0 1 ) (W 0 ), we have that ψ * 0 i(Y0)Ω0 = ψ * 0 i(Y 1 0 )Ω 0 + ψ * 0 i(Y 2 0 )Ω 0 = 0, because for Y 1 0 , the same reasoning as in (19) leads to ψ * 0 i(Y 1 0 )Ω 0 = (j 1 φ) * i(X 1 0 )Ω L = 0, (where X 1 0 = (ρ 1 1 ) −1 * Y 1 0 ) and, as j 1 φ is a holonomic section for Y 2 0 , following also the same reasoning as in (19), a local calculus gives ψ * 0 i(Y 2 0 )Ω 0 = (j 1 φ) * f α A (x) v A α − ∂y A ∂x α d m x = 0. The result for the sections FL • j 1 φ is a direct consequence of the equivalence Theorem 3 between the Lagrangian and Hamiltonian formalisms. Thus, equation (17) gives equations of three different classes: 1. Algebraic equations, determining W 1 → W 0 , where the sections solution take their values. These are the primary Hamiltonian constraints, and generate, byρ 0 2 projection, the primary constraints of the Hamiltonian formalism for singular Lagrangians. 2. Differential equations, forcing the sections solution ψ 0 to be holonomic. The Euler-Lagrange equations. Field equations in the unified formalism can also be stated in terms of multivector fields and connections in W 0 . In fact, the problem of finding sections solution to (17) can be formulated equivalently as follows: finding a distribution D 0 of T(W 0 ) such that it is integrable (that is, involutive), m-dimensional, ρ 0 M -transverse, and the integral manifolds of D 0 are the sections solution to the above equations. (Note that we do not ask them to be lifting of π-sections; that is, the holonomic condition). This is equivalent to stating that the sections solution to this problem are the integral sections of one of the following equivalent elements: • A class of integrable and ρ 0 M -transverse m-multivector fields {X 0 } ⊂ X m (W 0 ) satisfying that i(X0)Ω0 = 0, for every X 0 ∈ {X 0 }. • An integrable connection ∇ 0 in ρ 0 M : W 0 → M such that i(∇0)Ω0 = (m − 1)Ω 0 . Locally decomposable and ρ 0 M -transverse multivector fields and orientable connections which are solutions of these equations are called Lagrange-Hamiltonian multivector fields and jet fields for (W 0 , Ω 0 ). Euler-Lagrange and Hamilton-De Donder-Weyl multivector fields can be recovered from these Lagrange-Hamiltonian multivector fields (see [24]). Example As an example of application of these formalisms we consider a classical system which has been taken from [24]: minimal surfaces (in R 3 ). Other examples of application of the multisymplectic formalism are explained in detail in [39,43,91] as well as in many other references (see, for instance, [16,25,26,27,28,30,69,71] and quoted references). Geometric elements. Lagrangian and Hamiltonian formalisms The problem consists in looking for mappings ϕ : U ⊂ R 2 → R such that their graphs have minimal area as sets of R 3 , and satisfy certain boundary conditions. For this model, we have that M = R 2 , E = R 2 × R, and J 1 π = π * T * R 2 ⊗ R = π * T * M = π * T * R 2 , Mπ = π * (TM × M E), J 1 π * = π * TM = π * TR 2 . The coordinates in J 1 π, J 1 π * and Mπ are denoted (x 1 , x 2 , y, v 1 , v 2 ), (x 1 , x 2 , y, p 1 , p 2 ), and (x 1 , x 2 , y, p 1 , p 2 , p) respectively. If ω = dx 1 ∧ dx 2 , the Lagrangian density is L = 1 + (v 1 ) 2 + (v 2 ) 2 1/2 dx 1 ∧ dx 2 ≡ £dx 1 ∧ dx 2 , and the Poincaré-Cartan forms are Θ L = v 1 £ dy ∧ dx 2 − v 2 £ dy ∧ dx 1 + £ 1 − v 1 £ 2 − v 2 £ 2 dx 1 ∧ dx 2 , Ω L = −d v 1 £ ∧ dy ∧ dx 2 + d v 2 £ ∧ dy ∧ dx 1 − d £ 1 − v 1 £ 2 − v 2 £ 2 ∧ dx 1 ∧ dx 2 . The Euler-Lagrange equation of the problem are 0 = ∂p 2 ∂x 2 + ∂p 1 ∂x 1 dx 1 ∧ dx 2 = ∂ ∂x 1 v 1 £ + ∂ ∂x 2 v 2 £ dx 1 ∧ dx 2 = 1 £ 3 1 + ∂y ∂x 1 2 ∂ 2 y ∂x 2 ∂x 2 + 1 + ∂y ∂x 2 2 ∂ 2 y ∂x 1 ∂x 1 − 2 ∂y ∂x 1 ∂y ∂x 2 ∂ 2 y ∂x 1 ∂x 2 dx 1 ∧ dx 2 ,(20) and the associated Euler-Lagrange m-vector fields and connections which are the solutions to the Lagrangian problem are X L = f ∂ ∂x 1 + v 1 ∂ ∂y + ∂v 1 ∂x 1 ∂ ∂v 1 + ∂v 2 ∂x 1 ∂ ∂v 2 ∧ ∂ ∂x 2 + v 2 ∂ ∂y + ∂v 1 ∂x 2 ∂ ∂v 1 + ∂v 2 ∂x 2 ∂ ∂v 2 , ∇ L = dx 1 ⊗ ∂ ∂x 1 + v 1 ∂ ∂y + ∂v 1 ∂x 1 ∂ ∂v 1 + ∂v 2 ∂x 1 ∂ ∂v 2 + dx 2 ⊗ ∂ ∂x 2 + v 2 ∂ ∂y + ∂v 1 ∂x 2 ∂ ∂v 1 + ∂v 2 ∂x 2 ∂ ∂v 2 . The Legendre maps are given by FL(x 1 , x 2 , y, v 1 , v 2 ) = x 1 , x 2 , y, v 1 £ , v 2 £ , FL(x 1 , x 2 , y, v 1 , v 2 ) = x 1 , x 2 , y, v 1 £ , v 2 £ , £ − (v 1 ) 2 £ − (v 2 ) 2 £ , and then L is hyperregular. The Hamiltonian function is h = −[1 − (p 1 ) 2 − (p 2 ) 2 ] 1/2 , and so the Hamilton-Cartan forms are Θ h = p 1 dy ∧ dx 2 − p 2 dy ∧ dx 1 − hdx 1 ∧ dx 2 , Ω h = −dp 1 ∧ dy ∧ dx 2 + dp 2 ∧ dy ∧ dx 1 + dh ∧ dx 1 ∧ dx 2 . The Hamilton-De Donder-Weyl equations of the problem are ∂y ∂x 1 = − p 1 h , ∂y ∂x 2 = − p 2 h , ∂p 1 ∂x 1 = − ∂p 2 ∂x 2 ,(21) and the corresponding Hamilton-De Donder-Weyl m-vector fields and connections which are the solutions to the Hamiltonian problem are X h = f ∂ ∂x 1 − p 1 h ∂ ∂y + ∂p 1 ∂x 1 ∂ ∂p 1 + ∂p 2 ∂x 1 ∂ ∂p 2 ∧ ∂ ∂x 2 − p 2 h ∂ ∂y + ∂p 1 ∂x 2 ∂ ∂p 1 + ∂p 2 ∂x 2 ∂ ∂p 2 , ∇ h = dx 1 ⊗ ∂ ∂x 1 − p 1 h ∂ ∂y + ∂p 1 ∂x 1 ∂ ∂p 1 + ∂p 2 ∂x 1 ∂ ∂p 2 + dx 2 ⊗ ∂ ∂x 2 − p 2 h ∂ ∂y + ∂p 1 ∂x 2 ∂ ∂p 1 + ∂p 2 ∂x 2 ∂ ∂p 2 . Unif ied formalism For the unified formalism we have W = π * T * M × E π * (TM × M E), W r = π * T * M × E π * TM = π * (T * M × M TM ). If w = (x 1 , x 2 , y, v 1 , v 2 , p 1 , p 2 , p) ∈ W, the coupling form isĈ = (p 1 v 1 + p 2 v 2 + p)dx 1 ∧ dx 2 ; therefore W 0 = (x 1 , x 2 , y, v 1 , v 2 , p 1 , p 2 , p) ∈ W \| [1 + (v 1 ) 2 + (v 2 ) 2 ] 1/2 − p 1 v 1 − p 2 v 2 − p = 0 , and we have the forms Θ 0 = 1 + (v 1 ) 2 + (v 2 ) 2 1/2 − p 1 v 1 − p 2 v 2 dx 1 ∧ dx 2 − p 2 dy ∧ dx 1 + p 1 dy ∧ dx 2 , Ω 0 = −d 1 + (v 1 ) 2 + (v 2 ) 2 1/2 − p 1 v 1 − p 2 v 2 ∧ dx 1 ∧ dx 2 + dp 2 ∧ dy ∧ dx 1 − dp 1 ∧ dy ∧ dx 2 . Taking firstρ 0 2 -vertical vector fields ∂ ∂vα we obtain 0 = i ∂ ∂v α Ω 0 = p α − v α £ dx 1 ∧ dx 2 , which determines the submanifold W 1 = graph FL (diffeomorphic to J 1 π), and reproduces the expression of the Legendre map. Now, taking ρ 0 1 -vertical vector fields ∂ ∂p α , the contraction i ∂ ∂p α Ω 0 gives, for α = 1, 2, v 1 dx 1 ∧ dx 2 − dy ∧ dx 2 and v 2 dx 1 ∧ dx 2 + dy ∧ dx 1 respectively, so that, for a section ψ 0 = (x 1 , x 2 , y(x 1 , x 2 ), v 1 (x 1 , x 2 ), v 2 (x 1 , x 2 ), p 1 (x 1 , x 2 ), p 2 (x 1 , x 2 )) taking values in W 1 , we have that the condition ψ * 0 i ∂ ∂p α Ω 0 = 0 leads to v 1 − ∂y ∂x 1 dx 1 ∧ dx 2 = 0, v 2 − ∂y ∂x 2 dx 1 ∧ dx 2 = 0, which are the holonomy condition. Finally, taking the vector field ∂ ∂y we have i ∂ ∂y Ω 0 = −dp 2 ∧ dx 1 + dp 1 ∧ dx 2 and, for a section ψ 0 fulfilling the former conditions, the equation 0 = ψ * 0 i ∂ ∂y Ω 0 leads to the Euler-Lagrange equations (20). Now, bearing in mind the expressions of h and the Legendre map, from the Euler-Lagrange equations we get the Hamilton-De Donder-Weyl equations (21). The m-vector fields and connections which are the solutions to the problem in the unified formalism are X 0 = f ∂ ∂x 1 + v 1 ∂ ∂y + ∂v 1 ∂x 1 ∂ ∂v 1 + ∂v 2 ∂x 1 ∂ ∂v 2 + ∂p 1 ∂x 1 ∂ ∂p 1 + ∂p 2 ∂x 1 ∂ ∂p 2 ∧ ∂ ∂x 2 + v 2 ∂ ∂y + ∂v 1 ∂x 2 ∂ ∂v 1 + ∂v 2 ∂x 2 ∂ ∂v 2 + ∂p 1 ∂x 2 ∂ ∂p 1 + ∂p 2 ∂x 2 ∂ ∂p 2 , ∇ 0 = dx 1 ⊗ ∂ ∂x 1 + v 1 ∂ ∂y + ∂v 1 ∂x 1 ∂ ∂v 1 + ∂v 2 ∂x 1 ∂ ∂v 2 + ∂p 1 ∂x 1 ∂ ∂p 1 + ∂p 2 ∂x 1 ∂ ∂p 2 + dx 2 ⊗ ∂ ∂x 2 + v 2 ∂ ∂y + ∂v 1 ∂x 2 ∂ ∂v 1 + ∂v 2 ∂x 2 ∂ ∂v 2 + ∂p 1 ∂x 2 ∂ ∂p 1 + ∂p 2 ∂x 2 ∂ ∂p 2 , (f being a non-vanishing function) where the coefficients ∂vα ∂x ν = ∂ 2 y ∂x ν ∂x α are related by the Euler-Lagrange equations, and the coefficients ∂p α ∂x ν are related by the Hamilton-De Donder-Weyl equations (the third one). From these expressions we recover the Euler-Lagrange m-vector fields and connections which are the solutions to the Lagrangian problem, and the Hamilton-De Donder-Weyl m-vector fields and connections which are the solutions to the Hamiltonian problem obtained in the above paragraph. Discussion and outlook Multisymplectic geometry and its application to describe classical field theories have been fields of increasing interest in the last years. A lot of well-known results in the realm of symplectic geometry and symplectic mechanics have been generalized also for the multisymplectic case, but there are many other problems which remain open. Next we review some of these results and problems, and their current status. A fundamental result in symplectic geometry is the Darboux theorem. The analogous result also holds in some particular cases of multisymplectic forms (for instance, for volume forms). Nevertheless, in the general case, a multisymplectic manifold does not admit a system of Darboux coordinates for the multisymplectic form. In fact this is a problem arising from linear algebra: the classification of skew-symmetric tensors of degree greater than two is still an open problem. The kind of multisymplectic manifolds admitting Darboux coordinates has been identified [73], and they are those being locally multisymplectomorphic to bundles of forms (see also [33] for another approach to this problem). Another interesting subject concerns to the definition of Poisson brackets in multisymplectic manifolds. This is a relevant point, for instance, for the further quantization of classical field theories. This problem has been studied in the realm of polysymplectic manifolds [50,51] and for the multisymplectic case some recent contributions are [34,35,36]. However, the problem is not completely solved satisfactorily, and the research on this topic is still open. In the same way, approaches for generalizing symplectic integrators to this geometric framework (i.e., the so-called multisymplectic integrators) have been studied in recent years, and numerical methods have been developed for solving the field equations, which are based on the use of these multisymplectic integrators [77,79]. Research on this topic is in progress. Another field of increasing interest in the last years is the study of systems in classical field theories with nonholonomic constraints. This is a meeting topic between honholonomic mechanics and classical field theories. The construction of the Lagrangian and Hamiltonian formalism, as well as other problems such as the study of symmetries and reduction have been analyzed for the k-symplectic formulation [72] and for the multisymplectic models in several works [10,95,96,97,98]. Further developments have not been achieved. For instance, the generalization of the Marsden-Weinstein reduction theorem [80] to the multisymplectic framework. Concerning reduction theory in general, only partial results about reduction by foliations are currently being studied [49]. The corresponding reduction theorem has been stated and proved for the k-symplectic formulation [84], but the theory of reduction of multisymplectic Lagrangian and Hamiltonian systems under the action of groups of symmetries is still under research, and only partial results have been achieved [17,18,19,81]. The problem of quantization of classical field theories is another relevant topic to be developed. There are several works due to Kanatchikov devoted to geometric (pre)quantization of polysymplectic field theories [53,54,55,56,57,58,59], some attempts for the k-symplectic case [12,89], and other different approaches for the quantization of fields, in general (see, for instance, [8,90]). Nevertheless, the study of the geometric structures and obstructions to perform the geometric quantization program for covariant multisymplectic field theories is open to further research. As a final remark, many of the subjects that we have presented in this work have been studied also for higher-order field theories (see, for instance, [1,2,31,32,38,62,63,92,93]). One of the problems of the first multisymplectic models for these theories was that the definition of the corresponding multisymplectic structure (the Poincaré-Cartan form) was ambiguous. This trouble have been solved recently [13]. But, in general, the problem of stating complete and satisfactory geometrical models for the Lagrangian and Hamiltonian formalisms of these kinds of theories, as well as other related topics (symmetries, constraint algorithms for the singular cases, quantization, . . . ) are under development. One can expect to see more work on all these subjects in the future. A Appendix A.1 Multisymplectic manifolds Definition 7. Let M be a differentiable manifold, and Ω ∈ Ω k (M) (1 < k ≤ dim M). Ω is a multisymplectic form, and then (M, Ω) is a multisymplectic manifold, if 1. Ω ∈ Z k (M) (it is closed). 2. Ω is 1-nondegenerate; that is, for every p ∈ M and X p ∈ T p M, i(Xp)Ωp = 0 ⇔ X p = 0. If Ω is closed and 1-degenerate then it is a pre-multisymplectic form, and (M, Ω) is a premultisymplectic manifold. Multisymplectic manifolds of degree k = 2 are the usual symplectic manifolds, and manifolds with a distinguished volume form are multisymplectic manifolds of degree its dimension. Other examples of multisymplectic manifolds are provided by compact semisimple Lie groups equipped with the canonical cohomology 3-class, symplectic 6-dimensional Calabi-Yau manifolds with the canonical 3-class, etc. There are no multisymplectic manifolds of degrees 1 or dim M−1 because ker Ω is nonvanishing in both cases. Another very important kind of multisymplectic manifold is the multicotangent bundle of a manifold Q, Λ k (T * Q), that is, the bundle of k-forms in Q. This bundle is endowed with a canonical k-form Θ ∈ Ω k (Λ k (T * Q), and then Ω := −dΘ ∈ Ω k+1 (Λ k (T * Q) is a 1-nondegenerate form. Then the couple (Λ k (T * Q), Ω) is a multisymplectic manifold. A local classification of multisymplectic forms can be done only for particular cases [73,33]. If X , X ∈ X m (M) are non-vanishing multivector fields locally associated with the same distribution D, on the same connected open set U , then there exists a non-vanishing function f ∈ C ∞ (U ) such that X \| U = f X . This fact defines an equivalence relation in the set of nonvanishing m-multivector fields in M, whose equivalence classes will be denoted by {X } U . Then there is a one-to-one correspondence between the m-dimensional orientable distributions D in TM and the equivalence classes {X } M of non-vanishing, locally decomposable m-multivector fields in M. A.2 Multivector f ields A non-vanishing, locally decomposable multivector field X ∈ X m (M) is said to be integrable (resp. involutive) if its associated distribution is integrable (resp. involutive). If X ∈ X m (M) is integrable (resp. involutive), then so is every other in its equivalence class {X }, and all of them have the same integral manifolds. Moreover, Frobenius theorem allows us to say that a non-vanishing and locally decomposable multivector field is integrable if, and only if, it is involutive. If π : M → M is a fiber bundle, we are interested in the case where the integral manifolds of integrable multivector fields in M are sections of π. Thus, X ∈ X m (M) is said to be πtransverse if, at every point y ∈ M, (i(X )(π * β)) y = 0, for every β ∈ Ω m (M ) with ω(π(y)) = 0. Then, if X ∈ X m (M) is integrable, it is π-transverse if, and only if, its integral manifolds are local sections of π : M → M . Finally, it is clear that classes of locally decomposable and πtransverse multivector fields {X } ⊆ X m (M) are in one-to-one correspondence with orientable Ehresmann connection forms ∇ in π : M → M . This correspondence is characterized by the fact that the horizontal subbundle associated with ∇ is the distribution associated with {X }. In this correspondence, classes of integrable locally decomposable and π-transverse m multivector fields correspond to flat orientable Ehresmann connections. Theorem 3 . 3(equivalence theorem for sections) Let (J 1 π, Ω L ) be a hyper-regular Lagrangian system, and (J 1 π * , Ω h ) the associated Hamiltonian system.If a section φ ∈ Γ(M, E) is a solution to the Lagrangian variational problem (Hamilton principle), then the section ψ = FL • j 1 φ ∈ Γ(M, J 1 π * ) is a solution to the Hamiltonian variational problem (Hamilton-Jacobi principle).Conversely, if ψ ∈ Γ(M, J 1 π * ) is a solution to the Hamiltonian variational problem, then the section φ = τ • ψ ∈ Γ(M, E) is a solution to the Lagrangian variational problem. Fromĥ we recover a Hamiltonian sectionh : P → Mπ defined byh([p]) = (ρ 2 •ĥ)[(ρ r 2 ) −1 (([p]))], ∀ [p] ∈ P. (See the diagram.) See [ 27 ] 27for details. Let M be a n-dimensional differentiable manifold. Sections of Λ m (TM) are called m-multivector fields in M (they are the contravariant skew-symmetric tensors of order m in M). We denote by X m (M) the set of m-multivector fields in M. Then, X ∈ X m (M) is locally decomposable if, for every p ∈ M, there is an open neighbourhood U p ⊂ M and X 1 , . . . , X m ∈ X(U p ) such that X \| Up = X 1 ∧ . . . ∧ X m . A non-vanishing X ∈ X m (M) and a m-dimensional distribution D ⊂ TM are locally associated if there exists a connected open set U ⊆ M such that X \| U is a section of Λ m D\| U . AcknowledgementsI acknowledge the financial support of Ministerio de Educación y Ciencia, projects MTM 2005-04947, MTM 2008-00689/MTM and MTM 2008-03606-E/MTM. I wish to thank to Professors Miguel C. Muñoz-Lecanda and Xavier Gràcia for their comments, and to the referees, whose suggestions have allowed me to improve this work. Finally, thanks also to Mr. Jeff Palmer for his assistance in preparing the English version of the manuscript. Variational principles on rth order jets of fibre bundles in field theory. V Aldaya, J A De Azcárraga, J. Math. 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[]	[ "Junjing Xing \nCollege of Mathematical Sciences\nCollege of Intelligent Systems Science and Engineering\nHarbin Engineering University\n150001HarbinChina\n\nHarbin Engineering Uni-versity\n150001HarbinChina\n" ]	[ "College of Mathematical Sciences\nCollege of Intelligent Systems Science and Engineering\nHarbin Engineering University\n150001HarbinChina", "Harbin Engineering Uni-versity\n150001HarbinChina" ]	[]	Using the skew-Hopf pairing, we obtain R-matrix for the two-parameter quantum algebra U v,t . We further construct a strict monoidal functor T from the tangle category (OTa, ⊗, ∅) to the category (Mod, ⊗, Q(v, t)) of U v,t -modules . As a consequence, the quantum knot invariant of the tangle L of type (n, n) is obtained by the action of T on the closureL of L.Date: March 22, 2022.	null	[ "https://arxiv.org/pdf/2203.10478v1.pdf" ]	247,594,655	2203.10478	16a13ca087d3fd5a6e68e719c89ab0c9e6819917	20 Mar 2022 Junjing Xing College of Mathematical Sciences College of Intelligent Systems Science and Engineering Harbin Engineering University 150001HarbinChina Harbin Engineering Uni-versity 150001HarbinChina 20 Mar 2022THE KNOT INVARIANT ASSOCIATED TO TWO-PARAMETER QUANTUM ALGEBRASand phrases quantum algebraknot invariantR-matrix Using the skew-Hopf pairing, we obtain R-matrix for the two-parameter quantum algebra U v,t . We further construct a strict monoidal functor T from the tangle category (OTa, ⊗, ∅) to the category (Mod, ⊗, Q(v, t)) of U v,t -modules . As a consequence, the quantum knot invariant of the tangle L of type (n, n) is obtained by the action of T on the closureL of L.Date: March 22, 2022. Introduction Knot theory is the mathematical discipline with the unusually diverse applications, such as statistical mechanics [K11], symbolic logic and set theory [K16], quantum field theory [W], quantum computing [NSS], etc. Reshetikhin and Turaev [RT] related quantum algebras to knot invariants, often referred to as quantum invariants. They generalized the Jones polynomial of links and the related Jones-Conway and Kauffman polynomials to the case of graphs in R 3 . Inspired by the result, a large number of researchers began to pay attention to quantum invariants, such as Zhang [ZGB], Kauffman [KM] and Clark [C]. In [FL], the first author and Li provided a noval presentation of the two-parameter quantum algebra U v,t (g) by a geometric approach, where both parameters v and t have geometric meaning. Moreover, this presentation unifies the various quantum algebras in literature. By various specialization, one can obtain one-parameter quantum algebras [L], two-parameter quantum algebras [BW], quantum superalgebras [CFYW] and multi-parameter quantum algebras [HPR]. It is a natural question that whether this two-parameter quantum algebra provide a new knot invariant. This is what we will explore in this paper. In [C], Clark gave a partial answer to this question by proving that the quantum knot invariants for osp(1\|2n) and so(2n + 1) are essentially the same. The solution of the Yang-Baxter equation is called R-matrix which connects quantum algebras to the knot invariant theory. In the construction of quantum knot invariants, invariance under the Reidemeister III move holds naturally by attaching a copy of the R-matrix to each crossing. The construction of R-matrices for various quantum groups is very meaningful and interesting. In [L,Chapter 32], Lusztig provided a framework to construct the R-matrix of one-parameter quantum algebras via the quasi-R-matrix. In [FX], we defined a skew-Hopf pairing on the deformed quantum algebra which unifies various quantum algebras in literatures, such as oneparameter quantum algebras [L], two-parameter quantum algebras [BW], quantum superalgebras [CFYW] and multi-parameter quantum algebras [HPR]. This provided a tool to construct R-matrix for quantum algebras other than one-parameter one. For instance, in [BW], Benkart and Witherspoon constructed the R-matrix for two-parameter quantum algebra U r,s (sl n ) by the Hopf pairing which can be viewed as a special case of that in [FX]. In this paper, we construct the R-matrix for the two-parameter quantum algebra U v,t by using skew-Hopf pairing. This recovers the constructions of R-matrix in [L] and [BW] under certain assumptions. We further provide the functor T : (OTa, ⊗, ∅) → (Mod, ⊗, Q(v, t)), where OTa and Mod are the categories of tangles and U v,t -modules, respectively. This produces the machinery and correspondence for the construction of quantum invariants via representations of two-parameter quantum algebra U v,t . Furthermore, given a tangle L of type (n, n), we can get an endomorphism T(L) of the ground field Q(v, t), whereL is the closure of L. This paper is organized as follows. In Section 2, we recall the definition of twoparameter quantum algebra U v,t from [FL] and formulate the quasi-R-matrix Θ of U v,t . Part of the results are new for two-parameter quantum groups. In Section 3, we construct the R-matrix of two-parameter quantum algebras U v,t . In Section 4, we construct the functor between the categories of tangles and the categories of U v,t -modules. Acknowledgements. Z. Fan is partially supported by the NSF of China grant 11671108, the NSF of Heilongjiang Province grant JQ2020A001 and the Fundamental Research Funds for the central universities. 2. The two-parameter quantum algebra U v,t We briefly review the definition of the two-parameter quantum algebra U v,t in [FL]. Given a Cartan datum (I, ·), let Ω = (Ω ij ) i,j∈I be an integer matrix satisfying that (a) Ω ii ∈ Z >0 , Ω ij ∈ Z ≤0 for all i = j ∈ I; (b) Ω ij +Ω ji Ω ii ∈ Z ≤0 for all i = j ∈ I; (c) the greatest common divisor of all Ω ii is equal to 1. To Ω, we associate the following three bilinear forms on Z[I]. i, j = Ω ij , ∀i, j ∈ I. i, j = 2δ ij Ω ii − Ω ij , ∀i, j ∈ I. i · j = i, j + j, i , ∀i, j ∈ I. 2.1. The free algebra ′ f. For indeterminates v and t, we set v i = v i·i/2 and t i = t i·i/2 . Denoted by v ν = i v ν i i , t ν = i t ν i i and tr(ν) = i∈I ν i ∈ N, for any ν = ν i i ∈ N[I]. Let ′ f be the free unital associative algebra over Q(v, t) generated by the symbols θ i , ∀i ∈ I. By setting the degree of the generator θ i to be i, the algebra ′ f becomes an N[I]-graded algebra. For any ν ∈ N[I], we denote by ′ f ν the subspace of all homogenous elements of degree ν. We have ′ f = ⊕ ν∈N[I] ′ f ν and denote by \|x\| the degree of a homogenous element x ∈ ′ f. 2.1.1. The tensor product ′ f ⊗ ′ f. On the tensor product ′ f ⊗ ′ f, we define an associative Q(v, t)-algebra structure by (x 1 ⊗ x 2 )(y 1 ⊗ y 2 ) = v \|y 1 \|·\|x 2 \| t \|y 1 \|,\|x 2 \| − \|x 2 \|,\|y 1 \| x 1 y 1 ⊗ x 2 y 2 , for homogeneous elements x 1 , x 2 , y 1 and y 2 in ′ f. It is associative since the forms , and " · " are bilinear. Let r : ′ f → ′ f ⊗ ′ f be the Q(v, t)-algebra homomorphism such that r(θ i ) = θ i ⊗ 1 + 1 ⊗ θ i , for all i ∈ I. Proposition 2.1. [FL,Proposition 13] There is a unique symmetric bilinear form (,) on ′ f with values in Q(v, t) such that (a) (1, 1) = 1; (b) (θ i , θ j ) = δ ij 1 1−v −2 i , for all i, j ∈ I; (c) (x, y ′ y ′′ ) = (r(x), y ′ ⊗ y ′′ ), for all x, y ′ , y ′′ ∈ ′ f; (d) (x ′ x ′′ , y) = (x ′ ⊗ x ′′ , r(y)), for all x ′ , x ′′ , y ∈ ′ f. Here the bilinear form on ′ f ⊗ ′ f is defined by (x 1 ⊗ x 2 , y 1 ⊗ y 2 ) = t 2[\|x 1 \|,\|x 2 \|] (x 1 , y 1 )(x 2 , y 2 ). 2.1.2. Let σ : ′ f → ′ f op be a twisted anti-involution such that σ(θ i ) = θ i , and σ(xy) = t \|y\|,\|x\| − \|x\|,\|y\| σ(y)σ(x) (2.1) for any homogenous elements x, y ∈ ′ f. Let ρ : ′ f ⊗ ′ f → ′ f ⊗ ′ f be a linear map defined by ρ(x ⊗ y) = t \|y\|,\|x\| − \|x\|,\|y\| y ⊗ x, ∀x, y ∈ ′ f. We set t r = ρ • r. Lemma 2.2. We have r(σ(x)) = (σ ⊗ σ) t r(x), for all x ∈ ′ f. Proof. We show this lemma by induction on \|x\|. If x = θ i , it follows the definition of σ and t r. Assume that it holds for any homogenous elements x ′ and x ′′ . We shall show that it holds for x = x ′ x ′′ . Let's write r(x ′ ) = x ′ 1 ⊗ x ′ 2 and r(x ′′ ) = x ′′ 1 ⊗ x ′′ 2 ,\| = \|x ′ 1 \| + \|x ′ 2 \| and \|x ′′ \| = \|x ′′ 1 \| + \|x ′′ 2 \|, we have r(σ(x ′ x ′′ )) = (σ ⊗ σ) t r(x ′ x ′′ ). This finishes the proof. 2.1.3. Let · : Q(v, t) → Q(v, t) be the unique Q-algebra involution such that v = v −1 and t = t. Let · : ′ f → ′ f be the unique Q-algebra involution such that pθ i = pθ i , ∀p ∈ Q(v, t), i ∈ I. It's clear that \|x\| = \|x\| for any homogeneous element x ∈ ′ f. Let ′ f⊗ ′ f be the Q(v, t)-vector space ′ f ⊗ ′ f with the associative Q(v, t)-algebra structure given by (x 1 ⊗ x 2 )(y 1 ⊗ y 2 ) = v −\|y 1 \|\|x 2 \| t \|y 1 \|,\|x 2 \| − \|x 2 \|,\|y 1 \| x 1 y 1 ⊗ x 2 y 2 . Then ⊗ : ′ f ⊗ ′ f → ′ f⊗ ′ f is the Q(t)-algebra isomorphism. Let r : ′ f → ′ f⊗ ′ f be the Q(t)-algebra homorphism defined by r(x) = r(x), ∀x ∈ ′ f. Then we have r(θ i ) = θ i ⊗ 1 + 1 ⊗ θ i . Lemma 2.3. For any x ∈ ′ f, by setting r(x) = x 1 ⊗ x 2 , we have r(x) = v −\|x 1 \|·\|x 2 \| t \|x 2 \|,\|x 1 \| − \|x 1 \|,\|x 2 \| x 2 ⊗ x 1 . Proof. By the definition of r, we shall show that r(x) = v \|x 1 \|·\|x 2 \| t \|x 2 \|,\|x 1 \| − \|x 1 \|,\|x 2 \| x 2 ⊗ x 1 . The proof for Lemma 5 in [FL] works through if we replace v by v −1 . We note that the coassociativity property of r implies the coassociativity property of r, i.e., (r ⊗ 1)r = (1 ⊗ r)r. 2.1.4. The maps r i and i r. For any i ∈ I, let r i (resp. i r) : ′ f → ′ f be the unique linear map satisfying the following properties. r i (1) = 0, r i (θ j ) = δ ij , ∀j ∈ I and r i (xy) = v i·\|y\| t \|y\|,i − i,\|y\| r i (x)y + xr i (y); i r(1) = 0, i r(θ j ) = δ ij , ∀j ∈ I and i r(xy) = i r(x)y + v i·\|x\| t i,\|x\| − \|x\|,i x i r(y). By an induction on \|x\|, we can show that r(x) = r i (x) ⊗ θ i (resp. r(x) = θ i ⊗ i r(x)) plus other terms. 2.1.5. Quantum serre relations. Let J be the radical of the bilinear form (−, −). It is clear that J is a two-sided ideal of ′ f. Denote the quotient algebra of ′ f by f = ′ f/J. Recall the quantum integers from [FL]. For any n ∈ N, we have [n] v,t = (vt) n − (vt −1 ) −n vt − (vt −1 ) −1 , [n] ! v,t = n k=1 [k] v,t . Denote by θ (n) i = θ n i [n] ! v i ,t i . Proposition 2.4. [FL,Proposition 14] The generators θ i of f satisfy the following identities. p+p ′ =1−2 i·j i·i (−1) p t −p(p ′ −2 i,j i·i +2 j,i i·i ) i θ (p) i θ j θ (p ′ ) i = 0, ∀i = j ∈ I. 2.2. The presentation of the two-parameter quantum algebra U v,t . By Drinfeld double construction, we get the following presentation of the entire two-parameter quantum algebra U v,t , generated by symbols E i , F i , K ±1 i , K ′±1 i , ∀i ∈ I, and subjects to the following relations. (R1) K ±1 i K ∓1 i = K ′±1 i K ′∓1 i = 1. (R2) K i E j K −1 i = v i·j t j,i − i,j E j , K ′ i E j K ′−1 i = v −i·j t j,i − i,j E j , K ′ i F j K ′−1 i = v i·j t i,j − j,i F j , K i F j K −1 i = v −i·j t i,j − j,i F j . (R3) E i F j − F j E i = δ ij K i − K ′ i v i − v −1 i . (R4) p+p ′ =1−2 i·j i·i (−1) p t −p(p ′ −2 i,j i·i +2 j,i i·i ) i E (p) i E j E (p ′ ) i = 0, if i = j, p+p ′ =1−2 i·j i·i (−1) p t −p(p ′ −2 i,j i·i +2 j,i i·i ) i F (p ′ ) i F j F (p) i = 0, if i = j. The algebra U v,t has a Hopf algebra structure with the comultiplication ∆, the counit ε and the antipode S given as follows. ∆(K ±1 i ) = K ±1 i ⊗ K ±1 i , ∆(K ′±1 i ) = K ′±1 i ⊗ K ′±1 i , ∆(E i ) = E i ⊗ 1 + K i ⊗ E i , ∆(F i ) = 1 ⊗ F i + F i ⊗ K ′ i , ε(K ±1 i ) = ε(K ′±1 i ) = 1, ε(E i ) = ε(F i ) = 0, S(K ±1 i ) = K ∓1 i , S(K ′±1 i ) = K ′∓1 i , S(E i ) = −K −1 i E i , S(F i ) = −F i K ′−1 i . Let U + v,t (resp. U − v,t ) be a subalgebra of U v,t generated by E i (resp. F i ) . From the drinfeld double construction, we know that there are two well-defined algebra homomorphisms ι + : f → U + v,t (x → x + ) and ι − : f → U − v,t (x → x − , t → t −1 ) such that E i = θ + i , F i = θ − i for all i ∈ I. Let σ + : U + → U + (resp. σ − : U − → U − ) be an anti-involution such that σ + (x + ) = σ(x) + (resp. σ − (x − ) = σ(x) − ) for all x ∈ f. Lemma 2.5. For all x ∈ f, we have (i) S(x + ) = (−1) tr\|x\| v \|x\|·\|x\| 2 v −1 \|x\| K −1 \|x\| σ(x) + , (ii) S(x − ) = (−1) tr\|x\| v −\|x\|·\|x\| 2 v \|x\| σ(x) − K ′−1 \|x\| . Proof. The proofs of (i) and (ii) are similar. We shall only show (i) and left (ii) to readers. If x = θ i , (i) is straightforward by the definition of S(E i ). Assume that (i) holds for x 1 and x 2 . We shall show that it holds for x = x 1 x 2 . Let's write S(x + 1 ) = (−1) tr\|x 1 \| v \|x 1 \|·\|x 1 \| 2 v −1 \|x 1 \| K −1 \|x 1 \| σ(x 1 ) + and S(x + 2 ) = (−1) tr\|x 2 \| v \|x 2 \|·\|x 2 \| 2 v −1 \|x 2 \| K −1 \|x 2 \| σ(x 2 ) + . Then, we have S((x 1 x 2 ) + ) = S(x + 2 )S(x + 1 ) = (−1) tr(\|x 1 \|+\|x 2 \|) v −1 \|x 1 \|+\|x 2 \| v \|x 1 \|·\|x 1 \|+\|x 2 \|·\|x 2 \| 2 K −1 \|x 2 \| σ(x 2 ) + K −1 \|x 1 \| σ(x 1 ) + = (−1) tr(\|x 1 \|+\|x 2 \|) v −1 \|x 1 \|+\|x 2 \| v (\|x 1 \|+\|x 2 \|)·(\|x 1 \|+\|x 2 \|) 2 t \|x 2 \|,\|x 1 \| − \|x 1 \|,\|x 2 \| K −1 \|x 1 \|+\|x 2 \| σ(x 2 ) + σ(x 1 ) + = (−1) tr(\|x 1 \|+\|x 2 \|) v −1 \|x 1 \|+\|x 2 \| v (\|x 1 \|+\|x 2 \|)·(\|x 1 \|+\|x 2 \|) 2 K −1 \|x 1 \|+\|x 2 \| σ(x 1 x 2 ) + . This finishes the proof. Lemma 2.6. For all x, x ′ , x ′′ ∈ f, let r + i (x + ) = ι + • r i (x) and i r + (x + ) = ι + • i r(x). Then we have (i) r + i ((x ′ x ′′ ) + ) = v i·\|x ′′ \| t \|x ′′ \|,i − i,\|x ′′ \| r + i (x ′ + )x ′′ + + x ′ + r + i (x ′′ + ), (ii) i r + ((x ′ x ′′ ) + ) = i r + (x ′ + )x ′′ + + v i·\|x ′ \| t i,\|x ′ \| − \|x ′ \|,i x ′ + i r + (x ′′ + ), (iii) x + F i − F i x + = r + i (x + )K i − K ′ i ( i r + (x + )) v i − v −1 i , (iv) ∆(x + ) = x + ⊗1+ i r + i (x + )K i ⊗E i +· · · = K \|x\| ⊗x + + i E i K \|x\|−i ⊗ i r + (x + )+· · · . Proof. Statement (i) (resp. (ii)) directly follows the definition of ι + and r i (resp. i r). We now show (iii) by induction on \|x\|. The case that x = θ i is trivial. Assume that (iii) holds for x ′ and x ′′ . Then we have (x ′ x ′′ ) + F i − F i (x ′ x ′′ ) + = x ′+ (F i x ′′+ + r + i (x ′′+ )K i − K ′ i ( i r + (x ′′+ )) v i − v −1 i ) − F i x ′+ x ′′+ = r + i (x ′+ )K i − K ′ i ( i r + (x ′+ )) v i − v −1 i x ′′+ + x ′+ r + i (x ′′+ )K i − K ′ i ( i r + (x ′′+ )) v i − v −1 i = r + i (x ′+ )K i x ′′+ + x ′+ r + i (x ′′+ )K i v i − v −1 i − K ′ i ( i r + (x ′+ ))x ′′+ + x ′+ K ′ i ( i r + (x ′′+ )) v i − v −1 i = r + i ((x ′ x ′′ ) + )K i + K ′ i ( i r + ((x ′ x ′′ ) + )) v i − v −1 i . This proves (iii). By (i) and (ii), statement (iv) can be shown by induction on \|x\|. Lemma 2.7. For all y, y ′ , y ′′ ∈ f, let r − i (x − ) = ι − • r i (x) and i r − (x − ) = ι − • i r(x). Then we have (i) r − i ((y ′ y ′′ ) − ) = v i·\|y ′′ \| t i,\|y ′′ \| − \|y ′′ \|,i r − i (y ′− )y ′′− + y ′− r − i (y ′′− ), (ii) i r − ((y ′ y ′′ ) − ) = i r − (y ′− )y ′′− + v i·\|y ′ \| t \|y ′ \|,i − i,\|y ′ \| y ′− i r(y ′′− ), (iii) E i y − − y − E i = K i ( i r − (y − )) − r − i (y − )K ′ i v i − v −1 i , (iv) ∆(y − ) = y − ⊗K ′ \|y\| + i i r − (y − )⊗F i K ′ \|y\|−i +· · · = 1⊗y − + i F i ⊗r − i (y − )K ′ i +· · · . The proof of this lemma is similar to that of Lemma 2.6. 2.3. The quasi-R-matrix Θ. In this section, we shall simply write U instead of U v,t . We define a bar involution · : U → U such that E i = E i , F i = F i , K i = K ′ i , K ′ i = K i ; px = p · x, ∀p ∈ Q(v, t), x ∈ U. Let · : U ⊗ U → U ⊗ U be the Q(t)-algebra homomorphism given by · ⊗ · and ∆ : U → U ⊗ U the Q(t)-algebra homomorphism given by ∆(x) = ∆(x). Thus, we have ∆(E i ) = E i ⊗ 1 + K ′ i ⊗ E i , ∆(K ±1 i ) = K ±1 i ⊗ K ±1 i , ∆(F i ) = 1 ⊗ F i + F i ⊗ K i , ∆(K ′ ±1 i ) = K ′ ±1 i ⊗ K ′ ±1 i . (2.4) Let (U ⊗ U) ∧ be the completion of the vector space U ⊗ U with respect to the descending sequence of vector spaces H N = (U + U 0 ( trν≥N U − ν )) ⊗ U + U ⊗ (U − U 0 ( trν≥N U + ν )) for N = 1, 2, · · · . We set {i, j} = v i·j t j,i − i,j , ∀i, j ∈ I, which is a multiplicative bilinear form on Z[I] × Z[I]. Lemma 2.8. [FX,Proposition 4 ] Let U ≥0 (resp. U ≤0 ) be the subalgebra of U generated by E i and K i (resp. F i and K ′ i ) for all i in I. We denote K −µ (resp. K ′ −µ ) by K −1 µ (resp. K ′−1 µ ) for all µ ∈ N[I]. There is a skew-Hopf pairing (, ) φ : U ≥0 × U ≤0 → Q(v, t) such that (i) (1, 1) φ = 1, (ii) (E i , F j ) φ = δ ij (v −1 i − v i ) −1 , ∀i, j ∈ I, (iii)(K µ x, K ′ ν y) φ = {µ, ν}{µ, \|y\|}{\|x\|, ν}(x, y) φ , ∀µ, ν ∈ Z[I], x ∈ U + , y ∈ U − . For any homogenous elements x ∈ U + , y ∈ U − , we have (xK µ , yK ′ ν ) φ = (x, y) φ (K µ , K ′ ν ) φ , ∀µ, ν ∈ Z[I]. (2.5) By Lemma 2.6(iv) and Lemma 2.7(iv), we have (xK µ , yK ′ ν ) φ = (∆(x)∆(K µ ), y ⊗ K ′ ν ) φ = (xK µ , y) φ (K µ , K ′ ν ) φ = (x ⊗ K µ , ∆ op (y)) φ (K µ , K ′ ν ) φ = (x, y) φ (K µ , K ′ ν ) φ . This proves (2.5). Lemma 2.9. For all x ∈ U + and y ∈ U − , we have (i) (x, F i y) φ = (v −1 i − v i ) −1 ( i r + (x), y) φ , (ii) (x, yF i ) φ = (v −1 i − v i ) −1 (r + i (x), y) φ , (iii) (E i x, y) φ = (v −1 i − v i ) −1 (x, i r − (y)) φ , (iv) (xE i , y) φ = (v −1 i − v i ) −1 (x, r − i (y)) φ . Proof. The proofs of the four equations are similar. We shall only show (i) and left others to readers. By the definition of skew-Hopf pairing (, ) φ in [X97, Scetion 2.2], Lemma 2.6(iv) and Lemma 2.8(iii), we have (x, F i y) φ = (∆(x), F i ⊗ y) φ = (E i K \|x\|−i ⊗ i r + (x), F i ⊗ y) φ = (E i K \|x\|−i , F i ) φ ( i r + (x), y) φ = (E i , F i ) φ ( i r + (x), y) φ = (v −1 i − v i ) −1 ( i r + (x), y) φ . Lemma 2.10. For all x, y ∈ f, we have (x + , y − ) φ = (σ + (x + ), σ − (y − )) φ . Proof. It is straightforward to check it when x = θ i and y = θ j for some i, j ∈ I. Assume that the lemma holds for x 1 and x 2 . We shall show that it holds for x = x 1 x 2 . Let y ∈ f and r(y) = y 1 ⊗ y 2 with y 1 , y 2 homogeneous. Then we have ∆(y − ) = v −\|y 1 \|·\|y 2 \| t \|y 1 \|,\|y 2 \| − \|y 2 \|,\|y 1 \| y − 2 ⊗ K ′ \|y 2 \| y − 1 . By Lemma 2.2, r(σ(y)) = t \|y 2 \|,\|y 1 \| − \|y 1 \|,\|y 2 \| σ(y 2 ) ⊗ σ(y 1 ). Then we have ∆(σ(y) − ) = v −\|y 1 \|·\|y 2 \| σ(y 1 ) − ⊗ K ′ \|y 1 \| σ(y 2 ) − . (2.6) By (2.1), (2.6) and Lemma 2.8(iii), we have (σ + (x + 1 x + 2 ), σ − (y − )) φ =t \|x 2 \|,\|x 1 \| − \|x 1 \|,\|x 2 \| (σ + (x + 2 )σ + (x + 1 ), σ − (y − )) φ =t \|x 2 \|,\|x 1 \| − \|x 1 \|,\|x 2 \| (σ + (x + 2 ) ⊗ σ + (x + 1 ), ∆ op (σ − (y − ))) φ = t \|x 2 \|,\|x 1 \| − \|x 1 \|,\|x 2 \| v −\|y 1 \|·\|y 2 \| (σ + (x + 2 ) ⊗ σ + (x + 1 ), K ′ \|y 1 \| σ − (y − 2 ) ⊗ σ − (y − 1 )) φ = t \|x 2 \|,\|x 1 \| − \|x 1 \|,\|x 2 \| v −\|y 1 \|·\|y 2 \| (σ + (x + 2 ), K ′ \|y 1 \| σ − (y − 2 )) φ (σ + (x + 1 ), σ − (y − 1 )) φ = (σ + (x + 2 ), σ − (y − 2 )) φ (σ + (x + 1 ), σ − (y − 1 )) φ . ( 2.7) Similarly, we have (x + 1 x + 2 , y − ) φ = (x + 1 ⊗ x + 2 , ∆ op (y − )) φ = v −\|y 1 \|·\|y 2 \| t \|y 1 \|,\|y 2 \| − \|y 2 \|,\|y 1 \| (x + 1 ⊗ x + 2 , K ′ \|y 2 \| y − 1 ⊗ y − 2 ) φ = v −\|y 1 \|·\|y 2 \| t \|y 1 \|,\|y 2 \| − \|y 2 \|,\|y 1 \| (x + 1 , K ′ \|y 2 \| y − 1 ) φ (x + 2 , y − 2 ) φ = (x + 1 , y − 1 ) φ (x + 2 , y − 2 ) φ . (2.8) By (2.7) and (2.8), we have (x + 1 x + 2 , y − ) φ = (σ + (x + 1 x + 2 ), σ − (y − )) φ . This finishes the proof. By the relation (R2) in Section 2.2 , the subalgebra U + has the following decomposition U + = µ∈N[I] U + µ , where U + µ = {u ∈ U + \|K i u = v i·\|u\| t \|u\|,i − i,\|u\| uK i , K ′ i u = v −i·\|u\| t \|u\|,i − i,\|u\| uK ′ i , ∀i ∈ I}. The weight space U + µ is spanned by all the monomials E i 1 · · · E i l with grading µ. Similarly, the subalgebra U − has a decomposition U − = µ∈N[I] U − −µ and the spaces U + µ and U − −µ are nondegenerately paired under the skew-Hopf pairing (, ) φ . Then we may select a basis B of U − such that B µ = B ∩ U − −µ . Let {b * \|b ∈ B µ } be the basis of U + µ dual to B µ under (, ) φ . Lemma 2.11. Let x ∈ U + λ and y ∈ U − −λ for any λ ∈ N[I]. Then, (i) ∆(x) = 0≤µ≤λ b∈Bµ,b ′ ∈B λ−µ (x, b ′ b) φ b ′ * K µ ⊗ b * , (ii) ∆(y) = 0≤µ≤λ b∈Bµ,b ′ ∈B λ−µ (b ′ * b * , y) φ b ⊗ b ′ K ′ µ . Proof. The proofs of (i) and (ii) are similar. We shall only show (i). As x ∈ U + λ , we have ∆(x) ∈ 0≤µ≤λ U + λ−µ K µ ⊗ U + µ . Let h µ b,b ′ ∈ Q(v, t) be such that ∆(x) = 0≤µ≤λ b∈Bµ,b ′ ∈B λ−µ h µ b,b ′ b ′ * K µ ⊗ b * . Then for all b 1 ∈ B λ−µ , b 2 ∈ B µ and µ, we have (x, b 1 b 2 ) φ = (∆(x), b 1 ⊗ b 2 ) φ = 0≤µ≤λ b∈Bµ,b ′ ∈B λ−µ h µ b,b ′ (b ′ * K µ ⊗ b * , b 1 ⊗ b 2 ) φ = 0≤µ≤λ b∈Bµ,b ′ ∈B λ−µ h µ b,b ′ (b ′ * K µ , b 1 ) φ (b * , b 2 ) φ = h µ b 2 ,b 1 . This finishes the proof. For each x ∈ U + µ and y ∈ U − −µ , we have x = b∈Bµ (x, b) φ b * , y = b∈Bµ (b * , y) φ b. (2.9) For µ ∈ N[I], we define Θ µ = b∈Bµ b ⊗ b * ∈ U − −µ ⊗ U + µ . Set Θ µ = 0 if µ / ∈ N[I]. Lemma 2.12. For all i ∈ I, µ ∈ N[I], we have (i) (K i ⊗ K i )Θ µ = Θ µ (K i ⊗ K i ), (ii) (K ′ i ⊗ K ′ i )Θ µ = Θ µ (K ′ i ⊗ K ′ i ), (iii) (E i ⊗ 1)Θ µ + (K i ⊗ E i )Θ µ−i = Θ µ (E i ⊗ 1) + Θ µ−i (K ′ i ⊗ E i ), (iv) (1 ⊗ F i )Θ µ + (F i ⊗ K ′ i )Θ µ−i = Θ µ (1 ⊗ F i ) + Θ µ−i (F i ⊗ K i ). Proof. The first two are easy to check. We shall show (iii) and (iv) can be shown similarly. The proof of (iii) goes in a similar way as that for Lemma 4.10 in [BW]. For the convenience of the readers, we present it here. By Lemma 2.7(iii) , Lemma 2.9(iii)-(iv) and (2.9), we have (E i ⊗ 1)Θ µ − Θ µ (E i ⊗ 1) = b∈Bµ (E i b − bE i ) ⊗ b * =(v i − v −1 i ) −1 b∈Bµ (K i ( i r − (b)) − r −1 i (b)K ′ i ) ⊗ b * =(v i − v −1 i ) −1 ( b∈Bµ K i b ′ ∈B µ−i (b ′ * , i r − (b)) φ b ′ ⊗ b * − b∈Bµ b ′ ∈B µ−i (b ′ * , r − i (b)) φ b ′ K ′ i ⊗ b * ) = − b∈Bµ K i b ′ ∈B µ−i (E i b ′ * , b) φ b ′ ⊗ b * + b∈Bµ b ′ ∈B µ−i (b ′ * E i , b) φ b ′ K ′ i ⊗ b * = b ′ ∈B µ−i b ′ K ′ i ⊗ b∈Bµ (b ′ * E i , b) φ b * − b ′ ∈B µ−i K i b ′ ⊗ b∈Bµ (E i b ′ * , b) φ b * = b ′ ∈B µ−i b ′ K ′ i ⊗ b ′ * E i − b ′ ∈B µ−i K i b ′ ⊗ E i b ′ * =Θ µ−i (K ′ i ⊗ E i ) − (K i ⊗ E i )Θ µ−i . This finishes the proof. Proposition 2.13. Let Θ 0 = 1 ⊗ 1 and Θ = ν∈N[I] Θ ν ∈ (U ⊗ U) ∧ . Then we have ∆(u)Θ = Θ∆(u) for all u ∈ U (where this identity is in (U ⊗ U) ∧ ). This proposition follows from Lemma 2.12. The element Θ defined in this proposition is called the quasi-R-matrix. Corollary 2.14. We have ΘΘ = ΘΘ = 1 ⊗ 1 (equality in (U ⊗ U) ∧ ). The proof is similar to those for Corollary 4.1.3 in [L]. We define (, ) φ : U + × U − → Q(v, t) by (x, y) φ = (x, y) φ , ∀x ∈ U + , y ∈ U − . (2.10) It satisfies that (1, 1) φ = 1 and (E i , F j ) φ = δ ij (v i − v −1 i ) −1 . Lemma 2.15. (x + , y − ) φ = (−1) tr\|x\| v −\|x\|·\|y\|/2 v −\|x\| (x + , σ − (y − )) φ , ∀x, y ∈ f. Proof. It is straightforward to check it when x = θ i and y = θ j for some i, j ∈ I. Let x ∈ f and r(x) = x 1 ⊗ x 2 with x 1 , x 2 homogeneous. Assume that the lemma holds for y 1 and y 2 . We shall show that it holds for y = y 1 y 2 . By Lemma 2.3, we have ∆(x + ) = v \|x 1 \|·\|x 2 \| t \|x 2 \|,\|x 1 \| − \|x 1 \|,\|x 2 \| x + 2 K \|x 1 \| ⊗ x + 1 . Then, (x + , y − 1 y − 2 ) φ =(∆(x + ), y − 1 ⊗ y − 2 ) φ = v \|x 1 \|·\|x 2 \| t \|x 2 \|,\|x 1 \| − \|x 1 \|,\|x 2 \| (x + 2 K \|x 1 \| , y − 1 ) φ (x + 1 , y − 2 ) φ = v \|x 1 \|·\|x 2 \| t \|x 2 \|,\|x 1 \| − \|x 1 \|,\|x 2 \| (x + 2 , y − 1 ) φ (x + 1 , y − 2 ) φ . By (2.10), we have (x + , y − 1 y − 2 ) φ = v −\|x 1 \|·\|x 2 \| t \|x 2 \|,\|x 1 \| − \|x 1 \|,\|x 2 \| (x + 2 , y − 1 ) φ (x + 1 , y − 2 ) φ = (−1) tr(\|x 1 \|+\|x 2 \|) v −(\|x 1 \|·\|x 2 \|+(\|x 1 \|·\|y 2 \|+\|x 2 \|·\|y 1 \|)/2) v −(\|x 1 \|+\|x 2 \|) t \|x 2 \|,\|x 1 \| − \|x 1 \|,\|x 2 \| (x + 1 , σ − (y − 2 )) φ (x + 2 , σ − (y − 1 )) φ . (2.11) On the other hand, (x + , σ − (y − 1 y − 2 )) φ =t \|y 1 \|,\|y 2 \| − \|y 2 \|,\|y 1 \| (∆(x + ), σ − (y − 2 )σ − (y − 1 )) φ =t \|y 1 \|,\|y 2 \| − \|y 2 \|,\|y 1 \| (x + 1 K \|x 2 \| ⊗ x + 2 , σ − (y − 2 ) ⊗ σ − (y − 1 )) φ =t \|y 1 \|,\|y 2 \| − \|y 2 \|,\|y 1 \| (x + 1 , σ − (y − 2 )) φ (x + 2 , σ − (y − 1 )) φ . ( 2.12) This lemma follows from (2.11) and (2.12) with \|x 1 \| = \|y 2 \| and \|x 2 \| = \|y 1 \|. While Θ can be evaluated easily, it will be more convenient to have the following alternate description of Θ using the property of (, ) φ . Lemma 2.16. With the same notations as in Proposition 2.13, Θ = ν Θ ν is given by Θ ν = (−1) trν v ν·ν 2 v −ν b∈Bν b ⊗ σ + (b * ) ∈ U − −ν ⊗ U + ν . Proof. Since Θ is independent of the choice of basis, we let Θ ν = b∈Bν b ⊗ b * , ∀ν ∈ N[I], where b ∈ B = {b\|b ∈ B}. Then Θ ν = b∈Bν b ⊗ (b * ), ∀ν ∈ N[I]. (2.13) Note that b = b. We show the relation between b * and σ + (b * ). There exists an element b ′ ∈ B ν such that (b * , b ′ ) φ = δ b,b ′ . By (2.10) and Lemma 2.15, we have δ b,b ′ = (b * , b ′ ) φ = (b * , b ′ ) φ = (−1) trν v −ν·ν/2 v −ν (b * , σ − (b ′ )) φ . Therefore, we have b * = (−1) trν v ν·ν/2 v ν σ − (b) * . (2.14) By Lemma 2.10, we have σ − (b) * = σ + (b * ). (2.15) The Lemma follows from (2.13), (2.14) and (2.15). 3. The R-matrix for two-parameter quantum algebras 3.1. The module of U v,t . A U v,t -module M is called a weight module if it admits a decomposition M = λ∈N[I] M λ of vector spaces such that M λ = {m ∈ M\|K i · m = v i·λ c i,λ m, K ′ i · m = v −i·λ c i,λ m, ∀i ∈ I}, where c i,λ = t λ,i − i,λ . For any m ∈ M λ , we denote by \|m\| = λ. For all m ∈ Q(v, t), we define u · m = ε(u)m, ∀u ∈ U v,t , where ε is the counit of U v,t . Then, Q(v, t) is a trivial module of U v,t . Let M be U v,t -module and M * = Hom Q(v,t) (M, Q(v, t)). We define u · n * ∈ M * by u · n * (m) = n * (S(u) · m), ∀u ∈ U v,t , n * ∈ M * , m ∈ M, (3.1) then M * is also a U v,t -module. For any U v,t -modules M and N, we can construct the U v,t -module M ⊗ N = M ⊗ Q(v,t) N via the coproduct. In particular, we have U v,t -modules M * ⊗ M and M ⊗ M * . For any i ∈ I, λ ∈ N[I], we denote by v −λ = v −1 λ , and c i,−λ = c −1 i,λ . Lemma 3.1. Fix a U v,t -module M. (1) Let ev : M * ⊗ M → Q(v, t) be the Q(v, t)-linear map defined by m * ⊗ n → m * (n), ∀m * ∈ M * , n ∈ M. Then ev is a U v,t -module epimorphism. (2) Let qtr : Proof. The surjectivity and injectivity of the maps are clear. We shall show that these maps preserve the action of generators of U v,t . It is straightforward to verify these maps holds for K i and K ′ i . It's enough to check that these maps preserve the action of E i and F i for all i ∈ I. Show that M ⊗ M * → Q(v, t) be the Q(v, t)-linear map defined by m ⊗ n * → v 2 −\|m\| n * (m), ∀m ∈ M, n * ∈ M * . Then qtr is a U v,t -module epimorphism. (3) Let coev : Q(v, t) → M * ⊗ M be the Q(v, t)-linear map defined by 1 → w∈B v 2 \|w\| w * ⊗ w for some homogeneous Q(v, t)-basis B of M. Then coev is a U v,t -module monomor- phism.(ev(∆(E i )m * ⊗ n) = ev(∆(F i )m * ⊗ n) = 0, ∀m * ∈ M * , n ∈ M, (3.2) qtr(∆(E i )m ⊗ n * ) = qtr(∆(F i )m ⊗ n * ) = 0, ∀m ∈ M, n * ∈ M * , (3.3) ∆(E i ) w∈B v 2 \|w\| w * ⊗ w = ∆(F i ) w∈B v 2 \|w\| w * ⊗ w = 0, (3.4) ∆(E i ) w∈B w ⊗ w * = ∆(F i ) w∈B w ⊗ w * = 0. (3.5) We shall prove (3.3) and (3.4) for the action of E i , the remaining cases can be shown similarly. First, we show qtr(∆(E i )m⊗n * ) = 0. For homogenous elements m ∈ M, n * ∈ M * , we have qtr(∆(E i )m ⊗ n * ) = qtr(E i · m ⊗ n * + K i · m ⊗ E i · n * ) = v 2 −\|m\|−i n * (E i · m) + v 2 −\|m\| (E i · n * )(K i · m) = v 2 −\|m\|−i n * (E i · m) + v 2 −\|m\| n * (−K −1 i E i K i · m) = v 2 −\|m\|−i n * (E i · m) − v 2 −\|m\|−i n * (E i · m) = 0. Next, we show that ∆(E i ) w∈B v 2 \|w\| w * ⊗w = 0. Observe that x = m * ⊗n = 0 if and only if x(m ′ ) := m * (m ′ )n = 0 for all m ′ ∈ M. Let x = ∆(E i ) w∈B v 2 \|w\| w * ⊗w. Then we have x(m) = w∈B v 2 \|w\| (E i · w * (m)w + (K i · w * )(m)E i · w), ∀m ∈ M. For any w 0 ∈ B, we have x(w 0 ) = w∈B v 2 \|w\| (w * (−K −1 i E i · w 0 )w + w * (K −1 i · w 0 )E i · w) = − w∈B v 2 \|w\| v −i·(\|w 0 \|+i) c −1 i,\|w 0 \|+i w * (E i · w 0 )w + w∈B v 2 \|w\| v −i·\|w 0 \| c −1 i,\|w 0 \| w * (w 0 )E i · w = − v 2 \|w 0 \|+i v −i·(\|w 0 \|+i) c −1 i,\|w 0 \| E i · w 0 + v 2 \|w 0 \| v −i·\|w 0 \| c −1 i,\|w 0 \| E i · w 0 =0. Hence, x = 0. This finishes the proof. 3.2. The R-matrix of U v,t -module. We shall construct a U v,t -module isomorphism R M,M ′ : M ⊗ M ′ → M ′ ⊗ M for any finite dimensional weight modules M and M ′ , by the method used by Jantzen [Jan,Chap. 7] for the quantum algebras U q (g). The map R M,M ′ is the composite of three linear transformations P,f , Θ defined as follows. Let P : M ⊗ M ′ → M ′ ⊗ M be the Q(v, t)-linear bijection defined by P (m ⊗ m ′ ) = m ′ ⊗ m, ∀m ∈ M, m ′ ∈ M ′ . Recall that (, ) φ is a skew-Hopf pairing defined in Lemma 2.8. For any λ, µ ∈ Z[I], we define the map f : Z[I] × Z[I] → Q(v, t) × by f (λ, µ) = (K λ , K ′ µ ) −1 φ . (3.6) Then we have f (λ + µ, ν) = f (λ, ν)f (µ, ν) f (λ, µ + ν) = f (λ, µ)f (λ, ν) f (λ, µ) = f (−λ, −µ) f (i, µ) = v −i·µ c µ,i f (λ, i) = v −i·λ c i,λ , (3.7) where ν is also in Z[I]. We define a bijective linear mapf : M ⊗ M ′ → M ⊗ M ′ bỹ f (m ⊗ m ′ ) = f (λ, µ)m ⊗ m ′ , ∀m ∈ M λ , m ′ ∈ M ′ µ , λ, µ ∈ Z[I].′ of U v,t , we define R = R M,M ′ : M ⊗ M ′ → M ′ ⊗ M by R = Θ •f • P . Then R is a U v,t -module isomorphism. Proof. By Corollary 2.14, R is invertible transformation. Then, we shall show that R is a U v,t -module homomorphism, i.e., ∆(u)R(m ⊗ m ′ ) = R(∆(u)m ⊗ m ′ ), ∀u ∈ U v,t , m ∈ M, m ′ ∈ M ′ . By Proposition 2.13, we have ∆(u)R(m ⊗ m ′ ) = Θ∆(u)f • P (m ⊗ m ′ ) = Θ(f (\|m ′ \|, \|m\|)∆(u)(m ′ ⊗ m)). So it is suffices to show f • P (∆(u)m ⊗ m ′ )) = f (\|m ′ \|, \|m\|)∆(u)(m ′ ⊗ m) for all u ∈ U v,t . Hence it is enough to show that this equality holds all generators of U v,t . For u = K ν , K ′ ν , this is straightforward. The cases u = E i and u = F i are similar, so we shall prove the first case. By (2.4) and (3.7), we havẽ f • P (∆(E i )m ⊗ m ′ ) =f (\|m ′ \|, i + \|m\|)m ′ ⊗ E i m + f (i + \|m ′ \|, \|m\|)E i m ′ ⊗ K i m =f (\|m ′ \|, \|m\|)v −i·\|m ′ \| c i,\|m ′ \| m ′ ⊗ E i m + f (\|m ′ \|, \|m\|)E i m ′ ⊗ m =f (\|m ′ \|, \|m\|)(K ′ i m ′ ⊗ E i m + E i m ′ ⊗ m) =f (\|m ′ \|, \|m\|)∆(E i )(m ′ ⊗ m). This finishes the proof. For any finite dimension weight U v,t -module M 1 , M 2 , M 3 , we have maps R 12 , R 23 : M 1 ⊗ M 2 ⊗ M 3 → M 3 ⊗ M 2 ⊗ M 1 defined as R ⊗ Id and Id ⊗ R, respectively. We shall now verify that R satisfy the quantum Yang-Baxter equation R 12 • R 23 • R 12 = R 23 • R 12 • R 23 . We will need the following lemma. For 1 ≤ s, l ≤ 3, we definef sl on M 1 ⊗ M 2 ⊗ M 3 viaf sl (m 1 ⊗ m 2 ⊗ m 3 ) = f (\|m s \|, \|m l \|)m 1 ⊗ m 2 ⊗ m 3 . Let Θ op = µ b∈Bµ b * ⊗ b, Θ 12 = µ b∈Bµ b ⊗ b * ⊗ 1, and we define the other expressions in a similar way. Letting Θ f sl = Θ sl •f sl , we have the following identities for operators on M 1 ⊗ M 2 ⊗ M 3 . Lemma 3.3. (i) (∆ ⊗ 1)(Θ op ) •f 31 •f 32 = Θ f 31 • Θ f 32 . (ii)f 31 •f 32 • Θ 12 = Θ 12 •f 31 •f 32 . Proof. We shall give a detailed proof of (i). For any m 1 ∈ M 1 , m 2 ∈ M 2 , m 3 ∈ M 3 , by Lemma 2.11, we have (∆ ⊗ 1)(Θ op ) •f 31 •f 32 (m 1 ⊗ m 2 ⊗ m 3 ) =f (\|m 3 \|, \|m 1 \|)f (\|m 3 \|, \|m 2 \|)(∆ ⊗ 1)( µ b∈Bµ b * ⊗ b)(m 1 ⊗ m 2 ⊗ m 3 ) =f (\|m 3 \|, \|m 1 \|)f (\|m 3 \|, \|m 2 \|)( µ,b∈Bµ 0≤λ≤µ,b 1 ∈B λ ,b 2 ∈B µ−λ (b * , b 2 b 1 ) φ b * 2 K λ ⊗ b * 1 ⊗ b)(m 1 ⊗ m 2 ⊗ m 3 ) =f (\|m 3 \|, \|m 1 \|)f (\|m 3 \|, \|m 2 \|)( µ 0≤λ≤µ,b 1 ∈B λ ,b 2 ∈B µ−λ b * 2 K λ m 1 ⊗ b * 1 m 2 ⊗ b 2 b 1 m 3 . On the other hand, Θ f 31 • Θ f 32 (m 1 ⊗ m 2 ⊗ m 3 ) =f (\|m 3 \|, \|m 2 \|)Θ f 31 ( ν,b ′ ∈Bν m 1 ⊗ b ′ * m 2 ⊗ b ′ m 3 ) =f (\|m 3 \|, \|m 2 \|) ν,b ′ ∈Bν ζ,b ′′ ∈B ζ f (\|m 3 \| − ν, \|m 1 \|)b ′′ * m 1 ⊗ b ′ * m 2 ⊗ b ′′ b ′ m 3 . By Lemma 2.8 and (3.6), we have K ν · m 1 = f (−ν, \|m 1 \|)m 1 . We get the second expression by replacing the variables λ and µ − λ in the first expression with ν and ζ, respectively. This proves (i). Identity (ii) follows directly from (3.7). We thus obtain the following crucial property of R. Proposition 3.4. For any finite dimension weight U v,t -modules M 1 , M 2 , and M 3 , we have R 12 • R 23 • R 12 = R 23 • R 12 • R 23 : M 1 ⊗ M 2 ⊗ M 3 → M 3 ⊗ M 2 ⊗ M 1 . Proof. This proposition follows from Lemma 3.3, Proposition 3.2 and the following identities. P 12 • Θ f 12 = Θ f 12 • P 12 , P 23 • Θ f 23 = Θ f 32 • P 23 , P 23 • Θ f 31 = Θ f 21 • P 23 , P 23 • Θ f 12 = Θ f 13 • P 23 , P 12 • Θ f 13 = Θ f 23 • P 12 . Remark. By specialization of R-matrix, we recover the one for various quantum algebras in literatures. (1) By setting t = 1, the R-matrix in Proposition 3.2 degenerates into the one for one-parameter quantum algebras [L,Chapter 32]. (2) By setting v = (rs −1 ) 1 2 and t = (rs) − 1 2 , the R-matrix in Proposition 3.2 coincides with the one in [BW]. 4. Knot invariants associated to the two-parameter quantum algebra U v,t 4.1. Tangles. Recall the definition of the tangle from [Kas]. Let [0] be the empty set and [n] = {1, 2, · · · , n} for any integer n > 0. We denote by J the closed interval [0, 1] and by R 2 the real plane. Definition 4.1. [Kas,Section X.5] Let k and l be nonnegative integers. A tangle L of type (k, l) is the union of a finite number of pairwise disjoint simple oriented polygonal arcs in X = R 2 × J such that the boundary ∂L of L satisfies the condition ∂L = L ∩ (R 2 × {0, 1}) = ([k] × {0} × {0}) ∪ ([l] × {0} × {1}). For a tangle L of type (k, l), there exists two finite sequences s (L) and b(L) consisting of + and − signs. Let s(L) = (ε 1 , · · · , ε k ) and b(L) = (η 1 , · · · , η l ). For 1 ≤ i ≤ k, let ε i = + (resp. ε i = −) if the point (i, 0, 0) is an origin (resp. an endpoint) of L. On the contrary, for 1 ≤ i ≤ l, let η i = + (resp. η i = −) if the point (i, 0, 1) is an endpoint (resp. an origin) of L. , , From left to right, we denote the oriented tangles shown in Figure 1 by the symbols ↑, ↓, , , , , X + and X − , respectively. Example (1) For the tangles ↑ and ↓, we have s(↑) = (+), b(↑) = (+), s(↓) = (−) and b(↓) = (−). (2)For the tangles , we have s( ) = (−, +), b( ) = ∅. The product • and the tensor product ⊗ on the set of all isotopy types of tangles are defined as follows L • T = L T , L ⊗ T = L T , where L and T are tangles. The product L • T is well defined when s(L) = b(T ). 4.2. The strict monoidal category (OTa, ⊗, ∅) of tangles. Let OTa be a category. The objects of OTa are all finite sequences consisting of + and − including the empty sequence. Given two finite sequences ε = (ε 1 , · · · ε k ) and ν = (ν 1 , · · · ν l ), the oriented (k, l)-tangle L represents a morphism from ε to ν such that s(L) = ε and b(L) = ν. The tensor product of the objects ε and ν is the object (ε, ν). The composition (resp. tensor product) of morphisms is the product (resp. tensor product) of tangles. It is clear that (OTa, ⊗, ∅) is a strict monoidal category [Kas,Proposition XII.2.1]. In what follows, we shall omit the ⊗ sign between morphisms if there is no dangers of confusion. Theorem 4.2. [Tur90,Theorem 3.2] The category OTa is generated by the morphisms , , , , X + and X − , and is presented by them together with the relations: (i) ( ↑) • (↑ ) = ↑ = (↑ ) • ( ↑); (ii) ( ↓) • (↓ ) = ↓ = (↓ ) • ( ↓); (iii) (↓ ↓ ) • (↓ ↓ ↑ ↓) • (↓ ↓ X ± ↓ ↓) • (↓ ↑ ↓ ↓) • ( ↓ ↓) = ( ↓ ↓) • (↓ ↑ ↓ ↓) • (↓ ↓ X ± ↓ ↓) • (↓ ↓ ↑ ↓) • (↓ ↓ ); (iv) X + • X − = X − • X + = ↑ ↑ ; (v) (X + ↑) • (↑ X + ) • (X + ↑) = (↑ X + ) • (X + ↑) • (↑ X + ); (vi) (↑ ) • (X ± ↓) • (↑ ) = ↑ ; (vii) Y • T = ↓ ↑ , T • Y = ↑ ↓ , where Y = (↓ ↑ ) • (↓ X + ↓) • ( ↑ ↓) and T = ( ↑ ↓) • (↓ X − ↓) • (↓ ↑ ). 4.3. The strict monoidal category (Mod, ⊗, Q(v, t)) of U v,t -modules. Let {M(i)} i∈{+,−} be a collection of U v,t -modules where M(+) is a finite dimensional weight module of the two parameter quantum algebra U v,t with the highest weight λ and M(−) is its dual module. To each finite sequence j = (i 1 , · · · , i n ) with i 1 , · · · , i n ∈ {+, −} we associate the U v,t -module M(j) = (· · · ((M(i 1 ) ⊗ Uv,t M(i 2 )) ⊗ Uv,t M(i 3 )) ⊗ · · · ) ⊗ Uv,t M(i n )). For the empty sequence ∅, we set M(∅) = Q(v, t). Consider the category Mod whose objects are the pairs (j, M(j)) for all finite sequences j of elements of {+, −}. The morphisms (j, M(j)) → (j ′ , M(j ′ )) consist of all U v,t -linear homomorphisms M(j) → M(j ′ ). Composition of morphisms is the usual composition of homomorphisms. For simplicity, we denote the object (j, M(j)) by M(j). The tensor product ⊗ is defined by setting M(j) ⊗ M(j ′ ) = M(j, j ′ ). Knot invariants. Recall that ev, qtr, coev, and coqtr are the maps defined in Lemma 3.1. Likewise, let R be the map defined in Proposition 3.2. We denote the maps id + , id − , ev, qtr, coev, coqtr, R and R −1 by the symbols ↑, ↓, , , , , X + and X − , respectively. Then we have some substantial diagrammatic identities. Proof. The proofs of the four equalities are similar. We show the first equality in detail. In terms of morphisms, we wish to show (qtr ⊗ id + ) • (id + ⊗ coev) = id + . Let B be a homogeneous basis of M(+) and B * the dual basis of M(−). Then for any w 0 ∈ B, we have (qtr ⊗ id + ) • (id + ⊗ coev)(w 0 ) = w∈B v 2 \|w\| (qtr ⊗ id + )(w 0 ⊗ w * ⊗ w) = w∈B v 2 \|w\| v 2 −\|w 0 \| w * (w 0 )w = w 0 . To distinguish from , we let R +,+ = , R −,− = . Lemma 4.4. We have four equalities of diagrams (1) = ,(2)= ,(3)= , (4) = . Proof. The proofs of (1) − (4) are similar. We shall only show (1) in detail. It is equivalent to show that (1) holds for the following equality ϕ = ψ, where ϕ =qtr • (id + ⊗ qtr ⊗ id − ) • (id + ⊗ id + ⊗ R −,− ), ψ =qtr • (id + ⊗ qtr ⊗ id − ) • (R +,+ ⊗ id − ⊗ id − ). Let m 1 , m 2 ∈ M(+) and m * 3 , m * 4 ∈ M(−). On the left hand side, by Lemma 2.5, we have ϕ(m 1 ⊗ m 2 ⊗ m * 3 ⊗ m * 4 ) = ν b∈Bν v 2 −\|m 1 \|−\|m 2 \| f (\|m * 4 \|, \|m * 3 \|)(b · m * 4 )(m 2 )(b * · m * 3 )(m 1 ) = ν b∈Bν v 2 −\|m 1 \|−\|m 2 \| f (\|m * 4 \|, \|m * 3 \|)v ν·(\|m 2 \|−\|m 1 \|−ν) c −ν,\|m 1 \|+\|m 2 \| m * 4 (σ − (b)m 2 )m * 3 (σ + (b * )m 1 ) . On the right hand side, by Lemma 2.10, we rewrite the presentation of Θ in the basis σ − (B) such as Θ = ν b∈Bν σ − (b) ⊗ σ + (b * ). Then, we have ψ(m 1 ⊗ m 2 ⊗ m * 3 ⊗ m * 4 ) = ν b∈Bν v 2 −\|m 1 \|−\|m 2 \| f (\|m 2 \|, \|m 1 \|)m * 4 (σ − (b)m 2 )m * 3 (σ + (b * )m 1 ). It is enough to show that f (\|m 2 \|, \|m 1 \|) = f (\|m * 4 \|, \|m * 3 \|)v ν·(\|m 2 \|−\|m 1 \|−ν) c −ν,\|m 1 \|+\|m 2 \| . This equality holds for \|m 1 \| = −\|m * 3 \| − ν and \|m 2 \| = −\|m * 4 \| + ν. By this lemma, we have the following corollary. Lemma 4.6. We have two equalities of diagrams (f −1 (λ, λ)v 2 λ ) −1 = = f −1 (λ, λ)v 2 λ , where λ is the highest weight of M(+). Proof. We denote ϕ = = (id + ⊗ qtr) • (R +,+ ⊗ id + ) • (id + ⊗ coqtr), ψ = = (id + ⊗ qtr) • (R −1 +,+ ⊗ id + ) • (id + ⊗ coqtr). Since ϕ and ψ are U v,t -module homomorphisms from M(+) to M(+), both ϕ and ψ must be a multiple of the identity which is completely determined by the image of an extremal weight vector. Let m λ , m −λ ∈ M(+) be nonzero highest-weight and lowest-weight vectors. We have ϕ(m λ ) = (id + ⊗ qtr) • (R +, + ⊗ id + ) w∈B m λ ⊗ w ⊗ w * = (id + ⊗ qtr)( w∈B f (\|w\|, λ)w ⊗ m λ ⊗ w * ) = f (λ, λ)v 2 −λ m λ . Thus id + = f −1 (λ, λ)v 2 λ ϕ. By Corollary 2.14, we have Then we have id + = (f −1 (λ, λ)v 2 λ ) −1 ψ following (3.7). This finishes the proof. Lemma 4.7. We have two equalities of diagrams = ,(1) = . (2) Proof. The proofs of (1) and (2) are similar. We shall only show (1) in detail. Denoting by R +,− , (1) is equivalent to Then we compute R +,− = ϕ, where ϕ = (id − ⊗ id + ⊗ qtr) • (id − ⊗ R −1 +,+ ⊗ id − ) • (coev ⊗ id + ⊗ id − ).(id − ⊗ R −1 +,+ ⊗ id − ) • (coev ⊗ id + ⊗ id − )(m 1 ⊗ m * 2 ) = ν (−1) trν v ν·ν 2 v −ν b∈Bν w∈B v 2 \|w\| f −1 (\|w\| − ν, \|m 1 \| + ν) w * ⊗ b * m 1 ⊗ σ − (b)w ⊗ m * 2 . (4.2) By Lemma 2.5, we have By (4.1) and (4.4), we have R +,− (m 1 ⊗ m * 2 ) = ϕ(m 1 ⊗ m * 2 ). This finishes the proof. (id − ⊗ id + ⊗ qtr)( w∈B v 2 \|w\| f −1 (\|w\| − ν, \|m 1 \| + ν)w * ⊗ b * m 1 ⊗ σ − (b)w ⊗ m * 2 ) = w∈B v 2 ν f −1 (\|w\| − ν, \|m 1 \| + ν)m * 2 (σ(b)w)w * ⊗ b * m 1 = w∈B (−1) trν v ν·ν 2 v −ν v −ν·\|w\| c ν,\|w\| v 2 ν f −1 (\|w\| − ν, \|m 1 \| + ν)(bm * 2 )(w)w * ⊗ b * m 1 =(−1) trν v ν·ν 2 v ν v ν·(\|m * 2 \|−ν) c ν,−\|m * 2 \| f (\|m * 2 \|, \|m 1 \| + ν)bm * 2 ⊗ b * m 1 . By this lemma, we have the following corollary. Proof. To prove the proposition, it is enough to show that the evaluations of T at both sides of the relations of Theorem 4.2 coincide. The relations (i), (ii), (iii), (vi) and (vii) follow from Lemma 4.3, Corollary 4.5, Lemma 4.6 and Corollary 4.8, respectively. The relations (iv) and (v) hold for the properties of R. Remark Given a tangle L of type (n, n) for any n ∈ N, we get the closureL of L by connecting the origin and the endpoint one by one with no intersection. For example, see Figure 2. Furthermore, the evaluation of the functor T onL is an endomorphism of ground field Q(v, t). Therefore, T(L)(1) is a binary polynomial with the parameters v and t. That is called the quantum knot invariant. We expect this is a refinement of the knot invariant associated to the one-parameter case. L= ,L= . Figure 2 4 ) 4Let coqtr : Q(v, t) → M ⊗ M * be the Q(v, t)-linear map defined by 1 → w∈B w ⊗ w * for some homogeneous Q(v, t)-basis B of M. Then coqtr is a U v,t -module monomorphism. definition of Θ from Proposition 2.13. The linear transformation Θ = Θ M,M ′ : M ⊗ M ′ → M ⊗ M ′ is well-defined. Proposition 3.2. For any finite dimensional weight module M, M R − 1 1+,+ = P •f −1 • Θ. By Lemma 2.16, we compute ψ(m −λ ) = (id + ⊗ qtr) • (R −1 +,+ ⊗ id + ) w∈B m −λ ⊗ w ⊗ w * = (id + ⊗ qtr) w∈B f −1 (−λ, \|w\|)w ⊗ m −λ ⊗ w * = f −1 (−λ, −λ)v 2 λ m −λ . σ For any m 1 ∈ M(+) and m * 2 ∈ M(−), we have R +,− (m 1 ⊗ m * 2 ) = f (\|m * 2 \|, \|m 1 \|) 2.10, we have the representation of Θ in the basis σ(B) − (b) ⊗ b * . Corollary 4 . 8 . 48We have two equalities of diagrams= , = .Theorem 4.9. There exists a strict tensor functor T from the tangle category (OTa, ⊗, ∅) to the U v,t -module category (Mod, ⊗, Q(v, t)) such that T((+)) = M(+), T((−)) = M(−), andT(X + ) = (f (λ, λ)v 2 −λ ) −1 R, T( ) = coev, T( ) = coqtr, T(X − ) = f (λ, λ)v 2 −λ R −1 , T( ) = qtr, T( ) = ev,where λ is the highest weight of M(+). 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[ "Why should one compute periods of algebraic cycles?", "Why should one compute periods of algebraic cycles?" ]	[ "Hossein Movasati " ]	[]	[]	In this article we show how the data of integrals of algebraic differential forms over algebraic cycles can be used in order to prove that algebraic and Hodge cycle deformations of a given algebraic cycle are equivalent. As an example, we prove that most of the Hodge and algebraic cycles of the Fermat sextic fourfold cannot be deformed in the underlying parameter space. We then take a difference of two linear cycles inside the Fermat variety with intersection of codimension two in both cycles, and gather evidences that the Hodge locus corresponding to this is smooth and reduced. This implies the existence of new algebraic cycles in the Fermat variety whose existence is predicted by the Hodge conjecture for all hypersurfaces, but not the Fermat variety itself.... computer assisted proofs, as well as computer unassisted ones, can be good or bad. A good proof is a proof that makes us wiser, (Y. Manin).	null	[ "https://arxiv.org/pdf/1602.06607v4.pdf" ]	119,326,328	1602.06607	a59c262fe1c630e94a17219809ca24262e09e9fa	Why should one compute periods of algebraic cycles? Hossein Movasati Why should one compute periods of algebraic cycles? In this article we show how the data of integrals of algebraic differential forms over algebraic cycles can be used in order to prove that algebraic and Hodge cycle deformations of a given algebraic cycle are equivalent. As an example, we prove that most of the Hodge and algebraic cycles of the Fermat sextic fourfold cannot be deformed in the underlying parameter space. We then take a difference of two linear cycles inside the Fermat variety with intersection of codimension two in both cycles, and gather evidences that the Hodge locus corresponding to this is smooth and reduced. This implies the existence of new algebraic cycles in the Fermat variety whose existence is predicted by the Hodge conjecture for all hypersurfaces, but not the Fermat variety itself.... computer assisted proofs, as well as computer unassisted ones, can be good or bad. A good proof is a proof that makes us wiser, (Y. Manin). Introduction A quick answer to the question of the title is the following: if we compute such numbers, put them inside a certain matrix and compute its rank, then either we will be able to verify the Hodge conjecture for deformed Hodge cycles, or more interestingly, we will find a right place to look for counterexamples for the Hodge conjecture. In direction of the second situation, we collect evidences to Conjecture 1, and for the first situation we prove Theorem 1. In the present text all homologies with Z coefficients are up to torsion and all varieties are defined over complex numbers. Let n be an even number. For an integer −1 ≤ m ≤ n 2 let P n 2 ,P n 2 ⊂ P n+1 be projective spaces given by: (1) P n 2 :            x 0 − ζ 2d x 1 = 0, x 2 − ζ 2d x 3 = 0, x 4 − ζ 2d x 5 = 0, · · · x n − ζ 2d x n+1 = 0.P n 2 :                x 0 − ζ 2d x 1 = 0, · · · x 2m − ζ 2d x 2m+1 = 0, x 2m+2 − ζ 3 2d x 2m+3 = 0, · · · x n − ζ 3 2d x n+1 = 0. where ζ 2d := e 2π √ −1 2d . These are linear algebraic cycles in the Fermat variety X d n ⊂ P n+1 given by the homogeneous polynomial x d 0 + x d 1 + · · · + x d n+1 = 0, and satisfy P n 2 ∩P n 2 = P m . By convention P −1 means the empty set. In general we can take arbitrary linear cycles in the Fermat variety, see (19). Conjecture 1. Let n ≥ 6 be an even number, m := n 2 − 2 and let P n 2 andP n 2 be two linear cycles with P n 2 ∩P n 2 = P m inside the Fermat variety of degree d > 2(n+1) n−2 , and let Z ∞ be the intersection of a linear P n 2 +1 ⊂ P n+1 with X d n . There is a finite, nonempty set of pairs (r,ř) of coprime integers with the following property: there exists a semi-irreducible algebraic cycle Z of dimension n 2 in X d n such that 1. For some a, b ∈ Z, a = 0, the algebraic cycle Z is homologous to a(rP n 2 +řP n 2 ) + bZ ∞ . 1 Instituto de Matemática Pura e Aplicada, IMPA, Estrada Dona Castorina, 110, 22460-320, Rio de Janeiro, RJ, Brazil, www.impa.br/∼ hossein, [email protected] 2. The deformation space of the pair (X d n , Z), as an analytic variety, contains the intersection of deformation spaces of (X d n , P n 2 ) and (X d n ,P n 2 ) as a proper subset. An algebraic cycle Z = r i=1 n i Z i , n i ∈ Z in a smooth projective variety X is called semiirreducible if the pair (X, Z) can be deformed into (X t , Z t ) with Z t irreducible, for a precise definition see §10. Note that Z, a, b in the above conjecture depend on r andř. If d is a prime number or d = 4 or d is relatively prime with (n + 1)! then the Hodge conjecture for the Fermat variety X d n can be proved using only linear cycles, see [Ran81] and [Shi79a]. Therefore, the existence of the algebraic cycle Z in Conjecture 1 is not predicted by the Hodge conjecture for X d n . We have derived it assuming the Hodge conjecture for all smooth hypersurfaces of degree d and dimension n and few other conjectures with some computational evidences (Conjectures 8, Conjecture 10 and Conjecture 11). The number a is equal to 1 if the integral Hodge conjecture is true and the term bZ ∞ pops up because the relevant computations are done in primitive (co)homologies. Since the algebraic cycle Z is numerically equivalent to a(rP n 2 +řP n 2 ) + bZ ∞ this might be used to investigate its (non-)existence, at least for Fermat cubic tenfold. Our computations in this article suggest that (r,ř) = (1, −1) satisfies the property in Conjecture 1. Let C[x] d = C[x 0 , x 1 , · · · , x n+1 ] d be the set of homogeneous polynomials of degree d in n + 2 variables, and let T be the open subset of C[x] d parameterizing smooth hypersurfaces X of degree d and T 1 ⊂ T be its subset parameterizing those with a linear P n 2 inside X. We use the notation X t , t ∈ T and denote by 0 ∈ T the point corresponding to the Fermat variety, and so, X 0 = X d n . The algebraic variety T 1 is irreducible, however, as an analytic variety in a neighborhood (usual topology) of 0 ∈ T it has many irreducible components corresponding to deformations of a linear cycle inside X d n . Let us denote by V P n 2 the local branch of T 1 parameterizing deformations of the pair P n 2 ⊂ X d n . In general, for a Hodge cycle in H n (X d n , Z) we define the Hodge locus V δ 0 ⊂ (T, 0) which is an analytic scheme and its underlying analytic variety consists of points t ∈ (T, 0) such that the monodromy δ t ∈ H n (X t , Z) of δ 0 along a path in (T, 0) is still Hodge, see §4. For [P n 2 ] ∈ H n (X d n , Z) we know that V [P n 2 ] as analytic scheme is smooth and reduced and moreover V P n 2 = V [P n 2 ] , see the discussion after Theorem 6. This is not true for an arbitrary Hodge cycle. Conjecture 1 says that V P n 2 ∩ VP n 2 is a proper subset of the Hodge locus V r[P n 2 ]+ř[P n 2 ] , see Figure 1. In Conjecture 1 the case m = n 2 − 1 and r =ř = 1 is excluded, as the pair (X d n , P n 2 +P n 2 ) can be deformed into a hypersurface containing a complete intersection of type 1, 1, · · · , 1 n 2 times , 2. For small m's, the situation is not also strange. Theorem 1 ( [MV17]). Let (n, d, m) be one of the following triples (2, d, −1), 5 ≤ d ≤ 14, (4, 4, −1), (4, 5, −1), (4, 6, −1), (4, 5, 0), (4, 6, 0), (6, 3, −1), (6, 4, −1), (6, 4, 0), (8, 3, −1), (8, 3, 0), (10, 3, −1), (10, 3, 0), (10, 3, 1), and P n 2 andP n 2 be linear cycles in (1). The Hodge locus passing through the Fermat point 0 ∈ T and corresponding to deformations of the Hodge cycle r[P n 2 ] +ř[P n 2 ] ∈ H n (X d n , Z) with P n 2 ∩P n 2 = P m and r,ř ∈ Z, r = 0,ř = 0 is smooth and reduced. Moreover, its underlying analytic variety is simply the intersection V P n 2 ∩ VP n 2 . The cases (n, d) = (2, 4), (4, 3) are the only cases such that the ( n 2 + 1, n 2 − 1) Hodge number of X d n is equal to one, and these are out of our discussion as all Hodge loci V δ 0 are of codimension one, smooth and reduced. For the discussion of these cases and a baby version of Conjecture We conjecture that for a fixed n ≥ 6 and d > 2(n+1) n−2 , there is 0 ≤ M n,d < n 2 − 2 depending only on n and d such that for m ≤ M n,d , respectively M n,d < m < n 2 − 1, we have similar statements as in Theorem 1, respectively Conjecture 1. We do not have any idea how to describe M n,d in general. We expect that Theorem 1 for m = −1 is always true. In this case P n 2 andP n 2 do not intersect each other. The restriction on n and d in Theorem 1 is due to the fact that our proof is computer assisted, and upon a better computer programing and a better device, it might be improved. For now, the author does not see any theoretical proof. The first evidence for Conjecture 1 is the fact that for many examples of n and d, the codimension of the Zariski tangent space of the analytic scheme V r[P n 2 ]+ř[P n 2 ] is strictly smaller than the codimension of V P n 2 ∩VP n 2 which is smooth. In order to be able to investigate the smoothness and reducedness of this analytic scheme, we have worked out Theorem 14 which is just computing a Taylor series. Its importance must not be underestimated. The linear part of such Taylor series encode the whole data of infinitesimal variation of Hodge structures (IVHS) introduced by Griffiths and his coauthors in 1980's, and from this one can derive most of the applications of IVHS, such as global Torelli problem, see [CG80]. In particular, the proof of Theorem 1 uses just such linear parts. In a personal communication C. Voisin pointed out the difficulties on higher order approximation of the Noether-Lefschetz locus. This motivated the author to elaborate some of his old ideas in [Mov11] and develop it into Theorem 14. The second order approximations in cohomological terms (similar to IVHS), has been formulated in [Mac05], however it is not enough for the investigation of Conjecture 1, see Theorem 2, and it turns out one has to deal with third and fourth order approximations, see Theorem 3. We use Theorem 14 to check reducedness and smoothness of components of the Hodge loci. We break the property of being reduced and smooth into N -smooth for all N ∈ N, see §8, and prove the following theorem which is not covered in Theorem 1. Theorem 2. Let (n, d, m) be one of the triples (6, 3, 1), (6, 3, 0), (8, 3, 1) (2) (4, 4, 0), (8, 3, 2), (8, 3, 1), (10, 3, 3), (10, 3, 2), and P n 2 andP n 2 be linear cycles in (1). For all r,ř ∈ Z with 1 ≤ \|r\| ≤ \|ř\| ≤ 10 the analytic scheme V r[P n 2 ]+ř[P n 2 ] with P n 2 ∩P n 2 = P m is 2-smooth. It is 3-smooth in the cases (2) and for (n, d, m, r,ř) = (4, 4, 0, 1, −1). It is 4-smooth in the case (n, d, m, r,ř) = (6, 3, 1, 1, −1) and (n, d, m) = (6, 3, 0). Note that the triples in Theorem 2 are not covered in Theorem 1 and we do not know the corresponding Hodge locus. In order to solve Conjecture 1 we will need to identify non-reduced Hodge loci. We prove that: Theorem 3. Let P n 2 andP n 2 be linear cycles in (1) with P n 2 ∩P n 2 = P m . The analytic scheme V r[P n 2 ]+ř[P n 2 ] is either singular at the Fermat point 0 or it is non-reduced, in the following cases: 1. For all r,ř ∈ Z, 1 ≤ \|r\| ≤ \|ř\| ≤ 10, r =ř, m = n 2 − 1 and (n, d) in the list (2, d), 5 ≤ d ≤ 9,(4) (4, 4), (4, 5), (6, 3), (8, 3) (5) 2. For all r,ř ∈ Z, 1 ≤ \|r\| ≤ \|ř\| ≤ 10, r = −ř and (n, d, m) in the list (4, 4, 0), (6, 3, 1), (8, 3, 2) The upper bounds for \|r\| and \|ř\| is due to our computational methods, and it would not be difficult to remove this hypothesis. The verification of the case (n, d, m) = (8, 3, 2) in the second item by a computer takes more than 14 days! Theorem 3 in the case (n, d, m) = (2, 5, 0) and without the upper bound on \|r\|, \|ř\| follows from a theorem of Voisin in [Voi89], see Exercise 2, page 154 [Voi03] and its reproduction in [Mov17a] Exercise 16.9. Based on Theorem 2 and Theorem 3 we may conjecture that for (n, d, m, r,ř) = (4, 4, 0, 1, −1), (6, 3, 1, 1, −1), the analytic scheme V r[P n 2 ]+ř[P n 2 ] is smooth and reduced. If this is the case, its underlying analytic variety is bigger than V [P n 2 ] ∩ V [P n 2 ] (see §6), and so, we may try to formulate similar statements as in Conjecture 1 in these cases. However, one of the main ingredients of Conjecture 1 fails to be true in lower degrees, see Conjecture 8 and comments after this. The present article together with the book [Mov17a] is written during the years 2014-2017. One of the main aims of the book [Mov17a] has been to focus on computational aspects of Hodge theory. From this book we have just collected few results relevant to the content of this article, and in particular the study of the components of the Hodge locus passing through the Fermat point. The proof of Theorem 5, Theorem 6, Theorem 11, Theorem 12 and Theorem 14 are theoretical, whereas the proof of Theorem 1, Theorem 2, Theorem 3, Theorem 7, Theorem 8, Theorem 10 are computer assisted. These are partial verifications of many conjectures, for which we have to work with particular examples of d and n. In many cases we have just mentioned these as comments after each conjecture and have avoided producing more theorem-style statements. An undergraduate student in mathematics interested in challenging problems is invited to read conjectures in §9. We have to confess that we have not done our best to verify such conjectures as much as the computer performs the computations, and have contented ourselves to few special cases. There are few other results in the book [Mov17a] which are not announced here, and they might be useful for the investigation of Conjecture 1. The computer codes used in the present text are written as procedures in the library foliation.lib (version 2.20) of Singular, see [GPS01]. The reader who wants to get used to them is referred to [Mov17a] Chapter 18. This is mainly for codes used utill §6. From this section on, the name of procedures appears in the foot note of the pages where they are used. A different computer implementation of the proofs would be essential for two main reasons: first, it will be another confirmation of the results of the present paper, second, it will produce more results that the author was not able to obtain by his own primitive codes. This may produce precise conjectures for arbitrary dimension n and degree d. The organization of the text is as follows. Sections 2,3,4,5 are essentially the first version of the article which appeared in the Arxiv in 2015. These are the announcement of some of the author's results in the book [Mov17a]. In §2 we reformulate the Hodge conjecture using integrals. In §3 we introduce an alternative Hodge conjecture. This compares the deformation space of both algebraic and Hodge cycles. In §4 we recall the missing ingredient in the formulation of infinitesimal variation of Hodge structures. This is namely periods of Hodge/algebraic cycles. We then relate it to the alternative Hodge conjecture. In §5 we focus on Hodge cycles in the Fermat variety which cannot be deformed to nearby hypersurfaces. We then present the formula of periods of linear cycles inside the Fermat variety. From §6 we start to examine Conjecture 1. In this section we also prove Theorem 1. We first observe that the Zariski tangent space of the Hodge locus corresponding to the Hodge cycle [P n 2 +P n 2 ] has codimension strictly less than the codimension of the locus corresponding to deformations of the algebraic cycle P n 2 +P n 2 . This indicates the existence of a strange component of the Hodge locus provided that such a component is smooth and reduced. For this reason in §7 we introduce Conjecture 8 which ensures us that such components exists for certain linear combination of P n 2 andP n 2 . In order to investigate this conjecture, in §8 we announce our main result on the full power series expansion of periods. This might be used in order to investigate the smoothness and reducedness of the components of the Hodge loci. In this section we also prove Theorem 2 and Theorem 3. In §9 we introduce few other conjectures purely of linear algebraic nature. These are the last missing pieces in the proof of Conjecture 1. Finally, in §11 we explain how to handle Conjecture 1. My heartfelt thanks go to P. Deligne for all his emails in January and February 2016 which motivated me and gave me more courage and inspiration to work on my book [Mov17a] and the present article. This was in a time I was getting many disappointments and complains. I would like to thank C. Voisin for her comments on higher order approximation of Noether-Lefschetz locus. This research has not been possible without the excellent ambient of my home institute IMPA in Rio de Janeiro and the hospitality of MPIM at Bonn during many short visits. My sincere thanks go to both institutes. The last version of the article was written during a visit of Paris VII. I would like to thank H. Mourtada and F. El Zein for the invitation and CNRS for financial support. Finally, I would like to dedicate this article to two women, one in my memories and the other by my side: Rogayeh Mollayipour, my mother, who thought me lessons of life no other could do it, Sara Ochoa, my wife, whose contribution to the existence of this article is not less than mine. Hodge conjecture For a complex smooth projective variety X, an even number n, an element ω of the algebraic de Rham cohomology ω ∈ H n dR (X) and an irreducible subvariety Z of dimension n 2 in X, by a period of Z we simply mean (6) 1 (2π √ −1) n 2 [Z] ω, where [Z] ∈ H n (X, Z) is the topological class induced by Z. All the homologies with integer coefficients are modulo torsions, and hence they are free Z-modules. We have to use a canonical isomorphism between the algebraic de Rham cohomology and the usual one defined by C ∞ -forms in order to say that the integration makes sense, see Grothendieck's article [Gro66]. However, this does not give any clue how to compute such an integral. In general, integrals are transcendental numbers, however, in our particular case if X, Z, ω are defined over a subfield k of complex numbers then (6) is also in k, see Proposition 1.5 in Deligne's lecture notes in [DMOS82], and so it must be computable. In the C ∞ context many of integrals (6) are automatically zero. This is the main content of the celebrated Hodge conjecture: Conjecture 2 (Hodge Conjecture). Let X be a smooth projective variety of even dimension n and δ ∈ H n (X, Z) be a Hodge cycle, that is, δ ω = 0, for all closed (p, q)-form in X with p > n 2 , p + q = n. Then there is an algebraic cycle s i=1 n i Z i , n i ∈ Z, dim(Z i ) = n 2 and a natural number a ∈ N such that a · δ = n i [Z i ]. Using Poincaré duality our version of the Hodge conjecture is equivalent to the official one, see for instance Deligne's announcement of the Hodge conjecture [Del06], however, we wrote it in this format in order to point out that the Hodge decomposition is not needed in its announcement and bring it to its origin which is the study of integrals due to Abel, Poincaré, Picard among many others. For a prehistory of the Hodge conjecture see [Mov17a], Chapters 2 and 3. An alternative conjecture The Hodge conjecture does not give any information about non-vanishing integrals (6). In this article we show that explicit computations of (6) lead us to verifications of the following alternative for the Hodge conjecture: Conjecture 3 (Alternative Hodge Conjecture). Let {X t } t∈T be a family of complex smooth projective varieties of even dimension n, and let Z 0 be a fixed irreducible algebraic cycle of dimension n 2 in X 0 for 0 ∈ T. There is an open neighborhood U of 0 in T (in the usual topology) such that for all t ∈ U if the monodromy δ t ∈ H n (X t , Z) of δ 0 = [Z 0 ] is a Hodge cycle, then there is an algebraic deformation Z t ⊂ X t of Z 0 ⊂ X 0 such that δ t = [Z t ]. In other words, deformations of Z 0 as a Hodge cycle and as an algebraic cycle are the same. Before explaining the relation of this conjecture with integrals (6), we say few words about the importance of Conjecture 3. First of all, Conjecture 3 might be false in general, therefore, it might be called a property of Z 0 . P. Deligne pointed out that there are additional obstructions to the hope that algebraic cycles could be constructed by deformation (personal communication, 31 January 2016). For instance, the dimension of the intermediate Jacobian coming from the largest sub Hodge structure of H n−1 (X 0 , Q) ∩ (H n 2 , n 2 −1 ⊕ H n 2 , n 2 −1 ) might jump down by deformation. This observation does not apply to a smooth hypersurface, for which only the middle cohomology is non-trivial. We are interested in cases in which Conjecture 3 is true, see Theorem 4 below. Both Hodge conjecture and Conjecture 3 claim that a given Hodge cycle must be algebraic, however, note that Conjecture 3 provides a candidate for such an algebraic cycle, whereas the Hodge conjecture doesn't, and so, it must be easier than the Hodge conjecture. Verifications of Conjecture 3 support the Hodge conjecture, however, a counterexample to Conjecture 3 might not be a counterexample to the Hodge conjecture, because one may have an algebraic cycle homologous to, but different from, the given one in Conjecture 3. In [Gro66] page 103 Grothendieck states a conjecture which is as follows: let X → S be a smooth morphism of schemes and let S be connected and reduced. A global section α of H 2p dR (X/S) is algebraic at every fiber s ∈ S if and only if it is a flat section with respect to the Gauss-Manin connection and it is algebraic for one point s ∈ S. Conjecture 3, for instance for complete intersections inside hypersurfaces, implies this conjecture in the same context, however the vice versa is not true. The variety T d defined in §4 might be a proper subset of a component of the Hodge locus. This would imply that Z is homologous to another algebraic cycle with a bigger deformation space. This cannot happen for the linear case d = (1, 1, · · · , 1), see Theorem 4 below, and many examples of n and d and d, see [MV17]. The article [Blo72] is built upon the Grothendieck's conjecture explained above and it considers semi-regular algebraic cycles, that is, the semi-regularity map π : H 1 (Z, N X/Z ) → H n 2 +1 (X, Ω n 2 −1 ) is injective. The semi-regularity is a very strong condition. For instance, for curves inside surfaces, [Blo72] only considers the semi-regular curves with H 1 (Z, N X/Z ) = 0. Using Serre duality, one can easily see that this is not satisfied for curves with self intersection less than 2g − 2, where g is the genus of Z. A simple application of adjunction formula shows that apart from few cases, complete intersection curves inside surfaces do not satisfy this condition. In situations where the Hodge conjecture is true, for instance for surfaces, Conjecture 3 is still a non-trivial statement. For a smooth hypersurface X ⊂ P 3 of degree d ≥ 4 and a line P 1 ⊂ X, deformations of P 1 as a Hodge cycle and as an algebraic curve are the same. This follows from classical IVHS techniques introduced in [CGGH83]. In [Gre88,Gre89], [Voi88], Green and Voisin prove a stronger statement which says that the space of surfaces X ⊂ P 3 containing a line P 1 is the only component of the Noether-Lefschetz locus of minimum codimension d − 3. In order to reproduce the full statement of Green and Voisin's results in our context and in a neighborhood of the Fermat point, see Conjecture 9 and the comments after. In a similar way some other results of Voisin on Noether-Lefschetz loci, see [Voi90], fit into the framework of Conjecture 3. A weaker version of the mentioned statement in higher dimensions is generalized in the following way: Theorem 4 ([Mov17b] Theorem 2). For any smooth hypersurface X of degree d and dimension n in a Zariski neighborhood of the Fermat variety with d ≥ 2 + 4 n and a linear projective space P n 2 ⊂ X, deformations of P n 2 as an algebraic cycle and Hodge cycle are the same. Infinitesimal variation of Hodge structures for Fermat variety The relation between integrals (6) and Conjecture 3 is established through the so-called infinitesimal variation of Hodge structures developed in [CGGH83]. This is explained in [Mov17b], where the author has tried to keep the classical language of IVHS, and so we do not reproduce it here. The main application is going to be on Hodge and Noether-Lefschetz loci. The reader is referred to Voisin's expository article [Voi13] which contains a full exposition and main references on this topic. In order to keep the content of this text elementary, we explain this for complete intersection algebraic cycles inside hypersurfaces, and in particular, the Fermat variety. Let T be the parameter space of smooth hypersurfaces of degree d in P n+1 . A hypersurface X = X t , t ∈ T is given by the projectivization of f ( x 0 , x 1 , · · · , x n+1 ) = 0, where f is a homogeneous polynomial of degree d. Fix integers 1 ≤ d 1 , d 2 , . . . , d n 2 +1 ≤ d and d := (d 1 , d 2 , . . . , d n 2 +1 ). Let T d ⊂ T be the parameter space of smooth hypersurfaces with f = f 1 f n 2 +2 + · · · + f n 2 +1 f n+2 , deg(f i ) = d i , deg(f n 2 +1+i ) = d − d i , where f i 's are homogeneous polynomials. The algebraic cycle (7) Z := P{f 1 = f 2 = · · · = f n 2 +1 = 0} ⊂ X is called a complete intersection (of type d) in X. Note that this cycle is a complete intersection in P n+1 and it is not a complete intersection of X with other hypersurfaces. Let (8) ω i := Resi x i · n+1 j=0 (−1) j x j dx 0 ∧ · · · ∧ dx j ∧ · · · ∧ dx n+1 f k with k := n+2+ n+1 e=0 ie d , where Resi : H n+1 dR (P n+1 − X) → H n dR (X) is the residue map and x i = x i 0 0 · · · x i n+1 n+1 . After Griffiths [Gri69], we know that δ ∈ H n (X, Z) is a Hodge cycle if and only if (9) δ ω i = 0, ∀i with n + 2 + n+1 e=0 i e d ≤ n 2 . A cycle δ ∈ H n (X, Z) is called primitive if its intersection with [Z ∞ ] is zero. Recall that Z ∞ is the intersection of a linear P n 2 +1 ⊂ P n+1 with X. The Z-module H n (X, Z) 0 by definition is the set of primitive cycles. We denote by Hodge n (X, Z) ⊂ H n (X, Z) the Z-modules of n-dimensional Hodge cycles in X, and by Hodge n (X, Z) 0 its submodule consisting of primitive cycles. All the Z-modules in this text are up to torsions, and hence they are free. Let us now focus on the Fermat variety X d n which is obtained by the projectivization of (10) X d n : x d 0 + x d 1 + · · · + x d n+1 = 0. We denote by 0 ∈ T the point corresponding to X d n , that is, X 0 = X d n . Hodge cycles of the Fermat variety have been extensively studied by Shioda in his seminal works [Shi79b,Shi79a,Shi81]. We are mainly interested in the Hodge cycles [Z], where Z is a complete intersection of type d in P n+1 which lies in X d n . This is because all the examples of n and d in which the Hodge conjecture is known for X d n , one has only used this type of algebraic cycles, see [Mov17a] Chapter 17. The periods of a Hodge cycle δ ∈ Hodge n (X d n , Z) are defined in the following way (11) p i = p i (δ) := 1 (2π √ −1) n 2 δ ω i , n+1 e=0 i e = ( n 2 + 1)d − (n + 2). Using Deligne's result in [DMOS82] Proposition 1.5, we know that p i 's are in an abelian extension of of Q(ζ d ). If p i 's are all zero then δ is necessarily in the one dimension Q-vector space generated by [Z ∞ ]. We are going to explain the role of these numbers in the deformation of Hodge cycles. Definition 1. For natural numbers N , n and d let us define (12) I N := (i 0 , i 1 , . . . , i n+1 ) ∈ Z n+2 \| 0 ≤ i e ≤ d − 2, i 0 + i 1 + · · · + i n+1 = N Assume that n is even and d ≥ 2 + 4 n . Consider complex numbers p i indexed by i ∈ I ( n 2 +1)d−n−2 . For any other i which is not in the set I ( n 2 +1)d−n−2 , we define p i to be zero. Let [p i+j ] be a matrix whose rows and columns are indexed by i ∈ I n 2 d−n−2 and j ∈ I d , respectively, and in its (i, j) entry we have p i+j . The numbers #I d , #I n 2 d−n−2 , #I ( n 2 +1)d−n−2 are respectively, the dimension of the moduli space, ( n 2 +1, n 2 −1) Hodge number and ( n 2 , n 2 ) Hodge number minus one, of smooth hypersurfaces of dimension n and degree d. The following theorem justifies the importance of the algebraic numbers p i 's in (11). Theorem 5. Let X d n be the Fermat variety of dimension n and degree d parameterized by the point 0 ∈ T. Let also δ 0 ∈ Hodge n (X d n , Z) be a Hodge cycle. The kernel of the matrix [p i+j ] is canonically identified with the Zariski tangent space of the Hodge locus V δ 0 passing through 0 ∈ T and corresponding to δ 0 . The Hodge locus mentioned in the above theorem is actually the analytic scheme defined by (13) O V δ 0 := O T,0 δt ω 1 , δt ω 2 , · · · , δt ω a where ω 1 , ω 2 , · · · , ω a are reindexed ω i 's in (9). These are sections of the cohomology bundle H n dR (X t ), t ∈ (T, 0) such that for t ∈ (T, 0) they form a basis of F n 2 +1 H n dR (X t ), where F i 's are the pieces of the Hodge filtration of H n dR (X t ). Its points are all t in a small neighborhood of 0 such that the monodromy δ t ∈ H n (X t , Z) of δ 0 is a Hodge cycle, or equivalently, δt ω 1 = δt ω 2 = · · · = δt ω a = 0. This is a local analytic subset of T and by a deep theorem of Cattani-Deligne-Kaplan in [CDK95] we know that it is algebraic. This together with the fact that Hodge cycles of the Fermat variety are absolute and Deligne's Principle B in [DMOS82] implies that such an algebraic set is defined overQ, for details see [Voi13] Proposition 5.7. The Hodge locus in T is the union of all such local loci defined as before for all t ∈ T (one might take different ω i 's as in (9)). Theorem 5 follows from Voisin's result [Voi03] 5.3.3 on the Zariski tangent space of the Hodge locus and the computations of the infinitesimal variation of Hodge structures for the Fermat variety in [Mov17b]. An alternative proof using some ideas of holomorphic foliations is given in the later reference. n + 1 + d − a i 1 − a i 2 − · · · − a i k n + 1 where (a 1 , a 2 , . . . , a n+2 ) = (d 1 , d 2 , . . . , d n 2 +1 , d−d 1 , d−d 2 , . . . , d−d n 2 +1 ) and the second sum runs through all k elements (without order) of a i , i = 1, 2, . . . , n + 2, then T d is a component of the Hodge locus. In particular Conjecture 3 is true for smooth hypersurfaces X ⊂ P n+1 containing a complete intersection of type d, and in a non-empty Zariski open subset of T d . The number in the right hand side of (14) is actually the codimension of T d in T, see [Mov17a] Proposition 17.5, and so Theorem 6 is a consequence of this fact and Theorem 5, see [Mov17a] Theorem 17.6. For arbitrary d and n the hypothesis of Theorem 6 is verified for projective spaces Z = P n 2 ⊂ X, that is, for the case d = (1, 1, . . . , 1). This is (15) rank([p i+j ]) = n 2 + d d − ( n 2 + 1) 2 , see [Mov17b]. In this way we have derived Theorem 4. For this particular class of algebraic cycles, it is possible to prove the identity (15) without computing p i 's. We may expect or conjecture that the equality (14) is always true. This is the case for many examples of complete intersection algebraic cycles worked out in [MV17]. This includes the author's favorite example (n, d) = (4, 6), that is, the sextic Fermat fourfold: (16) X 6 4 : x 6 0 + x 6 1 + x 6 2 + x 6 3 + x 6 4 + x 6 5 = 0. The Fermat cubic tenfold X 3 10 : x 3 0 + x 3 1 + · · · + x 3 11 = 0 has the Hodge numbers 0, 0, 0, 1, 220, 925, 220, 1, 0, 0, 0 and the Q-vector space of its Hodge cycles has the maximum dimension which is 925. In this case the Hodge conjecture can be verified using linear cycles P 5 , see Theorem 9. We have only one possibility for T d . This is namely d = (1, 1, 1, 1, 1, 1). Its codimension is 20. General Hodge cycles for Fermat variety We say that a Hodge cycle δ ∈ Hodge n (X d n , Q) is general if rank[p i+j ] attains the maximal rank, that is, rank[p i+j ] = minimum{#I d , #I n 2 d−n−2 }. Note that #I n 2 d−n−2 (resp, #I d ) is the number of rows (resp. columns) of [p i+j ]. If there exists a general Hodge cycle then the subvariety of Hodge n (X d n , Q) given by rank[p i+j ] < min{#I d , #I n 2 d−n−2 } is proper and so there is a Zariski open subset U of Hodge n (X d n , Q) such that all δ ∈ U are general. This will hopefully justify the name. Moreover, Theorem 5 implies that for a general Hodge cycle δ the Hodge locus V δ is always smooth and reduced. n−2 then the right hand side of (17) is #I d which is also the dimension of the moduli of hypersurfaces of degree d and dimension n. Therefore, Conjecture 4 in this case implies that general Hodge cycles of the Fermat variety cannot be deformed in the moduli space of hypersurfaces of degree d and dimension n, in other words, any deformation of a general Hodge cycle of the Fermat variety to a nearby hypersurface X ⊂ P n+1 implies that X is obtained from X d n by a linear transformation of P n+1 . For a moment assume that we have a collection of algebraic cycles Z i , i = 1, 2, . . . , s such that [Z i ]'s generate the Q-vector space Hodge n (X d n , Q) of Hodge cycles, and so, we know the Hodge conjecture for X d n is valid. This together with Conjecture 4 implies that a general algebraic cycle s i=1 n i Z i , n i ∈ Z has a deformation space of the expected codimension which is the right hand side of (17). In particular, for d > 2(n+1) n−2 such an algebraic cycle cannot be deformed at all if we consider the parameter space T parameterizing the homogeneous polynomials of the type (18) f := x d 0 + x d 1 + · · · + x d n+1 − j∈I d t j x j . Any smooth hypersurface in a Zariski open neighborhood of the Fermat point 0 after a linear transformation of P n+1 , can be written as the zero set of some f in this format. The equalities (14) and (17) The upper bound on d is just due to the limitation of our computer and it might be improved if one uses a better computing machine. Theorem 7 for n = 4, d = 6 says the following: Theorem 8. A general Hodge cycle δ 0 ∈ H 4 (X 6 4 , Q) is not deformable, that is, the monodromy δ t ∈ H 4 (X t , Z), t ∈ (T, 0) of δ 0 to X t is no more a Hodge cycle. Note that we are using the parameter space in (18), otherwise, we should have stated that X t is obtained by a linear transformation of X 6 4 . For the computations of periods of Hodge cycles and proof of Theorem 7 and Theorem 8 see [Mov17a] Chapter 15 and 16. See also §18.8 for details of the computer codes used for the proofs. The same codes for the Fermat cubic tenfold runs out of memory. In this case one might use Theorem 11. Theorem 9 ([Ran81], [Shi79a], [AS83]). Suppose that either d is a prime number or d = 4 or d is relatively prime with (n + 1)!. Then Hodge n (X d n , Q) is generated by the homology classes of the linear cycles P n 2 , and in particular, the Hodge conjecture for X d n is true. This theorem is the outcome of many efforts in order to prove the Hodge conjecture for X d n using linear projective cycles. The cases (n, d) = (2, 6), (4, 6) are not covered by this theorem because such algebraic cycles are not enough in these cases. N. Aoki in [Aok87], inspired by his work with Shioda [AS83], has introduced more algebraic cycles and in this way he has been able to verify the Hodge conjecture for many other Fermat varieties, and in particular for the sextic Fermat fourfold. In this case we can determine the homology classes of linear cycles P n 2 explicitly, [Mov17a] Section 16.7. This together with Theorem 8 gives us: Theorem 10. A general Z-linear combination of projective linear cycles P 2 is not deformable in the moduli space of degree 6 hypersurfaces in P 5 . We propose two different methods in order to compute integrals (6). The first method is purely topological and it is based on the computation of the intersection numbers of algebraic cycles with vanishing cycles. In the case of the Fermat variety, we are able to write down vanishing cycles explicitly, however, they are singular, even though they are homeomorphic to spheres, and many interesting algebraic cycles of the Fermat variety intersect them in their singular points. This makes the computation of intersection numbers harder. The second method is purely algebraic and it is a generalization of Carlson-Griffiths computations in [CG80]. One has to compute the restriction of differential n-forms in X to the top cohomology of Z, and then, one has to compute the so-called trace map. The second method is the main topic of the Ph.D. thesis of R. Villaflor, see [Vil18]. For a = (a 1 , a 3 , . . . , a n+1 ) ∈ {0, 1, 2              x b 0 − ζ 1+2a 1 2d x b 1 = 0, x b 2 − ζ 1+2a 3 2d x b 3 = 0, x b 4 − ζ 1+2a 5 2d x b 5 = 0, · · · x bn − ζ 1+2a n+1 2d x b n+1 = 0. We call it a linear cycle inside the Fermat variety. Hopefully, this a, b notation will not be confused with the integers a, b in Conjecture 1. In order to avoid repetitions, we may assume that b 0 = 0 and for i an even number b i is the smallest number in {0, 1, . . . , n+1}\{b 0 , b 1 , b 2 , . . . , b i−1 }. In this way the number of linear cycles is (20) (n + 1) · (n − 1) · · · 3 · 1 · d n 2 +1 . For linear cycles the computation of periods is a direct consequence of a theorem of Carlson and Griffiths in [CG80]: Theorem 11. For i ∈ I ( n 2 +1)d−n−2 we have 1 (2π √ −1) n 2 P n 2 a,b ω i =    sign(b)·(−1) n 2 d n 2 +1 · n 2 ! ζ 2d if i b 2e−2 + i b 2e−1 = d − 2, ∀e = 1, ..., n 2 + 1, 0 otherwise. where ζ 2d is the 2d-th primitive root of unity and = n 2 e=0 (i b 2e + 1) · (1 + 2a 2e+1 ). This is done in [MV17] and it is the main ingredient of Theorem 1 Theorem 4 follows from the verification of the equality (15) for periods of linear cycles computed in Theorem 11. This verification turns out to be an elementary problem. Using Theorem 11 we can make Theorem 10 more concrete. We have verified the conjecture for (n, d) in: 2 Sum of two linear cycles (2, d), 5 ≤ d ≤ 8, (4, 4), (6, 3). 2 The procedure ndm is used for this purpose. We can use the automorphism group G d n of the Fermat variety and we can assume that P n 2 is (19) with a = (0, 0, · · · , 0) and b = (0, 1, · · · , n + 1). In order to avoid Conjecture 5 we will fix our choice of linear cycles: P n 2 = P n 2 a,b with a = (0, 0, · · · , 0), b = (0, 1, · · · , n + 1) P n 2 = P n 2 a,b with a = (0, 0, · · · , 0 m+1 times , 1, 1, · · · , 1), b = (0, 1, · · · , n + 1) which are those used in Introduction. For examples of H d n (m) see Table 1. For a sequence of natural numbers a = (a 1 , . . . , a s ) let us define (22) C a = n + 1 + d n + 1 − s k=1 (−1) k−1 a i 1 +a i 2 +···+a i k ≤d n + 1 + d − a i 1 − a i 2 − · · · − a i k n + 1 , where the second sum runs through all k elements (without order) of a i , i = 1, 2, . . . , s. By our convention, the projective space P −1 means the empty set. By abuse of notation we write a b := a, a, · · · , a b times . Hopefully, there will be no confusion with the exponential a b . Theorem 12. Let P n 2 ,P n 2 be two linear algebraic cycles in a smooth hypersurface of dimension n and degree d and with the intersection P m . We have (23) K d n (m) := codim(V P n 2 ∩ VP n 2 ) = 2C 1 n 2 +1 ,(d−1) n 2 +1 − C 1 n−m+1 ,(d−1) m+1 . In particular, if P n 2 does not intersectP n 2 then V P n 2 intersects VP n 2 transversely. The proof is a simple application of Koszul complex and can be found in Section 17.9 of [Mov17a]. We are now going to analyze the number H d n (m) for m = n 2 , n 2 − 1, · · · . Let P (the second equality follows from Theorem 12). One of the by-products of the proof is that V P n 2 as an analytic scheme is smooth and reduced. For m = n 2 − 1, we have H d n ( n 2 − 1) = C 1 n 2 ,2,(d−1) n 2 ,d−2 ≤ K d n ( n 2 − 1) = 2C 1 n 2 +1 ,(d−1) n 2 +1 − C 1 n 2 +2 ,(d−1) n 2 . The first equality is conjectural and we can verify it for special cases of n and d by a computer, see [MV17], Section 5. In this case the algebraic cycle P n 2 +P n 2 can be deformed into a complete intersection algebraic cycle of type (1 n 2 , 2), and so, the inequality is justified. Since the underlying complex variety of the Hodge locus V [P holds for arbitrary m between −1 and n 2 . We conjecture that H d n (−1) = 2 · C 1 n 2 +1 ,(d−1) n 2 +1 which is the value of K d n (m) (note that C 1 n+1 = 0). This is the same as to say that: Conjecture 6. Let P n 2 ,P n 2 be two linear algebraic cycles in the Fermat variety and with no common point. The only deformations of P n 2 +P n 2 as an algebraic or Hodge cycle is again a sum of two linear cycles. Particular cases of this conjecture has been announced in Theorem 1 (those with m = −1). It might happen that in (24) we have a strict inequality, see for instance Table 1. Conjecture 7. For n ≥ 6 we have (25) H d n ( n 2 − 2) < K d n ( n 2 − 2). Our favorite examples for verifying Conjecture 7 are cubic Fermat varieties, that is d = 3. For n ≥ 4 we have the following range: (26) n 2 + 1 3 ≤ rank([p i+j ]) ≤ n + 2 min{3, n 2 − 2} and in Table 1 We were also able to compute the five-tuples (n, d, m\|H d n (m), K d n (m)) in the list below: (4, 4, 0\|11, 12), (4, 4, −1\|12, 12), We were not able to compute more data such as ? in (4, 7, 0\|?, 54). For n = 2 and 4 ≤ d ≤ 14 we were also able to check Conjecture 6. Note that for the quartic Fermat fourfold we have the range 6 ≤ rank([p i+j ]) ≤ 21 and T 1,1,2 has codimension 8. Proof of Theorem 1 for r =ř = 1. This is just the outcome of above computations in which H d n (m) = K d n (m). The full proof will be given after Theorem 13. For r =ř = 1 we have Theorem 1 for (n, d, m) in (27) (12, 3, −1), (12, 3, 0), (12, 3, 1), (12, 3, 2), however, we were not able to verify Theorem 13 in these cases. For the convenience of the reader we have also computed the table of Hodge numbers for cubic Fermat varieties. Note that for d = 3, n = 4 the Hodge conjecture is well-known, see [Zuc77]. Theorem 13. For all pairs (n, d) in Theorem 1 with arbitrary −1 ≤ m ≤ n 2 and all x ∈ Q with x = 0, we have and so this number does only depend on (n, d, m) and not on x. Proof. Let a := H d n (m) be the number in the right hand side of (28) and let A(x) := [p i+j (P n 2 + xP n 2 )]. Except for a finite number of x ∈ Q, we have rank(A(x)) ≥ a and in order to prove the equality, it is enough to check it for a + 2 distinct values of x. This is because if rank(A(x)) > a then we have a (a + 1) × (a + 1) minor of A(x) whose determinant is not zero. This is a polynomial of degree at most a + 1 in x, and it has (a + 2) roots which leads to a contradiction. This argument implies that except for a finite number of values for x we have rank(A(x)) = a. These are the roots of det(B(x)) = 0, where B is any a × a minor of A(x) such that P (x) := det(B(x)) is not identically zero. We find such a minor and compute rank(A(x)) for all rational roots of P (x) and prove that this is a except for x = 0. 4 It seems interesting that only for (n, d, m) = (6, 3, 1), (6, 3, 0), (8, 3, 2),(8, 3, 1),(6, 4, 1),(10, 3, 3), (10, 3, 2) we find a rational root of P (x), and in all these cases it is x = −1. This seems to have some relation with Conjecture 1 for (r,ř) = (1, −1). Proof of Theorem 1: For all the cases in Theorem 1 rank rp i+j (P n 2 ) +řp i+j (P n 2 ) = rank p i+j (P n 2 ) +p i+j (P n 2 ) = K d n (m), where for the first equality we have used Theorem 13. We know that V P n 2 ∩ VP n 2 is the subset of the analytic variety underlying V r[P n 2 ]+ř[P n 2 ] and its codimension is K d n (m). This proves the theorem. Smooth and reduced Hodge loci Based on the computation in §6, we have formulated Conjecture 7, and we further claim that: In this case, the Hodge locus V [P n 2 ]+[P n 2 ] is smooth and reduced at 0 and it parameterizes hypersurfaces with a complete intersection of type (1 n 2 , 2), see the comments before Theorem 1. The proof can be found in [MV17]. The analytic scheme V r[P n 2 ]+ř[P n 2 ] is non-reduced or singular at 0 in the cases covered in Theorem 3. Other evidences to Conjecture 8 are listed in Theorem 2 and Theorem 3. Assuming the Hodge conjecture, the points of the Hodge locus V r[P n 2 ]+ř[P n 2 ] parametrizes hypersurfaces with certain algebraic cycles. We do not have any idea how such algebraic cycles look like. In order to verify Conjecture 8 without constructing algebraic cycles, we have to analyze the the generators δt ω i of the defining ideal of the Hodge locus in (13). These are integrals depending on the parameter t ∈ T and their linear part is gathered in the matrix [p i+j ]. If Conjecture 8 is true in these cases then we have discovered a new Hodge locus, different from V [P n 2 ] , V [P n 2 ] and their intersection. The whole discussion of §8 has the goal to provide tools to analyze Conjecture 8. The creation of a formula In this section we compute the Taylor series of the integration of differential forms over monodromies of the algebraic cycle P n 2 a,b inside the Fermat variety. Let us consider the hypersurface X t in the projective space P n+1 given by the homogeneous polynomial: (29) f t := x d 0 + x d 1 + · · · + x d n+1 − α t α x α = 0, t = (t α ) α∈I ∈ (T, 0), where α runs through a finite subset I of N n+2 0 with n+1 i=0 α i = d. In practice, we will take the set I of all such α with the additional constrain 0 ≤ α i ≤ d − 2. For a rational number r let [r] be the integer part of r, that is [r] ≤ r < [r] + 1, and {r} := r − [r]. Let also (x) y := x(x + 1)(x + 2) · · · (x + y − 1), (x) 0 := 1 be the Pochhammer symbol. For β ∈ N n+2 0 , β ∈ N n+2 0 is defined by the rules: 0 ≤β i ≤ d − 1, β i ≡ dβi . Theorem 14. Let δ t ∈ H n (X t , Z), t ∈ (T, 0) be the monodromy (parallel transport) of the cycle δ 0 := [P n 2 a,b ] ∈ H n (X 0 , Z) along a path which connects 0 to t. For a monomial x β = x β 0 0 x β 1 1 x β 2 2 · · · x β n+1 n+1 with k := n+1 i=0 β i +1 d ∈ N we have (30) C (2π √ −1) n 2 δt Resi x β Ω f k t = a:I→N 0 1 a! D β+a * · e π √ −1·E β+a * · t a , where the sum runs through all #I-tuples a = (a α , α ∈ I) of non-negative integers such that forβ := β + a * we have (31) β b 2e + 1 d + β b 2e+1 + 1 d = 1, ∀e = 0, ..., n 2 , and By definition a Hodge locus V δ 0 is 1-smooth. Theorem 2 and Theorem 3, and in particular their computational proof, must be considered our strongest evidence to Conjecture 8. Proof of Theorem 2 and Theorem 3 . The proof is done using a computer implementation of the Taylor series (30). 5 In order to be sure that this Taylor series and its computer implementation are mistake-free we have also checked many N -smoothness property which are already proved in Theorem 1. In Theorem 3 Item 1 we have proved that the corresponding Hodge locus is not 2-smooth except in the following case which we highlight it. Let P 1 andP 1 be two lines in the Fermat quintic surface intersecting each other in a point. The Hodge locus V rP 1 +řP 1 for all r,ř ∈ Z is 2-smooth. Moreover it is not 3-smooth for 0 < \|r\| < \|ř\| ≤ 10. In Theorem 3 Item 2 (resp. 3) we have proved that the corresponding Hodge locus is not 3-smooth (resp. 4-smooth). The property of being N -smooth for larger N 's is out of the capacity of my computer codes, see §12 for some comments. Uniqueness of components of the Hodge locus A Hodge cycle δ ∈ H n (X d n , Z) is uniquely determined by its periods p i (δ). This data gives the Poincaré dual of δ in cohomology, and hence, the classical Hodge class in the literature. Let Ho d n be the Z-module of period vectors p of Hodge cycles. We will also use its projectivization PHo d n (two elements p andp in the Z-module are the same if there are non-zero integers a anď a such that ap =ǎp). This Z-module can be described in an elementary linear algebra context without referring to advanced topics, such as homology and algebraic de Rham cohomology, see Chapter 16 of [Mov17a]. Therefore, the conjectures of the present section can be understood by any undergraduate mathematics student! If either d is a prime number or d = 4 or d is relatively prime with (n + 1)! then we may redefine Ho n d the Z-modules generated by p a,b , where p a,b i := ζ n 2 e=0 (i b 2e +1)·(1+2a 2e+1 ) 2d if i b 2e−2 + i b 2e−1 = d − 2,n 2 + d d − ( n 2 + 1) 2 ≤ rank([p i+j ]) ≤      n 2 d−1 n+1 if d < 2(n+1) n−2 , d+n n+1 − (n + 2) if d = 2(n+1) n−2 , d+n+1 n+1 − (n + 2) 2 if d > 2(n+1) n−2 . Before stating our main conjecture in this section, let us state a simpler one. Conjecture 9. Let n ≥ 2 be an even number and d ≥ 3 an integer with (n, d) = (2, 4), (4, 3). Let also p ∈ Ho d n such that (36) rank([p i+j ]) = n 2 + d d − ( n 2 + 1) 2 . Then p, up to multiplication by a rational number, is necessarily of the form p a,b . 5 See SmoothReduced and TaylorSeries. 6 The list of p a,b 's is implemented in the procedure ListPeriodLinearCycle. One can also formulate a similar conjecture for the next admissible rank. For n = 2 Voisin's result in [Voi88] tells us that this must be 2d − 7 := codim(T 1,2 ). For further discussion on this topic see [Mov17a] Chapter 19. It might happen that in Conjecture 9 one must exclude more examples of (n, d). Note that for (n, d) = (2, 4), (4, 3) both sides of (43) are equal to one for all non-zero p. We need to write down in an elementary language when the linear cycles P n 2 i andP n 2 j underlying two period vectors p i , i = (a, b) andp j , j = (ǎ,b), respectively, have the intersection P m . This is as follows: A bicycle attached to the permutations b andb is a sequence (c 1 c 2 . . . c r ) with c i ∈ {0, 1, 2, . . . , n + 1} and such that if we define c r+1 = c 1 then for 1 ≤ i ≤ r odd (resp. even) there is an even number k with 0 ≤ k ≤ n + 1 such that {c i , c i+1 } = {b k , b k+1 } (resp. {c i , c i+1 } = {b k ,b k+1 }) ) and there is no repetition among c i 's. By definition there is a sequence of even numbers k 1 , k 2 , · · · such that {c 1 , c 2 } = {b k 1 , b k 1 +1 }, {c 2 , c 3 } = {b k 2 ,b k 2 +1 }, {c 3 , c 4 } = {b k 3 , b k 3 +1 }, . . . . Bicycles are defined up to twice shifting c i 's, that is, (c 1 c 2 c 3 · · · c r ) = (c 3 · · · c r c 1 c 2 ) etc., and the involution (c 1 c 2 c 3 · · · c r−1 c r ) = (c r c r−1 · · · c 3 c 2 c 1 ). For example, for the permutations b = (0, 1, 2, 3, 4, 5),b = (1, 0, 5, 3, 4, 2) we have in total two bicycles (01), (2354). Note that bicycles give us in a natural way a partition of {0, 1, . . . , n + 1}. For such a bicycle we define its conductor to be the sum over k, as before, of the following elements: if c i = b k and c i+1 = b k+1 (resp. c i =b k and c i+1 =b k+1 ) then the element 1 + 2a k+1 (resp. 1 + 2ǎ k+1 ), and if if c i = b k+1 and c i+1 = b k (resp. c i =b k+1 and c i+1 =b k ) then −1 − 2a k+1 (resp. −1 − 2ǎ k+1 ). Because of the involution, the conductor is defined up to sign. In our example, the conductor of (01) and (2354) are respectively given by 1 + 2a 1 + 1 + 2ǎ 1 , 1 + 2a 3 − 1 − 2ǎ 3 − 1 − 2a 5 + 1 + 2ǎ 5 . A bicycle is called new if 2d divides its conductor, and is called old otherwise. Let m ij be the number of new bicycles attached to (i, j) minus one. 3. Zp a,b + Zpǎ ,b with m a,b,ǎ,b = n 2 − 2 and so rank[p i+j ] = H d n ( n 2 − 2). A complete analysis of Conjecture 10 would require an intensive search for the elements p ∈ Ho d n of low rank([p i+j ]). It might be true for n = 2 and large d's, and this has to do with the Harris-Voisin conjecture, see [Mov17b], and will be discussed somewhere else. Note that the numbers in items 1,2,3 of Conjecture 10 for n = 2 are respectively d − 3, 2d − 7 and 2d − 6 (for the last one see Conjecture 6). We just content ourselves with the following strategy for confirming Conjecture 10. Let p i , i = 1, 2, 3 be three distinct vectors of the form p a,b . We claim that for d > 3 we have (37) rank([p i+j ]) > H d n ( n 2 − 2), where p = p 1 + p 2 + p 3 . The number H d n ( n 2 − 2) is computed in §6 and so we check in total N 3 inequalities (37), where N is the number of p a,b 's in (20). This is too many computations and we have checked (37) for samples of p i 's for (n, d) = (4, 6). In this way we have also observed that the lower bound for d is necessary as (37) is not true for our favorite examples (n, d) = (4, 4), (6, 3). For d = 3, the vector p in (37) can be zero. 7 The final ingredient of Conjecture 1 is the following. In virtue of Theorem 5, it compares the Zariski tangent spaces of components of the Hodge locus passing through the Fermat point. Conjecture 11. Let n ≥ 6 and d ≥ 3. There is no inclusion between any two vector spaces of the form (38) ker([rp i+j +řp i+j ]) where p andp ranges in the set of all p a,b with m a,b,ǎ,b = n 2 − 1, n 2 − 2, r,ř ∈ Z coprime and m a,b,ǎ,b = n 2 , r = 1,ř = 0. Let δ,δ ∈ H n (X d n , Q) be two Hodge cycles with , V δ 's are of codimension one, smooth and reduced, and so, any inclusion (39) will be an equality and it implies that the period vectors ofδ,δ are the same. This implies that δ = aδ + b[Z ∞ ] for some a, b ∈ Q, and so, V δ = Vδ. We can verify Conjecture 11 in the following way. For simplicity we restrict ourselves to the pairs (n, d) in Theorem 1 and r =ř = 1. Let us take two matrices A and B as inside kernel in (38). Let also A * B be the concatenation of A and B by putting the rows of A and B as the rows of A * B. Therefore, A * B is a (2#I n 2 d−n−2 ) × (#I d ) matrix. In order to prove that there is no inclusion between ker(A) and ker(B) it is enough to prove that (20). This is a huge number even for small values of n and d. 8 Note that the vector space in (38) for (n, d, m)'s in Theorem 1 is equal to the Zariski tangent space of V P n 2 ∩ VP n 2 at the Fermat point, and hence it does not depend on r andř. This is the main reason why we restrict ourselves to the cases in Conjecture 11. 10 Semi-irreducible algebraic cycles Let X be a smooth projective variety and Z = r i=1 n i Z i , n i ∈ Z be an algebraic cycle in X, with Z i an irreducible subvariety of codimension n 2 in X. The following definition is done using analytic deformations and it would not be hard to state it in the algebraic context. Definition 3. We say that Z = r i=1 n i Z i , n i ∈ Z is semi-irreducible if there is a smooth analytic variety X , an irreducible subvariety Z ⊂ X of codimension n 2 (possibly singular), a holomorphic map f : X → (C, 0) such that 1. f is smooth and proper over (C, 0) with X as a fiber over 0. Therefore, all the fibers X t of f are C ∞ isomorphic to X. 2. The fiber Z t of f \| Z over t = 0 is irreducible and Z 0 = ∪ r i=1 Z i . 3. The homological cycle [Z] := r i=1 n i [Z i ] ∈ H n (X, Z) is the monodromy of [Z t ] ∈ H n (X t , Z). It is reasonable to expect that Item 3 is equivalent to a geometric phenomena, purely expressible in terms of degeneration of algebraic varieties. For instance, one might expect that n i layers of the algebraic cycle Z t accumulate on Z i , and hence semi-irreducibility implies the positivity of n i 's. Moreover, for distinct Z i and Z j , the intersection Z i ∩ Z j is of codimension one in both Z i and Z j , because Z i 's are irreducible and of codimension one in Z. In particular, the algebraic cycle rP . Recall that Z ∞ is the intersection of a linear P n 2 +1 with X d n . The following theorem can be considered as a counterpart of Conjecture 1. Theorem 15. Let (n, d) = (2, 4), (4, 3) and let Z be an algebraic cycle of dimension n 2 and with integer coefficients, in a smooth hypersurface of dimension n and degree d. If [Z] ∈ H n (X d n , Q) is not a rational multiple of [Z ∞ ] then there is a semi-irreducible algebraic cycleŽ of dimension n 2 in X d n such that aZ + bŽ + cZ ∞ is homologous to zero for some a, b, c ∈ Z with a, b = 0 . Proof. The algebraic cycle Z induces a homology class δ 0 = [Z] ∈ H n (X d n , Z) and the Hodge locus V δ 0 is given by the zero locus of a single integral f (t) := δt ω 0 , where ω 0 is given by (8) for i = (0, 0, · · · , 0). By our hypothesis on Z, f is not identically zero and since δ 0 = [Z] it vanishes at 0 ∈ T. We show that V δ 0 is smooth and reduced, and for this it is enough to show that the linear part of f is not identically zero. This follows from ∇ ∂ ∂t i ω 0 = ω i , i ∈ I d , I d = I ( n 2 +1)d−n−2 and the fact that ω 0 , ω i , i ∈ I d form a basis of F 1 of H n dR (X). Here, ∇ is the Gauss-Manin connection of the family of hypersurfaces given by (29). The Hodge conjecture in both cases is well-known. In the first case it is the Lefschetz (1, 1) theorem and in the second case it is a result of Zucker in [Zuc77]. This implies that δ t = [Z t ], where Z t := r i=1 n i Z i,t , Z i,t ⊂ X t , t ∈ V δ 0 , dim(Z i,t ) = n 2 , n i ∈ Q and for generic t, Z i,t is irreducible. Since V δ 0 ⊂ V [Z i,0 ] , we conclude that [Z i,0 ] = a i [Z] + b i [Z ∞ ] for some a i , b i ∈ Q. By our hypothesis on Z, one of a i 's is not zero let us call it a 1 . We get [Z] = a −1 1 [Z 1,0 ] − b 1 a −1 1 [Z ∞ ]. In Theorem 15 let us assume that Z is a sum of linear cycles. It would be useful to see whether the algebraic cycleŽ is a sum of linear cycles. One might start with the sum of two lines in the Fermat surface X 4 2 without any common points (the case (n, d, m) = (2, 4, −1)). 11 How to to deal with Conjecture 1? In this section we sketch a strategy to prove Conjecture 1 which follows the same guideline as of the proof of Theorem 15. Let δ 0 := r[P n 2 ] +ř[P n 2 ] ∈ H n (X d n , Z) with P n 2 ∩P n 2 = P m , m = n 2 − 2 and H d n (m) < K d n (m). Let also δ t ∈ H n (X t , Z), t ∈ (T, 0) be its monodromy to nearby fibers. Conjecture 8 implies that the intersection of V P n 2 and VP n 2 is a proper subset of the underlying n k Z k,t , Z k,t ⊂ X t , t ∈ V δ 0 , dim(Z k,t ) = n 2 , n k ∈ Z, such that Z k,t is irreducible for generic t and Z t is homologous to a non-zero integral multiple of δ t , see Figure 2. By Conjecture 8 we know that V δ 0 is smooth and reduced, and so, we have the inclusion of analytic schemes In order to proceed, we consider the cases of Fermat varieties such that linear cycles generates the the space of Hodge cycles over rational numbers (these are the cases in Theorem 9), or we assume Conjecture 10 for Ho d n being the the lattice of periods of all Hodge cycles and not just linear cycles. We apply Conjecture 10 and we conclude that for some linear cycles P Theorem 2 and Theorem 3. Such a Hodge locus is not 3-smooth except for (r,ř) = (1, ±1) for which we have even 4-smoothness in the case (8, 3, 2). The coefficients of the Taylor series in Theorem 14 seem to be defined in a reasonable ring, for instance, for (n, d) = (4, 3), (6, 3) and some sample truncated Taylor series, the ring of coefficients is Z[ 1 d , ζ 2d ]. If so, one may consider them modulo prime ideals, and in this way, study many related conjectures. The tools introduced in this article can be used in order to answer the following question which produces an explicit counterexample to a conjecture of J. Harris: determine the integer d (conjecturally less than 10) such that the Noether-Lefschetz locus of surfaces of degree d (resp degree < d) has infinite (resp. finite) number of special components crossing the Fermat point. Notice that Voisin's counterexample in [Voi91] is for a very big d. This problem will be studied in subsequent articles. For this and its generalization to higher dimensions one needs to classify linear combination of linear cycles in the Fermat variety which are semi-irreducible. The combinatorics of arrangement of linear cycles seems to play some role in this question. The author's favorite examples in this article have been cubic varieties, see Manin's book [Man86] for an overview of some results and techniques. Cubic surfaces carry the famous 27-lines which is exactly the number (20) of linear cycles for the Fermat cubic surface. Hodge conjecture is known for cubic fourfolds (see [Zuc77]), and for a restricted class of cubic 8-folds the Hodge conjecture is also known (see [Ter90]). In general the Hodge conjecture remains open for cubic hypersurfaces of dimension n ≥ 6. Conjecture 1 makes sense starting from cubic tenfolds whose moduli is 220-dimensional. It might be useful to review all the results in this case and to see what one can say more about the algebraic cycle Z in this conjecture. (42) V δ 0 ⊂ V [Z k, Figure 1 : 1Hodge locus of sum of linear cycles 1 see §10. Theorem 6 . 6Let X d n be the Fermat variety (10) and let Z be a complete intersection of type d inside X d n . Let also p i be the periods of δ = [Z] defined in (11). 1 +a i 2 +···+a i k ≤d Conjecture 4 . 4The Fermat variety X d n for d ≥ 2 + 4 n has always a general Hodge cycle.Let us discuss two extreme cases in the above conjecture. First, if n = 2 then a general Hodge cycle has rank[p i+j ] = d−1 3 . Conjecture 4 and Theorem 5 imply that there are infinite number of components of the Noether-Lefschetz locus of codimension d−1 3 passing through the Fermat point 0 ∈ T, provided that there are infinite number of general Hodge cycles with different ker[p i+j ]. This is compatible with the result in[CHM88] that the components of the Noether-Lefschetz locus with the maximal codimension are dense in T, both in the Zariski and usual topology. Second, if d > 2(n+1) Conjecture 5 . 5The number H d n (m) depends only on d, n, m and not on the choice of P as in Introduction. Let us first consider the case m = n 2 . For the proof of Theorem 4 we have verified the first equality in n (m) ≤ K d n (m) 3 3 3For the computations of H d n (m) and K d n (m) we have used the procedures SumTwoLinearCycle and Codim, respectively. Conjecture 8 .2 8Let P n 2 andP n 2 be two linear cycles in the Fermat variety X d n with d ≥ 2 + = P m with −1 ≤ m ≤ n 2 − 1 and H d n (m) < K d n (m). There is a finite number of coprime non-zero integers r,ř such that the analytic scheme V r[P n 2 ]+ř[P n 2 ] is smooth and reduced. 4 See GoodMinor and ConstantRank. Conjecture 10 . 10Let n ≥ 4 and d > 2(n+1) n−2 . If for some p ∈ Ho d n , p = 0 we haverank[p i+j ] ≤ H d n ( n 2 − 2),then p after multiplication with a natural number is in the set 1. Zp a,b and so rank([p i+j ]Zp a,b + Zpǎ ,b with m a,b,ǎ,b = n 2 − 1 and so rank([p i+j ]) [p i+j (δ)]) ⊂ ker([p i+j (δ)]),that is, the Zariski tangent space of V δ is contained in the Zariski tangent space of Vδ. The first trivial example to this situation is whenδ is a rational multiple of [Z ∞ ] for which we have p(δ) = 0 and Vδ = (T, 0). Let us assume that none ofδ and δ is a rational multiple of [Z ∞ ]. Next examples for this situation are in Theorem 1. In this theorem the Zariski tangent space of the Hodge locus V P m and r,ř ∈ Z, r = 0,ř = 0, at the Fermat point does not depend on r,ř. For larger m's such as n 2 − 1, the Zariski tangent spaces of V r[Pn 2 ]+ř[P n 2 ] at the Fermat point form a pencil of linear spaces and so there is no inclusion among its members. For (n, d) = (2, 4), (4, 3) A * B) > rank(A), rank(B). The number of verifications (40) is approximately N 4 , where N is the number of linear cycles given in P m , m ≤ n 2 − 2 in Conjecture 1 is not semi-irreducible. A smooth hypersurface of degree d and dimension n has the Hodge numbers h n,0 = h Figure 2 : 2Sum of linear cycles II analytic variety of V δ 0 . If the Hodge conjecture is true then there is an algebraic family of 0 ] 0, k = 1, 2, . . . , r which implies that(43) ker[p i+j (δ 0 )] ⊂ ker[p i+j (Z k,0 )], and so rank([p i+j (Z k,0 )]) ≤ rank[p i+j (δ 0 )]. The Hodge numbers of the fourth cohomology of a smooth sextic fourfold is 1,426,1752, 426,1, and the Fermat sextic fourfold has a very peculiar property that the Q-vector space of its Hodge cycles has the maximum dimension which is 1752. In this case the matrix [p i+j ] is a quadratic 426×426 matrix. We have only 10 possibilities for the locus T d of hypersurfaces with a complete intersection algebraic cycle. The corresponding data are listed in the table below:Codimension of the loci of complete intersection algebraic cycles (d 1 , d 2 , d 3 ) codim T (T d ) (d 1 , d 2 , d 3 ) codim T (T d ) (1, 1, 1) 19 (1, 3, 3) 71 (1, 1, 2) 32 (2, 2, 2) 92 (1, 1, 3) 37 (2, 2, 3) 106 (1, 2, 2) 54 (2, 3, 3) 122 (1, 2, 3) 62 (3, 3, 3) 141 can be checked computationally, as far as, we take particular examples of the degree d and the dimension n, compute the periods p i and the rank of [p i+j ]. Here, is the resultTheorem 7. The Fermat surface X d 2 , 4 ≤ d ≤ 8 has a general Hodge cycle. The Fermat fourfold X d 4 , 3 ≤ d ≤ 6 has also a general Hodge cycle. Table 2 : 2Hodge numbers For this computations we have used the procedure SumThreeLinearCycle. 8 For this proof we have used DistinctHodgeLocus. ] for r,ř coprime non-zero integers and \|r\|, \|ř\| ≤ 10 is 7-smooth and 4-smooth for (n, d, m) = (6, 3, 1) and (4, 4, 0), respectively. Therefore, it seems that we are in situations similar to Theorem 1. For (n, d, m) = (8, 3, 2), (10, 3, 3) the situation is similar to This theorem is the outcome of many computations in[Mov17a]. Its proof is obtained after a careful analysis of the Gauss-Manin connection of the full family of hypersurfaces around the Fermat point 0 ∈ T. For thus see Sections 13.9, 13.10, 17.11 of this book. In the next paragraph we are going to explain how to use Theorem 14 and give evidences for Conjecture 8.Recall the definition of the Hodge locus as an scheme in (13). Let f 1 , f 2 , · · · , f a ∈ O T,0 be the integrals such that f 1 = f 2 = · · · = f a = 0 is the underlying analytic variety of the Hodge locus V δ 0 . We take f 1 , f 2 , . . . , f k , k ≤ a such that the linear part of f 1 , f 2 , . . . , f k form a basis of the vector space generated by the linear part of all f 1 , f 2 , . . . , f a . By Griffiths transversality those of f i which come from For equivalentlybe the homogeneous decomposition of f , f i and g i , respectively. The identity (33) reduces to infinite number of polynomial identities:. . .Definition 2. For a Hodge locus V δ 0 as in (13) and N ∈ N we say that it is N -smooth if the first N equations in (34) holds for all f = f i , i = k + 1, k + 2, · · · , a. In other words (33) holds up to monomials of degree ≥ N + 1. Another important information about the algebraic cycle Z is a lower bound of the dimension of the Hilbert scheme parameterizing deformations of the pair (X d n , Z). One may look for the classification of the components of the Hilbert schemes of projective varieties in order to see whether such a Z exists or not. For instance, we know that if Z ⊂ P n+1 is an irreducible reduced projective variety of dimension n 2 and degree 2 then it is necessarily a complete intersection of type 1 n 2 , 2, see[EH87]. One might look for generalizations of this kind of results.Final commentsOne of the main difficulties in generalizing our main theorems in Introduction for other cases is that the moduli of hypersurfaces of dimension n and degree d is of dimension #I d = d+n+1 n+1 − (n + 2) 2 which is two big even for small values of n and d. One has to prepare similar tables as inTable 1with smaller number of parameters and then start to analyze N -smoothness. For some suggestions see[Mov17a]Exercises 15.13, 15.16, 15.17. The author has analyzed statements similar to Theorem 2 and Theorem 3 for hypersurfaces given by homogeneous polynomials of the form (48) f := A(x 0 , x 2 , . . . , x n ) + B(x 1 , x 3 , . . . , x n+1 ).The moduli of such hypersurfaces is of dimension 2 · d+ n 2 n 2 − 2( n 2 + 1) 2 and this makes the computations much faster. Here are some sample results mainly in direction of Theorem 3. The Hodge locus V r[P n 2 ]+ř[P n Some new algebraic cycles on Fermat varieties. Noboru Aoki, J. 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[]	[ "Jorge Soto-Andrade ", "M Francisca Yáñez ", "Jorge Soto-Andrade ", "M Francisca Yáñez " ]	[]	[]	We show that the Gelfand character χG of a finite group G (i.e. the sum of all irreducible complex characters of G ) may be realized as a " twisted trace" g → T r(ρg • T ) for a suitable involutive linear automorphism of L 2 (G), where ρ stands for the right regular representation of G in L 2 (G). We prove further that, under certain hypotheses, T may be obtained as T (f ) = f •L, where L is an involutive antiautomorphism of the group G so that T r(ρg • T ) = \|{h ∈ G : hg = L(h)}\|. We also give in the case of the group G = P GL(2, Fq) a positive answer to a question of K. W. Johnson asking whether it is possible to express the Gelfand character χG as a polynomial in a single irreducible character η of G.	10.1515/jgth-2020-0207	[ "https://arxiv.org/pdf/1303.4800v4.pdf" ]	55,176,299	1303.4800	6eeee2fb80e709ea7e06f6dd8bf0a650c5d6db59	20 Mar 2013 Jorge Soto-Andrade M Francisca Yáñez 20 Mar 2013arXiv:1303.4800v1 [math.GR] ON REALIZATIONS OF THE GELFAND CHARACTER OF A FINITE GROUPGelfand charactertwisted tracetotal characterSteinberg characterGelfand Model We show that the Gelfand character χG of a finite group G (i.e. the sum of all irreducible complex characters of G ) may be realized as a " twisted trace" g → T r(ρg • T ) for a suitable involutive linear automorphism of L 2 (G), where ρ stands for the right regular representation of G in L 2 (G). We prove further that, under certain hypotheses, T may be obtained as T (f ) = f •L, where L is an involutive antiautomorphism of the group G so that T r(ρg • T ) = \|{h ∈ G : hg = L(h)}\|. We also give in the case of the group G = P GL(2, Fq) a positive answer to a question of K. W. Johnson asking whether it is possible to express the Gelfand character χG as a polynomial in a single irreducible character η of G. Introduction The realization of the Gelfand character χ G of a finite group G, i.e. the sum of all ordinary irreducible characters of G is an old problem [6,3,9]. One approach to this problem is to try to obtain χ G by twisting the trace of some very natural representation (V, π) of G, like the regular representation, by a suitable linear automorphism T of its underlying space V, so as to obtain χ G (g) = T r(π g • T ) for all g ∈ G. Recall that twisted traces appear in many contexts in mathematics [2,3,4]. Another possible approach is to try to realize χ G as a polynomial in some remarkable character of G. In this vein, K. W. Johnson has asked whether it is possible to express the Gelfand character of G as a polynomial, with integer coefficients, in a single irreducible character η of G. In [5] an affirmative answer is given for the case of the dihedral groups. In this paper, we consider the case where G = P GL(2, q), q odd, and we take η to be the Steinberg character of G. We prove first that the Gelfand character of G cannot be expressed as a polynomial in the Steinberg character of G with integer coefficients. We prove then that it can be expressed however as a polynomial, of degree 2, in the Steinberg character of G with coefficients in authors partially supported by Fondecyt Grant 1120571. the ring R generated over Z by the linear characters of G, i. e. the unit character and the sign character. The Gelfand character χ G as a Twisted Trace Let G be a finite group and let (L 2 (G), ρ) and (L 2 (G), σ) be the right and left regular representation of G respectively; let (U k , π k ) (1 ≤ k ≤ r) denote all the irreducible unitary representations of G and I π k (1 ≤ k ≤ r) the isotypic component of type π k of ρ. Let U k be an orthonormal basis of U k and (e k ij (g)) 1≤i,j≤n k the matrix of the operator π k (g) (g ∈ G) with respect to the basis U k of U k , where n k denotes the dimension of U k . Moreover χ k denotes the character afforded by π k . The matrix coeficients e k ij (1 ≤ i, j ≤ n k , 1 ≤ k ≤ r) provide then an orthonormal basis B for the Hilbert space L 2 (G), and they satisfy the relations: (1) e k ij (g −1 ) = e k ji (g) and e k ij (gh) = n k l=1 e k il (g)e k lj (h) Proposition 1. Let (V, π 1 ) and (V, π 2 ) be two isomorphic representations of a finite group G such that π 1 h • π 2 g = π 2 g • π 1 h and let T be an involutive automorphism of V, that intertwines the representations π 1 and π 2 of G. Then the function T r(π 1 ? • T ) defined on G with values on C is a central function on G and so it is a linear complex combination of irreducible characters of G. Proof. For g, h in G, we have T r(π 1 g −1 hg • T ) = T r(π 1 g • T • π 1 g −1 h ) = = T r(T • π 2 g • π 1 g −1 h ) = T r(π 2 g • T • π 1 g −1 h ) = = T r(π 2 h • T ) = T r(T • π 1 h ) = = T r(π 1 h • T ). Recall that all irreducible representations π k of a finite group G are unitarizable. The isotypic component I π k of type (U, π k ) of ρ is isomorphic to U U * . Theorem 1. Let T the linear application of L 2 (G) defined by T (e k ij ) = e k ji for all e k ij ∈ B and the homomorphism σ from G to Aut(L 2 (G)) defined by σ g (e k ij ) = n k l=1 e k li (g)e k lj for all e k ij ∈ B and for each g ∈ G. Then T is an involutive automorphism of L 2 (G) and σ is a representation of G such that: i. ρ g • T = T • σ g , g ∈ G. ii. ρ g • σ h = σ h • ρ g , g, h ∈ G. iii. T r(ρ g • T ) = χ G (g), g ∈ G Proof. Since σ g (e k ij ) = T (ρ ' g (T (e k ij ) )) for all g ∈ G and e k ij ∈ B we obtain that σ g is an automorphism of L 2 (G) such that ρ g • T = T • σ g for each g ∈ G. Furthermore for g, h ∈ G and e k ij ∈ B verify that (ρ g • σ h )(e k ij ) = n k l=1 e k li (h) n k m=1 e k mj (g)e k lm = n k m=1 e k mj (g) n k l=1 e k li (h)e k lm = n k m=1 e mj (g) σ h (e k im ) = ( σ h • ρ g )(e k ij ). Finally, since (ρ g • T )(e k ij ) = n k l=1 e k li (g)e k jl for all g ∈ G and e k ij ∈ B then T r(ρ g • T ) = r k=1 ( n k i=1 e k ii (g) = r k=1 χ k (g) = χ G (g). Next we will prove that under certain conditions the central function T r(ρ g • T ) can be realized via an involutive anti-automorphism L of the group G. Theorem 2. Let L be an involutive anti-automorphism of G, such that χ k (L(g)) = χ k (g), for all g ∈ G and let L ⋆ be the automorphism of L 2 (G) defined by L ⋆ (f ) = f • L. Then for all g ∈ G, we have i. ρ g • L ⋆ = L ⋆ • σ L(g) −1 ii. ρ g • σ L(g) −1 = σ L(g) −1 • ρ g iii. T r(ρ g • L ⋆ ) = \|{h ∈ G : h −1 L(h)) = g}\|, g ∈ G iv. T r(ρ g • L ⋆ ) = T r(ρ g −1 • L ⋆ ) v. T r(ρ g • L ⋆ ) = Σ r k=1 ε k χ k (g) where ε k = ±1. . Proof. The proof of i. and ii. is a straightforward calculation. By computing the trace of ρ g • L ⋆ with respect to the canonical basis {δ g : g ∈ G} where δ g (h) = δ g,h , h ∈ g we obtain iii. If we notice that h −1 L(h) = g if and only if L(h) −1 h = g −1 and L is one to one we get iv. Let σ ⋆ the twisted representation of the left regular representation σ of G defined by σ ⋆ g = σ L(g) −1 . Due i) and ii) the representations (L 2 (G), ρ) and (L 2 (G), σ ⋆ ) of G satisfy all the conditions of the proposition 1, then we deduce that the complex function T r(ρ g • L ⋆ ) is central and T r(ρ g • L ⋆ ) = r k=1 ε k χ k (g). The antiautomorphism L induces an antiautomorphismL on the complex group algebra C[G]. Since χ k (L(g)) = χ k (g), g ∈ G, 1 ≤ k ≤ r,L acts as the identity on the center and therefore induces an antiautomorphismL k on each simple component of C[G] ∼ = 1≤k≤r M (n k , C). Due to Skolem-Noether theorem,L k is conjugated to the transposition:L k (a) = b −1 a t b, a ∈ M (n k , C) and b ∈ Gl(n k , C). Furthermore we have that a =L(L(a)) = b −1 b t ab −1 b t , then b −1 b t belongs to the center and therefore b t = ε k b with ε k = ±1. In this way, for each representation (U k , π k ) of G a symmetric form or a symplectic form b exists, with respect to which the linear operators π k (g) andL(π k (g)) are conjugated. Therefore if we consider the bilinear form u, v = v t bu then (2) L (π k (g))(u), v = v t (bL(π k (g)))u = v t (π k (g)) t bu = u, π k (g)(v) Let us suppose that ε k = 1 for some k. We choose an orthonormal basis U + k = {u i , 1 ≤ i ≤ n k } of U k ,respect to the symmetric form b and we denote by e k ij (g) = π k (g)(u j ), u i the matrix coefficients of π k (g) with respect to this basis. Due to eq.2 we have the following relations between the matrix coefficients ofL(π k (g)) and π k (g) (3) e k ij (L(g)) = u j , π k (g)(u i ) = e k ji (g). Let E k = e k ij (g) : 1 ≤ i, j ≤ n k and T r k (ρ g • L ⋆ ) the restriction of T r(ρ g • L ⋆ ) to the subspace E k . In order to compute T r k (ρ g • L ⋆ ) we note that (ρ g • L ⋆ )(e k ij )(h) = e k ij (L(hg)) = e k ij (L(g)L(h)) = n k l=1 e k li (L(g))e k lj (L(h)) = n k l=1 e k li (g)e k jl (h). Then (ρ g • L ⋆ )(e k ij ) = n k l=1 e k li (g)e k jl . Since T r k (ρ g • L ⋆ ) = 1≤i,j≤n k (ρ g • L ⋆ )(e k ij , e k ij = 1≤i,j≤n k n k l=1 e k li (g) e k jl (h), e k ij and e k jl , e k ij is equal to 1 when j = i, l = j and is equal to 0 otherwise, we get T r k (ρ g • L ⋆ ) = n k i=1 e k ii (g) = χ k (g) Let us suppose that ε k = −1 for some k. Let n = n k 2 , then we can find a basis U − k = {u i , 1 ≤ i ≤ n k } of U k such that u i , u i+n = 1 and u i , u j+n = u i+n , u j+n = u i , u j = 0, i = j, i, j = 1, ..., n k Analogously, equation 1 gives us the following relations for the matrix coefficients e k ij (g) of π k (g) with respect to this basis: π k (g)(u i ), u j = −e k j+ni (g), j ≤ n, π k (g)(u i ), u j = e k j−ni (g), j > n Therefore π k (L(g))(u i ), u j == − π k (g)(u j ), u i = e k i+nj (g), i ≤ n, and π k (L(g))(u i ), u j = −e k i−nj (g), i > n . Taking account these relations and equation 2 we get the following relations between the matrix coefficients ofL(π k (g)) and π k (g) (4) e k ij (L(g)) = e k j+ni−n (g), j ≤ n, i > n (5) e k ij (L(g)) = −e k j−ni−n (g), j > n, i > n (6) e k ij (L(g)) = −e k j+ni+n (g), j ≤ n, i ≤ n (7) e k ij (L(g)) = e k j−ni+n (g), j > n, i ≤ n In this case we get: For i, j ≤ n (8) (ρ g • L ⋆ )(e k ij ) = n l=1 −e k l+ni+n (g)e k j+nl+n + n k l=n+1 e k l−ni+n (g)e k j+nl−n . For i ≤ n, j > n (9) (ρ g • L ⋆ )(e k ij = n l=1 −e k l+ni+n (g)e k j−nl+n ) + n k l=n+1 e k l−ni+n (g)(−e k j−nl−n ). For i > n, j ≤ n (10) (ρ g • L ⋆ )(e k ij = n l=1 e k l+ni−n (g)(−e k j+nl+n ) + n k l=n+1 −e k l−ni−n )(g)e k j+nl−n . And for i > n, j > n Since n k > 0,we conclude that ε k = 1 for all k. Then using eq.3 it follows that L ⋆ (e k ij )(g) = e k ij (L(g)) = e k ji (g) ie L ⋆ (e k ij ) = e k ji . Proposition 3. If L is an involutive antiautomorphism of G such that : χ k (L(g)) = χ k (g), for 1 ≤ k ≤ r then (12) T r k (ρ g • L ⋆ ) = ( 1 \|G\| h∈G χ k (L(h)h−1))χ k (g) Proof. To prove equation (12) we compute the Fourier coefficients λ k , 1 ≤ k ≤ r of the central function T r(ρ g • L ⋆ ) = T r(L ⋆ • σ L(g) −1 ) with respect to the basis {χ k = n k i=1 e k ii : 1 ≤ k ≤ r} of Z[G]. First we observe that (L ⋆ • σ L(g) −1 )(e k ij )(h) = e k ij (L(g)L(h)) = n k l=1 e k il (L(g))(e k lj • L)(h) and (L ⋆ • σ L(g) −1 )(e k ij ), e k ij = 1 \|G\| h∈G n k l=1 e k il (L(g))e k il (L(h))e k ji (h −1 ). therefore T r(L ⋆ • σ L(g) −1 ) = r k=1 1≤i,j≤n k 1 \|G\| h∈G n k l=1 e k il (L(g))e k lj (L(h))e k ji (h −1 ) Since n k j=1 e k lj (L(h))e k ji (h −1 ) = e k li (L(h))h −1 ) we get that T r(L ⋆ • σ L(g) −1 ) = r k=1 1≤i,l≤n k 1 \|G\| h∈G e k li (L(h)h −1 )(e k il • L)(g) and then Indeed, just notice that the Steinberg character and all its powers are constant on hyperbolic elements (value 1) and on elliptic elements (value -1). There are however hyperbolic as well as elliptic elements whose sign (class of the determinant modulo squares) is +1 as well as -1. So it is impossible that the sign character sgn be a polynomial in the Steinberg character. λ k ′ = r k=1 1≤i,l≤n k 1 \|G\| h∈G e k li (L(h)h −1 )(e k il • L), χ k ′ By hypothesis χ k (L(g)) = χ k (g), 1 ≤ k ≤ r, then λ k ′ = r k=1 1≤i,l≤n k 1 \|G\| Proposition 2 . 2If i, j ≤ n then e k j+nl+n = e k ij and e k j+nl−n = e k ij b) If i ≤ n, j > n then e k j−nl+n = e k ij if and only if j = n + i > n and l + n = j > n and e k j−nl−n = e k ij c) If i > n, j ≤ n then e k j+nl−n ) = e k ij for j +n = i > n and l−n = j < n and e k j+nl+n = e k ij d) If i > n, j > n then e k j−nl+n = e k ij and e k j+nl−n = e k ij Therefore T r k (ρ g • L ⋆ ) = If L is an involutive antiautomorphism of G such that : χ k (L(g)) = χ k (g), for 1 ≤ k ≤ r and \|{g ∈ G : L(g) = g}\| = If we evaluate T r(ρ g • L ⋆ ) on e we obtain T r(L ⋆ ) = r k=1 ε k n k but T r(L ⋆ ) = \|{g ∈ G : L(g) = g}\| and by hypothesis \|{g ∈ G : L(g) = g}\| = r k=1 n k . .. If L be an involutive antiautomorphism of G such thatχ k (L(g)) = χ k (g)for all 1 ≤ k ≤ r then the following conditions are equivalent i.\|{g ∈ G : L(g) Neither the sign character sgn nor the Gelfand character of G belong to the ring Z[St]. Proof. This follows from propositions 2 and 3 Proposition 5. Let L an involutive antiautomorphism of G , τ : G → G defined by τ (g) = L(g −1 ) and c τ (χ) = \|G\| −1 g∈G χ(gτ (g)) the twisted Frobenius -Schur indicator defined by Kawanaka and Matsuyama. Thenand sinceand thereforeThis proves that 1 \|G\| h∈G χ k (hL(h −1 ) = ±1 if χ k (L(g −1 ) = χ k (g) ie χ k (L(g)) = χ k (g), and if the matrix representations R k afforded by χ k , satisfies R k (L(g)) = R k (g), then c τ (χ k ) = 1.The Gel'fand character as a polynomial in the Steinberg characterWe consider below the case of the group G = P GL(2, k), k a finite field of any characteristic. We denote by T the Coxeter torus of G. Proof. In[1]it was proved that St 2 = Ind G T 1 + St a fact that can be checked by a straightforward character calculation. Now the theorem follows from the explicit decomposition of Ind G T 1 obtained in[7]. Tensor Products of Irreducible Representations of the Groups GL(2,k) and SL(2,k), k a Finite Field. L Aburto, J Pantoja, Comm. in Algebra. 285L. Aburto, J. Pantoja, Tensor Products of Irreducible Representations of the Groups GL(2,k) and SL(2,k), k a Finite Field, Comm. in Algebra, 28(5) (2000), 2507-2514 A (very brief) history of the trace formula. A (very brief) history of the trace formula, retrieved from:www.claymath.org/cw/arthur/pdf/HistoryTraceFormula.pdf Generalized Frobenius-Schur Numbers. D Bump, D Ginzburg, Journal of Algebra. 278294313Bump, D., Ginzburg, D., Generalized Frobenius-Schur Numbers, Journal of Algebra 278 (2004) 294313 Twisted trace formula of the Brandt matrix. Hashimoto, Proc. Japan Acad. 53Ser. AHashimoto, Twisted trace formula of the Brandt matrix, Proc. Japan Acad., 53, Ser. A (1977), 98 -102. Total characters and Chebyshev Polynomials. E Poimenidou, H Wolfe, International J. Math. Math. Sciences. 3824472453E. Poimenidou, H. Wolfe, Total characters and Chebyshev Polynomials, International J. Math. Math. Sciences, 38 (2003), 24472453/ Geometrical Gel'fand Models, Tensor Quotients and Weil Representations. J Soto-Andrade, Proc. Symp. Pure Math. 47Amer. Math. SocJ. Soto-Andrade, Geometrical Gel'fand Models, Tensor Quotients and Weil Repre- sentations, Proc. Symp. Pure Math., 47 (1987), Amer. Math. Soc., 305-316. Twisted spherical functions on the finite Poincar Upper Half Plane. J Soto-Andrade, J Vargas, J. Algebra. 248J. Soto-Andrade, J. Vargas, Twisted spherical functions on the finite Poincar Upper Half Plane, J. Algebra 248 (2002), 724-746. A Terras, Fourier Analysis on Finite Groups and Applications. Cambridge, U.K.Cambridge U. PressA. Terras, Fourier Analysis on Finite Groups and Applications, Cambridge U. Press, Cambridge, U.K., 1999. A weakly geometrical Gel'fand model for GL(n, q) and a realization of the Gel'fand character of a finite group. M F Yañez, C. R. Acad. Sci. Paris Sr. I Math. 31611M. F. Yañez, A weakly geometrical Gel'fand model for GL(n, q) and a realization of the Gel'fand character of a finite group. C. R. Acad. Sci. Paris Sr. I Math. 316 (1993), no. 11, 1149-1154. Pregrado Escuela De, [email protected]. Cs. Básicas y Farmacéuticas. Univ. de ChileEscuela de Pregrado, Fac. Cs. Básicas y Farmacéuticas, Univ. de Chile, [email protected]	[]
[ "Imaging VLBI polarimetry data from Active Galactic Nuclei using the Maxi- mum Entropy Method", "Imaging VLBI polarimetry data from Active Galactic Nuclei using the Maxi- mum Entropy Method" ]	[ "Colm P Coughlan [email protected]:[email protected] \nDepartment of Physics\nUniversity College Cork\nIreland\n", "Denise C Gabuzda \nDepartment of Physics\nUniversity College Cork\nIreland\n" ]	[ "Department of Physics\nUniversity College Cork\nIreland", "Department of Physics\nUniversity College Cork\nIreland" ]	[]	Mapping the relativistic jets emanating from AGN requires the use of a deconvolution algorithm to account for the effects of missing baseline spacings. The CLEAN algorithm is the most commonly used algorithm in VLBI imaging today and is suitable for imaging polarisation data. The Maximum Entropy Method (MEM) is presented as an alternative with some advantages over the CLEAN algorithm, including better spatial resolution and a more rigorous and unbiased approach to deconvolution. We have developed a MEM code suitable for deconvolving VLBI polarisation data. Monte Carlo simulations investigating the performance of CLEAN and the MEM code on a variety of source types are being carried out. Real polarisation (VLBA) data taken at multiple wavelengths have also been deconvolved using MEM, and several of the resulting polarisation and Faraday rotation maps are presented and discussed.	10.1051/epjconf/20136107009	[ "https://arxiv.org/pdf/1310.6187v1.pdf" ]	9,297,380	1310.6187	a7c5e7164f3990d568406290d50bbbb2218f8401	Imaging VLBI polarimetry data from Active Galactic Nuclei using the Maxi- mum Entropy Method 23 Oct 2013 Colm P Coughlan [email protected]:[email protected] Department of Physics University College Cork Ireland Denise C Gabuzda Department of Physics University College Cork Ireland Imaging VLBI polarimetry data from Active Galactic Nuclei using the Maxi- mum Entropy Method 23 Oct 2013EPJ Web of Conferences will be set by the publisher DOI: will be set by the publisher c Owned by the authors, published by EDP Sciences, 2013 Mapping the relativistic jets emanating from AGN requires the use of a deconvolution algorithm to account for the effects of missing baseline spacings. The CLEAN algorithm is the most commonly used algorithm in VLBI imaging today and is suitable for imaging polarisation data. The Maximum Entropy Method (MEM) is presented as an alternative with some advantages over the CLEAN algorithm, including better spatial resolution and a more rigorous and unbiased approach to deconvolution. We have developed a MEM code suitable for deconvolving VLBI polarisation data. Monte Carlo simulations investigating the performance of CLEAN and the MEM code on a variety of source types are being carried out. Real polarisation (VLBA) data taken at multiple wavelengths have also been deconvolved using MEM, and several of the resulting polarisation and Faraday rotation maps are presented and discussed. The Maximum Entropy Method The Maximum Entropy Method (MEM) is an alternative deconvolution method to the CLEAN algorithm. It is a constrained optimisation method, and is based on a consideration of the function (1) where H is the entropy of a model map of the source, χ 2 is a measure of the difference between the model and the observed visibilities (there are two χ 2 terms, one for intensity and a second for polarisation), α and β are the Lagrangian optimisation parameters and other conditions can also be included to represent additional constraints, such as the positivity of the intensity in the model map. A form of entropy suitable for polarisation emission developed by Gull and Skilling [1] and used by Holdaway [2] and Sault [3] is J = H(I m , P m ) − αχ 2 (V Im , V d ) − βχ 2 (V Pm , V d ) −other conditionsH = − k I k (log( I k IB k e ) + 1 + m k 2 log( 1 + m k 2 ) + 1 − m k 2 log( 1 − m k 2 )) (2) where IB k is the flux at pixel k of a bias map (normally chosen to be a flat map with a total flux equal to the flux estimated for the source) and I k and m k are the Stokes intensity (I) flux and fractional polarisation, respectively, of pixel k. a e-mail: [email protected] b e-mail: [email protected] The Gull and Skilling entropy, H, is a form of Shannon entropy (often used to describe the information content of a dataset), which has been generalised to include information on the polarisation of the data. An examination of the form of H gives an indication as to how it will react to different types of sources. The Gull and Skilling entropy of a source that is described well by the bias map is high. A source which has low fractional polarisation (i.e. unordered magnetic field) will also have high Gull and Skilling entropy. The Gull and Skilling entropy is maximum for an unpolarised source that is identical to the bias map and this is the source that MEM will produce in the absence of any data that forces it to make a more complicated model (the χ 2 terms in Eqn. 1 force MEM to make a model that maximises the Gull and Skilling entropy, but also reproduces the data to within noise levels). By iteratively maximising J in Eqn. (1), the MEM method develops a model of the source which maximises the Gull and Skilling entropy of the model (the model has lowest possible polarisation, and looks as much like the bias map as the data allows), while also reproducing the observed data to within noise levels. This results in a balance between entropy (representing the effects of unsampled visibilities and thermal noise) and fidelity to the observed data. This method of deconvolution, while not as straightforward as the CLEAN algorithm, is statistically and mathematically well-founded and can produce extremely well deconvolved maps comparable to, and in some cases better than, the CLEAN algorithm. Unlike the standard CLEAN algorithm, MEM does not model the source as a series of delta functions. Instead MEM models the source as a continuous distribution -a more physically realistic model, but one which is computationally much more demanding. This increases the effective resolution of MEM, as it is not necessary to convolve the MEM model with the CLEAN beam. This means that the theoretical resolution of MEM is the Nyquist sampling theorem limit for the observation, although thermal and systematic noise may prevent drawing useful information at such small scales. It proves useful to convolve the MEM model map with a small beam to smoothen out these variations, although this limits the resolution of the resulting map. From experience, a beam of about 1 2 to 1 4 of the CLEAN beam works well for most sources. MEM is also known for its mathematical property of "super-resolution". Unlike the CLEAN algorithm, MEM's resolution can was directly derived from Eq. (1) (see [2]) to be x min = 1 4 u max (3) where x min is the resolution in the x direction and u max is the maximum baseline in the u direction (the same relation exists between the y direction in image space and the V direction in visibility space). This resolution is a factor of 4 below the best-case resolution expected from the Nyquist sampling limit, and therefore details at this resolution scale do not directly reflect information which has been recorded by the array. However, as MEM's model of the source as a continuous distribution is quite realistic, MEM can model the source at resolution levels below what has been observed. This modelling is done by creating a structure that can reproduce the data at the observed resolution levels while also having maximum Gull and Skilling entropy. In this way, MEM produces a conservative model of the source at resolutions below the Nyquist limit. How well this "super-resolution" models detail at these lower levels is unknown, however the maps produced at these levels do not contain any obvious spurious features and appear to be a faithful extrapolation of the observed data. Monte Carlo simulations are being performed to test how well it performs on model sources (where the details of the source are known at sub-Nyquist resolutions). Implementation and Testing of New MEM Software A C++ program was written to implement a version of MEM for polarised data based on the MIRIAD task "PMOSMEM" [4] and the AIPS task "VTESS" [5], both of which use the Cornwell-Evans implementation of MEM [6] but are unsuitable for deconvolving polarised VLBI data. VTESS does not support polarised data (although a AIPS related task, UTESS, implements a For further information about the data used see [9]. duces FITS files which can then be imported into AIPS or CASA for viewing or further processing. The low level of human input required to run the software on a source makes it very suitable for use in a CASA or AIPS pipeline. To ensure that the code was operating correctly and to characterise the behaviour of MEM based VLBI deconvolution, Monte Carlo simulations of the deconvolution of simulated sources are being performed. The code performs well deconvolving small Gaussians with realistic thermal noise, being able to recover the correct FWHM of the model Gaussian map. Further testing of the new polarisation MEM code using multiple Gaussian sources is currently underway. Results Markarian 501 Markarian 501 has an extended, bent jet with a "spinesheath" polarisation structure visible in some places (see Fig. 1a and [7]), which was fitted using a helical magneticfield model in [8]. The MEM map shows this spine-sheath polarisation structure about 8 mas from the core more clearly (Fig. 1b). The improved resolution of the MEM map separates the extended region of transverse polarisation (longitudinal magnetic field) about 4-mas from the core into several distinct regions with different polarisation orientations. The fan-like structure of the polarisation at the Southern edge of the jet in this region is suggestive of a longitudinal field component induced by local bending of the jet. The orthogonal orientation of the magnetic field in the inner jet is much more obvious in the MEM image. 1633+382 This jet also shows transverse polarisation structure, but the available CLEAN resolution is not sufficient to discern its nature (Fig. 2a). The higher resolution offered by MEM shows the polarisation structure more clearly, in particular, the presence of orthogonal polarisation (longitudinal B field) along the top half of the jet and longitudinal polarisation (orthogonal B field) along the bottom half of the jet 1.5-2 mas from the core (Fig. 2b). The CLEAN Faraday rotation measure (RM) map constructed using VLBA data at 2-6cm (see Fig. 2c) provided evidence for a transverse RM gradient across the jet, possibly due to a helical B field [9]. The higher-resolution MEM RM map (Fig. 2d) confirms this transverse RM structure, with a clear sign change in the RM from the Northern to the Southern part of the jet. Conclusions Software to implement a MEM-based deconvolution of VLBI polarimetry data has been written and is being tested with Monte Carlo simulations. Some first results using real, multi-wavelength VLBI polarisation data demonstrating its enhanced resolution over CLEAN have already been achieved. Future work will include multi-wavelength VLBI studies of a number of Active Galactic Nuclei at 2-6cm and 18-22cm. We intend to make our software available to the community once it has been fully tested and its operation well understood. Figure 1 :Figure 2 : 12Markarian 501 at 8.4 GHz. The contours are Stokes I, the ticks indicate the direction of the observed polarisation. 1a: CLEAN image with a beam of 1.46 x 1.18 mas, −33.85 • position angle. 1b: MEM image, convolved with ≈ 1 2 of the CLEAN beam.For further information about the data used see[7]. deconvolution method similar to MEM which can be used to deconvolve polarisation data), while PMOSMEM does support polarised data, but does not support VLBA data. The well-commented source codes of both tasks (PMOSMEM in particular) were extremely useful in writing a form of the algorithm which could handle polarised VLBA data. The software can be used by exporting dirty maps for each Stokes parameter and the dirty beam for the observation as FITS files. The code takes in some basic parameters, such as an estimate of the total flux and the final (post-deconvolution) desired root-mean-squared noise. If the deconvolution has been successful, it pro-1633+382 at 4.6 GHz. The contours are Stokes I and the ticks indicate the direction of the observed polarisation. The colour scale is Faraday rotation measure in rad/m 2 . The Faraday rotation measure maps were made with 6 frequencies running from 4.6 to 15.4 GHz. 2a: CLEAN image with a beam of 3.77 x 2.04 mas, −26.53 • position angle. 2b: MEM image, convolved with ≈ 1 3 of the CLEAN beam. 2c: CLEAN image with Faraday rotation, same beam. Note fewer contour lines are shown for clarity. 2d: MEM image with Faraday rotation, convolved with ≈ 1 2 of the CLEAN beam. AcknowledgementsThis research has been funded by the Irish Research Council (IRC). Indirect Imaging. S F Gull, J Skilling, Proc. IAU/URSI Symp. J.A. RobertsIAU/URSI SympCambridge Univ. Press267Gull, S.F., and Skilling, J., Indirect Imaging, Proc. IAU/URSI Symp., ed. J.A. Roberts, Cambridge Univ. Press, p.267 (1984). . M A Holdaway, J F C Wardle, Diego San, Int. Soc. for Optics and Photonics. 90Holdaway, M.A., Wardle J. F. C., San Diego'90, Int. Soc. for Optics and Photonics (1990). . R J Sault, The Astrophysical Journal. 354Sault, R.J., The Astrophysical Journal, 354, L61-L63 (1990). R J Sault, P J Teuben, M C H Wright, Astronomical Data Analysis Software and Systems IV. R. Shaw, H.E. Payne, J.J.E. Hayes77Sault R.J., Teuben P.J., Wright M.C.H., Astronomical Data Analysis Software and Systems IV, ed. R. Shaw, H.E. Payne, J.J.E. Hayes, ASP Conference Series, 77, 433-436 (1995). Information Handling in Astronomy -Historical Vistas. E W Greisen, 1-4040-1178-4Astrophysics and Space Science Library. Heck, A. ed.285109Kluwer Academic PublishersGreisen, E. W., Information Handling in Astronomy - Historical Vistas, Heck, A. ed., Kluwer Academic Pub- lishers, Dordrecht, ISBN 1-4040-1178-4, Astrophysics and Space Science Library, 285, 109 (2003). . T J Cornwell, K F Evans, Astronomy and Astrophysics. 143Cornwell, T.J., Evans, K.F., Astronomy and Astro- physics, 143, 77-83 (1985). . Pushkarev, MNRAS. 356859Pushkarev et al., MNRAS, 356, p. 859 (2005). . Murphy, MNRAS. 4301504Murphy et al., MNRAS, 430, p. 1504 (2013). . A Reichstein, D C Gabuzda, J.Phys.: Conf. Ser. 355Reichstein A., Gabuzda D.C., J.Phys.: Conf. Ser., 355 (2012).	[]
[ "Isotopic identification of engineered nitrogen-vacancy spin qubits in ultrapure diamond", "Isotopic identification of engineered nitrogen-vacancy spin qubits in ultrapure diamond" ]	[ "T Yamamoto \nNational Institute for Materials Science\n1-1 Namiki305-0044TsukubaIbarakiJapan\n\nJapan Atomic Energy Agency\n1233 Watanuki370-1292TakasakiGunmaJapan\n", "S Onoda \nJapan Atomic Energy Agency\n1233 Watanuki370-1292TakasakiGunmaJapan\n", "T Ohshima \nJapan Atomic Energy Agency\n1233 Watanuki370-1292TakasakiGunmaJapan\n", "T Teraji \nNational Institute for Materials Science\n1-1 Namiki305-0044TsukubaIbarakiJapan\n", "K Watanabe \nNational Institute for Materials Science\n1-1 Namiki305-0044TsukubaIbarakiJapan\n", "S Koizumi \nNational Institute for Materials Science\n1-1 Namiki305-0044TsukubaIbarakiJapan\n", "T Umeda \nInstitute of Applied Physics\nUniversity of Tsukuba\n1-1-1 Tennodai305-8573TsukubaIbarakiJapan\n", "L P Mcguinness \nInstitute for Quantum Optics\nUniversity of Ulm\nD-89081UlmGermany\n", "C Müller \nInstitute for Quantum Optics\nUniversity of Ulm\nD-89081UlmGermany\n", "B Naydenov \nInstitute for Quantum Optics\nUniversity of Ulm\nD-89081UlmGermany\n", "F Dolde \n3rd Physics Institute and Research Center SCoPE\nUniversity of Stuttgart\nD-70174StuttgartGermany\n", "H Fedder \n3rd Physics Institute and Research Center SCoPE\nUniversity of Stuttgart\nD-70174StuttgartGermany\n", "J Honert \n3rd Physics Institute and Research Center SCoPE\nUniversity of Stuttgart\nD-70174StuttgartGermany\n", "M L Markham \nElement Six Limited\nKing's Ride ParkSL5 8BPAscot, BerkshireUnited Kingdom\n", "D J Twitchen \nElement Six Limited\nKing's Ride ParkSL5 8BPAscot, BerkshireUnited Kingdom\n", "J Wrachtrup \n3rd Physics Institute and Research Center SCoPE\nUniversity of Stuttgart\nD-70174StuttgartGermany\n", "F Jelezko \nInstitute for Quantum Optics\nUniversity of Ulm\nD-89081UlmGermany\n", "J Isoya \nResearch Center for Knowledge Communities\nUniversity of Tsukuba\n1-2 Kasuga305-8550TsukubaIbarakiJapan\n" ]	[ "National Institute for Materials Science\n1-1 Namiki305-0044TsukubaIbarakiJapan", "Japan Atomic Energy Agency\n1233 Watanuki370-1292TakasakiGunmaJapan", "Japan Atomic Energy Agency\n1233 Watanuki370-1292TakasakiGunmaJapan", "Japan Atomic Energy Agency\n1233 Watanuki370-1292TakasakiGunmaJapan", "National Institute for Materials Science\n1-1 Namiki305-0044TsukubaIbarakiJapan", "National Institute for Materials Science\n1-1 Namiki305-0044TsukubaIbarakiJapan", "National Institute for Materials Science\n1-1 Namiki305-0044TsukubaIbarakiJapan", "Institute of Applied Physics\nUniversity of Tsukuba\n1-1-1 Tennodai305-8573TsukubaIbarakiJapan", "Institute for Quantum Optics\nUniversity of Ulm\nD-89081UlmGermany", "Institute for Quantum Optics\nUniversity of Ulm\nD-89081UlmGermany", "Institute for Quantum Optics\nUniversity of Ulm\nD-89081UlmGermany", "3rd Physics Institute and Research Center SCoPE\nUniversity of Stuttgart\nD-70174StuttgartGermany", "3rd Physics Institute and Research Center SCoPE\nUniversity of Stuttgart\nD-70174StuttgartGermany", "3rd Physics Institute and Research Center SCoPE\nUniversity of Stuttgart\nD-70174StuttgartGermany", "Element Six Limited\nKing's Ride ParkSL5 8BPAscot, BerkshireUnited Kingdom", "Element Six Limited\nKing's Ride ParkSL5 8BPAscot, BerkshireUnited Kingdom", "3rd Physics Institute and Research Center SCoPE\nUniversity of Stuttgart\nD-70174StuttgartGermany", "Institute for Quantum Optics\nUniversity of Ulm\nD-89081UlmGermany", "Research Center for Knowledge Communities\nUniversity of Tsukuba\n1-2 Kasuga305-8550TsukubaIbarakiJapan" ]	[]	Nitrogen impurities help to stabilize the negatively-charged-state of NV − in diamond, whereas magnetic fluctuations from nitrogen spins lead to decoherence of NV − qubits. It is not known what donor concentration optimizes these conflicting requirements. Here we used 10 MeV 15 N 3+ ion implantation to create NV − in ultrapure diamond. Optically detected magnetic resonance of single centers revealed a high creation yield of 40 ± 3% from 15 N 3+ ions and an additional yield of 56 ± 3% from 14 N impurities. High-temperature anneal was used to reduce residual defects, and charge stable NV − , even in a dilute 14 N impurity concentration of 0.06 ppb were created with long coherence times.	10.1103/physrevb.90.081117	[ "https://arxiv.org/pdf/1405.5837v2.pdf" ]	119,244,937	1405.5837	8d7775660b98f61b99cd17f03d4284c471d44b52	Isotopic identification of engineered nitrogen-vacancy spin qubits in ultrapure diamond 3 Sep 2014 T Yamamoto National Institute for Materials Science 1-1 Namiki305-0044TsukubaIbarakiJapan Japan Atomic Energy Agency 1233 Watanuki370-1292TakasakiGunmaJapan S Onoda Japan Atomic Energy Agency 1233 Watanuki370-1292TakasakiGunmaJapan T Ohshima Japan Atomic Energy Agency 1233 Watanuki370-1292TakasakiGunmaJapan T Teraji National Institute for Materials Science 1-1 Namiki305-0044TsukubaIbarakiJapan K Watanabe National Institute for Materials Science 1-1 Namiki305-0044TsukubaIbarakiJapan S Koizumi National Institute for Materials Science 1-1 Namiki305-0044TsukubaIbarakiJapan T Umeda Institute of Applied Physics University of Tsukuba 1-1-1 Tennodai305-8573TsukubaIbarakiJapan L P Mcguinness Institute for Quantum Optics University of Ulm D-89081UlmGermany C Müller Institute for Quantum Optics University of Ulm D-89081UlmGermany B Naydenov Institute for Quantum Optics University of Ulm D-89081UlmGermany F Dolde 3rd Physics Institute and Research Center SCoPE University of Stuttgart D-70174StuttgartGermany H Fedder 3rd Physics Institute and Research Center SCoPE University of Stuttgart D-70174StuttgartGermany J Honert 3rd Physics Institute and Research Center SCoPE University of Stuttgart D-70174StuttgartGermany M L Markham Element Six Limited King's Ride ParkSL5 8BPAscot, BerkshireUnited Kingdom D J Twitchen Element Six Limited King's Ride ParkSL5 8BPAscot, BerkshireUnited Kingdom J Wrachtrup 3rd Physics Institute and Research Center SCoPE University of Stuttgart D-70174StuttgartGermany F Jelezko Institute for Quantum Optics University of Ulm D-89081UlmGermany J Isoya Research Center for Knowledge Communities University of Tsukuba 1-2 Kasuga305-8550TsukubaIbarakiJapan Isotopic identification of engineered nitrogen-vacancy spin qubits in ultrapure diamond 3 Sep 2014(Dated: September 4, 2014) Nitrogen impurities help to stabilize the negatively-charged-state of NV − in diamond, whereas magnetic fluctuations from nitrogen spins lead to decoherence of NV − qubits. It is not known what donor concentration optimizes these conflicting requirements. Here we used 10 MeV 15 N 3+ ion implantation to create NV − in ultrapure diamond. Optically detected magnetic resonance of single centers revealed a high creation yield of 40 ± 3% from 15 N 3+ ions and an additional yield of 56 ± 3% from 14 N impurities. High-temperature anneal was used to reduce residual defects, and charge stable NV − , even in a dilute 14 N impurity concentration of 0.06 ppb were created with long coherence times. The realization of quantum registers, which are comprised of several quantum bits (qubits), is currently a central issue in quantum information and computation science. 1 Among many competing quantum systems, photoactive defect spins of negatively charged nitrogen vacancy (NV − ) centers in diamond are unique solid-state qubits, due in part to ambient pressure and temperature operation. [2][3][4] The NV − center is a single-photon emitter with zero-phonon-line (ZPL) at 637 nm, 5 where both of 3 A 2 electronic ground and 3 E excited states locate inside the diamond band-gap. The spin sublevels, \|m s = 0 and \|m s = ±1 , of the triplet (S = 1) ground state are separated by ∼ 2.87 GHz due to spin-spin interaction. 6 Arbitrary states including superpositions of spin levels may be created by resonant microwave pulses after optical initialization, and then readout by measuring fluorescence intensity. 3 Experimental proofs of strongly-coupled NV − spins, 7-9 magnetic coupling between a NV − spin and another electron spin 9,10 or nuclear spins, [11][12][13][14][15] , in addition to coupling to photons 16,17 or optical cavities, [18][19][20] exemplify the robust yet mutable nature of the NV scheme as well as the beginnings of scalability. The NV quantum coherence decays in time due to magnetic fluctuations from substitutional nitrogen (N 0 s ) electron spins and 13 C nuclear spins, and spin-lattice relaxation. [21][22][23] Thus, the use of high purity ([N 0 s ] ∼ ppb) type IIa diamonds with reduced 13 C content, and position controlled N ion implantation to create NV − centers, is a promising avenue towards a high quality multiqubit system. 8 Nevertheless, substitutional nitrogen impurities, which donate electrons to NV centers, are actually essential for stabilizing the NV − charge state. 24 The negative NV charge state is predominant at thermal equilibrium if [N s ] = [N 0 s + N + s ] > [NV] , and this is generally true for isolated NV centers in type Ib diamond ([N 0 s ] ∼ 20 − 200 ppm). [25][26][27] With decreasing N 0 s donor concentration, microscopic distributions of donors surrounding each NV center are significant for the charge state, rather than the Fermi position relative to the ground state of NV − . As a result locally inhomogenous distributions of either NV 0 or NV − are expected. 28 This may explain the relatively large reduction in NV − population observed by photoluminescence spectroscopy in type IIa diamonds ([N 0 s ] ∼ 30 − 300 ppb). 26 The presence of the neutral NV 0 charge state (S = 1/2) is undesirable as its applications are hindered by rapid dephasing in the ground state. Therefore, the understanding of a minimum concentration threshold of N 0 s impurities in order to form stable NV − spin qubits is of concern for reliable engineering and scalability. In this study we isotopically distinguish engineered 15 NV − spin qubits due to 15 N implantation from 14 NV − due to preexisting 14 N impurities in ultrapure diamond, both of which can be created by 15 N 3+ (10 MeV) implantation. Using a combination of confocal microscopy and spin resonance, we observe an implantation creation yield of ∼ 100%, of which 14 NV − centers comprise more than half this value. The nitrogen concentration of less than 0.1 ppb is low enough to attain long coherence times (∼ 2 ms) and sufficient to stabilize the charge state of NV − qubits, under reduced concentrations of residual defects by high-temperature anneal. In experiments, a high-purity, 99.99% 12 C-enriched (0.01%-13 C) homoepitaxial diamond (Element Six Ltd.) grown by chemical vapor deposition was used. The concentration of N impurities was expected to be less than arXiv:1405.5837v2 [cond-mat.mes-hall] 3 Sep 2014 0.1 ppb from the crystal growth condition, 29,30 which is far below the detection limit of secondary ion mass spectroscopy or electron spin resonance for the film thickness here. 15 N 3+ ions with an incident energy of 10 MeV per ion were implanted into the (100) crystal surface. By scanning a microbeam of full-width at half-maximum (FWHM) size ∼ 1.5 µm, a square grid of implantation sites separated by ∼ 8 µm was created [ Fig. 1(a)]. The average number of implanted ions was 2.8 per implantation site by measuring the beam flux before and after implantation. 31,32 To form NV centers, the sample was annealed at 1000 • C for 2 h in a vacuum of ∼ 10 −6 Torr. Observation of hyperfine structure of either 15 N (with a nuclear spin of I = 1/2, natural abundance 0.37%) or 14 N (I = 1, 99.63%) by optically detected magnetic resonance (ODMR) spectroscopy allowed determination of whether the investigated NV − centers were due to 15 N implants or 14 N impurities already present in the epitaxial layer [ Fig. 1(a)]. 33 The ODMR spectra were measured at room temperature, under a static magnetic field of ∼ 2 mT in order to separate the two transitions: \|m s = 0 ⇔\|m s = +1 and \|m s = 0 ⇔\|m s = −1 . ODMR was able to resolve NV − pairs with different axis orientations when their separation is below the confocal resolution [ Fig. 1 Room-temperature photoluminescence (PL) spectra were also measured for individual centers with 0.5 mW of 532 nm excitation (into the objective) and an accumulation time of 30 sec. Recent studies have shown that dynamical charge conversion between NV − and NV 0 occurs under illumination, and the controllable dynamics have been discussed. [34][35][36][37][38][39] Both NV − and NV 0 may be observed even for a single NV center, with time-averaged PL spectroscopy, if significant photoconversion appears during optical pumping (for example, see Ref. 40). However, we observed characteristic spectrum of the negative charge state from all 14 NV − and 15 NV − centers: 41 a weak zero-phonon-line (ZPL) at 638 nm accompanied with broad vibronic sidebands [ Fig. 1(c)]. No distinct signals of NV 0 charge states (575 nm ZPL) were found, at least within the accumulation time of 30 sec. Two centers that didn't show an ODMR signal, indicated by black arrows in Fig. 1(a), were unknown centers since their PL spectra were different from either NV − or NV 0 . Additional spins belonging to paramagnetic residual defects, resulting from the implantation and anneal process, may dominate the decoherence of implanted NV − spins. 42,43 Also, residual point defects such as divacancies may act as accepters, 44 to ionize NV − to NV 0 . To overcome these obstacles, high temperature anneal has been shown to be effective in reducing the concentration of 15 NV -5 μm S0 S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 (d) (b) Photons (kcps) (a) S0 S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 residual paramagnetic defects at ≥ 1000 • C, 42,45,46 with a concomitant increased population of NVs with long coherence times 43 and improved spectral stability 47 when compared to 800 • C anneal. Previous studies have also investigated NV charge instability due to residual defects after ion implantation, 10,33,40,48-53 neutron 24 or electron irradiation, 27 and anneal temperatures of 600-900 • C, while effects from the surface are additionally involved in shallow implantation studies. [54][55][56][57][58] In the present work, both high temperature anneal and high energy implantation were performed to provide a clean environment with minimal degradation of the NV − properties. Figure 1(d) shows the number of 14 NV − and 15 NV − centers at each implantation site in Fig. 1(a), labeled by S j (j = 0, 1, 2, · · · , 15). In total, eighteen 15 NV − and 25 14 NV − were created from 45 implanted 15 N ions (2.8×16) in the sixteen implantation sites. Dividing the number of created NV − centers by the number of implanted 15 N ions gives a creation yield of Y = 40 ± 3% for 15 NV − and 56 ± 3% for 14 NV − , where the NV − centers indicated by the white arrows in Fig. 1(a) were excluded for the counting since these centers are far from an implantation site and might be owing to a miss-hit. As reported in Ref. 42, the spin coherence times (T 2 ) in this implantation area were ∼ 2 ms at room temperature, which are the longest among implanted NV − qubits, and comparable to the longest recorded for naturally-formed NV − centers during crystal growth. 30,59,60 In addition to a total yield of 96%, we also obtained ∼ 100% yield for implanted NV − centers with T 2 times up to 1.6 ms in another high purity 12 C-99.99% enriched diamond (Element Six Ltd.) by similar implantation and annealing process (data not shown). Low-energy (10-30 keV) nitrogen ion implantation has provided creation yields of 20-21%, 9,50 which is just below the maximum expected value of 25%. 61 Compared to this, high-energy (18 MeV) implantation has exhibited 45% yield, which was interpreted due to the increased number of vacancies generated by increasing the implantation energy. 51 However, NV centers comprised of preexisting N impurities may also be counted in the yield, and thus the creation efficiency should change with the concentration of [N 0 s ] in each sample. To investigate further the creation of 14 NV − and 15 NV − centers, we measured the spatial distributions of each center by confocal microscopy. A diffraction-limited fluorescence spot from a single center has a lateral diameter (xy-plane) of ∼ 0.3 µm and a diameter of ∼ 0.7 µm along the optical axis (z-axis). To measure the coordinates, (x, y, z), of individual NV − centers with high precision, we used a Gaussian fit to find the position of maximal intensity of the fluorescence profile, giving an accuracy of < 0.1 µm in all axis-directions. NV − pairs were inseparable by fitting and thus measured as at the same position. 41 We compared the observed spatial distribution of NV − centers, to the computed statistical distributions of implanted 15 Experiments showed no obvious difference in the lateral (xy-plane) distributions between 15 NV − and 14 NV − centers. 41 On the other hand, we observed differences 0 S 0 S1 S2 S3 S4 S5 S6 S7 S8 S9 S1 0 S1 1 S1 2 S1 3 S1 4 S1 5 0 2 . in depth (z-axis) distributions between the two nitrogen sources: the mean depth of 3.5 ± 0.3 µm for 14 NV − was shallower than that of 3.8 ± 0.2 µm for 15 NV − . The depth distribution of 15 NV − centers shows no evidence of channeling, 41 and the mean depth agrees well with the computed depth range of implanted 15 N atoms by SRIM (3.82 ± 0.04 µm). Interestingly, all 14 NV − centers were located at the same or shallower depths than those of 15 NV − centers (Fig. 2), which is expected to result from the vacancy profile which trails towards surface (open circle in Fig. 2), as compared to the sharp Bragg peak of implanted 15 N atoms (solid circle). No 14 NV − centers were observed at depths shallower than 2.8 µm due to the low concentration of vacancies. Now we discuss the creation efficiency for implanted 15 NV − centers. The creation yield, Y (c v ), is written as Y (c v ) = N q (c v ) N i = N i × P r N i × P t (c v ),(1) where N q /N i is the ratio of the number of implanted 15 N ions (N i ) to the number of 15 NV − created by implantation (N q ). The product N i × P r gives the number of substitutional 15 N atoms among the implanted 15 N ions, dependent on the replacement probability P r at a carbon lattice site, and P t (c v ) is the probability of trapping a vacancy at a neighboring site of substitutional 15 N atom, which is proportional to concentrations of vacancies, c v . 51 For P t (c v ) ≈ 1, the obtained yield of 40% gives a replacement probability of P r ≈ 0.4, which agrees well with recent molecular dynamics simulations show-ing that 37% of implanted ions are substituted to lattice sites after implantation. 61 This suggests that our 10 MeV implantation and annealing process provides enough vacancies to form 15 NV − with a high trapping probability of P t (c v ) ≈ 1, and the creation efficiency is limited by those 15 N implants which remain as interstitial nitrogen after implantation (∼ 60% of implanted ions). Next we consider 14 NV − formation. Preexisting 14 N impurities near an implanted 15 N can be converted to 14 NV − centers, as observed from the pairs of 14 NV − and 15 NV − . In addition to this, shallower 14 N impurities near vacancy cascades generated by 15 N ion implantation, can be also transformed into 14 NV centers, and the creation probability depends on concentrations of substitional 14 N impurities ([N 0 s ]) and vacancies (c v ). Hence, Eq. 1 can be modified as Y ([N 0 s ], c v ) = [N 0 s ] × V v × P r N i × P t (c v ) × c v c v ,(2) where V v is an effective volume containing enough vacancies to form 14 NV − centers, and the replacement probability into carbon sites is given as P r = 1 since 14 N impurities are substitutional atoms. The probability of trapping a vacancy, P t (c v ) ≈ 1 in Eq. 1, is replaced by P t (c v ) × c v cv , where the factor of c v cv results from a smaller vacancy concentration, c v , for 14 N impurities than c v for implanted 15 N atoms. Figure 3(a) and (b) show the simulated statistical distributions of vacancies (after diffusion) and 15 N atoms, respectively, where z-axis is the ion implantation direction and the white solid lines indicate isolines of area density in each xz-plane projection. Calculating a volume of revolution (V v ) by rotating the area of more than 420 vacancies/µm 2 /ion around x = 0 axis in Fig. 3(a), and counting the number of vacancies (n v ) inside this volume, yields an average concentration of c v = n v V v = 28 ppb, where the selected depth range corresponds to the experimental range of 2.8 ≤ z ≤ 3.9 for 14 NV − centers (Fig. 2). Similarly, calculating the volume V v containing more than 2 atoms/µm 2 /ion from Fig. 3(b), and counting the number of vacancies inside V v from Fig. 3(a), gives the average concentration of c v = nv Vv = 42 ppb. Here V v is the volume in which implanted 15 N atom is found with the probability of 90%. Assigning those values and the observed yield of Y = 0.56 into Eq. 2, we obtain a concentration of 14 N impurities as 0.06 ppb, which agrees with the expected value (< 0.1 ppb) from the growth condition. 29,30 Photoconversion between NV − and NV 0 , observed in PL spectra has been reported for shallow NV − centers (< 200 nm depths), in spite of a similar (∼ppb) or higher (∼ppm) concentrations of N donors as described here. 40,54,58 Ionization of shallow NV − to the NV 0 charge state has been attributed to surface effects such as electron depletion due to an acceptor layer 54,55 or hole accumulation due to upward band bending at the hydrogen-terminated surface. 57 The observation of the stable negatively-charged-state in the present study im- plies that surface effects are negligible for deep NV centers at 3-4 µm depths. On the other hand, low energy implantation (< 5 keV) 58 through nano-hole apertures 52,66 is a promising route to fabrication of arrays of NV centers with high positional precision, however short T 2 times due to surface spins are problematic for building quantum registers. One solution is to overgrow an additional diamond layer onto the surface, which has succeeded in prolonging T 2 times. 67 Our results show that NV − spin qubits at the depth ≥ 2.8 µm exhibit reliable properties of long coherence times and stable charge states. In summary, engineered NV − spins qubits by 15 N 3+ ion implantation into high-purity, 13 C-depleted diamond were studied in a reduced background concentration of residual defects after high-temperature anneal. We observed a creation yield of 15 NV − (40±3%) which is likely to be limited by the population of implants having an interstitial configuration (∼ 60%). 68 Even with a N impurity concentration estimated as 0.06 ppb, a considerable fraction of created NV − centers consisted of preexisting N impurities. The low concentration of nitrogen impurities, which allow for long coherence times (∼ 2 ms), still play a significant role for NV − charge stabilization. A provisional mechanism for charge stabilization will be required when fabricating NV − from only implanted ions by further lowering the N donor concentration. This study was carried out as 'Strategic Japanese-German Joint Research Project' supported by JST and DFG (FOR1482, FOR1493 and SFB716), ERC, DARPA (b)]: the triplet and doublet hyperfine structures show 14 NV − with hyperfine constant A = 2.2 MHz and 15 NV − with A = 3.1 MHz, respectively. No pairs consisted of two 15 NV − centers, rather, all pairs were formed by one each of 15 N and 14 N. This agrees with a numerically estimated formation probability of ∼ 1% for 15 NV − pairs locating within a ∼ 0.3 µm laser spot, when given our microbeam size (FWHM of 1.5 µm). FIG. 1 . 1(Color online) (a) Confocal microscope image of lateral (xy plane) distribution of NV − centers with 532 nm excitation at a depth of ∼ 3.8 µm (left). The white dashed lines indicate a calculated square grid of implantation sites (2.8 ions/site). 41 The fluorescent spots indicated by black arrows were unknown centers (see text). The map of NV − centers identified by ODMR (right): 15 NV − (solid circle), 14 NV − (open circle), and 14 NV − -15 NV − pairs (cross). (b) ODMR spectrum of a NV − pair comprised of 14 NV − (hyperfine splitting of A = 2.2 MHz) and 15 NV − (A = 3.1 MHz) in the transition of \|ms = 0 ⇔ \|ms = +1 (c) PL spectrum of a 15 NV − center with 532 nm excitation. (d) The number of 15 NV − (black bar) and 14 NV − (white bar) in each implantation site. N atoms and vacancies using stopping and range of ions in matter (SRIM) Monte Carlo 62 (a displacement energy of 37.5 eV, 63 a diamond density of 3.52 g/cm 3 , and 8 × 10 4 of incident 15 N ions were used). By using the vacancy distribution computed by SRIM, we then simulated a statistical vacancy distribution after diffusion with an isotropic diffusion length of √ 2Dt ≈ 0.08 µm, where D = D 0 exp[−E a /(k B T )], with diffusion coefficient D 0 = 3.7 × 10 −6 cm 2 /s (Ref. 64), Boltzmann's constant k B = 1.4 × 10 −23 T/K, activation energy E a = 2.3 eV (Ref. 65), temperature T = 1273 K, and time t = 7200 s. 41 online) (b) Depth (z-axis) distributions of observed NV − centers at each implantation site as histogram: 15 NV − (black), 14 NV − (gray), and 14 NV − -15 NV − pairs (blue). The graph in the back side shows the simulated depth distributions for implanted 15 N atoms (solid circle) and vacancies with isotropic diffusion length of ∼ 0.08 µm (open circle). FIG. 3 . 3(Color online) Simulated xz-plane distributions of (a) vacancies with an isotropic diffusion length of ∼ 0.08 µm and (b) implanted 15 N atoms with an incident energy of 10 MeV, where z-axis is the implantation direction. The area-density isolines of vacancies and 15 N atoms are shown as white lines in (a) and (b), respectively. Nizovtsev for valuable discussions. Kay D Jahnke, Pascal Heller, Alexander Gerstmayr, the Alexander von Humboldt Foundation. We thank Denis Antonov, Yoshiyuki Miyamoto, Brett C. Johnson, and Alexander Pand Andreas Häußler for assistance with the experiments. * [email protected] Alexander von Humboldt Foundation. We thank Denis Antonov, Yoshiyuki Miyamoto, Brett C. Johnson, and Alexander P. Nizovtsev for valuable discussions, and Kay D. 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[]	[]	[]	[ "Annales Henri Lebesgue" ]	Contrary to the finite-dimensional case, the Möbius group admits interesting self-representations when infinite-dimensional. We construct and classify all these self-representations.The proofs are obtained in the equivalent setting of isometries of Lobachevsky spaces and use kernels of hyperbolic type, in analogy with the classical concepts of kernels of positive and negative type.Résumé. -Contrairement au cas usuel de dimension finie, le groupe de Möbius admet des auto-représentations intéressantes lorsqu'il est de dimension infinie. Nous les construisons et classifions toutes.Les démonstrations sont conduites dans le cadre équivalent des groupes d'isométries des espaces de Lobatchevski et reposent sur le concept de noyau de type hyperbolique, en analogie avec la notion classique de noyau de type positif ou négatif.	10.5802/ahl.14	[ "https://ahl.centre-mersenne.org/article/AHL_2019__2__259_0.pdf" ]	53,967,210	1805.12479	655e0db30d5ea85e09b48a9c380bcb0d1b1f18f1	2019 Annales Henri Lebesgue 2201910.5802/ahl.14NIC O L A S M O N O D PIE R R E P Y S E L F -R E P R E S E N TATIO N S O F T H E M Ö BI U S G R O U P A U T O -R E P R É S E N TATIO N S D U G R O U P E D E M Ö BI U SMöbius groupLobatchevsky spacehyperbolic spaceinfinite-dimensional space 2010 Mathematics Subject Classification: 53A3557S2553C50 Contrary to the finite-dimensional case, the Möbius group admits interesting self-representations when infinite-dimensional. We construct and classify all these self-representations.The proofs are obtained in the equivalent setting of isometries of Lobachevsky spaces and use kernels of hyperbolic type, in analogy with the classical concepts of kernels of positive and negative type.Résumé. -Contrairement au cas usuel de dimension finie, le groupe de Möbius admet des auto-représentations intéressantes lorsqu'il est de dimension infinie. Nous les construisons et classifions toutes.Les démonstrations sont conduites dans le cadre équivalent des groupes d'isométries des espaces de Lobatchevski et reposent sur le concept de noyau de type hyperbolique, en analogie avec la notion classique de noyau de type positif ou négatif. Introduction Context For an ordinary connected Lie group, the study of its continuous self-representations is trivial in the following sense: every injective self-representation is onto, and hence an automorphism. In the infinite-dimensional case, another type of "tautological" self-representations presents itself. Namely, the group will typically contain isomorphic copies of itself as natural proper subgroups. For instance, a Hilbert space will be isomorphic to most of its subspaces. Remarkably, some infinite-dimensional groups also admit completely different selfrepresentations which are not in any sense smaller tautological copies of themselves. This phenomenon has no analogue in finite dimensions and the simplest case is as follows. Let E be a Hilbert space and Is(E) ∼ = E O(E) its isometry group. To avoid the obvious constructions mentioned above, we only consider cyclic self-representations (in the affine sense). It is well known that there is a whole wealth of such selfrepresentations. They are described by functions of conditionally negative type. More precisely, the question becomes equivalent to describing all radial functions of conditionally negative type on E because one can arrange, by conjugating, that O(E) maps to itself. Thus, upon identifying a ray with R + , the question completely reduces to the study of a fascinating space of functions Ψ : R + → R + . Moreover, recall that new such functions can be obtained by composing a given Ψ with any Bernstein function. (Reference monographs for affine actions and Bernstein functions are [BdlHV08] and [SSV12], respectively. ) We see that this first example, Is(E), has many -almost too many -self-representations for a precise classification. How about other infinite-dimensional groups? Are they too rigid to admit any, or again so soft as to admit too many? Considering that Is(E) sits in the much larger Möbius group Möb(E) of E, this article answers the following questions: • Does any non-tautological Is(E)-representation extend to Möb(E)? • Can the irreducible self-representations of Möb(E) be classified? • Among all Bernstein functions, which ones correspond to Möbius representations? In short, the answer is that the situation is much more rigid than for the isometries Is(E), but still remains much richer than in the finite-dimensional case. Specifically, there is exactly a one-parameter family of self-representations. This appears as a continuous deformation of the tautological representation, given by the Bernstein functions x → x t where the parameter t ranges in the interval (0, 1]. Formal statements Recall that the Möbius group Möb(E) is a group of transformations of the conformal sphere E = E ∪ {∞}; it is generated by the isometries of E, which fix ∞, and by the inversions v → (r/ v ) 2 v, where r > 0 is the inversion radius (see e.g. [Res89,I.3]). In particular it contains all homotheties. A first basic formalisation of the existence part of our results is as follows. Let E be an infinite-dimensional separable real Hilbert space. Theorem I. -For every 0 < t 1 there exists a continuous self-representation Möb(E) → Möb(E) with the following properties: • it restricts to an affinely irreducible self-representation of Is(E), • it maps the translation by v ∈ E to an isometry with translation part of norm v t , • it maps homotheties of dilation factor λ to similarities of dilation factor λ t . Before turning to our classification theorem, we recast the discussion into a more suggestive geometric context by switching to the viewpoint of Lobachevsky spaces. Recall that there exist three space forms: Euclidean, spherical, and real hyperbolic. In infinite dimensions, these correspond to Hilbert spaces, Hilbert spheres and the infinite-dimensional real hyperbolic space H ∞ . (For definiteness, we take all our spaces to be separable in this introduction.) The hyperbolic Riemannian metric of curvature −1 induces a distance d on H ∞ and we consider the corresponding (Polish) isometry group Is(H ∞ ), which also acts on the boundary at infinity ∂H ∞ . After choosing a point in ∂H ∞ , we can, just like in the finite-dimensional case, identify ∂H ∞ with the sphere E in a way that induces an isomorphism Is(H ∞ ) ∼ = Möb(E). As will be discussed in Section 2, there is a linear model for Is(H ∞ ), and hence the usual notion of irreducibility -which happens to coincide with a natural geometric notion. Here is the geometric, and more precise, counterpart to Theorem I. Theorem I bis . -For every 0 < t 1 there exists a continuous irreducible self-representation ∞ t : Is(H ∞ ) −→ Is(H ∞ ) and a ∞ t -equivariant embedding f ∞ t : H ∞ −→ H ∞ such that for all x, y ∈ H ∞ we have (1.1) cosh d f ∞ t (x), f ∞ t (y) = cosh d(x, y) t . Intuitively, the representations ∞ t are limiting objects for the representations n t : Is(H n ) −→ Is(H ∞ ) of the finite-dimensional Is(H n ) that we studied in [MP14], although the latter do not have a simple explicit expression like (1.1) for the distance. Using this relation to n t and our earlier results, we can show that these ∞ t , which are unique up to conjugacy for each t, exhaust all possible irreducible self-representations. Theorem II. -Every irreducible self-representation of Is(H ∞ ) is conjugated to a unique representation ∞ t as in Theorem I bis for some 0 < t 1. In this statement, conjugacy of some to ∞ t is the usual notion of isomorphism for representations, that is: there exists an isometry g such that g (·)g −1 = ∞ t (·). Even though we stated earlier that ∞ t is, in some sense, a limit of representations n t for Is(H n ) as n goes to infinity, there is a priori a difficulty in making this precise. Consider indeed a standard nested sequence of totally geodesic subspaces H n ⊆ H n+1 ⊆ · · · ⊆ H m ⊆ · · · ⊆ H ∞ with dense union. For m n, we can restrict m t to a copy of Is(H n ). After passing to the irreducible component, this creates a copy of n t . For each new m n, there is a new positive definite component that needs to be taken into account. Likewise, the associated harmonic map f m t : H m → H ∞ from [MP14], when restricted to H n , does not coincide with f n t and indeed is not harmonic on H n ; rather, it lies at a finite distance from f n t . Our strategy for eliminating all these difficulties is not to work with the groups, and not to work with the spaces either. Instead, we only keep track of the various distance functions and study their pointwise convergence. We then show that the spaces and groups can be reconstructed from this data after this easier limit has been established. This limiting behaviour of the distance functions is a basic instance of the phenomenon of concentration of measure on spheres. This strategy is very much the same as the one behind the use of kernels of positive type to construct orthogonal representations, and of kernels of conditionally negative type to construct affine isometric actions. Specifically, we use the notion of kernels of hyperbolic type and establish the necessary reconstruction results. Further considerations The reader will have noticed that there is no continuity assumption in Theorem II, although the representations constructed in Theorem I bis are continuous. Indeed this formulation of Theorem II necessitates the following result, which leverages the automatic continuity proved by Tsankov in [Tsa13] for the orthogonal group. Theorem III. -Every irreducible self-representation of Is(H ∞ ) is continuous. Such an automatic continuity phenomenon can fail in finite dimensions due to the fact that algebraic groups admit discontinuous endomorphisms induced by wild field endomorphisms (see Lebesgue [Leb07,p. 533] and [Kes51]). We do not know whether Is(H ∞ ) enjoys the full strength of the automatic continuity established by Tsankov for the orthogonal group. Problem. -Is every homomorphism from Is(H ∞ ) to any separable topological group continuous? In another direction, the existence of interesting self-representation of Is(H ∞ ) raises the possibility of composing them. The geometric description of Theorem I bis suggests that the composition of ∞ t with ∞ s should be related to ∞ ts . Indeed, the considerations of Section 5.1 show readily that ∞ t • ∞ s remains non-elementary in the sense recalled below and hence admits a unique irreducible sub-representation necessarily isomorphic to ∞ ts . In other words, Theorem II implies that upon co-restricting to the unique irreducible part, the semi-group of irreducible self-representations modulo conjugation is isomorphic to the multiplicative semi-group (0, 1]. We could not ascertain that it is really necessary to extract the irreducible part. Problem. -Is the composition of two irreducible self-representations of Is(H ∞ ) still irreducible? The last section of this article adds a few elements to the study in [MP14] of the representations n t of Is(H n ) for n finite. Notation and preliminaries Minkowski spaces We work throughout over the field R of real numbers. The scalar products and norms of the various Hilbert spaces introduced below will all be denoted by · , · and · . Given a Hilbert space H, we consider the Minkowski space R ⊕ H endowed with the bilinear form B defined by B(s ⊕ h, s ⊕ h ) = ss − h, h . This is a "strongly non-degenerate bilinear form of index one" in the sense of [BIM05], to which we refer for more background. We endow R ⊕ H with the topology coming from the norms of its factors. The adverb "strongly" refers to the fact that the corresponding uniform structure is complete. Given a cardinal α, the hyperbolic space H α can be realised as the upper hyperboloid sheet H α = x = s ⊕ h ∈ R ⊕ H : B(x, x) = 1 and s > 0 , where H is a Hilbert space of Hilbert dimension α. The visual boundary ∂H α can be identified with the space of B-isotropic lines in R ⊕ H. We simply write H ∞ for our main case of interest, namely the separable infinite-dimensional Lobachevsky space H ∞ = H ℵ 0 . The distance function associated with the hyperbolic metric is characterized by cosh d(x, y) = B(x, y) and is therefore compatible with the ambient topology and complete. Second model It is often convenient to use another model for the Minkowski space R ⊕ H, as follows. Suppose given two points at infinity, represented by isotropic vectors ξ 1 , ξ 2 such that B(ξ 1 , ξ 2 ) = 1. Define E to be the B-orthogonal complement {ξ 1 , ξ 2 } ⊥ . Then −B induces a Hilbert space structure on E. We can now identify (R ⊕ H, B) with the space R 2 ⊕ E endowed with the bilinear form B defined by B (s 1 , s 2 ) ⊕ v, (s 1 , s 2 ) ⊕ v = s 1 s 2 + s 2 s 1 − v, v in such a way that the isomorphism takes ξ 1 , ξ 2 to the canonical basis vectors of R 2 (still denoted ξ i ) and that H α is now realised as H α = x = (s 1 , s 2 , v) : B (x, x) = 1 and s 1 > 0 , noting that s 1 > 0 is equivalent to s 2 > 0 given the condition B (x, x) = 1. We can further identify ∂H α with E = E ∪ {∞} by means of the following parametrisation by B -isotropic vectors: (2.1) v −→ 1 2 v 2 , 1 ⊕ v, ∞ −→ ξ 1 . In particular, 0 ∈ E corresponds to ξ 2 . This parametrisation intertwines the action of Is(H α ) with the Möbius group of E. For instance, the Minkowski operator exchanging the coordinates of the R 2 summand corresponds to the inversion in the sphere of radius √ 2 around 0 ∈ E. Subspaces The hyperbolic subspaces of H α are exactly all subsets of the form H α ∩ N where N < R ⊕ H is a closed linear subspace of R ⊕ H. It is understood here that we accept the empty set, points and (bi-infinite) geodesic lines as hyperbolic subspaces. Definition 2.1. -The hyperbolic hull of a subset of H α is the intersection of all hyperbolic subspaces containing it. A subset is called hyperbolically total if its hyperbolic hull is the whole ambient H α . Thus the hyperbolic hull of a subset X ⊆ H α coincides with H α ∩ span(X). It follows that X is hyperbolically total if and only if it is total in the topological vector space R ⊕ H. There is a bijective correspondence, given by H → H α ∩(R ⊕H ), between Hilbert subspaces H < H and hyperbolic subspaces that contain the point 1 ⊕ 0. In the second model, E → H α ∩ (R 2 ⊕ E ) is a bijective correspondence between Hilbert subspaces E < E and hyperbolic subspaces of H α whose boundary contains both ξ 1 and ξ 2 . Horospheres We can parametrise H α by R × E in the second model as follows. For s ∈ R and v ∈ E, define σ s (v) =    1 2 (e s + e −s v 2 ) e −s e −s v    . For any given v, the map s → σ s (v) is a geodesic line; its end for s → ∞ is represented by ξ 1 and its other end s → −∞ by the isotropic vector ( 1 2 v 2 , 1) ⊕ v of the parametrisation (2.1) above. If on the other hand we fix s, then the map σ s is a parametrisation by E of a horosphere σ s (E) based at ξ 1 . Computing B (σ s (u), σ s (v)), we find that the hyperbolic distance on this horosphere is given by (2.2) cosh d(σ s (u), σ s (v)) = 1 + 1 2 e −2s u − v 2 (∀ u, v ∈ E). If E < E is a Hilbert subspace and H the corresponding hyperbolic subspace H α ∩ (R 2 ⊕ E ) considered in Section 2.3, then σ s (E ) is σ s (E) ∩ H and coincides with a horosphere in H based at ξ 1 ∈ ∂H . Kernels of hyperbolic type The notion of kernel of hyperbolic type Kernels of positive and of conditionally negative type are classical tools for the study of embeddings into spherical and Euclidean spaces respectively (see e.g. Appendix C in [BdlHV08]). The fact that a similar notion is available for hyperbolic spaces seems to be well known, see e.g. §5 in [Gro01]. We formalise it as follows. Definition 3.1. -Given a set X, a function β : X × X → R is a kernel of hyperbolic type if it is symmetric, non-negative, takes the constant value 1 on the diagonal, and satisfies (3.1) n i,j=1 c i c j β(x i , x j ) n k=1 c k β(x k , x 0 ) 2 for all n ∈ N, all x 0 , x 1 , . . . , x n ∈ X and all c 1 , . . . , c n ∈ R. Remarks 3.2. -(i) The case n = 1 of (3.1) implies β(x, y) 1 for all x, y, and β ≡ 1 is a trivial example. (ii) The set of kernels of hyperbolic type on X is closed under pointwise limits. (iii) One can check that for every kernel ψ of conditionally negative type, the kernel β = 1 + ψ is of hyperbolic type. However, the geometric interpretation established below shows that this only provides examples that are in a sense degenerate; see Proposition 4.2. Just as in the spherical and Euclidean cases, the above definition is designed to encapsulate the following criterion. Proposition 3.3. -Given a map f from a non-empty set X to a hyperbolic space, the function β defined on X × X by β(x, y) = cosh d(f (x), f (y)) is a kernel of hyperbolic type. Conversely, any kernel of hyperbolic type arises from such a map f that has a hyperbolically total image. A straightforward way to establish Proposition 3.3 is to use the relation with kernels of positive type, as discussed in Section 3.2. However, this is unsatisfactory in one very important aspect, namely the question of naturality. In particular, how do transformations of X correspond to isometries of the hyperbolic space? Indeed an important difference between the above definition of kernels of hyperbolic type and the classical spherical and Euclidean cases is that (3.1) is asymmetric. An additional argument allows us to answer this question as follows. Theorem 3.4. -Let X be a non-empty set with a kernel of hyperbolic type β. Then the space H α and the map f : X → H α granted by Proposition 3.3 are unique up to a unique isometry of hyperbolic spaces. Therefore, denoting by Aut(X, β) the group of bijections of X that preserve β, there is a canonical representation Aut(X, β) → Is(H α ) for which f is equivariant. A function F : G → R on a group G will be called a function of hyperbolic type if the kernel (g, h) → F (g −1 h) is of hyperbolic type. In other words, this is equivalent to the data of a left-invariant kernel of hyperbolic type on G. Therefore, the above results imply readily the following. Corollary 3.5. -For every function of hyperbolic type F : G → R there is an isometric G-action on a hyperbolic space and a point p of that space such that F (g) = cosh d(gp, p) holds for all g ∈ G and such that the orbit Gp is hyperbolically total. If moreover G is endowed with a group topology for which F is continuous, then the G-action is continuous. The following interesting example is provided by the Picard-Manin space associated to the Cremona group. Example 3.6. -Let Bir(P 2 ) be the Cremona group and deg : Bir(P 2 ) → N be the degree function. That is, deg(g) 1 is the degree of the homogeneous polynomials defining g ∈ Bir(P 2 ). It follows from the work of Cantat [Can11] that the function deg is of hyperbolic type, the associated hyperbolic space being the Picard-Manin space (see also Chapter 5 in [Man86]). The geometric characterization of kernels of hyperbolic type In order to prove Proposition 3.3, we recall that a function N : X ×X → R is called a (real) kernel of positive type if it is symmetric and satisfies i,j c i c j N (x i , x j ) 0 for all n ∈ N, all x 1 , . . . , x n ∈ X and all c 1 , . . . , c n ∈ R. If h : X → H is any map to a (real) Hilbert space H, then N (x, y) = h(x), h(y) defines a kernel of positive type. Conversely, the GNS construction associates canonically to any kernel of positive type N a Hilbert space H and a map h : X → H such that N (x, y) = h(x), h(y) holds for all x, y ∈ X, and such that moreover h(X) is total in H (we refer again to Appendix C in [BdlHV08]). Now the strategy is simply to identify the hyperboloid H α in a Minkowski space R⊕H with the Hilbert space factor H. This is the Gans model [Gan66]; the drawback is that any naturality is lost. Proof of Proposition 3.3. -We first verify that, given a map f from X to a hyperbolic space H α , the kernel defined by β(x, y) = cosh d(f (x), f (y)) is indeed of hyperbolic type. In the ambient Minkowski space R ⊕ H for H α , the reverse Schwarz inequality implies B(v, v) B(v, v 0 ) 2 ∀ v, v 0 ∈ R ⊕ H with B(v 0 , v 0 ) = 1. This can also be verified directly by using the transitivity properties of O(B) to reduce it to the case v 0 = 1 ⊕ 0, where it is trivial. Given now x 0 , x 1 , . . . , x n ∈ X and c 1 , . . . , c n ∈ R, we apply this inequality to v 0 = f (x 0 ) and v = n k=1 c k f (x k ) and the inequality (3.1) follows. We turn to the converse statement; thus let β be an arbitrary kernel of hyperbolic type on X. Pick x 0 ∈ X and consider the kernel N on X defined by (3.2) N (x, y) = β(x, x 0 )β(y, x 0 ) − β(x, y). Condition (3.1) is precisely that N is of positive type. Consider thus h : X → H as given by the GNS construction for N and the corresponding hyperbolic space H α in R ⊕ H. Define f : X → H α by f (x) = β(x 0 , x) ⊕ h(x). Using (3.2), we obtain the desired relation B(f (x), f (y)) = β(x, y). Finally, we prove that f (X) is hyperbolically total; thus let V ⊆ R ⊕ H be the closed linear subspace spanned by f (X) and recall that it suffices to show that V is all of R ⊕H. The definition of N implies N (x 0 , x 0 ) = 0. Therefore, we have h(x 0 ) = 0 and hence V contains 1 ⊕ 0. Thus V = R ⊕ H follows from the fact that h(X) is total in H. Remark 3.7. -The construction of f shows that when a topology is given on X, the map f will be continuous as soon as the kernel β is continuous. Indeed, the corresponding statement holds for kernels of positive type, see e.g. Theorem C.1.4 in [BdlHV08]. Functoriality and kernels We now undertake the proof of Theorem 3.4. We keep the notations introduced in the proof of Proposition 3.3 for the construction of the space H α ⊆ R ⊕ H and of the map f : X → H α . In order to prove Theorem 3.4, it suffices to give another construction of R ⊕ H, of B and of f that depends functorially on (X, β), and only on this. Remark 3.8. -The previous construction introduced a choice of x 0 , and hence of N in (3.2), to define f . We now argue more functorially, but the price to pay is that the nature of the constructed bilinear form is unknown until we compare it to the non-functorial construction. We record the following fact, wherein V denotes the completion with respect to the uniform structure induced by the given non-degenerate quadratic form of finite index. The statement follows from the discussion in §2 of [BIM05], although it is not explicitly stated in this form. Proposition 3.9. -Let (V, Q) be a real vector space endowed with a nondegenerate quadratic form of finite index. Then there is a vector space V with a strongly non-degenerate quadratic form Q of finite index equal to that of Q, such that V embeds densely in V with Q extending Q. The quadratic space (V , Q) is unique up to isometry; any isometry of (V, Q) extends to an isometry of (V , Q). We now start our functorial construction. We extend β to a symmetric bilinear form on the free vector space R[X] on X. We denote by W 0 the quotient of R[X] by the radical R of this bilinear form, and by B the symmetric bilinear form thus induced on W 0 . We further denote by f the composition of the canonical maps X → R[X] → W 0 . In particular, we have β(x, y) = B f (x), f (y) for all x, y ∈ X. To finish the proof, it suffices to establish the following two claims. First, W 0 is non-degenerate of index 1, and hence has a completion W by Proposition 3.9. Secondly, there is an identification of W with R ⊕ H that intertwines f with f and B with B, though we may of course now use f to construct this identification. In fact, we shall prove both claims at once by exhibiting an injective linear map ι : W 0 → R ⊕ H such that B ι(u), ι(v) = B(u, v) and ι f (x) = f (x) hold for all u, v ∈ W 0 and all x ∈ X. The latter property implies in particular that ι(W 0 ) is dense in R ⊕ H since f (X) is hyperbolically total, see Section 2.3. Although ι will be constructed using f , the first claim still holds because this construction implies in particular that B is a non-degenerate form of index one and that the completion W coincides with R ⊕ H. We turn to the construction of ι. Extend f by linearity to a map f : R[X] −→ R ⊕ H. Denote by K the kernel of f ; by construction, K is contained in the radical R. We now check that in fact K = R; thus let λ = x∈X λ x [x] be a finite formal linear combination of elements of X and assume λ ∈ R. If f (λ) did not vanish, then f (λ) ⊥ would be a proper subspace of R ⊕ H. However, this subspace always contains f (X) since λ ∈ R, and thus we would contradict the fact that f (X) is total in R ⊕ H. At this point, f induces a map ι with all the properties that we required. This completes the proof of Theorem 3.4. Proof of Corollary 3.5. -We apply Theorem 3.4 to the kernel β defined on G by β(g, h) = F (g −1 h). Viewing G as a subgroup of Aut(G, β), we obtain a homomorphism : G → Is(H α ) and a -equivariant map f : G → H α . This means that f is the orbital map associated to the point p = f (e). It only remains to justify the continuity claim. Since f (G) = Gp is hyperbolically total, the orbital continuity follows readily from the continuity of f , noted in Remark 3.7, because isometric actions are uniformly equicontinuous. The latter fact also implies that orbital continuity is equivalent to joint continuity for isometric actions. Powers of kernels of hyperbolic type The fundamental building block for exotic self-representations is provided by the following statement. Theorem 3.10. -If β is a kernel of hyperbolic type, then so is β t for all 0 t 1. After we established this result, another proof was found, purely computational; it will be presented in [Mon18]. Proof of Theorem 3.10. -We can assume t > 0 since the constant function 1 satisfies Definition 3.1 trivially. In view of Proposition 3.3, it suffices to prove that for any hyperbolic space H α with distance d, where α is an arbitrary cardinal, the kernel (cosh d) t : H α × H α −→ R is of hyperbolic type. Definition 3.1 considers finitely many points at a time, which are therefore contained in a finite-dimensional hyperbolic subspace of H α (see e.g. Remark 3.1 in [BIM05]). For this reason, it suffices to prove the above statement for H m with m ∈ N arbitrarily large -but fixed for the rest of this proof. Given an integer n m, we choose an isometric embedding H m ⊆ H n and consider the map f n t : H n −→ H ∞ that we provided in Theorem C of [MP14] (it was simply denoted by f t in that reference, but we will shortly let n vary). Consider the kernel β n : H m × H m −→ R, β n (x, y) = cosh d f n t (x) , f n t (y) obtained by restriction to H m ⊆ H n ; it is of hyperbolic type by Proposition 3.3. The proof will therefore be complete if we show that β n converges pointwise to (cosh d) t on H m × H m . Choose thus x, y ∈ H m . We computed an integral expression for the quantity β n (x, y) = cosh d f n t (x), f n t (y) in §3.B and §3.C of [MP14]. Namely, writing u = d(x, y), we established β n (x, y) = S n−1 cosh(u) − b 1 sinh(u) −(n−1+t) db, where db denotes the integral against the normalised volume on the sphere S n−1 , and b 1 is the first coordinate of b when b is viewed as a unit vector in R n . We further recall (see [MP14,(3.vi )]) that cosh(u) − b 1 sinh(u) −(n−1) is the Jacobian of some hyperbolic transformation g −1 u of S n−1 . We can therefore apply the change of variable formula for g u and obtain β n (x, y) = S n−1 ϕ(g u b) db, where ϕ(b) = cosh(u) − b 1 sinh(u) −t . (3.3) The transformation g u is given explicitly in [MP14], namely it is g u = g e u ,0,Id as defined in §2.A of [MP14]. These formulas show that the first coordinate of g u b is (g u b) 1 = sinh(u) + b 1 cosh(u) cosh(u) + b 1 sinh(u) . Entering this into (3.3), we readily compute β n (x, y) = S n−1 cosh(u) + b 1 sinh(u) t db. We are thus integrating on S n−1 a continuous function depending only upon the first variable b 1 , which is now independent of n. Therefore, when n tends to infinity, the concentration of measure principle implies that this integral converges to the value of that function on the equator {b 1 = 0}. Since this equatorial value is (cosh(u)) t , we have indeed proved that β n (x, y) converges to (cosh(d(x, y)) t , as was to be shown. On representations arising from kernels General properties Let G be a group and F : G → R a function of hyperbolic type. According to Corollary 3.5, this gives rise to an isometric G-action on a hyperbolic space H α together with a point p ∈ H α whose orbit is hyperbolically total in H α , and such that F (g) = cosh d(gp, p) (∀ g ∈ G) . We now investigate the relation between the geometric properties of this G-action and the properties of the function F . The Cartan fixed-point theorem, in the generality presented, for example in [BH99, II.2.8], implies the following. Fixed points at infinity are a more subtle form of elementarity for the G-action; we begin with the following characterization for kernels. Then f (X) is contained in a horosphere if and only if β − 1 is of conditionally negative type. In particular we deduce the corresponding characterization for the G-actions. -Suppose that f (X) is contained in a horosphere. We can choose the model described in Section 2.4 in such a way that this horosphere is σ 0 (E) in the notations of that section. Therefore, equation (2.2) implies for all x, y ∈ X the relation β(x, y) = cosh d(f (x), f (y)) = 1 + 1 2 σ −1 0 (f (x)) − σ −1 0 (f (y)) 2 , where · is the norm of the Hilbert space E parametrising the horosphere. This witnesses that β − 1 is of conditionally negative type. Conversely, if β − 1 is of conditionally negative type, then the usual affine GNS construction (see e.g. §C.2 in [BdlHV08]) provides a Hilbert space E and a map η : X → E such that η(X) is total in E , and such that β(x, y) − 1 = 1 2 η(x) − η(y) 2 holds for all x, y. Now σ 0 • η is a map to a horosphere in the hyperbolic space H corresponding to E in the second model (Section 2.2), centred at ξ 1 ∈ ∂H . Let H ⊆ H be the hyperbolic hull of σ 0 • η(X) and observe that its boundary contains ξ 1 since β is unbounded. Thus σ 0 • η(X) is contained in a horosphere of H . By Theorem 3.4, σ 0 • η can be identified with f and hence the conclusion follows. Individual isometries The type of an individual group element for the action defined by F can be read from F . Recall first that the translation length (g) associated to any isometry g of any metric space Y is defined by (g) = inf d(gy, y) : y ∈ Y . We now have the following trichotomy. Proposition 4.4. -For any g ∈ G, the action defined by F satisfies (g) = ln lim n→∞ F (g n ) 1 n . Moreover, exactly one of the following holds. (1) F (g n ) is uniformly bounded over n ∈ N; then g is elliptic: it fixes a point in H α . (2) F (g n ) is unbounded and (g) = 0; then g is neutral parabolic: it fixes a unique point in ∂H α and preserves all corresponding horospheres but has no fixed point in H α . (3) (g) > 0; then g is hyperbolic: it preserves a unique geodesic line in ∂H α and translates it by (g). Remark 4.5. -Consider the Picard-Manin space associated to the Cremona group Bir(P 2 ) as mentioned in Example 3.6. Recall that the limit lim n→∞ deg(g n ) 1/n is the dynamical degree of the birational transformation g. Thus we see that the translation length is the logarithm of the dynamical degree, which is a basic fact in the study of the Picard-Manin space. Proof of Proposition 4.4. -Since arcosh(F (g n )) = d(g n p, p), we see that g n p, p). Now the statements of the proposition hold much more generally. Recall that if p is a point of an arbitrary CAT(0) space Y on which G acts by isometries, then the translation length of g ∈ G satisfies (4.1) (g) = lim n→∞ 1 n d(g n p, p), see e.g. Lemma 6.6 (2) in [BGS85]. If in addition X is complete and CAT(−1), then the above trichotomy holds, see for instance §4 in [BIM05]. ln lim n→∞ F (g n ) 1 n = lim n→∞ 1 n d( Self-representations of Is(H ∞ ) Definition of ∞ t We choose a point p 1 ∈ H ∞ and consider the corresponding function of hyperbolic type F 1 given by the tautological representation of Is(H ∞ ) on H ∞ . We denote by O the stabiliser of p 1 , which is isomorphic to the infinite-dimensional orthogonal group. Fix 0 < t 1. By Theorem 3.10, the function F t = (F 1 ) t is still of hyperbolic type. Appealing to Corollary 3.5, we denote by ∞ t the corresponding representation on H α . Observe that α ℵ 0 since F t is continuous. It follows that α = ℵ 0 since Is(H ∞ ) has no non-trivial finite-dimensional representation (this is already true for O since SO(n) has no non-trivial representation of dimension < n for large n). Hence we write H ∞ for H α . Given g ∈ Is(H ∞ ), we write t (g) for its translation length as an isometry under the representation ∞ t so as not to confuse it with its translation length under the tautological representation -which we can accordingly denote by 1 (g). Now Proposition 4.4 has the following consequence. Corollary 5.1. -We have t = t 1 and the representation ∞ t preserves the type of each element of Is(H ∞ ). We can further deduce the following. Corollary 5.2. -The representation ∞ t is non-elementary. We shall also prove that ∞ t is irreducible, but it will be more convenient to deduce it later on. Proof of Corollary 5.2. -Suppose for a contradiction that ∞ t is elementary. It cannot fix a point since F t is unbounded. Thus it either fixes a point at infinity or preserves a geodesic line. Choose a copy of PSL 2 (R) in Is(H ∞ ); being perfect, it fixes a point at infinity in either case and preserves the corresponding horospheres. This is however impossible because a hyperbolic element of PSL 2 (R) must remain hyperbolic under ∞ t by Corollary 5.1. Restricting to finite dimensions We begin with a general property of Is(H ∞ ). Proof. -Let p be a point on the axis associated to L and let O be the stabiliser of p, which is isomorphic to the infinite-dimensional (separable) orthogonal group. Then we have the Cartan-like decomposition Is(H ∞ ) = OLO; indeed, this follows from the transitivity of O on the space of directions at the given point p. On the other hand, any isometric action of O on any metric space has bounded orbits, see [RR07,p. 190]. The statement follows. In order to restrict a representation to finite-dimensional subgroups, we choose an exhaustion of our Minkowski space R⊕H by finite-dimensional Minkowski subspaces, for instance by choosing a nested sequence of subspaces R n ⊆ H with dense union. This gives us a nested sequence of totally geodesic subspaces H n ⊆ H n+1 ⊆ · · · ⊆ H ∞ with dense union, together with embeddings Is(H n ) < Is(H ∞ ) preserving H n . Moreover, the union of the resulting nested sequence of subgroups Is(H n ) ⊆ Is(H n+1 ) ⊆ · · · ⊆ Is(H ∞ ) is dense in Is(H ∞ ): this can be deduced, for example, from the density of the union of all O(n) in O, together with a Cartan decomposition (as introduced above) for some L ⊆ Is(H 2 ). Proof. -It suffices to show that the representation remains non-elementary when restricted to the connected component Is(H 2 ) • ∼ = PSL 2 (R) of Is(H 2 ); suppose otherwise. We denote by g a non-trivial element of the one-parameter subgroup L of hyperbolic elements represented by t → e t 0 0 e −t , and by w, the involution represented by [ 0 −1 1 0 ]. By Proposition 5.3 and continuity, g cannot fix a point in H ∞ . Therefore our apagogical assumption implies that Is(H 2 ) • either fixes a point in ∂H ∞ or preserves a geodesic line. Since Is(H 2 ) • is perfect, the former case holds anyway; thus let ξ ∈ ∂H ∞ be a point fixed by Is(H 2 ) • . Now g cannot act hyperbolically because otherwise it would have exactly two fixed points at infinity exchanged by w because w conjugates g to g −1 ; this would contradict the fact that both g and w fix ξ. Thus g is parabolic. We claim that ξ is in fact fixed by Is(H n ) • for all n 2. First, we know that Is(H n ) acts elementarily, because otherwise Proposition 2.1 from [MP14] would imply that g acts hyperbolically. Next, Is(H n ) cannot fix a point in H ∞ since g does not. Thus Is(H n ) fixes a point at infinity or preserves a geodesic and we conclude again by perfectness of Is(H n ) • that Is(H n ) • fixes some point in ∂H ∞ . This point has to be ξ because g, being parabolic, has a unique fixed point at infinity. This proves the claim. Finally, no other point than ξ is fixed by Is(H n ) • since this already holds for g. Therefore, ξ is also fixed by Is(H n ) since the latter normalises Is(H n ) • . Therefore Is(H ∞ ) fixes ξ by density. This contradicts the assumption that the self-representation was non-elementary. Completion of the proofs A crucial remaining step is the following result, which relies notably on our classification from [MP14]. Notice that the parameter t is uniquely determined by (5.1); indeed this equation implies that t is the ratio of translation lengths ( (g)) (g) for any hyperbolic element g. Therefore, Theorem 5.5, combined with Theorem 3.4, already implies the uniqueness result stated in Theorem II in the Introductionexcept for two points. First, the existence result of Theorem I bis is still needed to give any substance to this uniqueness statement, namely, we must still establish the irreducibility of the representations ∞ t . Secondly, since Theorem II is stated without continuity assumption, we need to establish the automatic continuity of Theorem III. We defer this (independent) proof to Section 6. Proof of Theorem 5.5. -We choose the exhaustion by finite-dimensional spaces H n in such a way that they all contain p 0 ; in particular, the stabiliser O of p 0 meets each Is(H n ) in a subgroup isomorphic to O(n). By Proposition 5.4, the restriction of to Is(H n ) is non-elementary and hence admits a unique minimal invariant hyperbolic subspace, which we denote by H ∞ n ⊆ H ∞ . Note that H ∞ n ⊆ H ∞ n+1 holds. In view of the classification that we established in Theorem B of [MP14], we have 0 < t 1 such that ( (g)) = t (g) holds for every g ∈ Is(H n ). In particular, it follows that t does not depend on n. does not fix gp, we deduce that V cannot fix gp. Considering that d(gp, vgp) is in J when v ∈ V , and that V is connected, the claim follows. To establish the proposition, we shall prove that every h ∈ Is(H ∞ ) with d(p, hp) < lies in Og −1 U gO. By the claim, there is u ∈ U with d(gp, ugp) = d(p, hp). In other words, there is q ∈ g −1 U g with d(p, qp) = d(p, hp). By transitivity of O on any sphere centred at p, there is q ∈ O with q qp = hp. We conclude h ∈ q qO, as claimed. Proof of Theorem III. -Let : Is(H ∞ ) → Is(H ∞ ) be an irreducible self-representation and let O < Is(H ∞ ) be the stabiliser of a point in H ∞ for its tautological representation. Then (O) has bounded orbits by [RR07,p. 190] and hence fixes a point p ∈ H ∞ by the Cartan fixed-point theorem [BH99, II.2.8]. We claim that the function D : Is(H ∞ ) −→ R + , D(g) = d (g)p, p is continuous at e ∈ Is(H ∞ ). Thus let (g n ) be a sequence converging to e in Is(H ∞ ). We fix some g / ∈ O; then Proposition 6.1 implies that we can write g n = k n g −1 u n gk n for k n , u n , k n ∈ O such that the sequence (u n ) converges to e. Since (O) fixes p, we have D(g n ) = d (u n ) (g)p, (g)p . Tsankov [Tsa13] proved that every isometric action of O on a Polish metric space is continuous; therefore the right-hand side above converges to zero as n → ∞, proving the claim. It now follows that D is continuous on all of Is(H ∞ ), because if g n → g ∞ in Is(H ∞ ), we can estimate d (g n )p, p − d (g ∞ )p, p d (g n )p, (g ∞ )p = d (g −1 ∞ g n )p, p . Equivalently, the function of hyperbolic type cosh D is continuous. Since is irreducible, the orbit of p is hyperbolically total. It follows that is continuous, see Theorem 3.4 and Corollary 3.5. Finite-dimensional post-scripta Et je vais te prouver par mes raisonnements. . . Mais malheur à l'auteur qui veut toujours instruire ! Le secret d'ennuyer est celui de tout dire. -Voltaire, Sur la nature de l'homme, 1737 Vol. I p. 953 of the 1827 edition by Jules Didot l'aîné (Paris). Our previous work [MP14] focused on representations n t into Is(H ∞ ) of the finitedimensional groups Is(H n ) ∼ = PO(1, n). A main application was the construction of exotic locally compact deformations of the classical Lobachevsky space H n . Nonetheless, a significant part of that article was devoted to the analysis of an equivariant map f n t : H n −→ H ∞ canonically associated to the representation with parameter 0 < t < 1. We proved notably that f n t is a harmonic map whose image is a minimal submanifold of curvature −n t(t+n−1) . Furthermore, we investigated a quantity denoted by I u in [MP14, §3.C] which, in the language of the present article, is none other than the radial function of hyperbolic type associated to f n t . More precisely, we showed that H ∞ contains a unique point p fixed by O(n) under n t . Therefore, we have a bi-O(n)-invariant function of hyperbolic type F : Is(H n ) −→ R + , F (g) = cosh d ( n t (g)p, p) . Being bi-O(n)-invariant, F can be represented by F 0 : R + −→ R + , F 0 (u) = F (e uX ), where e uX represents any one-parameter group of hyperbolic elements for a Cartan decomposition with respect to O(n), normalised so that e X has translation length 1. Now I u = F 0 (u). It was proved in [MP14] that f n t is large-scale isometric; we now propose a more precise and more uniform statement. Lemma 4.1. -The function F is bounded if and only if G fixes a point in H α . Let β be an unbounded kernel of hyperbolic type on a set X and consider the map f : X → H α granted by Proposition 3.3. Suppose F unbounded. Then the orbit Gp is contained in a horosphere if and only if F − 1 is of conditionally negative type. Proof of Proposition 4.2. - Let L ∼ = R be a one-parameter subgroup of hyperbolic elements of Is(H ∞ ). Then an arbitrary isometric Is(H ∞ )-action on a metric space has bounded orbits if and only if L has bounded orbits. Proposition 5.4. -Any continuous non-elementary self-representation of Is(H ∞ ) remains non-elementary when restricted to Is(H n ) for any n 2. Theorem 5.5. -Choose a point p 0 ∈ H ∞ . For every irreducible continuous self-representation of Is(H ∞ ), there is 0 < t 1 and p ∈ H ∞ , such that (5.1) cosh d (g)p, (h)p = cosh d(gp 0 , hp 0 ) t holds for all g, h ∈ Is(H ∞ ). An isometric action on H α is called elementary if it fixes a point in H α or in ∂H α , or if it preserves a line. Any non-elementary action preserves a unique minimal hyperbolic subspace and H α is itself this minimal subspace if and only if the associated linear representation is irreducible [BIM05, §4].The group O(B) of invertible linear operators preserving B acts projectively on H α , inducing an isomorphism PO(B) ∼ = Is(H α ). Alternatively, Is(H α ) is isomorphic to the subgroup of index two O + (B) < O(B) which preserves the upper sheet H α . See Proposition 3.4 in [BIM05]. Manuscript received on 11th June 2018, accepted on 4th December 2018. Recommended by Editor S. Cantat. Published under license CC BY 4.0. This journal is a member of Centre Mersenne. Nicolas MONOD EPFL (Switzerland) [email protected] Pierre PY Instituto de Matemáticas, Universidad Nacional Autónoma de México (México) Current address: IRMA, Université de Strasbourg & CNRS 67084 Strasbourg (France) [email protected] Next we observe that (O) fixes some point p ∈ H ∞ . This follows of course from the much stronger result of[RR07]cited in the above proof of Proposition 5.3, but it can also be deduced, for example, from the fact that O has property (T) as a polish group[Bek03,Rem. 3 (i)].Let p n be the nearest-point projection of p to H ∞ n . Then p n is fixed by O(n) since the projection map is equivariant under Is(H n ). Since the union of all H ∞ n is dense in H ∞ by irreducibility of the representation, it follows that the sequence (p n ) converges to p. We deduce that cosh d (g)p, (h)p = lim n→∞ d (g)p n , (h)p n holds for all g, h ∈ Is(H ∞ ). We now claim that we havefor all g, h in the union of all Is(H k ). This follows from our classification of the irreducible representations of Is(H n ) on H ∞ (Theorem B in[MP14]), together with the fact that p n is the unique point of H ∞ n fixed by O(n) (Lemma 3.9[MP14]), and from the computation of the limit performed in the proof of Theorem 3.10.In conclusion, we have indeed established the relation (5.1) on the union of all Is(H n ), and hence on Is(H ∞ ) by density.Remark 5.6. -The above proof also shows that p is the unique fixed point of O under the (arbitrary) irreducible continuous self-representation .Proof of Theorem I bis . -By construction, ∞ t is a continuous self-representation such that (5.1) holds for some point p with hyperbolically total orbit. We need to argue that it is irreducible.By Corollary 5.2, ∞ t is non-elementary and therefore it contains a unique irreducible part, or equivalently preserves a unique minimal hyperbolic subspace H ⊆ H ∞ . We now apply Theorem 5.5 to the resulting representation on H , yielding some parameter t and some point p ∈ H . We observe that t = t; one way to see this is to read the translation lengths from the asymptotic equation (4.1), which are unaffected by a change of base-point.Since the orbit of p under ∞ t is hyperbolically total, it suffices to show that p ∈ H to deduce H = H ∞ , which completes the proof. By Remark 5.6, the nearest-point projection of p to H is p . Now choose any g ∈ Is(H ∞ ) not in O. Then ∞ t (g) does not fix p, and moreover the projection of ∞ t (g)p to H is ∞ t (g)p . Since (5.1) holds for both p and p , we have. The sandwich lemma (in the form of Ex. II.2.12 in[BH99]) now implies that the four points p, ∞ t (g)p, ∞ t (g)p , p span a Euclidean rectangle. Since we are in a CAT(−1) space, this rectangle is degenerate and we conclude p = p , as desired.Remark 5.7. -We could have shortened the above proof of Theorem I bis by defining ∞ t to be the irreducible part of the representation constructed from a kernel. However, we find that there is independent interest in knowing that a given kernel is associated to an irreducible representation.We finally turn to the Möbius group formulation of Theorem I bis .Proof of Theorem I. -We use the second model for the hyperbolic space H ∞ , as in Section 2.2. In particular H ∞ sits inside the linear space R 2 ⊕ E.Thinking of the representation ∞ t as a self-representation of the group Möb(E), we can assume up to conjugacy that the stabiliser of ∞ in Möb(E) is mapped to itself. Note that this stabiliser is precisely the group of similarities of E. The image under ∞ t of the group of homotheties, being made of hyperbolic element of Möb(E), must then fix a unique point q of E. Conjugating again, we assume that q = 0, i.e. that the images of homotheties are linear maps of E. This also implies that O(E) maps to itself. The assertion about homotheties in Theorem I is now a direct consequence of the fact that ∞ t multiplies translation lengths in H ∞ by t. It is also a formal consequence that the translation by v ∈ E is sent to an isometry whose translation part has norm c v t for some constant c > 0, independent of v. Conjugating once more by a homothety, we can assume that c = 1.It remains only to justify that the induced self-representation of Is(E) is affinely irreducible: suppose that F ⊆ E is a closed affine subspace invariant under ∞ t (Is(E)). As in the proof of Theorem 5.Automatic continuityTsankov[Tsa13]proved that every isometric action of O on a Polish metric space is continuous. We do not know whether Is(H ∞ ) enjoys such a strong property. However, we shall be able to prove the automatic continuity of Theorem III by combining Tsankov's result for O with the following fact about the local structure of Is(H ∞ ). It follows that for all x, y ∈ H n we haveIn other words, f n t fails to be an isometry after rescaling by t only by an additive error bounded by log 2, independent of the dimension n and of t. Of course, equation (7.1) also holds for f ∞ t instead of f n t . This follows either by applying the same proof as below to f ∞ t , or by taking the limit in equation (7.1) as n goes to infinity. For the local behaviour of f n t , we refer the reader to Proposition 3.10 in[MP14]. As for the local behaviour of f ∞ t , one can observe that equation (1.1) from the introduction immediately implies thatas d(x, y) goes to 0.Recalling that the usual spherical metric d S on S n−1 or S ∞ can be realised as a visual metric at infinity induced by H n or H ∞ , which is defined in terms of exponentials of distances in H n or H ∞ (see e.g. [BH99, p. 434]), the following is an immediate consequence.Corollary 7.2. -The map f n t induces a bi-Lipschitz embedding of the snowflake (S n−1 , d t S ) into the round sphere (S ∞ , d S ). Proof of Proposition 7.1. -Thanks to Proposition 5.4, the restriction of ∞ t to Is(H n ) has an irreducible part, which is necessarily isomorphic to n t due to the classification of[MP14]. Considering that the distances decrease when projecting to the corresponding minimal invariant hyperbolic subspace (just as in the proof of Theorem I bis ), we deduce F 0 (u) cosh t (u). On the other hand, arcosh F 0 (u) is always bounded below by the translation length of e uX under ∞ t. which is tu. This yields F 0 (u) cosh(tuOn the other hand, arcosh F 0 (u) is always bounded below by the translation length of e uX under ∞ t , which is tu. This yields F 0 (u) cosh(tu). Now the inequalities involving distances follow by elementary calculus methods since cosh d (f n t (x). f n t (y)) = F 0 (u) when u = cosh d(x, yNow the inequalities involving distances follow by elementary calculus methods since cosh d (f n t (x), f n t (y)) = F 0 (u) when u = cosh d(x, y). Kazhdan's property (T). Bachir Bekka, Pierre De La Harpe, Alain Valette, New Mathematical Monographs. 11271Cambridge University PressBachir Bekka, Pierre de la Harpe, and Alain Valette, Kazhdan's property (T), New Mathematical Monographs, vol. 11, Cambridge University Press, 2008. ↑260, 265, 267, 271 Kazhdan's property (T) for the unitary group of a separable Hilbert space. Bachir Bekka, 509-520. ↑275Geom. Funct. Anal. 133Bachir Bekka, Kazhdan's property (T) for the unitary group of a separable Hilbert space, Geom. Funct. Anal. 13 (2003), no. 3, 509-520. ↑275 Manifolds of nonpositive curvature. Werner Ballmann, Mikhail Gromov, Viktor Schroeder, Progress in Mathematics. 61272BirkhäuserWerner Ballmann, Mikhail Gromov, and Viktor Schroeder, Manifolds of nonpositive curvature, Progress in Mathematics, vol. 61, Birkhäuser, 1985. ↑272 Metric spaces of non-positive curvature. Martin Robert Bridson, André Haefliger, Grundlehren der Mathematischen Wissenschaften. 319278SpringerMartin Robert Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften, vol. 319, Springer, 1999. ↑270, 275, 277, 278 Equivariant embeddings of trees into hyperbolic spaces. Marc Burger, Alessandra Iozzi, Nicolas Monod, Int. Math. Res. Not. 26322272Marc Burger, Alessandra Iozzi, and Nicolas Monod, Equivariant embeddings of trees into hyperbolic spaces, Int. Math. Res. Not. (2005), no. 22, 1331-1369. ↑263, 268, 269, 272 Sur les groupes de transformations birationnelles des surfaces. Serge Cantat, Ann. Math. 1741Serge Cantat, Sur les groupes de transformations birationnelles des surfaces, Ann. Math. 174 (2011), no. 1, 299-340. ↑266 A new model of the hyperbolic plane. David Gans, Am. Math. Mon. 733David Gans, A new model of the hyperbolic plane, Am. Math. Mon. 73 (1966), no. 3, 291-295. ↑267 CAT(κ)-spaces: construction and concentration. Mikhail Gromov, 101-140. ↑265Zap. Nauchn. Semin. (POMI). 280Mikhail Gromov, CAT(κ)-spaces: construction and concentration, Zap. Nauchn. Semin. (POMI) 280 (2001), 101-140. ↑265 Automorphisms of the field of complex numbers. Hyman Kestelman, 1-12. ↑262Proc. Lond. Math. Soc. 53Hyman Kestelman, Automorphisms of the field of complex numbers, Proc. Lond. Math. Soc. 53 (1951), 1-12. ↑262 Sur les transformations ponctuelles, transformant les plans en plans, qu'on peut définir par des procédés analytiques (Existrait d'une lettre adressée à M. C. Segre). Henri Lebesgue, Torino Atti. 42Henri Lebesgue, Sur les transformations ponctuelles, transformant les plans en plans, qu'on peut définir par des procédés analytiques (Existrait d'une lettre adressée à M. C. Segre), Torino Atti 42 (1907), 532-539. ↑262 Yuri Manin, Cubic forms. algebra, geometry, arithmetic. M. Hazewinkel. ↑266AmsterdamNorth-Holland Publishing Co4North-Holland Mathematical LibraryYuri Manin, Cubic forms. algebra, geometry, arithmetic, second ed., North-Holland Mathematical Library, vol. 4, North-Holland Publishing Co., Amsterdam, 1986, trans- lated from the Russian by M. Hazewinkel. ↑266 Nicolas Monod, Notes on functions of hyperbolic type. 269Nicolas Monod, Notes on functions of hyperbolic type, https://arxiv.org/abs/1807. 04157v1, 2018. ↑269 An exotic deformation of the hyperbolic space. Nicolas Monod, Pierre Py, Am. J. Math. 1365278Nicolas Monod and Pierre Py, An exotic deformation of the hyperbolic space, Am. J. Math. 136 (2014), no. 5, 1249-1299. ↑261, 262, 263, 269, 270, 274, 275, 276, 277, 278 Space mappings with bounded distortion. G Yurii, Reshetnyak, Translations of Mathematical Monographs. H. H. McFaden. ↑26173American Mathematical SocietyYurii G. Reshetnyak, Space mappings with bounded distortion, Translations of Mathe- matical Monographs, vol. 73, American Mathematical Society, 1989, translated from the Russian by H. H. McFaden. ↑261 On the algebraic structure of the unitary group. Éric Ricard, Christian Rosendal, Collect. Math. 582277Éric Ricard and Christian Rosendal, On the algebraic structure of the unitary group, Collect. Math. 58 (2007), no. 2, 181-192. ↑273, 275, 277 L René, Renming Schilling, Zoran Song, Vondraček, Bernstein functions. theory and applications. Walter de Gruyter37260second ed.René L. Schilling, Renming Song, and Zoran Vondraček, Bernstein functions. theory and applications, second ed., De Gruyter Studies in Mathematics, vol. 37, Walter de Gruyter, 2012. ↑260 Automatic continuity for the unitary group. Todor Tsankov, Proc. Am. Math. Soc. 14110277Todor Tsankov, Automatic continuity for the unitary group, Proc. Am. Math. Soc. 141 (2013), no. 10, 3673-3680. ↑262, 276, 277	[]
[ "Abstract Syntax Networks for Code Generation and Semantic Parsing", "Abstract Syntax Networks for Code Generation and Semantic Parsing" ]	[ "Maxim Rabinovich [email protected] \nComputer Science Division\nUniversity of California\nBerkeley\n", "Mitchell Stern [email protected] \nComputer Science Division\nUniversity of California\nBerkeley\n", "Dan Klein [email protected] \nComputer Science Division\nUniversity of California\nBerkeley\n" ]	[ "Computer Science Division\nUniversity of California\nBerkeley", "Computer Science Division\nUniversity of California\nBerkeley", "Computer Science Division\nUniversity of California\nBerkeley" ]	[ "Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics" ]	Tasks like code generation and semantic parsing require mapping unstructured (or partially structured) inputs to well-formed, executable outputs. We introduce abstract syntax networks, a modeling framework for these problems. The outputs are represented as abstract syntax trees (ASTs) and constructed by a decoder with a dynamically-determined modular structure paralleling the structure of the output tree. On the benchmark HEARTHSTONE dataset for code generation, our model obtains 79.2 BLEU and 22.7% exact match accuracy, compared to previous state-ofthe-art values of 67.1 and 6.1%. Furthermore, we perform competitively on the ATIS, JOBS, and GEO semantic parsing datasets with no task-specific engineering. * Equal contribution.	10.18653/v1/p17-1105	[ "https://www.aclweb.org/anthology/P17-1105.pdf" ]	13,529,592	1704.07535	2c1e874c3b67510a3215e535f5646b362de5bc89	Abstract Syntax Networks for Code Generation and Semantic Parsing July 30 -August 4, 2017. July 30 -August 4, 2017 Maxim Rabinovich [email protected] Computer Science Division University of California Berkeley Mitchell Stern [email protected] Computer Science Division University of California Berkeley Dan Klein [email protected] Computer Science Division University of California Berkeley Abstract Syntax Networks for Code Generation and Semantic Parsing Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics the 55th Annual Meeting of the Association for Computational LinguisticsVancouver, Canada; Vancouver, CanadaJuly 30 -August 4, 2017. July 30 -August 4, 201710.18653/v1/P17-1105 Tasks like code generation and semantic parsing require mapping unstructured (or partially structured) inputs to well-formed, executable outputs. We introduce abstract syntax networks, a modeling framework for these problems. The outputs are represented as abstract syntax trees (ASTs) and constructed by a decoder with a dynamically-determined modular structure paralleling the structure of the output tree. On the benchmark HEARTHSTONE dataset for code generation, our model obtains 79.2 BLEU and 22.7% exact match accuracy, compared to previous state-ofthe-art values of 67.1 and 6.1%. Furthermore, we perform competitively on the ATIS, JOBS, and GEO semantic parsing datasets with no task-specific engineering. * Equal contribution. Introduction Tasks like semantic parsing and code generation are challenging in part because they are structured (the output must be well-formed) but not synchronous (the output structure diverges from the input structure). Sequence-to-sequence models have proven effective for both tasks (Dong and Lapata, 2016;Ling et al., 2016), using encoder-decoder frameworks to exploit the sequential structure on both the input and output side. Yet these approaches do not account for much richer structural constraints on outputs-including well-formedness, well-typedness, and executability. The wellformedness case is of particular interest, since it can readily be enforced by representing outputs as abstract syntax trees (ASTs) (Aho et al., 2006), an approach that can be seen as a much lighter weight show me the fare from ci0 to ci1 lambda $0 e ( exists $1 ( and ( from $1 ci0 ) ( to $1 ci1 ) ( = ( fare $1 ) $0 ) ) ) Figure 2: Example of a query and its logical form from the ATIS dataset. The ci0 and ci1 tokens are entity abstractions introduced in preprocessing (Dong and Lapata, 2016). version of CCG-based semantic parsing (Zettlemoyer and Collins, 2005). In this work, we introduce abstract syntax networks (ASNs), an extension of the standard encoder-decoder framework utilizing a modular decoder whose submodels are composed to natively generate ASTs in a top-down manner. The decoding process for any given input follows a dy-namically chosen mutual recursion between the modules, where the structure of the tree being produced mirrors the call graph of the recursion. We implement this process using a decoder model built of many submodels, each associated with a specific construct in the AST grammar and invoked when that construct is needed in the output tree. As is common with neural approaches to structured prediction (Chen and Manning, 2014;Vinyals et al., 2015), our decoder proceeds greedily and accesses not only a fixed encoding but also an attention-based representation of the input (Bahdanau et al., 2014). Our model significantly outperforms previous architectures for code generation and obtains competitive or state-of-the-art results on a suite of semantic parsing benchmarks. On the HEARTH-STONE dataset for code generation, we achieve a token BLEU score of 79.2 and an exact match accuracy of 22.7%, greatly improving over the previous best results of 67.1 BLEU and 6.1% exact match (Ling et al., 2016). The flexibility of ASNs makes them readily applicable to other tasks with minimal adaptation. We illustrate this point with a suite of semantic parsing experiments. On the JOBS dataset, we improve on previous state-of-the-art, achieving 92.9% exact match accuracy as compared to the previous record of 90.7%. Likewise, we perform competitively on the ATIS and GEO datasets, matching or exceeding the exact match reported by Dong and Lapata (2016), though not quite reaching the records held by the best previous semantic parsing approaches (Wang et al., 2014). Related work Encoder-decoder architectures, with and without attention, have been applied successfully both to sequence prediction tasks like machine translation and to tree prediction tasks like constituency parsing (Cross and Huang, 2016;Dyer et al., 2016;Vinyals et al., 2015). In the latter case, work has focused on making the task look like sequence-tosequence prediction, either by flattening the output tree (Vinyals et al., 2015) or by representing it as a sequence of construction decisions (Cross and Huang, 2016;Dyer et al., 2016). Our work differs from both in its use of a recursive top-down generation procedure. Dong and Lapata (2016) introduced a sequenceto-sequence approach to semantic parsing, includ-ing a limited form of top-down recursion, but without the modularity or tight coupling between output grammar and model characteristic of our approach. Neural (and probabilistic) modeling of code, including for prediction problems, has a longer history. Allamanis et al. (2015) and Maddison and Tarlow (2014) proposed modeling code with a neural language model, generating concrete syntax trees in left-first depth-first order, focusing on metrics like perplexity and applications like code snippet retrieval. More recently, Shin et al. (2017) attacked the same problem using a grammar-based variational autoencoder with top-down generation similar to ours instead. Meanwhile, a separate line of work has focused on the problem of program induction from input-output pairs (Balog et al., 2016;Liang et al., 2010;Menon et al., 2013). The prediction framework most similar in spirit to ours is the doubly-recurrent decoder network introduced by Alvarez-Melis and Jaakkola (2017), which propagates information down the tree using a vertical LSTM and between siblings using a horizontal LSTM. Our model differs from theirs in using a separate module for each grammar construct and learning separate vertical updates for siblings when the AST labels require all siblings to be jointly present; we do, however, use a horizontal LSTM for nodes with variable numbers of children. The differences between our models reflect not only design decisions, but also differences in data-since ASTs have labeled nodes and labeled edges, they come with additional structure that our model exploits. Apart from ours, the best results on the codegeneration task associated with the HEARTH-STONE dataset are based on a sequence-tosequence approach to the problem (Ling et al., 2016). Abstract syntax networks greatly improve on those results. Previously, Andreas et al. (2016) introduced neural module networks (NMNs) for visual question answering, with modules corresponding to linguistic substructures within the input query. The primary purpose of the modules in NMNs is to compute deep features of images in the style of convolutional neural networks (CNN). These features are then fed into a final decision layer. In contrast to the modules we describe here, NMN modules do not make decisions about what to generate or which modules to call next, nor do they 2 Data Representation Abstract Syntax Trees Our model makes use of the Abstract Syntax Description Language (ASDL) framework (Wang et al., 1997), which represents code fragments as trees with typed nodes. Primitive types correspond to atomic values, like integers or identifiers. Accordingly, primitive nodes are annotated with a primitive type and a value of that type-for instance, in Figure 3a, the identifier node storing "create minion" represents a function of the same name. Composite types correspond to language constructs, like expressions or statements. Each type has a collection of constructors, each of which specifies the particular language construct a node of that type represents. Figure 4 shows constructors for the statement (stmt) and expression (expr) types. The associated language constructs include function and class definitions, return statements, binary operations, and function calls. Composite types enter syntax trees via composite nodes, annotated with a composite type and a choice of constructor specifying how the node expands. The root node in Figure 3a, for example, is a composite node of type stmt that represents a class definition and therefore uses the ClassDef constructor. In Figure 3b, on the other hand, the root uses the Call constructor because it represents a function call. Children are specified by named and typed fields of the constructor, which have cardinalities of singular, optional, or sequential. By default, fields have singular cardinality, meaning they correspond to exactly one child. For instance, the ClassDef constructor has a singular name field of type identifier. Fields of optional cardinality are associ-ated with zero or one children, while fields of sequential cardinality are associated with zero or more children-these are designated using ? and * suffixes in the grammar, respectively. Fields of sequential cardinality are often used to represent statement blocks, as in the body field of the ClassDef and FunctionDef constructors. The grammars needed for semantic parsing can easily be given ASDL specifications as well, using primitive types to represent variables, predicates, and atoms and composite types for standard logical building blocks like lambdas and counting (among others). Figure 2 shows what the resulting λ-calculus trees look like. The ASDL grammars for both λ-calculus and Prolog-style logical forms are quite compact, as Figures 9 and 10 in the appendix show. Input Representation We represent inputs as collections of named components, each of which consists of a sequence of tokens. In the case of semantic parsing, inputs have a single component containing the query sentence. In the case of HEARTHSTONE, the card's name and description are represented as sequences of characters and tokens, respectively, while categorical attributes are represented as single-token sequences. For HEARTHSTONE, we restrict our input and output vocabularies to values that occur more than once in the training set. Model Architecture Our model uses an encoder-decoder architecture with hierarchical attention. The key idea behind our approach is to structure the decoder as a collection of mutually recursive modules. The modules correspond to elements of the AST grammar and are composed together in a manner that mirrors the structure of the tree being generated. A vertical LSTM state is passed from module to module to propagate information during the decoding process. The encoder uses bidirectional LSTMs to embed each component and a feedforward network to combine them. Component-and token-level attention is applied over the input at each step of the decoding process. We train our model using negative log likelihood as the loss function. The likelihood encompasses terms for all generation decisions made by the decoder. Encoder Each component c of the input is encoded using a component-specific bidirectional LSTM. This results in forward and backward token encodings ( − → h c , ← − h c ) that are later used by the attention mechanism. To obtain an encoding of the input as a whole for decoder initialization, we concatenate the final forward and backward encodings of each component into a single vector and apply a linear projection. Decoder Modules The decoder decomposes into several classes of modules, one per construct in the grammar, which we discuss in turn. Throughout, we let v denote the current vertical LSTM state, and use f to represent a generic feedforward neural network. LSTM updates with hidden state h and input x are notated as LSTM(h, x). Composite type modules Each composite type T has a corresponding module whose role is to select among the constructors C for that type. As Figure 5a exhibits, a composite type module receives a vertical LSTM state v as input and applies a feedforward network f T and a softmax output layer to choose a constructor: p (C \| T, v) = softmax (f T (v)) C . Control is then passed to the module associated with constructor C. Constructor modules Each constructor C has a corresponding module whose role is to compute an intermediate vertical LSTM state v u,F for each of its fields F whenever C is chosen at a composite node u. For each field F of the constructor, an embedding e F is concatenated with an attention-based context vector c and fed through a feedforward neural network f C to obtain a context-dependent field embedding:ẽ Constructor field modules Each field F of a constructor has a corresponding module whose role is to determine the number of children associated with that field and to propagate an updated vertical LSTM state to them. In the case of fields with singular cardinality, the decision and update are both vacuous, as exactly one child is always generated. Hence these modules forward the field vertical LSTM state v u,F unchanged to the child w corresponding to F: F = f C (e F , c) .v w = v u,F .(1) Fields with optional cardinality can have either zero or one children; this choice is made using a feedforward network applied to the vertical LSTM state: p(z F = 1 \| v u,F ) = sigmoid (f gen F (v u,F )) . (2) If a child is to be generated, then as in (1), the state is propagated forward without modification. In the case of sequential fields, a horizontal LSTM is employed for both child decisions and state updates. We refer to Figure 5c for an illustration of the recurrent process. After being initialized with a transformation of the vertical state, s F,0 = W F v u,F , the horizontal LSTM iteratively decides whether to generate another child by applying a modified form of (2): p (z F,i = 1 \| s F,i−1 , v u,F ) = sigmoid (f gen F (s F,i−1 , v u,F )) . If z F,i = 0, generation stops and the process terminates, as represented by the solid black circle in Figure 5c. Otherwise, the process continues as represented by the white circle in Figure 5c. In that case, the horizontal state s u,i−1 is combined with the vertical state v u,F and an attention-based context vector c F,i using a feedforward network f update F to obtain a joint context-dependent encoding of the field F and the position i: e F,i = f update F (v u,F , s u,i−1 , c F,i ). The result is used to perform a vertical LSTM update for the corresponding child w i : v w i = LSTM v (v u,F ,ẽ F,i ). Finally, the horizontal LSTM state is updated using the same field-position encoding, and the process continues: s u,i = LSTM h (s u,i−1 ,ẽ F,i ). Primitive type modules Each primitive type T has a corresponding module whose role is to select among the values y within the domain of that type. Figure 5d presents an example of the simplest form of this selection process, where the value y is obtained from a closed list via a softmax layer applied to an incoming vertical LSTM state: p (y \| T, v) = softmax (f T (v)) y . Some string-valued types are open class, however. To deal with these, we allow generation both from a closed list of previously seen values, as in Figure 5d, and synthesis of new values. Synthesis is delegated to a character-level LSTM language model (Bengio et al., 2003), and part of the role of the primitive module for open class types is to choose whether to synthesize a new value or not. During training, we allow the model to use the character LSTM only for unknown strings but include the log probability of that binary decision in the loss in order to ensure the model learns when to generate from the character LSTM. Decoding Process The decoding process proceeds through mutual recursion between the constituting modules, where the syntactic structure of the output tree mirrors the call graph of the generation procedure. At each step, the active decoder module either makes a generation decision, propagates state down the tree, or both. To construct a composite node of a given type, the decoder calls the appropriate composite type module to obtain a constructor and its associated module. That module is then invoked to obtain updated vertical LSTM states for each of the constructor's fields, and the corresponding constructor field modules are invoked to advance the process to those children. This process continues downward, stopping at each primitive node, where a value is generated but no further recursion is carried out. Attention Following standard practice for sequence-tosequence models, we compute a raw bilinear attention score q raw t for each token t in the input using the decoder's current state x and the token's encoding e t : q raw t = e t Wx. The current state x can be either the vertical LSTM state in isolation or a concatentation of the vertical LSTM state and either a horizontal LSTM state or a character LSTM state (for string generation). Each submodule that computes attention does so using a separate matrix W. A separate attention score q comp c is computed for each component of the input, independent of its content: q comp c = w c x. The final token-level attention scores are the sums of the raw token-level scores and the corresponding component-level scores: q t = q raw t + q comp c(t) , where c(t) denotes the component in which token t occurs. The attention weight vector a is then computed using a softmax: a = softmax (q) . Given the weights, the attention-based context is given by: c = t a t e t . Certain decision points that require attention have been highlighted in the description above; however, in our final implementation we made attention available to the decoder at all decision points. Supervised Attention In the datasets we consider, partial or total copying of input tokens into primitive nodes is quite common. Rather than providing an explicit copying mechanism (Ling et al., 2016), we instead generate alignments where possible to define a set of tokens on which the attention at a given primitive node should be concentrated. 2 If no matches are found, the corresponding set of tokens is taken to be the whole input. The attention supervision enters the loss through a term that encourages the final attention weights to be concentrated on the specified subset. Formally, if the matched subset of componenttoken pairs is S, the loss term associated with the supervision would be log t exp (a t ) − log t∈S exp (a t ),(3) where a t is the attention weight associated with token t, and the sum in the first term ranges over all tokens in the input. The loss in (3) can be interpreted as the negative log probability of attending to some token in S. 4 Experimental evaluation 4.1 Semantic parsing Data We use three semantic parsing datasets: JOBS, GEO, and ATIS. All three consist of natural language queries paired with a logical representation of their denotations. JOBS consists of 640 such pairs, with Prolog-style logical representations, while GEO and ATIS consist of 880 and 5,410 such pairs, respectively, with λ-calculus logical forms. We use the same training-test split as Zettlemoyer and Collins (2005) for JOBS and GEO, and the standard training-development-test split for ATIS. We use the preprocessed versions of these datasets made available by Dong and Lapata (2016), where text in the input has been lowercased and stemmed using NLTK (Bird et al., 2009), and matching entities appearing in the same input-output pair have been replaced by numbered abstract identifiers of the same type. Evaluation We compute accuracies using tree exact match for evaluation. Following the publicly released code of Dong and Lapata (2016), we canonicalize the order of the children within conjunction and disjunction nodes to avoid spurious errors, but otherwise perform no transformations before comparison. Code generation Data We use the HEARTHSTONE dataset introduced by Ling et al. (2016), which consists of 665 cards paired with their implementations in the open-source Hearthbreaker engine. 3 Our trainingdevelopment-test split is identical to that of Ling et al. (2016), with split sizes of 533, 66, and 66, respectively. Cards contain two kinds of components: textual components that contain the card's name and a description of its function, and categorical ones that contain numerical attributes (attack, health, cost, and durability) or enumerated attributes (rarity, type, race, and class). The name of the card is represented as a sequence of characters, while its description consists of a sequence of tokens split on whitespace and punctuation. All categorical components are represented as single-token sequences. Evaluation For direct comparison to the results of Ling et al. (2016), we evaluate our predicted code based on exact match and token-level BLEU relative to the reference implementations from the library. We additionally compute node-based precision, recall, and F1 scores for our predicted trees compared to the reference code ASTs. Formally, these scores are obtained by defining the intersection of the predicted and gold trees as their largest common tree prefix. Settings For each experiment, all feedforward and LSTM hidden dimensions are set to the same value. We select the dimension from {30, 40, 50, 60, 70} for the smaller JOBS and GEO datasets, or from {50, 75, 100, 125, 150} for the larger ATIS and HEARTHSTONE datasets. The dimensionality used for the inputs to the encoder is set to 100 in all cases. We apply dropout to the non-recurrent connections of the vertical and horizontal LSTMs, selecting the noise ratio from {0.2, 0.3, 0.4, 0.5}. All parameters are randomly initialized using Glorot initialization (Glorot and Bengio, 2010). We perform 200 passes over the data for the JOBS and GEO experiments, or 400 passes for the ATIS and HEARTHSTONE experiments. Early stopping based on exact match is used for the semantic parsing experiments, where performance is evaluated on the training set for JOBS and GEO or on the development set for ATIS. Parameters for the HEARTHSTONE experiments are selected based on development BLEU scores. In order to promote generalization, ties are broken in all cases with a preference toward higher dropout ratios and lower dimensionalities, in that order. Our system is implemented in Python using the DyNet neural network library (Neubig et al., 2017). We use the Adam optimizer (Kingma and Ba, 2014) with its default settings for optimization, with a batch size of 20 for the semantic parsing experiments, or a batch size of 10 for the HEARTHSTONE experiments. Results Our results on the semantic parsing datasets are presented in Table 1 Ling et al. (2016). Our nearest neighbor baseline NEAREST follows that of Ling et al. (2016), though it performs somewhat better; its nonzero exact match number stems from spurious repetition in the data. a new state-of-the-art accuracy of 91.4% on the JOBS dataset, and this number improves to 92.9% when supervised attention is added. On the ATIS and GEO datasets, we respectively exceed and match the results of Dong and Lapata (2016). However, these fall short of the previous best results of 91.3% and 90.4%, respectively, obtained by Wang et al. (2014). This difference may be partially attributable to the use of typing information or rich lexicons in most previous semantic parsing approaches (Zettlemoyer and Collins, 2007;Kwiatkowski et al., 2013;Wang et al., 2014;Zhao and Huang, 2015). On the HEARTHSTONE dataset, we improve significantly over the initial results of Ling et al. (2016) across all evaluation metrics, as shown in Table 2. On the more stringent exact match metric, we improve from 6.1% to 18.2%, and on tokenlevel BLEU, we improve from 67.1 to 77.6. When supervised attention is added, we obtain an additional increase of several points on each scale, achieving peak results of 22.7% accuracy and 79.2 BLEU. (1)), SelfSelector())) ]) Figure 7: For many cards with moderately complex descriptions, the implementation follows a functional style that seems to suit our modeling strategy, usually leading to correct predictions. Error Analysis and Discussion As the examples in Figures 6-8 show, classes in the HEARTHSTONE dataset share a great deal of common structure. As a result, in the simplest cases, such as in Figure 6, generating the code is simply a matter of matching the overall structure and plugging in the correct values in the initializer and a few other places. In such cases, our system generally predicts the correct code, with the Figure 8: Cards with nontrivial logic expressed in an imperative style are the most challenging for our system. In this example, our prediction comes close to the gold code, but misses an important statement in addition to making a few other minor errors. (Left) gold code; (right) predicted code. exception of instances in which strings are incorrectly transduced. Introducing a dedicated copying mechanism like the one used by Ling et al. (2016) or more specialized machinery for string transduction may alleviate this latter problem. The next simplest category of card-code pairs consists of those in which the card's logic is mostly implemented via nested function calls. Figure 7 illustrates a typical case, in which the card's effect is triggered by a game event (a spell being cast) and both the trigger and the effect are described by arguments to an Effect constructor. Our system usually also performs well on instances like these, apart from idiosyncratic errors that can take the form of under-or overgeneration or simply substitution of incorrect predicates. Cards whose code includes complex logic expressed in an imperative style, as in Figure 8, pose the greatest challenge for our system. Factors like variable naming, nontrivial control flow, and interleaving of code predictable from the description with code required due to the conventions of the library combine to make the code for these cards difficult to generate. In some instances (as in the figure), our system is nonetheless able to synthesize a close approximation. However, in the most complex cases, the predictions deviate significantly from the correct implementation. In addition to the specific errors our system makes, some larger issues remain unresolved. Existing evaluation metrics only approximate the actual metric of interest: functional equivalence. Modifications of BLEU, tree F1, and exact match that canonicalize the code-for example, by anonymizing all variables-may prove more meaningful. Direct evaluation of functional equivalence is of course impossible in general (Sipser, 2006), and practically challenging even for the HEARTHSTONE dataset because it requires integrating with the game engine. Existing work also does not attempt to enforce semantic coherence in the output. Long-distance semantic dependencies, between occurrences of a single variable for example, in particular are not modeled. Nor is well-typedness or executability. Overcoming these evaluation and modeling issues remains an important open problem. Conclusion ASNs provide a modular encoder-decoder architecture that can readily accommodate a variety of tasks with structured output spaces. They are particularly applicable in the presence of recursive decompositions, where they can provide a simple decoding process that closely parallels the inherent structure of the outputs. Our results demonstrate their promise for tree prediction tasks, and we believe their application to more general output structures is an interesting avenue for future work. Figure 1 : 1Example code for the "Dire Wolf Alpha" Hearthstone card. The root portion of the AST. from the same AST, corresponding to the code snippet Aura(ChangeAttack(1),MinionSelector(Adjacent())). Figure 3 : 3Fragments from the abstract syntax tree corresponding to the example code inFigure 1. Blue boxes represent composite nodes, which expand via a constructor with a prescribed set of named children. Orange boxes represent primitive nodes, with their corresponding values written underneath. Solid black squares correspond to constructor fields with sequential cardinality, such as the body of a class definition(Figure 3a)or the arguments of a function call(Figure 3b). maintain recurrent state. Figure 4 : 4A simplified fragment of the Python ASDL grammar. 1 An intermediate vertical state for the field F at composite node u is then computed as v u,F = LSTM v (v u ,ẽ F ) . Figure 5b 5billustrates the process, starting with a single vertical LSTM state and ending with one updated state per field. Figure 5 : 5The module classes constituting our decoder. For brevity, we omit the cardinality modules for singular and optional cardinalities. primitive types: identifier, object, ...stmt = FunctionDef( identifier name, arg * args, stmt * body) \| ClassDef( identifier name, expr * bases, stmt * body) \| Return(expr? value) \| ... expr = BinOp(expr left, operator op, expr right) \| Call(expr func, expr * args) \| Str(string s) \| Name(identifier id, expr_context ctx) \| ... ... Table 2 : 2Results for the HEARTHSTONE task. SU- PATT refers to the system with supervised atten- tion mentioned in Section 3.4. LPN refers to the system of Figure 6: Cards with minimal descriptions exhibit a uniform structure that our system almost always predicts correctly, as in this instance.class IronbarkProtector(MinionCard): def __init__(self): super().__init__( 'Ironbark Protector', 8, CHARACTER_CLASS.DRUID, CARD_RARITY.COMMON) def create_minion(self, player): return Minion( 8, 8, taunt=True) class ManaWyrm(MinionCard): def __init__(self): super().__init__( 'Mana Wyrm', 1, CHARACTER_CLASS.MAGE, CARD_RARITY.COMMON) def create_minion(self, player): return Minion( 1, 3, effects=[ Effect( SpellCast(), ActionTag( Give(ChangeAttack The full grammar can be found online on the documentation page for the Python ast module: https://docs.python.org/3/library/ast. html#abstract-grammar Alignments are generated using an exact string match heuristic that also included some limited normalization, primarily splitting of special characters, undoing camel case, and lemmatization for the semantic parsing datasets. 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[ "Towards a non-perturbative calculation of Weak Hamiltonian Wilson coefficients", "Towards a non-perturbative calculation of Weak Hamiltonian Wilson coefficients" ]	[ "Mattia Bruno \nPhysics Department\nBrookhaven National Laboratory\nUpton11973NYUSA\n", "Christoph Lehner \nPhysics Department\nBrookhaven National Laboratory\nUpton11973NYUSA\n", "Amarjit Soni \nPhysics Department\nBrookhaven National Laboratory\nUpton11973NYUSA\n", "( Rbc \nPhysics Department\nBrookhaven National Laboratory\nUpton11973NYUSA\n", "Ukqcd Collaborations \nPhysics Department\nBrookhaven National Laboratory\nUpton11973NYUSA\n" ]	[ "Physics Department\nBrookhaven National Laboratory\nUpton11973NYUSA", "Physics Department\nBrookhaven National Laboratory\nUpton11973NYUSA", "Physics Department\nBrookhaven National Laboratory\nUpton11973NYUSA", "Physics Department\nBrookhaven National Laboratory\nUpton11973NYUSA", "Physics Department\nBrookhaven National Laboratory\nUpton11973NYUSA" ]	[]	We propose a method to compute the Wilson coefficients of the weak effective Hamiltonian to all orders in the strong coupling constant using Lattice QCD simulations. We perform our calculations adopting an unphysically light weak boson mass of around 2 GeV. We demonstrate that systematic errors for the Wilson coefficients C1 and C2, related to the current-current four-quark operators, can be controlled and present a path towards precise determinations in subsequent works.	10.1103/physrevd.97.074509	[ "https://arxiv.org/pdf/1711.05768v1.pdf" ]	119,401,493	1711.05768	c59722d015d4f81cc30e965945b1376481f10b05	Towards a non-perturbative calculation of Weak Hamiltonian Wilson coefficients Mattia Bruno Physics Department Brookhaven National Laboratory Upton11973NYUSA Christoph Lehner Physics Department Brookhaven National Laboratory Upton11973NYUSA Amarjit Soni Physics Department Brookhaven National Laboratory Upton11973NYUSA ( Rbc Physics Department Brookhaven National Laboratory Upton11973NYUSA Ukqcd Collaborations Physics Department Brookhaven National Laboratory Upton11973NYUSA Towards a non-perturbative calculation of Weak Hamiltonian Wilson coefficients We propose a method to compute the Wilson coefficients of the weak effective Hamiltonian to all orders in the strong coupling constant using Lattice QCD simulations. We perform our calculations adopting an unphysically light weak boson mass of around 2 GeV. We demonstrate that systematic errors for the Wilson coefficients C1 and C2, related to the current-current four-quark operators, can be controlled and present a path towards precise determinations in subsequent works. I. INTRODUCTION Weak decays of hadrons, and in particular of mesons, play an important role in our understanding of the fundamental forces and having precise theoretical predictions to compare against the experimental results can either strenghten the solidity of the Standard Model or lead to discoveries of new physics [1]. The large scale separation between the mesons, strongly bounded particles with masses of order Λ QCD , and the weak mediators with masses around 100 GeV, is used to simplify theoretical predictions of these processes in the framework of effective field theories. By integrating out the heavier degrees of freedom, specifically the weak bosons and heavy quarks, from the Standard Model, it is possible to define a new effective Hamiltonian with new operators and new coupling constants usually called the Wilson coefficients. The coefficients capture the effect of the weak bosons and heavy quarks that are absent from the effective field theory (EFT), making them well-suited for a perturbative calculation. Instead, matrix elements of the operators involving mesonic external states require a nonperturbative calculation. In the last decade, thanks to algorithmic and computational advances, the Lattice QCD community has been able to cover a wide range of processes involving two mesons (see e.g. Ref. [2]) and also to complete the first two-body final state decays of K → ππ [3][4][5][6]. On the other hand perturbative calculations of the Wilson coefficients have been successfully carried out up to NLO (for a comprehensive review see Ref. [7]) and in some cases up to NNLO [8]. In this work we explore the possibility of a non-perturbative method to compute the Wilson coefficients to address the perturbative uncertainty of the analytic calculations. The perturbative truncation error is traded with the statistical and systematic errors usually present in lattice calculations and the purpose of this paper is to define a methodology to obtain a precise determination of the Wilson coefficients where all uncertainties have been addressed. The problem of defining the weak effective Hamiltonian non-perturbatively has already had some initial considerations by the authors of Ref. [9]: by separating two hadronic weak currents and studying their dependence as a function of this distance it is pos-sible to define properly normalized operators, where the effect of the Wilson coefficients has been fitted away by using their perturbative expansion. Our approach differs from the one considered in Ref. [9] as we plan to directly determine the Wilson coefficients using gauge fixed external quark states, rather than mesons, in momentum space and not in coordinate space. The rest of the manuscript is organized as follows: in the next section we give an overview of the main features of the EFT and we describe our strategy to measure the Wilson coefficients; in section III we show our results and address the various uncertanties of the calculation; in section IV we report our determination of the Wilson coefficients in the MS scheme and discuss the comparison against the known perturbative results, and finally we conclude and present further directions for this project. II. COMPUTATIONAL METHOD Before introducing the lattice observables and our main strategy, we review the most important features of the EFT. For concreteness let us restrict to transitions among hadrons. When the weak bosons and the heavy quarks are integrated out, the leading effective field theory that arises is based on operators of dimension 6 with fourquark vertices. This can be easily seen by considering the first term in the expasion of the weak propagator in the limit m W → ∞. The full expansion however contains other terms that can be related to operators with higher dimensions, in fact the most general form of the effective Hamiltonian is H eff = i V CKM i G F √ 2 C i Q i + j ,dj >6 c (dj ) j m dj −4 W Q (dj ) j .(1) In eq. (1) G F represents the dimensionful Fermi constant which is related to the SU (2) L coupling of the Standard Model g 2 , according to G F / √ 2 = g 2 2 /(8m 2 W ). V CKM i denotes a generic product of the usual CKM matrix elements that depends on the flavor structure of the process and consequently of the operators. Note that among the heavy particles that are removed from the theory there arXiv:1711.05768v1 [hep-lat] 15 Nov 2017 is also the top quark, whose presence affects the Wilson coefficients [7,8]. A reliable computation of the Wilson coefficients C i requires a good suppression of the higher dimensional terms, which is obtained by performing the calculation at energy scales sufficiently small compared to m W . Once the operators in the second sum of eq. (1) can be neglected, the Wilson coefficients are obtained by equating the amplitudes computed in the EFT with the the ones computed in the full theory, which in our case differs from the Standard Model as explained below. Previous perturbative calculations [7] were based on amplitudes computed with off-shell external massless quarks and in our calculation we proceed along these lines. As ultraviolet quantities, the Wilson coefficients are expected to be independent from the infrared regulators of the theory and the external states used in the amplitudes; checking this explicitly will be an important task of our work. Similarly we will also test the gauge invariance of our results in the limit where the amplitudes go on-shell, where any depedence on the weak gauge fixing parameter disappears. The amplitudes of the EFT require additional renormalization conditions, due to the new divergences that appear, leading to the mixing of the renormalized operators among themselves. Hence, it is important to consider a complete basis of operators that closes under renormalization, such that Q R i (µ) = [Z(µ)] ij Q bare j .(2) The basis of operators depends on the details of the process considered. For instance, a 2 → 2 transition where the four external quarks have different flavors (e.g. c → sud) requires only two current-current operators. Instead, e.g., a basis of 7 independent operators is needed in the 3-flavor EFT (i.e. H ∆S=1 eff ) to describe the K → ππ process in the zero isospin channel, involving also disconnected contributions that are typically more difficult to compute on the lattice. Therefore, in this exploratory study we will focus only on the simpler current-current operators and consequently only on the Wilson coefficients C 1 and C 2 . A third remark concerns the running of the Wilson coefficients, as some care is required in computing them at low scales. In fact, if we naively match the two theories at scales µ m W we may encounter large logarithms in the form of log(m 2 W /µ 2 ). Therefore, in order to cancel these terms, the matching is performed exactly at µ = m W , thus defining the so-called initial conditions of the Wilson coefficients, and their values at scales lower than m W are obtained by solving their corresponding renormalization group equations. This leads to a resummation of all logarithms of the form α n α log m 2 W µ 2 k for any power k at fixed loop order n. The step-scaling matrix involved in the running of the Wilson coefficients is given by the ratio of Z at two different scales and is a well known and studied problem on the lattice (see e.g. Refs. [10,11]). Instead, in our work we focus on the initial conditions of the Wilson coefficients. More precisely, we compute the matching between a theory with 3 light dynamical quarks in the sea, playing the role of our 3-flavor EFT, with the full theory where we also include the W boson exchange. In this study we do not consider the problem of removing a heavy quark from the theory, which is relevant to match the Standard Model, with a top quark, onto an EFT with 5 dynamical quarks. Finally, our last remark concerns the feasibility of this study. When we introduce the lattice spacing a as a regulator for our theory we are explicitly introducing an ultraviolet cutoff of order a −1 . The Wilson coefficients, in essence, encode the information of momenta around and above m W , making them potentially very sensitive to discretization effects. We address this question by varying both m W and the lattice spacing. However, given the current limitations on the availability of fine lattice spacings, we perform our calculations in an unphysical scenario, where we take m W ≈ 2 GeV, but where we can control the other systematic uncertainties. Nevertheless we discuss, before concluding, how our results may have an impact on the determination of the Wilson coefficients with physical values of the weak boson mass. Once the theory is discretized the path integral can be solved using numerical simulations, limited by the present computational technologies: this translates into being able to simulate only finite quark masses and finite lattice boxes, which constitute the infrared regulators of the theory. Therefore in order to compute the Wilson coefficients the following limits need to be fulfilled m , L −1 p m W a −1 ,(3) where p represents the typical momentum of the external states used in the evaluation of the amplitudes. If we now consider the limit where p goes to zero, in the infinitevolume theory with mass-less quarks, contributions from higher dimensional operators should vanish. However, due to dimensional transmutation, the strong interactions possess a low intrisic scale, Λ QCD , responsible for the creation of condensates that could still contribute to eq. (1) through some operators Q (dj ) j . Nevertheless in Nature, where Λ QCD m W , these contributions should be suppressed and eventually one may neglect them. In our exploratory study we achived only Λ QCD /m W ≈ 0.2 and in the next sections we describe a strategy to quantify remaining non-perturbative contaminations and to take them into account in the systematic uncertanties. In this work we have adopted a momentum subtraction scheme as a prescription to subtract the divergences of the theory. Alternatively, position-space techniques [12,13] could be used in a similar way to directly estimate the Wilson coefficients. In principle the window problem sketched in eq. (3) can be partly circumvented with finite-size techniques [14]: in a finite and small box mass-less quarks can be simulated and the renormalization scale, in our case p, can be identified with the box size itself, thus removing the left-most inequality of eq. (3). In addition to that, very fine lattice spacings are accessible with present resources if the physical volume is small, thus imposing a large hierarchy between m W and Λ QCD . Furthermore, imposing boundary conditions on the gauge and quark fields in the temporal direction, such as Schrödinger Functional or Twisted BC [15][16][17][18][19], has the additional advantage of producing a well-behaved perturbative expansion. Calculations with this formalism (see e.g. Refs. [20,21]) have been carried out successfully for different quantities where all systematic uncertainties have been taken into account. These approaches constitute a valid alternative to further advance this project towards a physical value of the weak boson mass. A. The observables As mentioned earlier we restrict ourselves to the simpler current-current operators Q 1 and Q 2 Q 1 =(s i c j ) V−A (ū j d i ) V−A , Q 2 =(s i c i ) V−A (ū j d j ) V−A ,(4) differing only in the color index routing (i, j) between the two weak currents, with ( ū j d j ) V−A ≡ū j γ µ (1 − γ 5 )d j . The fields in eq. (4) are understood to reside all at the same space-time point x. In the following we introduce a compact notation which slightly differs from the one present in the literature, e.g. Ref. [7]. Let us define the generic operators O i (x, y) as O 1 (x, y) =s i (x)γ L µ c j (x) f µν (x, y)ū j (y)γ L ν d i (y) , O 2 (x, y) =s i (x)γ L µ c i (x) f µν (x, y)ū j (y)γ L ν d j (y)(5) with f a generic real-valued function, γ L µ ≡ γ µ (1 − γ 5 ) and Einstein summation rule on repeated indices. In this language the choice f µν (x, y) = δ µν δ(x − y) reproduces exactly the operators Q 1 and Q 2 in eq. (4). Then let us introduce the following four-point function with a single insertion of these operators [Γ(O i )] αβγδ abcd (y 1 , y 2 , y 3 , y 4 ) = s α a (y 2 )u γ c (y 4 ) O i (x, z)c β b (y 1 )d δ d (y 3 )(6) with greek and roman symbols denoting spin and color indices respectively. By inserting the operators in eq. (5) and computing the Wick contractions we end up with a Green's function that we later transform to momentum space, thus obtaining the diagram reported in Figure 1. In our notation all the four momenta are in-coming. To simplify the notation we will omit the flavor indices in the rest of the paper since we will use degenerate quarks; following the subscript 1, 2, 3, 4 of the y coordinate or of the momenta will allow the reader to trace the flavors back. To compute the quark propagators we invert the Dirac operator D(x, y) on plane waves with momentum p (we later refer to them as momentum sources) G(x, p) = y D −1 (x, y)e ipy .(7) By saturating the x-dependence with the appropriate phase factor, we obtain the propagator in momentum space and its expectation valuẽ G(x, p) = e −ipx G(x, p) ,(8)S(p) = 1 V x e −ipx G(x, p) .(9) The periodicity of the lattice in the temporal and spatial directions imposes a constraint on the accessible momenta p µ = 2πn µ /L µ , with n µ a positive integer. Moreover the breaking of the group of continuous rotations to the hypercubic ones leads to additional discretization effects that spoil the smoothness of the propagators as a function of p. To overcome both issues we employ twisted boundary conditions (BC) in the valence sector [22][23][24], that allow to access a dense set of momenta 1 p µ = 2πn µ L µ + θ µ L µ , θ µ ∈ [−π/2, π/2] .(10) For simplicity we give below the explicit expression of Γ(O 2 ), where we omit the index contractions inside the square brackets and where we use the γ 5 -hermiticity of the Dirac operator [Γ(O 2 )] αβγδ abcd (p 1 , p 2 , p 3 , p 4 ) = µν x,y γ 5G (x, −p 2 ) † γ 5 γ L µG (x, p 1 ) αβ ab f µν (x, y) × γ 5G (y, −p 4 ) † γ 5 γ L νG (y, p 3 ) γδ cd .(11) The amputated amplitudes Λ are easily obtained by inverting the expectation value of quark propagators S(p i ) in eq. (9) [Λ(Γ)] αβγδ abcd =[S(p 2 ) −1 ] αα aa [S(p 1 ) −1 ] ββ bb [S(p 4 ) −1 ] γγ cc [S(p 3 ) −1 ] δδ dd Γ α β γ δ a b c d .(12) At this point we define the amputated amplitudes Λ i ≡ Λ(Γ(O i )) with f µν (x, y) = δ µν δ(x−y). On the full theory side, only a single operator with color diagonal structure is needed to describe this process, namely O 2 in eq. (5) with f µν (x, y) = W µν (x, y), the tree-level propagator of the weak charged bosons in coordinate space (Euclidean metric), which we obtain by fourier transforming 2 W µν (p) = 1 p 2 + m 2 W δ µν − p µ p ν m 2 W .(13) Eqs. (6,11,12) hold in this case as well and we define Λ SM as the amputated amplitude obtained from O 2 with f replaced by the W boson propagator given above. The choice of the unitary gauge simplifies the calculation in the full theory to a single diagram. In the next section we present results also for the Feynman gauge (R ξ gauge with ξ = 1) where the contribution of the Goldstone boson needs to be included. The δ(x − y) function in the insertion of the local operators Q i simplifies the double sum over x and y in eq. (11) to a single one. Choosing plane waves at the source of the propagators allows us to perform such a sum over the entire volume, thus sampling the operators Q i on the full lattice and reducing the statistical fluctuations of the final amplitudes. When f is replaced by the W boson propagator, the double sum needs to be performed. Also in this case the usage of momentum sources gives us the freedom (at the sink of the propagators) to evaluate both sums over x and y explicitly, thus significantly reducing the noise of Λ SM . The only drawback of the combined use of momentum sources and twisted BCs is that a separate inversion is necessary for each momentum configuration that is considered, up to a total of 4 per configuration if we choose the four external legs with different momenta. Therefore, 2 In our calculation we use the lattice momenta apµ = 2 sin(apµ/2) in eq. (13). in an effort to balance the cost of the inversions against the benefit of the volume average, we have explored a second strategy to sample the W boson propagator. From the simple observation that a point-source propagator can be fourier transformed to any continuous momentum, up to small finite-size errors, we have studied the possibility to stochastically sample W µν (x, y), with x and y being the sources of the inversions rather than the sinks as in the case of momentum sources. The sampling technique is essentially borrowed from the calculation of the lightby-light contribution to the anomalous magnetic moment of the muon by the RBC/UKQCD collaboration [25] and it proceeds in two steps: • for a fixed x, the sum over y in eq. (11) is achieved by randomly sampling W µν with a probability distribution falling off rapidly for large separations \|x − y\|; to recover the flat sum over y the appropriate reweighting factor is applied and to further decrease the cost, the hypercubic symmetries are taken into account by randomly sampling only one element per equivalence class, defined by all the points y with the same distance from x, \|x − y\|; this procedure defines a cloud of points stochastically sampled around the center x; • to reduce the noise of the observables a second sum over the center of the cloud, x, is performed, which is again stochastic and with a flat distribution. Even though a continuous set of momenta is now accessible through the usage of the point sources (exceptional and non-exceptional kinematics can also be explored simultaneously), the statistical noise grows quickly: in our test we have used 40 points inside the cloud and this led to a controlled approximation of the sum over y (for the different values of the input W mass considered in this work); however the second sum over x, for which we used 16 different clouds, turns out to be the crucial one in further reducing the noise. Although the presence of a finite correlation length in our system decreases the number of useful points to reduce the noise, we have verified that the stochastic sampling leads to results that are at least 10 times noisier compared to the momentum source method, for approximately the same cost. For this reason we leave all the details of this secondary approach to the Appendix A and we concentrate in the rest of the manuscript on the analysis of the results from momentum sources. Nevertheless this technique has been very useful in an early stage of the work when we explored a vast range of momenta and kinematic configurations, on which we based our decision for the final strategy. B. The Wilson coefficients The appearance of new divergences in the EFT requires the introduction of additional renormalization conditions. In our study we adopt a variety of regularization independent schemes, called RI/MOM and RI/SMOM, which were introduced in Refs. [26][27][28], that are entirely defined by the choice of the external states and projectors: this translates into the momentum combination used in the calculation of the propagators and the projectors that we apply on the amputated Green's functions to obtain definite spin-color states. In this paper we have explored two combinations of the four momenta on the external legs: the nonexceptional case called RI/SMOM where (p i + p j ) = 0 for all pairs i = j, and the exceptional one (RI/MOM) where at least one linear combination of the external momenta vanishes. If the amplitude under study possesses the same symmetries of the light mesons, at momenta comparable with their Compton wave length, the smooth perturbative behavior of the amplitude may be spoiled by non-perturbative contaminations, referred to as Goldstone-pole contaminations [26,27]. Using non-exceptional kinematics (RI/SMOM) suppresses these unwanted effects [27,28] and we extensively discuss them in the next section. To define specific spin-color states we use two projectors P i (i = 1, 2) that we apply to the amputated amplitudes to define the matrix M ij = Tr P j Λ i ,(14) such that M is invertible. To achieve this we fix the color structure of the projectors to one of the operators P 1 = δ bc δ da Γ 1 ⊗ Γ 2 , P 2 = δ ba δ dc Γ 1 ⊗ Γ 2 ,(15) and we explore two options for the Dirac part, with different parities (even and odd) Γ 1 ⊗ Γ 2 = γ µ ⊗ γ µ + γ µ γ 5 ⊗ γ µ γ 5 γ µ γ 5 ⊗ γ µ + γ µ ⊗ γ µ γ 5 .(16) Eq. (16) defines the so-called γ (or γ µ ) projectors. Alternatively, the replacement γ µ → / q/\|q\| and γ µ γ 5 → / qγ 5 /\|q\| defines the / q projectors. Computing the Wilson coefficients using both parities and γ or / q Dirac structures turns out to be a crucial test of the calculation, as explained in the next section. In the rest of the paper we will refer to the two parities as VV + AA and VA + AV, with V and A labeling vector and axial Dirac structures. The renormalization conditions that we impose on the four-quark operators read as follows lim mq→0 Z RI ij (Z RI q ) 2 M lat jk \| µ 2 =p 2 = M RI ik \| µ 2 =p 2 ≡ M tree ik(17) with Z RI q S(p) lat = S(p) RI ,(18) lat indicating bare lattice quantities and M tree defined by replacing the amplitudes in eq. (14) with their tree-level counterparts. In the full theory no additional renormalization conditions are required for the projected amplitude, which we denote with W i = Tr (P i Λ SM ), beyond the usual wave function renormalization introduced above. It is important to note that since we are considering the weak theory at tree level, m W does not renormalize, since self-energy diagrams of the W boson propagator appear at higher orders in the weak coupling constant, g 2 . In the continuum theory, vector and axial current conservations (for mass-less quarks) guarantee that the same is true for g 2 as well. On the lattice however the usage of local vector and axial currents dictates the presence of the finite renormalization factors Z V and Z A , thus leading to g R 2 = Z V g 2 . Note that the V − A current can be renormalized with either Z A or Z V due to the excellent chiral properties of the Domain Wall formulation and to the employment of non-exceptional kinematics, as described later. Eventually we opt for Z V to avoid additional chiral symmetry breaking effects and a larger quark mass dependence compared to Z A . Now we have all the basic ingredients to impose the matching between the two theories G F √ 2 C RI i (µ)M RI ij (µ) = W RI j (µ) = g 2 2 8 Z 2 V (Z RI q ) 2 W lat j . (19) We have already simplified the usual CKM factors that appear in the same form on both sides of the equation. The weak coupling constant g 2 simplifies as well with the Fermi constant G F , leaving only a factor m 2 W on the right hand side. By expanding M RI and W RI with their corresponding bare lattice counterparts we obtain the definition of the Wilson coefficients C lat i ≡ m 2 W W lat j [M lat ] −1 ji , C RI i (µ) = C lat j [Z RI (µ)] −1 ji Z 2 V .(20) Eq. (20) nicely separates two basic ingredients and consequently two different problems: • the bare lattice Wilson coefficients C lat i which can be used to inspect the size of the higher dimensional operators and other systematic errors, and that we compute at small momenta for a variety of external states; • the renormalization factors that we compute at high momenta and use to renormalize C lat i and eventually connect to the MS scheme, by means of the one-loop conversion matrix Z RI→MS given in the Appendix B. Our analysis proceeds along these two steps: we first examine the dependence of the bare Wilson coefficients on the momentum scale, the quark mass and the finite box size. Then we briefly describe the results for [34,35]. The three ensembles have been generated with the same Domain Wall discretization for the sea quarks, the Shamir formulation with fifth dimension length Ls = 16, and Iwasaki gauge action, with bare couplings g 2 0 = 6/β reported in the second column. In the third column we quote the lattice spacings measured in Ref. [36] and in the last ones we provide the lattice dimensions and the bare sea quark masses. the renormalization matrices and discuss discretization errors, and finally present the comparison against the known perturbative results in the MS scheme. III. RESULTS In our study we have used the Domain Wall formalism, which retains excellent chiral properties, even at finite lattice spacing [29,30] (which become exact in the limit of infinite 5th dimension [31,32]), simplifying the renormalization pattern of the theory and suppressing the mixing among operators belonging to different representations of the chiral group [33]. We have measured the amplitudes described in the previous section on three different ensembles, reported in Table I, with 2+1 Domain Wall Fermions (DWF) in the sea. We have used a unitary setup by promoting the same discretization (Shamir DWF) also to the valence sector. Ensembles 16I and 24I have been used to study the dependence of the Wilson coefficients on the volume of the lattice. The additional ensemble 32I allows us to take the continuum limit, with two lattice spacings differing approximately by a factor of 2 in a 2 . In all our calculations we have fixed the (QCD) gauge to the Landau gauge. Our measurements require the calculation of quark propagators for which we have used a mass (in lattice units) of 0.04 on the 24I and 16I ensemble, and 0.03 on the 32I. We utilize 27 independent configurations for both 24I and 32I, separated by 100 and 200 Molecular Dynamics Units. In our analysis we use the jackknife method with bin size of 1, after checking the stability of the error of the Wilson coefficients for larger bin sizes. On our coarser ensembles, the 16I and 24I, we measure our propagators up to momenta of O(0.8 GeV), with data points evenly spaced in p 2 . On each configuration we perform 4 different inversions at fixed p 2 for the four different legs required in non-exceptional kinematics. Moreover on the 16I and 24I we also measure the same amplitudes with momentum injected explicitly along the time direction to test finite volume effects as explained later. On the 24I we repeat those measurements for three values of the quark mass, with am = 0.02 , 0.04 and 0.08. Finally on the 32I we compute the amplitudes for four different momenta up to 0.4 GeV. Our calculations of Λ SM cover a range for am W that goes from 0.6 to 1.334 on the 24I and 0.6 < am W < 1.0 on the 32I: the masses on the 32I have been tuned to match those on the 24I according to the ratio of lattice spacings computed in Ref. [36]. When the momentum of the external quark states becomes comparable to m W , the four-quark EFT is expected to deviate from the full theory, due to the lack of higher dimensional operators that eventually become relevant. Therefore for different choices of the external states in our definition of the amplitudes, we expect to observe different behaviors with respect to the dominant scale p 2 . However, in the limit where p 2 /m 2 W 0 they should all agree and give a consistent and unique value for the Wilson coefficients, up to O(Λ QCD ) contributions. This suggests that we should be able to fit C lat i with a polynomial function of p 2 /m 2 W and we turn to this now. In the next sections we address the problems related to the usage of small momenta such as finite volume and finite quark mass effects and possible remaining non-perturbative effects of order Λ QCD /m W . A. Fitting strategy In order to address the size of higher dimensional operators we study the momentum dependence of the C lat i s for several choices of the external states: we adopt a nonexceptional momentum configuration given by p 1 = (x, −x, 0, 0) p 2 = (0, 0, −x, x) p 3 = (−x, 0, x, 0) p 4 = (0, x, 0, −x)(21) with p 2 i = p 2 , ∀i, and transfer momentum q 2 = 2p 2 . In eq. (21) x denotes a continuous parameter obtained according to eq. (10). The exceptional momentum configuration is easily obtained by re-using the same propagator with one of the four momenta in eq. (21) on all the four legs. The momentum configuration proposed in eq. (21) leads to momentum conservation in the amplitudes and to exact equivalence between γ and / q projectors. In Figure 2 we show the results of C lat 1 and C lat 2 as a function of p 2 /m 2 W from the 24I ensemble. We observe a good convergence of the two sets of data points (exceptional and non-exceptional) for small values of the expansion variable p 2 /m 2 W . Nevertheless higher dimensional operators are quite sizable, given the high accuracy that we are able to achieve, which forces us to explore a particularly small range of momenta and eventually The two sets of data points correspond to the exceptional and non-exceptional kinematics described in eq. (21) measured on the 24I ensemble with am = 0.04. We perform combined polynomial fits where the constant term is constrained by the two data sets, which have been obtained from / q parity odd projectors. consider extrapolations to p 2 /m 2 W → 0. However, data obtained with non-exceptional kinematics show a milder dependence on the external momentum p 2 . To extract the bare values of the Wilson coefficients showed in Figure 2, we adopt combined polynomial fits to the exceptional and non-exceptional points with a common constant term C lat,ex i (x) = C lat i + N k=1 A ik x k , i = 1 , 2 C lat,nonex i (x) = C lat i + N k=1 B ik x k , x = p 2 m 2 W .(22) We base our decision for a combined fit on the fact that independent fits to the two data sets reproduce results for C lat i well compatible within 1 standard deviation (on a given fit range). To estimate the systematic uncertainties associated with these extrapolations we vary the upper limit of the fit range and the degree of the polynomial (N = 1, 2, 3) and we consider the fits with good χ 2 per d.o.f: we take as our final value the result from the highest polynomial, whose larger statistical error covers the discrepancy among the several fits 3 . With the lattice spacings that we have studied in this work, higher dimensional operators seems to be sufficiently suppressed only for momenta around and below Λ QCD . Therefore studying the dependence on the infrared regulators of the lattice Wilson coefficients is important to control their extraction. In general we expect them to be small in C lat i due to their cancellation between W and M , especially if the quark momentum is well below m W . However to what degree this is realized in practical non-perturbative simulations is not clear a priori and we study that below. B. Finite Volume Errors The first infrared regulator that we investigate is the box size. For this purpose we have measured the lattice Wilson coefficients on the 16I and 24I ensembles using exceptional kinematics. In Figure 3 we present the results for C lat 2 : note that the two plots, corresponding to the two ensembles, share the same y-axis to facilitate a visual comparison. Both plots contain four sets of points obtained by combining γ and / q projectors with amplitudes where the momentum is injected along xy and time directions. The parity of the projectors corresponds to VA + AV. For the 24I ensemble the various measurements converge to a unique point at small momenta, thus remarking the universality of the Wilson coefficients in that limit. However for the 16I ensemble a different behavior is observed: in this case only the combination of / q projectors with amplitudes with momentum along the time direction agrees with the correponding measurements in the larger volume, while the other sets of points converge to a different value. We interpret such a behavior as a finite volume error, that is largely suppressed when the momentum is injected along the time direction, which is twice as big as the spatial ones. From our numerical analysis we have noticed that the finite volume effect measured in the 16I ensemble decreases with m −1 W . In agreement with other observations presented later, this may be a consequence of a dimension-7 operator involving a non-perturbative condensate, whose strong dependence on the box volume eventually generates the spread that we observed. A similar behavior is observed also in the other Wilson coefficient, C lat 1 , where the measurements on the 16I do not agree at small momenta, in constrast with the 24I ensemble. C. Non-perturbative effects In our study we have not been able to achieve a large separation between the strong and weak scales, since Λ QCD /m W ≈ 0.2. This means that our calculation, based on RI/MOM techniques, might potentially suffer from non-perturbative contaminations often re- ferred to as Goldstone-pole contaminations [26,27]: in general varying the quark mass in the measurements of the amplitudes is useful to address this issue and has been previously used to non-perturbatively subtract them away [37]. Such contaminations are present mostly in observables that share the same quantum numbers of the lighter mesons, such as the axial or pseudo-scalar bilinear operators (see for example Ref. [38]). If we extend the discussion to fermionic formulations that do not preserve chiral symmetry, such as Wilson fermions, also the mixing with wrong chiralities is allowed and at small momenta can lead to pole behavior as well. For our study we have checked how well chiral symmetry is realized by repeating some measurements of the Wilson coefficients with a larger separation of the Domain Walls along the fifth dimension: we have obtained an excellent agreement for all combinations of projectors between results obtained with the Shamir formulation with L s = 32 and L s = 16 (the latter being the same one used for the sea quarks). Hence, we expect any mixing with wrong chiralities to be predominantly of infrared origin, due to the spontaneous breaking of chiral symmetry via quark condensates, and to vanish at high momenta. Based on CPS symmetry arguments [39][40][41], we also expect the parity odd projectors to show less contaminations compared to the parity even case. To quantify this problem we start from the difference between the Wilson coefficients computed with parity even and parity odd projectors, defined in eq. (16), for several values of the valence quark mass. In Figure 4 we show the results for the lattice Wilson coefficient C lat 2 from the exceptional momentum configuration, where we vary the quark mass and the parity of the (γ) projectors used to compute M lat ij and W lat i . For parity odd projectors we observe an excellent agreement for fixed input W mass at 1.8 GeV on the 24I ensemble. Crosses refer to parity even projectors and dots represent the opposite parity, all in the γ scheme. No difference is observed if γ projectors are replaced with the corresponding / q ones. The VV + AA projectors produce results very sensitive to the quark mass, contaminated by infrared effects proportional to 1/p 2 . A small quark mass dependence is observed in C lat 1 for parity odd projectors as well. among the different quark masses for all points, down to the smallest momentum. On the other hand parity even projectors lead to a strong dependence of C lat 2 on the quark mass, well compatible with the expectation of a Goldstone-pole contamination: in fact, by increasing the quark mass from 0.02 to 0.08 the data is driven towards the points obtained from parity odd projectors, due to the suppression of non-perturbative effects which we expect to be proportional to (p 2 + m 2 ) −1 , with m the mass of the light state coupling to the amplitude. Changing from γ to / q projectors does not affect the general trend showed in Figure 4, as well as turning to the non-exceptional kinematics in eq. (21). For C lat 1 we draw similar conclusions on the behavior of the parity even results, with the only exception that a quark mass depedence is visible also for parity odd projectors: given the different nature and size of this Wilson coefficient it turns out to be large, on the 10% level, but only 0.015 in absolute units (with C lat 1 ≈ 0.15). We account for this effect in our systematic error. To better understand the origin of the discrepancy between the two parities, we have projected our amputated Green's functions on the so-called wrong chiralities. For odd parity no significant mixing has been found. Instead we have measured strong contributions at small momenta from the projectors with form 1 ⊗ 1, γ 5 ⊗ γ 5 and σ µν ⊗ σ µν , which quickly vanish above 1 GeV. This provides another indication on the pollution of infrared effects in the parity even sector, as expected from CPS symmetry. In Appendix C we describe an alternative approach consisting of changing the definition of the projectors to suppress the overlap with the unwanted chiralities. Checking the gauge independence of our calculation is a second approach that we exploit to quantify nonperturbative contributions of O(Λ QCD ). Specifically, we have computed the W lat i amplitudes in Feynman gauge. In this case the weak boson propagator simplifies to the diagonal form W µν = δ µν p 2 + m 2 W(23) but the amplitude, even at tree-level, requires also the contribution of the Goldstone boson arising after the Electro-Weak Symmetry Breaking. 1 p 2 +m 2 W g 2 2 √ 2 2mγ 5 m W φ ± φ − d u FIG. 5. Goldstone boson propagator and coupling to the quark current. In the latter we have already assumed degenerate quark masses and simplified the contribution proportional to the identity. In our case, where all the external legs have the same mass, the vertex between the Goldstone and two quarks simplifies to the form reported in Figure 5. In the limit of mass-less quarks it should vanish identically thus leaving only eq. (23) to contribute to the amplitude. This may be spoiled again by non-perturbative effects, which may be very pronounced due to the γ 5 ⊗ γ 5 structure of this diagram. Therefore studying the two contributions separately could shed some light on the size of these effects and we examine this in Figure 6, which shows the Wilson coefficients C lat i computed only from the Goldstone boson diagram and with parity even projectors. Note that the full Wilson coefficients are obtained simply by adding 4 the result of Figure 6 to the C lat i s computed from the W exchange alone. For parity odd projectors the contribution from the Goldstone boson is negligible, numerically below 10 −6 and compatible with zero within 1 sigma. For parity even projectors, in contrast, we can measure a signal from the Goldstone boson diagram. However the small prefactor (2m/m W ) 2 makes this contribution negligible also in this case, as plotted in Figure 6, although it moves both C lat 1 and C lat 2 towards the corresponding parity odd data. Next we compare the Wilson coefficients measured in Unitary and Feynmann gauge with odd projectors exclusively, for which the Goldstone boson diagram can be neglected. The difference that we observe between the sets of data points comes from the gauge depedent part of the weak propagator in eq. (13) proportional to p µ p ν . The dependence on the weak gauge fixing condition is expected to vanish for on-shell quanti-ties, which we approach by injecting smaller and smaller momenta in the external quark legs of our amplitudes. As seen already above, in this limit non-perturbative effects produce contaminations that in this case let gauge-dependent terms survive: results between Feynman and Unitary gauge do not agree at the 2% level, even for the preferred parity odd projectors, and we may consider this difference as one source of systematic uncertainties δC gauge i (m W , p 2 ) = \|C lat,unit i (m W , p 2 ) − C lat,feyn i (m W , p 2 )\|. However, a second systematic error that we include in our calculation is taken from the difference between the even and odd projectors in Unitary gauge, δC proj In general, all the systematic uncertanties quoted in eq. (24) are estimated from the combined fits defined in eq. (22) where we exclude the points with p 2 < p 2 cut . As we increase p cut in our fit ranges, Λ QCD -type contributions are expected to vanish and this is reflected by our systematic uncertainty, which decreases also for larger values of m W . We demonstrate both behaviors in the two panels of Figure 7: the left one confirms essentially the left-most inequality in eq. (3) and the fact the natural variable to control non-perturbative effects is precisely p cut ; the right plot also confirms the general expectation that uncertainties are reduced with a larger separation between the QCD and weak scales. An additional indication of the non-perturbative origin of these contributions resides in the linear behavior in 1/m W showed in Figure 7: the presence of a condensate (a pure nonperturbative effect) could be captured by a higher dimensional operator in the form of qq p 2 m W O (6-dim) ,(25) whose functional form describes well the data. To summarize, we have collected evidence, from the spread between different projectors and gauge fixing conditions to the volume dependence, that lead to infrared contaminations vanishing with 1/m W . In the right plot of Figure 7 we also demonstrate how a future calculation with m W ≈ 4 GeV would significantly improve the precision well below the percent level. Finally, for the central values of the Wilson coefficients we use a p cut of 0.3 GeV for the 24I ensemble and 0.24 GeV for the 32I ensemble and we perform cubic and quadratic fits respectively according to eq. (22). We use amplitudes computed in unitary gauge projected on the odd sector with / q type projectors. No significant effect, beyond the ones already considered, is obtained from the difference with γ projectors. IV. NON-PERTURBATIVE DETERMINATION OF THE WILSON COEFFICIENTS The remaining systematic uncertanties that we need to address are related to discretization errors for which we need the renormalization factors. In the following we present non-perturbative results for the following values of the W boson mass: 1.4, 1.8, 2.1 and 2.4 GeV. A. Renormalization factors As before twisted boundary conditions turn out to be very useful, as they give us the possibility to tune the momentum to the desidered value. In our computation of the renormalization factors we have adopted the conventional RI/SMOM scheme described in Ref. [42]. Here a momentum 2p leaves the operator, as opposed to eq. (21), and only two inversions per configuration are required with momentum p 1 = (x, −x, 0, 0) and p 2 = (0, x, −x, 0) (with p 3 = p 1 and p 4 = p 2 ). Since the renormalization conditions in eq. (17) are imposed in the chiral limit we repeated the measurements for two values of the quark mass (am = 0.04 and 0.02) and we extrapolate linearly to zero quark masses. We computed the quark propagators such that \|ap\| ≈ am W , to obtain the Wilson coefficients renormalized at a scale coinciding with the mass of the weak boson utilized in their definition: in this way we are reproducing the so-called initial conditions, where large logarithms are avoided as explained in section II. Finally, we deal with the renormalization of the local currents on the lattice. Therefore from the same set of propagators we also compute the following quantities Γ V µ = x γ 5G (x, −p 2 ) † γ 5 γ µG (x, p 1 ) , Γ A µ = x γ 5G (x, −p 2 ) † γ 5 γ µ γ 5G (x, p 1 ) .(26) After the usual amputation with the inverse propagators, GeV FIG. 7. Left: δC2, as a function of pcut, decreases as more points at small momenta are excluded from the fit, suppressing non-perturbative effects. Right: measured difference of C2 from odd and even projectors in the γ scheme as function of the W boson mass. The lines are linear fits forced to pass through the origin with excellent χ 2 . The non-perturbative contaminations which we quantify with δC proj i given in eq. (24) seem to vanish only with one inverse power of mW. In the plot we present the data for the exceptional kinematics, but practically no difference is observed in the non-exceptional case. \|C VV+AA 2 − C VA+AV 2 \| p 2 → 0 m W = 4 GeV p 2 → 0 m W = 4 GeV p 2 → 0 m W = 4 GeV p 2 → 0 m W = 4 GeV p 2 → 0 m W = 4 GeV p 2 → 0 m W = 4 GeV p 2 → 0 m W = 4 we impose the renormalization conditions lim mq→0 Z V Z RI q Tr [Λ V µ γ µ ] = Tr [Λ V,tree µ γ µ ] ,(27)lim mq→0 Z A Z RI q Tr [Λ A µ γ µ γ 5 ] = Tr [Λ A,tree µ γ µ γ 5 ] .(28) In principle the appropriate renormalization condition should involve the Γ V −A µ , obtained by replacing γ µ with γ L µ in the first line of eq. (26). However we have explicitly checked from our measurements that Tr [Λ A µ γ µ ] 10 −3 (the same holds if A → V and γ µ → γ µ γ 5 ) which means that for the choice of projectors in eqs. (27,28) only Z V alone (or Z A ) renormalizes the weak coupling g 2 , thus leading to eq. (19). Replacing γ µ → ( / qq µ )/q 2 and γ µ γ 5 → ( / qγ 5 q µ )/q 2 in eqs. (27,28) defines the / q scheme as before. For simplicity, we do not combine Z q and the four-quark Z matrix from two schemes. Let us briefly examine the properties of the renormalization factor of the Wilson coefficients in eq. (20) Z MS,RI ≡ Z 2 V [Z RI ] −1 [Z RI→MS ] −1 ,(29) which we explicitly expand in terms of lattice observables for the reader's convenience (in the γ scheme) Z 2 V [Z RI ] −1 ik = Tr [Λ V,tree µ γ µ ] Tr [Λ V µ γ µ ] 2 · M lat ij [M tree ] −1 jk . (30) Firstly the quark mass dependence is negligible, as the four elements of the matrixZ slightly differ when the quark mass is changed from 0.04 to 0.02. Nevertheless we take this into account by linearly extrapolating to the chiral limit. Secondly we examine the difference between the renormalization matrixZ computed from two intermediate schemes, defined by the two classes of projectors, γ and / q. As we change the renormalization scale from 1.4 to 2.4 GeV the conversion factor Z RI→MS , which is only perturbative, becomes more precise. Nonetheless the ratioZ MS,γ [Z MS,/ q ] −1 significantly deviates from the identity matrix, the signature of missing terms of size α 2 s . The largest departures from 1 are observed in the diagonal terms, specifically from 23 to 18% for the (1, 1) element (that contributes mostly to C MS 1 ), and from 5 to 4% for the (2, 2) element that defines C MS 2 . These higher order effects are relatively large and can be reduced only by pushing the calculation to higher renormalization scales, where α s becomes smaller. Thirdly, at these relatively high momenta, both parity even and odd projectors give results well compatible with each other. B. Discretization Errors The lattice cutoff is the main limitation of the current results preventing the weak boson mass from being well separated from lower QCD scales. As we increase it, larger discretization errors are expected and values of m W of the order of the cutoff may sound already dangerous: in Figure 8 we demonstrate that the parameter space explored in this work is still in a region where discretization errors are reasonably under control. By plotting the continuum extrapolations of C RI 1 for the four values of m W considered and normalized at the finer ensemble 32I, we show how the size of the a 2 coefficient slightly changes with m W , making the m Wdependence practically negligible compared to the overall size of cutoff effects. The scaling violations that we observe for C RI 1 range from 10 to 17%; instead they are only at 1% level for C RI 2 , where the differences among the several values of m W are also irrelevant. The different magnitudes can be easily understood from the fact that in the free theory C 1 = 0 and C 2 = 1, meaning that discretization errors start at order α s , which on our lattices is approximately 0.3. Nevertheless we perform extrapolations in a 2 rather than α s (a)a 2 , since the numerical change is irrelevant. C. Final results and discussions The last aspect of our work consists of changing renormalization scheme from RI to the more common MS. This is the only step where perturbation theory enters in our calculation since the conversion matrices are known to 1-loop and reported in Appendix B. In Figure 9 we plot our results together with the known 1-loop analytic formulae that we take from Ref. [7]. For the latter we use α s from the 1 and 4-loop β function with 3 flavors and Λ MS N f =3 = 341(12)MeV taken from Ref. [20]. The error from the Λ parameter is propagated to the analytic Wilson coefficients and it is represented by the two shaded regions. The non-perturbative results include the various systematic errors described in the previous sections. The systematic uncertainty associated with the perturbative error of the RI → MS conversion, which is an O(α 2 s ) effect, can be read off from the difference between the two intermediate RI schemes studied in the paper, the γ and / q scheme. Such a difference turns out to be relatively large, following the previous discussion on the renormalization matrixZ. We remark that the RI → MS conversion matrix for the / q scheme given in Appendix B contains a large one-loop coefficient, whereas the γ scheme shows a much better convergence with a very small correction from RI to MS. In Figure 9 we compare our results for the C MS i s against the perturbative ones as a function of the W mass. Recall that we are focusing on the initial conditions of the Wilson coefficients, which means that their renormalization scale coincides with m W . The discrepancy between our values and the 1-loop results decreases for larger values of the weak mass as expected from the running of the strong coupling constant, since the perturbative expansion of the Wilson coefficients reads C i (m W ) = N n=0 α n s (m W )k (i) n + O(α N +1 s )(31) with tree-level values k Given the limitations of the RI → MS conversion factor to one-loop, even with higher precision, fitting our MS results beyond O(α s ) would be incorrect. In fact, we turn now to our non-perturbative determination by sticking to the RI scheme, where we can convert the analytic results of the Wilson coefficients without loss of generality. In Table II we report various results from different polynomial fits to the γ scheme, following eq. (31), which we also augumented with a term k Λ /m W . Table II show the potential of the method: the possibility to extract higher loop coefficients can be used to bound the error of C i (80 GeV) by considering, for example, the difference between the known 1-loop result and our 2-loop prediction. This approach has the potential to reduce the current systematic uncertainties where these Wilson coefficients are used and are relevant, such as the real part of the amplitudes of K → ππ for isospin 0 and 2 channels. One caveat that we need to remember is that any prediction of coefficients beyond one-loop depends on the number of flavors used in the simulations. For this reason we intend to continue our study by reusing the methodology developed in this paper on the finer ensembles generated by the RBC/UKQCD Collaboration [43] with 3 and 4 active flavors in the sea. Such a calculation will provide multiple benefits, from the possibility to push m W up to 4 GeV and substantially reduce the systematic uncertainties (see Figure 7), to controlling the flavor dependence of the coefficients of higher loops and reducing the systematic error from intermediate RI scheme, thus providing a solid prediction in the MS scheme. Finally, future extensions of this work will include the top quark contribution. V. CONCLUSIONS In this paper we have presented a method to compute the initial conditions of the Wilson coefficients to all orders in the strong coupling constant and leading order in g 2 . We have described the limitations of our exploratory study, mostly related to the presence of large infrared and non-perturbative effects. By looking at different observables we quantified those effects and took them into accout in our systematic uncertainties, that dominate the final errors. Nevertheless, we have demonstrated how precise statistical results can be achieved with the combined use of momentum sources and twisted boundary conditions. Therefore we expect to obtain excellent results with the next iteration of this calculation repeated on finer lattices, where the systematic errors will significantly decrease. Despite the limitations imposed by the lattice cutoff, we observed reasonable scaling violations for the values of m W that we explored. Moreover we discussed a strategy to extend the relevance of our study to place a bound on the error of the Wilson coefficients for the physical scenario with m W ≈ 80 GeV. VI. ACKNOWLEDGEMENTS We would like to thank our RBC and UKQCD collaborators for helpful discussions and support. M.B. is particularly indebeted to N. Christ for a critical reading of the manuscript and to T. Izubuchi and P. Boyle for many stimulating discussions. Computations for this work were carried out at the Fermilab cluster pi0 as part of the USQCD Collaboration, which is funded by the Office of Science of the U.S. Department of Energy. M.B., C.L. and A.S. were supported by the United States Department of Energy under Grant No. de-sc0012704. In addition C.L. is supported in part through a DOE Office of Science Early Career Award. In this Appendix we present further details on the alternative stochastic sampling method for the W boson propagator described in the main text. The key ingredient is the possibility to approximate the propagator in momentum space x e ip(y0−x) D −1 (x, y 0 ) with pointsource propagators transformed to any continuous momentum, up to finite-size errors S app (p) = x D −1 (x, y 0 ) ×      e ipµ(y0−x)µ \|y 0 − x\| µ < L µ /2 e ipµ(y0−x+L)µ (y 0 − x) µ ≤ −L µ /2 e ipµ(y0−x−L)µ (y 0 − x) µ ≥ L µ /2 (A1) It is easy to check that if the momentum is an allowed fourier mode the equation above amounts to a simple translation of the source to the origin. However if the momentum is not quantized, S app approximates well the propagator in infinte volume: the mass gap of QCD ensures that finite volume errors are exponentially small. The second feature of the stochastic sampling relies on the approximation of the sum over the W propagator. Let us fix one end to the origin and call r the second end. Due to the fast fall-off of the weak boson propagator, only small separations contribute to the signal. Hence, we start by dividing all distances r in classes defined by their absolute value \|r\| with multiplicities d \|r\| . Then we randomly choose one representative per class and we cover all distances up to a certain cutoff R inner . Up to lattice symmetries this sum is exact. Finally we sample the remaining classes starting from R inner with probability p(r) ∝ \|r\| −3 and we evaluate the appropriate reweighting probability w(r) to obtain the flat sum again. With am W = 1.0 on the 24I ensemble, sampling 30 classes exactly (below R inner ) and 10 classes stochastically (above R inner ) produces controlled approximations with stochastic errors around 1%. This sampling strategy defines what we call a "cloud" of points around the origin. As explained already in the text, the problem resides in the second sum over different clouds, each centered around a randomly chosen point. Although the noise of the final amplitudes and Wilson coefficients scales with the number of clouds, compared to the momentum sources, which amount to summing all of them, it is still from 5 to 10 times larger, when going from high (1 GeV) to small momenta. The access to all momenta has the additional advantage of averaging over different orientations, such as (p, 0, 0, 0), (0, p, 0, 0), (−p, 0, 0, 0), etc. . . , an effect already included in the numerical factors quote above. To reduce the cost of this method we have also employed an All-Mode-Averaging [44] strategy by comput-ing the point-source propagators of the clouds sloppily and adding the corresponding correction term, which we tuned to be well below the statistical error throughout the entire range of momenta. In our final measurements we have used up to 16 total clouds, each containing 40 points as described above. For the same cost we have obtained the data points showed in Figure 2 and in the right panel of Figure 3. µ u i )(ū j γ L µ d j ) , Q 3 =(s i γ L µ d i ) q (q j γ L µ q j ) .(3) Then we substitute u with c whenever it appears in eq. (B1) and we compute the Green's functions, similarly to eq. (6), [Γ(Q(3) i )] αβγδ abcd = s α a u γ c Q With the substitution u → c advocated above, we are explicitly eliminating all the disconnected diagrams of the three-flavor theory. At this point it is easy to demonstrate that Γ(Q 1 ) = Γ(Q 1 ) , Γ(Q (3) 2 ) = Γ(Q 2 ) , Γ(Q (3) 3 ) = Γ(Q 1 ) ,(3) where the operators on the r.h.s. correspond to eq. (4). In the three-flavor theory the 10 operator basis is redundant, leading to a smaller basis, with linear independent operators, often called the chiral basis (we consider again solely the linear combinations involving the three operators in eq. (B1)) Q 1 =3Q (3) 1 + 2Q (3) 2 − Q(3) 3 , Q 2 = 1 5 (2Q 1 − 2Q 2 + Q(3) 3 ) , Q 3 = 1 5 (−3Q 1 + 3Q 2 + Q(3) 3 ) . (B4) If we now combine eq. (B4) with eq. (B3) we can relate the chiral basis to our two operators as follows Γ(Q i ) = R ij Γ(Q j ) , R ij =    2 2 3/5 −2/5 −2/5 3/5    . (B5) In Ref. [42] the conversion matrices Z RI→MS are given in the chiral basis, which is the reason behind the introduction of the matrix R. We stress that Q 1 transforms under the (27,1) representation of the chiral group, whereas Q 2 and Q 3 under the (8,1) representation. This prevents any mixing between the two sectors once the penguin operators are discarded, as in our case. Finally let us introduce three ad hoc matrices T such that T (i) R = 1 0 0 1 , ∀i .(B7) To relate the renormalization factors of Ref. [42] to our specific case we need to recall the renormalization conditions in eq. (17) and the definition of the matrix M . Similarly to eq. (14) we can define a matrix M for the chiral basis M ij = Tr (P j Γ(Q i )) = (RM U T ) ij , with the matrix U being the rotation of the projectors P i = U ij P j . Similar to the matrices T (i) , we introduce three ad hoc matrices S (i) such that their product with U returns the 2 × 2 identity. Starting from the usual renormalization conditions for M in the RI sense, with the help of eq. (B8) and a few algebraic steps we finally obtain Z RI→MS 2×2 = T (i) · Z RI→MS, 3×3 · R (B9) The universality of γ scheme is such that for any choice of U (and consequently of S (i) ) eq. (B9) is always valid for i = 1, 2 and 3 and we verified that. Instead for / q projectors the situation is different, since a naive choice of projectors can lead to mixing between the (27,1) and (8,1) operators even with the explicit suppression of the penguin diagrams. An accurate choice of projectors that forbids this is provided in the Appendix B of Ref. [42]. For simplicity we consider eq. (B9) for T (2) which corresponds to selecting Q 2 and Q 3 alone. The final results for the conversion matrices reported below have been obtained from Table III of Ref. [42] for the γ scheme and from Table XIII for the / q scheme where in both cases we have set the penguin contributions to zero As explained in the main text, the bare lattice Wilson coefficients can be obtained from any reasonable choice of projectors in the definition of M and W . Below we present an alternative definition of the projectors defined in eq. (16) which holds both for γ and / q types. Since the aim of this digression is to reduce the discrepancy of the results obtained in the parity even and odd sectors, we focus on the former, the most problematic one. Let us introduce the "subtracted projectors" Z RI→MS = 1 2×2 + α s 4π ∆r ,(B10)P i = P VV+AA i + b α ij P α j ,(C1) with i, j = 1, 2 referring as usual to the color diagonal and mixed case and the index α labelling the wrong chiralities that could potentially mix in the parity even sector, namely SS ± PP, VV − AA and TT. The coefficients b α ij are fixed by solving the following system Tr (P i Q α j ) = 0 , α = SS ± PP, VV − AA, TT , which minimizes the overlap of the projectors on the operators with the corresponding wrong chiralities (the index i in the operator refers again to the color structure). From a numerical experiment we found that the Dirac structures SS − PP and VV − AA contribute only to the noise without producing any effect. Instead the remaining two chiralities have a beneficial effect on the Wilson coefficients: in particular the SS + PP alone reduces the discrepancy for both C lat 1 and C lat 2 by 20-30 %; the inclusion of the tensor structure further increases the agreement with the odd projectors to less than 1 sigma for C lat 2 , but has the opposite effect on C lat 1 . In conclusion, the best result for the subtracted even projectors has been obtained with α = SS + PP alone. Since the benefit was marginal, in our estimate of δC proj i we decided to use the un-subtracted projectors, giving a more conservative and safer error. FIG. 1 . 1Connected diagram of the generic four-point function described in eq. (6) and fourier transformed with phases e ip k y k . Disconnected contributions are forbidden by the flavor structure of the operators Oi. In our notation the four external quarks have all in-coming momenta. FIG. 2 . 2Dependence of the bare lattice Wilson coefficients on the momentum of the external quark states for a fixed value of mW ≈ 1.8 GeV. FIG. 3 . 3Dependence of the lattice Wilson coefficient C lat 2 on the momentum of the external states for fixed mW ≈ 1.8 GeV. The labels in the legend denote the choice of projectors (γ or / q) and momentum (injected along xy spatial directions or time t). Both panels show points obtained from parity odd projectors. Left: results from the 16I ensemble; if the momentum is along the time direction, twice as big as the spatial ones, C lat 2 from / q projectors coincides with the measurements performed on the larger lattice. Right: results from the 24I ensemble where all combinations of momentum and projectors converge to the same point in the limit of small momenta. FIG. 4 . 4Quark mass dependence of C lat 2 FIG. 6 . 6Contribution of the Goldstone boson exchange to the Wilson coefficients with parity even γ projectors. The two lines are linear fits of log C lat i in p 2 /m 2 W , which describe well the data. The suppression given by the prefactor in the vertices, (2m/mW) 2 = 0.0064, makes the two contributions negligible, even though they slightly reduce the discrepancy from the C lat i s obtained with parity odd projectors. \| (for better readability we omit the m W and p 2 dependence). The latter turns out to be much larger than δC gauge i and similarly accounts for nonperturbative contaminations. Therefore we have decided to discard δC gauge i . Finally we also consider the quark mass dependence as a source of systematic uncertainties, δC mass i = \|C lat,am=0.04 i − C lat,am=0.02 i \|, measured for exceptional kinematics only from parity odd projectors. Now that we have established how to quantify remaining non-perturbative effects, as our last step in the extraction of the Wilson coefficients we study the dependence of the total systematic uncertainty on the lower end (p cut ) of the fit range of C lati δC lat i (p cut ) = (δC proj i ) 2 + (δC mass i ) 2 + (δC poly i ) 2 ,(24)where we also quote the extrapolation error δC poly i as a function of p cut by taking the difference of the intercepts of two different polynomial fits. uncertainties plotted inFigure 9are all correlated with each other, allowing us to fit the data with eq. (31) and still be able to estimate the one-loop coefficients: from the γ intermediate scheme alone we obtain k −0.158(78), that agree within 1 sigma with the analytic results taken from Ref.[7], 0.44 and -0.15 respectively 5 . FIG. 9 . 9Results in the continuum limit for 3-flavor QCD of the Wilson coefficients in the MS scheme as a function of the W boson mass. The shaded regions represent the perturbative one-loop result with error propagated from the Λ-parameter. The difference between the two intermediate RI schemes accounts for higher order effects in the RI → MS conversion factors. The plotted error bars include both statistical and systematic uncertainties. The latter dominate and gradually decrease with mW. α s computed at 80 GeV. With the current large systematic uncertainty we are only sensitive to the 1-loop coefficient, but the results in Id β a [GeV −1 ] L/a × T /a am u,d amsTABLE I. List of the ensembles used in this work. In the table we report the most important physical parameters, the remaining details can be found in Refs.16I 2.13 1.78 16 3 × 32 0.01 0.04 24I 2.13 1.78 24 3 × 64 0.01 0.04 32I 2.25 2.38 32 3 × 64 0.008 0.03 The slope of the extrapolations slightly changes with mW, but remains a subdominant effect. The points have been horizontally displaced for better visibility.0.0000 0.0025 0.0050 0.0075 0.0100 0.0125 a 2 [fm 2 ] 0.95 1.00 1.05 1.10 C RI 1 (a)/C RI 1 (a ref ) m W ≈ 1.4 GeV m W ≈ 1.8 GeV m W ≈ 2.1 GeV m W ≈ 2.4 GeV FIG. 8. Relative continuum extrapolations of C RI 1 , renormal- ized at the value of the W boson mass used to compute them. We normalize the y-axis with the measurement of C RI 1 from 32I. TABLE II . IIThe coefficients reported inTable IIcould be used, in principle, to provide an estimate of the Wilson coefficients for physical values of m W by using eq. (31) withIn the first lines we report results for the Wilson coefficient C1, whereas the last ones refer to C2. All fits have very small χ 2 per d.o.f. due to the large systematic errors. The errors quoted in the table are both statitstical and sys- tematic. In our fits we always use all the four values of mW explored in this work. 1.00 1.25 1.50 1.75 2.00 2.25 m W [GeV] 0.85 0.90 0.95 1.00 C M S 2 NLO α (1−loop) s NLO α (4−loop) s q / γ 1.00 1.25 1.50 1.75 2.00 2.25 m W [GeV] Appendix A: Stochastic sampling of the weak propagator With twisted BC we can restrict our calculation to a single irreducible representation of the hypercubic group, say ap = (x, −x, 0, 0) with x ∈ R, thus obtaining smooth functions. This excludes only linear fits with apmax > 0.15 on the 24I, the ensemble for which we have a wider number of measurements. In the 32I case all extrapolations are compatible with each other. The multiplicative renormalization factor of this diagram is Z A (from the PCAC relation) and some care is required when considering the renormalized amplitude W RI i . 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[ "Theory of Zero-Bias Anomaly in Magnetic Tunnel Junctions: Inelastic Tunneling via Impurities", "Theory of Zero-Bias Anomaly in Magnetic Tunnel Junctions: Inelastic Tunneling via Impurities" ]	[ "L Sheng \nNational Laboratory of Solid State Microstructures\nNanjing University\n210093NanjingChina\n", "D Y Xing \nNational Laboratory of Solid State Microstructures\nNanjing University\n210093NanjingChina\n", "D N Sheng \nDepartment of Physics and Astronomy\nCalifornia State University\n91330NorthridgeCAUSA\n" ]	[ "National Laboratory of Solid State Microstructures\nNanjing University\n210093NanjingChina", "National Laboratory of Solid State Microstructures\nNanjing University\n210093NanjingChina", "Department of Physics and Astronomy\nCalifornia State University\n91330NorthridgeCAUSA" ]	[]	Using the closed-time path integral approach, we nonperturbatively study inelastic tunneling of electrons via magnetic impurities in the barrier accompanied by phonon emission in a magnetic tunnel junction. The spectrum density of phonon emission is found to show a power-law infrared singularity ∼ ω −(1−g) with g the dimensionless electron-phonon coupling. As a consequence, the tunneling conductance G(V ) increases with bias voltage \|V \| as G(V ) − G(0) ∼ \|V \| 2g , exhibiting a discontinuity in slope at V = 0 for g ≤ 0.5. This theory can reproduce both cusp-like and noncusp-like feature of the zero-bias anomaly of tunneling resistance and magnetoresistance widely observed in experiments.	10.1103/physrevb.70.094416	[ "https://arxiv.org/pdf/cond-mat/0408552v1.pdf" ]	26,758,001	cond-mat/0408552	a02e39f4b5c6a3420ec4d050041dc374118ac3c3	Theory of Zero-Bias Anomaly in Magnetic Tunnel Junctions: Inelastic Tunneling via Impurities 26 Aug 2004 L Sheng National Laboratory of Solid State Microstructures Nanjing University 210093NanjingChina D Y Xing National Laboratory of Solid State Microstructures Nanjing University 210093NanjingChina D N Sheng Department of Physics and Astronomy California State University 91330NorthridgeCAUSA Theory of Zero-Bias Anomaly in Magnetic Tunnel Junctions: Inelastic Tunneling via Impurities 26 Aug 2004numbers: Number: 7225Ba7547-m7340Rw7210Di Using the closed-time path integral approach, we nonperturbatively study inelastic tunneling of electrons via magnetic impurities in the barrier accompanied by phonon emission in a magnetic tunnel junction. The spectrum density of phonon emission is found to show a power-law infrared singularity ∼ ω −(1−g) with g the dimensionless electron-phonon coupling. As a consequence, the tunneling conductance G(V ) increases with bias voltage \|V \| as G(V ) − G(0) ∼ \|V \| 2g , exhibiting a discontinuity in slope at V = 0 for g ≤ 0.5. This theory can reproduce both cusp-like and noncusp-like feature of the zero-bias anomaly of tunneling resistance and magnetoresistance widely observed in experiments. The tunneling magnetoresistance (TMR) effect in a magnetic junction composed of two ferromagnetic (FM) electrodes separated by an insulating layer is presently of great interest because of its potential of device applications such as random access memories and magnetic sensors 1,2,3,4,5,6,7,8,9,10,11 . While the TMR effect has been attributed to spin polarization of conduction electrons in the FM electrodes 1,2,3 , its novel voltage dependence is still puzzling. The tunneling resistance and TMR ratio generally decrease with increasing bias voltage, often exhibiting a cusp-like feature at zero bias 1,5,6,7,8,9,10,11 . This effect, usually called the zero-bias anomaly (ZBA) 6 , was not only widely observed in transition-metal magnetic junctions, but also exists in semiconductor GaMnAs/AlAs/GaMnAs magnetic junctions 12 . It is found that the ZBA is very sensitive to the insulating material of the barrier. In earlier measurements 1 on Fe/Ge/Co junctions, the characteristic voltage V 1/2 that halves the TMR ratio is about 3 mV. It was reported recently that for FM/Al 2 O 3 /FM with Fe, Co, Ni, or their alloys as the FM electrodes, V 1/2 is usually one hundred to several hundred mV 5,6,7,8,9 , and for Ni/NiO/Co junctions V 1/2 is a few ten mV 11 . Junctions made of the same FM electrodes and identical but differently prepared insulator vary considerably concerning the voltage dependence of the resistance and TMR 9 . It was reported that the ZBA in the semiconductor magnetic tunnel junctions is much more striking with V 1/2 about 3mV 12 . Interestingly, in a very recent experiment 13 of the TMR between a Co sample and a CoFeSiB tip through a vacuum barrier, the ZBA did not occur for bias voltage upto a volt. Understanding of the ZBA is not only important for optimizing the TMR in practical applications, but also provides critical insight into the underlying physics of spin-dependent transport. There exist several possible explanations for the anomaly. The simplest one may involve the energy dependence of the electron density of states (DOS) and elastic transmission matrix elements 3,14,15 . Unfortunately, by comparison with experimental data, the TMR ratio calculated from such a model decreases too slowly at low bias 7 . The second mechanism is that the hot electrons from the emitting electrode may be scattered by local magnetic moments at the interfaces through s-d interaction 6 , which contributes to the conductance a term ∆G ∼ \|V \| at low bias voltage V . This theory succeeded in explaining some experimental data. The third mechanism might be based upon the electron tunneling assisted by magnons and phonons 7,16,17,18 , which was found to contribute ∆G ∼ \|V \| 3/2 for FM magnons, \|V \| 2 for surface antiferromagnetic magnons, and \|V \| 4 for phonons. This mechanism may explain some experimental data without the cusp-like feature at zero bias 18 , but has the difficulty to interpret the cusp-like feature of the ZBA observed in many other experiments 5,6,7,10,12 . On contrary to these intrinsic mechanisms, the effect of extrinsic nonmagnetic and magnetic impurities was emphasized in many works 8,10,11,14,18,19,20,21,22,23 . The voltage dependence caused by the elastic electron tunneling via impurities was studied theoretically in a few works 11,14,21 . However, none of them reproduced the essential features of the ZBA. Based upon the results of an earlier theory 24 , some authors 10,23 argued that, in the presence of inelastic scattering, while single-impurity tunneling processes are still elastic, electron tunneling via multiple-impurity chains may contribute an inelastic power-law conductance ∆G ∼ \|V \| p with p = 4/3 (5/2) for two-impurity (three-impurity) processes. Presently, the relevant mechanism for the ZBA is still under debate. In this work, a combined effect of the magnetic impurities in the barrier and the electronphonon interaction is proposed as a possible mechanism for the ZBA. What we consider in this model is not a simple addition of the elastic tunneling of electrons via the impurities and the inelastic one assisted by phonons 18 , but a novel inelastic electron tunneling due to the strong interplay between them, in which the electron tunneling via impurities is always accompanied by phonon emission. It is found that the spectrum density of phonon emission has a power-law infrared singularity ∼ ω −(1−g) with g the dimensionless electronphonon coupling. As a consequence, the conductance via impurities G I (V ) obeys a powerlaw voltage dependence G I (V ) ∼ \|V \| 2g with a discontinuity in slope at V = 0 for g ≤ 0.5. The conductance exponent 2g may be either smaller or greater than unity, being different from the earlier theories 6,16,17,18,24 where the conductance exponents are never smaller than unity. The calculated results can reproduce both cusp-like and non-cusp-like features of ZBA observed in experiments. An experiment based upon the isotope effect is suggested to further verify this theory. The model Hamiltonian of a magnetic tunnel junction is written as H = H L + H R + H C + H T . Here, H L(R) = ks ǫ L(R) ks a L(R) † ks a L(R) ks is the Hamiltonian of the electrons in the left (right) electrodes. The third term is the Hamiltonian of the electrons on the magnetic impurities, H C = lss ′ (E 0 δ ss ′ − Jσ ss ′ · S l )d † ls d ls ′ + qλ ω qλ b † qλ b qλ + qλls M qλ e iq·R l b qλ + b † −qλ d † ls d ls .(1) Here Since the impurities are also a part of the lattice, electrons scattering with the impurities will generally cause emission or absorption of phonons of the system, which is described by the electron-phonon interaction of the last term in Eq. (1). It is worth mentioning that a similar impurity-mediated electron-phonon interaction has been used to account for the leading T 2 correction to residue resistivity of simple metals 25,26 . Here, the direct scattering of electrons by phonons is neglected, which was found to open new tunneling channels, and yield even, positive correction to the conductance 27 . It does not account for the ZBA. The electron tunneling Hamiltonian is given by H T = kk ′ s T D kk ′ s a L † ks a R k ′ s + h.c. + lks T IL lks a L † ks d ls + h.c. + lks T IR lks a R † ks d ls + h.c. ,(2) where the first term represents direct tunneling of electrons between the electrodes, and the second and third terms represent transfer of electrons between the electrodes and the impurities. To treat the electron-phonon interaction in a nonperturbative manner, the Lang-Firsov transformation e S with S = − qλls (M qλ /ω qλ )d † ls d ls e iq·R l (b qλ −b † −qλ ) 28 is performed to diagonalize the electron-phonon interaction. The transformed Hamiltonian e S He −S is the same as the original one, except that the last term in H C vanishes, and operators d ls and d † ls in H T are replaced with d ls X l and d † ls X † l , respectively, with X l = exp[ qλ (M qλ /ω qλ )e iq·R l (b qλ − b † −qλ )]. In this nonperturbative treatment, we see clearly that electron tunneling via the impurities causes phonon emission or absorption, which reversely renormalizes the electron tunneling matrix elements via the impurities. The path-integral formulism on the closed-time path 29,30 is employed to study this nonequilibrium problem. By integrating out the electron variables in the two electrodes, the retarded (advanced) Green's function for electrons on impurity l is obtained in frequency domain asĜ C l,r(a) (ω) = [ω − E 0 + Jσ · S l −Σ l,r(a) (ω)] −1 . Here, 2 × 2 matrix representation in electron spin space has been employed. The retarded self-energyΣ l,r (ω) =Σ L l,r (ω) +Σ R l,r (ω) is a diagonal matrix with diagonal elements given by B > (ω) is the Fourier transform of −i X l (t)X † l (0) , whose general expression can be found in reference 26 . Σ L(R) ls,r (ω) = 1 2 ρ L(R) s \|T IL(R) l \| 2 dω ′ [1 − f L(R) (ω + ω ′ )]B > (−ω ′ ) + f L(R) (ω + ω ′ )B > (ω ′ ) ,(3) The tunneling current I is calculated from the rate of the charges flowing out of the emitting electrode. The total conductance G(V ) = I/V is obtained as G(V ) = G D + G I (V ), where G D = 2πe 2 \|T D \| 2 Tr(ρ LρR ) is the direct tunneling conductance independent of bias voltage, and G I (V ) = − N I e 2πV dǫ dω 1 dω 2 B > (ω 1 )B > (ω 2 ) f L (ǫ + ω 1 ) [1 − f R (ǫ − ω 2 )] − f R (ǫ + ω 1 ) [1 − f L (ǫ − ω 2 )] \|T IL l T IR l \| 2 Tr ρ LĜC l,r (ǫ)ρ RĜC l,a (ǫ) ,(4) comes from the tunneling via N I impurities. The bias dependence of the tunneling conductance is embodied in G I (V ). In Eq. (4), the last factor is averaged over the positions and spin orientations of the impurities,ρ L = [(ρ ↑ + ρ ↓ ) + (ρ ↑ − ρ ↓ )σ z ]/2, and ρ R = [(ρ ↑ + ρ ↓ ) ± (ρ ↑ − ρ ↓ )σ z ]/2 with sign + for parallel alignment and − for antiparallel alignment. The two electrodes are considered of the same material with electron DOS ρ ↑ and ρ ↓ for majority and minority spins, respectively. There are two distinct resonant energies E − = E 0 − JS and E + = E 0 + JS, at which the tunneling conductance given in Eq. (4) will be substantially enhanced. In this work, we confine ourselves to the off-resonance case, which is more probable in reality. The energy differences \|E F − E ± \| with E F the Fermi level are considered to be much greater than the broadenings of the resonant states. Besides, the bias voltage V is also relatively small \|eV \| < \|E F − E ± \|. In this case,Σ l,r(a) (ǫ) inĜ C l,r(a) (ǫ) can be neglected. The last factor in Eq. (4) is then divided into two parts: \|T IL l T IR l \| 2 and Tr[ρ LĜC l,r (ǫ)ρ RĜC l,a (ǫ)]. The former is a constant after averaged over the impurity position R l , and the latter involves only the average over the orientation of S l . The energy of the magnetic field used to produce the parallel alignment of the FM electrodes, usually several ten or hundred oersteds (g L µ B SB < ∼ 0.01meV), is much smaller than the entropy gain due to randomizing an impurity spin even at experimentally low temperatures, e.g., 1K [k B T ln(2S + 1) ∼ 0.1meV]. Therefore, it is assumed that the impurity spins are randomly oriented for both parallel and antiparallel alignments. Moreover, one can verify that the results will not change significantly, if the magnetic field is strong enough to align all the spins ferromagnetically in the parallel alignment. The function B > (ω) represents the spectrum density of phonon emission and absorption. For simplicity, three-dimensional Debye's model is used to describe the acoustic phonons. A dimensionless electron-phonon coupling constant g is defined as g = 3 qλ \|M qλ \| 2 ω qλ /ω 3 D with ω D the Debye frequency. Thus, at zero temperature we have iB > (ω) = dt exp {iωt − g [Cin(ω D t) + iSi(ω D t)]} ,(5) where Cin(x) = x 0 dt[1 − cos(t)]/t and Si(x) = x 0 dt sin(t)/t are the cosine and sine integral functions 31 . G I (V ) via impurities calculated by use of Eqs. (4) and (5) is plotted in Fig. 1 for nonmagnetic tunnel junctions. Since lim g→0 [iB > (ω)] = 2πδ(ω), lim g→0 G I (V ) = G I 0 is a constant independent of V , which is actually the maximum of G I (V ). From Fig. 1, it is clear that G I (V ) vanishes at V = 0 for any nonzero g, indicating that tunneling via impurities is totally suppressed, and G I (V ) increases as the voltage \|V \| is elevated. The stronger is the electron-phonon coupling g, the slower the conductance G I (V ) increases with \|V \|. To get some insight into low bias behavior of the tunneling conductance, we derive the low-ω approximation of Eq. (5) iω D B > (ω) ≃ αθ(ω)(ω/ω D ) −(1−g) ,(6) where α = 2Γ(1 − g) sin(gπ)e −gγ with γ = 0.5772 the Euler constant, and Γ(z) = ∞ 0 dtt z−1 e −t is the ordinary gamma function 31 . The unit step function θ(ω) indicates that at zero temperature only emission of phonons is possible. Substituting Eq. (6) into Eq. (4), we obtain an interesting result G I (V ) ∼ \|V \| 2g . For g ≤ 0.5 (g > 0.5), its slope dG I (V )/dV ∼ \|V \| 2g /V is discontinuous (continuous) at V = 0. In Fig. 2, the resistances for parallel alignment (R P ) and antiparallel alignment (R AP ) as well as TMR = (R AP − R P )/R AP of magnetic tunnel junctions are shown as functions of the bias voltage. It is found that R AP always decreases faster than R P , resulting in a decline of TMR. This behavior can be understood by the following argument. At V ≃ 0, the direct tunneling dominates. For antiparallel alignment, the majority-spin (minority-spin) band in the left electrode is only connected to the minority-spin (majority-spin) band in the right electrode, leading to a relatively high resistance. With increasing \|V \|, the impurity tunneling increases. Electron hopping via the magnetic impurities causes a mixing of the two spin channels, which effectively connects up the majority-spin bands in the two electrodes. As a consequence, the resistance drops rapidly. For g = 0.4, the TMR is discontinuous in slope at V = 0. According to Fig. 2, either increasing the electron-phonon coupling g or decreasing relatively the tunneling via impurities weakens the ZBA. In this theory, the energy scale of the ZBA is from zero to some ω D . In view of that ω D in most materials ranges from 10meV to 100meV, the characteristic energies of the ZBA observed in tunnel junctions with Ge 1 , NiO 11 and AlAs 12 as insulating barriers are consistent with our theory. For FM/Al 2 O 3 /FM, V 1/2 is greater than 100meV 5,6,7,8,9 . The impurity density might be low in these later junctions so that the ZBA is not prominent. According to a simplified model for impurity tunneling 32 , we estimate that the impurity effect and the ZBA will become observable as the impurity number per unit cross section reaches 1/µm 2 or higher. The present theory as well as the earlier theories 6,16,17,18,24 all predict that G(V ) for the tunnel junctions has a power-law dependence on small bias voltage, i.e., G(V )−G(0) ∼ \|V \| p . The low-temperature curves of the conductance (resistance) and TMR exhibit clear cusp- , d ls annihilates an electron with spin s on magnetic impurity l, and b qλ annihilates an acoustic phonon of wave vector q and polarization λ in the system. The magnetic impurities in the barrier are presumably the atoms of FM metals of the electrodes mixed into the barrier. Some of their outer-shell electrons are itinerant, and the others form localized spins. J represents the s-d like interaction between the itinerant electrons and localized spin S l withσ the Pauli matrix. The spins of the transition-metal ferromagnets under consideration are usually greater than 1/2, and can be treated as classical approximately. and the advanced self-energyΣ l,a (ω) = [Σ l,r (ω)] † . Here, as usual, the momentum and spin dependence in the transmission matrix element T IL(DOS and Fermi distribution function of the left (right) electrode, respectively. FIG. 1 : 1Normalized impurity conductance of nonmagnetic tunnel junctions as a function of bias voltage at zero temperature. Here J = 0 and ρ ↑ = ρ ↓ . FIG. 2 : 2like feature at zero bias in most experiments5,6,7,10,12 , and non-cusp-like feature in a few experiments18 . Mathematically, the cusp-like (non-cusp-like) feature requires that the slope Calculated results for R P , R AP , and TMR as functions of bias voltage for different g. The resistances are normalized by 2R AP (V = 0). Here, E 0 = E F , and spin polarization (ρ ↑ − ρ ↓ )/(ρ ↑ + ρ ↓ ) = 0.6.of the conductance dG(V )/dV ∼ \|V \| p /V be discontinuous (continuous) at V = 0, and so p ≤ 1 (p > 1). As mentioned in the introduction, all the earlier theories16,17,18,24 predicted constant p > 1, except for the theory of Zhang et al. 6 , where p ≡ 1. In the present theory, p = 2g may be either smaller or greater than unity, depending on the strength of the impurity-mediated electron-phonon interaction, which is more flexible for understanding both the cusp-like and non-cusp-like features within a unified theoretical framework. Our theory might be further verified directly by measurements of isotope effect. Let us assume that the insulating barrier is made of certain oxide. 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[ "On the pulse-width statistics in radio pulsars. I. Importance of the interpulse emission", "On the pulse-width statistics in radio pulsars. I. Importance of the interpulse emission" ]	[ "Krzysztof Maciesiak \nKepler Institute of Astronomy\nUniversity of Zielona Góra\nLubuska 265-265Zielona GóraPoland\n", "Janusz Gil \nKepler Institute of Astronomy\nUniversity of Zielona Góra\nLubuska 265-265Zielona GóraPoland\n", "Valério A R M Ribeiro \nAstrophysics Research Institute\nLiverpool John Moores University\nTwelve Quays House, Egerton WharfCH41 1LDBirkenhead\n" ]	[ "Kepler Institute of Astronomy\nUniversity of Zielona Góra\nLubuska 265-265Zielona GóraPoland", "Kepler Institute of Astronomy\nUniversity of Zielona Góra\nLubuska 265-265Zielona GóraPoland", "Astrophysics Research Institute\nLiverpool John Moores University\nTwelve Quays House, Egerton WharfCH41 1LDBirkenhead" ]	[ "Mon. Not. R. Astron. Soc" ]	We performed Monte Carlo simulations of different properties of pulsar radio emission, such as: pulsar periods, pulse-widths, inclination angles and rates of occurrence of interpulse emission (IP). We used recently available large data sets of the pulsar periods P , the pulse profile widths W and the magnetic inclination angle α. We also compiled the largest ever database of pulsars with interpulse emission, divided into the double-pole (DP-IP) and the single-pole (SP-IP) cases. We identified 31 (about 2%) and 13 (about 1%) of the former and the latter, respectively, in the population of 1520 normal pulsars. Their distribution on the P −Ṗ diagram strongly suggests a secular alignment of the magnetic axis from the originally random orientation. We derived possible parent distribution functions of important pulsar parameters by means of the Kolmogorov-Smirnov significance test using the available data sets (P , W , α and IP), different models of pulsar radio beam ρ = ρ(P ) as well as different trial distribution functions of pulsar period P and the inclination angles α. The best suited parent period distribution function is the log-normal distribution, although the gamma function distribution cannot be excluded. The strongest constraint on derived model distribution functions was the requirement that the numbers of interpulses generated by means of Monte Carlo simulations (both DP-IP and SP-IP cases) were exactly (within 1σ errors) at the observed level of occurrences. We found that a suitable model distribution function for the inclination angle is the complicated trigonometric function which has two local maxima, one near 0 • and the other near 90 • . The former and the latter implies the right rates of IP, occurrence, single-pole (almost aligned rotator) and double-pole (almost orthogonal rotator), respectively. It is very unlikely that the pulsar beam deviates significantly from the circular cross-section. We found that the upper limit for the average beaming factor f b describing a fraction of the full sphere (called also beaming fraction) covered by a pulsar beam is about 10%. This implies that the number of the neutron stars in the Galaxy might be underestimated. 33 and NN 203 3919 34. We thank R.N. Manchester for invitation of K.M. to Australia, support and access to the ATNF pulsar data. We also thank M. Kramer for sharing with us unpublished results concerning new interpulse cases. We thank M. Kolodziejczyk for linguistic help. We also thank the anonymous referee for useful comments.	10.1111/j.1365-2966.2011.18471.x	[ "https://arxiv.org/pdf/1102.3348v1.pdf" ]	118,466,796	1102.3348	0a5f088685988d8041caac0db09464b7033b4656	On the pulse-width statistics in radio pulsars. I. Importance of the interpulse emission 2010. January 12. 2013 Krzysztof Maciesiak Kepler Institute of Astronomy University of Zielona Góra Lubuska 265-265Zielona GóraPoland Janusz Gil Kepler Institute of Astronomy University of Zielona Góra Lubuska 265-265Zielona GóraPoland Valério A R M Ribeiro Astrophysics Research Institute Liverpool John Moores University Twelve Quays House, Egerton WharfCH41 1LDBirkenhead On the pulse-width statistics in radio pulsars. I. Importance of the interpulse emission Mon. Not. R. Astron. Soc 0002010. January 12. 2013Accepted . Received ; in original form(MN L A T E X style file v2.2)stars: pulsars: general -stars: neutron -stars: rotation We performed Monte Carlo simulations of different properties of pulsar radio emission, such as: pulsar periods, pulse-widths, inclination angles and rates of occurrence of interpulse emission (IP). We used recently available large data sets of the pulsar periods P , the pulse profile widths W and the magnetic inclination angle α. We also compiled the largest ever database of pulsars with interpulse emission, divided into the double-pole (DP-IP) and the single-pole (SP-IP) cases. We identified 31 (about 2%) and 13 (about 1%) of the former and the latter, respectively, in the population of 1520 normal pulsars. Their distribution on the P −Ṗ diagram strongly suggests a secular alignment of the magnetic axis from the originally random orientation. We derived possible parent distribution functions of important pulsar parameters by means of the Kolmogorov-Smirnov significance test using the available data sets (P , W , α and IP), different models of pulsar radio beam ρ = ρ(P ) as well as different trial distribution functions of pulsar period P and the inclination angles α. The best suited parent period distribution function is the log-normal distribution, although the gamma function distribution cannot be excluded. The strongest constraint on derived model distribution functions was the requirement that the numbers of interpulses generated by means of Monte Carlo simulations (both DP-IP and SP-IP cases) were exactly (within 1σ errors) at the observed level of occurrences. We found that a suitable model distribution function for the inclination angle is the complicated trigonometric function which has two local maxima, one near 0 • and the other near 90 • . The former and the latter implies the right rates of IP, occurrence, single-pole (almost aligned rotator) and double-pole (almost orthogonal rotator), respectively. It is very unlikely that the pulsar beam deviates significantly from the circular cross-section. We found that the upper limit for the average beaming factor f b describing a fraction of the full sphere (called also beaming fraction) covered by a pulsar beam is about 10%. This implies that the number of the neutron stars in the Galaxy might be underestimated. 33 and NN 203 3919 34. We thank R.N. Manchester for invitation of K.M. to Australia, support and access to the ATNF pulsar data. We also thank M. Kramer for sharing with us unpublished results concerning new interpulse cases. We thank M. Kolodziejczyk for linguistic help. We also thank the anonymous referee for useful comments. INTRODUCTION Although the number of pulsars discovered recently in modern search campaigns increased enormously, the observed pulsar population is still a small fraction of the neutron star population in the Galaxy. Most of them will never be detected as radio pulsars due to misalignment of their beams with our line-of-sight (l-o-s). However, many of those whose beams point towards the Earth still await detection in future, more sensitive pulsar surveys. Therefore, a more or less complete knowledge about Galactic pulsar population can be obtained only by means of statistical considera- * E-mail:[email protected] tions. Statistical studies of the pulse-width in mean profiles of radio pulsars is an important tool for investigations of the geometry of pulsar radiation. One especially important parameter that can be derived from such studies is the inclination angle between the magnetic and the spin pulsar axes. Early studies were carried out by Henry & Paik (1969), Roberts & Sturrock (1972, 1973, Backer (1976) and Manchester & Lyne (1977, hereafter ML77). Since the amount of the available data was small, these papers suffered from problems of small number statistics. A more complete work was performed by Prószyński (1979) and Lyne & Manchester (1988), who analyzed samples of about 200 pulse-width data measured near 400 MHz. Although the database used in these papers was quite rich, the pulse-width measurements were contaminated by the interstellar scattering dominating at low radio frequencies. More recently Gil & Han (1996; hereafter GH96) compiled a new database of 242 pulse-widths W10 (corresponding to about 10 per cent of the maximum intensity) measured at a higher radio frequency (near 1.4 GHz), which was relatively unbiased compared to the lower frequency data. GH96 used their pulse-width database to perform Monte Carlo simulations in an attempt to derive the distribution statistics of pulsar periods, pulse-widths, magnetic inclination angles and rates of interpulses. By comparing the simulated and observed (or observationally derived) quantities they concluded that the observed distribution of the inclination angles resembles a sine function following from the flat (random) distribution in the parent population, and that the probability (beaming fraction) of observing a pulsar was about 0.16. GH96 also pointed out that the rates of interpulse occurrence should be considered as an important aspect of pulsar population studies. On the other hand, Tauris & Manchester (1998, hereafter TM98) using a different method based on an analysis of the indirectly derived polarization position angles and magnetic inclination angles concluded that the observed distribution of the latter is cosine-like rather than sine-like as suggested by GH96. They also obtained the beaming fraction 0.10 ± 0.02, considerably lower than 0.16 obtained by GH96. TM98 pointed out a likely source of this discrepancy, namely the incorrect assumption used by GH96 that the observed distribution and the parent distribution of pulsar periods are similar. Recently, Zhang, Jiang & Mei (2003, hereafter ZJM03) followed the Monte Carlo simulation scheme developed by GH96. ZJM03 argued that both the parent distribution function and the observed distribution of pulsar periods can be modelled by the gamma function but with different values of the free parameters, and their Monte Carlo simulations included searching for a 2-D grid of these parameters. As a result, ZJM03 concluded that indeed the cosine-like distribution (suggested by TM98) is much more suitable to model the inclination angles in the parent pulsar population than the flat distribution (suggested by GH96). They argued that the most plausible parent distribution is a modified cosine function (see Section 2.1.1), which has a peak around 25 • and another weaker peak near 90 • . They also obtained the beaming fraction ∼ 0.12, consistent with the result of TM98. As emphasized by ZJM03 in the conclusions of their paper, neither they nor TM98 considered potentially important constraints related to the observable interpulse emission. Kolonko et al. (2004, KGM04 hereafter) corrected this shortcoming of analysis of ZJM03 and included the rates of occurrences of the interpulse emission, divided into categories of single-pole and double-pole origin. In this paper we follow the scheme developed by KGM04 but with a few important improvements. We used much richer databases of pulsar periods, pulse-widths and interpulse occurrences. Moreover, we use broader spectrum of trial distribution functions in our Monte Carlo simulations. Figure 1. Geometry of the pulsar radiation presented schematically on the "celestial" hemisphere centred on the neutron star (NS) with the radius R NS . The polar cap is marked as a black area on the shadow NS surface. The radio emission region is marked at the altitude rem. The fiducial plane ϕ 0 = 0 contains the rotation Ω and the magnetic m axes as well as the observer's direction. The following angles are marked: the longitudinal phase ϕ measured from the fiducial phase ϕ = 0, the inclination angle α between the magnetic m and the spin Ω axes, the impact angle β of the closest approach of the observer to the magnetic axis, the observer's angle ξ = α + β, the opening angle of the radiation beam ρ and the polarisation position angle ψ. Two exemplary line-of-sights are marked: l-o-s 1 passing through the magnetic axis (β = 0) and l-o-s 2 corresponding to a grazing impact angle β ∼ ρ. GEOMETRY OF THE PULSAR RADIATION The basic condition for the pulsar to be detected is that it should be bright enough for sensitivity of the observing system used in the radio observatory. In this paper we assume that this condition is always satisfied (some consequences of this assumption are discussed in Appendix A (in the on-line materials)) and our main concern is geometrical detection conditions. Pulsar can be detected if its narrow beam sweeps through the observer. We use this geometrical detection method in our Monte Carlo simulations, without taking into account the intrinsic pulsar luminosity (although we briefly discuss this problem in Section 5). However, we restrict our parameter space to quantities that should not be strongly affected by the luminosity problem. The geometry of pulsar radiation is schematically shown in Fig. 1. The observed pulse-width W (ignoring dispersive and scattering broadening; see Section 3.2) depends on the intrinsic angular radius of the beam ρ, the inclination angle α and the impact angle β. Purely geometrical pulse-width W l on the l-th level of the maximum intensity of the profile is W l = 2 · ϕ l = 4 · arcsin sin((ρ l +β)/2)·sin((ρ l −β)/2) sin α·sin(α+β) (Gil 1981). For the impact angle β = 0 • (corresponding to a very rare situation when the line-of-sight cuts through the centre of the beam) and the inclination angle α = 90 • (corresponding to the case of the orthogonal rotator), we get W = 2ρ. For non-orthogonal rotator this gives a very well known approximation W = 2ρ/cos α. Thus, roughly speaking the pulse-width is proportional to the angular radius of the beam ρ. We would like to emphasize that Eq. (1) assumes symmetry of the pulsar beam (and thus symmetry of the pulse-width but not necessarily the pulse shape) with respect to the fiducial phase ϕ0 (Fig. 1). We paid special attention to selecting pulsars with relatively low dispersion measure DM , free from broadening features (see Section 3.2) that would introduce significant asymmetry into pulsar profiles. 1/2 ,(1) Probability density distribution functions The main aim of this work is to carry out statistical studies of emission properties of the normal radio pulsars. On the one hand, we have samples of different measurable (directly or indirectly) parameters obtained for a large number of pulsars. On the other, we can simulate these parameters and compare the simulated values with the measured ones. In each case the important question we try to answer is what the statistical distributions of these parameters are in the observed sample, and more generally in the whole pulsar population. Below we consider possible trial distribution functions of selected pulsar parameters. We present the socalled parent trial distribution functions, which should be distinguished from the observed distributions. Inclination angle α and impact angle β In an arbitrary coordinate system the spin axis of the neutron star and the viewer's line-of-sight are both randomly chosen uniformly on the surface of the sphere, resulting in an isotropic distribution of both. The correlated probability distribution of the observer's angle ξ = α + β relative to the spin axis is given by f (ξ) = sin(ξ) = sin(α + β),(2) where α and β are the inclination and the impact angles 1 , respectively ( Fig. 1). However, the actual distribution of the inclination angle α may depend on number of unknown factors. Therefore, we considered a number of trial probability density functions for the parent distribution of α, including flat distribution f (α) = 2 π ,(3) sine function f (α) = sin α,(4) 1 Range of the observer's angle ξ, inclination angle α and impact angle β is as follows: 0 ξ π, 0 α π/2 and −π/2 β π/2. The observer's ξ angle is uniformly distributed on a sphere, however the inclination angle α is drawn from one of the trial probability density functions (Eqs. (3) - (7)). Thus, the impact angle β distribution is not uniform but depends on distribution of the inclination angle. Let us keep in mind that for the observed pulsars the impact angle β ρ, where ρ is the angular width of the emission beam (Fig. 1). Although distributions of both α and β are unknown, their sum is uniformly distributed on a sphere. or cosine function f (α) = cos α. (5) Apart from the simple functions presented above some more complicated probability density functions were considered as well. As a parent distribution function of the inclination angle α GH96 proposed the following function f (α) = 5 4 + cos 5 2 α . Later ZJM03 proposed even more complicated function f (α) = 0.6 cosh(3.5(α − 0.43)) + 0.15 cosh(4.0(α − 1.6)) which they called the modified cosine function. This function is characterised by two local maxima, around α ≈ 25 • and another weaker one around α ≈ 90 • . Interestingly ZJM03 did not consider rates of interpulse occurrence in their paper and their complicated cosine function was introduced to improve modelling of the parent inclination angles by means of Monte Carlo simulations. Later, KGM04 demonstrated that this function is also responsible for the occurrence of the proper amount of interpulses. We do not confirm this conclusion in the present paper, using much larger databases of the observed pulsar parameters and interpulses. We find that the most suitable parent distribution is that proposed by GH96 (6). Period P GH96 showed that the distribution function of the observed 516 pulsars with periods 0.05 < P < 4.2 s can be fitted 2 by the gamma function f (P ) = N0 x a−1 e −x ,(8) where N0 is the normalisation constant, x = P/m and, m and a are values of the free parameters. ZJM03 used this function to fit much larger sample of the 1165 period values from the same period range and obtained good result. They argued that the parent distribution of periods is different from the observed distribution and it is very convenient to use the gamma function with the values of m and a treated as free parameters of the model. We will use this approach, generalized by including a number of other trial probability density functions, like the Lorentz function f (P ) = C0 1 + (P − x0) 2 /a 2 0 ,(9) where C0 is the normalisation constant, x0 and a0 are free parameters. Another trial function that we considered was the Gauss function representing normal distribution f (P ) = 1 σ √ 2π exp −(P − x0) 2 2σ ,(10) 2 Fit of this function to the observed data was obtained with the least square method, minimalising i N (x i ) − N 0 x i+1 x i x a−1 e −x dx 2 . They found values of free parameters m and a as well as the normalising constant N 0 (details in GH96). where x0 and σ are free parameters. We also tried the lognormal distribution f (P ) = 1 P σ √ 2π exp − (log P − x0) 2 2σ 2(11) (suggested by Lorimer et al. (2006)), where x0 and σ are free parameters of period probability distribution function whose logarithm is normally distributed. Opening angle ρ The opening angle (radius) of the pulsar beam can be calculated from the pulse-width W if α and β angles are known, either from the polarisation data (Manchester & Taylor 1977) or from the width of the core component (using Eq. (29) taken from Rankin (1990; hereafter R90)), or both. Inverting Eq. (1) one obtains the opening angle ρ as a function of α, β and W (Gil et al. 1984) ρ l = 2 sin −1 sin α sin(α + β) sin 2 W l 4 + sin 2 β 2 1/2 . (12) Lyne & Manchester (1988) were the first who applied this equation to a large number of 10 per cent pulse-width data W10 measured at 408 MHz. They argued that ρ10(408 MHz) ≈ 6 • .5P −1/3 , which scaled to the 1.4 GHz was ρ10 = 5 • .8 P −1/3 .(13) Biggs (1990) reanalysed the same sample of data and argued that ρ10 = 5 • .6 P −1/2 . Rankin (1993a) analysed a large sample of pulse-widths data taken at different frequencies in different world radio observatories over a period of several years. She has interpolated all available data to frequency of about 1 GHz and divided them into different profile classes (according to Rankin (1983)). In each class she obtained a bimodal ∝ P −1/2 opening angle distribution. This result clearly indicated that pulsar beams consist one or two coaxial cones centred on the magnetic axis, with the opening angle ρ of each cone following P −1/2 period dependence. Gil, Kijak & Seiradakis (1993;GKS93) and Kramer et al. (1994) have confirmed this result at frequency 1.4 GHz, using data obtained with the Effelsberg 100 m radiotelescope. Instead of dividing pulsars into different classes to reveal the bimodal P −1/2 distribution of ρ, GKS93 performed a careful error analysis and rejected all data subject to large errors (broadening the apparent distribution). As a result they obtained that for given period P , the opening angle ρ can have two possible values: ρ10 = 6 • .3 P −1/2 4 • .9 P −1/2(15) (see Fig. 2 in GKS93). Kramer et al. (1994) obtained exactly the same result, using an independent method for both the pulse-width measurements and error analysis. Examining Fig. 2 in GKS93 we can notice that the inner cone with ρ = 4 • .9 P −1/2 seems to be preferred at shorter periods P < 0.7 s, while the outer cone with ρ = 6 • .3 P −1/2 dominates at longer periods P > 1.2 s. However, the exact model of transition between cones is not known. This observational feature is crucial, and it has to be taken into account in the statistical analysis to calculate the pulse-width in the synthetic population. We use Eq. (15) in two model variants: a) based on Fig. 2 in GKS93 we established the period value P = 0.7 s below which the inner cone (4 • .9 P −1/2 ), and above this value the outer cone (6 • .3 P −1/2 ), is always chosen, b) like in case a) but below period P = 0.7 s there is a 20 per cent chance to choose the outer cone and an 80 per cent chance for the inner cone. It is worth noting that Eq. (15) was derived by GKS93 by means of geometrical analysis of a large number of conal profiles. We believe that it describes well the low intensity pulse-width measurements used in this paper. Interpulse emission At the time of writing the manuscript of this paper there were 1520 normal pulsars (with periods longer than 20 ms) known. In nearly 3 per cent of them the so-called interpulse (IP) emission could be identified, by which we understand features separated by about 180 • (possible deviation could amount to about 40 per cent) from the main pulse (MP). The canonical lighthouse pulsar model naturally predicts the occurrence of interpulses. In this model two beams are collimated along the open lines of dipolar magnetic field. When the inclination angle α ≈ 90 • (almost orthogonal rotator) the observer can detect both beams, associated with two opposite magnetic poles. This is the so-called double-pole interpulse model (hereafter DP-IP). In this case both pulse components are clearly separated (by about 180 • of longitude) and there is not any kind of low level emission between them. Duty cycles of each component are small, typically several per cent of the pulsar period. Another possibility of generating the interpulse is described by the so-called singlepole model (SP-IP hereafter). This model requires a small inclination angle α (almost aligned rotator). In the SP-IP case pulse-widths are much broader than in DP-IP model, to the extent that they often fill the entire or most of the pulsar period (360 • ). Even if both components are separated, usually there is a low intensity bridge of emission between them. The first version of this model (Rickett & Lyne (1968)) assumes that MP and IP occur when the observer's line-of-sight cuts the wide hollow cone of radiation twice (ML77) at a distance of about 180 • of longitude ( Fig. C1 (Appendix C in the on-line materials)). In the other version of SP-IP model ( Fig. C2 (Appendix C in the on-line materials)) the line-of-sight stays in a pulsar beam for the entire pulsar period and MP and IP correspond to cuts through two nested conical beams or through the arrangement of the core beam surrounded by the cone (Gil (1983); hereafter G83, Gil (1985)). In the latter version the separation between MP and IP is naturally close to 180 • , and it does not depend on the observational frequency, while in the former version these properties are not natural. In addition to information about profile shape and/or widths one can usually use polarisation angle (PA) curves to distinguish between different kinds of interpulses. In case of DP-IP (almost orthogonal rotators) swings of PA are steep across both MP and IP components, while for SP-IP cases PA curve is typically flat over the entire profile (including MP, IP and weak emission bridge between them). Beam shape It is commonly assumed that the pulsar beam is circular in shape. Some authors however consider elliptical shapes with meridional compression or equatorial elongation. For example, Narayan & Vivekanand (1983) argued that the pulsar beam is elongated with the ratio of north-south (N-S) to east-west (E-W) dimension depending on pulsar period as R ≈ 1.8P −0.65 ,(16) so the effect is predominant at shorter periods P < 0.25 s. On the other hand, Biggs (1990) and McKinnon (1993) considered the opposite tendency of the beam compression in meridional direction, with the ratio of minor (N-S) to major (E-W) ellipse axes depending on the inclination angle α R ≈ cos α 3 cos 2 3 α ,(17) so this effect is predominant at large α close to π/2. Equations (1) and (12) are independent from the beam shape (if symmetry to the fiducial plane is conserved). The opening angle is always described by Eq. (12) or by its observational representations (Eq. (13),(14) or (15)). Taking into account ellipse geometry the following equation could be shown r = ρ 2 0 + β 2 1 R 2 − 1(18) and rR = R 2 ρ 2 0 + β 2 (1 − R 2 ).(19) Equation (19) describes circular beam when R = 1 and r = ρ. In this paper we argue that pulsar beam shape is circular or almost circular (a possible ellipticity can not be excluded but if it exists it is very small). Detection conditions For simplicity let us consider one hemisphere with 0 α π/2 (Fig. 1). Pulsar is detectable geometrically when the observer's l-o-s passes through its beam (independent of the actual beam shape). For circular or almost circular beam the detection condition is \|β\| < ρ0,(20) where ρ0 is the opening angle of the beam corresponding to the last open field lines. The above condition is valid for the main pulse emission. For interpulse emission within DP-IP model (almost orthogonal rotator α ∼ π/2) the detection condition is ρ0 > π − 2α − β.(21) In the SP-IP model, in which both MP and IP originate from single magnetic pole (almost aligned rotator α ∼ 0), we will consider two different versions of this model. In the first MP and IP represent two cuts through one conical beam (ML77) and the detection condition is ρ0 > 2α 2 + β 2 + 2αβ.(22) This version is presented in more details in Fig. C1. The second version corresponds to the case when the l-o-s remains inside the beam for the entire pulsar period, so the occurrence of both MP and IP result from the internal beam structure in the form of nested hollow-cones (G83). The detection condition for this case is ρ0 > 2α + β.(23) This version is presented in more details in Fig. C2. For elliptical pulsar beams the detection conditions expressed by equations (20), (21), (22) and (23) transform into equations (24), (25), (26) and (27), respectively, presented below: \|β\| < [R 2 ρ 2 0 + β 2 (1 − R 2 )] 1/2 ,(24)rR > π − 2α − β,(25)rR > 2α 2 + β 2 + 2αβ (26) rR > 2α + β.(27) In the case of SP-IP model, beside detection conditions (Eq. (22) or Eq. (26)) it is also important to apply some kind of morphological definition to avoid a danger of classifying just a broad double-peaked profiles as interpulses. By carefully reviewing our interpulse database we decided that the interpulse case has to have at least 100 • of longitude separation between the component peaks, that is 100 • < ∆ϕ IP −M P sep < 260 • .(28) Beam edge correction KGM04 used the detection condition ρ > \|β\| (Eq. (20)), where ρ was approximated by ρ10 obtained from observations (e.g. from models Eqs. (13) -(15)). Such an approach was justified only by the fact that it is difficult to measure pulse-width at the level lower than about 10% of the maximum intensity. It seems that this condition is too restrictive from the geometrical point of view, because the intensity level corresponding to the ρ10 beam radius could be much higher than the one that should be adopted as the edge of the beam. A simple model of the beam envelope is presented in Fig. 2. The two circles with different radii ρ0 and ρ10 correspond to two different intensity levels with respect to the maximum intensity level Imax in the centre of the beam, namely 10 % and low intensity level corresponding to the beam edge. It is important to notice that the maximum of the observed profile (upper panel) does not generally correspond to the maximum intensity of the beam and in reality depends on the impact angle β. Thus I10 does not necessarily correspond to the cut of the beam at 10% of its maximum (it is just 10% of the maximum of the observed profile). If we use ρ = ρ10 in detection conditions we can lose ∼ 10% pulsars with intensity less than 0.1 of the absolute maximum Figure 2. Schematic representation of the gaussian envelope of the beam intensity distribution. Two concentric circles centred on the magnetic axis m with radii ρ 0 and ρ 10 represent two intensity levels: edge of the beam and 10% of the maximum intensity in the beam centre, respectively. Two lines of sight trajectories l-o-s 1 and l-o-s 2 corresponding to different impact angles β 1 and β 2 are marked. One can notice that in the case of l-o-s 1 the detection independent of the actual values of ρ (ρ 0 or ρ 10 ) in the detection condition (Eq. (20)) is possible because β < ρ 10 < ρ 0 . However, in the case of l-o-s 2 the condition β < ρ 10 does now guarantee the detection, and new detection condition β < ρ 0 = 1.1ρ 10 should be introduced. of the beam. To improve this situation, let us consider Tables 1a and 1b in Gil & Kijak (1993). These tables are very useful since they present the pulse-width measurements at 10% and 1% maximum intensity of the profile for a group of pulsars. The latter one can be considered as being close to the level representing the edge of the beam. The values of the inclination angle α and the impact angle β are known, so using Eq. (12) we can calculate corresponding values of ρ10 and ρ1, which for available data give the mean value close to 1.1. If we now adopt ρ0 = ρ1 we can estimate the average value of the ratio of ρ0/ρ10 = 1.1, which we can use as the edge beam correction in detection conditions (Eqs. (13) -(15) and Eqs. (18) -(27)). DATABASES Periods In this paper we consider only the so-called normal radio pulsars. All magnetars, millisecond pulsars and pulsars in binary systems are excluded, since they constitute different evolutionary groups on the P −Ṗ diagram. At present there are 1830 known radio pulsars, with periods that range from 1.4 ms to 11.8 s. Following arguments expressed by KGM04 we consider only those with periods ranging from 20 ms to 8.5 s, both in our period database and in Monte Carlo simulations. The upper limit of 8.5 s is set by PSR J2144-3933 (Manchester et al. (1996 and Young et al. (1999)) with the longest period observed in radio wavelengths, whereas the lower limit we set at 20 ms. The estimates of initial pulsar period range from 14 ms (Migliazzo et al. 2002) to 140 ms (Kramer et al. 2003a). If the actual shortest period is indeed 14 ms instead of being close to 20 ms, then we miss only two normal pulsars, which is statistically irrelevant. All in all, our period database contains 1520 normal pulsars, 355 more than in the KGM04 database. The distribution of all selected periods is presented in the upper panel of Fig. 3. This sample can be best fitted by the log-normal function expressed by Eq. (11) with parameters x0 = −0.30 and σ = 0.80, although gamma function (Eq. (8)) with parameters m = 0.43 and a = 2.12 also gives a reasonable fit (see Table 2 for comparison). Pulse-widths In our simulations we attempt to reproduce the observed distribution of pulse-widths measured at 10% of the maximum intensity level. The observed pulse-width W depends on a number of geometrical (discussed in Section 2) and non-geometrical factors (like scattering, dispersion smearing, sampling or receiver time constant). The influence of the emission geometry (inclination angle α, impact angle β, opening angle ρ(P )) on the observed pulse-width W (Eq. (1)) is dominant, however some influence of non-geometrical factors cannot be neglected. Both GH96 and KGM04 used database that contained about 240 (242 and 238, respectively) pulse-widths that were carefully selected to avoid broadening effects. The trailing part of many profiles with large dispersion measure DM suffer from broadening caused by scattering of radio waves on free electrons in ISM. Our simulations do not take into account scattering, so our sample should be limited to pulsars whose profiles are not significantly broadened. The easiest selection method is to limit the values of dispersion measure but the question is the maximum permissible DM that warrants it? To answer this question we analysed the Fig. 7 in Cordes & Lazio (2003). We found that the limiting value of DM is about 150 pc cm −3 and we present our reasoning below. The typical duty cycle of normal pulsars is ∼ 5%, which for our shortest period (20 ms) gives a pulse window of about 1 ms. We assumed arbitrarily the value of 10 per cent (0.1 ms) of this value as the maximum possible broadening due to high DM . According to Fig. 7 in Cordes & Lazio (2003) this corresponds to dispersion measure value DM ∼ 150 pc cm −3 . Of course, the number of pulsars with period close to 20 ms in the whole sample is comparatively small, so probably the DM limit can be slightly higher. Increasing the maximum DM value to 170 pc cm −3 would extend the number of pulse-widths by less than 10 per cent, which seems to be irrelevant in our statistical analysis. However, for DM ∼ 200 . Formal fits of the log-normal (Eq. (11)) and the gamma (Eq. (8)), solid and dashed line respectively, distribution functions to the histogram of 1520 observed (upper panel) and simulated (lower panels) pulsar periods. The ordinate represents the number of observed (upper panel) or simulated (lower panels) pulsars. The upper panel shows also the distribution of interpulses according to the right-hand side scale, where blue and red colour corresponds to SP-IP and DP-IP cases, respectively, (as in Fig. 6)). Distribution of the simulated observed pulsar periods generated with the log-normal (middle panel) and the gamma (lower panel) parent period distribution functions with parameters x 0 = −0.30, σ = 0.80 and m = 0.43, a = 2.12, respectively. In both simulation cases the inclination angle distribution function was that of GH96 (Eq. (6)). The parameter N describes the number of observed and simulated periods. pc cm −3 the pulse broadening would be about 1 ms, which is evidently too much. Thus, we construct our new pulsewidth database using the value of 150 pc cm −3 as the limit for DM in our sample. The new database contains pulse-widths used by GH96 and KGM04, extended by pulse-widths from Swinburne In Rankin (1993a) and Rankin (1993b). The histogram of simulated inclination angles is presented in the lower panel. The simulations were performed using the log-normal period distribution (Eq. (11)) and the GH96 inclination angle distribution (Eq. (6)), as this combination gives the best results of the K-S significance test. had DM 150 pc cm −3 . Thus, our new pulse-width database contains 414 measurements, including 190 from GH96/KGM04 database. Their distribution is presented in the upper panel of Fig. 4. Inclination angles The inclination angle α between the magnetic and the spin pulsar axes is a parameter which cannot be observed but can 1998)). Thus, the pulse-width can be roughly scaled with the observational frequency ν as ν −0.1 . be derived indirectly. One method is based on the RVM and polarisation data (Lyne & Manchester (1988), R90, Rankin (1993a,b)). It is well know that this method is effective only in a small number of cases with very broad highly polarized profiles. Another, much more effective method was proposed by R90, in which α was derived from the width of the core component measured at the level of 50% maximum intensity (W core 50 ) α = sin −1 2 • .45P −1/2 /W core 50 .(29) In our simulation we used database of 149 inclination angles compiled by Rankin (1993a,b). Their distribution is presented in Fig. 5 (upper panel). Unfortunately, the inclination angle database is the only one which was not developed since Rankin's compilation. We would like to advertise here that the validity of the above Equation (29) will be examined statistically in the forthcoming Paper II. Interpulses As a result of discovery of large number of pulsars in recent multibeam surveys a number of cases with interpulse emission also increased from 14 in Taylor et al. (1993) catalogue to 27 in Weltevrede and Johnston (2008a; thereafter WJ08a) up to 44 presented in this paper. The new interpulse database was derived from the sample of 1520 normal pulsars. This database (Table 1) contains 31 doublepole interpulsars (DP-IP) and 13 single-pole interpulsars (SP-IP), which is 2.039% and 0.855% of the total number of normal pulsars, respectively 5 . A clear division into DP and SP interpulses was possible in most cases, based on the pulse profile morphology and/or polarization angle variations (see Section 2.2). It is important to note that the ratio of NSP −IP /NIP = 0.30 and NDP −IP /NIP = 0.70 is almost the same as in KGM04 (0.36 and 0.64). The total percentage ∼ 2.90% of interpulsars in the sample of normal pulsars is also unchanged. Thus, it seems that our new database is quite representative for interpulse occurrences in normal pulsars. To be consistent with our selection of pulsar periods we excluded all millisecond and other recycled pulsars from both the total pulsar sample and from the sample of pulsars with interpulses (this was not done in GH96, who incorrectly estimated rates of interpulse occurrences as a result of this mistake). All interpulsars (see Table 1) are marked on the P −Ṗ diagram presented in Fig. 6. As one can see from this figure DP-IP cases (represented by red dots) lie above 10 Myr line. This means that almost orthogonal rotators are rather young pulsars. On the other hand, SP-IP cases (blue dots) lie between ∼5 Myr and ∼500 Myr lines. Thus, the almost aligned rotators have a tendency to be older pulsars. In fact, the average values of P ,Ṗ , τ and B are 0.51 s, 3.49 × 10 −15 , 1.55 × 10 8 yr and 6.88 × 10 11 G for SP-IP cases and for DP-IP cases are 5 More precisely, the rates of occurrence of SP-IP and DP-IP are (0.86 ± 0.24)% and (2.04 ± 0.37)%, respectively. The errors were estimated 1σ = √ N /Ntot, which for SP-IP is √ 13/1520 = 0.24% and for DP-IP is √ 31/1520 = 0.37% (Ntot = 1520 is the number of normal pulsars from which the interpulses were extracted). Table 1. Table of 0.39 s, 2.86 × 10 −14 , 4.64 × 10 6 yr and 2.17 × 10 12 G. This strongly implies a secular alignment of the magnetic axis towards the spin axis, in agreement with the two humped distribution function of parent inclination angles expressed by Eq. (6). To some extent this is an observational support for results of simulation of the inclination angle α evolution made by WJ08a. These authors argued that the magnetic axis is likely to align from a random distribution at birth with a timescale of ∼ 10 7 years. It is not clear whether this conclusion is fully consistent with Eq. (6), although the alignment of the magnetic axis seems to be reproduced well in Fig. 6. MONTE CARLO SIMULATIONS OF CONAL EMISSION We performed Monte Carlo simulation of the pulse-widths W10, pulsar periods P and inclination angles α following a technique developed by KGM04, with a number of important improvements: 1. We used new, generally more numerous databases of pulsar periods (1520 as compared with 1165 in KGM04), pulse-widths (414 versus 238) and interpulses (44 versus 14). The inclination angle database was unchanged with respect to KGM04. 2. The pulse-widths measurements were selected with additional criterion that DM 150 pc cm −3 . This allows avoiding a significant pulse broadening contaminating pulsewidth measurements at the low intensity level. 3. The log-normal period distribution proposed by Lorimer et al. (2006) was added to a set of possible trial functions. 4. Beam edge correction in detection conditions was introduced in the form ρ0 = 1.1ρ10 where ρ10 is expressed by Eqs. (13) -(15). For more details see Section 2.5. 5. The separation between MP and IP within the ML77 version of SP-IP model is computed exactly from the detection geometry. As one can see from Tab. 1, the minimum (maximum) separation between MP and IP is about 100 • (260 • ). On the other hand, we have the empirical relation for component peaks ρs = 3 • .7 P −1/2 forP > 0.7s 4 • .6 P −1/2 forP 0.7s(30) found by GKS93 (see their Fig. 3), which can be used in Monte Carlo simulations. While using the wide hollow cone version of the SP-IP model (ML77) one can have a problem judging whether the pulse represents just a broad profile or it is already an interpulse case. Thus, beside satisfying the detection condition for the SP-IP (Eq. (22) or Eq. (26)) the simulated pulse-width W (ρs) should be larger than 100 • (smaller than 260 • ). The first test for our software was to reproduce results from KGM04 (using their databases). Results of this test are presented in Table B1 (Appendix B in the on-line materials). After making sure that our programs work correctly, we started new Monte Carlo simulations. They involve a number of subsequent steps described below: 1. Generate the inclination angle α as a random number with one of the trial parent probability density function f (α) corresponding to Eqs. (3) -(7). 2. Generate the pulsar period of P as a random number with the one of the trial parent probability density function f (P ) corresponding to Eqs. (8) -(11). 3. Generate the observer angle ξ as random number with the parent probability density function f (ξ) = sin ξ. 4. Calculate the impact angle β = ξ − α. 5. Calculate the beam opening angle ρ10(P ) at the level of 10% of the maximum intensity with one of the trial distribution function (Eqs. (13) -(15)). 6. Calculate the opening angle ρ0 corresponding to the edge of the beam ρ0 = 1.1ρ10. 7. Check the detection conditions for each simulated object (Eqs. (20) -(23) for circular and Eqs. (24) -(27) for elliptical beam). If the pulsar is detected then it is added the to total number of detected pulsars N det . 8. Calculate the pulse-width W10 according to Eq. (1) for each set of parameters ρ10(P ), α and β of the detected pulsar. 9. For each pulsar detected in a simulation run check the detection conditions for occurrence of an interpulse. For DP-IP model Eq. 10. Record all relevant information like inclination angle α, period P , pulse-width W10 and occurrence of the interpulses from one or two magnetic poles. 11. For each set of the probability density function of the inclination angle α, period P and the model of the opening angle ρ10, 50000 objects were created (each having a set of parameters such as: P , α, β, ρ and W ). After checking the detection conditions we obtained database of the observed simulated pulsars and distribution of their parameters α, P and W10 as well as occurrence of the interpulses from one NSP −IP or two NDP −IP magnetic poles. In order to obtain the statistical significance each simulation run was repeated 10 times and the results were averaged. 12. Judge the statistical significance using the Kolmogorov-Smirnov (K-S) test for the simulated observed pulsars parameters as α, P and W10 and observed distribution of these parameters. Record the values of D(α), P(α), D(P ), P(P ), D(W10) and P(W10), where D describes the maximum distance between both cumulative distribution functions and P is the probability that the two compared distributions are drawn from the same parent distribution. 13. Calculate the beaming fraction defined as f b = N det /Ntot (where N det is the number of detected pulsars in total population of Ntot simulated pulsars) and the occurrence of interpulses from one magnetic pole NSP −IP /N det or two magnetic poles NDP −IP /N det , respectively. 14. Change values of free parameters of a given trial period distribution function. 15. Repeat simulations with all possible combinations of distribution probability functions of α, P and ρ10. Table 1). Black dots represent 1476 normal pulsars, while red and blue circles correspond to 31 double-pole (DP-IP) and 13 single-pole (SP-IP) interpulsars, respectively. Lines of constant magnetic field B, characteristic age τ , spin-down luminosityĖ are shown. The death line (10 29 erg s −1 ) derived by Contopoulos & Spitkovsky (2006) is marked by the thick line. It is easy to notice that SP-IP cases are generally located much closer to the death line than DP-IP cases. 16. Search for good solutions satisfying simultaneously all demanded criteria, that is P(α) 0.5%, P(P ) 0.5%, P(W10) 0.5% as well as the rates of occurrence of SP-IP and DP-IP are at the observed level (0.86 ± 0.24)% and (2.04 ± 0.37)% (errors correspond to 1σ level). We performed Monte Carlo simulation of conal emission of radio pulsars according to procedure described in items 1-16 listed above. We checked 160 (80 for circular and 80 for elliptical beams) different combinations of trial functions for parent distribution periods P , inclination angles α and opening angles ρ10. For each combination of the above distribution functions we searched the two-dimensional grid of free parameters of the period distribution function. It gave the total number of 641 360 single simulation runs. In all solutions we identified "detectable" cases in which all prob-abilities (of P , α and W10) were greater than 0.005 (0.5%) and occurrence of interpulses were at the observed levels (2.04 ± 0.37)% for DP-IP and (0.86 ± 0.24)% for SP-IP. Such solutions we called "a good solution". In summary, each simulation run including 50000 detection attempts resulted in about 5500 detections (satisfying geometrical detection conditions (Section 2.4)). There were 641 360 simulation runs including all possible combinations of the probability distribution functions (Section 2.1). Among the resulting ∼ 4 × 10 9 geometrical detections we found 827 good solutions (as described in item 16 above). These solutions are listed in Tables B2 -B7 and 9 best examples with relatively high K-S probability values are shown in Table 2 (items 1a -9a). Items 1b -9b correspond to tests of the luminosity problem described in the Appendix A in the on-line materials. Table 2. Examples of 9 best simulation results for different distribution functions (Eqs. (6) -(15)) and circular beams. D and P are parameters of the Kolmogorov-Smirnov significance test. For symbol description see item 12 in Section 4. Lines denoted by letters a and b correspond to simulations without and with the luminosity problem included, respectively. Notice that for combination (6), (8) and (14) of distribution functions there is only one good solution that could be presented (see Fig. C6). No. Distributions f (P ) W 10 P α DP-IP SP-IP f b Equation numbers D P D P D P [%] [%] x 0 σ 1a (6) DISCUSSION Our new statistical analysis is based on a number of new databases that we compiled from the recently published data. We followed methodology developed by KGM04, which relies on comparison of synthetic and real pulsar data. Our period P database contains 1520 items (we excluded all recycled and binary pulsars) in the range of 0.02 -8.51 seconds. This database includes 355 more pulsars than recently analysed database compiled by KGM04. We also compiled a new database of pulse-widths W10 measured at 10% intensity level. This database contains 414 items, which is 176 more than that of KGM04. There are many more pulsewidth measurements available these days, but our database is restricted to pulsars with DM < 150 pc cm −3 , which guarantees avoiding significant external pulse broadening. Most importantly, we created the largest ever database of interpulse occurrence in pulsar emission. Our database contains 44 pulsars (compared with 14 pulsars in Taylor et al. (1993)/KGM04 database), including 31 cases of DP-IP and 13 cases of SP-IP. Although our IP database is more numerous than any of the previous ones (e.g. KGM04, WJ08a), it seems that the frequencies of occurrence are similar to those occurring in previous smaller databases. In fact, we have 2.90% of total number of IP cases, divided into 2.04% DP-IP and 0.86% SP-IP cases, respectively, in a population of 1520 pulsars. This can be compared with 2.71%, 1.94% and 0.78%, respectively, found in KGM04 database. One can therefore firmly state that there should be about 3% of IP cases in the population of normal pulsars, including about 2% and 1% of DP-IP and SP-IP cases, respectively. All pulsars from the period P database for which the value ofṖ is known are presented on the P −Ṗ diagram in Fig. 6. Black dots represent 1476 normal pulsars (without IP emission) while red and blue dots correspond to 31 DP-IP and 13 SP-IP cases, respectively. It is easy to notice that SP-IP cases are much older than DP-IP cases (with mean characteristic age 155 Myr versus 4.6 Myr, respectively). Also SP-IP cases represent weaker magnetic fields than DP-IP cases (6.9×10 11 versus 2.2×10 12 G). Assuming reasonable that DP-IP and SP-IP cases represent almost orthogonal (α close to 90 • ) and almost aligned (α close to 0 • ) rotators, respectively, one can conclude that the distribution of IP cases on the P −Ṗ diagram presented in Fig. 6 reveals a secular alignment of the magnetic axis towards the spin axis with a random initial value of the inclination angle. (11)) for periods, GH96 Eq. (6) for inclination angles and Eq. (13) for opening angles. Both axes describe free parameters of the period distribution log-normal function: parameter x 0 on the horizontal axis and the parameter σ on the vertical axis (these parameters are responsible for the location of the maximum and the width of the log-normal function, respectively). Below the main window the narrow bar with the Kolmogorov-Smirnov probabilities P(W ) for the pulse-width W 10 coded in shades of grey is presented (the highest value ∼ 0.95 corresponds to K-S probability obtained in this case). On the right hand the scales for K-S probabilities for inclination angles P(α) and for periods P(P ) are presented by different colours. Red rectangles in the solution area correspond to solutions simultaneously satisfying all constraints (K-S probabilities P > 0.005 for periods, inclination angles and pulse-widths as well as the occurrence of interpulses at the observed levels, i.e. (0.86 ± 0.24)% for SP-IP and (2.04 ± 0.37)% for DP-IP). Thus all acceptable solutions are represented by red rectangles and their distribution describes errors of the log-normal functions x 0 = −0.30 +0.13 −0.09 and σ = 0.80 +0.16 −0.11 (see also Tab.B2). The SP-IP cases are very interesting. In our simulations we used two models of SP-IP emission. In the classical model of ML77 (Fig. C1) the MP and the IP components result from two cuts of a very wide hollow emission cone. In this model the separation between MP and IP should be frequency dependent (following a spread of dipolar field lines), and its value being close to 180 • should be considered accidental (Weltevrede, Wright & Stappers 2007). An alternative SP-IP model was proposed by G83 (Fig. C2). This model is based on the assumption of the double conal structure of the pulsar beam. The line-ofsight of the nearly aligned rotator cuts one cone for the MP and the other one for the IP emission (see Fig. 8 in Kloumann & Rankin (2010) for schematic diagram for PSR B1944+17). This model naturally predicts two important properties: frequency independence of MP -IP separation (equal to 180 • ) and existence of the bridge of emission between MP and IP. Interestingly, in our Monte Carlo simula- Table 2. tions we detected more cases corresponding to ML77 model than to G83 model. Therefore, if these simulations are correct, most of the 14 SP-IP cases (about 1% of a total population) should show tendency to frequency dependent separation between MP and IP. Unfortunately, we are not able to verify this conclusion, as most of the available data correspond to a single frequency. However, this should be performed in some suitable project in the future. A convenient method of graphical representation of each set of trial distribution functions is "a colour map of solutions", example of which is presented in Fig. 7. More plots of this kind are presented in Appendix C. Colour contours represent levels of conformity of the observed and the simulated distributions. The "green" set of colours is used for inclination angles and the "blue" one is used for periods. Their legends are presented on the right-hand side of Fig. 7, which describe the corresponding probability levels P(α) and P(P ). The grey bar below the plot represents conformity of the observed and the synthetic distributions for the pulsewidth W10 in the form of K-S probability P(W ). The white colour represents zero probability and the black corresponds to the maximum probability occurring in a given plot. Axes of each plot represent the free parameters of period distribution function (i.e. m and a for the gamma function, x0 and σ for the log-normal function). The space for good solutions is restricted to the areas where all three probabilities P simultaneously exceed 0.5%. That is why in Fig. 7 and Fig. C3 -C6 all good solutions must lay in darkest blue contour. Moreover, the occurrence of interpulsars in this region must be on the observed level (specified at the end of the previous paragraph), which is marked by red rectangles in Fig. 7. In fact, the good solutions are represented by red rectangles because they are marked only if all required probabilities occur simultaneously. The numerical details of all good solutions are presented in Tables B2 -B6. Table 3. Number of good solutions for different combination of distribution functions described by Equations: (6) -inclination angle, (11) -log-normal period distribution, (8) -gamma period distribution; (14), (13) and (15b) 1. Solutions satisfying all criteria (P > 0.5% for all variables P , α and W10 as well as the definition and the occurrence of interpulses at the observed levels) were obtained mainly for the log-normal period distribution function (Eq. (11)) with parameters x0 = −0.30 +0.13 −0.09 and σ = 0.80 +0.16 −0.11 (see Fig. 7). Six best examples corresponding to this function are presented in Table 2 (items 1 -6). Also the gamma distribution period function (Eq. (8)) with parameters m = 0.45 +0.07 −0.08 and a = 2.04 +0.34 −0.16 (see Fig. C5) seems to be quite good, however the corresponding K-S probabilities are lower than in the log-normal distribution period function case. Four best examples are presented in Table 2 (items 7 -9). None of the period trial functions (Eqs. (8) -(11)) is perfectly suited to reproduce the observed distributions of our observables. However, the gamma and the log-normal functions are by far the best, with the latter being slightly better, most likely because it reproduces the tail of long periods (see middle panel in Fig. 3) very well. 2. Both the rotation axis and the observer's direction are randomly distributed in space (see Eq. (2)). 3. The only trial distribution function of the inclination angle that satisfies all constrains is the complicated trigonometric function of GH96 (represented by Eq. (6)). This function has two local maxima, one near 0 • (almost aligned rotator) and the other near 90 • (almost orthogonal rotator). We do not find support for the modified cosine function of ZJM03 (Eq. (7)), which was derived without taking into account a problem of frequency of occurrence of the IP emission. It is clear that non of the functions represented by Eqs. (3) -(7) reproduces the observed distribution of simulated variables P , W and α as well as the frequency of IP occurrence. However, the function of GH96 represented by Eq. (6) suits the best and is recommended as the model function for the parent distribution of the inclination angles. 4. As a result of simulations performed with suitable parent distribution functions of periods and inclination angles we obtained good solutions for most of the trial model functions for the opening angle. It is not possible to discriminate the opening angle functions using the statistical tools available to us. This is illustrated in Table 3. 5. The average beaming fraction f b = N det /Ntot ≈0.1, where Ntot = 50000 is the total number of simulated pulsar candidates and N det is the number of detected pulsars in our simulations. This value is close to the one obtained by TM98 and ZJM03, but lower than those obtained by GH96 and KGM04. One should realise that this factor corresponds to averaging over many pulsar parameters, the most important being pulsar period (for more details see the end of this Section). Moreover, since the pulsar luminosity is not taken into account (some distant pulsars will be too weak to be detected) this value of the beaming fraction should be considered as an upper limit. Once the luminosity problem is considered (see Appendix A for some details) the value of beaming fraction largely decreases, perhaps even to the value as low as 0.02. As already mentioned this paper is a continuation of our previous attempt to resolve the pulse-width statistics based on the modeling of pulsar geometry (KGM04). Although we improved the numbers and the quality of the observational databases as well as methods of data processing, we are still lacking an analysis of the possible effect of the intrinsic luminosity of radio pulsars on our results. This problem is, however, very difficult and complicated and we will discuss it only superficially, postponing a full treatment to a future paper (see also Appendix A). The proper approach would be to compare the synthetic radio luminosity with the minimum detectable flux achieved in a given pulsar survey, and thus it can be applied only to uniform data sets of pulsars detected in single survey. Our data do not have such a degree of uniformity. However, most surveys were less sensitive to long-period pulsars, as it stems from the nature of the applied Fourier-transform method. Since the interpulse emission (which appears to be the most restrictive constrain in our analysis) occurs mainly at shorter periods (see upper panel in Fig. 3), a possible under-representation of pulsars with longer periods should not significantly affect our general results. We assumed that the intrinsic pulsar luminosity does not depend significantly on the inclination angle (geometry) but can depend on P andṖ (pulsar evolution). We performed simple test to check possible influence of pulsar geometry and evolution. The result of this test is presented and discussed in Appendix A. However, the main results of this paper were obtained using only geometrical detection conditions, that is every pulsar in the field of view of our hypothetical radiotelescope was detected. It means that each detectable pulsar would be close and bright enough so its radio energy flux would exceed the detection threshold of our hypothetical radio pulsar search campaign. As we argue below (see also Appendix A), omitting the luminosity problem does not affect correctness of our results, at least significantly. However, taking it under consideration would vastly complicate the research programme (especially its computational part) and introduce additional uncertainties decreasing reliability of our conclusions. Also, it is worth emphasizing once more that our statistical studies were in principle limited to geometrical features (inclination angles, pulse-width and structure of the beam), without drawing conclusions depending on the luminosity problem, like the initial periods or the neutron stars birth rates. The absolute luminosity of pulsar can be written in general form as Lr = f (P,Ṗ ) = AP α 1Ṗ α 2 , where A, α1 and α2 are free parameters of a model (e.g. Arzoumanian, Chernoff & Cordes, 2002; ACC02 hereafter). Some authors even argue that for given P andṖ the luminosity is constant i.e. pulsars can be considered as standard candles. Without a detailed discussion we would like to emphasise that each particular approach depends on an adopted model and leads to some problems. For example, from the analysis of observational data ACC02 deduced that luminosity function is Lr = P −1.3Ṗ 0.4 /10 −15 10 29.3 erg s −1 . This could be considered as an empirical function, but one should realise that the authors used some controversial assumptions about the shape and size of the beam to obtain it, which are not really true in our opinion. However, more important are problems reported by other authors (Gonthier et al. 2004), who tried to use this function in their statistical studies. They noticed that this function gives too many bright pulsars as compared with observations and too high the NS birth rate. For example, Gonthier et al. (2004) used the above function in simulations of γ-ray pulsars detected by EGRET (Thompson 2008) but they had to reduce the amplitude A = 10 29.3 by factor of 60 (leaving values of α1 and α2 unchanged for unknown reasons). Anyway, it is important to realize that a form of the function of absolute pulsar luminosity is not known. The flux density of a pulsar with luminosity L erg s −1 and distance d is S = (f b /4π)L [mJy kpc 2 ], where f b is the beaming fraction describing a part of a full sphere illuminated by a pulsar beam. This value of S, calculated with high uncertainty, should be compared with the sensitivity of a radiotelescope expressed as the so-called minimum detectable flux densitySmin. Detection of pulsar is possible if S > Smin. We used data from pulsars discovered in more than a dozen surveys, with Smin changing from one campaign to another. It is impossible to find a universal, theoretical value of Smin corresponding to all pulsars in our sample, an thus it is impossible to find a consistent detection criterion based only on a distance to the pulsar and its luminosity. Apart from uncertainties introduced by this method, the parameter space would be extended enormously by many dimensions such as: distance, direction in the space to the pulsar, inhomogeneities in ISM, luminosity function and beaming fraction. Thus, computationally the problem greatly complicates and the expected benefits are not that high to justify these efforts. Therefore, we used the assumption that each pulsar detected in our "geometrical search" simulations corresponds to a flux density S greater than a hypothetical Smin, the value of which we do not specify. Instead, we use geometrical criterion ρ0 > β where ρ = 1.1ρ10 and ρ10 is calculated from empirical formulae (Eqs. (13) -(15)). In Appendix A we present a simple test to check how the luminosity problem can affect a validity of our conclusions. We do not intend to find any new model of the intrinsic pulsar luminosity. This is beyond the scope of our paper. Instead, we use a very convenient existing luminosity probability density function presented by RL10, which is best suited for our non-uniform (with respect to sensitivity of different pulsar search campaigns) database. The results of this check are presented in Table 2 (lines 1b -9b in comparison with lines 1a -9a) and shortly explained below. We found that most of our conclusions were not significantly changed but the number of pulsar detections dropped by a factor of several (see Appendix A). As a result the actual (but still period averaged) beaming fraction f b dropped to a value about 2% (compared with 10% obtained without luminosity). Therefore, we conclude that the actual beaming fraction f b (describing a part of the full sphere illuminated by an average pulsar beam) can be significantly lower than 10 %, perhaps even as low as about 2% (see Table 2). This is by a factor of few to several smaller than estimates of the beaming fraction value that can be found in the literature (0.17 -Gunn & Ostriker (1970), 0.16 -GH96, 0.14 -KGM04, 0.12 -ZJM03, 0.1 -TM98, 0.084 -WJ08a). This would suggest that the number of the neutron stars in the Galaxy is much larger than currently estimated. Recent discoveries of Rotating Radio Transients (McLaughlin et al. 2006) seem to support such a point of view. Indeed, it seems that there are 2 -3 times more RRATs than pulsars in the Galaxy (Keane 2010). RRATs seem to be just a normal radio pulsars in which we detect only strongest subpulses. More than 30 RRATs have been detected so far (Keane et al. 2010), thus an interesting question arises when one should expect first interpulse case in those objects. Our paper shows that there are about 3% of interpulses in normal pulsar population. Thus, among 30 RRATs one should expect one IP case. However, the situation is more complicated than that since in many cases the interpulse is much weaker than the main pulse. From careful analysis of all our interpulse cases we can predict that the first interpulse RRAT will occur when the population of these objects will rise to about 100. Finally, we would like to comment on the new interpulse case detected in PSR J2007+2722 using Einstein@Home global computing technique (Knispel et al. 2010). This is a 24 ms isolated pulsar with magnetic axis almost aligned to the spin axis. The mean profile covers full 360 • pulse window, with two equal amplitude components separated by about 180 • (see Fig. 1 in Knispel et al. (2010)). Thus, this case seems to be an ideal example of SP-IP object. However, the authors of the discovery paper argue that this is likely a disrupted recycled pulsar. For this reason we do not include it to our interpulse database. The global computing technique should result in more pulsar discoveries in the near future and we hope that some of them will contain the interpulse emission. CONCLUSIONS The main results and conclusions of this paper can be summarized as follows: 1. We compiled the largest ever database of 44 pulsars with interpulses, in which we identified 31 (about 2%) double-pole cases and 13 (about 1%) single-pole cases. 2. We found a strong evidence that the magnetic axis aligns with time towards the rotation axis from the originally random orientation. 3. The parent period distribution density function is most likely the log-normal distribution, although the gamma distribution cannot be excluded. 4. The most suitable model distribution function for the inclination angle is the complicated trigonometric function which has two local maxima, one near 0 • and the other near 90 • . The former and the latter implies the right rates of IP occurrence, single-pole (almost aligned rotator) and double-pole (almost orthogonal rotator), respectively. 5. The average pulsar beam has an almost perfect circular cross-section. 6. The upper limit for period averaged beaming fraction describing a fraction of the full sphere covered by pulsar beam is about 10%. To test how the luminosity problem can affect the validity of results and conclusions obtained in our paper we followed the results of Faucher-Giguere & Kaspi (2006), recently presented by Ridley & Lorimer (2010; hereafter RL10). As shown in this paper the radio luminosity can be calculated from logL = logL0 − logP + 0.5 log(Ṗ /10 −15 ) + δL,(A1) where L0 is 0.18 mJy kpc 2 and δL is randomly chosen from a normal distribution with σ δ L = 0.8. Thus the luminosity L can be calculated when the values of P andṖ are known. We generated pulsar period P as a random deviate with a parent probability density function f (P ) corresponding to Eqs. (8) -(11) in Section 2.1. The value of period derivative can be calculated from either Eqs. (9) and (10) in RL10 or by random generation with a distribution in agreement with the observed P −Ṗ diagram (Fig. 6). We used both methods and made sure that they lead to similar simulation results. For example, we found that L depends very weakly on the inclination angle α which enters into Eq. (9) in RL10. Once the luminosity L(P,Ṗ ) is calculated we can obtain the probability density function Fig. A1. Now, we can extend our Monte Carlo simulation procedure by adding to the list of 16 steps listed in Section 4 the detection condition based on the luminosity described above. The procedure was as follows. First, to simplify calculations the function was normalised to unity at the luminosity value equal to 0.1 mJy kpc 2 (see Fig. A1). This gave the proportionality constant C = 0.1 19/15 = 0.05 in Eq. (A2). For a generated P anḋ P the values of L and f (L) were calculated according to Eqs. (A1) and (A2), respectively. The pulsar was counted as detected when the value of f (L) lied below the thick line presented in Fig.A1 (and outside the shadowed area). The results of the luminosity test are presented in Table 2, where we compare good solutions with (lines 1b -9b) and without (lines 1a -9a) luminosity included, respectively. As one can see the only parameter that changed drastically is the beaming fraction f b , which dropped by a factor of about 8. This is a consequence of much smaller number of detections. However, the interpulse detection at the requested levels still determines the statistical structure of detected pulsar population. This is illustrated in Fig. 8, as compared with Fig. 7. In this Appendix we attempted to estimate an influence of the pulsar luminosity problem on the geometrical conclusion derived in our paper. We used a luminosity model represented by Eq. (A2) derived by Faucher-Giguere & Kaspi (2006) and presented by RL10. If this equation describes the luminosity density probability function for pulsars in our Galaxy, then the beaming fraction can be as low as about 2%. The advantage of this rather crude model is a convenience in use and independence of details of different pulsar search campaigns. Anyway, at this point we can firmly conclude that the average pulsar beaming fraction is a number between 0.02 and 0.10 (2% -10%). Table B2: Simulation results of the pulsar beam for opening angle distribution Eq. (13), inclination angle distribution function Eq. (6) and period distribution function Eq. (11). Graphical representation of this class of solutions is represented in Fig. 7. Table B3: Simulation results of the pulsar beam for opening angle distribution Eq. (14), inclination angle distribution function Eq. (6) and period distribution function Eq. (11). Graphical representation of this class of solutions is represented in Fig. C3. Table B4: Simulation results of the pulsar beam for opening angle distribution Eq. (15b), inclination angle distribution function Eq. (6) and period distribution function Eq. (11). Graphical representation of this class of solutions is represented in Fig. C4. Figure C1. Model of the single-pole interpulse -ML77 version. Two cones of average emission corresponding to frequencies ν 1 (blue) and ν 2 > ν 1 (red) with an opening angle ρ(ν 1 ) > ρ(ν 2 ) are marked schematically. For a given α and β angles the l-o-s (marked by the black circle) cuts the radiation cones twice. In this figure it is assumed that at frequency ν 1 MP and IP are separated about W 1 = 180 • of longitude. However with increasing frequency the separation W will be different (W 2 > W 1 for the presented geometry). This frequency dependence of MP-IP separation is the main difference between ML77 and G83 (presented in Fig. C2) versions of SP-IP models. Figure C2. Model of the single-pole interpulse for two nested hollow cones (or inner core surrounded by the cone) -G83 version. In this case the mean pulsar beam consist of core and cone or two nested, coaxial cones (blue circles for frequency ν 1 and red circles of frequency ν 2 > ν 1 ). For a given α and β angles the l-o-s (black circle) cuts through both cones. The inner and the outer cone is responsible for the occurrence of the MP and the IP, respectively. For different (higher) observational frequency ν 2 (red circles) the opening angle ρ(ν 1 ) > ρ(ν 2 ). However, this version of the SP-IP model the MP-IP separation W is frequency independent and equal to 180 • of longitude. APPENDIX B: TABLES No. f (P ) W 10 P α DP-IP SP-IP f b x 0 σ D P D P D P [%] [%No. f (P ) W 10 P α DP-IP SP-IP f b x 0 σ D P D P D P [%] [%No. f (P ) W 10 P α DP-IP SP-IP f b x 0 σ D P D P D P [%] [% APPENDIX C: FIGURES Figure C3. As in Fig. 7 but for opening angles described by Eq. (14). All acceptable solutions are represented by red rectangles and their distribution describes errors of the log-normal functions x 0 = −0.18 +0.16 −0.11 and σ = 0.81 +0.13 −0.12 (see also Tab.B3). . Figure C7. As in Fig. 7 but for period distribution function Eq. (7) and opening angles described by Eq. (13). Notice that no interpulses were found. Figure C8. As in Fig. C7 but for period distribution function Eq. (9).Notice that no interpulses were found. Figure C9. As in Fig. C7 but for opening angles described by Eq. (15b). Notice that no interpulses were found. This paper has been typeset from a T E X/ L A T E X file prepared by the author. Figure 3 3Figure 3. Formal fits of the log-normal (Eq. (11)) and the gamma (Eq. (8)), solid and dashed line respectively, distribution functions to the histogram of 1520 observed (upper panel) and simulated (lower panels) pulsar periods. The ordinate represents the number of observed (upper panel) or simulated (lower panels) pulsars. The upper panel shows also the distribution of interpulses according to the right-hand side scale, where blue and red colour corresponds to SP-IP and DP-IP cases, respectively, (as in Fig. 6)). Distribution of the simulated observed pulsar periods generated with the log-normal (middle panel) and the gamma (lower panel) parent period distribution functions with parameters x 0 = −0.30, σ = 0.80 and m = 0.43, a = 2.12, respectively. In both simulation cases the inclination angle distribution function was that of GH96 (Eq. (6)). The parameter N describes the number of observed and simulated periods. termediate -Latitude Pulsar Survey (Edwards et al. 2001), Parkes Southern Pulsar Survey I -III (Manchester et al. (1996), Lyne et al. (1988), D'Amico et al. (1998)), Parkes Figure 4 . 4Distribution of 414 pulse-widths W 10 measurements for pulsars with DM 150 pc cm −3 (upper panel). The distribution of simulated values is presented in the lower panel. The simulations were performed using the log-normal period distribution (Eq. (11)) and the GH96 inclination angle distribution (Eq. (6)), as this combination gives the best results of the K-S significance test. The parameter N describes the number of observed and simulated periods.Multibeam Pulsar Survey I -VI (Manchester et al. (2001), Morris et al. (2002), Kramer et al. (2003b), Hobbs et al. (2004), Faulkner et al. (2004), Lorimer et al. (2006) and Keith et al. (2009)). Parkes Southern Pulsar Survey I and II campaigns were carried out at 436 MHz and others on 1.4 GHz, so we had to scale their pulse-widths to 1.4 GHz 4 . After rejecting of all magnetars, milliseconds and binary pulsars 768 normal pulsars remained, among which 4 Scaling factor is 0.89 (= 1.4 GHz/0.436 GHz) −0.1 . This factor was obtained in the following way: the pulse-width is roughly proportional to the angular radius of the beam ρ, which is proportional to square root of the emission altitude r 1/2 em , which in turn depends on observational frequency ν (rem = (400 ± 80) km ν −0. Figure 5 . 5The upper panel shows the distribution of 149 inclination angles α derived from the pulsar polarisation characteristics (data taken from 44 known interpulses divided into double-pole (DP-IP) and single-pole (SP-IP) cases. Bibliographic marks are following: T/K -KGM04 using Taylor et al. (1993), WJ -Weltevrede & Johnston (2008b), A -D'Amico et al. (1998), M02 -Morris et al. (2002), K03 -Kramer et al. (2003b), H -Hobbs et al. (2004), L -Lorimer et al. (2006), R -Ribeiro (2008) † , M01 -Manchester et al. (2001), J -Janssen et al. (2009), K09 -Keith et al. (2009), K10 -Keith et al. (2010), C -Camilo et al. (2009), N -new interpulse identified in this work using data from H and L. (21) for circular beam or Eq. (25) for elliptical beam were used. For SP-IP model in ML77 version Eq. (22) for circular or Eq. (26) for elliptical beam were used while for G83 version Eq. (23) for circular or Eq. (27) for elliptical beam were used. Moreover, for the ML77 version of the SP-IP model, components separation (Eq. (30)) is checked whether it is in the range of 100 • -260 • . If an interpulse is detected then it is added to the number of interpulses NDP −IP or NSP −IP . Figure 6 . 6Diagram P −Ṗ for normal (see Section 3.1) pulsars, including 44 cases with interpulse emission (see Similar conclusion was reached by Weltevrede & Johnston (2008a) and most recently by Young et al. (2010), although using different statistical arguments. It is interesting to note that Young et al. (2010) reached their conclusion without using interpulse pulsars at all, while these objects were crucial for our analysis. Moreover, Young et al. (2010) argued that the best suitable solution for their simulation is Model II (see their Fig. 9) whereas our simulations seem to prefer their Model III. Figure 7 . 7Colour graphic representation of simulation of the best solution of the Monte Carlo simulations corresponding to the following set of trial probability density function: log-normal (Eq. Figure 8 . 8Same as inFig. 7but with luminosity problem included in simulations (Appendix A). This figure corresponds to the items 1b and 2b, whileFig. 7corresponds to the items 1a and 2a in ∈ [0 mJy kpc 2 , 0.1 mJy kpc 2 ) L −19/15 L ∈ [0.1 mJy kpc 2 , 2.0 mJy kpc 2 ) L −2 L ∈ [2.0 mJy kpc 2 , ∞ mJy kpc 2 ), (A2) (Eq.(17) in Faucher-Giguere & Kaspi (2006) and Eq.(15) in RL10). This function is illustrated in Figure A1 . A1The luminosity probability density function described by Eq. (A2). No pulsar detections are expected in shadowed rectangular area. Figure C4 . C4As in Fig. 7 but for opening angles described by Eq. (15b). All acceptable solutions are represented by red rectangles and their distribution describes errors of the log-normal functions x 0 = −0.29 +0.13 −0.09 and σ = 0.79 +0.13 −0.10 (see also Tab.B4). Figure C5 . C5As in Fig. 7 but for period distribution function Eq. (8) and opening angles described by Eq. (13). All acceptable solutions are represented by red rectangles and their distribution describes errors of the gamma functions m = 0.45 +0.07 −0.08 and a = 2.04 +0.34 −0.16 (see also Tab.B5). Figure C6 . C6As in Fig. C5 but for opening angles described by Eq. (14). The only one acceptable solution is represented by red rectangle with the gamma function parameters m = 0.38 and a = 2.51 (see also Tab.B6). -opening angle distribution functions. Careful studies of all contour plots (see examples in Fig. 7 and Fig. C3 -C9) and corresponding numerical solutions (Tables B2 -B6) lead to the following conclusions:No. Distributions No. of solutions 1 (6) (11) (13) 326 2 (6) (11) (14) 170 3 (6) (11) (15b) 183 4 (6) (8) (13) 147 5 (6) (8) (14) 1 Table B1 : B1Reproduction of the results from KGM04Original results from KGM04 Reproduction of KGM04 results in this work m = 0.35 a = 2.51 Parameter Average σ % Average σ % NOP 6385.3501 70.9845 6399.0098 66.5424 DP-IP 143.2600 11.8651 2.244 141.2800 11.1410 2.208 SP-IP 41.5600 6.7962 0.651 41.7100 6.4075 0.652 D(W 10 ) 0.1270 0.0066 0.1280 0.0060 P(W 10 ) 0.0016 0.0013 0.0012 0.0009 D(P ) 0.0600 0.0067 0.0581 0.0063 P(P ) 0.0038 0.0052 0.0053 0.0077 D(α) 0.1009 0.0055 0.1001 0.0055 P(α) 0.1075 0.0344 0.1061 0.0362 b f 12.771 12.798 m = 0.35 a = 2.53 NOP 6352.7300 71.2812 6364.2798 72.6219 DP-IP 141.3400 11.2045 2.225 136.4900 10.5605 2.145 SP-IP 41.0500 5.5766 0.646 42.6500 6.9969 0.670 D(W 10 ) 0.1217 0.0066 0.1267 0.0068 P(W 10 ) 0.0029 0.0025 0.0015 0.0013 D(P ) 0.0685 0.0063 0.0626 0.0070 P(P ) 0.0006 0.0015 0.0023 0.0034 D(α) 0.1013 0.0058 0.0996 0.0059 P(α) 0.1056 0.0369 0.1095 0.0373 b f 12.705 12.729 On the pulse-width statistics in radio pulsars.I. Importance of the interpulse emission 21] 1 -0.39 0.79 0.044 0.484 0.053 0.008 0.084 0.255 2.17 0.92 0.111 2 -0.38 0.74 0.041 0.533 0.051 0.007 0.089 0.191 2.21 1.00 0.111 3 -0.38 0.75 0.040 0.574 0.052 0.005 0.090 0.188 2.15 0.90 0.109 4 -0.38 0.76 0.039 0.593 0.049 0.016 0.087 0.218 2.00 0.89 0.110 5 -0.38 0.77 0.042 0.512 0.047 0.014 0.086 0.238 2.22 0.97 0.110 6 -0.38 0.78 0.039 0.604 0.046 0.018 0.085 0.244 2.18 0.87 0.110 7 -0.38 0.79 0.047 0.370 0.048 0.010 0.087 0.216 2.20 0.94 0.110 8 -0.38 0.80 0.046 0.414 0.049 0.012 0.086 0.232 2.22 0.97 0.110 9 -0.38 0.81 0.049 0.360 0.050 0.014 0.091 0.182 2.21 1.01 0.110 10 -0.38 0.82 0.048 0.362 0.051 0.007 0.084 0.254 2.27 0.96 0.111 11 -0.37 0.73 0.040 0.572 0.051 0.013 0.088 0.216 2.09 0.85 0.109 12 -0.37 0.74 0.036 0.685 0.047 0.013 0.087 0.218 2.26 0.91 0.109 13 -0.37 0.75 0.038 0.619 0.045 0.022 0.086 0.229 2.12 0.91 0.109 14 -0.37 0.76 0.041 0.535 0.047 0.019 0.088 0.208 2.19 0.94 0.109 15 -0.37 0.77 0.039 0.603 0.044 0.035 0.086 0.232 2.13 0.96 0.109 16 -0.37 0.78 0.040 0.588 0.045 0.020 0.086 0.237 2.07 0.91 0.110 17 -0.37 0.79 0.046 0.417 0.044 0.040 0.089 0.194 2.20 0.99 0.109 18 -0.37 0.80 0.045 0.449 0.041 0.064 0.089 0.201 2.08 0.97 0.110 19 -0.37 0.81 0.047 0.382 0.045 0.020 0.088 0.208 2.20 0.98 0.110 20 -0.37 0.82 0.045 0.438 0.047 0.015 0.085 0.250 2.21 0.96 0.110 21 -0.36 0.72 0.033 0.785 0.048 0.013 0.087 0.220 2.12 0.97 0.108 22 -0.36 0.73 0.033 0.793 0.048 0.009 0.087 0.218 2.21 0.88 0.109 23 -0.36 0.74 0.038 0.617 0.044 0.038 0.091 0.185 2.21 0.89 0.109 24 -0.36 0.75 0.038 0.622 0.045 0.021 0.089 0.198 2.10 0.93 0.109 25 -0.36 0.76 0.038 0.644 0.041 0.055 0.089 0.202 2.19 0.91 0.110 26 -0.36 0.77 0.036 0.694 0.043 0.047 0.083 0.263 2.16 0.96 0.110 No. f (P ) W 10 P α DP-IP SP-IP f b x 0 σ D P D P D P [%] [%] 27 -0.36 0.78 0.041 0.555 0.040 0.065 0.086 0.229 2.20 0.99 0.110 28 -0.36 0.79 0.045 0.430 0.043 0.049 0.088 0.208 2.14 1.01 0.110 29 -0.36 0.80 0.044 0.463 0.041 0.043 0.084 0.254 2.08 0.97 0.110 30 -0.36 0.81 0.044 0.458 0.041 0.052 0.088 0.220 2.22 0.94 0.110 31 -0.36 0.82 0.041 0.524 0.044 0.033 0.084 0.253 2.08 0.97 0.111 32 -0.36 0.83 0.046 0.403 0.043 0.030 0.087 0.221 2.25 1.08 0.110 33 -0.36 0.84 0.048 0.338 0.049 0.008 0.088 0.209 2.22 1.04 0.111 34 -0.35 0.69 0.032 0.800 0.050 0.010 0.091 0.178 2.07 0.88 0.108 35 -0.35 0.70 0.032 0.802 0.051 0.010 0.093 0.170 2.22 0.96 0.109 36 -0.35 0.71 0.028 0.908 0.045 0.021 0.086 0.237 2.16 0.93 0.108 37 -0.35 0.72 0.031 0.840 0.045 0.029 0.088 0.210 2.18 0.89 0.108 38 -0.35 0.73 0.033 0.772 0.044 0.028 0.088 0.208 2.20 0.91 0.109 39 -0.35 0.74 0.035 0.729 0.041 0.053 0.092 0.177 2.21 0.87 0.109 40 -0.35 0.75 0.033 0.796 0.042 0.033 0.087 0.223 2.21 0.87 0.109 41 -0.35 0.76 0.036 0.693 0.039 0.062 0.090 0.186 2.16 0.88 0.109 42 -0.35 0.77 0.036 0.692 0.040 0.048 0.086 0.222 2.17 0.95 0.110 43 -0.35 0.78 0.042 0.521 0.038 0.070 0.092 0.166 2.18 1.00 0.109 44 -0.35 0.79 0.038 0.612 0.036 0.127 0.085 0.252 2.16 0.96 0.108 45 -0.35 0.80 0.038 0.623 0.035 0.135 0.088 0.210 2.10 0.89 0.109 46 -0.35 0.81 0.041 0.539 0.037 0.092 0.086 0.225 2.26 0.87 0.109 47 -0.35 0.82 0.040 0.574 0.038 0.071 0.086 0.236 2.22 0.98 0.110 48 -0.35 0.83 0.041 0.525 0.042 0.034 0.085 0.243 2.14 0.96 0.110 49 -0.35 0.84 0.045 0.426 0.046 0.016 0.088 0.214 2.28 0.97 0.109 50 -0.35 0.85 0.046 0.410 0.052 0.008 0.086 0.234 2.14 1.04 0.110 51 -0.34 0.69 0.031 0.838 0.050 0.007 0.087 0.221 2.19 0.93 0.108 52 -0.34 0.70 0.025 0.946 0.047 0.018 0.084 0.245 2.16 0.92 0.108 53 -0.34 0.71 0.028 0.897 0.043 0.033 0.083 0.268 2.15 0.85 0.109 54 -0.34 0.72 0.027 0.931 0.040 0.049 0.082 0.278 2.13 0.89 0.108 55 -0.34 0.73 0.032 0.803 0.039 0.058 0.091 0.172 2.12 0.99 0.109 56 -0.34 0.74 0.033 0.773 0.040 0.060 0.088 0.207 2.19 0.95 0.108 57 -0.34 0.75 0.029 0.880 0.038 0.101 0.083 0.263 2.11 0.84 0.109 58 -0.34 0.76 0.034 0.759 0.035 0.159 0.086 0.229 2.03 0.94 0.109 59 -0.34 0.77 0.035 0.704 0.033 0.171 0.084 0.258 2.15 0.90 0.109 60 -0.34 0.78 0.038 0.622 0.034 0.158 0.087 0.212 2.13 0.94 0.108 61 -0.34 0.79 0.041 0.531 0.031 0.248 0.089 0.192 2.11 0.93 0.110 62 -0.34 0.80 0.041 0.534 0.034 0.154 0.088 0.218 2.01 1.01 0.109 63 -0.34 0.81 0.038 0.640 0.035 0.147 0.086 0.233 2.10 0.89 0.109 64 -0.34 0.82 0.041 0.546 0.037 0.089 0.087 0.224 2.23 0.94 0.109 65 -0.34 0.83 0.042 0.514 0.038 0.075 0.083 0.278 2.22 0.99 0.110 66 -0.34 0.84 0.042 0.515 0.042 0.042 0.088 0.211 2.21 1.00 0.109 67 -0.34 0.85 0.043 0.477 0.046 0.022 0.087 0.217 2.19 0.99 0.110 68 -0.34 0.86 0.048 0.354 0.053 0.007 0.087 0.214 2.18 1.01 0.109 69 -0.33 0.70 0.029 0.901 0.046 0.020 0.090 0.180 2.03 0.89 0.108 70 -0.33 0.71 0.028 0.902 0.044 0.020 0.091 0.174 1.95 0.91 0.109 71 -0.33 0.72 0.029 0.887 0.042 0.036 0.087 0.214 2.07 0.86 0.109 72 -0.33 0.73 0.029 0.893 0.038 0.077 0.089 0.203 2.02 0.87 0.108 73 -0.33 0.74 0.031 0.830 0.036 0.103 0.087 0.216 2.10 0.89 0.108 74 -0.33 0.75 0.033 0.784 0.037 0.103 0.090 0.192 2.18 0.91 0.108 75 -0.33 0.76 0.032 0.816 0.031 0.234 0.088 0.213 2.16 0.86 0.108 76 -0.33 0.77 0.029 0.887 0.030 0.273 0.087 0.227 2.06 0.96 0.108 77 -0.33 0.78 0.032 0.802 0.029 0.297 0.089 0.203 2.14 0.97 0.108 78 -0.33 0.79 0.038 0.648 0.028 0.323 0.092 0.169 2.10 0.91 0.109 79 -0.33 0.80 0.037 0.668 0.031 0.211 0.089 0.198 2.10 0.89 0.109 80 -0.33 0.81 0.034 0.764 0.028 0.346 0.086 0.226 2.07 0.94 0.109 81 -0.33 0.82 0.039 0.609 0.030 0.267 0.092 0.168 2.11 0.90 0.109 82 -0.33 0.83 0.041 0.547 0.039 0.062 0.088 0.207 2.31 1.01 0.110 83 -0.33 0.84 0.043 0.479 0.039 0.077 0.088 0.208 2.21 0.95 0.109 84 -0.33 0.85 0.042 0.526 0.046 0.020 0.087 0.219 2.22 0.95 0.109 85 -0.33 0.86 0.045 0.418 0.051 0.010 0.089 0.198 2.16 0.98 0.108 86 -0.32 0.70 0.027 0.937 0.052 0.005 0.084 0.252 2.09 0.82 0.108 87 -0.32 0.71 0.029 0.894 0.047 0.013 0.083 0.257 2.09 0.86 0.107 88 -0.32 0.72 0.027 0.914 0.042 0.034 0.085 0.252 2.04 0.84 0.107 89 -0.32 0.73 0.029 0.899 0.038 0.105 0.083 0.263 2.05 0.98 0.108 90 -0.32 0.74 0.030 0.864 0.037 0.109 0.087 0.220 2.07 0.99 0.107 91 -0.32 0.75 0.030 0.852 0.033 0.170 0.084 0.258 2.16 0.81 0.108 No. f (P ) W 10 P α DP-IP SP-IP f b x 0 σ D P D P D P [%] [%] 92 -0.32 0.76 0.031 0.835 0.028 0.312 0.085 0.239 2.21 0.86 0.108 93 -0.32 0.77 0.034 0.767 0.029 0.298 0.086 0.231 2.18 0.91 0.108 94 -0.32 0.78 0.036 0.693 0.027 0.389 0.090 0.196 2.06 0.99 0.108 95 -0.32 0.79 0.031 0.842 0.024 0.501 0.085 0.248 2.15 0.99 0.108 96 -0.32 0.80 0.038 0.623 0.027 0.361 0.085 0.243 2.23 0.98 0.109 97 -0.32 0.81 0.036 0.706 0.028 0.364 0.087 0.227 2.07 0.99 0.108 98 -0.32 0.82 0.037 0.645 0.028 0.341 0.084 0.259 2.14 0.97 0.109 99 -0.32 0.83 0.037 0.651 0.032 0.224 0.083 0.270 2.15 0.89 0.110 100 -0.32 0.84 0.041 0.530 0.038 0.087 0.085 0.238 2.20 0.99 0.109 101 -0.32 0.85 0.040 0.565 0.042 0.037 0.090 0.194 2.18 1.02 0.109 102 -0.32 0.86 0.045 0.444 0.045 0.021 0.084 0.254 2.19 0.96 0.108 103 -0.32 0.87 0.042 0.499 0.050 0.018 0.084 0.252 2.17 0.96 0.109 104 -0.31 0.72 0.026 0.952 0.047 0.014 0.090 0.188 1.98 0.86 0.108 105 -0.31 0.73 0.028 0.900 0.041 0.037 0.090 0.189 2.20 0.95 0.108 106 -0.31 0.74 0.029 0.872 0.038 0.078 0.092 0.170 2.15 0.89 0.108 107 -0.31 0.75 0.033 0.785 0.031 0.227 0.089 0.192 2.17 0.92 0.107 108 -0.31 0.76 0.031 0.834 0.028 0.333 0.092 0.165 2.18 0.90 0.107 109 -0.31 0.77 0.029 0.871 0.025 0.480 0.085 0.246 2.21 0.87 0.107 110 -0.31 0.78 0.035 0.726 0.023 0.541 0.092 0.176 2.12 0.92 0.108 111 -0.31 0.79 0.030 0.872 0.021 0.658 0.087 0.212 2.14 0.90 0.109 112 -0.31 0.80 0.032 0.804 0.024 0.537 0.089 0.203 2.24 0.96 0.108 113 -0.31 0.81 0.033 0.777 0.024 0.480 0.091 0.179 2.16 0.93 0.108 114 -0.31 0.82 0.036 0.677 0.026 0.428 0.089 0.204 2.14 0.94 0.108 115 -0.31 0.83 0.036 0.688 0.029 0.330 0.086 0.233 2.21 1.07 0.108 116 -0.31 0.84 0.034 0.757 0.032 0.210 0.088 0.201 2.20 0.95 0.109 117 -0.31 0.85 0.040 0.573 0.037 0.097 0.087 0.224 2.27 0.96 0.108 118 -0.31 0.86 0.043 0.482 0.042 0.041 0.091 0.184 2.20 1.00 0.108 119 -0.31 0.87 0.041 0.539 0.045 0.023 0.093 0.160 2.16 0.98 0.108 120 -0.31 0.88 0.043 0.479 0.053 0.005 0.087 0.212 2.12 1.06 0.109 121 -0.30 0.73 0.027 0.936 0.043 0.034 0.088 0.208 2.16 0.81 0.107 122 -0.30 0.74 0.029 0.875 0.038 0.076 0.087 0.229 2.11 0.86 0.108 123 -0.30 0.75 0.027 0.897 0.035 0.127 0.087 0.225 2.03 0.89 0.107 124 -0.30 0.76 0.027 0.915 0.032 0.250 0.084 0.260 2.13 0.81 0.107 125 -0.30 0.77 0.028 0.903 0.028 0.355 0.087 0.218 2.18 0.85 0.108 126 -0.30 0.78 0.030 0.860 0.026 0.398 0.090 0.196 2.12 0.91 0.108 127 -0.30 0.79 0.029 0.889 0.023 0.579 0.089 0.201 2.16 0.88 0.108 128 -0.30 0.80 0.032 0.821 0.018 0.813 0.089 0.196 2.10 0.89 0.108 129 -0.30 0.81 0.035 0.724 0.021 0.674 0.093 0.166 2.22 0.88 0.108 130 -0.30 0.82 0.034 0.752 0.021 0.650 0.088 0.207 2.25 0.92 0.107 131 -0.30 0.83 0.038 0.635 0.026 0.407 0.091 0.179 2.13 0.94 0.108 132 -0.30 0.84 0.037 0.657 0.034 0.150 0.086 0.227 2.25 1.01 0.108 133 -0.30 0.85 0.039 0.601 0.035 0.150 0.089 0.202 2.11 0.94 0.108 134 -0.30 0.86 0.040 0.587 0.040 0.055 0.091 0.176 2.19 0.98 0.109 135 -0.30 0.87 0.040 0.592 0.044 0.021 0.089 0.208 2.21 0.90 0.109 136 -0.30 0.88 0.044 0.461 0.049 0.009 0.088 0.206 2.26 1.00 0.108 137 -0.29 0.72 0.029 0.887 0.053 0.006 0.091 0.197 2.05 0.82 0.106 138 -0.29 0.73 0.026 0.951 0.050 0.007 0.089 0.209 2.08 0.87 0.107 139 -0.29 0.74 0.027 0.938 0.043 0.037 0.088 0.203 2.11 0.89 0.107 140 -0.29 0.75 0.029 0.892 0.042 0.038 0.092 0.171 2.03 0.83 0.106 141 -0.29 0.76 0.029 0.872 0.035 0.145 0.089 0.198 2.18 0.95 0.107 142 -0.29 0.77 0.029 0.887 0.030 0.303 0.090 0.184 2.08 0.91 0.107 143 -0.29 0.78 0.029 0.894 0.025 0.483 0.087 0.227 2.10 0.88 0.107 144 -0.29 0.79 0.030 0.853 0.023 0.552 0.090 0.179 2.19 0.91 0.107 145 -0.29 0.80 0.030 0.860 0.021 0.670 0.089 0.205 2.11 0.90 0.107 146 -0.29 0.81 0.035 0.734 0.019 0.797 0.089 0.195 2.26 0.89 0.107 147 -0.29 0.82 0.033 0.780 0.022 0.630 0.087 0.223 2.06 0.91 0.108 148 -0.29 0.83 0.033 0.778 0.027 0.392 0.089 0.203 2.13 0.93 0.108 149 -0.29 0.84 0.034 0.759 0.024 0.504 0.088 0.213 2.10 1.00 0.108 150 -0.29 0.85 0.041 0.534 0.036 0.119 0.092 0.164 2.19 0.93 0.107 151 -0.29 0.86 0.037 0.668 0.036 0.103 0.089 0.207 2.20 0.98 0.108 152 -0.29 0.87 0.041 0.537 0.042 0.044 0.093 0.162 2.22 1.03 0.108 153 -0.29 0.88 0.039 0.602 0.042 0.037 0.089 0.200 2.17 0.94 0.108 154 -0.29 0.89 0.043 0.476 0.049 0.014 0.091 0.183 2.31 0.98 0.108 155 -0.28 0.73 0.027 0.914 0.054 0.005 0.087 0.221 2.15 0.88 0.105 156 -0.28 0.74 0.030 0.860 0.049 0.017 0.086 0.231 2.01 0.84 0.106 No. f (P ) W 10 P α DP-IP SP-IP f b x 0 σ D P D P D P [%] [%] 157 -0.28 0.75 0.028 0.895 0.044 0.030 0.088 0.205 2.09 0.95 0.107 158 -0.28 0.76 0.028 0.908 0.040 0.061 0.088 0.206 2.09 0.89 0.106 159 -0.28 0.77 0.029 0.883 0.036 0.115 0.087 0.224 2.15 0.87 0.107 160 -0.28 0.78 0.028 0.902 0.032 0.217 0.091 0.179 1.97 0.84 0.107 161 -0.28 0.79 0.031 0.855 0.025 0.462 0.089 0.192 2.03 0.82 0.106 162 -0.28 0.80 0.029 0.897 0.024 0.528 0.084 0.258 2.14 0.97 0.107 163 -0.28 0.81 0.028 0.906 0.023 0.577 0.086 0.231 2.00 0.87 0.107 164 -0.28 0.82 0.030 0.836 0.019 0.757 0.086 0.234 2.10 0.94 0.107 165 -0.28 0.83 0.034 0.752 0.022 0.641 0.087 0.217 2.07 0.89 0.107 166 -0.28 0.84 0.034 0.761 0.024 0.489 0.087 0.211 2.17 0.91 0.108 167 -0.28 0.85 0.036 0.684 0.028 0.350 0.085 0.245 2.31 1.00 0.108 168 -0.28 0.86 0.034 0.747 0.034 0.178 0.086 0.230 2.14 0.95 0.108 169 -0.28 0.87 0.039 0.607 0.039 0.076 0.084 0.253 2.16 1.01 0.108 170 -0.28 0.88 0.036 0.686 0.043 0.033 0.090 0.181 2.13 0.93 0.108 171 -0.28 0.89 0.040 0.573 0.046 0.024 0.088 0.205 2.20 1.01 0.108 172 -0.28 0.90 0.039 0.596 0.050 0.009 0.086 0.224 2.09 0.94 0.109 173 -0.27 0.74 0.028 0.906 0.053 0.005 0.086 0.229 2.14 0.85 0.106 174 -0.27 0.75 0.031 0.840 0.045 0.024 0.089 0.211 2.12 0.86 0.105 175 -0.27 0.76 0.030 0.847 0.046 0.018 0.089 0.208 2.03 0.90 0.106 176 -0.27 0.77 0.029 0.865 0.039 0.060 0.090 0.180 2.09 0.89 0.106 177 -0.27 0.78 0.032 0.815 0.037 0.116 0.089 0.201 2.02 0.92 0.107 178 -0.27 0.79 0.031 0.848 0.034 0.173 0.091 0.176 2.10 0.88 0.107 179 -0.27 0.80 0.032 0.832 0.030 0.292 0.089 0.199 2.15 0.90 0.106 180 -0.27 0.81 0.032 0.818 0.027 0.438 0.094 0.154 2.11 0.84 0.106 181 -0.27 0.82 0.031 0.830 0.023 0.543 0.089 0.191 2.21 0.94 0.107 182 -0.27 0.83 0.032 0.802 0.023 0.572 0.091 0.179 2.16 0.92 0.106 183 -0.27 0.84 0.034 0.752 0.026 0.406 0.088 0.203 2.26 0.97 0.107 184 -0.27 0.85 0.032 0.818 0.024 0.494 0.089 0.201 2.12 0.95 0.107 185 -0.27 0.86 0.035 0.711 0.033 0.206 0.093 0.156 2.11 0.98 0.107 186 -0.27 0.87 0.038 0.650 0.036 0.119 0.094 0.149 2.04 0.93 0.107 187 -0.27 0.88 0.040 0.577 0.040 0.069 0.092 0.170 2.15 0.97 0.108 188 -0.27 0.89 0.039 0.607 0.045 0.023 0.093 0.156 2.14 0.96 0.108 189 -0.27 0.90 0.040 0.582 0.046 0.014 0.089 0.205 2.17 0.99 0.107 190 -0.26 0.75 0.029 0.881 0.052 0.006 0.086 0.224 2.20 0.88 0.105 191 -0.26 0.76 0.031 0.838 0.051 0.008 0.086 0.235 2.05 0.86 0.106 192 -0.26 0.77 0.032 0.801 0.045 0.041 0.080 0.307 2.13 0.89 0.106 193 -0.26 0.78 0.030 0.870 0.038 0.088 0.083 0.268 2.05 0.96 0.106 194 -0.26 0.79 0.029 0.893 0.033 0.207 0.087 0.223 2.03 0.91 0.106 195 -0.26 0.80 0.031 0.835 0.031 0.243 0.090 0.198 2.06 0.91 0.106 196 -0.26 0.81 0.033 0.788 0.030 0.277 0.089 0.197 2.11 0.96 0.107 197 -0.26 0.82 0.032 0.826 0.029 0.317 0.085 0.237 2.13 0.88 0.106 198 -0.26 0.83 0.033 0.783 0.022 0.610 0.088 0.215 2.17 0.87 0.107 199 -0.26 0.84 0.032 0.830 0.027 0.416 0.088 0.222 2.12 0.89 0.106 200 -0.26 0.85 0.032 0.816 0.026 0.430 0.087 0.225 2.10 0.93 0.106 201 -0.26 0.86 0.034 0.742 0.028 0.353 0.088 0.209 2.12 0.92 0.107 202 -0.26 0.87 0.036 0.704 0.033 0.204 0.087 0.225 2.01 1.03 0.107 203 -0.26 0.88 0.031 0.853 0.038 0.091 0.081 0.288 2.06 0.91 0.107 204 -0.26 0.89 0.036 0.709 0.039 0.060 0.085 0.243 2.19 0.99 0.107 205 -0.26 0.90 0.035 0.710 0.046 0.017 0.085 0.242 2.11 0.97 0.108 206 -0.26 0.91 0.039 0.607 0.051 0.006 0.087 0.230 2.29 1.01 0.106 207 -0.25 0.76 0.031 0.840 0.052 0.008 0.088 0.206 2.10 0.86 0.105 208 -0.25 0.77 0.030 0.847 0.049 0.011 0.087 0.215 2.18 0.90 0.106 209 -0.25 0.78 0.031 0.810 0.044 0.041 0.086 0.228 2.11 0.83 0.106 210 -0.25 0.79 0.029 0.876 0.040 0.104 0.087 0.216 2.12 0.88 0.106 211 -0.25 0.80 0.030 0.872 0.036 0.108 0.090 0.182 2.04 0.88 0.106 212 -0.25 0.81 0.032 0.825 0.035 0.149 0.091 0.185 2.13 0.86 0.106 213 -0.25 0.82 0.035 0.714 0.034 0.189 0.086 0.228 2.15 0.87 0.106 214 -0.25 0.83 0.031 0.846 0.030 0.339 0.091 0.183 2.15 0.89 0.105 215 -0.25 0.84 0.032 0.791 0.026 0.451 0.084 0.257 2.05 0.93 0.106 216 -0.25 0.85 0.033 0.794 0.028 0.320 0.083 0.259 2.18 0.90 0.107 217 -0.25 0.86 0.032 0.806 0.030 0.255 0.086 0.234 2.05 0.99 0.106 218 -0.25 0.87 0.035 0.720 0.029 0.299 0.088 0.214 2.03 0.90 0.107 219 -0.25 0.88 0.037 0.675 0.032 0.222 0.086 0.229 2.14 0.91 0.107 220 -0.25 0.89 0.033 0.785 0.039 0.071 0.085 0.244 2.05 0.97 0.107 221 -0.25 0.90 0.035 0.737 0.043 0.043 0.086 0.224 2.08 0.98 0.107 No. f (P ) W 10 P α DP-IP SP-IP f b x 0 σ D P D P D P [%] [%] 222 -0.25 0.91 0.039 0.604 0.047 0.031 0.087 0.223 2.23 0.94 0.107 223 -0.25 0.92 0.038 0.623 0.051 0.007 0.086 0.239 2.09 0.95 0.107 224 -0.25 0.93 0.042 0.520 0.056 0.006 0.087 0.222 2.09 1.01 0.108 225 -0.24 0.77 0.037 0.658 0.057 0.005 0.090 0.183 2.05 0.80 0.106 226 -0.24 0.78 0.033 0.784 0.052 0.009 0.093 0.164 2.12 0.82 0.105 227 -0.24 0.79 0.032 0.825 0.044 0.036 0.094 0.154 2.13 0.91 0.105 228 -0.24 0.80 0.033 0.772 0.044 0.032 0.090 0.184 2.08 0.87 0.106 229 -0.24 0.81 0.032 0.803 0.038 0.094 0.090 0.188 2.14 0.88 0.106 230 -0.24 0.82 0.030 0.860 0.039 0.074 0.089 0.204 2.16 0.88 0.106 231 -0.24 0.83 0.035 0.735 0.033 0.216 0.088 0.207 2.06 0.84 0.106 232 -0.24 0.84 0.031 0.831 0.031 0.281 0.090 0.184 2.11 0.89 0.107 233 -0.24 0.85 0.034 0.768 0.031 0.233 0.091 0.186 2.03 0.99 0.105 234 -0.24 0.86 0.034 0.750 0.029 0.282 0.088 0.221 2.08 0.90 0.106 235 -0.24 0.87 0.033 0.790 0.029 0.287 0.093 0.159 2.10 0.93 0.106 236 -0.24 0.88 0.035 0.715 0.033 0.196 0.090 0.185 2.21 0.94 0.107 237 -0.24 0.89 0.034 0.761 0.034 0.147 0.091 0.177 2.03 0.98 0.107 238 -0.24 0.90 0.036 0.685 0.037 0.092 0.088 0.215 2.11 0.91 0.106 239 -0.24 0.91 0.038 0.646 0.046 0.019 0.088 0.211 2.19 0.99 0.108 240 -0.24 0.92 0.039 0.592 0.050 0.011 0.091 0.170 2.13 0.95 0.107 241 -0.24 0.93 0.040 0.567 0.052 0.005 0.087 0.214 2.15 0.97 0.107 242 -0.23 0.78 0.034 0.761 0.053 0.007 0.089 0.199 2.11 0.91 0.105 243 -0.23 0.80 0.035 0.732 0.052 0.009 0.091 0.175 2.08 0.92 0.105 244 -0.23 0.81 0.036 0.687 0.045 0.030 0.085 0.233 2.03 0.90 0.105 245 -0.23 0.82 0.031 0.847 0.043 0.034 0.090 0.192 2.07 0.91 0.105 246 -0.23 0.83 0.034 0.762 0.039 0.099 0.091 0.178 2.07 0.89 0.106 247 -0.23 0.84 0.032 0.819 0.034 0.183 0.088 0.218 2.15 0.88 0.105 248 -0.23 0.85 0.031 0.835 0.034 0.139 0.089 0.200 2.07 0.89 0.106 249 -0.23 0.86 0.033 0.786 0.036 0.132 0.090 0.191 2.19 0.94 0.106 250 -0.23 0.87 0.034 0.753 0.030 0.245 0.088 0.212 2.09 0.96 0.105 251 -0.23 0.88 0.032 0.810 0.034 0.135 0.088 0.211 2.19 0.96 0.106 252 -0.23 0.89 0.035 0.729 0.035 0.121 0.088 0.209 2.07 0.91 0.106 253 -0.23 0.90 0.036 0.681 0.039 0.069 0.087 0.217 2.20 0.93 0.106 254 -0.23 0.91 0.035 0.737 0.042 0.039 0.089 0.196 2.15 0.90 0.107 255 -0.23 0.92 0.040 0.587 0.046 0.014 0.089 0.196 2.21 0.99 0.106 256 -0.23 0.93 0.038 0.619 0.051 0.011 0.086 0.238 2.25 0.97 0.106 257 -0.22 0.79 0.035 0.735 0.054 0.010 0.088 0.211 1.98 0.85 0.105 258 -0.22 0.80 0.037 0.668 0.054 0.011 0.088 0.213 2.07 0.90 0.105 259 -0.22 0.81 0.035 0.719 0.047 0.014 0.088 0.209 2.02 0.99 0.105 260 -0.22 0.82 0.036 0.706 0.049 0.008 0.085 0.244 2.07 0.88 0.106 261 -0.22 0.83 0.031 0.826 0.048 0.021 0.086 0.230 2.19 0.90 0.106 262 -0.22 0.84 0.036 0.696 0.044 0.050 0.083 0.265 2.21 0.91 0.105 263 -0.22 0.85 0.034 0.753 0.037 0.142 0.084 0.252 2.15 0.87 0.105 264 -0.22 0.86 0.036 0.701 0.035 0.150 0.089 0.199 2.08 0.97 0.105 265 -0.22 0.87 0.033 0.801 0.035 0.139 0.084 0.255 2.14 0.97 0.105 266 -0.22 0.88 0.034 0.750 0.034 0.178 0.088 0.213 2.10 0.91 0.106 267 -0.22 0.89 0.034 0.739 0.036 0.098 0.083 0.268 2.22 0.98 0.106 268 -0.22 0.90 0.035 0.734 0.041 0.052 0.084 0.247 2.10 0.95 0.106 269 -0.22 0.91 0.036 0.712 0.038 0.084 0.087 0.224 2.16 0.97 0.106 270 -0.22 0.92 0.034 0.759 0.042 0.040 0.085 0.239 2.17 0.95 0.106 271 -0.22 0.93 0.036 0.699 0.047 0.016 0.085 0.243 2.13 0.96 0.106 272 -0.21 0.82 0.038 0.626 0.054 0.006 0.093 0.166 2.02 0.85 0.105 273 -0.21 0.83 0.035 0.731 0.050 0.027 0.090 0.192 2.13 0.81 0.105 274 -0.21 0.84 0.035 0.721 0.051 0.009 0.094 0.149 2.06 0.91 0.104 275 -0.21 0.85 0.034 0.769 0.043 0.077 0.091 0.175 2.10 0.89 0.106 276 -0.21 0.86 0.034 0.759 0.040 0.059 0.088 0.209 2.09 0.91 0.105 277 -0.21 0.87 0.033 0.781 0.040 0.069 0.090 0.188 2.13 0.88 0.106 278 -0.21 0.88 0.035 0.730 0.038 0.069 0.091 0.184 2.03 0.95 0.106 279 -0.21 0.89 0.036 0.697 0.041 0.051 0.087 0.220 2.11 0.95 0.106 280 -0.21 0.90 0.035 0.716 0.038 0.075 0.091 0.185 2.12 0.92 0.106 281 -0.21 0.91 0.036 0.694 0.041 0.043 0.090 0.198 2.04 0.90 0.106 282 -0.21 0.92 0.038 0.633 0.042 0.038 0.092 0.168 2.18 0.98 0.106 283 -0.21 0.93 0.037 0.678 0.045 0.019 0.092 0.167 2.15 1.04 0.107 284 -0.21 0.94 0.036 0.709 0.049 0.010 0.091 0.188 2.07 0.92 0.106 285 -0.21 0.95 0.034 0.759 0.049 0.008 0.086 0.235 2.19 0.98 0.107 286 -0.20 0.83 0.037 0.661 0.053 0.007 0.087 0.226 1.99 0.86 0.105 No. f (P ) W 10 P α DP-IP SP-IP f b x 0 σ D P D P D P [%] [%] 287 -0.20 0.84 0.037 0.670 0.049 0.030 0.089 0.196 2.15 0.92 0.105 288 -0.20 0.85 0.035 0.729 0.047 0.013 0.083 0.263 2.24 0.91 0.105 289 -0.20 0.86 0.037 0.672 0.046 0.031 0.088 0.211 2.04 0.84 0.105 290 -0.20 0.87 0.036 0.705 0.043 0.033 0.091 0.188 2.09 0.93 0.105 291 -0.20 0.88 0.037 0.665 0.041 0.046 0.086 0.231 2.09 0.93 0.105 292 -0.20 0.89 0.038 0.627 0.039 0.088 0.091 0.185 2.12 1.01 0.105 293 -0.20 0.90 0.034 0.743 0.040 0.069 0.088 0.214 2.21 0.96 0.106 294 -0.20 0.91 0.037 0.670 0.042 0.054 0.092 0.171 2.14 0.88 0.105 295 -0.20 0.92 0.036 0.701 0.042 0.041 0.088 0.205 2.21 0.89 0.105 296 -0.20 0.93 0.038 0.635 0.044 0.028 0.091 0.182 2.05 0.93 0.106 297 -0.20 0.94 0.035 0.719 0.047 0.017 0.089 0.203 2.16 0.94 0.106 298 -0.20 0.95 0.038 0.622 0.049 0.009 0.089 0.201 2.14 0.97 0.106 299 -0.19 0.85 0.037 0.668 0.050 0.008 0.091 0.170 2.10 1.01 0.104 300 -0.19 0.86 0.039 0.618 0.051 0.007 0.087 0.214 2.16 0.86 0.104 301 -0.19 0.87 0.038 0.621 0.049 0.018 0.085 0.238 2.16 0.95 0.105 302 -0.19 0.88 0.037 0.682 0.047 0.016 0.088 0.218 2.19 0.94 0.104 303 -0.19 0.89 0.040 0.591 0.046 0.032 0.083 0.275 2.06 0.96 0.104 304 -0.19 0.90 0.037 0.655 0.043 0.042 0.085 0.247 2.14 0.87 0.105 305 -0.19 0.91 0.037 0.672 0.045 0.029 0.086 0.231 2.06 0.95 0.105 306 -0.19 0.92 0.038 0.641 0.045 0.022 0.089 0.203 2.04 0.93 0.105 307 -0.19 0.93 0.036 0.700 0.044 0.022 0.087 0.221 2.25 0.87 0.105 308 -0.19 0.94 0.036 0.704 0.048 0.011 0.086 0.229 1.92 0.90 0.106 309 -0.19 0.95 0.038 0.620 0.052 0.005 0.085 0.239 2.14 1.02 0.106 310 -0.18 0.87 0.037 0.661 0.050 0.010 0.089 0.192 2.10 0.86 0.105 311 -0.18 0.88 0.037 0.649 0.050 0.013 0.090 0.189 2.14 0.88 0.104 312 -0.18 0.89 0.039 0.599 0.051 0.009 0.090 0.197 2.10 0.90 0.104 313 -0.18 0.90 0.038 0.651 0.047 0.025 0.087 0.214 2.05 0.91 0.105 314 -0.18 0.91 0.040 0.577 0.049 0.011 0.086 0.229 2.11 0.91 0.105 315 -0.18 0.92 0.036 0.712 0.047 0.020 0.091 0.177 2.07 0.96 0.105 316 -0.18 0.93 0.035 0.724 0.046 0.019 0.086 0.233 2.21 0.95 0.105 317 -0.18 0.94 0.036 0.696 0.049 0.008 0.088 0.207 2.03 0.96 0.106 318 -0.18 0.95 0.040 0.585 0.051 0.008 0.087 0.224 2.13 0.93 0.105 319 -0.18 0.96 0.036 0.701 0.052 0.007 0.089 0.199 1.98 0.91 0.106 320 -0.17 0.88 0.040 0.571 0.051 0.006 0.089 0.198 2.20 0.90 0.104 321 -0.17 0.90 0.040 0.590 0.052 0.007 0.087 0.225 2.14 0.93 0.105 322 -0.17 0.91 0.037 0.678 0.048 0.013 0.088 0.210 2.00 0.89 0.104 323 -0.17 0.92 0.038 0.619 0.050 0.006 0.090 0.190 2.13 0.97 0.105 324 -0.17 0.93 0.040 0.562 0.052 0.006 0.086 0.229 2.14 0.92 0.104 325 -0.17 0.94 0.039 0.588 0.049 0.009 0.087 0.218 2.12 0.95 0.105 326 -0.17 0.95 0.041 0.538 0.049 0.010 0.088 0.204 2.13 0.89 0.105 Table B5 : B5Simulation results of the pulsar beam for opening angle distribution Eq. (13), inclination angle distribution function Eq. (6) and period distribution function Eq. (8). Graphical representation of this class of solutions is represented in Fig. C6.On the pulse-width statistics in radio pulsars.I. Importance of the interpulse emission 31No. f (P ) W 10 P α DP-IP SP-IP f b x 0 σ D P D P D P [%] [%] 1 0.37 2.32 0.036 0.687 0.049 0.008 0.088 0.206 2.13 0.96 0.110 2 0.37 2.33 0.038 0.634 0.050 0.005 0.090 0.189 2.18 0.94 0.110 3 0.37 2.35 0.035 0.729 0.049 0.007 0.091 0.177 2.07 0.92 0.109 4 0.37 2.36 0.036 0.690 0.050 0.006 0.090 0.181 2.06 0.89 0.109 5 0.37 2.37 0.032 0.797 0.049 0.009 0.086 0.231 2.21 0.98 0.109 6 0.37 2.38 0.033 0.760 0.051 0.007 0.086 0.234 2.22 0.96 0.109 7 0.38 2.26 0.041 0.554 0.051 0.005 0.088 0.206 2.20 0.99 0.110 8 0.38 2.27 0.039 0.601 0.049 0.007 0.089 0.208 2.20 0.97 0.110 9 0.38 2.28 0.041 0.549 0.049 0.007 0.092 0.173 2.19 0.97 0.110 10 0.38 2.29 0.041 0.547 0.048 0.010 0.090 0.180 2.20 0.98 0.109 11 0.38 2.30 0.036 0.696 0.049 0.008 0.090 0.184 2.14 0.98 0.110 12 0.38 2.31 0.035 0.731 0.047 0.014 0.089 0.200 2.19 0.98 0.109 13 0.38 2.32 0.033 0.778 0.047 0.012 0.091 0.179 2.07 0.88 0.109 14 0.38 2.33 0.034 0.736 0.048 0.010 0.093 0.155 2.08 0.99 0.109 15 0.38 2.34 0.033 0.776 0.050 0.008 0.091 0.176 2.18 0.91 0.108 16 0.38 2.35 0.034 0.749 0.051 0.008 0.093 0.167 2.21 0.90 0.109 17 0.38 2.36 0.030 0.851 0.052 0.006 0.094 0.150 2.09 0.93 0.108 18 0.39 2.19 0.051 0.287 0.050 0.006 0.096 0.138 2.20 1.03 0.111 19 0.39 2.22 0.039 0.609 0.048 0.010 0.091 0.173 2.15 0.95 0.110 20 0.39 2.23 0.040 0.573 0.047 0.012 0.093 0.159 2.16 0.98 0.110 21 0.39 2.24 0.039 0.603 0.047 0.012 0.090 0.182 2.16 0.95 0.110 22 0.39 2.25 0.034 0.766 0.046 0.016 0.089 0.202 2.16 1.01 0.109 23 0.39 2.26 0.036 0.700 0.046 0.014 0.089 0.207 2.14 0.94 0.109 24 0.39 2.27 0.034 0.735 0.044 0.022 0.090 0.185 2.15 0.98 0.109 25 0.39 2.28 0.034 0.744 0.045 0.020 0.092 0.164 2.14 1.02 0.109 26 0.39 2.29 0.033 0.787 0.045 0.017 0.089 0.204 2.09 1.05 0.108 27 0.39 2.30 0.030 0.873 0.049 0.010 0.085 0.234 2.11 0.88 0.108 28 0.39 2.31 0.030 0.842 0.052 0.005 0.088 0.220 2.07 0.94 0.108 29 0.40 2.13 0.050 0.289 0.050 0.006 0.088 0.209 2.19 1.03 0.112 30 0.40 2.14 0.050 0.316 0.051 0.006 0.090 0.186 2.20 1.01 0.111 31 0.40 2.15 0.051 0.282 0.050 0.007 0.094 0.153 2.15 1.00 0.112 32 0.40 2.16 0.045 0.442 0.048 0.009 0.093 0.160 2.05 1.08 0.111 33 0.40 2.17 0.041 0.532 0.047 0.011 0.090 0.185 2.13 0.93 0.110 34 0.40 2.18 0.044 0.461 0.044 0.018 0.092 0.170 2.24 0.96 0.111 35 0.40 2.19 0.039 0.589 0.044 0.023 0.088 0.205 2.21 1.01 0.111 No. f (P ) W 10 P α DP-IP SP-IP f b x 0 σ D P D P D P [%] [%] 36 0.40 2.20 0.040 0.559 0.044 0.020 0.091 0.182 2.23 0.99 0.109 37 0.40 2.21 0.037 0.656 0.045 0.018 0.092 0.170 2.16 0.99 0.110 38 0.40 2.22 0.039 0.602 0.045 0.019 0.095 0.142 2.20 0.97 0.109 39 0.40 2.23 0.038 0.640 0.043 0.028 0.091 0.183 2.14 0.97 0.109 40 0.40 2.24 0.033 0.774 0.045 0.024 0.089 0.198 2.13 0.96 0.109 41 0.40 2.25 0.033 0.784 0.049 0.011 0.093 0.158 2.07 0.98 0.109 42 0.40 2.26 0.034 0.770 0.047 0.020 0.090 0.183 2.18 0.96 0.108 43 0.40 2.27 0.034 0.737 0.049 0.013 0.091 0.182 2.16 1.05 0.109 44 0.40 2.28 0.030 0.871 0.053 0.005 0.091 0.172 2.06 0.92 0.108 45 0.41 2.10 0.048 0.338 0.049 0.007 0.089 0.203 2.23 1.05 0.111 46 0.41 2.11 0.045 0.432 0.047 0.010 0.090 0.180 2.23 1.01 0.112 47 0.41 2.13 0.047 0.375 0.044 0.025 0.089 0.203 2.14 1.06 0.110 48 0.41 2.14 0.044 0.451 0.042 0.029 0.090 0.188 2.23 0.98 0.109 49 0.41 2.15 0.044 0.433 0.043 0.030 0.091 0.178 2.23 0.96 0.110 50 0.41 2.16 0.041 0.540 0.042 0.030 0.091 0.181 2.18 0.98 0.110 51 0.41 2.17 0.038 0.633 0.041 0.037 0.088 0.206 2.23 0.96 0.109 52 0.41 2.18 0.038 0.641 0.042 0.033 0.089 0.201 2.09 1.01 0.109 53 0.41 2.19 0.039 0.604 0.042 0.032 0.093 0.166 2.11 1.02 0.110 54 0.41 2.20 0.034 0.746 0.043 0.030 0.090 0.191 2.11 0.97 0.109 55 0.41 2.21 0.036 0.698 0.045 0.031 0.090 0.190 2.20 0.92 0.109 56 0.41 2.22 0.037 0.656 0.047 0.016 0.095 0.147 2.10 1.04 0.108 57 0.41 2.23 0.031 0.832 0.050 0.023 0.088 0.209 2.14 0.94 0.108 58 0.41 2.24 0.031 0.842 0.053 0.006 0.090 0.182 2.19 0.96 0.108 59 0.42 2.06 0.045 0.413 0.051 0.006 0.083 0.270 2.25 1.05 0.113 60 0.42 2.07 0.047 0.386 0.049 0.008 0.085 0.251 2.29 1.08 0.110 61 0.42 2.08 0.047 0.377 0.048 0.009 0.085 0.248 2.22 1.04 0.111 62 0.42 2.09 0.045 0.422 0.045 0.019 0.085 0.240 2.19 1.03 0.110 63 0.42 2.11 0.040 0.555 0.042 0.033 0.083 0.257 2.25 1.02 0.110 64 0.42 2.12 0.043 0.488 0.041 0.041 0.082 0.279 2.29 0.99 0.110 65 0.42 2.13 0.041 0.542 0.041 0.040 0.086 0.238 2.21 1.03 0.109 66 0.42 2.14 0.038 0.623 0.039 0.053 0.090 0.187 2.17 0.96 0.109 67 0.42 2.15 0.037 0.649 0.041 0.047 0.083 0.267 2.14 1.05 0.109 68 0.42 2.16 0.037 0.649 0.041 0.040 0.086 0.231 2.24 0.99 0.108 69 0.42 2.17 0.037 0.663 0.042 0.041 0.084 0.257 2.16 1.03 0.109 70 0.42 2.18 0.036 0.675 0.046 0.025 0.085 0.238 2.13 0.88 0.109 71 0.42 2.19 0.032 0.805 0.054 0.006 0.083 0.265 2.20 0.92 0.108 72 0.42 2.20 0.034 0.766 0.051 0.008 0.087 0.218 2.26 0.95 0.108 73 0.43 2.05 0.048 0.354 0.045 0.018 0.093 0.163 2.24 1.03 0.111 74 0.43 2.06 0.045 0.433 0.044 0.023 0.089 0.192 2.10 1.02 0.111 75 0.43 2.07 0.046 0.409 0.042 0.029 0.090 0.188 2.17 0.95 0.110 76 0.43 2.08 0.042 0.520 0.042 0.038 0.091 0.174 2.17 0.97 0.110 77 0.43 2.09 0.038 0.621 0.039 0.050 0.084 0.252 2.07 1.05 0.110 78 0.43 2.10 0.038 0.615 0.041 0.038 0.083 0.261 2.21 1.00 0.109 79 0.43 2.11 0.040 0.571 0.043 0.035 0.088 0.210 2.16 0.88 0.109 80 0.43 2.12 0.038 0.618 0.040 0.052 0.089 0.197 2.15 1.00 0.109 81 0.43 2.13 0.037 0.646 0.047 0.023 0.090 0.187 2.20 1.00 0.109 82 0.43 2.14 0.032 0.815 0.047 0.021 0.087 0.222 2.07 0.97 0.108 83 0.43 2.15 0.034 0.759 0.048 0.018 0.089 0.194 2.19 0.98 0.109 84 0.43 2.16 0.037 0.675 0.050 0.011 0.091 0.176 2.19 0.96 0.109 85 0.44 2.01 0.051 0.294 0.050 0.014 0.091 0.180 2.13 1.03 0.111 86 0.44 2.02 0.046 0.408 0.048 0.012 0.092 0.165 2.23 1.01 0.111 87 0.44 2.03 0.048 0.349 0.046 0.016 0.090 0.185 2.38 1.08 0.111 88 0.44 2.04 0.042 0.502 0.044 0.024 0.092 0.172 2.20 1.03 0.111 89 0.44 2.05 0.043 0.480 0.043 0.032 0.089 0.197 2.17 1.08 0.110 90 0.44 2.06 0.040 0.576 0.042 0.034 0.093 0.160 2.16 0.98 0.110 91 0.44 2.07 0.040 0.568 0.041 0.040 0.091 0.179 2.21 1.04 0.110 92 0.44 2.08 0.037 0.644 0.043 0.033 0.089 0.193 2.11 0.96 0.110 93 0.44 2.09 0.033 0.771 0.040 0.049 0.086 0.227 2.18 0.98 0.109 94 0.44 2.10 0.035 0.715 0.042 0.034 0.092 0.173 2.13 0.97 0.109 95 0.44 2.11 0.038 0.625 0.045 0.026 0.091 0.182 2.15 0.93 0.109 96 0.44 2.12 0.034 0.754 0.049 0.011 0.091 0.175 2.25 1.04 0.109 97 0.44 2.13 0.037 0.665 0.054 0.005 0.093 0.167 2.18 1.00 0.108 98 0.45 1.99 0.047 0.366 0.051 0.005 0.092 0.169 2.22 1.01 0.110 99 0.45 2.01 0.044 0.457 0.048 0.014 0.093 0.158 2.12 1.03 0.110 100 0.45 2.02 0.043 0.466 0.046 0.017 0.092 0.170 2.29 1.01 0.111 No. f (P ) W 10 P α DP-IP SP-IP f b x 0 σ D P D P D P [%] [%] 101 0.45 2.03 0.042 0.514 0.043 0.026 0.095 0.142 2.11 1.08 0.109 102 0.45 2.04 0.037 0.674 0.039 0.066 0.088 0.208 2.11 0.96 0.109 103 0.45 2.05 0.041 0.555 0.040 0.055 0.092 0.172 2.11 1.02 0.110 104 0.45 2.06 0.040 0.584 0.047 0.015 0.095 0.146 2.19 1.00 0.109 105 0.45 2.07 0.038 0.623 0.043 0.037 0.092 0.173 2.19 0.96 0.109 106 0.45 2.08 0.035 0.739 0.050 0.013 0.094 0.152 2.14 1.01 0.109 107 0.45 2.09 0.033 0.771 0.049 0.019 0.089 0.199 2.09 1.04 0.108 108 0.46 1.97 0.045 0.426 0.048 0.010 0.089 0.192 2.18 1.00 0.111 109 0.46 1.98 0.043 0.477 0.049 0.007 0.086 0.237 2.11 1.09 0.110 110 0.46 1.99 0.042 0.504 0.048 0.011 0.090 0.193 2.14 1.01 0.109 111 0.46 2.00 0.041 0.544 0.045 0.020 0.089 0.206 2.27 0.97 0.109 112 0.46 2.01 0.039 0.613 0.043 0.034 0.086 0.228 2.12 0.99 0.109 113 0.46 2.02 0.042 0.502 0.044 0.026 0.090 0.191 2.09 1.05 0.110 114 0.46 2.03 0.038 0.639 0.044 0.030 0.090 0.188 2.11 0.96 0.110 115 0.46 2.04 0.037 0.654 0.046 0.018 0.092 0.166 2.17 0.97 0.109 116 0.46 2.05 0.033 0.778 0.048 0.014 0.088 0.210 2.09 1.00 0.109 117 0.47 1.95 0.045 0.440 0.050 0.007 0.087 0.236 2.19 0.97 0.110 118 0.47 1.96 0.045 0.437 0.051 0.006 0.089 0.198 2.15 1.04 0.110 119 0.47 1.97 0.039 0.603 0.047 0.014 0.083 0.264 2.09 0.93 0.109 120 0.47 1.98 0.042 0.520 0.046 0.017 0.088 0.205 2.25 1.01 0.109 121 0.47 1.99 0.041 0.530 0.045 0.018 0.089 0.198 2.21 0.99 0.110 122 0.47 2.00 0.042 0.514 0.048 0.016 0.085 0.236 2.13 1.05 0.109 123 0.47 2.01 0.039 0.603 0.046 0.017 0.089 0.214 2.19 0.97 0.109 124 0.47 2.02 0.036 0.704 0.051 0.007 0.087 0.226 2.10 1.05 0.108 125 0.47 2.03 0.036 0.701 0.050 0.013 0.087 0.218 2.06 0.96 0.109 126 0.47 2.05 0.037 0.678 0.056 0.006 0.086 0.241 2.16 0.99 0.108 127 0.48 1.91 0.048 0.355 0.053 0.009 0.095 0.140 2.23 1.07 0.110 128 0.48 1.93 0.043 0.467 0.051 0.008 0.089 0.200 2.17 1.04 0.110 129 0.48 1.95 0.042 0.525 0.046 0.019 0.088 0.211 2.25 0.97 0.109 130 0.48 1.96 0.040 0.560 0.046 0.016 0.087 0.218 2.34 0.97 0.110 131 0.48 1.97 0.040 0.582 0.045 0.023 0.090 0.191 2.17 1.00 0.109 132 0.48 1.98 0.037 0.678 0.046 0.020 0.089 0.196 2.24 1.02 0.109 133 0.48 2.00 0.034 0.754 0.052 0.007 0.085 0.251 2.07 1.03 0.108 134 0.48 2.01 0.035 0.721 0.055 0.008 0.087 0.225 2.17 0.99 0.108 135 0.49 1.93 0.042 0.518 0.047 0.013 0.087 0.220 2.29 0.92 0.109 136 0.49 1.94 0.043 0.466 0.047 0.013 0.089 0.203 2.24 0.96 0.109 137 0.49 1.95 0.035 0.716 0.046 0.015 0.083 0.275 2.17 1.00 0.109 138 0.49 1.97 0.038 0.643 0.049 0.011 0.089 0.206 2.22 1.01 0.108 139 0.50 1.91 0.042 0.515 0.049 0.008 0.091 0.183 2.21 0.96 0.109 140 0.50 1.92 0.041 0.531 0.047 0.012 0.087 0.215 2.19 1.04 0.109 141 0.50 1.93 0.038 0.625 0.050 0.009 0.089 0.200 2.19 0.97 0.108 142 0.51 1.88 0.044 0.455 0.052 0.005 0.094 0.153 2.10 1.04 0.110 143 0.51 1.89 0.041 0.548 0.050 0.007 0.091 0.180 2.20 1.08 0.108 144 0.51 1.90 0.038 0.619 0.051 0.006 0.090 0.195 2.20 1.02 0.108 145 0.51 1.91 0.036 0.685 0.052 0.005 0.089 0.198 2.22 0.93 0.109 146 0.52 1.89 0.039 0.601 0.054 0.006 0.090 0.191 2.11 1.02 0.108 147 0.52 1.90 0.040 0.589 0.052 0.005 0.086 0.234 2.19 0.99 0.108 Table B6 : B6Simulation results of the pulsar beam for opening angle distribution Eq. (14), inclination angle distribution function Eq. (6) and period distribution function Eq. (8). Graphical representation of this class of solutions is represented inFig. C5.No. f (P ) W 10 P α DP-IP SP-IP f b x 0 σ D P D P D P [%] [%] 1 0.38 2.51 0.071 0.047 0.049 0.009 0.083 0.266 2.29 1.07 0.110 P 0.30±0.05 ; Kijak & Gil (1997, ACKNOWLEDGMENTS Lyne A.G., Manchester R. N., 1988, MNRAS, 234, 477 Lyne A.G., Manchester R.N., Lorimer D. R., et al., 1998, MNRAS, 295, 743L Manchester R.N., Lyne A.G., 1977, MNRAS, 181, 761 (ML77) Manchester R.N., Taylor J.H., 1977, Pulsars Manchester R.N., Lyne A.G., D'Amico N., et al., 1996, MNRAS, 279, 1235 Manchester R.N., Lyne A.G., Camilo F., et al., 2001 , 1972, ApJ, 172, 435 Roberts D.H., Sturrock P.A., 1973, ApJ, 181, 161 Taylor H., Manchester R.N., Lyne A.G., 1993, ApJS, 88, 529 Tauris T.M., Manchester R.N., 1998	[]
[ "Strangeness Excitation Function in Heavy Ion Collisions", "Strangeness Excitation Function in Heavy Ion Collisions" ]	[ "Johann Rafelski ", "Jean Letessier ", "\nDepartment of Physics\nUniversity of Arizona\n85721TucsonArizonaUSA\n", "\nLaboratoire de Physique Théorique et Hautes Energies\nCERN-TH\n1211-Geneva 23Switzerland\n", "\nUniversité Paris\n7, 2 place JussieuF-75251Cedex 05\n" ]	[ "Department of Physics\nUniversity of Arizona\n85721TucsonArizonaUSA", "Laboratoire de Physique Théorique et Hautes Energies\nCERN-TH\n1211-Geneva 23Switzerland", "Université Paris\n7, 2 place JussieuF-75251Cedex 05" ]	[]	We study as function of energy strangeness created in relativistic heavy ion collisions. We consider statistical hadronization with chemical freeze-out in both equilibrium and nonequilibrium. We obtain strangeness per baryon and per entropy in the energy range 8.75 < ∼ √ sNN < ∼ 200 A GeV. A baryon density independent evaluation of the kaon to pion ratio is presented. PACS numbers: 12.38.Mh,24.10.Pa, Production of strange hadrons in heavy ion collisions has been predicted to be sensitive to deconfinement. Firstly, there is the establishment by way of gluon fusion reaction gg → ss of an abundant supply of strange quarks and antiquarks. Once the quark-gluon plasma (QGP) cools to the point of hadronization, in a second and independent step the final state hadrons are made in a recombination-fragmentation process[1]. A specific deconfinement feature is the enhanced production of strange antibaryons, increasing with strangeness content, a feature seen in recent experiments[2].We analyze here experimental results in search for discontinuities in excitation of strangeness as function of reaction energy. The production yields of (strange) hadrons are studied in several experiments at the RHIC and at the SPS. Understanding of hadron yields in terms of phase space densities allows us to evaluate the global properties of all particles produced [3]. The particle production can be well described in a very large range of yields solely by evaluating the accessible phase space size, when including many hadron resonances[4].In this work, we consider chemical equilibrium and non-equilibrium, i.e., we allow quark pair phase space occupancies, for light quarks γ q = 1, and/or strange quarks γ s = 1 [5] and we require balance of strange and antistrange quark content[6]. There are two independent fit parameters when we assume complete chemical equilibrium, the chemical freeze-out temperature T and µ b the baryochemical potential (or equivalently, the quark fugacity λ q = e µ b /3T ). Adding the possibility that the number of strange quark pairs is not in chemical equilibrium, γ s = 1, we have 3 parameters, and allowing also that light quark pair number is not in chemical equilibrium, we have 4 parameters. These three alternatives will be coded as open triangles (green online), open squares (violet) and filled squares (red), respectively, in the results we present graphically.Statistical hadronization cannot be modeled completely today, as we neither know all hadron resonances, nor do we know the required branching ratios of resonance decays. This introduces arbitrariness in the model which can lead to discrepancies between hadronization analysis results. To estimate the systematic error, we have performed a study varying the pion yield artificially by a factor 0.8 < f π < 1.2. One can infer from this study that, if a relatively large hadronization temperature is reported, the presumption must be made that the statistical hadronization program used does not produce as many pions as required by an extrapolation of resonance mass spectrum and resonance decay pattern. Our computed yields have been cross-checked (for the chemical equilibrium variant) as noted in acknowledgments.At RHIC, the baryon yield is found to contain many strange baryons. Not all results have been corrected for ensuing weak decays. We assume in our approach that 50% of weak decays from Ξ to Λ and from Ω to Ξ are inadvertently included in the yields, when these had not been corrected for such decays. We further assume that pions from such weak decays are not included in the experimental yields, as these pions can clearly be shown to originate outside the interaction vertex.We have carried out a RHIC-200 (i.e.,	10.1063/1.1843597	[ "https://arxiv.org/pdf/hep-ph/0308154v2.pdf" ]	118,468,491	hep-ph/0308154	258315f2cbaae468ce6eef10357c6699f9e4b527	Strangeness Excitation Function in Heavy Ion Collisions Oct 2003 Johann Rafelski Jean Letessier Department of Physics University of Arizona 85721TucsonArizonaUSA Laboratoire de Physique Théorique et Hautes Energies CERN-TH 1211-Geneva 23Switzerland Université Paris 7, 2 place JussieuF-75251Cedex 05 Strangeness Excitation Function in Heavy Ion Collisions Oct 2003(Dated: August 14, 2003)arXiv:hep-ph/0308154v2 6 We study as function of energy strangeness created in relativistic heavy ion collisions. We consider statistical hadronization with chemical freeze-out in both equilibrium and nonequilibrium. We obtain strangeness per baryon and per entropy in the energy range 8.75 < ∼ √ sNN < ∼ 200 A GeV. A baryon density independent evaluation of the kaon to pion ratio is presented. PACS numbers: 12.38.Mh,24.10.Pa, Production of strange hadrons in heavy ion collisions has been predicted to be sensitive to deconfinement. Firstly, there is the establishment by way of gluon fusion reaction gg → ss of an abundant supply of strange quarks and antiquarks. Once the quark-gluon plasma (QGP) cools to the point of hadronization, in a second and independent step the final state hadrons are made in a recombination-fragmentation process[1]. A specific deconfinement feature is the enhanced production of strange antibaryons, increasing with strangeness content, a feature seen in recent experiments[2].We analyze here experimental results in search for discontinuities in excitation of strangeness as function of reaction energy. The production yields of (strange) hadrons are studied in several experiments at the RHIC and at the SPS. Understanding of hadron yields in terms of phase space densities allows us to evaluate the global properties of all particles produced [3]. The particle production can be well described in a very large range of yields solely by evaluating the accessible phase space size, when including many hadron resonances[4].In this work, we consider chemical equilibrium and non-equilibrium, i.e., we allow quark pair phase space occupancies, for light quarks γ q = 1, and/or strange quarks γ s = 1 [5] and we require balance of strange and antistrange quark content[6]. There are two independent fit parameters when we assume complete chemical equilibrium, the chemical freeze-out temperature T and µ b the baryochemical potential (or equivalently, the quark fugacity λ q = e µ b /3T ). Adding the possibility that the number of strange quark pairs is not in chemical equilibrium, γ s = 1, we have 3 parameters, and allowing also that light quark pair number is not in chemical equilibrium, we have 4 parameters. These three alternatives will be coded as open triangles (green online), open squares (violet) and filled squares (red), respectively, in the results we present graphically.Statistical hadronization cannot be modeled completely today, as we neither know all hadron resonances, nor do we know the required branching ratios of resonance decays. This introduces arbitrariness in the model which can lead to discrepancies between hadronization analysis results. To estimate the systematic error, we have performed a study varying the pion yield artificially by a factor 0.8 < f π < 1.2. One can infer from this study that, if a relatively large hadronization temperature is reported, the presumption must be made that the statistical hadronization program used does not produce as many pions as required by an extrapolation of resonance mass spectrum and resonance decay pattern. Our computed yields have been cross-checked (for the chemical equilibrium variant) as noted in acknowledgments.At RHIC, the baryon yield is found to contain many strange baryons. Not all results have been corrected for ensuing weak decays. We assume in our approach that 50% of weak decays from Ξ to Λ and from Ω to Ξ are inadvertently included in the yields, when these had not been corrected for such decays. We further assume that pions from such weak decays are not included in the experimental yields, as these pions can clearly be shown to originate outside the interaction vertex.We have carried out a RHIC-200 (i.e., We study as function of energy strangeness created in relativistic heavy ion collisions. We consider statistical hadronization with chemical freeze-out in both equilibrium and nonequilibrium. We obtain strangeness per baryon and per entropy in the energy range 8.75 < ∼ √ sNN < ∼ 200 A GeV. A baryon density independent evaluation of the kaon to pion ratio is presented. Production of strange hadrons in heavy ion collisions has been predicted to be sensitive to deconfinement. Firstly, there is the establishment by way of gluon fusion reaction gg → ss of an abundant supply of strange quarks and antiquarks. Once the quark-gluon plasma (QGP) cools to the point of hadronization, in a second and independent step the final state hadrons are made in a recombination-fragmentation process [1]. A specific deconfinement feature is the enhanced production of strange antibaryons, increasing with strangeness content, a feature seen in recent experiments [2]. We analyze here experimental results in search for discontinuities in excitation of strangeness as function of reaction energy. The production yields of (strange) hadrons are studied in several experiments at the RHIC and at the SPS. Understanding of hadron yields in terms of phase space densities allows us to evaluate the global properties of all particles produced [3]. The particle production can be well described in a very large range of yields solely by evaluating the accessible phase space size, when including many hadron resonances [4]. In this work, we consider chemical equilibrium and non-equilibrium, i.e., we allow quark pair phase space occupancies, for light quarks γ q = 1, and/or strange quarks γ s = 1 [5] and we require balance of strange and antistrange quark content [6]. There are two independent fit parameters when we assume complete chemical equilibrium, the chemical freeze-out temperature T and µ b the baryochemical potential (or equivalently, the quark fugacity λ q = e µ b /3T ). Adding the possibility that the number of strange quark pairs is not in chemical equilibrium, γ s = 1, we have 3 parameters, and allowing also that light quark pair number is not in chemical equilibrium, we have 4 parameters. These three alternatives will be coded as open triangles (green online), open squares (violet) and filled squares (red), respectively, in the results we present graphically. Statistical hadronization cannot be modeled completely today, as we neither know all hadron resonances, nor do we know the required branching ratios of resonance decays. This introduces arbitrariness in the model which can lead to discrepancies between hadronization analysis results. To estimate the systematic error, we have performed a study varying the pion yield artificially by a factor 0.8 < f π < 1.2. One can infer from this study that, if a relatively large hadronization temperature is reported, the presumption must be made that the statistical hadronization program used does not produce as many pions as required by an extrapolation of resonance mass spectrum and resonance decay pattern. Our computed yields have been cross-checked (for the chemical equilibrium variant) as noted in acknowledgments. At RHIC, the baryon yield is found to contain many strange baryons. Not all results have been corrected for ensuing weak decays. We assume in our approach that 50% of weak decays from Ξ to Λ and from Ω to Ξ are inadvertently included in the yields, when these had not been corrected for such decays. We further assume that pions from such weak decays are not included in the experimental yields, as these pions can clearly be shown to originate outside the interaction vertex. We have carried out a RHIC-200 (i.e., √ s N N = 200 A GeV) Au-Au reaction analysis based on reported BRAHMS [7,8], PHENIX [9], PHOBOS [10], STAR [11,12,13], results for π, h − , p,p, K, K, K * , φ, Ω, Ω yields. We have further reanalyzed the extensive RHIC-130 Au-Au results (compare [14]) both in order to account for latest resonance yields and to make sure that there is no significant change introduced by the refinements made in the hadronization program. For the SPS, we take results obtained with Pb beams reacting with Pb stationary target at √ s N N = 8.75, 12.25, 17.2 GeV (projectile energy 40, 80, and 158 A GeV). We use here the SPS NA49-experiment 4π particle multiplicity results [15,16], which include π ± , K, K, Λ, Λ, φ at 40, 80, 158A GeV. We also fit (relative) yields of Ξ, Ξ, Ω, Ω when available. Since we fit 4π-particle yields, no information about the collective flow velocity is obtained. baryochemical potential µ b , strangeness chemical potential µS, the quark occupancy parameters γq and γs/γq, and the statistical significance of the fit. The star () indicates that there is an upper limit on the value of γ 2 q < e mπ /T (on left), and/or that the value is set (on right). We present the RHIC freeze-out statistical parameters in table I. These results are incorporating statistical and systematic errors in the data, along with errors in the statistical hadronization theory, arising from the above described uncertainty of the pion yield. The bottom line in table I presents the statistical significance. The introduction of the full chemical nonequilibrium (on left) reduces by factor two the value of χ 2 . Even when allowing for all systematic uncertainties and theoretical uncertainty regarding the pion yield, the nonequilibrium fit is a more compelling considering the (presently) more data rich RHIC-130 system. The confidence level of the chemical equilibrium approach remains here at 15%. At RHIC-200, the current data can be interpreted within a chemical equilibrium model, since the decisive results on relative yields of (multi)strange (anti)hyperons are not yet available. √ sNN [GeV] 200 130 200 130 T [MeV] 143 ± 7 144 ± 3 160 ± 8 160 ± 4 µ b [MeV] 21.5 ± 1 29.2 ± 1.5 24.5 ± 1 31.4 ± 1.5 µS [MeV] 4.7 ± 0.4 6.6 ± 0.4 5.3 ± 0.4 6.9 ± 0.4 γq 1.6 ± 0.3 1.6 ± 0.2 * 1 * 1 * γs/γq 1.2 ± 0.15 1.3 ± 0.1 1.0 ± 0 The statistical hadronization is expected to describe particle production well in presence of a sudden QGP breakup. Statistical hadronization approach is not appropriate when there is a lot of hadron-hadron rescattering in the final state, which may be the case at low SPS energies and below. In this case, kinetic models need to be applied, introducing a multitude of freeze-out conditions depending on the nature of particle considered. Accordingly, when we fit the low energy SPS results, a less favorable χ 2 is found than at RHIC, or at the top SPS energy. However, the stability of the physical properties we extract when we subject the system to perturbations regarding pion yield indicate that results concerning the physical properties of the hadron source are reliable. With the statistical parameters fixed, the properties of the fireball can be computed, evaluating the contributions made by each of the hadronic particle species. We first show how this works considering the energy stopping in figure 1. This result allows us to represent below our further findings as function of E th iN N , the specific per baryon pair thermal energy available at the time of hadronization. This dependence substitutes for the de- pendence on √ s N N , the initial energy per baryon pair brought into the reaction. We see in figure 1, that the chemical equilibrium fit (open triangles) shows counterintuitive behavior. We note that the fraction of the energy per baryon pair which is initially thermalized is obtained from this result by adding the kinetic energy of the collective matter flow. Using at RHIC-200 as the average transverse flow velocity v 2 ⊥ = 0.45 2 (and smaller values at smaller collision energies) we find a ≃ 20% correction. Thus, we see that 50 ± 5% of the energy per baryon carried into the reaction is initially made available to the thermal degrees of freedom, and this value is independent of the collision energy in the entire SPS and RHIC range 9 < √ s N N < 200 GeV. This means that the baryon stopping is two times larger then energy stopping. We are now ready to consider the final state specific per baryon strangeness yield. We evaluate strangeness yield s, the number of produced strange quark pairs and divide it by the similarly computed thermal fireball baryon number content b. b is obtained summing the net (particle minus antiparticle) yields of all nucleons, (multi strange) hyperons and their resonances. The baryon number b is naturally conserved. Strangeness is predominantly produced in the early stage of the reaction, when the density and temperature are highest, and there is little change in this yield during the late fireball evolution. Thus, s/b ratio probes directly the extreme initial conditions. Consideration of this yield ratio eliminates the absolute yield normalization parameter (sometimes but not always 'reaction' volume), as well as some uncertainties originating in the experimental particle yield data, which propagate through the analysis of experimental data. In figure 2, we show s/b as function of the thermal specific energy content E th i N N . Strangeness yield is continuous in the entire energy domain as is shown by the solid line drawn to guide the eye. The rise of specific strangeness yield is slightly faster than linear with energy, as is indicated in the figure insert. (The appearance of an exponential shape is due to logarithmic energy scale in figure 2). The results for SPS energy range are in quantitative agreement with predictions made assuming a QGP state of matter and gluon fusion strangeness formation mechanism, compare figure 38 in Ref. [20]. The SPS specific strangeness yield extrapolates smoothly to the RHIC energy range. The reaction mechanism producing strangeness and stopping baryon number are evolving in parallel yielding a smooth change in the ratio of both variables. The highest energy √ s N N =200 GeV data point in figure 2 seems to be slightly lower than the presented extrapolation predicts. We think that this yield will increase once we have included in the statistical hadronization analysis the hyperon yield ratios Ξ/Λ, Λ/p. The presence, in the global fit, of these particle ratios will increase the fitted strangeness yield, provided that their measured values are comparable in magnitude to those reported at √ s N N =130 GeV. An important feature of the RHIC experimental results, shown in figure 2, is the large strangeness yield per participating baryon. Although it has been early on noted that at RHIC-130, up to 8 strange quark pairs per baryon are produced [17], little attention has been given to this high yield. At RHIC-2000 each interacting baryon pair produces about 20 strange quark-antiquark pairs. An interesting question is how this great increase in yield compares to the production of particles in general. The Wróblewski ratio W s ≡ 2 ss / uū+ dd is often used in such a comparison. Recently it has been realized that this ratio can be artificially enhanced by high baryon density which can suppress light quark pair yields [18]. Therefore we here compare the strangeness production to the global entropy S yield. We evaluate S in the same way as we have obtained other global properties. There is an additional nonequilibrium entropy term to be allowed for when chemical non-equilibrium prevails. Both entropy S and strangeness s are nearly conserved in the hydrodynamical expansion of QGP and increase only moderately in the hadronization of QGP. The observed ratio s/S is established by microscopic reactions operational in the early stages of the heavy ion collision. Strangeness per entropy, s/S, is presented in figure 3 as function of the specific thermal energy content. The horizontal line is the maximum SPS yield base. We note the modest smooth rise in the SPS energy domain. Most interestingly, at RHIC, as compared to SPS, an unexpected 50% increase in s/S is noted, allowing for chemical nonequilibrium. The excitation of strangeness seems to rise faster with energy than the production of entropy. A possible explanation of this phenomenon is that the hot initial state, in which the threshold in energy for strangeness formation has been overcome, lives longer when formed at RHIC conditions. However, it appears important to fill the energy range between SPS and RHIC with data in search of a new physics energy threshold. Based on the study of the K + /π + particle yield ratio [16], such a new physics energy threshold had been expected below 40A GeV, at the lowest SPS energy, i.e., below the energy range we could consider in the global hadron freeze-out analysis. Inspecting available particle yields, we note that their dependence on the unknown baryon density is significant, mimicking new physics. This baryon density effect can be greatly reduced considering the product of particle and antiparticle yields. In this case, the chemical potential cofactors in particle yields cancel, being inverse of each other, and the baryon density effect largely disappears. To study kaon to pion ratio we thus form: K/π ≡ (K + K − )/(π + π − ), and effectively 'remove' baryon density effect present in the individual ratios K + /π + and K − /π − . Using the NA49 data set [19], we present the K/π ratio in figure 4. The filled squares are for the 4-π full phase space data set. The filled circles are for the central rapidity results, including RHIC-130 and RHIC-200 data [8]. The central rapidity NA49 results are from figure 6 in [19]. The pp charged K + /π + > K/π background is shown by open squares, and we indicate for √ s = 1800 the p-p TEVA-TRON point [21] which is at an energy beyond the range considered. The lines, in figure 4, guide the eye to the trends of the alternate gradient synchrotron (AGS) and SPS results. These trends of behavior intersect within the SPS low energy range below the lowest NA49 SPS point which was obtained at 30AGeV. However, this appears here to be a smooth transition from a rise to saturation of the K/π production. There is a clear enhancement of the K/π ratio at RHIC compared both with the AGS/SPS trend and with the K + /π + > K/π measured in elementary pp collisions. This enhancement coincides with the specific strangeness per entropy, s/S enhancement seen in figure 3. We have shown that the strangeness per baryon excitation in the fireball of dense matter formed at SPS and RHIC energies is a smooth function of energy, but we find a step-up in strangeness per entropy between the SPS and RHIC energy ranges, also visible in the rise of the K/π ratio. Thus there are two energy domains to investigate for threshold behavior, the 35+35 GeV RHIC range and the 20A GeV on fixed target low SPS energy, where the K/π ratio saturates. We further noted the quantitative agreement in the strangeness yield with predictions made for the SPS energy range, assuming QGP production mechanisms. We have shown that 20 strange-antistrange quark pairs are made per colliding baryon pair at the top RHIC energy. Note added More technical details are now available in Ref. [22]. A systematic study of SPS results by a yet another group has just appeared [23], and the numerical results shown there agree with our SPS results. Fig 10 shows that γ q = 1 is actually a local fit maximum for the top SPS energy. PACS numbers: 12.38.Mh,24.10.Pa,25.75.-q FIG. 1 : 1Fraction of energy stopping at SPS and RHIC: results are shown for 40, 80, 158A GeV Pb-Pb, 200A GeV S-W/Pb reactions and at RHIC for 65+65A GeV Au-Au interactions. Connecting lines guide the eye (color online). FIG. 2 : 2Strangeness per thermal baryon participant, s/b as function of thermal specific energy content E th i NN . FIG. 3 : 3Strangeness per entropy, s/S, as function of the thermal specific energy content E th i NN . FIG. 4 : 4K/π ratio as function of collision energy. Filled symbols are for nuclear and open for K + /π + measured in elementary pp collisions. Squares denote the full multiplicity ratio and circles the central rapidity yield ratio. TABLE I : IThe chemical freeze-out statistical parameters found for nonequilibrium (left) and semi equilibrium (right) fits to RHIC results. We show √ sNN , the temperature T , R. 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[ "Robust Observer Design for Polytopic Discrete-Time Nonlinear Descriptor Systems ⋆", "Robust Observer Design for Polytopic Discrete-Time Nonlinear Descriptor Systems ⋆" ]	[ "T J Meijer \nDepartment of Mechanical Engineering\nEindhoven University of Technology\nEindhovenThe Netherlands\n", "V S Dolk \nASML\nDe Run 66655504 DTVeldhovenThe Netherlands\n", "M. SChong \nDepartment of Mechanical Engineering\nEindhoven University of Technology\nEindhovenThe Netherlands\n", "W P M H Heemels \nDepartment of Mechanical Engineering\nEindhoven University of Technology\nEindhovenThe Netherlands\n" ]	[ "Department of Mechanical Engineering\nEindhoven University of Technology\nEindhovenThe Netherlands", "ASML\nDe Run 66655504 DTVeldhovenThe Netherlands", "Department of Mechanical Engineering\nEindhoven University of Technology\nEindhovenThe Netherlands", "Department of Mechanical Engineering\nEindhoven University of Technology\nEindhovenThe Netherlands" ]	[]	This paper considers the design of robust state observers for a class of slope-restricted nonlinear descriptor systems with unknown time-varying parameters belonging to a known set. The proposed design accounts for process disturbances and measurement noise, while allowing for a trade-off between transient performance and sensitivity to noise and parameter mismatch. We exploit a polytopic structure of the system to derive linear-matrix-inequality-based synthesis conditions for robust parameter-dependent observers for the entire parameter set. In addition, we present (alternative) necessary and sufficient synthesis conditions for an important subclass within the considered class of systems and we show the effectiveness of the design for a numerical case study. (M. S. Chong), [email protected] (W. P. M. H. Heemels).based joint parameter and state estimation schemes, see, e.g.,[5,18]. As such, the presented approach is relevant for a wide range of applications, especially, where the true parameters and/or the parameter estimates vary over time. Since the parameters belong to a known set, we aim to design observers that guarantee robustness for any possible (future) values of the time-varying true/estimated parameters. We will exploit a polytopic structure of the system, see also[12,24], to develop LMI-based conditions to synthesize such observers offline. The proposed observer is shown to guarantee ISS with respect to a general form of model mismatch. For some applications, e.g., joint parameter and state estimation[5,18], we are interested in ISS with respect to the parameter mismatch itself, which, as we will show, can also be guaranteed under (natural) additional assumptions. Observer designs have been proposed for descriptor systems, see, e.g.,[10,11]; for several classes of slope-restricted nonlinear (non-descriptor) systems, see, e.g.,[6,13,23]. These works consider nominal models, i.e., without uncertain parameters[6,13,23], or assume that the exact parameters are known[10,11]. Our design considers three types of uncertainties, namely (a) process disturbances, (b) measurement noise and (c) parameter mismatch, and allows for a trade-off between transient performance and sensitivity to each type, which therefore differs from these existing works, see also Section 5.1.1. Moreover, we propose alternative LMI-based	null	[ "https://arxiv.org/pdf/2207.04290v1.pdf" ]	250,426,659	2207.04290	9085c2701d0c44adae12672b56de2cc419fe5f0f	Robust Observer Design for Polytopic Discrete-Time Nonlinear Descriptor Systems ⋆ T J Meijer Department of Mechanical Engineering Eindhoven University of Technology EindhovenThe Netherlands V S Dolk ASML De Run 66655504 DTVeldhovenThe Netherlands M. SChong Department of Mechanical Engineering Eindhoven University of Technology EindhovenThe Netherlands W P M H Heemels Department of Mechanical Engineering Eindhoven University of Technology EindhovenThe Netherlands Robust Observer Design for Polytopic Discrete-Time Nonlinear Descriptor Systems ⋆ arXiv:2207.04290v1 [math.OC] 9 Jul 2022Robust state observernonlinear estimationlinear matrix inequalities (LMIs)circle criterionslope-restricted This paper considers the design of robust state observers for a class of slope-restricted nonlinear descriptor systems with unknown time-varying parameters belonging to a known set. The proposed design accounts for process disturbances and measurement noise, while allowing for a trade-off between transient performance and sensitivity to noise and parameter mismatch. We exploit a polytopic structure of the system to derive linear-matrix-inequality-based synthesis conditions for robust parameter-dependent observers for the entire parameter set. In addition, we present (alternative) necessary and sufficient synthesis conditions for an important subclass within the considered class of systems and we show the effectiveness of the design for a numerical case study. (M. S. Chong), [email protected] (W. P. M. H. Heemels).based joint parameter and state estimation schemes, see, e.g.,[5,18]. As such, the presented approach is relevant for a wide range of applications, especially, where the true parameters and/or the parameter estimates vary over time. Since the parameters belong to a known set, we aim to design observers that guarantee robustness for any possible (future) values of the time-varying true/estimated parameters. We will exploit a polytopic structure of the system, see also[12,24], to develop LMI-based conditions to synthesize such observers offline. The proposed observer is shown to guarantee ISS with respect to a general form of model mismatch. For some applications, e.g., joint parameter and state estimation[5,18], we are interested in ISS with respect to the parameter mismatch itself, which, as we will show, can also be guaranteed under (natural) additional assumptions. Observer designs have been proposed for descriptor systems, see, e.g.,[10,11]; for several classes of slope-restricted nonlinear (non-descriptor) systems, see, e.g.,[6,13,23]. These works consider nominal models, i.e., without uncertain parameters[6,13,23], or assume that the exact parameters are known[10,11]. Our design considers three types of uncertainties, namely (a) process disturbances, (b) measurement noise and (c) parameter mismatch, and allows for a trade-off between transient performance and sensitivity to each type, which therefore differs from these existing works, see also Section 5.1.1. Moreover, we propose alternative LMI-based Introduction In this paper, we propose a state observer for a class of slope-restricted nonlinear descriptor systems with time-varying uncertain parameters that is robust in the sense that it is input-to-state stable (ISS) [15] with respect to process disturbances, measurement noise and parameter mismatch. We consider a problem setting in which the true parameter values are unknown, however, they belong to a given compact set and a (possibly) time-varying estimate of the parameters (also in the set) is available. The model class, in particular with a parameter-dependent descriptor matrix, emerges when using implicit discretization schemes, even if the underlying continuous-time system does not feature a parameter-dependent descriptor matrix. The considered robust state estimation problem is relevant in its own right, for instance, to enable the synchronization of digital twins to their physical counterparts, see, e.g., [20]. In addition, state observers are essential, for instance, in the context of output-based feedback stabilization, see, e.g., [12,24], and in certain sampling-synthesis conditions for an important special case of systems, with a parameter-independent descriptor matrix, which is also considered in, e.g., [2,4]. For the latter class of systems, we show that the proposed conditions are not only sufficient, but also necessary thereby forming a strong indicator for the non-conservativeness of our design. To the best of our knowledge, no necessary and sufficient LMI-based synthesis conditions were available in the literature for this subclass of systems featuring uncertain parameters as well as slope-restricted nonlinearities. Interestingly, our results also reveal that for the considered subclass of systems, unlike other observer designs for polytopic systems in the literature, the structure obtained from our synthesis conditions does not lead to any additional conservatism compared to using observer gains which can depend arbitrarily on the parameters. The remainder of this paper is organized as follows. Section 3 states the problem definition and Section 4 motivates the considered model class. Section 5 presents our observer design, both of the aforementioned synthesis conditions and a numerical example, which demonstrates the effectiveness of the design. Finally, Section 6 provides some conclusions. All proofs are provided in the Appendix. Preliminaries The following notation will be used throughout this paper. The sets of real and natural numbers are denoted by R = (−∞, ∞) and N = {0, 1, 2, . . .}, respectively. Moreover, we denote R 0 = [0, ∞), N n = {n, n + 1, n + 2, . . .} and N [n,m] = {n, n + 1, . . . , m}, n, m ∈ N. The set of symmetric (positive definite) matrices of size n × n is denoted by S n (S n ≻0 ). The notation (u, v) stands for [u ⊤ v ⊤ ] ⊤ , where u ∈ R m and v ∈ R n with n, m ∈ N. Let e i ∈ R n , i ∈ N [1,n] , be the i-th n-dimensional unit vector. For a vector x ∈ R n , x = √ x ⊤ x denotes its Euclidean norm. B n r := {x ∈ R n \| x r} is the set of vectors with their norm bounded by r and Co{p 1 , . . . , p N } ⊂ R m denotes the convex hull of the points {p i } i∈N [1,N ] with p i ∈ R m , i ∈ N [1,N ] . For a matrix A, let He(A) = A + A ⊤ , A ⊤ denotes its transpose, A −⊤ the inverse of its transpose and σ min (A) denotes its smallest singular value. The symbol ⋆ completes a symmetric matrix, e.g., [ A ⋆ B C ] means [ A B ⊤ B C ]. M ≺ 0, M 0 and M ≻ 0 mean, respectively, that M ∈ S n is negative definite, negative semi-definite and positive definite. Let A := sup x =0 Ax / x denote the 2-norm of A, which equals λ max (A ⊤ A) with λ max (A) the largest eigenvalue of A. The symbol I is an identity matrix of appropriate dimensions and diag(D 1 , D 2 , . . . , D n ) is a block diagonal matrix with n diagonal blocks D i , i ∈ N [1,n] . Finally, y ← x means substituting x for y. A continuous function α : R 0 → R 0 is a K-function (α ∈ K), if it is strictly increasing and α (0) = 0. A continuous function β : R 0 × R 0 → R 0 is a KLfunction (β ∈ KL), if β (·, s) ∈ K for each s ∈ R 0 , β (r, ·) is non-increasing and β (r, s) → 0 as s → ∞ for each r ∈ R 0 . For a function x : N → R n , let x k denote the function evaluated at time k ∈ N, i.e., x k := x(k). We sometimes use x + to refer to x k+1 = x(k + 1). Finally, for a sequence {x k } k∈N we denote x ∞,k := max j∈N [0,k] x j and we denote the space of all bounded sequences taking values in R n with n ∈ N by ℓ n ∞ := {{x k } k∈N \| x k ∈ R n , k ∈ N and sup k∈N x k < ∞}. Problem definition Consider a discrete-time nonlinear descriptor system E(p k+1 )x k+1 = A(p k )x k + G(p k )φ(Hx k )+ B(p k )u k + F (p k )v k ,(1)y k = Cx k + Dw k , where x k ∈ R nx , u k ∈ R nu and y k ∈ R ny denote the state, input and output at discrete time k ∈ N. The process noise at time k ∈ N is denoted by v k ∈ R nv and the measurement noise by w k ∈ R nw . The unknown parameter p k ∈ P, k ∈ N, belongs to a known compact set P ⊂ R np and we have E, A : P → R nx×nx , G : P → R nx×n φ , B : P → R nx×nu , F : P → R nx×nv , H ∈ R n φ ×nx , C ∈ R ny×nx , D ∈ R ny×nw and φ : R n φ → R n φ . We assume that E(p) is non-singular for all p ∈ P and, thus, the system (1) admits a unique solution defined for all k ∈ N. In Section 4, we motivate the consideration of a parameter-dependent descriptor matrix E and, in particular, the dependence on the next parameter p k+1 . Assumption 1 The system (1) can be expressed in polytopic form, i.e., there exist functions ξ i : P → R 0 , i ∈ N := N [1,N ] , N ∈ N 1 , such that the mapping ξ : = (ξ 1 , . . . , ξ N ) satisfies ξ(P) ⊂ {µ ∈ R N 0 \| i∈N µ i = 1}, and matrices E i , A i ∈ R nx×nx , G i , H ⊤ i ∈ R nx×n φ , B i ∈ R nx×nu and F i ∈ R nx×nv , i ∈ N , such that, for all p ∈ P, E(p) A(p) B(p) F (p) G(p) = i∈N ξ i (p) Ei Ai Bi Fi Gi .(2) The above holds, e.g., if (1) depends affinely on the parameters and P is (embedded in) a convex polytope, see, e.g., [12] for a class of uncertain LPV systems. Assumption 2 The nonlinearity φ in (1) is slope- restricted, i.e., there exists Λ ∈ S n φ ≻0 such that Φ(x,y) y−x ⊤ 2I ⋆ −Λ 0 Φ(x,y) y−x 0, for all x, y ∈ R n φ ,(3)where Φ(x, y) = φ(y) − φ(x), x, y ∈ R n φ . If, in addition, φ(0) = 0, the slope restriction (3) implies that the nonlinearity is sector-bounded [22]. Given the measured output y k , input u k and an estimatê p k of the unknown parameter p k , our objective is to design an observer for the system (1) that produces, at time k ∈ N, a state estimatex k , which is robust in the sense that the error e k =x k − x k is ISS with respect to the process noise v k , measurement noise w k and the model (and parameter) mismatch (p k =p k − p k ). For the precise definition of ISS, see Definition 1 in Section 5. A motivating example We motivate the consideration of models of the form (1) by showing how they arise when discretizing underlying continuous-time models of the form E c (p(t)) d dt x(t) = A c (p(t))x(t),(4) where p(t) ∈ P, t ∈ R 0 , denotes the continuoustime unknown time-varying parameter vector and E c , A c : P → R nx×nx . For notational compactness, we removed the nonlinearity, input and noise in (4), but they can be included in the developments below, e.g., using a semi-implicit approach [1]. Let t k = kT s , k ∈ N, denote the time at the k-th sampling instance with sampling period T s ∈ R 0 . We discretize (4) using Tustin's method, which combines two approximations of E(p(τ k ))x(τ k ): E c (p(τ k ))x(τ k ) ≈ E c (p k )x k + θE c (p k )ẋ(t k ), (5) E c (p(τ k ))x(τ k ) ≈ E c (p k+1 )x k+1 − θE c (p k+1 )ẋ(t k+1 ), where τ k = (t k + t k+1 )/2, θ = T s /2 and x k ≈ x(t k ), k ∈ N. Equating both approximations and evaluating (4) at both time instances yields (E c (p k+1 ) − θA c (p k+1 )) x k+1 = (E c (p k ) + θA c (p k )) x k .(6) This brief derivation shows that discrete-time descriptor models of the form (1) arise naturally, when discretizing a parameter-dependent continuous-time model, and, in particular, motivates the dependence of the E-matrix in (1) on p k+1 . We stress that models with a parameterdependent E-matrix are prominent in discrete time in the sense that, even if E c is parameter-independent, an implicit discretization scheme yields a parameterdependent E-matrix in discrete time, see (6). The Ematrix of the discrete-time system is non-singular (as required in this paper) under the assumption that θ does not coincide, for any k ∈ N, with any generalized eigenvalue(s) of the pair (A(p k ), E(p k )), i.e., det(E(p k )− θA(p k )) = 0. From this point on we consider discrete-time models of the form (1), which includes the nonlinearity, input and noise. As mentioned before, these terms can be incorporated, e.g., by combining the above with forward Euler for the additional terms in a so-called semi-implicit approach, see, e.g., [1]. Example 1 By applying semi-implicit discretization, as discussed above, to [13, Example 1] (with added noise) we obtain a system of the form (1) with p k = (p 1 k , p 2 k ) ∈ P = [p 1 , p 1 ] × [p 2 , p 2 ] and, for p ∈ P, E(p) = 1+p 2 /2 p 2 /2−2p 1 p 2 1+p 2 , H ⊤ = 1 1 , A(p) = 1−p 2 /2 2p 1 −p 2 /2 −p 2 1−p 2 , G(p) = p 2 2p 2 ,(7)B(p) = F (p) = p 1 1 1 , C = 1 0 , D = 1,k = γ(t k )T s /2 ∈ [p 2 , p 2 ] = [0.0475, 0.0525], k ∈ N. The parameter set P can be taken as a polytope with N = 4 vertices ν 1 = (p 1 , p 2 ), ν 2 = (p 1 , p 2 ), ν 3 = (p 1 , p 2 ) and ν 4 = (p 1 , p 2 ). Subdivide P into two simplices, i.e., triangles, denoted ∆ i := Co{ν i , ν 3 , ν 4 }, i ∈ {1, 2}. The functions ξ i in Assumption 1, i ∈ N , are obtained by transforming to barycentric coordinates within each simplex. Let∆ 1 := ∆ 1 and∆ 2 := ∆ 2 \ Co{ν 3 , ν 4 }, such that∆ 1 ∩∆ 2 = {0}, to obtain piecewise linear functions ξ3(p) ξ4(p) := T −1 i (p − ν i ), for p ∈∆ i , ξ i (p) := 1 − ξ 3 (p) − ξ 4 (p), p ∈∆ i , 0, p / ∈∆ i ,(8) with T i = [ν 3 − ν i ν 4 − ν i ], i ∈ {1, 2}. Since P = ∪ i∈{1,2}∆i , the functions ξ i (p), i ∈ N , are defined for all p ∈ P. Evaluate the matrix-valued functions in (7) at each vertex, e.g., E i = E(ν i ), to obtain E i , A i , G i , B i , F i , i ∈ N . Since E(p) is non-singular for all p ∈ P, it seems convenient to pre-multiply the dynamics in (1) with E −1 (p) to avoid dealing with the descriptor structure. This may destroy the polytopic structure, however, resulting in a more challenging synthesis problem, see, e.g., [10]. Consider, for instance, the matrix inverse, for all p ∈ [0, 1], E −1 (p) = 1−p p p 1−p −1 = 1 1 − 2p 1−p −p −p 1−p ,(9) which is not polytopic despite E(p) being polytopic. We will see that preserving the polytopic structure allows for the derivation of LMI-based synthesis conditions. Robust polytopic observer design We assume that estimatesp k ∈ P, k ∈ N, of the unknown parameter vector p k are available at least one step into the future, i.e.,p k andp k+1 are known/estimated at time k ∈ N. Based on these estimates, we aim to design a state observer of the form E(p k+1 )x k+1 = A(p k )x k − L(p k+1 ,p k )(Cx k − y k )+ B(p k )u k + G(p k )φ(Hx k − K(p k+1 ,p k )(Cx k − y k )),(10) wherex k ∈ R nx denotes the state estimate at time k ∈ N and L : P × P → R nx×ny and K : P × P → R n φ ×ny are to be designed. We will construct these observer gains in terms of polytopic functions, i.e., for allp + ,p ∈ P, L(p + ,p) = i,j∈N ξ i (p)ξ j (p + )L ij ,(11)K(p + ,p) = Z(p + ,p) τ (p + ,p) ,(12) with τ (p + ,p) = i,j∈N ξ i (p)ξ j (p + )τ ij ,(13)Z(p + ,p) = i,j∈N ξ i (p)ξ j (p + )Z ij ,(14) where τ (p + ,p) ∈ R >0 for allp + ,p ∈ P, such that they can be constructed by synthesizing a finite number of variables L ij ∈ R nx×ny , τ ij ∈ R >0 and Z ij ∈ R n φ ×ny , i, j ∈ N . Here, ξ i are the functions in Assumption 1. The observer gain L is polytopic in bothp andp + here. The observer gain K in (12) with (13)- (14) is not necessarily polytopic, however, by taking τ (p + ,p) to be constant, i.e., τ (p + ,p) =τ for someτ ∈ R >0 for allp + ,p ∈ P, any polytopic observer gain can be constructed. Hence, this observer design is more general than when we restrict K to be polytopic a priori as in, e.g., [4,10,12]. In Section 5.2, we will deviate from the structure in (10) and (11) by designing an observer with L, which is polytopic inp, but not necessarily inp + (the dependence may be of a general nature). This introduces additional flexibility in L which we exploit to show that the proposed observer design conditions are necessary and sufficient for an important subclass of systems with a constant E. Below we present conditions to synthesize observer gains as in (11)- (14) that guarantee robust stability, where we take the following procedure: First, we consider robustness in terms of ISS with respect to process disturbances and measurement noise as well as a model mismatch induced by the parameter mismatch, which is often sufficient for output-based feedback control, see, e.g., [12]. In fact, we provide sufficient design conditions for the observer (10)- (14) and general system setup (1), in Section 5.1. Moreover, we present, in Section 5.2, an alternative set of observer design conditions that is not only sufficient, but also necessary for the case with a constant descriptor matrix thereby forming a strong indicator for the non-conservativeness of our design. Second, in Section 5.3, we show that, under extra assumptions, an ISS property with respect to the parameter mismatch itself can be obtained for observers synthesized using either of the proposed conditions. Robustness with respect to model mismatch Let e k :=x k − x k be the state estimation error corresponding to (1) and (10). We group all terms that constitute the model mismatch (induced bỹ p k :=p k − p k ), and denote, for compactness, ψ k = ψ(p k+1 ,p k , p k+1 , p k , x k+1 , x k , u k , v k ) := −(E(p k+1 ) − E(p k+1 ))x k+1 +(A(p k )−A(p k ))x k +(B(p k )−B(p k ))u k + (G(p k ) − G(p k ))φ(Hx k ) + (F (p k ) − F (p k ))v k . We also denoteφ k =φ(p k+1 ,p k , e k , x k , w k ) := φ (H − K(p k+1 ,p k )C)e k + K(p k+1 ,p k )Dw k + Hx k − φ(Hx k ). The error dynamics are then described by E(p k+1 )e k+1 = (A(p k ) − L(p k+1 ,p k )C)e k + G(p k )φ k − F (p k )v k + L(p k+1 ,p k )Dw k + ψ k . (15) Next, we give the definition of ISS [15], which incorporates our aim to achieve robustness for the entire parameter set. Definition 1 The system (15) is said to be ISS with respect to v, w and ψ, if there exist functions β ∈ KL and γ v , γ w , γ ψ ∈ K such that, for all e 0 ∈ R nx , v ∈ ℓ nv ∞ , w ∈ ℓ nw ∞ , ψ ∈ ℓ nx ∞ , {p k } k∈N and {p k } k∈N with p k ,p k ∈ P, e k β( e 0 , k) + γ v ( v ∞,k−1 )+ (16) γ w ( w ∞,k−1 ) + γ ψ ( ψ ∞,k−1 ), k ∈ N. To synthesize observers (10) that render the error system (15) ISS with respect to disturbances, measurement noise and model mismatch, we construct suitable ISS-Lyapunov functions [15]. Inspired by [7,12], we focus on a class of polytopic parameter-dependent quadratic ISS-Lyapunov functions, which leads to a systematic design procedure. Definition 2 The system (15) is said to be absolutely poly-quadratically ISS with respect to v, w and ψ, if it admits a poly-quadratic ISS-Lyapunov function V : R nx × P → R 0 of the form V (e,p) = i∈N e ⊤ ξ i (p)P i e (with P i ∈ S nx , i ∈ N ), satisfying 1 e 2 V (e,p) a e 2 ,(17)V (e + ,p + ) − V (e,p) − e 2 + κ v v 2 + κ w w 2 + κ ψ ψ 2 ,(18)for allφ(p + ,p, e, x, w) = φ((H − K(p + ,p)C)e + K(p + ,p)Dw + Hx) − φ(Hx) with φ satisfying As- sumption 2,p + ,p, p + , p ∈ P, e, x + , x, ψ ∈ R nx , u ∈ R nu , v ∈ R nv and w ∈ R nw , with a ∈ R 1 , κ v , κ w , κ ψ ∈ R >0 and e + satisfying E(p + )e + = (A(p) − L(p + ,p)C)e + G(p)φ(p + ,p, e, x, w) − F (p)v + L(p + ,p)Dw + ψ(p + ,p, p + , p, x + , x, u, v). The existence of a poly-quadratic ISS-Lyapunov function for (15) implies that the error system (15) is ISS, see [12]. By constructing poly-quadratic ISS Lyapunov functions, we derive sufficient LMI-based conditions for (15) to be (absolutely) poly-quadratically ISS, as stated in the theorem below. Theorem 1 Consider system (1) and observer (10) satisfying Assumptions 1 and 2. Let σ := min p∈P σ min (E(p)) and suppose there exist matrices P i ∈ S nx , X i ∈ R nx×nx , Y ij ∈ R nx×ny , Z ij ∈ R n φ ×ny and scalars τ ij , κ v , κ w , κ ψ ∈ R >0 , i, j ∈ N , such that ⋆ ⋆ −(XiFi) ⊤ 0 0 κv I ⋆ ⋆ (Yij D) ⊤ 0 (ΛZij D) ⊤ 0 κwI ⋆ X ⊤ i 0 0 0 0 κ ψ I      =:Mij ≻ 0,(19) for all i, j ∈ N . Then, the matrices X i are non-singular, P i ≻ 0, i ∈ N , and the state estimation error system (15) for system (1) and observer (10) with (11)- (14) and L ij = X −1 i Y ij , i, j ∈ N , is (absolutely) poly-quadratically ISS with respect to v, w and ψ with β(s, r) := √ κ ψ σ √ ρ r s, for some ρ ∈ (0, 1), γ v (s) := √ κ ψ κ v σs, γ w (s) := √ κ ψ κ w σs and γ ψ (s) := κ ψ σs, r, s ∈ R 0 . Moreover, V (e,p) = i∈N e ⊤ ξ i (p)P i e is a poly-quadratic ISS-Lyapunov function for (15) satisfy- ing (17)-(18) with a = κ ψ σ 2 > 1. Theorem 1 provides an LMI condition, with dimensions that scale with N 2 (recall that N is the number of vertices of the polytope from Assumption 1), which enables us to synthesize observer gains L(p + ,p) and K(p + ,p) such that (15) is ISS. When the parameter estimate is 1 Without loss of generality, we can obtain (18) by scaling V to get a coefficient −1 in − e 2 . The lower bound in (17) is guaranteed by (18), since V (e+,p+) 0 and (18) holds with v = w = ψ = 0, and, hence, a 1. constant, i.e.,p k+1 =p k , the condition can be relaxed by setting j = i in (19), such that the dimensions of the LMI scale linearly in N . The ISS-gains can, for example, be minimized via κ v , κ w and κ ψ by the optimization regime minimize c v κ v + c w κ w + c ψ κ ψ , subject to (19),(20) where c v , c w , c ψ ∈ R 0 with c v + c w + c ψ = 1. By tuning the weights c v , c w , c ψ , the proposed conditions allow us to make trade-offs between the transient performance, sensitivity to the different noise levels and model mismatch. In the nominal case (p k = 0, v k = 0, w k = 0, k ∈ N), the observer (10) with the obtained polytopic gains renders the error dynamics (15) globally exponentially stable. Discussion of Theorem 1 relative to other works We discuss Theorem 1 in relation to relevant existing results regarding two aspects: (a) the class of nonlinearities φ and (b) the conservatism of the LMI condition (19). Starting with (a): For systems with a scalar nonlinearity, i.e., where n φ = 1, the variables τ ij fulfill the same role as the diagonal multiplier matrix introduced in [6]. The class of vector-valued (n φ > 1) slope-restricted nonlinearities considered in Assumption 2 is very general and guaranteeing absolute stability with respect to such a general class of nonlinearities may for many applications yield conservative results. As a result, Theorem 1 applies to a wide range of nonlinear systems, however, it typically pays off to exploit specific properties/structure of the nonlinearity if such information is available. For instance, if the entries of the vector-valued nonlinearity are decoupled, i.e., φ i (z) can be written as φ i (z i ) for φ i : R → R, i ∈ N [1,n φ ] , we can deal with each decoupled nonlinearity separately by associating each with its own scalar τ k,ij ∈ R >0 , k ∈ N [1,n φ ] , i, j ∈ N , which leads to a diagonal multiplier matrix T ij = diag{τ 1,ij , . . . , τ n φ ,ij } such as in [6]. Other relaxations for specific classes of nonlinearities that may reduce conservatism are proposed in, for instance, [23]. Regarding (b): In the noiseless linear non-descriptor case (φ = 0, v = 0, w = 0 and E(p) = I for all p ∈ P), (19) reduces to the conditions in [12,Theorem 2], which were shown to also be necessary in [19]. In the next section, we restrict the considered class of systems to feature a constant E-matrix, which allows us to introduce more flexibility in how the observer gains depend on the parameters as well as introduce more slack variables. As a result, we can provide necessary and sufficient LMIbased synthesis conditions in this setting. The necessity indicates the non-conservativeness of our synthesis conditions. The conditions in Theorem 1 can be relaxed by applying the dilated LMI techniques used in, e.g., [2,8,9], which would allow us to introduce slack variables X ij , i, j ∈ N , (instead of just X i as in (19)) thereby reducing conservativeness. Necessary and sufficient conditions for the case E(p) =Ē for all p ∈ P Next, we provide necessary and sufficient conditions to synthesize observers of the form (10), that guarantee absolute poly-quadratic ISS, for systems (1) with a constant descriptor matrix, as formalized below. Assumption 3 It holds that E(p) =Ē, for all p ∈ P, for some non-singularĒ ∈ R nx×nx . A significant class of systems satisfies Assumption 3 including, for instance, standard state-space models, but also models obtained through forward Euler discretization of some underlying continuous-time descriptor system with a parameter-independent E-matrix. Note that E is still non-singular and, hence, pre-multiplying the dynamics byĒ −1 yields a non-descriptor system. However, compared to existing works, e.g., [12,19,24], we still show necessity for a more general class of nonlinear systems for which, to the best of our knowledge, such results were not available in the literature. In addition, our results reveal that the obtained observer gains are no more conservative than observer gains which arbitrarily depend on bothp k+1 as well asp k , as detailed below. For the above class of systems, we synthesize observers (10) with arbitrary observer gains L : P × P → R nx×ny and K : P × P → R n φ ×ny . In other words, we do not impose any restrictions on how L and K depend onp k andp k+1 . In doing so, we not only provide necessary and sufficient LMI-based synthesis conditions for the considered class of observers (10), but we will also uncover a necessary structure for the observer gains L and K. Finally, we adopt the additional assumption on the functions ξ i , i ∈ N , from Assumption 1 that {e i } i∈N ⊂ ξ(P) (which holds for the functions ξ i in Example 1). This assumption is mild/natural when studying necessary stability conditions for uncertain systems, since it means that we are not overapproximating the parameter set in order to satisfy Assumption 1 which inherently leads to conservatism. In the case where we do have to over approximate the parameter set and, thereby, introduce conservatism, we can redefine P to satisfy this assumption and apply the results below to guarantee necessity for overapproximating the parameter set. Theorem 2 Consider system (1) satisfying Assumptions 1-3 with {e i } i∈N ⊂ ξ(P). There exists an observer of the form (10), with some observer gains L : P × P → κvI ⋆ ⋆ (Yij D) ⊤ 0 (ΛZij D) ⊤ 0 κwI ⋆ X ⊤ ij 0 0 0 0 κ ψ I      =:Nij ≻ 0,(21) for all i, j ∈ N . In fact, then, P i ≻ 0, the matrices X ij are non-singular, i, j ∈ N , and the observer (10) with L(p + ,p) = i∈N ξ i (p)X −1 i (p + )Y i (p + ),p + ,p ∈ P,(22) where Xi(p+) Yi(p+) = j∈N ξ j (p + ) Xij Yij ,(23) and K as defined in (12)-(14) renders the error system (15) absolutely poly-quadratically ISS with respect to v, w and ψ with β, γ v , γ w and γ ψ as defined in Theorem 1. Moreover, V (e,p) = i∈N e ⊤ ξ i (p)P i e is a poly-quadratic ISS-Lyapunov function for (15) satisfying (17)-(18) with a = κ ψ σ 2 > 1 and σ := σ min (Ē). The conditions in Theorem 2 are LMIs in the decision variables X ij , Y ij , P i , Z ij , τ ij , κ v , κ w and κ ψ . Since they are necessary and sufficient, these conditions can be used to synthesize observers of the form (10) without introducing additional conservatism. In fact, Theorem 2 shows that restricting L and K to be of the form in (22)- (23) and (12)- (14), respectively, also does not introduce any further conservatism. As before, the synthesized observer gain K satisfies (12)- (14), however, considering systems (1) with a constant descriptor matrix allowed us to derive LMIs for synthesizing observers with a more general L as given in (22)- (23), which is polytopic inp, but no longer inp + . To see that (22) with (23) is indeed more general than (11), note that by setting X ij = X i , for all j ∈ N , the structure from Theorem 1 is recovered. Note that, in general, it is not possible to invert X i (p + ) analytically and, hence, this inverse needs to be computed on-line. In some applications it may be desirable to set X ij = X i , for all j ∈ N , thereby, avoiding the need to invert X i (p + ) online and, hence, reducing the on-line computational burden at the cost of introducing conservatism. Robustness with respect to parameter mismatch Under some extra assumptions, ISS with respect to the model mismatch implies ISS with respect to the parameter mismatch, as formalized next. Corollary 1 Suppose that the conditions in Theorem 1 or Theorem 2 hold and that the functions ξ i , i ∈ N , are continuous. Given K x , K u , K v ∈ R 0 , there exists a Kfunction γp such that, for all solutions (x, u, v) to (1) that satisfy x k ∈ B nx Kx , u k ∈ B nu Ku and v k ∈ B nv Kv and, for all k ∈ N, e k β( e 0 , k) + γ v ( v ∞,k−1 )+ γ w ( w ∞,k−1 ) + γp( p ∞,k−1 ),(24) for all e 0 ∈ R nx , w ∈ ℓ nw ∞ , {p k } k∈N and {p k } k∈N with p k ,p k ∈ P, k ∈ N. Here, β, γ v , γ w are as in Theorem 1. Corollary 1 states that, if u is bounded and keeps x bounded for all admissible initial conditions x 0 and disturbances v, the error system (15) corresponding to system (1) with observer (10) obtained from Theorem 1 or Theorem 2, satisfies an ISS-like property with respect to v, w andp. Illustrative example We demonstrate the effectiveness of our results using a case study based on Example 1. Consider again the system (1) with (7) and its polytopic representation derived in Example 1. The nonlinearity φ satisfies Assumption 2 with Λ = 2. Hence, the conditions of Theorem 1 are satisfied. The functions ξ i , i ∈ {3, 4}, are continuous at the boundary Co{ν 3 , ν 4 } and, thus, ξ i , i ∈ N , are continuous on P. We minimize (κ v + 5κ w + 0.01κ ψ )/6.01 subject to (19). The optimal solution achieves κ v = 0.5258, κ w = 6.6402 and κ ψ = 1.3535 · 10 3 . Computations show that σ = 1.01. We implement two observers (10): One with the constant (inaccurate) parameter estimatep k = ν 1 = (9.5 · 10 −3 , 4.75 · 10 −2 ) and the other with the exact parameter, i.e.,p k = p k , k ∈ N. We apply both observers to the system (1) with p k = (0.01, 0.11+0.11 sin(0.001πk)), u k = 3 sin(0.01πk). On the time interval k ∈ N [0,200] , v k is a uniformly distributed random variable in the in-terval [−1, 1] and the entries of w k are uniformly distributed random variables in the interval [−0.1, 0.1]. For k ∈ N 201 , no more noise is present, i.e., v k = 0 and w k = 0 for k ∈ N 201 . We simulate system (1) with x 0 = (1, 2) and both observers (10) withx 0 = (0, 0). Fig. 1 shows that the state estimates initially converge to a neighbourhood of the true states. We also see that after k = 200 (when the noise is absent), the state estimate of observer 2 converges asymptotically to the true state whereas observer 1 retains some error due to the parameter mismatch. This behaviour nicely illustrates that the state estimation error system is ISS with respect to the process disturbances, measurement noise and model/parameter mismatch. We can verify, using Lyapunov-based tools [16] and the results in Corollary 1, that ISS with respect to parameter mismatch is achieved. Conclusions We derived LMI-based conditions to synthesize polytopic observers for a class of nonlinear descriptor systems with uncertain parameters. The proposed approach leads to observers that can be synthesized a priori and which guarantee ISS with respect to process disturbances, measurement noise and model/parameter mismatch for the entire parameter set. This is particularly useful in applications where the parameter estimates are time-varying, such as in the joint parameter and state estimation scheme using a multi-observer in [5,18]. The presented LMI conditions allow trade-offs between transient performance and sensitivity to noise and model/parameter mismatch. For the class of systems with parameter-independent E-matrix, we show that we can exploit additional flexibility in the structure of the observer gains to derive alternative conditions that are both necessary and sufficient, being an indicator for the non-conservativeness of our synthesis conditions. For the latter class of systems, we also show that the obtained structure for the observer gains does not lead to additional conservatism with respect to using observer gains which depend arbitrarily on the parameters. Finally, the proposed conditions were used to synthesize observers for a numerical example, showing their strengths. Appendix First, we introduce some preliminaries which we use in proving our results. b) For all p ∈ P, there exists µ(p) ∈ R such that Q(p)− µ(p)B ⊤ (p)B(p) ≺ 0. Lemma 2 (Projection lemma [21, Lemma 4.15]) Let Q ∈ S n , A ∈ R m×n and B ∈ R p×n . There exists X ∈ R m×p such that Q + A ⊤ XB + B ⊤ X ⊤ A ≻ 0,(25) if and only if A ⊤ ⊥ QA ⊥ ≻ 0, and B ⊤ ⊥ QB ⊥ ≻ 0,(26) where A ⊥ and B ⊥ denote arbitrary matrices whose columns form a basis of ker A and ker B, respectively. Lemma 3 ((Lossless) S-procedure [3]) Let Q i ∈ S n , i ∈ N [0,N ] . Suppose that ∃τ i ∈ R 0 , i ∈ N [1,N ] , such that Q 0 − i∈N [1,N ] τ i Q i ≻ 0,(27) then, it holds that x ⊤ Q 0 x > 0, for all x ∈ R n \ {0} such that x ⊤ Q i x 0 for all i ∈ N [1,N ] .(28) Conditions (27) and (28) are equivalent if N = 1 and Q 1 satisfies Slater's condition, i.e., there exists some x 0 ∈ R n such that x ⊤ 0 Q 1 x 0 > 0. Proof of Theorem 1. Let M ij , i, j ∈ N , be as defined in (19). Suppose there exist symmetric matrices P i ∈ S nx , matrices X i ∈ R nx×nx , Y ij ∈ R nx×ny , Z ij ∈ R n φ ×ny , i, j ∈ N , and scalars τ ij , κ v , κ w , κ ψ ∈ R >0 , i, j ∈ N , such that M ij ≻ 0 for all i, j ∈ N . First, we note that it follows immediately from (19) that P i ≻ I ≻ 0 for all i ∈ N . Next, we show that the matrices X i , i ∈ N , are non-singular. To this end, let, for any i ∈ N , u ∈ R nx be such that X ⊤ i u = 0, then, by (19), u ⊤ (He(X i E j ) − P j )u = −u ⊤ P j u 0 for all j ∈ N . It follows, since P (p) ≻ 0 for all p ∈ P, that u = 0 and, thus, the matrices X i , i ∈ N , are non-singular. Using the fact that, by Assumption 1, ξ i (p) ∈ R 0 , i ∈ N , and i∈N ξ i (p) = 1 for all p ∈ N , we have P (p) := i∈N ξ i (p)P i ≻ 0, for all p ∈ P. We now show that a poly-quadratic ISS-Lyapunov function can be constructed. Let Γ := diag(κ v I nv , κ w I nw , κ ψ I nx ). Using Assumption 1, we have, for all i ∈ N andp + ∈ P, 0 ≺ j∈N ξ j (p + )M ij = (29)     He(X i E(p + )) − P (p + ) ⋆ ⋆ (X i A i − Y i (p + )C) ⊤ P i − I ⋆ ⋆ − (X i G i ) ⊤ Λ(τ i (p + )H − Z i (p + )C) 2τ i (p + )I − (X i F i ) ⊤ 0 0 (Y i (p + )D) ⊤ 0 (ΛZ i (p + )D) ⊤ Γ X ⊤ i 0 0     , where τ i (p + ) := j∈N ξ j (p + )τ ij , Y i (p + ) := j∈N ξ j (p + )Y ij and Z i (p + ) := j∈N ξ j (p + )Z ij , p + ∈ P and i ∈ N . It follows that X i E(p) + E ⊤ (p)X ⊤ i − P (p) ≻ 0 for all i ∈ N and p ∈ P. Therefore, (P (p) − X i E(p))P −1 (p)(P (p) − X i E(p)) ⊤ 0 for all i ∈ N and p ∈ P and, hence, for all i ∈ N and p ∈ P, X i E(p)P −1 (p)E ⊤ (p)X ⊤ i He(X i E(p)) − P (p) ≻ 0. (30) Using (30) followed by a congruence transformation with the non-singular matrix T i = diag(X −⊤ i , I nx , −I n φ , I nx+nv+nw ), i ∈ N ,p + ∈ P, we have, for all i ∈ N andp + ∈ P, that     E(p + )P −1 (p + )E ⊤ (p + ) ⋆ ⋆ A ⊤ i (p + ) P i − I ⋆ ⋆ G ⊤ i −ΛH i (p + ) 2τ i (p + )I − F ⊤ i 0 0 (L i (p + )D) ⊤ 0 −(ΛZ i (p + )D) ⊤ Γ I 0 0     ≻ 0, (31) where A i (p + ) := A i − L i (p + )C, H i (p + ) := τ i (p + )H − Z i (p + )C, L i (p + ) := j∈N ξ j (p + )L ij with L ij := X −1 i Y ij , i, j ∈ N andp + ∈ P. Since (31) is affine in all terms depending on i, we multiply by ξ i (p) and sum over all i ∈ N to find, for allp + ,p ∈ P,     E(p + )P −1 (p + )E ⊤ (p + ) ⋆ ⋆ A ⊤ (p + ,p) P (p) − I ⋆ ⋆ G ⊤ (p) −ΛH(p + ,p) 2τ (p + ,p)I − F ⊤ (p) 0 0 (L(p + ,p)D) ⊤ 0 −(ΛZ(p + ,p)D) ⊤ Γ I 0 0     ≻ 0,(32) with A(p + ,p) := A(p) − L(p + ,p)C, H(p + ,p) := τ (p + ,p)H − K(p + ,p)C,p + ,p ∈ P, and L, Z and τ as in (11) and (12)- (14). Using the Schur complement and the fact that Z(p + ,p) = τ (p + ,p)K(p + ,p), we find, for allp + ,p ∈ P,   P (p)−I ⋆ ⋆ ⋆ ⋆ 0 0 ⋆ ⋆ ⋆ 0 0 κvI ⋆ ⋆ 0 0 0 κwI ⋆ 0 0 0 0 κ ψ I   − (33) B ⊤ (p + ,p)E −⊤ (p + )P (p + )E −1 (p + )B(p + ,p)− τ (p + ,p)   0 ⋆ ⋆ ⋆ ⋆ Λ(H−K(p+,p)C) −2I ⋆ ⋆ ⋆ 0 0 0 ⋆ ⋆ 0 (ΛK(p+,p)D) ⊤ 0 0 ⋆ 0 0 0 0 0   ≻ 0, where, for p + , p ∈ P, B(p + , p) := A(p+,p) G(p) −F (p) L(p+,p)D I .(34) Using pointwise application of Lemma 3, (33) implies that, for all x + , x, e ∈ R nx , v ∈ R nv , w ∈ R nw , u ∈ R nu andp + ,p, p + , p ∈ P, e ⊤ + P (p + )e + − e ⊤ P (p)e − e 2 + (35) κ v v 2 + κ w w 2 + κ ψ ψ 2 , for allφ subject to φ 2 −φ ⊤ Λ((H − K(p + ,p)C)e + K(p + ,p)Dw) 0, which holds for anyφ satisfying (3) with φ(y) − φ(x) ←φ, x ← Hx and y ← (H − K(p + ,p)C)e + K(p + ,p)Dw + Hx, and e + subject to E(p + )e + = B(p + ,p)q with q := (e,φ, v, w, ψ), φ =φ(p + ,p, e, x, w) and ψ = ψ(p + ,p, p + , p, x + , x, u, v). In other words, V (e, p) = e ⊤ P (p)e satisfies (18). V also satisfies (17), since P (p) ≻ I, and, from the bottom-right of (33), κ ψ E ⊤ (p + )E(p + ) ≻ P (p + ) for allp + ∈ P, which implies I ≺ P (p) ≺ κ ψ σ 2 I, with σ as in Theorem 1, for all p ∈ P. Thus, κ ψ > σ −2 and V is a poly-quadratic ISS-Lyapunov function and the system is (absolutely) poly-quadratically ISS. Finally, we show ISS and derive the ISS-gains. Denote V k = V (e k ,p k ) and apply (17) and (18) to find V k+1 (1 − 1 κ ψ σ 2 )V k + κ v v k 2 + κ w w k 2 + κ ψ ψ k 2 . Apply- ing repetitively yields V k ρ k V 0 + κ ψ σ 2 (κ v v 2 ∞,k + κ w w 2 ∞,k + κ ψ ψ 2 ∞,k ), with ρ := (1 − (κ ψ σ 2 ) −1 ) ∈ (0, 1) since 1 < κ ψ σ 2 . Using (17) and √ a + b √ a + √ b for any a, b ∈ R 0 , we have ISS with respect to v, w and ψ with β ∈ KL and γ v , γ w , γ ψ ∈ K as in Theorem 1. Proof of Theorem 2. We prove sufficiency following similar steps as in the proof of Theorem 1. Let N ij , i, j ∈ N , be as defined in (21), and suppose there exist matrices P i ∈ S nx , X ij ∈ R nx×nx , Y ij ∈ R nx×ny , Z ij ∈ R n φ ×ny and scalars τ ij , κ v , κ w , κ ψ ∈ R >0 , i, j ∈ N , such that N ij ≻ 0 for all i, j ∈ N . First, it follows immediately from (21) that P i ≻ I ≻ 0, for all i ∈ N , and, hence, P (p) ≻ 0 for all p ∈ P. Following the same reasoning as in the proof of Theorem 1, we can show that the matrices X ij , i, j ∈ N , are non-singular and, thus, X i (p + ), i ∈ N , as in (23) are also non-singular for all p + ∈ P. Next, we show that a poly-quadratic ISS-Lyapunov function can be constructed. Let Γ := diag(κ v I nv , κ w I nw , κ ψ I nx ). Using Assumption 1, ξ i (p) ∈ R 0 , i ∈ N , and i∈N ξ i (p) = 1 for all p ∈ P, we have, for all i ∈ N andp + ∈ P, 0 ≺ j∈N ξ j (p + )N ij = (37)     He(X i (p + )Ē) − P (p + ) ⋆ ⋆ (X i (p + )A i − Y i (p + )C) ⊤ P i − I ⋆ ⋆ − (X i (p + )G i ) ⊤ Λ(τ i (p + )H − Z i (p + )C) 2τ i (p + )I − (X i (p + )F i ) ⊤ 0 0 (Y i (p + )D) ⊤ 0 (ΛZ i (p + )D) ⊤ Γ X ⊤ i (p + ) 0 0     , where Z i (p + ) := j∈N ξ j (p + )Z ij ,p + ∈ P, i ∈ N , and X i (p + ) and Y i (p + ) are as defined in (23). It follows that P (p) := i∈N ξ i (p)P i ≻ I ≻ 0 and (P (p) − X iĒ )P −1 (p)(P (p)− X iĒ ) ⊤ 0 for all i ∈ N and p ∈ P. Thus, it holds, for all p ∈ P and i ∈ N , that X i (p)ĒP −1 (p)Ē ⊤ X ⊤ i (p) He(X i (p)Ē) − P (p) ≻ 0. (38) Using (38) and congruence transformations with the non-singular matrices T i (p + ) = diag(X −⊤ i (p + ), I nx , −I n φ , I nx+nv +nw ), i ∈ N ,p + ∈ P, we obtain, for allp + ∈ P and i ∈ N , ⋆ ⋆ 0 0 κvI ⋆ ⋆ 0 0 0 κwI ⋆ 0 0 0 0 κ ψ I   − (41) B ⊤ (p + ,p)Ē −⊤ P (p + )Ē −1 B(p + ,p)− τ (p + ,p)   0 ⋆ ⋆ ⋆ ⋆ Λ(H−K(p+,p)C) −2I ⋆ ⋆ ⋆ 0 0 0 ⋆ ⋆ 0 (ΛK(p+,p)D) ⊤ 0 0 ⋆ 0 0 0 0 0   ≻ 0, where, for p + , p ∈ P, B(p + ,p) := A(p+,p) G(p) −F (p) L(p+,p)D I . Using pointwise application of Lemma 3, it follows that, for all x + , x, e ∈ R nx , v ∈ R nv , w ∈ R nw , u ∈ R nu and p + ,p, p + , p ∈ P, (35) holds for allφ subject to φ 2 − φ ⊤ Λ((H −K(p + ,p)C)e+K(p + ,p)Dw) 0, which holds for anyφ satisfying (3) with φ(y) − φ(x) ←φ, x ← Hx and y ← (H − K(p + ,p)C)e + K(p + ,p)Dw + Hx, and e + subject toĒe + = B(p + ,p)q with q as in (36), φ =φ(p + ,p, e, x, w) and ψ = ψ(p + ,p, p + , p, x + , x, u, v). From the bottom-right of (41), we have that P (p) ≺ κ ψ σ 2 I with σ as in Theorem 2, for all p ∈ P. Thus, V is a poly-quadratic ISS-Lyapunov function (as shown in more detail in the proof of Theorem 1) and the error system is (absolutely) poly-quadratically ISS. The precise ISS-gains can also be derived as in Theorem 1. To show necessity, suppose there exist matrix-valued functions L : P × P → R nx×ny and K : P × P → R n φ ×ny for which the observer (10) renders the error system (15) absolutely poly-quadratically ISS with respect to v, w and ψ. Then, by Definition 2, there exist symmetric matricesP i , i ∈ N , such thatV (e,p) = e ⊤P (p)e withP (p) := i∈N ξ i (p)P i , p ∈ P, satisfies 0 ≺ I P (p) āI, for allp ∈ P and for someā ∈ R >0 . Since {e i } i∈N ⊂ ξ(P), it follows that 0 ≺ I P i āI for all i ∈ N . Then, from (18), we have, for allp + ,p ∈ P, q ⊤ B ⊤ (p + ,p)Ē −⊤P (p + )Ē −1 B(p + ,p)q − e ⊤P (p)e − e 2 +κ v v 2 +κ w w 2 +κ ψ ψ 2 ,(43) with B as defined in (42), for someκ v ,κ w ,κ ψ ∈ R >0 and for all q ∈ R nq , with n q = 2n x + n u + n v + n w , subject to q ⊤   0 ⋆ ⋆ ⋆ ⋆ Λ(H−K(p+,p)C) −2I ⋆ ⋆ ⋆ 0 0 0 ⋆ ⋆ 0 (ΛK(p+,p)D) ⊤ 0 0 ⋆ 0 0 0 0 0   =:W (p+,p) q 0. (44) Note that (44) is Assumption 2 with φ(y) − φ(x) ←φ, x ← Hx and y ← (H −K(p + ,p)C)e+K(p + ,p)Dw+Hx expressed in terms of q. We proceed to show necessity by taking the following steps: a) We construct P : P → S nx ≻0 and κ v , κ w , κ ψ ∈ R >0 such that, for allp + ,p ∈ P, q ⊤ V(p + ,p)q > 0 for all q ∈ R nq \ {0} when q ⊤ W(p + ,p)q 0, where V(p + ,p) := diag(P (p) − I, 0, κ v I, κ w I, κ ψ I)− B ⊤ (p + ,p)Ē −⊤ P (p + )Ē −1 B(p + ,p), forp + ,p ∈ P. We achieve this using the following lemma, for which a proof is provided later in the Appendix. Lemma 5 ConsiderP : P → S nx ≻0 andκ v ,κ w ,κ ψ ∈ R >0 such that, for allp + ,p ∈ P, (43) holds for all q ∈ R nq subject to (44). For any ǫ ∈ (0, 1), we have, for allp + ,p ∈ P, q ⊤V (p + ,p)q > 0, withV(p + ,p) := diag(P (p) − (1 − ǫ)I, 0, (κ v + ǫ)I, (κ w + ǫ)I, (κ ψ + ǫ)I) − B ⊤ (p + ,p)Ē −⊤P (p + )Ē −1 B(p + ,p), for all q ∈ R nq \ {0} subject to (44). b) Next, we use the following S-procedure-like result, for which a proof is provided later in the Appendix, to obtain a single matrix inequality that is equivalent to q ⊤ V(p + ,p)q > 0 for all q ∈ R nq \ {0} when q ⊤ W(p + ,p)q 0 with V and W as in (45) and (44), respectively. Lemma 6 Consider V : P × P → S nq and W : P × P → S nq as in (45) and (44), respectively. There exists a function τ : P × P → R >0 such that, for all p + ,p ∈ P, V(p + ,p) − τ (p + ,p)W(p + ,p) ≻ 0, if and only if, for allp + ,p ∈ P, it holds that q ⊤ V(p + ,p)q > 0, for all q ∈ R nq \ {0} such that q ⊤ W(p + ,p)q 0. At first glance, Lemma 6 might appear to simply be a pointwise application of Lemma 3. However, although necessity follows immediately from Lemma 3, sufficiency is not immediately clear since W(p + ,p) does not necessarily satisfy Slater's condition for allp + ,p ∈ P. In fact, we distinguish, for eachp + ,p ∈ P, two cases: (i) (H − K(p + ,p)C) = 0 or K(p + ,p)D = 0, or (ii) (H − K(p + ,p)C) = 0 and K(p + ,p)D = 0. In the proof of Lemma 6, we apply pointwise for eachp + ,p ∈ P either (i) Lemma 3 since Slater's condition holds in this case, or (ii) Lemma 1, to construct τ : P × P → R >0 satisfying (47). c) Finally, we exploit the fact that {e i } i∈N ⊂ ξ(P) to evaluate the matrix inequality obtained in the previous step, which depends onp + ,p ∈ P, to obtain a finite set of conditions corresponding to each of the vertices. We then apply Lemma 2 to arrive at the conditions in Theorem 2 and, hence, complete the necessity part of the proof. Let us now proceed along the steps outlined above. As for step a), we directly obtain (45) using Lemma 5 by taking P (p) :=P (p)/(1−ǫ), p ∈ P, κ v := (κ v +ǫ)/(1−ǫ), κ w := (κ w + ǫ)/(1 − ǫ) and κ ψ := (κ ψ + ǫ)/(1 − ǫ). Next, we can directly apply Lemma 6 to conclude that there exists a function τ : P × P → R >0 such that (47) holds for allp + ,p ∈ P. K(p + ,p)D = 0) and since case (i) and (ii) together cover allp + ,p ∈ P, we conclude that there exists a function τ : P × P → R such that (47) holds for all p + ,p ∈ P. Finally, one of the diagonal blocks of (47) reads 2τ (p + ,p)I ≻ 0 and, hence, we have τ (p + ,p) > 0 for allp + ,p ∈ P, thus, there exists τ : P × P → R >0 such that (47) holds for allp + ,p ∈ P. Proof of Corollary 1. From either Theorem 1 or Theorem 2, we already have ISS with respect to v, w and ψ. Under the additional assumptions in this corollary, we now show ISS with respect top. Firstly, (3) implies that φ is globally Lipschitz continuous, i.e., there exists c ∈ R 0 such that φ(y) − φ(x) c y − x for all x, y ∈ R n φ . By the boundedness theorem, there exists K φ ∈ R 0 such that φ(x) K φ for all x ∈ B nx Kx . Since ξ i , i ∈ N , are continuous and P is compact, there exist K-functions α i , i ∈ N , such that ξ i (p + p) − ξ i (p) α i ( p ) for all p ∈ P andp ∈ D := {p − p \| p,p ∈ P} [17, Corollary III.10]. By substitution in γ ψ and using the boundedness of x, u and v, we obtain (24) with γp(s) = γ ψ (αp(s)), where αp(s) = i∈N α i (s)( E i + A i )K x + B i ∆ u + G i K φ + F i K v ) , and γ v , γ w as in Theorem 1 (if the system satisfies Assumption 3 as in Theorem 2 we substitute E i =Ē, for all i ∈ N , in αp). and φ(z) = sin(z) + z, z ∈ R. The parameters are the (constant) sampling time p1 k = T s ∈ [p 1 , p 1 ] = [9.5, 10.5] · 10 −3 and the time-varying Lipschitz constant of the system p Fig. 1 . 1State estimates and corresponding estimation errors. Lemma 1 ( 1Finsler's lemma[14, Lemma 3]) Let P ⊆ R d , Q : P → S n and B : P → R m×n . Then, the following statements are equivalent: a) For all p ∈ P, it holds that x ⊤ Q(p)x < 0 for allx ∈ R nx \ {0} such that B(p)x = 0. where A i (p + ) := A i − L i (p + )C, H i (p + ) := τ i (p + )H − Z i (p + )C, L i (p + ) := X −1 i (p + )Y i (p + ),p + ∈ P, i ∈ N and p + ∈ P. Since (39) is affine in all terms depending on i, multiplying by ξ i (p) and summing over all i ∈ N yields, for allp + ,p ∈ P,We express (49) equivalently as, for all i, j ∈ N ,It follows that Q ij ≻ B ⊤ ijĒ −⊤ P jĒ −1 j B ij 0 and, thus, it holds that, for all i, j ∈ N ,Applying Lemma 2 to (50) and (51), there exist matrices(52) Performing a congruence transformation with diag(I 2nx , −I n φ , I nv +nw+nx ), we have, for all i, j ∈ N ,Note that (53) implies that τ ij > 0 for all i, j ∈ N . Thus, we have completed the proof of necessity by construction of the matricesProof of Lemma 5. LetP : P → S nx ≻0 andκ v ,κ w ,κ ψ ∈ R >0 be such that, for allp + ,p ∈ P, (43) holds for all q ∈ R nq subject to (44). To show that, for any ǫ ∈ (0, 1), (46) holds for all q ∈ R nq \ {0} subject to (44), suppose, to the contrary, that there exist someq ∈ R nq \ {0} and somep + ,p ∈ P such thatq ⊤ W(p + ,p)q 0 and q ⊤V (p + ,p)q 0. Since we know already, using (43) and ǫ > 0, that, for allp + ,p ∈ P,q ⊤V (p + ,p)q 0 for allq ∈ R nq subject toq ⊤ W(p + ,p)q 0, it must hold thatq ⊤V (p + ,p)q = 0 for somep + ,p ∈ P. By inspection of (46) using (43) (particularly, how we added ǫ to specific diagonal blocks), we see that anyq = 0 for whichq ⊤ W(p + ,p)q 0 andq ⊤V (p + ,p)q = 0, for somê p + ,p ∈ P, must satisfȳand, hence, we can expressq asq = Mv for somev ∈ R n φ \ {0}. Substitution inq ⊤ W(p + ,p)q 0 shows that 0 v ⊤ M ⊤ W(p + ,p)Mv = −2 v 2 , which implies that v = 0 and, hence,q = 0. This contradicts our initial statement in whichq = 0, which completes our proof that (46) holds for all q ∈ R nq \ {0} subject to (44).Proof of Lemma 6. Since necessity readily follows from pointwise application of Lemma 3, we direct our attention to proving sufficiency, i.e., the existence of τ : P × P → R >0 satisfying (47). To this end, consider V and W as in(45)and(44), respectively, and suppose that q ⊤ V(p + ,p)q > 0 for all q ∈ R nq \ {0} such that q ⊤ W(p + , p)q 0. For eachp + ,p ∈ P, we distinguish two cases: (i) (H − K(p + ,p)C) = 0 or K(p + ,p)D = 0, or (ii) (H − K(p + ,p)C) = 0 and K(p + ,p)D = 0.First, we show that W(p + ,p) satisfies Slater's condition for allp + ,p ∈ P that fall under case (i). To this end, let ξ = (ē,12 Λ((H − K(p + ,p)C)ē + K(p + ,p)Dw), 0,w, 0), for someē ∈ R nx andw ∈ R nw , then, 2ξ ⊤ W(p + ,p)ξ = Λ((H − K(p + ,p)C)ē + K(p + ,p)Dw) 2 . It follows that, for allp + ,p ∈ P for which H − K(p + ,p)C = 0 and/or K(p + ,p)D = 0 (i.e., case (i)), there existē andw such that ξ ⊤ W(p + ,p)ξ > 0. Hence, by pointwise application, for each suchp + ,p ∈ P, of Lemma 3, there existsτ : P × P → R 0 such thatfor allp + ,p ∈ P for which H − K(p + ,p)C = 0 and/or K(p + ,p)D = 0. We now proceed with case (ii), i.e., the pathological case where H − K(p + ,p)C = 0 and K(p + ,p)D = 0 for somep + ,p ∈ P. In this case Slater's condition clearly does not hold, however, we can use Lemma 1 to show that there existsμ : P × P → R such that V(p + ,p) −μ(p + ,p)W(p + ,p) ≻ 0. To see this, note that by definition ofφ (above(15)) and by(3), H − K(p + ,p)C = 0 and K(p + ,p)D = 0 together imply thatφ = 0 and, hence M ⊤ q = 0 with M as defined in (54). By Lemma 1, there existsμ : P × P → R such that V(p + ,p) − 2MM ⊤ ≻ 0 for allp + ,p ∈ P for which H − K(p + ,p)C = 0 and K(p + ,p)D = 0. 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[ "LEGAL DECOMPOSITIONS ARISING FROM NON-POSITIVE LINEAR RECURRENCES", "LEGAL DECOMPOSITIONS ARISING FROM NON-POSITIVE LINEAR RECURRENCES" ]	[ "Minerva Catral ", "Pari L Ford ", "Pamela E Harris ", "ANDSteven J Miller ", "Dawn Nelson " ]	[]	[]	Zeckendorf's theorem states that any positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers; this result has been generalized to many recurrence relations, especially those arising from linear recurrences with leading term positive. We investigate legal decompositions arising from two new sequences: the (s, b)-Generacci sequence and the Fibonacci Quilt sequence. Both satisfy recurrence relations with leading term zero, and thus previous results and techniques do not apply. These sequences exhibit drastically different behavior. We show that the (s, b)-Generacci sequence leads to unique legal decompositions, whereas not only do we have non-unique legal decompositions with the Fibonacci Quilt sequence, we also have that in this case the average number of legal decompositions grows exponentially. Another interesting difference is Date: July 4, 2016. 2010 Mathematics Subject Classification. 60B10, 11B39, 11B05 (primary) 65Q30 (secondary). Key words and phrases. Zeckendorf decompositions, Fibonacci quilt, non-uniqueness of representations, positive linear recurrence relations, Gaussian behavior, distribution of gaps.	null	[ "https://arxiv.org/pdf/1606.09312v2.pdf" ]	53,582,746	1606.09312	82e284a240a3b7c9999ea2e5fcc02264442abc91	LEGAL DECOMPOSITIONS ARISING FROM NON-POSITIVE LINEAR RECURRENCES 1 Jul 2016 Minerva Catral Pari L Ford Pamela E Harris ANDSteven J Miller Dawn Nelson LEGAL DECOMPOSITIONS ARISING FROM NON-POSITIVE LINEAR RECURRENCES 1 Jul 2016arXiv:1606.09312v2 [math.CO] Zeckendorf's theorem states that any positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers; this result has been generalized to many recurrence relations, especially those arising from linear recurrences with leading term positive. We investigate legal decompositions arising from two new sequences: the (s, b)-Generacci sequence and the Fibonacci Quilt sequence. Both satisfy recurrence relations with leading term zero, and thus previous results and techniques do not apply. These sequences exhibit drastically different behavior. We show that the (s, b)-Generacci sequence leads to unique legal decompositions, whereas not only do we have non-unique legal decompositions with the Fibonacci Quilt sequence, we also have that in this case the average number of legal decompositions grows exponentially. Another interesting difference is Date: July 4, 2016. 2010 Mathematics Subject Classification. 60B10, 11B39, 11B05 (primary) 65Q30 (secondary). Key words and phrases. Zeckendorf decompositions, Fibonacci quilt, non-uniqueness of representations, positive linear recurrence relations, Gaussian behavior, distribution of gaps. INTRODUCTION A beautiful result of Zeckendorf describes the Fibonacci numbers as the unique sequence from which every natural number can be expressed uniquely as a sum of nonconsecutive terms in the sequence [Ze]. Zeckendorf's theorem inspired many questions about this decomposition, and generalizations of the notions of legal decompositions of natural numbers as sums of elements from an integer sequence has been a fruitful area of research [BBGILMT,BCCSW,BDEMMTTW,BILMT,CFHMN1,DDKMMV,DDKMV,DFFHMPP,GTNP,Ha,KKMW,MW1,MW2]. Much of previous work has focused on sequences given by a Positive Linear Recurrence (PLR), which are sequences where there is a fixed depth L > 0 and non-negative integers c 1 , . . . , c L with c 1 , c L non-zero such that a n+1 = c 1 a n + · · · + c L a n+1−L . (1.1) The restriction that c 1 > 0 is required to gain needed control over roots of polynomials associated to the characteristic polynomials of the recurrence and related generating functions, though in the companion paper [CFHMNPX] we show how to bypass some of these technicalities through new combinatorial techniques. The motivation for this paper is to investigate whether the positivity of the first coefficient is needed solely to simplify the arguments, or if fundamentally different behavior can emerge if the said condition is not met. To this end, we investigate the legal decompositions arising from two different sequences which we introduce in this paper: the (s, b)-Generacci sequence and the Fibonacci Quilt sequence. Both satisfy recurrence relations with leading term zero, hence previous results and techniques are not applicable. Moreover, although both have non-positive linear recurrences (as their leading term is zero), they exhibit drastically different behavior: the (s, b)-Generacci sequence leads to unique legal decompositions, whereas not only do we have non-unique legal decompositions with the Fibonacci Quilt sequence, we also have that the average number of legal decompositions grows exponentially. Another interesting difference is that while in the (s, b)-Generacci case the greedy algorithm always leads to a legal decomposition, in the Fibonacci Quilt setting the greedy algorithm leads to a legal decomposition (approximately) 93% of the time. We conclude the introduction by first describing the two sequences and their resulting decomposition rules and then stating our results. Then in §2 we determine the recurrence relations for the sequences, in §3 we prove our claims on the growth of the average number of decompositions from the Fibonacci Quilt sequence, and then analyze the greedy algorithm and a generalization (for the Fibonacci Quilt) in §4. 1.1. The (s, b)-Generacci Sequence and the Fibonacci Quilt Sequence. 1.1.1. The (s, b)-Generacci Sequence. One interpretation of Zeckendorf's Theorem [Ze] is that the Fibonacci sequence is the unique sequence from which all natural numbers can be expressed as a sum of nonconsecutive terms. Note there are two ingredients to the rendition: a sequence and a rule for determining what is a legal decomposition. An equivalent formulation for the Fibonacci numbers is to consider the sequence divided into bins of size one and decompositions can use the element in a bin at most once and cannot use elements from adjacent bins. A generalization of this bin idea was explored by the authors in [CFHMN1], where bins of size 2 with the same non-adjacency condition were considered; the sequence that arose was called the Kentucky sequence. The Kentucky sequence is what we now refer to here as the (1, 2)-Generacci sequence. This leads to a natural extension where 2 we consider bins of size b and any two summands of a decomposition must come from distinct bins with at least s bins between them. We now give the technical definitions of the (s, b)-Generacci sequences and their associated legal decompositions. Definition 1.1 ((s, b)-Generacci legal decompositions). For fixed integers s, b ≥ 1, let an increasing sequence of positive integers {a i } ∞ i=1 and a family of subsequences B n = {a b(n−1)+1 , . . . , a bn } be given (we call these subsequences bins). We declare a decomposition of an integer m = a ℓ 1 + a ℓ 2 + · · · + a ℓ k where a ℓ i > a ℓ i+1 to be an (s, b)-Generacci legal decomposition provided {a ℓ i , a ℓ i+1 } ⊂ B j−s ∪ B j−s+1 ∪ · · · ∪ B j for all i, j. (We say B j = ∅ for j ≤ 0.) Thus if we have a summand a ℓ i ∈ B j in a legal decomposition, we cannot have any other summands from that bin, nor any summands from any of the s bins preceding or any of the s bins following B j . Definition 1.2 ((s, b)-Generacci sequence). For fixed integers s, b ≥ 1, an increasing sequence of positive integers {a i } ∞ i=1 is the (s, b)-Generacci sequence if every a i for i ≥ 1 is the smallest positive integer that does not have an (s, b)-Generacci legal decomposition using the elements {a 1 , . . . , a i−1 }. Using the above definition and Zeckendorf's theorem, we see that the (1, 1)-Generacci sequence is the Fibonacci sequence (appropriately normalized). Some other known sequences arising from the (s, b)-Generacci sequences are Narayana's cow sequence, which is the (2, 1)-Generacci sequence, and the Kentucky sequence, which is the (1, 2)-Generacci sequence. Theorem 1.3 (Recurrence Relation and Explicit Formula). Let s, b ≥ 1 be fixed. If n > (s+1)b+1, then the n th term of the (s, b)-Generacci sequence is given by the recurrence relation a n = a n−b + ba n−(s+1)b . (1.2) We have a generalized Binet's formula, with a n = c 1 λ n 1 [1 + O ((λ 2 /λ 1 ) n )] (1.3) where λ 1 is the largest root of x (s+1)b − x sb − b = 0, and c 1 and λ 2 are constants with λ 1 > 1, c 1 > 0 and \|λ 2 \| < λ 1 . Remark 1.4. The (s, b)-Generacci sequence also satisfies the recurrence a n = a n−1 + a n−1−f (n−1) , (1.4) where f (kb + j) = sb + j − 1 for j = 1, . . . , b. While this representation does have its leading coefficient positive, note the depth L = f (n − 1) + 1 is not independent of n, and thus this representation is not a Positive Linear Recurrence. The proof of Theorem 1.3 is given in §2.1. We note that the leading term in the recurrence in (1.2) is zero whenever b ≥ 2, and hence this sequence falls out of the scope of the Positive Linear Recurrences results. Fibonacci Quilt Sequence. The Fibonacci Quilt sequence arose from the goal of finding a sequence coming from a 2dimensional process. We begin by recalling the beautiful fact that the Fibonacci numbers tile the plane with squares spiraling to infinity, where the side length of the n th square is F n (see Figure 1; note that here we start the Fibonacci sequence with two 1's). Inspired by Zeckendorf decomposition rules and by the Fibonacci spiral we define the following notion of legal decompositions and create the associated integer sequence which we call the Fibonacci Quilt sequence. The spiral depicted in Figure 1 can be viewed as a log cabin quilt pattern, such as that presented in Figure 2 (left). Hence we adopt the name Fibonacci Quilt sequence. Definition 1.5 (FQ-legal decomposition). Let an increasing sequence of positive integers {q i } ∞ i=1 be given. We declare a decomposition of an integer m = q ℓ 1 + q ℓ 2 + · · · + q ℓt (1.5) (where q ℓ i > q ℓ i+1 ) to be an FQ-legal decomposition if for all i, j, \|ℓ i − ℓ j \| = 0, 1, 3, 4 and {1, 3} ⊂ {ℓ 1 , ℓ 2 , . . . ℓ t }. This means that if the terms of the sequence are arranged in a spiral in the rectangles of a log cabin quilt, we cannot use two terms if they share part of an edge. Figure 2 shows that q n + q n−1 is not legal, but q n + q n−2 is legal for n ≥ 4. The starting pattern of the quilt forbids decompositions that contain q 3 + q 1 . We define a new sequence {q n }, called the Fibonacci Quilt sequence, in the following way. Definition 1.6 (Fibonacci Quilt sequence). An increasing sequence of positive integers {q i } ∞ i=1 is called the Fibonacci Quilt sequence if every q i (i ≥ 1) is the smallest positive integer that does not have an FQ-legal decomposition using the elements {q 1 , . . . , q i−1 }. From the definition of an FQ-legal decomposition, the reader can see that the first five terms of the sequence must be {1, 2, 3, 4, 5}. We have q 6 = 6 as 6 = q 4 + q 2 = 4 + 2 is an FQ-legal decomposition. We must have q 6 = 7. Continuing we have the start of the Fibonacci Quilt sequence displayed in Figure 2 (right). Note that with the exception of a few initial terms, the Fibonacci Quilt sequence and the Padovan (see entry A000931 from the OEIS) sequence are eventually identical. Theorem 1.7 (Recurrence Relations). Let q n denote the n th term in the Fibonacci Quilt. Then for n ≥ 6, q n+1 = q n + q n−4 , (1.6) for n ≥ 5, q n+1 = q n−1 + q n−2 , (1.7) n i=1 q i = q n+5 − 6. (1.8) The proof is given in §2.2. Remark 1.8. At first the above theorem seems to suggest that the Fibonacci Quilt is a PLR, as (1.6) gives us a recurrence where the leading coefficient is positive and, unlike the alternative expression for the (s, b)-Generacci, this time the depth is fixed. The reason it is not a PLR is subtle, and has to do with the second part of the definition: the decomposition law. The decomposition law is not from using (1.6) to reduce summands, but from the geometry of the spiral. It is worth remarking that (1.7) is the minimal length recurrence for this sequence, and the characteristic polynomial arising from (1.6) is divisible by the polynomial from (1.7). 1.2. Results. Our theorems are for two sequences which satisfy recurrences with leading term zero. Prior results in the literature mostly considered Positive Linear Recurrences and results included the uniqueness of legal decompositions, Gaussian behavior of the number of summands, and exponential decay in the distribution of the gaps between summands [BBGILMT,BILMT,DDKMMV,DDKMV,MW1,MW2]. In [CFHMN1], a first example of a non-positive linear recurrence appeared and the aforementioned results were proved using arguments technically similar to those already present in the literature. What is new in this paper are two extensions of the work presented in [CFHMN1]. The first is the (s, b)-Generacci sequence, whose legal decompositions are unique but where new techniques are required to prove its various properties. The second is the more interesting newly discovered Fibonacci Quilt sequence, which displays drastically different behavior, one consequence being that the FQ-legal decompositions are not unique (for example, there are three distinct FQ-legal decompositions of 106: 86+16+4, 86+12+7+1, and 65+37+4). As Theorem 1.9 follows from a similar argument to that in the appendix of [CFHMN1], we omit it in this paper. Remark 1.10. We could also prove this result by showing that our sequence and legal decomposition rule give rise to an f -decomposition. These were defined and studied in [DDKMMV], and briefly a valid f -decomposition means that for each summand chosen a block of consecutive summands before are not available for use, and that number depends solely on n. The methods of [DDKMMV] are applicable and yield that each positive integer has a unique legal decomposition. These results are not available for the Fibonacci Quilt sequence, as the FQ-legal decomposition is not an f -decomposition. The reason is that in an f -decomposition there is a function f such that if we have q n then we cannot have any of the f (n) terms of the sequence immediately prior to q n . There is no such f for the Fibonacci Quilt sequence, as for n ≥ 8 if we have q n we cannot have q n−1 and q n−3 but we can have q n−2 . We have already seen that the Fibonacci Quilt leads to non-unique decompositions; this is just the beginning of the difference in behavior. The first result concerns the exponential number of FQ-legal decompositions as we decompose larger integers. First we need to introduce some notation. Let {q n } denote the Fibonacci Quilt sequence. For each positive integer m let d FQ (m) denote the number of FQ-legal decomposition of m, and d FQ;ave (n) the average number of FQ-legal decompositions of integers in I n := [0, q n+1 ); thus d FQ;ave (n) := 1 q n+1 q n+1 −1 m=0 d FQ (m). (1.9) In §3 we prove the following. Theorem 1.11 (Growth Rate of Average Number of Decompositions). Let r 1 be the largest root of r 7 − r 6 − r 2 − 1 = 0 (so r 1 ≈ 1.39704) and let λ 1 be the largest root of x 3 − x − 1 = 0 (so λ 1 = 1 3 27 2 − 3 √ 69 2 1/3 + 3 −2/3 1 2 9 + √ 69 1/3 ≈ 1.32472) , and set λ = r 1 /λ 1 ≈ 1.05459. There exist computable constants C 2 > C 1 > 0 such that for all n sufficiently large, C 1 λ n ≤ d FQ;ave (n) ≤ C 2 λ n . (1.10) Thus the average number of FQ-legal decompositions of integers in [0, q n+1 ) tends to infinity exponentially fast. Remark 1.12. At the cost of additional algebra one could prove the existence of a constant C such that d FQ;ave (n) ∼ Cλ n ; however, as the interesting part of the above theorem is the exponential growth and not the multiplicative factor, we prefer to give the simpler proof which captures the correct growth rate. We end with another new behavior. For many of the previous recurrences, the greedy algorithm successfully terminates in a legal decomposition; that is not the case for the Fibonacci Quilt sequence. In §4 we prove the following. Theorem 1.13. There is a computable constant ρ ∈ (0, 1) such that, as n → ∞, the percentage of positive integers in [1, q n ) where the greedy algorithm terminates in a Fibonacci Quilt legal decomposition converges to ρ. This constant is approximately .92627. Interestingly, a simple modification of the greedy algorithm does always terminate in a legal decomposition, and this decomposition yields a minimal number of summands. 6 Definition 1.14 (Greedy-6 Decomposition). The Greedy-6 Decomposition writes m as a sum of Fibonacci Quilt numbers as follows: • if there is an n with m = q n then we are done, • if m = 6 then we decompose m as q 4 + q 2 and we are done, and • if m ≥ q 6 and m = q n for all n ≥ 1, then we write m = q ℓ 1 + x where q ℓ 1 < m < q ℓ 1 +1 and x > 0, and then iterate the process with input m := x. We denote the decomposition that results from the Greedy-6 Algorithm by G(m). Theorem 1.15. For all m > 0, the Greedy-6 Algorithm results in a FQ-legal decomposition. Moreover, if G(m) = q ℓ 1 + q ℓ 2 + · · ·+ q ℓ t−1 + q ℓt with q ℓ 1 > q ℓ 2 > · · · > q ℓt , then the decomposition satisfies exactly one of the following conditions: ( 1) ℓ i − ℓ i+1 ≥ 5 for all i or (2) ℓ i − ℓ i+1 ≥ 5 for i ≤ t − 3 and ℓ t−2 ≥ 10, ℓ t−1 = 4, ℓ t = 2. Further, if m = q ℓ 1 + q ℓ 2 + · · · + q ℓ t−1 + q ℓt with q ℓ 1 > q ℓ 2 > · · · > q ℓt denotes a decomposition of m where either (1) ℓ i − ℓ i+1 ≥ 5 for all i or (2) ℓ i − ℓ i+1 ≥ 5 for i ≤ t − 3 and ℓ t−2 ≥ 10, ℓ t−1 = 4, ℓ t = 2, then q ℓ 1 + q ℓ 2 + · · · + q ℓ t−1 + q ℓt = G(m). That is, the decomposition of m is the Greedy-6 decomposition. Let D(m) be a given decomposition of m as a sum of Fibonacci Quilt numbers (not necessarily legal): m = c 1 q 1 + c 2 q 2 + · · · + c n q n , c i ∈ {0, 1, 2, . . . }. (1.11) We define the number of summands by #summands(D(m)) := c 1 + c 2 + · · · + c n . In attacking this problem we developed a new technique similar to ones used before but critically different in that we are able to bypass technical assumptions that other papers needed to prove a Gaussian distribution. We elaborate on this method in [CFHMNPX], where we also determine the distribution of gaps between summands. We have chosen to concentrate on the Fibonacci Quilt results in this paper, and just state many of the (s, b)-Generacci outcomes, as we see the same behavior as in other systems for the (s, b)-Generacci numbers, but see fundamentally new behavior for the Fibonacci Quilt sequence. Using the methods of [BDEMMTTW], these results can be extended to hold almost surely for sufficiently large sub-interval of [a (n−1)b+1 , a bn+1 ). and variance of Y n , we have µ n = An + B + o(1) (1.14) σ 2 n = Cn + D + o(1) (1.15) for some positive constants A, B, C, D. Moreover if we normalize Y n to Y ′ n = (Y n − µ n )/σ n , then Y ′ n converges in distribution to the standard normal distribution as n → ∞. Unfortunately, the above methods do not directly generalize to Gaussian results for the Fibonacci Quilt sequence. Interestingly and fortunately there is a strong connection between the two sequences, and in [CFHMNPX] we show how to interpret many questions concerning the Fibonacci Quilt sequence to a weighted average of several copies of the (4, 1)-Generacci sequence. This correspondence is not available for questions on unique decomposition, but does immediately yield Gaussian behavior and determines the limiting behavior of the individual gap measures. RECURRENCE RELATIONS 2.1. Recurrence Relations for the (s, b)-Generacci Sequence. Recall that for s, b ≥ 1, an (s, b)-Generacci decomposition of a positive integer is legal if the following conditions hold. (1) No term a i is used more than once. (2) No two distinct terms a i , a j in a decomposition can have indices i, j from the same bin. (3) If a i and a j are summands in a legal decomposition, then there are at least s bins between them. The terms of the (s, b)-Generacci sequence can be pictured as follows: a 1 , . . . , a b B 1 , a 1+b , . . . , a 2b B 2 , . . . , a 1+nb , . . . , a (n+1)b B n+1 , a 1+(n+1)b , . . . , a (n+2)b B n+2 , a 1+(n+2)b , . . . , a (n+3)b B n+3 , . . . . Proof. This follows directly from the definition of the (s, b)-Generacci sequence. That is, we note that at the (s + 1) th -bin, we clearly have s-many bins to the left, yet we are unable to use any elements from those bins to decompose any new integers. Thus a i = i, for all 1 ≤ i ≤ (s + 1)b + 1. Lemma 2.2. If k can be decomposed using summands {a 1 , . . . , a p }, then so can k − 1. Proof. Let k = a ℓ 1 + a ℓ 2 + · · · + a ℓt with ℓ 1 > ℓ 2 > · · · > ℓ t be a legal decomposition of k. So k − 1 = a ℓ 1 + a ℓ 2 + · · · + (a ℓt − 1). It must be the case that either a ℓt − 1 is zero or it has a legal decomposition with summands indexed smaller than ℓ t , as a ℓt was added because it was the smallest integer that could not be legally decomposed with summands indexed smaller than ℓ t . If ℓ t was sufficiently distant from ℓ t−1 for the decomposition of k to be legal, using summands with even smaller indices does not create an illegal interaction with the remaining summands a ℓ 1 , . . . , a ℓ t−1 . This lemma allows us to conclude that the smallest integer that does not have a legal decomposition using {a 1 , . . . , a n } is one more than the largest integer that does have a legal decomposition using {a 1 , . . . , a n }. Lemma 2.3. If s, b, n ≥ 1 and 1 ≤ j ≤ b + 1, then a j+nb = a 1+nb + (j − 1)a 1+(n−s)b . ( 2.2) Proof. The term a 1+nb is the first entry in the (n + 1) st bin and trivially satisfies the recursion relation for j = 1. Recall a legal decomposition containing a member of the (n + 1) st bin would not have other addends from any of bins {B n−s+1 , B n−s+2 , . . . , B n , B n+1 }. So by construction we have a 2+nb = a 1+nb + a 1+(n−s)b , as the largest integer that can be legally decomposed using addends from bins B 1 , B 2 , . . . , B n−s is a 1+(n−s)b − 1. Using the same argument we have a 3+nb = a 2+nb + a 1+(n−s)b = a 1+nb + 2a 1+(n−s)b . (2.3) We proceed similarly for j = 4, . . . , b. For j = b + 1, the term a b+1+nb = a 1+(n+1)b is the first entry in the (n + 2) nd bin. By construction a 1+(n+1)b = a (n+1)b + a 1+(n−s)b . Using Equation (2.2) with j = b we have a 1+(n+1)b = a (n+1)b +a 1+(n−s)b = a 1+nb +(b−1)a 1+(n−s)b +a 1+(n−s)b = a 1+nb +ba 1+(n−s)b . (2.4) Proof of Theorem 1.3. Fix s, b ≥ 1. We proceed by considering i of the form j+nb, j ∈ {1, . . . , b}, so a i = a j+nb is the j th entry in the (n + 1) st bin. Using Lemma 2.3, a j+nb = a 1+nb + (j − 1)a 1+(n−s)b = a 1+(n−1)b + ba 1+(n−s−1)b + (j − 1)a 1+(n−s)b = a 1+(n−1)b + (j − 1)a 1+(n−s−1)b + (b − j + 1)a 1+(n−s−1)b + (j − 1)a 1+(n−s)b = a j+(n−1)b + (b − j + 1)a 1+(n−s−1)b + (j − 1)a 1+(n−s)b . (2.5) Again using the construction of our sequence we have a 1+(n−s)b = a (n−s)b + a 1+(n−2s−1)b . This substitution gives a j+nb = a j+(n−1)b + (b − j + 1)a 1+(n−s−1)b + (j − 1)a (n−s)b + (j − 1)a 1+(n−2s−1)b = a j+(n−1)b + a 1+(n−s−1)b + (j − 1)a 1+(n−2s−1)b + (b − j)a 1+(n−s−1)b + (j − 1)a (n−s)b = a j+(n−1)b + a j+(n−s−1)b + (b − j)a 1+(n−s−1)b + (j − 1)a (n−s)b . (2.6) Note that by Lemma 2.3, a (n−s)b = a 1+(n−s−1)b + (b − 1)a 1+(n−2s−1)b , so the last two terms in (2.6) may be simplified as (b − j)a 1+(n−s−1)b + (j − 1)a 1+(n−s−1)b + (j − 1)(b − 1)a 1+(n−2s−1)b = (b − 1) a 1+(n−s−1)b + (j − 1)a 1+(n−2s−1)b = (b − 1)a j+(n−s−1)b . (2.7) Substituting (2.7) into Equation (2.6) yields a j+nb = a j+(n−1)b + a j+(n−s−1)b + (b − 1)a j+(n−s−1)b = a j+(n−1)b + ba j+(n−s−1)b , (2.8) which completes the proof of the first part of Theorem 1.3. For the proof of the second part, we have from Lemma 2.3 a j+nb = a j−1+nb + a 1+(n−s)b , (2.9) thus a j+nb = a j−1+nb + a j−1+nb−(sb+j−2) (2.10) for j = 2, . . . , b + 1. The result now follows if we define f (j + nb) = sb + j − 1, for j = 1, . . . , b. We prove the Generalized Binet Formula and the approximation in Appendix A. Recurrence Relations for Fibonacci Quilt Sequence. Proof of Theorem 1.7. The proof is by induction. The basis cases for n ≤ 11 can be checked by brute force. By construction, we can legally decompose all numbers in the interval [1, q n−4 − 1] using terms in {q 1 , . . . , q n−5 }; q n−4 was added to the sequence because it was the first number that could not be decomposed using those terms. So, using q n , we can legally decompose all numbers in the interval, [q n , q n + q n−4 − 1]. In fact, we can decompose all numbers in the interval [1, q n + q n−4 − 1] using {q 1 , . . . , q n }. The term q n+1 will be the smallest number that we cannot legally decompose using {q 1 , . . . , q n }. The argument above shows that q n+1 ≥ q n + q n−4 . Notice q n + q n−4 = (q n−1 + q n−5 ) + q n−4 = q n−1 + (q n−4 + q n−5 ) = q n−1 + q n−2 . (2.11) It remains to show that there is no legal decomposition of m = q n + q n−4 = q n−1 + q n−2 . If q n were in the decomposition of m, the remaining summands would have to add to q n−4 . But that is a contradiction as q n−4 was added to the sequence because it had no legal decompositions as sums of other terms. Similarly, we can see that any legal decomposition of m does not use q n−1 , q n−2 , q n−4 . Notice that q n−3 must be part of any possible legal decomposition of m: if it were not, then m < n−5 i=1 q i = q n − 6 < q n < q n + q n−4 = m. Hence any legal decomposition would have m = q n−3 + x, where the largest possible summand in the decomposition of x is q n−5 . Now assume we have a legal decomposition of m = q n−3 + x. There are two cases. Case 1: The legal decomposition of x uses q n−5 as a summand. So m = q n−3 + x = q n−3 + q n−5 + y (2.12) and y can be legally decomposed using summands from {q 1 , q 2 , . . . , q n−10 }. Then using Equation (1.8), y < n−10 i=1 q i = q n−5 − 6. This leads us to the following: q n + q n−4 = m < q n−3 + q n−5 + q n−5 − 6 < q n−3 + q n−4 + q n−5 − 6 = q n−1 + q n−5 − 6 = q n − 6 < q n , (2.13) a contradiction. Case 2: The largest possible summand used in the legal decomposition of x is q n−8 . Thus q n + q n−4 = m < q n−3 + n−8 i=1 q i = q n−3 + q n−3 − 6 < q n−2 + q n−3 < q n , (2.14) another contradiction. So m cannot be legally decomposed using {q 1 , . . . , q n } and q n+1 = q n + q n−4 . The proof of Equation (1.7) follows from the work done in Equation (2.11). To prove Equation (1.8), note n+1 i=1 q i = q n+1 + n i=1 q i = q n+1 + q n+5 − 6 = q n+6 − 6. (2.15) Proposition 2.4 (Explicit Formula). Let q n denote the n th term in the Fibonacci Quilt sequence. Then q n = α 1 λ n 1 + α 2 λ n 2 + α 3 λ 2 n , (2.16) where α 1 ≈ 1.26724, λ 1 = 1 3 27 2 − 3 √ 69 2 1/3 + 1 2 9 + √ 69 1/3 3 2/3 ≈ 1.32472 (2.17) and λ 2 ≈ −0.662359 − 0.56228i (which has absolute value approximately 0.8688). Proof. Using the recurrence relation from Equation (1.6) in Theorem 1.7, we have the characteristic equation x 3 = x + 1. (2.18) Hence q n = α 1 λ n 1 + α 2 λ n 2 + α 3 λ 2 n , where λ 1 , λ 2 and λ 2 are the three distinct solutions to the characteristic equation, which are easily found by the cubic formula. We solve for the α i using the first few terms of the sequence. Straightforward calculations reveal α 1 ≈ 1.26724 α 2 ≈ −0.13362 + 0.128277i α 3 ≈ −0.13362 − 0.128277i, (2.19) completing the proof. GROWTH RATE OF NUMBER OF DECOMPOSITIONS FOR FIBONACCI QUILT SEQUENCE We prove Theorem 1.11 by deriving a recurrence relation for the number of FQ-legal decompositions. Specifically, consider the following definitions. • d n : the number of FQ-legal decompositions using only elements of {q 1 , q 2 , . . . , q n }. Note we include one empty decomposition of 0 in this count. Further, some of the decompositions are of numbers larger than q n+1 (for example, for n large q n + q n−2 + q n−20 > q n+1 ). We set d 0 = 1. • c n : the number of FQ-legal decompositions using only elements of {q 1 , q 2 , . . . , q n } and q n is one of the summands. We set c 0 = 1. • b n : the number of FQ-legal decompositions using only elements of {q 1 , q 2 , . . . , q n } and both q n and q n−2 are used. By brute force one can compute the first few values of these sequences; see Table 1. n d n c n b n q n 1 2 1 0 1 2 3 1 0 2 3 4 1 0 3 4 6 2 1 4 5 8 2 1 5 6 11 3 1 7 7 15 4 1 9 8 21 6 2 12 9 30 9 3 16 10 42 12 4 21 11 59 17 6 28 12 82 23 8 37 13 114 32 11 49 TABLE 1. Values of the first few terms of d n , c n and b n ; for ease of comparison we have included q n as well. We first find three recurrence relations interlacing our three unknowns. d n = c n + c n−1 + · · · + c 0 = c n + d n−1 c n = d n−5 + c n−2 − b n−2 b n = d n−7 , (3.1) which implies d n = d n−1 + d n−2 − d n−3 + d n−5 − d n−9 . (3.2) Proof. The relation for d n in (3.1) is the simplest to see. The left hand side counts the number of FQ-legal decompositions where the largest element used is q n , which may or may not be used. The right hand side counts the same quantity, partitioning based on the largest index used. It is important to note that c 0 is included and equals 1, as otherwise we would not have the empty decomposition (corresponding to an FQ-legal decomposition of 0). We immediately use this relation with n − 1 for n to replace c n−1 + · · · + c 0 with d n−1 . Our second relation comes from counting the number of FQ-legal decompositions where q n is used and no larger index occurs, which is just c n . Since q n occurs in all such numbers we cannot use q n−1 , q n−3 or q n−4 , but q n−2 may or may not be used. If we do not use q n−2 then we are left with choosing FQ-legal decompositions where the largest index used is at most n − 5; by definition this is d n−5 . We must add back all the numbers arising from decompositions using q n and q n−2 . Note that if n − 2 was the largest index used then the number of valid decompositions is c n−2 ; however, this includes b n−2 decompositions where we use both q n−2 and q n−4 . As we must use q n , we cannot use q n−4 and thus these b n−2 decompositions should not have been included; thus c n equals d n−5 + c n−2 − b n−2 . (Note: alternatively one could prove the relation c n = d n−5 + b n .) 12 Finally, consider b n . This counts the times we use q n (which forbids us from using q n−1 , q n−3 and q n−4 ) and q n−2 (which forbids us from using q n−3 , q n−5 and q n−6 ). Note all other indices at most n − 7 may or may not be used, and no other larger index can be chosen. By definition the number of valid choices is d n−7 . We now easily derive a recurrence involving just the d's. The first relation yields c n = d n − d n−1 while the third gives b n = d n−7 . We can thus rewrite the second relation involving only d's, which immediately gives (3.2). Armed with the above, we solve the recurrence for d n . Lemma 3.2. We have d n = β 1 r n 1 [1 + O ((r 2 /r 1 ) n )] ,(3. 3) where β 1 > 0, r 1 ≈ 1.39704 and r 2 ≈ 1.07378 are the two largest (in absolute value) roots of r 7 − r 6 − r 2 − 1 = 0. Proof. The characteristic polynomial associated to the recurrence for d n in (3.2) factors as r 9 − r 8 − r 7 + r 6 − r 4 + 1 = (r − 1)(r + 1)(r 7 − r 6 − r 2 − 1). (3.4) The roots of the septic are all distinct, with the largest r 1 approximately 1.39704 and the next two largest being complex conjugate pairs of size r 2 ≈ 1.07378; the remaining roots are at most 1 in absolute value. Thus by standard techniques for solving recurrence relations [Gol] (as the roots are distinct) there are constants such that d n = β 1 r n 1 + β 2 r n 2 + · · · + β 7 r n 7 + β 8 1 n + β 9 (−1) n . (3.5) To complete the proof, we need only show that β 1 > 0 (if it vanished, then d n would grow slower than one would expect). As the roots come from a degree 7 polynomial, it is not surprising that we do not have a closed form expression for them. Fortunately a simple comparison proves that β 1 > 0. Since d n counts the number of FQ-legal decompositions using indices no more than q n , we must have d n ≥ q n . As q n grows like λ n 1 with λ 1 ≈ 1.3247, if β 1 = 0 then d n < q n for large n, a contradiction. Thus β 1 > 0. We can now determine the average behavior of d FQ (m), the number of FQ-legal decompositions of m. Proof of Theorem 1.11. We have d FQ;ave (n) = 1 q n+1 q n+1 −1 m=0 d FQ (m). (3.6) We first deal with the upper bound. The summation on the right hand side of Equation (3.6) is less than d n , because d n counts some FQ-legal decompositions that exceed q n+1 . Thus d FQ;ave (n) ≤ d n q n+1 . (3.7) For n large by Lemma 3.2 we have d n = β 1 r n 1 [1 + O ((r 2 /r 1 ) n )] (3.8) with β 1 > 0 and r 1 ≈ 1.39704, and from Proposition 2.4 and λ 2 ≈ −0.662359 − 0.56228i (which has absolute value approximately 0.8688). Thus there is a C 2 > 0 such that for n large we have d FQ;ave (n) ≤ C 2 (r 1 /λ 1 ) n . We now turn to the lower bound for d FQ;ave (n). As we are primarily interested in the growth rate of d FQ;ave (n) and not on optimal values for the constants C 1 and C 2 , we can give a simple argument which suffices to prove the exponential growth rate, though at a cost of a poor choice of C 1 . Note that for large n the sum on the right side of Equation (3.6) is clearly at least d n−2016 . To see this, note d n−2016 counts the number of FQ-legal decompositions using no summand larger than q n−2016 , and if q n−2016 is our largest summand then by (1.8) our number cannot exceed q n = α 1 λ n 1 [1 + O ((λ 2 /λ 1 ) n )](3.n−2016 i=1 q i = q n−2011 − 6 ≤ q n . (3.11) Thus d FQ;ave (n) ≥ d n−2016 q n+1 . (3.12) We now argue as we did for the upper bound, noting that for large n we have d n−2016 = r −2016 1 · β 1 r n 1 [1 + O ((r 2 /r 1 ) n )] . (3.13) Thus for n sufficiently large d FQ;ave (n) ≥ C 1 (r 1 /λ 1 ) n , (3.14) completing the proof. GREEDY ALGORITHMS FOR THE FIBONACCI QUILT SEQUENCE 4.1. Greedy Decomposition. Let h n denote the number of integers from 1 to q n+1 − 1 where the greedy algorithm successfully terminates in a legal decomposition. We have already seen that the first number where the greedy algorithm fails is 6; the others less than 200 are 27, 34, 43, 55, 71, 92, 113, 120, 141, 148, 157, 178, 185 and 194. Table 2 lists h n for the first few values of n, as well as ρ n the percentage of integers in [1, q n+1 ) where the greedy algorithm yields a legal decomposition. We start by determining a recurrence relation for h n . Lemma 4.1. For h n as above, h n = h n−1 + h n−5 + 1, (4.1) with initial values h k = k for 1 ≤ k ≤ 5. Proof. We can determine the number integers in [1, q n+1 ) for which the greedy algorithm is successful by counting the same thing in [1, q n ) and in [q n , q n+1 ). The number of integers in [1, q n ) for which the greedy algorithm is successful is just h n−1 . Integers m ∈ [q n , q n+1 ) for which the greedy algorithm is successful must have largest summand q n . So m = q n + x. We claim x ∈ [0, q n−4 ). Otherwise m = q n + x ≥ q n + q n−4 = q n+1 , which is a contradiction. If x = 0, then m = q n can be legally decomposed using the greedy algorithm and we must add 1 to our count. If m is to have a successful legal greedy decomposition then so must n q n h n ρ n 1 1 1 100. Values of the first few terms of q n , h n and ρ n . x. Hence it remains to count how many x ∈ [1, q n−4 ) have successful legal greedy decompositions, but this is just h n−5 . Combining these counts finishes the proof. We now prove the greedy algorithm successfully terminates for a positive percentage of integers, as well as fails for a positive percentage of integers. Proof of Theorem 1.13. Instead of solving the recurrence in (4.1), it is easier to let g n = h n + 1 and first solve g n = g n−1 + g n−5 , g k = k + 1 for 1 ≤ k ≤ 5. (4.2) The characteristic polynomial for this is r 5 − r 4 − 1 = 0, or (r 3 − r − 1)(r 2 − r + 1). (4.3) By standard recurrence relation techniques, we have g n = c 1 λ n 1 + c 2 λ n 2 + · · · + c 5 λ n 5 , (4.4) where λ 1 = 1 3 27 2 − 3 √ 69 2 1/3 + 1 2 9 + √ 69 1/3 3 2/3 ≈ 1.32472 (4.5) is the largest root of the recurrence for g n (the other roots are at most 1 in absolute value). By Proposition 2.4 we have q n = α 1 λ n 1 + α 2 λ n 2 + α 3 λ n 3 , (4.6) where λ 1 , λ 2 , λ 3 are the same as in Equation (4.4) and α 1 ≈ 1.26724. We must show that c 1 α 1 = 0, as this will imply that g n and q n both grow at the same exponential rate. As g n ≥ 2g n−5 implies g n ≥ c2 n/5 we have that g n is growing exponentially, thus c 1 = 0. Unfortunately writing c 1 in closed form requires solving a fifth order equation, but this can easily be done numerically and the limiting ratio ρ n = h n /(q n+1 − 1) can be approximated well. That ratio converges to c 1 α 1 1 λ 1 ≈ 0.92627. 4.2. Greedy-6 Decomposition. Lemma 4.2. For ℓ ≥ 1 + 5k and k ≥ 0, we have q ℓ + q ℓ−5 + · · · + q ℓ−5k < q ℓ+1 . Proof. We proceed by induction on k. For the Basis Step, note q ℓ + q ℓ−5 < q ℓ + q ℓ−4 = q ℓ+1 . (4.7) For the Inductive Step: By inductive hypothesis and the recurrence relation stated in Theorem 1.7, q ℓ + (q ℓ−5 + · · · + q ℓ−5k ) < q ℓ + q ℓ−4 = q ℓ+1 , (4.8) completing the proof. Proof of Theorem 1.15. For the first part, we verify that if m ≤ 151 = q 17 the theorem holds. Define I n := [q n , q n+1 ) = [q n , q n+1 − 1]. Assume for all m ∈ ∪ n−1 ℓ=1 I ℓ , m satisfies the theorem. Now consider m ∈ I n . If m = q n then we add done. Assume m = q n + x with x > 0. Since q n+1 = q n + q n−4 , we know x < q n−4 . Then by the inductive hypothesis we know the x satisfies the theorem. Namely, G(x) = q k 1 + q k 2 + · · · + q ks is a FQ-legal decomposition which satisfies either Condition (1) or (2) but not both. Then G(m) = q n +q k 1 +q k 2 +· · ·+q ks and lastly n−k 1 ≥ 5. For the second part, let m = q ℓ 1 + q ℓ 2 + · · · + q ℓ t−1 + q ℓt be a decomposition that satisfies either Condition (1) or (2) but not both. Note that in both cases, this decomposition is legal. If t = 1, then m is a Fibonacci Quilt number and the theorem is trivial. So we assume t ≥ 2. Hence by construction of the sequence, m is not a Fibonacci Quilt number. Let G(m) = q k 1 + q k 2 + · · · + q ks . Note that s ≥ 2. For contradiction we assume the given decomposition is not the Greedy-6 decomposition. Without loss of generality we may assume q ℓ 1 = q k 1 . Since q k 1 was chosen according to the Greedy-6 algorithm, q ℓ 1 < q k 1 . Case 1: Using Lemma 4.2, m = q ℓ 1 + q ℓ 2 + · · · + q ℓ t−1 + q ℓt ≤ q ℓ 1 + q ℓ 1 −5 + · · · + q ℓ 1 −5(t−1) < q ℓ 1 +1 ≤ q k 1 < m (4.9) which is a contradiction. Case 2: Again using Lemma 4.2, m = q ℓ 1 + q ℓ 2 + · · · + q ℓ t−2 + q 4 + q 2 = q ℓ 1 + q ℓ 2 + · · · + q ℓ t−2 + q 5 + q 1 ≤ q ℓ 1 + q ℓ 1 −5 + · · · + q ℓ 1 −5(t−2) + q 1 < q ℓ 1 +1 + q 1 ≤ q k 1 + q 1 ≤ m (4.10) which is a contradiction. In order to prove Theorem 1.16 we will need several relationships between the terms in the Fibonacci Quilt sequence. The following lemma describes those relationships. (1) If n ≥ 7, then 2q n = q n+2 + q n−5 . (2) If n ≥ 8, then q n + q n−2 = q n+1 + q n−5 . (3) If n ≥ 10, then q n + q n−3 = q n+1 + q n−8 . Proof. The proof follows from repeated uses of the recurrence relations stated in Theorem 1.7: 2q n = q n + q n−1 + q n−5 = q n+2 + q n−5 , (4.11) q n + q n−2 = q n + q n−4 + q n−5 = q n+1 + q n−5 , (4.12) and q n + q n−3 = q n + q n−4 + q n−3 − q n−4 = q n+1 + q n−8 . (1) Replace 2q n with q n+2 + q n−5 (for n ≥ 7). (If n ≤ 6, replace 2q 6 with q 8 + q 2 , replace 2q 5 with q 7 + q 1 , replace 2q 4 with q 6 + q 1 , replace 2q 3 with q 5 + q 1 , replace 2q 2 with q 4 , and replace 2q 1 with q 2 .) (2) Replace q n−1 + q n−2 with q n+1 (for n ≥ 5). In other words, if we have two adjacent terms, use the recurrence relation to replace. (If n ≤ 4, replace q 3 + q 2 with q 5 and replace q 2 + q 1 with q 3 .) (3) Replace q n + q n−2 with q n+1 + q n−5 (for n ≥ 8). (If n ≤ 7, replace q 7 + q 5 with q 8 + q 2 , q 6 + q 4 with q 7 + q 2 , q 5 + q 3 with q 7 + q 1 , q 4 + q 2 with q 5 + q 1 , and q 3 + q 1 with q 4 .) (4) Replace q n + q n−3 with q n+1 + q n−8 (for n ≥ 10). (If n ≤ 9, replace q 9 + q 6 with q 10 + q 2 , q 8 +q 5 with q 9 +q 1 , q 7 +q 4 with q 8 +q 1 , q 6 +q 3 with q 7 +q 1 , q 5 +q 2 with q 6 , and q 4 +q 1 with q 5 .) (5) Replace q n + q n−4 with q n+1 (for n ≥ 6). In other words, if we have two adjacent terms, use the recurrence relation to replace. Notice that in all moves, the number of summands either decreases by one or remains unchanged. In addition, the sum of the indices either decreases or remains unchanged. There are three situations where neither the index sum nor the number of summands decreases; q 5 + q 3 = q 7 + q 1 , q 4 + q 2 = q 5 + q 1 , and 2q 3 = q 5 + q 1 . But in these situations, the number of q 5 , q 4 , q 3 , q 2 decrease. Therefore this process eventually terminates because the index sum and the number of summands cannot decrease indefinitely. Let m = q ℓ 1 + q ℓ 2 + · · · + q ℓ t−1 + q ℓt be the decomposition obtained after all possible moves. Each move either decreases the number of summands or replaces two summands with two that are farther apart in the sequence. In fact, closer examination of the moves reveals ℓ i − ℓ i−1 ≥ 5 except maybe ℓ t−1 = 5 and ℓ t = 1. If ℓ t−1 = 5 and ℓ t = 1, replace q 5 + q 1 with q 4 + q 2 . By Theorem 1.15 this is the Greedy-6 decomposition of m. APPENDIX A. GENERALIZED BINET FORMULA FOR (s, b)-GENERACCI SEQUENCE We now prove the Generalized Binet Formula for the (s, b)-Generacci sequence. The argument is almost standard, but the fact that the leading coefficient in the recurrence relation is zero leads to some technical obstructions. We resolve these by first passing to a related characteristic polynomial where the leading coefficient is positive (and then Perron-Frobenius arguments are applicable), and then carefully expand to our sequence. Proof of the Generalized Binet Formula in Theorem 1.3. The recurrence in (1.2) generates the characteristic polynomial x (s+1)b − x sb − b = 0. (A.1) Letting y = x b in (A.1), we are able to pass to studying q(y) = y s+1 − y s − b = 0. (A.2) The polynomial q(y) has the following properties. (1) The roots are distinct. (2) There is a positive root r satisfying r > \|r j \| where r j is any other root of q(y). (3) The positive root r described in (2) satisfies r > 1 and is the only positive root. To prove property (1), consider q ′ (y) = (s + 1)y s − sy s−1 = (s + 1)y s−1 y − s s + 1 . (A.3) If a repeated root y exists then q(y) = q ′ (y) = 0. Clearly, y = 0 is not a root, so y = s s + 1 < 1. In this case, b = s s + 1 s s s + 1 − 1 < 0, (A.4) which is a contradiction as b is a positive integer. Property (2) follows from the same argument used in the proof of Theorem A.1 in [BBGILMT], or by using the Perron-Frobenius Theorem for non-negative irreducible matrices. Furthermore, since the root r satisfies r s (r − 1) = b and b > 0, necessarily r > 1. Now, q(0) = −b, q(r) = 0, q ′ (y) < 0 for y < s s + 1 , and q ′ (y) > 0 for y > s s + 1 , implies that q(y) > 0 for all y > r. Hence, r is the only positive root of q(y), completing the proof of property (3). Let ω 1 > 0 be chosen so that ω b 1 = r, and let the (distinct) roots of (A.2) be denoted by ω b 1 , ω b 2 , . . . , ω b s+1 , where ω b 1 > 1 is the only positive root and ω b 1 > \|ω b j \|, for all j = 2, . . . , s + 1. For convenience, we arrange the roots so that ω b 1 > \|ω b 2 \| ≥ · · · ≥ \|ω b s+1 \|. Then the roots of (A.1) are given by is a primitive b th root of unity. Now, using standard results on solving linear recurrence relations (see for example [Gol,Section 3.7]), the n th term of the sequence has an expansion a n = b−1 k=0 s+1 j=1 α k,j (ω j ζ k b ) n = for j = 1, . . . , s + 1. Note that c 1 must be a real number, as otherwise a ℓb+v is non-real for large ℓ (since ω bℓ 1 > 0 is the dominant term in the expansion). The final step is to prove that c 1 > 0. If c 1 < 0, then for large ℓ, a ℓb+v < 0 (again since ω bℓ 1 > 0 is the dominant term in the expansion). If c 1 = 0, then a ℓb+v = s+1 j=m c j ω bℓ j , where m is the smallest index greater than 1 such that c m = 0. Then the dominant term in the expansion is ω bℓ m , where, by property (3) of the polynomial q(y), the root ω b m is either negative or complex non-real. If ω b m < 0, then ω bℓ m alternates in sign which violates a ℓb+v > 0 for all ℓ. If ω b m is complex nonreal, then ω bℓ m is not always real, again violating a ℓb+v > 0 for all ℓ. Thus, c j = 0 for all j > 1, and since c 1 = 0 this implies that a ℓb+v = 0, a contradiction. FIGURE 1 . 1The (start of the) Fibonacci Spiral. FIGURE 2 . 2(Left) Log Cabin Quilt Pattern. (Right) First few terms of the Fibonacci Quilt sequence. 1. 2 . 1 . 21Decomposition results. Theorem 1.9 (Uniqueness of Decompositions for (s, b)-Generacci). For each pair of integers s, b ≥ 1, a unique (s, b)-Generacci sequence exists. Consequently, for a given pair of integers s, b ≥ 1, every positive integer can be written uniquely as a sum of distinct terms of the (s, b)-Generacci sequence where no two summands are in the same bin, and between any two summands there are at least s bins between them. ( 1 . 12 ) 112Theorem 1.16. If D(m) is any decomposition of m as a sum of Fibonacci Quilt numbers, then #summands(G(m)) ≤ #summands(D(m)). (1.13) 1.2.2. Gaussian Behavior of Number of Summands in (s, b)-Generacci legal decompositions. Below we report on the distribution of the number of summands in the (s, b)-Generacci legal decompositions. Theorem 1. 17 ( 17Gaussian Behavior of Summands for (s, b)-Generacci). Let the random variable Y n denote the number of summands in the (unique) (s, b)-Generacci legal decomposition of an integer picked at random from [0, a bn+1 ) with uniform probability. 1 Then for µ n and σ 2 n , the mean 1 prove the following results related to the elements of the (s, b)-Generacci sequence. 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[ "Double Hodge Theory for a particle on Torus", "Double Hodge Theory for a particle on Torus" ]	[ "Vipul Kumar Pandey \nDepartment of Physics\nBanaras Hindu University\n221005VaranasiINDIA\n", "Bhabani Prasad Mandal \nDepartment of Physics\nBanaras Hindu University\n221005VaranasiINDIA\n" ]	[ "Department of Physics\nBanaras Hindu University\n221005VaranasiINDIA", "Department of Physics\nBanaras Hindu University\n221005VaranasiINDIA" ]	[]	We investigate all possible nilpotent symmetries for a particle on torus. We explicitly construct four independent nilpotent BRST symmetries for such systems and derive the algebra between the generators of such symmetries. We show that such a system has rich mathematical properties and behaves as double Hodge theory. We further construct the finite field dependent BRST transformation for such systems by integrating the infinitesimal BRST transformation systematically. Such a finite transformation is useful in realizing the various theories with toric geometry.	10.1155/2017/6124189	[ "https://arxiv.org/pdf/1708.06625v1.pdf" ]	55,692,324	1708.06625	d516fc2d7671f416e3597c8474ea4c7b312b23de	Double Hodge Theory for a particle on Torus 19 Aug 2017 Vipul Kumar Pandey Department of Physics Banaras Hindu University 221005VaranasiINDIA Bhabani Prasad Mandal Department of Physics Banaras Hindu University 221005VaranasiINDIA Double Hodge Theory for a particle on Torus 19 Aug 2017arXiv:1708.06625v1 [physics.gen-ph] We investigate all possible nilpotent symmetries for a particle on torus. We explicitly construct four independent nilpotent BRST symmetries for such systems and derive the algebra between the generators of such symmetries. We show that such a system has rich mathematical properties and behaves as double Hodge theory. We further construct the finite field dependent BRST transformation for such systems by integrating the infinitesimal BRST transformation systematically. Such a finite transformation is useful in realizing the various theories with toric geometry. I. INTRODUCTION The formulation based on BRST symmetry [1,2] plays crucial role in the discussion of quantization, renormalization, unitarity and other aspects of gauge theories. The nilpotency nature of BRST transformation is mainly responsible for simplified treatment in all these discussions. Thus it is extremely important to find more and more nilpotent symmetry associated with any system to study, particularly the systems with constraints. Toric geometry which is generalization of the projective identification that defines CP n corresponding to the most general linear sigma model provides a scheme for constructing Calabi-Yau manifolds and their mirrors [3]. Recently, on the basis of boundary string field theory [4], the brane-antibrane system was exploited [5] in the toroidal background to investigate its thermodynamic properties associated with the Hagedorn temperature [6,7]. The Nahm transform and moduli spaces of CP n models were also studied on the toric geometry [8]. In a four dimensional, toroidally compactified heterotic string, the electrically charged BPS-saturated were shown to become massless along the hyper surfaces of enhanced gauge symmetry of a two-torus moduli subspace [9]. In the present work we investigate various possible nilpotent symmetries for a particle on torus. Usual BRST symmetry for a particle on torus has already been constructed [10]. In this work we construct four different nilpotent symmetries associated with this system, namely BRST symmetry, anti-BRST symmetry, dual BRST (also known as co-BRST) symmetry and anti-dual BRST (also known as anti-co-BRST) symmetry [11][12][13]. We further construct two different bosonic symmetries using these nilpotent BRST symmetries and some discrete symmetries associated with ghost number are also written for such systems. Complete algebra satisfied by charges, which generate these symmetries are derived. Deep mathematical connections of such system with Hodge theory [33][34][35][36] are established in this work. We found that the system of particle on a torus is realized as Hodge theory w.r.t. to two different set of operators. The generators for BRST, dual-BRST symmetries and generator for corresponding bosonic symmetries constructed out of BRST and dual-BRST symmetries are analogous to exterior derivative, co-exterior derivative and Laplace operator in Hodge theory [14][15][16][17][18][19][20][21][22]. On the other hand the charges corresponding to anti-BRST symmetry, anti-dual-BRST symmetry and bosonic symmetry constructed out of these two BRST symmetries also from set of de-Rham co-homological operators. This indicates the mathematical foundation of the theory of a particle on a torus is extremely rich. We further extend the BRST transformation for this system by considering the BRST parameter as finite and field dependent. More than two decades ago Joglekar and Mandal introduced for the first time the concept of finite field dependent BRST(FFBRST)transformation [37], which had similar structure and properties of usual BRST transformation. However the path integral measure is not invariant due to finite nature of such transformation. It has been shown that by constructing suitable finite parameter one can calculate desirable Jacobian factor which under certain condition is added to the effective action of the theory . Thus FFBRST is capable of connecting generating functionals of two different effective theories. Because of these remarkable properties, FFBRST has become an useful tool of studying various field theoretic systems with BRST symmetry and it has found many applications [38][39][40][41][42][43][44][45][46][47]. We have constructed FFBRST transformation for the system of particle on torus to show the connection between two theories on torus with different gauge fixing. Now we present the plan of this manuscript. We start with the brief introduction about the free particle on the the surface of torus in Sec. II. Hamiltonian formulation for this theory is presented in sec. III. In Sec. IV the BFV formulation for this model has been discussed and BRST symmetry for such model has been constructed. In Sec. V the other nilpotent symmetry transformations for same system have been constructed. Co-BRST and anti co-BRST have been discussed in sec. VI. Other symmetries have been discussed in section VII. The connection between algebra satisfied by the nilpotent charges and de Rham co-homological operators of differential geometry is shown in Sec. VIII. In section IX we introduce FFBRST transformation and in next Sec. we connect theory in different gauges using FFBRST transformations. We conclude our results in Sec. XI. II. FREE PARTICLE ON SURFACE OF TORUS A particle moving freely on the surface of a torus is described by Lagrangian L 0 = 1 2 mṙ 2 + 1 2 mr 2θ2 + 1 2 m(b + r sin θ) 2φ2(1) where (r, θ, φ) are toroidal co-ordinates related to Cartesian coordinates as x = (b + r sin θ) cos φ, y = (b + r sin θ) sin φ, z = r cos θ(2) Here we have considered a torus with axial circle in the x − y plane centered at the origin, of radius b, having a circular cross section of radius r. The angle θ ranges from 0 to 2π, and the angle φ from 0 to 2π. Since the particle moves on the surface of torus of radius r, it is constrained to satisfy Ω 1 = r − a ≈ 0(3) The canonical Hamiltonian corresponding to the Lagrangian in Eq.(1) with the above constraint is then written as H 0 = p 2 r 2m + p 2 θ 2mr 2 + p 2 φ 2m(b + r sin θ) 2 + λ(r − a)(4) where p r , p θ and p φ are the canonical momenta conjugate to the coordinate r, θ and φ, respectively, given by p r = mṙ, p θ = mr 2θ , p φ = m(b + r sin θ) 2(5) The time evolution of the constraint Ω 1 yields the secondary constraint as Ω 2 = p r ≈ 0 (6) III. WESS-ZUMINO TERM AND HAMILTONIAN FORMULATION To construct a gauge invariant theory corresponding to the gauge non-invariant model in Eq.(4), we introduce the Wess-Zumino term [23] in the Lagrangian density L. For this purpose we enlarge the Hilbert space of the theory by introducing a new quantum field η, called as Wess-Zumino field, through the redifinition of fields r and λ in the original Lagrangian density L as follows r → r − η; λ → λ +η (7) With this redefinition of the fields, the modified Lagrngian density becomes L I = 1 2 (ṙ −η) 2 + 1 2 m(r − η) 2θ2 + 1 2 m(b + (r − η) sin θ) 2φ2 − (λ +η)(r − a − η)(8) Canonical momenta corresponding to this modified Lagrangian density are then given by p r = m(ṙ −η), p η = (m(ṙ −η) + (r − a − η)), p λ = 0 p θ = m(r − η) 2θ , p φ = m(b + (r − η) sin θ) 2φ(9) The primary constraints for this extended theory is ψ 1 ≡ p λ ≈ 0(10) The Hamiltonian density corresponding to L I is written as, H I = p rṙ + p ηη + p θθ + p φφ + p λλ − L I(11) The total Hamiltonian density after the introduction of a Lagrange multiplier field u corresponding to the primary constraint ψ 1 is then obtained as H I T = p 2 r 2m + p 2 θ 2m(r − η) 2 + p 2 φ 2m(b + (r − η) sin θ) 2 + λ(p r + p η ) + up λ(12) Following the Dirac's method of constraint analysis [22][23][24][25], we obtain secondary constraint ψ 2 ≡ (p η + p r ) ≈ 0(13) In next two sections, we extend this constrained theory to study the nilpotent symmetries associated with this theory. IV. BFV FORMULATION FOR FREE PARTICLE ON THE SURFACE OF TORUS To discuss all possible nilpotent symmetries we further extend the theory using BFV formalism [28][29][30][31]. In the BFV formulation associated with this system, we introduce a pair of canonically conjugate ghost fields (c,p) with ghost number 1 and -1 respectively, for the primary constraint p λ ≈ 0 and another pair of ghost fields (c,p) with ghost number -1 and 1 respectively, for the secondary constraint, (p η + p r ) ≈ 0. The effective action for a particle on surface of the torus in extended phase space is then written as S ef f = d 4 x p rṙ + p ηη + p θθ + p φφ − p λλ − p 2 r 2m − p 2 θ 2m(r − η) 2 − p 2 φ 2m(b + (r − η) sin θ) 2 +ċp +ċp − {Q b , ψ}(14) where Q b is the BRST charge and ψ is the gauge fixed fermion. This effective action is invariant under BRST transformation generated by Q b which is constructed using constraints in the theory as Q b = ic(p r + p η ) − ipp λ(15) The canonical brackets for all dynamical variables are written as [r, p r ] = [θ, p θ ] = [φ, p φ ] = [η, p η ] = [λ, p λ ] = {c,ċ} = i, {c,ċ} = −i(16) where rest of the brackets are zero. Now, the nilpotent BRST transformation, using the relation s b φ = [φ, Q b ] ± (± sign represents the fermionic and bosonic nature of the fields φ), are explicitly written as s b r = −c, s b λ =p, s bp = 0, s b θ = −c s b p φ = 0, s b p θ = 0, s b p = (p r + p η ) s bc = p λ , s b p λ = 0, s b c = 0(17) In BFV formulation the generating functional is independent of gauge fixed fermion [33], hence we have liberty to choose it in the convenient form as ψ = pλ +c(r + η + p λ 2 )(18) Putting the value of ψ in Eq. (14) and using Eqs., (15) and (16), we obtain S ef f = d 4 x p rṙ + p ηη + p θθ + p φφ − p λλ − p 2 r 2m − p 2 θ 2m(r − η) 2 − p 2 φ 2m(b + (r − η) sin θ) 2 +ċp +˙cp + λ(p r + p η ) + 2cc −pp + p λ (r + η + p λ 2 )(19) and the generating functional for this effective theory is represented as Z ψ = Dφ exp iS ef f(20) Now integrating this generating functional over p andp, we get Z ψ = Dφ ′ exp i d 4 x p rṙ + p ηη + p θθ + p φφ − p λλ − p 2 r 2m − p 2 θ 2m(r − η) 2 − p 2 φ 2m(b + (r − η) sin θ) 2 +ċċ + λ(p r + p η ) + 2cc + p λ (r + η + p λ 2 )(21) where Dφ ′ is the path integral measure for effective theory when integrations over fields p andp are carried out. Further integrating over field p λ we obtain an effective generating functional as Z ψ = Dφ ′′ exp i d 4 x p rṙ + p ηη + p θθ + p φφ − p 2 r 2m − p 2 θ 2m(r − η) 2 − p 2 φ 2m(b + (r − η) sin θ) 2 +˙cċ + λ(p r + p η ) − 2cc − (λ − r − η) 2 2(22) where Dφ ′′ is the path integral measure corresponding to all the dynamical variables involved in the effective action.The expression for effective action in above equation is similar to BRST invariant effective action in [34]. The BRST symmetry transformation for this effective theory is written as s b r = −c, s b λ =ċ, s b η = −c s b p r = 0, s b p η = 0 s bc = −(λ − η − r), s b c = 0(23) These transformations are on shell nilpotent. V. NILPOTENT SYMMETRIES In this section we will study various other nilpotent symmetries of this model with particle on a torus. For this purpose it is convenient to work using Nakanishi-Lautrup type auxiliary field B which linearize the gauge fixing part of the effective action in Eq. (22). The first order effective action is then given by S ef f = d 4 x p rṙ + p ηη + p θθ + p φφ − p 2 r 2m − p 2 θ 2m(r − η) 2 − p 2 φ 2m(b + (r − η) sin θ) 2 +ċċ + λ(p r + p η ) − 2cc − B(λ − r − η) + B 2 2(24) We can easily show that this action is invariant under the following off-shell nilpotent BRST transformation s b r = −c, s b λ =ċ, s b η = −c s b p r = 0, s b p η = 0, s b θ = 0 s bc = B, s bc = 0, s b p φ = 0 s b φ = 0, s b p θ = 0(25) Corresponding anti-BRST transformation for this theory is then written by interchanging the role of ghost and anti-ghost field as s ab r = −c, s ab λ =ċ, s ab η = −c s ab p r = 0, s ab p η = 0, s ab p φ = 0 s ab c = −B, s abc = 0, s ab θ = 0 s ab φ = 0, s ab p θ = 0(26) The conserved BRST and anti-BRST charges Q b and Q ab which generate above BRST and anti-BRST transformations are written for this effective theory as Q b = ic(p r + p η ) − ip λċ(27) and Q ab = ic(p r + p η ) − ip λċ(28) Further by using following equation of motion B +ṗ r = 0, B +ṗ η = 0ṙ − p r + λ = 0 B = p r + p η ,ċ + 2c = 0, c + 2c = 0, B +λ − r − η = 0(29) it is shown that these charges are constants of motion i.e.Q b = 0,Q ab = 0, and satisfy following relations, Q b Q ab + Q ab Q b = 0(30) To arrive on these relations, the canonical brackets [Eq. (16)] of the fields and the definition of canonical momenta have been used p λ = B, pc =ċ, p c = −ċ(31) The physical states of theory are annihilated by the BRST and anti-BRST charges, leading to (p r + p η )\|phys = 0 (32) and p λ \|phys = 0(33) This implies that the operator form of the first class constraint p λ ≈ 0 and (p r + p η ) ≈ 0 annihilates the physical state of the theory. Thus the physicality criteria is consistent with Dirac's method of quantization. VI. CO-BRST AND ANTI CO-BRST SYMMETRIES In this section, we investigate two other nilpotent transformations, namely co-BRST and anti co-BRST transformation which are also the symmetry of the effective action in Eq. (24). Further these transformations leave the gauge-fixing term of the action invariant independently and the kinetic energy term (which remains invariant under BRST and anti-BRST transformations) transforms under it to compensate for the transformation of the ghost terms. These transformations are also called as dual and anti dual-BRST transformation [11][12][13]. The nilpotent co-BRST transformation (s 2 d = 0) and anti co-BRST transformation (s 2 ad = 0) which leave the effective action [in eq. (24)] for a particle on torus invariant, are given by s d r = − 1 2ċ , s d λ = −c, s d η = − 1 2ċ s d p r = 0, , s d p η = 0, s dc = 0 s d c = 1 2 (p r + p η ), s d B = 0,(34) and s ad r = − 1 2ċ , s ad λ = −c, s ad η = − 1 2ċ s ad p r = 0, s ad p η = 0, s adc = 0 s ad c = − 1 2 (p r + p η ), s ad B = 0(35) These transformations are absolutely anti-commuting as {S d , S ad } = 0. The conserved charges for above symmetries are found using Noether's theorem and are written as Q d = i 1 2 (p r + p η )ċ + ip λc(36) and Q ad = i 1 2 (p r + p η )ċ + ip λ c(37) which generate the symmetry transformations in Eqs. (34) and (35) respectively. It is easy to verify the following relations s d Q d = −{Q d , Q d } = 0 s ad Q ad = −{Q ad , Q ad } = 0 s d Q ad = −{Q ad , Q d } = 0 s ad Q d = −{Q d , Q ad } = 0(38) which reflect the nilpotency and anti-commutativity property of s d and s ad (i.e. s 2 d = 0,s 2 ad = 0 and s d s ad + s ad s d = 0). VII. OTHER SYMMETRIES In this section, we construct other symmetries related to this system. Two different bosonic symmetries are constructed out of four nilpotent symmetries. Discrete symmetry related to ghost number is also constructed. A. Bosonic Symmetry In this part we construct the bosonic symmetry out of these nilpotent BRST symmetries of the theory. The BRST (s b ), anti-BRST (s ab ), co-BRST (s d ), and anti co-BRST(s ad ) symmetry operators satisfy the following algebra {s d , s ad } = 0, {s b , s ab } = 0 {s b , s ad } = 0, {s d , s ab } = 0(39) and we define bosonic symmetries, s w and sw as s w ≡ {s b , s d }, sw ≡ {s ab , s ad }(40) The fields variables transform under bosonic symmetry s w as s w r = − 1 2 (Ḃ + p r + p η ), s w λ = − 1 2 (2B −ṗ r −ṗ η ) s w η = − 1 2 (Ḃ + p r + p η ), s w p r = 0, s w p η = 0 s w c = 0, s w B = 0, s wc = 0(41) On the other hand transformation generated by sw is swr = − 1 2 (Ḃ + p r + p η ), swλ = 1 2 (2B −ṗ r −ṗ η ) swη = − 1 2 (Ḃ + p r + p η ), swp r = 0, swp η = 0 swc = 0, swB = 0, swc = 0(42) However the transformation generated by s w and sw are not independent as it is easy to see from Eq. (41) and (42) that the operators s w and sw satisfy the relation s w + sw = 0. This implies from Eq. (40), that {s b , s d } = s w = −{s ab , s ad }(43) It is clear from above algebra that the operator s w analogous of the Laplacian operator in the language of differential geometry and the conserved charge for the above symmetry transformation is calculated as Q w = −i[B 2 + 1 2 (p r + p η ) 2 ](44) which generates the transformation in Eq.(41). Using equation of motion, it can readily be checked that dQ w dt = −i dx[2BḂ + (p r + p η )(ṗ r +ṗ η )] = 0(45) Hence Q w is the constant of motion for this theory. B. Ghost Symmetry and Discrete Symmetry Now we consider yet another kind of symmetry of this system called ghost symmetry. The ghost numbers of the ghost and anti-ghost fields are 1 and -1 respectively. Rest of the variables in the action of this theory have ghost number zero. Keeping this fact in mind we can introduce a scale transformation of the ghost field, under which the effective action is invariant, as c → e Λ c c → e −Λc χ → χ(46) where χ = {r, η, θ, φ, u, λ, p r , p η , p θ , p φ , p u , B} and Λ is a global scale parameter. The infinitesimal version of the ghost scale transformation can be written as s g χ = 0 s g c = c s gc = −c(47) The Noether's conserved charge for above symmetry transformation is calculated as Q g = i[ċc +ċc](48) In addition to above continuous symmetry transformation, the ghost sector respects the following discrete symmetry transformations c → ±ic,c → ±ic (49) VIII. GEOMETRICAL COHOMOLOGY In this section we study the de Rham cohomological operators and their realization in terms of conserved charges which generate the nilpotent symmetries for the theory of a particle on the surface of torus. In particular we point out the similarities between the algebra obeyed by de Rham co-homological operators and that by different BRST conserved charges. A. Hodge-de Rham decomposition theorem and differential operators Before we proceed to discuss the analogy, we briefly review the essential features of Hodge theory. The de Rham cohomological operators in differential geometry obey the following algebra d 2 = δ 2 = 0, ∆ = (d + δ) 2 = dδ + δd ≡ {d, δ} [∆, δ] = 0, [∆, d] = 0(50) Where d, δ and ∆ are exterior, co-exterior and Laplace-Beltrami operator respectively. The operator d and δ are adjoint or dual to each other and ∆ is self-adjoint operator [35]. It is well known that the exterior derivative raises the degree of form by one when it operates on forms (i.e.df n ∼ f n+1 ), whereas the dual-exterior derivative lowers the degree of a form by one when it operates on forms (i.e.δf n ∼ f n−1 ). However ∆ does not change the degree of form (i.e.∆f n ∼ f n ). f n denotes an arbitrary n-form object. Let M be a compact, orientable Riemannian manifold, then an inner product on the vector space E n (M ) of n-forms on M can be defined as [36] (α, β) = M α ∧ * β where α, β ∈ E n (M ) and * represents the Hodge duality operator [37]. Suppose that α and β are forms of degree n and (n + 1) respectively, the following relation for inner product will be satisfied (dα, β) = (α, δβ)(52) Similarly, if β is form of degree n − 1, then we have the relation (α, dβ) = (δα, β). Thus the necessary and sufficient condition for α to be closed is that should be orthogonal to all co-exact forms of degree n. The form ω ∈ E n (M ) is called harmonic if dω = 0. Now let β be a n-form on M and if there exists another n-form α such that ∆α = β, then for a harmonic form γ ∈ H n , (β, γ) = (∆α, γ) = (α, ∆γ) = 0 (53) where H n (M ) denotes the subspace of E n (M ) of harmonic forms on M. Therefore, if a form α exist with the property that ∆α = β, then Eq. (53) is necessary and sufficient condition for β to be orthogonal to the subspace H n . This reasoning leads to the idea that E n (M ) can be partitioned into three distinct subspaces Λ n d , Λ n δ and H n which are consistent with exact, co-exact and harmonic forms respectively. Therefore, the Hodge-de Rham decomposition theorem can be stated as A regular differential form of degree n(α) may be uniquely decomposed into a sum of the harmonic form (α) H , exact form (α d ) and co-exact form (α δ ) i.e. α = α H + α d + α δ (54) where α ∈ H n , α s ∈ Λ n δ and α d ∈ Λ n d B. Hodge-de Rham decomposition theorem and conserved charge The generators of all the nilpotent symmetry transformations satisfy the following algebra Q 2 b = 0, Q 2 ab = 0, Q 2 d = 0, Q 2 ad = 0 {Q b , Q ab } = 0, {Q d , Q ad } = 0, {Q b , Q ad } = 0 {Q d , Q ab } = 0, [Q g , Q b ] = Q b , [Q g , Q ad ] = Q ad [Q g , Q d ] = −Q d , [Q g , Q ab ] = −Q ab , [Q w , Q r ] = 0 {Q b , Q d } = −{Q ad , Q ab } = Q w(55) Here the relations between the conserved charges Q b and Q ad as well as Q ab and Q ad can be found using equation of motions only. This algebra is similar to the algebra satisfied by de Rham co-homological operators of differential geometry given in Eq.(53). Comparing (53) and (58) we obtain following analogies (Q b , Q ad ) → d, (Q d , Q ab ) → δ, Q w → ∆(56) Let n be the ghost number associated with a given state \|ψ n defined in the total Hilbert space of states, i.e. iQ g \|ψ n = n\|ψ n (57) Then it is easy to verify the following relations Q g Q b \|ψ n = (n + 1)Q b \|ψ n Q g Q ad \|ψ n = (n + 1)Q ad \|ψ n Q g Q d \|ψ n = (n − 1)Q b \|ψ n Q g Q ab \|ψ n = (n − 1)Q ad \|ψ n Q g Q w \|ψ n = nQ w \|ψ n which imply that the ghost numbers of the states Q b \|ψ n ,Q d \|ψ n and Q w \|ψ n are (n+1),(n-1) and n respectively. The states Q ab \|ψ n and Q ad \|ψ n have ghost numbers (n-1) and (n+1) respectively. The properties of set (Q b , Q ad ) and (Q d , Q ab ) are same as of operators d and δ. It is evident from Eq. (58) that the set Q b , Q ad raises the ghost number of a state by one and the set Q d , Q ab lowers the ghost number of the same state by one. Keeping the analogy between charges of different nilpotent symmetries and Hodgede Rham differential operators, we express any arbitrary state \|ψ n in terms of the sets (Q b , Q d , Q w ) and (Q ad , Q ab , Qw) as \|ψ n = \|w n + Q b \|χ (n−1) + Q d \|φ (n+1) ψ n = \|w n + Q ad \|χ (n−1) + Q ab \|φ (n+1)(59) where the most symmetric state is the harmonic state\|w n that satisfies Q w \|w n = 0, Q b \|w n = 0, Q d \|w n = 0 Q ab \|w n = 0, Q ad \|w n = 0 (60) analogous to the Eq. (53). Therefore the BRST charges for a particle on a torus forms two separate set of de-Rham co-homological operator, namely {Q b , Q ab , Q w } and {Q d , Q ad , Qw}.Thus we call the theory of a particle on torus as double Hodge theory. Fermionic charges Q b , Q ab , Q d and Q ad follow following physicality criteria Q b \|phys = 0, Q ab \|phys = 0 Q d \|phys = 0, Q ad \|phys = 0(61) which lead to p λ \|phys = 0 (P r + P η )\|phys = 0 (62) This is the operator form of the first class constraint which annihilates the physical state as a consequence of physical criteria, which further is consistent with the Dirac's method of quantization of a system with first class constraints. IX. FINITE FIELD BRST TRANSFORMATIONS In this section we show that these nilpotent symmetries can be generalized by making the parameter finite and field dependent following the work of Joglekar and Mandal [38]. The BRST transformations can be generated from BRST charge using relation δφ = [φ, Q]δΛ where δΛ is infinitesimal anti-commuting BRST parameter under which effective action remains invariant. Joglekar and Mandal generalized the anti-commuting BRST parameter δΛ to be finite-field dependent instead of infinitesimal but space time independent parameter Θ[φ]. Under this generalization the path integral measure varies non-trivially. The Jacobian for these transformations for certain Θ[φ] can be calculated by following way. Dφ = J(k)Dφ ′ (k) = J(k + dk)Dφ ′ (k + dk)(63) Where k is a numerical parameter whose value lies between 0 and 1 (0 < k < 1 ). Here all the fields are taken to be k dependent. For a field φ(x, k), φ(x, 0) = φ(x) and φ(x, k = 1) = φ ′ (x). The invariance of the S ef f under φ(x, 0) → φ(x, k) is a BRST transformation given by φ(0) = φ(k) − δ b φ(k)Θ[φ, k].(64) J(k) can be replaced by e iS1[φ(k);k] for a certain functional S 1 which can be determined in each individual case using following condition Dφ(k) 1 J(k) dJ(k) dk − i dS 1 dk e i(S1+S ef f ) = 0 (65) where dS1 dk is a total derative of S 1 with respect to k in which dependence on φ(k) is also differentiated and the Jacobian can be expressed as e iS1 where S 1 is local functional of fields which satisfies the Eq.(63) where change in Jacobian is calculated as J(k) J(k + dk) = φ ± δφ(x, k + dk) δφ(x, k) = 1 J(k) dJ(k) dk dk (66) ± sign for bosonic and fermionic fields (φ) respectively. X. FFBRST FOR FREE PARTICLE ON SURFACE OF TORUS The effective action for the free particle on surface of torus using BFV formulation is written in Eq. (19) and its BRST transformation is given by Eq. (23).In BRST transformation given by Eq. (23), δΛ is global, infinitesimal and anti-commuting parameter. FFBRST transformation corresponding to this BRST transformation is written as s b r = cΘ, s b λ = −ċΘ, s b η = cΘ s b p r = 0, s b p η = 0, s b c = 0 s bc = (λ − η − r)Θ(67) where Θ is finite field dependent, global and anti-commuting parameter. Under this transformation too, effective action is invariant. Generating functional for this effective theory can be written as Z ψ = DΦ exp[i d 4 x p rṙ + p ηη + p θθ + p φφ − p λλ − p 2 r 2m − p 2 θ 2m(r − η) 2 − p 2 φ 2m(b + (r − η) sin θ) 2 +ċp +ċp + λ(p r + p η ) + 2cc −pp + p λ (r + η + p λ 2 ) ](68) where, DΦ = drdp r dθdp θ dφdp φ dηdp η dλdp λ dpdpdcdc(69) where DΦ is the path integral measure integrated over total phase space. The finite BRST transformation given above leaves the effective action invariant but path integral measure in generating functional is not invariant under this transformation. It gives rise to a Jacobian in the extended phase space which can be calculated as DΦ = drdp r dθdp θ dφdp φ dηdp η dλdp λ dpdpdcdc = J(k)dr(k)dp r (k)dθ(k)dp θ (k)dφ(k)dp φ (k)dη(k)dp η (k)dλ(k)dp λ (k)du(k)dp u (k)dp(k)dp(k)dc(k)dc(k) = J(k + dk)dr(k + dk)dp r (k + dk)dθ(k + dk)dp θ (k + dk)dφ(k + dk)dp φ (k + dk)dη(k + dk)dp η (k + dk) dλ(k + dk)dp λ (k + dk)dp(k + dk)dp(k + dk)dc(k + dk)dc(k + dk) Writing it in compact form as = d 4 x ψ δΨ(x, k + dk) δΨ(x, k)(71) Where Ψ = (r, p r , θ, p θ , φ, p φ , η, p η , λ, p λ , p,p, c,c). Which can be written as = 1 + dk c δΘ ′ (x, k + dk) δr(x, k) −ċ δΘ(x, k + dk) δλ(x, k) + c δΘ(x, k + dk) δη(x, k) +(λ − η − r) δΘ(x, k + dk) δc(x, k) = J(k) J(k + dk) = 1 − 1 J(k) dJ(k) dk dk(72) Now we consider an example to illustrate the FFBRST formulation. For that purpose we construct finite BRST parameter Θ obtained Θ ′ = iγ d 4 yc(y, k)p λ (y, k) (73) through Θ = Θ ′ (k) dk(74) The Jacobian change is calculated 1 J(k) dJ(k) dk = iγ d 4 y p λ 2(75) We make an ansatz for S 1 as, S 1 = i d 4 x ξ 1 (k)p λ 2(76) Where ξ 1 (k) is a k dependent arbitrary parameter. Now, dS 1 dk = i d 4 x ξ ′ 1 (k)p λ 2(77) Using condition in Eq. (65), we will get ξ 1 (k) = γk. Now the modified generating functional can be written as, Z = Dχ ′ (k) e i(S1+S ef f ) = Dφ ′ exp i d 4 x p rṙ + p ηη + p θθ + p φφ − p λλ − p 2 r 2m − p 2 θ 2m(r − η) 2 − p 2 φ 2m(b + (r − η) sin θ) 2 +ċp +˙cp + λ(p r + p η ) + 2cc −pp + p λ (r + η) + ( λ ′ 2 + γk)p λ 2(78) Here generating functional at k = 0 is the theory for a free particle on a surface of torus with a gauge parameter λ ′ and at k = 1, the generating functional for same theory with a different gauge parameter λ ′′ = λ ′ + 2γ. These two effective theories with two different gauge parameters on the surface of a torus are related through the FFBRST transformation with finite parameter given in Eq. (73). FFBRST transformation is thus helpful in showing the gauge independence of physical quantities. XI. CONCLUSIONS BFV formulation is a very powerful technique for quantization of gauge systems with constraints. It plays important role in analyzing the constraints and symmetries of the system. We have used this technique to study all the symmetries of a free particle on the surface of torus. We have constructed nilpotent BRST, dual-BRST, anti-BRST and anti-dual BRST transformations for this system. Dual-BRST transformations are also the symmetry of effective action and leaves gauge fixing part of the effective action invariant. Interchanging the role of ghost and anti-ghost fields the anti-BRST and antidual BRST symmetry transformations are constructed. We have shown that the nilpotent BRST and anti daul-BRST charges are analogous to the exterior derivative operators as the ghost number of the state \|ψ n on the total Hilbert space is increased by one when these charges operate on this state and algebra followed by these operators is same as the algebra obeyed by the de Rham co-homological operators. Similarly the dual BRST and anti-BRST charges are analogous to co-exterior derivative. The anti-commutators of BRST and dual BRST and anti-BRST and anti dual-BRST charges lead to bosonic symmetry. The corresponding charges are analogous to Laplacian operator. Further, this theory has another nilpotent symmetry called ghost symmetry under which the ghost term of the effective action is invariant. We further have shown that this theory behaves as double Hodge theory as the charges for BRST (Q b ), dual BRST (Q d ) and the charges for the bosonic symmetry generated out of these two symmetries (Q w ) form the algebra for Hodge theory. 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[ "Kondo Phase Transitions of Magnetic Impurities in Carbon Nanotubes", "Kondo Phase Transitions of Magnetic Impurities in Carbon Nanotubes" ]	[ "Tie-Feng Fang \nCenter for Interdisciplinary Studies\nLanzhou University\n730000LanzhouChina\n\nInstitute of Physics\nChinese Academy of Sciences\n100080BeijingChina\n", "Qing-Feng Sun \nInstitute of Physics\nChinese Academy of Sciences\n100080BeijingChina\n" ]	[ "Center for Interdisciplinary Studies\nLanzhou University\n730000LanzhouChina", "Institute of Physics\nChinese Academy of Sciences\n100080BeijingChina", "Institute of Physics\nChinese Academy of Sciences\n100080BeijingChina" ]	[]	We propose carbon nanotubes (CNTs) with magnetic impurities as a versatile platform to achieve unconventional Kondo physics, where the CNT bath is gapped by the spin-orbit interaction and surface curvature. While the strong-coupling phase is inaccessible for the special case of half-filled impurities in neutral armchair CNTs, the system in general can undergo quantum phase transitions to the Kondo ground state. The resultant position-specific phase diagrams are investigated upon variation of the CNT radius, chirality, and carrier doping, revealing several striking features, e.g., the existence of a maximal radius for nonarmchair CNTs to realize phase transitions, and an interference-induced suppression of the Kondo screening. We show that by tuning the Fermi energy via electrostatic gating, the quantum critical region can be experimentally accessed. 73.20.Hb, 72.15.Qm, 64.70.Tg Carbon nanotubes (CNTs) are formed by wrapping a graphene sheet into a cylinder of nanometer radius [1]. Their exceptional electronic structure has allowed the exploration of various fascinating Kondo phenomena, including the singlet-triplet Kondo resonance [2], the enhanced shot noise [3] due to the SU (4) Kondo effect[4], and the competitions with ferromagnetism [5] as well as superconductivity[6]. These studies have utilized short CNTs to construct quantum dots behaving as artificial magnetic impurities. Long CNTs, on the other hand, can play the role of one-dimensional (1D) host for a real magnetic impurity, which may be either a magnetic adatom on the top of a carbon atom (T site) or at the center of a hexagon (C site), or a substitutional dopant in a carbon vacancy (S site). Indeed, the Kondo effect for cobalt clusters adsorbed on metallic CNTs has already been observed[7]. This has spurred several theoretical works [8-10] to address related issues. However, a generic Kondo model of a CNT-hosted magnetic impurity, pertaining to arbitrary positions at the atomic scale, has not yet been established. More importantly, while these theories have all considered the metallic-CNT host as 1D normal metal to yield conventional Kondo physics[11], recent experiments [12] and calculations[13][14][15]have demonstrated that metallic CNTs can not be normal metal, always having rich low-energy band structures due to the spin-orbit interaction (SOI) and curvature effect. Interesting Kondo physics then arises when these ingredients in the CNT host are included.In this paper, after establishing a generic Hamiltonian for magnetic impurities in metallic CNTs, we show that depending on explicit impurity positions, the system can maps onto two kinds of hard-gap Kondo models whose host density of states (DOS) are identically gapped by the SOI and curvature effect, but scale distinctly outside the gap region due to the quantum interference between different hybridization paths. We combine renormalization group (RG) arguments and slave boson (SB) techniques to demonstrate that the local-moment (LM) state per-sists for half-filled impurities in neutral armchair CNTs due to the particle-hole (p-h) symmetry. Away from this special case, quantum phase transitions (QPTs) exist in these gapped systems, separating the Kondo and LM ground states. The resultant phase diagrams are characterized by the CNT radius, chirality, carrier doping, and the impurity position. The interference is found to reduce the Kondo regime, making S/C configurations unfavorable for Kondo screening as compared with T sites. For sufficiently deep impurity levels, two quantum critical regions are accessible by scanning tunneling probes and gating the CNT host, with signatures characterizing the nonarmchair from armchair chilarities.Our starting point is the Anderson Hamiltonian of a magnetic impurity in graphene,σ dσ models the impurity as usual[11]. H g is the graphene tight-binding Hamiltonian reading H g = i,j ,σ ta † σ (R ai )b σ (R bj ) + H.c., here a σ (R ai ) [b σ (R bj )] annihilates an π-band electron on sublattice A (B) at position R ai (R bj ), and t is the nearestneighbor hopping energy. The hybridization term H og =, where j stands for the A and/or B sublattice sites nearest to the impurity, and V xj (x=a, b) represent the corresponding hybridization amplitudes. In particular, g † σ =V a1 a † σ (0) for a T -site adatom. C-site impurities can hybridize with six surrounding carbon atoms, yieldingfor S-site impurities on sublattice A. In momentum space, the fermionic basis c kσα ≡ 1 √ 2 (αa kσ + φ k \|φ k \| b kσ ) diagonalizes the graphene Hamiltonian as H g = k,σ,α ε α (k)c † kσα c kσα , where φ k = 3 j=1 e ik·Xj , ε α (k) = αt\|φ k \|, and α = ±1. Close to the Dirac points K, the dispersion is linear, i.e., ε α (K + κ) ≃ α v F \|κ\| for \|κ\| ≪ \|K\|, with v F the Fermi velocity. In this basis, the hybridization becomes H og = k,σ,α V α (k)c † kσα d σ + H.c., where V α (k)=(αΦ ak + Φ bk \|φ k \|/φ * k )/ √ 2N , with Φ xk = j V xj e −ik·Rxj and N	10.1103/physrevb.87.075116	[ "https://arxiv.org/pdf/1208.2325v1.pdf" ]	119,282,352	1208.2325	fe82e658e5b04f26a91b925f4423b8dd1e6fe00f	Kondo Phase Transitions of Magnetic Impurities in Carbon Nanotubes 11 Aug 2012 Tie-Feng Fang Center for Interdisciplinary Studies Lanzhou University 730000LanzhouChina Institute of Physics Chinese Academy of Sciences 100080BeijingChina Qing-Feng Sun Institute of Physics Chinese Academy of Sciences 100080BeijingChina Kondo Phase Transitions of Magnetic Impurities in Carbon Nanotubes 11 Aug 2012 We propose carbon nanotubes (CNTs) with magnetic impurities as a versatile platform to achieve unconventional Kondo physics, where the CNT bath is gapped by the spin-orbit interaction and surface curvature. While the strong-coupling phase is inaccessible for the special case of half-filled impurities in neutral armchair CNTs, the system in general can undergo quantum phase transitions to the Kondo ground state. The resultant position-specific phase diagrams are investigated upon variation of the CNT radius, chirality, and carrier doping, revealing several striking features, e.g., the existence of a maximal radius for nonarmchair CNTs to realize phase transitions, and an interference-induced suppression of the Kondo screening. We show that by tuning the Fermi energy via electrostatic gating, the quantum critical region can be experimentally accessed. 73.20.Hb, 72.15.Qm, 64.70.Tg Carbon nanotubes (CNTs) are formed by wrapping a graphene sheet into a cylinder of nanometer radius [1]. Their exceptional electronic structure has allowed the exploration of various fascinating Kondo phenomena, including the singlet-triplet Kondo resonance [2], the enhanced shot noise [3] due to the SU (4) Kondo effect[4], and the competitions with ferromagnetism [5] as well as superconductivity[6]. These studies have utilized short CNTs to construct quantum dots behaving as artificial magnetic impurities. Long CNTs, on the other hand, can play the role of one-dimensional (1D) host for a real magnetic impurity, which may be either a magnetic adatom on the top of a carbon atom (T site) or at the center of a hexagon (C site), or a substitutional dopant in a carbon vacancy (S site). Indeed, the Kondo effect for cobalt clusters adsorbed on metallic CNTs has already been observed[7]. This has spurred several theoretical works [8-10] to address related issues. However, a generic Kondo model of a CNT-hosted magnetic impurity, pertaining to arbitrary positions at the atomic scale, has not yet been established. More importantly, while these theories have all considered the metallic-CNT host as 1D normal metal to yield conventional Kondo physics[11], recent experiments [12] and calculations[13][14][15]have demonstrated that metallic CNTs can not be normal metal, always having rich low-energy band structures due to the spin-orbit interaction (SOI) and curvature effect. Interesting Kondo physics then arises when these ingredients in the CNT host are included.In this paper, after establishing a generic Hamiltonian for magnetic impurities in metallic CNTs, we show that depending on explicit impurity positions, the system can maps onto two kinds of hard-gap Kondo models whose host density of states (DOS) are identically gapped by the SOI and curvature effect, but scale distinctly outside the gap region due to the quantum interference between different hybridization paths. We combine renormalization group (RG) arguments and slave boson (SB) techniques to demonstrate that the local-moment (LM) state per-sists for half-filled impurities in neutral armchair CNTs due to the particle-hole (p-h) symmetry. Away from this special case, quantum phase transitions (QPTs) exist in these gapped systems, separating the Kondo and LM ground states. The resultant phase diagrams are characterized by the CNT radius, chirality, carrier doping, and the impurity position. The interference is found to reduce the Kondo regime, making S/C configurations unfavorable for Kondo screening as compared with T sites. For sufficiently deep impurity levels, two quantum critical regions are accessible by scanning tunneling probes and gating the CNT host, with signatures characterizing the nonarmchair from armchair chilarities.Our starting point is the Anderson Hamiltonian of a magnetic impurity in graphene,σ dσ models the impurity as usual[11]. H g is the graphene tight-binding Hamiltonian reading H g = i,j ,σ ta † σ (R ai )b σ (R bj ) + H.c., here a σ (R ai ) [b σ (R bj )] annihilates an π-band electron on sublattice A (B) at position R ai (R bj ), and t is the nearestneighbor hopping energy. The hybridization term H og =, where j stands for the A and/or B sublattice sites nearest to the impurity, and V xj (x=a, b) represent the corresponding hybridization amplitudes. In particular, g † σ =V a1 a † σ (0) for a T -site adatom. C-site impurities can hybridize with six surrounding carbon atoms, yieldingfor S-site impurities on sublattice A. In momentum space, the fermionic basis c kσα ≡ 1 √ 2 (αa kσ + φ k \|φ k \| b kσ ) diagonalizes the graphene Hamiltonian as H g = k,σ,α ε α (k)c † kσα c kσα , where φ k = 3 j=1 e ik·Xj , ε α (k) = αt\|φ k \|, and α = ±1. Close to the Dirac points K, the dispersion is linear, i.e., ε α (K + κ) ≃ α v F \|κ\| for \|κ\| ≪ \|K\|, with v F the Fermi velocity. In this basis, the hybridization becomes H og = k,σ,α V α (k)c † kσα d σ + H.c., where V α (k)=(αΦ ak + Φ bk \|φ k \|/φ * k )/ √ 2N , with Φ xk = j V xj e −ik·Rxj and N We propose carbon nanotubes (CNTs) with magnetic impurities as a versatile platform to achieve unconventional Kondo physics, where the CNT bath is gapped by the spin-orbit interaction and surface curvature. While the strong-coupling phase is inaccessible for the special case of half-filled impurities in neutral armchair CNTs, the system in general can undergo quantum phase transitions to the Kondo ground state. The resultant position-specific phase diagrams are investigated upon variation of the CNT radius, chirality, and carrier doping, revealing several striking features, e.g., the existence of a maximal radius for nonarmchair CNTs to realize phase transitions, and an interference-induced suppression of the Kondo screening. We show that by tuning the Fermi energy via electrostatic gating, the quantum critical region can be experimentally accessed. 73.20.Hb,72.15.Qm,64.70.Tg Carbon nanotubes (CNTs) are formed by wrapping a graphene sheet into a cylinder of nanometer radius [1]. Their exceptional electronic structure has allowed the exploration of various fascinating Kondo phenomena, including the singlet-triplet Kondo resonance [2], the enhanced shot noise [3] due to the SU (4) Kondo effect [4], and the competitions with ferromagnetism [5] as well as superconductivity [6]. These studies have utilized short CNTs to construct quantum dots behaving as artificial magnetic impurities. Long CNTs, on the other hand, can play the role of one-dimensional (1D) host for a real magnetic impurity, which may be either a magnetic adatom on the top of a carbon atom (T site) or at the center of a hexagon (C site), or a substitutional dopant in a carbon vacancy (S site). Indeed, the Kondo effect for cobalt clusters adsorbed on metallic CNTs has already been observed [7]. This has spurred several theoretical works [8][9][10] to address related issues. However, a generic Kondo model of a CNT-hosted magnetic impurity, pertaining to arbitrary positions at the atomic scale, has not yet been established. More importantly, while these theories have all considered the metallic-CNT host as 1D normal metal to yield conventional Kondo physics [11], recent experiments [12] and calculations [13][14][15] have demonstrated that metallic CNTs can not be normal metal, always having rich low-energy band structures due to the spin-orbit interaction (SOI) and curvature effect. Interesting Kondo physics then arises when these ingredients in the CNT host are included. In this paper, after establishing a generic Hamiltonian for magnetic impurities in metallic CNTs, we show that depending on explicit impurity positions, the system can maps onto two kinds of hard-gap Kondo models whose host density of states (DOS) are identically gapped by the SOI and curvature effect, but scale distinctly outside the gap region due to the quantum interference between different hybridization paths. We combine renormalization group (RG) arguments and slave boson (SB) techniques to demonstrate that the local-moment (LM) state per-sists for half-filled impurities in neutral armchair CNTs due to the particle-hole (p-h) symmetry. Away from this special case, quantum phase transitions (QPTs) exist in these gapped systems, separating the Kondo and LM ground states. The resultant phase diagrams are characterized by the CNT radius, chirality, carrier doping, and the impurity position. The interference is found to reduce the Kondo regime, making S/C configurations unfavorable for Kondo screening as compared with T sites. For sufficiently deep impurity levels, two quantum critical regions are accessible by scanning tunneling probes and gating the CNT host, with signatures characterizing the nonarmchair from armchair chilarities. Our starting point is the Anderson Hamiltonian of a magnetic impurity in graphene, H=H o +H g +H og , where H o = σ ε d d † σ d σ + U 2 σ d † σ d σ d † σ dσ models the impurity as usual [11]. H g is the graphene tight-binding Hamiltonian reading H g = i,j ,σ ta † σ (R ai )b σ (R bj ) + H.c., here a σ (R ai ) [b σ (R bj )] annihilates an π-band electron on sublattice A (B) at position R ai (R bj ), and t is the nearestneighbor hopping energy. The hybridization term H og = σ g † σ d σ +H.c. with g † σ = j V aj a † σ (R aj )+V bj b † σ (R bj ) , where j stands for the A and/or B sublattice sites nearest to the impurity, and V xj (x=a, b) represent the corresponding hybridization amplitudes. In particular, g † σ =V a1 a † σ (0) for a T -site adatom. C-site impurities can hybridize with six surrounding carbon atoms, yielding g † σ = 3 j=1 V aj a † σ (X j ) + V bj b † σ (−X j ) with X j the lattice nearest-neighbor vectors, while g † σ = 3 j=1 V bj b † σ (X j ) for S-site impurities on sublattice A. In momentum space, the fermionic basis c kσα ≡ 1 √ 2 (αa kσ + φ k \|φ k \| b kσ ) diagonalizes the graphene Hamiltonian as H g = k,σ,α ε α (k)c † kσα c kσα , where φ k = 3 j=1 e ik·Xj , ε α (k) = αt\|φ k \|, and α = ±1. Close to the Dirac points K, the dispersion is linear, i.e., ε α (K + κ) ≃ α v F \|κ\| for \|κ\| ≪ \|K\|, with v F the Fermi velocity. In this basis, the hybridization becomes H og = k,σ,α V α (k)c † kσα d σ + H.c., where V α (k)=(αΦ ak + Φ bk \|φ k \|/φ * k )/ √ 2N , with Φ xk = j V xj e −ik·Rxj and N the number of sublattice sites. We now roll up the graphene sheet along the chiral vector C h =n 1 a 1 + n 2 a 2 to create a (n 1 , n 2 ) CNT [1], where n 1 , n 2 ∈ Z and a 1 , a 2 are the primitive lattice vectors. While κ's component parallel to the tube axis, p≡κ , remains continuous for CNTs of long length L, the periodic boundary condition, (K+κ)·C h =2πm, m ∈ Z, quantizes κ's perpendicular component, q τ ≡ κ ⊥ = (m + τ ν/3)/R. Here R = a 2π n 2 1 + n 2 2 + n 1 n 2 is the tube radius with a = \|a 1 \| ≃ 2.46Å the lattice constant, the valley index τ = ±1 denotes the two inequivalent K + , K − Dirac points, and ν =mod(n 1 − n 2 , 3) characterizes the metallic (ν = 0) or semiconducting (ν = ±1) CNTs. Restricting the graphene quantities, ε α (k), V α (k), c kσα , only to these allowed Bloch states near the Dirac points yields corresponding quantities for the CNT: the π-band spectrum ε pτ α = α v F p 2 + q 2 τ , the hybridization V pτ α = V α (K τ + κ) κ=(p, qτ ) , and the operator c pστ α . The surface curvature of CNTs induces the π band hybridizing with other high-energy bands (CIH), and enhances the effect of intrinsic SOI, V so , of carbon atoms. At second order in perturbation theory based on a double expansion of V so and a/R [14,15], the SOI gives a spindependent shift σα1Vsoa vF R of q τ and directly shifts the energy dispersion by −στ α 2 V so (a/R) cos 3θ, while the CIH only causes a valley-dependent q τ shift τ βa 2 cos 3θ vF R 2 . Here, the spin σ = ±, the parameters, α 1 , α 2 , β, relate to some unperturbed hopping amplitudes between carbon orbitals [16], and θ is the angle between the chiral vector and the zigzag direction along a 1 . Due to the hexagonal symmetry, this chiral angle is constrained to 0 θ 30 • , as calculated by θ=arctan √ 3n2 2n1+n2 for 0 n 2 n 1 only. These corrections result in q τ → q στ = τ ∆cv vF + σ∆so1 vF for the lowest metallic π subband, ε pτ α → ε pστ α =α v F p 2 + q 2 στ − στ ∆ so2 , and V pτ α → V pστ α = V α (K τ + κ) κ=(p, qστ ) , by setting ∆ so1 = α 1 V so a/R, ∆ so2 = α 2 V so (a/R) cos 3θ, and ∆ cv =β(a/R) 2 cos 3θ. Our generic Anderson Hamiltonian for a magnetic impurity coupled to the metallic CNT host then reads H=H o + H c , H c = p,σ,τ,α ε pστ α c † pστ α c pστ α + V pστ α c † pστ α d σ +H.c. ,(1) with the host DOS, ρ στ (ε)≡ p,α δ(ε − ε pστ α ), given by ρ στ (ε) = ρ 0 ε + στ ∆ so2 Θ ε + στ ∆ so2 − ∆ στ (ε + στ ∆ so2 ) 2 − ∆ 2 στ ,(2) where ρ 0 =L/(hv F ) and ∆ στ = ∆ cv + στ ∆ so1 . Note that the BCS-like gap opens even in metallic CNTs. It is allowed to replace V pστ α and ε pστ α in the Hamiltonian (1) by a proper constant coupling V 0 and an effective spectrum ε pστ α , respectively, as long as the resultant [17]. Remarkably, ρ(ε) must be spin-independent due to the nonmagnetic nature of the CNT. By performing a Schrieffer-Wolff transformation [18], the system can then be readily mapped onto the Kondo model H K =JŜ·ŝ(0) which describes the antiferromagnetic exchange interaction J=−2V 2 0 U/[ ε d ( ε d +U )]>0 of the impurity spinŜ with the host spinŝ(0) at the im- effective DOS ρ(ε) ≡ p,τ,α δ(ε − ε pστ α ) satisfies ρ(ε) = p,τ,α \|V pστ α /V 0 \| 2 δ(ε − ε pστ α )purity site 0, where ε d = ε d − E F < 0 is the impurity level measured from the Fermi energy E F . The effective DOS ρ(ε) seen by the impurity is essentially a renormalization of the bare CNT DOS, emerging from the quantum interference between different hybridization paths the electrons can take to hop in and out of the impurity. Hereafter, we focus on a particular class of impurity orbitals that hybridizes equally with the nearest carbon atoms on a given sublattice, i.e., V xj = V x . In this case, when the impurity is located on the S or C site, constructive interference renormalizes the CNT DOS as ρ SC (ε)= τ (ε + στ ∆ so2 ) 2 ρ στ (ε)/(2N t 2 ) by defining V 0 =V b for S site or V 0 = V 2 a + V 2 b for C site, whereas ρ T (ε) = τ ρ στ (ε)/(2N ) for T -sites adatoms where V 0 =V a and the interference is absent. These DOS represent two distinct classes of hard-gap Kondo models promising for unconventional Kondo physics. (i) A half-filled (U = −2 ε d ) impurity coupled to the neutral armchair (θ = 30 • ) CNT, where since ∆ so2 = ∆ cv = 0, no CIH effect exists and the SOI opens a gap of width, 2∆ so1 , at the Fermi level that exactly crosses the Dirac point. The system then exhibits strict p-h symmetry, which prohibits the appearance of even powers of the local level ε d in its renormalization by successively integrating out high energy states with energy ±Λ in the band edges [11,19]. The lowest contribution arising from two-loop vertex renormalizations, up to the leading order in a double expansion of V 0 and ε d , gives rise to the RG beta function β( ε d ) = 4 ρ(Λ)V 2 0 ε d /Λ. Consequently, the corresponding flow of the Kondo coupling J = −4V 2 0 / ε d reads β(J) = −4 ρ(Λ)V 2 0 J/Λ. Solving this RG equation yields J(Λ) = J(Λ 0 )exp − 4V 2 0 Λ Λ0 [ ρ(Λ)/Λ 2 ] dΛ , where Λ 0 is the initial band cutoff. It is evident that consecutive RG transformations increase the effective coupling. However, as the scaling of the gapped Kondo model characterized by ρ SC (ε) or ρ T (ε), terminates at the gap edge Λ = ∆ so1 , J(Λ) flows to a finite value rather than infinity, signaling the absence of the strong coupling (SC) Kondo phase. Therefore, the impurity ground state is always a local moment, consistent with previous numerical RG calculations on the rectangularly gapped band [20]. (ii) An infinite-U impurity in the neutral armchair CNT. Here the p-h symmetry of the whole system is violated, while the CNT bath remains p-h symmetric. RG transformations show that already at one-loop order, vertex renormalization of ε d and J occurs as the band width is reduced [11,19]. The resultant RG equation for the dimensionless Kondo coupling J = −2 ρ(Λ)V 2 0 / ε d scales according to β( J) = [ln ρ(Λ)] ′ Λ J − J 2 . We solve the beta function as J(Λ) = ρ(Λ) J(Λ 0 ) ρ(Λ 0 ) + J(Λ 0 ) Λ Λ0 [ ρ(Λ)/Λ] dΛ ,(3) where the denominator being vanishing or nonvanishing during scaling determines the impurity ground state. For T -site adatoms, the DOS ρ T (ε) = ρ 0 Θ(\|ε\| − ∆ so1 )\|ε\|/ ε 2 − ∆ 2 so1 is of BCS-type, with ρ 0 ≡ ρ 0 N . As the scaling proceeds, the denominator of Eq. (3) vanishes at the critical band width Λ c = T 0 K + ∆ 2 so1 /4T 0 K when 2T 0 K > ∆ so1 , directing the RG flow towards the SC Kondo fixed point. Here the Kondo temperature T 0 K ≡ Λ 0 exp[−1/ J(Λ 0 )] is defined as a scaling invariant [11] of the normal metallic model [realized by setting ρ(ε) = ρ T (Λ 0 ) ≃ ρ 0 for Λ 0 ≫ ∆ so1 ]. By contrast, for 2T 0 K < ∆ so1 , or equivalently, J(Λ 0 ) < [ln (2Λ 0 /∆ so1 )] −1 , the adatom flows to the unscreened LM state since the coupling J(Λ) already vanishes as the scaling enters into the gap region before it reaches the SC limit.We thus find a quantum-critical point of the impurity level ε dc = −2V 2 0 ρ 0 ln 2Λ0 ∆so1 across which, upon lowering ε d , the impurity undergoes a QPT from a screened to an unscreened moment. The explicit R dependence of this phase boundary can be written as ε dc = ε c1 −2V 2 0 ρ 0 ln R a , with ε c1 = −2V 2 0 ρ 0 ln 2Λ0 α1Vso . The interference-induced additional scaling (ε/t) 2 , imposed on the effective host DOS ρ SC (ε) = (ε/t) 2 ρ T (ε) for substitutional dopants or C-site adatoms, dramatically changes the RG flow of Eq. (3). It features a different phase boundary at ε dc = ε c2 − f ( R a ) separating the Kondo and LM phases, where ε c2 = −V 2 0 ρ 0 Λ 2 0 t 2 and f ( R a ) = V 2 0 ρ 0 α 2 1 V 2 so a 2 t 2 R 2 ln 2Λ0R α1Vsoa , by taking ρ SC (Λ 0 ) ≃ ρ 0 Λ 2 0 t 2 for Λ 0 ≫ ∆ so1 . For realistic parameters, this boundary is always much shallower than in the T -site case, reflecting a reduction of the Kondo regime by the interference. (iii) An infinite-U impurity coupled to the doped nanotube with arbitrary chiralities. In this general case, the gaps in the two valley sectors of the CNT DOS are different, being centered at ε = ±∆ so2 with width W 1 = 2\|∆ cv −∆ so1 \| and W 2 = 2(∆ cv +∆ so1 ), respectively. After summation over the two valleys, their overlap constitutes a net gap of width W = W1+W2 2 − 2∆ so2 centered at ε = min(∆ cv , ∆ so1 ) ≡ ε 0 , in the effective DOS ρ T (ε) and ρ SC (ε), provided that W > 0. The presence of ∆ so2 and ∆ cv for the nonarmchair chirality, and the deviation of the Fermi level from the Dirac point in the doped nanotube, definitely break the p-h symmetry of the CNT baths. This renders the previous RG arguments insufficient because all vertex functions will develop structures on a scale E F . The SB technique [21] accounts for this complication by introducing an auxiliary boson field to describe the empty state, together with a Lagrange multiplier λ to exclude double occupancy in the impurity. The boson field is further condensed to its saddle-point value r which minimizes the free energy, obeying 2 π EF −Λ0 dε Im[Σ(ε)G(ε)] = λ, where Σ(ε) = 1 π Λ0 −Λ0 dε ′ Γ(ε ′ )(ε − ε ′ + i0 + ) −1 , Γ(ε) = πV 2 0 ρ(ε) , and the impurity Green's function G(ε) = [ε − ε d − λ − r 2 Σ(ε)] −1 . After λ and r are self-consistently determined [21], the localized level ε d is renormalized to a Kondo resonance at ε d + λ. This mean-field treatment correctly describes the Kondo fixed point for impurity levels deep below E F , where charge fluctuations are frozen out. At the critical point, the Kondo temperature, defined as T K = ( ε d + λ) 2 + r 4 Γ 2 (ε d + λ) [21], must vanish. The saddle-point equation then yields the critical value of the impurity level, ε dc = 1 π EF −Λ0 dε Γ(ε) ε − E F + 1 π Λ0 EF dε Γ(ε) E F − ε .(4) This result is applicable to the doped case as well as to the nonarmchair chirality. Below we present numerical results for realistic CNT parameters [14,16]: α 1 =0.055, α 2 =0.217, β=93.75meV, V so =6meV, and t=2.5eV. Figure 1 presents impurity phase diagrams in the ( ε d , R) plane, when the Fermi energy is tuned to the gap center, E F = ε 0 . The QPTs discussed here exist only if the CNT bath is gapped, such that ρ(E F ) vanishes exactly. Solving the inequation W > 0, we find a critical chiral angle θ 0 = 1 3 arccos(α 1 /α 2 ) ≈ 25.1 • and an upper limit of CNT radius R 0 = βa/(V so α 2 ) ≈ 17.7nm. For θ 0 < θ 30 • , CNTs with arbitrary radius are always gapped (W > 0), resulting in that the impurity exhibits Kondo and LM ground states separated by transitions at ε dc in the whole range of R [see, e.g., Figs. 1(a) and 1(b) for the armchair case]. On the other hand, when 0 θ θ 0 , the Kondo-LM transition can occur only for R < R 0 . Beyond this upper limit R R 0 , one has W 0, leaving always a screened impurity state [see, e.g., Figs. 1(c) and 1(d) for the zigzag case]. The specific R-dependence of the phase boundary is also very sensitive to explicit impurity positions, despite that ρ T (ε) and ρ SC (ε) feature the same gap structure. When the CNT radius increases, the Kondo regime of impurities on T (S or C) sites gradually widens (narrows), with the boundary eventually decreasing to −∞ (increasing to ε c2 ) as R → ∞ for θ > θ 0 chiralities [see, e.g., Figs. 1(a) and 1(b)] and R → R 0 for θ θ 0 chiralities [see, e.g., Figs. 1(c) and 1(d)]. This sensitiveness stems from the quantum interference effect, which dramatically changes the scaling behavior of ρ SC (ε) outside the gap region, as compared with ρ T (ε). By noting ε c1 ≃ 400ε c2 ≃ −10 8 f (1) < 0 for the parameters used here, an interference-induced overall shrinking of the Kondo regime in the S/C configurations is also evident, as already emphasized before. Gating the CNT host to tune its Fermi energy away from the gap center renders E F closer to electronic states near one of the gap edges, in favor of screening the impurity. Therefore, the SC fixed point can be reached for CNTs, as a function of the impurity level ε d and discrete CNT radius R/a = √ 3n/2π (armchair), 3n/2π (zigzag), with n ∈ Z. The high-energy cutoff Λ0=0.5eV. f(1) ( d - c1 ) (V ) ( d - c1 ) (V ) (d) (c) (b) T / arm. smaller Kondo couplings (deeper impurity levels), widening the Kondo regime. This is confirmed by our calculations shown in Fig. 2(a) for T -site adatoms [16], where the Kondo and LM phases are bounded by an arched borderline peaked at E F = ε 0 . As E F moves further out of the gap region, arbitrary small J > 0 can always drive the impurity into the Kondo phase due to ρ(E F ) = 0. Interestingly, while the armchair CNT features a p-h symmetric phase diagram when it is tuned from hole doping (E F < 0) to electron doping (E F > 0), the arched phase boundary of nonarmchair CNTs always deviates from the hole-doped side because of ε 0 > 0, and can even fully enter into the electron-doped region for large CNT radius. The minimal radius R 1 needed for accessing this maximal p-h asymmetry can be determined by solving ε 0 > W 2 to obtain R 1 = R 0 α2 cos 3θ α1+α2 cos 3θ when θ θ 0 , thereby R 1 ≈ 14.1nm for zigzag CNTs. Obviously, for deep impurity levels, two consecutive QPTs occur whenever the Fermi energy sweeps over the two gap edges. These can be experimentally observed by scanning tunneling probes [7] which directly measure the impurity spectral densities, A(ε) ≡ − 1 π ImG(ε). By placing the Fermi energy far away from the gap region [ Fig. 2(b)], the smooth host DOS around E F gives rise to conventional Kondo resonances in A(ε). While the CNT chiralities are indistinguishable in these resonances, the interference inbuilt in the S/C configurations greatly narrows the resonances as compared with T sites, signaling a suppression of the Kondo effect which is not favorable for experimental observations, regardless of the existing universal scaling with the Kondo temperature [inset of Fig. 2(b)]. This scaling is violated when E F is tuned to access the quantum critical region around the gap edges [Figs. 2(c) and 2(d)]. Specifically, although the scaling of different impurity positions persists to some extent, the zigzag CNT hosts two-peak Kondo resonances distinct from the armchair one. We attribute the two-peak structure to a distortion by the distinct DOS of nonarmchair CNTs, which possess two singularities around each gap edge, arising from the two valley sectors of the bare DOS of Eq. (2). Finally, as E F shifts into the gap region, the SB equations break down and the Kondo resonances immediately collapse into the featureless LM spectra. In conclusion, we have addressed the Kondo problem of magnetic impurities in CNTs, demonstrating the existence of distinct QPTs in the impurity's ground state, which crucially depend on the characteristics of CNT and explicit impurity positions. Support from NBRP of China (2012CB921303 and 2009CB929100) and NSF-China is acknowledged. Supplementary information to "Kondo phase transitions of magnetic impurities in carbon nanotubes" Tie-Feng Fang and Qing-feng Sun 1) The parameters α 1 , α 2 , and β In the low-energy theory for carbon nanotubes, the effects of spin-orbit interaction and surface curvature on π electrons are well described in second-order perturbation theory [14,15]. The effects are equivalent to shift the dispersion relation by −στ α 2 V so (a/R) cos 3θ, to shift the perpendicular wave vector by σα 1 Vsoa vF R + τ β a 2 cos 3θ vF R 2 , and to shift the parallel wave vector by τ β ′ a 2 sin 3θ vF R 2 . Assuming sufficiently long nanotubes, the last shift is irrelevant, we thus drop it. The explicit forms of the parameters, α 1 , α 2 , and β, appearing in the remaining terms are [14]: α 1 = − √ 3ε s (V π pp + V σ pp ) 18V 2 sp , (S1) α 2 = √ 3V π pp 3(V π pp − V σ pp ) , (S2) β = V π pp (V π pp + V σ pp ) 8(V π pp − V σ pp ) . (S3) Here ε s is the energy of the carbon s orbital relative to the p orbital energy. The latter (i.e., the on-site energy of π electrons) is set to zero in our manuscript. V sp represents the unperturbed hopping amplitude between nearest-neighbor s and p orbitals. V π(σ) pp is the unperturbed hopping amplitude between nearest-neighbor p orbitals, giving rise to the π(σ) band. In this work, we use the parameter set: ε s = −8.9eV, V sp = 5.6eV, V π pp = −3.0eV, and V σ pp = 5.0eV [R. Saito, G. Dresselhaus, and M. S. Dresselhaus, Physical Properties of Carbon Nanotubes (Imperial College Press, London, 1998)], as it is also used in Ref. [14]. This give us, α 1 ≃ 0.055, α 2 ≃ 0.217, and β ≃ 93.75meV, to carry out numerical calculations. Using other sets of parameters [e.g., D. Tománek andM. A. Schluter, Phys. Rev. Lett. 67, 2331 (1991); J. W. Mintmire and C. T. White, Carbon 33, 893 (1995)] does not change our numerical results qualitatively. 2) Phase diagrams for Sand C-site impurities in the ( ε d , E F ) plane As shown in Fig. S1, the Kondo and LM phases of S/C impurities are also bounded by an arched borderline, showing features qualitatively same with T -site adatoms [see, Fig. 2(a) in the paper]. For example, the boundary is p-h symmetric for armchair nanotubes, but becomes ph asymmetric for nonarmchair nanotubes. The minimal radius R 1 derived in the paper for accessing the maximal p-h asymmetry also applies to this case. This is because ρ sc (ε) and ρ T (ε) share the same gap structure. They scale differently only outside the gap region due to the quantum interference effect. The effect of quantum interference is mainly reflected i) in the R-dependence of the boundary (see Fig. 1 in the paper), ii) in the fact that the arched LM region of S/C impurities are much sharper than T adatoms [compare Fig. 2(a) in the paper with Fig. S1 here], and iii) in the fact that for realistic nanotube parameters, the Kondo boundary of S/C impurities, ε dc , is always much shallower than the boundary of T adatoms, signaling the reduction of Kondo regime by interference. FIG. S1: Phase diagrams for substitutional dopants or C-site adatoms in armchair and zigzag nanotubes in the ( ε d , EF ) plane. The parameters used are the same as in Fig. 2(a). FIG. 1 : 1(Color online) Phase diagrams of Kondo-LM transitions for impurities sitting on T [(a),(c)], S or C [(b),(d)] sites in the (n,n) armchair [(a),(b)] and (3n,0) zigzag [(c),(d)] online) (a) Phase diagrams for T -site adatoms upon variation of the Fermi energy EF , with Λ0 = 0.5eV. 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[ "Skyrmions in coupled spin torque nano-oscillator structures", "Skyrmions in coupled spin torque nano-oscillator structures" ]	[ "H Vigo-Cotrina \nCentro Brasileiro de Pesquisas Físicas\nCentro Brasileiro de Pesquisas Físicas\n22290-180, 22290-180Rio de Janeiro, Rio de JaneiroRJBrazil, Brazil\n", "A P Guimarães \nCentro Brasileiro de Pesquisas Físicas\nCentro Brasileiro de Pesquisas Físicas\n22290-180, 22290-180Rio de Janeiro, Rio de JaneiroRJBrazil, Brazil\n" ]	[ "Centro Brasileiro de Pesquisas Físicas\nCentro Brasileiro de Pesquisas Físicas\n22290-180, 22290-180Rio de Janeiro, Rio de JaneiroRJBrazil, Brazil", "Centro Brasileiro de Pesquisas Físicas\nCentro Brasileiro de Pesquisas Físicas\n22290-180, 22290-180Rio de Janeiro, Rio de JaneiroRJBrazil, Brazil" ]	[]	In the present work, using micromagnetic simulation, we show that the magnetic coupling effect plays a very important role in the process of creation of skyrmions in a coupled system of spin-torque nano-oscillators (STNO). First, we have determined the magnetic ground state in an isolated STNO for different values of perpendicular uniaxial anisotropy (PUA) and Dzyaloshinskii-Moriya interaction (DMI). Next, we have applied a perpendicular pulse polarized current density (J) and found that it is possible to create a metastable Néel skyrmion from a disk whose ground state is a single magnetic domain. From these results, we obtained a phase diagram of polarized current intensity vs. time of application of the current pulse, for different values of parameters such as PUA, DMI, and distance between the STNOs. Our results show that, depending on the separation distance between the STNOs, the current density required to create a skyrmion changes due to the magnetic interaction.	null	[ "https://arxiv.org/pdf/1812.04648v1.pdf" ]	118,897,889	1812.04648	20ecc0c55a1b8aca9386eb93ae4c49a16db0da10	Skyrmions in coupled spin torque nano-oscillator structures 11 Dec 2018 H Vigo-Cotrina Centro Brasileiro de Pesquisas Físicas Centro Brasileiro de Pesquisas Físicas 22290-180, 22290-180Rio de Janeiro, Rio de JaneiroRJBrazil, Brazil A P Guimarães Centro Brasileiro de Pesquisas Físicas Centro Brasileiro de Pesquisas Físicas 22290-180, 22290-180Rio de Janeiro, Rio de JaneiroRJBrazil, Brazil Skyrmions in coupled spin torque nano-oscillator structures 11 Dec 2018Néel skyrmionpulse polarized currentspin-torque oscillatormicromagnetic simulation In the present work, using micromagnetic simulation, we show that the magnetic coupling effect plays a very important role in the process of creation of skyrmions in a coupled system of spin-torque nano-oscillators (STNO). First, we have determined the magnetic ground state in an isolated STNO for different values of perpendicular uniaxial anisotropy (PUA) and Dzyaloshinskii-Moriya interaction (DMI). Next, we have applied a perpendicular pulse polarized current density (J) and found that it is possible to create a metastable Néel skyrmion from a disk whose ground state is a single magnetic domain. From these results, we obtained a phase diagram of polarized current intensity vs. time of application of the current pulse, for different values of parameters such as PUA, DMI, and distance between the STNOs. Our results show that, depending on the separation distance between the STNOs, the current density required to create a skyrmion changes due to the magnetic interaction. Introduction Magnetic skyrmions are non-trivial spin textures that can appear in ferromagnets (FM) where there is lacking inversion symmetry [1][2][3][4]. In ultrathin film multilayer systems (FM/substrate), the Dzyaloshinskii-Moriya interaction (DMI) is the main responsible for the creation of skyrmions [1,2,5]. This DMI has its origin in the interface of the multilayer due to strong spin-orbit coupling between FM and the substrate [1,4]. There are two skyrmion types: Bloch skyrmion and Néel skyrmion [1,2,4]. The first is generally encountered in bulk systems, while the latter is present in multilayer systems [3,4]. Mathematically, a skyrmion is quantified by The topological Charge Q, which is defined by Q = (1/4π) m.(∂ x m × ∂ y m)dxdy, where m is the reduced magnetization. Q = ±1 for skyrmions [1,6]. Several works have shown that a skyrmion can be stabilized in geometries as, for example, disks [3,7,8], ultrathin films [9] and racetracks [10,11]. However, it is also possible to create a multiskyrmion cluster in a single nanodisk [12,13]. Skyrmions can also be created with the help of external perturbations, such as magnetic field [14][15][16], spin polarized current [3,17,18], or local heating [19]. Due to the fact that skyrmions are topologically protected stable magnetic structures [2,3,20] and can be moved with small current densities of the order of 10 6 A/m 2 [21], these have many potential applications, e.g., as logic gate devices [10], racetrack memories [11,22], or spin-torque nano-oscillators (STNOs) [23,24]. STNOs with magnetic skyrmions have been object of many studies [12,23,25], since these systems can be used to produce microwave [26] or spin wave based computing and logic devices [25]. For technological applications, the nanostructures are normally organized in arrays. This leads to the question of magnetic interactions between them. There are in the literature several works about the creation and stabilization of skyrmions in isolated structures [3,7,8,18,23,24,27], but work is still lacking on the role of magnetic interaction in the process of creation of a skyrmion. The aim of this work is to study the effect of the magnetic coupling in the creation of a Néel skyrmion in a spin torque nano-oscillator system. For this purpose, we tailored the perpendicular uniaxial anisotropy (PUA), Dzyaloshinskii-Moriya interaction (DMI) and polarized current density (J) in the free layer (ultrathin disk) of the STNO (isolated and coupled systems). We used the open source software Mumax3 [28], with cell size of 1 × 1 × L nm 3 , where L is the thickness of the free layer. The material used is Cobalt with parameters [3,9,18]: saturation magnetization M s = 5.8 × 10 5 A/m, exchange stiffness A = 1.5 × 10 −11 J/m, and damping constant α = 0.3 (for faster convergence). The perpendicular uniaxial anisotropy constant (K z ) and Dzyaloshinskii-Moriya exchange constant (D int ) varied from 0.4 to 1.8 MJ/m 3 and from 3 to 4.5 mJ/m 2 , respectively [3,7,27]. Results and discussion Isolated spin torque oscillator We considered a STNO (see Fig. 1), where the free layer has thickness L = 0.4 nm and diameter D = 80 nm. The spacer has parameters: Λ = 2 (Slonczewski asymmetry) and ǫ = 0.2 [18]. The polarizer has magnetization m p = (1,0,0) Figure 1: Schematic representation of a spin torque nano-oscillator (STNO). Blue region is the free layer, gray region is the spacer and yellow region is the polarizer. J 1 and J 2 are the densities of the spin currents flowing through the STNO. First, in order to determine the magnetic ground state of the free layer (J = 0 A/m 2 ), we considered in our micromagnetic simulations two initial magnetic configurations: perpendicular magnetic domain and Néel skyrmion. The energies of the final magnetic states are shown in Fig. 2 for values of DMI, from D int = 3 mJ/m 2 to D int = 4.5 mJ/m 2 . We have considered only cases where both perpendicular magnetic domain (PMD) and magnetic Néel skyrmion (MNS) can be obtained as final magnetic states. For example, for D int = 3 mJ/m 2 (see Fig. 2(a)), the region where it is possible to obtain PMD and MNS as final magnetic states is between K z = 0.4 MJ/m 3 and K z = 1 MJ/m 3 . For values less than K z = 0.4 MJ/m 3 (not shown here), there is only one single final magnetic state, that is the magnetic skyrmion, and for values greater that 1 kM/m 3 (not shown here), the only final magnetic state is the perpendicular magnetic domain. The same behavior occurs for all other cases shown in Fig Once established the ranges of values of quantities such as K z and D int , we proceed to create a metastable Néel skyrmion from the state of lower energy (perpendicular magnetic domain) following the methodology used by Yuan et al. [18]. There are two currents to be used (see Fig. 1): -J 1 (current flowing in the +z direction) and +J 2 (current flowing in the -z direction). We used \| J 1 , J 2 \| between 0.8 A/m 2 and 3 A/m 2 . The first was applied in order to flip the magnetization to the plane of the free layer, in the -x direction, and the last to change the direction to +x. Depending on the value of J 1 and the time duration and the value of the current density pulse +J 2 , it is possible to create a metastable Néel skyrmion in the free layer. Unlike the work done by Yuan et al. [18] where only one case is shown, for \| J 1 \| = \| J 2 \| = 2×10 12 A/m 2 , we here show that considering \| J 1 \| \| J 2 \|, it is also possible to obtain a MNS. The topological charge Q was measured in every case to verify 1 The cases for D int = 2.5 mJ/m 2 and D int = 5 mJ/m 2 (not shown here) have presented the same behavior. if the structure was a skyrmion. First, all the micromagnetic simulations start with perpendicular single domain ( Fig. 3 (a) and Fig. 3 (f)). This magnetic configuration is allowed to relax for a time interval of t = 0.5 ns. After that, in order to flip the magnetization in the -x direction ( Fig. 3 (b)), -J 1 was applied with duration of t = 0.5 ns (fixed time for all simulations). Next, -J 1 was switched off, and the current +J 2 was applied to change the direction of magnetization to +x. However, if the duration of J 2 is shorter than that required for this purpose, it is possible to see the formation of a deformed Néel skyrmion during this process ( Fig. 3 (c)). Once obtained this, the system is allowed to relax with J 2 = 0, until finally a Néel skyrmion is obtained as the final magnetic configuration (3 (e)). For this objective, we have considered duration times for J 2 from t = 0.1 ns to 0.3 ns, in 0.05 ns steps. For each value of J 1 , we used values of J 2 from J 2 = 0.8 A/m 2 to J 2 = 3 A/m 2 in 0.2 A/m 2 steps. The phase diagrams for the cases K z = 0.6 MJ/m 3 , K z = 0.8 MJ/m 3 and D int = 3 mJ/m 2 are shown in Fig.4 for different values of J 1 . It is possible to observe that the phase diagram is also modified by the value of J 1 , since J 1 controls the flipping of the magnetic moments to the plane of the free layer. For example, for K z = 0.8 MJ/m 3 and J 1 = -1.4×10 12 A/m 2 (Fig. 4 (c-d)), we have obtained a metastable Néel skyrmion 2 for almost the entire range of current duration used here. However, when this value was modified to J 1 = -2.8×10 12 A/m 2 , the phase diagram changes drastically. A similar result is obtained for the case K z = 0.6 MJ/m 3 ( Fig. 4 (a-b)). For all values of K z and D int used here, we have obtained phase diagrams that are dependent on J 1 , J 2 and duration of the current pulse. However, we have not encountered a direct relation between the values J 1 , J 2 and K z , D int in the creation of a metastable skyrmion. The phase diagrams are highly nonlinear, nevertheless , it is possible to find some general patterns for the formation of a skyrmion (see Fig. 11 in the Supplementary material 3 ). For example, for the cases where D int = 3.5 mJ/m 2 , a magnetic skyrmion can be created from a current density threshold 4 Fig. 11 in SP), but when K z increases to K z = 1 MJ/m 3 , the current density threshold to increase to \| J 1 \| = 1.8 A/m 2 (Fig. 11 in SP). For K z = 1.2 MJ/m 3 , it is not possible anymore to create a skyrmion with the values of density currents used in this work. The same behavior was found for D int = 4 mJ/m 2 , the current density threshold increases from \| J 1 \| = 1.2 A/m 2 (for K z = 0.8 MJ/m 3 ) to \| J 1 \| = 1.6 A/m 2 (for K z = 1 MJ/m 3 ) and for K z = 1.4 MJ/m 3 , the current density threshold increases up to \| J 1 \| = 2.8 A/m 2 . This increase in the values of current density threshold it is due to the fact that higher values of J 1 are necessary in order to flip the magnetization to the plane in order to overcome the perpendicular alignment due to the presence of K z . of \| J 1 \| = 1.4 A/m 2 for K z = 0.8 MJ/m 3 ( On the other hand, it is possible to observe how the increase of D int favors the decrease of the current density threshold \| J 1 \|. For example, for K z = 0.8 MJ/m 3 (Fig. 11 in SP), \| J 1 \| decreases from \| J 1 \| = 1.4 A/m 2 (D int = 3 mJ/m 2 ) to \| J 1 \| = 1.2 A/m 2 (D int = 4 mJ/m 2 ). The similar behavior is observed for K z = 1 MJ/m 3 , where \| J 1 \| decreases from \| J 1 \| = 2.6 A/m 2 (D int = 3 mJ/m 2 ) to \| J 1 \| = 1.4 A/m 2 (D int = 4.5 mJ/m 2 ). In all cases, the skyrmions have topological charge Q ≈ 1. Q is not exactly one, because our disk have finite size. An interesting case occur for K z = 0.8 MJ/m 3 , D int = 3 mJ/m 2 , J 1 = -3 A/m 2 , J 1 = 2.4 A/m 2 an duration time of J 2 of t = 10 ns: we have obtained two skymions, as final magnetic configuration, in the free layer, both with polarity p = +1 (in this case the topological charge Q is approximately two). For some combinations of K z , D int , J 1 and J 2 , we have obtained as final states a strange magnetic configurations (similar to show in Fig. 3 (i)) where a skyrmion coexists with a mag- The minimum values of D N , for all combinations of D int and K z , tend to a value in a range between approximately 5 nm and 7 nm (see Fig. 5), whereas the maximum value obtained of D N was of approximately 36 nm, which occupies approximately 40% of the diameter of the disk (free layer). In the cases where two skyrmions were obtained in the free layer, both skyrmions have the same D N , whose value is the same as in the case of obtaining a single Skyrmion of approxi-mately D N ≈ 13 nm. Although the Néel skyrmions are metastable, we can see that the values of D N follow the same behavior as in the cases where the skyrmion is a ground state magnetic configuration [30]. We have found that the values of D N do not depend on either current densities (J 1 and J 2 ) or duration times of pulse currents. These are dependent only on K z and D int . This case contrasts with the case where the polarizer has magnetization m p = (0,0,1) [7,18]. Coupled oscillators In order to study the effect of the magnetic interaction on the creation of skyrmions, we considered a pair of coupled spin torque oscillators (see Fig. 6). We used the same values of K z and D int shown in Fig. 2. The currents J 1 and J 2 were applied simultaneously in both STNOs in the same way as in the case of an isolated spin torque oscillator (previous section). We have considered a separation center to center distance d = 85 nm, 90 nm and 95 nm. The new phase diagrams for the case D int = 3 mJ/m 2 , K z = 0.8 MJ/m 3 and J 1 = -2.4×10 12 A/m 2 are shown in Fig. 7. It is possible to observe how the phase diagrams change for different values of the distance d. In some cases the magnetic interaction favors the creation of skyrmions and in other cases it does not. For example, using a value of J 2 = 2.2×10 12 A/m 2 and for any value of duration time of J 2 , it is impossible to create a skyrmion ( Fig. 7(a)) in an isolated STNO, but in a coupled system, it is possible to create a skyrmion for separation distances of d = 5 nm (Fig. 7(b)) and d = 10 nm (Fig. 7(c)), using a duration time of J 2 of t = 10 ns. The opposite behavior occurs when J 2 = 1.8×10 12 A/m 2 . In this case, it is possible to create the skyrmion in an isolated STNO (Fig. 7(a)) for duration time of J 2 from t = 20 ns, 25 ns and 30 ns, but it is impossible to create the skyrmion when the STNOs are coupled (Fig. 7(b), (Fig. 7(c)), (Fig. 7(d))) for any value of duration time of J 2 . The average Néel skyrmion diameters (D N ) neither depend on the magnetic interaction nor on the values of J 1 and J 2 . They have almost the same values as in the case of isolated STNO and have the same dependence on K z and D int as shown in Fig. 5. The phase diagrams for the coupled systems are also dependent on the values of J 1 . In Fig. 8, we show the phase diagrams for a value of J 1 = -1.8×10 12 A/m 2 . Its phase diagram is different from the one obtained for the case J 2 = -2. The phase diagrams are shown in Fig. 8. They are more complex than the phase diagrams shown in Fig. 7. Besides the cases where skyrmions were present or absent in both STNOs, there are cases where it was possible to create a skyrmion only in one of the STNOs. The behavior of the current density threshold of \| J 1 \| (see Fig.12 in SP) follows the same way that in the case of isolated STNO. We can see that the threshold \| J 1 \| decreases with the increase of D int and \| J 1 \| increases with the increase of K z . The quantitative values of \| J 1 \| are almost the same as those in the case of isolated STNO. The results shown in Fig. 7 and Fig. 8 prove that the magnetic interaction plays an important role in the process of creation of skyrmions in STNOs. In order to further explore the effects of the magnetic interaction, we additionally, have also considered one triangular array 5 of STNOs as shown in Fig. 9. The phase diagrams for K z = 0.8 MJ/m 3 , D int = 3 mJ/m 2 , J 1 = -2.4×10 12 A/m 2 are shown in Fig. 10. In this figure, it is possible to observe that the creation of skyrmions in the STNOs is different for different distances d. The STNOs in the triangular array, interacting in a different form from that in the case of a pair of coupled STNOs. For example, for a pair of coupled STNOs, with parameters: J 2 = 2.2×10 12 A/m 2 , t = 0.10 ns and distance d = 85 nm, the skyrmions can be created in the two STNOs ( Fig. 7 (a)), while for the same parameters, for a triangular array, the skyrmions cannot be created in the three STNOs ( Fig. 10 (a)). Also, a similar behavior can be seen for J 2 = 2×10 12 A/m 2 and distance d = 85 nm (Fig. 7 (b)), while that for a pair of coupled STNOs, the magnetic interaction prevents the creation of skyrmions in the whole range of values of duration of J 2 , but this magnetic inter- 5 The procedure to obtain the skyrmion is the same as in the previous cases. Conclusions In this work, we have studied the influence of the magnetic interaction in the creation of a metaestable skyrmion in isolated and coupled systems of STNOs using pulsed spin current densities, using micromagnetic simulation. We have built phase diagrams to know in what conditions it is possible to obtain a skyrmion. Although our phase diagrams do not show a direct relationship between the parameters used here, we have been able to find some general behavior such as the fact that the threshold J 1 increases with the increase of the anisotropy and decreases with the increase of D int . Our results show that the diameters of the metastable skyrmion have the same behavior as that of stable skyrmions. We have demonstrated that, in the case of coupled systems, the magnetic interaction plays an important role in the creation of metastable skyrmions, modifying the phase diagrams when the distance between the STNOs is changed. However, the magnetic interaction does not modify the values of the threshold J 1 . Our results show a point not sufficiently studied, the influence of the magnetic interaction in the creation of skyrmions. . 2. 1 , for D int = 3.5, 4 and 4.5 mJ/m 2 . Figure 2 : 2Energy of the magnetic final states of the free layer vs. of perpendicular uniaxial anisotropy constant (K z ). Figure 3 : 3Evolution of the z-component of the magnetization configuration from the initial perpendicular single domain to the creation of a metastable Néel skyrmion, for D int = 3 mJ/m 2 and (a-e) K z = 0.8 MJ/m 3 and (f-j) K z = 0.6 MJ/m 3 Figure 4 : 4Phase diagram for K z = 0.6 MJ/m 3 , K z = 0.8 MJ/m 3 and D int = 3 mJ/m 2 , and different values of the current density J 1 . Red squares indicate no creation of skyrmion, and green circles indicate creation of skyrmion. netic domain. These cases, can not be considered as a Néel Skyrmion. Similar configurations have been obtained in cobalt disks under application of perpendicular magnetic fields, as already showed by Talapatra et al. [29]. We have obtained the average Néel skyrmion diameters (D N ), considering as D N diameter, the region where the z-component of the magnetization (m z ) is equal to zero. These values were obtained measuring the Full-Width at Half-Maximum of the profile of m z along the diameter of the disk, and their values are shown in Fig. 5. The values of the diameters decrease with the increase of the K z . For example, for D int = 3 mJ/m 3 , we have obtained a decrease of approximately 80%, from D N ≈ 36 nm (K z = 0.4 MJ/m 3 ) to D N ≈ 8 nm (K z = 1 MJ/m 3 ). The same behavior occurs for the different values of D int . However, the values of D N increase with the values of D int for the same values of K z . For example, for K z = 1 MJ/m 3 , D N increases from D N ≈ 7.7 nm (D int = 3 mJ/m 3 ) to D N ≈ 34.68 nm (D int = 3 mJ/m 3 ). These behaviors are expected to the fact that the increase of K z favors the perpendicular alignment of the magnetic moments, thus decreasing the region where m z = 0, whereas the increase of D int favors the tilting of magnetic moments, increasing the region where m z = 0. Figure 5 : 5Skyrmion diameter D N as a function of the values of the perpendicular anisotropy constant K z , and D int . Figure 6 : 6Schematic representation of a pair of coupled spin torque oscillators separated by a center to center distance d. Figure 7 : 7Phase diagram for K z = 0.8 MJ/m 3 and D int = 3 mJ/m 2 for a) one isolated STNO; b) two STNOs, separation d = 85 nm; c) two STNOs, d = 90 nm; d) two STNOs, d = 95 nm. Red squares indicate no creation of skyrmions in either of the two STNOs, and green circles indicate creation of skyrmions in both STNOs. Figure 8 : 8Phase diagram for K z = 0.4 MJ/m 3 , K z = 0.8 MJ/m 3 and D int = 3 mJ/m 2 for a) one isolated STNO; b) two STNOs, separation d = 85 nm; c) two STNOs, d = 90 nm; d) two STNOs, d = 95 nm. Red squares indicate no creation of skyrmion in either of the two STNOs, blue stars indicate the creation of skyrmion in one of the two STNOs, and green circles indicate creation of skyrmions in both STNOs. Figure 9 : 9Schematic representation of an equilateral triangular array of coupled STNOs separated by a center to center distance d. action favors the creation of skyrmions in two of three STNOs (Fig. 10 (a)) Figure 10 : 10Phase diagram, for a triangular array, for K z = 0.8 MJ/m 3 and D int = 3 mJ/m 2 . Red circles indicate the STNO where there is no creation of skyrmions, green circles indicate the STNO where there is creation of skyrmions The skyrmions may have polarity p = +1 or p = -1. 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[ "Transport properties of Brownian particles confined to a narrow channel by a periodic potential", "Transport properties of Brownian particles confined to a narrow channel by a periodic potential" ]	[ "Xinli Wang \nDepartment of Chemical and Biomolecular Engineering\nThe Johns Hopkins University\n21218BaltimoreMDUSA\n", "German Drazer \nDepartment of Chemical and Biomolecular Engineering\nThe Johns Hopkins University\n21218BaltimoreMDUSA\n" ]	[ "Department of Chemical and Biomolecular Engineering\nThe Johns Hopkins University\n21218BaltimoreMDUSA", "Department of Chemical and Biomolecular Engineering\nThe Johns Hopkins University\n21218BaltimoreMDUSA" ]	[]	We investigate the transport of Brownian particles in a two-dimensional potential under the action of a uniform external force. The potential is periodic in one direction and confines the particle to a narrow channel of varying cross-section in the other direction. We apply the standard long-wave asymptotic analysis in the narrow dimension and show that the leading order term is equivalent to that obtained previously from a direct extension of the Fick-Jacobs approximation.We also show that the confining potential has similar effects on the transport of Brownian particles to those induced by a solid channel. Finally, we compare the analytical results with Brownian dynamics simulations in the case of a sinusoidal variation of the width of the parabolic potential in the cross-section. We obtain excellent agreement for the marginal probability distribution, the average velocity of the Brownian particles and the asymptotic dispersion coefficient, over a wide range of Péclet numbers.	10.1063/1.3226100	[ "https://arxiv.org/pdf/0905.3774v1.pdf" ]	119,286,931	0905.3774	2eb64578310d6c54253fa5e7068812fc7b2dc35c	Transport properties of Brownian particles confined to a narrow channel by a periodic potential 22 May 2009 Xinli Wang Department of Chemical and Biomolecular Engineering The Johns Hopkins University 21218BaltimoreMDUSA German Drazer Department of Chemical and Biomolecular Engineering The Johns Hopkins University 21218BaltimoreMDUSA Transport properties of Brownian particles confined to a narrow channel by a periodic potential 22 May 2009(Dated: May 22, 2009)PACS numbers: Keywords: We investigate the transport of Brownian particles in a two-dimensional potential under the action of a uniform external force. The potential is periodic in one direction and confines the particle to a narrow channel of varying cross-section in the other direction. We apply the standard long-wave asymptotic analysis in the narrow dimension and show that the leading order term is equivalent to that obtained previously from a direct extension of the Fick-Jacobs approximation.We also show that the confining potential has similar effects on the transport of Brownian particles to those induced by a solid channel. Finally, we compare the analytical results with Brownian dynamics simulations in the case of a sinusoidal variation of the width of the parabolic potential in the cross-section. We obtain excellent agreement for the marginal probability distribution, the average velocity of the Brownian particles and the asymptotic dispersion coefficient, over a wide range of Péclet numbers. INTRODUCTION Recent progress in microfluidic devices has led to the development of novel separation strategies that take advantage of the unprecedented control on the geometry and chemistry of the stationary phase at scales that are comparable to the size of the transported species [1,2,3]. A fundamental problem that is at the core of several of the proposed separation devices is the transport of Brownian particles through entropy barriers. A representative example is the motion of a suspended particle in a channel with periodically varying cross section. The purely-diffusive transport in the absence of an external force has been studied extensively [4,5,6,7], and a well-known approach is to reduce the dimensionality of the problem via the Fick-Jacobs approximation. In this approximation the motion in the cross section is reduced to an entropic contribution to the longitudinal transport [8]. The case of biased diffusion in the presence of a driving force has also received considerable attention due to its relevance to separation devices. Hänggi and coworkers examined the validity of a direct extension of the Fick-Jacobs equation to describe the biased motion of a Brownian particle in a narrow channel of varying cross section. The authors showed that using an ad-hoc position-dependent diffusivity provides a good approximation, for Péclet numbers below a critical value [9,10]. Dorfman and coworkers [11] also investigated the biased motion of a Brownian particle in a periodic channel. Specifically, they performed an asymptotic perturbation analysis to obtain simple expressions for the macroscopic transport coefficients that remain valid at relatively large Péclet numbers. Here, we also consider the transport of Brownian particles confined to a channel of periodically varying cross section but, in the present case, the confinement is induced by a potential energy landscape and not the solid boundaries of a channel. This type of spatial confinement occurs, for example, in the transport of suspended particles in microfluidic channels. Depending on the density of the particles, the van der Waals and electrostatic forces between the particle and the channel walls, and the chemical composition of the media, the particles could become confined to the secondary minima of the particle-wall interaction potential in the vertical direction [12]. In that case, the presence of a periodic pattern on the bottom wall, such as that created by the deposition of thin metal stripes perpendicular to the flow, would lead to the confinement of the Brownian particles to a periodically varying channel parallel to the wall, analogous to the one considered here. We shall show that the effect on the average velocity of the particles (and the probability distribution in general) induced by this type of soft confinement of the suspended particles to an energy minima is the same as that caused by geometric confinement between solid walls. Understanding the effect that this type of confinement has on the transport of suspended particles is particularly important for the development of recently proposed separation techniques in microfluidic devices that are based on partitioning [13]. TRANSPORT OF BROWNIAN PARTICLES IN A CONFINING PERIODIC PO-TENTIAL Let us consider the transport of Brownian particles in a potential that is periodic in the x-direction, V (x = x 0 , z) = V (x = x 0 + L, z) , and that confines the particles in the z-direction, that is V (x, z) → +∞ for z → ±∞. In the limit of negligible inertia effects the motion of the particles is described by the Smoluchowski equation for the probability density P (x, z, t), ∂P ∂t + ∇ · J = δ(x, z)δ(t).(1) The probability flux, J(x, z, t), is given by J = 1 η F P − ∂V ∂x P − k B T ∂P ∂x i + 1 η − ∂V ∂z P − k B T ∂P ∂z k,(2) where F is a uniform external force in the x-direction, η is the viscous friction coefficient, and we have used the Stokes-Einstein equation to write the diffusion coefficient as a function of η, D = k B T /η. In order to obtain the asymptotic distribution of particles within a single period of the potential we first introduce the reduced probability density and the reduced probability current (see Refs. [14,15] or the analogous approach presented in Ref. [16]), P (x, z, t) = +∞ nx=−∞ P (x + n x L, z, t),(3)J(x, t) = +∞ nx=−∞ J(x + n x L, z, t).(4) The reduced probability is obtained by solving the Smoluchowski equation with periodic boundary conditions in x. In particular, the long-time asymptotic probability density, P ∞ (x, z) = lim t→∞P (x, z, t), is governed by the equation (dropping the tilde), ∇ · J ∞ = 0.(5) The far-field condition in z is a vanishingly small probability density and flux due to the confining potential, J z ∞ = 1 η − ∂V ∂z P ∞ − k B T ∂P ∞ ∂z − −−− → z→±∞ 0.(6) Finally, the reduced probability is obtained by imposing the periodic boundary conditions in x, P ∞ (x = 0, z) = P ∞ (x = L, z),(7) and the normalization condition, P ∞ = L 0 dx ∞ −∞ P ∞ dz = 1.(8) In the case of narrow or slender channels, we assume that the characteristic length scale in the cross-section, i. e. perpendicular to the channel centerline, is given by ǫL, where ǫ ≪ 1 is the slenderness ratio. We can then write the governing equation and boundary conditions using the following dimensionless variables, x = x/L, z = z/(ǫL), V = V /(k B T ), where k B is the Boltzmann constant and T is the absolute temperature, and the re-scaled probability density P ∞ = ǫL 2 P ∞ . For the sake of simplicity all bars are dropped after nondimensionalization. The governing equation for the reduced probability (Eq. (5)) then becomes: ǫ 2 ∂ ∂x Pe − ∂V ∂x P ∞ − ∂P ∞ ∂x + ∂ ∂z − ∂V ∂z P ∞ − ∂P ∞ ∂z = 0,(9) where the Péclet number is defined as Pe = F L/k B T . Simple inspection of this equation shows that the leading order approximation for ǫ ≪ 1 is the local equilibrium in z, which corresponds to J z 0 (x, z) = 0, due to the no-flux far-field condition. In general, an accurate description of the long-time transport of Brownian particles can be obtained from the first two moments of the asymptotic probability, which describe the average velocity and the broadening of the distribution. Applying macrotransport theory we can write the average velocity in terms of the asymptotic probability distribution [16], v = 1 0 dx ∞ −∞ J x ∞ dz = 1 0 dx ∞ −∞ Pe − ∂V ∂x P ∞ − ∂P ∞ ∂x dz.(10) The dispersion coefficient D * , can also be calculated from the asymptotic distribution via the so-called B−field, which is the solution of the following differential equation [16], ∂ ∂z P ∞ ∂B ∂z − − ∂V ∂z P ∞ − ∂P ∞ ∂z ∂B ∂z + ǫ 2 ∂ ∂x P ∞ ∂B ∂x − Pe − ∂V ∂x P ∞ − ∂P ∞ ∂x ∂B ∂x = ǫ 2 P ∞ v .(11) The boundary conditions for the B−field are, ∂B ∂z − −−− → z→±∞ 0,(12)B(x = 1, z) − B(x = 0, z) = −1. Finally, the dispersion coefficient is given in terms of B(x, z) by D * = 1 0 dx ∞ −∞ P ∞ ∂B ∂x 2 + 1 ǫ 2 ∂B ∂z 2 dz.(13) ASYMPTOTIC ANALYSIS IN THE NARROW CHANNEL APPROXIMATION We apply the standard long-wave asymptotic analysis to obtain an approximate solution to the problem described above. First, we propose a solution to the stationary probability distribution in the form of a regular perturbation expansion in the slenderness parameter, P ∞ (x, z) ∼ p 0 + ǫ 2 p 1 + ǫ 4 p 2 + · · · .(14) Analogously, we write a regular perturbation expansion for the probability flux, J ∞ (x, z) ∼ J 0 + ǫ 2 J 1 + ǫ 4 J 2 + · · · .(15) Substituting these expansions into Eq. (9) it is straightforward to determine the governing equation for the leading order terms, ∂ ∂z − ∂V ∂z p 0 − ∂p 0 ∂z = ∂J z 0 ∂z = 0,(16) The corresponding leading order boundary and normalization conditions, derived from Eqs. (6)(7)(8), are: J z 0 (x, z = ±∞) = 0 (17) p 0 (x = 0, z) = p 0 (x = 1, z) p 0 = 1. Then, integrating Eq. (16) and taking into account the no-flux condition we obtain: p 0 (x, z) = f 0 (x)e −V (x,z) ,(18) where f 0 (x) is an unknown function that is to be determined from the second order O(ǫ 2 ) balance of Eq. (9), ∂ ∂z ∂V ∂z p 1 + ∂p 1 ∂z = ∂ ∂x Pe − ∂V ∂x p 0 − ∂p 0 ∂x .(19) The corresponding boundary and normalization conditions are: J z 1 (x, z = ±∞) = 0,(20)p 1 (x = 0, z) = p 1 (x = 1, z) p 1 = 0. Substituting the solution obtained for p 0 into Eq. (19) we obtain ∂ ∂z ∂V ∂z p 1 + ∂p 1 ∂z = ∂ ∂x Pef 0 − df 0 dx e −V (x,z) .(21) In steady state, the total flux in the x-direction is constant along the channel. Therefore, by integrating the equation above over the cross-section, and taking into account the zero flux condition in the z-direction, we obtain d dx Pef 0 − df 0 dx I(x) = 0,(22) where I(x) = ∞ −∞ e −V (x,z) dz.(23) We note that I(x) plays the role of the width w(x) of a solid channel, that is, the integral in the previous equation becomes equal to the width of the channel if we model the rigid boundaries by a potential field that is zero (infinite) inside (outside) the channel. In fact, replacing I(x) by w(x) in Eq. (22) we obtain Eq. (25) in Ref. [11]. This suggests that the confining potential V (x, z) has similar effects on the transport of Brownian particles to those induced by a channel with solid walls. Before we proceed to solve the equation for f 0 (x) it is also interesting to compare it with the equation obtained from the direct extension of the Fick-Jacobs approximation in the presence of an external force. In fact, in the narrow geometry that we are considering here, we can assume that the particle will reach local equilibrium in the cross section fairly rapid compared to its diffusive or convective motion along the channel. This separation of time scales suggests that the conditional probability P (z/x, t) could be approximated by the equilibrium distribution P eq (z/x). Then, the total probability density takes the form, P (x, z, t) = P (z/x, t)P (x, t) ≈ P eq (z/x)p(x, t) = e −V (x,z) I(x) p(x, t),(24) where p (x, t) is the marginal probability distribution, p(x, t) = ∞ −∞ P (x, z, t)dz,(25) Considering the long time limit, we can then use the same approximation for the asymptotic distribution, P ∞ (x, z) ≈ e −V (x,z) I(x) p(x).(26) Finally, we can project the problem into the longitudinal direction by substituting this expression into the governing equation for the probability density and integrating over the cross-section, d dx Pe p I(x) − d dx p I(x) I(x) = 0.(27) This is the form of the Fick-Jacobs equation used in the presence of an external force in previous studies [5,9]. It is interesting to point out that, by making the substitution p(x) = f 0 (x)I(x), the previous equation becomes identical to Eq. (22), which was derived from the asymptotic analysis. Therefore, the leading order term of the asymptotic analysis is equivalent to the the proposed extension of the Fick-Jacobs approximation in the presence of an external field. Then, the validity of the ad-hoc position-dependent effective diffusivity proposed in Ref. [10] could, in principle, be tested by extending the present approach to higher orders in the slenderness parameter. We now go back to equation (22) to obtain the general solution for f 0 (x), f 0 (x) = e Pe x C 1 e −Pe x I(x) dx + C 2 ,(28) where C 1 and C 2 are the constants of integration. The leading order contributions to the average velocity and the dispersion coefficient can then be calculated from the probability distribution. First, we obtain the general expression for the average velocity following a derivation analogous to that presented in Ref. [10] (appendix A) for the particle current, v = (1 − e −Pe ) 1 0 dxe Pe x I(x) x+1 x dx ′ e Pe x ′ I(x ′ ) −1 .(29) perturbation expansion for the B-field, analogous to those proposed for the probability density and flux, B ∼ B 0 + ǫ 2 B 1 + ǫ 4 B 2 + · · · .(30) Substituting the expansion for both the probability density and the B-field into Eq. (11) it is straightforward to show that the zeroth-order balance is an homogeneous equation that is satisfied by an arbitrary function B 0 (x). The governing equation for B 0 (x) is obtained from the O(ǫ 2 ) balance, after integrating over the cross-section and taking into account the no-flux boundary condition for the B-field in the z-direction, d 2 B 0 dx 2 + p ′ 0 − v p 0 dB 0 dx = v .(31) The remaining boundary condition is, B 0 (1) − B 0 (0) = −1.(32) The above two equations determine B 0 uniquely to within an arbitrary additive constant, which does not affect the calculation of the effective dispersion coefficient [16], D * = 1 0 dx ∞ −∞ p 0 (x, z) dB 0 dx 2 dz.(33) TRANSPORT OF BROWNIAN PARTICLES CONFINED BY A PARABOLIC PO-TENTIAL In this section, we discuss a specific example of the confining potential that illustrates our previous results and allows direct comparison with numerical results. Consider the transport of Brownian particles confined to a narrow channel by a parabolic potential of periodically varying width, V (x, z) = cos 2π L x + d 2 z − z 0 ǫL 2 k B T,(34) where z 0 is the position of the center of the channel, which we assume to be constant (straight centerline), and d > 1 determines the minimum opening of the channel. The slenderness of the geometry is given by ǫ ≪ 1, which is the ratio of the characteristic width of the potential in the z-direction, ǫL, to the length of one period, L. Using the same dimensionless variables introduced before and dropping again all bars for the sake of simplicity, we obtain, V (x, z) = (z − z 0 ) 2 2δ 2 (x) ,(35) where δ (x) = 1 √ 2 (cos 2πx + d) .(36) It is clear then that the particles will be confined in the z direction by a parabolic potential and that the width of the confining region is determined by δ(x). In fact, the equilibrium distribution of particles is given by the Boltzmann distribution with variance σ(x) = δ(x). In Fig. 1 we plot the equipotential lines z − z 0 = ±2δ(x), with d = 1.2 and z 0 = 8. In equilibrium, approximately 95% of the particles are confined to the region enclosed by these lines. For this specific potential we obtain I(x) = √ 2πδ(x) = √ π(cos 2πx + d) −1 , and f 0 (x) = − C 1 √ π 2π sin 2πx − Pe cos 2πx 4π 2 + (Pe) 2 − d Pe + C 2 e Pe x ,(37) Where C 1 and C 2 are integration constants. In particular, C 1 is the first constant of integration of Eq. (22), which corresponds to the total flux in the x-direction, C 1 = J x 0 . Thus, imposing periodicity and the normalization condition we obtain the leading order probability density, p 0 (x, z) = − J x 0 √ π e −V (x,z) 2π sin 2πx − Pe cos 2πx 4π 2 + Pe 2 − d Pe ,(38) and average velocity, J x 0 = v = Pe 2 + 4π 2 Pe Pe 2 + 4π 2 d √ d 2 −1 .(39) Finally, the leading order term of the marginal probability distribution, corresponding to the distribution in the heuristic extension to the Fick-Jacobs approximation, is given by p(x) = Pe 2 Pe 2 + 4π 2 d √ d 2 −1 1 − 1 Pe 2 2πPe sin 2πx − 4π 2 d d + cos(2πx)(40) Note that in the limiting case in which diffusive transport is dominant (vanishingly small Péclet number) we recover the equilibrium Boltzmann distribution, lim Pe→0 p 0 (x, z) = d 2 − 1 π e −V (x,z) .(41) On the other hand, in the limit of deterministic motion we obtain, lim Pe→∞ p 0 (x, z) = d + cos(2πx) √ π e −V (x,z) .(42) In this case, there is no boundary layer developing due to the large magnitude of the driving force, as it would be the case in the presence of solid walls and permeating forces [15]. Therefore, the diffusive fluxes are negligible compared to convective transport, and the average velocity tends to its bulk value, something that is also clear from Eq. (39). In addition, the integral of the asymptotic distribution over a cross section is uniform, which is also clear from Eq. (22). Note, however, that this limit is valid if the slenderness condition satisfies ǫ 2 Pe ≪ 1, as discussed in Ref. [11]. BROWNIAN DYNAMICS SIMULATIONS We performed Brownian dynamics simulations of particle transport under confinement by a potential landscape to examine the previous asymptotic results in more detail. The motion of Brownian particles in a viscous solvent in the limit of vanishingly small inertia is governed by the overdamped Langevin equations, η dx dt = F − ∂V ∂x + ηk B T ζ(t),(43) and η dz dt = − ∂V ∂z + ηk B T ζ(t),(44) where ζ(t) is a zero-mean Gaussian white noise with the correlation ζ i (t 1 )ζ j (t 2 ) = 2δ ij δ(t 1 − t 2 ). The value of the slenderness parameter used in the numerical simulations is ǫ = 0.01. between the simulations and the asymptotic analysis. CONCLUSIONS We have investigated the transport of Brownian particles driven by a uniform external force. The particles are confined to a narrow periodic channel by a parabolic potential in the cross-section. We used asymptotic methods to obtain the leading order solution of the two-dimensional Smoluchowski equation for the long-time probability distribution of particles reduced to a single period of the potential. We first showed that the leading order analysis reproduces a previously proposed extension of the Fick-Jacobs approximation to biased transport. We thus provide a systematic method to improve on the Fick-Jacobs approximation through higher order analysis. We also showed that the leading order equation is equivalent to that obtained for solid channels and thus demonstrated that a confining potential has analogous effects on the distribution of particles and their transport parameters, such as the average velocity and the asymptotic dispersion coefficient. We then analyzed the case of a cosine variation in the aperture of the confining channel and compared the results with Brownian dynamics simulations. We compared the long-time marginal distribution as well as its first moments (average velocity and dispersion coefficient) and obtained excellent agreement with the first order in the asymptotic analysis over a wide range of Péclet numbers. This material is partially based upon work supported by the National Science Foundation under Grant No. CBET-0731032. FIG. 1 : 1Equipotential lines corresponding to z = z 0 = ±2δ(x) for a periodic potential V (x, z) = (cos 2πx + d) 2 (z − z 0 ) 2 with d = 1.2 and z 0 = 8. FIG. 2 :FIG. 3 : 23Marginal probability density p(x) for different values of the Péclet number (Pe). The solid lines indicate the asymptotic results and the symbols correspond to the results of the Brownian dynamics simulations. In figure 2 we show the marginal probability distribution p(x) in a single period of the potential, for different values of the Péclet number Pe = 1, 10, 30, 50. The leading order term of the distribution, given by Eq. (40), agrees well with the results of the Brownian dynamics simulations for all Péclet numbers. The figure also shows that the marginal probability distribution tends to a uniform distribution as the Péclet number increases. This indicates that, as discussed before, the effect of the potential on the particle distribution, as well as the contribution of diffusive transport to the average velocity of the particles, decreases as the Péclet number increases. Figure 3 shows that the average velocity of the particles obtained in the simulations is accurately described by the leading order term in the perturbation expansion. Finally, in figure 4 we compare the effective dispersion coefficient obtained from the leading order term in the B-field (numerically solving Eq. (31) by means of finite differences) with that computed directly from the simulations. We observe good agreement Average velocity along the channel as a function of the Péclet number. The solid line corresponds to the leading order term of the asymptotic results and the solid circles represent the values computed from the Brownian dynamics simulations. FIG. 4 : 4Asymptotic dispersion coefficient as a function of the Péclet numbers. The solid line corresponds to the leading order term in the asymptotic analysis. The solid circles corresponds to the dispersion coefficient calculated from the Brownian dynamics simulations. . T Duke, Curr. Opin. Chem. Biol. 2592T. Duke, Curr. Opin. 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[ "Friend-based Ranking", "Friend-based Ranking" ]	[ "Francis Bloch [email protected] \nParis School of Economics\nUniversité Paris\n\n", "Matthew Olckers [email protected] ", "We Thank ", "Abhijit Banerjee ", "Leonie Baumann ", "Bhaskar Dutta ", "Matt Elliott ", "Ben Golub ", "Sanjeev Goyal ", "Matt Jackson ", "Matt Leister ", "Neil Lloyd ", "Brian Rogers ", "Leeat YaarivJonathan Weinstein ", "Yves Zénou ", "\nPanthéon-Sorbonne\nUniversité\nParis 1 Panthéon-Sorbonne\n" ]	[ "Paris School of Economics\nUniversité Paris\n", "Panthéon-Sorbonne\nUniversité\nParis 1 Panthéon-Sorbonne" ]	[]	We analyze the design of a mechanism to extract ordinal information disseminated in a social network. We show that friend-based ranking-the report by agents on the characteristics of their neighbors-is a necessary condition for ex post incentive-compatible and efficient mechanism design. We characterize the windmill network as the sparsest social network for which the planner can construct a complete ranking. When complete rankings cannot be achieved, ex post incentive-compatible and efficient mechanisms arise when social networks are bipartite or composed of triangles. We illustrate these findings using real social networks in India and Indonesia. for helpful discussions on the project. We are also grateful to seminar participants at several institutions for their comments.	10.2139/ssrn.3213311	[ "https://arxiv.org/pdf/1807.05093v3.pdf" ]	49,743,778	1807.05093	e125f2c3683bd8a751f388e96417c872a06284bd	Friend-based Ranking October 18, 2018 17 Oct 2018 Francis Bloch [email protected] Paris School of Economics Université Paris Matthew Olckers [email protected] We Thank Abhijit Banerjee Leonie Baumann Bhaskar Dutta Matt Elliott Ben Golub Sanjeev Goyal Matt Jackson Matt Leister Neil Lloyd Brian Rogers Leeat YaarivJonathan Weinstein Yves Zénou Panthéon-Sorbonne Université Paris 1 Panthéon-Sorbonne Friend-based Ranking October 18, 2018 17 Oct 20181social networksmechanism designpeer rankingtargeting JEL Classification numbers: D85D82O12D71 We analyze the design of a mechanism to extract ordinal information disseminated in a social network. We show that friend-based ranking-the report by agents on the characteristics of their neighbors-is a necessary condition for ex post incentive-compatible and efficient mechanism design. We characterize the windmill network as the sparsest social network for which the planner can construct a complete ranking. When complete rankings cannot be achieved, ex post incentive-compatible and efficient mechanisms arise when social networks are bipartite or composed of triangles. We illustrate these findings using real social networks in India and Indonesia. for helpful discussions on the project. We are also grateful to seminar participants at several institutions for their comments. Introduction In many social networks, individuals gradually acquire information about their neighbors through repeated interactions. Pupils in a class learn about the ability of other pupils with whom they write joint projects, workers in a firm learn about the productivity of the coworkers in their teams, and members of a community in a developing country learn about the needs of their close friends. This information, which is disseminated in the social network, may be of great use to an external planner who wants to extract information about members of the community. A teacher wants to learn about the ability of her pupils; an employer, the productivity of her workers; a funding agency, the needs of villagers in a developing country. In the classical literature on mechanism design, the principal designs a mechanism which asks individuals to report on their own types. However, a large number of mechanisms is used in practice which ask individuals to report not on their own type but on the type of others. Pupils are asked to assess the performance of other pupils, workers are asked to measure the productivity of their coworkers, or villagers are asked to rank other individuals in the community. The objective of our paper is to analyze these mechanisms, that ask individuals to report about their neighbors in the social network-mechanisms that we term "friend-based ranking mechanisms." In particular, we want to understand how the architecture of the fixed social network affects the planner's ability to construct a mechanism having desirable properties. While we treat the social network as exogenous, we note that in some situations the planner can design the network endogenously. Consider, for example, peer selection problems, e.g., when editors of scientific journals ask scholars to review the work of their peers, or scientific funding agencies ask applicants to review the projects of other applicants. By assigning reviewers to projects, the planner designs a network of observation which plays exactly the same role as the exogenous social networks described above. We study a setting with three characteristics. First, we assume that information is local. An individual may make comparisons only among his direct neighbors. Second, information is ordinal. Individuals lack the ability to quantify characteristics and can only assess whether one individual has a higher characteristic than another. Third, we assume that the planner has only one instrument at her disposal: she constructs a (complete) ranking of the members of the community. Hence, the number of outcomes that the planner can select from is very restricted. The planner cannot use transfers, and cannot punish individuals by excluding them from the ranking. In particular, she cannot impose punishments for inconsistent reports, as in the classical literature on implementation with correlated types (Crémer and McLean, 1985). We require that the planner's mechanism satisfy two properties. First, individuals must have an incentive to report information truthfully. In the ordinal setting we consider, the natural choices for implementation concepts are dominant-strategy and ex post implementation. However, we notice that the limited number of outcomes in our setting means that dominant-strategy implementation is too strong, leading to impossibility results. We adopt instead ex post incentive compatibility as the desirable incentive property of the mechanism. Second, we require the mechanism to be ex post efficient from the point of view of the planner, whose objective is to recover the true ranking of individuals in the community. More precisely, the ranking chosen by the planner must match the ranking that society would construct by aggregating all local information. If society can construct a complete ranking of individuals for any realization of types (a situation we label "completely informative"), the ranking chosen by the planner must match the true ranking. Otherwise, we may face different situations according to the realization of types. For some type realizations, even if the information aggregated by society is not complete, transitivity ensures that all individuals can be completely ranked. For other type realizations, society will be able to construct only a partial order on the individuals. In the latter case, the complete ranking chosen by the planner will be a completion of the partial order that the community is able to construct, and this completion will involve an arbitrary ranking across individuals who cannot be compared. We first analyze mechanism design in completely informative societies. We show that a society is completely informative if every pair of individuals can be compared, either through "self-reports" (the two individuals involved in the pair report on each other) or through "friend-based reports" (a third individual observes both individuals in the pair). Our main theorem shows that self-reports can be used only if they are backed by the report of a third individual. A mechanism satisfying ex post incentive compatibility and efficiency exists if and only if every pair of individuals has a common neighbor. We then characterize the sparsest network which satisfies this property. When the number of individuals is odd, this is the "friendship network" of Erdős, Rényi, and Sós (1966), the only network in which every pair of individuals has only one common friend. This network, also known as the windmill, has one individual as a hub who connects all other individuals who form pairs. When the number of individuals is even, this is a variant of the windmill in which one of the "sails" contains three individuals instead of two. In this network, one individual-the hub-is responsible for a large number of comparisons, so we also investigate other networks where every pair of agents has a common neighbor when the number of comparisons performed by every individual is capped. We then turn our attention to societies which are not completely informative; to study these, we add another requirement to the mechanism: to guarantee that whenever two individuals are incomparable, the mechanism ranks them in the same way independent of (irrelevant) information from other comparisons. Under this independence requirement, we show that any comparison based solely on self-reports must be discarded by the planner, as both individuals have an incentive to misreport. Hence, the planner can rely only on friend-based comparisons, and we construct a "comparison network" by linking two individuals if and only if they have a common neighbor. We find that there exist two network architectures for which the planner can construct a mechanism satisfying independence, ex post incentive compatibility, and efficiency. In the first architecture, the social network is bipartite (which is equivalent to the comparison network being disconnected). We use the bipartite structure to partition the set of individuals, so that individuals in one group rank individuals in the other group, and individuals are ranked across groups in an arbitrary way. In the second architecture, all links form triangles, and we can construct a mechanism exploiting the fact that any unsupported report is surrounded by supported links. However, we also note that there exist network architectures for which mechanisms satisfying all three properties named above cannot be constructed. The simplest example is a network of four individuals with one triangle and one additional link. We then highlight three aspects of our findings using social network data from villages in Karnataka, India, and neighborhoods in Indonesia. First, information, as measured by the share of unique comparisons the planner receives, depends on network structure. Two social networks of similar density may provide very different levels of information. Second, we decompose comparisons into those within a triangle, those across triangles, and a remainder. Low-density networks have a large share of across-triangle comparisons. Third, we simulate the process of capping the number of comparisons each individual provides. If the cap is small relative to the community size, the capped network is close to the upper bound in information (as measured by the number of unique comparisons). Finally, we consider different variants and robustness checks of our model. We show that dominant-strategy implementation is too strong, leading to an impossibility result in triangles. We analyze the robustness of our mechanism to joint deviations by groups. We study whether coarser rankings are easier or harder to implement than complete rankings. We study the impact of homophily. If individuals of similar characteristics are more likely to form friendships, the planner is more likely to extract the necessary and sufficient friend-based comparisons to find the complete ranking. This relationship is reversed when the probability of across-group links is close to zero. In practical terms, our analysis points to two important facts. First, it shows that it may be useful to partition the set of individuals into different groups and ask individuals in one group to rank those in another. For example, one may want to let men rank women and women rank men in a community. This procedure will result in a truthful and efficient ranking, but the price to be paid is that interrankings among individuals in the two groups will be arbitrary. Second, our analysis highlights the importance of triangles. Truthful and efficient comparisons will be possible if all links form triangles, suggesting that friend-based ranking should be used only in societies with high clustering. As high clustering is also associated with high density and low average distances, we conclude that friend-based ranking should be used only in communities with dense social networks with low diameters. Finally, our analysis can be used to help design review systems in peer-selection problems. It suggests that using asymmetric networks of observation, with central reviewers observing a large number of projects, may be a way to construct efficient and incentive-compatible peer-review systems. Literature Review We first discuss the relationship of our paper with the literature on community-based ranking in development economics. This literature documents experiments in which members of a community are gathered to collectively agree on a ranking in order to identify the poorest or the most able individuals. Rai (2002) discusses the indi-vidual incentives to lie in poverty targeting. Alatas, Banerjee, Hanna, Olken, and Tobias (2012) and Alatas, Banerjee, Chandrasekhar, Hanna, and Olken (2016) report on an experiment in Indonesia in which community members were asked to collectively identify recipients of benefits of social programs. They compare the accuracy of community-based targeting with traditional proxy-means testing, and argue that community-based targeting results in consensus, and brings higher satisfaction to all members of the community. Hussam, Rigol, and Roth (2017) report on a recent field experiment in Maharashtra, India where entrepreneurs were asked to rank their peers according to the profitability of their businesses. They prove that this is a more accurate method of ranking than using observable information about the entrepreneurs. We see friend-based ranking as a complement to targeting methods that are currently popular such as proxy-means tests. Although widely used, the proxy means test has been shown to perform only slightly better than universal transfers at reducing poverty (Brown, Ravallion, and Van de Walle, 2016) and to lack adjustment to transitory shocks (Coady, Grosh, and Hoddinott, 2004). The theoretical analysis of the paper is closely related to the limited literature in computer science and social choice theory studying peer selection. Alon, Fischer, Procaccia, and Tennenholtz (2011) analyze the design of mechanisms to select a group of k individuals among their peers. Alon, Fischer, Procaccia, and Tennenholtz (2011) prove a strong negative result: no deterministic efficient strategy-proof mechanism exists. Approximately efficient, stochastic, impartial mechanisms can be constructed, which are based on the random partition of individuals into clusters of fixed size such that individuals inside a cluster rank individuals outside the cluster. Holzman and Moulin (2013) analyze impartial voting rules when individuals nominate a single individual for office. They identify a class of desirable voting rules as two-step mechanisms, by which voters are first partitioned into districts which elect local winners, who then compete against one another to select the global winner. However, Holzman and Moulin (2013) also highlight a number of impossibility results, showing that there is no impartial voting procedure which treats voters symmetrically, nor any impartial voting procedure which guarantees (i) that an individual whom nobody considers best will be elected and (ii) that an individual whom everybody considers best will always be elected. Kurokawa, Lev, Morgenstern, and Procaccia (2015) and Aziz, Lev, Mattei, Rosenschein, and Walsh (2016) improve on the partition algorithm proposed in Alon, Fischer, Procaccia, and Tennenholtz (2011). They consider a more general setting, inspired by the new peer-review system instituted by the National Science Foundation to fund the Sensors and Sensing System program. Kurokawa, Lev, Morgenstern, and Procaccia (2015) propose the "credible subset mechanism," a process which first identifies candidates who are likely to win, and assigns ratings only to these candidates. Aziz, Lev, Mattei, Rosenschein, and Walsh (2016) propose a mechanism combining the insights of the partition mechanism of Alon, Fischer, Procaccia, and Tennenholtz (2011) with the impartial "divide the dollar" mechanism of De Clippel, Moulin, and Tideman (2008). Our model departs from all these models of peer selection in a number of ways. First, we consider ordinal rather than cardinal information as inputs to the mechanism. In our model, individuals do not assign grades to other individuals, but can only make bilateral comparisons. Second, we consider as output a complete ranking of individuals rather than a coarse ranking into two sets of acceptable and non acceptable candidates. (However, in Section 6, we also consider coarser rankings as a possible extension of our model.) Third, because dominant-strategy mechanisms do not exist, we weaken the incentive requirement to ex post implementation, thereby obtaining positive results which differ from the results obtained in the peer-selection literature. Fourth, and most importantly, we do not assume a specific assignment of proposals to reviewers, but consider an arbitrary network of observations captured by a social network. Our main objective is then to characterize those social networks (or structures of observability) for which mechanisms satisfying desirable properties can be constructed. The paper which is probably the more closely connected to ours is a recent paper by Baumann (2017) which analyzes network structures for which it is possible to identify the individual with the highest characteristic. Baumann (2017) constructs a specific multitier mechanism identifying the top individual from the reports of his neighbors. The mechanism admits multiple equilibria, but there are some social network architectures (e.g., the star) for which all equilibria result in the identification of the top individual. Our paper differs from Baumann's, however, in many dimensions. First, we consider an ordinal rather than a cardinal setting, giving rise to the possibility of incompleteness of the social ranking. Second, we assume that the objective of the planner is to rank all individuals rather than identify the top individual. Third, we do not assume an exogenous bound on the way in which individuals can misreport, in contrast to Baumann (2017), in which this exogenous bound plays a crucial role in the construction of equilibria. Model Individuals and communities We consider a community N of n individuals indexed by i = 1, 2, ..., n. Each individual i has a characteristic θ i ∈ R. Examples of θ i include wealth, aptitude for a job, or quality of a project. Characteristics are privately known and are drawn from a nonatomic continuous distribution F . Members of the community are linked by a connected, undirected graph g. The social network g is common knowledge among the individuals and the planner. The characteristic of individual i, θ i , can be observed by individual i and by all his direct neighbors in the social network g. We suppose that individuals cannot provide an accurate value for the characteristic θ i . Either the characteristic cannot be measured precisely, or individuals do not have the ability or the language to quantify θ i precisely. Instead, we assume that individuals possess ordinal information and are able to compare the characteristics of two individuals. For any individual i and any pair of individuals (j, k) that individual i can observe, we let t i jk = 1 if individual i observes that θ j > θ k , and t i jk = −1 if individual i observes that θ j < θ k . The ordinal comparison is assumed to be perfect: individual i always perfectly observes whether individual j's characteristic is higher than that of individual k. Given that the characteristics are drawn from a nonatomic continuous distribution, we ignore situations in which the two characteristics are equal. Individual i's information (and type) can thus be summarized by a matrix T i = [t i jk ], where t i jk ∈ {−1, 0, 1} and t i jk = 0 if and only if i observes the comparison between j and k, namely either i = j or i = k or g ij g ik = 1. When i = j or i = k, we call the comparison t i jk a self-comparison. When g ij g ik = 1, we call the comparison t i jk a friend-based comparison. The vector T = (T 1 , .., T n ) describes the information possessed by the community on the ranking of the characteristics of all the individuals. Obviously, because individual observations are perfectly correlated, individual types T i and T j will be correlated if there exists a pair of individuals (k, l) such that t i kl = 0 and t j kl = 0. Hence, if the planner could construct a punishment for contradictory reports, as in Crémer and McLean (1985), she would be able to induce the individuals to report their true type. However, we rule out arbitrary punishments. The information contained in the vector T = (T 1 , .., T n ) results in a partial ranking of the characteristics of the individuals, which we denote by . We let i T j if the information contained in T allows us to conclude that θ i > θ j . For a fixed social network g, the information contained in the vector T = (T 1 , .., T n ) may not be the same for different realizations of (θ 1 , .., θ n ). This is due to the fact that (i) new comparisons can be obtained by transitivity but (ii) the transitive closure of an order relation depends on the initial order relation. To illustrate this point, consider four individuals i = 1, 2, 3, 4 organized in a line as in Figure 1 If θ 1 < θ 2 < θ 3 < θ 4 , then given that t 1 12 = t 2 12 = −1, t 2 23 = t 3 23 = −1, t 3 34 = t 4 34 = −1, t 2 13 = −1, and t 3 24 = −1, the comparisons result in a complete ranking 1 ≺ 2 ≺ 3 ≺ 4. However, for other possible realizations of (θ 1 , θ 2 , θ 3 , θ 4 ), the ranking generated by the types T may be incomplete. For example, if θ 1 < θ 4 < θ 2 < θ 3 , we obtain 1 ≺ 2 ≺ 3 and 4 ≺ 2 ≺ 3, but 1 and 4 cannot be compared. A social network g is called completely informative if, for any realization of the characteristics (θ 1 , .., θ n ), the information contained in T results in a complete ranking of the members of the community. The following lemma characterizes completely informative social networks. Lemma 1. A social network g is completely informative if and only if, for any pair of individuals (i, j) either g ij = 1 or there exists an individual k such that g ik g jk = 1. A social network is completely informative if and only if every pair of individuals can be compared either by self-comparisons or by friend-based comparisons. Planner and mechanism design The objective of the planner is to construct a ranking of individuals according to the value of the characteristic θ i . For example, a charity wishes to rank potential beneficiaries by need, an employer wants to rank workers according to their ability, a bank wants to rank projects according to their profitability. We let ρ denote the complete order chosen by the planner. The set of all complete orders is denoted by P. The rank of individual i is denoted by ρ i . The planner wishes to construct a ranking as close as possible to the true ranking of the values of the characteristic θ i . We do not specify the preferences of the planner. In the ordinal setting that we consider, different measures of distances between rankings can be constructed. Instead of describing explicitly the loss function associated with differences in rankings, we focus attention on efficient mechanisms. Efficiency requires that the ranking ρ coincides with the ranking generated by T for any pair of individuals (i, j) who can be compared under T. Individuals care only about their rank ρ i and have strict preferences over ρ i . By convention, individuals prefer higher values of the ranking. Hence, ρ i is preferred to ρ i if and only if ρ i > ρ i . In particular, we assume that there are no externalities in the community, and thus, individuals do not derive any reward from high rankings of friends or low rankings of foes. A direct mechanism associates to any vector of reported matrices T ∈ T n a complete ranking ρ ∈ P. We impose the following two conditions on the mechanism: Ex post incentive compatibility. For any individual i, for any vector of types T = (T i , T −1 ), any type T i , the following holds ρ i (T) ≥ ρ i (T i , T −i ). Ex post efficiency. For any vector of types T, and for any pair of individuals i and j, the following holds if i T j, then ρ i (T) > ρ j (T). We focus on ex post implementation for two reasons. First, because we consider an ordinal setting, we select a robust implementation concept which does not depend on the distribution of types. Second, as we show in section 6, the alternative robust implementation concept-dominant-strategy implementation-is too strong for our setting. Ex post efficiency requires that the planner's ranking coincide with the true ranking of characteristics in a very weak sense. Whenever two individuals i and j can be ranked using the information contained in T, the ranking ρ i must be consistent with the ranking between i and j. As the order relation induced by T, T , may be very incomplete, the requirement may be very weak. The ranking ρ must be a completion of the ranking T . If T is a very small subset of N 2 , the ranking ρ may end up being very different from the true ranking of the values of the characteristic θ i . However as the true ranking of characteristics cannot be constructed using the local information from the social network, the difference between ρ and the true ranking should not be a matter of concern, since the planner chooses an efficient mechanism given the information available to the community. Completely informative rankings 3.1 Importance of common friends We first analyze conditions under which an ex post incentive-compatible and efficient mechanism can be constructed when the information available in the community results in a complete ranking. By Lemma 1, all pairs of individuals must either be directly connected, or observed by a third individual. The next theorem shows that for an ex post incentive-compatible and efficient mechanism to exist, all pairs of individuals must be observed by a third individual. Theorem 1. Suppose that the social network g is completely informative. An ex post incentive-compatible and efficient mechanism exists if and only if, for all pair of individuals (i, j), there exists a third individual k who observes both i and j, i.e., g ik g jk = 1. Theorem 1 shows that an ex post incentive-compatible and efficient mechanism exists in completely informative communities if and only if every pair of individuals (i, j) has a common friend k. Self-comparisons cannot be used. Every comparison requires the presence of a third party. If the two individuals i and j are connected, the link ij must be 'supported' by a third individual, following the terminology of Jackson, Rodriguez-Barraquer, and Tan (2012). The intuition underlying Theorem 1 is easy to grasp. If the comparison between θ i and θ j can be reported only by i and j, in an ex post efficient mechanism, one of them has an incentive to lie. Consider a ranking which places i and j as the two individuals with the lowest characteristics in the community. If both announce that θ i is smaller than θ j , then ρ i = 1, ρ j = 2. Similarly, if both announce that θ j is smaller than θ i , then ρ j = 1, ρ i = 2. But by incentive compatibility, neither of the individuals can improve his rank by changing his report on t ij . Hence i must still be ranked at position 1 when she announces θ i > θ j and j announces θ i < θ j , and similarly individual j must still be ranked at position 1 when she announces θ j > θ i and individual i announces θ i > θ j . As two individuals cannot occupy the same position in the ranking, this contradiction shows that there is no ex post incentive-compatible and efficient mechanism relying on self-reports. Notice that this impossibility result stems from the fact that the planner has a very small number of outcomes at her disposal. If she could impose any arbitrary punishment (for example by excluding all individuals who provide inconsistent reports), she could implement an ex post efficient mechanism in dominant strategies, as in Crémer and McLean (1985), for any network architecture. The construction of an ex post incentive-compatible and efficient mechanism when all links are supported is very intuitive. First consider a comparison between i and j which is observed by at least three individuals. The mechanism disregards the report of any individual who deviates from the reports of all other individuals. Hence no individual can unilaterally change the outcome of the mechanism when all other individuals report the truth. Next suppose that the comparison between i and j is dictated by a third party, a common friend k of i and j. A change in reports could not improve the rank of k given that all other individuals tell the truth and that the social network is completely informative. If the change in report creates an inconsistency in the ranking, the planner can detect if a single individual has cheated and punish him by ranking him at the worst position in the ranking. If the change in report does not create a violation in transitivity, because the social network is completely informative, the rank of individual k is fully determined by the reports of other individuals in the community. The rank of individual k is fixed and no change in report can improve the position of individual k in the ranking. This "friend-based" ranking mechanism is ex post incentive-compatible and efficient. Theorem 1 characterizes communities for which friend-based ranking mechanisms can be constructed. Clearly the complete network satisfies the conditions. However, the condition is also satisfied by many other social networks, which are less dense than the complete network. Our next result characterizes the sparsest networks for which the condition of Theorem 1 holds. This characterization is based on the "friendship theorem" of Erdős, Rényi, and Sós (1966). Theorem 2. (The "friendship theorem") If G is a graph of order n in which any two vertices i and j have one neighbor in common, then n = 2m + 1 and G contains m triangles which are connected at a common vertex. The "friendship theorem", initially stated and proved in Erdős, Rényi, and Sós (1966), asserts that in any community where every pair of individuals has exactly one friend in common, one individual is friends with everyone and is the common friend of all other individuals. 1 The "friendship graph" is illustrated in Figure 2 for n = 7. For obvious reasons, it is also called the "windmill graph". The friendship graph has exactly 3m edges. Our next theorem shows that this is actually the smallest number of edges for which a completely informative mechanism can be constructed when n is odd. When n is even, the graph which minimizes the number of edges is a variation of the friendship graph, where one of the sails of the windmill contains three vertices, as illustrated in Figure 2 for n = 8. (i) n = 7. For odd number of nodes, the windmill is also called a friendship graph. (ii) n = 8. For even number of nodes the windmill is modified and one sail has three nodes. Figure 2: Windmill graphs Theorem 3. Suppose that n ≥ 3. Let g be a social network for which friend-based ranking generates a complete ranking. Then g must contain at least 3n 2 −1 links if n is even and 3(n−1) 2 links is n is odd. If n is odd, the unique sparsest network architecture is the friendship network. If n is even, the unique sparsest network architecture is a modified windmill network where one of the sails contains three nodes i, j, k such that i, j and k are connected to the hub, i is connected to j and j is connected to k. Theorem 3 establishes a lower bound on the number of edges needed to obtain a complete ranking of the community. It also identifies the unique network architecture which reaches this lower bound: a windmill network where one of the nodes, the hub, connects all other nodes which form pairs. 2 This network architecture implies a very unequal distribution of degrees. The hub is connected to all nodes, whereas the remaining nodes have degree two or three. If agents have a limited capacity to compare other agents, the windmill network cannot be used, and one needs to resort to other more symmetric network architectures involving a larger number of links. An exact characterization of the minimal degree of a regular network for which all links can be supported remains an open question in graph theory. 3 4 Incomplete rankings and friend-based comparisons Comparison networks We now consider communities where the condition of Theorem 1 fails. The condition fails either when the community is completely informative but some comparisons are based only on self-reports or because the community is not completely informative. In the latter case, there exist some type profiles T for which individuals collectively cannot construct a complete ranking. We let i T j denote the fact that i and j cannot be compared using the information contained in T. As the mechanism ρ defines a complete ranking, it must choose an arbitrary ranking between i and j at T. We first define a condition on the mechanism ρ guaranteeing that the arbitrary ranking chosen between i and j is independent of the reported type profile T. Independence: The ranking ρ satisfies independence if for any two type profiles T and T such that i T j and i T j, then ρ i (T) > ρ j (T) ⇔ ρ i (T ) > ρ j (T ). When the independence condition is satisfied, if the comparison between some pair of individuals i and j relies on self-reports, the mechanism ρ cannot simultaneously satisfy ex post incentive compatibility and efficiency. Proposition 1. Suppose that there exists a pair of individuals (i, j) such that g ij = 1 but there is no individual k such that g ik g jk = 1. Then there exists no mechanism satisfying independence, ex post incentive compatibility and efficiency. Proposition 1 extends the necessity argument of Theorem 1 to show that the planner cannot construct an ex post incentive-compatible and efficient mechanism when two individuals provide self-reports. Hence truthful comparisons based on selfreports cannot be elicited by the planner. We thus ignore comparisons based on self-reports. We now modify the type of individual i, T i by removing any comparison t i ij which is not supported by a third individual, i.e., we let t i ij = 0 if there exists no k = i, j such that t k ij = 0. We search for mechanisms satisfying independence, ex post incentive compatibility and efficiency in the community where self-comparisons are ignored. Second, we construct a comparison network h which captures all comparisons that can be obtained using friend-based comparisons. Formally, we let h ij = 1 if and only if there exists k such that g ik g jk = 1. The network h collects all pairs of individuals which can be compared by a third individual. It differs from the social network g in two ways: (i) pairs of individuals which are linked in g but do not have a common friend appear in g but not in h, (ii) pairs of individuals which are not directly linked in g but have a common friend appear in h but not in g. Figure 3 illustrates a social network g and the corresponding comparison network h. The comparison network h is the minimal set of comparisons that the planner can guarantee for any possible realization of the characteristics. The planner complements the comparisons contained in h by taking their transitive closure. If the conditions of Theorem 1 hold, the comparison network h is the complete network. When the conditions fail, we characterize comparison networks which can be supported by a mechanism satisfying independence, ex post incentive compatibility and efficiency. Connected comparison networks and bipartite social networks We first provide a characterization of social networks which generate connected comparison networks. Proposition 2. Suppose that n ≥ 3. The comparison network h is connected if and only if g is not bipartite. Proposition 2 establishes that the network h is connected if and only if the social network g is not bipartite. If the network g is not bipartite, we construct a path connecting any pair of individuals i and j in the comparison network. If the network g is bipartite, and the nodes partitioned into the two sets A and B, the comparison network h is disconnected into two components: individuals in A rank individuals in B and individuals in B rank individuals in A. Individuals can be ranked inside the two sets A and B but rankings of individuals across the two sets must be arbitrary. Notice however that an individual in A cannot improve his ranking by lying about the ranking of individuals in B. Hence, when the social network g is bipartite (and the comparison network h disconnected), it is easy to construct a mechanism satisfying independence, ex post incentive compatibility and efficiency. Proposition 3. Suppose that the social network g is bipartite with two sets of nodes A and B. Then there exists a mechanism satisfying independence, ex post incentive compatibility and efficiency, which generates a ranking which coincides with the comparison network h on its two components A and B. Proposition 3 characterizes one situation where the planner can elicit information about comparisons: when the set of individuals in the community can be partitioned into two subsets where members of one subset observe members of the other subset. For example, one could survey separately men and women and ask men about the characteristics of women and women about the characteristics of men. However, this design would not allow the planner to obtain information about the ranking of individuals across the two sets. The mechanism completes the partial ranking by an arbitrary ranking across individuals in the two sets, possibly resulting in a final ranking which is very different from the true ranking. We observe that the partition of the set of nodes into groups which rank each other is the basis of most algorithms proposed in the computer science literature on peer selection. Social quilts in incomplete communities We now consider communities for which the comparison network h is connected but not complete. We provide a sufficient condition under which the planner can construct a mechanism satisfying independence, ex post incentive compatibility and efficiency. The mechanism is an extension of the mechanism constructed in the proof of Proposition 1 for completely informative communities. Proposition 4. Suppose that all links in g are supported (for all i, j such that g ij = 1, there exists a k such that g ik g jk = 1). Then there exists a mechanism satisfying independence, ex post incentive compatibility and efficiency. Proposition 4 identifies social networks which allow the planner to construct an incomplete ranking of the individuals: all links must be supported and the social network is thus formed of a collection of triangles. Following Jackson, Rodriguez-Barraquer, and Tan (2012), we call these communities "social quilts". Figure 4 illustrates one of these networks. Notice that some comparisons are supported as links within the triangles, and other comparisons are supported as links across triangles. Links across traingles do not play a role in Jackson, Rodriguez-Barraquer, and Tan's (2012) favor exchange context. Whether there exist other social networks g generating connected comparison networks h for which the planner can construct a mechanism satisfying independence, ex post incentive compatibility and ex post efficiency remains an open question. However, there are social networks for which the planner will not be able to construct a mechanism satisfying these three properties, as shown in the following example. In this example, the links (i, j), (i, k), (j, k) are supported, but the link (i, l) is not supported. Consider a realization of the characteristics such that θ l > θ j > θ i > θ k . If individual i announces θ l > θ j > θ k , by ex post efficiency, the planner constructs the rankings k, i, j, l and the rank of individual i must be equal to 2. If on the other hand individual i announces θ j > θ k > θ l , the planner constructs the ranking l, k, i, j and the ranking of individual i is now equal to 3. Hence individual i has an incentive to lie and announce θ j > θ k > θ l . In Example 1, the planner's ranking of i depends on his announcement on the rankings (j, l) and (k, l). Given that θ j > θ i > θ k , and that (i, j, k) form a triangle, the planner must rank i between j and k. Hence she cannot rank all three individuals j, k and l on the same side of individual i, as in the mechanism constructed in the proof of Proposition 4. But then, the announcement of individual i on (j, k, l), by changing the rank of l with respect to j and k, will also affect the ranking of individual i. Because individual i can manipulate his rank by his announcements on the unsupported links (j, k) and (j, l), there is no mechanism satisfying ex post incentive compatibility and efficiency in this community. Real-life social networks In the two previous sections, we analyzed conditions on social networks under which the planner can construct rankings that satisfy ex post incentive compatibility and efficiency. We use real social network data from India (Banerjee, Chandrasekhar, Duflo, and Jackson, 2013) and Indonesia (Alatas, Banerjee, Chandrasekhar, Hanna, and Olken, 2016) to highlight three implications of our theoretical results. We first observe that the information obtained in a social network does not depend only on the number of links. For a given density of the social network, we witness a large variation in the information obtained by friend-based ranking, depending on the exact structure of the network. Second, we analyze the role played in friend-based comparisons by supported links and links across triangles. For low-density networks, we show that links across triangles provide the majority of the friend-based comparisons. Third, we analyze how cognitive limitations affect the number of comparisons elicited by the planner. We cap the number of comparisons per individual and observe that when the cap is small relative to the community size, the information loss due to the cap is also small. The data from India and Indonesia are particularly useful because they contain multiple independent networks: 75 villages from Karnataka, India, and 622 neighborhoods from three provinces in Indonesia. Indonesian networks are smaller and denser than the Indian networks. We focus on the giant component of each network. Table 1 provides summary statistics of the networks. We report the mean, minimum, and maximum for each measure. The combined sample of networks provides a large range in network size and structure. We measure information using the density of the comparison network, which is simply the count of unique comparisons as a share of the n(n−1) 2 possible comparisons. Figure 7 provides further detail on the distribution of network characteristics. Notice that the Indonesian networks are denser and more clustered, and display shorter average distances than the Indian networks. As a result, the Indonesian networks contain more information, i.e., result in denser comparison networks. Figure 7 shows a tight relationship between average distance and the quantity of information. Since every comparison (i, j) is provided by a path of length 2 between i and j, this relationship is not surprising. In contrast to the relation between information and average distance, the relationship between information and density is not tight. In the following section we use an example to highlight two reasons for the variation in the quantity of information at a given density. Large variation in the quantity of information for a given density Dense social networks provide many comparisons, but density is not a good proxy for the quantity of information that the planner can extract from a social network. The windmill of Theorem 3 is completely informative but its density is only 3 n . This insight also applies to the data here. In the bottom left panel of Figure 8, we plot the density against the quantity of information for social networks of more than 50 households. We highlight two networks and plot their corresponding network diagrams. The orange network, from India, has 75 nodes and a density of 0.12. The green network, from Indonesia, has 69 nodes and the same density of 0.12. Despite having equal density, the orange network provides a quantity of information of 0.62nearly double the green network with 0.33. Two factors contribute to the greater amount of information in the orange network. First, as shown in the top right panel of Figure 8, the degree distribution of the orange is spread more widely than the green. The number of comparisons provided by a single node is a convex function of degree. For a given density, the greater the spread in the degree distribution, the more comparisons the social network provides. Second, the green network is a combination of cliques that are weakly connected to each other. Cliques repeat comparisons. Take a clique of seven nodes as an example. These seven nodes provide 105 comparisons yet 84 of these comparisons are repeated ones. Since the green and orange networks have a similar number of nodes and equal density, they each produce a total number of comparisons similar to the other's. The difference is that a greater share of the green network's comparisons are repeated. This example shows that the success of friend-based ranking depends not only on the number of links, but, more importantly, on how those links are structured. Decomposition of information Proposition 4 (in Section 4) shows the importance of triangles in constructing incentivecompatible and efficient mechanisms. Both the links within and across triangles are used to obtain truthful comparisons. For a given network we can decompose information into comparisons provided within and across triangles. We approach the decomposition by removing all unsupported links from the network and recalculating the quantity of information. The resulting supported network is incentive-compatible but information is reduced-from 0.37 to 0.27 on average for India and from 0.78 to 0.75 on average for Indonesia. A greater share of the Indonesian links is supported, which is due to the fact that those networks are denser and more clustered. We decompose the comparisons in the supported network into those within and across triangles. If a comparison appears both within and across a triangle, we categorize the comparison as "within." Figure 9 shows the decomposition for 50 Indian and 50 Indonesian networks. Each bar corresponds to a network and the bar is split between within triangles, across triangles, and a remainder (i.e., the comparisons which appear only in the unsupported network). At lower densities the majority of comparisons are provided across triangles, while as density of the social network increases, the share of comparisons within triangles increases. Capping comparisons For a given degree distribution, we can define a simple upper bound on the number of bilateral comparisons. Since we measure information by counting links in the comparison network, an upper bound on information is reached when none of the bilateral comparisons produced in the social network is repeated. With the degree of individual i denoted as d i , the upper bound is n i=1 d i (d i −1) n(n−1 , which is simplified to d(d−1) n−1 for regular networks of degree d. To analyze the effect of capping the degree of individuals on the quantity of information, we use simulations on the social network data. For each individual i, we randomly pick five friends whom i will compare to each other. The resulting network is directed. Suppose j picks i and i has more than five friends. There is no guarantee that i will also pick j. Figure 10 uses the mean from 100 iterations to measure the number of nonrepeated comparisons in the capped network. The standard deviation is less than .01 for any given network in our sample. This variation depends on the starting network. The capped information is close to the upper bound. In Figure 10 we contrast the upper bound to the mean information provided by the capped networks. Each bar represents a network capped at degree 5. When the cap is small relative to the community size (i.e., the number of households), only a small share of the comparisons in the capped network is repeated, so that the capped information is close to the upper bound. Robustness and extensions The analysis of friend-based ranking mechanisms relies on specific assumptions on the model. In this section, we relax some of these assumptions to test the robustness of our results. Dominant strategy implementation We first strengthen the incentive compatibility requirement to dominant-strategy implementation. The following proposition shows that dominant-strategy implementation is too strong in our setting. The outcome set is not rich enough to permit the construction of strategy-proof mechanisms. We first recall the definition of strategyproofness: Strategy-proofness. For any individual i, for any vector of announcements (T −i ) and any types T i , T i , ρ i (T i ,T −i ) ≥ ρ i (T i ,T −i ). Proposition 5. Let g be a triangle. There exists no mechanism satisfying strategyproofness and ex post efficiency. Proposition 5 is an impossibility result, highlighting a conflict between strategyproofness and efficiency in a very simple network architecture. As shown in the proof, the impossibility stems from the coarseness of the outcome space, which limits the power of the planner. There are only three possible outcomes corresponding to the three possible ranks. Strategy-proofness imposes a large number of constraints on the mechanism. We show, using a combinatorial argument, that if all the constraints are satisfied, two individuals must be occupying the same rank for some vectors of announcement. Hence, it is impossible to elicit truthful information in a complete network with three individuals. 4 Coarse rankings We next relax the assumption that the planner chooses a complete ranking and that individuals have strict preferences over ranks. We consider a setting where the planner selects only broad indifference classes. This is the typical situation in which the planner selects a set of recipients of the benefits of social programs, or of research funds. If the planner only chooses broad categories, she might be able to construct ex post incentive-compatible and efficient mechanisms even if self-reports are not supported by a third individual. The intuition is immediate: if there exist two "worst spots" in the ranking, the planner can punish individuals who send conflicting selfreports by placing both of them on the worst spot. We formalize this intuition in the following proposition. Proposition 6 thus shows that it is easier to construct ex post incentive-compatible and efficient mechanisms when the planner does not construct a complete ranking of the individuals. This observation raises new possibilities. It may be possible to construct strategy-proof and efficient mechanisms when the planner only assigns agents to broad categories. Group incentive compatibility We now allow for individuals to jointly deviate from truth-telling. We let individuals coordinate their reports and jointly misreport their types. Consider a triangle with three individuals. Each individual reports on the three links. The mechanism that we constructed in Theorem 1 of Section 3 assigns a ranking ρ(i) > ρ(j) when at least two of the individuals report that i is higher than j. This creates an incentive for any pair of individuals to misrepresent their types. For example, if the true ranking is θ 3 > θ 2 > θ 1 , individuals 1 and 2 have an incentive to misreport and announce that 2 is higher than 1, and 1 is higher than 3, so that in the end, ρ(2) = 3 > 2 and ρ(1) = 2 > 1. tion. This intuition can be exploited to show that there does not exist any mechanism satisfying ex post group-incentive compatibility and efficiency when n = 3. We first provide a formal definition of ex post group-incentive compatibility: Ex post group incentive compatibility. For any vector of types T, there does not exist a coalition S and a vector of types T S such that for all individuals i in S, ρ i (T S , T S ) ≥ ρ i (T) and ρ i (T S , T S ) > ρ i (T). for some i ∈ S. Proposition 7. Let g be a triangle. There does not exist a mechanism satisfying ex post group-incentive compatibility and efficiency. Homophily In this last extension, we analyze the effect of homophily on friend-based ranking. Using Golub and Jackson's (2012) islands model of network formation, we show that the probability of finding the full ranking of characteristics initially increases and then decreases in the homophily parameter. In a community of n individuals, n − 1 comparisons are necessary and sufficient to determine the full ranking of characteristics. The lowest is compared to the second lowest, the second to the third, and so on. We call this subset of comparisons C. We will consider a random model of network formation and ask: how likely is it that individuals form links that generate the set of bilateral comparisons C? Suppose that pairs of individuals form friendships with a given probability p (which is the Erdős-Rényi random graph model) and through friend-based ranking the realized network provides a set of comparisons T . What is the probability Pr[C ∈ T ]? For simplicity, we order the individuals in the community {1, 2, ..., n − 1, n} so that the private characteristic θ i > θ j if and only if i > j. We then define C = {(1, 2), (2, 3), ..., (n − 2, n − 1), (n − 1, n)}. Consider a pair (i, j). Pr[(i, j) ∈ T ] = Pr[∃ k = i, j : g ik = g jk = 1] = 1 − (1 − p 2 ) n−2 . Since there are n − 1 pairs in C, Pr[C ∈ T ] = 1 − (1 − p 2 ) n−2 n−1 . We now divide the community into two groups of equal size, N L for low and N H for high such that θ l < θ h ∀ l ∈ N L , h ∈ N H . Individuals form friendships within their group with probability p w and outside of their group with probability p o (which is Golub and Jackson's (2012) islands model with 2 groups). For p w ≥ p o , the gap between p w and p o is a measure of homophily. Keeping p o constant, if we increase p w , Pr[C ∈ T ] will increase since we have raised the expected density of the social network. We need to keep the expected density of the network constant to isolate the impact of homophily. Since we disregard selfcomparisons by Proposition 1, there are more outside group links than within group links. The number of outside group links is ( n 2 ) 2 while the number of within group links is ( n 2 ) 2 − n 2 . The ratio simplifies to n − 2 within group links for every n outside group links. Starting from a zero homophily base of p = p w = p o , we can analyze the impact of homophily by increasing p w and decreasing p o to keep the expected number of links constant. Let p w = p + η, where η is the homophily parameter. To keep the number of links constant, p o = p − η n n−2 . From a base of η = 0 we can increase η and observe how Pr[C ∈ T ] responds. This is represented graphically in Figure 6 for a community size n = 250 and a base probability of friendship p = 0.15. Along the horizontal axis, as η increases from 0 to around 0.1 the probability that the realized comparisons contain the comparisons needed to derive the full ranking increases. The intuition is simple. As homophily increases the probability that some individual k is friends with two individuals in the same group increases while the probability that k is friends with two individuals in different groups decreases. Since nearly all of the comparisons in C are pairs within the same group, Pr[C ∈ T ] rises with homophily. However, when the probability of outside group friendships p o approaches zero, Pr[C ∈ T ] approaches zero since there is little chance that the comparison of the highest in N L to the lowest in N H is in T . In Figure 6, Pr[C ∈ T ] drops sharply as η is above 0.12 and p o approaches zero. Low levels of homophily improve friendbased ranking whereas extreme homophily reduces the power of friend-based ranking as agents across groups cannot be compared. Conclusion This paper analyzes the design of mechanisms to rank individuals in communities in which individuals have only local, ordinal information on the characteristics of their neighbors. In these communities, pooling the information of all individuals may not be sufficient to obtain a complete ranking, and so we distinguish between completely informative communities and communities where only incomplete social rankings can be obtained. In completely informative communities, we show that the planner can construct an ex post incentive-compatible and ex post efficient mechanism if and only if each pair of individuals is observed by a third individual, i.e., the individuals in each pair have a common friend. We use this insight to characterize the sparsest social network for which a complete ranking exists as constituting a "friendship network" (or "windmill network") in the sense of Erdős, Rényi, and Sós (1966). When the social network is not completely informative, we show that any selfreport which is not supported by a third party must be discarded. We provide two sufficient conditions on the social network under which an ex post incentivecompatible and ex post efficient mechanism may be constructed. First, in bipartite networks, individuals on one side of the network can be used to rank individuals on the other side, resulting in an ex post efficient but incomplete ranking. Second, in "social quilts," where all links are supported in triangles, the planner can use the congruence of reports to construct truthful rankings over any pair of individuals. We use data on social networks from India and Indonesia to illustrate the results of the theoretical analysis. We measure information provided by the social network as the share of unique comparisons which can be obtained by friend-based comparisons (which corresponds to the density of the comparison network) and show that (i) information varies greatly even for a given density, (ii) across-triangle comparisons are important at low densities, and (iii) information is close to an upper bound when the degree is capped at a small value relative to the community size. Finally, we discuss robustness and extensions of the model, focusing on strategyproofness, group-incentive compatibility, coarse rankings, and homophily. To the best of our knowledge, this is the first paper to analyze this intriguing theoretical problem-the design of a mechanism constructing a complete ranking when individuals have local, ordinal information based on a social network. In future work, we would like to further our understanding of the problem, by considering in more detail the difference between ordinal and cardinal information, between complete and coarse rankings, and between different concepts of implementation. We also plan to extend the empirical and policy implications of the theoretical model by analyzing specific institutional settings in more detail. Hussam, Reshmaan, Natalia Rigol, and B Roth. 2017. "Targeting high ability entrepreneurs using community information: Mechanism design in the field.", Working paper. 75 92 96 107 115 116 123 132 133 134 137 140 142 144 148 153 155 159 161 166 170 173 174 175 178 187 192 193 194 201 211 213 219 226 227 231 236 239 243 246 248 269 270 272 274 286 287 298 310 Note: Each bar represents a single network on which we simulate a cap on degree. Each individual provides comparisons of at most 5 friends, selected uniformly at random when the cap is binding. An upper bound equal to sum of unique and repeated comparisons is given as a function of the degree distribution. The split between unique and repeated comparisons is calculated as the mean after 100 iterations. A Tables C Proofs Proof of Lemma 1 The condition is obviously sufficient, as it guarantees that for any pair (i, j) there exists an individual k such that t k ij = 0, Hence the matrix generated by (T 1 , .., T n ) contains nonzero entries everywhere outside the diagonal. Conversely, suppose that there exists a pair of individuals (i, j) who is observed by no other player and such that g ij = 0. Consider a realization of the characteristics such that θ i and θ j are two consecutive values. No individual can directly compare i and j. In addition, because there is no k such that θ k ∈ (θ i , θ j ), there is no k such that θ i ≺ θ k ≺ θ j or θ j ≺ θ k ≺ θ i . Hence the social network g is not completely informative. Proof of Theorem 1 Sufficiency. Suppose that for any pair of individuals (i, j), there exists a third individual k for whom g ik = g jk = 1. We define the mechanism ρ by constructing comparisons. Let r ij denote the comparison between i and j chosen by the planner. First consider a pair of individuals (i, j) who observe each other, g ij = 1. By assumption, there are at least three reports on the ranking of i and j. If all individuals transmit the same report t ij , let r ij = t ij . If all individuals but one transmit the same report t ij and one individual reports t ij = −t ij , ignore the ranking t ij and let r ij = t ij . In all other cases, let r ij = 1 if and only if i > j. Second consider a pair of individuals (i, j) who do not observe each other, g ij = 0. By assumption, there exists at least one individual k who observes them both. If there are at least three individuals who observe i and j, use as above a mechanism such that r ij = t ij if all individuals agree on t ij or only one individual chooses t ij = −t ij , and let r ij = 1 if i > j otherwise. If one or two individuals observe i, j, pick the individual k with the highest index. Consider the vector of announcementsT −k where one disregards the announcements of individual k. Let T −k be the binary relation created by letting t ij = 1 if and only if t l ij = 1 for all l = k. If there exists a directed path of length greater or equal to 2 between i and j in T −k , and for all directed paths between i and j in T −k , i 0 , .., i L we have t i l i l+1 = 1, then r ij = 1. If on the other hand for all directed paths between i and j in T −k , t i l i l+1 = −1, then r ij = −1. In all other cases, let individual k dictate the comparison between i and j, r ij = t k ij . Now consider all comparisons r ij . If they induce a transitive binary relation on N , let ρ be the complete order generated by the comparisons. Otherwise, consider all shortest cycles generated by the binary relation r ij . If there exists a single individual i who dictates at least two comparisons in all shortest cycles, individual i is punished by setting ρ i = 1 and ρ j > ρ k if and only if j > k for all j, k = i. If this is not the case, pick the arbitrary ranking where ρ i > ρ j if and only if i > j. We now show that the mechanism ρ is ex post incentive-compatible and ex post efficient. Suppose that all individuals except k report their true type, and consider individual k's incentive to report T k = T k . On any link (i, j) such that g ij = 1, as all other individuals make the same announcement, individual k cannot change the comparison r ij by misreporting. Consider a link (i, j) such that g ij = 0 and g ik g jk = 1. If there are at least three individuals who observe i and j, individual k cannot affect the outcome. Otherwise, if there is a directed path of length greater than equal to 2 in T −k , individual k's report cannot change the ranking. If individual k is not the highest ranked individual who observes i and j, then she cannot change the comparison r ij by misreporting. Hence we only need to focus attention on pairs (i, j) such that k is the highest index individual who observes i and j and there is no directed path between i and j in T −k . Suppose that all individuals l = k announce the truth, so that T −k = T −k . We first show that individual k cannot gain by making an announcement which induces cycles in the ranking generated by the comparisons r ij . Suppose that the ranking generated by r ij exhibits cycles. We first claim that the shortest cycles must be of length 3. Suppose that there exists a cycle of length L, i 0 , i 1 , .., i L . Because the community is completely informative, the binary comparisons generated by the announcements are complete, so that for any l, m, either r i l i m = 1 or r i l i m = −1. Now consider i 0 , i 1 , i 2 . If r i 0 i 2 = −1, i 0 , i 1 i 2 i 0 forms a shortest cycle of length 3. If not, r i 0 i 2 = 1 and we can construct a cycle of length L − 1, i 0 i 2 , .., i L . By repeating this argument, we either find shortest cycles of length 3 or end up reducing the initial cycle to a cycle of length 3. Consider next a shortest cycle of ijli. We claim that individual k must dictate at least two comparisons in the cycle. First note that if k does not dictate the comparison between i and j, there must be a directed path between i and j in T −k . To see this, notice that either g ij = 1 and then i T −k j or g ij = 0 but i and j are not observed by k or are observed by k and another individual with a higher index than k, in which case i T −k j. Finally, it could be that g ij = 0, i and j are observed by k, k is the highest index individual observing i and j, but then as k does not dictate the comparison (i, j), there must exist a directed path of length 2 between i and j in T −k . Now suppose first that k does not dictate any comparison in the cycle. There must exist a directed path between i and j, j and l and l and i in T −k , a contradiction since, as all individuals tell the truth, the binary relation generated by T −k is transitive. Next suppose that k dictates a single comparison (i, j) in the cycle but not the comparisons (j, l) and (l, i). Then there exists a directed path between j and l and a directed path between l and i in T −k . Hence there exists a directed path of length greater than or equal to 2 between j and i in T −k . Furthermore, as all individuals tell the truth, for all directed paths between j and i, r ji = 1. Hence the mechanism cannot let individual k dictate the choice between i and j, yielding a contradiction. We conclude that all shortest cycles are of length 3, and that in any cycle of length 3, individual k must dictate at least two of the three comparisons. Hence the mechanism assigns ρ(k) = 1 and individual k cannot benefit from inducing a cycle. Finally suppose that all comparisons r ij result in a transitive relation so that ρ can be constructed as the complete order generated by these comparisons. We claim that the comparisons generated by T −k are sufficient to compute the rank of k. In fact, for any i = k, either g ik = 1 and as all other individual tell the truth, r ik is independent of the report t k ik , or g ik = 0 and the report on (i, k) is truthfully made by another individual l. In both cases, the information contained in T −k is sufficient to construct the comparison r ik . Hence ρ k is independent of the announcement T k , concluding the proof that the mechanism is ex post incentive-compatible. To show that the mechanism is ex post efficient notice that, when all individuals truthfully report their types, the rankings r ij induce a transitive relation, and yield the complete ranking generated by T . Necessity. Suppose that the social network g satisfies the conditions of Lemma 1 but that there exists a pair of individuals (i, j) who observe each other but are not observed by any third individual k. Consider a realization of the characteristics such that θ i and θ j are the two lowest characteristics. Let T 1 be the type profile if θ i < θ j and T 2 the type profile if θ j < θ i . By ex post efficiency, because the rankings generated by T 1 and T 2 are complete, ρ i (T 1 ) = ρ j (T 2 ) = 1, ρ i (T 2 ) = ρ j (T 1 ) = 2. Because there are only two announcements t i ij and t j ij on the link (i, j), ex post incentive compatibility requires that individuals i and j cannot improve their ranking by changing their reports on the link (i, j). Let T −ij denote the announcements on all links but link ij. We must have ρ i (T −ij , t i ij = 1, t j ij = −1) = ρ i (T 1 ) = 1, ρ j (T −ij , t i ij = 1, t j ij = −1) = ρ j (T 2 ) = 1. resulting in a contradiction as i and j cannot both be ranked at position 1. Proof of Theorem 3 We establish the Theorem through a sequence of claims. Let (g) be the number of links in the social network g. Claim 1. If the social network is completely informative, then every individual must have at least 2 friends. Proof. Let d i be the number of friends of individual i. As g is connected, d i ≥ 1 for all i ∈ N . Suppose that d i = 1, and consider the unique neighbor j of i. As d i = 1, there is no k = j which is connected to i and can draw a comparison between i and j. Hence the network g is not completely informative, establishing a contradiction. Claim 2. If for any (i, j) there exists k such that g ik g jk = 1, then (g) ≥ 3(n−1) 2 if n is odd and (g) ≥ 3n 2 − 1 if n is even. Proof. Consider the following problem: For a fixed number of links L, compute the maximal number of comparisons of neighbors that can be generated by a social network g when all nodes have degree contained in [2, n − 1]. More precisely, let (d 1 , .., d n ) denote the degree sequence of g with the understanding that d i−1 ≥ d i for all i = 1, .., n. Then consider the problem: max (d 1 ,...,dn) d 1 (d 1 − 1) 2 + d 2 (d 2 − 1) 2 + ... + d n (d n − 1) 2 subject to 2 ≤ d i ≤ n − 1 ∀i , d 1 + d 2 + ... + d n = 2L . Notice that the objective function V (d 1 , .., d n ) = d 1 (d 1 −1) 2 + d 2 (d 2 −1) 2 + ... + dn(dn−1) 2 is strictly increasing and convex in (d 1 , ..., d n ). Assume first that n is odd. Then pick L = 3(n−1) 2 and d 1 = n − 1, d 2 = .. = d n = 2. Because V is strictly convex, V (n − 1, 2, ..., 2) = (n − 1)(n − 2) 2 + n − 1 = n(n − 1) 2 > V (d 1 , ..., d n ) for any (d 1 , .., d n ) = (n − 1, 2, ...2) such that d 1 + .. + d n = 3(n − 1) and d i ≥ 2 for all i. Now n(n−1) 2 is the total number of comparisons. So, as V (d 1 , .., d n ) is strictly increasing in n, the social network g must contain at least 3(n−1) 2 links for all comparisons to be constructed. Assume next that n is even. Pick L = 3n 2 − 1 and d 1 = n − 1, d 2 = 3, d 3 = .. = d n = 2. Because V is strictly convex, V (n − 1, 3, 2, ..., 2) = (n − 1)(n − 2) 2 + 3 + n − 2 = n(n − 1) 2 + 1 > V (d 1 , ..., d n ) for any (d 1 , .., d n ) = (n − 1, 3, 2, ...2) such that d 1 + .. + d n = 3n − 2 and d i ≥ 2 for all i. In addition notice that for L = 3n 2 − 2, V (n − 2, 2, 2, ..., 2) = (n − 2)(n − 3) 2 + n − 1 = (n − 2) 2 − n 2 > V (d 1 , ..., d n ) for any (d 1 , .., d n ) = (n − 2, 2, 2, ...2) such that d 1 + .. + d n = 3n − 4 and d i ≥ 2 for all i. Hence, the maximum of V i is smaller than n(n−1) 2 when (g) = 3n 2 − 2 and greater than n(n−1) 2 when (g) = 3n 2 − 1, establishing that the social network g must contain at least 3n 2 − 1 links for all comparisons to be constructed. Next we observe that the friendship network and the modified windmill network generate all comparisons. Claim 3. If n is odd, the friendship network containing exactly 3(n−1) 2 links, generates all comparisons. If n is even, the windmill with sails of size 2 and one sail of size 3 with an additional link, containing exactly 3n 2 − 1 links, generates all comparisons. Proof. The hub of the network, node n h , provides the comparisons between all other (n − 1) nodes. If n is odd, in any petal (i, j), i provides the comparison between j and n h and j provides the comparison between i and n h . If n is even, in any sail of size 2, (i, j),i provides the comparison between j and n h and j provides the comparison between i and n h . In the unique sail of size 3, (i, j, k), i provides the comparison between j and n h , j provides the comparisons between i and n h and k and n h and k provides a (redundant) comparison between j and n h . Finally we establish that the friendship network and the modified windmill network are the only network architectures generating all comparisons with the minimal number of edges. Claim 4. If n is odd, the friendship network is the only network with degree sequence (n − 1, 2, ..., 2). If n is even, the modified windmill network with n 2 − 2 sails of size 2 and one sail of size 3 with an additional link is the only network with degree sequence (n − 1, 3, 2, ..2). Proof. Let n be odd. Because one node has degree n − 1, the network is connected and this node is a hub. All other nodes must be connected to the hub, and if they have degree 2, they must be mutually connected to one other node. Let n be even. The same argument shows that all nodes with degree 2 must be connected to the hub and one other node. These nodes are mutually connected except for the petal of size 3, where one node is connected to the two other nodes in the sail. Proof of Proposition 1 As in the necessity part of the proof of Proposition 1, consider a realization of the characteristics such that θ i and θ j are the two lowest characteristics. Fix two type profiles T and T which agree on all comparisons except that θ i < θ j in T and θ j < θ i in T . Clearly, for any k = i, j, if k T j then k T i, as i and j can be compared under T. Similarly, if k T i then k T j. Furthermore, as θ i and θ j are the two smallest characteristics, all individuals k which can be compared to i and j have higher rank than i and j. Next consider k = i, j such that k T i. Then we must also have k T j, as otherwise k T j which implies k T i. Similarly, if k T j then k T i. Hence if an individual k cannot be compared to i under T, it cannot be compared to j under T, nor to j under T nor to i under T . By ex post efficiency, for all k which can be compared to i, j, ρ k (T) > ρ j (T) > ρ i (T). Similarly, by ex post efficiency, for all k which can be compared to i, j, ρ k (T ) > ρ i (T) > ρ j (T). By independence, for all k which cannot be compared to i, j, ρ k (T) > ρ i (T) if and only if ρ k (T ) > ρ i (T ) and ρ k (T) > ρ j (T) if and only if ρ k (T ) > ρ j (T ). Hence the set of individuals who are incomparable to i, j and are ranked below i and j under T and T are identical. But this implies that ρ i (T) = ρ i (T ) − 1, ρ j (T ) = ρ j (T) − 1 As we also have ρ i (T) < ρ j (T) and ρ j (T ) < ρ i (T ), we must have ρ i (T + 2 > ρ j (T) > ρ i (T), so that ρ j (T) = ρ i (T) + 1. Hence ρ i (T) = ρ j (T ), ρ j (T) = ρ i (T ). Because there are only two announcements t i ij and t j ij on the link (i, j), ex post incentive compatibility requires that individuals i and j cannot improve their ranking by changing their reports on the link (i, j). Let T −ij denote the announcements on all links but link ij. We must have ρ i (T −ij , t i ij = 1, t j ij = −1) = ρ i (T) ρ j (T −ij , t i ij = 1, t j ij = −1) = ρ j (T ), resulting in a contradiction as i and j cannot both be ranked at the same position. Proof of Proposition 2 We first prove the following Claim. Claim 5. The comparison network is connected if and only if for all i, j ∈ N , there exists an even walk between i and j. Proof. Suppose first that h is connected. Pick any two nodes i, j ∈ N and a walk i = i 0 , ..., i m = j in h. By definition, for any (i k , i k+1 ) in the walk, there exists j k ∈ N such that i k , i k+1 ∈ N j k . But this implies that there exists a walk in g connecting i to j given by i 0 , j 0 , i 1 , j 1 , ..., i m−1 , j m−1 , i m . 5 This walk contains an even number of edges, proving necessity of the claim. Next suppose that h is not connected and let i and j be two nodes in different components of h. We want to show that all walks between i and j in g are odd. Consider first a path between i and j. If the path is even, there exists a sequence of nodes i = i 0 , i 1 , .., i m = j where m = 2l is even such that g i k ,i k+1 = 1 for all k. But then, for any l = 0, m 2 − 1, h i 2l ,i 2l+1 = 1, and hence there exists a path i = i 0 , i 2 , ..., i m = j ∈ f , contradicting the fact that i and j belong to two different components in h. Hence all paths between i and j are odd. If there exists an even walk between i and j in g, it must thus involve an odd cycle starting at i or starting at j. Without loss of generality, suppose that there exists an odd cycle starting at i, i = i 0 , ..., i m = i, where m = 2l + 1 is odd. Consider any even path between i and j, where we index i = i m , ...i r = j and r = 2p is even. We construct a path in h between i and j as follows. Because m is odd, we first construct the sequence of connected nodes in h, i = i 2l+1 , i 2l−1 , i 2l−3 , .., i 1 . Because i is connected in g both to i 1 and i m+1 , we then link i 1 to i m+1 in h. Now m + 1 is even, so we can use the path between i and j to construct a sequence i m+1 , .., i r = j in h. Concatenating the two sequences, we construct a sequence i, i m−2 , ..., i 1 , i m+1 , .., i r = j in h, contradicting the fact that i and j belong to two different components in h. Hence if h is not connected, there exists a pair of nodes i, j such that all walks between i and j are odd, proving the necessity of the claim. We now prove the second claim Claim 6. For all i, j ∈ N there exists an even walk between i and j if and only if g is not bipartite. Proof. Suppose that g is bipartite with sets A and B. As N ≥ 3, at least one of the two sets has more than one element. Pick i, j such that i ∈ A and j ∈ B, then we claim that all walks between i and j must be odd. Any walk between i and j must contain an even number of edges alternating between nodes in A and B and a single edge between a node in A and a node in B. Hence the total number of edges must be odd, proving the necessity of the claim. Conversely, suppose that there exists a pair of nodes i, j such that all walks between i and j are odd. Consider the sets of nodes A = {k\|δ(i, k) is even} and B = {k\|δ(i, k) is odd }, where δ(i, k) denotes the geodesic distance between i and k in the graph. We first claim that if k ∈ A, all walks between i and k must be even. Suppose not, then there exist two different walks between i and k, one w 1 which is even (the shortest path between i and k) and one w 2 which is odd. Pick one particular path p between k and j. If this path is odd, then the walk between i and j containing w 2 followed by p is even, contradicting the assumption. If the path is even, then the walk between i and j containing w 1 followed by p is even, contradicting the assumption again. Hence all walks between i and nodes in A are even and all walks between i and nodes in B are odd. Next notice that there cannot be any edge between nodes in A. Suppose by contradiction that there exists an edge between k and l in A, and consider a walk between i and k, w 1 followed by the edge kl. This forms an odd walk between i and l, contradicting the fact that all walks between i and l must be even. Hence, there is no edge between nodes in A and similarly no edge between nodes in B, showing that the graph g is bipartite. Proof of Proposition 3 Consider a mechanism where all individuals in A are ranked above individuals in B. For any two individuals i and j in A, let ρ(i) > ρ(j) if i T j. If i T j or if the reports on i and j are incompatible, construct an arbitrary ranking by letting ρ(i) > ρ(j) if and only if i > j. Similarly, for any two individuals i and j in B, let ρ(i) > ρ(j) if i T j. If i T j or if the reports on i and j are incompatible, let ρ(i) > ρ(j) if and only if i > j. We will show that the mechanism satisfies the three properties. Clearly if i and j are incomparable under two profile types T and T , either one belongs to A and the other to B (in which case the mechanism ranks them in the same way under T and T ), or they belong to the same set, and the mechanism ranks them identically under T and T as it only uses the index to rank them. Hence independence is satisfied. The mechanism satisfies strategy-proofness, a stronger incentive compatibility notion than ex post incentive compatibility. Consider an individual i in A. Then we claim that if t i jk = 0 it must be that both j and k are in B. To see this notice that as g is bipartite it does not contain any triangle. Hence no self-report can be supported by a third individual, and hence t i ij = 0 for all j = i. The only case where t i jk = 0 is thus when g ij g ik = 1 and j, k ∈ B. Hence, by changing his report t i jk , individual i can only affect the ranking of individuals in B. As all individuals in B are ranked below individuals in A, this does not affect the rank of individual i, and hence individual i's ranking is independent of his announcement, proving that the mechanism is strategy-proof. Finally, notice that by construction, the mechanism ρ achieves an ex post efficient ranking separately on each of the two components A and B. by Proposition 2, the comparison network h is disconnected into two components A and B. Hence the mechanism ρ is also ex post efficient. Proof of Proposition 4 We consider the same mechanism as in the proof of Theorem 1: We define the mechanism ρ by constructing comparisons. Let r ij denote the comparison between i and j chosen by the planner. First consider a pair of individuals (i, j) who observe each other, g ij = 1. By assumption, there are at least three reports on the ranking of i and j. If all individuals transmit the same report on (i, j), let r ij = t ij . If all individuals but one transmit the same report t ij and one individual reports t ij = −t ij , ignore the ranking t ij and let r ij = t ij . In all other cases, let r ij = 1 if and only if i > j. Second consider a pair of individuals (i, j) who do not observe each other, g ij = 0. If there are at least three individuals who observe i and j, use as above a mechanism such that r ij = t ij if all individuals agree on t ij or only one individual chooses t ij = −t ij , and let r ij = 1 if i > j otherwise. If one or two individuals observe i, j, pick the individual k with the highest index. Consider the vector of announcementsT −k where one disregards the announcements of individual k. Let T −k be the binary relation created by letting t ij = 1 if and only if t l ij = 1 for all l = k. If there exists a directed path of length greater or equal to 2 between i and j in T −k , and for all directed paths between i and j in T −k , i 0 , .., i L we have t i l i l+1 = 1, then r ij = 1. If on the other hand for all directed paths between i and j in T −k , t i l i l+1 = −1, then r ij = −1. In all other cases, let individual k dictate the comparison between i and j, r ij = t k ij . Now consider all comparisons r ij . If they induce a transitive binary relation on N , let ρ be the complete order generated by the comparisons. Otherwise, consider all shortest cycles generated by the binary relation r ij . If there exists a single individual i who dictates at least two comparisons in all shortest cycles, individual i is punished by setting ρ i = 1 and ρ j > ρ k if and only if j > k for all j, k = i. If this is not the case, pick the arbitrary ranking where ρ i > ρ j if and only if i > j. We now prove that this mechanism satisfies all three conditions. Consider two type profiles T and T , and two individuals i and j such that i T j and i T j. Because T and T generate identical truthful reports and result in a transitive partial order, i and j must be ranked at the final completion phase of the mechanism, using the same ranking ρ(i) > ρ(j) if and only if i > j. Hence independence holds. Next consider any pair (i, j) such that i T j. There must exist a sequence of comparisons (i, i 1 , .., i t , .., i T , j) such that h i t−1 i t = 1 and i t−1 T i t . For any of these pairs, we must have r i t−1 i t = 1 and hence, because the announcement T generates a transitive partial order, ρ(i t−1 ) > ρ(i t ). But this implies that ρ(i) > ρ(j), establishing that the mechanism satisfies ex post efficiency. Finally, we show that the mechanism is ex post incentive-compatible. Consider individual k's incentive to change his announcement on a link ij when all other individuals tell the truth. If the link ij is supported, this change does not affect the outcome of the mechanism. So consider an unsupported link ij and let individual k be the highest index individual observing i and j. Suppose that all individuals l = k announce the truth, so that T −k = T −k . We first show that individual k cannot gain by making an announcement which generates cycles in the ranking r ij . Suppose that the binary relation generated by r ij exhibits a cycle i 0 i 1 ...i L By the same argument as in the proof of Theorem 1, individual k must dictate at least two comparisons in the cycle. We will show that the initial cycle must contain a cycle of length 3. Suppose that the initial cycle has length greater than or equal to 4. Let ij and lm be two comparisons dictated by individual k. Suppose first that the rank of l is strictly higher than the rank of j. As individual k observes both (i, j) and (l, m), he also observes both i and l. Hence i and l must be compared under r and either r il = 1 or r il = −1. Now if r il = −1, one can construct a shorter cycle by replacing the path lm..i by the path li. If r il = +1, one can construct a shorter cycle by replacing the path ij, , l by the path il. Next suppose that j = l so that the two comparisons (i, j) and (l, m) are adjacent in the cycle. Again because individual k observes both i and m, then i and m must be compared under r and either r im = +1 or r im = −1. If r im = +1, one can construct a shorter cycle by replacing ijm with im. If r im = −1, one can construct a cycle of length 3 ijmi. We conclude that if the binary relation r exhibits a cycle, there must exist a subcycle of length 3, so that all shortest cycles are of length 3. Furthermore, individual k must dictate at least two of the comparisons in all cycles. Hence, individual individual k has no incentive to make an announcement generating a cycle in r, as he will be punished and obtain the lowest rank. We finally assume that the ranking generated by r is acyclic and show that the rank of individual k must remain the same if he changes his report on any pair (i, j). Notice first that, if k dictates the ranking between i and j, i and j there does not exist a l such that i, j and l can be ranked under T −k . In fact, if i, j and l can be ranked using the reports of individuals in N \ k, they unanimously rank i and j through a path of length equal or greater than 3 and the mechanism does not let k dictate the choice between i and j. But this implies that whenever k is a dictator over the Now let t i denote the announcement t i ij = t i jk = t i ik = 1 and t i the announcement t i ij = −1, t i jk = 1, t i ik = −1. By Claim 7, ρ i (t i , t j , t k ) = ρ i (t i , t j , t k ) = 3, ρ j (t i , t j , t k ) = ρ j (t i , t j , t k ) = 2. ρ k (t i , t j , t k ) = ρ k (t i , t j , t k ) = 2. Hence we conclude that, at (t i , t j , t k ) either ρ i = 3, ρ j = 1 or ρ i = 1, ρ j = 3. But ρ j = 3 is impossible, as, by claim 7, ρ j (t i , t j , t k ) = ρ j (t i , t j , t k ) and ρ j (t i , t j , t k ) = ρ i (t i , t j , t k ) = 3. Hence we conclude that ρ i (t i , t j , t k ) = 3, ρ j (t i , t j , t k ) = 1, ρ k (t i , t j , t k ) = 2. (1) A similar reasoning shows that ρ j (t i , t j , t k ) = 3, and hence either ρ i = 2, ρ k = 1 or ρ i = 1, ρ k = 3 at (t i , t j , t k ). But ρ i (t i , t j , t k ) = ρ i (t i , t j , t k ) = ρ k (t i , t j , t k ) = ρ k (t i , t j , t k ) = 1. So we conclude that ρ i (t i , t j , t k ) = 2, ρ j (t i , t j , t k ) = 3, ρ k (t i , t j , t k ) = 1. Now, ρ j (t i , t j , t k ) = ρ j (t i , t j , t k ). By equation 1, ρ j (t i , t j , t k ) = 1 , so that ρ j (t i , t j , t k ) = 1. As ρ i (t i , t j , t k ) = ρ i (t i , t j , t k ) = 1, ρ k (t i , t j , t k ) = 2. Similarly, ρ k (t i , t j , t k ) = ρ k (t i , t j , t k ) and by equation 2, ρ k (t i , t j , t k ) = 1 so that ρ k (t i , t j , t k ) = 1. establishing a contradiction. Proof of Proposition 6 If individuals strictly prefer being ranked at ρ(i) = 2 to being ranked at ρ(i), the necessity part of the proof of Theorem 1 shows that whenever there exists a pair of individuals who are not observed by a third individual, there cannot exist an ex post incentive-compatible and efficient mechanism. Conversely, if individuals are indifferent between being ranked at ρ(i) = 1 and ρ(i) = 2, let ρ(i) = 1 and ρ(j) = 2 whenever i and j are the only two individuals observing the ranking between i and j and t i ij = t j ij . This guarantees that individuals have no incentive to send conflicting reports, and hence that this mechanism, completed by the mechanism constructed in the sufficiency part of Theorem 1, satisfies ex post incentive compatibility and efficiency. Proof of Proposition 7 Consider a vector of announcements where all three individuals agree on t 13 = −1, t 23 = −1, t 12 = 1. By ex post efficiency, ρ(1) = 2, ρ(2) = 1, ρ(3) = 3. We claim that ex post group-incentive compatibility implies that, whenever individual 3 announces t 3 13 = −1, t 3 23 = −1, t 3 12 = 1, the rank of individual 1 must be different from 3. If that were not the case, there would exist an announcement (t 1 , t 2 ) for individuals 1 and 2 resulting in a rank ρ(1) = 3 > 2, ρ(2) ≥ 1, contradicting ex post group-incentive compatibility. By a similar reasoning, whenever individual 1 announces t 1 12 = 1, t 1 13 = 1, t 1 23 = 1, the rank of individual 2 must be different from 3. Finally, when individual 2 announces t 2 12 = −1, t 2 23 = 1, t 2 13 = −1, the rank of individual 3 must be different from 3. So consider the announcement t 1 = (t 1 12 = 1, t 1 13 = 1, t 1 23 = 1), t 2 = (t 2 12 = −1, t 2 23 = 1, t 2 13 = −1), t 3 = (t 3 13 = −1, t 3 23 = −1, t 3 12 = 1). For this announcement, neither of the three individuals can be in position 3, a contradiction which completes the proof of the Proposition. Figure 1 : 1A line of four individuals Figure 3 : 3Social and comparison networks Figure 4 : 4A supported social network g Example 1 .Figure 5 : 15Let n = 4. individuals i, j, k are connected in a triangle and individual l is connected to iA social network g where a mechanism does not exist Proposition 6 . 6There exists an ex post incentive-compatible and efficient mechanism in any completely informative community if and only if the planner can place two individuals in the worst spot, i.e., if and only if any individual i is indifferent between ρ(i) = 1 and ρ(i) = 2. Figure 6 : 6Impact of homophily (p = 0.15, n = 250) Figure 7 :Figure 8 : 78Distribution of social network measuresNote: Social networks from India (in orange) and Indonesia (in green). Information is measured as the density of the comparison network. Large variation of information for a given density Note: The bottom left panel shows a scatter-plot of information and density for networks of more than 50 households. Two networks of similar density are highlighted by orange and green points on the scatter plots. The network diagrams corresponding to those two points are plotted in the top left and bottom right panel. The degree distribution of the highlighted networks is shown in the top right panel. Figure 9 : 9Decompose informationNote: We decompose information (unique comparisons) into comparisons which are provided within triangles, across triangles, and a remainder. By Proposition 4 all within and across triangle comparisons are incentive-compatible. Comparisons within the remainder may not be incentive-compatible. Figure 10 : 10Capping comparisons at 5 friends Wilf, Herbert S. 1971. "The friendship theorem." In Combinatorial Mathematics and itsApplications (Proc. Conf., Oxford, 1969). 307-309.Jackson, Matthew O, Tomas Rodriguez-Barraquer, and Xu Tan. 2012. "Social capital and social quilts: Network patterns of favor exchange." American Economic Review 102 (5):1857-97. Kurokawa, David, Omer Lev, Jamie Morgenstern, and Ariel D Procaccia. 2015. "Im- partial Peer Review." In IJCAI. 582-588. Longyear, Judith Q and Torrence D Parsons. 1972. "The friendship theorem." In Indagationes Mathematicae (Proceedings), vol. 75. Elsevier, 257-262. Rai, Ashok S. 2002. "Targeting the poor using community information." Journal of Development Economics 69 (1):71-83. Table 1 : 1Summary statistics of social networks Notes: Means are reported with minimum and maximum in brackets. Information is measured by the density of the comparison network. All statistics (except the number of households) are calculated on the giant component. Data is sourced from Banerjee et al. (2013) for India and Alatas et al. (2016) for Indonesia.India Indonesia Networks 75 622 Number of households 198.72 [77, 356] 52.85 [11, 263] Share in giant component .95 [.85, .99] .65 [.22, 1.00] Average degree 9.34 [6.82, 13.83] 17.96 [2.00, 218.00] Density .05 [.02, .12] .53 [.10, 1.00] Average clustering .26 [.16, .45] .82 [.48, 1.00] Average distance 2.75 [2.30, 3.32] 1.77 [1.00, 4.32] Information .37 [.18, .62] .78 [.25, 1.00] Different proofs of the friendship theorem have been proposed, often using complex combinatorial arguments(Wilf, 1971; Longyear and Parsons, 1972;Huneke, 2002). The proof of the theorem is very different from known proofs of the Friendship Theorem, mostly because we focus attention on the minimization of the number of edges rather than on the construction of a graph where any intersection of neighborhoods is a singleton.3 A family of regular graphs, called the rook graphs, satisfy the property. For an integer m ≥ 2, rook graphs are regular graphs of degree 2(m − 1) among m 2 nodes, and have the property that any two connected nodes have m − 2 nodes in common and every pair of unconnected nodes has two common neighbors. SeeBrouwer and Haemers (2011) for more details on rook graphs. The extension of this impossibility result to more than three individuals remains an open ques- Note that this walk is not necessarily a path even if the initial walk in h is a path, as the same node j k can be used several times in the walk. Network structure and the aggregation of information: Theory and evidence from Indonesia. Vivi Alatas, Abhijit Banerjee, G Arun, Rema Chandrasekhar, Benjamin A Hanna, Olken, The American Economic Review. 1067Alatas, Vivi, Abhijit Banerjee, Arun G Chandrasekhar, Rema Hanna, and Ben- jamin A Olken. 2016. "Network structure and the aggregation of information: The- ory and evidence from Indonesia." The American Economic Review 106 (7):1663- 1704. Targeting the poor: evidence from a field experiment in Indonesia. Vivi Alatas, Abhijit Banerjee, Rema Hanna, Julia Benjamin A Olken, Tobias, American Economic Review. 1024Alatas, Vivi, Abhijit Banerjee, Rema Hanna, Benjamin A Olken, and Julia Tobias. 2012. "Targeting the poor: evidence from a field experiment in Indonesia." Amer- ican Economic Review 102 (4):1206-40. Sum of us: Strategyproof selection from the selectors. Noga Alon, Felix Fischer, Proceedings of the 13th Conference on Theoretical Aspects of Rationality and Knowledge. the 13th Conference on Theoretical Aspects of Rationality and KnowledgeACMAriel Procaccia, and Moshe TennenholtzAlon, Noga, Felix Fischer, Ariel Procaccia, and Moshe Tennenholtz. 2011. "Sum of us: Strategyproof selection from the selectors." In Proceedings of the 13th Conference on Theoretical Aspects of Rationality and Knowledge. ACM, 101-110. Strategyproof Peer Selection: Mechanisms, Analyses, and Experiments. Haris Aziz, Omer Lev, Nicholas Mattei, Toby Jeffrey S Rosenschein, Walsh, AAAI. Aziz, Haris, Omer Lev, Nicholas Mattei, Jeffrey S Rosenschein, and Toby Walsh. 2016. "Strategyproof Peer Selection: Mechanisms, Analyses, and Experiments." In AAAI. 397-403. The diffusion of microfinance. Abhijit Banerjee, G Arun, Esther Chandrasekhar, Matthew O Duflo, Jackson, Science. 34161441236498Banerjee, Abhijit, Arun G Chandrasekhar, Esther Duflo, and Matthew O Jackson. 2013. "The diffusion of microfinance." Science 341 (6144):1236498. Identifying the Best Agent in a Network. Leonie Baumann, Working paper.Baumann, Leonie. 2017. "Identifying the Best Agent in a Network.", Working paper. Spectra of graphs. Andries E Brouwer, Willem H Haemers, Springer Science & Business MediaBrouwer, Andries E and Willem H Haemers. 2011. Spectra of graphs. Springer Science & Business Media. A poor means test? Econometric targeting in Africa. Caitlin Brown, Martin Ravallion, Dominique Van De Walle, National Bureau of Economic ResearchBrown, Caitlin, Martin Ravallion, and Dominique Van de Walle. 2016. "A poor means test? Econometric targeting in Africa.", National Bureau of Economic Research. Targeting outcomes redux. David Coady, Margaret Grosh, John Hoddinott, The World Bank Research Observer. 191Coady, David, Margaret Grosh, and John Hoddinott. 2004. "Targeting outcomes redux." The World Bank Research Observer 19 (1):61-85. Optimal Selling Strategies under Uncertainty for a Discriminating Monopolist When Demands Are Interdependent. Jacques Crémer, Richard Mclean, Econometrica. 532Crémer, Jacques and Richard McLean. 1985. "Optimal Selling Strategies under Un- certainty for a Discriminating Monopolist When Demands Are Interdependent." Econometrica 53 (2):345-61. Impartial division of a dollar. De Clippel, Herve Geoffroy, Nicolaus Moulin, Tideman, Journal of Economic Theory. 1391De Clippel, Geoffroy, Herve Moulin, and Nicolaus Tideman. 2008. "Impartial division of a dollar." Journal of Economic Theory 139 (1):176-191. On a Problem of Graph Theory. Paul Erdős, Alfréd Rényi, Vera Sós, Studia Scientiarium Mathematicarum Hungarica. 11Erdős, Paul, Alfréd Rényi, and Vera Sós. 1966. "On a Problem of Graph Theory." Studia Scientiarium Mathematicarum Hungarica 1 (1):215-235. How homophily affects the speed of learning and best-response dynamics. Benjamin Golub, Matthew O Jackson , The Quarterly Journal of Economics. 1273Golub, Benjamin and Matthew O Jackson. 2012. "How homophily affects the speed of learning and best-response dynamics." The Quarterly Journal of Economics 127 (3):1287-1338. Impartial nominations for a prize. Ron Holzman, Hervé Moulin, Econometrica. 811Holzman, Ron and Hervé Moulin. 2013. "Impartial nominations for a prize." Econo- metrica 81 (1):173-196. The friendship theorem. Craig Huneke, The American mathematical monthly. 1092Huneke, Craig. 2002. "The friendship theorem." The American mathematical monthly 109 (2):192-194. (i L , j L ) be the pairs on which k is a dictator and let J + denote the set of pairs (i l , j l ) such that k T −k i l , j l and J − the set of pairs such that k ≺ T −k i l , j l . Let T = (T −k , T k ) the announcement obtained when i changes his report on some of the pairs in J while keeping a transitive partial order. For any m such that k T −k m, k T m and k T m. Hence ρ k (T) > ρ m (T) and ρ k (T ) > ρ m (T ). Similarly, for any m such that k ≺ T −k m, k ≺ T m and k ≺ T m. Hence ρ k (T) < ρ m (T) and ρ k (T ) < ρ m (T ). We also have, for any m ∈ J + , ρ k (T) > ρ m (T) and ρ k (T ) > ρ m (T ). For any m ∈ J − , ρ k (T) < ρ m (T) and ρ k (T ) < ρ m (T ). J + , then k T m and k T m so that ρ k (T) > ρ m (T) and ρ k (T ) > ρ m (T ). Similarly, if m T −k i l for some i l ∈ J − , then k ≺ T m and k ≺ T m so that ρ k (T) < ρ m (T) and ρ k (T ) < ρ m (T ). Finally, if m T −i i l ∀i l ∈ J + , m ≺ T −k i l ∀i l ∈ J − and k T −k m, then k T m and k T m. Whenever k T m and k T m, then the ranking between k. T −k . Now let J = (i 1 , j 1 ), ..., (i l , j l ), ..and m is independent of the type profile. Hence, in all cases the ranking between k and m is identical under T and T . This argument completes the proofpair (i l , j l ) both individuals i l and j l are either both ranked above or below k by the reports T −k . Now let J = (i 1 , j 1 ), ..., (i l , j l ), ..(i L , j L ) be the pairs on which k is a dictator and let J + denote the set of pairs (i l , j l ) such that k T −k i l , j l and J − the set of pairs such that k ≺ T −k i l , j l . Let T = (T −k , T k ) the announcement obtained when i changes his report on some of the pairs in J while keeping a transitive partial order. For any m such that k T −k m, k T m and k T m. Hence ρ k (T) > ρ m (T) and ρ k (T ) > ρ m (T ). Similarly, for any m such that k ≺ T −k m, k ≺ T m and k ≺ T m. Hence ρ k (T) < ρ m (T) and ρ k (T ) < ρ m (T ). We also have, for any m ∈ J + , ρ k (T) > ρ m (T) and ρ k (T ) > ρ m (T ). For any m ∈ J − , ρ k (T) < ρ m (T) and ρ k (T ) < ρ m (T ). J + , then k T m and k T m so that ρ k (T) > ρ m (T) and ρ k (T ) > ρ m (T ). Similarly, if m T −k i l for some i l ∈ J − , then k ≺ T m and k ≺ T m so that ρ k (T) < ρ m (T) and ρ k (T ) < ρ m (T ). Finally, if m T −i i l ∀i l ∈ J + , m ≺ T −k i l ∀i l ∈ J − and k T −k m, then k T m and k T m. Whenever k T m and k T m, then the ranking between k and m is independent of the type profile. Hence, in all cases the ranking between k and m is identical under T and T . This argument completes the proof We first establish the following simple general claim. We first establish the following simple general claim: If ρ is strategy-proof, ρ i (T i ,T −i ) = ρ i (T i ,T −i ) for all. Claim 7Claim 7. If ρ is strategy-proof, ρ i (T i ,T −i ) = ρ i (T i ,T −i ) for all i, T i , T i ,T −i . Let T i be the true type of individual i. Then, individual i has an incentive to announce T i , contradicting the fact that ρ is strategy-proof. T I , T I ,T −i Such That Ρ I (t I ,T −i ) > Μ I (t I ,T −i, Consider next two vectors of types: • T 1 : t ij = t jk = t ik = 1. Suppose by contradiction that there exists i,Proof. Suppose by contradiction that there exists i, T i , T i ,T −i such that ρ i (T i ,T −i ) > µ i (T i ,T −i ). Let T i be the true type of individual i. Then, individual i has an incentive to announce T i , contradicting the fact that ρ is strategy-proof. Consider next two vectors of types: • T 1 : t ij = t jk = t ik = 1 As the mechanism is ex post efficient, it must assign ranks ρ i (T 1 ) = 3, ρ j (T 1 ) = 2, ρ k (T 1 ) = 1, ρ i (T 2 ) = 1. ρ j (T 2 ) = 3, ρ k (T 2 ) = 2As the mechanism is ex post efficient, it must assign ranks ρ i (T 1 ) = 3, ρ j (T 1 ) = 2, ρ k (T 1 ) = 1, ρ i (T 2 ) = 1, ρ j (T 2 ) = 3, ρ k (T 2 ) = 2.	[]
[ "Unit Ball Graphs on Geodesic Spaces", "Unit Ball Graphs on Geodesic Spaces" ]	[ "Masamichi Kuroda ", "Shuhei Tsujie " ]	[]	[]	Consider finitely many points in a geodesic space. If the distance of two points is less than a fixed threshold, then we regard these two points as "near". Connecting near points with edges, we obtain a simple graph on the points, which is called a unit ball graph. If the space is the real line, then it is known as a unit interval graph. Unit ball graphs on a geodesic space describe geometric characteristics of the space in terms of graphs. In this article, we show that chordality and (claw, net)-freeness, which are combinatorial conditions, force the spaces to be R-trees and connected 1-dimensional manifolds respectively, and vice versa. As a corollary, we prove that the collection of unit ball graphs essentially characterizes the real line and the unit circle.	10.1007/s00373-020-02231-3	[ "https://arxiv.org/pdf/1809.08608v1.pdf" ]	119,289,435	1809.08608	f166686243dcd035aa1160e5e412b160d956ecdb	Unit Ball Graphs on Geodesic Spaces 23 Sep 2018 Masamichi Kuroda Shuhei Tsujie Unit Ball Graphs on Geodesic Spaces 23 Sep 2018geodesic spaceR-treereal tree0-hyperbolic spaceunit ball graphunit interval graphchordal graphstrongly chordal graph(clawnet)-free graphHamiltonian hereditary graph 2010 MSC : 51D2005C6205C75 Consider finitely many points in a geodesic space. If the distance of two points is less than a fixed threshold, then we regard these two points as "near". Connecting near points with edges, we obtain a simple graph on the points, which is called a unit ball graph. If the space is the real line, then it is known as a unit interval graph. Unit ball graphs on a geodesic space describe geometric characteristics of the space in terms of graphs. In this article, we show that chordality and (claw, net)-freeness, which are combinatorial conditions, force the spaces to be R-trees and connected 1-dimensional manifolds respectively, and vice versa. As a corollary, we prove that the collection of unit ball graphs essentially characterizes the real line and the unit circle. Introduction Let (X, d) be a metric space. Consider finitely many points x 1 , . . . , x n in X and fix a threshold δ > 0. If d(x i , x j ) ≤ δ, we regard x i and x j as "near". We can construct a simple graph on the set {x 1 , . . . , x n } with edges between near points. It might be expected that we could obtain some information about X from graphs constructed in such a way. However, it seems difficult to study metric spaces with unit ball graphs without any other assumptions. For example, let (X, d) be a metric space defined by X := (cos θ, sin θ) ∈ R 2 π 6 ≤ θ ≤ 11π 6 and d is the restriction of the Euclidean metric in R 2 . Take points x = (cos π 6 , sin π 6 ), y = (cos 11π 6 , sin 11π 6 ), and z = (−1, 0) and suppose that δ = 1 (see Figure 1). Then x is "near" to y but not to z, which seems counterintuitive. It is natural to regard x and y as the "furthest" points in X and z is the midpoint between x and y. If we define a metric on X by arc length, then it fits our intuition. Thus, from now on, our interest focus on geodesic spaces defined as follows. x y z Figure 1: Intuitively x and y should be furthest but not with the Euclidean metric Definition 1.1. Let (X, d) be a metric space and x, y ∈ X. A geodesic from x to y is a distance-preserving map γ from a closed interval [0, d(x, y)] ⊆ R to X with γ(0) = x and γ(d(x, y)) = y. Its image is said to be a geodesic segment with endpoints x and y. We say that (X, d) is a geodesic space if every two points are joined by a geodesic. Note that a geodesic segment between two points is not necessarily unique. We will write a geodesic segment whose endpoints are x and y as [x, y]. If there exists a unique geodesic segment for every pair of points, then we say that (X, d) is uniquely geodesic. Example 1.2. The n-dimensional Euclidean space R n and its convex subsets are geodesic spaces. The n-dimensional sphere S n with the great-circle metric is a geodesic space. The vector space R n with L p -norm \|\| * \|\| p (1 ≤ p ≤ ∞) is also a geodesic space. Next we formulate the graphs in which we are interested. Definition 1.3. Let (X, d) be a (geodesic) metric space. A simple graph G = (V G , E G ) is said to be a unit ball graph on (X, d) if there exist a threshold δ > 0 and a map ρ, called a realization, from the vertex set V G to X such that {u, v} ∈ E G if and only if d(ρ(u), ρ(v)) ≤ δ. Let UBG(X, d) denote the collection of the unit ball graphs on (X, d). When there is no confusion with the metric, we may write it as UBG(X). Remark 1.4. In this article, the term "graph" refers an undirected simple graph on finite vertices. We frequently identify the vertices of a unit ball graph with the realized points in the space. Remark 1.5. A unit ball graph is the intersection graph of finitely many closed balls of the same size in a geodesic space. If we scale the metric, then the graph can be the intersection graph of unit balls, that is, balls of radius 1. When we consider the Euclidean spaces, we may always assume that a unit ball graph is the intersection graph of finitely many unit balls. Let H be a graph. A graph is said to be H-free if it has no subgraph isomorphic to H. A graph is called chordal if it is C n -free for all n ≥ 4, where C n denotes the cycle graph on n vertices. Note that a graph is chordal if and only if every cycle in it of length four or more has a chord, which is an edge connecting non-consecutive vertices of the cycle. For an integer n ≥ 3, the (complete) n-sun (or a trampoline) is a graph on 2n vertices { v i \| i ∈ Z/2nZ } such that the even-indexed vertices induce a complete graph, the oddindexed vertices form an independent set, and an odd-indexed vertex v i is adjacent to an even-indexed vertex v j if and only if i − j = ±1 (See Figure 2 for example). A graph is called sun-free if it is n-sun-free for all n ≥ 3. A graph is called strongly chordal if it is chordal and sun-free. Farber [9] investigated strongly chordal graphs and gave some characterizations. The definition above is one of such characterizations. A graph in UBG(R) is called a unit interval graph also known as an indifference graph, which is a very important object in combinatorics. There are several linear-time algorithms for recognizing unit interval graphs. Also, they are characterized by forbidden induced subgraphs as explained below. Theorem 1.6 (Wegner [14], Roberts [12]). A graph is unit interval if and only if it is chordal and (claw, net, 3-sun)-free (see Figure 3). A graph in UBG(S 1 ) is called a unit circular-arc graph, which is one of a natural generalization of unit interval graphs and this class is also well investigated (see [10], for example). Tucker [13, Theorem 3.1 and Theorem 4.3] characterized unit circular-arc graphs in terms of forbidden induced subgraphs. A graph in UBG(R 2 ) is called a unit disk graph. Breu and Kirkpatrick [4] showed that the recognition of unit disk graphs is NP-hard. Therefore it seems very difficult to characterize unit disk graphs in terms of forbidden induced subgraphs. Recently, Atminas and Zamaraev [2] discovered infinitely many minimal non-unit disk graphs. A graph in UBG(R 2 , \|\| * \|\| ∞ ) = UBG(R 2 , \|\| * \|\| 1 ) is called a unit square graph. Breu [3, Corollary 3.46.2] proved that the recognition of unit square graphs is also NP-hard. Neuen [11] proved that the graph isomorphism problem for unit square graphs can be solved in polynomial time and investigated a lot of properties of unit square graphs. For instance, Neuen showed that every unit square graph is (K 1,5 , K 2,3 , 3K 2 )-free (see [11, class space C 4 claw net 3-sun K 1,5 unit interval R ± ± ± ± ± unit circular-arc Notice that the collection of unit ball graphs can distinguish these four geodesic spaces (see Table 4). However, we can find easily non-isometric geodesic spaces whose unit ball graphs coincide. For example, UBG([0, 1]) = UBG(R). More generally, when X is a convex subset of R n with non-empty interior, we have that UBG(X) = UBG(R n ). In Section 2, we will give a sufficient condition for the coincidence of the collections of unit ball graphs (Corollary 2.5). S 1 ± ± ± ± unit disk R 2 unit square (R 2 , \|\| * \|\| ∞ ) ± The main contribution of this article is to characterize geodesic spaces whose unit ball graphs are chordal and geodesic spaces whose unit ball graphs are (claw, net)-free. In order to state the results, we define R-trees and tripods. There are several equivalent definitions for R-trees. Here we give one of them (see Section 3, for other conditions and details). Definition 1.7. A geodesic space is said to be an R-tree (or a real tree) if it is uniquely arc-connected, that is, every pair of points in it is joined by a unique arc. Definition 1.8. A subset Y of a geodesic space X is said to be a tripod (see Figure 5) if there exist four distinct points x 1 , x 2 , x 3 , y ∈ Y and geodesic segments [ x i , y] (i = 1, 2, 3) such that Y = [x 1 , y] ∪ [x 2 , y] ∪ [x 3 , y] and [x i , y] ∩ [x j , y] = {y} if i = j. The main results are as follows. Theorem 1.9. Let (X, d) be a geodesic space. Then the following are equivalent: (1) Every unit ball graph on X is strongly chordal. (2) Every unit ball graph on X is chordal. (3) X is an R-tree. Theorem 1.10. Let (X, d) be a geodesic space. Then the following are equivalent: (1) Every unit ball graph on X is (claw, net)-free. (2) X has no tripod. (3) X is homeomorphic to a manifold of dimension at most 1, that is, X is similar to S 1 or isometric to an interval, that is, a convex subset of R. Remark 1.11. According to [8], every (claw, net)-free graph has a Hamiltonian path. Hence we can deduce that a graph is (claw, net)-free if and only if it is a Hamiltonianhereditary graph, that is, every induced connected component of it has a Hamiltonian path as stated in [7]. Clearly, unit ball graphs on intervals and S 1 are Hamiltonianhereditary graphs. Theorem 1.10 asserts that spaces whose unit ball graphs are Hamiltonianhereditary graphs are exactly intervals and S 1 . Theorem 1.9, 1.10, and Table 4 lead to the following corollaries. Corollary 1.12. Let (X, d) be a geodesic space. Then the following are equivalent: (1) UBG(X) = UBG(R). (2) X is isometric to an interval which is not the single-point space. Corollary 1.13. Let (X, d) be a geodesic space. Then the following are equivalent: (1) UBG(X) = UBG(S 1 ). (2) X is similar to S 1 . The organization of this article is as follows. In Section 2, we study basic properties of unit ball graphs. In Section 3, we review the theory of R-trees and prove Theorem 1.9. In Section 4, we will prove Theorem 1.10. Basic properties First of all, we begin with the following proposition, which is easy to prove. Proposition 2.1. The class of unit ball graphs on a metric space (X, d) is hereditary. Namely, if G ∈ UBG(X), then every induced subgraph of G belongs to UBG(X). Therefore every class UBG(X) has a characterization in terms of forbidden induced subgraphs. However, as mentioned in the introduction, it is difficult to characterize UBG(X) in general. Next, we treat the most trivial case, that is, the single-point space { * }. Proposition 2.2. The following statements hold true: (1) UBG({ * }) consists of complete graphs, or equivalently 2K 1 -free graphs. Proof. The assertion (1) is trivial. To show (2), take two distinct points x, y ∈ X. Then the unit ball graph on {x, y} with threshold d(x, y)/3 is 2K 1 . Thus the assertion holds. By definition, a unit ball graph is the intersection graph of finitely many closed balls of the same size. Next we show that we may use open balls instead of closed ones. Proposition 2.3. A simple graph G = (V G , E G ) is a unit ball graph on a metric space (X, d) if and only if there exist δ > 0 and a map ρ : V G → X such that {u, v} ∈ E G if and only if d(ρ(u), ρ(v)) < δ. Proof. Assume that G is a unit ball graph on X. Then, by definition, there exist δ > 0 and a map ρ : V G → X such that {u, v} ∈ E G if and only if d(ρ(u), ρ(v)) ≤ δ. Let δ ′ = min { d(ρ(u), ρ(v)) \| {u, v} ∈ E G }. Then we have {u, v} ∈ E G if and only if d(ρ(u), ρ(v)) < δ ′ . The converse can be proven in a similar way. Next, we give a sufficient condition for inclusion of the classes of unit ball graphs. Proposition 2.4. Let (X, d X ) and (Y, d Y ) be metric spaces. Suppose that every finite subset in X is similarly embedded into Y . Namely, assume that, for any finite subset S, there exist r > 0 and a map f : S → Y such that d Y (f (a), f (b)) = rd X (a, b) for any a, b ∈ S. Then UBG(X) ⊆ UBG(Y ). Proof. Let G ∈ UBG(X) with a realization ρ and a threshold δ. Since ρ(V G ) is finite, by the assumption, there exist r > 0 and f : From Proposition 2.4, we have that UBG(R m ) ⊆ UBG(R n ) whenever m ≤ n. Therefore, intuitively, higher dimensional spaces could have more unit ball graphs. However, the converse is not true in general as follows. ρ(V G ) → Y such that d Y (f (ρ(u)), f (ρ(v))) = rd X (ρ(u), ρ(v)) for any u, v ∈ V G . Hence we conclude that G ∈ UBG(Y ) with a realiza- tion f • ρ and a threshold δ/r. Proposition 2.6. There exists a geodesic space X such that its Lebesgue covering dimension is 1 and UBG(X) consists of all graphs. Proof. Let G be a connected graph and X G the geodesic space obtained by replacing the edges of G with a copy of the unit interval [0, 1] (so X G is the underlying space of a 1-dimensional simplicial complex). Clearly, G ∈ UBG(X G ). Choose a vertex of G and connect a geodesic segment of length 1 to the corresponding point of X G . Let X be a geodesic space obtained by gluing the other endpoints with respect to each connected graph and its countably many copies. Then every graph belongs to UBG(X). Obviously, the Lebesgue covering dimension of X is 1. It is not clear whether the converse of Corollary 2.5 holds true or not. However, for geodesic spaces R, S 1 , and { * }, it is true by Corollary 1.12, 1.13, and Proposition 2.2 (2). Chordal graphs and R-trees In this section, we will give a proof of Theorem 1.9. Note that the implication (1) ⇒ (2) follows by definition. As mentioned before, an R-tree is a geodesic space in which every pair of points is joined by a unique arc, that is, the image of a topological embedding of a closed interval. Note that every arc in an R-tree is a (unique) geodesic segment and hence an R-tree is uniquely geodesic. Obviously, the real line R and intervals are R-trees. The underlying space of a 1-dimensional connected acyclic simplicial complex is also an R-tree. (1) X is an R-tree (2) X has no subspace homeomorphic to S 1 . (3) For any x, y, z ∈ X, whenever [ 3.1 Proof of Theorem 1.9 (2) ⇒ (3) The following two propositions are required. The first one is very famous. Proof. Assume that the intersection graph of U is disconnected. Then there exist nonempty subsets U 1 and U 2 of U such that U = U 1 ∪ U 2 , U 1 ∩ U 2 = ∅, and U 1 ∩ U 2 = ∅ for any U i ∈ U i (i = 1, 2). For each i ∈ {1, 2}, let O i := U ∈U i U . Then S ⊆ O 1 ∪ O 2 and S ∩ O i = ∅ for each i ∈ {1, 2} . Therefore S is disconnected. Thus the assertion holds. Proof of Theorem 1.9 (2) ⇒ (3). We assume that X is not an R-tree and show that there exists G ∈ UBG(X) such that G is non-chordal. By Proposition 3.1, there exists a topological embedding φ : S 1 → X. We consider S 1 as R/4Z and put p i := φ(i), and S i := φ([i, i + 1]) for i ∈ {1, 2, 3, 4} ⊆ R/4Z (see Figure 6). By Proposition x. Note that U r (x) ∩ U r (y) = ∅ whenever (x, y) ∈ S 1 × S 3 or (x, y) ∈ S 2 × S 4 . For every i, { U r (x) } x∈S i is an open covering of S i . Since S i is compact, we have a finite subset V i ⊆ S i such that { U r (x) } x∈V i is a finite open covering of S i . Adding two points p i , p i+1 if necessary, we may suppose that p i , p i+1 ∈ V i . Let G be the intersection graph of { U r (x) } x∈V 1 ∪···∪V 4 and we will identify the vertices of G with the corresponding points. We will show that G is not chordal. By Proposition 3.3, each induced subgraph G[V i ] is connected. Take a shortest path π i from p i to p i+1 in G[V i ]. Since p i ∈ S i−1 and p i+1 ∈ S i+1 , they are not adjacent. Therefore the length of the path π i is at least two and every intermediate vertex of π i is an interior point of S i . Connecting the paths π 1 , . . . , π 4 , we have a cycle C = (V C , E C ) satisfying the following conditions. (i) V C ∩ V i = ∅ and C[V C ∩ V i ] is a chordless path for each i. (ii) C has a vertex corresponding to an interior point of S i for each i. Suppose that C 0 is a minimal cycle satisfying these conditions. By the condition (ii), the length of C 0 is at least four. Let i ∈ {1, 2, 3, 4}. From (i), there exists no chord between two vertices in V C 0 ∩ V i . By minimality of C 0 , there exists no chord connecting V C 0 ∩ V i and V C 0 ∩ V i+1 . By choice of r, there exists no chord joining V C 0 ∩ V i and V C 0 ∩ V i+2 . Thus C 0 is a chordless cycle of length at least four and hence G is a non-chordal unit ball graph on X. Proof of Theorem 1.9 (3) ⇒ (1) Proposition 3.4. Let (X, d) be a geodesic space. Then the following conditions hold. (1) Let x 1 , . . . , x n , y 1 , . . . , y n ∈ X. Suppose that n i=1 [x i , y i ] = ∅. Then n i=1 d(x i , y i ) ≥ n i=1 d(x i , y σ(i) ) for any permutation σ. (2) Let G ∈ UBG(X). Suppose that {x i , y i } ∈ E G for each i ∈ {1, . . . , n} and there exists a permutation σ such that {x i , y σ(i) } ∈ E G for any i. Then n i=1 [x i , y i ] = ∅. Proof. (1) Take a point p ∈ n i=1 [x i , y i ]. Then we have n i=1 d(x i , y i ) = n i=1 (d(x i , p) + d(p, y i )) = n i=1 d(x i , p) + n i=1 d(p, y i ) = n i=1 d(x i , p) + n i=1 d(p, y σ(i) ) = n i=1 d(x i , p) + d(p, y σ(i) ) ≥ n i=1 d(x i , y σ(i) ), where we apply the triangle inequality. (2) By the assumption, there exists a threshold δ > 0 such that d(x i , y i ) ≤ δ and d(x i , y σ(i) ) > δ for any i ∈ {1, . . . , n}. Assume that n i=1 [x i , y i ] = ∅. Then by the assertion (1) we have nδ ≥ n i=1 d(x i , y i ) ≥ n i=1 d(x i , y σ(i) ) > nδ. This contradiction proves the assertion. Lemma 3.7. Suppose that the n-cycle C n is a unit ball graph of a geodesic space X. Let {x i } i∈Z/nZ be the vertex set of C n with {x i , x i+1 } ∈ E Cn (i ∈ Z/nZ). Suppose that {x i , x i+1 }, {x j , x j+1 } are non-adjacent edges. Then [x i , x i+1 ] ∩ [x j , x j+1 ] = ∅. Proof. Since {x i , x i+1 }, {x j , x j+1 } ∈ E Cn and {x i , x j }, {x i+1 , x j+1 } ∈ E Cn , we have im- mediately [x i , x i+1 ] ∩ [x j , x j+1 ] = ∅ by Proposition 3.4 (2). Lemma 3.8. Suppose that a geodesic space X admits an n-sun as a unit ball graph. Let {x i } i∈Z/2nZ be the vertex set such that even-indexed vertices induce a clique, odd-indexed vertices are independent, and two consecutive vertices form an edge. Then the following assertions hold. (1) Suppose that {x i , x i+1 } and {x j , x j+1 } are non-adjacent edges with j = i ± 2. Then [x i , x i+1 ] ∩ [x j , x j+1 ] = ∅. (2) When {x i , x i+1 }, {x j , x j+1 }, and {x k , x k+1 } are non-adjacent edges, we have that [x i , x i+1 ] ∩ [x j , x j+1 ] ∩ [x k , x k+1 ] = ∅. Proof. (1) Recall that an even-indexed vertex and an odd-indexed vertex in the n-sun are adjacent if and only if their indices are consecutive. If i ≡ j (mod 2), then neither {x i , x j } nor {x i+1 , x j+1 } is an edge. Otherwise, neither {x i , x j+1 } nor {x j , x i+1 } is an edge since j = i ± 2. In any cases, we may conclude that the assertion holds true by Proposition 3.4 (2). (2) For each s ∈ {i, j, k}, let p s and q s be the odd-indexed and even-indexed veritices of {x s , x s+1 }. We will show that neither {p i , q j }, {p j , q k }, nor {p k , q i } is an edge or neither {p i , q k }, {p k , q j }, nor {p j , q i } is an edge. Suppose that the former does not hold. Without loss of generality we may assume that {p i , q j } is an edge. Then the indices of the vertices q i , p i , q j , p j are consecutive in this order. Since the cardinality of the vertex set is at least 6, we have {p j , q i } is not an edge. Moreover, neither {p i , q k } nor {p k , q j } is an edge. Thus the latter condition holds. Using Proposition 3.4 (2), we have proved the assertion. Proof of Theorem 1.9 (3) ⇒ (1). We will show that every unit ball graph on an R-tree is chordal and sun-free. Assume that there exists n ≥ 4 such that C n ∈ UBG(X). Let {x i } i∈Z/nZ be the vertices of C n with {x i , x i+1 } ∈ E Cn . Then A := [x 1 , x 2 ] and B := n−1 i=3 [x i , x i+1 ] are disjoint closed connected subsets by Lemma 3.7. By Proposition 3.5, both of the segments [x 1 , x n ] and [x 2 , x 3 ] contains the bridge between A and B. In particular, we have [x 1 , x n ] ∩ [x 2 , x 3 ] = ∅, which contradicts to Lemma 3.7. Thus we conclude that every unit ball graph on X is chordal. Next, suppose that an n-sun is a unit ball graph on X. Let {x i } i∈Z/2nZ be the vertex set of the n-sun with the conditions mentioned in Assume that y 3 ∈ A and y 2n ∈ B. Then [x 3 , x 4 ] ∩ [x 2n , x 1 ] ⊇ [a, b] , which contradicts to Lemma 3.8 (1). The condition y 3 ∈ B and y 2n ∈ A also leads to a contradiction. If y 3 , y 2n ∈ A, then [x 3 , x 4 ] ∩ [x 2n , x 2n−1 ] ⊇ [a, b]. Therefore A ∩ [x 3 , x 4 ] ∩ [x 2n , x 2n−1 ] ∋ a. This is a contradiction to Lemma 3.8 (2). Finally assume that y 3 , y 2n ∈ B. Then we have B ∩ [x 3 , x 2 ] ∩ [x 2n , x 1 ] ∋ b, which is again a contradiction to Lemma 3.8 (2). Therefore we conclude that every unit ball graph on X is sun-free. Thus the proof has been completed. (claw, net)-free graphs and 1-dimensional manifolds In this section, we will prove Theorem 1.10. As mentioned Remark 1.11, it is well known that the implication (3) ⇒ (1) holds true. Proof of Theorem 1.10 (1) ⇒ (2) Lemma 4.1. Let G be a graph on vertex set {a i } l i=0 ∪ {b i } m i=0 ∪ {c i } n i=0 with positive integers l, m, n satisfying the following conditions. (i) {a i } l i=0 , {b i } m i=0 , {c i } n i=0 induce chordless paths. (ii) a 0 , b 0 , c 0 are leaves. (iii) {a l , b m , c n } induces a triangle. Then G has an induced subgraph isomorphic to a claw or a net. Proof. We proceed by induction of the number of the vertices of G. The initial case where l = m = n = 1 is trivial since G itself is a net. Hence we may assume that l ≥ 2 by symmetry. If the neighborhood of a 1 coincides with {a 0 , a 2 }, then G \ {a 0 } satisfies the assumptions. Therefore, by the induction hypothesis, G \ {a 0 } and hence G have a claw or a net. Without loss of generality, we may assume that there exists a minimal integer i such that {a 1 , b i } is an edge of G. By the assumption (ii) we have 1 ≤ i ≤ m. Assume that i < m. If {a 1 , b i+1 } is not an edge of G, then the four vertices a 1 , b i−1 , b i , b i+1 form a claw by the minimality of i and the assumption (i). Now suppose that {a 1 , b i+1 } is an edge. Note that {a 1 , b i , b i+1 } induces a triangle. Take a shortest path from b i+1 to c 0 in the induced subgraph on {b i+1 , . . . , b m , c n , . . . , c 0 }. This path together with two paths on {a 0 , a 1 } and {b 0 , . . . , b i } induce a subgraph of G satisfying the assumptions. Making use of the induction hypothesis, we have that G has a claw or a net. Hence we may assume that i = m. If {a 1 , c n } is an edge of G, then the graph G \ {a 2 , . . . , a l } satisfies the assumptions with the triangle {a 1 , b m , c n }. Therefore we may assume that {a 1 , c n } is not an edge of G. Proof of Theorem 1.10 (1) ⇒ (2). Assume that there exist four points x 1 , x 2 , x 3 , y ∈ X forming a tripod with center y. By Proposition 3.2, there exists r > 0 such that d({x i }, [x i+1 , y] ∪ [x i+2 , y]) > 2r for every i ∈ Z/3Z. Let γ be a geodesic from x 1 to y and l the greatest integer less than or equal to d(x 1 , y). Define a sequence {a i } l i=0 by a i := γ(ir). Note that U r (a i ) ∩ U r (a j ) = ∅ if and only if \|i − j\| ≤ 1. Define sequences {b i } m i=0 and {c i } n i=0 with respect to x 2 and x 3 in a similar way. By the choice of r, we have U r (a 0 ) ∩ U r (b i ) = ∅, U r (a 0 ) ∩ U r (c i ) = ∅, and so on. Moreover we have U r (a l ) ∩ U r (b m ) ∩ U r (c n ) ∋ y. Therefore the intersection graph of open balls of radius r with center points in {a i } l i=0 ∪ {b i } m i=0 ∪ {c i } n i=0 satisfies the assumptions of Lemma 4.1 and hence it has a claw or a net as an induced subgraph. φ : X → R by φ(x) :=      0 if x = q 0 , d(q 0 , x) if x ∈ X + , −d(q 0 , x) if x ∈ X − . Now we will show that φ preserves the distance. We will treat only the case where x, y ∈ X + since the other cases is clear or similar and show that one of [q 0 , x] and [q 0 , y] contains the other. If x ∈ [q 0 , q + ] or y ∈ [q 0 , q + ], then it is clear. Thus we may assume that [q 0 , q + ] ⊆ [q 0 , x] and [q 0 , q + ] ⊆ [q 0 , y]. By Lemma 4.2 (1) we have [q 0 , y] ⊆ [q 0 , x] or [q 0 , y] ⊇ [q 0 , x]. Therefore we may conclude that d(x, y) = \|d(q 0 , x) − d(q 0 , y)\| = \|φ(x) − φ(y)\|. Hence X is isometric to an interval. Second we assume that X is not an R-tree and show that X is similar to S 1 . By Proposition 3.1, there exist distinct three points x, y, z ∈ X and segments [ Figure 2 Figure 3 : 23Obstructions for unit interval graphs Figure 5 5Figure 5: A tripod ( 2 ) 2Let (X, d) be a metric space. Then UBG(X) = UBG({ * }) if and only if X = { * }. Corollary 2 . 5 . 25Let (X, d X ) and (Y, d Y ) be metric spaces. Suppose that every finite subset in X is similarly embedded into Y , and vice versa. Then UBG(X) = UBG(Y ). Proposition 3. 1 ( 1See [6, Proposition 2.3] and [5], for example). Let (X, d) be a geodesic space. Then the following conditions are equivalent. Proposition 3. 2 .. 3 . 23Suppose that A and B be disjoint compact subsets of a metric space (X, d). Then d(A, B) Let S be a connected subset of a topological space and U a finite open covering of S in which each member intersects with S. Then the intersection graph of U is connected. Figure 6 : 6The image of S 1 3.2, there exists r > 0 such that d(S 1 , S 3 ) > 2r and d(S 2 , S 4 ) > 2r. Let U r (x) denote the open ball of radius r with center Proposition 3 . 5 ( 35Alperin-Bass [1, Proposition 2.17], Chiswell [6, Lemma 1.9(b)]). Let A and B be non-empty closed connected subsets of an R-tree such that A ∩ B = ∅. Then there exist unique points a ∈ A and b ∈ B such that [a, b] ∩ A = {a} and [a, b] ∩ B = {b}. Definition 3 . 6 . 36We call [a, b] in Proposition 3.5 the bridge between A and B. Let Y be a non-empty closed connected subset of an R-tree and x a point. Call the other endpoint y of the bridge [x, y] between {x} and Y the closest point in Y to x when x ∈ Y . If x ∈ Y , define the closest point as x itself. Lemma 3.8. Let A := [x 1 , x 2 ] and B := 2n−2 i=4 [x i , x i+1 ]. By Lemma 3.8 (1), we have A and B are disjoint closed connected subsets of X. Let [a, b] be the bridge between A and B and set Y := A ∪ [a, b] ∪ B. Let y 3 and y 2n be the closest points in Y to x 3 and x 2n .We will show that y 3 , y 2n ∈ A ⊔ B. Assume that y 3 ∈ A ⊔ B. Then we havey 3 ∈ [a, b] ⊆ [x 1 , x 2n−1 ] ⊆ [x 2n , x 1 ] ∪ [x 2n−1 , x 2n ] byProposition 3.5 and 3.1. Furthermore, by Proposition 3.5 again, we have y 3 ∈ [x 3 , x 4 ]∩[x 2 , x 3 ]. Therefore y 3 ∈ ([x 2n , x 1 ]∩[x 3 , x 4 ])∪ ([x 2n−1 , x 2n ] ∩ [x 2 , x 3 ]). However, by Lemma 3.8 (1), we have [x 2n , x 1 ] ∩ [x 3 , x 4 ] = [x 2n−1 , x 2n ] ∩ [x 2 , x 3 ] = ∅, a contradiction. Hence y 3 ∈ A ⊔ B. We can show that y 2n ∈ A ⊔ B in a similar way. If {b m−1 , c n } is an edge of G, then the paths on {a 0 , a 1 , b m }, {b 0 , . . . , b m−1 }, {c 0 , . . . , c n } induce a subgraph satisfying the assumptions with the triangle {b m , b m−1 , c n }. Hence we may assume that {b m−1 , c n } is not an edge of G. Then the induced subgraph on {a 1 , b m−1 , b m , c n } is a claw. Thus the assertion holds true. Table 4 : 4Geodesic spaces and their unit ball graphsy x 1 x 2 x 3 x, y]∩[y, z] = {y}, the union [x, y]∪[y, z] is a geodesic segment joining x and z.(4) X is a Gromov 0-hyperbolic space, that is, for any geodesic segments [x, y], [x, z], [y, z], we have [x, z] ⊆ [x, y] ∪ [y, z]. x, y], [x, z], [y, z] such that [x, y] ∩ [y, z] = {y} and [x, z] = [x, y] ∪ [y, z]. We show that [x, z] ∩ [y, z] = {z} and [x, y] ∩ [x, z] = {x}. To prove the former, assume [x, z] ∩ [y, z] {z}. Lemma 4.2 (1) asserts that x ∈ [y, z] or y ∈ [x, z]. If x ∈ [y, z], then x ∈ [x, y] ∩ [y, z] = {y}. Hence x = y, which is a contradiction. When y ∈ [x, z], apply Lemma 4.2 (2), we obtain [x, y] and [y, z] are subsegments of [x, z] and hence [x, z] = [x, y] ∪ [y, z], which is again a contradiction. The latter case can be proved by symmetry. If X = [x, y] ∪ [x, z] ∪ [y, z], then it is easy to prove that X is similar to S 1 . Assume that there exists a point p ∈ X \ [x, y] ∪ [x, z] ∪ [y, z]. Take a geodesic segment [p, z] and consider it together with the geodesic segments [x, z], [y, z]. Since X has no tripod, we have [x, z] ∩ [p, z] {z} or [y, z] ∩ [p, z] {z}. Without loss of generality we may assume that the latter condition holds. Since p ∈ [y, z], we have y ∈ [p, z] by Lemma 4.2 (1). The segments [y, z] is a subsegment of [p, z] by Lemma 4.2 (2). Take the subsegment [p, y] [p, z]. Note that [p, y] ∩ [y, z] = {y}. We may deduce that x ∈ [p, y] in a similar way. Moreover, take the subsegment [p, x] [p, y] and we may show that z ∈ [p, x], which is a contradiction. Therefore we can conclude that X = [x, y] ∪ [x, z] ∪ [y, z] and it is similar to S 1 . Then a geodesic segment between x and y is unique and hence it is a subsegment of. Let x, y, z be distinct points with y ∈ [x, z. x, zLet x, y, z be distinct points with y ∈ [x, z]. Then a geodesic segment between x and y is unique and hence it is a subsegment of [x, z]. Assume that x ∈ [y, z] and y ∈ [x, z. We will prove that X has a tripod. Let E := [x, z] ∩ [y, z] and γ the geodesic corresponding to [x, zProof. (1) Assume that x ∈ [y, z] and y ∈ [x, z]. We will prove that X has a tripod. Let E := [x, z] ∩ [y, z] and γ the geodesic corresponding to [x, z]. Note that q = x, y, z since x, y ∈ E and E {z}. Take a geodesic segments [x, q] from [x, z] and two segments. Note that a geodesic segment joining two points is not necessarily unique). Then we may conclude that. x, q], [y, q] and [z, q] form a tripod by the choice of qSince E is compact, there exists a point q ∈ E such that d(x, q) = min p∈E d(x, p). Note that q = x, y, z since x, y ∈ E and E {z}. Take a geodesic segments [x, q] from [x, z] and two segments [y, q], [z, q] from [y, z] (Note that a geodesic segment joining two points is not necessarily unique). Then we may conclude that [x, q], [y, q] and [z, q] form a tripod by the choice of q. the geodesic segments in [x, z] and assume that there exists another geodesic segment [x, y] ′ between x and. Let [x, y], [y, z] denote. If [x, y] ⊆ [x, y] ′ , then [x, y] = [x, y] ′ , which is a contradiction. Hence there exists p ∈ [x, y] \ [x, y] ′ . Let [p, z] be the geodesic segment in [x, z] and [x, z] ′ the geodesic segment obtained by connecting [x, y] ′ and [y, zLet [x, y], [y, z] denote the geodesic segments in [x, z] and assume that there exists another geodesic segment [x, y] ′ between x and y. If [x, y] ⊆ [x, y] ′ , then [x, y] = [x, y] ′ , which is a contradiction. Hence there exists p ∈ [x, y] \ [x, y] ′ . Let [p, z] be the geodesic segment in [x, z] and [x, z] ′ the geodesic segment obtained by connecting [x, y] ′ and [y, z]. First we assume that X is an R-tree and construct a distance-preserving map from X to R. We may assume that X is not the single-point space. Take two distinct points q + , q − ∈ X and let q 0 be the midpoint between q + and q − . Next we show that. Then, Proof of Theorem. 12Each case yields a contradiction. x, q 0 ] ⊇ [q − , q 0 ], x ∈ [q − , q + ], or [q 0 , q + ] ⊆ [q 0 , x] for each x ∈ X. In order to prove this statement, consider geodesic segments [q − , q + ] and [q − , x]. If [q − , q + ] ∩ [q − , x] = {q − }, then [x, q + ] = [x, q − ] ∪ [q − , q + ] ⊇ [q − , q + ] by Proposition 3.1 and hence [x, q 0 ] ⊇ [q − , q 0 ]. When [q − , q + ] ∩ [q − , x] {q − }, we have x ∈ [q − , q + ] or [q − , q + ] ⊆ [q − , x] by Lemma 4.2. The latter condition implies [q 0 , q + ] ⊆ [q 0 , xThen [p, z] ∩ [x, z] ′ ⊇ [y, z] {z}. Hence, by (1), we have p ∈ [x, z] ′ or x ∈ [p, z]. Each case yields a contradiction. Proof of Theorem 1.10 (2) ⇒ (3). First we assume that X is an R-tree and construct a distance-preserving map from X to R. We may assume that X is not the single-point space. Take two distinct points q + , q − ∈ X and let q 0 be the midpoint between q + and q − . Next we show that [x, q 0 ] ⊇ [q − , q 0 ], x ∈ [q − , q + ], or [q 0 , q + ] ⊆ [q 0 , x] for each x ∈ X. In order to prove this statement, consider geodesic segments [q − , q + ] and [q − , x]. If [q − , q + ] ∩ [q − , x] = {q − }, then [x, q + ] = [x, q − ] ∪ [q − , q + ] ⊇ [q − , q + ] by Proposition 3.1 and hence [x, q 0 ] ⊇ [q − , q 0 ]. When [q − , q + ] ∩ [q − , x] {q − }, we have x ∈ [q − , q + ] or [q − , q + ] ⊆ [q − , x] by Lemma 4.2. The latter condition implies [q 0 , q + ] ⊆ [q 0 , x]. . X + Define, X − By X + , = { x ∈ X \| d(x, q − ) > d(x, q + ) } , X − := { x ∈ X \| d(x, q − ) < d(xDefine subspaces X + and X − by X + := { x ∈ X \| d(x, q − ) > d(x, q + ) } , X − := { x ∈ X \| d(x, q − ) < d(x, q + ) } . Note that for every x ∈ X \ {q 0 } we have that x ∈ X + if and only if x ∈. q 0 , q + ] or [q 0 , q + ] ⊆ [q 0 , x], and also we have that x ∈ X − if and only if x ∈ [q − , q 0 ] or [x, q 0 ] ⊇ [q − , q 0Note that for every x ∈ X \ {q 0 } we have that x ∈ X + if and only if x ∈ [q 0 , q + ] or [q 0 , q + ] ⊆ [q 0 , x], and also we have that x ∈ X − if and only if x ∈ [q − , q 0 ] or [x, q 0 ] ⊇ [q − , q 0 ]. . Suppose That X ∈ X \ (x + ∪ X −, that is, d(x, q − ) = d(x, q + ). Then x ∈ [q − , q + ] andSuppose that x ∈ X \ (X + ∪ X − ), that is, d(x, q − ) = d(x, q + ). Then x ∈ [q − , q + ] and R Alperin, H Bass, Length functions of group actions on Λ-trees, Combinatorial group theory and topology. Princeton, NJPrinceton Univ. PrR. Alperin and H. Bass, Length functions of group actions on Λ-trees, Combinatorial group theory and topology, Annals of mathematics studies, no. 111, Princeton Univ. Pr, Princeton, NJ, 1987, pp. 265 -378. On Forbidden Induced Subgraphs for Unit Disk Graphs. A Atminas, V Zamaraev, Discrete & Computational Geometry. A. Atminas and V. Zamaraev, On Forbidden Induced Subgraphs for Unit Disk Graphs, Discrete & Computational Geometry (2018). Algorithmic aspects of constrained unit disk graphs. H Breu, University of British ColumbiaPh.D. thesisH. Breu, Algorithmic aspects of constrained unit disk graphs, Ph.D. thesis, Univer- sity of British Columbia, 1996. Unit disk graph recognition is NP-hard. H Breu, D G Kirkpatrick, Computational Geometry. 91H. Breu and D. G. Kirkpatrick, Unit disk graph recognition is NP-hard, Computa- tional Geometry 9 (1998), no. 1, 3-24. Metric Spaces of Non-Positive Curvature. M R Bridson, A Haefliger, 10.1007/978-3-662-12494-9Grundlehren der mathematischen Wissenschaften. 319Springer Berlin HeidelbergM. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren der mathematischen Wissenschaften, vol. 319, Springer Berlin Hei- delberg, Berlin, Heidelberg, 1999, DOI: 10.1007/978-3-662-12494-9. Introduction to Λ-trees. I , World Scientific. 47032651I. Chiswell, Introduction to Λ-trees, World Scientific, Singapore ; River Edge, N.J, 2001, OCLC: ocm47032651. Hamiltonian-hereditary graphs. P Damaschke, Unpublished. P. Damaschke, Hamiltonian-hereditary graphs, Unpublished (1990). Forbidden subgraphs and the Hamiltonian theme. D Duffus, M S Jacobson, R J Gould, The theory and applications of graphs, 4th int. Conf., Kalamazoo/Mich. 1980. D. Duffus, M. S. Jacobson, and R. J. Gould, Forbidden subgraphs and the Hamil- tonian theme., 1981, Published: The theory and applications of graphs, 4th int. Conf., Kalamazoo/Mich. 1980, 297-316 (1981). M Farber, Characterizations of strongly chordal graphs. 43M. Farber, Characterizations of strongly chordal graphs, Discrete Mathematics 43 (1983), no. 2, 173-189. Characterizations and recognition of circular-arc graphs and subclasses: A survey. M C Lin, J L Szwarcfiter, Discrete Mathematics. 30918M. C. Lin and J. L. Szwarcfiter, Characterizations and recognition of circular-arc graphs and subclasses: A survey, Discrete Mathematics 309 (2009), no. 18, 5618- 5635. Graph Isomorphism for Unit Square Graphs. D Neuen, Leibniz International Proceedings in Informatics (LIPIcs). P. Sankowski and C. ZaroliagisGermanyDagstuhl5717Schloss Dagstuhl-Leibniz-Zentrum fuer InformatikD. Neuen, Graph Isomorphism for Unit Square Graphs, 24th Annual European Sym- posium on Algorithms (ESA 2016) (Dagstuhl, Germany) (P. Sankowski and C. Zaro- liagis, eds.), Leibniz International Proceedings in Informatics (LIPIcs), vol. 57, Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2016, pp. 70:1-70:17. Indifference graphs, Proof Techniques in Graph Theory. F S Roberts, Academic PressNew YorkF. S. Roberts, Indifference graphs, Proof Techniques in Graph Theory, Academic Press, New York; . London, London, 1969, pp. 139 -146. Structure theorems for some circular-arc graphs. A Tucker, Discrete Mathematics. 71A. Tucker, Structure theorems for some circular-arc graphs, Discrete Mathematics 7 (1974), no. 1, 167-195. Eigenschaften der Nerven homologisch-einfacher Familien R n. G Wegner, Göttingen UniversityPh.D. thesisG. Wegner, Eigenschaften der Nerven homologisch-einfacher Familien R n , Ph.D. thesis, Göttingen University, 1967. . Japan Japan Kurodamm@nbu, Japan Japan [email protected] [email protected]	[]
[ "DESIGN AND IMPLEMENTATION OF 5G E-HEALTH SYSTEMS, TECHNOLOGIES, USE CASES AND FUTURE CHALLENGES", "DESIGN AND IMPLEMENTATION OF 5G E-HEALTH SYSTEMS, TECHNOLOGIES, USE CASES AND FUTURE CHALLENGES" ]	[ "Senior Member, IEEEDi Zhang [email protected] \nSchool of Information Engineering\nInstituto de Telecomunicações\nis with the Federal University of Piauí (UFPI)\nZhengzhou University\nTeresina -PIChina. Joel J. P. C. Ro-, Brazil, Portugal\n", "Fellow, IEEEJoel J P C Rodrigues \nSchool of Information Engineering\nInstituto de Telecomunicações\nis with the Federal University of Piauí (UFPI)\nZhengzhou University\nTeresina -PIChina. Joel J. P. C. Ro-, Brazil, Portugal\n", "Yunkai Zhai [email protected] \nSchool of Information Engineering\nInstituto de Telecomunicações\nis with the Federal University of Piauí (UFPI)\nZhengzhou University\nTeresina -PIChina. Joel J. P. C. Ro-, Brazil, Portugal\n", "Di Zhang \nSchool of Management Engineering\nSchool of Fundamental Science and Eingineering\nNational Telemedicine Center, the National Engineering Laboratory for In-ternet Medical Systems and Applications,\nZhengzhou University\n450001ZhengzhouChina. Takuro Sato\n", "Yunkai Zhai \nWaseda University\nOokubo 3-4-1169-8555TokyoJapan\n" ]	[ "School of Information Engineering\nInstituto de Telecomunicações\nis with the Federal University of Piauí (UFPI)\nZhengzhou University\nTeresina -PIChina. Joel J. P. C. Ro-, Brazil, Portugal", "School of Information Engineering\nInstituto de Telecomunicações\nis with the Federal University of Piauí (UFPI)\nZhengzhou University\nTeresina -PIChina. Joel J. P. C. Ro-, Brazil, Portugal", "School of Information Engineering\nInstituto de Telecomunicações\nis with the Federal University of Piauí (UFPI)\nZhengzhou University\nTeresina -PIChina. Joel J. P. C. Ro-, Brazil, Portugal", "School of Management Engineering\nSchool of Fundamental Science and Eingineering\nNational Telemedicine Center, the National Engineering Laboratory for In-ternet Medical Systems and Applications,\nZhengzhou University\n450001ZhengzhouChina. Takuro Sato", "Waseda University\nOokubo 3-4-1169-8555TokyoJapan" ]	[]	Fifth generation (5G) aims to connect massive devices with even higher reliability, lower latency and even faster transmission speed, which are vital for implementing the e-health systems. However, the current efforts on 5G e-health systems are still not enough to accomplish its full blueprint. In this article, we first discuss the related technologies from physical layer, upper layer and cross layer perspectives on designing the 5G e-health systems. We afterwards elaborate two use cases according to our implementations, i.e., 5G e-health systems for remote health and 5G e-health systems for Covid-19 pandemic containment. We finally envision the future research trends and challenges of 5G e-health systems.	null	[ "https://arxiv.org/pdf/2106.05086v2.pdf" ]	235,377,005	2106.05086	2e95aa59a97f230df98a98cbea4f29633b1c0f33	DESIGN AND IMPLEMENTATION OF 5G E-HEALTH SYSTEMS, TECHNOLOGIES, USE CASES AND FUTURE CHALLENGES 10 Jul 2021 Senior Member, IEEEDi Zhang [email protected] School of Information Engineering Instituto de Telecomunicações is with the Federal University of Piauí (UFPI) Zhengzhou University Teresina -PIChina. Joel J. P. C. Ro-, Brazil, Portugal Fellow, IEEEJoel J P C Rodrigues School of Information Engineering Instituto de Telecomunicações is with the Federal University of Piauí (UFPI) Zhengzhou University Teresina -PIChina. Joel J. P. C. Ro-, Brazil, Portugal Yunkai Zhai [email protected] School of Information Engineering Instituto de Telecomunicações is with the Federal University of Piauí (UFPI) Zhengzhou University Teresina -PIChina. Joel J. P. C. Ro-, Brazil, Portugal Di Zhang School of Management Engineering School of Fundamental Science and Eingineering National Telemedicine Center, the National Engineering Laboratory for In-ternet Medical Systems and Applications, Zhengzhou University 450001ZhengzhouChina. Takuro Sato Yunkai Zhai Waseda University Ookubo 3-4-1169-8555TokyoJapan DESIGN AND IMPLEMENTATION OF 5G E-HEALTH SYSTEMS, TECHNOLOGIES, USE CASES AND FUTURE CHALLENGES 10 Jul 2021Takuro Sato, Life Fellow, IEEE Manuscript accepted by IEEE Communications Magazine on April 26, 2021. 978-1-5090-6008-5/17/$31.00Index Terms-remote healthCovid-19e-health5G Fifth generation (5G) aims to connect massive devices with even higher reliability, lower latency and even faster transmission speed, which are vital for implementing the e-health systems. However, the current efforts on 5G e-health systems are still not enough to accomplish its full blueprint. In this article, we first discuss the related technologies from physical layer, upper layer and cross layer perspectives on designing the 5G e-health systems. We afterwards elaborate two use cases according to our implementations, i.e., 5G e-health systems for remote health and 5G e-health systems for Covid-19 pandemic containment. We finally envision the future research trends and challenges of 5G e-health systems. I. INTRODUCTION The initial e-health system was introduced in 1955 by Cecil Wittson, M. D., to share lectures between University of Nebraska Medical Center (UNMC) and mental hospitals located in different places. In 1959, Albert J. Jutras reported remote diagnostic consultations based on the fluoroscopy images transmitted by coaxial cable. After that, e-health was adopted widely to provide e-diagnosis and e-treatment services. Dominant benefits of e-health attribute to the reduced time and manpower costs. In literature, e-health can be classified into two categories, i.e., outdoor scenario and indoor scenario. However, due to the weakness of prior generations of wireless technologies, such as fourth generation long-term evolution (4G LTE), the ambition of e-health was not fully accomplished. Different from existing generations of wireless technologies, fifth generation (5G) is an ideal solution for e-health's ambition. This is mainly because of the even faster transmission speed (especially the even faster uplink transmission speed) and the ultra-reliable and low latency communications (URLLC) of 5G [1], [2]. Thanks to the 5G technologies, we may provide 24 hours in-home health monitoring, timely e-diagnosis and etreatment services, which are of significant importance to the patient with chronic diseases such as cerebral stoke and myocardial infarction [3]. The 5G e-health systems can also play a significant role on containing the acute infectious diseases. This is achieved by the reduced physical contacts, accelerated disease information flow, and enhanced therapy capacity and efficiency. While deploying 5G e-health systems, we first need to identify the requirement of the specific 5G e-health service, and then tailor the necessary 5G technologies to cater to it. Under this premise, an identity management (IdM) framework for 5G e-health systems was proposed to connect different types of devices with different wireless access technologies [4]. It was found that the IdM framework can achieve the required security properties efficiently in the e-health systems. In [5], cloud computing-based Internet of things (C-IoT) was introduced to e-health systems, wherein the authors further investigated the energy efficiency performances of various proposals. However, as is known, existing studies about 5G e-health systems are mostly constrained to the network and application layers, perspective from the physical layer is limited [6]. In this article, we first investigate the 5G technologies on designing the 5G e-health systems from both physical layer and upper layers. We afterwards introduce two use cases according to our implementations, i.e., 5G e-health systems for remote health and 5G e-health systems for Covid-19 pandemic containment. On this basis, we exploit a series of field trials to compare the quality of service (QoS) and quality of experience (QoE) performances among the 4G LTE, 5G non-standalone (NSA) and 5G standalone (SA) deployments. The noticeable results about 5G from our field trials are: 1) The current 5G uplink transmission speed is about 100 Mbps, which is almost enough to transmit the high-definition medical streaming data; 2) The current 5G latency is about 10 ms, which is far away from the claimed less than 1 ms latency. We finally discuss the challenging issues on the worldwide deployment of 5G e-health systems. This article is a comprehensive survey about the 5G e-health systems from the cutting-edge enabling technologies to actual deployments, field trials and future challenges. We expect our work can attract more research attentions to accelerate its world-wide implementations, so that more people can share equal-quality medical service in spite of their living places and conditions. II. VERTICALS AND ENABLING TECHNOLOGIES OF 5G E-HEALTH SYSTEMS In this section, we first introduce the verticals and key performance indicators (KPIs) of 5G e-health systems. We afterwards elaborate the 5G enabling technologies of 5G ehealth systems. A. Verticals and KPIs of 5G e-Health Systems As depicted in Fig. 1 whereas in-hospital e-health (e.g., e-ward-round), remote health are the widely deployed indoor applications. The latest e-health vertical requirements are given in Table I, and their related acronyms are listed as follows 1 . • H1A, "educational surgery", H1B, "remote ultrasound examination", H1C, "paramedic support", H1D, "critical health event". • H2A, "the pillcam", which aims to test real-time transmission with feedback control of a pill camera (capsule video endoscopy) in order to improve the diagnosis effects. • H3A, "vital-sign patch prototype", H3B, "localizable tag". They use vital-sign patches with advanced geo-localisation to explore direct-to-cloud disposable vital-sign patches to enable continuous monitoring of ambulatory patients at anytime and anywhere. As we can see from Table I, most of the KPIs are beyond the reach of prior generations of wireless communications. Besides, it is also unpractical and unnecessary to connect all medical devices with optical fiber access networks. For instance, 24hours health monitoring service greatly relies on the wireless connections but not the optical-based wired connections. Furthermore, we cannot immediately establish a optical-based ehealth system while encountering some emergencies. Because of all these reasons, 5G becomes an ideal option for e-health systems. B. Enabling 5G Technologies for e-Health Systems 1) Physical Layer Technologies for 5G e-Health Systems: Massive multi-input multi-output (massive MIMO), millimeter wave (mmWave), full duplex (FD), non-orthogonal multiple access (NOMA) and intelligent reflecting surface (IRS) are symbolic 5G physical layer technologies. They can increase the spectrum and energy efficiencies (SEEs) of 5G e-health systems. This is achieved by scaling up the transceivers, allocating wider carrier bandwidth, enabling the simultaneous transmission/reception, sharing the same carrier frequency among multiple users and twisting the wireless channels with highly controllable and intelligent signal reflections. FD and NOMA also can reduce the latency and improve the throughput Recently, sparse vector coding (SVC)-based superposition is introduced to 5G e-health systems [7], which brings in lower block error rate (BLER) performance while maintaining a massive number of connected health monitoring devices. However, we cannot always get higher throughput or faster transmission speed simply by allocating wider carrier bandwidth. According to Shannon theory, carrier bandwidth parameter is involved at both the multiplier and the denominator sides, the benefit of wider carrier bandwidth thus becomes marginal as its value growing large. Due to blocking and path-loss effects, ultra-dense deployment (e.g., small/micro cell) is inevitable while implementing the massive MIMO and wider-bandwidthbased 5G e-health systems [8]. In addition, NOMA receiver needs to employ some sophisticated decoding mechanisms, such as successive interference cancellation (SIC). In this case, although 3rd generation partnership project (3GPP) has adopted NOMA as an optional uplink transmission scheme in 5G, it is still not a matured option, subsequent study on advanced decoding algorithm with low complexity will be needed [7]. 2) Upper Layer Technologies for 5G e-Health Systems: Cloud computing is one of the widely used upper layer technologies in 5G e-health systems. While applying, we can also invoke the big data, artificial intelligence (AI), and other state-of-the-art technologies to analyse the acquired data for precise diagnosis and treatment [9]. However, as a centralized computing technology, cloud computing might result in heavy burden to the core network. In order to remedy this, decentralized technologies such as distributed computing, fog (or edge) computing, are exploited to enable privilege access to entry point of the whole networking area to compute, store, communicate, and process data [10]. Apart from the widely used centralized and decentralized computing technologies in 5G e-health systems, nowadays increasing studies are on the medium access control (MAC) layer optimizations. Representative technologies are the age of information (AoI) and grant-free transmission (GFT), which are applied to the time-sensitive 5G e-health applications, for example, remote surgery and emergency rescue [11]. In literature, AoI aims to measure and optimize the information updates in a network to reduce the consumed time in terms of queuing and scheduling. GFT, on the other hand, enables user to immediately transmit data using the nearest pre-configured resources. Furthermore, AI-based network activity learning and decision making strategies can be adopted for different vertical requirements and appropriate performance requirements of complex mobility scenarios. 3) Cross Layer Technologies for 5G e-Health Systems: A cooperative architecture from physical, network, MAC and application layers is a prerequisite for cross layer technologies in 5G e-health systems. Among cross layer technologies, network slicing that can create different isolated end-to-end (E2E) slices [12] for medical verticals with shared physical infrastructures is an indispensable one. Network slicing can be divided into WAN slicing, service provider network (SPN) slicing and the core network (CN) slicing parts. While implementing, G.mtn (SPN technology proposed for 5G transport and approved by the international telecommunication union's telecommunication standardization sector (ITU-T), which is compatible with the Ethernet ecosystem and based on slicing the Ethernet core) and flexible Enternet (FlexE) are used to create SPN slices. Software-defined networking (SDN) and network functions virtualization (NFV) technologies are used to create, isolate, and recycle the CN slices from the core network. Besides, wireless caching and mobile edge computing (MEC) proactively cache the hot contents for subsequent sharing to reduce the latency and network burden, which is also widely used in WAN slices. In general, 5G e-health cannot be solely achieved by any technology mentioned above, a joint cross layer force is a better choice. For instance, mmWave-massive-MIMO opens up a new solution for 5G e-health systems [8], and the full duplex-NOMA-based on a decentralized network architecture has better performance in terms of capacity and latency while compared to other schemes [1]. Machine learning framework with both model-based method and data-based method can be applied in 5G network to estimate the channel condition with or without prior information, which yields better system feasibility [13]. Recently, intelligent communications is emerging as another indispensable topic of 5G e-health systems. It enables the network to predict the forthcoming activities from previous activities, and to allocate the eligible network resources according to the needs of these forthcoming activities. III. USE CASES AND FIELD-TRIALS OF 5G E-HEALTH SYSTEMS In this section, we first elaborate two typical use cases in line with our implementations. We afterwards introduce our field trials and the main test results of our implementations. For them, seeing a doctor is a challenge not only for the money cost but also for the long-distance traveling expenses. In order to provide them some high-quality medical infrastructures, we have connected most of their community clinics and prefecture-level hospitals to the center hub located in the first affiliated hospital of Zhengzhou university (FAHZZU) with 5G wireless networks. In this case, patients from these remote or mountainous areas do not need to go to the capital city to see a doctor while enjoying the equal quality medical service as in FAHZZU. A. 5G e-Health Systems for Remote Health In our implementation of 5G e-health systems for remote health, high-definition (HD) medical videos and images are transmitted among remote hospital/clinic, hub hospital and mega hospital. Medical experts from these hub and mega hospitals thus can remotely join in the diagnosis, and provide professional suggestions, diagnoses and treatments for patients in remote clinics, as in Fig. 2. However, due to the technical bottlenecks and potential risks of URLLC (as shown in this figure, the latency is much higher than the claimed 1 ms 2 ), we didn't demonstrate the critical remote health applications of 5G e-health systems, such as remote surgery. B. 5G e-Health Systems for Covid-19 Pandemic Containment It is widely agreed that reducing physical contract is needed to block the spread of Covid-19. Staying at home is adopted world widely as a common sense. Most of the schools, stores and entertainment places are closed during this pandemic. It 2 Here we adopted the 5G NSA deployment, more details about the current 5G field trial results can be found in section III-C. might be easy for ordinary people to keep the social distance, yet how to reduce the physical contract between medical staff and the infected citizen is a challenge. Thanks to 5G e-health systems, we can combine the vital-sign sensors with connectivity to other monitoring devices to remotely monitor the infected citizens. In this case, physical contacts between medical staffs and infected citizens are greatly reduced. Additionally, 5G e-health systems also enables the communication capacity from remote negative pressure isolation wards, prefecture-level hospitals, communities, mega hospitals and municipal control centers to accelerate the information exchanges about Covid-19, so that correct and effective containment methods can be immediately and widely adopted. From the first day Wuhan was closed (January 29), we activated our 5G e-health systems, and connected more prefecturelevel hospitals, community clinics to the mega hospital, FAHZZU, with 5G wireless networks. Thanks to the 5G ehealth systems, infected citizens could be remotely monitored and diagnosed by the respiratory and pneumology medical experts from our hub and mega hospitals. In our implementation case, 5G e-health systems provided a direct communication channel for the medical doctors and their infected patients with reduced physical contacts, and it helped the medical doctor to timely diagnose a large number of infected citizens from different places. It also offered e-training and online discussion functions to expedite the latest information flow about Covid-19. We traced the infected citizens in Henan from January 27, 2020 to February 05, 2020 during this Covid-19 pandemic. There were 161 infected cases including 38 severe cases were remotely treated and diagnosed. With the help of 5G e-health systems, although infected cases grew quickly, severe cases and passed away cases were contained. In addition to the 5G e-health systems, hierarchical diagnosis and treatment is another indispensable method to contain the Covid-19 pandemic. In Henan province, we left the mild cases to medical doctors of the communities, clinics, and prefecturelevel hospitals, whereas medical experts from the hub and mega hospitals were devoted to remotely monitor and offer actual treatment for the severe cases if needed. According to our experience, giving enough isolation wards, the pandemic can be well contained. However, if the test-positive citizen number exceeded that of isolation ward, outbreak subsequently came. C. Field Trials of 5G e-Health Systems In order to evaluate the technical effectiveness of 5G technologies in e-health systems, we employed a series of field trials to compare the 4G LTE, 5G NSA and 5G SA deployments. As depicted in Fig. 4, 5G NSA adopted the option-3x, and 5G SA used the option-2 in line with 3GPP Release-15. We first divided the field trials into static scene and moving scene. In the static scene, distance between BS and user terminal is about 50 m, and the moving speed of user terminal is 30 Km/s in the moving scene. In addition to that, NC was to denote the network congestion, and 4.9 GHz carrier frequency with a total 1 GB carrier bandwidth was employed in the field trials. Field trial results of our 5G e-health systems are given in Table II. As shown here, the performances of 5G in terms of transmission speed and latency outperform the 4G LTE's. Additionally, downlink speed of the moving scene is slightly lower than that with static scene, whereas the uplink speeds are almost of the same value. On the other hand, while encountering NC, downlink speed of the shared SA is greatly reduced, whereas NC has less effect to the private SA deployment. In this case, dedicated privacy SA might be needed while implementing the 5G e-health systems. Moreover, as we can see from this table, the claimed less than 1 ms latency still cannot be achieved by the current 5G technologies. Subsequent studies and optimizations are still needed in the forthcoming evolution of 5G. IV. FUTURE RESEARCH TRENDS AND CHALLENGES Prior discussions demonstrate that the current 5G technologies still cannot fully sustain the e-health systems, especially for the URLLC related applications. Besides, medical application, ethic issue, and international standardization, are some future challenges while deploying the 5G e-health systems. A. URLLC As mentioned in Fig. 2 and Table. II, latency ability of the current 5G technologies is still far away from the claimed less than 1 ms latency requirement of URLLC applications. In addition to that, less studies have been done on the communication reliability. Although re-transmission and GFT can enhance the communication reliability, it will bring in higher transmission latency performance. On the contrary, applications of 5G ehealth systems sometimes rely on both ultra-reliable and low latency wireless connections. For future studies, some methods that can improve both latency and reliability performances will be needed. To this end, a combination of multiple technologies from a cross layer perspective might be a feasible solution. For instance, in order to achieve performance targets of low latency and high reliability communications, combining the physical layer technologies (e.g., SVC, mmWave) and upper layer technologies (e.g., AoI, customized network slicing, MEC), and some other emerging technologies might be a practicable method. B. Medical Applications In literature, image processing-based precision medicine technology is adopted to remedy the weak-points of human eyes on detecting the invisible negative lesions. Medical image processing technologies also enable the doctor to trace-back, compare and forecast potential diseases. Due to large scale and even fast growing data volume, effective and fast medical image processing algorithms are required to reduce the process complexity, and to enhance the accuracy in terms of detection and forecast. To cope with these trends, endeavors on high capacity processing chips will be needed in the future. Precious medical treatment with 5G e-health systems relies on the multi-sources heterogeneous health and clinic data fusing and mining technologies. Medical data fusing and mining ask us to identify the specific fusion level, and extract the feature information. This is a challenging task due to the uncertainly, incompleteness and unstable features of multi-source medical data. Current solutions such as information processing and estimation, statistical inference, decision theory, etc., are incapable to cope with the massive and even growing trends of medical data. For future studies, big data and artificial intelligenceassisted processing methods that can process the large volume data more effectively will be some interesting topics. On the other hand, molecular communication technology that investigates the nano-scale molecular communication mechanisms, is emerging as another hot topic of 5G e-health applications. We may also invoke the molecular communication technology for precious targeted drug delivery, health monitoring, regenerative medicine and genetic engineering [14]. C. Ethical Challenges While sharing the high-quality medical resources to achieve individual opportunities and exercise individual autonomy with 5G e-health systems, fairness legitimacy and criterion are needed. It is the social obligation to offer open discussion, technical training and education with real information about the 5G e-health systems. For instance, according to the European convention on human rights (ECHR), personal data is a distinct right and needs to be controlled over and be free from intrusion into the person's life [15]. In future, while using the 5G e-health systems for e-diagnosis and e-treatment, patient's dignity and autonomy need to be considered for the storage, access and share of the health information. Personal information and health data are collected and transmitted with the massive connected smart devices in 5G e-health systems. The data might be leaked out or eavesdropped by malicious hack attacks during the collection and transmission periods. Information protection thus is emerging as a significant issue of 5G e-health systems. Previous studies on secured transmission and information protection greatly rely on the passive exogenous security methods. Therefore, attackers can always update its attack tools or bypass these protective methods. Different from these passive exogenous security methods, endogenous security uses the intrinsic property of wireless communications to enhance the transmission security, which can defense the escalating attack methods. In endogenous security studies, physical layer technologies such as adaptive beamforming, fingerprint-based transceiver classification and verification, upper layer technologies such as mimicry security and AI-based physical layer authentication, are emerging topics for future studies. D. International Standardization International standard has a key role for interoperability, efficiency and safety issues of 5G e-health systems across nations. Global consensus on 5G e-health systems from both design and implementation perspectives needs to be reached first before its world widely deployment. As we know that, adoption of 5G ehealth systems requires extensive sharing of information from hospitals to hospitals, organizations to organizations. However, conditions vary from country to country, region to region, which makes the uniform standard across nations a challenging issue. Secondly, legitimacy and ethical law's diversities from places to places, geographical and landscape differences across countries, are making the international standardization of 5G e-health systems an even challenging issue. For future studies, international agreements on unite efforts from different international associations (e.g., ITU, 3GPP) might be required, and the international standardization actions should recognize the differences across nations and regions. V. CONCLUSION This article is a humble attempt to provide a general scope on design and implementation of 5G e-health systems. We discussed the 5G technical solutions for e-health systems, elaborated two use cases according to our implementations, i.e., 5G e-health systems for remote health and 5G e-health systems for Covid-19 pandemic containment. We employed some field trials to compare the network performance among 4G LTE, 5G NSA, 5G shared SA and 5G private SA, and found that substantial studies on URLLC are still needed. We finally outlined the medical applications, ethical and international standardization challenges about the 5G e-health systems. Fig. 2 : 25G e-health systems for remote health. Fig. 2 2is our implementation of 5G e-health systems for remote health. Statistics show that there are 96.4 million residents in Henan province, and about half of them living in remote or mountainous areas with limited high-quality medical infrastructures (according to the statistical bureau of Henan province, 2018). Fig. 4 : 4Field trials of 5G e-health systems with 4G LTE, NSA and SA deployments. , health monitoring, emergency rescue are typical outdoor applications of 5G e-health systems,0 In-hospital health Remote diagnosis and treatment Emergent rescue In-home health health monitoring Telemedicine vehicle · Physical layer: Massive MIMO, mmWave, FD, NOMA, IRS, D2D, etc. · Upper layer: Cloud/fog computing, MAC layer optimizations, etc. · Cross Layer: HetNets, network slicing, MEC, etc. Related 5G technologies: Typical outdoor applications Typical indoor applications Small cell Small cell Macro cell Edge server 0 0 Fig. 1: 5G e-health applications and related technologies. TABLE I : I5G e-health vertical KPIs.Requirements H1A H1B H1C H1D H2A H3A H3B Unit Downlink speed 60 200 10 10 1 \ \ Mbps Uplink speed 60 200 10 10 1 \ \ Mbps Latency 25 25 200 200 3 1000 1000 Msec Mobility 50 \ 160 50 \ 200 200 Km/h Interactivity 1 1000 1 1 \ \ \ Transaction/sec Area traffic capacity 60 200 10 10 100 1 1 Mbit/sec/m 2 performance [1]. With the aid of beamforming and directional antenna technologies, we may further extend the coverage area and improve the transmission speed of 5G e-health systems. Besides, device-to-device communications (D2D) can create some localized direct-communication area with or without the help of base station (BS) to offload the wireless access network (WAN) traffic load, and to reduce the transmission latency. 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[ "Covariate-Balancing-Aware Interpretable Deep Learning models for Treatment Effect Estimation", "Covariate-Balancing-Aware Interpretable Deep Learning models for Treatment Effect Estimation" ]	[ "Kan Chen ", "Qishuo Yin \nGraduate Group of Applied Mathematics and Computational Science\nSchool of Arts and Sciences\nUniversity of Pennsylvania\nPhiladelphiaPennsylvaniaU.S.A\n", "Qi Long \nDivision of Biostatistics\nPerelman School of Medicine\nUniversity of Pennsylvania\nPhiladelphiaPennsylvaniaU.S.A\n" ]	[ "Graduate Group of Applied Mathematics and Computational Science\nSchool of Arts and Sciences\nUniversity of Pennsylvania\nPhiladelphiaPennsylvaniaU.S.A", "Division of Biostatistics\nPerelman School of Medicine\nUniversity of Pennsylvania\nPhiladelphiaPennsylvaniaU.S.A" ]	[]	Estimating treatment effects is of great importance for many biomedical applications with observational data. Particularly, interpretability of the treatment effects is preferable to many biomedical researchers. In this paper, we first give a theoretical analysis and propose an upper bound for the bias of average treatment effect estimation under the strong ignorability assumption. The proposed upper bound consists of two parts: training error for factual outcomes, and the distance between treated and control distributions. We use the Weighted Energy Distance (WED) to measure the distance between two distributions. Motivated by the theoretical analysis, we implement this upper bound as an objective function being minimized by leveraging a novel additive neural network architecture, which combines the expressivity of deep neural network, the interpretability of generalized additive model, the sufficiency of the balancing score for estimation adjustment, and covariate balancing properties of treated and control distributions, for estimating average treatment effects from observational data. Furthermore, we impose a so-called weighted regularization procedure based on non-parametric theory, to obtain some desirable asymptotic properties. The proposed method is illustrated by re-examining the benchmark datasets for causal inference, and it outperforms the state-of-art.	10.48550/arxiv.2203.03185	[ "https://arxiv.org/pdf/2203.03185v1.pdf" ]	247,292,053	2203.03185	2f1700a1f6511d4957674e9242340cde3fda0df0	Covariate-Balancing-Aware Interpretable Deep Learning models for Treatment Effect Estimation 7 Mar 2022 Kan Chen Qishuo Yin Graduate Group of Applied Mathematics and Computational Science School of Arts and Sciences University of Pennsylvania PhiladelphiaPennsylvaniaU.S.A Qi Long Division of Biostatistics Perelman School of Medicine University of Pennsylvania PhiladelphiaPennsylvaniaU.S.A Covariate-Balancing-Aware Interpretable Deep Learning models for Treatment Effect Estimation 7 Mar 20221Covariate Balancing PropertyTreatment Effect EstimationInterpretable Deep Learning * Estimating treatment effects is of great importance for many biomedical applications with observational data. Particularly, interpretability of the treatment effects is preferable to many biomedical researchers. In this paper, we first give a theoretical analysis and propose an upper bound for the bias of average treatment effect estimation under the strong ignorability assumption. The proposed upper bound consists of two parts: training error for factual outcomes, and the distance between treated and control distributions. We use the Weighted Energy Distance (WED) to measure the distance between two distributions. Motivated by the theoretical analysis, we implement this upper bound as an objective function being minimized by leveraging a novel additive neural network architecture, which combines the expressivity of deep neural network, the interpretability of generalized additive model, the sufficiency of the balancing score for estimation adjustment, and covariate balancing properties of treated and control distributions, for estimating average treatment effects from observational data. Furthermore, we impose a so-called weighted regularization procedure based on non-parametric theory, to obtain some desirable asymptotic properties. The proposed method is illustrated by re-examining the benchmark datasets for causal inference, and it outperforms the state-of-art. Introduction We consider the problem of estimating treatment effect using observational data in this paper. In causal inference, observational data refers to the information obtained from a sample or population that independent variables cannot be controlled over by the researchers because of ethical concerns or logistical constraints (Rosenbaum et al. (2010)). In contrast to observational data, experimental data or randomized control trials (RCT) are viewed as a "gold standard" in causal inference but expensive and somewhat impossible to get in many circumstances. However, to estimate causal effect from observational data, we first need to address all possible confounders or confounding variables that influence both treatment and outcome variables. For instance, a certain therapy may tend to be given to the younger patients ( < 30 years old) because the therapy is more effective to junior group. Hence, failure to adjust the confounding variables "age" may lead a biased even incorrect estimation of the treatment effect. Specifically, in this paper, we work under the "no-hidden confounding" or "strong ignorability" assumption, that is, there is no unobserved confounding variables. We consider the estimation of effect of treatment A (e.g. assignment of a certain therapy) on an outcome (e.g. recover or not) adjusting for covariates X (e.g. demographic status of patients). Conventional approach for estimation of treatment effects consists of two steps: first, fitting models for expected outcomes and propensity score, respectively; second, plugging the fitted models into a downstream estimator of the treatment effects. Neural network is a natural choice for the first step because of its strong predictive performance ; Johansson et al. (2016); Louizos et al. (2017); ; Alaa and van der Schaar (2017); Schwab et al. (2018); Yoon et al. (2018); Farrell et al. (2018); Shi et al. (2019); Bica et al. (2020)) but it is widely being regarded as a "black-box" predictor which lacks interpreability (Agarwal et al. (2020)). The main contributions of this paper can be summarized as follows. By viewing the error in predicting outcomes under the observed treatment assignment actually received as the training error and the error in predicting outcomes under the counter-factual treatment assignment (i.e. the opposite of the observed treatment assignment) received as the testing error, we first derive a bound on the error in estimating average treatment effect. We then propose a novel approach for estimating average treatment effect via exploiting the insights gained from the bound as well as leveraging recent-developed neural additive models (NAMs) to balance the trade-off between its predictive performance and interpretability. Motivated by the downstream weighted estimator of average treatment effect and non-parametric estimating theory, we impose a so-called weighted regularization process in the training process to reduce the bias of estimating average treatment effect and obtain some desired asymptotic properties such as efficiency and double robustness. And we demonstrate the power of our proposed model across different methods by using two causal inference benchmark datasets: IHDP (Hill (2011)) and ACIC (Mathews and Atkinson (1998)). 2 Related Works and Outline Shalit et al. (2017) gave a novel generalization-error bound on the error of individual treatment effect. Later, Johansson et al. (2020) proposed error bounds on the bias in estimated conditional average treatment effect based on the Integral Probability Metrics (IPM) distance between groups receiving different level of treatment. Our paper differs from the prior work in the sense that we implement energy balancing weights (EBWs, Huling and Mak (2020)) based on the weighted energy distance (WED) metrics in the bounds. The bounds we obtain are asymptotically tighter than the prior works because the distance between groups receiving different level of treatment can be ignorable if optimal weights are found. See Section 4 for details. Furthermore, the energy balancing weights can be viewed as a type of balancing score (Rosenbaum and Rubin (1983)). Prior works such as Shi et al. (2019) implemented propensity score in estimating average treatment effect. In comparison to propensity score, energy balancing weights have several advantages: 1.) energy balancing weights are non-parametric, hence, we may remove the potential caveat of miss-specification of propensity models; 2.) energy balancing weights take into account the balancing property of the observed covariates from treated group and control group under finite sample, which leads the reduction of bias in estimating average treatment effect. We exploit energy balancing weights in the process of estimating average treatment effect. The rest of this paper is organized as follows. Section 3 introduces the problem setup, notation, and some preliminary definitions for conducting theoretical analysis. Section 4 states our theoretical results and proposes a methodology implementing these results. Section 5 demonstrates the simulation results. Section 6 discusses some potential extensions. Preliminaries Throughout this paper, we consider the estimation of average treatment effect of a binary treatment. Consider a sample {(Y i , A i , X i )} of size n from a population, where Y i is the observed outcome of i-th unit, A i ∈ {0, 1} is a binary indicator of receiving treatment, and X i = (x 1i , x 2i , · · · , x pi ) ∈ R p is a p-dimensional covariate vector of unit i. We also let n 1 be the number of treated units, and n 0 be the number of control units. Hence, n 0 = n − n 1 . Following the potential outcome framework Rubin (1974); Splawa-Neyman et al. (1990), each unit i has two potential outcomes, Y i (1) and Y i (0) that unit i would have under the treatment and control, respectively. It follows that the observed outcome Y i = A i Y i (1)+(1−A i )Y i (0) . We make the standard stable unit treatment value assumption (SUTVA), that is, "the observation on one unit should be unaffected by the particular assignment of treatment to other units" (Cox (1958)). Under SUTVA, the observed outcome is consistent with the potential outcome in the sense that Y i = Y i (A i ). We further assume that the treatment assignment mechanism is strongly ignorable and positivity in the propensity score: {Y i (0), Y i (1)} \|= A i \|X i , 0 < P(A i = 1\|X i = x) < 1. In other words, this is saying that there is no unmeasured confounders and every unit has a chance to receive the treatment. Using the notation of potential outcome framework, the average treatment effect (ATE) is defined as τ ≡ E(Y (1) − Y (0)) which could be estimated asτ = 1 n n i=1 Y i (1) − Y i (0) if both potential outcomes were observed. Additionally, we let F 1 (x) ≡ P(X ≤ x\|A = 1), F 0 (x) ≡ P(X ≤ x\|A = 0) denote the cumulative distribution function (CDF) of covariate x in the treatment group and the control group, respectively. It follows that the CDF of X in the entire population is F (x) ≡ P(X ≤ x) = F 1 (x)P 1 +F 0 (x)P 0 , where P 1 ≡ P(A = 1)andP 0 ≡ P(A = 0). If we further let µ 1 (X i ) ≡ E(Y (1)\|X i ), µ 0 (X i ) ≡ E(Y (0)\|X i ) be the expected conditional treated and control outcomes based on the observed covariate X i for unit i, then the average treatment effect (ATE) can also be written as τ = R p µ 1 (x) − µ 0 (x) dF (x). And the individual treatment effect (ITE) for unit i is: φ(X i ) = µ 1 (X i ) − µ 0 (X i ). Hence, the estimated ITE and ATE can be written aŝ φ(X i ) =μ 1 (X i ) −μ 0 (X i ) τ = R pφ (x)dF n (x) whereμ 1 (X i ) andμ 0 (X i ) are the predicted expected outcomes in the treated and control groups given observed covariate X i and F n (x) = 1 n n i=1 1(X ≤ x) is the empirical CDF. Of note, the ATE measures the difference in average outcomes between units assigned to the treatment and units assigned to the control and captures population-level causal effects, whereas the ITE captures the individual level causal effect using the difference Y i (1) − Y i (0). The difference between Y i (1) − Y i (0) and µ 1 (X) − µ 0 (X) is that Y i (1) − Y i (0) is unique to an individual and may not be described exactly by a set of units. µ 1 (X) − µ 0 (X) can be used to describe the average treatment effect within a subgroup of units if X represents the observed covariates from a set of units. We also call it conditional average treatment effect (CATE). Definitions To facilitate the theoretical analysis for the upper bound of individual treatment effect, We state the following definitions. Definition 1. Let L : Y × Y → R + be the squared loss function L(y, y ) = y − y 2 . Then the expected pointwise loss is lμ a (x) ≡ Y L(µ a (x),μ a (x))p(Y (a)\|x)dY (a). The expected factual and counterfactural losses are: R F (μ a ) ≡ R p ×{0,1} lμ a (x)dF a (x)da R CF (μ a ) ≡ R p ×{0,1} lμ a (x)dF 1−a (x)da. R F (μ a , w) ≡ R p ×{0,1} lμ a (x)dF n,a,w (x)da R CF (μ a , w) ≡ R p ×{0,1} lμ a (x)dF n,1−a,w (x)da whereμ a (x) is the predicted expected conditional outcomes at treatment level A = a given observed covariate x, and F n,a,w (x) = n i=1 w i 1(X i ≤ x, A i = a)/n a is the weighted ECDF of data points at treatment level a with weights w = (w 1 , · · · , w n ) that satsify n i=1 w i A i = n 1 , n i=1 w i (1−A i ) = n 0 , and w i ≥ 0. To help understand Definition 1, consider the following example. If x is the demographic status of patients, a is the treatment level, and Y (a) is the potential outcome such as recovery or not given treatment level a, then R F measures the predictive performance under the treatment actually received. On the other hand, R CF measures the predictive performance under a counter-factual treatment assignment that is the opposite of the treatment actually received. From the perspective of machine learning, R F can be viewed as training error and R CF can be viewed as testing error. Definition 2. The expected loss in estimation of average treatment effect is R AT E (μ a ) ≡ R p (τ (x) − τ (x)) 2 dF (x). R AT E,n (μ a ) ≡ R p (τ (x) − τ (x)) 2 dF n (x). And when n → ∞, R AT E (μ a ) = R AT E,n (μ a ). Our proof relies on the notion of a Weighted Energy Distance metric by Huling and Mak (2020), which is a distance metric between two probability distributions. For two distributions G(x), H(x) defined on R p , we define the energy distance as follows, Definition 3 (Energy Distance (Cramér (1928))). The energy distance between two distributions G and H is E(G, H) ≡ 2 R p (G(x) − H(x)) 2 dx. (1) It can also be written as Székely (2003) for the equivalence of (1) and (2). E(G, H) = 2E Z − V 2 − E Z − Z 2 − E V − V 2 (2) where Z, Z i.i.d ∼ G, V, V i.i.d ∼ H. See proofs in When both G and H are empirical cumulative distribution functions (ECDF), i.e. G n is the ECDF of {Z i } n i=1 , H m is the ECDF of {V i } m i=1 , then E(G n , H m ) = 2 nm n i=1 m j=1 Z i − V j 2 − 2 n 2 n i=1 m j=1 Z i − Z j 2 − 2 m 2 n i=1 m j=1 V i − V j 2 . Motivated by the definition of energy distance, Huling and Mak (2020) proposed a weighted modification of this distance metric as follows. Definition 4. The weighted energy distance between F n,a,w and F n is defined as E(F n,a,w , F n ) = 2 n a n n i=1 n j=1 w i 1(A i = a) X i − X j 2 − 2 n 2 a n i=1 n j=1 w i w j 1(A i = A j = a) X i − X j 2 − 2 n 2 n i=1 n j=1 X i − X j 2 . In other words, E(F n,a,w , F n ) is the energy distance between the ECDF of a sample {X i } n i=1 and a weighted ECDF of {X i } i:A i =a . The weight w here can be viewed as a balancing score which can be used for substituting propensity score. More discussion is included in Section 4. Main Result Bias in Estimation of Average Treatment Effect We first state the results bounding the counter-factual loss, which serve as the foundation for bounding the bias for estimation of average treatment effect. We consider both the unweighted version and the weighted version. The proofs and details can be found in the supplementary material. Lemma 1. Following the notation defined in Section 3, we have R CF (μ a ) ≤ P 0 R F (μ 1 ) + P 1 R F (μ 0 ) + C lμ a E(F 1 , F ) + E(F 0 , F ) . (3) where C lμ a = ∂ p [−P 0 lμ 1 + P 1 lμ 0 ]/∂x p ) 2 / √ 2 is a constant. Furthermore, if we equip Inequality (3) with weights w such that n i=1 w i A i = n 1 , n i=1 w i (1 − A i ) = n 0 , w i ≥ 0, when n → ∞, we then have R CF (μ a , w) ≤ P 0 R F (μ 1 , w) + P 1 R F (μ 0 , w) + C lμ a E(F n,1,w , F n ) + E(F n,0,w , F n ) .(4) Theorem 1. Under the conditions of Lemma 1, we have R AT E (μ a ) ≤ 2 R F (μ 1 ) + R F (μ 0 ) + C lμ a E(F 1 , F ) + E(F 0 , F ) .(5) Furthermore, if we equip Inequality (5) with weights w such that n i=1 w i A i = n 1 , n i=1 w i (1−A i ) = n 0 , w i ≥ 0, when n → ∞, we then have R AT E (μ a ) = R AT E,n (μ a ) ≤ 2 R F (μ 1 , w) + R F (μ 0 , w) + C lμ a E(F n,1,w , F n ) + E(F n,0,w , F n ) .(6) Following the similar analysis approach in Shalit et al. (2017); Johansson et al. (2020), the basic idea of the proof of Theorem 1 is to bound R CF (·) using R F (·) and distance between the treated and control distributions, the latter of which leverages properties of energy distance investigated in Huling and Mak (2020). To better understand Theorem 1, we can decompose the bound into two parts: training error for factual outcomes, and the distance between treated and control distributions. We highlight the differences between our result and the prior result. First, our target estimator is for the average treatment effect (ATE) in comparison to the individual treatment effect (ITE) from Shalit et al. (2017) and the conditional average treatment effect (CATE) from Johansson et al. (2020). Second, we use weighted energy distance between F n,a,w and F n in the analysis of our bound. The benefit of using weighted energy distance comes from the key observation that if w * satisfies w * ∈ arg min w E(F n,1,w , F n ) + E(F n,0,w , F n ) n i=1 w i A i = n 1 , n i=1 w i (1 − A i ) = n 0 , w i ≥ 0 (7) then lim n→∞ E(F n,1,w * , F n ) = 0, lim n→∞ E(F n,0,w * , F n ) = 0 almost surely. See proof of Theorem 3.1 with slightly variation in the objective function in Huling and Mak (2020). This implies that Inequality (6) becomes R AT E (μ a ) = R AT E,n (μ a ) ≤ 2 R F (μ 1 , w * ) + R F (μ 0 , w * ) . which generates much tighter bound when the training error for factual outcomes is minimized. Motivated by Theorem 1, we propose a new objective function equipped with interpretable deep learning model to estimate the average treatment effect using observational data. Proposed Objective Function Before proposing our objective function, we need to state the definition of balancing score and a classic result from Rosenbaum and Rubin (1983). Definition 5 (Balancing Score). A balancing score b(X) is a function of the observed covariate X such that the conditional distribution of X given b(X) is the same for treated and control units (i.e. A \|= X\|b(X)). The weights w defined in Definition 4 is a type of balancing score since A \|= X\|w. Based on the definition of balancing score, we can state the classic result about the sufficiency of balancing score. Theorem 2 (Rosenbaum and Rubin (1983)). If the average treatment effect is identifiable from observational data by adjusting for X (i.e. τ = E E(Y (1)\|X) − E(Y (0)\|X) ), then we have τ = E E(Y (1)\|X, w) − E(Y (0)\|X, w) . Propensity score P (A = a\|X = x) is a popular balancing score used for estimation of average treatment effect. However, propensity score does not directly takes into account the balancing property of treated and control distributions under finite sample without further modification. If the treated distribution differs from the control distribution or the propensity score model is incorrectly specified, it may lead a biased estimation of propensity score. Hence, the estimation of average treatment effect is likely to be biased. To handle this issue, we propose to use the weights w from the weighted energy distance as our balancing score in the objective function. The sufficiency of the balancing score tells us that to obtain an unbiased estimation of average treatment effect, it suffices to adjust for the observed covariate X that is relevant to the estimation of balancing score. Therefore, we train the model by minimizinĝ θ,ŵ = arg min θ,wR (X, A; θ, w) s.t. n i=1 w i A i = n 1 , n i=1 w i (1 − A i ) = n 0 , w i ≥ 0 wherê R(X, A; θ,w) = 1 n n i=1 Y i − Q NN (X i , A i ; θ) 2 + α E(F n,1,w , F n ) + E(F n,0,w , F n ) .(8) In Equation (8), α ∈ R + is a hyperparameter. The fitted modelQ = Q NN represents neural network models. AndQ implicitly contains w. With the fitted modelQ in hand, we can then estimate the average treatment effect using some downstream estimators such aŝ τ = 1 n n i=1 Q (X i , 1; θ) −Q(X i , 0; θ) . Under some mild assumptions, the algorithmic convergence of training error for factual outcomes predictions is guaranteed under Neural Tangent Kernel (NTK) regime (Du et al. (2018)). Theorem 3 (Du et al. (2018)). Let w * be as defined in (7). We consider a neural network of the form Q NN (X i , A; θ) = 1 √ m m r=1 a r σ((θ r ) (X i , A)) where θ r ∈ R p is the weight vectors in the first hidden layer connecting to the r-th neuron, a r ∈ R is the output weight, σ(·) is the ReLU activation function, and m is the width of hidden layer. Assuming that ∀i, (X i , A i ) 2 = 1, \|Y i (a)\| < C for some constant C, m = Ω(n 6 /λ 4 0 δ 3 ), and we i.i.d initialize θ r ∼ N (0, I), a r ∼ Unif{−1, 1} for r ∈ [m], then with at least probability at 1 − δ over the initialization, we have the linear convergence rate forR(X, A; θ, w * ) under gradient descent algorithm. Here λ 0 = λ min (H ∞ ), where (H ∞ ) ij = 1 2 − arccos((X i , A i ) (X i , A i )) 2π (X i , A i ) (X i , A i ) . Detail explanation and implementation of the NTK theory can be found in Du et al. (2018);Bu et al. (2021). The algorithmic convergence of training error provides much tighter bound for the error in estimation of average treatment effect, which leads an impressive estimating performance. While deep learning models are very impressive in terms of predictive performance, the power of the models often comes at the expense of lack of interpretability. To balance the trade-off between predictive performance and interpretability, Agarwal et al. (2020) proposed the neural additive models (NAMs) which combine some of the expressivity of DNNs with the interpretability of generalized additive models. Such models can be particularly useful for high stakes decisionmaking domains such as medicine where interpretability is highly desired or even necessary. We use NAMs to predict factual and counterfactual outcomes as follows, Y i (a) = Q NAM (X i , a; θ) = p j=1 q j (x ij , a; θ j ) where X i = (x i1 , · · · , x ip ) ∈ R p , and Q NAM (·, ·; θ) is the fitted model, q j is a univariate shape function. Figure 1 shows the architecture of additive neural network, where z ij ∈ R is the shared representation for q j (·, 1; θ j ) and q j (·, 0; θ j ). We may replace the Q NN with the Q NAM in the objective function (8) for the interpretability under certain tasks. Covariate Balancing Score Weighted Regularization We now turn to weighted regularization, that is an modification of objective function used for training. This modified objective is motivated by the weighted downstream estimator of average treatment effect:τ w = 1 n 1 n i=1 w i Y i A i − 1 n 0 n i=1 w i Y i (1 − A i ).(9) For construction of weighted regularization process, we first introduce an extra model parameter and a regularization term γ(Y i , A i , X i ; θ, ) defined as follows: Q(X i , A i ; θ) = Q NN (X i , A i ; θ) + w i A i n n 1 − w i (1 − A i )n n 0 γ(Y i , A i , X i ; θ, ) = (Y i −Q(X i , A i , θ)) 2 We then train the model by minimizing the modified objectivê θ,ŵ,ˆ = arg min θ,w, R (X, A; θ, w) + β 1 n n i=1 γ(Y i , A i , X i ; θ, ) s.t. n i=1 w i A i = n 1 , n i=1 w i (1 − A i ) = n 0 , w i ≥ 0(10) whereR(X, A; θ, w) is defined in (8), and β ∈ R + is a hyperparameter. Then, our downstream estimator for average treatment effect becomeŝ τ wreg = 1 n n i=1 Q (X i , 1; θ) −Q(X i , 0; θ) . The philosophy behind the design of the weighted regularization process can be understood as follows. Recall the general procedure for estimating treatment effect has two steps: 1.) fit models for the conditional outcome and balancing score 2.) Plug the fitted models into a downstream estimator. A lot of alternative estimators are studied by tons of literature in semi-parametric statistics (Kennedy (2016)). In this paper, we focus on the non-parametric estimator in (9) because we don't need to make further model specification for the balancing score (weighted energy balancing) in comparison to Dragonnet model and target regularization used by Shi et al. (2019). The key results from non-parametric theory are: ifQ,ŵ,τ satisfy a certain equation, then the estimatorτ will have some desirable asymptotic properties such as • double robustness of the estimator: just one of the two models (i.e.Q,ŵ) needs to be correctly specified, for the estimator to be unbiased (Chernozhukov et al. (2017(Chernozhukov et al. ( , 2018); • efficiency:τ has the lowest variance of any consistent estimator of τ . This means thatτ is the most data efficient estimator. In order to ensure those nice asymptotic properties to hold, two conditions are needed to be satisfied: 1)Q andŵ are consistent estimators of outcomes and balancing score; 2) the following non-parametric estimating equation is satisfied: 1 n n i=1 Q (X i , 1; θ) −Q(X i , 0; θ) + w i A i n n 1 − w i (1 − A i )n n 0 (Y i −Q(X i , A i ; θ)) − τ = 0 (11) Details can be found in Van der Laan and Rose (2011); Chernozhukov et al. (2017). Since w * is derived from (7) which is non-parametric, the unbiasedness of the estimatorτ wreg automatically holds. The most important observation here is that minimizing the modified objective will forcẽ Q,ŵ,τ wreg to satisfies the estimating equation (11) because ∂ ∂  R (X; θ, w) + β 1 n n i=1 γ(Y i , A i , X i ; θ, )   = 0. As long as theQ,ŵ are consistent, by imposing the weighted regularization process, the estimatorτ wreg will have such nice asymptotic properties. Detail discussions about the design of such regularization process can be found in Shi et al. (2019). Numerical Experiments We conduct numerical experiments to demonstrate the advantages of our proposed model compared to current state-of-the-art. Since ground truth causal effects are in general unavailable in real-world data, our experiments are conducted using semi-synthetic data derived from the Infant Health and Development Program (IHDP) and from the 2018 Atlantic Causal Inference Conference (ACIC) competition, two popular benchmark datasets for assessing causal inference methods. IHDP. IHDP is a semi-synthetic dataset constructed from the Infant Health and Development Program, a randomized experiment began in 1985. Hill (2011) introduced this data in their 2011 "Bayesian nonparametric modeling for causal inference" Journal of Computational and Graphical Statistics paper. In this program, the treated group was provided with intensive high-quality child care and home visits from trained specialists. At the end of the intervention, when the children were three years old, the program was highly successful in considerably raising cognitive test scores of the treatment children in comparison to controls. The data collects measurements on the child such as birth weight, neonatal health index, first born weeks born preterm, head circumference, sex and twin status. The data also contains information of mother's behaviors engaged in during the pregnancy such as smoking status, alcohol usage and drug usage. The data generating process of the synthetic part of the data follows Hill (2011). The data has 747 observations with 26 features. ACIC 2018. The ACIC 2018 was developed from the 2018 Atlantic Causal Inference Conference competition data. It comes from IBM causal inference benchmark framework. The data is based on real-world clinical measurements taken from the Linked Birth and Infant Death Data (LBIDD) (Mathews and Atkinson (1998)). This dataset contains infant mortality statistics from the linked birth/infant death data set (linked file) for the 1999 period, broken down by a variety of maternal and infant characteristics. The features that it includes are demographics of mothers and infants such as education, prenatal care, race, birth weight and days of born preterm. The outcome is the mortality rate. The treatment is the race of mothers (black or white). The synthetic part of the data is generated from 63 distinct data generating process settings, and the data are relatively large. We follow the same selection criterion as Shi et al. (2019), that is, we randomly pick 3 datasets of size either 5,000 and 10,000. Interpreability The interpreability of neural additive models (NAMs) is due, in part, to how simple they can be visualized. Since each feature is handled independently by a learned shape function parameterized by a neural net, graphing the individual shape functions provides a comprehensive view of the model. It is feasible to have an intelligible explanation of the model's behavior visualized completely on a single page for data with a modest number of inputs. These shape function plots can provide an exact description of how NAMs estimate the average treatment effect, not merely just an explanation. This helps, for example, an decision-maker from public health to understand how to interpret the models and understand exactly how they estimate the average treatment effect from a certain risk factor. We use the data from IHDP and Figure 2 to demonstrate how it works. First, by removing the mean score for each graph (i.e., each feature) across the entire training dataset, we set the average score for each subplot in Figure 2 (i.e., each feature) to zero. A single bias term is then added to the model to make individual shape functions identifiable, so that the average prediction across all data points matches the observed baseline. This leads the interpretation of each term much easier. On the same graph, we plot each shape function. More specifically, we plot each learned shape function q j (x ij ) against x ij for an ensemble of neural additive model using blue line. This enables us to determine when the ensemble's models learned the same shape function and when they diverged. By plotting each shape function, we are able to interpret how model estimates treatment effect based on the contributions from each observed covariate such as birth weight, birth height and neonatal health index. Figure 2: Estimation of treatment effect on cognitive test scores of three year-old children from intensive, high-quality child care and home visit from trained provider. The selected plots are learned by NAM. Treatment Effect Estimation We compare our proposed approach (with/without weighted regularization process) with the fol- Covariate balancing propensity score (CBPS) can be viewed as an adjustment for propensity score model based on the distributions of treated and control units. To equip deep neural network models with covariate balancing propensity score, let Q NN (X i , A i ; θ) denote the predicted outcomes using neural network, and g(X i ; θ) be the estimated propensity score, then we train our neural network by minimizing the objective function θ = arg min θR (θ; X), wherê R(θ; X) = 1 n n i=1 (Q NN (X i , A i ; θ) − Y i ) 2 +h θ (A, X) Σ θ (A, X) −1h θ (A, X) h θ (A, X) = 1 n n i=1 h θ (A i , X i ) h θ (A i , X i ) = ∂g ∂θ (X i ; θ) A i −g(X i ;θ) g(X i ;θ)(1−g(X i ;θ)) X i Σ θ (A, X) = 1 n N i=1 E g(X i ; θ)g(X i ; θ) \|X i . Discussion and Conclusion In this paper, we provides a meaningful and straightforward generalization error bound for the bias of estimation of average treatment effect, Our bound connects the classic terms from machine There are some promising future extensions of our work. First, we may improve the expressivity and predictive performance of the neural additive model used for estimation of treatment effect by incorporating higher-order feature interactions. In the Bayesian causal forest by Hahn et al. (2020), they introduce an interaction term between propensity score and observed covariates to provide an adequate control over the strength of regularization over effect heterogeneity so that the bias of treatment effect is reduced. This idea could be applied to our setup by introducing the interaction term between balancing score and observed covariates. Another possible extension is to incorporate instrumental variables for the explanation for hidden confounders. Recall our work is under "strong ignorability" assumption. Hartford et al. (2017) equipped their model with instrumental variables for counterfactual prediction under additive hidden bias model assumption. However, such assumption is somehow not applicable in most real cases in comparison to traditional instrumental variables assumptions such as mean independence assumption or monotonicity assumption. Conducting theoretical analysis for the bias of interval estimation of average treatment effect under instrumental variable framework will be our next goal. Supplementary Materials for "Covariate-Balancing-Aware Interpretable Deep Learning models for Treatment Effect Estimation" .1 Proof of Lemma 1 Proof. According to the concept of expected factual and counterfactual losses in Definition 1, we will immediately have the concept of expected factual (counterfactual) treated and control losses: R F (μ 1 ) = R p lμ 1 (x) dF 1 (x) R F (μ 0 ) = R p lμ 0 (x) dF 0 (x) R CF (μ 1 ) = R p lμ 1 (x) dF 0 (x) R CF (μ 0 ) = R p lμ 0 (x) dF 1 (x) The relations between expected factual (counterfactual) losses and factual (counterfactual) expected / control losses are: R F (μ a ) =P 0 R F (μ 0 ) + P 1 R F (μ 1 ) R CF (μ a ) =P 1 R CF (μ 0 ) + P 0 R CF (μ 1 ) where P 1 ≡ P(A = 1), P 0 ≡ P(A = 0). Then if we do subtraction between R CF (μ a ) and P 0 R F (μ 1 ) + P 1 R F (μ 0 ) , R CF (μ a ) − P 0 R F (μ 1 ) + P 1 R F (μ 0 ) = P 0 R CF (μ 1 ) + P 1 R CF (μ 0 ) − P 0 R F (μ 1 ) + P 1 R F (μ 0 ) =P 0 R CF (μ 1 ) − R F (μ 1 ) + P 1 R CF (μ 0 ) − P 1 R F (μ 0 ) =P 0 R p lμ 1 (x) dF 0 (x) − R p lμ 1 (x) dF 1 (x) + P 1 R p lμ 0 (x) dF 1 (x) − R p lμ 0 (x) dF 0 (x) = − P 0 R p lμ 1 (x) d[F 1 − F 0 ](x) + P 1 R p lμ 0 (x) d[F 1 − F 0 ](x) = R p [−P 0 lμ 1 + P 1 lμ 0 ](x) d[F 1 − F 0 ](x) = R p [−P 0 lμ 1 + P 1 lμ 0 ](x) d[F − F 0 ](x) − R p [−P 0 lμ 1 + P 1 lμ 0 ](x) d[F − F 1 ](x) Let g(x) = [−P 0 lμ 1 + P 1 lμ 0 ](x), accoring to Theorem 4 (Koksma-Hlawka) by Mak and Joseph (2018): R p [−P 0 lμ 1 + P 1 lμ 0 ](x) d[F − F 0 ](x) = R p g(x) d[F − F 0 ](x) ≤ ∂ p g/∂x p 2 R p (F (x) − F 0 (x)) 2 dx 1/2 = ∂ p g/∂x p 2 √ 2 E(F, F 0 ) =C lμ a E(F, F 0 ) Similarly, R p [−P 0 lμ 1 + P 1 lμ 0 ](x) d[F − F 1 ](x) ≤ ∂ p g/∂x p 2 √ 2 E(F, F 1 ) =C lμ a E(F, F 1 ) where C lμ a = ∂ p [−P 0 lμ 1 +P 1 lμ 0 ](x)/∂x p ) 2 / √ 2 is a constant. According to the triangle inequality, R CF (μ a ) − P 0 R F (μ 1 ) + P 1 R F (μ 0 ) ≤ R CF (μ a ) − P 0 R F (μ 1 ) + P 1 R F (μ 0 ) = R p [−P 0 lμ 1 + P 1 lμ 0 ](x) d[F − F 0 ](x) − R p [−P 0 lμ 1 + P 1 lμ 0 ](x) d[F − F 1 ](x) ≤ R p [−P 0 lμ 1 + P 1 lμ 0 ](x) d[F − F 0 ](x) + R p [−P 0 lμ 1 + P 1 lμ 0 ](x) d[F − F 1 ](x) ≤C lμ a E(F, F 0 ) + E(F, F 1 ) Furthermore, we will prove similar equation when n → ∞ with weights. The expected factual (counterfactual) treated and control losses and their relations are given as: R F (μ 1 , w) = R p lμ 1 (x) dF n,1,w (x) R F (μ 0 , w) = R p lμ 0 (x) dF n,0,w (x) R CF (μ 1 , w) = R p lμ 1 (x)dF n,0,w (x) R CF (μ 0 , w) = R p lμ 0 (x)dF n,1,w (x) R F (μ a , w) =P 0 R F (μ 0 , w) + P 1 R F (μ 1 , w) R CF (μ a , w) =P 1 R CF (μ 0 , w) + P 0 R CF (μ 1 , w) where P 1 ≡ P(A = 1), P 0 ≡ P(A = 0). Then we can do subtraction between R CF (μ a , w) and P 0 R F (μ 1 , w) + P 1 R F (μ 0 , w) and get a similar result as what we did without weight: R CF (μ a , w) − P 0 R F (μ 1 , w) + P 1 R F (μ 0 , w) = R p [−P 0 lμ 1 + P 1 lμ 0 ](x) d[F n − F n,0,w ](x) − R p [−P 0 lμ 1 + P 1 lμ 0 ](x) d[F n − F n,1,w ](x) Here we still write g(x) = [−P 0 lμ 1 + P 1 lμ 0 ](x). Then according to Lemma 3.3 by Huling and Mak (2020), R p [−P 0 lμ 1 + P 1 lμ 0 ](x) d[F n − F n,0,w ](x) = C lμ a E(F n , F n,0,w ) and R p [−P 0 lμ 1 + P 1 lμ 0 ](x) d[F n − F n,1,w ](x) = C lμ a E(F n , F n,1,w ) Finally, we give the upper bound according to the triangular inequality similarly as we did before: R CF (μ a , w) − P 0 R F (μ 1 , w) + P 1 R F (μ 0 , w) ≤ R CF (μ a , w) − P 0 R F (μ 1 , w) + P 1 R F (μ 0 , w) = R p [−P 0 lμ 1 + P 1 lμ 0 ](x) d[F n − F n,0,w ](x) − R p [−P 0 lμ 1 + P 1 lμ 0 ](x) d[F n − F n,1,w ](x) ≤ R p [−P 0 lμ 1 + P 1 lμ 0 ](x) d[F n − F n,0,w ](x) − R p [−P 0 lμ 1 + P 1 lμ 0 ](x) d[F n − F n,1,w ](x) ≤C lμ a E(F n , F n,0,w ) + C lμ a E(F n , F n,1,w ) .2 Proof of Theorem 1 Definition 6. The expected variance of Y (a) with respect to the distribution p(x, a) is σ 2 Y (a) (p(x, a)) = R p ×Y (Y (a) − µ a (x)) 2 p(Y (a)\|x)dY (a)dF a (x) and we further define σ 2 Y (a) = min{σ 2 Y (a) (p(x, a)), σ 2 Y (a) (p(x, 1 − a))} σ 2 Y = min{σ 2 Y (1) , σ 2 Y (0) }. Claim 1. Given all conditions of Lemma 1, the following two inequalities hold: R p ×{0,1} μ a (x) − µ a (x) 2 2 dF a (x) da = R F (μ a ) − σ 2 Y (a) (p(x, a)) and R p ×{0,1} μ a (x) − µ a (x) 2 2 dF 1−a (x) da = R CF (μ a ) − σ 2 Y (a) (p(x, 1 − a)) Proof of Claim 1. R F (μ a ) = R p ×{0,1}×Y μ a (x) − Y (a) 2 2 p(Y (a)\|x) dF 1−a (x) dY (a) da = R p ×{0,1}×Y μ a (x) − µ a (x) 2 2 p(Y (a)\|x) dF 1−a (x) dY (a) da + R p ×{0,1}×Y µ a (x) − Y (a) 2 2 p(Y (a)\|x) dF 1−a (x) dY (a) da = R p ×{0,1} μ a (x) − µ a (x) 2 2 dF 1−a (x) da + σ 2 Y (1) (p(x, a)) + σ 2 Y (0) (p(x, a)) = R p ×{0,1} μ a (x) − µ a (x) 2 2 p(Y (a)\|x) dF 1−a (x) da + σ 2 Y (a) (p(x, a)) Therefore, R p ×{0,1} μ a (x) − µ a (x) 2 2 dF a (x) da = R F (μ a ) − σ 2 Y (a) (p(x, a)) Similarly, R p ×{0,1} μ a (x) − µ a (x) 2 2 dF 1−a (x) da = R CF (μ a ) − σ 2 Y (a) (p(x, 1 − a)) Claim 2. Given all conditions of Lemma 1, the following inequality also holds: R AT E (μ a ) ≤ 2 R F (μ a ) + R CF (μ a ) − 2σ 2 Y proof of claim 2 Proof. R AT E (μ a ) = R p τ (x) − τ (x) 2 2 dF (x) = R p μ 1 (x) −μ 0 (x) − µ 1 (x) − µ 0 (x) 2 2 dF (x) = R p μ 1 (x) − µ 1 (x) − μ 0 (x) − µ 0 (x) 2 2 dF (x) ≤2 R p μ 1 (x) − µ 1 (x) 2 2 dF (x) + 2 R p μ 0 (x) − µ 0 (x) 2 2 dF (x) =2 R p μ 1 (x) − µ 1 (x) 2 2 dF 1 (x) + 2 R p μ 1 (x) − µ 1 (x) 2 2 dF 0 (x) + 2 R p μ 0 (x) − µ 0 (x) 2 2 dF 1 (x) + 2 R p μ 0 (x) − µ 0 (x) 2 2 dF 0 (x) =2 R p ×{0,1} μ a (x) − µ a (x) 2 2 dF a (x) da + 2 R p ×{0,1} μ a (x) − µ a (x) 2 2 dF 1−a (x) da ≤2 R F (μ a ) − σ 2 Y + 2 R CF (μ a ) − σ 2 Y The last line is proved by Claim 1 given before. Proof of Theorem 1. According to Lemma 1 and its proof, we have R CF (μ a ) ≤ P 0 R F (μ 1 ) + P 1 R F (μ 0 ) + C lμ a E(F, F 0 ) + E(F, F 1 ) and R F (μ a ) = P 0 R F (μ 0 ) + P 1 R F (μ 1 ) then R F (μ a ) + R CF (μ a ) ≤P 0 R F (μ 1 ) + P 1 R F (μ 0 ) + C lμ a E(F, F 0 ) + E(F, F 1 ) + P 0 R F (μ 0 ) + P 1 R F (μ 1 ) =R F (μ 1 ) + R F (μ 0 ) + C lμ a E(F, F 0 ) + E(F, F 1 ) Therefore, R AT E (μ a ) ≤2 R CF (μ a ) − σ 2 Y + 2 R CF (μ a ) − σ 2 Y =2 R F (μ 1 ) + R F (μ 0 ) + C lμ a ( E(F, F 0 ) + E(F, F 1 )) − 2σ 2 Y When it comes to the proof of expected loss in estimation of average treatment effect with weight, Claims 3 still hold. The proof of the claim 3 basically follows the same proof of Claim 1 and Claim 2. Claim 3. Under conditions of Lemma 1, the following inequalities holds R p ×{0,1} μ a (x) − µ a (x) 2 2 dF n,a,w (x) da =R F (μ a , w) − σ 2 Y (a) (p(x, a)) R p ×{0,1} μ a (x) − µ a (x) 2 2 dF n,1−a,w (x) da =R CF (μ a , w) − σ 2 Y (a) (p(x, 1 − a)), and R AT E (μ a ) =R AT E,n (μ a ) ≤ 2 R F (μ 1 , w) + R F (μ 0 , w) − 2σ 2 Y According to the result in Lemma 1 with weight, Claim 1, Claim 2 and Claim 3, R F (μ 1 , w) + R CF (μ 0 , w) ≤R F (μ 1 , w) + R F (μ 0 , w) + C lμ a E(F n , F n,0,w ) + C lμ a E(F n , F n,1,w ) Therefore, R AT E (μ a ) ≤2 R F (μ 1 , w) + R F (μ 0 , w) − 2σ 2 Y ≤2 R F (μ 1 , w) + R F (μ 0 , w) + C lμ a E(F n , F n,0,w ) + C lμ a E(F n , F n,1,w ) Figure 1 : 1Architecture of additive neural network for predicting factual and counterfactual outcomes. lowing existing methods: a.) Bayesian Additive Regression Tree (BART) by Chipman et al. (2010); b.) deep Neural network structure by Shalit et al. (2017); c.) deep Neural network structure proposed by Shi et al. (2019) with target regularization; d.) causal forest model by Sharma et al. ( 2019 ) 2019motived by generalized random forest fromAthey et al. (2019) and e.) deep Neural network structure with covariate balancing propensity score (CBPS) byImai and Ratkovic (2014). This method is motivated by the Method of Moments, we match the first moment of the distributions of treated units and control units. Details of CBPS can be found inImai and Ratkovic (2014).For all deep neural network models, there are 3 hidden layers with 100 hidden units. To conduct a fair comparison, all the deep neural network models used are set to be NAMs. The hyperparameter α, β are 0.05 and 1. We train the deep neural network models using stochastic gradient descent with momentum. For causal forest, we implement the package from Sharma et al. (2019). And we use package from Sparapani et al. (2021) to implement BART.ResultsOur simulation results are summarized in Tables 1. For IHDP datasets, we randomly split the data into training set, testing set and validating set with the equal proportion 1/3. For ACIC 2018 dataset, we use all the data as training set and estimate the average treatment effect. We report the mean absolute error \|τ − τ \| in our summary tables. From the experimental results from IHDP and ACIC 2018, we identify two consistent patterns, those are, 1.) Weighted regularization process consistently improves the performance of our base model (8); 2.) our proposed model with weighted regularization outperforms all the existing models included as it has the lowest bias of estimation for average treatment effect in both the IHDP and ACIC 2018 datasets. Table 1 : 1Performance across different models for IHDP and ACIC 2018 dataset. We report the mean absolute error \|τ − τ \| in the summary tablesMethod IHDP ACIC DNN (Shi et al. (2019)) 0.29 0.56 BART (Chipman et al. (2010)) 0.95 1.32 CBPS (Imai and Ratkovic (2014)) 0.24 0.45 CF (Sharma et al. 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[ "Non-Smooth Backfitting for Excess Risk Additive Regression Model with Two Survival Time-Scales", "Non-Smooth Backfitting for Excess Risk Additive Regression Model with Two Survival Time-Scales" ]	[ "Munir Hiabu .*[email protected]†[email protected]‡[email protected] \nSchool of Mathematics and Statistics\nUniversity of Sydney\nCamperdown NSW 2006Australia\n", "Jens P Nielsen \nCass Business School, City\nUniversity of London\n106 Bunhill RowEC1Y 8TZLondonUnited Kingdom\n", "Thomas H Scheike \nDepartment of Public Health\nUniversity of Copenhagen\nØster Farimagsgade 5B1014Copenhagen KDenmark\n" ]	[ "School of Mathematics and Statistics\nUniversity of Sydney\nCamperdown NSW 2006Australia", "Cass Business School, City\nUniversity of London\n106 Bunhill RowEC1Y 8TZLondonUnited Kingdom", "Department of Public Health\nUniversity of Copenhagen\nØster Farimagsgade 5B1014Copenhagen KDenmark" ]	[]	We present a new backfitting algorithm estimating the complex structured non-parametric survival model of Scheike (2001) without having to use smoothing. The considered model is a non-parametric survival model with two time-scales that are equivalent up to a constant that varies over the subjects. Covariate effects are modelled linearly on each time scale by additive Aalen models. Estimators of the cumulative intensities on the two time-scales are suggested by solving local estimating equations jointly on the two time-scales. We are able to estimate the cumulative intensities solving backfitting estimating equations without using smoothing methods and we provide large sample properties and simultaneous confidence bands. The model is applied to data on myocardial infarction providing a separation of the two effects stemming from time since diagnosis and age.	10.1093/biomet/asaa058	[ "https://arxiv.org/pdf/1904.01202v1.pdf" ]	91,184,578	1904.01202	1114eaadcb7d8c693cab76ff3d52099be6dad78d	Non-Smooth Backfitting for Excess Risk Additive Regression Model with Two Survival Time-Scales April 3, 2019 2 Apr 2019 Munir Hiabu .*[email protected]†[email protected]‡[email protected] School of Mathematics and Statistics University of Sydney Camperdown NSW 2006Australia Jens P Nielsen Cass Business School, City University of London 106 Bunhill RowEC1Y 8TZLondonUnited Kingdom Thomas H Scheike Department of Public Health University of Copenhagen Øster Farimagsgade 5B1014Copenhagen KDenmark Non-Smooth Backfitting for Excess Risk Additive Regression Model with Two Survival Time-Scales April 3, 2019 2 Apr 20191Aalen modelcounting processdisability modelillness- death modelgeneralized additive modelsmultiple time-scalesnon- parametric estimationvarying-coefficient models We present a new backfitting algorithm estimating the complex structured non-parametric survival model of Scheike (2001) without having to use smoothing. The considered model is a non-parametric survival model with two time-scales that are equivalent up to a constant that varies over the subjects. Covariate effects are modelled linearly on each time scale by additive Aalen models. Estimators of the cumulative intensities on the two time-scales are suggested by solving local estimating equations jointly on the two time-scales. We are able to estimate the cumulative intensities solving backfitting estimating equations without using smoothing methods and we provide large sample properties and simultaneous confidence bands. The model is applied to data on myocardial infarction providing a separation of the two effects stemming from time since diagnosis and age. Introduction In many bio-medical applications in survival analysis it is of interest and needed to use multiple time-scales. A medical study will often have a followup time (for example time since diagnosis) for patients of different ages, and here both time-scales will contain important but different information about how the risk of, for example, dying is changing. We therefore consider the situation with two time-scales that are equivalent up to a constant for each individual, such as for example follow-up time and age. One may see this as arising from the the illness-death model, or the disability model, where the additional time-scale may be duration in the illness state of the model; see Keiding (1991) for a general discussion of these models. There is rather limited work on how to deal with multiple time-scales in a biomedical context, see for example Oakes (1995); Iacobelli & Carstensen (2013) and Duchesne & Lawless (2000) and references therein. We present a non-parametric regression approach with two time-scales where each time-scale contribute additively to the mortality. The regression setting models the effect of covariates by additive Aalen models on each time-scale (Aalen, 1989;Huffer & McKeague, 1991;Andersen et al., 1993;Martinussen & Scheike, 2006). This allows covariates to have effects that vary on two different time-scales. In a motivating example we consider patients that experience myocardial infarction, and aim at predicting the intensity considering the two time-scales age and time since myocardial infarction. As a consequence, we can make survival predictions for patients given their age at diagnosis. This model was considered previously by Scheike (2001) where estimation was based on smoothing for one of the time-scales. A study closely related to ours is Kauermann & Khomski (2006) who studied the two most common time scales: age and duration. The underlying technical setting of Kauermann & Khomski (2006) was a multiplicative hazard model without covariates that is estimated via splines. In contrast our approach is an additive hazard model including covariates and estimating without smoothing. Alternative smoothing methodologies to multiplicative hazard estimation includes Linton et al. (2003); Huang (1999); Hastie & Tibshirani (1986); Lin et al. (2016). None of the known multiplicative hazard approaches including the ones mentioned above are able to estimate without smoothing, include time varying covariate-effects, or are able to provide simultaneous confidence bands as the additive approach of this paper does provide. We do know that smoothing improves efficiencies of cumulatively estimated quantities, see Guillen et al. (2007) for the simplest possible case. However, smoothing is also a complexity and experts applying survival analysis have developed a practical way of smoothing by eye the underlying rough non-parametric estimators of Kaplan & Meier (1958); Nelson (1972). The advantage of providing estimators without smoothing is that there can be no confusion from the complicated process of picking the smoothing procedure first and the amount of smoothing after that. Even if a smoothing approach is eventually used, then the smoothing free procedure would always count as a benchmark approach to check whether something went wrong during the smoothing. Our backfitting approach is different from standard backfitting in regression, see for example the smooth additive backtiffing approach of Mammen et al. (1999), where data is projected down via a smoothing kernel onto an additive subspace. In the backfitting approach of this paper, the non-parametric dynamics is only taking place in the two time directions, and the end result is therefore closer to the classical approach of Nelson (1972) with a non-smooth estimator of the dynamics in the one-dimensional time axis. What is obtained through Aalen's additive hazard regression model on two time axis is that the dynamics of the two time effects are adjusted for covariaties in a way that keep the one-dimensional structure of the non-parametric dynamics. The expert user of survival methodology can therefore use the well developed intuition from looking at Nelson-Aalen estimators and Kaplan-Meier estimators when interpreting the empirical results based on the new methodology of this paper. Another advantage of estimating directly the cumulative hazards is that we are able to obtain a simple uniform asymptotic description of our estimators. We are thus able to construct confidence bands and intervals, that are based on bootstrapping the underlying martingales. The paper is organised as follows. Section 2 presents the model via counting processes. Section 3 gives some least squares based local estimating equations that are solved to give simple explicit estimators of the nonparametric effects of the model. Based on these explicit estimators we are able to derive asymptotic results and provide the estimators with asymptotic standard errors. Sections 4-6 discusses how to solve the equations and compute the estimator practically and how deal with identifiability issues. Section 7 shows how the large sample properties may be derived and in Section 8 we construct confidence bands. Section 9 demonstrates the finite sample properties supporting Section 10 where we use our proposed methods in a worked example. Finally, Section 10 discusses some possible extensions. Aalen's Additive Hazard Model for Two Time-Scales Let N i (t) i = 1, ..., n be n independent counting processes that do not have common jumps and are adapted to a filtration that satisfy the usual conditions (Andersen et al., 1993). We assume that the counting processes have intensities given by λ i (t) = p j=1 X ij (t)α j (t) + q k=1 Z ik (t)β k (t + a i ) = X i (t)α(t) + Z i (t)β(t + a i ), (0 ≤ t ≤ t max ),(1) where α = (α 1 , . . . , α p ) and β = (β 1 , . . . , β q ) are tupels of one dimensional deterministic functions, X T i (t) ∈ p and Z T i (t) ∈ q are predictable cadlag covariate vectors with X(t) and Z(t) having almost surely full rank, and a i is a real-valued random variable observed at time t = 0. If Z i (t) = 0 for all t, a i does not need to be observed. The model is the sum of two Additive Alalen Models running on two different time scales, see also Scheike(2001). The two time-scales are t and a = t+a i ∈ [a 0 , a max ] where the latter time-scale is specific to each individual and a 0 is some lower-limit that depends on the observed range of the second time-scale. Note, that no indicator variables are introduced but are absorbed in the covariates. In the illness-death model, say, t might be time since diagnosis (duration) among subjects that have entered the illness stage of the model and a i could be the age when the transition to the illness stage occurred, such that t + a i is the age of the subject. After introducing some notation we present an estimation procedure that leads to explicit estimators of A(t) = t 0 α(s)ds = ( t 0 α 1 (s)ds, . . . , t 0 α p (s)ds) T and B(a) = a a 0 β(u)du = ( a a 0 β 1 (u)du, . . . , a a 0 β q (u)du) T . The cumulative effects have the advantage compared to α(s) and β(a) that they may be used for inferential purposes since a more satisfactory simultaneous convergence can be established for these processes. We derive the asymptotic distribution for these estimators and a bootstrapping procedure quantifying the estimation uncertainty. Based on the cumulative intensity A(t) one may estimate the intensity α(t) by smoothing techniques. Notation Let Λ i (t) = t 0 λ i (s)ds such that M i (t) = N i (t) − Λ i (t) are martingales. Let further N (t) = (N 1 (t), ..., N n (t)) T be the n-dimensional counting process, Λ(t) = (Λ 1 (t), ..., Λ n (t)) T is its compensator, such that M (t) = (M 1 (t), ..., M n (t)) T is an n-dimensional martingale, and define matrices X(t) = (X 1 (t), . . . , X n (t)) T and Z(t) = (Z 1 (t), . . . , Z n (t)) T , with di-mensions n × p and n × q, respectively. The individual entry times are summarised in one vector a • = (a 1 , . . . , a n ). A superscript a > 0 denotes a shift in the argument, i.e, for a generic function f , f a (y) = f (y + a). For a generic matrix C(t), with n rows C i (t), and a n-dimensional vector v, C v (t) is defined through shifting the rows: C v i (t) = C i (t + v i ). For a generic matrix C, a minus superscript, C − , denotes the Moore-Penrose inverse. An integral, , with no limits denotes integration over the whole range. Identification of the entering nonparametric parameters In many cases some covariates will enter both the X and the Z design. If this is the case, then the functions α and β are not identified in model (1) -constants can be shifted for the components that share the same covariate without altering the intensity. Without loss of generality we assume that X and Z share the first d (0 ≤ d ≤ min(p, q)) columns, i.e., for all i = 1, . . . , n, X il = Z il , l ≤ d. We formulate the problem using group-theoretic arguments, see also Carstensen (2007); Kuang et al. (2008). Fix constants c 1 , . . . , c d and define f l as p+q valued function having all entries but the l th and the (d + l) th equal zero: f l (s, u) = (0, · · · , 0, c l s, 0, · · · , 0, −c l (u − a 0 ), 0, · · · , 0)) T , (l = 1, . . . , d). We define the group G by G = g : A B → A B + h \| h ∈ Lin(f 1 , . . . f d ) . The identification problem can be rephrased as that the intensity defined in (1) is a function of (A, B) T , which is invariant to transformations g ∈ G. In the sequel we circumvent the identification issue by adding the following constraint A l (t max ) = tmax 0 α l (s)ds = 0, (l = 1, . . . , d),(2) noting that for any solution (A 0 , B 0 ) of model (1), there exists a unique solution (A, B) = g(A 0 , B 0 ) that fulfills (2). Clearly other choices are also possible. Least squares minimisation ignoring the identificating of the nonparametric parameters We split the identification challenge in two. First we estimate ignoring identification of the parameters, and then we show in next section how to identify the estimated parameters. In this section we therefore ignore the identification problem keeping in mind that the solutions below are hence not unique. We motivate our estimator ( A, B) via the following least squares criteria. arg min A,B i    t 0 dN i (s) − j t 0 X ij (s)dA j (s) − k t 0 Z ik (s)dB a i k (s)    2 dt, where the integrals can be understood as Stieltjes integrals, noting that X i and Z i are left continuous. Minimisation runs over all possible integrators. One can already see that the minimiser, if it exists, will be a step-function, since t 0 dN i (s) is a step function. To simplify notation we will generally work in matrix notation so that above minimisation criteria can also be written as arg min A,B i t 0 dN i (s) − t 0 X i (s)dA(s) − t 0 Z i (s)dB a i (s) 2 dt. Straight forward computations utilzing calculus of variations lead to ( A, B) solving the following first order conditions for all t ∈ [0, t max ], a ∈ [a 0 , a max ]: i X i (t) T dN i (t) − X i (t)d A(t) − Z i (t)d B a i (t)dt = 0, i Z −a i i (a) T dN −a i i (a) − Z −a i i (a)d B(a) − X −a i i (a)d A −a i (a) = 0. Rearranging yields i X i (t) T dN i (t) − i X i (t) T Z i (t)d B a i (t) = X(t) T X(t)d A(t), i Z −a i i (a) T dN −a i i (a) − i Z −a i i (a) T X −a i i (a)d A −a i (a) = Z −a• (a) T Z −a• (a)d B(a). The last set of equations can be further rewritten to the backfitting equations A(t) = t 0 X(s) − dN (s) − E 1 (t\|u)d B(u) (3) B(a) = a a 0 Z −a• (u) − dN −a• (u) − E 2 (a\|s)d A(s),(4)6 where E 1 (s\|u) = i {X T (u − a i )X(u − a i )} −1 X −a i ,T i (u)Z −a i i (u)I(a i ≤ u ≤ a i + s), E 2 (u\|s) = i {Z −a•,T (s + a i )Z −a• (s + a i )} −1 Z T i (s)X i (s)I(a 0 − a i ≤ s ≤ u − a i ). Remark 1 In the case with no covariates, i.e., λ i (t) = Y i (t){α(t) + β(a i + t)}, with X i (s) = Z i (s) = Y i (s) ∈ , the risk indicators are E 1 (s\|u) = i 1 i Y i (u − a i ) Y −a i i (u)I(a i ≤ u ≤ a i + s), E 2 (u\|s) = i 1 i Y −a i i (s + a i ) Y i (s)I(a 0 − a i ≤ s ≤ u − a i ). Establishing existence, identification and uniqueness of the estimator In section 3 we outlined the identification problem but ignored it when establishing the estimator in the previous section. In this section we provide a fully identified estimator of our problem. When aiming to solve equations (3) and (4) the identification problem can no longer be ignored. In order to get a better grip of the situation we will now rewrite the backfitting equations as a linear operator equation. We can compress equations (3) and (4) into one matrix equation: A B = t 0 X(s) − dN (s) a a 0 Z −a• (u) − dN −a• (u) + 0 −E 1 −E 2 0 × A B , where with some miss-use of notation E l f (·) = E l (·, y)f (y)dx, (l = 1, 2). Or even simpler θ = m + E θ,(5) with obvious notation, and linear operator E: θ = A B , m = t 0 X(s) − dN (s) a a 0 Z −a• (u) − dN −a• (u) , E = 0 −E 1 −E 2 0 . Note that m is composed of the marginal Aalen estimators of the two time scales, t and a. Additionally, the operator E is compact because it is the composition of an integral operator, which is compact, and a derivative operator, which is bounded. The operator E being compact means that it can be arbitrarily close approximated by a finite dimensional matrix which simplifies both the numerical and theoretical considerations. If the eigenvalues of E are bounded away from one, then, (I − E) is invertible and we have θ = (I − E) −1 m. Hence existence and uniqueness of our proposed estimator can be translated to properties of the eigenvalues of E. One can for instance easily verify that if some covariates are both in the X and the Z design, then E will have an eigenvalue equal to one -as discussed in the following remark. Remark 2 Consider the most simple case 1 = d = p = q, i.e., λ i (t) = Y i (t){α(t) + β(a i + t)}. Given a constant c ∈ , consider the pair of linear function f 1 = (f 11 , f 12 ) T with f 11 (s) = cs, f 12 (u) = −c(u−a 0 ), as defined in Section 3. Assuming that Y i (s) and Y i (u − a i ) are bounded away from zero on the whole range s ∈ [0, t max ], u ∈ [a 0 , a max ], one can easily verify that E 2 f 11 (u) = c E 2 (u\|s)ds = c(u − a 0 ), E 1 f 12 (s) = −c E 1 (s\|u)du = −cs. To see this, e.g., for the second equation, note E 1 (s\|u)du = i a i +s a i 1 i Y i (u − a i ) Y −a i i (u)du = s 0 i Y i (t) i Y i (t) dt = s. Hence, we have E f 11 f 12 = −E 1 f 12 −E 2 f 12 = f 11 f 12 . So that one is clearly an eigenvalue of E with corresponding eigenfunction f 1 = (f 11 , f 12 ) T . In other words the identification issue of the model carries over to the estimator. With analogue arguments one can show that in the more general case the eigenspace corresponding to eigenvalue equal one includes the functions in Lin(f 1 , . . . f d ). Functions f 1 , . . . , f d are defined in Section 3. We now utilize constraint (2) and incorporate it into new backfitting equations: A(t) = t 0 X(s) − dN (s) − E 1 (t\|u)d B(u),(6)B(a) = a a 0 Z −a• (u) − dN −a• (u) − E 2 (a\|s)d A(s) + A dq (t max ) t max (a − a 0 ),(7) where A dq is the q-dimensional vector A dq = (A 1 , . . . , A d , 0, . . . , 0) T . This translates to the new operator equation θ = m + E θ, E = 0 −E 1 −E 2 0 ,(8) where E 2 h(a) = E 2 (a\|s)dh(s) − (a − a 0 )h dq (t max )t −1 max . The next proposition states that the solutions of (8) include all relevant solutions of (5) and that every solution of (8) is a solution of (5). Proposition 1 For every solution θ of (5), define θ 0 = (I − Π) θ, where Π h 1 (t) h 2 (a) = th dp 1 (t max )t −1 max −(a − a 0 )h dq 1 (t max )t −1 max . Then θ 0 is a solution of (8) and θ 0 + Lin(f 1 , . . . f d ),(9) are further solutions of (5). Reversly, for every solution θ 0 of (8), all functions of the form (9) are solutions of (5). The proof can be found in the appendix. With Proposition 1 at hand it is justified to define our estimator as the solution of (8). We will now discuss existence and uniqueness of the solution of (8). Note that E is known and hence one can calculate a numerical approximation of its eigenvalues by working on a grid. Consider the sub-space K = {h = (h 1 , . . . , h d , 0, . . . , 0)\| h l : → , x → c l x, c l ∈ , l = 1, . . . , d}. It holds that E 2 = E 2 (I − Π), where Π is a projection into K. We have K ⊆ kern(I − E 2 ). We can check whether K equals kern(I − E 2 ). This can be done by calculating the dimension of the eigenspace of E 2 corresponding to an eigenvalue equal one. The dimension will be at least d. If it is exactly d, then K = kern(I − E 2 ). The next proposition states that if kern(I −E 2 ) = K, and kern(I −E) = Lin(f 1 , . . . , f d ), then both I − E 2 and I − E are bijective. Proposition 2 Assume that E 2 has Eigenvalue 1 with multiplicity d. Then, (I − E 2 ) will be bijective. If furthermore E has Eigenvalue 1 with multiplicity d, then (I − E) is bijective and hence invertible. In particular a solution of equations (8) exists and it is unique. The proof can be found in the Appendix. Calculating the estimator There are two major ways of calculating the proposed estimator. Either one directly calculates (I − E) −1 and applies it on θ or something closer to an iterative procedure. For the latter, by iterative application of (8) we derive that θ = ∞ r=0 E r ( m) + E ∞ ( θ).(10) If the absolute values of the eigenvalues of E are bounded from above by a constant strictly smaller than 1, then (10) is well defined with E ∞ = 0, and the converging series θ = ∞ r=0 E r ( m), so that the iterative algorithm θ (r) = m + E θ (r−1)(11) converges from any starting point. Note that (11) is the usual way the backfiting equations (6),(7) or equivalently (8) are solved. Another way is to calculate the finite sum θ = r r=0 E r ( m), 10 with some stopping criteria r. We conclude that the proposed estimator can be calculated in a straight forward manner from the compound Aalen estimator m and the operator E. We now briefly discuss how E can be calculated in the simple case 1 = d = p = q. Here, E can be approximated by a j × k matrix where j, k are the number of grid points in [0, t max ] and [a 0 , a max ], respectively. This is done by first calculating the values E 1 (s 0 , a 0 ) and E 2 (a 0 , s 0 ) for every grid point; see Remark 1 for the definitions of the the functions. We call the resulting matrices E mx 1 and E mx 2 . Afterwards, E mx 2 is derived from E mx 2 , via E mx 2 = E mx 2 +      0 · · · 0 s 1 /s j 0 . . . 0 s 2 /s j . . . . . . . . . 0 . . . 0 1      . The matrices are then transformed to the desired operator via ∆ =          1 −1 0 · · · 0 0 . . . . . . . . . . . . . . . . . . . . . . . . 0 . . . . . . . . . . . . −1 0 · · · · · · 0 1          , E op 1 = E mx 1 × ∆, E op 2 = E mx 2 × ∆. Finally, E op = 0 −E op 1 −E op 2 0 . So that given a function h : [0, t max ]×[a 0 , a max ] → , one calculates its values on the grid and summarises it in a vector h grid . The function Eh is then approximated via E op h grid where the latter is a simple matrix multiplication. Asymptotics Note that we have θ = m + Eθ,(12) where m arises from m by replacing N by Λ. It is hereby quite remarkable that E is the observable operator from the previous sections and not some asymptotic limit. We further conclude that the least square solution (6) and (7) is a plug-in estimator of (12). The estimation error is then given as θ − θ = m − m + E( θ − θ).(13) As in the last section, If E has eigenvalues all bounded away from one, then θ − θ = (I − E) −1 ( m − m). So the asymptotic behaviour of θ − θ can be deduced from the asymptotic behaviour of (I − E) −1 and ( m − m), with the latter being the compound estimation error of two additive Aalen models on different time-scales. Theorem 1 Under assumptions (A)-(G), the estimator θ exists. Furthermore the estimator θ is n 1/2 consistent: n −1/2 ( θ − θ) → (I − E) −1 U, in Skorohod space D p+q [0, a max ]. Here, ( θ − θ) is treated as one stochastic process defined on [0, a max ] by setting for j = 1, . . . , p and ν ∈ [t max , a max ], ( θ − θ) j (ν) = ( θ − θ) j (t max ) . And similarly, for j = p + 1, . . . , p + q and ν ∈ [0, a 0 ], ( θ − θ) j (ν) = 0. The process U is a p + q dimensional mean-zero Gaussian process with covariation matrix Σ(ν 1 , ν 2 ) described in the Appendix, and E is the limit of E. The proof can be found in the Appendix. Confidence Bands While we could use the central limit theorem of the previous section to construct confidence bands, it has been suggested that better small sample performance can be achieved by directly bootstrapping the estimation error. We propose a wild bootstrap approach based on the relationship θ − θ = (I − E) −1 ( m − m) = (I − E) −1 t 0 X(s) − dM (s) a a 0 Z −a• (u) − dM −a• (u) = (I − E) −1 M 1 M 2 Since (I − E) −1 is known, it is enough to to only approximate M. We do this via the wild bootstrap version M (1) = t 0 X(s) − d N (s) a a 0 Z −a• (u) − d N −a• (u) , N i (s) = G i N i (s), or M (2) = t 0 X(s) − d M (s) a a 0 Z −a• (u) − d M −a• (u) , t 0 M i (s)ds = G i t 0 N i (s)ds − t 0 (X i (s)d A(s) + t 0 Z i (s)d B(s + a i ) , where G i is a mean zero random variable with unit variance. The random variable G i is generated such that for fixed i, it is independent to all other variables. It is straight forward to confirm that M (r) , r = 1, 2 is a mean zero process that has the same covariance as M (The covariance of M is given in the appendix). Hence, we directly derive the following proposition. Proposition 3 Under assumptions (A)-(G), the bootstrapped estimation error is uniformly consistent, i.e., for r = 1, 2 n −1/2 ((I − E) −1 M (r) ) → (I − E) −1 U, in Skorohod space D p+q [0, T ], where U is is described in Theorem 1. The proof can be found in the Appendix. One useful consequence of this is that we can estimate standard errors of our estimatorθ based on the approximation from the bootstrap. We denote these estimators asσ r (t) for the two components r = 1, 2. Corollary 1 Under assumptions (A)-(G), the bootstrapped errors lead to confidence bands CB (r) for θ(ν) over ν ∈ [ν 1 , ν 2 ] providing an asymptotic coverage probability of 1 − α, where CB (r) (ν) = θ(ν) + / − c 1−ασr (ν), and c 1−α = (1 − α) quantile of L    sup [ν 1 ,ν 2 ] n −1/2 (I − E) −1 M (r) σ r \|X, Z, N    We explore the performance of the estimator of the standard error and the uniform bands in the next section. 13 We generated data from the simple two-time scale model with age and duration that resemble the data we consider in worked example in the next section. Thus assuming that the hazard for those under risk is given as β(t + a i ) + α(t), Bias of backfitting We considered sample sizes 100, 200 and 400 and show the bias for the two-components in Table 1 Bootstrap uncertainty Secondly, we demonstrate that our bootstrap seems to work well to describe the uncertainty of the estimates. We simulated data as before and based on 1000 realisations with 100 bootstrap's based on G i dN i we estimated: a) the point-wise standard error for the two-components; b) computed the pointwise coverage baed on these; c) and constructed uniform confidence bands, as described in Corollary 1, for the the two components and its coverage. Table 2 around here We note that the standard error is well estimated by the bootstrapped standard deviation across all sample sizes and for both components. In addition the pointwise coverage is close to the nominal 95 % level for the larger sample sizes. But even for n = 100 the coverage is reasonable for most time-points for the two components. Finally, we also considered the performance of the confidence bands based on our bootstrap approach. Table 3 around here When n gets larger these bands are quite close to the nominal 95 % level, but for n = 100 the asymptotics have not quite set in to make the entire band work well. Application to the TRACE study The TRACE study group (see e.g. Jensen et al. (1997) ) has collected information on more than 4000 consecutive patients with acute myocardial infarction (AMI) with the aim of studying the prognostic importance of various risk factors on mortality. We here consider a subset of 1878 of these patients that are available in the timereg R package. At the age of entry (age of diagnosis) the patients had various risk factors recorded, but we here just show the simple model with the effects of the two-time-scales age and duration. It is expected that the duration time-scale has a strong initial effect of dying that then disappears when patients survive the first period right after their AMI. We then estimated the two-time-scale model α(t) + β(t + a i ) under the identifiability condition that 5 0 α(s)ds = 0. Restricting attention to patients more than 40 years of age, and within the first 5 duration years after the diagnosis. First we estimate the mortality on the two time-scales separately, the two marginal estimates, see Figure 1. Panel (a) shows the cumulative hazard on the age time-scale with the marginal estimate (full line) and the one with adjustment for duration effects (broken line), and panel (b) the mortality on the duration time-scale with the marginal estimate (full line) and with adjustment for age effects (broken line). We note that on the duration time-scale the cumulative hazard is quite steep. In addition we show 95 % confidence bands based on our bootstrap (regions), and the pointwise confidence intervals (dotted line). Figure 1 about here Taking out the duration effect slightly alters the estimate of the ageeffect. In contrast the duration effect is strongly confounded by age effect estimates, and here the two-time scale model more clearly demonstrates what is going on on the duration time-scale. The duration effect is strong initially and then after surviving the first 220 days we see a protective effect (dotted vertical line). We stress that the interpretation of the hazards on the two-time scales are difficult, due to, for example, the constraint that needs to be imposed to identify a specific solution. Nevertheless, it very useful to see the components from the two time-scales that jointly make up the hazard for an individual, and can be used for the prediction purposes as we demonstrate further below. Note also that due to the additive structure the duration effect can be interpreted as giving relative survival due to the duration time-scale. Figure 2 about here In Figure 2 we show the survival predictions for subjects that are 60, 70, or 80, respectively, using the two-time scale model. Thus computing exp(−(B(a 0 + t) −B(a 0 )) +Â(t)) and constructing the confidence bands using the bootstrap approach for (B(a 0 + t) −B(a 0 )) +Â(t) for t ∈ [0, 5]. These curves are a direct consequence of having the two-components and are directly interpretable. Discussion By utilising the additive structure we have demonstrated that one can estimate the effect of two time-scales directly by a backfitting algorithm that does not involve smoothing. By working on the cumulative this also lead to uniform asymptotic description and a simple bootstrap procedure for getting estimates of the uncertainty and for constructing for example confidence intervals. These cumulative may form the basis for smoothing based estimates when the hazard are of interest, but often the cumulative are the quantities of key interest for example when interest is on survival predictions. Clearly, the model could also be fitted by a more standard backfitting approach working on the hazard scale as in ... for multiplicative hazard models. Our backfitting approach can be extended for example the age-periodcohort model but here identifiability conditions are more complex to build into the estimation. A Proofs A.1 Proof of Proposition 1 With f k (s, u) = (0, · · · , 0, c k s, 0, · · · , 0, −c k (u − a 0 ), 0, · · · , 0)) T , (k = 1, . . . , d), the proposition directly follows from Lin(f 1 , . . . f d ) ⊆ Kern(I − E), and the fact that Π is a projection into Lin(f 1 , . . . f d ). A.2 Proof of Proposition 2 Since the eigenspace of E 2 corresponding to the eigenvalue equal 1 has dimension d, we know its exact form: {h = (h 1 , . . . , h d , 0, . . . , 0)\| h l is linear, l = 1, . . . , d}; see also Remark 2. One can then verify that kern(I −E 2 ) = kern(I −E 2 ) l , l = 2, 3, . . . . This is because linear functions cannot be constructed as sum of a linear and non-linear functions. Noting that E 2 is a compact operator, we conclude that I − E 2 is an isomorphism from Im(I − E 2 ) to Im(I − E 2 ). We introduce the operator E 2 = E 2 (I − Π), where Π is the projection onto ker(I −E 2 ). The condition E(h 1 , h 2 ) = (h 1 , h 2 ) is equivalent to E 2 E 1 h 2 = h 2 and E 1 E 2 h 1 = h 1 . Since the eigenspace of E corresponding to an eigenvalue of 1 has dimension d, E(h 1 , h 2 ) = (h 1 , h 2 ) is not true for non-linear h 1 , h 2 . This is because E = E when restricted on non-linear functions h 1 , h 2 . When considering a linear h 1 , then E 2 h 1 = 0. We conclude that the solution of E(h 1 , h 2 ) = (h 1 , h 2 ) is trivial. Hence the kern of (I − E) is trivial. Since E is compact this means (I − E) is bijective, in particular invertible. A.3 Assumptions We first define a few quantities. For every ν in, [0, a max ], we define the following matrices R(ν) = X(ν) 0 0 Z −a• (ν) , V (ν) = (X(ν), Z(ν)), as well as R (1) j (ν) = i R ij (ν) R (2) jk (ν) = i R ij (ν)R ik (ν), V (2) jk (ν) = i V ij (ν)V ik (ν), V(3)jkl (ν) = i V ij (ν)V ik (ν)V il (ν). We further define { E 1 (s\|u)} jk = h(x) l {r (2) (u − x)} −1,T q (2) (u − x))I(x ≤ u ≤ x + s)dx j,p+k , for j = 1, . . . , p, k = 1, . . . , q { E 2 (u\|s)} jk = h(x) l {r (2) (s + x)} −1,T q (2) (s)I(a 0 − x ≤ s ≤ u − x)dx p+j,k , for j = 1, . . . , q, k = 1, . . . , p. The limiting operator E is then defined analogue to E by replacing E 1 (s\|u), E 2 (u\|s) by E 1 (s\|u), E 2 (u\|s). We make the following assumptions. (A) There exist continuous functions r (1) j , r (2) jk , v (2) jk , v(3) jkl , (j, k, l = 1, . . . , p + q), such that for n → ∞ ij (ν))is non-singular. sup ν n −1 R (1) j (ν) − r (1) j (ν) = o p (1) sup ν n −1 R (2) jk (ν) − r (2) j (ν) = o p (1) sup ν n −1 V (2) jk (ν) − v (2) j (ν) = o p (1) sup ν n −1 V (3) jkl (ν) − v(D) sup m 2 ≤1 (I − E) −1 m ∞ < ∞,(E) The random variables (a i ) i=1,...,n are iid, independent of (X,Z) and are absolutely continuous with continuous density h. A.4 Proof of Theorem 1 We first prove the central limit theorem: n −1/2 ( m − m) → U. We write n −1/2 ( m − m)(ν) = n −1/2 ν 0 R(y) − dM (y) dM −a• (y) = M 1 M 2 = M. Since M 1 and M 2 are square integrable martingales (each with respect to its natural filtration), M is tight under the condition that its jumps are uniformly bounded. This follows from assumption (B), so M is indeed tight. Furthermore, under assumption (A), M is asymptotically uniformly close to M = n −1/2 i ν 0 r (1) (y) − dM i (y) dM −a• i (y), which is the sum of n iid random processes. So the limit of M, if it exists, must be Gaussian. Hence convergence of M to U is verified by establishing point-wise convergence of the covariance matrix of M to the covariance matrix of U . For two points ν 1 , ν 2 in [0, a max ] with ν 1 ≤ ν 2 , Cov (M(ν 1 ), M(ν 2 )) is a (p + q) × (p + q) matrix. We have Cov (M(ν 1 ), M(ν 2 )) = Cov(M 1 (ν 1 ), M 1 (ν 2 )) Cov(M 1 (ν 1 ), M 2 (ν 2 )) Cov(M 2 (ν 1 ), M 1 (ν 2 )) Cov(M 2 (ν 1 ), M 2 (ν 2 )) . With entry (j, k) given by i,l,m Cov ν 0 (ν 1 ,ν 2 ,a i ,j,k) 0 (R (2) (ν) −1 ) jl V il (ν)dM i (ν), ν 0 (ν 1 ,ν 2 ,a i ,j,k) 0 (R (2) (ν) −1 ) km (ν)V im (ν)dM i (ν) , 20 where ν 0 (ν 1 , ν 2 , a i , j, k) =            ν 1 for j ≤ p, k ≤ p min(ν 1 , ν 2 − a i ) for j ≤ p, k > p ν 1 − a i for j > p, k ≤ p ν 1 − a i for j > p, k > p . The two processes in the covariance are running in in the same time-interval. This is because we could eliminate the non-intersecting time points due to independence. Under assumption (B), the entries converge to i,l,m Cov ν 0 (ν 1 ,ν 2 ,a i ,j,k) 0 (r (2) (ν) −1 ) jl V il (ν)dM i (ν), ν 0 (ν 1 ,ν 2 ,a i ,j,k) 0 (r (2) (ν) −1 ) km (ν)V im (ν)dM i (ν) , so that the two process in the covariance are now even martingales with respect to the same filtration F i (ν 0 ) = σ{V i (u), N i (u), u ≤ ν 0 }. We can hence first calculate the conditional covariance, given F i , i.e., the predictable covariation process. Afterwards, the covariance is given as the expectation of predictable covariation process. For the predictable covariation process we get g ν 0 (ν 1 ,ν 2 ,a i ,j,k) 0 (r (2) (ν) −1 ) jl (r (2) (ν) −1 ) km V il (ν)V im (ν)V ig (ν) α(ν) β a i (ν) dν. From assumptions (A),(C), (E) we conclude that Cov (M(ν 1 ), M(ν 2 )) → Σ(ν 1 , ν 2 ) with entries Σ jk = l,m,g h(x) ν 0 (ν 1 ,ν 2 ,x,j,k) 0 (r (2) (ν) −1 ) jl (r (2) (ν) −1 ) km v (3) lmg (ν) α(ν) β x (ν) dνdx. Since the integral is well defined, we conclude convergence of M to U . We now need to handle the operator E. We have sup m 2 ≤1 (E − E)m ∞ = o p (1),(14) sup m 2 ≤1 Em ∞ < ∞.(15) Equation (14) follows directly from the uniform convergence of the kernel functions E 1 (s\|u), E 2 (u\|s) to E 1 (s\|u), E 2 (u\|s) which is ensured via Assumptions (A)-(C), (E). Inequality (15) is ensured, since the kernel functions are bounded using the same assumptions. Together with Assumption (D) it follows that the operator (I − E) −1 converges to the linear and bounded operator (I − E) −1 which gives the desired central limit theorem. A.5 Poof of proposition 3 For two points ν 1 , ν 2 in [0, a max ] with ν 1 ≤ ν 2 , the covariance of M (1) is given by i,l,m Cov ν 0 (ν 1 ,ν 2 ,a i ,j,k) 0 (R (2) (ν) −1 ) jl V il (ν)G i dN i (ν), ν 0 (ν 1 ,ν 2 ,a i ,j,k) 0 (R (2) (ν) −1 ) km (ν)V im (ν)G i dN i (ν) . Under assumption (B) this is uniformly close to i,l,m Cov ν 0 (ν 1 ,ν 2 ,a i ,j,k) 0 (r (2) (ν) −1 ) jl V il (ν)G i dN i (ν), ν 0 (ν 1 ,ν 2 ,a i ,j,k) 0 (r (2) (ν) −1 ) km (ν)V im (ν)G i dN i (ν) . As in the proof of Theorem 1, the two processes in the covariance are martingales with respect to F i , so we can calculate the covariance as expectation of the predictable covariation process. Hence, Cov M (1) (ν 1 ), M (1) (ν 2 ) → Σ(ν 1 , ν 2 ). subject that is 60,70, and 80, respectively (full lines). Predicted survival using only age for the three ages (broken lines), and survival using only duration (dotted line). 26 where β(a) ≡ 0.067 and the entry ages where drawn uniformly from [0, 25] but making sure that 10 % of the data started in 0 to (to avoid difficulties with left truncation in the estimation). The α(t) component was piecewise constant with rate 0.32 in the time-interval [0, 0.25], then 0.48 in (0.25, 0.5] and then finally to satisfy our constraint −0.044 in (0.5, 5], so that 5 0 α(s)ds = 0. All subjects were censored after 5 years of follow up. In all simulations we used a discrete approximation based on a time-grid of either 100 points in both the age direction [0, 30] and on the duration time-scale [0, 5]. j = 0, . . . , p and k = 0, . . . , q, and n → ∞n −1/2 sup s,i=1,...,n \|X ij (s)\| = o p (1) n −1/2 sup s,i=1,...,n \|Z ik (s)\| = o p (1) (C) For every ν, the matrix (r (2) Figure 1 :Figure 2 : 12Cumulative baseline on the two time-scales estimated marginally (full line) and in the two-time-scale model (broken line). Confidence bands (regions) and pointwise confidence intervals (dotted lines)Predicted survival with 95 % confidence bands (regions) for a based on 1000 realizations.We note that the the backfitting algorithm is almost unbiased across all sample size and improves as the sample size increases. This is despite the fact that the simulated component in the time-direction really is quite wild.age n=100 n=200 n=400 6.717 −0.001 0.006 −0.004 13.788 0.009 0.003 −0.006 20.859 0.018 0.001 0.002 27.929 0.027 0.004 0.010 35 0.078 0.006 0.013 time n=100 n=200 n=400 0.96 0.018 0.009 0.006 1.97 0.015 0.007 0.005 2.98 0.009 0.005 0.003 3.99 0.005 0.002 0.002 5 0 0 0 Table 1: Bias of backfitting algorithm for sample sizes n = 100, 200, 400 for the age and time component for selected ages and time points. Based on 1000 realisations. n age mean se sd cov time mean se sd cov 100 6.717 0.224 0.231 0.912 0.96 0.044 0.045 0.954 100 13.788 0.297 0.298 0.935 1.97 0.039 0.04 0.946 100 20.859 0.351 0.357 0.943 2.98 0.032 0.034 0.951 100 27.929 0.391 0.402 0.938 3.99 0.024 0.024 0.966 100 35 0.460 0.464 0.932 5 0.016 0.017 0.874 200 6.717 0.158 0.155 0.94 0.96 0.031 0.031 0.951 200 13.788 0.207 0.206 0.942 1.97 0.027 0.027 0.960 200 20.859 0.243 0.237 0.948 2.98 0.022 0.022 0.966 200 27.929 0.271 0.262 0.945 3.99 0.017 0.017 0.972 200 35 0.328 0.329 0.933 5 0.011 0.012 0.933 400 6.717 0.114 0.118 0.948 0.96 0.022 0.022 0.951 400 13.788 0.148 0.153 0.946 1.97 0.019 0.019 0.957 400 20.859 0.173 0.18 0.937 2.98 0.015 0.015 0.960 400 27.929 0.192 0.196 0.943 3.99 0.012 0.012 0.970 400 35 0.235 0.245 0.934 5 0.008 0.008 0.950 Table 2 : 2Uncertainty estimated from bootstrap for sample sizes n = 100, 200, 400 for the age and time component for selected ages and time points. Based on 1000 realisations and a bootstrap with 100 repetitions. mean of estimated standard errors (mean se), standard deviation of estimates (sd) and 95 % pointwise coverage (cov). Table 3 : 3Coverage of confidence bands estimated from bootstrap for sample sizes n = 100, 200, 400 for the age and time component. Based on 1000 realisations and a boostrap with 100 repetitions. A linear regression model for the analysis of life times. O O Aalen, Statist. Med. 8Aalen, O. O. (1989). A linear regression model for the analysis of life times. Statist. Med. 8, 907-925. P K Andersen, Ø Borgan, R D Gill, N Keiding, Statistical Models Based on Counting Processes. New YorkSpringerAndersen, P. K., Borgan, Ø., Gill, R. D., & Keiding, N. (1993). Statistical Models Based on Counting Processes. Springer, New York. Age-period-cohort models for the lexis diagram. B Carstensen, Statist. Med. 26Carstensen, B. (2007). Age-period-cohort models for the lexis diagram. Statist. Med. 26, 3018-3045. 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F W Huffer, I W Mckeague, J. Amer. Statist. Assoc. 86Huffer, F. W. & McKeague, I. W. (1991). Weighted least squares estimation for Aalen's additive risk model. J. Amer. Statist. Assoc. 86, 114-129. Multiple time scales in multistate models. S Iacobelli, B Carstensen, Statist. Med. 32Iacobelli, S. & Carstensen, B. (2013). Multiple time scales in multi- state models. Statist. Med. 32, 5315-5327. Does in-hospital ventricular fibrillation affect prognosis after myocardial infarction?. G V Jensen, C Torp-Pedersen, P Hildebrandt, L Kober, F E Nielsen, T Melchior, T Joen, P K Andersen, European Heart Journal. 18Jensen, G. V., Torp-Pedersen, C., Hildebrandt, P., Kober, L., Nielsen, F. E., Melchior, T., Joen, T., & Andersen, P. K. (1997). Does in-hospital ventricular fibrillation affect prognosis after myocardial infarction? European Heart Journal 18, 919-924. Nonparametric estimation from incomplete observations. E L Kaplan, P Meier, J. Amer. Statist. Assoc. 53Kaplan, E. L. & Meier, P. 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[ "Petrov-Galerkin method and Gauss collocation method for 1D general elliptic problems and superconvergence", "Petrov-Galerkin method and Gauss collocation method for 1D general elliptic problems and superconvergence" ]	[ "Waixiang Cao ", "Lueling Jia ", "Zhimin Zhang " ]	[]	[]	In this paper, we present and study C 1 Petrov-Galerkin and Gauss collocation methods with arbitrary polynomial degree k (≥ 3) for one-dimensional elliptic equations. We prove that, the solution and its derivative approximations converge with rate 2k − 2 at all grid points; and the solution approximation is superconvergent at all interior roots of a special Jacobi polynomial of degree k + 1 in each element, the first-order derivative approximation is superconvergent at all interior k − 2 Lobatto points, and the second-order derivative approximation is superconvergent at k − 1 Gauss points, with an order of k + 2, k + 1, and k, respectively. As a by-product, we prove that both the Petrov-Galerkin solution and the Gauss collocation solution are superconvergent towards a particular Jacobi projection of the exact solution in H 2 , H 1 , and L 2 norms. All theoretical findings are confirmed by numerical experiments.	10.3934/dcdsb.2020327	[ "https://arxiv.org/pdf/2002.02266v1.pdf" ]	211,044,015	2002.02266	21fc94c0b4363fab3f0522b68e49063c8d4c0925	Petrov-Galerkin method and Gauss collocation method for 1D general elliptic problems and superconvergence 6 Feb 2020 Waixiang Cao Lueling Jia Zhimin Zhang Petrov-Galerkin method and Gauss collocation method for 1D general elliptic problems and superconvergence 6 Feb 2020A C 1Hermite interpolationC 1 elementsSuperconvergenceGauss collocation methodsPetrov-Galerkin methodsJacobi polynomials AMS: 65N3065N3565N1265N15 In this paper, we present and study C 1 Petrov-Galerkin and Gauss collocation methods with arbitrary polynomial degree k (≥ 3) for one-dimensional elliptic equations. We prove that, the solution and its derivative approximations converge with rate 2k − 2 at all grid points; and the solution approximation is superconvergent at all interior roots of a special Jacobi polynomial of degree k + 1 in each element, the first-order derivative approximation is superconvergent at all interior k − 2 Lobatto points, and the second-order derivative approximation is superconvergent at k − 1 Gauss points, with an order of k + 2, k + 1, and k, respectively. As a by-product, we prove that both the Petrov-Galerkin solution and the Gauss collocation solution are superconvergent towards a particular Jacobi projection of the exact solution in H 2 , H 1 , and L 2 norms. All theoretical findings are confirmed by numerical experiments. Introduction Superconvergence phenomenon means that the convergent rate exceeds the best possible global rate at some special points. Those points are called superconvergent points. During the past several decades, the subject has attracted much attention from the scientific and engineering computing community, and it is well understood for the C 0 finite element method (see, e.g., [4,7,14,15,21,23,24,25,26,28,29,36]), the C 0 finite volume method (see, e.g., [8,11,13,19,31]), the discontinuous Galerkin method (see, e.g., [1,2,3,17,18,22,30,32,35]), and the spectral Galerkin method (see, e.g., [33,34]). Here by C 0 element methods we mean that the approximation space is continuous while its derivative function space is not continuous. As comparison, the relevant study for C 1 element methods (i.e., both the approximation space and its derivative function space are continuous) is lacking. Only very special and simple cases have been discussed (see. e.g., [29,6,5]). Comparing with continuous Galerkin (or C 0 element) and discontinuous Galerkin (DG) methods, the most attractive feature of C 1 element methods is the continuity of the derivative approximation across the element interface. As early as 1995, Wahlbin investigated the superconvergence of C 1 Galerkin (not Petrov Galerkin) and spline Galerkin methods in [29] for two-point boundary value problems and established a mathematical theory to find superconvergence points for the C 1 finite element solution under the locally uniform mesh assumption. It was proved in [29] that the function value approximation of the k-th C 1 Galerkin method is superconvergent with order k + 2 at zeros of a special polynomial, and the derivative error is k + 1-th order superconvergent at grid points as well as element mid-point when k is odd. While for even k, the superconvergence behavior changes: the function value approximation is superconvergent at interior Lobatto points, mesh points, and element mid-points, and the derivative is superconvergent at the Gauss points. All those superconvergence rates are one order higher than the counterpart optimal convergence rates and the superconvergence results are valid in case that the mesh is locally uniform. However, the generalization of the superconvergence analysis to quasi-uniform meshes is not straightforward. In 1999, Bialeck [6] studied piecewise Hermite bi-cubic orthogonal spline collocation solution of the Poisson equation on rectangular mesh and proved a fourth-order accuracy of the first order partial derivatives of the collocation solution at the partition nodes. Only recently, Bhal and Danumjaya in [5] presented a cubic spline collocation method for the one dimensional Helmholtz equation with discontinuous coefficients, and proved a fourth-order accuracy for the function value approximation and for the first-order derivative value approximation at the grid points. In this paper, we present and study a C 1 Petrov-Galerkin method and Gauss collocation method for elliptic equations in 1D. The trail space is taken as the C 1 polynomial space of degree not more than k, while the test space of the C 1 Petrov-Galerkin method is chosen as the L 2 polynomial space of degree not more than k − 2. As the reader may recall, the total degrees of freedom for the C 1 Petrov-Galerkin method is the same as that for the counterpart C 0 element method. The main purpose of our current work is to provide a unified mathematical approach to establish the superconvergence theory of C 1 element methods. We prove that, for general 1D elliptic equations, the solution of the C 1 Petrov-Galerkin method is superclose to a particular Jacobi projection of the exact solution and thus establish the following supreconvergece results at some special points: 1) both the function value and the first-order derivative approximations are superconvergent with order 2k − 2 at mesh nodes; 2) the function value approximation is superconvergent with order k +2 at roots of a generalized Jacobi polynomial; 3) the first-order derivative approximation is superconvergent with order k+1 at interior Lobatto points; 4) the second-order derivative approximation is superconvergent with order k at interior Gauss points. By interpreting the Gauss collocation method as a Petrov-Galerkin method up to some higher-order numerical integration errors, we also prove that the Gauss-collocation solution inherits almost all the superconvergence properties from the counterpart Petrov-Galerkin solution. The main contribution of this paper lies in that: in one hand, we provide a unified approach to establish the superconvergence theory of C 1 element methods and discover some new superconvergence phenomena, especially the (2k − 2)-th convergence rate of the derivative approximation at grid points and the superconvergence for the second order derivative approximation, which is greatly different from the C 0 element method and DG method, even the C 1 finite element method in [29]; on the other hand, all our superconvergence results are valid for non-uniform meshes. In other words, we improve the mesh condition from locally uniform meshes in [29] to quasi-uniform meshes. Furthermore, the superconvergence results for the C 1 Gauss collocation method can be viewed as the generalization of the one presented in [5]. Actually, the cubic spline collocation method in [5] is a special case of our current C 1 Gauss collocation method in case of k = 3. The rest of the paper is organized as follows. In section 2, we present a C 1 Petrov-Galerkin method and Gauss collocation method for elliptic equations under the one-dimensional setting. In section 3, we investigate approximation properties and superconvergence properties of a special Jacobi projection of the exact solution, which is the basis to establish the superconvergence theory for C 1 element methods. In section 4 and section 5, we separately study the superconvergence behavior of C 1 Petrov-Galerkin and Gauss collocation methods, where superconvergence at the grid points (function and first order derivative value approximations), at interior roots of Jacobi polynomials (function value approximation), at interior Lobatto points (first order derivative value approximation) and Gauss points (the second order derivative value approximation) are investigated. Numerical experiments supporting our theory are presented in section 6. Some concluding remarks are provided in section 7. Throughout this paper, we adopt standard notations for Sobolev spaces such as W m,p (D) on sub-domain D ⊂ Ω equipped with the norm · m,p,D and semi-norm \| · \| m,p,D . When D = Ω, we omit the index D; and if p = 2, we set W m,p (D) = H m (D), · m,p,D = · m,D , and \| · \| m,p,D = \| · \| m,D . Notation A B implies that A can be bounded by B multiplied by a constant independent of the mesh size h. A ∼ B stands for A B and B A. C 1 Petrov-Galerkin methods and Gauss collocation methods We consider the following two-point boundary value problem − (αu ′ ) ′ + βu ′ + γu = f, x ∈ Ω = (a, b), u(a) = u(b) = 0, (2.1) where α > α 0 > 0, γ − β ′ 2 ≥ 0, γ ≥ 0, α, β, γ ∈ L ∞ (Ω) , and f is real-valued function defined onΩ. For simplicity, we assume that α, β, γ are all constants. Other than technical complexity, there is no essential difficulty in analysis for variable coefficients as long as the above conditions are satisfied. Let a = x 0 < x 1 < . . . < x N be N + 1 distinct points on the intervalΩ. For all positive integers r, we define Z r = {1, . . . , r} and denote by τ j = (x j−1 , x j ), j ∈ Z N . Let h j = x j − x j−1 , and h = max j h j . We assume that the mesh is quasi-uniform, i.e., there exists a constant c such that h ≤ ch j , j ∈ Z N . Define V h := {v ∈ C 1 (Ω) : v\| τ j ∈ P k (τ j ), j ∈ Z N } to be the C 1 finite element space, where P k , k ≥ 3 denotes the space of polynomials of degree not more than k. Let V 0 h := {v ∈ V h : v(a) = v(b) = 0}. We adopt two numerical methods to solve the problem (2.1), i.e., the Petrov-Galerkin method and the Gauss collocation method. To establish the Petrov-Galerkin method, we choose V 0 h as our trail space and the piecewise polynomial space of degree k − 2 as the test space, which is defined as follows: W h := {w ∈ L 2 (Ω) : w\| τ j ∈ P k−2 (τ j ), j ∈ Z N }. Petrov-Galerkin method: The Petrov-Galerkin method for solving (2.1) is to find a u h ∈ V 0 h such that (−αu ′′ h , v h ) + (βu ′ h + γu h , v h ) = (f, v h ), ∀v h ∈ W h . (2.2) Gauss collocation method: Given any i ∈ Z N , we denote by g im , m ∈ Z k−1 the k − 1 Gauss points in the interval τ i . That is, {g im } k−1 m=1 are zeros of the Legendre polynomial of degree k − 1. Then the Gauss collocation method to (2.1) is: Find aū h ∈ V 0 h such that (−αū ′′ h + βū ′ h + γū h )(g im ) = f (g im ), (i, m) ∈ Z N × Z k−1 . (2.3) 3 Approximation and superconvergence properties of the truncated Jacobi projection In this section, we define a C 1 Jacobi projection of the exact solution and study the approximation and superconvergence properties of the Jacobi projection, which is of great importance to establish superconvergence results for the C 1 numerical solution, especially the discovery of superconvergence points. We begin with some preliminaries. We first introduce the Jacobi polynomials. The Jacobi polynomials, denote by J r,l n , r, l > −1, are orthogonal with respect to the Jacobi weight function ω r,l (s) := (1 − s) r (1 + s) l over I := (−1, 1). That is, 1 −1 J r,l n (s)J r,l m (s)ω r,l (s)ds = κ r,l n δ mn , where δ denotes the Kronecker symbol and κ r,l n = J r,l n 2 ω r,l := 2 r+l+1 Γ(n + r + 1)Γ(n + l + 1) (2n + r + l + 1)Γ(n + 1)Γ(n + r + l + 1) . Here Γ(n) denotes the Gamma function. Note that when r = l = 0, the Jacobi polynomial J r,l n is reduced to the standard Legendre polynomial. That is J 0,0 n (s) = L n (s) with L n (s) being the Legendre polynomial of degree n over [ −1, 1]. We extend the definition of the classical Jacobi polynomials to the cases where both parameters r, l ≤ −1 It was proved in [27] (see Lemma 6.2) that the Jacobi polynomials satisfy the following derivative recurrence relation ∂ s J r,l n (s) = C r,l n J r+1,l+1 n−1 (s). (3.2) where C r,l n =    −2(n + r + l + 1), if r, l ≤ −1 −n, if r ≤ −1, l > −1, or r > −1, l ≤ −1, 1 2 (n + r + l + 1), if r, l > −1. (3.3) By taking r = l = −2 in (3.1) and using the derivative recurrence relation (3.2), we obtain J −2,−2 n (s) = (1 − s) 2 (1 + s) 2 J 2,2 n−4 (s) = 2 n (1 − s) 2 (1 + s) 2 ∂ s J 1,1 n−3 (s) = 4 n(n − 1) (1 − s) 2 (1 + s) 2 ∂ 2 s J 0,0 n−2 (s) = 4 n(n − 1) (1 − s) 2 (1 + s) 2 ∂ 2 s L n−2 (s). On the other hand, we have, from (3.2) ∂ 2 s J −2,−2 n (s) = −2(n − 3)∂ s J −1,−1 n−1 (s) = c n J 0,0 n−2 (s) = c n L n−2 (s), c n = 4(n − 3)(n − 2).(3.4) The above Jacobi polynomial plays an important role in our later superconvergence analysis. Given any function u ∈ C 1 (Ω), suppose u(x) has the following Jacobi expansion in each 1] is the Jacobi polynomial of degree n over τ i , and H 3 u ∈ P 3 denotes the Hermite interpolation of u, i.e., element τ i , i ∈ Z N u(x)\| τ i = H 3 u(x) + ∞ n=4 u nĴ −2,−2 n (x), (3.5) whereĴ −2,−2 n (x) = J −2,−2 n ( 2x−x i −x i−1 h i ) = J −2,−2 n (s), s ∈ [−1,∂ m x H 3 u(x i ) = ∂ m x u(x i ), ∂ m x H 3 u(x i−1 ) = ∂ m x u(x i−1 ), m = 0, 1. By (3.4) and the orthogonality properties of the Legendre polynomial, we have u n = h 2 i 4c n τ i (∂ 2 x uL i,n−2 )(x)dx/ τ i L i,n−2 (x)L i,n−2 (x)dx. (3.6) Here c n is the same as that in (3.4) and L i,n (x) denotes the Legendre polynomial of degree n over τ i , that is, L i,n (x) = L n ( 2x − x i − x i−1 h i ) = L n (s), s ∈ [−1, 1]. Now we define a truncated Jacobi projection u I ∈ V h of u as follows: u I (x)\| τ i :=    H 3 u(x) + k n=4 u nĴ −2,−2 n (x), if k ≥ 4,H 3 u(x), if k = 3. (3.7) We have the following orthogonal and approximation properties for u I . Proposition 1 Assume that u ∈ W k+2,∞ (Ω) is the solution of (2.1), and u I is the Jacobi truncation projection of u defined by (3.7). Then the following orthogonality and approximation properties hold true. 1. Orthogonality: τ i (u− u I ) ′′ vdx = 0, τ i (u− u I ) ′ v ′ dx = 0, τ i (u− u I )v ′′ = 0, ∀v ∈ P k−2 (τ i ). (3.8) 2. Optimal error estimates: u − u I 0,∞,τ i + h u − u I 1,∞,τ i h k+1 \|u\| k+1,∞,τ i . (3.9) 3. Superconvergence of function value approximation on roots of J −2,−2 k+1 : (u − u I )(x i ) = (u − u I )(x i−1 ) = 0, \|(u − u I )(l im )\| h k+2 \|u\| k+2,∞,τ i , (3.10) where l im , m = 1, · · · , k − 3 for k ≥ 4 are interior roots of J −2,−2 k+1 in τ i . 4. Superconvergence of first order derivative value approximation on Gauss-Lobatto points: (u − u I ) ′ (x i ) = (u − u I ) ′ (x i−1 ) = 0, \|(u ′ − u I ) ′ (gl in )\| h k+1 \|u\| k+2,∞,τ i , (3.11) where gl in , n = 1, · · · , k − 2 are interior roots of ∂ xĴ −2,−2 k+1 (x) = c kĴ −1,−1 k (x) on τ i . That is, gl in , i ≤ k − 2 are interior Gauss-Lobatto points of degree k − 2. 5. Superconvergence of second order derivative value approximation on Gauss points: \|(u − u I ) ′′ (g in )\| h k \|u\| k+2,∞,τ i , (3.12) where g in , n ≤ k − 1 are interior roots of L i,k−1 (x), i.e., the Gauss points of degree k − 1. Proof. First, subtracting (3.7) from (3.5) yields that ∂ m x (u − u I )(x) = ∞ n=k+1 u n ∂ m xĴ −2,−2 n (x), m = 0, 1, 2. (3.13) Using (3.4) and the orthogonal properties of Legendre polynomials, we derive τ i (u − u I ) ′′ v = 0, ∀v ∈ P k−2 (τ i ). (3.14) Noticing that ∂ m xĴ −2,−2 n (x i ) = ∂ m xĴ −2,−2 n (x i−1 ) = 0, m = 0, 1, we easily get ∂ m x u I (x i ) = ∂ m x u(x i ), ∂ m x u I (x i−1 ) = ∂ m x u(x i−1 ), m = 0, 1. (3.15) Consequently, a simple integration by parts and (3.14) lead to τ i (u − u I ) ′ v ′ = 0, τ i (u − u I )v ′′ = 0, ∀v ∈ P k−2 (τ i ). (3.16) That is, (u − u I )⊥P k−4 , (u − u I ) ′ ⊥P k−3 , (u − u I ) ′′ ⊥P k−2 . Then (3.8) follows. We now prove the approximation and superconvergence properties (3.9)-(3.12). By a scaling from τ i to [−1, 1] and a simple integration by parts for (3.6), we have u n = (2n − 3) 2c n 1 −1 ∂ 2 s u(s)L n−2 (s)ds = γ n 1 −1 ∂ 2 s u(s) d n−2 (1 − s 2 ) n−2 ds n−2 ds = (−1) n−2 γ n 1 −1 ∂ n s u(s)(1 − s 2 ) n−2 , ∀n ≥ 4, where u(s) = u( 2x − x i − x i−1 h i ) = u(x), s ∈ [−1, 1], x ∈ τ i , γ n = (−1) n (2n − 3) c n 2 n−1 (n − 2)! . Noticing that ∂ n s u(s) = ( h i 2 ) n ∂ n x u(x) = O(h n ), we have \|u n \| h n \|u\| n,∞ , ∀n ≥ 4. (3.17) Then (3.9) follows. At roots of J −2,−2 k+1 (x), there holds \|(u − u I )(l im )\| = ∞ n=k+2 u nĴ −2,−2 n (x) h k+2 \|u\| k+2,∞ . This finishes the proof of (3.10). Similarly, we can prove (3.11)-(3.12). The proof is complete. Superconvergence for C 1 Petrov-Galerkin methods In this section, we study superconvergence properties of the C 1 Petrov-Galerkin method for (2.1). To this end, we begin with the introduction of the bilinear form of the finite element method and some Green functions. First, we denote by a(·, ·) the bilinear form of the finite element method, which is defined as a(u, v) = (αu ′ , v ′ ) − (βu, v ′ ) + (γu, v), ∀u, v ∈ H 1 (Ω). Second, given any x ∈ Ω, let G(x, ·) be the Green function for the problem (2.1). Then for any v ∈ H 1 (Ω), v(x) = a(v, G(x, ·)), ∀x ∈ Ω. (4.1) Especially, if v(x) ∈ H 1 0 (Ω), then the Green function G(x, ·) satisfies G(x, a) = G(x, b) = 0. Let S h be the C 0 finite element space, i.e., S h = {v ∈ C 0 (Ω) : v\| τ i ∈ P k , v(a) = v(b) = 0, i ∈ Z N }. Denote by G h ∈ S h the Galerkin approximation of G(x, ·), that is, v h (x) = a(v h , G h ) = a(v h , G(x, ·)), ∀v h ∈ S h . (4.2) Finally, we use the following notations in the rest of this paper e h := u − u h = ξ + η, ξ := u I − u h , η := u − u I . We have the following optimal error estimates for the C 1 Petrov-Galerkin method. Lemma 1 Assume that u ∈ W k+1,∞ (Ω) is the solution of (2.1), and u h is the solution of (2.2). Then u − u h 0,∞ h k+1 \|u\| k+1,∞ , u − u h 1,∞ h k \|u\| k+1,∞ , u − u h 2 h k−1 \|u\| k+1,∞ . (4.3) Proof. First, noticing that the exact solution u also satisfy (2.2), we have (−αe ′′ h , v h ) + (βe ′ h + γe h , v h ) = 0, ∀v h ∈ W h . (4.4) Especially, we choose v h = −ξ ′′ in the above equation and using the orthogonal property of η in (3.8) and (3.9) to get (αξ ′′ , ξ ′′ ) + (γξ ′ , ξ ′ ) − β 2 (\|ξ ′ (b)\| 2 − \|ξ ′ (a)\| 2 ) = (−αη ′′ + βη ′ + γη, ξ ′′ ) (4.5) (\|β\|h k + γh k+1 ) ξ ′′ 0 \|u\| k+1,∞ . (4.6) On the other hand, noticing that e ′ h (x) ∈ C 0 (Ω) ⊂ H 1 (Ω), we take v = e ′ h in (4.1) and use the integration by parts to obtain e ′ h (x i ) = a(e ′ h , G(x i , ·)) = (αe ′′ h − βe ′ h , G ′ (x i , ·)) + (γe ′ h , G(x i , ·)) = (αe ′′ h − βe ′ h − γe h , G ′ (x i , ·)) = (αe ′′ h − βe ′ h − γe h , G ′ (x i , ·) − I k−2 G ′ (x i , ·)). Here I k−2 v denotes the L 2 projection of v onto P k−2 . Since the Green function G(x i , ·) ∈ C k (τ j ), j ∈ Z N is bounded, we have \|ξ ′ (x i )\| = \|e ′ h (x i )\| h k−1 e h 2 h 2(k−1) \|u\| k+1 + h k−1 ξ 2 , i = 0, . . . , N. (4.7) Substituting (4.7) into (4.6) and using the Cauchy-Schwartz inequality yields (αξ ′′ , ξ ′′ ) + (γξ ′ , ξ ′ ) ≤ C(βh k + γh k+1 + h 2(k−1) ) 2 \|u\| 2 k+1,∞ + ( 1 2 + C 1 h 2(k−1) ) ξ 2 2 , where C, C 1 are some positive constants independent of h. Consequently, when h is sufficiently small, there holds \|ξ\| 2 + \|ξ\| 1 (\|β\|h k + γh k+1 + h 2(k−1) )\|u\| k+1,∞ . (4.8) Similarly, we choose v h = ξ ∈ H 1 0 (Ω) in (4.2) and again use the integration by parts to obtain \|ξ(x)\| = \|a(ξ, G h )\| = (−αξ ′′ + βξ ′ + γξ, G h −Ḡ h ) + (−αξ ′′ + βξ ′ + γξ,Ḡ h ) ≤ h ξ 2 G h 1 + \|(−αη ′′ + βη ′ + γη,Ḡ h )\| ≤ h ξ 2 G h 1 + h k+1 \|u\| k+1,∞ Ḡ h 1,1 . HereḠ h \| τ j ∈ P 0 (τ j ) denotes the cell average of G h . It has been proved in [14] that G h 2,1 1, which yields, together with the embedding theory G h 1 G h 2,1 1. Consequently, \|ξ(x)\| h k+1 \|u\| k+1,∞ + h ξ 2 h k+1 \|u\| k+1,∞ , and thus, ξ 0,∞ h k+1 \|u\| k+1,∞ , ξ 1,∞ h −1 ξ 0,∞ h k \|u\| k+1,∞ . Then (4.3) follows from the triangle inequality and the standard approximation theory. Now we are ready to present the superconvergence of the solution for the C 1 Petrov-Galerkin method. Theorem 1 Assume that u ∈ W k+2,∞ (Ω) is the solution of (2.1), and u h is the solution of (2.2). The following superconvergence properties hold true. 1. Supercloseness between the numerical solution and truncation projection in the H 2 norm: u h − u I 2 h k \|u\| k+1,∞ , if β = 0, u h − u I 2 h k+1 \|u\| k+1,∞ , if β = 0, γ = 0. (4.9) 2. if β = γ = 0, then u h (x) = u I (x), (u − u h )(x i ) = 0, (u − u h ) ′ (x i ) = 0. (4.10) 3. Superconvergence for both function value and derivative value approximations at nodes: \|(u − u h )(x i )\| h 2(k−1) \|u\| k+1,∞ , \|(u − u h ) ′ (x i )\| h 2(k−1) \|u\| k+1,∞ , i ∈ Z N . (4.11) Superconvergence of function value approximation on interior roots of J −2,−2 k+1 for k ≥ 4: \|(u − u h )(l im )\| h k+2 u k+2,∞ , (4.12) where l im , m = 1, · · · , k − 3 are k − 3 interior roots of J −2,−2 k+1 in τ i . 5. Superconvergence of first order derivative value approximation on Gauss-Lobatto points: \|(u − u h ) ′ (gl in )\| h k+1 u k+2,∞ , (4.13) where gl in , n ≤ k − 2 are interior Gauss-Lobatto points of degree k − 2. 6. Superconvergence of second order derivative value approximation on Gauss points: \|(u − u h ) ′′ (g in )\| h k u k+2,∞ ,(4. 14) where g in , n ≤ k − 1 are interior roots of L i,k−1 (x), i.e., the Gauss points of degree k − 1. Proof. First, (4.9) follows directly from (4.8). Furthermore, there holds from (4.6), \|u h − u I \| 2 + \|u h − u I \| 1 = 0, if β = γ = 0, which indicates that u h − u I is a constant. Noticing that (u h − u I )(a) = 0, we have u I = u h . Then (4.10) follows. Now we consider the superconvergence at nodes. In light of (4.1) and (4.4), we obtain \|e h (x i )\| = \|a(e h , G(x i , ·))\| = (−αe ′′ h + βe ′ h + γe h , G(x i , ·) − I k−2 G(x i , ·)) h 2(k−1) \|u\| k+1,∞ , where in the last step, we have used the fact the Green function G( x i , ·) ∈ C k (τ j ), j ∈ Z N is bounded. Similarly, we have from (4.7) and (4.3) \|e ′ h (x i )\| h k−1 e h 2 h 2(k−1) \|u\| k+1,∞ . Then (4.11) follows. We next prove (4.12)-(4.14). We consider two cases, i.e., k ≥ 4 and k = 3. Case 1: k ≥ 4 For any function v ∈ L 2 (Ω), we denote by I k−2 v the L 2 projection of v onto P k−2 and define ∂ −1 x v(x) := x a v(t)dt. In light of (4.1), we have from the integration by parts, the orthogonality (3.8) and (4.4), ξ(x) = a(ξ, G h ) = (−αξ ′′ + βξ ′ + γξ, G h ) = (−αξ ′′ , I k−2 G h ) + (βξ ′ + γξ, G h ) = (−αe ′′ h , I k−2 G h ) + (βξ ′ + γξ, G h ) = −(βe ′ h + γe h , I k−2 G h ) + (βξ ′ + γξ, G h ) = (βe ′ h + γe h , G h − I k−2 G h ) − (βη ′ + γη, G h ) = I 1 − I 2 . Now we estimate the two terms I 1 , I 2 , respectively. In light of (4.3) and the fact that G h 2,1 1 (see, e.g., [14]), we have \|I 1 \| = \|(βe ′ h + γe h , G h − I k−2 G h )\| h 2 ( e h 1,∞ + e h 0,∞ ) G h 2,1 h k+2 \|u\| k+1,∞ . On the other hand, by (3.8), there holds for k ≥ 4, τ i (u − u I )(x)dx = 0, and thus (∂ −1 x η)(x i ) = 0, i ∈ Z N , (∂ −1 x η)(x) = x x i−1 η(t)dt, ∀x ∈ τ i . Then by the integration by parts, \|I 2 \| = \|(βη ′ + γη, G h )\| = \|(β∂ −1 x η, G ′′ h ) − (γ∂ −1 x η, G ′ h )\| ∂ −1 x η 0,∞ G h 2,1 h η 0,∞ . Consequently, \|ξ(x)\| ≤ \|I 1 \| + \|I 2 \| h k+2 \|u\| k+1,∞ , ∀x ∈ Ω, which yields, together with the inverse inequality, ξ 0,∞ h k+2 \|u\| k+1,∞ , ξ 1,∞ h k+1 \|u\| k+1,∞ , ξ 2,∞ h k \|u\| k+1,∞ . Then the desired results (4.12)-(4.14) follow from the triangle inequality and the approximation properties of u I in Theorem 1. Case 2: k = 3 To prove (4.13) and (4.14) for k = 3, we first construct a special function w h ∈ P 3 ∩C 1 (Ω) satisfying the following condition: (αw ′′ h , v) = (βη ′ + γη, v) ∀v ∈ P 1 (τ j ) \ P 0 (τ j ), (4.15) w ′ h (x i ) = 0, w h (a) = 0 ∀i = 0, . . . , N. (4.16) We can prove that the function w h is uniquely defined. Actually, if the right hand side of (4.15) equals to zero, we can easily obtain that w ′′ h = 0. Then the boundary condition (4.16) indicates that w h = 0. We next estimate the function w h . We suppose w h (x) = N i=1 c i φ i (x) with φ i (x) ∈ P 3 ∩ C 1 (Ω) being the basis function associated with the node x i , that is, φ i (x) =      1 h 3 i+1 (x i+1 − x) 2 (2x + x i+1 − 3x i ), if x ∈ τ i+1 1 h 3 i (x − x i−1 ) 2 (3x i − 2x − x i−1 ), if x ∈ τ i , 0, else We choose v = x in (4.15) to obtain 12(c j − c j−1 ) h 3 j (x j − x, x) j = (βη ′ + γη, x) j , where (w h , v) j = τ j w h vdx,x j = x j +x j+1 2 . Consequently, \|c j − c j−1 \| h k+3 j \|u\| k+1,∞ , and thus, w ′′ h 0,∞,τ j \|c j − c j−1 \| h 3 j h k \|u\| k+1,∞ . Moreover, there hods for all x ∈ τ j \|w ′ h (x)\| = w ′ h (x j−1 ) + x x j−1 w ′′ h (x)dx ≤ h j w ′′ h 0,∞,τ j h k+1 \|u\| k+1,∞ . Then w ′ h 0,∞ h k+1 \|u\| k+1,∞ , w h 0,∞ w ′ h 0,∞ h k+1 \|u\| k+1,∞ . Now we are ready to prove (4.13) and (4.14) for k = 3. Let e h = u − u h =ξ +η,ξ := u I − u h − w h ,η := u − u I + w h . Choosing v h = −ξ ′′ in (4.4) following the same arguments as that in (4.6), we obtain (αξ ′′ ,ξ ′′ ) + (γξ ′ ,ξ ′ ) − β 2 (\|ξ ′ (b)\| 2 − \|ξ ′ (a)\| 2 ) = (−αη ′′ + βη ′ + γη,ξ ′′ ) = (βη ′ + γη,ξ ′′ ) + (−αw ′′ h + βw ′ h + γw h ,ξ ′′ ) = I. We now estimate the term I. Sinceξ ′′ \| τ j ∈ P 1 (τ j ), we have the following decompositioñ ξ ′′ = ξ 0 + ξ 1 , ξ 0 ∈ P 0 (τ j ), ξ 1 ∈ P 1 (τ j ) \ P 0 (τ j ). By (4.15) and the integration by parts, we get \|I\| = (βη ′ + γη, ξ 0 ) + (−αw ′′ h + βw ′ h + γw h , ξ 0 ) + (βw ′ h + γw h , ξ 1 ) = (η, ξ 0 ) + (βw ′ + γw h , ξ 0 ) + (βw ′ h + γw h , ξ 1 ) ( η 0 + w h 1 ) ξ ′′ 0 . In light of the estimates for w h and η, we get (αξ ′′ ,ξ ′′ ) + (γξ ′ ,ξ ′ ) − β 2 (\|ξ ′ (b)\| 2 − \|ξ ′ (a)\| 2 ) h k+1 \|u\| k+1,∞ \|ξ\| 2 . (4.17) By (4.11), there holds \|ξ ′ (b)\| = \|ξ ′ (b)\| h 2(k−1) \|u\| k+1,∞ . Substituting the above estimate into (4.17) and using the Cauchy-Schwartz inequality yields ξ ′′ 0 h k+1 \|u\| k+1,∞ . By the triangle inequality and the inverse inequality, ξ ′′ 0,∞ ≤ ξ ′′ 0,∞ + w ′′ h 0,∞ h − 1 2 ξ ′′ 0 + h k+1 \|u\| k+1,∞ h k \|u\| k+1,∞ . Furthermore, there holds for all x ∈ τ j \|ξ ′ (x)\| = ξ ′ (x j−1 ) + x x j−1 ξ ′′ (x)dx h 2(k−1) \|u\| k+1,∞ + h ξ ′′ 0,∞ h 2(k−1) \|u\| k+1,∞ + h k+1 \|u\| k+1,∞ . Then (4.13) and (4.14) follows from the triangle inequality and the approximation properties of u I for k = 3. This finishes our proof. Remark 1 As we may observe from the above theorem, for problems with constant coefficients, the convergence rate of the error u h − u I 2 is two order higher than the optimal convergence rate k − 1 in case of β = 0, γ = 0. However, this superconvergence result may not hold true for problems with variable coefficients. Actually, in case of β = 0, α = 0, γ = 0 with α a variable function, we have from (5.8) (αξ ′′ , ξ ′′ ) + (γξ ′ , ξ ′ ) − β 2 (\|ξ ′ (b)\| 2 − \|ξ ′ (a)\| 2 ) = (−αη ′′ + βη ′ + γη, ξ ′′ ) = ((ᾱ − α)η ′′ + βη ′ + γη, ξ ′′ ) h k ξ ′′ 0 \|u\| k+1,∞ , whereᾱ denotes the cell average of α, i.e.,ᾱ\| τ j = h −1 j τ j α(x)dx. Then we follow the same argument as that in Lemma 1 to obtain ξ ′′ h k \|u\| k+1,∞ . In other words, the convergence rate of u h − u I 2 for problems with variable coefficients is always k, only one order higher than the optimal convergence rate. This is the difference between the constant coefficients and variable coefficients. Our numerical examples will demonstrate this point. Superconvergence for Gauss Collocation methods This section is dedicated to the superconvergence analysis of the Gauss collocation method. Our analysis is along this line: we first prove that the Gauss collocation solutionū h is superclose to the Petrov-Galerkin solution u h ; then due to the supercloseness betweenū h and u h , the numerical solutionū h shares the same superconvergence results with that of u h , and finally we establish all superconvergence results for the solution of the Gauss collocation method. We begin with some preliminaries. We first denote by ω im , (i, m) ∈ Z N × Z k−1 the wight of Gauss quadrature. For any function u, v, we define the following discrete L 2 inner product (·, ·) * as (u, v) * := N i=1 k−1 m=1 (uv)(g im )ω im . For any v h ∈ W h , we multiply v h (g im )ω im , (i, m) ∈ Z N × Z k−1 on both sides of (2.3) and sum up all m from 1 to k − 1 to derive k−1 m=1 (−αū ′′ h + βū ′ h + γū h )(g im )v h (g im )ω im = k−1 m=1 f (g im )v h (g im )ω im . (5.1) As we may observe, the C 1 Gauss collocation method can be viewed as the counterpart Petrov-Galerkin method up to a Gauss numerical integration error. Note that the (k − 1)point Gauss quadrature is exact for all polynomials of degree not less than 2k − 3. Then a * (ū h , v h ) := (−αū ′′ h , v h ) + (βū ′ h , v h ) + (γū h , v h ) * = (f, v h ) * , ∀v h ∈ W h . (5.2) Denoteē h = u h −ū h . Subtracting (5.2) from (2.2), we have (−αē ′′ h , v h ) + (βē ′ h , v h ) − (γū h , v h ) * + (γu h , v h ) = (f, v h ) − (f, v h ) * , ∀v h ∈ W h ,(5.3) or equivalently, (−αē ′′ h + βē ′ h + γē h , v h ) = (γū h , v h ) * − (γū h , v h ) + (f, v h ) − (f, v h ) * , ∀v h ∈ W h . (5.4) We note that up to a Gauss numerical quadrature error, the right hand side of the above equation equals to zero. We have the following supercloseness result for the errorē h . Theorem 2 Assume that u ∈ W 2k,∞ (Ω) is the solution of (2.1), and u h andū h is the solution of (2.2) and (2.3), respectively. Then u h −ū h 0,∞ + h u h −ū h 1,∞ + h 2 u h −ū h 2 h k+2 u 2k,∞ . (5.5) Proof. Noticing thatē h ∈ V 0 h , we choose v h = −ē ′′ h ∈ W h in (5. 3) and use the integration by parts to obtain (αē ′′ h ,ē ′′ h ) + (γē ′ h ,ē ′ h ) * − β 2 (\|ē ′ h (b)\| 2 − \|ē ′ h (a)\| 2 ) = (f − γu h ,ē ′′ h ) * − (f − γu h ,ē ′′ h ). (5.6) For any function w, we denote I h w ∈ P k−1 the Gauss interpolation function of w satisfying I h w(x i ) = w(x i ), I h w(g im ) = w(g im ), m ∈ Z k−1 . Since v h I h w ∈ P 2k−3 for all v h ∈ W h , we have \|(w, v h ) − (w, v h ) * \| = \|(w − I h w, v h )\| h k w k v h 0 , ∀v h ∈ W h . (5.7) Plugging the above estimate into (5.6) gives (αē ′′ h ,ē ′′ h ) + (γē ′ h ,ē ′ h ) * h k (\|f \| k + \|u h \| k ) ē ′′ h 0 + \|ē ′ h (b)\| 2 h k (\|f \| k + \|u\| k+1,∞ ) ē ′′ h 0 + \|ē ′ h (b)\| 2 ,(5.8) where in the last step, we have used (4.3), the inverse inequality and the triangle inequality to get \|u h \| k \|u I \| k + h −k \|u I − u h \| 0 \|u\| k + \|u − u I \| k + h −k ( u I − u 0 + u h − u 0 ) \|u\| k+1,∞ . (5.9) To estimateē ′ h (b), we choose v =ē ′ h in (4.1) and using the integration by parts to obtain e ′ h (x i ) = a(ē ′ h , G(x i , ·)) = (αē ′′ h − βē ′ h , G ′ (x i , ·)) + (γē ′ h , G(x i , ·)) = (αē ′′ h − βē ′ h − γē h , G ′ (x i , ·) −Ḡ ′ ) + (αē ′′ h − βē ′ h − γē h ,Ḡ ′ ) = (αē ′′ h − βē ′ h − γē h , G ′ (x i , ·) −Ḡ ′ ) − (f,Ḡ ′ ) + (f,Ḡ ′ ) * , whereḠ ′ ∈ P 0 denotes the cell average of G ′ (x i , ·) and in the last step, we have used (5.4) and the fact that (ū h , v) − (ū h , v) * = 0, ∀v ∈ P k−3 . Using the fact that G( x i , ·) ∈ C k (τ j ) is bounded, we get (αē ′′ h − βē ′ h − γē h , G ′ (x i , ·) −Ḡ ′ ) h ē h 2 . On the other hand, by (5.7), we have \|(f,Ḡ ′ ) − (f,Ḡ ′ ) * \| h k \|f \| k . Consequently, \|ē ′ h (x i )\| h ē h 2 + h k \|f \| k . Substituting the above inequality into (5.8) and using the Cauchy-Schwartz inequality yields (αē ′′ h ,ē ′′ h ) + (γē ′ h ,ē ′ h ) * ≤ ( α 4 + Ch 2 ) ē ′′ h 2 0 + C 1 h 2k (\|f \| k + \|u\| k+1,∞ ) 2 for some positive C, C 1 . Therefore, when h is sufficient small, there holds ē ′′ h 0 h k (\|u\| k+1,∞ + \|f \| k ) h k u k+2,∞ . We next estimate ē h 0,∞ . Choosing v =ē h in (4.2) and using (5.4), we get e h (x) = a(ē h , G h ) = (−αē ′′ h + βē ′ h + γē h , G h ) = (−αē ′′ h + βē ′ h + γē h , G h − I k−2 G h ) + (−αē ′′ h + βē ′ h + γē h , I k−2 G h ) = (βē ′ h + γē h , G h − I k−2 G h ) + (f − γū h , I k−2 G h ) − (f − γū h , I k−2 G h ) * . (5.10) Here again I k−2 G h denotes the L 2 projection of G h onto P k−2 . By using the error of Gauss quadrature (see, e.g., [20], P.98 (2.7.12)), there exists some θ j ∈ τ j such that (f, I k−2 G h ) − (f, I k−2 G h ) * = N j=1 h 2k−1 [(k − 1)!] 4 (2k − 1)[(2k − 2)!] 3 (f I k−2 G h ) (2k−2) (θ j ) h 2k−1 f 2k−2,∞ N j=1 I k−2 G h k−2,∞,τ j h k+2 f 2k−2,∞ G h 2,1 . Here in the last step, we have used the inverse inequality v h m,p h n−m+ 1 p − 1 q v h n,q , ∀n < m. Similarly, there holds (γū h , I k−2 G h ) − (γū h , I k−2 G h ) * = N j=1 h 2k−1 [(k − 1)!] 4 (2k − 1)[(2k − 2)!] 3 (ū h I k−2 G h ) (2k−2) (θ j ) h 2k−1 ū h k,∞ G h k−2,∞ h k+2 ū h k,∞ G h 2,1 h k+2 ( ē h k,∞ + u h k,∞ ) G h 2,1 . On the other hand, \|(βē ′ h + γē h , G h − I k−2 G h )\| h 2 G h 2,1 ē h 1,∞ h ē h 0,∞ G h 2,1 . Since G h 2,1 is bounded, we have \|ē h (x)\| h k+2 f 2k−2,∞ + h ē h 0,∞ + h k+2 ( ē h k,∞ + u h k,∞ ) h k+2 u 2k,∞ + h ē h 0,∞ . Here in the last step, we have used (5.9) and the inverse inequality ē h k,∞ h −k ē h 0,∞ . Consequently, ē h 0,∞ h k+2 u 2k,∞ , ē h 1,∞ h −1 ē h 0,∞ h k+1 u 2k,∞ . This finishes our proof. ✷ Using the conclusions in the above theorem and the superconvergence results for the Petrov-Galerkin method, we have the following superconvergence properties for the solution of Gauss collocation methods. Theorem 3 Assume that u ∈ W 2k,∞ (Ω) is the solution of (2.1), andū h is the solution of (2.3). The following superconvergence properties hold true. Superconvergence of function value approximation on interior roots of J −2,−2 k+1 for k ≥ 4: \|(u −ū h )(l im )\| h k+2 u 2k,∞ , (5.11) where l im , m = 1, · · · , k − 3 are interior roots ofĴ −2,−2 k+1 (x) in τ i . 2. Superconvergence of first order derivative value approximation on Gauss-Lobatto points: \|(u −ū h ) ′ (gl in )\| h k+1 u 2k,∞ , (5.12) where gl in , i ≤ k − 2 are interior Gauss-Lobatto points of degree k − 2. 3. Superconvergence of second order derivative value approximation on Gauss points: \|(u −ū h ) ′′ (g in )\| h k u 2k,∞ , (5.13) where g in , n ≤ k − 1 are interior roots of L k−1 , i.e., the Gauss points of degree k − 1. Supercloseness between the numerical solution and the truncation projection of the exact solution in the H 2 norm: ū h − u I 2 h k u 2k,∞ . (5.14) Superconvergence for both function value and derivative value approximations at nodes: \|(u −ū h )(x i )\| h 2(k−1) u 2k,∞ , \|(u −ū h ) ′ (x i )\| h 2(k−1) u 2k,∞ , i ∈ Z N . (5.15) Proof. We only prove (5.15) since (5.11)-(5.14) follow directly from Theorem 1 and Theorem 2. In light of (5.10), we have \|ē h (x i )\| = (−αē ′′ h + βē ′ h + γē h , G h (x i , ·) − I k−2 G h (x i , ·)) + I h k−1 e h 2 + \|I\|, where we used that G h ∈ C k (τ j ) and I = (γū h − f, I k−2 G(x i , ·)) * + (γū h − f, I k−2 G(x i , ·)). Again we use the error of Gauss numerical quadrature and the fact I k−2 G h k−2,∞ 1 to get \|I\| h 2k−2 ( f 2k−2,∞ + ū h k,∞ ) I k−2 G h k−2,∞ h 2k−2 u 2k,∞ , and thus \|ē h (x i )\| h 2k−2 u 2k,∞ . Similarly, we can prove \|ē ′ h (x i )\| h 2(k−1) ( u k+1,∞ + f 2k−2,∞ ) h 2k−2 u 2k,∞ . Then the proof is complete. Remark 2 As we may observe from Theorem 1 and Theorem 3, to achieve the same superconvergence result, the regularity assumption of the exact solution u for the Gauss collocation method is much more stronger than that for the counterpart Petrov-Galerkin method. Numerical experiments In this section, we present some numerical examples to demonstrate the method and to verify the theoretical findings established in previous sections. In our numerical experiments, we solve the model problem (2.1) by the C 1 Petrov-Galerkin method (2.2) and the Gauss collocation method (2.3) with k = 3 and k = 4. We test various errors in our examples, including the H 2 error of u h − u I denoted as u h − u I 2 , the maximum errors of u − u h and (u − u h ) ′ at mesh points, the maximum errors of u − u h at interior roots ofĴ −2,−2 k+1 (x), (u − u h ) ′ and (u − u h ) ′′ at interior Gauss-Lobatto and Gauss points, respectively. They are defined by e un = max i \|(u − u h )(x i )\|, e u ′ n = max i \|(u − u h ) ′ (x i )\|, e u = max i,m \|(u − u h )(l im )\|, e u ′ = max i,n \|(u − u h ) ′ (gl in )\|, e u ′′ = max i,n \|(u − u h ) ′′ (g in )\|. Here l im , 1 ≤ m ≤ k − 3 are interior roots ofĴ −2,−2 k+1 (x), and gl in , 1 ≤ n ≤ k − 2 are interior Lobatto points, and g in , 1 ≤ n ≤ k − 1 are interior Gauss points in τ i . For simplicity, we do not distinguish the error symbols when the method is clearly stated in the following tables. Example 1. We consider the following equation with Dirichlet boundary condition: −αu ′′ (x) + βu ′ (x) + γu(x) = f (x), x ∈ [0, 1], u(0) = u(1) = 0. (6.1) We take the constant coefficients as α = β = γ = 1, and choose the right-hand side function f such that the exact solution to this problem is u(x) = sin(πx). Non-uniform meshes of N elements are used in our numerical experiments with N = 2, . . . , 32, which are obtained by randomly and independently perturbing each node of a uniform mesh by up to some percentage. To be more precise, x j = j N + 0.01 1 N sin( jπ N )randn(), 0 ≤ j ≤ N, where randn() returns a uniformly distributed random number in (0, 1). We list in Table 1 various approximation errors calculated by the C 1 Petrov-Galerkin method for k = 3, 4. As we may observe, both the convergence rates of the error e un and e u ′ n are 2k − 2, and the convergence rate of e u , e u ′ , e u ′′ is k + 2, k + 1, k, respectively. All these results are consistent with our theoretical findings in Theorem 1. We next test the superconvergence behavior of the C 1 Gauss collocation method. We present in Table 2 the numerical data for various errors and the corresponding convergence rates calculated by the C 1 Gauss collocation method. We observe a convergence rate of 2k − 2 for e un and e u ′ n , k + 2 for e u , k + 1 for e u ′ , and k for e u ′′ , which confirms the theory established in Theorem 3. Furthermore, we also test the supercloseness result between the C 1 numerical solution and the Jacobi truncation projection of the exact solution under the H 2 norm for two different choices of parameters: i.e., β = 0, γ = 0 and β = 0, γ = 0. Listed in Table 3 are the approximation errors of u h − u I 2 and their corresponding convergence rates. From Table 3 we observe that, for both Petrov-Galerkin and Gauss collocation methods, the convergence rate of u h − u I 2 is k in case of β = 0, γ = 0. However, when β = 0, γ = 0, the convergence rate is k for the Gauss collocation method, and k + 1 for the Petrov-Galerkin method, which is one order higher than that for the Gauss collocation method. u h − uI 2 C 1 Petrov-Galerkin C 1 Gauss collocation α = β = γ = 1 α = γ = 1, β = 0 α = β = γ = 1 α = γ = 1, β = 0 k N−(α(x)u ′ (x)) ′ + β(x)u ′ (x) + γ(x)u(x) = f (x), x ∈ [0, 1], u(0) = u(1) = 0. (6.2) In our experiments, we test the problems of variable coefficients and consider the following three cases: • Case 1: α(x) = e x , β(x) = cos(x), γ(x) = x; • Case 2: α(x) = e x , β(x) = 0, γ(x) = x; • Case 3: α(x) = e x , β(x) = 0, γ(x) = 0. The right-hand side function f is chosen such that exact solution is u(x) = sin x(x 12 − x 11 ). We use the piecewise uniform meshes, which are constructed by equally dividing each interval [0, 2 3 ] and [ 2 3 , 1] into N/2 subintervals with N = 4, ..., 64. We present various approximation errors and the corresponding convergence rates in Tables 4-5 for the C 1 Petrov-Galerkin method, and in Tables 6-7 for the C 1 Gauss collocation method, for three different cases with k = 3, 4, respectively. Again, we observe the same superconvergence results as those for the constant coefficients in Example 1, i.e., both errors e un and e u ′ n converge with a rate of 2k − 2, and the convergence rates of e u , e u ′ , e u ′′ are k + 2, k + 1, k, respectively. In other words, superconvergence results in Theorem 1 and Theorem 3 are still valid for the case of variable coefficients. We also test the error u I − u h 2 in the above three cases of variable coefficients. We list in Table 8 and Table 9 the numerical data u I − u h 2 and the convergence rate for C 1 Petrov-Galerkin approximation and Gauss collocation approximation with k = 3, 4. We observe that the convergence rate is always k in different choices of variable coefficients, including the case β = 0, Recall that for the constant coefficients case, the convergent rate of the C 1 Petrov-Galerkin approximation is k + 1 when β = 0. To sum up, superconvergence phenomena for problems with variable coefficients under non-uniform meshes still exist, and the superconvergence behavior for variable coefficients problems is similar with that for the constant coefficients problems. Conclusion In this work, we have presented a unified approach to study superconvergence properties of C 1 Petrov-Galerkin and Gauss collocation methods for one-dimensional elliptic equations. Our main theoretical results include the proof of the 2k − 2 superconvergence rate for both solution and its first order derivative approximations at grid points, the k + 2-th order function value approximation at roots of the Jacobi polynomialĴ −2,−2 k+1 (x), the k + 1-th order derivative approximation at roots of (Ĵ −2,−2 k+1 ) ′ (x) (i.e., the Lobatto points), and the k-th order second order derivative approximation at the Gauss points. An unexpected discovery is that the superconvergence rate of the first order derivative error at mesh points can reach as high as 2k − 2, which almost doubles the optimal convergence rate k. The superconvergence points for the second order derivative approximation is also novel. Our analysis indicate that for constant coefficients, the Gauss collocation method is essentially equivalent to the Petrov-Galerkin method up to practically neglect able numerical quadrature errors, see (5.2). Indeed, we always use numerical quadrature instead of exact integration in practice. Comparing with the traditional C 0 Galerkin method, the major gain of the C 1 Petrov-Galerkin method discussed in this work is the 2k − 2 convergence rate of the derivative approximation at nodes, with the sacrifice of function value convergence rate at nodes dropping from 2k to 2k − 2. Comparing with the C 1 Galerkin method studied in [29], the C 1 Petrov-Galerkin method discussed in this work has equal or better convergence rates in all respect. It seems that that the L 2 test function is superior to the C 1 test function. Therefore, the C 1 Petrov-Galerkin method is a method to recommend if one is also interested in derivative approximations. Based on the analysis, extension of our results to the higher dimensional tensor-product space is feasible. J r,l n (s) := (1 − s) −r (1 + s) −l J −r,−l n+r+l (s), r, l ≤ −1. (3.1) Table 1 : 1Errors, corresponding convergence rates for C 1 Petrov-Galerkin method, α = β = γ = 1.eun e u ′ n eu e u ′ e u ′′ k N error order error order error order error order error order 2 8.03e-04 - 6.11e-03 - - - 7.59e-03 - 1.31e-01 - 4 7.02e-05 3.49 4.92e-04 3.61 - - 5.61e-04 3.73 1.66e-02 2.98 3 8 4.59e-06 3.95 3.02e-05 4.04 - - 3.73e-05 3.92 2.18e-03 2.93 16 2.91e-07 4.04 1.90e-06 4.05 - - 2.66e-06 3.87 2.67e-04 3.03 32 1.80e-08 4.00 1.18e-07 3.99 - - 1.77e-07 3.90 3.38e-05 2.98 2 2.88e-05 - 2.23e-05 - 7.54e-05 - 8.72e-04 - 1.30e-02 - 4 4.25e-07 6.10 2.47e-07 6.51 1.36e-06 5.81 2.44e-05 5.17 8.88e-04 3.88 4 8 6.53e-09 6.21 5.38e-09 5.69 2.14e-08 6.17 7.91e-07 5.10 5.47e-05 4.02 16 1.04e-10 5.96 1.04e-10 5.67 3.45e-10 5.94 2.64e-08 4.89 3.73e-06 3.87 32 1.62e-12 6.00 1.84e-12 5.83 5.50e-12 5.97 8.62e-10 4.94 2.30e-07 4.02 Table 2 : 2Errors, corresponding convergence rates for C 1 Gauss collocation method, α = β = γ = 1.eun e u ′ n eu e u ′ e u ′′ k N error order error order error order error order error order 2 5.25e-03 - 1.36e-02 - - - 1.44e-02 - 8.32e-02 - 4 2.88e-04 4.13 7.26e-04 4.18 - - 8.35e-04 4.06 1.16e-02 2.84 3 8 1.82e-05 4.07 4.66e-05 4.05 - - 5.89e-05 3.91 1.61e-03 2.85 16 1.16e-06 3.94 2.91e-06 3.96 - - 4.01e-06 3.84 1.91e-04 3.08 32 7.18e-08 4.01 1.81e-07 4.01 - - 2.65e-07 3.92 2.45e-05 2.96 2 1.32e-05 - 1.04e-04 - 1.98e-04 - 9.54e-04 - 7.12e-03 - 4 2.92e-07 5.48 1.79e-06 5.85 3.14e-06 5.96 3.32e-05 4.83 5.77e-04 3.63 4 8 4.62e-09 5.97 2.80e-08 5.99 5.09e-08 5.94 1.05e-06 4.98 3.89e-05 3.89 16 7.56e-11 6.06 4.40e-10 6.12 8.61e-10 6.01 3.40e-08 5.04 2.46e-06 3.98 32 1.18e-12 6.08 6.87e-12 6.08 1.28e-11 6.15 1.05e-09 5.08 1.52e-07 4.02 Table 3 : 3u h − u I 2 and the corresponding convergence rates, constant coefficients. Table 4 : 4Errors and corresponding convergence rates for C 1 Petrov-Galerkin method, variable coefficients, k = 3.eun e u ′ n e u ′ e u ′′ k N error order error order error order error order Case 1 4 4.24e-04 - 4.42e-04 - 1.45e-02 - 4.98e-01 - 8 2.75e-05 3.94 2.94e-05 3.91 1.38e-03 3.39 8.75e-02 2.51 3 16 1.75e-06 3.97 1.87e-06 3.98 1.03e-04 3.74 1.25e-02 2.81 32 1.10e-07 4.00 1.17e-07 3.99 6.85e-06 3.91 1.63e-03 2.94 64 6.86e-09 4.00 7.34e-09 4.00 4.36e-07 3.97 2.05e-04 2.99 Case 2 4 3.39e-04 - 3.42e-04 - 1.37e-02 - 4.88e-01 - 8 2.25e-05 3.92 2.69e-05 3.67 1.31e-03 3.39 8.56e-02 2.51 3 16 1.44e-06 3.96 1.88e-06 3.84 9.74e-05 3.74 1.22e-02 2.81 32 9.03e-08 4.00 1.23e-07 3.93 6.46e-06 3.91 1.58e-03 2.95 64 5.64e-09 4.00 7.87e-09 3.97 4.11e-07 3.98 2.00e-04 2.99 Case 3 4 3.36e-04 - 3.53e-04 - 1.37e-02 - 4.88e-01 - 8 2.26e-05 3.89 2.82e-05 3.65 1.30e-03 3.39 8.56e-02 2.51 3 16 1.44e-06 3.97 1.97e-06 3.84 9.72e-05 3.75 1.22e-02 2.81 32 9.05e-08 4.00 1.29e-07 3.93 6.45e-06 3.91 1.59e-03 2.95 64 5.66e-09 4.00 8.26e-09 3.97 4.09e-07 3.98 2.00e-04 2.99 Table 5 : 5Errors and corresponding convergence rates for C 1 Petrov-Galerkin method, variable coefficients, k = 4.eun e u ′ n eu e u ′ e u ′′ k N error order error order error order error order error order Case 1 4 2.56e-06 - 1.71e-06 - 4.10e-05 - 1.19e-03 - 6.21e-02 - 8 4.06e-08 5.98 3.46e-08 5.62 8.78e-07 5.55 4.83e-05 4.63 4.91e-03 3.66 4 16 6.25e-10 6.02 6.31e-10 5.78 1.35e-08 6.03 1.38e-06 5.13 2.90e-04 4.08 32 1.05e-11 5.89 1.03e-11 5.94 2.25e-10 5.90 4.78e-08 4.85 1.94e-05 3.90 64 1.63e-13 6.01 1.65e-13 5.96 4.02e-12 5.81 1.52e-09 4.98 1.24e-06 3.97 Case 2 4 1.25e-06 - 8.07e-07 - 3.91e-05 - 1.16e-03 - 6.07e-02 - 8 2.01e-08 5.96 2.10e-08 5.27 8.31e-07 5.56 4.62e-05 4.65 4.79e-03 3.66 4 16 3.10e-10 6.02 3.99e-10 5.71 1.26e-08 6.05 1.33e-06 5.12 2.84e-04 4.08 32 5.12e-12 5.92 6.70e-12 5.90 2.13e-10 5.89 4.59e-08 4.86 1.89e-05 3.90 64 8.02e-14 6.00 1.05e-13 6.00 3.89e-12 5.77 1.46e-09 4.98 1.21e-06 3.97 Case 3 4 9.26e-07 - 5.75e-07 - 3.93e-05 - 1.16e-03 - 6.07e-02 - 8 1.48e-08 5.96 1.54e-08 5.22 8.33e-07 5.56 4.62e-05 4.65 4.79e-03 3.66 4 16 2.29e-10 6.02 2.95e-10 5.71 1.26e-08 6.05 1.33e-06 5.11 2.84e-04 4.08 32 3.81e-12 5.91 4.94e-12 5.90 2.13e-10 5.89 4.58e-08 4.86 1.89e-05 3.90 64 5.74e-14 6.05 9.17e-14 5.75 3.91e-12 5.77 1.46e-09 4.98 1.21e-06 3.97 Table 6 : 6Errors and corresponding convergence rates for C 1 Gauss collocation method, variable coefficients, k = 3.eun e u ′ n e u ′ e u ′′ k N error order error order error order error order Case 1 4 2.12e-03 - 3.30e-03 - 1.29e-02 - 7.15e-02 - 8 1.43e-04 3.89 5.10e-04 2.69 1.33e-03 3.28 1.35e-02 2.41 3 16 8.87e-06 4.01 4.71e-05 3.44 1.01e-04 3.72 2.03e-03 2.73 32 5.49e-07 4.02 3.42e-06 3.78 6.67e-06 3.92 2.76e-04 2.88 64 3.42e-08 4.00 2.24e-07 3.93 4.22e-07 3.98 3.59e-05 2.94 Case 2 4 2.30e-03 - 2.97e-03 - 1.42e-02 - 9.18e-02 - 8 1.56e-04 3.89 5.01e-04 2.57 1.45e-03 3.30 1.71e-02 2.43 3 16 9.62e-06 4.02 4.72e-05 3.41 1.09e-04 3.72 2.55e-03 2.74 32 5.95e-07 4.01 3.45e-06 3.77 7.25e-06 3.92 3.46e-04 2.88 64 3.71e-08 4.00 2.26e-07 3.93 4.59e-07 3.98 4.49e-05 2.94 Case 3 4 2.37e-03 - 2.84e-03 - 1.45e-02 - 9.13e-02 - 8 1.59e-04 3.89 4.85e-04 2.55 1.47e-03 3.30 1.70e-02 2.43 3 16 9.82e-06 4.02 4.60e-05 3.40 1.11e-04 3.73 2.54e-03 2.74 32 6.08e-07 4.01 3.37e-06 3.77 7.33e-06 3.92 3.45e-04 2.88 64 3.79e-08 4.00 2.21e-07 3.93 4.64e-07 3.98 4.49e-05 2.94 Table 7 : 7Errors and corresponding convergence rates for C 1 Gauss collocation method, variable coefficients, k = 4.eun e u ′ n eu e u ′ e u ′′ k N error order error order error order error order error order Case 1 4 1.45e-05 - 1.16e-04 - 8.66e-05 - 1.00e-03 - 8.32e-03 - 8 4.69e-07 4.95 3.01e-06 5.27 1.53e-06 5.82 3.87e-05 4.69 7.68e-04 3.44 4 16 1.25e-08 5.23 4.84e-08 5.96 1.64e-08 6.55 1.14e-06 5.09 5.70e-05 3.75 32 2.23e-10 5.81 7.53e-10 6.01 4.01e-10 5.35 3.76e-08 4.92 3.73e-06 3.94 64 3.61e-12 5.95 1.18e-11 6.00 7.73e-12 5.70 1.18e-09 4.99 2.34e-07 4.00 Case 2 4 1.60e-05 - 1.15e-04 - 9.17e-05 - 1.09e-03 - 1.06e-02 - 8 4.82e-07 5.05 3.09e-06 5.22 1.63e-06 5.81 4.18e-05 4.70 9.85e-04 3.42 4 16 1.31e-08 5.20 5.06e-08 5.93 1.78e-08 6.52 1.25e-06 5.06 7.21e-05 3.77 32 2.35e-10 5.80 7.92e-10 6.00 4.08e-10 5.45 4.10e-08 4.93 4.68e-06 3.94 64 3.80e-12 5.95 1.24e-11 6.00 7.92e-12 5.69 1.28e-09 5.00 2.93e-07 4.00 Case 3 4 1.61e-05 - 1.15e-04 - 9.18e-05 - 1.09e-03 - 1.05e-02 - 8 4.84e-07 5.06 3.08e-06 5.22 1.63e-06 5.82 4.18e-05 4.70 9.86e-04 3.42 4 16 1.32e-08 5.20 5.07e-08 5.92 1.78e-08 6.52 1.25e-06 5.06 7.21e-05 3.77 32 2.37e-10 5.80 7.96e-10 5.99 4.06e-10 5.45 4.10e-08 4.93 4.68e-06 3.95 64 3.84e-12 5.95 1.25e-11 6.00 7.89e-12 5.69 1.28e-09 5.00 2.93e-07 4.00 Table 8 : 8u h − u I 2 and corresponding convergence rates, variable coefficients, k = 3.u h − uI 2 k N error order error order error order Case 1 Case 2 Case 3 4 2.35e-02 - 1.89e-02 - 1.89e-02 - 8 3.46e-03 2.77 2.82e-03 2.75 2.82e-03 2.75 C 1 Petrov-Galerkin 3 16 4.49e-04 2.94 3.67e-04 2.94 3.67e-04 2.94 32 5.66e-05 2.99 4.64e-05 2.99 4.64e-05 2.99 64 7.10e-06 3.00 5.81e-06 3.00 5.81e-06 3.00 4 1.86e-01 - 1.93e-01 - 1.93e-01 - 8 2.65e-02 2.81 2.75e-02 2.81 2.75e-02 2.81 C 1 Gauss collocation 3 16 3.37e-03 2.97 3.51e-03 2.97 3.51e-03 2.97 32 4.22e-04 3.00 4.39e-04 3.00 4.39e-04 3.00 64 5.27e-05 3.00 5.49e-05 3.00 5.49e-05 3.00 Table 9 : 9u h − u I 2 and corresponding convergence rates, variable coefficients, k = 4.u h − uI 2 k N error order error order error order Case 1 Case 2 Case 3 4 5.03e-03 - 4.48e-03 - 4.48e-03 - 8 3.53e-04 3.83 3.17e-04 3.82 3.17e-04 3.82 C 1 Petrov-Galerkin 4 16 2.24e-05 3.98 2.01e-05 3.98 2.01e-05 3.98 32 1.40e-06 4.00 1.26e-06 4.00 1.26e-06 4.00 64 8.75e-08 4.00 7.86e-08 4.00 7.86e-08 4.00 4 2.09e-02 - 2.18e-02 - 2.18e-02 - 8 1.32e-03 3.99 1.38e-03 3.99 1.38e-03 3.99 C 1 Gauss collocation 4 16 7.74e-05 4.09 8.12e-05 4.08 8.12e-05 4.08 32 4.86e-06 3.99 5.09e-06 4.00 5.09e-06 4.00 64 3.06e-07 3.99 3.20e-07 3.99 3.20e-07 3.99 Superconvergence of discontinuous Galerkin solutions for a nonlinear scalar hyperbolic problem. 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[ "Facial Expressions Tracking and Recognition: Database Protocols for Systems Validation and Evaluation", "Facial Expressions Tracking and Recognition: Database Protocols for Systems Validation and Evaluation" ]	[]	[]	[]	Each human face is unique. It has its own shape, topology, and distinguishing features. As such, developing and testing facial tracking systems are challenging tasks. The existing face recognition and tracking algorithms in Computer Vision mainly specify concrete situations according to particular goals and applications, requiring validation methodologies with data that fits their purposes. However, a database that covers all possible variations of external and factors does not exist, increasing researchers' work in acquiring their own data or compiling groups of databases.To address this shortcoming, we propose a methodology for facial data acquisition through definition of fundamental variables, such as subject characteristics, acquisition hardware, and performance parameters. Following this methodology, we also propose two protocols that allow the capturing of facial behaviors under uncontrolled and real-life situations. As validation, we executed both protocols which lead to creation of two sample databases: FdMiee (Facial database with Multi input, expressions, and environments) and FACIA (Facial Multimodal database driven by emotional induced acting).Using different types of hardware, FdMiee captures facial information under environmental and facial behaviors variations. FACIA is an extension of FdMiee introducing a pipeline to acquire additional facial behaviors and speech using an emotion-acting method. Therefore, this work eases the creation of adaptable database according to algorithm's requirements and applications, leading to simplified validation and testing processes.	null	[ "https://arxiv.org/pdf/1506.00925v1.pdf" ]	12,288,676	1506.00925	74408cfd748ad5553cba8ab64e5f83da14875ae8	Facial Expressions Tracking and Recognition: Database Protocols for Systems Validation and Evaluation 2 Jun 2015 Facial Expressions Tracking and Recognition: Database Protocols for Systems Validation and Evaluation 2 Jun 2015Preprint submitted to Computers & Graphics June 3, 2015Computer VisionHuman-Computer InteractionPerformanceDatabaseAlgorithms ValidationDatabase Protocols Each human face is unique. It has its own shape, topology, and distinguishing features. As such, developing and testing facial tracking systems are challenging tasks. The existing face recognition and tracking algorithms in Computer Vision mainly specify concrete situations according to particular goals and applications, requiring validation methodologies with data that fits their purposes. However, a database that covers all possible variations of external and factors does not exist, increasing researchers' work in acquiring their own data or compiling groups of databases.To address this shortcoming, we propose a methodology for facial data acquisition through definition of fundamental variables, such as subject characteristics, acquisition hardware, and performance parameters. Following this methodology, we also propose two protocols that allow the capturing of facial behaviors under uncontrolled and real-life situations. As validation, we executed both protocols which lead to creation of two sample databases: FdMiee (Facial database with Multi input, expressions, and environments) and FACIA (Facial Multimodal database driven by emotional induced acting).Using different types of hardware, FdMiee captures facial information under environmental and facial behaviors variations. FACIA is an extension of FdMiee introducing a pipeline to acquire additional facial behaviors and speech using an emotion-acting method. Therefore, this work eases the creation of adaptable database according to algorithm's requirements and applications, leading to simplified validation and testing processes. Introduction In the field of Computer Vision (CV), there are several existing databases that contain a wide range of facial expressions and behaviors, developed based on specific scenarios. The data contained in these databases is usually used for validation and performance tests, as well as training of facial models in CV algorithms [1,2,3]. To date, computational works include only a limited number of features though, representing typical facial extraction elements [4,5,6]. In fact, there is no single database that integrates a full set of situations: some are dedicated only to expressions, others to lighting conditions, some are just for extracting facial patterns used to define training models, others for emotion classification, etc. This means the information is split across a variety of databases, making it impossible to validate a facial tracking system under numerous specific situations (e.g. partial face occlusions from hardware or glasses, changes in background, variations in illumination, head pose variations, etc...) or train emotion classifier systems capable of capturing the subtleties of the face using only one database. This drawback usually leads to systems over-fitting to data, presenting a high specificity to a certain environment or limiting facial features recognized [7]. Therefore, every time it is required to design validation and performance tests or training sets, researchers struggle to find databases that fit all system's requirements [7]. As example, to deploy the recent face tracking system [8] it was needed the compilation of three different databases. In alternative, researchers define and setup their own procedures to acquire own databases, collecting subjects, defining protocols, and preparing capture equipment -which are all time-consuming processes. This "database customization" requirement exists since databases require specific features or formats (e.g. high-resolution videos and infra-red pictures) according to CV system's profile and goal. These features and formats contain a wide range of variations in external and facial behavior parameters to simulate real-life situations and provide information that would reproduce the scenario accurately where the system is going to be applied [7]. In this work, we designed two generic protocols and developed a methodology for data acquisition for face recognition systems, as well as for tracking and training of CV algorithms. Our methodology defines each acquisition protocol to be composed of three basic variables: i) subject characteristics, ii) acquisition hardware and iii) performance parameters. These variables are classified as flexible (i.e. can be altered according to system requirements, not influencing protocol guidelines) and fixed (i.e. defined and constrained b the protocol guidelines). The flexible variables are connected to system requirements, and the fixed ones to the information recorded and simulated. As performance variables, we define the following parameters: external (e.g. environment changes in lightning and background) and facial (e.g. variations in facial expressions and their intensity). To test the accuracy and performance of algorithms in facial features tracking or to train face models, used databases need to contain a broad set of external and facial behavior variations. Setting up these variables through our proposed methodology and adopting our protocols eases the process of acquisition of databases with facial information under real-life scenarios and realistic facial behaviors. To validate this process, we followed both protocols and acquired two sample databases. We also analysed the obtained results to establish proof-of-concept. We dubbed the first protocol Protocol I that generated Fd-Miee "Facial database with multi input, expressions and environments". Protocol I aims to guide researchers through acquiring data using three capture hardware while varying the performance variable, giving special focus to external parameters variation. As Protocol I's extension, Protocol II introduces variations in performance variable regarding facial behaviors. Validation of this protocol generated FACIA "Facial Multimodal database driven by emotional induced acting". Figure 1 represents our overall contribution schematically, regarding the types of data captured in the protocols. It represents the Facial databases' universe through Environment situations and conditions, where we include the group of available facial behaviors, with a small part reserved to introduce behaviors ( Figure 1 -A). Taking this scheme into account, we can mirror the domain of our database protocol and represent the contributions of FdMiee and FACIA diagrammatically ( Figure 1 -B). Background To develop guidelines for database acquisition, we researched the literature for methodologies and variance parameters required to test and evaluate CV systems. We analyzed state-ofthe-art databases, and classified them into two groups, according to their output format: video and image-based. The most commonly-used video databases are as follows: • BU-4DFE (3D capture + temporal information): A 3D Dynamic Facial Expression Database [1]; • BP4D-Spontaneous: a high-resolution spontaneous 3D dynamic facial expression database [9]; • MMI Facial Expression Database [2]; • VidTIMIT Audio-Video Database [10]; • Face Video Database of the Max Plank Institute [11]. Comprehensive and well-documented video databases exist, for example [12] and [13]. However, to access them, a very strict license must be procured and a payment provided. BU-4DFE [1] presents a high-resolution 3D dynamic facial expression database. Facial expressions are captured at 25 framesper-second while performing six basic Ekman's emotions. Each expression sequence contains about 100 frames spread through 101 subjects. More recently, this database was extended to create a 3D spontaneous facial expressions [9]. Another facial expressions database commonly used is the MMI database [2]. It is an ongoing project that holds over 2000 videos and more than 500 images from 50 subjects. Also information of displayed AU's is given with the samples. The VidTIMIT Audio-Video [10] contains video and audio recordings from 43 people reciting 10 short sentences per person. Each person also performs a head rotation sequence per session, which in facial recognition can allow pose independence. Finally, Face Video Database from the Max Planck Institute provides videos of facial action units, used for Face and Object Recognition, though no more information is given [11]. Usage of videos instead of images on the model training allows a better detection of spontaneous and subtle facial movements. However, available databases are limited to standard facial expressions detection [1,2] or do not explore situations with different lighting levels. Regarding image-based databases, we came across a comparison study in the table VIII of [3]. This table describes the commonly-used image-based databases for validation of face tracking systems. It also exposes their limitations. As examples of current image-based databases, we analyzed the following databases: • Yale [14]; • Yale B [15]; • the FERET [16]; • CMU Pose, Illumination and Expression (PIE) [17]; • Oulu Physics [18]. Regarding Yale [14] and Yale B [15] database, it contains a limited number of grayscale images with well-documented variations on lighting, facial expressions, and pose variations. In contrast, the FERET database [16] has a high number of subjects with a complete pose variation. However, no information about lighting is given. Another interesting database is the CMU PIE [17] which also tests extreme lighting variations for 68 subjects. These three databases are frequently used for facial recognition, not only for model training but also for validation. Finally, we also highlight the Oulu Physics [18] database, since it presents a variation on lighting color (horizon, incandescent, fluorescent, and daylight) on 125 faces. Based on this research, we concluded that there is a wide range of databases that explore and simulate diverse facial expressions under different environment conditions. However, the available information is spread throughout many databases. In other words, a single database that combines all these facial and environment behaviors and variations providing a complete tool for validation of facial expressions tracking and classification is still non-existent. In [19], a complete state-of-the-art on emotional databases available nowadays can be found. We searched for a facial expressions database that would simultaneously provide color and depth video (3D data stream) as well as speech information, along with emotional data. Our search criteria, however, were not fulfilled. The increase of affect recognition CV methods [20] lead to a necessity of databases generation containing spontaneous expressions. To establish how to induce these expressions in participants, we analyzed the review paper on Mood Induction Procedures (MIP's) [21] and investigated which resource materials could be used to enhance and introduce realism in expressed emotions [22]. We concluded that the most commonlyused emotion induction procedure is the Velten method, characterized by a self-referent statement technique. However, the most powerful techniques are combinations of different MIPs, such as Imagination, Movies/Films instructions or Music [21] . Therefore, the technique chosen for our experiment was a combination of the Velten technique with imagination, where we proposed an emotional sentence enacting, similar to the one presented by Martin et al. [22]. Some available databases that use similar MIP's induce emotions in the users by asking them to imagine themselves in certain and pre-defined situations [23,24]. However, the usage of this procedure without complementary material (e.g. sentences) does not guarantee facial expressivity from the user [23,21]. Since we intended to record speech, we analysed state-of-theart multimodal databases [19] and found that there was none containing Portuguese speech. Therefore, we decided to explore this potential research avenue. Protocol Methodology Analysing the background and details of facial data acquisition setups, we propose that to create a protocol, three fundamental variables need to be characterized: subject characteristics, acquisition hardware and performance parameters (Table 1). These variables are classified as being either flexible or fixed, according to their impact on the protocol guidelines. Subject characteristics and acquisition hardware are flexible variables, as they can be changed according to system requirements. For example, use male subjects captured with a high-speed camera or other kind of hardware available, since they do not influence the guidelines of acquisition itself, but only interfere with the acquisition setup. In contrast, fixed variables such as performance parameters, influence guidelines definitions, i.e. different performance parameters require us to take different steps for their simulation and acquisition. Subject characteristics include gender, age, race, and other features that can be extrapolated from the subjects' samples. [25,26,27]. These variables are almost infinite [28] due to their uncontrolled nature in real-life environments. Facial behaviours should contain facial expressions data triggered by emotions, such as macro, micro, subtle, false, and masked expressions [29,30,31] or even speech information. Ekman et al. [29] defines six universal emotions: anger, fear, sadness, disgust, surprise and happiness. These universal emotions are expressed in different ways according to a person's mood and intentions. The way they are expressed leads us to an expressions-classification: • Macro: These expressions last between half a second and 4 seconds. They often repeat and fit what is being said as well as the speech. Facial expressions of high intensity are usually connected to six universal emotions [29,30]; • Micro: Brief facial expressions (e.g. milliseconds) related to emotion suppression or repression [29,30]; • False: Mirrors an emotion that is deliberately performed, ans is not being felt [29,30]; • Masked: False expression created to mask a felt macroexpression [29,30]; • Subtle: Expressions of low intensity that occur when a person starts to feel an emotion or shows an emotional response to a certain situation, another person, or surrounding environment. This is usually of low intensity [31]. Facial behaviors generated by speech usually contain a combination of the above expressions [32]. Following this methodology, we developed two protocols. We dubbed the first protocol to generate FdMiee Protocol I. To validate this protocol, we acquired data from eight subjects with different characteristics. We applied low-resolution, highresolution, and Infra-red cameras as acquisition hardware variables. As performance parameter variables, we simulated multiinput expressions and environments to test the invariance and accuracy of facial tracking systems exposed to changes, e.g. different lighting conditions, universal-based and speech facial expressions. To validate the results, we executed 360 acquisitions and demonstrated the protocol's potentials to acquire data containing uncontrolled scenarios and facial behaviors. We dubbed the second protocol to create FACIA database Protocol II. This is an extension of Protocol I's performance parameters variables, introducing induced facial behaviors. To validate the results, we studied the protocol's effectiveness for acquiring multimodal databases of induced facial expressions with speech, color, and depth video (3D data stream) data. To achieve this validation goal, we presented a novel induction method using emotional acting to generate facial behaviors inherent to expressions. We also provided emotional speech in the Portuguese language, since currently there is not any 3D facial database that uses this language. Similar to FdMiee, in FACIA we created proof-of-concept through an experiment with eighteen participants, in a total of 504 acquisitions. As a typical protocols' usage example, a research team has available database of 10 female subjects aged between 20-22. They would like to compile a database to test the head rotation tracking accuracy of a CV algorithm using a HD camera. Therefore, they define as subject characteristics the female gender and age range. Then, they choose a HD camera as acquisition hardware and afterward need to pick the Facial parameter: head rotation as Performance parameter. Finally, they need to follow our validated FdMiee protocol. In summary, to follow the protocols, we first choose the parameters to simulate as fixed Performance variables. This allow us to define the acquisition guidelines. Secondly, we determine the hardware variable and generate an acquisition setup. It is important to note that this variable is flexible, and thus changing this variable will not impact the guidelines. The same is verified using different subject characteristics. Protocols and Validation In this section, we describe in detail the two protocols that follow our proposed methodology. Protocol I resulted in the FdMiee sample database that contains facial data from uncontrolled scenarios. FdMiee focuses essentially on performance variable guidelines of external parameters. The obtained data was recorded with three types of acquisition hardware. As its extension, Protocol II focuses on testing and simulation of facial parameters of the performance variables, using Microsoft Kinect as hardware. Protocol I Facial recognition and tracking systems are highly dependent on external conditions (i.e. environment changes) [7]. To reduce this dependency, we developed a protocol based on our proposed methodology, for database creation with changes in terms of external parameters, such as light, background, occlusions, and multi-subject. For facial parameters, we setup guidelines to capture variations in head rotation, as well as universalbased, contempt and speech facial expressions. Table 1 summarizes the performance parameters acquired through this protocol. Requirements As protocol requirements, we setup the acquisition hardware and equipment to simulate the selected external and facial parameters. Acquisition Hardware The chosen acquisition hardware simulates realistic scenarios captured using three types of hardware. To test the protocol guidelines, we chose the following equipment: • Low-Resolution (LR) camera • High-Resolution (HR) camera • Infra-Red (IR) camera The first two cameras (LR and HR) allow us to study the influence of resolution on face tracking, face recognition, and expression recognition [33]. The IR camera allows us to disregard lighting variations [34,35,36] and provides a different kind of information than HR and LR cameras. The hardware used in this protocol should be aligned with one another to ensure future comparison between data acquired with different hardware. Environment-Change Generation Equipment To generate data with the defined parameters, we stabilize the following environment elements: Background A solid color and static background ease the process of detecting facial features and extracting information from the surrounding environment. The background should ideally be black (or very dark) to prevent interference with the IR camera (black color has lowere reflectance compared to lighter colors) Lighting The room must be lit up by homogeneous light, and not produce shadows or glitters in the subject's face. By taking these measure, we ensure that the skin color will have no variations throughout the acquisition process. Acquisition Setup The subject sits in front of the acquisition hardware. This hardware setup is composed of three cameras (LR, HR and IR). The subject's backdrop should be black with some space between them, to have the possibility of moving objects or subjects behind the main scene. This setup is exemplified in Figure 2 Protocol Guidelines To perform the acquisition, we suggested the presence of two members: one to perform the acquisitions (A) and the other to perform environment variations (B). The subject sits in front of the computer monitor and one of the team members aligns them with the cameras. During the entire acquisition procedure, the subject should remain as still as possible, to avoid producing changes during the various acquisition procedures. Before starting the experiment, each subject has access to a printed copy of the protocol. This reduces the acquisition time, since the subject already knows what is going to take place during the experiment. Each performance parameter simulated and introduced in the scenario has its own guidelines: Control Team member A takes a photo with the subject in the neutral face. Lighting Team member A takes 3 photos with different exposures (High, Medium, Low). This variable was only acquired in HR camera, because it is the only where it is possible to change the exposure level. Background Team prepare the background to the acquisition. Multi-Subject While subject is being record, team member B appear in the scene during 10 seconds. Occlusions For total occlusion, subject will start in the center of the scene and will slowly move to a point out of the scene. For partial occlusions, a photograph is taken with a plain color surface, like a piece of paper covering the following parts of the face: • Top; • Left; • Bottom; • Right. Head Rotation For each head pose (Yaw, Pitch and Roll) subject performs the movement in both directions while being recorded through the complete movement. Universal-Based Facial Expressions, plus Contempt Subject repeat during 10 seconds the following emotion expressions, starting from the neutral pose to a full pose: • Joy; • Anger; • Surprise; • Fear; • Disgust; • Sadness; • Contempt. Speech Facial Expressions The subject reads a cartoon or text and is encouraged to express his feelings about it. Obtained Outputs This protocol generates the following output data: • HR and LR Photographies (.jpeg) • LR camera videos -15fps (.wmv) • HR camera videos -25fps (.mov) • IR camera videos -100fps (.avi) The emotions generated through variation of facial parameters are expected to contain a mixture of macro and micro (i.e. subjects can be repressing and suppressing feelings) as well as false (i.e. subject is making an effort to express certain emotions) and subtle (i.e. when subject cannot generate a high intensity expression) plus speech-based expressions. Data Organization and Nomenclature For standardization purposes and further analysis, a folder for each acquisition hardware was created. Inside these folders exist sub-folders for each of the tested performance parameters. The output files were placed in the respective folder with the following template naming convention: CaptureModeVolunteer0X SimulationName take0Y.format , where CaptureMode is the type of hardware, X is the number of the subject, SimulationName is the name of the performance parameter acquired and respective information and Y is the take's identification number. FDMiee Acquisition & Protocol Validation Following the described protocol guidelines, we acquired data from eight volunteers with the following subject characteristics: Figure 3 it is possible to see sample results from some of the performance parameters with the different acquisition hardware. Protocol II The definition and extraction of induced facial behaviors and speech features inherent to spontaneous expressions is still a challenge for CV systems. To develop and subsequently evaluate a CV algorithm that achieves this goal, we proposed these two protocols to acquire a database containing, simultaneously, spontaneous facial expressions and speech information inherent to induced emotions, such as Ekman's universal emotions [29,30]. Therefore, in this experiment we focus on the definition of guidelines to capture facial parameters changes in the performance variable 1. Requirements We define two types of requirements: emotion induction method and equipment requirements. Emotion induction method is used as basis to define the protocol guidelines inherent to facial parameters simulation. The Emotion Induction Method The majority of spontaneous facial expressions are generated in real-life situations. To simulate these facial behaviors, we proposed a protocol where the system would ask for emotional acting in order to trigger facial responses from a subject. For this purpose, we combined a Mood Induction Techniques 1 (MIT 1) described by Hesse A.G. et al. [21] with mood induction sentences suggested by Pitas I. et al. [22]. As an application example, we could have a system that asks for certain user emotions expression through facial or speech features. The user must pronounce certain sentences with a particular tone and facial expression, matching the required emotional state. According to expression classification introduced in Section I, using this method we are able to induce macro, micro, false, masked, and subtle expressions. Macro expressions are implicit, since we ask for expression of the six of the Ekman universal emotions (i.e. anger, fear, sadness, disgust, surprise and happiness). However, since we are in an induced emotions context, subjects can have difficulty engaging in the proposed situation and generating micro, false and masked expressions. Also subtle expressions are triggered because subjects' engaging intensity can be low in the induced sentence or context. As expected, the produced facial expressions depend of subjects' interpretation and how they emerge themselves in the simulated situation. Our induction approach presents a novel view on emotion acting and their applications, though the domain still remains unexplored in state-of-the-art databases. We used common persons as subjects, instead of actors, to maintain the natural-ness of real-life scenarios and also achieve a larger diversity of facial behaviors. Actors gain, over time, professional skills that common population cannot reproduce, thus they might introduce features that cannot match the realworld human performance. Some available databases that use MIT 1, try to induce emotions in the users, asking them to imagine themselves in certain general and predefined situations [22,23,24]. We also avoided this approach, since suggesting certain situations will not guarantee certain emotion expressions as output by the subject. This is due to the fact that different individuals have different reactions as responses. Therefore, in our protocol, we asked the subjects to imagine and create for themselves some personal mental situation, while they enact the pre-defined sentence. This aims to ensure an engaging adaptation and natural response from the subject. As mentioned before, the chosen emotions were the six basic Ekman emotions [29], due to their scientific acceptance and applicability in realworld situations. The sentences are pronounced in the European Portuguese language to match the users mother tongue. This is another contribution of our work, since currently there is not a Multimodal European Portuguese database available. Equipment and Environment Requirements The acquisition setup uses the Microsoft Kinect as acquisition hardware variable. Kinect records 3D data stream as well as speech information. The illumination is not controlled however, as acquisitions were executed during different day periods under uncontrolled lighting conditions. The background is static and white, and there is no sound isolation, since speech signal can be affected by external noise. Sentences are displayed on a screen positioned in front of the subject. To allow further synchronization or re-synchronization, a sound and light emitter is used in the beginning of each recording (see example of Figure 4). For this experiment in our protocol validation, we developed a software that allows simultaneous recording of color and depth video with speech from Microsoft Kinect, in .bin, video, and audio formats. This software includes the Facetracker's Microsoft SDK, and also saves the information retrieved from this algorithm. Acquisition Setup The subject sits in front of the capture hardware. Distance between subject and Microsoft Kinect should be more than 1 meter to enable facial depth capture. A screen displays the sentence that is currently going to be "acted". The subject did not watch the recordings neither observe their own acting, to avoid auto-evaluation or influence their acting performance and expressivity. In FACIA protocol we propose the acquisition setup of Figure 5. Protocol Guidelines Each subject sits in front of the screen and acts out the two sentences per emotion 2 while their voice and face expression are recorded. Per sentence we execute the procedure two times. This ensures the integrity of final results. We suggest a minimum of two members (A and B) in the acquisition team. Before starting the experiment, a protocol describing the experiment is given to the subject. The experiment starts by a neutral sentence [37] (that can be used as baseline for further experiments). Therefore, to each sentence of Table 2 the following pipeline is repeated two times: 1. Acquisition team member A says 1,2,3 I will record!. Obtained Outputs Using our acquisition protocol, we obtained the following data per sentence enacted: • Video Color Resolution -30fps (.bin); • Depth image Resolution -30fps (.bin); • Audio pcm format 16000 Hz (.bin). • Facetracker SDK and Action Units detected (.bin). • Audio file (.wave). • Color Video file (.avi). As explained, regarding facial behaviors we are able to generate data containing macro, micro, false, masked and subtle expressions. Data organization and Nomenclature Similarly to procedure adopted in FdMiee protocol, we predefine how data acquire is going to be organized. To each subject is created a folder called Volunteer0X, where X is the number associated to the subject. Inside each subject folder are created eight additional folders: one per emotional sentence. Inside of each emotion folder we will have two folders numbered with corresponding sentence, where we will place three data types obtained. Regarding file names, we will use the following template: Volunteer0XEmotionSentence0YTake0Z.format Where X is the subject number, Y the sentence number and Z the take number. FACIA acquisition & Protocol validation To validate Protocol II, we follow it for eighteen subjects, in a total of 130 files per subject (total of 504 acquisitions). As subject characteristics variable we have seven female and eleven male; ages are in a range of 20-35 years old and they were all caucasian. As already explain, we require depth information so a Microsoft Kinect was used as acquisition hardware. As sample of results acquired during validation we can observed the Figure 6. Discussion and Conclusions In this paper, we presented a methodology to facilitate the development of two facial data acquisition protocols. Following this methodology, we presented the protocols for simulation and capturing of real-life scenarios and facial behaviors. To validate the protocols, two sample databases were created: FdMiee and FACIA. They contain comprehensive information on facial variations inherent to both spontaneous and non-spontaneous facial expressions under a wide range of realistic and uncontrolled situations. Generated databases can be used in a variety of applications, such as CV systems evaluation, testing, and training [7]. They also serve as proof-of-concept. Adopting our methodology and following our protocols reduces the time required for customized database acquisition. Throughout the protocol creation process, we characterized two groups of variables: flexible variables (subjects' characteristics and capture hardware) and fixed performance variables (external and facial parameters). The first protocol focuses on external parameters' simulation as variation of the fixed performance variable. As an extension, the second protocol provides guidelines to induce and capture real-life facial behaviors as fixed performance variables. Protocol I allows the acquisition of a facial database containing a large number of fixed parameters' variations (external and facial): lightning, background, multi-subject, occlusions, head rotation, universal-based, and speech facial expressions (Table 1). Lighting variations introduce changes in facial features (e.g. contrast and brightness) [15]. These variations enable us to test how CV systems react to and detect, and how tracking is affected. Static and dynamic variations in the background usually interfere with CV systems' performance while detecting and tracking faces [38]. Therefore, in this protocol, we simulate different background contexts, as well as introduce static and dynamic features in the environment. Similar to background variables, we simulate multi-subject environments, since this situation usually interferes with, and at times, disables CV systems' feature detection [7]. Occlusions generated by glasses or hardware are also common in real-life scenarios, influencing face recognition and emotion classification accuracy [25,26,27]. The increase of Head-Mounted-Displays usage in Virtual Reality applications makes it crucial to test systems invariance while using these variables. Regarding facial behaviors, we reproduced and captured two kinds of facial behaviors -universal-based and speech-based facial expressions. Universal-based Facial Expressions are related to pure emotions [29]. They provide data for emotion recognition systems and enable the testing of systems invariance while subjects' faces change expressions. Speech Facial Expressions, on the other hand, are inherent to all types of expressions [32] (as showed in the image 1) and enable the measuring of systems accuracy and precision. To validate Protocol I, we performed an acquisition on eight subjects with different subject characteristics, leading to the creation of FdMiee database. FdMiee contains facial behaviors under different environment contexts. Hence, this protocol enables the generation of databases that are useful for a wide range of CV systems performance tests. Protocol II extends the first protocol regarding facial behaviors and performance variables, by introducing induced facial features. To achieve this, we proposed an emotion induction method, where facial expressions were induced through emotional acting. Analysing FACIA generated in the validation process, we verified that facial behaviors inherent to certain emotional acting are indeed different among individuals; i.e. subjects performed different acts to realize identical emotional states. Analysing subjects' facial behaviors, we were able to simulate all types of expressions according to subjects interpretations and engaging in induction sentences. Hence, this protocol provides a large and heterogeneous set of facial behaviors, useful for determining the accuracy of tracking and recognition systems. This was intuitively expected, since expressions inherent to emotional states share some action units [29]. This mixing of expressions can compromise database usage to train a machine learning classifier in pure expressions recognition, increasing classification error. Microsoft Kinect was chosen as the acquisition hardware variable, so that we could record three kinds of data: color, depth (3D facial information) and speech. Introducing depth in the stored data provides valuable information [9]. However, recent studies point out that acquisition rate of Kinect is not sufficient for micro and subtle expressions capturing [29,30]. This argument explains the poor component of micro and subtle expressions present in FACIA. However, in our methodology we classify this variable as flexible, to ensure that protocol guidelines can be used with other acquisition hardware, i.e. guidelines can be applied with high frame rate cameras and improve the capture of these facial behaviors. The speech recording also allows the Portuguese emotional data collection, opening novel research lines in emotion classification and recognition present in the European Portuguese language speech. In conclusion, our proposed methodology facilitate the generation of facial data acquisition protocols. This methodology provides a tool for researchers to develop their own facial databases. It also enable performance tests, validation and training processes in CV systems in a wide range of life-like scenarios and facial behaviors, being adaptable to different subject characteristics and acquisition hardware. Future Work Our further work will focus on the following key tasks: First, we aim to enlarge our proof-of-concept sample databases and, subsequently, perform a statistical validation of the two protocols presented in this paper. Enlarging the databases will provide sufficient data for statistical validation, using various CV systems. The statistical validation will also provide more measurable information regarding data significance and impact. Second, we aim to devise more parameters for methodology variables to refine the validation process. Third, we aim to introduce a more heterogeneous subject samples, with a wider age range (thus greater presence of wrinkles and facial pigments), skin colors, and make-up. Fourth, we intend to carry out tests with more sophisticated acquisition hardware, such as high-speed cameras. And finally, to increase our work applicability, we intend to extend the fixed variable of performance parameters, providing more guidelines to generate novel situations. Figure 1 : 1Summary of database protocols' contributions (B) in facial database universe (A). Figure 2 : 2Acquisition setup proposed for Protocol I. Figure 3 : 3FDMiee samples results for HD Camera (A), Webcam (B) and IR Camera (C) Figure 4 : 4Example of our video-audio synchronizer (left). Figure 5 : 5Acquisition setup proposed by FACIA protocol Figure 6 : 6Sample of results obtained for fear (A) and (B) disgust emotion acting Table 1 : 1Protocol flexible and fixed variables.Protocol Variables Flexible Fixed Subjects Acquisition Performance Characteristics Hardware Parameters Gender Webcam External Parameters: Age HD Camera Background Race Infra-Red Camera Lightning (...) Microsoft Kinect Multi-Subject High-Speed Camera Occlusions (...) Facial Parameters: Head Rotation Expressions: Macro Micro False Masked Subtle Speech This variable introduces specific facial behaviors (e.g. cultural variations in emotion expressions) in the database. Regarding, acquisition hardware, we enabled the usage of any type of in- put hardware according to acquisition specifications. Differ- ent combinations of these flexible variables can be applied to any of the fixed performance parameters guidelines. Perfor- mance variables describe the procedures for acquiring the data required for performance tests of CV algorithms. They are split into External and Facial categories, according to what we want to test. External parameters are related to changes in the envi- ronment, such as background, lightning, number of persons in a scene (i.e. multi-subject), and occlusions Gender Male/Female; Glasses With/Without; Beard With different formats/Without.Age 20-35 years In Table 2 : 2FACIA emotion induction method: Sentences pronounced and acted by the subjects.Emotion Sentences Neutral A jarra está cheia com sumo de laranja Anger O quê? Não, não, não! Ouve, eu preciso deste dinheiro! Tués pago para trabalhar, não para beberes café. Disgust Ah, uma barata! Ew, que nojo! Fear Oh meu deus, est alguém em minha casa! Não tenho nada para si, por favor, não me magoe! Joy Que bom, estou rico! Ganhei! Que bom, estou tão feliz! Sadness A minha vida nunca mais será a mesma. Ele(a) era a minha vida. Surprise E tu nunca me tinhas contado isso?! Eu não estava nadaà espera!. 2. Acquisition team member B uses the light/sound synchronizer. 3. Subject performs the emotion acting. 4. 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[ "Absolutely Minimizing Lipschitz Extensions and Infinity Harmonic Functions on the Sierpinski gasket", "Absolutely Minimizing Lipschitz Extensions and Infinity Harmonic Functions on the Sierpinski gasket" ]	[ "Fabio Camilli ", "Raffaela Capitanelli ", "Maria Agostina Vivaldi " ]	[]	[]	Aim of this note is to study the infinity Laplace operator and the corresponding Absolutely Minimizing Lipschitz Extension problem on the Sierpinski gasket in the spirit of the classical construction of Kigami for the Laplacian. We introduce a notion of infinity harmonic functions on pre-fractal sets and we show that these functions solve a Lipschitz extension problem in the discrete setting. Then we prove that the limit of the infinity harmonic functions on the pre-fractal sets solves the Absolutely Minimizing Lipschitz Extension problem on the Sierpinski gasket.MSC2010: 31C20, 28A80.	10.1016/j.na.2017.07.005	[ "https://arxiv.org/pdf/1608.03715v2.pdf" ]	119,175,176	1608.03715	adb4d1f0467f42fd5b80f6b3c170071e79a51ab7	Absolutely Minimizing Lipschitz Extensions and Infinity Harmonic Functions on the Sierpinski gasket June 20, 2018 Fabio Camilli Raffaela Capitanelli Maria Agostina Vivaldi Absolutely Minimizing Lipschitz Extensions and Infinity Harmonic Functions on the Sierpinski gasket June 20, 2018Sierpinski gasketMcShane-Whitney extensionsAbsolute Minimizing Lipschitz Exten- sioninfinity Laplacian Aim of this note is to study the infinity Laplace operator and the corresponding Absolutely Minimizing Lipschitz Extension problem on the Sierpinski gasket in the spirit of the classical construction of Kigami for the Laplacian. We introduce a notion of infinity harmonic functions on pre-fractal sets and we show that these functions solve a Lipschitz extension problem in the discrete setting. Then we prove that the limit of the infinity harmonic functions on the pre-fractal sets solves the Absolutely Minimizing Lipschitz Extension problem on the Sierpinski gasket.MSC2010: 31C20, 28A80. Introduction The theory of Absolutely Minimizing Lipschitz Extension ( [1,4]) concerns the classical problem of extending a Lipschitz continuous function f defined on the boundary of an open set U ⊂ R d to the interior of U without increasing the Lipschitz constant. In other words, to find a Lipschitz continuous function u : U → R such that u = f on ∂U and Lip(u, U ) = Lip(f, ∂U ) (Lip denotes the Lipschitz constant). The previous problem has several solutions, all in between a maximal and a minimal one called respectively the McShane's extension and the Whitney's extension. But, among all these possible solutions, the "canonical" one is the so called Absolutely Minimizing Lipschitz Extension (AMLE in short). This function is characterized by satisfying the extension problem not only in U , but also in any open subset V of U , that is Lip(u, V ) = Lip(u, ∂V ) for any open V ⊂ U . The relevance of the notion of AMLE relies in the several additional properties that this function satisfies, for example it is the unique viscosity solution of the Dirichlet problem for the infinity Laplace equation ∆ ∞ u(x) = 0, x ∈ U. (1.1) Here ∆ ∞ u = d i,j=1 ∂ x i u∂ x j u∂ 2 x i x j u, the infinity Laplacian, is a nonlinear degenerate second order operator and a function satisfying (1.1) is said infinity harmonic. The theory of AMLE has been extended to lenght spaces (see [1,3,9,10,15]), hence it applies in particular to the Sierpinski gasket S endowed with its geodesic distance. Prescribed a boundary data f on the vertices of the initial simplex from which S is obtained via iteration, there exists a unique AMLE of f to S. Moreover this function can be characterized as in the Euclidean case by an intuitive geometric property, called Comparison with Cones. After the seminal work of Kigami [11], a standard way to define a harmonic function on the Sierpinski gasket, and more in general for the class of post-critically finite fractals, is to consider the uniform limit of solutions of suitable scaled finite difference differences on the pre-fractals. For the infinity Laplace operator this approach has been pursued in [5]. In this thesis, a graph infinity Laplacian on pre-fractal sets is defined and an algorithm to compute explicitly infinity harmonic functions is studied. By means of a constructive approach based on the previous algorithm, it is proved that the sequence of the infinity harmonic functions on the pre-fractal sets converges to a function defined on the limit fractal set. It is worth noting that the same graph infinity Laplacian is used in the Euclidean case to approximate the viscosity solution of (1.1) (see [12,13,14]). Following an approach similar to [5], we aim to define an infinity harmonic function on S as the limit of solutions of finite difference equations on pre-fractals. We study the Lipschitz extension problem on the pre-fractal sets and we show that an appropriate notion of AMLE can be introduced in this framework. We also prove that an AMLE is a solution of the graph infinity Laplacian and it satisfies a Comparison with Cone property with respect to the path distance. The Comparison with Cone property on the pre-fractals is crucial since it allows us to show that the limit of the AMLEs on the pre-fractals is an AMLE on the Sierpinski. The convergence result allows us to define an infinity harmonic function on the Sierpinski as the limit of infinity harmonic functions on the pre-fractals and to conclude the equivalence, as in Euclidean case, between AMLE and infinity harmonic functions. Hence, besides giving a simpler proof of the convergence result in [5], we also obtain as in the Euclidean case the equivalence among the various properties which characterize the AMLE theory. We remark that the previous construction on the Sierpinski gasket can be readily extended to the class of postcritically finite fractals since it is only based on the convergence of the path distance defined on the pre-fractal sets to the path distance (intrinsic length) on the limit fractal set ([2], [7]). The paper is organized as follows. In Section 2 we introduce notations and definitions and we prove some preliminary properties for the graph infinity Laplacian. Section 3 is devoted to the AMLE problem on pre-fractals. In Section 4 we recall the definition of AMLE in metric spaces and we prove the convergence result for the pre-fractals invading the Sierpinski gasket. Finally, in Section 5 we describe the algorithm in [5] and we study the relation between infinity and p-harmonic function in the pre-fractals. Notations and preliminary results In this section we introduce notations and definitions and we collect some preliminary results on infinity harmonic functions on graphs. Consider an unitary equilateral triangle V 0 of vertices {q 1 , q 2 , q 3 } in R 2 and the maps ψ i : R 2 → R 2 , i = 1, 2, 3, defined by ψ i (x) := q i + 1 2 (x − q i ). Iterating the ψ i 's, we get the set V ∞ = ∪ ∞ n=0 V n where each V n is given by the union of the images of V 0 under the action of the maps ψ w = ψ w 1 • · · · • ψ wn with w = (w 1 , . . . , w n ), w i ∈ {1, 2, 3}, a word of length \|w\| = n. Then the Sierpinski gasket S is the closure of V ∞ (with respect to the Euclidean topology) and it is the unique non empty compact set F which satisfies F = ∪ 3 i=1 ψ i (F ). For any n, we can identify V n with the graph (V n , ∼ n ), where ∼ n is the following relation on V n : for x, y ∈ V n , x ∼ n y if and only if the segment connecting x and y is the image of a side of the starting simplex under the action of some ψ w with \|w\| = n. If x, y ∈ V n and x ∼ n y, then we will say that x, y are adjacent in V n . Given a set K ⊂ V n \ V 0 , we define the boundary and the closure of K by ∂K = {y ∈ V n \ K : ∃x ∈ K s.t. y ∼ n x}, K = K ∪ ∂K. The distance between two adjacent vertices x, y ∈ V n is d n (x, y) = 1 2 n =: δ n , while the distance between x, y ∈ V n is the vertex distance d n (x, y) := min{d n (x 0 , x 1 ) + d n (x 1 , x 2 ) + · · · + d n (x N −1 , x N )} (2.1) where the minimum is over all the finite path {x 0 = x, x 1 , . . . , x N = y} with x i ∼ n x i+1 , i = 0, . . . , N − 1, connecting x to y. We also consider the distance d n,K between x, y ∈ K defined as in (2.1) where the minimum is taken over all the finite paths restricted to stay inside K, i.e. {x = x 0 , . . . , x N = y} with x i ∼ n x i+1 for i = 0, . . . , N − 1 and x i ∈ K for i = 1, . . . , N − 1 (if there is no path in K connecting x to y we set d n,K (x, y) = +∞). Observe that in general d n (x, y) ≤ d n,K (x, y) for x, y ∈ K. Definition 2.1. A set K ⊂ V n \ V 0 is said to be connected if d n,K (x, y) < +∞ for any x, y ∈ K. For u : V n → R and a non empty subset K ⊂ V n \ V 0 , we define Lip n (u, K) = max x,y∈K,x =y \|u(x) − u(y)\| d n,K (x, y) , (2.2) Lip n (u, ∂K) = max x,y∈∂K,x =y \|u(x) − u(y)\| d n,K (x, y) , (2.3) (note that in (2.2) the maximum is taken over the setK = K ∪ ∂K). For x ∈ V n \ V 0 ∆ n ∞ u(x) = max y∼nx {u(y) − u(x)} + min y∼nx {u(y) − u(x)}, (2.4) F n (u, x) = max y∼nx \|u(x) − u(y)\| δ n . (2.5) We give some basic properties of the previous operators. Definition 2.2. A function u : V n → R is said infinity harmonic in K ⊂ V n \ V 0 if ∆ n ∞ u(x) = 0 for all x ∈ K. Proposition 2.3. Let K ⊂ V n \ V 0 be a connected set. (i) If u, v : V n → R satisfy ∆ n ∞ u ≥ 0, ∆ n ∞ v ≤ 0 in K and u ≤ v on ∂K, then u ≤ v in K. (ii) For any g : ∂K → R, there exists a unique infinity harmonic function u in K and such that u = g on ∂K. Moreover min ∂K g ≤ u ≤ max ∂K g and either min y∼nx {u(y) − u(x)} < 0 < max y∼nx {u(y) − u(x)} or u(y) = u(x) ∀y ∼ n x. (2.6) Proof. If K is connected, then it is connected with the boundary, i.e. for any x ∈ K there is y ∈ ∂K such that d n,K (x, y) < ∞. Under this assumption, the comparison principle in (i) is proved in [13,Theorem 4]. Existence can be proved either as in [13,Theorem 5] by means of a fixed point argument or via the constructive approach given by the Lazarus algorithm, see Section 5.1 for details. The estimate at the boundary is again consequence of the comparison principle and since the constants are infinity harmonic. For property (2.6), see [13,Lemma 3]. In the next proposition we consider properties of the distance function with respect to the graph infinity Laplacian (2.4). Proposition 2.4. Let K ⊂ V n \ V 0 be connected and x 0 ∈ ∂K. Then the function u(x) = d n,K (x 0 , x) (respectively, v(x) = −d n,K (x 0 , x)) satisfies ∆ n ∞ u ≤ 0 (respectively, ∆ n ∞ v ≥ 0) in K. Proof. Consider the function u(x) = d n,K (x 0 , x). Given x ∈ K, then the minimum of u(y) − u(x), y ∼ n x, will be a negative one, along an adjacent pointȳ which is contained in the shortest path from x to x 0 . Since the path from y to x 0 will be one vertex shorter, then min y∼nx {u(y) − u(x)} = u(y) − u(x) = −δ n . If u does reach the maximum in K at x, then max y∼nx {u(y) − u(x)} ≤ 0; otherwise the maximum of u(y) − u(x) , y ∼ n x, will be positive attained at a point y such that x is contained in the shortest path from y to x 0 and max y∼nx {u(y) − u(x)} = u(y) − u(x) = δ n . Hence ∆ n ∞ u(x) = max y∼nx {u(y) − u(x)} + min y∼nx {u(y) − u(x)} ≤ δ n − δ n = 0. In a similar way it is possible to prove that v( x) = −d n,K (x 0 , x) satisfies ∆ n ∞ v ≥ 0 in K. In the next proposition we prove that a function u is linear along any minimal path joining two points which realizes the maximum of the slope at the boundary, see (2.3). This is a crucial property that will be exploited in Section 5.1 to explicitly compute an infinity harmonic function on V n . Proposition 2.5. Let K ⊂ V n \ V 0 be connected, u : K → R be such that Lip n (u, K) = Lip n (u, ∂K) and x, y ∈ ∂K such that Lip n (u, ∂K) = \|u(x) − u(y)\| d n,K (x, y) . If γ = {x = x 0 , x 1 , . . . , x N = y} is a minimal path for d n,K joining x to y, then u is linear along γ, i.e. u(x i ) = d n,K (x i , x)u(y) + d n,K (x i , y)u(x) d n,K (x, y) , i = 0, . . . , N Proof. Set L = Lip n (u, K). If u is not linear along γ and since Lip n (u, ∂K) = L there exists an index i ∈ {1, . . . , N } such that \|u(x i ) − u(x i−1 )\| < Ld n,K (x i , x i−1 ). (2.7) Then, by Lip n (u, K) = L and (2.7) Ld n,K (y, x) = \|u(y) − u(x N −1 ) + · · · + u(x i+1 ) − u(x i ) + u(x i ) − u(x i−1 ) + · · · + u(x 1 ) − u(x)\| < L d n,K (y, x N −1 ) + · · · + Ld n,K (x i , x i−1 ) + · · · + d n,K (x 1 , x) = Ld n,K (y, x) and therefore a contradiction. The next proposition connects the functional F n (u, x), which can be interpreted as the Lipschitz constant of u at x, with the Lipschitz constant Lip n (u, K) in K. Proposition 2.6. For u : V n → R and for any connected set K ⊂ V n \ V 0 , Lip n (u, K) = max x∈K F n (u, x) (2.8) Proof. It is clear that F n (u, x) ≤ Lip n (u, K) for any x ∈ K, hence max x∈K F n (u, x) ≤ Lip n (u, K). Let x, y ∈ K be such that Lip n (u, K) = \|u(x) − u(y)\|/d n,K (x, y). Consider a minimal path γ = {x = x 0 , x 1 , . . . , x N = y} from x to y. Hence \|u(x) − u(y)\| d n,K (x, y) ≤ N −1 i=0 \|u(x i+1 ) − u(x i )\| d n,K (x, y) ≤ δ n d n,K (x, y) ( N −1 i=1 F n (u, x i ) + F n (u, x 1 )) Since γ is composed by N arcs, then d n,K (x, y) = N 1 2 n = N δ n , hence \|u(x) − u(y)\| d n,K (x, y) ≤ 1 N N −1 i=0 max x∈K F n (u, x) = max x∈K F n (u, x) and therefore (2.8). The Absolutely Minimizing Lipschitz Extension problem in V n In this section we fix n ∈ N and we consider the Lipschitz extension of a function g : V 0 → R to V n . The following result is the analogous of the the classical Whitney-McShane solution to the Lipschitz extension problem. Proposition 3.1. Given K ⊂ V n \ V 0 connected and a function g : ∂K → R, set L 0 = Lip n (g, ∂K) and define M * (x) = max y∈∂K {g(y) − L 0 d n,K (x, y)}, M * (x) = min y∈∂K {g(y) + L 0 d n,K (x, y)}. Then Lip n (M * , K) = Lip n (M * , K) = L 0 and for any u : V n → R such that u = g on ∂K and Lip(u, K) = L 0 , we have M * (x) ≤ u(x) ≤ M * (x), x ∈ K. (3.1) Proof. By the very definition of Lip(u, K), for any x ∈ K, y ∈ ∂K g(y) − L 0 d n,K (x, y) = u(y) − L 0 d n,K (x, y) ≤ u(x) u(x) ≤ u(y) + L 0 d n,K (x, y) = g(y) + L 0 d n,K (x, y) and therefore (3.1). Let us prove that Lip(M * (x), V n ) = L 0 . Given x 1 , x 2 ∈ K, let y 1 ∈ ∂K be such that M * (x 1 ) = g(y 1 ) − L 0 d n (x 1 , y 1 ). Hence, M * (x 1 ) − M * (x 2 ) ≤ g(y 1 ) − L 0 d n,K (x 1 , y 1 ) − g(y 1 ) + L 0 d n,K (x 2 , y 1 ) ≤ L 0 d n,K (x 1 , x 2 ). The proof that M * (x 1 ) − M * (x 2 ) ≥ −L 0 d n,K (x 1 , x 2 ) is similar. Our approach to the Absolutely Minimizing Lipschitz Extension problem on V n is based on the following properties/observations: 1. If ∆ n ∞ u(x) = 0, then t = u(x) minimizes the functional I(t) = max y∼nx \|t−u(y)\| δn giving the Lipschitz constant of u at x (see [14,Theorem 5] and [12]). (i) A function u : V n → R is said an absolute minimizer for the functional Lip n on V n if for any connected set K ⊂ V n \ V 0 and for any v : V n → R such that u = v on ∂K, then Lip n (u, K) ≤ Lip n (v, K). We denote by AMLE(V n ) the set of the absolute minimizers for Lip n in V n . 2. Lip n (u, K) = max x∈K F n (u, x) (see Prop. 2.6). 3. If u solves ∆ n ∞ u(x) = 0 in K ⊂ V n \ V 0 connected, (ii) A function u : V n → R satisfies the Comparison with Cones property (noted CC property) in V n if for any connected set K ⊂ V n \ V 0 , for any x 0 ∈ ∂K, λ ≥ 0 and α ∈ R u ≤ λd n,K (x 0 , ·) + α on ∂K implies u ≤ λd n,K (x 0 , ·) + α on K, u ≥ −λd n,K (x 0 , ·) + α on ∂K implies u ≥ −λd n,K (x 0 , ·) + α on K. (iii) A function u is said an absolute minimizer for the functional F n on V n if for any x ∈ V n \ V 0 and for any v : V n → R such that v(y) = u(y) for all y ∼ n x, then F n (u, x) ≤ F n (v, x). We denote by AM(V n ) the set of the absolute minimizer for F n in V n . We have the following result. (ii) u satisfies the CC property in V n ; (iii) u is infinity harmonic in V n \ V 0 ; (iv) u ∈ AM(V n ). Proof. To show that (i) implies (ii), assume by contradiction that there exists a connected set K ⊂ V n \ V 0 , x 0 ∈ ∂K, α ∈ R and λ ≥ 0 such that u(x) ≤ α + λd n,K (x 0 , x) ∀x ∈ ∂K and that the set W = {y ∈ K : u(y) > α+λd n,K (x 0 , y)} is not empty (we assume that W is connected otherwise we consider a connected component of W ). Observe that ∂W ⊂ K and u(x) ≤ α + λd n,K (x 0 , x) ∀x ∈ ∂W. (3.2) Let y 1 , y 2 ∈ ∂W such that, defined L := Lip n (u, ∂W ), then L = u(y 2 ) − u(y 1 ) d n,W (y 1 , y 2 ) . Let γ be a minimal path for d n,W (y 1 , y 2 ), y 0 ∈ γ ∩W (this point exists by the definition of Lip n (u, ∂W )) and z 0 ∈ ∂W such that d n,K (y 0 , x 0 ) = d n,K (y 0 , z 0 ) + d n,K (z 0 , x 0 ) = d n,W (y 0 , z 0 ) + d n,K (z 0 , x 0 ). Hence by (3.2) u(x) ≤ β + λd n,W (z 0 , x) ∀x ∈ ∂W, u(y 0 ) > β + λd n,W (z 0 , y 0 ),(3.3) where β = α + λd n,K (z 0 , x 0 ). Since u ∈ AMLE(V n ), by Prop. 2.5 u is linear along γ and therefore u(y 2 ) − u(y 0 ) = Ld n,W (y 0 , y 2 ). Hence by (3.3) β + λd n,W (z 0 , y 2 ) ≥ u(y 2 ) = u(y 0 ) + Ld n,W (y 0 , y 2 ) > β + λd n,W (z 0 , y 0 ) + Ld n,W (y 0 , y 2 ) ≥ β + λd n,W (z 0 , y 2 ) + (L − λ)d n,W (y 0 , y 2 ). Since d n,W (y 0 , y 2 ) > 0, then L < λ. On the other hand, by (3.3) and Lip n (u, ∂W ) = Lip n (u, W ), we have u(z 0 ) ≤ β and u(z 0 ) + Ld n,W (y 0 , z 0 ) ≥ u(y 0 ) > β + λd n,W (y 0 , z 0 ) and therefore λ < L, hence a contradiction to W not empty. To prove that (ii) implies (i), assume now that u satisfies (ii) and set L = Lip n (u, ∂K). Fix x ∈ K, then by the CC property Since d n,J i (x, y) = d n,K (x, y) we obtain \|u(x) − u(y)\| ≤ Ld n,K (x, y). u(z) − Ld n,K (x, z) ≤ u(x) ≤ u(z) + Ld n,K (x, z) As K is connected, then for any z 1 ∈ J 1 and for any z 2 ∈ J 2 (where J 1 and J 2 denote two different connected component of K \ {x}) we have d n,K (z 1 , z 2 ) = d n,J 1 (z 1 , x) + d n,J 2 (x, z 2 ) and therefore, since x is arbitrary in K, Lip n (u, K) = L. We prove that (iii) implies (ii). If u is infinity harmonic in K, x 0 ∈ ∂K, α ∈ R and λ ≥ 0 are such that u(y) ≤ α + λd n,K (x 0 , y), ∀y ∈ ∂K, u(x) ≤ α + λd n,K (x 0 , x), ∀x ∈ K. Similarly for the other relation in the definition of the CC property. To show that (ii) implies (iii), assume by contradiction that there exists x ∈ V n \ V 0 such that ∆ n ∞ u(x) = 2η > 0. (3.5) Consider the set K = {x}. Hence ∂K = {y i } 4 i=1 where y i are the four points adjacent to x in V n . Let y 1 , y 2 ∈ ∂K be the points where u attains its maximum, respectively minimum, on ∂K and λ ≥ 0 be such that u(y 1 ) − u(y 2 ) = λd n,K (y 1 , y 2 ) = 2λδ n . Hence u(y j ) ≥ u(y 2 ) = u(y 1 ) − λd n,K (y 1 , y 2 ) = u(y 1 ) − λd n,K (y 1 , y j ), j = 1, 2, 3, 4, and therefore by the CC property in K u(x) ≥ u(y 1 ) − λd n,K (y 1 , x) = u(y 1 ) − λδ n . {u(y i )} = u(y 1 ) + u(y 2 ) 2 = u(y 1 ) 2 + u(y 1 ) − λd n,K (y 1 , y 2 ) 2 = u(y 1 ) − λδ n = u(y 1 ) − λd n,K (y 1 , x) It follows that u(x) + η = u(y 1 ) − λd n,K (y 1 , x) and therefore a contradiction to (3.6). The equivalence between (iii) and (iv) is proved in [14,Theorem 5] observing that ∆ n ∞ u(x) = 0 if and only if u(x) is such that The Absolutely Minimizing Lipschitz Extension problem in S Following [1,9,10], we introduce absolute minimizing Lipschitz extensions on S. We consider the length space (S, d) where d is the path distance defined by d(x, y) := inf { (γ) : γ is a path joining x to y } with (γ) the length of γ (see [7]). We consider as boundary of S, as for all the sets V n , the initial set V 0 = {q 1 , q 2 , q 3 } and we assume that a function g : V 0 → R is given. u ≤ λd(x 0 , ·) + α on ∂A implies u ≤ λd(x 0 , ·) + α on A u ≥ −λd(x 0 , ·) + α on ∂A implies u ≥ −λd(x 0 , ·) + α on A In the following proposition we summarize the results in [9,10] concerning the existence of AMLE in metric spaces which in particular applies to (S, d). (i) A function u is of class AMLE(S) if and and only if satisfies the Comparison with Cones property. (ii) For any given g : V 0 → R, there exists a unique function u ∈ AMLE(S) such that u = g on V 0 . Proof. For the proof of (i), see [10,Proposition 4.1], for the one of (ii), we refer to [9,Theorem 4.3] for existence and to [15,Theorem 1.4] for the uniqueness. Proof. Since the sequence {u n } n∈N is uniformly bounded and equi-Lipschitz continuous, there exists a function u : S → R such that, up to a subsequence, lim n→∞ u n (x) = u(x) and the convergence is also uniform. Moreover u is Lipschitz continuous with Lipschitz constant L. We prove that u satisfies the CC property on S. Given a proper, open set any proper, open, connected set A ⊂ S \ V 0 , let x 0 ∈ S \ A, α ∈ R, λ > 0 (if λ = 0 the argument is similar) be such that u(y) ≤ α + λd(x 0 , y), ∀y ∈ ∂A, and assume by contradiction that the set W = {y ∈ A : u(y) > α + λd(x 0 , y)} is not empty (we assume that W is connected otherwise we consider a connected component of W ). Observe that u(y) = α + λd(x 0 , y) ∀y ∈ ∂W. Let y 0 ∈ W and η > 0 be such that u(y 0 ) ≥ α + λd(x 0 , y 0 ) + 4η. (4.1) Defined W n = W ∩ V n , let ε = η L+λ . As W is open, there exists an integer n 0 and y * 0 ∈ W n 0 such that d(y 0 , y * 0 ) < ε. Since W n ⊂ W n+1 , then y * 0 ∈ W n for any n ≥ n 0 . Therefore by (4.1) and the continuity of u, we have u(y * 0 ) ≥ u(y 0 ) − Lε ≥ α + λd(x 0 , y * 0 ) + 3η. Similarly, there exists a point x * 0 ∈ V n 1 such that d(x 0 , x * 0 ) < η 2λ : then u(y * 0 ) ≥ α+λd(x 0 , y * 0 )+3η ≥ α+λ(d(y * 0 , x * 0 )+d(x 0 , x * 0 )−2d(x 0 , x * 0 ))+3η ≥ α+λ(d(y * 0 , x * 0 )+d(x 0 , x * 0 ))+2η. Defined β = α + λd(x 0 , x * 0 ), we have u(y * 0 ) ≥ β + λd(y * 0 , x * 0 ) + 2η. Moreover u(y) ≤ β + λd(x * 0 , y) ∀y ∈ A \ W. Let γ n ⊂ V n be a path {x * 0 = x n 0 , x n 1 , . . . , x n N = y * 0 } with x n i ∼ n x n i+1 , i = 0, . . . , N − 1, connecting x * 0 to y * 0 such that d n (x * 0 , y * 0 ) = (γ n ). Since d n (x * 0 , y * 0 ) → d(x * 0 , y * 0 ) for n → ∞ (see for example [2, Corollary 5.1], [7]), then d(x * 0 , y * 0 ) ≥ (γ n ) − η λ for n sufficiently large, where (γ n ) is the length of γ n . Denoted by z n the first point x n i ∈ γ n , i = 1, . . . , N − 1, such that x n i ∈ W , d n (x * 0 , y * 0 ) = d n (x * 0 , z n ) + d n (z n , y * 0 ). and by (4.4) d(x * 0 , y * 0 ) ≥ d n (x * 0 , z n ) + d n (z n , y * 0 ) − η λ . Set β n = β + λd n (x * 0 , z n ), then u(y) ≤ β + λd(x * 0 , y) ≤ β + λd n (x * 0 , y) ≤ β + λd n (x * 0 , z n ) + λd n (z n , y) = β n + λd n (z n , y) ∀y ∈ A \ W, (4.2) u(y * 0 ) ≥ β + λd(x * 0 , y * 0 ) + 2η ≥ β + λd n (x * 0 , z n ) + λd n (z n , y * 0 ) − η + 2η = β n + λd n (z n , y * 0 ) + η. (4.3) For any y ∈ ∂W n we have in particular that y ∈ A \ W and by the uniform convergence of u n to u and (4.2), there exists ε n → 0 for n → ∞ such that u n (y) ≤ β n + ε n + λd n,W n (z n , y) ∀y ∈ ∂W n . Then by the CC property for u n , we get (in particular) u n (y * 0 ) ≤ β n + ε n + λd n,W n (z n , y * 0 ) . (4.4) Passing to the limit for n → ∞ in (4.4) we get a contradiction to (4.3). In fact u(y * 0 ) ≤ lim inf(β n + 2ε n + λd n,W n (z n , y * 0 )) and by definition of z n , we have d n,W n (z n , y * 0 ) ≤ d n (z n , y * 0 ) so u(y * 0 ) ≤ lim inf(β n + 2ε n + λd n (z n , y * 0 )) = lim inf(β n + λd n (z n , y * 0 )) while from (4.3) we can deduce u(y * 0 ) ≥ lim sup(β n + λd n (z n , y * 0 )) + η Arguing in a similar way for the other relation, we conclude that u satisfies the CC property on S and therefore u ∈ AMLE(S). By the uniqueness of u, see Prop. 4.2, we get that all the sequence u n converges uniformly to u. Definition 4.4. We say that a function u : S → R is infinity harmonic if it is the limit of infinity harmonic functions u n : V n → R such that u n = u on V 0 . By Theorem 4.3, we have Corollary 4.5. The following properties are equivalent (i) u ∈ AMLE(S); (ii) u satisfies the Comparison with Cones property; (iii) u is infinity harmonic. Remark 4.6. As for the Laplacian on the Sierpinski gasket (see [16]), one would be tempted to say that, given f : V n−1 → R,f is an infinity harmonic extension of f to V n iff ∈ AMLE(V n ) and f = f on V n−1 . But a functionf with such property could not exist. This can be seen with the following example. Let g : V 0 → R such that g(q 1 ) = 0, g(q 2 ) = e ∈ [0, 1/7] and g(q 3 ) = 1 and denote by q ij = q ji , i, j = 1, 2, 3 and i = j, the point in V 1 \ V 0 on the segment of vertices q i and q j . By the Lazarus algorithm (see Section 5.1) the unique solution of ∆ 1 ∞ u(x) = 0 in V 1 \ V 0 and u = g on V 0 is given by u 1 (q 12 ) = (1 + e)/4 and u 1 (q 23 ) = (1 + e)/2, u 1 (q 13 ) = 1/2. If we consider the problem ∆ 2 ∞ u(x) = 0 in V 2 \ V 0 and u = g on V 0 , by applying again the Lazarus algorithm we find u 2 (q 12 ) = (3+4e)/12 and therefore u 2 (q 12 ) = u 1 (q 12 ) if e = 0. Hence the function u 2 does not satisfy ∆ 1 ∞ u 2 (x) = 0 in V 1 , since otherwise u 2 ≡ u 1 on V 1 . Note that the set K = V 1 \ V 0 is not connected as a subset of V 2 and ∂K = V 2 \ V 0 , hence the values of u 2 on V 2 are used to compute ∆ 2 ∞ u 2 (x) = 0 at x ∈ V 1 . This is a main difference with the case of the Laplacian where the harmonic extensionf of a function f to V n still satisfies the Laplace equation on V n−1 . The main consequence of this observation is that we cannot define an infinity harmonic function on S as a continuous function whose restriction to V n is infinity harmonic for all n ∈ N. The following property can be useful in order to characterize the limit of the sequence u n . x∈V n \V 0 F n (u, x) is increasing in n ∈ N. Proof. Let x ∈ V n \ V 0 be such that F n (u, V n ) = F n (u, x), y ∈ V n such that F n (u, x) = \|u(x) − u(y)\|/δ n and z the point in V n+1 in between x and y. Then \|u(x) − u(y)\| δ n ≤ 1 2 \|u(x) − u(z)\| δ n+1 + 1 2 \|u(z) − u(y)\| δ n+1 ≤ 1 2 F n+1 (u, z) + 1 2 F n+1 (u, z) ≤ F n+1 (u, V n+1 ) hence F n (u, V n ) ≤ F n+1 (u, V n+1 ). Some complements to the AMLE problem in V n We discuss in this section two further properties of the AMLE problem in V n : • an algorithm to compute explicitly an infinity harmonic function on V n ; • the relation between p-harmonic and infinity harmonic functions on V n . A constructive approach: The Lazurus algorithm We describe an algorithm introduced in [17] and extensively studied in [5] which allows to compute an infinity harmonic function on V n . Observing that the set of infinity harmonic functions is invariant by addition and multiplication for constants, if u is infinity harmonic, then v(x) = u(x) − min V 0 {u} max V 0 {u} − min V 0 {u} (5.1) is also infinity harmonic. Moreover v assumes the boundary values 0, e, 1 for some e ∈ We start with the boundary values v(q 1 ) = 0 v(q 2 ) = e, v(q 3 ) = 1 (see figure 5.1.(a)) and we compute the corresponding infinity harmonic function on V 1 . We denote by q ij = q ji , i, j = 1, 2, 3 and i = j, the point in V 1 \ V 0 on the segment of vertices q i and q j . Exploiting Prop.2.5, since Lip 1 (v, ∂(V 1 \ V 0 )) = v(q 3 ) − v(q 1 ) d 1 (q 1 , q 3 ) = 1, the function v is linear along the minimal path {q 1 , q 13 , q 3 } for d 1 (q 1 , q 3 ), hence v(q 13 ) = 1/2. To compute the other values we consider the connected set K = {q 12 , q 23 } with ∂K = V 0 ∪ {q 13 }. We distinguish two cases: and a minimal path for d 1,K (q 2 , q 3 ) is given by {q 2 , q 23 , q 3 }. Hence v(q 23 ) = (1 + e)/2. To determine v(q 12 ), we consider the connected set J = {q 12 } with boundary ∂J = {q 1 , q 13 , q 23 , q 2 }. Since Lip 1 (v, ∂J) = v(q 23 ) − v(q 1 ) d 1,K (q 1 , q 23 ) = (1 + e)/2 1 and a minimal path for d 1,K (q 1 , q 23 ) is given by {q 1 , q 12 , q 23 }, we get v(q 12 ) = (1 + e)/4 (see figure 5.1.(c)). Arguing as above it is possible (in principle) to compute infinity harmonic function on V n for any n ∈ N (the case n = 2 is detailed in [5]). Alternatively it is possible to use the iterative scheme developed in [14] in the framework of numerical approximation of infinity harmonic functions in Euclidean space. Since this scheme works for general graph, it can be applied to V n . It is worth noticing that a somewhat stronger result than Theorem 4.3 is obtained in [5]. In fact it is proved that the restriction of an infinity harmonic function u n on V n is eventually unchanging on V k for n sufficiently larger than k. This result is based on the explicit construction of the optimal paths for the Lazarus algorithm and the argument is rather involved. Infinity and p-harmonic function in V n Another important point of view of the AMLE theory is the relation between p-harmonic and infinity harmonic functions. In the Euclidean case it can be proved that the limit of p-harmonic functions for p → ∞ is an infinity harmonic function [1,4], while this property may fail in general metricmeasure spaces [10]. We start to prove a similar result for the pre-fractal V n . Fixed n ∈ N and given g : V 0 → R, we introduce for p ∈ [1, ∞) the p-energy functional A function which achieves the minimum in (5.2) is said a p-harmonic function on V n . The existence of a p-harmonic function on graphs is studied in [8]. then u is linear along a minimal path for d n,K joining two points which realize the maximal slope Lip n (u, ∂K) at the boundary ([5,17]). The same property holds also if Lip n (u, K) = Lip n (u, ∂K) (see Prop. 2.5).4. The functions M * (x), M * (x), defined by means of the distance function d n,K , give the minimal and maximal solution to the Lipschitz extension problem on K. Moreover, for x 0 ∈ ∂K, d n,K (x 0 , ·) and −d n,K (x 0 , ·) are a supersolution and a a subsolution of ∆ n ∞ u = 0 in K (Prop.2.4)Indeed it is clear that all the previous concepts are strictly related and, in analogy with the Euclidean case, we introduce the following definitions Definition 3.2. Proposition 3. 3 . 3The following properties are equivalent (i) u ∈ AMLE(V n ); z ∈ ∂K. Set J i a connected component of K \ {x}, then ∂J i ⊂ ∂K ∪ {x} and by (3.4) Lip n (u, ∂J i ) ≤ L. Again by the CC property we have u(x) − Ld n,J i (x, y) ≤ u(y) ≤ u(x) + Ld n,J i (x, y), for all y ∈ J i . e. u(x) minimizes the functional I(t) = max y∼nx \|t − u(y)\|/δ n .Corollary 3.4. Given g : V 0 → R and defined L 0 = max i,j=1,2,3 {\|g(q i ) − g(q j )\|}, where {q 1 , q 2 , q 3 } are the vertices of V 0 , then for any n ∈ N there exists a unique u n ∈ AMLE(V n ) such that u = g on V 0 .Proof. The previous result is an immediate consequence of Prop. 2.3 and Prop. 3.3.Remark 3.5. The definition of Lipschitz constant at the boundary, which is computed considering paths staying inside the domain, see (2.3), rules out the pathological example considered in [9, Example 2.2] where a function defined on the boundary does not have an absolutely minimizing extension. Note that the theory developed in this section applies to a generic graph. Let (X, d) be the metric space with X = {x, y, z} and d(x, y) = 3/2, d(x, z) = d(y, z) = 1. Consider K = {z}, hence ∂K = {x, y}, and f : ∂K → R defined by f (x) = 0, f (y) = 1.Hence Lip(f, ∂K) = 1/2 (consider the path inside K given by x 0 = y, x 1 = z, x 2 = x). If u is infinity harmonic in K, then u(z) = max{u(x), u(y)}/2 + min{u(x), u(y)}/2 = 1/2 and Lip(u, K) = Lip(u, ∂K) = 1/2. If the Lipschitz constant of f at the boundary is computed as \|f (y) − f (x)\|/d(x, y) = 2/3, then it is shown in[9] that an absolutely minimizing Lipschitz extension of f to X does not exist. Given A ⊂ S and f : A → R, we define the Lipschitz constant of f on A to be Lip(f, A) := sup x,y∈A,x =y \|f (y) − f (x)\| d(x, y) Definition 4.1. • A continuous function u : S → R is said an absolute minimizer for the functional Lip on S if for any proper, open, connected set A ⊂ S \ V 0 and for any v : S → R such that u = v on ∂A, then Lip(u, A) ≤ Lip(v, A). We denote by AMLE(S) the set of the absolute minimizer for Lip in S. • A function u : S → R satisfies the Comparison with Cones property (noted CC property) if for any proper, any proper, open, connected set A ⊂ S \ V 0 , for any x 0 ∈ S \ A, λ ≥ 0 and α ∈ R Theorem 4. 3 . 3Given g : V 0 → R, let u n be the AMLE(V n ) of g to V n . Then lim n→∞ u n (x) = u(x), uniformly in S,i.e. lim n→∞ sup x∈V n \|u n (x) − u(x)\| = 0, where u is the AMLE(S) of g to S. Proposition 4. 7 . 7For u : S → R, the functional F n (u, V n ) =: max [0, 1]. Since the values at the boundary determine univocally an infinity harmonic function, to compute u is sufficient to consider the case of the boundary values 0, e, 1 and then to inverte the affine transformation (5.1). It is possible to further reduce the computation by considering e ∈ [0, 1/2]. In fact if e ∈ [1/2, 1], then w(x) = 1 − v(x) is infinity harmonic and assume the boundary values 0, 1 − e ∈ [0, 1/2], 1 (obviously the order of the vertices of V 0 where these values are assumed is not relevant). Computed w, we have v(x) = 1 − w(x). minimal path for d 1,K (q 1 , q 3 ) is given by {q 1 , q 12 , q 23 , q 3 }. Applying again Prop. 2.5, we get v(q 12 ) = 1/3, v(q 23 ) = 2/3 (seefigure 5.1.(b)). (ii): If e ∈ [0, 1/3], thenLip 1 (v, ∂K) = v(q 3 ) − v(q 2 ) d 1,K (q 2 , q 3 ) = 1 − e 1 Figure 1 : 1infinity harmonic function on V 1 : (a) boundary condition, (b) e ∈ [1/3, 1/2], (c) e ∈ [0, 1/3] Dip. di Scienze di Base e Applicate per l'Ingegneria, "Sapienza" Università di Roma, via Scarpa 16, 00161 Roma, Proposition 5.1. Let {u n p } p≥1 be the family of the p-harmonic functions on V n such that u n p = g on V 0 . Then u n p → u n for p → ∞ where u n ∈ AMLE(V n ) and u n = g on V 0 .We need a preliminary lemma, expressing the local character of p-harmonic functions. For x ∈ V n \ V 0 and v : V n → R, setProof. If by contradiction there exists v : V n → R such that v(y) = u n p (y) for y ∼ n x and such that I n p (v, x) ≤ I n p (u n p , x) − ε for some ε > 0, then the functionū : V n → R defined bȳis such that I n p (ū) < I n p (u n p ). Hence a contradiction to the definition of p-harmonic function.Proof of Prop. 5.1. Let u n p be a p-harmonic function and u n ∈ AMLE(V n ) such that u n = g on V 0 . Thenwhere F n is defined as in (2.5) and q i , i = 1, 2, 3 are the vertices of V 0 . Hence, by (2.8). It follows that for any x ∈ V n \ V 0 and y ∼ n x, \|u n p (y) − u n p (x)\| is bounded, uniformly in p ∈ [1, ∞). Since u n p = g on V 0 , then the functions u n p are uniformly bounded in p on V n . Passing to a subsequence, we get that there exists a sequence {u p j } j∈N which converges to a functionū on V n such thatū = g on V 0 . We claim thatū ∈ AM n (V n ) (see Def. 3.2.(iii)). Let x ∈ V n and v : V n → R such that v(y) = u n p j (y) for y ∼ n x. Then by Lemma 5.2 we havePassing to the limit for j → ∞ in the previous inequality, since the p-norm in R N converges to the ∞-norm we get16Hence F n (ū, x) ≤ F n (v, x) and therefore the claim. Since there exists a unique u n ∈ AM n (V n ) such that u n = g on V 0 , see Prop. 3.3 and Cor. 3.4, it follows thatū = u n . Moreover by the uniqueness of u n , any convergent sequence of {u n p } p≥1 tends to u n . Therefore we conclude that lim p→∞ u n p = u n .We proved the convergence u n p → u n for p → ∞ in Proposition 5.1 and the convergence u n → u for n → ∞ in Theorem 4.3. A natural question is if it possible to invert the order of the limits. Actually we have only partial results. A notion of p-harmonic functions on V n is studied in[6]with the aim of defining a p-Laplace operator on the Sierpinski gasket in the spirit of Kigami's approach. The discrete p-energy E (n) p considered in[6]is different from (5.2), even if it is dominated from below and from above by (I n p ) p . The sequence of the discrete energies E (n) p is increasing, hence it is possible to define the p-energy E p on the Sierpinski gasket S as the limitNote that a minimizer v n p of the energy E (n) p defined in[6]such that v n p = g on V 0 could be different from a minimizer u n p of the p-energy in (5.2) with the same boundary datum. In[6,Cor.2.4], it is proved that, for p fixed, any sequence {v n p } of the p-harmonic-extensions with respect to the energy E (n) p on V n (such that v n p = g on V 0 ) converges uniformly as n → ∞ to a function v p that is a p-harmonic extension of g on S with respect the limit p-energy defined in (5.3). Passing to the limit for p → ∞ we can prove that any sequence v p of p-harmonic extensions with respect the p-energy E p on S such that v p = g on V 0 converges (up to a subsequence) to a function v : S → R uniformly in S but up to now we are not able to prove that v is a AMLE of g to S. A tour of the theory of absolutely minimizing functions. 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[ "ON THE ABELIANIZATION OF CONGRUENCE SUBGROUPS OF Aut(F 2 )", "ON THE ABELIANIZATION OF CONGRUENCE SUBGROUPS OF Aut(F 2 )" ]	[ "Daniel Appel " ]	[]	[]	Let F n be the free group of rank n and let Aut + (F n ) be its special automorphism group. For an epimorphism π : F n → G of the free group F n onto a finite group G we call Γ + (G, π) = {ϕ ∈ Aut + (F n ) \| πϕ = π} the standard congruence subgroup of Aut + (F n ) associated to G and π. In the case n = 2 we fully describe the abelianization of Γ + (G, π) for finite abelian groups G. Moreover, we show that if G is a finite non-perfect group, then Γ + (G, π) ≤ Aut + (F 2 ) has infinite abelianization.	null	[ "https://arxiv.org/pdf/0910.0090v2.pdf" ]	16,790,608	0910.0090	87973e2e25c5ef7943bffb77181570a23a02b727	ON THE ABELIANIZATION OF CONGRUENCE SUBGROUPS OF Aut(F 2 ) 12 Feb 2010 Daniel Appel ON THE ABELIANIZATION OF CONGRUENCE SUBGROUPS OF Aut(F 2 ) 12 Feb 2010 Let F n be the free group of rank n and let Aut + (F n ) be its special automorphism group. For an epimorphism π : F n → G of the free group F n onto a finite group G we call Γ + (G, π) = {ϕ ∈ Aut + (F n ) \| πϕ = π} the standard congruence subgroup of Aut + (F n ) associated to G and π. In the case n = 2 we fully describe the abelianization of Γ + (G, π) for finite abelian groups G. Moreover, we show that if G is a finite non-perfect group, then Γ + (G, π) ≤ Aut + (F 2 ) has infinite abelianization. 1. Introduction 1.1. Main Results. Let F n be the free group on n generators and Aut(F n ) its group of automorphisms. Moreover, let π : F n → G be an epimorphism of F n onto a finite group G. The automorphism group Aut(F n ) acts in a natural way on the (finite) set of all epimorphisms from F n onto G. By Γ(G, π) := {ϕ ∈ Aut(F n ) \| πϕ = π} we denote the stabilizer of π under this action. The group Γ(G, π) is called the standard congruence subgroup of Aut(F n ) associated to G and π. This is a finite index subgroup of Aut(F n ). A subgroup of Aut(F n ) containing some Γ(G, π) is called a congruence subgroup of Aut(F n ). A classical question is whether every finite index subgroup of Aut(F n ) is a congruence subgroup. Quite recently it has been shown that every finite index subgroup of Aut(F 2 ) is a congruence subgroup. See [2] and [3]. For n ≥ 3 this question is still open. Date: February 12, 2010. The author would like to thank F. Grunewald for proposing the topic and B. Klopsch for many helpful discussions. The groups Γ(G, π) have been studied by various authors. For instance, in [5], F. Grunewald and A. Lubotzky use the groups Γ(G, π) to construct linear representations of the automorphism group Aut(F n ). Our work is related to results of T. Satoh [11,12], see also Section 1.2. The joint work [1] of E. Ribnere and the author can be seen as an accompanying paper to the present one. The automorphism group Aut(F n ) has a well-known representation onto GL n (Z) given by ρ : Aut(F n ) −→ Aut(F n /F ′ n ) ∼ = GL n (Z), where F ′ n denotes the commutator subgroup of F n , see [8,Sec. 3.6] for details. Its kernel is denoted by IA n and called the group of IA nautomorphisms or sometimes also the classical Torelli group. For an interesting generalization see [13]. As one classically considers SL n (Z) instead of GL n (Z), we focus on the special automorphism group Aut + (F n ) := ρ −1 (SL n (Z)), which is a subgroup of index 2 in Aut(F n ). We set Γ + (G, π) := Γ(G, π) ∩ Aut + (F n ). This is a subgroup of index at most 2 in Γ(G, π). The term congruence subgroup of Aut + (F n ) is defined in the obvious way. In this paper we study the abelianizations Γ + (G, π) ab of the groups Γ + (G, π) in the case n = 2. We remark that if G is abelian, then, up to conjugation, Γ + (G, π) depends only on G but not on the particular epimorphism π : F 2 → G, see [1,Lem. 3.1]. Moreover, every finite abelian group generated by two elements can be written as Z/mZ × Z/nZ where n, m ∈ N such that n \| m. The following theorem therefore covers all possible choices for an epimorphism π : F 2 → G onto a finite abelian group. Theorem 1.1. Let m, n ∈ N such that m ≥ 3, n \| m and (m, n) = (3, 1). Let G := Z/mZ × Z/nZ and π : F 2 → G be an epimorphism. Then Γ + (G, π) ab ∼ = G × Z 1+12 −1 nm 2 p\|m (1−p −2 ) where the product runs over all primes p dividing m. Furthermore, we have Γ + (Z/2Z, π) ab ∼ = Z/2Z × Z/4Z × Z, Γ + (Z/3Z, π) ab ∼ = Z/3Z × Z/3Z × Z, Γ + (Z/2Z × Z/2Z, π) ab ∼ = Z/2Z × Z/2Z × Z/2Z × Z 2 . Theorem 1.2. Let π : F 2 → G be an epimorphism of F 2 onto a finite non-perfect group G. Then Γ + (G, π) has infinite abelianization. For n, m ∈ N with n \| m we define a subgroup of SL 2 (Z) by Γ(m, n) := {( a b c d ) ∈ SL 2 (Z) \| a ≡ m 1, b ≡ m 0 and c ≡ n 0, d ≡ n 1} . By PΓ(m, n) we denote the image of Γ(m, n) in PSL 2 (Z) under the natural projection. One of the main ingredients in our proofs is Proposition 1.3. Let m, n ∈ N such that m ≥ 3, n \| m and (m, n) = (3, 1). Then Γ(m, n) and PΓ(m, n) are free of rank 1 + nm 2 12 p\|m p prime 1 − 1 p 2 . In particular, for primes p ≥ 5, the groups Γ(p, 1) ≤ SL 2 (Z) are free of rank 1 + 1 12 p 2 (1 − p −2 ) so that the rank of Γ(p, 1) grows quadratically in p. In contrast, for n ≥ 3, one can show that the corresponding subgroups in SL n (Z) can always be generated by n(n − 1) matrices. See [5, Lem. 6.1]. 1.2. Related Results and Open Problems. In [5] Grunewald and Lubotzky use the groups Γ(G, π) to construct linear representations of the automorphism group Aut(F n ). In their concluding Section 9.4 they present, for some explicit G of small order, the indices of the groups Γ + (G, π) in Aut + (F n ) and also the abelianizations of the groups Γ + (G, π) which they obtain by MAGMA computations. Besides the case that G is finite abelian, they also consider the case that G = D r is a dihedral group. Their observations are explained by another result of the author, which says Γ + (D r , π) ab ∼ = Z/2Z × Z 2 , r odd Z/2Z × Z 3 , r even . The proof of this result is elementary but rather long. It shall therefore be postponed to the author's Ph.D. Thesis. Actually, Grunewald and Lubotzky present computational results for some more finite groups G, e.g., G = A 5 . In all considered cases Γ + (G, π) has infinite abelianization. For G non-perfect we now know by Theorem 1.2 that Γ + (G, π) ab is infinite. However, our proof does not work for perfect groups G. Hence we state Conjecture 1.4. For every epimorphism π : F 2 → G onto a nontrivial finite group G the group Γ + (G, π) ≤ Aut + (F 2 ) has infinite abelianization. The situation in the case n ≥ 3 looks different. Indeed, Grunewald and Lubotzky show that for every epimorphism π : F n → G from F n , n ≥ 3, onto a finite abelian group, the group Γ(G, π) ≤ Aut(F n ) has finite abelianization. This is Proposition 8.5 in [5]. Computational results [5,Sec. 9.4] indicate that Γ + (G, π) always has finite abelianization if n ≥ 3. This leads to Problem 1.5. Does Γ + (G, π) ≤ Aut + (F n ), where n ≥ 3, have finite abelianization for every epimorphism π : F n → G onto a non-trivial finite group G? Our work is related to results of Satoh [11,12]. In his papers Satoh considers the kernel T n,m of the composition Aut(F n ) ρ −→ GL n (Z) −→ GL n (Z/mZ). One easily sees that for m ≥ 3 we have T n,m = Γ + ((Z/mZ) n , π) where π : F n → (Z/mZ) n is the obvious epimorphism. Satoh shows that for n, m ≥ 2 one has T ab n,m ∼ = (IA ab n ⊗ Z Z/mZ) × Γ n (m) ab where Γ n (m) is the kernel of the natural epimorphism GL n (Z) → GL n (Z/mZ). Since IA 2 is free of rank 2, see [8, Sec. 3.6, Cor. N4], for n = 2 this reads T ab 2,m ∼ = (Z/mZ) 2 × Γ 2 (m) ab . Observe that for m ≥ 3 we have Γ 2 (m) = Γ(m, m) ≤ SL 2 (Z). This result therefore corresponds to our result in Theorem 1.1 for the special case G = (Z/mZ) 2 . Satoh also gives the integral homology groups of T 2,p for odd primes p. In particular, he shows that H 1 (T 2,p , Z) = (Z/pZ) 2 × Z 1+12 −1 p 3 (1−p −2 ) . Since the first integral homology group is just the abelianization, this corresponds to our result in Theorem 1.1 for the special case G = (Z/pZ) 2 . 2. Congruence Subgroups of SL 2 (Z) 2.1. Introduction. Let π : F 2 → G be an epimorphism of the free group F 2 onto a finite group G. In order to understand the image ρ(Γ + (G, π)) in SL 2 (Z), we introduce some families of finite index subgroups of SL 2 (Z). Recall that for two natural numbers m, n ∈ N such that n \| m we define the group m) is called the principal congruence subgroup of SL 2 (Z) of level m. It is the kernel of the natural epimorphism SL 2 (Z) → SL 2 (Z/mZ). A subgroup of SL 2 (Z) containing some Γ(m) is called a congruence subgroup of SL 2 (Z). It is a classical result that not all finite index subgroups of SL 2 (Z) are congruence subgroups. See for example [7]. Γ(m, n) = {( a b c d ) ∈ SL 2 (Z) \| a ≡ m 1, b ≡ m 0 and c ≡ n 0, d ≡ n 1}. For m ∈ N the group Γ(m) := Γ(m, In [1] it is shown that (2.1) [SL 2 (Z) : Γ(m, n)] = nm 2 p\|m 1 − 1 p 2 where the product runs over all primes p dividing m. It is convenient to use the following notation. Γ 0 (m) := {A ∈ SL 2 (Z) \| A ≡ ( * 0 * * ) mod m}, Γ 1 (m) := {A ∈ SL 2 (Z) \| A ≡ ( 1 0 * * ) mod m}. All the above subgroups of SL n (Z) are clearly congruence subgroups. We denote the images in PSL 2 (Z) of Γ 0 (m), Γ 1 (m) and Γ(m) under the natural projection SL 2 (Z) → PSL 2 (Z) by PΓ 0 (m), PΓ 1 (m) and PΓ(m), respectively. Note that this argument does not work for the case m = 2. Indeed, in contrast to PΓ(2), the group Γ(2) is not free but the direct product of a rank 2 free group and a cyclic group of order 2. This is the reason for the third exceptional case in Theorem 1.1 Lemma 2.2. Let m ≥ 3. Then Γ(m) ∼ = PΓ(m) is free of rank 1 + m 3 12 p\|m 1 − 1 p 2 where the product runs over all primes p dividing m. Proof. Observe that for m 1 , m 2 ∈ N such that m 1 \| m 2 we have Γ(m 2 ) ≤ Γ(m 1 ). Accordingly, the main point in our proof is that a subgroup of a free group is again free and its rank is given by the Schreier Formula [8, Thm. 2.10]. Since Γ(2) is not free, we consider two cases. Case 1: We have m = 2 a for some a ≥ 2. One can verify that PΓ(4) has index 4 in PΓ (2). Since the latter group is free of rank 2, we find that PΓ(4) is free of rank 5 and hence so is Γ(4). From (2.1) we know that Γ(2 a ) has index 2 3a−6 in Γ(4). We thus find that Γ(2 a ) is free of rank 1 + 1 12 2 3a (1 − 2 −2 ), as claimed. Case 2: We have p 0 \| m for some prime p 0 > 2. Again by (2.1), we see that [Γ(p 0 ) : Γ(m)] = m 3 p 3 0 p\|m p =p 0 1 − 1 p 2 . Since, by Proposition 2.1, Γ(p 0 ) is free of rank 1 + 1 12 p 3 0 (1 − p −2 0 ), we obtain the desired result. We next wish to generalize this result even further to the groups Γ(m, n) where m ≥ 3, n \| m and (m, n) = (3, 1). The first step is, of course, to show that these groups are free. Here we use the well-known description of PSL 2 (Z) as a free product PSL 2 (Z) = 0 1 −1 0 * 0 −1 1 1 where the first factor has order 2 and the second one has order 3. From the Kurosh Subgroup Theorem [8,Cor. 4.9.1] it follows that every nontrivial element of finite order in PSL 2 (Z) has either order 2 or 3. Using this observation and considering some minimal cases, we obtain Lemma 2.3. Let m ≥ 4, then PΓ 1 (m) is a free group. Proof. First we consider the case that m has a prime factor p ≥ 5. Note that it suffices to show that PΓ 1 (p) is free. To this end, we show that PΓ 1 (p) does not contain a non-trivial element of finite order. Then the Kurosh Subgroup Theorem yields the desired result. Assume that A ∈ PΓ 1 (p) is a non-trivial element of finite order. By definition of PΓ 1 (p) we have A ≡ 1 0 k 1 mod p for some 0 ≤ k ≤ p − 1. It follows that A p ≡ 1 0 0 1 mod p. Now we consider two cases. Case 1: A p = 1. Then the order of A divides p and we have a contradiction, since A has either order 2 or 3. Case 2: A p = 1. Then A p is a non-trivial element of PΓ(p), which is, by Theorem 2.1, a free group. Hence A p does not have finite order, contradiction. To prove the result for m not having a prime factor ≥ 5, it suffices to consider the cases where m = 4, 6, 9. By an explicit computation, using the Reidemeister-Schreier Method, one verifies that PΓ 1 (m) is also free in these cases. From Lemmas 2.2, 2.3, the formulas (2.1) and the Schreier Formula, we obtain Proposition 1.3. H. Rademacher [10] gives a very explicit description of the groups PΓ 0 (p) for p prime. Observe that for p ∈ {2, 3} the groups PΓ 0 (p) and PΓ 1 (p) coincide. By two examples of Rademacher we have PΓ 1 (2) ∼ = Z * Z/2Z PΓ 1 (3) ∼ = Z * Z/3Z. In particular, PΓ 1 (2) and PΓ 1 (3) are not free. This is the reason for the first two exceptional cases in Theorem 1.1. (G, π) only depends on G but not on the particular choice of the epimorphism π : F 2 → G. We may thus suppose that π(x) = (1, 0) and π(y) = (0, 1). We have an exact sequence Proofs of the Main Results 1 −→ IA 2 −→ Γ + (G, π) ρ −→ Γ(m, n) −→ 1. By Proposition 1.3, the group Γ(m, n) is free of rank r := 1 + nm 2 12 p\|m 1 − 1 p 2 . Let {M i = a i b i c i d i \| 1 ≤ i ≤ r} be a set of free generators of Γ(m, n) and write F 2 = x, y . Moreover, let ϕ i ∈ Aut + (F 2 ) such that ρ(ϕ i ) = M i , that is (3.1) ϕ i (x) ≡ a i x + c i y, ϕ i (y) ≡ b i x + d i y mod F ′ 2 where F ′ 2 denotes the commutator subgroup of F 2 . By a classical result of J. Nielsen [8, Sec. 3.6, Cor. N4] the group IA 2 is free on α x , α y , the inner automorphisms given by conjugation with x and y, respectively. Now a result of P. Hall, see [6,Ch. 13,Thm. 1], yields that Γ + (G, π) admits a presentation α x , α y , ϕ 1 , . . . , ϕ r \| ϕ i α x ϕ −1 i = w i , ϕ i α y ϕ −1 i = v i for 1 ≤ i ≤ r where the w i and v i are suitable words in α x , α y . We have ϕ i α x ϕ −1 i = α ϕ i (x) , ϕ i α y ϕ −1 i = α ϕ i (y) for 1 ≤ i ≤ r. Hence, from (3.1) it follows that (3.2) α x = a i α x + c i α y , α y = b i α x + d i α y in the abelianization of Γ + (G, π). This yields that Γ + (G, π) ab is the abelian group generated by α x , α y and ϕ i , 1 ≤ i ≤ r, subject to the relations (3.2). Observe that ( 1 0 n 1 ) ∈ Γ(m, n). Hence this matrix is a product of the M i and one consequence of the relations (3.2) is α x = α x + nα y . We thus find that nα y = 0. Similarly we find that mα x = 0. Obviously we can rewrite the defining relations (3.2) as (3.3) (a i − 1)α x = c i α y , (1 − d i )α y = b i α x . By definition of Γ(m, n) we have (a i − 1) ≡ b i ≡ 0 mod m, (1 − d i ) ≡ c i ≡ 0 mod n for 1 ≤ i ≤ r so that all relations in (3. 3) are consequences of mα x = 0 and nα y = 0. Hence we obtain a presentation Γ + (G, π) ab = α x , α y , ϕ 1 , . . . , ϕ r \| abelian, mα x = 0, nα y = 0 . This proves Γ + (G, π) ab ∼ = G × Z r . The three special cases can be obtained by an explicit computation. 3.2. Proof of Theorem 1.2. Let G be a finite non-perfect group. First we consider the case that G/G ′ ∼ = Z/2Z where G ′ denotes the commutator subgroup of G. Then we naturally obtain an epimorphism π : F 2 π −→ G −→ Z/2Z. One easily verifies that (3.4) Γ + (G, π) ≤ Γ + (Z/2Z,π). By [1, Lem. 3.1] there is some ϕ ∈ Aut + (F 2 ) such thatπϕ(x) = 1 andπϕ(y) = 0. Observe that Γ + (G, πϕ) = ϕ −1 Γ + (G, π)ϕ. We may therefore assume thatπ(x) = 1 andπ(y) = 0. We set ρ : Aut + (F 2 ) ρ −→ SL 2 (Z) −→ PSL 2 (Z) where the second epimorphism is the natural projection. Note thatρ is onto. Sinceρ induces an epimorphism Γ + (G, π) ab →ρ(Γ + (G, π)) ab , it suffices to show thatρ(Γ + (G, π)) has infinite abelianization. By (3.4) we haveρ (Γ + (G, π)) ≤ρ(Γ + (Z/2Z,π)) = PΓ 1 (2). Henceρ(Γ + (G, π)) is a finite index subgroup of PΓ 1 (2). Note that PΓ 1 (2) = PΓ 0 (2). By an example of Rademacher [10, Sec. 8], we have PΓ 0 (2) = 1 0 −1 1 * 1 −2 1 −1 where the the first factor is infinite cyclic and the second one has order 2. The Kurosh Subgroup Theorem yields thatρ(Γ + (G, π)) is the free product of (i) a possibly trivial free group, (ii) certain subgroups of conjugates of ( 1 0 −1 1 ) , (iii) certain conjugates of 1 −2 1 −1 . We shall prove that a free factor of type (i) or (ii) actually appears. For a contradiction, let us assume thatρ(Γ + (G, π)) is the free product of factors of type (iii) only. Thenρ(Γ + (G, π)) is generated by elements of order 2 andρ(Γ + (G, π)) ab ∼ = (Z/2Z) m for some m ∈ N. Let k be the order of G. Then the automorphism ϕ ∈ Aut + (F 2 ) given by ϕ(x) = xy k , ϕ(y) = y is an element of Γ + (G, π). Hence M := 1 0 k 1 ∈ρ(Γ + (G, π)). Since M ∈ ( 1 0 −1 1 ) , one easily sees that the image of M in PΓ 0 (2) ab has infinite order. On the other hand, the image of M inρ(Γ(G, π)) ab must have finite order. Observe that the inclusion mapρ(Γ + (G, π)) ֒→ PΓ 0 (2) induces a homomorphismρ(Γ + (G, π)) ab → PΓ 0 (2) ab such that the following diagram commutes. ρ(Γ + (G, π)) / / PΓ 0 (2) ρ(Γ + (G, π)) ab / / PΓ 0 (2) ab This implies that the image of M in PΓ 0 (2) ab has finite order, contradiction. The proof for G ab = Z/3Z is almost the same. In the remaining cases, we find thatρ(Γ + (G, π)) is a finite index subgroup of a free subgroup of PSL 2 (Z). In particular, it has infinite abelianization and so must have Γ + (G, π). 2. 2 . 2Free Congruence Subgroups. In[4] H. Frasch gives the following description of the groups PΓ(p) for p prime. Theorem 2. 1 ( 1Frasch). Let p ≥ 3 be a prime. Then PΓ(p) is free of rank 1 + 1 12 p 3 (1 − p −2 ). Moreover PΓ(2) is free of rank 2. We shall now generalize his result. Consider the natural projection SL 2 (Z) → PSL 2 (Z). By definition it maps Γ(m) onto PΓ(m). For m ≥ 3 the kernel − 3. 1 . 1Proof of Theorem 1.1. Let G = Z/mZ × Z/nZ where m ≥ 3, n \| m and (m, n) = (3, 1). Then, by [1, Lem. 3.1], up to conjugation, the group Γ + On the index of congruence subgroups of Aut(F n ). D Appel, E Ribnere, J. Alg. 321D. Appel, E. Ribnere, On the index of congruence subgroups of Aut(F n ). J. Alg. 321 (2009), 2875-2889. The faithfulness of the monodromy representations associated with certain families of algebraic curves. M Asada, J. Pure Appl. Algebra. 1592-3M. Asada, The faithfulness of the monodromy representations associated with certain families of algebraic curves. J. Pure Appl. Algebra 159 (2001), no. 2- 3, 123-147. The congruence subgroup property of Aut(F 2 ): A group-theoretic proof of Asada's theorem. K.-U Bux, M V Ershov, A S Rapinchuk, arXiv:0909.0304v1K.-U. Bux, M. V. Ershov, A. S. Rapinchuk, The congruence sub- group property of Aut(F 2 ): A group-theoretic proof of Asada's theorem arXiv:0909.0304v1 (2009). Die Erzeugenden der Hauptkongruenzuntergruppen für Primzahlstufen. H Frasch, Math. Ann. 108H. Frasch, Die Erzeugenden der Hauptkongruenzuntergruppen für Primzahlstufen. Math. Ann. 108 (1933), 229-252. Linear Representations of the Automorphism Group of a Free Group. F Grunewald, A Lubotzky, Geom. Funct. Anal. 18F. Grunewald, A. Lubotzky, Linear Representations of the Automorphism Group of a Free Group. Geom. Funct. Anal. 18 (2009), 1564-1608. D L Johnson, Presentations of Groups. CambridgeCambridge University PressD. L. Johnson, Presentations of Groups. Cambridge University Press, Cam- bridge, 1976. Congruence and non-congruence subgroups of the modular group: a survey. G Jones, Proceedings of groups -St. Andrews. groups -St. AndrewsCambridgeCambridge University Press121G. Jones, Congruence and non-congruence subgroups of the modular group: a survey. Proceedings of groups -St. Andrews 1985, 223-234, London Math. Soc. Lecture Note Ser. 121, Cambridge University Press, Cambridge, 1986. W Magnus, A Karrass, D Solitar, Combinatorial Group Theory. New YorkJohn Wiley & Sons, IncW. Magnus, A. Karrass, D. Solitar, Combinatorial Group Theory. John Wiley & Sons, Inc., New York, 1966. What do we know about the product replacement algorithm?. I Pak, Groups and Computation III. Columbus, Ohio; Berlinde Gruyter8Ohio State UnivI. Pak, What do we know about the product replacement algorithm? Groups and Computation III (Columbus, Ohio, 1999), Ohio State Univ. Math. Res. Inst. Publ. 8, de Gruyter, Berlin (2001), 301-347. Über die Erzeugenden von Kongruenzuntergruppe der Modulgruppe. H Rademacher, Abhandlungen Hamburg. 7H. Rademacher,Über die Erzeugenden von Kongruenzuntergruppe der Mod- ulgruppe. Abhandlungen Hamburg 7 (1929), 134-148. The abelianization of the congruence IA-automorphism group of a free group. T Satoh, Math. Proc. Camb. Phil. Soc. 142T. Satoh, The abelianization of the congruence IA-automorphism group of a free group. Math. Proc. Camb. Phil. Soc. 142 (2007), 239-248. The abelianization of the congruence IAautomorphism group of a free group. T Satoh, Corrigendum , Math. Proc. Camb. Phil. Soc. 143T. Satoh, Corrigendum: The abelianization of the congruence IA- automorphism group of a free group. Math. Proc. Camb. Phil. Soc. 143 (2007), 255-256. M Siegmund, Generalized Torelli Groups. Dissertation. GermanyHeinrich-Heine-Universität DüsseldorfM. Siegmund, Generalized Torelli Groups. Dissertation (Heinrich-Heine- Universität Düsseldorf, Germany, 2007). Current address: Mathematisches Institut der Heinrich-Heine-Universität. Egham, Surrey, TW20 0EX, United Kingdom40225Department of Mathematics, Royal Holloway College, University of LondonDepartment of Mathematics, Royal Holloway College, University of London, Egham, Surrey, TW20 0EX, United Kingdom. Current address: Mathematisches Institut der Heinrich-Heine-Universität, 40225 E-mail address: Daniel.Appel@uni-duesseldorf. Germany Düsseldorf, deDüsseldorf, Germany. E-mail address: [email protected]	[]
[ "arXiv:astro-ph/0509250v1 9 Sep 2005 Friedmann cosmology with a generalized equation of state and bulk viscosity", "arXiv:astro-ph/0509250v1 9 Sep 2005 Friedmann cosmology with a generalized equation of state and bulk viscosity" ]	[ "Xin-He Meng \nDepartment of physics\nNankai University\n300071TianjinChina\n", "Jie Ren \nDepartment of physics\nNankai University\n300071TianjinChina\n", "Ming-Guang Hu \nDepartment of physics\nNankai University\n300071TianjinChina\n" ]	[ "Department of physics\nNankai University\n300071TianjinChina", "Department of physics\nNankai University\n300071TianjinChina", "Department of physics\nNankai University\n300071TianjinChina" ]	[]	The universe media is considered as a non-perfect fluid with bulk viscosity and described by a more general equation of state. We assume the bulk viscosity is a linear combination of the two terms: one is constant, and the other is proportional to the scalar expansion θ = 3ȧ/a. The equation of state is described as p = (γ − 1)ρ + p0, where p0 is a parameter. This model can be used to explain the dark energy dominated universe. Different choices of the parameters may lead to three kinds of fates of the cosmological evolution: no future singularity, big rip, or Type III singularity of Ref. [S. Nojiri, S.D. Odintsov, and S. Tsujikawa, Phys. Rev. D 71, 063004 (2005)].	10.1088/0253-6102/47/2/036	[ "https://arxiv.org/pdf/astro-ph/0509250v1.pdf" ]	119,515,694	astro-ph/0509250	b862600bbe74b893604a2da0fe28a1e854bb8f0f	arXiv:astro-ph/0509250v1 9 Sep 2005 Friedmann cosmology with a generalized equation of state and bulk viscosity Xin-He Meng Department of physics Nankai University 300071TianjinChina Jie Ren Department of physics Nankai University 300071TianjinChina Ming-Guang Hu Department of physics Nankai University 300071TianjinChina arXiv:astro-ph/0509250v1 9 Sep 2005 Friedmann cosmology with a generalized equation of state and bulk viscosity PACS numbers: 9880Cq, 9880-k The universe media is considered as a non-perfect fluid with bulk viscosity and described by a more general equation of state. We assume the bulk viscosity is a linear combination of the two terms: one is constant, and the other is proportional to the scalar expansion θ = 3ȧ/a. The equation of state is described as p = (γ − 1)ρ + p0, where p0 is a parameter. This model can be used to explain the dark energy dominated universe. Different choices of the parameters may lead to three kinds of fates of the cosmological evolution: no future singularity, big rip, or Type III singularity of Ref. [S. Nojiri, S.D. Odintsov, and S. Tsujikawa, Phys. Rev. D 71, 063004 (2005)]. I. INTRODUCTION The cosmological observations indicate that the expansion of our universe accelerates [1]. Recently lots of work on extended gravity [2]such as modifying equation of state or by introducing the so called dark energy is to explain the cosmic acceleration expansion observed. To overcome the drawback of hydrodynamical instability, a linear equation of state of a more general form, p = α(ρ − p 0 ) is proposed [3], and this form incorporated into cosmological model can describe the hydrodynamically stable dark energy behaviors. The observations also indicate that the universe media is not a perfect fluid [4] and the viscosity is concerned in the evolution of the universe [5,6,7]. In the standard cosmological model, if the equation of state parameter ω is less than -1, the universe shows the future finite singularity called Big Rip [8,9]. Several ideas are proposed to prevent the big rip singularity, like by introducing quantum effects terms in the action. In this paper, we show that the Friedmann equations can be solved with both a more general equation of state and bulk viscosity detailed as follows. The equation of state is p = (γ − 1)ρ + p 0 ,(1) where p 0 and γ are two parameters. The bulk viscosity is expressed as ζ = ζ 0 + ζ 1ȧ a .(2) where ζ 0 and ζ 1 are two constants conventionally. The ω = p/ρ is constrained as −1.38 < ω < −0.82 [10] by present observation data, so the inequality in our case should be The parameter p 0 can be positive (attractive force) or negative (repulsive force), and conventionally ζ 0 and ζ 1 are regarded as positive. To choose the parameters properly, it can prevent the Big Rip problem or some kind of singularity for the cosmology model, like in the phantom energy phase, as shown below. −1.38 < γ − 1 + p 0 ρ < −0.82.(3) This paper is organized as follows. In Sec. II we describe our model and give out the exact solution. In Sec. III we consider some special cases of the solution. In Sec. IV we discuss the acceleration phase and the future singularities in this model, and in the last section (Sec. V) we summarize our conclusions. II. MODEL AND CALCULATIONS We consider the Friedamnn-Roberson-Walker metric in the flat space geometry (k=0) ds 2 = −dt 2 + a(t) 2 (dr 2 + r 2 dΩ 2 ),(4) and assume that the cosmic fluid possesses a bulk viscosity ζ. The energy-momentum tensor can be written as T µν = ρU µ U ν + (p − ζθ)H µν ,(5) where in comoving coordinates U µ = (1, 0), θ = U µ ;µ = 3ȧ/a, and H µν = g µν + U µ U ν [11]. By defining the effective pressure asp = p−ζθ and from the Einstein equation R µν − 1 2 g µν R = 8πGT µν , we obtain the Friedmann equationsȧ 2 a 2 = 8πG 3 ρ, (6a) a a = − 4πG 3 (ρ + 3p). (6b) The conservation equation for energy, T 0ν ;ν , yieldṡ ρ + (ρ +p)θ = 0. Using the equation of state to eliminate ρ and p, we obtain the equation which determines the scale factor a(t)ä a = − 3γ − 2 2ȧ 2 a 2 + 12πGζ 0ȧ a − 4πGp 0 ,(8) where the effective equation of state parameter is shifted from the original one as γ = γ − 8πGζ 1 .(9) So we can see that the equivalent effect of the second term in ζ is to change the parameter γ toγ in the equation of state. As shown in Ref. [5], the barrier ω = −1 between the quintessence region (ω > −1) and the phantom region (ω < −1) can be crossed, as a consequence of the bulk viscosity available. Since the dimension of the two terms 12πGζ 0 and −4πGp 0 is [time] −1 and [time] −2 , respectively, we define 12πGζ 0 = 1 T 1 ,(10)−4πGp 0 = 1 T 2 2 ,(11) then Eq. (8) becomes a a = − 3γ − 2 2ȧ 2 a 2 + 1 T 1ȧ a + 1 T 2 2 .(12) Here T 1 and T 2 are criteria to determine whether we should concern the ζ and p 0 . If T 1 >> t, thecosmictimescale, the effect of ζ can be neglected, and if T 2 >> t, the effect of p 0 can be neglected likewise. Concerning the initial conditions of a(t 0 ) = a 0 and θ(t 0 ) = θ 0 , ifγ = 0, the solution can be obtained as a(t) = a 0 1 2 1 +γθ 0 T − T T 1 exp t − t 0 2 1 T + 1 T 1 + 1 2 1 −γθ 0 T + T T 1 exp − t − t 0 2 1 T − 1 T 1 2/3γ . (13) And we obtain directlyȧ a = 1 3γ (1 +γθ 0 T − T T1 )( 1 T + 1 T1 )exp( t−t0 T ) − (1 −γθ 0 T + T T1 )( 1 T − 1 T1 ) (1 +γθ 0 T − T T1 )exp( t−t0 T ) + (1 −γθ 0 T + T T1 ) .(14) Here we define T = T 1 1 + 6γ(T 1 /T 2 ) 2 .(15) We can see that when T 2 → ∞, T = T 1 ; when T 1 → ∞, T = T 2 / √ 6γ. III.γ = 0 AND SPECIAL CASES Forγ = 0, we should use the mathematical L'Hospital's rule to calculate the limit of Eq. (13) rigously and note lim γ→0 dT dγ = − 3T 3 1 T 2 2 .(16) The limit of solution a(t) whenγ → 0 is a(t) = a 0 exp[ 1 3 θ 0 T 1 + T 2 1 T 2 2 e (t−t0)/T1 − 1 − T 1 (t − t 0 )T 2 gives the same result. So Eq. (13) is consistent forγ crossing zero. The T 2 → ∞ limit of Eq. (17) is a(t) = a 0 exp 1 3 θ 0 T 1 e (t−t0)/T1 − 1 .(19) and the T 1 → ∞ limit of Eq. (17) is a(t) = a 0 exp 1 3 θ 0 (t − t 0 ) + (t − t 0 ) 2 2T 2 2 .(20) These two special limits are also consistent with directly solving Eq. (12), by checking. Let us discuss two special cases in the following. When the constant term in the equation of state is not concerned, i.e. T 2 → ∞, a(t) = a 0 1 + 1 2γ θ 0 T e (t−t0)/T + 1 2/3γ .(21) and when the constant term in the bulk viscosity is not concerned, i.e. T 1 → ∞, a(t) = a 0 cosh t − t 0 2T +γθ 0 T sinh t − t 0 2T 2/3γ .(22) When T → ∞, the two cases become a(t) = a 0 1 + 1 2γ θ 0 (t − t 0 ) 2/3γ .(23) Forγ → 0, the limit case is a(t) = a 0 e θ0(t−t0)/3 ,(24) which corresponds to the de Sitter universe with accelerating cosmic expansion. Additional notions: Eqs. (6a) and (6b) can be rewritten as H 2 = 8πG 3 ρ,(25a)H = −4πG(p + ρ).(25b) From these equations, the relation among viscosity, the scalar factor a and Hubble parameter H is aH dH da = − 3γ 2 H 2 + 12πGζH − 6πGp 0 .(26) which reflects the viscosity functions for dark energy and matter dominated universe evolution. IV. ACCELERATION AND BIG RIP If the universe accelerates, then mathematicallÿ a a > 0. From Eq. (8), we can qualitatively see that the bulk viscosity and a negative density p 0 can cause the universe to accelerate. Since the expression ofä/a is too complicated in this situation, now we only discuss a special case, with p 0 = 0. Hereä/a > 0 yields ω = γ − 1 > 2 3 e (t−t0)/T − 1 + 2 θ 0 T + 8πGζ 1 .(28) As we know, if the bulk viscosity is zero as in the standard FriedammnRobertson-Walker cosmology model, an accelerating expansion universe corresponds to ω < −1/3. Inequality (28) tells us that if the bulk viscosity is large enough, the universe expansion can accelerate even if ω > −1/3. According to [9], the future singularities can be classified in the following way: • Type I ("Big Rip"): For t → t s , a → ∞, ρ → ∞ and \|p\| → ∞ • Type II ("sudden"): For t → t s , a → a s , ρ → ρ s and \|p\| → ∞ • Type III: For t → t s , a → a s , ρ → ∞ and \|p\| → ∞ • Type IV: For t → t s , a → a s , ρ → 0, \|p\| → 0 and higher derivatives of H diverge. In this paper, ρ → ∞ means p → ∞ (we assume γ = 1 generally). In the following we show that different choices of the parameters may lead to three fates of the universe evolution: no future singularity, big rip, or the Type III singularity. A.γ < 0 From Eq. (6a), we see √ ρ ∝ȧ a .(29) If the denominator of Eq. (14) is zero, 1 +γθ 0 T − T T 1 exp t − t 0 T + 1 −γθ 0 T + T T 1 = 0. (30) then a → ∞, ρ → ∞, so the big rip occurs. The solution for t is t s = T ln − 1 −γθ 0 T + T T1 1 +γθ 0 T − T T1 + t 0(31) If we want to prevent the big rip, there should be no real solution for t > t 0 , so − 1 −γθ 0 T + T T1 1 +γθ 0 T − T T1 < 1.(32) The inequality is equivalent to 1 +γθ 0 T − T T 1 > 0.(33) The above inequality can be satisfied in some conditions for the phantom energy, so the big rip will not occur. Furthermore, we can see that even if the dark energy is in the quintessence region, there also can be future singularity. For example, ifγ > 0 and p 0 < 0, it is possible that inequality (33) is not satisfied, so the future singularity may occur, which will be discussed in the next subsection below. The more explicit form of inequality (33) is 1 +γ θ 0 T 1 1 + 6γ(T 1 /T 2 ) 2 − 1 1 + 6γ(T 1 /T 2 ) 2 > 0. (34) From this inequality, we obtain that (i) p 0 < 0, i.e. T 2 2 > 0: inequality (33) is always unsatisfied, so there will be a big rip at time t s . (ii) p 0 > 0, i.e. T 2 2 < 0: inequality (33) is not always unsatisfied. If it is unsatisfied, there will be a big rip at time t s ; if it is satisfied, there is no future singularity. B.γ > 0 If the denominator of inequality (14) is zero, then a → 0, ρ → ∞, so the Type III singularity occurs. Following the same steps as before, we obtain that (i) p 0 < 0, i.e. T 2 2 > 0: inequality (33) is always satisfied, so there is no future singularity. (ii) p 0 > 0, i.e. T 2 2 < 0: inequality (33) is not always satisfied. If it is satisfied, there is no future singularity; if it is unsatisfied, there will be Type III singularity at time t s . For the caseγ = 0, a a = 1 3 θ 0 e (t−t0)/T1 + T 1 T 2 2 e (t−t0)/T1 − 1 .(35) So there is no future singularity in this case. To illustrate the parameters in the general solution Eq (13) more clearly, we draw some graphics in Fig. 1-4. The initial condition is [11] t 0 = 1000s, θ 0 = 1.5 × 10 −3 s −1 . At this time, the bulk viscosity ζ = 7.0 × 10 −3 g/cm · s, the corresponding T 1 = 5.1 × 10 28 s. We assume ζ 1 = 0 for simplicity. V. CONCLUSION In conclusion, we have solved the Friedmann equations with both a more general equation of state and bulk viscosity, and discussed the acceleration expansion of the universe evolution and the future singularities for this model. Compared with the standard model of cosmology, this model has had three additional parameters, ζ 0 , ζ 1 and p 0 : choices of ζ 0 and negative p 0 can cause the universe accelerate; ζ 1 can drive the cosmic fluid from the quintessence region to the phantom one [5], and positive p 0 may both prevent the big rip for phantom phase and lead to the Type III singularity of Ref. [9] for the quintessence phase. The relation between the choices of parameters and the future singularities of the cosmological evolution in this extended model is summarized as in the following table and we expect more detail investigations on viscosity effects to be carried out. FIG. 1 :FIG. 2 : 12p0 = 0, γ = 1, the dash line corresponds to T1 = 5.1×10 28 s, and another curve corresponds to T1 = 5.1×10 17 s. ζ = 0, p0 = 0, the dash line corresponds to γ = 0.181, another curve corresponds to γ = 0.18 case.C.γ = 0 FIG. 4 : ζ = 0 , 40ζ = 0, γ = 1, the dash line corresponds to T2 = 5.1 × 10 16 is, while another curve corresponds to T2 = 5.1 × 10 17 s case. T2 = 5.1 × 10 16 is, γ = 0.1. * Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: hu˙[email protected] ParametersFuture singularity (at t → t s )We thank Prof. I. Brevik for reading the manuscript with helpful comments and Profs. S.D.Odintsov and Lewis H.Ryder for lots of interesting discussions. 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[ "Lagrangian structure for two dimensional non-barotropic compressible fluids ", "Lagrangian structure for two dimensional non-barotropic compressible fluids " ]	[ "Pedro Nel \nSchool of Mathematics and Statistics\nIMECC-Institute of Mathematics, Statistics and Scientific Computing UNICAMP-Universidade Estadual de Campinas. Rua Sérgio Buarque de Holanda\nUPTC-Universidad Pedagógica y Tecnológica de Colombia Avenida Central del Norte\n651. 13083-859 Campinas39-115. 150003Tunja, BoyacáSPColombia, Brazil\n", "Maluendas Pardo [email protected] \nSchool of Mathematics and Statistics\nIMECC-Institute of Mathematics, Statistics and Scientific Computing UNICAMP-Universidade Estadual de Campinas. Rua Sérgio Buarque de Holanda\nUPTC-Universidad Pedagógica y Tecnológica de Colombia Avenida Central del Norte\n651. 13083-859 Campinas39-115. 150003Tunja, BoyacáSPColombia, Brazil\n", "Marcelo M Santos [email protected] \nSchool of Mathematics and Statistics\nIMECC-Institute of Mathematics, Statistics and Scientific Computing UNICAMP-Universidade Estadual de Campinas. Rua Sérgio Buarque de Holanda\nUPTC-Universidad Pedagógica y Tecnológica de Colombia Avenida Central del Norte\n651. 13083-859 Campinas39-115. 150003Tunja, BoyacáSPColombia, Brazil\n" ]	[ "School of Mathematics and Statistics\nIMECC-Institute of Mathematics, Statistics and Scientific Computing UNICAMP-Universidade Estadual de Campinas. Rua Sérgio Buarque de Holanda\nUPTC-Universidad Pedagógica y Tecnológica de Colombia Avenida Central del Norte\n651. 13083-859 Campinas39-115. 150003Tunja, BoyacáSPColombia, Brazil", "School of Mathematics and Statistics\nIMECC-Institute of Mathematics, Statistics and Scientific Computing UNICAMP-Universidade Estadual de Campinas. Rua Sérgio Buarque de Holanda\nUPTC-Universidad Pedagógica y Tecnológica de Colombia Avenida Central del Norte\n651. 13083-859 Campinas39-115. 150003Tunja, BoyacáSPColombia, Brazil", "School of Mathematics and Statistics\nIMECC-Institute of Mathematics, Statistics and Scientific Computing UNICAMP-Universidade Estadual de Campinas. Rua Sérgio Buarque de Holanda\nUPTC-Universidad Pedagógica y Tecnológica de Colombia Avenida Central del Norte\n651. 13083-859 Campinas39-115. 150003Tunja, BoyacáSPColombia, Brazil" ]	[]	We study the lagrangian structure for weak solutions of two dimensional Navier-Stokes equations for a non-barotropic compressible fluid, i.e. we show the uniqueness of particle trajectories for two dimensional compressible fluids including the energy equation. Our result extends partially the previous result obtained for barotropic fluids by D. Hoff and M. M. Santos[10].	10.1016/j.jmaa.2018.12.081	[ "https://arxiv.org/pdf/1810.11182v1.pdf" ]	119,736,027	1810.11182	3c9adb0ec121485c4f507ad5eacaa604ecbfb9f0	Lagrangian structure for two dimensional non-barotropic compressible fluids * 26 Oct 2018 Pedro Nel School of Mathematics and Statistics IMECC-Institute of Mathematics, Statistics and Scientific Computing UNICAMP-Universidade Estadual de Campinas. Rua Sérgio Buarque de Holanda UPTC-Universidad Pedagógica y Tecnológica de Colombia Avenida Central del Norte 651. 13083-859 Campinas39-115. 150003Tunja, BoyacáSPColombia, Brazil Maluendas Pardo [email protected] School of Mathematics and Statistics IMECC-Institute of Mathematics, Statistics and Scientific Computing UNICAMP-Universidade Estadual de Campinas. Rua Sérgio Buarque de Holanda UPTC-Universidad Pedagógica y Tecnológica de Colombia Avenida Central del Norte 651. 13083-859 Campinas39-115. 150003Tunja, BoyacáSPColombia, Brazil Marcelo M Santos [email protected] School of Mathematics and Statistics IMECC-Institute of Mathematics, Statistics and Scientific Computing UNICAMP-Universidade Estadual de Campinas. Rua Sérgio Buarque de Holanda UPTC-Universidad Pedagógica y Tecnológica de Colombia Avenida Central del Norte 651. 13083-859 Campinas39-115. 150003Tunja, BoyacáSPColombia, Brazil Lagrangian structure for two dimensional non-barotropic compressible fluids * 26 Oct 2018Lagrangian structurelog-lipschitzian vector fieldscompress- ible fluidnon-barotropic * Financial support by Fapespgrant 2009/15515-0 We study the lagrangian structure for weak solutions of two dimensional Navier-Stokes equations for a non-barotropic compressible fluid, i.e. we show the uniqueness of particle trajectories for two dimensional compressible fluids including the energy equation. Our result extends partially the previous result obtained for barotropic fluids by D. Hoff and M. M. Santos[10]. Introduction We consider the model equations for the non-barotropic (and polytropic) compressible fluids ρ t + div (ρu) = 0 (1) (ρu) t + div (ρu ⊗ u) + ∇P = µ△u + λ∇ (div u) (2) (ρe) t + div (ρeu) = K△e − P div u + µ \|∇u\| 2 + u k x j u j x k + (λ − µ) (div u) 2 ,(3) with t > 0 and x = (x 1 , x 2 ) ∈ R 2 , where repeated indexes mean summation from 1 to 2, and ρ, u = (u 1 , u 2 ), e, P and µ, λ > 0 (constants) denote, respectively, the density, velocity, specific internal energy, pressure and viscosities of the fluid. We assume that the fluid is ideal, i.e. P (ρ, e) = (γ − 1)ρe, where γ > 1 is a constant (the adiabatic constant). The symbol K denotes some positive constant related to the heat flow. These three equations describe, respectively, the conservation of mass, the conservation of momentum and the balance of energy (see e.g. [4], [2] or [1]). In terms of the convective derivative˙:= ∂ t +u·∇, assuming that (ρ, u, e) is sufficiently regular and using the equation (1) in the the equations (2) and (3), we can write the system (1)-(3) aṡ ρ = −ρdiv u (4) ρu = −∇P + µ△u + λ∇(div u) (5) ρė = K△e − P div u + µ \|∇u\| 2 + u k x j u j x k + (λ − µ) (div u) 2 . To the system (1)-(3) we add the initial conditions ρ(0, ·) = ρ 0 , u(0, ·) = u 0 , e(0, ·) = e 0 , which can be discontinuous functions. We shall assume that u 0 belongs to the Sobolev space H 1 (R 2 ), and, for constantsρ,ẽ, and l > 0, that the initial "energy" C 0 := ρ 0 −ρ 2 L ∞ (R 2 ) + u 0 2 H 1 (R 2 ) + (ρ 0 −ρ) 2 + \|u 0 \| 2 + \|e 0 −ẽ\| 2 + \|∇e 0 \| 2 (1 + \|x\| 2 ) l dx,(8) is sufficiently small. Our goal in this paper is to show the uniqueness of particle paths (trajectories of the velocity field u) of a weak solution (ρ, u, e) of the system (1)- (3) together with initial condition (7), following the plan of [10] for barotropic fluids. Briefly, a key idea in [10] is to write u = u P + u F,ω , where u P is a vector field associated with the pressure P and u F,ω is associated with the vorticity ω and the so called effective viscous flux, i.e. the quantity F := (µ + λ)divu − (P −P ), whereP := P (ρ). By energy estimates, some properties of regularity of the solution and some estimates on the convective derivative of u and using classical arguments of elliptic equations, it is shown that u P is a log-lipschitzian vector field in space, for each positive time. In addition, the log-lipschitzian seminorm of u P is locally bounded with respect to time. On the other hand, by classical Sobolev estimates and also by some estimates on the convective derivative of u, it is possible to show that the vector field u F,ω is lipschitzian in space, also for each positive time. Then, assuming that the initial velocity is in the Sobolev space H s (R 2 ), for some arbitrary s > 0, it is shown that the lipschitzian seminorm of u F,ω is locally integrable with respect to time. This is perhaps the most difficult part. Putting together the results for u P and u F,ω , one has that the log-lipschitzian seminorm of the velocity field u is locally integrable with respect to time. Therefore, the uniqueness of particle paths follows from Osgood's lemma. We shall show that this procedure is applicable to the nonbarotropic case (1)-(3), under the hypothesis that the initial velocity u 0 is in the Sobolev space H 1 with sufficiently small norm u 0 H 1 . The nonlinearity P (ρ, e) turns the problem very difficult. We shall use the solution to the system (1)-(3) obtained by [8]. More precisely, in this paper we show that under the above additional hypothesis ( u 0 H 1 (R 2 ) << 1), a similar theorem to Theorem 2.5 of [10] holds true for the equations (1)-(3), i.e. we shall prove the following result: Theorem 1. Let (ρ, u, e) be a weak solution of the system (1)-(3) and initial conditions (7), as in [8,Theorem 1.1]. There is a positive number ε such that if E 0 < ε then 1. for each x 0 ∈ R 2 there exits a unique map X(·, x 0 ) ∈ C([0, ∞]; R 2 ) ∩ C 1 ((0, ∞); R 2 ) satisfying X(t, x 0 ) = x 0 + t 0 u(X(τ, x 0 ), τ )dτ, ∀t ≥ 0;(9) 2. for each t > 0, the flux map x ∈ R 2 → X(t, x) ∈ R 2 is a homeomorphism; 3. for each compact set K in R 2 and any 0 ≤ t 1 < t 2 < ∞, the map X(t 1 , x) → X(t 2 , x), x ∈ K, is bijective and Hölder continuous; 4. for each t > 0, the map x ∈ R 2 → X(t, x) takes Hölder continuous curves into Hölder continuous curves, i.e. if C is a curve of class C α in R 2 , for some α ∈ [0, 1) then X(t, C) is a curve of classe C αe −Lt , where L is a positive constant depending on ρ and s. To the best of our knowledge, up to now only a few results are established regarding the lagrangian structure for compressible fluids. For barotropic fluids we can mention the following papers which prove the lagrangian structure: [9], in dimension two with the initial velocity in the Sobolev space H s , for an arbitrary s > 0, and with a piecewise Hölder continuous initial density across a single C 1,α curve; [10], in dimension two and three, with the initial velocity in H s , with s > 0 in dimension two and s > 1/2 in dimension three; [13], in dimension two, with a viscosity coefficient depending on the fluid density and the initial velocity in H 1 (R 2 ); [6], for spherically symmetric fluids, in dimension two and three, with both viscosity coefficients depending on the density; [12], in the half-space in dimension three with the Navier boundary condition and the initial velocity in H 1 . For the non-baratropic fluids, i.e. when the pressure depends also on the energy (in addition to the dependency on the density) extra difficulties appear to obtain the need estimates to show the lagrangian structure. It is necessary to conveniently extend some estimates presented in [8]. We use techniques showed in [10] and [11] to get estimates in L 2 spaces for the material derivatives of speed and internal energy. For convenience, let us state the mains properties we shall use here of the solution to the system (1)-(3) obtained by [8]: Theorem 2. [8] . Let C 0 be the quantitity defined in (8) but without the norm u 0 H 1 , and assume that λ < (1+ √ 2)µ. Let positive constants ρ >ρ > ρ > 0 andẽ > e 1 > e > 0 be given. Then there are positive constants C, ǫ such that if the initial data (ρ 0 , e 0 , u 0 ) satisfies C 0 ≤ ǫ and essinf e 0 ≥ e 1 , then the initial value problem (1)-(3), (7), has a global weak solution (ρ, e, u) with the following properties: ρ −ρ ∈ C([0, ∞); H −1 (R 2 )), ρ(·, t) −ρ ∈ (L 2 ∩ L ∞ )(R 2 )) t ≥ 0, u, e −ẽ ∈ C((0, ∞); L 2 (R 2 )), ρ ∈ [ρ, ρ] a.e., e(·, t) ≥ e a.e., u(·, t), F (·, t), ω(·, t), e(·, t) −ẽ ∈ H 1 (R 2 ), t > 0 (10) where F (as already said in the Introduction) denotes the so called effective viscous flux, i.e. the quantity F := (µ+λ)div u−(P −P ), beingP := P (ρ,ẽ), and ω denotes the vorticity matrix (i.e. ω = (ω i,j ), ω i,j = u i x j − u j x i ); sup 0<τ <t [(ρ −ρ) 2 + \|u\| 2 + (e −ẽ) 2 ] (x, τ )W dx + t 0 [\|∇u\| 2 + \|∇e\| 2 ] (x, τ )W dxdτ ≤ CC 0 (11) for any t > 0, where W ≡ W (x, τ ) := (1+\|x\| 2 ) l if τ ≤ 1 and W:=1 elsewhere; sup 0<t<1 t \|∇u\| 2 dx + 1 0 t\|u\| 2 dxdt ≤ CC 0 ; (12) sup 0<t<1 t 2 \|u\| 2 dx + 1 0 t 2 \|∇u\| 2 dxdt ≤ CC 0 .(13) In addition, for global positive constants θ, q, e(·, t) −ẽ L ∞ (R 2 ) ≤ CC θ 0 t −q , 0 < t ≤ 1; (14) u α,α/(2+2α) R 2 ×[t,∞) , e α,α/(2+2α) R 2 ×[t,∞) ≤ CC θ 0 t −q , 0 < t ≤ 1,(15) where C may depend additionally on t and α, and · α,β denotes the Hölder semi-norm with exponent α in the x variable and exponent β in the t variable; Furthermore, the solution (ρ, u, e) is the limit of smooth approximate solutions (ρ δ , u δ , e δ ), δ → 0, which satisfy the estimates (11)-(15) with the constants C, θ, q on the right hand side of these estimates independent of δ. Assuming the initial velocity u 0 in H 1 , the estimates (12), (13) can be improved such that the powers in t decrease by one, and we have also a similar estimate to (12) for the internal energy e, i.e. we have the following result which we prove in Section 3: Theorem 3. Under the same hypothesis and notations in Theorem 2, if the initial velocity u 0 is in the Sobolev space H 1 (R 2 ) and C 0 ≤ ǫ (possibly with a smaller ǫ than that in Theorem 2, and with C 0 , defined in (8), including now the norm u 0 H 1 (R 2 ) ), then we have the following estimates on the approximated solutions (ρ δ , u δ , e δ ) stated in Theorem 2, with the constant C as above (in particular, independent of δ): sup 0<t<1 \|∇u δ \| 2 dx + 1 0 \|u δ \| 2 dxdt ≤ CC θ 0 ; (16) sup 0<t<1 t \|u δ \| 2 dx + 1 0 t\|∇u δ \| 2 dxdt ≤ CC θ 0 ; (17) sup 0<t<1 \|∇e δ \| 2 dx + 1 0 \|ė δ \| 2 dxdt ≤ CC θ 0 .(18) The estimates (16) and (17) imply the lagrangian structure (uniqueness of particle paths) in initial time t = 0, as we show in Section 4, following [10]. The estimates (16), (17) were obtained in [9] in the case that the pressure is a function of the density only (more precisely, of the form P (ρ) = Aρ γ , for constants A > 0 and γ > 1). Here, since the pressure depends also on the energy, we have extra difficulties to obtain them. For instance, in our arguments (see Section 3) we need to use (18) to obtain (16), (17), i.e. due to the pressure term the three estimates (16)-(18) are entailed to each other. The remainder of this paper is organized as follows: in Section 2 we collect some facts we shall use in the next Sections 3, 4. In Section 3 we prove the estimates (16)-(18) and in Section 4 we prove Theorem 1. Preliminaries Throughout this paper we shall use some classical estimates which we recall for the convenience of the reader. The Morrey's inequality for a function f in the Sobolev space W 1,p (R 2 ) with p > 2 is f α ≤ C ∇f L p (R 2 )(19) where α = 1 − 2 p , · α denotes the Hölder semi-norm and C is a constant depending only on p. As a consequence, we have the estimate f L ∞ (R 2 ) ≤ C ∇f L p (R 2 ) + f L 2 (R 2 ) ,(20) which can be obtained from (19) by writing f (x) = − B 1 (0) \|x − y\| α ((f (x) − f (y)/\|x − y\| α ))dy + − B 1 (0) \|f (y)\|dy and properly estimating these integrals. We shall use several times the interpolation inequality f p L p (R 2 ) ≤ C f 2 L 2 (R 2 ) ∇f p−2 L 2 (R 2 ) .(21) A very useful expedient introduced by Hoff (see e.g. [7]) is to write the momentum equation (see (5)) as ρu = ∇F + µdivω (22) i.e. ρu j = F x j + µω j,k x k , j = 1, 2, where F is the effective viscous flux, i.e. the quantity F := (µ + λ)div u − (P −P ) (mentioned earlier),P := P (ρ,ẽ), and ω ≡ (ω j,k ), j, k = 1, 2, is the vorticity matrix, i.e. ω j,k = u j x j − u k x j . Indeed, applying the div and the curl operators to (22) we obtain ∆F = div(ρu), µ∆ω = curl(ρu)(23) where the last equation means µ∆ω j,k = ρu j x k − ρu k x j , j, k = 1, 2. Then by elliptic theory, given any p ∈ (1, ∞), there is a constant C such that ∇F (·, t) L p (R 2 ) , ω j,k (·, t) L p (R 2 ) ≤ C ρu(·, t) L p (R 2 ) , t > 0.(24) On the other hand, from the identity ∆u j = (λ + µ) −1 F x j + ω j,k x k + (λ + µ) −1 (P −P ) x j it follows that ∇u(·, t) L p (R 2 ) ≤ C( F (·, t) L p (R 2 ) + ω(·, t) L p (R 2 ) + (P −P )(·, t) L p (R 2 ) ), t > 0. (25) Furthermore, writing (λ + µ) −1 F x j + ω j,k x k = ∆u j F,ω and (λ + µ) −1 (P −P ) x j = ∆u j P , we have u = u F,ω + u P , with u F,ω satisfying the estimate D 2 u F,ω (·, t) L p (R 2 ) ≤ C ρu(·, t) L p (R 2 ) , t > 0,(26) in virtue of (24). The inequalities (24)-(26) will be used in the next sections. Regarding the part u P we have the following (leading to (29) below): Let us denote by LL the the space of log-lipschitzian functions in R 2 , i.e., the space of functions (or vector functions) f defined in R 2 such that the norm f LL := f LL + f L ∞ (R 2 ) is finite, where · LL denotes log-Lipschitzian seminorm defined by f LL := sup 0<\|x−y\|≤1 \|f (x) − f (y)\| m(\|x − y\|) being m(r) := r(1 − log r), if 0 < r ≤ 1 r, if r > 1. Then we have the following result: Lemma 1. Let Γ denote the fundamental solution of the laplacian in R 2 . 1. If 1 ≤ p 1 < 2 < p 2 ≤ ∞ and f ∈ L p 1 (R 2 ) ∩ L p 2 (R 2 ) then the vector field f * ∇Γ, where * denotes the standard convolution in R 2 (i.e. (f * ∇Γ)(x) = R 2 f (x − y)∇Γ(y)dy, x ∈ R 2 ) belongs to L ∞ (R 2 ) and f * ∇Γ L ∞ (R 2 ) ≤ C( f L p 1 (R 2 ) + f L p 2 (R 2 ) ),(27) where C is a constant depending only on p 1 and p 2 . 2. If 1 ≤ p < 2 e f ∈ L p (R 2 ) ∩ L ∞ (R 2 ) then f * ∇Γ ∈ LL and f * ∇Γ LL ≤ C( f L p (R 2 ) + f L ∞ (R 2 ) ),(28) where C is a constant depending only on p. Remark 1. This lemma holds true in R n , with the same proof, replacing the conditions on the p ′ s by p 1 < n < p 2 and p < n. Proof. To the first estimate, separate the integral f * ∇Γ in the ball B 1 (0) and in its complementary. Then, estimate each integral by applying the Hölder's inequality in a convenient way. As for the second estimate, take x, y ∈ R n with ε = \|x − y\| ≤ 1,x = x − y 2 , and separate the integral in (f * Γ x j )(x) − (f * Γ x j )(y), j = 1, 2, ..., n, in the balls B ε (x), B 2 (x) \ B ε (x) and in B 2 (x) c . Then it is possible to estimate the integral in these sets, respectively, by ε f L ∞ (R n ) , ε(ln 3 − ln ε) f L ∞ (R n ) and ε f L p (R n ) , times some constant. As a corollary of Lema 1, given any p ∈ [1, 2), we obtain the following estimate for the second part u P in the decomposition u = u F +ω + u P introduced above: u P (·, t) LL ≤ C( (P −P )(·, t) L p (R 2 ) + (P −P )(·, t) L ∞ (R 2 ) ), t > 0. (29) Indeed, in the above decomposition we can take u P as u P = Γ * ∇(P −P ) = (∇Γ) * (P −P ) = (P −P ) * ∇Γ. Thus, (29) is a consequence of (28), since by (11) we have (P −P )(·, t) ∈ L p (R 2 ), for any p ∈ [2/(1 + l), 2], and (P −)(·, t) ∈ L ∞ (R 2 ) by (14). In fact, we shall need u P (·, t) LL to be locally integrable in time. The estimate (11) gives that (P −P )(·, t) L p (R 2 ) has this property, so by (29), to have that it is enough that (P −P )(·, t) L ∞ (R 2 ) ) to be locally integrable. Fortunately, we have Lemma 2. [8, Lemma 4.4] 1 0 e −ẽ(·, t) L ∞ (R 2 ) dt ≤ CC θ 0 .(30) Then, combining (30) with ρ ≤ ρ ≤ ρ (see (10)), it follows that (P − P )(·, t) L ∞ (R 2 ) is locally integrable. Estimates of convective terms In this Section we show the estimates (16)-(18). For convenience we omit the superscript δ in the approximate solution (ρ δ , u δ , e δ ). We begin by defining the functionals B 0 (t) = sup 0≤τ ≤t \|∇u\| 2 dx + t 0 ρ\|u\| 2 dxdτ, B 1 (t) = sup 0≤τ ≤t \|∇e\| 2 dx + t 0 ρ\|ė\| 2 dxdτ, B(t) = B 0 (t) + B 1 (t). Lemma 3. If t ≤ 1, then B 0 (t) ≤ C(C θ 0 + k>1 B(t) k ) where k>1 is a finite sum over real indexes k > 1. Proof. Multiplying (5) byu j , we have ρ(u j ) 2 = −P x ju j + µ△u j + λ(div u) x j u j t + µ△u j + λ(div u) x j ∇u j · u and integrating by parts and summing on j = 1, 2, we obtain t 0 ρ\|u\| 2 dxdτ + 1 2 (µ\|∇u\| 2 + λ(div u) 2 )dx\| t 0 = + t 0 (P −P )divudx + t 0 µ△u j + λ(div u) x j ∇u j · udxdτ.(31) The term above with the pressure can be written as t 0 (P −P )divudx = t 0 (P −P )∂ t (div u)dxdτ + t 0 (P −P )div ((∇u)u) dxdτ = (P −P )div udx \| t 0 + t 0 (P −P )div (udiv u − (∇u)u) dxdτ + t 0 (P ρ ρ(div u) 2 − P eė div u)dxdτ = (P −P )div udx + t 0 (P −P )div ((∇u)u − udiv u) dxdτ + t 0 (P ρ ρ(div u) 2 − P eė div u)dxdτ (32) and, by the identity div ((∇u)u − udiv u) = (div u) 2 − u k x j u j x k , we have that the modulus of the second term on the last expression above is bounded by t 0 \|P −P \|\|∇u\| 2 dx. In addition, the fourth term in (32) can be written as t 0 P ρ ρ(div u) 2 dxdτ = (γ − 1) t 0 ρe(div u) 2 dxdτ = t 0 P (div u) 2 dxdτ = t 0 (P −P )(div u) 2 dxdτ + t 0 P (div u) 2 dxdτ. Then, since that the last integral in (31) is easily bounded by C \|∇u\| 3 dxdτ , we have that (31) can be bounded as t 0 ρ\|u\| 2 dxdτ + 1 2 µ\|∇u\| 2 + λ(div u) 2 dx ≤ C ∇u 0 L 2 (R 2 ) + (P −P )div udx +C t 0 \|∇u\| 2 dxdτ + C t 0 \|∇u\| 3 dxdτ +C t 0 \|P −P \|\|∇u\| 2 dxdτ − (γ − 1) t 0 ρėdiv udxdτ (33) Next, let us estimate each integral on the right hand side in this inequality The two first integrals in (33) are easily bounded using the energy estimates (11) for u. The third integral is estimated by C t 0 \|∇u\| 3 dxdτ ≤ C t 0 \|∇u\| 4 dxdτ 1/2 t 0 \|∇u\| 2 dxdτ 1/2 ≤ CC 1/2 0 t 0 \|∇u\| 4 dxdτ 1/2 = CC 1/2 0 t 0 ∇u 4 L 4 (R 2 ) dτ 1/2 Using that ρ is bounded from above and below (see (10)) and (11), we have t 0 \|P −P \|\|∇u\| 2 dxdτ ≤ C t 0 \|∇u\| 2 dxdτ + C t 0 \|e −ẽ\|\|∇u\| 2 dxdτ ≤ CC 0 + C t 0 e −ẽ L 2 (R 2 ) ∇u 2 L 4 (R 2 ) dτ ≤ CC 0 + CC 0 t 0 ∇u 2 L 4 (R 2 ) dτ ≤ CC 0 + CC 0 t 1/2 t 0 ∇u 4 L 4 (R 2 ) dτ 1/2 The terms in the two first integrals on the right hand side of (32)are easily bounded using the energy estimates or can be combined with the left hand side. The last integral is bounded by CC 1/2 0 B 1 (t) 1/2 , by Hölder's inequality and the definition of B 1 . Therefore, we just need to estimate the term of the L 4 norm: t 0 \|∇u\| 4 dxdτ ≤ t 0 (F 4 + \|ω\| 4 + \|P −P \| 4 )dxdτ ≤C t 0 F 2 dx \|∇F \| 2 dx + \|ω\| 2 dx \|∇ω\| 2 dx dτ + (CC 0 t + CC 2 0 ) ≤C t 0 ∇u 2 L 2 (R 2 ) + P −P 2 L 2 (R 2 ) ρ 1/2u 2 L 2 (R 2 ) dτ + (CC 0 t + CC 2 0 ) ≤C(B 0 (t) 2 + C 0 B 0 + C 0 + C 2 0 ). Finally, we have B 0 (t) ≤ C ∇u 0 2 L 2 (R 2 ) +CC 0 + CC 1/2 0 (B 0 (t) 2 + C 0 B 0 + C 0 + C 2 0 ) 1/2 + C(B 0 (t) 2 + C 0 B 0 + C 0 + C 2 0 ) + CC 1/2 0 B 1 (t) 1/2 ≤ C(C θ 0 + k>1 B(t) k ) where the sum over k is a finite sum and θ > 0 is a convenient constant. Lemma 4. B 1 (t) ≤ C C θ 0 + k>1 B(t) k , where k>1 is a finite sum over real indexes k > 1. Proof. Analogously to the proof of Lemma 3, from (3) we have B 1 (t) ≤ C ∇e 0 2 L 2 (R 2 ) + C t 0 \|∇e\| 2 \|∇u\|dxdτ + C t 0 (e −ẽ) 4 + \|∇u\| 4 dxdτ As mentioned above, t 0 ∇u 4 L 4 (R 2 ) dτ ≤ B 0 (t) 2 + CC 0 B 0 (t) + CC 0 + CC 2 0 . In addition, t 0 (e −ẽ) 4 dxdτ ≤ t 0 e −ẽ 2 L 2 (R 2 ) ∇e 2 L 2 (R 2 ) dτ ≤ CC 0 . Then, it remains only to estimate the term \|∇e\| 2 \|∇u\|dxdτ . Notice that t 0 \|∇e\| 2 \|∇u\|dxdτ ≤ C t 0 \|∇e\| 8/3 + \|∇u\| 4 dxdτ and, since, t 0 \|∇e\| 8/3 dxdτ ≤ C t 0 ∇e 2 L 2 (R 2 ) D 2 e 2/3 L 2 (R 2 ) ≤ sup 0≤τ ≤t ∇e 2/3 L 2 (R 2 ) t 0 ∇e 2 L 2 (R 2 ) 2/3 t 0 \|D 2 e\| 2 1/3 ≤ CC 2/3 0 B 1 (t) 2/3 t 0 \|D 2 e\| 2 1/3 ≤ CC 2/3 0 B 1 (t) 2/3 t 0 \|ė\| 2 + \|e\| 2 \|∇u\| 2 + \|∇u\| 4 1/3 ≤ CC 0 B 1 (t) 2/3 + CC 2/3 0 B 1 (t) 2/3 t 0 \|ė\| 2 + \|e −ẽ\| 2 \|∇u\| 2 + \|∇u\| 4 1/3 ≤ CC 0 B 1 (t) 2/3 + CC 2/3 0 B 1 (t) 2/3 B 1 (t) + B 0 (t) 2 + CC 0 B 0 (t) + CC 0 + CC 2 0 1/3 , choosing θ > 0 conveniently and applying the Young inequality, we can write B 1 (t) ≤ C C θ 0 + k>1 B(t) k Combining the previous lemmas, we obtain the estimate B(t) ≤ CC θ 0 ,(34) for any 0 ≤ t ≤ 1. We observe that by the above calculations, we have also that the 4-norm \|∇u\| 4 dxdτ is bounded by CC θ 0 . Next, we define a new function B 2 (t) = sup 0≤τ ≤t τ ρ\|u\| 2 dx + t 0 τ \|∇u\| 2 dxdτ, and show Lemma 5. B 2 (t) ≤ CC θ 0 Proof. We begin by applying the operator τu j (∂ t + div (·u)) to the moment equation (5) to get 1 2 ρ ∂ ∂τ τ (u j ) 2 + 1 2 τ ρu · ∇(u j ) 2 = 1 2 ρ(u j ) 2 − τu j P tx j + div P x j u + µτu j △u j t + div △u j u + λτu j (div u) tx j + div (div u) x j u . Calculating the integral in the variable x and t and using (4), we have 1 2 ρτ \|u j \| 2 = 1 2 t 0 ρ\|u j \| 2 dxdτ − t 0 τu j P tx j + div P x j u dxdτ + µ t 0 τu j △u j t + div △u j u dxdτ + λ t 0 τu j (div u) tx j + div (div u) x j u dxdτ Then, summing in j we get four integrals, which we estimate as follows. The first integral is bounded CC θ 0 by (34). As for the second, using (1), we have t 0 τu j P tx j + div P x j u dxdτ = − t 0 τ divuP t dxdτ − t 0 τ P x ju j x k u k dxdτ = − t 0 τ divu(P ρ ρ t + P e e t )dxdτ − t 0 τ P (u j x k x j u k +u j x k u k x j )dxdτ = t 0 τ P divudiv udxdτ − t 0 τ divu(u · ∇P )dxdτ − (γ − 1) t 0 τ ρėdivudxdτ − t 0 τ P (u j x k x j u k +u j x k u k x j )dxdτ =2 t 0 τ P divudiv udxdτ − t 0 τ Pu j x k u k x j dxdτ − (γ − 1) t 0 τ ρdivuėdxdτ In the case of the viscosity terms, we can get t 0 τu j △u j t + div (△u j u) dxdτ = − t 0 τ \|∇u\| 2 dxdτ + O τ \|∇u\|\|∇u\| 2 dxdτ . Similary, t 0 τu j (div u) tx j + div ((div u) x j u) dxdτ = − t 0 τ (divu) 2 dxdτ + O τ \|∇u\|\|∇u\| 2 dx.dτ With all this, we manage to conclude that τ ρ\|u\| 2 dx + 1 0 τ \|∇u\| 2 dxdτ ≤ CC θ 0 + C t 0 τ \|P −P \|\|∇u\|∇u\|dxdτ + C t 0 τ \|∇u\|∇u\| 2 dxdτ + C t 0 τ \|∇u\|\|ė\|dxdτ Terms with ∇u can be absorved on the left side, the L 2 norm ofė for the last integral above was estimated in (34) and the others are estimated as follows: first of all we have that P −P = (γ − 1)(ρ(e −ẽ) +ẽ(ρ −ρ)), so that, by interpolation inequality and the energy estimates, P −P L 4 (R 2 ) ≤ C( ρ −ρ L 4 (R 2 ) + e −ẽ L 4 (R 2 ) ) ≤ C( ρ −ρ L ∞ (R 2 ) ρ −ρ L 2 (R 2 ) + e −ẽ 2 L 2 (R 2 ) ∇e 2 L 2 (R 2 ) ). So, we get t 0 P −P 4 L 4 (R 2 ) ≤ CC θ 0 . Using this, along with the observation before the definition of B 2 , the energy estimates form [8] and (34), we obtain the desired result, i.e. B 2 (t) ≤ CC θ 0 . Lagrangian structure In this section we show Theorem 1. We recall that all the previous estimates were obtained uniformly with respect to the approximate solutions. We shall perform the decomposition u δ = u δ P + u δ F,ω for the approximate solution (u δ , ρ δ , e δ ) and shall write u δ P = u δ P δ , for simplicity. Proof of Theorem 1. The integral curve for the approximate field u δ starting at x 0 ∈ R 2 is given by X δ (x 0 , t) = x 0 + t 0 u δ (X δ (x 0 , τ ), τ )dτ, t ≥ 0.(35) The map t → X δ (t, x 0 ) is Hölder continuous, uniformly with respect to δ. Indeed, for any 0 ≤ t 1 < t 2 , by (20) we have \|X δ (x 0 , t 1 ) − X δ (x 0 , t 2 )\| ≤ t 1 t 2 u δ (·, t) L ∞ (R 2 ) dτ ≤ C t 1 t 2 u δ (·, τ ) L 2 (R 2 ) + ∇u δ (·, τ ) L p (R 2 ) dτ and we can bound the L 2 norm above by CC θ 0 (t 2 − t 1 ) γ , for some γ ∈ (0, 1)(independent of δ) exactly as in [8, (3.9)]. Using (25) and (20), the L p norm above can be bound as ∇u δ L p (R 2 ) ≤ F δ L p (R 2 ) + ω δ L p (R 2 ) + P δ −P L p (R 2 ) ≤ CC θ 0 1 + \|∇u δ \| 2 dx (1−η)/2 ρ δ \|u δ \| 2 dx η/2 + C ρ δ −ρ L p (R 2 ) + C e δ −ẽ L p (R 2 ) ≤ CC θ 0 1 + \|∇u δ \| 2 dx (1−η)/2 ρ δ \|u δ \| 2 dx η/2 + C ρ δ −ρ L 2 (R 2 ) + C e δ −ẽ 1−η L 2 (R 2 ) ∇e δ η L 2 (R 2 ) ≤ CC θ 0 1 + \|∇u δ \| 2 dx (1−η)/2 ρ δ \|u δ \| 2 dx η/2 + CC θ 0 (1 + ∇e δ η L 2 (R 2 ) ). where η = (p − 2)/p. Then t 2 t 1 ∇u δ (·, τ ) L p (R 2 ) dτ ≤ CC θ 0 t 2 t 1 1 + \|∇u δ \| 2 dx (1−η)/2 ρ δ \|u δ \| 2 dx η/2 dτ + CC θ 0 t 2 t 1 (1 + ∇e δ η L 2 (R 2 ) ) ≤ CC θ 0 [(t 2 − t 1 ) + (t 2 − t 1 ) γ 1 ], for some constant γ 1 ∈ (0, 1) (independent of δ). The above bounds show that X δ (x 0 , ·) is Hölder continuous with respect to δ, which guarantees the existence of a sequence X δn (x 0 , ·) converging (as δ n → 0) to a Hölder continuous map X(x 0 , ·), uniformly in compacts in [0, ∞). Then passing to the limit in (35) we get (9). To prove the uniqueness of the map X(·, x 0 ) satisfying (9), we restrict t close to zero, since the uniqueness of a trajectory starting in t > 0 is considerably simpler. We omit some details facilitating the exposure, which can be seen in [10]. Let X 1 (y 1 , ·) and X 2 (y 2 , ·) two integral curves starting at y 1 and y 2 , respectively, when t = 0, i.e. X 1 , X 2 satisfy (9) with X replaced by X 1 , X 2 and x 0 by y 1 , y 2 , respectively. Following [10], we introduce the function g(t) := \|u(X 2 (t, y 2 ), t) − u(X 1 (t, y 1 ), t)\| m(\|X 2 (t, y 2 ) − X 2 (t, y 2 )\|) and prove first the following lemma: Lemma 6. There is a constant C, independent of y 2 ∈ R 2 such that 1 0 g(τ )dτ ≤ C.(36) Proof. By Fatou's lemma, it is enough to prove (36) with u δ in place of u and with associated integral curves X δ 1 , X δ 2 in place of X 1 , X 2 . Recalling the decomposition u δ = u δ F,ω + u δ P , we observe that g δ (t) ≤ u δ P LL + u δ F,ω LL . By (29), (11) and (30), we have 1 0 u δ P LL dt ≤ C.(37) As for u δ F,ω LL , we can estimate ∇u δ F,ω L ∞ (R 2 ) using (20) and then (26). Thus, since the LL-semi norm is bounded by the Lipschitzian semi norm, we obtain u δ F,ω LL ≤ C( ∇u δ F,ω L 2 (R 2 ) + u δ L p (R 2 ) ). Now, ∇u δ F,ω L 2 (R 2 ) ≤ C( F δ L 2 (R 2 ) + ω δ L 2 (R 2 ) ) ≤ C( ∇u δ L 2 (R 2 ) + P δ −P L 2 (R 2 ) ), and P δ −P L 2 (R 2 ) ≤ CC θ 0 , then, using (21) it follows that u δ F,ω LL ≤ C(C θ 0 + ∇u δ L 2 (R 2 ) + u δ 1−η L 2 (R 2 ) ∇u δ η L 2 (R 2 ) ) where η = p−2 p . At this point the estimates (16), (17) are crucial to obtain that 1 0 ρ δuδ 1−η L 2 (R 2 ) ∇u δ η L 2 (R 2 ) dt ≤C( 1 0 t −η dt) 1/2 ( 1 0 ρ δ \|u δ \| 2 dt) 1−η ( t 0 t\|∇u δ \| 2 dt) η < ∞, since we can choose p > 2 sufficiently close to 2. Therefore, this and the above estimates, together with the energy estimate, show that 1 0 u δ F,ω LL dt < ∞. Returning to the integral curves X 1 e X 2 , we have that \|X 2 (t, y 2 ) − X 1 (t, y 1 )\| ≤ \|y 2 − y 1 \| + t 0 g(τ )m(\|X 2 (t, y 2 ) − X 1 (t, y 1 )\|)dτ so, by Osgood's Lemma ( [3], [5]) and by Lemma 6, we obtain that \|X 2 (t, y 2 ) − X 1 (t, y 1 )\| ≤ exp(1 − e − t 0 gdτ )\|y 2 − y 1 \| exp(− t 0 gdτ ) ,(38) which, in particular, implies X 1 = X 2 if y 1 = y 2 . Next, we show the second claim of Theorem 1. First, we show that for fixed t > 0, the map X(t, ·) : x → X(t, x) is injective. Suppose that X(t, y 1 ) = X(t, y 2 ) for some y 1 , y 2 ∈ R 2 . For t ′ ∈ (0, t), writing X(t ′ , y 1 ) − X(t ′ , y 2 ) = X(t ′ , y 1 ) − X(t, y 1 ) + X(t, y 2 ) − X(t ′ , y 2 ) = t t ′ u(X(τ, y 2 ), τ ) − u(X(τ, y 1 ), τ )dτ, we have, as above, that \|X(t ′ , y 1 ) − X(t ′ , y 2 )\| ≤ t t ′ g(τ )m(X(τ, y 1 ) − X(τ, y 2 ))dτ. Therefore, X(t ′ , y 1 ) = X(t ′ , y 2 ) for all 0 < t ′ ≤ t. Then, by the continuity of the maps X(·, y 1 ), X(·, y 2 ), it follows that y 1 = X(0, y 1 ) = X(0, y 2 ) = y 2 . To prove that the map x → X(t, x) is onto, we use the uniqueness of the particle paths. Indeed, for y ∈ R 2 , there is a curve Y (s) = X(s; y, t) with Y (t) = y, s ∈ [0, t]. The curves Y (s) and X(s, Y (0)) satisfy the problem d ds Z(s) = u(Z(s), s) Z(0) = Y (0) thus, Y (s) = X(s, Y (0)) for all s ∈ [0, t]. In particular, y = X(t, Y (0)). To conclude the proof of the claim 2 of Theorem 1, it remains to show that the map x → X(t, x) is open. Let A be an open set in R 2 . We fix a z 1 = X(t, y 1 ) with y 1 ∈ A. We know that for any z ∈ R 2 , there exists a curve Y (s) = X(s; x, t) defined for s ∈ [0, t] and such that Y (t) = z. In fact, Y (s) = z + t s u(X(τ ; z, t), τ )dτ = z + t s u(Y (τ ), τ )dτ (39) As done before, we can see that t ′ 0 u(·, τ ) L ∞ (R 2 ) dτ ≤ CC θ 0 t ′γ 1 and fixing r > 0 such that B r (y 1 ) ⊂ A we can chose t ′ ≤ [(r−r 1 )/(2CC θ 0 )] 1/γ 1 with 0 < r 1 < r to obtain t ′ 0 u(·, τ ) L ∞ (R 2 ) dτ ≤ r − r 1 2 .(40) On the other hand, since d dτ X(τ, y 1 ) = u(X(τ, y 1 ), τ ); X(t, y 1 ) = z 1 we have that This, together with (40) shows that \|y 1 − Y (0)\| ≤ r, which assures that Y (0) ∈ B r (y 1 ) when z ∈ B a (z 1 ). Finally, by the the uniqueness of the trajectories, we have that z = Y (t) = X(t, Y (0)), so B a (z 1 ) ⊂ X(t, A), as wanted. Proof of claim 3 of Theorem 1: The proof of 3 is also an application of the Osgood's inequality. Let y 1 , y 2 ∈ R 2 and consider the functioñ g(τ ) = \|u(X(τ, y 2 ), τ ) − u(X(τ, y 1 ), τ )\| m(\|X(τ, y 2 ) − X(τ, y 1 )\|) As in Lemma 6, we can show that t 0g (τ )dτ ≤ Ct γ . On the other hand, for t ∈ [t 1 , t 2 ], we can write X(t, y 2 )−X(t, y 1 ) = X(t 1 , y 2 )−X(t 1 , y 1 )+ t t 1 u(X(τ, y 2 ), τ )−u(X(τ, y 1 ), τ )dτ. Thus, \|X(t, y 2 ) − X(t, y 1 )\| ≤ \|X(t 1 , y 2 ) − X(t 1 , y 1 )\| + t t 1g (τ )m(\|X(τ, y 2 ) − X(τ, y 1 )\|)dτ. Then \|X(t, y 2 ) − X(t, y 1 )\| ≤ exp 1 − e − t t 1g (τ )dτ \|X(t 1 , y 2 ) − X(t 1 , y 1 )\| e − t t 1g (τ )dτ ≤ C(t)\|X(t 1 , y 2 ) − X(t 1 , y 1 )\| e −Lt γ , (42) for all t ∈ [t 1 , t 2 ]. In particular, for t = t 2 , as desired. Proof of claim 4 of Theorem 1: Let the curve C be parametrized by a Hölder continuous ϕ with exponent α. Defining ψ t (s) = X(t, ϕ(s)) and using (42), we obtain \|ψ t (s 2 ) − ψ t (s 1 )\| ≤ C(t)\|ϕ(s 2 ) − ϕ(s 1 )\| e −Lt γ ≤ C(t)\|s 2 − s 1 \| αe −Lt γ . This proves the claim 4. , substracting (39) of (41)\|X(s, y 1 ) − Y (s)\| ≤ \|z 1 − z\| + t s \|u(Y (τ ), τ ) − u(X(τ, y 1 ), τ )\|dτ ≤ \|z 1 − z\| + t s g(τ )\|m(Y (τ ) − X(τ, y 1 ))\|dτand using again the Osgood's inequality\|X(s, y 1 ) − Y (s)\| ≤ exp 1 − e − t s g(τ )dτ \|z 1 − z\| e t s g(τ )dτ for all s ∈ [t ′ , t). If we chose a ≤ r 1 exp 1 − e − t t ′ g(τ )dτ 1/γtwhere γ t is a suitable constant depending possibly of t, we get\|X(t ′ , y 1 ) − Y (t ′ )\| ≤ r 1 . Modern compressible flow with historical perspective. D Jhon, Anderson, 3tf ed. Mc Graw HillJhon D. Anderson. Modern compressible flow with historical perspective, 3tf ed. Mc Graw Hill, 2003. An Introduction to Fluids Dynamics. George Keith Batchelor, Cambridge University PressGeorge Keith Batchelor. An Introduction to Fluids Dynamics. Cam- bridge University Press, 1967. Jean Yves Chemin, Perfect Incompressible Fluids. Oxford University PressJean Yves Chemin. Perfect Incompressible Fluids. Oxford University Press, 1998. Dynamics of Viscous Compressible Fluids. Oxford University PressEduard Feireisl. Dynamics of Viscous Compressible Fluids. Oxford University Press, 2004. Differential analysis: differentiation, differential equations and differential inequalities. T M Flett, Cambridge University PressT. M. Flett. Differential analysis: differentiation, differential equations and differential inequalities. Cambridge University Press, 1980. Lagrange structure and dynamics for solutions to the spherically symmetric compressible navier-stokes equations. Zhenhua Guo, Hai-Liang Li, Zhouping Xin, Comm. Math. Phys. 3092Zhenhua Guo, Hai-Liang Li, and Zhouping Xin. Lagrange structure and dynamics for solutions to the spherically symmetric compressible navier-stokes equations. Comm. Math. Phys., 309 (2):371-412, 2012. Global solutions of the navier-stokes equations for multidimensional compressible flow with discontinuous initial data. David Hoff, Journal of Differential Equations. 120David Hoff. Global solutions of the navier-stokes equations for multidi- mensional compressible flow with discontinuous initial data. Journal of Differential Equations, 120:215-254, 1995. Discontinuous solutions of the navier-stokes equations for a multidimensional flows of heat-conducting fluids. David Hoff, Arch. Rational Mech. Anal. 139David Hoff. Discontinuous solutions of the navier-stokes equations for a multidimensional flows of heat-conducting fluids. Arch. Rational Mech. Anal, 139:303-354, 1997. Dynamics of singularity surfaces for compressible, viscous flows in two space dimensions. David Hoff, Communicatuions on Pure an Aplied Mathematics. 55David Hoff. Dynamics of singularity surfaces for compressible, viscous flows in two space dimensions. Communicatuions on Pure an Aplied Mathematics, 55:1365-1407, 2002. Lagrangean structure and propagation of singularities in multidimensional compressible flows. David Hoff, Marcelo M Santos, Arch. Rational Mech. Anal. 188David Hoff and Marcelo M. Santos. Lagrangean structure and prop- agation of singularities in multidimensional compressible flows. Arch. Rational Mech. Anal, 188:509-543, 2008. Global well-posedness of classical solutions with large oscillations and vacuum to the threedimensional isentropic compressible navier-stokes equations. Xiangdi Huang, Jing Li, Zhouping Xin, Comm. Pure Appl. Math. 654Xiangdi Huang, Jing Li, and Zhouping Xin. Global well-posedness of classical solutions with large oscillations and vacuum to the three- dimensional isentropic compressible navier-stokes equations. Comm. Pure Appl. Math., 65 (4):549-585, 2012. J Edson, Marcelo M Teixeira, Santos, arXiv:1805.00052Lagrangian structure for compressible flow in the half-space with the navier boundary condition. Edson J. Teixeira and Marcelo M. Santos. Lagrangian structure for compressible flow in the half-space with the navier boundary condition. arXiv:1805.00052, 2018. Compressible flows with a densitydependent viscosity coefficient. Ting Zhang, Daoyuan Fang, Siam Journal of Mathematical Analisys. 41Ting Zhang and Daoyuan Fang. Compressible flows with a density- dependent viscosity coefficient. Siam Journal of Mathematical Analisys, 41:2453-2488, 2010.	[]
[ "Stellar diffusion in barred spiral galaxies", "Stellar diffusion in barred spiral galaxies" ]	[ "Maura Brunetti \nGeneva Observatory\nUniversity of Geneva\nCH-1290SauvernySwitzerland\n", "Cristina Chiappini \nLeibniz-Institut für Astrophysik Potsdam (AIP)\nAn der Sternwarte 16D -14482PotsdamGermany\n", "Daniel Pfenniger \nGeneva Observatory\nUniversity of Geneva\nCH-1290SauvernySwitzerland\n" ]	[ "Geneva Observatory\nUniversity of Geneva\nCH-1290SauvernySwitzerland", "Leibniz-Institut für Astrophysik Potsdam (AIP)\nAn der Sternwarte 16D -14482PotsdamGermany", "Geneva Observatory\nUniversity of Geneva\nCH-1290SauvernySwitzerland" ]	[]	We characterize empirically the radial diffusion of stars in the plane of a typical barred disk galaxy by calculating the local spatial diffusion coefficient and diffusion time-scale for bulge-disk-halo N-body self-consistent systems which initially differ in the Safronov-Toomre-Q T parameter. We find different diffusion scenarios that depend on the bar strength and on the degree of instability of the disk. Marginally stable disks, with Q T ∼ 1, have two families of bar orbits with different values of angular momentum and energy, which determine a large diffusion in the corotation region. In hot disks, Q T > 1, stellar diffusion is reduced with respect to the case of marginally stable disks. In cold models, we find that spatial diffusion is not constant in time and strongly depends on the activity of the bar, which can move stars all over the disk recurrently. We conclude that to realistically study the impact of radial migration on the chemical evolution modeling of the Milky Way the role of the bar has to be taken into account.	10.1051/0004-6361/201117566	[ "https://arxiv.org/pdf/1108.5631v1.pdf" ]	53,404,420	1108.5631	026463164b71c742ca791c91a6fcd1a86b05b34c	Stellar diffusion in barred spiral galaxies 29 Aug 2011 April 28, 2013 Maura Brunetti Geneva Observatory University of Geneva CH-1290SauvernySwitzerland Cristina Chiappini Leibniz-Institut für Astrophysik Potsdam (AIP) An der Sternwarte 16D -14482PotsdamGermany Daniel Pfenniger Geneva Observatory University of Geneva CH-1290SauvernySwitzerland Stellar diffusion in barred spiral galaxies 29 Aug 2011 April 28, 2013Astronomy & Astrophysics manuscript no. diffusionGalaxies: kinematics and dynamicsGalaxies: stellar contentGalaxies: spiralGalaxy: diskbulgeMethods: numerical We characterize empirically the radial diffusion of stars in the plane of a typical barred disk galaxy by calculating the local spatial diffusion coefficient and diffusion time-scale for bulge-disk-halo N-body self-consistent systems which initially differ in the Safronov-Toomre-Q T parameter. We find different diffusion scenarios that depend on the bar strength and on the degree of instability of the disk. Marginally stable disks, with Q T ∼ 1, have two families of bar orbits with different values of angular momentum and energy, which determine a large diffusion in the corotation region. In hot disks, Q T > 1, stellar diffusion is reduced with respect to the case of marginally stable disks. In cold models, we find that spatial diffusion is not constant in time and strongly depends on the activity of the bar, which can move stars all over the disk recurrently. We conclude that to realistically study the impact of radial migration on the chemical evolution modeling of the Milky Way the role of the bar has to be taken into account. Introduction Disk galaxies are highly nonlinear systems which are driven by external forcing (satellites, interaction with nearby galaxies), internal instabilities (related to the formation of internal structures such as spiral arms, rings, bars) or a combination of both. Chaos and complexity, which are two different aspects of nonlinear response, dominate the dynamics of galactic systems. Disk galaxies are 'complex' in the sense that they are made of many components (stars, gas, dark matter in the bar-bulge, disk and halo components) whose interactions can give rise to spontaneous self-organization and to the emergence of coherent, collective phenomena. Examples of emergent behavior in disk galaxies are the formation of a central bar or the onset of spiral arms and warps, which can develop in dynamically cold disks (Revaz & Pfenniger 2004). Disk galaxies are not only complex systems, but even display chaotic behavior, since tiny differences in initial orbits of stars can exponentially blow up, as indicated by numerical simulations. Since the chaotic orbits are more sensitive to perturbations than regular (periodic) ones, external/internal forcing are more effective on this chaotic component. In this paper, we investigate chaotic and complex phenomena related to the formation of a central bar which give rise to the diffusion of the stellar component in the disk. The bar has a time-dependent activity, with a pattern speed which typically decreases in isolated galaxies (Sellwood 1981). However, the system can be cooled or heated by energy dissipation or infall of gas, or by forming stars on low-velocity dispersion orbits, with the net effect of impacting the amplitude of spiral waves and the strength of the bar, or even destroying it. In this way bars (and spiral waves) can be seen as recurrent patterns which can be rebuilt during their long history until the present configuration at redshift z = 0 (Bournaud & Combes 2002). Under the action of these non-axisymmetric patterns, stars move in the disk and gradually increase their velocity dispersion, as suggested by observations in the Solar neighborhood (see Holmberg et al. 2009 and references therein) and in external galaxies (Gerssen et al. 2000, Shapiro et al. 2003. The origin and the amount of disk heating are still open to debate. First attempts to explain such a heating process in disk galaxies were made by empirically modeling the observed increase of the stellar velocity dispersion with age in the solar neighborhood. Wielen (1977) suggested a diffusion mechanism in velocity space, which gives rise to typical relaxation times for young disk stars of the order of the period of revolution and to a deviation of stellar positions of 1.5 kpc in 200 Myr. The result was obtained without making detailed assumptions on the underlying local acceleration process responsible for the diffusion of stellar orbits. Global acceleration processes, such as the gravitational field of stationary density waves or of central bars with constant pattern speed, were ruled out since their contribution to the velocity dispersion of old stars was found to be negligible and concentrated in particular resonance regions (Wielen 1977;Binney & Tremaine 2008, p. 693). In isolated galaxies, different local accelerating mechanisms have been investigated, such as the gravitational encounters between stars and giant molecular clouds (Spitzer & Schwarschild 1951, 1953Lacey 1984), secular heating produced by transient spiral arms (Barbanis & Woltjer 1967;Carlberg & Sellwood 1985;Fuchs 2001) or the combination of the two processes (Binney & Lacey 1988;Jenkins & Binney 1990). Another heating mechanism was suggested by Minchev and Quillen (2006), who showed that the stellar velocity dispersion can increase with time due to the nonlinear coupling between two spiral density waves. Such local acceleration mechanisms suggest the existence of a significant component of the galactic gravitational field with a rather chaotic behavior. Pfenniger (1986) investigated the relation between diffusion and chaotic orbits. The latter typically react promptly to small perturbations. He pointed out that the effect of perturbations on regular orbits, such as epicyclic orbits, underestimates strongly the stellar diffusion rate when a stellar system becomes non-integrable, as for example in the presence of a central bar. As the central bar develops, reaches its maximal amplitude and then settles down to an almost steady state, its gravitational potential changes in time. In time-dependent po-tentials, the number of chaotic orbits typically decreases while the system secularly evolves toward a quasi-steady state through collective effects. The system is then ready again to respond (mainly through the remaining irregular orbits) to external perturbations, such as new infall of gas, and to recurrently restore a strong bar. The bar is thus able to perturb orbits of stars born or passing through its region, which can visit at later times the Solar neighborhood (Raboud et al. 1998). Indeed, most of the observational signatures of radial mixing reported in the literature (Grenon 1972(Grenon , 1999Castro et al. 1997) point to stars coming from a region next to the bulge/bar intersection, suggesting the bar to be a key player in the radial migration process. The subject of radial migration of stars was revived when a large scatter in the observed age-metallicity relation (AMR) was reported by Edvardsson et al. (1993), later confirmed by the larger Geneva-Copenhagen Survey sample (Nordström et al. 2004, Holmberg et al. 2007; see also Casagrande et al. 2011). It must be said that even though the AMR has been extensively studied in the solar vicinity, the results are still controversial due to the large uncertainties in stellar ages (see Pont & Eyer 2004). For instance, using a sample of stars for which it was possible to obtain chromospheric ages, Rocha-Pinto et al. (2000) have reported a much tighter AMR. The mechanisms driving radial diffusion and heating are still hotly debated, and in many cases the role of the bar is not taken into account. Assuming the whole scatter seen in Edvardsson et al. (1993) data was real, Sellwood & Binney (2002) pointed out that the radial excursion predicted by Wielen (1977) was not sufficient to explain the weakness of the AMR in the solar neighborhood. In order to explain both the large scatter in the AMR and the evidence that even old disk stars today have nearly circular orbits, Sellwood & Binney (2002) suggested a new mechanism based on the resonant scattering of stars under the effect of transient spiral waves. In this process, a star initially on a nearly circular orbit resonates with a rotating wave and changes its angular momentum. If the duration of the peak amplitude of the perturbing potential is less than the period of the 'horseshoe' orbits, i.e. orbits of particles trapped at the corotation radius of the spiral wave, the star can escape from the potential well without changing its eccentricity. The net effect of this scattering mechanism is that stars migrate radially without heating the disk. In other words, the overall distribution of angular momentum is preserved, except near the corotation region of the transient spiral wave, where stars can have large changes of their angular momenta. Haywood (2008) estimated upper values for the migration rate from 1.5 to 3.7 kpc/Gyr, which agree with the values in Lèpine et al. (2003) for the radial wandering due to the scattering mechanism assumed by Sellwood & Binney (2002). Radial diffusion of stars (and gas) could have important implications for the interpretation of key observational constraints for the formation of the Galaxy, such as the AMR, metallicity distributions, or the metallicity gradients, since old, probably more metal rich stars that formed at small galactocentric radii, as well as young metal-poor stars formed at large radii are enabled to appear in Solar-neighborhood samples (e.g. Haywood 2008). Due to the lack of detailed information on the processes driving stellar radial migration, models of the Galactic chemical evolution have evaluated past history of the solar neighborhood and the formation and evolution of the abundance gradients assuming that the Galaxy can be divided into concentric wide (∼1-2 kpc) cylindrical annuli, which evolve independently (van den Bergh 1962;Schmidt 1963;Pagel 1997;Chiappini et al. 1997, Chiappini et al. 2001. Schönrich & Binney (2009a) explored the consequences of mass exchanges between annuli by taking into account the effect of the resonant scattering of stars described before. This approach appears to be successful to replicate many properties of the thick disk in the Solar neighborhood without requiring any merger or tidal event (Schönrich & Binney 2009b). High resolution cosmological simulations (Roškar et al. 2008, Loebman et al. 2010 give support to the view that such scattering mechanism determines a significant migration in the stellar disk. However, the strong mixing driven by bar resonances was not taken into account (see below), casting thus doubts on some of the conclusions in the papers quoted above. The Milky Way (MW) is a barred galaxy and it is clear that the process above, not accounting for the existence of the bar, is probably just one of the processes at play among others. Indeed, the role of resonant couplings between bars and spirals (Tagger et al. 1987) in the distribution of energy and angular momentum in disk galaxies could also play a major role. Recently, Minchev and collaborators (Minchev & Famaey 2010, Minchev et al. 2011) have further analyzed this mixing mechanism finding that resonances between the bar and the spiral arms can act much more efficiently than transient spiral structures, dramatically reducing the predicted mixing time-scales. Moreover, while for the Sellwood & Binney (2002) mechanism to work short-lived transient spirals are required, in barred galaxies, such as the MW, spirals are most likely coupled with the bar as shown by Sparke & Sellwood 1987, and thus longer lived (Binney & Tremaine 2008, Quillen et al. 2010. As a consequence the radial migration process in the MW could have been different than currently predicted. In order to include the effect of radial migration in chemical evolution models and to gain a global (chemical and kinematic) understanding of the processes at play in the galactic disks, many dynamical aspects need to be further investigated and in particular the role of the bar, that is the strongest non-axisymmetric component in disk galaxies. In this paper, we present N-body simulations of barred spiral galaxies, and study how disks with different degrees of stability, ranging from marginally stable disks with Safronov-Toomre parameter Q T ∼ 1 to hot disks with Q T > 1, respond to the presence of bar patterns. Our aim is to estimate the time and length scales of stellar diffusion in the radial direction and to relate these quantities to the strength of the bar and to the number of hot particles in the disk, i.e. generally chaotic particles which are susceptible to cross the corotation barrier and to explore all space, being characterized by values of the Jacobi integral H larger than the value at the Lagrangian points, H > H(L 1,2 ) (Sparke & Sellwood 1987;Pfenniger & Friedli 1991). We investigate how these characteristic scales evolve in time, and depend on the activity of the bar. We consider different scenarios of diffusion, and discuss their implications for chemical evolution constraints in our Galaxy. The paper is organized as follows. In Sect. 2 we describe the simulations and the relevant parameters. In Sect. 3 we solve the diffusion equation in axisymmetric systems, we define the diffusion coefficient, the diffusion time-scale and the diffusion length-scale, and the methods used to estimate these quantities from the simulation results. In Sect. 4 we present our results. In Sect. 5 we discuss the implications for chemical evolution models of the MW and we summarise our findings. N-body simulations We have run self-consistent N-body simulations starting from a bar-unstable axisymmetric model. We have analyzed initial configurations with disk, bulge and dark halo components which differ on the initial value of the Safronov-Toomre parameter Q T = σ r κ/(3.36 G Σ) (Safronov 1960;Toomre 1964), where σ r is the radial velocity dispersion of the disk component, G is the gravitational constant, Σ is the disk surface density, κ is the epicycle frequency defined by κ 2 = RdΩ 2 /dR + 4Ω 2 , where Ω is the circular frequency related to the global gravitational potential Φ(R, z, t) in the disk plane z = 0 by Ω 2 = (1/R) ∂Φ/∂R. The initial mass distribution in our simulations corresponds to a superposition of a pair of axisymmetric Miyamoto-Nagai disks of mass M B , M D , horizontal scales A B + B, A D + B, and identical scale-height B, Φ MN (R, z) = i=B,D −GM i R 2 + (A i + √ B 2 + z 2 ) 2 (1) The first component represents the bulge (B), while the second the disk (D) (Pfenniger & Friedli 1991). The parameters have been set to A B = 0.07 kpc, A D = 1.5 kpc, B = 0.5 kpc, M B /M D = 3/17. The initial particle positions and velocities are found by a Monte-Carlo draw following the density law corresponding to Eq. (1), truncated to a spheroid of semi-axes R = 30 kpc, z = 10 kpc. The number of particles in the disk-bulge component is N = 4 · 10 6 and the total mass is M BD = 4.2 · 10 10 M ⊙ . In order to progressively heat the disk, we have added to this bulge-disk component an oblate pseudo-isothermal halo with the following density distribution (except in models m1 and m2): ρ H (R, z) = ρ h 1 + R 2 /R 2 h + z 2 /z 2 h (2) The number of particles in this halo component is N H = 2 · 10 6 , which is a value in the range suggested in Dubinski et al. 2009) in order to obtain convergent behavior in studies of bar formation and evolution. The length-scales are R h = 7.5 kpc and z h = 3.5 kpc. The density distribution has been truncated to R = 30 kpc, z = 15 kpc. We set the total mass in the dark halo M H to be four times the total mass in the bulge-disk component M BD , except in the model m3, where M H = 2M BD (see Table 1). The effect of adding the halo component is that the bar becomes progressively smaller and with higher pattern speed, the disk is hotter and less sensitive to the bar perturbations. We then impose the equilibrium of the first and second moments of velocities by solving Jeans' equations (see e.g., Binney & Tremaine 2008) with a constant Q T . The resulting distribution is relaxed for a couple of rotations until ripples spreading through the disk from the center disappear. We use this as the initial condition for the N-body simulations performed by using the Gadget-2 free source code (Springel et al. 2001, Springel 2005. The initial Gadget-2 configurations considered in this work differ on the values of the Safronov-Toomre parameter, which ranges from Q T ∼ 5 at two scale lengths from the center for hot disks to Q T ∼ 1 for marginally stable disks. These Q T values are listed in Table 1, along with the initial values of the radial and vertical velocity dispersions at two scale lengths from the center. We have considered these initial values in order to investigate how the radial diffusion depends on the disk sensitivity to perturbations. Thus, we have considered two extreme cases: in one case the disk is marginally stable and spiral waves develop (models m1 and m2), with the main global effect of heating in the radial direction (the Araki parameter σ z /σ r ∼ 0.5 at two scale lengths from the center at the end of the simulations and it decreases at larger radii, see the middle panel of Fig. 1). In the other case (model m6), the disk is heated and the velocity dispersions Table 1. Initial configuration of the N-body simulations: name of the model, initial Safronov-Toomre parameter at two scale lengths from the center, ratio between halo and disk-bulge masses, initial radial and vertical velocity dispersions at two scale lengths from the center, ratio between hot particles and total visible particles. increase in both the radial and the vertical direction (the Araki parameter σ z /σ r ∼ 0.8 at large radii at the end of the simulation, see the corresponding curve in the middle panel of Fig. 1). Model Q T M H /M BD σ r [km/s] σ z [km/s] N hot /N m1 1 - 20 18 12% m2 1.5 - The other models considered have intermediate values of Q T and σ z /σ r between these two extreme cases. The Safronov-Toomre and Araki parameters at the final times are shown in the left and middle panels, respectively, of Fig. 1. In the right panel of Fig. 1 the rotation curves for all the N-body models listed in Table 1 are shown at the final time. When the halo component is included (in models m3, m4, m5, m6), the rotation curve is flatter and V c ∼ 220 km s −1 at two scale lengths from the center, that is in the region between 6 and 10 kpc, depending on the model. The rotation curve of model m3, where the ratio between halo and disk-bulge mass is M H /M BD = 2, has intermediate values between that of models without the halo (m1, m2) and the others. We classify stellar orbits into three dynamical categories (Sparke & Sellwood 1987;Pfenniger & Friedli 1991). The first two dynamical categories are the bar and the disk orbits with the Jacobi integral H = E − Ω p L z smaller than the value at the Lagrangian points L 1,2 , H < H(L 1,2 ), where E is the total energy and L z is the z-component of the angular momentum. The separation of particles in the bar or disk component can be easily done since bar orbits typically have smaller values of L z and E than disk orbits. The third category includes hot orbits for which H ≥ H(L 1,2 ). The models considered here differ in the number of hot particles after the formation of the bar (the ratio between the number of stars in the hot component and the total number of the visible stars is listed in the last column of Table 1). The bar pattern speed Ω P ≡ dθ dt (t), where θ is the azimuthal angle of the bar major axis (in the inertial frame) calculated by diagonalising the moment of inertia tensor of the bar particles, ranges from 35 km s −1 kpc −1 for model m1 to 40 km s −1 kpc −1 for m6, at final times. These values are comparable to those found by Fux (1997 -see his Fig. 5), while recent estimates of the MW bar pattern speed are of the order of 50 − 60 km s −1 kpc −1 (Dehnen 2000, Minchev et al. 2007. The pattern speed is typically slowly decreasing in time with a rate of a few km/s/kpc/Gyr (Fux 1997;Bournaud & Combes 2002), so that the values at the beginning of the simulations are 60 km s −1 kpc −1 and 80 km s −1 kpc −1 for models m1 and m6, respectively. The corresponding corotation radius R c , obtained by the intersection of Ω P with the circular frequency, Ω P (t) = Ω(R c , t), increases in time (it typically ranges from R c = 2 to 5 kpc at final times in our models). In order to understand the role played by the central bar on the distribution of stars in the disk, we follow the evolution of the C m = j exp(i mθ j ) ,(3) the bar's strength is defined as the mode C 2 when the stars j are restricted to the bar component. We normalise C 2 with respect to the number of stars in the bar component, C 0 . This quantity is shown in Fig. 2 for models m1 and m6. Since we do not include the gas component in our models, we are not able to follow their evolution for times longer than a few Gyrs. After this typical time-scale, the bar amplitude saturates and the systems reach a quasi-steady state. Since the inclusion of the gas component necessarily requires the introduction of other less controlled parameters, such as the cooling rate or star formation, we prefer in this first study to limit the integration time over a couple of Gyrs, which is already enough to study the role of the bar in the radial migration process. The face-on and edge-on views of the density distribution of models m1 (upper panels) and m6 (bottom pannels) are shown in Fig. 3 at time t ∼ 550 Myr, that is just after the maximum strength of the bar (cf. Fig. 2). In the first case, both bar and spiral arms develop since the disk is sufficiently cold, while in the second case the disk is hot and only a bar (with a smaller corotation radius than in the previous case) develops in the central region. In the external regions of models m1 and m6, other patterns can be observed, the dominant being the pattern with m = 1. This mode is related to asymmetric distributions of mass pushed by the system rotation and it can give rise to filaments of stars or ring-like structures on long time-scales. Diffusion equation in axisymmetric systems In this paper we model the diffusion along the radial direction in the galactic plane by introducing a distribution function F(R, t) which satisfies the phenomenological spatial diffusion equation (from Fourier's law) in cylindrical coordinates: ∂ t F = 1 R ∂ R (RD∂ R F) ,(4) where D is the diffusion coefficient (in unit of area per time). This diffusion model has already been more succinctly presented in Brunetti et al. (2010). We suppose that for a limited time all the stochastic processes ongoing in a spiral galaxy can be described by such an equation characterized locally by a single positive diffusion coefficient D, to be empirically measured in the full dynamical simulations for a range of R and t. The parameter D conveys the notion that the diffusion process is a surface increase per unit time. For the sake of simplicity we choose D to be constant and independent of F, which makes Eq. (4) a linear partial differential equation for F. Another choice could have been to take C ≡ Dρ as a local parameter, where ρ is the mass density. The unit of C is mass per length per time which conveys then the notion of a mass flux gradient. However this choice would have led to a more difficult empirical determination of C since the equation involves then ρ and its gradient. Thus, the model considered here gives a lower limit on the diffusion happening in a barred galaxy, since other more complex diffusion processes are neglected in the present analysis. The general solution of Eq. (4) with constant D, which is non-singular at R = 0 is given by: F(R, t) = ∞ 0 A(s) e −Dts 2 J 0 (sR) s ds ,(5) where J 0 (x) = J 0 (−x) is the Bessel function of the first kind. The function A can be determined by taking the Hankel transform of F(R, 0): A(s) = ∞ 0 F(R, 0) J 0 (sR) R dR(6) By inserting Eq. (6) into Eq. (5) and assuming that the particles are initially localized at a certain radius R 0 at time t 0 = 0, F(R, 0) = F 0 R 0 δ(R − R 0 ), we obtain: formula 6.633.2). The time-evolution of an initial set of localized particles reads: F(R, t) = R 2 0 F 0 ∞ 0 s e −Dts 2 J 0 (sR) J 0 (sR 0 ) ds(7)F(R, t) = R 2 0 F 0 2Dt exp       − R 2 0 + R 2 4Dt       I 0 RR 0 2Dt ,(8) where I 0 (x) is the modified Bessel function of the first kind, which is finite at the origin, I 0 (0) = 1. In Fig. 4 two initial distributions with D = 1 and centered in R 0 = 0.5 and 2 (solid lines) evolve in time, as described by Eq. (8) (dashed and dotted lines, respectively). At large radii, the distributions are essentially Gaussian, while at small radii they are strongly modified from the contributions of particles at the center of the cylinder. Eq. (8) describes the distribution of the radial positions of stars in the disk at the initial time t i which diffuse toward position R 0 at time t 0 , such that t 0 − t i = ∆t ≤ T D , where T D is the diffusion time-scale or, equivalently, the distribution of stars which initially are in R 0 at t 0 and then diffuse toward R with ∆t ≤ T D . When R goes to zero, Eq. (8) reduces to: F(0, t) = R 2 0 F 0 2Dt exp −R 2 0 /(4Dt)(9) For large values of the argument the modified Bessel function I 0 (x) → (2πx) −1/2 exp(x) and thus F(R, t) reduces to: lim R→∞ F(R, t) = R 3/2 0 F 0 √ 4πDtR exp − (R − R 0 ) 2 4Dt = R 3/2 0 F 0 √ R N(µ, σ)(10) where N(µ, σ) is the Gaussian distribution with mean value µ = R = R 0 and standard deviation σ = √ 2Dt. In order to obtain a simple model of the distribution of the stars in a galactic disk, one can consider to envelop Eq. (10) by an exponential surface density Σ(R) ∝ exp(−R/R d ) (see, for example, Sellwood & Binney 2002), where R d is the disk scale length, thus obtaining: p d (R, t) = R 3/2 0 F 0 √ 4πDtR C exp − (R − R 0 ) 2 4Dt exp(−R/R d ) = C ′ √ R 0 √ 4πDtR exp − (R − R 0 + σ 2 /R d ) 2 4Dt = C ′ R 0 R N(µ ′ , σ)(11) where C and C ′ are normalization constants and N(µ ′ , σ) is the Gaussian distribution with mean value µ ′ = R = R 0 − σ 2 /R d and standard deviation σ = √ 2Dt. It is important to remember that the diffusion model used for obtaining Eq. (10) is only valid for times smaller than the diffusion time-scale. We set R 0 = R ⊙ = 8 kpc and R d ∼ 3 kpc. We consider the distribution of stars in R ∼ R ⊙ at the initial time. The previous expression, Eq. (11), can be used to estimate the relative fraction of stars which remain in a diffusion time-scale within the local volume \|R − R ⊙ \| ≤ d. This is given by: p(\|R − R ⊙ \| ≤ d) = R ⊙ +d R ⊙ −d N(µ ′ , σ) dR(12) The integral can be written as: p(\|R − R ⊙ \| ≤ d) = 1 √ π x + x − e −x 2 dx = 1 2 [erf(x + ) − erf(x − )] (13) where x ± = (σ/R d ± d/σ)/ √ 2. If d ≪ σ, we get x + ∼ x − and thus the probability of staying in the local volume is nearly zero. If d ∼ σ ≪ R d , we have p(\|R − R ⊙ \| ≤ d) ∼ 1 2 [erf(1/ √ 2) − erf(−1/ √ 2)] = erf(1/ √ 2) = 0.68. If d ∼ σ ∼ R d , we have p(\|R − R ⊙ \| ≤ d) ∼ 1 2 erf( √ 2) ∼ 0.48. Thus, the fraction of stars which remain in the local volume in a diffusion time-scale strongly depends on the ratio d/σ and on the value of R d . The diffusion coefficient D in Eqs. (4)-(8) can be regarded as an instantaneous coefficient which depends on the position at which particles are initially localised and on the diffusion time, since, as already mentioned, modeling the stellar migration as a diffusion process is valid only for time intervals less than the diffusion time-scale, ∆t ≤ T D . As we described in Brunetti et al. (2010), the diffusion time-scale can be estimated from the simulation results and it turns out to be of the same order of the rotation period, T D ∼ T rot = 2π/Ω(R, t). The diffusion coefficient is calculated by applying the nonlinear least-square method which minimizes the difference between the numerical results and the general solution of the diffusion equation described by Eq. (8) for times less than T D . At each time and radial position in our N-body simulations, we estimate the instantaneous diffusion coefficient and related quantities, thus obtaining a description of stellar migration along the whole simulation. Results The equations and the numerical methods described in the previous section and in Brunetti et al. (2010) allow us to calculate the diffusion coefficient D(R, t) which is shown in the contour maps of Fig. 5, top row, for the models m1 (left panel) and m6 (right panel). The others models m2, . . . , m5 have intermediate values of the diffusion coefficient between these two extreme cases. The bar's corotation radius is shown in Fig. 5 as a dashed white line. It can be seen that the diffusion coefficient is not constant in time nor in radius. If the disk is not too hot, D has the largest values D ∼ 0.12 kpc 2 Myr −1 outside the corotation radius of the bar (which increases in time in our simulations), where the density is strongly perturbed by a m = 2 pattern created by the bar and the transient spiral arms, and in the external regions R > 8 kpc, where the density is modulated by a m = 1 pattern. The stars respond collectively to these modulations and the process of migration corresponds to a diffusion in an axisymmetric system. In hot disks, the diffusion coefficient is large only in the external region R > 10 kpc, where the m = 1 mode appears, with values of the order of D ∼ 0.08 kpc 2 Myr −1 . In this latter case, the disk is not sufficiently cold to respond to the m = 2 perturbation created by the central bar. When the disk is marginally stable with initial Safronov-Toomre parameter Q T ∼ 1, the diffusion coefficient has the largest values just outside the corotation region, where two different families of orbits are present, as can be inferred by the total energy and angular momentum values of the particles in this region. We consider three different bins of particles located respectively in R = (1.5 ± 0.5) kpc, R = (3.0 ± 0.5) kpc and R = (8.0±0.5) kpc at t = 2.2 Gyr in the model m1. In the first bin, particles are mainly inside the bar. Their energies E span small negative values and the z-component of the angular momentum is nearly centered in L z ∼ 0 (see the two panels of Fig. 6, blue lines, labeled as 'bin 1'). Particles in the second bin centered in R = (3.0 ± 0.5) kpc at t = 2.2 Gyr belong to two different types of orbits: one family can migrate only inside the bar, the other can go outside the bar, in the disk. Large values of the diffusion coefficient D near the corotation region (see the left panel of Fig. 5, top row) are related to this superposition of two families of orbits. The corresponding values of E and L z are shown in the panels of Fig. 6 (red lines, labeled as 'bin 2'). The two peaks in the energy distribution correspond to the two families of orbits: bar particles have large negative energies, while particles which can go into the disk have small negative energies. Intermediate values correspond to the so called hot particles, which as already mentioned before have a Jacobi integral H larger than the value of H at the Lagrangian points L 1,2 , H ≥ H(L 1,2 ). We will discuss the hot particles later on in this section. It is important to note that the two peaks are increasingly less evident in hotter disks and the number of hot particles decreases as Q T increases (see the last column in Table 1). In the third bin, particles belong to the disk component: these disk particles have small negative energies and large values of L z (black lines, labeled as 'bin 3'). From D we can estimate the radial dispersion σ in a rotation period (which is of the order of the diffusion time-scale), σ = √ 2 D T rot . In the middle row of Fig. 5 we show the contour maps of the radial dispersion for models m1 (left panel) and m6 (right panel). If the disk is sufficiently cold, the radial dispersion is high near the corotation region of the bar, where it can recurrently assume values of the order of σ ∼ 6 kpc. This implies that internal stars can recurrently be forced by the activity of the bar to migrate in the external region of the disk. At an intermediate radius, such as R = 6 kpc, the radial dispersion is of the order of σ ∼ 3 kpc and it increases in the external less dense regions where the pattern m = 1 dominates. In the external region, the diffusion of the stellar component is related to the presence of both patterns m = 1 and m = 2 and it can be enhanced in regions where the bar's outer Lindblad resonance overlaps with the spiral arms resonances (Minchev et al. 2011). The radial migration driven by the bar seems to be efficient in cold disks. According to our results for model m1, the number of stars which stay always in a local volume of 100 pc around R = 8 kpc is low, since d ∼ 100 pc ≪ σ ∼ 5 kpc (see discussion after Eq. (13)). If the disk is hot, the values of σ are lower than the corresponding values for cold disks (at R = 6 kpc the radial dispersion is of the order of σ ∼ 1 kpc for model m6 (see Fig. 5, middle row, right panel) and consequently radial migration is less effective than before. Another interesting quantity is the diffusion velocity in a rotation period, defined by v D = σ/T rot = √ 2 D/T rot . We show the contour maps of this quantity in the bottom panels of Fig. 5 for The fact that the corotation region plays a crucial role in stellar diffusion can be also seen from Fig. 7. Here it is shown how the radial position of stars at some final time, R now at t now = 2.2 Gyr (on the vertical axis), depends on the corresponding radial distribution R past at time t past = 300 Myr (on the horizontal axis). The different colors in the map are related to the ratio of the number of stars at the two times, N(R past , t past )/N(R now , t now ). It can be seen that in the case of model m1 (left panel) particles near the corotation radius, R now ∼ R c = 4 kpc (which is the value of the corotation radius at t now = 2.2 Gyr in the considered model) came from inside and outside the corotation region, since they belong to two different families of orbits, as discussed before (see Figs. 6). Particles outside the corotation at the final time were spread over the disk in the past, with R c < R past ≤ 10 kpc, while particles well inside the corotation radius, R now < 2 kpc, were confined into the bar also in the past. On the contrary, in the hot model m6 (see the right panel of Fig. 7) particles were essentially located at the same positions in the past, with R past ∼ R now ± ∆R, where ∆R ∼ 1 kpc and it slightly increases with R now . In this case, no particular activity is observed in the corotation region, R c ∼ 2 kpc. As a further example, let us consider a bin of stars located in R now = (8.0 ± 0.1) kpc at time t now = 2.2 Gyr. Their evolution history is shown in Fig. 8, for the two models, m1 (left panel) and m6 (middle panel). When the bar's strength is maximum (at t ∼ 350 Myr, cf. Fig. 2), stars localized near the corotation radius are forced to move toward larger radii, since there the diffusion coefficient is large (cf. Fig. 5, top row). Most of the stars bounce then back and forth in the region R c < R < 10 kpc, to reach the final bin position. The evolution history in heated disks such as that of model m6 (see Fig. 8, middle panel) is very different, since stars are always localized in the region R = R now ± σ, with R now = 8 kpc and σ ∼ 2 kpc, in agreement with the corresponding value of the radial dispersion which can be inferred from the right panel of Fig. 5 (middle row). Hot particles, characterized by a Jacobi integral H > H(L 1,2 ), have a distribution which is maximum in the corotation region. In order to investigate the role of such hot particles in the diffusion process, we have removed them from the bin of stars localized in R now = (8.0 ± 0.1) kpc at time t now = 2.2 Gyr which we have considered just before. In the right panel of Fig. 8, the evolution history of the bin where the hot particles have been eliminated is shown. It must be compared with the evolution of the bin which includes them, shown in the left panel of the same figure. It can be seen that the main difference between the two panels is in the relative number of stars which are able to reach the corotation region. The hot particles migrate radially much more than the other particles in the disk. The number of hot particles in cold disks (such as m1) is nearly three times larger than the number in hot disks (see last column in Table 1). Discussion and conclusions From the previous sections it is clear that the amount of radial migration in disk galaxies is strongly dependent on the bar and spiral strengths. As we analyzed N-body simulations without gas, in order to focus our study on the pure effect of the bar, we can only trace radial migration for ∼2-3 Gyr. It is out of the scope of the present study to compare our simulations with the observations of the Solar neighborhood, which are the result of 10 Gyr of evolution. Indeed, our models were not intended to reproduce the conditions in our Galaxy, thus the precise values of, for example, the diffusion time-scale T D and the radial dispersion σ due to radial migration obtained from these models do not correspond to those valid for the Milky Way. However, the physical processes at play in cold disks, like our model m1, should be similar to those happening in our Galaxy, where the Safronov-Toomre-parameter in the Solar vicinity is Q T ∼ 2. Thus, we expect that the order of magnitude of the relevant quantities obtained from our numerical results are reasonably valid also for the earliest phases of the thin disk of our Galaxy (within ∼ 2-3 Gyr from the bar formation). Implications for chemical evolution models of the Milky Way Chemical evolution models of our Galaxy traditionally assume that the majority of stars do not migrate over large distances, and model the Galaxy by introducing independent radial annuli which are wide enough (around 1-2 kpc wide) so that this approximation would be a valid one (van den Bergh 1962;Schmidt 1963;Pagel 1997;Chiappini et al. 1997, Chiappini et al. 2001). The expectation is that intruders from other galactocentric distances would not represent more than a few percent of the stars in the local samples. However, as discussed in Sect. 1, there are recent claims that radial migration was more efficient than previously assumed. This, in turn, is driven by the large scatter in the AMR of the Geneva-Copenhagen sample. In the particular model of Chiappini et al. (2001), each annulus is 2 kpc wide (i.e. d = 2 kpc). Thus, in this case in the Solar vicinity d is nearly half the value of the radial dispersion obtained in model m1 at intermediate radii 6-8 kpc, d ∼ 2σ. Thus, the relation between the different length-scales which is approximately valid in the Solar vicinity is d ∼ σ ∼ R d (see discussion after Eq. (13)) and the percentage of stars which stayed in a volume \|R − R ⊙ \| ≤ d = 2 kpc turns out to be of the order of 50% in a diffusion time-scale, T D ∼ T rot = 2πR ⊙ /V c ∼ 223 Myr near the Sun. The region from which the rest of stars comes from depends on the activity of the bar which is not a constant pattern, as assumed in the past (Wielen 1977). If we consider for example the recurrent bar scenario described in Bournaud & Combes (2002), the bar can be rebuilt several times in a Hubble time in galaxies with significant gas accretion. When the bar strength is high, such as at the beginning of our simulations and in general when the bar is rebuilt by episodes of dissipative infall of gas, stars come mainly from the corotation region (see Fig. 8, left panel, t < 500 Myr), which is closer to the center since the corotation radius typically becomes smaller after each reformation episode (Bournaud & Combes 2002). When the bar strength saturates to a constant value during a quiescent phase, stars can span the region between the corotation radius and ∼10-11 kpc from the galactic center (see Fig. 8, left panel, t > 500 Myr). In Fig. 9 we show two examples of stellar diffusion. Stars localized in R = (8 ± 1) kpc and R = (3 ± 1) kpc at the end of the simulations are in the black bins, in the left and right panel, respectively. The red and blue distributions correspond to the radial positions of the stars 2.2 Gyr before, for model m1 and m6, respectively. As can be seen from the left panel in Fig. 9, when the disk is sufficiently cold (model m1, red distribution), the radial dispersion is higher than the corresponding values for the hot disk (model m6, blue distribution). In 2.2 Gyr (which is much larger than the diffusion time-scale), only 25% of the stars remain in the black bin, all the others come from outside. When the disk is hot, the percentage is nearly twice. In the right panel we show the case of stars near the corotation region (remember that the corotation radius is R c = 2 kpc for model m1 and 1 kpc for model m6). Stars come mainly from the corotation radius when the disk is cold (red distribution), while more than 60% stay at the same position if the disk is hot. We can conclude that the dynamical effects of stellar migration should be included in chemical evolution models of our Galaxy insofar as the distance d between each annulus is smaller than the radial dispersion σ, which is related to the diffusion coefficient D and to the diffusion time-scale T D by σ = √ 2 D T D . The radial dispersion σ depends on the degree of marginality of the disk. Conclusions Disk galaxies are complex systems where collective phenomena give rise to the emergence of nonlinear structures, such as central bars and spiral arms. We have investigated the role of bars in marginally stable disks and overheated disks, and their forcing effect on the hot (chaotic) particle component, which is more sensitive to external/internal perturbations. Modeling the migration of stars in marginally stable disks as a diffusion process in the radial direction is a powerful tool which allows us to estimate quantitatively two crucial parameters, the diffusion coefficient and the diffusion time-scale. With these quantities we are able to compute two other fundamental quantities which are the radial dispersion and the diffusion velocities at different radii and at different times. It is important to note that such a diffusion model makes sense only if there is a stochastic microscopic component at the origin of the diffusion. Ideally, the diffusion time-scale should not be much shorter than the microscopic e-folding time due to chaos, which in Nbody systems is typically of the order of the dynamical time (Miller 1964). Over longer time-scales, the diffusion model becomes influenced by the mere global dynamical evolution of the disk, so its behavior may depart from a simple linear diffusion equation with constant coefficient. Thus, the present diffusion calculation is useful for diffusion time-scales in a range around the rotational period. At each time and radial position in N-body simulations, we are able to estimate the instantaneous diffusion coefficient and the related quantities (i.e., diffusion time-scale, radial dispersion and diffusion velocity), thus obtaining a description of the stellar diffusion on the whole simulation. We have found that the diffusion time-scale is of the order of one rotation period and that the diffusion coefficient D depends on the evolution history of the disk and on the radial position. Larger values D are found in cold disks near the corotation region, which evolves in time, and in the external region, where asymmetric patterns develop. Marginally stable disks, with Q T ∼ 1, have two different families of bar orbits with different values of angular momentum L z and energy E, which determine a large diffusion in the corotation region. In hot disks, Q T > 1, stellar diffusion is much more reduced than in the case of marginal disks. The calculations of both the diffusion coefficient and the diffusion time-scale give us a quantitative measure of the migration process in the disk. Another advantage of studying the diffusion of stars in real space, rather than in velocity space, is that it can be more easily related to the evolution of chemical elements, which can be modeled as tracers which follow the evolution of the stellar component. The diffusion process of stars and tracers can be directly implemented in chemical evolution codes, which we plan to do in the near future. It is interesting to compare our results with those obtained in a recent analysis of Shevchenko (2011), where the Lyapunov and the diffusion times are estimated for the Quillen's model (Quillen 2003) which describes the Hamiltonian motion in the Solar neighborhood due to the interactions of bar and spiral arms resonances. He found that the Lyapunov time, of the order of 10 Galactic years, depends weakly on the model parameters, which can radically change the extent of the chaotic domain (as we obtain by varying the Safronov-Toomre parameter Q T ). The diffusion time, which characterizes the transport in the chaotic domain of the phase space, is calculated as the inverse of the diffusion rate in the energy variable (thus differing from our definition), with upper bounds of the order of 10 Gyr. It strongly depends on the radial position in the Galaxy, in agreement with our findings. We call attention to the fact that although there are good reasons (both theoretical and observational) to expect that radial process took place during the evolution of our Galaxy, it could have been much weaker than what has been proposed so far, as implied by the existence of radial abundance gradients, and its mild time-variation. Unfortunately, the uncertainties in the observed abundance gradient evolution are still large and we need to wait until better ages and distances will be available, which will happen in the near future thanks to GAIA and asteroseismology. Meanwhile, the theoretical work should focus on the relative importance of the main drivers of radial migration, and in the case of the Milky Way, on the role of the bar. Here we have shown that the radial migration process is not only timedependent but also changes with galactocentric distance, in connection to the bar, plus spiral arms. We plan to analyze simulations with bar but including also gas accretion, where the radial migration process can be traced for several Gyrs. In this way we will be able to answer if radial migration could have had repeated peaks along the MW history or if it faded away after the first 2-3 Gyr. These models coupled with the chemical information will be essential to interpret the radial mixing effects on the local age-metallicity relation, the metallicity distributions of the local thick and thin disks, and on the evolution of the abundance gradients in both disks. Fig. 1 . 1Left: Safronov-Toomre-parameter at the final time. Middle: Araki parameter at the final time. Right: rotation curves. Fig. 2 . 2Evolution in time of the bar's strength for models m1 and m6. strength of the bar in time for each model. If C m is the amplitude of the mode m in the density distribution, Fig. 3 . 3Thus, the diffusion of this distribution can be expressed in terms of Bessel and elementary functions(Gradsteyn & Ryzhik 2007, Density maps at t ∼ 550 Myr. Right: face-on views, left: edge-on views. Top panels: model m1, bottom panels: model m6. Fig. 4 . 4Two distributions with D = 1 and centered in R 0 = 0.5 and 2 (solid lines) evolve in time (dashed and dotted lines, respectively), as described by Eq. (8). Fig. 5 . 5Top row: Contour maps of the diffusion coefficient D. Middle row: Contour maps of the radial dispersion σ = √ 2 D T rot . Bottom row: Contour maps of the diffusion velocity v D = σ/T rot = √ 2 D/T rot . Left: model m1, right: model m6. models m1 (left) and m6 (right). The diffusion velocity can reach values of v D ∼ 40 km/s near the corotation region in the model m1, while it is always less than v D ∼ 20 km/s in the model m6. Fig. 6 . 6Energy values (left panel) and L z -values (right panel) of the particles at time 2.2 Gyr within the radial ranges R = (1.5 ± 0.5) kpc (bin 1, blue lines), R = (3.0 ± 0.5) kpc (bin 2, red lines) and R = (8.0 ± 0.5) kpc (bin 3, black lines) for model m1. 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